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Robustness Analysis of Simultaneous Stabilization and its Applications in Flight Control by Yasaman Saeedi A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Graduate Department of Aerospace Science and Engineering University of Toronto Copyright c 2011 by Yasaman Saeedi

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Robustness Analysis of Simultaneous Stabilization and itsApplications in Flight Control

by

Yasaman Saeedi

A thesis submitted in conformity with the requirementsfor the degree of Master of Applied Science

Graduate Department of Aerospace Science and EngineeringUniversity of Toronto

Copyright c© 2011 by Yasaman Saeedi

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Abstract

Robustness Analysis of Simultaneous Stabilization and its Applications in Flight

Control

Yasaman Saeedi

Master of Applied Science

Graduate Department of Aerospace Science and Engineering

University of Toronto

2011

Simultaneous stabilization is an important problem in the design of robust controllers.

It is the problem of designing a single feedback controller which will simultaneously

stabilize every member of a finite collection of liner time-invariant systems. This provides

simplicity and reliability which is desirable in aerospace applications. It can be used as

a back up control system in sophisticated airplanes, or an inexpensive primary one for

small aircraft. In this work the robustness of the simultaneous stabilization problem,

known as the Robust Simultaneous Stabilization (RSS) problem, is addressed. First, an

optimization methodology for finding a solution to the Simultaneous Stabilization (SS)

problem is proposed. Next, in order to provide simultaneous stability while maximizing

the stability robustness bounds, a multiple-robustness optimization design methodology

for the RSS problem is presented. The two proposed design methodologies are then

compared in terms of robustness of the designed controller.

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Acknowledgements

I would like to express my extreme gratitude to my supervisor, Dr. Hugh H.T. Liu,

for providing me with the opportunity to pursue my interests in this research field. His

insight and perspective on the topic was of great value to me and none of this would

be possible without his extreme support and guidance throughout the past few years. I

would also like to thank him for his kind understanding and support while I was going

through a difficult time in the past few months.

I would also like to thank my co-supervisor, Dr. Ruben Perez, for his much valued

support and insight. His knowledge on the topic of simultaneous stabilization was of

great value to me, he always made time to answer my questions and provide guidance

and suggestions, and I definitely owe much of what I know to him.

I would like to thank all my friends in the FSC lab, all the past and present members.

They made my experience at UTIAS enthusiating and inspiring and their appreciated

friendship and support is what I will take away from this.

Last, but not least, I would like to thank my family for their never ending love and

support, and for always being there by my side. I also truly want to thank my circle of

friends for their spiritual support and friendship whenever I needed it.

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Contents

1 Introduction 1

1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Motivation & Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Thesis Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Simultaneous Stabilization 6

2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Necessary and Sufficient Condition . . . . . . . . . . . . . . . . . . . . . 7

2.3 Simultaneous Stabilization by Linear State Feedback Control . . . . . . . 8

2.4 Optimal Stabilization via Linear State Feedback Control . . . . . . . . . 12

2.5 Bi-Level Decomposition-Based Strategy . . . . . . . . . . . . . . . . . . . 15

2.5.1 Decomposition formulation . . . . . . . . . . . . . . . . . . . . . . 15

2.5.2 Decomposed equivalent of necessary and sufficient conditions . . . 17

2.6 Parameter Optimization Approach . . . . . . . . . . . . . . . . . . . . . 17

2.6.1 Cost Function Definitions . . . . . . . . . . . . . . . . . . . . . . 18

2.6.2 Multiple Objective Design . . . . . . . . . . . . . . . . . . . . . . 19

2.7 Proposed SS Optimization Methodology . . . . . . . . . . . . . . . . . . 20

3 Robustness Analysis 22

3.1 Kharitonov’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Extreme Point Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.1 Stability as a Nonsingularity Problem via the ‘Kronecker Lyapunov

Matrix’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

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3.2.2 Necessary and Sufficient Vertex Solution for Robust Stability . . . 26

3.3 Stability Robustness Bounds . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4 Robustness Analysis of a Numerical Example . . . . . . . . . . . . . . . 29

3.4.1 Extreme Point Solution Application . . . . . . . . . . . . . . . . . 30

3.4.2 Kharitonov’s Theorem Application . . . . . . . . . . . . . . . . . 34

3.4.3 Stability Robustness Bound Application . . . . . . . . . . . . . . 35

4 Robust Simultaneous Stabilization Problem 38

4.1 An Extended Decomposition-Based Strategy for the RSS Problem . . . . 39

4.2 Multi-Objective Optimization . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.1 Formulation & the Concept of Pareto Optimality . . . . . . . . . 42

4.2.2 Weighted-Sum Method . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2.3 Multiple Robustness Optimization . . . . . . . . . . . . . . . . . . 46

5 Linear Simulation: An F4-C Flight Control Case Study 49

5.1 Introduction of the Test Case . . . . . . . . . . . . . . . . . . . . . . . . 49

5.2 Robustness Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.2.1 Perturbations Due to CLα uncertainties . . . . . . . . . . . . . . . 52

5.2.2 Robustness Optimization . . . . . . . . . . . . . . . . . . . . . . . 59

6 Non-Linear Simulation: A CRJ-200 Flight Control Case Study 67

6.1 Modelling of the CRJ-200 . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.1.1 Nonlinear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.1.2 Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.2 Introduction of the Test Case . . . . . . . . . . . . . . . . . . . . . . . . 72

6.3 Results: Ordinary and Gust-Encountered Flight . . . . . . . . . . . . . . 75

6.4 Robustness Investigation & Optimization . . . . . . . . . . . . . . . . . . 77

7 Conclusion and Future Developments 86

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

7.2 Future Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

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Bibliography 90

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List of Tables

3.1 F4-E flight operating conditions . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Allowed perturbation resulting in a stable matrix family . . . . . . . . . 33

3.3 Allowed perturbation resulting in an unstable matrix family . . . . . . . 33

3.4 Kharitonov’s polynomials coefficients . . . . . . . . . . . . . . . . . . . . 35

3.5 Stability robustness bounds . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.1 F4-C flight operating conditions . . . . . . . . . . . . . . . . . . . . . . . 50

5.2 Simultaneous Stabilization solution . . . . . . . . . . . . . . . . . . . . . 51

5.3 Closed-loop system eigenvalues . . . . . . . . . . . . . . . . . . . . . . . 51

5.4 F4-C characteristics at different flight conditions . . . . . . . . . . . . . . 55

5.5 Maximum allowable deviation in ∆CLα . . . . . . . . . . . . . . . . . . . 56

5.6 Kharitonov’s polynomials coefficients . . . . . . . . . . . . . . . . . . . . 57

5.7 Solutions to the SS and RSS problems . . . . . . . . . . . . . . . . . . . 60

5.8 Effect of relaxing the robustness on |∆CLα|max . . . . . . . . . . . . . . . 61

5.9 Results from different RSS optimization methodologies and objective func-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.10 A comparison of the closed-loop eigenvalues for different optimization

methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.1 CRJ-200 flight operating conditions . . . . . . . . . . . . . . . . . . . . . 74

6.2 Open-loop system eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . 75

6.3 Closed-loop system eigenvalues . . . . . . . . . . . . . . . . . . . . . . . 76

6.4 Maximum allowable deviation in ai,j . . . . . . . . . . . . . . . . . . . . 81

6.5 Effect of relaxing the robustness on |∆ai,j|max . . . . . . . . . . . . . . . 83

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6.6 A comparison of the closed loop eigenvalues for different optimization

methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

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List of Figures

2.1 Multidisciplinary optimization and simultaneous stabilization problem [27] 16

3.1 response to initial condition of the perturbed nominal plant . . . . . . . . 34

3.2 response to initial condition of the perturbed nominal plant . . . . . . . . 36

4.1 Robust Simultaneous Stabilization solution approach . . . . . . . . . . . 41

4.2 Pareto set in a convex objective space . . . . . . . . . . . . . . . . . . . . 44

4.3 Pareto Set in a non-convex objective space . . . . . . . . . . . . . . . . . 44

4.4 Pareto sets in a weighted-sum optimization problem . . . . . . . . . . . . 45

5.1 Response to initial condition . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.2 Kharitonov’s robust stability graphical check for different flight conditions 58

5.3 Maximum Eigenvalue of the closed-loop system vs. CLα . . . . . . . . . . 59

5.4 Maximum Eigenvalue of the closed-loop system vs. CLα . . . . . . . . . . 63

5.5 Response to initial condition: SS problem . . . . . . . . . . . . . . . . . . 65

5.6 Response to initial condition: Multiple Robustness Optimization solution 66

5.7 Response to initial condition: Decomposition-Based Strategy solution . . 66

6.1 The Flight Training Device (FTD) Facility . . . . . . . . . . . . . . . . . 68

6.2 Pitch angle tracker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.3 Time history of the states at different flight conditions subject to a 5-deg

step input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.4 Time history of the states at different flight conditions subject to a 5-deg

step input, when encountered with gust . . . . . . . . . . . . . . . . . . . 79

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6.5 Time history of the states at different flight conditions subject to a 5-deg

step input, under the RSS problem solution . . . . . . . . . . . . . . . . 85

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Chapter 1

Introduction

1.1 Overview

Simultaneous stabilization is an open and important problem in the design of robust

controllers. It is the problem of designing a single feedback controller which will simulta-

neously stabilize every member of a finite collection of liner time-invariant systems. The

simultaneous stabilization problem is defined as follows: Given n proper, linear time-

invariant plants P1(s), P2(s), ..., Pn(s), does there exist a single controller, C(s), such

that the closed loop (unitary feedback) system is internally stable for each of the given

plants? Also, under what conditions can such a controller be found? This has been a

problem of interest for many years now and various techniques have been proposed to

solve it.

The simultaneous stabilization problem of finding a single controller, which stabilizes

a finite set of different plants, is of practical interest. One motivation comes from the

stability requirements of a system operating in different modes. A common application is

the desire to control a system under normal operating conditions as well as under several

different failure modes, for example an industrial plant that has to operate in different

modes due to the possible sensor or actuator failure. A system may also have time-varying

parameters or several different normal modes of operation. For example, the dynamics of

the aircraft vary greatly with its altitude and speed. Hence, the attitude response of an

aircraft would be represented by a different mathematical model depending on different

1

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Chapter 1. Introduction 2

flight conditions across the flight envelope. Moreover, ensuring the stable operation

of a non-linear system at several different steady states may be desirable in aerospace

applications. A linearized model of such a non-linear system operating at different points

may have time-varying parameters and the dynamics of the system will certainly change.

Thus, for slowly changing plants, a linear controller could potentially stabilize a nonlinear

system if it were to simultaneously stabilize the plants linearized about several different

points of operation. A single stabilizing controller provides simplicity and reliability, as

well, which is of outmost necessity in aerospace applications. It can be used as a back-up

control system in sophisticated airplanes, as well as an inexpensive primary one for small

aircraft.

Simultaneous stabilization is also a subtopic of robust control. Robust stabiliza-

tion simultaneously stabilizes a continuous range of plants, whose parameters lie within

predefined regions, subject to possible performance constraints. The major distinction

between robust control and simultaneous stabilization is in the number of plants they

are attempting to stabilize. Robust stabilization contends with an infinite (uncountable)

number of plants, whereas simultaneous stabilization deals only with a finite number

of plants. Nevertheless, the simultaneous stabilization of a finite number of systems is

difficult. Unlike robust stabilization in which the continuum of plants must not vary

too far from a nominal plant, there may be no assumptions on the interrelatedness of

the finite number of distinct plants. To date, there is a complete, tractable solution to

the simultaneous stabilization problem only when there are no more than two plants to

simultaneously stabilize.

Simultaneous stabilization was first studied more than three decades ago and has re-

ceived considerable attention since. Numerous authors have considered the simultaneous

stabilization problem for different types of systems and controllers. Youla et al. (1974)

[39] provided necessary and sufficient conditions for the problem of strong stabilization

of one plant, but it was Ackermann (1980) [1] who first considered the problem of si-

multaneous stabilization of multiple plants and presented a mathematical formulation

for it. He also proposed a solution using the concept of state feedback. Later, Franklin

and Ackermann (1981) [17] considered the problem of stabilization of the longitudinal

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Chapter 1. Introduction 3

mode of an aircraft, and proposed a solution for design of a controller using two gyros

and accelerometer. Ackermann (1984) [3] was then able to propose a more efficient solu-

tion, using only two gyros. Vidyasagar and Viswanadham (1982) [34], Ghosh and Byrnes

(1983) [18], and Kale (1990) [20], studied the problem of designing a controller that would

simultaneously stabilize a collection of multiple-input multiple-output systems described

by transfer functions. It was shown by Saeks and Murray (1982) [32], and Vidyasagar

and Viswanadham (1982) [34] that the simultaneous stabilization of two systems reduces

to the problem of strong stabilization of one plant, considered previously by Youla et

al. (1974) [39]. A set of sufficient conditions was derived by Ghosh and Byrnes (1983)

[18] for simultaneous pole-assignability by dynamic output feedback. The single-input

single-output case was considered in detail by Debowski and Kurylowicz (1986) [13], who

developed a necessary and sufficient condition for the existence of a stabilizing compen-

sator and proposed an algorithm for its design. Petersen (1987) [30] studied the problem

of stabilizing a collection of single-input systems represented by state-space models via

non-linear state feedback control, and obtained a sufficient condition for the existence

of a stabilizing non-linear controller. Schmitendorf and Hollot (1989) [33] also consid-

ered the simultaneous stabilization of a collection of linear single-input systems via linear

state feedback control and obtained a sufficient condition for the existence of a stabiliz-

ing linear state feedback controller, which is later proposed in Wu et al. (1990) [35].

Later, Howitt and Luss (1991) [19] provided a necessary and sufficient condition for the

existence of a linear state feedback controller for the simultaneous stabilization problem.

Both Wu et al. (1990) and Howitt and Luss (1991) obtained such a controller by solving

a non-smooth optimization problem, where the objective was to minimize the function

representing the largest real part of the eigenvalues in order to increase the stability

margin, overlooking the transient behaviour of the system. Chow (1990) [10] uses the

definition of a “multimode” system controllability matrix (simultaneously describing the

controllability of all the systems) to provide a sufficient condition for simultaneously

placing the closed-loop system poles in specific locations. Boyd et al. (1993) [7] derived

a set of linear matrix inequalities and demonstrated that if a single solution exists, si-

multaneous stabilization can be guaranteed. It is then shown in Blondel (1994) [5] that

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Chapter 1. Introduction 4

it is not possible to rationally decide whether a set of three or more systems is simulta-

neously stabilizable or not, although available sufficient conditions can be used. Here,

the term “rationally undecidable” means that it is not possible to find a necessary and

sufficient general criterion for simultaneous stability of the systems, involving only the

coefficients of the linear systems, rational or logical operations, and sign test operations.

Finally, Paskota et al. (1994) [23] provided a solution to the problem by solving nonlin-

ear Lienard-Chipart constraints, while Dorato et al. (1995) [14] applied the “quantifier

elimination” computational technique to verify Lienard-Chipart stability constraints.

The problem of simultaneously stabilizing a final set of n plants is equivalent to

strongly stabilizing (i.e. using a stable controller or compensator) a set of n− 1 plants.

When n = 2, a necessary and sufficient condition exists, known as the parity interlac-

ing property. The problem, however, becomes harder when n ≥ 3. As said before,

simultaneous stabilization of more than two plants is rationally undecidable and due to

the problem’s nature, an analytical solution is very difficult to find. However, it can

be shown that while the simultaneous stabilization problem is generally rationally in-

tractable, in most cases, it can be tackled numerically by an optimization algorithm, and

simultaneously stabilizing controllers can be found accordingly.

1.2 Motivation & Contribution

In this work, the robustness of the simultaneous stabilization problem known as the Ro-

bust Simultaneous Stabilization (RSS) problem is addressed. At first, a new optimization

methodology for finding a solution to the Simultaneous Stabilization (SS) problem is pro-

posed, based on the previous work presented in the literature. It is preferable for such

a controller to be able to simultaneously stabilize a set of plants and maintain that sta-

bility when encountered with uncertainties. More specificly in aerospace applications, it

is desirable to control an aircraft under normal flight conditions as well as under several

different failure modes and uncertainties, such as structure failure or sudden upset of the

flight conditions. Such a single stabilizing controller provides a back-up control system

with simplicity and reliability. Hence, in order to provide simultaneous stability for a set

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Chapter 1. Introduction 5

of systems while maximizing the stability robustness bounds, a new multiple-robustness

optimization design methodology for the RSS problem is presented based on the concept

of multi-objective modelling. The two proposed design methodologies, i.e. the SS and

the RSS problem solutions, are then compared in terms of robustness of the designed

controller.

1.3 Thesis Layout

This dissertation is organized as following. Chapter 2 addresses the background required

for this thesis, including the formulation of the simultaneous stabilization problem as well

as the necessary and sufficient conditions required for the existance of a solution. This

chapter also provides a summary of several approaches from the literature for finding a

solution, and at the end of the chapter a new optimization methodology is presented.

Chapter 3 discusses the robustness of the simultaneous stabilization problem. Several

robustness analysis approaches, namely the extreme point solution, the Kharitonov’s

theorem, and the stability robustness bounds, are introduced for providing the physical

bounds of allowable perturbations. For comparison purposes, these approaches are then

applied to a numerical example. The Robust Simultaneous Stabilization (RSS) prob-

lem is described in Chapter 4, where the concept of multi-objective modelling is used

for providing a new multiple-robustness optimization design methodology for the RSS

problem. The final design is an improved controller in terms of robustness. Chapters 5

and 6 focus on the presentation and discusssion of the obtained results when the design

methodologies presented in the previous chapters are applied to a linear and a non-linear

flight control case study. Finally, conclusions and future work are made in Chapter 7.

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Chapter 2

Simultaneous Stabilization

In this chapter, a brief review of various simultaneous stabilization methods is presented.

First, the problem of simultaneous stabilization is formulated, and a necessary and suf-

ficient condition for solving this problem is stated. Later, several approaches from the

literature for finding the solution to the simultaneous stabilization problem are summa-

rized. Finally, based on the previous work in the field, a new simultaneous stabilization

design methodology is proposed.

2.1 Problem Formulation

As described before, simultaneous stabilization is the problem of finding a single unique

control law that can stabilize a finite set of plants simultaneously. In mathematical terms,

consider a collection of m different systems described by the state-space equations:

xk = Akxk (t) +Bkuk (t) , k = 1, 2, ...,m, (2.1)

yk = Ckxk (t)

where in the case of single-input single-output linear systems shown above, xk ∈ Rn is

the state vector of the kth system, and uk is a scalar control. It is assumed that each

system is controllable. A single feedback gain vector f ∈ Rn is sought such that when

the control

uk(t) = −fTxk(t) (2.2)

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Chapter 2. Simultaneous Stabilization 7

is implemented, each of the closed-loop systems

xk = (Ak −BkfT )xk, k = 1, 2, ...,m (2.3)

will be stable.

2.2 Necessary and Sufficient Condition

An equivalent to the above statement is that all the eigenvalues of each closed-loop

system must have negative real parts in order for the systems to be stable. Therefore,

there exists a solution to the simultaneous stabilization problem if and only if

minf

I = max︸ ︷︷ ︸1≤i≤n, 1≤k≤m

Re(λi,k)

≤ 0, (2.4)

where λi,k is the ith eigenvalues of the kth closed-loop system given by Eq. (2.3). If

the minimum value of I after the optimization is negative, the control law f which

minimizes the objective function will be a solution to the simultaneous stabilization

problem. However, if the minimum value of I is non negative, it can be concluded that

no solution exists to the simultaneous stabilization problem [19].

On the other hand, the solution of this problem may be unbounded. This problem

can be solved simply by imposing upper and lower bounds on the magnitudes of the

elements of the feedback gain vector f of the form

−µ ≤ fi ≤ µ i = 1, ..., n, (2.5)

where µ > 0 is positive and finite. Therefore f can not be arbitrarily large. It is also

preferred to choose a feedback gain that yields good performance. A large imaginary

part with respect to the real part of the eigenvalues will result in poor performance due

to insufficient damping. Therefore, in order to limit the minimum damping ratio, it is

desirable to limit the magnitude of the imaginary part with respect to that of the real

part [19]. To impose this constraint, a new parameter η ≥ 0 is introduced and the

following constraint is stated, where αi,k is the value of the real part of the closed-loop

systems’ eigenvalues and βi,k is that of the imaginary part.

η |βi,k| ≤ |αi,k| . (2.6)

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Chapter 2. Simultaneous Stabilization 8

To make the objective function continuously differentiable, the above simultaneous

stabilization problem can be translated into an equivalent optimization problem:

minf I = γ < 0

subject to

Re (λi,k) ≤ γ,

η |βi,k| ≤ |αi,k| , i = 1, ..., n, k = 1, ...,m

−µ ≤ fi ≤ µ.

The following sections focus on a number of different approaches taken from the liter-

ature to solve such a problem. Needless to say, the problem of simultaneous stabilization

has been tackled from many different points of view. First, two optimization method-

ologies for solving the simultaneous stabilization problem are presented. Next, a bi-level

decomposition-based strategy and a parameter optimization approach are introduced.

Finally, an alternate optimization methodology is proposed.

2.3 Simultaneous Stabilization by Linear State Feed-

back Control

As a means of determining whether or not a particular simultaneous stabilization problem

has a solution (and thus evaluating the necessary and sufficient condition stated before),

and of constructing a suitable controller when one does indeed exist, an optimization

problem was proposed in [19].

Problem P1. Choose f ∈ Rn to minimize the objective function

I = max︸ ︷︷ ︸1≤i≤n,1≤k≤m

Re (λi,k) . (2.7)

If the minimum value of I after the optimization is negative, the f which minimizes the

objective function will be a solution to the problem. However, if the minimum value of I is

non-negative, it can be concluded that no solution exists to the simultaneous stabilization

problem. Unfortunately, the vast majority of numerical methods for minimization are

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Chapter 2. Simultaneous Stabilization 9

suited only to problems with an objective function which is continuously differentiable.

As stated before, this problem can hence be transformed into an equivalent problem

whose objective function is continuously differentiable.

Problem P2. Choose f ∈ Rn and γ ∈ R to minimize the objective function

I = γ, (2.8)

subject to the constraint

Re (λi,k) ≤ γ, i = 1, ..., n, k = 1, ...,m. (2.9)

The eigenvalues λi,k are implicit functions of the feedback gain vector f . This rela-

tionship must be more explicit for computational purposes. Therefore, the eigenvalues

were allowed to be free parameters to be chosen along with f and γ, such that the ob-

jective function I is minimized. It is hence necessary to introduce additional constraints

which will explicitly describe the mathematical relationship between the feedback vector

and the eigenvalues. In order to develop this relationship, consider the kth single-input

single-output system. The vector f and the eigenvalues, λ1,k, λ2,k,..., λn,k of the matrix

Ak −BkfT are related in the following way. Let

pk (s) =n∏i=1

(s− λi,k) = δ1,k + δ2,ks+ ...+ δn,ksn−1 + sn (2.10)

be the characteristic polynomial of the kth closed-loop system. The closed-loop charac-

teristic polynomial coefficients vector ck ∈ Rn, defined by

ck =

δ1,k

δ2,k

...

δn,k

(2.11)

is a function of f since it is dependant on the feedback gain vector. This relationship

takes the form

Gkf + hk = ck, (2.12)

where Gk ∈ Rn×n and hk ∈ Rn can be defined by the following algorithm:

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Chapter 2. Simultaneous Stabilization 10

Step 1. Let eTk be the last row of[bk Akbk · · · An−1

k bk

]−1

.

Step 2. Gk = −[ek ATk ek · · ·

(An−1k

)Tek

]−1

.

Step 3. hk = Gk (Ank)T ek.

Although the eigenvalues will be treated as free parameters in the optimization proce-

dure, they must be chosen to satisfy constraints of the form (2.12). Note that some of the

eigenvalues may be complex, and it is assumed that they will take one of the following

two forms:

• if n is even, then

λi,k = αi,k + jβi,k, i = 1, ..., n, (2.13)

• if n is odd, then

λi,k = αi,k + jβi,k, i = 1, ..., n− 1,

λn,k = αn,k.

This type of parametrization of the eigenvalues is chosen because, when n is odd, at

least one real eigenvalue must exist. Since we are dealing with real systems, it is clear

that the eigenvalues are either real or they must occur in complex conjugate pairs. To

generalize this to any value of n, let us first define the even integer

N =

n, if n is even,

n− 1, if n is odd.(2.14)

Since the coefficients of the characteristic equations are real values, the following con-

straint should also be imposed:

gk =

β1,k + β2,k

α1,kβ2,k + α2,kβ1,k

β3,k + β4,k

α3,kβ4,k + α4,kβ3,k

...

βN−1,k + βN,k

αN−1,kβN,k + αN,kβN−1,k

= 0. (2.15)

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Chapter 2. Simultaneous Stabilization 11

Another consideration is that the solution to problem P2 may be unbounded. In a

particular simultaneous stabilization problem, it may be possible to shift the real part of

the greatest eigenvalue and thus the objective function I all the way to negative infinity,

and therefore a minimum will not exist. This difficulty can be easily overcome, however,

by placing simple bounds on the magnitudes of the elements in the feedback gain vector

f of the form

−µ ≤ fi ≤ µ, i = 1, ..., n , (2.16)

where µ > 0 is chosen to be real and finite. With these constraints, f can not be

made arbitrarily large, and therefore the objective function is always bounded. If, for a

particular value of µ there is no solution to problem P2, µ can be increased until a solution

is found, or until it is clear that no solution exists to the simultaneous stabilization

problem. This has an added advantage in limiting the magnitude of the control law

through a single parameter.

The performance of the controller is another issue to be considered, since it is desirable

to find a simultaneously stabilizing controller with acceptable performance. Even in a

stabilized system, a very large imaginary part βi,k compared to the real part αi,k will

result in rapid oscillation and poor performance. One solution is to limit the magnitude

of the imaginary part with respect to that of the real part. A new parameter η ≥ 0 is

now introduced and the constraints

βi,k ≥ 0, i = 1, 3, ..., N − 1, k = 1, ...,m

βi,k ≤ 0, i = 2, 4, ..., N, k = 1, ...,m

αi,k + ηβi,k ≤ 0, i = 1, 3, ..., N − 1, k = 1, ...,m

are imposed. Now if the solution to the simultaneous stabilization problem does not

yield good performance, η can be increased in order to improve the performance of the

controller.

Finally, the general optimization problem which gives a solution to the simultaneous

stabilization problem is stated as follows:

Problem P3. Given parameters µ > 0 and η ≥ 0, Choose f ∈ Rn, γ ∈ R, αi,k ∈ R, and

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Chapter 2. Simultaneous Stabilization 12

βi,k ∈ R to minimize the objective function

I = γ, (2.17)

subject to the constraints

−µ ≤ fi ≤ µ, i = 1, ..., n,

αi,k ≤ γ, i = 1, ..., n, k = 1, ...,m,

Gkf + hk = ck, k = 1, ...,m,

gk = 0, k = 1, ...,m,

βi,k ≥ 0, i = 1, 3, ..., N − 1, k = 1, ...,m,

βi,k ≤ 0, i = 2, 4, ..., N, k = 1, ...,m,

αi,k + ηβi,k ≤ 0, i = 1, 3, ..., N − 1, k = 1, ...,m.

If after optimization I is negative, the f that makes it negative is the solution to the

simultaneous stabilization problem, and if I is greater than or equal to zero, no solution

can be found within the given boundary for the elements of the control law f . It was also

seen that when applied to a number of case studies, the optimization process results in

a single best solution regardless of the chosen initial point. Hence, one can assume that

the obtained simultaneously stabilizing controller is the global optimum.

2.4 Optimal Stabilization via Linear State Feedback

Control

In this section, another optimization methodology is introduced as was originally pre-

sented in [23], which focuses on designing a control law with optimal transient response.

Again, consider the collection of m linear time-invariant systems:

x (t) = Akxk (t) +Bkuk (t) ; k = 1, 2, ...,m (2.18)

where x (t) ∈ Rn is the state and u (t) ∈ R is the control, and Ak ∈ Rn×n, Bk ∈ Rn are

independent of time. We assume that [Ak, Bk] is controllable for each k = 1, 2, ...,m. Let

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Chapter 2. Simultaneous Stabilization 13

us define the objective function for the kth system as follows:

Jk =1

2

∫ ∞0

(xTkQxk + uTkRuk

)dt, (2.19)

where Q ∈ Rn×n is symmetric and positive definite, and R > 0. This objective function

will ensure that a good transient response is obtained without using an unnecessary

amount of control. We consider linear state feedback controllers of the form:

uk = −fTxk, (2.20)

where f is known as the feedback gain vector. We require f to be the same for all the

systems, and −µ ≤ fi ≤ µ for all i = 1, ..., n where µ is a given positive real constant.

Substituting (2.20) into (2.18) and (2.19), one obtains

xk (t) = Ak (f)xk (t) , k = 1, 2, ...,m (2.21)

and

Jk (f) =1

2

∫ ∞0

[xTk(Q+ fRfT

)xk]dt, k = 1, 2, ...,m, (2.22)

where

Ak (f) =(Ak −Bkf

T), k = 1, 2, ...,m. (2.23)

Let F be a subset of Rn such that if f ∈ F , then the matrix Ak (f) is stable for each

k. F is hence called the set of simultaneously stabilizing feedback gain vectors (i.e. if

there is not such a simultaneously stabilizing controller, F is a null subset and the SS

problem has no solution). For brevity, a vector in F will also be called a feasible vector,

since it is a feasible potential solution to the problem of simultaneous stabilization. Let

Fµ be the subset of F consisting of all the vectors that satisfy:

−µ ≤ fi ≤ µ, i = 1, ..., n. (2.24)

The problem is now formally stated as follows:

Problem 1 Find a feasible vector f such that the cost function S (f) =∑mk=1 Jk (f) is

minimized over Fµ.

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Chapter 2. Simultaneous Stabilization 14

Note that a solution to this optimization problem will give us a controller which will

simultaneously stabilize all the systems with a limited control law gain, and further-

more due to the choice of the objective function, a desirable transient behaviour will be

obtained without using unnecessary control effort.

If the closed-loop system Ak − BkfT is stable, it is known (Choi and Sirisena, 1974

[9]) that

JK (f) =1

2

∫ ∞0

[xTkQxk +

(−fTxk

)TR(−fTxk

)]dt

=1

2

∫ ∞0

[xTk(Q+ fRfT

)xk]dt =

1

2tr (Pk) ,

where matrix Pk is the positive definite solution of the matrix Lyapunov equation

X(Ak −Bkf

T)

+(Ak −Bkf

T)TX +Q+ fRfT = 0. (2.25)

Since f ∈ F , the matrix Ak − BkfT is stable for each k = 1, 2, ...,m. Thus, for each

k = 1, 2, ...,m, the matrix Lyapunov equation has a unique positive definite solution. This

leads to an easy way of calculating Jk (f). Also, for a given matrix Ak (f) = Ak −BkfT ,

its characteristic polynomial is given by

det(λI − Ak (f)

)= λn + a1λ

n−1 + ...+ an. (2.26)

Then, Ak (f) is stable if and only if the following Hurwitz conditions are satisfied:

an > 0, an−2 > 0, ...

H1 > 0, H3 > 0, ... ,(2.27)

where Hi denotes the ith leading principal minor of the n× n Hurwitz matrix

H =

a1 a3 a5 · · · a2n−1

1 a2 a4 · · · a2n−2

0 a1 a3 · · · a2n−3

0 1 a2 · · · a2n−4

......

.... . .

...

0 0 0 · · · an

(ar = 0, r > n) . (2.28)

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Chapter 2. Simultaneous Stabilization 15

Note that there are several alternative versions of Hurwitz’s necessary and sufficient

conditions.

It is concluded that f is a feasible vector, f ∈ F , if and only if the above Hurwitz

stability constraints are satisfied for each k = 1, 2, ...,m. Moreover f ∈ Fµ if, in addition,

constraints −µ ≤ fi ≤ µ for i = 1, 2, ..., n are satisfied as well. Hurwitz’s necessary and

sufficient conditions for matrix stability can also be rewritten as a set of inequalities.

Hence, f ∈ F if and only if

gki (f) < 0, (2.29)

where gki (f) represents elements and leading principal minors of the Hurwitz matrix H

for matrix Ak −BkfT . In the light of this, Problem 1 can now be rewritten as:

Problem 2 Given parameter µ > 0, minimize S (f) =∑mk=1 Jk (f) with respect to f ,

subject to the constraints

−µ ≤ fi ≤ µ i = 1, 2, ..., n,

gkj (f) ≤ 0 k = 1, 2, ...,m, j = 1, 2, ...,M,

where M is the number of Hurwitz stability constraints.

2.5 Bi-Level Decomposition-Based Strategy

In the approach presented in [27], a bi-level design optimization architecture is adopted

in which design of each individual plant is taking place at the bottom level, and the top

level optimization aims for single control convergence of those individual controllers.

2.5.1 Decomposition formulation

An analogy between the simultaneous stabilization problem and the general multidisci-

plinary design optimization (MDO) problem formulation can be established by realizing

that each plant-stabilizing process is equivalent to a discipline in the general MDO pro-

cess, as shown in Fig. 2.1, in which the common design variable among all disciplines is

the simultaneous control gain K.

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Chapter 2. Simultaneous Stabilization 16

Figure 2.1: Multidisciplinary optimization and simultaneous stabilization problem [27]

A bi-level optimization strategy that enables decoupling and decomposition is used to

solve the simultaneous stabilization problem. At the system level (SL) the optimization

problem is stated as:

minzSL,ySLf(zSL, ySL)

subject to G∗j [zSL, z∗j , ySL, y

∗j (x∗j , y∗k, z∗j )] = 0

j, k = 1, ...,m, k 6= j

where f(zSL, ySL) represents the system-level objective function and Gjs are the compat-

ibility constraints, one for each subsystem. The lower subsystem-level (SSL) objective

function is formulated such that it minimizes the discrepancy between the given system

level variables and the subsystem variables that meet the local disciplinary constraints.

Thus, the jth subsystem optimization is stated as:

minzj ,yj ,xjGj[zSL, zj, ySL, yj(xj, yk, zj)]

=∑

(zSL − zj)2 +∑

(ySL − yj)2

subject to gj[xj, zj, yj(xj, yk, zj)] ≤ 0

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Chapter 2. Simultaneous Stabilization 17

2.5.2 Decomposed equivalent of necessary and sufficient condi-

tions

In simultaneous stabilization process, each plant-stabilization effort will be decomposed

at the subsystem levels, and the system level ensures stabilization with a unique control

gain K. using above formulation, a decomposed equivalent of Eq. (2.4) is:

SL|KSL,γSL min I = γSL (2.30)

subject to G∗j = 0 j = 1, ...,m

SSLj|Kj ,γj min Gj = (Kj −KSL)2 + (γj − γSL)2 (2.31)

subject to Re(λi,j) ≤ γj i = 1, ..., n

A solution to the problem will exist if and only if the system level converges with

an optimal control gain K∗, which minimizes the system-level objective function I and

makes it negative. Furthermore, at convergence, the subsystem compatibility constraints

(G∗j = 0) are met, which means each plant is stabilized by an optimal gain.

2.6 Parameter Optimization Approach

In the approach presented in [12], a parameter optimization methodology is used for

controller design. Since it is difficult to optimize a problem over multiple specifications

directly, a series of cost function are defined such that an improvement in a cost function

results in an improvement in the related specification.

In the parameter optimization framework, the controller elements are the variables

of optimization and hence it is very similar to a multiple objective parameter synthesis

(MOPS) method. However, instead of optimizing over the specifications directly (e.g.,

settling time, rise time, overshoot), cost functions are defined. This method then converts

a multiple-objective problem into a single-objective problem, by use of the weighted-sums

method.

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Chapter 2. Simultaneous Stabilization 18

2.6.1 Cost Function Definitions

In what follows, a number of cost function definitions associated with different types of

specifications are obtained. Later, a parameter optimization is performed to minimize a

weighted sum of these cost functions.

There are a number of different types of specifications that must be satisfied in the

design of a control system, some of which are listed in the following:

1. Tracking

To satisfy the tracking specifications, the designer may wish to minimize both the

transient and steady-state errors for a step command response. Hence, the following

cost functions are defined:

δJt =r∑i=1

∫ ∞0

δeidTδeiddt = tr

{BT A−TL0A

−1B}, (2.32)

Jtss =r∑i=1

eidssTeidss = tr

{(D − CA−1B

)T (D − CA−1B

)}, (2.33)

where L0 ≥ 0 satisfies the Lyapunov equation ATL0 + L0A+ CT C = 0.

2. Controller Limitations

It is also desirable to minimize the control response to a step command. To satisfy

the limitation of the control law, the designer may wish to minimize both the

transient and steady-state control effort, as well as the control rate. Thus, the

following cost functions are defined:

δJu =r∑i=1

∫ ∞0

δuiTδuidt = tr

{BTclA−Tcl LuA

−1cl Bcl

}, (2.34)

Juss =r∑i=1

uissTuiss = tr

{(Hcl −GclA

−1cl Bcl

)T (Hcl −GclA

−1cl Bcl

)}, (2.35)

and

Ju =r∑i=1

∫ ∞0

uiT

uidt = tr{BTclLuBcl

}, (2.36)

where Lu ≥ 0 satisfies the Lyapunov equation ATclLu + LuAcl +GTclGcl = 0.

3. Disturbance Rejection

The transient and steady-state response to disturbance can be calculated and the

corresponding cost functions can be defined.

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Chapter 2. Simultaneous Stabilization 19

4. Gain Limitations

To ensure that the controller gains do not become too large, we may penalize them

in the cost function by adding the following term:

Jgain = tr{KTK

}. (2.37)

5. Pole Location Constraint

The simplest constraint of this kind is to limit the closed-loop eigenvalues to a

region inside the left half plane (e.g., the closed loop poles are desired to lie to the

left of −δ, δ ≥ 0). The following LQ cost function ensures this specification:

JPL = E[∫ ∞

0xTQxdt

]= tr {PPLM0} , (2.38)

where x(t) is the unforced response of x = (Acl + δI)x, M0 = E[x(0)x(0)T ] > 0,

and PPL is the solution of (Acl + δI)TPPL + PPL(Acl + δI) +Q = 0.

6. Robustness and Fault Tolerance

One way to consider the robustness is to define a finite set of plants which reflects

the allowable deviations of the actual plant. For robust stability and robust per-

formance of this set of plants, LQ cost functions, as well as tracking, controller

limitations, and disturbance rejection cost functions can be added, respectively. In

the case of fault tolerance, the finite set of plants refers to the closed-loop systems

corresponding to allowable component failures.

2.6.2 Multiple Objective Design

Now the controller is designed by solving the single-objective optimization problem.

minKJ = wδtδJt + wtssJtss + wδuδJu + wussJuss

+wuJu + wδdδJd + wdssJdss + wPLJPL

+wgainJgain +W, (2.39)

where w ≥ 0 is the specific weight for each cost function, W is the weighted sum of any

additional cost function added for robustness consideration, and K is the control law.

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Chapter 2. Simultaneous Stabilization 20

The variable parameters are hence the weights, the damping ratios, and the un-damped

natural frequencies in the desired system. By finding a local minimum to (2.39), an

optimal solution is found with respect to the cost functions. This strategy is further

explained in [12].

2.7 Proposed SS Optimization Methodology

Based on the literature review, the following methodology for solving the problem of

simultaneous stabilization is proposed and further used for the work presented. The

motivation was to decrease the number of design variables and constraints in the opti-

mization problem. The optimization problem could later be solved using the MATLAB

function fminimax which finds a constrained minimum of the largest element of an ar-

ray of non-linear multivariable functions, starting at an initial estimate. In other words,

it minimizes the worst-case (largest) value of a set of multivariable objective functions.

This is generally referred to as the minimax problem. fminimax internally reformulates

the minimax problem into an equivalent Nonlinear Linear Programming problem by ap-

pending additional (reformulation) constraints of the form Fi (x) ≤ γ to the constraints,

and then minimizing γ over the design variable x. It then uses respective functions to

calculate the value of the objective functions and that of the constraints, and finds the

minimum using a sequential quadratic programming (SQP) method.

Again, consider the same collection of m linear time-invariant systems:

x (t) = Akx (t) +Bku (t) , k = 1, 2, ...,m (2.40)

where x (t) ∈ Rn is the state and u (t) ∈ R is the control, and Ak ∈ Rn×n, Bk ∈ Rn

are state and input matrices and independent of time. Also, consider the linear state

feedback controller of the form:

uk = −fTxk (2.41)

where f is known as the feedback gain vector. To limit the elements of the controller, f is

required to be the same for all the systems, and −µ ≤ fi ≤ µ, i = 1, 2, ..., n, where µ is

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Chapter 2. Simultaneous Stabilization 21

a given positive real constant. We may now state the simultaneous stabilization problem

as follows:

Problem Given parameter µ > 0 and η ≥ 0, minimize I = max Re (λi,k) with respect

to f , subject to the constraints

−µ ≤ fi ≤ µ, i = 1, 2, ..., n

αi,k + η |βi,k| ≤ 0, k = 1, 2, ...,m, i = 1, 2, ..., n.

Note that a solution to this optimization problem will give us a controller that will

simultaneously stabilize all the systems without using unnecessary control effort. The

application of this design procedure is further shown in the robustness analysis of a nu-

merical example in section 3.4, as well as the linear and nonlinear case studies introduced

in Chapters 5 and 6.

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Chapter 3

Robustness Analysis

In the previous chapter, the problem of simultaneous stabilization was introduced and

necessary and sufficient condition for the existance of such a controller was presented,

along with several techniques introduced in the control theory literature to design such a

controller. However, a less explored aspect is the robustness of such a control law. Since

uncertainties and parameter variations can certainly occur in real life aerospace applica-

tions, the designed controller should be able to provide stability in case of uncertainties

such as gusts, sudden change of flight conditions across the flight envelope, aircraft con-

figuration, and aerodynamic changes due to damages and failures. For instance, gusts

or winds influence aircraft dynamics. Low frequency wind has an effect on tracking per-

formance whereas high frequency wind affects the flight stability of an aircraft. That

being said, this thesis aims at investigating the robustness of the simultaneous stabi-

lization design methods for systems represented by a group of linear state-space models

with structured uncertainties. In the following, several robustness analysis approaches,

namely the extreme point solution, the Kharitonov’s theorem, and the stability robust-

ness bounds, are introduced to provide the physical bounds of allowable perturbations.

Furthermore, their application is shown using a numerical example.

22

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Chapter 3. Robustness Analysis 23

3.1 Kharitonov’s Theorem

In the simultaneous stabilization problem, the stability of the uncertain plants in the

imaginary space bounded by the given set of simultaneously stabilized plants is not

guaranteed. Hence, a method for analyzing the robustness of the design is discussed,

an application of which can be found in [26]. This approach, namely the Kharitonov’s

theorem, is presented as follows:

Definition 1 A Polynomial p(s, q) = {∑ni=0 δi(q)s

i|q ∈ Q} is an interval polynomial if

p(s, q) has an independent uncertainty structure, each coefficient depends continuously on

the uncertainty vector q, and the uncertainty bounding set Q is an n-dimensional box.

Associated with the interval polynomial are four fixed Kharitonov polynomial defined

from the upper and lower bounds of the interval polynomial coefficient as:

K1(s) = δ−0 + δ−1 s+ δ+2 s

2 + δ+3 s

3 + δ−4 s4 + δ−5 s

5 + ..., (3.1)

K2(s) = δ+0 + δ+

1 s+ δ−2 s2 + δ−3 s

3 + δ+4 s

4 + δ+5 s

5 + ..., (3.2)

K3(s) = δ+0 + δ−1 s+ δ−2 s

2 + δ+3 s

3 + δ+4 s

4 + δ−5 s5 + ..., (3.3)

K3(s) = δ−0 + δ+1 s+ δ+

2 s2 + δ−3 s

3 + δ−4 s4 + δ+

5 s5 + .... (3.4)

Theorem 1 An interval polynomial with invariant degree is robustly stable (Hurwitz for

every point in the uncertainty space) if and only if its four Kharitonov polynomials are

stable.

Proof Several proofs of Kharitonov’s Theorem are available in the literatures (see [16],[31],

and [25]).

The Kharitonov polynomials also provide the basis for a graphical check of robust

stability, using Zero Exclusion Condition.

Definition 2 Associated with the Kharitonov polynomials is a Kharitonov rectangle

whose four vertices are obtained by evaluating the four Kharitonov polynomials at s =

jw0. The size and position of this rectangle change with w, while its sides always remain

parallel to the real and imaginary axis.

If an interval polynomial with an invariant degree of n is robustly stable, the Kharitonov

polynomials will move in a counter-clockwise direction through n quadrants of the com-

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Chapter 3. Robustness Analysis 24

plex plane without touching or passing through the origin (Zero Exclusion Condition).

Theorem 2 An interval polynomial having invariant degree is robustly stable if and

only if the origin of the complex plane is excluded from the Kharitonov rectangle at all

nonnegative frequencies, i.e. 0 /∈ p(jw, q),∀w ≥ 0.

The Kharitonov’s theorem allows for significant reduction in the number of polyno-

mials to be checked for stability in order to assure robust stability of the design. For

the controller design purposes, an interval polynomial is defined from the closed-loop

characteristic equation.

3.2 Extreme Point Solution

In this approach, robustness of the designed controller can be investigated by checking

robust stability of a polytope of matrices. This method, as presented in [38] and [37],

provides a solution to the problem in the form of extreme points, using the fact that robust

stability problem can be converted to the robust nonsingularity problem involving the

Kronecker Lyapunov Matrix. Consider the following linear state-space description:

x = A(q)x(t) q ∈ Q, (3.5)

where x(t) ∈ Rn and q is an s vector of uncertain parameters varying in the prescribed

compact set Q. Let qi be given upper and lower bounds such that qi ≤ qi ≤ qi for

i = 1, ..., s. Now the matrix A(q) can be written as

A(q) = A0 +s∑i=1

qiAi (3.6)

where A0 is the nominal matrix and Ai are constant specified matrices, reflecting the

structure of the uncertainty. The set of possible matrices A(q) = [A(q) : q ∈ Q] forms a

polytope of matrices in Rn×n. Denoting qi as the ith extreme point of the set Q, generated

by each element in q taking its minimum or maximum value, and the extreme matrix

A(qi) as Ai, the above matrix family can be written as

A =

{A =

l=2s∑i=1

αiAi, αi ≥ 0,

∑αi = 1

}. (3.7)

It is assumed that all the vertex matrices Ai are asymptotically stable.

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Chapter 3. Robustness Analysis 25

3.2.1 Stability as a Nonsingularity Problem via the ‘Kronecker

Lyapunov Matrix’

As said before, the robust stability problem can be converted to a robust nonsingularity

problem involving the Kronecker Lyapunov matrix denoted by L in the ‘Dagger’ space,

defined as follows:

L = A† = A⊗ In + In ⊗ A, (3.8)

where In is the identity matrix and L is a square matrix of dimension m = 12n(n + 1).

By defining Li as Li = (Ai)†, any member of the matrix polytope can simply be written

as a member of the ‘dagger polytope L’ as:

L =

{L =

l=2s∑i=1

αiLi, αi ≥ 0,

∑αi = 1

}. (3.9)

It is explained in [38] that the polytopic matrix family ‘A’ with all its vertex matrices

being asymptotically stable, is robustly stable if and only if the matrix family ‘L’ is

robustly ‘real axis nonsingular’ and thus is robustly asymptotically stable. Here, ‘real

axis nonsingularity’ means the same sign nature of the real eigenvalues and the real parts

of the complex pair eigenvalues of the ‘Kronecker Lyapunov’ matrix family.

Now let λj be the jth eigenvalue (j = 1, 2, ...,m) of the ‘dagger space’ matrix L with

r real eigenvalues, and let D(L) denote the ‘Weighted Determinant’ of matrix L as:

D(L) = (−1)mλ1λ2...λm. (3.10)

By defining the concepts of ‘Weighted Real Axis Determinant’ and ‘Real Axis Non-

singularity Scalar’ as below, it is clear that they are constrained to be positive and real

for a real axis nonsingular matrix.

Γ(L) =([(−1)rλ1λ2...λr]

1r

) ([(−1)m−rλr+1λr+2λm

] 1r

)= ∆(L)σ(L). (3.11)

Thus, the polytopic matrix family ‘A’ is robustly stable if and only if each member

of the matrix family ‘L’ possesses a positive real Γ.

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Chapter 3. Robustness Analysis 26

3.2.2 Necessary and Sufficient Vertex Solution for Robust Sta-

bility

The interior matrices belonging to the matrix polytope L can be expressed as a con-

vex combination of not just the vertex matrices Li, but of a set of special matrices

labelled as ‘Virtual Center Anchored Virtual Rays’, namely Lvc,k(ρc, i1, i2, .., ik, j) =

[Lj + ρcLvc,k(i1, i2, ..ik)]. For Example:

L = γ(α1L1 + α2L

2 + α3L3) = γ1L

1 + γ2L2 + γ3L

3 =

α1[L1 + ρ1(L1 + L2 + L3)] + α2[L2 + ρ2(L1 + L2 + L3)] + α3[L3 + ρ3(L1 + L2 + L3)].

Now define a set of ‘Real Axis Nonsingularity’ matrices as follows:

Lrn,k(i1, i2, ..ik, j) = −[(Lvc,k(i1, i2, ..ik))

−1 Lj], (3.12)

where Lvc,k(i1, i2, ..ik) denotes the ‘virtual center’ matrix formed with k matrices taken

at a time (i1 = 1, 2, ..l; ..., ik = 1, 2, ..l) as:

Lvc,k(i1, i2, ..ik) = (L1 + L2 + ...+ Lk). (3.13)

A necessary and sufficient ‘extreme point’ condition for checking the robust stability

of a polytope of matrices, as further explained in [38], is presented in the following.

Theorem All the matrices belonging to the polytopic matrix family ‘A’ are asymptot-

ically stable if and only if for all (i1 = 1, 2, ..l; i2 = 1, 2, ..l; ..., il = 1, 2, ..l) the ‘Real

Axis Nonsingularity Matrices’ Lrn,k(i1, i2, ..ik, j) with k taking on values from 2 thru l

are ‘Real Axis Nonsingular’ (i.e., possess positive real Γ) and thus are ‘Asymptotically

Stable’.

3.3 Stability Robustness Bounds

As discussed before, the problem of maintaining the stability of a nominally stable system

subjected to perturbations has been of considerable interest to researchers. In this section,

the focus is on the analysis of system stability with parameter uncertainty in state-

space models. In particular, we are interested in obtaining some bounds on the system

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Chapter 3. Robustness Analysis 27

uncertainties that guarantee the stability of the perturbed system, assuming that the

nominal system is already stable.

Stability analysis in the area of time domain stability conditions has been available for

some time. However, explicit bounds on the perturbation of a linear system in order to

maintain stability is fairly a new topic which has first been reported only by Patel, Toda,

and Sridar [22], and Patel and Toda [24]. These bounds are given for ‘’highly structured

perturbations‘’ as well as for ‘’weakly structured perturbations‘’. For a given model

structure, highly structured perturbations are those for which only a magnitude bound

on individual matrix elements is known. Weakly structured perturbations are those for

which only a spectral norm bound for the error is known. Mathematical approaches

presented here will provide a bound for highly structured perturbations.

In what follows, the problem of robust stability analysis of linear systems in state-

space models is considered. It is known that every possible state matrix or the polytope

of the matrices can be written as the sum of a nominal stable plant plus some uncertainty

matrices. Considering a system with structured uncertainty and using a Lyapunov matrix

equation solution, an upper bound on allowable structured perturbations can be found

which maintains the stability of the nominal system. Different methods in driving this

upper bound is proposed in the literatures with different levels of conservatism. In [24],

the following state space representation of a perturbed dynamic system is given:

x = Ax(t) + Ex(t) = (A+ E)x(t), (3.14)

where x is the n-dimensional state vector, A is an n × n time invariant, asymptotically

stable nominal matrix, and E is an n × n error matrix. Moreover, the entries of E are

such that

|Eij| ≤ ε, (3.15)

where ε is the magnitude of the maximum deviation allowed. It was shown in [24] that

the perturbed system is stable if

ε <1

nσmax [P ]= µεP , (3.16)

where σmax [P ] represents the largest singular value of P , and P is the solution of the

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Chapter 3. Robustness Analysis 28

Lyapunov matrix equation:

ATP + PA+ 2In = 0, (3.17)

where In is an n× n identity matrix.

Patel and Toda [24] derived the stability robustness bounds with the assumption that

every element of the system matrix is perturbed independently of the other. Yedavalli

[36] on the other hand, obtained a less conservative robustness bound by assuming more

structure on the perturbation, as presented in the following. It was shown in [36] that

the system matrix A+ E of Eq. (3.14) is stable if:

|Eij|max = ε <1

σmax [|P |Un]s= µεY , (3.18)

where |P | denotes a matrix formed by taking the absolute value of every element of P ,

[P ]s denotes the symmetric part of P , and Un is an n × n matrix whose entries are

unity, i.e., Unij = 1 for all i, j = 1, ..., n if the corresponding element in A is subject to

perturbation. Moreover, P satisfies the Lyapunov equation given in Eq. (3.17).

That being said, it has been assumed in both approaches presented in the above that

the perturbations in the various elements of the system matrix are independent of one

another, which will introduce additional conservatism in the perturbation bounds. In

order to take this fact into account, another approach for finding the stability robustness

bounds for systems with structured uncertainty was proposed in [40]. This method is

hence less conservative than the previously presented approaches. Assume the error

matrix E has the following form:

E =r∑i=1

kiEi, (3.19)

where Ei are constant matrices, and ki are uncertain parameters assumed to vary in

intervals around zero, i.e., ki ∈ [−εi, εi]. It is seen that this type of formulation allows

for perturbations in different elements of the system to be dependent one another. The

proposed theorem in [40] is as follows.

Theorem Consider the linear system in Eq. (3.14) with A being nominal and stable,

and E of the form of Eq. (3.19). Let P be the solution to the Lyapunov equation (3.17).

Define Pi as:

Pi = (ETi P + PEi)/2 = [PEi]s , i = 1, 2, ...r. (3.20)

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Chapter 3. Robustness Analysis 29

The perturbed system (A+ E) is stable if

|Kj| < 1/σmax

(r∑i=1

|Pi|), j = 1, 2, ...r. (3.21)

3.4 Robustness Analysis of a Numerical Example

In this section, a numerical example as introduced in [19] has been chosen in order to

show the application of the previously introduced robustness investigation techniques.

It is the problem of stabilizing an F4E fighter aircraft as was done by Petersen (1987)

and Wu et al. (1990), where there are four three-dimensional systems to be stabilized,

and each system corresponds to the aircraft’s travelling at a different altitude and speed.

The numerical problem of stabilizing an F4-E fighter jet aircraft is described by the

state-space model asx1,k

x2,k

x3,k

=

a11 a12 a13

a21 a22 a23

0 0 −30

x1,k

x2,k

x3,k

+

b1

0

30

uk. (3.22)

where x1,k represents the normal acceleration a, x2,k represents the pitch rate q, and

x3,k represents the elevator angle δe. This state-space model includes the dynamics of

the actuator with the input being the stick position and the output being the elevator

deflection. The values of the parameters of the above model are given in Table 3.1, and

each system states the dynamics of the aircraft at a different altitude and speed. Using

the proposed optimization strategy in section 2.7, a simultaneously stabilizing controller

has been found. Later, three different approaches for investigating the robustness of the

designed controller, namely the extreme point solution, the stability robustness bounds,

and the Kharitonov’s theorem, have been applied to investigate the robustness of this

simultaneously stabilizing controller. The following simultaneously stabilizing controller

is hence constructed as specified in section 2.7, which ensures the stable operation of the

aircrfat at four different flight conditions.

f =

0.13241

2

−0.27757

. (3.23)

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Chapter 3. Robustness Analysis 30

Table 3.1: F4-E flight operating conditions

Operating Point 1 2 3 4

Altitude, ft 5000 35000 5000 35000

Mach number 0.5 0.9 0.85 1.5

a11 -0.9896 -0.6607 -1.702 -0.5162

a12 17.41 18.11 50.72 26.96

a13 96.15 84.34 263.5 178.9

a21 0.2648 0.08201 0.2201 -0.6896

a22 -0.8512 -0.6587 -1.418 -1.225

a23 -11.89 -10.81 -31.99 -30.38

b1 -97.78 -272.2 -85.09 -175.6

In the following sections, the robustness of the controller constructed for the numerical

example is initially investigated by applying the extreme point solution method. Then,

Kharitonov’s theorem is applied to the same example and again robustness of the design is

investigated. Finally, the stability robustness bounds are found to give an understanding

of the robustness of the design.

3.4.1 Extreme Point Solution Application

In the following, the extreme point solution approach is used for analyzing the robustness

of the controller designed for the numerical example. For the robustness investigation,

the first plant out of the set of 4 plants for which the simultaneously stabilizing controller

has been designed, was considered and the closed-loop system was obtained. The state

matrix of the stable closed-loop system has then been chosen as the nominal stable plant,

assuming that every element of the state and input matrices is subject to perturbations.

Uncertainty vector and constant specified matrices Ai were hence formulated and max-

imum and minimum bounds on the elements of the uncertainty vector has also been

found according to the boundary of perturbations for the elements of the state and input

matrices. The machinery behind this kind of formulation is briefly presented here. The

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Chapter 3. Robustness Analysis 31

closed-loop system in state-space model has the form of:

x (t) = (A−BfT )x (t) ,

where A is the state matrix, B is the input matrix, and f = [f1 f2 f3] is the controller.

Hence:

A−BfT =

a11 a12 a13

a21 a22 a23

a31 a32 a33

−b1

b2

b3

[f1 f2 f3

]

=

a11 − b1f1 a12 − b1f2 a13 − b1f3

a21 − b2f1 a22 − b2f2 a23 − b2f3

a31 − b3f1 a32 − b3f2 a33 − b3f3

= A0,

where ai,j ≤ ai,j ≤ ai,j for i, j = 1, 2, 3, and bi ≤ bi ≤ bi for i = 1, 2, 3. Therefore the

perturbed closed loop system will have the form of

Acl = A0 + ∆A = A0 +

∆a11 − (∆b1)f1 ∆a12 − (∆b1)f2 ∆a13 − (∆b1)f3

∆a21 − (∆b2)f1 ∆a22 − (∆b2)f2 ∆a23 − (∆b2)f3

∆a31 − (∆b3)f1 ∆a32 − (∆b3)f2 ∆a33 − (∆b3)f3

,

where ai,j − ai,j = ∆ai,j ≤ ∆ai,j ≤ ∆ai,j = ai,j − ai,j for i, j = 1, 2, 3, and bi − bi =

∆bi ≤ ∆bi ≤ ∆bi = bi − bi for i = 1, 2, 3. Now the uncertainty vector q and the constant

specified matrices can be defined as:

q =

q1

q2

...

q9

=

∆a11 − (∆b1)f1

∆a12 − (∆b1)f2

...

∆a33 − (∆b3)f3

,

and

A1 =

1 0 0

0 0 0

0 0 0

, A2 =

0 1 0

0 0 0

0 0 0

, · · · A9 =

0 0 0

0 0 0

0 0 1

,

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Chapter 3. Robustness Analysis 32

where the lower and upper bounds on the uncertainty vector elements, qs, can be found

as ∆ai,j − (∆bi)fj = qs ≤ qs ≤ qs = ∆ai,j − (∆bi)fj, for i, j = 1, .., 3, and s = 1, .., 9.

It can be seen that using this formulation, the uncertainty matrix has 9 elements and

hence there are nine constant specified matrices. Applying the extreme point solution

method, all the 29 different vertices, Ai, can be found by substituting 29 different combi-

nations of lower and upper values of the uncertainty vector elements in the closed-loop

state matrix. By transforming these vertices into the dagger space and constructing

Li matrices, and then generating real axis nonsingularity matrices, we can check their

asymptotic stability. If all of the real axis nonsingularity matrices as well as vertex ma-

trices are stable, it can be concluded that all the matrices belonging to the polytopic

matrix family of A, which is the set of all the possible perturbed closed-loop systems,

are asymptotically stable as well.

In the analysis process, using the first plant as the nominal system, a code was

developed to generate the uncertainties, their corresponding constant specified matrices,

and the real axis nonsingularity matrices in the dagger space. Arbitrary bounds for

the elements of the state and input matrices A1, B1 were chosen and in each case the

performance of the closed-loop system subjected to perturbations was investigated. For

some cases of the arbitrarily chosen boundaries, there are a number of vertices that are

not asymptotically stable. Thus, some of the matrices in the perturbed matrix family

A are not stable as well. In other words, the nominal stable plant is not guaranteed

to maintain its stability when it is subjected to perturbation in its elements within the

selected boundary. On the other hand, it is possible to find a boundary for perturbations

in the system elements that allows the nominal plant to maintain its stability when

subjected to disturbances in that specific system element.

In this investigation, the initial effort was to find a boundary for the state and input

matrix elements that would contain those of the four primary plants, while maintaining

the stability of every possible perturbed closed-loop system within that range. However,

it was not a realistic expectation since the four initial plants are located far away from

each other across the flight envelope.

In Table 3.2, a region of perturbation in the state and input matrix elements has been

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Chapter 3. Robustness Analysis 33

chosen arbitrarily, for which all of the real axis nonsingularity matrices are assymptoti-

cally stable. Thus, any perturbed closed-loop matrix lying inside the region specified by

those boundaries, is also stable. Hence, it is guaranteed that by imposing these bound-

aries on the system uncertainties, the perturbed closed-loop system remains stable. The

simulation results in Fig. 3.1 show the response to initial condition of the first system

when it is arbitrarily perturbed within the allowable boundaries of change given in Table

3.2. It is obvious that since every perturbed closed-loop system is proved to be stable

within the given range, the state response to initial condition should be converging to

zero as well. This is clearly shown in Fig. 3.1, where the response to initial condition is

in fact converging to zero.

Table 3.2: Allowed perturbation resulting in a stable matrix family

∆a11 ∆a12 ∆a13 ∆a21 ∆a22 ∆a23 ∆a31 ∆a32 ∆a33 ∆b1 ∆b2 ∆b3

min 0.1 0.5 0.5 0.2 0.2 1 0.2 0.2 1 0.7 0.2 1

max -0.1 -0.5 -0.5 -0.2 -0.2 -1 -0.2 -0.2 -1 -0.7 -0.2 -1

Table 3.3 shows another case of arbitrarily chosen boundaries which will result in

some of the vertices being unstable. Therefore, it is not guaranteed that all of the

perturbed closed-loop systems will maintain stability when disturbed within the chosen

ranges shown in Table 3.3. This is equivalent to saying that the designed controller is not

robust for the perturbations in the chosen boundary around the first nominal plant. This

can be regarded as an example of a perturbed nominal system for which stability within

the chosen boundaries is not guaranteed. It is hence concluded that the simultaneously

stabilizing controller is not robust if the first nominal system is subjected to perturbation

within the given boundaries shown in Table 3.3.

Table 3.3: Allowed perturbation resulting in an unstable matrix family

∆a11 ∆a12 ∆a13 ∆a21 ∆a22 ∆a23 ∆a31 ∆a32 ∆a33 ∆b1 ∆b2 ∆b3

min 0.5 0.5 0.5 0.2 0.2 1 0.2 0.2 1 0.7 0.2 1

max -0.5 -0.5 -0.5 -0.2 -0.2 -1 -0.2 -0.2 -1 -0.7 -0.2 -1

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Chapter 3. Robustness Analysis 34

Figure 3.1: response to initial condition of the perturbed nominal plant

3.4.2 Kharitonov’s Theorem Application

In the following, the Kharitonov’s Theorem has been used for investigating the robust-

ness of the simultaneously stabilizing controller designed for the numerical example. For

this investigation, the first plant out of the set of 4 plants was considered again and the

closed loop system was generated. The characteristic equation of this closed loop system

is found by solving the equation det(A − sI) = 0. Assuming that every element of the

state and input matrices can be perturbed, the coefficients of this characteristic equa-

tion is subject to changes accordingly. By varying the closed-loop system elements in a

respective boundary and finding the perturbed closed-loop systems, the corresponding

characteristic equation and its perturbed coefficients can easily be obtained. The last

step is to find the minimum and maximum values for each of these coefficients and gen-

erate the four Kharitonov’s polynomials, as introduced before. If the four Kharitonov’s

polynomials are stable, the family of the perturbed closed-loop systems is also stable.

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Chapter 3. Robustness Analysis 35

To investigate the effect of uncertainties on the system’s stability, the first element

of the state matrix (i.e. a11) was subjected to perturbations such as −1.2 ≤ a11 ≤ −0.6.

For different values of a11, the closed-loop system A1 − B1fT was generated and using

the ss2tf command in MATLAB, the characteristic equation and its coefficients were

obtained. Later, four Kharitonov’s polynomials were constructed and checked for sta-

bility. Table 3.4 presents the four Kharitonov’s polynomials obtained for the numerical

example, and Fig. 3.2 shows the response to initial condition of the closed-loop sys-

tem with an arbitrary value of a11 in the given allowable boundary. The stability of

Kharitonov’s polynomials was investigated and confirmed, and it was concluded that the

perturbed closed-loop system maintains stability for the given arbitrary disturbance in

the a11 element. Therefore, the response to initial condition of the system is expected to

be converging to zero, as plotted in Fig. 3.2.

Table 3.4: Kharitonov’s polynomials coefficients

Kh = a3s3 + a2s

2 + a1s+ a0

a3 a2 a1 a0

Kh1 1 8.1 766.5 1149.7

Kh2 1 7.5 768.8 1575.4

Kh3 1 7.5 766.5 1575.4

Kh4 1 8.1 768.8 1149.7

3.4.3 Stability Robustness Bound Application

In this section, the approaches proposed in [24] and [36] are applied to the numerical

example in order to find an upper bound on allowable perturbations, while maintaining

the stability of the disturbed nominal system. Again, The first closed-loop system has

been chosen as the nominal stable plant and the error matrix due to the introduced

perturbations is found. Following the results of [24], the Lyapunov matrix equation is

solved for the nominal plant A1 − B1fT , and the solution P is found. By finding the

maximum singular value of matrix P and using Eq. (3.16), an upper bound on the entries

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Chapter 3. Robustness Analysis 36

Figure 3.2: response to initial condition of the perturbed nominal plant

of the error matrix (or uncertainty vector elements) is obtained. In another approach, the

method presented in [36] uses the same Lyapunov matrix P , but results in a different and

less conservative upper bound (Eq. 3.18). These bounds represent the range of allowable

perturbation in the state and input matrix elements, which guarantees the stability of

the perturbed nominal plant.

The results obtained using the two methods presented in [24] and [36], are shown

in Table 3.5. These results have been obtained following the same previous formulation

for robustness investigation, with the first closed-loop system being the nominal plant

A0, and its perturbation ∆A being the error matrix. It has been assumed that the

perturbations in the elements of the state and input matrices are independent, due to

the assumptions in [24]. However, if the formulation accounts for the perturbations of

these elements to be dependent on each other, it is expected that a less conservative

robustness bound should be obtained.

It is hence concluded that the two techniques, namely the Extreme Point Solution

and the Kharitonov’s Theorem, work with an introduced range of perturbation applied to

the elements of a stable system, and determine whether or not the perturbed system will

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Chapter 3. Robustness Analysis 37

Table 3.5: Stability robustness bounds

Patel and Toda [24] Yedavalli [36]

µεP µεY

0.0418 0.0540

remain stable under uncertainties. The extreme point solution method, however, obtains

an upper bound on the allowable perturbation for which the stability of the disturbed

system is guaranteed. Finally as it is seen in Table 3.5, the method provided in [36] gives

a less conservative result compared to the one proposed in [24].

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Chapter 4

Robust Simultaneous Stabilization

Problem

In the previous chapters, the concept of Simultaneous stabilization problem and several

methods to solve the problem were presented, along with techniques and tools that can be

used to measure the robustness of the designed controller and the level of the closed-loop

system’s vulnerability to disturbances. However, it is now clear that such a controller

has to be able to not only simultaneously stabilize a set of plants, but also maintain

that stability when encountered with perturbations and uncertainties. It is therefore

important to address the robustness of the simultaneous stabilization, now known as

the Robust Simultaneous Stabilization (RSS) problem. Hence, the new objective is to

provide stability for all of the systems simultaneously, while maximizing the stability

robustness bounds.

In this chapter, the robustness tools introduced before are implemented in the design

process such that the final design is an improved controller in terms of robustness. First,

such a design approach presented in [29] is introduced which aims at improving the

robustness. Then the concept of multi-objective modelling is briefly addressed as shown in

[28], which is later used when a new design approach for the problem of RSS is presented.

More specifically, a combination of optimal stabilization and parameter optimization

methods (introduced in Chapter 2) has contributed in the development of this design

approach, with the emphasis of the cost function not on the performance or transient

38

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Chapter 4. Robust Simultaneous Stabilization Problem 39

response of the design, but rather on its robustness. The resulted multiple-objective

problem is later converted into a single-objective problem.

4.1 An Extended Decomposition-Based Strategy for

the RSS Problem

A decomposition approach for the solution of the simultaneous stabilization and simul-

taneous optimal control problem has been proposed in [27], as described in Chapter 2.

The decomposition approach was later extended in [29] to consider stability robustness

bounds. As presented here, This strategy was proposed to solve the problem of simul-

taneously stabilizing a set of plants while maximizing the upper bound on the allowable

structured uncertainty, which preserves the stability of each closed-loop system.

Again, consider the set of m different plants described by the state-space equation

xk (t) = Akxk (t) +Bkuk (t) , k = 1, 2, ...,m (4.1)

yk (t) = Ckxk (t) (4.2)

where xk ∈ Rn and u is a scalar control. The goal of the robust simultaneous stabilization

is to simultaneously stabilize a set of different systems by designing a single feedback

controller which can maintain stability in the presence of bounded uncertainties for each

plant. The structured uncertainty for each plant is represented by an n× n error matrix

Ek and the state-space equations:

xk (t) = (Ak + Ek)xk (t) +Bkuk (t) , (4.3)

yk (t) = Ckxk (t) , k = 1, 2, ...,m (4.4)

The entries of the uncertainty matrix Ek is bounded as

|Ek| ≤ ε (4.5)

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Chapter 4. Robust Simultaneous Stabilization Problem 40

where |Ek| denotes the absolute value of the entries of matrix Ek, and ε is the magnitude

of the maximum deviation in matrix entries. Again, a single feedback control law

uk (t) = −fTxk (t) (4.6)

is sought such that each one of the perturbed closed-loop system

xk (t) =(Ak −Bkf

T)xk (t) + Ekxk (t) = (ACLk + Ek)xk (t) (4.7)

remains stable under uncertainties.

The decomposition approach to the simultaneous stabilization problem originally pro-

posed in [27] was based on its analogy to a general multi-disciplinary optimization (MDO)

problem. Now an extended version of that approach, aimed at solving the RSS problem

[29], is proposed:

SL

∣∣∣∣∣∣∣∣min max

fSL,γSL− µ∗eYk

s.t. G∗k = 0, k = 1, 2, ...,m

SSLk

∣∣∣∣∣∣∣∣minfk,γk

Gk = (fk − fSL)2

s.t. Re (λi,k) ≤ γk < 0, i = 1, ..., n

(4.8)

where µ∗eYk is the upper bound on the structured uncertainties maintaining the stability

of the kth closed-loop system, corresponding to the subsytem optimal solution gain of

f ∗k . Such upper uncertainty bound is defined in [36] as:

|Ek|max = ε <1

σmax [|P ∗k |Un]s= µ∗eYk (4.9)

where Un is an n× n matrix whose entries are unity for each element of the closed-loop

system which is subjected to perturbations, and zero otherwise. P ∗k is the solution to the

Lyapunov matrix equation:

P ∗k(Ak −Bkf

∗T)

+(Ak −Bkf

∗T)TP ∗k + 2In = 0 (4.10)

where In is an n× n identity matrix.

Figure 4.1 shows the concept of the solution that simultaneously stabilize a set of m

plants while improving the robustness. The system-level optimization process maximizes

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Chapter 4. Robust Simultaneous Stabilization Problem 41

stability robustness bounds while the constraint enforces a unique gain solutionn for all of

the systems. The subsystem-level optimization process minimizes the discrepancy with

the system level gain solution while enforcing internal stabilization of each system under

the feedback control law.

Figure 4.1: Robust Simultaneous Stabilization solution approach

This methodology tries to increase the value of µ∗eYk for the system with the lowest

stability robustness bound at each step, which is obviously the most vulnerable system to

disturbances at that time. This may result in the overall optimization of the robustness

bounds being hurt. The reason is that the value of µ∗eYk for a single vulnerable plant can

be increased at the cost of decreasing that of the other plants substantially. As a result,

some of the plants may see a huge drop in their stability robustness boundaries and if

this outcome is not desirable, another optimization methodology should be presented.

The problem of increasing the robustness boundary of the more vulnerable plants while

improving or maintaing the same level of robustness for the others, can be treated as a

multi-objective modelling problem.

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Chapter 4. Robust Simultaneous Stabilization Problem 42

4.2 Multi-Objective Optimization

Many real-world search and optimization problems are naturally posed as non-linear pro-

gramming problems having multiple objectives. A multi-objective optimization problem

(MOOP) deals with more than one objective function and in most practical decision-

making problems, multiple objectives or multiple criteria are evident. Due to the lack

of suitable solution techniques, such problems were artificially converted into a single-

objective problem and solved. However, there exist a number of fundamental differences

between the working principles of single- and multi-objective optimization algorithms.

In a single-objective optimization problem, the task is to find one solution which op-

timizes the sole objective function. A multi-objective optimization is much more than

this simple idea. A MOO problem also has two different search spaces. The objective

functions constitute a multi-dimensional space, in addition to the usual decision variable

space common to all optimization problems.

The difficulty arises because such problems give rise to a set of trade-off optimal

solutions (known as Pareto-optimal solutions), instead of a single optimum solution. All

such trade-off solutions are optimal solutions to a multi-objective optimization problem.

Often, such trade-off solutions provide a front on an objective space plotted with the

objective values. This front is called the Pareto-optimal front and all such trade-off

solutions are called Pareto-optimal solutions.

4.2.1 Formulation & the Concept of Pareto Optimality

Where ‘many-wish’ attributes are involved in the design process, a multi-objective opti-

mization formulation can be applied to the design process where every ’wish attribute’

is modelled by an objective function. This formultion is presented mathematically as:

minimize[f1 (x) , f2 (x) , · · · , fNobjs (x)

]Twith respect to x ∈ Rn

subject to g (x) ≤ g0,

xl ≤ x ≤ xu

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Chapter 4. Robust Simultaneous Stabilization Problem 43

where the optimal solution x∗ can be found using already known optimization algo-

rithms. Most multi-objective optimization algorithms use the concept of dominance in

their search. To better understand this term, the concept of ’Pareto Optimality’ must

first be defined.

Definition A design vector x∗ ∈ D is Pareto optimal if and only if there is no vector

x ∈ D with the characteristics

fi (x) ≤ fi (x∗) for all i = 1, 2, ..., Nobjs (4.11)

and

fi (x) < fi (x∗) for at least one i, 1 ≤ i ≤ Nobjs (4.12)

By applying this definition to the decision variable space, there will be a set of solu-

tions, any two of which do not dominate each other. This set also has another property.

For any solution outside of this set, we can always find a solution in this set which will

dominate the former. Thus, this particular set has a property of dominating all other so-

lutions which do not belong to this set. In simple terms, this means that the solutions of

this set are better compared to the rest of solutions. This set, the resulted non-dominated

set for the given set of solutions, is given a special name called the Pareto-optimal set.

More importantly, the Pareto-optimal set always consists of solutions from a particular

edge of the feasible search region.

In design optimization, Pareto optimality represents a rational choice of solutions

and provides more information as the tradeoffs in the design process are now visible.

However, a design supercriterion or preference is necessary to choose the best solution

from a set of Pareto-optimal solutions.

4.2.2 Weighted-Sum Method

In this section, a number of commonly-used classical multi-objective optimization meth-

ods are briefly mentioned. In most cases, multi-objective problems can be converted into

a single-objective problem or a series of constrained optimization problems by modelling

design preferences and objective tradeoffs.

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Chapter 4. Robust Simultaneous Stabilization Problem 44

Figure 4.2: Pareto set in a convex objective space

Figure 4.3: Pareto Set in a non-convex objective space

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Chapter 4. Robust Simultaneous Stabilization Problem 45

(a) Weighted-sum method in a convex objec-

tive space

(b) Weighted-sum method in a non-convex ob-

jective space

Figure 4.4: Pareto sets in a weighted-sum optimization problem

Generally, if a relative preference factor among the objectives is known for a specific

problem, there is no need to solve a multi-objective optimization problem. A simple

method would be to form a composite objective function as the weighted sum of the ob-

jectives, where a weight for an objective is proportional to the preference factor assigned

to that particular objective. The values of the weights depend on the importance of each

objective in the context of the problem and a scaling factor.

This method of scalarizing an objective vector into a single composite objective func-

tion converts the multi-objective optimization problem into a single-objective optimiza-

tion problem. When such a composite objective function is optimized, in most cases it

is possible to obtain one particular trade-off optimal solution. However, the trade-off so-

lution obtained by using the preference-based strategy is largely sensitive to the relative

preference vector w used in forming the composite function, and a change in this prefer-

ence vector will result in a different trade-off solution. Having said that, this procedure

cannot be used to find Pareto-optimal solutions which lie on the non-convex portion of

the Pareto-optimal front, as shown in Figure 4.4.

In the most common case of weighted-sum method, a composite objective function

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Chapter 4. Robust Simultaneous Stabilization Problem 46

can be formed by summing the weighted objectives and the problem is then converted

to a single-objective optimization problem as follows:

minimize f (x, p) =Nobjs∑i=1

wifi (x, p) (4.13)

A few other commonly-used classical multi-objective optimization methods, such as

the Global Criterion Method or the Compromise Programming Method, are also ad-

dressed in [28]. Each one of them could have been used in the process of proposing a

new design approach to the problem of RSS.

4.2.3 Multiple Robustness Optimization

The designed controller has a certain degree of robustness in case of uncertainties, de-

pending on the closed-loop system for which it is beeing used. Assuming a system with

structured uncertainty and a stable nominal plant, an upper bound on the allowable

perturbations can be found, which maintains the stability of the uncertain closed-loop

system. As before, assume the following perturbed state-space equation:

x = ACLx (t) + Ex (t) = (ACL + E)x (t) (4.14)

where x is the n-dimensional state vector, A is an n×n time invariant and asymptotically

stable matrix, and E is the n× n error matrix such that

|Ei,j| ≤ ε

and ε is the magnitude of the maximum allowable deviation.

As it was stated before, the perturbed closed-loop system ACL + E is stable if:

|Ei,j|max = ε <1

σmax [|P |Un]s= µeY (4.15)

where σmax [.] represents the largest singular value of a matrix, |.| is formed by taking

the absolute value of every element of that matrix, and [.]s denotes the symmetric part

of it. Here, P is the solution of the Lyapunov matrix equation:

ATCLP + PACL + 2In = 0 (4.16)

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Chapter 4. Robust Simultaneous Stabilization Problem 47

In the approach presented in Chapter 2, the goal is to simultaneously stabilize all the

systems by minimizing the maximum real part of the eigenvalues as much as possible and

making it negative, within a given range for the controller enteries and a performance in-

dex. However, the following solution to the RSS problem tries to simultaneously stabilize

all the systems by imposing it as a constraint, while maximizing the stability robustness

bounds as the new objective function. Therefore, The wish attribute for each system

is to maximize its stability robustness bounds, µeYk . To avoid any confusion in Multi-

Objective Optimization problems, most applications convert a maximization problem

into a minimization problem and treat every problem as a combination of minimizing all

objectives. Now in order to simultaneously increase the robustness bound of each sys-

tem, µeYk , as much as possible, let us define the following wish attribute as an objective

function for the k-th system:

Jk =1

µeYk, k = 1, 2, ...,m (4.17)

where minimizing the above objective function is equall to maximizing the robustness

bound of each closed-loop system. Now the problem of maximizing µeYk is converted to

the problem of minimizing the objective function Jk = 1µeYk

, as the greater the stability

robustness boundary, the lower the new objective function Jk. Since the goal is to

increase the robustness of all of the systems under a single stabilizing controller f , this

problem can be treated as a ”many wish-many attributes” problem. In the previous

section several methods for solving such a problem were discussed. Here, the Wighted-

Sum Method is applied to convert this multi-objective problem to a single-objective

optimization problem. As for the weights, it has been assumed that there is no specific

preferences between the systems in terms of which one is more desired to have a higher

µeYk . Therefore, every objective function Jk corresponding to different systems has been

assigned the same weights of wk = 1.

In what follows, a new optimization methodology for the Robust Simultaneous Sta-

bilization Problem is proposed which yields a better solution in terms of robustness, as

seen in the follwoing investigations. This will ensure that all of the closed-loop systems

remain stable under a unique controller while increasing the robustness bounds in case

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Chapter 4. Robust Simultaneous Stabilization Problem 48

of uncertainties. The proposed optimization problem is stated as:

Problem Given parameters µ > 0 and η ≥ 0, choose the controller f ∈ Rn to

minimize the objective function

I =m∑k=1

1

µeYk(4.18)

with respect to f , and subject to the constraints

−µ ≤ fi ≤ µ, i = 1, ..., n (4.19)

αi,k = Re (λi,k) < 0, i = 1, ..., n, k = 1, ...,m (4.20)

αi,k + ηβi,k ≤ 0, i = 1, ..., n, k = 1, ...,m (4.21)

The simultaneous stability of the systems is insured by constrainting the real part of

all of the eigenvalue, Re (λi,k), to be negative at all times. As before, constraints on the

value of the controller entries and performance of the closed loop systems are imposed.

The controller which minimizes the new objective function, will also stabilize all the

plants simultaneously and yields a solution with the desired performance depending on

the value of η, without using unnecessary control effort. The problem formulation shown

in Eq. (4.18) is believed to be convex since Eqs. (4.15) through (4.17) are also convex

in nature. Hence, the weighted-sum method can be used to find the set of pareto-

optimal solutions to this problem. This methodology also enforces a way to increase

or decrease the robustness boundary of each system depending on its importance, by

assigning different weights to their respective objective functions in the multi-objective

optimization problem.

The application of the multiple robustness optimization and bi-level optimization

solutions to the RSS problem, is shown using two linear and nonlinear case studies

introduced in Chapters 5 and 6.

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Chapter 5

Linear Simulation: An F4-C Flight

Control Case Study

In this chapter, a simultaneously stabilizing feedbcak control law is designed for an F4C

fighter aircraft operating at 4 different flight conditions using the method described earlier

in section 2.7. Later, its robustness is investigated using methods described in [40], [38],

and [31]. The robustness is then relaxed and a new controller is obtained using the

multiple objective optimization methodology described in Chapter 4. The solution to

the RSS problem is further compared to the previous SS problem solution in terms of

robustness. Moreover, this design is again compared to the solution obtained using a

bi-level optimization methodology introduced in section 4.1 as was presented in [29].

5.1 Introduction of the Test Case

This test case considers the problem of simultaneous stabilization of the longitudinal

dynamics Short-Period mode of an F4C fighter aircraft at 4 distinct operating points, as

was originally proposed by Ackermann [2]. For this test case a simultaneously stabiliz-

ing controller is designed and then using several robustness analysis tools provided, its

robustness is later evaluated in the following section. The system is described in linear

time-invariant state-space form for all flight conditions as:

x(t) = Ax(t) +Bu, y(t) = Cx(t), (5.1)

49

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Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 50

where

A =

a11 a12 b1

a21 a22 b2

0 0 −20

, B =

0

0

20

, C = I (5.2)

The original Short-Period state vector (angle of attack α, and pitch rate q) is also ex-

tended to include the elevator actuator dynamics, represented by a first-order low-pass

filter with a time constant Te of 0.05 s. Thus:

x =[α q δe

]T(5.3)

Four distinct flight conditions have been chosen for this study, with variation in both

altitude and speed, as shown in Table 5.1.

Table 5.1: F4-C flight operating conditions

Operating Point 1 2 3 4

Altitude, ft Sea Level Sea Level 35000 45000

Mach number 0.206 1.1 0.6 2.15

a11 -0.4535 -2.112 -0.2978 -0.484

a12 0.9792 1.0 0.9866 0.9997

a21 -0.3693 -772.08 -1.803 -42.75

a22 -0.4615 -3.126 -0.4436 -0.3718

b1 -0.0290 -0.2098 -0.0411 -0.0419

b2 -1.459 -63.48 -4.989 -17.72

Using the optimization approach introduced in section 2.7, and with assumed values

of µ = 2 (limiting the feedback gain vector entries) and η = 0.2 (limiting the damping

ratio by introducing a performance factor), a simultaneously stabilizing controller is ob-

tained, as presented in Table 5.2. Note the negative value of the maximum real part of

the eigenvalues, showing that a controller has been found that minimizes the maximum

real part of the eigenvalues and makes it negative, thus assuring that simultaneous stabi-

lization has been achieved. Table 5.3 also shows the closed loop system eigenvalues and

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Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 51

all pole locations for all plants are in the negative real half of the complex plane, stating

the fact that all of the closed loop systems reached stability under the same design.

u (t) = −fTx (t) (5.4)

Table 5.2: Simultaneous Stabilization solution

Proposed SS optimization methodology (sec 2.7)

f -2.0000

-1.8404

0.1423

I = max Re(λ) -1.7721

Table 5.3: Closed-loop system eigenvalues

System Eigenvalues

A1 −B1fT -20.2173, and -1.7721 ± 1.2421i

A2 −B2fT -6.2202, and -10.9322 ± 54.6610i

A3 −B3fT -1.7721, and -10.9079 ± 6.8181i

A4 −B4fT -2.9612, and -10.3706 ± 23.3970i

For further illustration, the initial response simulation of the obtained closed loop

systems is presented here. Figure 5.1 shows the time history of the states subject to

the initial condition X = [ 1 1 1]T at time t = 0 for all four flight conditions. It can

be seen from the simulation that when the feedback gain is applied, the states of the

closed-loop system decay to the steady state and thus the closed loop systems are stable.

5.2 Robustness Investigation

It was shown that the controller is able to simultaneously stabilize all four different

flight conditions. However, any introduced perturbation in speed, altitude, airplane’s

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Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 52

Figure 5.1: Response to initial condition

configuration or aerodynamic shape, or other flight conditions and characteristics due

to damage, changes the closed loop system and therefor it is desired to check if the

perturbed closed loop system is still stable. Failures such as Loss of effectiveness of a

control surface or aerodynamic shape change due to loss of some parts can be modeled as

a change in aerodynamic derivatives, and it can be assumed that the derivatives change

suddenly. To make the investigation easier it is assumed that flight conditions (speed and

altitude), and all other stability derivatives and flight characteristics remain the same at

each operating point, except for one which preferably has a strong and obvious effect on

the aircraft’s performance.

5.2.1 Perturbations Due to CLαuncertainties

It is seen by looking at the state space representation of the short-period mode given

below that CLα has an obvious effect on the longitudinal dynamics of the aircraft. This

aerodynamic parameter directly affects the lift on the aircraft and the resulting moment

and hence it changes the longitudinal dynamics short-period behavior more significantly

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Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 53

than other derivatives. α

q

=

VT − Zα 0

−Mα 1

−1 Zα VT + Zq

Mα Mq

α

q

+

VT − Zα 0

−Mα 1

−1 Zδe

Mδe

δe(5.5)

α

q

=

a11 a12

a21 a22

α

q

+

b1

b2

δe (5.6)

Where the matrix entries can be calculated as:

a11 = ZαVT−Zα

a12 = VT+ZqVT−Zα

a21 = Mα + MαZαVT−Zα

a22 = Mq + Mα(VT+Zq)VT−Zα

b1 = ZδeVT−Zα

b2 = MαZδeVT−Zα

+Mδe

(5.7)

and the dimensional stability derivatives are introduced in terms of dimensionless deriva-

tives as

Zα = − qSm

(CD + CLα)

Mα = qScJyCmα

Zα = − qSc2mVT

CLα

Mα = qScJy

c2VT

Cmα

Zq = − qSc2mVT

CLq

Mq = qScJy

c2VT

Cmq

Zδe = − qSmCLδe

Mδe = qScJYCmδe

(5.8)

The dimensionless stability derivatives at each flight condition along with the geo-

metric characteristics of the aircraft are given in Table 5.4 as was originally presented in

[6], [11], and [8].

Now to investigate the robustness of the controller designed for this case study, assume

that the only source of perturbation is changes in the aerodynamic derivative CLα and

all of the other derivatives keep their initial value. Therefore, the new perturbed closed

loop system can be represented as:

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Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 54

A =(A−BfT

)+

−∆CLαqS/mVT 0 0

−∆CLαq2S2c2Cmα/2mJY V

2T 0 0

0 0 0

(5.9)

Effect of ∆CLα on the robust stability bounds can now be investigated.

Stability Robustness Bounds

Assuming that the only source of perturbation comes from changes in CLα results in:

ACL = A0 + E =(A−BfT

)+ E , (5.10)

where

E =

−∆CLαqS/mVT 0 0

−∆CLαq2S2c2Cmα/2mJY V

2T 0 0

0 0 0

.

Therefore r = 1, k1 = ∆CLα , and

E1 =

−qS/mVT 0 0

−q2S2c2Cmα/2mJY V2T 0 0

0 0 0

(5.11)

Using the strategy presented in Ref. [40], the magnitude of the maximum allowable

deviation in CLα is obtained for each flight condition and is presented in Table 5.5.

It is thus guaranteed that if CLα is disturbed within these ranges at each operating

point, the perturbed closed-loop system will remain stable.

The Extreme Point Solution

The extreme point solution strategy [38] can also be used to verify the robust stability

bounds. Assuming that at each operating point only CLα is allowed to perturb in the

maximum deviation range given in Table 5.5, the matrix familyA for each flight condition

can be defined as:

A (q) = A0 +S∑i=1

qiAi =(A−BfT

)+ E (5.12)

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Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 55

Tab

le5.

4:F

4-C

char

acte

rist

ics

atdiff

eren

tflig

ht

condit

ions

Fligh

tP

rop

erty

Con

dit

ion

Alt

itude,

ftM

CLα

qS

mVT

cJy

Cmα

1Sea

Lev

el0.

206

2.8

62.6

530

3892

423

016

.04

1221

86-0

.95

2Sea

Lev

el1.

13.

317

9253

038

925

1228

16.0

412

2193

-1.2

335

000

0.6

2.8

126

530

3892

558

416

.04

1221

93-1

.3

445

000

2.15

2.3

1004

530

3892

520

8116

.04

1221

930.

25

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Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 56

Table 5.5: Maximum allowable deviation in ∆CLα

Operating Point 1 2 3 4

|∆CLα |max < µεY = 3.5065 0.0474 4.9685 0.9716

E =

−∆CLαqS/mVT 0 0

−∆CLαq2S2c2Cmα/2mJY V

2T 0 0

0 0 0

where s = 1, l = 2s = 2, ∆CLα ≤ q1 = ∆CLα ≤ ∆CLα , and

A1 =

−qS/mVT 0 0

−q2S2c2Cmα/2mJY V2T 0 0

0 0 0

.

For the four different flight conditions, generate two vertex matrices A1 = A0 +(∆CLα .A1

)and A2 = A0 +

(∆CLα .A1

), and their respective vertices in dagger space,

L1 and L2. For each flight condition, ‘Real Axis Nonsinularity’ matrices can now be

defined:

KN1 = −[(L1 + L2

)−1.L1

],

KN2 = −[(L1 + L2

)−1.L2

]. (5.13)

The above matrices have been checked to be asymptotically stable for all four oper-

ating points and thus all of the matrix families A are stable. This verifies the results

obtained in the previous analysis, that if the aerodynamic derivative CLα is bounded

to perturb in the range given in Table 5.5 the resulting perturbed closed-loop system

remains stable.

The Kharitonovs Theorem

Assume CLα is allowed to perturb in the maximum deviation range provided in Table

5.5 for each operating point, and for each value of CLα the closed loop system and

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Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 57

its characteristic equation is found. The maximum and minimum of the characteristic

polynomial coefficients is later found and used to generate Kharitonov’s polynomials for

each flight condition, as presented in Table 5.6.

Table 5.6: Kharitonov’s polynomials coefficients

Operating Point s3 s2 s1 s0

1

K1(s) 1 24.1794 66.6442 69.0873

K2(s) 1 23.3426 86.0316 120.2817

K3(s) 1 23.3426 66.6442 120.2817

K4(s) 1 24.1794 86.0316 69.0873

2

K1(s) 1 28.1143 3242.6 19255

K2(s) 1 28.0537 3244.1 19400

K3(s) 1 28.0537 3242.6 19400

K4(s) 1 28.1143 3244.1 19255

3

K1(s) 1 24.0574 193.2548 203.7089

K2(s) 1 23.1174 215.0013 382.7542

K3(s) 1 23.1174 193.2548 382.7542

K4(s) 1 24.0574 215.0013 203.7089

4

K1(s) 1 23.9073 711.5970 1803.4

K2(s) 1 23.4963 721.1680 2075.5

K3(s) 1 23.4963 711.5970 2075.5

K4(s) 1 23.9073 721.1680 1803.4

Each set of Kharitonov’s polynomials was then checked for stability and found to be

stable, that is, all the roots of the polynomials have strictly negative real parts and are

in the left half plane. Therefor, it is guaranteed that if CLα changes in the range pro-

vided in Table 5.5, all four interval polynomials corresponding to four distinct perturbed

flight conditions are stable and thus the pitch-tracking controller has a certain degree of

robustness with respect to changes in CLα .

More over, the controller’s robustness is stablished graphically as all Kharitonov rect-

angles are plotted for different values of w, and it is seen that the Zero Exclusion Con-

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Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 58

dition is met as no rectangle touches the point z = 0, as shown in Figure 5.2. It is also

clearly seen in Figure 5.2 that the value sets for each flight condition move from the first

quadrant to the second and eventually to the third (the interval polynomial has a degree

of 3) without touching or passing through the origin, thus providing a graphical confir-

mation of the robust stability of the perturbed closed-loop system for the four different

flight conditions.

(a) Kharitonov rectangles, operating point 1 (b) Kharitonov rectangles, operating point 2

(c) Kharitonov rectangles, operating point 3 (d) Kharitonov rectangles, operating point 4

Figure 5.2: Kharitonov’s robust stability graphical check for different flight conditions

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Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 59

5.2.2 Robustness Optimization

Figure 5.3 shows the maximum real part of the closed-loop system eigenvalues for all four

flight conditions, when CLα is allowed to change in the maximum allowable deviation

range obtained before in Table 5.5. Note that the maximum real part of the eigenvalues

remains negative in all cases which means that the closed-loop system remains stable

despite large perturbations in CLα . These methods provide sufficient yet conservative

robust stability conditions and as long as the derivative changes in the given range, the

closed-loop system’s stability is guaranteed. However, because the stability robustness

bound theory provides a sufficient condition and not a necessary one, it is not possible

to draw any conclusion regarding the stability of a disturbed plant with CLα perturbing

outside of the obtained boundaries in Table 5.5. In other words, the perturbed system

may or may not maintain its stability, depending on its closed-loop pole placement. As a

result of this conservatism and as it is seen in Figure 5.3, it is still possible for the system

to remain stable even when moving out of the obtained robust stability bounds.

Figure 5.3: Maximum Eigenvalue of the closed-loop system vs. CLα

Note the general trend in Figure 5.3. The stability of the perturbed close-loop system

is increased (i.e. the maximum of the real parts of the eigenvalues gets more negative)

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Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 60

as CLα increases and thus provides more lift and control.

As said before, the robustness analysis provides a sufficient condition and not a nec-

essary one, thus no assumption can be made about the stability of a perturbed system

with the uncertainties outside of the given bounds. Therefore, and due to the fact that

these results are also highly conservative, it can be desirable to increase the robustness

bounds for each system. This was done using the methodologies presented in Chapter

4. The new requirement to increase the robustness introduces a new problem, previously

presented as the Robust Simultaneous Stabilization Problem (RSS) which is solved using

a multi-objective optimization method. For this test case the following wish attribute

has been formulated as the new objective function, converting a maximization prob-

lem to a minimization one, using the wighted sum method to solve the multi-objective

optimization problem of increasing the robustness bounds.

I =4∑j=1

1

|∆CLαj |max=

(1

|∆CLα1|max+

1

|∆CLα2|max+

1

|∆CLα3 |max+

1

|∆CLα4|max

)(5.14)

Note that in this investigation, the robustness of the design is determined with respect

to the value of the maximum allowable deviation in the stability derivative CLα . Finally,

a multiple robustness optimization problem as formulated in section 4.2.3 is solved and

the results are presented in Table 5.7.

Table 5.7: Solutions to the SS and RSS problems

SS Problem RSS Problem: RSS Problem:

Multiple Robustness Optimization Bi-level Optimization

f [−2 − 1.8404 0.1423]T [0.3123 − 2 1.5933]T [1.4481 − 2 1.6869]T

The new control law identified as the solution to the ”RSS problem: Multiple Ro-

bustness Optimization” is now compared to the previously obtained solution to the SS

problem, in terms of robustness of the design. Table 5.8 shows this comparison. For

each distinct flight condition, the stability robustness bounds or |∆CLα|max is obtained

for both the SS and RSS solutions. The 4th column also shows how much the maximum

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Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 61

allowable deviation range has been increased when the optimization goal is to expand the

robustness bounds. It is seen that for the first, second, and the forth operating condi-

tions, the maximum allowable deviation, |∆CLα |max, has increased significantly as shown

in Table 5.8, whereas there has been a slight decrease in the value for the third operating

point. This, however, is acceptable if not desirable since it is only a %1 decrease and the

robustness of the other systmes were significantly increased at this cost.

Table 5.8: Effect of relaxing the robustness on |∆CLα |maxunrelaxed robustness relaxed robustness

SS Problem RSS Problem: Multiple Robustness Optimization

Flight Condition |∆CLα|max |∆CLα|max % of increase in robustness

1 3.5065 4.4500 % 27

2 0.0474 0.0599 % 26

3 4.9685 4.8953 % -1

4 0.9716 1.0488 % 7

Now to give a sense of how desirable these results are, they are compared to the

solution of the ”RSS problem: Bi-level Optimization” which was presented in section 4.1.

Therefore, the following objective function was constructed where the goal is to maximize

the robustness boundary of the system with the most vulnerability to perturbations, i.e.

maximizing the minimum of the robustness bounds at each time step.

I = max −∣∣∣∆CLαj ∣∣∣max = max (−|∆CLα1|max,−|∆CLα2|max,−|∆CLα3 |max,−|∆CLα4|max)

(5.15)

The previously proposed extended decomposition based problem is solved as formu-

lated in Eq. 4.8 and the designed controller is shown in Table 5.7. The robustness of

this design is investigated as well and compared to that of the previous RSS solution,

as shown in Table 5.9. Having the smallest robustness bounds, both the second and the

forth operating points experienced an increase in their |∆CLα|max values. Whereas that

of the first and third operating points have seen a significant drop. Having said that, it

is seen that maximizing the minimum value of ∆CLα has been acheived, but only at the

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Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 62

cost of loosing the robustness of the other systems to a great extent. It can be said that

the multiple objective optimization problem (MOOP) methodology introduced before,

delivers a better result as it provides almost the same level of increase to ∆CLα for the

2nd and 4th systems while increasing that of the other two operating points as well.

Table 5.9: Results from different RSS optimization methodologies and objective functions

RSS Problem: RSS Problem:

Multiple Robustness Optimization Bi-level Optimization

Flight |∆CLα|max robustness increased by |∆CLα|max robustness increased by

Condition

1 4.4500 % 27 1.4883 % -57

2 0.0599 % 26 0.0606 % 27

3 4.8953 % -1 0.6901 % -86

4 1.0488 % 7 1.0795 % 11

For a more graphical comparison between these two approaches, Figure 5.4 shows the

values of the maximum real part of the eigenvalues for the 4 different closed-loop systems

versus the respective maximum allowable deviations in CLα as presented in Table 5.9. In

all cases, the maximum of the real part of the eigenvalues are negative, indicating that all

systems under different controllers have poles in the left half plane and thus simultaneous

stabilization was achieved. The set of pictures also clearly shows that the solution to the

Multi-objective optimization problem does a much better job overall in terms of increasing

the maximum allowable range of perturbations in CLα , when compared to the solution of

the bi-level decomposition problem (where robustness of the most critical case is increased

at the cost of reducing the robustness of some of the other plants significantly). This

behaviour can be explained due to the nature of the weighted-sum method optimization

technique, which aims for a more balancing effect when minimizing a multi-objective

problem. Note that the bi-level strategy yields a more conservative result in terms of

the allowable perturbation in CLα compared to that of the multi-objective optimization

problem, although conservatism is a characteristic of the stability robustness bounds

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Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 63

technique introduced in Chapter 3.

Figure 5.4: Maximum Eigenvalue of the closed-loop system vs. CLα

Results presented in Table 5.10 show all four closed-loop system’s poles under three

different control laws given in Table 5.7. As suggested by the pole locations, all of

the controllers provide simultaneous stability since the real part of the eigenvalues were

negative for all of the operating points.

Figures 5.5, 5.6, and 5.7 show the initial response of the unperturbed systems (i.e.

∆CLα = 0) to the initial condition x = [1 0 0]T for the SS problem solution, Multiple

Robustness Optimization solution, and Decomposition-Based Strategy solution, respec-

tively. In all three plots, states are decaying to the steady state and the systems are

stable. However, states are converging much faster to the steady states under the S.S.

control law than under the two other problem solutions. In other words, the other two

problem solutions are slightly overdamped and sluggish in reaching the steady states.

This is specially true for the decomposition-based bi-level optimization problem which is

sluggish in decaying the initial condition and converging to the steady states. This can

be expected since the primary goal of the RSS problem is not to optimize the pole place-

ment or performance, but rather to optimize the robustness of the design. Moreover, the

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Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 64

Tab

le5.

10:

Aco

mpar

ison

ofth

ecl

osed

-loop

eige

nva

lues

for

diff

eren

top

tim

izat

ion

met

hodol

ogie

s

Clo

sed-l

oop

Eig

enva

lues

Syst

emSS

Pro

ble

mR

SS

Pro

ble

m:

RSS

Pro

ble

m:

(Fligh

tC

ondit

ion)

Mult

iple

Rob

ust

nes

sO

pti

miz

atio

nB

i-le

vel

opti

miz

atio

n

A1−B

1fT

-20.

2173

,an

d-1

.772

1.24

21i

-50.

7045

,-0

.640

7,an

d-1

.435

8-5

2.62

00,

-0.1

510,

and

-1.8

820

A2−B

2fT

-6.2

202,

and

-10.

9322±

54.6

610i

-12.

8762

,an

d-2

2.11

39±

50.2

965i

-12.

9720

,an

d-2

3.00

20±

49.6

732i

A3−B

3fT

-1.7

721,

and

-10.

9079±

6.81

81i

-47.

6306

,-0

.598

0,an

d-4

.378

8-4

9.64

96,

-0.0

680,

and

-4.7

618

A4−B

4fT

-2.9

612,

and

-10.

3706±

23.3

970i

-3.9

594,

and

-24.

3812±

2.88

62i

-3.2

880,

-20.

1510

,an

d-3

1.15

49

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Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 65

decomposition-based formulation does not consider the performance of the controller as

a design criteria. This issue can be addressed later by imposing additional constraints

on the bi-level optimization problem which will improve the damping characteristics of

the design.

Figure 5.5: Response to initial condition: SS problem

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Chapter 5. Linear Simulation: An F4-C Flight Control Case Study 66

Figure 5.6: Response to initial condition: Multiple Robustness Optimization solution

Figure 5.7: Response to initial condition: Decomposition-Based Strategy solution

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Chapter 6

Non-Linear Simulation: A CRJ-200

Flight Control Case Study

In this chapter, a simultaneously stabilizing pitch tracking controller for a CRJ-200 re-

gional jet aircraft, operating at 4 different flight conditions, is designed with the elevator

deflection being the control input. The design of this control law is based on the linear

model of the plane’s longitudinal dynamics, using the design process proposed before in

section 2.7. Later, the effectiveness of this controller under the introduced normal op-

erating conditions as well as gust encountered flight conditions is investigated, with the

help of a state of the art CRJ-200 flight training device and simulation platform. The

robustness of the designed controller will also be further investigated using the method

described in section 3.3. Moreover, The robustness of the designed controller is relaxed

and a new control law is obtained using the optimization methodology proposed in Chap-

ter 4, which is later compared to previously obtained results. This new control law design

is further compared to a controller obtained using the method presented in [29], which is

also designed to have a more relaxed robustness using a bi-level optimization methodology

with a different objective function.

The state-of-art Flight Training Device (FTD) which can be used for pilot training,

presents a more realistic airplane model, as shown in Figure 6.1. It features a generic

regional twin jet aircraft cockpit, hydraulic sticks, cockpit panels and aircraft compo-

nents, flight systems, navigation and communication systems, and other electrical and

67

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Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study68

(a) (b)

Figure 6.1: The Flight Training Device (FTD) Facility

mechanical systems, as well as a visual system providing 150◦ (horizontal) x 35◦ (verti-

cal) field of view and an instructor operating station running a model of a Bombardier

CRJ-200. When using the FTD as a test bed for interactive pitch tracking control design

and simulation, the designed controller takes over as soon as the Simulink simulation is

started, overwriting the built-in autopilot.

6.1 Modelling of the CRJ-200

In order to derive the aircraft’s equations of motion, some principal reference frames used

in the flight dynamics should be listed in the following.

Earth–Fixed Reference Frame, FE: In many applications, any reference frame fixed

to the earth can be used as an inertial frame. The Earth–fixed reference frame is

an earth-surface frame with it’s origin located at any point of the earth surface

near the vehicle. The xE–axis points north, the zE–axis is directed vertically down

towards the center of the earth, and the yE–axis completes the right-handed system,

pointing towards the east.

Body–Fixed Reference Frame, FB: The origin of this frame is the center of mass.

The xB–axis lies in the plane of symmetry and is directed towards the aircraft’s

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Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study69

nose. The yB–axis is perpendicular to the plane of symmetry and is positive to-

wards the starboard side. The zB–axis completes the right–handed system, directed

downward.

Air–Trajectory Reference Frame or Wind Axes, FW : This frame has its origin fixed

to the plane at the center of mass. The xW is directed along the velocity vector

V of the vehicle relative to the atmosphere, and the zW axis lies in the plane of

symmetry. The yW–axis completes the right–handed system.

Stability Axes, FS: The stability axis system can be derived from the body–fixed sys-

tem by rotating the coordinate frame by −α0 around the yB–axis. Here, α0 is the

angle of attack during a reference flight condition. This system is commonly used

for flight control purposes. In particular, throughout this document the stability

coordinate system is used unless otherwise specified. For simplicity of notation, the

index S has often been omitted.

6.1.1 Nonlinear Model

In general, the mathematical models describing the motion of an aircraft are subject

to few assumptions. For example, the earth is a sphere rotating on an axis fixed in

inertial space with g being a radial vector, the atmosphere is at rest relative to the earth,

and the vehicle is a rigid body with no elastic mode taken into account. Moreover,

for a rigid vehicle having a plane of symmetry and with the flat–earth approximation

(the simplification of treating the earth as a flat stationary plane in inertial space), the

equations of motion describing the plane’s unsteady behaviour are reduced to nine scalar

differential equations, including six Euler equations of motion and three equations for

calculating the Euler angles φ, θ and ψ. These reduced equations obtained by neglecting

the spherical rotation of the earth are collected below:

X = mg sin θ +m (u+ q w − r v), (6.1)

Y = −mg cos θ sinφ+m (v + r u− pw), (6.2)

Z = −mg cos θ cosφ+m (w + p v − q u), (6.3)

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Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study70

L = Ix p− Ixz r + q r (Iz − Iy)− Ixz p q, (6.4)

M = Iy q + r p (Ix − Iz) + Ixz(p2 − r2), (6.5)

N = Iz r − Ixz p+ p q (Iy − Ix) + Ixz q r, (6.6)

φ = p+ (q sinφ+ r cosφ) tan θ, (6.7)

θ = q cosφ− r sinφ, (6.8)

ψ = (q sinφ+ r cosφ) sec θ, (6.9)

where the state vectore is defined as x = [u, v, w, p, q, r, φ, θ, ψ]T , and the control input

influencing the air reaction forces and moments is u = [δe, δp, δa, δr]T , representing eleva-

tor deflection, throttle setting, aileron, and rudder deflection respectively. The nonlinear

equations of motions are derived in detail and can be found in [15] as well as many other

references.

6.1.2 Linear Model

In most cases, a nonlinear system can be locally described by a linearized model about a

reference condition. Here, the reference steady-state is the symmetric steady rectilinear

flight over a flat earth with no angular velocity, meaning that v0 = p0 = q0 = r0 =

φ0 = ψ0 = 0. The steady-state values are denoted by subscript 0 and deviation from the

steady-state is represented by the prefix ∆, that is: x = x0 + ∆x. For vehicles with a

plane of symmetry, two uncoupled sets of longitudinal and lateral equations are found.

When using the small–disturbance theory, it is convenient to use wind axes for the lift-

force and drag-force equations, and the stability axes for the presentation of longitudinal

equations. Therefore, and because of the choice of the coordinates, w0 = 0 and u0 is

equal to the reference flight velocity. The forces and moments acting on the body is also

splitted into a steady-state reference value and a deviation from the reference condition:

X = X0 + ∆X, (6.10)

Y = Y0 + ∆Y, (6.11)

Z = Z0 + ∆Z, (6.12)

L = L0 + ∆L, (6.13)

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Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study71

M = M0 + ∆M, (6.14)

N = N0 + ∆N. (6.15)

For a symmetric rectilinear flight, it is obvious that X0 = mg sin θ0, Z0 = −mg cos θ0,

and Y0 = L0 = M0 = N0 = 0. To arrive at an approximation for the air reactions forces

and moments, the stability derivatives can now be used:

∆X = Xu∆u+Xw∆w +Xq∆q +Xu∆u+Xw∆w +Xq∆q +

Xδe∆δe+Xδp∆δp, (6.16)

∆Y = Yv∆v + Yp∆p+ Yr∆r + Yv∆v + Yp∆p+ Yr∆r +

Yδa∆δa+ Yδr∆δr, (6.17)

∆Z = Zu∆u+ Zw∆w + Zq∆q + Zu∆u+ Zw∆w + Zq∆q +

Zδe∆δe+ Zδp∆δp, (6.18)

∆L = Lv∆v + Lp∆p+ Lr∆r + Lv∆v + Lp∆p+ Lr∆r +

Lδa∆δa+ Lδr∆δr, (6.19)

∆M = Mu∆u+Mw∆w +Mq∆q +Mu∆u+Mw∆w +Mq∆q +

Mδe∆δe+Mδp∆δp, (6.20)

∆N = Nv∆v +Np∆p+Nr∆r +Nv∆v +Np∆p+Nr∆r +

Nδa∆δa+Nδr∆δr. (6.21)

Since the stability derivatives are found to be negligible with respect to time derivatives

of the state variables, they can be ommitted with the exception of Zw and Mw. The

longitudinal dynamics obtained by linearizing the non-linear equations of motion and

substituting the air reaction forces and moments accordingly, is as follows:

Along =

Xum

Xwm

0 −gcosθ0

Zum−Zw

Zwm−Zw

Zq+mu0

m−Zw−mgsinθ0m−Zw

1Iy

[Mu + MwZu

m−Zw

]1Iy

[Mw + MwZw

m−Zw

]1Iy

[Mq + Mw(Zq+mu0)

m−Zw

]−Mwmgsinθ0

Iy(m−Zw)

0 0 1 0

,

(6.22)

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Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study72

Blong =

Xδem

Xδpm

Zδem−Zw

Zδpm−Zw

Mδe

Iy+ MwZδe

Iy(m−Zw)

Mδp

Iy+

MwZδpIy(m−Zw)

0 0

, (6.23)

and

xlong = [u w q θ]T , (6.24)

ulong = [δe δp]T . (6.25)

The stability derivatives of the CRJ-200 in different flight conditions can be calculated

using the Aircraft Conceptual Design Toolbox for Matlab, written by Ruben Perez, Uni-

versity of Toronto. Furthermore, since the FTD is not equipped with an automatic

throttle control, it is reasonable to set δp = 0. Finally, a first order lag filter was found

to be adequate for modeling the behaviour of the actuators, as shown in [4]. The time

constants used are:

Televator = 1.6 s

Taileron = 2.75 s

Trudder = 0.1 s

6.2 Introduction of the Test Case

The aircraft’s longitudinal dynamics is represented by the following system for each flight

condition:

x = Ajx+Bjδe, (6.26)

δe = − 1

Teδe+

1

Teδec. (6.27)

where the elevator dynamics is given by a first order differential equation in the time

domain. Therefore for this case study, the longitudinal control augmented system (CAS)

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Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study73

of the aircraft, including both short period and phugoid modes and elevator actuator

dynamics, is represented here:

∆u

w

q

∆θ

δe

︸ ︷︷ ︸

x

=

Aj Bj

0 − 1Te

︸ ︷︷ ︸

A

∆u

w

q

∆θ

δe

︸ ︷︷ ︸

x

+

0

1Te

︸ ︷︷ ︸

B

δec︸︷︷︸u

, (6.28)

where δec is the elevator command input, the actuator system has a time constant Te of

1.6 s and the plant state coefficient matrices and control coefficient vectors are given as:

Aj =

a11j a12j 0 a14j

a21j a22j a23j a24j

a31j a32j a33j a34j

0 0 1 0

, Bj =

0

b2j

b3j

0

Cj = I. (6.29)

The control input is δec, and the state vector is defined as x = [u w q θ δe]T .

Four different flight conditions have been chosen with variation in both altitude and

speed, as shown in Table 6.1. The values of matrices parameters are also given in Table

6.1.

In order to design a controller to track a pitch angle command as illustrated in Figure

6.2, let us define Cz = [0 0 0 1 0] and extend the state vector x by the integral state

xI , where xI = e and e is the tracking error. The extended system now has the form of:

x

xI

︸ ︷︷ ︸

˙x

=

A 0

−Cz 0

︸ ︷︷ ︸

A

x

xI

︸ ︷︷ ︸

x

+

B

0

︸ ︷︷ ︸

B

u+

0

1

︸ ︷︷ ︸W

θc, (6.30)

where

u = −[Kx KI ]x = −Kx. (6.31)

Using the proposed optimization methodology introduced in section 2.7, with the

controller enteries being limited by a factor of µ = 2 and a performance index of η = 1,

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Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study74

Table 6.1: CRJ-200 flight operating conditions

Flight Condition

Property 1 2 3 4

Total Weight 40,000 40,000 40,000 40,000

xcg, ft 0.16c 0.16c 0.16c 0.16c

Ix, slug.ft2 55,717 55,717 55,717 55,717

Iy, slug.ft2 369,830 369,830 369,830 369,830

Iz, slug.ft2 411,017 411,017 411,017 411,017

Ixz, slug.ft2 17,789 17,789 17,789 17,789

Mach number 0.74 0.55 0.48 0.42

Altitude, ft 33,000 33,000 19,000 19,000

angle of attack, deg 0.7 0.7 0.7 0.7

Flight path angle, deg 0 0 0 0

a11 -0.0084 -0.0002 -0.0001 -0.00001

a12 0.0066 0.0146 0.0171 0.0206

a14 -32.198 -32.198 -32.198 -32.198

a21 -0.1221 -0.1375 -0.1425 -0.1584

a22 -0.8338 -0.5586 -0.8251 -0.7095

a23 718.744 533.551 489.778 428.541

a24 -0.3935 -0.3927 -0.3921 -0.3919

a31 0.0001 0.0001 0.0001 0.0001

a32 -0.0135 -0.0095 -0.0142 -0.0123

a33 -0.6610 -0.4397 -0.6490 -0.5564

a34 0.0001 0.0001 0.0001 0.0001

b2 -52.179 -26.656 -36.754 -36.884

b3 -7.0588 -3.6180 -4.9911 -5.0157

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Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study75

Figure 6.2: Pitch angle tracker

a simultaneously stabilizing controller is obtained as:

K = [0.0006 0.0010 − 0.2932 − 1.5976 1.0112 0.4683] . (6.32)

Table 6.2 presents the aircraft open-loop system eigenvalues and it can be seen that

except for the first flight condition, the longitudinal system is phugoid unstable.

Now, the closed loop system is given by

˙x =(A− BK

)x+ Wθc. (6.33)

Table 6.3 shows the pitch tracking closed-loop system eigenvalues, and it is seen that

all pole locations for all of the plants are in the negative real axis, and hence the closed

loop system is stable.

Table 6.2: Open-loop system eigenvalues

System Eigenvalues

A1 -0.7478 ± 3.1074i, and -0.0037 ± 0.0745i

A2 -0.4999 ± 2.2539i, and 0.0007 ± 0.0912i

A3 -0.7380 ± 2.6316i, and 0.0009 ± 0.0955i

A4 -0.6343 ± 2.2887i, and 0.0013 ± 0.1073i

6.3 Results: Ordinary and Gust-Encountered Flight

The effectiveness and performance of the pitch-tracking controller given in (6.32) is now

investigated under ordinary operating conditions for which it was designed, as well as

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Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study76

Table 6.3: Closed-loop system eigenvalues

System Eigenvalues

A1 −B1K -0.6051 ± 3.2348i, -0.2483 ± 0.2879i, -1.0447, and -0.0088

A2 −B2K -0.3355 ± 2.3586i, -0.1643 ± 0.2340i, -1.2519, and -0.0039

A3 −B3K -0.5537 ± 2.7338i, -0.2149 ± 0.2819i, -1.1908, and -0.0032

A4 −B4K -0.4436 ± 2.3843i, -0.1857 ± 0.2653i, -1.2598, and -0.0047

gust encountered flight conditions. This was done with the help of our integrated system

design and flight simulation platform, as discussed in [4] and [21], which is currently

being used for control system design, simulation, and testing. It consists of a state of the

art flight training device research simulator that is interconnected either with a real time

system simulator or with Matlab/Simulink in a computer terminal. This testbed repre-

sents a more complete and accurate aircraft model and hence the obtained simulation

results are much more accurate and precise than those obtained by linear or nonlinear

offline simulations in Simulink. For flying under ordinary operating conditions, the pitch

tracking control performance for a step input of 5 deg pitch angle at four different flight

conditions, is obtained from the testbed and is shown in Figure 6.3. It can be seen that

the designed pitch tracking controller provides adequate tracking at all flight conditions.

A sudden change in flight conditions, aircraft configuration or any other flight char-

acteristics due to damage results in nonlinear behavior, and although previously intro-

duced robustness investigation approaches are practical, due to the nonlinear nature of

the testbed a mathematical investigation of robust stability bounds is difficult. It was

found that these methods result in a very conservative region of stability which is also a

sufficient condition and not a necessary one. Hence, no conclusion can be made about the

stability outside of this region. Therefore to better illustrate the robustness of the con-

troller at each flight condition, the implemented ‘’Moderate Turbulence‘’ option within

the flight simulation platform was activated, which introduces sinusoidal gust to the sim-

ulation. The pitch-tracking control performance at four different flight conditions under

gust encounter is shown in Figure 6.4. Although highly oscillatory, and although the

controller is not able to damp out the external disturbances because it was not initially

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Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study77

desigend with disturbance rejection criteria in mind, the controller is still able to track

the step pitch command. This oscillation is due to the fact that the gust alleviation

characteristics was not taken into account at the time of designing the controller.

6.4 Robustness Investigation & Optimization

As it was shown, the controller is able to simultaneously stabilize all four different flight

conditions. However, any introduced perturbation in speed, altitude, airplane’s config-

uration or aerodynamic shape, or other changes in flight conditions and characteristics

due to damage, disturbes the closed loop system and therefore it is desired to check if the

perturbed closed loop system is still stable. This was also shown in the simulation and

as presented before, the controller was still able to stabilize the aircraft when encoun-

tered with the incorporated medium gust settings on the flight test bed. However, to

get more accurate results in terms of how much perturbation the state matrix’s elements

can toplerate while maintaining the stability of the closed-loop systems, a theoretical

investigation is needed with the help of the method presented in [36].

As discussed before, the state space representation of the aircraft’s longitudinal dynamics

is:

xlong =

Xum

Xwm

0 −gcosθ0

Zum−Zw

Zwm−Zw

Zq+mu0

m−Zw−mgsinθ0m−Zw

1Iy

[Mu + MwZu

m−Zw

]1Iy

[Mw + MwZw

m−Zw

]1Iy

[Mq + Mw(Zq+mu0)

m−Zw

]−Mwmgsinθ0

Iy(m−Zw)

0 0 1 0

xlong

+

Xδem

Zδem−Zw

Mδe

Iy+ MwZδe

Iy(m−Zw)

0

δe,

where stability derivatives are defined as

Xu = qSwCxuV

, Cxu = − (CDu + 2CD) ,

Xw = qSwCxαV

, Cxα = (CL − CDα) ,

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Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study78

(a) Pitch angle tracking for a 5-deg pitch angle step com-

mand

(b) Elevator deflection for a 5-deg pitch angle step com-

mand

(c) Pitch rate for a 5-deg pitch angle step command (d) Forward velocity changes for a 5-deg pitch angle step

command

(e) Upward velocity changes for a 5-deg pitch angle step

command

Figure 6.3: Time history of the states at different flight conditions subject to a 5-deg

step input

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Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study79

(a) Pitch angle tracking for a 5-deg pitch angle step com-

mand

(b) Elevator deflection for a 5-deg pitch angle step com-

mand

(c) Pitch rate for a 5-deg pitch angle step command (d) Forward velocity changes for a 5-deg pitch angle step

command

(e) Upward velocity changes for a 5-deg pitch angle step

command

Figure 6.4: Time history of the states at different flight conditions subject to a 5-deg

step input, when encountered with gust

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Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study80

Zu = qSwCzuV

, Czu = − (CLu + 2CL) ,

Zw = qSwCzαV

, Czα = − (CLα + CD) ,

Zq =qSwMACwCzq

2V, Czq = −CLq ,

Zw =qSwMACwCzα

2V 2 , Czα = −CLα ,

Mu = qSwMACwCmuV

, Cmu = (Cmu + 2Cm),

Mw = qSwMACwCmαV

, Cmα = Cmα ,

Mq =qSwMAC2

wCmq2V

, Cmq = Cmq ,

Mw =qSwMAC2

wCmα2V 2 , Cmα = Cmα ,

Xδe = qSwCxδe , Cxδe = −CDδe ,

Zδe = qSwCzδe , Czδe = −CLδe ,

Mδe = qSwMACwCmδe , Cmδe = Cmδe .

Assuming that the perturbation is due to sudden changes in altitude or flight mach

number, every matrix element which is a function of these variables will also change as

a result. Therefore, the perturbed closed loop system is formulated as:

ACL = A0 + E =(A− BK

)+ E ,

E =

∆a11 ∆a12 0 0 0 0

∆a21 ∆a22 ∆a23 ∆a24 ∆b2 0

∆a31 ∆a32 ∆a33 ∆a34 ∆b3 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

.

Hence:

U =

1 1 0 0 0 0

1 1 1 1 1 0

1 1 1 1 1 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

. (6.34)

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Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study81

Applying the methodology in [36] to this test case, the magnitude of the maximum

allowable deviation in aij is obtained for each flight condition, which is presented in

Table 6.4.

Table 6.4: Maximum allowable deviation in ai,j

Operating Point 1 2 3 4

|∆ai,j|max < µεY = 9.0093e-006 8.7701e-006 1.5187e-005 1.5141e-005

As it is seen from this table, these results are far too conservative. Moreover, this

method provides sufficient yet conservative robust stability conditions and as long as the

state matrix elements change in the given range, the closed-loop system’s stability is

guaranteed. However, because the stability robustness bound theory provides a sufficient

condition and not a necessary one, it is not possible to conclude anything regarding

the stability of the disturbed state matrix with it’s elements perturbing outside of the

obtained boundaries in Table 6.4, and depending on the closed-loop pole placement, the

perturbed system may or may not maintain its stability. As a result, it is still possible

for a perturbed system to remain stable even when disturbance is out of the obtained

robust stability bounds, and no assumption can be made regarding the stability of such a

perturbed plant. Since these results are highly conservative, it can be desirable to increase

the robustness bounds for each system. Hence, in order to deal with the conservatism

and relax the robustness bounds, the optimization problem must be looked at from a

different perspective. The new objective to increase the robustness bounds, introduces

a new problem previously discussed as the Robust Simultaneous Stabilization Problem

(RSS), which is solved using a multi-objective optimization methodology. Note that in

this investigation, the robustness of the design is addressed in terms of the values of

the maximum allowable deviation in the enteries ai,j of the state matrix. For this case

study, the following wish attribute is formulated as the new objective function, converting

a maximization problem to a minimization one and using the wighted sum method to

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Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study82

solve the multi-objective optimization problem of increasing the robustness bounds.

I =4∑j=1

1

µεY j=

1

µεY 1

+1

µεY 2

+1

µεY 3

+1

µεY 4

. (6.35)

Finally, By solving the above multiple robustness optimization problem proposed in sec-

tion 4.2.3, a new simultaneously stabilizing controller with a generally improved robust-

ness boundary is found:

K = [0.0001 0.0001 − 1.0000 − 0.1480 − 0.9999 0] . (6.36)

The robustness of this new controller identified as the solution to the ‘’RSS problem:

Multiple Robustness Optimization‘’ is now compared to the robustness of the previously

obtained solution to the SS problem. Table 6.5 shows this comparison. For each flight

condition, the stability robustness bounds, µεY j, is obtained for both the SS and RSS

control law solutions. The 4th column also shows how much the maximum allowable

deviation range has been increased when the optimization objective is to expand the

robustness bounds. As it is seen, for all of the four operating conditions, the maximum

allowable deviation in ai,j has increased significantly overall. The first plant correspond-

ing with the aircraft flying at the first operating condition, saw a significant improvement

in it’s stability robustness bounds by %88. The second, third, and fourth systems also ex-

perienced great improvement in their stability robustness values, by %14, %48, and %23

respectively. The results can again be explained due to the nature of the weighted-sum

method optimization technique, which introduces a more balancing effect when minimiz-

ing a multi-objective problem.

Table 6.6 presents closed-loop eigenvalues for all four closed-loop systems under the

two feedback control laws obtained by solving the SS and RSS problems. It is suggested

by the negative real part of the eigenvalues, indicating that all systems corresponding to

different flight conditions have poles in the left half plane, that both SS and RSS control

law solutions provide simultaneous stability.

The effectiveness and performance of the RSS pitch-tracking control law obtained

in (6.36) is further investigated under four flight conditions for which it was designed.

As explained before, the simultaion was performed using the integrated system design

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Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study83

Table 6.5: Effect of relaxing the robustness on |∆ai,j|maxunrelaxed robustness relaxed robustness

SS Problem RSS Problem: Multiple Robustness Optimization

Flight Condition |∆ai,j|max |∆ai,j|max % of increase in robustness

1 9.0093e-6 1.6951e-5 % 88

2 8.7701e-6 1.0051e-5 % 14

3 1.5187e-5 2.2439e-5 % 48

4 1.5141e-5 1.8692e-5 % 23

and flight simulation platform (FTD). Figure 6.5 shows the pitch tracking controller

performance for a step input of 5 degree pitch angle at four different flight conditions,

as obtained from the testbed. It is seen that the aircraft flying under the RSS problem

solution is stable and able to track the pitch angle command, however, the pitch angle

is converging much faster to the pitch angle command under the S.S. control law than

under the RSS control solution. After a long enough simulation time, which is not

satisfactory, the states are just starting to converge to their steady value. Overall, the RSS

solution is overdamped and sluggish in reaching the input command. This, however, is

expected, since the main objective of the RSS problem formulation is not to optimize the

performance of the controller, but rather to maximize the robustness of the design at every

flight condition. Clearly, this consideration takes away from the tracking performance

of the controller. Since the RSS problem formulation does not take the performance

into account as a design criteria, the issue can later be addressed by imposing additional

constraints which will improve the damping and tracking characteristics of the design.

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Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study84

Tab

le6.

6:A

com

par

ison

ofth

ecl

osed

loop

eige

nva

lues

for

diff

eren

top

tim

izat

ion

met

hodol

ogie

s

Clo

sed-l

oop

Eig

enva

lues

Syst

emSS

Pro

ble

mR

SS

Pro

ble

m:

(Fligh

tC

ondit

ion)

Mult

iple

Rob

ust

nes

sO

pti

miz

atio

n

A1−B

1fT

-0.6

051±

3.23

48i,

-0.2

483±

0.28

79i,

-1.0

447,

and

-0.0

088

-0.6

191±

3.72

70i,

-0.1

227±

0.14

35i,

-0.0

196,

and

-0.0

001

A2−B

2fT

-0.3

355±

2.35

86i,

-0.1

643±

0.23

40i,

-1.2

519,

and

-0.0

039

-0.4

085±

2.69

31i,

-0.0

850±

0.14

10i,

-0.0

115,

and

-0.0

001

A3−B

3fT

-0.5

537±

2.73

38i,

-0.2

149±

0.28

19i,

-1.1

908,

and

-0.0

032

-0.6

116±

3.14

22i,

-0.1

193±

0.15

19i,

-0.0

126,

and

-0.0

001

A4−B

4fT

-0.4

436±

2.38

43i,

-0.1

857±

0.26

53i,

-1.2

598,

and

-0.0

047

-0.5

225±

2.73

14i,

-0.1

043±

0.15

57i,

-0.0

123,

and

-0.0

001

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Chapter 6. Non-Linear Simulation: A CRJ-200 Flight Control Case Study85

(a) Pitch angle tracking for a 5-deg pitch angle step com-

mand

(b) Elevator deflection for a 5-deg pitch angle step com-

mand

(c) Pitch rate for a 5-deg pitch angle step command (d) Forward velocity changes for a 5-deg pitch angle step

command

(e) Upward velocity changes for a 5-deg pitch angle step

command

Figure 6.5: Time history of the states at different flight conditions subject to a 5-deg

step input, under the RSS problem solution

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Chapter 7

Conclusion and Future

Developments

This chapter summarizes the conclusions derived from the different optimization method-

ologies, investigations and simulations presented in this thesis. A discussion on several

possible improvements for future research is presented as well.

7.1 Conclusions

Simultaneous Stabilization addresses the stability of a number of distinct plants under a

single feedback controller. Such a controller stabilizes the closed loop system and provides

simplicity and reliability. However, the less explored aspect is the robustness of the

controller designed for simultaneous stabilization. Since uncertainties such as parameter

variations can occur in flight, the designed controller should be able to provide some

degree of stability in case of perturbations such as gust, structural damage and failure,

etc. For instance, low frequency wind has an effect on tracking performance whereas

high frequency wind affects flight stability. It is therefore important to address the

robustness of the simultaneous stabilization or the Robust Simultaneous Stabilization

(RSS) problem, and to provide simultaneous stability for all systems while maximizing

their stability robustness bounds.

The objective of this thesis was to develop an optimization methodology which can

86

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Chapter 7. Conclusion and Future Developments 87

deliver such a feedback control law. The optimization methodology proposed initially for

designing a simultaneously stabilizing controller was based on the previous research on

the topic, borrowing the same concept of a necessary and sufficient condition and con-

straints. The difference, however, was in the number of design variables and the imposed

constraints. The main addition to this initial optimization algorithm was to account for

the stability robustness bounds in the design process, hence finding a solution to the

RSS problem. The new developed optimization solution includes a different objective

function formulated to address systems’ robustness, using the concept of stability ro-

bustness bounds and multi-objective optimization. The optimization process is further

expanded to include the simultaneous stabilization condition as a constraint along with

the imposed constraints on the pole placement and the performance.

Two flight control case studies were considered for robustness optimization. The

investigation was performed through numerical and flight simulations, using a research

flight training device (FTD). The first case was the simultaneous stabilization of the

short-period mode of an F4-C fighter jet longitudinal dynamics, and the second case was

the simultaneously stabilizing pitch-tracking control for a CRJ-200 regional jet aircraft.

Several robustness analysis approaches were introduced to provide and test the physical

bounds of allowable perturbations. The proposed optimization solution to the SS problem

was applied to both case studies and simultaneously stabilizing controllers were designed.

These controllers were then compared to the Multiple-Robustness optimization solution

to the RSS problem, in terms of robustness bounds. To give a better comparison, both

control law solutions were also compared to the Bi-level optimization solution to the RSS

problem, in terms of stability robustness bounds.

In the first case study, the SS problem solution had a degree of robustness to varia-

tions in the aerodynamic derivative CLα , which was also verified by two other different

robustness investigation strategies. When expanding the robustness bounds through the

Multiple-Robustness optimization solution, the first, second, and the forth operating

conditions experienced a significant increase in their corresponding maximum allowable

deviation, |∆CLα|max, by %27, %26, and %7 respectively. There was a slight decrease of

%1 in the maximum allowable deviation for the third operating point, and the robustness

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Chapter 7. Conclusion and Future Developments 88

of the other three systems was significantly increased at this cost. When the robustness

was to improve through the Bi-level optimization solution to the RSS problem, the sec-

ond and the forth operating points experienced an increase in their maximum allowable

deviation by %27 and %11 respectively; whereas that of the first and third operating

points saw a significant drop of %-57 and %-86. The multiple-objective optimization

problem (MOOP) methodology delivered a better result in terms of increasing the sta-

bility robustness bounds.

In the second case study, a simultaneously stabilizing pitch-tracking controller was

designed and the simulation results were obtained using a state of the art flight simula-

tion platform. Later, the robust behaviour of the controller was shown by activating the

incorporated “Moderate Turbulence” function. When expanding the robustness bounds

through the Multiple-Robustness optimization solution, the maximum allowable devia-

tion in ai,j saw significant improvement. The first plant corresponding with the aircraft

flying at the first operating condition saw an improvement in its stability robustness

bounds by %88. The second, third, and fourth systems also experienced great improve-

ment in the stability robustness values by %14, %48, and %23, respectively.

7.2 Future Developments

Several improvements can be made to the design optimization strategy. It could be

interesting to investigate the controller’s stability robustness with respect to variations

in other prominent stability and aerodynamic derivatives such as Cmq , CLδe , etc., as well

as altitude and Mach number across the flight envelope. This would be more realistic

and of practical interest.

The behaviour of the closed loop systems when encountered with gust can also be a

topic of further investigation. It would be interesting to investigate the effect of different

gust models on the stability of the system at different operating points and the robustness

bounds.

One interesting addition to the optimization methodology would be to optimize for

a combination of robustness bounds and performance, in order to improve the transient

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Chapter 7. Conclusion and Future Developments 89

response of the design. The desired transient behaviour of the system could be included

in the objective function or it could be imposed as a constraint.

The proposed multiple-robustness optimization solution to the RSS problem could in-

clude other classical multi-objective optimization methods, which are more complicated

than the weighted-sum method in terms of formulation. A change in the objective func-

tion to represent more complex trade-offs is usually accompanied by a different optimum

solution to the problem. Performance criteria could also be implemented in the Bi-level

optimization solution to the RSS problem. This would result in improved stability while

maintaining the performance of the design.

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