Lecture 1

Chapter 1: Introduction

Eugenio Schuster

schuster@lehigh.edu

Mechanical Engineering and Mechanics

Lehigh University

Lecture 1 – p. 1/22

Book

MULTIVARIABLE FEEDBACK CONTROLAnalysis and designSigurd Skogestad

Norwegian University of Science and Technology

Ian Postlethwaite

University of Leicester

Lecture notes are based on material provided by Prof. Skogestad

Lecture 1 – p. 2/22

Content

1. Introduction [Chapter 1]

2. Classical feedback control [Chapter 2]

3. Performance limitations SISO [Chapter 5]

4. Uncertainty and robustness SISO [Chapter 7]

5. Elements of linear systems theory [Chapter 4]

6. Introduction to multivariable control [Chapter 3]

7. Performance limitations MIMO [Chapter 6]

8. Robust stability and performance MIMO [Chapter 8]

9. Controller design [Chapter 9]

Lecture 1 – p. 3/22

Another Books

C. Chen, Linear System Theory and Design, OxfordUniversity Press, 1999.G. Dullerud and F. Paganini, A Course in RobustControl Theory, Springer, 2000.M. Green and D. Limebeer, Linear Robust Control,Prentice Hall, 1995.J. Helton and O. Merino, Classical Control UsingH-infinity Methods, SIAM, 1998.J. Maciejowski, Multivariable Feedback Design, AddisonWesley, 1989.R. Sanchez-Pena and M. Sznaier, Robust SystemsTheory and Applications, John Wiley & Sons, 1998.K. Zhou, Robust and Optimal Control, Prentice Hall,1996.K. Zhou, Essentials of Robust Control, Prentice Hall,1998.

Lecture 1 – p. 4/22

Web Page

http://www.lehigh.edu/˜eus204/Teaching/

ME450_MRC/ME450_MRC.html

Lecture 1 – p. 5/22

Introduction

1. Study the “plant" to be controlled and learn about control objectives

2. Model the system and simplify the model (if necessary)

3. Analyze the resulting model; determine its properties

4. Decide which variables are to be controlled (controlled outputs)

5. Decide on measurement (sensors)/manipulated (actuators) variables

6. Select the control configuration

7. Decide on the type of controller to be used

8. Decide on performance specifications (based on control objectives)

9. Design a controller

10. Analyze resulting controlled system (redesign if necessary)

11. Simulate the resulting controlled system

12. Repeat from Step 2, if necessary.

13. Choose hardware and software and implement the controller

14. Test and validate control system and tune on-line if necessary

Lecture 1 – p. 6/22

Introduction

⇒ Much of the course will be spent on input-output“controllability analysis” of the plant/process.

Always keep in mind

Power of control is limited.

Control quality depends controller AND onplant/process.

Lecture 1 – p. 7/22

Introduction

1.1 The control problem [1.2]

y = Gu+Gdd(1.1)

y : output/controlled variable

u : input/manipulated variable

d : disturbance

r : reference/setpoint

Regulator problem : counteract d

Servo/Tracking problem : let y follow r

Goal of control: make control error e = y − r “small”.

Lecture 1 – p. 8/22

Introduction

Major difficulties:

Model (G,Gd) inaccurate⇒ RealPlant: Gp = G+ E ;

E = “uncertainty” or “perturbation” (unknown but bounded)

Nominal stability (NS) : system is stable with no modeluncertainty.

Nominal Performance (NP) : system satisfies performancespecifications with no model uncertainty.

Robust stability (RS) : system stable for “all” perturbedplants

Robust performance (RP) : system satisfies performancespecifications for all perturbed plants

Lecture 1 – p. 9/22

Introduction

The course will be make extensive use of the transferfunction representation, i.e., we will work on the frequencydomain.

Given a state-space representation

x = Ax+Bu,

y = Cx+Du,

the transfer-function representation is given by

y(s)

u(s)= G(s) = C(sI −A)−1B +D, G(s) =

[A B

C D

]

Lecture 1 – p. 10/22

Introduction

1.2 Transfer functions [1.3]

G(s) =βnz

snz + · · · + β1s+ β0sn + an−1sn−1 + · · · + a1s+ a0

(1.2)

For multivariable systems, G(s) is a matrix of transferfunctions.n = order of denominator (or pole polynomial) or order ofthe systemnz = order of numerator (or zero polynomial)n− nz = pole excess or relative order.

Lecture 1 – p. 11/22

Introduction

Definition

A system G(s) is strictly proper if G(s) → 0 as s → ∞.

A system G(s) is semi-proper or bi-proper ifG(s) → D 6= 0 as s → ∞.

A system G(s) which is strictly proper or semi-proper isproper.

A system G(s) is improper if G(s) → ∞ as s → ∞.

Lecture 1 – p. 12/22

Introduction

1.3 Scaling [1.4]

Proper scaling simplifies controller design and performanceanalysis.SISO :

unscaled:

y = Gu+ Gdd; e = y − r(1.3)

scaled:

d = d/dmax, u = u/umax(1.4)

where:

dmax — largest expected change in disturbance

umax — largest allowed input change

Lecture 1 – p. 13/22

Introduction

Scale y, e and r by:

emax — largest allowed control error, or

rmax — largest expected change in reference value

Usually:

y = y/emax, r = r/emax, e = e/emax(1.5)

MIMO :

d = D−1

dd, u = D−1

u u, y = D−1e y(1.6)

e = D−1e e, r = D−1

e r(1.7)

where De = emax, Du = umax, Dd = dmax and Dr = rmax arediagonal scaling matrices

Lecture 1 – p. 14/22

Introduction

Substituting (1.6) and (1.7) into (1.3):

Dey = GDuu+ GdDdd; Dee = Dey −Der

and introducing the scaled transfer functions

G = D−1e GDu, Gd = D−1

e GdDd(1.8)

Model in terms of scaled variables:

y = Gu+Gdd; e = y − r(1.9)

Often also:r = r/rmax = D−1

r r(1.10)

so that:

r = Rr where R∆= D−1

e Dr = rmax/emax(1.11)

Lecture 1 – p. 15/22

Introduction

❡ ❡✲ ✲ ✲❄ ❄

❄ ❄

✲u G

Gd

d

yr-

+

+

+

r

e

R

Figure 1: Model in terms of scaled variables

Objective:for |d(t)| ≤ 1 and |r(t)| ≤ 1,manipulate u with |u(t)| ≤ 1such that |e(t)| = |y(t)− r(t)| ≤ 1.

Lecture 1 – p. 16/22

Introduction

1.4 Linearization [1.5]

Given the general nonlinear model

x = f(x, u)(1.12)

where (x∗, u∗) is an equilibrium, i.e., f(x∗, u∗) = 0. Thelinearization around this equilibrium is given by

˙x ≈

(∂f

∂x

)∗

x+

(∂f

∂u

)∗

u , Ax+Bu(1.13)

where x , x− x∗ and u , u− u∗. Therefore,

x(s) = (sI − A)−1Bu(1.14)

Lecture 1 – p. 17/22

Introduction

1.5 Notation [1.6]

❜

❜

❜ ♣

✻

✻✛

✲❄✲✲✲

❄

✲ ++

ym-

n

y

d

Gd

++

+u GKr

❜

❜

✻

✲ ♣

✛

✲❄✲✲✲

❄

+

n(b) Two degrees-of-freedom control configuration

++

+ym

y

d

Gd

K Gur

(a) One degree-of-freedom control configuration

Lecture 1 – p. 18/22

Introduction

✲

✛

✲✲

(c) General control configuration

P

K

u v

zw

Figure 2: Control configurations

Lecture 1 – p. 19/22

Introduction

Table 1: Nomenclature

K controller, in whatever configuration. Sometimes bro-ken down into parts. For example, in Figure 2(b),K = [Kr Ky ] where Kr is a prefilter and Ky is the

feedback controller.

Conventional configurations (Fig 2(a), 2(b)):

G plant model

Gd disturbance model

Lecture 1 – p. 20/22

Introduction

r reference inputs (commands, setpoints)

d disturbances (process noise)

n measurement noise

y plant outputs. ( include the variables to be controlled(“primary” outputs with reference values r) and possiblyadditional “secondary” measurements to improve con-trol)

ym measured y

u control signals (manipulated plant inputs)

Lecture 1 – p. 21/22

Introduction

General configuration (Fig 2(c)):

P generalized plant model. Includes G and Gd and the in-terconnection structure between the plant and the con-troller.May also include weighting functions.

w exogenous inputs: commands, disturbances and noise

z exogenous outputs; “error” signals to be minimized,e.g. y − r

v controller inputs for the general configuration, e.g.commands, measured plant outputs, measured distur-bances, etc. For the special case of a one degree-of-freedom controller with perfect measurements we havev = r − y.

u control signals

Lecture 1 – p. 22/22