Add on material of the book Dynamical Systems - …1 Model Reference Adaptive Control Systems (MRAS)...

University of Twente

Intelligent Controlpart 1 - MRAS

Faculty EE-Math-CSDepartment of Electrical Engineering

Control Engineering

Author:prof.dr.ir. Job van Amerongen

102CE2004March 2004

264

Adaptation

Controller

y

process

Liapunov

K____s+a

Ki

Kw

p22

p21

Kd

Kp

2n�k

(s+2z s+ )�n2n

�

bp �

i

Intelligent Control

Part 1 Model Reference Adaptive Control

ii

University of Twente Control Engineering Author: Prof.dr.ir. Job van Amerongen

iii

University Twente

Part 1

Intelligent Control Model Reference Adaptive Control Systems

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© 2004 J. van Amerongen Save exceptions stated by the law no part of this publication may be reproduced in any form, by print, photoprint, microfilm or other means, included a complete or partial transcription, without the prior written permission of the publisher.

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Course intelligent control

1 Model Reference Adaptive Control Systems

1 2 3 4 5 6 7 8 9 10 11

Introduction The adaptation mechanism Sensititvity models The stability approach – method of Liapunov The hyperstability method Identification and state estimation Practical problems, noise, non-linearities Noise Discrete MRAS Conclusions References Appendix

1 4 8 10 19 24 29 31 32 36 36 37

2 Indirect Adaptive Control, Self tuning regulators

3 Learning control, Fuzzy control

1

Model Reference Adaptive Control Systems (MRAS)

I N T R O D U C T I O N

In these lecture notes various forms of model reference adaptive systems are discussed. We start with an intuitive approach, showing that basic feedback ideas help to find algorithms for parameter adjustment. We notice that two questions arise. The first is how to find proper signals that adjust the right parameter at the right moment. The second is how to guarantee stability for an adaptive system that is inherently non linear due to the multipliers present in the system. More insight in the first question is obtained by considering the sensitivity model approach. Stability can be guaranteed by using Liapunov’s stability theory for the design of the adaptive system. L E A R N I N G G O A L S After completing these notes you are expected to know – which signals play a role in an adaptive system – how an adaptive system can be designed based on a sensitivity approach – how an adaptive system can be designed based on a Liapunov (stability)

approach C O R E

1 Introduction

There are various structures that may give a control system the possibility to react to variations in its parameters or to changing characteristics of the disturbances. A normal feedback system also has the objective of decreasing the sensitivity for these types of variations. However, when the variations are large, even a well-designed constant-gain feedback system will not operate satisfactorily. Then a more complex controller structure is required and certain adaptive properties have to be introduced. An adaptive system may be defined as follows:

Definition of adaptive control

“An adaptive system is one in which in addition to the basic (feedback) structure, explicit measures are taken to automatically compensate for variations in the operating conditions, for variations in the process dynamics or for variations in the disturbances, in order to maintain an optimal performance of the system”. Many other definitions have been given in the literature; most of them only describe a typical class of adaptive systems. The definition given here assumes as a base an ordinary feedback structure for the primary reaction to disturbances and parameter variations. On a secondary level an adaptation mechanism tunes the gains of the primary controller, changes its structure, and generates additional signals and so on. In such an adaptive system the settings which are adjustable by the user are at the secondary level.

University of Twente Model Reference Adaptive control Systems Control Engineering

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According to the definition automatically changing from one mode of operation to a different one is considered as an adaptive feature. Use of the knowledge about the influence of an external variable on the behaviour of the system is an adaptive feature as well. This type of adaptation can be realized in two different ways: either by measuring particular disturbances and generating signals to compensate for them (feedforward control), or by adjusting the feedback controller gains according to a schedule based on knowledge about the influence

Gain scheduling of the variables on the system’s parameters (gain scheduling). Another possibility is using a bank of controllers and select the best controller in a similar

Mode switching way as gain scheduling. This is called mode switching. A variation on this idea is a multi-model approach. The outputs of all models in a bank of models are compared with the output of the process to be controlled. A controller can be designed and implemented based on the model of which the output has the best resemblance with the process output. In practice it is impossible to apply feed-forward control or gain scheduling to a lot of different variables. Several types of adaptive systems, in a more narrow sense, have been developed which allow a system to be optimized without any knowledge of the causes of changing process dynamics. Often the term adaptive control is restricted to these types of adaptive systems. There is no clear distinction between adaptive control and learning control. The term learning control is often used for more complex systems where a lot of memory is involved and for problems that cannot be solved by means of standard controllers, based on transfer functions, because they require another form of knowledge representation, e.g. in neural-network-like structures. These lecture notes deal with a particular kind of adaptive control, known as Model Reference Adaptive Control. Adaptive control systems can be classified in various ways. One possibility is to make a distinction between:

Direct and indirect adaptive control

– systems with direct adjustment of the controller parameters, without explicit identification of the parameters of the process (direct adaptive control)

– systems with indirect adjustment of the controller parameters, with explicit identification of the parameters of the process (indirect adaptive control)

Model Reference Adaptive Control Systems, mostly referred to as MRAC or MRAS, are mainly applied for direct adaptive control. However, in the following the application of MRAS to system identification will also be demonstrated. The basic philosophy behind the application of MRAS is that the desired performance of the system is given by a mathematical model, the reference model. When the behavior of the process differs from the ‘ideal’ behavior, which is determined by the reference model, the process is modified, either by adjusting the parameters of a controller (figure 1a) or by generating an additional input signal for the process (figure 1b). This can be translated into an optimization problem, i.e. minimization of the criterion:

2

0

= ∫T

C e dt (1)

where

Intelligent Control

3

= −m pe y y (2)

Instead of minimizing only the error between the output signals of the process and the reference model, all the state variables of the process and the reference model can be taken into account. When the state variables of the process are denoted as (xp) and those of the reference model as (xm), the error vector (e) can be defined:

m p= −e x x (3)

In that case the optimization problem can be translated into minimization of the criterion:

0

TTC dt= ∫ e Pe (4)

where P is a positive definite matrix.

Controller

AdaptiveController

Reference Model

yProcess

u

AdaptiveController

Reference Model

yuController Process

FIGURE 1a Parameter adaptive system FIGURE 1b Signal adaptive system

The following considerations may play a role in the choice between adaptation of the parameters and signal adaptation. An important property of systems with parameter adaptation is that such systems have a memory. As soon as the parameters of the process have been adjusted to their correct values, and there are no new changes, the adaptive loop is in fact not necessary anymore: process and reference model show the same behavior. In general memory is not present in systems with signal adaptation. Therefore, the adaptive loop remains necessary in all cases, in order to generate the right input signal. Consequently, signal adaptive systems must react faster to changing process dynamics than systems with parameter adaptation because no knowledge from the past can be used. In systems where the parameters constantly vary over a wide range this is advantageous. However, in a stochastic environment, i.e. in systems with a lot of noise, this may be a disadvantage. High gains in the adaptive loop may introduce a lot of noise in the input signal of the process as well.

As we will see later on, the multiplications in the adaptive controller always lead to a non-linear system. It could be argued what adaptive control is more than non-linear feedback.

When the parameters of the process vary slowly or only now and then, systems with parameter adaptation will give a better performance because of their memory. There are also adaptive algorithms which combine the advantages of both methods. In the following attention will mainly be focused on parameter adaptive systems, although the combination of parameter and signal adaptation will also be discussed.


4

Another way of looking at the system is the following. The ‘standard feedback control loop’ is considered as a fast reacting primary control system that has to reject ‘ordinary’ disturbances. Large variations in the parameters or large disturbances are dealt with by the slower reacting secondary (adaptive) control system (Figure 2).

secondary control system

primary control system

Reference Model

AdaptiveController

yuController Process

FIGURE 2 Primary and Secondary control 2 The adaptation mechanism

In the literature several methods have been described for designing adaptive systems. But you can get more insight into a method by thinking about how to find the algorithms yourself. This helps to really understand what is going on. Therefore, for the time being we will postpone the mathematics and examine the basic ideas of MRAS with a simple example. When we try to design an adaptive controller for this simple system we will automatically encounter the problems which require more theoretical background. Common properties of the various design methods as well as their differences will become clear. In figure 3 a block diagram is given of the system which will be used as an example throughout this text.

reference model

process

?

yubp

Kb Ka

ap

∫ ∫

FIGURE 3 Process and reference model

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In this example the (linear) process is described by the transfer function:

2 1+ +p

p

bs a s

and the model by 2 1m

m

bs a s+ +

or 2

2 22n

n n

Ks s

ωζω ω+ +

(5)

Of course the ‘controller’ with parameters Ka and Kb is not a real controller. In fact we assume at this stage that the process parameters can be adjusted directly.

Variations in the parameter ap can be compensated by adjusting Ka and variations in bp can be adjusted by adjusting Kb. This follows directly from the transfer of process plus controller in Figure 3:

( )2 1

+

+ + +b p

p a

K b

s a K s (6)

The (linear) reference model has the same order as the process. The following numerical values are chosen:

1nω = , 0.7z = , 1.4pa = , 0.5pb = (7)

In this case only the DC-gain of the process and the reference model differ by a factor of two. This can be seen in the step responses of this system (figure 4).

model

0 10 20 30 40 50 60 70 80 90 100time {s}

Err

orY

_pro

cess

Y_m

odel

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

FIGURE 4 Responses of process, reference model and error Because m pe y y= − and 1

2p my y= , in this case the error e is equal to yp. In order to get two identical responses the parameter Kb has to be adjusted. It is obvious that Kb should be increased. A reasonable choice for adjustment of Kb seems to be:

( ) ( )0b bK t K e dtβ= + ∫ (8)

With the ‘adaptive gain’ β the speed of adjustment can be set. The desired memory function is realized by means of the integration which also guarantees that a constant difference between (Kb + bp) and bm converges to zero. This ‘adaptive law’ with 0.5β = gives the results shown in figure 5.


6

Kb adaptation

0 10 20 30 40 50 60 70 80 90 100time {s}

Kp

0

0.2

0.4

0.6

0.8

model

0 10 20 30 40 50 60 70 80 90 100time {s}

Err

orY

_pro

cess

Y_m

odel

-0.5

0

0.5

1

1.5

-2

-1

0

1

2

-2

-1

0

1

2

FIGURE 5 Result of adaptation of Kb. At the left the parameter,

at the right responses of process, reference model and error Although the result is impressive, it quickly becomes clear that there are still a few problems left. When the input signal u is inverted the adjustment of Kb is going in the wrong direction, because of the negative sign of e. An unstable system is the result. However, the solution to this problem is simple. When the sign of the input signal is taken into account, for instance by multiplying e and u, the result of the parameter adjustment conforms again to figure 5. This yields the

MIT rule adjustment law known as the MIT rule:

( ) ( ) ( )0b bK t K eu dtβ= + ∫ (9)

A second problem is encountered when not only variations of bp have to be compensated for, but also variations of the parameter ap. A similar reasoning as in the case of adjustment of Kb could lead to an adjustment law for the parameter Ka, based on e and on the sign of u. But this would lead to the same adjustment law for each parameter. Apparently not only the direction of adjustment of the parameter has to play a role, but also the amount of adjustment of each parameter, relative to the others. Such a “dynamic speed of adjustment” may be realized by adjusting each parameter, depending upon the effect this adjustment has on decreasing the error. This reasoning leads to the candidate adjustment laws:

( ) ( ) ( )0b bK t K eu dtβ= + ∫ (10)

( ) ( ) ( )20a aK t K ex dtα= + ∫ (11)

The parameter Kb is adjusted when u, the signal which is directly influenced by Kb, is large and the parameter Ka is adjusted when x2, the signal which directly influenced by Ka, is large. In figure 6 simulation results are given. It appears that our intuitive reasoning yields a system where a reasonably fast adaptation takes place. In the simulation of figure 6 the values of bp and ap are taken as zero, which is compensated by appropriate initial values of Ka and Kb (0.5 and 0.7, respectively). The parameters converge to the correct values of 1 and 1.4 and as a result the responses of the process and the reference model become equal. The speeds of adaptation are chosen as α = 12 and β = 2.

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Kb and Ka adaptation

0 10 20 30 40 50 60 70 80 90 100time {s}

Kb

Ka

0

0.5

1

1.5

2

model

0 10 20 30 40 50 60 70 80 90 100time {s}

Err

orY

_pro

cess

Y_m

odel

-0.5

0

0.5

1

1.5

-3

-2

-1

0

1

-3

-2

-1

0

1

FIGURE 6 Result of adaptation of Ka and Kb In figure 6 the speed of adaptation, determined by the adaptive gains α and β, is still small. In order to speed up the system, the adaptive gains are increased to α = 60 and β = 10. This yields the disappointing results of figure 7.

Kb and Ka adaptation

0 10 20 30 40 50 60 70 80 90 100time {s}

Kb

Ka

-50

0

50

100

model

0 10 20 30 40 50 60 70 80 90 100time {s}

Err

orY

_pro

cess

Y_m

odel

-0.5

0

0.5

1

1.5

-3

-2

-1

0

1

-3

-2

-1

0

1

FIGURE 7 Adjustment of Ka and Kb with a higher speed of adaptation The adaptive system, which was stable with low adaptive gains, becomes unstable with higher adaptive gains. When the block diagram of this system is examined (figure 8) it is clear that this stability problem cannot easily be solved, due to the non-linearities which have been introduced into the system.


8

α

alpha

∫ Ka

β Beta

∫ Kb

2nωk

(s+2z s+ )ωn2nω

bp

ap

∫ ∫

FIGURE 8 Non-linearities in an adaptive control system Until now we have thus faced two main problems: 1. A kind of ‘dynamic speed of adaptation’ is needed in order to realize that

each parameter is only adjusted when the resulting error is sensitive to variation of that parameter.

2. A stability problem exists when the adaptive gains are increased as a result of the desire to speed up the adaptation. This stability problem cannot easily be solved by linear analysis methods because adaptation makes the system non-linear.

These two problems clarify the origin of various methods for designing MRAS. Two methods will be discussed in more detail in the following: - the sensitivity approach. This method emphasizes the determination of the

‘dynamic speed of adaptation’ with the aid of sensitivity coefficients. - the stability approach. This method emphasizes the stability problem. Due to

the non-linear character of an adaptive system it is essential that stability theory of non-linear systems be used. It will be shown that, together with a proof of stability, useful adaptive laws can be found.

3 Sensitivity models

In the sensitivity approach the first step is to translate the adaptation problem into an optimization problem by introducing the criterion:

2

0

12

t

C e dτ= ∫ (12)

In order to minimize C, the adjustable parameters Ki are varied. The direction of these variations is determined by the gradient ∂C/∂Ki, thus:

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i ii

CKK

α ∂∆ = −

∂ (13)

By differentiating eq. (13) to t, a continuous adaptive law can be found:

ii

i

dK d Cdt dt K

α ∂= −

∂ (14)

From eqs. (12) and (14) it follows that

212

ii

i

dKe

dt Kα ∂ = − ∂

(15)

or

ii

i

dK eedt K

α ∂= −

∂ (16)

In eq. (2) the error has been defined as:

m pe y y= − (17)

Because / 0m iy K∂ ∂ = , it follows from eqs. (16) and (17) that

pii

i

ydKe

dt Kα

∂=

∂ (18)

This algorithm is more or less similar to the algorithms of (10) and (11). The direction of adjustment and the amount of adjustment relative to other parameters is now determined by the error e and the sensitivity coefficient ∂yp/∂Ki. The latter can be determined by means of a sensitivity model. The sensitivity coefficient explicitly sees to it that adjustment of Ki only takes place when the error between process and reference model is sensitive to variations in this particular parameter.

Example The process in figure 3 can be described by the differential equation:

( ) ( )p p a p p b py a K y y K b u+ + + = + (19)

where y denotes dy/dt and 2 2/y d y dt= . After the parameter Kv is introduced eq. (19) can be rewritten into eq. (21).

v p aK a K= + (20)

( )p v p p by K y y K b u+ + = + (21)

Differentiation of this equation to Kv yields:


10

p p p

v pv v v

y y yK y

K K K∂ ∂ ∂

+ + = −∂ ∂ ∂

(22)

The differential equation (22) is identical to eq. (21), except for the input signal. Sensitivity model Eq. (22) is called a sensitivity model. When an estimation is made for the value

of Kv, e.g. by selecting it equal to the desired value am, the sensitivity coefficient ∂yp/∂Kv can be measured. From eq. (18) it follows that

pv

v

ydKe

dt Kα

∂=

∂ (23)

Assuming that the process parameter ap varies slowly compared with the adjustable parameter Ka due to the adaptation, it follows from eq. (20) that

v adK dKdt dt

≈ and p p

v a

y yK K

∂ ∂≈

∂ ∂ (24)

In figure 9 the resulting adaptive system has been given.

Sensitivitymodel

process

y∂ p∂ Ka____

y

bm

am

∫ ∫

ubp

Kb Ka

ap

∫ ∫

FIGURE 9 Example of an adaptive control system based on a sensitivity model. Ka is adjusted to compensate for variations in ap.

EXERCISE Derive the adaptive law for adjustment of the parameter Kb

The sensitivity method is simple and straightforward. The major disadvantage is that stability can only be demonstrated by simulation or tests in practice. An analytical proof of stability cannot be given. 4 The stability approach – method of Liapunov

The design of adaptive systems based on stability theories has originated because of the problems with stability in designs such as those based on sensitivity models. The second method of Liapunov is the most popular approach. A related approach is based on ‘hyperstability’ theory. Both approaches may produce the same results, so that there is no direct preference for one of the two with respect to the resulting algorithms. The use of Liapunov’s stability theory for the design of adaptive systems was introduced by Parks in 1966. Derivation of the adaptive laws is done most easily

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when process and reference model are described in state space form. The process can be written as:

p p p p= +x A x B u (25)

with

p p a′= +A A K (26)

and

p p b′= +B B K (27)

where p′A and p

′B are the varying process parameters which can be compensated by the controller parameters Ka and Kb. The reference model can be written as:

m m m m= +x A x B u (28)

Subtracting eq. (25) from eq. (28) yields, after defining e

m p= −e x x (29)

m p= + +e A e Ax Bu (30)

with

m p= −A A A (31)

m p= −B B B (32)

Adjustment of A and B necessitates non-linear adaptive laws like eqs. (10) and (11). Consequently, the resulting differential equation (30) is non-linear. To ensure that for t → ∞, e = 0 it is necessary and sufficient to prove that e = 0 is a stable equilibrium solution. According to Liapunov’s stability theory this can be done when we can find a (scalar) Liapunov function V(e), with the following properties: V(e) is definite positive (this means that V > 0 for e ≠ 0, eventually V = 0 for e = 0.

( )V e is definite negative (this means that V < 0 for e ≠ 0, eventually 0V = for e = 0.

V(e) → ∞, if → ∞e . When the Liapunov function V(e) is correctly chosen the adaptive laws follow directly from the conditions under which ( )V e is negative definite. The main


12

(theoretical) problem is the choice of a suitable V(e). Many suitable Liapunov functions can be found. Different Liapunov functions lead to different adaptive laws. Searching the Liapunov function which belongs to a certain candidate algorithm is a difficult procedure. However, in the literature several ‘standard’ Liapunov functions have been given which yield useful adaptive laws. Simple and generally applicable adaptive laws are found when we use the Liapunov function: ( ) T T TV α β= + +e e Pe a a b b (33)

Where – P is an ‘arbitrary’ definite positive symmetrical matrix, – a and b are vectors which contain the non-zero elements of the A and B

matrices, – α and β are diagonal matrices with positive elements which determine the

speed of adaptation. The choice of the Liapunov function as given in eq. (33) is a quite straightforward one. The Liapunov function represents a kind of ‘energy’ that is present in the system and when this energy contents goes to zero, the system reaches a stable equilibrium. In dynamic systems the energy is present in the integrators which can also be considered as the state variables of the system. The components of e , a and b are the state variables of the system described by eq. (30). The components of a and b are the parameter errors and can be seen as wrong initial conditions of the adaptive controller parameters. It is thus desirable that all state variables e , a and b go to zero. With this choice of P, α and β, V(e) is a definite positive function. Differentiation of V(e) yields:

2 2T T T TV α β= + + +e Pe e Pe a a b b (34)

Together with eq. (30) this yields:

( ) ( )2 2

2 2

T Tm m

T Tp

T T

V

α

β

= + +

+ + +

+ +

A e Pe e P A e

e PAx a a

e PBu b b

(35)

Let:

Tm m+ = −A P A P Q (36)

Because the matrix Am belongs to a stable system (the reference model), it follows from the theorem of Malkin that Q is a definite positive matrix. This implies that the first part of eq. (35):

( )T T Tm m+ = −e A P PA e e Qe (37)

is definite negative. Stability of the system can be guaranteed if the last part of eq. (35) is set equal to zero. Let:

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0T Tp α+ =e PAx a a (38)

0T T β+ =e PBu b b (39)

This yields, after some mathematical manipulations, the general form of the adjustment laws:

1

1 n

ni nk k ikni

a p e xα =

= − ∑ (40)

1

1 n

i nk k iki

b p e uβ =

= −

∑ (41)

where n is the order of the system. These algorithms have the same basic structure as the algorithms which were earlier found in a heuristic way. The main difference is that all elements of the error vector e (with weighting factors pnk) are used in the adaptive law, instead of only the error signal e. The factors pnk are the elements from the n-th row and the k-th column of the P-matrix. These elements can be found with the aid of eq. (36). An arbitrary definite positive matrix Q is chosen, after which P can be solved from eq. (36). In a simulation P can easily be solved from eq. (36) by writing eq. (36) as

0Tm m+ + =A P A P Q

and consider this expression as the equilibrium solution of the differential equation

Tm m

dPdt

= + +A P A P Q

This equation can easily be solved in e.g. 20-sim. The speed of convergence can be increased by introducing a speed up factor λ > 1, e.g. λ = 10: 1 T

m mdPdtλ

= + +A P A P Q

The following steps are thus necessary to design an adaptive controller with the method of Liapunov:

1. determine the differential equation for e 2. choose a Liapunov function V 3. determine the conditions under which V is definite negative 4. solve P from T

m m+ = −A P A P Q . The whole procedure will be clarified with an example.

Example In this example we will again use the system of figure 3. Step 1 Determine the differential equation for e.

The process can be described by the following differential equations:


14

1 2x x= (42)

( ) ( )2 1 2p a p bx x a K x b K u= − − + + + (43)

This yields:

( )0 1 01p p

p bp a b Ka K

= = +− − + A b (44)

The reference model (with ωn = 1 and 2zωn = 1.4) is described by:

0 1 01 1.4 1m m

= = − −

A b (45)

The differential equation for e is:

m p= + +e A e Ax bu (46)

with

( ) ( )0 0 0

0 1.4 1p a b pa K K b

= = − + + − + A b (47)

Only one element of the A-matrix, a22, and one element of the b-vector, b2, are not equal to zero.

Step 2 Choose a Liapunov function V. In this example eq. (33) will be used as a Liapunov function.

Step 3 Determine the conditions under which dV/dt is definite negative. These are the adjustment laws (40) and (41) which in this case simplify to:

( )22 21 1 22 2 222

1a p e p e xα

= − + (48)

( )22 21 1 22 22

1b p e p e uβ

= − + (49)

From eq. (47) it follows that:

( )22 1.4 p aa a K= − + + (50)

and

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( )2 1 b pb K b= − + (51)

With the assumption that the speed of adaptation is fast compared with the speed of variations in the process parameters, it follows from eqs. (48) and (49) that:

( )21 1 22 2 222

1aK p e p e x

α= − + (52)

( )21 1 22 22

1bK p e p e u

β= + (53)

Step 4 Solve P from Tm m+ = −A P PA Q

Any positive

definite matrix Q leads to a positive definite matrix P. The opposite is not true.

This fourth step starts with the choice of an ‘arbitrary’ matrix Q. It is apparent that, although the choice of Q is arbitrary, the performance of the adaptive system is influenced by the choice of Q. The first choice of Q is not necessarily the best one. But whatever the choice of Q is, as long as Q is positive definite, the stability of the resulting adaptive system can be guaranteed. For simple systems P can be found manually, for higher order systems, the computer can be used. As an example select:

4 0.80.8 1.6

=

Q (54)

which yields the following matrix equation:

11 22 11 22

22 22 21 22

0 1 0 1 4 0.81 1.4 1 1.4 0.8 1.6

p p p pp p p p

− + = − − −

(55)

This can be rewritten as:

21 11 22 22

11 21 22 21 22

2 1.4 4 0.81.4 2 2.8 0.8 1.6

p p p pp p p p p

− − − = − − − −

(56)

from which it follows that:

4 22 2

=

P (57)

Both factors p21 and p22 are thus 2 in this case. The complete adaptive laws in integral form are:

( ) ( )12 1 22 2 222 0

1 0t

a aK p e p e x dt Kα

= − + +∫ (58)

( ) ( )12 1 22 22 0

1 0t

b bK p e p e u dt Kβ

= + +∫ (59)


16

From a (theoretical) stability point of view α22 and β2 may be freely chosen. Stability is always guaranteed. In our earlier experiments it appeared that a choice of 1/α22 = 60 and 1/β2 = 10 yielded an unstable system with the adaptive laws (10) and (11). Repeating the experiment with the adaptive laws (58) and (59) as in figure 10a gives the results depicted in figure 10b. The system now remains stable and the error between process and reference model is decreasing quickly. The not-connected block Liapunov solves the Liapunov equation (36). The 20-sim code is as follows: 0.1 * ddt(P,Po) = transpose (Am)*P + P*Am + Q; p21=P[2,1]; p22=P[2,2];

y

process

Liapunov

p22

p21

KaKb

2nωk

(s+2z s+ )ωn2nω

bp

ap

∫ ∫

FIGURE 10a Adaptive system designed with Liapunov

Ka and Kb

0 10 20 30 40 50 60 70 80 90 100time {s}

Ka

Kb

-0.5

0

0.5

1

model

0 10 20 30 40 50 60 70 80 90 100time {s}

Pro

ces

Ref

eren

ce M

odel

erro

r

-4

-3

-2

-1

0

1

-2

-1

0

1

2

3

0.01

0.06

0.11

0.16

0.21

0.26

FIGURE 10b Responses Remarks - The stability conditions found with the method of Liapunov are sufficient

conditions, they are not necessary. This implies, for instance, that there is still some freedom in varying the relative values of the elements of the P-matrix, without directly affecting the stability of the system.

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- The speed of adaptation, which can be varied by the adaptive gains α (= 1/α22) and β (= 1/β2), may in principle be chosen freely. In a practical system the adaptive gains are limited. There will always be structural differences between the process and the reference model which implies that the ‘proof of stability’ does not hold anymore.

- In some situations the adaptation can be speeded up by selecting other adaptive laws. Instead of the ‘integral adaptive laws’ (40) and (41), ‘proportional plus integral adaptive laws’ are also generally used. In the latter case, eq. (40), for instance, changes into:

( ) ( )1, ,0

1 10t n

ni ni nk k i nk k ikni I ni P

a a p e x d p e xτα α−

= + − − ∑ ∑∫ (60)

The proportional term in eq. (60) can also be considered as a kind of signal adaptation. It speeds up the adaptation when the process parameters are rapidly changing and may improve the transient behavior of the parameters. Instead of the proportional term a relay function may also be used. This results in a ‘relay plus integral adaptive law’. Figure 10c gives the responses with a proportional plus integral adaptive law. It is clear that the effect of adding the proportional term is more damping of the transients of the adjusted parameters, while also the error signal converges faster than without the proportional term.

Ka and Kb

0 10 20 30 40 50 60 70 80 90 100time {s}

Ka

Kb

-0.2

0

0.2

0.4

0.6

model

0 10 20 30 40 50 60 70 80 90 100time {s}

Pro

ces

Ref

eren

ce M

odel

erro

r

-4

-3

-2

-1

0

1

-2

-1

0

1

2

3

0.002

0.012

0.022

0.032

0.042

0.052

FIGURE 10c Results with a proportional plus integral adaptive law αI = 60, αP = 120, βI = 10, βP = 20,

- On the other hand, one should be careful when applying complex adaptive

laws. In order to be able to apply a more complex algorithm, generally meant to speed up the adaptation, a good resemblance of the structures of process and reference model is essential. If there is such a resemblance the performance of the system may be improved. If there is more uncertainty, the more simple algorithms, like those of eqs. (40) and (41), may give good results. Complex laws also require that more parameters be chosen, while automatic tuning was just one of the aims of using adaptive control.

- A mathematical proof of stability with other adaptive laws, requires the availability of other Liapunov functions. Several Liapunov functions can be found in the literature. However, in principle the test of whether a certain candidate adaptive law yields a stable system can be done more easily with the hyperstability method as we will see later on.


18

EXERCISE Design an adaptive controller for the system of figure 11.

Controller

y

process

K____s+a

Ki

Kw

Kd

Kp bp ∫

Figure 11 A process and a controller

A second-order process with transfer function

( )K

s s a′

+ (61)

is controlled with the aid of a ‘PD’-controller (the differentiating action is realized by means of state feedback). The parameters of this controller are Kp and Kd .Variations in the process parameters K ′ and a can be compensated for by variations in Kp and Kd . Low-frequency disturbances of the input of the process (with amplitude Kw) can be compensated by an additional input with gain Ki, which may be considered as a kind of adaptive integrating action. In the stationary state Kw = Ki should hold. The desired performance of the complete feedback system is described by the transfer function:

21,

2 2 1 en 0.72

m nn

n n

Xz

U s z sω

ωω ω

= = =+ +

(62)

Hint Describe the system in state variables:

( )0 1 0 0

1p d p w v

uK K a K K K K K K K

= + ′ ′ ′ ′− − − +

x x (63)

For the design of this adaptive system it is easier to describe the system with the aid of the state variables ε and x2, where ε = u - xi:

( )2 2

0 1 0 00 1p d w v

uK K a K K K K Kx x

ε ε = + ′ ′ ′− − +

(64)

The description of the reference model is:

22, 2,

0 1 0 02 0 0 1

m m

m mn n

ux xzε ε

ω ω

= + − − (65)

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Derive the adaptive law for this system, using the procedure treated before. 5 The hyperstability method

Based on the work of Popov, Landau introduced the hyperstability concept for the design of adaptive control systems. The hyperstability method is, just as the second method of Liapunov, suitable for proving the stability of a non-linear

Hyperstability and passivity

system. The concept of hyperstability is closely related to the concept of passivity introduced by Willems. Passive elements such as capacitors, inductances and resistors will never store more energy than the energy that is stored in the element at t = 0 plus the energy supplied to it by the environment. The same holds for systems that consist of a network of such passive elements. If all the components in such a system are linear, the transfer function between the two energy-conjugated variables at the port of such a passive system is positive real. This means that the phase shift of such a system is always between plus and minus 90 degrees.

Example Consider, as an example, the arbitrary electric network and bond graph of figure 12a.

passive systempassive system

I CR

1

I

C

R

C

Se 1 0

FIGURE 12a Electric network and corresponding bond graph of a passive system

We compute the transfer function between the effort and flow, H(s) = f(s)/e(s) (or between the voltage and current, H(s) = I(s)/U(s)) at the port of this passive system and draw the nyquist and bode plots of this 4th order transfer function. When we observe the phase shift ϕ, we see that -90º < ϕ < 90 º (see figure 12b). The transfer function itself contains a number of zero’s that is equal to the number of poles minus one:


20

Linear System : Nyquist Diagram

-0.05 0 0.05 0.1 0.15Linear System Re

Line

ar S

yste

m Im

-0.1

-0.05

0

0.05

0.1

Se1\p.e -> C1\p.f

0.001 0.01 0.1 1 10 100Frequency (Hz)

Mag

nitu

de (d

B)

-100

-70

-40

-10

0.001 0.01 0.1 1 10 100Frequency (Hz)

Pha

se (d

eg)

-100

-50

0

50

100

FIGURE 12b Nyquist plot and bode plot of a 4th order passive system

The following statement holds for a scalar linear passive (or hyperstable) system:

Passivity of a linear system

Passivity of hyperstability implies that the real part of the polar plot (nyquist plot) is always positive:

( )Re ( 0H jω ≥ Any transfer function that does not have this property can be made positive real by adding a number of properly chosen zero’s. We can use this property for the design of an adaptive controller. We reconsider the error equation eq. (30)

m p= + +e A e Ax Bu (30)

the linear part of this system is given by:

m=e A e (66)

The last two non-linear and time varying terms of (30) can be considered as an input –w to this linear system:

m= −e A e w (67)

We have seen before that the components of w ––among others the adjustable parameters Ki –– depend on e. This implies that w can be seen as a (non-linear, time-varying) feedback of the system given by eq. (66). We have also seen that the adjustable parameters do not necessarily depend on the state vector e itself. They depended on a combination of one or more components of e. Therefore, we extend the system of eq. 67 with the output equation that defines the signal v:

m= −

=

e A e wv Ce

(68)

where C is the output matrix of the linear system. The adaptive system, seen as a linear system with a non-linear feedback can thus be drawn as figure 13.

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-w vlinear

part

non-linear part

FIGURE 13 A non-linear system split into two parts

If both parts of figure 13 are hyperstable, the complete feedback system will be a hyperstable (passive) system. For the linear part this implies that its transfer function should have a number of zero’s that is equal to the number of poles minus one, such that the system effectively behaves as a first-order system. Such zero’s can be introduced by a proper choice of the output matrix C. It can be demonstrated that such a positive real transfer function can be guaranteed if C is chosen equal to P, where P is the solution of the Liapunov equation:

Tm m+ = −A P PA Q (69)

This implies that: v = Pe (70)

To proof the passivity of the non-linear, time-varying part we cannot use an approach that uses transfer functions. We have to consider the basic concept of a passive system, i.e. a system that never delivers more energy to the outside world than the (finite) amount of energy that is stored in the system at t = 0 plus the amount of energy delivered to the system for t > 0. Energy flow to the passive element is considered a positive energy flow. The demand that an element is passive and does not supply more net energy to the environment than the positive and finite amount of energy E(0), present in the element at t = 0, can be mathematically expressed as:

0

(0) 0t

ef d Eτ + ≥∫ , or 0

(0)t

ef d Eτ ≥ −∫ (71)

where e and f are power-conjugated variables. In other words, such a passive element can only store or dissipate energy.

Example As an example we consider a (possibly non-linear) capacitor C.

The energy which goes into this capacitor is described by:

( ) ( )0

t

e i dτ τ τ⋅∫ (72)

This expression is positive if the environment delivers energy to the capacitor. For a passive system it is therefore necessary that not more energy is delivered to the environment than the energy present at t = 0 plus the net stored energy, given by eq. (72).

( ) ( ) ( ) ( )2 21 12 2

0 0

(0) 0 or (0)t t

Ce e i d e i d Ceτ τ τ τ τ τ+ ≥ ≥ −∫ ∫ (73)

In terms of our adaptive system: when we require that the non-linear part of figure 13 with the input and output variables v and w is passive, we have to choose w such that the following condition holds:


22

( ) 20

0

tTt dη τ γ= ≥ −∫ v w (74)

where 20γ < ∞ is a function of the initial conditions A(0) and B(0). Because w

contains the adaptive laws, this will enable that candidate adaptive laws are checked on their stability properties. When we split the system of eq. (30) into a linear and a non-linear part and we apply the adaptive laws (58) and (59), we find the block diagram of figure 14. In order to proof that the non-linear part is passive we must demonstrate that with these adaptive laws condition (74) is satisfied.

Axp

non-linear part

linear part

Bu

u

px

v

-w

B(0)

A(0)

β1_ __

α1_ __

∫

Am

∫

∫

P

FIGURE 14 Adaptive control system split into a linear and a non-linear part

Proof of passivity of the non-linear part

We want to check condition (74):

0

tT dη τ= ∫ v w (75)

with

p p− = +w Ax Bu (76)

m p= −A A A (77)

m p= −B B B (78)

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Without loss of generality, we assume that

( ) 00 = −A A and ( ) 00 = −B B (79)

When the matrices A and B are adjusted by means of the adaptive laws –Φ(t) and –Ψ(t), respectively, it follows that

( ) 00

t

A d Aφ τ τ= − −∫ (80)

( ) 00

t

B d Bψ τ τ= − −∫ (81)

Together with eq. (75) and (76) it follows that

( ) ( )0 00 0 0

t t tT

p d A d B dη φ τ τ ψ τ τ τ = + + +

∫ ∫ ∫v x u (82)

Any candidate adaptive law can now be substituted in this equation. When we try, for instance, the adaptive laws (40) and (41) used earlier, such that:

( ) Tptφ α= vx (83)

( ) Ttψ β= vu (84)

then it follows that:

0 00 0 0

t t tT T T

p p d A d B dη α τ β τ τ = + + +

∫ ∫ ∫v x vx u vu (85)

This can be split into two parts and be rewritten in a quadratic form as:

2

21 0 0

02

tTp d A Aαη τ

= + − ∫ vx (86)

2

22 0 0

02

tT d B Bβη τ

= + − ∫ vu (87)

The first terms of both equations are quadratic terms and are therefore positive. The conditions necessary to satisfy eq. (74) are thus:

21 02

Aαη ≥ − (88)

22 02

Bβη ≤ − (89)


24

When 20A and 2

0B are smaller than ∞ the condition (74) is satisfied. When P is selected according to eq. (69) the complete non-linear feedback system is hyperstable. For the system of figure 10a this leads with the aid of eqs. (50) and (51) to the adaptive laws (52) and (53). Remarks The difference between the hyperstability method and the method of Liapunov is not that different adaptive laws are found. Both methods differ in the way these laws are found. Instead of searching for Liapunov functions which belong to a certain candidate algorithm, the hyperstability method requires the proof that eq. (74) is satisfied. In principle, this is easier than finding a proper Liapunov function. When we compare the adaptive laws which are found from the stability approach with those found from the sensitivity approach, the following differences become clear. Instead of using only the error signal e, the difference between the outputs of the process and the reference model, the stability approach uses the signal Pe. In the cases we considered until now, Pe corresponded to the signal p12e1 + p22e2. This expression also contains the derivative of the error e with a positive effect on the system’s stability. By introducing the hyperstability method it has been demonstrated that due to this derivative, and in higher-order systems, due to higher derivatives as well, the maximum phase lag never exceeds ±90 degrees. In addition, the stability approach uses the states of the process or the input signal of the process, instead of the sensitivity coefficients. When we compare the signals e with p12e1 + p22e2 and ∂yp/∂Kv with x1, it appears that the corresponding signals of the stability approach and the sensitivity approach have a similar shape, but that the signals of the sensitivity approach (e and ∂yp/∂Kv) have a phase lag . One could consider the sensitivity coefficients as a kind of estimated states. The sensitivity model is a state estimator from this point of view. It will be clear that the observed phase lag has a deteriorating effect on the system’s stability. This explains the superiority of the designs following the stability approach.

EXERCISE Show that a proportional and a proportional plus integral adaptive law as used before, also lead to a hyperstable adaptive system.

6 Identification and state estimation

In the previous sections it has been indicated how MRAS can be used for direct adaptation of the parameters of a controller. In this case the process must follow the response of the reference model. However, when the process and the reference model change places, the reference model, in this case referred to as the ‘adjustable model’, will follow the response of the process (figure 15). This is realized by adjusting the parameters of the adjustable model. In doing so two things are achieved: 1 Identification of the process. The adjustment of the model parameters,

primarily intended to get identical responses from the process and the

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25

reference model, leads, after some time, to a situation where the parameters of the process and the reference model are identical.

2 State estimation. When the adjustment of the adjustable model is successful, states of the adjustable model will, after some time, be identical to the states of the process. The model states can be considered as estimates of the process states.

xpu

Adjustable Model

AdaptiveController

Controller Process

FIGURE 15 MRAS applied for identification The adaptive laws for identification and for (direct) adaptation are identical except that the states xp,i are replaced by the states xm,i. It should be noted that the proof of stability in this case involves that the following Liapunov equation be solved:

Tp p+ = −A P PA Q (90)

where instead of the A-matrix of the reference model Am the matrix Ap is used. The process has to be stable in order to get a stable adaptive system. Ap is also unknown. An estimate of Ap has to be made in order to be able to solve the Liapunov equation. Another issue relevant in the case of identification is the following. The design procedure according to the method of Liapunov yields a proof of asymptotic stability of e. This means that e → 0, when t → ∞. For the parameter errors (A and B) only stability in the sense of Liapunov can be guaranteed. That means that it may happen that A ≠ 0 and B ≠ 0 for t → ∞. It can easily be seen that such a situation may happen. When, for instance u = 0, for t → ∞, xp and xm will both approach zero and thus e as well. However, no parameter adjustment will take place anymore. In the case of adaptation this is hardly a problem because the primary goal is then to ensure that e → 0. The values of A and B do not really matter. But in the case of identification A and B must approach zero. However, it can be demonstrated that when the input signal is sufficiently ‘rich’, that means that the input signal has sufficiently power at all frequencies which are within the bandwidth of the system, it can be guaranteed that A and B both approach to zero. When the states of the process are corrupted with noise, the structure of figure 15 can be used to get filtered estimates of the process states. When the input signal itself is not very noisy, the model states will also be almost free of noise. By selecting adaptive gains which are not too large, noise on e can be prohibited


26

from affecting the model states too much. It is important to notice that in this case the filtering is realized without phase lag. The MRAS structure acts as an adaptive observer. This will be illustrated by means of an example.

Example Figure 16 illustrates this principle used in an autopilot for ships. The ship’s parameters are identified in a closed loop system. The transfer of the ship, from the rudder angle δ to the rate of turn ψ , can be described by the first order transfer function:

1s

s

Ks

ψδ τ

=+

(91)

The influence of waves on the motions of the ship can be modeled as an extra input to this model. However, it is not desirable to react on these waves with steering signals. One way to achieve this is to consider these disturbances as measurement noise, rather than process noise. Therefore, the influence of waves is simulated by adding colored noise to the signal ψ . This yields the signal ψ .

After integration of ψ the heading ψ is obtained. This process is controlled by an autopilot which consists of a proportional control action with gain Kp and a rate feedback with gain Kd. In order to prevent the rudder from reacting to each wave instead of ψ , the estimated signal ψ̂ is

used in the controller. The signal ψ̂ is obtained with the above-mentioned state estimator. The rudder angle is measured and is used as an input signal for a model with the transfer function

ˆ

1m

m

Ks

ψδ τ

=+

(92)

waves

adjustable model

controller ship

Kb

Ka

∫

Kd

coloring

kτs + 1

Kp ∫

FIGURE 16 MRAS applied to suppression of noise in an autopilot for ships

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Because in general Ks and τs will be unknown and may vary when, for instance, the speed of the ship changes, it is necessary to adjust Km and τm with the aid of an adaptation mechanism. In figure 17 responses are given of a simulation of this system, where

( )0 1/ 2m sK K= (93)

( )0 2m sτ τ= (94)

The following signals are given in this figure: ψ the rate of turn signal with noise due to the waves ψ the rate of turn signal without this noise

ψ̂ the estimated rate of turn signal Km the model gain 1/τm the inverse of the time constant of the model ψ the heading

( )ˆδ ψ the rudder angle based on ψ̂

( )δ ψ the rudder angle based on ψ

State estimation performance

0 50 100 150 200 250 300 350 400time {s}

iden

tific

atio

n er

ror

psi_

dot_

estim

ated

mea

sure

rate

of t

urn

-1

-0.5

0

0.5

1

1.5

2

-2

-1.5

-1

-0.5

0

0.5

1

-2

-1.5

-1

-0.5

0

0.5

1

Parametes Ka and Kb

0 50 100 150 200 250 300 350 400time {s}

Kb

Ka

0

0.05

0.1

0.15

model

0 50 100 150 200 250 300 350 400time {s}

Ref

eren

ceP

si

Rud

der

-40

-30

-20

-10

0

10

-40

-30

-20

-10

0

10

-25

0

25

50

75

100

FIGURE 17 Identification and state estimation with the aid of MRAS


28

Parametes Ka and Kb

0 200 400 600 800 1000 1200time {s}

Kb

Ka

0

0.05

0.1

0.15

FIGURE 18 Effect of decreasing adaptive gains

The choice of the speed of adaptation is a compromise between a fast parameter adjustment and suppression of fluctuations in the parameters due to the noise. By introducing decreasing adaptive gains identification can be fast in the beginning and slow down later on (see figure 18, where the speed of adaptation is slowly decreased for t > 400). This will result in parameters that do not fluctuate even when noise is present and in smooth estimates of the rate-of-turn signal. In the literature (Landau, 1979) variations on this principle have been described. By adding a second adjustable model the speed of the parameter estimation can be increased without affecting the suppression of fluctuation of the parameters (Hirsch and Peltie, 1973). It is also possible to improve the state estimation by adding a second adjustable model (Van Amerongen, 1982). The latter idea has had good results in an autopilot for ships. A summary of this application has been given by Van Amerongen (1984). This publication can be found in appendix A. The result of the identification can be used to adjust the controller parameters, e.g. by minimizing –with the aid of the Ricatti equation– the criterion

( )2 2J dtε λδ= +∫ (95)

Where ε is the course error and δ the rudder angle. The resulting structure is shown in figure 19 and the responses in figure 20.

waves

adjustable model

controller ship

optimalcontroller design

Kb

Ka

∫

Kd

coloring

kτs + 1

Kp ∫

FIGURE 19 Indirect adaptive control: the identification result is used to compute an optimal controller

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Parameters Kp and Kd

0 50 100 150 200 250 300 350 400time {s}

Kp

Kd

-10

0

10

20

30

40

model

0 50 100 150 200 250 300 350 400time {s}

Ref

eren

ceP

si

Rud

der

-40

-30

-20

-10

0

10

-40

-30

-20

-10

0

10

-60

-30

0

30

60

90

FIGURE 20 Simulation results. At the left: results of the on-line controller parameter optimization

7 Practical problems, noise, non-linearities

Until now the practical problems have been put aside. In this section a few practical problems and their solutions will be discussed: - not completely matching process and reference model structures - non-linearities in the process or the reference model - noise (in general, disturbances in the process)

Structural differences When the structures of the process and the reference model do not completely match, a formal proof of stability does not in theory hold anymore. If this were immediately disastrous, a practical application of MRAS would be hardly possible. It has already been mentioned that in this case simple algorithms have advantages over more complex algorithms. However, it is still important that the dominating dynamics of the process are present in the reference model as well. In general, it appears to be possible to compensate for small remaining differences between the structure of the process and the reference model by relatively fast parameter variations.

Example This is illustrated with the following example. A DC-motor with a flexible transmission is controlled by means of an adaptive controller, based on a second-order reference model. Because feedback of the motor angle and motor angular velocity is used, the badly damped poles, due to the flexible transmission do not hurt the stability. They are compensated by two complex zero’s (collocated control). As a result the adaptive gains vary during each transient but overall a good control performance is achieved. Figure 21 gives the adaptive control system and some detail of the model of the motor and load. Figure 22 gives some relevant simulation results.


30

Adaptation

Controller

Liapunov

Setpointgenerator Process

K i

p22

p21

Kd

Kp

(s+2z s+ )ωn2nω

2nω

V P

ME

omega_motor

phi_motor

FIGURE 21 Adaptive control system and a more detailed model of the process

Kp, Kd, Ki

0 5 10 15 20 25time {s}

Kd

Kp Ki

-1

0

1

2

3

4

model

0 5 10 15 20 25time {s}

phi_

mot

or {r

ad}

Ref

eren

ce M

odel

erro

r

-2

-1.5

-1

-0.5

0

0.5

1

-2

-1.5

-1

-0.5

0

0.5

1

-0.05

-0.025

0

0.025

0.05

0.075

0.1

FIGURE 22 Responses of the process, reference model and error and adjusted parameters

Non linearities Although the design methods which have been applied, such as the method of

Liapunov, are especially suited to non-linear systems, in the adaptive systems described it is essential that the non-linearities be restricted to the adaptation mechanism. Process and reference model must in principle be linear. Most non-linearities can be translated, however, into variations of the other parameters of the system. Such variations may be compensated by the adaptation mechanism. It is possible to give a proof for stable adaptive laws for non-linear systems as well, but those laws are in general more complex than the algorithms given

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before. It appears that these types of algorithms fail when there are only small structural differences between the process and the reference model.

out

in

FIGURE 23 Saturation

Saturation-type non-linearities have a more detrimental effect (figure 23). This type of non-linearity is often found in various types of actuators, such as hydraulic valves, which are completely open, electronic amplifiers etc. In this category a distinction can be made between non-linearities which limit the amplitude of the input signal of the process and those limiting the rate of change of this input signal. The first category can be dealt with simply. In principle there are two possibilities:

1. Switch off the adaptation mechanism as long as the process is in saturation

2. Modify the input signal of the process and the reference model such that the saturation is never reached.

This latter method is preferred, because in that case the adaptive system can remain unchanged. This approach is not only suited to ‘eliminating’ any saturation type of non-linearities from the process, but to dealing with saturations in the reference model as well. The proof of stability is not affected by this approach because the non-linearities are in principle removed from the control loop. The paper in Appendix A gives an example of this solution. 8 Noise

In practical applications of MRAS an important problem is the presence of noise on the states of the process. These difficulties arise due to the multiplication of the error signal:

e= +e e σ (96)

with

p p p= +x x σ (97)

This yields, for instance, for the components ie and ix :

, , , , , ,i i p i i p i i p i i e i e i pe x e x e xσ σ σ σ⋅ = + + + (98)

Assuming the mean value of ie and ix to be zero, integration of eq. (98), such as in the adaptive laws, yields in the presence of noise one non-zero term.

, ,0

t

i e i p dtσ σ∫ (99)

Because m p= −e x x and mx is noise-free, it follows that

e p= −σ σ (100)

and thus


32

2

, , ,i e i p i pσ σ σ⋅ = − (101)

Ka and Kb

0 20 40 60 80 100time {s}

Ka

Kb

-2

0

2

4

6

8

FIGURE 24 Parameter Ka drifts away because of noise on the signal xp

Equation (99) indicates that the adjusted parameters will drift away when the signals in the adaptive system are too small to compensate for this term with a non-zero mean. This is only a problem in the case of adaptation and not for identification. When MRAS is applied for identification instead of px , the noise-free signal xm is used. This implies that identification with MRAS yields unbiased estimates of the process parameters. However, in an adaptive system it is essential that measures be taken to prevent the parameters from drifting away. A choice can be made from one or more of the possibilities mentioned below: – the adaptation may be switched off when there are no set-point changes – the same can be realized more smoothly by multiplying the adaptive gains

with:

11 T+

(102)

where T denotes the time interval after the last set-point change. The principle of decreasing adaptive gains was already illustrated in figure 18.

- instead of using the signal xp, in the case of adaptation the signal xm may also be used. Although this is theoretically not correct, such a system may give good results in practice

- it is also possible to use the estimated states of the process, obtained by means of the earlier-mentioned adaptive state estimator

- it is possible to measure the terms σe and σp on-line. The product of both terms can be used to compensate for the drift. The usefulness of this method is determined by the accuracy with which both terms can be measured

- low-pass filters as well as dead band may be used in the adaptive loops. The use of filters is limited, because of the detrimental phase lag which they introduce. Application of a dead band is simple and effective. The only requirement for its effective application is that it must be possible to determine an upper bound of the signal which causes the drift.

Depending upon the type of application, a choice can be made from one or more of these measures. 9 Influence of discretization

In the foregoing process and reference model have been considered as continuous processes. Of course, the continuous algorithms may be used without any modification for those processes where the sampling interval can be chosen small enough to allow the discrete system to be described by the continuous equations. But for discrete systems discrete MRAS-based adaptation laws can be derived as well. With the hyperstability approach it is possible to derive discrete adaptive laws which have a shape similar to the continuous ones. The method of Liapunov may be applied when a small change is made in the structure of the adaptive system. We start with the following equations:

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33

( ) ( ) ( )1m m p mk k k+ = +x A x B u (103)

( ) ( ) ( )1p p p pk k k+ = +x A x B u (104)

The matrices Am, Ap, Bm and Bp are found from a transformation of the continuous-time equations. Their elements are functions of the sampling rate T. Note that the structure of the system has been changed in such a way that in eq. (103) xp is used instead of xm. This is illustrated in figure 25.

reference model

process e

x m

xpu

Bm

Am

Z-1

Bp

Ap

Z-1

FIGURE 25 Series-parallel structure for discrete MRAS The structure of figure 25 is referred to as ‘series-parallel’ MRAS. This is due to the fact that the reference model is not a ‘normal’ parallel reference model, but that it is placed partly in series (via Am) and partly parallel (via Bm) to the process. For this structure we can find adaptive laws as follows:

( ) ( ) ( )m pk k k= −e x x (105)

( ) ( ) ( ) ( ) ( )1 pk k k k k+ = +e A x B u (106)

with

( ) ( ) ( )m pk k k= −A A A (107)

( ) ( ) ( )m pk k k= −B B B (108)

At this stage a Liapunov function V(k) is selected and ∆V(k) is determined from:

( ) ( ) ( )1V k V k V k∆ = + − (109)

with

( ) ( ) ( ) ( ) ( ) ( ) ( )T T TV k k k k k k kα β= + +e Pe a a b b (110)

and a, b, α and β defined in conformity with eq. (33) this yields:


34

( ) ( ) ( )( ) ( ) ( ) ( ) ( ){ } ( ) ( ){ }

( ) ( ) ( ) ( ) ( ){ } ( ) ( ){ }1 1 1

1 1 1

T

Tp

TT

V k k k

k k k k k k k

k k k k k k k

α

β

∆ = − +

+ + + + + + −

+ + + + + + −

e Pe

e PA x a a a a

e PB u b b b b

(111)

The first term of eq. (111) is negative definite when P is a positive definite matrix. Let the last four terms of eq. (111) be equal to zero. This yields the adaptive laws:

( ) ( ) ( ),1

11 1α =

∆ + = +

∑

n

i n i pi

a k p e k x k (112)

( ) ( ) ( )1

11 1β =

∆ + = + ∑

n

i n ii

b k p e k u k (113)

where it has been assumed that:

( ) ( ) ( )1 2 1+ + ≈ +a k a k a k (114)

When the structure of figure 25 is applied for identification the advantage of unbiased parameter estimates disappears. By replacing xm by xp the term which causes drift in the parameters is introduced. This is a well known problem in system identification in general, where often such a series parallel structure is used. A complete adaptive system for the standard example used throughout these notes is given in figure 26. A continuous-time process is controlled by means of a discrete adaptive controller. Simulation results are given in figure 27.

xp(k)

xm(k)

e (k)

computer

process

}

reference modelH(z)

bp

Ka

AD

Kb

Pe

AD

AD

AD

ap

∫ ∫

FIGURE 26 Discrete adaptive controller for the standard process

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35

Parameters Ka and Kb

0 10 20 30 40 50 60 70 80 90 100time {s}

Kb

Ka

0

0.5

1

1.5

Prcoess, reference model and error

0 10 20 30 40 50 60 70 80 90 100time {s}

Y_p

roce

ssxm

[1]

erro

r

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

FIGURE 27 Results of discrete adaptive controller Because at each sample the reference-model states are made equal to the process states, the error between process and reference model remains small. When we zoom in, we clearly see the discrete nature of the sampled process state x1,p, the reference model state x1,m, and the error between these signals (figure 28).

Parameters Ka and Kb

0 5 10 15 20 25time {s}

Kb

Ka

0

0.5

1

1.5

Prcoess, reference model and error

0 5 10 15 20 25time {s}

Y_p

roce

ssxm

[1]

erro

r

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

FIGURE 28 First 25 seconds of the responses of figure 27


36

10 Conclusions

MRAS-based adaptation was a popular research topic in the period 1960-1990. It results in relatively simple adaptive laws that can be applied in practice. An example of such a practical application is described in Appendix A, where direct and indirect MRAS are used for direct and indirect parameters adaptation as well as state estimation. These algorithms form also the basis for the rudder roll stabilization system that is used for fast container ships and at the M-frigates of the Royal Netherlands Navy. In the book of Narendra and Annaswamy (1989) a series of practical applications has been described. In addition to the application on ship steering applications in process control (distillation columns), power systems, robot manipulators, blood pressure control and so on are treated. Besides from the practical applicability of these methods, the concepts that play a role in the design of MRAS, are similar to those used in system identification and, to a certain extend, learning control. 11 References

Classical book on model reference adaptive control

Landau, I.D., Adaptive Control – the model reference approach, Control and System theory series, 8, Marcel Dekker Inc., New York and Basel, 1979

Includes several examples of real applications

Narendra, K.S. and A.M. Annaswamy, Stable Adaptive Systems, Prentice Hall, Englewood Cliffs, New Jersey, 1989

Applications of MRAS to ship steering

Amerongen, J. van, Adaptive steering of ships: a model- reference approach to improved manoeuvring and economical course keeping, Ph.D thesis, Delft University of Technology, p. XII, 195, 1982 Amerongen, J. van, Adaptive steering of ships - a model reference approach, Automatica, vol. 20, no. 1, Pergamon Press, pp. 3-14, 1984 (available in Appendix A) Amerongen, J. van, P.G.M. van der Klugt and H.R. van Nauta Lemke, "Rudder roll stabilization for ships" Automatica, vol. 26 no. 4, pp. 679-690, 1990 Amerongen, J. van, Ships: Rudder Roll Stabilization, Systems and Control Encyclopedia, vol.1, Pergamon press, p 6, 1990

Paper on mode switching

Hilhorst R.A., J. van Amerongen, P. Löhnberg and H.J.A.F. Tulleken, Supervisory control of mode-switch processes, Automatica, no.30, pp 1319-1331, ISSN: 1319-1331, 1994

Overview of adaptive control systems, emphasis on identification -based systems

K. J. Astrom and B. Wittenmark, Adaptive control (2nd Ed.), Addison-Wesley, 1995

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