1

SYSTEM IDENTIFICATIONThe System Identification Problem is to estimate a model of a system based on input-output data.Basic Configuration

continuous

System

disturbance (not observed)v(t)

y(t)u(t)output (observed)input (observed)

discrete

System

{v(k)}

{y(k)}{u(k)}

2

We observe an input number sequence (a sampled signal)

{u(k)} = {u(0), u(1), ..., u(k), ..., u(N)}and an output sequence{y(k)} = {y(0), y(1), ..., y(k), ..., y(N)}

using standard z-transform notation

If we assume the system is linear we can write:-

Y G U V( ) ( ) ( ) ( )z z z z

3

G(z)U(z) Y(z)

V(z)

+

The disturbance v(k) is often considered as generated by filtered white noise :-

G(z)U(z) Y(z)

V(z)

+

H(z)(z)white noise filter disturbance

outputinput process

giving the description:Y G U H( ) ( ) ( ) ( ) ( )z z z z z

4

Parametric ModelsARX model (autoregressive with exogenous variables)

whereA

B

( )

( )

z a z a z

z b z b z b zn

n

nn

a

a

b

b

11

1

11

12

2

1

G(z)

U(z) Y(z)

V(z)

+

H(z)

(z)

z zz

n

BA

( )( )

1

1

11A( )z

5

giving the difference equation:y k a y k a y k n

b u k n b u k n b u k n n kn a

n b

a

b

( ) ( ) ( )

( ) ( ) ( ) ( )

1

1 2

1

1 2

and represents an extra delay of n sampling instants.

z n

identification problemdetermine n, na, nb (structure)

estimate

a a a

b b bn

n

a

b

1 2

1 2

, , ,

, , ,

(parameters)

6

ARMAX model (autoregressive moving average with exogenous variables)

where A

B

C

( )

( )

( )

z a z a z

z b z b z b z

z c z c z

nn

nn

nn

a

a

b

b

c

c

11

1

11

12

2

11

1

1

1

G(z)

U(z) Y(z)

V(z)

+

H(z)

(z)

z zz

n

BA

( )( )

1

1

C zz

( )( )

1

1A

7

giving the difference equation:y k a y k a y k n

b u k n b u k n b u k n n

k c k c k n

n a

n b

n c

a

b

c

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1

1 2

1

1

1 2

1

identification problemdetermine n, na, nb, nc (structure)

estimate

a a a

b b b

c c c

n

n

n

a

b

c

1 2

1 2

1 2

, , ,

, , ,

, , ,

(parameters)

8

General Prediction Error ApproachProcess

Predictor withadjustable

parameters

+

-

Algorithm forminimising

some function of e(t,)

u(t) y(t)

e(t,)

Predictor based on a parametric modelAlgorithm often based on a least squares method. 2

0

min ( )N

k

e k

9

ConsistencyA desirable property of an estimate is that it converges to the true parameter value as the number of observations N increases towards infinity.This property is called consistencyConsistency is exhibited by ARMAX model identification methods but not by ARX approaches (the parameter values exhibit bias).

10

Example of MATLAB Identification Toolbox Session

Input and Output Data of Dryer Model

0 5 10 15 20 25-2

-1

0

1

2OUTPUT #1

0 5 10 15 20 25

-1

0

1

INPUT #1

input and output data

11

th = arx(z2,[2 2 3]); % z2 contains datath = sett(th,0.08); % Set the correct sampling interval.present(th)

Results:

Loss fcn: 0.001685Akaike s FPE: 0.001731 Sampling interval 0.08The polynomial coefficients areB = 0 0 0 0.0666 0.0445A = 1.0000 -1.2737 0.3935

MATLAB statements and results:(ARX n, na = 2, nb = 2)

12

64 65 66 67 68 69 70 71 72-1.5

-1

-0.5

0

0.5

1

1.5ARX Simulated (solid) and measured (dashed) outputs - error = 6.56

Time

ARX model:

G z z zz z

( ) . .. .

3

1

1 2

0 0666 0 04451 12737 0 3935

13

MATLAB Demo

14

ADAPTIVE CONTROL

PERFORMANCEASSESSMENT &

UPDATINGMECHANISM

REGULATOR PROCESS

parameters slowlyvarying

ref + _

outputs (fast varying)

disturbancesfastvarying

regulatorparameters

K J Astrom

15

Adaptive control is a special type of nonlinear control in which the states of the process can be separated into two categories:-

(i) slowly varying states (viewed as parameters

(ii) fast varying states (compensated by standard feedback)In adaptive control it is assumed that there is feedback from the system performance which adjusts the regulator parameters to compensate for the slowly varying process parameters.

16

Adaptive Control ProblemAn adaptive controller will contain :-

•characterization of desired closed-loop performance (reference model or design specifications)

•control law with adjustable parameters

•design procedure

•parameters updating based on measurements

•implementation of the control law (discrete or continuous)

17

Overview of Some Adaptive Control Schemes

Gain Scheduling

regulator processoutput

ycontrolsignal

ucommandsignal

gainschedule

regulatorparameters

operatingconditions

The regulator parameters are adjusted to suit different operating conditions. Gain scheduling is an open-loop compensation.

18

Auto-tuning

PIDcontroller Process

KT s

T si

d1 1

FHG IKJ+

_

parameters K, Ti, Td

PID controllers are traditionally tuned using simple experiments and empirical rules. Automatic methods can be applied to tune these controllers.

(i) experimental phase using test signals; then:-

(ii) use of standard rules to compute PID parameters.

19

Model Reference Adaptive Systems MRAS

regulator process actualoutput

yuuc

modelidealoutputym

adjustmentmechanism

regulatorparameters

20

The parameters of the regulator are adjusted such that the error e = y - ym becomes small. The key problem is to determine an appropriate adjustment mechanism and a suitable control law.

d eedt

where determines the adaptation rate. This rule changes the parameters in the direction of the negative gradient of e2

MIT rule adjustment mechanism

21

Combining the MIT rule with the control law:

u u yc ( )and computing the sensitivity derivatives

eproduces the scheme:

process umultiplieruc y_+

e

filter

s

integrator

e

model

+

_

ym

multiplier

Note: steady-state will be achieved when the input to the integrator becomes zero. That is when y = ym

22

Self Tuning Regulators STR

regulator process actualoutput

yuuc

regulatorparameters

design

process parameters

estimation

23

The process parameters are updated and the regulator parameters are obtained from the solution of a design problem. The adaptive regulator consists of two loops:-

(i) inner loop consisting of the process and a linear feedback regulator(ii) outer loop composed of a parameter estimator (recursive) and a design calculation. (To obtain good estimates it is usually necessary to introduce perturbation signals)

Two problems:-(i) underlying design problem(ii) real time parameter estimation problem

24

MODEL REFERENCE ADAPTIVECONTROL

parameters

Mux

Mux1

referenceerror

reference,output,

command

Mux

Mux

1/sIntegrator1

1/sIntegrator

1s+2

filter_

0.5s+1

process

*

mult_

-K-

g2

-K- filter

-+e

*

mult

-K-

g1

2s+2

referencemodel

*

so

*

to

+-

feedbackerrorInput

Example - SIMULINK Simulation of MRAS

25

0 50 100 150-0.2

0

0.2

0.4

0.6

0.8

1

Time (second)

Input, Reference and Actual Outputs

26

MATLAB Demo

27

INTRODUCTION TO THE KALMAN FILTERState Estimation Problem

Vectors w(t) and v(t) are noise terms, representing unmeasured system disturbances and measurement errors respectively. They are assumed to be independent, white, Gaussian, and to have zero mean. In mathematical terms:-

x Ax Bu Gw y Cx Du v

w(t) v(t)

u(t) x(t) y(t)

SYSTEM

28

E t t

E t

E t

T

T

T

v w 0

w w Q

v v R

( ) ( )

( ) ( )

( ) ( )

FHG IKJFHG IKJFHG IKJ

for all and

(assumed constant)

(assumed constant)

where Q and R are symmetric and non negative definite covariance matrices. (E is the expectation operator)

Only u(t) and y(t) are assessable.

The state estimation problem is to estimate the states x(t) from a knowledge of u(t) and y(t). (and assuming we know A, B, G, C, D, Q, and R).

29

Construction of the Kalman-Bucy FilterSYSTEMu(t) y(t)

x(t)

( )( )x Ax Bu L y Cx Du tFilter equation :-

z CB

A

D

L(t)

u(t) y(t)

( )x t

xx ( )y t+ _ +

FILTER

+

30

The estimation problem is now to find L(t) such that the error between the real states x(t) and the estimated states is minimized. This can be formulated as:

( )x t

min [ ( ) ( )] [ ( ) ( )]( )L

x x x xt

E t t t tT FHG IKJ

( )( )x Ax Bu L y Cx Du tFilter equation :-

L(t) is a time dependent matrix gain.

R E Kalman

31

Duality Between the Optimum State Estimation Problem and the Optimum Regulator Problem

It can be shown that the optimum state estimation problem:min [ ( ) ( )] [ ( ) ( )]

( )Lx x x x

tE t t t tT FHG IKJ

subject to:

( )( )ˆ ˆ ˆ( )= , ( )=T T

tE E

x Ax Bu Gwy Cx Du vx Ax Bu L y Cx Du

ww Q vv R

is the dual of the optimum regulator problem:min

( )Lx GQG x u Ru

tT T TT

dt 12 0

FHG IKJzsubject to:

( )x A x C uu L x

T T

t

32

Thus L(t) can be obtained by solving the matrix Ricatti equation:

S AS SA SC R CS GQGT T T1

L R CS( ) ( )t t 1

Furthermore for large measurement times L(t) converges to:

Lim t LT

L( )

a constant matrix gain.

33

Linear Quadratic Estimator Design Using MATLAB

LQE Linear quadratic estimator design. For the continuous-time system:.

x = Ax + Bu + Gw {State equation} z = Cx + Du + v {Measurements} with process noise and measurement noise covariances: E{w} = E{v} = 0, E{ww'} = Q, E{vv'} = R, E{wv'} = 0 L = LQE(A,G,C,Q,R) returns the gain matrix L such that the stationary Kalman filter: . x = Ax + Bu + L(z - Cx - Du) produces an LQG optimal estimate of x.

34

Example:

produces:

A=[0 1;-1 0];G=[0;1];C=[1 0];Q=1;R=3;L=lqe(A,G,C,Q,R)

L =

0.5562 0.1547

( ) ( ( ))

( ) ( ( ))

x x

x x w t E w t

y x v t E v t

1 2

2 12

12

1

3

,

,

35

giving the filter equations:

( )

( )

x x l y y

x x l y yy x

1 2 1

2 1 2

1

where l1 = 0.5562, l2 = 0.1547

36

1s

1s

-1

+u(t) = 0

w(t) v(t)

+y(t)x1x2

SYSTEM

( )x t2 ( )x t1

1s

1s+ +

-1

_ +

l1

l2FILTER

x2 x1

y y

37

SIMULINK SIMULATION

1/sx2

+--w(t)

1/sx1

++y

v(t)

1.7

sqrt(R)

+-

y-Cx

++__

1/sx2hat

0.556

l1

-+_

0.155

l2

x1/x1hat

x2/x2hat

Mux

Mux1

Mux

Mux

PLANT

KALMAN FILTER

meas(y)

vtWS1

wtWS2

+-e1

+-e2

e1tWS3

e2tWS4

1

sqrt(Q)

1/sx1hat

38265 270 275 280

-6

-4

-2

0

2

4

6

Time (second)

Comparison of actual (solid) and measured (dash) states

x1

39

Comparison of actual (solid) and measured (dash) states

265 270 275 280

-6

-4

-2

0

2

4

6

Time (second)

x2

40

265 270 275 280-10

-5

0

5

Measurement signal y(t)

41

MATLAB Demo