Reduced Variance Pole-Assignment Servo Self-Tuning

Copyright © IF AC Adaptive S)'stellls in Control and Signal Processing, Lund, Sweden, 1986

REDUCED VARIANCE POLE-ASSIGNMENT SERVO SELF-TUNING M. B. Zarrop* and A. Karafakioglu** *Control Systems CentTe, University of Manchester Institute of Science and Technology, p.a. Box 88, Manchester M60 IQD , UK **Etibank, Ankara, Turkey

Abstract, A standard explicit pole-assignment servo controller is modified to allow enhancement of its noise-rejecting properties. This is achieved by a suitable overparameterization of the controlle r polynomials allowing scope for optimization and generalises the previously reported results for the regulator case (Zarrop and Fischer, 1985). Within this framework, a self-tuning servo controller is developed consisting of two r ec ursiv e estimation procedures operating in parallel. The resulting algorithm can give significant reduction in variance for little extra computational e'ffort and without modifying significantly the closed-loop response to set-point changes . Keywords. 1.

Adaptive control; optimization; pole placement; stochastic control.

INTRODUCTION

(i i)

(ii i) The pole-assignment approach to self-tuning regulation and control for single· input/single-output (SISO) systems is well known (Wellstead, Prager and Zanker 1979; Wellstead and Zanker 1979; Sanoff and Wellstead 1982; Sanoff 1984) and is inspired by a classical rather than an optimization approach to controller synthesis. This approach, however, can lead to large output variances so that, in achieving the specified closed-loop characteristics, the noise-rejecting properties of the control system are impaired. It was shown by Zarrop and Fischer (1985) that controller overparameterization can be used to introduce extra degrees of freedom to reduce output variance while preserving the required closed-loop pole set in a self-tuning mode. In this paper, the method is applied to the combined tracking/regulation problem. In addition, variance reduction is achieved by on-line estimation of a set of auxiliary parameters operating in parallel with the explicit model estimation .

A, B coprime C(z) 0 ="> Iz l

1

(Throughout this paper, the degree of a pOlynomial X is denoted by nx and its coefficients by x O"" .•. ,x All estimated quantities are capped by - .) n x

If it is required that the system output tracks a reference signal {r(t)} then anyone of a number of linear controllers may be implemented (Sanoff 1984; Tuffs and Clarke 1985). Here one particular poleassignment controller involving integral action is chosen as a test bed for the optimization technique discussed below. It must be emphasized, however, that the technique can be used in conjunction with any other pole-assignment controller. Omitting polynomial arguments for brevity, the controller

l7(z -1)

1 -

(2.2a)

Hr(t)

I7Fu ( t) + Gy ( t) where

2.

>

z

-1

(2 . 2b)

A POLE-ASSIGNMENT SERVO CONTROLLER (2.2c)

A brief review of pole assignment for known systems is now given to fix the notation and formulation. Systems and controllers are represented in discrete time by polynomial models.

leads (on average) to zero steady-state tracking error for constant reference signals and the closedloop equation

A known linear SISO randomly disturbed system is represented by the difference equation y (t)

A(z

-1

r(t-k) +

I7F -r

e(t)

(2.3)

)y(t) provided that the polynomials F,G satisfy the polynomial identity

(2.1)

where u(t), y(t) (t = ... ,-2,-1 ,0, 1 ,2, ... ) are the input and output sequences respectively, {e( t)} is a zero-mean white noise source of variance 0 2 and d is a constan t offset. The time delay k is an integer number of sample intervals (k ~ 1) and z is the unit forward shift operator (zry(t)=y(t+r». The polynomials A,B,C satisfy the conditions (i)

T(l) B

BTTY T

I7AF + z

-k

BG

CT

(2.4)

Thus the closed-loop poles are determined by the zeros of the chosen polynomial T. Essentially the controller incorporates an integrator to eliminate the offset and to ensure zero steady-state tracking error and this appears in the polynomial identity

A(O) = 1 = C(O)

403

404

M. B. Zarrop and A. Karafakioglu

(2.4) as an imposed factor

~(z-I).

var(u)

A unique solution of (2 . 4) is guaranteed if the polynomial degrees satisfy the conditions n n n

f

nb + k - 1

(2.5a)

g

na

(2.5b)

<

t

n

together with condition (ii) above. In this case, (2.4) is usually solved via a matrix inversion to yield the vector (f 1"" ,f ,g, . .. ,g )T. If both nf 0 ng nf,n g exceed the values in (2.5), i.e. the polynomials F,G are ' overparameterized', then, in general, an infinite number of solutions is possible, a property that can be used for optimization purposes (Zarrop and Fischer, 1985) and will be further exploited below. 3.

REDUCED OUTPUT VARIANCE

For deterministic reference trajectories, (2.2) and (2 . 3) yield the following expressions for var(y) and var(u), the closed-loop output and input variances respectively, var(y)

E(~F e(t»2

var(u)

E(r e(t»

2

(3.2)

A simple example is sufficient to show that the output variance may be considerably higher than the minimum achievable. The latter is the minimum of E{y(t) - r(t)}2 and is obtained through a standard manipulation (Astrom, 1970).

(3 . 6)

p

J

n

var(y)n +ovar(u)n

(0)

p

p

(3 . 7)

0)

(0 >

p

is quadratic in the coefficients of P and has a unique minimum given by

o

i

(3.8)

0,1, . . . ,np

For zero 0, (3.8) leads to the smallest output variance for the given degree of overparametrization . In cases where this may lead to excessive controller action, non zero values of a can be chosen. The stationarity conditions (3.8) are a set of np+l linear equations for the optimal P coefficients and may be solved by a matrix inversion (3.9a) where (Po,Pl'''·,Pn)

(3.1)

T

G

Go -~AP 2 E(-T-- e(t»

where the suffix np is attached to discriminate between variances corresponding to different deg~s of overparametrization. Hence the cost function

(2.5c)

n a + nb + k - c

n

T

(3.9b)

p

M..

E{ z

1J

-i VB

-' VB

-r(t»(z J -r(t»+o(z

E{o(z

1

-i ~A

Go

-r(t»(~(t» ~F

Example 1

~A

-r(t»

(z-j ~(t»)

x

d.

-i

- (z

(3.9c)

-(k+i) VB -r(t»

o

(~(t»}

x

(3 . 9d)

(i, j

y(t) = 1.ly(t-l)+u(t- l)+e(t)+0 . 5e(t - l)+0 . 01 1 - O.Olz- 1 This yields F(z - l) = 1, G(z - l) = 2.59-1. 105z- 1 and an ou t put variance of 1.980 2 compared to the minimum-variance value of 0 2 . Ove r parameterizing F and freedom that can be used as in Zarrop and Fischer the standard solution of n

f

G gives extra degrees of for optimization purposes (1985). Let Fo,G denote (2 . 4) satisfying ?2.5a,b):

-

nb + k

1,

0

n

go

n

a

(3 . 3a)

then the general solution of (2.4) is given by + z

F

F

G

G -

0

0

-k

BP

Table 1 shows the effect of selecting P optimally for Example 1. Table 1 Example 1 (0

=

1)

0

0

n

2

-1

p

0

1

50 2

2

3

2 . 0 1.5 1.3 1.2 1.2

1.6 1.7 1.8 2 . 0

7 . 9 4 . 7 3.9 3.5 3 . 3

4.3 3.0 2 . 4 2.0

p(z-l) (3 . 4a)

p

(3.4b)

n + n+l gap

Comparing (3.4) with (3.3a), the standard case can be conside r ed as corresponding t o np = - 1. For np = 0 , there is a single degree of freedom and so on .

0

0

0

T

n : p

0.495

(3 . 10a) -1

F(z -1)

1 + 0 . 495z

G(z -1)

-1 -2 2.095-0.66z -0 . 545z

4.

(3 . 10b)

Using (3.5) and (3.9), the minimum output variance achievable for the case of zero 0 is var(y)_l - d M

e(t»2

(3 . 5)

(3 . 10c)

MAGNITUDE OF VARIANCE REDUCTION

T -1

~F +z - k VSp

E(

1

var(u)

In particular, for

Using (3.1) - (3 . 3) ,

0

var(y)

(3.3c)

~AP

+ n

3

(3.3b)

where P(z - 1) is an arbitrary polynomial and

n

The matrix M is a symmetric positive definite Toeplitz matrix with np+l distinct elements, each of which can be expressed as a contour integral.

d

(4.1)

Pole-Assignment Servo Self-Tuning

405

For general n p ' the ratio r(n p ) defined by

T -1

~ M ~/var(Y)_l

r(n ) p

(5.2b) (4 . 2)

lies between 0 and 1 and measures the possible variance reduction achievable for given np'

and forgetting factors can be incorporated if required. Example 2

(Near pole-zero cancellation)

y(t) = 0 . 085y(t-l)+u(t-l )-0.lu(t-2)+e(t) For np=O, the matrix inversion is avoided and +0.8e(t-I)+0.01 r(0)

1-0.91z- 1 ; 0 2 = 0.01 (4.3)

x

It appears difficult to analyse this expression in general but certain special cases are of interest. and

Assuming na = ne = n t = k includes Example I), then

nb = 0

l+t 1 2 (-2-)

r(0)

ON-LINE COST REDUCTION

2

•

For simplicity consider the case a

o and

write

VF +z -k VBP J

n

E(

(0)

p

0

T

0) Variances

o

-1

p

x

100

2

var(y)

30.0

1 .3

1 .0

1 .0

var(u)

20.6

1.0

0.9

0.9

For np = 0, the optimal Po has the value -5.1844. Figure 1 shows the output behaviour when the desired trajectory is a square wave switching between +1 and -1 every 500 steps . The single degree of overparametrisation is switched in at t = 1000 and Po is estimated recursively using (5.2) with o~ set to 0.25. Figure 2 shows the evolution of po(t). In line with Table 2, a substantial reduction in output variance is rapidly achieved . 6.

EXTENDED SELF-TUNING CONTROL

In the remaining sections, it is assumed that the system (2.1) is unknown and that the controller synthesis is carried out based on a process model, constructed from the input-output data generated by (2.1). A linear process model is assumed of the form

The solution (3.9) for the optimal polynomial P involves the inversion of a matrix whose elements can be expressed as contour integrals. This can be avoided by recursively minimising the cost function (3.7) while noting that (3.5) and (3.6) remain unchanged if {e(t)} is replaced by a known zero mean white noise sequence {w(t)}, variance-w

Example 2 (a n

(4.4)

For arbitrary A,C,k, but retaining the conditions nb = 0, n t = I, the fast sampling situation implies that T ~ V. The identity (2.4) then implies that G ~ G1V where Gl is of degree n a -l. Noting that nf has the value k-l in this case, (2.4) becomes equivalent to the standard polynomial partition for the minimum variance controller and VF/T ~ F. This rough argument indicates that the output variance will be close to its minimum value without further optimization .

0

p

Table 2

(which

independent of the system, where tl lies in the range (-1,0) to correspond to a first order response in continuous time. Hence 0 < r(O) < 0.25 so that no more than 25% reduction in output variance can be achieved with a single degree of overparametrization. Further, fast sampling (t 1 ~ -1) implies r(O) ~ 0 and therefore negligible scope for variance reduction.

5.

Table 2 shows the input and output variances for -1 < n < 2 and zero a .

w(t»2

(5. la)

E[y (t)-x (t)p)2 w -w -

A(z

-1

)y(t) (6.1)

where A(O) = 1 = C(O) and the (n +n +n +2) - vector A B C e of parameters is estimated using a suitable estimation algorithm (RELS or RML). The following self-tuning pole-assignment control algorithm is proposed based on the controller in Section 2 and the minimization of I (0) discussed n in Section 5. p

(5 .1b ) Self-tuning Algorithm:

where (5. le)

Set and

~w(t)

',:SS

the row p-vector whose ith component given by - VBw( t-k- i)

nA, nB' n C' n p ' T, 0w

At time t: Estimate

0, . .. ,np (5.1d)

(6.1)

By analogy with RLS estimation, a sequence of estimates {p(t)} of the optimal E is generated as follows: A

E(t-I)+Pw(t)~w(t)

T

A

[Yw(t)-~w(t)E.(t-I))

(5.2a) ASC-N

( ii)

Solve for

vAF

o

where

A,B,2,6 Fo ,6 0

+ z-l BG

0

using RELS on the model from the polynomial identity eT

(6.2a) (6.2b)

406

M. B. Zarrop and A. Karafakioglu

(iii) Estimate P using i(t) from (5.2) where B,Fo in (5.1c,d) are replaced by current estimates . (iv) Compute u(t) using the control law VFu(t) + Gy(t) = Hr(t)

(6.3a)

parameter case. Table 3

Example n

Sample variances II I -1

p

0

0

x

100

where

(v)

,

-1--

F

Fo+z

H

CT(1)/B(1)

,

PB, G

Go -VpA

(6.3c)

Input u(t) into the system (2.1) and generate y(t+1).

Remark Using (6.3b). the control law (6.3a) can be cast in the easily i mpl ementabl e form (6.4) where the residual sequence {£(t)} isobtained from the estimation procedure applied to (6.1). Control performance is not significantly affected by dropping 6 and replacing £(t) by the prediction error e(t) used in RELS: VF u(t)+G y(t) = Hr(t)+VPCe(t) o

(6.5)

0

Note that (6.5) is the standard controller (corresponding to np = - 1) augmented by an auxiliary input signal linked to the estimation procedure . The extra input is necessarily strongly correlated with the input-output data in order to have the desired effect of variance reduction. 7.

A SIMULATED EXAMPLE

The first order system of Example 2 (Section 5) exhibits a high output variance using the standard pole assignment controller (Table 2) and therefore is very suitable for illustrating graphically the operation of the self-tuning algorithm discussed in Section 6 . The model was assumed to have the correct structure and the estimated polynomials were set to unity initially with the initial para meter covariance matrix set to 10001. For the on-line estimation of the optimal P(z), the initial estimate was set to zero, Pw(O) set to 10001 and 0 2 to 0 . 25 . A variable forgetting factor A(t) was w used during the transient period as follows : A(O)

0

A(t)

0.985A(t- l) + 0 . 015

var(y s)

30.8

1.2

1.2

var(u ) s

25.0

1.0

1.4

(6.3b)

(t<450)

(7.1)

Run time

1.00

1. 35 1. 14

The columns I,ll correspond to the two methods of generating the optimal P(z) on - line . Method I corresponds to (3.9) and involves constructing and inverting a matrix at each sample time. Method 11 corresponds to the P estimation (5.2). The corresponding trajectories for Po are shown in Figs. 7 and 8 respectively. Predictably, Method 11 involves less computation time than I and generates a smoother trajectorx ~o~ ~o~t) ~ !n ~h~ appendix the final estimates A,B,C,6,F o ,G o ,po,F,G are listed and it is clear that the closeness of var(ys) to its true value has low sensitivity to the estimation error that is incurred . 8.

CONCLUSIONS

A standard explicit self-tuning pole-assignment algorithm for tracking/regulation has been extended to allow output and/or input variances to be reduced . This is achieved by a suitable overparameterization of the controller polynomials, allowing scope for optimization, a method introduced by Zarrop and Fischer (1985) for the regulator problem . Simulation results indicate that the predicted variance reduction is ach ieved for little extra computational effort. No rigorous analysis of the stability/convergence properties of this adaptive algorithm is available at this time . ACKNOWLEDGMENTS The authors would like to thank Dr . P.E.Wellstead and Mr . G. Wagner for their help during the period leading up to the preparation of this paper. The work was carried out as part of a research gr ant financed by the Science and Engineering Research Council and the second author was given financial support by Etibank during her year of postg r aduate study in the U.K .

(t>450) The algorithm was run for 10000 iterations and the sample variances calculated as follows: 1 10000 T(1) B 2 rr(t - k)] var(ys) = 9500 L [y(t)- B(j) t=501 (7.2) 10000 T( 1) A 2 1 var(u ) L [u(t) - B(1) r r(t)] 9500 s t =501 (7.3) The reference signal is a square wave switching between +1 and -1 every 500 steps . Figures 3- 6 show the input and output signals for the standard self - tuner (n p = -1) and for a single degree of overparametrerization (n p = 0) . The degree of variance reduction is clear. For zero np only the effect of on- line P estimation is shown because the graphs corresponding to implementing (3.9) on- line appear to be almost identical. Table 3 displays the sample variances (7.2) and (7.3) and should be compared to the numerical values in Table 2, corresponding to the known

REFERENCES Astrom, K. J. (1970). Introduction to Stochastic Control. Academic Press, New York. Chapter 6. Sanof~ (1984). Incremental pole- assignment self - tuners. Control Systems Centre Report 619, UMIST . Sanoff , S. P. and P . E . Wellstead (1982). Extended self - tuning-practical aspects. Proc. 6th IFAC Symposium on Identification & System Parameter Estima t ion, Pergamon, Oxford. Tuffs , P.S. and D. W. Clarke (1985) . Self - tuning control of offset : a unified approach. lEE Proceedings, ~, Pt. D, 3, 100- 110. Wellstead, P.E. and P . M. Zanker (1979). Servo self- tuners. Int. J . Control, 30, 1, 27- 36. Wellstead, P . E . , D. Prager and P.M. Zanker (1979) . Self-tuning pole/zero assignment regulators. Int. J. Control, 30, 1, 1-26 . Zarrop , M.B . and M. Fischer (1985) . Reduced variance pole- assignment self - tuning regulation . Int. J. Control, 42, 5, 1013- 1033 .

407

Pole-Assignment Ser\'o Self-Tuning APPENDIX

Se 1£ -Tun ing

(t

10000)

-

In thi s appendix, further details are given for the simulated exampl e discussed in Section 7; see also Tables 2 and 3.

A(z)

1-0 .0448z,

B(z)

C(z)

1+0 .8177z,

d = 0.01 1

(a)

Po = - 2 . 2283

np = -1

(standa rd algorithm)

Known parameters: G (z) = -4.325+4.505z o

Self-Tuning A (z)

1-0.0536z,

C(z)

1+0.8225z,

Fo (z) (b)

(t

n

p

=

=

1+4 .5343z,

=

F(z)

1+ 0591z + 0 .1 510/

G( z)

0.8932 - 0 . 8175z +0 . 0997z

Self -Tuning

B(z) = 0 . 9981 - 0.0648 z

A(z)

1-0.0758z,

B(z)

d = 0 . 008

C(z)

1+0.8076z,

d = 0 .010

G0 (z)

Po = -4.5712

=

p

o

=

- 3.5749 +3.7507z

-5.1844

F(z) = 1+0. 1156z + 0.5184z G(z)

1.0002-0.0678z

10000) :

0

Known parameters:

=

Method I:

(t = 10000)

2 Method II:

=

0 . 9986 -0 .0922z

2

F(z)

1-0.1386z+0.4215z

G(z)

1.1135-1.2804z+0.3464z

2

2 FIGURES

0.8594 - 1. 120 1z + 0.4407z

(Example 2)

2

L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _~

Fig.l

Output.

On-line reduction (np=O) from step 500

408

M. B. Zarrop and A. Karafakioglu

-18

-g -8

-7 -6 -:5

-4

-3 -2

-1 8 +1

".,11,,1'

2

"4" '" "'6''''' "9. . . . . . ..-.+...0 ..........12". ·"'.. '14" 00' 'I'S

T8~8

~\182 steps

+2 +3

-14

Fig.2

Po estimate

(ef Fig.l)

r - - - - - - - - -- - - - - - - -- - - - - - - - - - - - - . - - - - - ,

4

Fig.3

Output.

n =-1 P

Po le· Assignm e nt Servo Self·Tu nin g

Fig.4

,---- _._ -_..

__

Output.

409

n =0 p

_ _ .....__ ...._. __.__._------ ---- ---- ---- ,

..... .. .

-8

. ... _ _ _ ......... _

Fig.5

.. _

.. _ .. _ _ _ _ _ _ _ _ _

Input.

n =-1 p

_ _ _ _ _- . . . L

M. B. Zarrop a nd A. Karafa kiog-Iu

410 ----1 - -_

...-- -------. -

--- - - - - - - - - - - - - - - - - - - - - - - - - - - ,

2

8

Fig.6

Input.

n =0 p

~1-----------------­

,-,

.j .

.::... ..

(I

-(

·' -r ~""'i"il"

_~

i3

lfl

"...'\"(....~'-

-1 - )

- ::: -1 _ e:-

'w '

-E; _

-7.

-8.

'---------------- .- -----------------------------_...

Fig.7

Po estimate

(Method I)

Pole-Assignment Servo Self-Tuning A

'"

5

4 1I

1

0 2

4

""{"'" "8""""'16 "'1'2"'" 14"""'1'6

-5 -1\3 -15

- 20 -25

-313 -35

-40 -15

-50

Fig .S

Po estimate (Method II)

"'18"""'20

x5,;

Shps