A design of a strongly stable self-tuning controller using coprime factorization approach

A DESIGN OF A STRONGLY STABLE SELF-TUNING CONTROLL...

14th World Congress ofIFAC

C-2a-06-6

Copyright © 1999 IF AC 14th Triennial World Comtress. Beiiing. P.R. China

A DESIGN OF A STRONGLY STABLE

SELF-TUNING CONTROLLER USING COPRlME FACTORIZATION APPROACH

Akira Inoue ~, Akira Yanou and Yoichi Hirashima

* Department of Systems Engineering, Faculty of Engineering, Okayama University, Okayama, 700-8530, Japan

Phone +81-86-251-8233, Fax +81-86-251-8233 E-mail [email protected]

Abstract: This paper proposes. a new self-tuning controller having new design parameters. In selecting the design parameters, the controller gives a. strongly stable self-tuning controller, that is, the closed-loop system is not only stable, but also the controller itself is stable. The controller consists of an extended minimum variance controller and a parameter identification law. The extended minimum variance controller is extended by including newly introduced design parameters into the generalized minimum variance controller given by Clarke and others. The parameters are introduced by applying the coprime factorization approach and comparing Youla parametrization of stabiliZing compensators with the minimum variance controller. Copyright © 1999 IFAC Keywords: minimum variance control, self-tuning controller, strongly stable, coprime factorization, adaptive control

1. INTRODUCTION

To control plants with uncertainty, self~tuning controllers are widely applied inindustry(Astrom and Wittenmark, 1995). Self-tuning controller is proposed by Astrom and Wittenmark(Astrom and Wittenmark, 1973) first, which is designed to minimize the variance of the plant output. Clarke and others(Clarke.D. W. et al., 1979) generalized the self-tuning controller to minimize the variance of a. generalized output instead of minimizing the variance of the plant output and the generalized self-tuning controller can be applied to a wider class of plants such as, unstable plants or nonminimum phase plants. After them, many design schemes are proposed (Astrom and Wittenmark, 1995). But they all based on the concept of Clarke and others' method. Still, there remain two problems on the self-tuuing controllers: First one is that the polynomials defining the generalized output is che-

sen to place the poles of the closed-loop system and once these polynomials are determined then the poles of the compensator are determined and could not be assigned independently to the closedloop poles. So, it is impossible to design a stable compensator in addition to the c1osed.;.loop stability, that is, to design a strongly stable controller. The second is that the transfer function from noise to output is also fixed by the polynomials defining the generalized output and the poles of the transfer function could not be designed. This paper proposes a method to solve these problems by using coprime factorization approach. Using coprimely factorized rational functions to express a controlled system, the most general form (Youla parametrization) for stabilizing compensator is obtained(Vidyasagar, 1985). Using the general form of compensators, the authors have proposed a unified design method of Model Reference Adaptive Control (MRAC)(Inoue and Masuda, 1993). In this paper, the same coprime

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A DESIGN OF A STRONGLY STABLE SELF-TUNING CONTROLL...

14th World Congress ofTFAC

{I)(t + km.}

factorization approach is applied to the design of self-tuning controller. In comparing the selftuning controller to Youla parametrization of stabilizing compensators, we introduce new design parameters in self-tuning controller. By choosing newly introduced parameters, we can design the poles of the compensator and the transfer function from noise to output and we can design a strongly stable controller.

(4)

_R[Z-l]W(t)

and p[z-l], Q[z- 11 and R[z-ll are polynomials given by a controller designer with degrees of n p , n q , n,... Usually these polynomials are selected to obtain stable desirable closed-loop poles. In this section and the next section, the coefficients of A[Z-l], B[Z-I] and C[Z-l] are assumed to be known, and the generalized. minimum variance control law {Clarke. D. W. et al., 1979) and an extended minimum variance control law are given.

Notations: Z-l denotes backward shift operator: z-ly(t) = y(t - 1). A polynomial function and a rational function are distinguished byA[z-l] and A(Z-l ).

In the generalized minimum variance control law, first, the following three Diophantine equations are solved;

2. PROBLEM STATEMENT AND GENERALIZED MINIMUM VARIANCE CONTROL LAW

= A[Z-l]E[z-l] + z-k", F[Z-l] (5) G[z-l] = E[Z-I]B[z-l] + C[z-l]Q[Z-I] (6) T[Z-l] = P[z-I]B[z-l] + Q[z-l]A[z-l] (7)

P[Z-l]C[Z-l]

Consider a single-input single-output system given. by the difference equation

where E[Z-l] is a monic polynomial with degree km -1 and F[z-l] is a polynomial with degree nl;

t= 0, 1,2,···

where u{t) is the input, y(t) is the output, k-m is the time delay, t{t) is a white Gaussian noise with zero mean. A[Z-l], B[Z-l] and C[Z-I] are polynomials;

+ al z-l + ... anz- n B[Z-l] = bo + b1z-l + ... b-rnz-m C[Z-l] = 1 + CIZ - I + ... qz-i

= P[z-lly(t + km) + Q[z-l]U(t)

nl

= max{n -

1, np+ 1- km}

(8)

Then, the !?eneralized output (I)(t + k tn ) and its prediction (I)(t + kmlt) are given(Clarke. D. W. et al., 1979) by

A[Z-l] = 1

(2)

iJ?(t + km) = (t + kmlt) + kmlt) ~ (F[Z-l]y(t)

~(t

+ E[z-11~(t + k + G[Z-l]U(t)

tn )

(9)

On the system (I), we put the following assumptions

-C[z-l]R[z-l]w(t»)fC[z-l] (10)

Assumption 1. The degrees n, m, I of A[z-l], B[z-l], C[Z-l] and the time delay km are known.

Since the estimate ci>(t + km.lt) and the noise term + kmJ have no correlation with each other, the control u(t) to minimize the variance J is obtained by choosing u(t) to make E[z-ll~(t

Assumption 2. The coefficients aI, "', an, bo , ... , b"'t) Cl, . . . , ct of A[z-l], B[Z-l], C[Z-l] are unknown.

4>(t

Assumption 3. The polynomials A[z-l] and B[z-l], A[Z-l] and C[Z-l] are coprime.

(3)

where iJ?(t + km) is the generalized output;

(11)

Making the numerator of eq (10) equal to 0, the control law is u(t) =

Assumption 4. The polynomial C[z-l] is stable

The control objective is that the output yet) has a desirable response to the reference input w(t). To accomplish this objective, the generalized minimum variance control given by Clarke and others(Clarke. D. W. et al., 1979) minimizes the following variance of a generalized output

+ kmlt) = 0

C[z-I]R[z-l] F[z-l] G[r l ] wet) - G[rl]y(t) (12)

Substituting the control (12) into the system (1), we get the closed-loop system

The control law (12) which is derived by Clarke and others(Clarke. D. W. et al., 1979) as the generalized minimum variance control has two problems:

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ISBN: 008 0432484

A DESIGN OF A STRONGLY STABLE SELF-TUNING CONTROLL...

14th World Congress ofTFAC

Problem 1. The polynomials P[Z-lJ, Q[z-l] and R[Z-l] defining the generalized output .p(t) are determined by controller designers to get the stable closed-loop characteristics, T[Z-l]. Then, the characteristic polynomial G[z-l] ofthe control law (12) is uniquely fixed and can not be designed independently to T[Z-l].

If the polynomials p[z-l], Q[z-l] and R[z-l] are chosen for T[z-l] to be stable, comparing transfer function (1) to (15), we can choose N(z-l) and D(Z-l) as

Problem 2. The effect of the noise on the output is given by the transfer function G[z-l]/T[z-l] and is also determined by the poLynomials P[z-l], Q[Z-l] and R[Z-l].

And N(Z-l), D(Z-l) E RHoo. Substituting (20) and (21) into Bezout equation (19) and comparing the equation to Diophantine equation (5), we get the solution of Bezout equation X(z-l), y(z-l) as

N(Z-l)

= z-k~ B[z-llfT[z-l]

D( Z-l) = A[Z-l]fT[z-l]

In order to solve these problems, we will introduce new design polynomials in the generalized minimum variance controller by using the coprime factorization approach and comparing the most general stabilizing compensator, Youla parametrization form, to the control law (12).

X(Z-l)

(21)

= F[z-l)fC[z-l]

y(z-l) = G[Z-l]fC[z-l]

(22)

Then the control law (12) is obtained from the Youla parametrization (16) by selecting the design parameters as

3. EXTENDED MINIMUM VARIANCE CONTROL LAW USING COPRIME

(23) In order to extend the control law (12), instead of choosing U(Z-l) as in (23), we keep U(Z-l) as a newly introduced design parameter for the control law (12). To simplify the description of the control law, using new two design polynomials Un[z-l] and stable U.l! Z-l], we express the rational function U(z-l) in the form of

FACTORIZATION APPROACH This section extends the generalized minimum variance control law in order to include design polynomials under the assumption that the coefficients of A[Z-l], B[z-l] and C[Z-l] are known. For the cop rime factorization approach, consider the family of stable rational functions:

RH

=

U(Z-l) = g:f;=:?T[z-l]

= {G( -1) = G,,[z-l] z Gd[z-l] ,

(24)

Then substituting (20)-(24) into (17) and (18), we get

Gd[z-l] : stable polynomial} (14) The transfer function is expressed by a ratio of rational function in RHoo ,

Cl(Z-l)

= (Ud[Z-l]G[Z-l]

- Un[z-l]z-kmC[z-l]

.B[z-l] )-1 U d [z-l]C[z-1 JR[z-l] C 2 (z-1)

u(t) = Cl (z-l )w(t) - C 2 (Z-1 )y(t)

= (y(z-l)

2

x (X (Z-l)

»-1

+ U(Z - l )D( Z-I»

- Un [z-l]z- km C[z-l]

+Un [z-l]C[z-l]A[z-lJ)

(26)

Then using C1(z-1) and C 2 (z-1) of (25) and (26) in (16), a newly extended minimum variance control law is

(16)

(Ud [Z-l]G[Z-l] - Un [Z-l]z-k m C[z-l]B[z-l DuCt)

_ U(z-l )N(z-l »-1 K(z-l)

C (z-1) = (y(z-l) _ U(Z-l )N(Z-l

= (Ud[z-l]G[z-l]

(25)

.B[Z-l ])-1 (Ud[z-l]F[z-l]

where N(z-l ),D(z-l) are rational functions in RH00 and are coprime in each other. Then, all the stabilizing two-degree-of-freedom compensator is given in Youla parametrization form (Vidyasagar , 1985):

Cl (z-l)

(20)

= -(Ud[Z-l]F[z-l] + Un [Z-l]C[Z-l]A[z-l])y(t)

(17)

+Ud [Z-l]C[Z-l]R[z-l]w(t)

(27)

To calculate this control, we express the polynomial operating on u(t) in the left-hand side of (27) by the leading term Udo90 and the rest of the polynomial;

(IB)

where U(z-l) and K(z-l) are rational functions in RHoo and are design parameters. X(Z-l) and y(z - l) are also in RHoo and the solutions of the following Bezout equations;

U,J[Z-l]G[Z-l] - Un [ z-l]z-k", C[z-l ]B[z-l]

g, Udogo + z-lG'[z-l]

(28)

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A DESIGN OF A STRONGLY STABLE SELF-TUNING CONTROLL...

Using this expression, the control law (27) calculated by

u{t) = __ 1_(Ud[Z-1]F[Z-lj

14th World Congress ofTFAC

be stable, second Un[Z-lj and Ud[Z-lj for the controller poles (the roots of (33)) to be stable.

IS

+ Un[z-llC[z-l]

Proof. Equations (30) and (31) are straightforwardly derived from (5)-(7) and (15)~(29). 0

udogo

1

.Alz-1])y(t) - --G1[Z-lJU(t - 1) udogo

+ _1_ Ud [Z-l }C[z-l ]R[z-l ]w(t)

4. SELF-TUNING CONTROLLER BY USING THE EXTENDED MINIMUM VARIANCE CONTROL

(29)

udogo

Theorem 1. Using the control law (27) or (29), the followings hold;

In this section, we assume that only the nominal values of the coefficients of the system (1) are known and the true or varying values are unknown, a.nd propose a self-tuning controller by adding a parameter identification law to the ex- . tended minimum variance controller (29). First choose the design parameter Ud[Z-l] and Un[z-l] so as that the poles of controller, that is, the roots of equation (33) are stable using the nominal values of A[z-lj, B[z- l l and C[z-I]. Then the self-tuning controller consists of the following 3 steps calcuLated in each sampling period.

(i) The closed-loop system is given by

y(t) =

z-~~:~:]-lJ R[z-l]W(t) + (Y(z-l) -U(z-l)N(z-l)) C[Z-l] ~(t) T[z-l]

(30)

(H) The generalized output (4) is U [

.p(t+km)=(zk"'E[z-l]-

-1]

U:[;_I]C[z-l])~(t)

Step 1. To identify the coefficients, the next recursive least square method is used;

(31) Since the noise ~(t) is white, from (31), the optimal prediction of
ii(t) = jj(t - 1) r(t - 1)"p(t - 1) 1)r(t - 1)"l/I(t _ 1) e(t) (34)

+ 1 + "(VCt -

(32)

r(t) that is, the control objective (11) holds,

_ oXr(t - l)"lji(t - l)1f1T(t - 1)r(t - 1) (35) 1 + )..1jJT(t - l)r(t - 1)'1>(t - 1) r(O) = aI, 0 < a < 00

(ill) From (27), the poles of the compensator are the roots of equation, Ud[Z-I]G[Z-I] - U~[z-l]z-k"'C(z-l]B[z-lJ

= r(t -1)

E(t) = yet) - (jT(t - 1)"l/I(t - 1) ry(t) = yet) - OT(t)"l/I(t - 1) eet) = [0.1 (t), ... , an(t), bo(t}, ... , bm(t),

=0 (33)

(36)

(37)

Cl Ct),"', Cj(t)]T (38) 'IjJ(t) = [-yet - 1), ... , -yet - n), u(t - km), .. "

Remark 1. (i) From (30), it is clear. that the tra.nsfer function from the reference input w(t) to the output is equal to the one by the control law of Clarke and others(Clarke. D. W. et al., 1979), and is independent of the choice of the design parameter Ud[Z-lj and Un[Z-lJ.

u(t - km. - m),1J(t -1)",' ,1J(t - 1)]T(39)

where Ul(t),.",CI(t) are the identified value, oX (0 < A S 1) is a forgetting factor, e(t) and ry(t) are a priori and a posteriori estimation error.

(H) The effect of the noise ~(t) on the output yet) may be reduced by choosing the parameter U(z-l) in (30), whereas the effect of noise by the control law (12) could not be changed in (13).

A[t : z-l], B[t: Z-l] and C[t : Z-I] by substituting the identified coefficients 0,1 (t), ... , CI(t) into polynoStep 2. Obtain the estimated polynomials

mials (2). Then calculate the solutions Elt: Z-lJ, p[t: z-lj, C[t: Z-l] and t[t: Z-l] ofDiophantine equations (5)~(7) using the estimates Alt : z-lJ, B[t: z-l] and O[t: z-l] instead of Aiz-1J, B[z-l] and Clz- 1 ].

(iii) The poles of the compensator can be designed by selecting polynomials Ud[Z-l] and Un[z-lj in (33), not changing the poles of the dosed-loop system from the roots of T[z-l] = O. We may design a strongly stable controller, that is, a controller by which both of the closed-loop system and the controller itself are stable, by sequentially designing first P[Z-l J, Q[z-l] and R[Z-l] for the closed-loop poles (the roots of T[Z-lj = 0) to

Step 3. Calculate the control input u(t) from (27) using Alt : z-l], B[t : z-l], C[t : z-l], F[t : z-l\ and G[t : Z-l].

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A DESIGN OF A STRONGLY STABLE SELF-TUNING CONTROLL...

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5. EXAMPLE

As for the convergence of the generalized output cll(t), the following theorem holds.

Consider a system and a generalized output

Theorem 2. If the assumptions [A.I] ~ [A.4] and the following assumptions hold, 1

(1 + O.6z- 1

+ 0.7z- 2 )y(t) = Z-l (0.5 - 1.5z- 1 )u( t)

,\

+ ~(t)

(46)

Assumption 5. Transfer function C[z- ] - - is 2 strongly positive reaL

CP{t

Assumption 6. The signal1jJ(t) used in parameter identification satisfies PE(Persistently Spanning) condition; that is, the symmetric matrix

The reference input wet) is a rectangular wave with amplitude 1.0 and period of 100 steps.

+ 1) =

yet

+ 1) + 1.5u(t) -

z-2. (-O.15)w{t)

(47)

The control law (12) by Clarke and others{Clarke. D. W. et al., 1979) is 2 1 ( ) = -O.15z- wet) _ -0.6 - O.7z- (t) (48) 1 _ 1.5z- 1 y

u t I _ 1.5z-1

is positive definite,

and the dosed-loop system is then the error of the generalized output;

u [

3 4 ( ) = -O.075z- + O.225z- wet) y t 1 _ 1.2z 1 + O.35z 2 1 - 1.5z- 1 + 1 _ 1.2z-1 + O.35z-2 €(t)

~ll

e(t) = ({l(t) - (E[z-l] - U:[:-l] C[z-l]z-k m )e(t)

(41)

The poles of the closed-loop system are Z-l = 1/0.5, 1/0.7 and stable. But the pole of the controller is z-l = 1/1.5 and unstable, so this controller is not strongly stable.

converges to zero a.s.(almost surely, converge except on a set having probability zero) lim e(t) = 0

t-->oo

a.s.

(42)

Selecting the design parameters in the extended miniIllum variance controller (29) as Ud = 1.0, Un = 0.6, the controller is

Proof. Since the signal 'Ij;(t) satisfies PE conA dition and the transfer function C[z-l] -"2 is

-O.15z- 2 u(t) = 1 _ 1.8z-1 + 0.9z- 2 wet)

strongly positive real, the error of parameter identification (jet) = (} - (jet) and the error of output estimation, z(t) = 1](t) -~(t) converge to zero a.s.; Hm (j(t)

t---+oo

= 0,

lim z(t) =

t~oo

° a.s.

_ -O.34z-1 + 0.42z- 2 (t) 1 - 1.8z- 1 + O.9z- 2 y

() Y t

U d [Z-l]

-{(E[t: z-l]_ E[z-l]) - (C[t: z-l] U [z-l] n

Ud[Z-I]

z-k",

}!( t)

1 _ 1.2z- 1

+ O.225z- 4 ( ) + O.35z- 2 w t +

The poles of the closed-loop system by this controller from wet) to yet) are not changed from the poles by the controller (48) and the poles of the controller are Z-l = 1/(0.9 ± O.3i), Iz-11 = 1/0.9487 and are improved to be stable, that is, the controller is strongly stable. The effect of noise on the output yet) is also changed from (49) to (51).

(44)

z-l] - C[t : z~l] Un[Z-lj z-k", )z( t)

_C[z-l])

-0.075z-3

(51)

Then the error of the generalized output is given by

= (Elt:

=

1 - 3z- 1 + 3.05z- 2 - O.525z- 3 1 - 2.4z- 1 + 2.14z-2 - O.84z- 3 + O.1225z-4 ~(t)

A[t: Z-l]y(t) = B[t: Z-l]U(t - km,}

e(t)

(50)

and the closed-loop system is

(43)

This statement can be proved in a similar way by (Goodwin and Sin, 1984). The equation (37) is rewritten as

+C[t : z- 1 11](t)

(49)

Assuming the coefficients of the plant (46) are unknown and the variance of noise is (T = 0.0036, computer simulations are conducted. In the simulations, the least square identification law (34)~ (39) with the forgetting factor). = 0.99 is used

(45)

Using (43) to (45), we get the convergence of e(t).

o

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A DESIGN OF A STRONGLY STABLE SELF-TUNING CONTROLL...

14th World Congress ofTFAC

<:

0.'

~ -<>.•

0

0.'

i'

Fig. 1. Control result by STC method by Clarke and others

0

D

~ -0.

-. "op

Fig. 3. Parameter identification processes in the case of the solid line in Fig.2 reference input, the poles of the compensator and the effect of noise on the output can be changed by selecting the newly introduced design polynomials in the proposed controller. Fig. 2. Control result by STC parametrized method proposed in this paper

By designing the poles of the controller, a strongly stable controller is obtained.

and the initial value of the estimated covariance matrix r(O) is Cl:' = 1.0 and the initial values of identified coefficients are set to be equal to 0.5 of the true values.

Also proposed a self-tuning controller for systems with unknown coefficients by adding coefficient identification law to the extended minimum variance controller and proved the convergence of the controlled output.

Simulation results are shown in Fig.1 for the case of using the controller (48) by Clarke and others(Clarke. D. W. et al., 1979), and Fig.2 with the controller (50) proposed in this paper. In both figures, dotted lines show the output without noise and are sa.me in Fig.l and Fig.2. This fact indicates that the output response to reference input is not changed. by introducing the design parameters Un [z-l] and Ud[z-l].

7. REFERENCES Astrom, K. J. and B. Wittenmark (1973). On selftuning regulators. Automatica 9, 185-199. Astrom, K. J. and B. Wittenmark (1995). Adaptive Control. 2nd ed .. Addison-Wesley. Clarke. D. W., M. A. D. Phil, P. J. Gawthrop et al. (1979). Self-tuning control. PrQc. lEE 126, 6, 633~40. Goodwin, G. C. and K S. Sin (1984). Adaptive Filtering Prediction and Control. Prentice Hail. New Jersey. Inoue, A., V. T. Kroumovand S. Ma.suda (1993). A unified design method for model reference approach. In: Preprints of the [FAC World Congress 1993. Vol. 2. Sydney. pp. 83-88. Vidyasagar, M. (1985). Control System Synthesis: A Factorization Approach. The MIT Press. Cambridge.

In the simulations, the solid lines give the output responses with noise and in the simulation at step = 100, the feedback loop is cut. Fig.l shows the output by the controller (48) is divergent, whereas, the output in Fig.2 by the controller (50) stays bounded. t

Fig.3 shows parameter identification processes in the case with noise and feedback break(the solid line in Fig.2).

6. CONCLUSION In this paper, the generalized minimum variance control by Clarke and others(Clarke. D. ,V. et al., 1979) is extended to an extended minimum variance. controller including new design polynomials by using coprime factorization approach and comparing the most general two-degree-offreedom compensator in Youla parametrization form. \Vithout changing the response of output to

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