A self-tuning controller for mimo non-linear systems

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, VOL. 1 , 129- 142 (1987)

A SELF-TUNING CONTROLLER FOR MIMO NON-LINEAR SYSTEMS*

MUKUL AGARWAL Department of Chemical and Nuclear Engineering, University of California, Santa Barbara, C A 93106. U.S .A.

A N D

DALE E. SEBORGt Department of Chemical and Nuclear Engineering, University of California, Santa Barbara, CA 93106, U.S .A.

SUMMARY

A new self-tuning control (STC) strategy for non-linear, multi-inputlmulti-output control problems is proposed. This strategy is applicable to a broad class of non-linear systems, which can include any non- linear functions of the old inputs and outputs as well as the products of these functions and any powers of the most recent inputs. Simulation results for a two-link robot manipulator demonstrate that the new non-linear STC strategy provides significant improvement over conventional STCs based on linear models.

K E Y WORDS Adaptive control Non-linear control systems Multivariable control systems Self-tuning regulators Robots

1. INTRODUCTION

Self-tuning control systems based on linear process models have been remarkably successful in controlling many non-linear processes by treating them as time-varying linear processes. However, linear-model-based controllers are far from optimal for rapid, large changes in the operating point of the system. In practical applications, the major sources of non-linearity are often known from first principles of are readily identified by empirical means. Hence improved control can be expected by exploiting the non-linear structure of the model. Goodwin and Sin6 have mentioned the possibility of developing self-tuning regulators based on certain non-linear models. In few special situations, self-tuners based on non-linear models have provided significant improvements over standard self-tuners based on linear models. ’- lo However, the ap- plicability of these algorithms is limited to processes with specific non-linearities, such as polynomials in the manipulated input and bilinear products of output and input.

This paper presents a new self-tuning controller for multiple-input/multiple-output (MIMO) processes based on a general process model containing a broad class of non-linear terms. The

* The original version of this paper was presented at the 2nd IFAC Workshop on Adaptive Systems in Control and Signal Processing, Lund, Sweden, July 1986. t Author to whom correspondence should be sent.

This paper was recommended for publication by editor E. Mosca

0890-6327/87/020129- 14$07 .OO 0 1987 by John Wiley & Sons, Ltd.

Received October I986 Revised I3 April I987

130 M. AGARWAL AND D. E. SEBORG

analysis provides an extension of the non-linear STC for SISO systems, developed in an earlier paper, I ' to MIMO systems.

The paper is organized as follows. Section 2 presents the general non-linear model used to describe the process and the quadratic performance criterion. Section 3 develops the control scheme for the new self-tuning controller and then combines it with on-line estimation of the controller parameters. Section 4 provides an evaluation of the new algorithm based on simulations of a robot manipulator. Finally, Section 5 states the conclusions.

2. SYSTEM DESCRIPTION

The new self-tuning controller minimizes a quadratic cost-function objective based on a very general class of non-linear discrete-time models. It is assumed that the non-linear multi- input/multi-output process can be described adequately by a discrete-time model, which is linear in system parameters and allows for different number of inputs and outputs and different time delays between different input-output pairs. Such a model can be represented in the following general form:

A(z- ' )y( t ) = C K ( ~ ~ ' ) [ R , u ( ' ~ ) ( i ) Y , ( r ) l + d + C ( i Y ' ) [ ( t ) N

1" I

Y,(t)= g l [ y q ( t - j ) , j = O , 1,2 ,..., 4 = 1,2 ,... ,n; u p ( f - h ) , h = 1 , 2 , 3 , . . . , p = 1 , 2 ,... , m ] i = 1 , 2 ,..., N (1)

u(rg)(f) = [ U ? ( f ) , u ? ( l ) , '.., u%t(t)] i = 1,2, ..., N K(z- ' ) = diag(z-'l, ..., z - k n )

where, t is the sampling instant, I = 1,2,3, ...; N and ri are known integers (r; 2 0); y E R " is the measured output; u € R"' is the manipulated input; d E R" is the unknown disturbance; f E R" and ( f ( f ) ) is a sequence of equally distributed random vectors, with zero mean and unknown covariance, independent of previous inputs and outputs; gil i = 1,2, .. . , N are known single-valued time-invariant functions involving known parameters; kq, q = 1,2, ..., n is the known minimum time delay between any input and the qth output

k,=min(k,,,p= 1,2 ,..., m) 2 I , q = 1.2 ,..., n with kqp the time delay between the qth output and the pth input, expressed as one plus the largest integer multiple of the sampling period smaller than or equal to the time delay; and z-I is the backward-shift operator with the properties

z-'p(t) = d t - j ) z - " v d w ( ~ ) l = V!df - j ) W - A j = 1,2 ,3 , ... (2)

z-'p(tl I 1 2 ) = O(tl - j I f2 - j )

where V is a constant matrix, cp is a vector of appropriate dimensions, $ is a scalar and t , f l , t z are any discrete instants.

In equation ( l ) , Bi, i = 1,2, ..., N are n x rn matrices and A is an n x n polynomial matrix given by

A(2-l) = I + AIZ- ' + a * * + A,,z-""

It is assumed that the element in the rth row and the cth column of any matrix Ai is equal to zero if k, - k, > i. This assumption implies that yr(t + k,) is not a function of y,(t + k, + j ) ,

MIMO NON-LINEAR SYSTEMS 131

j = 1,2, ... for any c for which k, < k r ; otherwise, unknown future inputs would be required in making predictions of yr(t + kr ) . Such predictions will be required in the self-tuning control that is derived in Section 3. In other words, i t is assumed that the polynomial matrix K-'AK does not contain elements involving positive powers of z . Furthermore, C is an n x n polynomial matrix given by

c(z-') = I + C'z- ' + * a * + Cn,--"" with the assumption that all roots of det C ( z - ' ) lie inside the unit circle in the z-plane.

Remark 1. The non-linear system model in equation (1) is linear with respect to the system parameters Bi(i = 1,2, ..., N), A and C. Further, it does not require the numbers of inputs and outputs to be equal.

To simplify the notation, define k such that for any vector x t R",

~ ( t + k ) [ x l ( t + k l ) , xz(t + kz), e e . 9 xn(t + kn)]

Then the performance index to be considered can be written as

I=&(ii P' (z - ' )y ( t+k j+ P"(t-')Ay(t+k)-R(z-')w(f)1i2+ l lQ'(~-1)u(t)l121 (3)

where A y ( t ) = y ( t ) - y ( t - l ) , the notation 11 v I t 2 = vTv is used, & is the expectation operator conditioned with respect to data up to time t , w E R" is the setpoint, R is an n x n diagonal rational transfer function matrix, Q' is an m x m diagonal rational transfer function matrix, and P ' and P" are polynomials given by

P I ( z - I ) = 1 + Piz- ' + * * *p~, , .z " p '

p"(Z-1) = pd'+ ptz-1 + ... + p;p..z-"p"

Remark 2. As suggested by Waller and Gustafsson" in their analysis of continuous-time optimal control problems, in many situations the control system performance is improved by including the Ay term in the performance index. This is demonstrated in Section 4.

The polynomials P' and P" can be combined such that the performance index in equation

(4)

(3) becomes

I = &[ )I P ( z - ' ) Y ( ~ + k ) - R(z-')w(t) 1 1 2 + IIQ'(Z-')U(t) 1 1 2 )

P(z- ' ) = Po + P'z-' + 1 . . + P",Z-"'

where P is a polynomial given by

However, it may be more convenient for the user to specify P' and PI' separately.

Remark 3. In the above, P, P' , and P" are specified to be polynomials and not polynomial matrices. This restriction is required by the derivation of the predictive model in the Appendix, where equation (27) would not follow if P were a polynomial matrix.

3. A NEW SELF-TUNING CONTROLLER

The new STC algorithm utilizes a predictive model that expresses y ( t + k ) in terms of past and

132 M. AGARWAI. A N D D. E. SEBORG

present values of y and u at time 1. This predictive model can be expressed as N

C(Z- I)P(z-')y(t + k ) = i ( Z - ' ) y ( f ) + C E(z-~)[B,u('~)(~)Y;(~)] i = 1

+ E(z-')K-l(z-')d + ~ ( z - ' ) K - ' ( z - ' ) ~ ( z - ' ) € ( f ) (5)

where E, c, E and C are defined in the Appendix. Define the k-step-ahead value of y predicted at time t , y*(t + k 1 f), such that

P ( t - ' ) y ( t + k) = P ( z - ' ) y * ( f + k I f ) + K-'(Z-')F(Z-')E(~) (6)

Notice that the predicted value y*(f + k I I ) depends on values of y and u only at times prior to and equal to f. This is a consequence of the assumption made in Section 2 that the polynomial matrix K - 'AK does not contain positive powers of z. With this assumption, equation (26) implies that ~ ( z - I ) , and therefore C-'(z-'), do not contain positive powers of z.

Substituting equation (6) into the performance index of equation (4) yields

/ = R [ I I P ( z - 1 ) y * ( f + k I t ) + K ~ ' ( z ~ ' ) ~ ( z - ' ) ~ ( f ) - R ( z - ' ) w ( f ) I I 2 + I IQ'(z ')u(t)I[') (8)

In the above equation, the term K-'(z-')E(z-')((f) is independent of y(t), u(t) and previous values of y and u. This is because the polynomial matrix K-IE does not contain z-', j = 0, I , ... . Thus K-l(z-l)E(z-l)t(t) is not correlated with the remaining terms on the RHS of equation (8), which depend only upon the system conditions at time t and previous sampling instants. Hence equation (8) can be written as

I= !I P(Z-')y*(f + k 1 f) - R(Z-')W(f) ) I2 t I ) Q ' ( Z - ' ) u ( f ) I ) '+ &ill K -'(z -')E(z-')[(f)11')

The minimization of I can now be considered as a deterministic optimization problem. A necessary condition for an optimum is

aqau(t) = o or

Differentiating equation (7) with respect to u(t), and noting that c(0) = I , B(0) = POI, gives

where U'ri-l' is an m x m diagonal matrix with the elements u i r i - ' ) , u i r i - ' ) , ..., u,',7-') com-

MlMO NON-LINEAR SYSTEMS 133

prising the diagonal. From equations (9). (10) and (7), the resulting control law is

r,#O

This can be rearranged as N C [ PoBi~""(t)Y;(t)] + q

I = I r, # 0

+ c ( z - ' ) [ (2 i = I

r,#O

T - I

[P~B,riU''~-"(r)Y.(r)i)] (Q' (o))TQ' (z-')u(t) = O

. . r , = 0

Here q E R" is the sum of u(t)-independent terms on the LHS of equation (1 1). The control law (12) gives the control action u(f) in the form of polynomials involving

elements of u(t). If the model parameters A(z-I) , B;(i = I , 2, ..., N ) , C ( Z - ' ) and d are known, E(z-'), F ( z - ' ) and c ( z - ' ) can be calculated as outlined in the Appendix; and the control law (12) can be solved using standard root-finding techniques for polynomial equations, provided that the polynomial equation has at least one real solution. Only those real solutions are relevant that satisfy any physical constraints on u, such as upper and lower limits on the torque provided by a motor. If there is more than one relevant solution, the one closest to the previous control action, u ( t - l), can be selected for u(f) . In case there is no relevant solution, the setpoint w(t) can be altered temporarily as suggested by Lachmann.'

I f the system model in equation (1) is linear with respect to the most recent control input (i.e., r, E { 0, 1 ) , i = 1 , 2, ..., N ) , then the control law in equation (12) has the explicit representation

u(t) = - [ CP + C(Z-')(@~)-'(Q' ( 0 ) ) T Q ' ( ~ - l ) l N (13)

CP C [PoBiY,(r)] i = I r,fO

provided that the inverses exist.

Remark 1. Since Q'(z- ' ) , which is defined in equation (4), appears in the control law of equation (12) only in conjunction with Q'(O), the user need only specify

Q(z-') = (Q' (0 ) )TQ' (~ - l ) (14)

134 M . AGARWAL AND D. E. SERORG

On-line parameter estimation

If the model parameters in equation ( I ) are unknown, they can be estimated on-line from input-output data. However, it is more convenient to use the input-output data to estimate directly the parameters required in the control law of equation (12). For this implicit estimation problem, rearrange equation (7) as

h'

P(z- ' )y* ( t +kl t ) = C(~-l)y(f) + C ~ ( ~ - ~ ) I B ~ u ( " ) ( t ) Y ~ ( t ) l 1 - I

+ E(z-I)K-'(z-')d + [ I - C ( Z - ~ ) ] [ P ( z ' )y*(t + k 1 I ) ]

Adding equation (6) to the above equation, and premultiplying the resulting equation by K(z I ) , gives

P ( z - ' ) y ( t ) = K(z-')P(z-I)y(t) + C K ( Z ~ ' ) " ( Z - ' ) [ U ( ' ~ ) ( ~ ) Y ~ ( ~ ) J + 6 N

I - I

+ K(z- ' )H(z- ' ) [ P(z-')y*(f + k 1 t ) l + e ( f ) (15) where

GI(,?-') = E(z- ' )B1 i - 1,2, .... N H(z-1) = I1 - C ( z - 9 1

(16) 6 = K(z ')I?(z I)K-'(z-')d = k(l)d

e ( t ) = E(z-')€(f) Due to the nature of [ ( I ) and the orders of the polynomials in E ( t - I ) , & ( t ) is not correlated with any of the other terms on the RHS of equation (15). Hence parameters @(z-'), Cj(Y1) ( i - 1,2, ..., N), H ( z - ' ) and 6 can be estimated from equation (15) and input-output data using the standard recursive extended least-squares (RELS) method. The KELS method is required because it generates the values of the unknown elements y * ( t + k \ 1 ) in the data vector.

Then, the control law in equation (12) can be rewritten in terms of the estimated parameters $ ( z - ' ) , G I ( z - ' ) ( i = 1,2, .... N ) , H(2-I) and 6. Using the definitions in equations (14) and (16), and noting that C,(O) - PoB,, i = 1,2, ..., N, since E(0) = Pol, equation (12) becomes

r# = 0

Again, for a system model with rl € (0, I ] , i = I, 2, ..., N, the control law in equation (13) can be rewritten as

u ( f ) = - [++ [ ~ - H ( z l)J(*l)-'Q(~-')l-l~ N (18)

+= C [C; i (~)~j ( t ) l 1 = l

r. # 0

MlMO NON-LINEAR SYSTEMS 135

Remark 2. Due to the generalized formulation of the system model in equation (1). several elements of the matrices Bi may be known to be zero in a particular application. Let n ~ , be the total number of unknown elements in the n h rows of the matrices Bi, i = 1,2, ..., N. Further- more, let n ~ , be the number of those unknown elements in the n h rows of Bi for which the argument corresponds to a one-step-ahead value of the argument of another unknown element in the rth row of any Bi. Then the number of parameters that need to be estimated for the implicit STC, nimp, is

nimp = C [ (nj, + ~ ) n + nK, + ( n B , - n K , ) k r ] + ncn2 + n r = I

where n ~ , is as defined in the Appendix. For the typical situation where n/, = n~ + kmax - 1 - k,, this becomes

n

n i m p = C [ ( n A + k m a x - kr)n + n K , + (ne, - ~ K , W ~ I + ncn2 + n (19)

By comparison, i f the parameters of the system model ( I ) are estimated explicitly, the number of parameters is

r = I

where the inequality results if k, < k m a x - 1 for any r , due to one or more parameters in A(2-I) being known to be zero according to the assumption stated in Section 2. Thus the explicit STC requires estimation of fewer parameters. However, the explicit STC also involves the extra computational burden of obtaining I?(f1), F ( 2 - I ) and e(,-’) from the estimated parameters. Therefore a preference between the two implementations would depend on the specific nature of the system that is to be controlled. In particular, the choice will be affected by the values of the time delays, the nature of the non-linear functions, the available computational speed in relation to the sampling time, and the utility for other purposes of the estimates of the model parameters.

4. SIMULATION RESULTS

Previous researchers have reported self-tuning position and velocity control of robotic manipulators, However, they have been restricted to the use of linear models. l 3 This simulation example evaluates the self-tuning position control of a robotic manipulator based on a non- linear model. Consider the two-link robot manipulator analysed by Slotine and Sastry. l 4 The manipulator consists of two rigid links of equal length and mass (both normalized to unity) in the horizontal plane. Choosing as the measured outputs the angle 01 made by the first link with the x-axis and the angle 02 made by the second link with respect to the first, and as manipulated inputs the torques TI and T2 applied to the two joints, the dynamics of the manipulator is given by

6, = [ (2/3)TI - (2/3 + cos &)Ti]/(16/9 - cos2 ( 3 2 )

6 2 = [ - (2/3 + cos 82)T; + 2(5/3 + cos @)Ti]/(16/9 - COS’ 0 2 )

T( = 2TI + sin 8262(261 + 62) Ti = 2T2 - sin Q26:

(21)

The above system is unstable in the open-loop condition and has infinite steady-state gain and no time delay. The system was simulated using the fourth-order Runge-Kutta numerical

136 M . AGARWAL AND D. E. SEBORG

integration method with a time step of 0.001 s. Random measurement noise uniformly distributed on the interval [ -0*05", +0.05"] was added to the outputs. The torque inputs were constrained to an absolute value of 10 N m in each direction.

The system was controlled using the new non-linear STC based on the model

el([+ I ) = a l e l c t ) + a 2 e l ( t - i ) + b l ~ l ( t ) ~ l ( t ) + ~ 2 ~ 2 ( t )

ez(t + 1) = a3ez(t) + &2(t - 1) + b 5 ~ l ( t ) ~ 5 ( t ) + bay*(?) + b3T2(t)Y3(t) + b4Y4(f) + dl + 41(t + 1)

+ b7T~(f)Y7(f) + bEYE(t) + d2 t 4 2 ( 1 + 1) yl(t) = I / [ 1619 - COS* e2(t)] Y 2 ( t ) = Yl(t)sin e2(t)[e2(t) - 8 2 ( r - l ) ] 12eI(t) - 2 @ ( r - 1) + & ( t ) - e2(t - 1)1 y3(t) = ~ , ( 1 ) [ 2 / 3 + cos ez(t)l ~ ~ ( 1 ) = Y3(t)sin ez(t)[el(t) - el(t - 111 Y55(f) = Y3(t)

(22)

Y6(f)= Y2(1)[2/3 +cos el([)] Y7(t) = YI ( t ) [ 5/3 -t COS ez(t)] Y 8 ( t ) = y7(r)sin ez(t)[el(t)-el(t- 1 ) 1 2

The performance of the new STC was compared with the performance of a linear STC that was based on the model

e,(t + 1) = a l e l ( t ) + a2el(t- I ) + a3ez( t ) + a4e2(t - 1 ) + bl TI ( t ) + br Tz ( I ) + dl + .El ( t + 1 )

&(I + 1) = a5el(t) + a6e1(t - 1) + a,ez(t) + asez(t - 1)

+ b3Tl ( t ) + b4T2(f) + d2 + 42( t + 1) (23)

The sampling time was chosen to be 0.02 s. All parameters were initialized to zero and estimated using the recursive least-squares method. The UDU covariance factorization method of Thornton and Bierman" was employed, with a forgetting factor of 0.98. The diagonal elements of the covariance matrix were initialized to lo6. For both the linear and the non-linear controllers, the best results were obtained when the covariance matrix was not reset sub- sequently. The outputs el and & were initially set to -90" and 170" respectively. The initial perturbation to the system was provided by setting the torques T I and T2 to 0.1 and - 0- 1 N m respectively for one instant before closing the loop.

Slotine and Sastry l4 report results for tracking slow setpoint changes when the angular velocities 6, and 6 2 are measured as well as the angles 81 and 8 2 . Both the linear and non- linear STCs gave comparable results for identical setpoint changes without requiring measurement of the angular velocities. The tuning parameters for both the STCs were P ' = 1 , PI'= 0, R = I and Q = 10-6J.

A set of larger and faster setpoint changes was then specified. The response of the non-linear STC with P ' = 1 - O-~Z-', P " = 1, R = 0.11 and Q = 10-61 is shown in Figure 1. It is seen that the outputs track the setpoints satisfactorily. Since the angular velocities are penalized in the performance index via the A y term, the measurement noise leads to rapid fluctuations in the inputs, as shown in Figure 2. By contrast, Figure 3 shows the response of the linear STC with P ' = 1 - 0 - 9 5 z - ' , P " = 1, R = 0.051 and Q = Considerable interaction between the outputs is apparent even though the controller has been detuned significantly, leading to an overall

MlMO NON-LINEAR SYSTEMS

90

0 -

-90

137

-

- c) PI S I-

ru ID u al K I-

Figure 1 . Servo

- E

Z - w al Y U L 0 I-

- E

Z

N al 3 U L 0 I-

I

a""- ~ ~~ ~

0 . 0 4 . 0 8 . 0 12.0 16.0

0 . 0 4 . 0 6 .0 12.0 16.0

Time (sec)

response of the non-linear STC for nominal design parameters

10

¶

0

-¶

-10 0 . 0 4 . 0 8 . 0 12.0 16.0

10

0

Y , 1 l . [II I I -5

- ¶ O _- 0 .0 4 . 0 (1.0 12.0 16.0

Time k e c )

Figure 2. The control inputs for Figure 1


c1 a U m e

(u

Y

c +

m m

-1w 0 . 0 4.0 6 . 0 12.0 16.0

-1w 0 .0 4.0 0 . 0 12.0 $6.0

Time ( m c )

Figure 3 . Servo response of the linear STC

0 . 0 4 .0 0 . 0 12.0 16.0

0 . 0 4 . 0 6 . 0 10.0 i O . 0

Tim. (8.C)


MIMO NON-LINEAR SYSTEMS

.. (P Y

139

(u m

I-

--- 0 . 0 4.0 0 . 0 12.0 18.0

0.0 4 . 0 a.0 12.0 1e.o

Time ( s a c )

Figure 5 . Servo response of the non-linear STC with increased penalty on u

-101 I , I 0.0 4.0 1 .0 12.0 18.0

0.0 4 . 0 1 . 0 12.0 11.0

Tim. (see)


140 M . AGARWAL AND D. E. SEBORG

0 . 0

A

0 0

N 10 CI al s -80

-

I- l l

4 . 0 1.0 12.0 lS.0

-160 0 . 0 4 . 0 6 . 0 12.0 1 6 . 0

Time k c c )

Figure 7. Servo response of the non-linear STC with no penalty on Ay

10

I

9 5

a l o

z I

4

3 0

F 0 -I

-10 0 0 4 . 0 1.0 12.0 16.0

0 . 0 4 . 0 0 . 0 12.0 16.0

Time (eec)


MIMO NON-LINEAR SYSTEMS 141

sluggish response. The inputs, shown in Figure 4, are also more oscillatory than for the non- linear STC.

Figures 5 and 6 show the effect of increasing the penalty on the inputs to Q = 10-51 for the non-linear STC. Although this leads to far less oscillatory inputs than in Figure 2, the output response becomes worse than in Figure 1 , as expected.

When the run in Figures 1 and 2 is repeated without penalizing the angular velocities, i.e., P" = 0, the inputs become less oscillatory, as shown in Figure 8. However, as Figure 7 shows, the output response becomes too oscillatory. This shows that P" is an important tuning parameter that can lead to improved performance.

5 . CONCLUSIONS

A self-tuning controller has been developed for a general class of non-linear multi-input/multi- output process models. These models can include any known non-linear functions of the old inputs and outputs as well as the products of these functions and any powers of the most recent inputs. In order to allow the use of well-known linear estimation techniques, the models have been assumed to be linear in the unknown parameters. In particular, the parameters contained within the known non-linear functions are assumed to be known apriori. However, this is not a serious limitation as, in many practical applications, the specific nature of the non-linearities can often be characterized accurately based on physical or empirical grounds.

A very general algorithm has been proposed that allows for different numbers of inputs and outputs and for different time delays for each input-output pair. The time delays are allowed to be fractional with respect to the sampling time and not all of them need to be time-invariant or to be known precisely. Only the minimum time delay between each output and any input is required to be known and to vary only within the range of one sampling period. A new performance index for adaptive control has been introduced that penalizes incremental changes in the outputs. However, the development requires specification of identical penalties for all outputs. This limitation can largely be circumvented by rescaling the outputs appropriately.

Simulations for the self-tuning control of a robot manipulator have been presented. The results demonstrate the significant advantage of using the non-linear STC over the linear STC and the usefulness of the new performance index.

APPENDIX. DERIVATION OF THE k-STEP-AHEAD PREDICTIVE MODEL

Consider the identity

C ( ~ - ~ ) P ( Z - ' ) = A ( Z - ~ ) E ( Z - ~ ) + K ( z - ~ ) F ( z - ' )

E(z - ' ) = POI + [el(z-') , e2(z-'), ..., e , , ~ ' ) ] '

where E and F are unique polynomial matrices such that

F(z - ' ) = [fi(Z-') , f 2 ( ~ - ' ) , ..., f n ( Z - ' ) l

and e , ( z - ' ) and f ,(z-') are column VeCtOrS

e , ( z - ' ) = e , , l z - ' +e, ,zz- '+ ... + e,,k, ~ ( ~ a - ~ )

f , (z- ' ) = f , ,o + fl , 'z-' + ... + f,,.,,z-"', i = 1,2, ..., n nf, = max(nA + kmax - l , n c + PI - k ,

k,,, = max(k,, i = 1,2, ..., n )


There exist non-unique polynomial matrices and F, l6 such that

e ( z - ')F(z- I ) = F(z- )E(z- ' ) where

E ( z - I ) = POI+ [i?t(z-t), & ( z - t ) , .** , i%(z- t ) l P(z-1) = [ i , ( z - I ) , f 2 ( Z 4 ) , ..., k ( z r 1 ) 1 i ? ; ( z - ' ) = i j iJz-' + c;,zz-2 + ' a . + 6;,&,- l z - ( k ~ - ' )

i i (Z-1) = f;, 0 + i,,tz-' + ... + ff.",,z-fl"

i = 1,2, ..., n

Define a matrix c(z-') satisfying the condition

C(z-' ) P ( z - ' ) = E(z- ' )K- ' ( z - ' )A (z- ' )K (z-') + P(z- ')K(z- ' ) (26)

Then, comparing equation (24) premultiplied by E(z- ' )K- I (z- ' ) with equation (26) postmultiplied by K-'(z-')E(z-'), and using equation (25), gives

(27)

It can be shown that there exists at least one set of matrices E(z- ' ) and F(z- ' ) such that C ( z - ' ) is a polynomial matrix and has ordcr equal to nc + k,,, - k m i n . " Then, premultiplying equation (1) by e ( z - ' ) K - ' ( z - ' ) , adding to i t equation (26) postmultiplied by K - ' ( z - ' ) y ( O , and using equation (27), yields equation ( 5 ) .

e ( z - I ) K - ' ( ~ - ' ) E ( z - ' ) = E(z - ' )K- ' (z-I)C(Z- ' )

ACKNOWLEDGEMENTS

Financial support from NSF Engineering Research Center grant CDR-842 14 15 is gratefully acknowledged.

REFERENCES

I . Astrom, K . J . and B. Wittenmark, 'On self-tuning regulators', Aufomafica, 9, 185-199 (1973). 2. Clarke, D. W. and P. J . Gawthrop, 'Self-tuning controller', Proc. IEE, 122, 929-934 (1975). 3. Clarke, D. W. and P . J . Gawthrop, 'Self-tuning control', Proc. IEE. 126, 633-640 (1979). 4. Koivo, H. N., 'A multivariable self-tuning controller', Automatica, 16, 351 -366 (1980). 5 . Seborg, D. E., 'I. F. Edgar and S. L. Shah, 'Adaptive control strategies for process control: A survey', AIChE

6. Goodwin. G. C. and K. S. Sin, 'Adaptive Filfering, Prediction and Control, Prentice-Hall. Englewood Cliffs, NJ.

7. Keviczky, L., I . Vajk and J . Hetthessy, 'Self-tuning extremal controller for the generalized Hammerstein model',

8. Anbumani. K., L. M. Patnaik and 1. G. Sarma, 'Self-tuning minimum-variance control of nonlinear systems of

9. Lachmann, K . H., Parameter-adaptive control of a class of non-linear processes', 6th IFAC Symp. on Identificu-

10. Svoronos, S., G. Slephanopoulos and R. Aris, 'On bilinear estimation and control'. Inr. J. Control, 34. 651-684

1 1 . Agarwal, M . and I). E. Seborg, 'Self-tuning controllers for nonlinear systems', lFAC Workshop on Adupfive

12. Waller, K . V. T. and S. E. Custafsson. 'Performance indices for multivariable PI-control with incomplete feed-

13. Koivo, A. J . and T. H. Guo, 'Adaptive linear controller for robotic manipulators', IEEE Trans. Automatic Con-

14. Slotine, J . J . and S . S. Sastry, 'Tracking control of nonlinear systems using sliding surfaces, with application to

IS. Thornton. C. I- . and G. J . Bierman, 'Filtering and error analysis via the 1 J D U covariance factorization', IEEE

16. Wolovich, W. A, , Linear Multivariable Sysfems, Springer, New York. 1974. 17. Agarwal, M., Ph.D. Dissertution. University of California, Santa Barbara (in preparation), 1987.

J., 32, 881-913 (1986).

1984.

5rh IFAC Symp. on Idenrijcafion Sysfem Parameter Esfimafron. Darmstadt, FRC, 1979.

the Hammerstein model', IEEE Trans. Aufomatic Control, AC-26, 959-961 (1981).

tion and S-vstem Parameter Estimation, Arlington, VA, 1982, pp. 372-378.

(1981).

Control oJ Chemicul Processes, Frankfurt, 1985. Also, Aulomutica (in press).

back', Chem. Eng. Sci., 30, 1265-1271 (1975).

trol, AC-28, 162-170 (1983).

robot manipulators'. Int. J . Control, 38, 465-492 (1983).

Trans. Automatic Control, AC-23, 901 -907 (1978).

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A self-tuning controller for mimo non-linear systems