6
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 36, NO. IO, OCTOBER 1991 1 I99 I Acoust., Speech, Signal Process., vol. ASSP-38, pp. 191-193, Jan. 3.5 0.0 0.d 0;2 013 0:4 015 Fig. 1. Power spectrum estimate. The power spectrum obtained from the above AR and MA parame- ter estimates is shown in Fig. 1. From the spectrum, we see that the nine sinusoids have clearly been detected. Unlike the I-D case where k sinusoids in white noise correspond to a (2k, 2k)th-order ARMA process, several 2-D sinusoids in white noise can be mod- eled by using a lower order 2-D ARMA process. V. CONCLUSIONS In this note, we have addressed the estimation problem of the MA parameters of a 2-D ARMA model. This problem has been shown to be equivalent to solving a set of overdetermined 2-D transcenden- tal equations. In order to solve such equations, an iterative algo- rithm based on some extensions of the Newton-Raphson method has been developed. The performance of the new algorithm has been demonstrated by a numerical example. For several 2-D sinusoids in white noise, it is believed to be the first time to have provided the corresponding whole 2-D ARMA model, including the AR and MA parameters. In our example, we assumed the known autocorrelations. In principle, if r(k, rn) is replaced by the sample autocorrelation estimated from the observed 2-D data, the algorithm is also avail- 1990. J. A. Cadzow and K. Ogino, “Two-dimensional spectral estimation,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-29, pp. 396-401, June 1981. R. L. Kashyap, “Characterization and estimation of two-dimensional ARMA models,” IEEE Trans. Tnformat. Theory, vol. IT-30, pp. 736-745, Sept. 1984. X.-D. Zhang and J. Cheng, “High resolution two-dimensional AR- MAspectral estimation,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-39, Mar. 1991. [9] A. Rosenfeld, Ed., Image Modeling. New York: Acadmic, 1981. X.-D. Zhang and H. Takeda, “An order recursive generalized least squares algorithm for system identification,” IEEE Trans. Automat. Contr., vol. AC-30, pp. 1224-1227, Dec. 1985. J. A. Cadzow, “High performance spectral estimation-A new ARMA method,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-28, pp. 524-529, Aug. 1980. M. Kaveh, “High resolution spectral estimation for noise signal,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-27, pp. 286-287, June 1979. X.-D. Zhang and H. Takeda, “An approach to time series analysis and ARMA spectral estimation,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-35, pp. 1303-1313, Sept. 1987. A. K. Jain, “Advances in mathematical models for image processing,” Proc. IEEE, vol. 69, pp. 502-528, May 1981. G. E. P. Box and G. M. Jenkins, Time Series Analysis- Forecast- ing and Control. J. C. Chow, “On the estimation of the moving-average parameters,” IEEE Trans. Automat. Contr., vol. AC-17, pp. 268-269, Apr. 1972. [6] [7] [8] [lo] [l 1) [12] [13] [14] [15] (161 San Francisco, CA: Holden-Day, 1970. Feedforward Digital Tracking Controller Dong H.Chyung Abstract-A digital tracking controller is presented for linear time invariant control systems. It is based on feeding forward the reference input and its delayed values. A necessary and sufficient condition for the existence of such a controller is also given. able. Some important problems remain to be solved associated with I. INTRODUCTION applications of the algorithm. For instance, an efficient approach to the AR parameter estimation based on the sample autocorrelations is not available at the moment; the order determination of 2-D ARMA ( p l , p2; ql, q2) models and the colored noise case need to be studied further. In this note, the problem of synthesizing a tracking controller for linear digital control systems is considered. The objective is to force the system output to follow a given reference input with zero steady-state error. The reference input is assumed to be a polyno- mial function. The proposed tracking controller is based on directly ACKNOWLEDGMENT feeding forward the reference input and its delayed values. A necessary and sufficient condition for the existence of the tracking controller is also obtained, An example is given to illustrate the proposed method. The author would like to thank the reviewers for valuable corn- ments which helped to improve the note. REFERENCES [l] [2] J. W. Woods, “Two-dimensional Markov spectral estimation,” IEEE Trans. Informat. Theory, vol. IT-22, pp. 552-559, Sept. 1976. J. S. Lim and N. A. Malik,“A new algorithm for two-dimensional maximum entropy power spectrum estimation,” IEEE Trans. Acoust., Speech, Signal Proccess., vol. ASSP-29, pp. 401-413, June 1981. S. W. Lang and J. H. McClellan,“Multidimensional MEM spectral estimation,” IEEE Trans. Acoust., Speech, Signal Process., vol. H. Kimura and Y. Honoki, “A hybrid approach to high resolution two-dimensional spectrum analysis,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-35, pp. 1024-1036, July 1987. X.-D. Zhang and D.-Y. Cui, “Performance analysis of Kimura and Honoki’s hybrid approach to 2-D spectrum estimation,” IEEE Trans. [3] ASSP-30, pp. 880-886, Dec. 1982. [4] [5] Continuous time tracking controller for linear control systems has been studied by many investigators [I], [3], [4], and a discrete time version of [l] is considered in [2]. In these references, the control system is first augmented by error integrals, and hence integrals of both state and control variables, and then a stabilizing tracking controller is designed for the augmented system. One of the major advantages of the augmentation method is that the resulting con- troller is robust with respect to system parameter variations. On the other hand, because the system is augmented with additional integra- Manuscript received December 22, 1989; revised April 13, 1990. The author is with the Department of Electrical and Computer Engineer- IEEE Log Number 9101754. ing, University of Iowa, Iowa City, IA 52242. 0018-9286/91/1000-1199$01.oO 0 1991 IEEE

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 36, NO. IO, OCTOBER 1991 1 I99

I Acoust., Speech, Signal Process., vol. ASSP-38, pp. 191-193, Jan.

3.5

0.0 0.d 0;2 013 0:4 015

Fig. 1. Power spectrum estimate.

The power spectrum obtained from the above AR and MA parame- ter estimates is shown in Fig. 1. From the spectrum, we see that the nine sinusoids have clearly been detected. Unlike the I-D case where k sinusoids in white noise correspond to a (2k , 2k)th-order ARMA process, several 2-D sinusoids in white noise can be mod- eled by using a lower order 2-D ARMA process.

V. CONCLUSIONS

In this note, we have addressed the estimation problem of the MA parameters of a 2-D ARMA model. This problem has been shown to be equivalent to solving a set of overdetermined 2-D transcenden- tal equations. In order to solve such equations, an iterative algo- rithm based on some extensions of the Newton-Raphson method has been developed.

The performance of the new algorithm has been demonstrated by a numerical example. For several 2-D sinusoids in white noise, it is believed to be the first time to have provided the corresponding whole 2-D ARMA model, including the AR and MA parameters.

In our example, we assumed the known autocorrelations. In principle, if r ( k , rn) is replaced by the sample autocorrelation estimated from the observed 2-D data, the algorithm is also avail-

1990. J. A. Cadzow and K. Ogino, “Two-dimensional spectral estimation,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-29, pp. 396-401, June 1981. R. L. Kashyap, “Characterization and estimation of two-dimensional ARMA models,” IEEE Trans. Tnformat. Theory, vol. IT-30, pp. 736-745, Sept. 1984. X.-D. Zhang and J. Cheng, “High resolution two-dimensional AR- MAspectral estimation,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-39, Mar. 1991.

[9] A. Rosenfeld, Ed., Image Modeling. New York: Acadmic, 1981. X.-D. Zhang and H. Takeda, “An order recursive generalized least squares algorithm for system identification,” IEEE Trans. Automat. Contr., vol. AC-30, pp. 1224-1227, Dec. 1985. J. A. Cadzow, “High performance spectral estimation-A new ARMA method,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-28, pp. 524-529, Aug. 1980. M. Kaveh, “High resolution spectral estimation for noise signal,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-27, pp. 286-287, June 1979. X.-D. Zhang and H. Takeda, “An approach to time series analysis and ARMA spectral estimation,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-35, pp. 1303-1313, Sept. 1987. A. K. Jain, “Advances in mathematical models for image processing,” Proc. IEEE, vol. 69, pp. 502-528, May 1981. G. E. P. Box and G. M. Jenkins, Time Series Analysis- Forecast- ing and Control. J. C. Chow, “On the estimation of the moving-average parameters,” IEEE Trans. Automat. Contr., vol. AC-17, pp. 268-269, Apr. 1972.

[6]

[7]

[8]

[lo]

[ l 1)

[12]

[13]

[14]

[15]

(161 San Francisco, CA: Holden-Day, 1970.

Feedforward Digital Tracking Controller

Dong H.Chyung

Abstract-A digital tracking controller is presented for linear time invariant control systems. It is based on feeding forward the reference input and its delayed values. A necessary and sufficient condition for the existence of such a controller is also given.

able. Some important problems remain to be solved associated with I. INTRODUCTION applications of the algorithm. For instance, an efficient approach to the AR parameter estimation based on the sample autocorrelations is not available at the moment; the order determination of 2-D ARMA ( p l , p 2 ; q l , q2) models and the colored noise case need to be studied further.

In this note, the problem of synthesizing a tracking controller for linear digital control systems is considered. The objective is to force the system output to follow a given reference input with zero steady-state error. The reference input is assumed to be a polyno- mial function. The proposed tracking controller is based on directly

ACKNOWLEDGMENT feeding forward the reference input and its delayed values. A necessary and sufficient condition for the existence of the tracking controller is also obtained, An example is given to illustrate the proposed method.

The author would like to thank the reviewers for valuable corn- ments which helped to improve the note.

REFERENCES [l]

[2]

J . W. Woods, “Two-dimensional Markov spectral estimation,” IEEE Trans. Informat. Theory, vol. IT-22, pp. 552-559, Sept. 1976. J. S . Lim and N. A. Malik,“A new algorithm for two-dimensional maximum entropy power spectrum estimation,” IEEE Trans. Acoust., Speech, Signal Proccess., vol. ASSP-29, pp. 401-413, June 1981. S. W . Lang and J. H . McClellan,“Multidimensional MEM spectral estimation,” IEEE Trans. Acoust., Speech, Signal Process., vol.

H. Kimura and Y . Honoki, “A hybrid approach to high resolution two-dimensional spectrum analysis,” IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-35, pp. 1024-1036, July 1987. X.-D. Zhang and D.-Y. Cui, “Performance analysis of Kimura and Honoki’s hybrid approach to 2-D spectrum estimation,” IEEE Trans.

[3]

ASSP-30, pp. 880-886, Dec. 1982. [4]

[5]

Continuous time tracking controller for linear control systems has been studied by many investigators [I], [ 3 ] , [4], and a discrete time version of [ l ] is considered in [ 2 ] . In these references, the control system is first augmented by error integrals, and hence integrals of both state and control variables, and then a stabilizing tracking controller is designed for the augmented system. One of the major advantages of the augmentation method is that the resulting con- troller is robust with respect to system parameter variations. On the other hand, because the system is augmented with additional integra-

Manuscript received December 22, 1989; revised April 13, 1990. The author is with the Department of Electrical and Computer Engineer-

IEEE Log Number 9101754. ing, University of Iowa, Iowa City, IA 52242.

0018-9286/91/1000-1199$01.oO 0 1991 IEEE

1200 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 36, NO. I O , OCTOBER 1991

tors, the dimension of the augmented system is increased, and also the controller becomes a dynamic controller. As a consequence, the resulting system becomes more complex. In addition, since the error integral acts as a low-pass filter, in certain cases, the rise time of the closed loop system becomes too long and as a result the stiffness of the servosystem suffers. Also, if the system actuator saturates, as is the case in most of real systems, the augmentation based controller causes a large overshoot, and even an instability in some cases, when the error is large, a phenomenon commonly referred to as "integrator windup" [ 5 ] .

Because no error integrals are involved in the proposed con- troller, it is relatively simple and avoids most of the problems mentioned above. On the other hand, it has a major disadvantage of being sensitive to system parameter variations. It should be noted, however, that there are many practical systems where a feedforward tracking controller is perfectly satisfactory. For step reference in- puts, the feedforward controller has been used in single-input, single-output classical control systems for a long time in an ad hoc manner. No systematic methods for multi-input multi-output sys- tems with general polynomial reference input, however, are cur- rently available. Clearly the two types of tracking controllers have their respective advantages and disadvantages. A suitable controller should be chosen only after carefully examining the given system characteristics and its performance requirements.

11. PROBLEM STATEMENT

Consider the linear time-invariant digital control system

x ( k + 1 ) = A x ( k ) + B u ( k )

Y ( k ) = CX( k ) (SI where x = ( x , , x 2 ; . ., x , , ) ~ is the ( n x 1 ) state vector, U =

(U', u 2 ; . * , u , ) ~ is the (rn x 1 ) control vector, y =

( yl, y,, * , Y , )~ is the ( r X 1) output vector, and A , B , and C are nonzero constant matrices with compatible dimensions. It is assumed that the matrices B and C have full rank. The reference input vector y* is given by a qth-order polynomial vector function of k

y * ( k ) = U, + ~ , k + a ,k2 + .. . +u,k4 ( 1 )

where a,, i = 0, 1 , a , q are ( r x 1) constant vectors. The prob- lem is to design a controller U such that the corresponding output y ( k ) follows the reference input y* (k ) in steady state without error, that is, y ( k ) - y * ( k ) as k + 00.

111. FEEDFORWARD CONTROLLER

For the given reference input y*, let x* and U* satisfy the equations

X*(k + 1 ) = A X * ( k ) + BU*(k)

U*( k ) = CX*( k ) . ( 2 ) If such x* and U* do not exist, then it is not possible for the output to follow the given reference input y*. Let

A x = x - X *

AU = U - U*. (3)

(4)

Then, from the system (S) and ( 2 )

A x ( k + 1 ) = A A x ( k ) + B A u ( k ) .

Suppose the linear feedback control

stabilizes system (4), that is, A x ( k ) + 0 as t + CO. Then, ~ ( k ) +

x * ( k ) as t + 00, and so y ( k ) + y * ( k ) as k - CO. Thus, from (3)

U( k ) = U*( k ) + Au( k )

= u * ( k ) - K ( x ( k ) - x * ( k ) ) (6)

is a desired tracking controller. Note that the control given by (5) stabilizes the system (4) if and only if the control

u ( k ) = - K x ( k )

stabilizes the original system (S). The stabilizing feedback gain K can be obtained by any available method, such as optimal control, pole zero assignment, or even a classical method. It must be chosen, however, such that the desired transient performance is achieved.

Define the delay operator V by

V x ( k ) = x ( k - 1 ) .

Then, since y* is a qth-order polynomial,

4 + 1

( 1 - v ) q + ' y * ( k ) = ( - l ) ; + I C ; V ' y * ( k ) i = O

4+ 1

i = O = ( - l ) i + I C , y * ( k - i ) = 0 (7)

= ( q + l ) ! / ( q + 1 - i)! i! . Sup- for all k. In the equation, pose x* is given by

4

X * ( k ) = F,y*(k - i ) (8) i = O

for some ( n x r ) constant matrices F,, i = 0, 1,2;. ., q. Then,

( 1 - V ) ' + l x * ( k + 1 ) = 4

F , ( 1 - V ) ' + ' y * ( k - i + 1 ) = 0 . I=o

(9)

On the other hand,

4+ 1

i = O (1 - V ) Q + l X * ( k + I ) = ( - I ) ; + , C ; V ' x * ( k + 1 )

4+ I = ( - I ) f + , C , x * ( k - i + 1 ) .

i = O

Thus, 4+ I

i = O 1 (- ~ ) k + ~ ~ , x * ( k - i + I ) = 0 .

From ( 2 ) , together with the above equation,

4 + I ( - l ) ; + ' C ; x * ( k - i + 1 ) = o

i = O

- z x * ( j + 1 ) + A X * ( j ) + BU*(j) = 0 ,

C x * ( j ) = y * ( j ) , j = k , k - I ; . . , k - q (11)

j = k , k - l : . . , k - q

where I is the ( n X n) identity matrix. Since y * ( j ) is known for j = k , k - 1, . . . , k - q, the above equations represent ( q + l)(n + r ) + n scalar simultaneous equations for the ( q + l)(n + rn) + n scalar unknown elements of x*(k + 1). x*( A. U*( i ) . i = k . k

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 36, NO. 10, OCTOBER 1991

- 1 , . . e , k - 4. In matrix form,

( 4 + 2)"

-

O 0

B 0 O

0 -

d o l d , I d 2 1 . . . d q + , I 0 - I A 0 . . . O B 0 - I A . . . 0 0

0 0 0 ' . * A 0 0 c 0 0 0 0 0 c . . . 0 0

. . .

. . .

X*(k + 1 )

x * ( k - 1 )

x * ( k )

x * ( k - q )

U * ( k ) U * ( k - 1 )

- u * ( k - 4) 0 0 0 * . . C O

. . .

. . .

. . .

. . .

. . .

. . .

. . .

1201

where d , = ( - l ) k + , C r # 0 for all i. Let Q be the ( n + ( 4 + l ) (n + r ) ) x (n + ( q + l ) (n + m)) coefficient matrix in (12). Obviously

d o l d , I d , I . . . dq+,I 0 0 . . . 0

B I A 0 . . . 0 B 0 * . . -1 A . . . 0 0 B . . .

rank 1 .. . . . .

0 0 0 * . . A 0 0 . . *

= n + ( 4 + 1)n . (13)

Equation (12) must be satisfied for all reference input y*. Thus, every row vector of the matrix

: : : ::: :] I 0 0 0 * . . c 0 0 . . . 0

0 c 0 . . . 0 0 c . . . . . .

. . .

must be linearly independent of every row vector in the matrix given in (13). Since the rank of the above matrix is ( 4 + I)r, the rank of the coefficient matrix Q in (12) must be n + ( 4 + l ) ( n + r ) , that is, rank(Q) = n + ( 4 + l ) (n + r ) is a necessary condition for the existence of a tracking controller. In this case, there are exactly n + ( q + l ) ( n + r) linearly independent scalar equations in (12). On the other hand, a solution of (12) exists if the number of independent equations is not greater than the number of unknowns, that is, n + ( 4 + l ) (n + r ) 5 n + ( 4 + l ) (n + m). If rank(Q) = n + ( 4 + I)(n + r ) , then obviously n + ( q + l ) (n + r ) 5 n + ( q + l ) (n + m). Thus, rank(Q) = n + ( 4 + l ) (n + r) is a necessary and sufficient condition for the existence of the proposed tracking controller. Although checking the rank of matrix Q by hand is tedious, it is a matter of writing a simple program when a computer is used. If r = m, then the matrix Q is a square matrix, and hence the solution is unique. If r < m then the solution is not unique.

When 4 = 0, the necessary and sufficient condition becomes

- I I 0 rank(Q) = r a n k [ ;I A B ] = ( n + r ) + n.

C O Now

rank - I A B = r a n k 0 A - I B . lo' : :l lo' 1 :I

Since rank ( I ) = n, the rank of the above matrix is ( n + r) + n if and only if

Thus, the above condition is also a necessary condition for the existence of the proposed tracking controller.

Because (12) is a linear equation, the solution is in the form 4

x * ( k ) = x F , y * ( k - i) (15)

(16)

r = O

4

u * ( k ) = 1 G , y * ( k - i )

where F, and G, are constant matrices. Indeed, the solution for x* is in the form assumed in (8). In case the solution is not unique, it is advantageous to find all solutions first and then select the set which is most convenient for actual implementation.

The tracking controller is now given by

U( t ) = U * ( t ) - K ( x ( t) - x*( t ) )

r = O

= 5 { G l y * ( t - i) +KF,y*(t - i)} - K x ( t ) . r = O

4

= 1 H r y * ( t - i) - Kx( t ) . (17) r = O

where HI = GI + KF,, i = 0, l ; . . , 4. The control given by (17) is a tracking controller for any feed-

back gain K which stabilizes the given system (S), that is U = - Kx stabilizes the system. Thus, as long as the given system is stabilized, it could be a partial state, or even an output, feedback control as well.

VI. EXAMPLE

Consider the system shown in Fig. 1 . Let

x , = y , , x 2 = y 2 , x 3 = x , , x 4 = x * >

The problem is to design a digital tracking controller for the system which will track any ramp reference input y* with zero

1202 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 36, NO. 10. OCTOBER 1991

- - 1

0 0 0 0 0 0 0 0 0 0 0 1 0 0

- 0

steady state error. Suppose the sampling time for the digital control is 0.1 s. Then the discrete system equation is given by

4.9958e-3 4.1639e-6 4.1639e-6 4.9958e-3 9.9833e-2 1.6650e-4 1.6650e-4 9.9833e-2

9.9501e-1 4.9917e-3 9.9833e-2 1.6650e-4 4.9917e-3 9.9551e-1 1.6650e-4 9.9833e-2

- 9.9667e-2 9.9667e-2 9.9501e- 1 4.9917e -3 9.9667e-2 - 9.9667e-2 4.99 17e-3 9.950 1 e - 1

Equation (12) for the system is

0 0 0 1 0 0 0 - 1 0 0 0 - 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1

0 0 0 0

9.9501e-1 4.9917e-3 9.9667e-2 9.9667e-2

- 1 0 0 0 1 0 0 0 2 0 0 0

0 0 0 0

4.9917e-3 9.9501e-1 9.9667e-2

- 9.9667e-2 0

- 1 0 0 0 1 0 0 0 2 0 0

0 0 0 0

9.9833e-2 1.6650e-4 9.9501 e- 1 4.9917e-3

0 0

- 1 0 0 0 0 0 0 0 2 0

0 0 0 0

1.6650e-4 9.9833e-2 4.99 17e- 3 9.9501 e- 1

0 0 0

- 1 0 0 0 0 0 0 0

- 2

4.9958e-3 4.1639e-6 0 0 4.1639e-6 4.9958e-3 0 0 9.9833e-2 1.6650e-4 0 0 1.6650e-4 9.9833e-2 0 0

9.9501e-1 4.9917e-3 9.9833e-2 1.6650e-4 0 0 4.9958 e- 3 4.1639 e-6 4.9917e-3 9.9501e-1 1.6650e-4 9.9833e-2 0 0 4.1639e-6 4.9958e-3

-9.9667e-2 9.9667e-2 9.9501e-1 4.9917e-3 0 0 9.9833e-2 1.6650e-4 9.9667e-2 -9.9667e-2 4.9917e-3 9.9501e-1 0 0 1.6650e-4 9.9833e-2

0 0 0 0 0 0 0 0

YT(k Y,*(k

YT(k - 1)

Y a k - 1) 0 0 0 0

1203 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 36, NO. 10, OCTOBER 1991

ME1 u1,

00 Fig. 1 , Two-input two-output control system.

and the solution is given by

9.9917 0.0834 0.0834 9.9917

r o 0 1 - 1 9.90917 0.0834 O lY*(k - 1 )

L0.0834 9.9917 1

To find the stabilizing feedback gain matrix, let y* = 0, and consider the quadratic cost functional

m

J = 2 { x T ( k ) Q x ( k ) + u ‘ ( k ) R u ( k ) } . r = O

For the matrices,

the optimal feedback gain matrix K for the system is given by

0.4363 0.4486 0.1525 2.0615 1 ’ 0.4486 0.4363 2.0615 0.1525 K = [ For the above feedback gain, the tracking controller is given by

U( k ) = U*( k ) + K ( x*( k ) - X( k ) )

22.5477 0.4776]y*(k) = [ 0.4776 22.5477

- [21.0992 I .0413]y*(k - 1 ) 1.0413 21.0922

- 0.4486 0.4363 2.0615 0.1525 [ 0.4363 0.4486 0.1525 2.0615]x(‘)’

Several step and ramp responses of the resulting feedforward system are shown in Fig. 2.

CONCLUSIONS

A new digital tracking controller is presented for linear time invariant control systems. It is based on feeding forward delayed reference input. A necessary and sufficient condition for the exis- tence of the proposed controller is also given. The choice between the proposed feedforward tracking controller and the one based on error integral augmentation depends on the characteristics and re- quirements of the system under consideration. In certain cases, a better result can be achieved by employing a combination of both methods.

REFERENCES

[l] B. Porter and A. Bradshaw, “Design of linear multivariable continu- ous-time tracking systems,” Int . J . Syst. Sci., vol. 5, no. 12, pp. 1155-1164, 1974.

1

I I L - 0 2 4 6 8 10 12 14 16 18 20

(a) 2 1

1 5

oc

0 2 4 6 8 10 12 14 16 18 20

(C) 30 1 25 ~

20 -

15 -

1

5 5 15 10 20 25

(d) Fig. 2. System responses. (a) y:(k) = 0, y ; (k ) = 1. (b) y: (k) = -1, y ; ( k ) = 1. (c) y r ( k ) = 1, y ; (k ) = 2. (d) y f ( k ) = 0.1 k, $(k) = 5 + 0.1 k.

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 36, NO. 10, OCTOBER 1991

A. Bradshaw and B. Porter, “Design of linear multivariable discrete- time tracking systems,” Int. J . Syst. Sci. , vol. 6, no. 2, pp. 117-125, 1975. E. J. Davison, “The robust control of a servomechanism problem for linear time-invariant multivariable systems,” IEEE Trans. Au- tomat. Contr., vol. AC-21, pp. 25-34, Feb. 1976. C.-T. Chen, Linear Systems Theory and Design. New York: Holt, Rinehart. and Winston, 1984. G. F. Franklin, J. D. Powell, and A. Emami-Naeini, Feedback Control of Dynamic Systems, 2nd ed. Reading, MA: Addison- Wesley, 1988.

H , Controller Design for the SISO Case Using a Wiener Approach

D. Fragopoulos, M. J. Grimble, and U. Shaked

Abstract-A solution to the H , mixed sensitivity problem for the SlSO case is obtained using a Wiener approach to parameterize all equalizing and stabiliizing controllers. The controller which incorporates the LQG solution has a structure similar to that of Youla et al. The system of equations thus obtained is square and has some degree advantage over previious solutions.

I. INTRODUCTION

Polynomial solutions of the H, control problem have been developed in continuous-time by Kwakernaak, [5] and in discrete- time by Grimble [3]. The latter approach was based on a lemma ( [ 5 , Lemma 31, which ‘converts the H, problem into an L, minimiza- tion problem. However, in some special cases (nongeneric) a link between the H, and L, problems cannot be established as noted by Kwakernaak. In thiis special case, the solution procedure developed in [3] needs to be modified but the basic form of the equations is virtually unchanged.

The H, solutioii presented here is for both the generic and the nongeneric cases, in discrete-time for single-input single-output systems. A transfer-function formulation is used, from which subse- quent polynomial equations are obtained. The alternative set of equations obtained can be manipulated to give the equations ob- tained in [3].

Notes on the numerical solution of the equations are included. This has previously been considered by Saeki [6] and by Brown [ 11.

11. SYSTEM DESCRIITION

The discrete-tim single-input single-output system under consid- eration is shown in Fig. 1 . The system output y ( t ) is to be regulated against the effect of the disturbance d( t ) . Here .$ is an H , bounded signal. Also A , B , C are polynomials in the delay operator z - I .

Assumptions I: 1) The polynom.ia1 C should not have roots on the unit circle.

Without loss of generality C can be assumed to be a stable polynomial and also C(0) # 0. If it is not it can be replaced by the stable factor of C*C. Thus C will be assumed to be strictly stable.

Manuscript received January 5 , 1990; revised August 31, 1990. D. Fragopoulos arid M. J. Grimble are with the Industrial Control Unit,

University of Strathclyde Marland House Glasgow, England, G1 IQE. U. Shaked is with the Department of Elecironic Communications, Con-

trol, and Computer Systems, School of Engineering, Tel-Aviv University, Ramat-Aviv, Tel Aviv 69978, Israel.

IEEE Log Number 9101753.

Fig. 1. The system.

2 ) The plant W = A ’ B is stabilizable and has at least one pure

3) All the blocks in Fig. 1 are causal systems. The controller is denoted by CO := C;:C,, with CO,, CO, are

delay.

polynomial in z - I . Let

s := ( 1 + WC,) - I sensitivity

M : = COS control sensitivity.

The disturbance power spectrum is written as +d,,. Let the con- troller input e := - y = - Sd have a power spectral density ace := S+,,S*; the control signal U := -Md, and the control signal power spectrum +uu := M+ddM*. Here S* denotes the adjoint of s: S*( z - I ) = S( z ) .

111. THE H, OPTIMAL CONTROL PROBLEM

A . Cost Function

The following cost function is to be minimized:

J,= sup I x ( z - l ) ( = \ l X ( z - ’ ) \ l , (1)

( 2 )

/ i I = I

where

X ( Z - ’ ) := Qaee + RauU

and the weighting transfer functions are defined as

(3)

Note the the above cost function is equivalent to the usual mixed sensitivity problem [5] .

Assumptions 2: 1 ) The weightings A , , A , , B,, B, are stable polynomials in

2 ) The polynomial ( B , BA,)*( B,BAr) + ( B , A,)*( B, A q ) has Z - 1 .

no roots on the unit circle.

B. Stabilizing Solutions

As a preparatory stage for the parameterization of all stabilizing terms of the H2 optimal solution.

The H2 solution minimizing

J 2 = 6

H, optimal control problem a controllers will be obtained in

dz X - (4)

J 1 i 1 = 1 z

is given by Theorem I . Theorem 1: Define stable polynomial factor D,

D,*D, = ( A B , A , ) * ( A B , A , ) + ( B B ~ A , ) * ( B B , A , ) . ( 5 )

0018-9286/9111000-1204$01.00 0 1991 IEEE