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180 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 2, FEBRUARY 2000

A New Approach to the Solution of the ` 

1 

ControlProblem: The Scaled- Q  Method

Mustafa Khammash

 Abstract—In this paper, we explore a new approach for solvingmultiple input–multiple output (MIMO)

1 

optimal controlproblems. This approach, which we refer to as the scaled-approach, is introduced to alleviate many of the difficulties facingthe numerical solution of optimal

1 

control problems. In partic-ular, the computations of multivariable zeros and their directionsare no longer required. The scaled- method also avoids thepole-zero cancellation difficulties that existing methods based onzero-interpolation face when attempting to recover the optimalcontroller from an optimal closed-loop map. Because the scaled-approach is based on solving a regularized auxiliary problem forwhich the solution is always guaranteed to exist, it can be usedno matter where the system zeros are (including the stabilityboundary). Upper and lower bounds that converge to the optimalcost are provided, and all solutions are based on finite-dimensionallinear programming for which efficient software exists.

 IndexTerms—Discrete-time systems,1 

control, optimal control,robust control.

I. INTRODUCTION

THE optimal control problem addresses the design of 

optimal controllers that minimize the peak errors caused

by unknown bounded disturbances. The mathematical formu-

lation of this objective requires the design of a linear time-in-

variant (LTI) controller that minimizes the induced norm of 

the transfer function relating the disturbances to the error signals

of interest (or any other output to be regulated). Because the in-

duced normof a transfer function corresponds to the norm

of the impulse response, the resulting optimization problem is a

norm minimization problem.

The norm minimization problem was posed by Vidyasagar

[1] along similar lines to the problem. This problem was

also addressedin [2]. The first general solution of the problem

based on linear programs was provided by Dahleh and Pearson

[3]. This work utilized duality theory to reduce the infinite-di-

mensional problem to a finite-dimensional linear program-

ming problem, in the one block case. In [4], Mendlovitz showed,

that once the dual problem indicated the solution is finite-di-

mensional, the primal problem can be used directly to obtain thesolution. In [5], and later in [6] and [7], approximate solutions

to the multiblock problem were proposed. These methods are

based on characterizing the closed-loop maps using zero inter-

polation conditions obtained from the Smith–McMillan form.

Manuscript received October 23, 1997; revised August 19, 1998. Recom-mendedby AssociateEditor, E. Feron.This work wassupported by theNationalScience Foundation under Grant ECS-9457485.

The author is with the Department of Electrical and Computer Engineering,Iowa State University, Ames,IA 50011 USA (e-mail: Khammash@iastate.edu).

Publisher Item Identifier S 0018-9286(00)01061-8.

The work reported in [6] and [7] also addressed the issue of 

redundancy of constraints originating from the zero interpola-

tion constraints. Up to this point, the methods available pro-

vided only converging upper bounds to test for optimality. Du-

ality notions were first used to obtain converging lower bounds

that can be used in combination with upper bounds to guar-

antee nearness to optimality in [8] and [9]. The approach giving

rise to converging upper bounds to the optimum was termed

the finitely many variables (FMV) approach, whereas that pro-

viding lower bounds was termed the finitely many equations

(FME) approach. In [10] and [11], a geometric approach based

on dynamic programming was proposed. The delay augmen-tation method for the problem was proposed by Diaz–Bo-

billo and Dahleh in [12] to obtain further information about the

structure of the optimal controller, hence avoiding unnecessarily

high-order controllers.

In the approaches mentioned above based on linear program-

ming, computation of zeros and zeros directions is used to place

interpolation conditions on the closed-loop transfer function.

Aside from the issue of constraint redundancy, computation of 

the zero directions can cause numerical difficulties, especially

when certain eigenvalues are close to the unit circle [12].

Perhaps the most serious drawback of utilizing zero interpo-

lation conditions is in recovering the optimal controller once

the optimal closed-loop function has been constructed. Thisclosed-loop function must satisfy the interpolation conditions

exactly to ensure that the appropriate cancellations take place

when solving for the controller. Given that equality constraints

in linear programs are only satisfied up to certain tolerances,

the task of recovering the optimal controller is complicated by

the need to determine which nearby poles and zeros should

cancel each other.

Recently, two new methods that avoid zero interpolation alto-

gether have been introduced in [14]and[15]. The method in[14]

is based on solving a standard problem and a sequence of 

finite-dimensional semidefinite quadratic programming prob-

lems. In this paper, the scaled- approach is presented and its

features explored. This method is motivated by our numericalexperience, which indicates that solving the problem by di-

rectly approximating the optimal parameter has nice numer-

ical properties, even in the presence of zeros on the stability

boundary. In principle, this method is not unlike the design

approach in [16]. We can easily show that, by truncating the

parameter in the problem, finite linear programs whose op-

timal solution converges to the optimal norm are obtained.

By far, the biggest shortcoming of truncation approaches has

been the lack of converging lower bounds that indicate how

close the converging upper bounds are to the optimal solution.

0018–9286/00$10.00 © 2000 IEEE

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186 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 2, FEBRUARY 2000

Fig. 2. Example problem.

Fig. 3. Upper and lower bound plots versus N  for various values of    .

10, 100, and , all of which can be shown to be upper

bounds for some optimal . It is clearly seen from the

figure that the upper and lower bounds converge to , whose

value is 71.1146. It can also be seen that in this case the upper

bounds do not change for the selected values of . It is also

interesting to observe that, although the lower bounds

depend on the size of (with lower values giving faster con-

vergence), even bad upper bound estimates for , which

are reflected by large , still give reasonably fast convergence

for the lower bounds. Indeed, when is used in this

example, errors between and of less than 1% are

observed for . The desirable convergence properties

observed here, and the extent to which they apply in general,warrant further investigation.

  A. Controller Order 

We now address the issue of controller order. Using an op-

timal from any of the upper bounds, we can construct a con-

troller that yields the performance level corresponding to that

upper bound. Such is finite impule response (FIR) and has

length equal to . As is increased, the upper bound ap-

proaches the optimal value , and in the process, the length

of the resulting will be increased as well, which does not nec-

essarily lead to large-order controllers. In fact, in this example,

as in several others solved using this approach, the long FIR

TABLE ITABLE SHOWING THE APPROXIMATION ERROR FOR SEVERAL RATIONAL

APPROXIMATIONS TO THE 29th-ORDER OPTIMAL FIR SOLUTION Q 

CORRESPONDING TO  

can be well approximated by a much lower order rational stable

system. To demonstrate, for = 30, the FIR obtained from

the upper bound has length 30 and, hence, a McMillan degree

of 29. Using the Hankel SVD method proposed by Kung [im-

plemented in MATLAB imp2ss() command], however, ex-

cellent low-order rational approximations have been obtained.

These approximations are given in Table I. The order of the con-

troller itself will be equal to the order of the used plus four.

Of course, the error in the closed-loop norm resulting from the

approximation error in can be no larger than (and typically

much less than) times the norm of the approxima-tion error of .

VIII. CONCLUSIONS

In this paper we have introduced a new approach to solving

the control problem. Like many other solutions to the

problem, the underlying solution utilizes finite-dimensional

linear programming as a method for computation. The solution

is obtained through introducing an auxiliary problem with

desirable properties and then relating its solution to that of 

the problem. The auxiliary problem can be viewed as a

regularization of the problem characterized by introducing

the parameter directly in the optimization and by introducinga scaled norm of in the objective (or, equivalently, by adding

to the constraints a norm constraint). This “regularization” of 

the problem in combination with the fact that the parameter

variables are explicitly included in the optimization problem

leads to a number of desirable features that characterize the

auxiliary problem. First, it is possible to solve the problem

without the need for computing system zeros or including

zero interpolation conditions at all. Such conditions can, and

frequently do, complicate the numerical computations, both

in the construction of the linear program constraints and in

the controller recovery stage after the solution is obtained.

Another feature of the proposed approach is that the auxiliary

problem always has a solution, regardless of the location of system zeros. In fact, in cases in which unit circle zeros are

present, solving the auxiliary problem is the best approach to

take because solutions that get arbitrarily close to the optimal

index often result in parameters with arbitrarily large norms,

a situation avoided by the auxiliary problem setup. A third

feature of the auxiliary problem is one whose absence in

FIR -approximation methods motivated this whole research

direction, namely, obtaining lower bounds to the solution that

converge to the optimal index. Using ideas from duality theory,

it has been shown that we can indeed obtain such bounds for

the auxiliary problem. Furthermore, it has been shown that such

bounds can be obtained by appropriately truncating constraints

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KHAMMASH: A NEW APPROACH TO THE SOLUTION OF THE CONTROL PROBLEM: THE SCALED- METHOD 187

from the auxiliary problem. Such procedure simply does not

yield any results if applied to the original problem formulated

for FIR approximation.

In the proposed approach, it was suggested that one way to

try avoid large-order controllers is to approximate a long FIR

that is close to optimal with a low-order rational . The

reason this approach often yields very good results is that FIR

sequences that are obtained from solving an upper boundproblem are approaching a rational optimal solution . One

problem that remains open is how to arrive at an optimal rational

directly (when one exists) without attempting to reconstruct

it from FIR sequences, while avoiding zero interpolaton. An-

other problem of interest is to generalize these ideas for prob-

lems in which multiple objectives are of interest, including pos-

sibly other norms in addition to the norm. Some initial re-

sults along these line have been obtained [18], although much

remains to be explored.

REFERENCES

[1] M. Vidyasagar, “Optimal rejection of persistent bounded disturbances,”  IEEE Trans. Automat. Contr., vol. AC-31, June 1986.

[2] A. A. Barabanov and O. N. Granichin, “An optimal controller for linearplant with bounded noises,” Automat. and Remote Contr., vol. 5, pp.41–46, 1984.

[3] M. A. Dahleh and J. B. Pearson, “  ̀ optimal feedback controllersfor MIMO discrete-time systems,” IEEE Trans. Automat. Contr., vol.AC-32, Apr. 1987.

[4] M. A. Mendlovitz, “A simple solution to the  ̀ optimization problem,”Syst. Contr. Lett., vol. 12, 1989.

[5] M. A. Dahleh and J. B. Pearson, “Optimal rejection of persistent dis-turbances, robust stability, and mixed sensitivity minimization,” IEEE Trans. Automat. Contr., Aug. 1988.

[6] J. S. McDonald and J. B. Pearson, “̀  optimal control of multivariablesystems with output norm constraints,” Automatica, vol. 27, Mar. 1991.

[7] O. J. Staffans, “On the four-block model matching problem in  ̀ ,”Helsinki Univ. of Technol., Espoo, Finland, 1990.

[8] , “Mixed sensitivity minimization problems with rational  ̀ op-timal solutions,” J. Optim. Theory Applicat., vol. 70, 1991.

[9] M. A. Dahleh, “BIBO stability robustness in the presence of coprimefactor perturbations,” IEEE Trans. Automat. Contr., vol. 37, Mar. 1992.

[10] A. E. Barabanov and A. A. Sokolov, “Minimax controllers for linearplants under  ̀ -bounded disturbances,” in Proc. Europ. Contr. Conf.,vol. 2, pp. 754–759.

[11] , “Geometrical solutions to  ̀ -optimization problem with com-bined conditions,” in Proc. Asian Contr. Conf., vol. 3, pp. 331–334.

[12] I. J. Diaz-Bobillo and M. A. Dahleh, “Minimization of the maximumpeak-to-peak gain: The general multiblock problem,” IEEE Trans. Au-

tomat. Contr., vol. 38, Oct. 1993.[13] M. A. Dahleh and I. J. Diaz-Bobillo, Control of Uncertain Systems: A  Linear Programming Approach. Englewood Cliffs, NJ: Prentice-Hall,1995.

[14] N. Elia and M. A. Dahleh, “A quadratic programming approach forsolving the  ̀ multi-block problem,” in Proc. 1996 Conf. Decision and Contr., Kobe, Japan, 1996, pp. 4028–4033.

[15] M.Khammash, “Solutionof the  ̀ optimal control problem without zerointerpolation,” in Proc. 1996 Conf. Decision Contr., Kobe, Japan, 1996,pp. 4040–4045.

[16] S. Boyd and C. Barratt, Linear Controller Design: Limits of Perfor-mance. Englewood Cliffs, NJ: Prentice-Hall, 1991.

[17] D. J. Luenberger, Optimization by Vector Space Methods. New York:Wiley, 1969.

[18] M. Salapaka, M. Khammash, and M. Dahleh, “Solution of MIMO H  ̀

problem without zero interpolation,” SIAM J. Optimiz. Contr., vol. 37,no. 6, pp. 1865–1873, 1999.

Mustafa Khammash received the B.S. degree(magna cum laude) from Texas A&M University,College Station, in 1986 and the Ph.D. degreefrom Rice University, in 1990, both in electricalengineering.

He has been with the Electrical and ComputerEngineering Department at Iowa State University,Ames, since 1990, where he currently holds thepostion of Associate Professor. His research interestsinclude linear and nonlinear robust control, systemtheory, power systems stability and control, flight

control, and feedback control in biological systems.Dr. Khammash received the National Science Foundation Young Investigator

Award, the ISU Foundation Early Achievement in Research and ScholarshipAward, and the Ralph Budd Best Engineering Ph.D. Thesis Award. He is anAssociate Editor of  Systems and Control Letters.