Diophantine equations in control—A survey

Amomatica, Vol. 29, No. 6, pp. 1361-1375, 1993

{XXgi-1098/93 $6.(1(I+ 0.(X) ~ 1993 Pergamon Press Ltd

Printed in Great Britain.

Survey Paper

Diophantine Equations in Control A Survey*t VLADIMIR

KU~ERA~:

Fractional representations are a useful algebraic tool for control system design. They result in diophantine equations for the controller. All desirable controllers are obtained in parametric form. Key Words---Algebraic system theory; control system synthesis; feedback control; linear systems; polynomials; transfer functions.

that evolve in discrete time. The correspondence of algebraic and time-domain quantities is particularly simple and transparent in this case, and can be captured by polynomial rings. The first attempts to use polynomial equations in the design of control systems date back to Volgin (1962), Strejc (1967), A,str6m (1970) and Peterka (1972). The algebraic background of these polynomial manipulations was explained by K u ~ r a (1973, 1974a, 1978b) and the polynomial equation approach to the analysis and synthesis of discrete-time linear control systems was fully developed in Ku~era (1979). This approach was extended to cover continuoustime systems by Hautus (1975), Pernebo (1981), Callier and Desoer (1982), Ku~era (1983a, 1986a, 1986b) and especially by Vidyasagar (1985). A particular impetus was provided by the descriptor (or semi-state, or implicit or singular) linear systems, which can easily be studied by means of transfer functions and which have brought attention to the impulsive behaviour of linear systems (Verghese, L6vy and Kailath, 1981; Lewis, 1986). In the algebraic setting, the corresponding notion of properness can be handled in a way that is similar to handling stability in linear systems (Ku~ra, 1984). Although this approach was originally applied to solve the very simplest control problems like stabilization and pole placement, it was gradually extended and found useful in solving a broad variety of problems. These include dead-beat control (Ku~era, 1980a; K u ~ r a and .~ebek, 1984b), model matching (A,str6m, 1980; Kui~era, 1981), disturbance rejection (Ku/~era, 1983b; Sternad, 1987), minimum variance control (/~str6m, 1970; Peterka, 1972), shortest correlation control strategy (Ku~era, 1977), LQG or HE optimal control (Kui~era, 1979, 1980b, 1986a; Hunt, Sebek and Grimble, 1987; Hunt, 1989; Mosca, Giarr6 and Casavola, 1990; Hunt and Ku6era, 1992), H~ optimization (Kwakernaak, 1985, 1991; Grimble, 1986) and several types of tracking problem (KuiSera, 1980a; Sebek, 1982; Ku~era and Sebek, 1984a, b). Self-tuning and predictive control problems have also been addressed using this approach (.~strOm and Wittenmark, 1973; Clarke and Gawthrop, 1975; Peterka, 1984; Grimble, 1984; Sternad, 1987; Hunt, 1989), and it is most interesting to see these methods

Abstract--This survey is also a tutorial whose aim is to explain the role of diophantine equations in the synthesis of feedback control systems. These are linear equations in a ring and result from a fractional representation of the systems involved. The cornerstone of the exposition is a simple parametrization of all stabilizing controllers for a given plant. One can then choose, in principle, the best controllers for various applications. These ideas evolved from early attempts to use polynomial equations in the design of discrete-time linear systems. By now they have been extended to continuous-time, infinitedimensional, time-varying and non-linear systems. INTRODUCTION THE FEATURES of modern control theory is the growing presence of algebra. Polynomial modules were found to be useful in describing the dynamics and structure of linear systems (Kalman, Falb and Arbib, 1969; Rosenbrock, 1970; Hautus and Heymann, 1978; Blomberg and Ylinen, 1983; Fliess, 1990). In contrast to other approaches, in particular those based on state space (Kalman, 1963) and trajectory behaviour (Willems, 1986, 1989), these results revived interest in fractional representations of transfer functions. In this approach, the transfer function of a system is regarded as an element of the field of fractions associated with an appropriate ring, depending on the property of the system under study (Desoer, Liu, Murray and Saeks, 1980). The property then corresponds to a divisibility condition in the ring. As a result, the mathematical synthesis of a control system having a desired property leads to the solution of a linear diophantine equation over that ring. Quite naturally, these ideas originated in the realm of linear finite-dimensional and time-invariant systems ONE

OF

*Received 23 April 1992; revised 27 January 1993; received in final form 16 April 1993. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Editor K. J. /~str6m. t The original version of this paper was presented at the First European Control Conference, Grenoble, France, 2-5 July 1991. ~:Institute of Information Theory and Automation, Academy of Sciences, P.O. Box 18, 182 08 Prague, Czech Republic. Tel. (42) 2 6641 4669; Fax (42) 2 6641 4903; Telex 12 20 18 atom c. e-mail [email protected] 1361

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V. KU~ERA

working also in fields related to control, namely in signal processing (Ku~era, 1978a; Grimble, 1985, 1991; Roberts and Newmann, 1988; Ahl6n and Sternad, 1989, 1991; Sebek, 1992). Another step forward in extending the algebraic approach was the case of time-varying linear systems. The field of coefficients is replaced by a differential ring of functions over an interval of time (Kamen, Khargonekar and Poola, 1985; Je~ek and Nagy, 1989). The resulting rings are no longer commutative, for multiplication and differentiation do not commute. Nevertheless, the same master ideas can be used to reduce the mathematical synthesis of control systems to the solution of linear diophantine equations (Je~ek and Nagy, 1989). Some control strategies, when applied over a finite horizon, lead to time-varying control laws no matter whether the properties of the underlying system vary in time or not. So these control problems naturally fall within the same framework. A more delicate extension of this algebraic approach is towards infinite-dimensional linear systems. Transfer functions of such systems belong to fields of fractions over more general rings (Kamen, 1975) in which suitable factorizations may not exist (Callier and Desoer, 1978; Vidyasagar, Schneider and Francis, 1982). If they do exist, however, one can apply the algebraic approach to advantage (Vidyasagar, 1985). New control strategies for distributed systems (Callier and Desoer, 1980; Je~ek, 1989) and, in particular, for time-delay systems (Morse, 1976; Kamen, Khargonekar and Tannenbaum, 1985; Sebek, 1987) were formulated and solved in this way. Another class of systems that can be studied by introducing more general rings are n-D systems, or systems whose behaviour evolves in a vector-valued time set (Kaczorek, 1985). The diophantine equations over the ring of polynomials in n indeterminates were studied by Sebek (1988). Perhaps the most surprising feature of the fractional approach is that it is not restricted to linear systems. In fact, suitable fractions for the input-output maps of non-linear systems were defined by Hammer (1987) and used to solve some control problems of practical importance, including that of stabilization (Hammer 1986; Paice and Moore, 1990). These results clearly demonstrate that it is the existence of appropriate fractions rather than the linearity of the underlying system that enables application of the diophantine equation approach to the synthesis of control systems. On the other hand, the resulting diophantine equations are always linear because the part of the control system to be designed, while entering the overall transfer function in a non-linear fashion, enters linearly its numerator and/or denominator. This paper is a tutorial whose aim is to explain the fundamentals and the rationale of the algebraic approach and guide the reader through various examples of control system analysis and synthesis. The emphasis is on the ideas, not on the technicalities. The paper will hopefully motivate the reader to pursue the details in the references and help him or

her to master the power and appreciate the elegance of the algebraic methods in control theory. The actual design of control systems, in contrast to mathematical synthesis, is an engineering task that cannot be reduced to algebra. Design contains many additional aspects which have to be taken into account: sensor placement, computational constraints, actuator constraints, redundancy, performance robustness, among many others. There is a need for understanding of the controlled process, a feeling for what kinds of performance objectives are unrealistic, or even dangerous, to ask for. Some of the issues involved are exposed in the recent books by ,~str6m and Wittenmark (1984), Maciejowski (1989) and Boyd and Barratt (1991). The fractional representations and diophantine equations, nevertheless, are a very useful tool for controller design.

ALGEBRAIC PRELIMINARIES We explain briefly some notions from algebra which will be used throughout the paper. For a rigorous and comprehensive treatment the reader is referred to Jacobson (1953) or Zariski and Samuel (1958). The appendices in Vidyasagar (1985) make a good introduction to the subject. A ring is a set equipped with two operations, addition and multiplication, that forms a commutative group with respect to addition, a semigroup with respect to multiplication and satisfies distributive laws connecting the two operations. Loosely speaking, one can add, subtract and multiply any two elements of a ring; moreover, the addition is commutative. If the multiplication is commutative, the ring is called commutative. If the ring has an identity, then its elements having a muitiplicative inverse are called units of the ring. A ring in which every non-zero element is a unit is called a field. Examples of fields: real numbers R, rational functions R(q) in the indeterminate q over R. Examples of rings: integers Z, rational functions from R(q) that are analytic in a subset of the complex plane, such as proper rational functions Rp(q), analytic at q -- ~; proper and Hurwitz-stable rational functions RH(q), analytic in Re q -> 0 including q = oo; proper and Schur-stable rational functions Rs(q), analytic in Iql --- 1; proper and finite-expansion rational functions Rf(q), analytic for every q :# 0 (polynomials in q- 1); polynomials R[q] in q, analytic for every q 4: 0o. The units of Rp(q) are the rational functions of relative degree zero, those of R.(q) and Rs(q) are minimum-phase rational functions and those of Rr(q) and R[q] are non-zero constants. An element A of Rp(q) is said to be strictly proper if A ( ~ ) = 0.

A ring is said to be a domain if the product of every pair of non-zero elements is non-zero. A subset of a ring is said to be an ideal if it is a group with respect to addition and is closed with respect to multiplication by elements of the ring. An ideal is principal if it is generated by multiples of a

Diophantine equations in control single element. A domain in which every ideal is principal is a principal ideal domain; one in which every finitely generated ideal is principal is called a Bezout domain. We note that our example rings are all commutative and principal ideal domains. The linear equation A X = B in a commutative ring has a solution X if and only if the ideal generated by A contains that generated by B, or A divides B in that ring. An equation in X and Y of the type AX + BY = C is called the diophantine equation, and it is solvable in a principal ideal (or even a Bezout) domain if and only if every common divisor of A and B divides C. If X ' , Y' is a solution pair of this equation, then X = X ' + BW,

Y -- Y' - A W

is also a solution for an arbitrary element W of the ring (Ku~era, 1979). Two elements of a commutative ring with identity are relatively prime if their only common divisors are units of the ring. For A and B relatively prime, the above formula for X and Y generates all solution pairs of the diophantine equation. SYSTEMS AND STABILITY The algebraic approach to be presented is based on the input-output properties of systems. We shall represent the systems by their input-output maps; in particular, linear time-invariant systems will be described by transfer functions. The fundamentals of the algebraic approach will be explained for causal linear systems with rational transfer functions and whose input u and output y are scalar quantities. We suppose that u and y live in a space of functions mapping a time set into a value set. The time set is a subset of real numbers bounded on the left, say R+ (the non-negative reals) in the case of continuous-time systems and Z+ (the non-negative integers) for discrete-time systems. The value set is taken to be the field of real numbers R; later we shall generalize. The transfer function of a continuous-time system is the Laplace transform of its impulse response g(t), G(s) =

g(t)e-" dt.

For discrete-time systems, the transfer function is defined as the z-transform of its unit pulse response

(he, h, ....

),

H(z) = ~ hiz-i, i=o and it is always proper. Systems having the desirable property that a "bounded input" produces a "bounded output" are usually called stable. More precisely, let the input and output spaces of a continuous-time system be the spaces of locally (Lebesgue) integrable functions f from R + into R and define a norm

Ilfll® = ess sup If(t)l. t~eO

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The corresponding normed space is denoted by L ®. Then the system is said to be bounded-input bounded-output (BIBO) stable if any input u e L ® produces an output y • L ®. It is a well-known fact (Desoer and Vidyasagar, 1975) that a system with a rational transfer function G(s) is BIBO-stable if and only if G(s) is proper and Hurwitz-stable, i.e. belongs to Rn(s). In the study of discrete-time systems, we let the input and output spaces be the spaces of infinite sequences f = (fo, fl . . . . ) in R and define a norm Ilfll® = sup Ifd. i=0,1 ....

The corresponding normed space is denoted by l ®. Then a system is said to be BIBO-stable if any input u • l ® gives rise to an output y e l ®. As shown in Desoer and Vidyasagar (1975), a system with a proper rational transfer function H ( z ) is BIBO-stable if and only if H ( z ) is Schur-stable, i.e. belongs to Rs(z). FRACTIONAL DESCRIPTION Consider a rational transfer function G(s); it is an element of the field R(s). This field is the field of fractions associated with several rings, including all the example rings introduced above. Hence one can write B

G=-A

where A and B belong to one of the example rings. When A and B are relatively prime, the fractional representation of G is unique up to the units of the ring. The choice of the ring can be matched with the ultimate goal of our investigations (Desoer, Liu, Murray and Saeks, 1980). In particular, when we are going to study a certain property of the system we can choose the underlying ring so that all systems having this property will give rise to transfer functions residing in that ring. Through this choice the irrelevant information evaporates from A and B and the investigations will be more transparent. With this in mind, the example rings can be used naturally in studying the following properties. Rp(s), the impulse-free behaviour of continuous-time systems; Rp(z), the causality of discrete-time systems; RH(s) and Rs(Z), the BIBO-stability of continuous-time and discrete-time systems; Rf(z), the finite impulse response in discrete-time systems; R[s] and R[z], the modal properties (pole placement) of continuous-time and discrete-time systems. For example, neither an integrator with the transfer function 1(s) =-

1 S

nor a differentiator with the transfer function D(s) =s is a BIBO-stable system: one can write them in terms

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iV. KU(~ERA

of RH(s) as follows

from ul, u2 to y~, Y2

t(s) =

s+a

,

b D(s) = "~

S S

s-ka

where a > 0 and b are some real numbers. A specific feature of discrete-time systems results from the inclusion Rf(z) c Rs(z). We observe that the ring Rf(z) of rational functions analytic outside z = 0 is isomorphic with the ring of polynomials in the delay operator z -j, denoted by R[z-l]. In fact, polynomials in z -j have been used in the analysis and design of discrete-time systems since the early days (Tou, 1959; Voigin, 1962; Strejc, 1967; Ku~era, 1979, 1991), for instance b z

b z+a

z+a

-BY Y ' = A X + B Y u2

-BY Y ~ = A X + B Y u~'

Y2

-AY A X + B Y u2

(of which two are the same) reside in Rn(s) or Rs(z), respectively. This is the case if and only if the common denominator A X + B Y is a unit of Rn(s) or Rs(z). In other words, 1 / ( A X + B Y ) should be proper and stable. We illustrate with the example where Sj is a differentiator and $2 is an invertor such that T~ = s,

T2 = - 1 .

We can take A=

1

s+a'

B=

S

s+a'

X=I,

Y=I

for any real a > 0. Then bz I

AX + BY =

l+az-"

Z

Now we better understand why: the polynomials in z -~ are in fact special, proper and Schur-stable rational functions in z and hence suitable for the study of finite as well as stable and causal processes (Ku~era, 1991). FEEDBACK SYSTEMS To control a system means to alter its dynamics so that a desired behaviour is obtained. This can be done through feedback. A typical feedback system consists of two subsystems, S~ and Sz, connected as shown in Fig. 1. In many applications, it is desirable that the feedback system be internally BIBO-stable in the sense that whenever the exogenous inputs u, and u2 are bounded in magnitude so too are the internal signals y, and Y2 (Desoer and Chan, 1975). In order to study this property, we express the transfer functions of S, and $2 in the form B

BX Y ' = A X + B Y u''

Y

where A, B and X, Y are two couples of relatively prime elements of R , ( s ) or Rs(z), according to the nature of the system. By definition, the feedback system will be internally BIBO-stable if and only if the four transfer functions

s+l s+a

is a unit of RH(S). Hence, the resulting feedback system is internally BIBO-stable. Any loop closed around a discrete-time system involves some information delay, however small (Kui~era, 1991). Indeed, a control action applied to S, cannot affect the measurement from which it was calculated in $2. Therefore, either S, or $2 must include a one-step delay; we shall always think of it as having been included in $1. PARAMETRIZATION OF STABILIZING CONTROLLERS The design of feedback control systems consists of the following: given one subsystem, say $1, we are to determine the other subsystem, $2, so that the resulting feedback system shown in Fig. 1 has a desirable property. We call S, the plant and $2 the controller. Our focus is on achieving the desirable property of internal BIBO-stability. Any controller $2 that BIBO-stabilizes the plant S, will be called a stabilizing controller for the plant. Suppose S, is a continuous-time plant that gives rise to the transfer function B TI=-A for some relatively prime elements A and B of RH(S). It follows from the foregoing analysis that a stabilizing controller exists and that all controllers that stabilize the given plant are generated by all solution pairs X, Y with X :/: 0 of the equation AX + BY = 1

U2

FIG. 1. Feedback system.

over RH(S). This particular diophantine equation is called the Bezout equation. This is a result of fundamental importance (Ku~era, 1974b; Ku~era, 1975; Youla, Bongiorno and Jabr, 1976; Youla, Jabr and Bongiorno, 1976; Desoer, Liu,

D i o p h a n t i n e e q u a t i o n s in control Murray and Saeks, 1980) which can be considered to have launched this entire area of research. It provides a parametrization of the family of all stabilizing controllers for St in a particularly simple form, I"2=

1365

The plant uncertainty can be modelled conveniently in terms of its fractional description (Vidyasagar, 1985). Let St be a nominal plant giving rise to the transfer function B

Y' - A W X' + BW

Tt=-A

where X ' , Y' are any elements of RH(s) such that A X ' + B Y ' = 1 and W is a parameter that varies over RH(S) while satisfying X ' + B W ~ O. In order to determine the set of all controllers $2 that stabilize St, one needs to do two things: (i) express Tt as a ratio of two coprime elements from RH(s) and (ii) find a particular solution in RH(s) of a Bezout equation. The first step is very easy for rational systems. The second step is equivalent to finding one controller that stabilizes St. Once these two steps are completed, the formula above provides a parametrization of the set of all stabilizing controllers for St. The condition X ' + BW--/: 0 is not very restrictive, as X ' + B W can equal zero for at most one choice of W. As an example, we shall stabilize an integrator plant St. Its transfer function can be written over R . ( s ) as 1

where A and B are relatively prime elements of Ru(s). We define a norm of an element F(s) e Rn(s) by IIFII = sup IF(s)l. Res~O

We consider the neighbourhood of St which consists of all plants S; having the transfer function B r

T; =A--7

where A' and B' are relatively prime elements of RE(S) such that IIA - m'll < e,,

IIB - B'II < e2

for some specified reals et > 0 and E2 > 0. Now, let $2 be a BIBO-stabilizing controller for St whose transfer function

s+l

T,=

T2 ~

S

s+l

_D

Y X

satisfies

Suppose that, using some design procedure, we have found a stabilizing controller for St, namely

AX+BY=I.

T2-- - 1 .

Then $2 will BIBO-stabilize all plants S~ in the neighbourhood of St if and only if

This leads to a particular solution X, Y of the equation

et IIXII + e2 IIYII -< 1

1

slX+ s+

Y=I,

say X ' = 1, Y' = 1. The solution set in RH(S) of this equations is X=I+

1

s+l

W,

s

Y=I-

s+l

IIFII = sup IF(z)l

T~=

S

s+l

Izl~t

W.

Hence all controllers $2 that BIBO-stabilize St possess the transfer function 1-

(see Doyle and Stein, 1981; Chen and Desoer, 1984; Vidyasagar, 1985). When we work in discrete time, we define a norm of an element F ( z ) e Rs(z) by

W

1 +s--~l W where W is any function in RH(S). ROBUST STABILIZATION We shall now consider the design of stabilizing controllers for imprecisely known plants. Thus a nominal plant description is available, together with a description of the plant uncertainty, and the objective is to design a controller that stabilizes all plants lying within the specified band of uncertainty. Such a controller is said to stabilize robustly the family of plants.

and proceed alone the same lines. PARAMETRIZATION OF CLOSED-LOOP TRANSFER FUNCTIONS The utility of the Bezout equation A X + B Y = 1 for the study of internal BIBO-stability derives not merely from the fact that it provides a parametrization of all controllers that stabilize a given plant in terms of a "free" parameter W, but also from the simple manner in which this parameter enters the resulting (BIBO-stable) closed-loop transfer functions. In fact, Yt = B ( X ' + B W ) u t , y2=-B(r'-AW)ut,

Yt = - B ( Y ' - A W ) u 2 y2=-A(Y'-AW)u2

and we observe that all these transfer functions are a n n e in W. For example, the integrator stabilized in the last

1366

V. KUt~ERA

example yields the closed-loop transfer functions

1 (1 1

Yl = S---~

$+ 1

)'2 =

APoXo + B Y = 1.

s

W

___ s+ 1

s

W

The disturbance rejection is said to be robust if the conditions (i) and (ii) hold even as the plant S, is slightly perturbed, that is to say, if S, is replaced by any S; with transfer function

_ __L__s

s

W

B' r:=~7

Y' = - s + 1 Y2--

where Xo, Y is any solution pair such that Xo ~ 0 of the equation

s+l(1-s--~

)u2

where W is arbitrary in RH(s). This result serves to parametrize the performance specifications and it is the starting point for the selection of the best controller for the application at hand. The crucial point is that the resulting optimization/selection problem is attine in the parameter W while it is non-linear in the controller $2. DISTURBANCE REJECTION Let us show how further specifications, beyond those of internal BIBO-stability, can be handled. Suppose we are given a plant $1, with transfer function B

such that I I m - A ' I I < e.,

for some reals e. > 0 and e2 > 0. Whenever the disturbance can be rejected, it can be rejected in a robust way (Francis and Vidyasagar, 1983; Vidyasagar, 1985). All that is needed is to make X divisible by P rather than by P0. Thus, if ~ is any controller that stabilizes the plant S, = S2/P then $2 = S2/P achieves the robust disturbance rejection. The presence of the 1/P term within the controller is referred to elsewhere (Francis and Wonham, 1976; Wonham, 1979) as the "internal model principle". As an example, we shall asymptotically eliminate the effect of a periodic disturbance

T~ A in fractional form over R . ( s ) , say, whose output is corrupted by a disturbance v of the form

Q

as+b V=s2+l for any real a and b, which is injected at the output of an integrator plant S,. We take, for instance,

1J=--

P

where only P is specified. The objective is to design a controller $2 such that (i) the feedback system shown in Fig. 2 is internally BIBO-stable and (ii) the effect of the disturbance v is asymptotically eliminated from the plant output y. This means that

AX Q Y

A=

s s+l'

to obtain X. = 1 +

Y=

Y XoPo

~v

P=

s2+l (s+l) z

1

s+l

W

3s2+2s + 1 s $2+1 (s + 1)2 s+l(s+l) 2W

for any W from R.(s). All BIBO-stabilizing controllers that reject the disturbance have the transfer function

Since A X + B Y is a unit for every stabilizing controller and Q is unspecified, Po must divide X in R.(s). Therefore, $2 exists if and only if P and B are relatively prime in R . ( s ) and has the transfer function

7'2 =

1 s+l'

s s2+l X 1 s+l(s+l) 2 "+s--~Y=I

must belong to R . ( s ) , i.e, both A X + B Y and P should cancel. We define relatively prime elements Ao, Po of R . ( s ) by Ao Po

B=

and solve the equation

AX + BYP

A P

l I B - B'II < e2

T~=

3s2+2s + 1 s --W' (s + 1) 2 s+ 1 s2+l 1 (s + 1)2 + s - - ~ w '

where s2+ 1 W' = ($ + 1) 2 W. They all achieve robust rejection since A and P are relatively prime in R,,(s).

Flc~. 2. Disturbance rejection.

REFERENCE TRACKING By now it is clear how the algebraic synthesis works: the performance specifications call for

D i o p h a n t i n e equations in control y

1367

and I I A - A ' I I < el,

l i B - B'II < e2

for some reals el > 0 and t2 > 0. Now A' and B' are not specified but A ' X + B ' Y is still a unit, say U, of Rs(z). We have

FIG. 3. Reference tracking.

{A'X B'(Y- Z)\ G e = ~--~- + U ) F" divisibility conditions in the underlying ring, and these are equivalent to diophantine equations. Another control problem of importance is that of reference tracking. Suppose we are given a plant S,, with transfer function B TI=-A

in fractional form over Rs(z), say, together with a reference r of the form G y~--.-F where only F is specified. We recall that S~ contains a one-step delay, hence B is strictly proper. The objective is to design a controller $2 such that (i) the feedback system in Fig. 3 is internally BIBO-stable and (ii) the plant output y exactly asymptotically tracks the reference signal r. The controller can operate on both r (feedforward) and y (feedback), so it is described by two transfer functions

Hence, two conditions are necessary and sufficient for robust reference tracking (Francis and Vidyasagar,

1983; Vidyasagar, 1985), namely F divides X,

F divides Y - Z

in Rs(z). We illustrate on a discrete-time plant S, given by 1 T1=-1 +0.5z

whose output is to track every sinusoidal sequence of the form a+bz r . ~ . - 1-z+z 2 where a, b are unspecified reals. Taking 1 +0.5z A = - - ,

B=-

Z

1-z+z F = - -

2

Z2

and solving the pair of diophantine equations l+0"5Zx+ly= Z

Y

1 Z'

1

Z

Z

r~,=- 2, r~,=~.

1-z+z 2 - - V +

1Z

Z2

The requirement of tracking imposes (Ku~era, 1984) that the tracking error

2+ BZ

=1

yields the tracking controllers in parametric form

e=r-y - (1

Z

G

1 +0.5z Z

W,

2+1w,

AX--~ B--Y)

Z

belong to Rs(z). Since A X + B Y is a unit for every stabilizing controller and G is unspecified, F must divide 1 - B Z in Rs(z). Hence, there must exist an element V of Rs(z) such that 1 - B Z = FV. Therefore $2 exists if and only if F and B are relatively prime in Rs(z) and its two transfer functions evolve from solving the two diophantine equations

AX + BY = 1 FV+BZ=I where the element V of Rs(z) serves to express the tracking error as

e = VG. The reference tracking is said to be robust if both conditions (i) and (ii) hold even as the plant $1 is slightly perturbed to S~, with transfer function B' TI'=-A'

factorized over Rs(z) such that B' is strictly proper

Z,--1

1--Z+Z 2

Z

Z2

T2, =

W2

1 2+-W~ Z

for any elements WI, W2 of Rs(z). The resulting error is

e=(l+~W2)

a+bzz2

Not all of these controllers, however, achieve a robust tracking of the reference. The two divisibility conditions are fulfilled if and only if W~ is restricted to W, =

2-2z 1-z+z z2 Z +

2 W

where W is free in Rs(z). MODEL MATCHING It is nice to see how the primary requirement of BIBO-stability decouples from the secondary require-

1368

V. KU~ERA for arbitrary W from Rn(s). A particularly simple controller is obtained for

iy

FIG. 4. Model matching.

W =(1-a)s

namely T2y = - a ,

merits imposed on the control system. To illustrate further this point, we consider the problem of (exact) model matching. We are given a plant St with transfer function B Tt----A

Y

Z =

the transfer functions of the feedback and the feedforward parts of $2, respectively, the overall transfer function relating v and y reads

Of course, there are other controllers $2 that generate M, for example T2y = O,

IIFIh = ( F ' F ) ''2 where the asterisk denotes conjugation, F * ( z ) = F(z-t), and (-) denotes taking the term independent of z. The corresponding normed space is the rational Hardy space/-/2. We are given a discrete-time plant & having two inputs: the control input u and the exogenous input v. We describe the plant by two transfer functions B Tt,=~,

BZ --M. AX + BY

The requirement of BIBO-stability for the feedback system then decouples this model matching equation into two linear diophantine equations AX + BY = 1 B Z = M.

It follows that the model matching by a stabilizing controller is possible if and only if B divides M in R.(s). Any and all controllers are obtained by solving the two equations (Ku~era, 1986b). As an example, we take again the integrator plant St. In order to generate from St a target system with transfer function

y, T2 ~ - - - S t'

if and only if A, B are relatively prime in Rs(z). Let

T~

be a parametrization of all stabilizing controllers in terms of a free parameter W from Rs(z). The closed-loop transfer functions are then parametrized

b =

s+a

u

'1

in R.(s). All matching controllers are given by

1+

s+ 1

s+l

W

1

T2y-

b--

,

w

Y' - A W X'+BW

Y=I

1 - - Z

s+l

s

C A

s+a

1 s+l

s--~

Tt,,

where A, B, C are elements from Rs(z) and A is a least common denominator. We recall that St contains a one step-delay, hence B is strictly proper. The objective is to determine a controller $2 such that (i) the feedback system shown in Fig. 5 is internally BIBO-stable and (ii) the closed-loop transfer function from v to y, denoted by T, has minimal H2-norm. There exists a stabilizing controller for &, say

b

for some positive reals a and b, we solve the diophantine equations

1

T~,,=sM.

H2 OPTIMAL CONTROL The performance specifications often involve a norm minimization. Let us consider the ring Rs(z) of rational functions F(z) analytic within the set Izl > 1 and define a norm by

-

s --X+ s+l

Tz,, = b.

They are not stabilizing ones, however.

in fractional form over R . ( s ) , say, and a model (proper and stable) rational function M. We suppose that A and B are relatively prime in R.(s). The objective is to design an internally BIBO-stabilizing controller $2, having a feedback and a feedforward part, such that the transfer function of the closed loop system from v to y is M (see Fig. 4). Denoting

M=

s+l + a,

T~,,= 1+

s +a 1

7-71

w

Fro. 5. Optimal control.

Diophantine equations in control

1369

is P = - 1 we obtain the minimizing parameter

as

r --

T~ 1 + T~,T2

Bw)

= c(x'+

W

and the task is to find the parameter W that minimizes

=

1 2_z-i •

Therefore the//2 optimal controller is

IITIh. Suppose there exist two units FA and Fc of Rs(z), the so-called spectral factors, such that B*B = FT~FB, C*C = F~Fc.

Then

IITll~-- ( ( x ' + BW)*C*C(X' + BW)) --

((F.FcW +B-2"X'Fc;( F.FcW + ~.

X'F~

F~

)) .

Let P be the proper part of the second term in the parentheses, i.e.

B*

Q*

-~a X ' F c = P +-~s

for some Q in Rs(z). Then the cross-terms contribute nothing to the norm and we have

The first term being independent minimizing parameter is seen to be

of

W,

the

P W

~

- - -

FaFcY' + A P FsFcX' - B P

while the minimum norm is

[ITll2mi"

= Q

2"

As an example, consider a summator plant XI+I = I t

1 2

----

and the corresponding norm I[Tl[2m~, = 2. The /-/2 optimization problem is close to the L Q G or minimum variance control problem. In fact, if v is a stationary, zero mean and unit variance, white noise sequence, then [[T[[2 is the variance of y in the steady state. FINITE IMPULSE RESPONSE Transients in discrete-time linear systems can settle in finite time. Systems having the property that a "finite input sequence" produces a "finite output sequence" will be called finite-input finite-output (FIFO) stable. It is a well-known fact (Ku6era, 1980a) that a system with a proper rational transfer function H ( z ) is FIFO-stable if and only if H ( z ) is a polynomial in z -~, i.e. belongs to Rf(z). Equivalently, the unit pulse response of the system is finite. Let us consider the feedback configuration shown in Fig. 6 and focus on achieving the desirable property of internal FIFO-stability. To this end, we write the transfer function of the plant as

F~Fc

By construction, it belongs to Rs(z). It follows that the optimal controller is unique and given by

72=

T2 =

B Tt=-A where A and B are relatively prime functions from Rf(z). We recall that S~ contains a one-step delay, hence B is strictly proper• Repeating the arguments concerning BIBO-stability, we conclude that a FiFO-stabilizing controller exists and that all controllers that internally FIFO-stabilize the given plant have the transfer function T2~

---

Y X

+ UI - - ~Jl

where X, Y is the solution class of the equation yt = xt + vt AX + BY = 1

described by the transfer functions

T~=

1 z-l'

z-2 T,~=-z-l"

We take the following elements of Rs(z) to describe the plant A=l-z

-1,

B = z -~,

C=l-2z

-~

in Rf(z). In particular, if X ' , Y' defines any FIFO-stabilizing controller for $1, the set of all such controllers can be parametrized as

T~=

Y' - A W X' + BW

and calculate the spectral factors FB=I,

Fc=2-z

-l.

One particular BIBO-stabilizing controller is given by T 2 = - l , hence X ' = I and Y ' = I . Since the proper part of

B*

-~BX'Fc=2Z-1

ntIMI

FIG. 6. Feedback system redrawn.

1370

V. KU~ERA

where the parameter W varies over Rf(z). We note that W is completely free because strict properness of B prevents X ' + B W from turning zero. The parameter W can be chosen so as to meet further requirements on the closed-loop system (Ku~era and Kraus, 1993). For reasons of simplicity, economy or aesthetics we may require that T2 have a smallest McMillan degree. Then W must be taken so that X ' + B W and Y ' - A W have least degrees as elements of R[z-1]. This particular controller achieves also the shortest impulse responses within the closed-loop system. A well-known example is the dead-beat controller. We consider the plant Z

possess also impulsive modes associated with poles at s = ~ , which of course are not reflected in its characteristic polynomial. As long as C has a high enough degree, these modes can be eliminated by choosing a least-degree solution of the characteristic equation. In particular, if d e g A - < deg B we seek to minimize d e g X and if d e g A - > d e g B we seek to minimize deg Y. For example, if a double integrator plant xl = x2, y=xt

is to be converted into an harmonic oscillator .~l = x2,

i

Tl=-1 -z

we take A = s 2, B = I , equation

Y=l-(1-z

~)W

we obtain all dead-beat (or FIFO-stabilizing) controllers as 1 -

T2=

(1 -

z

l+z

l)W ~W

The dead-beat controller of least McMillan degree is obtained for W = 0:

T~= -1. MODAL CONTROL If one wants the control system to have certain modes, that is, to have its finite poles located at certain positions, it is necessary to compensate the given plant so that the closed-loop system (see Fig. 6) has a prespecified characteristic polynomial. Quite naturally, we shall employ the ring of polynomials, say R[s] for definiteness. Suppose the plant S~ gives rise to the transfer function B Tt---A and the controller S: to

T~=

Y X'

both factorized over R[s]. Provided A is the characteristic polynomial of St and X is that of $2, then the characteristic polynomial, C, of the feedback system equals A X + BY.

Thus, given the plant St and the required polynomial C we obtain a polynomial equation that determines the controller $2. The resulting system, however, can

and solve the

in R[s]. The solutions read X = I + W,

in Rf(z) read

C=s2+l

s2X + Y = s ~ + 1

(1- z-~)X + z ~Y=I

~W,

.fcz = u - x I

y=x~

-~

and interpret the exogenous signals w~ and w2 as accounting for the effect of the initial conditions of S~ and $2. The requirement of FIFO-stability is then equivalent to achieving finite responses u and y for all initial conditions. Since all solutions of the equation

X=l+z

x2 = u

Y= l-sZW

for any polynomial W ~ e - 1 . The plant has order 2 and the resulting harmonic oscillator has order 2 as well; hence the controller must be of order 0. This is secured by minimizing the degree of Y. So we put W = 0 and obtain the transfer function of the controller

T,= -1. MULTIVARIABLE SYSTEMS Up until now we have considered only single-input single-output systems. In the case of multiple inputs and/or multiple outputs, the input-output properties of the system are represented by a matrix of transfer functions. The additional intricacies introduced by these systems stem mainly from the fact that the matrix multiplication is not commutative. Consider a rational transfer function matrix G ( s ) whose dimensions are, say, m x n. Then it is always possible (Vidyasagar, 1985) to factorize G as follows: G=BA

i=,~

l~

where the factors are matrices over a Bezout domain of desirable functions, say R~(s), such that A, B are relatively right prime ,4, /~ are relatively left prime. These "matrix fractions" are unique except for the possibility of multiplying the "numerator" and "denominator" matrices by a unimodular matrix over R~(s). Furthermore, B and /~ have the same invariant factors and A and ,4 have the same non-unit invariant factors (note that A is n × n while ,4 is re×m).

We illustrate the use of left and right matrix fractions in the problem of internal BIBOstabilization. We consider the feedback configuration shown in Fig. 1, where S~ and $2 give rise to transfer function matrices Tt and ~ , respectively. We factorize

D i o p h a n t i n e e q u a t i o n s in c o n t r o l T~ and T2 in terms of proper-stable rational matrices as T~ = B A - ' = ,4-' B 1"2= - X - ' Y

= - ?~-L

The transfer function matrices that relate u t, u2 with y~, Y2 in the feedback system then read Yl = B ( X A + Y B ) - I X u l , Yt = - B ( X A

+ YB)-IYu2

y2 = It - A ( X A + Y B ) 'X]ut, y2 = - A ( X A

+ YB)-lyu2

or alternatively y, = 2 ( A X

1371

The diversity of infinite-dimensional systems cannot be captured by a single notion of stability. In the context of discrete-time systems, the BIBO-stability defined earlier is termed the /®-stability, because a BIBO-stable system transforms l ® input sequences into l ® output sequences. The set of transfer functions of causal l~-stable systems is the set of the z-transforms of absolutely summable sequences. In some applications it is more natural to use 12-stability. A discrete-time causal system is/2-stable if it transforms 12 input sequences into l 2 output sequences, where 12 is the space of infinite sequences f = (f0, fl . . . . ) with a norm defined by

+ ~?)-'Bu,,

y, = [X(A,~" + Bl?)-',4 - l]u2 Y2 = - ? ( , 4 X + B17")-'/~u,, Y2 = -I7"(,~-~" +/312)-',4u2. The feedback system is well-posed if the two indicated inverses exist. By definition, the feedback system will be internally BIBO-stable if the four transfer function matrices above are proper-stable rational. Provided the two pairs A, B and X, Y are relatively right prime and the two pairs ,4,/~ and X, Y are relatively left prime, this will be the case if and only if (Ku~era, 1979; Vidyasagar, 1985; Maciejowski, 1989) the common denominator X A + YB, or alternatively ,4,(" + BY, is a unimodular matrix over RH(s). A parametrization of all controllers $2 that internally BIBO-stabilize the plant S~ is now at hand. Given the relatively (left and right) prime, properstable rational factorizations Ti = B A i = , ~ - i B ,

we select proper-stable rational matrices X, Y and ,(', 17"such that XA+YB=I,

A,¢ + B ? = I.

Then the family of all stabilizing controllers is given by

T2 = - ( X + W B ) - I ( Y - Will) -(~"- A W)(f( + BW)-'

where W is a proper-stable rational matrix parameter such that X + B W is non-singular and 1~" is a proper-stable rational matrix parameter such that X + W B is non-singular. It is easy to see that all four closed-loop transfer function matrices are attine in the parameters W and I~'. Thus, control synthesis problems beyond stabilization can be handled by determining the parameters W or if' as described for single-input single-output systems. INFINITE-DIMENSIONAL SYSTEMS If the value set of a system is a normed function space, rather than the set of real numbers, we speak of an infinite-dimensional system. The transfer function of such a system is no longer rational.

The set of transfer functions of causal 12-stable systems is the Hardy space H~ of functions bounded on the unit circle with analytic continuation outside the circle. A good choice for the set of desirable transfer functions is the ring D of functions that are analytic on the open set Izl > 1 and continuous on the closed set Izl->l. It corresponds to causal 12-stable systems whose frequency response is continuous. The ring D is not a principal ideal domain, it is not even a Bezout domain (Vidyasagar, Schneider and Franicis 1982). This means that the systems whose transfer functions belong to the field of fractions of D cannot be assumed a priori to have a relatively prime fractional representation over D. The transfer functions whose denominators are restricted to having only a finite number of zeros in the closed set Izl -> 1 and none on the unit circle Izl = 1, however, do have a relatively prime fractional representation (Vidyasagar, 1985). The algebraic approach developed for rational systems carries over completely to systems with these transfer functions. Another desirable property of control systems is related to the ring E of functions that are analytic in the entire complex plane but at z = 0; this point can be an essential singularity. A system whose transfer function belongs to E features a unit-pulse response (ho, h, . . . . ) that converges to zero faster than any exponential. This property is a generalization of finite impulse response in finite-dimensional systems. The ring E is a principal ideal domain and its field of fractions is the set of meromorphic functions in z -I. Therefore, a "rapid descent" control strategy (Je~ek, 1989) can be formulated and solved for a fairly broad class of infinite-dimensional systems: given a plant with meromorphic transfer function B Tl=-A where A and B are relatively prime functions from E, we seek a feedback controller with meromorphic transfer function T2 such that the impulse response of the closed-loop system shown in Fig. 6 descend faster than any exponential. All such controllers are given

1372

V. Ku~ERA

by T2 ~

---

for relatively fight prime elements A, B and X, Y and for relatively left prime elements A,/~ and X, Y of Is(z). Then, drawing from the preceding discussion, the feedback system is internally BIBO-stable if and only if X A + YB, or equivalently .4~" +/~Y, is a unit of ls(z). In order to determine any and all BIBO-stabilizing controllers $2 for the plant S~ we solve the equation

Y X

where X, Y satisfy the diophantine equation AX = BY = 1

in E and X ¢ O.

X A + YB = 1

TIME-VARYING SYSTEMS We shall now outline the extension of the algebraic, transfer function approach to time-varying linear, say, discrete-time systems. We consider the set l0 of all infinite real sequences x = ( x o , x~ . . . . ). With pointwise addition and pointwise multiplication, lo is a ring (commutative, with identity). With the right-shift operator o defined by (ox)~ = x H the ring 1o is called a difference ring. Time-varying causal systems are defined by unit-pulse responses (ho, h, . . . . ) whose members are infinite sequences from a difference subring of 1o. Particular classes of time-varying systems can be studied by fixing a particular difference subring. Examples include the set of all periodic sequences p s with period N or the set of all bounded sequences 1~; we shall consider the latter. When

i=o

exists, it is called the (formal) transfer function of the system. We observe that H resides in l®(z), the ring of rational transfer functions in z over l ®, with multiplication defined by

x z = z(ox),

A g + Bf" = 1 for ,(', 17"in ls(z) and put = -(?-AW)(g

+ BW)-'

for an arbitrary parameter W in Is(z) such that f f + B W 4:O. To illustrate, we consider the plant (Poola, 1984) given by x,+~ = x , + a , u , ,

t=O, 1, . . .

y,=a~x,,

ho=O,

hi=a(o'a),

i=1,2 .....

Since oZa = a, a ( o a ) = O, the transfer function of S~ reads +....

It admits left and right factorizations

x in l ®.

Tz=-X-'Y=-~'f(

for an arbitrary parameter I~' in Is(z) such that X + WB ~s O. Equivalently, we can proceed by solving the equation

TI = z - 2 a + z - 4 a

Z m+n

This non-commutative multiplication captures in a very natural way the time-variance of our systems (Kamen, Khargonekar and Poola, 1985). We shall study the problem of stabilization: given a plant S,, design a controller $2 such that the closed-loop system shown in Fig. 6 is internally BIBO-stable. A bounded linear system, whose coefficients are in 1®, is BIBO-stable if and only if its transfer function H ( z ) is stable, in the sense that Ilh, ll®~0 as i---~oo. The set of stable rational functions from l®(z) is a ring and will be denoted by Is(z). To solve the problem of stabilization, we shall express the transfer functions of the systems involved in terms of elements from Is(z). Since this ring is non-commutative and since it is not a domain, not all transfer functions in l~(z) can be expressed as relatively prime fractions of elements from Is(z); moreover, we have to distinguish left and right fractions (see Poola and Khargonekar, 1987). Accordingly, we assume that the transfer functions of S, and $2 admit both left and right fractional representations T~= B A - ' = A - ' B ,

T2 = - ( X + ff'/~)-'(Y - I~A)

where a = (a,)7=o is an 1= sequence such that a, = 1 if t is even and a, = 0 if t is odd. The unit-pulse response of Si is (ho, h, . . . . ) where

H ( z ) = ~ z ihi

zmz n ~

for X, Y in Is(z) and put

-'

T~ = B A - ' = A - ' B

over ls(z), for instance 2a,

A=l-z

A = l - z-2a,

B=z-2a B = z-2a.

These can be verified by applying the multiplication rules z-2az-2 a

= z-4(o-2a)

=

z-4a.

Now A.('+/}12= 1 admits a solution pair ,~" = 1, 1) = 1. Therefore, all stabilizing controllers possess the transfer function Tz = - [ 1 - (1

-

z-2a)W][1 + z-2aW] -j

where W ranges over ls(z). NON-LINEAR SYSTEMS As a final example we illustrate how the fraction approach can be applied to the stabilization of non-linear systems. Owing to the abundance of unstable non-linear systems, the stabilization problem is probably one of the most commonly encountered problems in engineering science and practice.

D i o p h a n t i n e e q u a t i o n s in control We shall work in discrete time, with the signal space lo, the set of all infinite real sequences x = ( x o , xt . . . . ). We can define a norm on 1o by Ilxll =sup,2-' Ix, I. A non-linear system is simply a map S:lo-*lo, transforming input sequences into output sequences. The problem of stabilizing a non-linear plant St consists in finding a (non-linear) controller $2 such that the feedback system shown in Fig. 6 is internally BIBO-stable in that the internal signals u and y of the closed loop are bounded in the presence of suitably bounded, but otherwise arbitrary, signals wt and w2 injected around the loop. The basic concept that facilitates our development is that of rationality. A system S is right rational if there are BIBO-stable maps A and B with common domain 1~-1o, where A is invertible, such that S = B A - t ; the system S is left rational if there are BIBO-stable maps A and /~ with common domain 1~-1o, where A is invertible, such that S = ~ - t ~ . We say that A and B are relatively right prime if the set of all bounded sequences w ¢ 1o such that Sw is bounded, but A - t w is unbounded, is the empty set. Similarly and /~ are relatively left prime if the set of all unbounded sequences W¢lo such that Sw is unbounded and/~w is bounded is the empty set. Consider now a strictly causal plant St :1o'-* 1o that is right rational, with a relatively right prime fraction representation St

= BA-'

where A : l - - * l o and B:l--*lo are BIBO-stable maps. Then there exist (Hammer, 1987) BIBO-stable maps X : lo--~ l and Y: lo--~ l, where X is invertible, such that the following identity holds: XA + YB=I:I--~L

These maps define the causal controller $2:1o-* !o as

Suppose that X and Y satisfy a Lipschitz condition. Then (Paice and Moore, 1990) the controller S2 will internally BIBO-stabilize the plant St in that the signals u and y are bounded for suitably bounded inputs wt and We, or equivanetly all the closed-loop transfer maps are BIBO-stable. Let the plant St be also left rational, with a relatively left prime fraction representation S, = A-t/~ where A : 7--~ lo and B : i---~lo are BIBO-stable maps. Then the class of all BIBO-stable maps X w and Yw satisfying XwA + YwB = I

is characterized in terms of an arbitrary BIBO-stable non-linear map W : [--* l as Yw = Y-

WA.

Therefore, having found one controller $2 that internally stabilizesSt, the class of all controllers S2w that internallystabilizeSt is given by Sz tv = - X ~v'Yw

provided that Xw and Yw satisfy a Lipschitz condition (Palce and Moore, 1990). A broad example of non-linear systems are the recursive systems. A system S : l o - * l o is recursive if there exist non-negative integers 0~,/~ and a function f such that for every input sequence u ¢ l o , the elements of the output sequence y • S u can be computed recursively from the relation y,+~+, --f(y,..... y,+,~,u ...... u,+#) for all t=O, I .... given the initial conditions Yo, Yt ..... y,. If S is an injective system with a continuous recursion function f then it is both leftand right rationalprovided it is operated by bounded input sequences (Hammer, 1987). If the set of bounded input sequences for which S produces bounded output sequences is a rich enough set, in a suitable sense (Hammer, 1986), then S can be internally stabilized. For example, the system y,+,=eY'+u,,

t--O, l,...

is stabilizedby u, = - e y' provided the input sequences are suitably bounded, whereas no stabilizer exists for the system y,+, = 2)', + e u' as no input sequence produces a bounded output sequence. To illustrate, let us determine all non-linear controllers $2 that internally BIBO-stabilize the plant St given by Y,+t=eY'+u,,

t = O , l .....

Left and right fraction representations of St are easily found to be A:u,=xt-e

$2 = - X - t Y .

Xw = X + WB,

1373

x'-~,

A :.f,+, =y,+t - e",

B:y,+t=x, /~ :~,+, = u,.

One pair of maps X, Y that satisfy the Bezout identity is X:x,=u,,

Y : x , = e y'.

Letting W : x , = w ( ~ , ) , t = 0 , 1 . . . . be an arbitrary BIBO-stable map satisfying a Lipschitz condition, we obtain all internally BIBO-stabilizers for S. in the form Xw

= X +

W B :x, = u, + w(u,_O

Yw = Y - W / i :x, = e ~' - w(y, - e y'-')

and these will work for every bounded sequence. CONCLUSIONS The diophantine equation approach is a transfer function-based control theory in which the transfer functions are viewed and handled as algebraic objects. While conceived for linear finite-dimensional, time-invariant systems, it has been generalized to extend the scope of the theory and includes some time-varying, infinite-dimensional and even non-linear

1374

V. KUt~ERA

systems. The approach is based on the factorization of transfer functions over an appropriate ring, thus reducing the mathematical synthesis of control systems to the solution of linear equations in that ring. The starting point of this approach is to obtain a simple parametrization of all controllers that stabilize the given plant. O n e can then choose, in principle, the best controller for various applications. Examples have been given in the paper to illustrate the synthesis. O n the practical side, this approach yields new computational algorithms which are simple, efficient and easily implementable. The interested reader is referred to Ku~era (1979) for details. Acknowledgement--The author acknowledges support from the Czechoslovak Academy of Sciences through Grant No. 27 501. REFERENCES Ahl6n, A. and M. Sternad (1989). Optimal deconvolution based on polynomial methods. IEEE Trans. Acoustics, Speech and Signal Processing, ASSP-37, 217-226. Ahl6n, A. and M. Sternad (1991). Wiener filter design using polynomial equations. IEEE Trans. Signal Processing, SP-39, 2387-2399. /~str6m, K. J. (1970). Introduction to Stochastic Control Theory. Academic Press, New York. ,~str6m, K. J. (1980). Robustness of a design method based on assignment of poles and zeros. IEEE Trans. Aut. Control, AC-25, 588-591. ~str6m, K. J. and B. Wittenmark (1973). On self-tuning regulators. Automatica, 9, 185-199. /~str6m, K.J. and B. Wittcnmark (1984). Computer Controlled Systems: Theory and Design. Prentice-Hall, Englewood Cliffs, NJ. BIomberg, H. and R. Ylinen (1983). Algebraic Theory for Multivariable Linear Systems. Academic Press, London. Boyd, S. P. and C. H. Barratt (1991). Linear Controller Design: Limits of Performance. Prentice-Hall, Englewood Cliffs, NJ. Callier, F. M. and C. A. Desoer (1978). An algebra of transfer functions of distributed linear time-invariant systems. IEEE Trans. Circuits and Systems, CAS-25, 651-662. Callier, F. M. and C. A. Desoer (1980). Stabilization, tracking and disturbance rejection in multivariable convolution systems. Annales de la Societ~ Scientifique de Bruxelles, 94, 7-51. Callier, F. M. and C. A. Desoer (1982). Multivariable Feedback Systems. Springer-Verlag, New York. Chen, M. J. and C. A. Desoer (1984). Algebraic theory of robust stability of interconnected systems. IEEE Trans. Aut. Control, AC-29, 511-519. Clarke, D. W. and P. J. Gawthrop (1975). Self-tuning controller. Proc. lEE, 122, 929-934. Desoer, C. A. and W. S. Chan (1975). The feedback interconnection of linear time-invariant systems. J. Franklin Inst., 300, 335-351. Desoer, C. A. and M. Vidyasagar (1975). Feedback Systems: Input-Output Properties. Academic Press, New York. Desoer, C. A., R. W. Liu, J. Murray and R. Saeks (1980). Feedback system design: the fractional representation approach to analysis and synthesis. IEEE Trans. Aut. Control, AC-25, 399-412. Doyle, J. C. and G. Stein (1981). Muitivariable feedback design: concepts for a classical/modern synthesis. IEEE Trans. Aut. Control, AC-26, 4-16. Fliess, M. (1990). Some basic structural properties of generalized linear systems. Systems Control Lett., 15, 391-396. Francis, B. A. and W. M. Wonham (1976). The internal model principle of control theory. Automatica, 12, 457-465.

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