FIFO Stable Control Systems

Copyright © IFAC 12th Triennial World Congress, Sydney, Australia, 1993

FIFO STABLE CONTROL SYSTEMS V.

Ku~era*,l

and FJ. Kraus**

*Institute of Information Theory and Automation, Academy of Sciences, p.a. Box 18, 18208 Praha, Czech Republic **Automatic ControL Laboratory, Swiss FederaL Institute of TechnoLogy, ETH-Zentrum, CH-8092 ZUrich, Switzerland

Abstract. Motivated by the dead-beat and shortest-correlation control strategies, the paper introduces the notion of finite-input finite-output (FIFO) stability for linear discretetime control systems. A simple parametrization of all controllers that FIFO-stabilize a given plant is obtained. The best controller is then chosen for various applications, including H2 optimization, disturbance rejection and transfer function shaping. Key Words. Linear systems; discrete-time systems; stability; finite transients

of our development. We show how various additional requirements can be accommodated in the FIFO-stable design by choosing the parameter. These include either requirements on the controller (least order or partial pole placement) or requirements on the closed-loop transfer function (amplitude bounds or H2 optimization). The system can also be tuned to reject persistent disturbances in finite time.

1. INTRODUCTION

The dead-beat control is a well-known example of linear control strategies for systems working in discrete time. It is a deterministic control problem where we wish to make ·the error vanish in finite time, usually as short as possible. Various subtle modifications of the problem are described in Kucera (1980), O'Reilly (1981), Astrom and Wittenmark (1984) and Kucera (1991).

Some of these problems have already been studied in the literature (Kucera 1979). The design requirements, however, were imposed on particular signals in the control systems (the error, for example) and consequently resulted in open-loop strategies. In contrast, the FIFO-stability is a property of the system. Also the additional design requirements studied in this paper are imposed on the system rather than on its responses. This leads to genuine closed-loop control strategies.

The stochastic counterpart of the dead-beat control is the shortest-correlation control strategy, introduced by Kucera (1977). The objective is to render the error sequence as white as possible, thereby squeezing out all the information it contains for control purposes. Motivated by these control strategies, we introduce the notion of finite-input finite-output (FIFO) stability for linear discrete-time systems described by rational transfer functions. Roughly speaking, a system is FIFO-stable if every input sequence of finite length gives rise to an output sequence whose length is also finite.

2. STANDARD CONTROL SYSTEM We consider the control system configuration shown in the block diagram

A simple parametrization of all controllers that FIFO-stabilize a given plant is obtained. This result is based on a factorization approach, similar to the one used by Vidyasagar (1985) to study the bounded-input bounded-output stability of linear systems.

v

w M

The parametrization result is the starting point

y

1 This author acknowledges support from the ETH Ziirich through Program 0-15-103-90

where M represents the synthesis model of the 405

poles but those at z = 0 are in fact polynomials in z-l. It is therefore natural to express all transfer functions under consideration as quotients of polynomials in z-l . In particular, we write for the plant

generalized plant and K is the controller. The generalized plant contains what is usually called the plant in a control problem plus all weighting functions that reflect the design objectives. The signal v denotes all exogenous inputs to the plant, the output w is an error signal, y incorporates the measured variables, and u is the control input.

M

_B 11-A

M 12 -- A C

M

_D 21 - A

M 22 -- A E

(1)

We shall confine our attention to discrete-time systems represented by proper rational transfer functions . Such systems are causal, linear, timeinvariant and finite-dimensional. For ease of presentation we further assume that u, v and w, y are scalar quantities.

where A, B, C, D, E are polynomials in z-1 and A is the least common denominator of the four transfer functions; similarly for the controller

As a result, the plant will be described by the equations y = M 11 u + M l2 V W = M 2l U + M 22 V and the controller by

where Q has zero constant term in order for K to be strictly proper.

Q

(2)

K=-P

The nine transfer functions from v, Vl, V2 to w, y, u are given as

u= Ky where M 11 , M l 2, M 2l , M22 and K are all proper real-rational functions of the complex variable z. The discrete-time controllers always introduce some information delay (however small) in the closed control loop. Therefore we shall require K to be strictly proper .

T.v,u --

AP AP+BQ

T.

v'u -

AQ - AP+BQ

T.v,y --

BP AP+BQ

T.v,y --

BQ - AP+BQ

T.vy --

_ Tvu -

CP AP+BQ

T.v,W --

_

T.v,W --

DP AP+BQ

CQ - AP+BQ

DQ - AP+BQ

and 3. FIFO STABILITY: DEFINITION

T.vw

We want to endow the closed-loop system with the following desirable property : whenever all inputs applied to the system at rest are finite sequences then so too are all internal signals in the system. We shall call this property (internal) finite-input finite-output (FIFO) stability.

+

BE-CD Q A AP+BQ'

(3)

5. FIFO STABILITY: INTERPRETATION The plant gives rise to the transfer function

1[BDEC] '

M= A

w u

EP AP+BQ

These are all polynomials in z-l, i.e. the closedloop system is FIFO-stable, if and only if (i) AP+ BQ divides 1 and (ii) A divides BE - CD.

More precisely, we consider the closed-loop system and inject signals Vl and V2 around the loop : v

-

Then A defines the roots outside the origin of the minimal polynomial of M . Let F be a polynomial in Z-1 that defines the roots outside the origin of the characteristic polynomial of M. Then

y

We say the closed-loop system is FIFO-stable if the nine transfer functions from v, Vl, V2 to w, y, u have no poles outside the origin z = O.

- BE-CD _ G det M A2 - F for some polynomial G.

The FIFO-stability does not imply that the state of the closed-loop system is brought to zero in finite time from all initial values. This will be the case, however, if both the plant and the controller are free of hidden modes, except of those associated with z = o.

Now FIFO-stability requires that the closed-loop system have no poles outside the origin. Condition (i) ensures that A and Bare coprime, so the offending roots of the minimal polynomial of M (given by A) can be shifted to the origin by some controller. Condition (ii) secures that F divides A, so that M has no other poles (which cannot be shifted anyway) but possibly at z = O. To summarize: the characteristic and the minimal polyno-

4. FIFO STABILITY: ANALYSIS We note that the rational functions having no 406

and put

mials of M have the same roo~s outside the origin, and these can be shifted at will.

Xo = X

+ WOBz-1 .

Then 6. FIFO STABILITY: CONTROLLERS

P Q

Based on the stability analysis, it is now a routine matter to find a FIFO-stabilizing controller. We shall do much more, however. We shall obtain a parametrization of all controllers that FIFOstabilize the given plant. This will allow us to handle, in a systematic manner, further requirements on the control system by choosing the parameter.

=

max(deg X o,

The transfer function Tllw of a FIFO-stable closed-loop system is a polynomial in z-l . Suppose we want to make its degree as small as possible.

(5)

Denoting

G=BE-CD A and substituting (6) in (3) we obtain a parametrization of all closed-loop transfer functions

+ W Bz- 1) + G(Y (EX + GY) + WCDz- 1. E(X

(6)

where W is a free polynomial in z-l . Therefore the set of all controllers that FIFO-stabilize the plant is given as Y - WAz-1 K = - X + W Br1 .

~ degA .

8. TRANSFER FUNCTION OF LEAST DEGREE

The set of all FIFO-stabilizing controllers can be parametrized as follows. Let X, Y be a particular solution of (5) such that Y has zero constant term . Then the solution set P, Q of (5) with Q constrained to have zero constant term is given as P = X + WBz- 1 Y - WAz- 1

degYo

Ko = _ Yo . Xo It is strictly proper and 6K o deg Yo).

and Q has zero constant term.

Q

~ degB,

As a result, the controller of least McMillan degree is unique and given by

(4)

=1

1

We note that both P and Q achieve their least degrees simultaneously, for deg AP = deg BQ.

where P, Q is the solution set of the polynomial equation

AP+BQ

W O)B z -

Yo - (W - W O )Az- 1

degXo

The set of all FIFO-stabilizing controllers for M is given by

Q

+ (W -

and Xo, Yo is the unique least-degree solution of equation (5); it corresponds to W = Wo. In fact,

It falls out from the stability analysis that there exists a (strictly proper) FIFO-stabilizing controller for the plant M if and only if (i) A and Bare coprime and (ii) A divides BE - CD .

K =-P

Xo

WAz- 1) (8)

Suppose that CD i= 0 and divide CDz- 1 into EX + GY to get the quotient W 1 and the remainder Tllw 1,

(7)

EX

+ GY = W 1CDz- 1 + Tllw1 '

Then 7. CONTROLLER OF LEAST MCMILLAN DEGREE

Tllw = T Ilw1

+ (W + Wt}CDz- 1

and T Ilw1 is the least-degree closed-loop transfer function, obtained on setting W = - W 1 , and

The set of FIFO-stabilizing controllers contains controllers of arbitrarily high McMillan degree . For reasons of simplicity, economy or aesthetics it is reasonable to look for controllers of least McMillan degree within the set.

deg T Ilw1

~

deg CD .

The corresponding controller is

,

In view of (4) the McMillan degree of K is given by 61{ = max( deg P, deg Q).

/\1

Y + W 1Az-1

=- X

- W 1Bz-1 '-

Y1 Xl

(9)

If CD = 0 then Tllw does not depend on K . This degenerate case is of no practical interest.

Hence we are after the least-degree solution of (5). Let us divide Az-1 into Y,

9. MINIMUM H2 NORM

Y = W oAz- 1 + Yo

Suppose we wish to minimize the H 2 -norm of the transfer function T llw of a FIFO-stable closed-loop

where Wo is the quotient and Yo the remainder, 407

which in turn defines W2 and

system. Since Tvw is a polynomial in Z-l, it does belong to the real-rational Hardy space H2 and its norm is given by

All other things being equal, the larger n is allowed the smaller H 2-norm can be expected. For increasing n, however, this solution need not converge to the unconstrained H 2-optimal controller because of the a priori assumption that n is finite.

(10) where the asterisk denotes conjugation, T:w(z-l) = Tvw(z), and ( . ) denotes taking the constant term.

f 0 and write = Tvwl + WCDz- l

1(2.

10. TRANSFER FUNCTION SHAPING

Suppose that CD Tvw

Any particular transfer function

(11)

Tvw = TO

where Tvwl is the least-degree Tvw and W is a polynomial parameter. Let us now bound the degree of Tvw from above, say deg Tvwl

of a FIFO-stable closed-loop system may have some coefficients that are too high for the application considered . Then we allow a higher-degree Tvw and shape it to satisfy

< deg Tvw < n,

which corresponds to W having degree deg W

ITil ::; /3i,

< m:= n -1- degCD

Tvw = Tvwl

We use (10) and (11) to obtain

+ (W· D·C·zTvwl )

and we minimize this expression with respect to W. The minimum is achieved for some W = W2 that satisfies the set of linear equations

}'

for i = 0,1, .. ., m-I.

\3

The corresponding H 2-controller is given by l 1(2 = _ Y - (W2 - WI)Az-

+ (W2

+ WCDz- l

If W = W3 is a solution to these inequalities, the corresponding controller is given as

(zi D*C*CDW2) = _(i+ l D*C*Tvwl) (12)

X

(14)

as in (11) where Tvwl is the least-degree Tvw and W is a polynomial parameter. For W of any fixed degree, inequalities (14) lead to a set of linear inequalities for the coefficients of W. These may be consistent for W of high enough degree.

= (T:wlTvwl) + (W· D·C·CDW) +(T:wlz-1CDW)

i = 0,1, ...

where /30, /31, ... are desired bounds. To this end, we write

and determine W so as to minimize the H 2-norm (10).

IITvwll~

+ T1Z- l + T2 Z - 2 + ...

---------=-

Yl - W3Az-l = - Xl + W3 B Z- l

where Xl, Yl come from (9).

- WI)Bz-l

11. DISTURBANCE REJECTION

or, in terms of (9), as 1(2

= _ Yl

In a FIFO-stable control system, w is a finite sequence for any finite sequence v. Persistent disturbances v are not rejected at w . In case the dynamics of v are known, however, one can choose a FIFO-stabilizing controller that rejects this v in finite time.

l

W2 AzXl + W2Bz-l -

There is an alternative approach. Write (10) as IITvw II~

= Vf

Let the transfer function of a FIFO-stable control system be parametrized as in (8),

where f is the coefficient vector of the polynomial Tvw, and (11) as

Tvw = (EX

where W is a free polynomial in z-l. Suppose that 1 v= -v' R where R is a polynomial in z-l that describes the (partially) known dynamics of v. Then w = Tvwv will be a polynomial in z-l if and only if R divides Tvw, i.e. Tvw = RTR

where tl and ware the coefficient vectors of Tvwl and W, respectively, and fI is the n x m matrix whose i-th column is the coefficient vector of z-iCD. Then IITvwll~ =

+ GY) + WCDz- l

iffl + 2wT fITfl + wT fIT fIw.

Minimizing this expression with respect to w leads to a set of linear equations for the optimal w, namely

for some polynomial TR. 408

described by the transfer functions

In view of (8) this goal can be achieved whenever CD and Rare coprime. In this case the two unknown polynomials TR and W satisfy the equation RTR - WCDz- 1 = EX

+ GY.

Mll = z~1 M 21

-I - z-1

M 22

2-z z-1

-I - z-1

(15) Let us determine the FIFO-stabilizing controllers. We define

Let T 4 , W 4 be one solution pair of (15). Then the solution set is given by TR T4 + W'CDz- 1

W

M 12 =

A=1-z-1,

W 4 +W'R

-

1,

The conditions (i) and (ii) are verified and

where W' is now the free polynomial parameter .

G=2z- 1.

The set of transfer functions Tvw4 that absorb the disturbance dynamics R is seen to be

Tvw4 = R(T4 + W'CDz- 1)

C = 2z- 1 E = z-l.

B=l, D = z-l,

We choose a particular solution X, Y of equation (5), say

(16)

X

and the corresponding set of controllers

(Y - W 4Az-l) - W'RAz- 1 K4 = - (X + W4Bz-l) + W'RBz-l . (17)

= 1 + z-l,

= z-2.

Y

All FIFO-stabilizing controllers are given by (7) as Z-1 - (1 - z-I)W I(=-Z-I----~~--~-(19) 1+Z-1+Z-1W

If Tvw4 of least degree is required, we divide CDz- 1 into T4 and substitute the quotient for -W' in (16). If K4 of least McMillan degree is desired, we divide RAz- 1 into Y - W 4Az- 1 and substitute the quotient for W' in (17).

where W is a polynomial parameter. The FIFO-stabilizing controller having least McMillan degree is obtained from (19) on setting W= -1: Ko = _z-l.

12. CONTROLLER DYNAMICS It is sometimes desirable to prescribe some of the controller poles. This leads to

The set of transfer functions from v to w in the closed-loop system that is FIFO - stabilized via (19) is given by (8) as

P=SPs

Tvw = z-1

+ z-2 + 2z- 3

-

z-2(1- 2z-1)W. (20)

where S is a polynomial in z-1 specified by the poles chosen and Ps is left free to secure FIFOstability.

The transfer function of least degree is obtained from (20) on putting W = -1,

Using (7) we end up with the polynomial equation

(21)

SPs - W Bz- 1 = X

(18)

The corresponding controller (9) is

for Ps and W. It is obvious that the partial pole assignment can be achieved if and only if Band S are cop rime polynomials. Then, if Ps and Ws is one solution of (18), the solution set is given by Ps Ps + W' Bz- 1

W

Kl = _z-1 and it happens to coincide with Ko. Let us allow for a longer Tvw, say of degree less than n = 4, and mimimize its H 2-norm. Parametrizing around (21) we have

Ws+W'S

Tvw = z-l

where W' is a free polynomial parameter. The corresponding set of controllers is

z-2(1 - 2z- 1)W

(22)

where W = Wo, a constant. Equation (12) for i = 0 reads

1

K

+ 2z- 2 -

(Y - WSAz-l) - W'SAz= - -'----=-:-=----::~:__-:-:--S S(Ps + W' Bz-l)

(-2z+5-2z- 1)wo) = -(z(2z+3-2z- 1)) that is

Of course we are free to use W' to achieve still further design requirements.

5wo = 2. Thus the minimizing parameter is W = ~ and (13) yields the optimal controller

13. EXAMPLE

_1

K2 = -z

To illustrate, we consider the generalized plant 409

3 + 2z -1 5 + 2Z-1

The solution set of (25) is obtained as 1 + z-1 + z- 1W' Ps -1 - z-1 + (1 - z-1)W' W

The resulting transfer function is T"w2 = -z

8

-1

+ Sz

-2

4_ 3

+ Sz

with W' a parameter. It is seen that W' results in the controller

and

_1

/(s=-z

Suppose we need a polynomial T"w whose coefficients Ti are bounded by

3

ITil:S2,

2 - z -1 l-z-1

that enjoys the desired property and gives rise to the transfer function

(23)

i=O,I , ...

= -1

Starting with the parametrization (22) we see at once that (23) is not satisfied for W = o. So we try W = Wo, a constant. Then (22) reads T"w = z-1

+ (2 -

wO)z-2

+ 2woz-3

14. CONCLUDING REMARKS We have introduced the notion of FIFO-stability as a tool for the design of linear discrete-time feedback systems in which transients are bound to vanish in finite time. All controllers that FIFO stabilize the given plant have been parametrized and the degrees of freedom then used to accommodate further design specifications.

and Wo must satisfy the inequalities 3

3

12 - wol :S 2' 12w ol :S 2' These are consistent for ~ ~, say, we have

~. Taking W

:S Wo :S

'-' _ _1 } ~3 - - z

=

1 + z -1 2 + Z-1

Finite transients may lead to large excursions of the regulated variables. Some of the design procedures, namely the H2 optimization and the transfer function shaping, can be used to eliminate this inconvenience. The price we have to pay, however, is an increase of the system complexity.

and T"w3

=

3 -2 + 2z + z -3 .

-1 Z

Suppose the disturbance v has a pole at z = ~ that should be rej ected at w in finite time . So we set R = 1 - ~Z-1 and solve equation (15) (1- ~z-1)TR - z-2(1- 2z- 1)W

15. REFERENCES

=

Astrom K. J. and B. Wittenmark (1984). Computer Controlled Systems: Theory and Design, Prentice-Hall, Englewood Cliffs, NJ . Kucera V. (1977) . Shortest correlation control stragegy, IEEE Trans. Automatic Control, 22, 463-465. Kucera V. (1979). Discrete Linear Control: The Polynomial Equation Approach, Wiley, Chichester. Kucera V. (1980). A dead-beat servo problem, Int. J. Control, 32, 107-113. Kucera V. (1991) . Analysis and Design of Discrete Linear Control Systems, Prentice-Hall , London . O 'Reilly J. (1981). The discrete linear time invariant time-optimal control problem: An overview, A utomatica, 17, 363-370. Vidyasagar M. (1985). Control System Synthesis: A Factorization Approach, MIT Press, Cambridge, MA.

(24) The solution set of (24) is TR W

z-1

22 15

+ 14 z-2 + z-2(1- 2z- 1)W' +

5 (1

1 -1)W'

- '3 z

where W' is a polynomial parameter. This yields the controller set (17) -1 '-' _ } ~4 --z

-22 + 37z- 1 - 5(1- z-1)(3 - z-1)W' 15 + 37z- 1 + 5z- 1(3 - Z-1)W'

where the controller /(4 of least McMillan degree corresponds to W' = O. This controller gives rise to -1 1 -1 14 -1 T"w4 = z (1 - '3 z )(1 + S-Z ). Finally let us try to make our controller into a proportional-plus-integral one. This requires setting P = (1 - z-1 )Ps and solving the equation (18)

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