Automatica, Vol. 33, No. 6, pp. 1163-1168, 1997

Pergamon PIk !3ooo5-1098(97)ooo19-8 0 1997 Elsevier Science Ltd. All rights reserved

Printed in Great Britain was-1098/97 $17.00 + 0.00

Brief Paper

Disturbance Attenuation in Finite-spectrum-assignment Controllers*

ALEXANDER MEDVEDEVT

Key Words-Delays; stability; multivariable systems; disturbance rejection.

Abstract-Two finite-spectrum-assignment controllers for continuous linear dynamic systems with multiple time delays in input signal are introduced. The controllers are shown to comprise least-squares state estimators. Disturbance attenua- tion properties stipulated by the choice of shift operator in the observers are studied. Q 1997 Elsevier Science Ltd.

1. Introduction Dynamic systems with input delays represents a simple but interesting and practically significant class of infinite- dimensional systems. A wide range of control methods have been developed to handle stabilization, reference input following and disturbance attentuation in plants with time delays. A special place among these methods belongs to the finite-spectrum-assignment (FSA) controllers introduced by Olbrot (1978). Under spectral controllability of the plant, an FSA controller places the roots of the closed-loop system characteristic polynomial at a finite number of arbitrarily predefined points in the complex plane. In this way, one can view the FSA controllers as generalizations of the Smith predictor.

Originally developed as state-feedback controllers, the FSA controllers combined with observers have been shown to solve the FSA problem of time-delay systems with unmeasurable state (Watanable and Ito, 1981). More recently, the same structure has been derived by application of standard results in distributed parameter systems (Pandolfi, 1991).

To be implemented, the FSA controller in Olbrot, (1978) demands, apart from the current value of the state vector, only evaluation of a convolution integral over a sliding window. It is interesting to see whether such an integral operator and/or time delays suffice for implementation of an FSA controller using system output. Naturally, to this end, one needs a dynamic system (observer) parameterized by means of the above-mentioned operators and reconstructing the signal to be fed back out of measurements of the plant inputs and outputs.

In most cases, Luenberger-type observers are exploited in output-feedback controllers. Since the Luenberger observer includes an internal loop, it contributes an extra factor to the closed-loop system characteristic polynomial and increases the system dynamic complexity. However, there is an

* Received 26 June 199.5; revised 1 April 1996; received in final form 4 December 1996. An abridged version of this paper was presented at the 13th IFAC World Congress, which was held in San Francisco, U.S.A. during 30 July-5 July 1996. The Published Proceedings of this IFAC Meeting may be ordered from: Elsevier Science Limited, The Boulevard, Langford Lane, Kiddington, Oxford OX5 lGB, U.K. This paper was recommended for publication in revised form by Associate Editor Y. Yamamoto under the direction of Editor Ruth F. Curtain. Corresponding author Professor A. Medvedev. Tel. +46 920 91302; Fax +46 920 91558; E-mail alexander.medvedev@sm.luth.se.

t Control Engineering Group, Lulefi University, S-971 87, Luleg, Sweden.

alternative in the form of so-called least-squares state estimators (observers) reconstructing the plant state vector in a feedforward manner. Fairly general results on the existence and properties of least-squares observers have been obtained by applying the notion of a pseudodifferential operator (Medvedev, 1996).

In this paper, two new output-feedback FSA controllers are constructed by taking advantage of two specific pseudodifferential operators: continuous time delay and the convolution operator over a sliding window. The latter is shown to improve disturbance attenuation properties of the controller with respect to 2-norm-bounded measurement noise.

2. Problem statement Consider a linear, continuous, time-invariant dynamic

system with multiple time delays in the control signal

i(t) = Ax(t) + 2 B,u(t - hi), , =,I

YW = W) + u(t), (1)

wherexERn, u~R”,yeR’, h,,=O,hi<hi+,, i=l,..., p, A, C and Bi, i = 0,. . , p, are real matrices of appropriate dimensipns, and u E L: is the measurement disturbance.

Let B = &1 exp (-Ahi)Bi. Assume now that the pair (A, C) is observable and the pair (A, B) is controllable.

The finite-spectrum-assignment (FSA) problem (Olbrot, 1978; Manitius and Olbrot, 1979) is formulated to design a feedback controller u = %(y, u) so that the characteristic polynomial of the closed-loop system is given by

det (sl - A - BF) = 0, (2)

where the controller gain matrix F can be chosen arbitrarily.

3. Finite-spectrum-assignment controllers In this section, two new output feedback controller

structures solving the FSA problem of the plant (1) are introduced. The controllers are shown to be based on continuous least-squares (LS) observers. Two different shift operators are used to parameterize the observers.

Consider the Gramian matrix

w(k) = c exp (-AT?,) CTC exp (-A 7i), r,r’.T(*)

evaluated on the ordered set of time delays

Y((k)={7, ,..., a}, 7i<7,+,, i=l,..., k.

Let a(.) denote the spectrum of the matrix (.), so that o(A)=&r, . . . . CL,} and all pi, i=l,... ,r, rsn, are distinct.

Lemma 1. Assume that the pair (A, C) is observable. Then any finite interval 9 E [0, x) contains a set of time delays T(n) c $ such that rank IV(n) =n. Furthermore, for an arbitrary T(n), rank W(n)<n only if the following two conditions are satisfied

1163

1164 Brief Papers

(i) o(A) includes complex eigenvalues pi, pj E u(A);

(ii) there is a r E T((n) for some k satisfying the equality

7 _ jWk) Pi-P,’

k=*l,*2,... .

Proof. See the Appendix.

As long as q(r(k) is positive-definite, the state vector in (1) can always be chosen so that w(k) = I (Medvedev and Toivonen, 1994). In Theorem 1 below, the plant is assumed to be in such form, for the sake of convenience. A more tangible motivation for choosing this particular realization will be given in the next section. Note also that if the inequality kl zn holds, the parameter k becomes an extra degree of freedom in the design procedure. Let

(Yv)(D, h; t) = 1” exp [-A(f3 + h)]Du(r + 0) dt? -h

TheErem 1. If the pair (A, C) is observable and the pair (A, B) is controllable then in any finite interval 4 c [0, 01) almost all sets of time delays T((n) are such that the controller

u(t) = F z exp (-ATrj)CT

’ (Ytr - zj) + c,$l (yu)(Bi~ zj + hi; f))] t4)

yields a solution to the FSA problem for the plant given by (1). All the excluded sets of time delays satisfy conditions (i) and (ii) of Lemma 1.

Proof. Let Z{.} denote the Laplace transform. In the sequel, when it agrees with the context, capital letters are used to represent Laplace transforms of the corresponding time functions, i.e. Z{z(t)} = Z(s). Then

= (sl - A)-‘[exp (-Ah) - exp (-hsZ)]DV(s). (5)

Taking the Laplace transform of the first equation in (1) under zero initial conditions gives

(sl - A)X(s) - 5 B,e-*“U(s) = 0. i=l,

(6)

Substituting (6) into the Laplace transform of (4) results in

U(s) = F 2 exp ( -AT.rj)CTC 5 E .3-0i )

X (~1 - A)-l2 /jil/(~)e-(h,+rds ( ,=”

+ (sl -A)-’ fi {exp [-A(hi + z,)] - e-(hf+5)sl)BiZ/(s))], i=O

or, after cancelling the delay terms,

U(s) = F c exp ( -AT7,)CTC(sl - A)-’ exp (-Az,)BU(s). r, E ‘(n )

Then, taking into account the special choice of plant realization, one arrives at

[I - F(s1 - A)-‘B]U(s) = 0. (7)

Note now, that the stability properties of the null solution of the closed-loop system (1) (4) are defined by the singularities of the Laplace transforms (6) and (7) (Manitius and Olbrot, 1979; Pandohi, 1991):

sl - A X(s) = o,

0 I - F(sI - A)-‘ii 1 [ 1 WI

Evaluation of the matrix determinant gives

det (sl -A) det [I - F(sl -A)-‘@

= det (sf - A) det [I - (sl - A)-‘BF] = det (sl -A - BF),

where the first equality comes from the property det (I - BA) = det (I - AB) for any rectangular matrices A and B allowing multiplication. 0

It is known from Manitius and Olbrot (1979) that a solution to the FSA problem is provided by a state feedback of the form

u(t) = l%(r),

x(r) =x(r) + 2 (Yu)(& h,; r). ,=”

(8)

When the plant state vector is not available, an observer is commonly employed for implementing the control law (8). Observer-based FSA controllers can be found in Watanable and Ito (1981) and Furukawa and Shimemura (1983). In contrast, the controller (4) does not contain any single Luenberger-type observer, and generates the desired control signal directly from the measurements of the plant inputs and outputs. In fact, apart from the gain matrix F, (4) represents a so-called continuous LS observer (Medvedev, 1996). In this case, the continuous delay operator is used for observer derivation. Indeed, since f satisfies the differential equation (for a thorough study of this kind of transformations, see Fiagberdzi and Pearson, 1986)

the observer

x’ = Ax(r) + k%(r),

f,(r) = c exp (-ATTj)CT r, E ‘(?I )

x (y(r - 7j) + C,go (yU)(Bi, rj + hii r)) (9)

yields an estimate for X. Along the lines of Medvedev and Toivonen (1994) it can be shown that f, is an exact (deadbeat) estimate for F(r), r 2 q, +h,, provided the measurements are available starting from r = 0.

The generalization of the LS state estimation to the case of the pseudodifferential operator of a certain class reported in Medvedev (1996) makes it possible to derive FSA controllers other than those based on the delay operator.

Given a nontrivial time delay r and a set of real distinct non-zero constants

A(r) = {Ai, j = 1, , r},

introduce the linear operator L: Lr+ L2 such that for u E L, and some r and A E A

(Lv)(A, r: r) = c’ eA(‘-%(e) de. ‘1-t

Another way of defining (L.)(h, t;r) is to use the Laplace inversion integral

(Lv)(A, ?; 1) = & r_ii_p(A, 7, s)V(s)e” ds, L F

where c is a suitable real constant and

p(A, 5, s) = s_~. (10)

In the context of pseudodifferential operators, p(A, r, s) is called the symbol of the operator &.)(A, r;r) (Egorov, 1986). In the sequel, the operator (L.)(A, z;r) is also applied to vectors, and is understood componentwise.

Consider the Gramian matrix

V(r) = x p(A,, 7, AT)CTCp(A,. c A). A, t A(r)

Lemma 2. Assume that A is chosen such that A iI a(A) = 0.

Brief Papers 1165

Then, for an observable pair (A, C) and any real set A of n elements, rank (V(n)) = n.

Proof: See Theorem 1 and Example 3 in Medvedev (1996).

Exactly as in the case of the delay operator, without loss of generality, it is assumed that the state-space representation for (1) is chosen so that V((n) = I.

Denote u(m) = exp(A.) and let (. *.) be the convolution integral

(v *u) = ‘~(r - @u(e) de

Theorem 2. Assume that all the conditions of Lemma 2 hold. Then the controller

u(t) = F 1

C ~(h,, r, AT)CT (LY)(Aj, r: t) A,cAW) [

+ C 2 (‘I’Lu)(B,, hi: t) +P(Ajt r, A)(u *Bu)(~) ( ,=O

-(u *BLa)(t)]] (II)

yields a solution to the FSA problem for the plant given by (I).

Proof. Taking Laplace transforms of the individual terms in (11) under zero initial conditions gives

Z{(Ly)(A,, r;t)} =p(Aj, r, s)C(sl - A)-‘2 B,e-h+U(s), . ,=O

Y { 5 (YLu)(B,, h,:r) ,=,I I

=p(Aj, s;s)(s~ - A)-l(B - 2 Bie-“+)U(s), 2-o

cY{(u *i&)(r)} = p(A,, 7, s)(sl - A)-‘&J(s),

LE{(p(A,, 7, A)@*&)(r)} =p(A,, z, A)(sl- A)-‘kJ(s).

Now, using the relationships above and taking into account the special choice of the plant realization, the Laplace transform of the control signal satisfies

[I - F(s1 - A)-‘B]U(s) = 0. (12)

This coincides with the Laplace transform of (4), given by (7). Therefore, following the proof of Theorem 1, one arrives at the conclusion that the controller (11) assigns the modes of the closed-loop system (l), (11) at the roots of the characteristic polynomial

det (sl - A - BF) = 0.

This completes the proof. 0

Once again, it is possible to show that

f(r) = 2 P(Aj, z, A’)CT (LY)(A,t T; t) A,EAW) [

+ C 2 (YLu)(Bi, hi; r) + P(Aj, 7, A)(v *k)(r) ( ,=O

-(v *BLu)(r))] (13)

is a continuous deadbeat observer estimate for x(t), r 2 7,

provided that the measurements are available starting from r = 0 (Medvedev, 1996). Thus it can be concluded that (4) and (11) are parameterizations of an observer-based FSA controller exploiting two different operators: continuous delay and sliding-window convolution (L.)(A, r;r). It is

noteworthy that the latter, along with its matrix generaliza- tion, naturally appears in the formulae for optimally robust controllers for delay systems (Dym er al., 1995).

Inspection of (4) and (11) shows that the controller dynamics are given by pseudodifferential equations (Egorov, 1986) instead of the usual differential ones. The choice of shift operator used for controller parameterization appears to be an additional degree of freedom in the controller design.

4. Disturbance arrenruarion in continuous LS observers In this section, measurement disturbance attenuation

properties of the continuous LS observers (9) and (13) are studied and compared. The plant with the output measurement corrupted by disturbance is given by (1) with u +O. In many practical situations, u is a high-frequency noise. If unfiltered by the observer, it could produce spiky control signals and course actuator wear-off.

Consider the symbol of (L.)(A, r; r). It can also be interpreted as the transfer function of an infinite-dimensional filter,

p(A, 7, jo) = 1 - eAr[cos (wr) - j sin (or)]

jo - A

whose gain-frequency characteristic is

Ip(h, 7, jw)l = m, f(o) = ’ + eZ”” - 2eAT ‘OS (wr) A” + 02

(14)

Note that using a series expansion of the exponential in (10). one gets

p(A,7,s)=~(A-s)‘-‘r’,

i=, i!

which implies filter stability for A > 0 as well as for A < 0. For low frequencies (o < [Al). the transfer function is

approximately a constant gain (see Fig. l),

whereas for high frequencies (o > IA/), the transfer envelope has the same slope as that of a pure integrator,

Let II *II2 and 1). IIx be the 2-norm and r-norm, respectively.

Lemma 3. For p(A, r, jw) the following relation holds:

IIdA, 5. jo)llz = L’ e2hr _ 1

7. (15)

1041 ..’ b .‘.....I ......‘I “.‘.... . . . ..J lo4 lo-' 10" 10' Id 10'

Fig. 1. Bode plots for the transfer functions (1 - eAr)/(s - A) (dashed line) (eZAr + l)/(s - A) (dash-dotted line) and

(1 - e(A-r)r)/(.s - A) (solid line).

1166 Brief Papers

Proof: Applying the definition of the 2-norm and taking (14), into account one gets

Observing that

direct evaluation of the integral in (16) yields (15). 0

Let xi E L2, i = 1,. . . , n, form a vector x = [x1,. . , _rnlT. Using the conventional notation

IIxll2 = ((~XWW dl)‘“?

the following upper bound on the estimation error in (13) can be derived.

Theorem 3. Assume that all the conditions of Lemma 2 are fuhilied and therefore there exists an LS observer (13) estimating f evaluated for the plant (1). Then the estimation error e(t) = ff - X satisfies the inequality

(17)

Proof Denoting the standard scalar product in R’” by (e, .), the observer (13) can be rewritten in the simpler form

where

Y, = (Ly)(Aj, T; t) + C

+&A,, r, A)(u *h)(t) - (v *h)(r)), j = 1, . , n.

Thus the estimation error e caused by a disturbance u is given by

(18)

Consider an element of e. Then, according to (18).

Ile, IL = syp lei(t)l = sy Ih -0 5 y{Iwl2 lZl21

where l.12 is the Euclidean norm in R”‘. The last step above follows from the Cauchy-Schwarz inequality. Furthermore, note that the vectors w,, i = 1,. . . , n, comprise an orthonormal set in R”‘, since their Gramian matrix ‘V(v(n) is a unit matrix. Therefore, denoting the elements of Z by Zi, i = 1, . , In,

“SP 4sl2 I421 = “‘tP 142 = sup (,Z z:)“2

Utilizing the well-known system gain (Doyle ef al., 1992)

IIZillx~ IIW)(A,, 7;4112 II412, i E [(i - 111 + LiO

It is noteworthy that /(L.)(Aj, 7:r)l12 exists and is well

defined in spite of the mapping’s Fourier transform not being strictly proper (see Lemma 3). Now, taking into account the composition of Z and summarizing the above argument, one obtains

Ihllx 5 (2 f II(L.)(Aj, 7; Oll$)

112 IbJll2.

j=I

Logically, this bound is’ valid for any element ei, and simultaneously is independent of i. Thus

IlelL~ (,$ 1 lI(~~)(A,, r;t)ll~)“2 I1412.

Taking advantage of (15) completes the proof. 0

It becomes obvious from Theorem 3 that smaller values of A should be preferred to lessen observer sensitivity to measurement noise. In this manner, IIp(A, r, s)l12 is decreasing (see Fig. 2). and so does the right-hand side of the inequality (17). The choice of r is a trade-off between the observer’s convergence rate and control-signal limitations. Smaller values of r might lead to the generation of high-amplitude control signals, whilst greater values of r tend to prolong observer transients.

Another important observation is that the inequality (17) is valid only when the plant realization is chosen to yield 4r= I. Thus, balancing (1) with respect to V is an essential part of the controller design.

Evidently, the structure of the controller (11) is more complicated than that of (4). and therefore a natural question to ask is whether the increased dynamic complexity implies, in some sense, a better performance. Regarding Theorem 3, one can note that there is no similar bound on the estimation error in (9). Indeed, the 2-norm for the delay operator is infinite, which means that a disturbance signal bounded in the 2-norm sense might generate signals of infinite magnitude. This, of course, is a clear shortcoming of the delay-operator-based controller.

5. Numerical example To illustrate the techniques developed above, a simple

simulation example is treated in this section. Consider a second-order plant with a time delay in the control signal:

m(r) = Ax(r) + h(r) + Bu(t - h,),

y(t) = C.W), (19)

where

A=[“0 _:,6], B=[;,787], C=[l 01, h,=0.2.

The controller (4) is implemented using T(, = 0 and T, = 0.1, placing the poles of the closed-loop system at s, = -3 and s2 = -4. A reference input r(r) =5sin(5r) is added to the

0.7

0.6

0.5

0.4 T = 0.3

02

0.1

0

lmJ 0 -30 hmba

Fig. 2. Ilp(h, r, s)j12 as a function of A and T.

Brief Papers 1167

1.0

0.0

-1.0

-2.0

-3.0

-4.0

0 2 4 6 8 10

Fig. 3. FSA controller for the plant with time delays: solid line, x,; dashed line, xa.

controller-generated feedback signal in order to demonstrate the closed-system response. Simulation results are shown in Fig. 3. As a reference, simulation results for the plant without delay but controlled by the state-feedback controller with the same F matrix i.e.

i.,,(t) = (A + BF)x,,(t) + Br(r), l? = B + exp (-Ah,) B,

(20)

are shown in Fig. 4. Note that the transfer functions of these systems are different, and only the denominators coincide.

A FSA controller of the type (11) is implemented, employing (L.)(-3,0.3, t). In principle, it takes two such operators to handle a second-order system (see Theorem 2). However, this can be overridden by directly using the system output signal y, which corresponds to zero multiplicity of the operator L. In the simulation of the controller, a different realization of (19) is used in order to achieve ‘V = I. Simulation results are shown in Fig. 5. As a reference, a state-feedback FSA controller of the form u(r) = F?(t) providing the same pole placement as the observer-based ones is designed and simulated (Fig. 6).

1.0

0.0

-1.0

-2.0

-3.0

4.0

t 0 2 4 6 8

Fig. 4. State-feedback_controller for the plant without time delays, i = Ax + Bu: solid line, x,: dashed line, x2.

1.0

0.0

-1.0

-2.0

-3.0

-4.0

t 0 2 4 6 8 10

Fig. 5. FSA controller with L operator for the plant (18): solid line, xl; dashed line, x2.

0 2 4 6 8

t

Fig. 6. State-feedback FSA controller for the plant (18): solid line, xi; dashed line, x2.

6. Conclusions Two novel finite-spectrum-assignment controllers for linear

continuous systems with time delays have been derived, employing two alternative pseudodifferential operators, namely time delay and the finite-memory convolution operator. The latter has been shown to provide an effective mean of enhancing measurement disturbance attentuation in the controller. A simulation example has been considered to illustrate the controller performances.

Acknowfedgmenr-Financial support by the Swedish Na- tional Board for Industrial and Technical Development, Grant F2558-1, is gratefully acknowledged.

References Doyle, J. C., Francis, B. A. and Tannenbaum. A. R. (1992)

Feedback Control Theory. Macmillan, New York. Dym, H., Georgiou, T. and Smith, M. (1995) Explicit

formulars for optimally robust controllers for delay systems. IEEE Trans. Autom. Control AC-48,656-669.

Egorov, Yu. (1986) Linear Differential Equations of Principal Type. Consultants Bureau, New York.

1168 Brief Papers

Fiagberdzi, Y. I. and Pearson, A. E. (1986) Feedback stabilization of linear autonomous time lag systems. IEEE Trans. Autom. Control AC-31,&V-855.

Furukawa, T. and Shimemura, E. (1983) Predictive control for systems with time delay. Int. J. Control 37,399-412.

Gohberg, I., Lancaster, P. and Rodman, L. (1986) Invariant Subspaces of Matrices with Applications. Wiley- Interscience, New York.

Medvedev, A. (1996) Continuous least-squares observers with applications. IEEE Trans. Autom. Control AC-41, 1530-1536.

Manitius, A. Z. and Olbrot, A. W. (1979) Finite spectrum assignment problem for systems with delays. IEEE Trans. Autom. Control AC-24,541-553.

Medvedev, A. and Toivonen, H. (1994). Feedforward time-delay structures in state estimation. Finite memory smoothing and continuous deadbeat observers. IEE Proc., ft D 141,121-129.

Olbrot, A. W. (1978) Stabilizability, detectability, and spectrum assignment for linear systems with general time delays. IEEE Trans. Autom. Control AC-23,887-890.

Pandolfi, L. (1991) Dynamic stabilization of systems with input delays. Automatica 27, 1047-1050.

Watanabe, K. and Ito, M. (1981) An observer for linear feedback control laws of multivariable systems with multiple delays in controls and outputs. Syst. Control. Lett. 1,54-59.

Appendix

Lemma 4, Assume that p(s) is an analytic function in a neighborhood of each eigenvalue p,, . . . , pLr of A (EL I,..., p, are assumed to be distinct). Then p(A) has exactly the same invariant subspaces as A if and only if the following conditions hold:

(i) P&i) #P&j) if Pi + Pj

(ii) dp(s)/ds IsXcli#O f or every eigenvalue pi with height of the Jordan block greater than 1.

Proof: See Theorem 2.11.3 in Gohberg et al. (1986).

Proof of Lemma 1. Define the unobservable space of the pair (A, C) as

X= ,fi Ker (CA’-‘).

Because of the observability assumption on A and C, it follows that X=0. Lemma 4 stipulates two necessary and sufficient conditions for preserving invariant subspaces of a matrix under a functional transformation. Since X is an invariant subspace of A, to obtain the necessary and sufficient conditions for the equality

fiKer(CAi-‘)=,fiKer(Cexp(-A?i))=B (A.l) i=l

to hold, it suffices to investigate when the transformation p(s) = exp ( -sri), i = 1, . . . , n, satisfies (i) and (ii) in Lemma 4. Condition (i) yields

eV;(l - e+,-!-%)) # 0

for all T E F(n) and all pi, pj E o(A). Since an exponential is never equal to zero, the above condition is equivalent to (3). Obviously, (ii) is always true for the function in question. Noting that (A.l) implies

Cexp t-AfJ rank W(n)=n, W(n) =

[ 1 ! C exp (-A?,)

and using the parameterization w(n) = WT(n)W(n) com- pletes the proof of the necessary and sufficient conditions for w(n) to become singular. Since w(n) loses rank at isolated points, it follows that nonsingularity of the Gramian can be achieved inside any arbitrary real interval 0