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Linear Periodic Control for ρ-Stabilization and Asymptotic Tracking Under Unbounded Output Multiplicative Perturbations2
Copyright @ IFAC Periodic Control Systems, Cemobbio-Como, Italy, 2001
LINEAR PERIODIC CONTROL FOR p-STABILIZATION AND ASYMPTOTIC TRACKING UNDER UNBOUNDED OUTPUT MULTIPLICATIVE PERTURBATIONS 2 Sergio Galeani,Osvaldo Maria Grasselli and Laura Menini . , 1
• Dip. Informatica, Sistemi e Produzione Universita di Roma Tor Vergata Via di Tor Vergata, 110 -00133 Roma, Italy
Abstract: In this paper the problem of the asymptotic tracking and stabilization with infinite gain margin, is addressed for linear time-invariant discrete-time multivariable systems in the case when unknown different scalar gains act on the outp~ts. Nece~sa~y and sufficient conditions for the solvability of the problem by means of a hnear penodlc discrete-time error feedback dynamic controller are derived. A procedure is given for designing the proposed periodic controller (containing an internal model of the reference signals). 'Copyright @2001 IFAC Keywords: Robustness , tracking, gain margins , MIMO , discrete-time system
1. INTRODUCTION
odic control is strongly helpful to face some structured uncertainties (Khargone~r et al., 1985) .
The problem of the robust asymptotic tracking and stabilization of a linear time-invariant multivariable system was studied by several authors both for continuous-time and for discrete-time . plants (see, e.g., (Davison, 1976),(Francis and Wonham, 1976), (Grasselli and Longhi, 1991b), (Grasselli et al., 1993), (Kimuraand Tanaka, 1981) and the references therein) . In such papers, both independent perturbations of the matrices characterizing the state-space description of the plant, and perturbations of some physical parameters, were considered. Here, the same kind of problem is dealt with for a system whose outputs are subject to unknown multiplicative perturbations of scalar gains whose values may be unbounded. This kind of problem, for the mere asymptotic stability, is called the gain margin problem. For SISO systems, and for the case when the unknown scalar gain may vary in a finite interval and a linear time-invariant controller is used, the problem was studied and solved in (Tannenbaum, 1980; Tannenbaum, 1982; Khargonekar and Tannenbaum, 1985) . If the unknown scalar gain is allowed to vary in an infinite interval the problem is usually called the infinite gain margin (IGM) problem. Periodic controllers to obtain an arbitrarily large gain margin were described in (Khargonekar et al., 1985) for discrete-time plants, in (Lee et al., 1987) for continuous-time plants, since peri-
1
2
E-mail: {galeani.grasselli.menini}Gdisp.uniroma2.it This work was supported by ASI, CNR and MURST
139
For MIMO systems, the same problem can be extended in two different ways: by considering a scalar unknown constant gain acting on every output (or input) of the system, or by considering different independent unknown constant gains acting on each scalar output (or each scalar input) of the system. Note that, in the second case, in general a controller designed to solve the problem when the unknown gains act on the input is not able to solve the problem when the unknown gains act on the output. An arbitrarily large gain margin for the first kind of extension by linear periodic controllers was obtained in (Francis and Georgiou, 1988; Yan et al., 1994) both for discrete-time systems and for continuous-time systems by digital control schemes (in (Francis and Georgiou, 1988) by means of a conventional periodic digital controller, in (Yan et al., 1994) by the use of Generalized Sampled-data Hold Functions (GSHF)) . For the second kind of extension, the authors are aware of two kinds of results for continuous-time systems: the well known LQR, state feedback results and some sufficient conditions (which become necessary if decoupling is required too) in (Maeda and Vidyasagar, 1985) for dynamic output feedback; for the same kind of extension, the IGM problem is solved in (Galeani et al., 2000) for discretetime systems, in the case when the unknown scalar gains act on the inputs of the plant, making use of linear periodic controllers.
Here the same kind of problem, i.e., the IGM problem for discrete-time linear time invariant (LTI) MIMO systems, will be addressed in the case of different independent unknown scalar gains acting on each scalar output of the system, with the additional requirement of the asymptotic tracking of reference signals belonging to a general known class of polynomial-exponential kind. Necessary and sufficient conditions, and a design procedure, will be given for the solution of the problem via a linear periodic error feedback controller, in the case of a prescribed rate of convergence of the free responses of the closed-loop system. The proposed compensator is the series connection of a LTI subcompensator, which guarantees the internal model needed to asymptotically track the exogenous signals, and of a linear periodically timevarying (LPTV) subcompensator whose role is to guarantee the prescribed robust asymptotic stability of the control system. Notations For a linear discrete-time system ~ its input, state a!1d output will be denoted by usU , x S ( .) and yS ( . ), respectively, unless otherwise specified, and the matrices giving its state-space description will be denoted by ASO , B S(-) , C S(.) , and DS(-), so that S = (AS(k), BS(k), CS(k) ,
D S(k)) will be characterized by:
+ 1) = AS(k)xS(k) + BS(k)uS(k), (la) yS(k) = CS(k)xS(k) + DS(k)uS(k) . (lb)
xS(k
and (M){s ,t) E lRa x b are defined , where the (i, j) element of (M){s ,t) is equal to the element (Si, tj) of M for each i E {1, ... , e} , j E {1, . .. , f} , whereas all the elements of (M) (s,t) are equal to the corresponding elements of M with the exception of the elements in positions (Si, tj) for i E {1 , .. . , e},j E {1, ... , f} which are equal to zero. For singletons like {sd, the shorthand (M){'1,t) will be used instead of (M){{st} ,t) and a similar notation will be used if t or, possibly, both sand t are singletons. For any given square matrix M E IRv x v , the spectral radius of M (i.e. the maximum of the moduli of the eigenvalues of M) will be denoted by r(M) . Note that the matrix norm IIMlloo' defined by
IIMlloo
:=
.
I:;=II(M){i,j)l , IIMlloo '
max
t E{I , ... ,v}
the relation r(M) ~
satisfies
For any matrix N E ~ x v of rank 11, NU will denote its right pseudo inverse, expressed by NU = N'(NN,)-l . For j , k E Z , k < j, the symbol I:~=j s(i), for any kind of argument s(i), will have the meaning of zero, even when the argument s(i ) cannot be computed for i < j or for i > k . 2. PRELIMINARlES In order to formally state the problem studied in this paper, some definitions are needed, related with LT! systems and LPTV systems. Then, let S = (A, B , C, D) be a LTI discrete-time system, whose state space description is given by:
Throughout the paper, Z+ will denote the set of non negative integers, and, for non-dynamic systems, either LTI or LPTV, the same symbol will be used for the matrix of the gains which characterizes its input-output behaviour and for the system itself.
x(k + 1) = Ax(k) + BU(k) , y(k) = Cx(k) + Du(k) , where k E and A , B , definitions referred to controlled,
The identity matrix of dimension 11 will be denoted by Iv or, when confusion cannot arise, simply by I. Zero vectors, of dimension s, and matrices, of dimensions s x r, will be denoted by Os and OS XTl respectively, whenever specifying such dimensions will be useful. Further, given a vector v E lR" and an ordered set of positive integers S = {S 1 , ... ,Se} of cardinality e ~ a and such that 1 ~ SI < Se ~ a, two new vectors (v){s) E IRe and (v){s) E r are defined, where the i-th component of (v){s) is equal to the si-th component of v for
(2a) (2b)
Z, x(k) E IRn , u(k) E IRP, y(k) E IRq , C and D are constant. Later on, the and properties stated for S will be any LTI system, e.g., the plant to be which will be introduced in Section 3.
In addition, since a periodic compensator will be used, consider a LPTV discrete-time system = (.A(k), B(k), C(k), D(k» , whose state space description is characterized by the equations:
e
x(k
+ 1) =
.A(k)x(k) y(k) = C(k)x(k)
+ B(k)u(k), + D(k)u(k),
(3a) (3b)
where k E Z, x(k) E IR'''', u(k) E IRP, y(k) E IRq, and .AO, BO, CO, DO, are w-periodic matriceswith entries in III Later on, the definitions and notations given for will be referred to any LPTV system, e.g., the compensator to be synthesized. Specifically, from now on, denote by ~(i,j) the state transition matrix of from time j to time i ~ j (where ~(i,j) = .A(i - l).A(i - 2)·· · .A(j) if i > j, and ~(j,j) = In). It is well known that, for any initial time ko E Z, and for all initial states
each i E {1 , ... ,el, whereas (v){s) is equal to v with the exception of the components in positions SI , ... , Se which are equal to zero. For singletons like {S I}, the shorthand (v) (S1) will be used instead of (v){{sd) ' Given a matrix M E lR"xb and two ordered sets of positive integers S = {SI,. " ,Se} of cardinality e ~ a and t = {tl,'" ,tt} of cardinality f ~ b, and such that 1 ~ SI < Se ~ a, 1 ~ tl < tt ~ b, two new matrices (M){s ,t) E lRe xt
e
e
140
e,
x(ko ) E IRn and all input functions u(·), the output response y( .) of the system for k ~ ko can be obtained from the output response of a lifted and specifically from that representation of introduced i!.1 (Meyer and Burrus, 1975), that is a LTI system eko whose input and output sequences are the lifted representations of the input and output sequences of The state space description of system eko is characterized by the equations:
stability of and also the rate of convergence of the state free motions of according to the following definition. Definition 1. For a given positive constant p ~ 1, the LPTV system of period w, is said to be p-stable if there exist positive p < p and c > 1 such that for all initial times ko E Z, and for all initial states x(ko ) E IRn, the state free motions of satisfy the following relation:
e
e,
e,
e.
xko(h + 1) y ko (h)
e,
e
IIx(k)1I < cp(k-ko)lIx(ko)ll ,
= Akoxko(h) + thoUko(h), (4a) = C koX ko (h) + D ko Uko (h), (4b)
Vk ~ ko, k
Z.
E
The form of Definition 1 is such that for p = 1 it implies the mere exponential stability of characterized by some p < 1. By applying Definition 1 to the LTI system eko described by (4), whose period is 1 and whose time variable is h, it is easy to derive the following proposition. Proposition 1. The w-periodic system is pstable if and only if all the characteristic multipliers of ..1.(.) are smaller than pW , in modulus, or, equivalently, if and only if, for an arbitrary ko E Z, eko is pW -stable.
where xko(h) E IRn, uko(h) E ]RWP, Yko(h) E ]RW<1 , Ako = ~(ko + w, ko), tho = [~(ko + w, ko + l).8(k o) . .. ~(ko +w, ko +w).8(ko +w -1)], C ko = [(C(ko)~(ko, ko))' ... (C(ko + w - l)~(ko + w 1, ko})']', and Dko is a (w x w) block matrix whose (i,j) block is zero if i < j, is equal to D(k o + j - 1, ko + j - 1) if i = j and is equal to C(k o + i - l)~(ko + i - 1, ko + j).8(ko + j - 1) otherwise. In fact , if Xko(O) = x(k o ) and uko(h) = [u'(k o + hw) .. . u'(ko + w - 1 + hw)]' for all h E Z+, then xko(h) = x(ko + hw) and Yko(h) = [y'(k o + hw) .. . Y'(ko +w -1 + hw)]' for all h E Z+. From now on, eko will be called the associated system of at (the initial) time ko.
e,
e
In the following, the characteristic mUltipliers of ..1.0 will be called also the characteristic multipliIn addition, the w-periodic system ers of system will be said to be p-stabilizable if there exists an w-periodic linear map Q(k) E IRpxn such that all the characteristic multipliers of A(.) + .80Q(-) are smaller than pW, in modulus; it will be said to be p-detectable if a dual property holds. Proposition 2. (Grasselli and Longhi, 1991a) 8ystem is p-stabilizable (or p-detectableJ if and only if, for an arbitrary ko E Z,
e
e.
e
e
Now consider the special case when system is time invariant, i.e., consider 8 = (A, B, C, D). In such a case, one can always look at system 8 as a LPTV system of arbitrary period w ~ 1; moreover, for every choice of the period w, the associated systems of 8 at all initial times ko have the same state space description of 8 0 = (Ao, B o, Co, Do) where Ao = AW, Bo = [AW-IB . .. B], Co = rC' ... (CAw-1 )']', and Do a (w x w)-block matrix whose (i,j) block is 0 if i < j, is equal to D if i = j and is equal to CAi-j-1 B if i > j.
e
rank [zIn - Ako .8ko]
(or
In the following, periodic systems will be denoted by a tilded capital letter, whereas LTI systems will be denoted by a capital letter with no tilde. However, the associated systems of LPTV systems like at a given time ko, despite the fact of being time-invariant, will be denoted by the tilded capital letter used to denote the LPTV system, with the subscript ko, like in the case of eko. Notice also that the general type of notation introduced in Section 1 through equations (1) could apply, in principle, also to systems 8 and 8 0 , so that symbols such as xS(k), uS(k), yS(k), xSO(h), uSO(h) and ySO(h) could actually be used instead of the symbols x(k), u(k), y(k), xo(h), uo(h) and yo(h), respectively; although this might seem confusing, sometimes it can be helpful.
= n, Vz E C, Izl ~ pW,(5)
_
rank [ z InC~oAko ] =
n,
Vz E C, Izl
~ pW].
(6)
Since the purpose of this paper is to make use of a LPTV dynamic compensator in order to control a LTI system, all the previous definitions, notations and discussion concerning the LPTV system can be applied to the LPTV connection of a LPTV (sub)compensator and a LTI subsystem. In particular, when referring to the LTI system 8 by itself (whose period w is 1), its characteristic multipliers are the eigenvalues of A (whose modulus is to be smaller than p for the p-stability of S) and conditions (5) and (6) reduce, respectively, to
e
e
rank [zIn - A and rank [zInc-
In addition, recall that the characteristic multipliers of A(-), i.e., the eigenvalues of Ako, are independent of ko, together with their algebraic multiplicities in the characteristic polynomial of Ako; therefore they characterize the asymptotic
B] = n,
A] = n,
Vz E C, Izl ~ p, (7) Vz E C, Izl
~ p.
(8)
Therefore conditions (5) and (6) are equivalent, respectively, to the pW -stabilizability and the pW_ detectability of the LTI system eko. It is also
141
recalled that the controllability and the reconstructibility of S can be tested by checking the rank of the matrices in (7) and (8), respectively, for all nonzero z E C (Grasselli, 1980; Grasselli and Longhi, 1991 a) .
to be all distinct and to satisfy l1?il ~ 1, im[1?d ~ 0, i = 1,2, .. . , IL, where· means complex conjugate, and, for i, j E Z+,
3. PROBLEM STATEMENT AND SOLUTION
Assume that, for each i = 1,2, .. . , ILl, im[1?d > 0 and, for each i = ILl + 1, ... , IL, im[1?iJ = 0, for some ILl E Z +, ILl ~ IL ·
The problem here studied of the asymptotic tracking with p-stability and infinite gain margin (to be formally defined) will be dealt with for the family P of LTI systems to be controlled defined in the subsequent Definition 3, on the basis of the nominal description of the plant P to be controlled - which will be assumed to be subject to output multiplicative perturbations. The state space description of P is given by: xP(k+ 1) = APxP(k) + BPuP(k) , (9a) yP(k)=CPxP(k) + DPuP(k) , (9b)
C):=O if i < j,
P
Z,8 j E
~ }, i = 1, 2, ..., IL,
. t
.
~ J.
The following theorem gives a solution to Problem 1. For the sake of brevity its proof is omitted. However, a procedure (i.e., the subsequent Procedure 1) for the design of a compensator that, under the conditions of Theorem 1, constitutes a solution to Problem 1, will be explicitly reported. Theorem 1. A LPTV dynamic error feedback compensator, having e(k) as input and uP(k) as output, that solves Problem 1 exists if and only if
(i) system P is p-stabilizable and p- detectable; (ii) rank [AP
;/iI ~:]
= nP +
q,
Vi
E
{I, ... ,IL}; (iii) each row of D P is nonzero. Remark 3. For the second kind of extension of the IGM problem to MIMO discrete-time systems, (Galeani et al. , 2000) is the only contribution of which the authors are aware, although there a stable controller is required and the unknown scalar gains act on each scalar input of the plant (instead of each scalar output as here); namely, if requirement (c) is substituted with the asymptotic stability of the required compensator k in the statement of Problem 1, a problem dual to the problem solved in (Galeani et al. , 2000) is obtained. Theorem 1 of (Galeani et al., 2000) gives the necessary and sufficient conditions for the solvability of the latter, under the assumption that all the inputs are needed for the p-stabilizability of the plant; such conditions are the dual ones of conditions (i) and (iii) of the above stated Theorem 1. However, the part of the compensator
(10)
~:= {r('): r(k)=~1[8j1?:-j+8;(1?;)k-jJ0), E
.
. : J." (t' _ J')'. If
(a) t is well-posed; (b) t is p-stable; (c) its error response e(k) := y(k) - r(k) asymptotically goes to zero for all reference signals r( .) E R, for all the initial states of t and for all initial times ko E Z.
The class R of reference signals r( ·) to be asymptotically tracked is assumed to be:
Vk
.,
t.
Now, the problem, here considered, of the asymptotic tracking with p-stability and infinite gain margin can be formally stated as follows . Problem 1. For the given nominal plant P , and for a prescribed positive p ~ 1, find (if any) a period w ~ 1 and a LPTV dynamic output feedback compensator k of period w , having y(k) and r(k) as inputs and uP(k) as output, such that the closed-loop LPTV control system t obtained by connecting k and P=. satisfies the following three requirements for all P=. E P, i. e., for all :=: E Xq :
where xP(k) E ]Rn , uP(k) E ]RP is the control input, yP(k) E ]Rq is the output, and A P , BP, C P , D P , are real matrices of suitable dimensions. Definition 2. The family Xq of admissible output multiplicative perturbations for P is defined as q Xq = {3 E ]Rqx , 3 = diag(6, .. · ,~q) : ~i E [l , +oo),ViE {I, . .. ,q}} . Definition 3. For the nominal plant P = (AP, BP, CP, DP), the family P of output perturbed LT! plants P=. is defined as the set of the LT! systems obtained as the series connection of P and an admissible output multiplicative perturbation 3 E X q , that is P = {P=. (APS:, BPa, CPs:, DPS:) : APa = AP,BPS: = BP,CPs: = 3C P , DPS: = 3D P , 3 E X q } . Remark 1. Throughout the paper the output yPs: of the actual plant to be controlled Ps will be assumed to be its controlled and measured output and will be denoted simply by y. 0 Remark 2. Notice that the choice of the interval [1, +00) for the admissible scalar gains ~i in Definition 2 does not imply any loss of generality. As a matter of fact, if more general intervals of the kind [~i.O , +00), with ~i.O =I 1 (and ~i.O > 0) , were to be considered, it would be sufficient to modify the nominal plant P by multiplying each row of c P by the corresponding value of ~i.O . 0
R := Rl EB R2 EB .... EB RI"
C) t
(11)
for some positive integers IL, Vi, i = 1,2, . . . ,IL, and some 1?i E C, i = 1,2, ... , IL, which are assumed
142
k
here proposed that is devoted to the mere pstabilization of the plant is much simpler than the dual of the compensator proposed in (Galeani et al., 2000) . For a comparison of our results with the other existing ones, mentioned in the introduction, 0 see Remark 4 in (Galeani et al., 2000). Remark 4. Condition (Hi) of Theorem 1 can be seen as a severe restriction, since it makes unsolvable Problem 1 for strictly causal plants (i.e., for when D P = 0) . The necessity of condition (Hi) of Theorem 1 when P is 8I80 (and then condition (Hi) is equivalent to the condition DP =f. 0) was conjectured in (Khargonekar et al., 1985). 0
. Fig. 1. The structure of the control system proposed to solve Problem 1. Remark 5. As opposed to the (obvious) necessity of conditions (i) (for output feedback p-stabilization) and (U) (for asymptotic tracking) of Theorem 1, the necessity of condition (iii) is linked to the error feedback structure chosen in Theorem 1 for the compensator, so that one could think that a weaker condition is necessary if a more general structure is chosen. However, it is well known that a tracking controller (based on the internal model principle) won't work in the presence of small uncertainties, unless an error feedback structure is chosen; on the other hand, if such a structure is used, the tracking capabilities given by the internal model will be preserved as long as the closed loop system is asymptotically stable, even for large uncertainties. 0
For each i E {I , ... , /J.d , set tP fij (z) = tP /ij (z) (z19 i )lIi (z - 19;)lIi, j = 1, ... , q. For each i E {/J.1 + 1, ... ,/J.} , set tPfij(Z) = tPfij(Z)(Z - 19 i )lIi, j 1, ... , q. For all j E {I, .. . , p} , set Cj(z) := zdeg(,pj(z)). Compute system KM = (AKM , BKM,CKM,DKM) as a minimal realization of the transfer matrix (z) cp(Z) ) dtag tPI (z) , . . . , tPp (z ) .
. (Cl
Step 2 (Computation of S) Compute system S =
(A, B, C, D) as the cascade connection of KM and P, by putting uP(k) = yKM (k), for all k E Z. IfS is reachable and observable, set S = (A, B, C, D) equal to S, else compute a minimal realization of S, and set S = (A, B, C, D) equal to it. Denote by n the dimension of the state space of system S (which has the same number of inputs and outputs than system P has). Step 3 (Fix the integer r and the sets of integers ri and Ci for i = O, ... , r - 1) Define 11" = {I, . .. ,p} and X = {I , . . . ,q}. Select r E N, 1 ~ r ~ q, such that it is possible to partition X into r ordered subsets ro = {rO,l,'" , rO,OIQ}' . . . , r r-l = {rr-l ,l, . .. ,rr-l,OIr _ , } , 2:~':-~ Qi = q, such that, for every set ri, i = 0, ... , r - 1, it is possible to .find an ordered subset Ci = {Ci,1,'" , Ci ,ni } of 11" such that matrix (D)(ri ,ci) is nonsingular. Step 4 (Fix matrices Li for i = 0, ... , r -1) For each i = 0 , . .. , r -1, define Li E IRq x q as follows : I (L) i (ri ,rd = OIi'
Step 5 (Fix matrices Ri for i = 0 , . . . , r -1) For each i = 0, ... , r - 1, define R; E IRpxq as follows : (Ri )(ci,rd = (D)(r i,cd) -1
-~; (~:)
(R;)(Ci,ri ) = O.
Qo := 0,
(13a) i-I
Qi:= -
L
[CAi-l-j BR j +
j=O
i:
CA i - l - k BRkQk] L j .
(13b)
k=j+1
Step 7 (Choice of the period w) Call v the reachability index of system S and set F = 2:jr ':-o1 LjCAJ.. Set w E Z equal to the smallest integer satisfying w ~ v + r such that the pair (AW, F) is reconstructible. Step 8 (Fix matrices W i for i = r, . . . , w - 1) Compute V E JR1lxq such that all the eigenvalues of AW + V F are equal to zero. Then,
Step 1 (Design of subcompensator KM (internal model of reference signals)) Set t/J! (z) = .. . = tPp(z) = 1. Fix /J. ordered subsets ft = {Ill, .. ' , ftq} , . . . , flJ. = {llJ.l, . .. , flJ.q} of {I, .. . ,p} such that for all i E {I, . . . ,/J.}
rank ([ A
,
Step 6 (Fix matrices Qi for i = 0, .. . , r -1) For each i = 0, . .. , r - 1, define Qi E IRqxq as:
The following design procedure of a solution to Problem 1 will refer to fig. 1, where the structure of the proposed control system is depicted. Procedure 1. (valid under the hypotheses that system P satisfies conditions i-iii of Theorem 1): Design of a solution to Problem 1.
P
(L.·)(ri,rd=O .
-Ro(Iq+Qo) :
define Ul
] , Bl
[
-14-1 (Iq + Qr-I) [AW-lB .. . AW-rB] andB2 = [Aw-r-lB ... AB
({I, ... ,np+q}.J,j)
nP
+ q.
Bl and compute U2 := B~ (V - BIUl ). Partition
(12)
143
U2 as U2
~r.
= [
l'
with W i E JRPxq " i
=
r
Ww-l
. .. ,w-l. Step 9 (Choice of matrix r) Set TJ = {I, .. . ,n}. Let T E JR"lxn such that T(AW + V F)T- 1 is in Jordan form, and let T be the degree of the minimal polynomial of (AW + VF) . Fix 11, ... ,Iq E IR+ such that, for each i E {1, . . . ,q}, li > 1 and 1
1 + li
11
(TV) (1'/,i)( FT
(pWT
11
-1
\ i ,1'/)
00<
Set r = diag(rl, ... 'Iq). Step 10 (Fix subcompensators £, R, H) For each i = r, ... , w - 1, set
+ 1)I/T q
Q,
1
.
(14) Wand
Qi := Oqxq . For each i = 0, ... ,r - 1, set W i := OpXq . Then, define £, R, Q and W as periodic non dynamic gain matrices £(k), R(k), Q(k) and W(k) such that £(hw + i) = L i , R(hw + i) = Ri, Q(hw + i) = Qi and W(hw + i) = W i , for all h E Z and for each i = 0, . . . ,w - 1. Furthermore, set H as the LPTV dynamic system whose state space description is characterized by the matrices; AH (k) = Iq, BH (k) = Iq } CH(k) = Iq, DH(k) = Oqxq for k
= hw + i,
hE Z, i
= 0, .. .
,w - 2, (15a)
A~ (k) = Oq xq, B~ (k) = Oqxq } CH(k) = Iq, DH(k) = Oqxq for k = hw
+ w - 1, h
E Z.
(15 b )
Step 11 Define the overall compensator k as the connection of r, £, H, Q, R, Wand KM according to the block diagram depicted in fig. 1.
<>
Remark 6. If, at step 9 of Procedure 1, 11, .. . , Iq E IR+ are fixed such that , Vi E {I, ... ,q}, li > 1 and the following relation is satisfied: -1-II(TV) li - 1
(FT-I) . (1'/,.)
("1'/)
11
< (pWT + 1)1/T 00
q
1
(where TJ = {I, ... ,n}), instead of (14), then the control system in fig . 1 satisfies all the requirements of Problem 1 for the whole family of output perturbations defined by Xq := {=: E IRqxq , =: = diag({1, . . . ,{q): {i E (-00,-1] u [+I,+oo) , Vi E
{1, .. . ,q}}.
0
4. REFERENCES Davison, E.J . (1976). The robust control of a servomechanism problem for linear timeinvariant multivariable systems. IEEE Trans. Aut. Control AC-21, 25-34. Francis, B.A. and T.T. Georgiou (1988) . Stability theory for linear time-invariant plants with periodic digital controllers. IEEE Trans. Automatic Control 33(9), 820-832.
Francis, B.A. and W.M. Wonham (1976) . The internal model principle of control theory. Automatica 12,457- 465. Galeani, S., O. M. Grasselli and L. Menini (2000) . Linear periodic control for strong p-stabilization with infinite gain margin. In: Proc. of the American Control Conference. Chicago, IL, USA. pp. 1164-1168. Grasselli, O.M. (1980). Conditions for controllability and reconstructibility of discrete-time linear composite systems. Int. J. of Control 31(3), 433-44l. Grasselli, O.M. and S. Longhi (1991a) . Finite zero structure of linear periodic discrete-time systems. Int.J.Syst.Science 22(10),1785-1806. Grasselli, O .M. and S. Longhi (1991b) . Robust output regulation under uncertainties of physical parameters. Syst. Contr. Lett. 16, 33-40 . Grasselli, O.M., S. Longhi and A. Tornambe (1993). Robust tracking and performance for multivariable systems under physical parameter uncertainties. A utomatica 29, 169-179. Khargonekar, P.P. and A. Tannenbaum (1985). Non-euclidian metrics and the robust stabilization of systems with parameter uncertainty. IEEE Trans . Automatic Control 30(10), 1005-1013. . Khargonekar, P.P., K. Poolla and A. Tannenbaum (1985). Robust control of linear time-invariant plants using periodic compensation. IEEE Trans. Automatic Control 30(11), 1088-1096. Kimura, H . and Y. Tanaka (1981) . Minimal-time minimal order deadbeat regulators with internal stability. IEEE Trans . Automatic Control AC-26, 1276-1282. Lee, S., S.M. Meerkov and T . Runolfsson (1987) . Vibrational feedback control: Zeros placement capabilities. IEEE Trans. Automatic Control 32(7), 604- 610. Maeda, H. and M. Vidyasagar (1985). Design of multivariable feedback systems with infinite gain margin and decoupling. Systemst'3Control Letters 6, 127-130. Meyer, R.A. and C .S. Burrus (1975) . A unified analysis of multirate and periodically timevarying digital filters . IEEE Trans. Circuit Systems 22, 162-168. Tannenbaum, A. (1980). Feedback stabilization of linear dynamical plants with uncertainty in the gain factor . Int. J. of Control 32(1), 116. Tannenbaum, A. (1982) . Modified NevanlinnaPick interpolation and feedback stabilization of linear plants with uncertainty in the gain factor . Int. J. of Control 36(2), 331-6. Yan, W .-Y., B .D.O. Anderson and R.R. Bitmead (1994). On the gain margin improvement using dynamic compensation based on generalized sampled-data hold functions . IEEE Trans. Automatic Control 39(11), 2347- 2354.
144
LINEAR PERIODIC CONTROL FOR p-STABILIZATION AND ASYMPTOTIC TRACKING UNDER UNBOUNDED OUTPUT MULTIPLICATIVE PERTURBATIONS 2 Sergio Galeani,Osvaldo Maria Grasselli and Laura Menini . , 1
• Dip. Informatica, Sistemi e Produzione Universita di Roma Tor Vergata Via di Tor Vergata, 110 -00133 Roma, Italy
Abstract: In this paper the problem of the asymptotic tracking and stabilization with infinite gain margin, is addressed for linear time-invariant discrete-time multivariable systems in the case when unknown different scalar gains act on the outp~ts. Nece~sa~y and sufficient conditions for the solvability of the problem by means of a hnear penodlc discrete-time error feedback dynamic controller are derived. A procedure is given for designing the proposed periodic controller (containing an internal model of the reference signals). 'Copyright @2001 IFAC Keywords: Robustness , tracking, gain margins , MIMO , discrete-time system
1. INTRODUCTION
odic control is strongly helpful to face some structured uncertainties (Khargone~r et al., 1985) .
The problem of the robust asymptotic tracking and stabilization of a linear time-invariant multivariable system was studied by several authors both for continuous-time and for discrete-time . plants (see, e.g., (Davison, 1976),(Francis and Wonham, 1976), (Grasselli and Longhi, 1991b), (Grasselli et al., 1993), (Kimuraand Tanaka, 1981) and the references therein) . In such papers, both independent perturbations of the matrices characterizing the state-space description of the plant, and perturbations of some physical parameters, were considered. Here, the same kind of problem is dealt with for a system whose outputs are subject to unknown multiplicative perturbations of scalar gains whose values may be unbounded. This kind of problem, for the mere asymptotic stability, is called the gain margin problem. For SISO systems, and for the case when the unknown scalar gain may vary in a finite interval and a linear time-invariant controller is used, the problem was studied and solved in (Tannenbaum, 1980; Tannenbaum, 1982; Khargonekar and Tannenbaum, 1985) . If the unknown scalar gain is allowed to vary in an infinite interval the problem is usually called the infinite gain margin (IGM) problem. Periodic controllers to obtain an arbitrarily large gain margin were described in (Khargonekar et al., 1985) for discrete-time plants, in (Lee et al., 1987) for continuous-time plants, since peri-
1
2
E-mail: {galeani.grasselli.menini}Gdisp.uniroma2.it This work was supported by ASI, CNR and MURST
139
For MIMO systems, the same problem can be extended in two different ways: by considering a scalar unknown constant gain acting on every output (or input) of the system, or by considering different independent unknown constant gains acting on each scalar output (or each scalar input) of the system. Note that, in the second case, in general a controller designed to solve the problem when the unknown gains act on the input is not able to solve the problem when the unknown gains act on the output. An arbitrarily large gain margin for the first kind of extension by linear periodic controllers was obtained in (Francis and Georgiou, 1988; Yan et al., 1994) both for discrete-time systems and for continuous-time systems by digital control schemes (in (Francis and Georgiou, 1988) by means of a conventional periodic digital controller, in (Yan et al., 1994) by the use of Generalized Sampled-data Hold Functions (GSHF)) . For the second kind of extension, the authors are aware of two kinds of results for continuous-time systems: the well known LQR, state feedback results and some sufficient conditions (which become necessary if decoupling is required too) in (Maeda and Vidyasagar, 1985) for dynamic output feedback; for the same kind of extension, the IGM problem is solved in (Galeani et al., 2000) for discretetime systems, in the case when the unknown scalar gains act on the inputs of the plant, making use of linear periodic controllers.
Here the same kind of problem, i.e., the IGM problem for discrete-time linear time invariant (LTI) MIMO systems, will be addressed in the case of different independent unknown scalar gains acting on each scalar output of the system, with the additional requirement of the asymptotic tracking of reference signals belonging to a general known class of polynomial-exponential kind. Necessary and sufficient conditions, and a design procedure, will be given for the solution of the problem via a linear periodic error feedback controller, in the case of a prescribed rate of convergence of the free responses of the closed-loop system. The proposed compensator is the series connection of a LTI subcompensator, which guarantees the internal model needed to asymptotically track the exogenous signals, and of a linear periodically timevarying (LPTV) subcompensator whose role is to guarantee the prescribed robust asymptotic stability of the control system. Notations For a linear discrete-time system ~ its input, state a!1d output will be denoted by usU , x S ( .) and yS ( . ), respectively, unless otherwise specified, and the matrices giving its state-space description will be denoted by ASO , B S(-) , C S(.) , and DS(-), so that S = (AS(k), BS(k), CS(k) ,
D S(k)) will be characterized by:
+ 1) = AS(k)xS(k) + BS(k)uS(k), (la) yS(k) = CS(k)xS(k) + DS(k)uS(k) . (lb)
xS(k
and (M){s ,t) E lRa x b are defined , where the (i, j) element of (M){s ,t) is equal to the element (Si, tj) of M for each i E {1, ... , e} , j E {1, . .. , f} , whereas all the elements of (M) (s,t) are equal to the corresponding elements of M with the exception of the elements in positions (Si, tj) for i E {1 , .. . , e},j E {1, ... , f} which are equal to zero. For singletons like {sd, the shorthand (M){'1,t) will be used instead of (M){{st} ,t) and a similar notation will be used if t or, possibly, both sand t are singletons. For any given square matrix M E IRv x v , the spectral radius of M (i.e. the maximum of the moduli of the eigenvalues of M) will be denoted by r(M) . Note that the matrix norm IIMlloo' defined by
IIMlloo
:=
.
I:;=II(M){i,j)l , IIMlloo '
max
t E{I , ... ,v}
the relation r(M) ~
satisfies
For any matrix N E ~ x v of rank 11, NU will denote its right pseudo inverse, expressed by NU = N'(NN,)-l . For j , k E Z , k < j, the symbol I:~=j s(i), for any kind of argument s(i), will have the meaning of zero, even when the argument s(i ) cannot be computed for i < j or for i > k . 2. PRELIMINARlES In order to formally state the problem studied in this paper, some definitions are needed, related with LT! systems and LPTV systems. Then, let S = (A, B , C, D) be a LTI discrete-time system, whose state space description is given by:
Throughout the paper, Z+ will denote the set of non negative integers, and, for non-dynamic systems, either LTI or LPTV, the same symbol will be used for the matrix of the gains which characterizes its input-output behaviour and for the system itself.
x(k + 1) = Ax(k) + BU(k) , y(k) = Cx(k) + Du(k) , where k E and A , B , definitions referred to controlled,
The identity matrix of dimension 11 will be denoted by Iv or, when confusion cannot arise, simply by I. Zero vectors, of dimension s, and matrices, of dimensions s x r, will be denoted by Os and OS XTl respectively, whenever specifying such dimensions will be useful. Further, given a vector v E lR" and an ordered set of positive integers S = {S 1 , ... ,Se} of cardinality e ~ a and such that 1 ~ SI < Se ~ a, two new vectors (v){s) E IRe and (v){s) E r are defined, where the i-th component of (v){s) is equal to the si-th component of v for
(2a) (2b)
Z, x(k) E IRn , u(k) E IRP, y(k) E IRq , C and D are constant. Later on, the and properties stated for S will be any LTI system, e.g., the plant to be which will be introduced in Section 3.
In addition, since a periodic compensator will be used, consider a LPTV discrete-time system = (.A(k), B(k), C(k), D(k» , whose state space description is characterized by the equations:
e
x(k
+ 1) =
.A(k)x(k) y(k) = C(k)x(k)
+ B(k)u(k), + D(k)u(k),
(3a) (3b)
where k E Z, x(k) E IR'''', u(k) E IRP, y(k) E IRq, and .AO, BO, CO, DO, are w-periodic matriceswith entries in III Later on, the definitions and notations given for will be referred to any LPTV system, e.g., the compensator to be synthesized. Specifically, from now on, denote by ~(i,j) the state transition matrix of from time j to time i ~ j (where ~(i,j) = .A(i - l).A(i - 2)·· · .A(j) if i > j, and ~(j,j) = In). It is well known that, for any initial time ko E Z, and for all initial states
each i E {1 , ... ,el, whereas (v){s) is equal to v with the exception of the components in positions SI , ... , Se which are equal to zero. For singletons like {S I}, the shorthand (v) (S1) will be used instead of (v){{sd) ' Given a matrix M E lR"xb and two ordered sets of positive integers S = {SI,. " ,Se} of cardinality e ~ a and t = {tl,'" ,tt} of cardinality f ~ b, and such that 1 ~ SI < Se ~ a, 1 ~ tl < tt ~ b, two new matrices (M){s ,t) E lRe xt
e
e
140
e,
x(ko ) E IRn and all input functions u(·), the output response y( .) of the system for k ~ ko can be obtained from the output response of a lifted and specifically from that representation of introduced i!.1 (Meyer and Burrus, 1975), that is a LTI system eko whose input and output sequences are the lifted representations of the input and output sequences of The state space description of system eko is characterized by the equations:
stability of and also the rate of convergence of the state free motions of according to the following definition. Definition 1. For a given positive constant p ~ 1, the LPTV system of period w, is said to be p-stable if there exist positive p < p and c > 1 such that for all initial times ko E Z, and for all initial states x(ko ) E IRn, the state free motions of satisfy the following relation:
e
e,
e,
e.
xko(h + 1) y ko (h)
e,
e
IIx(k)1I < cp(k-ko)lIx(ko)ll ,
= Akoxko(h) + thoUko(h), (4a) = C koX ko (h) + D ko Uko (h), (4b)
Vk ~ ko, k
Z.
E
The form of Definition 1 is such that for p = 1 it implies the mere exponential stability of characterized by some p < 1. By applying Definition 1 to the LTI system eko described by (4), whose period is 1 and whose time variable is h, it is easy to derive the following proposition. Proposition 1. The w-periodic system is pstable if and only if all the characteristic multipliers of ..1.(.) are smaller than pW , in modulus, or, equivalently, if and only if, for an arbitrary ko E Z, eko is pW -stable.
where xko(h) E IRn, uko(h) E ]RWP, Yko(h) E ]RW<1 , Ako = ~(ko + w, ko), tho = [~(ko + w, ko + l).8(k o) . .. ~(ko +w, ko +w).8(ko +w -1)], C ko = [(C(ko)~(ko, ko))' ... (C(ko + w - l)~(ko + w 1, ko})']', and Dko is a (w x w) block matrix whose (i,j) block is zero if i < j, is equal to D(k o + j - 1, ko + j - 1) if i = j and is equal to C(k o + i - l)~(ko + i - 1, ko + j).8(ko + j - 1) otherwise. In fact , if Xko(O) = x(k o ) and uko(h) = [u'(k o + hw) .. . u'(ko + w - 1 + hw)]' for all h E Z+, then xko(h) = x(ko + hw) and Yko(h) = [y'(k o + hw) .. . Y'(ko +w -1 + hw)]' for all h E Z+. From now on, eko will be called the associated system of at (the initial) time ko.
e,
e
In the following, the characteristic mUltipliers of ..1.0 will be called also the characteristic multipliIn addition, the w-periodic system ers of system will be said to be p-stabilizable if there exists an w-periodic linear map Q(k) E IRpxn such that all the characteristic multipliers of A(.) + .80Q(-) are smaller than pW, in modulus; it will be said to be p-detectable if a dual property holds. Proposition 2. (Grasselli and Longhi, 1991a) 8ystem is p-stabilizable (or p-detectableJ if and only if, for an arbitrary ko E Z,
e
e.
e
e
Now consider the special case when system is time invariant, i.e., consider 8 = (A, B, C, D). In such a case, one can always look at system 8 as a LPTV system of arbitrary period w ~ 1; moreover, for every choice of the period w, the associated systems of 8 at all initial times ko have the same state space description of 8 0 = (Ao, B o, Co, Do) where Ao = AW, Bo = [AW-IB . .. B], Co = rC' ... (CAw-1 )']', and Do a (w x w)-block matrix whose (i,j) block is 0 if i < j, is equal to D if i = j and is equal to CAi-j-1 B if i > j.
e
rank [zIn - Ako .8ko]
(or
In the following, periodic systems will be denoted by a tilded capital letter, whereas LTI systems will be denoted by a capital letter with no tilde. However, the associated systems of LPTV systems like at a given time ko, despite the fact of being time-invariant, will be denoted by the tilded capital letter used to denote the LPTV system, with the subscript ko, like in the case of eko. Notice also that the general type of notation introduced in Section 1 through equations (1) could apply, in principle, also to systems 8 and 8 0 , so that symbols such as xS(k), uS(k), yS(k), xSO(h), uSO(h) and ySO(h) could actually be used instead of the symbols x(k), u(k), y(k), xo(h), uo(h) and yo(h), respectively; although this might seem confusing, sometimes it can be helpful.
= n, Vz E C, Izl ~ pW,(5)
_
rank [ z InC~oAko ] =
n,
Vz E C, Izl
~ pW].
(6)
Since the purpose of this paper is to make use of a LPTV dynamic compensator in order to control a LTI system, all the previous definitions, notations and discussion concerning the LPTV system can be applied to the LPTV connection of a LPTV (sub)compensator and a LTI subsystem. In particular, when referring to the LTI system 8 by itself (whose period w is 1), its characteristic multipliers are the eigenvalues of A (whose modulus is to be smaller than p for the p-stability of S) and conditions (5) and (6) reduce, respectively, to
e
e
rank [zIn - A and rank [zInc-
In addition, recall that the characteristic multipliers of A(-), i.e., the eigenvalues of Ako, are independent of ko, together with their algebraic multiplicities in the characteristic polynomial of Ako; therefore they characterize the asymptotic
B] = n,
A] = n,
Vz E C, Izl ~ p, (7) Vz E C, Izl
~ p.
(8)
Therefore conditions (5) and (6) are equivalent, respectively, to the pW -stabilizability and the pW_ detectability of the LTI system eko. It is also
141
recalled that the controllability and the reconstructibility of S can be tested by checking the rank of the matrices in (7) and (8), respectively, for all nonzero z E C (Grasselli, 1980; Grasselli and Longhi, 1991 a) .
to be all distinct and to satisfy l1?il ~ 1, im[1?d ~ 0, i = 1,2, .. . , IL, where· means complex conjugate, and, for i, j E Z+,
3. PROBLEM STATEMENT AND SOLUTION
Assume that, for each i = 1,2, .. . , ILl, im[1?d > 0 and, for each i = ILl + 1, ... , IL, im[1?iJ = 0, for some ILl E Z +, ILl ~ IL ·
The problem here studied of the asymptotic tracking with p-stability and infinite gain margin (to be formally defined) will be dealt with for the family P of LTI systems to be controlled defined in the subsequent Definition 3, on the basis of the nominal description of the plant P to be controlled - which will be assumed to be subject to output multiplicative perturbations. The state space description of P is given by: xP(k+ 1) = APxP(k) + BPuP(k) , (9a) yP(k)=CPxP(k) + DPuP(k) , (9b)
C):=O if i < j,
P
Z,8 j E
~ }, i = 1, 2, ..., IL,
. t
.
~ J.
The following theorem gives a solution to Problem 1. For the sake of brevity its proof is omitted. However, a procedure (i.e., the subsequent Procedure 1) for the design of a compensator that, under the conditions of Theorem 1, constitutes a solution to Problem 1, will be explicitly reported. Theorem 1. A LPTV dynamic error feedback compensator, having e(k) as input and uP(k) as output, that solves Problem 1 exists if and only if
(i) system P is p-stabilizable and p- detectable; (ii) rank [AP
;/iI ~:]
= nP +
q,
Vi
E
{I, ... ,IL}; (iii) each row of D P is nonzero. Remark 3. For the second kind of extension of the IGM problem to MIMO discrete-time systems, (Galeani et al. , 2000) is the only contribution of which the authors are aware, although there a stable controller is required and the unknown scalar gains act on each scalar input of the plant (instead of each scalar output as here); namely, if requirement (c) is substituted with the asymptotic stability of the required compensator k in the statement of Problem 1, a problem dual to the problem solved in (Galeani et al. , 2000) is obtained. Theorem 1 of (Galeani et al., 2000) gives the necessary and sufficient conditions for the solvability of the latter, under the assumption that all the inputs are needed for the p-stabilizability of the plant; such conditions are the dual ones of conditions (i) and (iii) of the above stated Theorem 1. However, the part of the compensator
(10)
~:= {r('): r(k)=~1[8j1?:-j+8;(1?;)k-jJ0), E
.
. : J." (t' _ J')'. If
(a) t is well-posed; (b) t is p-stable; (c) its error response e(k) := y(k) - r(k) asymptotically goes to zero for all reference signals r( .) E R, for all the initial states of t and for all initial times ko E Z.
The class R of reference signals r( ·) to be asymptotically tracked is assumed to be:
Vk
.,
t.
Now, the problem, here considered, of the asymptotic tracking with p-stability and infinite gain margin can be formally stated as follows . Problem 1. For the given nominal plant P , and for a prescribed positive p ~ 1, find (if any) a period w ~ 1 and a LPTV dynamic output feedback compensator k of period w , having y(k) and r(k) as inputs and uP(k) as output, such that the closed-loop LPTV control system t obtained by connecting k and P=. satisfies the following three requirements for all P=. E P, i. e., for all :=: E Xq :
where xP(k) E ]Rn , uP(k) E ]RP is the control input, yP(k) E ]Rq is the output, and A P , BP, C P , D P , are real matrices of suitable dimensions. Definition 2. The family Xq of admissible output multiplicative perturbations for P is defined as q Xq = {3 E ]Rqx , 3 = diag(6, .. · ,~q) : ~i E [l , +oo),ViE {I, . .. ,q}} . Definition 3. For the nominal plant P = (AP, BP, CP, DP), the family P of output perturbed LT! plants P=. is defined as the set of the LT! systems obtained as the series connection of P and an admissible output multiplicative perturbation 3 E X q , that is P = {P=. (APS:, BPa, CPs:, DPS:) : APa = AP,BPS: = BP,CPs: = 3C P , DPS: = 3D P , 3 E X q } . Remark 1. Throughout the paper the output yPs: of the actual plant to be controlled Ps will be assumed to be its controlled and measured output and will be denoted simply by y. 0 Remark 2. Notice that the choice of the interval [1, +00) for the admissible scalar gains ~i in Definition 2 does not imply any loss of generality. As a matter of fact, if more general intervals of the kind [~i.O , +00), with ~i.O =I 1 (and ~i.O > 0) , were to be considered, it would be sufficient to modify the nominal plant P by multiplying each row of c P by the corresponding value of ~i.O . 0
R := Rl EB R2 EB .... EB RI"
C) t
(11)
for some positive integers IL, Vi, i = 1,2, . . . ,IL, and some 1?i E C, i = 1,2, ... , IL, which are assumed
142
k
here proposed that is devoted to the mere pstabilization of the plant is much simpler than the dual of the compensator proposed in (Galeani et al., 2000) . For a comparison of our results with the other existing ones, mentioned in the introduction, 0 see Remark 4 in (Galeani et al., 2000). Remark 4. Condition (Hi) of Theorem 1 can be seen as a severe restriction, since it makes unsolvable Problem 1 for strictly causal plants (i.e., for when D P = 0) . The necessity of condition (Hi) of Theorem 1 when P is 8I80 (and then condition (Hi) is equivalent to the condition DP =f. 0) was conjectured in (Khargonekar et al., 1985). 0
. Fig. 1. The structure of the control system proposed to solve Problem 1. Remark 5. As opposed to the (obvious) necessity of conditions (i) (for output feedback p-stabilization) and (U) (for asymptotic tracking) of Theorem 1, the necessity of condition (iii) is linked to the error feedback structure chosen in Theorem 1 for the compensator, so that one could think that a weaker condition is necessary if a more general structure is chosen. However, it is well known that a tracking controller (based on the internal model principle) won't work in the presence of small uncertainties, unless an error feedback structure is chosen; on the other hand, if such a structure is used, the tracking capabilities given by the internal model will be preserved as long as the closed loop system is asymptotically stable, even for large uncertainties. 0
For each i E {I , ... , /J.d , set tP fij (z) = tP /ij (z) (z19 i )lIi (z - 19;)lIi, j = 1, ... , q. For each i E {/J.1 + 1, ... ,/J.} , set tPfij(Z) = tPfij(Z)(Z - 19 i )lIi, j 1, ... , q. For all j E {I, .. . , p} , set Cj(z) := zdeg(,pj(z)). Compute system KM = (AKM , BKM,CKM,DKM) as a minimal realization of the transfer matrix (z) cp(Z) ) dtag tPI (z) , . . . , tPp (z ) .
. (Cl
Step 2 (Computation of S) Compute system S =
(A, B, C, D) as the cascade connection of KM and P, by putting uP(k) = yKM (k), for all k E Z. IfS is reachable and observable, set S = (A, B, C, D) equal to S, else compute a minimal realization of S, and set S = (A, B, C, D) equal to it. Denote by n the dimension of the state space of system S (which has the same number of inputs and outputs than system P has). Step 3 (Fix the integer r and the sets of integers ri and Ci for i = O, ... , r - 1) Define 11" = {I, . .. ,p} and X = {I , . . . ,q}. Select r E N, 1 ~ r ~ q, such that it is possible to partition X into r ordered subsets ro = {rO,l,'" , rO,OIQ}' . . . , r r-l = {rr-l ,l, . .. ,rr-l,OIr _ , } , 2:~':-~ Qi = q, such that, for every set ri, i = 0, ... , r - 1, it is possible to .find an ordered subset Ci = {Ci,1,'" , Ci ,ni } of 11" such that matrix (D)(ri ,ci) is nonsingular. Step 4 (Fix matrices Li for i = 0, ... , r -1) For each i = 0 , . .. , r -1, define Li E IRq x q as follows : I (L) i (ri ,rd = OIi'
Step 5 (Fix matrices Ri for i = 0 , . . . , r -1) For each i = 0, ... , r - 1, define R; E IRpxq as follows : (Ri )(ci,rd = (D)(r i,cd) -1
-~; (~:)
(R;)(Ci,ri ) = O.
Qo := 0,
(13a) i-I
Qi:= -
L
[CAi-l-j BR j +
j=O
i:
CA i - l - k BRkQk] L j .
(13b)
k=j+1
Step 7 (Choice of the period w) Call v the reachability index of system S and set F = 2:jr ':-o1 LjCAJ.. Set w E Z equal to the smallest integer satisfying w ~ v + r such that the pair (AW, F) is reconstructible. Step 8 (Fix matrices W i for i = r, . . . , w - 1) Compute V E JR1lxq such that all the eigenvalues of AW + V F are equal to zero. Then,
Step 1 (Design of subcompensator KM (internal model of reference signals)) Set t/J! (z) = .. . = tPp(z) = 1. Fix /J. ordered subsets ft = {Ill, .. ' , ftq} , . . . , flJ. = {llJ.l, . .. , flJ.q} of {I, .. . ,p} such that for all i E {I, . . . ,/J.}
rank ([ A
,
Step 6 (Fix matrices Qi for i = 0, .. . , r -1) For each i = 0, . .. , r - 1, define Qi E IRqxq as:
The following design procedure of a solution to Problem 1 will refer to fig. 1, where the structure of the proposed control system is depicted. Procedure 1. (valid under the hypotheses that system P satisfies conditions i-iii of Theorem 1): Design of a solution to Problem 1.
P
(L.·)(ri,rd=O .
-Ro(Iq+Qo) :
define Ul
] , Bl
[
-14-1 (Iq + Qr-I) [AW-lB .. . AW-rB] andB2 = [Aw-r-lB ... AB
({I, ... ,np+q}.J,j)
nP
+ q.
Bl and compute U2 := B~ (V - BIUl ). Partition
(12)
143
U2 as U2
~r.
= [
l'
with W i E JRPxq " i
=
r
Ww-l
. .. ,w-l. Step 9 (Choice of matrix r) Set TJ = {I, .. . ,n}. Let T E JR"lxn such that T(AW + V F)T- 1 is in Jordan form, and let T be the degree of the minimal polynomial of (AW + VF) . Fix 11, ... ,Iq E IR+ such that, for each i E {1, . . . ,q}, li > 1 and 1
1 + li
11
(TV) (1'/,i)( FT
(pWT
11
-1
\ i ,1'/)
00<
Set r = diag(rl, ... 'Iq). Step 10 (Fix subcompensators £, R, H) For each i = r, ... , w - 1, set
+ 1)I/T q
Q,
1
.
(14) Wand
Qi := Oqxq . For each i = 0, ... ,r - 1, set W i := OpXq . Then, define £, R, Q and W as periodic non dynamic gain matrices £(k), R(k), Q(k) and W(k) such that £(hw + i) = L i , R(hw + i) = Ri, Q(hw + i) = Qi and W(hw + i) = W i , for all h E Z and for each i = 0, . . . ,w - 1. Furthermore, set H as the LPTV dynamic system whose state space description is characterized by the matrices; AH (k) = Iq, BH (k) = Iq } CH(k) = Iq, DH(k) = Oqxq for k
= hw + i,
hE Z, i
= 0, .. .
,w - 2, (15a)
A~ (k) = Oq xq, B~ (k) = Oqxq } CH(k) = Iq, DH(k) = Oqxq for k = hw
+ w - 1, h
E Z.
(15 b )
Step 11 Define the overall compensator k as the connection of r, £, H, Q, R, Wand KM according to the block diagram depicted in fig. 1.
<>
Remark 6. If, at step 9 of Procedure 1, 11, .. . , Iq E IR+ are fixed such that , Vi E {I, ... ,q}, li > 1 and the following relation is satisfied: -1-II(TV) li - 1
(FT-I) . (1'/,.)
("1'/)
11
< (pWT + 1)1/T 00
q
1
(where TJ = {I, ... ,n}), instead of (14), then the control system in fig . 1 satisfies all the requirements of Problem 1 for the whole family of output perturbations defined by Xq := {=: E IRqxq , =: = diag({1, . . . ,{q): {i E (-00,-1] u [+I,+oo) , Vi E
{1, .. . ,q}}.
0
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