Discrete-Time Systems - Semantic Scholar

A Direct Method of Constructing H2 Suboptimal Controllers { Discrete-Time Systems Zongli Lin

Ali Saberi

Department of Applied Mathematics and Statistics State University of New York at Stony Brook Stony Brook, NY 11794-3600 [email protected]

School of Electrical Engineering and Computer Science Washington State University Pullman, WA 99164-2752 [email protected]

Peddapullaiah Sannuti

Yacov A. Shamash

Department of Electrical and Computer Engineering Rutgers University, P.O. Box 909 Piscataway, NJ 08855-0909 [email protected]

College of Engineering and Applied Sciences State University of New York at Stony Brook Stony Brook, NY 11794-2200 [email protected]

Abstract

For discrete-time systems, an H2 suboptimal control problem is de ned and analyzed. Then an algorithm called `H2 suboptimal state feedback gain sequence' (H2 SOSFGS) algorithm is developed. Rather than utilizing a `perturbation method' which is numerically sti and computationally prohibitive, the H2 SOSFGS algorithm utilizes a direct eigenvalue assignment method to come up with a sequence of H2 suboptimal state feedback gains. Also, although the sequence of H2 suboptimal state feedback gains constructed by the H2 SOSFGS algorithm depends on a parameter ", the construction procedure itself does not require explicitly the value of the parameter ". Next, attention is focused on constructing a sequence of H2 suboptimal estimator based measurement feedback controllers. Three dierent estimator structures, prediction, current, and reduced order estimators, are considered. For a given H2 suboptimal state feedback gain, a sequence of estimator gains for any of the three considered estimators can be constructed by merely dualizing the H2 SOSFGS algorithm. The direct method of constructing H2 suboptimal controllers developed here has a number of advantages over the `perturbation method', e.g., it has the ability to design all three types of estimator based controllers while still maintaining throughout the design the computational simplicity of it. This paper is the discrete-time version of [4]. There are some conceptual similarities as well as fundamental dierences between the H2 suboptimal control problems for continuous- and discrete-time systems. The fundamental dierences arise mainly from the fact that, in contrast to continuous-time, for discrete-time systems, the in mum of the H2 norm over the class of strictly proper controllers is in general dierent from the in mum of the H2 norm over the class of proper controllers.

2

1. Introduction Multivariable system analysis and design techniques have been studied in a variety of settings during the last three decades or so. H2 optimal control theory, or its stochastic interpretation well known as \linear quadratic Gaussian (LQG) control theory", is one of the earliest and powerful design tools that emerged. H2 optimal control theory focuses on the case when a control design problem is modeled as a problem of minimizing the H2 norm of a certain transfer function while utilizing a state or a measurement feedback controller. A recent text book [8] deals with various issues associated with a general H2 optimal control problem, whether it be regular or singular. These issues include: the existence and uniqueness of an H2 optimal solution, characterization and parameterization of all H2 optimal static and dynamic state feedback as well as measurement feedback controllers and in particular characterization and construction of H2 optimal estimator based measurement feedback controllers, pole/zero cancelations, H2 optimal xed modes, H2 optimal xed decoupling zeros, selection of an H2 optimal controller that places simultaneously the closed-loop poles at desired locations whenever possible, selection of an H2 optimal controller to meet some secondary considerations (e.g., obtaining an H2 optimal controller that satis es some robustness considerations posed in terms of an H1 constraint, etc.), suboptimal solutions, and computational issues. In a practical problem, it may turn out that an H2 optimal controller may not exist for a given plant. That is, a given plant may not satisfy the necessary and sucient conditions for the existence of an H2 optimal controller. Then a designer is forced to seek a suboptimal controller. In the absence of a formal de nition of a suboptimal controller, any controller which guarantees the internal stability of the closed-loop system can be construed as a suboptimal controller. A good de nition of suboptimality can be given through the notion of attaining an H2 norm (or any speci ed norm) of the chosen transfer function arbitrarily close to the in mum of such norms. In this regard, a sequence or a family of controllers can be called suboptimal, if one can select a controller from the family such that the resulting H2 norm is within an arbitrarily given value, say ", from the in mum. It turns out that a sequence or a family of suboptimal controllers as de ned above always exists as long as the given system is internally stabilizable. As alluded to earlier, in selecting an H2 optimal controller, several issues such as H2 optimal xed modes, pole/zero cancelations, H2 optimal xed decoupling zeros, available freedom in closed-loop pole assignment, etc., play important roles. These issues pertain to selecting a suboptimal controller as well. Thus, keeping such issues in perspective, we need to develop methods of constructing a sequences of H2 suboptimal controllers of a given architecture, e.g., an estimator based one. This paper is intended to develop such methods. The recent book [8] mentions two methods of constructing a sequence of H2 suboptimal controllers, one a perturbation method, and another a direct eigenvalue assignment method. Perturbation method uses standard \regularization via perturbation" technique and the \continuity argument". As indicated in [8], although perturbation method is useful in investigating certain theoretical aspects of a sequence of suboptimal controllers, it is not particularly suited for practical construction of such controllers. In fact, it is computationally inecient, and is numerically ill conditioned. Moreover, because of the way the \regularization" is done, the perturbation method leads to a model of the given plant dynamics in which all the measurements are contaminated with noise. This implies that

3 while constructing estimator based controllers, one always obtains only a prediction estimator based controller. That is, construction of current, and reduced order estimator based controllers are precluded owing to the way the \regularization" is done. In view of these shortcomings of the perturbation method, we focus our attention here on developing a direct method of constructing a sequence of H2 suboptimal controllers for any given system. The \direct method" developed here, besides removing numerical ill conditioning, has several advantages over the perturbation method. We will enumerate these advantages at the end of the paper when the conclusions are drawn. This paper is the discrete-time version of the paper [4]. Although there are some conceptual similarities between the H2 suboptimal control problems for continuous- and discretetime systems, there are as well several fundamental dierences between them. These fundamental dierences arise mainly from the fact that, in contrast to continuous-time, for discrete-time systems, the in mum of the H2 norm over the class of strictly proper controllers is in general dierent from the in mum of the H2 norm over the class of proper controllers. The paper is organized as follows. After establishing some preliminary notation, Section 2 recalls certain results on a special coordinate basis and on H2 optimal control. Section 3 de nes an H2 suboptimal control problem, and what we mean by a sequence of H2 suboptimal controllers. It also develops certain preliminary results on H2 suboptimal control. Our main contributions are in Sections 4 and 5 where in we develop algorithms of constructing respectively sequences of H2 suboptimal static state feedback controllers and estimator based measurement feedback controllers. Finally, Section 6 draws the conclusions of our work. Throughout the paper, A0 denotes the transpose of A, I denotes an identity matrix, while Ik denotes the identity matrix of dimension kk. C , C O , C and C respectively denote the whole complex plane, the unit circle, the open unit disc, and the complex plane outside the unit disc. (A) denotes the set of eigenvalues of A. A matrix is said to be stable if all its eigenvalues are in C . Similarly, a transfer function G(z) is said to be stable if all its poles are in C . Given a stable transfer function G(z), as usual, its H2 norm is de ned by l

l

l

l

l

l



kGk = 21 tr 2

Z

?

G(ej' )G0(e?j')d'

1=2

:

2. Notations and some preliminaries Even before we de ne an H2 suboptimal control problem, in this section we rst introduce some preliminary notations, then recall a special coordinate basis of linear systems, and then nally recall a connection between H2 optimal control problem of a given system and the disturbance decoupling problem of another related auxiliary system. The special coordinate basis of linear systems recalled here plays a signi cant role in our design of a sequence of H2 suboptimal controllers. On the other hand, as seen in [8], the connection between H2 optimal control and disturbance decoupling is at the heart of exploring and understanding several aspects of H2 optimal control. As such, recalling such a connection builds our intuition in exploring and understanding several aspects of H2 suboptimal control.

4

2.1. Preliminary Notations A system or plant  dealt with here is characterized by :

8 > < > :

x(k + 1) = Ax(k) + Bu(k) + Ew(k) y(k) = C x(k) + D w(k) z(k) = C x(k) + D u(k); 1

(2.1)

1

2

2

where x 2 IRn is a state, u 2 IRm is a control input, w 2 IRl is an exogenous disturbance input, y 2 IRp is a measured output, and z 2 IRq is a controlled output. Without loss of generality, we assume throughout the paper that the matrices [C2; D2], [C1; D1], [B 0; D20 ]0, and [E 0; D10 ]0 are of maximal rank. Also, we assume that the pair (A; B ) is stabilizable, and the pair (A; C1) is detectable. An arbitrary measurement feedback controller C is characterized by C :

(

v(k + 1) = A v(k) + B y(k); u(k) = C v(k) + D y(k): con

con

con

con

(2.2)

We note that C, as given in (2.2), is strictly proper when Dcon = 0. We use the following notations. The closed-loop system consisting of the plant  and a controller C is denoted by   C. A controller C is said to be internally stabilizing the system , if the closedloop system   C is internally stable, i.e., if   C has all its poles in C ?. Also, a controller C is said to be admissible if it provides internal stability for the closed-loop system   C . The transfer matrix from w to z of   C is denoted by Tzw (  C). Often in our development, we use two subsystems of the given system . These subsystems are: 1 which is characterized by the matrix quadruple (A; E; C1; D1), and 2 which is characterized by the matrix quadruple (A; B; C2; D2). l

2.2. A Special Coordinate Basis In this subsection, we recall a special coordinate basis (SCB) of linear time-invariant systems [9,7]. Such a coordinate basis has a distinct feature of explicitly displaying the nite and in nite zero structures of a given system, and as such plays a signi cant role in our development. Consider a discrete-time system  characterized by (A; B; C; D), 8 > < x(k + 1) > :

= Ax(k) + Bu(k)

(2.3) y(k) = Cx(k) + Du(k); where u and y are respectively some input (control input or disturbance) and output (measured or controlled output) of . It can then be easily shown that using singular value decomposition one can always nd an orthogonal transformation U and a nonsingular matrix V that render the direct feedthrough matrix D into the following form,   I 0 m 0 UDV = 0 0 ; (2.4)  :

5 where m0 is the rank of matrix D. Thus the system in (2.3) can be rewritten as  8 u0(k)  ; > ^ > x ( k + 1) = Ax ( k ) + [ B B ] > 0 1 < u^1(k)        > y (k) = C0 x(k) + Im0 0 u0(k)  ; > > : 0 y^ (k) 0 0 u^ (k) C^ 1

1

1

(2.5)

where the matrices B0, B^ 1, C0 and C^ 1 have appropriate dimensions. In the following lemma, whenever there is no ambiguity and in order to avoid notational clutter, the running time index k is omitted.

Lemma 2.1 (SCB). Consider the system  given in (2.3) and characterized by a matrix quadruple (A; B; C; D). Then, there exist

1. coordinate free non-negative integers na(), nb(), nc(), nd(), md()  m ? m0 and qi, i = 1;


; md(), and 2. non-singular state, output and input transformations ?S , ?O and ?I which take the given  into a special coordinate basis that displays explicitly both the nite and in nite zero structures of . The special coordinate basis which is referred to as the SCB is described by the following set of equations: x = ?S x; y = ?O y; u = ?I u x = [x0a; x0b; x0c; x0d]0 ; xd = [x01; x02;


; x0m ]0 z = [z00 ; zd0 ; zb0 ]0 ; zd = [z1; z2;


; zm ]0 u = [u00; u0d; u0c]0 ; ud = [u1; u2;


; um ]0 ; and xa(k + 1) = Aaaxa + B0az0 + Ladzd + Labzb (2.6) xb(k + 1) = Abbxb + B0bz0 + Lbdzd ; zb = Cbxb (2.7) xc(k + 1) = Acc xc + B0cz0 + Lcb zb + Lcdzd + Bc [Ecaxa + uc] (2.8) and for each i = 1;


; md, d

d

d

2 md X xi(k + 1) = Aqi xi + B0iz0 + Lid zd + Bqi 4Eiaxa + Eib xb + Eicxc + Eij xj j =1

3 + ui5

zi = Cq xi ; zd = Cdxd

(2.10)

i

and

z = C axa + C bxb + C cxc + 0

0

0

0

md X j =1

(2.9)

C j xj + u : 0

(2.11)

0

Here the states xa; xb; xc and xd are respectively of dimensions na(), nb(), nc (), and nd() = Pmi qi, while xi is of dimension qi for each i = 1;


; md(). The control vectors u , ud and uc are respectively of dimensions m , md() and mc() = m ? m ? md(), d

=1

0

0

0

6 while the controlled output vectors z , zd and zb are respectively of dimensions p = m , pd = md() and pb = p ? p ? pd . The matrices Aq , Bq and Cq have the following form: 0

0

0

i

"

Aq = 00 Iq0? i

i

#

1

"

i

i

#

; Bq = 01 ; Cq = [1; 0;


; 0]: i

0

i

(2.12)

(Obviously for the case when qi = 1, we have Aq = 0, Bq = 1 and Cq = 1.) Furthermore, the pair (Acc, Bc) is controllable and the pair (Abb, Cb) is observable. Also, assuming that xi are arranged such that qi  qi+1, the matrix Lid has the particular form, i

i

i

Lid = [Li ; Li ;


; Li i? ; 0; 0;


; 0]: 1

2

1

Also, the last row of each Lid is identically zero.

Proof : The proof of this lemma can be found in [9] and [7].

We can rewrite the SCB given by Lemma 2.1 in a more compact form as a system ~ B; ~ C; ~ D~ ) given by characterized by the quadruple (A;

LabCb 0 LadCd 3 Abb 0 Lbd Cd 777 ; Lcb Cb Acc Lcd Cd 5 BdEb BdEc Ad 2 Ba 0 0 3 6B 0 777 ; b 0 B~ := ??S [ B B^ ] ?I = 664 B c 0 Bc 5 B d Bd 0 3 2 2 3 C C a Cb Cc Cd C~ := ??O 64 75 ?S = 64 0 0 0 Cd 75 ; 0 Cb 0 0 C^

2 Aaa 6 A~ := ??S 1 (A ? B0C0)?S = 664 B 0E c ca Bd E a

0

1

0

0

1

0

0

1

and

0

0

0

0

0

1

3

2 Im0 6 ? 1 ~ D := ?O D?I = 4 0

0 0 0 0 75 : 0 0 0 In what follows, we state some important properties of the SCB which are pertinent to our present work.

Property 2.1. We note that (Abb, Cb) and (Aq , Cq ) form observable pairs. Unobservability i

i

could arise only in the variables xa and xc. In fact, the given system  is observable (detectable) if and only if (Aobs, Cobs ) is an observable (detectable) pair, where #

"

#

"

Aobs = BAEaa A0 ; Cobs = CE a CE c ; a c c ca cc 0

0

Ea = [ E 0 a E 0 a


Em0 a ]0 ; Ec = [ E 0 c E 0 c


Em0 c ]0 1

2

d

1

2

d

7 Similarly, (Acc , Bc) and (Aq , Bq ) form controllable pairs. Uncontrollability could arise only in the variables xa and xb. In fact,  is controllable (stabilizable) if and only if (Acon , Bcon ) is a controllable (stabilizable) pair, where i

i

#

"

#

"

Acon = A0aa LAabCb ; Bcon = BB a LLad : b bd bb 0

0

Property 2.2. The given system  is right-invertible if and only if xb and hence zb are nonexistent, left-invertible if and only if xc and hence uc are nonexistent, invertible if and only if both xc and xb are nonexistent. Property 2.3. The eigenvalues of Aaa are the invariant zeros of . We denote by na (), +

na(), and na? () the number of invariant zeros of  which are respectively outside the unit disc, on the unit circle, and inside the unit disc in the complex plane.

2.3. Connection Between H Optimal Control Problems and Disturbance Decoupling Problems 2

In this subsection, we recall two problems, one an H2 optimal control problem and another a disturbance decoupling problem. Then we recall certain interconnections between these two problems. We rst recall the following de nition regarding H2 optimal control.

De nition 2.1. Let a system  of the form (2.1) be given. The H2 optimal control problem is to nd an internally stabilizing proper (strictly proper) controller C which minimizes the H2 norm of the closed-loop transfer matrix Tzw (  C). The in mum of an H2 optimal control problem over the class of admissible proper controllers is denoted by p, that is (2.13)

p := inf fkTzw (  C)k j C is proper and internally stabilizes g: Similarly, we denote the in mum of an H optimal control problem over the class of admissible strictly proper controllers as sp , that is

sp := inf fkTzw (  C)k j C is strictly proper and internally stabilizes g: (2.14) An internally stabilizing proper (strictly proper) controller C is said to be an H optimal controller if it achieves a closed-loop H norm p (respectively, sp ). 2

2

2

2

2

The conditions for the existence of a strictly proper H2 optimal controller are dierent from those of a non-strictly proper H2 optimal controller. Moreover, it turns out that in the case of discrete-time systems (but not for continuous-time systems) p is in general smaller than

sp (see for details [11]). As discussed in detail in [8,11], the strictly proper H2 optimal control problem for a given system  can be reformulated as a disturbance decoupling problem via strictly proper measurement feedback with internal stability (DDPMS) for an auxiliary system denoted

8 here by PQ . In fact, a strictly proper controller that is H2 optimal for  solves the DDPMS for the auxiliary system PQ and vice versa. Next, the proper H2 optimal control problem can also be reformulated as a DDPMS for an auxiliary system denoted here by dPQ . In the latter case, we rst choose a preliminary static output feedback. Then, there is a one to one relationship between the H2 optimal controllers for  and the controllers that solve the DDPMS for the auxiliary system dPQ . However in this case, the controllers are not identical but are related via this preliminary feedback. In what follows, in order to facilitate the introduction of auxiliary systems PQ and dPQ , we rst introduce two other auxiliary systems P and Q and state their properties. Then, after recalling the de nition of a DDPMS, we relate the H2 optimal control problems for  to the DDPMS for PQ and dPQ . As a preliminary step before introducing P , we introduce a discrete-time linear matrix inequality ( DLMI) as follows: F (P )  0; (2.15) where  0 PA ? P + C20 C2 C20 D2 + A0PB  : F (P ) := A D 0 0 D20 D2 + B 0PB 2 C2 + B PA Here the matrices A, B , C2, and D2 are the data that characterizes the subsystem 2 of the given system  as in (2.1). It is shown in [8] that there exists a unique semi-stabilizing solution P of the DLMI (2.15) whenever the pair (A; B ) is stabilizable. Moreover, such a solution P is positive semi-de nite, strongly rank minimizing, and in fact is the largest among all strongly rank minimizing solutions. Assuming that (A; B ) is stabilizable, compute such a solution P . Next, de ne the maximal rank matrix [ CP DP ] where CP has n columns and DP has m columns, such that  0 CP [ C D ] : (2.16) F (P ) = D P P 0 P Next, we would like to introduce another DLMI, G(Q)  0; (2.17) where  0 ? Q + EE 0 ED0 + AQC 0  1 1 G(Q) := AQA D1 E 0 + C1QA0 D1D10 + C1QC10 : Here the matrices A, E , C1, and D1 are the data that characterizes the subsystem 1 of the given system  as in (2.1). Again, as shown in [8], there exists a unique semi-stabilizing solution Q of the DLMI (2.17) whenever the pair (A; C1) is detectable. Assuming that (A; C1) is detectable, compute such a solution Q. Next, de ne the maximal rank matrix [ EQ0 DQ0 ]0 where EQ has n rows and DQ has p rows, such that   E Q G(Q) = D [ EQ0 DQ0 ] : (2.18) Q Now we are ready to de ne two auxiliary systems P and Q . The system P is de ned by 8 x (k + 1) = AxP(k) + BuP(k) + EwP(k) > > < P P : > yP(k) = C1xP (k) + D1 wP(k) (2.19) > : zP(k) = CPxP (k) + DP uP(k);

9 where the matrices A, B , C1, D1, and E are the data pertaining to the system  of (2.1), and CP and DP are as in (2.16). Similarly, we de ne the auxiliary system Q as Q :

8 x (k + 1) > > < Q yQ(k) > > :

= AxQ(k) + BuQ(k) + EQwQ(k) = C1xQ(k) + DQ wQ(k) zQ(k) = C2xQ(k) + D2uQ (k);

(2.20)

where the matrices A, B , C1, C2, and D2 are the data pertaining to the system  of (2.1), and the matrices EQ and DQ are as in (2.18). The following lemma reveals some fundamental properties of P as de ned in (2.19).

Lemma 2.2. Consider the auxiliary system P as in (2.19) with the pair (A; B ) being stabi-

lizable. Let a subsystem 2P of P be characterized by the matrix quadruple (A; B; CP; DP). Recall a subsystem 2 of the given system , and note that 2 is characterized by the quadruple (A; B; C2; D2). Then an inter-relationship between 2P and 2 is described as follows: 1. 2P is right invertible. 2. 2P has a total of na-(2) + na(2) + na+ (2) + nb(2) + nd(2) invariant zeros which are given by: (a ) the na-(2) stable (i.e. those in C ) invariant zeros of 2, (b ) the na(2) invariant zeros of 2 which are on the unit circle C O, (c ) the na+ (2) mirror imagesy with respect to the unit circle of all the unstable (i.e. those in C ) invariant zeros of 2, and (d ) some nb (2) + nd(2) xed locations inside the open unit disc C which contain the stable input decoupling zeros (but not the invariant zeros) of 2. 3. 2P has no in nite zeros of order greater than or equal to one. 4. 2P is invertible if and only if 2 is left invertible. l

l

l

l

The following lemma which reveals some important properties of Q is analogous to Lemma 2.2.

Lemma 2.3. Consider the auxiliary system Q as in (2.20) with the pair (A; C ) being de1

tectable. Let a subsystem 1Q of Q be characterized by the matrix quadruple (A; EQ; C1; DQ ). Recall a subsystem 1 of the given system , and note that 1 is characterized by the quadruple (A; E; C1; D1). Then an inter-relationship between 1Q and 1 is described as follows: 1. 1Q is left invertible.

For continuous-time systems, the mirror image of a complex number  + j is de ned as ? + j , whereas in discrete-time systems, the mirror image of a complex number rej is de ned as r1 ej . y

10 2. 1Q has a total of na-(1) + na(1) + na+(1) + nb(1) + nd(1) invariant zeros which are given by: (a ) the na-(1) stable (i.e. those in C ) invariant zeros of 1, (b ) the na(1) invariant zeros of 1 which are on the unit circle C O, (c ) the na+(1) mirror images with respect to the unit circle of all the unstable (i.e. those in C ) invariant zeros of 1, and (d ) some nc (1) + nd (1) xed locations in the open unit disc C which contain the stable output decoupling zeros (but not the invariant zeros) of 1. 3. 1Q has no in nite zeros of order greater than or equal to one. 4. 1Q is invertible if and only if 1 is right invertible. We now de ne the auxiliary system PQ by combining appropriately the auxiliary systems P and Q, 8 x (k + 1) = AxPQ(k) + BuPQ(k) + EQwPQ (k) > > < PQ + DQwPQ (k) (2.21) PQ : > yPQ(k) = C1xPQ(k) > : zPQ (k) = CPxPQ(k) + DP uPQ (k): Here the matrices A, B , and C1 are the data pertaining to the system  of (2.1), and the matrices CP , DP, EQ , and DQ are as in (2.16) and (2.18). In order to de ne the auxiliary system dPQ , we rst de ne a set D as l

l

l

l

o

n

D := D 2 IRmp j DP D DQ = ?R ; con

(2.22)

con

(2.23) R := (DP0 )y(DP0 CPQC 0 + B 0PED0 )(DQ0 )y; where (
)y denotes the generalized inverse of (
). We now de ne the auxiliary system dPQ . For any given D 2 D , let 8 x (k + 1) = AdxPQ(k) + B u~PQ(k) + EQd wPQ (k) > > < PQ + DQ wPQ (k) dPQ : > yPQ (k) = C xPQ(k) (2.24) > : d zPQ(k) = CP xPQ(k) + DP u~PQ (k); where u~PQ is a new control signal, and where Ad = A + BD C ; EQd = EQ + BD DQ ; CPd = CP + DP D C : (2.25) In what follows, two subsystems of PQ , namely  PQ and  PQ, play important roles.  PQ is characterized by the matrix quadruple (A; EQ; C ; DQ) and is the same one as  Q. Similarly,  PQ is characterized by the matrix quadruple (A; B; CP; DP ) and is the same one as  P. Since  PQ is the same as  Q , and  PQ is the same as  Q, the zero structures of  PQ and  PQ are respectively as discussed in Lemmas 2.3 and 2.2. The following is the de nition of the DDPMS for a given system, say PQ . 1

1

1

con

1

con

1

con

con

1

1

1

2

1

1

2

2

1

1

1

2

2

2

To obtain dPQ , we apply to the auxiliary system PQ the static output feedback uPQ = Dcon yPQ + u~PQ with u~PQ as the new control signal, and then delete in it the feedthrough term from wPQ to zPQ . 1

11

De nition 2.2. Consider a system PQ as in (2.21). The disturbance decoupling problem with measurement feedback and internal stability (DDPMS) for PQ is the problem of nding a proper controller C of the form (2.2) such that the closed-loop system PQ C is internally stable, while the resulting closed-loop transfer function is identical to 0. We say that the strictly proper disturbance decoupling problem with measurement feedback and internal stability for PQ is solvable if there exists a strictly proper controller C of the form (2.2) with Dcon = 0 such that the closed-loop system PQ  C is internally stable, while the resulting closed-loop transfer function is identical to 0. As we said earlier, there are certain interconnections between the H2 optimal control problem for  and the DDPMS for either PQ or dPQ . The following lemmas recalled from [8,11] show such interconnections.

Lemma 2.4. Consider an H optimal control problem by strictly proper controllers as de2

ned by De nition 2.1 for a system  as in (2.1). Assume that (A; B ) is stabilizable and (A; C1) is detectable. Also, consider the auxiliary system PQ as given in (2.21), and a strictly proper controller C as in (2.2) with Dcon = 0. Then, the controller C is a strictly proper H2 optimal controller for  if and only if it solves the DDPMS for PQ .

Lemma 2.5. Consider a proper H optimal control problem as de ned by De nition 2.1 for a system  as in (2.1). Assume that (A; B ) is stabilizable and (A; C ) is detectable. Also, consider the auxiliary system dPQ de ned in (2.24) for some D 2 D . Then the following 2

1

statements are equivalent:

con

1. A proper controller C with state space realization (Acon; Bcon; Ccon; D~ con) solves the DDPMS for dPQ . 2. A proper controller C with state space representation (Acon; Bcon; Ccon; Dcon + D~ con) is a proper H2 optimal controller for the given system . Moreover, whenever the DDPMS for dPQ is solvable via a proper controller, it is also solvable via a strictly proper controller, and the solvability conditions of the DDPMS for dPQ are independent of the particular choice of Dcon as long as Dcon 2 D. We note that Lemmas 2.4 and 2.5 convert the task of nding an H2 optimal controller for a given system  to the task of nding a controller that solves the DDPMS for an auxiliary system constructed from the data of the given system . In particular, one can nd a strictly proper H2 optimal controller for , whenever it exists, by simply nding a a strictly proper controller characterized by (Acon; Bcon; Ccon; 0) that solves the DDPMS for PQ . Similarly, whenever it exists, one can nd a proper H2 optimal controller for  as follows: rst select a Dcon 2 D , next construct the system dPQ , then nd a a strictly proper controller characterized by (Acon; Bcon; Ccon; 0) that solves the DDPMS for dPQ , nally the controller characterized by (Acon; Bcon; Ccon; Dcon) is a proper H2 optimal controller for .

12

3. H2 suboptimal control { de nitions and preliminary results As discussed in the introduction, our goal in this paper is to develop a direct method by which one can construct a sequence of H2 suboptimal controllers of both state feedback as well as estimator based measurement feedback type. Before we do so, to start with we formally introduce in this section several de nitions related to suboptimal control. In this regard, we rst de ne the H2 suboptimal control problem for the given system , and various other de nitions associated with it. Next, as in H2 optimal control, we recognize that the H2 suboptimal control problem for the given system  is closely related to an H2 almost disturbance decoupling problem with measurement feedback and internal stability (H2-ADDPMS) for either the auxiliary system PQ or dPQ . We then establish clearly such a relationship. In all this development, we consider both measuremet feedback as well as its special case of state feedback.

3.1. De nitions We have the following de nitions regarding H2 suboptimal control.

De nition 3.1. (H2 suboptimal control problem) Consider a system  as given in (2.1). Then the H2 suboptimal control problem by proper (strictly proper) controllers is a problem of nding, if it exists, a sequence of proper (respectively, strictly proper) controllers f C(") j " > 0 g which satisfy the following two conditions: 1. There exists an " > 0 such that for any " 2 (0; "], the closed-loop system   C(") is internally stable. 2. As " ! 0, the H2 norm of the corresponding closed-loop transfer function Tzw (  C(")) approaches p (respectively, sp ) as given in De nition 2.1.

De nition 3.2. (Sequence of H2 suboptimal controllers) Consider the H2 suboptimal control problem as in De nition 3.1 for a given system  as in (2.1). Then, a sequence of proper (or strictly proper) controllers f C(") j " > 0 g that solves such a problem is referred to as a sequence of H2 suboptimal proper (respectively, strictly proper) controllers. Moreover, for the special case when the entire state is available for feedback (i.e. when C1 = I and D1 = 0), for simplicity of our presentation, a sequence of H2 suboptimal static state feedback controllers (in fact, gains) is denoted by F "(A; B; C2; D2; E ) which is a set of parameterized gain matrices F (") 2 IRmn . Furthermore, a controller from a sequence of H2 suboptimal proper (or strictly proper) controllers is simply called an H2 suboptimal proper (or strictly proper) controller or H2 suboptimal state feedback controller as the case may be. We introduce next the concept of a sequence of H2 -level suboptimal controllers.

De nition 3.3. (Sequence of H -level suboptimal controllers) Consider a system 2

 as given in (2.1), and a parameter > 0. Then, a sequence of H2 suboptimal proper (or strictly proper) controllers as in De nition 3.2 is said to be a sequence of H2 -level suboptimal proper (or strictly proper) controllers if every element C(") of it is stabilizing and kTzw (  C("))k2 < p + (respectively, kTzw (  C("))k2 < sp + ). Moreover,

13 for the special case when the entire state is available for feedback (i.e. when C = I and D = 0), for simplicity of our presentation, a sequence of H -level suboptimal static state feedback controllers (in fact, gains) is denoted by F " (A; B; C ; D ; E ) which is a set of parameterized gain matrices F (") 2 IRmn . Furthermore, a controller from a sequence of H -level suboptimal proper (or strictly proper) controllers is simply called an H -level suboptimal proper (or strictly proper) controller or H -level suboptimal state feedback controller as the case may be. 1

1

2

2

2

2

2

2

3.2. Connection Between H Suboptimal Control Problems and Almost Disturbance Decoupling Problems 2

As stated earlier, analogous to the case of an H2 optimal control problem, an H2 suboptimal control problem can be related to an almost disturbance decoupling problem for the auxiliary system for either PQ or dPQ . A formal de nition of an almost disturbance decoupling problem, say for PQ , is given below.

De nition 3.4. (H -ADDPMS) Consider a system PQ as in (2.21). Then, for PQ , the 2

H almost disturbance decoupling problem with measurement feedback and internal stability, in short (H -ADDPMS)p, is de ned as follows. For any  > 0, nd a proper controller C of the form (2.2) such that the following hold: 2

2

1. The closed-loop system PQ  C is internally stable; and 2. The resulting closed-loop transfer function TzPQ wPQ (PQ  C) has an H2 norm less than . Moreover, we label an (H2-ADDPMS)p as an (H2-ADDPMS)sp when the class of controllers considered is admissible and strictly proper. In our narrative, whenever we write H2-ADDPMS, it applies to both (H2-ADDPMS)p and (H2 -ADDPMS)sp. Also, for the special case when the entire state of  is available for feedback, i.e. when C1 = I and DQ = 0, the corresponding H2 -ADDPMS is referred to as H2 -ADDPSS. Furthermore, a sequence of proper (strictly proper) controllers f C(") j " > 0 g is said to solve the (H2-ADDPMS)p (respectively, (H2-ADDPMS)sp) for PQ if the following hold: 1. There exists an " > 0 such that for any " 2 (0; "], the closed-loop system PQ  C(") is internally stable; and 2. The H2 norm of the resulting closed-loop transfer function TzPQ wPQ (PQ  C(")) tends to zero as " tends to zero. Such a sequence is called a proper (strictly proper) sequence of H2 ADDPMS controllers. In particular, for the special case when the entire state is available for feedback, for simplicity, a sequence of static state feedback controllers that solve the H2 ADDPSS is simply called a sequence of H2 ADDPSS feedback gains. Also, a sequence of H2 ADDPSS feedback gains is called an H2 -level ADDPSS feedback gains if every element C(") of it is stabilizing and kTzPQ wPQ (PQ  C("))k2 < .

14 As we said earlier, there are de nite connections between the H2 suboptimal control problem for the system  and the H2-ADDPMS for either the auxiliary system PQ or dPQ . The following lemmas explore this connection, and as such play signi cant roles in our construction of a sequence of H2 suboptimal strictly proper (or proper) controllers for . Lemma 3.1. Consider an H2 suboptimal control problem as de ned by De nition 3.1 for a system  as in (2.1). Assume that (A; B ) is stabilizable and (A; C1) is detectable. Also, consider the auxiliary system PQ as given in (2.21). Moreover, consider a sequence of strictly proper controllers f C(") j " > 0 g (each one of them is of the form given in (2.2) with Dcon = 0). Then, the following two statements are equivalent: 1. The considered sequence f C(") j " > 0 g solves the H2 -ADDPMS for PQ . 2. The considered sequence f C (") j " > 0 g is a sequence of H2 suboptimal strictly proper controllers for . Proof: From [8] (pages 213-214), we note that for any member of the sequence f C(") j " > 0 g, say "C, as Dcon = 0 we have kTzw (  "C)k22 = kTzPQ wPQ (PQ  "C)k22 + tr [CPQCP0 ] + tr [E 0PE ] = kTzPQ wPQ (PQ  "C)k22 + ( sp )2: This implies that (3.1) kTzPQwPQ (PQ  "C)k22 = kTzw (  "C)k22 ? ( sp )2: Now, if f C (") j " > 0 g is a sequence of H2 suboptimal controllers for , then (3.2) kTzw (  "C)k2 ! sp as " ! 0; and thus, in view of (3.1), we have kTzPQwPQ (PQ  "C)k2 ! 0 as " ! 0: (3.3) Thus, the sequence f C (") j " > 0 g solves the H2-ADDPMS for PQ . To show the other way, again if f C(") j " > 0 g solves the H2-ADDPMS for PQ , we rst have (3.3) which, in view of (3.1), then implies (3.2).

Lemma 3.2. Consider an H suboptimal control problem as de ned by De nition 3.1 for a system  as in (2.1). Assume that (A; B ) is stabilizable and (A; C ) is detectable. Also, consider the auxiliary system dPQ de ned in (2.24) for some D 2 D . Moreover, consider a sequence of proper controllers f C(") j " > 0 g (each one of them is of the form given in 2

1

con

(2.2) with a state space realization (Acon("); Bcon("); Ccon("); D~ con("))). Then, the following two statements are equivalent: 1. The considered sequence of proper controllers f C(") j " > 0 g solves the H2-ADDPMS for dPQ . 2. The sequence of proper controllers f C(") j " > 0 g (each one of them is of the form given in (2.2) with a state space realization (Acon("); Bcon("); Ccon("); Dcon + D~ con("))) is a sequence of H2 suboptimal proper controllers for . Proof: It is omitted and will be given in the full version of the paper.

15

3.3. Existence of H Suboptimal Controllers 2

The previous subsection considered certain preliminary properties of H2 suboptimal control. In this subsection, we show that there always exist a solution to an H2 suboptimal control problem provided that the given system is internally stabilizable. We have the following theorem.

Theorem 3.1. Consider an H suboptimal control problem by proper or by strictly proper 2

controllers as de ned by De nition 3.1 for a system  as in (2.1). Then, it is solvable if and only if the pair (A; B ) is stabilizable and the pair (A; C1) is detectable.

Proof: It is omitted and will be given in the full version of the paper. We next consider the special case when the entire state is available for feedback, i.e., when C1 = I and D1 = 0. In this case, in view of Theorem 3.1, obviously a sequence of H2 suboptimal state feedback controllers exist if and only if the pair (A; B ) is stabilizable. In fact, in this case, we can have an additional result. That is, whenever a sequence of H2 suboptimal state feedback controllers exists, there exists as well a sequence of H2 suboptimal staic state feedback controllers, or equivalently a sequence F " (A; B; C2; D2 ; E ). This fact is established in the following lemma.

Lemma 3.3. Consider an H suboptimal control problem as de ned by De nition 3.1 for a 2

system  as in (2.1) for the special case when C1 = I and D1 = 0. Then, a sequence of H2 suboptimal state feedback gains F " (A; B; C2; D2; E ) and a sequence of H2 -level suboptimal state feedback gains F " (A; B; C2; D2 ; E ) exist if and only if the pair (A; B ) is stabilizable.

Proof: It is omitted and will be given in the full version of the paper.

4. Construction of a sequence of H2 suboptimal state feedback gains It is obvious that when the entire state of the given system  is available for feedback (i.e. C1 = I and D1 = 0),  is characterized by the matrix quintuple (A; B; C2; D2; E ). Our intention in this section is to develop an algorithm which takes the quintuple (A; B; C2; D2; E ) as its input parameters and yields as its output a sequence of H2 suboptimal state feedback gains F "(A; B; C2; D2 ; E ), and a sequence of H2 -level suboptimal state feedback gains F " (A; B; C2; D2; E ) whenever is given. The algorithm developed here can obviously be named as H2 suboptimal state feedback gain sequence algorithm, and is abbreviated as H2 SOSFGS. A schematic diagram of H2 SOSFGS is given in Figure 4.1. Among others, one can emphasize two main attributes of H2 SOSFGS algorithm, (1) the design is decentralized, i.e. the needed computations are performed on lower order subsystems of the given system, (2) although the sequence of H2 suboptimal state feedback gains constructed by the algorithm depends on a parameter ", the construction procedure itself does not require explicitly the value of the parameter ", and as such unlike in `perturbation methods', one faces in the design process neither numerical stiness nor computational complexity. We now recall an H2 low-gain design algorithm from [3]. This design algorithm is a main building block of the H2 SOSFGS design algorithm.

16

A; B; C ; D ; E 2

2

H2SOSFGS

F "(A; B; C 2 ; D2 ; E ) F " (A; B;-C2; D2; E )

Figure 4.1: A block diagram interpretation of H2 SOSFGS.

4.1. An H Low-Gain Design Algorithm 2

Consider the linear system

x(k + 1) = Ax(k) + Bu(k); x 2 IRn; u 2 IRm

(4.1)

where we assume that (A; B ) is stabilizable and all the eigenvalues of A are in the closed unit disc. The H2 low-gain design we are proposing is carried out in three steps.

Step 1 : Find the state transformation T (Chen, 1984) such that (T ? AT; T ? B ) is in the 1

following form,

2 B A


A ` 0 3 6 0 A


A ` 0 777 6 6 . . . . ? ? .. . . .. .. 77 ; T B = 66 ... T AT 6 4 0 0


A` 0 75 B 0 0 0 0 A (where ` is an integer) and for i = 1; 2;


; `, 3 2 0 1 0


0 6 0. 0. 1.
.

0. 777 6 6 .. .. . . .. 77 ; Ai = 66 .. 6 0 0


1 75 4 0 ?ain ?ain ? ?ain ?


?ai 1

2 A1 6 0 6 6 = 66 ... 6 4 0

12

1

2

2

1

1

01

0

i

i

1

i

2

1

0 B2 ... 0 B02

1




0 3


0  777 ...


B`


B0` ...

... 7 ; 7 7

5 

2 3 0 6 0. 777 6 6 Bi = 66 .. 77 : 6 7 405

1

Furthermore, the transformation T is such that all the eigenvalues of Ai are on the unit circle and all the eigenvalues of A0 are strictly inside the unit circle. Here the 's represent submatrices of less interest.

Step 2 : For each (Ai; Bi), let Fi(") 2 IR n be the state feedback gain such that (Ai ? Bi Fi(")) = (1 ? ")(Ai) 2 C 1

i

l

Such an Fi(") exists and is unique since (Ai; Bi) is a single input controllable pair.

Step 3 : Let

u(k) = ?F (")x(k)

(4.2)

17 where the state feedback gain matrix F (") is given as 2 F1("2`?1(r2+1)(r3+1)


(r`+1)) 6 6 0 6 ... 6 6 F (") = 66 0 6 6 0 4

F (" 2

0

0

?2 (r3 +1)(r4 +1)


(r`+1)

2`

... 0 0 0





.



)

03 0. 777 .. 77 ?1 0 777 T 05 0

0 0 0. 0. .. .. .. 2(r +1)


F`?1(" ) 0


0 F`(")


0 0 and where ri is the largest algebraic multiplicity of the eigenvalues of Ai. `

(4.3)

The parameterized state feedback gain F (") as given by (4.3) has the following prominent property.

Theorem 4.1. Consider the linear system as given by (4.1). Suppose that (A; B ) is stabi-

lizable and all the eigenvalues of A are located inside or on the unit circle. Consider also the state feedback control given by (4.2). Then we have the following properties: 1. The closed-loop system matrix A ? BF (") is Hurwitz for all " 2 (0; 1]. 2. There exists an " 2 (0; 1] such that for all " 2 (0; "] and for all k  0,

kF (")k  " k(A ? BF ("))kk  "



0

?1(r1 +1)(r2 +1)


(r` +1)

2`

kF (")(A ? BF ("))kk  `"(1 ? ")k= + `? "r 2

1

` +2

(4.4)





1 ? "2 ?1(r1+1)(r2+1)


(r +1) `

1 ? "2(r +1) `



`

k=2

+




+1"2 ?2(r2+1)(r3+1)


(r +1)+1 1 ? "2 ?1(r2+1)(r3+1)


(r +1) `

`

`

`

where  and i's are some " independent positive constant numbers.

Proof : See [3].

k=2

k=2

(4.5)

(4.6)

18

4.2. H SOSFGS Design Algorithm 2

The following algorithm results in an H2 SOSFGS for the quintuple (A; B; C2; D2; E ). Step 1 (Construction of 2P ): Following the procedure described in Section 2, construct the subsystem 2P as  ) + B uP(k) 2P : xz P((kk)+=1)C =x A(kx)P+(kD (4.7) u (k): P

P P

P P

We recall that P is given as, 8 > < xP (k + 1) = A xP (k ) + B uP (k ) + EwP (k ) P : > yP(k) = xP(k) : z (k ) = C x (k ) + D u (k ): P P P P P

(4.8)

Step 2 (SCB Transformation): Perform a nonsingular state, input and output transformation on the system 2P. That is, let

xP = ?PS xP ; zP = ?POzP; uP = ?PI uP such that the system  P can be written in the following SCB form, xP = [x0a; x0c]0; zP = z ; uP = [u0 ; u0c]0; 2

0

0

and

xa(k + 1) = APaaxa(k) + B Paz (k) (4.9) P P P P (4.10) xc(k + 1) = Acc xc(k) + B cz (k) + Bc [Ecaxa + uc(k)] P P z (k) = C axa(k) + C cxc(k) + u (k): (4.11) We note that the state xb is not present as  P is right invertible, and that xd is absent also since  P has no in nite zero of order greater than or equal to one. Step 3 (Construction of a low-gain Fa(")): By the Property 2.1 of the SCB, the pair (APaa; B Pa) is stabilizable. Also, by Property 2.3 of the SCB, the eigenvalues of APaa are the invariant zeros of the linear system P, and hence by Lemma 2.2 are all located in the closed unit disc. Hence, following the `Low-Gain design' method, one can design a feedback gain Fa(") for the pair (APaa; B Pa) such that, (a) The matrix Aaa ? B aFa(") is Hurwitz for all " 2 (0; 1]; (b) There exists an " 2 (0; 1] such that for all " 2 (0; "], kFa(")k  P " (4.12)  k= P P ?1 rP  P


rP P P k r 1 2 k(Aaa ? B aFa(")) k  P?1 r1P r2P


rP 1 ? " " (4.13) 0

0

0

0

0

0

0

0

2

2

0

2

0

0

0

2`

0

2`

(

+1)(

+1)

( ` +1)

(

+1)(

+1)

( ` +1)

2

19

kF)("aA( Paa ? B PaFa("))k k  `P"(1 ? ")k= + `P? "r (1 ? " rP )k= +


k=  P ?1 rP P ?2 rP P


rP P


rP r P r 2 3 2 3 1?" + " (4.14) 2

0

1

2`

(

+1)(

+1)

1

2`

( ` +1)+1

(

` +2

+1)(

2( ` +1)

+1)

2

2

( ` +1)

where ` and riP 's are integers, and P and iP's are positive constant numbers, all independent of ". Step 4 (Construction of a parameterized gain F (")): By the property of SCB, the pair (APcc; BcP) is controllable, hence one can choose a feedback gain matrix Fc such that APcc + BcPFc is stable and has a chosen set of eigenvalues. Next a composite static state feedback gain is formed for the system 2P. This state feedback gain takes the form of   (4.15) F (") = ?I FFu0 (") ??S 1 u where Fu0 = [ C0Pa + Fa(") C0Pc ] and Fu = [ EcaP Fc ]. This concludes the description of a low-gain based state feedback design method that leads to a parameterized gain F ("). Step 5 (Construction of F " (A; B; C2; D2; E )): Finally one can construct a sequence of static state feedback gains as follows: F " (A; B; C2; D2; E ) = fF (") j F (") as in (4.15); " > 0g: (4.16) c

c

The following theorem veri es that the sequence F " (A; B; C2; D2; E ) as given by (4.16) is an H2 SOSFGS.

Theorem 4.2. Consider the system  as in (2.1). Then the sequence F " (A; B; C2; D2; E ) as given by (4.16) is an H2 SOSFGS. Proof: This is to show that the closed-loop system comprising of the system  and the control law u = F (")x is asymptotically stable for suciently small ", and kTzPwP (P  C ("))k ! 0 as " ! 0. Equivalently, we need to show that the auxilliary system, 2

 ~P : xP(k + 1) = A xP(k) + B uP(k) + wP(k) (4.17) zP(k) = CP xP(k) + DP uP(k) with uP = F (")xP is asymptotically stable for suciently small ", and for any xP(0) 2 IRn ,

1 X

lim

"!0 k=0

kzP(k)k ! 0:

(4.18)

2

As proved in [3], this is indeed the case.

Remark 4.1. Once an F "(A; B; C ; D ; E ) is constructed, F " (A; B; C ; D ; E ) can easily be obtained as follows. Since F "(A; B; C ; D ; E ) is an H SOSFGS for the system , for any

> 0, there exists an " > 0 such that for any " 2 (0; " ], F (") is stabilizing and kTzw (  C ("))kH2 < p + . Hence, F " (A; B; C ; D ; E ) = fF (") 2 F "(A; B; C ; D ; E ) j " 2 (0; " ]g. 2

2

2

2

2

2

2

2

2

2

2

20

5. H2 suboptimal observer based measurement feedback controllers In this section, at rst three dierent estimator structures (prediction, current, and reduced order estimators) are reviewed. Then, for any particular H2 suboptimal state feedback gain, a sequence of gains for all the three dierent estimator structures are constructed in such a way that the cascade of one of these estimators and the given state feedback controller forms an H2 suboptimal measurement feedback controller. It turns out that the sequence of gains for any chosen estimator can also be constructed by dualizing the H2 SOSFGS algorithm. This development will be given in the full version of the paper.

6. Conclusions For discrete-time systems, we explore here a direct method of constructing a sequence of H2 suboptimal feedback controllers of either state feedback or estimator based measurement feedback type. An algorithm called H2 SOSFGS algorithm is developed to construct an H2 suboptimal state feedback gain sequence. The sequence of H2 suboptimal state feedback gains constructed by H2 SOSFGS algorithm depends on a parameter ". For any particular H2 suboptimal state feedback gain, a sequence of gains for any of three dierent estimator structures (prediction, current, and reduced order estimators) `suitable' to generate a sequence of H2 suboptimal measurement feedback controllers can also be constructed by dualizing the H2 SOSFGS algorithm (This will be developed in the full version of the paper). Some attributes of the design scheme presented above are enumerated below. 1. The state feedback gains are parameterized directly in terms of a tuning parameter ". The design equations can be solved without explicitly requiring a value for ". This implies that, unlike other methods such as `perturbation methods' in which parameterized algebraic Riccati equations (ARE) where parameterization is implicit are to be solved repititively, no `repetitive' solutions of the design equations developed here are necessary as " changes. In this sense, the design presented here is a `one-shot' design and thus " truly acts as a tuning parameter. A similar advantage also holds good in constructing estimator gains. 2. The design equations are developed using several subsystems of the given system. In this sense the design is decentralized. Such a decentralized method of design, reduces the computational complexity of designing a large scale system. Also, by adopting a standard method of design for each subsystem, the mechanics of performing the design are simpli ed. The computations required for each subsystem design do not involve arbitrarily small or large numbers. This implies that as the tuning parameter " decreases, the design does not face any `stiness' problem which inherently cripples other design methods such as `perturbation methods' owing to the interaction of various slow and fast dynamic phenomena. 3. The direct method developed here allows the construction of any of three dierent estimator structures (prediction, current, and reduced order estimators) unlike `perturbation methods' which preclude such a construction.

21

References [1] B. D. O. Anderson and J. B. Moore, Optimal Control: Linear Quadratic Methods, Prentice-Hall, New York, 1989. [2] C.T. Chen, Linear System Theory and Design, Holt, Rinehart and Winston, New York, 1984. [3] Z. Lin, A. Saberi, P. Sannuti and Y. Shamash, \Perfect regulation for linear discretetime systems { A low-gain based design approach," Automatica , Vol. 32, No. 7, pp. 1085-1091, 1996. [4] Z. Lin, A. Saberi, P. Sannuti and Y. Shamash, \A direct method of constructing H2 suboptimal controllers { continuous-time systems," submitted for publication. [5] A. Saberi, B. M. Chen and P. Sannuti, Loop transfer recovery: analysis and design, Springer-Verlag, London, 1993. [6] A. Saberi, Z. Lin and A Stoorvogel, \H2 and H1 almost disturbance decoupling problem with internal stability," Proceedings of 1995 ACC, also to be published in International Journal of Robust and Nonlinear Control. [7] A. Saberi and P. Sannuti, \Squaring down of non-strictly proper systems," International Journal of Control, vol. 51, no. 3, pp. 621-629, 1990. [8] A. Saberi, P. Sannuti and B.M. Chen, H2 Optimal Control, Prentice Hall International, London, 1995. [9] P. Sannuti and A. Saberi, \A special coordinate basis of multivariable linear systems { nite and in nite zero structure, squaring down and decoupling," International J. Control, Vol. 45, pp. 1655-1704, 1987. [10] A. A. Stoorvogel, A. Saberi and B.M. Chen, \Full and reduced order observer based controller design for H2-optimization," International Journal of Control, Vol. 58, No. 4, pp. 803-834, 1993. [11] H. L. Trentelman and A. A. Stoorvogel, \Sampled-data and discrete-time H2 optimal control," SIAM J. Contr. & Opt., Vol. 33, No. 3, pp. 834-862, 1995.