A Direct Method of Constructing H2 Suboptimal ... - Semantic Scholar

A Direct Method of Constructing H2 Suboptimal Controllers { Continuous-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 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 observer based measurement feedback controllers. Both full as well as reduced order observer based controllers are developed. For a given H2 suboptimal state feedback gain, a sequence of observer gains for either a full or a reduced order observer 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 both full and reduced order observer based controllers while still maintaining throughout the design the computational simplicity of it.

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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 [7] 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 observer or 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 observer based one. This paper is intended to develop such methods. The recent book [7] 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 [7], 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

3 which all the measurements are contaminated with noise. This implies that while constructing observer based controllers, one always obtains a full order observer based controller. That is, construction of a reduced order observer based controller is 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. 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 observer 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 ? , C and C + respectively denote the whole complex plane, the open left half complex plane, the imaginary axis, and the open right half complex plane. (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(s) is said to be stable if all its poles are in C ?. Given a stable transfer function G(s), as usual, its H2 norm is de ned by  1 Z 1 1=2 0 kGk2 = 2 tr G(j!)G (?j!)d! : l

l

l

O

l

l

l

?1

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 [7], 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.

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

8 > < x_ = Ax + Bu + Ew + D1 w  : > y = C1 x : z = C2 x + D2 u;

(2.1)

4 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  is characterized by  :

(

C

C

v_ = A v + B y; u = C v + D y: con

con

con

con

(2.2)

We note that  , 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  is denoted by    . A controller  is said to be internally stabilizing the system , if the closedloop system    is internally stable, i.e., if    has all its poles in C ?. Also, a controller  is said to be admissible if it provides internal stability for the closed-loop system    . The transfer matrix from w to z of    is denoted by Tzw (   ). 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). C

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C

C

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C

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C

C

C

2.2. A Special Coordinate Basis In this subsection, we recall a special coordinate basis (SCB) of linear time-invariant systems [8,6]. 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 system  characterized by a quadruple (A; B; C; D),  x_ = Ax + Bu  : y = Cx + Du;

(2.3)

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 D = UDV = 0 0 ; (2.4) 0

where m0 is the rank of D. Thus the system in (2.3) can be rewritten as u  8 0 > ^ > < x_ = A x + [ B0 B 1 ] u^1        > > : z0 = C^0 x + Im 0 u0 z^ C 0 0 u^

(2.5)

0

1

1

1

where the matrices B0, B^ 1, C0 and C^ 1 have appropriate dimensions. We have the following lemma.

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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 ; z = ?O z ; u = ?I u 0 x = [xa; x0b; x0c; x0d]0 ; xd = [x01; x02;


; x0md ]0 z = [z00 ; zd0 ; zb0 ]0 ; zd = [z1; z2;


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


; umd ]0 ; and x_ a = Aaaxa + B0az0 + Lad zd + Labzb (2.6) x_ b = Abbxb + B0bz0 + Lbdzd ; zb = Cbxb (2.7) x_ c = Accxc + B0cz0 + Lcb zb + Lcd zd + Bc[Ecaxa + uc ] (2.8) and for each i = 1;


; md, 2 3 md X x_ i = Aqi xi + B0iz0 + Lidzd + Bqi 4Eiaxa + Eibxb + Eicxc + Eij xj + ui5 j =1

and

zi = Cqi xi ; zd = Cdxd z = C axa + C bxb + C cxc + 0

0

0

0

md X j =1

(2.9) (2.10)

C j xj + u : 0

0

(2.11)

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


; md(). The control vectors u0, ud and uc are respectively of dimensions m0, md() and mc() = m ? m0 ? md(), while the controlled output vectors z0, zd and zb are respectively of dimensions p0 = m0, pd = md() and pb = p ? p0 ? pd . The matrices Aqi , Bqi and Cqi have the following form: # " # " 0 I q ? 1 i (2.12) Aqi = 0 0 ; Bqi = 01 ; Cqi = [1; 0;


; 0]: (Obviously for the case when qi = 1, we have Aqi = 0, Bqi = 1 and Cqi = 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, Lid = [Li1; Li2;


; Li i?1; 0; 0;


; 0]: Also, the last row of each Lid is identically zero.

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Proof : The proof of this lemma can be found in [8] and [6].

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; 2 Aaa LabCb 0 LadCd 3 7 6 A~ := ??S 1 (A ? B0C0)?S = 664 B 0E LAbbC A0 LLbd CCd 775 ; c ca cb b cc cd d BdEa BdEb BdEc Ad 2 B0a 0 0 3 6 B0b 0 0 77 B~ := ??S 1 [ B0 B^ 1 ] ?I = 664 B 0 Bc 75 ; 0c B0d Bd 0 2 3 2 3 C C C C C 0 0a 0b 0c 0d C~ := ??O1 64 75 ?S = 64 0 0 0 Cd 75 ; 0 Cb 0 0 C^ 1 and 3 2 Im 0 0 D~ := ??O1D?I = 64 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. 0

Property 2.1. We note that (Abb, Cb) and (Aqi , Cqi ) form observable pairs. Unobservability

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 # " " # C C A 0 0a 0c aa Aobs = B E A ; Cobs = E E ; c ca cc a c

Ea = [ E 0 a E 0 a


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


Em0 d c ]0 Similarly, (Acc , Bc) and (Aqi , Bqi ) 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 # # " " B L A L C a ad aa ab b ; Bcon = B L : Acon = 0 A b bd bb 1

2

2

1

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 in the open right half complex plane, on the imaginary axis, and in the open left half complex plane.

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2.3. Connection Between an H Optimal Control Problem and a Disturbance Decoupling Problem 2

In this subsection, we recall two problems, one an H2 optimal control problem and another a disturbance decoupling problem. Then we recall an interconnection 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 H optimal control problem is to nd an internally stabilizing proper (strictly proper) controller  which minimizes the H norm of the closed-loop transfer matrix Tzw (   ). The in mum of an H optimal 2

C

2

2

C

control problem over the class of admissible proper controllers is denoted by p, that is

p := inf fkTzw (   )k2 j  is proper and internally stabilizes g: (2.13) Similarly, we denote the in mum of an H2 optimal control problem over the class of admissible strictly proper controllers as sp , that is

sp := inf fkTzw (   )k2 j  is strictly proper and internally stabilizes g: (2.14) An internally stabilizing proper (strictly proper) controller  is said to be an H2 optimal controller if it achieves a closed-loop H2 norm p (respectively, sp ). C

C

C

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C

As shown in [7], although the conditions for the existence of a strictly proper H2 optimal controller are dierent from those of a non-strictly proper H2 optimal controller, it turns out that in the case of continuous-time systems (but not in discrete-time systems) the value of the in mum p is the same whether proper or strictly proper controllers are considered. Next, as discussed in detail in [7], the H2 optimal control problem for a given system  can be related to a disturbance decoupling problem via measurement feedback with internal stability (DDPMS) for an auxiliary system denoted here by  . In what follows, in order to facilitate the introduction of the auxiliary system  , we rst introduce two other auxiliary systems  and  and state their properties. Then, after recalling the de nition of a DDPMS, we relate the H2 optimal control problem for  to the DDPMS for  . As a preliminary step before introducing  , we introduce a continuous-time linear matrix inequality (CLMI) as follows: F (P )  0; (2.15) where  A0P + PA + C 0 C PB + C 0 D  2 2 F (P ) := B 0P + D0 C 2 2 0 D2 D2 : 2 2 Here the matrices A, B , C2, and D2 are the data that characterizes the subsystem 2 of the given system  as in (2.1). As shown in [7], whenever the pair (A; B ) is stabilizable, there exists a unique semi-stabilizing solution P of the CLMI (2.15). Moreover, such a solution P is positive semi-de nite, rank minimizing, and is the largest among all symmetric solutions. Assuming that (A; B ) is stabilizable, we compute such a solution P . Next, de ne the maximal rank matrix [ C D ] where C has n columns and D has m columns, such that  0 C [C D ]: F (P ) = D (2.16) 0 PQ

PQ

P

Q

PQ

P

P

P

P

P

P

P

P

P

8 Now we are ready to de ne the rst auxiliary system  as P

8 > < x_ = Ax + Bu + Ew + D1 w  : > y = C1 x : P

P

P

P

P

P

P

z =Cx + D u ; P

P

P

P

(2.17)

P

P

where the matrices A, B , C1, D1, and E are the data pertaining to the system  of (2.1), and C and D are as in (2.16). We need to be aware of certain fundamental properties of  , and the following lemma reveals them. P

P

P

Lemma 2.2. Consider the auxiliary system  as in (2.17) with the pair (A; B ) being stabiP

lizable. Let a subsystem 2 of  be characterized by the matrix quadruple (A; B; C ; D ). 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 2 and 2 is described as follows: P

P

P

P

P

1. 2 is right invertible. 2. 2 has a total of na-(2) + na(2) + na (2) + nb (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 imaginary axis, (c ) the na (2) mirror images with respect to the imaginary axis of all the unstable (i.e. those in C +) invariant zeros of 2, and (d ) some nb(2) xed locations in the open left-half complex plane which contain the stable input decoupling zeros (but not the invariant zeros) of 2. 3. 2 has the same in nite zero structure as 2 does. 4. 2 is invertible if and only if 2 is left invertible. P

P

+

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+

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P

P

Next, as a preliminary to introducing  , we introduce another CLMI as follows: Q

G(Q)  0;

where

(2.18)

 0 + EE 0 QC 0 + ED0  G(Q) := AQC+QQA : + D E0 D D0 Here the matrices A, E , C , and D are the data that characterizes the subsystem  of the given system  as in (2.1). Again, as shown in [7], whenever the pair (A; C ) is detectable, we know that there exists a unique semi-stabilizing solution Q of the CLMI (2.18). Moreover, such a solution Q is positive semi-de nite, rank minimizing, and is the largest among all symmetric solutions. Assuming that (A; C ) is detectable, compute such a solution Q. Next, 1

1

1

1

1

1

1

1

1

1

1

9 de ne the maximal rank matrix [ E 0 D0 ]0 where E has n rows and D has p rows, such that E  [ E 0 D0 ] : (2.19) G(Q) = D Q

Q

Q

Q

Q

Q

Q

Q

Now we introduce the auxiliary system  as Q

8 > < x_ = Ax + Bu + E w +D w  : > y = C1 x > : Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

(2.20)

z =Cx + Du ; where the matrices A, B , C , C , and D are the data pertaining to the system  of (2.1), and the matrices E and D are as in (2.19). The following lemma, which reveals some important properties of  , is analogous to Lemma 2.2. 2

Q

1

Q

2

2

Q

Q

2

Q

Q

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

Q

tectable. Let a subsystem 1 of  be characterized by the matrix quadruple (A; E ; C1; D ). 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 1 and 1 is described as follows: Q

Q

Q

Q

Q

1. 1 is left invertible. 2. 1 has a total of na-(1) + na(1) + na (1) + nc(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 imaginary axis, (c ) the na (1) mirror images with respect to the imaginary axis of all the unstable (i.e. those in C +) invariant zeros of 1, and (d ) some nc(1) xed locations in the open left-half complex plane which contain the stable output decoupling zeros (but not the invariant zeros) of 1. 3. 1 has the same in nite zero structure as 1 does. 4. 1 is invertible if and only if 1 is right invertible. Q

Q

+

l

+

l

Q

Q

We now de ne the auxiliary system  by combining appropriately the auxiliary systems  and  . The auxiliary system  is described by PQ

P

Q



PQ

8 > < x_ : >y :

PQ

= Ax + Bu + E w = C1 x +D w z =C x + D u : PQ

PQ

PQ

PQ

PQ

P

PQ

PQ

P

Q

PQ

Q

PQ

(2.21)

PQ

Here the matrices A, B , and C1 are the data pertaining to the system  of (2.1), and the matrices C , D , E , and D are as in (2.16) and (2.19). P

P

Q

Q

10 In what follows, two subsystems of  , namely 1 and 2 , play important roles. 2 is characterized by the matrix quadruple (A; B; C ; D ) and is the same one as 2 . Similarly, 1 is characterized by the matrix quadruple (A; E ; C1; D ) and is the same one as 1 . Since 1 is the same as 1 , and 2 is the same as 2 , the zero structures of 1 and 2 are respectively as discussed in Lemmas 2.3 and 2.2. The following is the de nition of the DDPMS for  . PQ

PQ

PQ

P

PQ

Q

P

P

Q

PQ

PQ

PQ

Q

Q

Q

PQ

PQ

PQ

De nition 2.2. Consider a system  as in (2.21). The disturbance decoupling problem with measurement feedback and internal stability (DDPMS) for  is the problem of nding a proper controller  of the form (2.2) such that the closed-loop system   is internally PQ

PQ

C

stable, while the resulting closed-loop transfer function is identical to 0.

PQ

C

The following lemma, recalled from [7,9] connects the H2 optimal control problem for  with the DDPMS for  . PQ

Lemma 2.4. Consider an H optimal control problem as de ned by De nition 2.1 for a 2

system  as in (2.1). Assume that (A; B ) is stabilizable and (A; C1) is detectable. Also, consider the auxiliary system  as given in (2.21), and a proper controller  as in (2.2). Then, the following two statements are equivalent. 1.  is an H2 optimal controller for , i.e. the closed-loop system    is internally stable, and the H2 norm of the closed-loop transfer function from w to z is equal to the in mum p. 2.  solves the DDPMS for  , i.e. the closed-loop system    is internally stable, and the resulting transfer function from w to z is equal to zero. Moreover, the above equivalence holds in particular if one considers strictly proper controllers, i.e., controllers  as in (2.2) with Dcon = 0. PQ

C

C

C

C

PQ

PQ

PQ

C

PQ

C

Necessary and sucient conditions under which an H2 optimal proper controller  or an H2 optimal strictly (Dcon = 0) proper controller  exists for the given system  can be found in [7,9]. C

C

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 observer 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 the auxiliary system  . 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. PQ

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

De nition 3.1. (H suboptimal control problem) Consider a system  as given in 2

(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  (") j " > 0 g which satisfy the following two conditions: C

1. There exists an " > 0 such that for any " 2 (0; "], the closed-loop system    (") is internally stable. 2. As " ! 0, the H2 norm of the corresponding closed-loop transfer function Tzw (   (")) approaches p (respectively, sp ) as given in De nition 2.1. C

C

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  (") 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. C

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  (") of it is stabilizing and kTzw (   ("))k2 < p + (respectively, kTzw (   ("))k2 < sp + ). 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 -level 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 -level suboptimal proper (or strictly proper) controllers is simply called an H2 -level suboptimal proper (or strictly proper) controller or H2 -level suboptimal state feedback controller as the case may be. C

C

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

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

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

PQ

PQ

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  of the form (2.2) such that the following hold: 2

2

C

1. The closed-loop system    is internally stable; and 2. The resulting closed-loop transfer function Tz w (   ) has an H2 norm less than . PQ

C

PQ

PQ

PQ

C

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 D = 0, the corresponding H2 -ADDPMS is referred to as H2 -ADDPSS. Furthermore, a sequence of proper (strictly proper) controllers f  (") j " > 0 g is said to solve the (H2-ADDPMS)p (respectively, (H2-ADDPMS)sp) for  if the following hold: Q

C

PQ

1. There exists an " > 0 such that for any " 2 (0; "], the closed-loop system    (") is internally stable; and 2. The H2 norm of the resulting closed-loop transfer function Tz w (   (")) tends to zero as " tends to zero. PQ

PQ

PQ

PQ

C

C

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  (") of it is stabilizing and kTz w (   ("))k2 < . C

PQ

PQ

PQ

C

As we said earlier, there is a de nite connection between the H2 suboptimal control problem for the system  and the H2-ADDPMS for the auxiliary system  . The following lemma explores this connection, and as such plays a signi cant role in our construction of a sequence of H2 suboptimal controllers for . PQ

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

a system  as in (2.1). Assume that (A; B ) is stabilizable and (A; C1) is detectable. Also, consider the auxiliary system  as given in (2.21), and a sequence of proper (strictly proper) controllers f  (") j " > 0 g each member of which is of the form  as in (2.2). Then, the following two statements are equivalent: PQ

C

C

1. The sequence of proper (strictly proper) controllers f  (") j " > 0 g is a sequence of H2 suboptimal controllers for . 2. The sequence of proper (strictly proper) controllers f  (") j " > 0 g solves the H2ADDPMS for  . C

C

PQ

13

Proof: From [7] (pages 163-164), we note that for any member of f  (") j " > 0 g, say " , C

C

we have kTzw (  " )k22 = kTz w (  " )k22 + tr [Q(A0P + PA + C20 C2)] + tr [E 0PE ] = kTz w (  " )k22 + ( p)2: This implies that kTz w (  " )k22 = kTzw (  " )k22 ? ( p)2: (3.1) Now, if f  (") j " > 0 g is a sequence of H2 suboptimal controllers for , then (3.2) kTzw (  " )k2 ! p as " ! 0; and thus, in view of (3.1), we have kTz w (  " )k2 ! 0 as " ! 0: (3.3) Thus, the sequence f  (") j " > 0 g solves the H2-ADDPMS for  . To show the other way, again if f  (") j " > 0 g solves the H2-ADDPMS for  , we rst have (3.3) which, in view of (3.1), then implies (3.2). We next consider the special case when the entire state is available for feedback. Here we deal only with static state feedback controllers. The following lemma which is a special case of Lemma 3.1 plays a central role in the next section where an algorithm is developed to construct a sequence of H2 suboptimal state feedback gains for . Lemma 3.2. Consider an H2 suboptimal control problem as de ned by De nition 3.1 for a system  as in (2.1) with C1 = I and D1 = 0, i.e., a system in which the entire state is available for feedback. Let (A; B ) be stabilizable. Also, consider the auxiliary system  as in (2.17). Then, the following properties hold: 1. Consider a sequence of H2 suboptimal state feedback gains F "(A; B; C2; D2; E ) which is constructed for the system . Then F "(A; B; C2; D2; E ) is also a sequence of H2 suboptimal state feedback gains for the system  , and equivalently it is a sequence of H2 ADDPSS feedback gains for  . 2. Consider a sequence of H2 suboptimal state feedback gains F "(A; B; C ; D ; E ) which is constructed for the system  . Then F " (A; B; C ; D ; E ) is a sequence of H2 ADDPSS feedback gains for  , and it is also a sequence of H2 suboptimal state feedback gains for the system . 3. Consider a sequence of H2 -level suboptimal state feedback gains F " (A; B; C2; D2; E ) which is constructed for the system . Then F " (A; B; C2; D2; E ) is also a sequence of H2 -level suboptimal state feedback gains for the system  , and equivalently it is a sequence of H2 -level ADDPSS feedback gains for  . 4. Consider a sequence of H2 -level suboptimal state feedback gains F " (A; B; C ; D ; E ) which is constructed for the system  . Then F " (A; B; C ; D ; E ) is a sequence of H2 -level ADDPSS feedback gains for  , and it is also a sequence of H2 -level suboptimal state feedback gains for the system . Proof: It is obvious from Lemma 3.1. C

PQ

PQ

PQ

PQ

C

PQ

PQ

PQ

C

PQ

PQ

C

C

C

C

PQ

PQ

PQ

C

C

PQ

C

PQ

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

P

14

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: From Lemma 3.1, we know that there exists a sequence of H suboptimal controllers 2

for  if and only if there exists a sequence of controllers that solves the H2 -ADDPMS for  . On the other hand, based on the properties of 1 and 2 (see Lemmas 2.2 and 2.3), it follows from [5] that the H2-ADDPMS for  is solvable if and only if the pair (A; B ) is stabilizable and the pair (A; C1) is detectable. 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. PQ

PQ

PQ

PQ

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 follows easily from Corollary 2.4 of [5].

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

15

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. 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.

4.1. An H Low-Gain Design Algorithm 2

Consider the linear system

x_ = Ax + Bu; 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-left half s-plane. The H low-gain design we are proposing is carried out in three steps. 2

Step 1: Find the state transformation T ([2]) such that (T ? AT; T ? B ) is in the following 1

form,

1

2 B1 0


0 03 66 0 B2


0 0 777 ... . . . ... ... 7 ; T ?1B = 66 ... 66 77 4 0 0


B` 05 B01 B02


B0` 0 0 0 0 A0 where ` is an integer and for i = 1; 2;


; `, 2 3 2 3 0 0 1 0


0 6 66 0 7 0. 1.
.

0. 77 66 0 777 66 .. . . .. 77 ; Bi = 66 ... 77 : .. .. Ai = 6 . 64 0 75 64 0 0 0


1 75 1 ?aini ?aini?1 ?aini?2


?ai1 2 A1 A12


A1` 66 0 A2


A2` 6 T ?1AT = 66 ... ... . . . ... 64 0 0


A `

3  777

... 7 ; 77

5 

Furthermore, the transformation T is such that (Ai; Bi) is controllable, all the eigenvalues of Ai are on the imaginary axis, and all the eigenvalues of A have strictly negative real parts. Here and elsewhere as usual 's represent submatrices of less interest. 0

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

l

16 Note that Fi(") is unique.

Step 3 : Let

u = F (")x where the state feedback gain matrix F (") is given as 2 `? F1 ("2 (r +1)


(r` +1)) 66 0 66 .. . F (")=6 66 0 64 0 0 1

2

0

(4.2)

0 2`? (r +1)


(r` +1) F2 (" ) 0 .. .. . . . . . 2(r` +1) 0 F`?1 (" ) 0 0 0 F` (") 0 0 0 2

0 0 .. .






3





















3 0 0 777 .. 7 ?1 . 77T (4.3) 0 77 05 0

and where ri is the largest algebraic multiplicity of the eigenvalues of Ai. 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 sat-

bilizable and all the eigenvalues of A are in the closed-left half s-plane. Then we have the following properties: 1. The closed-loop system matrix A + BF (") is stable for all " > 0. 2. There exists an " > 0 such that for all " 2 (0; "],

kF (")k  " ke A

BF (")tk 

( +

kF (")e A

" `?



r1 +1)(r2 +1)


(r` +1)

1(

2

0

(4.4)

e?"

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


(r` +1)

2

t=2

?"t=2 +  "r` +2 e?"2(r` +1) t=2 +


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


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


(r`+1) t=2

(4.5)

BF (")tk  

( +

1

(4.6)

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

Proof: See [3]. 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 2 ): Following the procedure described in Section 2, construct the subsystem 2 as  Bu 2 : xz_ == CA xx + (4.7) +D u : P

P

P

P

P

P

P

P

P

P

P

17 We recall that  is given as, P

8 > < x_ = A x + B u + Ew  : >y = x :z = C x + D u : P

P

P

P

P

P

P

P

P

P

P

(4.8)

P

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

x = ?S x ; z = ?Oz ; u = ?I u such that the system  can be written in the following SCB form, x = [x0a; x0c; x0d]0; xd = [x0 ; x0 ;


; x0md ]0; xi = [xi ; xi ;


; xiqi ]0 z = [z0 ; zd0 ]; zd = [z ; z ;


; zmd ]0; zi = xi u = [u0 ; u0c; u0d]0; ud = [u ; u ;


; umd ]0 P

P

P

P

P

P

P

P

P

2P

P

1

1

P

0

P

0

1

2

P

2

1

2

1

P

2

and

x_ a = Aaaxa + B az + Ladzd x_ c = Accxc + B cz + Lcd zd + Bc [Ecaxa + uc ] P

P

0

P

P

0

P

P

0

P

x_ i = Aqi xi + B iz + Li zd + Bqi [Eiaxa + Eicxc + P

P P

0

0

0

P

0

P P

P

z = C axa + C cxc + P

(4.9) (4.10)

P

0

0

md X P

C j xj + u P

0

j =1

P

P

md X P

j =1

Eij xj + ui]; i = 1; 2;


; md

P

P

(4.11) (4.12)

0

We also note that since 2 is right invertible the state xb is not present. Step 3 (Construction of a low-gain Fa(")): By the Property 2.1 of the SCB, the pair (Aaa; [B0a; Lad]) is stabilizable. Also, by Property 2.3 of the SCB, the eigenvalues of Aaa are the invariant zeros of the linear system2 , and hence by Lemma 2.2 are all located in the closed left half s-plane. Hence, following the `Low-Gain design' method, one can design a feedback gain Fa(") for the pair (Aaa; [B0a; Lad]) such that, (a) The matrix Aaa + [B0a; Lad]Fa(") is stable for all " > 0; (b) There exists an "a 2 (0; 1] such that for all " 2 (0; "a], P

P

P

P

P

P

P

P

P

P

P

P

kFa(")k  k "

(4.13)

P

ke Aaa B a;LPad]Fa("))tk  ( P +[ 0P

kFa(")e Aaa

"` 2

P



P

0

?1 (r1P +1)(r2P +1)


(rPP +1) `

`P ?1 (rP +1)(rP +1)


(rP +1) 1 2 `P t=2

2 e?"

(4.14)

B a ;LPad ]Fa ("))tk   P "e?"t=2 +  PP "r`PP +2 e?"2(r`P +1) t=2 +


` ` ?1

( P +[ 0P

+1 " P

P

P `P ?2 (r2 +1)(r3 +1)


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


(r`PP +1) t=2 ` e

2

(4.15)

18 where ` and ri 's are integers, and k > 0 and i > 0' are constants, all independent of ". For later use, denote P

P

P

P

ra = 2` ? (r + 1)(r + 1)


(r` + 1) and partition the matrix Fa(") as 0 (")]0; Fa(") = [Fa0 ("); Fad0 (")]0 = [Fa (")0; Fa0 ("); Fa0 (");


; Fam d P

P

1

P

P

1

P P

2

0

0

1

P

2

and for each i = 1; 2;


; md , Fai(") 2 IR1na . Step 4 (Construction of a parameterized gain F (")): By the property of SCB, the pair (Acc; Bc ) is controllable, hence one can choose a feedback gain matrix Fc such that Acc +Bc Fc is stable and has a chosen set of eigenvalues. Also, choose Fi such that Aqi + Bqi Fi is stable. The existence of such a gain matrix Fi is guaranteed by the special form of (Aqi ; Bqi ). For further use, let the rst element of Fi be Fi1. Next a composite static state feedback gain is formed for the system 2 . This state feedback gain takes the form of 2 3 F u (") F (") = ?I 64 Fuc 75 (?S )?1 (4.16) Fud (") where i h Fu (") = Fa0(") ? C0a ?C0c ?C01 ?C02


? C0md Fuc = [ ?Eca Fc 0 0


0 ] 2 F (") 3 u 66 Fu (") 77 Fud (") = 66 ... 77 4 5 Fum (") d and for i = 1 to md, F  F i 1 i Fui (") = qi Fai ? Eia ?Eic ?Ei1 ?Ei2


qi Sqi (~") ? Eii


?Eimd "~ "~ Sqi (~") = Diagf1; "~; "~2;


; "~qi ?1g; "~ = "6ra +1: Step 5 (Construction of F " (A; B; C2; D2; E )): Finally one can construct a sequence of static state feedback gains as follows: where Fa0 2 IRm

P 0

P

nPa

P

P

P

P

P P

P

P P

P P

P P

P

0

P

P

P

P

P

P

P

0

P

P

1

2

P

P

P

P

P

P

P

P

P

P

P

P

P

F " (A; B; C2; D2; E ) = fF (") j F (") as in (4.16); " > 0g:

(4.17)

The following theorem veri es that the sequence F " (A; B; C2; D2; E ) as given by (4.17) is an H2 SOSFGS.

19

Theorem 4.2. Consider the given system  as in (2.1). Then the sequence F " (A; B; C ; D ; E ) 2

as given by (4.17) is an H2 SOSFGS.

2

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 kTz w (  C ("))k ! 0 as " ! 0. Equivalently, we need to show that the auxilliary system, P

2

 ~ : x_ = A x + B u + w z =C x +D u P

P

P

P

P

P

P

P

P

P

(4.18)

P

P

with u = F (")x is asymptotically stable for suciently small ", and for any x (0) 2 IRn , P

P

lim "!0

Z1 0

P

kz (t)k dt ! 0:

(4.19)

2

P

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 ("))kH < 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

2

5. H2 suboptimal observer based measurement feedback controllers This section is devoted to construct a sequence of H2 suboptimal observer based measurement feedback controllers. Controllers with observer based architecture are used ubiquitously in the modern control literature. As is well known, the idea behind this type of controller is to implement a \desirable" static state feedback law via an observer of a given type (say, Luenberger or full order or reduced order) that provides an estimate of the state and consequently forms a measurement feedback law hopefully having the same \desirable" feature as the original state feedback law does. Thus the design of a \desirable" measurement feedback law is divided into two stages, (1) design of a \desirable" static state feedback law, and (2) design of a \suitable" observer. These two stages are in general coupled and should be dealt with sequentially. That is, for a given desired state feedback law, one must design an appropriate observer to ensure that the resulting measurement feedback law would have the same desired feature as the original state feedback law does. In the context of H2 suboptimal control theory, one constructs an H2 suboptimal controller with observer based architecture by implementing an H2 suboptimal static state feedback law via a \suitable" observer. A central issue we face here is the following. Once we are given an H2 -level suboptimal state feedback law (or gain), we need to design a full or a reduced order observer such that, when the given state feedback law is implemented by cascading it with the observer, the resulting controller is an H2 -level suboptimal measurement feedback controller. To do so, the observer must be designed judiciously and must be \suitable" for the given particular H2 -level suboptimal state feedback law. It turns out that in general the \suitability" of an observer depends on the choice of the H2 suboptimal

20 state feedback law. In other words, an observer that is \suitable" for a particular H2 suboptimal state feedback law might not be \suitable" for some other one. Also, a particular H2 -level suboptimal state feedback law might have many \suitable" observers. Once the basic issue of designing a \suitable" observer is resolved, we are then capable of constructing a sequence of H2 -level suboptimal measurement feedback controllers by implementing a particular H2 -level suboptimal state feedback law one at a time from a sequence of such laws via one of the observers of possibly many \suitable" observers. In what follows, Subsection 5.1 reviews the architecture of observer based controllers. This is done not only as a review but also to establish our notation. Then, Subsection 5.2 develops the theoretical aspects of how \suitable" observers can be constructed to implement a given H2 -level suboptimal state feedback law. It turns out that the H2 SOSFGS algorithm developed earlier can itself be used to construct a sequence of \suitable" observers for any given H2 -level suboptimal state feedback law.

5.1. Review of Controllers with Observer Based Architecture In this subsection we review some basics of observer based controllers. As is well known, the job of an observer is to produce an estimate x^ of the state x by utilizing only the measurement y as its input. In what follows, we rst recall the architecture of Luenberger observer based controllers which are of general type and subsume all other observer based controllers. Next, we recall the architecture of full order and reduced order observer based controllers which are particular cases of Luenberger observer based controllers. We do this since full and reduced order observer based controllers are used ubiquitously in the literature. Also, we note that observer theory can be developed for any speci ed system. In the following description of controllers, we use  as the given system. This is because, as seen from Lemma 3.1, in order to construct a sequence of H2 suboptimal controllers with observer based architecture for , one can equivalently construct a sequence of H2 suboptimal controllers with observer based architecture for the auxiliary system  of (2.21). This is what we intend to do subsequently. We now review brie y the following observer based controllers for  . PQ

PQ

PQ

Luenberger Observer Based Controller:

Consider the following Luenberger observer,  v_ = L~ v + G y + G u 1 2 (5.1a) x^ = 1v + 2y where v 2 IRr with r being the order of the controller and x^ 2 IRn. As seen from (5.1a), the Luenberger observer is characterized by the ve parameters L~ , G1, G2, 1, and 2 which are referred to as Luenberger observer parameters. It is known that, in the disturbance free case (i.e. w = 0), the variable x^ is an asymptotic estimate of the state x provided that the quintuple (L~ ; G1; G2; 1; 2) satis es the following conditions: for some 3, 9 (L~ ) 2 C ? ; > > ~ 3A ? L 3 = G1C1; = (5.1b) G2 = 3B; and > > ; 2C1 + 1 3 = In: PQ

PQ

PQ

PQ

PQ

PQ

PQ

PQ

l

21 A static state feedback control law u = Fx is implemented as u = F x^ : (5.1c) Equations (5.1a), (5.1b), and (5.1c) together de ne a Luenberger observer based controller. It is known that the poles of the Luenberger observer are given by (L~ ), while the poles of the closed-loop system comprising  and the above Luenberger observer based controller are given by the union of (A + BF ) and (L~ ). PQ

PQ

PQ

PQ

PQ

Full Order Observer Based Controller:

Consider a full order observer, x^_ = (A + Kf C1)^x + Bu ? Kf y : (5.2a) As seen easily, the full order observer is characterized by Kf which is referred to as the full order observer gain. The gain Kf is selected such that (A + Kf C1) is stable. We note that the poles of the full order observer are given by (A + Kf C1). A static state feedback control law u = Fx is implemented as u = F x^ : (5.2b) Equations (5.2a) and (5.2b) together de ne a full order observer based controller. It is straightforward to verify that the above full order observer based controller is a special case of the Luenberger observer based controller as in (5.1) with 9 L~ = A + Kf C1 > > > G1 = ?Kf > > = G2 = B (5.3) > 1 = In > > 2 = 0 > > ; 3 = In; and all the conditions given in (5.1b) merely imply that (5.4) (A + Kf C1) 2 C ? : Also, we note that the poles of the closed-loop system comprising  and the above full order observer based controller are given by the union of (A + BF ) and (A + Kf C1). PQ

PQ

PQ

PQ

PQ

PQ

PQ

PQ

l

PQ

Reduced Order Observer Based Controller:

In what follows, we develop a reduced order observer based controller of dynamic order n ? rank[C1 ; D ]+rank[D ]. At rst, without loss of generality, we assume that the matrices C1 and D have already been transformed to the following form,  0  D  C 02 C1 = I and D = 00 ; (5.5) 0 p?m where m0 is the rank of D . Thus, the system  as in (2.21) can be partitioned as follows: 8   E  B   x  > x_ 1 = A11 A12 1 1 + B u + E1 w > x A A x _ 2 2 2 21 22 > < 2      y   0 (5.6) D x C 0 1 02 0 > w + = > 0 > : zy1 = Ip?m 0 C xx2 + D u : Q

Q

Q

Q

0

Q

PQ

PQ

Q

Q

PQ

PQ

0

PQ

P

PQ

P

PQ

22 Here, by an abuse of notation but to simplify our presentation, we partitioned x as [x01; x02]0 rather than [x01 ; x02 ]0, and similarly y as [y00 ; y10 ]0 rather than [y00 ; y10 ]0. The idea behind the construction of a reduced order observer based controller is that we only need to build an observer for x2 as x1 (or equivalently y1) is available as a measurement. Our techniques to do so are based on the method discussed in Section 7.2 of Anderson and Moore [1]. Now, the dierential equation for x2 is given by PQ

PQ

PQ

PQ

PQ

PQ

!

i

h

y +E w x_ = A x + A B u where y is known, and u is temporarily assumed known. Observations of x are made via y and y~, where y~ := A x + E w = y_ ? A x ? B u : (5.7) If we do not worry about the dierentiation for a moment, we note that we have to build an observer for the following system, 8 i y ! h > x_ = A x + E w + A B > < u r : (5.8) ! " # " # > y C D > : y~ = A x + E w : It is shown in Proposition 2.2.2 on p.32 of Saberi, Chen and Sannuti [4] that r is detectable whenever the given system  is detectable. Next, we nd the observer for the system r ,    y C x^ ?  : x^_ = A x^ + A y + B u + Kr A y_ ? A x ? B u where Kr is the reduced order observer gain. We partition Kr = [Kr Kr ] (5.9) 2

1

22

2

21

1

2

2Q

PQ

PQ

2

PQ

1

12

2

1Q

2

22

0

02

2

22

2

21 1

2

2

2

12

1

PQ

11

2Q

0

PQ

21

PQ

02

1

2

PQ

0

2

12

0

1

PQ

1Q

PQ

1

1

11

1

1

PQ

1

h i

so as to be compatible with the partitioning of yy~ . Then, the use of change of variables v := x^2 + Kr1y1 results in a reduced order observer, 8 v_ = (A22 + Kr0 C02 + Kr1A12)v + (B2 + Kr1B1)u > > < (5.10a)  0  + [?0 KrI0 ; A21 + Kr1 A11 ? (A22 + Kr0 C02 + Kr1 A12)Kr1 ]y > n ? r > : x^ = I v + 0 ?K y ; 0

PQ

PQ

PQ

r

r1

PQ

where r is the dimension of x2 or equivalently the dimension of v, and is given by n ? rank[C1 ; D ] + rank[D ]. We use this reduced order observer to implement a static state feedback control law u = Fx as u = F x^ = F2v + [ 0 F1 ? F2Kr1 ] y ; (5.10b) where F is partitioned in conformity with the partitioning of x = [x01; x02]0, F = [F1 ; F2]: (5.11) Q

Q

PQ

PQ

PQ

PQ

PQ

PQ

23 Thus, equations (5.10a) and (5.10b) together de ne the reduced order observer based controller. Next, it is simple to verify that the above reduced order observer based controller is also a special case of the Luenberger observer based controller given in (5.1) with 9 > L~ = A22 + Kr0C02 + Kr1A12 > G1 = [?Kr0; A21 + Kr1A11 ? (A22 + Kr0C02 + Kr1A12)Kr1] > > > > G2 = B + K B 2 r 1 1 > 0 = (5.12) 1 = I > >   r > > 2 = 00 ?InK?r > > r1 > ; 3 = [ Kr1 Ir ] ; and all the conditions given in (5.1b) merely imply that (5.13) (A + Kr C + Kr A ) 2 C ?: Also, it is seen that the poles of the reduced order observer are (A +Kr C +Kr A ), while the poles of the closed-loop system comprising  and the above reduced order observer based controller are given by the union of (A + BF ) and (A + Kr C + Kr A ). 22

0

02

1

l

12

22

0

02

1

12

1

12

PQ

22

0

02

Remark 5.1. For the case when 1 is right invertible and the matrix D is of maximal rank, we have A22 = A, B2 = B , C02 = C1, and D0 = D . Using these facts, it is easy to see that both full and reduced order observer based controllers coalesce into one and the same. Q

Q

Q

Remark 5.2. We know that a full order observer based controller, when it is applied to  , PQ

yields a strictly proper closed-loop transfer function from w to z . In the same manner, we would like to point out that although the reduced order observer based controller is not strictly proper, however, when it is applied to  , it indeed yields a strictly proper closedloop transfer function from w to z . Thus, either a full or reduced order observer based controller described above, when applied to  , yields a strictly proper closed-loop transfer function from w to z . PQ

PQ

PQ

PQ

PQ

PQ

PQ

PQ

5.2. Construction of a Sequence of H Suboptimal Controllers with Observer Based Architecture 2

In this subsection, we pursue methods of constructing \suitable" observers to implement a given H2 -level suboptimal state feedback law. In this regard, the following lemma recalled from [7] plays a key role. It develops an expression for the closed-loop transfer function from w to z when a static state feedback gain or when an observer based controller is used to control  . PQ

PQ

PQ

Lemma 5.1. Consider a given system  as in (2.1). Assume that (A; B ) is stabilizable

and (A; C1) is detectable. Also, consider the auxiliary system  as given in (2.21). Let F be a static state feedback gain which renders A + BF stable. Let Ts; z w (s) be the transfer function from w to z when the static state feedback controller characterized by F is used to control  . Similarly, consider a general Luenberger observer based controller PQ

PQ

PQ

PQ

PQ

PQ

24 characterized by (5.1). Let Tl; z w (s) be the transfer function from w to z when the Luenberger observer based controller is used to control  . Then, we have Ts; z w (s) = (C + D F )(sI ? A ? BF )?1E ; (5.14) and Tl;z w (s) ? Ts; z w (s) = ?Ts;z u (s)l (s); (5.15) where l (s) = F [ 1(sI ? L~ )?1( 3E ? G1D ) ? 2D ]; (5.16) and where Ts; z u (s) is the transfer function from u to z when the static state feedback controller characterized by F is used to control  ; and Ts;z u (s) is given by Ts; z u (s) = (C + D F )(sI ? A ? BF )?1B + D : (5.17) PQ

PQ

PQ

PQ

PQ

PQ

PQ

P

PQ

PQ

P

PQ

Q

PQ

PQ PQ

Q

Q

PQ

PQ PQ

PQ

PQ

PQ PQ

P

Q

PQ PQ

P

P

Proof: See the proof of Lemma 9.4.1 of [7].

Equations (5.15) and (5.16) together show how the transfer function from w to z , when the Luenberger observer based controller is used, diers from the corresponding transfer function when a static state feedback controller is used. As such, they play a key role when comparing a state feedback controller with an observer based controller. In fact, in view of these expressions, it is clear that in order to recover the H2 suboptimality of a state feedback law, the observer must be designed such that the size of ?Ts; z u (s)l(s) can be made arbitrarily small. In what follows, we examine such a design for both the cases of traditional full order and reduced order observers, and show that such a design is always possible provided that the given system is stabilizable and detectable. Indeed, even in general, it turns out that one can always choose Luenberger observer parameters such that k ? Ts;z u (s)l(s)k2 can be made arbitrarily small. It is trivial to observe that in order to make k ? Ts;z u (s)l(s)k2 arbitrarily small one needs only to make kl(s)k2 arbitrarily small. This is true because Ts; z u (s) is independent of the choice of the observer parameters. We rst consider the case of full order observers. For this case, the corresponding l(s) can be rewritten as f (s), f (s) = F (sI ? A ? Kf C1)?1(E + Kf D ); (5.18) where the gain Kf is the observer parameter. One needs to choose Kf appropriately so that the size of f (s) can be made arbitrarily small. We have the following lemma. PQ

PQ

PQ PQ

PQ

PQ

PQ PQ

PQ PQ

Q

Q

Lemma 5.2. Consider a given system  as in (2.1). Also, consider the full order observer based controller given in(5.2a) and (5.2b). Assume that (A; B ) is stabilizable and (A; C1) is detectable. Then, for any  > 0, and for all Kf = Ff0 where Ff is any member of the set F " (A0; C10 ; E 0 ; D0 ; F 0), we have kf (s)k2 < : Q

Q

Proof: It follows from the properties of  , i.e., Lemma 2.3, and the solvability conditions 1Q

of H2 ADDPSS, i.e., Theorem 5.9.2 of [7]. The Kf determined by the above lemma depends on the state feedback gain F . On the other hand, the following lemma shows that it is possible to decouple the state feedback controller design with that of observer design.

25

Lemma 5.3. Consider a given system  as in (2.1). Also, consider the full order observer based controller given in(5.2a) and (5.2b). Assume that (A; B ) is stabilizable and (A; C1) is detectable. Then, for any  > 0, and for all Kf = Ff0 where Ff is any member of the set F " (A0; C10 ; E 0 ; D0 ; I ), we have kf (s)k2 < : Q

Q

Proof: It follows from the properties of  , i.e., Lemma 2.3, and the solvability conditions of H2 ADDPSS, i.e., Theorem 5.9.2 of [7].

1Q

We next consider the case of reduced order observers. For this case, the corresponding l(s) can be rewritten as r (s), r (s) = F2(sI ? A22 ? Kr0C02 ? Kr1A12)?1(E2 + Kr0D0 + Kr1E1 ); (5.19) where the gain Kr is the observer parameter and is partitioned as in (5.9). Obviously, again one needs to choose Kr appropriately so that the size of r (s) can be made arbitrarily small. The following lemma shows that this is always possible under the same assumptions as that of Lemma 5.2. Q

Q

Lemma 5.4. Consider a given system  as in (2.1). Assume that (A; B ) is stabilizable and

(A; C1) is detectable. Also, consider the reduced order observer based controller given in 0 (5.10a) and (5.10b). for  C Then,  Dany0  > 0, and for all Kr = Fr where Fr is any member of 0 the set F " (A022; A02 ; E20 ; E 0 ; F20), we have 12

Q

1Q

kr (s)k < : 2

Proof: It follows from the properties of the subsystem

(A0

 C 0

; E0

 D 0

; E ), i.e., ; A Lemma 9.3.1 of [7], and the solvability condition of H ADDPSS, i.e., Theorem 5.9.2 of [7]. As in the case of full order observer based controllers, the reduced order observer gain Kr determined by the above lemma depends on the particular state feedback gain F . On the other hand, the following lemma shows that it is possible to decouple the state feedback controller design with that of observer design. 22

02

12

2Q

0

1Q

2

Lemma 5.5. Consider a given system  as in (2.1). Assume that (A; B ) is stabilizable and

(A; C1) is detectable. Also, consider the reduced order observer based controller given in 0 (5.10a) and (5.10b). for  Then,  any0  > 0, and for all Kr = Fr where Fr is any member of 0 C02 ; E 0 ; D0 ; I ), we have the set F " (A022; A 2 E 12

Q

1Q

kr (s)k < : 2



0

 0 ; E 0 ; ED ), i.e., Lemma 9.3.1 of [7], and the solvability condition of H ADDPSS, i.e., Theorem 5.9.2 of [7].

Proof: It follows from the properties of the subsystem (A0 ; AC 22

2

02

12

2Q

0

1Q

26

6. Conclusions We explore here a direct method of constructing a sequence of H2 suboptimal feedback controllers of either state feedback or observer 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 either a full or reduced order type of observer `suitable' to generate a sequence of H2 suboptimal measurement feedback controllers can also be constructed by dualizing the H2 SOSFGS algorithm. 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 observer 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 reduced order observer based controllers unlike `perturbation methods' which preclude such a construction.

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 of linear multivariable systems { a low-and-high gain design," Proceedings the Workshop on Advances on Control and Its Applications, Computer and Systems Research Laboratory, University of Illinois at Urbana-Champaign, May 21, 1994. (Lecture Notes in Control and Information Sciences vol. 208), pp. 172-193, editors: H. Khalil, J. Chow and P. Ioannou.

27 [4] A. Saberi, B. M. Chen and P. Sannuti, Loop transfer recovery: analysis and design, Springer-Verlag, London, 1993. [5] 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. [6] A. Saberi and P. Sannuti, \Squaring down of non-strictly proper systems," International Journal of Control, vol. 51, no. 3, pp. 621-629, 1990. [7] A. Saberi, P. Sannuti and B.M. Chen, H2 Optimal Control, Prentice Hall International, London, 1995. [8] 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. [9] 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.