H_2 Optimal Output Feedback Control for a General

2014 European Control Conference (ECC) June 24-27, 2014. Strasbourg, France

H2 optimal output feedback control for a general discrete-time system* Sebastian Florin Tudor1,† and Cristian Oar˘a1 Abstract— For a generalized discrete–time system, possibly improper or polynomial, we give realization based formulas for the H2 optimal output feedback controller, under the same general hypotheses as in the proper case. The solution is provided in term of two descriptor algebraic Riccati equations and features the same elegant simplicity as the standard case.

I. INTRODUCTION Ever since it emerged during the 1960s in the works of Kalman in the form of a quadratic optimization (see e.g. [1]), the H2 optimal control problem played a keystone role in many fundamental topics in control theory and has encountered a wide range of applications - for a comprehensive treatment we refer to [2]. The problem is concerned with operating a dynamical system, described by differential equations, while minimizing a cost – usually defined as a quadratic functional and associated with the dissipated energy. When the optimal control is generated by an output feedback controller we obtain the genuine H2 optimal design problem, which actually consists in finding a stabilizing controller that minimizes the H2 norm of the resulting closed– loop system. The problem has been approached by using various mathematical techniques specific to the stochastic or the deterministic settings: the model matching technique applied for a multivariable system, either using state–space or transfer function matrix (TFM) formalisms [3], [4], [5], [6], the Diophantine equation for a scalar system, the generalized Popov based approach for continuous– and discrete–time systems [7], to mention just a few. Kucera was the first to consider the LQG / H2 control problem for singular systems [8], [9]. More recently, the model matching technique has been extended in [10] to cover the case of a generalized continuous–time system whose TFM may be improper. Improper (or singular) systems provide a great tool for modeling general physical systems, since they can include algebraic non-dynamic constraints, impulsive behavior and reversed time dynamics [11]. The range of applications of these systems varies from engineering, including power systems, electrical networks, aerospace industry, control of *This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS - UEFISCDI, project identification number PN-II-ID-PCE-2011-3-0235. 1 Sebastian Florin Tudor and Cristian Oar˘ a are with the Department of Automatic Control and Systems Engineering, Faculty of Automatic Control and Computer Science, ”Politehnica” University of Bucharest, Romania. † The work of the first author was supported by the European Social Fund, Program POSDRU, ID 132397. E-mail addresses: [email protected] (corresponding author), [email protected].

978-3-9524269-1-3 © EUCA

robots, modeling and control of algebraic dynamical systems, mechanical systems (see for example [12], [13], [14]), to economics [15]. Furthermore, cyber-physical systems under attack, mass/gas transportation networks and advanced communication systems can also be modeled as singular (descriptor) systems, see [16]. Motivated by this wide applicability, we extend in this paper the optimal H2 control theory to cover the class of generalized (singular) discrete-time systems using a novel approach, based on Popov’s positivity theory [17] and the original results in [7]. The main tool for the problem at hand is a special type of algebraic Riccati equation, investigated in [18]. Moreover, we provide a realization based solution for our main controller, featuring the same numerical easiness of the proper case. The paper is organized as follows. In Section II, we give the general framework for the problem at hand. Section III presents the main result. In Section IV we provide the proof of our main result, using the solutions of two special problems. In order to show the applicability of our results, two numerical examples are included in Section V. The paper ends with several conclusions. II. PRELIMINARIES By C, D, and ∂D we denote the complex plane, the open unit disk, and the unit circle, respectively. We will consider z ∈ C the complex variable. For a constant matrix A with elements in C we denote by A∗ its conjugate transpose. In stands for the identity matrix with dimension n. We will use Λ(A−zE) to denote the union of generalized eigenvalues of the matrix pencil A − zE (finite and infinite, multiplicities counting). The set of p × m TFMs with rational elements having coefficients in C is denoted by Cp×m (z). To represent an improper or polynomial discrete-time system T ∈ Cp×m (z), we will use a general type of realization called centered: T(z)

−1 D  + C(zE − A) B(z0 − z) A − zE B , =: C D z

=

(1)

0

where z0 ∈ C is fixed, n is called the order (or the dimension) of the realization, A, E ∈ Cn×n , B ∈ Cn×m , C ∈ Cp×n , D ∈ Cp×m , and the matrix pencil A − zE is regular, i.e., det(A − zE) 6≡ 0. A realization (1) is called minimal if its order is as small as possible among all realizations of this type. We call the realization normalized if z0 E − A = I. Centered realizations exhibit some nice features, due to the flexibility in choosing z0 always disjoint

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from the set of poles of T. For example, the order of a centered minimal realization always equals the McMillan degree δ(T) of T and the matrix D in (1) equals the value of T in z0 . Moreover, for a normalized realization we can recover a standard-like characterization of the TFM: −1  1 B. In − E T(z) = D + C z0 − z Therefore, centered realizations have been extensively used in the literature to solve problems for generalized systems whose TFMs is improper [19], [20], [21], [22]. Throughout the paper, we will consider realizations centered on the unit circle, i.e., z0 ∈ ∂D, with z0 not a pole of T. Conversions between descriptor realizations (defined in [11]) and centered realizations can be done by simple manipulations, see Section 5 in [23]. For T ∈ Cp×m (z), we show now a direct method to obtain a minimal realization centered at z0 ∈ C (which is not among its poles). An improper system has two parts: a differential (dynamical) part and an algebraic part, which contains the nondynamic modes (see, e.g., the decomposition in [24], Section 3). Therefore, we can always write T(z) = G(z) + P(z), where G(∞) = 0 and P(z) is a matrix polynomial. Minimal realizations for G(z) and P(z) may be obtained with numerically reliable algorithms in the form (see [11]):       1 N − zIn2 B2 A − zIn1 B1 ,P = , G(z) = C1 0 C2 D2 z (2) where N is a nilpotent matrix. Then it can be directly checked that   A − zIn1 0 (z0 In1 − A)−1 B1 0 In2 − zN (In2 − z0 N )−1 B2  T(z) =  C1 C2 D z0 (3) is a minimal realization centered at z0 , with n = n1 +n2 , and D := C1 (z0 In1 − A)−1 B1 + z0 C2 (In2 − z0 N )−1 B2 + D2 . We call the system (1) stable if its pole pencil A − zE has Λ(A − zE) ⊂ D, see e.g. [11]. The system pencil is   A − zE B(z0 − z) S(z) := . C D The realization (1) (or the pair (A − zE, B)) is called stabilizable if the following conditions hold:   (i) rank  A − zE B = n, ∀z ∈ C\D; (ii) rank E B = n. Analogously, we say that the realization (1) is detectable (or the pair (C, A−zE) is detectable) if the pair (A∗ −zE ∗ , C ∗ ) is stabilizable. Consider the change of variables u = F x + v, where F ∈ Cm×n is an arbitrary constant matrix and v ∈ Cm . With this, the pole pencil becomes A − zE + BF (z0 − z). Let (A−zE, B) be a stabilizable pair and fix z0 ∈ C\Λ(A−zE). m×n Then such that Λ A − zE + BF (z0 −
there is F ∈ C z) ⊂ D, see for example [22].

Finally, define the H2 −norm of a stable system (1) as Z 2π   1 kTk22 := Trace T∗ (ejθ )T(ejθ ) dθ. 2π 0 A useful evaluation of the H2 −norm of the system T(z), with input u and output y, is given by kTk22 =

m X

ky i k22 ,

(4)

i=1

where y i , i = 1, . . . , m is the componentwise output for the componentwise input ui := δei , i = 1, . . . , m, with δ the discrete-time unit impulse and ei , i = 1, . . . , m is the Euclidean basis of Rm . III. MAIN RESULT Consider a general discrete-time system T ∈ Cp×m (z), possibly improper or polynomial, with input u and output y, written in partitioned form as      y1 T11 T12 u1 = , (5) y2 T21 T22 u2 where Tij ∈ Cpi ×mj (z), with i, j ∈ {1, 2}, and m := m1 + m2 , p := p1 + p2 . The H2 optimization problem consists in finding an output feedback controller K ∈ Cp2 ×m2 (z), i.e., u2 = Ky2 , such that the closed-loop system TCL = LFT(T, K) := T11 + T12 K(I − T22 K)−1 T21 (6) is stable and the H2 −norm attains its minimum, i.e., K = arg

min

K stabilizing

kTCL k2 .

We state now our main result. Theorem 1. Let T ∈ Cp×m (z) having a minimal realization   A − zE B1 B2 (7) T(z) =  C1 D11 D12  . C2 D21 0 z 0

Assume that the following hypotheses hold: (H1 ) (A − zE, B2 ) is stabilizable and ∀θ ∈ [0, 2π),   A − ejθ E B2 (z0 − ejθ ) rank = n + m2 . (8) C1 D12 (H2 ) (C2 , A − zE) is detectable and ∀θ ∈ [0, 2π),   A − ejθ E B1 (z0 − ejθ ) rank = n + p2 . (9) C2 D21 Then the descriptor discrete–time algebraic Riccati equations (DDTAREs, see [18]) E ∗ XE − A∗ XA − ((z0 E − A)∗ XB2 + C1∗ D12 ) · ∗ (D12 C1 + B2∗ X(z0 E − A)) + C1∗ C1 = 0, (10) ∗ EY E ∗ − AY A∗ − ((z0 E − A)Y C2∗ + B1 D21 )· ∗ −1 · (D21 D21 ) (D21 B1∗ + C2 Y (z0 E − A)∗ ) + B1 B1∗ = 0, (11) have stabilizing hermitian solutions X and Y , respectively. Furthermore, the controller given in Box 1 solves the H2

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−1 ∗ · (D12 D12 )

 K(z) =

A − zE + (B2 F + L∗ C2 − B2 RC2 )(z0 − z) B2 R − L∗ F − RC2 R

 , z0

−1

∗ ∗ F := − (D12 D12 ) (D12 C1 + B2∗ X(z0 E − A)) , ∗ −1 L := − (D21 D21 ) (D21 B1∗ + C2 Y (z0 E − A)∗ ) , ∗ ∗ ∗ ∗ −1 R := −(D12 D12 )−1 (D12 D11 D21 )(D21 D21 ) .

Box 1  A − zE + B2 F (z0 − z) B2 (RC2 − F )(z0 − z) B1 + B2 RD21 0 A − zE + L∗ C2 (z0 − z) B1 + L∗ D21  TCL (z) =  C1 + D12 F D12 (RC2 − F ) 0 z 

0

Box 2

optimization problem, where F and L are the stabilizing Riccati feedbacks. Remark 2. The hypotheses (A − zE, B2 ) stabilizable and (C2 , A − zE) detectable are necessary conditions for the existence of a stabilizing controller. Indeed, it is well-known in the proper case that K stabilizes T iff it stabilizes T22 , see the paper [22] and Chapter 6 in [25]. For the general case, the proof follows straightforeword from the standard case. Remark 3. (H1 ) and (H2 ) are regularity assumptions, see [25], [26] for the standard case. The conditions (8) and (9) ensure that T12 and T21 have no zeros on the unit circle. In particular, (H1 ) shows that   0 A − ejθ0 E = n + m2 ⇒ rank D12 = m2 , rank C1 D12 ∗ D12 is invertible. where θ0 is such that z0 = ejθ0 . Thus D12 Further, this implies that the DDTARE (10) is well defined and has a stabilizing hermitian solution, see the Appendix in [18]. Moreover, numerically sound formulas for the stabilizing solution to the DDTARE (10) may be obtained from Theorems 10, 11 in [18]. Dual conclusions follow from (H2 ).

Remark 4. In Theorem 1 we have implicitly assumed that T22 (z0 ) = D22 = 0. The assumption is made for simplifying the formulas, with no loss of generality. Indeed, if K is a solution to the problem with D22 = 0, then K(I + D22 K)−1 is a solution for the original problem. Remark 5. The closed-loop system with T as in (7) and K as in Box 1 has a state-space realization given in Box 2. It is now obvious that TCL is stable, since F and L are the stabilizing feedbacks, i.e., Λ (A − zE + B2 F (z0 − z)) ⊂ D and Λ (A − zE + L∗ C2 (z0 − z)) ⊂ D. IV. SPECIAL PROBLEMS We proceed now with the proof of Theorem 1, which is based on the solution of two special problems, namely the Full Information problem and the two-block problem (we have borrowed the terminology from the model matching problem). Throughout this section we will consider that D11 = 0 in (7).

We begin with the Full Information (FI) problem. The corresponding system is given by   A − zE B1 B2    C1   0  D12  . (14) TF I (z) =    I 0 0 0 I 0 z 0

Note that for the FI problem the state and  the disturbances T are available for measurement, i.e., y2 = xT uT1 . Proposition 6. For the FI problem, assume that (H1 ) holds. Then the DDTARE (10) has a stabilizing solution X = X ∗ . Furthermore, the static controller   KF I = F 0 (15) solves the H2 optimization problem, where F is the stabilizing Riccati feedback given in Box 1, and   min kTCL k22 = 2 Trace Re(B1∗ XAF EF−1 B1 ) − B1∗ XB1 , K (16) with AF = A + z0 B2 F, EF = E + B2 F . Proof: From (H1 ) it follows that the DDTARE (10) has a stabilizing hermitian solution X, see Appendix in [18]. It is easy to see that the control signal is u2 = KF I y2 = F x. Thus, the closed-loop system has the pole pencil A − zE + B2 F (z0 − z) =: AF − zEF . Since F is the stabilizing Riccati feedback, we can conclude that TCL is stable. Moreover, since Λ(AF − zEF ) ⊂ D, we can compute a standard state–space realization for TCL . Therefore, the matrix EF is nonsingular. It remains to prove the optimality requirement. It can be easily checked that the existence of X = X ∗ for the DDTARE (10) is equivalent to the existence of a solution (X = X ∗ , V, W ) to the system of equations  ∗ D12 D12 = V ∗ V,  ∗ ∗ (z0 E − A) XB2 + C1 D12 = W ∗ V, (17)  E ∗ XE − A∗ XA + C1∗ C1 = W ∗ W, where V ∈ Cm1 ×m1 is invertible, W ∈ Cm1 ×n , and F := −V −1 W is the corresponding stabilizing feedback. Denote

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(17)

ky1,k k2 = (C1 xk + D12 u2,k )∗ (C1 xk + D12 u2,k ) = ke y1,k k2 + x∗k A∗ XAxk − x∗k E ∗ XExk − 2 Re [x∗k (z0 E − A)∗ XB2 u2,k ] = ke y1,k k2 − (Exk + B2 u2,k )∗ X(Exk + B2 u2,k ) + (Axk + z0 B2 u2,k )∗ X(Axk + z0 B2 u2,k ). (12)     (15) ky1i k22 = ke y1i k22 − 2e∗i B1∗ XB1 ei − 2 Re e∗i B1∗ X(Ax0 + z0 B2 u2,0 ) = −2e∗i B1∗ XB1 ei + 2 Re e∗i B1∗ XAF EF−1 ei . (13) Box 3

ye1,k = W xk + V u2,k . We evaluate successively in (12), Box 3, the 2−norm of the output signal y1,k (the quality output). Let u1,k = ui1,k := δk ei , for i = 1, . . . , m1 , k ≥ 0, where ei is the canonical base of Rm1 and δ the discrete–time unit impulse. Let the output be denoted in this case by y1i . From the dynamics of the system (14) we have Exk + B2 u2,k = Axk−1 + B1 ei ∆k + z0 B2 u2,k−1 , (18) with ∆k := z0 δk−1 − δk . Substitute (18) in (12) and sum the resulting expression from k = 0 to ∞ to get the first equality in (13), Box 3. Further, using the controller (15), we get ye1i = 0, since u2 = −V −1 W x. Thus the optimality is reached with the controller (15). Sum both sides in (13) from i = 1, . . . , m, and recall the evaluation (4) to arrive at (16). This ends the proof. Consier now the two-block problem, for which p2 = m1 , T12 fulfills the hypothesis (H1 ) in Theorem 1, and T21 fulfills hypothesis (A1 ): D21 ∈ Cm1 ×m1 is invertible and
−1 Λ A − zE − B1 D21 C2 (z0 − z) ⊂ D.

with K = LFT(G, KF I ) given in (20). Therefore the closed-loop system is identical for the FI problem and the two-block problem, and thus stable and optimal. The next result follows by duality from Proposition 8. We will need to assume (A2 ): D12 ∈ Cm2 ×m2 is invertible and
−1 Λ A − zE − B2 D12 C1 (z0 − z) ⊂ D. Proposition 9. Assume p1 = m2 , (A2 ) and (H2 ). Then the DDTARE (11) has a stabilizing hermitian solution Y , the H2 optimal controller is given by K(z)  =  −1 A − zE + (−B2 D12 C1 + L∗ C2 )(z0 − z) −L∗ , −1 −D12 C1 0 z0 (22) where L is the corresponding Riccati stabilizing feedback in Box 1, and   min kTCL k22 = 2 Trace Re(C1 EL−1 AL Y C1∗ ) − C1 Y C1∗ , K (23) where AL := A + z0 L∗ C2 and EL := E + L∗ C2 .

To proceed, we need the following lemma.

We finally proceed with: Proof of Theorem 1. The idea of the proof is to reduce Lemma 7. Consider the system T given in (7) with D11 = 0 the problem to a dual two-block one. With (H1 ) and (H2 ) it and assume that (A1 ) holds. We have that follows that the DDTAREs (10) and (11) have stabilizing TF I = T ⊗ G, (19) hermitian solutions. The DDTARE (10) has a stabilizing solution iff the matrix system (17) has a stabilizing solution where ⊗ is the Redheffer product (see [7]) and (X = X ∗ , V, W ), with F = −V −1 W . Replace the output y1 = C1 x + D12 u2 in (7) with ye1 = W x + V u2 to obtain G(z) =   the system −1 −1 A − zE − B1 D21 C2 (z0 − z) B1 D21 B2       0 0 A − zE B1 B2       I   . e e T T 12   e I I 0 T(z) = 11 :=  W 0 V  . (24) T T −1 −1 21 22 0 −D21 C2 −D21 C2 D21 0 z0 z 0

Proposition 8. For the two block problem the DDTARE (10) has a stabilizing hermitian solution X, the H2 optimal controller is given by

−1

Since V is invertible and Λ(A − zE − B2 V W (z0 − z)) ⊂ e satisfies (A2 ) and (H2 ). Therefore, we have a dual D, T e solved in Proposition 9. Write the two-bloc problem for T, e to obtain K in Box 1, with controller (22) for the system T K(z)  =  −1 −1 R = 0 (since we fixed D = 0). In other words, the 11 A − zE + (B2 F − B1 D21 C2 )(z0 − z) B1 D21 , controller F 0 z0   (20) A − zE + (B2 F + L∗ C2 )(z0 − z) −L∗ K(z) = where F is the corresponding stabilizing Riccati feedback in F 0 z0 2 Box 1, and min kTCL k2 is as in (16). (25) e The stability of the solves the two block problem for T. Proof: Notice that closed-loop system TCL = LFT(T, K) follows from the e CL = LFT(T, e K), since the two systems share stability of T TCL = LFT(TF I , KF I ) = LFT(T
⊗ G, KF I ) (21) the same pole pencil. = LFT T, LFT(G, KF I ) = LFT(T, K), 1585



z−4 T2 (z) = 2z 3 − 5z 2 + 3z + 1 2z 4 − 7z 3 + 5z 2 − 2    −11.2507 −74 −59 −31 −4  −28.8534  −59 −47.25 −24.75 −3.25    X=  −31 −24.75 −13.25 −1.75  , Y =  −45.3381 −9.8265 −4 −3.25 −1.75 −0.25    F = 8 6.5 3.5 0.5 , L = −1.5653 −3.6867 4

K2 (z) = 

−28.8534 −78.2077 −124.3853 −27.2597 −5.7929

 ,

 −45.3381 −9.8265 −124.3853 −27.2597  , −199.5417 −46.3979  −46.3979 −17.2944  −1.2555 , R = 2.

0.00435z 4 + 0.01024z 3 + 0.03504z 2 + 0.1098z − 0.1539 , z 4 − 1.327z 3 − 0.2854z 2 + 0.6157z

3z(z − 1.225)(z − 1.046)(z + 0.2338)(z 2 − 2.825z + 3.467)(z 2 + 3.384z + 5.677)

 T2CL (z) =

−z −2z 4 + 7z 3 − z 2 − 4z

2

2

 

−4z(z − 2.255)(z − 0.9932)(z + 0.4768)(z − 1.38z + 0.81)(z + 3.507z + 10.22) (z − 0.3297)(z + 0.4288)(z − 0.001108)(z + 0.001107)(z 2 + 9 × 10−7 z + 1.2 × 10−6 )(z 2 − 1.244z + 0.5774) Box 4: Example 2

It only remains to show that K in (25) is indeed optimal. The first equality in (13) holds, since no closedloop dynamics was used to derive it. To evaluate x0 and u2,0 consider the closed-loop dynamics (see Box 2) at k = 0 and use the dynamics of the controller (25). Finally, evaluate the closed-loop H2 −norm. We get kTCL k22 = e CL k22 + 2Trace [Re(B1∗ X(AE −1 + I)B2 F E −1 (B1 + kT F L L∗ D21 ) + AF EF−1 B1 ) − B1∗ XB1 ]. The equality holds for e satisfies the hypotheses all stabilizing controllers. Since T e CL k22 is reached for K of a dual two-block problem, min kT in (25) and threfore so is min kTCL k22 . The extension for D11 6= 0 follows by employing a technique similar to the one in Chapter 14.7, [25]. This concludes the proof.

We provide here two numerical examples. The first system has both polynomial and improper elements, while the second one is a polynomial matrix. Example 1. Let 20z 2 + 2z − 22 z+1

    10z 2 − z − 9 T1 (z) =     20z 3 + 8z 2 − 13z − 5 z+1

with the corresponding stablizing Riccati feedback   F = −0.4472 0.9615 1.2610 , for which the generalized eigenvalues

V. NUMERICAL EXAMPLES



Note that T1 (1) = D and that n = δ(T1 ) = 3. It is easy to check that T1 satisfies the hypotheses (H1 ) and (A1 ) of a two-block problem (Proposition
−1 8), since D21 = 5 and Λ A − zE − B1 D21 C2 (1 − z) = {0.8010, −0.8208, −0.3802} ⊂ D. The stablizing solution of the DDTARE (10) is   −1.4574 −1.3623 1.7877 3.2746  , X =  −1.3623 −2.7686 1.7877 3.2746 −5.3911

−4z 2 − z + 1 z+1

are inside the unit circle. With Proposition 8, we obtain the proper controller



K1 (z) =

    2 −2z + 3 ,   3 2 −4z − 2z + 5z + 1  z+1

with p1 = 2 and m1 = 1. The system is improper having one simple pole at −1 and one pole at ∞ with multiplicity two. A realization centered at z0 = 1 is   −1 − z 0 1 −13 1  0 −z 1 −1 0     0 −1 0 −10 2    T1 (z) =  . 1 1 0 0 −2     0 1 1 0 1  1 0 2 5 0 z =1 0

Λ (A − zE + B2 F (1 − z)) = {0.6633, −0.5851, −0.4861}

0.1994z 3 + 0.06138z 2 − 0.2135z − 0.04723 . z 3 − 0.575z 2 − 0.2573z − 0.009621

The closed-loop system is given by   −(z + 4.106)(z + 0.7224)(z − 1)   12.0398(z − 1)(z + 1)(z + 0.1541) T1CL (z) = . (z − 0.6633)(z + 0.5851)(z + 0.4861) Thus the closed-loop system is stable and proper. Moreover, the evaluation (16) is verified, i.e., min kTCL k2 = 
 1 2 Trace B1∗ XAF EF−1 B1 − B1∗ XB1 2 = 15.0623.

Example 2. Consider further the polynomial matrix T2 in Box 4 with m2 = p2 = 1. The system has all its poles

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at ∞ (one with multiplicity 3, and minimal realization with z0 = 1 is  1 0 0 0  −z 1 0 0   z −z 1 0 T2 (z) =   0 3z −2z 1   1 0 0 0 0 0 0 1

R EFERENCES

one simple). A centered  0 1 −1 1 0 0   0 0 2   0 0 0   4 −3 −1  1 −2 0 z

.

0 =1

Note that n = δ(T2 ) = 4. This is a completely general case, and therefore we will use Theorem 1. The solutions of the DDTAREs (10), (11), and the stabilizing Riccati feedbacks F, L are also given in Box 4. The controller K2 is proper, see Box 4. The closedloop system is stable and proper. The optimal value of the H2 −norm was computed to be min kTCL k2 = 155.079. VI. CONCLUSIONS We solved in this paper the H2 optimal control problem for generalized discrete–time systems (possibly improper or polynomial) using an output feedback controller. Explicit reduction to a model matching problem and inner-outer factorization has been avoided. A realization based formula for the optimal controller was given in terms of stabilizing solutions of two special Riccati equations. This leads to a numerical burden of the overall algorithm similar to the proper case. Two numerical examples showed the applicability of our results.

[1] R. Kalman, “When is a linear control system optimal?” ASME Trans. Series D: J. Basic Eng., vol. 86, pp. 51–60, 1964. [2] B. Anderson and J. Moore, Optimal Control: Linear Quadratic Methods. Prentice-Hall International, London, 1989. [3] B. A. Francis, “On the Wiener-Hopf approach to optimal feedback design,” System Control Lett., vol. 2, pp. 197–200, 1982. [4] J. C. Doyle, Lecture Notes in Advances in Multivariable Control. ONR/Honeywell Workshop, Minneapolis, 1984. [5] M. Vidyasagar, Control System Synthesis: A Factorization Approach. MIT Press, Cambridge, MA, 1985. [6] J. Doyle, K. Glover, P. Khargonekar, and B. Francis, “State–space solutions to standard H2 and H∞ control problems,” IEEE Transactions on Automatic Control, vol. 34, no. 8, pp. 831–847, 1989. [7] V. Ionescu, C. Oar˘a, and M. Weiss, Generalized Riccati Theory and Robust Control: A Popov Function Approach. John Wiley & Sons, New York, 1999. [8] V. Kucera, “Stationary LQG control of singular systems,” IEEE Trans. Automatic Control, vol. AC-31, pp. 31–39, 1986. [9] ——, “The rational H 2 control problem,” Proc. 31st IEEE Conf. on Decision and Control, pp. 3610–3613, 1992. [10] K. Takaba and T. Katayama, “H 2 output feedback control for descriptor systems,” Automatica, vol. 34, no. 7, pp. 841–850, 1998. [11] L. Dai, Singular Control Systems. Lecture Notes in Control and Information Sciences, 118. Springer-Verlag, Berlin, Heidelberg, 1989. [12] M. Gunther and U. Feldmann, “ CAD-based electric-circuit modeling in industry. I. Mathematical structure and index of network equations,” Surveys Math. Indust., vol. 8, no. 2, pp. 97–129, 1999. [13] P. Rabier and W. Rheinboldt, Nonholonomic Motion of Rigid Mechanical Systems from a DAE viewpoint. SIAM, Philadelphia, PA, 2000. [14] D. Lind and K. Schmidt, “Chapter 10, Symbolic and algebraic dynamical systems,” ser. Handbook of Dynamical Systems, B. Hasselblatt and A. Katok, Eds. Elsevier Science, 2002, vol. 1(A), pp. 765 – 812. [15] D. Luenberger, “Dynamic equations in descriptor form,” IEEE Trans. Automat. Control, vol. AC-22, no. 3, pp. 312–321, 1977. [16] F. Pasqualetti, F. Dorfler, and F. Bullo, “Attack Detection and Identification in Cyber-Physical Systems,” IEEE Trans. on Automatic Control, vol. PP, no. 99, p. 1, 2013. [17] V. Popov, Hiperstability of Control Systems. Berlin: Springer Verlag, 1973. [18] C. Oar˘a and R. Andrei, “Numerical Solutions to a Descriptor Discrete Time Algebraic Riccati Equation,” Systems and Control Letters, vol. 62, no. 2, pp. 201–208, 2013. [19] I. Gohberg and M. A. Kaashoek, Regular rational matrix functions with prescribed pole and zero structure. Topics in Interpolation Theory of Rational Matrix Functions, Boston, Birkhauser, 1988. [20] I. Gohberg, M. Kaashoek, and A. Ran, “Factorizations of and extensions to J-unitary rational matrix functions on the unit circle,” Integral Equations Operator Theory, vol. 15, pp. 262–300, 1992. [21] M. Rakowski, “Minimal factorizations of rational matrix functions,” IEEE Trans. Circuits Syst. I, vol. 39, no. 6, pp. 440–445, 1992. [22] C. Oar˘a and S¸. Sab˘au, “Parametrization of Ω−stabilizing controllers and closed-loop transfer matrices of a singular system,” System and Control Letters, vol. 60, no. 2, pp. 87–92, 2011. [23] ——, “All doubly coprime factorizations of a general rational matrix,” Automatica, vol. 45, no. 8, pp. 1960–1964, 2009. [24] C. Oar˘a and R. Marinic˘a, “J factorizations of a general discrete-time system,” Automatica, vol. 49, no. 7, pp. 2221 – 2228, 2013. [25] K. Zhou, J. Doyle, and K. Glover, Robust and Optimal Control. Prentice Hall, Englewood Cliffs, New Jersey, 1996. [26] A. Saberi, P. Sannuti, and B. M. Chen, H2 Optimal Control. Prentice Hall International, 1995.

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