# Indefinite LQ Problem for Irregular Singular Systems

Authors: Qingxiang Fang; Jigen Peng; Feilong Cao
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

The indefinite LQ problem for irregular singular systems is investigated. Under some general conditions, the optimal control-state pair is obtained by solving an algebraic Riccati equation. The optimal control is synthesized as state feedback. All the finite poles of the closed-loop system are located on the left-half complex plane. An example is given to show the validity of the proposed conclusion.


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## Body

## 1. Introduction

Singular systems have comprehensive practical background [1–3]. Great progress has been made in the theory and its applications since 1970s [4–8]. The treatment of the LQ problem for linear systems has been well studied since the early 1960s. A detailed discussion of this topic can be found in [9, 10]. For linear singular systems, there have also been a lot of excellent results about the LQ problem [11–13]. A linear quadratic control problem is indefinite when the cost weighting matrices for the state and the control are allowed to be indefinite. Although the indefinite LQ problem is meaningless for standard state space systems, it is solvable for singular systems under some conditions. To the best of our knowledge, [14] is the first article which studied a nonstandard quadratic regulator on a finite time horizon with indefinite weighting matrix for the control. Up to now, indefinite LQ theory has been extensively developed and has found interesting applications in finance, especially in dynamic macroeconomics theory [15, 16]. Many applications for this problem can be seen in [17, 18] and the references therein.A lot of literature about indefinite LQ control problem can be found. Trentelman [19] deals with the free-endpoint regular linear quadratic problem with indefinite cost functional in the context of linear quadratic optimal control and establishes necessary and sufficient conditions for the existence of optimal controls. Ran and Trentelman [20] consider the discrete time infinite-horizon linear quadratic problem with indefinite cost criterion and obtain necessary and sufficient conditions for the existences of optimal controls based on a geometric characterization of the set of all Hermitian solutions of the discrete time algebraic Riccati equation. Bucci and Pandolfi [21] study the finite horizon nonstandard LQ problem for an abstract dynamic system and provide necessary and sufficient conditions for finiteness of the value function corresponding to the control problem. Cui and Zhang [22] discuss the finite time horizon indefinite LQ problem for singular linear discrete time-varying systems and show that the indefinite LQ problem is dual to that of projection for backward stochastic systems. Cui et al. [23] study the indefinite LQ problem for singular discrete time-varying linear systems with multiple input delays and obtain an explicit controller by computing the gain of the smoothing estimation of dual systems. The above discussions were inducted in the premise that the singular system is regular. The indefinite LQ problem for irregular singular systems was rarely mentioned.In this paper, the indefinite LQ problem for irregular singular systems is discussed. It is shown that the indefinite LQ problem for irregular singular systems can be transformed to the regular LQ problem for standard state space systems by restricted system equivalence transformation under some conditions. The system state is decomposed into free state and restricted state and the input is decomposed into free input and forced input. The optimal control is synthesized as state feedback. The optimal control-state pair is derived by solving an algebraic Riccati equation. All the finite poles of the closed-loop system are located on the left-half complex plane.The remainder of the paper is organized as follows. In Section2, the indefinite LQ problem for irregular singular systems is transformed to the optimal problem for standard state space systems by restricted system equivalence transformation. In Section 3, we obtain the sufficient conditions for the unique existence of optimal control-state pair and solve the state feedback gain. A simulation example is exploited to demonstrate the effectiveness of the proposed results in Section 4. This paper is briefly concluded in Section 5.Notation. Throughout the paper, the superscript “T” stands for matrix transposition; Rn denotes the n-dimensional Euclidean space; Rn×m is the set of n×m real matrices; In is the n×n identity matrix; Rλ stands for the real part of λ; for real symmetric matrix A, A>0 means that A is definite-positive and A≥0 means that A is semidefinite-positive. The matrices, if not explicitly stated, are assumed to have compatible dimensions.


## 2. Statement and Transformation of Indefinite LQ Problem

Consider the optimal control problem of linear irregular singular systems(1)Ex˙=Ax+Bu,Ex(0)=x0
with quadratic cost
(2)J(u,x)=∫0∞‍[xT,uT]Q[xT,uT]Tdt,
where E,A∈Rm×n, B∈Rm×r, Q∈R(n+r)×(n+r), 0<rank⁡E=p<n, QT=Q, and x, u, and x0 are state, input, and initial state, respectively.System (1) is said to be regular if m=n and det⁡(sE-A)≢0; otherwise, it is irregular. For the irregular singular system, Zhu et al. [13] discussed the state feedback LQ problem.We denote the above LQ problem about (1) and (2) as problem P and define the set of admissible control-state pairs of problem P as
(3)Ω={J(u,x)∣(u,x)ispiecewisecontinuous,mmmmmmsatisfies(1)and|J(u,x)|<∞p}.ProblemP is called indefinite LQ problem if there are not only positive numbers but also negative numbers in the set of eigenvalues of matrix Q.The control objective of this paper is to find an optimal control-state pair(u*,x*)∈Ω such that
(4)J(u*,x*)=min⁡(u,x)∈ΩJ(u,x)
when problem P is indefinite LQ problem.Sincerank⁡E=p, there exist matrices M11∈Rp×m and N11∈Rn×p such that M11EN11=Ip, where M11 has full row rank and N11 full column rank. Suppose M12 is one of the left zero divisors of E and N12 one of the right zero divisors of E; then the matrices M1=[M11T,M12T]T and N1=[N11,N12] are nonsingular.Let(5)M1AN1=[A11A12A21A22],M1B=[B1B2],
where A11∈Rp×p, B1∈Rp×r.Definition 1 (see [24]).
System (1) is impulse controllable if there exists a smooth (impulse-free) control-state pair of system (1) for every initial condition.Obviously, it is necessary for the solvability of the problemP that system (1) is impulse controllable. The following lemma establishes two necessary and sufficient conditions for the impulse controllability of system (1).Lemma 2.
System (1) is impulse controllable if and only if
(6)rank⁡[A21A22B2]=rank⁡[A22B2]
or
(7)rank⁡[0E0EAB]=rank⁡[EAB]+rank⁡E.The proof is similar to that of Theorem  3 in [24], so it is omitted.In the following discussions, we always assume that (7) holds.Denote(8)d=rank⁡A22=rank⁡[0EEA]-2rank⁡E,
and then there exist matrices M21∈Rd×(m-p) and N21∈R(n-p)×d such that M21A22N21=Id, where M21 has full row rank and N21 full column rank. Suppose M22 is one of the left zero divisors of A22 and N22 one of the right zero divisors of A22; then the matrices M2=[M21T,M22T]T and N2=[N21,N22] are nonsingular.Let(9)A12N2=[A121,A122],M2A21=[A211A212],M2B2=[B21B22],
where A121∈Rp×d, A211∈Rd×p, and B21∈Rd×r.Denote(10)s=rank⁡[A22,B2]=rank⁡[0E0EAB]-2rank⁡E,
and then rank⁡[A212,B22]=rank⁡B22=s-d and there exist matrices M31∈R(s-d)×(m-p-d) and N31∈Rr×(s-d) such that M31B22N31=Is-d, where M31 has full row rank and N31 full column rank. Suppose M32 is one of the left zero divisors of B22 and N32 one of the right zero divisors of B22; then the matrices M3=[M31T,M32T]T and N3=[N31,N32] are nonsingular and M32A212=0.DenoteN=diag⁡([N1,Ir])diag⁡([Ip,N2,N3]), M31A212=A2121, B1N3=[B11,B12], B21N3=[B211,B212], and
(11)[x1T,x2T,x3T,u1T,u2T]T=N-1[xT,uT]T,
where B11∈Rp×(s-d), B211∈Rd×(s-d), x1∈Rp, x2∈Rd, x3∈Rn-p-d, u1∈Rs-d, and u2∈Rr+d-s; then the system (1) is restricted system equivalent (r.s.e.) to the following system:
(12)x˙1=A¯11x1+A122x3+B¯12u2,x1(0)=M11x0,x2=A¯21x1-B212u2,u1=A¯31x1,
where A¯11=A11-A121A211+A121B211A2121-B11A2121, B¯12=B12-A121B212, A¯21=B211A2121-A211, and A¯31=-A2121.In the system (12), the state variables x1, x2 and input variable u1 are determined uniquely by x3 and u2. Thus the state variable x3 is free and the input variable u1 is not free. We call x3 the free state and u1 the forced input. Accordingly, x1 and x2 are called restricted state and u2 is called free input. When B¯12=0 and B212=0, system (12) has a unique solution if and only if A122=0 or the dimension of x3 is zero.A system is usually required to have a unique solution when it is handled. The following lemma establishes a necessary and sufficient condition for the uniqueness of solutions of system (1).Lemma 3.
System (1) with B=0 has a unique solution if and only if
(13)rank⁡[0EEA]=rank⁡[EA]+rank⁡E
or
(14)rank⁡[0EEA]=n+rank⁡E.In the following discussions, we always assume that (13) or (14) holds; that is, we consider the control problem of the following system:
(15)x˙1=A¯11x1+B¯12u2,x1(0)=M11x0,x2=A¯21x1-B212u2,u1=A¯31x1.We have the following two conclusions about the input of system (1).Lemma 4.
System (1) has no forced input if and only if
(16)rank⁡[0EEA]=rank⁡[0E0EAB].Lemma 5.
If the equality(17)rank⁡[0E0EAB]=r+rank⁡[0EEA]
holds, then system (1) has no free input.It is meaningless to design the controller for a system without free input. In the following discussions, we always assume that (17) does not hold; that is, the system (1) satisfies that
(18)rank⁡[0E0EAB]<r+rank⁡[0EEA].Denote(19)P1=N12N21M21M12,P2=N31M31M22M12,P3=[P31,P32,P33]=[N11-P1AN11+P1BP2AN11N12N22-P1BN32-P2AN110N32],N4=[Ip00A¯210-B2120In-p-d0A¯310000Ir+d-s],
and Q¯=(Q¯ij)3×3=P3TQP3, where Q¯11∈Rp×p, Q¯22∈R(n-p-d)×(n-p-d), and Q¯33∈R(r+d-s)×(r+d-s); then P3=NN4 and the quadratic cost corresponding to system (15) is
(20)J1(u2,x1,x3)=∫0∞‍[x1T,x3T,u2T]N4TNTQNN4[x1T,x3T,u2T]Tdt,=∫0∞‍(x1TQ¯11x1+x3TQ¯22x3+u2TQ¯33u2mmmmmc+2x1TQ¯12x3+2x1TQ¯13u2+2x3TQ¯23u2)dt.Theorem 6.
If there exist matricesM11∈Rp×m, M12∈R(m-p)×m, M21∈Rd×(m-p), M22∈R(m-p-d)×(m-p), M31∈R(s-d)×(m-n), N11∈Rn×p, N12∈Rn×(n-p), N21∈R(n-p)×d, N22∈R(n-p)×(n-p-d), N31∈Rr×(p+s-n), and N32∈Rr×(n+r-p-s) satisfying the following(1)
M11, M12, M21, M22, and M31 have full row rank and N11, N12, N21, N22, N31, and N32 have full column rank,(2)
M11EN11=Ip, M12E=0, EN12=0, M22M12AN12=0, M31M22M12BN31=Ip+s-n, M21M12AN12N21=Id, M12AN12N22=0, M22M12BN32=0,(3)
M11(B-AP1B)N32≠0,(4)
P32TQP3=0,(5)
P33TQP33>0,(6)
[P31,P33]TQ[P31,P33]≥0,then problem P can be transformed to a regular LQ problem of the standard state space system.Proof.
WhenP32TQP3=0 holds, Q¯12=0, Q¯22=0, and Q¯23=0, and the quadratic cost is
(21)J2(u2,x1)=∫0∞‍(x1TQ¯11x1+2x1TQ¯13u2+u2TQ¯33u2)dt,=∫0∞‍[x1T,u2T][P31,P33]TQ[P31,P33][x1T,u2T]Tdt.
Therefore, when [P31,P33]TQ[P31,P33]≥0, the weighted matrix of x1 and u2 in the objective function is semidefinite-positive. P33TQP33>0 ensures that the optimal control problem of system (15) with cost (21) is a regular LQ problem. In addition, M11(B-AP1B)N32≠0, which is equivalent to B¯12≠0, constitutes the necessary and sufficient condition of existing free input in system (15) with (18).When the conditions in Theorem6 are satisfied, let
(22)u¯2=u2+Q¯33-1Q¯13Tx1,
and then system (15) can be written as
(23)x˙1=A~11x1+B¯12u¯2,x1(0)=M11x0x2=A~21x1-B212u¯2u1=A¯31x1
and the corresponding quadratic cost is
(24)J3(u¯2,x1)=∫0∞‍(x1TQ~11x1+u¯2TQ¯33u¯2)dt,
where A~11=A¯11-B¯12Q¯33-1Q¯13T, A~21=A¯21+B212Q¯33-1Q¯13T, and Q~11=Q¯11-Q¯13Q¯33-1Q¯13T.We denoteP1 as the optimal control problem of system (23) with cost (24) and define the set of admissible control-state pairs of problem P1 as
(25)Ω1={(u1,u¯2,x1,x2)∣(u1,u¯2,x1,x2)ispiecewisecontinuous,mmmmmmmmmmsatisfies(23)and|J3(u¯2,x1)|<∞}.According to the definition of an equivalent relation of the two optimal control problems [12], problem P is equivalent to problem P1. Obviously, P1 is an optimal problem of the standard state space system.


## 3. Design of the State Feedback Controller

In this section, we solve the problemP via solving P1.Before further discussion, we first give two lemmas.Lemma 7.
The pair(A~11,B¯12) is stabilizable if and only if
(26)rank⁡[λE-A,B]=rank⁡[0E0EAB]-rank⁡E,∀λ,Rλ≥0.Proof.
Denote(27)M=[Ip-A121-B1100Id-B211000Is-d0000Im-p-s][Ip+d00M3][Ip00M2]M1;
then
(28)M[A,B]N=[A¯11000B¯12-A¯21Id00B212A212100Is-d000000].
The pair(A~11,B¯12) is stabilizable if and only if
(29)rank⁡[λI-A~11,B¯12]=p,∀λ,Rλ≥0;
that is
(30)rank⁡[λI-A¯11,B¯12]=p,∀λ,Rλ≥0,
which is equivalent to (26) from (28).DenoteQ=[Q1,Q2], where Q1∈R(n+r)×n, Q2∈R(n+r)×r.Lemma 8.
When the conditions in Theorem6 are satisfied, the pair (A~11,Q~11) is detectable if and only if
(31)rank⁡[A-λEBP3TQ1P3TQ2]=n+r,∀λ,Rλ≥0.Proof.
DenoteQ^=[Q^1,Q^2,Q^3,Q^4,Q^5]=NTQN, where Q^1∈R(n+r)×p, Q^2∈R(n+r)×d, Q^3∈R(n+r)×(n-p-d),Q^4∈R(n+r)×(s-d), and Q^5∈R(n+r)×(r+d-s); then Q¯=N4TQ^N4 and N4TQ^3=P3TQ1N12N22=0.
The pair(A~11,Q~11) is detectable if and only if
(32)rank⁡[A~11-λIpQ~11]=p,∀λ,Rλ≥0
which is equivalent to
(33)rank⁡[A¯11-λIpB¯12Q¯11Q¯13Q¯13TQ¯33]=n+r-s,∀λ,Rλ≥0;
that is,
(34)rank⁡[A11-λIpA121B11B12A211IdB211B212A21210Is-d0N4TQ^1N4TQ^2N4TQ^4N4TQ^5]=n+r,∀λ,Rλ≥0.
An easy computation shows that the above equality is equivalent to (31).Definition 9 (see [6]).
The finiteλs satisfying det⁡(λE-A)=0 are called finite poles for the singular system Ex˙=Ax.In the following, we give the conclusion concerning problemP.Theorem 10.
Assume the conditions in Theorem6 are satisfied and the rank equalities (26) and (31) hold; then problem P has a unique optimal control-state pair and the optimal control can be synthesized as state feedback. The finite poles of closed-loop system all are located on the left-half complex plane and the optimal value is J*=x0TM11TPM11x0, where P is the unique semidefinite-positive solution of the Riccati equation
(35)PA~11+A~11TP-PB¯12Q¯33-1B¯12TP+Q~11=0.Proof.
Since the pair(A~11,B¯12) is stabilizable and the pair (A~11,Q~11) is detectable, Riccati equation (35) has a unique semidefinite-positive P and the eigenvalues of matrix A~11-B¯12Q¯33-1B¯12TP are located on the left-half complex plane.
According to Maximum Principle, the unique optimal control-state pair(u1*,u¯2*,x1*,x2*) of the optimal control problem of system (23) with cost (24) is
(36)[x1*x2*u1*u¯2*]=[IpA~21+B212Q¯33-1B¯12TPA¯31-Q¯33-1B¯12TP]x1*,
and the optimal value is J*=x0TM11TPM11x0, where x1* is the solution of equation
(37)x˙1=(A~11-B¯12Q¯33-1B¯12TP)x1,x1(0)=M11x0.
It follows from (11) and (22) that the unique optimal control-state pair (u*,x*) of problem P is
(38)[x*u*]=N[IpA~21+B212Q¯33-1B¯12TPA¯31-Q¯33-1(Q¯13T+B¯12TP)]x1*
and the optimal closed-loop system is
(39)Ex˙*=(A+BN3[A¯31-Q¯33-1(Q¯13T+B¯12TP)][Ip,0]N1-1)x*.
An easy computation shows that the finite poles of optimal closed-loop system (39) and the eigenvalues of matrix A~11-B¯12Q¯33-1B¯12TP are the same. Therefore, the finite eigenvalues of closed-loop system all are located on the left-half complex plane.


## 4. A Simulation Example

In this section, we give a simple example to illuminate the design method of state feedback controller and demonstrate its feasibility.Consider the irregular singular system (1) with quadratic cost (2), where
(40)E=[31220211021113221111],A=[10255431236446621545],B=[316626354315524],x0=[-11110],Q=[32121322400-22-1102-220-120-24-1-1-11-22-15-2-1320-1-24-22-1-1-1-1-21].Obviously,p=2, d=2, and s=3 and the eigenvalues of Q are -4.0310, 0.3589, 2.0810, 3.1335, 5.2966, 6.6965, 9.4645. Let
(41)M1=[0-0.500100.50000-11000-101-11100-3];N1=[10-0.5-0.501-0.5-0.500100001],M2=[0.2000.1-1300.8-231],N21=[1-101],M3=-1546,N3=[121461323010001],
then the system (1) is r.s.e. to the following system:
(42)x˙1=[-0.31541.98370.28265.2320]x1+[-3.4185-2.18484.99466.9457]u¯2,x1(0)=[-0.50.5]x2=[0.55260.57960.3102-0.5298]x1-[0.3261-0.73912.38042.8043]u¯2u1=[0.6304,-3.1957]x1
and the corresponding quadratic cost is
(43)J1(u¯2,x1)=∫0∞‍(x1T[1.8721-4.4330-4.433028.3724]x1mmmmc+u¯2T[65.460777.063177.0631110.1961]u¯2[1.8721-4.4330-4.433028.3724])dt
under the transformations (11) and (22).The unique positive semidefinite solution of Riccati equation (35) is
(44)P=[1.88352.02242.022428.1168],
the optimal performance index is J*=6.4889, and the simulation results are displayed in Figure 1.Figure 1
Curves of optimal control-state pair.


## 5. Conclusion

In this paper the indefinite LQ problem for irregular singular systems has been investigated. By restricted system equivalence transformation, we transformed the indefinite LQ problem for irregular singular systems to the regular LQ problem for standard state space systems. System state was decomposed into free state and restricted state and input was decomposed into free input and forced input. Based on optimization theory for standard state space systems, we have derived optimal control-state pair by solving an algebraic Riccati equation and the optimal control was synthesized as state feedback under some general conditions. The finite poles of closed-loop system are all located on the left-half complex plane.However, there are many problems unsolved about the indefinite LQ problem of the irregular singular systems. For example, in this paper the effect of external disturbances on the system has not been involved. More importantly, in the controller design, we need to solve an algebraic Riccati equation, which is still a challenge. Therefore, we think that the significance of the paper exists in theory more than in practice.



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