Singular Linear-Quadratic Control for Semistabilization
Singular Linear-Quadratic Control for Semistabilization Andrea L’Afflitto and Wassim M. Haddad
Abstract— Singular control has been extensively studied for the linear-quadratic regulator problem for achieving asymptotic stability of controlled linear systems with cheap control. In this paper, the singular control problem is extended to guarantee a weaker form of closed-loop stability, namely, semistability, which is of paramount importance for consensus control of network dynamical systems. Specifically, we show that the optimal state-feedback controller for the semistable linearquadratic regulator problem can be solved using an algebraic Riccati equation. This result allows us to address the semistable singular control problem and obtain several expressions for the minimum cost of a semistabilizing singular controller. In particular, after establishing connections between Qui and Davison’s formula for the minimum cost of a cheap controller and our results, we prove that the cost of a semistabilizing singular controller is zero if and only if the controlled system is minimum phase and right invertible.
I. I NTRODUCTION For the linear dynamical system G given by x(t) ˙ = Ax(t) + Bu(t),
x(0) = x0 ,
t ≥ 0,
y(t) = Cx(t),
(1) (2)
where x ∈ Rn , u ∈ Rm , y ∈ Rl , A ∈ Rn×n , B ∈ Rn×m , and C ∈ Rl×n , the classical singular control problem consists of finding a feedback control law u(·) = φ(x(·)) for l = m such that (1) is asymptotically stable and the performance measure Z ∞ J0 (x0 , u(·)) , lim (x(t) − xe )T R1 (x(t) − xe ) ε→0 0 + ε2 (u(t) − ue )T R2 (u(t) − ue ) dt (3) is minimized in the sense that J0 (x0 , φ(·)) =
min u(·)∈S0 (x0 )
J0 (x0 , u(·)),
(4)
where ue , φ(xe ), xe , limt→∞ x(t), R1 ∈ Rn×n is nonnegative definite, that is, R1 = R1T ≥ 0, R2 ∈ Rm×m is positive definite, that is, R2 = R2T > 0, and S(x0 ) , {u(·) : u(·) is measurable and x(·) given by (1) satisfies x(t) → xe as t → ∞}. (5) In this case, it can be shown that the optimal controller takes the form u = Kx, where K ∈ Rm×n [1], [2]. This problem has received considerable attention in the literature since it addresses a limiting case of the linearquadratic regulator problem [3], it can be used for system This work was supported in part by the Domenica Rea D’Onofrio Fellowship and the Air Force Office of Scientific Research under Grant FA9550-12-1-0192. A. L’Afflitto and W. M. Haddad are with the School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150, USA [email protected],
[email protected]
characterization, such as the invertibility problem [4], and it can be used in the design of high gain feedback systems [5], [6]. Furthermore, the singular control problem has been extended to non-square systems [7], that is, l 6= m, affine nonlinear systems [8], and discrete-time linear systems [9]. A form of stability which is weaker than asymptotic stability is semistability, that is, the property whereby every trajectory that starts in a neighborhood of a Lyapunov stable equilibrium converges to a (possibly different) Lyapunov stable equilibrium. Semistability implies Lyapunov stability, and is implied by asymptotic stability [10, p. 260]. Semistability arises naturally in dynamical network systems [11], [12], which cover a broad spectrum of applications including cooperative control of unmanned air vehicles, autonomous underwater vehicles, distributed sensor networks, air and ground transportation systems, swarms of air and space vehicle formations, and congestion control in communication networks, to cite but a few examples. A unique feature of the closed-loop dynamics under any control algorithm in dynamic networks is the existence of a continuum of equilibria representing a desired state of convergence. Under such dynamics, the desired limiting state is not determined completely by the system dynamics, but depends on the initial system state as well [11], [12]. In this paper, we address the singular control problem for semistabilization. Specifically, we address the problem of finding u(·) = φ(x(·)) such that the controlled system (1) is semistable and the performance measure (3) with R1 = C T C and R2 = Im , that is, Z ∞ J0 (x0 , u(·)) = lim (x(t) − xe )T C T C(x(t) − xe ) ε→0 0 + ε2 (u(t) − ue )T (u(t) − ue ) dt, (6) is minimized in the sense of (4). In Section II, we present notation, definitions, and the mathematical background necessary for addressing the singular semistabilization control problem, and we provide necessary and sufficient conditions for semicontrollability and semiobservability of linear dynamical systems. Section III discusses the linear-quadratic regulator problem for semistabilization. In Section IV, we discuss the main results of the paper, namely, we present sufficient conditions for semistability of singular controls and we derive several expressions for the minimum cost of a singular controller. Finally, in Section V we draw conclusions and highlight recommendations for future research. II. N OTATION , D EFINITIONS , AND M ATHEMATICAL P RELIMINARIES The notation used in this paper is fairly standard. Specifically, R denotes the set of real number, R+ denotes the set of positive real numbers, R+ denotes the set of nonnegative real numbers, Rn denotes the set of n × 1 real column vectors, Rn×m denotes the set of n × m real matrices, C denotes the set of complex numbers, C+ denotes the set of complex numbers with positive real part, and C+ denotes the set of complex numbers with nonnegative real part. Furthermore, Rprop (s) denotes the set of proper rational transfer functions
with coefficients in R and Rl×m prop (s) denotes the set of l × m matrices with entries in Rprop (s). Given the linear dynamical system G, x(t) ˙ = Ax(t) + Bu(t),
x(0) = x0 ,
t ≥ 0,
(7)
y(t) = Cx(t) + Du(t), (8) A B denotes a realization of where D ∈ Rl×m , G ∼ C D G. We write ||·|| for the Euclidean vector norm, ||·||F for the Frobenius matrix norm, S ⊥ for the orthogonal complement of a set S, R(A) and N (A) for the range space and the null space of a matrix A, respectively, spec(A) for the spectrum of the square matrix A, det A for the determinant of the square matrix A, rank A for the rank of the matrix A, (·)# for the group generalized inverse of a matrix, In or I for the identity matrix in Rn×n , and 0n×m or 0 for the zero matrix in Rn×m . Next, we define semistability for linear systems. Definition 2.1 ([13, Def. 11.6.1]): The system (1) with u ≡ 0 is semistable if spec(A) ⊂ {s ∈ C : Re s < 0} ∪ {0} and, if 0 ∈ spec(A), then 0 is semisimple. Note that if (1) with u ≡ 0 is semistable, then, for every x0 ∈ Rn , xe = limt→∞ x(t) = (In − AA# )x0 or, equivalently, limt→∞ eAt = In − AA# [13, Prop. 11.8.2]. Next, we introduce the definitions of semicontrollability and semiobservability for linear systems. Definition 2.2 ([14]): Let A ∈ Rn×n and B ∈ Rn×m . The pair (A, B) is semicontrollable if ⊥ \ n T k−1 T N (B (A ) ) = [N (AT )]⊥ , (9) k=1 0
where A , In . Definition 2.3 ([14]): Let A ∈ Rn×n and C ∈ Rl×n . The pair (A, C) is semiobservable if n \
N (CAk−1 ) = N (A).
(10)
k=1
Semicontrollability and semiobservability are extensions of controllability and observability. In particular, semicontrollability is an extension of null controllability to equilibrium controllability, whereas semiobservability is an extension of zero-state observability to equilibrium observability. Lemma 2.1: Consider the linear dynamical system (1). If the pair (A, B) is semicontrollable and 0 ∈ spec(A), then 0 is unstabilazable. Proof: Let w be a left-eigenvector of A with associated eigenvalue λ = 0 so that w ∈ N (AT ). By the definition of semicontrollability, it follows that w ∈ N (B T ). Now, recall that given a left eigenpair (µ, z) of A, (A, B) is uncontrollable if and only if z ∈ N (B T ) [15, Th. 6.2-5]. Now, the assertion follows immediately by noting that if λ ∈ spec (A)∩C+ is uncontrollable, then λ is unstabilizable. The following proposition provides a necessary and sufficient condition for verifying semicontrollability of the pair (A, B). Proposition 2.1: Consider the linear dynamical system (1) and suppose 0 ∈ spec (A). The pair (A, B) is semicontrollable if and only if there exists v ∈ Cn \ {0} such that v ∈ N (AT ) ∩ N (B T ).
Proof: Necessity is a direct consequence of Definition 2.2. To prove sufficiency, note that it follows from Theorem 12.6.8 of [13] that there exists an orthogonal matrix S ∈ Rn×n such that A1 A12 B1 −1 ˆ ˆ A , SAS = , B , SB = , (11) 0 A2 0 where A1 ∈ Rq×q , B1 ∈ Rq×n , and (A1 , B1 ) is controllable. Now, it follows from Lemma 2.1 that there exists z 6= 0 ˆT ˆ = 0 and B ˆ T zˆ = 0, such that z ∈ N (AT 2 ). Therefore, A z T T T where zˆ , [0 , z ] . The result now follows by noting that AT v = 0 and B T v = 0, where v , S T zˆ. Lemma 2.2: Consider the linear dynamical system G given by (1) and (2) with B = 0. If the pair (A, C) is semiobservable and 0 ∈ spec(A), then the eigenvalue λ = 0 is undetectable. Proof: The proof is dual to the proof of Lemma 2.1 and, hence, is omitted. The following proposition provides necessary and sufficient conditions for verifying semiobservability of the pair (A, C). Proposition 2.2: Consider the dynamical system G given by (1) and (2) with B = 0 and suppose 0 ∈ spec (A). The pair (A, C) is semiobservable if and only if there exists v ∈ Cn \ {0} such that v ∈ N (A) ∩ N (C). Proof: The proof is dual to the proof of Proposition 2.1 and, hence, is omitted. The following result is used later in the paper. Proposition 2.3 ([10], [8]): Consider the linear dynamical system G given by (1) and (2). If l = m and rank(CB) = m, then there exists a change of coordinates x 7→ (y, z) such that G is equivalent to y(t) ˙ A1 A2 y(t) B1 = + u(t), t ≥ 0, (12) z(t) ˙ B0 A0 z(t) 0 where z ∈ Rn−m , A1 ∈ Rm×m , A2 ∈ Rm×(n−m) , B0 ∈ R(n−m)×m , A0 ∈ R(n−m)×(n−m) , and B1 , CB. Consider the dynamical system G given by (7) and (8), and let G(s) , C(sIn − A)−1 B + D. Then, recall that T G(s) ∈ Rl×m prop (s) is inner if and only if G (−s)G(s) = Im , l×m G(s) ∈ Rprop (s) is minimum phase if and only if the zeros of G(s) are nonnpositive, where the zeros of G(s) ∈ Rl×m prop are the roots of the numerator polynomials in the nonzero entries of the Smith-McMillan form of G(s) [3], [15, p. 446]. Note that for the realization given in Proposition 2.3, the zeros of C(sI − A)−1 Bare the eigenvalues of A0 [8]. A B , the controllability Finally, recall that for G ∼ C D and observability Gramians Q and P of (7) and (8) are given by the solutions to the Lyapunov equations 0 = AQ + QAT + BB T , T
T
0 = A P + P A + C C,
(13) (14)
where Q ≥ 0 and P ≥ 0. If P = Q = In , then the realization G is a balanced realization [16]. If l = m, then G(s) is right invertible if and only if G(s) has full row rank for at least one s ∈ C [3]. Theorem 2.1 ([3], [17]): The transfer function G(s) ∈ Rl×m prop (s) can be factored as G(s) = G1 (s)G2 (s), where p×m G1 (s) ∈ Rl×p prop (s), p ≤ l, is inner and G2 (s) ∈ Rprop (s) is minimum phase and right invertible. The unstable poles of
G2 (s) are equal to the unstable poles of G(s). In addition, if G(s) is strictly proper, then G2 (s) is strictly proper. Corollary 2.1 ([3]): Let G(s) ∈ Rl×m prop (s) be right invertible and let G(s) = G1 (s)G2 (s) be a factorization as in Theorem 2.1. Then the system G1 , with transfer function G1 (s) ∈ Rl×l prop (s), is square, the zeros of G1 (s) are equal to the zeros of G(s) whose real part is positive, and the poles of G1 (s) are equal to the negatives of the zeros of G1 (s). Corollary 2.2 ([17]): Let G(s) ∈ Rl×m prop (s) be a nonminimum phase right invertible transfer function, let G(s) = G 1 (s)G2 (s) be a factorization as in Theorem 2.1, let G1 ∼ A1 B 1 be a balanced realization of G1 (s) ∈ Rl×l prop (s) C1 D 1 A2 B 2 and let G2 ∼ be a stabilizable and detectable C2 0 l×m realization of G2 (s) ∈ Rprop(s). Then, a stabilizable and A B detectable realization of G ∼ is given by C D A1 B1 C2 0 C1T T A= , B= , C = , D = 0. 0 A2 B2 C2T D1T (15)
that P = limt→∞ Ptf (t). These relevant issues are discussed in this paper and have been partially addressed in [14]. Theorem 3.2 ([14]): Consider the closed-loop system G given by (1) and (2) with feedback controller u(t) = Kx(t), where K ∈ Rm×n . Then G is semistable if and only if for every semicontrollable pair (A, B) and semiobservable pair (A, C) there exists an n × n matrix P = P T ≥ 0 such that
III. L INEAR -Q UADRATIC R EGULATOR P ROBLEM FOR S EMISTABILIZATION
It is worth noting that the authors in [14] a priori assume that the state feedback control law that minimizes the performance measure (23) and guarantees semistability of (1) is in the form of u(t) = Kx(t). This fact is shown in [19] using Hamilton-Jacobi-Bellman theory [10]. The following standard theorem provides necessary and sufficient conditions for guaranteeing the existence of a steady-state solution to the differential Riccati equation. To state this result, consider the performance measure Z tf T Jy (x0 , u(·)) , y (t)R3 y(t) + ε2 uT (t)R2 u(t) dt,
In this section, we formulate the linear-quadratic regulator problem for semistabilization. Proposition 3.1 ([14]): Consider the linear dynamical system G given by (1) with u ≡ 0. If G is semistable, then, for every n × n nonnegative definite matrix R, Z ∞ JR (x0 ) , [x(t) − xe ]T R[x(t) − xe ]dt < ∞, (16) 0
where xe = (I − AA# )x0 . The following classical result from optimal control theory is needed. Theorem 3.1 ([18, p. 91]): Consider the linear dynamical system (1) with performance measure Z tf T Jtf (x0 , u(·)) , x (t)R1 x(t)+uT (t)R2 u(t) dt. (17) 0
If Ptf (t) = PtTf (t) ≥ 0, t ∈ [0, tf ], is a solution to the differential Riccati equation −P˙ (t) = AT P (t) + P (t)A + R1 − P (t)BR2−1 B T P (t), P (tf ) = 0,
t ∈ [0, tf ],
(18)
0 = A˜T P + P A˜ + C T C + K T R2 K,
0
Finally, the feedback control u(·) = Kx(·) minimizes the performance measure Z ∞ J(x0 , u(·)) , (x(t) − xe )T R1 (x(t) − xe ) 0 + (u(t) − ue )T R2 (u(t) − ue ) dt, (23) in the sense that J(x0 , K) = min J(x0 , u(·)) = xT 0 PLS x0 . u∈S(x0 )
u(t) = −K(t)x(t) = −R2−1 B T Ptf (t)x(t)
(19)
minimizes the performance measure (17). Furthermore, the minimal cost for (17) is given by Jtf (x0 , K(·)) = xT 0 Ptf (0)x0 . PtTf (t)
(20)
The existence of Ptf (t) = ≥ 0, t ∈ [0, tf ], satisfying (18) such that (1) is semistable with u(t) = −R2−1 B T Ptf (t)x(t) and tf → ∞ has not been addressed in the literature. Furthermore, in this case, it is not clear whether (18) has a steady state solution, that is, it is not clear whether there exists P ∈ Rn×n , P = P T ≥ 0, such
(24)
0
(25) where R3 = > 0. Theorem 3.3 ([2, Th. 3.7]): Consider the system G given by (1) and (2), and performance measure (25) with ε = 1, and let Ptf (t) = PtTf (t) ≥ 0, t ∈ [0, tf ], be a solution to the differential Riccati equation R3T
−P˙ (t) = AT P (t) + P (t)A + C T R3 C − P (t)BR2−1 B T P (t), P (tf ) = 0, t ∈ [0, tf ]. (26) The system G has no poles that are unstable, uncontrollable, and observable if and only if limt→∞ Ptf (t) = P , where P = P T ≥ 0 is a solution to the algebraic Riccati equation 0 = AT P + P A + C T R3 C − P BR2−1 B T P.
then
(21)
where A˜ , A + BK and limt→∞ x(t) = xe ∈ N (K). Furthermore, the least squares solution of (21) is given by Z ∞ ˜T ˜ eA t (C T C + K T R2 K)eAt dt. PLS , (22)
(27)
Finally, in this case, the state-feedback control law u(t) = −R2−1 B T P x(t) guarantees Lyapunov stability of the closedloop linear dynamical system G. In general, as shown in [14], the feedback gain matrix K that minimizes (23) and guarantees semistability of (1) with u(t) = Kx(t) can be obtained as a solution to a linear matrix inequality. However, the following result provides semistabilization via a Riccati equation. Corollary 3.1: Consider the linear dynamical system G given by (1) and (2), and performance measure (23) with R1 = C T R3 C. Assume (1) with u(t) = Kx(t), where K ∈ Rm×n , is semistable and (23) is minimized in the sense
of (4). If the pair (A, B) is semicontrollable, the pair (A, E1 ) is semiobservable, R1 = E1T E1 , and xe ∈ N (K), then K = −R2−1 B T PLS ,
(28)
where PLS is the least squares solution to (27). Proof: It follows from Theorems 3.1 and 3.3 that the minimum of (23) is given by J(x0 , K) = xT 0 PLS x0 , T where PLS = PLS ≥ 0 is the least squares solution to (27). Furthermore, the optimal controller is given u(t) = −R2−1 B T PLS x(t) and (1) is Lyapunov stable. Now, by Theorem 3.2, the minimum of (23) is J(x0 , K) = xT 0 PLS x0 , where PLS is the least squares solution to (21). If K = −R2−1 B T P , then (27) and (21) are identical and the assertion is proven by the uniqueness of the minimum of (23). The following classical theorem for finding the minimal cost for a singular control is necessary for later developments. Theorem 3.4 ([20]): Consider the linear dynamical system given by (1) and (2) with performance measure (25), and assume rank B = m and rank C = l. Let tf → ∞ and let Pε = PεT ≥ 0 be a solution to the algebraic Riccati equation 0 = AT P + P A + C T R3 C −
1 P BR2−1 B T P. ε2
(29)
Then the following statements hold. i) If l > m, then limε→0 Pε 6= 0. ii) If l = m and the numerator polynomial of det C(sIn − A)−1 B is not identically equal to zero and has roots with nonpositive real part, then limε→0 Pε = 0. iii) If l < m and there exists a matrix M ∈ Rm×l such that the numerator polynomial of det C(sIn − A)−1 BM is not identically equal to zero and has roots with nonpositive real parts only, then limε→0 Pε = 0. The following theorem gives a converse to Statement ii) of Theorem 3.4. A similar result was proven by Kwakernaak and Sivan [20] for the case where A˜ = A + BK is Hurwitz. Theorem 3.5: Consider the linear dynamical system given by (1) and (2) with performance measure (25), and assume rank B = m and rank C = l. Let tf → ∞, let l = m, and let P¯ε be the least squares solution to (29). If u(t) = Kx(t) guarantees semistability of (1) and minimizes the performance measure (25), the numerator polynomial ψ(s) of det C(sI − A)−1 B is not identically equal to zero, and limε→0 P¯ε = 0, then the roots of ψ(s) are nonpositive. Proof: If limε→0 P¯ε = 0, then (29) specializes to
Now, it follows from Theorem 3.3 that det(sIn − A − BK) = det In − K(sI − A)−1 B det(sIn − A) 1 1 = det In + R2−1 B T P¯ε · ε ε (sIn − A)−1 B . (32) Thus, the roots of the closed-loop characteristic polynomial that stay as ε → 0+ approach the roots of det(sIn − finite −1 A) det R2 L(sIn − A)−1 B , that is, the roots approach the roots of 1 −1 det(sIn − A) det R2 2 U R32 C(sIn − A)−1 B 1 − 21 2 = det R2 U R3 ψ(s). (33) Now, the result follows using identical arguments as in [2, p. 308]. IV. S EMISTABILITY AND S INGULAR C ONTROL In order to address the singular control problem for semistabilization, we first need to show that the performance measure (6) is well-defined when (1) is semistable with u(t) = Kx(t). Proposition 4.1: Consider the system (1) with performance measure (6). If there exists K ∈ Rm×n such that (1) with u(t) = Kx(t) is semistable, then (6) is well-defined, that is, J0 (x0 , u(·)) < ∞. Proof: Consider the performance measure Z ∞ Jε (x0 , u(·)) , (x(t) − xe )T C T C(x(t) − xe ) 0 + ε2 (u(t) − ue )T (u(t) − ue ) dt. (34) Since C T C + ε2 K T K ≥ 0, it follows from Proposition 3.1 that Jε (x0 , u(·)) < ∞, which proves the assertion since J0 (x0 , u(·)) = limε→0+ Jε (x0 , u(·)) and Jε is a monotone function of ε that is bounded from below. Next, we give an expression for the state-feedback control law that minimizes the performance measure (6) and guarantees semistability of the system given by (1) and (2). Lemma 4.1: Consider the linear dynamical system given by (1) and (2) with performance measure (6). Suppose (A, B) is semicontrollable, (A, C) is semiobservable, xe ∈ N (K), (1) is semistable with u(t) = Kx(t) and (6) is minimized. Then, the minimum of the performance measure (6) is given by J0 (x0 , K) = lim Jε (x0 , K) = xT 0 PLS x0 , ε→0
− 12
− 12
1 2
1 2
(R2 L)T (R2 L) = (R3 C)T (R3 C),
(30)
where L , limε→0+ 1ε B T P¯ε . Note that, since R2 is nonsingular, L exists and, since R3 is positive definite, 1 rank C T R3 C = rank R32 C = rank C = m. Thus, there exists U ∈ Rm×m such that U T U = Im and
where PLS = solution of
R∞ 0
˜T
˜
eA t C T CeAt dt is the least squares
0 = AT P + P A + C T C − lim
ε→0
1 P BB T P. ε2
1 2
= U R3 C.
(31)
(36)
Furthermore, the optimal control is given by u(t) = −Lx(t),
−1 R2 2 L
(35)
where L = limε→0+
1 T ε B PLS
is well defined.
(37)
Proof: Equation (35) follows immediately from Theorem 3.2. Next, it follows from Theorem 3.2 that min Jε (x0 , K) = xT 0 Pε x 0 ,
u∈S(x0 )
(38)
where Pε is the least squares solution to (29) for R3 = Il and R2 = Im . Now, (35) and (38) guarantee the existence of PLS satisfying (29) as ε → 0, that is, limε→0 Pε = PLS . Furthermore, as shown in the proof of Theorem 3.5, limε→0+ 1ε B T PLS exists. The proof now follows as direct consequence of Corollary 3.1. The next theorem provides a closed-form expression for the optimal performance measure (6) extending a wellknown property of classical singular control to singular semistabilization. Theorem 4.1: Consider the linear dynamical system given by (1) and (2), with u(t) = Kx(t) such that (1) is semistable and the performance measure (6) is minimized. If l > m, then J0 (x0 , K) > 0. Alternatively, if l = m and the zeros of the numerator polynomial of det C(sIn − A)−1 B are not identically zero and have nonpositive real part, then J0 (x0 , K) = 0. Finally, if l < m and there exists a matrix M such that the numerator polynomial of det C(sIn −A)−1 BM is not identically zero and has zeros with nonpositive real parts, then J0 (x0 , K) = 0. Proof: The proof is a direct consequence of Lemma 4.1 and Theorem 3.4. Corollary 4.1: Consider the linear dynamical system given by (1) and (2), with u(t) = Kx(t) such that (1) is semistable and the performance measure (6) is minimized. Suppose l = m and the numerator polynomial ψ(s) of det C(sIn − A)−1 B is not identically equal to zero. If J0 (x0 , K) = 0, then the roots of ψ(s) are nonpositive. Proof: The proof is a direct consequence of Lemma 4.1 and Theorem 3.5. The following theorem provides an expression for J0 (x0 , K) in terms of a reduced-order system when the openloop system is not minimum phase. Theorem 4.2: Consider the linear dynamical system given by (1) and (2), with u(t) = Kx(t) such that (1) is semistable and the performance measure (6) is minimized. If l = m, rank(CB) = m, and all the roots of the numerator polynomial ψ(s) of det C(sI − A)−1 B have nonnegative real part, then the dynamical system G given by (1) and (2) with u(t) = Kx(t) is equivalent to (12) and the minimal performance measure (6) is given by J0 (x0 , K) = z T (0)P0 z(0),
(39)
where P0 is the least squares solution to T 0 = AT (40) 0 P0 + P0 A 0 − P 0 B 0 B 0 P0 . Proof: It follows from Proposition 2.3 that the system G given by (1) and (2) is equivalent to (12) and the roots of ψ(s) are the eigenvalues of A0 [8]. Next, it follows from Lemma 4.1 that T y(0) y(0) ˆ J0 (x0 , K) = xT P x = P , (41) LS 0 LS 0 z(0) z(0)
where PˆLS , limε→0 Pˆε and Pˆε is the least squares solution of T A1 A2 A1 A2 0= P +P + B 0 A0 B 0 A0 1 Im 0 B1 B1T 0 P. (42) − 2P 0 0(n−m)×(n−m) 0 0 ε
Letting any solution P to (42) be of the form εP1 εP2 P = + O(ε2 ), εP2T P0 + εP3
(43)
where P0 , P1 , P2 , and P3 are independent of ε [21], it follows from (42) with ε = 0 that 1
P1 = B1 B1T )− 2 , P2 = P3 =
(44)
1 B1 B1T )− 2 B0T P0 , 1 P0 B0 B1 B1T )− 2 B0T P0 ,
(45) (46) −1
and (40) holds. Since the zeros of det C(sI − A) B have nonnegative real part, the eigenvalues of −A0 are nonpositive, that is, −A0 is Lyapunov stable and, by Theorem 3.3, there exists a solution P0 to (40). Therefore, the assertion follows immediately from (41) and (43). Finally, we extend Qiu and Davison’s formula for the optimal singular control [3] to semistabilization. Theorem 4.3: Consider the linear dynamical system G given by (1) and (2), with u(t) = Kx(t) such that (1) is semistable and the performance measure (6) is minimized. If G has transfer function G(s) = C(sIn − A)−1 B that is nonminimum phase and right invertible, then the minimal performance measure (6) is given by l X 1 , J0 (x0 , K) = 2 λ j=1 j
(47)
where λ1 , . . . , λl , l ≤ n, are the zeros of G(s) whose real part is positive. Conversely, if l = m, the numerator polynomial ψ(s) of det C(sI − A)−1 B is not identically equal to zero, and (47) holds, then G(s) is minimum phase and right invertible. Proof: It follows from Corollary 2.1 that G(s) can be factored as G(s) = G1 (s)G2 (s), where G1 (s) ∈ Rl×l prop (s), −1 (s), G (s) , C (sI − A ) B + D G2 (s) ∈ Rl×m 1 1 1 1 1 , and prop −1 G2 (s) , C2 (sI − A2 ) B2 . Furthermore, by Corollary 2.1, the poles of G1 (s) are equal to the negatives of the zeros of G(s) whose real part is positive, that is, λ1 , . . . , λl , l ≤ n. Next, let Pε = PεT ≥ 0 denote the solution to the algebraic Riccati equation given by (29). Using the factorization G(s) = G1 (s)G2 (s), it follows that Pε = Il 0l×(n−l) T , where Pε2 = Pε2 ≥ 0 is a solution 0(n−l)×l Pε2 to the algebraic Riccati equation 1 T T 0 = AT (48) 2 P + P A2 + C2 C2 − 2 P B2 B2 P. ε Now, Theorem 3.4 implies that limε→0 Pε2 = 0. Thus, I 0 limε→0 Pε = , and hence, by Lemma 4.1 it follows 0 0 that I 0 T J0 (x0 , K) = xT (49) 0 0 0 x0 = x0,1 x0,1 , T T where x0 , [xT 0,1 , x0,2 ] is partitioned as A is in (15).
Next, since the system (1) with u(t) = Kx(t) is semistable, limt→∞ x(t) = xe . Thus, for t → ∞, the output of G1 (s) is ye = Cxe . Consequently, the output of G2 (s) −1 −1 is G−1 1 (0)ye , which implies that x0,1 = −A1 B1 G1 (0)ye . Therefore, since G1 (0) is unitary, it follows that −1 −1 T −T J0 (x0 , K) = tr B1T A−T 1 A1 B1 = tr A1 B1 B1 A1 . (50)
The transfer function G1(s) is inner, and hence, has a A1 B 1 balanced realization G1 ∼ . Thus, by Corollary C1 D1 2.2, it follows that B1 B1T = C1T C1 , P1 = Q1 = Il , and (13) and (14) hold for A = A1 , B = B1 , C = C1 , P = P1 , and Q = Q1 . Hence, −T T J0 (x0 , K) = −tr A−1 = −2 tr A−1 1 (A1 + A1 )A1 1 , (51)
which proves the assertion. Conversely, if l = m, the numerator polynomial ψ(s) of det C(sI − A)−1 B is not identically equal to zero, and (47) holds, then it follows from Corollary 4.1 that the zeros of ψ(s) are nonpositive, which implies that G(s) is right invertible [7]. V. C ONCLUSION In [14] the authors developed an optimal control framework that guarantees semistability via a state feedback controller and minimizes a quadratic performance measure using linear matrix inequalities. In this paper, we show that the optimal control law that guarantees semistability of a linear dynamical system and minimizes a quadratic performance measure can be found by solving an algebraic Riccati equation approach. Furthermore, as for asymptotically stable closed-loop systems, we show that the optimal singular control cost is zero if and only if the controlled semistable system is minimum phase and right invertible. R EFERENCES [1] Z. Gajic and M. T. Lim, Optimal Control of Singularly Perturbed Linear Systems and Applications. New York: Marcel Dekker, 2001. [2] R. S. H. Kwakernaak, Linear Optimal Control Systems. New York: Wiley, 1972. [3] L. Qiu and E. J. Davison, “Performance limitations of non-minimum phase systems in the servomechanism problem,” Automatica, vol. 29, pp. 337–349, 1993. [4] P. Sannuti, “Direct singular perturbation analysis of high-gain and cheap control problems,” Automatica, vol. 19, pp. 41–51, 1983. [5] P. Kokotovic, H. K. Khalil, and J. O’Reilly, Singular Perturbation Methods in Control: Analysis and Design. London, UK: Academic Press, 1986. [6] A. Saberi and P. Sannuti, “Cheap and singular controls for linear quadratic regulators,” IEEE Transactions on Automatic Control, vol. 32, no. 3, pp. 208–219, 1987. [7] B. Francis, “The optimal linear-quadratic time-invariant regulator with cheap control,” IEEE Transactions on Automatic Control, vol. 24, no. 4, pp. 616 – 621, 1979. [8] M. Seron, J. Braslavsky, P. Kokotovic, and D. Mayne, “Feedback limitations in nonlinear systems: from Bode integrals to cheap control,” IEEE Transactions on Automatic Control, vol. 44, no. 4, pp. 829 –833, 1999. [9] G. Marro, D. Prattichizzo, and E. Zattoni, “Geometric insight into discrete-time cheap and singular linear quadratic Riccati (LQR) problems,” IEEE Transactions on Automatic Control, vol. 47, pp. 102–107, 2002. [10] W. M. Haddad and V. Chellaboina, Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach. Princeton, NJ: Princeton Univ. Press, 2008. [11] Q. Hui and W. M. Haddad, “Distributed nonlinear control algorithms for network consensus,” Automatica, vol. 44, no. 9, pp. 2375 – 2381, 2008. [12] Q. Hui, M. Haddad, and S. Bhat, “Finite-time semistability and consensus for nonlinear dynamical networks,” IEEE Transactions on Automatic Control, vol. 53, no. 8, pp. 1887–1900. [13] D. S. Bernstein, Matrix Mathematics. Princeton, NJ: Princeton Univ. Press, 2008. [14] W. M. Haddad, Q. Hui, and V. Chellaboina, “H2 optimal semistable control for linear dynamical systems: An LMI approach,” J. Franklin Inst., vol. 348, pp. 2898–2910, 2011.
[15] T. Kailath, Linear systems. Prentice-Hall, 1980. [16] K. Glover, “All optimal Hankel-norm approximations of linear multivariable systems and their L∞ -error bounds.” International Journal of Control, vol. 39, pp. 1115–1193, 1984. [17] L. B. Jemaa and E. J. Davison, “Limiting performance of optimal linear discrete filters,” Automatica, vol. 39, no. 7, pp. 1221–1226, 2003. [18] D. Kirk, Optimal Control Theory: An Introduction. New Jersey: Prentice-Hall, 2004. [19] A. L’Afflitto and W. M. Haddad, “Optimal control, singular control, and limits of performance for nonlinear semistable dynamical systems,” in Proc. American Control Conference, submitted, Portland, OR, June 2014. [20] H. Kwakernaak and R. Sivan, “The maximally achievable accuracy of linear optimal regulators and linear optimal filters,” IEEE Transactions on Automatic Control, vol. 17, no. 1, pp. 79 – 86, feb 1972. [21] A. Jameson and R. E. O’Malley, “Cheap control of the time-invariant regulator,” Applied Mathematics and Optimization, vol. 1, pp. 337– 354, 1975.
Abstract— Singular control has been extensively studied for the linear-quadratic regulator problem for achieving asymptotic stability of controlled linear systems with cheap control. In this paper, the singular control problem is extended to guarantee a weaker form of closed-loop stability, namely, semistability, which is of paramount importance for consensus control of network dynamical systems. Specifically, we show that the optimal state-feedback controller for the semistable linearquadratic regulator problem can be solved using an algebraic Riccati equation. This result allows us to address the semistable singular control problem and obtain several expressions for the minimum cost of a semistabilizing singular controller. In particular, after establishing connections between Qui and Davison’s formula for the minimum cost of a cheap controller and our results, we prove that the cost of a semistabilizing singular controller is zero if and only if the controlled system is minimum phase and right invertible.
I. I NTRODUCTION For the linear dynamical system G given by x(t) ˙ = Ax(t) + Bu(t),
x(0) = x0 ,
t ≥ 0,
y(t) = Cx(t),
(1) (2)
where x ∈ Rn , u ∈ Rm , y ∈ Rl , A ∈ Rn×n , B ∈ Rn×m , and C ∈ Rl×n , the classical singular control problem consists of finding a feedback control law u(·) = φ(x(·)) for l = m such that (1) is asymptotically stable and the performance measure Z ∞ J0 (x0 , u(·)) , lim (x(t) − xe )T R1 (x(t) − xe ) ε→0 0 + ε2 (u(t) − ue )T R2 (u(t) − ue ) dt (3) is minimized in the sense that J0 (x0 , φ(·)) =
min u(·)∈S0 (x0 )
J0 (x0 , u(·)),
(4)
where ue , φ(xe ), xe , limt→∞ x(t), R1 ∈ Rn×n is nonnegative definite, that is, R1 = R1T ≥ 0, R2 ∈ Rm×m is positive definite, that is, R2 = R2T > 0, and S(x0 ) , {u(·) : u(·) is measurable and x(·) given by (1) satisfies x(t) → xe as t → ∞}. (5) In this case, it can be shown that the optimal controller takes the form u = Kx, where K ∈ Rm×n [1], [2]. This problem has received considerable attention in the literature since it addresses a limiting case of the linearquadratic regulator problem [3], it can be used for system This work was supported in part by the Domenica Rea D’Onofrio Fellowship and the Air Force Office of Scientific Research under Grant FA9550-12-1-0192. A. L’Afflitto and W. M. Haddad are with the School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150, USA [email protected],
[email protected]
characterization, such as the invertibility problem [4], and it can be used in the design of high gain feedback systems [5], [6]. Furthermore, the singular control problem has been extended to non-square systems [7], that is, l 6= m, affine nonlinear systems [8], and discrete-time linear systems [9]. A form of stability which is weaker than asymptotic stability is semistability, that is, the property whereby every trajectory that starts in a neighborhood of a Lyapunov stable equilibrium converges to a (possibly different) Lyapunov stable equilibrium. Semistability implies Lyapunov stability, and is implied by asymptotic stability [10, p. 260]. Semistability arises naturally in dynamical network systems [11], [12], which cover a broad spectrum of applications including cooperative control of unmanned air vehicles, autonomous underwater vehicles, distributed sensor networks, air and ground transportation systems, swarms of air and space vehicle formations, and congestion control in communication networks, to cite but a few examples. A unique feature of the closed-loop dynamics under any control algorithm in dynamic networks is the existence of a continuum of equilibria representing a desired state of convergence. Under such dynamics, the desired limiting state is not determined completely by the system dynamics, but depends on the initial system state as well [11], [12]. In this paper, we address the singular control problem for semistabilization. Specifically, we address the problem of finding u(·) = φ(x(·)) such that the controlled system (1) is semistable and the performance measure (3) with R1 = C T C and R2 = Im , that is, Z ∞ J0 (x0 , u(·)) = lim (x(t) − xe )T C T C(x(t) − xe ) ε→0 0 + ε2 (u(t) − ue )T (u(t) − ue ) dt, (6) is minimized in the sense of (4). In Section II, we present notation, definitions, and the mathematical background necessary for addressing the singular semistabilization control problem, and we provide necessary and sufficient conditions for semicontrollability and semiobservability of linear dynamical systems. Section III discusses the linear-quadratic regulator problem for semistabilization. In Section IV, we discuss the main results of the paper, namely, we present sufficient conditions for semistability of singular controls and we derive several expressions for the minimum cost of a singular controller. Finally, in Section V we draw conclusions and highlight recommendations for future research. II. N OTATION , D EFINITIONS , AND M ATHEMATICAL P RELIMINARIES The notation used in this paper is fairly standard. Specifically, R denotes the set of real number, R+ denotes the set of positive real numbers, R+ denotes the set of nonnegative real numbers, Rn denotes the set of n × 1 real column vectors, Rn×m denotes the set of n × m real matrices, C denotes the set of complex numbers, C+ denotes the set of complex numbers with positive real part, and C+ denotes the set of complex numbers with nonnegative real part. Furthermore, Rprop (s) denotes the set of proper rational transfer functions
with coefficients in R and Rl×m prop (s) denotes the set of l × m matrices with entries in Rprop (s). Given the linear dynamical system G, x(t) ˙ = Ax(t) + Bu(t),
x(0) = x0 ,
t ≥ 0,
(7)
y(t) = Cx(t) + Du(t), (8) A B denotes a realization of where D ∈ Rl×m , G ∼ C D G. We write ||·|| for the Euclidean vector norm, ||·||F for the Frobenius matrix norm, S ⊥ for the orthogonal complement of a set S, R(A) and N (A) for the range space and the null space of a matrix A, respectively, spec(A) for the spectrum of the square matrix A, det A for the determinant of the square matrix A, rank A for the rank of the matrix A, (·)# for the group generalized inverse of a matrix, In or I for the identity matrix in Rn×n , and 0n×m or 0 for the zero matrix in Rn×m . Next, we define semistability for linear systems. Definition 2.1 ([13, Def. 11.6.1]): The system (1) with u ≡ 0 is semistable if spec(A) ⊂ {s ∈ C : Re s < 0} ∪ {0} and, if 0 ∈ spec(A), then 0 is semisimple. Note that if (1) with u ≡ 0 is semistable, then, for every x0 ∈ Rn , xe = limt→∞ x(t) = (In − AA# )x0 or, equivalently, limt→∞ eAt = In − AA# [13, Prop. 11.8.2]. Next, we introduce the definitions of semicontrollability and semiobservability for linear systems. Definition 2.2 ([14]): Let A ∈ Rn×n and B ∈ Rn×m . The pair (A, B) is semicontrollable if ⊥ \ n T k−1 T N (B (A ) ) = [N (AT )]⊥ , (9) k=1 0
where A , In . Definition 2.3 ([14]): Let A ∈ Rn×n and C ∈ Rl×n . The pair (A, C) is semiobservable if n \
N (CAk−1 ) = N (A).
(10)
k=1
Semicontrollability and semiobservability are extensions of controllability and observability. In particular, semicontrollability is an extension of null controllability to equilibrium controllability, whereas semiobservability is an extension of zero-state observability to equilibrium observability. Lemma 2.1: Consider the linear dynamical system (1). If the pair (A, B) is semicontrollable and 0 ∈ spec(A), then 0 is unstabilazable. Proof: Let w be a left-eigenvector of A with associated eigenvalue λ = 0 so that w ∈ N (AT ). By the definition of semicontrollability, it follows that w ∈ N (B T ). Now, recall that given a left eigenpair (µ, z) of A, (A, B) is uncontrollable if and only if z ∈ N (B T ) [15, Th. 6.2-5]. Now, the assertion follows immediately by noting that if λ ∈ spec (A)∩C+ is uncontrollable, then λ is unstabilizable. The following proposition provides a necessary and sufficient condition for verifying semicontrollability of the pair (A, B). Proposition 2.1: Consider the linear dynamical system (1) and suppose 0 ∈ spec (A). The pair (A, B) is semicontrollable if and only if there exists v ∈ Cn \ {0} such that v ∈ N (AT ) ∩ N (B T ).
Proof: Necessity is a direct consequence of Definition 2.2. To prove sufficiency, note that it follows from Theorem 12.6.8 of [13] that there exists an orthogonal matrix S ∈ Rn×n such that A1 A12 B1 −1 ˆ ˆ A , SAS = , B , SB = , (11) 0 A2 0 where A1 ∈ Rq×q , B1 ∈ Rq×n , and (A1 , B1 ) is controllable. Now, it follows from Lemma 2.1 that there exists z 6= 0 ˆT ˆ = 0 and B ˆ T zˆ = 0, such that z ∈ N (AT 2 ). Therefore, A z T T T where zˆ , [0 , z ] . The result now follows by noting that AT v = 0 and B T v = 0, where v , S T zˆ. Lemma 2.2: Consider the linear dynamical system G given by (1) and (2) with B = 0. If the pair (A, C) is semiobservable and 0 ∈ spec(A), then the eigenvalue λ = 0 is undetectable. Proof: The proof is dual to the proof of Lemma 2.1 and, hence, is omitted. The following proposition provides necessary and sufficient conditions for verifying semiobservability of the pair (A, C). Proposition 2.2: Consider the dynamical system G given by (1) and (2) with B = 0 and suppose 0 ∈ spec (A). The pair (A, C) is semiobservable if and only if there exists v ∈ Cn \ {0} such that v ∈ N (A) ∩ N (C). Proof: The proof is dual to the proof of Proposition 2.1 and, hence, is omitted. The following result is used later in the paper. Proposition 2.3 ([10], [8]): Consider the linear dynamical system G given by (1) and (2). If l = m and rank(CB) = m, then there exists a change of coordinates x 7→ (y, z) such that G is equivalent to y(t) ˙ A1 A2 y(t) B1 = + u(t), t ≥ 0, (12) z(t) ˙ B0 A0 z(t) 0 where z ∈ Rn−m , A1 ∈ Rm×m , A2 ∈ Rm×(n−m) , B0 ∈ R(n−m)×m , A0 ∈ R(n−m)×(n−m) , and B1 , CB. Consider the dynamical system G given by (7) and (8), and let G(s) , C(sIn − A)−1 B + D. Then, recall that T G(s) ∈ Rl×m prop (s) is inner if and only if G (−s)G(s) = Im , l×m G(s) ∈ Rprop (s) is minimum phase if and only if the zeros of G(s) are nonnpositive, where the zeros of G(s) ∈ Rl×m prop are the roots of the numerator polynomials in the nonzero entries of the Smith-McMillan form of G(s) [3], [15, p. 446]. Note that for the realization given in Proposition 2.3, the zeros of C(sI − A)−1 Bare the eigenvalues of A0 [8]. A B , the controllability Finally, recall that for G ∼ C D and observability Gramians Q and P of (7) and (8) are given by the solutions to the Lyapunov equations 0 = AQ + QAT + BB T , T
T
0 = A P + P A + C C,
(13) (14)
where Q ≥ 0 and P ≥ 0. If P = Q = In , then the realization G is a balanced realization [16]. If l = m, then G(s) is right invertible if and only if G(s) has full row rank for at least one s ∈ C [3]. Theorem 2.1 ([3], [17]): The transfer function G(s) ∈ Rl×m prop (s) can be factored as G(s) = G1 (s)G2 (s), where p×m G1 (s) ∈ Rl×p prop (s), p ≤ l, is inner and G2 (s) ∈ Rprop (s) is minimum phase and right invertible. The unstable poles of
G2 (s) are equal to the unstable poles of G(s). In addition, if G(s) is strictly proper, then G2 (s) is strictly proper. Corollary 2.1 ([3]): Let G(s) ∈ Rl×m prop (s) be right invertible and let G(s) = G1 (s)G2 (s) be a factorization as in Theorem 2.1. Then the system G1 , with transfer function G1 (s) ∈ Rl×l prop (s), is square, the zeros of G1 (s) are equal to the zeros of G(s) whose real part is positive, and the poles of G1 (s) are equal to the negatives of the zeros of G1 (s). Corollary 2.2 ([17]): Let G(s) ∈ Rl×m prop (s) be a nonminimum phase right invertible transfer function, let G(s) = G 1 (s)G2 (s) be a factorization as in Theorem 2.1, let G1 ∼ A1 B 1 be a balanced realization of G1 (s) ∈ Rl×l prop (s) C1 D 1 A2 B 2 and let G2 ∼ be a stabilizable and detectable C2 0 l×m realization of G2 (s) ∈ Rprop(s). Then, a stabilizable and A B detectable realization of G ∼ is given by C D A1 B1 C2 0 C1T T A= , B= , C = , D = 0. 0 A2 B2 C2T D1T (15)
that P = limt→∞ Ptf (t). These relevant issues are discussed in this paper and have been partially addressed in [14]. Theorem 3.2 ([14]): Consider the closed-loop system G given by (1) and (2) with feedback controller u(t) = Kx(t), where K ∈ Rm×n . Then G is semistable if and only if for every semicontrollable pair (A, B) and semiobservable pair (A, C) there exists an n × n matrix P = P T ≥ 0 such that
III. L INEAR -Q UADRATIC R EGULATOR P ROBLEM FOR S EMISTABILIZATION
It is worth noting that the authors in [14] a priori assume that the state feedback control law that minimizes the performance measure (23) and guarantees semistability of (1) is in the form of u(t) = Kx(t). This fact is shown in [19] using Hamilton-Jacobi-Bellman theory [10]. The following standard theorem provides necessary and sufficient conditions for guaranteeing the existence of a steady-state solution to the differential Riccati equation. To state this result, consider the performance measure Z tf T Jy (x0 , u(·)) , y (t)R3 y(t) + ε2 uT (t)R2 u(t) dt,
In this section, we formulate the linear-quadratic regulator problem for semistabilization. Proposition 3.1 ([14]): Consider the linear dynamical system G given by (1) with u ≡ 0. If G is semistable, then, for every n × n nonnegative definite matrix R, Z ∞ JR (x0 ) , [x(t) − xe ]T R[x(t) − xe ]dt < ∞, (16) 0
where xe = (I − AA# )x0 . The following classical result from optimal control theory is needed. Theorem 3.1 ([18, p. 91]): Consider the linear dynamical system (1) with performance measure Z tf T Jtf (x0 , u(·)) , x (t)R1 x(t)+uT (t)R2 u(t) dt. (17) 0
If Ptf (t) = PtTf (t) ≥ 0, t ∈ [0, tf ], is a solution to the differential Riccati equation −P˙ (t) = AT P (t) + P (t)A + R1 − P (t)BR2−1 B T P (t), P (tf ) = 0,
t ∈ [0, tf ],
(18)
0 = A˜T P + P A˜ + C T C + K T R2 K,
0
Finally, the feedback control u(·) = Kx(·) minimizes the performance measure Z ∞ J(x0 , u(·)) , (x(t) − xe )T R1 (x(t) − xe ) 0 + (u(t) − ue )T R2 (u(t) − ue ) dt, (23) in the sense that J(x0 , K) = min J(x0 , u(·)) = xT 0 PLS x0 . u∈S(x0 )
u(t) = −K(t)x(t) = −R2−1 B T Ptf (t)x(t)
(19)
minimizes the performance measure (17). Furthermore, the minimal cost for (17) is given by Jtf (x0 , K(·)) = xT 0 Ptf (0)x0 . PtTf (t)
(20)
The existence of Ptf (t) = ≥ 0, t ∈ [0, tf ], satisfying (18) such that (1) is semistable with u(t) = −R2−1 B T Ptf (t)x(t) and tf → ∞ has not been addressed in the literature. Furthermore, in this case, it is not clear whether (18) has a steady state solution, that is, it is not clear whether there exists P ∈ Rn×n , P = P T ≥ 0, such
(24)
0
(25) where R3 = > 0. Theorem 3.3 ([2, Th. 3.7]): Consider the system G given by (1) and (2), and performance measure (25) with ε = 1, and let Ptf (t) = PtTf (t) ≥ 0, t ∈ [0, tf ], be a solution to the differential Riccati equation R3T
−P˙ (t) = AT P (t) + P (t)A + C T R3 C − P (t)BR2−1 B T P (t), P (tf ) = 0, t ∈ [0, tf ]. (26) The system G has no poles that are unstable, uncontrollable, and observable if and only if limt→∞ Ptf (t) = P , where P = P T ≥ 0 is a solution to the algebraic Riccati equation 0 = AT P + P A + C T R3 C − P BR2−1 B T P.
then
(21)
where A˜ , A + BK and limt→∞ x(t) = xe ∈ N (K). Furthermore, the least squares solution of (21) is given by Z ∞ ˜T ˜ eA t (C T C + K T R2 K)eAt dt. PLS , (22)
(27)
Finally, in this case, the state-feedback control law u(t) = −R2−1 B T P x(t) guarantees Lyapunov stability of the closedloop linear dynamical system G. In general, as shown in [14], the feedback gain matrix K that minimizes (23) and guarantees semistability of (1) with u(t) = Kx(t) can be obtained as a solution to a linear matrix inequality. However, the following result provides semistabilization via a Riccati equation. Corollary 3.1: Consider the linear dynamical system G given by (1) and (2), and performance measure (23) with R1 = C T R3 C. Assume (1) with u(t) = Kx(t), where K ∈ Rm×n , is semistable and (23) is minimized in the sense
of (4). If the pair (A, B) is semicontrollable, the pair (A, E1 ) is semiobservable, R1 = E1T E1 , and xe ∈ N (K), then K = −R2−1 B T PLS ,
(28)
where PLS is the least squares solution to (27). Proof: It follows from Theorems 3.1 and 3.3 that the minimum of (23) is given by J(x0 , K) = xT 0 PLS x0 , T where PLS = PLS ≥ 0 is the least squares solution to (27). Furthermore, the optimal controller is given u(t) = −R2−1 B T PLS x(t) and (1) is Lyapunov stable. Now, by Theorem 3.2, the minimum of (23) is J(x0 , K) = xT 0 PLS x0 , where PLS is the least squares solution to (21). If K = −R2−1 B T P , then (27) and (21) are identical and the assertion is proven by the uniqueness of the minimum of (23). The following classical theorem for finding the minimal cost for a singular control is necessary for later developments. Theorem 3.4 ([20]): Consider the linear dynamical system given by (1) and (2) with performance measure (25), and assume rank B = m and rank C = l. Let tf → ∞ and let Pε = PεT ≥ 0 be a solution to the algebraic Riccati equation 0 = AT P + P A + C T R3 C −
1 P BR2−1 B T P. ε2
(29)
Then the following statements hold. i) If l > m, then limε→0 Pε 6= 0. ii) If l = m and the numerator polynomial of det C(sIn − A)−1 B is not identically equal to zero and has roots with nonpositive real part, then limε→0 Pε = 0. iii) If l < m and there exists a matrix M ∈ Rm×l such that the numerator polynomial of det C(sIn − A)−1 BM is not identically equal to zero and has roots with nonpositive real parts only, then limε→0 Pε = 0. The following theorem gives a converse to Statement ii) of Theorem 3.4. A similar result was proven by Kwakernaak and Sivan [20] for the case where A˜ = A + BK is Hurwitz. Theorem 3.5: Consider the linear dynamical system given by (1) and (2) with performance measure (25), and assume rank B = m and rank C = l. Let tf → ∞, let l = m, and let P¯ε be the least squares solution to (29). If u(t) = Kx(t) guarantees semistability of (1) and minimizes the performance measure (25), the numerator polynomial ψ(s) of det C(sI − A)−1 B is not identically equal to zero, and limε→0 P¯ε = 0, then the roots of ψ(s) are nonpositive. Proof: If limε→0 P¯ε = 0, then (29) specializes to
Now, it follows from Theorem 3.3 that det(sIn − A − BK) = det In − K(sI − A)−1 B det(sIn − A) 1 1 = det In + R2−1 B T P¯ε · ε ε (sIn − A)−1 B . (32) Thus, the roots of the closed-loop characteristic polynomial that stay as ε → 0+ approach the roots of det(sIn − finite −1 A) det R2 L(sIn − A)−1 B , that is, the roots approach the roots of 1 −1 det(sIn − A) det R2 2 U R32 C(sIn − A)−1 B 1 − 21 2 = det R2 U R3 ψ(s). (33) Now, the result follows using identical arguments as in [2, p. 308]. IV. S EMISTABILITY AND S INGULAR C ONTROL In order to address the singular control problem for semistabilization, we first need to show that the performance measure (6) is well-defined when (1) is semistable with u(t) = Kx(t). Proposition 4.1: Consider the system (1) with performance measure (6). If there exists K ∈ Rm×n such that (1) with u(t) = Kx(t) is semistable, then (6) is well-defined, that is, J0 (x0 , u(·)) < ∞. Proof: Consider the performance measure Z ∞ Jε (x0 , u(·)) , (x(t) − xe )T C T C(x(t) − xe ) 0 + ε2 (u(t) − ue )T (u(t) − ue ) dt. (34) Since C T C + ε2 K T K ≥ 0, it follows from Proposition 3.1 that Jε (x0 , u(·)) < ∞, which proves the assertion since J0 (x0 , u(·)) = limε→0+ Jε (x0 , u(·)) and Jε is a monotone function of ε that is bounded from below. Next, we give an expression for the state-feedback control law that minimizes the performance measure (6) and guarantees semistability of the system given by (1) and (2). Lemma 4.1: Consider the linear dynamical system given by (1) and (2) with performance measure (6). Suppose (A, B) is semicontrollable, (A, C) is semiobservable, xe ∈ N (K), (1) is semistable with u(t) = Kx(t) and (6) is minimized. Then, the minimum of the performance measure (6) is given by J0 (x0 , K) = lim Jε (x0 , K) = xT 0 PLS x0 , ε→0
− 12
− 12
1 2
1 2
(R2 L)T (R2 L) = (R3 C)T (R3 C),
(30)
where L , limε→0+ 1ε B T P¯ε . Note that, since R2 is nonsingular, L exists and, since R3 is positive definite, 1 rank C T R3 C = rank R32 C = rank C = m. Thus, there exists U ∈ Rm×m such that U T U = Im and
where PLS = solution of
R∞ 0
˜T
˜
eA t C T CeAt dt is the least squares
0 = AT P + P A + C T C − lim
ε→0
1 P BB T P. ε2
1 2
= U R3 C.
(31)
(36)
Furthermore, the optimal control is given by u(t) = −Lx(t),
−1 R2 2 L
(35)
where L = limε→0+
1 T ε B PLS
is well defined.
(37)
Proof: Equation (35) follows immediately from Theorem 3.2. Next, it follows from Theorem 3.2 that min Jε (x0 , K) = xT 0 Pε x 0 ,
u∈S(x0 )
(38)
where Pε is the least squares solution to (29) for R3 = Il and R2 = Im . Now, (35) and (38) guarantee the existence of PLS satisfying (29) as ε → 0, that is, limε→0 Pε = PLS . Furthermore, as shown in the proof of Theorem 3.5, limε→0+ 1ε B T PLS exists. The proof now follows as direct consequence of Corollary 3.1. The next theorem provides a closed-form expression for the optimal performance measure (6) extending a wellknown property of classical singular control to singular semistabilization. Theorem 4.1: Consider the linear dynamical system given by (1) and (2), with u(t) = Kx(t) such that (1) is semistable and the performance measure (6) is minimized. If l > m, then J0 (x0 , K) > 0. Alternatively, if l = m and the zeros of the numerator polynomial of det C(sIn − A)−1 B are not identically zero and have nonpositive real part, then J0 (x0 , K) = 0. Finally, if l < m and there exists a matrix M such that the numerator polynomial of det C(sIn −A)−1 BM is not identically zero and has zeros with nonpositive real parts, then J0 (x0 , K) = 0. Proof: The proof is a direct consequence of Lemma 4.1 and Theorem 3.4. Corollary 4.1: Consider the linear dynamical system given by (1) and (2), with u(t) = Kx(t) such that (1) is semistable and the performance measure (6) is minimized. Suppose l = m and the numerator polynomial ψ(s) of det C(sIn − A)−1 B is not identically equal to zero. If J0 (x0 , K) = 0, then the roots of ψ(s) are nonpositive. Proof: The proof is a direct consequence of Lemma 4.1 and Theorem 3.5. The following theorem provides an expression for J0 (x0 , K) in terms of a reduced-order system when the openloop system is not minimum phase. Theorem 4.2: Consider the linear dynamical system given by (1) and (2), with u(t) = Kx(t) such that (1) is semistable and the performance measure (6) is minimized. If l = m, rank(CB) = m, and all the roots of the numerator polynomial ψ(s) of det C(sI − A)−1 B have nonnegative real part, then the dynamical system G given by (1) and (2) with u(t) = Kx(t) is equivalent to (12) and the minimal performance measure (6) is given by J0 (x0 , K) = z T (0)P0 z(0),
(39)
where P0 is the least squares solution to T 0 = AT (40) 0 P0 + P0 A 0 − P 0 B 0 B 0 P0 . Proof: It follows from Proposition 2.3 that the system G given by (1) and (2) is equivalent to (12) and the roots of ψ(s) are the eigenvalues of A0 [8]. Next, it follows from Lemma 4.1 that T y(0) y(0) ˆ J0 (x0 , K) = xT P x = P , (41) LS 0 LS 0 z(0) z(0)
where PˆLS , limε→0 Pˆε and Pˆε is the least squares solution of T A1 A2 A1 A2 0= P +P + B 0 A0 B 0 A0 1 Im 0 B1 B1T 0 P. (42) − 2P 0 0(n−m)×(n−m) 0 0 ε
Letting any solution P to (42) be of the form εP1 εP2 P = + O(ε2 ), εP2T P0 + εP3
(43)
where P0 , P1 , P2 , and P3 are independent of ε [21], it follows from (42) with ε = 0 that 1
P1 = B1 B1T )− 2 , P2 = P3 =
(44)
1 B1 B1T )− 2 B0T P0 , 1 P0 B0 B1 B1T )− 2 B0T P0 ,
(45) (46) −1
and (40) holds. Since the zeros of det C(sI − A) B have nonnegative real part, the eigenvalues of −A0 are nonpositive, that is, −A0 is Lyapunov stable and, by Theorem 3.3, there exists a solution P0 to (40). Therefore, the assertion follows immediately from (41) and (43). Finally, we extend Qiu and Davison’s formula for the optimal singular control [3] to semistabilization. Theorem 4.3: Consider the linear dynamical system G given by (1) and (2), with u(t) = Kx(t) such that (1) is semistable and the performance measure (6) is minimized. If G has transfer function G(s) = C(sIn − A)−1 B that is nonminimum phase and right invertible, then the minimal performance measure (6) is given by l X 1 , J0 (x0 , K) = 2 λ j=1 j
(47)
where λ1 , . . . , λl , l ≤ n, are the zeros of G(s) whose real part is positive. Conversely, if l = m, the numerator polynomial ψ(s) of det C(sI − A)−1 B is not identically equal to zero, and (47) holds, then G(s) is minimum phase and right invertible. Proof: It follows from Corollary 2.1 that G(s) can be factored as G(s) = G1 (s)G2 (s), where G1 (s) ∈ Rl×l prop (s), −1 (s), G (s) , C (sI − A ) B + D G2 (s) ∈ Rl×m 1 1 1 1 1 , and prop −1 G2 (s) , C2 (sI − A2 ) B2 . Furthermore, by Corollary 2.1, the poles of G1 (s) are equal to the negatives of the zeros of G(s) whose real part is positive, that is, λ1 , . . . , λl , l ≤ n. Next, let Pε = PεT ≥ 0 denote the solution to the algebraic Riccati equation given by (29). Using the factorization G(s) = G1 (s)G2 (s), it follows that Pε = Il 0l×(n−l) T , where Pε2 = Pε2 ≥ 0 is a solution 0(n−l)×l Pε2 to the algebraic Riccati equation 1 T T 0 = AT (48) 2 P + P A2 + C2 C2 − 2 P B2 B2 P. ε Now, Theorem 3.4 implies that limε→0 Pε2 = 0. Thus, I 0 limε→0 Pε = , and hence, by Lemma 4.1 it follows 0 0 that I 0 T J0 (x0 , K) = xT (49) 0 0 0 x0 = x0,1 x0,1 , T T where x0 , [xT 0,1 , x0,2 ] is partitioned as A is in (15).
Next, since the system (1) with u(t) = Kx(t) is semistable, limt→∞ x(t) = xe . Thus, for t → ∞, the output of G1 (s) is ye = Cxe . Consequently, the output of G2 (s) −1 −1 is G−1 1 (0)ye , which implies that x0,1 = −A1 B1 G1 (0)ye . Therefore, since G1 (0) is unitary, it follows that −1 −1 T −T J0 (x0 , K) = tr B1T A−T 1 A1 B1 = tr A1 B1 B1 A1 . (50)
The transfer function G1(s) is inner, and hence, has a A1 B 1 balanced realization G1 ∼ . Thus, by Corollary C1 D1 2.2, it follows that B1 B1T = C1T C1 , P1 = Q1 = Il , and (13) and (14) hold for A = A1 , B = B1 , C = C1 , P = P1 , and Q = Q1 . Hence, −T T J0 (x0 , K) = −tr A−1 = −2 tr A−1 1 (A1 + A1 )A1 1 , (51)
which proves the assertion. Conversely, if l = m, the numerator polynomial ψ(s) of det C(sI − A)−1 B is not identically equal to zero, and (47) holds, then it follows from Corollary 4.1 that the zeros of ψ(s) are nonpositive, which implies that G(s) is right invertible [7]. V. C ONCLUSION In [14] the authors developed an optimal control framework that guarantees semistability via a state feedback controller and minimizes a quadratic performance measure using linear matrix inequalities. In this paper, we show that the optimal control law that guarantees semistability of a linear dynamical system and minimizes a quadratic performance measure can be found by solving an algebraic Riccati equation approach. Furthermore, as for asymptotically stable closed-loop systems, we show that the optimal singular control cost is zero if and only if the controlled semistable system is minimum phase and right invertible. R EFERENCES [1] Z. Gajic and M. T. Lim, Optimal Control of Singularly Perturbed Linear Systems and Applications. New York: Marcel Dekker, 2001. [2] R. S. H. Kwakernaak, Linear Optimal Control Systems. New York: Wiley, 1972. [3] L. Qiu and E. J. Davison, “Performance limitations of non-minimum phase systems in the servomechanism problem,” Automatica, vol. 29, pp. 337–349, 1993. [4] P. Sannuti, “Direct singular perturbation analysis of high-gain and cheap control problems,” Automatica, vol. 19, pp. 41–51, 1983. [5] P. Kokotovic, H. K. Khalil, and J. O’Reilly, Singular Perturbation Methods in Control: Analysis and Design. London, UK: Academic Press, 1986. [6] A. Saberi and P. Sannuti, “Cheap and singular controls for linear quadratic regulators,” IEEE Transactions on Automatic Control, vol. 32, no. 3, pp. 208–219, 1987. [7] B. Francis, “The optimal linear-quadratic time-invariant regulator with cheap control,” IEEE Transactions on Automatic Control, vol. 24, no. 4, pp. 616 – 621, 1979. [8] M. Seron, J. Braslavsky, P. Kokotovic, and D. Mayne, “Feedback limitations in nonlinear systems: from Bode integrals to cheap control,” IEEE Transactions on Automatic Control, vol. 44, no. 4, pp. 829 –833, 1999. [9] G. Marro, D. Prattichizzo, and E. Zattoni, “Geometric insight into discrete-time cheap and singular linear quadratic Riccati (LQR) problems,” IEEE Transactions on Automatic Control, vol. 47, pp. 102–107, 2002. [10] W. M. Haddad and V. Chellaboina, Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach. Princeton, NJ: Princeton Univ. Press, 2008. [11] Q. Hui and W. M. Haddad, “Distributed nonlinear control algorithms for network consensus,” Automatica, vol. 44, no. 9, pp. 2375 – 2381, 2008. [12] Q. Hui, M. Haddad, and S. Bhat, “Finite-time semistability and consensus for nonlinear dynamical networks,” IEEE Transactions on Automatic Control, vol. 53, no. 8, pp. 1887–1900. [13] D. S. Bernstein, Matrix Mathematics. Princeton, NJ: Princeton Univ. Press, 2008. [14] W. M. Haddad, Q. Hui, and V. Chellaboina, “H2 optimal semistable control for linear dynamical systems: An LMI approach,” J. Franklin Inst., vol. 348, pp. 2898–2910, 2011.
[15] T. Kailath, Linear systems. Prentice-Hall, 1980. [16] K. Glover, “All optimal Hankel-norm approximations of linear multivariable systems and their L∞ -error bounds.” International Journal of Control, vol. 39, pp. 1115–1193, 1984. [17] L. B. Jemaa and E. J. Davison, “Limiting performance of optimal linear discrete filters,” Automatica, vol. 39, no. 7, pp. 1221–1226, 2003. [18] D. Kirk, Optimal Control Theory: An Introduction. New Jersey: Prentice-Hall, 2004. [19] A. L’Afflitto and W. M. Haddad, “Optimal control, singular control, and limits of performance for nonlinear semistable dynamical systems,” in Proc. American Control Conference, submitted, Portland, OR, June 2014. [20] H. Kwakernaak and R. Sivan, “The maximally achievable accuracy of linear optimal regulators and linear optimal filters,” IEEE Transactions on Automatic Control, vol. 17, no. 1, pp. 79 – 86, feb 1972. [21] A. Jameson and R. E. O’Malley, “Cheap control of the time-invariant regulator,” Applied Mathematics and Optimization, vol. 1, pp. 337– 354, 1975.
