Hamburger Beiträge zur Angewandten Mathematik

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1 Hamburger Beiträge zur Angewandten Mathematik The Kalman-Yakubovich-Popov inequality for differential-algebraic systems: existence of nonpositive solutions Timo Reis, Matthias Voigt Nr February 015

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3 The Kalman-Yakubovich-Popov inequality for differential-algebraic systems: existence of nonpositive solutions Timo Reis a,, Matthias Voigt b a Universität Hamburg, Fachbereich Mathematik, Bundestraße 55, 0146 Hamburg, Germany b Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, 1063 Berlin, Germany Abstract The Kalman-Yakubovich-Popov lemma is a central result in systems and control theory which relates the positive semidefiniteness of a Popov function on the imaginary axis to the solvability of a linear matrix inequality. In this paper we prove sufficient conditions for the existence of a nonpositive solution to this inequality for differential-algebraic systems. Our conditions are given in terms of positivity of a modified Popov function in the right complex half-plane. Our results also apply to non-controllable systems. A consequence of our results are bounded real and positive real lemmas for differential-algebraic systems. Keywords: bounded realness, differential-algebraic equation, Kalman-Yakubovich-Popov lemma, Popov function, positive realness 010 MSC: 34A09, 15A, 15A39, 93B17, 93C05 1. Introduction We consider linear time-invariant differential-algebraic control systems Eẋ(t)=Ax(t)+ Bu(t), (1) where se A K[s] n n is assumed to be regular (i.e., det(se A) is not the zero polynomial) and B K m n. The set of such systems is denoted byσ n,m (K) and we write [E, A, B] Σ n,m (K). The function u : R K m is a control input, whereas x : R K n denotes the state of the system. The set of all solution trajectories (x, u) : R K n K m induces the behavior B [E,A,B] := { (x, u) L loc (R,Kn ) L loc (R,Km ) : Eẋ=Ax+ Bu }, where ẋ denotes the distributional derivative of x. We consider the so-called modified Popov function Ψ : C\σ(E, A) C m m, (a) whereσ(e, A)={λ C:det(λE A) 0} (see the subsequent section for the notation) and (λe A) Ψ(λ)= 1 B Q S (λe A) 1 B I m S T (b) R I m with Q=Q K n n, S K n m, and R=R K m m. Note thatψ( ) is takes Hermitian values and coincides on ir with the classical Popov function (where theλin the first factor is Corresponding author addresses: timo.reis@math.uni-hamburg.de (Timo Reis), mvoigt@math.tu-berlin.de (Matthias Voigt) replaced by λ).ψ( ) is, in contrast to the Popov function, neither rational nor meromorphic. We characterize Ψ( ) 0 on the open complex right half plane. This is strongly related to the feasibility of infinite time horizon linear-quadratic optimal control problems with zero final state [1]. For standard state space systems (i.e., with E=I n ), the above property can be checked by the famous Kalman-Yakubovich- Popov (KYP) lemma, see [, 3, 4, 5]. The lemma states that if [I n, A, B] is controllable, thenψ(iω) 0 holds for all iω σ(a) if, and only if, the so-called KYP inequality A P+PA+ Q PB+S B P+S 0, (3) R has a Hermitian solution P K n n. On the other hand, there are modifications of this lemma for special choices of Q, S, and R. For instance, for Q=0 n n, S = C, and R=D+ D, one can show that if [I n, A, B] is controllable, then with G(s)=C(sI n A) 1 B+ D it holds Ψ(λ)=G(λ)+ G(λ) 0 λ C + \σ(a) if, and only if, the KYP inequality (3) with Q=0 n n, S= C, and R=D+D has a solution P 0. This is called positive real lemma and is of great importance in the context of passivity [6]. The KYP lemma states that positive semi-definiteness of Ψ( ) on ir\σ(a) is equivalent to the existence of a solution of the KYP inequality, whereas for the positive real lemma, positive semi-definiteness ofψ( ) in C + \σ(a) is equivalent to the existence of a nonpositive solution of the KYP inequality. Thus, a natural question is whetherψ( ) 0 in C + \σ(a) is equivalent to the existence of a solution P 0 of the KYP inequality. The great Jan C. Willems first casually claimed in his seminal article [1] that this statement holds for controllable Preprint submitted to Systems and Control Letters February 16, 015

4 systems. However, in a quite immediately successional erratum [7], this claim has been disproved by himself with the aid of a counter-example. Willems further stated in this erratum [7] that the equivalence holds, if an inertial decomposition [ Q S C S = 1 C 1 C1 D ] [ 1 C R D 1 C 1 D 1 D C C D ] 1 D C D D, (4a) is admitted, where C 1 K m n, C K p n, D 1 K m m, D K p m, and G 1 (s) := C 1 (si n A) 1 B+ D 1 Gl m (K(s)). (4b) However, no proof of this statement has been carried out. A further condition for the larger class of behavioral systems has been examined by Trentelman and Rapisarda in [8, 9]. Under an additional assumption which translates toψ( )>0 on ir, the existence of nonpositive solutions has been characterized by means of an associated Pick matrix. In this paper, we revisit Willems condition (4) and prove it for differential-algebraic systems. Thereby we are also dealing with non-controllable systems. We further present conditions for all solutions of the KYP inequality being nonnegative. We will apply our results to formulate positive real and bounded real lemmas for differential-algebraic systems. Notation We use the standard notations i,λ, A, A, I n, 0 m n for the imaginary unit, the complex conjugate of λ C, the conjugate transpose of a complex matrix and its inverse, the identity matrix of size n n and the zero matrix of size m n (subscripts may be omitted, if clear from context). The symbol K stands for either the fieldrof real numbers, or the fieldcof complex numbers. The closure of S C is denoted by S. By writing A ( ) B we mean that for two Hermitian matrices A, B K n n, the matrix A B is positive semidefinite (negative semidefinite). The following concept, namely equality and semidefiniteness on some subspace will be frequently used in this article. Definition 1.1 (Equality and semidefiniteness on a subspace). LetV K n be a subspace and A, B K n n be Hermitian. Then we write A= V ( V, V ) B, if v (A B)v=(, ) 0 holds for all v V. The following sets are further used in this article: N 0 the set of natural numbers including zero C +, C the open sets of complex numbers with positive and negative real parts, resp. K[s], K(s) Gl n (K) σ(a) σ(e, A) the ring of polynomials and the field of rational functions with coefficients ink, resp. the group of invertible n n matrices with entries in a fieldk spectrum of A K n n ={λ C:det(λE A)=0}, the set of generalized eigenvalues of the matrix pencil se A K[s] n n RH p m L loc (I,Kn ). Preliminaries the space of rational p m matrix-valued functions which are bounded inc + the set of measurable and locally square integrable functions f : I K n on the set I R.1. Differential-Algebraic Systems We first introduce some systems theoretic concepts for differential-algebraic systems [E, A, B] Σ n,m (K). First we consider notions related to controllability and stabilizability, see also [10, 11] and [1, Def. 5..] for the definition and the respective algebraic conditions in terms of the system matrices. Definition.1 (Controllability and stabilizability). A system [E, A, B] Σ n,m (K) is called a) behaviorally (beh.) stabilizable if for all (x 1, u 1 ) B [E,A,B], there exists some (x, u) B [E,A,B] with (x(t), u(t))=(x 1 (t), u 1 (t)), if t<0, and lim t (x(t), u(t))=0; b) behaviorally (beh.) controllable if for all (x 1, u 1 ), (x, u ) B [E,A,B], there exists some (x, u) B [E,A,B] and some T> 0 with (x 1 (t), u 1 (t)), if t<0, (x(t), u(t))= (x (t), u (t)), if t>t; c) completely controllable if for all x 0, x f K n, there exists some (x, u) B [E,A,B] and some T> 0 with x(0)= x 0 and x(t)= x f. Moreover, we also consider differential-algebraic systems [E, A, B] Σ n,m (K) which have an additional output equation y(t)=cx(t)+du(t), where C K p n and D K p m. We denote the set of all such systems byσ n,m,p (K) and we write [E, A, B, C, D] Σ n,m,p (K) (or [E, A, B, C] Σ n,m,p (K) if D=0). The behavior is given by B [E,A,B,C,D] := { (x, u, y) B [E,A,B] L loc (R,Kp ) : and the expression G(s)=C(sE A) 1 B+ D K(s) p m y=cx+du }, is called the transfer function of [E, A, B, C, D] Σ n,m,p (K). Behavioral detectability means that the state can be asymptotically reconstructed from the knowledge of input and output, cf. [1, Def ]. See also [1, Thm ] for an equivalent algebraic criterion. Definition. (Behavioral detectability). The system [E, A, B, C, D] Σ n,m,p (K) is called behaviorally (beh.) detectable if (x 1, u, y), (x, u, y) B [E,A,B,C,D] lim t x 1 (t) x (t)=0.

5 .. System Equivalence Form, System Space, and Space of Consistent Initial Differential Veriables In this paper we consider systems [E, A, B] Σ n,m (K) under system equivalence which is defined as follows. Definition.3 (System equivalence). [10] Two systems [E i, A i, B i ] Σ n,m (K), i=1,, are called system equivalent, if there exist W, T Gl n (K) such that T 0 se A B = W se1 A 1 B 1. 0 I m Remark.4. Beh. stabilizability, beh. controllability and complete controllability are invariant under system equivalence. The following system equivalence form is basically consisting of a Kalman decomposition of the infinite eigenvalues of se A and will be of great importance in our proofs. Proposition.5 (System equivalence form). [13] Let [E, A, B] Σ n,m (K) be given. Then there exist W, T Gl n (K) such that [ sẽ Ã B ] :=W [ se A B ] T 0 0 I m si n1 A 11 0 se 13 A 13 B 1 = 0 se I n se 3 A 3 B, 0 0 se 33 I n3 0 (5) where E K n n and E 33 K n 3 n 3 are nilpotent and [ ] In1 0 A11 0 B1,, Σ 0 E 0 I n B n1 +n,m(k) is completely controllable. Remark.6. The system [E, A, B] is beh. stabilizable (beh. controllable) if, and only if, the ODE system [I n1, A 11, B 1 ] is stabilizable (controllable) [14]. Next we introduce two fundamental spaces of a system [E, A, B], namely the system space and the space of consistent initial differential variables. Definition.7 (System space, space of consistent initial differential variables). Let [E, A, B] Σ n,m (K) be given. a) The system space of [E, A, B] is the smallest subspace V sys K n+m, such that for all (x, u) B [E,A,B] it holds ( ) x(t) V u(t) sys for almost all t R. b) The space of consistent initial differential variables of [E, A, B] is defined by V diff := { x 0 K n : (x, u) B [E,A,B] with Ex(0)=Ex 0 }. In the next lemma we consider the behavior of the two above spaces under systems equivalence. 3 Lemma.8. [15, 14] Assume that [E, A, B] Σ n,m (K) with the system spacev sys K n+m and the space of consistent initial differential variablesv diff K n is given. Moreover, let W, T Gl n (K) and letṽsys K n+m andṽdiff K n be the system space and the space of consistent initial differential variables of [WET, WAT, WB] Σ n,m (K), respectively. Then it holds a) T 0 V sys = Ṽ 0 I sys ; m b) V diff = T Ṽdiff. Finally, we give an explicit form of the system space and the space of consistent initial differential variables for systems given in system equivalence form. Lemma.9. Let the system [E, A, B] Σ n,m (K) be given in system equivalence form (5). Letν N 0 be the index of nilpotency of E. Then it holds a) V sys = im M Vsys, where I n B M Vsys = E B... E ν 1B (6) I m b) V diff = K n 1+n ker E 13 E 3 E 33 Proof. Statement a) can be concluded from the fact that the equation E 33 ẋ 3 (t) = x 3 (t) implies x 3 (t) = 0, whereas from E ẋ (t)= x (t)+b u(t) we obtain ν 1 x (t)= E B u (i) (t). i=0 Statement b) is a consequence of [15, Prop..9]..3. Differential-Algebraic Kalman-Yakubovich-Popov Lemma We recall a version of the KYP lemma for differentialalgebraic equations which has recently been shown in [14, Thm. 4.1]. Theorem.10 (KYP lemma for differential-algebraic systems). Let [E, A, B] Σ n,m (K) with the system spacev sys K n+m, and Q=Q K n n, S K n m, R=R K m m be given. Further, let the modified Popov function Ψ( ) be defined as in (). Then the following statements hold: a) If the KYP inequality A PE+E PA+ Q E PB+S B PE+ S R Vsys 0, P=P (7) has a solution P K n n, then Ψ(iω) 0 ω R with iω σ(e, A). (8).

6 b) If (8) and at least one of the two properties i) rankψ(iω)=m for someω R with iω σ(e, A), and max { rank [ λe A B ], rank [ λe A B ]} = n for allλ C; ii) [E, A, B] is beh. controllable is satisfied, then there exists some P K n n that solves the KYP inequality (7). 3. Main results In this section we investigate the existence of nonpositive solutions of the KYP inequality (7). First, we define what we mean by nonpositivity. Definition 3.1 (Nonpositive solution). Let [E, A, B] Σ n,m (K) with the space of consistent initial differential variablesv diff K n be given. Then we call a solution P of the KYP inequality (7) nonpositive, if E PE Vdiff 0. (9) First we show the simpler part: The positive semidefiniteness ofψ( ) in C + \σ(e, A) is necessary for the existence of a nonpositive solution. This requires the following technical lemma. Lemma 3.. Let [E, A, B] Σ n,m (K) with the systems space V sys K n+m and the space of consistent initial variables V diff K n be given. Then for allλ C\σ(E, A) the following statements hold: a) b) c) (λe A) 1 B A B =λ [ E 0 ] (λe A) 1 B I m I m (λe A) im 1 B V I sys m im(λe A) 1 B V diff Proof. Statements a) and b) have been shown in [14]. It remains to show c): Assume thatλ C\σ(E, A) and let W, T Gl n (K) be transformation matrices leading to system equivalence form (5). Then im(λe A) 1 B=im T ( λẽ Ã ) 1 B (λi n1 A 11 ) 1 B 1 = im T (λe I n ) 1 B 0 T Kn 1+n ker E 13 E 3 E 33 =V diff. Proof. By using Lemma 3. a), for allλ C\σ(E, A) we have (λe A) 1 B A PE+E PA E PB (λe A) 1 B I m B PE 0 I m (λe A) = 1 ( B A [PE ] E P [A ] ) 0 + B I m B 0 (λe A) 1 B I m (λe A) = 1 ( B E [PE ] E λ P [E ] ) 0 +λ 0 I m 0 0 (λe A) 1 B I m = Re(λ) B ( λe A ) 1 E PE ( λe A ) 1 B. The KYP inequality (7) and Lemma 3. b) yield the inequality Ψ(λ) Re(λ) B ( λe A ) 1 E PE ( λe A ) 1 B λ C\σ(E, A). Now plugging in someλ C + \σ(e, A) and using (9) and Lemma 3. c), we obtain the desired result. We now show that, under an additional assumption, the nonnegativity of the modified Popov function also implies the existence of a nonpositive solution. For this we need the following two lemmas. Lemma 3.4. Let [E, A, B, C, D] Σ n,m,p (K) be beh. stabilizable and beh. detectable and assume that its transfer function fulfills G(s) RH p m. Then it holdsσ(e, A) C. Proof. Since generalized state-space transformations preserve beh. stabilizability and beh. detectability, we can w.l.o.g. assume that se A is given in quasi-weierstraß form [16], i.e., se A= sir A 11 0, B= 0 se I n r B1, C= C B 1 C where E K n r n r is nilpotent, B 1 K r m, B K n r m, C 1 K p r, and C K p n r. This corresponds to a decomposition G(s)=G sp (s)+ G poly (s) where G sp (s)=c 1 (si r A 11 ) 1 B 1 K(s) p m is strictly proper and G poly (s) = C (se I n r ) 1 B K[s] p m. The assumption G(s) RH p m then implies G sp (s) = C 1 (si r A 11 ) 1 B 1 RH p m. Beh. stabilizability and beh. detectability of [E, A, B, C, D] imply stabilizability and detectability of [I r, A 11, B 1, C 1, D]. Then [17, Lem. 8.3] yieldsσ(a 11 ) C, and henceσ(e, A) C. Theorem 3.3. Let [E, A, B] Σ n,m (K) with the systems space V sys K n+m and the space of consistent initial variables V diff K n be given and assume that Q = Q K n n, S K n m, and R=R K m m. Suppose that P K n n is a nonpositive solution of the KYP inequality (7). Then the modified Popov function () fulfills Ψ(λ) 0 λ C + \σ(e, A). (10) 4 Lemma 3.5. Let two systems [E, A, B, C 1, D 1 ] Σ n,m,m (K), [E, A, B, C, D ] Σ n,m,p (K) given with transfer functions G 1 (s)= C 1 (se A) 1 B+ D 1 Gl m (K(s)), G (s)= C (se A) 1 B+ D K(s) p m Then the pencil se+a B C 1 D 1

7 is regular and the transfer function of the system [E e, A e, B e, C e ] [ E 0 :=, 0 0 A B, C 1 D 1 is G e (s)=g (s)g 1 1 (s). [ 0 I m ], [ C D ] ] Σ n+m,m,p (K) (11) Proof. Regularity of se e A e has been shown in [18, Lem. 8.9] for E=I n. Our more general result follows by the same argumentation. The remaining result follows from 1 [ se A B 0 (se A) = 1 BG 1 1 (s) ] C 1 D 1 I m G 1 1 (s), and consequently G e (s)=c e (se e A e ) 1 B e = [ (se A) C D 1 BG 1 1 (s) ] G 1 1 (s) = G (s)g 1 1 (s). Theorem 3.6. Let [E, A, B] Σ n,m (K) with system space V sys K n+m and space of consistent initial differential variablesv diff K n be given and assume that Q=Q K n n, S K n m, and R=R K m m. Suppose that the modified Popov function () fulfills (10). Further, assume that at least one of the following two assumptions holds: i) [E, A, B] is beh. stabilizable and the modified Popov function () satisfies rankψ(iω) = m for someω R with iω σ(e, A); ii) [E, A, B] is beh. controllable. Let C 1 K m n, C K p n, D 1 K m m, and D K p m be given such that [ Q S C S = 1 C 1 C1 D ] [ 1 C R Vsys D 1 C 1 D 1 D C C D ] 1 D C D D, (1) and G 1 (s) := C 1 (se A) 1 B+ D 1 Gl m (K(s)). Then the solution set of the KYP inequality (7) is nonempty. Furthermore, the following holds: a) There exists a nonpositive solution of the KYP inequality (7). b) If, furthermore, λe+a B rank C 1 D 1 = n+m λ C+, (13) C D then all solutions of the KYP inequality (7) are nonpositive. Proof. Define G (s) := C (se A) 1 B+ D K(s) p m. In the first part of the proof we show the theorem for a special case and then turn to the more general situation. Step 1: We show that Theorem 3.6 holds under the additional assumption that C 1 = 0 and D 1 = I m : 5 Step 1.1: We show that under assumptions of Theorem 3.6 and C 1 = 0, D 1 = I m we have G (s) RH p m : By C 1 = 0 and D 1 = I m we obtain G 1 (s)=i m. Hence, Ψ(λ)=I m G (λ)g (λ) 0 λ C + \σ(e, A). As a consequence, we obtain G (s) RH p m. Step 1.: We show that under the additional assumption that C 1 = 0, D 1 = I m and (13), the solution set of the KYP inequality (7) is nonempty. Moreover, all its solutions are nonpositive: First, by a continuity argument and the fact that the Popov functions coincides with the modified Popov function on ir, we obtain that the Popov function is pointwisely positive semidefinite on ir. Then, by Theorem.10, the KYP inequality (7) has a solution P K n n. We now show that this solution fulfills E PE 0: C 1 = 0 and D 1 = I m implies that λe+a B λe A n+m=rank 0 I m = m+rank λ C. C C D (14) Then, by [1, Thm ], [E, A, B, C, D ] is beh. detectable. We further know from Step 1.1 that G (s) RH p m. Since [E, A, B] is beh. stabilizable by assumption, Lemma 3.4 gives σ(e, A) C. Let W, T Gl n (K) be matrices leading to system equivalence form (5). Thenσ(A 11 )=σ(e, A) C. Since P and C can be accordingly transformed, and, in particular, nonpositivity is preserved under this transformation, we see that it is no loss of generality to assume that W= T= I n. Let P 11 P 1 P 13 P= P 1 P P 3, C = C 1 C C 3 P 13 P 3 P 33 be partitioned according the block structure of (5). Assume that ν N 0 is the index of nilpotency of E. WithM Vsys as in (6), we obtain from Lemma.9 a) thatv sys = imm Vsys. Thus, the KYP inequality (7) reduces to [ A 0 M PE+E PA C C E PB C D ] V sys B PE D C I m D D M Vsys K 11 K 1 0 N 1,1... N 1,ν 1 K1 K 0 N,1... N,ν = N1,1 N,1 0 M 1,1... M 1,ν 1 (15) N1,ν 1 N,ν 1 0 M1,ν 1... M ν 1,ν 1 where K 11 = A 11 P 11+ P 11 A 11 C 1 C 1, K 1 = P 11 B 1 C 1 D + C 1 C B A 11 P 1E B, K = B C C B + B C D +D C B + I m D D B E P 1 B 1 B 1 P 1E B,

8 and, for 1 i j ν 1, N 1,i = P 1 E i B + C 1 C E i B A 11 P 1E i+1 B, N,i = B C C E i B +D C E i B + B E P 1 Ei B 1 B 1 P 1E i+1 B, M i, j = B (E )i+1 P E j B +B (E )i P E j+1 B B (E )i C C E j B. We will now show by induction that 1) P 1 E i B = 0 for i=1,...,ν 1, and ) B (E )i P E j B = 0 for i, j=1,...,ν 1. First note that by properness of G (s) we obtain C E B =...=C E ν 1 B = 0. Together with the fact that E ν = 0 this implies M ν 1,ν 1=0. Since (15) is positive semidefinite, this yields N 1,ν 1 = 0, N,ν 1 = 0, M 1,ν 1 =...= M ν,ν 1 = 0. This however implies P 1 E ν 1B = 0 and B (E )i P E ν 1B = 0, i=1,...,ν 1. Repeating this process inductively leads to statements 1) and ). Now define the controllability matrix C := [ B E B... E ν 1 B ]. By the system equivalence form (5) the system [E, I n, B ] Σ n,m(k) is completely controllable. Thus by [19, Thm. -.1] we obtain rankc = n. (16) By statements 1) and ) it holds P 1 E C = 0 and C E P E C = 0, hence by (16) this already implies Moreover, since P 1 E = 0, E P E = 0. (17) 0 K 11 = A 11 P 11+ P 11 A 11 C 1 C 1, the Lyapunov inequality A 11 ( P 11)+( P 11 )A 11 0 holds. Usingσ(A 11 ) C, we obtain from [0, Lem. 3.18] that P 11 0, i.e., P This, together with (17) and the representation ofv diff as in Lemma.9 b) implies I n1 0 0 P 11 P 1 P 13 I n1 0 0 E PE= Vdiff 0 E 0 P 1 P P 3 0 E P 13 P 3 P P = Vdiff V diff Step 1.3: We prove that statement a) holds for C 1 = 0 and D 1 = I m : By the Kalman decomposition [19, Sect..5], it is no loss of generality to assume that se11 A se A= 11 0 B1, B=, C se 1 A 1 se A B = [ C 1 0 ], 6 where [E 11, A 11, B 1, C 1 ] Σ n1,m,p (K) with the system space V sys,1 K n1+m and the space of consistent initial differential variablesv diff,1 K n1+m is beh. detectable. Then we obtain G (s)=c 1 (se 11 A 11 ) 1 B 1 + D. By the assumptions and Theorem.10, there exists some solution P 11 K n 1 n 1 of the KYP inequality [ A 11 P 11 E 11 + E 11 P 11 A 11 C1 C 1 E11 P 11B 1 C1 D ] B 1 P 11E 11 D C 1 I m D D. Vsys,1 0. Since [E, A, B, C, D ] is beh. stabilizable and beh. detectable, we can make use of the results of Step 1. to see that E 11 P 11E 11 Vdiff,1 0. Using the block triangular structure of E, A, and B, we obtain x 1 V sys Ṽsys := x Kn+m : u V diff Ṽdiff := {( ) } x1 K x n : x 1 V diff,1. ( ) x1 V u sys,1, (18) Now define P11 0 P= K 0 0 n n. Then E PE Ṽdiff 0 and [ A PE+E PA C C E PB C D ] B PE D C I m D D. A 11 P 11E 11 + E11 P 11A 11 C1 C 1 0 E11 P 11B 1 C1 D = B 1 P 11E 11 D C 1 0 I m D D. Ṽsys 0. Hence, by (18), we see that P solves the KYP inequality (7). Step : We infer the overall result: Using Lemma 3. b), we obtain that the modified Popov function fulfills (λe A) Ψ(λ)= 1 B Q S (λe A) 1 B I m S R I m [ (λe A) 1 B (λe A) 1 B = ] [ C 1 C 1 C1 D 1 I m D 1 C 1 D 1 D 1 (λe A) 1 [ B C C C D I m D C D D = G 1 (λ)g 1(λ) G (λ)g (λ). ] I ][ m ] (λe A) 1 B I m In particular, the function G e (s)=g (s)g 1 1 (s) K(s)p m fulfills I m G e (λ)g e(λ) 0 (19) for allλ C + \σ(e, A) which are no poles of G e (s). Consider the matrices E e, A e, B e, C e as in (11). By Lemma 3.5 we obtain that the pencil se e A e K[s] n+m n+m is regular and C e (se e A e ) 1 B e = G (s)g 1 1 (s)=g e(s).

9 The structure of E e, A e, and B e yields rank λe e A e B e = rank λe A B + m λ C. Hence, we can use [1, Thm ] and beh. stabilizability of [E, A, B] to see that [E e, A e, B e ] is as well beh. stabilizable. LetV sys,e andv diff,e be the system space and the space of consistent initial differential variables of the system (11), respectively. Consider the KYP inequality A e P e E e + EeP e A e + Q e EeP e B e + S e B T e P e E e + S e R Vsys,e 0, P e = P e, e (0) where [ C Q e = C C D ] D C D D, S e = 0, R e = I m. The modified Popov function associated to the KYP inequality (0) then readsψ e (λ)=i m G e(λ)g e (λ). By (19) and the results from Step 1, there exists a nonpositive solution P e K n+m n+m of (0). Partition P Pe,1 P e = P e,1 P e, with P K n n. Then the equation Ee P ee e Vdiff,e 0 implies E PE Vdiff 0. Hence, it suffices for the desired statement a) to prove that P indeed solves the KYP inequality (7). By definition ofv sys,e, E e, A e, and B e, we have x ( x V sys,e = u Kn+m : V u) sys and z=c 1 x+ D 1 u z. Thus, the matrix I n 0 T e := 0 I m C 1 D 1 fulfills T e V sys =V sys,e. Thereby, we obtain [ A 0 Vsys Te e P e E e + Ee P ea e + Q e Ee P ] eb e + S e B ep e E e + S e T R e e [ A = PE+E PA+ C1 C 1 C C E PB+C1 D 1 C D ] Vsys B PE+D 1 C 1 D C D 1 D 1 D D A = PE+E PA+ Q E PB+S Vsys B PE+ S. R Finally, for statement b) it remains to be shown that all solutions of the KYP inequality are nonpositive, if (13) holds. Now let P K n n be a solution of the KYP inequality (7). Then P 0 P e = 0 0 solves the KYP inequality (0). Further, by elementary column operations, we obtain λe e + A e B e λe+a B rank C e 0 = rank C 1 D 1 + m 0 I m C D = n+m λ C +. 7 Hence we are in the situation of Step 1., which leads to E PE 0 = E 0 0 ep e E e Vdiff,e 0, and therefore, E PE Vdiff 0. This completes the proof. 4. Positive real and bounded real lemma Now we apply our results to characterize positive realness and bounded realness of transfer functions. Definition 4.1 (Positive realness, bounded realness). A rational function G(s) K(s) p m is called a) positive real if p=m, G(s) has no poles in C + and G(λ)+ G (λ) 0 λ C + ; b) bounded real if G(s) RH p m and Remark 4.. I m G (λ)g(λ) 0 λ C +. a) Bounded realness is equivalent to theh norm [0, Sec 4.3] of G(s) being less or equal to one. b) The notions of positive realness and bounded realness are quite traditional in linear systems theory [6]. Note that, though the terminology suggests, we actually do not impose any realness assumption in the previous definition. Realness actually does not feature any noteworthy additional property, whence we use the traditional notion also for the complex case. Theorem 4.3 (Differential-algebraic positive real lemma). Let [E, A, B, C, D] Σ n,m,m (K) with the system spacev sys K n+m, the space of consistent initial differential variablesv diff K n, and the transfer function G(s) K(s) m m be given. a) If there exists some Hermitian P K n n with E PE Vdiff 0 and A PE+E PA E PB C B PE C D D Vsys 0, (1) then G(s) is positive real. b) Assume that G(s) is positive real and least one of the following properties holds: (i) [E, A, B] is beh. stabilizable and G(iω) + G(iω)>0 for someω R with iω σ(e, A); (ii) [E, A, B] is beh. controllable. Then there exists some Hermitian P K n n fulfilling (1). c) If, in addition to the assumptions in b), [E, A, B, C, D] is beh. detectable, then all Hermitian P K n n with (1) fulfill E PE Vdiff 0. Proof. Statement a) follows from Theorem 3.3 with Q = 0, S= C, R=D+D and a substitution P P. To prove b), we simply check whether we meet the assumptions of Theorem 3.6 after the above substitution: In the terminology of Theorem 3.6 we have C 1 = C = 1 C, D 1 = 1 (D+I m )

10 and D = 1 (D I m ), G 1 (s)= 1 (G(s)+I m ), and G (s)= 1 (G(s) I m ). Invertibility of G 1 (s) is a consequence of the fact that for allλ C + and u C m it holds u (I m + G(λ))u= u + 1 u (G (λ)+ G(λ))u u u. To prove c), it suffices to translate and check that (13) is true. This however is a consequence of [1, Thm ] together with λe+a B λe+a B rank C 1 D 1 = rank 1 1 C D C (D+ I m ) 1 1 C (D I m ) λe+a = m+rank λ C. C Remark 4.4. We note that in [1], a similar version of the positive real lemma has been elaborated. For the existence of a solution of some P 0 fulfilling (1), minimality (which implies complete controllability and the respectively dual property for observability) of [E, A, B, C, D] has however been presumed. By using substitutions P P, C 1 = 0 m n, D 1 = I m, C = C, D = D, G 1 (s)=i m and G (s)= G(s) in Theorem 3.3, and further making use of (14), we can analogously conclude the differential-algebraic bounded real lemma. Theorem 4.5 (Differential-algebraic bounded real lemma). Let [E, A, B, C, D] Σ n,m,p with the system spacev sys K n+m, the space of consistent initial differential variablesv diff K n, and the transfer function G(s) K(s) p m be given. a) If there exists some Hermitian P K n n with A PE+E PA+ C C E PB+C D B PE+D C D D I Vsys 0, () m then G(s) is bounded real. b) Assume that G(s) is positive real and least one of the following properties hold: (i) [E, A, B] is beh. stabilizable and I m G(iω) G(iω)>0 for someω R with iω σ(e, A); (ii) [E, A, B] is beh. controllable. Then there exists some Hermitian P K n n fulfilling (). c) If, in addition to the assumptions in b), [E, A, B, C, D] is beh. detectable, then all Hermitian P K n n with () fulfill E PE Vdiff Conclusions In this paper we have proven a new sufficient condition for the existence of nonpositive solutions of the Kalman- Yakubovich-Popov (KYP) inequality. 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