On Positive Quadratic Forms and the Stability of Linear Time-Delay Systems

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1 On Positive Quadratic Forms and the Stability of Linear Time-Delay Systems Matthew Peet,1, Antonis Papachristodoulou,and Sanjay Lall 1 Abstract We consider the problem of constructing Lyapunov functions for linear differential equations with delays. For such systems it is known that stability implies that there eists a quadratic Lyapunov function on the state space, although this is in general infinite dimensional. We give an eplicit parametrization of a finite-dimensional subset of the cone of Lyapunov functions using positive semidefinite matrices. This allows the computation to be formulated as a semidefinite program. Key words: Time-delay, nfinite dimensional systems, Stability, Semidefinite programming, Parametric uncertainty 1 ntroduction functional of the form n this paper we present an approach to the construction of Lyapunov functions for systems with time-delays. Specifically, we are interested in systems of the form ẋ(t) A i (t h i ) + i1 h 0 B(s)(t s)ds, (1) where (t) R n. n the simplest case we are given information about the the delays h 0,..., h k, the matrices A 0,..., A k, and a matri of polynomials, B, and we would like to determine whether the system is stable. For such systems it is known that if the system is stable, then this property can be proven using a Lyapunov Department of Aeronautics and Astronautics 4035, Stanford University, Stanford CA , U.S.A. Department of Engineering Science, University of Oford, Oford, OX1 3PJ, UK addresses: mmpeet@stanford.edu (Matthew Peet ), antonis@eng.o.ac.uk ( Antonis Papachristodoulou), lall@stanford.edu (Sanjay Lall ). 1 Authors were partially supported by the Stanford UR Architectures for Secure and Robust Distributed nfrastructures, AFOSR DoD award number V (φ) + M(s) ds T N(s, t)φ(t) ds dt, where M and N are piecewise-continuous matri-valued functions. Here φ :, 0] R n is an element of the state space, which for delayed systems is the infinitedimensional space of continuous functions mapping, 0] to R n. The derivative of V has a similar structure to V and is also defined by matri functions which are affine transformations of M and N. The goal of this paper is to show how to use semidefinite programming to construct piecewise-continuous functions M and N such that the functional V is positive. Roughly speaking, in Section 3, for a piecewisecontinuous function M, we show that V 1 (φ) M(s) ds is positive for all φ if and only if there eists a piecewise Preprint submitted to Automatica 16 January 2007

2 continuous matri-valued function T such that T (t) 0 M(t) T (t) dt 0 for all t That is, we convert positivity of the integral to pointwise positivity. This result is stated precisely in Theorem 6. Pointwise positivity may then be easily enforced, and in particular in this case is equivalent to a sum-of-squares constraint. The constraint that T integrates to zero is a simple linear constraint on the coefficients. The condition that the derivative of the Lyapunov function be negative is similarly enforced. Notice that the sufficient condition that M(s) be pointwise nonnegative is conservative, and as the equivalence above shows it is easy to generate eamples where V 1 is nonnegative even though M(s) is not pointwise nonnegative. n Section 4, we show that if N is defined by polynomial N ij on the interval i j, then V 2 (φ) T N(s, t)φ(t) ds dt is positive if and only is there eists a Q 0 such that N ij (s, t) Z(s) T Q ij Z(t). Here Z is a monomial basis. This result is stated precisely in Theorem 9. This result allows us to epress our positivity conditions directly in terms of positive matrices. Note that simple positivity of N in this case is not even sufficient for positivity of V 2 as illustrated by a simple countereample such as N(s, t) (s t) 2, h 2 and s Notation Let N denote the set of nonnegative integers. Let S n be the set of n n real symmetric matrices, and for S S n we write S 0 to mean that S is positive semidefinite. For X any Banach space and R any interval let Ω(, X) be the space of all functions Ω(, X) { f : X } and let C(, X) be the Banach space of bounded continuous functions C(, X) { f : X f is continuous and bounded } equipped with the norm f sup f(t) X t We will omit the range space when it is clear from the contet; for eample we will simply write Ca, b] to mean C(a, b], X). A function f Ωa, b] is called piecewise C 1 if there eists a finite number of points a h 1 < < h k b such that f is continuous at all a, b]\{h 1,..., h k }. We will write, as usual f(a+) lim t a+ f(t). Define also the projection H t : Ω, ) Ω, 0] for t 0 and h > 0 by (H t )(s) (t + s) for all s, 0]. We follow the usual convention and denote H t by t. We consider polynomials in n variables. For α N n define the monomial in n variables α by α α 1 1 α n n. We say M is a real matri polynomial in n variables if for some finite set W N n we have M() α W A α α 1.1 Prior Work The use of Lyapunov functionals on an infinite dimensional space to analyze differential equations with delay originates with the work of Krasovskii 9]. For linear systems, quadratic Lyapunov functions were first considered by Repin 15]. The book of Gu, Kharitonov, and Chen 4] presents many useful results in this area, and further references may be found there as well as in Hale and Lunel 5], Kolmanovskii and Myshkis 8] and Niculescu 12]. The idea of using sum-of-squares polynomials together with semidefinite programming to construct Lyapunov functions originates in Parrilo 13]. where A α is a real matri for each α W. For Banach spaces X and Y, define an integral operator A with kernel function k : {f : X Y } to be the map A : Ω(, X) Ω(, Y ) such that A(s) k(s, t)(t)dt is in Ω(, Y ). Similarly, a multiplier operator B with multiplier m : {f : X Y } is defined to be the map B : Ω(, X) Ω(, Y ) such that B(s) m(s)(s) 2

3 f X Y and Ω(, X) is a Hilbert space with inner product,, then an operator, C, is positive with respect to Ω(, X) if, C 0 for all Ω(, X) 2 System Formulation Suppose 0 h 0 < h 1 < < h k h, H { 0,..., k }, and let H c, 0]\H. Suppose A 0,..., A k R n n and B is continuous. We consider linear differential equations with delay, of the form ẋ(t) A i (t i )+ i0 h 0 B(s)(t s)ds for all t 0 (2) where the trajectory :, ) R n. The boundary conditions are specified by a given function φ :, 0] R n and the constraint (t) φ(t) for all t, 0]. (3) f φ C, 0], then there eists a unique differentiable function satisfying (2) and (3). The system is called eponentially stable if there eists σ > 0 and a R such that for every initial condition φ C, 0] the corresponding solution satisfies (t) ae σt φ for all t 0. We write the solution as an eplicit function of the initial conditions using the map G : C, 0] Ω, ), defined by (Gφ)(t) (t) for all t where is the unique solution of (2) and (3) corresponding to initial condition φ. Also for s 0 define the flow map Γ s : C, 0] C, 0] by Γ s φ H s Gφ which maps the state of the system t to the state at a later time t+s Γ s t. 2.1 Lyapunov Functions Suppose V : C, 0] R. We use the notion of derivative as follows. Define the Lie derivative of V with respect to Γ by V (φ) lim sup r 0+ 1( V (Γr φ) V (φ) ) r We will use the notation V for both the Lie derivative and the usual derivative, and state eplicitly which we mean if it is not clear from contet. We will consider the set X of quadratic functions, where V X if there eists bounded piecewise C 1 functions M :, 0) S 2n and N :, 0), 0) R n n such that V (φ) + M(s) ds T N(s, t)φ(t) ds dt (4) The following important result shows that for linear systems with delay, the system is eponentially stable if and only if there eists a quadratic Lyapunov function. Theorem 1 The linear system defined by equations (2) and (3) is eponentially stable if and only if there eists a Lie-differentiable function V X and ε > 0 such that for all φ C, 0] V (φ) ε 2 V (φ) ε 2 (5) Further V X may be chosen such that the corresponding functions M and N of equation (4) have the following smoothness property: M(s) and N(s, t) are continuous at all s, t such that s H c and t H c. PROOF. See Kharitonov and Hinrichsen 6] for a recent proof. 3 Positivity of ntegrals We would like to be able to computationally find functions V X which satisfy the positivity conditions (5). n this section, we address the first part of the functional, V 1 (y) y(0) y(t) y(0) M dt 0. (6) y(t) The following three lemmas are necessary to prove the main result, Theorem 6. Lemma 2 Suppose f :, 0] R is piecewise continuous. Then the following are equivalent. (i) f(t) dt 0 (ii) there eists a function g :, 0] R which is piecewise continuous and satisfies f(t) + g(t) 0 g(t) dt 0 for all t 3

4 PROOF. The direction (ii) (i) is immediate. To show the other direction, suppose (i) holds, and let g be g(t) f(t) + 1 h Then g satisfies (ii). f(s) ds for all t Lemma 3 Suppose H { 0,..., k } and let H c, 0]\H. Let f :, 0] R n R be continuous on H c R n, and suppose there eists a bounded function z :, 0] R, continuous on H c, such that for all t, 0] f ( t, z(t) ) inf f(t, ) Further suppose for each bounded set X R n the set is bounded. Then { f(t, ) X, t, 0] } inf f ( t, y(t) ) dt y C,0] PROOF. Let K t is easy to see that inf f(t, ) dt inf f ( t, y(t) ) dt K y C,0] inf f(t, ) dt (7) since if not there would eist some continuous function y and some interval on which which is clearly impossible. f ( t, y(t) ) < inf f(t, ) We now show that the left-hand side of (7) is also less than or equal to K, and hence equals K. We need to show that for any ε > 0 there eists y C, 0] such that f ( t, y(t) ) dt < K + ε To do this, for each n N define the set H n R by k 1 H n (h i α/n, h i + α/n) i1 and choose α > 0 sufficiently small so that H 1 (, 0). Let z be as in the hypothesis of the lemma, and pick M and R so that M > sup z(t) t R sup { f(t, ) t, 0], M } For each n choose a continuous function n :, 0] R n such that n (t) z(t) for all t H n and sup n (t) < M t,0] This is possible, for eample, by linear interpolation. Now we have, for the continuous function n f ( t, n (t) ) dt K + K + H n K + 4Rα(k 1)/n This proves the desired result. ( f ( t, n (t) ) f ( t, z(t) )) dt ( f ( t, n (t) ) f ( t, z(t) )) dt Lemma 4 Suppose f :, 0] R n R and the hypotheses of Lemma 3 hold. Then the following are equivalent. (i) For all y C, 0] f ( t, y(t) ) dt 0 (ii) There eists g :, 0] R which is piecewise continuous and satisfies f(t, z) + g(t) 0 g(t) dt 0 for all t, z PROOF. Again we only need to show that (i) implies (ii). Suppose (i) holds, then inf f ( t, y(t) ) dt 0 y C,0] and hence by Lemma 3 we have r(t) dt 0 where r :, 0] R n is given by r(t) inf f(t, ) for all t 4

5 The function r is continuous on H c since f is continuous on H c R n. Hence by Lemma 2, there eists g such that condition (ii) holds, as desired. Remark 5 Lemma 4 is stated in a way which applies to general functions given specific conditions are satisfied. The following main result shows that these conditions are satisfied in the quadratic case. Theorem 6 Suppose M :, 0] S m+n is piecewise continuous, and there eists ε > 0 such that for all t, 0] we have M 22 (t) ε M(t) ε 1 Then the following are equivalent. (i) For all R m and continuous y :, 0] R n y(t) M y(t) ] dt 0 (8) (ii) There eists a function T :, 0] S m which is piecewise continuous and satisfies T (t) 0 M(t) + 0 for all t, 0] 0 0 T (t) dt 0 PROOF. Again we only need to show (i) implies (ii). Suppose R n, and define f(t, z) z M(t) z for all t, z Since by the hypothesis M 22 has a lower bound, it is invertible for all t and its inverse is piecewise continuous. Therefore z(t) M 22 (t) 1 M 21 (t) is the unique minimizer of f(t, z) with respect to z. By the hypothesis (i), we have that for all y C, 0] f ( t, y(t) ) dt 0 Hence by Lemma 4 there eists a function g such that g(t) + f(t, z) 0 g(t) dt 0 for all t, z (9) The proof of Lemma 2 gives one such function as We have g(t) f ( t, z(t) ) + f(s, z(s)) dt f ( t, z(t) ) T ( M 11 (t) M 12 (t)m 1 22 (t)m 21(t) ) and therefore g(t) is a quadratic function of, say g(t) T T (t), and T :, 0] S m is continuous on H c. Then equation (9) implies T T (t) + as required. z M(t) 0 z for all t, z, We have now shown that the conve cone of functions M such that the first term of (4) is nonnegative is eactly equal to the sum of the cone of pointwise nonnegative functions and the linear space of functions whose integral is zero. Note that in (8) the vectors and y are allowed to vary independently, whereas (4) requires that y(0). t is however straightforward to show that this additional constraint does not change the result, using the technique in the proof of Lemma 3. The key benefit of this is that it is easy to parametrize the latter class of functions, and in particular when M is a polynomial these constraints are semidefinite representable constraints on the coefficients of M. 4 Positivity of Double ntegrals The previous section dealt only with the first part of the functional. We now give results which allow us to treat the second part of the functional. We first define the following functions. Let Z d be the vector of monomials in variable y of degree d or less, for eample Define the function Z n d by ] Z 4 (y) T 1 y y 2 y 3 y 4 Z n d (y) Z d (y) This means the entries of Zd n are monomials in y. The rows of Zd n define a basis for the degree d polynomials in n dimensions. Theorem 7 Suppose N : R 2 R n n is a polynomial of degree 2d. Then the following are equivalent: 5

6 (1) The following hold for all C (s)n(s, t)(t)dsdt 0 (2) There eists a Q 0 such that N(s, t) + N(t, s) T Z n d (s) T QZ n d (t) PROOF. That 2 implies 1 is clear. Now suppose N is a polynomial of degree d, then N can be represented as N(s, t) Now we have 2d i,j0 A ij s i t j Z n 2d(s) T AZ n 2d(t). N(s, t) + N(t, s) T Z n 2d(s) T (A + A T )Z n 2d(t). Let Q A + A T. Now suppose there eists some vector u such that u T Qu < 0. Let B Z2d(s)Z n 2d(s) n T ds. Since the rows of Z2d n are linearly independent functions, we have that B is invertible. Now let (s) Z 2d (s) T B 1 u. Then we have the following. 2 (s) T N(s, t)(t)dsdt (s) T (N(s, t) + N(t, s) T )(t)dsdt (s) T Z2d(s) n T (A + A T )Z2d(t)(t)dsdt n (Z2d(s)Z n 2d(s) n T B 1 u) T dsq Z2d(t)Z n 2d(t) n T B 1 udt (BB 1 u) T QBB 1 u u T Qu < 0 Thus we have by contradiction that Q 0. Now, divide Q as Q11 Q 12 Q, Q T 12 Q 22 where Q 22 R d d. Then Q 22 0 since N is of degree 2d. Therefore Q 12 Q T 12 0 since Q 0. Thus Q 11 0 and N(s, t) + N(t, s) T Z n d (s) T Q 11 Z n d (t) 4.1 Piecewise-Continuous Kernels We now deal with piecewise-continuous functions. First, some notation. Define the intervals { 1, 0] if i 1 H i i, i 1 ) if i 2,..., k Let, 0] and define the vector of indicator functions g : R k by g i (t) { 1 if t Hi 0 otherwise for all i 1,..., k and all t, 0]. Define a i and b i to be the scalars such that {a i s + b i : s, 0]} h i, h i 1 ] for i 1,..., k. Lemma 8 Suppose the function N : R 2 R n n is continuous on H c H c. Define N ij (s, t) N(s, t)g i (s)g j (t). Suppose R : R 2 R nk nk is given by R(s, t) N ij (a i s + b i, a j t + b j ) e i e T j i,j1 N 11 N 12 N 1k N N k1 N kk where e i are the unit vectors in R k. Then N defines a positive integral operator if and only if R defines a positive integral operator. PROOF. Suppose R defines a positive integral operator. Let i (s) g i (s)(s) and i (s) a i i (a i s+b i ). Let s i a i s + b i and t i a i t + b i. Then since R is positive, we have (s) T N(s, t)(t)dsdt i,j1 i,j1 H i H j i (s i ) T N ij (s i, t j ) j (t j )ds i dt j i (s) T N ij (a i s + b i, a j t + b j ) j (t)dsdt (s) T R(s, t) (t)dsdt 0 since is continuous on. 6

7 For the converse, suppose N defines a positive operator, then for any C, partition as T 1,, T k ] T. and define i (s) i (s/a i b i /a i )/a i and (s) i (s)g i (s). i1 Then (s) is continuous on H c and (s) T R(s, t)(t)dsdt i,j1 i (s) T N ij (a i s + b i, a j t + b j ) j (s)dsdt i,j1 i,j1 i,j1 H i H i H i i (s i ) T N ij (s i, t j ) j (t j )ds i dt j H j i (s) T g i (s)n(s, t)g j (t) j (t)dsdt H j (s) T N(s, t) (t)dsdt H j (s) T N(s, t) (t)dsdt 0. The last follows since it can be shown that positivity for continuous functions on, 0] implies positivity for continuous functions on H c. Theorem 9 Suppose the function N : R 2 R n n is piecewise-continuous with N ij polynomials of degree 2d. Then the following are equivalent: (1) For all C, (s)n(s, t)(t)dsdt 0 (2) There eists a Q 0 such that if Q ij is the i, jth block of Q, then N ij (s, t) Z n d (s) T Q ij Z n d (t) PROOF. Suppose N satisfies condition 1. Let R be given by R(s, t) N ij (a i s + b i, a j t + b j ) e i e T j i,j1 By Lemma 8 and Theorem 7, for some Q 0 and so N ij (s, t) Z n d R(s, t) Zd nk (s) nk QZ d (t) ( s b ) T ( i t QijZ d n b ) j, a i a i a j a j where Q ij is the i, jth block of Q. Now Define T i to be the matri such that ( s Zd n b ) i T i Zd n (s). a i a i Eistence and invertibility of T i can be shown by simple construction. Define T T i e i e T i i1 where e i are the unit vectors in R k. Then T i are the block-diagonal elements of T. Let Q T T QT. Then Q 0, Q ij Ti T Q ij T j and N ij (s, t) Z n d ( s a i b i a i ) T ( t QijZ d n Zd n (s) T Ti T Q ij T j Zd n (t) Zd n (s) T Q ij Zd n (t). Therefore, we have condition 2. a j b j a j Now suppose that condition 2 holds. Let Q T T QT 1 0 and define Then R(s, t) Zd nk i,j1 R(s, t) Z nk d (s) T nk QZ d (t). (s) T T T QT 1 Zd nk (t) Z n d (s) T T T i Q ij T 1 j Zd n (t) e i e T j Zd n (a i s + b i ) T Q ij Zd n (a j t + b j ) e i e T j i,j1 N ij (a i s + b i, a j t + b j ) e i e T j i,j1 Therefore, by Theorem 7 and Lemma 8, N defines a positive operator and thus condition 2 holds. ) 7

8 5 Lie Derivatives 5.1 Single Delay Case We first present the single delay case, as it will illustrate the formulation in the more complicated case. Suppose that V X is given by (4), where M :, 0] S 2n and N :, 0], 0] R n n. Since there is only one delay, if the system is eponentially stable then there always eists a Lyapunov function of this form with C 1 functions M and N. Then the Lie derivative of V is V (φ) φ() + Partition D and M as M(t) T D(s) φ() ds M11 M 12 (t) M 21 (t) M 22 (t) T E(s, t)φ(t) ds dt (10) D(t) D11 D 12 (t) D 21 (t) D 22 (t) so that M 11 S n and D 11 S 2n. Without loss of generality we can assume M 11 and D 11 are constant. The functions D and E are linearly related to M and N by D 11 AT 0 M 11 + M 11 A 0 M 11 A 1 A T 1 M M 12(0) + M 21 (0) M 12 () h M 21 () 0 M22 (0) h 0 M 22 () D 12 (t) AT 0 M 12 (t) Ṁ12(t) + N(0, t) A T 1 M 12 (t) N(, t) D 22 (t) Ṁ22(t) E(s, t) N(s, t) s 5.2 Multiple-delay case + N(s, t) t We now define the class of functions under consideration for the Lyapunov functions. Define the intervals { 1, 0] if i 1 H i i, i 1 ) if i 2,..., k For the Lyapunov function V, define the sets of functions Y 1 { M :, 0] S 2n M 11 (t) M 11 (s) for all s, t, 0] } M is C 1 on H i { for all i 1,..., k Y 2 N :, 0] S n N(s, t) N(t, s) T for all s, t, 0] } N is C 1 on H i H j for all i, j 1,..., k and for its derivative, define Z 1 { D :, 0] S (k+2)n D ij (t) D ij (s) for all s, t, 0] for i, j 1,..., 3 D is C 0 on H i { Z 2 E :, 0] S n for all i 1,..., k E(s, t) E(t, s) T for all s, t, 0] } E is C 0 on H i H j for all i, j 1,..., k Here D Z 1 is partitioned according to D 11 D 12 D 13 D 14 (t) D 21 D 22 D 23 D 24 (t) D(t) D 31 D 32 D 33 D 34 (t) D 41 (t) D 42 (t) D 43 (t) D 44 (t) } (11) where D 11, D 33, D 44 S n and D 22 S (k 1)n. Let Y Y 1 Y 2 and Z Z 1 Z 2. Notice that if M Y 1, then M need not be continuous at h i for 1 i k 1, however, it must be right continuous at these points. We also define the derivative Ṁ(t) at these points to be the right-hand derivative of M. We define the continuity and derivatives of functions in Y 2, Z 1 and Z 2 similarly. We define the jump values of M and N at the discontinuities as follows. M(h i ) M( i +) M( i ) for each i 1,..., k 1, and similarly define N(h i, t) N( i +, t) N( i, t) Definition 10 Define the map L : Y Z by (D, E) 8

9 L(M, N) if for all t, s, 0] we have D 11 A T 0 M 11 + M 11 A ( M12 (0) + M 21 (0) + M 22 (0) ) h ] D 12 M 11 A 1... M 11 A k 1 ] M 12 (h 1 )... M 12 (h k 1 ) D 13 1 ( M11 A k M 12 () ) h D 22 1 h diag( M 22 (h 1 ),..., M 22 (h k 1 ) ) D 23 0 D 33 1 h M 22() D 14 (t) N(0, t) + A T 0 M 12 (t) Ṁ12(t) N( 1, t) + A T 1 M 12 (t) D 24 (t). N( k 1, t) + A T k 1 M 12(t) D 34 (t) A T k M 12 (t) N(, t) D 44 (t) Ṁ22(t) and N(s, t) N(s, t) E(s, t) + s t Here D is partitioned as in (11) and the remaining entries are defined by symmetry. The map L is the Lie derivative operator applied to the set of functions specified by (4); this is stated precisely below. Notice that this implies that L is a linear map. Lemma 11 Suppose M Y 1 and N Y 2 and V is given by (4). Let (D, E) L(M, N). Then the Lie derivative of V is given by V (φ) φ( 0 ). φ( k ) + T φ( 0 ). D(s) ds φ( k ) T E(s, t)φ(t) ds dt (12) PROOF. The proof is straightforward by differentiation and integration by parts of (4). 6 Matri Sums of Squares n this paper we use polynomial matrices as a conveniently parametrized class of functions to represent the functions M and N defining the Lyapunov function (4) and its derivative. Theorem 6 has reduced nonnegativity of the first term of (4) to pointwise nonnegativity of a matri polynomial in one variable. A matri polynomial in one variable is pointwise nonnegative semidefinite if and only if it is a sum of squares; see Choi, Lam, and Reznick 1]. We now briefly describe the matri sum-ofsquares construction. Suppose z is a vector of N monomials in variables y, for eample ] z(y) T 1 y 1 y 1 y2 2 y3 4 For each y, we can define K(y) : R n R Nn by ( K(y) ) z(y) This means K(y) z(y), and the entries of K are monomials in y. Now suppose U S Nn is a symmetric matri, and let M(y) K T (y)uk(y) Then each entry of the matri M is a polynomial in y. The matri M is called a matri sum-of-squares if U is positive semidefinite, in which case M(y) 0 for all y. Given a matri polynomial G(y), we can test whether it is a sum-of-squares by finding a matri U such that G(y) K T (y)uk(y) (13) U 0 (14) Equation (13) is interpreted as equating the two polynomials G and K T UK. Equating their coefficients gives a family of linear constraints on the matri U. Therefore to find such a U we need to find a positive semidefinite matri subject to linear constraints, and this is therefore a semidefinite program. See Scherer and Hol 16] and Kojima 7] for more details on matri sums-of-squares and their properties, and Vandenberghe and Boyd 19] for background on semidefinite programming. 6.1 Piecewise matri sums-of-squares. Recall we define the vector of indicator functions g :, 0] R k by g i (t) { 1 if t Hi 0 otherwise for all i 1,..., k and all t, 0]. Now let z(t) be the vector of monomials ] z(t) T 1 t t 2... t d 9

10 Define the function Z n,d :, 0] R (d+1)kn n by Z n,d (t) z(t) g(t) Then, define the sets P n,d and Σ n,d by P n,d { Z T n,d(t)uz n,d (t) U S (d+1)nk } Σ n,d { Z T n,d(t)uz n,d (t) U S (d+1)nk, U 0 } These sets are defined so that if M P n,d then each entry of the n n matri M is a piecewise polynomial in t. f M Σ n,d, then it is called a piecewise matri sum-ofsquares. We will omit the subscripts n, d where possible in order to lighten the notation. f we are given a function G :, 0] R n which is piecewise polynomial, and polynomial on each interval H i, then it is a piecewise sum of squares of degree 2d if and only if there eists a matri U such that G(t) Z T n,d(t)uz n,d (t) (15) U 0 (16) which, as for the non-piecewise case discussed above is a semidefinite program. 6.2 Kernel functions. We can also consider functionals of the form of the second term of the Lyapunov function (4) using the notation presented here. We define the set of piecewise polynomial positive kernels in two variables s, t by Γ n,d { Z T n,d(s)uz n,d (t) U S (d+1)nk, U 0 } For any function N Γ n,d, by Theorem 9, we have for all φ C(, 0], R n ). T N(s, t)φ(t) ds dt Stability Conditions Theorem 12 Suppose there eist d N and piecewise matri polynomials M, T, N, D, U, E such that T 0 M + Σ 2n,d 0 0 U 0 D + Σ (k+2)n,d 0 0 N Γ n,d E Γ n,d (D, E) L(M, N) T (s) ds 0 U(s) ds 0 M 11 0 D 11 0 Then the system defined by equations (2) and (3) is eponentially stable. PROOF. Assume M, T, N, D, U, E satisfy the above conditions, and define the function V by (4). Then Lemma 11 implies that V is given by (12). The function V is the sum of two terms, each of which is nonnegative. The first is nonnegative by Theorem 6 and the second is nonnegative since N Γ n,d. The same is true for V. The strict positivity conditions of equations (5) hold since M 11 > 0 and D 11 < 0, and Theorem 1 then implies stability. Remark 13 The feasibility conditions of Theorem 12 are semidefinite-representable. n particular the condition that a piecewise polynomial matri lie in Σ is a set of linear and positive semidefiniteness constraints on its coefficients. Similarly, the condition that T and U integrate to zero is simply a linear equality constraint on its coefficients. Standard semidefinite programming codes may therefore be used to efficiently find such piecewise polynomial matrices. Most such codes will also return a dual certificate of infeasibility if no such polynomials eist. As in the Lyapunov analysis of nonlinear systems using sum-of-squares polynomials, the set of candidate Lyapunov functions is parametrized by the degree d. This allows one to search first over polynomials of low degree, and increase the degree if that search fails. There are various natural etensions of this result. The first is to the case of uncertain systems, where we would like to prove stability for all matrices A i in some given 10

11 semialgebraic set. This is possible by etending Theorem 12 to allow Lyapunov functions which depend polynomially on unknown parameters. A similar approach may be used to check stability for systems with uncertain delays. t is also straightforward to etend the class of Lyapunov functions, since it is not necessary that each piece of the piecewise sums-of-squares functions be nonnegative on the whole real line. To do this, one can use techniques for parameterizing polynomials nonnegative on an interval; for eample, every polynomial p() f() ( 1)( 2)g() where f and g are sums of squares is nonnegative on the interval 1, 2]. 7 Numerical Eamples 7.1 Eample with a Single Delay n this eample, we compare our results with the discretized Lyapunov functional approach used by Gu et al. in 4] in the case of a system with a single delay. Although numerous other papers have also given sufficient conditions for stability of time-delay systems, e.g. 3, 10, 11], the approach introduced by Gu has demonstrated a particularly high level of precision. When we are comparing results we will only consider eamples which have been presented in the literature. We use SOSTOOLS 14] and SeDuMi 17] for solution of all semidefinite programming problems. Consider the following system of delay differential equations ẋ(t) (t) + (t τ) The problem is to estimate the upper and lower bounds on τ for which the differential equation remains stable. Using the algorithm presented in this paper and by gridding the parameter space, we can determine the approimate region of stability. We then use a bisection method to find the minimum and maimum values of this region. Our results are summarized in Table 1 and are compared to the analytical limit and to the results obtained in 4]. Our results d τ min τ ma Analytic Piecewise Functional N 2 τ min τ ma Table 1 τ ma and τ min for discretization level N 2 using the piecewiselinear Lyapunov functional and for degree d using our approach and compared to the analytical limit The results for this test case illustrate a high rate of convergence of the accuracy as the parameter d increases. Note that the algorithm only proves stability at the input points. t makes no claim for points not tested. To provide a stability proof over certain parameter regions, the parameter-dependent algorithm should be used. n the net eamples, we now include τ as an uncertain parameter and search for parameter-dependent Lyapunov functionals which prove stability over the region. The proposed regions are based on data from the deterministic algorithm. Results from this test are given in Table 2. d in τ d in θ τ min τ ma Analytic Table 2 Stability on the interval τ min, τ ma] vs. degree using a parameter-dependent functional 7.2 Eample with Multiple Delays Consider the following system of delay-differential equations ẋ(t) (t) + (t τ ) + (t τ) Again, the problem is to estimate the stable region. These results are summarized in Table 3. For the piecewise functional method, N 2 is the level of both discretization and subdiscretization. Our Approach d τ min τ ma Analytic Piecewise Functional N 2 τ min τ ma Table 3 τ ma and τ min using the piecewise-linear Lyapunov functional of Gu et al. and our approach and compared to the analytical limit 7.3 Eample with Parametric Uncertainty n this eample, we illustrate the fleibility of our algorithm with a simplistic control design and analysis problem. Suppose we are controlling a simple inertial mass remotely using a PD controller. Now suppose that the derivative control is half of the proportional control. Then we have the following dynamical system. ẍ(t) a(t) a 2 ẋ(t) 11

12 t is easy to show that this system is stable for all positive values of a. However, because we are controlling the mass remotely, some delay will be introduced due to, for eample, the fied speed of light. We assume that this delay is known slowly varying. Now we have the following delay-differential equation with uncertain, timeinvariant parameters a and τ. ẍ(t) a(t τ) a ẋ(t τ) 2 Whereas before the system was stable for all positive values of a, now, for any fied value of a, there eists a τ for which the system is unstable. n order to determine which values of a are stable for any fied value of τ, we apply the algorithm from this paper to a gridded parameter space. Based on these results, we propose stability regions of the form a a min, a ma ] and τ τ min, τ ma ]. This type of region is compact and can be represented as a semi-algebraic set using the polynomials p 1 (a) (a a min )(a a ma ) and p 2 (τ) (τ τ min )(τ τ ma ). By now applying the parameter-dependent version of the algorithm, we are able to construct Lyapunov functionals which prove stability over a number of parameter regions. These regions are illustrated in Figure 1. delay(s) PD control with delay - stable control gain(a) Fig. 1. Regions of Stability for Eample 2 8 Conclusion The general question of stability of linear differential equations with delay is NP-hard 18]. n this paper, we have shown that the question of stability can be epressed as a conve optimization problem within the function space C0, 1]. Furthermore, through the use of polynomial programming techniques and projections from C0, 1] onto the space of polynomials, we have shown how to use semidefinite programming to compute solutions to this optimization problem. Thus we have created a sequence of sufficient conditions for stability of linear time-delay systems, indeed by polynomial degree, d, and whose compleity is of order k(n(d + 1)(k + 2)) 2. As d, the validity of the polynomial approimation increases and the conditions become increasingly accurate, as adequately illustrated by the numerical eamples. Aside from the accuracy of the results, the approach taken in this paper has an important advantage. Because of the fleibility of the polynomial approach, we are able to make various etensions of our results to other problems which can be epressed using conve optimization in C0, 1]. Specifically, in this paper we have addressed the problem of systems with parametric uncertainty using a variant of the S-procedure. Additionally, etension to nonlinear systems with time-delay is easily addressed, although in this case conservatism arises from a number of sources. A more etensive treatment of nonlinear time-delay systems is the topic of another paper currently in preparation. We conclude by mentioning that the methods of this paper would seem to allow us to synthesize stabilizing controllers for linear time-delay systems. This observation is motivated by viewing the functions M and R as defining a full rank operator on L 2 0, 1]. We have observed that for given functions, the inverse of such an operator can be computed numerically. By computing Lyapunov functionals for the adjoint system constructed by Delfour and Mitter 2], this invertibility result seems to imply that one can construct stabilizing controllers. This work is ongoing. References 1] M.-D. Choi, T.-Y. Lam, and B. Reznick. Real zeros of positive semidefinite forms. i. Mathematische Zeitschrift, 171:1 26, ] M. C. Delfour and S. K. Mitter. Hereditary differential systems with constant delays.. general case. Journal of Differential Equations, 12: , ] E. Fridman and U. Shaked. An improved stabilization method for linear time-delay systems. EEE Transactions on Automatic Control, 47(11): , ] K. Gu, V. L. Kharitonov, and J. Chen. Stability of Time-Delay Systems. Birkhäuser, ] J. K. Hale and S. M. Lunel. ntroduction to Functional Differential Equations, volume 99 of Applied Mathematical Sciences. Springer-Verlag, ] V. L. Kharitonov and D. Hinrichsen. Eponential estimates for time delay systems. Systems and control letters, 53(5): , ] M. Kojima. Sums of squares relaations of polynomial semidefinite programs. Research Report B- 397, Department of Mathematical and Computing Sciences, Tokyo nstitute of Technology, ] V. Kolmanovskii and A. Myshkis. ntroduction to the Theory and Applications of Functional Differential Equations. Kluwer Academic Publishers, ] N. N. Krasovskii. Stability of Motion. Stanford University Press, ] C. Y. Lu, J. S.-H. Tsai, and T. J. Su. On improved delay-dependent robust stability criteria for uncertain systems with multiple-state delays. EEE 12

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