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Control System Analysis on Symmetric Cones

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Control System Analysis on Symmetric Cones Submitted, 2015 Conference on Decision and Control (CDC) http://www.cds.caltech.edu/~murray/papers/pm15-cdc.html Control System Analysis on Symmetric Cones Ivan Papusha Richard M. Murray Abstract— Motivated by the desire to analyze high dimen- Autonomous systems for which A is a Metzler matrix are sional control systems without explicitly forming computation- called internally positive,becausetheirstatespacetrajec- ally expensive linear matrix inequality (LMI) constraints, we tories are guaranteed to remain in the nonnegative orthant seek to exploit special structure in the dynamics matrix. By Rn using Jordan algebraic techniques we show how to analyze + if they start in the nonnegative orthant. As we will see, continuous time linear dynamical systems whose dynamics are the Metzler structure is (in a certain sense) the only natural exponentially invariant with respect to a symmetric cone. This matrix structure for which a linear program may generically allows us to characterize the families of Lyapunov functions be composed to verify stability. However, the inclusion that suffice to verify the stability of such systems. We highlight, from a computational viewpoint, a class of systems for which LP ⊆ SOCP ⊆ SDP stability verification can be cast as a second order cone program (SOCP), and show how the same framework reduces to linear motivates us to determine if stability analysis can be cast, for programming (LP) when the system is internally positive, and to semidefinite programming (SDP) when the system has no example, as a second order cone program (SOCP) for some special structure. specific subclass of linear dynamics, in the same way that it can be cast as an LP for internally positive systems, or an I. INTRODUCTION SDP for general linear systems. Using Jordan algebraic techniques, which have proved to In this paper we wish to verify the stability of a linear be effective in unifying interior point methods for conic dynamical system programming, we will discuss a generalization of the Metzler x˙(t)=Ax(t),A∈ Rn×n,x(t) ∈ Rn. (1) property known as cross-positivity to characterize dynamics that are exponentially invariant with respect to a symmetric It is well known that if A has no special structure, then cone. In so doing, we demonstrate the strong unifying power system (1) is stable if and only if there exists a symmetric of the Jordan product on linear systems and discuss a rich × matrix P = P T ∈ Rn n satisfying the linear matrix class of systems that admit SOCP-based analysis. inequalities II. JORDAN ALGEBRAS AND SYMMETRIC CONES P = P T ≻ 0,AT P + PA ≺ 0. (2) The following background is informal and mostly meant Such a matrix P corresponds to a quadratic Lyapunov to set out notation, adopting conventions familiar to system function V : Rn → R, theory and optimization [4], [5]. Jordan algebras in the context of symmetric cones are rich and well studied field, V (x)=⟨x, P x⟩, with an excellent reference [6]. For more recent reviews, which is positive definite and decreasing along trajectories see [7], [8], [9]. of (1). A. Cones on a vector space. Searching for a matrix P satisfying (2) is in general a semidefinite program (SDP), but if A has a special structure, Let V be a vector space with inner product ⟨·, ·⟩.Asubset this semidefinite program can sometimes be cast as a lin- K ⊆ V is called a cone if it is closed under nonnegative ear program (LP)—with advantages in numerical stability, scalar multiplication: for every x ∈ K and θ ≥ 0 we have opportunities for parallelism, and better scaling to high θx ∈ K.AconeK is pointed if it contains no line, or dimensions than the general SDP. For example, if A is a equivalently Metzler matrix, i.e.,itsoff-diagonalentriesarenonnegative, x ∈ K, −x ∈ K =⇒ x =0. Aij ≥ 0, for all i ̸= j, Aconeisproper if it is closed, convex, pointed, and has then it suffices to search for a diagonal matrix P satis- nonempty interior. Every proper cone induces a partial order fying (2). Equivalently, it is enough to find an entrywise ≼ on V given by int Rn int Rn positive vector p ∈ + such that −Ap ∈ + is entrywise positive. Finding such a vector p is, of course, an x, y ∈ V, x ≼ y ⇐⇒ y − x ∈ K. LP, [1], [2], [3]. We have x ≺ y if and only if y − x ∈ int K.Similarly The authors are with the Control and Dynamical Systems Department, we write x ≽ y and x ≻ y to mean y ≼ x and y ≺ x, California Institute of Technology, Pasadena, CA USA. respectively. B. Jordan algebras. decomposition, and are (in finite dimensions, [6]) isomorphic A(Euclidean)Jordanalgebraisaninnerproductspace to a Cartesian product of (V,⟨·, ·⟩) endowed with a Jordan product ◦ : V × V → V , • n × n self-adjoint positive semidefinite matrices with which satisfies the following properties: real, complex, or quaternion entries, 1) Bilinear: x ◦ y is linear in x for fixed y and vice-versa • 3 × 3 self-adjoint positive semidefinite matrices with 2) Commutative: x ◦ y = y ◦ x octonion entries (Albert algebra), and 3) Jordan identity: x2 ◦ (y ◦ x)=(x2 ◦ y) ◦ x • Lorentz cone. 4) Adjoint identity: ⟨x, y ◦ z⟩ = ⟨y ◦ x, z⟩. In practice, symmetric cones have a differentiable log-det In particular, a Jordan product need not be associative. When barrier function, allowing numerical optimization via interior 2 Rn S1 we interpret x = x◦x,theJordanidentityallowsanypower point methods [7], [10]. The symmetric cones +(= + × k S1 n Sn x , k ≥ 2 to be inductively defined. An identity element e ··· × +), L+ and + give rise to LP, SOCP, and SDP, satisfies e ◦ x = x ◦ e = x for all x ∈ V ,anddefinesa respectively. We now define these three cones. number r = ⟨e, e⟩ called the rank of V . D. Three important symmetric cones. The cone of squares K corresponding to the Jordan Rn product ◦ is defined as 1) Nonnegative orthant +: the cone of squares asso- ciated with the standard Euclidean space Rn and Jordan K = {x ◦ x | x ∈ V }. product Every element x ∈ V has a spectral decomposition (x ◦ y)i = xiyi,i=1,...,n, r i.e.,theentrywise(orHadamard)product.Theidentityele- x = λ e , i i ment is e = 1 =(1,...,1),andtheJordanframecomprises !i=1 the standard basis vectors. The quadratic representation of where λ ∈ R are eigenvalues of x and the set of eigenvec- n i an element z ∈ R is given by the diagonal matrix Pz = tors {e1,...,er}⊆V ,calledaJordan frame,satisfies diag(z)2 r 2 e = ei,ei ◦ ej =0for i ̸= j, ei = e. i x !i=1 0 We have x ∈ K (written x ≽K 0,orsimplyx ≽ 0 when the context is clear) provided λi ≥ 0.Similarly,x ∈ int K if and only if λi > 0 (written x ≻K 0). The spectral decomposition allows us to define familiar concepts like trace, determinant, and square root of an element x by taking the corresponding function of the eigenvalue. That is, x1 x2 r r r tr det 1/2 1/2 Fig. 1. Second-order (Lorentz) cone L3 . x = λi, x = λi,x = λi ei, + !i=1 i"=1 !i=1 n Rn where the last quantity only makes sense if x ≽ 0. 2) Lorentz cone L+: partition every element of as R Rn−1 Finally, every element z ∈ V has a quadratic representa- x =(x0,x1) ∈ × and define the Lorentz cone (also known as second-order or norm cone) as tion, which is a map Pz : V → V given by 2 n R Rn−1 Rn Pz(x)=2(z ◦ (z ◦ x)) − z ◦ x. L+ = {(x0,x1) ∈ × |∥x1∥2 ≤ x0}⊆ . C. Symmetric cones. This cone is illustrated in Figure 1 for the case n =3.It n The closed dual cone of K is defined as can be shown that L+ is the cone of squares corresponding to the Jordan product K∗ = {y ∈ V |⟨x, y⟩≥0, for all x ∈ K}, x0 y0 ⟨x, y⟩ ∗ ◦ = . AconeK is self-dual if K = K.Theautomorphismgroup #x1$ #y1$ #x0y1 + y0x1$ Aut(K) of an open convex cone K is the set of invertible linear transformations g : V → V that map K to itself, The rank of this algebra is 2, giving a particularly simple spectral decomposition Aut(K)={g ∈ GL(V ) | gK = K}. x = λ1e1 + λ2e2, An open cone K is homogeneous if Aut(K) acts on the cone transitively, in other words, if for every x, y ∈ K where the eigenvalues and Jordan frame are there exists g ∈ Aut(K) such that gx = y.Finally,K λ = x + ∥x ∥ ,λ= x −∥x ∥ , is symmetric if it is homogeneous and self-dual. Symmetric 1 0 1 2 2 0 1 2 1 1 1 1 cones are an important object of study because they are e1 = ,e2 = . the cones of squares of a Jordan product, admit a spectral 2 #x1/∥x1∥2$ 2 #−x1/∥x1∥2$ T The identity element is e =(1, 0) and the quadratic repre- • Lorentz cone: L(x)=Ax,whereA Jn + JnA ≽ ξJn sentation of an element z ∈ Rn is the matrix for some ξ ∈ R. T zT J z • Positive semidefinite matrices: L(X)=AX +XA for P = zzT − n J , z 2 n any matrix A. In fact, due to a result in [13], the examples above precisely where J =diag(1, −1,...,−1) ∈ Rn×n is a n × n inertia n characterize cross-positive operators on Rn and Ln (but not matrix with signature (1,n− 1).NotethatP is a positive + + z Sn ). semidefinite matrix if and only if z ≽ 0. + Sn 3) Positive semidefinite cone +: consider the vector B. A class of Lyapunov functions space Sn of real, symmetric n × n matrices with the trace inner product.
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