Analysis of Control Systems on Symmetric Cones

Ivan Papusha Richard M. Murray

Abstract— It is well known that exploiting special structure is Finding a vector p ∈ Rn that satisfies this second condi- a powerful way to extend the reach of current optimization tools tion (3) can be formulated as an LP with fewer decision to higher dimensions. While many linear control systems can be variables than the corresponding SDP [1], [2], [3]. treated satisfactorily with linear matrix inequalities (LMI) and semidefinite programming (SDP), practical considerations can Systems for which A is a Metzler matrix are called in- still restrict scalability of general methods. Thus, we wish to ternally positive, because their state stays in the nonnegative Rn work with high dimensional systems without explicitly forming orthant + if it starts in the nonnegative orthant. As we SDPs. To that end, we exploit a particular kind of structure will see, the Metzler structure is (in a certain sense) the only in the dynamics matrix, paving the way for a more efficient natural matrix structure for which an LP may generically be treatment of a certain class of linear systems. We show how second order cone programming (SOCP) can be used instead composed to verify stability. However, the inclusion of SDP to find Lyapunov functions that certify stability. This LP ⊆ SOCP ⊆ SDP framework reduces to a famous linear program (LP) when the system is internally positive, and to a semidefinite program interpreted as an expressiveness ranking of popular conic (SDP) when the system has no special structure. programming methods, begs the question of whether stability analysis can be cast, for example, as a second order cone pro- I. INTRODUCTION gram (SOCP) for some specific subclass of linear dynamics, This paper is concerned with stability of the linear dynam- in the same way that it can be cast as an LP for internally ical system positive systems, and an SDP for unstructured linear systems. x˙(t)= Ax(t), A ∈ Rn×n, x(t) ∈ Rn, (1) More generally, we wish to know when certain conic programming techniques can be used to verify stability of under certain conditions on the dynamics matrix A. When certain linear systems. In answering this question we will this matrix has no special structure, the system is stable if discuss a generalization of the Metzler property known and only if there exists a symmetric matrix P = P T ∈ Rn×n as cross-positivity and the notion of a Jordan algebra to satisfying the linear matrix inequalities characterize dynamics that are exponentially invariant with

T T respect to a symmetric cone. In so doing, we demonstrate P = P ≻ 0, A P + P A ≺ 0. (2) the strong unifying power of the Jordan product on linear The existence of such a matrix P corresponds to the exis- systems and discuss a rich, little-known class of systems that tence of a quadratic Lyapunov function V : Rn → R, admit SOCP-based analysis. II. JORDAN ALGEBRAS AND SYMMETRIC CONES V (x)= hx,Pxi, The following background is informal and meant to set out which is positive definite (V (x) > 0 for all x 6= 0) and notation, adopting conventions familiar to system theory and decreasing (V˙ < 0 along trajectories of (1)). optimization [4], [5], [6]. Jordan algebraic techniques have To find a matrix P that satisfies (2), one generally needs to proved to be effective in unifying interior point methods for formulate and solve a semidefinite program (SDP), however, conic programming [7]. In the context of symmetric cones, if A has special structure, this semidefinite program can they are are rich and well studied field, with an excellent sometimes be cast as a simpler linear program (LP)—with reference [8]. For more recent reviews, see [9], [10]. advantages in numerical stability, opportunities for paral- lelism, and better scaling to high dimensions (n ≫ 1). A. Cones on a vector space. For example, if A is a given Metzler matrix, i.e., its off- Let V be a vector space over the reals with inner product diagonal entries are nonnegative, h·, ·i. A subset K ⊆ V is called a cone if it is closed under nonnegative scalar multiplication: for every x ∈ K and θ ≥ 0 Aij ≥ 0, for all i 6= j, we have θx ∈ K. A cone K is pointed if it contains no line, or equivalently then it suffices to search for a diagonal matrix P satisfy- ing (2). Equivalently, condition (2) can be replaced with the x ∈ K, −x ∈ K =⇒ x = 0. simpler vector condition A cone is proper if it is closed, convex, pointed, and has pi > 0, (Ap)i < 0, for all i = 1,...,n. (3) nonempty interior. Every proper cone induces a partial order  on V given by The authors are with the Control and Dynamical Systems Department, California Institute of Technology, Pasadena, CA USA. x,y ∈ V, x  y ⇐⇒ y − x ∈ K. We have x ≺ y if and only if y − x ∈ int K. Similarly there exists g ∈ Aut(K) such that gx = y. Finally, K we write x  y and x ≻ y to mean y  x and y ≺ x, is symmetric if it is homogeneous and self-dual. Symmetric respectively. cones are an important object of study because they are the cones of squares of a Jordan product, admit a spectral B. Jordan algebras. decomposition, and are (in finite dimensions, [8]) isomorphic A (Euclidean) Jordan algebra is an inner product space to a Cartesian product of (V, h·, ·i) 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) Bilinearity: x◦y is linear in x for fixed y and vice-versa • 3 × 3 self-adjoint positive semidefinite matrices with 2) Commutativity: x ◦ y = y ◦ x octonion entries (Albert algebra), and 2 2 3) Jordan identity: x ◦ (y ◦ x)=(x ◦ y) ◦ x • Lorentz cone. 4) Adjoint identity: hx,y ◦ zi = hy ◦ x,zi. In this work, we pay attention to symmetric cones because Note that a Jordan product is commutative, but it need not be they have a differentiable log-det barrier function, allowing 2 associative. When we interpret x = x◦x, the Jordan identity numerical optimization with interior point methods [7], [11]. k allows any power x , k ≥ 2 to be inductively defined. An The symmetric cones Rn (= S1 ×···× S1 ), Ln , and Sn e e e + + + + + identity element satisfies ◦ x = x ◦ = x for all x ∈ V , give rise to LP, SOCP, and SDP, respectively. We now define e e and defines a number r = h , i called the rank of V . these three cones. The cone of squares K corresponding to the Jordan product ◦ is defined as D. Examples of symmetric cones. Rn K = {x ◦ x | x ∈ V }. 1) Nonnegative orthant +: the cone of squares asso- ciated with the standard Euclidean space Rn and Jordan Every element x ∈ V has a spectral decomposition product r (x ◦ y)i = xiyi, i = 1,...,n, x = λiei, Xi=1 i.e., the entrywise (or Hadamard) product. The identity ele- e 1 where λi ∈ R are eigenvalues of x and the set of eigenvec- ment is = = (1,..., 1), and the Jordan frame comprises tors {e1,...,er}⊆ V , called a Jordan frame, satisfies the standard basis vectors. The quadratic representation of Rn r an element z ∈ is given by the diagonal matrix Pz = 2 e diag(z)2 ei = ei, ei ◦ ej = 0 for i 6= j, ei = . Xi=1

We have x ∈ K (written x K 0, or simply x  0 when x0 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, r r r x1 x2 tr det 1/2 1/2 x = λi, x = λi, x = λi ei, Xi=1 iY=1 Xi=1 3 Fig. 1. Second-order (Lorentz) cone L+. where the last quantity only makes sense if x  0. Finally, every element z ∈ V has a quadratic representa- n Rn 2) Lorentz cone L+: partition every element of as tion, which is a map Pz : V → V given by n−1 x =(x0,x1) ∈ R × R and define the Lorentz cone (also 2 Pz(x)=2(z ◦ (z ◦ x)) − z ◦ x. known as second-order or norm cone) as n R Rn−1 Rn C. Symmetric cones. L+ = {(x0,x1) ∈ × | kx1k2 ≤ x0}⊆ . The closed dual cone of is defined as K This cone is illustrated in Figure 1 for the case n = 3. It ∗ n K = {y ∈ V |hx,yi≥ 0, for all x ∈ K}, can be shown that L+ is the cone of squares corresponding to the Jordan product A cone K is self-dual if K∗ = K. The automorphism group Aut x y hx,yi (K) of an open convex cone K is the set of invertible 0 ◦ 0 = . linear transformations g : V → V that map K to itself, x1 y1 x0y1 + y0x1 Aut(K)= {g ∈ GL(V ) | gK = K}. The rank of this algebra is 2, giving a particularly simple spectral decomposition for a given element x, An open cone K is homogeneous if Aut(K) acts on the cone transitively, in other words, if for every x,y ∈ K x = λ1e1 + λ2e2, × where the eigenvalues and Jordan frame are • Nonnegative orthant: L(x)= Ax, where A ∈ Rn n is Metzler. λ1 = x0 + kx1k2, λ2 = x0 − kx1k2, T • Lorentz cone: L(x)= Ax, where A Jn + JnA  ξJn 1 1 1 1 for some ξ ∈ R. e1 = , e2 = . 2 x1/kx1k2 2 −x1/kx1k2 • Positive semidefinite matrices: L(X)= AX +XAT for The identity element is e = (1, 0) and the quadratic repre- any matrix A. sentation of an element z ∈ Rn is the matrix In fact, the examples above precisely characterize all cross- Rn n Sn zT J z positive operators on + and L+ (but not +) [13]. P = zzT − n J , z 2 n B. A class of Lyapunov functions n×n where Jn = diag(1, −1,..., −1) ∈ R is the n×n inertia Let L : V → V be a linear operator, and consider the matrix with signature (1, n − 1). Note that Pz is a positive linear dynamical system n semidefinite matrix if and only if z L+ 0. Sn x˙(t)= L(x), x(0) = x0, (4) 3) Positive semidefinite cone +: consider the vector space Sn of real, symmetric matrices with the n × n where x0 ∈ V is an initial condition. We make the as- trace inner product. If we define the Jordan product as the sumption that eL(K) ⊆ K, or equivalently, that L is cross- symmetrized matrix product, positive. Systems that obey this this assumption are, of 1 course, very special, because any trajectory that starts in the X ◦ Y = (XY + YX), 2 cone of squares K will remain within K for all time t ≥ 0, then the cone of squares is Sn , the positive semidefinite + x0 ∈ K =⇒ x(t) ∈ K for all t ≥ 0, matrices with real entries. The spectral decomposition of an element X ∈ Sn follows from the eigenvalue decomposition in other words, such systems are exponentially invariant with n respect to the cone K. We consider a generalization of the T T class of quadratic Lyapunov functions on V given by X = λivivi , =⇒ ei = vivi , i = 1,...,n. Xi=1 Vz : V → R, Vz(x)= hx,Pz(x)i, The quadratic representation of an element Z ∈ Sn is given n n where z ∈ V is a parameter, and Pz is the quadratic by the map PZ : S → S , representation of z in the Jordan algebra V . The following PZ (X)=2(Z ◦ (Z ◦ X)) − (Z ◦ Z) ◦ X = ZXZ. theorem gives a necessary and sufficient condition that Vz is a Lyapunov function [9], [10], [14]. III. ALINK TO QUADRATIC LYAPUNOV FUNCTIONS Instead of discussing the original dynamics (1) on Rn, we Theorem 1 (Gowda et al. 2009). Let L : V → V be a consider linear dynamics on a Jordan algebra V . This will linear operator on a Jordan algebra V with corresponding Rn n symmetric cone of squares K, and assume that eL(K) ⊆ K. allow us to present a unified treatment of the cones +, L+, as well as restate familiar results from classical state space The following statements are equivalent: Sn theory by specializing to +. (a) There exists p ≻K 0 such that −L(p) ≻K 0 (b) There exists z ≻ 0 such that LP + P LT is negative A. Cross-positive linear operators. K z z definite on V . A linear operator L : V → V is cross-positive with respect (c) The system x˙(t) = L(x) with initial condition x0 ∈ K to a proper cone K if is asymptotically stable. ∗ x ∈ K,y ∈ K , and hx,yi =0 =⇒ hL(x),yi≥ 0. Proof. See [14, Theorem 11]. Exponentials of cross-positive operators leave the cone in- This result allows the existence of a distinguished element variant: if L : V → V is cross-positive with respect to K, p ≻K 0 in the interior of K, where the vector field points in tL then e (K) ⊆ K for all t ≥ 0. Interestingly, the converse the direction of −K, or equivalently −L(p) ≻K 0, to certify is also true [12, Theorem 3]. the existence of a quadratic Lyapunov function of the form For example, a matrix A ∈ Rn×n, thought of as a linear R transformation A : Rn → Rn, is cross-positive with respect Vz : V → , Vz(x)= hx,Pz(x)i. to the cone Rn if and only if is a Metzler matrix, i.e., + A In fact, the derivative of Vz along trajectories of (4) is for all Aij ≥ 0, i 6= j, V˙z(x)= hx,P˙ z(x)i + hx,Pz(˙x)i or equivalently trajectories x(t) of the system = hL(x),Pz(x)i + hx,Pz(L(x))i T x˙(t)= Ax(t) = hx,L (Pz(x))i + hx,Pz(L(x))i T Rn Rn = hx, (PzL + L Pz)(x)i remain in + whenever they enter +. Therefore, cross- Rn Rn positivity is a generalization of the Metzler property. For the algebra and the corresponding cone +, Examples of operators L that satisfy eL(K) ⊆ K are Theorem 1 translates as follows. The quadratic representation Rn of an element z ≻ + 0 is a diagonal matrix D with strictly B. Diagonal dominance. positive diagonal. Let A be a cross-positive (i.e., Metzler) A square matrix A ∈ Rn×n is (weakly) diagonally matrix. The system x˙(t)= Ax(t) is stable (i.e., Hurwitz), if dominant if its entries satisfy and only if there exists an entrywise positive vector p such that |Aii|≥ |Aij |, for all i = 1,...,n. Rn Rn p ≻ + 0, −Ap ≻ + 0, Xj=6 i which happens if and only if there exists a diagonal Lya- As a simple consequence of Gershgorin’s circle theorem, punov function V (x)= xT Dx, diagonally dominant matrices with nonnegative diagonal T entries are positive semidefinite. D ≻Sn 0, AD + DA ≺Sn 0. + + If there exists a positive diagonal matrix D ∈ Rn×n Finding such a p is an LP for a fixed A. In addition, if such that AD is diagonally dominant, then the matrix A we know (or impose, through a feedback interconnection) is generalized or scaled diagonally dominant. Equivalently, that A has the Metzler structure, then a diagonal Lyapunov there exists a positive vector y ≻Rn 0 such that function candidate suffices to ensure stability. This trick has + been used in, e.g., [3] to greatly simplify (and in certain cases |Aii|yi ≥ |Aij |yj , for all i = 1,...,n. parallelize) stability analysis and controller synthesis. Xj=6 i Now consider the algebra Sn and the corresponding cone Sn Note that generalized diagonally dominant matrices are +. One can define many cross-positive operators, but one comes to mind: the Lyapunov transformation L : Sn → Sn, also positive semidefinite, and include diagonally dominant matrices as a special case. Symmetric generalized diagonally T L(X)= AX + XA , dominant matrices with nonnegative diagonal entries are also + where A ∈ Rn is a given matrix. (By construction, L known as H -matrices, and have a very nice characterization is cross-positive). Following the theorem, we now restate as matrices with factor width of at most two, see [17, some widely known facts about linear systems. The matrix Theorem 9]. differential equation One consequence of this characterization is that H+ T matrices can be written as a sum of positive semidefinite X˙ (t)= AX + XA , X(0) = X ≻Sn 0 0 + matrices that are nonzero only on a 2 × 2 principal sub- is asymptotically stable if and only if there exists a matrix matrix, see [18]. For example, a 3 × 3 H+-matrix A = AT P ≻ 0 such that L(P ) = AP + P AT ≺ 0, if and only if is the sum of terms of the form there exists a matrix Z ≻ 0 such that the function x1 x2 0 y1 0 y2 0 0 0 Tr 2 VZ (X)= hX,PZ (X)i = (XZXZ)= kXZkF A = x2 x3 0 +  0 0 0  + 0 z1 z2 , 0 0 0 y 0 y 0 z z is a Lyapunov function. This happens if and only if A is a    2 3  2 3 (Hurwitz) stable matrix. where the sub-matrices are all positive semidefinite, IV. DYNAMICS ON A LORENTZ CONE x1 x2 y1 y2 z1 z2 Rn Sn  0,  0,  0. The cones + and + have been well studied in the liter- x2 x3 y2 y3 z2 z3 ature. The “intermediate case”—the cone Ln —is, however, + Recently, this fact has been exploited in [19], [20] to quite strange. This cone has received relatively little attention extend the reach of sum-of-squares techniques to high di- in the control community. Recent efforts have been in the mensional dynamical systems without imposing full LMI context of model matching [15]. Our work here can be seen constraints on the Gram matrix, and in [18] to preprocess as a complement. SDPs for numerical stability. n A. Enforcing L+-invariance To apply the main theorem we require that eL(K) ⊆ K. C. Rotated quadratic constraints. n For the case K = L+, this means A must satisfy the LMI Real, symmetric, positive semidefinite matrices of size , as they occur in the characterization of +-matrices AT J + J A − ξJ  0, ξ ∈ R. (5) 2 × 2 H n n n above, are special, because they satisfy a restricted hyper- If A were to affinely depend on optimization variables (as it bolic constraint; hence their definiteness can be enforced with would if it were a closed loop matrix in a linear feedback an SOCP rather than SDP [7]. Specifically, we have, n synthesis problem), then enforcing L+-invariance would also be an LMI—we might as well dispense with any special x1 x2 2  0 ⇐⇒ x1 ≥ 0, x3 ≥ 0, x1x3 − x2 ≥ 0 structure, and resort to algebraic Riccati, bounded-real, or x2 x3 general LMI-based analysis, e.g., [16]. 2x2 ⇐⇒ ≤ x1 + x3 However, if A has additional structure, for example, it x1 − x3 2 has an embedded internally positive block transverse to the 3 ⇐⇒ (x1 + x3, 2x2,x1 − x3) ∈L , Lorentz cone axis, then the LMI (5) can be simplified. This + simplification can be performed with diagonal dominance. for scalars x1, x2, and x3. V. EXAMPLES Here, the state xi represents the amount of material at A. Embedded shearless positive block. node i, ℓij ≥ 0 the transfer rate between nodes i and j in the base system, a ∈ R the self-degradation rate of the catch- We show how these techniques can be used to consider 0 all buffer, and h the rate of transfer between node i and the the simple, augmented (n + 1)-dimensional dynamics i catch-all node. Note that A is Metzler by construction, but x˙ 0 a0 0 x0 n×n the augmented system (8) need not be. = , a0 ∈ R, A ∈ R , x˙ 1  0 Ax1 Several problems are now readily solved: by checking (6) n (or (7)), with the treated as variables and fixed, the with trajectory x(t)=(x0(t),x1(t)) ∈ R × R , on the ℓij h = 0 n+1 augmented system is L5 -invariant only if a ≥−1.25, which cone L+ . After writing out the LMI (5) in block form, + 0 n+1 is an upper bound on the Metzler eigenvalue of A. Thus the we determine that the augmented system is L+ -invariant if and only if catch-all node must consume material no faster than with T rate constant . In addition, if and are known, and 2a0I − (A + A )  0. (6) 1.25 ℓij hi 5 the augmented system (8) is L+-invariant, then it is stable It is stable if and only if A is Hurwitz and a0 < 0. Roughly n+1 provided a0 < 0. Of course, these types of problems can be speaking it is L+ -invariant and stable if the stability degree cast as an LP or SOCP. of A (i.e., minus the maximum real part of the eigenvalues n of A) is at least |a0|. See Figure 3. C. Other examples on L+. In general, the stability degree constraint (6) is an LMI in the variables (A, a ), however, if A is Metzler then the LMI 0 x x can be replaced with the constraint 0 0 T + 2a0I − (A + A ) is an H -matrix, (7) without any loss. From previous sections, the H+-matrix constraint (7) is an SOCP. x1 x2 x1 x2 x2 x0 Fig. 3. Embedded focus along x0-axis

1) Twist system: Suppose A+AT = 0 (skew-symmetric). The matrix system x4 x1 x˙ a hT x 0 = 0 0 , x˙ 1  0 A x1 n is L+-invariant if and only if khk2 ≤ a0, i.e., the point (a ,h) is in the Lorentz cone Ln . Note if a = 0, the x3 0 + 0 dynamics correspond to a twist in homogeneous coordinates,

Fig. 2. Transportation network x1,...,x4 from [3, Figure 2], augmented see, e.g., [21]. with a catch-all buffer x0. 2) Proper orthochronous Lorentz transformations: The restricted Lorentz group SO+(1, n − 1) is generated by B. Augmented transportation network. spatial rotations and Lorentz boosts, and consists of linear T We consider the linear transportation network shown in transformations that keep the quadratic form x Jnx invari- + Figure 2, which is inspired by the one studied in [3]. The ant. In particular, elements of SO (1, n − 1) correspond n network might represent a base system of four buffers (solid to L+-invariant dynamics matrices, and thus fall within our framework. nodes x1,...,x4) exchanging and consuming material in a way that preserves the network structure. The base system VI. CONCLUSION is augmented with an extra catch-all buffer (grayed out node In this work we analyzed linear systems with a special x ), leading to the augmented dynamics 0 structure by Jordan algebraic techniques. In particular, we put T x˙ 0 a0 h x0 4 analysis of internally positive systems, which have recently = , x0(t) ∈ R, x¯(t) ∈ R , (8)  x¯˙   0 A  x¯  been in vogue, in the same theoretical framework as systems where the base dynamics x¯˙ = Ax¯ are given by the internally that are exponentially invariant with respect to the Lorentz positive system cone, as well as general linear systems. It is evident that Lyapunov functions for such systems can be significantly x˙ 1 −1 − ℓ31 ℓ12 0 0 x1 simpler, computationally and representationally, than if the x˙ 2  0 −ℓ12 − ℓ32 ℓ23 0  x2 = . special cone-invariant dynamic structure were not exploited. x˙ ℓ ℓ −ℓ − ℓ ℓ x  3  31 32 23 43 34   3 In this framework, summarized in Table I, the relevant x˙   0 0 ℓ −4 − ℓ  x   4  43 34  4 structure is cross-positivity of the dynamics matrix on a x¯˙ (t) A x¯(t) symmetric cone. | {z } | {z } | {z } Algebra: Real Lorentz Symmetric V Rn Rn Sn Rn n Sn K + L+ + hx,yi xT y xT y Tr(XY T ) T 1 x ◦ y xiyi (x y,x0y1 + y0x1) 2 (XY + YX) T int 2 T z Jnz Pz, z ∈ K diag(z) zz − 2 Jn X 7→ ZXZ T T 2 T T z Jnz 2 V (x)= hx,Pz(x)i x diag(z) x x zz − 2 Jn x kXZkF Free variables in V (x) n n n(n + 1)/2 dynamics L x 7→ Ax x 7→ Ax X 7→ AX + XAT L is cross-positive A is Metzler A satisfies (5) by construction T −L(p) ≻K 0 (Ap)i < 0 k(Ap)1k2 < (−Ap)0 AP + PA ≺ 0 Stability verification LP SOCP SDP TABLE I SUMMARY OF DYNAMICS PRESERVING A CONE

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