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 . 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 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 entries (), 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 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. If we define the Jordan product as the Let L : V → V be a linear operator, and consider the symmetrized matrix product, linear dynamical system 1 x˙(t)=L(x),x(0) = x , (3) X ◦ Y = (XY + YX), 0 2 where x ∈ V is an initial condition. We make the as- Sn 0 then the cone of squares is +,thepositivesemidefinite sumption that eL(K) ⊆ K,orequivalently,thatL is cross- matrices with real entries. The spectral decomposition of an positive. Systems that obey this this assumption are, of Sn element X ∈ follows from the eigenvalue decomposition course, very special, because any trajectory that starts in the n cone of squares K will remain within K for all time t ≥ 0, T T X = λivivi , =⇒ ei = vivi ,i=1,...,n. !i=1 x0 ∈ K =⇒ x(t) ∈ K for all t ≥ 0, Sn The quadratic representation of an element Z ∈ is given in other words, such systems are exponentially invariant with Sn Sn by the map PZ : → , respect to the cone K.Weconsiderageneralizationofthe

PZ (X)=2(Z ◦ (Z ◦ X)) − (Z ◦ Z) ◦ X = ZXZ. class of quadratic Lyapunov functions on V given by

III. COMPUTATIONAL APPROACH TO DYNAMICS ON A Vz : V → R,Vz(x)=⟨x, Pz(x)⟩, SYMMETRIC CONE where z ∈ V is a parameter, and Pz is the quadratic Rn Instead of discussing the original dynamics (1) on , representation of z in the V .Thefollowing we consider instead general linear dynamics on a Jordan theorem gives a necessary and sufficient condition that Vz is algebra V .Thiswillallowustopresentaunifiedtreatmentof aLyapunovfunction Rn n the cones +, L+,aswellasrestatefamiliarresultsfrom Sn Theorem 1 (Gowda et al. 2009). Let L : V → V be a classical state space theory by specializing to +.Welist most notable results by Gowda et al. [8], [9], [11]. linear operator on a Jordan algebra V with corresponding symmetric cone of squares K,andassumethateL(K) ⊆ K. A. Cross-positive linear operators. The following statements are equivalent:

AlinearoperatorL : V → V is cross-positive with respect (a) There exists p ≻K 0 such that −L(p) ≻K 0 T to a proper cone K if (b) There exists z ≻K 0 such that LPz + PzL is negative x ∈ K, y ∈ K∗, and ⟨x, y⟩ =0 =⇒⟨L(x),y⟩≥0. definite on V . (c) The system x˙(t)=L(x) with initial condition x0 ∈ K Exponentials of cross-positive operators leave the cone in- is asymptotically stable. variant: if L : V → V is cross-positive with respect to K, then etL(K) ⊆ K for all t ≥ 0.Infact,theconverseisalso Proof. See [11, Theorem 11]. true [12, Theorem 3]. This result allows the existence of a distinguished element Rn×n For example, a matrix A ∈ ,thoughtofasalinear p ≻ 0 in the interior of K,wherethevectorfieldpointsin Rn Rn K transformation A : → ,iscross-positivewithrespect the direction of −K,orequivalently−L(p) ≻ 0,tobea Rn K to the cone + if and only if A is a Metzler matrix, i.e., certificate for the existence of a quadratic Lyapunov function of the form Aij ≥ 0, for all i ̸= j, R or equivalently that trajectories x(t) of the system Vz : V → ,Vz(x)=⟨x, Pz(x)⟩.

x˙(t)=Ax(t) In fact, the derivative of Vz along trajectories of (3) is precisely Rn Rn remain in + whenever they enter +.Therefore,cross- positivity can be thought of as a generalization of the Metzler V˙z(x)=⟨x,˙ Pz(x)⟩ + ⟨x, Pz(˙x)⟩ property. = ⟨L(x),Pz(x)⟩ + ⟨x, Pz(L(x))⟩ L Examples of operators L that satisfy e (K) ⊆ K are T × = ⟨x, L (Pz(x))⟩ + ⟨x, Pz(L(x))⟩ • Nonnegative orthant: L(x)=Ax,whereA ∈ Rn n is T Metzler. = ⟨x, (PzL + L Pz)(x)⟩ Rn Rn For the algebra and the corresponding cone +, structure, and resort to algebraic Riccati, bounded-real, or Theorem 1 translates as follows. The quadratic representation general LMI-based analysis, e.g.,[15]. Rn of an element z ≻ + 0 is a diagonal matrix D with strictly However, if A has some extra structure, for example, it positive diagonal. Let A be a cross-positive (i.e.,Metzler) has an embedded positive block transverse to the Lorentz matrix. The system x˙(t)=Ax(t) is stable (i.e.,Hurwiz),if cone axis, then the LMI (4) can be simplified, as we discuss and only if there exists an entrywise positive vector p such below after introducing the concept of diagonal dominance. that Rn Rn p ≻ + 0, −Ap ≻ + 0, B. Diagonal dominance. which happens if and only if there exists a diagonal Lya- AsquarematrixA ∈ Rn×n is (weakly) diagonally punov function V (x)=xT Dx, dominant if its entries satisfy

T Sn Sn D ≻ + 0,AD+ DA ≺ + 0. |Aii|≥ |Aij |, for all i =1,...,n. !j≠ i Finding such a p is a LP for a fixed A.Inaddition,ifwe know (or impose, through a feedback interconnection) that A As a simple consequence of Gershgorin’s circle theorem, has the Metzler structure, then a diagonal Lyapunov function diagonally dominant matrices with nonnegative diagonal candidate suffices to ensure stability. This trick has been entries are positive semidefinite. used in, e.g.,[3]togreatlysimplify(andincertaincases If there exists a positive diagonal matrix D ∈ Rn×n parallelize) stability analysis and controller synthesis. such that AD is diagonally dominant, then the matrix A Now consider the algebra Sn and the corresponding cone is generalized or scaled diagonally dominant. Equivalently, n S .Onecandefinemanycross-positiveoperators,butone Rn + there exists a positive vector y ≻ + 0 such that comes to mind: the Lyapunov transformation L : Sn → Sn, |A |y ≥ |A |y , for all i =1,...,n. L(X)=AX + XAT , ii i ij j !j≠ i where A ∈ Rn is a given matrix. (By construction, L Note that generalized diagonally dominant matrices are is cross-positive). Following the theorem, we now restate also positive semidefinite, and include diagonally dominant some widely known facts about linear systems. The matrix matrices as a special case. Symmetric generalized diagonally differential equation dominant matrices with nonnegative diagonal entries are also T + ˙ Sn known as H -matrices, and have a very nice characterization X(t)=AX + XA ,X(0) = X0 ≻ + 0 as matrices with factor width of at most two, see [16, is asymptotically stable if and only if there exists a matrix Theorem 9]. T P ≻ 0 such that L(P )=AP + PA ≺ 0,ifandonlyif One consequence of this characterization is that H+ there exists a matrix Z ≻ 0 such that the function matrices can be written as a sum of positive semidefinite Tr 2 matrices that are nonzero only on a 2 × 2 principal sub- VZ (X)=⟨X, PZ (X)⟩ = (XZXZ)=∥XZ∥F matrix, see [17]. For example, a 3 × 3 H+-matrix A = AT is a Lyapunov function. This happens if and only if A is a is the sum of terms of the form (Hurwiz) stable matrix. x1 x2 0 y1 0 y2 00 0 IV. CASE STUDY: DYNAMICS ON A LORENTZ CONE A = ⎡x2 x3 0⎤ + ⎡ 000⎤ + ⎡0 z1 z2⎤ , Rn Sn 000 y2 0 y3 0 z2 z3 The cones + and + have been well studied in the liter- ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ n ature. The “intermediate case”—the cone L+—is, however, where the sub-matrices are all positive semidefinite, quite strange. This cone has received relatively little attention in the control community. Recent efforts have been in the x x y y z z 1 2 ≽ 0, 1 2 ≽ 0, 1 2 ≽ 0. context of model matching [14]. Our work here can be seen #x2 x3$ #y2 y3$ #z2 z3$ as a complement. Recently, this fact has been exploited in [18], [19] to n A. Enforcing L+-invariance extend the reach of sum-of-squares techniques to high di- To apply the main theorem, we require that eL(K) ⊆ K. mensional dynamical systems without imposing full LMI n constraints on the Gram matrix, and in [17] to preprocess In the case K = L+,thedynamicsmatrixA should apriori satisfy the LMI SDPs for numerical stability. T R A Jn + JnA − ξJn ≽ 0,ξ∈ . (4) C. Rotated quadratic constraints. If A were to affinely depend on optimization variables (as it Real, symmetric, positive semidefinite matrices of size would if it were a closed loop matrix in a linear feedback 2 × 2,astheyoccurinthecharacterizationofH+-matrices n synthesis problem), then enforcing L+-invariance would also above, are special, because they satisfy a restricted hyper- be an LMI—we might as well dispense with any special bolic constraint; hence their definiteness can be enforced with aSOCPratherthanSDP[7].Specifically,wehave, where the base dynamics x¯˙ = Ax¯ are given by the internally x x positive system 1 2 ≽ 0 ⇐⇒ x ≥ 0,x≥ 0,xx − x2 ≥ 0 x x 1 3 1 3 2 # 2 3$ x˙ 1 −1 − ℓ31 ℓ12 00x1 2x2 ⎡x˙ 2⎤ ⎡ 0 −ℓ12 − ℓ32 ℓ23 0 ⎤ ⎡x2⎤ ⇐⇒ ≤ x1 + x3 = . )#x1 − x3$) x˙ 3 ℓ31 ℓ32 −ℓ23 − ℓ43 ℓ34 x3 ) )2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ) ) 3 ⎢x˙ 4⎥ ⎢ 00ℓ43 −4 − ℓ34⎥ ⎢x4⎥ ⇐⇒ )(x1 + x3, 2x)2,x1 − x3) ∈L+, ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ for scalars x , x ,andx . x¯˙ (t) A x¯(t) 1 2 3 , -. / , -. / , -. / D. Embedded positive block. Here, the state xi represents the amount of material at node i, ℓ ≥ 0 the transfer rate between nodes i and j in the We are now equipped to consider the simple, augmented ij base system, a ∈ R the self-degradation rate of the catch- (n +1)-dimensional dynamics 0 all buffer, and hi the rate of transfer between node i and the x˙ 0 a0 0 x0 n×n catch-all node. Note that A is Metzler by construction. = ,a0 ∈ R,A∈ R , #x˙ 1$ # 0 A$#x1$ Several problems are now readily solved: by checking (5) n with trajectory x(t)=(x0(t),x1(t)) ∈ R × R ,onthe (or (6)), with the ℓij treated as variables and h =0fixed, the 5 n+1 augmented system is L -invariant only if a0 ≥−1.25,which cone L+ .AfterwritingouttheLMI(4)inblockform, + n+1 is an upper bound on the Metzler eigenvalue of A.Thusthe 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 1.25.Inaddition,ifℓ and h are known, and 2a0I − (A + A ) ≽ 0. (5) ij i 5 the augmented system (7) is L+-invariant, then it is stable It is stable if and only if A is Hurwiz and a0 < 0.Roughly provided a < 0.Ofcourse,thesetypesofproblemscanbe n+1 0 speaking it is L+ -invariant and stable if the stability degree cast as LP or SOCP. of A (i.e.,minusthemaximumrealpartoftheeigenvalues of A)isatleast|a |. n 0 F. O t h e r ex a m p l e s o n L+. In general, the stability degree constraint (5) is an LMI in the variables (A, a0),however,ifA is Metzler then the LMI can be replaced with the constraint x0 x0 T + 2a0I − (A + A ) is a H -matrix, (6) without any loss. From previous sections, the H+-matrix constraint (6) is a SOCP.

x1 x2 x1 x2 x2 x0

Fig. 3. Embedded focus along x0-axis

1) Embedded block: For the dynamics x4 x 1 x˙ a 0 x 0 = 0 0 , #x˙ 1$ # 0 A$#x1$

n T This system is L+-invariant if and only if A + A ≼ 2a0I. It is stable if and only if A is Hurwiz and a0 < 0.Figure3 x3 illustrates embedded block dynamics without a shear term h. T Fig. 2. Transportation network x1,...,x4 from [3, Figure 2], augmented 2) Twist system: Suppose A+A =0(skew-symmetric). with a catch-all buffer x0. The system x˙ a hT x 0 = 0 0 , E. Example in 5D. #x˙ 1$ # 0 A $#x1$ We consider the linear transportation network shown in n is L+-invariant if and only if ∥h∥2 ≤ a0, i.e.,thepoint Figure 2, which is inspired by the one studied in [3]. The n (a0,h) ∈L+.Furthermore,ifa0 =0,thisisatwistin network might represent a base system of four buffers (solid homogeneous coordinates. nodes x1,...,x4)exchangingandconsumingmaterialina 3) Proper orthochronous Lorentz transformations: The way that preserves the network structure. The base system restricted Lorentz group SO+(1,n− 1),whichisgenerated is augmented with an extra catch-all buffer (grayed out node by spatial rotations and Lorentz boosts, consists of linear x0), leading to the augmented dynamics T transformations that keep the quadratic form x Jnx invari- T + x˙ 0 a0 h x0 R R4 ant. In particular, elements of SO (1,n− 1) correspond to = ,x0(t) ∈ , x¯(t) ∈ , (7) n # x¯˙ $ # 0 A $#x¯ $ L+-invariant dynamics matrices. Algebra: Real Lorentz Symmetric V Rn Rn Sn Rn n Sn K + L+ + ⟨x, y⟩ 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 &→ ZXZ T T 2 T T z Jnz 2 V (x)=⟨x, Pz(x)⟩ x diag(z) x x !zz − 2 Jn" x ∥XZ∥F Free variables in V (x) n n n(n +1)/2 dynamics L x &→ Ax x &→ Ax X &→ AX + XAT L is cross-positive A is Metzler A satisfies (4) by construction T −L(p) ≻K 0 (Ap)i < 0 ∥(Ap)1∥2 ≤ (−Ap)0 AP + PA ≺ 0 Stability verification LP SOCP SDP TABLE I SUMMARY OF DYNAMICS PRESERVING A CONE

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