Stability Optimization for Polynomials and Matrices

Suppose that the coefficients of a monic polynomial or entries of a square matrix depend affinely on parameters, and consider the problem of minimizing the root radius (maximum of the moduli of the roots) or root abscissa (maximum of their real parts) in the polynomial case and the spectral radius or spectral abscissa in the matrix case. These functions are not convex and they are typically not locally Lipschitz near minimizers. We first address polynomials, for which some remarkable analytical results are available in one special case, and then consider the more general case of matrices, focusing on the static output feedback problem arising in control of linear dynamical systems. We also briefly discuss some spectral radius optimization problems arising in the analysis of the transient behavior of a Markov chain and the design of smooth surfaces using subdivision algorithms. 1 Optimization of Roots of Polynomials Optimization of roots of polynomials can arise in many contexts, but perhaps the most important application area is feedback control in the frequency domain [Dor99]. Consider the problem min p∈P max λ∈C {Re λ | p(λ) = 0} where P = {(λ + 2λ)(x0 + x1λ+ λ ) + y0 + y1λ+ y2λ 2 | x0, x1, y0, y1, y2 ∈ R}. This arises in maximizing, over all linear feedback controllers of order two, the asymptotic decay rate for a two-mass-spring dynamical system with one input (an actuator positioning the first mass) and one output (the measured position of the second mass) [HO06]. The polynomials in P are those that are admissible as the denominators of the relevant rational closed-loop system transfer function, and their roots must be in the left half of the complex plane for the system to be stable. Note that P is a set of monic polynomials with degree 6 whose coefficients depend affinely on 5 parameters. A construction was given in [HO06] of a polynomial in P of the form (λ−λ0) 6, with just one distinct negative real root λ0 of multiplicity 6, and its local optimality (the property that no nearby polynomial in P has all its roots to the left of λ0) was established. It was subsequently discovered that the solution constructed in [HO06] is globally optimal (no polynomial ∗Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA. Supported in part by the U.S. National Science Foundation Grant DMS-1016325. 1 in P has all its roots to the left of λ0). This fact is a special case of a remarkable general theory that we summarize below, extracting the main results from a recent paper by Blondel, Gürbüzbalaban, Megretski and the author [BGMO12]. As noted there, it turns out that part of this general theory was first given in a little-known 1979 Ph.D. thesis of Raymond Chen [Che79b]. 1.1 Root Optimization over a Polynomial Family with a Single Affine Constraint Every monic polynomial of degree n may be represented by a point in Cn, representing the coefficients of the monomials λn−1, . . . , λ, 1. Let r denote the root radius of such a polynomial: r(p) = max {|λ| : p(λ) = 0, λ ∈ C} . The polynomial p is said to be Schur stable if r(p) < 1. Let a denote the root abscissa: a(p) = max {Re(λ) : p(λ) = 0, λ ∈ C} . The polynomial p is said to be Hurwitz stable if a(p) < 0. As functions of the polynomial coefficients, the radius r and abscissa a are not convex. They are continuous, but not Lipschitz continuous near a polynomial p with a multiple root whose modulus or real part respectively equals r(p) or a(p). So, in general, global minimization of the radius or abscissa over an affine family of monic polynomials, pushing the roots as far as possible towards the origin or left in the complex plane, seems hard. Indeed, variations on the question of whether a given polynomial family contains one that is stable (has roots inside the unit circle or in the lefthalf plane) have been studied for decades [BGL95]. But if an affine family of monic polynomials of degree n has n− 1 free parameters, this question can be answered efficiently. Equivalently, there is a single affine constraint on the coefficients. 1.2 The Root Radius We begin with optimizing the root radius over real coefficients. Informally, we want to push all the roots as close to zero as possible. By compactness, an optimal polynomial must exist; the following result states an explicit form for the solution. Theorem RRR. [BGMO12, Theorem 1] Let b0, b1, . . . , bn be real scalars (with b1, . . . , bn not all zero) and consider the affine family P = {λ + a1λ n−1 + . . .+ an−1λ+ an : b0 + n