Long-time average cost control of polynomial systems: A sum of squares approach

Abstract

Abstract-This paper provides a numerically tractable approach for long-time average cost control of nonlinear dynamical systems with polynomials of system state on the right-hand side. First, a recently-proposed method of obtaining rigorous bounds of long-time average cost is outlined for the uncontrolled system, where the polynomial constraints are relaxed to be sumof-squares and formulated as semi-definite programs. As such, it allows to use any general (polynomial) functions to optimize the bound. Then, a polynomial type state feedback controller design scheme is presented to further suppress the long-time average cost. The derivation of state feedback controller is given in terms of the solvability conditions of state-dependent bilinear matrix inequalities. Finally, the mitigation of oscillatory vortex shedding behind a cylinder is addressed to illustrate the validity of the proposed approach.

Optimization With Affine Homogeneous Quadratic Integral Inequality Constraints

“…The existence of L n follows from ( 25), (21), and (18). Moreover, given four integers a ≤ b and c ≤ d, let…”

“…) and its derivatives up to order k ≤ m − 1 be expanded as in (17), and let B k−1 u ∈ R 2k be defined according to (3). Moreover, let M be as in (19), and let ǔM ∈ R k+M +1 be defined according to (21). There exists a matrix…”

“…Other input-to-state/output properties such as passivity, reachability, and input-to-state stability can be studied in a similar way using dissipation inequalities for integral functionals of the state variable [2,4]. Finally, the computational cost of designing optimal control policies for systems with complex dynamics, such as turbulent flows, may be reduced by requiring the control law to minimize an upper bound on the objective function rather than the objective itself [20,21,23,24], and in the case of PDEs such upper bounds can be found by solving suitable integral inequalities [8,9,[11][12][13]19].…”