Nonlinear static output feedback controller design for uncertain polynomial systems: An iterative sums of squares approach

2015 European Control Conference (ECC)

Chernyshenko

Lasagna

et al. 2015

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.

Section: B An Iterative Algorithmmentioning

confidence: 99%

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

2015 European Control Conference (ECC)

Chernyshenko

Lasagna

et al. 2015

“…For any non-negative integers i 1 , i 2 satisfying i 1 < i 2 , the tunable variables in F i1 are always a subset of the tunable variables in F i2 . Hence, (17) can be resolved by solving a sequence of convex optimization problems:…”

Section: The Structure Of Small-feedback Controllermentioning

confidence: 99%

“…However, using SOS techniques for optimal control, as for example in [13][14][15], is subject to a generic dif culty: the problem of simultaneously optimizing both the control and the Lyapuniov function is non-convex. Iterative procedures were proposed for overcoming this dif culty [14,16,17]. However, the iterations can diverge and seeking the global extremum is often difcult.…”

Section: Introductionmentioning

confidence: 99%

Low-order state-feedback controller design for long-time average cost control of fluid flow systems: A sum-of-squares approach

2015 34th Chinese Control Conference (CCC)

Chernyshenko

2015

This paper presents a novel state-feedback controller design approach for long-time average cost control of uid ows. Due to the high dimensionality of uid dynamical system, direct controller design renders to high-order state feedback that might not be applicable in practice. To resolve this, the original system is rst transformed into a reduced-order uncertain system, where polynomial bounds for the involved uncertainties are evaluated analytically. Meanwhile, instead of minimizing the time-averaged cost itself, we use its upper bound as the objective function for controller design. Then, under the framework of sum-of-squaresbased optimization, the control law and a tunable function similar to the Lyapunov function are optimized simultaneously. The inherent non-convexity of the optimization is overcome by assuming that the controller takes a small-feedback structure, which actually is a series in a small parameter with all the coef cients being nite-order polynomials of the reduced-order system state. The main characteristics of the induced controller lie in its low-order structure and robustness.

“…These advances have emerged as a promising and numerically tractable basis to solve many computationally hard problems in control for systems whose dynamics are described by polynomial functions, e.g. [5][6][7]. The application of such methods to various problems in fluid dynamics, from stability analysis to bounds on time averaged quantities, is discussed in Ref.…”

Section: Introductionmentioning

confidence: 99%

Controlling fluid flows with positive polynomials

Lasagna

2016 35th Chinese Control Conference (CCC)

Tutty

et al. 2016

A novel nonlinear feedback control design methodology for incompressible fluid flows aiming at the optimisation of long-time averages of key flow quantities is presented. The key idea, first outlined in Ref. [1], is that the difficulties of treating and optimising long-time averages are relaxed by shifting the analysis to upper/lower bounds for minimisation/maximisation problems, respectively. In this setting, control design reduces to finding the polynomial-type state-feedback controller that optimises the bound, subject to a polynomial inequality constraint involving the cost function, the nonlinear system, the controller itself and a tunable polynomial function. A numerically tractable approach, based on Sum-of-Squares of polynomials techniques and semidefinite programming, is proposed. As a prototypical example of control of separated flows, the mitigation of the fluctuation kinetic energy in the unsteady two-dimensional wake past a circular cylinder at a Reynolds number equal to 100, via controlled angular motions of the surface, is investigated. A compact control-oriented reduced-order model, resolving the long-term behaviour of the fluid flow and the effects of actuation, is first derived using Proper Orthogonal Decomposition and Galerkin projection. In a full-information setting, linear state-feedback controllers are then designed to reduce the long-time average of the resolved kinetic energy associated to the limit cycle of the system. Controller performance is then assessed in direct numerical simulations.