Robust Optimal Switching Control for Nonlinear Systems

Abstract

Abstract. We formulate a robust optimal control problem for a general nonlinear system with finitely many admissible control settings and with costs assigned to switching of controls. We formulate the problem both in an L 2 -gain/dissipative system framework and in a game-theoretic framework. We show that, under appropriate assumptions, a continuous switching-storage function is characterized as a viscosity supersolution of the appropriate system of quasivariational inequalities (the appropriate generalization of the Hamilton-Jacobi-Bellman-Isaacs equation for this context), and that the minimal such switching-storage function is equal to the continuous switching lower-value function for the game. Finally we show how a prototypical example with one-dimensional state space can be solved by a direct geometric construction.

H∞ control for discrete‐time nonlinear switching systems

“…The left-hand side of the gain inequality (5) is the hybrid cost that penalizes both continuous evolutions and discrete transitions caused by the external disturbances. This generalized l 2 gain formulation is similar to that defined by Ball et al [2].…”

“…Switching plays a central role in many practical complex dynamical systems such as chemical processes, automotive processes, electrical systems. The study of switching systems recently has attracted much attention among the control scientists and engineers [1][2][3][4], and a theory of switched systems is being developed. A switched system consists of multiple subsystems and a mechanism for switching between them.…”

“…To date, little research [2,16,17] has been reported to study the H ∞ control problem for hybrid systems in general, nor for the important subclass of switching systems. One of the difficulties in such a generalization is in the formulation of a meaningful l 2 gain for a system involving multiple subsystems.…”

A probabilistic verification theorem for the finite horizon two-player zero-sum optimal switching game in continuous time

“…If a sufficiently regular solution of the system (1) exists then v i (s, x) is nothing else but the optimal profit that can be generated by switching, with initial conditions i for the system's mode and x for the process X at time s ∈ [0, T]. Probabilistic solution methods for optimal switching problems have been investigated since the 1970s and 1980s in various degrees of generality (see, for instance, [4], [26], [28], [34], and [35]), and most of the recent research in this area has been a combination of the martingale approach via Snell envelopes [11], [24], and the theory of backward stochastic differential equations (BSDE) [7], [14], [20].…”

Finite Set Control Transcription for Optimal Control Applications

“…5 Other nonlinear systems employ optimal control strategies involving a generalized Bellman equation, impulse control, and linear programming. 1,12 Classical optimal control theory is also used to address, at least, the continuous aspects of a problem. 13,14 The solution process presented in this study relies on numerical optimization techniques rooted in sequential quadratic programming.…”