In this paper we consider several problems involving control with limited actuation and sampling rates. Event-based control has emerged as an attractive approach for addressing the problems of control system design under rate limitations. In event-based control, a system is actuated or a control signal is changed only when certain events occur. For example, a control signal might be applied only when some measure of deviation of the system state from equilibrium is exceeded. Thus, control action is only applied when it is needed, keeping control performance satisfactory while reducing the rate that the system must be sensed and actuated. In principle, the problem of determining how to optimally schedule the sensing or actuation of a system can be cast as a Markov decision process. However, the optimal value function for these Markov decision processes generally does not have a simple structure. So, determining a closed-form expression or simple parametrization of the optimal value function is generally not possible. In this paper we develop new computational methods for event-based control. Under a given policy, one can obtain an upper bound on control performance using an approximate value function for the associated Markov decision process.We will consider performance bounds that can be obtained using quadratic approximate value functions. We will find the policy that minimizes the upper bound obtainable over all possible quadratic approximate value functions. This policy and the associated performance bound can be obtained by solving a sequence of semidefinite programs indexed by a scalar parameter.
