Closed-loop stability of systems driven by real-time, dynamic optimization algorithms

The receding horizon control (RHC) scheme uses online optimization to find a finite-horizon control input to a constrained dynamic system. This paper examines the relationship between the optimization algorithm and the closed-loop dynamic system in RHC. Past research on RHC has assumed that the optimization algorithm provides an optimal solution in a fixed time interval. Since RHC typically employs quadratic programming, which is usually solved only approximately, this presupposition is not valid. Instead of making the traditional optimality assumption, this paper supposes that the provided solutions are only suboptimal. A sufficient condition is derived for closed-loop stability given control sequences which are optimal with tolerance ε. Also, a bound is derived for the number of computations to find an ε-optimal solution from a warm start using an interior-point method. As long as this number of computations can be carried out in less than the time step of the dynamic system, the closed-loop is guaranteed to be stable