Sum-of-Squares Optimization Based Approximate Dynamic Programming for Time-Varying Systems and Its Application in Suboptimal Guidance Law Design

In this paper, we propose a method for sub-optimal control of time-varying polynomial systems and use it for pursuits guidance law design. Since, engagement equations between pursuit and target are depend on the range between them and this range is varying during the flight, guidance law designer is faced with a time-varying system. The developed methods for control of time-invariant systems are not directly applicable for time-varying systems. One of the conventional approaches for pursuits guidance law design is the optimal control. Approximate dynamic programming is a well-known method for solving the optimal control problem. One of the challenges of using this method for control of nonlinear time-varying systems is the difficulty of solving the Bellman equation. In the proposed method of this paper, solving the Bellman equation has been relaxed with solving a sum-of-squares optimization problem. It will be proved that the designed control policy with this method is globally exponentially stabilizing. Finally, performance of the proposed method for pursuits guidance will be illustrated with numerical simulations.