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This paper considers a constrained co-design problem for linear time-invariant (LTI) systems. With a practical vision of real-world problems, constraints on the control signal and system states are involved in the proposed algorithm. The main goal is to design a sub-optimal controller by simultaneously obtaining the control policy and the model parameters. To this end, the conventional problem of solving the Hamiltonian-Jacobi-Bellman (HJB) equation is transformed into a nonlinear non-convex optimisation problem. Then, by reformulating the constraints of the optimisation problem, it is relaxed into a convex Semi-Definite Programming (SDP). By presenting an iterative method, the control cost, the performance, and the number of iterations are improved compared to conventional methods, and a closer result to the optimal solution is obtained. The performance and efficacy of the proposed algorithm are investigated through a case study on the physical load positioning system.

LQR based optimal co-design for linear control systems with input and state constraints