In the design problem of linear discrete-time control systems with input and state constraints, a feedback gain is admissible if all the constraints are satisfied. This paper presents the design method for obtaining an optimal linear admissible control law using a criterion for admissible gain when the set of initial states is given by a convex polyhedron.
This problem has attracted special interest recently. The positively invariant condition, which assures that the trajectories of control system from the set of initial states stay within the set, is often used to design the control law. In fact, this condition is too conservative, so that it can cover only a narrow region of admissible gains, and the resulting control system is in general not optimal in the range of admissible linear laws.
A method for computing the maximum value of constrained variables, α(f), is derived from the study in the dual space; this gives the necessary and sufficient condition for admissible gain and is more efficient than the method proposed by Kiendl because it requires less trajectory computations.
The design problem is formulated as the following programming problem: find an admissible gain minimizing a given performance index. As for the performance index, either a weighted quadratic sum of the states or a size of the state at a specific time is considered. It is also shown that the solvability of the problem can be examined by solving the minimization problem of α(f). Gain gradients on the function α(f) and the performance indices are derived, which are used in the optimization algorithms. The nonconvex property of these problems is pointed out and for each problem an algorithm finding a suboptimal gain is proposed. A 2nd-order control system design problem with constraints is treated as a numerical example, and satisfactory results are obtained, which prove the effectiveness of the design algorithm.
