# Article

**Keywords:**

discrete-time system; feedback dynamic system; polyhedral constraints; conical approach; linear feedback

**Summary:**

A previous paper by the same authors presented a general theory solving (finite horizon) feasibility and optimization problems for linear dynamic discrete-time systems with polyhedral constraints. We derived necessary and sufficient conditions for the existence of solutions without assuming any restrictive hypothesis. For the solvable cases we also provided the inequative feedback dynamic system, that generates by forward recursion all and nothing but the feasible (or optimal, according to the cases) solutions. This is what we call a dynamic (or automatic) solution. The crucial tool for the development of the theory was the conical approach to linear programming, illustrated in detail in a recent book by the first author. Here we extend this theory in two different directions. The first consists in generalizations for more complex constraint structures. We carry out two cases of mixed input state constraints, yielding the dynamic solution for both of them. The second case is particularly interesting because it appears at first sight hopeless, but, again, resort to the conical approach provides the key to overcome the difficulty. The second direction consists in evaluating the possibility of obtaining at least one solution to problems in the present class, by means of linear, instead of inequative, feedback. We illustrate three mechanisms that exclude any linear solution. In the first the linear feedback cannot handle cases where the origin is in the constraining set for the state. In the second the linear feedback lacks the initial condition independence of the inequative solution. In the third the linear feedback cannot control the geometric multiplicity of eigenvalues of the system, and this prevents stabilization, when the constraint structure is such that we cannot allow the state to converge to the origin. These results clearly strengthen the significance and relevance of the theory of linear (optimal) regulator.

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