2.4-1. Steady-state behavior. In this problem we consider a rather unrealistic discrete system because it is...

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2.4-1. Steady-state behavior. In this problem we consider a rather unrealistic discrete system because it is simple enough to allow an analytic treatment. Thus, let the plant Xk+1 have performance index of =[J] [o o xe +[i] Xk+ N-1 T 91 92 Jo = { XXN [22 22]x + ruz). +22 1 _92 91 Uk (1) a. Find the optimal steady-state (i.e., N ) Riccati solution S* and show that it is positive definite. Find the optimal steady-state gain K* and determine when it is nonzero. -K**k (2) b. Find the optimal steady-state closed-loop plant and demonstrate its stability. c. Now the suboptimal constant feedback Uk = - (3) is applied to the plant. Find scalar updates for the components of the sub- optimal cost kernel Sk. Find the suboptimal steady-state cost kernel S and demonstrate that S S** = 2.4-1. Steady-state behavior. In this problem we consider a rather unrealistic discrete system because it is simple enough to allow an analytic treatment. Thus, let the plant Xk+1 have performance index of =[J] [o o xe +[i] Xk+ N-1 T 91 92 Jo = { XXN [22 22]x + ruz). +22 1 _92 91 Uk (1) a. Find the optimal steady-state (i.e., N ) Riccati solution S* and show that it is positive definite. Find the optimal steady-state gain K* and determine when it is nonzero. -K**k (2) b. Find the optimal steady-state closed-loop plant and demonstrate its stability. c. Now the suboptimal constant feedback Uk = - (3) is applied to the plant. Find scalar updates for the components of the sub- optimal cost kernel Sk. Find the suboptimal steady-state cost kernel S and demonstrate that S S** =