One-step memory closed-loop control of linear interconnected subsystems (Q2718819)

Consider the problem \(J_i=x_n'Q_ix_n+\sum_{k=0}^{n-1}(x_k'Q_ix_k+u_{1,k}'R_{i1}u_{1,k} +u_{i,k}'R_iu_{i,k})\), \(i=1,\dots ,m,\) s.t. \(x_{k+1}=Ax_k+B_1u_{1,k}+\sum_{i=2}^mB_iu_{i,k}\), where \(A, B_i, Q_i\geq 0, R_i>0, R_{i1}>0\) for \(i=2,\dots m\), and \(R_{11}=0\) are constant matrices of appropriate dimensions. The paper deals with the additional possibilities of the first subsystem to control \(x_k\), \(k=0,\dots ,n-1\) by changing feedback representation of \(u_{1,k}\) in \(J_i\). Suppose the closed-loop system with the first subsystem controller generates one of feasible trajectories \(x_k^0\), \(k=0,\dots ,n\), where \(x_k^0=A_{k-1}x_{k-1}^0\), \(x^0=x_0\), \(A_{k-1}=A-\sum_{i=1}^mB_iK_{i,k-1}^0\). The one-step memory representation \(u_{1,k}^{i0}=-K_{1,k}^0x_k+P_{i,k}(x_k-A_{k-1}x_{k-1})\) is adopted, where \(P_{i,k}\) is an arbitrary nonzero weighting matrix. Its optimal selection is mentioned. A numerical example is supplied.