The closed-loop optimal feedback model predictive control policy for systems with stochastic correlated parameters

We consider the following discrete-time with stochastic parameters system on the probabilistic space (Q, F ,P): x(k +1) = Ax(k) + B[n(k +1), k + 1]u(k), (1) where x(k) e Kⁿ* is the vector of state, u(k) e Kⁿ" is the vector of control inputs; n(k) e К^q is assumed to be stochastic time series. The matrices A e Кⁿx ^xnx, B [п (k), k] e К ^x u are the system matrix and the input matrix, respectively. All the elements of B[n(k),k] are assumed to be linear functions of n(k). Let F =( Fk )_k>i be the complete filtration with a-field Fk generated by the {n(s): s=0,1,2,... ,k} that models the flow of information to time k. We allow the time series n(k) is serially correlated. Let assume that we know the first- and the second-order conditional moments for the stochastic vector n(k) about Fk : E {n(k + i)/ Fk } = n(k + i), E {n(k + i)n^T (k + j)/ Fk } = © j (k ),(k = 0,1,2,...),(i, j = 1,2,..., l). We define the following cost function with receding horizon, which is to be minimized at every time k J (k + m / k) = E j j x^T (k + i) R₁(k, i) x(k + i) + u ^T (k + i -1/ k) R(k, i)u(k + i -1/ k) / x(k), F_k j, (2) on trajectories of system (1) over the sequence of predictive control inputs u(k/k),.,u(k+m-1/k) dependent on information up to time k, where R₁(k,i) > 0, R(k,i) > 0 are given symmetric weight matrices of corresponding dimensions; m is the prediction horizon. The closed-loop optimal feedback law minimizing criterion (2) was derived via dynamic programming. Conditions that guarantee the stability of the infinite horizon formulation are given.

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