Model predictive control for nonlinear stochastic systems with serially correlated parameters under constraints

Let the control object is described by the equation: x(k +1) = Ax(k) + B[r(k +1), k + 1]u(k) + f (x(k), u(k), w(k +1)), (1) where x(k) is the nx-dimensional vector of state; u(k) is the nu-dimensional vector of control; w(k) is the nw-dimensional vector of white noses with zero-mean and identity covariance matrix; n(k) is the ^-dimensional stochastic vector; w(k) is independent of n(k) (k = 0, 1, 2...); A, B[n(k),k] are the matrices of corresponding dimensions. All of the elements of B[n(k),k] are assumed to be linear functions of n(k). The function f is defined by its statistical properties as follows: E { f (x(k ),u(k ),w(k+1))/ x(k )} = 0, ^{E {f} (x(k )u(k ),^w(k+¹⁾⁾f T (x(k )u(k ),^w(k+1)>/^x(k )}=^T 0+IT ^(xT(k )Wx(k )+uT(k )Mu(k)) for all x(k), where r = n(n + 1)/2, T^l (i = 0,r), W' and M' (i = 1,r) are positive semidefinite and symmetric matrices. Let F = (§£be the complete filtration with a-field generated by the {t|(,s): 5 = 0, 1,2, ...,k} that models the flow of information to the moment k. We allow the parameters n(k) to be serially correlated. Let us assume that we know the first- and second-order conditional moments for the stochastic vector n(k) about F : E {r(k + i)/F_k } = r(k + i), E{r(k + i)r^T (k + j) / Fk } = (k), (k = 0,1,2,...), (i, j = 0,1,2,..., d). We impose the following inequality constraints on the control inputs (element-wise inequality): u_mm(k) < S(k)u(k) < u_max(k), (2) where S(k) is the matrix of corresponding dimension. For control of system (1) we synthesize the strategies with a predictive control model. At each step k we minimize the quadratic criterion with a receding horizon m J(k+mjk) = E{ ^ x^T (k + i)R (k + i)x(k + i) - R(k + i)x(k + i) + u^T (k + i - 1/k)R(k + i - 1)u(k + i - 1/k) /x(k), F}, i=1 on trajectories of system (1) over the sequence of predictive controls u(kk), ..., u(k + m- 1/k) dependent on information up to moment k, under constraints (2), where R (k + i) > 0, R(k + i) > 0 are given symmetric weight matrices of corresponding dimensions, R (k + i) is a given vector of corresponding dimension, m is the prediction horizon, k is the current moment. The synthesis of predictive control strategies is reduced to the sequence of quadratic programming tasks.

Keywords

Authors

References