Model predictive control for discrete-time systems with serially correlated parameters and multiplicative and additive noises under constraints

Let the control object is described by the equation: (1) where x(i)el"* is the vector of state; u(k)eMⁿ' is the vector of control; r|(£)eR^?is the stochastic vector; A^k) eR"'^y"', В-х(к'),к]еШ"'^Уп" = Цц(к),к] e М^и»^хи» . All of the elements of Bi [{k), k] (z = 0, 1, ..., n), D[r(k), k] are assumed to be linear functions of r| (k% {v(k) eK";4=0, 1,...}, {^w(k) e К"" ; к = 0, 1, ...} are white noise vectors with zero mean and unique co-variance matrix, E{w(k)v^T(s)} = 0, i?{r|(fc)v^T(5)} = 0, i?{r|(fc)w^T(.s)} =0 for all k, s. Let F = be the complete filtration with a-field generated by the {t|(,s): 5 = 0, 1,2, ...,k} that models the flow of in formation to time 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_k : E {л(к + i) / Fk} = n(k + i), E {_n(k + iW (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_msx(ky,S(k) _S R™ ;u_min(k),u_msx(k) s R*. (2) For control of system (1), we synthesize the strategies with a predictive control model. At each step k we minimize the following criterion with a receding horizon m m J(k + m / k) = Z E{x (k + i)R (k + i)x(k + i) / x(k), F} ~Z E{x (k + i) / x(k), F (k + i)E{x(k + i) / x(k), F} - (3) i=1 i=1 m m-1 R(k + i)E{x(k + i) / x(k),F} + Z E{u^T (k + i / k)R(k + i)u(k + i / k) / x(k), F}, i=1 i=0 on trajectories of system (1) over the sequence of predictive controls u(k/k), ..., u(k + m - 1/k) dependent on the system state x(k) and information up to time k F_k , under constraints (2); where R₁(k + i) > 0,R₂(k + i) > 0,R(k + i) > 0 are given symmetric weight matrices of corresponding dimensions; R₃ (k + i) is a given vector of corresponding dimension; m is the prediction horizon. Different cost functions can be obtained from criterion (3) after setting the coefficients Ri(k + i), R2(k + i), and R3(k + i) to some appropriate values. Problem 1. Taking R₂(k + i) = 0, we have the MPC problem with quadratic criterion. Problem 2. Let system (1) have a scalar output y(k) = L(k)x(k), where L(k) is a vector of appropriate dimension. Taking R (k + i) = R (k + i) = |j(k + i)L (k + i)L(k + i), R (k + i) = X(k + i)L(k + i), (i = 1, m), where ^(k + i) > 0, X(k + i) > 0 are scalar values, we have a mean-variance optimization problem.

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