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Guaranteed cost control for discrete-time linear systems under controller gain perturbations. (English) Zbl 0960.93036

The authors consider the problem of guaranteed cost control for discrete-time linear systems described by the equation \(x_{k+1} = Ax_k + Bu_k,\) where \(x_k\in\mathbb R^n\) is the state and \(u_k\in\mathbb R^m\) is the control input, \(A\) and \(B\) are known constant matrices. The cost function associated with this system is \(J=\sum_{k=0}^\infty(x_k^TQx_k+u_k^TRu_k)\), where \(Q > 0\) and \(R > 0\) are given weighting matrices. For a given controller \(u_k= Kx_k,\) the actual controller implemented is assumed to be \(u_k = (K +\Delta K)x_k,\) where \(K\) is the nominal controller gain, and \(\Delta K\) represents the gain perturbations. Two classes of perturbations are considered: (a) \(\Delta K\) is of the additive form \(\Delta K=H_1F_1E_1, F_1^TF_1\leqslant \rho I, \rho\geqslant 0,\) with \(H_1\) and \(E_1\) being known constant matrices, and \(F_1\) the uncertain parameter matrix. (b) \(\Delta K\) is of the multiplicative form \(\Delta K=H_2F_2E_2K, F_2^TF_2\leqslant \rho I, \rho\geqslant 0,\) with \(H_2\) and \(E_2\) being known constant matrices, and \(F_2\) the uncertain parameter matrix. The state feedback control designs are given in terms of solutions to algebraic Riccati equations. The designs are such that the cost of the closed-loop system is guaranteed to be within a certain bound for all admissible uncertainties. A numerical example is given to illustrate the design procedures.

MSC:

93C73 Perturbations in control/observation systems
93C55 Discrete-time control/observation systems
49N10 Linear-quadratic optimal control problems
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