## The explicit linear quadratic regulator for constrained systems.(English)Zbl 0999.93018

A technique to compute the explicit state-feedback solution of a discrete-time linear quadratic control problem subject to state and input constraints is presented. First the quadratic program (QP), which must be solved to determine the optimal control action, is derived. The original QP is viewed as a multi-parametric QP (mp-QP). The properties of an mp-QP are analysed, and an efficient algorithm to solve it is developed. It is shown that the closed form solution is piecewise affine and continuous for both the finite horizon problem (model predictive control, MPC) and the usual infinite time measure (constrained linear quadratic regulation). The controller can be implemented with substantially reduced on-line calculations preserving all performance and stability properties of MPC. The special on-line QP solvers are no longer required, only the evaluation of an explicitly defined piecewise linear function must be performed on-line. The proposed technique is attractive for a wide range of practical problems in which the computational complexity of on-line optimization is prohibitive.

### MSC:

 93B40 Computational methods in systems theory (MSC2010) 93B51 Design techniques (robust design, computer-aided design, etc.) 93C55 Discrete-time control/observation systems 49N10 Linear-quadratic optimal control problems 65Y20 Complexity and performance of numerical algorithms

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