Constrained stabilization of discrete‐time systems

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Abstract
Based on the growth rate of the set of states reachable with unit‐energy inputs, we show that a discretetime controllable linear system is globally controllable to the origin with constrained inputs if and only if all of its eigenvalues lie in the closed unit disk. These results imply that the constrained Infinite‐Horizon Model Predictive Control algorithm is stabilizing for a sufficiently large number of control moves if and only if the controlled system is stabilizable and all its eigenvalues lie in the closed unit disk.In the second part of the paper, we propose an implementable Model Predictive Control algorithm and show that with this scheme a discrete‐time linear system withnpoles on the unit disk (with any multiplicity) can be globally stabilized if the number of control moves is larger thann. For pure integrator systems, this condition is also necessary. Moreover, we show that global asymptotic stability is preserved for any asymptotically constant disturbance entering at the plant input.

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