Overapproximation of NCS Model Sample Clauses

Overapproximation of NCS Model. ∈ Direct controller synthesis based on (142) is difficult, due to the infinite number of possible values of the sampling intervals and delays (hk, τk) Θ, and to the nonlinear appearance of these uncer- tain parameters in the matrices A˜hk ,τk , B˜hk ,τk of the discrete time NCS model. A way to make the system (142) amenable for controller synthesis is to overapproximate it by a system in which the uncertainties appear in a polytopic and/or additive manner. This can be achieved by using one of the available overapproximation methods (see [153] for an overview and thorough comparison of all the existing overapproximation techniques). Here, we take a method derived in [150], that is based on the real Jordan form of the continuous-time system matrix A, although other techniques can be used as well. In the following this method is briefly summarized. Let the state matrix A = TJT—1, with J the real Jordan form of A, and T an invertible matrix. The integrals in (142) are computed by substituting eAs = TeJsT—1, in order to obtain a model in which the uncertain parameters hk and τk appear explicitly. This leads to a model of the form ξk+1 = A˜hk ,τk ξk + B˜hk ,τk uk, (144) Σ with (hk, τk) ∈ Θ, for all k ∈ N, where we can rewrite A˜hk ,τk and B˜hk ,τk in (142) as A˜hk,τk = F0 + 2ν Σ αi(hk, τk )Fi,