LMI-based robust sampled-data stabilization of polytopic LTI systems: A truncated power series expansion approach

Abstract

For continuous-time linear time-invariant (LTI) systems with polytopic uncertainties, we develop a robust sampled-data state-feedback control design scheme in terms of linear matrix inequalities (LMIs). Truncated power series expansions are used to approximate a discretized model of the original continuous-time system. The system matrices obtained by using the power series approximations are then expressed as homogeneous polynomial parameter-dependent (HPPD) matrices of finite degrees, and conditions for designing the controller are formulated as a HPPD matrix inequality, which can be solved by means of a recent LMI relaxation technique to test the positivity of HPPD matrices with variables in the simplex. To take care of the errors induced by the remainder terms of the truncated power series, the terms are considered as norm bounded uncertainties and then incorporated into the proposed LMI conditions. Finally, examples are used to illustrate the approach.

Analytical Upper Bound for the Error on the Discretization of Uncertain Linear Systems by using the Tensor Product Model Transformation

“…. Contrast this sampling time with the maximum ones obtained by the methods of [30], = ms, and of [10], = ms, with error bounds of = . and = .…”

“…To illustrate the advantages of the proposed discretization procedure, we synthesize digital state feedback controllers based on the derived discretized model. For that, we adapted the controller design condition provided in [30], which is given in the following theorem.…”

“…We compare our discretization procedure with the ones of [10] and [30] in terms of the maximum sampling period and the bounds of the discretization error. Notice that our approach and the one of [30] provide distinct upper bounds for matrices ̂( ) and ̂( ), whereas the approach of [10] provides a single upper bound for both matrices. Two numerical examples are provided to this end.…”

“…For the method of [30], a truncated order Taylor series expansion was adopted, polynomial matrices of degree 8 and = (a parameter used by [30] to calculate the discretization error); whereas, for the approach of [10], a truncated order Taylor series expansion is considered with affine-dependent optimization variables. Notice that our discretization period is about 1.7 times greater than the one of [30] whereas it is the same as [10]. Nevertheless, our approach does not require determining the Taylor series expansion of each matrix in order to obtain a discretized model and, furthermore, the analytic upper bound for the discretization error is about 10 times smaller than in the other methods.…”

“…Table 1 summarizes the results attained for each aforementioned method. 178 ms 0.0043 0.0012 [30] 103 ms 0.0463 0.0051 [10] 178 ms 0.0473…”