New iterative linear matrix inequality based procedure for H₂ and H_∞ state feedback control of continuous‐time polytopic systems

Abstract

This article provides new linear matrix inequality (LMI) sufficient conditions for a generalized robust state feedback control synthesis problem for linear continuous-time polytopic systems. This generalized problem includes the robust stability, H 2-norm, and H ∞-norm problems as special cases. Using a novel general separation result, which separates the state feedback gain from the Lyapunov matrix but with the state feedback gain synthesized from the slack variable, then allows the formulation of LMI sufficient conditions for the generalized problem. Compared to existing parameterized LMI based conditions, where auxiliary scalar parameters are introduced in order to include the quadratic stability conditions (ie, assuming a constant Lyapunov matrix) as a special case, the proposed new conditions are true LMIs and contain as a particular case the optimal quadratic stability solution. Utilizing any initial solution derived by the quadratic or some existing methods as a starting solution, we propose an algorithm based on an iterative procedure, which is recursively feasible in each update, to compute a sequence of nonincreasing upper bounds for the H 2-norm and H ∞-norm. In addition, if no feasible initial solution can be found for some uncertain systems using any existing methods, another algorithm is presented that offers the possibility of obtaining a robust stabilizing gain. Numerical examples from the literature demonstrate that our algorithms can provide less conservative results than existing methods, and they can also find feasible solutions where all other methods fail. K E Y W O R D S H 2 and H ∞ norms, LMI relaxations, polytopic uncertainties, robust state feedback control, slack variables, uncertain linear systems 1 INTRODUCTION The design of robust H 2-norm and H ∞-norm controllers for linear time-invariant systems with polytopic uncertainties has received considerable attention in the past decades. Many studies devoted to investigating robust stability and This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Mixed performance for robust fuzzy control of nonlinear autonomous surface vehicle via T‐S model approach

“…From the simulation, the performance of the ASV cannot be guaranteed and is not satisfied for the requirement of motion. By using the input in (4) with K i , i {1, 2, 3}, in (17), and the above disturbance input, the regulated output vector y(t) ℜ 3 with zero initial state and the proposed compensator is shown in Figure 6. From Figure 6, the H ∞ performance γ = 0.1 can be observed.…”

“…performace has been an interesting research topic in recent years [16][17][18][19][20]. To the best of authors' knowledge, there are less results considering the mixed H 2 /H ∞ performance about ASV under consideration.…”

“…On the other hand, H 2 performance of system under consideration would be an another requirement to minimize a quadratic performance index about the initial state of system under no disturbance [15, 16]. Hence, the mixed performace has been an interesting research topic in recent years [16–20]. To the best of authors' knowledge, there are less results considering the mixed H 2 / H ∞ performance about ASV under consideration.…”