Analysis and Design of $$H_{\infty }$$ H ∞ Controllers for 2D Singular Systems with Delays

Abstract

The H ∞ control design problem is solved for the class of 2D discrete singular systems with delays. More precisely, the problem addressed is the design of state-feedback controllers such that the acceptability, internal stability and causality of the resulting closed-loop system are guaranteed, while a prescribed H ∞ performance level is simultaneously fulfilled. By establishing a novel version of the bounded real lemma, a linear matrix inequality condition is derived for the existence of these H ∞ controllers. They can then be designed by solving an iterative algorithm based on LMI optimizations. An illustrative example shows the applicability of the algorithm proposed.

Admissibility and robust H∞ controller design for uncertain 2D singular discrete systems with interval time‐varying delays

“…Denote x h (i, j) = T(iΔx − Δx, jΔt), x v (i, j) = T(iΔx, jΔt), assume that the disturbance input is given by 𝜔(i, j) and 𝜏 = Δt. It is easy to see that Equation ( 46) can be converted into the 2-D singular delayed system (1) without parameter uncertainty and with parameter matrices: We now compare Corollary 1 with the results proposed in Xu and Zou 36 and Kririm et al 37 Noting that the result given in Xu and Zou 36 and Kririm et al 37 are delay-independent and the result given in Corollary 1 is delay-dependent. Thus Table 3 shows a comparison results on minimum disturbance attenuation 𝛾 min achieved in each method, from which it can be seen that our results have less conservatism than those given in Xu and Zou 36 and Kririm et al 37 The obtained H ∞ controller is u(i, j) = [ − 7.4232 44.3762 ] x(i, j), after applying this controller, the closed-loop system is stabilized as depicted in the state responses of the closed-loop system given in Figure 1, which confirm that the designed controller is efficient, the frequency response is shown in Figure 2, and its maximum value is 0.0356 that is smaller than 𝛾.…”

“…1. The conditions given in Theorem 1 is delay-dependent, which is less conservative than delay-independent one given in Xu and Zou 36 and Kririm et al 37 It should be mentioned that the delay-independent conditions do not take the delay size in the consideration, so they are conservative for many systems. 2.…”

“…Consider the chemical reactors, heat exchange, and pipe furnaces given in Xu and Zou 36 and Kririm et al, 37 which can be expressed in the partial differential equation with time delays:…”

“…In the literature there are few work concerning the stability and stabilization of 2‐D singular systems with time‐varying delays, in the work of Wang et al 34 based on the Lyapunov‐Krasovskii's functional (LKF) method and by employing a Jensen‐type discrete inequality to manipulate the difference of a LKF candidate, a sufficient delay‐dependent condition is obtained to analyse the stability of 2‐D state‐varying delayed singular systems, the condition is expressed in terms of linear matrix inequalities, in the other hand in the work of Hien et al 35 an approach is proposed to analyse the stability for generalized directional Roesser delayed systems, by mean of 2‐D Lyapunov‐Krasovskii's functional candidate and by employing zero‐type free matrix equality, new delay‐dependent LMIs conditions are proposed to answer that, if the system under consideration is admissible. The problem of $$$ {H}_{\infty } $$$ control was also studied in (Xu and Zou; 36 Kririm et al 37 ) for 2‐D singular Roesser models with constant delays, by using the bounded real lemma approach, delay‐independent LMI‐based conditions were derived for the design of state feedback controllers guarantying the closed‐loop system to be admissible.…”

“…In the sequel to the above, there is no results in the literature that deals with the robust $$$ {H}_{\infty } $$$ control for uncertain 2‐D singular systems with time‐varying delays and all the work are restricted to studying the stability for 2‐D singular systems with time‐varying delay (see e.g., Wang et al 34 and Hien et al 35 ), in the other hand, in (Xu and Zou; 36 Kririm et al 37 ) the problem of $$$ {H}_{\infty } $$$ control for 2‐D singular systems with constant delay is studied, and all given conditions are delay‐independent. From this, we have motivated to study the problem of admissibility and robust $$$ {H}_{\infty } $$$ control for uncertain 2‐D singular systems with time‐varying delays and norm‐bounded parameter uncertainties, then by utilizing an augmented Lyapunov‐Krasovskii's functional and employing new version of Jensen inequality which allows us to introduce a zero‐type free matrix equations inspired from the work of Kim 38 …”