Bilinear matrix inequality‐based nonquadratic controller design for polytopic‐linear parameter varying systems

Abstract

This article proposes relaxed sufficient bilinear matrix inequality (BMI) conditions to design a gain-scheduling controller for nonlinear systems described by polytopic-linear parameter varying (LPV) representations. The obtained conditions are derived based on a nonquadratic Lyapunov function and a parallel distributed compensator scheme. The controller design procedure involves some novel null terms and leads to a BMI problem, which hardly has been solved in previous researches. Furthermore, to effectively solve the BMI conditions, a novel sequential approach is proposed which convert the overall BMI problem into linear matrix inequality (LMI) constraints and some simpler BMI conditions with fewer dimensions than the original one. Initially, the LMI conditions are solved as a convex optimization problem. Second, the BMI terms are iteratively linearized near the feasible solutions of the LMIs and each solution candidates for the BMI constraints. Finally, the linearized condition is solved as an eigenvalue problem. To show the merits of the proposed approach, several numerical comparisons and simulations are provided.

Optimal gain‐scheduling control of proton exchange membrane fuel cell: An LMI approach

“…From (31), one concludes that the Lyapunov function is decreasing. This means X T (t )P X (t )| t →∞ = 0.…”

“…A polytopic-LPV is a class of LPV systems, which its main property is that it comprises linear systems as the vertices, and the stability analysis and control synthesis can be assured only based on the vertices [31,32]. Such a feature provides a systematic approach to design a proper controller.…”

Optimal Robust LPV Control Design for Novel Covid-19 Disease

“…Therefore, 𝐴, 𝐵 and 𝐶 matrices are quantized as follows: It can be noticed that in (8), A matrix has an entry with some uncertainties and B matrix is made of three entries that include some uncertainties. So, the polytopic model [27][28][29] of this system is like a multidimensional space which has 16 vertices. It means that there are 8 different B matrices and two different A matrices as: 𝐵 = 𝐵 1 𝛿 1 + 𝐵 2 𝛿 2 + ⋯ + 𝐵 8 𝛿 8 ,𝐴 = 𝐴 1 𝛿 1 𝐴 2 𝛿 2 (9) Where 𝛿 𝑖 represent uncertainties.…”

Optimal Frequency Regulation in AC Mobile Power Grids Exploiting Bilinear Matrix Inequalities

“…A challenge in the formulation of the S-LFC problem in literature is that the main problem results in Bilinear Matrix O Inequalities (BMI) and there are restricted approaches for dealing with BMIs. To solve BMIs, different approaches are presented including Alternate Minimization (AM) [18], [19], Inner Convex Approximation Method (ICAM) [20] Iterative linear matrix inequalities (ILMIs) [21], [22], two-step process [23], Sequential Parametric Convex Approximation (SPCA) [24], [25], Branch and Bound (BB) [26], path-following methods [27] and Method of Reduction of Variables (MRV) [28] . However, all existing approaches have some drawbacks.…”