Polynomial control design for polynomial systems: A non-iterative sum of squares approach

Abstract

This paper proposes a non-iterative state feedback design approach for polynomial systems using polynomial Lyapunov function based on the sum of squares (SOS) decomposition. The polynomial Lyapunov matrix consists of states of the system leading to the non-convex problem. A lower bound on the time derivative of the Lyapunov matrix is considered to turn the non-convex problem into a convex one; and hence, the solutions are computed through semi-definite programming methods in a non-iterative fashion. Furthermore, we show that the proposed approach can be applied to a wide range of practical and industrial systems that their controller design is challenging, such as different chaotic systems, chemical continuous stirred tank reactor, and power permanent magnet synchronous machine. Finally, software-in-the-loop (SiL) real-time simulations are presented to prove the practical application of the proposed approach.

Continuous Stability TS Fuzzy Systems Novel Frame Controlled by a Discrete Approach and Based on SOS Methodology

“…As shown in Table 1, δ increases by increasing m. By comparing our proposed idea with the existing results in the literature, the stability regions of the continuous model (22) in the plan a-b are presented in Figures 1-8. Figures 1-3 show the existing results:…”

“…Solving the conditions presented in Theorem 3 leads to feasible solutions for m 1 and b ≥ 1. To expand the solutions feasibility that guarantee the continuous stability model (22), the proposed idea was to apply the previously-described steps. Indeed, a ∈ [0, 25] was maintained, and parameter b was adjusted as much as possible to obtain the largest stabilization regions that guarantees the Euler discrete stabilization system.…”

“…Mathematics 2021, 9, x FOR PEER REVIEW 8 of 20 the continuous stability model (22), the proposed idea was to apply the previously-described steps. Indeed, [ ] 0, 25 a ∈ was maintained, and parameter b was adjusted as much as possible to obtain the largest stabilization regions that guarantees the Euler discrete stabilization system.…”

“…As shown in Table 1, δ increases by increasing m . By comparing our proposed idea with the existing results in the literature, the stability regions of the continuous model (22) in the plan ab are presented in Figures 1-8. Figures 1-3 show the existing results: , the discrete gains are given by: , the discrete gains are given by: , the discrete gains are given by: Based on the proposed idea, Figures 4-8 display the stability regions obtained for several values of m and δ in order to show their influences on the stability results.…”

“…While some studies focused on the LMI conditions, another trend studied the stability analysis of polynomial fuzzy models based on the sum of squares techniques (SOS). In this case, convex optimization problems as LMI conditions are reformulated as sum of square decomposition problems [17][18][19][20][21][22][23][24]. This approach was intended for modeling and controlling nonlinear systems using polynomial fuzzy models.…”