tailieunhanh - Switched Systems 2012 Part 3

Tham khảo tài liệu 'switched systems 2012 part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Stabilization of saturated switching systems 21 with Y AiK Gi BiK DisYiCiK BiK D Zi T 1 Zil - Gi gT Xi 0 84 Vk 1 . VCk CiK G . Pi V i j eZ2 Vs e 1 y Vl e 1 m with 85 Hi ZtGf1 Ki YiVfi1 Pi X 1 86 then the closed-loop uncertain saturated switching system 69 is asymptotically stable V x0 e Q and for all switching sequences aft . Proof The proof is similar to that given for Corollary These results can be illustrated with the following example. Example . Consider a SISO saturated switching discrete system with two modes given by the following matrices A1 q1 t 1 1 0 1 q11 B1 q1 t 10 5 C1 q1 t 1 q12 1 A2 q2 t 0 q21 1 1 B2 q2 t 2 q22 C2 q2 t 23 . The vertices of the domain of uncertainties that affect the first mode are V11 v12 v13 0-1 v14 . The vertices of the domain of uncertainties that affect the second mode are V21 v22 v23 v24 . Using Theorem a stabilizing controller for the nominal system is K1 K2 . To test the robustness we can use the Corollary which leads to the following results P1 P 2 On the other hand the use of Theorem leads to the following results K1 K2 . 22 Switched Systems Figures 5 6 and 7 concern the first method. In Figure 5 the switching signals a t and the evolution of uncertainties used for simulation are plotted. Figure 6 shows the obtained level set of stability Up 1 efPi 1 which is well contained inside the sets of saturations while Figure 7 presents some system motions evolving inside the level set starting from different initial states. Fig. 5. Switching signals a t and uncertainties evolution Fig. 6. Inclusion of the ellipsoids inside the polyhedral sets Fig. 7. Motion of the system with controllers obtained with and Corollary Figure 7 shows the level set of stability ui 1 fiPi 1 using the second method of Theorem which is well contained inside the sets of