Finsler's Lemma: Let x R, WE Sn, and F e Rmxn with rank(F) < n....

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Finsler's Lemma: Let x € R", WE Sn, and F e Rmxn with rank(F) < n. The following statements are equivalent: 1. x W x < 0 for all Fx = 0 (x = 0). 2. F-TWF <0. 3. There exists a scalar μ such that W - FTF <0. 4. There exists an X € Rn×m such that W + XF + FT XT < 0. To begin, write down the closed-loop dynamics under the control law u(t)= Kr(t) and apply the Lyapunov LMI condition. Then show that this can be expressed as item 4 in Finsler's lemma. Use the equivalence of 4 and 3 with a change of variables and a congruence transformation to obtain the desired result. [3] Finsler's Lemma: Let x € R", WE Sn, and F e Rmxn with rank(F) < n. The following statements are equivalent: 1. x W x < 0 for all Fx = 0 (x = 0). 2. F-TWF <0. 3. There exists a scalar μ such that W - FTF <0. 4. There exists an X € Rn×m such that W + XF + FT XT < 0. To begin, write down the closed-loop dynamics under the control law u(t)= Kr(t) and apply the Lyapunov LMI condition. Then show that this can be expressed as item 4 in Finsler's lemma. Use the equivalence of 4 and 3 with a change of variables and a congruence transformation to obtain the desired result. [3]

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The question provided outlines Finslers Lemma and sets up the task of showing an equivalence between a certain Lyapunov Linear Matrix Inequality LMI c... View the full answer