ECE 517
532 CHAPTER 13. FEEDBACK LINEARIZA'HON 6V1 T = Wfo&lt;ml+ Ziafume fo&lt;n.o&gt;} 15i 2/ TPE On any bounded neighborhood of the origin, we can use...
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# src="/qa/attachment/10483484/" alt="Screen Shot 2019-09-23 at 8.12.05 PM.png" />I am trying to recreate the models in questions 13.16 and 13.17(attached) don simulink. I found the solution(attached) but I'm having trouble recreating it myself. Specifically I'm not sure what to put for the boxes that say "eta2" and "xi21". I have also attached a pic of what I have so far below. Thanks Attachment 1 Attachment 2 Attachment 3 ATTACHMENT PREVIEW Download attachment Screen Shot 2019-09-23 at 8.11.52 PM.png 532 CHAPTER 13. FEEDBACK LINEARIZA’HON 6V1 T = Wfo&lt;m°l+ Zia—fume —fo&lt;n.o&gt;} — 15—i— 2\/ éTPE On any bounded neighborhood of the origin, we can use continuous diiferentiability of V1 and f0 to obtain V S —0ta(lll7ll) + killéll — kkzllﬁll for some positive constants k1 and k2. Choosing k &gt; 161 / k2 ensures that V is nega— tive deﬁnite. Hence, the origin is asymptotically stable. E1 The foregoing discussion shows that a minimum phase input—output linearizable system can be stabilized by the state feedback control u = a(:c) — ﬂ(:c)KT2(z) ‘ (13.43) The control (13.43) is independent of T1(z). Therefore, it is independent of the function (I) that satisﬁes the partial diﬂerential equation (13.15). The proof of Lemma 13.1 is valid only on bounded sets. Hence, it cannot be extended to show global asymptotic stability. We can show global asymptotic stability by requiring the system i] = fo(7],€) to be input-to—state stable when 5 is viewed as the input. Lemma 13.2 The origin of (13.41)—(13.42) is globally asymptotically stable if the system 1'7 = fo(n,f) is input-to-state stable 0 Proof: Apply Lemma 4.7. CI lnput-to—state stability of 7'; = fo(7h 5) does not follow from global asymptotic, or even exponential, stability of the origin of 1'] = fo(17, 0), as we saw in Section 4.10. Consequently, knowing that an input—output linearizable system is “globally” min- imum phase does not automatically guarantee that the control (13.43) will globally stabilize the system. It will be globally stabilizing if the origin of 7'] = fo(r/, 0) is globally exponentially stable and fo(n, 5) is globally Lipschitz in (17, 5), since in that case Lemma 4.6 conﬁrms that the system 7'; = f0 (17, 5) will be input-to—state stable. Otherwise, we have to establish input-to-state stability by further analysis. Global Lipschitz conditions are sometimes referred to as linear growth conditions. The next two examples illustrate some of the difﬁculties that may arise in the absence of linear growth conditions. Example 13.16 Consider the second-order system 7'7 = -'n + 1725 5 = v While the origin of i7 = —7] is globally exponentially stable, the system 7‘] = —71+n2€ is not input-to-state stable. This fact can be seen by noting that £(t) E 1 and 13.4. STATE FEEDBACK CONTROL 533 77(0) 2 2 imply that 17(t) Z 2. Therefore, ’1’] grows unbounded. On the other hand, by Lemma 13.1, we see that the linear control 1) = —k§, with k &gt; 0, stabilizes the origin of the full system. In fact, the origin will be exponentially stable. However, this linear control does not make the origin globally asymptotically stable. Taking V = 7/5 and noting that 9=n€+17€=-kn€-n€+n252=-(1+k)V+I/2 we see that the set {M &lt; 1 +lc} is positively invariant. On the boundary 175 = 1+k, the trajectory is given by ”(t) = ek‘n(0) and {(t) = (“5(0) Thus, 17(t)€(t) E 1+k. Inside the set {n5 &lt; 1 + k}, I/(t) will be strictly decreasing and after a ﬁnite time T, 11(t) S 1/2 for all t Z T. Then, 7W- S —(1/2)n2, for all t Z T, which shows that the trajectory approaches the origin as it tends to inﬁnity. Hence, the set {né &lt; 1 + k} is the exact region of attraction. While this conclusion shows that the origin is not globally asymptotically stable, it also shows that the region of attraction expands as k increases In fact by choosing k large enough, we can include any compact set in the region of attraction. Thus, the linear feedback control n = —lc§ can achieve semiglobal stabilization. A If the origin of 7'] = fo(17,0) is globally asymptotically stable, one might think that the triangular system (13.39)—(13.40) can be globally stabilized, or at least semiglobally stabilized, by designing the linear feedback control 1; = —K§ to assign the. eigenvalues of (A — BK) far to the left in the complex plane so that the solution of § = (A — BK )5 decays to zero arbitrarily fast. Then, the solution of E = fo(n, E) will quickly approach the solution of 'i] = fo(7],0), which is well behaved, because its origin is globally asymptotically stable. It may even appear that this strategy is the one used to achieve semiglobal stabilization in the preceding example. The next example shows why such strategy may fail.9 Example 13.17 Consider the third-order system «7 = -%(1+€2)773 51 = 52 £2 = ’U The linear feedback control v = —k”e — wee “=e‘ —K€ assigns the eigenvalues of 0 1 A—BK=[_k2 -21.] 9See, however, Exercise 13.20 for a special case where this strategy will work. ATTACHMENT PREVIEW Download attachment Screen Shot 2019-09-23 at 8.12.05 PM.png 5. Simulate the control systems described in Examples 13.16 and 13.17 in Khalil’s Nonlinear Systems book (3rd edition, pp. 532—534) and conﬁrm the unstable behavior of closed-loop solutions. ATTACHMENT PREVIEW Download attachment Screen Shot 2019-09-23 at 8.12.20 PM.png 2) In Examples 13.17, we consider the third-order system 1 n= —§(1+€2)n3, £1 =62: \$2 = v! and the linear feedback control 1) = 4851— 2M2, k &gt; o. If 17% &gt; 1, the system will have a ﬁnite escape time if k is chosen large enough. The Simulink diagram and the simulation result can be found in Fig. 2. Convergence after peaking with k = 4 Finite escape time with k = 4.22 data 0 2 4 6 B 10 0 0.1 0.2 0.3 0.4 0.5 DW’“ time (seconds) time (seconds) (a) Simulink diagram (b) Simulation result (17(0) = 4, {1(0) = {2(0) = 1) Fig. 2: Problem 5.2)

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