532 CHAPTER 13. FEEDBACK LINEARIZA’HON 6V1 T

= Wfo<m°l+ Zia—fume —fo<n.o>} — 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 > 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 > 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 < 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 < 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 < 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.