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1、Chapter 4External Stability:Notions and Characterizations4.1 IntroductionThis chapter is devoted to the analysis of external global stability notions used in mathematical control and system theories. The presented stability notions are de-veloped in the system-theoretic framework described in Chap.
2、1 so that one can obtain a wide perspective of the role of stability in various classes of deterministic systems. The results in this chapter are of both theoretic importance and practical relevance since almost all engineering and natural systems are subject to external input signals, which may tak
3、e diverse forms as reference signals and actuator and sensor disturbances.Another feature that distinguishes the “external” stability notions developed in this chapter from “internal” stability notions introduced in Chap. 2 is that the no-tions are not uniform with respect to the effect of external
4、inputs. Therefore, the notions are not direct extensions of the corresponding stability notions for dynam-ical systems with no external inputs, often referred to as the disturbance-free case. More specifically, the notions presented in this chapter are variations of the no-tion of Input-to-State Sta
5、bility (ISS), introduced by E.D. Sontag in his seminal work 22. The ISS property has been proved to be very useful for the study of nonlinear systems, since it captures two main stability notions: Lyapunov stabil-ity (describing the behavior of zero-input response with respect to nonzero initial con
6、ditions) and bounded-input bounded-state stability (describing the behavior of zero-state response with respect to nonzero inputs); see 24 for further details.A large part of this chapter is also devoted to the presentation of methods of proving external stability properties. In this chapter, we foc
7、us mainly on Lyapunov methods, although transformation methods and analytical solutions can be exploited as in Chap. 2. However, Lyapunov methods have been proved to be much more useful for the derivation of useful inequalities that show specific external stability properties. Small-Gain methods wil
8、l be the subject of the following chapter.In what follows, := (X , Y , MU , MD , , , H ) will be a control system with the BIC property and for which 0 X is a robust equilibrium point from the input u MU . Moreover, u0 MU will be the identically zero input, i.e., u0(t ) = 0 UI. Karafyllis, Z.-P. Jia
9、ng, Stability and Stabilization of Nonlinear Systems,143Communications and Control Engineering,DOI 10.1007/978-0-85729-513-2_4, Springer-Verlag London Limited 20111444External Stability: Notions and Characterizationsfor all t 0. For the output map H :+ X U Y , we assume that eitherH : + X U Y is con
10、tinuous or that there exists a partition = i i=0 of + with diameter r 0 such that H : + X U Y satisfies Hypothesis (L2) in Sect. 1.7 of Chap. 1.4.2 DefinitionsTo begin with, consider a RGAOS system . In other words, according to Theo-rem 2.1, there exist functions KL and , K+ such that the following
11、 esti-mate holds for all (t0, x0, d ) + X MD and t t0:Ht , (t , t0, x0, u0, d ), 0 Y+ (t ) (t , t0, x0, u0, d ) X(t0) x0 X , t t0As control engineers, a natural question to ask is whether or not this RGAOS prop-erty is robust in the face of nonzero external input u MU , i.e., u(t ) 0. While “robustn
12、ess” is mathematically interpreted as “structural stability,” the essence of the question is to understand the impact of external inputs on the behavior of sys-tem , or more precisely the output variables of system .The answer to the above question is critically important for many practical prob-lem
13、s, but the answer to the question in our context of general complex systems is far from obvious. For example, there is this well-known phenomenon of “finite es-cape” (i.e., solutions blow up in finite time) when is a highly nonlinear system. An elementary example of this kind is the scalar system x
14、= x + x2u with external input u. Clearly, the zero-input system x = x is globally exponentially stable, and thus RGAOS. However, for any arbitrarily small constant input u = u = 0, some solutions blow up in finite time. Indeed, for any initial condition x(0) = x0 such thatux0 1 and for t 0, the asso
15、ciated solution x(t ) =tx0+goes to +e (1u x0)u x0as t ln( u x0 ) +.u x01Even in the absence of the finite escape phenomenon for every admissible external input u MU , there is still no straightforward answer to the question we ask. This is because, even when the solutions are defined for each posi-t
16、ive time, it is still possible that certain inputs produce unbounded output re-sponses, i.e., lim supt + H (t , (t , t0, x0, u, d ), u(t ) Y = +. An interesting question to ask is how to characterize the class of external inputs that do notproduce unbounded output responses. An even stronger require
17、ment is the char-acterization of the class of inputs for which the output response converges, i.e., limt + H (t , (t , t0, x0, u, d ), u(t ) Y = 0.It is reasonable to expect that the magnitude of the external input u(t ) U will play a significant role. However, it is worth pointing out that the smal
18、lness of the size of external inputs (except the trivial case where u(t ) U 0) does not make the problem go away, as shown in the above-mentioned scalar example.The notion of (Uniform) (Weighted) Input-to-Output Stability property allows us to study the effect of the magnitude of the external input
19、to the output response. This4.2 Definitions145is the major reason that this stability notion has proved to be very useful in Math-ematical Control Theory. Of course, it should be emphasized that the (Uniform) (Weighted) Input-to-Output Stability property has a large number of theoretical and practic
20、al applications.Next, we present the (Uniform) (Weighted) Input-to-Output Stability property.Definition 4.1 Suppose that is RFC from the input u MU . If there exist functions KL, , K+ , and N such that the followingestimate holds for all u MU , (t0, x0, d ) + X MD m and t t0:Ht , (t , t0, x0, u, d )
21、, u(t )XYU (t0) x0, t t0sup ( ) u( )(4.1) then, we say that satisfies the Weighted Input-to-Output Stability (WIOS) prop-erty from the input u MU with gain N and weight K+ . Moreover, if (t ) 1, then we say that satisfies the Uniform Weighted Input-to-Output Stability (UWIOS) property from the input
22、 u MU with gain N and weight K+ . If there exist functions KL, K+ , and N such that the following estimate holds for all u MU , (t0, x0, d ) + X MD , and t t0:Ht , (t , t0, x0, u, d ), u(t )XYU (t0) x0, t t0sup u( )(4.2) then, we say that satisfies the Input-to-Output Stability (IOS) property from t
23、he input u MU with gain N . Moreover, if (t ) 1, then we say that satisfies the Uniform Input-to-Output Stability (UIOS) property from the input u MU with gain N . Finally, for the special case of the identity output mapping, i.e., H (t , x, u) := x, the (Uniform) (Weighted) Input-to-Output Stabilit
24、y property from the input u MU is called (Uniform) (Weighted) Input-to-State Stability (U)(W)ISS) property from the input u MU .Remark 4.1 Using the inequalities maxa, b a + b maxa + (a), b + 1(b) (which hold for all K and a, b 0), it should be clear that the WIOS property for := (X , Y , MU , MD ,
25、, , H ) can be defined by using an estimate based on “max,”Ht , (t , t0, x0, u, d ), u(t )XYt0Umax t(4.3)(t0) x0, tt0,sup( ) u( ) instead of (4.1). Notice that the functions and involved in (4.1) and (4.3) are not necessarily the same. Similarly, the IOS property for := (X , Y , MU , MD , , , H ) ca
26、n be defined by using an estimate of the form:1464External Stability: Notions and CharacterizationsHt , (t , t0, x0, u, d ), u(t )XYt0Umaxt(4.4) (t0) x0, tt0,sup u( ) instead of (4.2).We call estimate (4.1) “a Sontag-like estimate,” because E.D. Sontag invented the notion of ISS in 22 for finite-dim
27、ensional continuous-time systems, which is expressed using an estimate of the formX tXmax t0U (t , t0, x0, u, d )x0, tt0 ,sup u( ) Moreover, Sontag and Wang formulated IOS in 27, 28 (see also 8) for continuous-time finite-dimensional systems using an estimate of the form (4.1) with(t ) (t ) 1.Exactl
28、y as in the case of internal stability properties, for external stability notions, we have found the following methods of proving the (U)(W)IOS property in the literature:(1) Analytical SolutionsFor this line of research, basic estimates for the solutions of the system are extracted by actually solv
29、ing the differential (or difference) equations (or in-equalities).(2) Transformation MethodsIn this case, basic estimates for the solutions of the system are derived by trans-forming the system into a different system with special properties.(3) The method of Lyapunov functions and functionalsBasic
30、estimates for the solutions are derived by means of a (or many) Lyapunov functional(s) and comparison lemmas.(4) Small-Gain MethodsBasic estimates for the solutions of the system are derived by means of small-gain arguments.(5) Qualitative MethodsCertain qualitative properties of the solutions guara
31、ntee that the (U)(W)IOS property holds. Working in this way, usually we cannot have an explicit ex-pression for the gain functions or the weight functions.All the above methods, with the exception of Small-Gain methods, will be ex-plained in the present chapter. The Small-Gain methods will be explai
32、ned in detail in the following chapter. Again it should be emphasized that the methods of proving external stability properties can be (and usually are) combined.The following lemmas provide characterizations of the WIOS and UWIOS properties.Lemma 4.1 Suppose that is RFC from the input u MU . Furthe
33、rmore, suppose that there exist functions V : + X U + with V (t , 0, 0) = 0 for all t 0, N , and K+ such that the following properties hold:4.2 Definitions147P1. For all s 0 and T 0, it holds thatsup V t , (t , t0, x0, u, d ), u(t ) sup;t0( ) u( )tUt t0, x0 X s, t0 0, T , d MD , u MU 0 and T 0, ther
34、e exists a := (, T ) 0 such that;supVt , (t , t0, x0, u, d ), u(t )t0supt( ) u( )Ut t0, x0 X , t0 0, T , d MD , u MU .P3.For all 0, T 0, and R 0, there exists := (, T , R) 0 such that t0;sup Vt , (t , t0, x0, u, d ), u(t )sup( ) u( )tU .t t0 + , x0 X R, t0 0, T , d MD , u MUThen, there exist functio
35、ns KL and K+ such that the following estimate holds for all u MU , (t0, x0, d ) + X MD , and t t 0:V t , (t , t0, x0, u, d ), u(t )+t0UX t(t0) x0, t t0sup ( ) u( )(4.5) Moreover, if there exists a N such that H (t , x, u) Y a(V (t , x, u) for all (t , x, u) + X U , then for every K , satisfies the W
36、IOS prop-erty from the input u MU with gain N and weight K+ , where (s) := a( (s) + ( (s).Lemma 4.2 Suppose that is RFC from the input u MU . Furthermore, suppose that there exist functions V : + X U + with V (t , 0, 0) = 0 for all t 0, N, and K+ such that the following properties hold:P1. For every
37、 s 0, it holds that t0sup;supV t , (t , t0, x0, u, d ), u(t )( ) u( )tUt t0, x0 X s, t0 0, d MD , u MU 0, there exists := () 0 such that t0;supV t , (t , t0, x0, u, d ), u(t )sup( ) u( )tUt t0, x0 X , t0 0, d MD , u MU .P3.For all 0 and R 0, there exists := (, R) 0 such that t0;supV t , (t , t0, x0,
38、 u, d ), u(t )sup( ) u( )tU .t t0 + , x0 X R, t0 0, d MD , u MU1484External Stability: Notions and CharacterizationsThen, there exists a function KL such that estimate (4.5) holds for all u MU , (t0, x0, d ) + X MD , and t t0 with (t ) 1. Moreover, if there existsa N such that H (t , x, u) Y a(V (t , x, u) for all (t , x, u) + X U , then for every K , satisfies the UWIOS property from the input u MU with gain N and weight K+ , where (s) := a( (s) + ( (s).Remark 4.2 Notice that Lemmas 4.1 and 4.2 can be very useful for the demonstr