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Conditions for Fixed-Time Stability
Our first result presents a necessary and sufficient condition for fixed-time stability, here and in the theorems that follow the properties of strict Lyapunov functions (see Definition 1.2) play a crucial role. Theorem 2.1. Consider system (1.1). The following properties are equivalent:
i)The origin is fixed-time stable on .
ii)There exists a strict Lyapunov function V for system (1.1) satisfying for all x 2 x2 Z 0 V˙ ( x( (s))) < 1; V (x) sup ds + (2.2) x(t)).
where s 7! t is the inverse mapping of t 7! V ( Proof. i) ) ii). If the system (1.1) is fixed-time stable, then its settling-time function is such that T := supx2 T (x) < +1 and x(t) = 0 for all t T and for all x 2 . Since fixed-time stability implies asymptotic stability, according to Theorem 1.4, there exists a strict Lyapunov function V for (1.1) and therefore there exists a well defined application [0;T (x)) ! (0;V (x)], t 7! V ( x(t)) strictly decreasing and differentiable for all t 2 [0;T (x)). Hence, for any x 2 , there exists a differentiable inverse mapping (0;V (x)] ! [0;T (x)), s 7! t, also decreasing that satisfies for all s 2 (0;V (x)] 0(s) = 1 : V ( x( (s))).
The change of variables s = V ( x(t)) and the fact that V ( x(T (x))) = 0 for all x 2 lead to Z T (x) Z 0 T (x) = dt = 0(s)ds = 0 V (x) Then we have that + 1 > x2 T x x2 Z 0 ( ) V (x) sup sup 0 ZV (x) V˙ ( x( (s))) : ds ds (2.3) V ( x( (s))).
for all x 2 and the conclusion readily follows.
ii ) ) i). According to Theorem 1.4, because there exists a strict Lyapunov function V for system (1.1), its origin is asymptotically stable. The equation (2.3) implies, furthermore, that it is fixed-time
stable.
Note that no assumptions on the regularity of T (x) have been made, therefore, the conditions stated in Theorem 2.1 do not exclude the case of discontinuous T (x). Also, the equation (2.2) is in general difficult to verify since it involves the explicit calculation of the trajectories x and of the inverse mapping . In what follows, more constructive conditions will be presented and the case of discontinuous T (x) will be excluded. Sufficient Conditions for FxTS with Continuous T Theorem 2.2. Suppose that there exists a continuously differentiable strict Lyapunov function V : ! R 0 for system (1.1) such that S1 there exists a continuous positive definite function r : R 0 ! R 0 that verifies.
Necessary Conditions for FxTS
In order to obtain the first result of this section, we will make use of the next lemma, whose proof can be found at the end of the chapter.
Lemma 2.1. Suppose that the origin of system (1.1) is asymptotically stable on . Then there exist a
! R 0 and strict Lyapunov function V : ! R 0 for (1.1), a continuous positive definite function W : some 1; 2 2 K1 that satisfy ˙ M1 V (x) = W (x) 8x 2 . (x)) 8x 2 . M2 1(V (x)) W (x) 2 (V In fact, Lemma 2.1 is a corollary of Kurzweil’s theorem (Theorem 1.4) and it states, in words, that if a given system is AS, then there exists a class-K1 function that satisfies, instead of the well known inequality V (x) < 0, the equality M1.
Now we are ready to present a necessary condition for FxTS using a similar characterization as the one employed in Theorem 2.2. Theorem 2.3. Consider system (1.1) and suppose that the origin is fixed-time stable on . Then there exist a strict Lyapunov function V and a class-K1 function q that verifies N1 Z + 1 q(z) <+1; 0 dz N2 q(V (x)) ˙ V (x) 8x 2 .
Proof. Since fixed-time stability implies asymptotic stability we know, from Lemma 2.1, that there exists a strict Lyapunov function V and a continuous positive definite function W that satisfy the M conditions. Then we have that for all x 2 : I there exist a decreasing di erentiable mapping [0 ; T ( x )] ! (0 ; V ( x0 )], t 7! V ( ( )), with T ( x ) = ff x t inffT 0 : V ( x(T )) = 0g and its corresponding inverse mapping (0;V (x)] ! [0;T (x)], s 7! t, also decreasing and differentiable such that 0(s) = 1=V˙ ( x( (s))). II Since V (x) and W (x) := V (x) satisfy M, there exists some q 2 K1 such that q(V (x)) V (x) 8x 2 and N2 is satisfied.
FxT Stabilization of Nonlinear Affine Systems
In this section, we will give a sufficient conditions for fixed-time stabilization following a similar structure of well known results on asymptotic stabilization of autonomous systems. Consider the following affine in the input u system: m x˙ = f0(x) + Xfi (x)ui ; x 2 Rn and u 2 Rm; (2.7) i=1.
where f0(0) = 0, fi is continuous for all 0 i m and such that (2.7) has uniqueness of solutions in forward time. Its closed-loop representation is given by m x˙ = f0(x) + Xfi (x)ui (x); x 2 Rn: (2.8) i=1. Let us recall the definition of stabilization and propose a definition of fixed-time stabilization. In the latter, we will only consider fixed-time stabilization with continuous settling-time functions. Definition 2.2. The control system (2.7) is stabilizable (respectively fixed-time stabilizable) if there exists a nonempty neighborhood of the origin Rn and a C0 feedback control law u : ! Rm such that:
1. u(0) = 0;
2. the origin of the system (2.8) is asymptotically stable (respectively fixed-time stable with a continuous settling-time function). Such a feedback law u(x) is called a stabilizer (respectively fixed-time stabilizer) for system (2.7). A radially unbounded, positive definite, C1 function V : (CLF) for the system (2.7) if for all x 2 nf0g, ! R 0 is a control Lyapunov function inf (a(x) + hB(x); ui) < 0; (2.9) u2Rm where a(x) = Lf0 V (x), B(x) = (B1(x); : : : ; Bm(x)) with Bi (x) = Lfi V (x) for 1 i m. Such a control Lyapunov function satisfies the small control property (SCP) if for each > 0, there exists > 0 such that, if x 2 Bn, then there exists some u 2 Bm such that a(x) + hB(x); ui < 0: (2.10).
Table of contents :
Thanks
Table of Contents
Acronyms
Notations
Résumé Long
General Introduction
1 Theoretical Background
1.1 Stability Rates in Nonlinear Systems
1.2 Input-to-State Stability
1.3 Homogeneous Systems
1.4 The Implicit Lyapunov Function Approach
1.5 General Problem Statement
2 Conditions for Fixed-Time Stability
2.1 Conditions for Fixed-Time Stability
2.2 Necessary Conditions for FxTS
2.3 FxT Stabilization of Nonlinear Affine Systems
2.4 Conclusions
2.5 Proofs
3 NonA ISS Lyapunov Functions
3.1 Nonasymptotic Input-to-State Stability
3.2 Explicit Characterization
3.3 Implicit Characterization
3.4 Conclusions
3.5 Proofs
4 FT and FxT Observers for Linear MIMO Systems via ILF
4.1 Implicit Lyapunov Function Candidate
4.2 Observers’ Design
4.3 Numerical Simulations
4.4 Experimental Results
4.5 Conclusions
4.6 Proofs
5 Output Fixed-Time Stabilization of a Chain of Integrators
5.1 Output Feedback Control Design
5.2 Parameter Tuning
5.3 Conclusions
5.4 Proofs
Conclusions
Bibliography
