On the design of new classes of fixed-time stable systems with predefined upper bound for the settling time
R. Aldana-López, D. Gómez-Gutiérrez, E. Jiménez-Rodríguez, J. D. Sánchez-Torres, M. Defoort
OOn the design of new classes of fixed-time stable systemswith predefined upper bound for the settling time
R. Aldana-L´opez a , D. G´omez-Guti´errez a , b , E. Jim´enez-Rodr´ıguez c ,J. D. S´anchez-Torres d , M. Defoort e a Multi-agent autonomous systems lab, Intel Labs, Intel Tecnolog´ıa de M´exico, Jalisco, M´exico. b Tecnologico de Monterrey, Escuela de Ingenier´ıa y Ciencias, Jalisco, M´exico. c CINVESTAV, Unidad Guadalajara, Jalisco, M´exico. d Research Laboratory on Optimal Design, Devices and Advanced Materials -OPTIMA-, Department of Mathematics andPhysics, ITESO, Jalisco, M´exico. e LAMIH, UMR CNRS 8201, Polytechnic University of Hauts-de-France, Valenciennes, 59313 France.
Abstract
This paper aims to provide a methodology for designing autonomous and non-autonomous systems with a fixed-time stableequilibrium point where an Upper Bound of the Settling Time (
UBST ) is set a priori as a parameter of the system. Inaddition, some conditions for such an upper bound to be the least one are provided. This design procedure is a relevantcontribution when compared with traditional methodologies for the design of fixed-time algorithms satisfying time constraintssince current estimates of an
UBST may be too conservative. The proposed methodology is based on time-scale transformationsand Lyapunov analysis. It allows the presentation of a broad class of fixed-time stable systems with predefined
UBST , placingthem under a common framework with existing methods using time-varying gains. To illustrate the effectiveness of ourapproach, we design novel, autonomous and non-autonomous, fixed-time stable algorithms with predefined least
UBST . Key words:
Prescribed-time, fixed-time stability, Lyapunov analysis, time-base generators, time-scale transformation.
In recent years, dynamical systems exhibiting convergence to their origin in some finite time, independent of theinitial condition of the system, have attracted a great deal of attention. For this class of dynamical systems, theirorigin is said to be fixed-time stable, which is a stronger notion of finite-time stability [1,2], because in the latter thesettling time is, in general, an unbounded function of the initial condition of the system. A Lyapunov differentialinequality for an autonomous system to exhibit fixed-time stability was presented in [3], together with an UpperBound of the Settling Time (
UBST ) of the system trajectory. However, such an upper estimate is too conservative [4].Only recently, non-conservative
UBST have been derived [4, 5] for some scenarios. An alternative characterization,based on homogeneity theory, was proposed in [6, 7]. Although it is a powerful tool for the design of high orderfixed-time stable systems, it poses a challenging design problem for time constrained scenarios, since an
UBST isoften unknown. Thus, the design of fixed-time stable systems where an
UBST is set a priori explicitly as a parameterof the system, as well as the reduction/elimination of the conservativeness of an
UBST , is of great interest. This (cid:63)
Corresponding author: D. G´omez-Guti´errez.
Email addresses: [email protected] (R. Aldana-L´opez), [email protected] (D. G´omez-Guti´errez), [email protected] (E. Jim´enez-Rodr´ıguez), [email protected] (J. D. S´anchez-Torres), [email protected] (M. Defoort).
Preprint submitted to ArXiv June 21, 2019 a r X i v : . [ m a t h . O C ] J un roblem has been partially addressed for autonomous systems, mainly focusing on the class of systems proposedin [3, 8, 9], see, e.g., [4, 8, 10]; and for non-autonomous systems, mainly focusing on time-varying gains that eitherbecome singular [11–16] or induce Zeno behavior [17, 18] as the predefined-time is reached. Contributions:
We provide a methodology for designing new classes of autonomous and non-autonomous fixed-time stable systems, where an
UBST is set a priori explicitly as a parameter of the system. The main result is asufficient condition in the form of a Lyapunov differential inequality, for a nonlinear system to exhibit this property.Additionally, we show that for any fixed-time stable system with continuous settling time function there exists aLyapunov function satisfying such differential inequality. Based on this characterization, we show how a fixed-timestable system with predefined
UBST can be designed from a nonlinear asymptotically stable one, presenting sufficientconditions for such an upper bound to be the least one. To illustrate our approach, we provide examples showinghow to derive autonomous and non-autonomous fixed-time stable systems, with predefined least
UBST . This is asignificant contribution to the design of control systems satisfying time constraints since, even in the scalar case, theexisting
UBST estimates are often too conservative [4].
Notation: R is the set of real numbers, ¯ R = R ∪ {−∞ , + ∞} , R + = { x ∈ R : x ≥ } and ¯ R + = R + ∪ { + ∞} . TheEuclidean norm of x ∈ R n is denoted as || x || . h (cid:48) ( z ) = dh ( z ) dz denotes the first derivative of the function h : R → R . Fora continuosly differentiable function f : R n → R , the row vector ∂f /∂x is defined by ∂f /∂x = [ ∂f /∂x , . . . , ∂f /∂x n ]. C k ( I ) is the class of functions f : I → R with k ≥ I ⊆ R which has continuous k -th derivative in I . For z ∈ R + , Γ( z ) = (cid:82) + ∞ e − ξ ξ z − dξ is the Gamma function; for x, a, b ∈ R + , B ( x ; a, b ) = (cid:82) x ξ a − (1 − ξ ) b − dξ and B − ( · ; · , · ) are the incomplete Beta function and its inverse, respectively; for x ∈ R erf( x ) = (cid:82) x √ π e − ξ dξ is theError function. For x >
0, ci( x ) = − (cid:82) + ∞ x cos( ξ ) ξ dξ and si( x ) = − (cid:82) + ∞ x sin( ξ ) ξ dξ are the cosine and sine integralsrespectively. K ba is the class of strictly increasing C ((0 , a )) functions h : [0 , a ) → ¯ R with a, b ∈ ¯ R satisfying h (0) = 0and lim z → a h ( z ) = b . Consider the system ˙ x = − T c f ( x, t ) , ∀ t ≥ t , f (0 , t ) = 0 , (1)where x ∈ R n is the state of the system, T c > t ∈ [ t , + ∞ ) and f : R n × R + → R n is nonlinear,continuous on x and piecewise continuous on t (with a finite number of discontinuities on any finite interval).Thus, the solutions are understood in the sense of Caratheodory [19]. We assume that f ( · , · ) is such that the originof system (1) is asymptotically stable and, except at the origin, system (1) has the properties of existence anduniqueness of solutions in forward-time on the interval [ t , + ∞ ) [20]. The solution of (1) for t ≥ t with initialcondition x is denoted by x ( t ; x , t ), and the initial state is given by x ( t ; x , t ) = x . Remark 1
For simplicity, throughout the paper, we assume the origin is the unique equilibrium point of the systemsunder consideration. Thus, without ambiguity, we refer to the global stability (in the respective sense) of the originof the system as the stability of the system. The extension to local stability is straightforward.
Definition 2 [21](Settling-time function) The settling-time function of system (1) is defined as T ( x , t ) = inf { ξ ≥ t : lim t → ξ x ( t ; x , t ) = 0 } − t . For autonomous systems ( f in (1) does not depend on t ), the settling-time function is independent of t . Definition 3 [21] (Fixed-time stability) System (1) is said to be fixed-time stable if it is asymptotically stable [20]and the settling-time function T ( x , t ) is bounded on R n × R + , i.e. there exists T max ∈ R + \ { } such that T ( x , t ) ≤ T max if t ∈ R + and x ∈ R n . Thus, T max is an UBST of x ( t ; x , t ) . We are interested on finding sufficient conditions on system (1) such that an
UBST is given by the parameter T c ,i.e. T c = T max . Note that there are infinite choices for T max . Of particular interest is to find sufficient conditions suchthat T c is the least UBST . 2 .1 Time-scale transformations
As in [22], the trajectories corresponding to the system solutions are interpreted, in the sense of differentialgeometry [23], as regular parametrized curves. Since we apply regular parameter transformations over the timevariable, then without ambiguity, this reparametrization is sometimes referred to as time-scale transformation.
Definition 4 [23, Definition 2.1] A regular parametrized curve, with parameter t , is a C ( I ) immersion c : I → R ,defined on a real interval I ⊆ R . This means that dcdt (cid:54) = 0 holds everywhere. Definition 5 [23, Pg. 8] (Regular parameter transformation) A regular curve is an equivalence class ofregular parametrized curves, where the equivalence relation is given by regular (orientation preserving) parametertransformations ϕ , where ϕ : I → I (cid:48) is C ( I ) , bijective and dϕdt > . Therefore, if c : I → R is a regularparametrized curve and ϕ : I → I (cid:48) is a regular parameter transformation, then c and c ◦ ϕ : I (cid:48) → R are consideredto be equivalent. The methodology presented in this section to obtain new classes of fixed-time stable systems with predefined
UBST subsumes some existing results for designing fixed-time stable systems, in both, the autonomous and the non-autonomous cases.
Assumption 6 H : R → R is continuous in R , locally Lipschitz in R \ { } , and satisfies H (0) = 0 and ∀ z ∈ R \ { } , z H ( z ) > . Moreover, ˜ x ( τ ; x , is the unique solution of the asymptotically stable system d ˜ xdτ = −H (˜ x ) , ˜ x (0; x ,
0) = x , ˜ x ∈ R , (2) and T ( x , is its settling time. The following lemma presents the construction of the parameter transformation that will be used hereinafter.
Lemma 7
Under Assumption 6, suppose that ψ ( τ ) = T c (cid:90) τ x ( ξ ; x , , ψ ( ξ )) dξ, (3) has a unique solution ψ ( τ ) on I (cid:48) = [0 , T ( x , , where T c > , and Ψ( x, ˆ t ) is a function Ψ : R × R + → R + \ { } continuous for all x ∈ R \ { } and ˆ t ∈ J = [0 , lim τ →T ( x , ψ ( τ )) . Then, the map ψ : I (cid:48) → J is bijective. Moreover,the bijective function ϕ : I = [ t , t + lim τ →T ( x , ψ ( τ )) → I (cid:48) defined by ϕ − ( τ ) = ψ ( τ ) + t is a parametertransformation. PROOF.
Let ψ ( τ ) be the unique solution of (3) in I (cid:48) . Since ˜ x ( τ ; x , τ ∈ I (cid:48) , is continuous, thenΨ(˜ x ( τ ; x , , ψ ( τ )) − is continuous on τ ∈ I (cid:48) and ψ ( τ ) is C ( I (cid:48) ). Moreover, for all x ∈ R \{ } and ˆ t ∈ J , Ψ( x, ˆ t ) > dψdτ >
0, hence ψ is injective [24, Pg. 34]. On the other hand, note that lim τ → inf I (cid:48) ψ ( τ ) = inf J andlim τ → sup I (cid:48) ψ ( τ ) = sup J , hence, by the continuity of ψ , ψ is surjective. Thus, ψ : I (cid:48) → J is bijective. It follows that ϕ is C ( I ), satisfies dϕdτ > ϕ is a parameter transformation.The following lemma shows that if (2) has a known settling time function, then the parameter transformation givenin Lemma 7 induces a nonlinear system with known settling-time function.3 emma 8 Under Assumption 6, suppose that (3) has a unique solution on I (cid:48) = [0 , T ( x , , where T c > and Ψ : R × R + → R + \ { } is continuous for all x ∈ R \ { } and ˆ t ∈ J = [0 , lim τ →T ( x , ψ ( τ )) . Then, the system ˙ x = − T c Ψ( x, ˆ t ) H ( x ) , x ( t ; x , t ) = x , (4) where ˆ t = t − t , x ∈ R , has a unique solution x ( t ; x , t ) = (cid:40) ˜ x ( ϕ ( t ); x , for t ∈ I elsewhere, (5) where I = [ t , t + lim τ →T ( x , ψ ( τ )) and ϕ − ( τ ) = ψ ( τ ) + t with ψ ( τ ) the solution of (3) . Moreover, (4) isasymptotically stable and the settling time of x ( t ; x , t ) is given by T ( x , t ) = lim τ →T ( x , ψ ( τ ) . (6) PROOF.
First, we will show that, if ψ ( τ ) is a solution of (3), then x ( t ; x , t ) = ˜ x ( ϕ ( t ); x ,
0) where ϕ − ( τ ) = ψ ( τ ) + t , is a solution of (4) on I . To this end, notice that, by Lemma 7, ϕ : I → I (cid:48) is a parametertransformation. Thus, ˜ x ( τ ; x , τ ∈ I (cid:48) and x ( t ; x , t ) = (˜ x ◦ ϕ )( t ; x ,
0) = ˜ x ( ϕ ( t ); x , t ∈ I are equivalent (inthe sense of regular curves). Moreover, ˙ x = ddt ˜ x ( ϕ ( t ); x ,
0) = d ˜ xdτ (cid:12)(cid:12) τ = ϕ ( t ) dϕdt . Note that d ˜ xdτ (cid:12)(cid:12) τ = ϕ ( t ) = H (˜ x ( ϕ ( t ); x , dϕdt = (cid:18) dϕ − dτ (cid:12)(cid:12)(cid:12) τ = ϕ ( t ) (cid:19) − = (cid:18) dψdτ (cid:12)(cid:12)(cid:12) τ = ϕ ( t ) (cid:19) − = T c − Ψ(˜ x ( ϕ ( t ); x , , ψ ( ϕ ( t ))) = T c − Ψ( x ( t ; x , t ) , ˆ t ), since ψ ( ϕ ( t )) = ϕ − ( ϕ ( t )) − t . Hence, ˙ x = − T c − H ( x ( t ; x , t ))Ψ( x ( t ; x , t ) , ˆ t ). Therefore, for the solution ˜ x ( τ ; x , τ ∈ I (cid:48) of (2), there exists an equivalent curve x ( t ; x , t ), t ∈ I , under the parameter transformation ϕ , that issolution of (4) on I .The next part of the proof is to show that x ( t ; x , t ) is, in fact, the unique solution of (4) on I . Tothis end, note that ( x ◦ ϕ − )( τ ; x ,
0) satisfy (2) since d ˜ xdτ = ddτ ( x ◦ ϕ − )( τ ; x ,
0) = dxdt (cid:12)(cid:12) t = ϕ − ( τ ) dϕ − dτ = − T c − H (˜ x ( τ ; x , x ( τ ; x , , ϕ − ( τ ) − t ) dψdτ = −H (˜ x ( τ ; x , I , thereexists an equivalent curve on I (cid:48) , under the parameter transformation ϕ − , that is solution of (2). Now, assume bycontradiction that x ( t ; x , t ) and x ( t ; x , t ) are two different solutions of (4) on I , then x ( ϕ − ( τ ); x , t ) and x ( ϕ − ( τ ); x , t ) are their associated equivalent solutions of (2) on I (cid:48) . On the one hand, if x ( ϕ − ( τ ); x , t ) (cid:54) = x ( ϕ − ( τ ); x , t ) there is a contradiction since (2) satisfies the conditions of existence and uniqueness of solution inthe interval I (cid:48) [20, Theorem 3.1]. On the other hand, if x ( ϕ − ( τ ); x , t ) = x ( ϕ − ( τ ); x , t ) then x ( t ; x , t ) = x ( t ; x , t ). Hence, (4) has a unique solution on I .The last part of the proof is to show that (5) is the unique solution of (4) on [ t , + ∞ ). Note that since ˜ x ( τ ; x , τ = T ( x ,
0) then, x ( t ; x , t ) reaches the origin at t = t +lim τ →T ( x , ψ ( τ ). Moreover, since (4)has an equilibrium point at x = 0, then the solution of (4) remains at the origin for all t ∈ [ t +lim τ →T ( x , ψ ( τ ) , + ∞ ).Hence, (5) satisfies (4) ∀ t ∈ I , and we can conclude that (4) is asymptotically stable and (5) is the unique solutionin the interval [ t , + ∞ ) with settling time function given by (6). In the rest of the paper, we analyze the cases where Ψ( x, ˆ t ) is time invariant for autonomous systems or a functiononly of t for non-autonomous systems. In Lemma 10 and Lemma 12 we show that in these cases, (3) has a uniquesolution. Assumption 9
Let
Φ : R + → ¯ R + \ { } be a function satisfying Φ(0) = + ∞ , ∀ z ∈ R + \ { } , Φ( z ) < + ∞ and (cid:90) + ∞ Φ( z ) dz = 1 . (7)4 emma 10 Under Assumption 6, let Ψ( z, ˆ t ) = (Φ( | z | ) H ( | z | )) − for z ∈ R \ { } where Φ( · ) satisfies Assumption 9,then, (3) has a unique solution on I (cid:48) = [0 , T ( x , , given by ψ ( τ ) = T c (cid:90) τ Φ( | ˜ x ( ξ ; x , | ) H ( | ˜ x ( ξ ; x , | ) dξ. (8) PROOF.
Let Ψ( z, ˆ t ) = (Φ( | z | ) H ( | z | )) − and notice that Ψ(˜ x ( τ ; x , , ψ ) − is independent of ψ . Therefore, itfollows that dψdτ = T c Ψ(˜ x ( τ ; x , , ψ ) − , ψ (0) = 0 (9)has a unique solution given by ψ ( τ ) = T c (cid:82) τ Φ( | ˜ x ( ξ ; x , | ) H ( | ˜ x ( ξ ; x , | ) dξ . Moreover, by [25, Lemma 1.2.2], asolution of (9) is also a solution of (3) and vice versa. Assumption 11
Let
Φ : R + → ¯ R + \ { } be a continuous function on R + \ { } satisfying (7) and ∀ τ ∈ R + \ { } , Φ( τ ) < + ∞ . Moreover, Φ is either non-increasing or locally Lipschitz on R + \ { } . Lemma 12
Under Assumption 6, let I (cid:48) = [0 , T ( x , , and consider the first order ordinary differential equation dψ ( τ ) dτ = T c Φ( τ ) , ψ (0) = 0 , τ ∈ I (cid:48) , (10) where Φ( · ) satisfies Assumption 11, then (10) has a unique solution ψ : I (cid:48) → J = [0 , lim τ →T ( x , ψ ( τ )) , which isbijective and given by ψ ( τ ) = T c (cid:90) τ Φ( ξ ) dξ, τ ∈ I (cid:48) . (11) Moreover, let Ψ( z, ˆ t ) = Φ( ψ − (ˆ t )) then ψ ( τ ) is also the unique solution of (3) on I (cid:48) . PROOF.
It follows that (10) has a unique solution given by ψ ( τ ) = T c (cid:82) τ Φ( ξ ) dξ . Note that ∀ τ ∈ I (cid:48) \ { } , ψ : I (cid:48) → J is C ( I (cid:48) ) with dψdτ >
0, and ψ (0) = 0, hence ψ is injective [24, Pg. 34]. Note that lim τ → inf I (cid:48) ψ ( τ ) = inf J and lim τ → sup I (cid:48) = sup J and, by continuity of ψ , ψ is surjective. Hence, ψ : I (cid:48) → J is bijective. SinceΦ( ψ − ( ψ ( τ ))) = Φ( τ ), then ψ ( τ ) is a solution of (3). Now, on the one hand if Φ is non-increasing, then Φ ◦ ψ − is non-increasing. To show this let a > b , ψ − ( a ) > ψ − ( b ) and Φ( ψ − ( a )) < Φ( ψ − ( b )). On the other hand, if Φ is Lipschitzon [ (cid:15), + ∞ ) , ∀ (cid:15) >
0, then Φ ◦ ψ − is Lipschitz on J \ [0 , (cid:15) ). To show this, note that there exists a constant M Φ > | Φ( ψ − ( x )) − Φ( ψ − ( x )) | ≤ M Φ | ψ − ( x ) − ψ − ( x ) | ≤ M | x − x | where M = M Φ max x ∈ J \ [0 ,(cid:15) ) ( ψ − ) (cid:48) ( x ).Then, in the former (resp. in the latter) case it follows from Peano’s uniqueness Theorem [25, Theorem 1.3.1] (resp.from Lipschitz uniqueness Theorem [25, Theorem 1.2.4]) that dzdτ = T c Φ( ψ − ( z )) , z ( (cid:15) ) = T c (cid:90) (cid:15) Φ( ξ ) dξ, (12)has a unique solution z = ψ ( τ ), ∀ (cid:15), τ ∈ I (cid:48) \ { } . Since ψ (0) = 0, then (12) with (cid:15) = 0 has a unique solution z = ψ ( τ ), τ ∈ I (cid:48) . Moreover, by [25, Lemma 1.2.2], a solution of (12) is a solution of (3) and vice versa.In Lemma 13, we present a characterization for a map Ψ : R × R + → ¯ R + , in the autonomous case, such thatsystem (4) is fixed-time stable with T c as the least UBST . Lemma 13 (Characterization of Ψ( z, ˆ t ) for fixed-time stability of autonomous systems with predefined least UBST)Under Assumption 6, let Ψ( z, ˆ t ) be Ψ( z, ˆ t ) = (cid:40) (Φ( | z | ) H ( | z | )) − for z ∈ R \ { } . otherwise (13) where Φ( · ) satisfies Assumption 9. Then, system (4) is fixed time stable with T c as the predefined least UBST. ROOF.
Notice that, by Lemma 10, the solution of (3) is given by (8). Using the change of variables z = | ˜ x ( τ ; x , t ) | , the settling time function of (4) given in (6) can be expressed as T ( x , t ) = − T c (cid:82) | x | Φ( z ) H ( z ) (cid:16) − H ( z ) (cid:17) dz = T c (cid:82) | x | Φ( z ) dz . Since Φ( · ) > T ( x , t ) is increasing with respect to | x | .Since Φ( · ) satisfies (7), the settling-time function satisfiessup ( x ,t ) ∈ R n × R + T ( x , t ) = lim | x |→ + ∞ T c (cid:90) | x | Φ( z ) dz = T c . Thus, (4) is fixed-time stable with T c as the least UBST .The following result states the construction of fixed-time stable non-autonomous systems with predefined
UBST . Lemma 14 (Characterization of Ψ( z, ˆ t ) for fixed-time stability of non-autonomous systems with predefined leastUBST) Under Assumption 6, let ψ ( τ ) , τ ∈ I (cid:48) = [0 , T ( x , , be the solution of (10) and ψ − (ˆ t ) its inverse map,and let Ψ( z, ˆ t ) = ψ − (ˆ t )) for ˆ t ∈ [0 , T c ) , otherwise (14) where ˆ t = t − t and Φ( · ) satisfies Assumption 11. Then, system (4) is fixed-time stable with T c as the predefinedUBST. Furthermore,(1) the settling time is exactly T c for all x (cid:54) = 0 if T ( x ,
0) = + ∞ , for all x (cid:54) = 0 ;(2) T ( x , t ) < T c if T ( x , < + ∞ , but the least UBST is T c if, in addition, T ( x , is radially unbounded, i.e. T ( x , → + ∞ as | x | → + ∞ . PROOF.
By Lemma 12, the solution of (3) is given by (11). Then, the settling time function of (4) is given by T ( x , t ) = T c (cid:82) T ( x , Φ( ξ ) dξ . To show item (1) note that if T ( x ,
0) = + ∞ , then T ( x , t ) = T c (cid:82) + ∞ Φ( ξ ) dξ = T c , ∀ x ∈ R \ { } . Hence, the settling time is exactly T c , ∀ x ∈ R \ { } . To show item (2) note that, since T ( x , < + ∞ then T ( x , t ) = T c (cid:82) T ( x , Φ( ξ ) dξ < T c . However, if T ( x ,
0) is radially unbounded, thensup ( x ,t ) ∈ R n × R + T ( x , t ) = lim | x |→ + ∞ T c (cid:82) T ( x , Φ( ξ ) dξ = T c . Hence, T c is the least UBST . Remark 15
Fixed-time stability of non-autonomous systems has been applied for the design of stabilizingcontrollers [12], observers [26] and consensus algorithms [14, 18, 27, 28] with predefined settling-time at T c , whichuses time-varying gains that are either continuous in [ t , T c + t ) [11–14] or piecewise continuous requiring Zenobehavior [17, 18] as t approaches T c + t . Notice that, in this paper, we focus on the former case. Apart from theabove mentioned references, we refer the reader to [28, 29] for some engineering applications of fixed-time stabilityof non-autonomous systems. Remark 16
In the autonomous case, T c is the least UBST, whereas, in the non-autonomous case, if item (1) issatisfied, every nonzero trajectory converges exactly at T c . This feature has been referred in the literature as predefined-time [13], appointed-time [17] or prescribed-time [14]. However, note that for Φ( z ) in Lemma 14 to comply (7) , itmust vanish as z approaches + ∞ , therefore lim t → t + T − c Ψ( z, ˆ t ) = + ∞ since lim ˆ t → T − c ψ − (ˆ t ) = + ∞ . However, ifitem (2) in Lemma 14 is satisfied, then, for a finite x , the origin is reached before t + T c .3.3 Lyapunov analysis for fixed time stability with predefined UBST The above lemmas were focused on scalar fixed-time stable systems. As an extension to the stability analysis, thefollowing Theorem provides a sufficient condition for a (general) nonlinear system to be fixed-time stable withpredefined
UBST . This result follows from the comparison lemma [20, Lemma 3.4] and the application of the aboveresults on the time derivative of the Lyapunov candidate function.6 heorem 17 (Lyapunov characterization for fixed-time stability with predefined UBST) If there exists a continuouspositive definite radially unbounded function V : R n → R , such that its time-derivative along the trajectories of (1) satisfies ˙ V ( x ) ≤ − T c Ψ( V ( x ) , ˆ t ) H ( V ( x )) , x ∈ R n \ { } , (15) where Ψ( z, ˆ t ) is characterized by the conditions of Lemma 13 or the ones of Lemma 14, then, system (1) is fixed-timestable with T c as the predefined UBST. If the equality in (15) holds, then T c is the least UBST. Furthermore, inthe autonomous case, if system (1) is fixed-time stable and has a continuous settling time function T ( x , t ) suchthat sup x ∈ R n T ( x , t ) = T c , then there exists a continuous positive definite function V : R n → R , such that itstime-derivative along the trajectories of (1) satisfies (15) with Ψ( z, ˆ t ) characterized by the conditions of Lemma 13.If in addition, lim (cid:107) x (cid:107)→∞ T ( x , t ) = T c then V ( x ) is radially unbounded. PROOF.
Let w ( t ) be a function satisfying w ( t ) ≥ w = − T c Ψ( w, ˆ t ) H ( w ), and let V ( x ) ≤ w (0).Then, T c is the least UBST of w ( t ). Moreover, by the comparison lemma [20, Lemma 3.4], it follows that V ( x ( t ; x , t )) ≤ w ( t ). Consequently, V ( x ( t ; x , t )) will converge to the origin before T c . If (15) is an equalityand V ( x ) = w (0), then, V ( x ( t ; x , t )) = w ( t ) and T c is the least UBST . Moreover, assume that system (1)is an autonomous fixed-time stable system with a continuous settling satisfying sup x ∈ R n T ( x , t ) = T c , and let G ( z ) = T c (cid:82) z Φ( ξ ) dξ , with Φ( · ) satisfying Assumption 9. Note that G (cid:48) ( z ) > , ∀ z ≥ G : R + → [0 , T c )is a bijection [24, Pg. 34]. Moreover note that G (0) = 0 and lim z →∞ G ( z ) = T c . Hence, V ( x ) = G − ( T ( x, t ))is a continuous and positive definite function satisfying V (0) = 0. Note that ∂V∂x = (cid:0) G − (cid:1) (cid:48) ( T ( x, t )) ∂T ( x,t ) ∂x andthat the direct integration of ˙ V ( x ) = − T c ∂V∂x f ( x ) in the interval [0 , T ( x, t )) leads to T ( x, t ) = (cid:82) V ( x ) F ( ξ ) − dξ with F ( V ( x )) = − T c ∂V∂x f ( x ), thus, ∂T ( x,t ) ∂x = − F ( V ( x )) − ∂V∂x . Hence, (cid:0) G − (cid:1) (cid:48) ( T ( x, t )) = − F ( V ( x )). Therefore,˙ V ( x ) = − T c ∂V∂x f ( x ) = − (cid:0) G − (cid:1) (cid:48) ( T ( x, t )) = − T c Φ( V ( x )) − = − T c Ψ( V ( x ) , ˆ t ) H ( V ( x )) , ∀ x ∈ R n \ { } . It followsstraightforward that, if lim (cid:107) x (cid:107)→∞ T ( x , t ) = T c then V ( x ) = G − ( T ( x, t )) is radially unbounded. Remark 18
In [30, Proposition 1] it was stated without proof that if the system is fixed-time stable with continuoussettling time function then there exist a radially unbounded Lyapunov function satisfying the differential inequalityproposed in [3], but with k = 1 . In Theorem 17 we present, to our best knowledge, the first proof of a converse-liketheorem for [3, Lemma 1] (we show in Section 4, that the Lyapunov differential inequality in (15) , subsumes the onein [3]). The following theorem presents the design of fixed-time stable systems with predefined
UBST . This result allowsdesigning fixed-time stable vector systems with predefined
UBST from an asymptotically stable one.
Theorem 19 (Generating fixed-time stable systems with predefined UBST) Under Assumption 6, let the system ˙ y = − g ( y ) , (16) be asymptotically stable, where y ∈ R n , g : R n → R n is continuous and locally Lipschitz everywhere except, perhaps,at y = 0 with g (0) = 0 . If there exists a Lyapunov function V ( y ) for system (16) such that ˙ V ( y ) ≤ −H ( V ( y )) , ∀ y ∈ R n , (17) then, if Ψ( V ( x ) , ˆ t ) g ( x ) is continuous on x ∈ R n where ˆ t = t − t and Ψ( z, ˆ t ) is a function satisfying the conditionsof Lemma 13 or Lemma 14, the system ˙ x = − T c Ψ( V ( x ) , ˆ t ) g ( x ) , x ( t ; x , t ) = x (18) has a unique solution in the interval [ t , + ∞ ) and it is fixed-time stable with T c as the predefined UBST. PROOF.
Since the conditions of Lemma 13 or Lemma 14 are satisfied, then, (3) has a unique solution. Hence, theproof of the existence of a unique solution for (18) follows by the same arguments as those of the proof of Lemma 8.7 ( z ) ˙ x = − T c (Φ( (cid:107) x (cid:107) ) (cid:107) x (cid:107) ) − x Conditions (i) γ ( αh ( z ) p + βh ( z ) q ) − k h (cid:48) ( z ) ˙ x = − γT c h (cid:48) ( (cid:107) x (cid:107) ) ( αh ( (cid:107) x (cid:107) ) p + βh ( (cid:107) x (cid:107) ) q ) k x (cid:107) x (cid:107) γ = Γ (cid:16) − kpq − p (cid:17) Γ (cid:16) kq − q − p (cid:17)(cid:16) αβ (cid:17) − kpq − p α k Γ( k )( q − p ) , kp < kq > α, β, p, q, k > (ii) π (exp (2 h ( z )) − − / h (cid:48) ( z ) ˙ x = − π T c h (cid:48) ( (cid:107) x (cid:107) ) (exp(2 h ( (cid:107) x (cid:107) )) − / x (cid:107) x (cid:107) lim z → + h (cid:48) ( z ) = + ∞ (iii) exp ( − h ( z )) h (cid:48) ( z ) ˙ x = − T c h (cid:48) ( (cid:107) x (cid:107) ) exp( h ( (cid:107) x (cid:107) )) x (cid:107) x (cid:107) lim z → + h (cid:48) ( z ) = + ∞ (iv) ρ (sin( h ( z ) + α )(1 + h ( z )) − h (cid:48) ( z ) ˙ x = − ρ (1 + h ( (cid:107) x (cid:107) )) T c h (cid:48) ( (cid:107) x (cid:107) )(sin( h ( (cid:107) x (cid:107) ) + α ) (cid:107) x (cid:107) x ρ = α − ci(1) cos(1) − si(1) sin(1), α > z → + h (cid:48) ( z ) = + ∞ Table 1Examples of Φ( z ) satisfying Assumption 9, and fixed-time stable systems with predefined least UBST derived from them.
Let V ( y ) be a Lyapunov function candidate for (16) such that (17) holds. Therefore, ˙ V ( y ) = − ∂V∂y g ( y ) ≤ −H ( V ( y )), ∀ y ∈ R n . Hence, the evolution of V ( x ) is given by ˙ V ( x ) = − T c ∂V∂x Ψ( V ( x ) , ˆ t ) g ( x ) ≤ − T c Ψ( V ( x ) , ˆ t ) H ( V ( x )), ∀ x ∈ R n .Therefore, by Theorem 17, V ( x ( t ; x , t )) converges towards the origin in fixed-time with T c as the predefined UBST .Theorem 19 allows designing fixed-time stable systems with predefined
UBST from an asymptotically stable onesthat has a Lyapunov function satisfying (17). Note that exponentially stable, finite-time stable and fixed-time stablesystems usually have a Lyapunov function satisfying a differential inequality. Notice that by construction, such V ( x )will also be a Lyapunov function for system (18) with time derivative satisfying (15). Notice that, an arbitraryselection of V ( y ) and Ψ( z, ˆ t ) may lead to a right-hand side of (18) discontinuous at the origin. A constructionfrom a linear system, guaranteeing continuity of the right-hand side of (18) is provided in the following proposition.Additionally, a case where T c is the least UBST is exposed.
Corollary 20
Let Ψ( z, t ) defined as in (13) with Φ( z ) satisfying Assumption 9 and H ( z ) = (2 λ max ( P )) − z , P ∈ R n × n is the solution of A T P + P A = I with − A ∈ R n × n Hurwitz and λ max ( P ) is the largest eigenvalueof P . Then, Ψ( V ( x ) , ˆ t ) Ax , where V ( x ) = √ x T P x , is continuous, and the system ˙ x = − T c Ψ( V ( x ) , ˆ t ) Ax (19) is fixed-time stable with T c as the predefined UBST. Moreover, if A = αI + S with α a positive constant and S askew-symmetric matrix then T c is the least UBST. PROOF.
Consider system (16) with g ( y ) = Ay which has a Lyapunov function V ( y ) = (cid:112) y T P y satisfying˙ V ≤ − (2 λ max ( P )) − V ( y ) = −H ( V ( y )). Note that, V ( · ) is continuous and Ψ( · , ˆ t ) is continuous except at the origin.Therefore, since Ψ(0 , ˆ t ) A (0) = 0, to check continuity, it only suffices to show that lim (cid:107) x (cid:107)→ + (cid:107) Ψ( V ( x ) , ˆ t ) Ax (cid:107) =0 which follows from lim (cid:107) x (cid:107)→ + (cid:107) Ψ( V ( x ) , ˆ t ) Ax (cid:107) = 4 λ max ( P ) lim (cid:107) x (cid:107)→ + ( x T A T Ax )( x T P x ) − Φ( V ( x )) − ≤ λ max ( P ) λ max ( A T A ) λ min ( P ) lim (cid:107) x (cid:107)→ + Φ( V ( x )) − = 0. Hence, Ψ( V ( x ) , ˆ t ) Ax is continuous everywhere. It follows fromTheorem 19 that (19) is fixed-time stable with T c as the predefined UBST . Note that, if A = αI + S and P = α I ,then ˙ V ( y ) = −H ( V ( y )) = − αV ( y ) and ˙ V ( x ) = − T c Ψ( V ( x ) , ˆ t ) H ( V ( x )). Hence, by Theorem 17, (19) is fixed-timestable with T c as the least UBST . Remark 21
Similar methods as the ones presented in [3, 4] and [12] can be used to derive robust controllers withpredefined UBST, autonomous and non-autonomous, based on the systems proposed in this paper.
UBST
In this subsection, we present the construction of some examples of Φ( · ) satisfying Assumption 9 for the designof autonomous fixed time stable systems with predefined least UBST . The result is mainly obtained by applyingCorollary 20. For simplicity, we take A = I ∈ R n × n . 8 = . x = x = x = e x = . x = x = x = e x = . x = x = x = e time ( t ) time ( t ) time ( t ) Figure 1. Examples of autonomous fixed-time systems with t = 0 and T c = 1 with x ∈ R . From left to right: System (20)with α = 1 , β = 2 , p = 0 . , q = 2 and k = 1; System (21); System (22) with p = 1 / Proposition 22
Let h ( z ) be K ∞∞ . Then, functions Φ( z ) given in Table 1, satisfy Assumption 9. Moreover, the system ˙ x = − T c (Φ( (cid:107) x (cid:107) ) (cid:107) x (cid:107) ) − x , where x ∈ R n and − T c (Φ( (cid:107) x (cid:107) ) (cid:107) x (cid:107) ) − x shown in Table 1 is fixed-time stable with T c asthe least UBST. PROOF.
Note that γ (cid:82) + ∞ ( αz p + βz q ) − k dz = γ (cid:32) Γ ( − kpq − p ) Γ ( kq − q − p )( αβ ) − kpq − p α k Γ( k )( q − p ) (cid:33) = 1 [4], therefore, by Proposition 29then Φ( z ) given in Table 1- (i) satisfies (7). Moreover, since Φ(0) = + ∞ , then Φ( z ) satisfies Assumption 9. In asimilar way, Φ( z ) given in Table 1- (ii) satisfies Assumption 9. The proof that Table 1- (iii) and Table 1- (iv) satisfy (7)follows by applying Proposition 29 to the functions F ( z ) = exp( − z ), and F ( z ) = ρ (sin( z )+1)(1+ z ) − (which satisfy (cid:82) + ∞ F ( z ) dz according to Proposition 28), respectively. The proof that the system ˙ x = − T c (Φ( (cid:107) x (cid:107) ) (cid:107) x (cid:107) ) − x with − T c (Φ( (cid:107) x (cid:107) ) (cid:107) x (cid:107) ) − x shown in Table 1 is fixed-time stable with T c as the least UBST follows by applying Corollary 20with g ( x ) = x , V ( x ) = (cid:107) x (cid:107) , H ( V ( x )) = (cid:107) x (cid:107) and Φ( z ) given in Table 1. Remark 23
The system in Table 1-(i) with h ( z ) = z reduces to the system analyzed in [3, 30]. However, here T c is given as the predefined least UBST. This feature is a significant advantage with respect to [3, 30], because, asillustrated in [4], an UBST provided in [3] is too conservative. Notice that, the fixed-time stable system with predefinedUBST, analyzed in [8], is found from Table 1-(iii) with h ( z ) = z p with < p < . Thus, the algorithms in [3] and[8] are subsumed in our approach. Example 24
From Table 1 new classes of fixed-time stable systems with predefined UBST, not present in theliterature, can be derived. For instance, the systems ˙ x = − γT c (1 + (cid:107) x (cid:107) )( α log(1 + (cid:107) x (cid:107) ) p + β log(1 + (cid:107) x (cid:107) ) q ) k x (cid:107) x (cid:107) (20) and ˙ x = − π T c (exp(2 (cid:107) x (cid:107) ) − / x (cid:107) x (cid:107) (21) are obtained from Table 1-(i) and Table 1-(ii) with h ( z ) = log(1 + z ) and h ( z ) = z respectively. Moreover, the system ˙ x = − γ (sin( (cid:107) x (cid:107) p ) + 2) T c p (1 + (cid:107) x (cid:107) p ) x (cid:107) x (cid:107) p (22) is obtained from Table 1-(iv) with h ( z ) = z p with < p < . Simulations are shown in Figure 1.4.2 Examples of non-autonomous fixed-time stable systems with predefined least UBST In this subsection, we focus on the construction of functions Φ( ψ − (ˆ t )) satisfying the conditions of Lemma 14. Basedon these functions, we provide some examples of non-autonomous systems, with T c as the settling time for everynonzero trajectory as well as non-autonomous systems with T c as the least UBST .9 ( ψ − (ˆ t )) − Conditions (i) T c η (cid:48) (ˆ t ) (cid:12)(cid:12) − η (ˆ t ) (cid:12)(cid:12) − ( α +1) α ≥ η ( z ) is K T c and C ([0 , T c )) (ii) π sec (cid:16) π ˆ t T c (cid:17) η (cid:48) (cid:16) tan (cid:16) π ˆ t T c (cid:17)(cid:17) η ( z ) is K ∞∞ and C ([0 , + ∞ )) (iii) √ π η (cid:48) (cid:16) erf − (cid:16) ˆ tT c (cid:17)(cid:17) exp (cid:18) erf − (cid:16) ˆ tT c (cid:17) (cid:19) η ( z ) is K ∞∞ and C ([0 , + ∞ )) (iv) γ ( αP (ˆ t ) p + βP (ˆ t ) q ) k η (cid:48) ( P (ˆ t )) γ = Γ ( m p ) Γ ( m q ) (cid:16) αβ (cid:17) mp α k Γ( k )( q − p ) , kp < kq > α, β, p, q, k > m p = − kpq − p , m q = kq − q − p , η ( z ) is K ∞∞ , P ( z ) p − q = βα B − (cid:16) Γ( m p )Γ( m q ) z Γ( k ) T c ; m p , m q (cid:17) − − βα Table 2Examples of Φ( ψ − (ˆ t )), ˆ t = t − t satisfying conditions of Lemma 14, from which non-autonomous fixed-time stable systemswith predefined settling time can be designed. Notice that, in each case, lim t → T − c Φ( ψ − (ˆ t )) − = + ∞ . g ( x ) = xg ( x ) = x + (cid:98) x (cid:101) x = e x = e x = e x = e x = e x = e g ( x ) = xg ( x ) = x + (cid:98) x (cid:101) g ( x ) = xg ( x ) = x + (cid:98) x (cid:101) time ( t ) time ( t ) time ( t ) Figure 2. Examples of non-autonomous fixed-time stable system (18) with t = 0, T c = 1, (cid:98) x (cid:101) = | x | sign( x ) and x ∈ R .From left to right. Φ( ψ − (ˆ t )) − in (23); Φ( ψ − (ˆ t )) − in (24); and Φ( ψ − (ˆ t )) − in (25) with p = 0 . q = 2, α = 1 and β = 2. Proposition 25
With Φ( ψ − (ˆ t )) given in Table 2, Ψ( z, ˆ t ) given in (14) , satisfies the conditions of Lemma 14.Moreover, the system ˙ x = − T c Ψ( (cid:107) x (cid:107) , ˆ t ) x , x ∈ R n , is fixed-time stable with T c as the settling-time for every nonzerotrajectory. PROOF.
To show that Ψ( z, ˆ t ) given in Table 2- (i) satisfies the condition of Lemma 14, choose h ( z ) = α (cid:16) − η ( z )) α − (cid:17) . Note that h ( z ) is K ∞ T c for α ≥
0. Therefore, Proposition 30 can be used with α ≥
0. Hence,by choosing Φ( z ) as in Proposition 30, then Φ( ψ − (ˆ t )) − = T c h (cid:48) (ˆ t ) = T c η (cid:48) (ˆ t )(1 − η (ˆ t )) α +1 . Note that since ψ − (ˆ t ) is C ([0 , T c )) and η (ˆ t ) is C ([0 , T c )), then Φ( z ) is C ([0 , + ∞ )) and therefore satisfies Assumption 11. To show thatwith Φ( ψ − (ˆ t )) given in Table 2- (ii) –Table 2- (iv) , Ψ( x, ˆ t ) given in (14), satisfies the conditions of Lemma 14, let F ( z ) = π ( z +1) , F ( z ) = √ π exp (cid:0) − z (cid:1) and F ( z ) = γ ( αz p + βz q ) − k which satisfies (cid:82) + ∞ F ( z ) dz = 1. Let h ( z )be K ∞∞ and C ([0 , + ∞ )), hence, by Proposition 29, Φ( z ) = h (cid:48) ( z ) π ( h ( z ) +1) and Φ( z ) = h (cid:48) ( z ) √ π exp (cid:0) − h ( z ) (cid:1) satisfyAssumption 11. Furthermore, since Φ( z ) = γ ( αh ( z ) p + βh ( z ) q ) − k h (cid:48) ( z ) is non-increasing, it satisfies Assumption 11.Moreover, by Proposition 29, (11) leads to ψ ( τ ) = T c π arctan( h ( τ )), ψ ( τ ) = T c erf ( h ( τ )) and using Proposition 28, ψ ( τ ) = T c ( α/β ) mp γα k ( q − p ) B (cid:18)(cid:16) αβ h ( τ ) p − q + 1 (cid:17) − ; m p , m q (cid:19) , respectively. Hence, with η ( z ) = h − ( z ) and η (cid:48) ( z ) = h (cid:48) ( h − ( z )) ,we obtain Φ( ψ − (ˆ t )) − given in Table 2- (ii) –Table 2- (iv) . From the construction of Φ( ψ − (ˆ t )) − it follows thatΨ( z, ˆ t ) satisfies the conditions of Lemma 14. The proof that ˙ x = − T c Ψ( (cid:107) x (cid:107) , ˆ t ) x is a fixed-time stable system with T c as the settling-time for every nonzero trajectory follows from Lemma 14- (1) . Remark 26
With α = 0 , Ψ( z, ˆ t ) in Table 2-(i) reduces to the class of TBG proposed in [11]. Particular TBGs,which can be derived from Table 2-(i), were used in [12–16, 26–28] for the design of prescribed-time algorithms. xample 27 Taking η ( z ) = z/T c and α = 0 in Table 2-(ii) results in Φ( ψ − (ˆ t )) − = T c − ˆ t/T c (23) which corresponds to a TBG. Other time-varying gains, which are not a TBG are obtained from Table 2-(ii) andTable 2-(iv) by taking η ( z ) = z , which yields to Φ( ψ − (ˆ t )) − = π (cid:18) π ˆ t T c (cid:19) , (24) and Φ( ψ − (ˆ t )) − = γ ( αP (ˆ t ) p + βP (ˆ t ) q ) k . (25) It follows from Lemma 14, that taking g ( x ) = x leads to system (18) where all non-zero trajectories has T c asthe settling time (Figure 2 solid-line), whereas taking g ( x ) = x + (cid:107) x (cid:107) α − x with < α < leads to system (18) having T c as the least UBST (Figure 2 dotted-line). Thus, for finite initial conditions, the origin is reached before T c . Simulations for the system ˙ x = − T c Ψ( | x | , ˆ t ) g ( x ) using (23) , (24) and (25) , x ∈ R are presented in Figure 2. We presented a methodology for designing fixed-time stable algorithms such that an
UBST is set a priori explicitly asa parameter of the system, proving conditions under which such upper bound is the least one. Our analysis is basedon time-scale transformations and Lyapunov analysis. We have shown that this approach subsumes some existingmethodologies for the design of autonomous and non-autonomous fixed-time stable systems with predefined
UBST and allows to generate new systems with novel vector fields. Several examples are given showing the effectivenessof the proposed method. As future work, we consider the application/extension of these results to differentiators,control and consensus algorithms.
Acknowledgement
The authors would like to thank the anonymous reviewers for his/her valuable comments and suggestions onimproving the quality of the manuscript.
A Auxiliary identitiesProposition 28
The following identities are satisfied i ) (cid:90) + ∞ sin( z ) + a (1 + z ) dz = a − ci(1) cos(1) − si(1) sin(1) ii ) (cid:90) x ( αz p + βz q ) − k dz = ( α/β ) m p α k ( q − p ) B (cid:32)(cid:18) αβ x p − q + 1 (cid:19) − ; m p , m q (cid:33) for kp < , kq > , α, β, p, q, k > , m p = − kpq − p , m q = kq − q − p and a > . PROOF. i) It follows from (cid:82) + ∞ a (1+ z ) dz = a and the change of variables u = 1 + z with integration by parts andthe definition of ci( z ) and si( z ). ii) It follows by the change of variables u = (cid:16) αβ x p − q + 1 (cid:17) − using the definition of B ( · ; · , · ) similarly as in [4]. 11 Some results on designing Φ( z ) Proposition 29
Let h ( · ) be a K ∞∞ function and let F : R + → ¯ R { } a function satisfying (cid:82) + ∞ F ( z ) dz = M . Then, Φ( z ) = M F ( h ( z )) h (cid:48) ( z ) satisfies (7) . Furthermore, with such Φ( z ) , (11) becomes ψ ( τ ) = T c (cid:82) h ( τ )0 F ( ξ ) dξ . Moreover,if F ( z ) < + ∞ , ∀ z ∈ R + and lim z → + h (cid:48) ( z ) = + ∞ , then Φ( · ) satisfies Assumption 9. PROOF.
Using ξ = h ( z ), it follows (cid:82) + ∞ Φ( z ) dz = M (cid:82) + ∞ F ( h ( z )) h (cid:48) ( z ) dz = M (cid:82) + ∞ F ( ξ ) dξ = 1. Moreover, if F ( z ) < + ∞ , ∀ z ∈ R + and lim z → + h (cid:48) ( z ) = + ∞ , then lim z → + Φ( z ) = F (0) lim z → + h (cid:48) ( z ) = + ∞ . Hence, Φ( · )satisfies Assumption 9. The rest of the proof follows from (11) and the change of variables u = h ( ξ ). Proposition 30
Let h ( z ) be a K ∞ T c function. Then, the function Φ( z ) characterized by Φ( h ( z )) = T c (cid:16) dh ( z ) dz (cid:17) − satisfies Assumption 11. Moreover, let Ψ( z, ˆ t ) = Φ( ψ − (ˆ t )) then, the solution of (3) is given by ψ ( τ ) = h − ( τ ) and Φ( ψ − (ˆ t )) = Φ( h (ˆ t )) . PROOF.
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