Independent of delay stabilization using implicit Lyapunov function method

Abstract

Nonlinear implicit Lyapunov function based control algorithms with delay in control and/or measurement channel are presented. The proposed control algorithms provide global asymptotic stability for all delays less than a certain threshold value and global asymptotic stability with respect to a compact set containing the origin for any delay over the threshold value. The algorithms are also applicable for the fast-varying delay and sampled-data cases. The theoretical results are supported by numerical simulations and comparison.

Finite-Time Stability Tools for Control and Estimation

“…As we can conclude from the results of this section, FTS is not a natural type of behavior of time-delay systems, since the influence of past values of the state may block a finite-time settling of the trajectories at the origin, then additional structural conditions are needed. Control design approaches for finite-time stabilization of time-delay systems can be found in Polyakov et al, 2015b;Nekhoroshikh et al, 2020), and the simplest ways of obtaining accelerated regulation in this class of systems is by using prediction techniques to compensate the delays as in (Karafyllis, 2006), the theory of homogeneity (Zimenko et al, 2017;Zimenko et al, 2019) or the domination approach (Wang et al, 2020b).…”

Implicit Lyapunov Function-based Formation Control for Heterogeneous Vehicular Platoons with Time Delays

“…According to [19], if the following linear matrix inequalities ) 0, ( 1)( ) 0, ( ) 0, ( 1)( ) 0, , 0, 0 0, 0 0, 0,…”

Event-triggered Finite-time Control Using Inverse-optimal Implicit Lyapunov Function

“…Remark 3: To solve the implicit function (22) with respect to V (x), the results in [19] show how to explicitly compute the expression of V (x) under n = 2. For the more general case with n > 2, a numerical solution of the implicit equation can be solved directly by the bisection or Newton's method, as discussed in [32].…”

“…However, problems may arise when V → 0, since (i) the physical control command can not be infinity. Invoking the product of D µ (V )x of two signals, the latter of which goes to zero, while the former goes to infinity, some measurement noise of x can result in the product and the control u unbounded; (ii) a singularity of P (V ) occurs and numerical errors exist in the algebraic equation (22). For example, letting µ = 0.5, ρ = 0.9, ψ = (see Example 1 in Section 4), the obtained result through the 'fsolve' function is V = 1.1102×10 −16 which can not make the equality (22) strictly hold since a multiplication of very large and very small values creates numerical problems.…”