Constructions of Strict Lyapunov Functions for Discrete Time and Hybrid Time-Varying Systems

We provide explicit closed form expressions for strict Lyapunov functions for time-varying discrete time systems. Our Lyapunov functions are expressed in terms of known nonstrict Lyapunov functions for the dynamics and finite sums of persistency of excitation parameters. This provides a discrete time analog of our previous continuous time Lyapunov function constructions. We also construct explicit strict Lyapunov functions for systems satisfying nonstrict discrete time analogs of the conditions from Matrosov's Theorem. We use our methods to build strict Lyapunov functions for time-varying hybrid systems that contain mixtures of continuous and discrete time evolutions.Comment: 14 pages. Accepted for publication in Nonlinear Analysis: Hybrid Systems and Applications on September 6, 200

Further Remarks on Strict Input-to-State Stable Lyapunov Functions for Time-Varying Systems

We study the stability properties of a class of time-varying nonlinear systems. We assume that non-strict input-to-state stable (ISS) Lyapunov functions for our systems are given and posit a mild persistency of excitation condition on our given Lyapunov functions which guarantee the existence of strict ISS Lyapunov functions for our systems. Next, we provide simple direct constructions of explicit strict ISS Lyapunov functions for our systems by applying an integral smoothing method. We illustrate our constructions using a tracking problem for a rotating rigid body.Comment: 6 pages, submitted for publication in June 200

Further Results on Lyapunov Functions for Slowly Time-Varying Systems

We provide general methods for explicitly constructing strict Lyapunov functions for fully nonlinear slowly time-varying systems. Our results apply to cases where the given dynamics and corresponding frozen dynamics are not necessarily exponentially stable. This complements our previous Lyapunov function constructions for rapidly time-varying dynamics. We also explicitly construct input-to-state stable Lyapunov functions for slowly time-varying control systems. We illustrate our findings by constructing explicit Lyapunov functions for a pendulum model, an example from identification theory, and a perturbed friction model.Comment: Accepted for publication in Mathematics of Control, Signals, and Systems (MCSS) on November 20, 200

Further Results on Strict Lyapunov Functions for Rapidly Time-Varying Nonlinear Systems

We explicitly construct global strict Lyapunov functions for rapidly time-varying nonlinear control systems. The Lyapunov functions we construct are expressed in terms of oftentimes more readily available Lyapunov functions for the limiting dynamics which we assume are uniformly globally asymptotically stable. This leads to new sufficient conditions for uniform global exponential, uniform global asymptotic, and input-to-state stability of fast time-varying dynamics. We also construct strict Lyapunov functions for our systems using a strictification approach. We illustrate our results using a friction control example.Comment: 10 pages, 0 figues, revised and accepted for publication as a regular paper in Automatica in May 2006. To appear in October 2006 issu

Time-Varying Input and State Delay Compensation for Uncertain Nonlinear Systems

A robust controller is developed for uncertain, second-order nonlinear systems subject to simultaneous unknown, time-varying state delays and known, time-varying input delays in addition to additive, sufficiently smooth disturbances. An integral term composed of previous control values facilitates a delay-free open-loop error system and the development of the feedback control structure. A stability analysis based on Lyapunov-Krasovskii (LK) functionals guarantees uniformly ultimately bounded tracking under the assumption that the delays are bounded and slowly varying

3 sampled-data control of nonlinear systems

This chapter provides some of the main ideas resulting from recent developments in sampled-data control of nonlinear systems. We have tried to bring the basic parts of the new developments within the comfortable grasp of graduate students. Instead of presenting the more general results that are available in the literature, we opted to present their less general versions that are easier to understand and whose proofs are easier to follow. We note that some of the proofs we present have not appeared in the literature in this simplified form. Hence, we believe that this chapter will serve as an important reference for students and researchers that are willing to learn about this area of research

Global Stabilization of Triangular Systems with Time-Delayed Dynamic Input Perturbations

A control design approach is developed for a general class of uncertain strict-feedback-like nonlinear systems with dynamic uncertain input nonlinearities with time delays. The system structure considered in this paper includes a nominal uncertain strict-feedback-like subsystem, the input signal to which is generated by an uncertain nonlinear input unmodeled dynamics that is driven by the entire system state (including unmeasured state variables) and is also allowed to depend on time delayed versions of the system state variable and control input signals. The system also includes additive uncertain nonlinear functions, coupled nonlinear appended dynamics, and uncertain dynamic input nonlinearities with time-varying uncertain time delays. The proposed control design approach provides a globally stabilizing delay-independent robust adaptive output-feedback dynamic controller based on a dual dynamic high-gain scaling based structure.Comment: 2017 IEEE International Carpathian Control Conference (ICCC

Stability of time-varying systems in the absence of strict Lyapunov functions

When a non-linear system has a strict Lyapunov function, its stability can be studied using standard tools from Lyapunov stability theory. What happens when the strict condition fails? This paper provides an answer to that question using a formulation that does not make use of the specific structure of the system model. This formulation is then applied to the study of the asymptotic stability of some classes of linear and non-linear time-varying systems.Peer ReviewedPostprint (author's final draft

Adaptive Backstepping Control for Fractional-Order Nonlinear Systems with External Disturbance and Uncertain Parameters Using Smooth Control

In this paper, we consider controlling a class of single-input-single-output (SISO) commensurate fractional-order nonlinear systems with parametric uncertainty and external disturbance. Based on backstepping approach, an adaptive controller is proposed with adaptive laws that are used to estimate the unknown system parameters and the bound of unknown disturbance. Instead of using discontinuous functions such as the $\mathrm{sign}$ function, an auxiliary function is employed to obtain a smooth control input that is still able to achieve perfect tracking in the presence of bounded disturbances. Indeed, global boundedness of all closed-loop signals and asymptotic perfect tracking of fractional-order system output to a given reference trajectory are proved by using fractional directed Lyapunov method. To verify the effectiveness of the proposed control method, simulation examples are presented.Comment: Accepted by the IEEE Transactions on Systems, Man and Cybernetics: Systems with Minor Revision