1,755 research outputs found
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
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
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
Indirect Field Oriented Control of Induction Motors is Robustly Globally Stable
Field orientation, in one of its many forms, is an established control method for high dynamic performance AC drives. In particular, for induction motors, indirect fieldoriented control is a simple and highly reliable scheme which has become the de facto industry standard. In spite of its widespread popularity no rigorous stability proof for this controller was available in the literature. In a recent paper (Ortega et al, 1995) [Ortega, R., D. Taoutaou, R. Rabinovici and J. P. Vilain (1995). On field oriented and passivity-based control of induction motors: downward compatibility. In Proc. IFAC NOLCOS Conf., Tahoe City, CA.] we have shown that, in speed regulation tasks with constant load torque and current-fed machines, indirect field-oriented control is globally asymptotically stable provided the motor rotor resistance is exactly known. It is well known that this parameter is subject to significant changes during the machine operation, hence the question of the robustness of this stability result remained to be established. In this paper we provide some answers to this question. First, we use basic input-output theory to derive sufficient conditions on the motor and controller parameters for global boundedness of all solutions. Then, we give necessary and sufficient conditions for the uniqueness of the equilibrium point of the (nonlinear) closed loop, which interestingly enough allows for a 200% error in the rotor resistance estimate. Finally, we give conditions on the motor and controller parameters, and the speed and rotor flux norm reference values that insure (global or local) asymptotic stability or instability of the equilibrium. This analysis is based on a nonlinear change of coordinates and classical Lyapunov stability theory
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 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
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