Nonlinear Regulation Theory Research Articles

Behavior of a stable nonlinear infinite-dimensional system under the influence of a nonlinear exosystem

This paper considers a nonlinear infinite-dimensional system ΣN obtained by the feedback interconnection of a well-posed linear system ΣP with a globally Lipschitz (memoryless) nonlinear feedback operator N. First, under mild assumptions, we establish the global existence and uniqueness of a state trajectory and an output function for ΣN, for any initial state in its state space X and any input signal of class. Then we investigate the behavior of ΣN when it is driven by a nonlinear time-invariant exosystem with well defined dynamics forward and backward in time. Under the assumption that ΣP is exponentially stable, denoting the state space of the exosystem by W, we find that there exists a continuous map Π : W → X such that regardless of initial states limt→∞ ‖Πw(t) – x(t)‖ = 0, where w(t) is the state of the exosystem and x(t) is the state of ΣN. In particular, when w is T-periodic, then the state of the interconnection tends to a T-periodic limit cycle. The construction of Π can be viewed as an extension of the famous center manifold theorem, which lies at the basis of nonlinear regulator theory, to a class of infinite-dimensional systems.

Designing Nonlinear Zero Dynamics to Reject Periodic Waveforms

In linear systems, designers use zeros for loop-shaping and the attenuation of disturbances. For example, imaginary-axis zeros can be used to reject sinusoids at a specific frequency such as a 50 Hz or 60 Hz power line interference. If the disturbance is nonsinusoidal, repetitive control methods may be used but additional controller states for each harmonic are required. Thus, rejecting periodic disturbance that are nonsinusoidal with low controller order remains an open problem. In this paper, we constructively design low-order nonlinear dynamics that reject a nonsinusoidal disturbance. We identify a particular case of the nonlinear regulator theory wherein the Byrnes–Isidori PDE has a simple solution. This case leads to a constructive procedure for designing nonlinear zero dynamics in systems with input disturbances. The constructive procedure is accessible to designers who do not have experience with the nonlinear geometric control theory.

Tracking of Partially Unknown Trajectories for Permanent Magnet Synchronous Motors ,

In this paper, an adaptive internal model-based control architecture is designed to deal with an exogenous trajectory tracking problem for a Permanent Magnet Synchronous Motor (PMSM). More in detail, we show how to design a controller able to guarantee the asymptotic tracking of partially unknown exogenous trajectories belonging to a given family, embedding in the regulator the internal model of this family; the control algorithm is able to attain the asymptotic tracking without the knowledge of reference derivatives and exploiting only instantaneous error feedbacks. The theoretical machinery exploited is the nonlinear regulation theory, specialized for the energy-based port- Hamiltonian formalism. The same methodology is proved to be able to deal even with the presence of exogenous voltage disturbances superimposed to the control voltages.

Stabilization of Uncertain Nonlinear Systems via Immersion and Invariance

This paper presents a method for designing asymptotically stabilising and adaptive control laws for uncertain nonlinear systems. The method relies upon the notions of system immersion and manifold invariance and, in principle, does not require the knowledge of a (control) Lyapunov function. The construction of the stabilising control laws resembles the procedure used in nonlinear regulator theory to derive the (invariant) output-zeroing manifold and its friend, and is well suited, for instance, in situations where we know a stabilising controller of a nominal reduced-order model, which we would like to robustify with respect to higher-order dynamics. In adaptive control problems the method yields stabilising schemes that counter the effect of the uncertain parameters adopting a robustness perspective. This is in contrast with most existing adaptive designs that (relying on certain matching conditions) treat these terms as disturbances to be rejected. The construction does not invoke certainty equivalence, nor requires a linear parameterisation. Furthermore, viewed from a Lyapunov perspective, it provides a procedure to add cross terms between the parameter estimates and the plant states. To illustrate the concepts presented in this paper we introduce several academic and physical examples, including a mechanical system with flexibility modes, an electromechanical system with parasitic actuator dynamics, and an adaptive, nonlinearly parameterised, visual servoing problem.

Immersion and invariance: a new tool for stabilization and adaptive control of nonlinear systems

A new method to design asymptotically stabilizing and adaptive control laws for nonlinear systems is presented. The method relies upon the notions of system immersion and manifold invariance and, in principle, does not require the knowledge of a (control) Lyapunov function. The construction of the stabilizing control laws resembles the procedure used in nonlinear regulator theory to derive the (invariant) output zeroing manifold and its friend. The method is well suited in situations where we know a stabilizing controller of a nominal reduced order model, which we would like to robustify with respect to higher order dynamics. This is achieved by designing a control law that asymptotically immerses the full system dynamics into the reduced order one. We also show that in adaptive control problems the method yields stabilizing schemes that counter the effect of the uncertain parameters adopting a robustness perspective. Our construction does not invoke certainty equivalence, nor requires a linear parameterization, furthermore, viewed from a Lyapunov perspective, it provides a procedure to add cross terms between the parameter estimates and the plant states. Finally, it is shown that the proposed approach is directly applicable to systems in feedback and feedforward form, yielding new stabilizing control laws. We illustrate the method with several academic and practical examples, including a mechanical system with flexibility modes, an electromechanical system with parasitic actuator dynamics and an adaptive nonlinearly parameterized visual servoing application.

On robust regulation via sliding mode for nonlinear systems

A robust control scheme for the output tracking of nonlinear systems is proposed. The scheme is based on the nonlinear regulator theory and the sliding mode approach. It is shown that if perturbations and/or uncertainties appear on the system, the sliding mode controller is able to perform approximate asymptotic tracking, in the sense that the output tracking error remains bounded, provided the upper bounds of the uncertain terms are known.

Nonlinear regulator theory and an inverse optimal control problem

Nonlinear optimal regulators are discussed, and some useful properties are isolated. An inverse problem of nonlinear regulator design is posed and solved.

Nonlinear regulator theory and an inverse optimal control problem

Nonlinear regulator theory and an inverse optimal control problem