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.

http://eprints.imperial.ac.uk/retrieve/2034/license.txt

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

An integrated approach to the design of practical adaptive control algorithms is presented. Many existing ideas are brought together, and the effect of various design parameters available to a user is explored. The theory is extended by showing how the problem of stabilizability of the estimated model can be overcome by running parallel estimators. It is shown how asymptotic tracking of deterministic set points can be achieved in the presence of unmodeled dynamics. >

Design issues in adaptive control

This paper proposes a new direct adaptive control algorithm which is robust with respect to additive and multiplicative plant unmodeled dynamics. The algorithm is designed based on the reduced-order plant, which is assumed to be minimum phase and of known order and relative degree, but is analyzed with respect to the overall plant which, due to the unmodeled dynamics, may be nonminimum phase and of unknown order and relative degree. It is shown that if the unmodeled dynamics are sufficiently small in the low-frequency range, then the algorithm guarantees boundedness of all signals in the adaptive loop and "small" residual tracking errors for any bounded initial conditions. In the absence of unmodeled dynamics, the residual tracking error is shown to be zero.

A robust direct adaptive controller

Paper: Instability analysis and improvement of robustness of adaptive control

Online estimation of the frequency of a sinusoidal signal is a classical problem in systems theory that has many practical applications. In this paper the authors provide a solution to the problem of ensuring a globally convergent estimation. More specifically, they propose a new adaptive notch filter whose dynamic equations exhibit the following remarkable features: 1) all signals are globally bounded and the estimated frequency is asymptotically correct for all initial conditions and all frequency values; 2) the authors obtain a simple tuning procedure for the estimator design parameters, which trades-off the adaptation tracking capabilities with noise sensitivity, ensuring (exponential) stability of the desired orbit; and 3) transient performance is considerably enhanced, even for small or large frequencies, as witnessed by extensive simulations. To reveal some of the stability-instability mechanisms of the existing algorithms and motivate our modifications the authors make appeal to a novel nonlinear (state-dependent) time scaling. The main advantage of working in the new time scale is that they remove the coupling between the parameter update law and the filter itself, decomposing the system into a feedback form where the required modifications to ensure stability become apparent. Even though they limit their attention here to the simplest case of a single constant frequency without noise the algorithm is able to track time-varying frequencies, preserve local stability in the presence of multiple sinusoids, and is robust with respect to noise.

/pdf/a-globally-convergent-frequency-estimator-55p3afkx1e.pdf

A globally convergent frequency estimator

Conditions are given under which an integral manifold ("slow manifold") exists for a nonlinear system representing a broad class of adaptive algorithms. The parameter update equation restricted to the manifold is an exact description of the slow adaptation process which can be approximately analyzed by averaging. Stability properties of the slow motion in the manifold are extended off the manifold by a set of conditions under which the manifold is shown to be attractive. This two-time-scale analysis provides a geometrical visualization of instability phenomena observed in adaptive systems and generalizes earlier local stability results.

Integral manifolds of slow adaptation

The instability of adaptive schemes for small values of parameter adjustment gain has the form of a slow drift of adjustable parameters. With an averaging analysis we derive not only a sufficient condition for stability of this drift, but also a sufficient condition for it to be unstable. A sharp boundary between stability and instability is signal dependent and much less demanding than the usual strict positive realness property. The new positivity condition has a simple signal energy interpretation.

http://ieeexplore.ieee.org/iel5/9/24224/01103810.pdf

A stability-instability boundary for disturbance-free slow adaptation with unmodeled dynamics

Two types of disturbance instability of simple adaptive schemes are analyzed. A drift equation shows how stable equilibria are forced to infinity by infrequent but persistent switches of a constant disturbance. The σ-modification proposed by P. Ioannou counteracts these instabilities.

B. Riedle

Papers

Integral manifolds of slow adaptation

A stability-instability boundary for disturbance-free slow adaptation with unmodeled dynamics

Disturbance instabilities in an adaptive system

Hopf Bifurcation in an Adaptive System with Unmodeled Dynamics

Stability bounds for slow adaptation: an integral manifold approach