Laurent Burlion

Brief paper: On the Lyapunov-based adaptive control redesign for a class of nonlinear sampled-data systems

Stabilization of the exact discrete-time models of a class of nonlinear sampled-data systems, with an unknown parameter, is addressed. Given a Lyapunov-based continuous-time adaptive controller that ensures some stability properties for the closed-loop system, a sufficient condition for the design of high order discrete-time controllers is given. The stability analysis is carried out considering the truncated Fliess series of the Lyapunov difference equation. Due to the appearance of power terms of the unknown parameter, the problem is reparameterized in a convex-like form and an estimation law for the new unknown parameter is derived with no need of overparametrization or projection techniques. Then, assuming appropriate conditions hold, high order controllers can be designed. The boundedness of the extended state vector is ensured under some conditions, for a sufficiently small sampling period. It is shown how increasing the controller order can improve system performance.

Using exponential time-varying gains for sampled-data stabilization and estimation

This paper provides exponential stability results for two system classes. The first class includes a family of nonlinear ODE systems while the second consists of semi-linear parabolic PDEs. A common feature of both classes is that the systems they include involve sampled-data states and a time-varying gain. Sufficient conditions ensuring global exponential stability are established in terms of Linear Matrix Inequalities (LMIs) derived on the basis of Lyapunov-Krasovskii functionals. The established stability results prove to be useful in designing exponentially convergent observers based on sampled-data measurements. It is shown throughout simulated examples from the literature that the introduction of time-varying gains is beneficial to the enlargement of sampling intervals while preserving the stability of the system.

Adaptive Observer for a Class of Parabolic PDEs

The problem of state observation, based on spatially-sampled output measurements, is addressed for a class of infinite dimensional systems, modelled by a semi-linear heat equation augmented with a structured uncertain part involving a set of unknown parameters. An adaptive observer is designed that provides online estimates of the system (spatially distributed) state and unknown parameters based on sampled data (in space). Sufficient conditions for the observer to be exponentially convergent are established. These include an ad-hoc persistent excitation condition as well as a condition on how the observer gain must be selected in relation with the space sampling interval.

Sampled-Data Adaptive Observer for a Class of State-Affine Output-Injection Nonlinear Systems

The problem of observer design is addressed for output-injection nonlinear systems. A major difficulty with this class of systems is that the state equation involves an output-dependent term that is explicitly dependent on unknown parameters. As the output is only accessible to measurement at sampling times, the output-dependent term turns out to be (almost all time) subject to a double uncertainty, making previous adaptive observers inappropriate. Presently, a new hybrid adaptive observer is designed and shown to be exponentially convergent under ad-hoc conditions.

Adaptive boundary observer for parabolic PDEs subject to domain and boundary parameter uncertainties

We are considering the problem of state observation for a class of infinite dimensional systems modeled by parabolic type PDEs. The model is subject to parametric uncertainty entering in both the domain equation and the boundary condition. An adaptive boundary observer, providing online estimates of the system state and parameters, is designed using finite- and infinite-dimensional backstepping-like transformations. The observer is exponentially convergent under an ad hoc persistent excitation condition.

Adaptive observer for parabolic PDEs with uncertain parameter in the boundary condition

We are considering the problem of state observation for a class of infinite dimensional systems. The latter are modelled by a parabolic PDE and a boundary condition involving an unknown parameter. The problem is dealt with using an adaptive observer that provides online estimates of the system (spatially distributed) state and the unknown parameters. The observer is formally shown to be exponentially convergent under an ad-hoc persistent excitation condition.

Output to input saturation transformation: Demonstration and application to disturbed linear systems

In the case of linear systems, control law design is often performed so that the resulting closed-loop meets specific frequency requirements. However, in many cases, it may be observed that the obtained controller does not enforce time-domain requirements amongst which the objective of keeping an output variable in a given interval. In this article, a transformation is proposed to convert expected bounds on an output variable into time-varying saturations on the synthesized linear control law. It is demonstrated that the resulting closed-loop is stable and satisfies time-domain constraints in the presence of unknown bounded disturbance. An application to a linear ball and beam model is presented.

Longitudinal Manoeuvre Load Control of a Flexible Large-Scale Aircraft

Abstract This paper discusses the design and validation of an integrated long range flexible aircraft load controller, at a single flight/mass configuration. The contributions of the paper are in twofold: (i) first, a very recent frequency-limited model approximation technique is used to reduce the dimension of the large-scale aeroservoelastic aircraft model over a finite frequency support while guaranteeing optimal mismatch error, secondly, (ii) a structured controller is designed using an ℋ ∞ -objective and coupled with an output saturation strategy to achieve flight performance and load clearance, i.e. wing root bending moment saturation. The entire procedure - approximation and control - is finally assessed on the high fidelity large-scale aircraft model, illustrating the effectiveness of the procedure on a high fidelity model, used in the industrial context in the load control validation process.

On dynamic inversion with rate limitations

This paper investigates new ways of accounting for saturations in nonlinear dynamic inversion schemes. Because of its high practical interest, the particular case of rate limitations will receive a special attention here. The central contribution of the paper consists of showing how the nonlinear control problem can be rewritten in a standard form, strongly inspired by the robustness analysis framework. More precisely, standard dynamic-inversion-based controllers are viewed in this paper as particular solutions to a more general nonlinear compensation scheme. Various strategies are then proposed to solve the problem, among which multi-channel H∞ design approaches - for which new optimization tools were recently made available - will play a key role. In a second step, anti-windup optimization is also considered to further improve the control laws. The proposed methodology is illustrated by a realistic aircraft control design example.

Updating the gain of global finite-time high gain observers

Recently, new results about semi-global and global finite-time observation have been obtained by the use of continuous high gain observers. In this paper, we propose to extend these results by studying “time-varying high gain” observers and by providing new update laws: first, we adapt the law introduced by L. Praly in the case of asymptotic observation to the finite-time case and we prove that the updated high gain remains bounded. Secondly, we propose a new update law which guarantees the high gains value tends asymptotically to 1.