Boumediene Hamzi

Analysis and Control of Hopf Bifurcations

In this paper, control systems with two uncontrollable modes on the imaginary axis are studied. The main contributions include the local orientation control of periodic solutions and center manifolds, the quadratic normal form of systems with two imaginary uncontrollable modes, the stabilization of the Hopf bifurcation by state feedback, and the quadratic invariants that characterize the nonlinearity of a system and its Hopf bifurcation.

Nonlinear discrete-time control of systems with a Naimark-Sacker bifurcation

In this paper we study the problem of the stabilization of nonlinear control system with one complex uncontrollable mode. We find normal forms and quadratic invariants; then we compute center manifolds, and use bifurcation theory to synthesize quadratic controllers.

Ignored input dynamics and a new characterization of control Lyapunov functions

Our objective in this paper is to extend as much as possible the dissipativity approach for the study of robustness of stability in the presence of known/unknown but ignored input dynamics. This leads us to: give a new characterization of control Lyapunov functions (CLF) where LfV is upper-bounded by a function of LgV,; define the dissipativity approach as : - assuming the ignored dynamics are dissipative with storage function W and (known) supply rate w, - analyzing closed-loop stability with the sum of the storage function W and a CLF for the nominal part. Stability margin is given in terms of an inequality the supply rate should satisfy. Unfortunately this extension of the dissipativity approach cannot still cope with ignored dynamics which have non zero relative degree or are non minimum phase.

On the control of Hopf bifurcations

Linear and quadratic normal forms of nonlinear systems with a pair of imaginary uncontrollable modes are derived. Based on the normal form, formulae of feedbacks are found to control the bifurcation of the system. The Hopf bifurcation cannot be removed from the closed-loop system, because the imaginary eigenvalues are uncontrollable. However, is it proved that both the orientation and the stability of the periodic solution can be controlled by state feedback. It is proved that a linear feedback determines the orientation of the periodic solution around the bifurcation point, and the quadratic feedback controls the stability of the periodic solution. The explicit relation between the feedback and the performance of the periodic solution, such as the orientation and stability, is derived.

The Controlled Center Dynamics

The center manifold theorem is a model reduction technique for determining the local asymptotic stability of an equilibrium of a dynamical system when its linear part is not hyperbolic. The overall system is asymptotically stable if and only if the center manifold dynamics is asymptotically stable. This allows for a substantial reduction in the dimension of the system whose asymptotic stability must be checked. Moreover, the center manifold and its dynamics need not be computed exactly; frequently, a low degree approximation is sufficient to determine its stability. The controlled center dynamics plays a similar role in determining local stabilizability of an equilibrium of a control system when its linear part is not stabilizable. It is a reduced order control system with a pseudoinput to be chosen in order to stabilize it. If this is successful, then the overall control system is locally stabilizable to the equilibrium. Again, usually low degree approximation suffices.

Canonical forms for nonlinear discrete time control systems

In this paper we provide a canonical form for discrete-time control systems whose linear approximation around an equilibrium is controllable and prove that two systems are feedback equivalent if and only if their canonical forms coincide. This is a nice generalization of results obtained for continuous time control systems. We also compute the homogeneous invariants under the action of a homogeneous feedback group. Consequently, as for the continuous systems, we deduce that the discrete time systems in consideration do not admit nontrivial symmetries, i.e., a map preserving the dynamics.

Balanced reduction of nonlinear control systems in reproducing kernel Hilbert space

We introduce a novel data-driven order reduction method for nonlinear control systems, drawing on recent progress in machine learning and statistical dimensionality reduction. The method rests on the assumption that the nonlinear system behaves linearly when lifted into a high (or infinite) dimensional feature space where balanced truncation may be carried out implicitly. This leads to a nonlinear reduction map which can be combined with a representation of the system belonging to a reproducing kernel Hilbert space to give a closed, reduced order dynamical system which captures the essential input-output characteristics of the original model. Empirical simulations illustrating the approach are also provided.

Control of center manifolds

In this paper, we use a feedback to change the orientation and the shape of the center manifold of a system with uncontrollable linearization. This change directly affect the reduced dynamics on the center manifold, and hence change the stability properties of the original system.

Resonant terms and bifurcations of nonlinear control systems with one uncontrollable mode

In this paper we provide a simple algorithm of feedback design for systems with uncontrollable linearization with only quadratic degeneracy, such as transcritical and saddle-node bifurcations. This approach avoids the computation of nonlinear normal forms. It is based only on quadratic invariants which can be determined directly from the quadratic terms in the uncontrollable dynamics.

Normal forms for discrete time parameterized systems with uncontrollable linearization

We determine quadratic normal forms for discrete time parameterized nonlinear control systems which possess one uncontrollable mode in their linearization. These normal forms are the simplest elements of the equivalence class of the group of transformations by quadratic change of coordinates and quadratic feedback.