The Controlled Center Dynamics

Abstract

Abstract. 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. 1. Introduction. Center manifold theory plays an important role in the study of the stability of dynamical systems when the equilibrium point is not hyperbolic. The center manifold is an invariant manifold of the differential equation which is tangent at the equilibrium point to the eigenspace of the neutrally stable eigenvalues. For instance, as the local dynamic behavior "transverse" to the center manifold is relatively simple since it is the one of the flows in the local stable (and unstable) manifolds, the center manifold method isolates the complicated asymptotic behavior by locating an invariant manifold tangent to the subspace spanned by the eigenspace of eigenvalues on the imaginary axis. In practice, one does not compute the center manifold and its dynamics exactly, since this requires the resolution of a quasi-linear PDE which is not easily solvable. In most cases of interest, an approximation of degree two or three of the solution is sufficient. Then we determine the reduced dynamics on the center manifold, study its stability, and conclude about the stability of the original system [24,26,21,6,15].The combination of this theory with the normal form approach of Poincaré [25] was used extensively to study parameterized dynamical systems exhibiting bifurcations [27]. The center manifold theorem provides, in this case, a means of systematically reducing the dimension of the state spaces which need to be considered when analyzing bifurcations of a given type. In fact, after determining the center manifold, the analysis of these parameterized dynamical systems is based only on the restriction of the original system on the center manifold whose stability properties are the same as the ones of the full order system. This approach was also adopted in control theory. The combination of the normal form approach for control systems [20] and center manifold theory enabled the analysis and stabilization of systems with one or two uncontrollable modes in continuous and

Event-triggered Stabilization for Nonlinear Systems with Center Manifolds

“…When the overall system cannot be stabilized by just u = K 11 z, feedback of the form u = K 11 z + K 12 y must be employed with K 12 chosen to stabilize the dynamics on the center manifold. This may also be inadequate, following which an additive pseudo-control K n (y) must be introduced to yield u = K 11 z + K 12 y + K n (y), as in [16]. The pseudo-control K n : R k → R m is a continuously differentiable nonlinear function of y and is chosen to stabilize the dynamics on the center manifold.…”

“…The structure of the ISS Lyapunov functions chosen in our work is inspired by the nonsmooth Lyapunov function in [26]. In [16], where design of control laws in the presence of center manifolds is presented, also uses the same nonsmooth Lyapunov function.…”

“…The work presented here builds on the preliminary findings from our recent work [15], where an explicit Lyapunov-based characterisation of local input-to-state stability (LISS) for nonlinear systems with center manifolds was derived, assuming a general controller structure proposed in [16]. This LISS characterization is used in the present work to design eventtriggering conditions.…”

Stabilization design for nonlinear systems based on center manifolds

“…Hamzi et al [5] dealt with control systems with two uncontrollable modes on the imaginary axis. Hamzi et al [6,7] introduced the controlled center dynamics, by choosing a feedback law that stabilized the controlled center dynamics, then it stabilized the full order system. Thus this approach can also be viewed as a reduction technique for some classes of controlled differential equations.…”

Optimal control of nonlinear aeroelastic system with non‐semi‐simple eigenvalues at Hopf bifurcation points

“…7 The controlled center dynamics is a reduced order control system whose dimension is the number of controllable modes and whose stabilizability properties determine the stabilizability properties of the full system. This approach was generalized to the general class of nonlinear systems with any number of uncontrollable modes, 8 it was found that by changing the feedback, the stability properties of the control center dynamics will change, and the stability properties of the full order system will change too.…”