Wei Kang

Extended quadratic controller normal form and dynamic state feedback linearization of nonlinear systems

In this paper, a set of extended quadratic controller normal forms of linearly controllable nonlinear systems is given, which is the generalization of the Brunovsky form of linear systems. A set of invariants under the quadratic changes of coordinates and feedbacks is found. It is then proved that any linearly controllable nonlinear system is linearizable to second degree by a dynamic state feedback.

Locally Convergent Nonlinear Observers

We introduce a new method for the design of observers for nonlinear systems using backstepping. The method is applicable to a class of nonlinear systems slighter larger than those treated by Gauthier, Hammouri, and Othman [IEEE Trans. Automat. Control, 27 (1992), pp. 875--880]. They presented an observer design method that is globally convergent using high gain. In contrast to theirs, our observer is not high gain, but it is only locally convergent. If the initial estimation error is not too large, then the estimation error goes to zero exponentially. A design algorithm is presented.

Bifurcation Control via State Feedback for Systems with a Single Uncontrollable Mode

The state feedback control of bifurcations with quadratic or cubic degeneracy is addressed for systems with a single uncontrollable mode. Based on normal forms and invariants, the classification of bifurcations for systems with a single uncontrollable mode is obtained (Table 1). Using invariants, stability characterizations are derived for a family of bifurcations, including saddle-node bifurcations, transcritical bifurcations, pitchfork bifurcations, and bifurcations with a cusp or hysteresis phenomenon. Bifurcations in systems under perturbed feedbacks are also addressed. In the case of a saddle-node bifurcation, continuous but not differentiable feedbacks are introduced to locally remove the bifurcation and to achieve the stability.

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.

Extended normal forms of quadratic systems

A set of extended quadratic controller normal forms of linearly controllable systems with single input is given. These normal forms are considered as the extension of the form due to P. Brunovsky (1970) to the nonlinear systems. It is proved that, given a nonlinear system, there exists a dynamic feedback so that the extended system has a linear approximation which is accurate to the second or higher degree. All the results are restricted to the single-input nonlinear systems. The idea of finding quadratic normal forms and extending the state space is also successfully used in the problem of finding nonlinear observers.<<ETX>>

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.

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.

Observation of a rigid body from measurement of a principal axis

A method for observing the attitude of a freely rotating spacecraft by measuring one of its principal axes is given. A method of determining the angular velocity from the trajectory of one of its coordinates is also given.<<ETX>>

Resonant terms in a class of systems with stationary bifurcations

Bifurcation control of systems with a single parameter and a single input is addressed in this paper. We focus on systems with quadratic degeneracy, i.e. the quadratic part of the system fully determines local behavior. Three qualitative performances are identified for these systems, namely an unstable isolated equilibrium point, a transcritical bifurcation, and a saddle-node bifurcation. The characterization of these bifurcations are found based on invariant matrices and the coefficients of state feedback. The stability of all equilibrium points in a bifurcation is determined by the invariant matrices and the feedback. It is also proved that the qualitative performance of these bifurcations, such as the type of a bifurcation and the stability of an equilibrium point, is independent of quadratic and higher degree terms in the feedback. The approach in this paper does not require quadratic normal form. However, for systems with cubic degeneracy, the general formula for center manifold is too complicated.