Bifurcation Control and Feedback Stabilizatio

The problem of local stabilization of nonlinear control systems with linearizations that contain uncontrollable modes on the imaginary axis is considered. A methodology of designing a stabilizing control is investigated. It involves the following steps: (1) reduction of the stability problem to the stability of the center manifold system, (2) simplification of the vector field on the center manifold using the theory of normal forms, and (3) finding conditions under which the simplified vector field is asymptotically stable. Three cases of degeneracies in the linearized system are treated, and sufficient conditions for the existence of stabilizing controls are given in each case. A theorem is presented regarding the robustness of the above control strategies. >

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.

In this note, we deal with the stability and stabilization problems for nonlinear quadratic discrete-time periodic systems. By using the quadratic Lyapunov function, and a so called periodic invariant set, delay-independent sufficient conditions for local stability and local stabilization for nonlinear quadratic discrete-time periodic systems are derived in terms of linear matrix inequalities (LMIs). Based on these sufficient conditions, iterative linear matrix inequality algorithms are proposed for maximizing the stability regions of the systems. Finally, two examples are given to illustrate the effectiveness of the methods presented in this paper.

This paper is concerned with fuzzy-model based stabilization of nonlinear networked control systems (NCSs) with time-varying transmission delays and transmission intervals based on a random-delay approach. The real-time distribution of input delays resulting from transmission delays and intervals is modeled as a dependent and nonidentically distributed process. Then a randomly switched Takagi-Sugeno fuzzy system with multiple input-delay subsystems is proposed to model the nonlinear NCSs. Based on an improved Lyapunov-Krasovskii method, which takes into account the real-time distribution of input delays in estimating cross-product integral terms, new sufficient conditions are deriveed for the exponential stability of the overall systems. The resulting controller design method is equivalent to a nonlinear convex optimization problem with LMI constraints. Numerical examples are presented to substantiate the effectiveness of our results.

Bifurcation analysis and control of macro traffic flow models based on real-time traffic state influence

Characterization of stability limits of Ledinegg instability and density wave oscillations for two-phase flow in natural circulation loops

This paper presents the nonlinear analysis and control of a third-order model for longitudinal flight dynamics. Bifurcation analysis using numerical study for F-8 aircraft reveals that Hopf bifurcations may occur at some critical values of the elevator commands. These results agree with the well-known nonlinear phenomena existed in longitudinal flight dynamics. Furthermore, bifurcation control using both washout filter and direct state feedback are applied to develop the stabilizing laws for system equilibria and periodic solutions emerged from bifurcation points. Numerical simulations are providedto demonstrate these main results.

State-flipped control design for the stabilization of probabilistic Boolean control networks.

The finite-time stabilization and robust control of stochastic nonlinear systems was discussed in this paper. First, by transforming the considered system to its equivalent Hamiltonian system, we showed that the stochastic dissipative Hamiltonian system was finite-time stable in probability if the Hamiltonian function and internal damping matrix possessed some structures. Then, we utilized the energy shaping and damping injection techniques to reformulate the internal structures and Hamiltonian function of the stochastic Hamiltonian system and constructed a finite-time stabilization controller. Finally, a robust finite-time controller was put forward for uncertain stochastic nonlinear systems by using the Hamiltonian function to construct a solution of Hamiltonian-Jacobi Inequality. Numerical simulations results demonstrated the effectiveness of the proposed stabilization and robust stabilization methods.

The stabilization of two-dimensional non-linear control systems is treated, although the main technique is applicable to more general systems. It combines the theory of centre manifolds and singularities of algebraic functions to design continuous, but not necessarily Lipschitz, stabilizing feedback control laws. In particular, the devices known as the Newton diagram and the Newton or Puiseux polygon are employed to construct uniformizing transformations. Finally, remarks are made about systems for which no continuous stabilizing feedback control law exists.

We study the local feedback stabilization of Hopf bifurcations for nonlinear systems of infinite dimensions in the case where the linearized vector field has a pair of simple nonzero imaginary eigenvalues and all its other eigenvalues lie strictly in the left half-plane. Discussing the normal form of nonlinear systems obtained by making use of the integral averaging method, we obtain sufficient and necessary condition for controlling the stability of the systems even if the critical modes are uncontrollable. As an application, we apply the obtained results to the control of axial flow engine compressor.

In this paper, we study a classical Gierer-Meinhardt model with some degenerate conditions. The stability of the constant steady-state solution is first obtained. After that, the Hopf bifurcation and steady-state bifurcation are discussed in detail, especially under the degenerate cases such as invalid simple eigenvalue condition and violated crossing condition. Meanwhile, the bifurcation direction of simple steady-state bifurcation is given, and the local bifurcation can be extended to the global bifurcation. Finally, through numerical simulations, we clearly show that degenerate bifurcations can lead to richer patterns, such as the coupling of two eigenfunction modes. It is shown that the usual pitchfork Hopf and steady-state bifurcations are transformed into the transcritical bifurcations under the degenerate conditions. The results obtained on the degenerate cases can enrich the theoretical and numerical results on the Gierer-Meinhardt model in the existing work.

: Local feedback stabilization of bifurcated solution branches is studied. Two cases are considered: that in which the nominal system undergoes a Hopf bifurcation as a parameter is varied, and the case of a stationary bifurcation from a simple zero eigenvalue. For each case, results on the existence of a stabilizing feedback are given, Moreover, simple synthesis techniques for the stabilizing controllers are discussed. A concept of proximity stabilization is introduced as an alternative to stabilization in the ordinary sense for systems that are not locally stabilizable. A result is stated on the genericity of proximity stabilizability. Motivation for further research in several areas is given. Keywords: Control systems; Electrical engineering.

This paper proposes and develops a generalized sector bound approach to feedback stabilization of nonlinear control systems described by state-space models. This approach is inherited from the methodology of the sector bounded or passive nonlinearities, and influenced by the concept of absolute stability. It aims not only to regionally stabilize the nonlinear dynamics asymptotically but also to maximize the estimated region of quadratic attraction and to ensure nominal performance at each equilibrium. Similar to gain scheduling control, a path of equilibria is programmed based on the assumption of centered-∊-cover which leads to a sequence of linear controllers that regionally stabilize the desired equilibrium asymptotically. Simulation results are worked out to illustrate our proposed design method.

This paper investigates the problem of parameter-dependent state feedback control of continuous-time systems in the context of finite time stability. The controller is designed in order to guarantee that the closed-loop system is finite time stable. The system is considered time varying with the parameters modeled within a unit simplex. The design conditions obtained by means of Lyapunov functions are expressed as linear matrix inequalities. The finite time stability is assessed by using homogeneous polynomially parameter-dependent state feedback gains with arbitrary degree g. LMI relaxations are proposed based on Pólya's theorem. A controller is obtained by the solution of a factibility problem and the effect of the relaxation procedures analyzed by an optimization problem. Numerical examples are provided.