Modal exact linearization of a class of second-order switched nonlinear systems

This paper considers the problem of stabilizing single-input affine switched nonlinear systems. The main idea is to transform a switched nonlinear system to an equivalent controllable switched linear system. First, we define the notion of modal state feedback linearization. Then, we develop a set of conditions for modal state feedback linearizability of a certain class of second order switched nonlinear systems. Considering two special structures, easily verifiable conditions are proposed for the existence of suitable state transformations for modal feedback linearization. The results are constructive. Finally, the method is illustrated with two examples, including a Continuous Stirred Tank Reactor (CSTR) to demonstrate the applicability of the proposed approach.     

Problem of Stabilizing a Switching System Using a Piecewise-Linear Control System

The problem of stabilizing a mathematical hybrid system with switchings between the operating modes is solved. Each of these modes is associated with nonlinear differential equations that have control parameters. The switching instances (conditions) are control components. A stabilizer must be designed in positional form that allows the trajectory of the entire nonlinear system to reach the target set in the phase space for a (prescribed) finite time. To solve the problem, k]an apparatus of continuous piecewise-linear Lyapunov functions is used along with the corresponding piecewise-linear control functions. A theorem concerning the sufficient conditions for the stabilizability of a hybrid system in the considered class of controls is proved. An algorithm for constructing the Lyapunov functions and the stabilizer is given.

     

Nonlinear stability of an axial electric field: Effect of interfacial charge relaxation

We introduce a nonlinear perturbation technique to third order, to study the stability between two cylindrical inviscid fluids, subjected to an axial electric field. The study takes into account the relaxation of electrical charges at the interface between the two fluids. At first order, a linear dispersion relation is obtained. Analytical and numerical results for the overstability and incipient instability conditions are given. For perfect dielectric fluids, the electric field has a stabilizing influence, while for leaky dielectric fluids, the electric field can have either a stabilizing or a destabilizing influence depending on the conductivity and permittivity ratios of the two fluids. At higher order, a nonlinear dispersion relation (nonlinear Ginzburg–Landau equation) is derived, describing the evolution of wave packets of the problem. For leaky dielectric fluids near the marginal state, a nonlinear diffusion equation (nonlinear incipient instability) is obtained. For perfect dielectric fluids, two cubic nonlinear Schrödinger equations are obtained. One of these equations to determine a nonlinear cutoff electric field separating stable and unstable disturbance, whereas the other is used to analyze the stability of the system. It is found that the nonlinear stability criterion depending on the ratio of permittivity, Such effects can only be explained successfully in the nonlinear sense, as the linear analysis unsuccessful to inform about them.     

Solvability of nonlinear boundary-value problems arising in modeling plasma diffusion across a magnetic field and its equilibrium configurations

We study the simplest one-dimensional model of plasma density balance in a tokamak type system, which can be reduced to an initial boundary-value problem for a second-order parabolic equation with implicit degeneration containing nonlocal (integral) operators. The problem of stabilizing nonstationary solutions to stationary ones is reduced to studying the solvability of a nonlinear integro-differential boundary-value problem. We obtain sufficient conditions for the parameters of this boundary-value problem to provide the existence and the uniqueness of a classical stationary solution, and for this solution we obtain the attraction domain by a constructive method.     

Feedback Stabilization Methods for the Solution of Nonlinear Programming Problems

In this work, we show that, given a nonlinear programming problem, it is possible to construct a family of dynamical systems, defined on the feasible set of the given problem, so that: (a) the equilibrium points are the unknown critical points of the problem, which are asymptotically stable, (b) each dynamical system admits the objective function of the problem as a Lyapunov function, and (c) explicit formulas are available without involving the unknown critical points of the problem. The construction of the family of dynamical systems is based on the Control Lyapunov Function methodology, which is used in mathematical control theory for the construction of stabilizing feedback. The knowledge of a dynamical system with the previously mentioned properties allows the construction of algorithms, which guarantee the global convergence to the set of the critical points.     

Feedback stabilization and adaptive stabilization of stochastic nonlinear systems by the control Lyapunov function

Our aims of this paper are twofold: On one hand, we study the asymptotic stability in probability of stochastic differential system, when both the drift and diffusion terms are affine in the control. We derive sufficient conditions for the existence of control Lyapunov functions (CLFs) leading to the existence of stabilizing feedback laws which are smooth, except possibly at the equilibrium state. On the other hand, we consider the previous systems with an unknown constant parameters in the drift and introduce the concept of an adaptive CLF for stochastic system and use the stochastic version of Florchinger's control law to design an adaptive controller. In this framework, the problem of adaptive stabilization of nonlinear stochastic system is reduced to the problem of non-adaptive stabilization of a modified system.     

Observability of Nonlinear Systems

We consider the observability of systems of the form = Ax +Nx, y = Fx, where A is a linear operator and N and F are nonlinear.We show that if the system is linearized about an equilibriumpoint x_e and the linearized system is continuously initiallyobservable, then the nonlinear system is continuously initiallyobservable in some neighbourhood of x_e. We then look at conditionsunder which solutions of the nonlinear system can be extendedfor all time and consider the problem of stabilizing the systemby feedback controls such that the solutions are eventuallyin the observability neighbourhood of x_e. Finally, we applythese ideas to two systems: a wave equation and a diffusionequation with nonlinear perturbations and nonlinear observations.     

Optimal NPID Stabilization of Linear Systems

We consider the problem of finding the optimal, robust stabilization of linear systems within a family of nonlinear feedback laws. Investigation of the efficiency of full-state based and partial-state based so-called NPID feedback schemes proposed for the stabilization of systems in robotic applications has provided the motivation for our work. We prove that, for a given quadratic Lyapunov function and a given family of nonlinear feedback laws, there exist optimal piecewise linear feedbacks that make the generalized Lyapunov derivative of the closed-loop system minimal. The result provides improved stabilization over the nonlinear stabilizing feedback law proposed in Ref. 1 as demonstrated in simulations of the Sarcos Dextrous Manipulator.     

一类非线性系统的输出反馈控制及全局渐近稳定性研究

考虑不可测状态是非线性的非严格三角形非线性系统的全局渐近稳定性问题.提出了一种新的反馈控制设计方法,构造一个线性动态输出补偿器,并全局稳定所控制的非线性系统.     

Stabilization of nonlinear discrete-time dynamic control systems with a parameter and state-dependent coefficients

A numerical-analytical algorithm for designing nonlinear stabilizing regulators for the class of nonlinear discrete-time control systems is proposed that significantly reduces computational costs. The resulting regulator is suboptimal with respect to the constructed quadratic functional with state-dependent coefficients. The conditions for the stability of the closed-loop system are established, and a stability result is stated. Numerical results are presented showing that the nonlinear regulator designed is superior to the linear one with respect to both nonlinear and standard time-invariant cost functionals. An example demonstrates that the closed-loop system is uniformly asymptotically stable.     

Chaos control in AFM systems using nonlinear delayed feedback via sliding mode control

In this paper a nonlinear delayed feedback control is proposed to control chaos in an Atomic Force Microscope (AFM) system. The chaotic behavior of the system is suppressed by stabilizing one of its first-order Unstable Periodic Orbits (UPOs). At first, it is assumed that the system parameters are known, and a nonlinear delayed feedback control is designed to stabilize the UPO of the system. Then, in the presence of model parameter uncertainties, the proposed delayed feedback control law is modified via sliding mode scheme. The effectiveness of the presented methods is numerically investigated by stabilizing the unstable first-order periodic orbit of the AFM system. Simulation results show the high performance of the methods for chaos elimination in AFM systems.     

Some results on the internal finite‐time stabilization of vibrating strings with interior mass

In this paper, the problem of internal finite‐time stabilization for 1‐D coupled wave equations with interior point mass is handled. The nonlinear stabilizing feedback law leads, in closed‐loop, to nonlinear evolution equations where Kato theory is used to prove the well‐posedness. In addition, it is showed that in some cases, the solution of this hybrid system is constant in finite‐time if we use Neumann boundary conditions. This result can be improved (in complete finite‐time stability sense) if we change the above feedback.     

Convection in a variable gravity field

The problem of convection in a variable gravity field is studied by using methods of linear instability theory and nonlinear energy theory. It is shown that the decreasing or increasing of gravity in a specific direction can be stabilizing or destabilizing and it is further shown how to quantify this effect. Specific results are presented for the situation where gravity decreases linearly throughout a plane layer. The nonlinear results are found to be very close to the linear ones and define a small band where possible subcritical instabilities may arise.