Stabilization of nonlinear switched systems using control Lyapunov functions

In this paper, we study the stabilization of general nonlinear switched systems by using control Lyapunov functions. The concept of control Lyapunov function for nonlinear control systems is generalized to switched control systems. The first part of our contribution provides a necessary and sufficient condition of stabilization. The main idea is to use a common control Lyapunov function; this is achieved with the converse Lyapunov theorem dedicated to switched systems. In the second part, an explicit construction of a common control Lyapunov function is addressed with respect to a finite family of switched systems. The approach uses a family of control Lyapunov functions attached to the subsystems.     

Stabilization of generalized Hamiltonian systems with internally generated energy and applications to power systems

Passivity-based control (PBC) is a very powerful design methodology for dynamic systems. In this paper, the stabilization of generalized Hamiltonian control systems with internally generated energy is considered using PBC. Sufficient conditions concerning the passivation of this kind of Hamiltonian control systems are given. The results are applied to power systems, including multimachine power systems with steam valve control and single machine infinite bus with superconducting magnetic energy storage. Simulations are employed to demonstrate the effectiveness of the proposed control methodology.     

Isolation of the Trivial Part of a Nonlinear Control System by Factorization: II

The problem of constructing aggregated systems (quotient systems) of the simplest form for nonlinear control systems is considered. This factorization reduces the original control system to a decomposition, which permits one to reduce the dimension of control problems.     

On the Optimal Stabilisation for a set of Programmed Motions of the Bilinear Control System

During recent years there has been considerable interest in using bilinear systems [1, 2] as mathematical models to represent the dynamic behavior of a wide class of engineering, biological and economic systems. Moreover, there are some methods [3] which may approximate nonlinear control systems by bilinear systems. For the first time Zubov has proposed a method of stabilization control synthesis for a set of programmed motions in linear systems [4]. In papers [5, 6] this method has been developed and used to solve the same problem for bilinear systems. In the present paper the following problems are considered. First, synthesis of nonlinear control as feedback under which the bilinear control system has a given set of programmed and asymptotic stable motions. Because this control is not unique, the second problem concerns optimal stabilization. In this paper a method for the design of nonlinear optimal control is suggested. This control is constructed in the form of a convergent series. The theorem on the sufficient conditions to solve this problem is represented.     

Suboptimal Control of Time-Varying Neutral Control Systems

In this paper,the optimal control of time-varying neutral control systems is investigated by means of the Maxmium Principle of time-delay systems.optimal control law is given. Then sub-optimal control of time-varying neutral control systems is studied by using sensitivity approach. Algorithm for suboptimal control is given.     

Nonlinear open-plus-closed-loop (NOPCL) control of dynamic systems

A nonlinear open-plus-closed-loop (NOPCL) control strategy is proposed in this paper to entrain complex dynamic systems to arbitrarily given goal dynamics. It contains nonlinear closed-loop control actions as well as nonlinear open-loop control actions. Compared to the existing open-plus-closed-loop (OPCL) control, the NOPCL control usually results in much simpler error equations that determine the basins of entrainment of the controlled dynamic systems. For a large number of dynamic systems such as the Lorenz, Rossler, Duffing, Van der Pol, and Chua systems, the NOPCL control is shown to have global basins of entrainment for arbitrarily given goal dynamics.     

Model-reference control of chaotic systems

In this paper, the problem of controlling chaotic systems is studied. A control law is introduced for a chaotic system to follow a desired reference system. The control strategy is developed within the general framework of the nonlinear model-reference control systems. Lyapunov stability is used to ensure the global stability of the error dynamics represents the difference between the desired and chaotic systems.     

Research on mean square exponential stability of networked control systems with multi-step delay

A new control mode is proposed for networked control systems whose network-induced delay is longer than a sampling period. The proposed control mode can make full use of control information and improve the performance of the system. Under the control mode, the mathematical model of networked control systems is obtained. Markov characteristic of the transfer delay is discussed. Based on Markov chain theory, the infinite horizon controller is designed, which is shown to render corresponding networked control systems mean square exponentially stable. Simulation results show the validity of the proposed theory.     

Minimum-fuel control of high-order systems

The minimum-fuel control problem is of special interest in various space systems. To date, solutions of minimum-fuel control problems have been carried out for relatively low-order systems. Space structures, however, are generally characterized by a large number of degrees of freedom, so that minimum-fuel control of such systems requires a new approach. In the independent modal-space control (IMSC) method, the control laws are designed in the modal space for each mode independently. The minimum-fuel problem reduces to that of a set of independent second-order systems, so that minimum-fuel control is possible. This paper shows how the IMSC method can be used to control a space structure with a minimum amount of fuel. A numerical example is presented.This research was supported by NASA Research Grant No. NAG-1-225, sponsored by the Spacecraft Control Branch, Langley Research Center.     

Relaxation of nonlinear impulsive controlled systems on Banach spaces

Relaxation control for a class of semilinear impulsive controlled systems is investigated. Existence of mild solutions for semilinear impulsive controlled systems is proved. By introducing a regular countably additive measure, we convexify the original control systems and obtain the corresponding relaxed control systems. The existence of optimal relaxed controls and relaxation results is also proved.     

Contributions to flatness‐based control of systems with input‐dependent delays

Flatness‐based control of nonlinear systems has been generalized in [1] to systems with time‐delays of constant amplitude. Continuous stirred tank chemical reactors with recycle provide examples of such systems. However, if the volume flow‐rate in the recycle is used as control input the time delay depends on control. This can be avoided by a proper choice of the independent variable, i.e., time transformation: Transported volume is used instead of time. This leads to systems with a time delay of constant amplitude belonging to a class of systems which can be called a generalized type of π‐flat systems [2], barely studied so far. Motion planning and tracking control can be achieved by flatness‐based methods.     

一类具独立子系统的退化时滞控制系统的能控性

讨论退化时滞微分控制系统的能控性问题.首先将退化时滞微分控制系统化为标准形式,除去关联项,得到具独立子系统的退化时滞微分控制系统.然后就一般的退化时滞微分控制系统,得到其能控的充要条件为其可达集等于全空间.对于具独立子系统的广义时滞控制系统,给出其能控的充要条件为每个子系统的可达集等于其相应的子空间,并给出其能控的代数判据,最后举例说明主要结果的应用.     

Optimal controller placement in modal control of complex systems

Within the framework of modal control of large systems, a simple approach is advanced for the determination of optimal control configuration under an energy constraint, i.e., optimal locations of a limited number of controllers such that the total energy requirement for control is minimized. It is shown that the resulting design criterion is a simple function of projections of the control matrix onto components of eigenvectors associated with the affected eigenvalues. Furthermore, it is applicable to both single-input and multi-input systems. Systems possessing distinct complex eigenvalues are considered but the approach is equally applicable to other types of systems. Examples show that the minimum-energy control configuration also tends to be the most effective in terms of accomplishing control objectives.     

Suppression of chaotic behavior in horizontal platform systems based on an adaptive sliding mode control scheme

This work presents an adaptive sliding mode control scheme to elucidate the robust chaos suppression control of non-autonomous chaotic systems. The proposed control scheme utilizes extended systems to ensure that continuous control input is obtained in order to avoid chattering phenomenon as frequently in conventional sliding mode control systems. A switching surface is adopted to ensure the relative ease in stabilizing the extended error dynamics in the sliding mode. An adaptive sliding mode controller (ASMC) is then derived to guarantee the occurrence of the sliding motion, even when the chaotic horizontal platform system (HPS) is undergoing parametric uncertainties. Based on Lyapunov stability theorem, control laws are derived. In addition to guaranteeing that uncertain horizontal platform chaotic systems can be stabilized to a steady state, the proposed control scheme ensures asymptotically tracking of any desired trajectory. Furthermore, the numerical simulations verify the accuracy of the proposed control scheme, which is applicable to another chaotic system based on the same design scheme.     

Tracking control of nonlinear chaotic systems with dynamics uncertainties

This paper deals with the tracking control of nonlinear chaotic systems with dynamics uncertainties. A robust control strategy is developed to control a class of nonlinear chaotic systems with uncertainties. The proposed strategy is an input-output control scheme which comprises an uncertainty estimator and a linearizing-like feedback. The control time is explicitly computed. Computer simulations of the Duffing system are provided to verify the validity of the proposed control scheme.     

The observability of linear singularly perturbed systems in state space

The state-space method is used to investigate the complete x- and y-relative observability of linear stationary singularly perturbed (LSSP) systems. Criteria, necessary conditions and sufficient conditions, phrased in terms of matrix ranks, are obtained for the observability; they involve the solutions of the defining equations, which are recurrent algebraic matrix equations. Duality principles are established between the LSSP observed and the control systems with coefficients of varying scales, the LSSP observed systems and LSSP control systems, and the governing equations for the observed and control systems. An example is given.     

On Passivity-Based Control of Non-Classical Electromechanical Systems

Passivity-based control has exclusively been pursued for dynamical systems possessing energetic state functions that are quadratic in the generalized velocities. This assumption does not apply to many dynamical systems, for instance in electromechanics, thus passivity-based control has not been established for these classes of systems. This contribution presents an augmented PD control scheme for such systems. To this end the system dynamics is represented in the event space considered as a Finsler space. It is shown that in this setting the skew symmetry property is retained. A passivity-based augmented PD controller is designed in the event space, and restricted to the configuration space giving rise to a passivity-base control law. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Approximate controllability of a class of semilinear degenerate systems with boundary control

This paper concerns a class of control systems governed by semilinear degenerate equations with boundary control in one-dimensional space. The control is proposed on the ‘degenerate’ part of the boundary. The control systems are shown to be approximately controllable by Kakutani's fixed point theorem.     

年龄相关的非线性时变种群扩散系统最优分布控制的存在性

讨论了一类非线性时变种群扩散系统的最优分布控制问题,利用LionsJL的偏微控制理论和先验估计,证明了系统最优分布控制的存在性.所得结果可为非线性种群扩散系统中的最优控制问题的实际研究提供必要的理论基础.