Structure-preserving stabilization of Hamiltonian control systems

Sufficient conditions for nonasymptotical stabilization of equilibrium points in Hamiltonian control systems using the Lyapunov technique and feedback transformation which preserve the Hamiltonian structure are presented.     

A set oriented approach to optimal feedback stabilization

We present a numerical construction of an optimal control based feedback law for the stabilization of discrete time nonlinear control systems. The feedback is based on a recently developed numerical solution method for optimal control problems using set oriented and graph theoretic algorithms. We show how this method can be used to construct approximately optimal and stabilizing feedback laws and present an a posteriori error estimation technique for the adaptive generation of the underlying set oriented space discretization.     

A new approach to the cheap LQ regulator exploiting the geometric properties of the Hamiltonian system

The cheap LQ regulator is reinterpreted as an output nulling problem which is a basic problem of the geometric control theory. In fact, solving the LQ regulator problem is equivalent to keep the output of the related Hamiltonian system identically zero. The solution lies on a controlled invariant subspace whose dimension is characterized in terms of the minimal conditioned invariant of the original system, and the optimal feedback gain is computed as the friend matrix of the resolving subspace. This study yields a new computational framework for the cheap LQ regulator, relying only on the very basic and simple tools of the geometric approach, namely the algorithms for controlled and conditioned invariant subspaces and invariant zeros.     

Finite‐time robust simultaneous stabilization of a set of nonlinear time‐delay systems

This paper investigates the finite‐time robust simultaneous stabilization problem of a set of nonlinear time‐delay systems with general forms and proposes some new simultaneous stabilization results. First, by developing an equivalent form and applying augmented technique, this paper obtains an augmented equivalent form of the original systems. Secondly, based on the equivalent form, we study finite‐time simultaneous stabilization problem and present some new stabilization results by constructing some suitable Lyapunov functionals. Thirdly, using the simultaneous stabilization results obtained, this paper investigates the finite‐time robust simultaneous stabilization problem for the set systems and proposes a delay‐dependent robust simultaneous stabilization result. Finally, the study of an illustrative example shows that the results obtained by this paper work well in the finite‐time robust simultaneous stabilization the set systems. It is shown that, by using the method in this paper, the developed conditions do not contain delay terms, which can avoid solving nonlinear mixed matrix inequalities and reduce effectively computational burden in studying nonlinear time‐delay systems.     

An average optimal control approach to the set stabilization problem for boolean control networks

This paper addresses the set stabilization problem for deterministic Boolean control networks (BCNs). An optimal control approach is investigated to solve the problems by using the semi‐tensor product of matrices, where a policy iteration algorithm for the set stabilization problem is deduced. Finally, the intervention problem of a cAMP receptor protein is addressed in the framework of the set stabilization problem. The problem is solved to validate the effectiveness of the proposed policy iteration approach for a practical application.     

给定环境下稳定开放量子系统的哈密顿量方法

针对耗散已知情况下Lindblad主方程描述的开放量子系统,本文通过哈密顿量的设计实现了系统对于目标平衡态的稳定化.借助相干矢量体系,将矩阵形式下的原始系统模型转换为了一个矢量形式的线性系统,并证明了变换前后系统稳定属性的等价性.通过保证矢量化线性系统模型的稳定性,并使系统的唯一平衡态等于期望的目标态,得到了系统哈密顿量的设计框架.特别地,本文讨论了这两类条件下系统哈密顿量各元素的范围,并指出根据它们的交集即可构成所设计的系统哈密顿量.最后,在一个两能级系统上进行了数值仿真实验,验证了本文哈密顿量稳定化方案的有效性.     

Global output feedback stabilization of upper‐triangular nonlinear systems using a homogeneous domination approach

This paper addresses the problem of global output feedback stabilization for a class of upper‐triangular systems with perturbing nonlinearities that are higher‐order in the unmeasurable states. A new design method based on the homogeneous domination approach and finite‐time stabilization technique is developed, which leads to global output feedback stabilizers for the upper‐triangular nonlinear systems under a homogeneous growth condition. A new perspective shown in this paper is that the finite‐time stabilization, in addition to its faster convergence rate, can also be utilized to handle control problems that were previously unresolved under asymptotic stabilization. Copyright © 2006 John Wiley & Sons, Ltd.     

Exponential stability and stabilization of extended linearizations via continuous updates of Riccati‐based feedback

Many recent works on the stabilization of nonlinear systems target the case of locally stabilizing an unstable steady‐state solution against small perturbations. In this work, we explicitly address the goal of driving a system into a nonattractive steady state starting from a well‐developed state for which the linearization‐based local approaches will not work. Considering extended linearizations or state‐dependent coefficient representations of nonlinear systems, we develop sufficient conditions for the stability of solution trajectories. We find that if the coefficient matrix is uniformly stable in a sufficiently large neighborhood of the current state, then the state will eventually decay. On the basis of these analytical results, we propose a scheme that is designed to maintain the stabilization property of a Riccati‐based feedback constant during a certain period of the state evolution. We illustrate the general applicability of the resulting algorithm for setpoint stabilization of nonlinear autonomous systems and its numerical efficiency in 2 examples.     

Passivity‐based asymptotic stabilization for switched nonlinear systems using the sampled integral stabilization technique

The asymptotic stabilization problem is studied for a cascade connection of passive switched nonlinear systems and a passive switched nonlinear system in this paper. When each subsystem is asymptotically zero state detectable and passive on active time intervals, asymptotic stabilization is achieved via co‐design of switching laws and controllers without damping injection. First, an output‐feedback controller is designed to asymptotically stabilize a cascade connection of two passive switched systems if outputs are measurable. Second, when the output of the first switched system is noisy or unmeasurable, a sampled integral stabilization (SIS) technique is employed to investigate asymptotical stabilization of a cascade connection by measuring only the storage function of the second switched system. Finally, as a special case of a cascade connection, the SIS technique is used to stabilize a passive switched system without damping injection. Under this circumstance, the controller is designed by sampling the integral of the passive output. The two‐link robot manipulator is provided to illustrate the effectiveness of the SIS technique.     

Switched system approach to stabilization of networked control systems

Stabilization problems of networked control systems (NCSs) with bounded packet losses and transmission delays are addressed. We model such NCSs as a class of switched systems, and establish stabilizing conditions in the form of matrix inequalities by using packet‐loss dependent Lyapunov functions. By solving the inequalities, packet‐loss dependent controllers are designed for two types of packet‐loss processes: one is an arbitrary packet‐loss process, and the other is a Markovian packet‐loss process. Several numerical examples and simulations are worked out to demonstrate the effectiveness of the proposed design techniques. Copyright © 2010 John Wiley & Sons, Ltd.     

Input‐to‐output stabilization of nonlinear systems via backstepping

An extension of backstepping method for stabilization of nonlinear systems with respect to a set is presented. Robust control laws providing the system with input‐to‐output stability are proposed. Possibilities of non‐strict Lyapunov functions' application are discussed. The differences between a conventional backstepping method and an approach proposed in Kolesnikov (Synergetic Control Theory. Energoatomizdat: Moscow, 1994; 344) are analyzed. Performance of the obtained solutions is demonstrated by computer simulation for pendulum with actuator example. Copyright © 2008 John Wiley & Sons, Ltd.     

Stabilization of continuous‐time homogeneous bilinear systems by constant inputs through nonlinear minimization with applications to the static output feedback stabilization problem

An algorithm to provide constant‐input stabilizing control inputs for multi‐input continuous‐time bilinear systems is proposed in this paper. The algorithm is based on the formal discrete‐time approximation of the system and the unconstrained nonlinear minimization. The key features of the new algorithm are as follows. First, the formal discrete‐time approximation makes the set of stabilizing control inputs star‐shaped centred at the origin, hence the minimization is to be performed only in a neighbourhood of the origin selected by the designer. Second, the algorithm is always capable of finding a solution if one exists, as long as the minimization inside the neighbourhood is successful. Third, by a slight modification, the algorithm permits us to place all the eigenvalues of the system inside a rectangular region in the complex plane, as long as it is feasible. The algorithm is also applicable to the static output feedback stabilization problem of linear time‐invariant systems. Copyright © 2007 John Wiley & Sons, Ltd.     

Local stabilization of semilinear parabolic system by boundary control

This paper addresses local stabilization of a semilinear parabolic system by boundary control. Under a certain assumption on the nonlinearity of the parabolic equation, a linear boundary feedback control law is applied to control the semilinear system. Based on the operator theories and relations of different norms, locally exponential stabilization of the closed loop system is established. Simulations support the established stability result.     

Decentralized uniform finite-time stabilization of large-scale nonlinear networks by small-gain theorems

We solve the problem of decentralized uniform finite-time stabilization for a large-scale network of nonlinear systems in triangular form which are not feedback linearizable and whose input–output maps are not invertible. For this, we provide a new recursive design to satisfy the conditions of a certain modification of the small-gain theorems for the case of the uniform finite-time stability of large-scale networks. In the general case, we consider the general triangular form systems whose input–output links are surjections only. In addition, we consider the special case of the triangular form systems with polynomial integrators, which is also needed as the first step of the proof of our main result. In the latter case, the design becomes constructive, the decentralized stabilizers can be designed explicitly, and this is demonstrated by examples.     

Exponential stabilization of nonholonomic dynamic systems by smooth time-varying control

A general dynamic model is proposed for describing a large class of nonholonomic systems including extended chained systems, extended power systems, underactuated surface vessel systems etc. By introducing an assistant state variable and a time-varying state transformation based on the concept of minimal dilation degree, this class of nonholonomic systems is transformed into linear time-varying control systems, and the asymptotic exponential stability is thus achieved by using a smooth time-varying feedback control law. The existence and uniqueness of the minimal dilation degree for the discussed systems are also proved under certain conditions.     

Global output‐feedback stabilization for stochastic nonlinear systems: A double‐domination approach

This paper is concerned with global stabilization via output feedback for a class of stochastic nonlinear systems with time‐varying continuous output function. Under linear growth conditions, a new double‐domination method is proposed for the first time to construct an output‐feedback stabilizing controller. Different from the related results, the design of the observer is performed without using the information on the output function and nonlinearities. This paper also provides a viewpoint at the feedback stabilization to eliminate the continuous measurement error originating from inaccurate detection of system state. A simulation example is presented to demonstrate the effectiveness of control strategy.     

Feedback stabilization of a linear control system in Hilbert space with ana priori bounded control

This paper derives a feedback controlf(t), ‖f(t)‖_E≦r,r>0, which forces the infinite-dimensional control systemdu/dt=Au+Bf, u(0)=u _o ≠H to have the asymptotic behavioru(t)→0 ast→∞ inH. HereA is the infinitesimal generator of aC _o semigroup of contractionse ^At on a real Hilbert spaceH andB is a bounded linear operator mapping a Hilbert space of controlsE intoH. An application to the boundary feedback control of a vibrating beam is provided in detail and an application to the stabilization of the NASA Spacecraft Control Laboratory is sketched. This research was sponsored in part by the Air Force Office of Scientific Research, Air Force Systems Command, USAF Contract/Grants AFOSR 81-0172 and AFOSR 87-0315.     

Robust stabilization of nonlinear plants—an L2 approach

This paper extends the theory of H_∞ control for linear plants to input affine nonlinear plants without special output structure. This result is used to develop solutions to the robust stabilization problem for several classes of uncertain nonlinear plants.