Quantum control based on three forms of Lyapunov functions

This paper introduces the quantum control of Lyapunov functions based on the state distance, the mean of imaginary quantities and state errors.In this paper, the specific control laws under the three forms are given.Stability is analyzed by the La Salle invariance principle and the numerical simulation is carried out in a 2D test system.The calculation process for the Lyapunov function is based on a combination of the average of virtual mechanical quantities, the particle swarm algorithm and a simulated annealing algorithm.Finally, a unified form of the control laws under the three forms is given.

基于对数型障碍Lyapunov函数的俯仰驾驶仪设计

相控阵雷达制导拦截弹在拦截高速大机动目标时,弹体气动参数剧烈非线性变化影响控制系统对大攻角的响应效果,应设计考虑攻角指令跟踪误差约束的鲁棒自动驾驶仪控制律。基于双幂次趋近律和对数型障碍李雅普诺夫函数设计了鲁棒非线性自动驾驶仪控制律,可将攻角指令的跟踪误差快速收敛于约束范围内。通过降阶扩张状态观测器对系统中存在的气动参数不确定性和系统外部扰动进行在线估计和补偿。稳定性分析和数值仿真表明,攻角可在1 s内达到稳态且攻角响应的稳态误差<10-4°,所提出的自动驾驶仪控制律具有较强的鲁棒性,能够准确稳定地跟踪攻角指令,控制导弹产生所需过载进而精准拦截目标。

Construction of Control Lyapunov Functions for a Class of Nonlinear Systems

The construction of control Lyapunov functions for a class of nonlinear systems is considered. We develop a method by which a control Lyapunov function for the feedback linearizable part can be constructed systematically via Lyapunov equation. Moreover, by a control Lyapunov function of the feedback linearizable part and a Lyapunov function of the zero dynamics, a control Lyapunov function for the overall nonlinear system is established.

On stability of discontinuous systems via vector Lyapunov functions

This paper deals with the stability of systems with discontinuous righthand side （with solutions in Filippov＇s sense） via locally Lipschitz continuous and regular vector Lyapunov functions. A new type of “set-valued derivative” of vector Lyapunov functions is introduced, some generalized comparison principles on discontinuous systems are shown. Furthermore, Lyapunov stability theory is developed for a class of discontinuous systems based on locally Lipschitz continuous and regular vector Lyapunov functions.

A Survey on the Control Lyapunov Function and Control Barrier Function for Nonlinear-Affine Control Systems

This survey provides a brief overview on the control Lyapunov function(CLF)and control barrier function(CBF)for general nonlinear-affine control systems.The problem of control is formulated as an optimization problem where the optimal control policy is derived by solving a constrained quadratic programming(QP)problem.The CLF and CBF respectively characterize the stability objective and the safety objective for the nonlinear control systems.These objectives imply important properties including controllability,convergence,and robustness of control problems.Under this framework,optimal control corresponds to the minimal solution to a constrained QP problem.When uncertainties are explicitly considered,the setting of the CLF and CBF is proposed to study the input-to-state stability and input-to-state safety and to analyze the effect of disturbances.The recent theoretic progress and novel applications of CLF and CBF are systematically reviewed and discussed in this paper.Finally,we provide research directions that are significant for the advance of knowledge in this area.

Universal construction of control Lyapunov functions for a class of nonlinear systems

A method is developed by which control Lyapunov functions of a class of nonlinear systems can be constructed systematically. Based on the control Lyapunov function, a feedback control is obtained to stabilize the closed-loop system. In addition, this method is applied to stabilize the Benchmark system. A simulation shows the effectiveness of the method.

Stabilization of discrete nonlinear systems based on control Lyapunov functions

The stabilization of discrete nonlinear systems is studied. Based on control Lyapunov functions, a sufficient and necessary condition for a quadratic function to be a control Lyapunov function is given. From this condition, a continuous state feedback law is constructed explicitly. It can globally asymptotically stabilize the equilibrium of the closed-loop system. A simulation example shows the effectiveness of the proposed method.

Hybrid Predictive Control Based on High-Order Differential State Observers and Lyapunov Functions for Switched Nonlinear Systems

In this paper, a hybrid predictive controller is proposed for a class of uncertain switched nonlinear systems based on high-order differential state observers and Lyapunov functions. The main idea is to design an output feedback bounded controller and a predictive controller for each subsystem using high-order differential state observers and Lyapunov functions, to derive a suitable switched law to stabilize the closed-loop subsystem, and to provide an explicitly characterized set of initial conditions. For the whole switched system, based on the high-order differentiator, a suitable switched law is designed to ensure the whole closed-loop’s stability. The simulation results for a chemical process show the validity of the controller proposed in this paper.

Constructing Common Quadratic Lyapunov Functions for a Class of Stable Matrices

Narendra 和 Balakrishnan 建议了一个方法构造普通二次的 Lyapunov 功能(CQLF )[1 ] ，稳定的矩阵的 whena 集合是可交换的。这份报纸的目的是概括方法到非可交换、非可解决的大小写。一个修改构造算法被建议，某些条件被提供向是的矩阵保证结果 CQLF。下次，讨论的问题是一个稳定的矩阵什么时候能与 CQLF 被加到一套矩阵为扩大集合构造新 CQLF。

基于Lyapunov-MPC方法的非合作目标近距离抵近控制

针对航天器近距离视线抵近同时存在轨道与姿态机动的非合作目标特定方位控制问题,提出了一种基于控制Lyapunov函数(CLF)的模型预测控制方法 (LMPC)。首先,根据视线坐标系下的轨道动力学方程,建立了满足视线指向要求的航天器相对轨道动力学模型,并推导了期望轨道的解析表达式。其次,利用MPC方法设计控制器进行在线优化控制,并通过纳入基于李雅普洛夫方法的非线性反步控制的显式特征,构建收缩约束式,以确保闭环稳定性。接着,对基于LMPC的控制方法的递归可行性和闭环稳定性进行了证明。最后,仿真结果证明了所设计的LMPC轨迹跟踪方法的有效性和鲁棒性。

基于隐式Lyapunov函数的弹药传输机械臂变增益超螺旋滑模控制

为提高某弹药传输机械臂在传输弹药过程中受到负载不确定性影响下的定位控制鲁棒性,提出一种新型的基于隐式Lyapunov函数变增益超螺旋滑模控制方法。建立弹药传输机械臂系统的动力学方程,并对方程中摩擦力矩及支反力矩的重要参数进行辨识,减少系统不确定性。构造隐式Lyapunov函数并设计出一种新型的变增益超螺旋滑模控制策略,通过Lyapunov稳定性理论证明闭环系统全局稳定。超螺旋滑模控制增益的变化取决于系统的Lyapunov函数,是滑模变量的可微函数,具有控制参数易调节的优势。实验结果表明:摩擦力矩及支反力矩补偿可使系统定位时间由补偿前2.659 s缩短到1.157 s,提高了系统定位性能。在不同工况下,控制方法可有效地抑制系统负载不确定性的影响,保证系统的定位稳定性。

Synchronising chaotic Chua's circuit using switching feedback control based on piecewise quadratic Lyapunov functions

This paper investigates the chaos synchronisation between two coupled chaotic Chua＇s circuits. The sufficient condition presented by linear matrix inequalities （LMIs） of global asymptotic synchronisation is attained based on piecewise quadratic Lyapunov functions. First, we obtain the piecewise linear differential inclusions （pwLDIs） model of synchronisation error dynamics, then we design a switching （piecewise-linear） feedback control law to stabilise it based on the piecewise quadratic Laypunov functions. Then we give some numerical simulations to demonstrate the effectiveness of our theoretical results.

Relaxed delay-dependent stabilization criteria for discrete-time fuzzy systems based on a switching fuzzy model and piecewise Lyapunov functions

This paper presents delay-dependent stability analysis and controller synthesis methods for discrete-time Takagi-Sugeno （T-S） fuzzy systems with time delays. The T-S fuzzy system is transformed to an equivalent switching fuzzy system. Consequently, the delay-dependent stabilization criteria are derived for the switching fuzzy system based on the piecewise Lyapunov function. The proposed conditions are given in terms of linear matrix inequalities （LMIs）. The interactions among the fuzzy subsystems are considered in each subregion, and accordingly the proposed conditions are less conservative than the previous results. Since only a set of LMIs is involved, the controller design is quite simple and numerically tractable. Finally, a design example is given to show the validity of the proposed method.

Design of bounded feedback controls for linear dynamical systems by using common Lyapunov functions

For a linear dynamical system,we address the problem of devising a bounded feedback control,which brings the system to the origin in finite time.The construction is based on the notion of a common Lyapunov function.It is shown that the constructed control remains effective in the presence of small perturbations.

Robust L_1 filtering with pole constraint in a disk via parameter-dependent Lyapunov functions

The problem of robust L 1 filtering with pole constraint in a disk for linear continuous polytopic uncertain systems is discussed. The attention is focused on design a linear asymptotically stable filter such that the filtering error system remains robustly stable, and has a L 1 performance constraint and pole constraint in a disk. The new robust L 1 performance criteria and regional pole placement condition are obtained via parameter-dependent Lyapunov functions method. Upon the proposed multiobjective performance criteria and by means of LMI technique, both full-order and reduced-order robust L 1 filter with suitable dynamic behavior can be obtained from the solution of convex optimization problems. Compared with earlier result in the quadratic framework, this approach turns out to be less conservative. The efficiency of the proposed technique is demonstrated by a numerical example.

Robust D-stability LMI conditions of matrix polytopes via affine parameter-dependent Lyapunov functions

The problem of the robust D-stability analysis for linear systems with parametric uncertainties is addressed. For matrix polytopes, new conditions via the affine parameter-dependent Lyapunov function of uncertain systems are developed with the benefit of the scalar multi-convex function. To be convenient for applications, such conditions are simplified into new linear matrix inequality （LMI） conditions, which can be solved by the powerful LMI toolbox. Numerical examples are provided to indicate that this new approach is less conservative than previous results for Hurwitz stability, Schur stability and D-stability of uncertain systems under certain circumstances.

A new approach to stability analysis of neural networks with time-varying delay via novel Lyapunov-Krasovskii functional

In this paper, new delay-dependent stability criteria for asymptotic stability of neural networks with time-varying delays are derived. The stability conditions are represented in terms of linear matrix inequalities （LMIs） by constructing new Lyapunov-Krasovskii functional. The proposed functional has an augmented quadratic form with states as well as the nonlinear function to consider the sector and the slope constraints. The less conservativeness of the proposed stability criteria can be guaranteed by using convex properties of the nonlinear function which satisfies the sector and slope bound. Numerical examples are presented to show the effectiveness of the proposed method.

Nonlinear Control of Interior PMSM Using Control Lyapunov Functions

In this paper, we introduce a non-linear torque control for an interior permanent-magnet synchronous motor (IPMSM). The nonlinear control is based on a Control Lyapunov Function (CLF) technique. The proposed stabilizing feedback law for the IPMSM drive is a damping control method and is shown to be globally asymptotically stable. The CLF method takes the system nonlinearities into account in the control system design stage. Such nonlinearities are due to the cross coupling between the q and the q currents in addition to the system parameters like the inductances and the flux linkages. The complete IPMSM drive incorporating the proposed CLF has been successfully simulated in a plant model for both motor and inverter. The performance of the proposed drive is investigated in simulation at different operating conditions. It is found that the proposed control technique provides a good torque control performance for the IPMSM drive ensuring the global stability. In later work, we are planning to investigate other phenomena such as magnetic saturation, nonlinear loads, mechanical friction and flexibilities.

Control design and comprehensive stability analysis of acrobots based on Lyapunov functions

A design method for controllers and a comprehensive stability analysis for an acrobat based on Lyapunov functions are presented. Three control laws based on three Lyapunov functions are designed to increase the energy so as to move the acrobot into the unstable inverted equilibrium position, and solve the problem of posture and energy. The concept of a non-smooth Lyapunov function is employed to analyze the stability of the whole system. The validity of this strategy is demonstrated by simulations.

滞后型脉冲泛函微分方程有界性的Lyapunov逆定理

本文通过建立滞后型脉冲泛函微分方程饱和解的存在唯一性定理,在广义常微分方程与滞后型脉冲泛函微分方程等价的基础上,研究了滞后型脉冲泛函微分方程关于一致有界性的Lyapunov逆定理.