Balancing for a class of non-linear singularly perturbed systems

In this work, a balancing method is investigated for a class of non-linear singularly perturbed systems. The main result presented here shows that the well-known 'two-stages' strategy used with singular perturbations in control theory can be extended to compute a balancing form of non-linear singularly perturbed systems. So, an approximate balancing form is derived from the balancing forms of the slow and fast subsystems both computed separately. This two-stage method avoids the difficult task of solving high dimensional and ill-conditioned Hamilton-Jacobi equations due to the presence of the small singular perturbation parameter.     

Stabilization of discrete systems with multiple-time scales

A two-stage method is developed for the stabilization of linear time-invariant discrete systems with multiple-time scales. It is shown that the feedback gains are completely independent. The closed-loop system is asymptotically stable for all sufficiently small singular perturbation parameters which are constrained to be in a bounded set.     

Application of a Perturbation Method for Realistic Dynamic Simulation of Industrial Robots

This paper presents the application of a perturbation method for the closed-loop dynamic simulation of a rigid-link manipulator with joint friction. In this method the perturbed motion of the manipulator is modelled as a first-order perturbation of the nominal manipulator motion. A non-linear finite element method is used to formulate the dynamic equations of the manipulator mechanism. In a closed-loop simulation the driving torques are generated by the control system. Friction torques at the actuator joints are introduced at the stage of perturbed dynamics. For a mathematical model of the friction torques we implemented the LuGre friction model that accounts both for the sliding and pre-sliding regime. To illustrate the method, the motion of a six-axes industrial Stäubli robot is simulated. The manipulation task implies transferring a laser spot along a straight line with a trapezoidal velocity profile. The computed trajectory tracking errors are compared with measured values, where in both cases the tip position is computed from the joint angles using a nominal kinematic robot model. It is found that a closed-loop simulation using a non-linear finite element model of this robot is very time-consuming due to the small time step of the discrete controller. Using the perturbation method with the linearised model a substantial reduction of the computer time is achieved without loss of accuracy.     

Decentralized Adaptive Control of Interconnected Non-Linear Systems Using High Gain Observer

This paper investigates a decentralized adaptive control strategy for a class of interconnected unknown non-linear systems. The idea of the strategy is based on the feedback linearizing control and perturbation estimation. A high gain observer is designed in association with each sub-system to estimate the states and a fictitious state which is defined to represent the system perturbation including the combined effect of system non-linearities, unknown system dynamics, disturbances and interactions between sub-systems. Subject to the availability of sub-system states, two local controllers — decentralized non-linear adaptive state-feedback controller (DNASFC) and decentralized non-linear adaptive output-feedback controller (DNAOFC) — are developed using a high gain perturbation observer (HGPO) or a high gain state and perturbation observer (HGSPO) respectively. The stability and error analysis of the high gain observers and the closed-loop control systems are addressed in detail. The two proposed controllers are evaluated on an interconnected non-linear system involving two inverted pendulums on carts without velocity measurements.     

一种闭环微纳定位扫描系统设计

针对小推力压电陶瓷的特性,提出了一种闭环微纳定位扫描系统设计。该系统以单片机80C196KC为核心,使用12位A／D转换器MAX120和12位D／A转换器DA7521做D／A转换,模拟驱动部分采用前置低压误差运放和后置高压精密运放组成双级放大模块;控制部分采用Fuzzy-PID复合控制算法进行非线性处理,很好地改善了压电陶瓷的迟滞、蠕变、非线性等不足,且提高了控制系统的精度。将本系统应用于白光相移干涉检测,实验结果表明：闭环电压控制的最大误差为18mV,置位重复精度为6nm,系统相对误差为0．15％;在位移量为10μm的时间一位移阶跃响应曲线中阶越响应时间约为20ms,且在电压为80V的蠕动曲线中,3min内的蠕变为3．6％,3min以后变化开始缓慢。     

Robust stability and stabilization of discrete-time non-linear systems: The LMI approach

The purpose of this paper is to convert the problem of robust stability of a discrete-time system under non-linear perturbation to a constrained convex optimization problem involving linear matrix inequalities (LMI). The nominal system is linear and time-invariant, while the perturbation is an uncertain non-linear time-varying function which satisfies a quadratic constraint. We show how the proposed LMI framework can be used to select a quadratic Lyapunov function which allows for the least restrictive non-linear constraints. When the nominal system is unstable the framework can be used to design a linear state feedback which stabilizes the system with the same maximal results regarding the class of non-linear perturbations. Of particular interest in this context is our ability to use the LMI formulation for stabilization of interconnected systems composed of linear subsystems with uncertain non-linear and time-varying coupling. By assuming stabilizability of the subsystems we can produce local control laws under decentralized information structure constraints dictated by the subsystems. Again, the stabilizing feedback laws produce a closed-loop system that is maximally robust with respect to the size of the uncertain interconnection terms.     

A non-linear spacecraft attitude tracking controller for large non-constant rate commands

A new non-linear tracking control algorithm based on an attitude error quaternion is studied in this paper. The control law developed here uses the commanded attitude rate without transformation into the body frame. The direct use of the commanded attitude rate simplifies the calculation of its derivative, which is used in the control law. The solutions and the equilibrium points of the closed-loop system, which is a time-varying non-linear system, are obtained in different scenarios. In order to analyse the stability of the system and the tracking performance, two different forms of perturbation dynamics with seven state variables are introduced. Local stability and performance analysis shows that the eigenvalues of the linearized perturbation dynamics are determined only by the gain matrices in the control algorithm and the inertia matrix. The existence of globally stable tracking control is proved using a Lyapunov function. Simulation results show that the spacecraft can track the commanded attitude and rate quickly for a non-zero acceleration rate command.     

Dynamic inversion for multivariable non-affine-in-control systems via time-scale separation

This paper presents a tracking design methodology applicable to multivariable non-affine-in-control systems. The main focus is on solving the tracking problem for non-linear systems whose dynamics depend non-linearly on the control input. The latter is designed to be faster than the main system dynamics. Using singular perturbation theory along with the Lyapunov stability theorems, it is shown that the proposed controller approximates an unknown dynamic inversion based solution with bounded errors, provides closed-loop stability, and solves the tracking problem with bounded errors. Simulations illustrate the theoretical results.     

基于两步方法的闭环子空间辨识算法

闭环辨识算法具有广泛的工程应用前景,而子空间方法近年来应用于多个领域中,但子空间方法无法直接应用于闭环辨识,因此研究闭环子空间辨识算法具有重要意义.两步方法可用于辨识闭环系统,但计算量巨大,推导复杂,需要进一步改进.针对这种情况,提出了一种改进算法,使用将两步方法与正交投影相结合的方法,并利用QR分解实现,直接构建虚拟信号序列,大大减少了计算量,最后使用PI-MOESP辨识算法辨识模型.仿真实验将该算法与其他子空间辨识算法相比较,显示出该算法的有效性及计算量的显著减少.     

机械臂的自适应径向基函数神经网络双二次泛函最优控制

针对非线性机械臂系统中难以权衡控制能量与控制误差比重的最优控制问题,本文提出一种基于自适应径向基函数(RBF)神经网络二阶段叠加优化的双二次泛函最优求解模型,实现在非线性机械臂控制系统中用不大的控制能量来保持较小的控制误差的综合最优控制.在本文所提模型中,首先设计一种线性误差函数,作用于非线性控制方程,并采用自适应RBF网络逼近非线性控制方程中存在的不确定项,构成闭环反馈系统,实现对非线性系统的最优控制;其次,将待求参数复合成双二次泛函的解域,并设计一种新型的类递归神经网络求解该带约束条件的双二次型模型,实现模型求解的快速收敛并得其解.通过理论分析及数值仿真实例验证了所提模型能有效提高非线性系统的控制精度、稳定性、鲁棒性及自适应性,从而实现非线性系统的综合最优控制.     

Closed-loop control of a class of non-linear systems†

A technique is described for the closed-loop control of a class of non-linear systems iii a potentially large neighbourhood of a nominal trajectory. By the introduction of extra states it is shown how to reduce non-linear differential equations to a related canonical linear form. Since the linear form is not readily amenable to standard design techniques a new synthesis procedure is proposed for the construction of a closed-loop control. The proposed technique relies heavily on computer computation. The computation for the closed-loop control of a lifting body, a system not completely controllable, is given in detail to illustrate the applicability of the method.     

贪心线性推移负载平衡算法

针对环与线性阵列的负载平衡速度较慢与迁移量较大的问题,提出一种贪心线性推移平衡算法。该算法适用于任何具有哈密尔顿通路的图结构网络。其平衡过程的负载迁移量一般不大,平衡负载速度较快。对二维网状网等网络结构的贪心线性推移平衡算法进行改进,得到分二阶段的贪心线性推移平衡算法。实验结果表明,此类改进在平衡条件减弱时能较大地提高算法的时间性能。     

Transformation groups for weakly non-linear periodic systems†

The method of infinitesimal transformations is applied to weakly non-linear periodic systems described by n first-order coupled differential equations. The transformation groups, which leave the equations invariant, are determined by a perturbation method, Those transformations roduce the systems to non-linear autonomous form. The equivalence of the results of the method, within the first-order approximation, to other averaging and perturbation techniques is illustrated by an example of a second-order system.     

Closed-loop identification with a quantizer

This paper proposes a new closed-loop identification scheme for a single-input single-output (SISO) control loop based upon a quantizer inserted into the feedback path. The quantizer can be used to generate an equivalent persistently exciting signal to which the well known two-stage method of closed-loop identification can be applied. The paper examines the performance and behaviour of the quantizer-based closed-loop identification and gives suggestions for the choice of quantizer interval. Simulation and experimental examples are used to illustrate the proposed new CLID scheme.     

A systematic method to obtain ultimate bounds for perturbed systems

In this paper, we develop a systematic method to obtain ultimate bounds for both continuous- and discrete-time perturbed systems. The method is based on a componentwise analysis of the system in modal coordinates and thus exploits the system geometry as well as the perturbation structure without requiring calculation of a Lyapunov function for the system. The method is introduced for linear systems having constant componentwise perturbation bounds and is then extended to the case of state-dependent perturbation bounds. This extension enables the method to be applied to non-linear systems by treating the perturbed non-linear system as a linear system with a perturbation bounded by a non-linear function of the state. Examples are provided where the proposed systematic method yields bounds that are tighter or at least not worse than those obtained via standard Lyapunov analysis. We also show how our method can be combined with Lyapunov analysis to improve on the bounds provided by either approach.     

Memoryless H ∞ controller design for switched non-linear systems with mixed time-varying delays

This article deals with the H _∞ control problem for a class of switched non-linear systems with mixed time-varying delays. The novel features here are that the system in consideration is non-linear perturbation with discrete and distributed delays, the time-varying delay is also involved in the observation output, and the controllers to be designed satisfy some exponential stability constraints on the closed-loop poles. By using Lyapunov–Razumikhin functional approach, new sufficient conditions for the H _∞ control with exponential stability constraint are derived in terms of the solution of Riccati-type equations. The approach allows for simultaneous computation of the two bounds that characterise the stability rate of the solution.     

Coordinated orbit-tracking control of second-order non-linear agents with directed communication topologies

This paper deals with two-dimensional and three-dimensional cooperative control of multiple agents formation tracking a set of given closed orbits, where each agent has intrinsic second-order non-linear dynamics and the communication topology among agents is directed. By using our previous curve extension method, the cooperative control system can be regarded as a cascade system composed of the orbit-tracking subsystem and the formation subsystem with the orbit-tracking error as input. A novel solution is established by separatively designing the orbit-tracking control law and the formation control protocol ignoring the perturbation at first and then applying input-to-state stability theory to analyse the asymptotic stability of the cascade system. It is shown that the closed-loop system is asymptotic stability if the directed communication topology contains a directed spanning tree. The effectiveness of the analytical results is verified by numerical simulations.     

Feedback control of nonlinear systems by extended linearization

For single-input, multiple-output, nonlinear systems, we consider a design method based on the family of linearizations of the system, parameterized by constant operating points. Nonlinear state feedback control laws and observer/state feedback control laws are designed such that the eigenvalues of the family of linearized closed-loop systems are placed at specified values that are locally invariant with respect to the closed-loop operating point. The method is illustrated by application to the problem of automatically balancing an inverted pendulum.     

State-feedback control of non-linear systems†

A design method for state-feedback controllers for single-input non-linear systems is proposed. The method makes use of the transformations of the non-linear system into ‘controllable-like’ canonical forms. The resulting non-linear state feedback is designed in such a way that the eigenvalues of the linearized closed-loop model are invariant with respect to any constant operating point. The method constitutes an alternative approach to the design methodology recently proposed by Baumann and Rugh. Also a review of different transformation methods for non-linear systems is presented. An example and simulation results of different control strategies are provided to illustrate the design technique.     

An analysis and design method for systems with structural uncertainty

A new characterization of control Lyapunov functions is presented for non-linear systems with structural uncertainty. Based on this characterization, we deal with the problem of globally asymptotical stabilization of two classes of non-linear systems with structural uncertainty. Sufficient conditions for the globally asymptotical stabilization of the closed-loop system are acquired respectively. In addition, state feedback control laws are designed respectively to globally asymptotically stabilize the equilibrium of the closed-loop system. Finally, the simulations show the effectiveness of the method.