Feedforward digital tracking controllers for motion control applications

This paper reviews digital tracking control algorithms for motion control applications. In tracking control, the control objective is to steer the control object along the time-varying desired output. Two design approaches are presented for the case where the desired signal is known in advance, i.e. previewable. One approach is based on the mathematical inverse of a closed-loop system consisting of a controlled plant and a feedback controller. If the mathematical inverse is asymptotically stable, i.e. the closed-loop system does not possess zeros outside the unit circle (unstable zeros), it is an ideal feedforward controller for achieving perfect tracking under the preview assumption. For closed-loop systems with unstable zeros, a cancellation technique for the phase shift induced by unstable zeros is introduced. Another approach is based on linear quadratic optimal control and is known as finite optimal preview control. In this approach, the feedback controller and feedforward controller are determined simultaneously by minimizing a quadratic performance index which involves a tracking error term and a term related to the control effort. Applications of these tracking control algorithms to mechanical systems control are described.     

一种新的非最小相位系统的控制方法

非最小相位系统的控制中，需要抑制由不稳定零点引起的负调并同时缩短系统的调节时间．针对非最小相位系统负调与调节时间的相互影响及负调与不稳定零点的相互关系，提出将控制过程分为抑制负调阶段和跟踪输入阶段，并适时改变系统不稳定零点数，用遗传算法统一优化各阶段的控制器参数．仿真结果表明该控制方法大大减弱了负调，并同时缩短了调节时间，达到了良好的控制效果．     

多通道约束下的网络控制系统的最优跟踪性能

本文针对双通道约束下的线性时不变网络控制系统的随机信号跟踪性能极限问题进行了研究.网络通信包含通信噪声和通信带宽两种信道因素.被控系统考虑是非最小相位和不稳定系统,并且系统包含多个不同的非最小相位零点和多个不同的不稳定极点.对上行通道和下行通道都存在通信带宽约束及高斯白噪声影响的情形,从频域角度,通过采用双自由度控制器和尤拉参数化方法,获得了此类网络控制系统的最优可达的跟踪性能.研究结果表明网络控制系统的跟踪性能极限完全由被控对象的结构特征(非最小相位零点、不稳定极点以及被控对象的系统增益),参考输入信号和网络特性(高斯白噪声的统计特征、通信信道带宽)所决定.最后,仿真结果检证了所得结果的正确性.     

Best tracking and regulation performance under control energy constraint

This paper studies optimal tracking and regulation control problems, in which objective functions of tracking error and regulated response, defined by integral square measures, are to be minimized jointly with the control effort, where the latter is measured by the plant input energy. Our primary objective in this work is to search for fundamental design limitations beyond those known to be imposed by nonminimum phase zeros, unstable poles, and time delays. For this purpose, we solve the problems explicitly by deriving analytical expressions for the best achievable performance. It is found that this performance limit depends not only on the plant nonminimum phase zeros, time delays, and unstable poles-a fact known previously-but also on the plant gain in the entire frequency range. The results thus reveal and quantify another source of fundamental performance limitations beyond those already known, which are nonexistent when only conventional performance objectives such as tracking and regulation are addressed without taking into account the control energy constraint. Among other things, they exhibit how the lightly damped poles, the anti-resonant zeros, as well as the bandwidth of the plant may affect the performance.     

控制增益未知的多变量极值搜索系统神经网络自适应协同控制

针对一类控制增益未知的多变量极值搜索系统,提出了一种神经网络自适应协同控制方法.该方法利用协同控制实现状态变量之间的协同收敛,并确保对系统内部参数扰动和外界干扰具有不变性;以极值搜索控制方法得到的搜寻变量作为输入量,设计多层神经网络逼近状态变量的极值变化率和未知的变量与函数;采用Nussbaum函数解决系统控制增益未知的问题;同时运用自适应参数抵消神经网络逼近误差的影响.稳定性分析证明了系统的状态跟踪误差、输出量与其极值之间的误差、极值搜索变量的跟踪误差以及神经网络各参数的估计误差均指数收敛至原点的一个有界邻域.理论分析与仿真结果验证了该方法的有效性.     

A new tracking controller for discrete‐time SISO non minimum phase systems

A new tracking controller for discrete‐time Single Input Single Output (SISO) non‐minimum phase (NMP) systems is presented. In the proposed method, after cancelation of poles and cancelable zeros of the system, the controller adds some NMP zeros to compensate the effect of NMP zero (zeros) of the system. As a result, the phase of the overall transfer function will be almost linear and its magnitude approaches unity for all frequencies. The method can be applied even to the systems with complex conjugate NMP zeros. As well, it is applicable to the systems for which the conventional methods cannot properly be used. Furthermore, a generalization of method to continuous‐time systems is another given result. Several examples are provided to illustrate the effectiveness of the method. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Fractional-order unstable pole-zero cancellation in linear feedback systems

As a very well-known classical fact, non-minimum phase zeros of the process put some limitations on the performance of the feedback system. The source of these limitations is that non-minimum phase zeros cannot be cancelled by unstable poles of the controller since such a cancellation leads to internal instability. The aim of this paper is to propose a method for fractional-order cancellation of non-minimum phase zeros of the process and studying its properties. It is specially shown that the proposed cancellation strategy increases the phase and gain margin without leading to internal instability. Since the systems with higher gain and phase margin are easier to control, the proposed method can be used to arrive at more effective controls, which is also verified by the simulation results.     

Approximation-based adaptive control for a class of mobile robots with unknown skidding and slipping

An adaptive tracking control approach using function approximation technique is proposed for trajectory tracking of Type (2,0) wheeled mobile robots with unknown skidding and slipping in polar coordinates and at the dynamic level. The nonlinear disturbance observer (NDO) is used to estimate a nonlinear disturbance term including unknown skidding and slipping. The adaptive control system is designed via the function approximation technique using neural networks employed to compensate the NDO error. It is proved that all signals of the controlled closed-loop system are uniformly bounded and the point tracking errors converge to an adjustable neighborhood of the origin regardless of large initial tracking errors and unknown skidding and slipping. Simulation results are presented to validate the good tracking performance and robustness of the proposed control system against unknown skidding and slipping.     

State feedback with fractional integral control design based on the Bode’s ideal transfer function

State feedback technique through a gain matrix has been a well-known method for pole assignment of a linear system. The technique could encounter a difficulty in eliminating the steady-state errors in some states. Introducing an integral element can effectively eliminate these errors. State feedback with fractional integral control is proposed, in this work, for pole placement of a linear time invariant system. The proposed method yields simple gain formulae. The paper presents the derivation of the design formulae. The method is applied to stabilise an inherently unstable inverted pendulum-cart system. Simulation and experimental results show the effectiveness of the proposed method for set-point tracking, disturbance rejection and stabilising the inverted pendulum. Comparison with the results obtained from applying Achermann’s formula is also presented.     

Digital inverse model control using Generalised holds with extensions to the adaptive case

In this paper, a digital implementation of an inverse-model based control scheme is proposed using Generalised Sampling and Hold Functions. The implementation of the controller using this kind of holds allows overcoming the difficulties related to the presence of unstable zeros in the continuous-time model and the usual appearance of unstable discretisation zeros in the discrete model when a ZOH is applied. The Generalised Sampling and Hold Functions allows obtaining a discrete model of the plant with all its zeros stable which allows realizing an exact inverse model of the plant in comparison to the use of a classical ZOH which only allows, in general, an approximate inversion of the plant. The proposed approach is then extended to the adaptive case where the stability and tracking properties of the general scheme are fully proved. Simulation examples showing the scope and application of the method are also presented.     

Best Tracking Performance of Networked Control Systems Based on Communication Constraints

The best tracking problem for a single‐input‐single‐output (SISO) networked control system with communication constraints is studied in this paper. The tracking performance is measured by the energy of the error signal between the output of the plant and the reference signal. The communication constraints under consideration are finite bandwidth and networked induced‐delay. Explicit expressions of the minimal tracking error have been obtained for networked control systems with or without communication constraints. It is shown that the best tracking performance dependents on the nonminimum phase zeros, and unstable poles of the given plant, as well as the bandwidth and networked induced‐delay. It is also shown that, if the constraints of the communication channel do not exist, the best tracking performance reduces to the existing tracking performance of the control system without communication constraints. The result shows how the bandwidth and networked induced‐delay of a communication channel may fundamentally constrain a control system's tracking capability. Some typical examples are given to illustrate the theoretical results.     

Output‐feedback tracking in causal nonminimum‐phase nonlinear systems using higher‐order sliding modes

Asymptotic output‐feedback tracking in a class of causal nonminimum phase uncertain nonlinear systems is addressed via sliding mode techniques. Sliding mode control is proposed for robust stabilization of the output tracking error in the presence of a bounded disturbance. The output reference profile and the unknown input/disturbance are supposed to be described by unknown linear exogenous systems of a given order. Local asymptotic stability of the output tracking error dynamics along with the boundedness of the internal states are proven. The unstable internal states are estimated asymptotically via the proposed multistage observer that is based on the method of extended system center. A higher‐order sliding mode observer/differentiator is used for the exact estimation of the input–output states in a finite time. The bounded disturbance is reconstructed asymptotically. A numerical example illustrates the efficiency of the proposed output‐feedback tracking approach developed for causal nonminimum phase nonlinear systems. Copyright © 2009 John Wiley & Sons, Ltd.     

Distributed event‐triggered tracking control with a dynamic leader for multiple Euler‐Lagrange systems under directed networks

The distributed tracking control for multiple Euler‐Lagrange systems with a dynamic leader is investigated in this article via the event‐triggered approach. Only a portion of followers have access to the leader, and the communication topology among all agents is directed that contains a directed spanning tree rooted at the leader. The case that the leader's generalized velocity is constant is first considered, and a distributed event‐based control law is developed by using a velocity estimator. When the leader's generalized velocity is time‐varying, novel distributed continuous estimators are proposed to avoid the undesirable chattering effect while guaranteeing that the estimate errors converge to zeros. With the designed distributed estimators, another distributed event‐based control protocol is provided. Controller update frequency and resource consumption in our work can be reduced by applying the aforementioned two distributed control laws, and the tracking errors can converge to zeros. In addition, it is rigorously proved that no agent exhibits Zeno behavior. Finally, the effectiveness of the proposed distributed event‐based control laws is elucidated by a number of simulation examples.     

Two-degree-of-freedom ?2-optimal tracking with preview

A simple SISO two-degree-of-freedom pole-placement design method is presented that provides ?₂-optimal tracking of a given reference signal. The closed-loop pole locations are first chosen by the system designer. The closed-loop zeros are then placed in an optimal fashion by a computationally inexpensive algorithm to achieve asymptotic tracking with an optimal transient response. The preview approach, which has become a common method for dealing with systems which have non-minimum phase behavior, can then optionally be used to further improve the transient behavior for both minimum phase and non-minimum phase systems. Unlike previous results based on the preview approach, the solution presented here takes into consideration the closed-loop pole dynamics, and is ?₂ optimal with respect to all other two-degree-of-freedom preview controllers with the same closed-loop poles. A simple solution to the H₂ model matching problem, where the design parameter Q is not rational, but polynomial, is the heart of the solution method.     

Predictor-based neural dynamic surface control for distributed formation tracking of multiple marine surface vehicles with improved transient performance基于预估器神经网络动态面的海洋航行器分布式队形跟踪

In this paper, we investigate the distributed formation tracking problem of multiple marine surface vehicles with model uncertainty and time-varying ocean disturbances induced by wind, waves, and ocean currents. The objective is to achieve a collective tracking with a time-varying trajectory, which can only be accessed by a fraction of follower vehicles. A novel predictor-based neural dynamic surface control design approach is proposed to develop the distributed adaptive formation controllers. We use prediction errors, rather than tracking errors, to construct the neural adaptive laws, which enable the fast identification of the vehicle dynamics without incurring high-frequency oscillations in control signals. We establish the stability properties of the closed-loop network via Lyapunov analysis, and quantify the transient performance by deriving the truncated L₂ norms of the derivatives of neural weights, which we demonstrate to be smaller than the classical neural dynamic surface control design approach. We also extend the above result to the distributed formation tracking using the relative position information of vehicles, and the advantage is that the velocity information of neighbors and leader are required. Finally, we give the comparative studies to illustrate the performance improvement of the proposed method.     

Robust output tracking control of a class of non-minimum phase systems and application to VTOL aircraft

In this article, we study the output tracking control of a class of MIMO nonlinear non-minimum phase systems in the presence of input disturbances. In order to attenuate the effects of disturbances, the method of uncertainty and disturbance estimator (UDE) is extended to the controller design for non-minimum phase systems. Due to the fact that the accumulated disturbances is composed of internal states and external disturbances, a different stability analysis is given, and the overall closed-loop system is proved to be semi-globally stable. The proposed state-feedback controller not only forces system outputs to asymptotically track desired trajectories, but also drives the unstable internal dynamics to follow bounded and causal ideal internal dynamics (IID) solved via stable system centre (SSC) method. Simulation results demonstrate that the proposed controller achieves excellent tracking and disturbance rejection performance via the example of VTOL aircraft which has been the benchmark of nonlinear non-minimum phase systems.     

基于干扰观测器的机械臂广义模型预测轨迹跟踪控制

为实现对多自由度机械臂关节运动精确轨迹跟踪,提出一种基于非线性干扰观测器的广义模型预测轨迹跟踪控制方法。针对机械臂轨迹跟踪运动学子系统,采用广义预测控制（Generalized Predictive Control,GPC）方法设计期望的虚拟关节角速度。对于机械臂轨迹跟踪动力学子系统,考虑机械臂的参数不确定性和未知外界扰动,利用GPC方法设计关节力矩控制输入,基于非线性干扰观测器方法实时估计和补偿系统模型中的不确定性。在李雅普诺夫稳定性理论框架下证明了机械臂关节角位置和角速度的跟踪误差最终收敛于零的小邻域。数值仿真验证了所提出控制方法的有效性和优越性。     

Output feedback control of nonlinear systems using RBF neuralnetworks

An adaptive output feedback control scheme for the output tracking of a class of continuous-time nonlinear plants is presented. An RBF neural network is used to adaptively compensate for the plant nonlinearities. The network weights are adapted using a Lyapunov-based design. The method uses parameter projection, control saturation, and a high-gain observer to achieve semi-global uniform ultimate boundedness. The effectiveness of the proposed method is demonstrated through simulations. The simulations also show that by using adaptive control in conjunction with robust control, it is possible to tolerate larger approximation errors resulting from the use of lower order networks.     

Discrete-time output feedback sliding-mode control design for uncertain systems using linear matrix inequalities

An output feedback-based sliding-mode control design methodology for discrete-time systems is considered in this article. In previous work, it has been shown that by identifying a minimal set of current and past outputs, an augmented system can be obtained which permits the design of a sliding surface based upon output information only, if the invariant zeros of this augmented system are stable. In this work, a procedure for realising discrete-time controllers via a particular set of extended outputs is presented for non-square systems with uncertainties. This method is applicable when unstable invariant zeros are present in the original system. The conditions for existence of a sliding manifold guaranteeing a stable sliding motion are given. A procedure to obtain a Lyapunov matrix, which simultaneously satisfies both a Riccati inequality and a structural constraint, is used to formulate the corresponding control to solve the reachability problem. A numerical method using linear matrix inequalities is suggested to obtain the Lyapunov matrix. Finally, the design approach given in this article is applied to an aircraft problem and the use of the method as a reconfigurable control strategy in the presence of sensor failure is demonstrated.     

一类未知多变量非线性系统的动态神经网络自适应控制

对一类未知的非线性的多变量系统，提出了用动态神经网络实现直接自适应控制的策略，基于Ｌｙａｐｕｎｏｖ理论，获得一个稳定并且连续的学习律，避免了递归训练过程，闭环系统被证明是鲁棒稳定的，跟踪误差收敛到一个小的残集，这种方法的特点是即不需要离线学习阶段也不要求初始的参数误差足够小，仿真结果验证了提出的动态网络的自适应控制算法的有效性。