Event-triggered Networked H∞ Output Tracking Control Based on Dynamic Compensation Controller

In this paper, the event-triggered H_∞ output tracking problem is investigated for networked control systems. In order to reduce the output tracking error as well as to improve network resource utilization, we propose an idea of dynamic compensation controller with the discrete-time event-triggered mechanism, that is, the integral term of tracking error and the state of the reference system are introduced to form states of the augmented system. We first examine the dynamic compensation idea by the H_∞ output tracking control problem for linear time-invariant (LTI) systems. Then, we model the closed-loop event-triggered networked control system as a time-delay augmented linear system. By constructing a Lyapunov-Krasovskii functional with the delay fractioning technique, the stability conditions with lower conservatism are derived in the form of the linear matrix inequalities (LMIs). Furthermore, a method is proposed to design the H_∞ dynamic compensation controllers and the discrete-time event-triggered mechanisms. Finally, the satellite tracking control problem is used as an example to show that the dynamical compensation idea is effective in reducing the tracking error and that the proposed method in this paper can achieve better performance than that in the existing literature.

     

一种可保证瞬态特性的改进的鲁棒模型参考自适应控制

针对典型的鲁棒模型参考自适应控制中瞬态性能无法得到保障的问题, 提出一种改进的鲁棒模型参考自适应控制器. 该控制器在标准的鲁棒自适应控制中加入??_∞ 补偿器, 以抑制闭环自适应系统中参数估计误差和不确定扰动对系统输出跟踪性能造成的不利影响. 理论分析和仿真验证表明, 所提出的控制器不但保留了典型鲁棒模型参考自适应控制的理想特性, 并且通过设计适当的??∞ 补偿器使得闭环系统的瞬态性得到了较大的改善, 其改善的程度依赖于??∞ 补偿器性能指标的大小.

     

H∞ output tracking fuzzy control for nonlinear systems with time-varying delay

This paper proposes an observer-based output tracking control via virtual desired reference model for a class of nonlinear systems with time-varying delay and disturbance. First, the Takagi–Sugeno fuzzy model represents the nonlinear system with time-varying delay and disturbance. Then we design an observer to estimate immeasurable states and controller to drive the error between estimated state and virtual desired variables (VDVs) to zero such that the overall control output tracking system has H_∞ control performance. Using Lyapunov–Krasovskii functional, we derive sufficient conditions for stability. The advantages of the proposed output control system are (i) systematic approach to derive VDVs for controller design; (ii) relaxes need for real reference model; (iii) drops need for information of equilibrium; (iv) relaxed condition is provided via three-step procedure to find observer and controller gain. We carry out simulation using a continuous stirred tank reactor system where the effectiveness of the proposed controller is demonstrated by satisfactory numerical results.     

Adaptive control of linear time-varying systems

Single-input single-output uncertain linear time-varying systems are considered, which are affected by unknown bounded additive disturbances; the uncertain time-varying parameters are required to be smooth and bounded but are neither required to be sufficiently slow nor to have known bounds. The output, which is the only measured variable, is required to track a given smooth bounded reference trajectory. The undisturbed system is assumed to be minimum-phase and to have known and constant relative degree, known sign of the ‘high frequency gain’, known upper bound on the system order. An adaptive output feedback control algorithm is designed which assures: (i) boundedness of all closed-loop signals; (ii) arbitrarily improved transient performance of the tracking error; (iii) asymptotically vanishing tracking error when parameter time derivatives are L₁ signals and disturbances are L₂ signals.     

Robust model reference adaptive control for transient performance enhancement

To circumvent the potentially poor transient response induced by nonlinear uncertain dynamics in the adaptive control system, this article proposes a new model reference adaptive control design scheme to improve its transient control response. We first construct a compensator to online extract the undesired dynamics in the online learning, which is incorporated into the reference model and control simultaneously. Then, an error feedback term is incorporated into the reference model to speed up the convergence of both the compensator and tracking error. Moreover, a new leakage term containing the estimation error is constructed and then added in the adaptive law to guarantee the convergence of both the estimation error and tracking error. To further reveal the mechanisms behind these proposed methods, a new methodology to analyze the transient error bounds based on L₂‐norm and Cauchy‐Schwartz inequality is also developed. Based on the analysis results, we find that the proposed methods can effectively reduce the bound of the tracking error and thus achieve an improved transient control performance without violating the system stability even with high‐gain adaptation. In addition, the frequency‐domain analysis is resorted to show the comparative responses of different adaptive laws, which indicate that the proposed adaptive law can maintain the stability margin even with a high‐gain learning rate. A numerical example is given to demonstrate improved control responses of these proposed schemes.     

Characterization of zero tracking error references in the kinematic control of wheeled mobile robots

This work establishes the reference signal conditions for zero tracking error when controlling wheeled mobile robots under the kinematic framework, that is, when the low-level dynamics is neglected. The reference characterization is based on the classical decoupled robot control and the inverse kinematics of fixed, centered orientable, castor and Swedish wheels. Procedures to avoid tracking error when a particular condition is not satisfied are also indicated. Simulations are shown to illustrate the reference conditions for each type of mobile robot and their implications. Finally, an industrial forklift is considered in a real situation to validate the previous results and to highlight the limits of the kinematic framework assumption.     

Optimal Performance in Tracking Stochastic Signal Under Disturbance Rejection

The analysis method of optimal tracking performance is proposed for multiple‐input multiple‐output (MIMO) linear time‐invariant (LTI) systems under disturbance rejection. An H₂ criterion of the error signal between the output of the plant and the reference signal is used as a measure for the tracking performance. Spectral factorization is applied to obtain the optimal solution of the system tracking error. The explicit expressions are derived for this minimal tracking error with respect to random reference signals under disturbance rejection. It is shown that the nonminimum phase zeros, the zero direction, the unstable poles, the pole direction of a given plant, statistical characteristics of the reference input signal, and disturbance signal have a negative effect on a feedback system's ability to reduce the system error with disturbance rejection. The results show that the optimal tracking performance will further be damaged because of disturbance rejection. Some typical examples are given to illustrate the theoretical results.     

Intelligent mixed H2/H∞ adaptive tracking control system design using self-organizing recurrent fuzzy-wavelet-neural-network for uncertain two-axis motion control system

In this paper, an intelligent adaptive tracking control system (IATCS) based on the mixed H₂/H_∞ approach under uncertain plant parameters and external disturbances for achieving high precision performance of a two-axis motion control system is proposed. The two-axis motion control system is an X–Y table driven by two permanent-magnet linear synchronous motors (PMLSMs) servo drives. The proposed control scheme incorporates a mixed H₂/H_∞ controller, a self-organizing recurrent fuzzy-wavelet-neural-network controller (SORFWNNC) and a robust controller. The combinations of these control methods would insure the stability, robustness, optimality, overcome the uncertainties, and performance properties of the two-axis motion control system. The SORFWNNC is used as the main tracking controller to adaptively estimate an unknown nonlinear dynamic function that includes the lumped parameter uncertainties, external disturbances, cross-coupled interference and frictional force. Moreover, the structure and the parameter learning phases of the SORFWNNC are performed concurrently and online. Furthermore, a robust controller is designed to deal with the uncertainties, including the approximation error, optimal parameter vectors and higher order terms in Taylor series. Besides, the mixed H₂/H_∞ controller is designed such that the quadratic cost function is minimized and the worst case effect of the unknown nonlinear dynamic function on the tracking error must be attenuated below a desired attenuation level. The mixed H₂/H_∞ control design has the advantage of both H₂ optimal control performance and H_∞ robust control performance. The sufficient conditions are developed for the adaptive mixed H₂/H_∞ tracking problem in terms of a pair of coupled algebraic equations instead of coupled nonlinear differential equations. The coupled algebraic equations can be solved analytically. The online adaptive control laws are derived based on Lyapunov theorem and the mixed H₂/H_∞ tracking performance so that the stability of the proposed IATCS can be guaranteed. Furthermore, the control algorithms are implemented in a DSP-based control computer. From the experimental results, the motions at X-axis and Y-axis are controlled separately, and the dynamic behaviors of the proposed IATCS can achieve favorable tracking performance and are robust to parameter uncertainties.     

Path-following for linear systems with unstable zero dynamics

A constructive solution to the path-following problem for MIMO linear systems with unstable zero dynamics is developed. While the original control variable steers the system output along the path, the path parameter θ is used as an additional control to stabilize zero dynamics with a feedback law which is nonlinear due to the path constraint. A sufficient condition for solvability of the path-following problem is given in terms of the geometric properties of the path. When this condition is satisfied, an arbitrary small L₂ norm of path-following error can be achieved, thus avoiding performance limitations of the standard reference tracking problem imposed by unstable zero dynamics.     

Active fault tolerant control for nonlinear systems with simultaneous actuator and sensor faults

The goal of this paper is to describe a novel fault tolerant tracking control (FTTC) strategy based on robust fault estimation and compensation of simultaneous actuator and sensor faults. Within the framework of fault tolerant control (FTC) the challenge is to develop an FTTC design strategy for nonlinear systems to tolerate simultaneous actuator and sensor faults that have bounded first time derivatives. The main contribution of this paper is the proposal of a new architecture based on a combination of actuator and sensor Takagi-Sugeno (T-S) proportional state estimators augmented with proportional and integral feedback (PPI) fault estimators together with a T-S dynamic output feedback control (TSDOFC) capable of time-varying reference tracking. Within this architecture the design freedom for each of the T-S estimators and the control system are available separately with an important consequence on robust L ₂ norm fault estimation and robust L ₂ norm closed-loop tracking performance. The FTTC strategy is illustrated using a nonlinear inverted pendulum example with time-varying tracking of a moving linear position reference.     

State feedback tracking control for indirect field-oriented induction motor using fuzzy approach

This paper deals with the synthesis of fuzzy controller applied to the induction motor with a guaranteed model reference tracking performance. First, the Takagi-Sugeno (T-S) fuzzy model is used to approximate the nonlinear system in the synchronous d-q frame rotating with field-oriented control strategy. Then, a fuzzy state feedback controller is designed to reduce the tracking error by minimizing the disturbance level. The proposed controller is based on a T-S reference model in which the desired trajectory has been specified. The inaccessible rotor flux is estimated by a T-S fuzzy observer. The developed approach for the controller design is based on the synthesis of an augmented fuzzy model which regroups the model of induction machine, fuzzy observer, and reference model. The gains of the observer and controller are obtained by solving a set of linear matrix inequalities (LMIs). Finally, simulation and experimental results are given to show the performance of the observer-based tracking controller.     

Optimal tracking performance of MIMO control systems with communication constraints and a code scheme

This paper investigates the issue of the optimal tracking performance for multiple-input multiple-output linear time-invariant continuous-time systems with power constrained. An H₂ criterion of the error signal and the signal of the input channel are used as a measure for the tracking performance. A code scheme is introduced as a means of integrating controller and channel design to obtain the optimal tracking performance. It is shown that the optimal tracking performance index consists of two parts, one depends on the non-minimum phase zeros and zero direction of the given plant, as well as the reference input signal, while the other depends on the unstable poles and pole direction of the given plant, as well as on the bandwidth and additive white noise of a communication channel. It is also shown that when the communication does not exist, the optimal tracking performance reduces to the existing normal tracking performance of the control system. The results show how the optimal tracking performance is limited by the bandwidth and additive white noise of the communication channel. A typical example is given to illustrate the theoretical results.     

Output feedback LMI tracking control conditions with H∞ criterion for uncertain and disturbed T-S models

This work concerns the tracking problem of uncertain Takagi-Sugeno (T-S) continuous fuzzy model with external disturbances. The objective is to get a model reference based output feedback tracking control law. The control scheme is based on a PDC structure, a fuzzy observer and a H_∞ performance to attenuate the external disturbances. The stability of the whole closed-loop model is investigated using the well-known quadratic Lyapunov function. The key point of the proposed approaches is to achieve conditions under a LMI (linear matrix inequalities) formulation in the case of an uncertain and disturbed T-S fuzzy model. This formulation facilitates obtaining solutions through interior point optimization methods for some nonlinear output tracking control problems. Finally, a simulation is provided on the well-known inverted pendulum testbed to show the efficiency of the proposed approach.     

Robust iterative learning controller design using the performance weighting function of feedback control systems

Iterative learning controllers combined with existing feedback controllers have prominent capability of improving tracking performance in repeated tasks. However, the iterative learning controller has been designed without utilizing effective information such as the performance weighting function to design a feedback controller. In this paper, we deal with a robust iterative learning controller design problem for an uncertain feedback control system using its explicit performance information. We first propose a robust convergence condition in the ?₂-norm sense for an iterative learning control (ILC) scheme. We present a method to design an iterative learning controller using the information on the performance of the existing feedback control system such as performance weighting functions and frequency ranges of desired trajectories. From the obtained results, several design criteria for iterative learning controller are provided. Through analysis on the remaining error, the loop properties before and after learning are compared. We also show that, in the ?₂-norm sense, the remaining error can be less than the initial error under certain conditions. Finally, to show the validity of the proposed method, simulation studies are performed.     

Internal model based robust inversion feedforward and feedback 2DOF control for LPV system with disturbance

To reduce the adverse effects on the control performance and disturbance rejection caused by system uncertainty, a novel internal model based robust inversion feedforward and feedback 2DOF control approach was proposed for LPV system with disturbance. The proposed control approach combines the internal model control and robust inversion based 2DOF control, it utilizes internal model based control to reject external disturbance, utilizes robust inversion 2DOF control to enhance the control resolution and guarantee the system control performance. At first, a LMI synthesis approach for LPV system model identification and a disturbance compensator optimization design method which could minimize H_∞ norm of output error caused by disturbance are presented. Then, combined with internal loop for disturbance compensation, a robust inversion feedforward controller is designed by robust inversion approach and the feedback controller which could render the requirements of reference signal tracking performance and robustness satisfied is obtained by the H_∞ mixed sensitivity synthesis approach. Finally, atomic force microscopy (AFM) vertical positioning simulation experiments are conducted and the experiment results showed that the proposed control approach could achieve better output performance and disturbance rejection compared with conventional internal model based control and robust inversion based 2DOF control approach.     

Transverse function control with prescribed performance guarantees for underactuated marine surface vehicles

This paper studies the problem of stabilizing reference trajectories (also called as the trajectory tracking problem) for underactuated marine vehicles under predefined tracking error constraints. The boundary functions of the predefined constraints are asymmetric and time‐varying. The time‐varying boundary functions allow us to quantify prescribed performance of tracking errors on both transient and steady‐state stages. To overcome difficulties raised by underactuation and nonzero off‐diagonal terms in the system matrices, we develop a novel transverse function control approach to introduce an additional control input in backstepping procedure. This approach provides practical stabilization of any smooth reference trajectory, whether this trajectory is feasible or not. By practical stabilization, we mean that the tracking errors of vehicle position and orientation converge to a small neighborhood of zero. With the introduction of an error transformation function, we construct an inverse‐hyperbolic‐tangent‐like barrier Lyapunov function to show practical stability of the closed‐loop systems with prescribed transient and steady‐state performances. To deal with unmodeled dynamic uncertainties and external disturbances, we employ neural network (NN) approximators to estimate uncertain dynamics and present disturbance observers to estimate unknown disturbances. Subsequently, we develop adaptive control, based on NN approximators and disturbance estimates, that guarantees the prescribed performance of tracking errors during the transient stage of on‐line NN weight adaptations and disturbance estimates. Simulation results show the performance of the proposed tracking control.     

Lyapunov recursive design of robust adaptive tracking control with L 2-gain performance for electrically-driven robot manipulators

This paper develops a new Lyapunov recursive design for the tracking control problem of rigid-link electrically-driven robot manipulators with uncertainty by taking a tracking performance into account. The tracking performance is evaluated by L ₂-gain from a torque level disturbance signal to a penalty signal for the tracking error between outputs of the manipulator and desired trajectories. The novelty of our approach is in the strategy to construct such a Lyapunov function recursively that ensures not only stability of a tracking error system but also an L ₂-gain constraint, which provides a closed-form solution for non-linear H     

Robust neural network control system design for linear ultrasonic motor

Linear ultrasonic motor (LUSM) has much merit, such as high precision, fast control dynamics and large driving force, etc.; however, the dynamic characteristic of LUSM is nonlinear and the precise dynamic model of LUSM is difficult to obtain. To tackle this problem, this study presents a robust neural network control (RNNC) system for LUSM to track a reference trajectory with L ₂ robust tracking performance. The developed RNNC system is composed of a neural network controller and a robust controller. The neural network controller is the principal controller used to mimic an ideal controller and the robust controller is adopted to achieve L ₂ robust tracking performance. The developed RNNC system is then applied to control an LUSM. Experimental results show that the developed RNNC system can achieve favorable tracking performance with unknown of LUSM model.     

Tracking Performance Achievement for Continuous‐Time Delayed Linear Systems Subject to Actuator Saturation and output Disturbances

Controlling continuous‐time input‐delayed nonminimum‐phase linear systems is addressed in the presence of actuator saturation and output‐disturbances. Focusing on output‐reference tracking, the control design is dealt with in the pseudo‐polynomials ring. A quite appealing L₂ ‐tracking performance is shown to be achievable in the presence of arbitrary inputs i.e. the output reference and the output disturbance. The performance is formulated in terms of a well defined output‐reference mismatch error (ORME), depending on the inputs’ rate and their compatibility with the actuator saturation constraint.     

Adaptive learning control of linear systems by output error feedback

This paper addresses the problem of designing an output error feedback tracking control for single-input, single-output uncertain linear systems when the reference output signal is smooth and periodic with known period T. The considered systems are required to be observable, minimum phase, with known relative degree and known high frequency gain sign. By developing in Fourier series expansion a suitable unknown periodic input reference signal, an output error feedback adaptive learning control is designed which ‘learns’ the input reference signal by identifying its Fourier coefficients: bounded closed-loop signals and global exponential tracking of both the input and the output reference signals are obtained when the Fourier series expansion is finite, while global exponential convergence of the input and output tracking errors into arbitrarily small residual sets is achieved otherwise. The structure of the proposed controller depends only on the relative degree, the reference signal period, the high frequency gain sign and the number of estimated Fourier coefficients.