Robust prevention of limit cycle for nonlinear control systems with parametric uncertainties both in the linear plant and nonlinearity

A new method is proposed to compute all feasible robust stabilizing controllers for preventing the generation of limit cycle of nonlinear control systems with parametric uncertainties both in the linear plant and nonlinearity. The describing function analysis method is employed to approximate the behaviors of the nonlinearity. The Kharitonov theorem is utilized to characterize parametric uncertainties in the linear plant and nonlinearity. Necessary conditions for limit cycles are established. Boundaries for the generation of limit cycle and boundaries for asymptotic stability are portrayed exploiting the stability equation method. The region for prescribed limit cycle behavior and the region for asymptotic stability are located. An admissible specification-oriented Kharitonov region is found directly on the controller parameter plane. The region is non-conservative and constitutes all of the feasible controller gain sets to achieve robust prevention of limit cycle for the considered uncertain nonlinear control systems. The way to tune the controller gains is suggested. Finally, for comparison purpose, two illustrative examples proposed in the literature are given to show how the proposed algorithm can be effectively applied to tune a robust controller to achieve a prescribed limit cycle behavior and accomplish robust limit cycle amplitude suppression and prevention.     

Hopf bifurcation to a short porous journal-bearing system using the Brinkman model: weakly nonlinear stability

On the basis of the Brinkman model, the weakly nonlinear stability characteristics of short porous journal-bearing systems are presented. By applying the Hopf bifurcation theory, the weakly nonlinear behaviors near the critical stability boundary are predicted. According to results, the onset of oil whirl for porous bearings is a bifurcation phenomenon; it can exhibit supercritical limit cycles or subcritical limit cycles for journal speeds in the vicinity of the bifurcation point. With a fixed permeability parameter, such supercritical limit cycles for journal speeds in excess of the threshold speed are confined to a specific region in the (ω, ε_s) plane; and outside this region subcritical limit cycles exist for journal speeds below the threshold speed. In addition, increasing the value of system parameter, S_p, may change supercritical bifurcation into the more complicated subcritical bifurcation.     

Determination of all feasible robust PID controllers for open-loop unstable plus time delay processes with gain margin and phase margin specifications

This paper proposes a novel alternative method to graphically compute all feasible gain and phase margin specifications-oriented robust PID controllers for open-loop unstable plus time delay (OLUPTD) processes. This method is applicable to general OLUPTD processes without constraint on system order. To retain robustness for OLUPTD processes subject to positive or negative gain variations, the downward gain margin (GM_down), upward gain margin (GM_up), and phase margin (PM) are considered. A virtual gain-phase margin tester compensator is incorporated to guarantee the concerned system satisfies certain robust safety margins. In addition, the stability equation method and the parameter plane method are exploited to portray the stability boundary and the constant gain margin (GM) boundary as well as the constant PM boundary. The overlapping region of these boundaries is graphically determined and denotes the GM and PM specifications-oriented region (GPMSOR). Alternatively, the GPMSOR characterizes all feasible robust PID controllers which achieve the pre-specified safety margins. In particular, to achieve optimal gain tuning, the controller gains are searched within the GPMSOR to minimize the integral of the absolute error (IAE) or the integral of the squared error (ISE) performance criterion. Thus, an optimal PID controller gain set is successfully found within the GPMSOR and guarantees the OLUPTD processes with a pre-specified GM and PM as well as a minimum IAE or ISE. Consequently, both robustness and performance can be simultaneously assured. Further, the design procedures are summarized as an algorithm to help rapidly locate the GPMSOR and search an optimal PID gain set. Finally, three highly cited examples are provided to illustrate the design process and to demonstrate the effectiveness of the proposed method.     

Stability maps for PID control of multiple lag processes

PID controller parameter stability limits are determined for processes with multiple lags. The limits are graphed in the gain-integral plane. Approximations to the PI limits are developed. This permits rapid estimation of the stable region. It is an approximate function of the maximum gain and the square root of the sum of the lags.     

Computation of stabilizing PI and PID controllers for processes with time delay

In this paper, a new method for the computation of all stabilizing PI controllers for processes with time delay is given. The proposed method is based on plotting the stability boundary locus in the (kp, ki) plane and then computing the stabilizing values of the parameters of a PI controller for a given time delay system. The technique presented does not need to use Pade approximation and does not require sweeping over the parameters and also does not use linear programming to solve a set of inequalities. Thus it offers several important advantages over existing results obtained in this direction. Beyond stabilization, the method is used to compute stabilizing PI controllers which achieve user specified gain and phase margins. The proposed method is also used to design PID controllers for control systems with time delay. The limiting values of a PID controller which stabilize a given system with time delay are obtained in the (kp, ki) plane, (kp, kd) plane, and (ki, kd) plane. Examples are given to show the benefits of the method presented.     

Robust PID tuning strategy for uncertain plants based on the Kharitonov theorem

In this paper, the Kharitonov theorem for interval plants is exploited for the purpose of synthesizing a stabilizing controller. The aim here is to develop a controller to simultaneously stabilize the four Kharitonov-defined vortex polynomials. Different from the prevailing works, the controller is designed systematically and graphically through the search of a non-conservative Kharitonov region in the controller coefficient parameter plane. The region characterizes all stabilizing PID controllers that stabilize an uncertain plant. Thus the relationship between the Kharitonov region and the stabilizing controller parameters is manifest. Extensively, to further guarantee the system with certain robust safety margins, a virtual gain phase margin tester compensator is added. Stability analysis is carried out. The control system is proved to maintain robustness at least to the pre-specified margins. The synthesized controller with coefficients selected from the obtained non-conservative Kharitonov region can stabilize the concerned uncertain plants and fulfill system specifications in terms of gain margins and phase margins.     

Friction-Induced Vibration by Stribeck’s Law: Application to Wiper Blade Squeal Noise

This paper is concerned with the squeal noise of a wiper/windscreen contact. It is shown that squeal noise stems from friction-induced self-excited vibrations in the context of Stribeck’s law for friction coefficient. The study is specifically focussed on the instability range of velocities and not on the amplitude of limit cycles. The studied dynamic system consists of a single degree-of-freedom mass-spring-damper oscillator submitted to a velocity-dependent frictional force which follows the Stribeck law. The local stability is analyzed by the first Lyapunov method and results in a stability criterion. Experiments have been performed on a glass/elastomer contact lubricated with water. The tribometer ‘LUG’ provides measurements of the vibrational velocity and friction force versus sliding speed. It is found that the instability appears during the transition between boundary and elastohydrodynamic regimes where the negative gradient of the friction versus velocity curve is steep. The apparition and vanishing of instability are correctly predicted by the steady-state stability criterion.     

Improved automatic tuning of PID controller for stable processes

This paper presents an improved automatic tuning method for stable processes using a modified relay in the presence of static load disturbances and measurement noise. The modified relay consists of a standard relay in series with a PI controller of unity proportional gain. The integral time constant of the PI controller of the modified relay is chosen so as to ensure a minimum loop phase margin of 30. A limit cycle is then obtained using the modified relay. Hereafter, the PID controller is designed using the limit cycle output data. The derivative time constant is obtained by maintaining the above mentioned loop phase margin. Minimizing the distance of Nyquist curve of the loop transfer function from the imaginary axis of the complex plane gives the proportional gain. The integral time constant of the PID controller is set equal to the integral time constant of the PI controller of the modified relay. The effectiveness of the proposed technique is verified by simulation results.     

PID controller tuning for the first-order-plus-dead-time process model via Hermite-Biehler theorem

This paper discusses PID stabilization of a first-order-plus-dead-time (FOPDT) process model using the stability framework of the Hermite-Biehler theorem. The FOPDT model approximates many processes in the chemical and petroleum industries. Using a PID controller and first-order Padé approximation for the transport delay, the Hermite-Biehler theorem allows one to analytically study the stability of the closed-loop system. We derive necessary and sufficient conditions for stability and develop an algorithm for selection of stabilizing feedback gains. The results are given in terms of stability bounds that are functions of plant parameters. Sensitivity and disturbance rejection characteristics of the proposed PID controller are studied. The results are compared with established tuning methods such as Ziegler-Nichols, Cohen-Coon, and internal model control.     

A new design method for PI–PD control of unstable processes with dead time

PID controllers are still widely practiced in the industrial systems. In the literature, many publications can be found considering PID controller design for unstable processes. However, owing to the structural limitations of PID controllers, generally, good closed loop performance cannot be achieved with a PID for controlling unstable processes and usually a step response with a high overshoot and oscillation is obtained. On the other hand, PI–PD controllers are proved to give very satisfactory closed loop performances for unstable processes. The paper presents a simple design method to tune parameters of a PI–PD controller for the control of the unstable processes with time delay. The proposed method is based on plotting the stability boundary locus, which is a locus dependent on the parameters of the controller and frequency, in the parameter plane. The method uses a new concept named centroid of the convex stability region. Simulation examples and an experimental application are given to illustrate the superiority of the proposed method over some existing ones.     

Optimal Predicted Fuzzy PI Gain Scheduling Controller of Constant Turning Force Systems with Fixed Metal Removal Rate

The dynamic behaviour of the turning process is nonlinear and time-varying owing to variations in cutting depth. This paper proposes an optimal predicted fuzzy PI gain scheduling controller to control the constant turning force (CTF) process with a fixed metal removal rate (MRR) under various cutting conditions. The predicted fuzzy PI gain scheduling control scheme consists of two parts: the fuzzy PI gain scheduling controller; and the grey predictor. First, the optimal parameters of both the grey predictor and the optimal PI gains corresponding to each desired cutting depth in the range of operation, are designed off-line by using the proposed optimal combined method, i.e. Taguchi–RGA method, which integrates the Taguchi method and a real-coded genetic algorithm (RGA). Then, before the parameters of both the grey predictor and the PI gains are scheduled on-line, by fuzzy inference in terms of the changes of cutting depth, the optimal set of triangular-type membership functions of the fuzzy inference mechanism for scheduling the parameters of both the grey predictor and the PI gains are also designed off-line by using the Taguchi–RGA method. Computer simulations are performed to verify the applicability of this optimal predicted fuzzy PI gain scheduling control scheme for controlling the CTF process with a fixed MRR under various cutting conditions. It is shown that such an optimal predicted fuzzy PI gain scheduling control scheme can achieve satisfactory performance and better results than those reported recently in the literature.     

Intelligent phase plane switching control of a pneumatic muscle robot arm with Magneto-Rheological Brake

Pneumatic cylinders are one kind of low cost actuation sources which have been applied in industrial and robotics field, since they have a high power/weight ratio, a high-tension force and a long durability. To overcome the shortcomings of conventional pneumatic cylinders, a number of newer pneumatic actuators have been developed such as McKibben Muscle, Rubber Actuator and Pneumatic Artificial Muscle (PAM) Manipulators. However, some limitations still exist, such as the air compressibility and the lack of damping ability of the actuator bring the dynamic delay of the pressure response and cause the oscillatory motion. In addition, the nonlinearities in the PAM manipulator still limit the controllability. Therefore, it is not easy to realize motion with high accuracy and high speed and with respect to various external inertia loads. To overcome these problems, a novel controller which harmonizes a phase plane switching control method (PPSC) with conventional PID controller and the adaptabilities of neural network is newly proposed. In order to realize satisfactory control performance a variable damper, Magneto-Rheological Brake (MRB), is equipped to the joint of the robot. The mixture of conventional PID controller and an intelligent phase plane switching control using neural network (IPPSC) brings us a novel controller. The experiments were carried out in a robot arm, which is driven by two PAM actuators, and the effectiveness of the proposed control algorithm was demonstrated through experiments, which had proved that the stability of the manipulator can be improved greatly in a high gain control by using MRB with 1PPSC and without regard for the changes of external inertia loads.     

悬臂梁智能结构主共振响应的最优化控制

应用最优化控制方法研究悬臂梁主共振响应的减振控制。由主共振振动定常解的稳定条件得到反馈控制增益的参数范围。以主共振有控制振动峰值与无控制振动峰值的比值为减振的衰减率,以该衰减率为目标函数,反馈控制增益参数范围为约束条件,利用最优化方法计算得到衰减率最小的速度反馈控制参数。计算发现速度反馈控制参数数值越大,衰减率越小,控制效果越好。以非线性振动系统的能量函数为目标函数,计算得到能量函数值最小的位移反馈控制参数,实现智能结构体主共振响应的控制器设计。由悬臂梁主共振振动实验实测数据计算出最优化控制参数,设计非线性主共振振动实验控制系统,进行减振控制实验研究。     

Time–frequency analysis of friction-induced vibration under reciprocating sliding conditions

A time–frequency analysis can give an overall view of the behaviour of friction-induced vibration. In this paper, short-time Fourier transform (STFT), Wigner–Ville distribution (WVD), Choi–Williams distribution (CWD) and Zhao–Atlas–Marks distribution (ZAMD) are applied to analyze time–frequency characteristics of friction-induced vibration. The result shows that there is always a frequency change in the time–frequency presentation of vibration in the location where the vibration is bounded. The frequency changes in time–frequency presentations are associated with nonlinearity of vibration systems. The nonlinearity may be counted as the evidence to support the consideration that friction-induced vibrations are bounded by limit cycles due to the system nonlinearity. Based on the time–frequency presentations of vibrations, it may be concluded that the friction vibration system is generally a linear system in the phase of vibration initiation but is a nonlinear system in the phases of vibration being bounded and disappearance.     

Nominal and robust stability regions of optimization-based PID controllers

In recent decades, several optimization-based methods have been developed for the proportional-integral-derivative (PID) controller design, and the common feature of these methods is that the controller has only one adjustable parameter. To keep the closed-loop systems stable is an essential requirement for the optimization-based PID controllers. In almost all these methods, however, no exact stability region for the single adjustable parameter was sketched. In this paper, using the proposed analytical procedure based on the dual-locus diagram technique, explicit stability regions of the optimization-based PID controllers are derived for stable, integrating, and unstable processes with time delay in the nominal and perturbed cases, respectively. It is revealed that the proposed analytical procedure is effective for the determination of the nominal and robust stability regions and it offers simplicity and ease of mathematical calculations over other available stability analysis methods. The results in this paper provide some insight into the tuning of the optimization-based PID controllers.     

An adaptive PID like controller using mix locally recurrent neural network for robotic manipulator with variable payload

Being complex, non-linear and coupled system, the robotic manipulator cannot be effectively controlled using classical proportional-integral-derivative (PID) controller. To enhance the effectiveness of the conventional PID controller for the nonlinear and uncertain systems, gains of the PID controller should be conservatively tuned and should adapt to the process parameter variations. In this work, a mix locally recurrent neural network (MLRNN) architecture is investigated to mimic a conventional PID controller which consists of at most three hidden nodes which act as proportional, integral and derivative node. The gains of the mix locally recurrent neural network based PID (MLRNNPID) controller scheme are initialized with a newly developed cuckoo search algorithm (CSA) based optimization method rather than assuming randomly. A sequential learning based least square algorithm is then investigated for the on-line adaptation of the gains of MLRNNPID controller. The performance of the proposed controller scheme is tested against the plant parameters uncertainties and external disturbances for both links of the two link robotic manipulator with variable payload (TL-RMWVP). The stability of the proposed controller is analyzed using Lyapunov stability criteria. A performance comparison is carried out among MLRNNPID controller, CSA optimized NNPID (OPTNNPID) controller and CSA optimized conventional PID (OPTPID) controller in order to establish the effectiveness of the MLRNNPID controller.     

Design and stability analysis of impedance controller for bilateral teleoperation under a time delay

A new impedance controller is proposed for bilateral teleoperation under a time delay. The proposed controller does not need to measure or estimate the time delay in the communication channel using the force loop-back. In designing a stable impedance controller, absolute stability is used as a stability analysis tool, which results in a less conservative controller than the passivity concept. Moreover, in order to remove the conservatism associated with the assumption of infinite port impedances, the boundaries of human and environment impedance are set to finite values. Based on this, this paper proposes a parameter design procedure for stable impedance controllers. The validity of the proposed control scheme is demonstrated by experiments with a 1-dof master/slave system.     

基于QFT的开关阀控气动位置伺服系统鲁棒控制

考虑开关阀的滞后特性,建立了高速开关阀控气动位置控制系统的非线性模型,并在原模型的基础上,将系统模型处理为包含滞后环节的数学模型。针对该模型对象参数不确定、摄动量大和负载变化范围大等问题,采用包含小闭环的2自由度控制结构,用定量反馈理论,进行系统的鲁棒稳定性、干扰抑制和跟踪性能设计,设计了具有鲁棒性能的控制器,并且设计了有效的摩擦力补偿器。经过试验验证,系统不仅具有较好的稳定鲁棒性,还具有很好的动态性能和位置精度。     

Autotuning positive feedback time delay controller for dead time processes

Biased relay feedback tests are applied to dead time processes to obtain their ultimate gains and ultimate frequencies. First-order process with dead time models are then fitted to the estimated gains and frequencies. A time delay controller that incorporates a simple compensator with a delay element in positive feedback can be derived from the fitted model. The time delay controller gives better performance comparing with classical Ziegler and Nichols tuned PID controller. Experimental study is included to demonstrate the effectiveness of the proposed tuning scheme and the time delay control algorithm.     

时滞阻尼液固混合介质隔振系统的动力学研究

针对单一线性阻尼被动隔振无法兼顾共振区与隔振有效区的隔振性能问题,提出了时滞立方非线性主动阻尼控制策略。以具备分段线性刚度液固混合介质受控隔振系统为研究对象,运用多尺度摄动法对受控系统进行主共振分析,建立了等效刚度和等效阻尼的概念,并考察时滞对受控系统幅频特性与传递率特性的影响。研究结果表明:由于主动时滞的引入,立方非线性速度反馈增益既可调控阻尼,还可以调整隔振系统的等效刚度;与单一线性阻尼相比,立方速度反馈在不改变隔振有效区隔振效果的同时,显著降低了共振区的传递率;对于因鞍结分岔在主共振造成的幅值跳跃,通过选择合适的反馈时滞可以有效抑制跳跃的出现;通过稳定性分析发现,受控系统并非是全时滞稳定,因此最后给出控制器中反馈增益和时滞的参数区间设计方法。