时滞光电跟踪系统鲁棒内模PID控制器设计赵志诚;刘志远;张井岗【摘要】针对时滞光电跟踪提出了一种内模PID(IMC-PID)控制器设计与参数整定的解析方法.首先建立了系统的一阶时滞积分(FODI)模型,并用二阶加时滞(SOPDT)模型进行逼近,然后利用一阶Taylor表达式代替系统模型中的时滞项,导出了控制器参数的整定规则.特别是为了保证系统的鲁棒性,可以根据最大灵敏度解析计算内模PID控制器的可调参数λ_n.仿真结果表明,与常规方法相比,所提方法不仅提供了较好的设定值跟踪和扰动抑制特性,而且对于系统参数摄动具有更好的鲁棒性.另外,实验结果也证实了该方法能够提高系统跟踪性能和跟踪精度.【期刊名称】《光电工程》【年(卷),期】2010(037)001【总页数】7页(P30-36)【关键词】内模PID;光电跟踪系统;时滞系统【作者】赵志诚;刘志远;张井岗【作者单位】哈尔滨工业大学控制科学与工程系,哈尔滨,150001;太原科技大学自动化系,太原,030024;哈尔滨工业大学控制科学与工程系,哈尔滨,150001;太原科技大学自动化系,太原,030024【正文语种】中文【中图分类】V5560 IntroductionOpto-electronic tracking system is a complex equipment which mainly includes photoelectric detection,signal processing, control systems, precision machinery and other parts. It is widely applied in the filed of civilian and military industry such as NC machine tools, astronomical observation, shooting range measurement, weapons control, and flight simulators etc[1]. For opto-electronic tracking system, a good servo performance, such as high-accuracy, high-speed, non-overshoot and no vibration, is necessary [2]. Furthermore, in order to keep stability for parameter perturbation system, a high robustness is also very important.In other words, it is expected to achieve both performance and robustness in opto-electronic tracking system. However, considering the disturbance and uncertainty in the system, the conventional PID control is difficult to meet the requirement of the control performance effectively.So, based on the PID control, many improved PID type control schemes are researched in [1][3][4],respectively. A multi-mode control algorithm was proposed in [3]. The algorithm is expressed as follows: if the error is large, the saturation control is adopted; if the error is medium, the square root control is adopted; and if the error is small, the PID control is adopted. Paper [4]proposed a fuzzy-PID control approach, which could shorten the transient time, reduce the overshoot and improve the tracking accuracy and robustness of control system. Paper[1]proposed a single neuron fuzzy PID control method. Thus the trackingsystem not only has the capability of learning and self-adaptation, but also has better dynamic performance and steady-state performance than that of PID control. But in the above-mentioned schemes, the measurement time-delay of image tracker is not considered.To deal with the measurement time-delay, various advanced control approaches are presented. Paper [5]proposed an adaptive prediction and compensation method, which applied LMS algorithm and an adaptive filter with transverse structure to delay prediction compensation. The simulation results demonstrated the effectiveness of the method. Paper [6]introduced the method of predictive filtering in opto-electronic tracking system. Paper [7]adopted a state prediction and estimation method based on robust H∞ filter for opto-electronic tracking system.The experimental results show that the proposed method has high accuracy and good robustness. Whereas, the predictive filtering method needs complex calculation, and it easily causes the arithmetic to diverge.To find a simple design method of the PID type controller with a significant performance improvement has become an important research issue for control engineers. Because of the simplicity and improved performance of the IMC-based tuning rules, the analytically derived IMC-PID tuning methods have attracted the attention of industrial users [8]. The IMC-PID tuning rule has only one user-defined tuning parameter, which is directly related to the closed-loop performance and robustness of the system. The PID controller design has been discussed extensively in the literature for first-order plus delay time and second-order plus delay timestable/unstable process[8-10]. But the design of a simple and robust controller with improved performance has not yet been fully achieved. This paper focuses on the design of IMC-PID controller for an opto-electronic tracking system with time-delay. The system can be represented by a first-order delayed integrating (FODI)model. According to the principle of IMC, an analytical design approach of PID controller is proposed. Then, the tuning method of the controller parameter is given. Moreover, the maximum sensitivity can be applied to guarantee the robustness of the system. The simulation and experimental results show that this scheme is easy to be realized and has better performance than the conventional approach.1 Design of IMC-PID Controller for Time-delay SystemThe block diagram of IMC system is shown in Fig.1, where Q(s)is the internal model controller, G(s)is the process, M(s)is the model, and R(s),Y(s), D(s)is the set point, output and external disturbance of the system respectively. According to the design procedure for IMC system, the model is factorized asWhere M-(s) and M+(s)are the portions of the model inverted and not inverted, respectively. M+(s)is usually a non-minimum phase and contains delay time and/or right half plane zeros of M(s), while M-(s)is stable and of minimum phase with no predictors. The IMC controller Q(s)takes the formWhere f(s)is a user specified low-pass filter and usually chosen asWhere r is sufficiently large in order to guarantee that the IMC controller Q(s)is proper. Also, λ is the time constant, determined by the expected system performance. A smaller λ provides faster closed-loop response, while a l arger λ is also less sensitive to model mismatches.Fig.1 Block diagram of IMCFig.2 Equivalent block diagram of IMCIMC structure in Fig.1 can be reduced to the equivalent classic feedback structure shown in Fig.2. Gc(s)is a feedback controller. The relation between the feedback controller Gc(s)and the internal model controller Q(s)can be expressed as follows:System dynamics are often approximated by low order transfer function models for ease in controller design.The dynamics of a large number of industrial controlled objects can be represented by FOPDT and SOPDT transfer function models of the forms:The IMC filter structure exploited here is given asSo, the IMC controller can be obtained, and the corresponding feedback controller isIn order to make the resulting controller in Eq.(9)has a PID controller structure, the time-delay term is approximated by the simple first orderTaylor expansion.Let the forms of the PI and PID controllers beWhere Kp, Ti and Td are the proportional gain, the integral time constant and the derivative time constant,respectively. Table 1 shows the IMC-based PI/PID controller settings for the FOPDT and SOPDT models, where λn = λ/θ.Table 1 Settings of the controller for FOPDT and SOPDT modelsM(s)Kp Ti Td τ τ K s τs 1 e+-θK(λ+θ)=θ(λ+)k τ n 1-θ e s τs+2 22 K s ξτ+1K(λξτ 2ξτ 2+2ξτ θ)=θ(λ+)Kn 1ξ τ22 Design of IMC-PID Controller for Opto-electronic Tracking System2.1 Model of Opto-electronic Tracking SystemUsually, a high accuracy opto-electronic tracking system is a speed-position control system constituted of two closed-loops, which is annexed a position loop on the base of the speed governing system. The configuration of opto-electronic tracking system is shown in Fig.3. TheTV/IR tracker includes cameras and signal processing circuit. The module picks and separates the targets in visible-field according to the standard phase alternating line,and provides the target coordinates of current point. Taking into account the image processing, the measurement time-delay can not be ignored. So the image tracker can be depicted by a proportion plus time-delay model. The speed controller adopts proportion-integral (PI)algorithm, and the speed feedback device is an opto-electronicencoder. In practice the speed control loop can be approximated to a first-order inertia unit due to the high crossover frequency. The reduction ratio of the reducer is i: 1. The transfer function from the angular velocity of the motor to the output of the system can be represented by an integrator, and the integral time is i. So the dynamic structure of opto-electronic tracking system is shown in Fig.4. Namely, the position tracking system can be represented by a first-order delayed integrating (FODI)model:Fig.3 Block diagram of opto-electronic tracking systemFig.4 Dynamic structure of opto-electronic tracking system2.2 Design of IMC-PID controllerConsider the FODIP model of the opto-electronic tracking system as Eq.(13). It can be approximated as SOPDT model, and becomesWhere φ is an arbitrary constant with a sufficiently large value. Thus, the IMC-PID controller is the same as that for the SOPDT. According to the PID tuning rules listed in Table 1, the parameters of the PID are given asWhere λn = λ/θ.3 Tuning of the Parameter of the ControllerGain and phase margins are two well known measures of robustness and simple analytical formulas to tune PI/PID controller for stable/unstable FOPDT and SOPDT models to meet user defined gain and phase margins have been proposed. However, the gain and phase margin specificationsgive poor results for systems with unusual frequency response curve and may fail to give reasonable bounds on the sensitivity functions [11]. The maximum sensitivity (Ms)is the inverse of the shortest distance from the Nyquist curve of the open loop transfer function to the critical point (-1, j0), and is defined asMs measures the closeness of the Nyquist curve from the critical point at all frequencies and not just the two frequencies as associated with gain and phase margins, so it can serve as a better measure of system robustness. A small value of Ms indicates that the stability margin of the control system is large. Typical values of Ms are in the range of 1.2∼2.0.A first order Pade approximation is used to replace the delay term of the loop transfer function, and the sensitivity function can be written. For 1/ Ms to be the minimum distance of the Nyquist curve from the critical point, the Nyquist curve of the loop transfer function should touch the circle with centre (-1, j0)and radius 1/ Ms.According to the repeated roots condition of the sensitivity function, the relation between the adjustable parameter λn of IMC-PID controller and the maximum sensitivity Ms can be get [11]Where, λn = λ/θ. The λn can be obtained by solving (17)f or various values of Ms.4 Simulation and Experiment ResultsTo demonstrate the effectiveness of the proposed method, simulation studies are carried out. Suppose the model for an opto-electronic trackingsystem is represented as the following FODI model, which can be approximated by the SOPDT mode asThe maximum sensitivity Ms = 1.2 is chosen. Hence, the adjustable parameter λn of IMC-PID controller can be calculated via Eq.(17), and the parameters of the controller can be tuned via Eq.(15). In addition, the parameters of a conventional PID controller can be found by using the Z-N method. The system is simulated with a unit step reference at t= 0 and a step output disturbance with value of 0.2 at t = 2.When the model is accurate, the simulation result is shown in Fig.5. For the conventional method, although the response is faster than the proposed method, but the response has a large overshoot. Hence, the proposed method not only provides a better set-point tracking, but also has a steadier disturbance rejection response.Fig.5 Step response with a nominal modelFig.6 Step response with +50%mismatches in gainFig.7 Step response with +50%mismatches in time constantFig.8 Step response with +100%mismatches in delay timeThe robustness of the controller is evaluated by inserting a perturbation uncertainty of +50% in the static gain K, the time constant T and +100% in the delay time θ to yield the model mismatch, respectively. The simulation results for the proposed and conventional tuning rules are shown in Fig.6∼Fig. 8. When all three parameters vary simultaneously, the response shown in Fig.9 exhibits the worst-case model mismatchIn addition, the proposed method was applied to an opto-electronic tracking system to track an air target and the effect was compared with conventional approach. The target flies at an altitude of 1 000 meters with v=150 m/s. The range of azimuth and elevation is from 1 439 mil to 570 mil and from 542 mil to 1 385 mil,respectively. During this process, the maximum angular velocity and acceleration of azimuth tracking is 60 °/s and 80 °/s2, respectively.Correspondingly, the maximum angular velocity and acceleration of elevation tracking is 7.5 °/s and 7 °/s2,respectively. The sampling cycle of the control system is 20 ms. The tracking error curves of the proposed method are shown in Fig.10. Obviously, the error variety is steady and undistinguishable. Fig.11, which is got from conventional method, shows that the error variety is obvious and the tracking error is larger. So the conclusion can be drawn that the proposed method can bring better tracking property and higher accuracy.Fig.9 Step response with mismatches in all parameters simultaneously Fig.10 Tracking error with the proposed methodFig.11 Tracking error with the conventional method5 ConclusionAiming at an opto-electronic tracking system with time-delay, an analytical design method and parameters tuning approach of IMC-PID controller is presented. Firstly, a FODI model for the system is built, and the model can be approximated by the SOPDT mode. On the basis of the simple first-order Taylor approximant for the time-delay term, the tuning rules of the controller parameter are provided. Especially the adjustable parameter λnof IMC-PID controller can be calculated by choosing the maximum sensitivity to guarantee robustness of the system. 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