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52 Acta Electrotechnica et Informatica, Vol. 11, No. 2, 2011, 5257, DOI: 10.2478/v10198-011-0019-6 ISSN 1335-8243 (print) 2011 FEI TUKE ISSN 1338-3957 (online) www.aei.tuke.sk /aei THE MOST SUITABLE ALGORITHM FOR INVERTED PENDULUM CONTROL Peter PSTOR Department of Avionics, Faculty of Aeronautics, Technical University of Koice, Rampov 7, 041 21 Koice, Slovak Republic, e-mail: pastor_peto ABSTRACT The aim of the paper is to show comparison between three possible realizations of the PID regulator connection. In this case the regulated parameter is deviation from desired vertical position. The structure of the regulator is same like structure of autopilot used in aircraft for pitch angle stabilization. Three different structures of algorithms are described and these structures differ by proportional, integration and derivate gain connection. The goal is to find the most suitable structure for pendulum stabilization. The criteria for the best system selection are systems stability, wide range of stabilized angle, uncomplicated final structure and no overshooting of inputs limitations. Keywords: inverted pendulum, thrust vectoring nozzle, PID regulator, autopilot 1. INTRODUCTION Inverted pendulum is a typical example of the inherently unstable system and is widely used as benchmark for testing control algorithms (PID controllers, neural networks, fuzzy logic, etc.). This system approximates the dynamics of a rocket immediately after lift-off, or dynamics of a thrust vectored aircraft in unstable flight regimes in low dynamic pressure conditions. The objective of the rocket control problem is to maintain the rocket in a vertical attitude while it accelerates 1. Angular position of the inverted pendulum is controlled by input force. In this case the controlling force is generated by system of vectored nozzles, where force is directly proportional to nozzle deflection. Position limitation ( 20 deg), rate limitation ( 60 deg/sec) and nozzle dynamic representing by 2nd order transfer function are also considered. The model of the system of the vectored nozzles will be briefly described later. This model is connected with the nonlinear model of the inverted pendulum given by following equations: ()22222cosd xddMmmlmlFdtdtdt+= (1) ()22222cossindd xJmlmlmgldtdt+= + (2) where M mass of the cart, m mass of the pendulum, l length to pendulum centre of mass, J inertia of the pendulum, deviation from vertical position, x cart position coordinate, g acceleration of gravity, F input force. The construction of the model is described in more details in publication 2. This system will be applied for final nonlinear analyses of the selected controlling system. Transfer function, given by equation (3) will be utilized for controller design and regulators parameters setting: ( )( )5222201,2815 101,90836lsKJmlU ssssgJ=+ (3) where K gain of the system, 0 natural frequency of the system. This function is easy to analyse and the PID regulator design is also not complicated. 2. 1ST ALGORITHM The first algorithm structure is described by following control law 3: ( )( )( )( )ZF sPsssDs=+ (4) where F(s) force applied to the pendulum; Z(s) desired value of angle; P, D coefficients of the regulator. The structure of this autopilot consists of two loops outer and inner and is depicted in Fig. 1. Fig. 1 Structure of the 1st Algorithm The transfer function of the inner loop is: 2202202201KssKsKDssKDss+=+ (5) And transfer function of the whole system: ( )( )()220ZsKPssKDsKP=+ (6) The P, D parameters could be calculated by comparing the denominator of equation (6) with desired denominator shape: 222SPSPSPs + (7) Acta Electrotechnica et Informatica, Vol. 11, No. 2, 2011 53 ISSN 1335-8243 (print) 2011 FEI TUKE ISSN 1338-3957 (online) www.aei.tuke.sk /aei From previous formula you can find the similarity with aircraft short period mode and the same criteria for short period damping SP and frequency SP are useable for this purpose. The criteria for short period damping and frequency according 4,5,6 are: 0,351,3;1/ secSPSPrad Compare denominator of equation (6) with expression (7): ()222202SPSPSPsKDsKPs += Coefficient D can be calculated: 12SPSPJDkgmsl = (8) And coefficient P: 22SPJPmgkgmsl= + (9) Fig. 2 shows the step response, when input signal is step function with final value: /10. This value was approximately calculated from equations (1) and (2). Value of P coefficient is: P= -461049 and D coefficient: D=-312133. Fig. 2 Angle Time Response You can see from Fig. 3, that the input force in t=0 exceeds the limitation and this structure cannot be used for further design. Fig. 3 Input Force Time Response 3. 2ND ALGORITHM The second algorithm is given by following control law 3: ( )( )( )( )ZPsIF ssDssss+=+ (10) where I integration coefficient of the PID regulator and meaning of other parameters is same like in equation (4). The structure of the autopilot is depicted in Fig. 4. Fig. 4 Structure of the 2nd Algorithm The model consists also from 2 loops inner and outer and the transfer function valid for inner loop is given by equation (5). Including outer loop, the final transfer function is: ( )( )()3220ZsKPsKIssKDsKPsKI+=+ (11) Binomial standard form for 3rd order system describes desired time response 3: 322333ZZZsss+, where Z is desired value of natural frequency. The P, I, D coefficients are: 223ZJPmgkgmsl= + (12) 33ZJIkgmsl= (13) 13ZJDkgmsl= (14) The time of regulation can be approximately calculated by using formula: 7secrZt (15) The step response is shown in Fig. 5 and the PID regulator coefficients are: P = - 1085315,8; I = - 624267; D = - 468200. It can be observed in Fig. 5 the undesirable overshoot. Try to adjust P, I and D coefficients to eliminate the overshoot. The coefficients for different Z value are shown in Table 1. Table 1 Coefficients for different Z value Z P I D 1 -383015,8 -73033 -234100 2 -1085315,8 -624267 -468200 4 -3894515,8 -4994133 -936400 54 The Most Suitable Algorithm for Inverted Pendulum Control ISSN 1335-8243 (print) 2011 FEI TUKE ISSN 1338-3957 (online) www.aei.tuke.sk /aei Fig. 5 Angle Time Response It is possible to determine from Fig. 6 the relationship between overshoot and Z value. If the Z value is increasing, the overshoot is decreasing and vice versa. Fig. 6 Angle Time Response You can see in Fig. 7 that input force exceeds limitation again for all coefficients value setting. Fig. 7 Input Force Time Response 4. 3RD ALGORITHM The following control law is valid for third algorithm 3: ( )( )( )( )( )2ZsF ss DssPsIss=+ Lets divide the previous expression by 1/s: ( )( )( )( )( )ZIF ssDsPssss=+ (16) The structure of the autopilot is shown in Fig. 8. Fig. 8 Structure of the 3rd Algorithm The structure consists of the three loops inner, middle and outer. The form of the inner loop is the same like in previous examples and is given by equation (5). The transfer function including middle loop has form: ()220KsKDsKP+ And the transfer function of the whole system: ()3220KIsKDsKPsKI+ (17) The denominator of equation (17) has the same form like denominator of equation (11), so the same value of P, I, D coefficient are valid. Fig. 9 shows time response to input step function with final value /10. Fig. 9 Angle Time Response Fig. 10 Input Force Time Response Acta Electrotechnica et Informatica, Vol. 11, No. 2, 2011 55 ISSN 1335-8243 (print) 2011 FEI TUKE ISSN 1338-3957 (online) www.aei.tuke.sk /aei You can observe in Fig. 10 that input force does not exceed the limitation, which is depicted as red limiting line. This structure is the most suitable for angle control, because in previous examples the input force exceeds limited value. The Bode characteristic of transfer function given by equation (17) is shown in the following figure. Fig. 11 Bode Characteristic 5. NONLINEAR ANALYSES The structure for nonlinear analyses consists of two nonlinear models model of inverted pendulum described by equations (1) (2) and model of thrust vectoring system of aircrafts engine including dynamic of nozzles given by 2nd order transfer function similar like in publication 4: 240040400ss+ Its deflection is limited 20 deg in position and 60 deg/sec in rate 1. Model provides calculation 7 of the summary forces and moments generated by thrust system. In this example only force in pitch control is considered: ( )( )sinF sTs= (18) where T is the thrust produced by nozzle and its value is constant during simulation; angle between vectored nozzle deflection and longitudinal axes. Substitute equation (18) into control law (16): ( )( )( )( )( )sinZITssDsPssss=+ (19) Lets assume the simplification for small angle of nozzle deflection (approximately up to 20 deg) is valid: sin (s) = (s). Divide equation (19) by thrust T: ( )( )( )( )( )1ZDPIssssssTTTs=+ (20) Equation (20) represents control law for system mention above and shown in Fig. 12. Fig. 12 System for Nonlinear Analyses New P, I, D values can be calculated by applying equation (20) and assuming that nozzles generated thrust 148 916 N. It is necessary to emphasise, that these parameters are constant only if the thrust of the aircraft is constant. In case the thrust varies during simulation, these parameters have to be adjusted according actual thrust value. Note the P, I, D values are given as ratio. This is very important fact for practical realization of the similar system with same properties like mention above system. The system depicted in Fig. 12 was analysed. Fig. 13 shows step response when Z = 18,8 deg and this is the maximum value, when pendulum can be stabilized. This limitation can be also calculated from equations (2) and (1) for constant value. The motion of the pendulum above this limitation is unstable. Fig. 13 Angle Time Response If the input is impulse function with period 20 s and pulse width 50% then the maximum Z is limited to 16,1 deg. The time response of angle and angular velocity is depicted in the following figures. Fig. 14 Angle Time Response 56 The Most Suitable Algorithm for Inverted Pendulum Control ISSN 1335-8243 (print) 2011 FEI TUKE ISSN 1338-3957 (online) www.aei.tuke.sk /aei Fig. 15 Angular Velocity Time Response The following figure shows the response when Z = 0 and disturbance random force from region 50000N is in the input of the inverted pendulum model. Fig. 16 Angle Time Response Observe also the maximum value of the Z as function of input signal frequency in Fig. 17 and in Fig. 18. The condition to determine this value is system stability. Fig. 17 Maximum Z Value Fig. 18 Maximum Z Value in More Details The maximum disturbance force dependency on the frequency of input force is depicted in Fig. 19. Fig. 19 Disturbance Force Deviation from desired vertical position is caused by input disturbance. The absolute values of these errors are shown in Fig. 20. Fig. 20 Absolute Values of Errors 6. CONCLUSIONS The most suitable structure for inverted pendulum control is structure given by control law (16) and is depicted in Fig. 8. Other structures are not suitable for control, because value of input force exceed the limitation given by vectoring system model. Nonlinear analyses of selected structure also give good results, but other limitations need to be assumed. Envelope of given angle and disturbance force limitation is shown in Fig. 15 and in Fig. 16. Advantage of the PID system control is relatively convenient system design. Disadvantage is small range of controlled angles and sensitivity when frequency of input signal is between 2-7 rad/s. Control system is reliable when these limitations are not exceeded. REFERENCES 1 DABNEY, J. HARMAN, T.: Mastering Simulink. Upper Sanddle River: Pearson Prentice Hall, 2004, ISBN 0-13-142477-7. 2 WHITE, J. R.: System Dynamics. Online course, /system_dynamics/sdyn/s7/s7invp1/s7invp1.html#equations 3 LAZAR, T. ADAMK, F. LABUN, J.: Modelovanie vlastnost a riadenia lietadiel. Faculty of Aeronautics, Technical University of Koice, 2007, ISBN 978 80 8073 8396. Acta Electrotechnica et Informatica, Vol. 11, No. 2, 2011 57 ISSN 1335-8243 (print) 2011 FEI TUKE ISSN 1338-3957 (online) www.aei.tuke.sk /aei 4 ADAMS, R. J. BUFFINGTON, J. M. SPARKS, A. G. BANDA, S. S.: Robust Multivariable Flight Control. London: Springer-Verlag, 1994, ISBN 3-540-19906-3. 5 ROSKAM, J.: Airplane Flight Dynamics and Automatic Flight Controls. Lawrence: Design, Analysis and Research Corporation, 2001, ISBN 1-884885-17-9. 6 MIL-STD-1797A, Flying Qualities of Piloted Aircraft, Department of Defence Handbook, 19 December 1997. 7 TARANENKO, V. T.: Dinamika samoleta s vertikanym vzletom i posadkoj. Moskva: Mainostrojenije, 1978. Received November 10, 2010, accepted March 21, 2011 BIOGRAPHY Peter Pstor was born on 6.5.1981. In 2005 he graduated (MSc) with distinction at the department of Avionics of the Faculty of Aeronautics at Technical University in Koice. He is external doctorate student at the department of Avionics of the Faculty of Aeronautics at Technical University in Koice since 2006. His scientific research is focusing on Mathematical modelling of aircraft and aircrafts engines. In addition, he also investigates questions related with the stabilization of inherently unstable systems. 倒立摆的控制最合适的算法摘要 本文的主要内容是PID调节器三种算法之间的比较。在这种情况下调节来自于所需垂直位置参数的偏差。调节器的结构和自动驾驶仪是相同的，都用于飞机俯仰角稳定。三种不同的结构的算法描述和这些结构之间不同的比例、积分和微分的增益是相互连接的。目标是使倒立摆能够找到最合适的算法结构。使之获得最好的系统的稳定性，稳定的角度范围宽，结构比较简单，没有超出规定的输入的局限性。关键词：倒立摆；推力矢量喷管；PID调节器；自动驾驶仪1引言倒立摆是一个典型的不稳定系统的例子，它被广泛用作测试控制算法（PID控制器、神经网络、模糊逻辑等）的基准。本系统研究了在火箭起飞后的弹动力学，或者是在低动态压力不稳定的条件下推力矢量飞机飞行的条件。火箭的控制问题主要是使它保持在一个垂直的状态上当它的火箭加速时1。倒立摆的摆角位置通过输入力来控制。在这种情况下，控制力是在矢量喷管的系统中产生的，在那里力和喷管偏转是成正比的。位置限制（20）、速率限制（60度/秒）和喷嘴动力学通过第二阶传递函数都被表示出来2，在之后系统的模型向量喷嘴也被简要的描述出来。对这个倒立摆模型的非线性系统进行分析得到： (1) (2) 其中M小车的质量，m-倒立摆摆杆质量，L摆杆转动轴心到杆质心的长度，J-摆杆惯量，偏离垂直位置角度，X车位置坐标，g-重力加速度，F输入力。通过这些参数，可以得到系统的实际模型。该系统将用于所选择的控制系统的最终的非线性分析。方程（3)为系统的传递函数，它是利用控制器的设计和调节器的参数设置来 对系统进行分析的： (3)其中K增益的系统，wn固有频率系统。这个函数很容易分析，可以看出PID调节器设计也不复杂。2 第一算法第一种算法描述的结构如下3： (4)F（s）-倒立摆所受外界作用力；Z（s）所需的角度值；P，D调节器的系数。这种自动驾驶仪的结构包括两个循环外部和内部，如图1所示： 图1第一算法的结构传递函数的内循环是： (5)和整个系统的传递函数： (6) 通过对P、D参数的计算及方程（6）得出： (7)从以往的公式，你可以找到相似飞机的短周期模式和相同的标准，短期内阻尼sp和频率wsp是可用的为这个目的，标准的短周期阻尼和频率根据4，5，6是： 把方程(6)和表达式(7)合并得：系数D可以计算： (8)和系数P： (9)图2显示了阶跃响应，输入信号最终值阶跃函数为：/ 10。这个值是从方程的近似计算（1）和（2）得到的。系数P值为：P = 461049和系数D为：D = 312133。 图2 角的时间响应你可以看到图3，当t = 0的输入力超过这个结构限制时，它将不可用于进一步的设计。 图3输入力的响应时间3 第二算法 第二个算法给出了跟踪控制法【3】: (10)I系数的选择和PID调节器的积分系数及其他参数的选择是一样的，如公式(4)。自动驾驶仪的结构如图4所示。图4 第二算法的结构 该模型由2循环也从内在和外在循环及传递函数中给出了有效的内部循环方程(5)。包括外层循环，最终的转移函数是： (11)第三阶系统的二项标准的形式描述了所需的时间响应【3】：Wz是期望值的自然频率。 P，I，D系数： (12) (13) (14)时间的调节可以通过使用公式近似计算为： (15)阶跃