Modeling and Performance Comparison of Triple PID and LQR Controllers for Parallel Rotary Double Inverted Pendulum

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1 Bipin Krishna, Deepak Chandran, Dr. V. I George, Dr. I. hirunavukkarasu 45 Modeling and Performance Comparison of riple PID and LQ Controllers for Parallel otar Double Inverted Pendulum Bipin Krishna, Deepak Chandran, Dr. V. I George, Dr. I. hirunavukkarasu 4,,,4 Department of Instrumentation & Control Engineering Manipal Institute of echnolog Manipal, Karnataka, India bipin.monai@gmail.com bstract: his paper presents the modelling and controller design of nonlinear parallel otar Double inverted pendulum sstem using Proportional-Integral-Derivative (PID) controller and Linear Quadratic egulator (LQ). LQ, an optimal control technique, and PID control method have been used in this paper to control the nonlinear dnamical sstem Inverted pendulum, a highl nonlinear unstable sstem is used as a benchmark for implementing the control methods. In this paper, modelling of the sstem through otational geometr have been carried out, in which the entire inertia tensors are taken into account. he sstem dnamics for a pendulum with a full inertia tensor using a Lagrangian formulation are presented.. Fundamentals he phsical structure of the DIP is as shown in Figure, where the related phsical parameters of the sstem are listed in the able. he DC motor is used to appl a torque to the rotating arm and the link between rotating arm and Pendulum arm & is not actuated but free to rotate. Kewords: Nonlinear sstem, Parallel otar Double Inverted Pendulum, riple PID, LQ, otational Geometr. I. INODUCION Inverted pendulum is a well-known benchmark sstem in control sstem laboratories which is inherentl unstable. In this work full dnamics of the sstem is derived using classical mechanics and Lagrangian formulation. Dnamic performance are inspected and compared of the two controllers. his paper proves that the LQ controller can promise the inverted pendulum a quick and smoother stabilizing process and with less oscillation and better robustness than the riple-pid controller. he novelt of this paper is the modelling of the sstem through rotational geometr, Design and comparison of the two controllers for the otar double Inverted pendulum. II. MHEMICL MODELING OF PLLEL OY DOUBLE INVEED PENDULUM In this paper the dnamics of the DIP are derived using the rotational geometr [] of the sstem which is explained in section II. Man papers have onl considered the rotational inertia of the pendulum for a single principal axis or neglected it altogether []. he sstem dnamics for otar Double Inverted Pendulum with a full inertia tensor using a Lagrangian formulation are presented in this paper. lso a linearized model is obtained b neglecting the disturbance torques. Simulation results show the open loop sstem characteristics his work is supported b Defense esearch and Development Organization (DDO) Govt. of India (EIP/E/466/M//4). BLE I. Fig. : Schematic of otar double inverted pendulum sstem. DEFINION OF PMEES ELED O OY DOUBLE INVEED PENDULUM Para Definition Value meter m otal mass of otating rm.5kg m Centre of mass of pendulum.5 Kg m Centre of mass of pendulum. Kg L Distance from joint to Centre of mass of.4 m pendulum L Distance from joint to Centre of mass of. m pendulum m Mass of the rotating arm.5 Kg L Length of the rotating arm.5 m b Viscous coefficients of rotating arm.4 N-m-s b Viscous coefficients of pendulum. N-m-s b Viscous coefficients of pendulum.8 N-m-s K orque constant.5 Nm/ m K Back emf constant. Volt /rad b esistance of motor circuit Ω International ournal of Emerging rends in Electrical and Electronics (IEEE ISSN: -9569) Vol. Issue., une. 5.

2 Bipin Krishna, Deepak Chandran, Dr. V. I George, Dr. I. hirunavukkarasu 46 right hand coordinate sstem has been used to define the rotation of horizontal arm and the two pendulum arms. he coordinate axes of the rotating arm and two pendulum arms are the principal axes such that the inertia tensor are diagonal in the form as in (). x x z z x x z z x x z z,, () indicate the Inertia tensors of otating arm, Pendulum,Pendulum respectivel. x x, indicate rotation of the arm along x axis when a torque is applied along x axis. Similarl we can define and.he angular rotation of rotating arm α, is measured in the horizontal plane where a counterclockwise direction (when v iewed from above) is positive []. he angular rotation of pendulum, β, is measured in the vertical plane where a counterclockwise direction (when viewed from the front) is positive, same as in pendulum is β, when pendulums are hanging down in the stable equilibrium position β = β =. τ, τ are the disturbance torque experienced b pendulum and pendulum respectivel. Energies of the arms are defined as: For rotating arm, Potential energ is given b Kinetic energ is, () k V m V () ml zz () Potential energ of pendulum is given b P mgl cos() () nd Kinetic energ is, () k V mv ( msin()( L cos()) L l (4) l sin())( cos() x x sin()) zz z z Potential energ of pendulum is given b P m g l c o s() Kinetic energ is given b () K V m V ( sin()( cos()) sin()) m L L l l ( xx cos() sin()) z z otal potential energ, (6) = P +P +P otal kinetic energ, = We can write Lagrange equation L = KE PE as follows L ()( ml zsin()( z m cos()) L L l l sin())( cos() x x sin()) z z ( m sin()( L cos()) L sin()) l l (7) ( x co x s() sin()) cos() z z m gl m gl cos() he velocities of the pendulum arms are derived based on the rotation matrices and frame of references defined for each arm. Since the arms are long and slender moment of inertia is considered to be negligible, more over the arms have rotational smmetr such that the moment of inertia in the principal axes are equal thus inertia tensor can be approximated as follows, xx zz xx zz xx zz bove dnamic equation can be written in a little easier wa b making the following substitution: otal moment of inertia of rotating arm about the pivot point, ml Moment of inertia of pendulum about its pivot point, (9) ml Moment of inertia of pendulum, () (8) (5) International ournal of Emerging rends in Electrical and Electronics (IEEE ISSN: -9569) Vol. Issue., une. 5.

3 Bipin Krishna, Deepak Chandran, Dr. V. I George, Dr. I. hirunavukkarasu 47 () m l otal moment of inertia experience b the motor when pendulum and are in hanging position, m L m L = ml ml ml () Initiall the control input is the torque applied to the pivot of the rotating arm. DC motor is used to drive the rotating arm, so the voltage is taken as the control input. s far as we are neglecting the effect of inductor, the torque and the voltage can be related b the equation, K mv K mkb () Where are the motor parameters mentioned in the able. compact form can be obtained b using these simplifications, thus obtaining a coupled electro mechanical equation as follows, K m + = (4) Where, sin() sin() m l m l sin() m l Lcos() m l Lcos() m l Lcos() m l Lcos() sin() ' sin() sin() m l L m l sin() ml sin() m l L ' sin() ml sin() g b ' sin() mlsin() g b KmKb sin() sin()() b B. Linearized state equation (5) Linearized equation of the dnamic sstem for the upright position is derived below. Finding the linearized model using the equilibrium point where. fter expanding the alor series at x= Sin(x),, cos(x) We can make approximation of the nonlinear equation in (4) around the equilibrium point thus equation becomes, KmKb K mv ()() ml L ml L b ml L()() ml g b m l L()() m l g b (6) be Here x can be taken as x= [ ] the state variable and control input be the voltage, v. hen (6) can arrange in the form of P x Qx u x P Qx P u : x Bu Where, P m ll ml L m ll mll Q KmKb () b m gl b m gl b = Km (7) (8) Disturbance torque ma be neglected in the analsis, so the matrix became K [ m ] (9) Open loop responses of the sstem under various initial conditions are given in the fig International ournal of Emerging rends in Electrical and Electronics (IEEE ISSN: -9569) Vol. Issue., une. 5.

4 Bipin Krishna, Deepak Chandran, Dr. V. I George, Dr. I. hirunavukkarasu 48 Pendulum ngle (deg) beta ime (sec) Pendulum ngle (deg) (a) beta time (sec) (b) Fig. (a) Open loop response of β ()=75 (b) Open loop response of β ()=78 III. CONOLLE DESIGN In this paper riple PID and LQ control strategies has been used to stabilize parallel rotar inverted pendulum. PID control make use the effects of the proportional (Kp), derivative (Kd) and integral (Ki) control. Sometimes PID controllers does not provide sufficient performance for the control of the nonlinear and undefined sstems. n optimal control sstem is designed to balance the same designed successfull using LQ. his method can be adopted either to linear sstem or rather linearized sstem.. Designing and uning of riple PID controller In literature man method have been proposed to control the inverted pendulum, such as traditional PID control [], fuzz control [], genetic algorithm optimizing control [4], and linear quadratic regulator (LQ) control [5 ]. Even though man control algorithms are proposed for the control of inverted pendulum, PID control is the most widel used control method in the realization of the control sstem. However, the rotar double inverted pendulum sstem is a one input and three output sstem which challenges to the one input and one output control characteristic of the single PID controller. While we consider inverted pendulum, having two output, onl one can be controlled and mostl cart position control will be neglected. In this paper, we made an attempt to solve this multi-output problem b implementing a triple PID controller. Fig. Block Diagram of riple PID controller for parallel otar Double Inverted Pendulum In this proposed work the reference angle is taken as 9 deg.[6] Pulse signal is used as the position reference signal, with an amplitude of.5m, pulse period s and width 5% of period, b which the abilit of the controller to track the position is being checked. fter tuning the control parameters, we obtain a group of parameters for double-pid controller, with which the controller is capable of controlling the inverted pendulum sstem and providing a ver robust performance. lpha(rad) Pulse input lpha ime(sec) Fig 4: Horizontal arm angle when tracking pulse signal. Pendulum ngle (rad) ime(sec) Fig 5. Pendulum angle keep balancing while Horizontal rm tracking reference signal fter tuning, the control parameter are obtained as Beta International ournal of Emerging rends in Electrical and Electronics (IEEE ISSN: -9569) Vol. Issue., une. 5.

5 Bipin Krishna, Deepak Chandran, Dr. V. I George, Dr. I. hirunavukkarasu 49 4 [ K 8., K 5., K 4] p i D [ K 56.8, K.6, K 5] p i D [ K.6, K, K.56] p i D Desired Position (9 deg) Beta angle which is closel influenced b the three concerned states: Horizontal arm angle and pendulum angle and pendulum angle. fter trial and error we choose the weighting matrix as and corresponding feedback gain matrix as Q= diag [4, 8,,, ], and = K= [ ] Pendulum ngle (deg) ime (sec) Fig 6. PID control of pendulum ngle Pendulum ngle(deg) Desired Position(9 deg) Beta ngle Fig 8. Block diagram of the designed LQ controller C. ssessment of Dnamic response using designed Controllers Dnamic performance indices are chosen to reflect practical control effect. Performance indices include rise time, transition time, stead state error, peak overshoot. 5 Beta ime(sec) Fig. 7 PID control of pendulum ngle B. Designing and tuning of LQ controller he plant is linear time invariant and the state space equation is X X BU () Which minimize the performance index cost () X QX U U dt () Where Q is a semi definite matrix, is a positive definite matrix. State variable and input variables are penalized b Q and matrices respectivel for the smallest performance index function. o minimize the cost function the feedback control law is obtained as u ()()() t kx t B Px t () Where P is the onl positive definite smmetric solution which meets the iccati equation P P Q PB B P () he optimalit of the LQ control algorithm totall depends on choosing of Q and matrix. However, there is no resolving method to choose these two matrices. he usual method to choose Q and is b means of Simulation and trial.[8] In this section, the LQ controller is tuned b changing the nonzero elements of the Q matrix elements, Beta ngle(deg) ime(sec) Pendulum ngle (degree) (a) Beta ime (sec) (b) Fig 9. Dnamic response of the sstem using LQ for (a) β ()=75 (b) β ()=78 International ournal of Emerging rends in Electrical and Electronics (IEEE ISSN: -9569) Vol. Issue., une. 5.

6 Bipin Krishna, Deepak Chandran, Dr. V. I George, Dr. I. hirunavukkarasu 5 Pendulum ngle(deg) LQ riple PID ime(sec) optimizing a cart-double pendulum sstem, International ournal of utomation and Computing, 9, vol. 6, pp [5] H. M. Wang, S.. iang, LQ control of single inverted pendulum based on square root filter, dvanced Materials esearch,, pp [6] Musthafa INKI,Umit ONEN Pid and Interval pe- Fuzz Logic Control of Double Inverted Pendulum Sstem //$6. C IEEE vol [7] Lal Bahadur Prasad, Barjeev agi, Hari Om Gupta, Modelling & Simulation for Optimal Control of Nonlinear Inverted Pendulum Dnamical Sstem using PID Controller & LQ Sixth sia Modelling Smposium [8] aoa P Hespanha Lecture notes on LQ/LQG controller design evisions from version anuar 6, chapter 5 Pendulum ngle (deg) LQ riple PID Bipin Krishna has obtained his Diploma in Computer science from echnical Education Board of Kerala in and B.ech degree in Electronics & Instrumentation Engineering from nna Universit, Chennai in 6. In he finished his M.ech in Instrumentation Control sstems from Manipal Universit with honors. Presentl he is working as ssistant Professor (Sr.) in the department of Instrumentaion & control Engineering, MI, Manipal Universit. He has got publications including ournals & ConferecesHis areas of interests are Control sstems, sstem identification, Embedde sstem design etc ime (sec) Fig. Dnamic response comparison of designed controllers IV. CONCLUSION riple PID and LQ, an optimal control technique have been implemented in this paper to control parallel otar Double inverted pendulum. he dnamic response of the sstem for each controller have been compared. Simulations for the control schemes have been carried out using MLB-SIMULINK models. he simulation results justif the comparative advantages of optimal control using LQ method than riple PID control. Dnamic performance proves that the proposed LQ controller can guarantee the otar Double Inverted pendulum a faster and smoother stabilizing process and with better robustness than the riple-pid controller. Deepak Chandran finished his B.ech and pursuing M.ech in Control Sstems at Maniipal Institute of echnolog, Manipal Universit, Manipal, Karnataka. His area of interests are nonlinear control sstems, daptive control, Instrumentation sstem design etc. EFEENCES [] Benjamin Seth Cazzolato, Zebb Prime On the Dnamics of Futura Pendulum, ournal of Control Science and Engineering, Volume. [] Y.Xin,B.Xu,H Xin,L U Hu he computer simulation and real time control for the inverted pendulum sstem based on PID communication sstem and information echnolog Lecture notes in EE,Vol.,pp []. Yi, N. Yubazaki, Stabilization fuzz control of inverted pendulum sstems, rtificial Intelligence in Engineering,, pp.5 6. [4]. K. Liu, C. H. Chen, Z. S. Li,. H. Chou, Method of inequalities based multi-objective genetic algorithm for International ournal of Emerging rends in Electrical and Electronics (IEEE ISSN: -9569) Vol. Issue., une. 5.