Vol. 6, No., A Novel Ball on Beam Stailizing Platform with Inertial Sensors Part II: Hyrid Controller Design: Partial Pole Assignment & Rapid Control Prototyping Ali Shahaz Haider Control Systems Laoratory COMSATS Institute of Information Technology, Beta Facility for Advance Automatic Control Tech. Muhammad Bilal COMSATS Institute of Information Technology Samter Ahmed COMSATS Institute of Information Technology Saqi Raza COMSATS Institute of Information Technology Imran Ahmed COMSATS Institute of Information Technology Astract This research paper presents a novel controller design for one degree of freedom (-DoF) stailizing platform using inertial sensors. The plant is a all on a pivoted eam. Multi-loop controller design technique has een used. System dynamics is oservale ut uncontrollale. The uncontrollale polynomial of the system is not Hurwitz hence system is not stailizale. Hyrid compensator design strategy is implemented y partitioning the system dynamics into two parts: controllale susystem and uncontrollale susystem. Controllale part is compensated y partial pole assignment in the inner loop. Prediction oserver is designed for unmeasured states in the inner loop. Rapid control prototyping technique is used for compensator design for the outer loop containing the controlled inner loop and uncountale part of the system. Real-time system responses are monitored using MATLAB/Simulink that show promising performance of the hyrid compensation technique for reference tracking and roustness against model inaccuracies. Keywords stailizing platform; all on eam; multi-loop controller; inertial sensors; rapid control prototyping; partial pole assignment I. INTRODUCTION Stailizing platforms are among challenging control systems. One of such systems is the single degree of freedom (-DoF) all on eam mechanism. Plant of this control prolem consists of a all capale of rolling on a eam under the action of gravity due to the inclination of the eam. The control ojective is to stailize the positions of the all on the eam in the presence of external disturances and to achieve all position reference tracking. The system is open loop unstale so feedack is inevitale [], []. Owing to the significance of all on eam system a lot of research work has een dedicated to it. Classical PID controller has een implemented in [] treating system a single input single output plant without taking in to account the internal states of the system. The oserver-ased model reference adaptive iterative learning controller has een demonstrated in []. A new technique ased on geometric control has een implemented in [], which involves designing immersion and invariance ased speed and rotation angle oserver for the all and eam system. Decoupled neural fuzzy sliding mode control of the nonlinear all on eam system has een considered in []. Nonlinear model predictive control for a all and eam has een implemented in []. MATLAB ased modeling and modulation of nonlinear all-eam system controller has een demonstrated in [6]. A new adaptive state feedack controller for the all and eam system is presented in [7]. Augmented state estimation and LQR control for a all and eam system are implemented in [8]. Adaptive Neural Network for stailization of all on eam system has een studied in [9]. Human simulated intelligent control for all and eam system is implemented in []. The Lyapunov direct method for the stailization of the all is presented in [] and Energy-ased alance control approach to the all and eam system is presented in []. The majority of research work in the literature takes into account a reduced order model of the system y neglecting certain states in the system. In this research paper full order 86 P age
Vol. 6, No., model of the system is stailized using a novel method that is a hyrid of partial pole assignment and rapid control prototyping using feedack from inertial sensors. Rapid Control Prototyping is a controller testing and tuning strategy on the actual plant in the feedack loop. With the availaility of lowcost high processing capaility digital processors and software suits, responses of real plants can directly e otained and evaluated for a given control law. Nowadays rapid control prototyping is industry-wide adopted ecause the ehavior of control algorithm can directly e tested on real world plants. This research paper is the second part of two parts research. Part-I descried geometrically accurate and detailed nonlinear model of the all on eam system followed y linearization and state space conversion. In this part-ii of the research work, controller is designed for the model developed in part-i. Organization of the paper is as follows, section-ii gives a rief overview of system dynamics. Section-III comprises of multi-loop hyrid compensation design involving partial pole assignment for inner loop and rapid control prototyping for the outer loop. Section-IV presents simulation and experimental results followed y section-v descriing conclusions and future work. Fig.. Sensing mechanism Inclination of the eam is actuated y a permanent magnet DC motor (PMDC) with its shaft coupled to a rotary potentiometer. Motor is driven y driver oard. Control strategy is implemented y a digital micro controller and data acquisition card (DAQ) interfaced with MATLAB/Simulink for real time data monitoring and processing. The continuous time state space of the plant is given y (), which has een derived in []. II. OVERVIEW OF SYSTEM DYNAMICS Hardware platform is shown in Figure. Functional description for this plant is given in []. Position of a metallic all capale of rolling on a eam is to e controlled. Beam consists of two parallel rods. Both rods are hollow thin cylindrical. One rod is wound y a chromium wire and the other rod has metallic conducting surface. Position of the all is monitored y a linear potentiometer mechanism which consisting of aforementioned two rods shorted y metallic all hence producing a voltage proportional to position of all on the eam. An accelerometer and a rate gyro on an inertial measurement unit (IMU) oard measure eam inclination angle and angular velocity respectively as shown in Figure..8 9.8 A =.e.6e,.6.6 6.e.6 T B = [ ], C =.,.. T D =. [ ] () The model () is discretized in the MATLAB using cd command with zero-order-hold and.sec sampling interval. The discretized state space model is given y (). Fig.. Hardware platform III. CONTROLLER DESIN System dynamics in () are oservale ut uncontrollale. In order to stailize the system and to achieve control ojects, system in () is partitioned in lock upper triangular configuration given y (). The partitioning has created two susystems as shown in Figure. One of these susystems is completely controllale and oservale. This susystem is 87 P age
Vol. 6, No., named susystem given y (). The other susystem is termed susystem given y (). This partitioning into susystems is shown in Figure. Our controller design strategy involves hyrid compensation in multi-loop control topology. Susystem is controlled in inner loop y unmeasured state oservation followed y partial pole assignment. Controlled susystem along with susystem is compensated in outer loop using rapid control prototyping. x ( k+ ) = x + x + H e, a aa a a a a =, y. y =. y. () x ( k+ ) = x + H e, y = C x + D ek ( ), C. =, D =.. () xk ( + ) = xk ( ) + Hek ( ), y( k + ) = C x + De, x ( k + ) 9.99e-.8866e-.9e-9 -.89e-.8e-7 x ( k + ) 9.98e- 9.799e-.888e-7 -.89e-8 7.786e- x ( k + ) x ( k + ) x ( k + ) x ( k + ) 6 x ( k + ) 7 =.9e-6 -.66e-7.7e- -8.998e-.8e- 9.7e- 6.e- 8.7e- 6.e- -.66e- 9.679e- y y y =. y. x y x x x x +. x 6 x 7 ek ( ). x 6.78e- x.8e-7 x 7.98e-6 x +.7e- x.7e-7 x 6.e- 6 x 9.887e- 7 ek ( ), () 88 P age
Vol. 6, No., x ( k + ) ( ) a aa a x k a Ha = ek ( ), ( ) + ( ) H a x k + x k x a y = [ C C ] De, a + x x ( k + ) 9.99e-.8866e-.9e-9 -.89e-.8e-7x 6.78e- x ( k + ) 9.98e- 9.799e-.888e-7 -.89e-8 7.786e-x.8e-7 x ( k + ).9e-6 -.66e-7.7e-x 7.98e-6 x ( k + ) = -8.998e-.8e- x +.7e- ek ( ), x ( k + ) 9.7e- 6.e-x.7e-7 x ( k + ) 6 8.7e-.e-x 6.e- 6 x ( k + ) 7 -.66e- 9.679e- x 9.887e- 7 x y x y x y =. x + ek ( ). y. x y. x 6 x 7 () /. T =. /. (6) The new system given y (7). T TH, T H, C CT h h h = = = is Fig.. System partitioning into two susystems A. Prediction oserver for susystem In order to accomplish pole assignment for susystem, we have to design oserver for unmeasured states. State x 6 is unmeasured [] in vector x. Following the standard procedure for minimum order prediction oserver design in [], we define a similarity transformation matrix T for system in () C = C T = I and x = Tq. such that [ ] h haa ha, ha h H ha, H h e.7e h = 9.67e.e =.77e 8.7e H h 7.6e 7 =.e = 6.e C = = C C [ ] h ha h Let K [ α β] e. (7) = e the oserver state gain matrix. Placing the pole of oserver at origin puts condition (8) on oserver closed loop characteristic polynomial. 89 P age
Vol. 6, No., zi + K = z (8) h e ha B. Pole assignment to susystem We assign one pole at and two poles at.8, an experimental optimal for fast response within actuator capacity. The characteristic polynomial ecomes z( z.8)( z.8). Let K e the state gain for pole assignment then from [] we have K = φ( ) M [ ] T where M is the controllaility matric of susystem. This expression results in state gain given y (). [.98.6. ] K = e e e () Stailized closed loop susystem is shown in Figure 6 with new reference input vk ( ) and signal e given y (). Fig.. Block diagram representation of susystem & susystem ek ( ) = vk ( ) + Kx () Solution of (8) is non-unique. Assigning α = we get β =.88. Value K = [.88] is used in oserver e design algorithm (9). { } h e ha + { K } y ha e haa { K }{ qk ( ) + Ky } qk ( + ) = H KH ek ( ) h e ha e (9) Oserved state vector is given y (). C h C ha x = T qk ( ) + y K () e The procedure for minimum order prediction oserver design is presented diagrammatically in Figure. Fig. 6. Susystem stailized y pole assignment and prediction oserver Step response of inner loop system containing oserver and pole assignment is shown in Figure 7. C. Inner loop system dynamics with stailized susystem Using (), () and () we get the dynamics of the overall system given y equation (). xk ( + ) = xk ( ) + Hvk ( ) y = Cx s s () System matrix in () with susystem stailized is given y (). = aa s s a HK () Fig.. Minimum order prediction oserver for susystem To implement rapid control prototyping we treat system in () as single input single output system with input vk ( ) and distance covered y all on eam y as an output and we 9 P age
Vol. 6, No., consider it as inner loop system given y transfer function in (). + + + + + ( z) = K IL IL z z + z + z + z + z + az az az az az a 6 ( ) () Values of various parameters of transfer function () are taulated in Tale. TABLE I. INNER LOOP SYSTEM PARAMETERS Parameter Value Parameter Value a. -.98 a.98 8.6 a -.87-7.68 a -.68.96 a.67 -.68 a 76 K 6 IL e-7 Test Command eneration + - z ς C = K OL OL z ρ v[k] Compensator Model Response Evaluation y[k] Fig. 8. Rapid Control Prototyping implementation strategy Tuned parameter values for K OL Hardware Under Test Data Out port Data In port C are given y (7). OL = 6. ς =.99 (6) (7) ρ =.967 Overall implementation of multi loop control law that is hyrid of pole assignment and rapid control prototyping has een explained diagrammatically in Figure 9. Partial pole assignment is implemented on digital controller in inner loop followed y rapid control prototyping strategy implemented in outer loop using real time data acquisition, processing and monitoring in MATLAB. Figure shows simulation of the hyrid multi-loop control algorithm. This simulation is used to otain simulated responses in section IV. Figure shows actual implementation of RCP strategy in Simulink. MATLAB/Simulink (Outer Loop, Rapid Control Prototyping) Compensator - + Reference Command r[k] Real Time rk [ ] Monitoring Fig. 7. Step responses for inner loop system v[k] v[k] DAQ Interface y[k] Hardware-Software Boundary y[k] D. Rapid Control prototyping for inner loop RCP implementation strategy is elucidated in Figure 8. Using hardware/software interface module i.e. NI DAQ, real plant is put into the software control loop with model compensator to e tuned. Responses of the system against various test commands are evaluated and controller parameters are adjusted accordingly until satisfactory performance is achieved. Compensator model that has een used is given y (6). + - u[k] state weighted sum x [k] e[k] state oserver Digital Controller (Inner Loop, Partial Pole Placement) y [k] Hardware (Actual Plant) Fig. 9. Block diagram of multi loop hyrid control law implementaton 9 P age
Vol. 6, No., IV. SIMULATION AND EXPERIMENTAL RESULTS The proposed hyrid multi loop control law is simulated and experimentally tested. Figure shows the step response of position of the all on eam. Actual response nearly follows simulation result. Response settles down in.sec. Fig.. Simulation of hyrid multi-loop control algorithm in Simulink Fig.. Actual Rapid Control Prototyping implementation in MATLAB/Simulink.. [k] Simulation [k] Actual..... Fig.. Unit step response of position of all on the eam Unit step response in Figure has zero steady state error. Figure shows unit step response of the eam angle. The supply limitations result in the lag in the actual response during fast transients, however the steady state response well follows the simulation response. Fig.. Unit step response of the angle of the eam Figure shows unit step response of servo arm angle. The lag in the actual response during fast transients is due to the supply limitations. The steady state response follows simulation response. 9 P age
Vol. 6, No., 6. [k] Simulation [k] Actual e [k] Simulation a e [k] Actual a.. - -... Fig.. Unit step response of angle of the servo arm Figure shows unit step response of PMDC motor current. Actual current waveform is limited within ±A power supply current ounds. Figure 6 shows unit step response of eam angular velocity. Actual angular velocity of the eam is ounded y ±A current limits of supply as shown in Figure. Figure 7 shows unit step response of motor input voltage. Actual input voltage waveform is ounded y ±V power supply limits for PMDC motor driver oard. Figure 8 shows the unit step response of the control algorithm signal vk ( ) from Figure 9. The supply limitations are not included in the simulations so that we may compare actual response with ideal conditions of the simulation and monitor ideal compensator roustness against practical limitations. -... Fig.. Unit step response of the current of PMDC motor i a [k] Simulation i a [k] Actual -6... Fig. 7. Unit step response of the voltage applied to the PMDC motor v[k] Simulation v[k] Actual Fig. 8. Unit step response of signal v[k] The Sinusoidal and Sawtooth reference tracking responses are shown in Figure 9 and Figure. Trapezoidal reference tracking response is shown in Figure. Actual response well follows the simulation responses with a constant steady state error for the ramp part of the reference input signal. Despite actual model has saturation limits for current and voltage yet responses well follow the simulation results. This tantamount to roustness of proposed technique against model inaccuracies..... Sinusoidal Reference Tracking / t Simulation / t Actual.. -. -. -. D[k] Actual D[k] Simulation input... Fig. 6. Unit step response of the angular velocity of the eam 6 8 Fig. 9. Sinusoidal reference tracking response of the position of the all 9 P age
Vol. 6, No.,.. -. -. Sawtooth Reference Tracking Fig.. Sawtooth reference tracking response for the position of the all.. -. -. Trapizoidal Reference Tracking Fig.. Trapizoidal reference tracking response for the position of the all V. CONCLUSIONS D[k] Actual D[k] Simulation input D[k] Actual D[k] Simulation input A novel compensator for the all on eam platform is presented. Full order dynamic model of system is roken down into two parts. One part is controlled y partial pole assignment. The resulting system is compensated y rapid control prototyping. Experimental results validate that this hyrid compensator design strategy has given full control on all system outputs with system order reduction and it has given excellent results, especially regarding reference tracking and roustness against model inaccuracies. REFERENCES [] A. S. Haider, M. Nasir, B. Safir, and F. Farooq, A Novel Ball on Beam Stailizing Platform with Inertial Sensors, International Journal of Advanced Computer Science and Applications, vol. 6, no. 8, pp. 6, July. [] W. Y. Chung, and C. C. Ju, An oserver-ased model reference adaptive iterative learning controller for nonlinear systems without state measurement, Proc. of the International Conference on Fuzzy Theory and Its Applications, Taipei, Taiwan, pp. 7 76, Dec. [] P. Rapp, O. Sawodny, and C. Tarin, An immersion and invariance ased speed and rotation angle oserver for the all and eam system, Proc. of the American Control Conference, Washington DC, pp. 69 7, June. [] R. M. Nagarale, and B. M. Patre, Decoupled neural fuzzy sliding mode control of nonlinear systems, Proc. of the IEEE International Conference on Fuzzy Systems, Hyderaad, India, pp. 8, July. [] D. Martinez, and F. Ruiz, Nonlinear model predictive control for a Ball&Beam, Proc. of the IEEE th Colomian Workshop on Circuits and Systems, Barranquilla, Colomia, pp., Nov. [6] C. Wenzhuo, S. Xiaomei, and X, Yonghui, Modeling and modulation of nonlinear all-eam system controller ased on matla, Proc. of the 9th International Conference on Fuzzy Systems and Knowledge Discovery, Shenyang, China, pp. 88 9, May. [7] D. N. Kouya, and F. A. Okou, A new adaptive state feedack controller for the all and eam system, Proc. of the th Canadian Conference on Electrical and Computer Engineering, Niagara Falls, Canada, pp. 7, May. [8] P. Z. Hua, Z. eng, and L. C. Xiang, Augmented state estimation and LQR control for a all and eam system, Proc. of the 6th IEEE Conference on Industrial Electronics and Applications, Beijing, China, pp. 8, June. [9] W. Wei, and X. Peng, A Research on Control Methods of Ball and Beam System Based on Adaptive Neural Network, Proc. of the International Conference on Computational and Information Sciences, Chengdu, China, pp. 7 7, Dec. [] P. Xu, L. Lei, and Z. Tan, Human simulated intelligent control for all and eam system, Proc. of the 9th Chinese Control Conference, Beijing, China, pp. 96, July. [] A. I. Carlos, The Lyapunov direct method for the stailisation of the all on the actuated eam, International Journal of Control, vol, no. 8, pp. 69 78, Oct 9. [] E. Lia, Z. Z. Lianga, Z.. Hou, and M. Tana, Energy-ased alance control approach to the all and eam system, International Journal of Control, vol. 6, no. 8, pp. 98 99, May 9. [] K. Ogata. Modern Control Engineering. rd ed., New Jersey: Prentice Hall, 997. 9 P age