PERFORMANCE EVALUATION OF RELIABLE H INFINITY OBSERVER CONTROLLER WITH ROBUST PID CONTROLLER DESIGNED FOR TRMS WITH SENSOR, ACTUATOR FAILURE

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1 Far East Journal of Electronics and Communications 16 Pushpa Publishing House, Allahabad, India Published Online: April 16 Volume 16, Number, 16, Pages ISSN: PERFORMANCE EVALUATION OF RELIABLE H INFINITY OBSERVER CONTROLLER WITH ROBUST PID CONTROLLER DESIGNED FOR TRMS WITH SENSOR, ACTUATOR FAILURE Manipal Institute of Technology Manipal University Manipal-57614, Karnataka India rao.vidya@manipal.edu Received: September 9, 15; Revised: November, 15; Accepted: January, 16 Keywords and phrases: reliable H infinity observer controller, robust proportional integral and derivative (PID) controller, state feedback control, twin rotor MIMO system (TRMS). Communicated by Yeong Jeu Sun Abstract The partial or complete failure of sensors and actuators used in twin rotor MIMO system (TRMS) is considered as uncertainty, which may lead to control system failure resulting in severe damages. Hence, there is a need to design a controller which performs identically with and without sensor or actuator failures. The observer based controller approach is used to isolate these failures. The H infinity observer designed estimates all the states of the TRMS even under sensor, actuator failure. This is successfully used with a state feedback H infinity controller to isolate sensor, actuator fault. The novel implementation of robust proportional integral and derivative controller with H infinity observer is also done for twin rotor MIMO system with the same sensor, actuator failure. Its performance is compared with reliable H infinity observer controller. The real time implementation results are presented.

2 356 Nomenclature TRMS Twin Rotor Multiple Input Multiple Output System x (1 1) state vector of TRMS A, B, C, D TRMS state model parameters y ( 1) output vector of TRMS Controlled input to TRMS w 1 (1 1) vector, worst case process noise (with unknown statistics) v 1 (1 1) vector, worst case measurement noise (with unknown statistics) Q (1 1) vector, covariance of process noise w1 R ( ) vector, covariance of measurement noise v 1 z xˆ (1 1) vector, estimated state of TRMS (1 1) vector, output of TRMS to be estimated and regulated P, X (1 1) vector, solution for the Riccati equation P L (1 1) vector, initial value chosen for P by the designer (1 1) vector, positive definite matrix chosen by the designer K 1 (1 ) vector, H J q observer gain Cost function to be minimized Performance bound F j u F u = Failed actuator output a aj = a Fraction of failure

3 Performance Evaluation of Reliable H Infinity Observer 357 u j F b j a oj = a Normal actuator output The disturbance Gain through which the disturbance enters to the system g H controller performance index aj a j Upper bound of fraction of actuator failure Lower bound of fraction of actuator failure G Real Matrix of dimension (1 1) K H controller gain Reference Pitch angle Reference Yaw angle Pitch output Yaw output 1. Introduction TRMS and helicopter system are similar in the sense that they have a set of nonlinear system of equations. In both cases, there is a significant cross coupling between main and tail rotor dynamics. Difference between TRMS and helicopter system is that, in helicopter system, the aerodynamic forces are controlled by changing the angle of attack of the rotor blades, but in TRMS, the same is controlled by varying the angular velocities of two rotors. The hovering property of TRMS/helicopter is the main area of interest in this work. Station keeping, or hovering in spite of uncertainties like sensor or/and actuator failures, is important for a variety of light missions such as load delivery and air-sea rescue. TRMS is thus a good benchmark problem to test and explore flight control methodologies. Perturbations on helicopter system which are likely to affect the control system are, disturbances like gusty air, sudden rush of air or water, noises like tones generated by gear box, interference of high frequency signals, uncertainties like change in payloads, equipment wear, changes in operating

4 358 conditions and sensors or/and actuator failures [1]. Similarly, uncertainties on TRMS are modeling errors occurred due to linearization or stochastic modeling, measurement errors, equipment wear, sensors or/and actuator failures []. This work presented in the paper addresses mostly on the worst case uncertainties like sensor or/and actuator failure. Since experimenting on complex air vehicles like helicopter is a difficult task, TRMS is considered as a benchmark due to similarity and simplicity. Because of these nonlinearities and cross coupling, obtaining exact model of TRMS is difficult and hence system identification approach is used which gives the approximate model of the system [3, 4]. The approximation in modeling is considered as one of the uncertainties. The uncertainties like sensor and actuator failure are assumed to be worst case noise with unknown statistics. Also, in TRMS only two states (pitch angle and yaw angle) are measured. Hence, there is a need for controlling TRMS with sensor or/and actuator failure with only two states measured. Therefore, rest of the TRMS states are estimated using an observer under sensor, actuator working fine as well as failed conditions. H observer is chosen as an observer because it works well in presence of worst case noise with unknown statistics with uncertain plant dynamics [3, 13, ]. As an initial step H observer performance is compared with the Kalman observer performance [6, 7]. The state feedback H controller is designed for TRMS which takes up the estimated states from H observer and adjusts the controller gain and further determines the control signal which makes TRMS stable even in presence of sensor or/and actuator failure [17, 18]. As in [8, 9], a dynamic model for the characterization of two-degree-of-freedom Twin Rotor MIMO System in hover is extracted using a black-box system identification technique which is presented in [3]. In [1], Kalman filter is designed as an observer and LQG controller is designed for a TRMS. Similarly in [11], the Kalman observer designed is used for sensor fault isolation. [1] uses Chebyshev neural network (CNN), which estimates unknown nonlinearities, whose weights are adaptively adjusted. Similar methodology is followed in this paper to rectify the sensor fault occurring in

5 Performance Evaluation of Reliable H Infinity Observer 359 TRMS using H observer [13]. In [14, 7], a Linear Quadratic Gaussian (LQG) compensator is used as a state feedback inner loop controller. In [15], the LQR controller has been designed using the Computed Torque technique. In [16], Fuzzy PID controller is designed by taking nonlinearities into consideration. In [14-16, 7, 8], the controller is designed without considering the sensor failure or actuator failure. The papers [17, 18, 3] give an algorithm for controller design for sensor or/and actuator failure. Using [17, 18], the H observer controller is designed with the identified model for Twin Rotor System which isolates sensor fault as well as actuator fault. In the present work, it is shown that the identified model is a good model which is of 1th order whose complexity is very less compared to the TRMS model given in [19] whose order is 8. The design of H controller using graphical approach is given in [1, ]. In [3], a set of random variables are introduced to describe the case of multiple probabilistic sensors or/and actuator faults as well as the measurement distortion. That is achieved by the static gain adjustment of the H work aims at the reliable H controller. However, the present controller design using estimated states whereas the states are estimated dynamically with an H observer. H control is realized by a method based on the algebraic Riccati equation for designing robust reliable H control laws for plants with structured uncertainty. The design method consists of incorporating information on the plant uncertainty into the Algebraic Riccati Equations (ARE) used for nominal H disturbance rejection designs which ensures that the closed loop control system is stable and guarantees H performance even in presence of uncertainties like modeling errors and sensor, actuator failure. PID controller is designed in [4, 5] for MIMO system. But when uncertainties like sensor failure or actuator failure occur the algorithm fails to give the required stability. As given in [6], without sensor or actuator failure the algorithm works for TRMS also. But under sensor actuator failure for TRMS the logic fails. In the present work, robust PID controller logic is used along with H observer.

6 36 In Section and Section 3, reliable H observer controller design and implementation on TRMS is presented under sensor, actuator failure. In Section 4, the design of robust PID with and without an H TRMS, under sensor, actuator failure is presented. Results of H observer for observer controller implementation are given in Section 5. Section 6 shows the results of robust PID controller for TRMS. Conclusion of the work is given in Section 7.. Design of Reliable H Observer Controller for TRMS with Sensor, Actuator Failure Realization of Kalman observer needs the exact knowledge of Matrices A and C and the process and measurement noise have to be white noises. If worst case noises like sensor, actuator failure impact on the system, Kalman observer would not give satisfactory result [7, 13, 9]. There is a need for an observer which estimates the states with minimum error with uncertain A and C along with worst case noise like sensor, actuator failure. H observer is chosen because it takes care of these things. Due to nonlinearities and cross coupling in TRMS obtaining exact model is difficult. The system identification method is used to get a stochastic model of the system on performing experimentation on TRMS. System identification toolbox constructs mathematical models for TRMS from measured input-output data which is explained in [3, 7]. The models used in the present work are Auto Regressive Moving Average Exogenous (ARMAX) models. The high order model obtained for TRMS is reduced to lower order model without compromising on its dynamics. This reduced order model is then converted into continuous which is of the order 1 shown in (1) to (4). The state space representation is shown in (5). Main Pitch (Pitch angle/voltage supplied to main rotor): * 1 s -.169s +.15s s s s (1)

7 Performance Evaluation of Reliable H Infinity Observer 361 Main Yaw (Yaw angle/voltage supplied to tail rotor): s s s s s s () Cross Pitch (Pitch angle/voltage supplied to tail rotor): s s s (3) Cross Yaw (Yaw angle/voltage supplied to main rotor): s s s s (4), û ù ë é = A, û ù ë é = B, û ù ë é - = D (5) û ù ë é - = C

8 36 Thus, the 1th order model is used to realize a full order H observer so that the predicted output ŷ is as close to actual output y as possible, in spite of measurement noises, sensor failure that corrupts the final output measurement. From this observer, all 1 states of TRMS are estimated [3, 7, ]. Using the game theory approach, dynamic real time H designed with the goal of finding accurate observer gain Ko1 observer is which minimizes the difference between the predicted output and the true output. The Ko1 is so adjusted that less emphasis is placed on noisy measurements and more emphasis on actual measurements. Considering TRMS as a continuous time linear system, with A as system matrix, B as the input matrix, C as the output matrix and D as the direct transmission matrix. Then, x & = Ax + Bu + w 1, (6) y = Cx + Du + v 1, (7) z = Lx, (8) where L is the user-defined matrix and z is the vector that we want to estimate. Estimate of z is denoted by ẑ. In this case, both z and x are same since we have taken L as identity matrix. Estimate of state at time is ˆx ( ). The vectors w1 and v1 are disturbances with unknown statistics, they need not even be zero mean. y is the system output and x is the state matrix and xˆ is estimated state by the H observer. A, B, C, D are the TRMS matrices taken from (5). The cost function used is shown in (9) [13], ò T z - zˆ dt J =, (9) ( ) ˆ T x - x( ) + ( w + v ) dt ò J is the measure of the performance of the state estimator. The nature s goal 1 1

9 Performance Evaluation of Reliable H Infinity Observer 363 is to impose worst case disturbances and noises ( w1 and v 1 ) on the system and maximize J. But the observer s aim is to derive an observer gain which makes J very small in spite of worst w1 and v1. This is achieved by defining cost function as in (9). The goal is to find an estimator such that 1 J <. (1) q The estimator that solves this problem is given in (11) to (15): where P, P = P, (11) P & T T = AP + PA + Q - Ko1CP + qpl LP, (1) T T -1 Ko 1 = PC ( CPC + R), (13) xˆ & = Axˆ + Bu + Ko1( y - Cxˆ ), (14) z ˆ = Lxˆ, (15) Q, R are positive definite matrices chosen by the designer. Q is the covariance of process noise w1, noise, v1. R is the covariance of measurement P is the solution for the differential Riccati equation. Initial values of x and xˆ are assumed to be zero. Initial matrix P represents the level of confidence for the initial parameter value q, where q is the performance bound [7, 13]. P and x( ) have been assumed as identity and zero, respectively [13]. Initially very small value for q is chosen. In this work, q =. 1 was chosen initially. Simultaneously J 1 condition ( s). T zw 1 q is checked along with the performance and for the The performance of the observer was not satisfactory for this q. So q was increased to.1, iteratively (6) to (15) are executed. Same conditions were checked. This was repeated several

10 364 times and the best q found to be was.1. For q =.1, the observer stopped estimating the states. Using equations (6) to (15), H observer is realized using SIMULINK and implemented for TRMS as shown in Figure 3.1, Figure 3.4, Figure 3.8 and Figure 3.9. For choosing the Q, a trial and error method is adopted. Initially diagonal elements of Q and R are chosen to be very close to zero. The responses of estimated states of TRMS are simulated by increasing diagonal element of Q and R individually. This increase in Q and R is stopped when all estimated TRMS states are exactly same as the true TRMS states. Thus, a dynamic real time estimation has been done using H observer in which the gain of the observer changes according to sensor, actuator working or faulty. After many trials Q and R are obtained as in (16) and (17) for the best estimation: Q = eye( 1, 1), (16) é9 ù R =, 9 (17) ë û P = eye( 1, 1), (18) L = eye( 1, 1). (19) All the states are estimated under normal condition, faulty sensor condition, faulty actuator condition or both. All the states are fed to H which is designed as a feedback controller. H state feedback controller which enables the H controller control means designing a norm of the closed loop transfer function T zw1 ( s) to be smaller than a prescribed positive bound g, that is to say the resulting control law can retain disturbance rejection. The development of the reliable H controllers assumes that the sensor failures can be detected and the observer dynamics could be accordingly adjusted [5, 6, 17, 18].

11 Performance Evaluation of Reliable H Infinity Observer 365 Let us consider a model of the actuator along with sensor failure represented as shown in (): F u j F = a aj u j + a oj b j. () Here F u j is the failed actuator output, aaj is the fraction of failure and u j is the normal actuator output. The disturbance F b j enters to the system through the gain a oj : a J < aaj < a j, (1) where aj and a j are upper and lower bounds of the actuator failure fraction. In this work, H aaj is taken as a, aoj is taken as a o. For designing observer based reliable controller the uncertain plant of the form () is considered: x & = Ax + F Bu + Gw 1 and y = Cx + F Du + v 1, () where x is the state of the system, y is the measured output. w1 and v 1 disturbances and F u are is the control input. Matrix G is real with appropriate dimension. w1 enters the system through G. We assume that TRMS system matrix A has structured uncertainty. Values for A, B, C, D are taken from equation (5). Let the controller be of the form as shown in (3): where xˆ is estimation of the state x using H u F = REF - Kˆx, (3) observer under normal or sensor failure condition obtained from (9) to (15). X is solution for Riccati equation as given in (4) which is a positive definite matrix. Then for the

12 366 controller to give reliable performance in the presence of actuator failure should satisfy the following condition ( A, C) is a detectable pair. X and P should satisfy the Riccati equations given in (4), (1) and (13): T A X + XA + 1 g T XGG X T -1 - XB[( N N ) - N N ] B X + C C =, T T T (4) where N = diag[ t1, t,..., tm], (5) N = diag[ t11, t1,..., t1m ], (6) t j = maxa a a j J 1 J {( a a ) } = a + a - 1, i = 1,, 3,..., j (7) 1 1 tij J j = = max{( 1 - a, 1 - a }, j 1,, 3,..., m, (8) K T = NNB X. (9) For any sensor, actuator failures the closed loop system is asymptotically stable and T zw ( s) g. (3) 1 [5, 17, 18, 1, 3]. For the implementation, the values for g =.1, = 1, a j =, a =.5, a. 5 (that is, for 5 percent actuator failure). In this = work, g is chosen to be 1 percent of performance bound q. K is the H controller gain which also satisfies (31): T a J ( a P - X ) K = a C, (31) where a >, X > and P - ³ a X. The flow of the reliable H controller algorithm for TRMS is shown in Figure.1. observer

13 Performance Evaluation of Reliable H Infinity Observer 367 Figure.1. Flowchart of reliable H TRMS. observer controller algorithm for 3. Reliable H Observer Controller Implementation for TRMS with Sensor, Actuator Failure In this part of the paper, the reliable observer controller designed in Section is implemented on real TRMS. It is tested and validated for its reliability when sensor or/and actuator has failed partially or completely. The Simulink block diagram for TRMS control under both sensor, actuator failure is shown in Figure 3.1 to Figure 3.1. The sensor is made to fail at time t = 4s and actuator is made to fail at time t = 5s. The actuator failure

14 368 is rectified by switching to redundant actuator (in this work the same actuator is considered as redundant actuator at time t = 7s). The switching is done using multiple switches, the results of which are shown in Figure 5.1 and Figure 5.. Different failure cases are given in [3]. The set point r_pitch and r_yaw for pitch reference and yaw reference inputs, respectively are given as step input with amplitude 1 rad. y_pitch and y_yaw, measured angles are taken from the feedback encoder which is the sensor. The control inputs u_pitch and u_yaw from the controller are fed to the DAC systems as shown in Figure 3.1. yhat_pitch and yhat_yaw are the estimated outputs from the observer. Different subsystems included in implementing reliable H observer and controller is shown in Figure 3. to Figure 3.1. Failure detector subsystem is as shown in Figure 3.1. In the failure detector subsystem measured and estimated pitch and yaw angles are compared. If difference between them is greater than.5 rad, for more than 5s, a trigger is given for alarm, warning that there is a failure in sensor or/and actuator. Figure 3.1. Implementation of reliable observer and controller under sensor failure at t = 4s actuator failure at t = 5s.

15 Performance Evaluation of Reliable H Infinity Observer 369 Figure 3..a. Fail trigger circuit for TRMS pitch sensor. Figure 3..b. Fail trigger circuit for TRMS yaw sensor. Figure 3.3.a. Fail trigger circuit for TRMS pitch actuator. Figure 3.3.b. Fail trigger circuit for TRMS yaw actuator.

16 37 Figure 3.4. Reliable H-infinity observer and controller. Figure 3.5. Subsystem to get u_pitch and u_yaw. Figure 3.6. Subsystem to calculate controller gain, K.

17 Performance Evaluation of Reliable H Infinity Observer 371 Figure 3.7. Subsystem to calculate reliable H estimated output(yhat). controller output(u) and Figure 3.8. Subsystem to calculate design P.

18 37 Figure 3.9. Subsystem to calculate estimated state ( ˆx ). Figure 3.1. Failure detector system. 4. Robust PID Controller Design for TRMS Robust PID controller is designed for TRMS using [5] whose block diagram is shown in Figure 4.1. With both sensor and actuator failure the robust PID controller is implemented on real TRMS. As in [5] the control signal u is generated by summation of state feedback control signal and conventional PID control signal. Uncertainties like sensor, actuator failure make the robust PID controller fail as shown in Figure 6.1 and Figure 6.. This is overcome by the use of an observer. In this work, the H observer is used along with the robust PID controller designed for TRMS which is shown in Figure 4.. The robust PID controller with H observer is implemented on Real TRMS. This technique makes the robust PID controller reliable even for the uncertainties like

19 Performance Evaluation of Reliable H Infinity Observer 373 sensor, actuator failure. The stable TRMS pitch, yaw outputs are obtained and the results of which are shown in Figure 6.3 and Figure 6.4. Figure 4.1. Block diagram of TRMS with robust PID controller (without estimated output of TRMS fed back). Figure 4.. Block diagram of TRMS with robust PID controller (with estimated output of TRMS fed back). 5. Results of Reliable H Observer Controller for TRMS 5.1. TRMS control under both sensor and actuator failure As shown in Figure 5.1 and Figure 5., the sensor has been failed at time t = 4s. Measured pitch and yaw angles are zero since measurement system fails. Even if the measurement system failed the output continue to

20 374 be present and the TRMS is controlled because of H observer estimates all states which are same as true states under no failure condition. At time t = 5s, the actuator is made to fail. Since there is no driving source (no redundant actuator to take up this control signal) the output reduces to zero. But there is control signal present. The failure detector output is as shown in Figure 5.3. At time t = 7s, the actuator is replaced with a working actuator (same actuator), bringing back the output to the reference level. The failure detector shows a continuous failure from t = 45s, since sensor failure has not been rectified. Figure 5.1. Pitch angle-under both sensor, actuator failure. Figure 5.. Yaw angle-under both sensor, actuator failure.

21 Performance Evaluation of Reliable H Infinity Observer 375 Figure 5.3. Failure detector output/sensor and actuator failure. 6. Results of Robust PID Controller for TRMS 6.1. TRMS robust PID control under sensor, actuator failure (without H observer) When sensor actuator failure occurs, using robust PID controller with actual output fed back (without H observer) the TRMS pitch and yaw angles could not be controlled. These are shown in Figure 6.1 and Figure 6.. Figure 6.1. Pitch angle-under both sensor, actuator failure (without H observer).

22 376 Figure 6.. Yaw angle-under both sensor, actuator failure (without H observer). 6.. TRMS robust PID control under sensor, actuator failure with estimated output fed back (with H observer) As shown in Figure 6.3 and Figure 6.4, the TRMS sensors are failed at 4s and actuators are failed at 6s. Since H observer estimates all the states of TRMS with minimum error [7], the robust PID controller will be able to control the pitch output and yaw output without allowing the sensor, actuator fault to affect the TRMS output. Comparing Figure 5.1 with Figure 6.3 and Figure 5. and Figure 6.4, we obtain the results as shown in Table 1. Figure 6.3. Pitch angle-under both sensor, actuator failure (with H observer).

23 Performance Evaluation of Reliable H Infinity Observer 377 Figure 6.4. Yaw angle-under both sensor, actuator failure (with H observer). Table 1. Comparison between H observer controller performance and robust PID controller performance under sensor actuator failure Performance measure Reliable H controller observer Robust PID controller (with H observer) Rise time Less (approximately 3s) More (approximately 8s) Settling time Less (approximately 1s) More (approximately 35s) Overshoot No More (approximately 5%) 7. Conclusion The reliable H observer controller is designed and implemented for TRMS with sensor, actuator failure. The real time results show that even if the sensor, actuator of the TRMS fail the TRMS remains stable with the H observer controller technique. The H observer controller performance is compared with robust PID control technique. With the use of actual TRMS output (without H observer) the robust PID controller fails to provide stability for TRMS under sensor or actuator failure. It is shown that the same robust PID controller can be used for obtaining stable result with the use of

24 378 estimated TRMS output (with H observer) even under sensor, actuator failure. However, the performance of Robust PID controller with H observer is found to be inferior to the performance of reliable H controller. observer Acknowledgement The authors thank the anonymous referees for their valuable suggestions which let to the improvement of the manuscript. References [1] Comtrawing five, 748 USS Eterprise St Suite 5, Milton, Fl , Helicopter aerodynamic workbook, September. [] Twin Rotor MIMO System Manual, Feedback Instruments Ltd., U.K, S,. [3] Vidya S. Rao, Milind Mukerji, V. I. George, Surekha Kamath and C. Shreesha, System identification and observer design for TRMS, International Journal of Computer and Electrical Engineering 5(6) (13), [4] S. M. Ahmed, A. J. Chipperfield, M. O. Tokhi, Rahideh and M. H. Shaheed, Dynamic modeling of a twin-rotor multiple input-multiple output system, Proc. Instn. Mech. Engrs., Vol. 16, Part I: J. Systems and Control Engineering, (), [5] Bruce A. Francis, A Course in H Control Theory, Springer-Verlag, New York, [6] Doyle J. Grover, K. Kargonekar and B. Francis, State space solutions to standard H and H controlled problems, IEEE Transactions on Automatic Control 34(8) (1989), [7] Vidya S. Rao, V. I. George and Surekha Kamath, Comparison of Kalman observer and H infinity observer designed for TRMS, International Journal of Control and Automation, SERSC publishers, Accepted for publishing. [8] S. M. Ahmed, A. J. Chipperfield and Tokhi, Dynamic modelling and control of a DOF TRMS, IEEE Transactions on Automatic Control (),

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