Conference on Control and Tolerant Sytem Nice, France, October 68, WeA4. MRAC + H Tolerant Control for Linear Parameter Varying Sytem Adriana VargaMartínez, Vicenç Puig, Lui E. GarzaCatañón and Ruben MoraleMenendez Abtract Two different cheme for Tolerant Control (FTC) baed on Adaptive Control, Robut Control and Linear Parameter Varying (LPV) ytem are propoed. Thee cheme include a Model Reference Adaptive Controller for an LPV ytem (MRACLPV) and a Model Reference Adaptive Controller with a H Gain Scheduling Controller for an LPV ytem (MRACH GSLPV). In order to compare the performance of thee cheme, a CoupledTank ytem wa ued a tetbed in which two different type of fault (abrupt and gradual) with different magnitude and different operating point were imulated. Reult howed that the ue of a Robut Controller in combination with an Adaptive Controller for an LPV ytem improve the FTC cheme becaue thi controller wa Tolerant againt enor fault and had an accommodation threhold for actuator fault magnitude from to 6. G I. INTRODUCTION lobal market have increaed the demand for more and better product, which require higher level of plant availability and ytem reliability. Thi iue ha promoted that engineer and cientit give more attention to the deign of method and ytem that can handle certain type of fault (i.e. Tolerant Sytem). On the other hand, global crii create more competition between indutrie and production loe and lack of preence in the market are not an option. In addition, modern ytem and challenging operating condition increae the poibility of ytem failure which can caue lo of human live and equipment. In thee environment the ue of automation and intelligent ytem i fundamental to minimize the impact of fault. Therefore, Tolerant Control method have been propoed, in which the mot important benefit i that the plant continue operating in pite of a fault; thi trategy prevent that a fault develop into a more eriou failure. Although everal application have ued LPV ytem theory to develop FTC cheme ([], [], []) and alo MRACbaed approache for FTC have been explored ([4], [5], [6], [7], [8], [9]), none of them integrate the three methodologie propoed in thi paper: MRAC, LPV and H. The main intention of thi work i to develop a paive tructure of FTC able to deal with abrupt and gradual fault in actuator and enor of nonlinear procee repreented by LPV model. An MRAC controller wa choen a a FTC becaue guarantee aymptotic output tracking, it ha a direct phyical interpretation and it i eay to implement. The H Gain Scheduling Controller wa alo choen becaue it increae the robut performance and tability of the cloe loop ytem. Two different approache for FTC baed on Adaptive, Robut and LPV control are propoed. Firt, a Model Reference Adaptive Controller for an LPV ytem (MRAC LPV) i conidered and econd a combination of a MRAC with a H Gain Scheduling controller for an LPV ytem (MRACH GSLPV) i alo propoed. Reult howed that MRACH GSLPV ha a better performance than the MRACLPV approach, becaue wa Tolerant againt enor fault and had an accommodation threhold for actuator fault magnitude from to 6.. II. BACKGROUND A. LPV Control Theory The Linear Parameter Varying (LPV) ytem depend on a et of variant parameter over time. Thee ytem can be repreented in tate pace (continuou or dicrete). The principal characteritic of thi type of ytem i the matrix repreentation function of one or more variable parameter over time. The continuou repreentation of an LPV ytem i: x =A φ(t) x+b φ(t) u () y=c φ(t) x+d φ(t) u () where x R n repreent the tate pace vector, y R m i the meaurement or output vector, u R p i the input vector, φ repreent the parameter variation over time and A(.), B(.), C(.) and D(.) are the continuou function of φ. An LPV ytem can be obtained through different methodologie; if the phyical repreentation of the nonlinear ytem i obtained, the Jacobian Linearization method, the State Tranformation Method and the Subtitution Function method can be ued to obtain the LPV ytem. The main objective of thee methodologie i to occult the nonlinearity of the ytem in any variable in order to get the LPV ytem. On the other hand, if the experimental data model i obtained, the LPV ytem can be created uing the Leat Square Etimation for different operating point of the ytem [], []. 97844485//$6. IEEE 94
B. Model Reference Adaptive Control (MRAC) The MRAC, hown in Figure, implement a cloe loop controller where the adaptation mechanim adjut the controller parameter to match the proce output with the reference model output. The reference model i pecified a the ideal model behavior that the ytem i expected to follow. Thi type of controller behave a a cloe loop controller becaue the actuating error ignal (difference between the input and the feedback ignal) i fed to the controller in order to reduce the error to achieve the deired output value. The controller error i calculated a follow: e = yy m () where y i the proce output and y m i the reference output. To reduce the error, a cot function wa ued in the form of: J θ =/ e θ (4) where θ i the adaptive parameter inide the controller. The function above can be minimized if the parameter θ change in the negative direction of the gradient J, thi i called the gradient decent method and i repreented by: dθ dt = γ J θ = γ e θ e (5) where γ i the peed of learning. The implemented MRAC ued in thi experiment i a econd order ytem and ha two adaptation parameter: adaptive feed forward gain (θ ) and adaptive feedback gain(θ ). Thee parameter will be updated to follow the reference model. e θ = a r+a r u +a r +a c dθ r dt e θ =γ e θ e=γ a r+a r +a r +a r u c e (6) = a r+a r y dθ = +a r +a r dt γ e e = γ a r+a r y e (7) θ +a r +a r Reference Model u c u y + e Controller Proce Controller Parameter Adaptation Mechanim Fig.. Model Reference Adaptive Controller (MRAC) general cheme []. III. PROPOSED SCHEMES Two different FTC cheme were developed in thi work: a MRACLPV cheme and a MRACH GSLPV cheme. To tet thee approache, a econd order coupled twotank ytem wa choen. Thi coupledtank ytem i compoed by two cylindrical tank (ee Figure ): an upper and a lower tank (tank and tank ). A pump i ued to tranport water from the water reervoir to tank. Then, the outlet flow of tank flow to tank. Finally, the outlet flow of tank y m end in the water reervoir [4]. The water level of the tank are meaured uing preure enor located at the bottom of each tank. The differential dynamic model of thi ytem i []: h t = a A g h (t)+ k p A u(t) (8) h t = a A g h (t) a A g h (t) (9) y t =h (t) () In Table, the variable definition involve in the above ytem are explained. Table I Variable Definition Variable Definition Value h water level of tank h water level of tank A croection area of tank 5.579 cm A croection area of tank 5.579 cm a croection area of the outflow orifice of.78 cm tank a croection area of the outflow orifice of.78 cm tank U pump voltage k p pump gain. cm / V G gravitational 98 cm/ contant α 4 contant.98 x 7 α contant.659 x 5 α contant.7 x α contant 4.6 x α contant.58 An LPV model of the above ytem i computed by a polynomial fitting technique that approximate h i for h i cm with φ i h i, where [4]: φ i =α 4 h i 4 +α h i +α h i +α h i +α () The parameter φ and φ are bounded with the following value: The LPV end in: where:.=φ φ φ =.6 ().=φ φ φ =.6 () x =A φ x+bu (4) y=cx (5) x= h h (6) 95
A. MRACLPV Controller y= y y (7) A φ =.585φ.585φ.585φ (8) B=.7 (9) C= () D= () A Model Reference Adaptive Controller of the LPV ytem wa deigned (MRACLPV). Firt, the tatepace LPV model wa tranformed to a continuou verion: G LPV ()=C IA B+D () dθ = γ e e = γ.585 φ + φ +.5857 φ φ dt θ +.585 φ + φ +.5857 φ φ e (8) The adaptive feedback update rule θ i: dθ = γ e e = γ.585 φ + φ +.5857 φ φ e (9) dt θ +.585 φ + φ +.5857 φ φ Figure repreent the MRACLPV cheme, in thi figure the Reference Model, the Proce Model, the feed forward update rule (bottom left) and the feedback update rule (bottom right) are repreented a LPV ytem. The feed forward and the feedback update rule change in order to follow the reference model. + u c +.585.5857.585.5857 Reference Model.858.585.5857 y reference Proce Model u.858 y proce + e.585.5857.585.5857.585.5857 Fig.. MRACLPV Controller Structure. Pump Fig.. Coupledtank ytem deigned by [5]. G LPV ()=.585 φ.585 φ.585 φ G LPV ()=.858 φ +.585 φ +.585 φ.858 φ.7 () (4) G LPV ()= (5) +.585 φ + φ +.5857 φ φ The reference model i: Tank.858 φ Reference Model= (6) +.585 φ + φ +.5857 φ φ Thi model i the ame a the proce model when with no fault..858 φ Proce Model= (7) +.585 φ + φ +.5857 φ φ The adaptive feed forward update rule θ i: h h Water Reervoir Tank B. MRACH GSLPV Controller In order to deign the H Gain Scheduling LPV Controller for the MRACH GSLPV Controller (Figure 4), two weighting function were etablihed (W mi and W ai ). To obtain W mi and W ai the next procedure wa realized: Firt, 4 plant were calculated uing the extreme operation point (φ =., φ =.; φ =., φ =.6; φ =.6, φ =.; φ =.6, φ =.6) and a nominal plant (φ =.5, φ =.5) were obtained uing an average of the operation point. Then, the multiplicative uncertainty (W mi ) and additive uncertainty (W ai ) were calculated for each plant a follow: W mi = Plant inominal Plant Nominal Plant () W ai =Plant inominal Plant () The next tep i to plot a Bode diagram of the above uncertaintie and find a weighting function that include all the individual plant uncertaintie. With the Bode diagram, the multiplicative and additive uncertaintie function that include all the plant were computed: W mt =.75 4 +. +..88 4 +.56887 +.9878 +.9.89 4 +.49 +.757.8 () W at = () 6 +.947 5 +.6 4 +.57 +.984 +9.56e 5 After calculating W mt and W at the following procedure wa realized:. The value of the learning rate γ and the pecific deired operation point were etablihed a φ and φ. 96
. W mt and W at have to be tranformed into a Linear Time Invariant (LTI) ytem.. The parameter range ha to be pecified in order to obtain the variation range of value of a timevarying parameter or uncertain vector. In thi experiment there are dependent parameter, thi mean that the range of value of thee parameter form a multidimenional box..=φ φ φ =.6 (4).=φ φ φ =.6 (5) 4. The tate pace LPV model i tranformed into an LTI ytem and then the parameter dependent ytem i pecified. 5. The loop haping tructure of the LPV ytem i pecified. 6. The augmented plant i formed. 7. The H Gain Scheduling Controller wa calculated with the hinfg Matlab function. Thi function calculate an H gain cheduled control for parameter dependent ytem with an affine dependence on the time varying parameter. The parameter are aumed to be meaured in real time. To calculate the controller the function implement the quadratic H performance approach. 8. The deired operating point are pecified in order to return the convex decompoition of the parameter et of box corner. 9. The evaluation of the deired operating point in the polytopic repreentation of the gaincheduled controller i realized. From here, the tate pace matrice are extracted and then tranformed into a continuou time pace. + u c +.585.5857.585.5857 H inf Gain Scheduling Controller num den controller controller Fig. 4. MRAC H Gain Scheduling LPV Controller arquitecture. The input of both controller mut be peritently exciting in order to converge to the deired output value. IV. RESULTS Reference Model y reference Proce Model u.858 y proce + e.585.5857 Two different type of fault were imulated in the implemented cheme: abrupt and gradual fault. Abrupt fault in actuator repreent for intance a pump tuck and in enor a contant bia in meaurement. A gradual fault could be a progreive lo of electrical power in pump, and a drift in the meaurement for enor. For each of the two propoed cheme: MRACLPV.858.585.5857.585.5857.585.5857 Controller and MRAC H Gain Scheduling LPV Controller (MRAC H GSLPV) both fault were teted obtaining the reult hown in Table II. Thee reult explain the range of fault ize in which the methodologie are robut, fault tolerant or untable againt the fault. The next Table and Figure how the implementation of the fault in the above methodologie. Thee operation point were elected to demontrate the capabilitie of both controller, but any operation point between the range of φ and φ can be choen. TABLE II RESULTS OF EXPERIMENTS OF THE MRACLPV AND THE MRACH GSLPV APPROACHES Approach Senor Abrupt Gradual Abrupt Gradual MRAC LPV MRAC H LPV < f < f > +/ < f <+/ f >+/ < f < 6 f > 6 FT FT < f < 6 f > 6 +/ < f <+/ 6 f >+/ 6 +/ < f <+/ 6 f >+/ 6 f= % deviation from nom. value, f= % deviation, and o on FT = Tolerant, U = Untable. In Table II the accommodation ( Tolerant) and the untable range for the MRACLPV and the MRACH LPV are hown. For example for abrupt enor fault the MRACLPV ha a Tolerant threhold for fault from magnitude to. On the other hand, the MRACH LPV wa Tolerant for all magnitude of thi pecific fault type. Figure 5 how for abrupt fault cae, the bet cheme i the MRACH LPV becaue i robut againt enor fault of magnitude ( % deviation from nominal value) and i fault tolerant to actuator fault of magnitude 6 (6 % deviation from nominal value), for the actuator fault the real deviation from the nominal ytem performed by the controller wa of % from the nominal value at the time of the fault. On the other hand, the MRACLPV reulted to be fault tolerant for enor and abrupt fault, for example for enor fault the real deviation from the nominal value wa 8% and for the actuator fault the real deviation wa of 9% from the nominal value. Both controller are working in the operating point φ =. and φ =.5, the abruptenor fault wa introduced at time 5, and an abruptactuator fault wa introduced at time 5,. In addition, a change in the operating point wa performed at time,. Figure 6 how for abrupt fault cae, the bet cheme i the MRACH LPV becaue i fault tolerant againt enor 97
fault of magnitude and i fault tolerant to actuator fault of magnitude 6. The above mean that for the enor fault the ytem ha a real deviation of.8% and for the actuator fault the ytem ha a real deviation of % from the nominal value. On the other hand, the MRACLPV reulted to be untable for enor fault of magnitude and fault tolerant for abrupt fault of magnitude 6 becaue the real deviation from the nominal ytem wa more than %. It i important to mention that the fault wa accommodated after, becaue the MRAC controller continue to minimize the error over the time; thi i one of the advantage of thi controller. In thi example both controller are working in the operating point φ =. and φ =.5, the abruptenor wa introduced at time 5, and the abruptactuator fault wa introduced at time 5,. In addition a change in the operating point wa performed at time,. 5 4.5 4.5 MRACLPV operating point φ =.6 and φ =.6, the gradualenor fault wa introduced in time 5, and the gradualactuator fault wa introduced at time 5,. In addition a change in the operating point wa performed at time,...4.6.8..4.6.8.5.5 MRACLPV Senor Time (econd) MRACH GSLPV x 4.5.5.5 Senor Fig. 5..5..4.6.8..4.6.8.5.5.5 Senor Time (econd) MRACH GSLPV Robut againt Senor Comparion between MRACH LPV and MRACLPV Controller with an abruptenor fault of magnitude and an abruptactuator fault of magnitude 6, the operating point are φ =. and φ =.5. Figure 7 preent for gradual fault cae, the bet cheme i the MRACH LPV becaue i robut againt enor fault of magnitude (maximum deviation of % from nominal value with a %/ec ramp) and i fault tolerant to actuator fault of magnitude 6 (maximum deviation of 6% from nominal value with a %/ec ramp) change); the real ytem deviation at the time of the fault for the actuator time wa of 4.5% and wa accommodated immediately. On the other hand, the MRACLPV reulted to be fault tolerant for enor and actuator fault of magnitude and 6, repectively; in which the real deviation from the nominal value for the enor fault wa of 77% and for the actuator fault wa of 8%. Both controller are working in the x 4..4.6.8..4.6.8 Time (econd) x 4..4.6.8..4.6.8 Time (econd) Fig. 6. Comparion between MRACH LPV and MRACLPV Controller with an abruptenor fault of magnitude and an abruptactuator fault of magnitude 6, the operating point are φ =. and φ =.5. 4.5 4.5.5.5.5 Fig. 7. Comparion between MRACH LPV and MRACLPV Controller with a gradualenor fault of magnitude and a gradualactuator fault of magnitude 6, the operating point are φ =.6 and φ =.6. x 4..4.6.8..4.6.8.5.5.5 MRACLPV Senor Time (econd) MRACH GSLPV Robut againt Senor x 4..4.6.8..4.6.8 Time (econd) x 4 98
Figure 8 decribe that for gradual fault, the MRACH LPV cheme i fault tolerant againt enor fault of magnitude and it i fault tolerant to actuator fault of magnitude 6. The above reulted in a real deviation from the nominal value of % and 4.5% for enor and actuator fault, repectively. Alo, the MRACLPV reulted to be fault tolerant to enor and actuator fault of magnitude and 6, repectively; with a real deviation of 8% for enor fault and of 9% for actuator fault from the nominal value. Even though, both cheme are fault tolerant againt enor and actuator fault, the bet cheme i the MRACH LPV becaue the deviation of the proce model from the reference model of thi cheme i maller than the deviation of the MRACLPV cheme. Both controller are working in the operating point φ =.6 and φ =.6, the gradualenor fault wa introduced at time 5, and the gradualactuator fault wa introduce at time 5,. In addition a change in the operating point wa performed at time,. 4.5 4.5.5.5.5..4.6.8..4.6.8.5.5.5 MRACLPV Senor Time (econd) MRACH GSLPV Senor Fig. 8. Comparion between MRACH LPV and MRACLPV Controller with a gradualenor fault of magnitude and a gradualactuator fault of magnitude 6, the operating point are φ =.6 and φ =.6. V. CONCLUSIONS In the experiment, the MRACH GSLPV methodology behaved better than the MRACLPV cheme becaue wa fault tolerant againt enor fault of any magnitude (f= and f=). The MRACH GSLPV howed better reult becaue i a combination of two type of controller, one i a Model Reference Adaptive Controller (MRAC) and the other one i a H Gain Scheduling Controller, both controller were deigned for an LPV ytem giving them x 4..4.6.8..4.6.8 Time (econd) x 4 the poibility of controlling any deired operating condition between the operation range of the dependent variable (φ and φ ). On the other hand, the MRACLPV methodology reulted to be fault tolerant for enor fault magnitude between and and it wa fault tolerant for actuator fault magnitude between and 6 (the MRACH GSLPV approach had the ame fault tolerant threhold for actuator fault). ACKNOWLEDGMENT A. VargaMartínez, L. E. GarzaCatañón and R. MoraleMenendez thank to the Tecnológico de Monterrey Campu Monterrey, the Superviion and Advanced Control Reearch Chair and to the Univeridad Politécnica de Cataluña (UPC) for their upport during thi reearch. REFERENCES [] M. Rodrigue, D. Theilliol, S. Aberkane, and D. Sauter. Tolerant Control Deign for Polytopic LPV Sytem, Int. J. Appl. Math. Comput. Sci., 7(), pp. 77, 7. [] J. Boche, A. El Hajjaji, and A. Rabhi. tolerant control for vehicle dynamic, 7 th IFAC Sympoium on Detection, Superviion and Safety of Technical Procee, Barcelona, Spain, June 9, pp. 8. [] M. Luzar, M. Witczak, V. Puig, and F. Nejjari. Development of a Tolerant Control with MATLAB and It Application to the TwinRotor Sytem, 7 th Workhop on Advanced Control and Diagnoi, 9, pp. 8. [4] A. Abdullah, and M. Zribi. Model Reference Control of LPV Sytem, J of the Franklin Intitute, vol. 46, pp. 85487, April 9. [5] Y. Cho, K. Kim and Z. Bien. Tolerant Control uing a Redundant Adaptive Controller, 9 th Conf on Deciion and Control, Honolulu, Hawaii, December 99, pp. 467478. [6] M. Ahmed. Neural Net baed MRAC for a Cla of Nonlinear Plant, J of Neural Network, (), pp. 4, January. [7] K. Thanapalan, S. Vere, E. Roger, and S. Gabriel. Tolerant Controller Deign to Enure Operational Safety in Satellite Formation Flying, 45 th IEEE Conference on Deciion & Control, San Diego, California, December 6, pp. 56567. [8] W. Yu. Hinfinity Trackingbaed adaptive fuzzyneural control for MIMO uncertain robotic ytem with time delay, J of Fuzzy Set and Sytem, vol. 46, pp. 754, 4. [9] Y. Miyaato. Model Reference Adaptive Control of Polytopic LPV Sytem An Alternative Approach to Adaptive Control, Proceeding of the IEEE Int Symp on Intelligent Control, Munich, Germany, October 6, pp. 7. [] B. Bamieh, and L. Giarré. LPV Model: Identification for Gain Scheduling Control, European Control Conf, Porto, Italy,. [] A. Marco, and G. Bala. Development of LinearParameterVarying Model for Aircraft, J of Guidance, Control and Dynamic, vol. 7, no., March 4. [] J. Nagrath. Control Sytem Engineering, rd Ed., Anhan Ltd, 6, pp. 75. [] H. Pan, H. Wong, V. Kapila, and M. Queiroz. Experimental validation of a nonlinear back tepping liquid level controller for a tate coupled two tank ytem, J of Control Engineering Practice, vol., pp. 74, 5. [4] G. Forythe, M. Malcolm, and C. Moler. Computer Method for Mathematical Computation, Prentice Hall, USA, 977. [5] J. Apkarian. Coupled Water Tank Experiment Manual, Quaner Conulting Inc., Canada, 999. 99