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TO AVOID FINES return on or before date due. . “ DATE DUE DATE DUE DATE DUE 553754 L________ ‘L________. e 4 PET—’7 L.“ MSU Is An Affirmative ActiorVEqual Opportunity Institution CW ””1 _ .___________________._——-—-——_. ' MODELING A RADIO CONTROL SCALE HELICOPTER . FOR ROBUST CONTROL DESIGN By Chang-p0 Chao A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Mechanical Engineering 1993 ABSTRACT MODELING A RADIO CONTROL SCALE HELICOPTER FOR ROBUST CONTROL DESIGN By Chang-p0 Chao Helicopter flight control serves as an interesting and significant benchmark control design problem. In this study we look at the modeling of a scale helicopter and how it can be related to control design. Three nonlinear models of a radio control (RIC) scale helicopter, designed and fabricated in the Dynamic Systems and Control laboratory, are derived. Their only difference is the inflow velocity distribution assumed to exist over the rotor disk. After theses nonlinear models are determined we use the framework of uncertain linear system to represent them in a manner useful for robust control design. Our goal is to establish the effect of parameter variations in the nonlinear models on the uncertain linear system representations and their associated robust control designs. We use parametric and nonparametric uncertain linear system representations and the established theory of robust control. The 2-norm of the tracking error is our performance measure. Each controller is evaluated by considering the worst case tracking response of all possible compensated nonlinear models to step inputs of three different amplitudes (10°,20°,30°) . The merit of each modeling effort and control design is considered and recommendations are made regarding the approach that offers the greatest promise for application. Copyright by CHANG—PO CHAO 1993 DEDICATION To my parents and girlfriend who have given me the greatest love and support ACKNOWLEDGMENTS Over the past one and one half years a number of present and former professors and graduate students have contributed either directly or indirectly to my research. In particular, I would like to thank my major advisor Dr. Philip FitzSimons, who made me aware of standard procedures for system identification and convinced me of the advantages of the Internal Model Control structure. Without substantial advice and support, I could not have completed this project. I also owe special thanks to Dr. Clark Radcliffe who provided me timely help about my open-loop response measurement and Mr. Jerry Palazorro who designed the original helicopter hovering device a year ago. Finally, I would like to thank Xian Li Huang, Sachin Gogate, Parag Wakankar and Jonathan Iwamasa who have often assisted me in the Control Laboratory. iv TABLE OF CONTENTS LIST OF TABLES ................................................................................ viii LIST OF FIGURES ............................................................................... ix NOMENCLATURE ................................................................................ xi 1 INTRODUCTION 1.1 Modeling the Physical System ......................................................... 1 1.2 Modeling for Control System ......................................................... 2 1.3 Robust Control Design ................................................................. 2 2 MODELING THE PHYSICAL SYSTEM 2.1 Parallelogram Linkage Dynamics ..................................................... 4 2.2 Drive Train Dynamics .................................................................. 5 2.3 Rotor Thrust and Torque .............................................................. 6 3 PARAMETER IDENTIFICATION 3.1 Documented and Experimental Information Available .............................. 9 3.2 Nominal Linkage Parameters and How They were Determined ................... 9 3.3 Nominal Rotor Parameters and How They were Determined .................... 10 3.4 Nominal Motor/Drive Train Parameters and How They were Determined ..... 12 3.5 Nominal Nonlinear State Variable Models .......................................... 13 4 LINEARIZATION AND EXPERIMENTAL MODEL VALIDATION 4.1 Linearization ........................................................................... 15 4.2 Model Validation ....................................................................... l6 5 MODELING FOR CONTROL DESIGN 5.1 Linear State Variable Model .......................................................... 19 5 .2 Effect of Physical model parameter variations on linear model parameter variation ................................................................................. 19 5.3 Effect of Physical model nonlinearity on linear model parameter variation ..... 20 5.4 Effect of Physical model parameter measurements on linear model parameter V variations using Conic Sector Bound method ...................................... 21 6 CONTROL DESIGN AND EVALUATION ................................................ 23 6.1 Parametric Uncertainty ........................................................... 24 6.2 Nonparametric Uncertainty ...................................................... 26 6.3 Comprehensive Uncertainty ..................................................... 27 6.4 Nominal Plants Used for Robust Control Design ............................ 29 7 EVALUATION ................................................................................. 31 7.1 How is the Worst-Case Determined ........................................... 31 7.2 Worst-case responses for different robust controllers and the associated models ............................................................................ 32 8 CONCLUSION ................................................................................. 37 REFERENCE ...................................................................................... 38 APPENDIX A.l Mathematical Development A.1.l DC Motor Dynamics .............................................................. 39 A.1.2 Application of Momentum Theory for Rotor .................................. 40 A.1.3 Application of Blade Element Theory for Rotor ............................... 42 A.1.4 Linearization ....................................................................... 44 A.2 Experimental Data A.2.1 Experimental Measurements of the DC Motor Armature Resistance ....... 45 A.2.2 Relationship between the Thrust and Speed of Rotor ......................... 46 A23 Relationship among the Thrust, Torque, Speed of the Rotor, Current and Voltage ............................................................................ 47 A24 Relationship between output voltage of Hall Effect sensor and the angular displacement of the parallel linkages ........................................... 48 A3 Nominal System Parameters ......................................................... 49 A.4 Parameter Uncertainties ................................................................ 51 vi A.5 Simulation Results ............................................................................ 53 LIST OF TABLES Numberm Bag: 3.3.1 3.4.1 5.3.1 7.2.1 7.2.2 A.2.1 A.2.2 A.2.3 A.2.4 Optimal rotor parameters Optimal nominal parameters derived from power balance The Conic Sector Bounds for three different models Worst-case 2-norrns for the cases with IMC filter which has 2. = 0.1 Worst-case 2-norms for the cases with IMC filter which has 2. = 0.2 Measurements of the motor armature resistance, R, Measurements for the relationship between the thrust and rotor speed Measurements for the relationship among the thrust, torque, rotor, current and voltage Measurements for the relationship between Hall Effect sensor and the angular displacement of the parallelogram linkage 10 12 22 32 33 45 47 48 LIST OF FIGURES 1311mm: 1.1 2.1.1 2.3.1 3.2 3.3.1 3.3.2 3.4.1 4.2.1 4.2.2 4.2.3 6.0.1 6.1.1 6.1.2 6.1.3 6.2.1 6.2.2 6.3.1 6.3.2 6.4 7.2.1 7.2.2 7.2.3 7.2.4 7.2.5 A. l .1 Schematic diagram of the scale helicopter Nomenclature of the parallelogram linkage Velocity distributions Hall Effect sensor curve fitting The thrust with the associated optimized parameters The load torques with the associated optimized parameters Power balance with three different velocity distribution Schematic diagram of the experimental setup Experimental and theoretical Frequency Response Coherence of the Frequency Response The IMC structure Fuzzy Nyquist plot for cubic model Local Plot of p(ia))cc(ito) for cubic model Upper bounds of the sensitivities for parametric representation Robust stability consideration Upper bounds of the sensitivity functions Upper bounds of sensitivity fuctions for comprehensive representation Robust consideration for "Comprehensive nonparametric representation" Nominal plants Worse-case response for 300 input with A = 0.1t Worse-case response for 30° input with A = 0.2 Control effort for input 300 with 4 = 0-2 Rotor speed for input 30° with 4 = 0-1 Rotor speed for input 30° with A = 0.2 Simplified model of a DC motor ix 35 36 36 39 A.1.2 A.1.3 Control Volume used in Momentum Theory 2—D Airfoil Terms for a Rotor Blade Section 40 42 IMC ISE NOMENCLATURE Internal Model Control Integral Squared Error Airfoil lift curve s10pe Coefficient of Coulomb friction for the rotor Coefficient of viscous fi'iction for the rotor Coefficient of viscous friction for the parallelogram linkage Lift coefficient Drag coefficient Airfoil chord length Classic feedback controller Disturbance Upstream cross-sectional differential area of the stream tube Cross-sectional differential area of the stream tube at the rotor disk Downstream cross-sectional differential area of the stream tube Drag force Differential drag force Differential lift force Error signal IMC filter Feedback signal Offset of the acting line of the thrust xi Gravity Current of the DC motor armature circuit Total moment of inertia for the drive train, which referred to rotor speed Inertia of the parallel linkage Slope of the output equation for Hall Effect sensor Motor Back EMF constant Motor Torque constant Lift force Inductance of the DC motor armature circuit Length from O to C.G. of the parallelogram linkage Length of rotor-side arm of parallel linkages Length from the joint of the parallelogram linkage to center of mass of the counterweight Bound on the multiplicative uncertainty Slope of the output equation for Hall Effect sensor Mass of the parallelogram linkage and gear assembly Mass of the counterweight Atmospheric pressure Pressure on upper side of the rotor plane Pressure on lower side of the rotor plane Plant model Nominal model IMC controller lst step IMC controller for H2 - optimal control w(s) y(t).y(S) y’(S) yrs Millimeter: a At 8(3) Radius of the rotor DC Motor Armature resistance Radial position from the center of the rotor Physical input Normalized input Thrust generated by the rotation of the rotor Constant velocity Motor armature voltage Motor back EMF Output voltage of Hall Effect sensor Velocity of the rotor disk relative to the ambient air Velocity of the air flow through the rotor plane Velocity of the air flow through the rotor plane Wind velocity Weighting function Calculated nominal system output Actual process output Steady state Angle of attack of airfoil Sampling time interval Sensitivity Inflow angle xiii ‘Q Q Time constant of IMC filter Angular displacement of the parallelogram linkage Airfoil Pitch angle Air density Torque Motor frictional torque Motor output torque Load torque Rotor speed Blade Element Theory Cubic velocity distribution Linear velocity distribution Momentum Theory Nominal value Uniform velocity distribution xiv 1. INTRODUCTION 1.1 Modeling the Physical System ........ ...................................... .................................... .................................................................. ................................... ' ' 'u. \: ' :- -:. ................................ n‘t '.'-'-':'-' . e‘l ................................... ................... Output Signal Parallelogram linkage = -------------------- Gear box ..... ..... ..... ..... ..... '''''''''' ..... ... ....... ........ ''''''''' ........ r if Hall Effect sensor be motor Couterweight Base structure Input Signal Figure 1.1 Schematic Diagram of the Scale Heloopter A single-degree-of-freedom scale helicopter was designed and built in the Dynamic Systems and Control Laboratory to investigate the modeling and control of a rotor driven system. A schematic diagram of this device is shown in Figure 1.1. The mechanical part of the system has four components: a rotor system (DC motor, drive train and rotor), a parallelogram linkage, a base structure, and a counterweight. The rotor system is mounted on the parallelogram linkage, which is attached to the base structure, and is balanced by a counterweight that can be moved to adjust the equilibrium thrust level. The input signal corresponds to a voltage applied to the permanent magnet DC motor which powers the rotor through the drive train. The pitch of the individual rotor blades is fixed and the thrust of the rotor system is modulated by varying the rotor speed. The angle of the parallelogram linkage relative to the base is the controlled variable. A Hall Effect sensor provides the output signal which is used for control purposes. A nonlinear, lumped parameter model was used to describe the dynamics of the device. The electrical dynamics of the DC motor were neglected since they are much faster than the 2 mechanical dynamics of the system. The thrust produced and torque required by the rotor were obtained by combining Blade Element Theory and Momentum Theory (as in [2] and [3]). We assumed quasi-static aerodynamics and considered three different inflow velocity distributions over the rotor disk: uniform, linear, and cubic. The parameters required for each model were identified using the method of least squares to fit the theoretical results to static thrust measurements. Additional measurements and documented data were used to determine the nominal values and expected variations of the other model parameters. 1.2 Modeling for Control Design Once the three nominal nonlinear models were determined, we used the framework of uncertain linear systems (ULS) to represent the dynamics of the device in a manner applicable to robust control design. Both parametric and nonparametric ULS representations were considered. We found the parametric ULS representations first and then used disk shaped uncertainty regions in the Nyquist plane to form the nonparametric representations. The parametric uncertainties in the ULS representations were the result of unknown system parameters and system nonlinearities. We used the hard bound method discussed in [5] and the conic sector bound method discussed in [6] to find bounds for these parameters. We considered four parametric uncertainty descriptions and four nonparametric uncertainty descriptions. Three of the parametric ULS models follow directly from the uncertain nonlinear models derived and have a similar algebraic structure. They differ only in the ranges of the parameters which result from the nonlinear terms associated with the different assumed inflow distributions. The fourth description we chose to include all three of the other descriptions. After finding the four parametric ULS descriptions we then found their associated nonparametric representations. 1.3 Robust Control Design and Evaluation The robust control design procedure we used consisted of two steps. In the first step, an H2 Optimal control was designed to minimize the 2-norrn of the tracking error for the 3 nominal linear plant model assuming a step input was to be tracked. Once the nominal controller was designed it was augmented by a low-pass filter which was adjusted to achieve robust stability and performance. The main goal of this study was to determine how variations in the nonlinear model parameters affect the uncertain linear system models and the associated robust control designs. We considered eight different plant models, hence eight different controllers. To evaluate each controller we assumed that the “truth” model was described by one of the three nonlinear model structures considered and had parameters belonging to a prescribed allowable set. The performance of each controller was then judged by determining from among all of the possible “truth” models the tracking error with the largest 2-norm that resulted from a step up command held for ten seconds followed by a step down command held for ten seconds. In order to get a feel for the nonlinear behavior of the compensated systems three step amplitudes were used (10°,20°,30°). 2. MODELING THE PHYSICAL SYSTEM 2.1 Parallelogram Linkage Dynamics f.» Qua-run.-. 4;th "tr Figure 2.1.1 Nomenclature of the Parallelogram Linkage Using the nomenclature of Figure 2.1.1 and elementary dynamics we find the equation of motion for the parallelogram linkage is given by Jpé+bwé=(-Mg+TLP)cose+Tf, (2.1.1) 1,, = JP + £52,125 (2.1.2) M: LCGm-Lsms (2.1.3) where b,. P = Coefficient of viscous friction for the parallelogram linkage f, = Offset of the acting line of the thrust J p = Moment of inertia about 0 J p, = Moment of inertia of the parallelogram linkage + rotor system L5 = Length from the joint of the parallelogram linkage, O, to the center of mass of the counterweight LCG = Length from O to CC. of the parallelogram linkage L = Length fiom the joint of the parallelogram linkage, O, to the shaft of the DC motor 5 m = Mass of the parallelogram linkage and rotor assembly m, = Mass of the counterweight T = Thrust generated by the rotor 0 = Angular displacement of the parallelogram linkage 2.2 Drive Train Dynamics A permanent magnet DC motor is used to drive the rotor through a single stage drive train. The motor output torque balances the load torque which has three components. These three components include the torque required to accelerate the drive train/rotor system, the torque required to drive the rotor blades through the air, rm", and the torque required to overcome friction in the drive train, gem-m. The relationship among these torques is given by (see Appendix A.1.1) V — K a) . K1[_a";e—v_] = er + 7"rotor + Tfriction (2-2-1) 0 KILL-151:9): J,a') + rm, + bosign(co) + b,w (2.2.2) where bo = Coulomb friction coefficient D1 = Viscous friction coefficient J, = Total Moment of Inertia of the Drive Train + Rotor (referred to the rotor speed) K v = Back EMF Constant (referred to the rotor speed) K 1 = Torque Constant (referred to the rotor speed) Ra = Armature Resistance Va = Supply Voltage (0 = Rotor Speed and in the second expression we assume Tfrt'crion results from Coulomb and viscous friction terms. 2.3 Rotor Thrust and Torque To predict the thrust and torque generated by the rotor we combine Blade Element Theory and Momentum Theory [2], [3]. The basic assumptions required are that the air is incompressible, the flow is quasi-static and the inflow velocity distribution over the rotor disk, v1 = v1(r), is represented by a one parameter family of surfaces. Three commonly assumed inflow velocity distributions are uniform, linear, and cubic v, (r) = V“ (2.3.1) v1(r) = V, x r (2.3.2) v, (r) = Vc x r2 x (R — r) (2.3.3) where r = Radial distance from the rotor’s hub R = Radius of the rotor V“, V,,Vc = Parameters of the inflow velocity distributions Once the inflow velocity distribution is assumed we find two differential expressions for thrust. One results from Momentum Theory (see Appendix A.1.2) and the other from Blade Element Theory (see Appendix A.1.3). Integrating each of these expressions and then setting them equal to each other makes it possible to solve for the inflow velocity parameter. This, in turn, makes it possible to determine the thrust produced and the torque required by the rotor as a function of the rotor parameters, rotor speed, and velocity of the rotor relative to the air mass. The expressions that result are given below. (1) Rotor Model Assuming Uniform Inflow Distribution 3 2 Tm = pac[m20p%- tog-V.) (2.3.4) Tm = zztpltz(vu2 — Vuvo) (2.3.5) 7 2 Zacwze R V_ = (1°. _ __wac)+ 1J(_acw — v0) + ———L- (2.3.6) 2 87: 2 41r 3n 3 rm, .. = pat-[roan L— V“2 52-) (2.3.7) ’ 3 2 (2) Rotor Model Assuming Linear Inflow Distribution R3 Tm = pac(w’0p - am)? (2.3.8) 4 3 T, a, = 47rp(5—V,2 -R—voV,) (2.3.9) ' 4 3 2 4 20 V, =(fl-fl)+lJ(fl_fl) +£9.34 (2.3.10) 6R 67tR 2 37rR 3R 31tR 2 R“ TNWJ = pac(mOPV, - V, )7 (2.3. I I) (3) Rotor Model Assuming Cubic Inflow Distribution R3 V R5 r = c020 —- c 2.3.12 cube: paC( p 3 20 ] ( ) R8 R5 T =42: ——V’-— V 2.3.13 can: p(168 e 20 V0 e] ( ) 2 56 (0’6 V. =(21v: _ 210cm)+ _1_\/(21acc;) _ 421,) + ac 5 P (2.3.14) 5R 203R 2 lOrtR 5R 7tR R6 R' t = a wOV——V2— 2.3.15 rotor.c p C( p c 30 c 168) ( ) Note that the subscripts u,c,l refer to the uniform, cubic and linear velocity distributions, respectively. The subscript bet indicates the expression derived from Blade Element Theory and mt indicates that it derived from Momentum Theory. N tout» Velocity (misec) i2 boon-one .3 ooooooooooooo innocence-coco; ...... Cubicf ooooooooooo l E E 1.25 6.05 6.1 6.15 b. r (m) = Radial position from the center of rotor D Figure 2.3.1 Velocity distributions Three different velocity distributions are shown in Figure 2.3.1 that generating the same magnitude thrust (0.3 LB = 1.33 N). Among the three velocity distributions shown it appears that the cubic one is most appealing on physical grounds. However, the other two distributions are often used for historical reasons or due to their simplicity. 3 PARAMETER IDENTIFICATION 3.1 Documented and Experimental Information Available The moror armature resistance, R, , was measured at twenty-three different rotor rotative positions (Appendix A.2.1). The nominal value was determined to be the average of the above twenty measurements. The torque conStant. K ,, which has the same numerical value as the back EMF constant, K,, in 81 was obtained by referring to the catalog provided by the DC mator company (and factoring in the gear ratio since we use the rotor speed as our reference). To investigate the relationships among the thrusr, rotor speed, motor armature current and voltage, we performed a static thrust test (Appendix A.2.2 and A.2.3). The thrust was measured using a model DFG 50 force gauge [8] from the Chantillon Company. The rotor speed was measured using a B&K model strobescope type 4913. The armature current and the applied voltage were determined using two model Fluke 77 digital multimetcrs. 3.2 Nominal Linkage Parameters and How They were Determined The relationship between the output voltage of the Hall Effect Sensor and the angular displacement of the parallelogram linkage was also investigated by experiment (see Appendix A.2.4). 1 ‘5 A0.“““““'5; ......... ........ "*"Measurements =3 ' ' ' " - " Linear curve fitting 3, 0 ................................................................................ 0 i .3? ;o'-05.......... ......................................................................... '1-40 930 £20 #10 '0 lo 20 30 4o 0 , Angular displacement of the parallelogram linkage (degree) Figure 3.2 Hall Effect sensor curve fitting 10 The linear curve fitting in Figure 3.2 shows that over a certain range of the magnetic rotation, the relationship between angular position and the output voltage is linear and the nominal voltage output is nearly zero. This linear relationship can be approximated by V” =KH9 (3.2.1) where K H = 0.0235 (Volts/degree) and V” is the output voltage of the sensor. 3.3 Nominal Rotor Parameters and How They were Determined Among the rotor and aerodynamic parameters, the chord length, c, and the radius of the rotor, R , were measured directly. The variation of air density p in the laboratory is small enough to be ignored. Because the pitch angle 6, may be different from its static value when the rotor is rotating and the exact value of the characteristic lift curve slope a is hard to determine, the theoretical thrust can not predict very well if we use these parameter value. To solve this problem, the least square method is utilized to calculate the optimal rotor parameters 9, and a by fitting theoretical to experimental thrust (see Appendix A.2.2). The results are shown in Table 3.3.1. Table 3.3.1 Optimal rotor parameters m Uniform model Linear model Cubic model a (no unit) 6.05 5.99 5.72 0, (degree) 13.1 12.8 13.0 Figure 3.3.1 shows that the thrust predicted using the associated optimized parameters listed in Table 3.3.1 are almost the identical curves. It makes us ensured that all three thrust prediction can well predict the thrust well with the rotor speed from 0 RPM to 1400 RPM. 11 2.‘ 1. '0' cubic model 'x' linear model ”a '-' uniform model g 1- '*' experimental measurement 0 v . - . a ................................................................................. -i O. ........................................ 0 200 400 T500 800 {000 1200 1400 Rotor speed (RPM) Figure 3.3.1 The thmst with the associated optimized parameters 0.03‘ v 0.0 . .............................................................................. - ’E‘ 0.02 ....... t O. CllbiC model ............................................ i \z/ 00 'x' linearmodel 3 ; 5 , a . . - un1form model / {-9 001 ........... ............ g ............ g ...... - ..... = ........... _ : : : 1/ - 00 ........... .....' ....................... _ 5 3 / : O.m .......... ............ ....... 1 ............................................. .4 ‘0 “200 400 600 800 1000 1200 1400 Rotor speed (RPM) Figure 3.3.2 The load torques with the associated optimized parameters In Figure 3.3.2 the load torques predicted by three different set of optimal parameters receptively shown to be different especially in high rotor speed range. 12 3.4 Nominal Motor/Drive Train Parameters and How They were Determined Among motor/drive train parameters, the coefficient of Coulomb friction for the DC motor, b0, was obtained by measuring the largest current that could be applied to the motor without initiating rotor rotation. The coefficient of viscous friction, b1, was identified by the power balance iv, = tWtu + ifR, + rpmcu (3.4. 1) We assume the frictional terms result from the sum of the Coulomb's and viscous friction, then the above equation become iOV, = rmww + ifR, + bolwl + blur)2 (3.4.2) In (3.4.2), since 0, 9p used to predict rm, are set to fit the experimental results using the measurements of the thrust, we can assume 1,0,0, is well predicted and the nominal values of V“, in, R, and b0 can be measured, the only unknown in the power balance (3.4.2) is the coefficient of the viscous friction, bl . Therefore by fitting the output power to the input power and using least square method again, three nominal optimal bl are obtained in Table 3.4.1 for three different velocity distributions. Table 3.4.1 Optimal nominal parameters derived from power balance bl (Nm sec/rad) Uniform Distribution 3.41x10-4 Linear Distribution 3.29x10-4 Cubic Distribution 3.85x10-4 Table 3.4.1 reveals that the values of bl for uniform and linear distribution are smaller than the cubic one because the load torque predicted by uniform and linear distribution in Figure 3.3.2 is larger than the other's. 13 35 . 2 g 30 r *I ouput power, Trotorw+lazka +bdd 4'wa ................... E 25 , _, input power, i‘V‘ ................... 3 20. ........... 3 . O 154-... ........... Q Q. I ' ............ 104 ........ . i......;n...“..............E .......................... 51 .......... ............................ a, I “H... ................................ 0 + ‘ ' — ' . I I 0 200 400 600 800 1000 1200 1400 1600 5'5 E e. H E n. E ' .. ' : E E E 0 "' 800 1000 1200 1400 0 200 400 000 Rotor speed (RPM) Figure 3.4.1 Power balance with three different velocity distribution Figure 3.4.1 shows that the output power and the input power can well balanced for the three different velocity distributions with their associated optimal parameters b1 '3 and the fittings shown up are almost identical for three different velocity distributions. 3.5 Nominal Nonlinear State Variable Model With the equation of motion for the parallelogram linkage, the equation describing the drive train dynamics, the expressions used to predict the thrust and torque and the parameters determined in this section which are measurable nominal values or optimized l4 ones, the overall nonlinear system can be described by the following two nonlinear differential equations and one output equation. Jpé+b,_,é= (-Mg+Tx L,)cos0+f, X?“ (2.1.1) 0’): K W -bosign(a))-b1w- r / (2.2.1) . t Ra rotor Jr VH = 19,0 (3.2.1) In addition to some dimensions and weights which can be measured directly, the other parameters should have uncertainties associated with them such as (1) Motor armature resistance R. (2) Coulomb's friction b0 (3) Coefficient of the viscous friction b1 (4) Torque and Back EMF constant, K , and K , (5) Slope of the hall Effect output equation (3.2.1), K H (6) Lift curve slope, a (7) Pitch angle, 6,, In state variable form, we may combine the nominal nonlinear state variable model (2.1.1), (2.2.1) and (3.2.1) to be written as nonlinear state equations it = f(x,u) (3.5.1) y= Kflxl 0 x = = 9 where 12 (0 4 LINEARIZATION AND EXPERIMENTAL MODEL VALIDATION 4.1 Linearization In the laboratory, for the case the counterweight is set to be 2 lb (0.91 kg), the operating range for the angular displacement of the parallelogram linkage is from -40 degree to 50 degree and the input armature voltage V, is about from 6.2 volts to 7.2 volts to prevent the bottom of the gear box from hitting the ground. By setting the right hand side of the nonlinear state equations (3.5.1) to zeros and solving it, we find the nominal supply voltage and the nominal motor speed V“, = 6.40 (Volts), 000 = 853 (RPM), 0, = 0 (rad / sec) (4.1.1) needed to keep the parallelogram linkage at the equivalent position 90. This set of nominal values are especially for the case of the cubic velocity distribution model. Using these nominal values, the nonlinear state equations (3.5.4) can be linearized as (see Appendix A.1.5) x=Ax+Bu 412 y=Cx (°') '0 1 0 where A = a2, a22 a23 , .0 an an '0 B= 0 -ba C=[KH 0 O] For the case that the equivalent 00 is 5°, the linearized A, B, and C are 15 16 0 1 0 A = —0.0094 —0. 3979 0.05320 0 0.5763 -3. 6531 B=[0 0 49.6161]7 C=[1.4324 0 0] MA) = {-3. 6672 -1. 4795 —0.0064] We can see that the system have three non zero stable eigenvalues including one very slow eigenvalue which is close to imaginary axis. 4.2 Model Validation A experimental set-up shown in Figure 4.2.1 was used to investigate open-loop response for model validation. HP35660A Signal Analyzer SO 2 output cured) Hall Effect input sensor Nominal Voltage Supply, HP623SAJ HP 6825A Amplifer ‘ Figure 4.2.1 Schematic diagram of the experimental setup ......... .................. .............................................. ............................ .......................................................... .............................. ...................... ................................... + The HP 35660A signal analyzer [7] was used to investigate the frequency response. This analyzer can provide a source signal for excitation as input signal of the analyzer itself. In this experiment set-up this signal was added to the nominal voltage which is supplied by the HP 6235A power supply and is estimated off-line by the preceding experiments, which measured the armature DC motor voltage V“, required for keeping the parallelogram 17 linkage at the equilibrium position. HP 6825A amplifier provide the power to drive DC motor. The output signal of the Hall Effect sensor is quasi-proportional to the angular displacement of the parallelogram linkages, and this signal is connected to output channel of the signal analyzer. By setting the coefficient of the viscous friction for the shaft of parallelogram linkage as bkp = 0.33 (N sec/ rad) (4.2.1) and using periodic chirp of amplitude 0.4 volts for excitation, the experimental and the theoretical results are matched to each other well in Figure 4.2.2. In this Figure it shows when the frequency of the excitation goes beyond 0.8 Hz, the amplitude of the response is nearly zero and the noise become dominant, then the measurements are not reliable. By the plotting of coherence shown in Figure 4.2.3, this coherence is very low when the frequency goes below about 0.06 Hz. That means at this frequency region, the system nonlinearity become dominant. It is why the theoretical result can not predict the behavior very well at the very low frequency region. a 5 g ; 3 g a: Theoretical response .g . . . . . ..s ........... ............ ..... b: EXPCI'imcnlal response 0 ' ' - 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 '33: 3 1:" n. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Frequency(Hz) Figure 4.2.2 Experimental and theoretical Frequency Response 18 Coherence 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Frequency(Hz) Figure 4.2.3 Coherence oi the Frequency Response 5 MODELING FOR CONTROL DESIGN 5.1 Linear State Variable Model In the linear state model (4.1.2), several parameter variations involved in this model. Generally, they can be divided into two groups. In the first group, the uncertainties arise from system parameter variation. These parameters are (1) Motor armature resistance R, (2) Coefficient of the viscous fiiction b1 (3) Torque and Back EMF constant, K , and K , (4) Slope of the hall Effect output equation (3.5.3), K H Compared to those involved in nonlinear state variable model, Coulomb friction is not included here because this terms will eliminated after linearization. The second group is from nonlinearity and variation associated with the thrust and torque which have the nonlinear characteristics and parametric variation arising from (1) Lift curve 510pe, a (2) Pitch angle, 9, 5.2 Effect of Physical model parameter variations on linear model parameter variations The uncertainty for R, can be obtained by applying the statistic Hard Bound method [5] on the static measurements (see appendix A.2.4) by assuming uniform distribution over the uncertainty interval, thus Rafi = 2.25 (ohm), Rd E [2.10.2.40] (5.2.1) where R“.0 is the nominal DC motor armature resistance and subscript "0" denotes the nominal value. 19 20 The uncertainties of K ,, K,, b,, a and 6,, are approximated by 5% of their nominal values (see Appendix A3) 5.3 Effect of Physical model nonlinearity on linear model parameter variations In addition to the above three parametric uncertainties, in fact we neglect the nonlinearity due to linearization. In order to include the nonlinearity variation in the linear model, here we introduce " Conic Sector Bound " method [6]. The Conic Sector Bound Method Given The nonlinear function f (55) that is Lipschitz, where i" is n x1 vector Find if and 5 such that there exists a vector I; that can represent this nonlinear function as f (5?) = I; 3 within the compact domain Dl , (V i" e 01,), and this It has the bound where I? , g , andk are all 1x n vectors. The strategy to solve this problem is that first we grid the compact domain D, and take the finite number (relatively large enough) of sample points in D,. If the nonlinear function is known to be Lipschitz [9], then for every single sample point by minimizing the 2-norrn of the vector IE, we can find the most conservative IE to predict the nonlinear function. Secondly, comparing these all values of 15 resulting from every single point in , we choose the maximum of all upper bounds as a new upper bound and the minimum of all lower bounds as a new lower bound. In order to apply Conic Sector Bound method, the equations used to predict the thrust produced and the torque required to drive the rotor are assumed to have the following forms for deriving Conic Sector Bounds. 21 T(é,a)) — T(éo,wo) = (Ti,o +kn)0' + (T3 +k,,)(c0 - 000) = aué + (1,2(0) — coo) . . . , (5.3.1) t(9.a))— «00.00): (1",o +k.,,)6+(rm° + ana) - r00) = a2,0+ 0122(0) - coo) o _ fl 0 = d7. 0 = d7 0 = dt where To - déLm To, dw WM 70 d 9 ”M 1:0, do) Mm Because the above four nominal derivatives can be calculated. The remaining unknowns in (5.3.1) are krrvkrz 0r aura” for Tm, (5.3.2) [€21,192 or 0:21.05, for Torque By minimizing the weighted 2-norm of kaT (5.3.3) ’ 0 where D = [:1 d2] k =[krrrk12] 0"[kzrrk22] and using the form of (5.3.1), we can find Conic Sector Bounds for the nonlinear function. Note that the weighting matrix can be determined arbitrarily in order to obtain physically reasonable Conic Sector Bounds. Within the reasonable compact domain D = {(0,w):|é| s 2,600 RPM 5 a) $1400 RPM} (5.3.3) ,we consider the nonlinearity of the thrust and torque and the parametric uncertainties arising form a and 6p, Conic Sector Bounds are found by using the particular weighting that make the system stable. These Bounds are shown in Table 5.3.1. 5.4 Effect of Physical model parameter measurements on linear model parameter variations using Conic Sector Bound method In addition to nonlinearity, we also use this Conic Sector Bound method to accommodate the measurement uncertainty in linear variable model. 22 Table 5.3.1 The Conic Sector Bounds for three different models weighting an an % Eu 3.7 a 6; Cubic ' 30 -1.71x10-1 -9.14x10'2 1.25x10'2 2.47x10-2 Linear 30 -1.87x10-1 -9.90x10-2 1.13x10’2 2.65x102 Uniform 30 -2.10x10'1 -l.34x10'1 1.01x10-2 2.73x10-2 weightinL “zit “22 d _ _ 2 (11 9A “21 9212. “2: Cubic 20 -2.05x10-3 4.43x10-4 1.25x10-4 3.05x10-4 Linear 20 -1.47x10-3 1.42x10-3 1.95x104 3.72x10‘4 Uniform 20 -1.83x10-3 1.25x10-3 1.77x10-4 3.58x10-4 For the uncertainty associated with K H , it can be calculated by treating the finite number of experimental measurements as the nonlinear function values and applying the above Conic Sector Bound method. Thus this uncertainty can be found as v” = K” 0, K” e[0.0200,0.0250] . (5.4.1) 6 CONTROL DESIGN The dynamics of the parallelogram linkage is heavily influenced by the parameter variations. One method introduced by Morari and Zafiriou [7] uses the IMC controller to minimize the effect of model uncertainty. This IMC structure is shown in Figru'e 6.0.1. Figure 6.0.1 The IMC stniclure Here p denotes the plant and p' the nominal models. r;, r’ and w are physical input, normalized input and weighting function respectively. q is IMC controller. By comparison between the classic feedback controller structure and the IMC structure shown in Figure 6.0.1, the performance achievable with IMC is identical to that achievable with the classic feedback controller 6, shown selected such that c = 4 (6.0.1) The IMC controller design contain two steps. In the first step the controller 6 is selected for good response without regard for constraint and model uncertainty; that is 4' = 1; (6.0.2) P Note that our model inverse is an acceptable solution because the linear model (4.1.2) is minimum-phase plant. In the second step for satisfying the robustness requirement the controller 4" is augmented by a low-pass filter f q = (if (6.0.3) 24 Real Figure 6.1.1 Fuzzy Nyquist plot for cubic model to provide the roll-off necessary for robustness and milder action of the manipulated variable. If it is designed for asymptotically constant inputs (step input), then 1 e ..4 f (rr+0’ ‘60 ) This filter can make q proper in order to be causal. 6.1 Parametric Uncertainty Parametric uncertainty representation describes the model by using the linear state variable model (4.1.2) and letting all uncertain parameters vary in their associated uncertain interval including Conic Sector Bounds. The Fuzzy Nyquist in Figure 6.1.1 shows all these various linear models which use cubic velocity distribution asSumption. Using the nominal linear model (4.1.2) as nominal plant to design the IMC controller q, we plot the local Nyquist band of p(ia))cc(iw) near to (-1,0) with 2. from 0.05 to 10in Figure 6.1.2, which will never encircle or cover (-1,0) so that the closed-loop system is guaranteed to be stable. By plotting Nyquist bands of p(ia))cc(iw) for the other two models, the results also shows p(iw)cc(iw) does not cover (-1,0). The sensitivity function e is defined as 2 0 39-2 __E,-4 .5 -8 ~10 1'6 . 2222'222 222.22 1.4 ~ . 3.22.3 1.2 . " .. ' :5? 1' fingers = 2 IE 0'3 ” .zu' 0.4 . - 355.. y 111.123: :.::::::: ; . ,; ; ; ; Solrd- Cubrc 0.2 -' ~j— Dashed-Linear Dotted-Uniform 101 102 103 w(rad/sec) Figure 6.1.3 Upper bounds of the sensitivities for parametric representation 1 (1+ch’ In [7] the robust performance requires that the distance from all possible p(i 61))cc (i (o) to E: (6.1.1) the point (-1,0), i.e., |1+ pet] has to exceed the specified maximum weighting function w. By using the information revealed in the Fuzzy Nyquist plot of Figure 6.1.2 or the other p10t of p(ia))cc(iw) for the other two models, we find out the upper bounds of the sensitivity functions shown in Figure 6.1.3 for three different distribution models. It 0.7 . I. i.::::‘: ... :TOA ""E'j':z-szzs-5-...5,,5_I: .3: Solid - Cubic Dashed-Linear LI _ Dotted - Uniform ——o.a 0.2 0.1 w(rad/sec) Figure 6.2.1 Robust stability consideration appears that the bounds derived from three different models are almost identical to each other. The upper bounds of the weighting functions can be obtained by the inverses of the preceding sensitivity bounds. 6.2 Nonparametric Uncertainty In nonparametric representation, the family 17 shown in Nyquist domain, which represent all possible plants p , can be enclosed by a disk defined by H = {p: lp(iw) - “MM $11400} (6.2.1) 1500’) where 7,, is referred to as an multiplicative uncertainty and 15(1'60) is the nominal plant or the model defining the centers of all the disk shaped regions, which is used to design IMC controller q . In [7], it suggests that for nonparametric representation, if the values of UL] never exceed one over all frequency domain, the closed-loop system is guaranteed to be robustly stable. Figure 6.2.1 shows that the values of If LI never exceeds one over all physical reasonable frequency domain so that the closed-loop system is robustly stable. 27 Solid - Cubic Dashed- Linear 2 Dotted- Uniform $1.5 >- : IE 1 5 2‘005 m 0.5 vii-=5 1m E}—2 01 ()lm . . .... .. . ..i..ir(in+.....im Figure 6.2.2 Upper bounds of the sensitivity functions The robust performance criterion also can be given by |f1',,|+|(1- f)w| <1 (6.2.2) The above equation can be used to derive the upper bounds of weighting functions and sensitivity functions which are shown in Figure 6.2.2. 6.3 Comprehensive Uncertainty For the case of the comprehensive uncertainty representation we consider that the model must be able to represent the dynamics that could arise from any of the three linear variable models (differing by inflow distributions). Two controllers result from comprehensive uncertainty. The first one, called "Comprehensive parametric representation," uses the nominal plant that has associated nominal parameters with uncertainties which can represent all uncertainties arising from any of three models. The second one called, "Comprehensive nonparametric representation," uses nominal plant that is the centers of uncertainty disks in Nyquist domain, which encircle the uncertainty arising from any of three models. 28 Solid - Parametric mixed Dashed - Nonparametric mixed Sensitivity 9 u. w(rad/sec) Figure 6.3.2 Robust consideration for "Comprehensive nonparametric representation" Using the same previous criterions, for the case of "Comprehensive parametric representation," p(ia))cc(ico) still not encircle (-l,0) and the sensitivities are shown in Figure 6.3.1 to be smaller than " r ‘ I- , “EL" ones. For the case of "Comprehensive nonparametric representation," the values of I f I‘MI shown in Figure 6.3.2 does not exceed one so that the closed-loop system with this controller still be robustly stable. 29 6.4 Nominal Plants Used for Robust Control Design Eight nominal plants used for robust control design are plotted in Figure 6.4. These nominal plants are derived based on the uncertainties of (a) Parametric representation and Cubic model. The nominal plant is the linear nominal planet of the cubic model. (b) Parametric representation and Linear model. The nominal plant is the linear nominal plant of the linear model. (c) Parametric representation and Uniform model. The nominal plant is the linear nominal plant of the uniform model. (d) Comprehensive parametric representation. The nominal plant is the average of the above three nominal plants (e) Nonparametric representation and Cubic model. The nominal plant is the centers of uncertainty disk in Nyquist domain for the cubic model (f) Nonparametric representation and Linear model. The nominal plant is the centers of uncertainty disk in Nyquist domain for the linear model (3) Nonparametric representation and uniform model. The nominal plant is the centers of uncertainty disk in Nyquist domain for the uniform model (h) Comprehensive nonparametric representation. The nominal plant is the centers of uncertainty disk in Nyquist domain. This uncertainty can arise from any of the above three uncertainty linear models 3O 0 . Solid - Cubic E _10 ............ Dashed-Linear ..................... . Dotted - Uniform a ' I -20 ------------ - Dashdotted - Comprehensive --------- 5.1... 7 4 --------- 1°90 fl. I T . ' I «3 ~30 5 Figure 6.4 Nominal plants 7 EVALUATION We used the nonlinear state variable model (3.5.1) instead of linearized model and unit output feedback structure for simulation. The IMC filter with 2 = 0.1 or 2 = 0.2 was realized by controller canonical form [10]. The applicable eight different robust controller which use the nominal plant shown in Figure 6.4. The control effort was the sum of the output of the IMC filter and the nominal voltage supply that can keep the system at the equilibrium position when the feedback signal is zero. The simulation algorithm was using 4th and 5th order Runge-Kutta formulas which has tolerance set under le-S. 7.1 How is the Worst-Case Determined In order to define the worst case, we introduce an performance index called 2-norm which is ‘ “evil: '1 r 2 =1/Ile(t)| dt (7.1) g ‘/§[e(ti)2 +r:(t,.+1)2 )At i=0 2 where e,- = y,- — y” At = Sample time interval = 0.1 sec y = Time response trace y” = Steady state Using this 2-norm definition, the worse case occurs when 2-norm error reaches its maximum value. While using the three nonlinear state variable models (3.5.1), in addition to considering possible plants due to the parametric uncertainties, which result from lift curve slope , pitch angle, DC motor armature resistance, torque constant, viscous friction and Hall Effect sensor, we add the uncertainties of 5% of the Coulomb friction into 31 32 Table 7.2.1 Worst-case 2-norms tor the cases with IMC filter which has 2 = 0.1 Controller (1 Controller b Controller 6 Controller d 100 13.7233 *1 13.7111 *u 13.7273 *1 13.6879 *1 20° ' 29.6059 * u 29.5788 *u 29.5314 *c 29.5289 *1 A, e- __ 5.;8244 *u, _, 56-478 r- U403 *u Controller 8 Controller f Controller 3 Controller 11 10° 14.6214 *1 14.4197 *1 14.0989 *1 15.5336 *1 20° 30.4191 *u 30.1712 *u 30.0410 *u 31.8937 *1 8 513027 52-4556 *u . 55-732 , 503499 * " Nore that c. u. or I represent the worst case is conesponding to cubic, linear or uniform model respectively. account. Using line search method [11], we search for the worst-case responses with respect to each controller. 7.2 Worst-case responses for different robust controllers and the associated models In simulation, for each controller we search for worse case and the associate 2-norms which might corresponding to any of velocity model or system parameter. The reference signal is set to be the step input which is some certain degree during the 10 sec and 0° in the remaining 10 second. The simulation result shows that the worst-case parameters are always boundary parameters and these parameters are dependent on the inflow velocity distribution but independent of the controller used. Table 7.2.1 and Table 7.2.2 list the worst-case 2-norms for each controller, which are corresponding to different IMC filters. 33 Table 7.2.2 Worst-case 2-norms tor the cases with IMC filter which has 2 = 0.2 W Controller 0 Controller b Controller c Controller 61 10° 18.7924 *1 18.8056 *1: 18.7924 *1 18.7664 *1 20° 38.2131 *u 38.2131 *u 37.9692 *c 37.8170 *1 300 ‘ 58.8725 *u 58.7634 *u 58.5773 *1 58.4899 *u Controller e Controller f Controller g Controller h 10° 20.1400 *1 19.8789 *1 19.5240 *1 21.0060 *1 20° 40.9684 *u 40.4430 *u 39.7484 *u 42.6154 *1 -_3°° , __ 639947“ 5 , 6230*.“ *“_ 65.4545 ‘14 "' Note that c. u, or I represent the worst case is corresponding to cubic, linear or uniform model respectively. Form Table 7.2.1 with 2 :01, in 2-norm sense, the controller (1 derived from the comprehensive parametric uncertainty is proved better than the others for the smaller step input (e.g. 10° 20°) but for larger step input (e.g. 30°), the controller h derived from comprehensive nonparametric uncertainty is proved better. Form Table 7.2.2 with 2 = 0.2, in 2-norm sense, the controller d derived from the comprehensive parametric uncertainty is proved better than the others for different kinds of input In Figure 7.2.1, using the step input 30° we plot the eight worst-case traces corresponding to eight controllers with 2 = 0.1. It indicates that the upward motion is absolutely different from the downward motion. Figure 7.2.2 shows the same worst-case responses but with 2 = 0.2. g 30 ..................... ...................... ............ S tepinput:30b_ g 20 ' Solid - Controller a i 10 ...... Dashed - Controller b I e Dotted - Controller c 5 «r o ----- Dashdotted — Controller d .............. 3 ... go ’ : : c: : : g g '10 o 5 10 15 20 8 "l 50 z 3 E g- a ............. ..................... .. .................... - Q '3 .~ ' ‘7 : Step input: 30° g = 30 . .' I .................. ;. ..................... ‘2. .................. .................... .1 985; ' - 3 E i 2 0 20 . ..... SOlid- CODU'OHCI'C ,‘,, .................... q 0 f Dashed - Controller f ' 3. _ 3 ............... E O 10 ...... DOtth - Controller 8 ..... . ,_ ........... ..... . 0 ....... Dashdotted - Controller h .3 ........ V .- ._ : '10 o ‘5 ’ 10 15 20 Time History (sec) Figure 7.2.1 Worse-case response for 30° input with 2 = 0.1 Figure 7.2.3 shows the time trace of the control effort; i.e., the output voltage of the IMC filter for the worst—case for the input 30° with 2 = 0.2. It indicates that the amplifier used for implementation have to be powerful enough to generate as high as about sum of peak of output voltage of IMC filter, 10 Volts and nominal voltage about 6 Volts. Figure 7.2.4 shows the time traces corresponding to rotor speed using IMC filter with 2 = 0.1. Obviously, the rotor speed is out of reasonable physical range {0:600RPM S a) S 1400RPM} (7.2.1) which is the one we used to derive the Conic Sector Bounds. Figure 7.2.4 shows rotor speed will always in the physical range if we use IMC filter with 2 = 0.2. The Angular Displacement Output Voltage of IMC filter (V lots) 40 r ‘__ 3 g E 30 ............................................ in ...............s.'.t.e..:p..il.1..p.L.l.t.:....3.b.°....—r b0 20[ ...... , Solid- C'onmllcra 43-... ..................§ ..................... — é ....... Dashed-Conmllerb =. .......................................... t 10 Dotted - Controller c 3 “3, O ...... Dashdotted - Controller d .............. ii. i z 63-10 #5 10 15 to ..J a 40 - - ~ ~ - g s 30....... .................... E g Step Input: 30° g 20 Solid-Controllere . ...................... “r 10 .......... Dashed.é.Conml.l§r.f...... ...................... .‘5’ Dotted -§Controller g I «5 0 ........ ..Dashdoned.-.-.Controller:h. .......... '..-.;.__,,_-_ '10 :5 '10 is 0 Time History (sec) Figure 7.2.2 Worse-case response for 30° input with 2 = 0.2 10 maximum : s Dashed - Controller b E 5 Dotted _ Controllcrc ......................... ...................... - Dashdotted - Controller d g 0 ' : _ -5 - ..................... q -10 E i i 0 5 10 15 20 10 , - Solrd - Controller e 5 ...... Dashed. .-.C9nt.r91.l.¢rf ................................ . ...................... _ Dotted - Co'ntroller g : O . Dashdotted - Controller h -5 ........................................... ...................... _ -10 i i 0 5 15 20 10 Time History (sec) Figure 7.2.3 Control effort for input 30° with 2 = 0.2 36 2500 2m . ....................................................................... 1500 ............................................................................................ 1W ............‘.._..: ...... : ............................................... é. ....................... 500 """" Solid - Controller a A """" Dashed - Controller b g 2 Dotted - Controller 6 j 22 Dashdotted - Controller d 5 T3: 5 10 15 20 8. t t (I) ‘5 ............................................................................................ 1 5 ....................................................... Step. input. . 3.0.". .............. M ....................................................................... Solid - Controller e 0 -------- Dashed - Controller f SOOL ,,,,,,, Dotted - Controller 3 ' Dashdotted - Controller h 4 43 ‘1 5 10 15 20 Time History (see) Figure 7.2.4 Rotor speed for input 30° with 2 = 0.1 1400 Solid - Controller a . 12m ...... DaShed - CODIIOIICI' b ........... sfé.p. ififiijt” . “3.0. .0. .......... ... 1000 ..... Dotted - Controller c j ............................................. _ Dashdotted - Controller d ‘ 5 . ..................... .4 g ..... ..... ..................... .. O- I E 5 10 15 20 m ..................................................................... é ..................... _ 5 o .................... 3 ..................................................................... ‘3‘ 3 Solid - Controller e Dashed - Controller f Dotted - Controller g Dashdorted - Controller h 7 , E 5 10 15 20 Time History (sec) Figure 7.2.5 Rotor speed for input 30° with 2 = 0.2 8. CONCLUSION The physical system was described by a nonlinear model. This nonlinear model is able to predict the dynamics of drive train, parallelogram linkage and the thrust produced and the torque required by the rotor. We modeled a nonlinear system using the framework of Uncertain Linear System which has the uncertainty that results from parametric uncertainty or Conic Sector Bounds. By using this ULS, we can investigate the effect of three different inflow velocity distributions on both the uncertain linear system and the control design. It was found that for small parameter variations, the effect of inflow velocity distribution assumed was not very significant Eight robust controllers were designed to minimize the effect of the uncertainty. They are designed using parametric, nonparametric and comprehensive uncertainty representation for three different velocity distribution models. The stability analysis shows that the closed- 100p system is stable for any controller. A time domain analysis including simulation and the 2-norm error calculation were completed. The worst case always occurs for the plant with the boundary parameters. The simulation results shows that using the reasonable IMC filter with 2 = 0.2, the controller design using comprehensive parametric uncertainty yields better performance than the others. 37 REFERENCE [1] [2] [3] [4] [5] [6] [7] Benjamin C. UK, 1991, Automatic Control System, Prentice Hall, Englewood Cliffs, New Jersey. J. Seddon, 1990, Basic Helicopter Aerodynamics, American Institute of Aeronautics and Astronautics. Inc., Washington DC. Donald M. Layton,1984, Helicopter Performance, Matrix Publisher, Inc., Beaverton, Oregon. HP 35 660A Dynamic Signal Analyzer Getting Started Guide and F rant-Panel Reference, 1988, Hewllet-Packard Company, Everette, Washington. Bo Wahlberg and Lennart Ljung, 1992," Hard Frequency -Domain Model Error Bounds from Least-Square Like Identification Techniques," IEEE Transaction on Automatic Control, vol. 37, No. 37, pp. 900-912. Anderson Broan, 1990, Optimal Control .° Linear Quadratic Method, Prentice Hall, Englewood Cliffs, New Jersey Manfred Morrari and Evanghelos Zafiriou, 1989, Robust Process Control, Prentice Hall, Englewood Cliffs, New Jersey. [8] Menu of Digital Force Gauge Model D!“ G, 1986, Chatillon Force Measurement, [9] Greensboro, North Carolina. Hassan K. Khalil, 1992, Nonlinear Systems, Macmillan, New York. [10] Wilson J Rugh, 1993, Linear System Theory, Prentice Hall, Englewood Cliffs, New Jersey. [11] J asbir S. Arora, 1989, Introduction to Optimum Design, McGraw-Hill, New York. 38 APPENDIX A.l Mathematical Development Figure A.1.1 Simplified model of a DC motor In a running permanent magnet DC motor, the current i, flows through the armature which has resistance R, , inductance L, and supply armature voltage Va. Since the armature is a conductor rotating in a magnetic field, a voltage referred to as the back EMF, V,, is induced in the armature. A simple model shown in Figure A.1.1 describing this electric circuit is given by v, = V, + L, d" dt +R,i, (A.1.l.1) The relationships governing the behavior of the gyrator portion of the model in Figure A.1 are given by V,=K,a) (A.1.1.2) rm”, = Kg, (A.1.1.3) where K , is the back EMF constant, a) is the rotor speed, rmwis the output torque of the motor and K, is the torque constant (Note that for convenience we have chosen the rotor speed as our reference speed and this will affect the values of K v and K1). The term 39 40 La :- in (A.1.1.l) is small enough over the frequency range of interest to be considered zero. Therefore we can combine (A.1.1.1), (A.1.l.2), and (A.1.1.3) to find rm, = [([w] (A.1. 1.4) R This motor torque balances the torque required to accelerate the drive train/rotor system, the torque required to drive the rotor blades through the air mass, rm", and the torque resulting from friction in the drive train 1M.” . This relationship is given by V —K a) . KT[-'a—ls—v—] = 1,0) + Trotor + Tfriction (A.1.1.5) a Thrust dAz, va2 Figure A.1.2 Control Volume used in Momentum Theory In Figure A.1.2, we show the control volume used for this analysis. The areas, pressures, and velocities shown in this Figure are defined below dA0 = upstream cross-sectional differential area of the stream tube dA, = Cross-sectional differential area of the stream tube at the rotor disk dA2 = Downstream cross-sectional differential area of the stream tube P0 = Atmospheric pressrn'e 41 P1 = Pressure on upper side of the rotor disk P2 = Pressure on lower side of the rotor disk v0 = Upstream velocity of the air flow relative to the rotor disk v1 = Velocity of the air flow through the rotor disk v2 = Downstream velocity of the air flow relative to the rotor disk p = Air density We assume that the air is incompressible and apply Bernoulli's equation to streamlines above and below the rotor disk and find 1 l Po""§l"’o2 =Pr+§PV12 1 1 Po + 5m.” = P2 + 5M Subtracting (A.1.2.1) from (A.1.2.2) yields The Momentum Equation is given by (P 2 - Pr )d/‘r = Pvzzddz ‘onszo The Continuity Equation yields VodAo = vi“: = ”2% Substituting (A.1.2.5) into (A.1.2.4), we have (P2 - P1)dA1=PV1(V2 “'Vo)dA1 Multiplying (A.1.2.3) by dA1 leads to (P2 "' Pr)“ = %P(V2 “VoXVz +Vo)dAr Comparing (A.l.2.6) and (A.1.2.7), we find v2 = 2vl - v0 (A.1.2.1) (A.1.2.2) (A.1.2.3) (A.1.2.4) (A.1.2.5) (A.1.2.6) (A.1.2.7) (A.1.2.8) Substituting (A.1.2.8) into (A.1.2.6) we finally arrive at an expression for the differential thrust dT generated by the rotor 42 d7 = (P2 - P1)dA1 =pv1(2v1 - 2v0)dA1 (A.1.2.9) = 210V1(V1 - Vo)dA1 Note that v1 is the inflow velocity distribution assumed to exist over the rotor disk. HEEY' [EllEl II EB Blade Element Theory is introduced [2] here to find an alternative expression for the thrust and torque that are dependent on the rotor geometric design. In order to develop this expression, certain terms used in 2-D airfoil theory are shown in Figure A.1.3 below. Rotor disk plane Figure A.1.3 2-D Airfoil Terms tor a Rotor Blade Section Recalling from 2-D airfoil theory the drag and lift forces are dependent on the square of the resultant wind velocity v, . We can write the differential lift and drag equations as d0 = épvw2(CDc)dr (A.1.3.1) l dL = Epvw2(CLc)dr (A.1.3.2) where d represent the differential notation. D and L are the drag and the lift force, respectively. Note that dB is parallel and dL is perpendicular to the relative wind v,,. The lift coefficient is approximated well (as long as the section is within stall limits) by CL = aa (A.1.3.3) where a = Lift curve slope or = (6 - 4)) = Angle of attack 43 The inflow angle (in Fig A.1.3) 4: is typically small so that it can be approximated by 4, = tan-1 (.31) a A (A.1.3.4) (Of (Dr and the resultant wind velocity can be approximated by v" z (or (A.1.3.5) Here V1 is the inflow wind velocity, r, refers to the radial position on the rotor blade, and a) is the angular speed of the rotor. The angle of attack is the difference between the pitch angle 9,, and the inflow angle 4). a = 0,, - ¢ (A.1.3.6) By substituting (A.1.3.3), (A.1.3.4), (A.1.3.5) and (A.1.3.6) into (A.1.3.2), the Since dB is very small relative to dL, the differential thrust d1“ is well approximated by differential lift dL well. Thus dI‘ ... épwzrzac(9p —Zv;-)dr (A.1.3.8) The torque required to drive the blade element through the air mass can be approximately produced by the component of the lifting force dL in the rotor disk plane normal to the blade centerline. Therefore dr rotor == rsin(¢)dL= r¢dL=-:%dL (A.1-39) Substituting (A.1.3.7) into (A.1.3.9), we find dr rotor = pac(car20,,v1 - nfldr (A. 1.3.10) The differential thrust (A.1.3.8) and torque (A.1.3.10) are used to derive net thrust and torque. EIII° . . x=Ax+Bu y=Cx '0 1 o where A: an 022 023 . _0 “32 as. 70 B: 0 .ba C=[KH O 0] (—Mg+TxLP)xsinJy “21: J, P dT cosx, f (17” =—x—-xL +—’x—-b , a” 4x, J, P J, dx, ‘4’ =flxcosxle +-f-’-x£, drr3 JP ' JP 3 (4.1.2) 45 A.2 Experimental Data All experiments were designed and performed in Control Laboratory at Michigan State University. Some experiments apply high-tech equipments such as Force Gauge and Signal Analyzer. Standard experimental procedure and detailed technique are recorded in associated menus available in the laboratory. All measurements precision are recorded in maximum precision of the equipments. The armature resistance Ra is not possible to be measured when the rotor is running. We used the multimeter to measure the static motor armature resistance with respect to twenty- three different angular position of the rotor. Figure A.2.1 Measurements of the motor armature resistance. Ra 46 52281.1'1 II] lS IKE Taking off the counterweight, Force Gauge, which was used to measure lift force, was placed below the rotor assembly to support parallelogram linkage. By using Strobescope Type 4913, we can measure the rotor speed in unit of RPM. The reasonable operating range for rotor speed is from 0 RPM to 1400 RPM. The following measurements were calibrated based on the precision of digital multimeter. m (RPM) (N) 0 0 100 4.48x10‘2 200 8.96x10-2 300 8.96x10-2 400 8.96x10’2 500 1.79x10-T 300 2.69x10-1 700 4.48::10-I 800 5.38x10-1 900 8.06x10-T 1000 1.06 1100 1.25 1200 1.52—— 1300 1.79—— 1400 1.15 Table A.2.2 Measurements tor the relationship between the thrust and rotor speed 47 t. i‘ tl.ll'°."t.l° 1' In. ort.‘ 0161.00 1' (00 fin‘l 21¢ 0 :° We measured the Current and voltage by repeating the same experimental procedure and taking more samples within the reasonable operating range. The results is shown in Table A.2.3. Speed (RPM) Voltage (Volts) Current (Amp) Thrust (N) 40 1.00 0.150 2.22x102 50 2.00 0.200 2.22x102 405 3.00 0.290 8.96x10'2 491 3.50 0.360 1.86x10-1 598 4.00 0.450 2.91x10-1 669 4.50 0.540 - 3.72x10-l 749 5.00 0.630 5.40x101 834 5.50 . 0.740 6.50x10-1 886 6.00 0.890 7.84x10-1 944 6.50 0.990 8.96x10-1 995 7.00 1.12 1.00 1053 7.50 1.26 1.12 1110 8.00 1.40 1.25 1150 8.50 1.54 1.39 1210 9.00 1.73 1.55 1250 9.50 1.86 1.66 1310 10.0 2.00 1.79 1340 10.5 2.12 1.93 1380 11.0 2.26 1.99 1400 11.5 2.39 2.14 1420 __ 12.0 , 2.53 2.20 Table A.2.3 Measurements for the relationship among the thrust, torque. rotor, current and vohage a ‘ {'eQOIIUOI'I"l,°.|.. 0 t". It ‘ ‘10 9|. 1'=r' l. l E I I] H' I Using the split supply i 6 volts and adjusting the angular position of the Hall Effect sensor relative to the base support, we can set the nominal output voltage of Hall Effect sensor to nearly zero when the parallelogram linkage is at horizontal position. The following measurements were made with angular displacement of the parallelogram linkage from -32.5° to 40°. flular displacement (degree) Output voltage (Volts) 40.0 -0.753 35.0 -O.677 30.0 -0.612 25.0 -0.542 20.0 -0.47 15.0 -O.365 10.0 -0.276 5.00 -0. 167 0.00 -0.047 -5.00 0.108 -10.0 0.235 -15.0 0.391 -20.0 0.521 -25.0 0.713 -30.0 0.854 -32.5 0.903 Table A.2.4 Measurements for the relationship between Hall Effect sensor and the angular displacement of the parallelogram linkage 49 A.3 Nominal System Parameters 51mins a = 5.72 (Cubic) = 5.99 (Linear) = 6.05 (Uniform) b, = 6.31 x10-3 b, = 3.85 x104 (Cubic) = 3.29 x10’4 (Linear) = 3.85 x10‘4 (Uniform) b”, = 0.33 9 c = 3.00 x102 f, = 2.30 x10-2 g = 9.81 J,= 5.53 x104 K, = 6.57 x10‘3 K, = 6.57 x10-3 LCG = 4.95 x10-1 m = 7.30 x10-1 R = 2.20 x10-1 R, = 2.25 Warmers 9,, = 13.00 (Cubic) Characteristic lift curve slope Coefficient of Coulomb friction Coefficient of viscous friction Coefficient of viscous friction for the parallelogram linkage Chord length Offset of acting line of the thrust Magnitude of gravity Moment of inertia for the rotor and the gear assembly Back EMF constant Torque constant Length from O to CC. of the parallelogram linkage Mass of the parallelogram linkage and gear assembly Radius of the rotor Armature resistance Pitch angle (dimensionless) (N m) (N m sec/rad) T (N sec/rad) (m) (m) (m / sec?- ) (kg m2) (kg m2) (Volt sec/rad) (N m/amp) (m) (kg) (m) (ohm) (degree) 50 = 12.80 (Linear) = 13.19 (Uniform) p = 1.23 Air density (kg/m3) 51 A.4 Parameter Uncertainties All uncertainties which were derived from conic sector bounds, or hard-bound and used in chapters are listed below. (1) Armature resistance R..o = 2.27 (ohm) R‘ e [2.10.2.40] (2) Coefficient of Coulomb friction 60.0 = 6.31x10"3 (N m) b0 e[5.99,6.62]><10’3 (3) Coefficient of viscous friction in the drive train (i) Cubic bm = 3.84 x104 Nm(sec/ rad) b0 e[3.65,4.04]x10" (ii) Linear bL0 = 3.29 x10'4 Nm(sec/ rad) b0 e[3.12,3.45]x10" (iii) Uniform bl.() = 3.41 x104 N m(sec/ rad) b0 6 [3.24,3.58]x10" (4) Torque constant Km = 0.00657 (N m / Amp) K , e[624,689]x10‘3 (5) Back EMF constant K... = 0.00657 (Volt (sec/ rad)) KV e [6. 24.6.89] x 10'3 (6) Hall Effect output equation v, = 16,9, K,” = 0.0235 K” e[2.00,2.50]><10'2 (7) Lift curve slope ( no unit ) (i) Cubic a0=5.72 ae[5.44,6.01] (ii) Linear a0 = 5.99 a 6 [5.69.6 29] (iii) Uniform a0 = 6. 05 a e [5.75.6.35] (7) Pitch angle (i) Cubic 9p.0 = 1.30 x101 (degree) (ii) Linear , 970.0 = 1.28 x 101 (degree) (iii) Uniform 0,0 = 1.31 x 101 (degree) 52 0,, e [1. 23, 1. 36] x 101 (degree) 6,, 6 [1.21.1. 34] x 101 (degree) 0p 6 [1. 24, l. 37] x 101 (degree) 53 A.5 Simulation Results Three state traces and control effort which is corresponding to worst—case response are all plotted in the following eighty-four plots with the input 10°. 20° or 30° and 2 = 0.1 or 0.2 In each plot, the corresponding controller is (1) Solid line - Controller 0 or e (2) Dashed line - Controller b or f (3) Dotted line - Controller c or g (4) Dot dashed line - Controller d or h 54 (l) The first twenty-four plots are for the IMC controllers with 2 =0.1 Angular displacement of parallelogram linkage (degree) Angular velocity of parallelogram linkage (rad/sec) 50 Simulation of Parametric Conu'ollers for 30 degree input -10r Time History (sec) 1 5 Simulation of Parametric Controllers for 30 degree input ' 0 5 10 15 20 Time History (sec) 55 ’2‘ E1: 3; § . not. 0 : -500 i ~1000 E 0 5 10 15 20 Time History (sec) 25 Simulation of Parametric Controller for input: 30 degrees Output voltage of IMC filter (Volts) 5 10 15 20 Time History (sec) 56 Simulation of NoNParameuic Controller for input 30 degrees Angular displacement of parallelogram linkage (degree) -10 Time History (sec) 1 5 Simulation of NoNParametric Controller for input 30 degrees Angular velocity of parallelogram linkage (rad/sec) Time History (sec) 57 '82 8 Simulation of NonParameuic Controllers for input 30 degrees 3 § 3 Rotor Speed (RPM) U! 8 Time History (sec) 25 Simulation of NoNParametric Controller for input 30 degrees Output voltage of IMC filter (Volts) Time History (sec) 58 30 Simulation of Parametric Controller for input 20 degree 0 5 10 15 20 Time History (sec) Angular displacement of parallelogram linkage (degree) Simulation of Parametric Controller for input 20 degree O .5 N Angular velocity of parallelogram linkage (rad/sec) o N Time History (sec) 2000 1500 5 Rotor Speed (RPM) Ur S? O -500 59 Simulation of Parametric Controllers for input 20 degree ............................................................................................. 5 10 15 20 Time History (sec) Simulation of Parametric Controller for input 20 degre "é 2, ............................................. : .............................................. - is. 8": S L) H.HH.HH.HH.HH.HH.--.HH.UH.”3 ............................................. - E . '4— 2 ° 5 at z ...... i ]l O . > 5 a -5_ ........................................... ........................................... . :3 2 <3 3 -10 ‘ 0 5 10 15 20 Time History (sec) Angular displacement of parallelogram linkage (degree) Angular velocity of parallelogram linkage (rad/sec) Simulation of NonParametric Controller for input 20 degLree 30 -5 Time History (sec) Simulation of NoNParametric Controller for input 20 degree 0.8 f Time History (sec) 61 Simulation of NonParametric Controllers for input 20 degree 2000* 1500 a E 1000 g 2 5m .............................................. 1 K ........................................ g s M i 0 ............................................... 31 ............................................. él -500 E 0 5 10 15 Time History (sec) 15 Simulation of NoNParametric Controller for input 20 degre Output voltage of IMC filter (volts) 20 Time History (sec) 20 62 Simulation of Parametric Controllers for input 10 degree Angular displacement of parallelogram linkage (degree) -2 . Time History (sec) 0 4 Simulation of Parametric Controllers for input 10 degree ° 0 5 10 15 20 Time History (sec) Angular velocity of parallelogram linkage (rad/sec) 1400 1200 Rotor Speed (RPM) Output Voltage of IMC filter (Volts) 1000 63 Simulation of Parametric Controllers for input 10 degree 400 ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo ooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooo 200 10 15 20 Time History (sec) Simulation of Parametric Controllers for input 10 degre ooooooooooooooooooooooooooooooooooooooooooooooo ................................................ ................................................ c oooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooo 10 Time History (sec) Angular displacement of parallelogram linkage (degree) Angular velocity of parallelogram linkage (rad/sec) Simulation of NonParametric Controllers for input 10 degree Time History (sec) 0 4 Simulation of NonParametric Controllers for input 10 degre- Time History (sec) 65 Simulation of NonParametric Controllers for input 10 degree 1400 1200 ...................... ....................... ....................... a 1000 .............................................................................................. 3 800 - (I) H 2% 600 400 2 E E E 000 5 10 15 20 Time History (sec) 6 Simulation of NonParametric Controllers for input 10 degre- 4. ............................................................................................ Output voltage of IMC filter (Volts) 0 5 10 15 20 Time History (sec) 66 (2) The next twenty-four plots are for IMC controllers with 2 =0.2 Angular displacement of parallelogram linkage (degree) Angular velocity of parallelogram linkage (rad/sec) Simualtion of Parameuic controller for 30 degree Time History (sec) Simualtion of Parametric controller for 30 degree 20 o ........................................................................................... Time History (sec) 67 1400 1200 .5 Rotor Speed (RPM) 00 § 8 8 Simualtion of Parametric controller for 30 degree p—A CD 5 10 1'5 20 Time History (sec) Simualtion of Parameuic controller for 30 degree {It Output Voltage of IMC filter (V lots) t'a $3 2'3 O 5 10 15 20 Time History (sec) 68 Simualtion of NonParameuic controller for 30 degree Angular displacement of parallelogram linkage (degree) Time History (sec) 0 6 Simualtion of NonParametric controller for 30 degree Angular velocity of parallelogram linkage (rad/sec) Time History (sec) 69 Simualtion of NonParametric controller for 30 degree 1400 1200 ' .3 Rotor Speed (RPM) § °8° § 0 5 10 1.5 20 Time History (sec) Simualtion of NonParametric controller for 30 degre- ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo Output Voltage of IMC filter (Volts) _10 : : : 0 5 10 15 20 Time History (sec) Angular displacement of parallelogram linkage (degree) Angular velocity of parallelogram linkage (rad/sec) 70 Simualtion of Parameuic controller for 20 degree 0 5 10 15 20 Time History (sec) Simualtion of Parametric controller for 20 degre Time History (sec) 71 1200 1100 Simualtion of Parametric controller for 20 degre .3 300...... 700 Rotor Speed (RPM) 500 t u ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo Output voltage of IMC filter (Volts) -2, .............................................. ' [ ............................................. .4 ............................................... .............................................. -5 5 0 5 10 15 20 Time History (sec) 72 25 Simualtion of NonParametric controller for 20 degree /-v'.‘h.‘::.-.- -- v: Angulandgrpmtdnedmwmfim) -5 Time History (sec) 0 4 Simualtion of NonParametric controller for 20 degree Angular velocity of parallelogram linkage (rad/sec) Time History (sec) 1100 Rotor Speed (RPM) Output voltage of IMC filter (Volts) 73 Simualtion of NonParametric controller for 20 degree Time History (sec) Simualtion of NonParametric controller for 20 degre u ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo I oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo Time History (sec) 74 0 2 Simualtion of Parametric controller for 10 degree Time History (sec) 420864202 1111 32on own—q: Edema—0:256 mo 80882668 33mg 20 Simualtion of Parametric controller for 10 degree .................................................... 5 _ H.H. » O JUJMO J 2. 00.00. 0 0 89525 owe—c: Edema—2336 «o .0623 R~=w=< Time History (sec) Rotor Speed (RPM) Output voltage of IMC filter (Volts) 75 1000 Simualtion of Parametric controller for 10 degree 95o ....................... ....................... ..................... 850 800 750 700 65 Time History (sec) Simualtion of Parametric controller for 10 degre - I o ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo -1 ............................................... 1 .............................................. -2 ............................................... .............................................. -3 5 O 5 10 15 20 Time History (sec) 76 Simualtion of NonParametric controller for 10 degree Angular displacement of parallelogram linkage (degree) -2 Time History (sec) 0 2 Simualtion of NonParametric controller for 10 degree Angular velocity of parallelogram linkage (rad/sec) Time History (sec) Rotor Speed (RPM) Output voltage of [MC filter (Volts) 77 Simualtion of NonParametric controller for 10 degree 6500 5 10 1'5 20 Time History (sec) Simualtion of NonParametric controller for 10 degree oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo -1 ............................................................................................. -2r .............................................. .............................................. -3 3 O 5 10 15 20 Time History (sec) ..‘l