Self-Scheduled Control of a Gyroscope

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1 Preprint of the 9th World Congre The International Federation of Automatic Control Cape Town, South Africa. Augut 24-29, 24 Self-Scheduled Control of a Gyrocope Julian Thei Chritian Radich Herbert Werner Intitute of Control Sytem, Hamburg Univerity of Technology, Hamburg, Germany ( julian.thei@tuhh.de, h.werner@tuhh.de). Intitute of Mechanic and Ocean Engineering, Hamburg Univerity of Technology, Hamburg, Germany ( chritian.radich@tuhh.de) Abtract: A claical mixed enitivity minimization approach and a model matching formulation are compared with the goal to deign a linear parameter-varying augmented tate feedback control law for a laboratory-cale control moment gyrocope. Dynamic weighting filter are ued to impoe integral action and roll-off on the controller. Conequently, meaurement noie i effectively uppreed and teady tate accuracy i guaranteed even in the preence of input diturbance. Both deign are validated in real-time experiment and compared to a previou deign that ue tatic weight. With the new deign, control effort i reduced while tranient performance i maintained and tracking accuracy, a well a diturbance attenuation, i improved. Keyword: Linear parameter-varying ytem, Robut control application, Robut controller ynthei, Gyrocope, Mixed enitivity problem, Multivariable control ytem. INTRODUCTION Control moment gyrocope are inherently coupled nonlinear plant and therefore difficult to control by tandard linear deign method. They are ued, e. g., in attitude control of pacecraft and tability augmentation of hip. In thi paper, a laboratory-cale control moment gyrocope, the ECP model 75, i ued a a tetbed for linear parametervarying (LPV) gain cheduled controller deign. Recently, firt ucceful experimental validation of an LPV control trategy for the ECP 75 ha been reported in Abba et al. (23, 24), demontrating that the plant can be tabilized in a wide operating range. Still, ome pecification with practical relevance, uch a input diturbance attenuation, were not conidered and teady tate accuracy relied on a tatic prefilter. To addre thee iue, an LPV tate feedback controller augmented with dynamic weighting filter i deigned in thi paper. The dynamic weight allow to guarantee teady tate accuracy through integral control, which alo reult in effective rejection of input diturbance. Further, a rolloff i included in the controller to reduce enitivity to meaurement noie. Two different deign approache are purued: a claical S/KS mixed enitivity minimization and a model matching formulation. The new deign are experimentally validated and the reult are compared to the one reported by Abba et al. (24). The paper i tructured a follow: Section 2 decribe the plant and the LPV model ued in ynthei, detailing the predominant dynamic effect. Section 3 briefly review LPV ynthei machinery and the different deign approache are dicued in Section 4. Section 5 preent an experimental comparion of the different controller and draw concluion. See ctrlgyro.htm for detail. 2. PLANT DESCRIPTION The control moment gyrocope i a flywheel upended in a gimbal mounting and i modeled with Neweul-M 2 (Kurz et al. (2)) a a four-degree-of-freedom multibody ytem. The ytem i in chain tructure, i. e., without kinematic loop. Each body i linked to the previou body by a rotational joint perpendicular to the lat joint axi. A kinematic model i depicted in Fig.. x 2 z q q 3 Fig.. Kinematic of the control moment gyrocope The nonlinear equation of motion in minimal form are [ ] [ ] I T M(q) q + k(q, q) = f( q) +, () T 2 where q = [q q 2 q 3 q 4 ] T, M i the ma matrix, k the vector of generalized nondiipative force including gyrocopic term, and f i the vector of generalized diipative force. The input T and T 2 repreent torque applied by 4 q 2 y q 4 3 Copyright 24 IFAC 629

2 9th IFAC World Congre Cape Town, South Africa. Augut 24-29, 24 electric motor at axi and 2, repectively. The controlled output conidered in thi paper are the unactuated angle y = [ q 3 q 4 ] T. 2. Linear Parameter-Varying Model For ynthei purpoe, a implified model with tate x = [q 3 q 4 q 2 q 3 q 4 ] T i ued. It wa introduced and validated in Abba et al. (24) and i baed on linearization about a moving operating point ρ(t) = [ q (t) q 2 (t) q 3 (t)] T. Note that thi i not an equivalent formulation of the nonlinear ytem (), but rather repreent a continuou family of Jacobian linearization. The parametrization of a nonlinear ytem in term of a varying operating point i a claical approach for gain cheduling control, ee e. g. Rugh and Shamma (2). Formally, the operating point i defined by the timevarying parameter vector ρ: R P with a compact et P R nρ of admiible parameter trajectorie. A parameter-dependent tate pace repreentation i then ẋ = A(ρ(t)) x + B(ρ(t)) u (2a) y = C(ρ(t)) x + D(ρ(t)) u (2b) with the continuou matrix function A: P R nx nx, B : P R nx nu, C : P R ny nx, D : P R ny nu and the tate x: R R nx, input u: R R nu and output y : R R ny and with the cheduling ignal ρ. Since the operating point, and thu the parameter vector, cannot change arbitrarily fat, rate of change ρ: R V in the parameter are defined to lie within the polyhedron V = {v R nρ v i ν i, i =,..., n ρ }. Further, a the trajectorie are not known a priori, the parameter pace P V i repreented by auxiliary variable p P and v V. For brevity, the explicit dependence on time and parameter i dropped from thi point on. 2.2 Predominant Dynamic Variation The ditribution of control input to the controlled output varie trongly and depend mainly on the angle q 2. For q 2 =, the ytem i decoupled. Applying torque to axi then produce a reactive torque reulting in a negative motion around axi 3. Torque applied to axi 2, on the other hand, reult in preceion and rotate the device around axi 4. Thi however alo change q 2, which inevitably introduce cro coupling. The effect i illutrated in Fig. 2, where it i hown that the direction of the cro coupling depend on the ign of q 2. For q 2 = ±9, the allocation of input and output completely wap. The ytem i thu only diagonal dominant in the vicinity of q 2 =. The angle q 3 alo trongly influence the dynamic. Fig. 3 depict the input-output pole zero map from the econd input to q 4 for q 2 = and different frozen value of q 3. It how that for poitive q 3, there i non-minimum phae behavior which become more dominant, i. e., lower, with more inclination and for q 3 = ±9 reult in a zero at the origin. For q 3 =, the zero i located at infinity, leading to a change in the relative degree of the ytem. The dominant pole pair further approache the origin for q 3 ±9, indicating that the ytem bandwidth decreae while damping increae. Normalized impule repone Gimbal inclination q 2 ( ) Fig. 2. Normalized impule repone gain for different value of q 2 [ 9, 9 ]: T q 3 ( ) and T q 4 ( ); T 2 q 3 ( ) and T 2 q 4 ( ) Imaginary Axi Real Axi 9 9 Fig. 3. Input-output pole-zero map T 2 q 4 for different value of q 3 [ 9, 9 ] The angular velocity q of the flywheel alo change the location of the dominant pole pair: larger velocitie increae the reonant frequency, lower velocitie decreae it. Damping i only marginally affected by variation in q. 3. LPV GAIN SCHEDULING The controller ynthei procedure employed in thi paper wa developed in Wu (995); Wu et al. (996). It pecifie performance in term of the induced L 2 -norm of a generalized plant. A generalized LPV plant P i aid to have an induced L 2 -norm P if there exit a contant γ > uch that for all z = P w, w L 2, and (p, v) P V z 2 γ w 2, z 2 = z T (τ) z(τ) dτ. (3) A guaranteed induced L 2 -norm and tability of a cloedloop ytem can be achieved by tate feedback control with the following theorem. Theorem. (Wu (995)). Let a compact et P V, a performance pecification γ > and a generalized LPV plant of the form ẋ z z 2 = A B B 2 C C 2 I x w u (4) 63

3 9th IFAC World Congre Cape Town, South Africa. Augut 24-29, 24 be given. Further, let R : P R nx nx be a continuouly differentiable ymmetric matrix function uch that for all (p, v) P V R, (5a) Ψ R CT γ B C R I, (5b) γ B T I with Ψ = R (A B 2 C 2 ) T + (A B 2 C 2 ) R n ρ B 2 B2 T R v i. p i i= The tate feedback control law u = ( B T 2 R + C 2 ) x (6) render the cloed-loop ytem aymptotically table with an induced L 2 -norm le than γ. Since the linear matrix inequality (LMI) contraint (5) are infinite dimenional due to the functional dependence on the cheduling parameter p, it i propoed in Wu (995) to grid the parameter pace. The contraint are then only enforced on a finite dimenional et {p k } P. Conequently, ufficiency of the condition in between grid point i lot but due to continuity, the olution remain valid for a ufficiently dene grid. Further, the arbitrary dependence of the Lyapunov matrix R on p need to be retricted to arrive at a tractable formulation. Selecting bai function g i for the Lyapunov matrix R(p) = n R i= R i g i (p) (7) then lead to the emidefinite program min γ uch that γ,{r i} n R LMI (5) hold (p, v) {p k } vert (V) where vert(v) denote the vertice of the polyhedron V. 4. MIXED SENSITIVITY DESIGN Mixed enitivity minimization i a very veratile tool for control deign, ee e. g. Zhou et al. (995); Skogetad and Potlethwaite (). It general two-block repreentation i [ ] min W S K W W 2 K S 3, (9) where S = (I + G K) i the enitivity function, K denote the controller and G the plant model. The baic idea of mixed enitivity deign i to minimize the induced L 2 -norm of weighted cloed-loop tranfer function by tacking them together. There i a variety of weighting cheme in ue, each with certain propertie. It i however important to recognize that all mixed enitivity approache rely on the ame principle: to hape the enitivity function while achieving cloed-loop tability. Every weighting cheme i thu in principle capable of achieving the ame performance. Thi i due to the fact that the only degree of freedom in the minimization i the feedback controller K, on which all cloed-loop tranfer function depend. Thi fact i ometime mitaken in the mixed enitivity approach and (8) i eaier to recognize in an open loop haping procedure, cf. Skogetad and Potlethwaite (). Remark: For LPV ytem, the notion of a tranfer function doe not exit. Still a frozen parameter interpretation i ueful, epecially in conjunction with the gridding approach that treat the LPV ytem a a family of LTI ytem. 4. S/KS Deign Arguably, the mot common weighting cheme i the S/KS cheme, i. e., W 3 = I. It eek to directly hape the enitivity function while limiting the control enitivity. Thi make it a very tranparent weighting cheme: the enitivity function i the primary indicator for performance and the control enitivity retrict control authority and indicate robutne to untructured additive (high frequency) perturbation uch a unmodeled actuator dynamic. The weighting filter for the deign in thi paper are elected to be diagonal, with W coniting of one integrator per channel, whoe croover frequency determine the cloed-loop bandwidth. The filter W 2 i alo choen diagonal, but with firt order high-pa characteritic to enforce a roll-off in the controller. The S/KS cheme thu reult in 4 filter tate. Magnitude plot of the filter are given in Fig Model Matching Deign Another frequently encountered approach i model matching, where W 3 = T ref i elected according to the deired cloed-loop tranfer function. The idea of impoing the dynamic propertie of an ideal reference ytem on a cloedloop ytem ha a long hitory in control engineering, ee e. g. Limebeer et al. (993). It eemingly provide an intuitive method to addre tranient, which i generally difficult in frequency-domain-baed controller ynthei. With W 3 = T ref, it follow from (9) that the minimization objective become min K [ W S T ref W 2 K S T ref ] = min K [ ] W (T T ref T ref ), () W 2 (K S T ref ) where T = I S i the complementary enitivity function. It i important at thi point to note that the deign inherently lead to a two-degree-of-freedom controller, i. e., while all cloed-loop tranfer function maintain their interpretation, reference following i not anymore governed by the complementary enitivity T alone. Intead, the tranfer function for reference tracking become T T ref. The minimization () thu eek to minimize the difference between thi reference tranmiion and the deired tranfer function T ref. The natural choice of weighting filter reulting from thi interpretation i to elect T ref with low-pa characteritic and unit teady tate gain, W a an integral weight to enure model matching in teady tate and a high-pa filter W 2 to limit control action due to change in the reference ignal. A problem with thi choice i that the control enitivity KS i alo weighted with the low-pa filter T ref, which contradict claical deign guideline. 63

4 9th IFAC World Congre Cape Town, South Africa. Augut 24-29, 24 Singular Value (db) Fig. 4. Weighting filter for S/KS cheme: W ( W 2 ( ) Singular Value (db) ) and Fig. 5. Weighting filter for model matching cheme: W ( ), W 2 ( ) and W 3 ( ) Thi may lead to undeirably large control action in repone to output diturbance and may ignificantly reduce robutne. In exchange, tuning i very imple with thi cheme. In accordance with the relative degree of the plant, a econd order ytem with critical damping i elected a the reference model for each channel, i. e., T ref i diagonal. A diagonal integral weight W with a croover frequency half a decade above the bandwidth of the reference model i ued to aure model matching in a ufficiently large frequency range. Again, W 2 i elected a a diagonal firt order high-pa filter. Thi eventually lead to a total number of 8 filter tate for the model matching deign. The correponding magnitude plot are hown in Fig EXPERIMENTAL RESULTS In order to ufficiently cover the dynamic effect decribed in Section 2.2, it i neceary to include both negative and poitive value for the angle q 2 and q 3 in the grid. Mirroring a poitive grid till produced a tabilizing controller but with ignificant performance deterioration. An equally paced grid of point i thu ued to cover the operating range q [3 rad rad, 6 ], q 2 [, ], q 3 [ 75, 75 ]. The rate bound are elected a q < rad, q 2 2 < 2 rad and q 3 < 2 rad baed on experimental obervation. An affine parameter-dependent Lyapunov matrix i elected for implicity and the cheduling ignal are normalized with repect to their maximum value, i. e., q 3 rad q R(p) = R + R rad + R 2 + R 3 2 q () It i verified by uing a lower bound on the achievable performance (ee Saupe (23)), that higher order bai function do not ignificantly improve the reult. y r x T ref x T ref e r W x W x W 2 F W 2 Fig. 6. Augmented tate feedback controller with dahed line indicating part only applicable to the model matching deign; otherwie r = r. SDPT3 (Toh et al. (999)) i employed to olve the minimization problem (8) with YALMIP (Löfberg (24)) ued a a parer. A uboptimal algorithm, developed in Saupe (23), i ued to enure that the Lyapunov matrix R i well conditioned ince it i inverted in the controller recontruction tep (6). The experimental etup allow direct meaurement of all angle q by mean of encoder. In order to obtain the angular velocitie q, firt order differentiation filter D() = 6 π +6 π are ued. The controller are implemented in Simulink and deployed on a real time target. The gain calculation F (p) = ( B2 T (p) R ) (p) + C 2 (2) i performed online at a ampling rate of khz with the inverion of R efficiently carried out by LDL T factorization, exploiting poitive definitene of R. The tructure of the augmented tate feedback controller i hown in Fig. 6. The controller inherit the dynamic order of the weighting filter, i. e., 4 for the S/KS and 8 for the model matching deign. Prior to each experiment, the flywheel i brought to it initial velocity uing a PI regulator. Authority i then witched at t = 5 to the LPV controller. 5. Trajectory Tracking Both controller were manually tuned until atifactory performance wa achieved. For comparion, the repone reported in Abba et al. (24) are ued. Their LPV tate feedback deign employ the S/KS cheme with contant weight and ue an additional firt order prefilter with a bandwidth of rad on the reference ignal and a econd tatic prefilter to reduce the teady tate error. It i therefore alo a two-degree-of-freedom deign. All three evaluated controller achieve a imilar level of performance in term of the peed-of-repone (Fig. 7). Both S/KS and model matching controller yield repone that are almot identical. Compared to the deign of Abba et al. (24), cro coupling are ignificantly reduced and almot nonexitent with the dynamic controller. Tracking accuracy i alo clearly improved due to integral control. It i alo oberved that the dynamic controller deliver conitent reult regardle of the direction of the reference tep. The controller by Abba et al. (24) i fater for tep toward the origin, while it i lightly lower in the oppoite direction. u 632

5 gimbal inclination q4 ( ) gimbal inclination q3 ( ) 9th IFAC World Congre Cape Town, South Africa. Augut 24-29, input u (V) 5 5 input u2 (V) Fig. 7. Tracking of gimbal inclination q3 : reference ( ) controller model matching ( time () ), controller by Abba et al. (24) ( time () Fig. 8. Control input ignal u aturated at ±V: controller by Abba et al. (24) ( ) controller matching ( ), S/KS ( 6 ), S/KS ( 7 ) and 7 ) and model q ( rad ) q2 ( ) Fig. 9. Scheduling ignal q and q2 with S/KS ( time () ) and model matching ( ) controller 6 7

6 9th IFAC World Congre Cape Town, South Africa. Augut 24-29, 24 q3 ( ) q4 ( ) time () Fig.. Input diturbance rejection: reference ( ), controller by Abba et al. (24) ( ), S/KS ( ) and model matching ( ) controller A cloe examination further how that the model matching controller produce repone with higher order characteritic, i. e., fat rie time with viible phae lag compared to the S/KS controller. Thi i expected from the additional prefiltering of the reference ignal through T ref. The correponding input ignal are hown in Fig. 8. Input u i very imilar for all three controller, with the model matching controller reulting in maller peak for change in q 3 due to the prefiltered reference ignal. Input u 2 i alo very imilar for S/KS and model matching controller but differ greatly from the controller by Abba et al. (24) that lack a roll-off and thu amplifie meaurement noie. Fig. 9 how the cheduling ignal trajectorie for S/KS and model matching controller. They can be verified to remain within the predefined bound. The great reemblance of both trajectorie further ugget good repeatability of the experiment ince not only the controlled output but alo the internal variable evolve imilarly. 5.2 Input Diturbance Rejection A a econd experiment, an input diturbance i conidered. A zero reference i provided and at t =, a tep diturbance of V i applied to each of the two input channel. The inufficiency of a tatic deign for thi cenario become readily apparent from Fig : while the tatic controller maintain a teady tate offet of approximately 45 for q 3 and 6 for q 4, both dynamic controller completely reject the diturbance. The S/KS controller reult in a peak of 6 for both q 3 and q 4, and after 3, the input diturbance i compenated. The model matching controller achieve even lightly better reult, with a peak of 4 for q 3, 6 for q 4 and 2 until complete compenation. Thi i a conequence of the elected two-degree-offreedom tructure, in which the cloed-loop bandwidth for diturbance rejection i larger than for reference tracking. 5.3 Concluion The experiment demontrate the advantage to be gained by uing augmented over tatic tate feedback: the control effort i reduced through the ue of a dynamic weight W 2 and teady tate accuracy i guaranteed through integral control via W. Thi hold true even with input diturbance acting on the plant. Both, the S/KS and model matching approach achieve almot identical performance, where it hould be recalled that the model matching controller i a two-degree-of-freedom deign with twice a many dynamic tate a the ingle-degree-of-freedom S/KS controller. ACKNOWLEDGEMENTS The author thank Hoam Abba for haring hi experimental data. REFERENCES Abba, H., Ali, A., Hahemi, S., and Werner, H. (23). LPV gain-cheduled control of a control moment gyrocope. In American Control Conference, Abba, H., Ali, A., Hahemi, S., and Werner, H. (24). LPV tate-feedback control of a control moment gyrocope. Control Engineering Practice, 24, Kurz, T., Eberhard, P., Henninger, C., and Schiehlen, W. (2). From Neweul to Neweul-M 2 : Symbolical equation of motion for multibody ytem analyi and ynthei. Multibody Sytem Dynamic, 24(), 4. Limebeer, D.J.N., Kaenally, E.M., and Perkin, J.D. (993). On the deign of robut and two degree of freedom controller. Automatica, 29(), Löfberg, J. (24). YALMIP : A toolbox for modeling and optimization in MATLAB. In Proc. CACSD Conference. Taipei, Taiwan. URL e/johanl/yalmip. Rugh, W.J. and Shamma, J.S. (2). Reearch on gain cheduling. Automatica, 36, 4 4. Saupe, F. (23). Linear Parameter Varying Control Deign for Indutrial Manipulator. Number 223 in Fortchrittberichte VDI Reihe 8. VDI Verlag, Düeldorf, Germany. ISBN Skogetad, S. and Potlethwaite, I. (). Multivariable Feedback Control. Prentice Hall, Upper Saddle River, NJ, 2nd edition. Toh, K., Tuetuencue, R., and Todd, M. (999). SDPT3 a MATLAB oftware package for emidefinite programming. Optimization Method and Software,, URL Wu, F. (995). Control of Linear Parameter Varying Sytem. Ph.D. thei, Univ. California, Berkeley. Wu, F., Yang, X., Packard, A., and Becker, G. (996). Induced L 2 -norm control for LPV ytem with bounded parameter variation rate. Int. J. Robut Nonlinear Control, 6, Zhou, K., Doyle, J.C., and Glover, K. (995). Robut and Optimal Control. Prentice Hall, Upper Saddle River, NJ. 634