MODELLING AND CONTROL OF A SINGLE DEGREE-OF-FREEDOM DYNAMIC WIND TUNNEL RIG

Transcription

1 MODELLING AND CONTROL OF A SINGLE DEGREE-OF-FREEDOM DYNAMIC WIND TUNNEL RIG Pal M. Davison, Mario di Bernardo, Mark H. Lowenberg Departments of Aerospace Engineering and Engineering Mathematics, University of Bristol, Qeen s Bilding, University Walk, Bristol, BS8 1TR, UK, pal.davison@bristol.ac.k Department of Engineering Mathematics, University of Bristol, Qeen s Bilding, University Walk, Bristol, BS8 1TR, UK, m.dibernardo@bristol.ac.k Department of Aerospace Engineering, University of Bristol, Qeen s Bilding, University Walk, Bristol, BS8 1TR, UK, m.lowenberg@bristol.ac.k Keywords: model identification, wind tnnel, non-linear, controller design, bifrcation. Abstract Development of flight control laws for modern highlymanoevrable aircraft generally takes many years. Controller design is often only started after a detailed mathematical model of the aircraft exists, which may well already be many years into the development programme. This problem is being exacerbated by the need for modern aircraft to be controllable at extremely high angles of attack (p to 8 ), which increases control system complexity and reires more time to be spent in development and testing. A possible method for redcing the time spent in both aerodynamic and control system design phases is being developed at the University of Bristol. By sing a novel, mlti-axis dynamic wind tnnel rig it is hoped that control system design can be condcted in parallel with aerodynamic development and mathematical modelling to redce the overall aircraft procrement time. This paper aims to give a brief otline of the pilot rig at the University of Bristol and present modelling and experimental control reslts obtained on the rig in a single degree-of-freedom configration. 1 Introdction There have been relatively few attempts to design control laws for highly-manoevrable aircraft sing actively controlled models flying in mltiple degrees of freedom in wind tnnels. A single degree-of-freedom (DOF) rig (pitch only) with actated control srfaces was sed at the National Aerospace Laboratories in India to model time-dependent effects on highly swept delta wings [9]. A 2 DOF rig (roll and yaw) sing active control srfaces agmented with compressed-air blowing was developed at Cambridge University and sccessflly sed to develop lateral-directional controllers for the HHIRM model sing H methods [8]. In [] a 4 DOF rig was sed to develop gst alleviation controllers for a small, high-wing trboprop aircraft. The model was free to roll, pitch and yaw abot a central gimbal within the model, and cold slide p and down a vertical wire monted in the tnnel. Tore-motors were sed to actate elevators, ailerons, and trailing edge flaps. A similar 4 DOF rig was developed at Cranfield Institte of Technology [1] to extract aerodynamic models, develop control systems and perform wind tnnel simlations of dynamic motions. A fll 6 DOF free-flight rig was developed by NASA for the 3 by 6 Langley fll-scale tnnel (e.g. [4]). The model to be tested was free to fly within the tnnel working section, with electrical power, compressed air (for proplsion) and control and feedback signals being fed from the top of the tnnel via a slack mbilical chord. Controllers were implemented sing compters otside the tnnel, with three pilots providing control inpts and handling ality feedback. The cost of this type of rig is prohibitive in most cases de to the size of the wind tnnel reired, complexity of the model and nmber of operators needed. The Pendlm Spport (PSR) [3, 6, 7] was proposed by TsAGI in Rssia, and hopes to address some of the problems associated with obtaining aerodynamic derivatives and designing control laws, most notably time and cost. The rig is mltiprpose in natre; it can be sed for aerodynamic modelling (where periodic and arbitrary motions can be generated), control system design and evalation sing active control srfaces. A DOF pilot rig has been bilt and is crrently being tested at the University of Bristol, however the reslts presented here are for single degree of freedom tests carried ot dring development of the rig. The rest of the paper is arranged as follows. In section 2 the wind tnnel rig at the University of Bristol is described in more detail. Section 3 otlines the non-linear behavior of the rig, and describes the novel mathematical modelling technie developed to captre the observed experimental dynamics. Example model validation reslts are also presented in Section 3. In Section 4 some initial control system comparisons are presented and discssed, before conclsions and recommendations are given in Section. 2 Wind Tnnel Model The model being sed for the pilot rig is an approximate 1/16th scale BAe Hawk, constrcted mainly of fibreglass covered wood (Figre 1). The dimensions of the model are: wing span.612m, length.6m, wing srface area.78m 2 and mean aerodynamic chord.13m. When balanced abot the rota-

2 δe δe Figre 1: Wind tnnel rig in the Department 7 open-jet tnnel. tion point the weight of the model withot gimbal is 1.8kg. Alminim gimbals are sed to achieve rotation in p to five degrees of freedom (three degrees in the model (roll, pitch and yaw) and two at the tnnel monting point to give (constrained) heave and lateral motion), sing precision ball bearings to minimise friction. Precision carbon-film potentiometers are sed for anglar position feedback (accrate to ±. ), and solid-state rate gyros for anglar velocity (±.2 /s accracy). For the reslts presented here, the model is monted inverted in a single DOF only (pitch). Moment of inertia in pitch is.343kgm 2 when balanced abot the pivot point. The model has all-moving tailplanes which are directly driven by miniatre model aircraft servos. A dspace DS113 real-time control system is sed for data acisition and control, sing Matlab/Simlink and Real-Time Workshop for rapid control system prototyping. For all reslts presented here, the sampling rate was 1Hz. The wind tnnel sed for testing is a 1.1m diameter open-jet tnnel with a maximm speed of 4m/s, and a trblence level of approximately 1.% at 2m/s. All tests were performed at 2m/s (corresponding to Re = based on model wing chord). 3 Open-Loop Experimental Reslts and Modelling To aid the development of controllers for the rig it was necessary to first develop a mathematical model of the system. Even in a single DOF the model exhibits some interesting non-linear behavior in the form of stable and nstable limit cycle regions, with both sb- and spercritical Hopf bifrcations. Experimental bifrcation diagrams were derived to analyse the dynamics (Figre 2). These were constrcted by slowly varying the inpt (symmetric tailplane deflection) from one deflection limit to the other over a one hor period. Tests were performed sing both increasing and decreasing tailplane angle, and revealed a region of hysteresis associated with the bifrcations at δ e 2. The bifrcation diagrams were constrcted by only plotting Figre 2: Experimental bifrcation diagrams for the single degree-of-freedom wind tnnel model with decreasing and increasing tailplane deflection. points at which the absolte vale of pitch rate,, was less than 2 /s1. Figre 2 shows some interesting non-linear behavior in the form of a large amplitde limit cycle ( 2 < δe < 1 ), a small amplitde limit cycle at low angleof-attack ( 1 < δe < 1 ), and a region of hysteresis at 21 < δe < 19. An innovative method for accrately modelling the observed experimental dynamics was developed, making explicit se of the experimental bifrcation diagram, as well as time histories of the motion (for the sake of brevity, the reader is referred to [2] for a detailed description of the identification procedre). The model retrns pitch acceleration ( ) as a fnction of its states, the pitch angle and the pitch rate (also labelled ), and the control inpt corresponding to the tailplane deflection δe. The limit cycle amplitdes, stability and freencies observed in the experimental reslts were sed to identify the model parameters as described in [2]. Figre 3 shows the bifrcation diagram for the mathematical model of the rig which can be seen to captre well the reslts of the experiments. A second order rate and position limited actator model was developed from freency-sweep tests on the model servos. The actator model incldes a large deadband (±.7 ) and stiction, giving significant hysteresis. 3.1 Model Validation Open-loop validation of the model is presented in [2]. An example comparison between experimental time history and nmerical simlation is given in Figre 4, for a fixed tailplane deflection of 1. The model is released from rest at a point off the limit cycle, and the oscillatory decay shows very good agreement between experimental reslts and the mathematical model. Leaving to [2] frther details on the model derivation and its validation, focs is now trned to its se for the synthesis of ap1 de to experimental noise it is not possible to plot prely points at which is zero, as one wold wish.

3 Stable Fixed Point Unstable Fixed Point Stable Limit Cycle Unstable Limit Cycle Hopf Point (c) 2 (d) ( /s) sim ( /s) δ e 1 (e) 1 (f) Figre 3: Bifrcation diagram for the mathematical model. δ edemand 1 1 δ edemand sim 1 1 propriate control laws for the rig. After testing the performance of different control laws nmerically on the derived model, their experimental application to the rig is presented with the aim of (i) validating the se of the model for control design and (ii) assessing the performance of different controllers on the rig Figre : Experimental evoltion,(c),(e) and nmerical simlations,(d),(f) of the system response to a step reference inpt with an open-loop feed-forward controller with (t >3s) and withot (t <3s) the pitch rate feedback compensation. Here K ff = -.7, K = Experimental Data Model Figre 4: Comparison of mathematical model with experimental reslts; model released from rest at t 4.2s(δ e = 1 ). 4 Control System Design 4.1 Open-Loop with Pitch Rate Compensator As a starting point, an open-loop controller was considered consisting of jst a proportional feed-forward gain K ff. The reference signal is the demand on the pitch angle, which is simply mltiplied by the constant feed-forward gain to prodce the control inpt corresponding to the tailplane deflection demand δ e. Figre shows the simlated and experimental response of the state variables and nder the action of the open-loop controller ( <t<3). Note the good agreement between the control performance predicted by the model nmerical simlations (Fig. -(d)-(f)) and the actal experimental reslts (Figre -(c)-(e)). No trblence model was sed in the nmerical model. The fixed feed-forward gain vale was chosen to give good cor- relation between the reference and otpt for the step responses in Figre, however, when a reference signal of a different amplitde is sed the tracking response is significantly worse given the open-loop natre of the controller and the fact that the rig is a highly non-linear system. Moreover, the open-loop response shown in Figre for t < 3s also confirms the presence of highly oscillatory modes at high angles-of-attack (α). Therefore, pitch damping was added by inclding a proportional fixed-gain pitch rate feedback as shown in Figre 6. The reslting performance of the open-loop feed-forward control in the presence of the feedback pitch rate compensator is shown in Figre for t>3s. A feedback gain of K =.2 was fond to limit RMS amplitde in the limit cycle regions withot introdcing self-excited oscillations. In fact, a consistent redction of the oscillatory modes is observed, which redces the amplitde of the high α limit cycle from approximately to 1. As can be seen from Figre -(d)-(f), again the mathematical model shows good alitative agreement with the experimental reslts bt for a slight discrepancy in absolte vales. A possible reason for this is sensitivity of the rig to centre of gravity location of the model, casing a demand Kff Figre 6: Open-loop pitch angle controller with pitch rate compensator. K

4 fixed offset in. 4.2 Non-linear Feed-Forward with Pitch Rate Feedback demand Look p Table 4.3 Non-linear Feed-Forward with PID Pitch Angle Feedback demand Look p Table PID K Figre 7: Non-linear inverse trim crve feed-forward with pitch rate compensator. To improve the performance of the controller presented in the previos section, the fixed-gain forward path is replaced with a non-linear inverse trim crve derived from the rig model in order to have an improved correlation between the reference inpt (pitch angle demand) and the state being controlled (pitch angle). The look-p table gives tailplane deflection as a fnction of pitch angle and was derived by carrying ot an extensive analysis of the model presented in [2], briefly described in Sec. 3. The reslting control scheme is shown in Figre 7. Experimental time histories of the controlled rig to sinsoidal and step reference inpts are depicted in Figre 8. In both cases, note the presence of a non-zero steady-state error in the system response. This is to be expected becase of the mainly open-loop natre of the control on. In order to eliminate this nwanted residal error, the synthesis of an appropriate feedback loop on is now investigated. Figre 9: Feed-forward and pitch angle feedback control. To compensate for the steady-state error present in the response shown in Fig. 8, a PID controller on is added, as shown in Figre 9. The presence of a derivative action in the PID controller makes the pitch rate compensation redndant, while adding a predictive element by feeding forward the rate-ofchange of the reference signal. The gains of the PID controller were tned sing different methodologies, with the best performance being obtained by trial-and-error. Namely, a small proportional gain was sed (K p =.1) in order not to case nwanted oscillatory behavior. The integral gain, K i was set to -1 to give a good compromise between response and stability, while the derivative gain, K d, was set to.2 (i.e. to the same vales of the feedback gain K sed in the pitch rate compensator presented in Section 4.1). s for this control scheme are presented in Figres 1 and 11, where the experimental behavior of the controlled system is compared with the prediction of the nmerical simlation of the model Figre 8: of experimental system to step and sine wave reference sing non-linear feed-forward control. Figre 1: Comparison of experimental and nmerical simlation reslts for large amplitde sine wave reference (feed-forward pls PID).

5 Figre 11: Large-amplitde step response for experimental rig and nmerical simlation (feed-forward pls PID). Again, a very good agreement between experiments and nmerics can be observed, frther confirming the sitability of the model presented in [2] for control design applications. The control responses show good sine wave tracking performance, with deviations from the demand occrring mainly in the high and low angle-of-attack limit cycle regions. The step response, shown in Fig. 11, is also good with rapid transients and, when compared with Fig. 8, showing the absence of any residal steady-state error. A higher overshoot is observed in the downward direction. A non-linear nconstrained optimisation fnction (Matlab fnction fminsearch) was sed to tne the PID controller gains. A series of 3 second simlations were rn with a 2 amplitde freency-swept sine wave pitch angle demand, and a cost fnction calclated as the integral of pitch angle error over the simlation. The optimisation proved to be local, depending heavily on the vales passed to the fnction, hence the gains were tned initially by trial-and-error to get a good initial gess for the optimisation rotine. The gain vales fond were K p =.33, K i = 1.24 and K d =.2. (Note the relatively large integral action needed to compensate for the non-zero steady-state error de to the non-linear pertrbation acting on the system.) An example response of the system sing these feedback gains is shown in Figres 13 and 14. It is observed that, in the absence of the non-linear feed-forward action, the PID controller reslts in a highly oscillatory response and therefore an overall poorer tracking performance. This increase in oscillation is possibly de to the larger pitch rate feedback gain sed by the PID controller (K d ) introdcing self-excited oscillations, as mentioned in Section 4.1. The step response in Fig. 14 also shows a large amont of oscillation in the low angle-of-attack region. This might be overcome by re-tning the PID gains which were originally optimised sing a sinsoidal reference signal PID Control demand PID 1 Figre 12: PID pitch angle controller. For the prposes of comparison, the feed-forward non-linear term was removed to assess the performance of jst a PID feedback controller on (Figre 12). The controller gains were tned via nmerical simlations on the model and then tested on the rig. To implement this controller on the experimental rig, the pitch angle reference signal was differentiated and sed to give a pitch rate error signal which is then fed throgh the derivative gain. This is eivalent to differentiating the pitch angle error, bt avoids noise problems associated with differentiating the pitch angle signal on the experimental rig Figre 13: Comparison of experimental and nmerical simlation reslts for large amplitde sine wave reference (PID controller). Conclsions and Ftre Work The control of a novel dynamic wind tnnel configration has been discssed. In particlar, after presenting the experimental bifrcation diagrams of the rig, the derivation and valida-

6 freedom experimental simlator. The rig is expected to provide a method for rapid prototyping of control schemes and generation of aerodynamic data, especially applicable to the development of small Unmanned Aerial Vehicles (UAVs) Figre 14: Large-amplitde step response for experimental rig and nmerical simlations (PID controller). tion of an appropriate model based on sch diagrams has been briefly discssed. The model was then sed to design different controllers which have been experimentally implemented and tested. The system response in the presence of different control laws has been presented and evalated. It was shown that the best performance can be obtained by sing a feedforward non-linear action copled with a feedback PID controller which garantees good tracking properties characterised by zero steady-state error and relatively low oscillatory modes. Both the feed-forward inverse look-p table and the tning of the PID gains were achieved throgh the nmerical analysis of the model, confirming its sitability for control system design. Ongoing work is addressing the design of more sophisticated control laws inclding gain schedling, adaptive and robst controllers in order to address typical aircraft-related isses sch as handling alities. While carrying ot the experimental work described in the paper the se of the rig for control system design has been demonstrated, however, problems and limitations have also been identified. One sch limitation is imposed by the actators sed (off-the-shelf miniatre radio-controlled model type servos); they have a relatively poor resoltion and a large position deadband which limits the accracy of reslts. Using a larger model wold allow higher specification actators to be sed. A frther limitation is tnnel trblence, as shown in the time histories presented. Ths, a representative trblence model is to be inclded in the mathematical model at a later stage. Finally, a fll DOF rig has recently been demonstrated at the University of Bristol and it is hoped will allow the comparison of mltivariable control schemes on a mlti degree-of- Acknowledgements The fnding for the pendlm rig apparats by QinetiQ Bedford is grateflly acknowledged. Thanks is de also to Hilton Kyle for help with the experimental work. References [1] M.V. Cook. On the se of small scale aircraft models for dynamic wind tnnel investigation of stability and control. Trans Inst M C, 9(4):19 197, Oct Dec [2] P.M. Davison, M.H. Lowenberg, and M. Di Bernardo. Experimental analysis and modeling of limit cycles in a dynamic wind tnnel rig. Accepted for pblication, Jornal of Aircraft, 23. [3] M.G. Goman. Mathematical modelling of high angle of attack aerodynamics for solving flight dynamics problems. Technical report, De Monfort University, Leicester, Jne Research Monograph 13. [4] S.B. Grafton, J.R. Chambers, and P.L. Coe Jr. Wind-tnnel free-flight investigation of a model of a spin-resistant fighter configration. Technical Note TN D-7716, NASA, [] B. Krag. Gst-vehicle parameter identification by dynamic simlation in wind tnnels. AGARD-CP-23, November [6] M.H. Lowenberg and H.L. Kyle. Derivation of nsteady aerodynamic models from wind tnnel mlti-axis test rigs. In Proceedings of The International Concil of the Aeronatical Sciences, , September 22. [7] M.H. Lowenberg and H.L. Kyle. Development of a pendlm spport rig dynamic wind tnnel apparats. In Proceedings of Atmospheric Flight Mechanics Conference and Exhibition, AIAA , Agst 22. [8] G. Papageorgio and K. Glover. Design, development and control of the HIRM wind tnnel model. In Proceedings of IEEE Conference on Decision and Control, December [9] M.S. Rajamrthy. Generation of comprehensive longitdinal aerodynamic data sing dynamic wind-tnnel simlation. Jornal of Aircraft, 34(1):29 33, Janary-Febrary 1997.