in Truck Suspension - Design-by-Simulation O. Vaculn, W. Kortum, W. Schwartz Oberpfaenhofen, D Wessling, Germany

Transcription

1 Analysis and Design of Semi-Active Damping in Truck Suspension - Design-by-Simulation O. Vaculn, W. Kortum, W. Schwartz DLR, German Aerospace Research Establishment Institute for Robotics and System Dynamics Oberpfaenhofen, D Wessling, Germany

2 Abstract This paper presents an interconnection between the multibody system simulation package SIMPACK and the control design and analysis software MATLAB. The features of this interface are demonstrated by example of a semi-active suspension of a heavy duty vehicle. A highly nonlinear model of a dynamic system is derived from an existing truck. The model is implemented in SIMPACK and then a controller is designed with aid of the MATLAB/SIMULINK tools. Goal of the control design is rst to minimize dynamic tyre forces for increasing the service life of roads and second to reduce vertical acceleration for improving ride comfort. The controller design is performed with the aid of a multi-objective parameter optimization. Keywords/ Multibody Simulation, Control Design, Vehicle Suspension, Multi-Objective Parameter Optimization 1 Introduction Design of mechanical parts of many products is more and more accompanied by integrating various electronical control devices in order to improve the behaviour of the nal product. The importance of electronically based control in traditionally purely mechanical systems has resulted in the establishment of a new multidisciplinary research eld called MECHATRONICS. The formerly separated procedures of synthesising mechanical systems and its control limited the design process. The designer previously had to include control elements in multibody system (MBS) software as 'black boxes' without large support in the area of control synthesis. The second step was to describe a model in the control design software. It required a reduction of model complexity and often an additional linearization of the model. In all cases the design of such mechatronic systems was inconvenient in particular because mechanical analysis and design and control synthesis were performed within dierent and independent packages. The most comfortable solution of the problem is a connection of an MBS software and a control design software to enable easy modelling of mechanical systems and comfortable design of controllers in an integrated fashion. Furthermore, some control design tools support export and implementation of control design to a control hardware, to eectively complete the design process of the electronic controller in such a way.

3 The possibility to transfer simulation results from the control design tool to the MBS software is also very useful, because the MBS software packages usually provide better post-processing features including 3D visualization (animation). Because of the necessity to bridge the gap an interface between the multibody software tool SIMPACK and the control design package MATLAB has been prepared. Advantages of using the interface are demonstrated on the example of designing a heavy duty vehicle semi-active suspension controller. The design of vehicle suspension generally plays an important role in providing stability, handling and ride comfort. In the area of heavy vehicles the potential road damage should also be taken into account; this is, however, usually of a greater concern to the road maintenance authorities rather than to the vehicle industry. The optimal suspension should improve the following basic requirements: regulation of body movement, regulation of suspension movement, regulation of force distribution. It is dicult to obtain good results for the aforementioned demands, which are more or less conicting for suspensions being based on passive devices. Thus, in recent years many teams have been looking into the potential of controllable suspensions. Despite of the fact that proper mechanisms of pavement damage are not precisely known, it has been shown, that uctuation of tyre forces results into decreasing of pavement serviceability [1], which could depend on certain powers of the tyre forces. The European Community supports research projects called COPERNICUS. One of them is dealing with the inuence of a semi-active suspension on ride comfort and on road damage. The project is called Semi-Active Damping of Truck Suspension and its Inuence on Driver and Road Loads (SADTS). Within this project four institutions, the German Aerospace Establishment (DLR), the Czech Technical University of Prague (CTU), the Slovak Technical University of Bratislava (STU) and the Czech truck company LIAZ work closely together. The project is dealing with semi-active control concepts [2]. The advantages of the semi-active concept are not only good feasibility, but also fail safe, low energy demands and, last but not least, easy adaptation of existing components and vehicles. Dierent control concepts are investigated. The concept with the best predicted simulation results will be implemented to the real heavy vehicle - a three axle platform truck equipped with controllable shock absorbers, see Figure 1.

4 Figure 1: Truck Selected for SADTS Investigation (from LIAZ company) The control strategy of controllable shock absorbers should achieve two main goals, minimization of dynamic tyre force uctuation and increasing driver comfort. A number of design strategies exists for semi-active suspensions [3, 4, 5]. A controller can be derived for an active suspension using linear or nonlinear design methods and then "clipped" to the semi-active conditions, in which energy can be dissipated only. A dierent method of controller design is presented in this paper. The controller is derived by simulation together with multi-objective parameter optimization (MOPO). MOPO allows to nd a satisfactory compromise among dierent performance criteria despite the fact that they conict with each other: improvement of one often means deterioration of the others. Nonlinear models can be used for the MOPO design without any restrictions. 2 SIMPACK - MATLAB/SIMULINK Interface SIMPACK is a simulation package for the analysis and design of dynamic and kinematic systems like road and rail vehicles, robots and machines. SIMPACK provides a graphical user interface, interactive model setup, static and kinematic analysis, linear/nonlinear system simulation and analysis, symbolic code generation, parameter variation, 2D plots and 3D realistic animation, [6], see Figure 2. The extensive library contains complete suspension systems, tyre models, wheel/rail modules, drive units, joints and many more. SIMPACK also allows to dene external input and output vectors to enable a model communication with other tools.

5 User Interface Model Setup Nonlinear Optimization and Control Design Tools CAD Post-Processing external Parameter Variation System Analysis Kinematics/Dynamics Linear/Nonlinear Post-Processing FTP, Covariance,... Visualization 2D 3D MBS Formalism Explicit/Implicit Symbolic/Numerical MBS Library Suspensions, Tyre, Wheel/Rail,... Linear Control Design Tools FEM Figure 2: Basic Structure of SIMPACK and Interfaces MATLAB [7] is an environment for general numeric computation, in particular control system analysis and design, which has become a standard tool in many universities and research institutes. MATLAB has been extended by many application specic toolboxes. Originally being applicable to linear systems only (the core functions of MATLAB were linear algebra methods) more recently MATLAB has been extended to deal with nonlinear systems. SIMULINK, a MATLAB extension, is an eective tool for modelling, analyzing, and simulating general linear and nonlinear systems. In SIMULINK the model is represented graphically by block diagrams; for many applications no code needs to be written. Several possibilities how to connect independent simulation packages exist, [8]. In this case the numerical integration of the complete system by one integrator is preferred. Two basic ways are then possible: A controller could be exported from MAT- LAB/SIMULINK and linked to SIMPACK, or an MBS model could be exported from SIMPACK and linked to MATLAB/SIMULINK. The main disadvantage of an integration in MATLAB/SIMULINK is the absence of a dierential algebraic equation (DAE) solver. As a result only multibody systems in ordinary dierential equation (ODE) form can be exported presently to this en-

6 vironment. Despite this fact it was decided to start with the numerical integration and design in MATLAB. The analysis and design features of MATLAB as well as the possible export of control code from SIMULINK to hardware (DSP) are good reasons for testing this strategy more deeply. The model export from SIMPACK to MATLAB/SIMULINK can be realized in three principle levels: linear system interface, symbolic code interface, function call interface. The three interfaces similar to those existing for SIMPACK / MATRIX X, [9], are presented in the following paragraphs. 2.1 Linear System Interface One possible way to describe general linear systems is by linear system matrices A, B, C, D for state space representation of the system dynamics in the form of _x = Ax + Bu; (1) y = Cx + Du: (2) Systems represented by the matrices are easily interchangeable not only between program packages, but also between dierent computer platforms. The Linear System Interface oers a capability to generate a set of the linear system matrices A, B, C, D to a MATLAB-readable M-le. The matrices represent a linearized multibody system obtained by the SIMPACK features (see Figure 3). The Linear System Interface is suitable for linear system analysis and control design. 2.2 Symbolic Code Interface Nonlinear multibody systems can be described by sets of nonlinear dierential equations. A symbolic code generated by SIMPACK is a FORTRAN program representing the nonlinear dierential equations describing certain models of multibody systems. The symbolic code has to be linked together with an interface allowing the code to be called as a SIMULINK user dened S-function. Finally, the SIMPACK model is

7 SIM PACK Analysis and Design of General Mechanical Systems m s m u. x = f(x,u) y = g(x,u).m file A B C D SIMULINK. x = Ax + Bu y = Cx + Du Figure 3: Linear System Interface represented by one block with corresponding inputs and outputs in a SIMULINK block diagram. The Symbolic Code Interface is independent of a computer platform, but if the multibody system is modied, the FORTRAN code must be generated, compiled and linked again. 2.3 Function Call Interface Alternatively to the Symbolic Code Interface the Function Call Interface generates a set of dierential equations of the multibody system using the generic SIMPACK database le in which the multibody system is described. No additional les need to be generated by SIMPACK and no compilation is necessary if the model is changed. The function call interface enables even to write data les for SIMPACK post-processing including 3D animation. It makes the interface fully bidirectional. The Function Call Interface is also created as a SIMULINK S-function, but in this case the interface includes all SIMPACK libraries, in which SIMPACK elements, equation generation and le handling are programmed, see Figure 4. The S-function has two parameters: a SIMPACK model lename and a sampling period for saving the post-processing les. SIMPACK and SIMULINK must run on the same platform, while using the Function Call Interface. 3 Multi-Objective Parameter Optimization The multi-objective parameter optimization (MOPO), [10], has become more and more a central design strategy for complex dynamic systems, [11]. "Optimal" solu-

8 SIMULINK S-function Simulink Interface C Universal Interface Fortran SIMPACK Libraries Fortran SIMPACK Database Figure 4: Structure of the Function Call Interface between SIMPACK and SIMU- LINK tions are searched in the parameter space such that certain - sometimes conicting - performance specications have to be met. The design strategy for multi-objective parameter optimization is briey sketched in Figure 5. The rst block in the automatic synthesis iteration domain includes a plant model and a controller. They have to be dened by the designer and may contain both linear and nonlinear descriptions. A free synthesis parameter vector p is tuned to achieve the optimal design. For controller design the parameter vector p usually consists of feedback gains. However, it can also include other free system parameters of the model itself. The parameter vector should be varied with respect to performance criteria vector c which is calculated in the second block. The performance criteria are summarized in this vector: c = [c 1 ; c 2 ; :::; c i ; :::] T : (3) The performance criteria consist of some outputs (or states) of the controlled system to represent the system specications in a suitable way. Root mean square values together with maximal deviations are usually used. The performance criteria c i are dependent on the parameter vector p. However the optimization methods cannot improve all performance criteria simultaneously. Some are minimized at expense of others which may increase. After a sucient number of optimization steps an improved design is reached which is Pareto-optimal, i.e. no criterion can be further improved without deteriorating

9 Engineer s Decision Domain design problem, system modelling, final design decision Vehicle Model Controller synthesis parameters p Simulation Analysis performance criteria c Optimization design parameters d Evaluation Automatic Synthesis Iteration Domain Figure 5: The Synthesis Loop of Optimization others. A task of the designer is to weight the performance criteria according to their optimization priority by coecients of the design parameter vector d. It means the proper design strategy. Each criterion is associated with the value of the vector d. d = [d 1 ; d 2 ; :::; d i ; :::]T : (4) The problem of searching for the desired Pareto-optimum can be transformed into a min/max scalar problem. As a design comparator for detecting better designs the maximum function = max i (c i (p)=d i ) (5) can be chosen. The strategy is to nd a (local) minimum = (p ) in varying the synthesis parameter vector p: = minp max i (c i (p)=d i ): (6) If 1 then vector p presents an acceptable compromise to the multi-objective optimization problem. The cost function is also denoted as a preference function. The complete strategy consists of two imbedded loops. The outer loop is initialized by the system denition and criteria. The inner loop starts on a basis of simulation and analysis initialized by a provisional parameter vector p (0) with corresponding calculated criteria values c (0).

10 4 Design of Semi-Active Suspension Controller 4.1 Model and Control Concept Vehicle Model As a rst example for testing the interface a nonlinear two mass vehicle model is used, see Figure 6. The vehicle model is equipped with a nonlinear tyre model, a nonlinear air spring model and a realistic model of controllable shock absorber, which includes not only nonlinear characteristics but also its dynamics. Parameters of the vehicle model represent one half of a rear driven vehicle axle, the axle, on which the highest road forces are expected. This axle has maximal static load of kg. The front and the rear trailing axles have static loads of kg each. m s sprung mass m s Air Spring Tyre Force Force m u unsprung mass m u Deflection Deflection Figure 6: Two Mass Vehicle Model The present vehicle test model has ve states, two degrees of freedom form four states (positions and velocities of sprung and unsprung masses) and one state is due to the dynamics of the controllable shock absorber. The vehicle model including the controllable shock absorber is prepared in SIMPACK Model of Controllable Shock Absorber The proper parameter of the shock absorber which can be controlled is the pressure in the shock absorber which can be modied by opening and closing a proportional valve: The cross section can be tuned to each arbitrary size. This allows the continuously variable adaptation of the pressure and therefore of the damper characteristics from sti to soft within a range dened by completely closed and opened valve cross section. An electrical current between zero ampere and a maximal value commands the adjustment of the valve. The shock absorber force is then a function of the actual shock absorber velocity and the valve cross section, i.e. the actual current i act. Figure 7 shows the force law

11 of controllable shock absorber, the force as a function of shock absorber velocity and current. Adjusting to another current the reaction of the shock absorber starts after a time delay of ca. ve milliseconds in a phase of adaptation which is dependent on electrodynamics, valve dynamics and oil hydraulics. The time until the shock absorber is fully adjusted to the new commanded current diers whether the valve is being closed (total time ca. 35 ms) or being opened (ca. 15 ms). In the model this dynamics is implemented by a rst order transfer function with dierent time constants for increase and decrease of current x 10 F com 1 rebound i com = 0 A 1 0 A i act F damp force [N] bounce F com 2 i com 2 i = i com max 3 i max -0.5 F com v damp relative velocity vrel(m/s) Figure 7: Parameter Array of a Controllable Shock Absorber Essential for a semi-active damper in order to dissipate a maximum of vibration energy of the vehicle is a control concept which includes the dynamics of the controllable shock absorber. In this model the damper force is given as function of actual velocity and current. So the actuator command will be a commanded damper force F com which leads to a commanded current i com between zero ampere and the maximal value. The intersection point of the line of commanded force with the line of actual velocity in the parameter array of damper, Figure 7, denes the necessary commanded current: For F com1 the commanded current i com1 = 0 A, for F com2 within the range and for F com3 the maximum.

12 4.1.3 Model Excitation Two inputs are chosen for the vehicle model: deterministic road disturbance represented by a sine wave 0.1 m high and ca. 6 m long (see Figure 8). The vehicle speed is 20 m/s; stochastic excitation representing a good asphalt road from a SIMPACK excitation database recalculated for the speed 20m/s (see Figures 9, 10). The simulation is performed for a time of four seconds in both cases. 0.2 Road Excitation [m] Time [s] Figure 8: Sine Wave Road Excitation in Time Domain 0 Amplitude [db] Frequency [rad/s] Figure 9: Power Spectral Density of Stochastic Road Excitation

13 Road Excitation [m] Time [s] Figure 10: Realization of Stochastic Road Excitation in Time Domain Control Concept An extended state feedback is chosen as a controller structure. Four state variables, positions of sprung and unsprung masses and their absolute velocities, are extended with a road prole and a velocity of the road prole, see Figure 11. These six signals are amplied and summarized. This sum is a desired (commanded) force for the controllable shock absorber. Tyre forces are in this case represented by tyre deections. The tyre model is realised by a nonlinear spring, see Figure 6, but the main nonlinearity, a lift-o eect simulation, is not likely because of the selected road inputs. Root mean square (RMS) values of tyre deection and maximal tyre deection (instead of forces) are used as performance criteria in the process of MOPO together with RMS and maximal values of sprung mass acceleration. The goal is to set-up the controller gains in order to minimize performance criteria according to chosen design parameters. All six gains of the controller are selected as synthesis parameters. The feedback controller is designed in MATLAB/SIMULINK, in which the complete model is also numerically integrated. The Function Call Interface is used to import the SIMPACK model to the MATLAB/SIMULINK environment. The MATLAB Optimization Toolbox [12] is used for multi-objective parameter optimization.

14 sprung mass position Road Generator road excitation sprung mass velocity unsprung mass position unsprung mass velocity du/dt Derivative Sprung mass acceleration 2 Outport2 desired force road excitation road excitation velocity + - Sum Tyre deflection V6_Sum output in vector 1 Outport1 Vehicle Model K Gain Run Start Optimization Figure 11: Extended State Feedback SIMULINK Block Diagram 4.2 First Results of Optimization Initial results obtained from simulations of the vehicle model with a controlled semiactive suspension are compared to results of simulations of a vehicle model equipped with a passive shock absorber with nonlinear characteristics of a serial vehicle shock absorber. The results of the simulation with the deterministic input, sine wave, are shown in Figures 12 and 13. Figure 12 presents time responses of the tyre deection and Figure 13 shows the time responses of vertical acceleration. In both cases a signicant positive inuence of the semi-active suspension to the performance criteria is visible. Relative reduction of performance criteria is shown in Figure 15. Almost 60% decrease of the RMS value of the tyre deection is reached. The maximal tyre deection is suppressed to 76% of the passive situation. The RMS value of vertical acceleration is reduced to 34% of the original value and the maximal acceleration to 48%. In the case of the stochastic excitation the tyre deection performance criteria are emphasized. It means, that the synthesis parameters allow small increase of the

15 performance criteria of vertical acceleration, but in a dened range only semi active passive 0.01 Tyre Deflection [m] Time [s] Figure 12: Tyre Deection 8 6 semi active passive 4 Vertical Acceleration [m/s^2] Time [s] Figure 13: Vertical Acceleration of Sprung Mass The results of the simulations with a stochastic road input are shown in Figure 14, where a comparison of power spectral densities of the tyre deection of the passive model and the model with the controllable shock absorber are presented. For the

16 passive model there are two signicant maxima in the characteristics, resonance of sprung mass around 7.4 rad/s which is equivalent to the sprung mass eigenfrequency and resonance around 51 rad/s according to the unsprung mass eigenfrequency. Both maxima are decreased in the case of semi-active suspension. A new maximum occurs for the frequency about 29 rad/s. It could be caused by the dynamics of the controllable shock absorber. 3 2 x 10-5 Power Spectral Density passive semi-active Frequency [rad/s] Figure 14: Power Spectral Density of Tyre Deection Relative changes of performance criteria are shown in Figure 15. The RMS value of the tyre deection is reduced by 8.6% and the maximal value of the tyre deection is decreased by 28%. The change of the maximal tyre deection is even higher than in the case of the deterministic road. Because of the synthesis parameters, the performance criteria of the vertical acceleration are worse compared to the passive case. The RMS value is increased by almost 19% and the maximal value by 10.5%. Passive Suspension Semi-Active Suspension Deterministic Road Semi-Active Suspension Stochastic Road RMS MAX RMS MAX TYRE DEFLECTION VERTICAL ACCELERATION Figure 15: Normalized Performance Criteria (Passive = 1)

17 5 Conclusions and Open Problems The paper has presented an interface between the multibody system simulation package SIMPACK and the control design tool MATLAB/SIMULINK. This interface of two software packages has resulted in a mechatronic design tool. The main advantage is that designer is exible both in multibody system modelling as well as in controller design and optimization process. Absence of a dierential algebraic equation solver is considered as a relative drawback of the interface. The design by simulation of vehicle semi-active suspension controller has been provided with the aid of the interface. The controller has been derived using multiobjective parameter optimization. Signicant advantages of semi-active suspension for the minimization of the road forces has been veried for both deterministic and stochastic road by rst promising results. The small increase in vertical acceleration in the case of the stochastic road is not considered as critical, because the design is performed for the middle axle. The inuence of middle axle to driver comfort is not signicant. The results obtained so far should be taken as intermediate. During the next period of the project the following problems will be taken into account: robustness of the controller, because of possible variations of several vehicle parameters, e.g. types of road disturbances, velocity of the vehicle, load and center of mass position. The ISO 2631 weighting should be performed for vertical acceleration evaluation. A more complex vehicle model, see Figure 16, will be used for future multi-objective parameter optimization process. Figure 16: 3D Model of the LIAZ Truck

18 6 Acknowledgments The authors would like to thank gratefully to the European Community for supporting this research project and co-operation institutions CTU, STU and LIAZ for their assistance. References [1] D. Cebon. Vehicle generated road damage: A review. Vehicle System Dynamics, 18, pages 107 { 150, [2] K. Yi and J.K. Hedrick. The use of semi-active suspensions to reduce pavement damage. Technical report, Dpt. of Mech. Eng., University of California at Berkeley, [3] P.J.Th. Venhovens. The development and implementation of adaptive semiactive suspension control. Vehicle System Dynamics, 23, pages 211{235, [4] H.E. Tseng and J.K. Hedrick. Semi-active control laws - optimal and suboptimal. Vehicle System Dynamics, 23, pages 545 { 569, [5] A. Titli and S. Boverie. Fuzzy and neuro control for semi-active and active suspension. In 1-st IFAC Workshop on Advances in Automotive Control, pages 46 { 50, Ascona, [6] W. Kortum and R.S. Sharp. Multibody computer code in vehicle system dynamics. Supplement to Vehicle System Dynamics, 22, [7] The MathWorks Web Site. WWW page: [8] W. Kortum. Trends in der Rechnersimulation im Hinblick auf die Systemdynamik Fahrzeug - Fahrweg. In VDI, Simulation und Simulatoren fur den Schienenverkehr, Munchen, [9] SIMPACK - SIMAX Manual. DLR, [10] H.D. Joos. Informationstechnische Behandlung des mehrzieligen optimierungsgestutzten regelungstechnischen Entwurfs. Dissertation, Univ. Stuttgart, [11] H. Wentscher, W. Kortum, W.R. Kruger. Fuselage vibration control using semi-active nose gear. In AGARD'95, Ban, Canada, [12] A. Grace. Optimization toolbox user's guide. The MathWorks, 1992.