Simulação por Controle por Realimentação de Estados

See discussions stats and author profiles for this publication at httpswwwresearchgatenetpublication275040898 Modeling car suspension using SpaceState Variables Research April 2015 CITATIONS 0 READS 705 2 authors Jorge Guillermo Tecnológico de Monterrey 64 PUBLICATIONS 244 CITATIONS SEE PROFILE Andrés José Rodríguez Torres Tecnológico de Monterrey 14 PUBLICATIONS 23 CITATIONS SEE PROFILE All content following this page was uploaded by Jorge Guillermo on 22 April 2019 The user has requested enhancement of the downloaded file V CONGRESO INTERNACIONAL DE INGENIERÍA MECATRÓNICA Y AUTOMATIZACIÓN CIIMA 2016 Modelling and Response of a Car Suspension Modelado y Respuesta de una Suspensión de Carro Jorge Guillermo Díaz Rodriguez Departamento de Engenharia Mecânica PUCRio CEP 22451900 RJ Brasil jorgegdiazalunopucriobr Andres José Rodriguez Torres Departamento de Engenharia Mecânica PUCRio CEP 22451900 RJ Brasil anrodrigueztorresgmailcom ResumenSe dedujo un modelo para la suspensión de un carro simplificando a un modelo masaresorteamortiguador Fue hecho en Simulink función de transferencia y espacio estados La respuesta de los métodos en lazo abierto fue la misma hallando el sistema estable por el criterio de Nyquist Un controlador PID fue diseñado y la respuesta en lazo cerrado se comparó con lazo abierto con Nyquist Bodé y salto hallando estabilidad y el overshoot se redujo considerablemente Índice de TérminosSuspensión activa control en espacioestados modelado Abstract The modelling of a car suspension was deducted for one tire set simplifying it to mass spring damper system It was done using Simulink transfer function and state space method Response of the two methods in open loop was the same finding the system to be stable using Nyquist criteria A PID controller was designed and response in closed loop was compared to open loop with Nyquist Bode and step finding the system continued stable and the overshoot lowered considerably Keywords Active suspension control state space modelling 1 INTRODUCTION The paper shows modelling and controlling one quart of a car suspension Should roads were smooth and even suspensions would not need to exist That is just the case roads have bumps and holes that make tires loose contact with the ground therefore loosing friction stability and steering capability Car manufacturers have been building suspensions with an elastic springs since the 1920s and added a viscous damper later on 1 When a wheel passes over an imperfection it experiences a vertical acceleration The suspension will absorb that energy minimizing vibration and creating a comfort sensation for passengers Lately demands for improved ride comfort and controllability of vehicles and high availability of electronic systems has motivated the development of active and semi active suspension systems Most likely they are electronically controlled and improve comfort as well as road handling 1 2 An active suspension system has the capability to adjust itself continuously to changing road conditions 3 This paper focuses in obtaining a physical model detailing the process using mechanics laws and explores the model by performing frequency and state space transformations along with evaluation of the models In future work application of more advanced techniques will be performed 2 MODELLING For the simplification used the cars weight was partitioned in four equal masses each one attached to one suspension system as shown in Figure 1 where ms represents the mass of the car herein ms represents ¼ of the car mass mt the tires mass ks and bs the spring and dampers suspension and kt the tires elasticity The control variable is position x and the actuating variable is the hydraulic cylinder force Fa Figure 1 Suspension Schematics Although Lagrangian mechanics would be the first choice the presence of nonconservative forces makes Lagrange more difficult to use in this case Following a modelling example of 9 or 10 where a similar system is modelling in an analogous way that however introduced nonlinear terms when including vibration angles this paper uses Newtonian mechanics Remembering that a spring produces a force proportional to displacement the viscous damper produces a force proportional to the positions first derivative and the mass produces a force proportional to the positions second derivative the system is modelled as particles Figure 2 shows in a free body diagram FBD the acting forces in ms Figure 2 FDB for ms Showing Inertial Forces The balance of forces for ms can be described in 1 ddotxs fracFa bs dotxt dotxs ks xt xsms 1 For mt the FBD is show in Figure 3 Figure 3 FBD for mt Showing Inertial Forces The balance of forces for mt is shown in 2 which summarizes its movement ddotxt fracFa bs dotxi dotxs ks xt xs kt xt xrmt 2 where Fa is the hydraulic actuator force placed between ms and mt as shown in Figure 1 The values for 1 and 2 are ms 250 kg Ks 18600 Nm bs 1000 Nsm mt 50 kg Kt 196000 Nm A physical model was drafted in SolidWorks in order to import it to Mathworks Simmechanics However the imported model provided erratic behaviour No further work was done using that approach Equations 1 and 2 were modelled with Simulinks 5 To test the model an arbitrary constant force of 1500N was applied simulating Fa and an external disturbance uneven road modelled as a sinus wave as shown in Figure 4 The response for the model via Simulink to a step is shown in Figure 4 A passenger feels the movement of xs It can be seen that both masses experience a deviation from the road profile Such situation would give a passenger an uneasy feeling The system reaches 63 of the final value at 011 s aprox Figure 4 Model Response to a Step External Disturbance Analysis in Frequency Domain The systems transfer function TF was calculated from Newtons second law as shown in 1 and 2 They can be rewritten as 3 and 4 ms ddotxs bs dotxs dotxt ks xs xt 0 3 mt ddotxt bs dotxi dotxs ks xi xs kt xt xr 0 4 where ddotxs dotxs dotxt xr xt are ft Applying Laplace transform to 3 and 4 5 and 6 were obtained where ddotxs dotxt xr xt are gt ms S2 xs S bs S xsS xt S ks xsS xt S 0 5 mt S2 xt S bs S xt S xs S ks xt S xs S kt xt S xr S 0 6 For the suspension the output is the car position xs and input is the road profile xr and remembering the systems transfer function TF is the quotient of output over input 6 an already simplified TF is shown in 7 TFS frackt S bt ktms mt S4 mt ms S3 bs ms mt ks mt kt S2 bs ks S ks kt 7 Replacing numerical values and simplifying 7 becomes 8 TFS frac1569 E4 00538 S 1S4 24 S3 22896 S2 15680 S 291680 8 Equation 8 is a TF of 4th magnitude with 4 complex roots on the denominator Figure 5 shows the response via TF to a V CONGRESO INTERNACIONAL DE INGENIERÍA MECATRÓNICA Y AUTOMATIZACIÓN CIIMA 2016 10 cm bump The system has an overshoot of 176 cm and reaches 63 of that value 63 cm in 03 s Figure 5 Model Response to a 10 cm bump Bode Nyquist and root locus plots for open loop were obtained using Matlab Bode is shown in Figure 6 It shows a first natural frequency of 2223 Hz corresponding to the natural resonant frequency of the suspension and a second natural resonant frequency of 942 Hz corresponding to the tire the system is attenuating the input signal the bandwidth frequency is 56 rads a DC gain of 41 dB a phase margin of 32o and a roll off gain of 05786 dB Figure 7 shows the Nyquist Diagram which depicts a crossing at 0 0003 leaving good room from modifying the system Using trial and error K value was found so the TF became as shown below which made GsHs1 where Gs is the system and Hs is the controller to find 4 3 2 16600538 1 24 22896 15680 291680 S M S S S S S The actual point in Nyquist was 0994 0 as shown in Figure 8 a Checking the response in Bode Figure 8 b the gain at 180o is almost 0 Therefore K166 full fills the condition of magnitude and angle Figure 6 Bode Diagram for the system Figure 7 Nyquist Diagram for the system The root locus diagram zoom showed in Figure 9 displays the four roots for the TF denominator in open loop which are 1166 15076i 1166 15076i 034 356i 034 056i Because all the poles are negative it can be said the system is stable 6 Figure 8 a Nyquist Diagram with K b Bode with K Figure 9 Zoom of root locus Diagram for the system Statespace Analysis Because there are four energy accumulators in the suspension ms mt ks and kt the system has four state variables These are the variables that describe the systems energy 7 Where the first following two are potential and last two represent kinetic energy X1Xs Vert displacement of the car ms X2Xt Vert displacement of the tire mt X3Xsdot Vert speed of the car X4Xtdot Vert speed of the tire It is worth say that Xr cannot be a state variable because it does not affect directly the output xs Remembering that the equations for state space are dotXAxBu input YCxD output Because the output Xs is already known then Y must be equal to Cx making D equal to 0 and C1 0 0 0 Now deriving the proposed state variables Eq 10 are obtained dotx1dotxs dotx2dotxt dotx3ddotxsfracksmsx1fracksmsx2 fracbsmsx3 fracbsmsx4 10 dotx4ddotxt fracksmtx1 fracksktmtx2 fracbsmtx3 fracbsmtx4 fracktmtxrt Then 10 can be rewritten as matrix form 9 as shown in 11 and 12 beginbmatrix dotx1 dotx2 dotx3 dotx4 endbmatrix beginbmatrix 0 0 1 0 0 0 0 1 fracksms fracksms fracbsms fracbsms fracksmt frackt ksmt fracbsmt fracbsmt endbmatrix beginbmatrix x1 x2 x3 x4 endbmatrix beginbmatrix 0 0 0 fracktmt endbmatrix xr Y 1 0 0 0T x1 x2 x3 x4 Replacing numeric values 12 becomes beginbmatrix dotx1 dotx2 dotx3 dotx4 endbmatrix beginbmatrix 0 0 1 0 0 0 0 1 744 744 4 4 372 4292 20 20 endbmatrix beginbmatrix x1 x2 x3 x4 endbmatrix beginbmatrix 0 0 0 3920 endbmatrix xr The eigenvalues for matrix A indicate the type of system response If all have negative real part the system is stable If there is a positive real part the system is unstable and the response will grow without limit as time goes on If all eigenvalues are completely real the response is exponential If at least two eigenvalues form a complex conjugate pair the response will oscillate The eigenvalues found for matrix A are 103139 641988i 103139 641988i 16861 81326i 16861 81326i They are all have real component negative Therefore the system is stable The response of the system in open loop modelled in state space is shown in Figure 10 It can be seen the response is almost identical as the response when the system was modelled via TF Figure 5 Figure 10 State Space Response to a step disturbance Observability and Controllability According to the PopovBelevitchHautus test 7 a system is controllable if for a matrix V B AB A2B A3B An1B has rank n For this case matrix V has a rank of 4 equal to As order Therefore the system is controllable Now a system is observable if for SC CA CA2 CA3 CAn1 has rank n For this case matrix S has a rank of 4 equal to As order Therefore the system is observable Matrix gain K Recalling the system poles are the eigenvalues of A a matrix K is sought that can modify the input which will modify the eigenvalues that in turn will change the systems behaviour Now the response of a system with gain is given by Eq 14 dotxtAxtBr Kxt 14 A new Acl matrix can be defined as 15 Acl A B K xt 15 Leaving the response of a system as 16 xt Acl Br 16 Where K k1 k2 k3 k4T then the new Acl defined in 15 will be Acl beginbmatrix 0 0 1 0 0 0 0 1 3725 3725 4 4 372 3920 K1 3920 K3 4292 20 3920 K3 3920 K4 20 endbmatrix Which has determinant of S4 3920 k4 S3 24 S3 3920 k2 S2 15680 k3 S2 15680 k4 S2 frac218325 S2 15680 S 15680 k1 S 15680 k2 S 291648 k3 S 291648 k4 S 291648 k1 291648 k2 291648 On the other hand sI AS4 24 S3 43664 S2 15680 S 291648 Comparing terms of the same order to find K simplifying coefficients and writing them in extended matrix form it gives beginbmatrix 1 1 0 0 1 1 186 186 0 1 4 4 0 0 1 0 endbmatrix beginbmatrix 1 1 5569 15793 endbmatrix Solving it gives values for K of Kbeginbmatrix 45694 45694 15793 15793 endbmatrix The response of the system with gain matrix K is shown in Figure 11 for a 10 cm step disturbance One can see the peak has lowered to an accepted value of 12 cm as opposed as shown in Figure 10 where the peak reaches 16 cm Figure 11 Systems Response with Gain Matrix PID The ZiegerNichols method was used to find the PID constants From the step response the following values were obtained ks0063 Tu0001 Tr00631 In Table 1 are shown the values for constants for P PI and PID control Table 1 Values for P PI and PID controllers Kp Ki Kd PID 12Trkstu 56762 2Tu0002 05Tu0 0005 PI 09Trkstu 42571 103Tu00033 P Trkstu 47302 However modelling the step response none of this set of values gave an acceptable response making the system unstable or amplifying the response Changes using criteria from Table 2 which shows recommended changes for PID values to enhance the systems response 8 were used with negative response as well Table 2 Recommended actions to modify PID values CL RESP to RISE t OVERSHOOT t SET t Kp Ki Kd V CONGRESO INTERNACIONAL DE INGENIERÍA MECATRÓNICA Y AUTOMATIZACIÓN CIIMA 2016 A PID tuning in Simulink Figure 12 was used obtaining values for Kp2000 Ki 14009577 and Kd 005 The response with such values is shown in Figure 13 where one can see the overshoot decreased to about 9 mm despite having a noise step function Figure 12 Suspension in closed loop modelled in Simulink Figure 13 Response from CL with PID via Simulink With Simulinks linearization tool Bode Nyquist and root locus diagrams were obtained as shown in Figure 14 Figure 15 and Figure 16 respectively In Bode diagram one can see the natural frequencies did not change In Nyquist the system runs clockwise therefore it remains stable and the root locus diagram the poles did not change when comparing to values in Figure 9 Figure 14 Bode diagram for the system with PID Figure 15 Nyquist diagram for the system with PID Figure 16 Root locus diagram for the system with PID Modeling PID in Simulink Having the suspension modelled in blocks the selected type of controller was PID From Figures 4 and 5 can be seen that measured outputs are xs position of ms and xt position of mt Measured input is xr The PID block is fed by the error A PID controller has three independent parameters which can be interpreted in terms of time where Kp depends on the present error Ki on the accumulation of past errors and Kd is a forecast for error All of them are based on current rate of change 4 A PID controller gives an output to take the exit variable to a desired target by using a formula shown in Eq 3 A block for such equation was built as shown in 17 1 1 1 1 N P I D s N s 17 Constants for the PID controller were taken from 1 and are Kp 7955 Ki 500 Kd 0001 Four events were tested over the model built as blocks in Simulink A sinusoidal a step saw and a ramp The systems responses are shown from Figure 17 to Figure 19 for the sinus step and saw disturbances respectively The sinus signal represents a noneven road The step signal aims to reproduce a speed bump The saw signal V CONGRESO INTERNACIONAL DE INGENIERÍA MECATRÓNICA Y AUTOMATIZACIÓN CIIMA 2016 had a frequency of 2 s and an amplitude of 005 m and simulates an unpaved road All of the disturbs started at time zero but the ramp This was done to check the systems response to a disturb after it has stabilized Figure 17 Response with a Sinus Disturbance The sinus excitation reaches 63 of it stable value at 027 s approximately whereas the step one does it at 014s and ramp at 024 s Because the saw disruption is continuous over time the system does not have time to come a stable value Although last said in all cases the system hovers over a stable value or a range Figure 18 Response with a Step Disturbance Figure 19 Response with a Saw Disturbance Figure 20 Response with a Ramp Disturbance CONCLUDING REMARKS A partition of a car was made in order to simplify the modelling A disadvantage of doing so is that rotation between axis is ignored which would introduce two extra degrees of freedom on the suspension One rotation along connected wheels and more between front and rear axis Further work ought to introduce such rotations in order to reproduce more accurately a cars suspension The proposed model reproduces the behaviour of the selected problem modelled in time state space and frequency domain Tests were performed to check the models stability Gain eigenvalues roots observability and controllability being all of them favourable The system was stable before and after the control action showing an acceptable agreement with results from 2 Following example of response evaluation of several external disturbances as presented in 11 a couple of them were modelled as signals that represented bumps or even gravel The systems response showed a constant behaviour in form although of different values for all of them V CONGRESO INTERNACIONAL DE INGENIERÍA MECATRÓNICA Y AUTOMATIZACIÓN CIIMA 2016 The selected control method creates a fast and prompt response to the systems disturbances lowering the amplitude of oscillation therefore creating a more comfortable ride for the passenger For the modelling as blocks the PID parameters were adjusted doing finetunning At the end the error response was hovering around 20 However the movement of ms 14 mass of the car did not move This would be acceptable since it is the movement that a passenger and the cars structure would actually feel The ramp disturbance showed that after being stabilized the controller does not allow the system to deviate from the established error Without the use of advanced control techniques such us fuzzy control 12 a reduction of vibration was accomplished Nevertheless this paper produced a model which was tested compared with and its results validated under different methods for future use implementing new control methods Some of the mentioned methods such us fuzzy control are already implemented in Matlabs control toolbox 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Cantilever Beam RCTA Vol 2 No 10 2007 10 Camargo J Camacho F Fernandez A Controller Design for an Inverted Pendulum Using Nonlinear Model RCTA Vol 2 No 18 2011 11 Higuera O Salmanca J Continuous and Discrete Control Design Based on LMI RCTA Vol 2 No 18 2011 12 C H Valencia M Vellasco R Tanscheit K Figueiredo Magnetorheological Damper Control in a Leg Prosthesis Mechanical Robot Intelligence Technology and Applications ISBN 9783319055817 Springer USA v p805 818 2014 View publication stats