Simulation of an Active Suspension Using PID Control
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The document simulates an active vehicle suspension using PID control. It builds a quarter car model and analyzes the controllability and observability. Numerical simulations show the active suspension with PID controller improves performance over the passive suspension by reducing displacement peaks and settling times for step, harmonic, and noise road profiles. The active suspension improves comfort but has higher manufacturing costs than passive suspensions.
Simulation of an Active Suspension Using PID Control
- 1. Simulation of an Active Suspension Using PID Control Darlan Ferreira de Sousa Suzana Moreira Avila Faculdade UnB-Gama Universidade de Brasilia - Brazil
- 2. Summary Suspension Objective Mathematical Formulation Numerical Results Conclusion
- 3. Suspension A vehicular suspension system has the main purpose of adapting the car's behavior based on two main parameters, comfort and stability. This performance is related to features like stiffness and damping coefficient of the suspension components such as springs, dampers and actuators.
- 4. Objective This work studies the performance of an active suspension, with a quarter car vehicle model using a PID controller. Controllability and observability properties are analysed in a way to verify good control performance. The analysis is carried out using MATLAB and SIMULINK toolbox capabilities.
- 5. Mathematical Formulation Suspension systems can be modeled approximately by spring-mass-dashpot system of two degrees of freedom. It is called a quarter car model, where the sprung mass Ms is attached by the suspension, modelled as a spring and a damper, to the unsprung mass Mu. The spring stiffness is given by ks and the damping coefficient is bs. When considering an active suspension, the automatic actuator is connected and is modelled by a control force u. The tire is represented by a spring with stiffness kt. Quarter car model
- 6. Mathematical Formulation The motion governing equation can be rewritten in a state space form: Where A is the state space matrix, B is the input matrix, C is the output matrix, D is the direct transmission matrix and U is the input of system, to the quarter car system presented in matrices A and B
- 7. Mathematical Formulation Automatic Control - PID control (proportional-integral-derivative) compares the real value of the output quantity with the reference value (target value), determines the deviation and produces a control signal which will reduce the deviation to zero or a small value. The transfer function in this case is given by: 𝑃(𝑠) 𝐸(𝑠) = 𝐾 𝑝 1 + 1 𝑇𝑖 𝑠 + 𝑇𝑑 𝑠 Where Kp represents the proportional gain, Td represents the time derivative and Ti the integral time. The first term of the transfer function corresponds to proportional control, integral control to the second and so on
- 8. Mathematical Formulation The open loop block diagram used to analyze the passive suspension:
- 9. Mathematical Formulation In active suspension case the block diagram is a closed-loop diagram:
- 11. Numerical Results Controlability The controllability matrix rank was 4 indicating that the system is controllable.
- 12. Numerical Results Observability In the case of observability property, three different cases were studied: a) measuring only x1; b) measuring only x2; c) measuring both x1 and x2. In all cases the rank of the observability matrix was 4 indicating that the system is observable no matter measuring only one output or both of them.
- 13. Numerical Results – Passive Suspension It can be observed an overshoot of 53% and a suspension time response of 3,1 s. These values can be minimized to improve passengers comfort by the installation of an active suspension. Sprung mass displacement time history when subjected to a step load profile
- 14. Numerical Results – Active Suspension Sprung mass displacement time history – step road profile. Active suspension improve the system behavior with an overshoot of only 29%, reducing the maximum displacement on 45,3%. Suspension time response reduced from 3,1 s to 2,3 s (25,8%). -0,02 0 0,02 0,04 0,06 0,08 0,1 0,12 0,14 0,16 0,18 0 1 2 3 4 5 6 7 8 9 10 Displacament (meter) Time (sec) Step Open Loop PID Tune
- 15. Numerical Results Sprung mass displacement time history – harmonic road profile It can be observed that the maximum displacement reduced about 83,8% comparing active to passive case. And also on the steady state response a good improvement on performance is achieved. -0,015 -0,01 -0,005 0 0,005 0,01 0,015 0 1 2 3 4 5 6 7 8 9 10 Displacament (meter) Time (sec) Harmonic Open Loop PID Tune
- 16. Numerical Results Sprung mass displacement time history – White noise road profile it can be noticed that also in this case active PID suspension achieves a very good performance. When comparing maximum sprung mass displacement a 72,5% reduction is reached. -0,03 -0,02 -0,01 0 0,01 0,02 0,03 0 1 2 3 4 5 6 7 8 9 10 Displacament (meter) Time (sec) White Noise Open Loop PID Tune
- 17. Conclusion The system showed up to be controllable and observable. Comparing the performance of an active suspension, designed with a PID controller, with the passive one, it can be observed a considerable improvement on efficiency for all the road profiles considered. In the case of the harmonic road profile a reduction of 83,8% on the maximum sprung mass displacement was found out.
- 18. Conclusion Setting the PID parameters through Tune Simulink tool, is merely a basis for designing the controller. It is recommended verify what is the influence of the PID gains on the behavior of the controller to make a more detailed analysis. It has been found that the action of PID control showed satisfactory results improving the performance of the suspension system, however active suspensions still have a very high cost of manufacturing, installation and maintenance compared to passive suspensions, that is the reason it is not widespread in series productions of vehicles.