I am looking for techniques to approximate a nonlinear suspension model with friction of the form:
x_dot = f(x,u) and y = g(x,u)
to:
x_dot = Ax +Bu and y= Cx+Du.
I am currently using Matlab for my research. I am not sure if linearizing the ODEs using partial derivatives around the equilibrium point will serve the purpose. How do I ensure the robustness of this approximation? Are there any alternative approaches to linearization?
Thanks in advance!
Dear Guru,
maybe it is worth to have a look for Takagi-Sugeno (TS) modeling.
Best regards
Hi Guru,
If you are uncomfortable with the linearization method, you may try the model-free adaptive control approach to deal with the approximation of a nonlinear vehicle suspension model.
Alternatively, you can use Carleman Linearization method of different orders at various operating points to capture the range of effective dynamics of the nonlinear suspension system.
As mentioned by Dr. Dessort, the Takagi–Sugeno (T–S) fuzzy system is also another good approach to model the nonlinear suspension system in a set of local linear state-space systems.
It strongly depends on the type of nonlinearity.
"I am not sure if linearizing the ODEs using ..." If function f is not smooth (continuously differentiable), you cannot use the "cheap" differentiation.
Try to see some papers to deal with the problem:
1) Duy Anh Nguyen et all, Study on Control of Quarter Car Suspension Using Simple Force Actuator for Education Purpose / in AETA 2013: Recent Advances in Elect. Eng… (2014). doi.org/10.1007/978-3-642-41968-3_46
2) K. Hemanth, Vertical dynamic analysis of a quarter car suspension system with MR damper, J Braz. Soc. Mech. Sci. Eng.
DOI 10.1007/s40430-015-0481-7
3) https://doi.org/10.15282/ijame.14.3.2017.2.0349
For simple differential equstions; I use Laplace tranfirm to get algebraic equations. Assumptions are needed for time equal to zero. This is known in process control of dynamic systems.
You may consider an adaptation of "State Dependence" approach by updating the linearized state equations (at equilibrium/some point) at each time step. The following publications may help.
- Sağlam F., Ünlüsoy, Y. S., “State Dependent Riccati Equation Control of an Active Hydro-Pneumatic Suspension System”, Journal of Automation and Control Research, v.1, pp. 1-10, 2014.
-Çimen T. “State-Dependent Riccati Equation Control- A Survey”, Proceedings of the 17th World Congress, The International Federation of Automatic Control, Seoul, Korea, July 6-11, 2008.
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