Control of active suspension system using H and adaptive robust controls

This paper presents a control of active suspension system for quarter-car model with two-degree-of-freedom using  H and nonlinear adaptive robust control method. Suspension dynamics is linear and treated by  H method which guarantees the robustness of closed loop system under the presence of uncertainties and minimizes the effect of road disturbance to system. An Adaptive Robust Control (ARC) technique is used to design a force controller such that it is robust against actuator uncertainties. Simulation results are given for both frequency and time domains to verify the effectiveness of the designed controllers.


INTRODUCTION
Automotive suspension systems have been developed from the begin time of car industrial with a simple passive mechanism to the present with a very high level of sophistication.Suspensions incorporating active components are studied to improve the overall ride performances of automotive vehicle in recent years.Active suspension must provide a trade-off between several competing objectives: passenger comfort, small suspension stroke for packing and small tire deflection for vehicle handling.In the early studies, linear model of suspension are used with the assumption of ideal force actuator.The most applicable force actuator using in practice is hydraulic actuator that has a high non-linearity characteristic.Hence to solve completely problem, recently studies consider to the dynamics and the non-linearity of hydraulic actuator [2,7,9] .This paper presents a control of active suspension system for quarter-car model with two-degree-of-freedom by using  H and nonlinear adaptive robust control method.The system is divided into two parts: the linear part is whole system except actuator and nonlinear part is hydraulic actuator.The linear part is treated using  H control method that guarantees the robustness of closed loop system under the presence of uncertainties and minimizes the effect of disturbance.The variations of system parameters are solved by multiplicative uncertainty model.In hydraulic actuator, there are some unknown factors such as bulk modulus of hydraulic fluid that has strong effect to actuator dynamics.Hence, the nonlinear adaptive control is suitable for designing actuator controller.This paper applied the ARC technique to design a the controller robust against actuator uncertainties [3,4] .The error between desired acting force calculated from  H controller and actual force generated by hydraulic actuator is considered as the disturbance to the linear system.Simulations have been done in both frequency and time domains to verify the effectiveness of the designed controllers.

SYSTEM MODELING
The scheme of suspension system and hydraulic actuator used in this paper is described in Fig. 1.The governing dynamic equations of suspension system including hydraulic actuator can be presented as the following [9] 4 2 1   A : piston area  x is actual control force generated from actuator and d x 5 is the desired control force which is calculated from  H controller. Consider 5 x as the control input, the systems (1)-( 4) can be rewritten in the form and the measured output is the velocity of car body The augmented system

Fig. 2. Configuration of control system
The state space expression of the plant ) (s P with adding measurement noise n can be written in the following form The state space expression of the plant ) (s G can be written as follows The transfer function from disturbance to the state of the augmented system is

ADAPTIVE ROBUST CONTROL OF NONLINEAR PART
In this part we will derive the controller for hydraulic actuator used in suspension system.The controller is designed based on adaptive robust control technique proposed by Bin Yao [3]  .Consider hydraulic actuator dynamic equations ( 5)-( 6).The parameter is considered as unknown parameter . The main reason for choosing f  as unknown factor is that the bulk modulus of hydraulic fluid is known to change dramatically even when there is a small leakage between piston and cylinder.
The equation ( 5) can be written in the form The adaptive control law can be obtained as the following steps.
Step 1: Let's define Define the error variable: To find a virtual control law  for 6 x such that 5 x tracks its desired value d x 5 using the procedure suggested in [3].The term b , representing the nonlinear static gain between the flow rate and the valve opening 6 x , is a function of 6 x and also is non-smooth since 6 x appears through a discontinuous function ) sgn( 6x .So a smooth modification is needed [3]  .
Define the smooth projection The adaptive part a  and the robust control

SIMULATION RESULTS
The numerical values using in this simulation are given in the Table 1 [9] .The weighting function is chosen as

Frequency domain
The plot of uncertainties and weighting functions are given in Fig. 3. Figures ( 4)-( 6) show the gain plots for three transfer functions (10)-( 12) in cases of passive system, active system with desired force and actual force input.As shown in the figures, the designed nonlinear ARC controller can treat the nonlinearity and keep the  H frequency performance well.

Time domain
The responses of the system with step and sine wave disturbances are considered.Responses of the system in case of step disturbance are given in Figs. ( 7)-( 9).The step road velocity is of 0.1 m/s.Body acceleration and tire deflection are much reduced but the suspension deflection is higher.Responses of the system in case of sine wave disturbance are given in Figs. ( 10)-( 12).The road amplitude is assumed to be 0.1 m with frequency of 1 Hz .At this frequency, active system reduces considerably the effects of disturbance.

CONCLUSION
This paper presents a control of active suspension system using  H and nonlinear adaptive robust control method.

Fig
Fig.1 Suspension system and actuator Define parameters as the follows s m : sprung mass

Step 2 :
To find an actual control law for u

Table 1 .
Numerical values for simulation