Control of vehicle active suspensions by using PD+PI type fuzzy logic with sliding surface

A PD+PI type fuzzy logic controller with sliding surface is presented in this study. This controller consists of two parts which are PD type and PI type fuzzy logic units. Inputs to those fuzzy logic units are the sliding surface functions and their derivatives. The integrated controller is applied to two degrees of freedom vehicle active suspension model. Both time and frequency domain analysis are evaluated. Numerical results demonstrate that the proposed controller improves the vibration isolation of the vehicle body, without causing a suspension degeneration problem and without degrading road holding very much. 1. Introduction Semi-active suspensions in which electrorheological (ER) and magnetorheological (MR) fluid dampers are frequently preferred provide considerable improvements in ride comfort of passengers especially with very small power requirements [1]. On the other hand active suspensions offer significant improvements in ride comfort if compared with passive and semi-active suspensions. Therefore, extensive studies concerning active suspensions have been carried out during last decades. Hrovat [2] applied the optimal control laws on quarter car, half car and full car models and compared their performances with their passive counterparts. D’Amato and Viassolo [3] proposed a fuzzy logic controller (FLC) for a quarter car active suspension system. By using fuzzy logic (FL) the knowledge coming from experts can be expressed by fuzzy rules. FLC is applicable to systems with uncertain mathematical model and there are two types of FLCs. The first one is the PD type FLC in which the error and its derivative are used as inputs and the control signal is the output. The second one is the PI type FLC in which error and its derivative are used as inputs and the incremental change in control signal is the output. Transient response of the PD type FLC is better than the PI type FLC, but in some cases steady state error can not be removed for PD type FL controlled system [4]. 2. Design of the controller In this section, a PD+PI type FLC with sliding surface is presented (Figure 1). The sliding function σN and its derivative N σ are used as inputs. Here the sliding function (surface) is defined as e e  + α = σ where e is the error, e is derivative of error and α is the negative value of the slope of the sliding surface. In sliding mode control (SMC), by changing the control IC-MSQUARE 2012: International Conference on Mathematical Modelling in Physical Sciences IOP Publishing Journal of Physics: Conference Series 410 (2013) 012002 doi:10.1088/1742-6596/410/1/012002 Published under licence by IOP Publishing Ltd 1 input according to certain predefined rules, system states are driven to the sliding surface and then forced to remain there. Also it is known that conventional FLCs operate like SMCs with a boundary layer [5]. This is why sliding function and its derivative were chosen as inputs in this study. The outputs of the PD+PI type FLC are control signal uN and incremental change in control signal ΔuN. Figure 1 Block diagram of the PD+PI type FLC with sliding surface Triangular membership functions for the input and output variables are presented in Figure 2. Here NB, NM, NS, Z, PS, PM and PB denote negative big, negative medium, negative small, zero, positive small, positive medium and positive big, respectively. SFi (i=1,2,3,4) are the input scaling factors and SFu and SFΔu are the output scaling factors of the PD+PI type FLC. Figure 2 Membership functions for the a) inputs variables b) output variables Table 1 Rule table for computing uN and ΔuN N σ NB NS Z PS PB σN NB NB NB NM NS Z NS NB NM NS Z PS Z NM NS Z PS PM PS NS Z PS PM PB PB Z PS PM PB PB For the PD part output of the FLC it is thought that if the states are far from the sliding surface then control input should be big and vice versa in order to bring the states of he system to the that surface. Similarly, for the PI part output of the FLC it is thought that if the states are far from the sliding surface then incremental change in control input should be increased and vice versa. The rule table for the PD+PI type FLC is constructed by using this manner and it is given in Table 1. 3. Vehicle model Quarter car model, given in Figure 3, has two degrees of freedom which are body bounce y2 and displacement of the wheel y1 that are both in vertical directions. In order to have a realistic model, the spring and damper elements used in the suspension and the tire spring have nonlinear characteristics as seen in Figure 3 and the corresponding spring and damper forces are defined as [6]: (b) (a) N σ , N σ ) , ( N N σ σ μ  NB NS Z PS PB 1 -­‐1 0 1 N u , N u Δ ) , ( N N u u Δ μ NB NM NS Z PS PM PB 1 -­‐1 0 1 Fuzzy Logic Rule Base σ σ PD t u ) (