Chapter 7. Alternative Control Methods

Transcription

1 Chapter 7 Alternative Control Methods The purpose of this chapter is to survey other modifications to the skyhook control policy, known as the groundhook and hybrid control policies. These control policies are generally designed for two-degree-of-freedom systems and do not directly apply to the SDOF seat suspension used in this research. This chapter, however, will explore the possibility of further modeling the seat suspension as a 2DOF system. This chapter will also experimentally evaluate the merits of such control policies. 7.1 Groundhook Control Policy As was shown for an SDOF system, the optimal control policy for reducing the motion of the seat is the skyhook control policy. If we consider both the vehicle cab and seat suspension, then the system becomes a 2DOF system, as shown in Fig SEAT C 1 CAB K 2 C 2 CHASSIS Figure 7.1. A 2DOF Model of the Cab and Seat Suspension. 7

2 This configuration also represents a single suspension of a vehicle. We would like to apply the same concept as skyhook damping to form what is known as groundhook damping. To do this, let's review the skyhook configuration for a 2DOF suspension system, as shown in Fig C SKY C SKY K 2 K 2 Ideal Skyhook Realistic Skyhook Figure 7.2. Skyhook Formulation for the 2DOF System. We immediately notice a potential problem. With the skyhook damper only removing energy from, the motion of becomes excessive due to the fact that no damping is applied to it. Of course, in the realistic system, the semiactive damper will still remove some of the energy from, since the semiactive damper is mounted between the two masses. In vehicle suspensions, this problem is known as wheelhop, and can cause excessive axle (mass 2) bounce and loss of contact with the road surface. Now consider the case where we move the skyhook damper to mass 2, as shown in Fig This is known as the groundhook configuration, since we are attempting to "hook" the second mass to the ground. 71

3 C GND C GND K 2 K 2 X in, V in Ideal Groundhook Realistic Groundhook Figure 7.3. Groundhook Formulation for the 2DOF System. The potential problem with groundhook damping is that the motion of becomes undamped. However, as in the skyhook case, this will not truly happen since there is some residual damping between and due to the off-state and location of the damper. To derive the equations for the groundhook control policy, we proceed exactly as described for Eq. (2.6). The relative velocity between mass 1 and mass 2 is defined to be positive when separating. If mass 2 is moving with a positive velocity and mass 1 and mass 2 are approaching each other (V 12 is negative), the groundhook damping force should be negative (i.e., in the downward sense). Thus, V 2 V 12 < F SA = C GND V 12 (7.1) where F SA is the semiactive damper force. Again, if V 2 is positive but and are separating (V 12 is positive), the groundhook damping force is still in the negative direction. The semiactive damper, however, cannot produce a force in this direction, therefore the damping force must be made as small as possible, or ideally V 2 V 12 > F SA = (7.2) We can show the same results for all combinations of V 2 and V 12. Summarizing Eqs. (7.1) and (7.2) and rewriting them yields 72

4 -V2V > F = C V -V2V 12 < F SA = 12 SA GND 2 Similar to the skyhook system, applying the damper model to Eq. (7.3) gives (7.3) -V2V > i = βv -V2V 12 < i = 12 2 (7.4) where i is the damper current and β is a constant. Equation (7.4) is the functional representation of how groundhook control is programmed in a semiactive controller. 7.2 Hybrid Control Policy As observed earlier, the skyhook control policy minimizes seat motion (or vehicle body motion for a primary suspension) at the cost of cab motion (or axle motion), whereas groundhook control minimizes cab motion at the expense of seat motion. We would like to take advantage of the desirable aspects of each control policy and choose a more suitable compromise for a given application. This is the fundamental idea of hybrid control, shown in Fig C SKY C HYB C GND K 2 C 2 K 2 C 2 Ideal Hybrid Realistic Hybrid Figure 7.4. Hybrid Control. 73

5 The control policy is a combination of the skyhook and groundhook damper control signals: VV VV > < VV 2 12 > VV 2 12 < σ σ σ σ = V = i= H + (1- ) SKY 1 SKY { = V = GND 2 GND [ ασ SKY α σ GND ] (7.5) where σ SKY and σ GND are the skyhook and groundhook components of the current to the damper, respectively, α is the relative ratio between skyhook and groundhook control, and H is a constant gain. When α is 1, the control policy reduces to pure skyhook, while when α is, the control is purely groundhook. To practically implement hybrid control, we need to know V 1, V 2, and the relative velocity V 12. For ease of application in a research environment (i.e., proof of concept), we will measure V 1, and V 2 using an accelerometer and analog integrator exactly as the skyhook controller measures V 1. The relative velocity, V 12, is then computed using the difference between V 1 and V 2. This entire system was implemented using the dspace test system at Lord. Figure 7.5 shows a block diagram representation of the hybrid controller, while Fig. 7.6 shows the SIMULINK code for the hybrid controller. 74

6 Analog Integrator M Accelerometer Hybrid Control Policy MR Damper K Accelerometer Analog Integrator dspace Controller BASE Current Driver Same as the Lord Skyhook Controller Figure 7.5. Block Diagram of the Hybrid Controller. S Seat Acc S Base Acc ADC Unit DS112AD x(n+1)=ax(n)+bu(n) y(n)=cx(n)+du(n) Seat Accel 'Rate' Filter x(n+1)=ax(n)+bu(n) y(n)=cx(n)+du(n) Base Accel Rate Filter -.3 Abs Vel1 Scale factor * V12 V12 Gain Product Switch -.3 Abs Vel2 Scale Off State Current factor -1. Invert * Abs Abs.75 Alpha 1 One Abs Alpha * Product2 * Product3 + + Output Sum 5 Output Gain Saturation DAC Unit DS112DA S Current Product1 Switch1 Abs1 Off State Current1 Figure 7.6. SIMULINK Code for the Hybrid Controller. Using this system, any of the skyhook, groundhook, or hybrid control policies can be implemented. All of the gains internal to the SIMULINK code were adjusted so that the output in a skyhook control mode (α = 1) matched the original skyhook controller as closely as possible. 75

7 7.3 Experimental Results Figure 7.7 shows the swept sine acceleration transmissibility of the groundhook system for two gains. As expected, the resonance peak is relatively large for both cases. Figure 7.8 shows the time response for the groundhook system for H=5. The damper current often saturates at the maximum level of 1 Amp, indicating that the controller is attempting to control the system. There are, however, relatively low levels of jerk for the extreme current discontinuities. 3 Groundhook Controller, α= Acceleration Transmissibility H=25 H= Frequency, Hz Figure 7.7. Acceleration Transmissibility of Groundhook Control. 76

8 1 Groundhook Time Response, α=, H=5 Acceleration, m/s 2 Damper Current, Amps Time, Sec. Figure 7.8. Time Response of the Groundhook Controller. Figures 7.9 and 7.1 show the acceleration transmissibilities for the hybrid systems. Increasing the gain, H, shifts the transmissibility in frequency, whereas increasing α (moving towards a skyhook control) decreases the level of transmissibility. Comparing the hybrid control to a pure skyhook system shows that the skyhook system has much better high-frequency isolation. Acceleration Transmissibility Hybrid Controller, H = 32 α=.25 α=.5 α= Frequency, Hz Figure 7.9. Hybrid Transmissibility for H=32. 77

9 Acceleration Transmissibility Hybrid Controller, H = 5 α=.25 α=.5 α= Frequency, Hz Figure 7.1. Hybrid Transmissibility for H=5. Finally, Figs through 7.13 show the acceleration of the seat in the time domain for various levels of α and H. Increasing α increases the amount of jerk, which makes sense since for the pure groundhook, control there is very little jerk, while the pure skyhook control tends to have a great deal of jerk. Further, increasing the gain H also increases the amount of jerk. 1 Hybrid Controller, α=.25, H= Acceleration, m/s Time, Sec. Figure Acceleration Time Response for α=.25 and H=5. 78

10 1 Hybrid Controller, α=.75, H= Acceleration, m/s Time, Sec. Figure Acceleration Time Response for α=.75 and H=32. 1 Hybrid Controller, α=.75, H= Acceleration, m/s Time, Sec. Figure Acceleration Time Response for α=.75 and H=5. 79