Design and Simulation of Active Suspension System by Using Matlab

More documents

Recommendations

Info

Figure 8. Uncompensated Root LocusAs it was defined earlier, we require the overshoot to beless than 5%. The damping ratio z, can be found from theapproximation damping ratio equation:log( 0.05)z = −(21)2( π2 + log(0.05) )The command 'sgrid' is used to overlay desired percentovershoot line on the close-up root locus. From the Figure8., we see that there are two pairs of the poles and zeros thatare very close together. These pairs of poles and zeros arealmost on the imaginary axis and they might make the bussystem marginally stable, which might cause a problem.We have to make all of the poles and zeros move into theleft-half plane as far as possible to avoid the unstablesystem. We have to put two zeros very close to the twopoles on the imaginary axis of uncompensated system forpole-and-zero cancellation. Moreover, we put another twopoles further on the real axis to get fast response.ADDING A NOTCH FILTERThe notch filter diminishes the unfavourable dynamics ofthe object. Let's se if the notch filter (two-lead controller)will be sufficient to meet our requirements. We shall put thepoles at 30 and 60, and zeros at 3±3.5i. To our m-file, weshall add the following command and then run it:z1=3+3.5i; z2=3-3.5i; p1=30; p2=60;numc=conv([1 z1],[1 z2]);denc=conv([1 p1],[1 p2]);rlocus(conv(nump,numc),conv(denp,denc))Now that we have moved the root locus across the 5%damping ratio line, we can choose a gain that will satisfythe design requirements. Recall that we want the settlingtime and the overshoot to be as small as possible. Generally,to get a small overshoot and a fast response, we need toselect gain corresponding to a point on the root locus nearthe real axis and far from the complex axis or the point thatthe root locus crosses the desired damping ratio line. But inthis case, we need the cancellation of poles and zeros nearthe imaginary axis, so we need to select a gaincorresponding to a point on the root locus near zeros andpercent overshoot line. We can perform this in Matlab inthe following way:[k,poles]=rlocfind(conv(nump,numc),conv(denp,denc))Figure 10. Root Locus with Notch Filterselected point = -2.9559+3.4307ik = 2.2755e+008The value of k we use as the gain for the compensator in thefollowing way:numc=k*numc;Now let's see what the closed-loop step response looks likewith the compensator:Figure 11. Closed-loop Response with the Notch FilterFigure 9. Root Locus with Notch FilterFrom this plot we see that when the bus encounters a 0.1mstep on the road, the maximum oscillation amplitude of itsbody is about 1.65 mm, and the settling time isapproximately 2 sec. This response is satisfactory relativeto the initial requirements.6
FREQUENCY DESIGN METHODSince the phase and magnitude curve carry the completeinformation on the dynamic features of the system, it is firstnecessary to, in designing the method by the applicationfrequency region method, express the technicalrequirements regarding the quality of the system's behavior,in the language of numerous values of the parametres thattypify the appearance of these characteristics. This methodof a great importance also, because the magnitude andphase curve can be experimentally recorded without theknowing of the analytical expression of the transferfunction. The main idea of the frequency-based design is touse the Bode plot of the open-loop transfer function toestimate the closed-loop response. Adding a controller tothe system changes the open-loop Bode plot so that theclosed-loop response will also change. Let's first draw theBode plot for the original open-loop transfer function.Adding the following command into the m-file andrerunning it we get:w=logspace(-1,2);bode(nump,denp,w)Figure 12. Bode Plot of the Open Loop SystemFor easier representations of systems with naturalfrequencies of the system, we normalize and scale ourfinding before plotting the Bode plot, so that the lowfrequencyasymptote of each term is at 0 dB. Thisnormalization by adjusting the gain, K, makes it easier toadd the components of the Bode plot. The effect of K is themove of the magnitude curve up (increasing K) or down(decreasing K) by an amount 20 *logK, but the gain, K, hasno effect on the phase curve. Therefore from the previousplot, K must be equal to 100 dB or 100000 to move themagnitude curve up to 0 dB at 0.1 rad/sec. Let's go back toour m-file and add the following command before the 'bode'command and rerun the m-file:nump=100000*numpFigure 13. Bode Plot of the Open Loop SystemADDING TWO-LEAD CONTROLLERFrom the Bode plot above, we see that the phase curve isconcave at about 5 rad/sec. First, we will try to addpositive phase around this region, so that the phase willremain above the -180° line. Since a large phase marginleads to a small overshoot, we will want to add at least 150degrees of positive phase at the area near 5 rad/sec. Sinceone lead controller cannot add more than 90 degrees, wewill use a two-lead controller. As we want 150 degrees total,we will need 75 degrees from each controller:1−sin75a = = 0.01733241+sin75Now, let's determine the required space between the zeroand the pole for the desired maximum of the added phase:1T = = 1.51915W aThe value aT is significant because of the need to add themaximum phase at the desired frequency:a 0.0173324aT = == 0.02633W 5Now, let's put our two-lead controller into the system andsee what the Bode plot looks like. We shall add thefollowing command into the m-file:numc=conv([1.51915 1],[1.51915 1]);denc=conv([0.02633 1],[0.02633 1]);margin(conv(nump,numc),conv(denp,denc))Figure 14. Bode Plot of the System with Notch Filter7
Page 1 and 2: Seoul 2000 FISITA World Automotive
Page 3 and 4: When we want to consider input W(s)
Page 5: time is very short, but that the re

Design and Simulation of Active Suspension System by Using Matlab

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?