ECSE 404: CONTROL SYSTEMS ANALYSIS AND ACTIVE CONTROL OF HIGH-SPEED RAIL PANTOGRAPH SYSTEM MATTHEW BOWEN

Transcription

1 ECSE 44: CONTROL SYSTEMS ANALYSIS AND ACTIVE CONTROL OF HIGH-SPEED RAIL PANTOGRAPH SYSTEM MATTHEW BOWEN

2 1 Introduction Some high-speed rail systems are powered by electricity supplied to a pantograph on the train s roof from a catenary overhead as shorn in Figure 1. The force applied by the pantograph to the catenary is regulated to avoid loss of contact due to excessive transient motion. A proposed method to regulate the force uses a closed-loop feedback system, whereby a force F up, is applied to the bottom of the pantograph, resulting in an output force applied to the tatenary at the top. The transfer function of the simplified model is: G(s) = Y(s) U(s) =.7883(s ) (s s )(s s ) where y(t) is the difference between the pantograph head displacement and catenary displacement and u(t) is the upward force applied to the pantograph under active control. [1][2] Through both mathematical analysis and Matlab simulations, the above system will be modeled and simulated. Further, feedback techniques will be designed to enhance the system (using PD, PID controllers) while mainaining stability and proper steady state error margins. Offered in Appendices A and B are this analysis Matlab code and corresponding output. The formatting within this document was designed using LyX (MiKTeX).

3 2 State Space Model The above model, in the time domain, can be represented as: x (t) = A(t)x(t) + B(t)u(t) y(t) = C(t)x(t) + D(t)u(t) where, for the pantograph model whose transfer function is described in section 1, A =, B =, 1 1 C = [ ], D = [] 3 Time-Response In higher-order systems, system parameters may become more and more difficult to calculate by hand. We can exploit the fact that the poles nearest to the Im axis have the largest contribution in the response, and the system G(s) can be approximated (and normalized) to the second order as, Poles: Therefore, p 1,2 = ± j96.4 p 3,4 = ± j18.97 (dominant) G approx =.4593 s s Both the approximation s and original systems response information is tabulated in Figure 2, where, T s = 4, ζω n π T p = ω n 1 ζ 2, s 2 + 2ζω n s + ω n 2, P. O. = e ζπ 1 ζ 2 3

4 G approx G(s) Percent Overshoot 51% % Damping Ratio.29 N/A Natural Frequency 19.4 rad/s N/A Settling Time.88s.866s Peak Time.167s.134s Figure 2: Response Characteristics for G approx and G(s) In addition, a Matlab plot (Figure 3) of both responses is offered below, Figure 3:Response Plot of approximated (smoother) and actual G(s) Simply observing the real parts of the two pairs of poles ( Re{p 1 } < 1) we Re{p 2 } immediately notice that the second pair of poles is not sufficiently far away from the dominant pair to be completely ignored. The response information reflects this fact. 4 Closed-Loop System The interaction between the head of the pantograph and the catenary is realized by a spring. The output force of the spring is proportional to the displacement of this spring, which is the difference between vertical positions of the catenary and the pantograph head. The pantograph active-loop control is assuemed to be as shown in Figure 4, below: 4

5 Figure 4: Block diagram of active, closed-loop pantograph control system For gain K=1, the transfer function between F desired and F out, is found as: H(s) = F out K G(s).823 = F desired K G(s) = 823G(s) G(s) where G(s) was defined previously. The equivalent (and simplified) state space respresentation can then be found to be: x (t) = A(t)x(t) + B(t)u(t) y(t) = C(t)x(t) + D(t)u(t) where, A =, B =, 1 1 C = [ ], D = [] 5 Stability When considering closed-loop systems, it is important to define some range of stability. Here, it was found that for the system outlined in Figure 4, the system is stable for the range of controller gain, < K < where the lower bound constraint was imposed in the problem statement, and the upper bound was solved for by evaluating a Routh-table in Matlab. 5

6 6 Steady-State Error Simply observing the formula for steady-state error, e ss = 1 1 K allows us to conclude that e ss is minimized by increasing gain, K. Sweeping over the stable region of controller gains ( < K < ) using numerical methods in search for a minima confirms this fact, with e ss,min =.3471 for K = Root Locus Techniques and P-Controller The root locus plot for the system outlined in Figure 4 is shown below, in Figure 5, Figure 5: Root locus plot of feedback system of Figure 4 Assuming again a second-order approximation, in order to find a gain K that obtains an overshoot of 38%, we can make use of the percent overshoot formula provided in section 3 above and obtain ζ = In order to obtain the corresponding controller gain K, the intersection of ζ is taken with the root locus plot above, occurring at -9.4+j29 yields roughly K t =1.1*1 5. It is important to note this is the total gain, that is, K t /(all other constants in transfer function). The actual gain is then K=1.1* =

7 Using the equations for T s and T p defined earlier, it is possible to estimate these times for the second order approximation with the gain found above: T s = 4 ζω n =.42s T p = π ω n 1 ζ 2.11s These can then be compared to the actual response of the feedback system for an equivalent gain K. This response is plotted below, and its corresponding step response information is tabulated beneath it: Figure 6: Step response of feedback Percent Overshoot 58.3% Settling Time, T s.5979s Peak Time, T p.116s Figure 7: Response characteristics for feedback system Notice the second order apprpoximation has a lower % overshoot (as designed), as well as a slightly lower settling time. This is an indication that a system similar to the second order approximation may be a step in the right direction that is, eliminating the high-frequency component within G(s) would yield better results. This can be accomplished with the help of a notch filter, which will be employed in the sections below. 7

8 8 PD and PID Controllers At this stage, the settling time is greater than.5s. It is desired to have a settling time of.3s and to reduce the step response steady-state error to zero and eliminate the high-frequency oscillation in the step response, as observed above. The first step taken is to cascade a notch filter, G notch (s) = s2 + 16s + 92 (s + 6) 2 which yields a revised forward transfer function of, G (s) =.6487K(s )(s2 + 16s + 92) (s s )(s s ) In order to achieve a settling time of T s =.3s, and an overshoot of < 6%, a PD controller is designed to replace the constant-gain controller in Figure 4. Since the problem statement does not define a precise overshoot, 25% O.S. is selected arbitrarily. These design specifications yield an operating point, γ = ζω n ± jω n 1 ζ 2 = ± j3.216 Summing the angles between the poles and zeros of G (s) and the operating point, = Σz Σp + 18 o = 72.8 o θ γ G (s), for the PD controller to contribute 72.8 o, z PD = tan(72.8o ) z PD = This yields a new controller function G PD, G PD = K(s ) The corresponding gain, K, can be found from the root locus plot (as was done in the previous section). The intersection occurs at K 24. Since it is also desired to have an e ss of zero throughout the sytstem s step response, the PD controller above can be expanded to a PID controller to achieve this task. That is, we can make use of a popular design[3] to eliminate steady-state error, K(s )(s +.1) G PID = s with corresponding gain K 26. 8

9 In order to investigate whether or not we have achieved the desired specifications, two plots were generated. The first, the response is taken between -1s in order to demonstrate the T s =.3s and overshoot. The second response demonstrates the overall system functioning as planned: no error, low overshoot, and low settling times. Figure 8:Step response of G(s) feedback with PID controller and G notch, demonstrating a T s =.3s and O. S. < 6% Figure 9: Overall step response of G(S) feedback with PID controller and G notch 9

10 9 Frequency Response Techniques The bode plot is displayed below for the system with the notch filter cascaded: Figure 1: Bode diagram of system with notch filter included From this plot, the gain margin, phase margin and phase cross over frequency are tabulated below: Gain Margin, G m (96.743dB) Phase Margin, θ m Inf. Cross-over Frequency, ω cm 3.44, N/A Figure 11: Tabulated gain margin, phase margin and crossover frequencies for bode plot in figure 1. The stability, established by the above bode plot, has an upper bound at K= This is less than the previously determined value for the system that did not contain a notch filter. The discrepancy demonstrates the trade-off of cascading a notch-filter: the response no longer contains high frequency oscillations, but the range of stability was descreased as a result! 1

11 1 References [1] D.N. O Connor, S. D. Eppinger, W. P. Seering and D. N. Wormley. Active Control of a High-Speed Pantograph., Journal of Dynamic Systems, Measurement, and Control. Vol 1, March, [2] Michalska, H., ECSE 44: CONTROL SYSTEMS FINAL PROJECT. Obtained from WebCT. Fall 211. [3] Wolovich, William A. Automatic Control Systems: Basic Analysis and Design. Fort Worth: Saunders College Pub., Print. 11

12 11 Appendix A: Matlab Code A digital copy of this code* can be found at: * Be warned, because of some of the calculations required numerical methods, the code itself takes ~1 2 minutes to execute. format long g; s=tf('s'); G=.7883*(s+53.85)/((s^ *s )*(s^ *s )); P=pole(G); ApproxDen=poly([P(3) P(4)]); [y1,t1]=step(g); Gapprox=tf(1,ApproxDen); [y2,t2]=step(gapprox); Gapprox=Gapprox*(y1(length(y1))/(y2(length(t2)))); C=.4593; %Gapprox=.7883*(s+53.85)/(s^ *s ); % Gapprox2 = 1/(s^ *s ); [num,den] = tfdata(g); [A,B,C,D] = tf2ss(cell2mat(num),cell2mat(den)); disp('1. STATE SPACE MODEL '); A B C D clear A B C D num den; disp('2. TIME-RESPONSE '); disp('*** a) '); a = step(gapprox); Step = stepinfo(a); disp(sprintf('percent Overshoot: %d Peak Time: %d Settling Time: %d', Step.Overshoot, Step.PeakTime, Step.SettlingTime)); damp(gapprox) clear a Step; disp('*** b) '); a = step(g); Step = stepinfo(a); disp(sprintf('percent Overshoot: %d Peak Time: %d Settling Time: %d', Step.Overshoot, Step.PeakTime, Step.SettlingTime)); damp(g) clear a Step; disp('3. CLOSED-LOOP SYSTEM '); disp('*** a) '); K=1; H=K*G*.823/(1+K*G*.823); H disp('*** b) '); [num,den] = tfdata(h); [A,B,C,D] = tf2ss(cell2mat(num),cell2mat(den)); A B C D clear A B C D num den H K; disp('4. STABILITY '); endk=2; N=1; M=1; olds=1; news=1; oldk=1; lowsse=inf; for K=1:1:endK news = isstable(k*g*.823/(1+k*g*.823)); if olds ~= news olds=1; news=1; for i=oldk:k 12

13 end break; news = isstable(i*g*.823/(1+i*g*.823)); if olds~=news disp(sprintf('system is stable for < K < %d \n\n', oldk)); break; end sse=abs(1-dcgain(i*g*.823/(1+i*g*.823))); if (sse<lowsse) lowsse=sse; lowssek=i; end olds=news; oldk=i; end sse=abs(1-dcgain(k*g*.823/(1+k*g*.823))); if (sse<lowsse) lowsse=sse; lowssek=k; end olds=news; oldk=k; end stability=k; disp('5. STEADY STATE ERROR '); disp('*** a) & b) '); disp(sprintf('within the stable region, ie. < K < %d, the lowest steadystate error is %d at K = %d \n\n', oldk, lowsse, lowssek)); clear lowsse endk N M olds news oldk sse lowssek i K; disp('6. ROOT LOCUS TECHNIQUES & P-CONTROLLER '); disp('*** a) '); K=1; G1=K*G*.823; rlocus(g1) sgrid(.29435,); clear G1 K; disp('*** b) & c) '); disp('parts b) & c) require hand calculations, included in the report. '); disp('*** d) '); H=G*7135*823/1; step(feedback(h,1),:.1:1) stepinfo(feedback(h,1)) clear H; disp('7. PD & PID CONTROLLERS '); disp('*** a) '); Gnotch=.64877*(s+53.85)*(s^2+16*s+92)/((s^ *s+376.3)*(s^ *s+9 283)*(s+6)^2); Gnotch disp('*** b) '); K=2523; Gnotch2=K*Gnotch*(s ); Gnotch2 stepinfo(feedback(gnotch2,1)) disp('*** c) '); K=2682; Gnotch3=Gnotch*K*(s+.1)*(s )/s; Gnotch3 stepinfo(feedback(gnotch3,1)) disp('*** d) '); step(feedback(gnotch3,1),:.1:1) % step between -1s to show settling time step(feedback(gnotch3,1)) % step of entire system, looks good! disp('8. FREQUENCY RESPONSE TECHNIQUES '); bode(gnotch); [GM,PM,CF1,CF2]=margin(Gnotch) 13

14 12 Appendix B: Matlab Code Console Output* *graph screenshots excluded, since they have been included in the body of this report 1. STATE SPACE MODEL A = B = C = D = TIME-RESPONSE *** a) Percent Overshoot: e+1 Peak Time: 14 Settling Time: e+1 Eigenvalue Damping Freq. (rad/s) -4.6e e+1i 2.9e e+1-4.6e e+1i 2.9e e+1 *** b) Percent Overshoot: e+1 Peak Time: 42 Settling Time: e+2 Eigenvalue Damping Freq. (rad/s) -4.6e e+1i 2.9e e+1-4.6e e+1i 2.9e e e e+1i 8.3e e e e+1i 8.3e e+1 3. CLOSED-LOOP SYSTEM *** a) Transfer function: s^ e4 s^ e6 s^ e8 s^ e9 s e s^ s^ e4 s^ e5 s^ e8 s^ e9 s^ e1 s^ e11 s e13 *** b) A = Columns 1 through

15 Columns 6 through 8 B = C = Columns 1 through 5 Columns 6 through D = STABILITY System is stable for < K < STEADY STATE ERROR *** a) & b) Within the stable region, ie. < K < , the lowest steady-state error is e-1 at K = ROOT LOCUS TECHNIQUES & P-CONTROLLER *** a) *** b) & c) Parts b) & c) require hand calculations, included in the report. *** d) ans = RiseTime: SettlingTime: SettlingMin: SettlingMax: Overshoot: Undershoot: Peak: PeakTime: PD & PID CONTROLLERS *** a) Transfer function:.6488 s^ s^ s e s^ s^ e4 s^ e6 s^ e7 s^ e8 s e1 *** b) Transfer function: 15

16 1637 s^ e5 s^ e7 s^ e9 s e s^ s^ e4 s^ e6 s^ e7 s^ e8 s e1 ans = RiseTime: SettlingTime: SettlingMin: SettlingMax: Overshoot: Undershoot: Peak: PeakTime: *** c) Transfer function: 174 s^ e5 s^ e7 s^ e9 s^ e1 s e s^ s^ e4 s^ e6 s^ e7 s^ e8 s^ e1 s ans = RiseTime: SettlingTime: SettlingMin: SettlingMax: Overshoot: Undershoot: Peak: PeakTime: *** d) 8. FREQUENCY RESPONSE TECHNIQUES GM = PM = Inf CF1 = CF2 = NaN 16