Question 3*- Linearization Consider the system in Figure 3 Cart Position where s

Question

Question 3*- Linearization Consider the system in Figure 3 Cart Position where spring is at free length Figure 3: Cart and pendulum system The equations of motion of this system are nonlinear. Do the following Note that ???-r and r = 0 is an equilibrium (ie. ? = ? = 0) if F-0 Show that if we linearize these equations about this equilibrium we obtain a) where ?-?- o' (s) F(s) the system stable? You may use the parameters 150 N/m for this and the b) ind the transfer functon 0.15 Ns/m, k M-1 kg, m 0.1 kg, L 0.2m, b remainder of the problem. c) Use MATLAB to plot the root locus for the system you found in b). Canyo stabilize the system with a gain-only compensator?

Explanation / Answer

M = 1.0;

m = 0.1;

b = 0.1;

I = 0.006;

g = 9.8;

l = 0.3;

q = (M+m)*(I+m*l^2)- (m*l)^2;

s = tf('s');

P_pend = (m*l*s/q)/(s^3 + (b*(I + m*l^2))*s^2/q - ((M + m)*m*g*l)*s/q - b*m*g*l/q);

rlocus(P_pend)

title('Root Locus of Plant (under Relative Control)')

C = 1/s;

rlocus(C*P_pend)

title('Root Locus with Fundamental Control')

zeros = zero(C*P_pend)

posts = pole(C*P_pend)

z = [-3 - 4];

p = 0;

k = 1;

C = zpk(z,p,k);

rlocus(C*P_pend)

title('Root Locus with PID Controller')

K = 20;

T = feedback(P_pend,K*C);

impulse(T)

title('Response of Pendulum Point to a Drive Unsettling influence under PID Control');

P_cart = (((I+m*l^2)/q)*s^2 - (m*g*l/q))/(s^4 + (b*(I + m*l^2))*s^3/q - ((M + m)*m*g*l)*s^2/q - b*m*g*l*s/q);

T2 = feedback(1,P_pend*C)*P_cart;

t = 0:0.01:8.5;

impulse(T2, t);

title('Response of Truck Position to a Motivation Unsettling influence under PID Control')

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