Continuing Exercise CE7.2 parts (a) and (b). Only consider case i (input force,

Question

Continuing Exercise CE7.2 parts (a) and (b). Only consider case i (input force, output pendulum angle). Perform the analysis and design only for the equilibrium condition corresponding to = 0 (rad).

Submit a MATLAB® script m-file that produces the numerical and graphical results that you integrate into your solution.

CE7.2a Since this is a fourth-order system, we will need four desired eigenvalues for future control law design in CE2. Use a fourthorder ITAE approach with undamped natural frequency n = 3 rad/s to generate the four desired eigenvalues. Plot the desired system impulse response.

b For the desired closed-loop eigenvalues from CE7.2a, design state feedback control laws (i.e., calculate K) for all three cases from CE2.2. In each case, evaluate your results: Plot and compare the simulated open- versus closed-loop output responses for the same input cases as in CE2.2a. Be sure to scale the vertical axes so that the closed-loop responses are clearly visible.

CE2.2a Use the numerical parameters in Table 2.4 for this and all ensu ing CE2 assignments (see Figure 1.15) Simulate and plot the open-loop state variable responses for three cases (for this problem use the state-space realizations of CE1.2b); assume zero initial state for all cases [except Case i(b) below] i. Single-input, single-output: input f(t) and output (1) (a) unit impulse input f (t) and zero initial state (b) zero input -f(t) and an initial condition of (0) = 0.1 rad ii. Single-input, multiple-output: impulse input f (t) and two iii. Multiple-input, multiple-output: two unit step inputs f(t) Simulate long enough to demonstrate the steady-state behavior (zero initial conditions on all other state variables) outputs w(t) and (t) and (1) and two outputs w(t) and (1) What are the system eigenvalues? Based on these eigenvalues and the physical system, explain the system responses. TABLE 2.4 Numerical Parameters for CE2 System Parameter Value Units Name mi i 2 cart mass kg pendulum mass 0.75 m 9.8 m/s gravitational acceleration pendulum length

Explanation / Answer

m1 = 2;
m2 = 1;
b = 0.1;
I = 0.006;
g = 9.81;
l = 0.75;

A = [0 1 0 0;
0 0 (m2*g*)/m1 0;
0 0 0 1;
0 0 g*(m1+m2)/m1*l 0];
B = [ 0;
(I+m1*l^2)/m1;
0;
m2*l/m1];
C = [1 0 0 0;
0 0 1 0];
D = [0;
0];

states = {'x' 'x_dot' 'phi' 'phi_dot'};
inputs = {'u'};
outputs = {'x'; 'phi'};

sys_ss = ss(A,B,C,D,'statename',states,'inputname',inputs,'outputname',outputs);

poles = eig(A)

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