Continuing Exercise CE6.2. Perform the analysis for both equilibrium conditions corresponding to (1) = 0 (rad) and (2) = (rad). Produce phase portraits of x2 vs x1 and x4 vs x3. - Submit a MATLAB script m-file that produces the numerical and graphical results that you integrate into your solution.

CE6.2 Using Lyapunov analysis, assess the stability properties of the CE2 system; any case will dobecause the A matrix is identical for all input-output cases, stability does not change. Lyapunov stability analysis will not succeed (why?); therefore, assess system stability via eigenvalue analysis. Plot phase portraits to reinforce your results.

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