Hybrid vehicle. In Chapter 7, Figure P7.31 shows the block

Chapter 8, Problem 71

(choose chapter or problem)

Hybrid vehicle. In Chapter 7, Figure P7.31 shows the block diagram of the speed control of an HEV rearranged as a unity feedback system (Preit1, 2007).

Let the transfer function of the speed controller be

\(G_{S C}(s)\) = \(K_{P_{s e}}+\frac{K_{I_{s e}}}{s}=\frac{K_{P_{s c}}\left(s+\frac{K_{P_{s s}}}{K_{P_{s c}}}\right)}{s}\)

a. Assume first that the speed controller is configured as a proportional controller \((K_{I_{s C}}\) = 0 and \(G_{S C}(s)\) = \(K_{P_{S C}}\). Calculate the forward path open-loop poles. Now use MATLAB to plot the system's root locus and find the gain, \(K_{P_{s c}}\) that yields a critically damped closed-loop response. Finally, plot the time-domain response, c(t), for a unit-step input using MATLAB. Note on the curve the rise time, \(T_r\), and settling time, \(T_s\).

b. Now add an integral gain, \(K_{I_{S C}}\), to the controller, such that \(K_{I_{S C}} / K_{P_{S C}}\) = 0.4. Use MATLAB to plot the root locus and find the proportional gain, \(K_{P_{s c}}\), that could lead to a closed loop unit-step response with 108 overshoot. Plot c(t) using MATLAB and note on the curve the peak time, \(T_p\), and settling time, \(T_S\). Does the response obtained resemble a second-order underdamped response?