Question

Operation Research

Discuss the application of the Routh-Hurwitz stability criterion in control system analysis. What are its main limitations?I6 marke1

What is a transfer function? Discuss its main characteristics.

Consider the armature-controlled DC motor shown in Figure 2.1. The electrical system is represented by an armature circuit, in which ua is the applied armature voltage, ia the armature current, Ra the armature resistance, La the armature inductance and eb is the back electromotive force generated in the armature. The mechanical part is represented by a rotational system, in whichJ is the moment of inertia due to the rotor and the load, B the viscous rotational damping associated with the load, 0 is the angular position, Tm is the torque produced by the motor and TI, is the load torque.

The system is modelled by the following differential and algebraic equations

\begin{aligned} R_{a} i_{\alpha}(l)+L_{a} \frac{\mathrm{d} i_{a}(t)}{\mathrm{d} t}+e_{b}(t) &=v_{a}(t) \\ J \frac{\mathrm{d}^{2} \theta(t)}{\mathrm{d} t^{2}}+B \frac{\mathrm{d} \theta(t)}{\mathrm{d} t} &=\tau_{m}(t)-\tau_{L}(t) \\ e_{b}(t) &=K_{L} \omega(t) \\ \tau_{n}(t) &=K_{T} i_{a}(t) \end{aligned}

where KE is the back electromotive force constant of the motor, Kr is the torque constant of the motor, and w is the angular velocity.

Obtain a state-space representation of the armature-controlled DC motor if the state variables are the armature current and the angular velocity, the inputs to the system are the armature voltage and the load torque, and the only output is the angular velocity. Clearly identify matrices A, B, C and D.

[Hint: The derivative of angular position is angular velocity and the derivative of angular velocity is angular acceleration].