Exercise 4.2

Control design ∙ Laplace/Frequency domain

Fig. 25 Block diagram of the DC-servomotor.

In Fig. 25 the block diagram of a DC-servomotor is illustrated.

In this system, the input voltage \(u\) first goes through a isolated DC-DC convert[1], which applies the desired voltage \(u\) to the motor. This converter is modeled by the first-order transfer function with time constant \(T=10\) ms.

The voltage is then applied to the motor, described by a second order transfer function with gain \(k_m=10\), electrical time constant \(T_2 =25\) ms, and mechanical time constant \(T_1 = 50\) ms.

The servomotor is first tested with \(F(s) = 1\), but the closed-loop turns out to be too slow.

1. Find \(F(s) = K\) so thatthe closed-loop is twice as fast as for \(F(s)=1\).

2. Then, find \(K\) so that the following accuracy requirements are sastisfied:

   • When \(\theta_{ref}(t) = \textrm{step}(t)\), at steady state \(\lvert \theta - \theta_{ref} \lvert \leq 0.001\)

   • When \(\theta_{ref}(t) = 10 \cdot \textrm{ramp}(t)\), at steady state \(\lvert \theta(t) - \theta_{ref}(t) \lvert \leq 0.01\)

3. (Extra!) Assume that the motor current can be measured. With the help of MATLAB, design a cascade control scheme for \(G_2(s) = \frac{Z(s)}{U(s)} = \frac{1}{(1+sT)(1+sT_2)}\) and \(G_1(s) = \frac{Y(s)}{Z(s)} = \frac{10}{s(1+sT_1)}\), so that the phase margin is at least \(60^\circ\), and the bandwidth of the servomotr is at least \(100\) rad/s.

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[1]

A DC-DC isolated converter is an electronic device that applies the desired control signal \(u\) to the motor while keeping the controller board electrically insulated from the high-current/high-voltage motor circuit.