Control System Analysis and Derivative Feedback

This problem requires us to determine the value of the derivative rate feedback constant, \(K_t\), for a control system. We are given the overall transfer function of the system and a desired damping ratio.

Transfer Function Identification

The given overall transfer function of the control system is: \[ \dfrac{C(s)}{R(s)}= \dfrac{36}{s^2+3.6s+36} \] We will compare this with the standard form of a second-order system's transfer function, which is: \[ \dfrac{C(s)}{R(s)} = \dfrac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2} \] By comparing the given transfer function with the standard form, we can extract the system parameters.

• From the constant term in the denominator and the numerator, we find the natural frequency squared: \[ \omega_n^2 = 36 \] Therefore, the natural frequency \(\omega_n\) is: \[ \omega_n = \sqrt{36} = 6 \text{ rad/s} \]
• From the coefficient of the 's' term in the denominator, we get: \[ 2\zeta_{initial}\omega_n = 3.6 \] This term represents the initial damping characteristics of the system before the additional derivative rate feedback constant \(K_t\) is fully accounted for in achieving the desired damping ratio. We can calculate the initial damping ratio if needed, but the primary goal is to find \(K_t\) to achieve a new target damping.

Derivative Feedback Impact

Derivative rate feedback (with constant \(K_t\)) is typically used to increase the damping of a control system without significantly affecting its natural frequency. When derivative feedback is applied to a system, it modifies the characteristic equation (the denominator of the closed-loop transfer function).

For a system where the overall transfer function is in the form \( \dfrac{K_{gain}}{s^2 + A s + B} \), and derivative feedback \(K_t\) is added (e.g., in a unity feedback system where \(G(s) = \frac{K_{gain}}{s(s+a)}\) and the feedback is \(1+K_t s\)), the coefficient of the 's' term in the denominator is modified. The new 's' coefficient becomes \((A_{original} + K_{gain}K_t)\).

In our case, the "overall transfer function" given represents the system structure from which we derive \(\omega_n\) and the initial 's' coefficient, and then we apply \(K_t\) to modify this 's' coefficient to achieve the target damping ratio.

• The initial 's' coefficient (from the given transfer function) is \(3.6\). This is \(2\zeta_{initial}\omega_n\).
• The gain \(K_{gain}\) from the numerator is \(36\).
• The effect of derivative feedback \(K_t\) will add a term \(K_{gain}K_t\) to the 's' coefficient. So, the new 's' coefficient will be \(3.6 + 36 K_t\).

Desired Damping Ratio Calculation

The problem states that we should "Consider the Damping ratio 0.9". This is our desired damping ratio for the system after the derivative feedback is applied.

• Desired damping ratio, \(\zeta_{desired} = 0.9\).
• Natural frequency, \(\omega_n = 6 \text{ rad/s}\) (which remains unchanged by derivative feedback in this configuration).
• The desired 's' coefficient in the denominator of the standard second-order system form is \(2\zeta_{desired}\omega_n\). \[ 2\zeta_{desired}\omega_n = 2 \times 0.9 \times 6 \] \[ 2\zeta_{desired}\omega_n = 1.8 \times 6 \] \[ 2\zeta_{desired}\omega_n = 10.8 \]

Solving for Derivative Rate Feedback Constant K_t

Now, we equate the new 's' coefficient (from applying \(K_t\)) with the desired 's' coefficient to find \(K_t\).

The 's' coefficient after derivative feedback is \(3.6 + 36 K_t\). The desired 's' coefficient for \(\zeta = 0.9\) is \(10.8\).

Therefore, we set up the equation: \[ 3.6 + 36 K_t = 10.8 \] Now, we solve for \(K_t\): \[ 36 K_t = 10.8 - 3.6 \] \[ 36 K_t = 7.2 \] \[ K_t = \dfrac{7.2}{36} \] \[ K_t = \dfrac{72}{360} \] \[ K_t = \dfrac{1}{5} \] \[ K_t = 0.2 \]

Thus, the value of the Derivative rate feedback constant \(K_t\) is \(0.2\).