The Transfer Function Role Rate, Step Responses, Controller Gain Kp, Routh Hurwitz & Proportional Kp - IT Assignment Help

Assignment Task -

Figure 1.c shows the step responses of the direction control system to a 1o direction change command, for different values of controller gain Kp.

a) From the plot shown in figure 1.3, which controller gain provides the best response in terms of stability and speed of response taken together?

b)Derive the transfer function φ(s)/φc(s) from the block diagram of figure 1.1. 

c) Derive the overall transfer function ψ(s)/ψcom(s) taking g = 9.81 m/s2 and U0 = 150 m/s.

d)Using the Routh Hurwitz criterion finds the maximum value which Kp can take before the direction control system becomes unstable.

2. Consider the inverted pendulum system shown in figure 2.1 below. The input is the force F and the output is the angle θ between the vertical axis and the rod as shown in the figure below.

a) Calculate the poles of the transfer function G(s) and explain why this system is unstable. Describe the behavior of the output θ(t) in response to a unity impulse force  F(t) = δ(t). 

b) It is proposed to stabilize the closed-loop system with the introduction of a suitable proportional + derivative controller as shown in figure 2.2 below. 

The following transfer function is obtained when neglecting the influence of friction on the system. 

(i) Find the range of values of the proportional Kp and derivative Kd controller gains ensuring the stability of the closed-loop system.

(ii) Calculate the steady-state error to a unit step input.

c) The system specifications require a 2% settling time of 1 second or less, and overshoot to a step input of no more than 5%. 

(i) Translate the transient specification into a dominant pair of complex conjugate poles and calculate the damping ratio ζ and natural frequency ωn.

(ii) Find the proportional Kp and derivative Kd controller gains that achieve these transient specifications.

(iii) Calculate the steady-state error to a unit step input. 

d)The system specifications also require a steady-state error of 0.2 or less to a unit step input.  

 (i) Design an additional in-series lag compensator as shown in figure 2.3 to achieve this steady-state specification, whilst not adversely affecting the transient response from part (c).

 (ii) Calculate the steady-state error and verify whether it is below the target value of 0.2. 

4. (a) Consider the system shown below in figure 1.1.

(i) Plot the Bode diagram of the open-loop transfer function G(s) using MATLAB.  

(ii) Find the gain crossover frequency ω1 from the Bode plot of the magnitude.

(b) The system specifications require a ±2% settling time of 1 second or less and overshoot to a step input of no more than 5%. A lead compensator is introduced in the feedforward path as shown in figure 4.2 below.

(i) Translate the transient specification into a dominant pair of complex conjugate poles and Calculate the compensator parameters, Kc, α, zc and pc. 

(ii) Using MATLAB plot the Bode diagram of the open-loop transfer function  Gc(s)*G(s).

(iii) Find the gain crossover frequency ω2 from the Bode plot of the magnitude.  

(iv) Calculate the phase of Gc(jω)G(jω) at this gain crossover frequency ω2 and,  subsequently, calculate the phase margin.

(c) It is proposed to keep the overshoot at exactly 5% for the system with the compensator. 

(i) What is the corresponding damping ratio and subsequently, phase margin?  

(ii) Find from the Magnitude plot on the Bode diagram the increment in gain K(dB) required to obtain the required phase margin found in (i) and the new gain crossover frequency ωgc. 

(iii) Calculate the new gain of the compensator K Kc α and the new transfer function of the compensator. 

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