Reference no: EM133304747

I. General requirements for all tasks (performed once before solving the tasks): -

1) Generate your own random control system that satisfy the Control system specifications given in table 1 exactly.
a. It is not allowed to have any pole zero cancellation between the poles and zeros of Gp(s) and H(s).
b. There must be complex poles or zeros in the generated system whenever possible.
c. It is allowed to have one pole or one zero of the loop transfer function in the right-hand side of s-plane (optional but not compulsory). Placement of one pole and one zero of the loop transfer function in the right-hand side of s- plane is not allowed (prohibited).
d. The characteristic equation of the generated control system should satisfy the necessary conditions of the Routh stability.
2) Find the response of the system due to finite random initial conditions only mathematically in details.
3) Draw the response of the system due to finite random initial conditions only using MATLAB.
4) Find the response of the system due to unit step function only mathematically in details.
5) Draw the response of the system due to unit step function only using MATLAB.
6) Find the response of the system due to both the initial conditions and the unit step function at the same time mathematically in details.
7) Draw the response of the system due to both the initial conditions and the unit step function at the same time using MATLAB.
8) Find the steady state part and the transient part of the response of the system due to both the initial conditions and the unit step function mathematically.

II. Tasks description:-
1) Task 1. Stability analysis using Routh-Hurwitz.
1. Find the characteristic equation of the generated control system.
2. Apply the Routh-Hurwitz stability steps in details to the generated system.
3. Specify the overall stability of the system.
4. Support your results and find the roots of the characteristic equation.
5. Draw the roots of the characteristic equation in the s-plane.

2) Task 2. Design using Routh-Hurwitz stability.
1. Add a controller {Gc(s)} to the loop transfer function given in table 1 with constant gain K{Gc(s) = K}.

2. Apply the Routh-Hurwitz stability steps in details to the controlled system.
3. Find the range of k values within the range {0 ≤ k ≤ ∞} required for absolute stability of the system according to one of the following stability scenarios: -
a. System is stable for all k values {0 ≤ k ≤ ∞}.
b. System is stable within a limited range of k values, {kmin ≤ k ≤ kmax}, Such that kmin > 0, kmax < ∞.
c. System is stable for k values that is greater than a specific value{kmin ≤ k ≤ ∞}, Such that kmin > 0.
4. Prepare a table that shows roots of the characteristic equation for every range of stable and unstable k values within the range {0 ≤ k ≤ ∞} by substituting certain numerical value for k within each range of stability and unstability.

3) Task 3. Steady state error analysis.
1. Based on task 2 previously given, substitute a specific value of k that will lead to a stable closed loop system.
2. Apply the Routh-Hurwitz stability steps in details to the controlled system based on the substituted value of k.
3. In case of non-unity feedback transfer function, transform your system to the standard unity feedback system in details.
4. Find all error constants (kp, kv, ka) for the system using step, ramp and parabolic input functions of unity magnitude.
5. Find all steady state errors for the system using step, ramp and parabolic input functions of unity magnitude.
6. Draw all error signals for the system using step, ramp and parabolic input functions using MATLAB.

4) Task 4. Drawing the roots locus.
1. Add a controller to the loop transfer function given in table 1 with constant gain K{Gc(s) = K}.
2. Draw the roots locus of the control system in a step-by-step manner based on the 11-step procedure covered in the handouts. Special focus should be given to the following steps:-
a. Real axis segment supported with a figure in the s-plane illustrating the solution.
b. Angles and intersection point of the infinite asymptotes supported with a figure in the s-plane illustrating the solution.
c. Points where the root locus cross the imaginary axis supported with a figure in the s-plane illustrating the solution.
d. Branch points on the real axis supported with a figure in the s-plane illustrating the solution.
e. The angle of departure of root locus from a complex pole and the angle of arrival of root locus to a complex zero supported with a figure in the s-plane illustrating the solution.

3. Draw the final roots locus sketch using MATLAB.