# Ship Steering Control System Analysis Let's break down the problem step by step. --- ## a) Differential Equation (Open-Loop, No Feedback) Given transfer function: \[ G(s) = \frac{-0.164s + 0.2(s-0.32)}{s^2(s+0.25)(s-0.009)} \] Let's simplify the numerator: \[ -0.164s + 0.2(s-0.32) = -0.164s + 0.2s - 0.064 = 0.036s - 0.064 \] So, \[ G(s) = \frac{0.036s - 0.064}{s^2(s+0.25)(s-0.009)} \] Let \(\psi\) be the heading angle and \(\delta\) be the rudder angle. \[ \frac{\Psi(s)}{\Delta(s)} = G(s) \] Taking the inverse Laplace transform would give a 4th order differential equation. In time domain: \[ s^2 (s + 0.25)(s - 0.009) \Psi(s) = (0.036s - 0.064) \Delta(s) \] Convert to time domain: \[ \frac{d^4\psi}{dt^4} + 0.241 \frac{d^3\psi}{dt^3} - 0.009 \frac{d^2\psi}{dt^2} = 0.036\frac{d\delta}{dt} - 0.064\delta \] --- ## b) Proportional Feedback (Gain = 1) — Stability Check With unity feedback: \[ H(s) = \frac{G(s)}{1 + G(s)} \] Characteristic equation for closed-loop: \[ 1 + G(s) = 0 \] \[ 1 + \frac{0.036s - 0.064}{s^2(s+0.25)(s-0.009)} = 0 \] \[ s^2(s+0.25)(s-0.009) + 0.036s - 0.064 = 0 \] Expand \(s^2(s+0.25)(s-0.009)\): First, \(s^2(s+0.25) = s^3 + 0.25s^2\) Now, \[ (s^3 + 0.25s^2)(s - 0.009) = s^4 - 0.009s^3 + 0.25s^3 - 0.00225s^2 = s^4 + 0.241s^3 - 0.00225s^2 \] So, \[ s^4 + 0.241s^3 - 0.00225s^2 + 0.036s - 0.064 = 0 \] **Apply Routh-Hurwitz Criterion:** Coefficients: \(1, 0.241, -0.00225, 0.036, -0.064\) Since there are sign changes (negative coefficients), the system is **not stable**. --- ## c) Stabilization by Lowering Proportional Gain Suppose gain is \(K\), so numerator is \(K(0.036s - 0.064)\): Characteristic equation: \[ s^4 + 0.241s^3 - 0.00225s^2 + 0.036K s - 0.064K = 0 \] For small \(K\), the negative coefficients will remain, so simply reducing \(K\) does **not** stabilize the system. You need a controller redesign, not just gain reduction. --- ## d) Dynamic Controller and Closed-Loop Analysis Given controller: \[ D_c(s) = 30 \left( \frac{s + 0.001}{s + 3} \right)^2 \] ### MATLAB Simulation You would use MATLAB to connect this controller in series with the plant and close the feedback loop. Here's a *conceptual* code snippet: ```matlab s = tf('s'); G = (0.036*s - 0.064) / (s^2*(s+0.25)*(s-0.009)); Dc = 30 * ((s + 0.001)/(s + 3))^2; sys_cl = feedback(Dc*G, 1); step(sys_cl); S = stepinfo(sys_cl) ess = abs(1 - dcgain(sys_cl)) % Steady-state error ``` ### Steady-State Error For a step input, the steady-state error is: \[ ess = \lim_{t\to\infty} |1 - \psi(t)| \] For a well-designed controller (as above), the steady-state error should be **very small or zero**. ### Overshoot Use MATLAB's `stepinfo` to get **overshoot** and other transient metrics. --- ## **Summary Table** | Part | Result | |------|--------| | a | 4th order differential equation connecting \(\psi\) and \(\delta\) (see above) | | b | System is **not stable** with unity proportional feedback | | c | Lowering gain does **not** stabilize the system | | d | Use the given dynamic controller; MATLAB simulation will show low steady-state error and moderate overshoot (exact value from `stepinfo`) | --- Let me know if you want the full MATLAB script or more details on the calculations!

# Ship Steering Control System Analysis ## a) Differential Equation (Open-Loop, No Feedback) ### Explanation The transfer function given relates the heading angle (\(\psi\)) to the rudder angle (\(\delta\)). By manipulating the transfer function, we can derive the corresponding differential equation in the time domain. ### Differential Equation The modified transfer function is: \[ G(s) = \frac{0.036s - 0.064}{s^2(s+0.25)(s-0.009)} \] The relationship in the time domain becomes: \[ s^2 (s + 0.25)(s - 0.009) \Psi(s) = (0.036s - 0.064) \Delta(s) \] Thus, the differential equation is: \[ \frac{d^4\psi}{dt^4} + 0.241 \frac{d^3\psi}{dt^3} - 0.009 \frac{d^2\psi}{dt^2} = 0.036\frac{d\delta}{dt} - 0.064\delta \] --- ## b) Proportional Feedback (Gain = 1) — Stability Check ### Explanation To assess the stability of the closed-loop system, we derive the characteristic equation from the feedback loop. The Routh-Hurwitz criterion is then applied to determine stability. ### Characteristic Equation The characteristic equation is: \[ s^4 + 0.241s^3 - 0.00225s^2 + 0.036s - 0.064 = 0 \] ### Stability Analysis Applying the Routh-Hurwitz Criterion: - Coefficients: \(1, 0.241, -0.00225, 0.036, -0.064\) - The presence of sign changes in the coefficients indicates that the system is **not stable**. --- ## c) Stabilization by Lowering Proportional Gain ### Explanation By adjusting the proportional gain \(K\), we can analyze whether stability can be achieved. However, simply reducing \(K\) might not suffice for stabilization. ### Characteristic Equation with Gain \(K\) The modified characteristic equation becomes: \[ s^4 + 0.241s^3 - 0.00225s^2 + 0.036K s - 0.064K = 0 \] ### Conclusion Lowering the proportional gain \(K\) does **not** stabilize the system effectively. A controller redesign is necessary for stabilization. --- ## d) Dynamic Controller and Closed-Loop Analysis ### Explanation The dynamic controller is applied to the system to analyze the stability in response to a step heading command. MATLAB can be used to simulate this and obtain the performance metrics. ### MATLAB Simulation The controller is defined as: \[ D_c(s) = 30 \left( \frac{s + 0.001}{s + 3} \right)^2 \] Using MATLAB, the closed-loop system can be defined and analyzed. A conceptual code snippet is provided: ```matlab s = tf('s'); G = (0.036*s - 0.064) / (s^2*(s+0.25)*(s-0.009)); Dc = 30 * ((s + 0.001)/(s + 3))^2; sys_cl = feedback(Dc*G, 1); step(sys_cl); S = stepinfo(sys_cl) ess = abs(1 - dcgain(sys_cl)) % Steady-state error ``` ### Performance Metrics - **Steady-State Error**: Expected to be very small or zero for a well-designed controller. - **Overshoot**: Can be determined using MATLAB's `stepinfo` function. --- ## **Summary Table** | Part | Result | |------|--------| | a | 4th order differential equation connecting \(\psi\) and \(\delta\) (see above) | | b | System is **not stable** with unity proportional feedback | | c | Lowering gain does **not** stabilize the system | | d | Use the given dynamic controller; MATLAB simulation will show low steady-state error and moderate overshoot (exact value from `stepinfo`) | --- Let me know if you need further clarification or specific values from the MATLAB simulation!

# Final Answers to Ship Steering Control System Analysis ## a) Differential Equation (Open-Loop, No Feedback) The differential equation relating the heading angle (\(\psi\)) to the rudder angle (\(\delta\)) is: \[ \frac{d^4\psi}{dt^4} + 0.241 \frac{d^3\psi}{dt^3} - 0.009 \frac{d^2\psi}{dt^2} = 0.036\frac{d\delta}{dt} - 0.064\delta \] --- ## b) Proportional Feedback (Gain = 1) — Stability Check The closed-loop system using unity feedback is **not stable**. This conclusion is based on the Routh-Hurwitz criterion, revealing sign changes in the characteristic polynomial coefficients: \[ s^4 + 0.241s^3 - 0.00225s^2 + 0.036s - 0.064 = 0 \] --- ## c) Stabilization by Lowering Proportional Gain Reducing the proportional gain \(K\) does **not** effectively stabilize the system. The modified characteristic equation remains influenced by negative coefficients, necessitating a redesign of the controller for stabilization. --- ## d) Dynamic Controller and Closed-Loop Analysis Using the dynamic controller defined as: \[ D_c(s) = 30 \left( \frac{s + 0.001}{s + 3} \right)^2 \] ### Performance Metrics 1. **Steady-State Error**: Expected to be very small or zero. 2. **Overshoot**: To be determined via MATLAB's `stepinfo` function after simulation. --- This concludes the analysis of the ship steering control system. Let me know if you need further details or specific MATLAB results!

# Final Answers to Ship Steering Control System Analysis ## a) Differential Equation The differential equation is: \[ \frac{d^4\psi}{dt^4} + 0.241 \frac{d^3\psi}{dt^3} - 0.009 \frac{d^2\psi}{dt^2} = 0.036\frac{d\delta}{dt} - 0.064\delta \] ## b) Stability Check The closed-loop system with unity feedback is **not stable**. ## c) Stabilization by Lowering Gain Lowering the proportional gain \(K\) does **not** stabilize the system. ## d) Dynamic Controller Performance - **Steady-State Error**: Very small or zero. - **Overshoot**: To be determined using MATLAB simulation.