Final Exam Controls Fall 2017 9:30 Am – 12:11 Pm

Final Exameleg460 Controls Fall 2017930am 1045am 1211 2017this

This document contains instructions for an individual final exam in control systems engineering. Students are required to create a Word file that includes the following components:

  • The steps taken to calculate the parameters’ values for the control system modifications.
  • The MATLAB code used for simulation and analysis.
  • The relevant figures generated during the analysis.

The file should be named following the format: LastNameFirstName.doc, and submitted via email to the instructor at [email address] by 10:45 AM on the specified exam date. Only one email containing a single Word file will be accepted for grading. Cheating will result in a failing grade.

Paper For Above instruction

The control system design process involves careful analysis and modification to achieve desired stability and performance criteria. In this particular exam, the task is to design a phase lag compensator to enhance the phase margin of an existing control system, and to analyze the system's response before and after compensation. The process involves multiple steps, including Bode plot analysis, parameter determination, MATLAB implementation, and performance evaluation.

Analysis of the Uncompensated System

The first stage involves analyzing the original system without any compensation. Using the transfer function provided (assumed known from the figure), the Bode diagram is plotted to determine the current phase margin. The phase margin is identified at the gain crossover frequency where the magnitude of the open-loop transfer function is unity (0 dB). The Bode plot reveals the gain crossover point and the phase margin, which is the difference between the actual phase at that frequency and -180°. This analysis provides the baseline for designing the compensator.

Design of the Phase Lag Compensator

The goal is to improve the phase margin by designing a phase lag compensator using the Bode plot method. The key steps include selecting the new gain crossover frequency, calculating the required phase lag, and determining the compensator parameters z, p, and k. The phase lag compensator typically has the transfer function:

Gc(s) = (s + z) / (s + p)

where z and p are the zero and pole, respectively, with z

In MATLAB, the Bode plots of the uncompensated system and the compensator are generated to visualize the changes. The parameters z, p, and k are adjusted iteratively to meet the specifications, ensuring the new phase margin is approximately ±1° or as required.

Analysis of the Compensated System

Once the compensator parameters are selected, the compensated system transfer function is computed by multiplying the original open-loop transfer function with the compensator. The Bode plot of the compensated system is then plotted to verify the increased phase margin and desirable gain crossover frequency. The phase margin should now meet the specified requirement.

Closed-Loop System Analysis

The closed-loop transfer function is derived from the open-loop transfer function using:

T(s) = G(s)Gc(s) / (1 + G(s)Gc(s))

Poles of the closed-loop system are calculated by solving for the roots of the characteristic equation 1 + G(s)Gc(s) = 0. The system's step response is simulated in MATLAB to evaluate transient response characteristics, such as rise time, overshoot, and settling time, indicating the control system's performance. These metrics are essential to assess whether the control goals are achieved.

System Controllability and Observability

Controllability and observability are fundamental properties that determine whether the system states can be manipulated and measured effectively. Using the state-space representation, matrices A, B, C, and D are defined, and controllability and observability matrices are constructed:

  • Controllability matrix: [B, AB, A^2B, ..., A^{n-1}B]
  • Observability matrix: [C; CA; CA^2; ...; CA^{n-1}]

Full rank of these matrices indicates the system is controllable and observable, respectively. These properties are crucial for designing effective state feedback controllers and observers.

Performance Evaluation

The final step involves analyzing the system's step response post-compensation, including the settling time, overshoot, and steady-state error. MATLAB simulations provide visual confirmation of improved transient behavior. The overall goal is to ensure the system is stable, responsive, and meets performance specifications.

Summary

This comprehensive control system design process illustrates the practical application of Bode plot analysis, compensator design, pole-zero placement, and MATLAB simulation tools. It highlights the importance of phase margin enhancement in control stability and performance, especially in systems requiring precise regulation. Through systematic analysis and iterative adjustment, the control system is optimized to meet the design criteria effectively.

References

  1. Ogata, K. (2010). Modern Control Engineering (5th Edition). Prentice Hall.
  2. Nise, N. S. (2015). Control Systems Engineering (7th Edition). Wiley.
  3. Kuo, B. C., & Golnaraghi, F. (2003). Automatic Control Systems (8th Edition). Wiley.
  4. Franklin, G. F., Powell, J. D., & Emami-Naeini, A. (2015). Feedback Control of Dynamic Systems. Pearson.
  5. Ogata, K. (2009). Discrete-time Control Systems. Pearson.
  6. Stephan, T., & Zhang, Y. (2018). MATLAB for Control Engineers. CRC Press.
  7. Control System Toolbox User's Guide. MathWorks. (2022).
  8. Katsuhiko, H. (2020). Control System Design: An Introduction to State-Space Methods. Springer.
  9. Åström, K. J., & Murray, R. M. (2010). Feedback Systems: An Introduction for Scientists and Engineers. Princeton University Press.
  10. Devasia, S., Pantea, C., & Neely, C. (2014). Control of Systems with Nonlinearities. SIAM.

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