06. MG3217 Kendali Proses S04: Routh Test dan Root Locus

Summary

This video discusses methods for analyzing the stability of a system using the Routh-Hurwitz criterion. It explains how to determine the stability of a system by analyzing its characteristic equation and using a table to check the signs of coefficients. Through examples, the script highlights how to evaluate the system's stability by ensuring that all elements in the second column are positive and non-zero. The video emphasizes the importance of understanding root behavior, real vs. imaginary roots, and avoiding oscillations for maintaining stability in a system. It also touches on the limitations of the root method and the need for graphical analysis to confirm stability.

Takeaways

• 😀 The stability of a system can be analyzed by examining its characteristic equation, particularly through methods like the Routh-Hurwitz criterion.
• 😀 If the characteristic equation has a positive root, the system is unstable, and if all roots are located on the left half of the complex plane, the system is stable.
• 😀 The Routh-Hurwitz criterion involves creating a table of coefficients and calculating intermediate values to determine stability.
• 😀 For stability, all elements in the second column of the Routh table must be positive and non-zero. If any value is negative, the system is unstable.
• 😀 In practice, MATLAB or similar software can be used to quickly find the roots of a characteristic equation, simplifying the stability analysis process.
• 😀 Stability can be determined by calculating the coefficients of the characteristic equation, filling out a table, and ensuring all the calculated values are positive.
• 😀 In some cases, determining the value of a parameter (like K) can help achieve stability, as shown in the example with the system's characteristic equation and the chosen value for K.
• 😀 The stability of the system is also influenced by how close the roots are to the real axis in the complex plane, as imaginary components indicate potential oscillations.
• 😀 If the system has both positive and negative real parts for its roots, it may experience oscillations, leading to instability.
• 😀 The process of determining stability also involves visualizing the system's behavior graphically to observe how it deviates from the setpoint and whether it oscillates or remains stable.

Q & A

What method is discussed in the transcript to determine the stability of a system?

The transcript discusses using the Routh-Hurwitz criterion to determine the stability of a system. This method involves creating a table based on the coefficients of the characteristic equation to evaluate the stability of the system.

How does the Routh-Hurwitz criterion help in determining stability?

The Routh-Hurwitz criterion helps determine system stability by checking the signs of the coefficients in the second column of the Routh array. If all values are positive and non-zero, the system is stable. If any value is negative or zero, the system is unstable.

What happens if the coefficients in the second column of the Routh array are not all positive?

If any of the coefficients in the second column of the Routh array are negative or zero, the system is unstable. The presence of negative coefficients indicates that the system may have poles in the right half of the complex plane, leading to instability.

What is the significance of the first column in the Routh array?

The first column in the Routh array is crucial for determining the behavior of the system. It helps in identifying the number of right-hand plane poles, which directly influences the stability of the system. If all entries in the first column are positive, the system remains stable.

What is the process of constructing the Routh array as explained in the script?

To construct the Routh array, you start by filling the first row with the even-powered coefficients and the second row with the odd-powered coefficients of the characteristic equation. Then, subsequent rows are calculated using a specific formula involving the previous rows' coefficients.

Why is the Routh-Hurwitz criterion considered efficient for stability analysis?

The Routh-Hurwitz criterion is considered efficient because it allows for determining the stability of a system without having to explicitly find the roots of the characteristic equation. This is particularly useful for higher-order systems where finding the roots manually can be complex and time-consuming.

What does it mean if the system has both real and imaginary components in its roots?

If the system has both real and imaginary components in its roots, it indicates that the system may exhibit oscillatory behavior. If the real part is positive, the system is unstable, but if the real part is negative, the system can be stable or oscillatory depending on the imaginary component.

What does the example involving a characteristic equation and a table help demonstrate?

The example involving a characteristic equation and a table helps demonstrate how to apply the Routh-Hurwitz criterion in practice. It shows how to construct the table, calculate the coefficients, and determine the stability of the system based on the signs of the coefficients in the second column.

What is the significance of using Matlab or similar programs in solving stability problems?

Using programs like Matlab simplifies the process of finding the roots of the characteristic equation, as they automate the calculations involved in determining the stability. This allows for quick analysis without manual computations, making it easier to handle complex equations.

How does the discussion suggest dealing with unstable systems?

For unstable systems, the discussion suggests determining the values of parameters (like Kc) that would make the system stable. In some cases, adjusting parameters can move the system's poles into the left half of the complex plane, thus stabilizing the system.