Can A Positive Feedback System Become Stable?

Introduction

Positive feedback systems are often associated with instability and oscillations. However, the question remains whether a positive feedback system can become stable under certain conditions. In this article, we will delve into the world of transfer functions and stability analysis to explore the possibility of a stable positive feedback system.

The Transfer Function of a Positive Feedback System

The transfer function of a positive feedback system is given by the equation:

$$\frac{H(s)}{1-G(s)H(s)}$$

where $H(s)$ is the forward gain and $G(s)$ is the feedback gain. This equation represents the ratio of the output to the input of the system.

Approximation of the Transfer Function

When the feedback gain is much greater than the forward gain, i.e., $G(s)H(s) \gg 1$, the transfer function can be approximated to be:

$$\frac{1}{-G(s)}$$

This approximation is valid when the feedback gain dominates the forward gain, and the system behaves like a simple inverting amplifier.

Stability Analysis

To determine the stability of a positive feedback system, we need to analyze the poles of the transfer function. The poles are the values of $s$ that make the denominator of the transfer function equal to zero. In this case, the denominator is $1-G(s)H(s)$.

The Routh-Hurwitz Criterion

One of the most widely used methods for stability analysis is the Routh-Hurwitz criterion. This method involves constructing a table of coefficients from the transfer function and checking if the table has any sign changes. If the table has no sign changes, the system is stable.

The Nyquist Criterion

Another important method for stability analysis is the Nyquist criterion. This method involves plotting the Nyquist diagram of the transfer function and checking if the diagram encircles the critical point $-1+j0$. If the diagram encircles the critical point, the system is unstable.

Conditions for Stability

For a positive feedback system to be stable, the following conditions must be met:

• The feedback gain must be less than or equal to the forward gain, i.e., $G(s)H(s) \leq 1$.
• The transfer function must have no poles in the right half of the s-plane.
• The Routh-Hurwitz criterion must not have any sign changes.
• The Nyquist diagram must not encircle the critical point $-1+j0$.

Example of a Stable Positive Feedback System

Consider a simple positive feedback system with the following transfer function:

$$\frac{H(s)}{1-G(s)H(s)} = \frac{1}{s+1}$$

In this case, the feedback gain is $G(s) = 1$, and the forward gain is $H(s) = 1/s+1$. The transfer function can be approximated to be:

$$\frac{1}{-G(s)} = \frac{1}{-1} = -1$$

The poles of the transfer function are at $s = -1$, which is in the left half of the s-plane. Therefore, the system is stable.

Conclusion

In conclusion, a positive feedback system can become stable under certain conditions. The transfer function must have no poles in the right half of the s-plane, and the Routh-Hurwitz criterion must not have any sign changes. The Nyquist diagram must not encircle the critical point $-1+j0$. Additionally, the feedback gain must be less than or equal to the forward gain. By meeting these conditions, a positive feedback system can be designed to be stable and provide a desired output.

References

• [1] Feedback Systems: An Introduction to Feedback Principles by G. F. Franklin and J. D. Powell
• [2] Control Systems Engineering by N. S. Nise
• [3] Modern Control Systems by B. C. Kuo

Further Reading

• Stability Analysis of Feedback Systems
• Nyquist Criterion for Stability Analysis
• Routh-Hurwitz Criterion for Stability Analysis

Keywords

• Positive feedback system
• Transfer function
• Stability analysis
• Routh-Hurwitz criterion
• Nyquist criterion
• Feedback gain
• Forward gain
• Poles of the transfer function
  

Introduction

In our previous article, we explored the possibility of a stable positive feedback system. We discussed the transfer function, stability analysis, and conditions for stability. In this article, we will answer some frequently asked questions related to positive feedback systems and stability analysis.

Q: What is the main difference between a positive feedback system and a negative feedback system?

A: The main difference between a positive feedback system and a negative feedback system is the direction of the feedback loop. In a positive feedback system, the feedback loop is in the same direction as the input, whereas in a negative feedback system, the feedback loop is in the opposite direction of the input.

Q: What is the significance of the Routh-Hurwitz criterion in stability analysis?

A: The Routh-Hurwitz criterion is a widely used method for stability analysis. It involves constructing a table of coefficients from the transfer function and checking if the table has any sign changes. If the table has no sign changes, the system is stable.

Q: What is the Nyquist criterion, and how is it used in stability analysis?

A: The Nyquist criterion is a method for stability analysis that involves plotting the Nyquist diagram of the transfer function. The Nyquist diagram is a plot of the magnitude and phase angle of the transfer function as a function of frequency. If the Nyquist diagram encircles the critical point -1+j0, the system is unstable.

Q: What are the conditions for stability in a positive feedback system?

A: For a positive feedback system to be stable, the following conditions must be met:

• The feedback gain must be less than or equal to the forward gain.
• The transfer function must have no poles in the right half of the s-plane.
• The Routh-Hurwitz criterion must not have any sign changes.
• The Nyquist diagram must not encircle the critical point -1+j0.

Q: Can a positive feedback system be stable if the feedback gain is greater than the forward gain?

A: No, a positive feedback system cannot be stable if the feedback gain is greater than the forward gain. In this case, the system will be unstable and will exhibit oscillations.

Q: What is the significance of the poles of the transfer function in stability analysis?

A: The poles of the transfer function are the values of s that make the denominator of the transfer function equal to zero. If the poles are in the right half of the s-plane, the system is unstable.

Q: Can a positive feedback system be stable if the poles of the transfer function are in the right half of the s-plane?

A: No, a positive feedback system cannot be stable if the poles of the transfer function are in the right half of the s-plane. In this case, the system will be unstable and will exhibit oscillations.

Q: What is the relationship between the Routh-Hurwitz criterion and the Nyquist criterion?

A: The Routh-Hurwitz criterion and the Nyquist criterion are two different methods for stability analysis. However, they are related in that the Routh-Hurwitz criterion can be used to determine the stability of a system based on the poles of the transfer function, while the Nyquist criterion can be used to determine the stability of a system based on the Nyquist diagram.

Q: Can a positive feedback system be stable if the Routh-Hurwitz criterion has sign changes?

A: No, a positive feedback system cannot be stable if the Routh-Hurwitz criterion has sign changes. In this case, the system will be unstable and will exhibit oscillations.

Q: What is the significance of the critical point -1+j0 in stability analysis?

A: The critical point -1+j0 is a point on the Nyquist diagram that is used to determine the stability of a system. If the Nyquist diagram encircles the critical point, the system is unstable.

Q: Can a positive feedback system be stable if the Nyquist diagram encircles the critical point -1+j0?

A: No, a positive feedback system cannot be stable if the Nyquist diagram encircles the critical point -1+j0. In this case, the system will be unstable and will exhibit oscillations.

Conclusion

In conclusion, a positive feedback system can become stable under certain conditions. The transfer function must have no poles in the right half of the s-plane, and the Routh-Hurwitz criterion must not have any sign changes. The Nyquist diagram must not encircle the critical point -1+j0. Additionally, the feedback gain must be less than or equal to the forward gain. By meeting these conditions, a positive feedback system can be designed to be stable and provide a desired output.

References

• [1] Feedback Systems: An Introduction to Feedback Principles by G. F. Franklin and J. D. Powell
• [2] Control Systems Engineering by N. S. Nise
• [3] Modern Control Systems by B. C. Kuo

Further Reading

• Stability Analysis of Feedback Systems
• Nyquist Criterion for Stability Analysis
• Routh-Hurwitz Criterion for Stability Analysis

Keywords

• Positive feedback system
• Transfer function
• Stability analysis
• Routh-Hurwitz criterion
• Nyquist criterion
• Feedback gain
• Forward gain
• Poles of the transfer function