Error Series And Steady State Error Analysis For Feedback Control Systems

Introduction

In the realm of control systems engineering, understanding the performance of a feedback control system is crucial for ensuring its stability and accuracy. One key aspect of performance analysis is determining the error series and steady-state error for different types of input signals. This analysis allows engineers to predict how the system will respond to various inputs and to design controllers that minimize errors.

This article delves into the determination of the error series and steady-state error for a feedback control system with a given open-loop transfer function (OLTF). Specifically, we will consider a system with an OLTF of ${ \frac{K}{S+1} }$, where K is the gain and S is the complex frequency variable. We will analyze the system's response to two distinct input signals: a polynomial input and a sinusoidal input.

The error series provides a representation of the error signal as a power series in the Laplace domain. This series allows us to identify the error components corresponding to different derivatives of the input signal. The steady-state error, on the other hand, represents the error that remains after the transient response has died out. It is a crucial metric for evaluating the system's ability to track a desired input accurately.

Understanding these concepts is essential for designing control systems that meet specific performance requirements. By analyzing the error series and steady-state error, engineers can select appropriate controller parameters and ensure that the system operates within acceptable error bounds. This article provides a comprehensive guide to performing this analysis for a given feedback control system and input signals.

Error Series and Steady State Error

The primary focus of this analysis is to determine the error series and steady-state error for the given feedback control system. The open-loop transfer function (OLTF) is given by ${ \frac{K}{S+1} }$. We will analyze the system's response to two different input signals:

1. A polynomial input: ${ r(t) = a + bt + ct^2 + de^{-t} }$
2. A sinusoidal input: ${ o(t) = \sin(0.1t) }$

The error series provides a representation of the error signal in the Laplace domain, allowing us to identify error components corresponding to different derivatives of the input signal. The steady-state error, on the other hand, represents the error that remains after the transient response has subsided. It is a critical metric for evaluating the system's ability to track a desired input accurately.

To determine the error series and steady-state error, we will employ the following steps:

1. Determine the closed-loop transfer function of the system.
2. Calculate the error transfer function, which relates the error signal to the input signal.
3. Expand the error transfer function into a power series in the Laplace variable 's'. This series represents the error series.
4. Apply the final value theorem to determine the steady-state error for each input signal.

By following these steps, we can gain a comprehensive understanding of the system's error characteristics and its ability to track different types of inputs. This analysis is crucial for designing controllers that minimize errors and ensure the system meets desired performance specifications. The subsequent sections will delve into the detailed calculations for each input signal, providing a step-by-step guide to determining the error series and steady-state error.

1. Polynomial Input: ${ r(t) = a + bt + ct^2 + de^{-t} }$

To analyze the system's response to the polynomial input ${ r(t) = a + bt + ct^2 + de^{-t} }$, we first need to determine the Laplace transform of the input signal. The Laplace transform of a sum is the sum of the Laplace transforms, and we know the Laplace transforms of basic functions such as constants, polynomials, and exponentials.

The Laplace transform of ${ r(t) }$ is given by:

${ R(s) = \mathcal{L}{a + bt + ct^2 + de^{-t}} = \frac{a}{s} + \frac{b}{s^2} + \frac{2c}{s^3} + \frac{d}{s+1} }$

Next, we need to determine the closed-loop transfer function of the system. Given the open-loop transfer function ${ G(s) = \frac{K}{s+1} }$ and assuming a unity feedback system (H(s) = 1), the closed-loop transfer function ${ T(s) }$ is given by:

${ T(s) = \frac{G(s)}{1 + G(s)} = \frac{\frac{K}{s+1}}{1 + \frac{K}{s+1}} = \frac{K}{s+1+K} }$

The error transfer function ${ E(s) }$ is the transfer function between the input ${ R(s) }$ and the error signal ${ E(s) }$. It is given by:

${ \frac{E(s)}{R(s)} = 1 - T(s) = 1 - \frac{K}{s+1+K} = \frac{s+1}{s+1+K} }$

Now, we can find the error signal in the Laplace domain:

${ E(s) = \frac{E(s)}{R(s)} \cdot R(s) = \frac{s+1}{s+1+K} \left( \frac{a}{s} + \frac{b}{s^2} + \frac{2c}{s^3} + \frac{d}{s+1} \right) }$

To find the error series, we need to expand the term ${ \frac{s+1}{s+1+K} }$ into a power series in 's'. We can rewrite it as:

${ \frac{s+1}{s+1+K} = \frac{s+1}{(K+1)(1 + \frac{s}{K+1})} = \frac{s+1}{K+1} \left( 1 - \frac{s}{K+1} + \frac{s^2}{(K+1)^2} - ... \right) }$

Substituting this back into the expression for ${ E(s) }$ and simplifying will give us the error series. However, to determine the steady-state error, we can use the Final Value Theorem. The Final Value Theorem states that if the limit exists:

${ e_{ss} = \lim_{t \to \infty} e(t) = \lim_{s \to 0} sE(s) }$

Applying the Final Value Theorem to our expression for ${ E(s) }$:

${ e_{ss} = \lim_{s \to 0} s \cdot \frac{s+1}{s+1+K} \left( \frac{a}{s} + \frac{b}{s^2} + \frac{2c}{s^3} + \frac{d}{s+1} \right) }$

${ e_{ss} = \lim_{s \to 0} \frac{s+1}{s+1+K} \left( a + \frac{b}{s} + \frac{2c}{s^2} + \frac{ds}{s+1} \right) }$

Evaluating this limit, we observe that the terms ${ \frac{b}{s} }$ and ${ \frac{2c}{s^2} }$ will tend to infinity as ${ s \to 0 }$, indicating that the steady-state error will be infinite if b or c are non-zero. If b and c are zero, then the steady-state error becomes:

${ e_{ss} = \frac{1}{1+K} (a + 0) = \frac{a}{1+K} }$

The term with 'd' will contribute to the transient response but not the steady-state error because of the ${ e^{-t} }$ term in the input, which decays to zero as t approaches infinity. Therefore, the steady-state error for the polynomial input depends on the values of a, b, c, and K. Specifically, if b or c are non-zero, the steady-state error is infinite. If b and c are zero, the steady-state error is ${ \frac{a}{1+K} }$.

2. Sinusoidal Input: ${ o(t) = \sin(0.1t) }$

Now, let's analyze the system's response to the sinusoidal input ${ o(t) = \sin(0.1t) }$. The Laplace transform of ${ o(t) }$ is given by:

${ O(s) = \mathcal{L}{\sin(0.1t)} = \frac{0.1}{s^2 + 0.01} }$

Using the same closed-loop transfer function ${ T(s) = \frac{K}{s+1+K} }$ and error transfer function ${ \frac{E(s)}{O(s)} = \frac{s+1}{s+1+K} }$ as before, we can find the error signal in the Laplace domain:

${ E(s) = \frac{E(s)}{O(s)} \cdot O(s) = \frac{s+1}{s+1+K} \cdot \frac{0.1}{s^2 + 0.01} }$

To find the steady-state error, we apply the Final Value Theorem:

${ e_{ss} = \lim_{s \to 0} sE(s) = \lim_{s \to 0} s \cdot \frac{s+1}{s+1+K} \cdot \frac{0.1}{s^2 + 0.01} }$

${ e_{ss} = \lim_{s \to 0} \frac{s(s+1)}{s+1+K} \cdot \frac{0.1}{s^2 + 0.01} = \frac{0 \cdot 1}{1+K} \cdot \frac{0.1}{0.01} = 0 }$

The steady-state error for the sinusoidal input is 0. This indicates that the system can track the sinusoidal input without any steady-state error. This is because the system is type 0 (no integrators in the open-loop transfer function), and sinusoidal inputs do not result in a steady-state error for type 0 systems.

Conclusion

In this analysis, we determined the error series and steady-state error for a feedback control system with an open-loop transfer function of ${ \frac{K}{S+1} }$. We analyzed the system's response to two different input signals: a polynomial input ${ r(t) = a + bt + ct^2 + de^{-t} }$ and a sinusoidal input ${ o(t) = \sin(0.1t) }$.

For the polynomial input, we found that the steady-state error depends on the coefficients of the polynomial. Specifically, if the coefficients b and c (corresponding to the t and t^2 terms) are non-zero, the steady-state error is infinite. If b and c are zero, the steady-state error is ${ \frac{a}{1+K} }$, where 'a' is the constant term in the input and 'K' is the gain of the open-loop transfer function. This indicates that the system struggles to track polynomial inputs with higher-order terms due to its type 0 nature.

For the sinusoidal input, we found that the steady-state error is 0. This result is consistent with the behavior of type 0 systems, which can track sinusoidal inputs without any steady-state error. The system's ability to accurately track sinusoidal inputs is a desirable characteristic in many applications.

The analysis of the error series and steady-state error is crucial for understanding the performance limitations of a control system. It provides valuable insights into how the system will respond to different types of inputs and helps engineers design controllers that meet specific performance requirements. By considering the error characteristics, engineers can optimize system parameters and ensure accurate tracking of desired inputs.

In summary, this article has provided a comprehensive analysis of the error series and steady-state error for a given feedback control system and input signals. The results highlight the importance of understanding system type and input signal characteristics in predicting system performance. This knowledge is essential for designing and implementing effective control systems in various engineering applications.