June 8, 2023

# What is a steady state error?

Imagine you are controlling a temperature system and trying to maintain a steady temperature. You make adjustments, but the system still shows fluctuations around the set point. How can you reduce these oscillations and achieve a stable state? This is where the concept of steady state error comes into play.

In control engineering, steady state error refers to the difference between the desired output and the actual output of a control system under steady state conditions. It represents the error that remains even after prolonged operation, when transient effects have died down. In mathematical terms, steady state error is the limit of the system's error as time approaches infinity.

Steady state error is a crucial performance parameter of control systems, as it affects the accuracy, stability, and responsiveness of the system. It quantifies the deviation of the output from the set point, and it's a measure of the system's ability to track a reference signal. High steady state error means poor performance, as the system cannot achieve the desired output even after repeated attempts.

One important thing to note is that steady state error is not the same as transient error. Transient error is the error that occurs during the transition from one steady state to another. This error is temporary and will eventually die down, but steady state error persists even after the system has reached a steady state.

### Factors Influencing Steady State Error

The magnitude and type of the system's input signal are major factors affecting the steady state error. The type of control system, the order of the system, and the presence of disturbances or noise also impact the error. In general, complex or higher-order systems have higher steady state error than simple or lower-order systems, as they require more control effort to track the set point.

Another factor that can influence steady state error is the type of feedback used in the control system. There are two types of feedback: positive and negative. Positive feedback amplifies the input signal, while negative feedback reduces the input signal. Negative feedback is the most commonly used type of feedback in control systems, as it provides stability and reduces steady state error.

Disturbances or noise in the system can also affect steady state error. Disturbances are external factors that affect the system's output, while noise is random fluctuations in the system's output. Both can cause the system to deviate from the set point and increase steady state error.

### Types of System Inputs

The type of input signal used in the control system can also affect steady state error. There are several types of input signals, each with its own characteristics:

• Step Input: a sudden change in input value. This type of input signal is often used to test the system's response to a sudden change in the set point.
• Ramp Input: a gradual change in input value. This type of input signal is often used to test the system's ability to track a reference signal that changes over time.
• Parabolic Input: input value changes at a quadratic rate. This type of input signal is often used to test the system's response to a signal that accelerates or decelerates over time.
• Sinusoidal Input: input value varies sinusoidally over time. This type of input signal is often used to test the system's response to a periodic signal.

Each type of input signal has its own characteristics that can affect the system's steady state error. For example, a sinusoidal input signal can cause the system to oscillate around the set point, increasing steady state error. A ramp input signal can cause the system to have a constant steady state error, even if the system is able to track the reference signal accurately.

Overall, understanding steady state error is crucial for designing and optimizing control systems. By taking into account the factors that influence steady state error and selecting the appropriate input signal and feedback type, engineers can design control systems that are accurate, stable, and responsive.

Calculating steady state error is a crucial step in analyzing and designing control systems. It involves finding the limit of the system's error as time goes to infinity. This is important because it allows engineers to evaluate the performance of a system and make necessary adjustments to improve its behavior. Here is a detailed explanation of the process:

### Step-by-Step Process

1. Find the transfer function of the system: The transfer function is a mathematical representation of the relationship between the system's input and output. It is usually expressed in terms of Laplace transforms and can be obtained through experimental data or theoretical analysis.
2. Calculate the error transfer function: The error transfer function is defined as 1 - GcG, where Gc is the controller transfer function and G is the plant transfer function. This function represents the difference between the desired output and the actual output of the system.
3. Find the steady state gain: The steady state gain is the value of the transfer function as the frequency approaches zero. It is usually denoted as Kss and can be calculated by taking the limit of the transfer function as s approaches zero.
4. Calculate the steady state error: The steady state error is the difference between the desired output and the actual output of the system as time approaches infinity. It can be calculated using the formula ess = 1/(1+Kp), where Kp is the steady state gain.

### Examples and Solutions

For example, consider a unity feedback system with a transfer function G(s) = 10/(s+5)(s+8). If the input is a step signal with a magnitude of 1, the error will converge to a non-zero value. To calculate the steady state error, we can use the formula:

ess = 1/(1+Kp)

Where Kp is the steady state gain, which equals 10/40 = 0.25

Thus, ess = 1/(1+0.25)= 0.8

This means that the error will converge to 0.8 as time approaches infinity. This information can be used to adjust the system's parameters and improve its performance.

### Common Mistakes and How to Avoid Them

One common mistake in calculating steady state error is assuming that the gain of the system is equal to the steady state error. However, gain and error are not the same, and the system's response to different inputs may cause different steady state errors. To avoid this mistake, always use the correct formula and pay attention to the input signal. It is also important to consider the system's stability and the effects of external disturbances on its behavior.

Another common mistake is neglecting the dynamics of the system and assuming that it is a simple static process. This can lead to inaccurate results and poor performance. To avoid this mistake, always analyze the system's behavior over time and take into account its transient response and frequency characteristics.

By following these guidelines and using accurate mathematical models, engineers can calculate steady state error with confidence and design effective control systems that meet their performance requirements.

## Steady State Error in Control Systems

Control systems are used in various applications to regulate the output of a system and achieve a desired performance. However, in some cases, the output may not reach the desired value even after a long time, which is known as steady state error. To overcome this issue, different types of control actions are used. Here are some common types of control:

### Proportional Control

Proportional control is a simple and effective method for reducing steady state error. It adjusts the output of the system according to the difference between the set point and the current output. The proportional control gain determines the amount of change in the output for a given error. However, for higher-order systems, proportional control alone may cause overshoot or oscillations, which can affect the stability of the system.

### Integral Control

Integral control is a robust and reliable method that eliminates the steady state error for any input signal. It uses the previous errors to adjust the output and bring the system to the desired set point. The integral control gain determines the amount of correction applied to the output for a given error. However, integral control may cause overshoot or instability, especially for systems with significant time delays. Moreover, it may take a longer time to reach the set point compared to proportional control.

### Derivative Control

Derivative control is another method used to improve the system's response time and stability. It adjusts the output based on the rate of error change, which helps in reducing overshoot and oscillations. The derivative control gain determines the amount of correction applied to the output for a given rate of change of error. However, derivative control may cause noise amplification or instability, especially for systems with significant measurement noise.

### PID Control

PID control is the most commonly used control method in industrial applications. It combines proportional, integral, and derivative control to achieve optimal performance and stability for a wide range of systems. PID control uses the error, its integral, and derivative to compute the control output. The proportional gain determines the immediate correction, the integral gain eliminates the steady state error, and the derivative gain improves the system's response time. However, tuning the control parameters of PID control can be challenging for complex systems.

In conclusion, control systems use different types of control actions to reduce steady state error and improve performance. Each method has its advantages and disadvantages, and the choice of the control method depends on the system's characteristics and requirements.

Reducing steady state error requires a combination of control techniques and system modifications. Here are some ways to reduce error:

### Tuning Control Parameters

Controlling parameters such as gain, time constants, and bandwidth affect the system's response and performance. Tuning these parameters using feedback or trial-and-error methods can improve the accuracy and stability of the system.

### System Modifications

System modifications such as adding feedback loops, filters, or actuators can improve the control performance and reduce steady state error. System upgrades or replacements may be necessary for severely degraded or unreliable systems.