May 26, 2023

# What is feedback linearization? Feedback linearization is a powerful control technique that enables the transformation of a nonlinear system into a linear one, making it easier to design controllers that stabilize the system and achieve desired performance. In this article, we explore the definition, mathematical foundations, types, and implementation of feedback linearization in control systems. We also discuss the importance of feedback linearization in control system design and the practical challenges of implementing this technique.

## Understanding feedback linearization

In this section, we provide an overview of feedback linearization and its key components. We also discuss the importance of feedback linearization in control systems.

Feedback linearization is a powerful control technique that can transform a nonlinear system into a linear one through appropriate coordinate changes and feedback control. The key idea is to introduce a change of variables that makes the system look linear in the new coordinates, and then design a feedback controller to stabilize the transformed system.

This technique is particularly useful in situations where it is difficult to design a controller for a nonlinear system, as it enables the use of linear control methods, which are well-established. Feedback linearization can also improve the overall performance and stability of the system by reducing the nonlinearities that can generate unpredictable behaviors. This can help to achieve better control of complex systems and improve their overall efficiency and reliability.

### Definition and basics

Feedback linearization is a control technique that transforms a nonlinear system into a linear one through appropriate coordinate changes and feedback control. The key idea is to introduce a change of variables that makes the system look linear in the new coordinates, and then design a feedback controller to stabilize the transformed system.

One of the primary benefits of feedback linearization is that it allows us to use linear control techniques on nonlinear systems. This is important because linear control methods are well-established, and we have a deep understanding of how they work. By transforming a nonlinear system into a linear one, we can apply these well-established techniques to achieve better control of the system.

Another benefit of feedback linearization is that it can help to reduce the complexity of the control system. Nonlinear systems can be very complex, and it can be difficult to design a controller that can handle all of the nonlinearities. By transforming the system into a linear one, we can simplify the control problem and make it easier to design a controller that can handle the system's behavior.

### The importance of feedback linearization in control systems

Feedback linearization is an important tool for control system design because it makes it possible to apply linear control theory to nonlinear systems. This allows for the use of well-established control techniques and methods that have been extensively studied and validated.

By using feedback linearization, we can take advantage of the fact that linear control methods are well-established and widely used. This can help to simplify the control problem and make it easier to design a controller that can handle the system's behavior. Additionally, feedback linearization can improve the overall performance and stability of the system by reducing the nonlinearities that can generate unpredictable behaviors.

Overall, feedback linearization is an important tool for control system design because it allows us to apply well-established control techniques to nonlinear systems, which can help to simplify the control problem and improve the overall performance and stability of the system.

### Key components of feedback linearization

The key components of feedback linearization are the coordinate transformation and the feedback control design. To transform a nonlinear system into a linear one, we need to find an appropriate change of variables that makes the system look linear in the new coordinates.

The coordinate transformation is a critical step in feedback linearization because it allows us to simplify the control problem by transforming the system into a linear one. This transformation involves finding an appropriate change of variables that makes the system look linear in the new coordinates. Once we have transformed the system, we can apply linear control techniques to achieve better control of the system.

Next, we design a feedback control law that stabilizes the transformed linear system. This feedback control law is designed based on the transformed system's state variables and the desired closed-loop dynamics. The feedback control law is critical because it allows us to stabilize the system and achieve the desired performance.

Overall, the key components of feedback linearization are the coordinate transformation and the feedback control design. These components work together to transform a nonlinear system into a linear one and stabilize the system using feedback control.

## Types of feedback linearization

In control theory, feedback linearization is a technique used to transform a nonlinear system into a linear one, making it easier to design a control law to achieve desired system behavior. In this section, we discuss the different types of feedback linearization, including full-state feedback linearization, output feedback linearization, and partial feedback linearization.

### Full-state feedback linearization

Full-state feedback linearization involves using measurements of all the system state variables to design the feedback control law. This method is effective when all of the system state variables are measurable. By using feedback linearization, the nonlinear system can be transformed into a linear system, which makes it easier to design a control law to achieve desired system behavior. This technique is widely used in many applications, including aerospace, robotics, and automotive systems.

For example, consider the control of an inverted pendulum. The system is highly nonlinear, and it is difficult to design a control law to achieve desired system behavior. However, by using full-state feedback linearization, the system can be transformed into a linear one, making it easier to design a control law to stabilize the pendulum in the upright position.

### Output feedback linearization

Output feedback linearization involves using measurements of only the system output to design the feedback control law. This method is useful when some of the state variables are unmeasurable or difficult to measure, but the output can be measured. By using output feedback linearization, the nonlinear system can be transformed into a linear system, which makes it easier to design a control law to achieve desired system behavior.

For example, consider the control of a chemical reactor. The system is highly nonlinear, and it is difficult to measure all the state variables. However, the reactor output can be easily measured. By using output feedback linearization, the system can be transformed into a linear one, making it easier to design a control law to achieve desired system behavior.

### Partial feedback linearization

Partial feedback linearization involves using a subset of the state variables to design the feedback control law. This method is useful when some of the state variables are unmeasurable, and it is not possible to measure the output directly. By using partial feedback linearization, the nonlinear system can be transformed into a linear system, which makes it easier to design a control law to achieve desired system behavior.

For example, consider the control of a quadrotor. The system is highly nonlinear, and it is difficult to measure all the state variables. However, by using partial feedback linearization, a subset of the state variables can be used to design a control law to achieve desired system behavior, such as altitude control or attitude control.

## The mathematics behind feedback linearization

In this section, we delve into the mathematical foundations of feedback linearization. We discuss nonlinear systems, their representation, relative degree, and input-output linearization.

### Nonlinear systems and their representation

Nonlinear systems are systems that do not exhibit a proportional relationship between the inputs and outputs. Their dynamics are governed by nonlinear functions that represent the system's behavior.

A common way to represent nonlinear systems is with a set of differential equations that describe the system's dynamics. This representation is known as the state-space representation. In the state-space representation, the system's state is described by a set of variables, and the inputs and outputs are related to the state variables through the nonlinear functions that govern the system's behavior.

Nonlinear systems can also be represented using other mathematical models, such as transfer functions or difference equations. However, the state-space representation is the most common and general way to represent nonlinear systems.

### Relative degree and input-output linearization

The relative degree of a nonlinear system is a property that relates the input and output of the system. It is defined as the number of times the output must be differentiated to obtain an expression that includes the input directly.

Input-output linearization is a method that transforms a nonlinear system into a linear one by making the input appear explicitly in the output relative degree number of times, through an appropriate change of variables and feedback control. This transformation allows us to use linear control techniques to design controllers for nonlinear systems.

The input-output linearization technique is based on the concept of relative degree, which is used to determine the appropriate change of variables and feedback control that will transform the nonlinear system into a linear one. The relative degree of a system can be determined by examining the system's differential equations and identifying the highest order of differentiation of the output that includes the input directly.

### The feedback linearization algorithm

The feedback linearization algorithm involves using the relative degree of the system to choose an appropriate change of variables that makes the system look linear in the new coordinates. Next, a feedback control law is designed based on the transformed linear system and the desired closed-loop dynamics.

The feedback control law is designed to cancel out the nonlinear terms in the transformed system, leaving only the linear terms. This allows us to use standard linear control techniques to design a controller for the system. The feedback control law is typically designed using a Lyapunov stability analysis, which ensures that the closed-loop system is stable and converges to the desired state.

Overall, feedback linearization is a powerful technique for controlling nonlinear systems. By transforming a nonlinear system into a linear one, we can use well-established linear control techniques to design controllers that can stabilize the system and achieve desired performance.

## Implementing feedback linearization in control systems

In this section, we discuss the practical considerations and challenges of implementing feedback linearization in control systems. We also discuss the design and stability considerations of a feedback linearization controller.

### Designing a feedback linearization controller

The design of a feedback linearization controller involves transforming the nonlinear system into a linear one and designing a feedback control law that stabilizes the transformed linear system. The design must also consider the desired closed-loop dynamics and the system's practical limitations and constraints.

### Stability and robustness considerations

Feedback linearization controllers must maintain stability and robustness even in the presence of disturbances, parameter uncertainties, and other uncertainties that can affect the system's behavior. This requires designing controllers that consider these uncertainties and that are robust to them.

### Practical challenges and limitations

Implementing feedback linearization in control systems involves addressing practical challenges that can affect the technique's performance. These challenges include non-measurable or difficult-to-measure state variables, modeling errors, and other uncertainties that can affect the system's behavior.

Despite these challenges, feedback linearization remains a powerful and effective control technique that can improve the performance and stability of control systems in a variety of applications.

## Conclusion

Feedback linearization is an important control technique that transforms nonlinear systems into linear ones, enabling the use of well-established and validated control methods and techniques. Understanding feedback linearization's key components, types, and mathematical foundations is crucial for effective design and implementation of feedback linearization controllers. While practical challenges must be addressed, feedback linearization remains an effective and powerful tool for control system design and performance improvement.