10.5 Feedback linearization
11 min read•august 20, 2024
is a powerful control technique that transforms into equivalent linear systems. It enables the application of linear control methods to complex nonlinear systems, simplifying controller design and analysis.
This approach is particularly useful for systems with strong nonlinearities, like robotics and aerospace applications. By canceling out nonlinearities through and feedback control, it provides a systematic framework for controlling challenging nonlinear systems.
Feedback linearization overview
- Feedback linearization is a powerful technique in control theory that transforms a nonlinear system into an equivalent linear system through a change of coordinates and feedback control
- It enables the application of linear control methods to nonlinear systems, simplifying the design and analysis of controllers
- Feedback linearization is particularly useful for systems with strong nonlinearities, such as robotics, aerospace, and process control applications
Nonlinear systems vs linear systems
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- Nonlinear systems exhibit complex behaviors and interactions between variables, making them challenging to analyze and control compared to linear systems
- Linear systems follow the principle of superposition and have proportional relationships between inputs and outputs, allowing for simpler mathematical representations and control techniques
- Nonlinear systems often require advanced control strategies, such as feedback linearization, to achieve desired performance and
Feedback linearization definition
- Feedback linearization is a technique that cancels out the nonlinearities of a system through a combination of coordinate transformation and feedback control
- It involves finding a suitable change of coordinates () and designing a feedback control law that renders the closed-loop system linear in the new coordinates
- The resulting linear system can then be controlled using well-established linear control methods, such as pole placement or optimal control
Feedback linearization vs gain scheduling
- Gain scheduling is another approach to control nonlinear systems, where the controller gains are adjusted based on the operating conditions or system states
- Feedback linearization provides a more systematic and rigorous approach to nonlinear control, as it directly addresses the nonlinearities through coordinate transformation and feedback
- Gain scheduling may require extensive tuning and can be less effective for systems with strong nonlinearities, while feedback linearization offers a more general and powerful framework for nonlinear control design
Input-output linearization
- focuses on linearizing the input-output relationship of a nonlinear system, without necessarily linearizing the entire state-space dynamics
- It aims to achieve a linear mapping between the system input and a chosen output variable, enabling the application of linear control techniques to regulate the output
Input-output linearization process
- The input-output linearization process involves differentiating the output variable repeatedly until the input appears explicitly in the resulting equation
- The number of times the output needs to be differentiated to obtain a direct relationship with the input is called the of the system
- Once the input-output relationship is linearized, a feedback control law is designed to cancel out the nonlinearities and achieve the desired linear behavior
Relative degree of nonlinear system
- The relative degree of a nonlinear system is the minimum number of times the output needs to be differentiated for the input to appear explicitly in the resulting equation
- It determines the order of the linearized input-output dynamics and the feasibility of input-output linearization
- Systems with well-defined relative degrees are suitable candidates for input-output linearization, while systems with undefined relative degrees may require alternative control approaches
Lie derivatives in input-output linearization
- Lie derivatives are mathematical tools used in input-output linearization to compute the derivatives of the output variable along the system dynamics
- They capture the rate of change of a function (output) along the direction of a vector field (system dynamics)
- Lie derivatives are recursively computed until the input appears explicitly, determining the relative degree and the linearizing feedback control law
Zero dynamics of nonlinear system
- represent the internal dynamics of a nonlinear system that are not observable from the chosen output variable
- They describe the behavior of the system when the output is constrained to zero through feedback control
- The stability of zero dynamics is crucial for the overall stability and performance of the closed-loop system under input-output linearization
- Unstable zero dynamics can lead to internal instability, even if the input-output dynamics are linearized and stabilized
Input-state linearization
- Input-state linearization aims to linearize the entire state-space dynamics of a nonlinear system, not just the input-output relationship
- It involves finding a coordinate transformation (diffeomorphism) that transforms the nonlinear system into an equivalent linear system in the new coordinates
Input-state linearization process
- The input-state linearization process consists of two main steps: coordinate transformation and feedback control design
- The coordinate transformation is chosen to cancel out the nonlinearities in the state equations, resulting in a linear system in the new coordinates
- The feedback control law is then designed based on the linearized system to achieve the desired closed-loop performance and stability
Diffeomorphism in input-state linearization
- A diffeomorphism is a smooth and invertible coordinate transformation that preserves the topological properties of the state space
- In input-state linearization, the diffeomorphism is carefully chosen to cancel out the nonlinearities in the state equations and transform the system into a linear form
- The existence of a suitable diffeomorphism is a necessary condition for input-state linearization, and its construction often involves solving partial differential equations
Coordinate transformation for input-state linearization
- The coordinate transformation in input-state linearization maps the original nonlinear state variables to new state variables in which the system dynamics become linear
- It is typically composed of a combination of the original state variables and nonlinear functions of those variables
- The choice of the coordinate transformation is crucial for the success of input-state linearization and requires a good understanding of the system dynamics and nonlinearities
Feedback law design for input-state linearization
- Once the nonlinear system is transformed into a linear system through coordinate transformation, a feedback control law is designed based on the linearized dynamics
- The feedback law aims to achieve the desired closed-loop performance, such as stability, tracking, or regulation
- Linear control techniques, such as pole placement, linear quadratic regulator (LQR), or robust control methods, can be applied to design the feedback law in the transformed coordinates
Exact feedback linearization
- is a special case of input-state linearization where the entire nonlinear system dynamics are linearized through coordinate transformation and feedback control
- It requires the nonlinear system to satisfy certain conditions, such as being affine in control and having a well-defined relative degree
Conditions for exact feedback linearization
- For a nonlinear system to be exactly feedback linearizable, it must satisfy two main conditions:
- The system must be affine in control, meaning that the control input appears linearly in the state equations
- The system must have a well-defined relative degree equal to the dimension of the state space
- These conditions ensure that the nonlinearities can be completely canceled out through coordinate transformation and feedback, resulting in a linear closed-loop system
Single-input single-output (SISO) systems
- Exact feedback linearization is more straightforward for SISO systems, where there is only one input and one output variable
- For SISO systems, the relative degree condition simplifies to the requirement that the output variable must be differentiated a number of times equal to the system order for the input to appear explicitly
- The feedback linearization process involves designing a coordinate transformation and a feedback control law based on the Lie derivatives of the output variable
Multi-input multi-output (MIMO) systems
- Exact feedback linearization becomes more complex for MIMO systems, where there are multiple inputs and multiple outputs
- In MIMO systems, the relative degree is defined for each input-output pair, and the overall system relative degree is the sum of the individual relative degrees
- The coordinate transformation and feedback control law must be designed to simultaneously linearize all input-output channels while ensuring the compatibility of the individual relative degrees
Limitations of exact feedback linearization
- Exact feedback linearization has some limitations and challenges:
- It requires the system to satisfy strict conditions, such as being affine in control and having a well-defined relative degree, which may not hold for all nonlinear systems
- The linearization process relies on accurate system models and parameters, and model uncertainties or parameter variations can degrade the performance of the linearizing feedback controller
- The linearizing feedback control law may lead to high control efforts or actuator saturation, especially when dealing with strong nonlinearities or large disturbances
- Despite these limitations, exact feedback linearization remains a powerful tool for nonlinear control design when the system satisfies the required conditions
Partial feedback linearization
- is an approach that linearizes a subset of the system dynamics, rather than the entire system as in exact feedback linearization
- It is applicable when the system does not satisfy the conditions for exact feedback linearization or when only partial linearization is desired
Partial feedback linearization definition
- Partial feedback linearization involves finding a coordinate transformation and feedback control law that linearize a portion of the system dynamics, while leaving the remaining dynamics nonlinear
- The linearized portion is typically chosen based on the control objectives and the structure of the system nonlinearities
- Partial feedback linearization allows for a trade-off between the complexity of the linearizing controller and the achievable performance
Internal dynamics in partial feedback linearization
- In partial feedback linearization, the system dynamics are divided into two parts: the external dynamics and the internal dynamics
- The external dynamics represent the portion of the system that is linearized through coordinate transformation and feedback control
- The internal dynamics, also known as the zero dynamics, correspond to the remaining nonlinear dynamics that are not directly controlled or observed
Stability analysis of internal dynamics
- The stability of the internal dynamics is crucial for the overall stability and performance of the partially linearized system
- If the internal dynamics are stable, the partially linearized system can be controlled using linear techniques applied to the external dynamics
- Unstable internal dynamics can lead to undesirable behaviors or instability, even if the external dynamics are stabilized
- Lyapunov stability theory and other nonlinear analysis tools are used to assess the stability of the internal dynamics
Partial feedback linearization applications
- Partial feedback linearization finds applications in various domains, such as:
- Underactuated systems, where the number of control inputs is less than the number of degrees of freedom (robotics, )
- Systems with non-minimum phase dynamics, where exact feedback linearization may lead to unstable zero dynamics (process control, power systems)
- Hierarchical control structures, where partial feedback linearization is used for the inner control loop while outer loops handle the remaining nonlinearities (automotive systems, robotics)
- Partial feedback linearization provides a flexible framework for nonlinear control design, allowing for a balance between linearization, stability, and performance requirements
Feedback linearization examples
- Feedback linearization has been successfully applied to a wide range of nonlinear systems across different domains
- The following examples demonstrate the versatility and effectiveness of feedback linearization in controlling complex nonlinear systems
Feedback linearization of robotic systems
- Robotic systems, such as robot manipulators or mobile robots, often exhibit highly nonlinear dynamics due to the coupling between joints, inertia effects, and gravitational forces
- Feedback linearization can be used to linearize the nonlinear dynamics of robotic systems, enabling the application of linear control techniques for motion planning, trajectory tracking, or force control
- Examples include the feedback linearization of a multi-link robot arm for precise position control or the linearization of a mobile robot's dynamics for path following and obstacle avoidance
Feedback linearization of chemical processes
- Chemical processes, such as reactors or distillation columns, are characterized by nonlinear dynamics arising from complex chemical reactions, mass and heat transfer, and fluid flow
- Feedback linearization can be employed to linearize the nonlinear models of chemical processes, facilitating the design of controllers for process regulation, product quality control, or optimization
- Examples include the feedback linearization of a continuous stirred tank reactor (CSTR) for temperature control or the linearization of a distillation column for composition control
Feedback linearization of aerospace systems
- Aerospace systems, such as aircraft or spacecraft, exhibit nonlinear dynamics due to aerodynamic forces, gravitational effects, and the coupling between translational and rotational motions
- Feedback linearization can be applied to linearize the nonlinear dynamics of aerospace systems, enabling the design of controllers for attitude control, trajectory tracking, or landing maneuvers
- Examples include the feedback linearization of an aircraft's equations of motion for autopilot design or the linearization of a spacecraft's attitude dynamics for pointing and stabilization control
Feedback linearization of electrical systems
- Electrical systems, such as power converters or electric machines, often involve nonlinear dynamics due to the presence of switching elements, magnetic saturation, or cross-coupling effects
- Feedback linearization can be used to linearize the nonlinear models of electrical systems, facilitating the design of controllers for power regulation, voltage stabilization, or torque control
- Examples include the feedback linearization of a DC-DC converter for output voltage regulation or the linearization of an induction motor's dynamics for high-performance speed control
Feedback linearization implementation
- The implementation of feedback linearization involves several key steps and considerations to ensure the effective control of nonlinear systems
- This section discusses the design of feedback linearization controllers, the choice between and output feedback, robustness aspects, and adaptive techniques
Feedback linearization controller design
- The design of a feedback linearization controller involves two main components: the coordinate transformation and the feedback control law
- The coordinate transformation is chosen to cancel out the nonlinearities in the system dynamics, resulting in a linear system in the new coordinates
- The feedback control law is then designed based on the linearized system to achieve the desired closed-loop performance, such as stability, tracking, or regulation
- Linear control design techniques, such as pole placement, linear quadratic regulator (LQR), or robust control methods, can be applied to the linearized system
State feedback and output feedback
- Feedback linearization can be implemented using either state feedback or output feedback, depending on the available measurements and the system structure
- State feedback assumes that all the state variables are measurable and can be used for feedback control, allowing for the direct linearization of the state-space dynamics
- Output feedback relies on the measured output variables and their derivatives to linearize the input-output relationship, without requiring access to the full state information
- The choice between state feedback and output feedback depends on factors such as sensor availability, system observability, and implementation complexity
Robustness of feedback linearization
- Feedback linearization relies on the accurate cancellation of system nonlinearities through coordinate transformation and feedback control
- However, in practice, the system model may be subject to uncertainties, parameter variations, or external disturbances, which can affect the performance and stability of the feedback linearization controller
- Robustness analysis and design techniques, such as sliding mode control, adaptive control, or robust control methods, can be incorporated into the feedback linearization framework to handle uncertainties and enhance the controller's robustness
- Robust feedback linearization aims to maintain the desired closed-loop performance and stability in the presence of model uncertainties and external disturbances
Adaptive feedback linearization techniques
- combines feedback linearization with adaptive control techniques to handle systems with unknown or time-varying parameters
- The adaptive controller estimates the unknown system parameters online and updates the feedback linearization control law accordingly
- Adaptive feedback linearization can be implemented using various adaptive control schemes, such as model reference adaptive control (MRAC), adaptive pole placement, or adaptive backstepping
- The adaptive controller ensures that the feedback linearization remains effective and maintains the desired closed-loop performance, even in the presence of parameter uncertainties or variations
- Adaptive feedback linearization is particularly useful for systems with changing operating conditions, aging effects, or environmental disturbances that affect the system parameters
Key Terms to Review (25)
Adaptive Feedback Linearization: Adaptive feedback linearization is a control strategy that adjusts the nonlinear system dynamics into a linear form by applying a feedback controller that adapts in real time to changes in the system. This technique is particularly useful for systems with unknown or time-varying parameters, allowing for improved tracking and performance while maintaining stability. The adaptability aspect ensures that the control law can effectively handle uncertainties in system behavior.
Aerospace systems: Aerospace systems refer to the integrated technologies and processes involved in the design, development, and operation of aircraft and spacecraft. These systems encompass various elements including control systems, navigation, communication, and propulsion, all of which are critical for ensuring the safety, efficiency, and performance of aerospace vehicles.
Affine systems: Affine systems are a type of control system where the dynamics can be expressed as a linear function of the state and input variables, plus an additional constant term. This structure allows for straightforward analysis and design using techniques from linear control theory while still accommodating nonlinearities in certain elements. The behavior of affine systems can often be simplified, making them easier to work with in feedback linearization and other control methods.
Controllability: Controllability is a property of a dynamic system that determines whether it is possible to steer the system's state from any initial state to any desired final state within a finite amount of time using appropriate inputs. This concept is vital in the design and implementation of control strategies, as it informs how effectively a system can be manipulated through inputs, directly linking to state-space representation, feedback mechanisms, and system observability.
Coordinate transformation: Coordinate transformation is the process of changing from one coordinate system to another, allowing for different perspectives in representing and analyzing systems. This concept is essential in simplifying complex problems and is frequently used in control theory, especially when dealing with nonlinear systems. By transforming coordinates, it becomes easier to design controllers and analyze system behaviors under various conditions.
Diffeomorphism: A diffeomorphism is a special type of mapping between two manifolds that is smooth, has a smooth inverse, and preserves the structure of the manifold. This concept is crucial in understanding how systems can be transformed to simplify analysis, particularly in nonlinear control systems where feedback linearization is applied to achieve desired behavior. Diffeomorphisms ensure that the essential properties of the system are maintained during transformations, allowing for effective control strategies.
Exact Feedback Linearization: Exact feedback linearization is a control strategy used to transform a nonlinear system into an equivalent linear system through a specific feedback control law. This method allows for easier analysis and design of controllers since linear systems are generally simpler to manage. By applying this technique, one can achieve desired performance characteristics and stability by appropriately manipulating the system's input based on its state.
Feedback control laws: Feedback control laws are strategies used in control systems to adjust system inputs based on the difference between desired and actual outputs. This approach enables systems to automatically correct deviations from a desired behavior, enhancing stability and performance. Feedback control laws play a crucial role in various applications, enabling dynamic adjustments to ensure systems operate within specified limits.
Feedback linearization: Feedback linearization is a control technique used to transform a nonlinear system into an equivalent linear system through state feedback. By applying appropriate control inputs that depend on the state of the system, the dynamics of the original nonlinear system can be simplified, allowing for easier analysis and controller design. This approach is particularly useful in adaptive control, addressing the unique characteristics of nonlinear systems, and is often analyzed using Lyapunov's methods to ensure stability.
Hermann W. P. F. L. J. van der Schaft: Hermann W. P. F. L. J. van der Schaft is a prominent researcher in the field of control theory, particularly known for his work on feedback linearization and nonlinear systems. His contributions have significantly shaped the understanding and development of control strategies that allow complex nonlinear systems to be treated as linear systems under certain conditions, enhancing their controllability and stability.
Input-output behavior: Input-output behavior refers to the relationship between the inputs applied to a system and the outputs produced by that system, characterizing how the system reacts to external stimuli. This concept is crucial for understanding the dynamics of control systems, as it helps in analyzing how inputs affect system performance, stability, and efficiency.
Input-output linearization: Input-output linearization is a control technique that aims to transform a nonlinear system into an equivalent linear system through a suitable change of input. This method allows for the application of linear control strategies, making it easier to design controllers for systems that are inherently nonlinear. By redefining inputs and outputs, the system's dynamics can be manipulated, enabling the designer to achieve desired performance characteristics.
Jacobian Matrix: The Jacobian matrix is a matrix that represents the first-order partial derivatives of a vector-valued function. It provides crucial information about the local behavior of the function, particularly in the context of transformations and linear approximations, which are essential in methods like feedback linearization and various linearization techniques.
Lie Derivative: The Lie derivative is a mathematical concept that measures the change of a tensor field along the flow of another vector field. It provides a way to analyze how different geometrical and physical quantities evolve when a system is transformed by a given flow. This concept is particularly relevant in control theory, especially in feedback linearization, where understanding the dynamics of nonlinear systems through transformations is crucial.
Multi-input multi-output (MIMO) systems: MIMO systems are control systems that have multiple inputs and multiple outputs, allowing for complex interactions and feedback loops between various signals. These systems are essential in modern control applications, as they can handle the dynamic relationships between various inputs and outputs more effectively than single-input single-output (SISO) systems. By utilizing MIMO strategies, engineers can achieve better performance and stability in controlling a wide range of processes.
Nonlinear systems: Nonlinear systems are dynamic systems in which the output is not directly proportional to the input, leading to complex behavior that cannot be accurately predicted using linear approximations. These systems often exhibit phenomena such as multiple equilibrium points, limit cycles, and chaotic behavior. Their analysis requires different methods and tools compared to linear systems, particularly in understanding stability, applying describing functions, and utilizing feedback linearization techniques.
Partial Feedback Linearization: Partial feedback linearization is a control technique used to simplify the dynamics of a nonlinear system by applying a feedback control law that effectively cancels out certain nonlinearities, transforming the system into a linear form for specific states. This method allows for the design of controllers that can stabilize and control nonlinear systems by exploiting their structure while retaining some of their original characteristics. It is particularly valuable in scenarios where full feedback linearization is either not feasible or not necessary.
Relative degree: Relative degree refers to the difference between the number of poles and zeros of a system when expressed in a specific form, particularly in the context of feedback linearization. It plays a significant role in determining the system's controllability and stability, as well as how it can be transformed into a linear system for easier analysis and control design.
Robotic control: Robotic control refers to the methodologies and techniques used to manage the behavior and motion of robotic systems, ensuring they perform desired tasks with precision and reliability. This concept encompasses various strategies like state feedback, pole placement, and advanced algorithms to optimize control performance, allowing robots to interact effectively with their environment and achieve specific objectives.
Ruths: Ruths are specific structured functions that describe the input-output relationship in control systems, particularly in the context of feedback linearization. They play a crucial role in transforming nonlinear dynamic systems into a linear form, making them easier to analyze and control. This process relies on the system's dynamics and helps in stabilizing the system by applying appropriate feedback mechanisms.
Single-Input Single-Output (SISO) Systems: Single-input single-output (SISO) systems are control systems characterized by one input signal and one output signal. This simple structure allows for easier analysis and design, making it a fundamental concept in control theory. Understanding SISO systems provides a foundation for exploring more complex multi-input multi-output (MIMO) systems and lays the groundwork for various control techniques, including feedback linearization.
Stability: Stability refers to the ability of a system to maintain its performance over time and return to a desired state after experiencing disturbances. It is a crucial aspect in control systems, influencing how well systems react to changes and how reliably they can operate within specified limits.
State Feedback: State feedback is a control strategy that uses the current state of a system to compute the control input, allowing for the manipulation of system dynamics to achieve desired performance. This approach is pivotal in various control methodologies, enabling engineers to place poles of the closed-loop system in locations that ensure stability and performance, manage trade-offs between state regulation and cost, and facilitate robust control under uncertainties.
State Space Representation: State space representation is a mathematical model that describes a physical system using a set of input, output, and state variables, along with a set of first-order differential (or difference) equations. This approach provides a comprehensive framework for analyzing and designing control systems by capturing the dynamics of the system in a structured manner. It allows for the representation of both linear and nonlinear systems, making it versatile in control theory applications.
Zero Dynamics: Zero dynamics refer to the behavior of the internal states of a system when the output is held at zero. This concept plays a critical role in feedback linearization, allowing for the analysis and control of systems by separating the dynamics that influence the output from those that do not. Understanding zero dynamics helps in designing controllers that can stabilize a system even when certain outputs are not active.