Glossary

10.2 Linearization techniques

10 min read•august 20, 2024

Linearization techniques are crucial for simplifying complex nonlinear systems in control theory. By approximating nonlinear behavior with linear models, engineers can apply well-established linear control methods to analyze stability and design controllers.

This approach, while powerful, has limitations. Linear models are only accurate near the operating point and can't capture some nonlinear phenomena. Understanding these constraints is key to effectively applying linearization in real-world control systems.

Linearization of nonlinear systems

  • Linearization is a technique used to approximate nonlinear systems with linear models, enabling the application of linear control theory
  • Linearization simplifies the analysis and design of control systems by providing a tractable mathematical representation of the system dynamics
  • The linearized model is valid in the vicinity of an operating point, allowing for local and controller design

Importance of linearization

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  • Many real-world systems exhibit nonlinear behavior, making their analysis and control challenging
  • Linearization enables the use of well-established linear control techniques, such as frequency-domain analysis and state-space methods
  • Linearized models provide insights into system behavior and facilitate the design of controllers that ensure stability and performance

Limitations of linear models

  • Linear models are approximations and may not capture the full complexity of nonlinear systems
  • The validity of linearization is limited to a small region around the operating point, and the model accuracy decreases as the system moves away from this point
  • Nonlinear phenomena, such as multiple equilibrium points, limit cycles, and chaos, cannot be adequately represented by linear models

Taylor series expansion

  • is a mathematical tool used to approximate a nonlinear function by a series of terms involving its derivatives
  • The expansion represents the function as a sum of its value and derivatives at a specific point, multiplied by powers of the deviation from that point
  • The accuracy of the approximation depends on the number of terms included in the series

First-order approximation

  • The first-order Taylor series approximation, also known as linearization, considers only the first derivative of the function
  • It approximates the nonlinear function by a straight line tangent to the function at the point of interest
  • The first-order approximation is given by f(x)≈f(a)+f′(a)(x−a)f(x) \approx f(a) + f'(a)(x-a), where aa is the point of interest and f′(a)f'(a) is the first derivative evaluated at aa

Higher-order terms

  • Higher-order terms in the Taylor series expansion include second and higher-order derivatives of the function
  • These terms capture the curvature and other nonlinear effects of the function more accurately than the first-order approximation
  • The inclusion of higher-order terms improves the accuracy of the approximation but increases the complexity of the resulting model

Truncation error

  • Truncation error arises from neglecting the higher-order terms in the Taylor series expansion
  • The magnitude of the truncation error depends on the number of terms included and the distance from the point of expansion
  • As the system moves further away from the operating point, the truncation error increases, limiting the validity of the linearized model

Jacobian matrix

  • The is a matrix of partial derivatives that describes the local behavior of a vector-valued function
  • It captures the sensitivity of the function's output to changes in its input variables
  • The Jacobian matrix is a crucial tool in the linearization of nonlinear systems

Definition and properties

  • For a vector-valued function f(x)\mathbf{f}(\mathbf{x}), where x=[x1,x2,…,xn]T\mathbf{x} = [x_1, x_2, \ldots, x_n]^T, the Jacobian matrix J(x)\mathbf{J}(\mathbf{x}) is defined as:
\frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \cdots & \frac{\partial f_1}{\partial x_n} \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \cdots & \frac{\partial f_2}{\partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1} & \frac{\partial f_m}{\partial x_2} & \cdots & \frac{\partial f_m}{\partial x_n} \end{bmatrix}$$ - The Jacobian matrix is a square matrix if the input and output dimensions are the same ($m = n$) - The Jacobian matrix is used to approximate the nonlinear function $\mathbf{f}(\mathbf{x})$ by a linear function in the neighborhood of a point $\mathbf{x}_0$ ### Evaluating the Jacobian - To evaluate the Jacobian matrix at a specific point $\mathbf{x}_0$, the partial derivatives are computed by substituting the values of $\mathbf{x}_0$ into the expressions for each element of the matrix - Symbolic differentiation can be used to obtain the general expressions for the partial derivatives, which can then be evaluated at the desired point - Numerical differentiation techniques, such as finite differences, can also be employed to approximate the Jacobian matrix when analytical expressions are not available ### Relationship to linearization - The Jacobian matrix is the key component in the linearization of nonlinear systems - The linearized model of a nonlinear system $\dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}, \mathbf{u})$ around an operating point $(\mathbf{x}_0, \mathbf{u}_0)$ is given by: $$\dot{\mathbf{x}} \approx \mathbf{f}(\mathbf{x}_0, \mathbf{u}_0) + \mathbf{A}(\mathbf{x} - \mathbf{x}_0) + \mathbf{B}(\mathbf{u} - \mathbf{u}_0)$$ where $\mathbf{A} = \frac{\partial \mathbf{f}}{\partial \mathbf{x}}|_{(\mathbf{x}_0, \mathbf{u}_0)}$ and $\mathbf{B} = \frac{\partial \mathbf{f}}{\partial \mathbf{u}}|_{(\mathbf{x}_0, \mathbf{u}_0)}$ are the Jacobian matrices with respect to the state and input variables, respectively, evaluated at the operating point - The Jacobian matrices $\mathbf{A}$ and $\mathbf{B}$ capture the local dynamics of the system and are used to analyze stability and design controllers for the linearized system ## Equilibrium points - Equilibrium points are steady-state solutions of a dynamical system where the state variables remain constant over time - At an [equilibrium point](https://www.fiveableKeyTerm:equilibrium_point), the rate of change of the state variables is zero, i.e., $\dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}_e, \mathbf{u}_e) = \mathbf{0}$, where $\mathbf{x}_e$ and $\mathbf{u}_e$ denote the equilibrium state and input, respectively - Equilibrium points play a crucial role in the analysis and control of nonlinear systems, as they determine the long-term behavior of the system ### Definition and types - An equilibrium point $\mathbf{x}_e$ of a nonlinear system $\dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}, \mathbf{u})$ is a solution that satisfies $\mathbf{f}(\mathbf{x}_e, \mathbf{u}_e) = \mathbf{0}$ - Equilibrium points can be classified into several types based on their stability properties: - Stable equilibrium: Small perturbations from the equilibrium state result in the system returning to the equilibrium point (e.g., a pendulum at its lowest point) - Unstable equilibrium: Small perturbations from the equilibrium state cause the system to diverge from the equilibrium point (e.g., a pendulum at its highest point) - Saddle point: The equilibrium point is stable in some directions and unstable in others (e.g., a ball on a saddle-shaped surface) ### Stability of equilibrium points - The stability of an equilibrium point determines the system's behavior in its vicinity - Lyapunov stability theory provides a framework for analyzing the stability of equilibrium points without explicitly solving the differential equations - Lyapunov's direct method involves constructing a Lyapunov function, which is a positive definite function that decreases along the system trajectories in the neighborhood of the equilibrium point - If a Lyapunov function exists, the equilibrium point is stable; if the Lyapunov function strictly decreases, the equilibrium point is asymptotically stable ### Linearization around equilibrium - Linearization around an equilibrium point simplifies the stability analysis and control design for nonlinear systems - The linearized model captures the local dynamics of the system in the vicinity of the equilibrium point - The stability of the linearized system can be determined by analyzing the eigenvalues of the Jacobian matrix evaluated at the equilibrium point - If all eigenvalues have negative real parts, the equilibrium point is asymptotically stable; if at least one eigenvalue has a positive real part, the equilibrium point is unstable ## State-space representation - [State-space representation](https://www.fiveableKeyTerm:State-space_representation) is a mathematical model that describes a system using a set of first-order differential equations - It provides a compact and general framework for representing and analyzing both linear and nonlinear systems - The state-space model consists of two equations: the state equation, which describes the evolution of the system's state variables, and the output equation, which relates the state variables to the system's outputs ### Nonlinear state-space models - A nonlinear state-space model is represented by the following equations: $$\dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}, \mathbf{u})$$ $$\mathbf{y} = \mathbf{h}(\mathbf{x}, \mathbf{u})$$ where $\mathbf{x}$ is the state vector, $\mathbf{u}$ is the input vector, $\mathbf{y}$ is the output vector, $\mathbf{f}(\cdot)$ is the nonlinear state transition function, and $\mathbf{h}(\cdot)$ is the nonlinear output function - Nonlinear state-space models can capture complex system behaviors, such as multiple equilibrium points, limit cycles, and chaos - The analysis and control of nonlinear state-space models often require specialized techniques, such as Lyapunov stability theory, feedback linearization, and sliding mode control ### Linearized state-space models - Linearization of a nonlinear state-space model around an operating point $(\mathbf{x}_0, \mathbf{u}_0)$ results in a linear state-space model of the form: $$\dot{\mathbf{x}} = \mathbf{A}(\mathbf{x} - \mathbf{x}_0) + \mathbf{B}(\mathbf{u} - \mathbf{u}_0)$$ $$\mathbf{y} = \mathbf{C}(\mathbf{x} - \mathbf{x}_0) + \mathbf{D}(\mathbf{u} - \mathbf{u}_0)$$ where $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$, and $\mathbf{D}$ are the state, input, output, and feedthrough matrices, respectively, obtained by evaluating the Jacobian matrices of $\mathbf{f}(\cdot)$ and $\mathbf{h}(\cdot)$ at the operating point - Linearized state-space models enable the application of linear control techniques, such as pole placement, LQR, and Kalman filtering - The stability and performance of the linearized system can be analyzed using eigenvalue analysis and frequency-domain techniques ### Linearization of input-output models - Input-output models, such as transfer functions, describe the relationship between the system's inputs and outputs without explicitly considering the state variables - Linearization of input-output models involves approximating the nonlinear system by a linear [transfer function](https://www.fiveableKeyTerm:Transfer_Function) around an operating point - The linearized transfer function can be obtained by applying the Taylor series expansion to the nonlinear input-output relationship and retaining only the first-order terms - Linearized input-output models facilitate the design of controllers using classical control techniques, such as PID control and lead-lag compensation ## Validity of linearization - The validity of linearization refers to the extent to which the linearized model accurately represents the behavior of the nonlinear system - Linearization is an approximation technique, and its accuracy depends on several factors, such as the operating point, the magnitude of the perturbations, and the system's nonlinearity ### Local vs global validity - Linearization is a local approximation, meaning that the linearized model is valid only in a small neighborhood around the operating point - As the system moves away from the operating point, the accuracy of the linearized model decreases, and the model may no longer capture the essential dynamics of the nonlinear system - In contrast, global validity refers to the ability of a model to accurately represent the system's behavior over a wide range of operating conditions - Nonlinear models, such as Hammerstein-Wiener models and neural networks, can provide better global approximations of nonlinear systems ### Regions of attraction - The region of attraction (ROA) of an equilibrium point is the set of initial conditions from which the system's trajectories converge to the equilibrium point - For a stable equilibrium point, the ROA is an important concept in assessing the validity of the linearized model - If the system's operating conditions lie within the ROA, the linearized model can provide a good approximation of the system's behavior - Estimating the ROA can be challenging for nonlinear systems, and techniques such as Lyapunov functions and sum-of-squares programming can be used to compute conservative estimates ### Checking model accuracy - To assess the validity of a linearized model, it is essential to compare its predictions with the actual behavior of the nonlinear system - Simulation studies can be conducted to evaluate the model's accuracy over a range of operating conditions and perturbations - Model validation techniques, such as cross-validation and residual analysis, can be used to quantify the model's goodness-of-fit and identify regions where the model's accuracy is poor - If the linearized model's accuracy is insufficient, it may be necessary to consider higher-order approximations, such as quadratic or cubic terms in the Taylor series expansion, or to use nonlinear modeling techniques ## Feedback linearization - Feedback linearization is a nonlinear control technique that aims to transform a nonlinear system into an equivalent linear system through a change of variables and feedback control - The goal is to cancel out the nonlinearities in the system and enable the application of linear control techniques to the transformed system - Feedback linearization can be classified into two main categories: input-state linearization and input-output linearization ### Input-state linearization - Input-state linearization, also known as full-state feedback linearization, transforms the entire state equation of a nonlinear system into a linear form - The technique involves finding a coordinate transformation and a feedback control law that cancel out the nonlinearities in the state equation - For a nonlinear system $\dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}) + \mathbf{g}(\mathbf{x})\mathbf{u}$, input-state linearization seeks a coordinate transformation $\mathbf{z} = \mathbf{T}(\mathbf{x})$ and a feedback control law $\mathbf{u} = \mathbf{\alpha}(\mathbf{x}) + \mathbf{\beta}(\mathbf{x})\mathbf{v}$ such that the transformed system is linear in the new coordinates: $$\dot{\mathbf{z}} = \mathbf{A}\mathbf{z} + \mathbf{B}\mathbf{v}$$ where $\mathbf{v}$ is the new input vector, and $\mathbf{A}$ and $\mathbf{B}$ are the desired state and input matrices, respectively - Input-state linearization requires the system to be feedback linearizable, which imposes certain conditions on the system's structure and controllability ### Input-output linearization - Input-output linearization focuses on linearizing the input-output relationship of a nonlinear system, rather than the entire state equation - The technique involves finding a feedback control law that cancels out the nonlinearities in the output equation and results in a linear input-output map - For a nonlinear system with output $y = h(\mathbf{x})$, input-output linearization seeks a feedback control law $\mathbf{u} = \mathbf{\alpha}(\mathbf{x}) + \mathbf{\beta}(\mathbf{x})\mathbf{v}$ such that the input-output relationship is linear: $$y^{(r)} = \mathbf{v}$$ where $r$ is the relative degree of the system, which represents the number of

Key Terms to Review (16)

Equilibrium point: An equilibrium point refers to a state in a dynamic system where the system remains unchanged over time, meaning that the forces acting on it are balanced. In control theory, this point is crucial as it determines the stability and behavior of the system near this state. Understanding equilibrium points allows for effective analysis and design of control systems, especially when assessing stability and performance using various principles.
Frequency Response: Frequency response is the measure of a system's output spectrum in response to an input signal, revealing how the system reacts to different frequencies. It helps in analyzing the stability and performance of systems by illustrating gain and phase shifts across a range of frequencies, which is crucial for understanding system behavior in various applications.
Gain margin: Gain margin is a measure of system stability in control theory, specifically indicating how much gain can be increased before a system becomes unstable. It provides insight into how close a system is to the brink of instability, relating closely to various stability criteria and performance metrics. Understanding gain margin is essential for analyzing the robustness of control systems and their response to changes in parameters or disturbances.
Hendrik W. Bode: Hendrik W. Bode was an influential engineer and researcher known for his pioneering contributions to control theory, particularly in the areas of system analysis and frequency response. His work laid the foundation for techniques that are essential for understanding how systems behave under various conditions, and he is well-regarded for developing methods that link time-domain analysis to frequency-domain concepts. Bode's insights and methodologies have significantly shaped modern control engineering, particularly in areas like stability and performance analysis.
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.
Linear approximation: Linear approximation is a method used to estimate the value of a function at a point close to a known point by using the tangent line at that known point. This technique simplifies complex functions into linear equations, making calculations easier and more manageable. It relies on the concept that, near a specific point, a function can be closely approximated by its linearization, which is particularly useful in calculus and applied mathematics.
Linear modeling: Linear modeling is a mathematical approach used to represent a system or process using linear equations, which describe the relationship between variables in a proportional way. This technique simplifies complex systems, allowing for easier analysis and prediction of system behavior by approximating nonlinear dynamics within a defined operating range. Linear modeling is crucial for understanding system stability and performance in control theory.
Nyquist Criterion: The Nyquist Criterion is a graphical method used in control theory to determine the stability of a feedback control system based on its open-loop frequency response. By analyzing the Nyquist plot, which represents how the gain and phase of a system change with frequency, engineers can assess whether the closed-loop system will remain stable under various conditions. This criterion connects transient response, steady-state error, stability, digital controller design, and linearization by providing a framework to evaluate system performance across these areas.
Perturbation Methods: Perturbation methods are mathematical techniques used to find an approximate solution to a problem that cannot be solved exactly, by introducing a small parameter that represents a perturbation. These methods help simplify complex problems, making them more manageable by analyzing how a system responds to small changes or disturbances from a known solution. This approach is particularly useful in various fields, including physics and engineering, as it allows for the analysis of stability and behavior in dynamic systems.
Root locus: Root locus is a graphical method used in control theory to analyze how the roots of a transfer function change as a particular parameter, usually gain, varies. This technique provides insights into the stability and dynamic behavior of a system by mapping the location of the poles in the complex plane. It connects crucial aspects such as transient response, steady-state error, and system robustness across various applications.
Rudolf Kalman: Rudolf Kalman is a renowned mathematician and engineer best known for developing the Kalman filter, a powerful mathematical tool used for estimating the state of a dynamic system from noisy measurements. His work has had a profound impact on various fields, including control theory, robotics, and signal processing, enabling effective decision-making in systems affected by uncertainty.
Small-signal approximation: Small-signal approximation is a method used in control theory and circuit analysis to simplify the analysis of nonlinear systems by linearizing their behavior around an operating point. This technique assumes that the deviations from the operating point are small, allowing complex nonlinear equations to be approximated as linear ones. This approach makes it easier to analyze system stability, frequency response, and feedback effects without dealing with the complexities of the full nonlinear behavior.
Stability analysis: Stability analysis is the process of determining whether a system's behavior will remain bounded over time in response to initial conditions or external disturbances. This concept is crucial in various fields, as it ensures that systems respond predictably and remain operational, particularly when analyzing differential equations, control systems, and feedback mechanisms.
State-space representation: State-space representation is a mathematical framework used to model dynamic systems through a set of first-order differential (or difference) equations. This approach expresses the system's state variables and their relationships, providing a comprehensive way to analyze and design control systems across various domains.
Taylor Series Expansion: A Taylor series expansion is a representation of a function as an infinite sum of terms, calculated from the values of its derivatives at a single point. This mathematical tool allows us to approximate complex functions using polynomials, which can be particularly useful for simplifying analysis and solving problems in various fields, including control theory. By evaluating the function and its derivatives at a specific point, we can create a polynomial that closely matches the function's behavior near that point.
Transfer Function: A transfer function is a mathematical representation that relates the output of a system to its input in the Laplace domain, typically expressed as a ratio of polynomials. This concept allows for the analysis and design of control systems by capturing dynamic behavior and system characteristics, facilitating the understanding of stability, frequency response, and time-domain behavior.
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