5.1 State-space models

State-space models are powerful tools for analyzing and controlling complex systems. They use mathematical equations to describe a system's behavior over time, representing its internal state, inputs, and outputs.

These models are crucial in control theory, allowing engineers to design effective controllers for various applications. By capturing a system's dynamics in matrix form, state-space models enable the use of linear algebra techniques for analysis and control design.

State-space representation

• State-space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations or difference equations
• Provides a convenient and compact way to model and analyze the behavior of a system with multiple inputs and outputs
• Allows for the application of powerful mathematical tools from linear algebra and control theory to analyze and design complex systems

State variables

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• State variables are a set of variables that completely describe the state or condition of a system at any given time
• Represent the minimum amount of information needed to predict the future behavior of the system
• Examples include position and velocity of a mechanical system, voltage and current of an electrical circuit, or temperature and pressure of a thermal system

State equations

• State equations describe the dynamics of the system by relating the state variables to the inputs and the rate of change of the state variables
• Represented as a set of first-order differential equations (continuous-time) or difference equations (discrete-time)
• Capture the internal dynamics of the system and how the state variables evolve over time based on the current state and input

Output equations

• Output equations relate the state variables and the inputs to the outputs of the system
• Describe how the measurable or observable quantities of the system depend on the internal state and the external inputs
• Allow for the computation of the system outputs based on the current state and input values

Matrix notation

• State-space models are often represented using matrix notation for compactness and ease of manipulation
• The state equations are written as $\dot{x}(t) = Ax(t) + Bu(t)$ (continuous-time) or $x[k+1] = Ax[k] + Bu[k]$ (discrete-time), where $x$ is the state vector, $u$ is the input vector, $A$ is the state matrix, and $B$ is the input matrix
• The output equations are written as $y(t) = Cx(t) + Du(t)$ (continuous-time) or $y[k] = Cx[k] + Du[k]$ (discrete-time), where $y$ is the output vector, $C$ is the output matrix, and $D$ is the feedthrough matrix

Linear vs nonlinear models

• State-space models can be classified as linear or nonlinear based on the nature of the equations describing the system dynamics
• Linear models have state equations and output equations that are linear combinations of the state variables and inputs, resulting in the matrices $A$, $B$, $C$, and $D$ being constant
• Nonlinear models have state equations or output equations that contain nonlinear functions of the state variables or inputs, such as quadratic terms, trigonometric functions, or exponentials
• Linear models are easier to analyze and design controllers for, while nonlinear models can capture more complex behaviors but require specialized techniques for analysis and control

Continuous-time state-space models

• Continuous-time state-space models describe the behavior of a system using differential equations, where the state variables and outputs are functions of a continuous time variable $t$
• Commonly used for modeling physical systems that evolve continuously over time, such as mechanical, electrical, and thermal systems
• The state equations and output equations are expressed using derivatives of the state variables and inputs with respect to time

First-order differential equations

• In continuous-time state-space models, the state equations are represented as a set of first-order differential equations
• Each state variable is associated with a first-order differential equation that describes its rate of change with respect to time
• The right-hand side of the differential equation is a linear combination of the state variables and inputs, with coefficients given by the elements of the $A$ and $B$ matrices

Higher-order differential equations

• Some systems may be described by higher-order differential equations, such as second-order or third-order equations
• Higher-order differential equations can be converted into a set of first-order differential equations by introducing additional state variables
• For example, a second-order differential equation can be transformed into two first-order differential equations by defining the velocity as an additional state variable

Discrete-time state-space models

• Discrete-time state-space models describe the behavior of a system using difference equations, where the state variables and outputs are defined at discrete time instants $k$
• Used for modeling systems that are sampled or controlled at regular intervals, such as digital control systems or computer-controlled processes
• The state equations and output equations are expressed using differences of the state variables and inputs between consecutive time steps

Difference equations

• In discrete-time state-space models, the state equations are represented as a set of difference equations
• Each state variable is associated with a difference equation that describes its value at the next time step based on the current state and input
• The right-hand side of the difference equation is a linear combination of the state variables and inputs at the current time step, with coefficients given by the elements of the $A$ and $B$ matrices

Sampling and discretization

• Continuous-time systems can be converted into discrete-time models through a process called sampling and discretization
• Sampling involves measuring the continuous-time signals at regular intervals and representing them as a sequence of discrete-time values
• Discretization methods, such as the zero-order hold (ZOH) or the Tustin approximation, are used to approximate the continuous-time system dynamics in the discrete-time domain
• The choice of the sampling period and discretization method can affect the accuracy and stability of the resulting discrete-time model

State-space model properties

• State-space models possess certain properties that are crucial for the analysis, design, and control of systems
• These properties include controllability, observability, and stability, which provide insights into the fundamental characteristics of the system and its behavior
• Understanding and leveraging these properties is essential for designing effective control strategies and ensuring the desired performance of the system

Controllability

• Controllability is a property that determines whether a system can be steered from any initial state to any desired final state within a finite time by applying an appropriate input
• A system is said to be controllable if there exists an input sequence that can drive the system from any initial state to any desired state
• The controllability matrix, denoted as $\mathcal{C} = [B, AB, A^2B, \ldots, A^{n-1}B]$, is used to check the controllability of a system, where $n$ is the number of state variables
• If the controllability matrix has full rank (i.e., rank $n$), then the system is controllable

Observability

• Observability is a property that determines whether the initial state of a system can be determined from the observed outputs over a finite time interval
• A system is said to be observable if the initial state can be uniquely determined from the knowledge of the input and output sequences
• The observability matrix, denoted as $\mathcal{O} = [C^T, (CA)^T, (CA^2)^T, \ldots, (CA^{n-1})^T]^T$, is used to check the observability of a system, where $n$ is the number of state variables
• If the observability matrix has full rank (i.e., rank $n$), then the system is observable

Stability

• Stability is a property that characterizes the long-term behavior of a system and its response to perturbations or initial conditions
• A system is said to be stable if its state variables remain bounded and converge to an equilibrium point or a steady-state value over time
• The stability of a state-space model can be determined by analyzing the eigenvalues of the state matrix $A$
• If all the eigenvalues of $A$ have negative real parts (continuous-time) or lie within the unit circle (discrete-time), then the system is asymptotically stable

State-space model transformations

• State-space model transformations involve modifying the state variables, inputs, or outputs of a system to obtain an equivalent representation with desired properties or simplified structure
• These transformations can be used to convert a state-space model into a canonical form, decouple the system dynamics, or facilitate the design of controllers and observers
• Common types of state-space model transformations include similarity transformations, canonical forms, and coordinate transformations

Similarity transformations

• Similarity transformations involve applying a nonsingular matrix $T$ to the state variables of a system, resulting in a new set of state variables $z = Tx$
• The transformed state-space model has the same input-output behavior as the original model but may have a different state matrix $\tilde{A} = TAT^{-1}$, input matrix $\tilde{B} = TB$, and output matrix $\tilde{C} = CT^{-1}$
• Similarity transformations preserve the eigenvalues, controllability, and observability properties of the system
• They can be used to simplify the state-space model, decouple the system dynamics, or convert the model into a canonical form

Canonical forms

• Canonical forms are standardized representations of state-space models that have specific structures and properties
• They are obtained by applying appropriate similarity transformations to the original state-space model
• Canonical forms can simplify the analysis and design of controllers and observers by exploiting the special structure of the matrices
• Two commonly used canonical forms are the controllable canonical form and the observable canonical form

Controllable canonical form

• The controllable canonical form is a state-space representation in which the state matrix $A$ and input matrix $B$ have a specific structure that highlights the controllability properties of the system
• In the controllable canonical form, the state matrix $A$ is a companion matrix, and the input matrix $B$ has a simple form with ones and zeros
• The controllable canonical form can be obtained by applying a similarity transformation based on the controllability matrix
• It is useful for designing state feedback controllers and pole placement techniques

Observable canonical form

• The observable canonical form is a state-space representation in which the state matrix $A$ and output matrix $C$ have a specific structure that highlights the observability properties of the system
• In the observable canonical form, the state matrix $A$ is a companion matrix, and the output matrix $C$ has a simple form with ones and zeros
• The observable canonical form can be obtained by applying a similarity transformation based on the observability matrix
• It is useful for designing state observers and output feedback controllers

Coordinate transformations

• Coordinate transformations involve changing the basis or the reference frame in which the state variables are expressed
• They can be used to simplify the state-space model, decouple the system dynamics, or align the state variables with physical quantities of interest
• Examples of coordinate transformations include rotation matrices, scaling matrices, and linear combinations of state variables
• Coordinate transformations can be applied to the state variables, inputs, or outputs of the system, depending on the desired objectives

State-space model analysis

• State-space model analysis involves studying the properties, behavior, and performance of a system using the tools and techniques of linear algebra and control theory
• It aims to gain insights into the system dynamics, stability, and response characteristics, which are essential for designing effective control strategies
• Key aspects of state-space model analysis include eigenvalue and eigenvector analysis, modal decomposition, and Lyapunov stability

Eigenvalues and eigenvectors

• Eigenvalues and eigenvectors are fundamental concepts in linear algebra that play a crucial role in state-space model analysis
• Eigenvalues are scalar values $\lambda$ that satisfy the equation $Av = \lambda v$, where $A$ is the state matrix and $v$ is a nonzero vector called an eigenvector
• The eigenvalues of the state matrix $A$ determine the stability and dynamic behavior of the system
• If all the eigenvalues have negative real parts (continuous-time) or lie within the unit circle (discrete-time), the system is asymptotically stable
• The eigenvectors associated with each eigenvalue represent the modes or directions in which the system evolves

Modal decomposition

• Modal decomposition is a technique that expresses the state-space model in terms of its eigenvectors and eigenvalues
• It involves diagonalizing the state matrix $A$ using a modal matrix $V$ whose columns are the eigenvectors of $A$
• The resulting state-space model has a diagonal state matrix $\Lambda = V^{-1}AV$, where $\Lambda$ is a diagonal matrix containing the eigenvalues of $A$
• Modal decomposition decouples the system dynamics into independent modes, each associated with an eigenvalue and eigenvector pair
• It provides insights into the natural frequencies, damping ratios, and mode shapes of the system

Lyapunov stability

• Lyapunov stability is a powerful framework for analyzing the stability of nonlinear systems and designing stabilizing controllers
• It is based on the concept of Lyapunov functions, which are scalar functions that decrease along the system trajectories
• A system is said to be Lyapunov stable if there exists a Lyapunov function $V(x)$ that satisfies certain conditions, such as being positive definite and having a negative semidefinite time derivative
• Lyapunov stability can be used to determine the stability of equilibrium points, estimate the region of attraction, and design stabilizing feedback controllers
• Common Lyapunov functions include quadratic forms, sum-of-squares polynomials, and energy-like functions

State-space model design

• State-space model design involves developing control strategies and algorithms based on the state-space representation of a system
• It aims to achieve desired performance objectives, such as stabilization, tracking, disturbance rejection, or optimal control, by manipulating the system inputs based on the measured or estimated states
• Key techniques in state-space model design include pole placement, state feedback control, state observers, and optimal control

Pole placement

• Pole placement is a control design technique that aims to place the closed-loop poles (eigenvalues) of a system at desired locations in the complex plane
• It involves designing a state feedback controller $u = -Kx$, where $K$ is a gain matrix, such that the eigenvalues of the closed-loop system matrix $A-BK$ match the desired pole locations
• Pole placement allows for shaping the dynamic response of the system, such as achieving a desired settling time, overshoot, or damping ratio
• The desired pole locations are chosen based on performance specifications and constraints, such as stability margins or frequency-domain characteristics

State feedback control

• State feedback control is a control strategy that uses the measured or estimated states of a system to generate the control input
• It involves designing a feedback gain matrix $K$ such that the control input $u = -Kx$ stabilizes the system and achieves the desired performance objectives
• State feedback control can be combined with pole placement techniques to assign the closed-loop poles at desired locations
• It requires full state measurement or state estimation using observers if some states are not directly measurable
• State feedback control can be extended to include integral action, feedforward terms, or adaptive mechanisms to improve robustness and performance

State observers

• State observers are dynamical systems that estimate the unmeasured states of a system based on the available measurements and the system model
• They are used when some of the states cannot be directly measured or when the measurements are noisy or incomplete
• State observers combine the model predictions with the measured outputs to produce an estimate of the complete state vector
• Two common types of state observers are full-order observers and reduced-order observers

Full-order observers

• Full-order observers estimate all the states of a system, including the measured and unmeasured states
• They have the same order (number of states) as the original system and are designed to have stable error dynamics
• The observer gain matrix is chosen such that the observer poles are placed at desired locations, ensuring fast convergence of the state estimates to the true values
• Full-order observers are commonly used when all the states need to be estimated or when the system has a high degree of uncertainty

Reduced-order observers

• Reduced-order observers estimate only the unmeasured states of a system, assuming that the measured states are directly available
• They have a lower order than the original system, as they do not estimate the measured states
• Reduced-order observers are designed to have stable error dynamics for the unmeasured states and can be combined with the measured states to reconstruct the complete state vector
• They are computationally more efficient than full-order observers and are preferred when some states are already measured or when the system has a large number of states

Optimal control

• Optimal control is a control design approach that seeks to find the best control input that minimizes a specified performance criterion or cost function
• It involves formulating an optimization problem that balances the control effort, state deviations, and other performance metrics over a given time horizon
• Two widely used optimal control techniques are the linear quadratic regulator (LQR) and the Kalman filter

Linear quadratic regulator (LQR)

• The linear quadratic regulator (LQR) is an optimal control technique for linear systems that minimizes a quadratic cost function of the states and control inputs
• The cost function typically includes weighted terms for the state deviations and control effort, with the weights reflecting the relative importance of each term
• The LQR control law is given by $u = -Kx$, where $K$ is the optimal feedback gain matrix obtained by solving the algebraic Riccati equation
• LQR provides a systematic way to design state feedback controllers that balance performance and control effort, and it guarantees stability and robustness properties

Kalman filter

• The Kalman filter is an optimal state estimation technique for linear systems in the presence of process and measurement noise
• It recursively estimates the states of a system by combining the model predictions with the noisy measurements in a statistically optimal way
• The Kalman filter consists of two main steps: prediction and update, which are performed iteratively as new measurements become available
• The prediction step uses the system model to propagate the state estimate and its uncertainty (covariance) forward in time
• The update step corrects the