State feedback control is a powerful technique in modern control theory. It uses knowledge of a system's internal states to precisely regulate its behavior, making it particularly useful for complex systems with multiple inputs and outputs.
This approach involves measuring or estimating state variables and using this information to generate control inputs. It allows for precise control of system dynamics and can handle various constraints, making it a versatile tool in control engineering.
State feedback control overview
- State feedback control is a powerful technique in modern control theory that allows for the precise regulation of a system's behavior by utilizing the knowledge of its internal states
- It involves measuring or estimating the system's state variables and using this information to generate a control input that drives the system to a desired state or trajectory
- State feedback control is particularly useful for systems with multiple inputs and outputs (MIMO) and can handle complex system dynamics and constraints
State space representation
State variables
- State variables are a set of variables that completely describe the internal state and future behavior of a dynamic system at any given time
- The choice of state variables depends on the specific system and the control objectives, but they typically include physical quantities such as position, velocity, acceleration, temperature, or voltage
- The number of state variables determines the order of the system and the dimension of the state space
State transition matrix
- The state transition matrix, denoted as $A$, describes how the state variables evolve over time in the absence of external inputs
- It captures the inherent dynamics of the system and represents the linear relationships between the state variables
- The state transition matrix is a square matrix with dimensions equal to the number of state variables
- The input matrix, denoted as $B$, describes how the external inputs affect the state variables
- It maps the control inputs to the corresponding changes in the state variables
- The input matrix has dimensions of the number of state variables by the number of control inputs
Output matrix
- The output matrix, denoted as $C$, relates the state variables to the system's outputs or measurements
- It determines which combinations of state variables are observed or measured by the sensors
- The output matrix has dimensions of the number of outputs by the number of state variables
Feedthrough matrix
- The feedthrough matrix, denoted as $D$, represents the direct influence of the control inputs on the system's outputs
- It captures any immediate effect of the inputs on the outputs without passing through the state variables
- In many practical systems, the feedthrough matrix is assumed to be zero, indicating no direct input-output relationship
Pole placement
Desired pole locations
- Pole placement is a technique used to shape the dynamic response of a system by placing its poles at desired locations in the complex plane
- The desired pole locations are chosen based on the desired transient response characteristics, such as settling time, overshoot, and damping ratio
- By selecting appropriate pole locations, the system's response can be made faster, slower, more damped, or less damped
Characteristic equation
- The characteristic equation of a system is obtained by setting the determinant of the state transition matrix minus the identity matrix multiplied by the Laplace variable $s$ equal to zero
- The roots of the characteristic equation are the system's poles, which determine its stability and dynamic behavior
- The coefficients of the characteristic equation can be manipulated by state feedback to achieve the desired pole locations
Controllability
- Controllability is a fundamental property of a system that determines whether it is possible to steer the system from any initial state to any desired state in a finite amount of time
- A system is said to be controllable if the controllability matrix, formed by the input matrix and its successive powers multiplied by the state transition matrix, has full rank
- Controllability is a necessary condition for pole placement, as it ensures that the system's poles can be arbitrarily placed using state feedback
State feedback gain matrix
Feedback gain calculations
- The state feedback gain matrix, denoted as $K$, determines the control input based on the measured or estimated state variables
- The feedback gains are calculated by solving a set of linear equations that relate the desired pole locations to the coefficients of the characteristic equation
- The feedback gains are chosen to cancel out the system's original poles and replace them with the desired pole locations
- Ackermann's formula is a closed-form solution for calculating the state feedback gain matrix based on the desired pole locations and the system's controllability matrix
- It provides a direct way to compute the feedback gains without the need for iterative calculations or numerical methods
- Ackermann's formula is particularly useful for systems with a small number of state variables and distinct desired pole locations
Linear quadratic regulator (LQR)
Cost function
- The linear quadratic regulator (LQR) is an optimal control technique that minimizes a quadratic cost function while stabilizing the system
- The cost function typically includes weighted terms for the state variables and control inputs, representing the trade-off between control effort and system performance
- By choosing appropriate weights, the LQR design can balance the competing objectives of regulation, tracking, and control energy minimization
Riccati equation
- The solution to the LQR problem involves solving the algebraic Riccati equation, which is a nonlinear matrix equation
- The Riccati equation relates the optimal state feedback gain matrix to the system matrices and the cost function weights
- Solving the Riccati equation yields the optimal feedback gains that minimize the cost function while ensuring system stability
LQR gain matrix
- The LQR gain matrix, obtained from solving the Riccati equation, provides the optimal state feedback gains for the LQR problem
- The LQR gains are designed to minimize the quadratic cost function and achieve the desired system performance
- The resulting closed-loop system with LQR control exhibits optimal behavior in terms of regulation, disturbance rejection, and robustness
Observer design
Observability
- Observability is a property of a system that determines whether it is possible to estimate the system's internal states based on the available measurements or outputs
- A system is said to be observable if the observability matrix, formed by the output matrix and its successive powers multiplied by the state transition matrix, has full rank
- Observability is a necessary condition for the design of state observers or estimators
Luenberger observer
- A Luenberger observer is a dynamic system that estimates the state variables of a plant based on the available measurements and the system model
- It consists of a copy of the plant model and a feedback term that corrects the estimation error based on the difference between the actual and estimated outputs
- The observer gain matrix is designed to ensure the convergence of the estimated states to the true states and to achieve desired observer dynamics
Kalman filter
- The Kalman filter is an optimal state estimator that recursively estimates the state variables of a system in the presence of process and measurement noise
- It combines the system model and the available measurements to produce an optimal estimate of the states based on the minimum mean square error criterion
- The Kalman filter is widely used in applications such as navigation, tracking, and control systems where noise and uncertainties are present
Separation principle
Controller and observer design
- The separation principle states that the design of the state feedback controller and the state observer can be carried out independently for a linear system
- The controller is designed assuming that the state variables are available for feedback, while the observer is designed to estimate the states based on the available measurements
- The separation principle allows for a modular and systematic approach to the design of state feedback control systems
Closed-loop stability
- The separation principle guarantees that if the controller and observer are individually stable, then the overall closed-loop system combining the controller and observer will also be stable
- The stability of the closed-loop system can be analyzed by examining the eigenvalues of the state transition matrix of the combined controller-observer system
- The separation principle simplifies the stability analysis and design of state feedback control systems with state estimation
Integral control
Steady-state error
- Integral control is a technique used to eliminate steady-state errors in the system's response to constant reference inputs or disturbances
- The steady-state error is the difference between the desired and actual output values when the system reaches a steady state
- Integral control adds an additional state variable that accumulates the error over time and generates a control input proportional to the accumulated error
Augmented state space model
- To incorporate integral control, the state space model of the system is augmented with an additional state variable representing the integral of the error
- The augmented state space model includes the original state variables as well as the integral state variable
- The integral state variable is driven by the error signal and affects the control input through the feedback gain matrix
Robust control
Parameter uncertainties
- Robust control deals with the design of control systems that maintain desired performance and stability in the presence of uncertainties and variations in system parameters
- Parameter uncertainties can arise due to modeling errors, manufacturing tolerances, aging effects, or changes in operating conditions
- Robust control techniques aim to design controllers that are insensitive to parameter variations within a specified range
Sensitivity analysis
- Sensitivity analysis is a tool used in robust control to quantify the effect of parameter variations on the system's performance and stability
- It involves computing sensitivity functions that relate changes in system parameters to changes in closed-loop performance metrics
- Sensitivity analysis helps identify critical parameters and guides the design of robust controllers that minimize the impact of parameter uncertainties
H-infinity control
- H-infinity control is a robust control technique that minimizes the worst-case gain of the closed-loop system in the presence of uncertainties and disturbances
- It involves formulating the control problem as an optimization problem where the objective is to minimize the H-infinity norm of the closed-loop transfer function
- H-infinity control provides a systematic framework for designing controllers that achieve robust performance and stability guarantees
Digital implementation
Discretization methods
- Digital implementation of state feedback control requires the discretization of the continuous-time system model and controller
- Discretization methods, such as the zero-order hold (ZOH) or the Tustin approximation, convert the continuous-time system matrices into discrete-time equivalents
- The choice of discretization method depends on the desired accuracy, computational efficiency, and the characteristics of the system and controller
Sampling and reconstruction
- In digital control systems, the continuous-time signals are sampled at regular intervals to obtain discrete-time signals
- The sampling rate should be chosen based on the Nyquist-Shannon sampling theorem to avoid aliasing and ensure accurate representation of the continuous-time signals
- Reconstruction techniques, such as zero-order hold or interpolation, are used to convert the discrete-time control inputs back to continuous-time signals for actuation
Digital controller design
- Digital controller design involves the synthesis of discrete-time controllers based on the discretized system model and the desired performance specifications
- The state feedback gain matrix and observer gain matrix are computed using discrete-time equivalents of the continuous-time design methods, such as pole placement or LQR
- Digital controllers are implemented using computer algorithms and can take advantage of the flexibility and programmability of digital systems