A state-space model represents a system as a set of input, output, and state variables. it can be used to model physical systems, measurements, and noise. It uses a numerical integration scheme that operates on a state vector. The relationship between these variables can be described with a series of first-order linear differential equations. The state variables define the output variables. The state variables are the minimum set of variables that fully describe the system. The state-space system converts a set of differential equations into an equivalent set of 1st-order ODEs.
In its continuous linear form, the equation for a state space model is:
$$\begin{matrix} \dot{x} = Ax+Bu \\
y=Cx+Du
\end{matrix}$$
For the Nonlinear form, the equation for a state-space model is:
$$\begin{matrix}
\dot{x}=f(x,u) \\
y=g(x,u)
\end{matrix}$$
Where \(x\) is the state vector, \(u\) is the input vector, and \(y\) is the output vector. The state-space form is a type of linear model. State-space models are the ideal representation of systems for modern control system design.
\(A\) is called the state matrix. it has dimensions of \(N_x \times N_x\) where \(N_x\) is the number of states. The state matrix describes the internal dynamics of the system. $B$ is the input to state matrix. It has dimensions of \(N_x \times N_u\) where \(N_u\) is the number of inputs to the system. This matrix describes how inputs to the system affect the internal states. \(C\) is the state-to-output matrix, It has the dimensions of \(N_y \times N_x\) where \(N_y\) is the number of elements in the output vector. This matrix describes how the internal states affect observable parameters. The final matrix \(D\) is called the feedthrough matrix, it has the dimensions of \(N_y \times N_u\). This matrix describes how an input to the system affects the immediate observable outputs. The combination of these matrices along with the state vector and input vectors, describe how the state vector changes over time. The matrix representation remains the same while the input and state vectors change.
The eigenvalues of the system are simply the eigenvalues of the A matrix. Because the state-space model builds equations for the derivative vectors, you can use numerical integration methods to integrate over the derivative vectors to get the states, therefore solving for the time-series dynamics of the system.
State Variables – key to the state-space models
Discrete-Time State-Space Model
Rotating Cube Model
Static Gain State-Space Model
Converting Transfer Function to State-Space Model – number of states is equal to cumulative order of transfer functions
MIMO State-Space Models
Spring Mass Damper System
State-Space Simulation
Missile Longitudinal Dynamics
Numerical Integration Methods -used to solve the state-space models to get a time response
4DoF Dynamic Lateral Aircraft Model
State-Space Circuit Model – examples of circuits
State Observer Error Equation
Process and Measurement Noise
Equations for Dynamic Inversion Control
Transfer Function of a State-Space System
Wing Rock in Fighter Aircraft
Wing Rock Adaptive Controller
Nonlinear Simulation Methods – alternative to linear state-space
State-Space Closed-Loop Control – equation of matrices
David Conn – developed the state-space method for microwave systems.
F-15 Active Lateral Directional Model – a lateral-directional state-space model of F-15 Active
Matlab minreal() – removes uncontrollable or unobservable states.
Adding Damping to a Simple Harmonic Oscillator – eigenvalue analysis of state-space Model
Rotorcraft System Identification – the rotorcraft derivative models are typically in state-space form.
[[Simulink Block Diagram of State-Space Model]]
State-Space Bicycle Vehicle Model – a state-space model of bicycle vehicle
State Feedback – control mechanism
Eigenvalues of State-Space System
Kalman Filter – uses a state-space plant representation
LQR Controller – uses a state-space plant representation
LQR-PI Controller – also uses a state-space plant representation
Model-predictive Control – also uses a state-space plant representation
Modal Form – Used to re-bias the state variable vectors
Fibonacci Sequence – the Fibonacci sequence can be represented as a state-space model
Gyroplane Model– uses a state-space representation
Disadvantages of the State-Space Model
General Accelerometer State-Space Model – state-space model of a general accelerometer
Piezoelectric Accelerometer State-Space Model– model of piezoelectric accelerometer
Capacitive Accelerometer State-Space Model – model of capacitive accelerometer
Laplace Transform of a State-Space Model
Equivalent Reduced Wing Structure – uses either a lumped-parameter model or a state-space model.
State-Space Averaging – switching power converters use this method to average the state-space models of each switching condition.
State-Space Estimators
State-Space Regulators
State-Space Model of a Pendulum
State-Space Model with Sensor Bias
JDAM Baseline Guidance System – uses state-space representation of a combined plant and controller system
Python State-Space Spring Mass Damper System – example of state-space implementation with Python
Lead-Lag Compensator to State-Space – converting time-domain lead/lag compensator to state-space
Steady State Transfer Function Matrix – uses a state-space representation to derive the transfer matrix.
Second-Order Continuous System with Time Delay
Rate Saturation Equations – state-space model with rate saturation
Oblique Wing Aircraft Moment Equations – state-space moment equations
Eurofighter Control Law Control Surface and Integration Location – uses an inversion of the control (B) matrix
Second-Order Continuous System with Time Delay – includes state-space representation
MFA Control Law – state-space models are used for the research aircraft models
[[X-31A Flight Control Laws]] – use a state-space plant model for designing the feedback matrix
[[Nodal Analysis]] – is identical to state-space modeling for circuits
Generator and Converter Model – generator is a 6-order state-space model
MMC Motor Propulsion System Model – uses 4th and 2nd-order state-space models for the electrical and mechanical parts of the motor
[[Flywheel Energy Storage]] – has a PMSM that can be modeled by a 2nd-order state-space model
[[Proportional Navigation Seeker State-Space Model]]
[[Resistive Thevenin Battery Model 2RC]] – includes a state-space battery model
[[MCLAWS Simulink Model]] – actuators used 2nd-order dynamics in state-space form
[[LTI Systems]] – can be expressed in state-space form
[[N4SID]] – state-space subsystem identification algorithm
[[Combining State-Space Models for Ventilators]]
[[State-Space Pilot-Vehicle Model]]
[[F-18 Longitudinal Model]] – 10-state state-space model
[[F-22 Scale Model Parameters]] – uses the state-space form
[[ARES Simulation Model]] – 6DoF nonlinear state-space simulation
[[B-2 State-Space Conversion]] – converted NASTRAN frequency data to a state-space model
[[F-22 Lateral Directional Flight Control System]] – state-space matrices were provided for each analysis flight condition
[[F100 Model]] – uses a state-space form
[[F-16 Longitudinal State-Space Model]]
Point Kinetics Decomposition Method – uses a state-space model for the reactor neutronics
[[CIFER]]
Sources
- [1] “Control System Simulation.” Accessed: Apr. 11, 2023. [Online]. Available: https://cookierobotics.com/063/
- [2] “Introduction to State-Space Equations | State Space, Part 1 – YouTube.” Accessed: Jun. 04, 2023. [Online]. Available: https://www.youtube.com/
- [3] K. M. Legursky, “System Identification and the Modeling of Sailing Yachts,” in Day 2 Thu, October 25, 2012, Providence, Rhode Island, USA: SNAME, Oct. 2012, p. D021S002R002. doi: 10.5957/SMC-2012-A32.