Introduction to State Space Model
State Space Model is a mathematical model in control engineering. It is a state-space representation of a physical system of a set of inputs and outputs along with some set of state variables related by first-order differential equations. State variables in this model are a type of variable whose value changes over time and depends on the values that have been given for the input variables. The value of the output variables depends on the value of the state and input variables. Putting a model into state-space representation is the basis for many methods in control analysis and the dynamics process.
Representation of State Space Model
The continuous-time form of state-space model of Linear Time-Invariant(LTI) can be represented as below:
The first equation is the state equation and the second equation is the output equation respectively.
In the above equation,
X is the state vector.
X˙ is the differential state vector.
U is the input vector.
Y is called the output vector.
A is the system matrix
B is the input matrix and C is the output matrix respectively.
D is known as the feed-forward matrix.
Basic Terms related to State-Space Model
- State: It is a group of different variables, which concludes the whole history of the given system to predict the future values of the variables that are, output variables.
- State Space: State Space is known as the set of all possible and known states of a system. In state-space, each unique point represents a state of the system. For example, Take a pendulum moving in to and fro motion. The state of such an idealized pendulum is represented by its angle and its angular velocity. So the state-space here can be defined as the set of all possible pairs of angle and angular velocities(“(angle, velocity)”).
- State Variable: The state variables are one of the sets of state variables or system variables that represent the whole system at any given period. The state variables that are required to represent a given system are n and are equal to the order of the system that is defining the differential equation. In the case of transfer function, it is first reduced to proper fraction, then the denominator of the fraction denotes the minimum number of state variables that are required to represent a given system.
- State Vector: State Vector is a vector in which state variables are represented as elements. There are many mathematical models from which we can define the state-space model. Those are the transfer function model and differential equation model. We can define the model from any of the two existing models.
- Stability: This is one of the important and basic properties of state-space models. In General, in any state-space model, we can define the stability of the system using the eigenvalues of the state-space matrix A. If all the eigenvalues are negative, then we can say that the system is stable. If any of the eigenvalues are positive, then the system becomes unstable.
Example of State-Space Model by direct derivation (Mechanical Translating)
Consider the example shown below and derive a state-apace model for the below-shown figure. The input here is fa and the output is y.
we can derive free body equations at two points from the above-shown figure, x, and y.
m.x + k1.x + k2.x – k1.y = fa
b.y + k1.y – k1.x = 0
In the above figure, we have three energy storage elements, so we obtain three state equations. Here, the energy-storing elements are the spring k2, the spring k1, and mass m. So our state variables will be x and y.
q1 = x
q2 = x
q3 = x
Now, we need to find equations for these derivatives. The motion equations from the free body diagram when input u = fa are,
Advantages of state-space analysis
- The analysis is done based on the initial condition.
which means that we no need to know all the conditions of the system, just by knowing the initial conditions of the system we can estimate the future values or the output of the system. Based on some considerable initial values we can estimate the values of the system.
- More accurate than transfer function techniques.
We mostly consider differential function techniques to estimate the values of the system, but the result obtained when the transfer function technique is used are considered to be more accurate than the results obtained when the differential function technique is used.
- Analysis of multi-input and multi-output systems is made easy by using the state-space model. In the case of the transfer function, when there is single input and single output we can take the Laplace transform of input and the output and get the result. But, when there are multiple inputs and multiple outputs we cannot perform the same process on it. Since we have a state vector in our state-space model, we can represent all our input and output variables in the vector form and operate to get the desired output for the system.
- Gives information about controllability. It means that we can say to how much extent the given system is controllable. It tells us to how much extent it can control its functionality. The State-Space model gives us information about the functionality of a particular system.
- The state-space analysis applies to all dynamic systems, which means that by using this system we can analyze all dynamic systems like linear system, non-linear system, time-variant system, and time-invariant system.
- State Space Model is a mathematical model in control engineering which is a state-space representation of a physical system of a set of inputs and outputs along with some set of state variables related by first-order differential equations.
- Putting a model into state-space representation is the basis for many methods in control analysis and the dynamics process.
- A state is a group of different variables, which concludes the whole history of the given system
- State Space is known as the set of all possible and known states of a system.
- The state variables are one of the sets of state variables or system variables that represent the whole system at any given period.
- State Vector is a vector in which state variables are represented as elements.
This is a guide to State Space Model. Here we discuss the definition, examples of State-Space Model by direct derivation, advantages. You may also have a look at the following articles to learn more –