What do you understand by a state variable?

The state variable is expressed as the set of variables that indicates the mathematical "state" of any specific dynamical system. Also, in the absence of outside force influencing the system, the state can be utilized for evaluating the future characteristics of a certain state.

What is state variable model?

The state variable model of a system can be expressed as the model that includes first-order ordinary differential equations (ODEs). The ODEs signify the time derivatives of any given set of state variables. The quantity of state variables designates the order of the system, which is supposed to match the degree of the denominator polynomial in its transfer function description.

The natural variables connected with the energy reserving elements are generally utilized as state variables, although alternate variables can also be considered. Some of the examples of natural variables are as follow:
The capacitor voltage, inductor current, and others in the electric circuit.
The location as well as the velocity of the inertial mass in the mechanical systems.

Basic concepts of state variable model

The following are the basic terminology involved in this topic,

• State
• State variable
• State vector

State

A state is described as the group of variables that outline the system's history in order to predict future values.

State variable

In a system, the number of state variables needed is equivalent to the number of storage elements that exist in the given system.
Examples of state variables-current moving through the inductor and the voltage across a capacitor.

State vector

The state vector is described as the vector that includes the state variables in the form of elements.

Examples of state variable model

The position coordinates, as well as mechanical parts velocity, are standard state variables in a mechanical system. Also, if these variables are known, it is conceivable to estimate the future state of the objects.

• The state variable in thermodynamics is an independent variable such as entropy, internal energy, enthalpy, etc. Some other examples are temperature, pressure, and volume. Some examples of process functions are work, heat, and others.
• In electronic circuits, the node's voltages, as well as the currents in the circuit, are basically the state variables. Also, the quantity of state variables in the electrical circuit is identical to the number of storage elements like inductors and capacitors. The state variable for an inductor is the current through the inductor, and for a capacitor, it is the voltage across the capacitor.
• In ecosystem prototypes, population measures (or concentrations) of plants, animals, and resources like nutrients, organic material are standard state variables.

The state equations

Let $x\left(t\right)$ describe a vector of scalar input, and $u\left(t\right)$ describe a scalar input, and $y\left(t\right)$ describe a scalar output; then the state variable model of a linear time-invariant (LTI) single-input single-output (SISO) system is written in its generic form as:

$$\begin{gathered} \stackrel{·}{x}\left(t\right)=ax\left(t\right)+bu\left(t\right) \\ y\left(t\right)=c^{T}x\left(t\right)+du\left(t\right) \end{gathered}$$

In the above, $A$ is an $n\times n$ matrix, $b$ is a column vector that distributes the inputs, $c^T$ is a row vector that combines the state variables to form the output, and $d$ is a scalar feedforward term contributing to the output.

The state variable model of a multi-input multi-output (MIMO) system with $m$ inputs and $p$ outputs is described by the state and output equations given as:

$$\begin{gathered} \stackrel{·}{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right) \\ y\left(t\right)=C^{T}x\left(t\right)+Du\left(t\right) \end{gathered}$$

In the above equations, the variable dimensions are:

$x\in R^n, u\in R^m, \mathrm{and} y\in R^p$

And the matrices have the following dimensions:

$A\in R^{n\times n}, B\in R^{n\times m}, C\in R^{p\times n}, \mathrm{and} D\in R^{p\times m}$

In the following, we restrict our attention to prototypes of SISO systems unless noted. In other cases, we assume the system corresponds to a precisely proper transfer function description of the system.

State variable model in control systems

The state variables are represented as the states of a general system in most of the engineering or science fields. The conceivable combinations of state variable values are expressed as the system's state space. The state equations are defined as the type of equations that associate the current state of a system to its present input and past states. On the other hand, equations expressing the values of the output variables in terms of the state variables and inputs are called the output equations. As displayed below, for a linear time-invariant system, the state equations and output equations are indicated using coefficient matrices: A, B, C, and D.

$A\in R^{N\times N}, B\in R^{N\times L}, C\in R^{M\times N}, \mathrm{and} D\in R^{M\times L}$ where N, L, and M are the dimensions of the vectors describing the state, input, and output, respectively.

Developing a state-space model from a system diagram (mechanical translating)

In order to create a state-space model, the free-body diagrams of the system can be used. When the quantities that evaluate the system's energy are considered as your state variables, then a state-space system is usually simple to derive. In a mechanical system, for example if the extension of springs (potential energy, ½kx²) is selected and the velocity of masses (kinetic energy, ½mv²); for electrical circuit select voltage across capacitors, ½Ce² (e=voltage) and current through inductors (½Li²).

A state-space model from a system diagram can easily be formed. Nonetheless, there are several conditions in which it is not completely simple to design a state-space model from a system diagram.
There are many different ways to represent any physical system, differential equations, transfer function, state-space, etc. Since they all represent the same physical system, there must be ways to transform from one physical system to another.

Transformations to other forms

If converting from one representation to another, a simple alternative is to go through an intermediate representation. Consider you are changing from a single nth order differential equation to a state-space representation. It is easier to transform the differential equation to a transfer function representation as compared to transforming the transfer function to state space.

Common Mistakes

Students often confuse state variables with state functions. But there is a lot of difference between the two. The state variable is one of the sets of variables that are used to describe the mathematical "state" of a dynamic system whereas the state function is a property whose value doesn’t depend on the path taken to reach that specific value.

Context and Applications

This topic is significant in the professional exams for both graduate and postgraduate courses, especially for

• Bachelor of Technology in Mechanical Engineering
• Bachelor of Technology in Civil Engineering
• Master of Technology in Mechanical Engineering
• Bachelor of Science in Mathematics
• Master of Science in Mathematics
• Doctor of Philosophy in Mathematics

Related Concepts

• MATLAB
• Observability and controllability
• Mathematical modeling of mechanical and electrical systems
• Laplace transforms 
• Control systems

Practice Problems

Q1. State variable analysis has several advantages overall transfer function as:

1. It is applicable for linear and non-linear & variant and time-invariant systems.
2. Analysis of MIMO system.
3. It takes the initial conditions of the system into account.
4. All of the mentioned.

Correct Option: (d)

Explanation:

The advantages of state variables analysis are as follows:

• The state variable analysis is applied for examining the MIMO system.
• It is used in both types of systems, either linear or non-linear.
• Both time-variant and time-invariant are included in state variable analysis.
• The state variable analysis does consider the initial condition.

Q2. The minimum number of states require to describe the two-degree differential equation:

1. 1
2. 2
3. 3
4. 4

Correct Option: (b)

Explanation: The number of states needed in two-degree differential equation will be 2.

Q3. A transfer function of the control system does not have pole-zero cancellation. Which one of the following statements is true?

1. The system is neither controllable nor observable.
2. The system is completely controllable and observable.
3. The system is observable but uncontrollable.
4. The system is controllable but unobservable.

Correct Option: (b)

Explanation: In the case of having pole-zero cancellation, the control system is not observable as well as controllable. 

Q4. The analysis of multiple input and multiple outputs is conveniently studied by:

1. State-space analysis
2. Root locus approach
3. Characteristic equation approach
4. Nicholas chart

Correct Option: (a)

Explanation: The state-space analysis is the only approach among the other options that are utilized for examining the multiple inputs as well as outputs systems.

Q5. The state equation in the phase canonical form can be obtained from the transfer function by:

1. Cascaded decomposition
2. Direct decomposition
3. Inverse decomposition
4. Parallel decomposition

Correct Option: (d)

Explanation: Parallel decomposition will be used for achieving the state equation from the transfer function.