State Space Model | Control Systems - Electrical Engineering (EE) PDF Download

The state space model of Linear Time-Invariant (LTI) system can be represented as,
State Space Model | Control Systems - Electrical Engineering (EE)

The first and the second equations are known as state equation and output equation respectively. Where,

  • X and X˙ are the state vector and the differential state vector respectively.
  • U and Y are input vector and output vector respectively.
  • A is the system matrix.
  • B and C are the input and the output matrices.
  • D is the feed-forward matrix.

Basic Concepts of State Space Model

The following basic terminology involved in this chapter.

State

It is a group of variables, which summarizes the history of the system in order to predict the future values (outputs).

State Variable

The number of the state variables required is equal to the number of the storage elements present in the system.

Examples − current flowing through inductor, voltage across capacitor

State Vector

It is a vector, which contains the state variables as elements.

In the earlier chapters, we have discussed two mathematical models of the control systems. Those are the differential equation model and the transfer function model. The state space model can be obtained from any one of these two mathematical models. Let us now discuss these two methods one by one.

State Space Model from Differential Equation

Consider the following series of the RLC circuit. It is having an input voltage, vi(t) and the current flowing through the circuit is i(t) .
State Space Model | Control Systems - Electrical Engineering (EE)

There are two storage elements (inductor and capacitor) in this circuit. So, the number of the state variables is equal to two and these state variables are the current flowing through the inductor, i(t) and the voltage across capacitor, vc(t) . From the circuit, the output voltage, v0(t) is equal to the voltage across capacitor, vc(t) .
State Space Model | Control Systems - Electrical Engineering (EE)Apply KVL around the loop.

State Space Model | Control Systems - Electrical Engineering (EE)

The voltage across the capacitor is -
State Space Model | Control Systems - Electrical Engineering (EE)Differentiate the above equation with respect to time.
State Space Model | Control Systems - Electrical Engineering (EE)State Space Model | Control Systems - Electrical Engineering (EE)
We can arrange the differential equations and output equation into the standard form of state space model as,
State Space Model | Control Systems - Electrical Engineering (EE)Where,
State Space Model | Control Systems - Electrical Engineering (EE)

State Space Model from Transfer Function

Consider the two types of transfer functions based on the type of terms present in the numerator.

  • Transfer function having constant term in Numerator.
  • Transfer function having polynomial function of ‘s’ in Numerator.

Transfer function having constant term in Numerator

Consider the following transfer function of a system
State Space Model | Control Systems - Electrical Engineering (EE)Rearrange, the above equation as 
State Space Model | Control Systems - Electrical Engineering (EE)Apply inverse Laplace transform on both sides. 
State Space Model | Control Systems - Electrical Engineering (EE)Let 
State Space Model | Control Systems - Electrical Engineering (EE)and u(t)=u
Then, 
State Space Model | Control Systems - Electrical Engineering (EE)From the above equation, we can write the following state equation.
State Space Model | Control Systems - Electrical Engineering (EE)The output equation is - 
State Space Model | Control Systems - Electrical Engineering (EE)The state space model is - 
State Space Model | Control Systems - Electrical Engineering (EE)Here, D=[0].

Example: Find the state space model for the system having transfer function.
State Space Model | Control Systems - Electrical Engineering (EE)Rearrange, the above equation as, 
State Space Model | Control Systems - Electrical Engineering (EE)
Apply inverse Laplace transform on both the sides. 
State Space Model | Control Systems - Electrical Engineering (EE)Let 
State Space Model | Control Systems - Electrical Engineering (EE)
and u(t)=u
Then, the state equation is
State Space Model | Control Systems - Electrical Engineering (EE)The output equation is 
State Space Model | Control Systems - Electrical Engineering (EE)The state space model is 
State Space Model | Control Systems - Electrical Engineering (EE)

Transfer function having polynomial function of ‘s’ in Numerator

Consider the following transfer function of a system
State Space Model | Control Systems - Electrical Engineering (EE)The above equation is in the form of product of transfer functions of two blocks, which are cascaded.
State Space Model | Control Systems - Electrical Engineering (EE)Here, 
State Space Model | Control Systems - Electrical Engineering (EE)Rearrange, the above equation as
State Space Model | Control Systems - Electrical Engineering (EE)Apply inverse Laplace transform on both the sides. 
State Space Model | Control Systems - Electrical Engineering (EE)Let
State Space Model | Control Systems - Electrical Engineering (EE)

and u(t)=u
Then, the state equation is
State Space Model | Control Systems - Electrical Engineering (EE)Consider
State Space Model | Control Systems - Electrical Engineering (EE)Rearrange, the above equation as
State Space Model | Control Systems - Electrical Engineering (EE)Apply inverse Laplace transform on both the sides.
State Space Model | Control Systems - Electrical Engineering (EE)By substituting the state variables and y(t)=y in the above equation, will get the output equation as,
State Space Model | Control Systems - Electrical Engineering (EE)Substitute, x˙n value in the above equation.
State Space Model | Control Systems - Electrical Engineering (EE)The state space model is
State Space Model | Control Systems - Electrical Engineering (EE)
State Space Model | Control Systems - Electrical Engineering (EE)If bn=0 , then,
State Space Model | Control Systems - Electrical Engineering (EE)

The document State Space Model | Control Systems - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Control Systems.
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FAQs on State Space Model - Control Systems - Electrical Engineering (EE)

1. What are the basic concepts of a State Space Model?
Ans. The basic concepts of a State Space Model include representing a system's dynamics using state variables, input, output equations, and matrices for state transition and output observation.
2. How can a State Space Model be derived from a set of differential equations?
Ans. A State Space Model can be derived from a set of differential equations by expressing the system in state-space form, where the derivatives of state variables are equal to functions of the states, inputs, and time.
3. How can a State Space Model be obtained from a transfer function?
Ans. A State Space Model can be obtained from a transfer function by using methods like controllability and observability to convert the transfer function into a state-space representation.
4. What does it mean when a transfer function has a constant term in the numerator?
Ans. When a transfer function has a constant term in the numerator, it indicates that the system has a non-zero steady-state gain, which affects the system's overall response to input signals.
5. What is the significance of having a polynomial function of 's' in the numerator of a transfer function?
Ans. Having a polynomial function of 's' in the numerator of a transfer function signifies the presence of poles in the system, which determine the stability and transient response characteristics of the system.
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