# State Space Analysis Of Control Systems

#### Introduction

State space analysis of control system is based on the modern theory which is applicable to all types of systems like single input single output systems, multiple inputs and multiple outputs systems, linear and nonlinear systems, time varying and time invariant systems. Let us consider few basic terms related to state space analysis of modern theory of control systems

State: It refers to smallest set of variables whose knowledge at t = t0 together with the knowledge of input for t ≥ t0 gives the complete knowledge of the behavior of the system at any time t ≥ t0.

State Variables: It refers to the smallest set of variables, which help us to determine the state of the dynamic system. State variables are defined by x1(t), x2(t)…Xn(t).

State Vector : Suppose there is a requirement of n state variables in order to describe the complete behavior of the given system, then these n state variables are considered to be n components of a vector x(t). Such a vector is known as state vector.

State Space: It refers to the n dimensional space that has x1 axis, x2 axis …xn axis.

The state space model of Linear Time-Invariant (LTI) system can be represented as :

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.

Follow the given steps to create and analyze a state space model in MATLAB.

1 Open MATLAB

2. Define matrices A, B, C and D.

3. Use the command ‘ss’ to create a state space model as shown below:

The output for the aforementioned command is as follows:

4. In order to create a discrete state space model, use the following command:

The output to the above command is given below:

5. To convert the given state space model into its corresponding transfer function, use the command ‘ss2tf’ as illustrated.

The output is demonstrated below:

6. To convert the transfer function back to the state space model, use the command ‘tf2ss’ as illustrated.

The output is as follows:

#### Conclusion

The creation of a state space model was studied accompanied by its conversion from the transfer function form and vice versa in MATLAB.

##### Shivani Singh

A final year engineering student from Bharati Vidyapeeth's College of Engineering, New Delhi with great zeal and passion for writing and technology. I have great love for food, nature, reading and am a true Delhiite at heart !

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