# Controllability and Observability of a State Space Model

#### Introduction

The concept of controllability refers to the ability of a controller to arbitrarily alter the functionality of the system plant. The term observability describes whether the internal state variables of the system can be externally measured. They are dual aspects of the same problem.

#### Controllability

Complete state controllability describes the ability of an external input to move the internal state of a system from any initial state to any other final state during a finite time interval. Moreover, a system with an internal state vector x is called controllable if and only if the system states can be varied by transforming the input of the system.

Controllability Matrix

For LTI (linear time-invariant) systems, a system is reachable if and only if its controllability matrix, ζ, has a complete row rank of p, where p is the dimension of the matrix A, and p × q is the dimension of matrix B.

A system is controllable when any state x1 can be driven to the zero state x = 0 in a finite number of steps. A system is controllable when the rank of the system matrix A is p, and the rank of the controllability matrix is equal to:

If the second equation is not satisfied, the system is not controllable.

#### Observability

A system with an initial state, t0 , is observable if and only if the value of the initial state can be determined from the system output y(t) that has been observed through the time interval t0 < t < tf . If the initial state cannot be calculated, the system is unobservable.

State-Observability

A system is completely state-observable at time t0 or the pair (A, C) is observable at t0 if the only state that is unobservable at t0 is the zero state x = 0.

Observability Matrix

The observability of the system is dependent only on the system states and the system output, so we can simplify our state equations to remove the input terms:

Matrix Dimensions:
A: p × p
B: p × q
C: r × p
D: r × q

Therefore, we can show that the observability of the system is dependent only on the coefficient matrices A and C. We can show precisely how to determine whether a system is observable, using only these two matrices. If we have the observability matrix Q:

We can show that the system is observable if and only if the Q matrix has a rank of p.

#### Controllability of a System

Follow the given steps to find out whether a system is controllable or not.

1. Open MATLAB

2. Define the state space model you want to analyse.

The resulting system is as follows:

3. Determine the controllability matrix with the help of the command ‘ctrb’. This is demonstrated below.

4. Check the value of the determinant of the controllability matrix in order to determine whether the system is controllable or not.

#### Observability of a System

Follow the given steps to find out whether a system is observable or not.

1. Open MATLAB

2.  Define the state space model you want to analyse.

The output is as follows:

3.  Determine the observability matrix with the help of the command ‘obsv’ followed by its rank. This is illustrated below.

4. Check whether the system is observable or not using the condition given below:

#### Conclusion

Taking a step a further into state space analysis, we successfully observed the controllability and observability of a given state space model.

##### 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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