Control System Toolbox™

ctrb - Controllability matrix

Syntax

Co = ctrb(sys)

Description

ctrb computes the controllability matrix for state-space systems. For an n-by-n matrix A and an n-by-m matrix B, ctrb(A,B) returns the controllability matrix

(2-1)

where has n rows and nm columns.

Co = ctrb(sys) calculates the controllability matrix of the state-space LTI object sys. This syntax is equivalent to executing

Co = ctrb(sys.A,sys.B)

The system is controllable if Co has full rank n.

Example

Check if the system with the following data

A =
     1     1
     4    -2

B =
     1    -1
     1    -1

is controllable. Type

Co=ctrb(A,B);

% Number of uncontrollable states
unco=length(A)-rank(Co)

These commands produce the following result.

unco =
     1

Limitations

Estimating the rank of the controllability matrix is ill-conditioned; that is, it is very sensitive to roundoff errors and errors in the data. An indication of this can be seen from this simple example.

This pair is controllable if but if , where eps is the relative machine precision. ctrb(A,B) returns

which is not full rank. For cases like these, it is better to determine the controllability of a system using ctrbf.

See Also

ctrbf, obsv

covar

ctrbf