Control System Toolbox Function Reference

ctrb

Form the controllability matrix

Syntax

Co = ctrb(A,B)
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

(0-1)

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)

and MATLAB returns

unco =
     1

Limitations

Estimating the rank of the controllability matrix is ill-conditioned; that is, it is very sensitive to round-off 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       Compute the controllability staircase form

obsv        Compute the observability matrix

covar

ctrbf