ctrb

Controllability matrix

Syntax

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

Description

Co = ctrb(A,B) returns the controllability matrix:

$C o = [\begin{matrix} B & A B & A^{2} B & \dots & A^{n - 1} B \end{matrix}]$

where A is an n-by-n matrix, B is an n-by-m matrix, and Co 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:

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

The system is controllable if Co has full rank n.

Examples

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Check System Controllability

Open Live Script

Define A and B matrices.

A = [1  1;
     4 -2];
B = [1 -1;
     1 -1];

Compute controllability matrix.

Co = ctrb(A,B);

Determine the number of uncontrollable states.

unco = length(A) - rank(Co)

unco = 1

The uncontrollable state indicates that Co does not have full rank 2. Therefore the system is not controllable.

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.

$A = \begin{matrix} [\begin{matrix} 1 & δ \\ 0 & 1 \end{matrix}], & B = [\begin{array}{l} 1 \\ δ \end{array}] \end{matrix}$

This pair is controllable if $δ \neq 0$ but if $δ < \sqrt{e p s}$ , where eps is the relative machine precision. ctrb(A,B) returns

$[\begin{matrix} B & A B \end{matrix}] = [\begin{matrix} 1 & 1 \\ δ & δ \end{matrix}]$

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

Documentation

ctrb

Syntax

Description

Examples

Check System Controllability

Limitations

See Also

Control System Toolbox Documentation

Support