ctrb

Controllability of state-space model

Syntax

Co = ctrb(A,B)

Co = ctrb(sys)

Description

A dynamic system is said to be controllable if it is possible to apply control signals that drive the system to any state within a finite amount of time. This characteristic is also called reachability. ctrb computes a controllability matrix from state matrices or from a state-space model. You can use this matrix to determine controllability.

For instance, consider a continuous-time state-space model with Nx states, Ny outputs, and Nu inputs:

$$\begin{array}{l}\dot{x}=Ax+Bu\\ y=Cx+Du\end{array}$$

Here, x, u and y represent the states, inputs and outputs respectively, while A, B, C and D are the state-space matrices with the following sizes:

• A is an Nx-by-Nx real-valued or complex-valued matrix.

• B is an Nx-by-Nu real-valued or complex-valued matrix.

• C is an Ny-by-Nx real-valued or complex-valued matrix.

• D is an Ny-by-Nu real-valued or complex-valued matrix.

The system is controllable if the controllability matrix generated by ctrb $$Co=\left[\begin{array}{ccccc}B& AB& A^{2}B& \dots & A^{n-1}B\end{array}\right]$$ has full rank, that is, the rank is equal to the number of states in the state-space model. The controllability matrix Co has Nx rows and Nxu columns. For an example, see Controllability of SISO State-Space Model.

Co = ctrb(A,B) returns the controllability matrix Co using the state matrix A and input-to-state matrix B. The system is controllable if Co has full rank, that is, the rank of Co is equal to the number of states.

Co = ctrb(sys) returns the controllability matrix of the state space model sys. This syntax is equivalent to:

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

Examples

Check System Controllability

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.

Controllability of SISO State-Space Model

For this example, consider the following SISO state-space model with 2 states:

$$A=\left[\begin{array}{cc}-1.5& -2\\ 1& 0\end{array}\right]B=\left[\begin{array}{c}0.5\\ 0\end{array}\right]C=\left[\begin{array}{cc}0& 1\end{array}\right]D=1$$

Create the SISO state-space model defined by the following state-space matrices:

A = [-1.5,-2;1,0];
B = [0.5;0];
C = [0,1];
D = 1;
sys = ss(A,B,C,D);

Compute the controllability matrix and find the rank.

Co = ctrb(sys)

Co = 2×2

    0.5000   -0.7500
         0    0.5000

The size of the controllability matrix depends on the size of the A and B matrices. For instance, if matrix A is an Nx-by-Nx matrix and matrix B is an Nx-by-Nu matrix, then the resultant matrix Co has Nx rows and Nxu columns. Here, Nx is the number of states and Nu is the number of inputs.

rank(Co)

ans = 
2

Since the rank of the controllability matrix Co is equal to the number of states, the system sys is controllable.

Alternatively, you can also use just the A and B matrices to find the controllability matrix.

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

ans = 
2

Input Arguments

A — State matrix

Nx-by-Nx matrix

State matrix, specified as an Nx-by-Nx matrix where, Nx is the number of states.

B — Input-to-state matrix

Nx-by-Nu matrix

Input-to-state matrix, specified as an Nx-by-Nu matrix where Nx is the number of states and Nu is the number of inputs.

sys — State-space model or model array

ss model object | genss model object | uss model object

State-space model or model array, specified as:

• A state-space (ss) model object, when the inputs A, B, C and D are numeric matrices or when converting from another model object type.

• A generalized state-space model (genss) object, when one or more of the matrices A, B, C and D includes tunable parameters, such as realp parameters or generalized matrices (genmat). The function uses the current values for tunable parameters.

• An uncertain state-space model (uss) object, when one or more of the inputs A, B, C and D includes uncertain matrices. The function uses the nominal values for uncertain parameters. Using uncertain models requires Robust Control Toolbox™ software.

Output Arguments

Co — Controllability matrix

array

Controllability matrix, returned as an array. When sys is:

• A single state-space model with Nx states and Nu inputs, then the resultant array Co has Nx rows and Nxu columns.

• An array of state-space models sys(:,:,j1,...,jN), then Co is an array with N+2 dimensions, that is, Co(:,:,j1,...,jN).

References

[1] Paige, C. C. "Properties of Numerical Algorithms Related to Computing Controllability." IEEE Transactions on Automatic Control. Vol. 26, Number 1, 1981, pp. 130-138.

Version History

Introduced before R2006a