# obsv

Observability of state-space model

## Syntax

``Ob = obsv(A,C)``
``Ob = obsv(sys)``

## Description

A dynamic system is said to be observable if all its states can be known from the output of the system. `obsv` computes an observability matrix from state matrices or from a state-space model. You can use this matrix to determine observability.

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

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

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 observable if the observability matrix generated by `obsv` has full rank, that is, the rank is equal to the number of states in the state-space model. The observability matrix `Ob` has `Nx` rows and `Nxy` columns. For an example, see Observability of SISO State-Space Model.

example

````Ob = obsv(A,C)` returns the observability matrix `Ob` using the state matrix `A` and state-to-output matrix `C`. The system is observable if `Ob` has full rank, that is, the rank of `Ob` is equal to the number of states.```

example

````Ob = obsv(sys)` returns the observability matrix of the state space model `sys`. This syntax is equivalent to:Ob = obsv(sys.A,sys.C);```

## Examples

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Define `A` and C matrices.

```A = [1 1; 4 -2]; C = [-1 1; 1 -1];```

Compute observability matrix.

`Ob = obsv(A,C);`

Determine the number of unobservable states.

`unobsv = length(A) - rank(Ob)`
```unobsv = 1 ```

The unobservable state indicates that `Ob` does not have full rank 2. Therefore, the system is not observable.

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

`$A=\left[\begin{array}{cc}-1.5& -2\\ 1& 0\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}B=\left[\begin{array}{c}0.5\\ 0\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}C=\left[\begin{array}{cc}0& 1\end{array}\right]\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}D=1$`

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 observability matrix and find the rank.

`Ob = obsv(sys)`
```Ob = 2×2 0 1 1 0 ```

The size of the observability matrix depends on the size of the `A` and `C` matrices. For instance, if matrix A is an `Nx`-by-`Nx` matrix and matrix `C` is an `Nx`-by-`Ny` matrix, then the resultant matrix `Ob` has `Nx` rows and `Nxy` columns. Here, `Nx` is the number of states and `Ny` is the number of outputs.

`rank(Ob)`
```ans = 2 ```

Since the rank of the observability matrix `Ob` is equal to the number of states, the system `sys` is observable.

Alternatively, you can also use just the `A` and `C` matrices to find the observability matrix.

```Ob = obsv(sys.A,sys.C); rank(Ob)```
```ans = 2 ```

## Input Arguments

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

State-to-output matrix, specified as an `Ny`-by-`Nx` matrix where, `Nx` is the number of states and `Ny` is the number of outputs.

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

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Observability matrix, returned as an array. When `sys` is:

• A single state-space model with `Nx` states and `Ny` outputs, then the resultant array `Ob` has `Nx` rows and `Nxy` columns.

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

## Limitations

• `obsv` is not recommended for control design as computing the rank of the observability matrix is not recommended for observability testing. `Ob` will be numerically singular for most systems with more than a handful of states. This fact is well-documented under section III in .

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