Editing Controllability (section)

== Rank condition for controllability==
The [[Controllability Gramian]] involves integration of the [[state-transition matrix]] of a system. A simpler condition for controllability is a [[rank (linear algebra)|rank]] condition analogous to the Kalman rank condition for time-invariant systems.

Consider a continuous-time linear system  <math>\Sigma</math> smoothly varying in an interval <math>[t_0,t]</math> of  <math>\mathbb{R}</math>:

: <math>\dot{\mathbf{x}}(t) = A(t) \mathbf{x}(t) + B(t) \mathbf{u}(t)</math>
: <math>\mathbf{y}(t) = C(t) \mathbf{x}(t) + D(t) \mathbf{u}(t).</math>

The state-transition matrix <math>\phi</math> is also smooth. Introduce the n x m matrix-valued function <math>M_0(t) = \phi(t_0,t)B(t)</math> and define 
: <math>M_k(t)</math> = <math>\frac{\mathrm{d^k} M_0}{\mathrm{d} t^k}(t), k\geqslant 1</math>.
 
Consider the matrix of matrix-valued functions obtained by listing all the columns of the <math>M_i</math>, <math>i = 0,1, \ldots, k</math>:

<math>M^{(k)}(t) := \left[M_0(t), \ldots, M_k(t)\right] </math>.

If there exists a <math>\bar{t} \in [t_0,t]</math> and a nonnegative integer k such that <math>\operatorname{rank}M^{(k)}(\bar{t})=n</math>, then  <math>\Sigma</math> is controllable.<ref name=":0">Eduardo D. Sontag, [https://books.google.com/books?id=f9XiBwAAQBAJ Mathematical Control Theory: Deterministic Finite Dimensional Systems].</ref>

If  <math>\Sigma</math> is also analytically varying in an interval <math>[t_0,t]</math>, then <math>\Sigma</math> is controllable on every nontrivial subinterval of <math>[t_0,t]</math> if and only if there exists a <math>\bar{t} \in [t_0,t]</math> and a nonnegative integer k such that  <math>\operatorname{rank}M^{(k)}(t_i)=n</math>.<ref name=":0" />

The above methods can still be complex to check, since it involves the computation of the state-transition matrix <math>\phi</math>.  Another equivalent condition is defined as follow. Let <math>B_0(t) = B(t)</math>,  and for each <math>i \geq 0</math>, define

: <math>B_{i+1}(t) </math>= <math>A(t)B_i(t) - \frac{\mathrm{d}}{\mathrm{d} t}B_i(t). </math> 
In this case, each <math>B_i</math> is obtained directly from the data <math> (A(t),B(t)).</math> The system is controllable if there exists a <math>\bar{t} \in [t_0,t]</math> and a nonnegative integer <math>k</math> such that  <math>\textrm{rank}( \left[ B_0(\bar{t}), B_1(\bar{t}), \ldots, B_k(\bar{t}) \right]) = n </math>.<ref name=":0" />

=== Example ===

Consider a system varying analytically in <math> (-\infty,\infty) </math> and matrices

<math>A(t) = \begin{bmatrix}
t & 1 & 0\\ 
0 & t^{3} & 0\\ 
0 & 0 & t^{2} 
\end{bmatrix}</math>, <math>B(t) = \begin{bmatrix}
0 \\ 
1 \\ 
1  
\end{bmatrix}.</math>  
Then 
<math> [B_0(0),B_1(0),B_2(0),B_3(0)] = \begin{bmatrix}
0 & 1 & 0 &-1\\ 
1 & 0 & 0&0\\ 
1 & 0 & 0&2
\end{bmatrix}</math> and since this matrix has rank 3, the system is controllable on every nontrivial interval of  <math>\mathbb{R}</math>.

=== Continuous linear time-invariant (LTI) systems ===

Consider the continuous linear [[time-invariant system]]

: <math>\dot{\mathbf{x}}(t) = A \mathbf{x}(t) + B \mathbf{u}(t)</math>
: <math>\mathbf{y}(t) = C \mathbf{x}(t) + D \mathbf{u}(t)</math>

where 
: <math>\mathbf{x}</math> is the <math>n \times 1</math> "state vector",
: <math>\mathbf{y}</math> is the <math>m \times 1</math> "output vector",
: <math>\mathbf{u}</math> is the <math>r \times 1</math> "input (or control) vector",
: <math>A</math> is the <math>n \times n</math> "state matrix",
: <math>B</math> is the <math>n \times r</math> "input matrix",
: <math>C</math> is the <math>m \times n</math> "output matrix",
: <math>D</math> is the <math>m \times r</math> "feedthrough (or feedforward) matrix".

The <math>n \times nr</math> controllability matrix is given by

:<math>R = \begin{bmatrix}B & AB & A^{2}B & ...& A^{n-1}B\end{bmatrix}</math>

The system is controllable if the controllability matrix has full row [[Rank (linear algebra)|rank]] (i.e. <math>\operatorname{rank}(R)=n</math>).