Reachability

So there is a nice closed form expression of system trajectories given any input! By convolution of the input, we can express the state and output trajectory of any LTI system as $$ x(t) = e^{At}x_0 + \int_0^t e^{A(t-\tau)}Bu(\tau)d\tau$$ and $$ y(t) = Cx(t) + Du(t) = Ce^{At}x_0 + \int_0^t Ce^{A(t-\tau)}Bu(\tau)d\tau + Du(t).$$ Here, the first term is the effect of the initial condition, the second term is influenced by the input and the third term is the direct feedthrough.

This output trajectory expression answers some interesting questions too: what states are we able to control the system to? This is called the reachability of the system! Remember we have equilibria $(\bar x, \bar u) \leftarrow A\bar x + B\bar u = 0$? Well then if $A$ is invertible this means that $\bar x = -A^{-1}B\bar u$. So, if $A^{-1}B$ is full rank, we can attain any steady state we'd desire!

This only answers steady state reachability, what can we reach in transient manners though? That's slightly more involved, but not too bad. Lets look back at the expression $$ x(t) = e^{At}x_0 + \int_0^t e^{A(t-\tau)}Bu(\tau)d\tau,$$ then the interesting part is $\int_0^t e^{A(t-\tau)}Bu(\tau)d\tau = \int_0^t e^{At}Bu(t-\tau)d\tau$ and especially $e^{At}B$, because this factor we can't influence with the input. The term with the initial condition can be compensated for anyways.

Cayley–Hamilton theorem

Our bestest buddies Cayley and Hamilton came up with a theorem that we need to understand the factor $e^{At}$ better. Quick reminder of how Taylor series worked (around 0 specifically (also called MacLaurin series if you want to do some barefaced fame stealing)): $$ f(x) = \sum_{k=0}^{\infty}\frac{f^{(k)}(0)}{k!}x^k = \sum_{k=0}^{\infty}\alpha_k(x) f^{(k)}(0).$$ Also remember how exponential derivatives worked: $$e^{At} \overset{d/dt}{\rightarrow} Ae^{At} \overset{d/dt}{\rightarrow} A^2e^{At} \overset{d/dt}{\rightarrow} A^3e^{At} \overset{d/dt}{\rightarrow} ... \Rightarrow \frac{d^k}{d^kt}e^{At} = A^ke^{At}.$$

Looks like we're getting somewhere, what if we substitute that derivative expression into the Taylor series and recall that $e^{0A}=I$, then $$ e^{At} = \sum_{k=0}^{\infty}\alpha_k(x) A^k.$$ One step closer still! Now lets right multiply with $B$ and write the summation as a matrix multiplication $$ e^{At}B = \mathbf{\alpha}(t)\begin{bmatrix}B & AB & A^2B & A^3B & \cdots \end{bmatrix}, \qquad \alpha_k(t)=\frac{t^k}{k!}.$$

So if $\begin{bmatrix}B & AB & A^2B & A^3B & \cdots \end{bmatrix}$ has full row rank, we can use the input to make $\int_0^t e^{A(t-\tau)}Bu(\tau)d\tau$ anything we want, so we can make $x(t)$ anything we want!! This is the reachability criterion and we actually only have to check the first $n-1$ powers of $A$. This forms the so called reachability matrix (also called controllability matrix), $$ W_r = \begin{bmatrix}B & AB & A^2B & A^3B & \cdots & A^{n-1}B\end{bmatrix}.$$ So if this $W_r$ of the pair $(A,B)$ is full rank, the system is reachable (or controllable).

There is also a coordinate transform to something called the reachable canonical form. It's in the slides but not very useful in this explanation.