Controllability/Reachability of an LTI system

Controllability and reachability are defined for discrete-time systems as:

• A system is controllable if from any state $x_0\in {\mathbb{R}}^n$ one can achieve the state $x_N=0$ using a series of inputs $\{u_k{\}}_0^{N-1}$.
• A state $x_N\in {\mathbb{R}}^n$ is reachable if it can be achieved by applying a series of inputs $\{u_k{\}}_0^{N-1}$ when starting from the initial state $x_0=0$.

For continuous-time systems they are similarly defined as:

• A system is controllable if from any state $x(0)\in {\mathbb{R}}^n$ one can achieve the state $x(t)=0$ using a control policy $u(\tau )$, $\tau \in ⟨0,t)$.
• A state $x(t)\in {\mathbb{R}}^n$ is reachable if it can be achieved by applying a control policy $u(\tau )$, $\tau \in ⟨0,t)$ when starting from the initial state $x(0)=0$.

For linear systems all states are reachable if the system is controllable.

Discrete time

The first $n+1$ states of a linear time-invariant system $$x_{k+1}=A{x}_k+B{u}_k,x_k\in {\mathbb{R}}^n,u_k\in {\mathbb{R}}^m,$$ assuming an initial state $x_0$ and a series of inputs $\{u_k{\}}_0^{N-1}$, can be expressed as $$\begin{array}{rl}{x}_1& =A{x}_0+B{u}_0\\ x_2& =A^2{x}_0+AB{u}_0+B{u}_1\\ & ⋮\\ x_n& =A^n{x}_0+A^{n-1}B{u}_0+A^{n-2}B{u}_1+\dots +B{u}_{n-1}\\ x_{n+1}& =A^{n+1}{x}_0+A^{n}B{u}_0+A^{n-1}B{u}_1+A^{n-2}B{u}_2+\dots +B{u}_n\end{array}$$ For square matrices that satisfy their own characteristic equation the Cayley-Hamilton theorem states that for $N\ge n$ we may express $A^N$ as a linear combination of the lower matrix powers of $A$: $$A^N=a_{0}I+a_{1}A+\dots +a_{n-1}{A}^{n-1}.$$ Therefore, the state $x_{n+1}$ can be rewritten as $$x_{n+1}=A^{n+1}{x}_0+A^{n-1}B(u_1+a_{n-1}{u}_0)+A^{n-2}B(u_2+a_{n-2}{u}_0)+\dots +B(u_n+a_0{u}_0)$$ which can be manipulated into the form $$x_{n+1}=A^{n+1}{x}_0+R\left[\begin{array}{c}{u}_n+a_0{u}_0\\ u_{n-1}+a_1{u}_0\\ u_1+a_{n-1}{u}_0\end{array}\right].$$ where $$R=\left[\begin{array}{cccc}B& AB& \dots & A^{n-1}B\end{array}\right]$$ is the controllability matrix.

The same substitution can be applied also for subsequent timesteps up-to infinity, changing only the particular form of the vector of inputs' linear combinations. This has two consequences:

• All states reachable in $N>n$ steps are also reachable in $n$ steps (with unlimited inputs).
• If $R$ is rank deficient, some directions in the state-space cannot be effected by the inputs and therefore the system is uncontrollable.

Continuous time

The state of a linear time-invariant system $$\dot{x}(\tau )=Ax(\tau )+Bu(\tau ),x(\tau )\in {\mathbb{R}}^n,u(\tau )\in {\mathbb{R}}^m$$ at time $t$, starting from the initial state $x(0)$ and influenced by a continuous input $u(\tau )$, can be expressed as $$x(t)=e^{At}x(0)+{\int }_0^t{e}^{A(t-\tau )}Bu(\tau )d\tau .\text{\hspace{1em}(1)}$$ For square matrices that satisfy their own characteristic equation the Cayley-Hamilton theorem has a consequence that the inifinite series $$e^{At}=I+At+\frac{(At{)}^2}{2!}+\dots$$ can be expressed using a finite number of terms $$e^{At}={\varphi }_0(t)I+{\varphi }_1(t)A+\cdots +{\varphi }_{n-1}(t)A^{n-1}.$$ After substituting it back into the second term of (1) we attain $$x(t)=e^{At}x(0)+{\int }_0^t({\varphi }_0(t-\tau )I+{\varphi }_1(t-\tau )A+\cdots +{\varphi }_{n-1}(t-\tau )A^{n-1})Bu(\tau )d\tau$$ which can be further manipulated into the form $$x(t)=e^{At}x(0)+R{\int }_0^t\left[\begin{array}{c}{\varphi }_0(t-\tau )u(\tau )\\ {\varphi }_1(t-\tau )u(\tau )\\ ⋮\\ {\varphi }_{n-1}(t-\tau )u(\tau )\end{array}\right]d\tau ,$$ where $$R=\left[\begin{array}{cccc}B& AB& \dots & A^{n-1}B\end{array}\right]$$ is the controllability matrix. If $R$ is rank deficient, some directions in the state-space cannot be effected by the inputs and therefore the system is uncontrollable.