Controllable system

Definition

A system is called (completely) controllable if for any initial time $t_0$, any initial state $\mathbf{x}(t_0)$ can be transferred to any final state $\mathbf{x}^*$ (i.e. $\mathbf{x}(t) = \mathbf{x}^*(t)$) using some input $\underset{\sim}{\mathbf{u}}[t_0, t]$ over a finite time interval (i.e. $t$ is finite).

Refinements

Total controllability

If the system is completely controllable over every (or almost every) finite interval.

Strong controllability

If the system is controllable from each input terminal.

Output controllability

If the system output (rather than state) can be set arbitrarily at some finite time $t$, by using an appropriate input.

Controllability conditions for analog systems

Consider analog linear system $$\begin{align} \\ \dot{\mathbf{x}}(t) = A(t)\mathbf{x}(t)+B(t)\mathbf{u}(t) \\ \\ \mathbf{y}(t) = C(t)\mathbf{x}(t)+D(t)\mathbf{u}(t) \end{align}$$

with analog bounded $A(t)$.

Define $$P(t_0,t) = \int_{t_0}^t \phi(t,\tau) B(\tau)B^T(\tau)\phi^T(t,\tau)d\tau$$ where $\phi(t,\tau)$ denotes a transition matrix from time $t$ to time $\tau$.

Alternatively using $\phi(t,\tau) = \phi(t,t_0)\phi(t_0,\tau)$, $$P(t_0,t) = \phi(t_0,\tau)\int_{t_0}^t \phi(t_0,\tau) B(\tau)B^T(\tau)\phi^T(t_0,\tau)d\tau \phi(t,t_0)$$

Controllability from origin

The system is controllable if any state can be reached from the origin in finite time.

Condition 1

This system is controllable if and only if given any $t_0$, $P(t_0,t)$ is nonsingular for some finite $t > t_0$.

When $A(t)$ is bounded for all finite $t$, we know that $\phi(t,t_0)$ is nonsingular for all finite $t$, so $P(t_0,t)$ is nonsingular if and only if the matrix $$\hat{S}(t_0,t) \triangleq \int_{t_0}^t \phi(t_0,\tau) B(\tau)B^T(\tau)\phi^T(t_0,\tau)d\tau$$ is nonsingular. Equivalently, if for every constant vector $\mathbf{\mu} \neq \mathbf{0}$, $$\mathbf{\mu}^T \hat{S}(t_0,t)\mathbf{\mu} = \int_{t_0}^t \mathbf{\mu}^T \phi(t_0,\tau) B(\tau)B^T(\tau)\phi^T(t_0,\tau)\mathbf{\mu} d\tau > 0$$

Condition 2

This continuous-time system is controllable if and only if given any $t_0$, and for every $\mathbf{\mu} \neq \mathbf{0}$, the vector $\mathbf{z}(\tau) \triangleq B^T(\tau) \phi^T(t_0,\tau)\mathbf{\mu}$ is not identically zero for $\tau \geq t_0$.

Controllability test for fixed continuous-time system

Controllable if and only if for every $\mathbf{\mu} \neq \mathbf{0}$, $$B^T (A^k) \mathbf{\mu} \neq \mathbf{0}$$ for at least one $k = 0,1,2,...,n-1$.

Controllability test for fixed analog-time system

(In the case of the standard form state equations, this occurs when $A,B,C,D$ are not dependent upon $t$)

When the controllability grammian

Controllability conditions for discrete systems

Consider discrete linear system $$\begin{eqnarray} \mathbf{x}(k+1) = A(k) \mathbf{x}(k) + B(k) \mathbf{u}(k) \\ \mathbf{y}(k) = C(k) \mathbf{x}(k) + D(k) \mathbf{u}(k) \end{eqnarray}$$

Condition 1'

The system is controllable if and only if given any $k_0$, there exists a finite $k > k_0$ such that $$P'(k_0,k) \triangleq \sum_{i=k_0}^{k-1} \phi(k,i+1)B(i)B^T(i)\phi^T(k,i+1)$$ is nonsingular. Note $\phi(k,i+1)$ is the transition matrix between $k$ and $i+1$.

Condition 2'

The system is controllable if and only if given any $k_0$ there exists a finite $k>k_0$ such that for every $\mathbf{\lambda} \neq \mathbf{0}$, the vector $\mathbf{v}(i) = B^T(i)\phi^T(k,i+1)\mathbf{\lambda}$ is nonzero for some value $i = k_0,...,k-1$, i.e. $\underset{\sim}v \neq 0$ on $[k_0,k-1]$.

Condition 2''

The system is controllable if and only if given any $k_0$ and for every $\mathbf{\mu} \neq \mathbf{0}$, the vector $\mathbf{z}(i) = B^T(i) \phi^T(k_0,i+1)\mathbf{\mu}$ is nonzero for some value of $i \geq k_0$.

Here $\phi^T(k_0,i+1) \triangleq [A(i) A(i-1)...A(k_0)]^{-1}$ for $i \geq k_0$.

Controllability test for fixed discrete-time system

For fixed discrete-time systems, the state equation becomes

For

$\Gamma$

Condition 3

#incomplete

References

1. P. E. Sarachik, Principles of Linear Systems, Cambridge Press, 1996, pp. 151-158.