Controllability is an important property of a control system and plays a crucial role in many regulation problems, such as the stabilization of unstable systems using feedback, tracking problems, obtaining optimal control strategies, or, simply prescribing an input that has a desired effect on the state.
Controllability and observability are dual notions. Controllability pertains to regulating the state by a choice of a suitable input, while observability pertains to being able to know the state by observing the output (assuming that the input is also being observed).
Broadly speaking, the concept of controllability relates to the ability to steer a system around in its configuration space using only certain admissible manipulations. The exact definition varies depending on the framework or the type of models dealt with.
The following are examples of variants of notions of controllability that have been introduced in the systems and control literature:
- State controllability: the ability to steer the system between states
- Strong controllability: the ability to steer between states over any specified time window
- Collective controllability: the ability to simultaneously steer a collection of dynamical systems
- Trajectory controllability: the ability to steer along a predefined trajectory rather than just to a desired final state
- Output controllability: the ability to steer to specified values of the output
- Controllability in the behavioural framework: a compatibility condition between past and future input and output trajectories
State controllability
The state of a deterministic system, which is the set of values of all the system's state variables (those variables characterized by dynamic equations), completely describes the system at any given time. In particular, no information on the past of a system is needed to help in predicting the future, if the states at the present time are known and all current and future values of the control variables (those whose values can be chosen) are known.
' (or simply ' if no other context is given) describes the ability of an external input (the vector of control variables) to move the internal state of a system from any initial state to any final state in a finite time interval.
That is, we can informally define controllability as follows:
If for any initial state and any final state there exists an input sequence to transfer the system state from to in a finite time interval, then the system modeled by the state-space representation is controllable. For the simplest example of a continuous, linear time-invariant (LTI) system, the row dimension of the state space expression determines the interval; each row contributes a vector in the state space of the system. If there are not enough such vectors to span the state space of , then the system cannot achieve controllability. It may be necessary to modify and to better approximate the underlying differential relationships it estimates to achieve controllability.
Controllability does not mean that a reached state can be maintained, merely that any state can be reached.
Controllability does not mean that arbitrary paths can be made through state space, only that there exists a path within a finite time interval. When the time interval can also be specified, the dynamical system is often referred to as being strongly controllable.
Continuous linear systems
Consider the continuous linear system
<math display=block>\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}</math>
There exists a control from state at time to state at time if and only if is in the column space of
<math display=block>W(t_0,t_1) = \int_{t_0}^{t_1} \phi(t_0,t)B(t)B(t)^\mathsf{T}\phi(t_0,t)^\mathsf{T} \mathrm{d}t</math>
where is the state-transition matrix, and is the Controllability Gramian.
In fact, if is a solution to then a control given by would make the desired transfer.
Note that the matrix defined as above has the following properties:
{\mathrm{d}t}W(t,t_1) = A(t)W(t,t_1)+W(t,t_1)A(t)^\mathsf{T}-B(t)B(t)^\mathsf{T}, \; W(t_1,t_1) = 0</math>
| satisfies the equation
<math display=block>W(t_0,t_1) = W(t_0,t) + \phi(t_0,t)W(t,t_1)\phi(t_0,t)^\mathsf{T}</math>
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 condition analogous to the Kalman rank condition for time-invariant systems.
Consider a continuous-time linear system smoothly varying in an interval :
<math display=block>\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}</math>
The state-transition matrix is also smooth. Introduce the matrix-valued function and define
<math display=block>M_k(t) = \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 matrix (, for ):
<math display=block>M^{(k)}(t) := \left[M_0(t), \ldots, M_k(t)\right] .</math>
If there exists a and a nonnegative integer such that , then is controllable.
If is also analytically varying in an interval , then is controllable on every nontrivial subinterval of if and only if there exists a and a nonnegative integer such that .
:<math>R = \begin{bmatrix} \mathbf{g}_1 & \cdots & \mathbf{g}_m & [\mathrm{ad}^k_{\mathbf{g}_i}\mathbf{\mathbf{g}_j}] & \cdots & [\mathrm{ad}^k_{\mathbf{f\mathbf{\mathbf{g}_i}] \end{bmatrix}.</math>
Here, <math>[\mathrm{ad}^k_{\mathbf{f\mathbf{\mathbf{g]</math> is the repeated Lie bracket operation defined by
: <math>[\mathrm{ad}^k_{\mathbf{f\mathbf{\mathbf{g] = \begin{bmatrix} \mathbf{f} & \cdots & j & \cdots & \mathbf{[\mathbf{f}, \mathbf{g}]} \end{bmatrix}. </math>
The controllability matrix for linear systems in the previous section can in fact be derived from this equation.
Controllability via state feedback
When control authority on a linear dynamical system is exerted through a choice of a time-varying feedback gain matrix <math>K(t)</math>, the system
: <math>\dot{\mathbf{x = (A - BK(t))\mathbf{x}</math>
is nonlinear, in that products of control parameters and states are present. The accessibility distribution <math>R</math> is, as before,
: <math>R= \begin{bmatrix} B & AB & \cdots & A^{n-1}B \end{bmatrix}. </math>
It is clear that for the system to be controllable, it is necessary that <math>R</math> has full column rank. It turns out that this condition is also sufficient. However, the (optimal) control strategy explained earlier needs to be modified so that the trajectory when applying an optimal input to steer the system between the specified states, does not pass through the origin, else the regulating input cannot be written in feedback form <math>u=-K(t)\mathbf{x}</math>. The controllability, as well as the strong controllability of this bilinear system
was proven in
.
Collective controllability: feedback control of state transition
Collective controllability represents the ability to steer <math>n</math> linear dynamical systems that obey identical dynamics
: <math>\dot{\mathbf{x^{(i)}(t) = A \mathbf{x}^{(i)}(t) + B \mathbf{u}^{(i)}(t)</math>
where <math>n</math> equals the dimension of <math>\mathbf{x}</math>,
between specified starting and ending configurations by way of a common state feedback gain matrix <math>K(t)</math>, and thereby, each instantiating a control input
: <math>\mathbf{u}^{(i)}(t)=K(t){\mathbf{x^{(i)}(t) </math>
for <math>i\in\{1,\ldots,n\}</math>, respectively.
The accessibility distribution <math>R</math> having full column rank is trivially a necessary condition. It is also sufficient, and in fact, the collective is strongly controllable, in that it can be steered from an initial
configuration
: <math>\Phi(0)= \begin{bmatrix} \mathbf{x}^{(1)}(0)\ldots \mathbf{x}^{(n)}(0) \end{bmatrix} </math>
to any specified terminal configuration
: <math>\Phi(T)= \begin{bmatrix} \mathbf{x}^{(1)}(T)\ldots \mathbf{x}^{(n)}(T) \end{bmatrix}, </math>
provided <math>\det(\Phi(0)\Phi(T))>0 </math>, over any specified time interval <math>[0,T] </math> through a choice of a common time-varying feedback gain matrix <math>K(t)</math> provided <math>R</math> has full column rank.
Controllability under input constraints
In systems with limited control authority, it is often no longer possible to move any initial state to any final state inside the controllable subspace. This phenomenon is caused by constraints on the input that could be inherent to the system (e.g. due to saturating actuator) or imposed on the system for other reasons (e.g. due to safety-related concerns). The controllability of systems with input and state constraints is studied in the context of reachability and viability theory.
Controllability in the behavioral framework
In the so-called behavioral system theoretic approach due to Willems (see people in systems and control), models considered do not directly define an input–output structure. In this framework systems are described by admissible trajectories of a collection of variables, some of which might be interpreted as inputs or outputs.
A system is then defined to be controllable in this setting, if any past part of a behavior (trajectory of the external variables) can be concatenated with any future trajectory of the behavior in such a way that the concatenation is contained in the behavior, i.e. is part of the admissible system behavior.
Stabilizability
A slightly weaker notion than controllability is that of stabilizability. A system is said to be stabilizable when all uncontrollable state variables can be made to have stable dynamics. Thus, even though some of the state variables cannot be controlled (as determined by the controllability test above) all the state variables will still remain bounded during the system's behavior.
Reachable set
Let and (where is the set of all possible states and is an interval of time). The reachable set from in time is defined as: