Controllability

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Controllability is an important property of a control system, and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control.

Controllability and observability are dual aspects of the same problem.

Roughly, the concept of controllability denotes the ability to move a system around in its entire configuration space using only certain admissible manipulations. The exact definition varies slightly within the framework or the type of models applied.

The following are examples of variations of controllability notions which have been introduced in the systems and control literature:

  • State controllability.
  • Output controllability
  • Controllability in the behavioural framework

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[edit] State controllability

The state of a system, which is a collection of system's variables values, completely describes the system at any given time. In particular, no information on the past of a system will help in predicting the future, if the states at the present time are known.

Thus state controllability is usually taken to mean that it is possible - by admissible inputs - to steer the states from any initial value to any final value within some time window.

Note that controllability does not mean that once you reach a state that you will be able to keep it there, but merely that you can reach that state.

[edit] Continuous Linear Time-Invariant (LTI) Systems

Consider the continuous linear time-invariant system

\dot{\mathbf{x}}(t) = \mathbf{Ax}(t) + \mathbf{Bu}(t)
\mathbf{y}(t) = \mathbf{Cx}(t) + \mathbf{Du}(t)

where

\mathbf{x} is the "state vector",
\mathbf{y} is the "output vector",
\mathbf{u} is the "input (or control) vector",
\mathbf{A} is the "state matrix",
\mathbf{B} is the "input matrix",
\mathbf{C} is the "output matrix",
\mathbf{D} is the "feedthrough (or feedforward) matrix".

The controllability matrix is given by

R = \begin{bmatrix}B & AB & A^{2}B & ...& A^{n-1}B\end{bmatrix}

The system is controllable if the controllability matrix has a full rank.

[edit] Discrete Linear Time-Invariant (LTI) Systems

For a discrete-time linear state-space system (i.e. time variable k\in\mathbb{Z}) the state equation is

\textbf{x}(k+1) = A\textbf{x}(k) + B\textbf{u}(k)

Where A is an n \times n matrix. The test for controllability is that the matrix

C = \begin{bmatrix}B & AB & A^{2}B & ...& A^{n-1}B\end{bmatrix}

has full rank (i.e., rank(C) = n). The rationale for this test is that if n columns of C are linearly independent: in this case each of the n states is reachable giving the system proper inputs through the variable u(k).

[edit] Example

For example, consider the case when n = 2. If \begin{bmatrix}B & AB\end{bmatrix} has rank 2 (full rank). In this case B and AB are linearly independent and span the entire plane. If the rank is 1 then B and AB are collinear and cannot possibly span the plane.

Assume that the initial state is zero.

At time k = 0: x(1) = A\textbf{x}(0) + B\textbf{u}(0) = B\textbf{u}(0)

At time k = 1: x(2) = A\textbf{x}(1) + B\textbf{u}(1) = AB\textbf{u}(0) + B\textbf{u}(1)

At time k = 0 all of the reachable states are on the line formed by the vector B. At time k = 1 all of the reachable states are linear combinations of AB and B. If the system is controllable then these two vectors can span the entire plane and can be done so for time k = 2. The assumption made that the initial state is zero is merely for convenience. Clearly if all states can be reached from the origin then any state can be reached from another state (merely a shift in coordinates).

This example holds for all positive n, but the case of n = 2 is easier to visualize.

[edit] Analogy for example of n = 2

Consider an analogy to the previous example system. You are sitting in your car on an infinite, flat plane and facing north. The goal is to reach any point in the plane by driving a distance in a straight line, come to a full stop, turn, and driving another distance, again, in a straight line. If your car has no steering then you can only drive straight, which means you can only drive on a line (in this case the north-south line since you started facing north). The lack of steering case would be analogous to when the rank of C is 1 (the two distances you drove are on the same line).

Now, if your car did have steering then you could easily drive to any point in the plane and this would be the analogous case to when the rank of C is 2.

If you change this example to n = 3 then the analogy would be flying in space to reach any position in 3D space (ignoring the orientation of the aircraft). You are allowed to:

  • fly in a straight line
  • turn left or right by any amount (Yaw)
  • direct the plane upwards or downwards by any amount (Pitch)

Although the 3-dimensional case is harder to visualize, the concept of controllability is still analogous.

[edit] Nonlinear Systems

Nonlinear systems in the control-affine form

\dot{\mathbf{x}} = \mathbf{f(x)} + \sum_{i=1}^m \mathbf{g}_i(\mathbf{x})u_i

is locally accessible about x0 if the accessibility distribution R spans n space, when n equals the rank of x and R is given by:

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}

Here, [\mathrm{ad}^k_{\mathbf{f}}\mathbf{\mathbf{g}}] is the repeated Lie bracket operation defined by

[\mathrm{ad}^k_{\mathbf{f}}\mathbf{\mathbf{g}}] = \begin{bmatrix} \mathbf{f} & \cdots & j & \cdots & \mathbf{[\mathbf{f}, \mathbf{g}]} \end{bmatrix}

The controllability matrix for linear systems in the previous section can in fact be derived from this equation.

[edit] Output controllability

Output controllability means the ability to manipulate the outputs of a system by admissible inputs. For a system with several outputs, it might not be possible to manipulate these outputs independently by the admissible inputs, in which case the system is not output controllable.

[edit] Controllability in the behavioural framework

In the so-called behavioural system theoretic approach, due to Willems (see people in systems and control) the 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 behaviour (state trajectory) can be concatenated with any future part of a behaviour with which it shares the current state in such a way that the concatenation is contained in the behaviour, i.e. is part of the admissible system behaviour.

[edit] Stabilizability

A slightly weaker notion than controllability is that of Stabilizability. A system is determined to be stabilizable when all uncontrollable states have stable dynamics. Thus, even though some of the state cannot be controlled (as determined by the controllability test above) all the states will still remain bounded during the system's behaviour.

[edit] External links

The system described above or (A,B) is said to be stabilizable if there exists a feedback matrix F such that A+BF has all its eigenvalues in the open left-half plane C^{-}.