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 \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} There exists a control from state at time to state at time if and only if is in the column space of 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 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: {{blist \frac{\mathrm{d}}{\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 W(t_0,t_1) = W(t_0,t) + \phi(t_0,t)W(t,t_1)\phi(t_0,t)^\mathsf{T} }} == 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 : \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} The state-transition matrix is also smooth. Introduce the matrix-valued function and define M_k(t) = \frac{\mathrm{d^k} M_0}{\mathrm{d} t^k}(t), k\geqslant 1. Consider the matrix of matrix-valued functions obtained by listing all the columns of the matrix (, for ): M^{(k)}(t) := \left[M_0(t), \ldots, M_k(t)\right] . 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 . The above methods can still be complex to check, since it involves the computation of the state-transition matrix . Another equivalent condition is defined as follow. Let , and for each , define B_{i+1}(t) = A(t)B_i(t) - \frac{\mathrm{d}}{\mathrm{d} t}B_i(t). In this case, each is obtained directly from the data . The system is controllable if there exists a and a nonnegative integer such that . Example Consider a system varying analytically in and matrices \begin{align} A(t) &= \begin{bmatrix} t & 1 & 0\\ 0 & t^{3} & 0\\ 0 & 0 & t^{2} \end{bmatrix}, \\ B(t) &= \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}. \end{align} Then [B_0(0),B_1(0),B_2(0),B_3(0)] = \begin{bmatrix} 0 & 1 & 0 &-1\\ 1 & 0 & 0&0\\ 1 & 0 & 0&2 \end{bmatrix}, and since this matrix has rank 3, the system is controllable on every nontrivial interval of {{nowrap|\mathbb{R}.}} Continuous linear time-invariant (LTI) systems Consider the continuous linear time-invariant system \begin{align} \dot{\mathbf{x}}(t) &= A \mathbf{x}(t) + B \mathbf{u}(t) \\ \mathbf{y}(t) &= C \mathbf{x}(t) + D \mathbf{u}(t) \end{align} where • is the state vector; • is the output vector; • is the input (or control) vector; • is the state matrix; • is the input matrix; • is the output matrix; and • 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 full row rank (i.e. . == Discrete linear time-invariant (LTI) systems ==

For a discrete-time linear state-space system (i.e. time variable {{nowrap|k\in\mathbb{Z})}} the state equation is \textbf{x}(k+1) = A\textbf{x}(k) + B\textbf{u}(k) where is an matrix and is a matrix (i.e. is inputs collected in a vector). The test for controllability is that the matrix \mathcal{C} = \begin{bmatrix}B & AB & A^{2}B & \cdots & A^{n-1}B\end{bmatrix} has full row rank (i.e., ). That is, if the system is controllable, will have columns that are linearly independent; if columns of are linearly independent, each of the states is reachable by giving the system proper inputs through the variable . Derivation Given the state at an initial time, arbitrarily denoted as , the state equation gives , then , and so on with repeated back-substitutions of the state variable, eventually yielding \textbf{x}(n)=B\textbf{u}(n-1) + AB\textbf{u}(n-2) + \cdots + A^{n-1}B\textbf{u}(0) + A^n\textbf{x}(0) or equivalently \textbf{x}(n)-A^n\textbf{x}(0)= [B \, \, AB \, \, \cdots \, \, A^{n-1}B] [\textbf{u}^\mathsf{T}(n-1) \, \, \textbf{u}^\mathsf{T}(n-2) \, \, \cdots \, \, \textbf{u}^\mathsf{T}(0)]^\mathsf{T}. Imposing any desired value of the state vector on the left side, this can always be solved for the stacked vector of control vectors if and only if the matrix of matrices at the beginning of the right side has full row rank. Example For example, consider the case when and (i.e. only one control input). Thus, and are vectors. If has rank 2 (full rank), and so and are linearly independent and span the entire plane. If the rank is 1, then and are collinear and do not span the plane. Assume that the initial state is zero. At time : x(1) = A\textbf{x}(0) + B\textbf{u}(0) = B\textbf{u}(0) At time : x(2) = A\textbf{x}(1) + B\textbf{u}(1) = AB\textbf{u}(0) + B\textbf{u}(1) At time all of the reachable states are on the line formed by the vector . At time all of the reachable states are linear combinations of and . If the system is controllable then these two vectors can span the entire plane and can be done so for time . 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 , but the case of is easier to visualize. === Analogy for example of = 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 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 is 2. If you change this example to 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. == 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 are locally accessible about x_0 if the accessibility distribution R spans n space, when n equals the dimension 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. == Controllability via state feedback ==

When control authority on a linear dynamical system is exerted through a choice of a time-varying feedback gain matrix K(t), the system : \dot{\mathbf{x}} = (A - BK(t))\mathbf{x} is nonlinear, in that products of control parameters and states are present. The accessibility distribution R is, as before, : R= \begin{bmatrix} B & AB & \cdots & A^{n-1}B \end{bmatrix}. It is clear that for the system to be controllable, it is necessary that R 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 u=-K(t)\mathbf{x}. 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 n linear dynamical systems that obey identical dynamics : \dot{\mathbf{x}}^{(i)}(t) = A \mathbf{x}^{(i)}(t) + B \mathbf{u}^{(i)}(t) where n equals the dimension of \mathbf{x}, between specified starting and ending configurations by way of a common state feedback gain matrix K(t), and thereby, each instantiating a control input : \mathbf{u}^{(i)}(t)=K(t){\mathbf{x}}^{(i)}(t) for i\in\{1,\ldots,n\}, respectively. The accessibility distribution R 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 : \Phi(0)= \begin{bmatrix} \mathbf{x}^{(1)}(0)\ldots \mathbf{x}^{(n)}(0) \end{bmatrix} to any specified terminal configuration : \Phi(T)= \begin{bmatrix} \mathbf{x}^{(1)}(T)\ldots \mathbf{x}^{(n)}(T) \end{bmatrix}, provided \det(\Phi(0)\Phi(T))>0 , over any specified time interval [0,T] through a choice of a common time-varying feedback gain matrix K(t) provided R has full column rank. == Null controllability ==

If a discrete control system is null-controllable, it means that there exists a controllable u(k) so that x(k_0) = 0 for some initial state x(0) = x_0. In other words, it is equivalent to the condition that there exists a matrix F such that A+BF is nilpotent. This can be easily shown by controllable-uncontrollable decomposition. == Output controllability ==

'''' is the related notion for the output of the system (denoted in the previous equations); the output controllability describes the ability of an external input to move the output from any initial condition to any final condition in a finite time interval. It is not necessary that there is any relationship between state controllability and output controllability. In particular: • A controllable system is not necessarily output controllable. For example, if matrix and matrix does not have full row rank, then some positions of the output are masked by the limiting structure of the output matrix, and therefore unachievable. Moreover, even though the system can be moved to any state in finite time, there may be some outputs that are inaccessible by all states. A trivial numerical example uses and a matrix with at least one row of zeros; thus, the system is not able to produce a non-zero output along that dimension. • An output controllable system is not necessarily state controllable. For example, if the dimension of the state space is greater than the dimension of the output, then there will be a set of possible state configurations for each individual output. That is, the system can have significant zero dynamics, which are trajectories of the system that are not observable from the output. Consequently, being able to drive an output to a particular position in finite time says nothing about the state configuration of the system. For a linear continuous-time system, like the example above, described by matrices , , , and , the '''' \begin{bmatrix} CB & CAB & CA^2B & \cdots & CA^{n-1}B & D\end{bmatrix} has full row rank (i.e. rank ) if and only if the system is output controllable. == 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{{cite journal |author1=Claire J. Tomlin |author2=Ian Mitchell |author3=Alexandre M. Bayen |author4=Meeko Oishi|author4-link=Meeko Oishi | title = Computational Techniques for the Verification of Hybrid Systems | journal = Proceedings of the IEEE |volume=91 |issue=7 |pages=986–1001 | doi = 10.1109/jproc.2003.814621 | year = 2003 |bibcode=2003IEEEP..91..986T | url = http://www.cs.ubc.ca/~mitchell/Papers/publishedIEEEproc03.pdf | access-date = 2012-03-04 == 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: R^T{(x)} = \left\{ z \in X : x \overset{T}{\rightarrow} z \right\}, where denotes that there exists a state transition from to in time . For autonomous systems the reachable set is given by: \mathrm{Im}(R)=\mathrm{Im}(B)+\mathrm{Im}(AB)+....+\mathrm{Im}(A^{n-1}B), where is the controllability matrix. {{math theorem |name= Claim |math_statement= In terms of the reachable set, the system is controllable if and only if {{nowrap|\mathrm{Im}(R)=\mathbb{R}^n.}}}} {{math proof |title= Proof |proof= We have the following equalities: \begin{align} R &= [B\ AB ....A^{n-1}B] \\ \mathrm{Im}(R) &= \mathrm{Im}([B\ AB ....A^{n-1}B]) \\ \mathrm{dim(Im}(R)) &= \mathrm{rank}(R) \end{align} Considering that the system is controllable, the columns of should be linearly independent. So: \begin{align} \mathrm{dim(Im}(R)) &= n \\ \mathrm{rank}(R) &= n \\ \mathrm{Im}(R) &= \R^{n}\qquad \blacksquare \end{align} }} A related set to the reachable set is the controllable set, defined by: C^T{(x)} = \left\{ z \in X : z \overset{T}{\rightarrow} x \right\}. The relation between reachability and controllability is presented by Sontag: {{olist |type= lower-alpha R(0)=R^k{(0)=X} (where is the set of all possible values or states of and is the time step). \begin{align} R(0) &= R^e{(0)=X} \\ C(0) &= C^e{(0)=X} \end{align} for all . }} Example Let the system be an n dimensional discrete-time-invariant system from the formula: \phi(n,0,0,w)=\sum\limits_{i=1}^n A^{i-1}Bw(n-1), where (final time, initial time, state variable, restrictions) is defined as the transition matrix of a state variable from an initial time to a final time with some restrictions ). It follows that the future state is in if and only if it is in {{nowrap|\mathrm{Im}(R),}} the image of the linear map , defined as: R(A,B)\triangleq [B\ AB ....A^{n-1}B], which maps u^{n}\mapsto X. When and we identify with a matrix whose columns are in that order. If the system is controllable the rank of is . If this is true, the image of the linear map is all of . Based on that, we have: R(0)=R^k{(0)=X} with X\in\R^{n}. == See also ==