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Controllability is an important property of a
control system A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large industrial c ...
, 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 Observability is a measure of how well internal states of a system can be inferred from knowledge of its external outputs. In control theory, the observability and controllability of a linear system are mathematical duals. The concept of observ ...
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


State controllability

The
state State may refer to: Arts, entertainment, and media Literature * ''State Magazine'', a monthly magazine published by the U.S. Department of State * ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, United States * ''Our S ...
of a
deterministic system In mathematics, computer science and physics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will thus always produce the same output from a given star ...
, 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. ''Complete state controllability'' (or simply ''controllability'' 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 some initial state \mathbf and some final state \mathbf there exists an input sequence to transfer the system state from \mathbf to \mathbf in a finite time interval, then the system modeled by the state-space representation is controllable. For the simplest example of a continuous, LTI system, the row dimension of the state space expression \dot = \mathbf\mathbf(t) + \mathbf\mathbf(t) 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 \mathbf, then the system cannot achieve controllability. It may be necessary to modify \mathbf and \mathbf 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 the prescribed finite time interval.


Continuous linear systems

Consider the
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
linear Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
system A linear time-invariant system behaves the same but with the coefficients being constant in time. : \dot(t) = A(t) \mathbf(t) + B(t) \mathbf(t) : \mathbf(t) = C(t) \mathbf(t) + D(t) \mathbf(t). There exists a control u from state x_0 at time t_0 to state x_1 at time t_1 > t_0 if and only if x_1 - \phi(t_0,t_1)x_0 is in the
column space In linear algebra, the column space (also called the range or image) of a matrix ''A'' is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding mat ...
of : W(t_0,t_1) = \int_^ \phi(t_0,t)B(t)B(t)^\phi(t_0,t)^ dt where \phi is the
state-transition matrix In control theory, the state-transition matrix is a matrix whose product with the state vector x at an initial time t_0 gives x at a later time t. The state-transition matrix can be used to obtain the general solution of linear dynamical systems ...
, and W(t_0,t_1) is the
Controllability Gramian In control theory, we may need to find out whether or not a system such as \begin \dot(t)\boldsymbol(t)+\boldsymbol(t)\\ \boldsymbol(t)=\boldsymbol(t)+\boldsymbol(t) \end is controllable, where \boldsymbol, \boldsymbol, \boldsymbol and \boldsymb ...
. In fact, if \eta_0 is a solution to W(t_0,t_1)\eta = x_1 - \phi(t_0,t_1)x_0 then a control given by u(t) = -B(t)^\phi(t_0,t)^\eta_0 would make the desired transfer. Note that the matrix W defined as above has the following properties: * W(t_0,t_1) is
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
* W(t_0,t_1) is positive semidefinite for t_1 \geq t_0 * W(t_0,t_1) satisfies the linear
matrix differential equation A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. A matrix differential equation contains more than one funct ...
:: \fracW(t,t_1) = A(t)W(t,t_1)+W(t,t_1)A(t)^-B(t)B(t)^, \; W(t_1,t_1) = 0 * W(t_0,t_1) satisfies the equation :: W(t_0,t_1) = W(t_0,t) + \phi(t_0,t)W(t,t_1)\phi(t_0,t)^


Rank condition for controllability

The
Controllability Gramian In control theory, we may need to find out whether or not a system such as \begin \dot(t)\boldsymbol(t)+\boldsymbol(t)\\ \boldsymbol(t)=\boldsymbol(t)+\boldsymbol(t) \end is controllable, where \boldsymbol, \boldsymbol, \boldsymbol and \boldsymb ...
involves integration of the
state-transition matrix In control theory, the state-transition matrix is a matrix whose product with the state vector x at an initial time t_0 gives x at a later time t. The state-transition matrix can be used to obtain the general solution of linear dynamical systems ...
of a system. A simpler condition for controllability is a
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
condition analogous to the Kalman rank condition for time-invariant systems. Consider a continuous-time linear system \Sigma smoothly varying in an interval _0,t/math> of \mathbb: : \dot(t) = A(t) \mathbf(t) + B(t) \mathbf(t) : \mathbf(t) = C(t) \mathbf(t) + D(t) \mathbf(t). The state-transition matrix \phi is also smooth. Introduce the n x m matrix-valued function M_0(t) = \phi(t_0,t)B(t) and define : M_k(t) = \frac(t), k\geqslant 1. Consider the matrix of matrix-valued functions obtained by listing all the columns of the M_i, i = 0,1, \ldots, k: M^(t) := \left _0(t), \ldots, M_k(t)\right. If there exists a \bar \in _0,t/math> and a nonnegative integer k such that \operatornameM^(\bar)=n, then \Sigma is controllable.Eduardo D. Sontag
Mathematical Control Theory: Deterministic Finite Dimensional Systems
If \Sigma is also analytically varying in an interval _0,t/math>, then \Sigma is controllable on every nontrivial subinterval of _0,t/math> if and only if there exists a \bar \in _0,t/math> and a nonnegative integer k such that \operatornameM^(t_i)=n. The above methods can still be complex to check, since it involves the computation of the state-transition matrix \phi. Another equivalent condition is defined as follow. Let B_0(t) = B(t), and for each i \geq 0, define : B_(t) = A(t)B_i(t) - \fracB_i(t). In this case, each B_i is obtained directly from the data (A(t),B(t)). The system is controllable if there exists a \bar \in _0,t/math> and a nonnegative integer k such that \textrm( \left B_0(\bar), B_1(\bar), \ldots, B_k(\bar) \right = n .


Example

Consider a system varying analytically in (-\infty,\infty) and matrices A(t) = \begin t & 1 & 0\\ 0 & t^ & 0\\ 0 & 0 & t^ \end, B(t) = \begin 0 \\ 1 \\ 1 \end. Then
_0(0),B_1(0),B_2(0),B_3(0) 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
= \begin 0 & 1 & 0 &-1\\ 1 & 0 & 0&0\\ 1 & 0 & 0&2 \end and since this matrix has rank 3, the system is controllable on every nontrivial interval of \mathbb.


Continuous linear time-invariant (LTI) systems

Consider the continuous linear
time-invariant system In control theory, a time-invariant (TIV) system has a time-dependent system function that is not a direct function of time. Such systems are regarded as a class of systems in the field of system analysis. The time-dependent system function is ...
: \dot(t) = A \mathbf(t) + B \mathbf(t) : \mathbf(t) = C \mathbf(t) + D \mathbf(t) where : \mathbf is the n \times 1 "state vector", : \mathbf is the m \times 1 "output vector", : \mathbf is the r \times 1 "input (or control) vector", : A is the n \times n "state matrix", : B is the n \times r "input matrix", : C is the m \times n "output matrix", : D is the m \times r "feedthrough (or feedforward) matrix". The n \times nr controllability matrix is given by :R = \beginB & AB & A^B & ...& A^B\end The system is controllable if the controllability matrix has full row
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
(i.e. \operatorname(R)=n).


Discrete linear time-invariant (LTI) systems

For a
discrete-time In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "po ...
linear state-space system (i.e. time variable k\in\mathbb) the state equation is :\textbf(k+1) = A\textbf(k) + B\textbf(k) where A is an n \times n matrix and B is a n \times r matrix (i.e. \mathbf is r inputs collected in a r \times 1 vector). The test for controllability is that the n \times nr matrix :\mathcal = \beginB & AB & A^B & \cdots & A^B\end has full row
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
(i.e., \operatorname(\mathcal C) = n). That is, if the system is controllable, \mathcal C will have n columns that are
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
; if n columns of \mathcal C are
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
, each of the n states is reachable by giving the system proper inputs through the variable u(k).


Derivation

Given the state \textbf(0) at an initial time, arbitrarily denoted as ''k''=0, the state equation gives \textbf(1) = A\textbf(0) + B\textbf(0), then \textbf(2) = A\textbf(1) + B\textbf(1)= A^2\textbf(0)+AB\textbf(0)+B\textbf(1), and so on with repeated back-substitutions of the state variable, eventually yielding :\textbf(n)=B\textbf(n-1) + AB\textbf(n-2) + \cdots + A^B\textbf(0) + A^n\textbf(0) or equivalently :\textbf(n)-A^n\textbf(0)= \, \, AB \, \, \cdots \, \, A^B textbf^T(n-1) \, \, \textbf^T(n-2) \, \, \cdots \, \, \textbf^T(0)T. Imposing any desired value of the state vector \textbf(n) 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 n=2 and r=1 (i.e. only one control input). Thus, B and A B are 2 \times 1 vectors. If \beginB & AB\end has rank 2 (full rank), and so B and AB are
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
and span the entire plane. If the rank is 1, then B and AB are
collinear In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...
and do not span the plane. Assume that the initial state is zero. At time k=0: x(1) = A\textbf(0) + B\textbf(0) = B\textbf(0) At time k=1: x(2) = A\textbf(1) + B\textbf(1) = AB\textbf(0) + B\textbf(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.


Analogy for example of ''n'' = 2

Consider an
analogy Analogy (from Greek ''analogia'', "proportion", from ''ana-'' "upon, according to" lso "against", "anew"+ ''logos'' "ratio" lso "word, speech, reckoning" is a cognitive process of transferring information or meaning from a particular subject ( ...
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 Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building de ...
of the
aircraft An aircraft is a vehicle that is able to fly by gaining support from the air. It counters the force of gravity by using either static lift or by using the dynamic lift of an airfoil, or in a few cases the downward thrust from jet engines ...
). 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 + \sum_^m \mathbf_i(\mathbf)u_i are locally accessible about x_0 if the accessibility distribution R spans n space, when n equals the rank of x and R is given by: :R = \begin \mathbf_1 & \cdots & \mathbf_m & mathrm^k_\mathbf& \cdots & mathrm^k_\mathbf\end. Here, mathrm^k_\mathbf/math> is the repeated
Lie bracket In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
operation defined by : mathrm^k_\mathbf= \begin \mathbf & \cdots & j & \cdots & \mathbf \end. The controllability matrix for linear systems in the previous section can in fact be derived from this equation.


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

''Output controllability'' is the related notion for the output of the system (denoted ''y'' 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 ''D'' = 0 and matrix ''C'' 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 ''D''=0 and a ''C'' 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 In mathematics, zero dynamics is known as the concept of evaluating the effect of zero on systems. History The idea was introduced thirty years ago as the nonlinear approach to the concept of transmission of zeros. The original purpose of intr ...
, 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 A, B, C, and D, the m \times (n+1)r ''output controllability matrix'' :\begin CB & CAB & CA^2B & \cdots & CA^B & D\end has full row rank (i.e. rank m) 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 In graph theory, reachability refers to the ability to get from one vertex to another within a graph. A vertex s can reach a vertex t (and t is reachable from s) if there exists a sequence of adjacent vertices (i.e. a walk) which starts with s an ...
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 This is an alphabetical list of people who have made significant contributions in the fields of system analysis and control theory. Eminent researchers The eminent researchers (born after 1920) include the winners of at least one award of the IEE ...
), 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 T ∈ ''Т'' and x ∈ ''X'' (where X is the set of all possible states and ''Т'' is an interval of time). The reachable set from x in time T is defined as: R^T = \left\, where xz denotes that there exists a state transition from x to z in time T. For autonomous systems the reachable set is given by : :\mathrm(R)=\mathrm(B)+\mathrm(AB)+....+\mathrm(A^B), where R is the controllability matrix. In terms of the reachable set, the system is controllable if and only if \mathrm(R)=\mathbb^n. Proof We have the following equalities: :R= \ AB ....A^B/math> :\mathrm(R)=\mathrm( \ AB ....A^B :\mathrm(R))=\mathrm(R) Considering that the system is controllable, the columns of R should be
linearly independent In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
. So: :\mathrm(R))=n :\mathrm(R)=n :\mathrm(R)=\R^\quad \blacksquare A related set to the reachable set is the controllable set, defined by: :C^T = \left\. The relation between reachability and controllability is presented by Sontag: (a) An n-dimensional discrete linear system is controllable if and only if: :: R(0)=R^k (Where X is the set of all possible values or states of x and k is the time step). (b) A continuous-time linear system is controllable if and only if: :: R(0)=R^e for all e>0. if and only if C(0)=C^e for all e>0. Example Let the system be an n dimensional discrete-time-invariant system from the formula: ::Φ(n,0,0,w)=\sum\limits_^n A^Bw(n-1) (Where Φ(final time, initial time, state variable, restrictions) is defined is the transition matrix of a state variable x from a initial time 0 to a final time n with some restrictions w). It follows that the future state is in R^k ⇔ it is in the image of the linear map: ::Im(R)=R(A,B)≜ Im( \ AB ....A^B/math>), which maps, ::u^→X When u=K^ and X=K^ we identify R(A,B) with a n by nm matrix whose columns are the columns of B,\ AB, ....,A^B in that order. If the system is controllable the rank of \ AB ....A^B/math> is n. If this is truth, the image of the linear map R is all of X. Based on that, we have: ::R(0)=R^k with XЄ\R^.


See also

*
Observability Observability is a measure of how well internal states of a system can be inferred from knowledge of its external outputs. In control theory, the observability and controllability of a linear system are mathematical duals. The concept of observ ...
* State observer


Notes


References

{{reflist


External links

*
MATLAB function for checking controllability of a system
*

Classical control theory fr:Représentation d'état#Systèmes linéaires