Controllability

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

State controllabilityEdit

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.

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 other final state in a finite time interval.[1]:737

Controllability does not mean that a reached state can be maintained, merely that any state can be reached.

Continuous linear systemsEdit

Consider the continuous linear system [note 1]

 
 

There exists a control   from state   at time   to state   at time   if and only if   is in the column space of

 

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:

 
  •   satisfies the equation
 [2]

Rank condition for controllabilityEdit

The Controllability Gramian involves the integration of the state-transition matrix of the 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   of  :

 
 

The state-transition matrix   is also smooth. Introduce the n x m matrix-valued function   and define

  =  .

Consider the matrix of matrix-valued functions obtained by listing all the columns of the  ,  :

 .

If there exists a   and a nonnegative integer k such that  , then   is controllable.[3]

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 k such that   .[3]

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

 =  

In this case, each   is obtained directly from the data   The system is controllable if there exists a   and a nonnegative integer   such that  .[3]

ExampleEdit

Consider a system varying analytically in   and matrices

 ,   Then   and since this matrix has rank 3, the system is controllable on every nontrivial interval of  .

Continuous linear time-invariant (LTI) systemsEdit

Consider the continuous linear time-invariant system

 
 

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",
  is the   "feedthrough (or feedforward) matrix".

The   controllability matrix is given by

 

The system is controllable if the controllability matrix has full row rank (i.e.  ).

Discrete linear time-invariant (LTI) systemsEdit

For a discrete-time linear state-space system (i.e. time variable  ) the state equation is

 

where   is an   matrix and   is a   matrix (i.e.   is   inputs collected in a   vector). The test for controllability is that the   matrix

 

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  .

DerivationEdit

Given the state   at an initial time, arbitrarily denoted as k=0, the state equation gives   then   and so on with repeated back-substitutions of the state variable, eventually yielding

 

or equivalently

 

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.

ExampleEdit

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  :  

At time  :  

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 n = 2Edit

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 systemsEdit

Nonlinear systems in the control-affine form

 

are locally accessible about   if the accessibility distribution   spans   space, when   equals the rank of   and R is given by:[4]

 

Here,   is the repeated Lie bracket operation defined by

 

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

Null ControllabilityEdit

If a discrete control system is null-controllable, it means that there exists a controllable   so that   for some initial state  . In other words, it is equivalent to the condition that there exists a matrix   such that   is nilpotent.

This can be easily shown by controllable-uncontrollable decomposition.

Output controllabilityEdit

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. 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, 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   output controllability matrix

 

has full row rank (i.e. rank  ) if and only if the system is output controllable.[1]:742

Controllability under input constraintsEdit

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[5] and viability theory.[6]

Controllability in the behavioral frameworkEdit

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.[7]:151

StabilizabilityEdit

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.[8]

Reachable setEdit

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 [9]:

 , where xTz denotes that there exists a state transition from x to z in time T.

For autonomous systems the reachable set is given by :

 ,

where R is the controllability matrix.

In terms of the reachable set, the system is controllable if and only if  .

Proof We have the following equalities:

 
 
 

Considering that the system is controllable, the columns of R should be linearly independent. So:

 
 
 

A related set to the reachable set is the controllable set, defined by:

 .

The relation between reachability and controllability is presented by Sontag [9]:

(a) An n-dimensional discrete linear system is controllable if and only if:

  (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:

  for all e>0.

if and only if   for all e>0.

Example Let the system be an n dimensional discrete-time-invariant system from the formula:

Φ(n,0,0,w)=  (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   ⇔ it is in the image of the linear map:

Im(R)=R(A,B)≜ Im( ),

which maps,

 →X

When   and   we identify R(A,B) with a n by nm matrix whose columns are the columns of   in that order. If the system is controllable the rank of   is n. If this is truth, the image of the linear map R is all of X. Based on that, we have:

  with XЄ .

See alsoEdit

NotesEdit

  1. ^ A linear time-invariant system behaves the same but with the coefficients being constant in time.

ReferencesEdit

  1. ^ a b Katsuhiko Ogata (1997). Modern Control Engineering (3rd ed.). Upper Saddle River, NJ: Prentice-Hall. ISBN 978-0-13-227307-7.
  2. ^ Brockett, Roger W. (1970). Finite Dimensional Linear Systems. John Wiley & Sons. ISBN 978-0-471-10585-5.
  3. ^ a b c Eduardo D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems.
  4. ^ Isidori, Alberto (1989). Nonlinear Control Systems, p. 92–3. Springer-Verlag, London. ISBN 3-540-19916-0.
  5. ^ Claire J. Tomlin; Ian Mitchell; Alexandre M. Bayen; Meeko Oishi (2003). "Computational Techniques for the Verification of Hybrid Systems" (PDF). Proceedings of the IEEE. 91 (7): 986–1001. CiteSeerX 10.1.1.70.4296. doi:10.1109/jproc.2003.814621. Retrieved 2012-03-04.
  6. ^ Jean-Pierre Aubin (1991). Viability Theory. Birkhauser. ISBN 978-0-8176-3571-8.
  7. ^ Jan Polderman; Jan Willems (1998). Introduction to Mathematical Systems Theory: A Behavioral Approach (1st ed.). New York: Springer Verlag. ISBN 978-0-387-98266-3.
  8. ^ Brian D.O. Anderson; John B. Moore (1990). Optimal Control: Linear Quadratic Methods. Englewood Cliffs, NJ: Prentice Hall. ISBN 978-0-13-638560-8.
  9. ^ a b Eduardo D. Sontag (2013). Mathematical control theory: deterministic finite dimensional systems. Springer Science & Business Media.

External linksEdit