Controllability

Controllability From Wikipedia, the free encyclopedia Jump to: navigation, search 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 Contents [hide] * 1 State controllability * 2 Continuous Linear Time-Invariant (LTI) Systems * 3 Discrete Linear Time-Invariant (LTI) Systems o 3.1 Example o 3.2 Analogy for example of n = 2 * 4 Nonlinear Systems * 5 Output controllability * 6 Controllability in the behavioural framework * 7 Stabilizability * 8 See also * 9 References * 10 External links [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. Complete state controllability (or simply controllability if no other context is given) describes the ability of an external input to move the internal state of a system from any initial state to any other final state in a finite time interval.[1]:737 [edit] Continuous Linear Time-Invariant (LTI) Systems Consider the continuous linear time-invariant system /dot{/mathbf{x}}(t) = A /mathbf{x}(t) + B /mathbf{u}(t) /mathbf{y}(t) = C /mathbf{x}(t) + D /mathbf{u}(t) where /mathbf{x} is the n /times 1 "state vector", /mathbf{y} is the m /times 1 "output vector", /mathbf{u} 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 = /begin{bmatrix}B & AB & A^{2}B & ...& A^{n-1}B/end{bmatrix} The system is controllable if the controllability matrix has a full row rank (i.e. /operatorname{rank}(R)=n). [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 and B is a n /times r matrix (i.e. /mathbf{u} is r inputs collected in a r /times 1 vector. The test for controllability is that the n /times nr matrix /mathcal{C} = /begin{bmatrix}B & AB & A^{2}B & ...& A^{n-1}B/end{bmatrix} has full row rank (i.e., rank(C) = n). That is, if the system is controllable, C will have n columns that are linearly independent; if n columns of C are linearly independent, 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 and r = 1 (i.e. only one control input). Thus, B and AB are n /times 1 vectors. If /begin{bmatrix}B & AB/end{bmatrix} has rank 2 (full rank), and so B and AB are linearly independent and span the entire plane. If the rank is 1, then B and AB are collinear and do not 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[citation needed]: 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 is the related notion for the output of the system; 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. A controllable system is not necessarily output controllable, and an output controllable system is not necessarily controllable. 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{bmatrix} CB & CAB & CA^2B & /cdots & CA^{n-1}B & D/end{bmatrix} must have full row rank (i.e. rank m) if and only if the system is output controllable.[1]:742 [edit] Controllability in the behavioural 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 (state trajectory) can be concatenated with any future part of a behavior with which it shares the current state in such a way that the concatenation is contained in the behavior, i.e. is part of the admissible system behavior.[citation needed] [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 states cannot be controlled (as determined by the controllability test above) all the states will still remain bounded during the system's behavior.[citation needed] [edit] See also * Observability [edit] References Text document with red question mark.svg This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please improve this article by introducing more precise citations where appropriate. (May 2009) 1. ^ a b Katsuhiko Ogata (1997). Modern Control Engineering (3rd ed.). Upper Saddle River, NJ: Prentice-Hall. ISBN 0-13-227307-1. [edit] External links * Controllability on PlanetMath Retrieved from "http://en.wikipedia.org/wiki/Controllability" Categories: Control theory