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Recall from Section 13.1.1 that linear constraints restrict the velocity to an -dimensional

hyperplane. The linear model in (13.37) is in parametric form, which means that each action variable may allow

an independent degree of freedom. In this case, . In the extreme case of , there are no

actions, which results in . The phase velocity is fixed for every point . If , then at

every a one-dimensional set of velocities may be chosen using . This implies that the direction is

fixed, but the magnitude is chosen using . In general, the set of allowable velocities at a point is an

-dimensional linear subspace of the tangent space (if is nonsingular).

In spite of (13.37), it may still be possible to reach all of the state space from any initial state. It may be costly,

however, to reach a nearby point because of the restriction on the tangent space; it is impossible to command a

velocity in some directions. For the case of nonlinear systems, it is sometimes possible to quickly reach any point

in a small neighborhood of a state, while remaining in a small region around the state. Such issues fall under the

general topic of controllability, which will be covered in Sections 15.1.3 and 15.4.3.

Although not covered here, the observability of the system is an important topic in control [192,478]. In terms

of the I-space concepts of Chapter 11, this means that a sensor of the form is defined, and the task

is to determine the current state, given the history I-state. If the system is observable, this means that the

nondeterministic I-state is a single point. Otherwise, the system may only be partially observable. In the case of 

linear systems, if the sensing model is also linear,

(13.39)

then simple matrix conditions can be used to determine whether the system is observable [192]. Nonlinear 

observability theory also exists [478].

As in the case of discrete planning problems, it is possible to define differential models that depend on time. In

the discrete case, this involves a dependency on stages. For the continuous-stage case, a time-varying linear 

system is defined as

(13.40)

In this case, the matrix entries are allowed to be functions of time. Many powerful control techniques can be

easily adapted to this case, but it will not be considered here because most planning problems are time-

invariant (or stationary).

Next: 13.2.3 Nonlinear Systems Up: 13.2 Phase Space Representation Previous: 13.2.1.3 Higher order 

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differential

Steven M Lavalle 2010-04-24

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