Internet Control Course

Prisons and mysteries of a plain domain

Assume that we are driving a very simple cart (Fig. 1.1) along straight rails without any friction.

 
Figure 1: The cart is going along straight rails. is a position of the cart on the rails. is a velocity of the cart.

The dynamics of the cart, in accordance with Newton's second law, is described by the system

where denotes the derivative of with respect to time, i.e. the coordinate measures a position of the cart on the rail road, the coordinate is a velocity of the cart and u is a control force. The state of the cart is a pair The set of all possible cart states constitutes the state space which is - two-dimensional real space or, in other words, the state space is a plane.

Suppose it is permitted only to use controls from a class given in advance. For example,

1)

if controls continuously depend on time and act only when then where C(0,T ) is the set of real continuous functions defined on the interval [0,T] .

2)

if controls are supposed to have a finite number of discontinuities such as switches, then if u(t) is bounded for all there exists a finite set such that is continuous for all (the finite set may depend on

3)

if admissible controls have to be piecewise-constant, then

where if the derivative exists and is equal to zero for all except a finite subset in [0,T]. The finite subset may depend on

4)

Throughout this book the following classes of controls will be mostly used:

Assume that it is not permitted that our cart goes out of some domain It is quite a natural assumption which can be faced in many applications of system theory. Then it is important to know the conditions under which the cart is controllable on D by an admissible class of controls, say Throughout the first part of this book unless the contrary is explicitly stated, D is an open connected subset of

Remark. We will say that a subset is reachable (and/or accessible ) from if there exist two points and a control which steers the cart from to in a finite time T>0 and for all

Controllability property of the cart crucially depends on the geometry of the boundary of the domain D . To this respect, it is convenient to discuss some examples.

Example 1.1. Consider controllability of the cart on the domain D shown in Fig.1.2, i.e.

 
Figure 2: We cannot leave the subsets without leaving the domain D . That is why, and turn out to be prisons for the cart.

If our cart is in one of the subsets then we cannot leave the subset without leaving the domain D . For instance, as long as we have the cart drifting to the right because and on We cannot change the drift direction without leaving the domain D . Thus the system is not controllable on D .

Example 1.2. Let be the reflection and the image of D (Fig.1.2) under the reflection, i.e. Q = r(D) (see Fig 1.3).

 
Figure 3: are mysteries for the cart.

Then we cannot reach any point of the subsets as long as we are not in or respectively. Thus the cart is not controllable on Q .

In both examples we cannot control the system because of the existence of the subsets such as in Example 1.1 and in Example 1.2. The subsets look like prisons for our cart ( we are sentenced to live there whenever we find ourselves inside of a prison). After reversing the time scale the prisons become mysteries. In this sense one can call mysteries for the cart ( we cannot get into or into if we are not already inside them).

It is easy to see that the cart is controllable on D iff in D there are no prisons and mysteries for the cart. That explains why we are paying so much attention to prisons and mysteries of a plane domain. As soon as we have analytical descriptions for the mysteries and prisons we can answer the question: "What are the conditions under which the cart is controllable on D ?"