Internet Control Course

Controllability analysis in a convex domain.

We begin by defining the concept of convex set.

Thus, if and D is convex, then every point

must also be in the set.

The conditions (1.6) of Theorem 1.2 turn out to be necessary and sufficient for cart controllability on a convex domain.

Proof. Necessity has been proved in Theorem 1.2. By Theorem 1.1, in order to prove sufficiency we have to show that implies for all

Since D is a convex set, it is only necessary to prove that implies

If then

The equalities (1.7) imply

Hence

Q.E.D.

Equilibrium set of the linear system

is the set of the states in which the system can remain stationary under an appropriate control input. Equilibrium set of the cart is "rails", i.e. Equilibrium set of two-dimensional linear system is the straight line Let denote the function Then denotes the equilibrium set of

Corollary. A linear two-dimensional system is controllable by on a convex domain D iff

Proof. If the system is controllable in any domain, then and consequently is equivalent to the cart. Therefore using (1.3), (1.5), we can rewrite the equalities (1.8) in the form (1.7). Thus an application of Theorem 1.4 completes the proof.

Q.E.D.

Using Theorem 1.4 and its corollary we can investigate controllability of a linear two-dimensional system on a convex domain with the help of convex optimization methods.

Example 1.3(linear programming for the controllability analysis). Let us investigate controllability of the system

on the convex set D defined by the inequalities

Applying the simplex method (for the details see, e.g., [37]) we obtain that the function attains the minimum equal to -33 on D and the maximum equal to 15 on D .

Moreover the function p(x) takes the value -33 at the point and the value 15 at Thus, according to Theorem 1.4, the system (1.9) is controllable on D by iff

Therefore the set of linear systems being of the form (1.9) and controllable on D is the two-dimensional plane defined by (1.10) in

Example 1.4 Let us investigate controllability of the cart by on the convex domain D defined by the inequality

where is a real number. In order to find a maximum and a minimum of the function on we construct the Lagrange function

It is well known (see, e.g., [ 11 ]) that p can have the maximum or the minimum at the point only when

for some Therefore

and we obtain that the function attains the maximum at the point and the minimum at If then are not in and, by Theorem 1.4, the cart is not controllable on D . Thus the cart is controllable on D iff

So far we have assumed that controls can take arbitrary large values. From a practical point of view this is quite a restrictive assumption. More natural considerations imply that controls have to be bounded by some known functions. In other words, the problem is to study cart controllability on a domain D by the following class of admissible controls.

where are differentiable real functions defined on is the solution of (1.1) generated by the control u(t) and the initial condition

can be one of the following classes of controls : For instance, if and with then we are dealing with controls from

In general, control constraints change the life of the cart in a bad way. The cart having been controllable by on a domain D can lose its controllability after an introduction of control constraints. On the other hand, if the cart is controllable by on D , then the cart remains to be controllable on D by Since all necessary and sufficient conditions for controllability on a domain D by become necessary conditions for controllability on the domain D by where

Let us investigate controllability of the cart by with In this investigation an important role is played by the equilibrium set Indeed, let denote the parabola which is the cart trajectory passing through a point x under the control (see Figs.1.4, 1.5). Then the cart is not controllable on a simply connected domain D if there exists a point such that either or In fact, both and spoil the domain D so that D gets mysteries and prisons (Fig.1.16).

 
Figure 16: The arrows show in what direction the cart moves along the trajectories.

More precisely, the following theorem generalizes Theorem 1.1.