solve set of inequalities

I am working in state space where I have a triangle in R² with vertices v1 =(−1, 0)' , v2 = (1, 1)', and v3 = (1, −1)'

The normal vectors on the three facets F1, F2, and F3 of the triangle are

n1 = (1;0) , n2 =(1/sqrt(5))*[-1;-2]; n3=(1/sqrt(5))*[-1;2] respectively.

On the triangle T we consider the system : xdot=A.x+Bu+a.

with state x ∈ T and scalar input −1 ≤ u ≤ 1. I want to construct an affine feedback law u = Fx + g such that the state of the closed-loop system can only leave the Triangle T through the facet F1, the vertical line segment between the vertices v2 and v3.

So I should find an input u1 at vertex v1 , u2 at vertex v2 and u3 at v3 which are satisfying set of inequalities:

for u1 the following inequalities should hold:

(1) n2'.B.u1 ≤ −n'2. (A.v1 + a)

(2) n3'. B.u1 ≤ −n3'.(A.v1 + a)

(3) n2'.B.u1 + n3'. B.u1 < −n2'. (A.v1 + a) − n3'.(A.v1 + a)

and, additionally, −1 ≤ u1 ≤ 1.

for u2 the following inequalities should hold:

(1) n2'.Bu2 > −n1' (Av2 + a)

(2) n3'. B.u2 ≤ −n3'.(A.v1 + a) with −1 ≤ u2 ≤ 1

for u3 the following inequalities should hold:

(1) n1' Bu3 > −n1'(Av3 + a)

(2) n2'Bu3 ≤ −n2' (Av3 + a)

with −1 ≤ u3 ≤ 1

and after that I should solve the system ui = Fvi + g to find F and g.

I tried to solve these inequalities to find u2 (By hand i become an interval for u2 : [−1 6 , 1]), but when i run my code in i got :

Empty sym: 0-by-1

I have another qst please: I need the value of u2 for next step how can I get allowable value from this interval ? there is a function to use ?

close all;
clc;
v1=[-1;0]; v2=[1;1];v3=[1;-1];
line([v1(1) v3(1)],[v1(2) v3(2)],'Color','k');
hold on
line([v1(1) v2(1)],[v1(2) v2(2)],'Color','k');
line([v2(1) v3(1)],[v2(2) v3(2)],'Color','k');
axis([-2 2 -2 2]);
% define the normal vectors
n1=[1;0]; n2=(1/sqrt(5))*[-1;-2]; n3=(1/sqrt(5))*[-1;2];
A=[-1 -1;-2 1];
B=[2;-2];
a=[3;1];
syms u2 
const1=n1'*B*(u2)>-n1'*(A*v2+a);
const2=n3'*B*(u2)<=-n3'*(A*v2+a);
const3=u2 <= 1;
const4=u2 >= -1;
conds=[const1 const2 const3 const4];
sol=vpasolve(conds,u2)

[John D'Errico]

Impossible to answer. You describe a triangle, in terms of vertices v1,v2,v3. No problem. The triangle is clear. You even give us information we don't need, like the normal vectors to the edges.

But then you tell us you want to solve for u2. What is u2? How is it connected to this problem? We cannot read your mind, to then know what you are thinking. Then you give us code to do something that does not seem to work, so it is difficult to know what you need.

If you want an answer, then make it possible to get an answer. Be clear about your problem. Define what you want to solve for, rather than hoping we can guess what you might be asking.

[tty]

I edited the post. Sorry I didn't want to be long in my description

[Torsten]

Don't know if it helps:

Using

sol=solve(conds,u2);

instead of

sol=vpasolve(conds,u2);

gives

sol = 0

as one possible solution.

If you can set up your problem without using the symbolic toolbox, you can use "linprog" to get a feasible point for your system of inequalities.