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 ?
v1=[-1;0]; v2=[1;1];v3=[1;-1];
line([v1(1) v3(1)],[v1(2) v3(2)],'Color','k');
line([v1(1) v2(1)],[v1(2) v2(2)],'Color','k');
line([v2(1) v3(1)],[v2(2) v3(2)],'Color','k');
n1=[1;0]; n2=(1/sqrt(5))*[-1;-2]; n3=(1/sqrt(5))*[-1;2];
const1=n1'*B*(u2)>-n1'*(A*v2+a);
const2=n3'*B*(u2)<=-n3'*(A*v2+a);
conds=[const1 const2 const3 const4];
