Arthur Fleig, Marcus von Lossow, Tobias Ramming, Michael Schamel, Karl Worthmannrestart; with(LinearAlgebra):
A:=Matrix(6,6,[[0,1,0,0,0,0],[0,0,0,0,0,0],[0,0,0,1,0,0],[g/l[1],0,-g/l[1],0,0,0],[0,0,0,0,0,1],[g/l[2],0,0,0,-g/l[2],0]]):
B:=Matrix(6,1,[0,1/M,0,0,0,0]):
printf("The matrices A and B are defined. We consider the linear control system\134n\134n");
diff(x(t),t)=A.x(t)+B.u(t);R:=proc(A::Matrix,B::Matrix)
local i,n,S;
description "The procedure computes R(A,B) = [B, AB, A^2B, ..., A^(n-1)B]";
n:=Dimension(A)[1];
S:=Matrix(n,Dimension(B)[2],B);
for i from 2 to n do
S:=Matrix(n,i*Dimension(B)[2],[S,A.Column(S,i-1)]);
od;
end proc:
Describe(R);Normalform:=proc(A::Matrix,B::Matrix)
local i,n,S,T;
description "If the system is controllable, i.e., rk R(A,B) = n, the procedure will determine the normal form given in Theorem 3.2.3. Otherwise it computes the controllability decomposition";
n:=Dimension(A)[1];
if (Rank(R(A,B)) < n) then
printf("Since the pair (A,B) is not controllable, we compute the controllability decomposition\134n\134n");
S:=Matrix(n,Rank(R(A,B)),ColumnSpace(R(A,B)));
T:=Matrix(n,n,[S,Matrix(n,n-Rank(R(A,B)),[convert(NullSpace(R(A,B)),list)])]);
return(simplify(MatrixInverse(T).A.T),simplify(MatrixInverse(T).B));
else
S:=Matrix(1,n,Row(MatrixInverse(R(A,B)),n));
for i from 2 to n do
S:=Matrix(i,n,[[S],[Row(S,i-1).A]]);
od;
return(simplify(S.A.MatrixInverse(S)),simplify(S.B));
fi;
end proc:
Describe(Normalform);printf("We consider the case %a is not equal %a. Then the controllability matrix R(A,B) is given by\134n\134n", l[1], l[2]);
R(A,B);
printf("and has rank %d. ", Rank(%));
printf("Next, we determine, if possible, the normal form given in Theorem 3.2.3 via the procedure Normalform\134n\134n");
Normalform(A,B); N:=%[1]:printf("In order to verify the obtained normalform, we check the coefficients of the last row. To this end, we compute the characteristic polynomial of the matrix A and compare the corresponding coefficients.\134n\134n");
p:=s->CharacteristicPolynomial(A,s): sort(p(s),s);
check:=0:
for i from 1 to Dimension(A)[1] do
if (coeff(p(s),s,i-1)=-N(Dimension(A)[1],i)) then
#printf("Coefficient %a is correct\134n", alpha[i]);
check:=check+1;
else
printf("Coefficient %a = %a is not correct! The correct value is %a\134n", alpha[i], -N(Dimension(A)[1],i), coeff(p(s),s,i-1));
fi;
if (check=Dimension(A)[1]) then
printf("The normalform for the control system in consideration is computed succesfully.\134n\134n");
fi;
od;printf("We consider the case %a=%a. Then the controllability matrix R(A,B) is given by\134n\134n", l[1], l[2]);
R(subs(l[1]=l[2],A),B);
printf("and has rank %d. ", Rank(%));
printf("Next, we determine, if possible, the normal form given in Theorem 3.2.3 via the procedure Normalform.\134n\134n");
Normalform(subs(l[1]=l[2],A),B); LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=