If xdot=Ax+Bu is a state space representation, what would a column of zeros in the A matrix depict in particular other than the system being singular in nature.
It means that at least one eigenvalue is zero or in other words you have a pole at the origin, i.e. a pure integrator.
Hi, if your system contains n state variables (x in R^n) this means simply that you can reduce your system to a (n-1) dimension system to control.
It means that at least one eigenvalue is zero or in other words you have a pole at the origin, i.e. a pure integrator.
dear Srikanth,
Apart from the fact that the matrix may be singular, the nullity of a colony generates non-cyclicity by one of the vectors of B, which implies the existence of non-controllable sub-space and hence the non-controllability of the system.
For more information see links and attached file in subject.
Operators, Functions, and Systems - An Easy Reading: Model Operators ...
https://books.google.dz/books?isbn=0821852655
Linear and Nonlinear Multivariable Feedback Control: A Classical ...
https://books.google.dz/books?isbn=0470061049
Best regards
As said Adrian Gambier, only represent eigenvalues = 0. The systems with a integrator or integrator chain usually have that characteristic. Also you can check drift free systems, where A = 0.
Consider the following simple example: A = [0 a1; 0 a2], B = [b1 b2]'
1) det (sI - A)= s (s - a2) => p1 = 0 and p2 = a2
2) det Co = det ([B | A*B]) = b2 ( b1 * a2 - b2 * a1)
From (1): the system has a pure integrator, but the order is 2 and it cannot be reduced.
From (2): the controllability depend of B, if b2 ≠ 0 and b2 ≠ (a1/a2) b1, the system is completely controllable independently of the zero column.
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