"Modern" ( since 1980.. ) control theory often uses so-called linear state space models.
These models are especially useful in systems with more than one output, and sometimes also more than one input. These models can have many internal degrees of freedom, simple in themselves, but complicated in their interactions. Aircraft stabilty and control analysis is a prime example.
The general formulation of such a model is :
| x • = A . x + B . u | (1) |
The matrices A and B are blocks of constants. They represent the system, linearized around some basic state, usually an equilibrium condition like a pendulum hanging straight down or an airplane in steady flight.
The variables in the so-called "state vector" x, together describe the momentary motion of the system. The vector u lists the usually also time-varying input variables to the system.
The matrix equation gives the evolution over time, of the x under the influence of x itself, and of the inputs u. Both x and u can have more than element, which is why they are called "vectors" in the linear algebra involved. But they are not normally physical vectors in 3D space. They are vectors in an abstract sense only.
The raised dot means "derivative with respect to time". It can also be written as d/dt.
Equation (1) says that the momentary change of the state x with time is directly proportional to the elements of the state itself, and of the inputs u. There are no squares, no cubes or worse. These equations are strictly linear. This may sound very restrictive, but in fact for small movements around some steady state value, the real varying curves can always be approximated by their local slope.
We will find that A has to be a square matrix with sides the size of the state, and B has to be as wide as the size of the input vector u, and as high as the state vector.
TBW. First order system, scalar. Second order, refer to pendulum.
Higher orders, refer to airplane modes. And chains of springs.
The system diagonal contains all the damping. The rest is redistribution only.
Modes. Refer to coupled pendulums.
TBW.
TBW.
Refer to pendulum, x not orthogonal to v.
Refer to coupled pendulums for modes.
The choice of the state elements is unique in one way, because these are the elements needed to describe the dynamics of the system. But they can be scaled to different units, and even combined in different ways. This very useful trick is discussed here.
Matlab conventions.
Eigenvalues ( VE, E ), VE( #element, #eigenvalue), check that VE( :, 3:6) are the column eigenvectors for the eigenvalues 3:6. Eigenvalues are ordered ****. Eingevectors are normalzized to ****. The *** element is aligned with the real axis, and the whole thing can be scaled and rotated (==scaled with a unit imaginary number) at will.
In imaginary pairs, choose the eigenvalues with positive imaginary part. They rotate left (anti-clockwise) with time, because of iωt.
The accompanying eigenvector can have elements with positive and negative imaginary parts. They don't have absolute phase anyway, only relative phase difference between the elements.