Nonlinear Control Sy.. - Free

14 CHAPTER 1. INTRODUCTION
Figure 1.7: System trajectories of Example 1.9 (a) uncoupled system, (b) original system
diagonalizable. In this case, there always exist a similarity transformation P such that
P-1AP = J = Al
L 0
1
A2 ] .
The matrix J is in the so-called Jordan canonical form. In this case the transformed system
is
or
y = P-'APy
yl = Ayl + y2
y2 = 42
the solution of this system of equations with initial condition yo = [ylo, y20]T is as follows
yl = yloeAt +
y2 =
The shape of the solution is a somewhat distorted form of those encountered for diagonalizable
systems. The equilibrium point is called a stable node if A < 0 and unstable node if
A>0.
Example 1.10 Consider the system
0 -2][x2
The eigenvalues in this case are Al = A2 = A = -2 and the matrix A is not diagonalizable.
Figure 1.8 shows the trajectories of the system. In this example, the eigenvalue A < 0 and
thus the equilibrium point [0, 0] is a stable node.
y2oeAc
y2oteat