Nonlinear Control Sy.. - Free

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Figure 1.12: Nonlinear RLC-circuit.
Figure 1.13: Unstable limit cycle.
CHAPTER 1. INTRODUCTION
Figure 1.13 shows the vector field diagram of this system with µ = 1. As can be seen in the
figure, all trajectories diverge from the orbit and the limit cycle is unstable.
1.8 Higher-Order Systems
When the order of the state space realization is greater than or equal to 3, nothing
significantly different happens with linear time-invariant systems. The solution of the state
equation with initial condition x0 is still x = eATxo, and the eigenvalues of the A matrix
still control the behavior of the trajectories. Nonlinear equations have much more room in
which to maneuver. When the dimension of the state space realization increases from 2 to
3 a new phenomenon is encountered; namely, chaos.