Nonlinear Control Systems

Nonlinear behavior

Nonlinear systems do not satisfy superposition. Examples include actuator saturation, dead zones, friction and products or nonlinear functions of state. They can have multiple equilibria, amplitude-dependent responses and limit cycles.

Local linearization

For xdot = f(x,u), find an equilibrium with f(x0,u0) = 0. Small perturbations satisfy approximately Δxdot = AΔx + BΔu, where A and B are Jacobians at the equilibrium. A stable linearization with all eigenvalues strictly in the left half-plane gives local asymptotic stability for a sufficiently smooth system. Eigenvalues on the imaginary axis make this linear test inconclusive.

Worked example

Consider xdot = −x + x³ with zero input. Equilibria satisfy x(x² − 1) = 0, so x = 0, ±1. The derivative f'(x) = −1 + 3x² is −1 at zero and 2 at ±1. Thus zero is locally asymptotically stable and the other two equilibria are unstable. For V = x²/2, Vdot = x(−x + x³) = −x² + x⁴ < 0 when 0 < |x| < 1. This confirms attraction toward zero within that interval. It does not claim global stability: outside the interval the dynamics point away from zero.

Phase plane and describing functions

For two-state autonomous systems, phase-plane trajectories show state evolution and equilibrium behavior. Describing functions approximate a nonlinear element by its fundamental sinusoidal response at a specified amplitude. An intersection suggesting a limit cycle is an approximate prediction requiring further analysis or simulation, not a general proof of existence or stability.

Common mistakes

Do not extend a local linear conclusion to all initial conditions. Do not assume zero eigenvalues mean instability. Do not apply superposition to saturated systems.

Quick check

1. Can a nonlinear system have multiple equilibria? Yes.
2. Is a describing function exact for every waveform? No.
3. What does negative Vdot establish with a suitable positive-definite V? A stability conclusion within the domain where the required conditions hold.