2.3 Linear and Nonlinear Systems

Linear models expose structure¶

A linear time-invariant model has the form

\dot{x}=A(p)x+B(p)u+E(p)d, \qquad y=C(p)x+D(p)u.

Linear models support poles, zeros, transfer functions, controllability, observability, frequency response, and convex controller formulations. They are often the best first model for coupling diagnosis and local controller design.

But most engineered systems are nonlinear:

\dot{x}=f(x,u,d,p).

Geometry, aerodynamic forces, friction, contact, actuator saturation, fluid flow, and power electronics introduce nonlinearities. Linearization about $(x_0,u_0,p_0)$ gives

\delta\dot{x}=A\delta x+B\delta u, \quad A=\left.\frac{\partial f}{\partial x}\right|_0, \quad B=\left.\frac{\partial f}{\partial u}\right|_0.

Adjustable inverted pendulum¶

For pendulum angle $\theta$ from upright, length $l$ , mass $m$ , damping $b$ , and pivot torque $u$ ,

ml^2\ddot\theta+b\dot\theta-mgl\sin\theta=u.

Near upright, $\sin\theta\approx\theta$ , so

\ddot\theta\approx \frac{g}{l}\theta-\frac{b}{ml^2}\dot\theta+\frac{1}{ml^2}u.

Increasing $l$ reduces the open-loop divergence rate $\sqrt{g/l}$ but also reduces torque authority through $1/(ml^2)$ . Geometry changes instability and actuation simultaneously. A linear CCD study can reveal that trade locally, but it cannot certify recovery from large initial angles or account for torque saturation during swing-up.

When a linear model is adequate¶

A linear model is often sufficient when operating deviations are small, one equilibrium dominates, nonlinear constraints remain inactive, and the final design is validated over the required envelope. A nonlinear model is needed when the design exploits large motion, saturation, switching, contact, state-dependent aerodynamics, or multiple equilibria.

Multi-point and parameter-varying models¶

Between a single linearization and a full nonlinear model lie useful compromises: linearizations at several operating points, gain-scheduled models, linear parameter-varying models, and reduced nonlinear surrogates. A wind turbine, for example, may require different local models across below-rated and above-rated operation. The CCD problem must prevent improvement at one point from degrading another unacceptably.

Activity 2.3: locate the nonlinearity¶

Activity 2.3: locate the nonlinearity

Consider the adjustable inverted pendulum

ml^2\ddot\theta+b\dot\theta-mgl\sin\theta=\operatorname{sat}_{\tau_{\max}}\!\left(u-\tau_f\operatorname{sgn}\dot\theta\right),

with $m=2$ kg, $b=0.15$ N m s/rad, Coulomb friction $\tau_f=0.8$ N m, torque limit $\tau_{\max}=10$ N m, and candidate lengths $l\in\{0.40,0.80\}$ m. The local linear model replaces $\sin\theta$ with $\theta$ and omits saturation and Coulomb friction.

At $\theta=5^\circ,15^\circ,30^\circ,$ and $45^\circ$ , calculate

\varepsilon_{\sin}=100\frac{|\theta-\sin\theta|}{|\sin\theta|}

and identify the largest tested angle for which the gravity linearization error is below 2%. 2. For each length, compute the nonlinear and linearized static balancing torques at $30^\circ$ and $45^\circ$ . Mark cases in which the required nonlinear torque exceeds $\tau_{\max}$ . 3. Calculate the local open-loop divergence rate $\sqrt{g/l}$ and input coefficient $1/(ml^2)$ for both lengths. Quantify the geometry trade between slower instability and reduced torque authority. 4. At $\theta=30^\circ$ and $\dot\theta=0.10$ rad/s, determine the fraction of available torque consumed by gravity and Coulomb friction for each design. 5. Locate every nonlinearity in the governing equation and classify it as geometric, frictional, or actuator-related. Based on the calculations, state which nonlinearities can change the preferred link length or motor rating.

Use radians inside all trigonometric and error calculations and $g=9.81$ m/s $^2$ .