Physical Modeling, Operating Points & Linearization

A useful model preserves the mechanisms that determine the intended control decision. Start from storage and flow of mass, momentum, charge, energy, or information; add empirical constitutive relations such as drag or heat transfer; and expose sensors and actuators at the boundary. Only then decide which states and nonlinearities can be reduced.

The tangent is accurate only near the chosen operating point. A different equilibrium generally produces different gains and time constants.

From physical inventory to state equations #

For each candidate state, ask what is stored. A capacitor stores electric-field energy and its voltage cannot jump under finite current. A mass stores momentum and its velocity cannot jump under finite force. A tank stores volume and its level changes by net flow. These continuity facts help identify states before any algebra.

A general plant can be written

$$\dot x=f(x,u,d,p),\qquad y=h(x,u,d,p),$$

where $x$ is state, $u$ manipulated input, $d$ disturbance, and $p$ parameters. Record:

• units and positive directions for every variable;
• algebraic constraints and conservation boundaries;
• actuator and sensor dynamics, not only the nominal process;
• time delays, hysteresis, friction, saturation, dead zones, and switching;
• the parameter and operating envelope in which each law is valid.

Dimensional inconsistency is a model failure, not a numerical inconvenience. Each term added in one differential equation must have the same units.

Equilibrium and perturbation coordinates #

For a constant operating point $(x_0,u_0,d_0)$,

$$f(x_0,u_0,d_0,p_0)=0.$$

Define perturbations $\delta x=x-x_0$, $\delta u=u-u_0$, $\delta d=d-d_0$, and $\delta y=y-y_0$. A first-order Taylor expansion gives

$$\delta\dot x=A\delta x+B\delta u+E\delta d,$$

$$\delta y=C\delta x+D\delta u+F\delta d,$$

with Jacobians evaluated at the operating point:

$$A=\left.\frac{\partial f}{\partial x}\right|_0,\quad B=\left.\frac{\partial f}{\partial u}\right|_0,\quad C=\left.\frac{\partial h}{\partial x}\right|_0,\quad D=\left.\frac{\partial h}{\partial u}\right|_0.$$

Constant terms disappear because the equilibrium satisfies $f_0=0$. The omitted remainder is second order and higher in the perturbations. That is the mathematical reason the model is local.

Worked example: gravity-drained tank #

For tank cross-section $A_t$, inflow $q_{in}$, and outlet $q_{out}=k\sqrt h$,

$$A_t\dot h=q_{in}-k\sqrt h.$$

At equilibrium $q_0=k\sqrt{h_0}$. With $\delta h=h-h_0$ and $\delta q=q_{in}-q_0$,

$$A_t\delta\dot h=\delta q-\frac{k}{2\sqrt{h_0}}\delta h.$$

Therefore

$$\frac{\Delta H(s)}{\Delta Q(s)}=\frac{1}{A_ts+k/(2\sqrt{h_0})}=\frac{K_h}{\tau_hs+1},$$

where

$$K_h=\frac{2\sqrt{h_0}}{k},\qquad \tau_h=\frac{2A_t\sqrt{h_0}}{k}.$$

Both gain and time constant increase with $\sqrt{h_0}$. A controller tuned at a low level is not automatically equivalent at a high level. The nonlinear equation also forbids $h<0$, a fact the linear perturbation model cannot enforce.

Linearization about a trajectory #

An operating condition need not be static. Around nominal trajectories $x_0(t),u_0(t)$ satisfying the nonlinear equations,

$$\delta\dot x=A(t)\delta x+B(t)\delta u,$$

where the Jacobians vary with time. This linear time-varying model is appropriate for ascent, batch processes, robotic maneuvers, or scheduled operating regimes. Freezing $A(t)$ at one instant produces only a local snapshot; it is justified when parameter variation is slow relative to closed-loop dynamics and verified across the schedule.

What linearization preserves and loses #

Near a hyperbolic equilibrium, eigenvalues of $A$ usually determine local qualitative stability. The linearization also gives local gains, modes, controllability, observability, and frequency response. It loses or can misrepresent:

• saturation, dead zones, backlash, hysteresis, and relay switching;
• multiple equilibria and boundaries between their basins;
• amplitude-dependent frequency and damping;
• finite escape, impacts, and state constraints;
• behavior at a nonhyperbolic equilibrium with eigenvalues on the imaginary axis.

If the linearized model has a zero eigenvalue or a purely imaginary pair, the first-order test may be inconclusive. Return to nonlinear terms, center-manifold/normal-form reasoning, energy or Lyapunov methods, or direct nonlinear analysis.

Model reduction without hidden fiction #

A mode may be neglected only relative to a stated bandwidth and input/output path. Fast stable dynamics can sometimes be approximated by static gain, but their phase still matters near crossover. A small time constant is not small if the controller excites it. Pole-zero cancellation in a fitted transfer function can hide an internal unstable or weakly observable mode.

Validate a reduced model with data not used to derive it:

1. compare equilibrium and local gains across the intended envelope;
2. compare step, frequency, or multisine response over the intended bandwidth;
3. inspect residuals for bias and correlation with past inputs;
4. test the neglected dynamics under the controller's expected command spectrum;
5. document the state, input, parameter, and frequency validity region.

A model is adequate for a decision, not true in the abstract.