4.3 State and Algebraic Variables

Differential states carry memory¶

The state $x(t)$ is the smallest collection of quantities that, together with current inputs and data, determines future evolution under the model. For a positioner with motor electrical dynamics,

where $q$ is joint angle, $\omega$ angular speed, and $i$ motor current. A representative model is

Plant variables change the state equations. The state trajectory is therefore an analysis result in a nested simulation, or a set of optimization variables tied together by defect constraints in a simultaneous transcription.

Algebraic variables enforce instantaneous relations¶

The vector $z(t)$ contains quantities determined by algebraic equations rather than independent time integration. Examples include reaction forces, circuit currents under a quasi-static electrical approximation, power-flow variables, aerodynamic induction factors, and kinematic loop forces.

A differential-algebraic model can be written

Promoting an algebraic output to a decision variable requires retaining its consistency equation. Otherwise, an optimizer may choose physically impossible values.

States, measurements, and outputs differ¶

A sensor output $y_m=s(x,z)+n$ need not reveal every state. A performance output such as electrical power $P_e=vi$ may not be measured. Keep differential states, algebraic variables, measured outputs, and performance outputs distinct even when their numerical values coincide in a simplified example.

Activity 4.3: classify model quantities¶