Dynamic Systems in State Space, Transfer Functions & Frequency Response

A dynamic model explains how present state and inputs generate future behavior. The same physical process can be represented by differential equations, state updates, an input/output operator, a transfer function, an impulse response or a frequency response. These are not interchangeable labels: equivalence holds only under specific assumptions.

State-space is the most explicit dynamic form #

A general continuous-time model is

$$\dot{x}=f(x,u,d,t),\qquad y=g(x,u,d,t),$$

where $x$ is internal state, $u$ a manipulated input, $d$ a disturbance and $y$ an output. It can express nonlinearity, multiple inputs and outputs, time variation, constraints and nonzero initial state.

Near an operating point $(x_0,u_0)$, linearization gives

$$\delta\dot{x}=A\delta x+B\delta u+E\delta d,\qquad \delta y=C\delta x+D\delta u.$$

This model describes perturbations, not arbitrary large motion. If saturation, dead zones or mode switches are reached, the linear model may fail even when its matrices were estimated perfectly.

State coordinates need not be unique. Any invertible transformation $z=Tx$ produces an equivalent realization. What matters are invariant input/output properties and whether the chosen state has useful physical or computational meaning.

Transfer functions eliminate internal state #

For a causal linear time-invariant system with zero initial conditions, Laplace transformation gives

$$G(s)=\frac{Y(s)}{U(s)}=C(sI-A)^{-1}B+D.$$

Poles describe natural modes visible and excitable at the interface; zeros describe input patterns or frequencies suppressed by the input/output relation. A state-space model can contain uncontrollable or unobservable modes that cancel from $G(s)$. A minimal realization contains neither.

Transfer functions are excellent for single-loop stability, disturbance paths and frequency-domain design. They hide internal state constraints and become cumbersome for nonlinear, time-varying or large multivariable systems.

Time and frequency views #

The impulse response $h(t)$ is the output produced by an ideal impulse under zero initial state. For an LTI system,

$$y(t)=(h*u)(t)=\int_0^t h(t-\tau)u(\tau)\,d\tau.$$

The step response integrates the impulse response and reveals gain, delay, time constants, overshoot and settling. One trace does not uniquely identify a high-order model; different dynamics can look alike over a limited window.

The frequency response evaluates $G(j\omega)$. A Bode plot shows magnitude and phase versus logarithmic frequency. It reveals bandwidth, resonances, roll-off and phase lag that may be hard to see in one time trace. It assumes sinusoidal steady state and linearity around the tested operating point.

Worked first-order process #

For a thermal lump with temperature deviation $x$, heater command $u$, time constant $\tau$ and steady-state gain $K$,

$$\tau\dot{x}+x=Ku.$$

State-space form is $\dot{x}=-x/\tau+(K/\tau)u$, $y=x$. The transfer function is

$$G(s)=\frac{K}{\tau s+1}.$$

For a unit step, $y(t)=K(1-e^{-t/\tau})$. At $t=\tau$, the response reaches $1-e^{-1}\approx63.2\%$ of its final value. In frequency, magnitude begins rolling off near $\omega=1/\tau$ and phase approaches $-90^\circ$. These are three views of one model, each exposing a different inference.

Choosing a representation #

Every model should state input/output definitions, reference directions, units, time domain, initial-condition convention, parameter uncertainty, operating region and bandwidth. Without that contract, algebraically correct transformations can connect physically different quantities.