Feedback for physicists: A tutorial essay on control

Feedback and control theory are important ideas that should form part of the education of a physicist but rarely do. This tutorial essay aims to give enough of the formal elements of control theory to satisfy the experimentalist designing or running a typical physics experiment and enough to satisfy the theorist wishing to understand its broader intellectual context. The level is generally simple, although more advanced methods are also introduced. Several types of applications are discussed, as the practical uses of feedback extend far beyond the simple regulation problems where it is most often employed. Sketches are then provided of some of the broader implications and applications of control theory, especially in biology, which are topics of active research.

Figure 1

Low-pass electrical filter.

Figure 2

Driven simple harmonic oscillator.

Figure 3

(Color in online edition) Transfer function for a first-order, low-pass filter. (a) Bode magnitude plot $\left[\mid G\left(i\omega \right)\mid \right]$; (b) Bode phase plot $\left[\arg G\left(i\omega \right)\right]$.

Figure 4

(Color in online edition) Transfer function for a second-order system. $\zeta =0.1$ gives underdamped and $\zeta =1$ critically damped dynamics. (a) Bode magnitude plot; (b) Bode phase plot.

Figure 5

Block diagram illustrating signal flow from the input $u$ through the system dynamics $G$ to the output $y$ through the sensor $H$.

Figure 6

Block diagram illustrating closed-loop control of a system $G\left(s\right)$. Controller dynamics are given by $K\left(s\right)$.

Figure 7

Block diagram illustrating closed-loop control of a system $G\left(s\right)$ subject to disturbances $d\left(s\right)$ and measurement noise $\xi \left(s\right)$. The control signal $r\left(s\right)$ is assumed to be noise free. If present, the block $F\left(s\right)$ adds feedforward dynamics.

Figure 8

(Color in online edition) Phase-space plot of harmonic oscillator and its observer. The harmonic oscillator trajectory starts from the dot and traces a circle. The observer starts from the square with incorrect initial velocity but converges to track the oscillator. For simplicity, the control signal $u=0$.

Figure 9

Michelson interferometer with feedback element $K\left(s\right)$ added to linearize output for large displacements.

Figure 10

Op-amp based noninverting amplifier.

Figure 11

(Color in online edition) Two scenarios for a pitchfork bifurcation, showing steady-state solution $y$ as a function of the control parameter $\lambda$: (a) supercritical; (b) subcritical, where the arrows show the maximum observable hysteresis.

Figure 12

(Color in online edition) Transfer function for a degenerate second-order system with integral control, showing gain and phase margins for $K_i=1$. (a) Bode magnitude plot; (b) Bode phase plot.

Figure 13

Zero-order-hold digitization of an analog signal. Points are the measured values. Note that their average implies a sampling delay of half the sampling interval, $T_s$.

Figure 14

Block diagram of feedback system with nontrivial sensor dynamics, given by $H\left(s\right)$.

Figure 15

Control of the temperature of a probe located a distance $L$ inside a metal bar, with heater at the bar’s end.

Figure 16

(Color in online edition) Root-locus plot of the poles of Eq. (4.18) as a function of the gain, with $z=1$ and $p=-1+i$. The crosses denote the poles when $K_p=0$, while the circle denotes the zero. The two poles approach each other, meeting and becoming real when $K_p=-4+2\sqrt{5}\approx 0.47$. One root then approaches 1 as $K_p\to \infty$, crossing zero at $K_p=2$, while the other root goes to $-\infty$. The system becomes unstable at $K_p=2$.

Figure 17

Model of a hand balancing a vertical rod.

Figure 18

(Color in online edition) (a) Schematic of a desired loop shape $L\left(s\right)$. (b) Corresponding phase plot.

Figure 19

(Color in online edition) Bode plots of a lag compensator [Eq. (5.6)] with $a=10$.

Figure 20

(Color in online edition) Phase-space plot of dynamical trajectories for a free, frictionless particle acted on by a force $u=\pm 1$. The thrust switches as trajectories reach the curve $AOB$. The heavy curve shows an example trajectory (84).

Figure 21

(Color in online edition) Illustration of the Nyquist sampling theorem in frequency space. (a) The maximum frequency of the power spectrum of the continuous signal is less than the Nyquist frequency (positive and negative frequencies indicated by arrows). The spectrum of the sampled signal accurately represents the continuous signal. (b) The maximum frequency exceeds the Nyquist frequency ${\omega }_N$, and the aliased spectra overlap, distorting the estimate of the continuous spectrum from sampled data. The individual copies of the spectrum of the continuous signal are shown in the dashed lines. The overlap is apparent, particularly near multiples of ${\omega }_N$.

Figure 22

(Color in online edition) Aliasing of a high-frequency signal (solid curve, with period $T$) produced by sampling with a period $T_S$ that is greater than the Nyquist criterion. The apparent measured points (large dots) are interpreted as coming from a much lower frequency signal (dashed curve).

Figure 23

(Color in online edition) Time series for a discrete dynamical system showing a “ringing pole.” Generated from $y_{n+1}=(1-\lambda )y_n$, with $\lambda =1.9$.

Figure 24

(Color in online edition) Control of a first-order system in the presence of noise. (a) Open-loop fluctuations of the controlled variable in response to environmental noise; no sensor noise is present; (b) closed-loop proportional control; (c) sensor noise added; (d) Kalman filter added; actual system state and the Kalman estimate, which has smaller fluctuations, are both shown. In (a)–(d), $\sigma$ is the standard deviation of the measured signal.

Figure 25

Block diagram of an IMC controller. Shaded area is implemented either in a computer program or in control electronics.

Figure 26

(Color in online edition) Multiplicative bound for a variable delay. Bottom curves: magnitude of Eq. (5.48); top curve: magnitude of Eq. (5.49).

Figure 27

(Color in online edition) Illustration of a multiplicative uncertainty bound and of robust stability. The thick, central line is the nominal system $G_0$, with $\mathrm{Im}\left[G_0\left(i\omega \right)\right]$ plotted vs $\mathrm{Re}\left[G_0\left(i\omega \right)\right]$ over $0<\omega <\infty$ (“Nyquist plot”). The thin lines that shadow it are the uncertainty limits, meaning that the actual system follows a path somewhere within the shaded area. The shaded circle gives the multiplicative uncertainty bound at one frequency. If we now interpret the thick central line as a loop transfer function and if the system is closed in a feedback loop, then robust stability implies that the set of loop transfer functions defined by the light shaded region cannot touch the point $-1$, which is indicated by the large dot.

Figure 28

(Color in online edition) Constraints on the sensitivity function $S=1∕(1+L)$. (a) Logarithm of the sensitivity function $\ln \mid S\left(i\omega \right)\mid$ for a second-order system with proportional gain. The negative area where control is “robust” is just balanced by the positive area, where control is “fragile.” Note that the slow decay (here, $\ln \mid S\mid \sim {\omega }^{-2}$ at large frequencies) implies that a significant part of the positive area is off the plot. (b) Nyquist plot of the same system. Note that when the plot enters the shaded circle of radius 1, centered on $-1$, we have $\mid S\mid >1$.

Figure 29

(Color in online edition) Metastability: the ball is stable to small perturbations but unstable to larger ones.

Figure 30

(Color in online edition) Saturation of proportional gain. Thin dashed line is the proportional gain with no saturation. Thick lines show saturation. There is, in general, no reason for saturation to be symmetric about zero, as depicted here.

Figure 31

(Color in online edition) Illustration of the OGY chaos-control algorithm. (a) Time series of $x_n$. Control is initiated at $t=0$ about unstable fixed point $x^{*}$ (horizontal solid line). Dashed lines show the tolerance range $\pm \epsilon$ within which OGY algorithm is active. Here, the algorithm actives at $t=24$. (b) Control parameter ${\lambda }_n$, as adjusted by OGY algorithm to stabilize fixed point.

Figure 32

Light beams and pupil sizes. Light beam is shaded light; pupil is shaded dark. (a) The ordinary case: the beam is larger than the pupil. (b) Breaking the feedback loop: the beam is smaller than the pupil. (c) Inducing pupil oscillations: a small beam falls on the edge of the pupil.

Figure 33

(Color in online edition) Dynamics of a genetic toggle switch. (a) Dynamics in the $u\text{−}v$ concentration plane (arrows), with gray scale representing the magnitude of the vector field. (b) Equivalent dynamics on center manifold. The three equilibrium positions are represented by the two minima and the unstable intermediate state.