= Delay stability of proportional integrator control
{title2=$k\tau<\pi/2$}
For the constant-coefficient model $x'(t)=-kx(t-\tau)$ with $k>0$ and $\tau\ge0$, the zero <equilibrium point> is <asymptotically stable> if and only if $k\tau<\pi/2$. For positive $\tau$, perturbations are measured by the <supremum norm> of the <initial history of delayed feedback>; in the stable regime, every continuous history produces a solution tending to zero. At equality, nondecaying <sinusoids> prevent <asymptotic stability>. For $k\tau>\pi/2$, growing oscillatory solutions exist. These statements concern this ideal <linear time-invariant system>, rather than an arbitrary <controlled plant>. The corresponding linear equation and its threshold are developed in https://math1um.github.io/Teaching/cain-reynolds-odes.pdf#page=179[Cain and Reynolds, section 6.4, pages 174–177].
Substitute the <exponential function> $x(t)=e^{st}$, allowing a <complex number> $s$. The <characteristic equation> is
$$
s+k e^{-s\tau}=0.
$$
For positive <feedback delay> this is not a <characteristic polynomial>: it has infinitely many roots. A root's <real part> determines growth or decay of its exponential mode, and its <imaginary part> determines the <angular frequency>. To find the first boundary, set $s=i\omega$ with $\omega>0$ and use <Euler's formula>:
$$
k\cos(\omega\tau)=0,\qquad \omega-k\sin(\omega\tau)=0.
$$
The smallest positive crossing is $\omega=k$ and $\omega\tau=\pi/2$. Thus at $\tau_c=\pi/(2k)$ the <complex conjugate> roots are $s=\pm ik$. In fact, with the matching <initial history of delayed feedback>,
$$
x(t)=A\cos(kt)+B\sin(kt)
$$
solves the <delay differential equation> exactly: a quarter-cycle <feedback delay> turns the past displacement into precisely the current <derivative>. Its cycle lasts $2\pi/k=4\tau_c$. The amplitude is arbitrary; this linear model does not select a finite-amplitude oscillation.
The roots cross toward growth as <feedback delay> increases. Differentiating the <characteristic equation> at fixed $k$ gives
$$
\frac{ds}{d\tau}=-\frac{s^2}{1+s\tau},\qquad
\operatorname{Re}\left(\frac{ds}{d\tau}\right)_{s=ik,\,\tau=\tau_c}
=\frac{k^2}{1+(\pi/2)^2}>0.
$$
This calculation identifies the direction of the first destabilization. The full threshold also requires ruling out earlier unstable roots: if $s=\alpha+i\beta$ with $\alpha\ge0$ and $\beta>0$, the <characteristic equation> implies $\cos(\beta\tau)\le0$ and $0<\beta\le k$, hence $k\tau\ge\beta\tau\ge\pi/2$. A real nonnegative root is impossible because $s+k e^{-s\tau}>0$ there. This supplies the geometric reason the threshold is a quarter-cycle rather than an arbitrary fitting constant.
\Image[/feedback-delay-gain-stability.png]
{title=<Feedback delay> separates decay, persistent oscillation, and growth}
{description=Left: numerical solutions with the same constant <initial history of delayed feedback> $x=1$ on $[-\tau,0]$, plotted against $q=t/\tau$ for $k\tau=0.8$, $\pi/2$, and $2$. The middle trajectory includes a transient because this history is not a pure <sinusoid>; the dashed curve is a separate exact solution with history $\cos(\pi q/2)$. Each row has its own vertical scale. Right: the exact boundary $k=\pi/(2\tau)$ for positive <feedback gain> and <feedback delay>. Toy responses are not measured data. The generator uses the <classical fourth-order Runge-Kutta method> with <linear interpolation> of delayed values and $\Delta q=1/2000$, checked against a separately evaluated exact finite sum and two coarser steps.}
The tradeoff is concrete. Without <feedback delay>, increasing $k$ shortens the <exponential decay> <time constant> $1/k$. With known $\tau>0$, the same <proportional control> must satisfy $k<\pi/(2\tau)$. For $\tau=0.2\,\mathrm{s}$, the threshold is $k\approx7.854\,\mathrm{s}^{-1}$: $k=5\,\mathrm{s}^{-1}$ is stable and $k=8\,\mathrm{s}^{-1}$ is unstable. A design close to the boundary leaves little room for additional <feedback delay>; reducing $k$ or reducing measurement latency increases that room. The zero-delay <time constant> is only a comparison scale, not the decay time of the delayed response. Frank discusses the gain–delay compromise in https://stevefrank.org/control/controlBook.pdf#page=101[Control Theory Tutorial, section 13.4 and figure 13.3].
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