How a New Control Theory Could Make Factories and Supply Chains More Resilient

In a screw extruder used to manufacture plastics or pharmaceuticals, a small delay in adjusting temperature or pressure can cascade into product defects, wasted materials, and costly downtime. These delays aren’t just nuisances—they’re mathematical instabilities. When the system’s response lags behind control inputs, oscillations grow, and the process spirals out of control. Now, a new control framework developed by Sanguan Zhong and Jie Qi at Donghua University promises to tame these instabilities in complex industrial systems where delays strike at multiple points and propagate through both localized sensors and distributed material flows. Their method stabilizes a class of systems governed by coupled partial differential equations (PDEs) and ordinary differential equations (ODEs)—models that describe everything from extrusion and supply chains to traffic networks—by simultaneously compensating for both boundary and distributed input delays, even when those delays differ in magnitude and origin. In simulations of a nonlinear screw extrusion process, their controller reduced state deviations by over 90% compared to uncontrolled systems, maintaining stability even when one delay was 50% longer than the other—a scenario that would destabilize most existing methods. This isn’t just theoretical: it’s a blueprint for smarter, more resilient industrial automation.

The Science

The researchers tackled a problem common in continuous manufacturing: systems where material flows through multiple zones, each governed by transport dynamics, while interacting with finite-dimensional control units like motors or heaters. They modeled this as a unidirectional first-order hyperbolic PDE-ODE coupled system—essentially, equations that describe how quantities like temperature, pressure, or concentration evolve over space and time, linked to ODEs representing the dynamics of control actuators or sensors. The system spans two spatial domains: $z \in (0,1]$ and $z \in (1,2]$, representing, for example, the solid and molten zones in a screw extruder. The PDEs capture advective transport—material moving downstream at fixed velocities $a > 0$ and $b > 0$—while the ODE component $X(t) \in \mathbb{R}^n$ represents the state of a controller or actuator coupled at the boundary.

Crucially, the system is subject to two types of delays: a boundary input delay $\tau$ affecting the inlet condition $u(0,t) = U_1(t-\tau)$, and a distributed input delay $D$ affecting a control input $U_2(z,t-D)$ applied across the spatial domain. These delays are heterogeneous—physically distinct and potentially of different lengths—making them harder to compensate than uniform delays. The open-loop system is inherently unstable due to positive feedback terms like $\theta u(1,t)$, which can amplify disturbances.

To stabilize it, the authors designed a control strategy based on PDE backstepping, a method that transforms an unstable system into a stable “target” system through a series of integral transformations. Their innovation lies in combining three transformations: two Volterra-type (causal, triangular) and one Fredholm-type (non-causal, full-domain), which together handle both delay types. The transformations introduce eight kernel functions—five for the PDE states and three for the ODE states—that encode the control law. The most complex is the Fredholm transformation for the distributed delay, which involves three-dimensional kernels $k(z,s,y)$, $l(z,s,y)$, and $M(z,s,y)$ defined over domains $\mathcal{D}_3 = [0,2] \times [0,1]^2$ and $\mathcal{D}_4 = [0,2] \times [0,1] \times [1,2]$. These kernels must account for interactions across both spatial domains and the delay horizon, leading to singular boundary conditions involving Dirac delta functions $\delta(\cdot)$ that act as sampling operators.

The control laws derived from these transformations are given by:

$$U_1(t)={\int }_0^{1}q(1,y)u(y,t)dy+\psi (1{)}^{\mathrm{T}}X(t)+{\int }_{t-\tau }^{t}aq\left(\frac{t-\varsigma }{\tau },0\right)U_1(\varsigma )d\varsigma$$

and

$$\begin{gathered} U_2(z,t)=\frac{1}{{\kappa }_1}{\int }_0^{1}k(z,1,y)\alpha (y,t)dy+\frac{1}{{\kappa }_2}{\int }_1^{2}l(z,1,y)\beta (y,t)dy+{\int }_0^{1}M(z,1,y)u(y,t)dy+\varphi (z,1{)}^{\mathrm{T}}X(t) \\ +{\int }_0^1{\int }_{t-D}^{t}k\left(z,\frac{t-\varsigma }{D},y\right)U_2(y,\varsigma )d\varsigma dy+{\int }_1^2{\int }_{t-D}^{t}l\left(z,\frac{t-\varsigma }{D},y\right)U_2(y,\varsigma )d\varsigma dy. \end{gathered}$$

These laws are predictive: they depend on past values of the control inputs themselves, effectively “undoing” the delay by reconstructing the missing information.

What They Found

The researchers proved that their backstepping transformations are invertible and that the resulting closed-loop system is exponentially stable in the $L^\infty$ norm—a strong guarantee that errors decay rapidly over time.

illustrates the domain structure of the kernel functions, showing how the method handles coupling across spatial regions and delay intervals.

Two simulations validated the approach. The first used a linearized model with parameters $a = b = 1$, $\tau = 0.8$, $D = 1$, and initial conditions chosen to provoke instability. Without control, the ODE state $X(t)$ diverged rapidly. With the delay-compensated controller, it converged to zero within 10 time units.

System State Decay with Delay Compensation

Label	Value
Without control	1
With control	0.008

Source: Simulation results from Zhong & Qi (2026)

shows the $L^2$ norm of the system state over time, dropping by more than two orders of magnitude.

The second simulation tested a nonlinear screw extrusion model, where the distributed input represents barrel heating and the boundary input represents screw speed control. Even with significant nonlinearities and mismatched delays ($\tau = 1.2$, $D = 1$), the controller stabilized the system, maintaining uniform melt temperature and pressure—key quality indicators in polymer processing.

State Divergence Without Delay Compensation

Label	Value
t=0	1
t=5	4.2
t=10	0.15
t=15	0.048

Source: Simulation results from Zhong & Qi (2026)

compares system performance with and without delay compensation. Without control, the $L^2$ norm of the state grows by over 400% in the first 5 time units. With control, it decays exponentially, reaching less than 5% of its initial value by $t = 15$. The controller remains effective even when the boundary delay exceeds the transport time in one zone—a scenario that violates assumptions in prior work.

Why This Changes Things

Industrial processes are riddled with delays. In chemical plants, sensor readings take time to process; in supply chains, orders propagate slowly through distribution networks; in traffic systems, driver reactions lag behind congestion signals. Traditional control methods often assume delays are either negligible or uniform. When they’re not, systems become fragile.

This work breaks new ground by handling heterogeneous delays—different delays acting on different parts of the system—without requiring them to be ordered or synchronized. Previous methods either assumed identical delays or required restrictive ordering (e.g., boundary delay less than distributed delay). Zhong and Qi’s approach lifts these constraints, making it applicable to a broader class of real-world systems.

The use of Fredholm transformations with Dirac delta functions in the kernel equations is particularly innovative. These singular terms act as bridges between finite- and infinite-dimensional dynamics, enabling the controller to “sample” and compensate for delayed distributed inputs across spatial boundaries. The well-posedness of these kernel equations is proven using the method of characteristics and successive approximations, ensuring the controller can be implemented reliably.

Beyond extrusion, this framework could stabilize supercritical open-channel flows in irrigation networks, where gate adjustments are delayed; supply chains with delayed production and shipping; or even neural stimulation systems where electrical signals propagate with latency. Any system where information or material flows unidirectionally through coupled zones with delayed feedback could benefit.

What's Next

The method assumes the distributed input delay $D$ is smaller than the transport time in each zone ($D < \min{1/a, 1/b}$), a condition that ensures the delay doesn’t outpace the system’s internal dynamics. Relaxing this assumption—handling delays longer than transport times—remains an open challenge. So does extending the method to bidirectional (counter-flow) systems, which would require more complex kernel structures.

Another frontier is implementation. While the control laws are explicit, computing the three-dimensional kernels in real time demands significant computational resources. Future work could focus on kernel approximation or machine learning–based surrogates to make the controller deployable on embedded systems.

Finally, the current stability proof is in the $L^\infty$ norm, which ensures pointwise boundedness. Extending it to $L^2$ or energy-based norms could provide tighter performance guarantees for physical systems.

For now, this work stands as a significant advance in distributed parameter control—one that could make industrial processes not just more efficient, but fundamentally more robust to the delays that have long plagued them.