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Mathematical Challenges of Controlling Large-scale Complex Systems

By Paul Davis

An airplane’s environmental control system is a complex assembly of components having a broad remit. Passengers should be kept at a Goldilocks temperature — not too hot and not too cold. The plane’s electronics must not overheat, and beverages in the galley fridge should be chilled and ready to serve. Identifying an appropriate control architecture for aircraft is but one of the problems described by Andy Sparks of United Technologies in an invited address at the 2017 SIAM Conference on Control and Its Applications, held in Pittsburgh, Pa., this July.

The United Technologies corporate family manufactures significant and complex pieces of modern infrastructure, notably Pratt & Whitney aircraft engines (see Figure 1), Otis elevators, Carrier heating and cooling systems, and multiple sub-assemblies for aircraft. As the Group Leader in Control Systems at the United Technologies Research Center (UTRC), Sparks guides his team in finding better ways to control complicated systems throughout United Technologies’ portfolio.

Figure 1. Aerospace is but one source of problems in estimation and control within the United Technologies family. Image courtesy of United Technologies Research Center.

Sparks emphasized that none of his team’s target systems are “complex” in the sense of being capable of adaptation or self-organization. Rather, they are large, complicated assemblies that must be engineered to perform specified functions efficiently, safely, and reliably. He and his colleagues develop tools to design, control, and manage these systems and their intricate interactions.

From a control perspective, these complex products are high-dimensional. Hundreds or even thousands of components must work together seamlessly, and the couplings among them are dynamic and complex. Furthermore, each of these products encompasses multiple specialized technical domains, including—but hardly limited to—energy flow, mechanics, electronics of all sorts, thermodynamics, and fluids. To reinforce the depth of this complexity, Sparks observed that many more countries are capable of building a nuclear weapon than are capable of producing a gas turbine aircraft engine!

Control engineers need to model uncertainty. Safety is critical; e.g., elevators are fitted with multiple fail-safe systems to counter the full range of potential internal and external failures that could cause one to fall. Operational limits and stability margins can demand predictive control as well.

In the context of environmental control systems in passenger aircraft, Sparks described two specific challenges his team faces: identifying a control architecture for a given system and selecting gain variables to use in the controller. Topics left for another day include controller cost, estimation, fault detection and diagnosis, computation, communication, security — both cyber and physical, and verification and validation. 

These environmental systems are composites of components — air coolers and circulators for the passenger cabin, refrigeration in the galley, thermal protection for some of the avionics, etc. Since the components do not comprise an organic whole designed from scratch, the overall environmental system controller is commonly an assembly of single-loop, hand-tuned systems.

In an ideal world, a sophisticated multivariable controller would likely outperform such a piecemeal mix. However, the sophisticated option would increase the burdens of system design, cost, and service. For example, a customer might be persuaded that such added investments would pay for themselves in fuel savings over the life of an expensive and highly sophisticated aircraft engine, and those engines do indeed employ state-of-the-art, multivariable controllers. For cabin cooling, however, the perfect technical solution is economically infeasible.

So market forces, abetted by aircraft engineers’ well-founded, risk-averse instincts, combine to argue for a synthesis of the tried and true—typically individual proportional-integral controllers—to manage the compressor(s) and circulators that maintain temperatures throughout an aircraft’s cabin, in its galley, and within its electronics. Given those realities, Sparks and his colleagues have developed tools for selecting a sparse control architecture that is “efficient, scalable, and robust.” That process involved both direct collaboration with the engineers responsible for the product and publication in the open literature [1, 2].

Figure 2 illustrates four possible configurations for a candidate environmental system controller: single-input single-output, block diagonal, sparse, and multiple-input multiple-output (MIMO). Configuring an effective sparse controller requires a rigorous way to identify the variables requiring feedback. Given those variables, graph theory can guide selection of the system architecture. Actual design of the controller completes the job.

Figure 2. Four possible configurations for an aircraft’s environmental system controller: single-input single-output, block diagonal, sparse, and multiple-input multiple-output (MIMO). Image courtesy of United Technologies Research Center.

The framework for variable selection is a minimization problem with two principal terms. One is a traditional closed-loop control objective function. The other is a weighted “sparsity-promoting term;” e.g., the number of variables in the feedback term. The augmented Lagrangian of that problem is non-convex — the team used an alternating direction method of multipliers to solve the minimization. For the aircraft environmental control problem, this approach began with nine control loops and found six to be optimal. Those six had the same structure as the relative gain array for the corresponding MIMO system, but without the overhead of the extra variables.

In another application—a 39-bus, 10-generator model of the New England electric power distribution grid—the control objective is to maintain phase angle and frequency throughout the network. Attempting to suppress oscillations using sparse output feedback revealed an explicit trade-off between system performance and the sparsity of the controller. Similar analysis of a toy system containing a hundred or so connected spring-mass oscillators confirmed intuition about sparse control: feedback control must know the behavior of both the local mass and nearby neighbors.

In response to an audience question about redundancy and fault tolerance in a sparse control system, Sparks explained another virtue of the sparsity analysis; in essence, it identifies those components requiring redundancy. Redundancy can be added where it matters.

One of the team’s primary current concerns is verification and validation, which requires exhaustive Monte Carlo simulation. The time demanded by these simulations presents a significant barrier to the implementation of new systems. Considerable research is needed to develop more informed, faster random simulations.

When flying in the temperature-controlled comfort of a turbine-engined aircraft to your next SIAM meeting or riding an elevator inside the massive conference hotel, think about the control engineers who coordinated the complexity surrounding you. They are hardly sparse among the SIAM community!

Sparks’ presentation is available from SIAM either as slides with synchronized audio, or as a PDF of slides only.

Acknowledgments: My thanks to Andy Sparks and Bob LaBarre, Principal Mathematician and Associate Director, Systems in the System Dynamics and Optimization group at UTRC, for helpful conversations about the ways in which the varying needs and expectations of different business units, markets, and products inform scientific and engineering decisions.

[1] Lin, F., & Adetola, V. (2017). Co-design of sparse output feedback and row/column-sparse output matrix. In 2017 American Control Conference (ACC) (pp. 4359-4364). Seattle, WA.
[2] Lin, F., & Adetola, V. (2017). Sparse Output Feedback Synthesis via Proximal Alternating Linearization Method. Preprint, arXiv:1706.08191.

Paul Davis is professor emeritus of mathematical sciences at Worcester Polytechnic Institute.