# Towards Model-based Automatic Vehicle Transmissions

By Paul Davis

Volker Mehrmann of the Technische Universität Berlin received the W.T. and Idalia Reid Prize at the 2018 SIAM Annual Meeting, which took place in Portland, Ore., this July. The award recognized Mehrmann “for his fundamental contributions to the broad area of differential-algebraic equations (DAEs), their control and optimization, and their practical applications.”

Early in his lecture, Mehrmann displayed a photo of William T. Reid’s well-known 1972 monograph titled Riccati Differential Equations, which he first encountered as a postdoctoral researcher in 1984. Mehrmann’s overview of his own prize-winning work with DAEs hewed to the spirit of Reid’s classic publication: the search for mathematically rigorous and tractable approaches to increasingly broad families of optimal control problems.

DAEs—a generalization of ordinary differential equations—arise in many aspects of control. A standard tool for modeling multiphysics systems, researchers use them to express conservation laws. “Black-box” or automatic system modeling also generates DAEs that impose constraints at the interfaces of communication between coupled systems or solvers. Mehrmann observed that this widespread “black-box” DAE modeling approach overlooks all of the difficulties in numerical methods and control techniques, thus causing substantial challenges in numerical integration methods, stability analysis, scale discrepancies, consistent initialization, etc. These complications highlight the need for improved DAE modeling, simulation, and control procedures.

To set the stage for his talk’s punchline—the power of the so-called port-Hamiltonian formulation—Mehrmann described an industrial problem: development of a controller for a novel automatic transmission via a model-based approach (see Figure 1). This research project, which was sponsored by German automotive corporation Daimler AG, involved nine Ph.D. candidates, including Mehrmann’s former student Peter Hamann [1].

**Figure 1.** Views of a half-toroid automatic transmission. Traction (transmission) fluid and multiphysics modeling challenges are apparent. Image courtesy of [2].

Challenges included multiphysics modeling (e.g., multi-body systems, hydraulics, and the dynamics of the traction (or transmission) fluid as essential features), real-time simulation and control, modeling as a network of subcomponents, and model reduction. Ultimately, Daimler AG strove for improved fuel economy, smoother shifting on the road, and looser manufacturing tolerances on the production line.

These modeling requirements and the system’s overall complexity naturally led to an optimal control problem for a DAE formulation. For analysis purposes and enhanced formulation for simulation and control, researchers typically group control inputs, system state variables, and selected derivatives into one large “state” vector. They then extract an improved (regularized) model from an array of equations and some derivatives.

Using this framework, Mehrmann and his colleagues establish conditions sufficient for the local existence of a so-called regularized system: a pair of implicit equation systems. One set describes the modeled dynamics while the other captures all of the algebraic constraints, the solution manifold’s definition, and the consistency requirements for the initial conditions. The corresponding optimal control problem seeks the minimum of a standard cost functional involving the state and control, subject to DAE constraints presumed to exist in such a regularized form.

However, neither the classic results from Reid’s monograph (the optimal feedback control is lurking in a Riccati equation’s solution) nor Pontryagin’s maximum principle (the optimal control maximizes a Hamiltonian) apply directly; the DAEs may not be uniquely solvable, and researchers must constrain the solution variable to a locally constructed manifold.

Adding to these mathematical difficulties, the pure model-based approach to Daimler’s transmission control problem did not work as anticipated; e.g., modeling the traction fluid required a completely different timescale from the rest of the system. Researchers derived the controllers for the actual transmissions now in dealers’ showrooms by taking measurements from a prototype and realizing a model of the input-output behavior via system identification — an approach that is classical but time-consuming and expensive (due to the need for a prototype).

Nonetheless, when one door closes, another door—or perhaps more fittingly, “port,” as in “port-Hamiltonian,”—opens. Conceptually, a port-Hamiltonian formulation begins with a network of elements that can store (a capacitor in an electric network), dissipate (a resistor), or transfer (the network’s topology) energy. A port is a point where energy enters or exits the system. The Hamiltonian records the distribution of energy among the energy-storing components.

A dissipation inequality replaces the energy conservation requirement found in classical Hamiltonian systems. Port-Hamiltonian systems are closed under power-conserving interconnections; an assembly of port-Hamiltonian systems is itself port-Hamiltonian, and the structure is invariant under Galerkin discretization and model reduction.

These apparently modest ideas extend powerfully, even to infinite-dimensional spaces, to enable enormously complicated models of control systems that connect vastly different physical domains and various “black-box” simulators. Naturally, a potpourri of well-structured DAEs—some describing system behavior and some constraining it—can result from a port-Hamiltonian formulation.

Since port-Hamiltonian-DAE systems are ideal for energy-based, modular modeling, Mehrmann and his colleagues developed linear and nonlinear formulations. Their results prove a dissipation inequality as well as the power-conserving connections that enable modeling via interconnected modules. Furthermore, these representations are very robust to structural perturbations.

Mehrmann and his team’s current work is enriching the mathematical and computational infrastructure that supports port-Hamiltonian-DAE systems: modeling, data acquisition, mathematical analysis, numerical simulation, and optimal control techniques based on this structure. Thanks to these tools, a model-based, software-controlled transmission may not be too far down the road.

*Mehrmann’s Reid Prize lecture is available from SIAM as slides with synchronized audio or a PDF of slides only.*

*The W.T. and Idalia Reid Prize was established in 1994 and is awarded in the broadly-defined areas of differential equations and control theory. Since 2000, SIAM has presented the prize annually with support from a bequest from Idalia Reid in memory of her late husband, William T. Reid. The award includes a $10,000 monetary prize and an engraved medal. W.T. Reid worked in differential equations, the calculus of variations, and optimal control, sharing naming rights for the workhorse Gronwell-Reid-Bellman inequality. He held faculty appointments at the University of Chicago, Northwestern University, the University of Iowa, the University of Oklahoma, and the University of Texas. Reid was an important figure in the optimal control community and a beloved mentor to his students.*^{1}

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^{1} John Burns, a student of Reid and the 2010 recipient of the W.T. and Idalia Reid Prize, provided a personal account entitled “William T. and Idalia Reid: His Mathematics and Her Mathematical Family” as his Reid Lecture. A PDF of his lecture and slides with synchronized audio are available from SIAM.

**References**

[1] Hamann, P. (2009). *Analyse regelungstechnischer Methoden zur modellbasierten Regelung von Toroidgetrieben/Analysis of control methods for the model-based control of toroid transmissions*. (Doctoral dissertation). Technische Universität Berlin, Berlin, Germany.

[2] Nissan Motor Company, Ltd. (2000). Extroid CVT: For Application to Rear-Wheel-Drive Cars Powered by Large Engines. *Nissan’s CVT Technologies*.

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