2026 Theses Doctoral

Stable Inversion and Predictive Control of Nonminimum-Phase Discrete-Time Systems Using Second-Order Holds

Fritch, Joseph C.

High-precision trajectory tracking in flexible aerospace systems is limited not only by feedback bandwidth, lightly damped structural dynamics, and actuator constraints, but also by the sampled-data representation used in digital implementation. Although feedforward inverse control provides a mechanism for accurate trajectory generation, discretization of continuous-time systems with pole excess can introduce nonminimum-phase sampled-data zeros that render classical discrete-time inverses unstable or physically unrealistic. Building on prior generalized-hold results showing that hold choice can influence sampled-data zero structure and inverse behavior, this dissertation argues that the specific second-order-hold representation used to pose the inverse and predictive-control problems materially affects the resulting finite-horizon sampled-data operator.

This dissertation develops a hold-aware finite-horizon framework for inverse and predictive control based specifically on lifted sampled-data formulations induced by second-order hold (SOH) models. The work begins by establishing the classical sampled-data background, including zero-order hold discretization, discretization-induced zeros, lifted Toeplitz input-output operators, and the connection between unstable sampled-data inverse behavior and the small singular values of finite-horizon lifted maps. Within this framework, stable inversion is interpreted not only through transfer-function zero locations but also through rank, conditioning, and singular-value structure of the sampled-data operator.

A central contribution of the dissertation is the development of exact sampled-data and lifted input-output formulations for continuity-preserving second-order holds. Both backward and forward SOH models are derived from a common quadratic moment-integral framework, together with an augmented matrix-exponential construction that remains valid for singular continuous-time state matrices. The backward SOH model provides a causal sampled-data representation suitable for prediction and implementation, while the forward SOH model provides a preview-based representation suited to offline inverse design. These formulations show that, for the SOH constructions studied here, the hold affects more than intersample smoothness: it changes the sampled-data operator itself and therefore changes the structure, conditioning, and implementation of the finite-horizon inverse problem.

The dissertation further analyzes the asymptotic discretization-zero structure of the forward SOH formulation and compares it with the classical zero-order hold case. This comparison shows that the forward-SOH formulation produces a distinct sampled-data zero family, while the finite-horizon interpretation demonstrates that practical inverse behavior in the examples depends jointly on sampled-data zero locations and on the lifted operator induced by the hold model. These ideas are illustrated numerically using a third-order robot-link model and a fourth-order flexible-spacecraft model. In these examples, SOH improves either finite-horizon inverse solvability or continuous-time intersample tracking performance relative to zero-order hold.

The same sampled-data viewpoint is then extended beyond offline inversion. A generalized predictive control formulation is developed directly from the causal backward-SOH model, yielding a receding-horizon controller whose predictor reflects a richer intersample input representation than conventional zero-order hold. A two-stage tuning study on the flexible-spacecraft model shows that, after startup-history effects decay, the SOH-based predictor provides substantially improved continuous-time tracking relative to a comparable zero-order-hold-based controller.

The dissertation also extends the lifted inverse viewpoint to nonminimum-phase multi-input, multi-output systems. A block Toeplitz formulation is developed using matrix-valued Markov parameters, and a reduced addressed-sample inverse is introduced to improve conditioning by underspecification. Explicit actuator magnitude and slew-rate constraints are then incorporated through semidefinite programming, producing a constrained finite-horizon inverse design suitable for high-order flexible structures.

Finally, the practical realizability of the proposed framework is validated experimentally. A third-order RC hardware plant is identified experimentally, discretized using the forward-SOH formulation, and controlled through a finite-horizon lifted inverse implemented on an embedded microcontroller. Although the sampled-data model remains nonminimum phase, the lifted inverse is shown to be numerically well posed and physically realizable, with observed tracking errors dominated by quantization and implementation limits rather than by inverse instability.

Taken together, the results reinforce the sampled-data viewpoint that the hold model is not merely a reconstruction detail in digital control. In the SOH framework developed here, it is part of the mathematical definition of the finite-horizon inverse and predictive-control problems themselves. By developing exact SOH sampled-data models, finite-horizon lifted inverse formulations, predictive-control extensions, constrained MIMO designs, and experimental validation, this dissertation advances the theory and practical application of trajectory generation and control for flexible systems operating under digital implementation.