Sean Warnick

We show that two widely accepted model reduction techniques, Balanced Truncation and Balanced Singular Perturbation Approximation, can be derived as limiting approximations of a carefully constructed parameterization of Linear Time Invariant (LTI) systems by employing the Model Boundary Approximation Method (MBAM), a recent development in the Physics literature. This unifying framework of these popular model reduction techniques shows that Balanced Truncation and Balanced Singular Perturbation Approximation each correspond to a particular boundary point on a manifold, the "model manifold," which is associated with the specific choice of model parameterization and initial condition, and is embedded in a sample space of measured outputs, which can be chosen arbitrarily, provided that the number of samples exceeds the number of parameters. We also show that MBAM provides a novel way to interpolate between Balanced Truncation and Balanced Singular Perturbation Approximation, by exploring the set of approximations on the boundary of the manifold between the elements that correspond to the two model reduction techniques; this allows for alternative approximations of a given system to be found that may be better under certain conditions. The work herein suggests similar types of approximations may be obtainable in topologically similar places (i.e. on certain boundaries) on the model manifold of nonlinear systems if analogous parameterizations can be achieved, therefore extending these widely accepted model reduction techniques to nonlinear systems.

Performance of integrated production scheduling and advanced process control with disturbances is summarized and reviewed with four progressive stages of scheduling and control integration and responsiveness to disturbances: open-loop segregated scheduling and control, closed-loop segregated scheduling and control, open-loop scheduling with consideration of process dynamics, and closed-loop integrated scheduling and control responsive to process disturbances and market fluctuations. Progressive economic benefit from dynamic rescheduling and integrating scheduling and control is shown on a continuously stirred tank reactor (CSTR) benchmark application in closed-loop simulations over 24 h. A fixed horizon integrated scheduling and control formulation for multi-product, continuous chemical processes is utilized, in which nonlinear model predictive control (NMPC) and continuous-time scheduling are combined.

A novel formulation for combined scheduling and control of multi-product, continuous chemical processes is introduced in which nonlinear model predictive control (NMPC) and noncyclic continuous-time scheduling are efficiently combined. A decomposition into nonlinear programming (NLP) dynamic optimization problems and mixed-integer linear programming (MILP) problems, without iterative alternation, allows for computationally light solution. An iterative method is introduced to determine the number of production slots for a noncyclic schedule during a prediction horizon. A filter method is introduced to reduce the number of MILP problems required. The formulation’s closed-loop performance with both process disturbances and updated market conditions is demonstrated through multiple scenarios on a benchmark continuously stirred tank reactor (CSTR) application with fluctuations in market demand and price for multiple products. Economic performance surpasses cyclic scheduling in all scenarios presented. Computational performance is sufficiently light to enable online operation in a dual-loop feedback structure.

In this paper we study state-space realizations of Linear and Time-Invariant (LTI) systems. Motivated by biochemical reaction networks, Gon\c{c}alves and Warnick have recently introduced the notion of a {\em Dynamical Structure Functions} (DSF), a particular factorization of the system's transfer function matrix that elucidates the interconnection structure in dependencies between manifest variables. We build onto this work by showing an intrinsic connection between a DSF and certain sparse left coprime factorizations. By establishing this link, we provide an interesting systems theoretic interpretation of sparsity patterns of coprime factors. In particular we show how the sparsity of these coprime factors allows for a given LTI system to be implemented as a network of LTI sub-systems. We examine possible applications in distributed control such as the design of a LTI controller that can be implemented over a network with a pre-specified topology.

Network reconstruction, i.e., obtaining network structure from data, is a central theme in systems biology, economics and engineering. In some previous work, we introduced dynamical structure functions as a tool for posing and solving the problem of network reconstruction between measured states. While recovering the network structure between hidden states is not possible since they are not measured, in many situations it is important to estimate the minimal number of hidden states in order to understand the complexity of the network under investigation and help identify potential targets for measurements. Estimating the minimal number of hidden states is also crucial to obtain the simplest state-space model that captures the network structure and is coherent with the measured data. This paper characterizes minimal order state-space realizations that are consistent with a given dynamical structure function by exploring properties of dynamical structure functions and developing an algorithm to explicitly obtain such a minimal realization.

This paper presents a new approach to distributed controller design that exploits a partial-structure representation of linear time invariant systems to characterize the structure of a system. This partial-structure representation, called the dynamical structure function, characterizes the {\em signal structure}, or open-loop causal dependencies among manifest variables, capturing a significantly richer notion of structure than the sparsity pattern of the transfer function. The design technique sequentially constructs each link in an arbitrary controller signal structure, and the main result proves that the resulting controller is either stabilizing or no controller with the desired structure can stabilize the system.

Interconnected dynamic systems are a pervasive component of our modern infrastructures. The complexity of such systems can be staggering, which motivates simplified representations for their manipulation and analysis. This work introduces the complete computational structure of a system as a common baseline for comparing different simplified representations. Linear systems are then used as a vehicle for comparing and contrasting distinct partial structure representations. Such representations simplify the description of a system's complete computational structure at various levels of fidelity while retaining a full description of the system's input-output dynamic behavior. Relationships between these various partial structure representations are detailed, and the landscape of new realization, minimality, and model reduction problems introduced by these representations is briefly surveyed.