Invention Disclosure Form

Saviz Mowlavi, Mouhacine Benosman

The invention relates generally to modeling and control of systems governed by partial differential equations (PDEs). Potential applications include HVAC airflow control, thermal cooling of devices, LiDAR airflow reconstruction, robotics, etc. Some relevant patents are MERL-3457 and MERL-3368.

Describe in a few words the technical field of the invention without referencing the invention itself.

The present disclosure relates generally to system modeling, prediction and control, and more particularly to systems and methods of modeling and control of high dimensional physical systems using a polytopic reduced order model.

State problem(s) addressed by your invention. Describe known solutions for the stated problem(s) and their limitations. Identify the relevant publications, patents, and/or patent applications. Do not mention your own invention in this section.

Active control of fluid flows has many potential benefits, from drag reduction in aircrafts and ships to improved efficiency of heating and air-conditioning systems, among many other examples [brunton2015closed]. But real-time feedback control requires inferring the state of the system from sparse measurements using a state estimation algorithm, which typically relies on a model for the underlying dynamics [simon2006optimal]. Among estimation algorithms, the Kalman filter is by far the most well-known thanks to its optimality for linear systems, which has led to its widespread use in numerous applications [kalman1960, zarchan2005progress]. However, systems such as fluid flows are governed by partial differential equations (PDEs) which, when discretized, yield high-dimensional and oftentimes nonlinear dynamical models with hundreds or thousands of state variables. These high-dimensional models are too expensive to integrate with common state estimation techniques, especially in the context of embedded systems. Thus, for control purposes, a common practice is to design state estimators from a reduced-order model (ROM) of the system, in which the underlying dynamics are projected to a low-dimensional subspace that is computationally tractable [barbagallo2009closed, rowley2017]. In particular, recent papers have demonstrated the efficacy of combining a data-driven ROM constructed using dynamic mode decomposition (DMD) with a Kalman filter to estimate unsteady fluid flows using sparse sensor measurements in the absence of model uncertainties [gomez2019data, tsolovikos2020estimation].

The dynamics of physical systems such as fluid flows is oftentimes sensitive to various physical parameters such as wind speed, viscosity, etc. When these parameters are unknown, the accuracy of the estimate is adversely impacted by the uncertainty in the model and, thus, the ROM. To deal with these model uncertainties, an extension of the previously referenced studies was recently proposed, wherein a bank of local DMD models for multiple parameter values was utilized in a multiple model Kalman filter framework, in which the estimate is a weighted average of individual estimates produced by independent Kalman filters running separately for each parameter value [tsolovikos2022multiple]. This latter framework belongs to a broader class of multiple model techniques addressing parameter uncertainties [orjuela2008state, adeniran2016modeling], which also include polytopic models where a single state or estimate is produced by a weighted average of the local models themselves [takagi1985fuzzy, apkarian1995self, angelis2003system, fujimori2006model].

State the primary problem addressed by your invention and your solution to that problem. Describe primary (most essential) and secondary (important) features of the invention. Describe advantages that each feature provides over the conventional solutions.

Please supply at least one top-level figure in a separately uploaded file explaining a primary feature of your solution and/or basic system components or method steps. Powerpoint is the most convenient format.

FIG. 1 is a flow diagram of a method for constructing a polytopic reduced order model of a parametric dynamical system offline, and using the polytopic reduced order model to control the system online, according to according to some embodiments of the present disclosure.

FIG. 2 is a schematic diagram of the architecture of the polytopic reduced order model, according to according to embodiments of the present disclosure.

FIG. 2 is a schematic diagram of the architecture of the polytopic reduced order model 103, according to according to embodiments of the present disclosure. Starting from an initial condition $\mathbf{z}_{0}$ for the high-dimensional state at initial time $0$, the projection operation 108 defines a corresponding reduced state $\mathbf{x}_{0}$. The polytopic DMD model 109 is then used to evolve the reduced state up to an arbitrary time $T$ according to a time sequence of physical parameter values $\{p\}$ and control values $\{\mathbf{u}\}$ from $0$ to $T$, resulting in a predicted reduced state $\mathbf{x}_{T}$ at time $T$. Finally, the lifting operation 110 defines the high-dimensional state $\mathbf{x}_{T}$ corresponding to $\mathbf{x}_{T}$.

FIG. 3 is a schematic diagram illustrating the construction of the polytopic reduced order model, according to embodiments of the present disclosure.

FIG. 3 is a schematic diagram illustrating the construction of the polytopic reduced order model 103, according to embodiments of the present disclosure. Starting from a training dataset 107 of solution trajectories corresponding to $N$ different parameter values, step 301 first computes the global modes defining the projection operation 108 and the lifting operation 110. For each parameter value, step 302 then computes a local DMD model approximating the dynamics of the corresponding trajectory in the training dataset. Then, step 303 computes the polytopic DMD model as a weighted average of the local DMD models obtained in step 302.

FIG. 4 is a block diagram illustrating online control of the operation of the parametric dynamical system using the polytopic reduced order model, according to according to some embodiments of the present disclosure.

Your objective here is to explain your invention to a patent attorney. Invention is a concept, not a thing. At this stage, we encourage you to think conceptually, i.e., specify “what” is done to solve the problem, rather than “how” the embodiment happens to carry it out. After the invention is discussed and defined with help of a patent attorney, you might be asked to provide technical descriptions of your invention, i.e., to describe how various embodiments of your invention work, how to make them, and how to use them.

Claims are optional at this stage, but useful if you have them.

Please list the closest prior art here after performing a thorough patent and literature search. Do not include general background citations, but only prior art that is material to the novelty of your invention. Three to seven items is a rule of thumb. For U.S. Patents and Applications, document numbers are sufficient. For Foreign Patents also cite the country or patent office which issued the patent. For non-patent literature (NPL), give complete citations, and submit document in pdf, doc, or other editable form along with this disclosure. Point out relevant pages, line numbers and/or figures if document is over 10 pages. The Patent Office does not allow cites to web URLs.