Patent Description:
Control theory in control systems engineering is a subfield of mathematics that deals with the control of continuously operating dynamical systems in engineered processes and machines. The objective is to develop a control policy for controlling such systems using a control action in an optimum manner without delay or overshoot and ensuring control stability.

Conventionally, some methods of controlling the system are based on techniques that allow a model-based design framework in which the system dynamics and constraints may directly be considered. Such methods may be used in many applications to control the systems, such as the dynamical systems of various complexities. Examples of such systems may include production lines, car engines, robots, numerically controlled machining, motors, satellites, and power generators.

Further, a model of dynamics of a system or a model of a system describes dynamics of the system using differential equations. However, in a number of situations, the model of the system may be nonlinear and may be difficult to design, to use in real-time, or it may be inaccurate. Examples of such cases are prevalent in certain applications such as robotics, building control, such as heating ventilating and air conditioning (HVAC) systems, smart grids, factory automation, transportation, self-tuning machines, and traffic networks. In addition, even if a nonlinear model may be available, designing an optimal controller for control of the system may essentially be a challenging task.

Moreover, in absence of accurate models of the dynamical systems, some control methods exploit operational data generated by dynamical systems in order to construct feedback control policies that stabilize the system dynamics or embed quantifiable control-relevant performance. Typically, different types of methods of controlling the system that utilize the operational data may be used. In an embodiment, a control method may first construct a model of the system and then leverage the model to design the controllers. However, such methods of control result in a black box design of a control policy that maps a state of the system directly to control commands. However, such a control policy is not designed in consideration of the physics of the system.

In another embodiment, a control method may directly construct control policies from the data without an intermediate model-building step for the system. A drawback of such control methods is potential requirement of large quantities of data in the model-building step. In addition, the controller is computed from an estimated model, e.g., according to a certainty equivalence principle, but in practice the models estimated from the data may not capture the physics of dynamics of the system. Hence, a number of control techniques for the system may not be used with constructed models of the system.

<NPL>), proposes a framework to identify and construct bilinear models and their associated observables from data by simultaneously learning a bilinear Koopman embedding for the underlying unknown nonlinear control system as well as a Control Lyapunov Function (CLF) for the Koopman based bilinear model using a learner and falsifier.

The invention is defined in the appended independent claims. Embodiments of the invention are defined in the appended dependent claims.

In describing embodiments of the disclosure, the following definitions are applicable throughout the present disclosure. A "control system" or a "controller" may be referred to a device or a set of devices to manage, command, direct or regulate the behavior of other devices or systems. The control system can be implemented by either software or hardware and can include one or several modules. The control system, including feedback loops, can be implemented using a microprocessor. The control system can be an embedded system.

An "air-conditioning system" or a heating, ventilating, and air-conditioning (HVAC) system may be referred to a system that uses a vapor compression cycle to move refrigerant through components of the system based on principles of thermodynamics, fluid mechanics, and/or heat transfer. The air-conditioning systems span a very broad set of systems, ranging from systems which supply only outdoor air to the occupants of a building, to systems which only control the temperature of a building, to systems which control the temperature and humidity.

A "central processing unit (CPU)" or a "processor" may be referred to a computer or a component of a computer that reads and executes software instructions. Further, a processor can be "at least one processor" or "one or more than one processor".

<FIG> shows a block diagram 100A of two stages to train a neural network model in an offline stage to be used in an online stage of controlling an operation of a system, according to an embodiment of the present disclosure. The block diagram 100A includes the two stages, such as an offline stage <NUM> and an online stage <NUM>. The block diagram 100A depicts control and estimation of large-scale systems, such as the system having non-linear dynamics represented by partial differential equations (PDEs) using a two-stage apparatus, i.e., the offline stage <NUM> and the online stage <NUM>.

The offline stage <NUM> (or a stage I) may include a neural network model <NUM>. The neural network model <NUM> has an autoencoder architecture. The neural network model <NUM> comprises an autoencoder <NUM> that includes an encoder and a decoder. The neural network model <NUM> further comprises a linear predictor <NUM>. The offline stage <NUM> may further include a computational fluid dynamics (CFD) simulation or experiments module <NUM>, differential equations <NUM> for representation of the non-linear dynamics of the system, a digital representation of time series data <NUM> indicative of and collocation points <NUM>. The online stage <NUM> (or a stage II) may include a data assimilation module <NUM> and a control unit <NUM> to control the system.

In the offline stage <NUM>, an offline task for the control and estimation of the system may be carried out to derive the linear predictor <NUM>. In some embodiments, the linear predictor <NUM> may be based on a reduced-order model. The reduced-order model may be represented by a Koopman operator. Such reduced-order model may be referred as a latent-space model. In general, the dimension of the latent space may be equal, larger or smaller than the input. Details of an architecture of the Koopman operator to represent the linear predictor <NUM> are further provided, for example, in <FIG>.

Typically, the latent-space model may be a nonlinear and a high-dimensional model. The present disclosure enables designing of the latent-space model that conforms to desired properties, such as linearity and being of reduced order. Moreover, data for development of latent-space model may be generated by performing high fidelity CFD simulation and experiments by use of the CFD simulation or experiments module <NUM>.

Generally, the CFD refers to a branch of fluid mechanics that may utilize numerical analysis and data structures to analyze and solve problems that may involve fluid flows. For example, computers may be used to perform calculations required to simulate a free-stream flow of the fluid, and an interaction of the fluid (such as liquids and gases) with surfaces defined by boundary conditions. Further, multiple software have been designed that improves an accuracy and a speed of complex simulation scenarios associated with transonic or turbulent flows that may arise in applications of the system, such as the HVAC applications to describe the airflow in the system. Furthermore, initial validation of such software may typically be performed using apparatus such as wind tunnels. In addition, previously performed analytical or empirical analysis of a particular problem related to the airflow associated with the system may be used for comparison in the CFD simulations.

In some embodiments, the digital representation of the time series data <NUM> is obtained by use of the CFD simulation or experiments module <NUM>. The CFD simulation or experiments module <NUM> may output a dataset, such as the digital representation of the time series data <NUM> that may be utilized to develop the latent-space model (or the linear predictor <NUM>). The latent-space model may be constructed for several trajectories generated by the CFD simulations. The HVAC system may be installed in a room. The room may have various scenarios, such as a window may be open, a door may be closed, and the like. The CFD simulations may be performed for the room where the window is closed, the window is opened, the number of occupants is one, two or multiple, and the like. In such a case, the autoencoder <NUM> may be valid for all such conditions associated with the room. The tasks such as the CFD simulations may be carried in the offline stage <NUM>.

In some embodiments, the collocation points <NUM> associated with a function space of the system, may be generated based on the PDE, the digital representation of time series data <NUM> and a linearly transformed encoded digital representation (such as an output of the linear predictor <NUM>). The neural network model <NUM> may be trained based on the generated collocation points <NUM>. Specifically, the neural network model <NUM> may be trained based on a difference between the prediction of the latent-space model and the dataset such as the digital representation of the time series data <NUM> plus a physics-informed part i.e., the differential equations <NUM> for representation of the non-linear dynamics of the system, which generates the collocation points <NUM>.

Furthermore, an output of the neural network model <NUM> may be utilized by the data assimilation module <NUM> of the online stage <NUM>. The data assimilation module <NUM> may output, for example, reconstructed models of temperature and velocity in an area, such as the room associated with the system, such as the HVAC system. The reconstructed models of temperature and velocity may be utilized by the control unit <NUM>. The control unit <NUM> may generate control commands to control the operations (such as an airflow) of the system, such as the HVAC system.

The data assimilation module <NUM> utilizes a process of data assimilation that refers to assimilation of exact information from sensors with a possibly inexact model information. For example, the room may be installed with the sensors to monitor certain sensory data. Examples of the sensory data, installed within the room for the HVAC applications, include, but may not be limited to, thermocouple reading, thermal camera measurements, velocity sensor data, and humidity sensor data. The information from the sensors may be assimilated by the data assimilation module <NUM>.

Typically, the data assimilation refers to a mathematical discipline that may seek to optimally combine predictions (usually in the form of a numerical model) with observations associated with the system. The data assimilation may be utilized for various goals, for example, to determine an optimal state estimate of the system, to determine initial conditions for a numerical forecast model of the system, to interpolate sparse observation data using knowledge of the system being observed, to identify numerical parameters of a model from observed experimental data, and the like. Depending on the goal, different solution methods may be used.

It may be noted that the offline stage <NUM> and the online stage <NUM> are examples of development of simplified and robust neural network model <NUM>, that in turn may be used for estimation and control of the system having non-linear dynamics by the control unit <NUM>. Typically, the estimation and control of the system involves estimating values of parameters of the linear predictor <NUM> based on measured empirical data that may have a random component. The parameters describe an underlying physical setting in such a way that the value of the parameter may affect distribution of the measured data. Moreover, an estimator, such as the control unit <NUM> attempts to approximate unknown parameters using the measurements. Generally, two approaches are considered for the approximation. A first approach is a probabilistic approach that may assume that the measured data is random with probability distribution dependent on the parameters of interest. A second approach is a set-membership approach that may assume that the measured data vector belongs to a set which depends on the parameter vector. In the present disclosure, the probabilistic approach may be employed for the approximation.

It may be noted that by incorporating knowledge of the physics informed part or the differential equations associated with the system, a need for large training datasets, such as the digital representation of time series data <NUM> for identifying the latent-space model may be reduced. Moreover, since the neural network model <NUM> performs operator learning, it enables the neural network model <NUM> to predict beyond a training horizon, and it may further be used for compressed sensing, estimation, and control of the system.

The linear predictor <NUM> of the neural network model <NUM> may be represented by the Koopman operator. The architecture of the Koopman operator is further described in <FIG>.

<FIG> shows a schematic diagram 100B of architecture of a Koopman operator, according to some embodiments of the present disclosure. The schematic diagram 100B shows the Koopman operator in a finite dimensional space, represented by a matrix K, which induces a finite-dimensional linear system.

The Koopman operator is defined as a foundation of to describe the latent-space model. The Koopman operator may be based on Hamiltonian systems to formulate the Koopman operator in discrete time. In certain cases, a continuous time formulation may be considered to formulate the Koopman operator.

Typically, the Hamiltonian system is a dynamical system governed by Hamilton's equations. Such a dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. Formally, the Hamiltonian system is a dynamical system characterized by a scalar function H(q, p), also known as the Hamiltonian, wherein p and q are generalized coordinates. Further, a state of the system, r, is described by the generalized coordinates p and q, corresponding to generalized momentum and position respectively. Both the generalized coordinates p and q are real-valued vectors with a same dimension N. Thus, the state of the system is completely described by a 2N-dimensional vector r(q, p) and the evolution equations are given by Hamilton's equations as follows: <MAT> <MAT>.

The Hamiltonian system may be utilized to describe the evolution equations of a physical system such as the system with the non-linear dynamics. The advantage of the Hamiltonian system is that it gives important insights into the dynamics of the system, even if the initial value problem may be solved analytically.

In some embodiments the Koopman operator may be based on the continuous-time dynamical system. Considering the continuous-time dynamical system as follows: <MAT> with x ∈ X ⊆ Rn. Further, a time-t flow map operator Ft : X → X is defined as: <MAT>.

Moreover, an alternative description for the dynamical systems in terms of evolution of functions of possible measurements may be given as y = g(x). The function g :X→ R is called a measurement function and may belongs to some set of functions G(x). Generally, the set of functions is often not defined a-priori, and other functions, such as Hilbert spaces or reproducing kernel Hilbert spaces (RKHS) are common choices as the functions. In all cases, however, G(X) is of significantly higher dimension than X, thus, dimensionality may be traded for linearity. Furthermore, the Koopman operator K is an infinite-dimensional linear operator that acts on all observable functions such as to satisfy the following equation: <MAT>.

The equation <NUM> may further be utilized in the dynamical systems with continuous spectra. Thus, a transformation from a state-space representation of the dynamical system to the Koopman representation trades nonlinear, finite-dimensional dynamics for linear, infinite-dimensional dynamics. The advantage of such a trade-off is that the linear differential equations may be solved using the spectral representation. In a practice scenario, a sufficiently large, but finite, sum of modes is used to approximate the Koopman spectral solution.

If the dynamics is sufficiently smooth, an infinitesimal generator L of the Koopman operator family may be defined as: <MAT>.

From the equation <NUM>, following may be observed: <MAT>.

The generator L is sometimes referred to as a Lie operator. For example, the generator L is a Lie derivative of the function g along the vector field f(x) when the dynamics is given by dx/dt=f(x). On the other hand, the following equation is considered: <MAT>.

Based on the equation <NUM>, it may be concluded the following: <MAT>.

Moreover, an applied Koopman analysis seeks key measurement functions that behave linearly in time, and the eigenfunctions of the Koopman operator are functions that exhibit such behavior. A Koopman eigenfunction φ(x) corresponding to an eigenvalue λt satisfies the following equation: <MAT>.

In some embodiments, the Koopman eigenfunctions φ(x) may be demonstrated as eigenfunctions of the Lie operator L, although with a different eigenvalue, i.e., <MAT>.

In such a case, the equation <NUM> may be rewritten as follows: <MAT>.

Equation <NUM> is referred as a dynamical system constraint (DSC) equation. Once a set of eigenfunctions {φ1, φ2, · · · , φM } is obtained, observables that may be formed as a linear combination of the set of eigenfunctions, i.e., <MAT> have a particularly simple evolution under the Koopman operator as follows: <MAT>.

It may be implied from the equation <NUM> that <MAT> is an invariant subspace under the Koopman operator Kt and may be viewed as the new coordinates on which the dynamics of the system evolve linearly.

Since the goal of the disclosure is to is to study nonlinear dynamical systems using linear theory, the function g(x) may be generalized as follows: <MAT> where ψ is an arbitrary transformation parameterized by ω. Equation is referred as an observable reconstruction equation (ORE).

Thus, the Koopman eigenfunctions are an important basis based on which any observable may be expressed, such as the ORE. The Koopman eigen function themselves are given by the DSC. It is observed in the transformation is that the finite-dimensional, non-linear dynamical system defined by the function f and the infinite-dimensional, linear dynamics defined by the Koopman equation are two equivalent representations of the same fundamental behavior. Moreover, the observables g and the associated Koopman mode expansion may be linked successfully to the original evolution defined by the function f. Importantly, the Koopman operator captures everything about the non-linear dynamical system, and the eigenfunctions define a nonlinear change of coordinates in which the system becomes linear.

It may be noted that if the observable functions g is restricted to an invariant subspace spanned by eigenfunctions of the Koopman operator, then it may induce a linear operator K that is finite dimensional and advances the eigen observable functions on this subspace. Such subspace is represented in the <FIG>.

Moreover, asymptotic methods may be used to approximate certain eigenfunctions for simple dynamics (e.g., polynomial nonlinear dynamics), however, there is no analytical procedure to seek for the eigen-pairs of Koopman operator in general. Some computational methods, for example, a dynamic mode decomposition (DMD) technique may be used to approximate eigenfunctions of Koopman operator. Details of the DMD technique are further provided, for example, in <FIG>.

<FIG> illustrates a schematic overview 200A of principles used for controlling the operation of the system, according to some embodiments of the present disclosure. The schematic overview 200A depicts a control apparatus <NUM> and a system <NUM>. The system <NUM> may be the system with the non-linear dynamics. The control apparatus <NUM> may include a linear predictor <NUM>. The linear predictor <NUM> may be same as the linear predictor <NUM> of <FIG>. The control apparatus <NUM> may further include a control unit <NUM> in communication with the linear predictor <NUM>. The control unit <NUM> is analogous to the control unit <NUM> of <FIG>.

The control apparatus <NUM> may be configured to control continuously operating dynamical system, such as the system <NUM> in engineered processes and machines. Hereinafter, 'control apparatus' and 'apparatus' may be used interchangeable and would mean the same. Hereinafter, 'continuously operating dynamical system' and 'system' may be used interchangeably and would mean the same. Unclaimed examples of the system <NUM> include, but may not be limited to, light detection and ranging (LIDAR) systems, condensing units, production lines, self-tuning machines, smart grids, car engines, robots, numerically controlled machining, motors, satellites, power generators, and traffic networks. The control apparatus <NUM> or the control unit <NUM> may be configured to develop control policies, such as the estimation and control commands for controlling the system <NUM> using control actions in an optimum manner without delay or overshoot in the system <NUM> and ensuring control stability.

In some embodiments, the control unit <NUM> may be configured to generate the control commands for controlling the system <NUM> based on at least one of a model-based control and estimation technique or an optimization-based control and estimation technique, for example, a model predictive control (MPC) technique. The model-based control and estimation technique may be advantageous for control of the dynamic systems, such as the system <NUM>. For example, the MPC technique may allow a model-based design framework in which the dynamics of the system <NUM> and constraints may directly be considered. The MPC technique may develop the control commands for controlling the system <NUM>, based on the model of the latent space model or the linear predictor <NUM>. The linear predictor <NUM> of the system <NUM> refers to dynamics of the system <NUM> described using linear differential equations.

In some embodiments, the control unit <NUM> may be configured to generate the control commands for controlling the system <NUM> based on a data-driven based control and estimation technique. The based control and estimation technique may exploit operational data generated by the system <NUM> in order to construct feedback control policy that stabilizes the system <NUM>. For example, each state of the system <NUM> measured during the operation of the system <NUM> may be given as the feedback to control the system <NUM>.

Typically, use of the operational data to design the control policies or the control commands is referred as the data-driven based control and estimation technique. The data-driven based control and estimation technique may be utilized to design the control policy from data and the data-driven control policy may further be used to control the system <NUM>. Moreover, in contrast with such data-driven based control and estimation technique, some embodiments may use operational data to design a model, such as the linear predictor <NUM>. The data-driven model, such as the linear predictor <NUM> may be used to control the system <NUM> using various model-based control methods. Further, the data-driven based control and estimation technique may be utilized to determine actual model of the system <NUM> from data, i.e., such a model that may be used to estimate behavior of the system <NUM> that has non-linear dynamics. In an example, the model of the system <NUM> may be determined from data that may capture dynamics of the system <NUM> using the differential equations. Furthermore, the model having physics based PDE model accuracy may be learned from the operational data.

Moreover, to simplify the computation of model generation, an ordinary linear differential equation (ODE) for the linear predictor <NUM> may be formulated to describe the dynamics of the system <NUM>. In some embodiments, the ODE may be formulated using model reduction techniques. For example, the ODE may be reduced dimensions of the PDE, e.g., using proper orthogonal decomposition and Galerkin projection or DMD. Further, the ODE may be a part of the PDE, e.g., describing the boundary conditions. However, in some embodiments, the ODE may be unable to reproduce actual dynamics (i.e. the dynamics described by the PDE) of the system <NUM>, in cases of uncertainty conditions. Examples of the uncertainty conditions may be a case where boundary conditions of the PDE may be changing over a time or a case where one of coefficients involved in the PDE may be changing.

Further, an example of the data-driven based control and estimation technique is provided in <FIG> illustrates a schematic diagram 200B that depicts an exemplary method to approximate the Koopman operator, according to some embodiments of the present disclosure. In some embodiments, the Koopman operator may be approximated by use of the data-driven approximation technique. The data-driven approximation technique may be generated using numerical or experimental snapshots. For example, a dynamic mode decomposition (DMD) approximation technique may be used as the data-driven approximation technique. The schematic diagram 200B includes snapshots <NUM>, steps of algorithm <NUM>, a set of modes <NUM>, a predictive reconstruction <NUM> and shifted values <NUM> of the snapshots <NUM>.

The DMD approximation technique may be utilized to approximate the Koopman operator for example, of a fluid over a cylinder. The DMD approximation technique is a dimensionality reduction algorithm. Typically, given a time series of the data, the DMD approximation technique computes the set of modes <NUM>. Each mode of the set of modes <NUM> may be associated with a fixed oscillation frequency and a decay or growth rate. For linear systems, the set of modes <NUM> and the fixed oscillation frequency are analogous to normal modes of the system, but more generally, they may be analogous to approximations of the set of modes <NUM> and the eigenvalues of a composition operator (referred as the Koopman operator).

Furthermore, due to intrinsic temporal behaviors associated with each mode of the set of modes <NUM>, the DMD approximation technique differs from other dimensionality reduction methods such as principal component analysis, that may compute orthogonal modes that lack predetermined temporal behaviors. As the set of modes <NUM> are not orthogonal, the DMD approximation technique-based representations may be less parsimonious than those generated by the principal component analysis. However, the DMD approximation technique may be more physically meaningful than the principal component analysis as each mode of the set of modes <NUM> is associated with a damped (or driven) sinusoidal behavior in time.

In the DMD approximation technique, the method to approximate the Koopman operator may start with collection of the snapshots <NUM> (such as images of the CFD simulation and experiments) and the shifted values <NUM> of the snapshots <NUM>. For example, the snapshots <NUM> correspond to the digital representation of the time series data <NUM>. The steps of algorithm <NUM> of the DMD approximation technique are described as follows:.

In the steps of algorithm <NUM>, the singular value decomposition of snapshot matrix is taken, and linear operator is approximated as a best A matrix, which minimizes the following equation: <MAT>.

Optionally, the matrix A may further be reduced by dropping one or more modes of the set of modes <NUM>. The Eigen decomposition of such matrix A may provide the DMD eigen modes depicted in the set of modes <NUM>. The matrix A may be used to reconstruct the data corresponding to the predictive reconstruction <NUM>. The predictive reconstruction <NUM> may be output by the data assimilation module <NUM> of <FIG>. For example, the predictive reconstruction <NUM> may include the data associated with reconstruction of temperature and velocity of the room, for HVAC systems.

Typically, the DMD approximation technique utilizes a computational method to approximate the Koopman operator from the data. Advantageously, the DMD approximation technique possesses a simple formulation in terms of linear regression. Therefore, several methodological innovations have been introduced, for example, a sparsity promoting optimization may be used to identify the set of modes <NUM>, the DMD approximation technique may be accelerated using randomized linear algebra, an extended DMD approximation technique may be utilized to include nonlinear measurements, a higher order DMD that acts on delayed coordinates may be used to generate more complex models of the linear predictor <NUM>, a multiresolution DMD approximation technique with multiscale systems that exhibit transient or intermittent dynamics may be used, and the DMD approximation algorithm may be extended to disambiguate the natural dynamics and actuation of the system. The DMD approximation technique may further include a total least-squares DMD, a forward-backward DMD and variable projection that may improve the performance of DMD over noise sensitivity. Such methods may be utilized in various applications, such as fluid dynamics and heat transfer, epidemiology, neuroscience, finance, plasma physics, robotics and video processing.

In some embodiments, the Koopman operator may be approximated by use of a deep learning technique. In certain scenarios, the DMD approximation technique may be unable to represent the Koopman eigenfunctions. In such cases, the deep learning technique, such as neural network models may be utilized for approximating the Koopman operator, leading to linear embedding of the non-linear dynamics of the system <NUM>. The deep learning technique may be successful in long-term dynamic predictions and the fluid control for the HVAC systems. The deep learning technique may further be extended to account for uncertainties, modeling PDEs and for optimal control of the system <NUM>. Example of architecture of the neural network model, may include, but may not be limited to, neural ODEs for dictionary learning and graphical neural networks utilized for learning the compositional Koopman operators.

An example of the usage of the deep learning technique (or the neural network model) to approximate the Koopman operator is further provided in <FIG>.

<FIG> illustrates a schematic diagram 200C of an autoencoder architecture of the neural network model, according to some embodiments of the present disclosure. The deep neural network model may be utilized to learn linear basis and the Koopman operator using data of the snapshots <NUM>. The schematic diagram 200C includes the autoencoder <NUM>. The autoencoder <NUM> includes an encoder <NUM>, a decoder <NUM> and a linear predictor <NUM>. The linear predictor <NUM> may be same as the linear predictor <NUM> of <FIG>. The schematic diagram 200C further includes a linear predictor <NUM> and a linear predictor <NUM>.

The autoencoder <NUM> may be a special type of neural network model suitable for the HVAC applications. The encoder <NUM> may be represented as "ϕ". The encoder <NUM> learns the representation of the relevant Koopman eigenfunctions, that may provide intrinsic coordinates that linearize the dynamics of the system <NUM>. The decoder <NUM> may be represented as "ψ" or "ϕ-<NUM>". The decoder <NUM> may seek an inverse transformation to reconstruct the original measurements of the dynamics of the system <NUM>. Further, if the encoder <NUM> is defined as ϕ : x→ (φ1(x), φ2(x),. , φM (x))T , then up to a constant, the encoder <NUM> may learn such transformation "ϕ" and the decoder <NUM> may learn the transformation "ψ" as shown in the observable reconstruction equation (ORE), such as in equation <NUM>.

Moreover, within the latent space of the autoencoder <NUM>, such as the linear predictor <NUM>, the dynamics of the system <NUM> is constrained to be linear. Therefore, in some embodiments, a squared matrix "K" is used to drive the evolution of the dynamics of the system <NUM>. Generally, there is no invariant, finite dimensional Koopman subspace that captures the evolution of all the measurements of the system <NUM>, in such a case the squared matrix K may only approximate the true underlying linear operator.

Typically, the autoencoder <NUM> may be trained in a number of ways. Normally, the training dataset X is arranged as a three-dimensional (3D) tensor, with its dimensions to be (i) number of sequences (with different initial states), (ii) number of snapshots, and (iii) dimensionality of the measurements, respectively. Further, the constraint of linear dynamics may be enforced by a loss term resembling ∥ϕ(xn+<NUM>) - Kϕ(xn)∥, that may be represented by the linear predictor <NUM>, or linearity may be enforced over multiple steps resembling ∥ϕ(xn+p) - Kpϕ(xn)∥, that may be represented by the linear predictor <NUM>, generating recurrencies in the neural network architecture or the autoencoder architecture. It should be noted that the linear predictor <NUM> and linear predictor <NUM> are considered as examples of the linear predictor <NUM>.

<FIG> illustrates a block diagram <NUM> of an apparatus <NUM> for controlling the operation of a system, according to some embodiments of the present disclosure. The block diagram <NUM> may include the apparatus <NUM>. The apparatus <NUM> may include an input interface <NUM>, a processor <NUM>, a memory <NUM> and a storage <NUM>. The storage <NUM> may further include models 310a, a controller 310b, an updating module 310c and a control command module 310d. The apparatus <NUM> may further include a network interface controller <NUM> and an output interface <NUM>. The block diagram <NUM> may further include a network <NUM>, a state trajectory <NUM> and an actuator <NUM> associated with the system <NUM>.

The apparatus <NUM> includes the input interface <NUM> and the output interface <NUM> for connecting the apparatus <NUM> with other systems and devices. In some embodiments, the apparatus <NUM> may include a plurality of input interfaces and a plurality of output interfaces. The input interface <NUM> is configured to receive the state trajectory <NUM> of the system <NUM>. The input interface <NUM> includes the network interface controller (NIC) <NUM> adapted to connect the apparatus <NUM> through a bus to the network <NUM>. Moreover, through the network <NUM>, either wirelessly or through wires, the apparatus <NUM> receives the state trajectory <NUM> of the system <NUM>.

The state trajectory <NUM> may be a plurality of states of the system <NUM> that defines an actual behavior of dynamics of the system <NUM>. For example, the state trajectory <NUM> may act as a reference continuous state space for controlling the system <NUM>. In some embodiments, the state trajectory <NUM> may be received from real-time measurements of parts of the system <NUM> states. In some other embodiments, the state trajectory <NUM> may be simulated using the PDE that describes the dynamics of the system <NUM>. In some embodiments, a shape may be determined for the received state trajectory <NUM> as a function of time. The shape of the state trajectory <NUM> may represent an actual pattern of behavior of the system <NUM>.

The apparatus <NUM> further includes the memory <NUM> for storing instructions that are executable by the processor <NUM>. The processor <NUM> may be a single core processor, a multi-core processor, a computing cluster, or any number of other configurations. The memory <NUM> may include random access memory (RAM), read only memory (ROM), flash memory, or any other suitable memory system. The processor <NUM> is connected through the bus to one or more input and output devices. Further, the stored instructions implement a method for controlling the operations of the system <NUM>.

The memory <NUM> may be further extended to include storage <NUM>. The storage <NUM> may be configured to storage <NUM> models 310a, the controller 310b, the updating module 310c, and the control command module 310d.

The controller 310b may be configured to store instructions upon execution by the processor <NUM> that executes one or more modules in the storage <NUM>. Moreover, the controller 310b administrates each module of the storage <NUM> to control the system <NUM>.

Further, in some embodiments, the updating module 310c may be configured to update a gain associated with the model of the system <NUM>. The gain may be determined by reducing an error between the state of the system <NUM> estimated with the models 310a and an actual state of the system <NUM>. In some embodiments, the actual state of the system <NUM> may be a measured state. In some other embodiments, the actual state of the system <NUM> may be a state estimated with the PDE describing the dynamics of the system <NUM>. In some embodiments, the updating module 310c may update the gain using an extremum seeking. In some other embodiments, the updating module 310c may update the gain using a Gaussian process-based optimization technique.

The control command module 310d may be configured to determine a control command based on the models 310a. The control command module 310d may control the operation of the system <NUM>. In some embodiments, the operation of the system <NUM> may be subject to constraints. Moreover, the control command module 310d uses a predictive model-based control technique to determine the control command while enforcing constraints. The constraints include state constraints in continuous state space of the system <NUM> and control input constraints in continuous control input space of the system <NUM>.

The output interface <NUM> is configured to transmit the control command to the actuator(s) <NUM> of the system <NUM> to control the operation of the system <NUM>. Some examples of the output interface <NUM> may include a control interface that submits the control command to control the system <NUM>.

The control of the system <NUM> is further explained in <FIG> illustrates a flowchart <NUM> of principles for controlling the operation of the system <NUM>, according to some embodiments of the present disclosure. The flowchart <NUM> may include steps <NUM>, <NUM> and <NUM>.

In some embodiments, the system <NUM> may be modeled from physics laws. For instance, the dynamics of the system <NUM> may be represented by mathematical equations using the physics laws.

At step <NUM>, the system <NUM> may be represented by a physics-based high dimension model. The physics-based high dimension model may be the partial differential equation (PDE) describing the dynamics of the system <NUM>. The system <NUM> is considered to be the HVAC system, whose model is represented by Boussinesq equation. The Boussinesq equation may be obtained from the physics, which describes a coupling between airflow and the temperature in the room. Accordingly, the HVAC system model may be mathematically represented as: <MAT> <MAT> <MAT> where T is a temperature scalar variable, u is a velocity vector in three dimensions, µ is a viscosity and the reciprocal of the Reynolds number, k is a heat diffusion coefficient, p is a pressure scalar variable, g is gravity acceleration, and β is the expansion coefficient. The set of equations, such as equation <NUM>, equation <NUM> and equation <NUM> are referred to as Navier-Stokes equation plus conservation of energy. In some embodiments, such combination is known as Boussinesq equation. Such equations are valid for cases where the variation of temperature or density of air compared to absolute values of a reference point, e.g., temperature or density of air at the corner of the room, are negligible. Similar equations may be derived when such assumption is not valid, thus compressible flow model needs to be derived. Moreover, the set of equations are subjected to appropriate boundary conditions. For example, the velocity or temperature of the HVAC unit may be considered as boundary condition.

The operator Δ and ∇ may be defined in <NUM>-dimensional room as: <MAT> <MAT>.

Some embodiments, refers to the governing equations in more abstract from of as follows:.

where <MAT> and <MAT> are respectively the state and measurement at time k, f: <MAT> is a time-invariant nonlinear map from current to next state, and <MAT> is a linear map from state to measurement.

In some embodiments such abstract dynamics may be obtained from a numerical discretization of a nonlinear partial differential equation (PDE), that typically requires a large number n of state dimensions.

In some embodiments, the physics-based high dimension model of the system <NUM> needs to be resolved to control the operations of the system <NUM> in real-time. For example, the Boussinesq equation needs to be resolved to control the airflow dynamics and the temperature in the room. In some embodiments, the physics-based high dimension model of the system <NUM> comprises a large number of equations and variables, that may be complicated to resolve. For instance, a larger computation power is required to resolve the physics-based high dimension model in real-time. Thus, the physics-based high dimension model of the system <NUM> may be simplified.

At step <NUM>, the apparatus <NUM> is provided to generate the reduced order model to reproduce the dynamics of the system <NUM>, such that the apparatus <NUM> controls the system <NUM> in efficient manner. In some embodiments, the apparatus <NUM> may simplify the physics-based high dimension model using model reduction techniques to generate the reduced order model. In some embodiments, the model reduction techniques reduce the dimensionality of the physics-based high dimension model (for instance, the variables of the PDE), such that the reduced order model may be used to in real-time for prediction and control of the system <NUM>. Further, the generation of reduced order model for controlling the system <NUM> is explained in detail with reference to <FIG>. At step <NUM>, the apparatus <NUM> uses the reduced order model in real-time to predict and control the system <NUM>.

<FIG> illustrates a block diagram <NUM> that depicts generation of the reduced order model, according to some embodiments of the present disclosure. The linear predictor <NUM> is the reduced order model. The block diagram <NUM> depicts an architecture that includes the digital representation of the time series data <NUM>, and the autoencoder <NUM>. The autoencoder <NUM> includes the encoder <NUM>, the decoder <NUM> and the linear predictor <NUM>. The block diagram <NUM> further depicts an output <NUM> of the autoencoder <NUM>.

The snapshots <NUM> of the CFD simulation or experiments are the data needed for the autoencoders, such as the autoencoder <NUM>, which are neural network models as described in <FIG>. The latent space is governed by the linear ODE, that is to be learned based on both the snapshots <NUM> of the data and model information using the DSC equation, such as equation <NUM>.

Moreover, for a given time-dependent differential equation (for example, ODE or PDE), there may be a set of feasible initial conditions. Some embodiments define the feasible initial conditions as the ones that may fall into the domain of the system dynamics f.

Typically, the domain of a function is a set of inputs accepted by the function. More precisely, given a function f: X →Y, the domain of f is X. The domain may be a part of the definition of a function rather than a property of it. In such a case X and Y are both subsets of R, and the function f may be graphed in a Cartesian coordinate system. In such a case, the domain is represented on an x-axis of the graph, as the projection of the graph of the function onto the x-axis.

The collocation points <NUM> may be samples extracted from the domain of the system dynamics f, such that in case of the PDEs, the collocation points <NUM> may satisfy the boundary conditions. For example, if the boundary conditions of the system dynamics f are periodic, the collocation points <NUM> should be periodic. If the boundary conditions are Dirichlet, i.e. the system dynamics f equals to certain values at its boundary points, the collocation point <NUM> should also be equal to such values at the correposnding boundary points. Advantageously, the collocation points <NUM> may be much computationally cheaper to be evaluated compared to the computation of the snapshots <NUM>. The snapshots <NUM> may be generated either by a simulator or experiments, while the collocation points <NUM> may be generated simply by sampling them from a feasible function space.

Moreover, the function space is a set of functions between two fixed sets. Often, the domain and/or codomain may have additional that may be inherited by the function space. For example, the set of functions from any set X into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication. In other scenarios, the function space might inherit a topological or metric structure.

The autoencoder <NUM> may receive the digital representation of the time series data <NUM> and the collocation points <NUM> projected into the differential equations. The encoder <NUM> encode the digital representation into the latent space. The linear predictor <NUM> may propagate the encoded digital representation into the latent space with the linear transformation determined by values of parameters of the linear predictor <NUM>. Furthermore, the decoder <NUM> may the decode the linearly transformed encoded digital representation. The output <NUM> of the linearly transformed encoded digital representation may be the reconstructed snapshots or the decoded linearly transformed encoded digital representation.

A basic neural network model implemented for the architecture of the autoencoder <NUM> is described in <FIG> illustrates a schematic diagram <NUM> of the neural network model, according to some embodiments of the present disclosure. The neural network may be a network or circuit of an artificial neural network, composed of artificial neurons or nodes. Thus, the neural network is an artificial neural network used for solving artificial intelligence (AI) problems. The connections of biological neurons are modeled in the artificial neural networks as weights between nodes. A positive weight reflects an excitatory connection, while a negative weight values mean inhibitory connections. All inputs <NUM> of the neural network model may be modified by a weight and summed. Such an activity is referred to as a linear combination. Finally, an activation function controls an amplitude of an output <NUM> of the neural network model. For example, an acceptable range of the output <NUM> is usually between <NUM> and <NUM>, or it could be -<NUM> and <NUM>. The artificial networks may be used for predictive modeling, adaptive control and applications where they may be trained via a training dataset. Self-learning resulting from experience may occur within networks, which may derive conclusions from a complex and seemingly unrelated set of information.

The architecture of the blocks of the autoencoder <NUM> are described in <FIG>, <FIG> and <FIG>.

<FIG> illustrates a diagram 700A that depicts input of the digital representation in the encoder <NUM> of the neural network model (such as the autoencoder <NUM>), according to some embodiments of the present disclosure. The diagram 700A includes the encoder <NUM>, the snapsots <NUM>, the collocation points <NUM>, and a last layer <NUM> of the encoder <NUM>.

The input of encoder <NUM> may be either the snapshots <NUM> or the collocation points <NUM>. The snapshots <NUM> may be for example the digital representation of time series data <NUM>. The encoder <NUM> takes values of the the snapshots <NUM> or the collocation points <NUM>. The encoder <NUM> outputs to the latent space or the linear predictor <NUM> through the last layer <NUM> of the encoder <NUM>. The digital representation of time series data <NUM> indicative of the measurements of the operation of the system <NUM> at different instances of time may be collected. Further, for training of the neural network model (such as the autoencoder <NUM>) having the autoencoder architecture, the encoder <NUM> may receive encode the digital representation into the latent space. The process of encoding is the model reduction.

<FIG> illustrates a diagram 700B that depicts propagation of the encoded digital representation into the latent space by the linear predictor <NUM> of the neural network model, according to some embodiments of the present disclosure. The diagram 700A includes the last layer <NUM> of the encoder <NUM>, the linear predictor <NUM>, and a last iteration <NUM> of the linear predictor <NUM> or the latent space model.

The linear predictor <NUM> is configured to propagate the encoded digital representation into the latent space with linear transformation determined by values of parameters of the linear predictor <NUM>. The output of the last iteration <NUM> of the linear predictor <NUM> is passed to the decoder <NUM> of the neural network model. The process of propagating the encoded digital representation into the latent space is referred as reduced order model propagation or time integration.

<FIG> illustrates a diagram 700C depicting decoding of linearly transformd encoded digital representation by the decoder <NUM> of the neural network model, according to some embodiments of the present disclosure. The diagram 700C includes the decoder <NUM>, the last iteration <NUM> of the linear predictor <NUM>, and an output <NUM> of the decoder <NUM>.

The decoder <NUM>, puts forward the input and results in the output <NUM>. The decoder is configured to decode the linearly transformed encoded digital representation to generate the output <NUM>. The output <NUM> is the decoded linearly transformed encoded digital representation, such as the reconstructed snapshots as described in <FIG>. The process of the decoding is the reconstruction of the snapshots.

The neural network model identifies a few key coordinates spanned by the set of Koopman eigenfunctions {φ1, φ2, ···, φM}. The output of the encoder <NUM> is z = ϕ(x), where x is the input comprising in general as the summation of the snapshots <NUM> and the collocation points <NUM>. The dynamic within the latent space is linear and the output of the linear predictor <NUM> is given by <IMG> = Lz, where L is continuous Koopman operator and is parametrized by the neural network model. Furthermore, an inverse of x = ψ(z). The neural network model is trained to minimize a loss function including a prediction error between outputs of the neural network model decoding measurements of the operation at the instant of time and measurements of the operation collected at the subsequent instance of time. The loss function further includes the residual factor of the PDE having eigenvalues dependent on the parameters of the linear predictor <NUM>.

The loss function Jtotal of the neural network model is given by the following equation: <MAT> with following convention:.

The first term in the loss function is called the physics-informed part since it is a function of the system dynamics f. It is based on the DSC. Since it is associated with a differentiation (gradient) ∇ϕ, we may use automatic differentiation as to measure variation of the system <NUM> with respect to the differential equations <NUM>.

The physics-informed neural networks (PINNs) may seamlessly integrate the measurement data and physical governing laws by penalizing the residuals of the differential equation in the loss function using automatic differentiation. Such an approach alleviates the need for a large amount of data by assimilating the knowledge of the equations into the training process.

In some embodiments, the system <NUM> may be controlled by using a linear control law including a control matrix formed by the values of the parameters of the linear predictor <NUM>.

The loss function may further be described as follows: <MAT>.

The part ∥x - x̂∥<NUM> of the equation <NUM> refers to the reconstruction error. The part +∥x(t + Δt) - x̂(t + Δt)∥<NUM> of the equation <NUM> refers to a prediction error parametrized on ωi. The Lie operator PDE is the residual factor of the PDE having eigenvalues dependent on the parameters of the linear predictor <NUM> and parametrized on ωi.

In the physics informed Koopman networks (PIKNs), such knowledge of the dynamics of the system <NUM> is leveraged to enforce the linearity constraint. The neural network model is trained by minimizing the quantity, i.e. the loss function ∥φk(x). f - µkφk(x)∥, k = <NUM>, <NUM>, <NUM>, , M. The squared matrix L is used to approximate the Lie operator, which in turn is related to the Koopman operator, and term ∥Lϕ(x) - ∇ϕ(x). f∥ is minimized. In some embodiments, eigen-decomposition to the Lie operator is performed. The residual factor of the PDE is based on the Lie operator. For example, finding the eigenvalue and eigenfunction pairs of the Lie operator corresponds to performing eigen-decomposition to the squared matrix L.

<FIG> illustrates an exemplary diagram <NUM> for real-time implementation of the apparatus <NUM> for controlling the operation of the system <NUM>, according to some embodiments of the present disclosure. The exemplary diagram <NUM> includes a room <NUM>, a door <NUM>, a window <NUM>, a ventilation units <NUM>, and a set of sensors <NUM>.

The system <NUM> is an air conditioning system. The exemplary diagram <NUM> shows the room <NUM> that has the door <NUM> and at least one window <NUM>. The temperature and the airflow of the room <NUM> are controlled by the apparatus <NUM> via the air conditioning system through ventilation units <NUM>. The set of sensors <NUM> such as a sensor 810a and a sensor 810b are arranged in the room <NUM>. The at least one airflow sensor, such as the sensor 810a is used for measuring velocity of the air flow at a given point in the room <NUM>, and at least one temperature sensor, such as the sensor 810b is used for measuring the room temperature. It may be noted that other type of setting may be considered, for example a room with multiple HVAC units, or a house with multiple rooms.

The system <NUM> may be described by the physics-based model called the Boussinesq equation, as exemplary illustrated in <FIG>. However, the Boussinesq equation contains infinite dimensions to resolve the Boussinesq equation for controlling the air-conditioning system. The model comprising the ODE. Data assimilation may also be added to the ODE model. The model reproduces the dynamics (for instance, an airflow dynamics) of the air conditioning system in an optimal manner. Further, in some embodiments, the model of the air flow dynamics connects the values of the air flow (for instance, the velocity of the air flow) and the temperature of the air conditioned room during the operation of the air conditioning system. Moreover, the apparatus <NUM> optimally controls the air-conditioning system to generate the airflow in a conditioned manner.

<FIG> illustrates a flow chart <NUM> depicting a method for training the neural network model, according to some embodiments of the present disclosure.

At step <NUM>, the digital representation of time series data <NUM> indicative of measurements of the operation of the system <NUM> at different instances of time may be collected. Details of the collection of the digital representation of time series data <NUM> are further described, for example, in <FIG>.

At step <NUM>, the neural network model <NUM> may be trained. The neural network model <NUM> has the autoencoder architecture including the encoder <NUM> configured to encode the digital representation into the latent space, the linear predictor <NUM> configured to propagate the encoded digital representation into the latent space with linear transformation determined by values of parameters of the linear predictor <NUM>, and the decoder <NUM> configured to decode the linearly transformed encoded digital representation to minimize the loss function including the prediction error between outputs of the neural network model <NUM> decoding measurements of the operation at the instant of time and measurements of the operation collected at the subsequent instance of time, and the residual factor of the PDE having eigenvalues dependent on the parameters of the linear predictor <NUM>.

Specific details are given in the following description to provide a thorough understanding of the embodiments. However, if understood by one of ordinary skill in the art, the embodiments may be practiced without these specific details. For example, systems, processes, and other elements in the subject matter disclosed may be shown as components in block diagram form in order not to obscure the embodiments in unnecessary detail. In other instances, well-known processes, structures, and techniques may be shown without unnecessary detail to avoid obscuring the embodiments. Further, like reference numbers and designations in the various drawings indicated like elements.

Also, individual embodiments may be described as a process which is depicted as a flowchart, a flow diagram, a data flow diagram, a structure diagram, or a block diagram. Although a flowchart may describe the operations as a sequential process, many of the operations can be performed in parallel or concurrently. A process may be terminated when its operations are completed but may have additional steps not discussed or included in a figure. Furthermore, not all operations in any particularly described process may occur in all embodiments. A process may correspond to a method, a function, a procedure, a subroutine, a subprogram, etc. When a process corresponds to a function, the function's termination can correspond to a return of the function to the calling function or the main function.

Various methods or processes outlined herein may be coded as software that is executable on one or more processors that employ any one of a variety of operating systems or platforms. Additionally, such software may be written using any of a number of suitable programming languages and/or programming or scripting tools, and also may be compiled as executable machine language code or intermediate code that is executed on a framework or virtual machine. Typically, the functionality of the program modules may be combined or distributed as desired in various embodiments.

Individual embodiments above are described as a process which is depicted as a flowchart, a flow diagram, a data flow diagram, a structure diagram, or a block diagram. Although a flowchart shows the operations as a sequential process, many of the operations can be performed in parallel or concurrently. A process may be terminated when its operations are completed but may have additional steps not discussed or included in a figure. Furthermore, not all operations in any particularly described process may occur in all embodiments. A process may correspond to a method, a function, a procedure, a subroutine, a subprogram, etc. When a process corresponds to a function, the function's termination can correspond to a return of the function to the calling function or the main function.

Claim 1:
A control method for controlling an operation of a system (<NUM>), the system (<NUM>) being an air-conditioning system comprising ventilation units (<NUM>), an airflow sensor (810a) and a temperature sensor (810b), the system (<NUM>) having non-linear dynamics represented by partial differential equations (PDEs),
the control method including a computer-implemented method of training a neural network model (<NUM>) for controlling the operation of the system having non-linear dynamics represented by partial differential equations (PDEs), the computer-implemented method of training the neural network model (<NUM>) comprising:
collecting a digital representation of time series data (<NUM>) indicative of measurements of the operation of the system (<NUM>) at different instances of time; and
training the neural network model (<NUM>) having an autoencoder architecture including an encoder configured to encode the digital representation into a latent space, a linear predictor (<NUM>) configured to propagate the encoded digital representation into the latent space with linear transformation determined by values of parameters of the linear predictor (<NUM>), and a decoder configured to decode the linearly transformed encoded digital representation to minimize a loss function including a prediction error between outputs of the neural network model (<NUM>) decoding measurements of the operation at an instant of time and measurements of the operation collected at a subsequent instance of time, and a residual factor of the PDE having eigenvalues dependent on the parameters of the linear predictor (<NUM>), wherein the residual factor of the PDE is based on a Lie operator, the computer-implemented method further comprising performing eigen-decomposition to the Lie operator;
the control method further comprising controlling the system (<NUM>) to control a temperature and an airflow in a room through the ventilation units (<NUM>), by using a linear control law including a control matrix formed by the values of the parameters of the linear predictor (<NUM>).