Patent Description:
Vapor compression cycles represent a fundamental technology in contemporary society because of their wide use in air-conditioning and space heating applications. Role of the vapor compression cycles is expected to grow in future years as the vapor compression cycles provide an effective means for decarbonizing heating systems and utilizing electrical energy generated by renewable sources, such as photovoltaic or wind power. There is thus widespread interest in further developing vapor compression cycle technology so that they are both energy efficient and satisfy performance requirements related to user health and comfort in buildings.

A range of control technologies employed in the vapor compression cycles use a model that represents underlying physics/dynamics of the vapor compression cycle, for predicting behavior of the vapor compression cycle and controlling the vapor compression cycle. Accurate predictions of the behavior of vapor compression cycle can be used in control technologies, e.g., model predictive control and estimation of performance parameters such as cooling capacity delivered by the heat exchangers (HEXs). The model of the vapor compression cycle may also play an important role in development of fault detection and diagnosis algorithms, and are central to emerging digital twin technologies aiming to improve energy efficiency, streamline maintenance, and maximize user comfort.

Some approaches use a physics-based model for predicting the behavior of the vapor compression cycle. The physics-based model is derived from first principles of physics resulting in a high-dimensional set of nonlinear differential algebraic equations for a multiphysical system such as vapor-compression cycle. Such a model is generally formulated to satisfy application-specific physics-based or computational requirements and relate to physical processes most relevant to the application. Simplifying assumptions that accompany such requirements, e.g., lumped parameters and finite discretization, often lead to a mismatch between the model predictions and data collected from the system. Addressing the mismatch to improve accuracy of the model predictions is a non-trivial task that may require unjustifiable effort and complexity resulting in comparatively smaller prediction improvements.

Alternatively, a data driven model may be used for predicting the behavior of the vapor compression cycle. The data driven model is developed using data collected from the system. However, the data driven model need large datasets including full-state trajectories to achieve reasonable modeling accuracy. Such large datasets are unavailable due to limited sensor data. Further, the data-driven model suffers from a combination of other challenges, such as non-interpretability of the data driven model, large parameter search spaces, and do not explicitly account for fundamental physical laws that govern the system.

In the prior art, for example, Vedang M Deshpande et al. (<NPL> ) disclose smoothing methods in the extended and ensemble Kalman estimation frameworks that satisfy physical constraints and address practical limitations with standard implementations of these estimators.

Lei Zhao et al. (<NPL>) disclose a model-based optimization strategy for vapor compression refrigeration cycle, wherein through analyzing each component characteristics and interactions within the cycle, the optimization problem is formulated as minimizing the total operating cost of the energy consuming devices subject to the constraints of mechanical limitations, component interactions, environment conditions and cooling load demands. A MGA (modified genetic algorithm) together with a solution strategy for a group of nonlinear equations is proposed to obtain optimal set point under different operating conditions.

Xin Ma et al. (<NPL>) disclose a simple, yet accurate hybrid model of scroll compressor (SC) used in micro-compressed air energy storage system (MCAES). The modeling process starts with basic thermodynamic and fundamental physical principles but captures only few key operational characteristic parameters to predict the exhaust temperature, exhaust flow rate and the input torque of the SC, and then the appropriate input and output variables are selected according to the control demands of the system.

Yuan Yin et al. (<NPL>) disclose the APHYNITY framework, a principled approach for augmenting incomplete physical dynamics described by differential equations with deep data-driven models. It consists of decomposing the dynamics into two components: a physical component accounting for the dynamics for which we have some prior knowledge, and a data-driven component accounting for errors of the physical model. The learning problem is carefully formulated such that the physical model explains as much of the data as possible, while the data-driven component only describes information that cannot be captured by the physical model; no more, no less.

Mona Buisson-Fenet et al. (<NPL>, XP091232785) disclose a flexible framework to incorporate a broad spectrum of physical insight into neural ODE-based system identification, giving physical interpretability to the resulting latent space, wherein this insight is either enforced through hard constraints in the optimization problem or added in its cost function.

Accordingly, there is still a need for a model of the dynamics of the vapor compression cycle for controlling the vapor compression cycle.

Optional aspects of the invention are provided by the dependent claims.

The invention is based on the realization that modeling efforts for the vapor compression cycle may benefit from a hybrid modeling paradigm which combines domain expertise represented by the physics-based model with the data-driven model to learn the behavior of the vapor compression cycle. Accordingly, the present disclosure provides the hybrid model of the dynamics of the vapor compression cycle that includes the physics-based model and the data driven model. The physics-based model is configured to predict transitions of states of the vapor compression cycle in accordance with observed variables and a control input to the vapor compression cycle based on parameters of the physics-based model. The physics-based model includes one or a combination of governing partial differential equations discretized over spatial and temporal domains, look up tables, and interpolation splines for calculating thermodynamic properties of the refrigerant of the vapor compression cycle. The data driven model is trained with machine learning to estimate residual errors of the state transitions predicted by the physics-based model. The data driven model may include a neural network.

The physics-based model parameterized by parameter vector θ, which includes a number of finite control volumes for discretization of governing partial differential equations, geometric parameters of the vapor compression cycle, properties of materials used in the vapor compression cycle, a total mass of the refrigerant and the like. On the other hand, the data driven model, e.g. a neural network, is parameterized by a vector w, which includes weights and biases of each artificial neuron as well as parameters that define an architecture of the neural network.

It is an object of the invention to estimate model attributes of the hybrid model, based on observed variables collected over multiple instances of time. The model attributes of the hybrid model include parameters of the physics-based model, i. e, parameters θ, and parameters of the data driven model, i. e, parameters w. The observed variables include temperatures and pressures measured by sensors installed at different locations in the vapor compression cycle. It is desired that the estimated model attributes are such that contribution of the physics-based model is maximized in the hybrid model while the data driven model is utilized only to correct modeling errors that cannot be otherwise rectified with the physics-based model alone. Therefore, the data driven model is utilized to learn residual dynamics based on mismatches of the physics-based model with respect to the observed variables. Accordingly, in an embodiment of the present disclosure, the parameters of the physics-based model and the data driven model are determined by solving a joint optimization problem that minimizes a cost function comprising penalties on deviations between predicted and real outputs of the vapor compression cycle, and weighted norm of the data driven model outputs.

The invention is based on the recognition that simultaneously estimating the parameters θ of the physics-based model and the parameters w of the data driven model is a challenging problem because they are fundamentally different variables. On one hand, the parameters θ of the physics-based model, in general, has low dimension but derivative with respect to the parameters θ are hard to compute, as they require derivative of a solution of a differential equation. Moreover, the dynamics of the vapor-compression cycle are nonlinear, numerically stiff, and have derivative discontinuities due to phase changes that accompany evaporation or condensation. On the other hand, the parameters w have large dimension, but the derivatives can be efficiently computed using standard computing tools such as automatic differentiation.

The invention is based on a realization that the computational challenges associated with the joint optimization problem can be addressed by decomposing the joint optimization problem into two equivalent tractable unconstrained optimization problems that determine θ and w separately instead of solving the joint optimization problem. For instance, the joint optimization problem can be decomposed into a first optimization problem of joint estimation of the states of the vapor compression cycle and the parameters θ of the physics-based model, and a second optmization problem of determining the parameters w of the data driven model.

The invention is based on the realization that the first optimization problem can be interpreted as a joint state and parameter estimation problem based on the physics-based model of the vapor compression cycle. Such an interpretation allows utilization of optimal smoothing formulation to determine a solution. In particular, an approximate solution to the first optimization problem can be determined efficiently using wide range of nonlinear estimation methods. However, in general, estimation methods do not guarantee that physics-based constraints, such as monotonicity of pressures in refrigerant flow-direction in the vapor compression cycle, will be satisfied. It is important that such physics-based constraints are satisfied to ensure that state estimates are physically realizable. Accordingly, the invention uses a constrained Kalman smoother, such as a Constrained Extended Kalman Smoother (C-EKS) or Constrained Ensemble Kalman Smoother (C-EnKS), that is tailored for estimation of the vapor compression cycle and enforces the physics-based constraints during joint estimation of the states of the vapor compression cycle and the parameters of the physics-based model.

Therefore, the first optimization problem is solved using the constrained Kalman smoother to estimate the parameters of the physics-based model and the states of the vapor compression cycle. In particular, the constrained Kalman smoother is executed over the observed variables collected over multiple time instances to estimate the parameters of the physics-based model and the states of the vapor compression cycle.

There exists a difference between the states predicted by the physics based model and the states estimated by executing the constrained Kalman smoother. Such a difference is referred to as a residual error. To compensate for the residual error, the parameters w of the data driven model are determined such that a cumulative learning cost function, which is comprised of a difference between the residual errors and the data driven model outputs corresponding to the same inputs that generated the error residuals at that time instants, is minimized. The residual errors are known constants within the scope of the second optimization problem, thus, the data driven model essentially attempts to learn the residual errors for a given input state vector and control input vector.

To that end, some embodiments of the present disclosure are based on the realization that the second optimization problem of determining the parameters of the data driven model is a standard neural network training problem. According to an embodiment, the second optimization problem is solved using stochastic gradient descent to determine the parameters of the data driven model.

Further, the hybrid model is updated with the parameters of physics-based model estimated by executing the constrained Kalman smoother, and the parameters of the data driven model determined by solving the second optimization problem using the stochastic gradient descent. In other words, the updated hybrid model includes the physics-based model with the parameters estimated by executing the constrained Kalman smoother and the data driven model with the parameters determined by solving the second optimization problem using the stochastic gradient descent. Further, the operation of the vapor compression cycle is controlled using the updated hybrid model. For instance, based on the updated hybrid model, control inputs to actuators of the vapor compression cycle are determined. The control inputs, for example, include e.g., a speed of a compressor, a speed of a fan, and a position of an expansion valve. The operation of the vapor compression cycle is controlled according to the control inputs.

Accordingly, the invention discloses a controller for controlling an operation of a vapor compression cycle based on a hybrid model of dynamics of the vapor compression cycle including a physics-based model and a data driven model, wherein the physics-based model is configured to predict transitions of states of the vapor compression cycle in accordance with observed variables and a control input to the vapor compression cycle based on parameters of the physics-based model, and wherein the data driven model is trained with machine learning to estimate residual errors of the state transitions predicted by the physics-based model, the controller comprising: a processor; and a memory having instructions stored thereon that, when executed by the processor, cause the controller to: collect a digital representation of observed variables of the operation of the vapor compression cycle over multiple instances of time; execute a constrained Kalman smoother over the observed variables collected over multiple instances of time to jointly estimate the parameters of the physics-based model and states of the vapor compression cycle to minimize a cost function comprised of residual errors of the state transitions predicted by the physics-based model for the multiple instances of time, and residual errors between observed variables estimated by the constrained Kalman smoother for the multiple instances of time and the corresponding collected observed variables; update the data driven model to minimize a difference between the states estimated by executing the constrained Kalman smoother over the observed variables collected over multiple instances of time and the states predicted by the physics-based model; update the hybrid model with the estimated parameters of the physics-based model and the updated data driven model; and control the operation of the vapor compression cycle using the updated hybrid model.

Accordingly, the invention discloses a method for controlling an operation of a vapor compression cycle based on a hybrid model of dynamics of the vapor compression cycle including a physics-based model and a data driven model, wherein the physics-based model is configured to predict transitions of states of the vapor compression cycle in accordance with observed variables and a control input to the vapor compression cycle based on parameters of the physics-based model, and wherein the data driven model is trained with machine learning to estimate residual errors of the state transitions predicted by the physics-based model. The method comprises: collecting a digital representation of observed variables of the operation of the vapor compression cycle over multiple instances of time; executing a constrained Kalman smoother over the observed variables collected over multiple instances of time to jointly estimate the parameters of the physics-based model and states of the vapor compression cycle to minimize a cost function comprised of residual errors of the state transitions predicted by the physics-based model for the multiple instances of time, and residual errors between observed variables estimated by the constrained Kalman smoother for the multiple instances of time and the corresponding collected observed variables; updating the data driven model to minimize a difference between the states estimated by executing the constrained Kalman smoother over the observed variables collected over multiple instances of time and the states predicted by the physics-based model; updating the hybrid model with the estimated parameters of the physics-based model and the updated data driven model; and controlling the operation of the vapor compression cycle using the updated hybrid model.

Accordingly, the invention discloses a non-transitory computer readable storage medium embodied thereon a program executable by a processor for performing a method for controlling an operation of a vapor compression cycle based on a hybrid model of dynamics of the vapor compression cycle including a physics-based model and a data driven model, wherein the physics-based model is configured to predict transitions of states of the vapor compression cycle in accordance with observed variables and a control input to the vapor compression cycle based on parameters of the physics-based model, and wherein the data driven model is trained with machine learning to estimate residual errors of the state transitions predicted by the physics-based model. The method comprises: collecting a digital representation of observed variables of the operation of the vapor compression cycle over multiple instances of time; executing a constrained Kalman smoother over the observed variables collected over multiple instances of time to jointly estimate the parameters of the physics-based model and states of the vapor compression cycle to minimize a cost function comprised of residual errors of the state transitions predicted by the physics-based model for the multiple instances of time, and residual errors between observed variables estimated by the constrained Kalman smoother for the multiple instances of time and the corresponding collected observed variables; updating the data driven model to minimize a difference between the states estimated by executing the constrained Kalman smoother over the observed variables collected over multiple instances of time and the states predicted by the physics-based model; updating the hybrid model with the estimated parameters of the physics-based model and the updated data driven model; and controlling the operation of the vapor compression cycle using the updated hybrid model.

In the following description, for purposes of explanation, numerous specific details are set forth in order to provide a thorough understanding of the present disclosure. It will be apparent, however, to one skilled in the art that the present disclosure may be practiced without these specific details. In other instances, apparatuses and methods are shown in block diagram form only in order to avoid obscuring the present disclosure.

<FIG> illustrates a vapor compression cycle <NUM>, according to an embodiment of the present disclosure. The vapor compression cycle <NUM> includes a compressor <NUM>, a condensing heat exchanger <NUM>, an expansion valve <NUM>, and an evaporating heat exchanger <NUM> located in a space <NUM>. Heat transfer from the condensing heat exchanger <NUM> is promoted by use of a fan <NUM>, while heat transfer from the evaporating heat exchanger <NUM> is promoted by use of a fan <NUM>. The vapor compression cycle <NUM> may include variable actuators, such as a variable compressor speed, a variable expansion valve position, and variable fan speeds. There are many other alternate equipment architectures to which the present disclosure pertains with multiple heat exchangers, compressors, valves, and other components such as accumulators or reservoirs, pipes, and so forth, and the illustration of the vapor compression cycle <NUM> is not intended to limit the scope or application of the present disclosure to systems whatsoever.

In the vapor compression cycle <NUM>, the compressor <NUM> compresses a low pressure, low temperature vapor-phase fluid (a refrigerant) to a high pressure, high temperature vapor state, after which it passes into the condensing heat exchanger <NUM>. As the refrigerant passes through the condensing heat exchanger <NUM>, the heat transfer promoted by the fan <NUM> causes the high-temperature, high pressure refrigerant to transfer its heat to ambient air, which is at a lower temperature. As the refrigerant transfers the heat to the ambient air, the refrigerant gradually condenses until the refrigerant is in a high pressure, low temperature liquid state. Further, the refrigerant leaves the condensing heat exchanger <NUM> and passes through the expansion valve <NUM>, and expands to a low pressure boiling state from which it enters the evaporating heat exchanger <NUM>. As air passing over the evaporating heat exchanger <NUM> is warmer than the refrigerant itself, the refrigerant gradually evaporates as it passes through the evaporating heat exchanger <NUM>. The refrigerant leaving the evaporating heat exchanger <NUM> is at a low pressure, low temperature state. The low pressure, low temperature refrigerant re-enters the compressor <NUM> and the same cycle is repeated.

The vapor compression cycle <NUM> operates at a nominal set of input values for actuators, e.g., a speed of the compressor <NUM>, a speed of the fan <NUM>, a position of the expansion valve <NUM>, a speed of the fan <NUM>, and the like. It is desired or an objective that the vapor compression cycle <NUM> achieve performance metrics, for example, regulating variables such as a temperature or humidity in the space <NUM> or regulating process variables such as a temperature or a pressure at one or more points in the vapor compression cycle <NUM>. To achieve such objectives, one or more sensors are installed at various locations in the vapor compression cycle <NUM> to monitor variables of interest. The variables of interest may include the temperature, the humidity, and/or the pressure. For example, sensors, such as a sensor <NUM>, a sensor <NUM>, a sensor <NUM>, a sensor <NUM>, and a sensor <NUM> (collectively referred to as sensors <NUM>, <NUM>, <NUM>, <NUM>, and <NUM>), are located at different locations. The sensors <NUM>, <NUM>, <NUM>, and <NUM> monitor the temperature and/or the pressure at their respective locations. Alternatively or in addition, measurements of variables in the space <NUM>, such as temperature or humidity, may also be obtained via sensors such as a sensor <NUM>.

Information from the sensors <NUM>, <NUM>, <NUM>, <NUM>, and <NUM> is input to a controller <NUM> associated with the vapor compression cycle <NUM>. Based on the information from the sensors <NUM>, <NUM>, <NUM>, <NUM>, and <NUM>, the controller <NUM> may control an operation of the vapor compression cycle <NUM>. For example, based on the information from the sensors <NUM>, <NUM>, <NUM>, <NUM>, and <NUM>, the controller <NUM> may change the input values of the actuators, e.g., the speed of the compressor <NUM>, the speed of the fan <NUM>, the position of the expansion valve <NUM>, and the speed of the fan <NUM>, to achieve desired performance metrics.

<FIG> shows a block diagram of the controller <NUM>, according to an embodiment of the present disclosure. The controller <NUM> includes a processor <NUM> and a memory <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 systems. Additionally, in some embodiments, the memory <NUM> may be implemented using a hard drive, an optical drive, a thumb drive, an array of drives, or any combinations thereof.

The processor <NUM> is configured to collect a digital representation of observed variables of an operation of the vapor compression cycle <NUM> over multiple instances of time. In an embodiment, the processor <NUM> collects the digital representation of observed variables of the operation of the vapor compression cycle <NUM> from the sensors <NUM>, <NUM>, <NUM>, <NUM>, and <NUM>. The observed variables include, for example, measurements of one or more of the temperature and the pressure at different locations in the vapor compression cycle <NUM>.

The controller <NUM> may employ a model that represents underlying physics/dynamics of the vapor compression cycle <NUM>, for predicting behavior of the vapor compression cycle <NUM>. Based on the collected digital representation of observed variables, the model predicts the behavior of the vapor compression cycle <NUM>. Accurate predictions of the behavior of vapor compression cycle <NUM> can be used in a range of control technologies, e.g., model predictive control and estimation of performance parameters such as cooling capacity delivered by the heat exchangers (HEXs). The model of the vapor compression cycle <NUM> may also play an important role in development of fault detection and diagnosis algorithms, and are central to emerging digital twin technologies aiming to improve energy efficiency, streamline maintenance, and maximize user comfort.

The model of the vapor compression cycle <NUM> may be a physics-based model. The physics-based model is derived from first principles of physics resulting in a high-dimensional set of nonlinear differential algebraic equations for a metaphysical system such as vapor-compression cycle <NUM>. Such a model is generally formulated to satisfy application-specific physics-based or computational requirements and relate to physical processes most relevant to the application. Simplifying assumptions that accompany such requirements, e.g., lumped parameters and finite discretization, often lead to a mismatch between the model predictions and data collected from the system (e.g., Vapor Compression System (VCS)). Addressing the mismatch to improve accuracy of the model predictions is a non-trivial task that may require unjustifiable effort and complexity, resulting in comparatively smaller prediction improvements.

Alternatively, the model of the vapor compression cycle <NUM> may be a data driven model. The data driven model is developed using the data collected from the system. However, the data driven model need large datasets including full-state trajectories to achieve reasonable modeling accuracy. Such large datasets are unavailable due to limited sensor data. Further, the data-driven model suffers from a combination of other challenges, such as non-interpretability of the data driven model, large parameter search spaces, and do not explicitly account for fundamental physical laws that govern the system.

Some embodiments are based on the realization that modeling efforts for the vapor compression cycle <NUM> may benefit from a hybrid modeling paradigm which combines domain expertise represented by the physics-based model with the data-driven model to learn the behavior of the vapor compression cycle <NUM>. Accordingly, the present disclosure provides a hybrid model <NUM> of the dynamics of the vapor compression cycle <NUM>. In an embodiment, the hybrid model <NUM> is stored in the memory <NUM>. The hybrid model <NUM> is described below in <FIG>.

<FIG> illustrates the hybrid model <NUM> of the dynamics of the vapor compression cycle <NUM>, according to some embodiments of the present disclosure. The hybrid model <NUM> of the dynamics of the vapor compression cycle <NUM> includes a physics-based model <NUM> and a data driven model <NUM>. The physics-based model <NUM> is configured to predict transitions of states of the vapor compression cycle <NUM> in accordance with the observed variables and a control input to the vapor compression cycle <NUM> based on parameters of the physics-based model <NUM>. The physics-based model <NUM> includes one or a combination of governing partial differential equations discretized over spatial and temporal domains, look up tables, and interpolation splines for calculating thermodynamic properties of the refrigerant of the vapor compression cycle <NUM>. The data driven model <NUM> is trained with machine learning to estimate residual errors of the state transitions predicted by the physics-based model <NUM>. The data driven model <NUM> may include a neural network. Mathematically, the hybrid model <NUM> is given as <MAT> where xk is a state vector of the vapor compression cycle <NUM> at time tk and uk is a vector of control inputs to the vapor compression cycle <NUM> at time tk. The vector uk includes the control inputs commanded by the controller <NUM>, such as the speed of the compressor <NUM>, the speed of the fan <NUM>, the position of the expansion valve <NUM>, and the speed of the fan <NUM>. Additionally, the vector uk may also include variables which affect the behavior of the vapor compression cycle <NUM>, such as temperature and relative humidity in the space <NUM> and ambient outdoor environment. In equation (<NUM>), Fp represents the physics-based model parameterized by parameter vector θ, which includes a number of finite control volumes for discretization of governing partial differential equations, geometric parameters of the vapor compression cycle, properties of materials used in the vapor compression cycle <NUM>, a total mass of the refrigerant and the like. On the other hand, Fd denotes the data driven model, e.g. a neural network, parameterized by a vector w, which includes weights and biases of each artificial neuron as well as parameters that define an architecture of the neural network.

It is an object of some embodiments to estimate model attributes of the hybrid model <NUM>, based on the observed variables collected over multiple instances of time. The model attributes of the hybrid model <NUM> include parameters of the physics-based model <NUM>, i. e, parameters θ, and parameters of the data driven model <NUM>, i. e, parameters w.

The observed variables can be represented by the following measurement equation <MAT> where η is a normal variable N(<NUM>, R). Measurement model h(xk, uy, θ) takes as inputs state vector xk and the control inputs vector uk to predict the observed variables yk, and η accounts for random disturbances, e.g. sensor noise, that corrupt the measurements. The observed variables yk include temperatures and pressures measured by the sensors <NUM>, <NUM>, <NUM>, <NUM>, and <NUM> installed at different locations in the vapor compression cycle <NUM>.

The processor <NUM> is configured to estimate the model attributes of the hybrid model <NUM>, i.e., the parameters θ and w, by solving the following joint optimization problem <MAT> where the observed variables are collected and stored over multiple time instances k=<NUM>,<NUM>,.

First term in the problem (<NUM>) directly penalizes contribution of the data driven model Fd weighted by a positive definite matrix P, while enforcing equality constraints as defined by the hybrid model (<NUM>). The choice of the first term in cost function is based on the realization that naïve minimization of an obvious cost, e.g. ∥ xk+<NUM> - Fp(xk, uk, θ) - Fd (xk, uk, w) ∥ , can lead to undesired solutions. For example, in an extreme case, a combination of optimal parameters (θ*, w*) may exist such that Fp (xk, uy, θ*) = <NUM> and information of the dynamics of the vapor compression cycle <NUM> comes only from the data-driven model while discarding information from the physics-based model. Instead, it is desired that the physics-based model Fp provides most of the explanation of the behavior of the vapor compression cycle <NUM> because the physics-based model Fp tends to be more robust and generalizes better to new or unseen data. In the limiting case, if there exist parameters θ* such that xk+<NUM> - Fp(xk, uk, θ*) = <NUM> , then the formulation (<NUM>) determines such parameters and discard any contribution from the data-driven model.

Second term in the problem (<NUM>) penalizes deviations of the predicted measurements h(xk, uy, θ) from the observed variables yk weighted by a covariance matrix R of the sensor noise that corrupts the measurements. Finally, the last term in problem (<NUM>) accounts for initial state uncertainty as initial condition is generally not known and it is assumed to be normally distributed as N(x<NUM>, S).

In data-driven modeling approaches such as the neural networks, it is common to determine optimal parameters (θ*, w*) by minimizing the cost function over multiple trajectories of the system for improved modeling accuracy and better generalizing properties. Therefore, the problem (<NUM>) can be reformulated as following joint optimization problem <MAT> where superscript i denotes ith trajectory in training data set including a total of N trajectories.

Numerical optimization to solve (<NUM>) is challenging because the equality constraints need to be relaxed or incorporated via Lagrangian methods and the parameters θ and w are fundamentally different optimization variables. On one hand, the parameters θ in general has low dimension but derivative with respect to θ are hard to compute, as they require derivative of a solution of a differential equation. Moreover, the dynamics of the vapor-compression cycle <NUM> are nonlinear, numerically stiff, and have derivative discontinuities due to phase changes that accompany evaporation or condensation. On the other hand, w has large dimension, but the derivatives can be efficiently computed using standard computing tools such as automatic differentiation.

Some embodiments of the present disclosure are based on a realization that the computational challenges associated with the joint optimization (<NUM>) can be addressed by decomposing (<NUM>) into two equivalent tractable unconstrained optimization problems that determine θ and w separately instead of solving the joint optimization problem (<NUM>), as described below in <FIG>.

<FIG> shows a schematic illustrating decomposition of the joint optimization problem (<NUM>), according to some embodiments of the present disclosure. The joint optimization problem (<NUM>) <NUM> is decomposed into a first optimization problem <NUM> of joint estimation of the states of the vapor compression cycle <NUM> and the parameters θ of the physics-based model <NUM>, and a second optmization problem <NUM> of determining the parameters w of the data driven model <NUM>.

The first optimization problem <NUM> includes minimizing a cost function comprised of residual errors of the state transitions predicted by the physics-based model <NUM> for the multiple instances of time, and residual errors between observed variables estimated by the constrained Kalman smoother for the multiple instances of time and the corresponding collected observed variables. Mathematically, the first optimization problem <NUM> is given by <MAT>.

The second optimization problem <NUM> is given by <MAT>.

The first optimization problem (<NUM>) <NUM> is solved to estimate the parameters θ* of the physics-based model <NUM> and the states <MAT> at all time instances for the observed variables collected.

Some embodiments are based on the realization that the first optimization problem (<NUM>) <NUM> of joint estimation of the states and the parameters of the physics-based model <NUM> can be interpreted as a maximum a posteriori estimation of θ and <MAT>, where the VCS is subject to Gaussian disturbances. Therefore, the first optimization problem (<NUM>) <NUM> can be efficiently solved using <NUM>-dimensional variational (4DVar) data assimilation methods. In practice, computational costs and scalability concerns motivate use of optimal smoothing methods, such as Kalman smoother, for solving the first optimization problem (<NUM>) <NUM>.

Kalman smoothing algorithms are non-causal data assimilation methods in which measurement data available at a future time instance is used to improve an estimate of the state in the past using a Kalman estimation framework. Therefore, Kalman smoothers are well-suited for batch processing of measurements or observed variables collected over multiple instances of time. Kalman smoothing methods are guaranteed to be optimal for solving (<NUM>) if underlying functions Fp and h are linear in optimization variables. While underlying governing equations of the vapor compression cycle <NUM> are not linear, Kalman smoothing algorithms are still advantageous for obtaining an approximate solution to (<NUM>) due to their reasonably well accuracy, computational efficiency and simpler implementations compared to sophisticated solvers that might be otherwise needed to solve (<NUM>). Kalman smoothers for nonlinear systems can be implemented in various formulations, such as Extended Kalman Smoother and Ensemble Kalman Smoother. Extended Kalman Smoother, also known as Rauch-Tung-Striebel (RTS) smoother, linearizes nonlinear dynamics and uses Jacobian matrices for propagation and update steps of Kalman estimation process. Ensemble Kalman Smoother is Monte-Carlo type method in which an ensemble of random particles sampled from a state distribution is used for propagation phase, and empirically calculated covariance matrices are used in update steps. For a high-dimensional nonlinear system like the vapor compression cycle <NUM>, Ensemble Kalman Smoother is used as it circumvents need to store high-dimensional covariance matrices which are needed in Extended Kalman Smoother.

Moreover, Kalman smoothers can be customized for specific vapor compression applications for substantially faster solutions that also incorporate physics-based state constraints. For example, refrigerant pressure in a heat exchanger must decrease in a direction of flow to satisfy fundamental physical relationships in the vapor compression cycle <NUM>. If such a constraint on the refrigerant pressure is not satisfied after a state correction is applied at a given time, the solver may not be able to successfully integrate a dynamic model forward over the next time interval because perturbed pressures may cause nonphysical changes in a direction of the flow and violate fundamental assumptions of the dynamic model.

Therefore, the first optimization problem (<NUM>) <NUM> is solved using a constrained Kalman smoother <NUM> to estimate the parameters θ* of the physics-based model <NUM> and the states <MAT>. In particular, the processor <NUM> executes the constrained Kalman smother over the observed variables collected over multiple time instances to estimate the parameters θ* of the physics-based model <NUM> and the states <MAT> by minimizing the cost function of (<NUM>).

There exists a difference between the states predicted by the physics based model <NUM> and the states estimated by executing the constrained Kalman smoother. Such a difference is referred to as a residual error. The residual error can be calculated as follows <MAT>.

To compensate for the residual error, the parameters w of the data driven model <NUM> are determined such that a cumulative learning cost function, which is comprised of a difference between the residual errors and the data driven model <NUM> outputs corresponding to the same inputs that generated the error residuals at that time instants, is minimized. The residual errors <MAT> are known constants within the scope of the second optimization problem, thus, the data driven model <NUM> essentially attempts to learn the residual errors for a given input state vector and control input vector.

To that end, some embodiments of the present disclosure are based on the realization that the second optimization problem (<NUM>) <NUM> of determining the parameters w* is a standard neural network training problem. According to an embodiment, the second optimization problem (<NUM>) <NUM> is solved using stochastic gradient descent <NUM> to determine the parameters w* of the data driven model <NUM>. The data driven model <NUM> is trained with training data set comprising ordered tuples, ( <MAT>). Input to the data driven model <NUM> is ( <MAT>) and output of the data driven model <NUM> is the residual error <MAT>.

Further, the processor <NUM> updates the hybrid model <NUM> with the parameters θ* of physics-based model <NUM> and the parameters w*of the data driven model <NUM>. In other words, the updated hybrid model <NUM> includes the physics-based model <NUM> with the parameters θ* and the data driven model <NUM> with the parameters w*. Further, the processor <NUM> controls the operation of the vapor compression cycle <NUM> using the updated hybrid model <NUM>. For instance, based on the updated hybrid model <NUM>, the processor <NUM> determines the control inputs to the actuators of the vapor compression cycle <NUM>, e.g., the speed of the compressor <NUM>, the speed of the fan <NUM>, the position of the expansion valve <NUM>, and the speed of the fan <NUM>. Further, the processor <NUM> controls the actuators based on the determined control inputs.

In some embodiments, the digital representation of observed variables of the operation of the vapor compression cycle <NUM> collected from the sensors <NUM>, <NUM>, <NUM>, <NUM>, and <NUM>, is transmitted to a storage medium co-located at the same geographical site where the vapor compression cycle <NUM> is located.

<FIG> shows a schematic of a system architecture <NUM> including a storage medium <NUM>, according to some embodiments of the present disclosure. The digital representation of observed variables of the operation of the the vapor compression cycle <NUM> collected over multiple instances of time from the sensors <NUM>, <NUM>, <NUM>, <NUM>, and <NUM>, and control inputs provided by the controller <NUM> are stored in the storage medium <NUM>. Additionally, in some embodiments, other internal information associated with the controller <NUM>, such as internal controller variables, discrete variables from control logic, or other information produced by the controller <NUM>, may be stored in the storage medium <NUM>. The storage medium may be co-located at the same geographical site where the vapor compression cycle <NUM> is located.

Further, data from the storage medium <NUM> is periodically provided to the controller <NUM>, either at regular intervals or when there is an event that calls for the updating of the hybrid model <NUM>, for example, when a user requests for information related to the operation of the vapor compression cycle <NUM>. In an embodiment, the data provided from the storge medium <NUM> may include the digital representation of observed variables of the operation of the vapor compression cycle collected over multiple instances of time.

Some embodiments are based on the realization that a part or whole of the data stored in the storage medium <NUM> may be stored using cloud computing resources, and, in addition, the updating of the hybrid model <NUM> may be implemented on a remote server. Such an embodiment is described in <FIG>.

<FIG> shows a schematic of a cloud-based architecture <NUM>, according to some embodiments of the present disclosure. The vapor compression cycle <NUM>, the controller <NUM>, and the storage medium <NUM> is considered to be a system <NUM>. The system <NUM> is in communication with the remote server <NUM> (also referred to as a cloud computing system) via a network <NUM>. In this case, a size of the storage medium <NUM> may vary and, additionally or alternatively, may or may not be present, depending on ability and/or reliability of the remote server <NUM> to access data from the vapor compression cycle <NUM>.

The system <NUM> is configured to transmit a part or whole of data (e.g., the digital representation of the observed variables and control inputs) to the remote server <NUM> for storage, rather than maintaining the data in the storage medium <NUM> co-located with the vapor compression cycle <NUM>. Further, the remote server <NUM> may store the model attributes of the hybrid model <NUM>, i.e., the parameters θ of the physics-based model <NUM>, and the parameters w of the data driven model <NUM>.

The model attributes of the hybrid model <NUM> stored in the remote server <NUM> may be updated at regular intervals or when an update triggering event occurs. The update triggering event includes a user requests information related to the operation of the vapor compression cycle <NUM>, replacement of a failed component (e.g. compressor, fan) of the vapor compression cycle <NUM> with a new component, the user requests to update the hybrid model <NUM>, and the like. For example, a failed component (e.g. compressor, fan) of the vapor compression cycle <NUM> may be replaced during maintenance service by an independent contractor. The specification information of the newly installed component can be used to update the hybrid model <NUM> on the remote server <NUM> to mimic the physical vapor compression cycle <NUM> as closely as possible.

For instance, when an update triggering event occurs, the remote server <NUM> executes the constrained Kalman smoother <NUM> over the observed variables collected over multiple instances of time to jointly estimate the parameters of the physics-based model <NUM> and the states of the vapor compression cycle <NUM> to minimize the cost function comprised of the residual errors of the state transitions predicted by the physics-based model <NUM>, and the residual errors between observed variables estimated by the constrained Kalman smoother for the multiple instances of time and the corresponding collected observed variables.

Further, the remote server <NUM> updates the data driven model <NUM> to minimize a difference between the states estimated by executing the constrained Kalman smoother <NUM> over the observed variables collected over multiple instances of time and the states predicted by the physics-based model <NUM>. The remote server <NUM> further updates the hybrid model with the estimated parameters of the physics-based model <NUM> and the updated data driven model.

The remote server <NUM> further transmits the updated hybrid model to the controller <NUM> via the network <NUM>. The controller <NUM> receives the updated hybrid model and controls the operation of the vapor compression cycle using the received updated hybrid model.

The cloud-based architecture <NUM> is advantageous. For example, only limited computational resources are required to be co-located with the vapor compression cycle <NUM>, and appropriate computational resources can be easily adjusted and scaled in the remote server <NUM>, i.e., cloud. In addition, both the data and the estimates of the states of the vapor compression cycle <NUM> can be simultaneously used in a variety of different contexts, including but not limited to equipment service or maintenance scheduling, or use in development of next-generation systems. According to some embodiments, the estimates of the states of the vapor compression cycle <NUM> may indicate a need for equipment maintenance that is not readily apparent from measured data. The cloud-based architecture <NUM> may make such information readily and asynchronously available to service companies so that they can automatically follow-up with a user and schedule a maintenance call. In addition, the information provided to the service company enables use of precise diagnostics and service tools.

In an embodiment, the constrained Kalman smoother that is executed to estimate the parameters θ* of the physics-based model <NUM> and the state s <MAT> is a constrained ensemble Kalman smoother. The constrained ensemble Kalman smoother is described below in <FIG>.

<FIG> shows a block diagram of a method <NUM> for estimation of the states using the constrained ensemble Kalman smoother, according to some embodiments of the present disclosure.

The method <NUM> begins with data <NUM> from the operation of the vapor compression cycle <NUM> and a model <NUM> of the vapor compression cycle <NUM>. The data <NUM> includes a set of control inputs u and measurements, i.e. observed variables y from the sensors installed in the vapor compression cycle <NUM>. According to an embodiment, the model <NUM> of the vapor compression cycle <NUM> describes both evolution of state variables via a function f and the measurements as a function of the state variables via a second function h which may be potentially a nonlinear function.

At block <NUM>, the data <NUM> and the model <NUM> are used to initialize the unconstrained ensemble Kalman smoother with an initial ensemble of states <MAT> of size M. One approach for initializing the constrained ensemble Kalman smoother is to determine a consistent initialization of the model <NUM> given a user-specified set of initial conditions, and then perturb initial states with a stochastic set of perturbations with an estimated model covariance.

At block <NUM>, the state variables that are transformed into a range of the covariance r(i)* are updated over a smoothing window for every measurement in the data <NUM> by solving the constrained optimization problem. The transformed corrections are then transformed back into coordinate system of original state variables to obtain corrected augmented state estimates. Although this embodiment of the constrained smoothing method <NUM> is not constructed for real-time use, the solution of the constrained optimization problem in block <NUM> for every sample at each available data point may be computationally expensive. To this end, in some embodiments, the update step in block <NUM> can be replaced with a two-stage process, wherein in a first stage, the update is performed without constraints. After this update is applied, in a second stage, the constrained optimization problem in block <NUM> is solved only if the corrected augmented state estimates obtained in the first stage violate the constraints. The second stage is skipped if the constraints are satisfied after the corrections in the first stage.

While there may be N data points available, either all N data points may be used or a smaller set may be used in the smoothing window. For example, the state updates in the coordinate system of the range of the covariance at the first measurement time k = <NUM> are calculated while applying constraints and then transformed back into the coordinate system of the original state variables.

At block <NUM>, it is checked if the measurement time k = <NUM> is the time instant of the last data point if the end of the available data has been reached. Since the measurement time k = <NUM> is not equal to N, at block <NUM>, nonlinear model f is then solved forward from the first to the second measurement time k = <NUM>, and then the state variables are once again updated to account for the measurements at time k = <NUM>.

At time k = <NUM>, state corrections are applied to the state estimates at both times k = <NUM> and k = <NUM> and the constraints are enforced at both times k = <NUM> and k = <NUM>. This ensures that the state estimates at time k = <NUM> account for the measurements at time k = <NUM> and that the constraints will be satisfied at both times. In such an iteration, the length of data over which the state estimates will gradually increase as additional data is incorporated until all of the N data points are incorporated, and all of the state variables will be continually updated to reflect the new information provided by the data points that are sequentially added to the growing smoothing window. Increases in the length of the smoothing window as more data points are incorporated may pose serious computational challenges such as prohibitive memory requirements and computation time.

To this end, the length of a smoothing window, i.e., number of data points l = k - m + <NUM> in a smoothing window may be fixed to be a constant after sufficient number of data points are incorporated in the corrected state estimates in some embodiments, so that the data points available at times prior to m are not considered in the update step <NUM>. Once each data point is processed, at block <NUM>, a final set of smoothed state estimates is output. The final set of smoothed state estimates includes the states of the vapor compression cycle <NUM>.

In another embodiment, the constrained ensemble Kalman smoother <NUM> is adapted to jointly estimate the states and the parameters of the physics-based model. Within the scope of the joint estimation, the constrained ensemble Kalman smoother <NUM> estimates an augmented vector which includes both the states and the parameters of the physics-based model, while following a similar general flow of the method of the constrained ensemble Kalman smoother <NUM>.

Further, an overall method for controlling the operation of the vapor compression cycle <NUM> is described below in <FIG>.

<FIG> shows a block diagram of a method <NUM> for controlling the operation of the vapor compression cycle <NUM>, according to an embodiment of the present disclosure. At block <NUM>, the method <NUM> includes collecting a digital representation of observed variables and control inputs of the operation of the vapor compression cycle <NUM> over multiple instances of time. At block <NUM>, the method <NUM> includes executing the constrained Kalman smoother <NUM> over the observed variables collected over multiple instances of time to jointly estimate the parameters of the physics-based model <NUM> and the states of the vapor compression cycle <NUM> to minimize the cost function comprised of the residual errors of the state transitions predicted by the physics-based model <NUM>, and the residual errors between observed variables estimated by the constrained Kalman smoother for the multiple instances of time and the corresponding collected observed variables.

At block <NUM>, the method <NUM> includes updating the data driven model <NUM> to minimize a difference between the states estimated by executing the constrained Kalman smoother <NUM> over the observed variables collected over multiple instances of time and the states predicted by the physics-based model <NUM>. At block <NUM>, the method <NUM> includes updating the hybrid model <NUM> with the estimated parameters of the physics-based model <NUM> and the updated data driven model. At block <NUM>, the method <NUM> includes controlling the operation of the vapor compression cycle <NUM> using the updated hybrid model.

<FIG> shows a block diagram for controlling the operation of the vapor compression cycle <NUM> using the updated hybrid model, according to an embodiment of the present disclosure. At block <NUM>, unobserved variables of the vapor compression cycle <NUM> are determined using the updated hybrid model. The unobserved variables correspond to the variables that are difficult to measure or cannot be measured directly. For example, the unobserved variables may include the amount of refrigerant in the vapor compression cycle <NUM>, thermal energy delivered by one or more heat exchangers of the vapor compression cycle <NUM>, and a thermodynamic quality of the refrigerant flow at an inlet or outlet of one or more heat exchangers of the vapor compression cycle <NUM>. For instance, the unobserved variables <MAT> of the vapor compression cycle <NUM> are determined from the parameters θ* of the physics-based model <NUM> of the updated hybrid model, the states <MAT>, and known control inputs as follows <MAT> where g(. ) is a known function that maps its arguments to the unobserved variables.

Further, at block <NUM>, control inputs to the actuators of the vapor compression cycle <NUM> are determined based on the updated hybrid model and the unobserved variables. The control inputs may include one or more of the speed of the compressor <NUM>, the speed of the fan <NUM>, the position of the expansion valve <NUM>, and the speed of the fan <NUM>.

Furthermore, at block <NUM>, actuators of the vapor compression cycle <NUM> are controlled according to the determined control inputs to control the operation of the vapor compression cycle <NUM>.

<FIG> shows a schematic diagram of a computing device that can be used for implementing the controller <NUM> and the method <NUM> of the present disclosure. The computing device <NUM> includes a power source <NUM>, a processor <NUM>, a memory <NUM>, a storage device <NUM>, all connected to a bus <NUM>. Further, a high-speed interface <NUM>, a low-speed interface <NUM>, high-speed expansion ports <NUM> and low speed connection ports <NUM>, can be connected to the bus <NUM>. In addition, a low-speed expansion port <NUM> is in connection with the bus <NUM>. Further, an input interface <NUM> can be connected via the bus <NUM> to an external receiver <NUM> and an output interface <NUM>. A receiver <NUM> can be connected to an external transmitter <NUM> and a transmitter <NUM> via the bus <NUM>. Also connected to the bus <NUM> can be an external memory <NUM>, external sensors <NUM>, machine(s) <NUM>, and an environment <NUM>. Further, one or more external input/output devices <NUM> can be connected to the bus <NUM>. A network interface controller (NIC) <NUM> can be adapted to connect through the bus <NUM> to a network <NUM>, wherein data or other data, among other things, can be rendered on a third-party display device, third party imaging device, and/or third-party printing device outside of the computing device <NUM>.

The memory <NUM> can store instructions that are executable by the computing device <NUM> and any data that can be utilized by the methods and systems of the present disclosure. The memory <NUM> can include random access memory (RAM), read only memory (ROM), flash memory, or any other suitable memory systems. The memory <NUM> can be a volatile memory unit or units, and/or a non-volatile memory unit or units.

The storage device <NUM> can be adapted to store supplementary data and/or software modules used by the computer device <NUM>. The storage device <NUM> can include a hard drive, an optical drive, a thumb-drive, an array of drives, or any combinations thereof. Further, the storage device <NUM> can contain a computer-readable medium, such as a floppy disk device, a hard disk device, an optical disk device, or a tape device, a flash memory or other similar solid-state memory device, or an array of devices, including devices in a storage area network or other configurations. The instructions, when executed by one or more processing devices (for example, the processor <NUM>), perform one or more methods, such as those described above.

The computing device <NUM> can be linked through the bus <NUM>, optionally, to a display interface or user Interface (HMI) <NUM> adapted to connect the computing device <NUM> to a display device <NUM> and a keyboard <NUM>, wherein the display device <NUM> can include a computer monitor, camera, television, projector, or mobile device, among others. In some implementations, the computer device <NUM> may include a printer interface to connect to a printing device, wherein the printing device can include a liquid inkjet printer, solid ink printer, large-scale commercial printer, thermal printer, UV printer, or dye-sublimation printer, among others.

The high-speed interface <NUM> manages bandwidth-intensive operations for the computing device <NUM>, while the low-speed interface <NUM> manages lower bandwidth-intensive operations. Such allocation of functions is an example only. In some implementations, the high-speed interface <NUM> can be coupled to the memory <NUM>, the user interface (HMI) <NUM>, and to the keyboard <NUM> and the display <NUM> (e.g., through a graphics processor or accelerator), and to the high-speed expansion ports <NUM>, which may accept various expansion cards via the bus <NUM>. In an implementation, the low-speed interface <NUM> is coupled to the storage device <NUM> and the low-speed expansion ports <NUM>, via the bus <NUM>. The low-speed expansion ports <NUM>, which may include various communication ports (e.g., USB, Bluetooth, Ethernet, wireless Ethernet) may be coupled to the one or more input/output devices <NUM>. The computing device <NUM> may be connected to a server <NUM> and a rack server <NUM>. The computing device <NUM> may be implemented in several different forms. For example, the computing device <NUM> may be implemented as part of the rack server <NUM>.

The description provides exemplary embodiments only, and is not intended to limit the scope, applicability, or configuration of the disclosure. Rather, the following description of the exemplary embodiments will provide those skilled in the art with an enabling description for implementing one or more exemplary embodiments. Contemplated are various changes that may be made in the function and arrangement of elements without departing from the scope of the invention as set forth in the appended claims.

Specific details are given in the following description to provide a thorough understanding of the embodiments. However, understood by one of ordinary skill in the art can be that 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 in order to avoid obscuring the embodiments. Further, like reference numbers and designations in the various drawings indicated like elements.

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.

Further, embodiments of the present disclosure and the functional operations described in this specification can be implemented in digital electronic circuitry, in tangibly-embodied computer software or firmware, in computer hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. Further some embodiments of the present disclosure can be implemented as one or more computer programs, i.e., one or more modules of computer program instructions encoded on a tangible non transitory program carrier for execution by, or to control the operation of, data processing apparatus. Further still, program instructions can be encoded on an artificially generated propagated signal, e.g., a machine-generated electrical, optical, or electromagnetic signal, which is generated to encode information for transmission to suitable receiver apparatus for execution by a data processing apparatus.

According to embodiments of the present disclosure the term "data processing apparatus" can encompass all kinds of apparatus, devices, and machines for processing data, including by way of example a programmable processor, a computer, or multiple processors or computers.

A computer program (which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code) can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and it can be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a computing environment.

Claim 1:
A controller (<NUM>) for controlling an operation of a vapor compression cycle (<NUM>) based on a hybrid model (<NUM>) of dynamics of the vapor compression cycle (<NUM>) including a physics-based model (<NUM>) and a data driven model (<NUM>), wherein the physics-based model (<NUM>) is configured to predict transitions of states of the vapor compression cycle (<NUM>) in accordance with observed variables and a control input to the vapor compression cycle (<NUM>) based on parameters of the physics-based model (<NUM>), and wherein the data driven model (<NUM>) is trained with machine learning to estimate residual errors of the state transitions predicted by the physics-based model (<NUM>), the controller (<NUM>) comprising: a processor (<NUM>); and a memory (<NUM>) having instructions stored thereon that, when executed by the processor (<NUM>), cause the controller (<NUM>) to:
collect a digital representation of observed variables and control inputs of the operation of the vapor compression cycle (<NUM>) over multiple instances of time;
execute a constrained Kalman smoother over the observed variables collected over multiple instances of time to jointly estimate the parameters of the physics-based model (<NUM>) and states of the vapor compression cycle (<NUM>)by minimizing a cost function comprised of residual errors of the state transitions predicted by the physics-based model for the multiple instances of time, and residual errors between observed variables estimated by the constrained Kalman smoother for the multiple instances of time and the corresponding collected observed variables;
update the data driven model (<NUM>) to minimize a difference between the states estimated by executing the constrained Kalman smoother over the observed variables collected over multiple instances of time and the states predicted by the physics-based model (<NUM>);
update the hybrid model (<NUM>) with the estimated parameters of the physics-based model (<NUM>) and the updated data driven model (<NUM>); and
control the operation of the vapor compression cycle (<NUM>) using the updated hybrid model (<NUM>).