EP4670015A1

EP4670015A1 - Embedded sparse identification of nonlinear dynamics based on a recursive least square algorithm

Info

Publication number: EP4670015A1
Application number: EP23706724.4A
Authority: EP
Inventors: Iron TESSARO
Original assignee: Volvo Truck Corp
Current assignee: Volvo Truck Corp
Priority date: 2023-02-20
Filing date: 2023-02-20
Publication date: 2025-12-31
Also published as: WO2024175171A1

Abstract

A computer system (100) comprising processing circuitry (102) configured to: provide a sparse identification of nonlinear dynamics model SINDy, (104) embedded in the computer system, the SINDY-model includes a set of mathematical functions weighted by a set of coefficients (C), the SINDY-model is configured to define a relationship between sensor values corresponding to at least one first physical quantity and soft sensor values corresponding to a second physical quantity, acquire online sensor values (106) corresponding to the at least one first physical quantity while the soft sensor is operative and embedded in the computer system, apply a recursive least square algorithm, RLS-algorithm (108), to the online sensor values and the SINDY-model to find updated set of coefficients (110) for the SINDy-model, adjust the set of coefficients of the embedded SINDy-model according to the updated set of coefficients.

Description

EMBEDDED SPARSE IDENTIFICATION OF NONLINEAR DYNAMICS BASED ON A RECURSIVE LEAST SQUARE ALGORITHM

TECHNICAL FIELD

[0001] The disclosure relates generally to soft sensors. In particular aspects, the disclosure relates to embedded sparse identification of nonlinear dynamics based on a recursive least square algorithm. The disclosure can be applied to any embedded system but herein exemplified in embedded systems of heavy-duty vehicles, such as trucks, buses, and construction equipment, among other vehicle types. Although the disclosure may be described with respect to a particular vehicle, the disclosure is not restricted to any particular vehicle.

BACKGROUND

[0002] Soft sensors are software-based sensors that generate physical quantities without the need for direct sensor measurements. Soft sensors in embedded system software, for example, in automotive system software, use sensor measurements of different physical quantities to calculate an estimate of the desired physical quantity using mathematical functions and/or algorithms. One approach for creating soft sensors is to use neural networks and their variations to generate a “black box” solution where the dynamics, i.e., underlying equations, are unknown. These solutions can be unreliable and difficult to troubleshoot, however, and may also consume significant computing overhead, e.g., memory and processor power, in the embedded system. Furthermore, online training of such neural networks requires significant computational resources.

[0003] Thus, there is a need for a more customizable and serviceable soft sensor with lower computing overhead requirements.

SUMMARY

[0004] According to a first aspect of the disclosure, there is provided a computer system comprising processing circuitry configured to: provide a sparse identification of nonlinear dynamics model (SINDy) embedded in the computer system, the SINDY-model includes a set of mathematical functions weighted by a set of coefficients, the SINDY-model is configured to define a relationship between sensor values corresponding to at least one first physical quantity and soft sensor values corresponding to a second physical quantity, acquire online sensor values corresponding to the at least one first physical quantity while the soft sensor is operative and embedded in the computer system, apply a recursive least square algorithm (RLS-algorithm) to the online sensor values and the SINDY-model to find updated set of coefficients for the SINDy-model, adjust the set of coefficients of the embedded SINDy -model according to the updated set of coefficients.

[0005] The first aspect of the disclosure may seek to update the SINDY-model online to improve the embedded SINDY-model. A technical benefit may include that the SINDY- model is improved in a computationally efficient way. Accordingly, a method for allowing SINDy to learn online is presented, which saves lots of computational resources as there is less need for prior data for learning. By using the efficient learning algorithm recursive least squares, the SINDy-model can learn in real-time with data collected during the usage.

[0006] In some examples, including in at least one preferred example, optionally the processing circuitry further being configured to: receive a sensor value from at least one first sensor corresponding to the at least one first physical quantity; and generate, based on the received sensor value and the embedded SINDy-model, a soft sensor value. A technical benefit may include the generation of a computationally efficient soft sensor which reduces the need for physical sensors.

[0007] In some examples, including in at least one preferred example, optionally the second physical quantity may comprise at least one of: a temperature of a component of a mechanical system, a peak in-cylinder pressure (PCP) for a vehicle engine, and a turbocharger speed for a vehicle engine. Thus, the soft sensor may be applied to various quantities.

[0008] In some examples, including in at least one preferred example, optionally the RLS-algorithm may be embedded in the computer system. A technical benefit may include improved efficiency in updating the SINDY-model.

[0009] In some examples, including in at least one preferred example, optionally the set of mathematical function may comprise nonlinear functions. A technical benefit may include improved accuracy and complexity in the soft sensor.

[0010] In some examples, including in at least one preferred example, optionally the set of mathematical functions may comprise at least one of a polynomial function and a trigonometric function. A technical benefit may include improved accuracy and complexity in the soft sensor.

[0011] In some examples, including in at least one preferred example, optionally the processing circuitry may be configured to apply the online recursive least square algorithm each time new online senor values are acquired to update the set of coefficients in real-time. A technical benefit may include that the SINDY-model is continuously updated and improved every time new data is acquired.

[0012] In some examples, including in at least one preferred example, optionally the set of mathematical function may comprise nonlinear functions including at least one of a polynomial function and a trigonometric function, the second physical quantity may comprise at least one of: a temperature of a component of a mechanical system, a peak in-cylinder pressure (PCP) for a vehicle engine, and a turbocharger speed for a vehicle engine, the RLS- algorithm may be embedded in the computer system, and the processing circuitry may be configured to: receive a sensor value from at least one first sensor corresponding to the at least one first physical quantity; generate, based on the received sensor value and the embedded SINDy -model, a soft sensor value, and apply the online recursive least square algorithm each time new online senor values are acquired to update the set of coefficients in real-time.

[0013] There is further provided a vehicle comprising the computer system according to the first aspect.

[0014] According to a second aspect of the disclosure, providing, by a processor device of a computer system, a sparse identification of nonlinear dynamics model (SINDy) embedded in the computer system, the SINDY-model includes a set of mathematical functions weighted by a set of coefficients, the SINDY-model is configured to define a relationship between sensor values corresponding to at least one first physical quantity and soft sensor values corresponding to a second physical quantity, acquiring, by the processor device, online sensor values corresponding to the at least one first physical quantity while the soft sensor is operative and embedded in the computer system, applying, by the processor device, a recursive least square algorithm (RLS-algorithm) to the online sensor values and the SINDY-model to find updated set of coefficients for the SINDy-model, adjusting, by the processor device, the set of coefficients of the embedded SINDy-model according to the updated set of coefficients. [0015] The second aspect of the disclosure may seek to update the SINDY-model online to improve the embedded SINDY-model. A technical benefit may include the SINDY-model is improved in a computationally efficient way. Accordingly, a method for allowing SINDy to learn online is presented, which saves lots of computational resources as there is less need for prior data for learning. By using the efficient learning algorithm Recursive Least Squares, the SINDy can learn in real-time with data collected during the usage.

[0016] In some examples, including in at least one preferred example, optionally receiving, by the processor device, a sensor value from at least one first sensor corresponding to the at least one first physical quantity; and generating, by the processor device, based on the received sensor value and the embedded SINDy -model, a soft sensor value. A technical benefit may include the provision of a computationally efficient soft sensor which reduces the need for physical sensors.

[0017] In some examples, including in at least one preferred example, optionally the second physical quantity may comprise at least one of: a temperature of a component of a mechanical system or electrical system, a peak in-cylinder pressure (PCP) for a vehicle engine, and a turbocharger speed for a vehicle engine. Thus, the soft sensor may be applied to various quantities.

[0018] In some examples, including in at least one preferred example, optionally the RLS-algorithm may be embedded in the computer system. A technical benefit may include improved efficiency in updating the SINDY-model.

[0019] In some examples, including in at least one preferred example, the set of mathematical function may comprise nonlinear functions. A technical benefit may include improved accuracy and complexity in the soft sensor.

[0020] In some examples, including in at least one preferred example, optionally the set of mathematical functions may comprise at least one of a polynomial function and a trigonometric function. A technical benefit may include improved accuracy and complexity in the soft sensor.

[0021] In some examples, including in at least one preferred example, optionally the method may comprise applying, by the processor device, the online recursive least square algorithm each time new online senor values are acquired to update the set of coefficients in real-time. A technical benefit may include that the SINDY-model is continuously updated and improved every time new data is acquired. [0022] In some examples, including in at least one preferred example, optionally, the method may comprise embedding, by the processor device, machine readable instructions according to the RLS-algorithm and the SINDy-model in the computer system of a vehicle. [0023] In some examples, including in at least one preferred example, optionally, the set of mathematical function comprises nonlinear functions including at least one of a polynomial function and a trigonometric function, the second physical quantity comprises at least one of: a temperature of a component of a mechanical system, a peak in-cylinder pressure (PCP) for a vehicle engine, and a turbocharger speed for a vehicle engine, the method may comprise: embedding, by the processor device, the RLS-algorithm in the computer system, receiving, by the processor device, a sensor value from at least one first sensor corresponding to the at least one first physical quantity; generating, by the processor device, based on the received sensor value and the embedded SINDy-model, a soft sensor value, and applying, by the processor device, the online recursive least square algorithm each time new online senor values are acquired to update the set of coefficients in real-time.

[0024] Furthermore, a computer program product is provided comprising program code for performing, when executed by the processing circuitry, the method of any of the second aspect.

[0025] In addition, a non-transitory computer-readable storage medium is provided comprising instructions, which when executed by the processing circuitry, cause the processing circuitry to perform the method of any of the second aspect.

[0026] The disclosed aspects, examples (including any preferred examples), and/or accompanying claims may be suitably combined with each other as would be apparent to anyone of ordinary skill in the art. Additional features and advantages are disclosed in the following description, claims, and drawings, and in part will be readily apparent therefrom to those skilled in the art or recognized by practicing the disclosure as described herein.

[0027] There are also disclosed herein computer systems, control units, code modules, computer-implemented methods, computer readable media, and computer program products associated with the above discussed technical benefits.

BRIEF DESCRIPTION OF THE DRAWINGS

[0028] FIG. 1 is an exemplary system diagram of a computer system according to an example. [0029] FIG. 2 is an exemplary truck according to an example vehicle.

[0030] FIG. 3 is a flow-chart of an exemplary method to update a SINDy-model according to an example.

[0031] FIG. 4 is another view of FIG. 1, according to an example system.

[0032] FIG. 5 is a flow chart of an exemplary method to update a SINDy-model according to an example.

[0033] FIG. 6 is a schematic diagram of an exemplary computer system for implementing examples disclosed herein, according to an example.

DETAILED DESCRIPTION

[0034] The detailed description set forth below provides information and examples of the disclosed technology with sufficient detail to enable those skilled in the art to practice the disclosure.

[0035] In various systems where sensors are used for measuring a physical quantity, the system space requirements, complexity, and cost can be reduced if the number of sensors can be kept at a minimum. Although methods including e.g., neural networks seem suitable, they require a relatively high processing power.

[0036] In the present description, a computationally less resourceful method is presented that relies on sparse identification of nonlinear dynamics, SINDy, with an advantageous online training using recursive least squares to update coefficients of the SINDy model. The presented method provides for a soft sensor which saves computational resources and, at the same time, allows for online training or adaption of the SINDy model.

[0037] FIG. 1 is an exemplary system diagram of a computer system 100 according to an example. The computer system 100 comprises processing circuitry 102 configured to provide a SINDy-model 104 embedded in the computer system 100. The SINDy model 104 may thus be stored on a non-volatile memory 105 accessible to the processing circuitry 102.

[0038] The method described herein may be applied to any embedded system, i.e., a system having a combination of hardware including physical sensors, and software.

[0039] The SINDY-model 104 includes a set of mathematical functions weighted by a set of coefficients. The SINDY-model 104 is configured to define a relationship between sensor values corresponding to at least one first physical quantity and soft sensor values corresponding to a second physical quantity. [0040] The first physical quantity and the second physical quantity are correlated in some way such as belonging to the same dynamic system.

[0041] When the SINDy model 104 is initially generated, a library of mathematical functions is assumed. The library of functions may include polynomial functions, trigonometric functions, or other functions of varying complexity, as desired. The data- driven algorithm of SINDy leverages the fact many dynamical systems have relatively few terms on their governing equations. As a result, a library of candidate nonlinear functions, e.g., Q(X) = [i x X² • • • ^siⁿ (X) sin (2X) . . . ], where X is an input dataset, can be analyzed using a training dataset to generate an accurate and efficient mathematical function for the soft sensor. The library of candidate dynamics may be extended to include support for more general nonlinearities, such as polynomials, trigonometric functions, etc., and may also incorporate partial knowledge from the physics of the dynamic system.

[0042] For example, the training dataset may include measurement samples x grouped as columns in a matrix X, representing the sensor data input, and measurement samples y grouped as a column vector y, representing the physical quantity to be estimated by the soft sensor including the SINDy -model:

[0044] The SINDy -model is a linear combination, or weighted combination of the nonlinear functions Q(X). Thus, during a prior training step for generating the subsequently provided SINDy -model, the coefficients in the matrix C are determined to find the optimal coefficients C. For example, a dynamic system sensor can be written as a linear system, e.g., DSINDY ⁼ Q(X) • C, where C is a matrix of coefficients indicating how much of the candidate nonlinear functions in "Q(X)" must be combined (i.e., summed) to represent the dynamic system. With D_SINDY as the target dataset y, the system can be approximated for C by leftmultiplying both sides of the linear system by the Moore-Penrose pseudoinverse of "Q(X)", i.e., Q(X - y = Q(X - Q(X) • C, resulting in C ~ Q(X - y Q(X) • C ~ y. The coefficients C may be optimized to have as few terms as possible while still meeting predetermined performance and efficiency requirements. [0045] The SINDy model that is provided as the embedded model 104 is the product of the coefficients C by the inputs organized as in Q(X), but without the terms removed from C during its optimization.

[0046] The computer system 100 is connected to physical sensors 107 of the dynamic system that is being measured. When the soft sensor 104 is operative and embedded in the computer system 100, an online sensor value 106 may be acquired corresponding to the at least one first physical quantity. The sensor value 106 may be received from the physical sensor 107. The at least one sensor value 106 is input to the SINDy -model 104, and a soft sensor value 109 may be provided as an output.

[0047] The computer system 100 includes an embedded recursive least square algorithm, RLS-algorithm 108, stored on a non-volatile memory 111. The non-volatile memories 111 and 105 may be the same memory device. The RLS-algorithm 108 may be applied to the online sensor values 106 and the SINDY-model 104 to find updated set of coefficients C’ for the SINDy -model 104. The updated coefficients C’ replace the previous coefficients C so that the SINDy -model 104 is updated online.

[0048] The computer system 100 may receive new sensor values continuously as the soft sensor is operative, and each time new sensor values are received from the sensors 107, the online recursive least square algorithm is applied to update the set of coefficients C in realtime. The latest set of coefficients are stored on the memory 105.

[0049] The online recursive least square algorithm may be generally formulated as:

[0050] Initialize: (bo appears)

[0051] P_o = G4 )^-1>

[0052] x₀ = P₀ A^TM,

[0053] For each k=l, 2, 3,... do:

[0054] (6k appears)

[0056] K_k = P_kA^T _k

[0057] x_k = x_k- + K_k b_k - A_kx_k_ , end.

[0058] Here, x is the coefficients of the SINDy-model, b is newly received sensor data, and A is the weighted combination of the non-linear functions Q(X).

[0059] The computer system 100 may be implemented in a vehicle 1 such as a truck shown in fig. 2 further comprising at least one sensor configured to measure a physical quantity. Although the vehicle in fig. 1 is depicted as a heavy-duty truck, examples of the present disclosure may as well be implemented in other types of vehicles, such as in busses, light-weight trucks, passenger cars, construction equipment, industrial applications, and marine applications including e.g. vessels or ships.

[0060] The examples present disclosure may be applied to any embedded system involving physical sensors.

[0061] The first physical quantity and the second physical quantity are correlated in some way such as belonging to the same dynamic system. For example, the dynamic system may be a mechanical or electrical system, not necessarily related to a vehicle. However, some examples are provided below in the case of vehicles.

[0062] In one example implementation, the second physical quantity comprises at least one of a temperature of a component of a mechanical system, fuel flow, NOx, concentration in an aftertreatment system, a peak in-cylinder pressure (PCP) for a vehicle engine, and a turbocharger speed for a vehicle engine.

[0063] For example, in one possible implementation, the inputs related to first physical quantities may include: Engine speed [rad/s], Torque [Nm], Ambient air pressure [kPa], Boost pressure [kPa], Boost temperature [K], Exhaust manifold pressure [kPa], Intake airflow [kg/s]. For this combination of first physical quantities, the second physical quantity measured by the soft sensor may be turbo speed (rad/s).

[0064] In a further example, in one possible implementation, the inputs related to first physical quantities may include: Fuel rail pressure [kPa], and engine speed [rad/s]. For this combination of first physical quantities, the second physical quantity measured by the soft sensor may be fuel flow [kg/s],

[0065] In a further example, in one possible implementation, the inputs related to first physical quantities may include: Fuel rail pressure [kPa], Advance angle [rad], Boost temperature [K], and Burned air fraction [ratio]. For this combination of first physical quantities, the second physical quantity measured by the soft sensor may be NOx concentration [ratio],

[0066] In a further example, in one possible implementation, the inputs related to first physical quantities may include: Torque [Nm], Engine speed [rad/s], Boost pressure [kPa], Exhaust manifold pressure [kPa], Intake airflow [kg/s]. For this combination of first physical quantities, the second physical quantity measured by the soft sensor may be In-cylinder pressure [kPa],

[0067] The above examples relate to vehicle applications. However, the method described herein is not limited to vehicle applications, as it can be used in any application that contains an embedded system.

[0068] FIG. 3 is a flow-chart of method steps to update an SINDy-model according to examples.

[0069] In step SI 02, providing, by a processor device of a computer system, a sparse identification of nonlinear dynamics model (SINDy) embedded in the computer system, the SINDY-model includes a set of mathematical functions weighted by a set of coefficients. The SINDY-model is configured to define a relationship between sensor values corresponding to at least one first physical quantity and soft sensor values corresponding to a second physical quantity. The SINDy-model is added in the embedded software of the system.

[0070] For example, the SINDy-model may be applied as a soft sensor in an engine electronic control unit (EECU) for an automotive application, although it can be used for any other sensor connected to any embedded system. For instance, data from a temperature sensor, along with other sensors already connected to the embedded system, can be used to train the SINDy model, thus generating a soft sensor that mimics the temperature sensor based on a nonlinear dynamical system, represented by an equation (i.e., SINDy’ s algorithm output), that relates the input sensors used to the target temperature sensor. After reasonable validation tests, the hardwired temperature sensor can be removed, or the new soft sensor can be used to monitor its behavior. This type of already trained soft sensor is provided in step S102.

[0071] The set of mathematical functions of the SINDy-model may comprises nonlinear functions such as at least one of a polynomial function and a trigonometric function.

[0072] In step SI 04, acquiring, by the processor device, online sensor values corresponding to the at least one first physical quantity while the soft sensor is operative and embedded in the computer system. The online sensor values are acquired from physical hardware sensors of the system that is being monitored. The first physical quality and the second physical quantity are correlated and relate to the same dynamic mechanical or electrical system. [0073] In step SI 06, applying, by the processor device, a recursive least square algorithm (RLS-algorithm) to the online sensor values and the SINDY-model to find updated set of coefficients for the SINDy -model. Accordingly, variations of the behavior of the soft sensor may then be used by the recursive least squares algorithm to update the coefficients of the SINDy terms that represent the soft sensor in real time. The RLS-algorithm may be embedded in the computer system. Thus, the method may comprise embedding, by the processor device, machine readable instructions according to the RLS-algorithm and the SINDy-model in the computer system of a vehicle or another embedded system.

[0074] In step SI 08, adjusting, by the processor device, the set of coefficients of the embedded SINDy-model according to the updated set of coefficients. The online recursive least square algorithm is applied each time new online senor values are acquired to thereby update the set of coefficients in real-time.

[0075] When the dynamic system is turned off, for example when the vehicle stops and is turned off, or the embedded system is turned off, the method ends, and the most recent set of coefficients of the embedded SINDy-model are stored in a memory 105.

[0076] During use of the soft sensor, and concurrently with the updating of the coefficients of the SINDY-model using the RLS, the processor device receives a sensor value from at least one first sensor corresponding to the at least one first physical quantity.

Subsequently, generating, by the processor device, based on the received sensor value and the embedded SINDy-model, a soft sensor value.

[0077] In one possible implementation, the set of mathematical function comprises nonlinear functions including at least one of a polynomial function and a trigonometric function, the second physical quantity comprises at least one of a temperature of a component of a mechanical system, a peak in-cylinder pressure (PCP) for a vehicle engine, and a turbocharger speed for a vehicle engine, the method further comprises: embedding, by the processor device, the RLS-algorithm in the computer system, receiving, by the processor device, a sensor value from at least one first sensor corresponding to the at least one first physical quantity; generating, by the processor device, based on the received sensor value and the embedded SINDy-model, a soft sensor value, and applying, by the processor device, the online recursive least square algorithm each time new online senor values are acquired to update the set of coefficients in real-time. [0078] There is further provided a computer program product 120 comprising program code for performing the method, when executed by the processing circuitry 102.

[0079] A non-transitory computer-readable storage medium 122 comprising instructions, which when executed by the processing circuitry, cause the processing circuitry to perform the method.

[0080] FIG. 4 is another view of FIG. 1, according to an example. FIG. 4 is an exemplary system diagram of a computer system 100 according to an example. The computer system 100 comprises processing circuitry 102 configured to: provide a sparse identification of nonlinear dynamics model SINDy, 104 embedded in the computer system, the SINDY- model includes a set of mathematical functions weighted by a set of coefficients, the SINDY- model is configured to define a relationship between sensor values corresponding to at least one first physical quantity and soft sensor values corresponding to a second physical quantity, acquire online sensor values 106 corresponding to the at least one first physical quantity while the soft sensor is operative and embedded in the computer system, apply a recursive least square algorithm, RLS-algorithm 108, to the online sensor values and the SINDY-model to find updated set of coefficients 110 for the SINDy -model, adjust the set of coefficients of the embedded SINDy -model according to the updated set of coefficients.

[0081] FIG. 5 is a flow chart of a method according to an example. In step SI 02, providing, by a processor device of a computer system, a sparse identification of nonlinear dynamics model (SINDy) embedded in the computer system, the SINDY-model includes a set of mathematical functions weighted by a set of coefficients, the SINDY-model is configured to define a relationship between sensor values corresponding to at least one first physical quantity and soft sensor values corresponding to a second physical quantity. IN step SI 04, acquiring, by the processor device, online sensor values corresponding to the at least one first physical quantity while the soft sensor is operative and embedded in the computer system. In step SI 06 applying, by the processor device, a recursive least square algorithm to the online sensor values and the SINDY-model to find updated set of coefficients for the SINDy-model. In step S108, adjusting, by the processor device, the set of coefficients of the embedded SINDy-model according to the updated set of coefficients.

[0082] FIG. 6 is a schematic diagram of a computer system 600 for implementing examples disclosed herein. The computer system 600 is adapted to execute instructions from a computer-readable medium to perform these and/or any of the functions or processing described herein. The computer system 600 may be connected (e.g., networked) to other machines in a LAN, an intranet, an extranet, or the Internet. While only a single device is illustrated, the computer system 600 may include any collection of devices that individually or jointly execute a set (or multiple sets) of instructions to perform any one or more of the methodologies discussed herein. Accordingly, any reference in the disclosure and/or claims to a computer system, computing system, computer device, computing device, control system, control unit, electronic control unit (ECU), processor device, processing circuitry, etc., includes reference to one or more such devices to individually or jointly execute a set (or multiple sets) of instructions to perform any one or more of the methodologies discussed herein. For example, control system may include a single control unit or a plurality of control units connected or otherwise communicatively coupled to each other, such that any performed function may be distributed between the control units as desired. Further, such devices may communicate with each other or other devices by various system architectures, such as directly or via a Controller Area Network (CAN) bus, etc.

[0083] The computer system 600 may comprise at least one computing device or electronic device capable of including firmware, hardware, and/or executing software instructions to implement the functionality described herein. The computer system 600 may include processing circuitry 602 (e.g., processing circuitry including one or more processor devices or control units), a memory 604, and a system bus 606. The computer system 600 may include at least one computing device having the processing circuitry 602. The system bus 606 provides an interface for system components including, but not limited to, the memory 604 and the processing circuitry 602. The processing circuitry 602 may include any number of hardware components for conducting data or signal processing or for executing computer code stored in memory 604. The processing circuitry 602 may, for example, include a general-purpose processor, an application specific processor, a Digital Signal Processor (DSP), an Application Specific Integrated Circuit (ASIC), a Field Programmable Gate Array (FPGA), a circuit containing processing components, a group of distributed processing components, a group of distributed computers configured for processing, or other programmable logic device, discrete gate or transistor logic, discrete hardware components, or any combination thereof designed to perform the functions described herein. The processing circuitry 602 may further include computer executable code that controls operation of the programmable device. [0084] The system bus 606 may be any of several types of bus structures that may further interconnect to a memory bus (with or without a memory controller), a peripheral bus, and/or a local bus using any of a variety of bus architectures. The memory 604 may be one or more devices for storing data and/or computer code for completing or facilitating methods described herein. The memory 604 may include database components, object code components, script components, or other types of information structure for supporting the various activities herein. Any distributed or local memory device may be utilized with the systems and methods of this description. The memory 604 may be communicably connected to the processing circuitry 602 (e.g., via a circuit or any other wired, wireless, or network connection) and may include computer code for executing one or more processes described herein. The memory 604 may include non-volatile memory 608 (e.g., read-only memory (ROM), erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), etc.), and volatile memory 610 (e.g., randomaccess memory (RAM)), or any other medium which can be used to carry or store desired program code in the form of machine-executable instructions or data structures and which can be accessed by a computer or other machine with processing circuitry 602. A basic input/output system (BIOS) 612 may be stored in the non-volatile memory 608 and can include the basic routines that help to transfer information between elements within the computer system 600.

[0085] The computer system 600 may further include or be coupled to a non-transitory computer-readable storage medium such as the storage device 614, which may comprise, for example, an internal or external hard disk drive (HDD) (e.g., enhanced integrated drive electronics (EIDE) or serial advanced technology attachment (SATA)), HDD (e.g., EIDE or SATA) for storage, flash memory, or the like. The storage device 614 and other drives associated with computer-readable media and computer-usable media may provide nonvolatile storage of data, data structures, computer-executable instructions, and the like.

[0086] Computer-code which is hard or soft coded may be provided in the form of one or more modules. The module(s) can be implemented as software and/or hard-coded in circuitry to implement the functionality described herein in whole or in part. The modules may be stored in the storage device 614 and/or in the volatile memory 610, which may include an operating system 616 and/or one or more program modules 618. All or a portion of the examples disclosed herein may be implemented as a computer program 620 stored on a transitory or non-transitory computer-usable or computer-readable storage medium (e.g., single medium or multiple media), such as the storage device 614, which includes complex programming instructions (e.g., complex computer-readable program code) to cause the processing circuitry 602 to carry out actions described herein. Thus, the computer-readable program code of the computer program 620 can comprise software instructions for implementing the functionality of the examples described herein when executed by the processing circuitry 602. In some examples, the storage device 614 may be a computer program product (e.g., readable storage medium) storing the computer program 620 thereon, where at least a portion of a computer program 620 may be loadable (e.g., into a processor) for implementing the functionality of the examples described herein when executed by the processing circuitry 602. The processing circuitry 602 may serve as a controller or control system for the computer system 600 that is to implement the functionality described herein. [0087] The computer system 600 may include an input device interface 622 configured to receive input and selections to be communicated to the computer system 600 when executing instructions, such as from a keyboard, mouse, touch-sensitive surface, etc. Such input devices may be connected to the processing circuitry 602 through the input device interface 622 coupled to the system bus 606 but can be connected through other interfaces, such as a parallel port, an Institute of Electrical and Electronic Engineers (IEEE) 1394 serial port, a Universal Serial Bus (USB) port, an IR interface, and the like. The computer system 600 may include an output device interface 624 configured to forward output, such as to a display, a video display unit (e.g., a liquid crystal display (LCD) or a cathode ray tube (CRT)). The computer system 600 may include a communications interface 626 suitable for communicating with a network as appropriate or desired.

[0088] The operational actions described in any of the exemplary aspects herein are described to provide examples and discussion. The actions may be performed by hardware components, may be embodied in machine-executable instructions to cause a processor to perform the actions, or may be performed by a combination of hardware and software. Although a specific order of method actions may be shown or described, the order of the actions may differ. In addition, two or more actions may be performed concurrently or with partial concurrence.

[0089] Example 1 : a computer system comprising processing circuitry configured to: provide a sparse identification of nonlinear dynamics model (SINDy) embedded in the computer system, the SINDY-model includes a set of mathematical functions weighted by a set of coefficients (C), the SINDY-model is configured to define a relationship between sensor values corresponding to at least one first physical quantity and soft sensor values corresponding to a second physical quantity, acquire online sensor values corresponding to the at least one first physical quantity while the soft sensor is operative and embedded in the computer system, apply a recursive least square algorithm (RLS-algorithm) to the online sensor values and the SINDY-model to find updated set of coefficients for the SINDy -model, adjust the set of coefficients of the embedded SINDy-model according to the updated set of coefficients.

[0090] Example 2: the computer system of example 1, the processing circuitry further being configured to: receive a sensor value from at least one first sensor corresponding to the at least one first physical quantity; and generate, based on the received sensor value and the embedded SINDy-model, a soft sensor value.

[0091] Example 3: The computer system of any of examples 1-2, wherein the second physical quantify comprises at least one of: a temperature of a component of a mechanical system, a peak in-cylinder pressure (PCP) for a vehicle engine, and a turbocharger speed for a vehicle engine.

[0092] Example 4: The computer system of any of examples 1-3, wherein the RLS- algorithm is embedded in the computer system.

[0093] Example 5: The computer system of any of examples 1-4, wherein the set of mathematical function comprises nonlinear functions.

[0094] Example 6: The computer system of any of examples 1-5, wherein the set of mathematical functions comprises at least one of a polynomial function and a trigonometric function.

[0095] Example 7: The computer system of any of examples 1-6, wherein the processing circuitry is configured to: apply the online recursive least square algorithm each time new online senor values are acquired to update the set of coefficients in real-time.

[0096] Example 8: The computer system of example 1, wherein: the set of mathematical function comprises nonlinear functions including at least one of a polynomial function and a trigonometric function, the second physical quantify comprises at least one of: a temperature of a component of a mechanical system, a peak in-cylinder pressure (PCP) for a vehicle engine, and a turbocharger speed for a vehicle engine, the RLS-algorithm is embedded in the computer system, and the processing circuitry is configured to: receive a sensor value from at least one first sensor corresponding to the at least one first physical quantity; generate, based on the received sensor value and the embedded SINDy -model, a soft sensor value, and apply the online recursive least square algorithm each time new online senor values are acquired to update the set of coefficients in real-time.

[0097] Example 9: A vehicle comprising the computer system of any of examples 1-8.

[0098] Example 10: A computer-implemented method, comprising: providing, by a processor device of a computer system, a sparse identification of nonlinear dynamics model (SINDy) embedded in the computer system, the SINDY-model includes a set of mathematical functions weighted by a set of coefficients (C), the SINDY-model is configured to define a relationship between sensor values corresponding to at least one first physical quantity and soft sensor values corresponding to a second physical quantity, acquiring, by the processor device, online sensor values corresponding to the at least one first physical quantity while the soft sensor is operative and embedded in the computer system, applying, by the processor device, a recursive least square algorithm (RLS-algorithm) to the online sensor values and the SINDY-model to find updated set of coefficients for the SINDy -model, adjusting, by the processor device, the set of coefficients of the embedded SINDy-model according to the updated set of coefficients.

[0099] Example 11 : The method of example 10, further comprising: receiving, by the processor device, a sensor value from at least one first sensor corresponding to the at least one first physical quantity; and generating, by the processor device, based on the received sensor value and the embedded SINDy-model, a soft sensor value.

[0100] Example 12: The method of any of examples 10-11, wherein the second physical quantify comprises at least one of: a temperature of a component of a mechanical system, a peak in-cylinder pressure (PCP) for a vehicle engine, and a turbocharger speed for a vehicle engine.

[0101] Example 13: The method of any of examples 10-12, wherein the RLS-algorithm is embedded in the computer system.

[0102] Example 14: The method of any of examples 10-13, wherein the set of mathematical function comprises nonlinear functions.

[0103] Example 15: The method of example 10-14, wherein the set of mathematical functions comprises at least one of a polynomial function and a trigonometric function. [0104] Example 16: The method of any of examples 9-15, comprising: applying, by the processor device, the online recursive least square algorithm each time new online senor values are acquired to update the set of coefficients in real-time.

[0105] Example 17: The method of any of examples 10-16, comprising: embedding, by the processor device, machine readable instructions according to the RLS-algorithm and the SINDy -model in the computer system of a vehicle.

[0106] Example 18: The method of example 10, wherein: the set of mathematical function comprises nonlinear functions including at least one of a polynomial function and a trigonometric function, the second physical quantify comprises at least one of: a temperature of a component of a mechanical system, a peak in-cylinder pressure (PCP) for a vehicle engine, and a turbocharger speed for a vehicle engine, the method further comprises: embedding, by the processor device, the RLS-algorithm in the computer system, receiving, by the processor device, a sensor value from at least one first sensor corresponding to the at least one first physical quantity; generating, by the processor device, based on the received sensor value and the embedded SINDy -model, a soft sensor value, and applying, by the processor device, the online recursive least square algorithm each time new online senor values are acquired to update the set of coefficients in real-time.

[0107] Example 19: A computer program product comprising program code for performing, when executed by the processing circuitry, the method of any of examples 9-18. [0108] Example 20: A non-transitory computer-readable storage medium comprising instructions, which when executed by the processing circuitry, cause the processing circuitry to perform the method of any of examples 9-18.

[0109] The terminology used herein is for the purpose of describing particular aspects only and is not intended to be limiting of the disclosure. As used herein, the singular forms "a," "an," and "the" are intended to include the plural forms as well, unless the context clearly indicates otherwise. As used herein, the term "and/or" includes any and all combinations of one or more of the associated listed items. It will be further understood that the terms "comprises," "comprising," "includes," and/or "including" when used herein specify the presence of stated features, integers, actions, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, actions, steps, operations, elements, components, and/or groups thereof. [0110] It will be understood that, although the terms first, second, etc., may be used herein to describe various elements, these elements should not be limited by these terms. These terms are only used to distinguish one element from another. For example, a first element could be termed a second element, and, similarly, a second element could be termed a first element without departing from the scope of the present disclosure.

[OHl] Relative terms such as "below" or "above" or "upper" or "lower" or "horizontal" or "vertical" may be used herein to describe a relationship of one element to another element as illustrated in the Figures. It will be understood that these terms and those discussed above are intended to encompass different orientations of the device in addition to the orientation depicted in the Figures. It will be understood that when an element is referred to as being "connected" or "coupled" to another element, it can be directly connected or coupled to the other element, or intervening elements may be present. In contrast, when an element is referred to as being "directly connected" or "directly coupled" to another element, there are no intervening elements present.

[0112] Unless otherwise defined, all terms (including technical and scientific terms) used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this disclosure belongs. It will be further understood that terms used herein should be interpreted as having a meaning consistent with their meaning in the context of this specification and the relevant art and will not be interpreted in an idealized or overly formal sense unless expressly so defined herein.

[0113] It is to be understood that the present disclosure is not limited to the aspects described above and illustrated in the drawings; rather, the skilled person will recognize that many changes and modifications may be made within the scope of the present disclosure and appended claims. In the drawings and specification, there have been disclosed aspects for purposes of illustration only and not for purposes of limitation, the scope of the disclosure being set forth in the following claims.

Claims

Claims What is claimed is:

1. A computer system (100) comprising processing circuitry (102) configured to: provide a sparse identification of nonlinear dynamics model SINDy, (104) embedded in the computer system, the SINDY-model includes a set of mathematical functions weighted by a set of coefficients (C), the SINDY-model is configured to define a relationship between sensor values corresponding to at least one first physical quantity and soft sensor values corresponding to a second physical quantity, acquire online sensor values (106) corresponding to the at least one first physical quantity while the soft sensor is operative and embedded in the computer system, apply a recursive least square algorithm, RLS-algorithm (108), to the online sensor values and the SINDY-model to find updated set of coefficients (110) for the SINDy -model, adjust the set of coefficients of the embedded SINDy-model according to the updated set of coefficients.

2. The computer system of claim 1, the processing circuitry further being configured to: receive a sensor value from at least one first sensor corresponding to the at least one first physical quantity; and generate, based on the received sensor value and the embedded SINDy-model, a soft sensor value.

3. The computer system of any of claims 1-2, wherein the second physical quantity comprises at least one of: a temperature of a component of a mechanical system, a peak incylinder pressure (PCP) for a vehicle engine, and a turbocharger speed for a vehicle engine.

4. The computer system of any of claims 1-3, wherein the RLS-algorithm is embedded in the computer system.

5. The computer system of any of claims 1-4, wherein the set of mathematical function comprises nonlinear functions.

6. The computer system of any of claims 1-5, wherein the set of mathematical functions comprises at least one of a polynomial function and a trigonometric function.

7. The computer system of any of claims 1-6, wherein the processing circuitry is configured to: apply the online recursive least square algorithm each time new online senor values are acquired to update the set of coefficients in real-time.

8. The computer system of claim 1, wherein: the set of mathematical function comprises nonlinear functions including at least one of a polynomial function and a trigonometric function, the second physical quantity comprises at least one of: a temperature of a component of a mechanical system, a peak in-cylinder pressure (PCP) for a vehicle engine, and a turbocharger speed for a vehicle engine, the RLS-algorithm is embedded in the computer system, and the processing circuitry is configured to: receive a sensor value from at least one first sensor corresponding to the at least one first physical quantity; generate, based on the received sensor value and the embedded SINDy -model, a soft sensor value, and apply the online recursive least square algorithm each time new online senor values are acquired to update the set of coefficients in real-time.

9. A vehicle comprising the computer system of any of claims 1-8.

10. A computer-implemented method, comprising: providing (SI 02), by a processor device of a computer system, a sparse identification of nonlinear dynamics model (SINDy) embedded in the computer system, the SINDY-model includes a set of mathematical functions weighted by a set of coefficients (C), the SINDY- model is configured to define a relationship between sensor values corresponding to at least one first physical quantity and soft sensor values corresponding to a second physical quantity, acquiring (SI 04), by the processor device, online sensor values corresponding to the at least one first physical quantity while the soft sensor is operative and embedded in the computer system, applying (SI 06), by the processor device, a recursive least square algorithm (RLS- algorithm) to the online sensor values and the SINDY-model to find updated set of coefficients for the SINDy -model, adjusting (SI 08), by the processor device, the set of coefficients of the embedded SINDy -model according to the updated set of coefficients.

11. The method of claim 10, further comprising: receiving, by the processor device, a sensor value from at least one first sensor corresponding to the at least one first physical quantity; and generating, by the processor device, based on the received sensor value and the embedded SINDy -model, a soft sensor value.

12. The method of any of claims 10-11, wherein the second physical quantity comprises at least one of a temperature of a component of a mechanical system, a peak in-cylinder pressure (PCP) for a vehicle engine, and a turbocharger speed for a vehicle engine.

13. The method of any of claims 10-12, wherein the RLS-algorithm is embedded in the computer system.

14. The method of any of claims 10-13, wherein the set of mathematical function comprises nonlinear functions.

15. The method of claim 10-14, wherein the set of mathematical functions comprises at least one of a polynomial function and a trigonometric function.

16. The method of any of claims 9-15, comprising: applying, by the processor device, the online recursive least square algorithm each time new online senor values are acquired to update the set of coefficients in real-time.

17. The method of any of claims 10-16, comprising: embedding, by the processor device, machine readable instructions according to the RLS-algorithm and the SINDy -model in the computer system of a vehicle.

18. The method of claim 10, wherein: the set of mathematical function comprises nonlinear functions including at least one of a polynomial function and a trigonometric function, the second physical quantity comprises at least one of: a temperature of a component of a mechanical system, a peak in-cylinder pressure (PCP) for a vehicle engine, and a turbocharger speed for a vehicle engine, the method further comprises: embedding, by the processor device, the RLS-algorithm in the computer system, receiving, by the processor device, a sensor value from at least one first sensor corresponding to the at least one first physical quantity; generating, by the processor device, based on the received sensor value and the embedded SINDy -model, a soft sensor value, and applying, by the processor device, the online recursive least square algorithm each time new online senor values are acquired to update the set of coefficients in real-time.

19. A computer program product comprising program code for performing, when executed by the processing circuitry, the method of any of claims 9-18.

20. A non-transitory computer-readable storage medium comprising instructions, which when executed by the processing circuitry, cause the processing circuitry to perform the method of any of claims 9-18.