A device for controlling a system using polynomial mechanics.

Abstract

To provide devices for controlling a system described with polynominal dynamics by efficiently solving a polynominal optimization problem.SOLUTION: A device for controlling an operation of a system performing a task determines a current control input based on a feedback signal by solving a polynomial optimization of a polynomial function with a reformulation derived by introducing additional variables reducing the degree of the polynomial function till a target degree, the reformulation being subject to constraints on a structure of the additional variables. The device solves a mixed-integer optimization problem to find a subset of encodings among all possible encodings of factorizations of the polynomial function that reduce the degree of the polynomial function to the target degree with a predetermined minimum number of additional variables, and selects an optimal encoding from the subset of encodings with an optimal relaxation bound.SELECTED DRAWING: Figure 1

Description

Technical Field

[0001] Related Applications This application is related to U.S. Patent Application No. 17 / 366,718.

[0002] This disclosure generally relates to system control, and more specifically to an apparatus for controlling a system using polynomial dynamics.

Background Art

[0003] Controllers such as model predictive control (MPC) are used to control complex dynamical systems in many applications. Examples of such systems include production lines, vehicles, engines, robots, generators, and other numerically controlled machines.

[0004] The dynamics of these systems may be defined by polynomial functions. Therefore, the dynamics of the above systems are sometimes referred to as polynomial dynamics. The control of a system described by polynomial dynamics leads to stability analysis problems, optimal control problems, and control design problems, which can be called polynomial optimization problems. In the case of real-time applications of control, these polynomial optimization problems generally need to be solved on embedded hardware with limited computing resources under strict timing constraints.

Summary of the Invention

Problems to be Solved by the Invention

[0005] Currently, there are few optimization techniques that attempt to solve this polynomial optimization problem under strict timing constraints by converting the polynomial optimization problem into a linear program in a higher-dimensional space. However, these available optimization techniques may not be able to solve this polynomial optimization problem within strict timing constraints, even when the polynomial optimization problem is solved by converting it into a linear program.

[0006] Therefore, what is needed is a device that controls systems explained by polynomial mechanics by efficiently solving polynomial optimization problems. [Means for solving the problem]

[0007] The objective of some embodiments is to provide a device for solving polynomial optimization problems of polynomial functions under timing constraints by transforming the polynomial optimization problem into a higher-dimensional space. Some embodiments are based on the understanding that the failure to satisfy the timing constraints lies in one or a combination of (i) the size of the resulting linear transformation and (ii) the weak relaxation bound derived from the linear transformation. Indeed, a large size of the resulting transformation increases the computation time required to solve the transformation. Weak relaxation resulting from the linear transformation leads to searching a large number of nodes in an iterative optimization, such as a branch-and-bound search tree. This also contributes to increased computation time.

[0008] The objective of some embodiments is to provide a device for solving the polynomial optimization of a polynomial function using a reformulation, which is derived by introducing a minimum number of additional variables that reduce the degree of the polynomial function to a target degree, and is constrained by the structure of these additional variables.

[0009] To derive the reformulation, the objective of some embodiments is to find the optimal solution from all possible codings of the factorization of a polynomial function by solving a mixed-integer optimization problem, where the optimal solution reduces the degree of the polynomial function to a target degree at the optimal relaxation limit among the possible transformations to a linear program, given the constraints of the number of additional variables introduced. Alternatively, the objective of some embodiments is to control a system performing a task by solving the polynomial optimization of a polynomial function using a reformulation that achieves the optimal relaxation limit when limited to the introduction of a certain number of additional variables.

[0010] To control a system performing a task, the objective of some embodiments is to transmit a sequence of control inputs to the system. As a result, the system may change a state associated with the system according to the task. For this purpose, some embodiments receive a feedback signal containing a sequence of corresponding measurements. In one example of an embodiment, each measurement in the sequence of corresponding measurements indicates a state of the system caused by the corresponding control input. For example, the system may be a power grid and / or an electric elevator. In these examples, the sequence of control inputs may be transmitted to the power grid and / or the electric elevator.

[0011] To transmit a sequence of control inputs, an objective of some embodiments is to determine the current control input for controlling the system based on a feedback signal containing current measurements of the system's current state, for example, by solving a polynomial optimization of a polynomial function at each control step. The polynomial optimization used herein may be the minimization of a polynomial function. For example, the polynomial function may be a multilinear program, a mixed-integer nonlinear program, and the like.

[0012] Some embodiments are based on the recognition that polynomial optimization of a polynomial function can be a complex and time-consuming process to solve. Therefore, the objective of some embodiments is to derive a reformulation for polynomial optimization. In other words, the objective of some embodiments is to reduce the degree of the polynomial function to a target degree. The reformulation used herein may be equivalent to a polynomial function with a reduced degree. To derive the reformulation, some embodiments factorize the polynomial function by introducing additional variables. The factorization used herein may be a reduction in the degree of the polynomial function to a target degree by introducing additional variables. Since the reformulation is derived by introducing additional variables, the reformulation may be constrained by the structure of the additional variables. According to some embodiments, by setting a target degree, the polynomial function may be factorized into at least one of a bilinear programming reformulation or a trilinear programming reformulation.

[0013] Some embodiments are based on the understanding that computation time can be significantly reduced by appropriately introducing additional variables. If the transformation results in the introduction of many additional variables, the transformation will increase the computation time required to solve the transformation. Solving the transformation is a step that is repeated many times as part of solving the polynomial optimization problem. Linear transformations are employed to calculate the lower bound of the global optimal solution of the polynomial optimization problem. However, if the result of the linear transformation is a lower bound that is far removed from the global optimal solution of the polynomial optimization, the computation time required for the solving process will be long.

[0014] Some embodiments are based on the understanding that there can be multiple ways to factorize a polynomial function by introducing additional variables, which leads to multiple possible factorizations. Therefore, the objective of some embodiments is to evaluate all possible factorizations of a polynomial function, encode a set of all possible factorizations, and select the factorization that yields the maximum relaxation limit from among all possible factorizations limited by the number of additional variables introduced. The size-constrained optimal limiting factorization used herein may correspond to a factorization that is determined to have the maximum relaxation limit when limited to introducing no more than a specified number of additional variables. In some embodiments, the additional variables may be the product of intermediate additional variables ending in the product of at least two of the original variables of the polynomial function. In some other embodiments, the additional variables may be the product of the original variables of the polynomial function and the intermediate additional variables. The additional variables are at least one of a bilinear or trilinear product.

[0015] To find all possible factorizations, some embodiments define each possible factorization of a monomial in a polynomial function as an in-tree having the following properties: the root of the in-tree corresponds to the original monomial of the polynomial function; the leaf nodes of the in-tree correspond to the original variables of the original monomial; all nodes other than the leaf nodes have two child nodes, each representing a monomial; and the product of the monomials of the child nodes is equal to the monomial represented by the parent node of the child node.

[0016] Therefore, each possible factorization of a monomial is read from the in-tree representation. To encode all possible factorizations, the objective of some embodiments is to represent each monomial in a polynomial function as a decision diagram, which encodes all possible factorizations of the monomial through a set of paths in this decision diagram. For this purpose, some embodiments generate a set of decision diagrams, each of which corresponds to a monomial in a polynomial function. According to some embodiments, each path may define different additional variables that factorize the monomial. According to some embodiments, the decision diagram encoding all possible linearizations of a multilinear monomial is an in-tree with a root containing the original variables of the multilinear monomial.

[0017] To select the size-constrained optimal limit factorization, the objective of some embodiments is to solve a mixed-integer optimization problem. According to some embodiments, the mixed-integer optimization problem may be formulated and solved to find an optimal solution from all possible codings of the polynomial function that reduces the degree of the polynomial function to a target degree at the maximum relaxation limit, given that the number of additional variables is restricted to a given number or less. The given number can be specified as a design parameter imposed as a constraint on the mixed-integer optimization.

[0018] In some embodiments, a mixed-integer optimization problem solves a two-level optimization problem, where the outer level restricts a set of arcs (arcs) selected in a decision diagram to a specified number or less, and the selected arcs actually define the reformulation of each monomial into a linear program. The inner-level problem calculates the relaxation limits by solving a linear program for the variables introduced as a result of the specific selection of arcs in the decision diagram. A key insight is the novel method of transforming a two-level mixed-integer optimization problem into a one-level mixed-integer optimization. This allows for the identification of size-constrained optimal limit factorizations using various mixed-integer optimization algorithms. Since the reformulation is derived by identifying those with optimal relaxation limits when restricted to the introduction of a certain number of additional variables, the computation required to solve polynomial optimizations using reformulation is reduced compared to other optimization techniques available. Thus, the polynomial optimization of polynomial functions is solved in an efficient manner.

[0019] Accordingly, one embodiment discloses a device for controlling the operation of a system performing a task. The device includes a transceiver that modifies the state of the system according to the task by transmitting a sequence of control inputs to the system and receives a feedback signal containing a corresponding sequence of measurements, each measurement indicating the state of the system caused by the corresponding control input. The device further includes a feedback controller, which determines the current control input for controlling the system by solving a polynomial optimization of a polynomial function using a reformatting based on the feedback signal containing current measurements of the system's current state, the reformatting being derived by introducing an additional variable that reduces the degree of the polynomial function to a target degree, and is constrained by the structure of the additional variable. Furthermore, the device includes an optimally factorizable reduction module, which finds an optimal solution that reduces the degree of the polynomial function to a target degree from all possible encodings of the polynomial function, resulting in a size-constrained optimal limit factorization, by solving a mixed-integer optimization problem.

[0020] Accordingly, another embodiment discloses a method for controlling the operation of a system performing a task. This method includes the steps of changing the state of the system according to the task by transmitting a sequence of control inputs to the system, and receiving a feedback signal containing a sequence of corresponding measurements, each measurement indicating the state of the system caused by the corresponding control input. The method further includes the step of determining the current control inputs for controlling the system by solving a polynomial optimization of a polynomial function using a reformatting based on the feedback signal containing current measurements of the system's current state, the reformatting being derived by introducing additional variables that reduce the degree of the polynomial function to a target degree, and being constrained by the structure of the additional variables. The method further includes the step of finding an optimal solution that reduces the degree of the polynomial function to a target degree from all possible encodings of the polynomial function, resulting in a size-constrained optimal limit factorization, by solving a mixed-integer optimization problem.

[0021] Accordingly, another embodiment discloses a non-temporary computer-readable storage medium in which a processor-executable program is realized to perform a method for controlling the operation of a system performing a task. The method includes the steps of changing the state of the system according to a task by transmitting a sequence of control inputs to the system, and receiving a feedback signal including a sequence of corresponding measurements, each measurement indicating the state of the system caused by the corresponding control input. The method further includes the step of determining the current control inputs for controlling the system by solving a polynomial optimization of a polynomial function using a reformatting based on the feedback signal including current measurements of the system's current state, the reformatting being derived by introducing additional variables that reduce the degree of the polynomial function to a target degree, and being constrained by the structure of the additional variables. The method further includes the step of finding an optimal solution that reduces the degree of the polynomial function to a target degree from all possible encodings of the polynomial function's factorization, resulting in a size-constrained optimal limit factorization, by solving a mixed-integer optimization problem.

Brief Description of Drawings

[0022] [Figure 1] It is a diagram for explaining a block diagram showing a controller for controlling the operation of a system according to some embodiments of the present disclosure. [Figure 2] It is a diagram for explaining a block diagram of a controller according to some embodiments of the present disclosure. [Figure 3A] It is a diagram for explaining a schematic diagram showing the formulation of the first level and the second level of a decision diagram according to some embodiments of the present disclosure. [Figure 3B] It is a diagram for explaining a schematic diagram showing the formulation of the third level of a decision diagram according to some embodiments of the present disclosure. [Figure 3C] It is a diagram for explaining a schematic diagram showing the formulation of the fourth level of a decision diagram according to some embodiments of the present disclosure. [Figure 3D] It is a diagram for explaining a schematic diagram showing a decision diagram of a monomial according to some embodiments of the present disclosure. [Figure 3E] It is a diagram for explaining a subgraph as a specific example representing a possible factorization of a monomial according to some embodiments of the present disclosure. [Figure 3F] It is a diagram for explaining a subgraph as a specific example representing another possible factorization of a monomial according to some embodiments of the present disclosure. [Figure 3G] It is a diagram for explaining a schematic diagram showing a unified decision diagram according to some embodiments of the present disclosure. [Figure 3H] It is a diagram for explaining a schematic diagram showing the minimum factorization of a polynomial function according to some embodiments of the present disclosure. [Figure 3I] It is a diagram for explaining a schematic diagram showing the optimal boundary factorization with size constraints of a polynomial function according to some embodiments of the present disclosure. [Figure 4] It is a diagram for explaining a schematic diagram showing the minimum factorization of a polynomial function according to examples of some other embodiments of the present disclosure. [Figure 5]This figure illustrates a schematic diagram illustrating the minimum factorization of a polynomial function, relating to some further embodiments of the present disclosure. [Figure 6A] A schematic diagram illustrating different relaxations of the function as specific examples used in several embodiments is shown. [Figure 6B] Block diagrams of methods for selecting the optimal factorization of a polynomial function according to several embodiments are shown. [Figure 6C] A schematic diagram illustrating the relaxation principle used in several embodiments is shown. [Figure 6D] This figure illustrates a method 600 for determining the current control input, according to some examples of embodiments of the present disclosure. [Figure 7] This figure illustrates a schematic diagram showing a controller connected to a power grid according to some embodiments of the present disclosure. [Figure 8] This figure illustrates a schematic diagram showing a controller connected to an electric elevator according to some embodiments of the present disclosure. [Figure 9] This diagram illustrates the process flow of a controller according to some embodiments of this disclosure. [Modes for carrying out the invention]

[0023] The following description includes numerous specific details to ensure that the disclosure is fully understood. However, it will be apparent to those skilled in the art that the disclosure can be implemented without these specific details. Where else, the apparatus and methods are shown in block diagram form, solely to avoid obscuring the disclosure.

[0024] As used herein and in the claims, the terms “for example,” “for instance,” and “such as,” as well as “comprising,” “having,” “including,” and other forms of these verbs, when used with an enumeration of one or more components or other items, should be interpreted as open-ended, meaning that the enumeration should not be considered to exclude any further components or items. The term “based on” means based at least partially. Furthermore, it should be understood that the style and terminology used herein are for illustrative purposes only and should not be considered restrictive. Any headings used herein are for convenience only and have no legal or restrictive effect.

[0025] Figure 1 illustrates a block diagram showing a controller 100 that controls the operation of system 102 according to some embodiments of the present disclosure. According to one embodiment, the controller 100 may be associated with system 102 and control the operation of system 102 performing a task. In other words, the controller 100 may be operationally connected to system 102 and control the operation of system 102.

[0026] In one embodiment, the controller 100 may be a model predictive control (MPC) programmed according to a model 104 of the system 102 (hereinafter referred to as the dynamic model 104). The dynamic model 104 may be a set of equations that express the changes in the state and output of the system 102 over a period of time as a function of the current input, previous input, and previous output. The dynamic model 104 may include constraints 106 that represent the physical and operational limitations of the system 102. While the system 102 is operating, the controller 100 receives a command 108 that indicates the desired behavior of the system 102. The command 108 may be, for example, an action command. In response to receiving the command 108, the controller 100 generates a sequence of control inputs that function as input signals 110 to the system 102. The system 102 changes its state according to the task in response to the input signals 110. Furthermore, the system 102 may output a sequence of measurements as a feedback signal 112. For example, one or more sensors installed in the system 102 may output the feedback signal 112. In the sequence of measurements, each measurement can indicate the state of system 102 caused by the corresponding control input.

[0027] The system 102 described herein may be any machine or device that is controlled by specific operating input signals 110 (i.e., sequences of control inputs) which may be associated with physical quantities such as voltage, pressure, force, and torque, and which may return feedback signals 112 which may be associated with physical quantities such as current, flow rate, velocity, and position that indicate the state of the system 102. The measured values ​​are partly related to previous measured values ​​of the system 102 and partly related to previous and current control input values. The degree of dependence on previous control inputs and previous measured values ​​is encoded in the state of the system 102. The operation of the system 102, for example, the motion of the components of the system 102, may include sequences of measured values ​​generated by the system 102 according to a given specific control input value.

[0028] The dynamic model 104 of system 102 may include a set of mathematical equations that describe how the measurements of system 102 change over time as a function of current and previous control inputs and previous measurements. The state of system 102 is generally any set of information that changes over time, for example, a suitable subset of current and previous control inputs and measurements that, together with the dynamic model 104 and future control inputs, can uniquely define the future motion of system 102.

[0029] System 102 may be subject to physical and specification constraints 106 that limit the range in which System 102's measurements, control inputs, and potentially states can function.

[0030] The controller 100 can be implemented in hardware or as a software program executed on a processor, such as a microprocessor. The controller 100 receives feedback signals 112 and commands 108 at a fixed or variable controlled period sampling interval and uses the feedback signals 112 and commands 108 to determine input signals 110 for controlling the operation of the system 102.

[0031] Figure 2 shows a block diagram of a controller 100 according to some embodiments of the present disclosure. The controller 100 includes a memory 200, at least one processor 202, a transceiver 204, and a bus 206. The memory 200 may be implemented as a storage medium such as random access memory (RAM), read-only memory (ROM), a hard disk, or any combination thereof. For example, the memory 200 may store instructions that can be executed by at least one processor 202. In addition, the memory 200 may store the constraints 106 of system 102 and the dynamics model 104 of system 102. The at least one processor 202 may be implemented as a single-core processor, a multi-core processor, a computing cluster, or any number of other configurations. The at least one processor 202 may be operationally connected to the memory 200 and / or the transceiver 204 via the bus 206. According to one embodiment, at least one processor 202 may be configured as an optimally factorizable reduction module 202b and / or a feedback controller 202a. Thus, the optimally factorizable reduction module 202b and / or the feedback controller 202a may be implemented within a single-core processor, a multi-core processor, a computing cluster, or any number of other configurations.

[0032] The feedback controller 202a is configured to solve the polynomial optimization of a polynomial function. The polynomial optimization used herein may represent the minimization of a polynomial function. The polynomial function used herein may be a function in the form of a polynomial (one or more equations). For example, a function may be a polynomial if it does not contain trigonometric terms, exponential terms, and / or logarithmic terms. As an example, the polynomial function may be the control policy of system 102. In this example, the feedback controller 202a may solve the minimization of the control policy of system 102. The control policy of system 102 used herein may be a set of equations defined based on system model 104, or a set of equations selected as options for controlling system 102. As an example, the control policy is a function of the current state and / or previous state of system 102. As an example, the control policy of system 102 may be a multilinear program or a mixed integer nonlinear program.

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[0036] Furthermore, the objective of the feedback controller 202a is to solve the polynomial optimization of a polynomial function using a reformatting derived by introducing additional variables. The reformatting used herein corresponds to a polynomial function with reduced degree. Since the reformatting is derived by introducing additional variables, the reformatting may be constrained by the structure of the additional variables.

[0037] In one embodiment, the optimally factorizable reduction module 202b may be configured to derive a reformatting of a polynomial function. According to one embodiment, in order to derive a reformatting of a polynomial function, the optimally factorizable reduction module 202b may be configured to factorize a polynomial function. The factorization used herein may correspond to reducing the degree of a polynomial function to a target degree by introducing additional variables.

[0038] Some embodiments are based on the understanding that there can be multiple ways to factorize a polynomial by introducing additional variables, which can lead to multiple factorizations. For this reason, the optimal factorizable reduction module 202b may be configured to discover all possible codings for the factorization of the polynomial function and to discover the optimal solution from among all possible codings for the factorization of the polynomial function. In one embodiment, the optimal factorizable reduction module 202b may be configured to solve a mixed-integer optimization problem in order to find the optimal solution such that the optimal solution reduces the degree of the polynomial function to a target degree and yields a size-constrained optimal limit factorization. Thus, the mixed-integer optimization problem may be formulated as a problem for discovering the optimal solution from among all possible codings for the factorization of a polynomial function such that the optimal solution reduces the degree of the polynomial function to a target degree and yields a size-constrained optimal limit factorization.

[0039] Furthermore, the objective of the feedback controller 202a is to determine the current control input for controlling system 102 based on a feedback signal containing current measurements of the current state of system 102, by solving a polynomial optimization using a reformulation obtained from the optimal factorizable reduction module 202b, for example, in each control step or in some control steps. In other words, the feedback controller 202a may use a feedback signal containing current measurements of the current state of system 102 and solve a polynomial optimization using a reformulation obtained from the optimal factorizable reduction module 202b to determine the current control input for controlling system 102. For this reason, at least one processor 202 may be configured to determine a sequence of control inputs for system 102 based on a feedback signal containing a sequence of measurements.

[0040] In one embodiment, the transceiver 204 may be configured to transmit a sequence of control inputs to the system 102 as an input signal 110. The system 102 may then change its state according to the task. Furthermore, the system 102 may output a feedback signal 112. For this reason, the transceiver 204 may also be configured to receive the feedback signal 112. Furthermore, at least one processor 202 may be configured to determine the input signal 110. In this way, the controller 100 may control the operation of the system 102 performing the task by transmitting the input signal 110 and receiving the feedback signal 112. As an example, the optimally factorizable reduction module 202b may derive a reformatting of a polynomial function, as described in the detailed description of Figures 3A to 3I.

[0041] Figure 3A illustrates a schematic diagram showing the formulation of the first level 300 and second level 306 of the decision diagram according to some embodiments of the present disclosure. Figure 3A is described in conjunction with Figure 2. According to one embodiment, the optimal factorizable reduction module 202b may be configured to set a target degree to reduce the degree of the polynomial function to a target degree in order to derive a reformatting of the polynomial function. In one embodiment, the degree of the polynomial function may be greater than the target degree. Thus, the optimal factorizable reduction module 202b may derive the reformatting of the target degree by setting the target degree. According to one embodiment, the optimal factorizable reduction module 202b may set the target degree based on one or more of the dynamics of system 102, the number of variables in each multilinear monomial of the polynomial function (e.g., control policy), and / or user input. For example, the optimally factorizable reduction module 202b may set the target degree to a value such as "1", "2", or "3 or less", and thus may be configured to factorize a polynomial function into at least one of bilinear programming reformulation and trilinear programming reformulation. The bilinear programming reformulation used herein may be an approximation of a polynomial function whose degree is 2 or less. The trilinear programming reformulation used herein may be an approximation of a polynomial function whose degree is 3 or less. For example, the optimally factorizable reduction module 202b may set the target degree of a polynomial function to the value "2" in order to derive a bilinear programming reformulation.

[0042] In one embodiment, the optimally factorizable reduction module 202b may be configured to represent each multilinear monomial in a polynomial function as a decision diagram in order to derive a bilinear programming reformulation. In one example of the embodiment, the optimally factorizable reduction module 202b may represent each multilinear monomial (fi(x)) as a decision diagram such that the decision diagram encodes all possible factorizations of the corresponding multilinear monomial.

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[0045] Figure 3C illustrates a schematic diagram showing the formulation of the fourth level 316 of the decision diagram according to some embodiments of the present disclosure. Figure 3C will be described in conjunction with Figure 3B. For illustrative purposes, Figure 3C does not show the arcs 304a, 304b, 304c, and 304d and arcs 314a, 314b, 314c, 314d, 314e, and 314f that define the intermediate monomials represented by multiple parent nodes 312a, 312b, 312c, and 312d, but Figure 3C may also include the arcs 304a, 304b, 304c, and 304d and arcs 314a, 314b, 314c, 314d, 314e, and 314f that define the intermediate monomials represented by multiple parent nodes 312a, 312b, 312c, and 312d.

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[0071] Finally, the size of the RML is the cardinality of the relevant set of appropriate triplets T.

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[0089] Once a set of decision diagrams is generated, the optimally factorizable reduction module 202b may be configured to use this set of decision diagrams to select the minimum number of additional variables. According to one embodiment, the optimally factorizable reduction module 202b may be configured to solve a consistent path problem in a set of decision diagrams connected by the consistency of intermediate monomials in order to select the minimum number of additional variables. The intermediate monomials used herein may be monomials represented by the parent nodes of the decision diagrams in the set of decision diagrams. The consistency of intermediate monomials used herein may indicate that if there are intermediate monomials with a similar structure in one or more decision diagrams in the set of decision diagrams, then intermediate monomials with a similar structure can be selected for factorizing a polynomial function and can be expressed with the same additional variables. The consistent path problem used herein may be defined as finding intermediate monomials in a set of decision diagrams that follow the consistency of intermediate monomials, such that the intermediate monomials factorize a polynomial function.

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[0096] Some embodiments are based on the understanding that the optimally factorizable reduction module 202b can derive multiple bilinear programming reformulations for polynomial optimization using the same number of minimum additional variables. For this reason, the optimally factorizable reduction module 202b may identify a bilinear programming reformulation that leads to a maximum lower bound LP relaxation.

[0097] For illustrative purposes, in Figures 3A to 3I, the optimal factorizable reduction module 202b can be configured to set the target degree of the polynomial function to the value "2". For example, when the target degree of the polynomial function is set to the value "2", the optimal factorizable reduction module 202b may be configured as described in the detailed explanation in Figure 4.

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[0102] Since the degree of the monomial in equation (11a) and the degree of the product term in equation (11b) are equal to the value "2", equations (11a) and (11b) may be bilinear programming reformulations of a polynomial function f(x). Furthermore, in one example of the embodiment, the optimally factorizable reduction module 202b may reduce the bilinear programming reformulation to an LP relaxation using McCormick convex and concave hulls.

[0103] For illustrative purposes, Figure 4 considers the case where the optimal factorizable reduction module 202b sets the target degree of the polynomial function to the value "2". In several other embodiments, the optimal factorizable reduction module 202b may set the target degree of the polynomial function to a value of "3 or less". For example, when the target degree of the polynomial function is set to a value of "3 or less", the optimal factorizable reduction module 202b may be configured as described in detail in Figure 5.

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[0107] Since the degree of the monomial in equation (12a) and the degree of the product term in equation (12b) are equal to a value of "3 or less", equations (12a) and (12b) may be trilinear programming reformulations of a polynomial function f(x). Furthermore, in some embodiments, the optimally factorizable reduction module 202b may reduce the trilinear programming reformulation to an LP relaxation using one or more of the McCormick convex and concave hulls and / or polyhedral envelopes. For example, the trilinear product may be expressed as a linear function using a polyhedral envelope and a set of inequalities.

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[0110] Figure 6A shows a schematic diagram illustrating different relaxations of function 612 as specific examples used in several embodiments. As can be seen from Figure 6A, function 612 is non-convex and difficult to optimize. To reduce the complexity of optimization, function 612 can be approximated by convex functions such as functions 614, 616, and 618. All of these approximation functions are convex with global optima and are therefore easy to solve. However, the solutions of these functions do not provide a solution to the original function 612, but rather provide relaxation limits for the solution. For example, if the objective is to minimize function 612, the solutions of functions 614, 616, and 618 provide lower bounds for the solution to the minimization of function 612. As can be seen from Figure 6A, different functions provide different relaxation limits. For example, the relaxation limit of function 614 is higher than that of function 618. If the objective of optimizing function 612 is to find the global minimum, relaxation 616 is better or more optimal than relaxation 618.

[0111] Some embodiments are based on the recognition that different codings of the factorization of a polynomial function can have different relaxation limits. Therefore, the number of additional variables is one criterion for optimizing the factorization, and the relaxation limit of the factorization is another criterion for optimization.

[0112] Figure 6B shows a block diagram of a method for selecting the optimal factorization of a polynomial function according to several embodiments. This method discovers all possible codings for the factorization of the polynomial function (622). An example of such all possible codings is shown in Figure 3D. Next, this method discovers a subset of possible codings that satisfy the optimization criteria by solving a mixed-integer optimization problem (624). Different embodiments can use different optimization criteria. For example, one embodiment places all codings obtained by introducing fewer than a predetermined number of additional variables in the subset. In addition to or instead of this, another embodiment selects a limited number of codings in the subset. An example of such a subset of all possible codings is shown in Figure 3I.

[0113] Next, this method finds relaxation limits for all codings within a subset (626) and selects the coding with the optimal relaxation limit as the optimal factorization of the polynomial function (628). The optimality of the factorization used herein depends on the objective of optimizing the polynomial function. If the objective is minimization of the polynomial function, the optimal relaxation limit will be the highest. If the objective is maximization of the polynomial function, the optimal relaxation limit will be the lowest.

[0114] Figure 6C shows a schematic diagram illustrating the relaxation principle used in several embodiments. The factorization of the polynomial function 632 is expressed using a non-convex equality arising from the introduction of an additional variable. Embodiments relax these non-convex problems by replacing the equality with a linear inequality 634 (636). The solution to system 634 provides the relaxation limit of factorization 632.

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[0116] Figure 6D illustrates a method 600 for determining the current control input, according to some examples of embodiments of the present disclosure. Method 600 may be used in conjunction with the controller 100 described in the detailed description of Figure 2. For example, at least one processor 202 may be configured to execute blocks of method 600.

[0117] Initially, in block 602, the controller 100 may be configured to test for the existence of a Lyapunov function by solving a polynomial optimization for the control policy. For example, the controller 100 may test for the existence of a Lyapunov function by computing a minimization for the control policy using a reformulation obtained from the optimally factorizable reduction module 202b. For example, the Lyapunov function may be a set of equations that define the stability of system 102 using a feedback signal that includes current measurements of the current state of system 102 and / or previous measurements of previous states of system 102.

[0118] In block 604, the controller 100 may determine the current control command according to the control policy. The current control command used herein may define the desired behavior of the system 102. For example, if the solution to the polynomial optimization of the control policy indicates that the system 102 is not directed toward stability, the controller 100 may determine the current control command to direct the system 102 toward stability.

[0119] In block 606, the controller 100 may determine the current control input based on the current command it has obtained. The system 102 may then be further controlled using this current control input.

[0120] Figure 7 illustrates a schematic diagram showing a controller 100 connected to a power grid 700 according to some embodiments of the present disclosure. The power grid 700 can transmit power from a generator 702 to one or more loads 704 and / or a transformer 706 via a transmission line 708. The transformer 706 may be connected to another power grid and / or one or more other loads. Some embodiments are based on the recognition that if the load in the power grid increases rapidly, the difference between the peak load power and the trough load power will also increase, which can lead to a voltage violation problem. For this reason, transmission line switching technology may be used as a voltage control method. For example, voltage may be controlled by switching transmission lines. In fact, the selection of transmission lines to switch and their coordination with other voltage control measures may represent an optimal power switching problem.

[0121] Some embodiments are based on the understanding that the optimal power transmission switching problem can be formulated as a mixed-integer nonlinear program using binary variables corresponding to the on / off states of the transmission lines. For this reason, the power grid 700 may be operationally coupled to the controller 100. The controller 100 may be configured to obtain a sequence of control inputs indicating the on / off states of the transmission lines in the power grid 700 by solving a polynomial optimization of a polynomial function corresponding to the mixed-integer nonlinear program. In this way, the controller 100 may avoid voltage violation problems in the power grid.

[0122] Figure 8 illustrates a schematic diagram showing a controller 100 connected to an electric elevator 800 according to some embodiments of the present disclosure. The electric elevator 800 includes an elevator car 802 which is operatively connected to different components of the electric elevator 800 via at least one elevator rope. For example, the elevator car 802 and a counterweight 804 may be connected to each other by main ropes 806 and 808 and a compensating rope 810. The elevator car 802 may include a crosshead 812 and a safety plank 814. A pulley 816 for moving the elevator car 802 and a counterweight 804 which passes through the elevator shaft 818 may be located at the front (or bottom) of the elevator shaft 818 in a machine room (not shown). The electric elevator 800 may also include a compensating pulley 820. The elevator shaft 818 includes a front wall 822, a rear wall 824, and a pair of side walls 826.

[0123] The elevator car 802 and counterweight 804 have their center of gravity at the point where the sum of the moments in the x, y, and z directions is zero. In other words, all moments around the center of gravity cancel each other out, so that the elevator car 802 or the counterweight 804 can theoretically be supported and balanced at its center of gravity (x,y,z). Typically, the main ropes 806 and 808 are connected to the crosshead 812 of the elevator car 802, onto which the coordinates of the elevator car 802's center of gravity are projected. The main ropes 806 and 808 are connected to the ends of the counterweight 804, onto which the coordinates of the counterweight 804's center of gravity are projected.

[0124] During the operation of the electric elevator 800, various components of the elevator are affected by internal and external disturbances, such as lateral sway caused by wind, resulting in lateral motion of these components. Such lateral motion of the components can cause lateral sway of the elevator rope, which needs to be measured. Therefore, one or a set of lateral sway sensors 828 can be placed in the electric elevator 800 to determine the lateral sway of the elevator rope.

[0125] A set of sensors may include at least one lateral sway sensor 828. For example, the lateral sway sensor 828 is configured to detect lateral sway of the elevator rope at a sway position corresponding to the position of the lateral sway sensor.

[0126] During the operation of the electric elevator 800, the position of the sway is determined and sent to the controller 100. The controller 100 determines the sway state of the elevator rope, such as the amplitude and velocity of the sway. The controller 100 can determine the sway state based solely on the sway measurements. The controller 100 may determine a control policy, which may be a function of the amplitude and velocity of the sway. Furthermore, the controller 100 may determine the sequence of control inputs to the electric elevator 800 by solving a polynomial optimization of a polynomial function (i.e., the determined control policy). The sequence of control inputs may correspond to damping forces. Furthermore, the controller 100 may change the damping force on the elevator rope by controlling the semi-active damper actuator 830 of the electric elevator 800 based on the sequence of control inputs.

[0127] Figure 9 shows a process flow 900 of the controller 100 according to some embodiments of the present disclosure. This process flow 900 may be used in conjunction with the controller 100 described in detail in Figure 2. First, in block 902, the controller 100 may obtain a polynomial function. For example, the polynomial function may be a mixed-integer nonlinear program, a multilinear program, etc. In block 904, the controller 100 may set a target order for the polynomial function to reduce the order of the polynomial function to the target order. For example, the controller 100 may set the target order as described in the detailed description of Figure 3A. In block 906, the controller 100 may find an optimal solution from all possible codings of the factorization of the polynomial function such that the optimal function reduces the order of the polynomial function to the target order with the minimum number of additional variables. For example, the controller 100 may find an optimal solution as described in the detailed description of Figures 3A to 5.

[0128] In block 908, the controller 100 may reduce the degree of the constraint product term in the polynomial optimization to a target degree. For example, the controller 100 may reduce the degree of the constraint product term using one or more of the McCormick convex and concave hulls and / or polyhedron envelopes. In block 910, the controller 100 may obtain a global optimal solution by solving the polynomial optimization of the polynomial function using a branch-and-bound algorithm.

[0129] The above description provides only specific embodiments and is not intended to limit the scope, applicability, or configuration of the present disclosure. Rather, the above description of specific embodiments will provide a description that enables the implementation of one or more specific embodiments. Various modifications are intended to be made to the function and configuration of the elements without departing from the spirit and scope of the subject matter disclosed in the appended claims.

[0130] Specific details are provided in the above description to ensure a full understanding of the embodiments. However, those skilled in the art will understand that embodiments can be carried out even without these specific details. For example, systems, processes, and other elements in the disclosed subject matter may be shown as components in the form of block diagrams to avoid obscuring the embodiments with unnecessary details. In other cases, well-known processes, structures, and techniques may be shown without unnecessary details to avoid obscuring the embodiments. Furthermore, similar reference numbers and names in different drawings refer to similar elements.

[0131] Furthermore, individual embodiments may be described as processes shown as flowcharts, flow diagrams, data flow diagrams, structural diagrams, or block diagrams. While flowcharts may describe operations as sequential processes, many operations can be performed in parallel or simultaneously. In addition, the order of operations may be rearranged. A process may terminate when its operations are complete, but it may have additional steps that are not discussed or included in the diagrams. Moreover, not all operations in any process specifically described can occur in all embodiments. A process may correspond to a method, function, procedure, subroutine, subprogram, etc. If a process corresponds to a function, the termination of the function may correspond to returning the function to the calling function or the main function.

[0132] Furthermore, embodiments of the disclosed subject matter may be implemented either manually or automatically, at least in part. Manual or automatic implementation may be performed, or at least assisted, through a machine, hardware, software, firmware, middleware, microcode, hardware description language, or any combination thereof. If implemented in software, firmware, middleware, or microcode, the program code or code segments for performing the required tasks may be stored in a machine-readable medium. A processor(s) may perform the required tasks.

[0133] The various methods or processes outlined herein may be encoded as software executable on one or more processors employing any one of a variety of operating systems or platforms. In addition, such software may be written using any of several suitable programming languages ​​and / or programming or scripting tools, and may be compiled as executable machine language code or intermediate code that runs on a framework or virtual machine. Typically, the functions of program modules may be combined or distributed as desired in various embodiments.

[0134] Embodiments of this disclosure may be implemented as methods, and an example thereof is provided. The order of operations performed as part of this method may be determined in any suitable manner. Thus, embodiments may be configured such that operations are performed in an order different from the order illustrated, which may include performing some operations simultaneously, although they are shown as a series of operations in the illustrated embodiments. While this disclosure has been described with reference to certain preferred embodiments, it should be understood that various other modifications and alterations are possible within the spirit and scope of this disclosure. Accordingly, it is an aspect of the appended claims to cover all such variations and alterations that fall within the true spirit and scope of this disclosure.

Claims

1. A device for controlling the operation of a system that performs a task, the device is The device further comprises a transceiver configured to change the state of the system according to the task by transmitting a sequence of control inputs to the system, and to receive a feedback signal including a sequence of corresponding measurements, each measurement indicating the state of the system caused by the corresponding control input, and the device further The device comprises a feedback controller, which is configured to determine the current control input for controlling the system by solving a polynomial optimization of a polynomial function using a reformulation based on the feedback signal, which includes current measurements of the system's current state, the reformulation being derived by introducing additional variables that reduce the order of the polynomial function to a target order, subject to the constraints of the structure of the additional variables, and the device further, An apparatus comprising an optimally factorizable reduction module, wherein the optimally factorizable reduction module is configured to solve a mixed-integer optimization problem to discover a subset of codings from all possible codings for factorizing the polynomial function that reduce the degree of the polynomial function to the target degree using a predetermined minimum number of additional variables, and to select the optimal coding having an optimal relaxation limit from the subset of codings.

2. The apparatus according to claim 1, wherein the additional variable in the reformulation is the product of intermediate additional variables that end in the product of at least two of the original variables of the polynomial function.

3. The apparatus according to claim 1, wherein the additional variable includes the product of the intermediate additional variable and the original variable of the polynomial function.

4. The apparatus according to claim 1, wherein the additional variable is at least one of a bilinear product or a trilinear product.

5. The apparatus according to claim 1, wherein the optimally factorizable reduction module is configured to factorize the polynomial function into at least one of bilinear programming reformulation or trilinear programming reformulation by setting the target degree.

6. The apparatus according to claim 1, wherein each of the factorizations of the polynomial function is read from a representation of an in-tree having a characteristic, the characteristic being that the root of the in-tree corresponds to the original monomial of the polynomial function, the leaf nodes of the in-tree correspond to the original variables of the original monomial, all nodes other than the leaf nodes have two child nodes, each child node representing a monomial, and the product of the monomials of the child nodes is equal to the monomial represented by the parent node of the child node.

7. The aforementioned optimally factorizable reduction module is, The optimal factorizable reduction module is configured to generate a set of decision diagrams for the polynomial function by representing each multilinear monomial in the polynomial function as a decision diagram, the decision diagram encodes all possible factorizations of the multilinear monomial through a set of paths in the decision diagram, each path defines different additional variables that factorize the multilinear monomial, and the optimal factorizable reduction module further, The apparatus according to claim 1, configured to select the minimum number of additional variables by solving a consistent path problem in a set of decision diagrams connected by the consistency of the intermediate monomials, such that intermediate monomials in different decision diagrams having the same structure are represented by the same additional variables.

8. The apparatus according to claim 7, wherein the mixed integer optimization problem is solved by minimizing the consistent path problem with respect to the nodes in the set of decision diagrams and the arcs connecting the nodes, subject to the consistency of the intermediate monomials.

9. The apparatus according to claim 7, wherein the decision diagram encoding all possible factorizations of the multilinear monomial is an in-tree having a root containing the original variable of the multilinear monomial.

10. The apparatus according to claim 1, wherein the optimally factorizable reduction module is further configured to identify the reformulation associated with the greatest lower bound.

11. The apparatus according to claim 1, wherein the feedback controller solves the polynomial optimization by relaxation using a branch-bound algorithm, and the relaxation provides a lower bound on the polynomial optimization at all nodes of the branch-bound algorithm.

12. The apparatus according to claim 1, wherein the feedback controller determines the current control command according to the control policy and tests for the existence of a Lyapunov function by solving the polynomial optimization for the control policy.

13. The apparatus according to claim 1, wherein the feedback controller is configured to determine the sequence of control inputs to the power grid, and the power grid is operationally connected to the apparatus.

14. The apparatus according to claim 1, wherein the feedback controller is configured to determine the sequence of control inputs to the electric elevator, and the electric elevator is operationally connected to the apparatus.

15. The apparatus according to claim 1, wherein for each coding in the subset of codings, the optimal factorizable reduction module is configured to obtain a polyhedral relaxation of the coding by replacing the equation resulting from the introduction of additional variables with a linear inequality, to obtain a solution as the relaxation limit of the coding by solving the polyhedral relaxation, and to select the optimal coding from the subset of codings by comparing the corresponding relaxation limits based on the purpose of the polyhedral relaxation.

16. A method for controlling the operation of a system that performs a task, the method is A step of changing the state of the system according to the task by transmitting a sequence of control inputs to the system, The method further includes the step of receiving a feedback signal containing a sequence of corresponding measurements, each measurement indicating the state of the system caused by the corresponding control input, and the method further includes The method includes the step of determining the current control input for controlling the system by solving the polynomial optimization of a polynomial function using a reformulation based on the feedback signal, which includes the current measured value of the current state of the system, wherein the reformulation is derived by introducing an additional variable that reduces the degree of the polynomial function to a target degree, subject to the constraints of the structure of the additional variable, and the method further includes A method comprising the steps of: solving a mixed-integer optimization problem to discover a subset of codings from all possible codings for factoring the polynomial function that reduces the degree of the polynomial function to the target degree using a predetermined minimum number of additional variables; and selecting the optimal coding with an optimal relaxation limit from the subset of codings.

17. The method according to claim 16, wherein for each coding in the subset of codings, the method includes the steps of obtaining a polyhedral relaxation of the coding by replacing the equation resulting from the introduction of an additional variable with a linear inequality, finding a solution as a relaxation limit of the coding by solving the polyhedral relaxation, and selecting the optimal coding from the subset of codings by comparing the corresponding relaxation limits based on the purpose of the polyhedral relaxation.

18. A non-temporary computer-readable storage medium on which a processor-executable program is implemented for performing a method for controlling the operation of a system that performs a task, wherein the method is A step of changing the state of the system according to the task by transmitting a sequence of control inputs to the system, The method further includes the step of receiving a feedback signal containing a sequence of corresponding measurements, each measurement indicating the state of the system caused by the corresponding control input, and the method further includes The method includes the step of determining the current control input for controlling the system by solving the polynomial optimization of a polynomial function using a reformulation based on the feedback signal, which includes the current measured value of the current state of the system, wherein the reformulation is derived by introducing an additional variable that reduces the degree of the polynomial function to a target degree, subject to the constraints of the structure of the additional variable, and the method further includes A non-temporary computer-readable storage medium comprising the steps of: solving a mixed-integer optimization problem to discover a subset of codings from all possible codings for factoring the polynomial function that reduces the degree of the polynomial function to the target degree using a predetermined minimum number of additional variables; and selecting the optimal coding with an optimal relaxation limit from the subset of codings.

19. The medium according to claim 18, wherein for each coding in the subset of codings, the method includes the steps of obtaining a polyhedral relaxation of the coding by replacing the equation resulting from the introduction of an additional variable with a linear inequality, finding a solution as a relaxation limit of the coding by solving the polyhedral relaxation, and selecting the optimal coding from the subset of codings by comparing the corresponding relaxation limits based on the objective of optimality of the polyhedral relaxation.

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