Apparatus for controlling systems using polynomial dynamics

Abstract

To provide a device, a method and a storage medium that control an action of a system for executing a task.SOLUTION: A method is configured to: change a state of a system according to a task by transmitting a sequence of a control input to the system; receive a feedback signal; solve polynominal optimization of a polynominal expression function using reformulation, on the basis of the feedback signal including a current measurement value of a current state of the system, and thereby obtain a current control input for controlling the system. The reformulation is derived by introducing an additional variable reducing a degree of the polynominal expression function to a target degree and is subject to a constraint of a structure of the additional variable. The method is further configured to solve a mixed integer optimization problem, and thereby discover an optimal solution reducing the degree of the polynominal expression function to the target degree, using an additional variable of a minimum number, from all possible coding of factorization of the polynominal expression function.SELECTED DRAWING: Figure 9

Description

[Technical Field]

[0001] The present disclosure relates generally to system control, and more particularly to an apparatus for controlling a system using polynomial dynamics. [Background technology]

[0002] Controllers such as model predictive control (MPC) are used in many applications to control complex dynamical systems. Examples of such systems include production lines, vehicles, engines, robots, generators, and other numerically controlled machines.

[0003] The dynamics of these systems may be defined by polynomial functions. Hence, the dynamics of such systems are sometimes called polynomial dynamics. Control of systems described by polynomial dynamics leads to stability analysis problems, optimal control problems, and control design problems, which can be called polynomial optimization problems. For real-time applications of control, these polynomial optimization problems must be solved under strict timing constraints, often on embedded hardware with limited computational resources. Summary of the Invention [Problem to be solved by the invention]

[0004] Currently, only a few optimization techniques attempt to solve polynomial optimization problems under strict timing constraints by transforming them into linear programs in higher dimensional spaces. However, these available optimization techniques may not be able to solve the polynomial optimization problem within the strict timing constraints, even when solving it by transforming it into a linear program.

[0005] Therefore, what is needed is an apparatus for controlling systems described by polynomial mechanics by solving polynomial optimization problems in an efficient manner. [Means for solving the problem]

[0006] An object of some embodiments is to provide an apparatus for solving polynomial optimization of a polynomial function using a reformulation, where the reformulation is derived by introducing a minimum number of additional variables that reduces the degree of the polynomial function to a target degree, subject to constraints on the structure of the additional variables. To derive the reformulation, an object of some embodiments is to find an optimal solution among all possible encodings of the factorization of the polynomial function by solving a mixed integer optimization problem, where the optimal solution reduces the degree of the polynomial function to the target degree using a minimum number of additional variables. An object of some embodiments is also to control a system that performs a task by solving the polynomial optimization of a polynomial function using a reformulation derived by introducing a minimum number of additional variables.

[0007] To control a system to perform a task, some embodiments aim to send a sequence of control inputs to the system, which may then cause the system to change a state associated with the system in accordance with the task. To this end, some embodiments receive a feedback signal that includes a sequence of corresponding measurements. In one example embodiment, each measurement in the sequence of corresponding measurements indicates a state of the system caused by the corresponding control input. By way of example, the system may be an electric power grid and / or an electric elevator. In these examples, the sequence of control inputs may be sent to the electric power grid and / or the electric elevator.

[0008] To transmit a sequence of control inputs, an objective of some embodiments is to determine, at each control step, a current control input for controlling the system based on a feedback signal that includes a current measurement of the current state of the system, e.g., by solving a polynomial optimization of a polynomial function. Polynomial optimization, as used herein, may be the minimization of a polynomial function. By way of example, the polynomial function may be a multilinear program, a mixed-integer nonlinear program, etc.

[0009] Some embodiments recognize that polynomial optimization of a polynomial function can be a complex and time-consuming process to solve. Therefore, an objective of some embodiments is to derive a reformulation for polynomial optimization. In other words, an objective of some embodiments is to reduce the degree of a polynomial function to a target degree. A reformulation, as used herein, may correspond to a reduction in the degree of a polynomial function. To derive the reformulation, some embodiments factorize the polynomial function by introducing additional variables. Factorization, as used herein, may refer to reducing the degree of a polynomial function to a target degree by introducing additional variables. Because the reformulation is derived by introducing additional variables, the reformulation may be subject to constraints on 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.

[0010] Some embodiments recognize that there may be multiple ways to factor a polynomial function by introducing additional variables, leading to multiple factorizations. Therefore, an objective of some embodiments is to determine all possible factorizations of a polynomial function, encode all possible factorizations in a set, and be able to select a minimal factorization from among all possible factorizations. As used herein, a minimal factorization may correspond to a factorization determined by introducing a minimal number of additional variables. In some embodiments, the additional variables may be a product of intermediate additional variables that end up being a product of at least two of the original variables of the polynomial function. In other embodiments, the additional variables may be a product of an original variable of the polynomial function and an intermediate additional variable. The additional variables are at least one of a bilinear product and a trilinear product.

[0011] To find all possible factorizations, some embodiments define each possible factorization of a monomial in a polynomial function as an in-tree with the following properties: the root of the in-tree corresponds to an original monomial in the polynomial function; the leaf nodes of the in-tree correspond to the original variables of the original monomial; every node other than the leaf nodes has 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 child node's parent node. Thus, each possible factorization of the monomial is read from the in-tree representation. To encode all possible factorizations, some embodiments aim to represent each monomial in the polynomial function as a decision diagram, which encodes all possible factorizations of the monomial through a set of paths in the decision diagram. To this end, some embodiments generate a set of decision diagrams, each corresponding to a monomial in the polynomial function. According to some embodiments, each path may define a different additional variable that factors the monomial. According to some embodiments, a decision diagram that encodes all possible linearizations of a multilinear monomial is an in-tree with a root that contains the original variables of the multilinear monomial.

[0012] To select the minimal factorization, some embodiments aim to solve a mixed-integer optimization problem. According to some embodiments, a mixed-integer optimization problem may be formulated and solved to find an optimal solution among all possible encodings of the factorization of the polynomial function that reduces the degree of the polynomial function to a target degree using a minimal number of additional variables. According to some embodiments, the mixed-integer optimization problem is solved by minimizing the number of nodes and arcs connecting the nodes in a set of decision diagrams that obey the consistency of intermediate variables. Some embodiments are based on the recognition that solving the consistent path problem by minimizing the number of nodes and arcs connecting the nodes in a set of decision diagrams that obey the consistency of intermediate variables leads to selecting a minimal number of additional variables. Because the reformulation is derived by introducing a minimal number of additional variables, the computation for solving the polynomial optimization using the reformulation is reduced compared to other available optimization techniques. Thus, polynomial optimization of a polynomial function is solved in an efficient manner.

[0013] Accordingly, one embodiment discloses an apparatus for controlling the operation of a system performing a task. The apparatus includes a transceiver for transmitting a sequence of control inputs to the system to change the state of the system according to the task and for receiving a feedback signal including a sequence of corresponding measurements, each measurement indicating a state of the system caused by the corresponding control input. The apparatus further includes a feedback controller for determining a current control input for controlling the system by solving a polynomial optimization of a polynomial function using a reformulation based on the feedback signal including current measurements of a current state of the system, the reformulation being derived by introducing additional variables that reduce the degree of the polynomial function to a target degree, subject to a structure constraint of the additional variables. The apparatus further includes an optimal factorizable reduction module for solving a mixed-integer optimization problem to find an optimal solution that reduces the degree of the polynomial function to the target degree using a minimum number of additional variables from among all possible encodings of the factorization of the polynomial function.

[0014] Accordingly, another embodiment discloses a method for controlling the operation of a system performing a task. The method includes altering a state of the system according to the task by sending a sequence of control inputs to the system and receiving a feedback signal including a corresponding sequence of measurements, each measurement indicating a state of the system caused by the corresponding control input. The method further includes determining a current control input for controlling the system by solving a polynomial optimization of a polynomial function using a reformulation based on the feedback signal including current measurements of a current state of the system, the reformulation being derived by introducing additional variables that reduce the degree of the polynomial function to a target degree, subject to constraints on the structure of the additional variables. The method further includes solving a mixed integer optimization problem to find an optimal solution that reduces the degree of the polynomial function to the target degree using a minimum number of additional variables from among all possible encodings of a factorization of the polynomial function.

[0015] Accordingly, another embodiment discloses a non-transitory computer-readable storage medium having a program executable by a processor to execute a method for controlling the operation of a system to perform a task. The method includes: modifying a state of the system according to the task by sending a sequence of control inputs to the system; and receiving a feedback signal including a sequence of corresponding measurements, each measurement indicating a state of the system caused by the corresponding control input. The method further includes determining a current control input for controlling the system by solving a polynomial optimization of a polynomial function using a reformulation based on the feedback signal including current measurements of the current state of the system, the reformulation being derived by introducing additional variables that reduce the degree of the polynomial function to a target degree, subject to constraints on the structure of the additional variables. The method further includes solving a mixed-integer optimization problem to find an optimal solution that reduces the degree of the polynomial function to the target degree using a minimum number of additional variables from among all possible encodings of the factorization of the polynomial function. [Brief explanation of the drawings]

[0016] [Figure 1] FIG. 1 illustrates a block diagram showing a controller that controls the operation of the system, according to some embodiments of the present disclosure. [Figure 2] FIG. 2 illustrates a block diagram of a controller according to some embodiments of the present disclosure. [Figure 3A] FIG. 1 illustrates a schematic diagram showing first-level and second-level formulations of decision diagrams, according to some embodiments of the present disclosure. [Figure 3B] FIG. 1 illustrates a schematic diagram showing a third level formulation of a decision diagram, according to some embodiments of the present disclosure. [Figure 3C] FIG. 1 illustrates a schematic diagram showing a fourth level formulation of a decision diagram, according to some embodiments of the present disclosure. [Figure 3D] FIG. 1 illustrates a schematic diagram showing a decision diagram for a monomial, according to some embodiments of the present disclosure. [Figure 3E] FIG. 2 illustrates an exemplary subgraph representing one possible factorization of a monomial, according to some embodiments of the present disclosure. [Figure 3F] FIG. 10 illustrates an exemplary subgraph representing another possible factorization of a monomial, according to some embodiments of the present disclosure. [Figure 3G] FIG. 1 illustrates a schematic diagram showing a set of decision diagrams, according to some embodiments of the present disclosure. [Figure 3H] FIG. 1 illustrates a schematic diagram showing minimal factorization of a polynomial function, according to some embodiments of the present disclosure. [Figure 3I] FIG. 1 illustrates an algorithm for determining additional variables to an optimal solution, according to some embodiments of the present disclosure. [Figure 4] FIG. 10 illustrates a schematic diagram showing minimal factorization of a polynomial function, according to some other example embodiments of the present disclosure. [Figure 5] FIG. 10 illustrates a schematic diagram showing minimal factorization of a polynomial function, according to some examples of still other embodiments of the present disclosure. [Figure 6] FIG. 2 illustrates a method for determining a current control input according to some example embodiments of the present disclosure. [Figure 7] FIG. 1 illustrates a schematic diagram showing a controller connected to a power grid, according to some embodiments of the present disclosure. [Figure 8] FIG. 1 illustrates a schematic diagram showing a controller connected to an electric elevator, according to some embodiments of the present disclosure. [Figure 9] FIG. 10 illustrates a process flow of a controller according to some embodiments of the present disclosure. DETAILED DESCRIPTION OF THE INVENTION

[0017] In the following description, for purposes of explanation, numerous specific details are set forth in order to provide a thorough understanding of the present disclosure. However, it will be apparent to those skilled in the art that the present disclosure may be practiced without these specific details. In other instances, devices and methods are shown in block diagram form solely to avoid obscuring the present disclosure.

[0018] As used in this specification and claims, the terms "for example," "for instance," and "such as," as well as "comprising," "having," "including," and other forms of these verbs, when used in conjunction with a list of one or more components or other items, should be construed as open-ended, meaning that the list should not be considered to exclude further components or items. The term "based on" means based at least in part on. Furthermore, it should be understood that the style and terminology used herein are for purposes of description and should not be considered to be limiting. Any headings used herein are for convenience only and have no legal or limiting effect.

[0019] 1 illustrates a block diagram showing a controller 100 controlling the operation of a system 102, according to some embodiments of the present disclosure. According to an embodiment, the controller 100 may be associated with the system 102 and control the operation of the system 102 to perform tasks. In other words, the controller 100 may be operatively connected to the system 102 and control the operation of the system 102.

[0020] In an example embodiment, the controller 100 may be a model predictive control (MPC) programmed according to a model 104 of the system 102 (hereinafter, the dynamic model 104). The dynamic model 104 may be a set of equations that describe the change in state and output of the system 102 over time as a function of current inputs, previous inputs, and previous outputs. The dynamic model 104 may include constraints 106 that represent physical and operational limitations of the system 102. During operation of the system 102, the controller 100 receives commands 108 that indicate a desired behavior of the system 102. The commands 108 may be, for example, motion commands. In response to receiving the commands 108, the controller 100 generates a sequence of control inputs that serve as input signals 110 to the system 102. In response to the input signals 110, the system 102 changes the state of the system 102 according to a task. Additionally, 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 indicates a state of the system 102 caused by a corresponding control input.

[0021] The system 102 described herein may be any machine or device that is controlled by particular operational input signals 110 (i.e., a sequence of control inputs), possibly associated with physical quantities such as voltage, pressure, force, torque, etc., and that returns feedback signals 112, possibly associated with physical quantities such as current, flow, speed, position, etc., that indicate the state of the system 102. Measurements are related in part to previous measurements of the system 102 and in part to previous and current control input values. Dependence on previous control inputs and previous measurements is encoded in the state of the system 102. Operation of the system 102, e.g., movement of components of the system 102, may include a sequence of measurements generated by the system 102 according to specific given control input values.

[0022] The dynamic model 104 of the system 102 may include a set of mathematical equations that describe how measurements of the system 102 change over time as a function of current and previous control inputs and previous measurements. The state of the system 102 is generally any collection of time-varying information, for example, a proper 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 the system 102.

[0023] The system 102 may be subject to physical limitations and specification constraints 106 that limit the range over which the measurements, control inputs, and possibly states of the system 102 can function.

[0024] The controller 100 may be implemented in hardware or as a software program running on a processor, e.g., a microprocessor. The controller 100 receives feedback signals 112 and commands 108 at fixed or variable control period sampling intervals and uses the feedback signals 112 and commands 108 to determine input signals 110 for controlling the operation of the system 102.

[0025] 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 a random access memory (RAM), a read-only memory (ROM), a hard disk, or any combination thereof. By way of example, the memory 200 may store instructions executable by the at least one processor 202. Additionally, the memory 200 may store constraints 106 of the system 102 and a dynamic model 104 of the 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 operatively connected to the memory 200 and / or the transceiver 204 via the bus 206. According to an embodiment, at least one processor 202 may be configured as an optimal factorizable reduction module 202b and / or a feedback controller 202a. Accordingly, the optimal 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.

[0026] The objective of the feedback controller 202a is to solve polynomial optimization of a polynomial function. Polynomial optimization, as used herein, may refer to the minimization of a polynomial function. A polynomial function, as used herein, may be a function (one or more equations) in polynomial form. For example, a function may be polynomial if it does not contain trigonometric terms, exponential terms, and / or logarithmic terms. As an example, the polynomial function may be a control policy for the system 102. In this example, the feedback controller 202a may solve the minimization of the control policy for the system 102. As used herein, the control policy for the system 102 may be a set of equations determined based on the system model 104 or may be a set of equations selected as options for controlling the system 102. As an example, the control policy is a function of the current state of the system 102 and / or a previous state of the system 102. As an example, the control policy for the system 102 may be a multilinear program or a mixed-integer nonlinear program.

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[0030] Additionally, the objective of the feedback controller 202a is to solve the polynomial optimization of a polynomial function using a reformulation derived by introducing additional variables. A reformulation, as used herein, corresponds to a reduced-order version of the polynomial function. Because the reformulation is derived by introducing additional variables, the reformulation may be subject to constraints on the structure of the additional variables.

[0031] In one example embodiment, the optimal factorizable reduction module 202b may be configured to derive a reformulation of the polynomial function. According to an embodiment, to derive the reformulation of the polynomial function, the optimal factorizable reduction module 202b may be configured to factorize the polynomial function. Factorization, as used herein, may correspond to reducing the degree of the polynomial function to a target degree by introducing additional variables.

[0032] Some embodiments are based on the recognition that there may be multiple ways to factor a polynomial by introducing additional variables, leading to multiple factorizations. Thus, the optimal factorizable reduction module 202b may be configured to find all possible encodings of the factorization of the polynomial function and to find an optimal solution among all possible encodings of the factorization of the polynomial function. In one example embodiment, the optimal factorizable reduction module 202b may be configured to solve a mixed-integer optimization problem to find the optimal solution, whereby the optimal solution reduces the degree of the polynomial function to a target degree with a minimal number of additional variables. Thus, the mixed-integer optimization problem may be formulated as a problem to find an optimal solution among all possible encodings of the factorization of the polynomial function, such that the optimal solution reduces the degree of the polynomial function to a target degree with a minimal number of additional variables.

[0033] Additionally, a purpose of the feedback controller 202a is to determine current control inputs for controlling the system 102 based on feedback signals including current measurements of the current state of the system 102, e.g., at each control step or at some control steps, by solving a polynomial optimization using a reformulation obtained from the optimal factorizable reduction module 202b. In other words, the feedback controller 202a may use the feedback signals including current measurements of the current state of the system 102 and solve a polynomial optimization using a reformulation obtained from the optimal factorizable reduction module 202b to determine current control inputs for controlling the system 102. Thus, for a feedback signal including a sequence of measurements, the at least one processor 202 may be configured to determine a sequence of control inputs for the system 102.

[0034] According to one embodiment, the transceiver 204 may be configured to transmit a sequence of control inputs to the system 102 as input signals 110, thereby causing the system 102 to change its state in accordance with the task. Furthermore, the system 102 may output a feedback signal 112. To this end, the transceiver 204 may be further configured to receive the feedback signal 112. Furthermore, the at least one processor 202 may be configured to determine the input signals 110. In this manner, the controller 100 may control the operation of the system 102 to perform the task by transmitting the input signals 110 and receiving the feedback signal 112. As an example, the optimal factorizable reduction module 202b may derive a reformulation of the polynomial function, as described in the detailed description of FIGS. 3A-3I.

[0035] FIG. 3A illustrates a schematic diagram showing formulations of the first level 300 and the second level 306 of a decision diagram according to some embodiments of the present disclosure. FIG. 3A is described in conjunction with FIG. 2. According to an embodiment, the optimal factorizable reduction module 202b may be configured to set a target order for reducing the order of the polynomial function to the target order to derive a reformulation of the polynomial function. In an example embodiment, the order of the polynomial function may be greater than the target order. Thus, the optimal factorizable reduction module 202b may derive a target order reformulation by setting the target order. According to an embodiment, the optimal factorizable reduction module 202b may set the target order based on one or more of the dynamics of the system 102, the number of variables in each multi-linear monomial of the polynomial function (e.g., the control policy), and / or user input. As an example, the optimal factorizable reduction module 202b may set the target degree as a value such as “1,” “2,” “3 or less,” etc., and may therefore be configured to factorize the polynomial function into at least one of a bilinear programming reformulation and a trilinear programming reformulation. A bilinear programming reformulation, as used herein, may be an approximation of a polynomial function whose degree is equal to or less than the value “2.” A trilinear programming reformulation, as used herein, may be an approximation of a polynomial function whose degree is equal to or less than the value “3.” As an example, the optimal factorizable reduction module 202b may set the target degree of the polynomial function to the value “2” to derive a bilinear programming reformulation.

[0036] According to one embodiment, the optimal factorizable reduction module 202b may be configured to represent each multilinear monomial in the polynomial function as a decision diagram to derive a bilinear programming reformulation. In one example embodiment, the optimal factorizable reduction module 202b represents each multilinear monomial (f i (x)) may be represented as a decision diagram.

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[0033] Figure 3C illustrates a schematic diagram showing a formulation of the fourth level 316 of a decision diagram according to some embodiments of the present disclosure. Figure 3C will be described in conjunction with Figure 3B. For convenience of illustration, 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 are not shown in Figure 3C, but Figure 3C may also include 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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[0061] Once the set of decision diagrams is generated, the optimal factorizable reduction module 202b may be configured to select a minimum number of additional variables using the set of decision diagrams. According to an example embodiment, the optimal factorizable reduction module 202b may be configured to solve a consistent path problem in a set of decision diagrams connected by intermediate monomial consistency to select a minimum number of additional variables. As used herein, an intermediate monomial may be a monomial represented by a parent node of a decision diagram in the set of decision diagrams. As used herein, intermediate monomial consistency may indicate that if there are intermediate monomials with similar structure in one or more decision diagrams in the set of decision diagrams, the intermediate monomials with similar structure can be selected for factoring a polynomial function and can be expressed with the same additional variables. As used herein, the consistent path problem may be defined as finding intermediate monomials in a set of decision diagrams that obey intermediate monomial consistency such that the intermediate monomials factor a polynomial function.

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[0065] 31 illustrates an algorithm 334 for finding additional variables to the optimal solution, according to some embodiments of the present disclosure. In one example embodiment, the optimal factorizable reduction module 202b may execute the algorithm 334 for finding additional variables to the optimal solution. By way of example, the optimal solution can be mathematically expressed as (u * ,z * )

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[0071] In this manner, the optimal factorizable reduction module 202b may derive an LP relaxation by introducing a minimum number of additional variables. Because the LP relaxation is derived by introducing a minimum number of additional variables, the computations required to solve the polynomial optimization using the LP relaxation are reduced compared to other available optimization techniques. Thus, the controller 100 can solve the polynomial optimization in an efficient manner. In one example embodiment, the feedback controller 202a may determine the sequence of control inputs by solving the LP relaxation using a branch and bound algorithm.

[0072] Some embodiments are based on the recognition that the optimal factorizable reduction module 202b may derive multiple bilinear programming reformulations for polynomial optimization with the same number of minimal additional variables. Thus, the optimal factorizable reduction module 202b may identify a bilinear programming reformulation that leads to an LP relaxation of the maximum lower bound.

[0073] 3A-3I, it is assumed that the optimal factorizable reduction module 202b may set the target degree of the polynomial function to a value of 2. By way of example, if the target degree of the polynomial function is set to a value of 2, the optimal factorizable reduction module 202b may be configured as described in the detailed description of FIG.

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[0078] Because the degree of the monomial in Equation (6a) and the degree of the product term in Equation (6b) are equal to the value 2, Equations (6a) and (6b) may be a bilinear programming reformulation of the polynomial function f(x). Furthermore, in an example embodiment, the optimal factorizable reduction module 202b may reduce the bilinear programming reformulation to an LP relaxation using McCormick convex and concave hulls.

[0079] For purposes of illustration, consider the case in FIG. 4 where the optimal factorizable reduction module 202b sets the target degree of the polynomial function to a value of "2." In yet other embodiments, the optimal factorizable reduction module 202b may set the target degree of the polynomial function to a value of "3 or less." By way of 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 the detailed description of FIG. 5.

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[0083] Because the degree of the monomial in Equation (7a) and the degree of the product term in Equation (7b) are equal to a value "less than or equal to three," Equations (7a) and (7b) may be a trilinear programming reformulation of the polynomial function f(x). Furthermore, in some example embodiments, the optimal 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. By way of example, the polyhedral envelope may be used to express the trilinear product as a linear function using a set of inequalities.

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[0086] 6 illustrates a method 600 for determining a current control input according to some example embodiments of the present disclosure. Method 600 may be used in conjunction with controller 100 described in the detailed description of FIG. 2. By way of example, at least one processor 202 may be configured to perform the blocks of method 600.

[0087] 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. By way of 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 optimal factorizable reduction module 202b. For example, the Lyapunov function may be a set of equations that define the stability of the system 102 using feedback signals that include current measurements of the current state of the system 102 and / or previous measurements of previous states of the system 102.

[0088] At block 604, the controller 100 may determine current control commands according to the control policy. As used herein, a current control command may define a desired behavior of the system 102. As an example, if the solution to a polynomial optimization of the control policy indicates that the system 102 is not oriented toward stability, the controller 100 may determine current control commands to oriented the system 102 toward stability.

[0089] In block 606, the controller 100 may determine current control inputs based on the determined current commands. Such current control inputs may be used to further control the system 102.

[0090] FIG. 7 illustrates a schematic diagram showing the 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 via a transmission line 708 to one or more loads 704 and / or a transformer 706. The transformer 706 may be connected to another power grid and / or one or more other loads. Some embodiments recognize that if the load on the power grid increases suddenly, the difference between the peak load power and the valley load power also increases, potentially leading to a voltage violation problem. Therefore, power line switching techniques may be used as a voltage control method. As an example, voltage may be controlled by switching power lines. In fact, the selection of which power lines to switch and coordination with other voltage control strategies may represent an optimal power line switching problem.

[0091] Some embodiments are based on the recognition that the optimal transmission switching problem can be formulated as a mixed-integer nonlinear program with binary variables corresponding to the on / off states of transmission lines. To this end, power grid 700 may be operatively coupled to controller 100. Controller 100 may be configured to determine a sequence of control inputs indicative of the on / off states of transmission lines in power grid 700 by solving a polynomial optimization of a polynomial function corresponding to the mixed-integer nonlinear program. In this manner, controller 100 may avoid voltage violation problems in the power grid.

[0092] FIG. 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 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 sheave 816 for moving the elevator car 802 and the counterweight 804 through an elevator shaft 818 may be located at the top (or bottom) of the elevator shaft 818 within a machine room (not shown). The electric elevator 800 may also include a compensating sheave 820. The elevator shaft 818 includes a front wall 822, a rear wall 824, and a pair of side walls 826.

[0093] The elevator car 802 and counterweight 804 have a center of gravity at a point where the sum of the moments in the x, y, and z directions is zero. In other words, all of the moments about the center of gravity cancel out, so the elevator car 802 or counterweight 804 can theoretically be supported and balanced at the 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 center of gravity of the elevator car 802 are projected. The main ropes 806 and 808 are connected to the tip of the counterweight 804, onto which the coordinates of the center of gravity of the counterweight 804 are projected.

[0094] During operation of the electric elevator 800, various components of the elevator are subject to lateral sway caused by internal and external disturbances, such as wind, resulting in lateral movement of these components. Such lateral movement of the components can cause lateral sway of the elevator ropes that needs to be measured. Therefore, one or a set of sway sensors 828 can be placed in the electric elevator 800 to determine the lateral sway of the elevator ropes.

[0095] The set of sensors can include at least one sway sensor 828. For example, the sway sensor 828 is configured to detect sway of the elevator rope at a sway position associated with the position of the sway sensor.

[0096] During operation of the electric elevator 800, the sway position is determined and sent to the controller 100. The controller 100 determines the elevator rope sway state, for example, the sway amplitude and velocity. The controller 100 may determine the sway state based solely on the sway measurement. The controller 100 may determine a control policy, which may be a function of the sway amplitude and velocity. Furthermore, the controller 100 may determine a sequence of control inputs to the electric elevator 800 by solving 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.

[0097] FIG. 9 illustrates a process flow 900 for a controller 100 according to some embodiments of the present disclosure. This process flow 900 may be used with the controller 100 described in the detailed description of FIG. 2. Initially, in block 902, the controller 100 may obtain a polynomial function. By way of example, the polynomial function may be a mixed integer nonlinear program, a multilinear program, or the like. In block 904, the controller 100 may set a target order for the polynomial function and reduce the order of the polynomial function to the target order. By way of example, the controller 100 may set the target order as described in the detailed description of FIG. 3A. In block 906, the controller 100 may find an optimal solution among all possible encodings of a 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. By way of example, the controller 100 may find an optimal solution as described in the detailed description of FIGS. 3A-5.

[0098] In block 908, the controller 100 may reduce the order of the product terms of the constraints in the polynomial optimization to a target order. By way of example, the controller 100 may reduce the order of the product terms of the constraints using one or more of McCormick convex and concave hulls and / or polyhedral envelopes. In block 910, the controller 100 may obtain a global optimum by solving the polynomial optimization of the polynomial function using a branch and bound algorithm.

[0099] The above description provides exemplary embodiments only and is not intended to limit the scope, applicability, or configuration of the present disclosure. Rather, the above description of exemplary embodiments will provide those skilled in the art with an enabling description for implementing one or more exemplary embodiments. Various changes are contemplated that may be made in the function and arrangement of elements without departing from the spirit and scope of the disclosed subject matter as set forth in the appended claims.

[0100] Specific details are provided in the above description to provide a thorough understanding of the embodiments. However, those skilled in the art will understand that the embodiments may be practiced without these specific details. For example, systems, processes, and other elements of the disclosed subject matter may be shown as components in block diagram form so as not to obscure the embodiments in unnecessary detail. In other instances, well-known processes, structures, and techniques may be shown without unnecessary detail so as not to obscure the embodiments. Furthermore, like reference numbers and names in the various drawings indicate like elements.

[0101] Also, particular embodiments may be described as a process that is depicted as a flowchart, a flow diagram, a data flow diagram, a structure diagram, or a block diagram. While a flowchart may describe operations as a sequential process, many of the operations may be performed in parallel or simultaneously. Additionally, the order of operations may be rearranged. A process may be terminated when its operations are completed, but may have additional steps not discussed or included in the diagram. Moreover, not all operations in any specifically described process may occur in all embodiments. A process may correspond to a method, a function, a procedure, a subroutine, a subprogram, etc. When a process corresponds to a function, the termination of the function may correspond to a return of the function to the calling function or the main function.

[0102] Furthermore, embodiments of the disclosed subject matter may be implemented, at least in part, either manually or automatically. The manual or automated implementation may be performed or at least assisted by machine, hardware, software, firmware, middleware, microcode, hardware description languages, or any combination thereof. When implemented in software, firmware, middleware, or microcode, the program code or code segments to perform the necessary tasks may be stored on a machine-readable medium. A processor(s) may perform the necessary tasks.

[0103] The various methods or processes outlined herein may be coded as software executable on one or more processors employing any one of a variety of operating systems or platforms. Additionally, such software may be written using any of a number of suitable programming languages ​​and / or programming or scripting tools, and compiled as executable machine language code or intermediate code that runs on a framework or virtual machine. Typically, the functionality of the program modules may be combined or distributed as desired in various embodiments.

[0104] Embodiments of the present disclosure may be implemented as a method, an example of which is provided. The order of operations performed as part of this method may be arranged in any suitable manner. Thus, embodiments may be configured to perform operations in an order different from that illustrated, which may include performing some operations simultaneously, even though they are shown as a series of operations in the illustrated embodiments. While the present disclosure has been described with reference to certain preferred embodiments, it should be understood that various other variations and modifications are possible within the spirit and scope of the present disclosure. Therefore, it is the object of the appended claims to cover all such variations and modifications that fall within the true spirit and scope of the present disclosure.

Claims

1. 1. An apparatus for controlling the operation of a system to perform a task, said apparatus comprising: a transceiver configured to send a sequence of control inputs to the system to change a state of the system in accordance with the task, and to receive a feedback signal including a sequence of corresponding measurement values, each measurement value indicative of a state of the system caused by a corresponding control input; a feedback controller configured to determine a current control input for controlling the system by solving a polynomial optimization problem of a polynomial function using a reformulation based on the feedback signal including a current measurement value of a current state of the system, the reformulation being derived by introducing an additional variable that reduces the degree of the polynomial function to a target degree, subject to a structure constraint of the additional variable; and the apparatus further comprising:

1. An apparatus comprising: an optimal factorizable reduction module configured to solve a mixed integer optimization problem by encoding a factorization of the polynomial function with additional variables and minimizing, among all possible encodings, a number of additional variables such that an encoding is achieved that reduces the degree of the polynomial function to the target degree.

2. The apparatus of claim 1 , wherein the additional variables in the reformulation are products of intermediate additional variables that end with a product of at least two of the original variables of the polynomial function.

3. The apparatus of claim 1 , wherein the additional variables include products of intermediate additional variables and original variables of the polynomial function.

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

5. 2. The apparatus of claim 1, wherein the optimal factorizable reduction module is configured to factorize the polynomial function into at least one of a quadratic linear programming reformulation or a cubic linear programming reformulation by setting the target degree.

6. Each of the factorizations of the polynomial function is represented by an in-tree representation having the following properties: The characteristics are: the root node corresponds to the primitive monomial of said polynomial function; the leaf nodes correspond to the primitive variables of the primitive monomials; every node other than the leaf nodes has two child nodes, each child node representing a monomial; and The apparatus of claim 1 , further comprising: a product of the monomials of the child nodes equals a monomial represented by a parent node of the child nodes.

7. The optimal factorizable reduction module comprises: configured to represent each multilinear monomial in the polynomial function as a decision diagram to generate a set of decision diagrams for the polynomial function, the decision diagram encoding all possible factorizations of the multilinear monomial through a set of paths in the decision diagram, each path defining a different additional variable that factors the multilinear monomial; The optimal factorizable reduction module further comprises: configured to solve a consistent path problem in the set of decision diagrams connected by intermediate monomial consistency; 2. The apparatus of claim 1, wherein the optimal factorizable reduction module is configured to solve the consistent path problem by selecting a minimum number of additional variables such that intermediate monomials with the same structure in different decision diagrams are represented by the same additional variables.

8. 8. The apparatus of claim 7, wherein the mixed integer optimization problem solves the consistent path problem by minimizing over the nodes and arcs connecting the nodes in the set of decision diagrams that obey the consistency of the intermediate monomials.

9. the decision diagram encoding all possible factorizations of the multilinear monomial is represented in an in-tree representation with a root containing the original variable of the multilinear monomial; The properties of the in-tree representation are: the root node corresponds to the primitive monomial of said polynomial function; the leaf nodes correspond to the primitive variables of the primitive monomials; every node other than the leaf nodes has two child nodes, each child node representing a monomial; and The apparatus of claim 7 , further comprising: a product of the monomials of the child nodes equals a monomial represented by a parent node of the child nodes.

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

11. The apparatus of claim 1 , wherein the feedback controller solves the polynomial optimization problem by relaxation using a branch and bound algorithm, the relaxation providing a lower bound for the polynomial optimization problem at every node of a branch and bound algorithm.

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

13. The apparatus of claim 1 , wherein the feedback controller is configured to determine a sequence of the control inputs to an electrical grid, the electrical grid being operatively connected to the apparatus.

14. The apparatus of claim 1 , wherein the feedback controller is configured to determine a sequence of the control inputs to an electric elevator, the electric elevator being operatively connected to the apparatus.

15. 1. A method for controlling the operation of a system to perform a task, the method comprising: changing the state of the system in accordance with the task by sending a sequence of control inputs to the system; receiving a feedback signal including a sequence of corresponding measurements, each measurement indicative of a state of the system caused by a corresponding control input, the method further comprising: determining a current control input for controlling the system by solving a polynomial optimization problem of a polynomial function using a reformulation based on the feedback signal comprising a current measurement value of a current state of the system, the reformulation being derived by introducing additional variables that reduce the degree of the polynomial function to a target degree, subject to constraints on the structure of the additional variables; and the method further comprising: encoding a factorization of the polynomial function with additional variables and solving a mixed integer optimization problem by minimizing the number of additional variables among all possible encodings that results in an encoding that reduces the degree of the polynomial function to the target degree.

16. 1. A non-transitory computer-readable storage medium having embodied thereon a processor-executable program for performing a method for controlling the operation of a system to perform tasks, the method comprising: changing the state of the system in accordance with the task by sending a sequence of control inputs to the system; receiving a feedback signal including a sequence of corresponding measurements, each measurement indicative of a state of the system caused by a corresponding control input, the method further comprising: determining a current control input for controlling the system by solving a polynomial optimization problem of a polynomial function using a reformulation based on the feedback signal comprising a current measurement value of a current state of the system, the reformulation being derived by introducing additional variables that reduce the degree of the polynomial function to a target degree, subject to constraints on the structure of the additional variables; and the method further comprising: a step of solving a mixed integer optimization problem by encoding a factorization of the polynomial function with additional variables and minimizing the number of additional variables from all possible encodings that results in an encoding that reduces the degree of the polynomial function to the target degree.

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