Reduced-order split measurement device and method for singularly perturbed systems

Abstract

The present invention discloses a reduction-order partitioning measurement device and method for a singular perturbation system. The reduction-order partitioning measurement device for a singular perturbation system according to the present invention includes: a slow observer that derives a slow state variable from the output of the first gain operator; a fast observer that derives a slow state variable from the output of the second gain operator; and an inverse converter that inversely converts the slow state variable output from the slow observer (230) and the fast state variable output from the fast observer.

Description

Technology Field

[0001] The present invention relates to a reduced-order split measurement technique for a singular perturbation system in which slow state variables and fast state variables are mixed, and more specifically, to an apparatus and method for a reduced-order split measurement for a singular perturbation system that enables the estimation of measurement gain values ​​for the singular perturbation system through split measurement considering time delay conditions. Background Technology

[0002] Recently, sensorless observation (Observer estimation) methods are being actively researched, in which state variables that cannot be measured by sensors are modeled and estimated by a program system, whereas state variables that cannot be measured by sensors are read through the sensors. A traditional sensorless observation method related to this is the Luenberger observer algorithm.

[0003] Generally, industrial systems consist mainly of a mixture of electrical and mechanical elements, and when modeled, mechanical elements exhibit slow state changes while electrical elements exhibit fast state changes. A dynamic system in which slow and fast state changes exist simultaneously in such a mixed state is called a singularly perturbed system.

[0004] When modeling a singular perturbation system, using a general measurement algorithm causes severe delays in simulation calculation speed. Furthermore, when using existing measurement algorithms to find gain values ​​for a singular perturbation system, it becomes impossible to find them due to numerical issues within the program.

[0005] Regarding the modeling of a singular perturbation system, using a mobile robot's autonomous driving platform as an example and referring to Fig. 1, the explanation is as follows.

[0006] When linear velocity (v) and angular velocity (w) are provided as inputs, the state variables of the autonomous driving platform—latitude (x), longitude (y), and heading angle (h)—are outputs. Here, if the initial state of the heading angle (h) is set to 0 and the autonomous driving platform is followed by an elliptical or straight path as the target path, the change in longitude (y) may be greater than the change in latitude (x). In this case, large and small values ​​exist in the state equation matrix of the system model, and the singular perturbation parameter epsilon can be found.

[0007] As described above, when a mobile robot follows a path, that is, when an elliptical target path is generated and followed, the change in state x(t) is small and the change in state y(t) is large. The state equation can be expressed by the following [Equation 1].

[0008] [Mathematical Formula 1]

[0009]

[0010] Here, using MATLAB’s tfest (transfer estimation) and tf2ss (transfer function to state space model) codes between inputs u(t) and y(t) yields matrix A and B values. When following a path in the form of an ellipse or a straight line, small and large values ​​exist within matrix A, allowing the singular perturbation parameter epsilon to be found and enabling the modeling of a singular perturbation system.

[0011] Figure 2 is a block diagram of a reduction order measuring instrument for a singular perturbation system according to the prior art.

[0012] Referring to FIG. 2, the reduction order measuring instrument (20) for a singular perturbation system according to the prior art is intended to model only the state where no measurement is taken from the sensor, and for this purpose includes gain calculators (21), (22), summers (23), (26), integrators (24), and feedback calculators (25).

[0013] The gain operator (21) applies a gain ( to the output o(t) of the singular perturbation system (10). ) multiplies, and the gain operator (22) adds a gain ( to the output o(t) ) multiplies. Here, the output o(t) represents the output of the singular perturbation system (10) that is not measured by the sensor, and there coexist a slowly changing state variable and a rapidly changing state variable.

[0014] The summer (23) sums the output of the gain operator (21) and the output of the feedback operator (25).

[0015] The integrator (24) integrates the output of the summer (23) and outputs the result. This becomes. Here, the output is a state variable of the system that is not measured from the sensor that is reconstructed in the reduction order measuring instrument (20).

[0016] The feedback operator (25) is the output of the integrator (24). Matrix for modeling for ( Performs the operation.

[0017] The summer (26) is the output of the integrator (24). and the output of the gain calculator (22) are added together, at which time the output This state transition coefficient ( It can be reconstructed into the state variables of the original system by ).

[0018] The summing result of the above summing device (26) is the output of the reduction order division measurement estimation device (20). It becomes.

[0019] However, the reduction-order split measurement estimation device for a singular perturbation system according to conventional technology modeled the reduction-order meter under the assumption that the output is a slow variable of a mechanical element that cannot be measured.

[0020] Therefore, there are difficulties in finding the measurement gain values ​​for the singular perturbation system, and there is a problem with slow calculation speed. The problem to be solved

[0021] The objective of the present invention is to provide a device and method for measuring a reduced-order partitioned measurement device for a singular perturbation system, which allows for the measurement of a purely slow state and a purely fast state in parallel and independently partitioned when a slow state and a fast state coexist in unmeasurable states using state transitions during singular perturbation system modeling. means of solving the problem

[0022] A reduction order division measurement device for a singular perturbation system according to the present invention for achieving the above objective comprises: a first gain operator that multiplies the output of the singular perturbation system by a slow gain and a second gain operator that multiplies the output of the singular perturbation system by a fast gain; a slow observer that derives a slow state variable from the output of the first gain operator; a fast observer that derives a fast state variable from the output of the second gain operator; an inverse converter that inversely converts the slow state variable output from the slow observer (230) and the fast state variable output from the fast observer; a third gain operator that multiplies the output of the singular perturbation system by an output gain; and an adder that adds the output of the gain operator and the output of the inverse converter and provides the result as a final output.

[0023] Additionally, the slow observer comprises: a first summer that sums the output of the first gain operator and the output of the first feedback operator; a first integrator that integrates the output of the first summer and outputs a slow state variable accordingly; and a first feedback operator that performs a first submatrix operation on the slow state variable and provides the result as the other input of the first summer.

[0024] In addition, the first submatrix mentioned above is, It is characterized by being. In the first submatrix above, , is the modeled system matrix, and is the gain value of the measuring instrument.

[0025] Additionally, the fast observer is characterized by including: a second summer that sums the output of the second gain operator and the output of the second feedback operator; a second integrator that integrates the output of the second summer and outputs a fast state variable accordingly; and a second feedback operator that performs a second submatrix operation on the fast state variable and provides the result as the other input of the first summer.

[0026] In addition, the second submatrix mentioned above is, It is characterized by being. In the second submatrix above, is the singular perturbation parameter epsilon, and , is the modeled system matrix, and is the gain value of the measuring instrument.

[0027] A method for measuring a reduced-order division of a singular perturbation system according to the present invention for achieving the above objective comprises: a step in which a first gain operator multiplies the output of the singular perturbation system by a slow gain and a second gain operator multiplies the output of the singular perturbation system by a fast gain; a step in which a slow observer derives a slow state variable from the output of the first gain operator; a step in which a fast observer derives a fast state variable from the output of the second gain operator; a step in which an inverse converter inversely converts the slow state variable output from the slow observer and the fast state variable output from the fast observer; a step in which a third gain operator multiplies the output of the singular perturbation system by an output gain; and a step in which an adder adds the output of the gain operator and the output of the inverse converter and provides the result as a final output.

[0028] Additionally, the step of deriving the slow state variable comprises: a step in which a first summer adds the output of the first gain operator and the output of the first feedback operator; a step in which a first integrator integrates the output of the first summer and outputs a slow state variable accordingly; and a step in which a first feedback operator performs a first submatrix operation on the slow state variable and provides the result as the other input of the first summer.

[0029] Additionally, the step of deriving the fast state variable comprises: a step in which a second summer adds the output of the second gain operator and the output of the second feedback operator; a step in which a second integrator integrates the output of the second summer and outputs a fast state variable accordingly; and a step in which a second feedback operator performs an operation on a second submatrix with respect to the fast state variable and provides the result as the other input of the first summer. Effects of the invention

[0030] The reduction order division measurement device and method for a specific perturbation system according to the present invention have the following effects.

[0031] First, by implementing a state meter capable of independently dividing and measuring the slow and fast states in parallel when they coexist in a state not measured by the sensor using various state transitions, the computation speed can be significantly reduced compared to a state meter that measures the mixed state all at once. Consequently, this has the effect of drastically shortening the simulation time for high-order systems.

[0032] Second, simulation time is minimized, and it has the effect of comparing with sensor data after system estimation or estimating states that are unmeasurable from the sensor. Brief explanation of the drawing

[0033] Figure 1 is a graph of the state change for path following of a mobile robot. FIG. 2 is a block diagram of a reduction order split measurement estimation device for a singular perturbation system according to the prior art. FIG. 3 is a block diagram of a reduction order split measurement estimation device for a singular perturbation system according to the present invention. Figure 4 is a state transition flowchart for splitting a slow system and a fast system in the present invention. Figures 5(a) and 5(b) are graphs showing that the error between the slow state variable reconstructed in the instrument and the state variable in the system, and the error between the fast state variable reconstructed in the fast observer and the fast state variable in the system converge to a value of 0 in the simulation. FIG. 6 is a flowchart of a reduction order division measurement method for a specific perturbation system according to the present invention. Specific details for implementing the invention

[0034] Embodiments of the present invention will be described in detail below with reference to the accompanying drawings. First, it should be noted that in assigning reference numerals to the components of each drawing, the same components are given the same reference numeral whenever possible, even if they are shown in different drawings. Furthermore, in describing the present invention, detailed descriptions of related known components or functions are omitted if they are deemed obvious to those skilled in the art or could obscure the essence of the invention.

[0035] FIG. 3 is a block diagram of a reduction order partitioned measurement estimation device for a singular perturbation system according to the present invention, and

[0036] FIG. 4 is a state transition flowchart for splitting a slow system and a fast system in a reduction order split measurement estimation device according to the present invention, and

[0037] Figures 5(a) and 5(b) are graphs showing that the error between the slow state variable reconstructed in the instrument and the state variable in the singular perturbation system, and the error between the fast state variable reconstructed in the fast observer and the fast state variable in the singular perturbation system converge to a value of 0 in the simulation.

[0038] Referring to FIG. 3, the reduction order division measurement device (200) for a singular perturbation system according to the present invention includes first and second gain calculators (210), (220), a slow observer (230), a fast observer (240), an inverse converter (250), a third gain calculator (260), and an adder (270).

[0039] The operation of the reduction order division measurement device for the singular perturbation system according to the present invention configured as described above is explained with reference to FIGS. 4 and 5 as follows.

[0040] The singular perturbation system (100) is input It receives and generates an output o(t). Here, the input of the singular perturbation system (100) It is not specifically limited, but may include, for example, the linear velocity (v) and angular velocity (w) of the mobile robot. Also, the output o(t) of the singular perturbation system (100) may be in the form of a combination of slow state variables and fast state variables.

[0041] The singular perturbation system (100) is modeled in the form of the following [Equation 2].

[0042] [Mathematical Formula 2]

[0043]

[0044] Here, is relatively Since the value is large compared to, a fast state can be created. Using the tfest (transfer estimation) code in the Matlab program, matrices A and B representing the input and output relationship can be obtained.

[0045] A measuring device (measuring instrument) that measures all states, from those measured by the sensor to those not measured, by taking into account the entire order of the specific perturbation system (100), is called a full-order observer.

[0046] In contrast, the reduced-order division measurement device (200) corresponds to a measuring instrument for unmeasurable states that are not measured by a sensor. That is, reduced-order measurement means a method of modeling only the states that are not measured by a sensor. Here, the meaning of division is that the unmeasurable states are mixed with slow states and fast states and divided into a slow observer (230) and a fast observer (240).

[0047] The first gain operator (210) applies a slow gain ( Multiply by ) and output the result, and the second gain operator (220) applies fast gain ( Multiply by ) and output the result. Here, the above slow gain ( ) and Fast Gain( ) controls the speed of reconstructing the state of the system in the reduction order division measurement device (200).

[0048] The slow observer (230) receives the output of the first gain calculator (210) and from it obtains a purely slow state variable (hereinafter referred to as the 'slow state variable') ( ) is derived. To this end, the slow observer (230) includes a first summer (231), a first integrator (232), and a first feedback operator (233).

[0049] The first summing unit (231) sums the output of the first gain unit (210) and the output of the first feedback unit (233).

[0050] The first integrator (232) integrates the output of the first summer (231) and the slow state variable ( Prints ).

[0051] The first feedback operator (233) is the slow state variable ( The first submatrix ( It calculates ) and provides the result as the other input of the first summer (231). Here, , is the modeled system matrix, and is the gain value of the measuring instrument.

[0052] The fast observer (240) receives the output of the second gain calculator (220) and from it obtains a purely fast state variable (hereinafter referred to as 'fast state variable') ( ) is derived. To this end, the fast observer (240) includes a second summer (241), a second integrator (242), and a second feedback operator (243).

[0053] The second summer (241) sums the output of the second gain operator (220) and the output of the second feedback operator (243).

[0054] The second integrator (242) integrates the output of the second summer (241) and the fast state variable ( Prints ).

[0055] The second feedback operator (243) is the fast state variable ( ) targeting the second submatrix ( It calculates ) and provides the result as the other input of the second summer (231). Here, is the singular perturbation parameter epsilon, and , is the modeled system matrix, and is the gain value of the measuring instrument.

[0056] The inverse converter (250) is a slow state variable (output from the slow observer (230) ) and a fast state variable output from the above fast observer (240) Inversely transform ) and the resulting state variable( Prints ).

[0057] The third gain calculator (260) applies an output gain ( Multiply by ) and output the result.

[0058] The summer (270) adds the output of the third gain calculator (260) and the output of the inverse converter (250) and the result is the final output ( Provides as ).

[0059] Accordingly, the final output of the reduction order division measurement device (200) ) includes the result of splitting and measuring each in parallel independently when slow state variables and fast state variables coexist in unmeasurable states using various state transitions.

[0060] Ultimately, in the measurement gain value (AK*C) for the singular perturbation system (100), A and C represent the modeled system matrix, and K represents the instrument gain value. In the case of the singular perturbation system (100), large and small values ​​coexist in matrix A, so there are many cases where this gain value K cannot be found in the entire system. Taking this into account, the reduction order division measurement device (200) divides the system into a slow subsystem and a fast subsystem, respectively, and the measurement gain value of the slow subsystem and Because it is found by dividing, the instrument gain value K for the entire singular perturbation system, which is not found due to numerical problems, can be found through the inverse transform.

[0061] Referring to FIG. 4, in order to design a slow and fast reduction order partitioned measuring device (200) by modeling a singular perturbation system (100) in a reduction order partitioned measuring device (200), a transformed assumed system for the singular perturbation system (100) must be found. Then, a sequential form for the assumed system is found through state transformation, and then the real system for the assumed sequential form is transformed and found. Afterward, a block diagonal form is created using state transformation again, and parallel observers, namely a slow observer (230) and a fast observer (240), are each implemented independently. Among the mathematical equations in the following description, the measuring device for the slow and fast states can be implemented in a parallel form through the equation in the block diagonal form.

[0062] The reduction order division measurement device (200) is the input of the singular perturbation system (100). After estimating the output o(t), the singular perturbation system (100) is modeled using the code tfest(transfer estimation) in the MATLAB program.

[0063] The reduction order division measurement device (200) is a slow state variable (as described above) ) and fast state variables( After estimating ), the unmeasurable state from the sensor can be estimated through the inverse transformation and summation processes.

[0064] The reduction order division measurement device (200) is an input of the singular perturbation system (100). Design a transfer function based on the relationship between and the output o(t), convert it into a state equation, and then the singular perturbation parameter epsilon(epsilon: The singular perturbation system (100) is modeled by finding the value of ). In order to model the singular perturbation system (100), there must be numerically large elements within the matrix A of the above [Equation 3], and the above Finding a value is It means modeling the value (see Equation 3).

[0065] [Mathematical Formula 3]

[0066]

[0067]

[0068]

[0069]

[0070]

[0071]

[0072]

[0073] If the mathematical formula is redefined for the design (modeling) of the reduction order division measurement device (200), it is as follows [Mathematical Formula 4].

[0074] [Mathematical Formula 4]

[0075]

[0076]

[0077]

[0078]

[0079]

[0080]

[0081]

[0082]

[0083]

[0084]

[0085]

[0086] The modeling of the reduction order division measurement device (200) can be expressed by the following [Equation 5].

[0087] [Mathematical Formula 5]

[0088]

[0089] [Equation 6] for locating eigenvalues ​​for a singular perturbation system (100) can be expressed as follows.

[0090] [Mathematical Formula 6]

[0091]

[0092]

[0093]

[0094]

[0095]

[0096] From the above formula, a hypothetical system is constructed, which can be expressed by the following [Equation 7].

[0097] [Mathematical Formula 7]

[0098]

[0099]

[0100]

[0101]

[0102] Using the above [Equation 7], the following coordinate transformation is performed through a state transformation process as shown in the above Fig. 4.

[0103] Designing a sequential measuring instrument, which can be expressed by the following [Equation 8].

[0104] [Mathematical Formula 8]

[0105]

[0106]

[0107] A purely slow observer (230) and a purely fast fast observer (240) are designed, which can be expressed by the following [Equation 9].

[0108] [Mathematical Formula 9]

[0109]

[0110]

[0111] From the above [Equation 9], it can be seen that the equation is in the form of a block diagonal where there are elements only on the diagonal and 0 in non-diagonal cases. Based on this [Equation 8], equations for the slow subsystem and the fast subsystem can be derived.

[0112] State conversion formula , It can be expressed by the following [Mathematical Formula 10].

[0113] [Mathematical Formula 10]

[0114]

[0115]

[0116] The above [Equation 10] is a state transformation equation to allow the system to return to the coordinates of the original singular perturbation system through inverse transformation.

[0117] As shown in Figures 5(a) and 5(b), it can be seen that the error of the slow state variable and the error of the fast state variable according to the present invention converge to a reference value of 0 in the simulation. That is, it means that the slow observer and the fast observer reconstruct the state variables of the system.

[0118] Meanwhile, FIG. 6 is a flowchart of a reduction order division measurement method for a specific perturbation system according to the present invention.

[0119] Referring to FIG. 6, the reduction order partitioning measurement method for a singular perturbation system according to the present invention includes a gain calculation step (S61), a slow state variable derivation step (S62), a fast state variable derivation step (S63), an inverse transformation step (S64), an output gain calculation step (S65), and a summation step (S66).

[0120] The method for measuring the reduction order division of the specific perturbation system of the present invention is described as follows with reference to the attached Fig. 3.

[0121] In the gain operation step (S61), the first gain operator (210) applies a slow gain ( ) is multiplied and the result is output. In the slow state variable derivation step (S62), the second gain operator (220) applies the fast gain ( Multiply by ) and output the result. Here, the above slow gain ( ) and Fast Gain( ) controls the speed of reconstructing the state of the system in the reduction order division measurement device (200).

[0122] In the slow state variable derivation step (S62), the slow observer (230) receives the output of the first gain calculator (210) and derives the slow state variable ( derives ).

[0123] In the fast state variable derivation step (S63), the fast observer (240) receives the output of the second gain calculator (220) and derives a fast state variable ( derives ).

[0124] In the inverse transformation step (S64), the inverse transformer (250) is a slow state variable (output from the slow observer (230) ) and a fast state variable output from the above fast observer (240) Inversely transform ) and the resulting state variable( Prints ).

[0125] In the output gain calculation step (S65), the inverse converter (250) is a slow state variable output from the slow observer (230). ) and a fast state variable output from the above fast observer (240) Inversely transform ) and the resulting state variable( Prints ).

[0126] In the summing step (S66), the summing unit (270) adds the output of the third gain calculator (260) and the output of the inverse converter (250) and the result is the final output ( Provides as ).

[0127] Although preferred embodiments illustrating the technical concept of the present invention have been described and illustrated above, the present invention is not limited to the configuration and operation as illustrated and described, and those skilled in the art will understand that numerous changes and modifications can be made to the present invention without departing from the scope of the technical concept. Accordingly, all such appropriate changes and modifications and equivalents should be considered to be within the scope of the present invention. Explanation of the symbols

[0128] 100: Singular perturbation system 200: Reduced-order split measurement device 210: 1st gain arithmetic unit 220: 2nd gain arithmetic unit 230 : Slow Observer 231 : First Adder 232 : 1st integrator 233 : 1st feedback operator 240 : Fast Observer 241 : Second Additive 242 : Second integrator 243 : Third feedback operator 250: Inverse converter 260: Third gain operator 270 : Adder

Claims

Claim 1 A reduction order partitioning measurement device for a singular perturbation system comprising: a first gain operator that multiplies the output of the singular perturbation system by a slow gain and a second gain operator that multiplies the output of the singular perturbation system by a fast gain; a slow observer that derives a slow state variable from the output of the first gain operator; a fast observer that derives a fast state variable from the output of the second gain operator; an inverse converter that inversely converts the slow state variable output from the slow observer (230) and the fast state variable output from the fast observer; a third gain operator that multiplies the output of the singular perturbation system by an output gain; and an adder that adds the output of the gain operator and the output of the inverse converter and provides the result as a final output. Claim 2 A reduction order partitioning measurement device for a singular perturbation system, characterized in that, in claim 1, the slow observer comprises: a first summer that sums the output of the first gain operator and the output of the first feedback operator; a first integrator that integrates the output of the first summer and outputs a slow state variable accordingly; and a first feedback operator that performs a first submatrix operation on the slow state variable and provides the result as the other input of the first summer. Claim 3 In paragraph 2, the first submatrix is A reduced-order partitioning measurement device for a singular perturbation system characterized by being, in the first submatrix, , is the modeled system matrix, and is the gain value of the measuring instrument. Claim 4 A reduction order partitioning measurement device for a singular perturbation system, characterized in that, in paragraph 3, the fast observer comprises: a second summer that sums the output of the second gain operator and the output of the second feedback operator; a second integrator that integrates the output of the second summer and outputs a fast state variable accordingly; and a second feedback operator that performs a second submatrix operation on the fast state variable and provides the result as the other input of the first summer. Claim 5 In paragraph 4, the second submatrix is, A reduced-order partitioning measurement device for a singular perturbation system characterized by being, in the second submatrix, is the singular perturbation parameter epsilon, and , is the modeled system matrix, and is the gain value of the measuring instrument. Claim 6 A method for measuring a reduced-order partition of a singular perturbation system, comprising: a step in which a first gain operator multiplies the output of a singular perturbation system by a slow gain, and a second gain operator multiplies the output of the singular perturbation system by a fast gain; a step in which a slow observer derives a slow state variable from the output of the first gain operator; a step in which a fast observer derives a fast state variable from the output of the second gain operator; a step in which an inverse converter inversely converts the slow state variable output from the slow observer and the fast state variable output from the fast observer; a step in which a third gain operator multiplies the output of the singular perturbation system by an output gain; and a step in which an adder adds the output of the gain operator and the output of the inverse converter and provides the result as a final output. Claim 7 A method for measuring reduced order partitioning for a singular perturbation system, characterized in that, in claim 6, the step of deriving the slow state variable comprises: a step in which a first summerizer sums the output of the first gain operator and the output of the first feedback operator; a step in which a first integrator integrates the output of the first summerizer and outputs a slow state variable accordingly; and a step in which a first feedback operator performs a first submatrix operation on the slow state variable and provides the result as the other input of the first summerizer. Claim 8 A method for measuring reduced order partitioning for a singular perturbation system, characterized in that, in claim 7, the step of deriving the fast state variable comprises: a step in which a second summer combines the output of the second gain operator and the output of the second feedback operator; a step in which a second integrator integrates the output of the second summer and outputs a fast state variable accordingly; and a step in which a second feedback operator performs a second submatrix on the fast state variable and provides the result as the other input of the first summer.

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