Classifications

• • G—PHYSICS
  • G05—CONTROLLING; REGULATING
  • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
  • G05B11/00—Automatic controllers
  • G05B11/01—Automatic controllers electric
  • G05B11/36—Automatic controllers electric with provision for obtaining particular characteristics, e.g. proportional, integral, differential
• • G—PHYSICS
  • G05—CONTROLLING; REGULATING
  • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
  • G05B19/00—Programme-control systems
  • G05B19/02—Programme-control systems electric
  • G05B19/18—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form
  • G05B19/4155—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form characterised by programme execution, i.e. part programme or machine function execution, e.g. selection of a programme
• • G—PHYSICS
  • G05—CONTROLLING; REGULATING
  • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
  • G05B2219/00—Program-control systems
  • G05B2219/30—Nc systems
  • G05B2219/42—Servomotor, servo controller kind till VSS
  • G05B2219/42033—Kind of servo controller

Definitions

• the invention relates to the field of correctors for the control of systems, and more particularly correctors of the PID (proportional-integral-derivative) type.
• the control of a quantity consists of the measurement, in real time, of the difference, called error, between the real value of this quantity and the setpoint to be reached.
• a corrector then applies a transfer function to this error.
• the result of this transfer function forms a command to be applied to reduce the error between the setpoint and the actual value of the quantity to be controlled.
• PID correctors are characterized by an operating law comprising a proportional term, a derivative term and an integral term, hence the name.
• PID correctors are linear, that is, their transfer function is a linear function. Tuning a linear PID corrector requires the adjustment of three coefficients to tune that PID corrector, which makes linear PID correctors simple to implement.
• the control carried out with a well-adjusted linear PID corrector is robust (i.e. resistant to disturbances), fast (i.e. with low response time) and precise (i.e. the asymptotic error is substantially zero).
• a PID corrector is very effective for slaving a drone as part of a model with low pitch and roll disturbances around the horizontal position of the drone.
• the nonlinear effects of this type of model are relatively small, and can be treated as uncertainties and minimized by adjusting the corrector, for example using an H-infinite criterion.
• Nonlinear correctors of the sliding mode control type, or SMC are known, as described in the article by Xu, R., & Ozguner, U. (2006, December) “Sliding mode control of a quadrotor helicopter ”, In Proceedings of the 45th IEEE ConfInteron Decision and Control (pp. 4957-4962). IEEE.
• a digital control device for the servoing of a system to be controlled digitally, operating in discrete time steps, from an error vector received as input at each time step, comprising a memory arranged to receive control parameter data comprising a uniformity factor selected from the range [-1; 0], a feedback gain matrix linked to the system to be controlled, a dilation generator matrix, a Lyapunov matrix defining a homogeneous canonical norm, a lower bound, an upper bound, and a proportional coefficient, a derivative coefficient and a integral coefficient characteristics of the system to be controlled, an estimator designed to determine a range of homogeneous canonical norm value for a current time step from the estimation ranges of the previous time steps, of the error vector of the current time step , the expansion generator matrix, the Lyapunov matrix, the lower limit and the upper limit, which define the estimation range for the first time step, a computer arranged to return a command of the system to be controlled for the current time step from the sum between the error vector
• the Applicant has designed a new type of corrector, with improved performance compared to a linear PID corrector.
• the Applicant has implemented this new corrector in a system initially slaved with a linear PID corrector by drawing profile from the adjustments already made by a manufacturer on this linear PID corrector.
• This device is particularly advantageous because it improves the response time and the robustness of the system compared to a linear corrector characterized by the coefficients kp, ki and kd. These improvements are very easy to implement, because the system in its overall is little changed. Finally, compared to a linear PID corrector, the additional cost in complexity and in computation load is very low.
• the invention may have one or more of the following characteristics:
• the computer is arranged to calculate the factor calculated from the feedback gain matrix, the range of homogeneous canonical norm value for a current time step, the proportional coefficient, the derivative coefficient and the generator matrix of expansion according to the formula
• K 0 is the feedback gain matrix
• m is the homogeneity factor
• P is a value drawn from the range of values of homogeneous canonical norm for the current time step
• K is a matrix associating the proportional coefficient and the derivative coefficient
• G is the matrix of generator of dilation
• the computer is arranged to calculate the value representing the integral between the first time step and the current time step of a product associating the integral coefficient, the expansion generator matrix, the range of homogeneous canonical norm value of the time step and the error vector according to the formula
• K i is the integral coefficient
• G is the expansion generator matrix
• P is a value drawn from the range of homogeneous canonical norm value for the time step T
• the computer is designed to draw as a value in the range of homogeneous canonical norm value for a given time step the upper limit of this range,
• the computer is designed to use the upper limit of the range of homogeneous canonical norm value for the current time step as a value drawn from the range of homogeneous canonical norm value for the current time step,
• the estimator determines the range of homogeneous canonical norm value for a current time step by initializing a lower range value and an upper range value with the bounds of the homogeneous canonical norm value range of the previous time step and applying : - a first test determining if the product of the transpose of the product of the error vector of the current time step and of the transpose of the exponential of the product of the expansion generator matrix by the negative of the logarithm of the range value upper range, the Lyapunov matrix, and the product of the error vector of the current time step and the transpose of the exponential of the product of the expansion generator matrix by the negative of the logarithm of the upper range value is greater than 1, and, if applicable, the definition of the range of homogeneous canonical norm value of the time step of the current time step between the upper range value and the minimum between the double of the upper range value and the upper bound,
• the invention also relates to a quadrotor which comprises a device according to the invention for calculating a command from a setpoint received as an input and a computer program product designed to implement the estimator and the calculator of the device. according to the invention.
• Figure 1 shows a schematic view of a coordinate system of a quadrotor
• FIG. 2 shows a schematic view of a device according to the invention
• FIG. 3 represents an example of the operating loop of the device of FIG. 2
• Figure 4 shows an example of a particular implementation of a function of Figure 3
• Figure 5 shows an example of a particular implementation of another function of Figure 3,
• Figure 6 shows a comparison of the accuracy between a setpoint, a conventional PID control, and the homogeneous PID control of the invention in the x dimension
• FIG. 7 represents a comparison of the precision between a setpoint, a conventional PID control, and the homogeneous PID control of the invention in the dimension y
• FIG. 8 represents a comparison of the precision between a setpoint, a conventional PID control, and the homogeneous PID control of the invention in the dimension z
• FIG. 9 represents a comparison of the precision between a setpoint, a conventional PID control, and the homogeneous PID control of the invention in the Y dimension
• FIG. 10 Figure 10 shows the robustness of the conventional PID control
• FIG. 11 Figure 11 shows the robustness of the homogeneous PID control according to the invention
• FIG. 12 quantifies the improvements represented in FIGS. 6 to 11 in the L2 standard.
• the drawings and the description below essentially contain elements of a certain nature. They can therefore not only serve to better understand the present invention, but also contribute to its definition, if necessary.
• Figure 1 shows a schematic view of a coordinate system of a quadrotor. This system will be used to explain the conventional operation of a system controlled by the device according to the invention.
• rotors exert respective forces f 1 to f 4 , for example to make a drone fly.
• the control of the rotors is carried out by means of a control u which is a vector of dimension 4.
• R 4 ® R 4 is an invertible mapping, an example of which can be found in article by Wang, S., Polyakov, A., & Zheng, G .. “Quadrotor Control Design under Time and State Constraints: Implicit Lyapunov Function Approach” in 2019 18th European Control Conference (pp. 650-655). IEEE.
• the matrices A, B and C are known for a given system to be controlled, and the control is controlled by a linear derivative integral proportional control according to the following formula:
• K p is the proportional coefficient of the command
• K d is the derivative coefficient of the command
• K i is the integral coefficient of the command
• This matrix must be a matrix of dimension d * d which respects the anti-Hurwitz criterion, that is to say that all its eigenvalues have a positive real part, and respect the following constraint:
• / is the identity matrix
• m is a homogeneity factor in the range [-
• the matrix P must be a symmetric matrix of dimension d * d which respects the following constraints:
• exp () is the exponential function
• P is a homogeneous canonical norm defined implicitly by the matrices G and P according to the following constraints:
• Figure 2 shows a schematic view of a device according to the invention.
• the device 2 comprises a memory 4, an estimator 6 and a computer 8.
• the device 2 receives operating data 12 from a system to be controlled 10, and the computer 8 sends the system 10 a command 14.
• the memory 4 can be any type of data storage suitable for receiving digital data: hard disk, hard disk with flash memory, flash memory in any form, random access memory, magnetic disk, storage distributed locally or in the cloud, etc.
• the data calculated by the device can be stored on any type of memory similar to the memory 4, or on the latter. This data can be erased after the device has performed its tasks or retained.
• the memory 4 receives the parameters described above, namely the homogeneity factor m chosen from the range [-1; 0], the feedback gain matrix K 0 , the expansion generator matrix, the Lyapunov matrix defining the homogeneous canonical norm, the proportional coefficient K p , the derivative coefficient K d and the integral coefficient characteristic of the system to be controlled.
• the memory 4 also receives a lower bound a, an upper bound b, and a number of loops N.
• the Applicant has discovered an algorithm making it possible to estimate a range of values approaching the value of the standard. canonical. This algorithm will be described with figure 3. However, in order to guarantee its convergence, it is necessary to threshold part of the calculations with the lower bound a and the upper bound b, and to define a number of loops N.
• the estimator 6 and the computer 8 are elements which directly or indirectly access the memory 4. They can be produced in the form of an appropriate computer code executed on one or more processors. By processors, it must be understood any processor adapted to the calculations described below.
• Such a processor can be produced in any known manner, in the form of a microprocessor for a personal computer, of a dedicated chip of FPGA or SoC type, of a computing resource on a grid or in the cloud, of a microcontroller, or any other form suitable for providing the computing power necessary for the embodiment described below.
• a processor can also be made in the form of specialized electronic circuits such as an ASIC.
• a combination of processor and electronic circuits can also be envisaged.
• estimator 6 and the calculator 8 are described separately from the system 10 in FIG. 2, they are intended to be integrated into the latter, either by modifying the controller of the system 10 or by adding to it.
• device 2 although shown and described separately, aims to be intimately integrated within system 10 and replace its linear derivative integral proportional control. It will nevertheless remain possible to achieve this in the form of an addition which is grafted onto a system 10 and derives the order from it.
• FIG. 3 represents an operating loop of the device 2.
• the device 2 is digital and makes use of the fact that the systems 10 for which it is designed are also digital. This means that it operates in discrete time steps. Also, instead of talking about a time variable t, one will use thereafter an index i which designates a number of time steps, and therefore an instant t equivalent to the product of the index i by the duration of the step of device time 2.
• FIG. 3 is therefore presented in the form of an update loop which receives as input in an operation 300 data 12 of error vector E and which outputs in an operation 399 control data 14 u to system 10.
• all the elements calculated in the various loops will be stored in memory 4.
• these elements could not be stored, and, when a function refers to a passed value, the latter could be recalculated.
• the estimator 6 performs an Est () function to determine a range of values that approximates the value of the homogeneous canonical norm of the Math 11 formula.
• FIG. 4 represents an exemplary embodiment of the function Est ().
• the Is () function relies on testing a value in order to determine directly or dichotomously the bounds of the range that approaches the value of the homogeneous canonical norm.
• this range is fixed by the lower bound a and the upper bound b, that is to say that nc [0] is equal to [a; b]. Then, for each time step, this function begins in an operation 400 by initializing the lower bound value variable A and the upper bound value variable B with the values of the bounds of the range determined at the previous step nc [il] .
• the first test is performed in an operation 410 with the execution of a test () function with the upper bound value variable B as an argument.
• the test function is as follows: [Math 12]
• test value (B) is greater than 1, then the range nc [i] is determined in an operation 420 to be between the upper bound variable B and the minimum between twice the upper bound variable B and the upper bound b. Then the function ends in an operation 499. If the test value (B) is less than 1, then the second test is performed in an operation 430. Here, it is the lower bound value variable A which is given as an argument to the test () function.
• test value (A) is less than 1
• range nc [i] is determined in an operation 440 as being between the maximum between half of the lower bound variable A and the lower bound b of a part and the lower bound variable A on the other. Then the function ends in operation 499.
• test value (A) is greater than 1, then the range nc [i] is calculated by dichotomy in a loop that begins by initializing a loop index j to 0.
• an end-of-loop condition is tested in an operation 455 by comparing the index j to the number of loops N. If the latter is not reached, then the loop begins in an operation 460 by calculating a dichotomy variable V which is set to half the sum of the lower bound value variable A and the variable of upper bound value B.
• the dichotomy variable V is then given as an argument to the test () function in an operation 465. If the test value (V) is less than 1, then the upper bound value variable B is updated with the variable dichotomy V in an operation 470. If the test value (V) is greater than 1, then the lower bound value variable A is updated with the dichotomy variable V in an operation 475.
• the index j is incremented in an operation 480 and the loop resumes with the test of operation 455.
• the range nc [i] is defined in an operation 490 as between the value variable of lower bound A and the variable of upper bound value B and the function Is () ends in operation 499.
• N the number of loops N is an important factor and constitutes a compromise to be chosen for device 2. In fact, it will regularly happen that the tests of operations 410 and 430 are negative, in which case the loop is executed, this being the case. which is more costly in computing time and in energy consumed. Also, it will be advisable to choose the value of N as a function of the desired precision / calculation time and energy consumed compromise.
• the function Est () could be implemented differently, for example by the use of gradient.
• the Est () function can return the value ncv [i] directly instead of the range nc [i].
• operation 310 of FIG. 3 is followed by determining the value of the integral term of the Math formula 10, with the execution in an operation 320 of a function Int ( ) by computer 8.
• FIG. 5 represents an example of the implementation of the function Int ().
• the numerical nature of the implementation is exploited by replacing the computation of the integral term of the Math 10 formula with the addition of the increment to the previous value.
• the first integral is initialized to 0, and for the following ones the function begins in an operation 500 by initializing the integral value I [i] of the current step with the integral value of the previous step I [i-1] .
• the value of the homogeneous canonical norm for the current step is chosen by a function cho () in an operation 510. As described above, in a preferred version, it is the upper bound of the range nc [i] which is retained. Finally, in an operation 520, the integral value I [i] is calculated by adding the contribution of the term of the current time step, then the function Int () ends in an operation 599. As a variant, the function Int ( ) could recalculate the value of the integral I [i] starting from zero.
• the command u is calculated in an operation 330 of FIG. 3.
• the computer 8 executes a function Cont ().
• the Cont () function performs the calculation of the following formula:
• Figures 6 to 11 show comparisons between a linear integral derivative proportional control and the homogeneous integral derivative proportional control of the device 2.
• FIGS. 6 to 9 represent a comparison of the precision between a setpoint, a conventional PID control, and the homogeneous PID control of the invention in the dimensions x, y, z and Y, each time a closer view of the stable area.
• Figures 10 and 11 show the robustness respectively of the conventional PID control and the homogeneous PID control according to the invention.
• FIG. 12 quantifies these improvements in the L2 standard. Note, the energy cost is only 1.1% more compared to conventional PID control.

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• 2020
  • 2020-05-12 FR FR2004684A patent/FR3110257A1/fr active Pending
• 2021
  • 2021-05-11 US US17/925,148 patent/US20230176545A1/en active Pending
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