WO2019076881A2

WO2019076881A2 - Methods of proportional-integral-derivative control

Info

Publication number: WO2019076881A2
Application number: PCT/EP2018/078211
Authority: WO
Inventors: José Miguel Silva Do Carmo NOGUEIRA; André Miguel Alves FARINHA; Filipe Da Conceição Fernandes THOMAZ; Tiago Filipe Pinto SILVA
Original assignee: Biosurfit, S.A.
Priority date: 2017-10-18
Filing date: 2018-10-16
Publication date: 2019-04-25
Also published as: WO2019076881A3

Abstract

A method of proportional-integral-derivative, PID, control of a controlled system is provided. The method comprises: determining an error between a setpoint of the controlled system and a state variable of the controlled system; determining a control signal based on a sum of a proportional term, an integral term, and a derivative term, each of the terms being a function of the error and being weighted by a respective gain; and applying the control signal to the controlled system. At least one of a gain of the integral term or a gain of the derivative term is based on the setpoint and the state variable.

Description

METHODS OF PROPORTIONAL-INTEGRAL-DERIVA I E CONTROL

Field

[0001] The present disclosure relates generally to methods of proportional-integral-derivative (PID) control. In particular, but without limitation, the present disclosure relates to methods of controlling a speed of a motor.

Background

[0002] Proportional-integral-derivative (PID) controllers are commonly used in control systems, eg, for controlling temperature, pressure, force, position, or speed of a process. PID controllers can, in particular, be used to control the speed of a motor.

[0003] PID controllers generally calculate an error value which is representative of a difference between a (desired) setpoint and a

(measured) process variable, and apply a correction based on proportional, integral, and derivative terms, each of which is a function of the error value and is weighted by a respective gain.

Summary

[0004] Aspects and features of the present disclosure are defined in the accompanying independent claims .

[0005] A method of proportional-integral-derivative (PID) control of a controlled system is provided. The method comprises

determining an error between a setpoint of the controlled system and a state variable of the controlled system. A control signal is determined based on a sum of a proportional term, an integral term, and a derivative term. Each of the terms is a function of the error and is weighted by a respective gain. The control signal is applied to the controlled system.

[0006] At least one of a gain of the proportional term, a gain of the integral term, or a gain of the derivative term may be based on the setpoint and the state variable. [0007] The at least one of the gain of the proportional term, the gain of the integral term, or the gain of the derivative term may be based on the error.

[0008] The at least one of the gain of the proportional term, the gain of the integral term, or the gain of the derivative term may be based on a ratio of the error and a reference value.

[0009] The at least one of the gain of the proportional term, the gain of the integral term, or the gain of the derivative term may be based on an exponential decay function of the error.

[0010] The exponential decay function of the error may be of the form + c,-^A-< 0

Ao

/(Δ) = ae ^b&° + c, 0 < < 1,

Ao

ae^~b + c, - > 1

A₀ where Δ is the error, Δ₀ is the reference value, and a, b , and c are predetermined coefficients.

[0011] Consequent to a switching event in the controlled system, a contribution of the sum to the control signal applied to the controlled system may be temporarily reduced.

[0012] Temporarily reducing the contribution of the sum to the control signal may comprise reducing the contribution of the sum to the control signal, and increasing the contribution of the sum to the control signal according to a continuous transition function.

[0013] The contribution of the sum to the control signal applied to the controlled system may be temporarily reduced for a predetermined amount of time.

[0014] The controlled system may be powered by a plurality of power sources. The switching event may comprise switching from one of the plurality of power sources to another of the plurality of power sources . [0015] The method may further comprise limiting the control signal to a first predetermined range of values.

[0016] The control signal at a given time instant may be based on the sum at that time instant and a value of the control signal at a previous time instant.

[0017] The method may further comprise limiting a sum, over time, of values of the sum of the proportional, integral, and derivative terms to a second predetermined range of values .

[0018] The method may further comprise limiting the sum of the proportional, integral, and derivative terms to a third

predetermined range of values.

[0019] The method may further comprise determining that a sign of the error has changed, and temporarily reducing the gain of the integral term consequent to determining that the sign of the error has changed.

[0020] The controlled system may be an electric motor. In

particular, the electric motor may be a brushless direct current, DC, motor.

[0021] The setpoint and state variable may respectively be

indicative of a desired and a measured position of a point on the motor .

[0022] The setpoint and state variable may respectively be

indicative of a desired and a measured speed of the motor.

[0023] There is also provided an apparatus arranged to perform any of the methods described herein.

[0024] The apparatus may comprise a processor configured to perform any of the methods described herein. The apparatus may also comprise a sensor arranged to determine the state variable of the controlled system.

[0025] There is also provided a non-transitory computer-readable medium carrying computer-readable instructions which, when executed by a processor, cause the processor to carry out any of the methods described herein.

[0026] An effect of the present disclosure is the provision of a PID control method having improved performance.

[0027] An effect of the present disclosure is the provision of a PID control method where the gains of the proportional, integral and derivative terms may not need to be manually tuned for each

particular controlled system, and may instead be tuned

automatically .

[0028] An effect of the present disclosure is to avoid the effects of integral error accumulation on the output of a PID controller.

[0029] The sensorless control of the speed of a brushless direct current (BLDC) motor based on back electromotive force may

inherently present a measurement delay that is much higher than the controller actuation timeframe, thus leading to significant velocity overshoots and deviation from the desired movement of the motor when using a standard PID controller. An effect of the present

disclosure is to reduce the controller deviations, due to controller delay, when the controlled system comes closer to the desired setpoints .

[0030] Two voltage sources may be used to power a motor in order to increase the range of speeds at which the motor can be operated, without compromising on the resolution of the speeds which can be selected (ie, allowing the speed to be increased or decreased in small increments) . At a given desired speed, switching between those voltage sources may be appropriate so as to ensure stability of the motor. However, it may take multiple seconds to switch between the sources, while the controller may have a much shorter cycle time. An effect of the present disclosure is that the controller may be aware of the switch, and may smoothen the

transition between sources so as to mitigate temporary deleterious effects of the switch. Brief description of the drawings

[0031] Examples of the present disclosure will now be explained with reference to the accompanying drawings in which:

Fig. 1 illustrates an arrangement in which a controller is operable to control a controlled system;

Fig. 2 shows a block diagram of a controller;

Figs. 3a and 3b show a flow chart of the steps of a method of PID control;

Fig. 4 illustrates one example of the arrangement; and

Fig. 5 illustrates an exponential decay function.

[0032] Throughout the description and the drawings, like reference numerals refer to like parts.

Detailed description

[0033] Fig. 1 shows a control arrangement 100 comprising a

controlled system 150 (or 'process' , or 'plant' ) , which is

controlled by a controller 200. The controller 200 receives a setpoint SP(t) as an input, where t represents time, and outputs a control signal OP(t) (or 'control update signal', or 'correction signal', or 'control output signal'), which is fed to the controlled system 150. A state variable PV(t) (or 'process variable') of the controlled system 150 is determined, and fed back to the controller 200 to be used in the determination of the control signal OP(t) . A feedback loop is thus established, thereby enabling accurate control of the controlled system 150.

[0034] The controlled system 150 may be an electric motor, such as a brushless direct current (BLDC) motor. In that case, the setpoint SP(t) and the state variable PV(t) may be respectively indicative of a desired and a measured speed of the motor (eg, in revolutions per minute, RPM) , or of a desired and a measured position of a point on the motor (eg, a point on the rotor of the motor) . [0035] Fig. 2 shows a block diagram of the controller 200, which may be used for implementing elements of the methods described herein. The controller 200 comprises a processor 210 arranged to execute computer-readable instructions, which may be stored in a memory 220, for example a random access memory. The memory 220 may also store previous values of any of the signals described below. The processor 210 may receive data, eg, the setpoint SP(t), via an analog-to-digital (A/D) converter. The processor 210 may also output data, eg, the control signal OP(t), via a digital-to-analog

(D/A) converter. A sensor 250 may be arranged to determine the state variable PV(t) of the controlled system 150 and to communicate that state variable to the A/D converter 230 and/or to the processor 210. Although the controller 200 of Fig. 2 comprises a computer processor, a person skilled in the art will understand that the methods described herein may alternatively be implemented using analog circuitry.

[0036] A method of PID control of the controlled system 150 is explained with respect to Figs. 3a and 3b. This method may be implemented by the controller 200, or indeed by any processor or processing means.

[0037] In the following disclosure, the steps of this method are primarily explained as steps that are performed repeatedly at each time instant n, ie, in an iterative manner. However, it will be understood that these steps could alternatively be performed continuously, eg, using analog circuitry.

[0038] In step S305, an error A(t) [or e (t) ] between a setpoint SP(t) of the controlled system 150 and a state variable PV(t) of the controlled system 150 is determined (or ^calculated' ) , ie,

A(t) = SP(t) - PV(t) .

[0039] In step S310, a control signal OP(t) is determined based on a sum UP(t) of a proportional term UP_p(t), an integral term UPi (t) , and a derivative term UPd(t) . Each of these terms is a function of the error Δ and is weighted by a respective gain (or ^weight' ) Kp(t), Ki(t), and K_d(t) . That is, UP(t) = K_p{t)U_p{) + Ki(t)Ui(t) + K_d(t)U_d(t) .

[0040] In one example, the control signal OP(n) at a given time instant n may be based on the sum UP (n) at that time instant, as well as a value of the control signal OP(n-l) at a previous time instant n-1. In particular, the control signal OP(n) at a given time instant n may be given by

OP(n) = OP(n - 1) + UP n) .

In this way, the control signal may be changed (or adjusted) more slowly. In other words, a form of smoothing is applied.

[0041] In another example, the control signal OP(t) and the sum UP(t) may be equal. That is, the control signal OP (n) at a given time instant n may be given by

OP(n) = UP(n).

In this way, the control signal may be changed more quickly.

[0042] The proportional, integral and derivative terms are

respectively given by i/_p (t) = A(t) , i/_i(t) = /_z ^t ₌₀ A(z)dz , dA(t)

U_d(t) = dt

Thus, the proportional term is proportional to the error, the integral term is an integral of the error, and the derivative term is a derivative of the error.

[0043] At least one of the gain K_p(t) of the proportional term, the gain Ki(t) of the integral term, or the gain Kd(t) of the derivative term may be based on the setpoint SP(t) and the state variable PV(t) . In particular, at least one of these gains may be based on the error Δ (t ) .

Δ

[0044] At least one of these gains may be based on a ratio — of the error Δ and a reference value (or ^bandwidth value') Δο . The reference value Δο may be determined based on a response time of the controlled system 150. The reference values used to determine the gains of each of the terms may be the same, or different.

[0045] In one example, the gain K_p(t) of the proportional term may not be a function of the setpoint SP(t), the state variable PV(t), or the error A(t), and may simply be given by

where c_p is a predetermined coefficient.

[0046] At least one of the gains may be based on an exponential decay function of the error Δ. An example of an exponential decay function is shown in Fig. 5. In particular, the exponential decay function may be of the form + c,-^A-< 0

Ao

b Δ

/(Δ) = ae^{~ ~}o+c, 0 <— < 1,

Ao

ae^~b + c,^-> 1

A₀ where Δ is the error, Δ₀ is the reference value, and a, b, and c are predetermined coefficients.

In one example, the exponential decay function (Δ) satisfies the following requirements:

• when - > 1, /(Δ) « 0.0,

Ao

• when - < 0, /(Δ) « 1.0,

Ao

• when = 0.1, /(Δ) * 0.6,

Ao

• when = 0.2, /(Δ) * 0.35.

Ao

[0047] The gain Ki(t) of the integral term may be given by if_i(t) = c_i-f₁(A(t)), where fi(A) is of the form given in paragraph [0046], and C; is a predetermined coefficient.

[0048] The gain Kd(t) of the derivative term may be given by ¾(t) = c_dl + c_d2 - f₂ (A(t)) , where f₂ (A) is of the form given in paragraph [0046], and c_dl and c_d2 are predetermined coefficients. Functions fi(A) and f₂ (A) may be the same, or different.

[0049] Although a state of the controlled system 150 may be changed in the controller 200 in a quasi-instantaneous manner, such a change may take some time to actually be reflected in the controlled system 150. In other words, the controlled system 150 may take some time to transition to the new state. As a result, the control signal OP(t) applied shortly after such a change occurs in the controller 200 may be incorrect. This may be particularly problematic in cases where the time between iterations of the method of Figs. 3a and 3b is significantly shorter than the time it takes for a change of state to be reflected in the controlled system 150. This may, for example, be the case when the controlled system 150 also has its own control circuitry, which responds more slowly than the controller 200.

[0050] As one example, where the controlled system 150 is a BLDC motor, and the BLDC motor may be powered by multiple voltage sources, the controlled system 150 may include a separate controller that switches between those sources. Switching between voltage sources may take 3 seconds, while the controller 200 may perform the method of Figs. 3a and 3b every 0.1 seconds. When the switch between voltage sources occurs, the controlled system 150 is temporarily in an unknown state as far as the controller 200 is concerned: the change of state has already been made in the

controller 200 (in the firmware) , but has not yet occurred in the controlled system 150 (in the hardware) .

[0051] This issue may be mitigated by temporarily reducing a contribution of the sum UP(t) to the control signal OP(t) . Such an approach may be particularly useful to accommodate the feedback error from slower measurement systems, or external control systems.

[0052] Accordingly, in step S315, consequent to a switching event in the controlled system 150 (or a change of state of the controlled system 150), a contribution of the sum UP(t) to the control signal OP(t) may be temporarily reduced. In the case where the controlled system 150 is powered by a plurality of (power) sources, the switching event may comprise switching from one of the plurality of power sources to another of the plurality of power sources.

[0053] The contribution may be temporarily reduced by reducing the contribution of the sum UP(t) to the control signal OP(t), and subsequently increasing the contribution of the sum UP(t) to the control signal OP(t) according to a continuous transition function

(or ^hysteresis function') g. In one example, reducing the

contribution of the sum UP(t) to the control signal OP(t) may comprise reducing the contribution to zero.

[0054] The contribution may be temporarily reduced for a

predetermined amount of time T, which may be selected according to a time required for the switching event to occur (or end) in the controlled system 150.

[0055] In order to temporarily reduce the contribution of the sum UP(t) to the control signal OP(t), the control signal OP(n) may be determined, at a time instant n, as

OP(n) = OP{n - 1) + UP'(n), where UP'(n) is then given by

UP'(n) = g(t_h -UP(n , g is the continuous transition function, and t_h is the time since the switching event. The continuous transition function g may be given by

where is a continuous function (eg, linear, n-degree

polynomial, exponential, etc.) . This continuous function /i(t_¾) may be selected according to a transition response of the controlled system 150. As one example, the function may be a linear function, ie, given by h(t_h) = mt_h ,

where m is a predetermined coefficient.

[0056] The incorporation of previous values of the control signal OP(n-l) in the value of the control signal OP (n) at a given time instant may cause integral error accumulation. In order to mitigate this effect, in step S320, the gain Ki (t) of the integral term may be temporarily reduced consequent to determining that a sign of the error A(t) has changed (from positive to negative, or from negative to positive) . At a time instant n, such a determination may be expressed as determining whether

[Δ(η) > 0 Λ Δ(η - 1) < 0] V [Δ(η) < 0 Λ Δ(η - 1) > 0] is true. The gain Ki(t) of the integral term may, in particular, be set to 0 for one or more time instants .

[0057] In steps S325 to S335, one or more saturation controls may be applied to the control signal OP(t) and/or the sum UP(t) . As a result, desynchronization of the controlled system 150 and the controller 200 may be avoided or reduced.

[0058] In particular, in step S325, the control signal OP(t) may be limited to a first predetermined range of values s_{1 min} to s_{1 max} .

That is, sl,min— — ^sl,max ·

[0059] In step S330, a sum, over time, of values of the sum UP(t) of the proportional, integral and derivative terms may be limited to a second predetermined range of values s_2imin to s_{2 rnax} . In

particular, the sum over time may be taken over N consecutive time samples. That is, s2,min—∑z=n-W+l^(^z) — ^s2,max^■

In a continuous time system, the sum may be an integral taken over a duration T. [0060] In step S335, the sum UP(t) of the proportional, integral and derivative terms may be limited to a third predetermined range of values s_3min to s_3max. That is, s3,min — UP(t ≤ S_{3 max} .

[0061] In step S340, the control signal OP (t) is applied to the controlled system 150.

[0062] Once the steps of Figs. 3a and 3b have been performed, they may be repeated. The method is thereby able to adapt to changes in the setpoint SP(t) and/or the state variable PV(t) .

[0063] A number of the equations set out above include

predetermined coefficients (eg, a , b, c, c_p, c_ir c_dl, c_d2) . Any of these coefficients may be determined according to the controlled system 150. In particular, they may depend on the power source used for the controlled system 150, and/or on whether the state variable PV(t) of the controlled system is ascending or descending at a given time instant. Any of the predetermined coefficients may be

determined using a curve optimisation method, such as least squares.

[0064] As will be apparent to a person skilled in the art, unless otherwise indicated, the steps of the methods set out above need not all be performed, and some steps may be omitted without departing from the scope of the disclosure. Furthermore, unless otherwise indicated, the steps need not be performed sequentially, and may be performed in any order.

[0065] An example of the process of Figs. 3a and 3b is set out in Fig. 4. In this example, steps S305-315 and S325-340 are performed, while step S320 is not shown/performed.

[0066] The present disclosure may be applied to motor speed control (eg, BLDC motors), positional control (eg, robot manipulators), or industrial control systems for controlling parameters such as temperature, pressure, force, flow rate, etc.

[0067] The methods described herein may be embodied on a computer- readable medium, which may be a non-transitory computer-readable medium. The computer-readable medium carries computer-readable instructions arranged for execution upon a processor so as to make the processor carry out any or all of the methods described herein.

[0068] The term "computer-readable medium" as used herein refers to any medium that stores data and/or instructions for causing a processor to operate in a specific manner. Such storage medium may comprise non-volatile media and/or volatile media. Non-volatile media may include, for example, optical or magnetic disks. Volatile media may include dynamic memory. Exemplary forms of storage medium include, a floppy disk, a flexible disk, a hard disk, a solid state drive, a magnetic tape, or any other magnetic data storage medium, a CD-ROM, any other optical data storage medium, any physical medium with one or more patterns of holes, a RAM, a PROM, an EPROM, a FLASH-EPROM, NVRAM, and any other memory chip or cartridge.

[0069] The above description has been made in terms of specific examples for the purpose of illustration and not limitation. Many modifications and combinations of, and alternatives to, the features described above will be apparent to a person skilled in the art and are intended to fall within the scope of the invention, which is defined by the claims that follow.

Claims

1. A method of proportional-integral-derivative, PID, control of a controlled system, the method comprising: determining an error between a setpoint of the controlled system and a state variable of the controlled system; determining a control signal based on a sum of a proportional term, an integral term, and a derivative term, each of the terms being a function of the error and being weighted by a respective gain; and applying the control signal to the controlled system, wherein at least one of a gain of the integral term or a gain of the derivative term is based on the setpoint and the state variable .

2. The method of claim 1, wherein the at least one of the gain of the integral term or the gain of the derivative term is based on the error .

3. The method of claim 2, wherein the at least one of the gain the integral term or the gain of the derivative term is based on ratio of the error and a reference value.

4. The method of any of claims 2 to 3, wherein the at least one of the gain of the integral term or the gain of the derivative term is based on an exponential decay function of the error.

5. The method of claim 4 when dependent on claim 3, wherein the exponential decay function of the error is of the form a + c,— < 0

__b_A_

/(Δ) = ^Δ + c, 0 <— <

e^_¾ + c,^- > 1 where Δ is the error, Δ₀ is the reference value, and a , b , and c are predetermined coefficients.

6. A method of proportional-integral-derivative, PID, control of a controlled system, the method comprising: determining an error between a setpoint of the controlled system and a state variable of the controlled system; determining a control signal based on a sum of a proportional term, an integral term, and a derivative term, each of the terms being a function of the error and being weighted by a respective gain; applying the control signal to the controlled system; and consequent to a switching event in the controlled system, temporarily reducing a contribution of the sum to the control signal applied to the controlled system.

7. The method of claim 6, wherein temporarily reducing the contribution of the sum to the control signal comprises: reducing the contribution of the sum to the control signal; and increasing the contribution of the sum to the control signal according to a continuous transition function.

8. The method of any of claims 6 to 7 , wherein the contribution of the sum to the control signal applied to the controlled system is temporarily reduced for a predetermined amount of time.

9. The method of any of claims 6 to 8 , wherein the controlled system is powered by a plurality of power sources, and the switching event comprises switching from one of the plurality of power sources to another of the plurality of power sources.

10. The method of any preceding claim, further comprising: limiting the control signal to a first predetermined range of values .

11. The method of any preceding claim, wherein the control signal at a given time instant is based on the sum at that time instant and a value of the control signal at a previous time instant.

12. The method of claim 11, further comprising: limiting a sum, over time, of values of the sum of the proportional, integral, and derivative terms to a second

predetermined range of values.

13. The method of any of claims 11 to 12, further comprising: limiting the sum of the proportional, integral, and derivative terms to a third predetermined range of values.

14. The method of any preceding claim, further comprising: determining that a sign of the error has changed; and temporarily reducing the gain of the integral term consequent to determining that the sign of the error has changed.

15. The method of any preceding claim, wherein the controlled system is an electric motor.

16. The method of claim 15, wherein the setpoint and state variable are respectively indicative of a desired and a measured position of a point on the motor.

17. The method of claim 15, wherein the setpoint and state variable are respectively indicative of a desired and a measured speed of the motor.

18. The method of any of claims 15 to 17, wherein the electric motor is a brushless direct current, DC, motor.

19. An apparatus arranged to perform the method of any preceding claim .

20. The apparatus of claim 19, wherein the apparatus comprises: a processor configured to perform the method of any of claims 1 to 18; and a sensor arranged to determine the state variable of the controlled system.

21. A non-transitory computer-readable medium carrying computer- readable instructions which, when executed by a processor, cause the processor to carry out the method of any of claims 1 to 18.