ZA201005422B - Method of and system for determining proportional and integral control parameter values for a feedback control system - Google Patents

Description

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BACKGROUND OF THE INVENTION =

THIS invention relates to at least a method of and system for determining proportional and [integral control parameter values for a feedback control system, as well as a proportional and integral control system.

In feedback control systems PI (Proportional + Integral) or PID (Proportional + Integral + . Derivative) control is typically used for at least reducing uncertainty in the feedback control systems. However, in PI control, two parameters of a Pl section of the controller, namely proportional gain k and integral time Ti, are not determined uniquely by a single performance : specification.

Practitioners have known that, in some respects, the proportional and integral components could compensate each other, but there appears to be no direct method to calculate the multitude of the good or best combinations.

The present invention seeks at least to provide a more convenient way to determine a maximum proportional gain and minimum integral time combinations. The present invention also seeks to determine or calculate accurately at least a single pair of Pl parameters for given sensitivity bound, without iterations.

SUMMARY OF THE INVENTION

According to a first aspect of the invention there is provided a method of determining proportional and integral control parameter values for a feedback control system of a plant having an associated plant frequency response, the method comprising: determining a set of maximum potential proportional controller gain values for which proportional plant values do not violate defined sensitivity bounds, wherein the proportional plant values comprises pre-selected frequency response values of the plant frequency response with proportional only control applied thereto by way of a plurality of potential proportional controller gain values; and selecting at least one minimum proportional controller gain value from the determined set of maximum potential proportional controller gain values for use as the maximum proportional controller gain value in the feedback control system.

The method may comprise pre-selecting frequency response values of the plant frequency response. Instead, or in addition, the method may comprise receiving pre-selected frequency response values.

The method may comprise applying a plurality of potential proportional controller gain values to the plant frequency response thereby to obtain a plant frequency response with proportional only control.

The method may comprise determining an integral time value for the feedback control system, which method comprising:

selecting a smaller proportional controller gain value, wherein the smaller proportional controller gain value is less than the determined maximum proportional controller gain value; determining a set of minimum potential integral time values for which a closed-loop system approximates or reaches but does not violate a defined sensitivity bound, : wherein the closed-loop system comprises pre-selected frequency response values of the plant frequency response with proportional control applied thereto by the ) selected smaller proportional controller gain value and integral time values applied thereto by way of a plurality of potential integral time values; and selecting at least one maximum integral time value from the determined set of minimum potential integral time values for use as the minimum integral time value in the feedback control system. .

Instead, or in addition, the method may comprise receiving a selected smaller proportional controller gain value.

The method may comprise pre-selecting frequency response values of the plant frequency response to apply the smaller proportional controller gain value and plurality of potential ) integral time values thereto. The pre-selected frequency response values may be the same as the pre-selected frequency response values for determining the maximum proportional controller gain value as hereinbefore described.

The method may comprise: ] applying the smaller proportional controller gain value to the plant frequency response; and applying the plurality of potential integral time values to the plant frequency response with smaller proportional controller gain value applied thereto. :

The method may comprise determining optimal proportional and integral parameter ranges, which method comprises:

“ selecting a set of proportional controller gain values less than the determined - maximum proportional gain value, for each proportional controller gain value in the selected set, determining an associated integral time value thereby to obtain a plurality of corresponding proportional controller gain value and integral time value pairs; evaluating a sensitivity function for each proportional controller gain value and integral time value pair, wherein the proportional controller gain value and integral time value pair comprise a proportional controller gain value and its associated integral time value; and selecting a plurality of optimal proportional controller gain values and integral time value pairs which minimise the sensitivity function at one or more relevant frequencies.

Selecting a set of proportional controller gain values may comprise selecting a set of proportional controller gain values from between the determined maximum proportional - controller gain value and zero. In an example embodiment, selecting a set of proportional controller gain values may comprise selecting a set of proportional controller gain values from between ten percent of the determined maximum proportional controller gain value and 90 percent of the determined maximum proportional controller gain value. In a preferred example embodiment the method may comprise selecting a set of proportional controller gain values from between 50 percent of the determined maximum proportional controller gain value and 100 percent of the determined maximum proportional controller gain value.

According to a second aspect of the invention there is provided a method of determining an integral time value for a feedback control system of a plant having an associated plant frequency response, the method comprising: } selecting a smaller proportional controller gain value, wherein the smaller proportional controller gain value is less than a maximum proportional controller gain value for the plant;

determining a set of minimum potential integral time values for which a closed-loop system approximates or reaches but does not violate a defined sensitivity bound, : wherein the closed-loop system comprises pre-selected frequency response values of the plant frequency response with proportional control applied thereto by the selected smaller proportional controller gain value and integral time values applied thereto by way of a plurality of potential integral time values; and selecting at least one maximum integral time value from the determined set of minimum potential integral time values for use as the minimum integral time value in the feedback control system.

According to a third aspect of the invention there is provided a system for determining proportional and integral control parameter values for a feedback control system of a plant having an associated plant frequency response, the system comprising: a memory for storing data; a processor operatively connected to the memory, the processor comprising: a gain determining module configured to determine a. set of maximum potential proportional controller gain values for which proportional plant i values do not violate defined sensitivity bounds, wherein the proportional plant values comprises pre-selected frequency response values of the plant frequency response with proportional only control applied thereto by way of a plurality of potential proportional controller gain values; and a gain selection module configured to select at least one minimum proportional controller gain value from the determined set of maximum potential proportional controller gain values for use as the maximum proportional controller gain value in the feedback control system.

The gain determining module may be configured to pre-select the frequency response . values of the plant frequency response.

The gain determining module may be further configured to apply a plurality of potential proportional controller gain values to the plant frequency response thereby to obtain a plant frequency response with proportional only control.

The gain selection module may be configured to select a smaller proportional controller gain value, wherein the smaller proportional controller gain value is less than the determined maximum proportional controller gain value. C

The processor may be further arranged to determine an integral time value for the feedback control system. In this regard, the processor may further comprise: an integral time determining module configured to determine a set of minimum potential integral time values for which a closed-loop system approximates or . reaches but does not violate a defined sensitivity bound, wherein the closed-loop system comprises pre-selected frequency response values of the plant frequency with proportional control applied thereto by the selected smaller proportional controller gain value and integral time values applied thereto by way of a plurality of potential integral time values; and an integral time selection module configured to select at least one maximum integral time value from the determined set of minimum potential integral time values for use as the minimum integral time value in the feedback control system.

The processor may be configured to determine optimal proportional and integral parameter ranges. : The gain selection module may be configured to select a set of proportional controller gain ~ values less than the determined maximum proportional gain value. ~The processor may preferably comprise a ranging module configured to: . determine, for each proportional controller gain value in the selected set, an associated integral time value thereby to obtain a plurality of corresponding proportional controller gain value and integral time value pairs;

~ evaluate a sensitivity function for each proportional controller gain value and integral time value pair; and select a plurality of optimal proportional controller gain value and integral time value : pairs which minimise the sensitivity function at one or more relevant frequencies.

According to a fourth aspect of the invention, there is provided a method of determining optimal proportional and integral parameter ranges for a feedback control system of a plant having an associated plant frequency response, which method comprises: receiving information indicative of a maximum proportional controller gain value of a feedback control system; selecting a set of proportional controller gain values less than the determined maximum proportional gain value; . for each proportional controller gain value in the selected set, determining an associated minimum integral time value thereby to obtain a plurality of corresponding proportional controller gain value and integral time value pairs; evaluating a sensitivity function for each proportional controller gain value and : integral time value pair; and selecting a plurality of optimal proportional controller gain value and integral time value pairs which minimise the sensitivity function at one or more relevant frequencies. ’

According to a fifth aspect of the invention there is provided a proportional and integral control system with proportional and integral control parameter values determined by way of the method or system as hereinbefore described.

According to a sixth aspect of the invention, there is provided an electronic storage means } storing a set of computer executable instructions which when executed by way of a machine, : causes the machine to perform any one of the methods as hereinbefore described.

BRIEF DESCRIPTION OF THE DRAWINGS :

Figure 1 shows a schematic block diagram of a system for determining proportional and integral control parameter value/s for a feedback control system in accordance with an example embodiment;

Figure 2 shows a schematic block diagram of a processor of the system of Figure 1 in accordance with an example embodiment;

Figure 3 shows a Nyquist plot illustrating determination of a proportional controller gain value in accordance with an example embodiment;

Figure 4 shows a Nyquist plot illustrating determination of an integral time value in accordance with an example embodiment;

Figure 5 shows a Bode magnitude plot illustrating sensitivity magnitude over logarithmic frequency for a range of proportional gain and optimal integral time values in accordance with an example embodiment; )

Figure 6 shows a flow diagram of a method of determining a maximum proportional controller gain value for a feedback control system in accordance with an example embodiment;

Figure 7 shows a flow diagram of a method of determining a minimum integral time value for a feedback control system in accordance with an example embodiment;

Figure 8 shows a flow diagram of a method of determining an optimal proportional and integral parameter ranges in accordance with an example embodiment; and

Figure 9 shows a diagrammatic representation of a machine in the example form of a computer system in which a set of instructions for causing the machine to perform any one or more of the methodologies discussed herein, may be executed.

DESCRIPTION OF PREFERRED EMBODIMENTS

In the following description, for purposes of explanation, numerous specific details are set forth in order to provide a thorough understanding of an embodiment of the present disclosure. It will be evident, however, to one skilled in the art that the present disclosure may be practiced without these specific details.

Referring to Figures 1 to 5 of the drawings where a system for determining proportional and integral control parameters for a feedback control system is generally indicated by reference ) numeral 10. For brevity, the proportional and integral control parameters are the proportional controller gain values k and integral time values T; respectively. In this regard, the system 10 is accessible, by a control engineer 12 for example, to determine the proportional controller gain values and integral time values respectively for a feedback control system or controller of a plant (not shown), which plant having a corresponding plant transfer function. oo

It will be understood by those skilled in the art that one general justification for feedback control is the reduction of uncertainty in a control system. This uncertainty reduction is measured by the sensitivity function S. Any uncertainty is reduced when the magnitude of the sensitivity is less than 1 or, differently stated, less than 0dB. Under practical conditions, logarithmic sensitivity integral for all stable feedback systems is exactly 0. That means that sensitivity reduction in some frequency range is balanced by sensitivity increase in some other frequency range. (In that sense, feedback does both — it reduces uncertainty and increases it.). In practice, therefore one has to specify the maximum allowable sensitivity increase M at any frequency. This increase may be somewhere around the range of 1.4 (+3dB) to 2 (+6dB). The same specification defines a very useful margin to instability and is, } therefore, a fundamental feedback control specification.

If L denotes the negative feedback loop transfer function, then the sensitivity function is given as: :

S=—— (1)

The loop transfer function, in turn is a product of the plant, sensor, actuator, signal processing and controller transfer functions. Of these, only the controller transfer function is subject to design or tuning at the controller design or tuning stage of the overall system : design by the control engineer 12 for example. In this regard, the system 10 is used to assist or facilitate design or tuning of the controller.

Often, some parts or aspects of the controller are determined for reasons other than sensitivity reduction. In many practical situations, the so-called derivative action of the PID (Proportional Integral and Derivative) controller is, or can be, determined before the optimisation of the PI section of the controller. Pre-selection of the ‘derivative’ (in practice, of the lead-lag) action is assumed in the present invention.

For simplicity of notation, all given, pre-existing, or pre-selected sections of the loop L are combined into a single transfer function P. Hence, the frequency response of the loop is } given as: 1

Lio) = K 1+ 1) Pla) @

It follows that the abovementioned sensitivity (and stability margin) specification may be written as 1

CY < Vo sie) + 2G) = ? 3)

Where the frequency, measured in radians per time-unit, is denoted by w and j is the imaginary unit. :

It will be appreciated that equation (3) may be written as a condition on the loop frequency response: , 1 1+ (jo) 2—, Vo te Li) 2 5; "

Also, it may be convenient to define the feedback loop with proportional only control as .

Lp (jo) = kP(jo) | (5)

Sometimes, proportional-only-control is sufficient. This would avoid the problem of integrator (or reset) windup and is used in cascaded control systems. In general however, that is not sufficient and the term with integrator needs to be added as will be discussed below: . 1 .

L(jo) = k + = L,(jo) ©

Returning to the discussion of the system 10, the system 10 comprises a memory 14 for storing data and a processor 16. To this end, the memory 14 may be a machine-readable medium and may for example be memory in the processor 16, main memory, and/or hard disk drive, which carries a set of instructions to direct the operation of the processor 16. It is to be understood that the processor 16 may be one or more microprocessors, controllers, or - any other suitable computing device, resource, hardware, software, or embedded logic.

In an example embodiment, the processor 16 comprises a plurality of components or modules which correspond to the functional tasks to be performed by the system 10 or processor 16. In this regard, “module” in the context of the specification will be understood to include an identifiable portion of code, computational or executable instructions, data, or computational object to achieve a particular function, operation, processing, or procedure. It follows that a module need not be implemented in software; a module may be implemented in software, hardware, or a combination of software and hardware. Further, the modules need not necessarily be consolidated into one device but may be spread across a plurality of devices.

The processor 16 is configured to pre-select frequency response values of the plant frequency response P(jw) which would assist in the determination of the control parameters, particularly the proportional controller gain values k. Graphically, turning to Figure 3, these values or portion of the frequency response arefis the thicker solid line portion of P(jw) indicated by reference numeral 18. oo

Differently defined, the pre-selected plant frequency response values refers to a discrete set of plant frequency response values and is necessary in the present discrete calculation of something that is in principle a continuum. The plant frequency response is a continuous function of frequency — hence there are infinitely many values. This discretisation arises from two sources. One is that a manageable discrete set of frequency values (determined by experience generally) is needed and the other is optional and relevant when there is uncertainty in the plant model itself. The uncertainty (such as not precisely known friction, environment temperature, wind-speed, and the like) needs to be represented by a finite set of discrete values (determined by experience generally). The inclusion of uncertainty in this . method leads to the possibility of more than one plant value for each frequency.

It follows that the plant frequency response values are selected by the engineer or designer 12 accordingly. The processor 16 is therefore configured to receive this selection, apart from others, from the designer 12.

The processor 16 advantageously comprises a gain determining module 20 configured to determine a set of maximum potential proportional controller gain values k for which proportional plant values 22 do not violate defined sensitivity bounds, wherein the proportional plant values 22 comprises pre-selected frequency response values 18 of the plant frequency response with proportional only control applied thereto by way of a plurality of potential proportional controller gain values k i.e. L, or more particularly L,.(jw). In an example embodiment, the sensitivity bound of M is indicated in Figure 3 and 4 with a dotted circle 24 around (-1 +0j). The radius of the circle 24 is 1/M, in the present example embodiment 1/M = 0.5.

The k values are conveniently determined or calculated as the radial distance from plant = frequency response 18 to the sensitivity circle 24. There is no iteration or recursion, hence no initial values or ranges. :

In an example embodiment, the system 10 may be configured to accept sensitivity bound selections from the control engineer 12. In other words, the system 10 allows for the control engineer 12 to define desired sensitivity bounds or more particularly maximum sensitivity : bounds.

Differently defined, it will be noted that the module 20 effectively determines or selects proportional controller gain values k so that all relevant sections of P(jw) 18 within the angle between two straight dash-dotted lines 26 are contracted (or expanded) to the right of the sensitivity circle 24. It will be appreciated that the lines 26 define a sensitivity angle in which the sensitivity circle 24 is seen from the origin of the complex plane. The thick portions illustrated are important for the present invention and the thin portions of P(jw) do not affect the present calculation.

It will be appreciated that the initial loop consists only of the plant and possibly other predetermined parts of instrumentation, that is denoted by P. Therefore, the initial Lp=k*P, with k=1. Subsequently, k<1 means contraction and k>1 means expansion. In many cases the proportional gain has to be negative. The negative sign is a pre-selected feature and is accordingly associated with P in this method. This justifies why the proposed invention works with positive k values only.

The processor 16 further comprises a gain selection module 28 configured to select at least one minimum proportional controller gain value k from the determined set of maximum potential proportional controller gain values k for use as the maximum proportional controller gain value kns in the feedback control system. It will be noted that kn... is the single . maximum proportional controller gain value for the proportional only feedback loop L, arising from the selected relevant plant frequency response values. In this regard, the illustrated plant values or loop 22 may be described by equation (5) and is the plant frequency response with proportional only control applied thereto by way of kn... No integral action can be added to the latter. it will be appreciated that the processor 16 may be arranged to apply a plurality of potential proportional controller gain values k to the plant frequency response thereby to obtain a plant frequency response with proportional only control L,.

In this regard, the processor 16 may be configured also to monitor L, for each k so as to determine whether the particular L, satisfies the sensitivity bounds. oo

By way of example, the processor 16 may be configured to implement a method as described by the following, commented, Matlab pseudo-code: % Define the maximum permitted sensitivity MdB in dB, e.g. MdB=6; .

M=10~(MdB/20); % Define a set of relevant frequency values w; % (Experience is needed for conditionally stable designs.) % (Multiple values of w -- for example, to cater for uncertainty -- % are permitted.) w=jtw; :

% Obtain the corresponding plant frequency response P(jw); % A sensitivity related number that is used repetitively: }

Mim=1/M/M-1; % An array of numbers that are used repetitively:

Gamma=1+(imag(P)./real(P)).A2; % Number of elements, nn, in all arrays: | : nn=max(size(P)); % Pre-allocation of memory for array k: k=zeros(1,nn); % Calculation of the array of all frequency dependent gains k: ) for n=1:nn % An array of numbers permitting evaluation of whether % points of P(jw) are within the angle in Figure 1: anglecond=Gamma*Mim+1; if real(P(n))<0&&anglecond(n)>=0 % Within the relevant angle k(n)=(sqrt(anglecond(n))-1)./Gamma(n)./real(P(n)); else % Not within the relevant angle k(n)=inf; end end : : % The maximum possible proportional gain: kmax=min(k); % Check that no available frequency response value of the resulting loop transfer function violates the given sensitivity bound M

As previously mentioned, the processor 16 is further advantageously arranged to determine ) an integral time value T; for the feedback control system. In order to add integral action, the proportional gain Kk... has to be reduced further such that the plant transfer function with proportional only control 22 is substantially as illustrated in Figure 4. In this regard, the gain . selection module 28 may be further configured to select a smaller proportional controller gain value Ksmai, Wherein the smaller proportional controller gain. value Kgnay, is less than the determined maximum proportional controller gain value Kp.

In an example embodiment, the module 28 may be configured to receive a selection of Ksman from the designer 12 according to what s/he is trying to achieve. It must be smaller for any i integral component to be possible. The larger the difference (knax-k) the stronger the integral action for the same sensitivity-margin specification. This is where the flexibility comes in and the proposed invention leaves it open to allow this flexibility to be used for whatever is important in a particular application. The last, ranging, step in the method (discussed below) determines the bound of up to how much smaller makes sense. What the designer 12 eventually uses depends on what else the designer wants to achieve.

It will be noted that the processor 16 is also arranged to select or receive a selection of a set of frequency response values P(jw) 18 in a similar fashion as hereinbefore described which will affect the calculation. However, it will be noted that the values 18 may be the same or may differ from the values 18 selected by the processor for determination of km. as . hereinbefore described.

In any event, the processor 16 further comprises an integral time determining module 30 configured to determine a set of minimum potential integral time values T; for which a closed- loop system L 32 approximates or reaches but does not violate a defined sensitivity bound, wherein the close-loop system L 32 comprises pre-selected frequency response values of : the plant frequency response with proportional control applied thereto by the selected smaller proportional controller gain value ksnay, and integral time values T; applied thereto by way of a plurality of potential integral time values T;, in other words L, with respective T; applied thereto. The 1/(wTi) values are calculated as the perpendicular distance from proportional loop frequency response to the sensitivity circle 24. There is no iteration or recursion, hence no initial values or ranges. :

Effectively, referring to Figure 4, the module 30 extends every point of L, (jw) clockwise at a right angle toward the sensitivity bound circle 24, because: tlio) = (12) 2p 0) = pli) po) ;

The processor 16 also includes an integral time selection module 34 configured to select at least one maximum integral time value Ti, from the determined set of minimum potential integral time values T; for use as the minimum integral time value in the feedback control . system. It will be noted that T;,, may be the optimal integral time value or Tj, for the loop arising from the selected smaller proportional controller gain value kya and the relevant plant frequency response values. :

In an example embodiment, with reference to equation 7, the frequency dependent extension factor ¥ (oT) is determined such that the thick section of Lplio) 22 in Figure 4 is shifted to be outside the sensitivity bound circle 24, except at one (or more) point(s) where they touch. The thin portions of Lp(jo) do not affect this calculation. The result is indicated by the thick section L 32 of the dashed curve in Figure 4.

Similarly as hereinbefore described, the processor 16 may be arranged to apply a plurality of ’ potential integral time values T; to the plant frequency response with proportional only control thereby to obtain a plant frequency response with proportional and integral control L.

The processor 16 is therefore also configured to monitor desired values of L for each T; so as to determine whether those particular values of L satisfy the sensitivity bounds.

Also by way of example, the processor 16 may preferably be configured to implement a method as described by the following, commented, Matlab pseudo-code: % This section normally follows the section for calculating kmax. : % However, the range of the frequencies, in the array w, may differ. | . % In conditionally stable designs, lower w-values have to be added. % Select a value for the proportional gain kp less than kmax; % The corresponding loop with proportional only control:

LP=kp*P; % ... and the array of its magnitude squares:

LP2=abs(LP).A2;

A17- % An array of numbers permitting evaluation of whether % points of LP(jw) can reach the M-circle in Figure 2: ’ solcond=LP2/M/M-LP2.A2-real(LP).*(2*LP2+real(LP)); % Number of elements, nn, in all arrays: : nn=max(size(P)); % Calculation of the array of all frequency dependent % clockwise-perpendicular extension factors wTi = wclnt: for n=1:nn . if imag(LP(n))<=0&&solcond(n)>=0 % Can reach the M-circle welnt(n)=w(n).*(-imag(LP(n))-sqrt(solcond(n)))./LP2(n); else % Cannot reach the M-circle i wclnt(n)=inf; end end % The optimal integral time:

Ti=1/min(wcint); ) % Check that no available frequency response value of the resulting loop transfer function violates the given sensitivity bound M

In a preferred example embodiment, the processor 16 comprises a ranging module 36 configured to determine optimal proportional and integral parameter ranges.

To this end, the gain selection module 28 is configured to select or receive a selection of a set of proportional controller gain values kgm. less than the determined maximum proportional gain value kms. [In particular, the module 28 is configured to select the set of proportional controller gain values k,n. from between ten percent and 90 percent of the determined km... In a preferred example embodiment, the module 28 is configured to . selecting a set of proportional controller gain values ksnas from between 50 percent and 100 percent of kn.x. The smaller values of k are suboptimal, but the minimum optimal gain does depend on the particulars of the system.

The ranging module 36 is therefore configured to determine, for each proportional controller gain value ksmay in the selected set, an associated optimal or minimum integral time value

Tiopt @s hereinbefore described, thereby to obtain a plurality of corresponding proportional controller gain value and integral time value pairs. In this regard, the module 36 may be communicatively coupled to one or more of the modules 22, 28, 30 and 34 so as to obtain the pairs. It will be appreciated that, all determinations made by the processor 16 for example knax Tip, @and the respective pairs may be stored in the memory 14 for access by the processor or control engineer 12.

The ranging module 36 is configured to evaluate a sensitivity function for each proportional controller gain value and integral time value pair, and select a plurality of optimal proportional controller gain value and integral time value pairs (kop, Tiopr) Which minimise the sensitivity function at one or more relevant frequencies.

The present “relevant frequencies” or frequency are selected by the designer 12 according to a specific need ~ they must be within a range of frequency in which the control system is meant to regulate uncertainty, this is what is meant by the term “relevant”. The parameters - are calculated usually at higher frequencies, although the frequency values are almost not used in the calculations. This depends on specific system and what the designer 12 tries to achieve. For example, the frequency values are not used at all for the calculation of k and they are used only in the last moment for evaluating the numerical value of Ti from the multitude of values (1/wTi). The calculation of (1/wTi) does not require or use the knowledge of the frequency w.

In any event, the ranging module 36 is configured to determine or evaluate the sensitivity equation 1. Each of these evaluated sensitivities satisfies equation 3, because k is selected so that inequality 3 is given and the integral time is calculated to reach M in equation 3 exactly for each pair of k and Ti.

Further by way of example, the processor 16 may preferably be configured to implement a method as described by the following, commented, Matlab pseudo-code to determine optimal k and T; ranges: % This section normally follows the section for calculating kmax. | . % However, the range of the frequencies, in the array w, may differ.

% In conditionally stable designs, lower w-values have to be added. % Select the array of proportional gain values, less than kmax; kp=linspace(0.1*kmax,0.9*kmax,9); % Pre-allocation of memory for array Ti:

Ti=0"kp; : % Number of controller parameter pairs, kmd, to be considered: kmd=max(size(kp)); % Number of elements, nn, in most other arrays: nn=max(size(P)); } % Repeat for each selected value of proportional gain in array kp: for k=1:kmd % Loop with proportional only control and its magnitude squares:

LP=kp(kmd+1-k)*P; LP2=abs(LP).*2; % An array of numbers permitting evaluation of whether : % points of LP(jw) can reach the M-circle in Figure 2: solcond=LP2/M/M-LP2.22-real(LP).*(2*LP2+reail(LP)), % Calculation of the array of all frequency dependent % clockwise-perpendicular extension factors wTi = wclnt: for n=1:nn if imag(LP(n))<=0&&solcond(n)>=0 % Can reach the M-circle welnt(n)=w(n).*(-imag(LP(n))-sqrt(solcond(n)))./LP2(n); else % Cannot reach the M-circle wcelnt(n)=inf; . end end % Array of the optimal integral times:

Ti(kmd+1-k)=1/min(wclint);

oi . 20107 054,, % The (sub-) optimal loop and its low-frequency value:

L=LP.*(1+1/Ti(kmd+1-Kk)./jw); Slo(kmd+1-k)=1/(1+L(1)); - end : % The lowest low-frequency sensitivity and % the corresponding number of the parameter pair: [Sbest,n}=min(abs(Slo)); % The optimal integral time:

TiOpt=Ti(n) % ... and the corresponding optimal proportional gain: kOpt=kp(n)

It will be appreciated the whole range of the optimal pairs of proportional gain and integral time is based on repeating the above Ti calculation code for a range of k values, all less than Fmax All values of k between kopt and Kmax are optimal together with the corresponding values of Ti between Tiopt and =. Figure 5 indicates a typical sequence of the optimal and sub-optimal sensitivities with the maximum sensitivity specification at 6 dB.

All displayed sensitivity functions are ‘optimal’ in the sense that the integral time Ti is minimised for the corresponding proportional gain k. However the optimal pair k, Ti) is the one that minimises the sensitivity in the loop bandwidth. There are two clear optima in this sense. One of them, (ko, Tiopr), Minimises the sensitivity function S for frequencies between 0 and about 2 in Figure 5. The other corresponds to proportional only control, (kyax =), and minimises S for frequencies above 2. The corresponding Lp reduces sensitivity over the maximum possible frequency range, but is by far the worst at low frequencies. The thin solid lines between these two extremes, in Figure 5, are compromises between the two extremes. )

The overall selection should be made between the two extreme pairs, (kop, Tiopt) @Nd (Kmaxs =), However, in some situations, a designer or control engineer 12 may prefer parameters that yield the sub-optimal dashed curves in Figure 5.

In an example embodiment, the system 10 may advantageously be a computer implemented system which comprises a user interface or GUI (Graphical User Interface) with which the designer or control engineer can access the system 10 thereby to design or facilitate design of a PI control system.

In other example embodiments, the system 10 may be advantageously arranged to optimise a designed PI control system. :

Example embodiments will now be further described in use with reference to Figures 6 to 8.

The example methods shown in Figures 6 to 8 are described with reference to Figures 1 to 5, although it is to be appreciated that the example methods may be applicable to other ) systems (not illustrated) as well.

Referring now to Figure 6 of the drawings where a flow diagram of a method of determining proportional and integral control parameter values for a feedback control system is generally indicated by reference numeral 40. In a typical example embodiment, a control engineer 12 } desiring to design or optimise the design of a PI control system or more particularly the controller would use the system 10 to obtain proportional controller gain values and integral time values.

In any event, the method 40 comprises selecting, at block 42, the desired frequency response values P(jw) 18 which would assist in the determination of at least the proportional . . gain k. This may advantageously be done by way of the processor 16 of the system 10. In other example embodiments, these values may be selected by the engineer 12, and received by the system 10 accordingly for example via the user interface.

The method 40 then comprises determining or selecting, at block 44 by way of the module 22 as hereinbefore described, a set of maximum potential proportional controller gain values ) k for which proportional plant values L, 22 do not violate defined sensitivity bounds 24.

The method 40 further comprises selecting, at block 46 via the module 28, kn. as hereinbefore described.

Although not illustrated, the method 40 may also comprise applying a plurality of potential proportional controller gain values k to the plant frequency response P(jw) 18 thereby to obtain L, and monitoring L, for each k so as to determine whether the particular L, satisfies the sensitivity bounds.

Referring to Figure 7 of the drawings where a method of determining an integral time value

T; for the feedback control system is generally indicated by reference numeral 50. The method 50 may conveniently form part of or may follow on from the method 40. However, should kn,.x be known a priori, the method 50 may therefore conveniently be a standalone method.

In any event, the method 50 comprises determining or selecting, at block 52, the desired frequency response values P(jw) 18 which would assist in the determination of at least the integral time values 7. These values may be the same values as described above with reference to block 42 of method 40. However, this may not always be the case though as these values may differ from those selected in the method 40.

The method 50 comprises selecting, at block 54 by way of module 28, a smaller proportional controller gain value Ksmar, Wherein Ksmay is less than the determined k..x for example as determined in method 40.

The method 50 comprises determining, at block 56 via module 30 as hereinbefore : described, a set of minimum potential integral time values T; for which the closed-loop system L 32 (Figure 4) approximates or reaches but does not violate a defined sensitivity bound.

The method 50 may then comprise selecting, at block 58 via module 34, T, as hereinbefore described. )

Although not illustrated the method 50 may also comprise applying a plurality of potential integral time values T;to L, thereby to obtain a L as hereinbefore described.

The method 50 may also comprise monitoring desired values of L for each T; so as to determine whether those particular values of L satisfy the sensitivity bounds.

Referring now to Figure 8 of the drawings where a method of determining optimal proportional and integral parameter ranges in accordance with an example embodiment is generally indicated by reference numeral 60. It will be appreciated that the method 60 may follow on from one or both of methods 40 and 50. However, this method 60 may also be a standalone method if k and T; values are known a priori for example.

In any event, the method 60 comprises selecting a set of plant frequency response values, at block 62 by way of the processor 16 or receiving the selection from the designer 12 as hereinbefore described.

The method 60 then comprises determining, selecting, or receiving a selection, by way of module 28, a set of proportional controller gain values Ksmay less than Kya.

The method 60 comprises determining, at block 64 by way of the ranging module 36 a maximum potential proportional controller gain value from the set. The method then comprises using the determined maximum potential proportional controller gain value to determine an associated T..

The method 60 then comprises evaluating, at block 66 by way of the ranging module 36, a low frequency sensitivity function for each proportional controller gain value kn and integral time value T; pair. _

The method 60 comprises selecting, at block 68 by way of the ranging module 36, those plurality or range/s of optimal proportional controller gain value and integral time value pairs (kopt, Tiopr) Which minimise the low frequency sensitivity function at one or more relevant frequencies. :

It follows that should a proportional controller gain value and integral time value pair not minimise the sensitivity function, the method 60 comprises selecting, at block 70, another } proportional controller gain value and determine another corresponding T; for evaluation.

The method 60 is conveniently repeated for all ks, in the set and corresponding T;

It will be noted that the methods illustrated in Figures 6 to 8 may preferably be computer implemented method for assisting the control engineer to design or optimise a PI control : system, particularly the PI controller. In this regard, the invention extends to an electronic storage medium or machine-readable means comprises a set of computer or machine readable and executable instructions to perform any of the methodologies described herein.

Figure 9 shows a diagrammatic representation of machine in the example form of a computer system 100 within which a set of instructions, for causing the machine to perform any one or more of the methodologies discussed herein, may be executed. In alternative embodiments, the machine operates as a standalone device or may be connected (e.g., networked) to other machines. In a networked deployment, the machine may operate in the capacity of a server or a client machine in server-client network environment, or as a peer ; machine in a peer-to-peer (or distributed) network environment. The machine may be a personal computer (PC), a tablet PC, a set-top box (STB), a Personal Digital Assistant (PDA), a cellular telephone, a web appliance, a network router, switch or bridge, or any machine capable of executing a set of instructions (sequential or otherwise) that specify actions to be taken by that machine. Further, while only a single machine is illustrated, the term “machine” shall also be taken to include any collection of machines that individually or jointly execute a set (or multiple sets) of instructions to perform any one or more of the methodologies discussed herein.

The example computer system 100 includes a processor 102 (e.g., a central processing unit (CPU), a graphics processing unit (GPU) or both), a main memory 104 and a static memory X 106, which communicate with each other via a bus 108. The computer system 100 may further include a video display unit 110 (e.g., a liquid crystal display (LCD) or a cathode ray tube (CRT)). The computer system 100 also includes an alphanumeric input device 112 (e.g., a keyboard), a user interface (Ul) navigation device 114 (e.g., a mouse), a disk drive unit 116, a signal generation device 118 (e.g., a speaker) and a network interface device 120. :

The disk drive unit 116 includes a machine-readable medium 122 on which is stored one or more sets of instructions and data structures (e.g., software 124) embodying or utilized by any one or more of the methodologies or functions described herein. The software 124 may also reside, completely or at least partially, within the main memory 104 and/or within the processor 102 during execution thereof by the computer system 100, the main memory 104 and the processor 102 also constituting machine-readable media.

The software 124 may further be transmitted or received over a network 126 via the network interface device 120 utilizing any one of a number of well-known transfer protocols (e.g.,

HTTP). : }

While the machine-readable medium 122 is shown in an example embodiment to be a single medium, the term "machine-readable medium" should be taken to include a single medium or multiple media (e.g., a centralized or distributed database, and/or associated caches and servers) that store the one or more sets of instructions. The term "machine-readable : - medium" shall also be taken to include any medium that is capable of storing, encoding or carrying a set of instructions for execution by the machine and that cause the machine to perform any one or more of the methodologies of the present invention, or that is capable of storing, encoding or carrying data structures utilized by or associated with such a set of instructions. The term "machine-readable medium" shall accordingly be taken to include, but not be limited to, solid-state memories, optical and magnetic media, and carrier wave signals.

The present invention provides a convenient way for parameterisation of a Pl controller uniquely by a single performance specification, for example the maximum permitted sensitivity. The present invention calculates directly the maximum proportional gain and } minimum integral time combinations that satisfy the maximum permitted sensitivity specification. The present invention only uses plant frequency response numerical values without regard to how they were obtained and does the proportional gain and integral time calculations directly, substantially instantaneously. The invention further gives the range of these pairs that optimise the sensitivity in the loop bandwidth. A specific pair of parameters may then be either selected by the application engineer, or determined by an ASOC . (application specific optimisation criterion).

Claims (28)

cL "26- Claims Co

1. A method of determining proportional and integral control parameter values for a : feedback control system of a plant having an associated plant frequency response, the method comprising: determining a set of maximum potential proportional controller gain values for ’ which proportional plant values do not violate defined sensitivity bounds, wherein the proportional plant values comprises pre-selected frequency response values of the plant frequency response with proportional only control applied thereto by way of a plurality of potential proportional controller gain values; and selecting at least one minimum proportional controller gain value from the determined set of maximum potential proportional controller gain values for use as the maximum proportional controller gain value in the feedback control system.

2 A method as claimed in claim 1, comprising pre-selecting frequency response values of the plant frequency response.

3. A method as claimed in claim 1, comprising receiving pre-selected frequency response values of the plant frequency response.

4, A method as claimed in claimed in any one of the preceding claims, the method comprising applying a plurality of potential proportional controller gain values to the plant frequency response thereby to obtain a plant frequency response with proportional only control.

5. A method as claimed in any one of the preceding claims, the method comprising ’ determining an integral time value for the feedback control system by: : selecting a smaller proportional controller gain value, wherein the smaller proportional controller gain value is less than the determined maximum proportional controller gain value; Co determining a set of minimum potential integral time values for which a closed-loop system approximates or reaches but does not violate a defined sensitivity bound, wherein the closed-loop system comprises pre-selected : frequency response values of the plant frequency response with proportional control applied thereto by the selected smaller proportional controller gain value and integral time values applied thereto by way of a plurality of potential : integral time values; and selecting at least one maximum integral time value from the determined set of . minimum potential integral time values for use as the minimum integral time value in the feedback control system.

6. A method as claimed in claim §, the method comprising receiving the selected smaller proportional controller gain value.

7. A method as claimed in either claim 5 or 6, the method comprising pre-selecting frequency response values of the plant frequency response to apply the smaller proportional controller gain value and plurality of potential integral time values thereto.

8. A method as claimed in claim 7, the method comprising: applying the smaller proportional controller gain value to the plant frequency response; and applying the plurality of potential integral time values to the plant frequency . response with smaller proportional controller gain value applied thereto.

9. A method as claimed in any one of the preceding claims, the method comprising determining optimal proportional and integral parameter ranges by: selecting a set of proportional controller gain values less than the determined maximum proportional gain value;

for each proportional controller gain value in the selected set of proportional . controller gain values, determining an associated integral time value thereby to obtain a plurality of corresponding proportional controller gain value and integral time value pairs; oo evaluating a sensitivity function for each proportional controller gain value and integral time value pair, wherein the proportional controller gain value and ; integral time value pair comprise a proportional controller gain value and its associated integral time value ; and selecting a plurality of optimal proportional controller gain values and integral time value pairs which minimise the sensitivity function at one or more relevant frequencies.

10. A method as claimed in claim 9, wherein selecting the set of proportional controller gain values comprises selecting a set of proportional controller gain values from between the determined maximum proportional controller gain value and zero.

“11. A method as claimed in either claim 9 or 10, wherein selecting the set of proportional controller gain values comprises selecting a set of proportional controller gain values from between ten percent of the determined maximum proportional controller gain value and 90 percent of the determined maximum proportional controller gain value.

12. A method as claimed in claim 11, comprising selecting a set of proportional controller . gain values from between 50 percent of the determined maximum proportional controller gain value and 100 percent of the determined maximum proportional controller gain value.

13. A method of determining an integral time value for a feedback control system of a plant having an associated plant frequency response, the method comprising: : selecting a smaller proportional controller gain value, wherein the smaller proportional controller gain value is less than a maximum proportional controller gain value for the plant; a a determining a set of minimum potential integral time values for which a closed-loop system approximates or reaches but does not violate a defined sensitivity bound, wherein the closed-loop system comprises pre-selected frequency response values of the plant frequency response with proportional control applied thereto by the selected smaller proportional controller gain value and integral time values applied thereto by way of a plurality of potential integral time values; and selecting at least one maximum integral time value from the determined set of minimum potential integral time values for use as the minimum integral time value in the feedback control system.

14. A system for determining proportional and integral control parameter values for a feedback control system of a plant having an associated plant frequency response, the system comprising: a memory for storing data; . a processor operatively connected to the memory, the processor comprising: a gain determining module configured to determine a set of maximum potential proportional controller gain values for which proportional plant values do not violate defined sensitivity bounds, wherein the ) proportional plant values comprises pre-selected frequency response values of the plant frequency response with proportional only control applied thereto by way of a plurality of potential proportional controller gain values; and a gain selection module configured to select at least one minimum proportional controller gain value from the determined set of maximum potential proportional controller gain values for use as the maximum proportional controller gain value in the feedback control system.

15. A system as claimed in claim 14, wherein the gain determining module is configured . to pre-select the frequency response values of the plant frequency response.

a £2010/05499

16. A system as claimed in either claim 14 or 15, wherein the gain determining module is configured to apply a plurality of potential proportional controller gain values to the plant frequency response thereby to obtain a plant frequency response with proportional only control.

17. A system as claimed in any one of claims 14 to 16, wherein the gain selection module is configured to select a smaller proportional controller gain value, wherein . the smaller proportional controller gain value is less than the determined maximum proportional controller gain value.

18. A system as claimed in any one of claims 14 to 17, wherein the processor is further arranged to determine an integral time value for the feedback control system.

19. A system as claimed in any one of claims 14 to 18, wherein the processor further comprises: an integral time determining module configured to determine a set of minimum potential integral time values for which a closed-loop system approximates or reaches but does not violate a defined sensitivity bound, : wherein the closed-loop system comprises pre-selected frequency response values of the plant frequency response with proportional control applied thereto by the selected smaller proportional controller gain value and integral time values applied thereto by way of a plurality of potential integral time values; and an integral time selection module configured to select at least one maximum integral time value from the determined set of minimum potential integral time values for use as the minimum integral time value in the feedback control system.

20. A system as claimed in any one of claims 14 to 18, wherein the processor is configured to determine optimal proportional and integral parameter ranges.

21. A system as claimed in any one of claims 14 to 20, wherein the gain selection module is configured to select a set of proportional controller gain values less than : the determined maximum proportional gain value.

22. A system as claimed in claim 21, wherein the processor comprises a ranging module configured to: : determine, for each proportional controller gain value in the selected set, an associated integral time value thereby to obtain a plurality of corresponding proportional controller gain value and integral time value pairs; evaluate a sensitivity function for each proportional controller gain value and . integral time value pair, and : select a plurality of optimal proportional controller gain value and integral time value pairs which minimise the sensitivity function at one or more relevant frequencies. }

23. A method of determining optimal proportional and integral parameter ranges for a feedback control system of a plant having an associated plant frequency response, which method comprises: receiving information indicative of a maximum proportional controller gain : value of a feedback control system; selecting a set of proportional controller gain values less than the determined maximum proportional gain value; for each proportional controller gain value in the selected set, determining an associated minimum integral time value thereby to obtain a plurality of corresponding proportional controller gain value and integral time value pairs; evaluating a sensitivity function for each proportional controller gain value and integral time value pair; and }

selecting a plurality of optimal proportional controller gain value and integral time value pairs which minimise the sensitivity function at one or more relevant frequencies.

24. A proportional and integral control system with proportional and integral control parameter values determined by way of the method as claimed in any one of claims 1 to 12, 13, or 23 or system claimed in any one of claims 14 to 22.

25. An electronic storage means storing a set of computer executable instructions which ~ when executed by way of a machine, causes the machine to perform any one of the : methods as claimed in any one of claims 1 to 12, 12 or 23.

26. A method substantially as described herein with reference to the accompanying drawings.

27. A system substantially as described herein with reference to the accompanying : drawings.

28. A storage means substantially as described herein with reference to the accompanying drawings. DATED THIS 29™ DAY OF JULY 2010 SPOOR & FISHER APPLICANT'S PATENT ATTORNEYS