WO2006041415A1

WO2006041415A1 - A method and system for measuring a process frequency response

Info

Publication number: WO2006041415A1
Application number: PCT/SG2005/000350
Authority: WO
Inventors: Kok Kiong Tan; Raihana Ferdous
Original assignee: National University Of Singapore
Priority date: 2004-10-11
Filing date: 2005-10-11
Publication date: 2006-04-20

Abstract

A method and system for measuring a frequency response for tuning of a process controller. The method comprises providing a process functional block having a transfer function representative of a process; providing a feeback signal from an output of the process functional block and forming a feedback loop around the process functional block; inducing self-oscillations based on a reference input signal and the feedback signal utilising a relay element; boosting a fundamental frequency component of the self-oscillation relative to other harmonics in the self-oscillations; and measuring the frequency response at the output of the process functional block.

Description

A METHOD AND SYSTEM FOR MEASURING A PROCESS FREQUENCY

RESPONSE

FIELD OF INVENTION

The present invention relates to a method and system for measuring a process frequency response.

BACKGROUND

Industrial processes typically utilise numerous control elements or controllers, such as electrical, magnetic, or electro magnetic controllers, which may be implemented as programmable circuits. A typical factory floor application may implement hundreds of such controllers. One common controller used is a proportional-integral-derivative (PID) controller.

In process control system design, process model estimation is a fundamental and important component as the estimated model in a non-parametric or parametric form provides key input parameters for the tuning of the controllers. The tuning of the controllers to their best performance directly affects the quality, quantity and cost of the product. Traditional methods of process model estimation are, in general, a fairly time-consuming procedure, involving persistently injecting excitation inputs to the process control system and applying various time-consuming techniques. Fortunately, knowledge of an extensive full-fledged dynamical model is often not necessary in many of the controllers used in the process industry. The estimation of a critical point in a frequency response at the output of a process to be controlled (for example critical frequency and gain) is often sufficient.

The use of a conventional relay feedback technique disclosed in US Patent Number 4,549,123 for estimation of the critical point has been widely adopted in the process control industry. In particular, the relay feedback technique has been used widely in the process control industries for PID controller tuning. Due to the success of this technique, there has been much research work to extend the application domain. However, there are potential problems associated with the relay feedback-based estimation techniques in relation to the accuracy of the estimation. These problems arise as a result of the approximations used in the development of the procedures for estimating the critical point. In particular, the basis of most existing relay-based procedures of critical point estimation is the describing function method which is based on a fundamental assumption that the higher harmonics induced in the relay feedback loop are negligible compared to the fundamental frequency. This assumption may not be true in certain cases, and the existing relay-based procedures could result in estimates of the critical point that are significantly different from their real values. Such problematic circumstances arise particularly in under damped processes and processes with significant time-delay, and poorly tuned control loops would result if the critical point estimates were used for controller tuning.

Moreover, the conventional relay feedback technique becomes inapplicable for unstable processes where the process time-delay is long. The control performance achievable for an unstable process is particularly sensitive to the accuracy of the process model. In simple cases of unstable processes with short time-delay, a stable limit cycle oscillation may exist, but if the estimate of the critical point is inaccurate and it is used as the basis for the tuning of a controller, the control performance may be very unsatisfactory or even unstable.

Other known constraints of the relay feedback technique include inapplicability to certain classes of processes, a long time duration to settle to stationary oscillations in some cases, and inapplicability to extract other points of the process frequency response.

An adaptive method has been proposed to achieve near-zero error in the critical point estimation. However, this improved accuracy is achieved at the expense of a more complicated implementation procedure over the conventional relay feedback technique. The additional implementation cost is an obstacle to the acceptance of this improved method, since one key reason for the success of the relay feedback technique in industrial applications is the simple and direct approach the relay feedback technique adopts. A need therefore exist for a critical point estimation technique that addresses at least one of the above mentioned problems.

SUMMARY

In accordance with a first aspect of the present invention there is provided a method of measuring a process frequency response, the method comprising providing a process functional block having a transfer function representative of a process; providing a feeback signal from an output of the process functional block and forming a feedback loop around the process functional block; inducing self- oscillation based on a reference input signal and the feedback signal utilising a relay element; boosting a fundamental frequency component of the self-oscillation relative to other harmonics in the self-oscillation; and measuring the frequency response at the output of the process functional block.

The boosting of the fundamental frequency component may be provided by providing a proportional gain to the fundamental frequency component.

The method may comprise utilising a preload relay functional block connected before the process functional block, the preload relay functional block comprising a parallel connection of a relay and a proportional gain element.

The method may further comprise determining one or more parameters from the frequency response at the output of the process functional block once the self- oscillation has reached a steady state.

The method may further comprise utilising the parameters for directly tuning a process controller for controlling the process.

The method may further comprise utilising the parameters for deriving a model to indirectly tune a process controller for controlling the process.

In accordance with a second aspect of the present invention there is provided a system for measuring a process frequency response, the system comprising a process functional block having a transfer function representative of a process; a feedback loop around the process functional block providing a feeback signal from an output of the process functional block; a relay element for inducing a self-oscillation of the system based on a reference input signal and the feedback signal; a gain element for boosting a fundamental frequency component of the self- oscillation relative to other harmonics in the self-oscillation; and an analyser element for measuring the frequency response at the output of the process functional block.

The gain element for the boosting of the fundamental frequency component may provide a proportional gain to the fundamental frequency component of the self-oscillation.

The gain element for boosting of the fundamental frequency component and the relay element may be provided as a preload relay functional block connected before the process functional block, the preload relay functional block comprising a parallel connection of a relay and a proportional gain element.

The analyser elment may be arranged for determining one or more parameters of the frequency response at the output of the process functional block once the self-oscillation has reached a steady state.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the invention will be better understood and readily apparent to one of ordinary skill in the art from the following written description, by way of example only and in conjunction with the drawings, in which:

Figure 1 is a block diagram of a configuration of a preload relay feedback system according to an example embodiment.

Figure 2 shows plots of input and output limit cycle oscillation of a transfer function of the preload relay feedback system according to an example embodiment.

Figure 3 shows plots of output limit cycle oscillation of the preload relay feedback system according to an example embodiment and a conventional relay feedback technique. Figure 4 shows a comparison of critical point estimation in of the preload relay feedback system according to an example embodiment and a conventional relay feedback technique.

Figure 5 shows a comparison of critical point estimation in of the preload relay feedback system according to an example embodiment and a conventional relay feedback technique.

Figure 6 shows a comparison table of critical point estimation in the preload relay feedback system according to an example embodiment and a conventional relay feedback technique.

Figure 7 shows a comparison table of critical point estimation in the preload relay feedback system according to an example embodiment and a conventional relay feedback technique.

Figure 8 shows a comparison table of critical point estimation in the preload relay feedback system according to an example embodiment and a conventional relay feedback technique.

Figure 9 shows a comparison table of critical point estimation in the preload relay feedback system according to an example embodiment and a conventional relay feedback technique.

Figure 10 shows a comparison table of critical point estimation in the preload relay feedback system according to an example embodiment and a conventional relay feedback technique.

Figure 11 shows a flowchart illustrating a method of measuring a frequency response for tuning of a process controller, according to an example embodiment.

DETAILED DESCRIPTION Figure 1 shows an example embodiment of the present invention, there is provided a schematic circuit diagram of a preload relay feedback system 100 for critical point estimation of a controller.

The preload feedback system 100 comprises a preload relay functional block 102, process functional block 104 and a feedback loop 128. The preload relay functional block 102 is utilised such that the fundamental frequency in the forced oscillations induced are boosted to increase the relative amplitude of the fundamental frequency over other harmonics. The process functional block 104 represents the process to be controlled. The process functional block 104 is represented by a transfer function, G_p, which describes the characteristics of the process.

During operation, an input signal 112 is passed into the preload relay feedback system 100. At the summing point 110 a feedback signal 108 from the feedback loop 128 is subtracted from the input reference signal 112, to produce a relay input signal 126. The feedback signal 108 is comprised of the output signal 106. The relay input signal 126 is passed through the preload relay functional block 102, producing a process input signal 124. Finally, the process input signal 124 is passed into the process functional block 104 producing the output signal 106. An analyser 103 is connected at the output of the process functional block 104 for measuring the output signal 106. The input reference signal 112 is a reference signal which represents the desired value of the process variable the control system is supposed to track. For example, if the variable is a temperature control system, the reference signal will represent the desired temperature etc.

The preload relay functional block 102 comprises of a parallel connection of a relay functional block 116 with a proportional gain functional block 114 of gain value, K, for increasing the relative amplitude of the fundamental frequency of the relay input signal 126 over other harmonics. In the example embodiment, the relay functional block 116 functions as an ideal relay. It is appreciated that the relay functional block 116 may function as a relay with hysteresis, saturation relay or the like. The proportional gain functional block 114 provides the gain value K, to the signal processed by it. A periodic signal is produced by the relay functional block 116. The relay functional block 116 switches in response to the relay input (or error) signal 126, a periodic signal at the phase-crossover frequency is produced. The proportional gain functional block 114 increases the fundamental harmonics to better satisfy the describing function assumption in the example embodiment.

During operation, the relay input signal 126 is passed through the relay functional block 116 and the proportional gain functional block 114 to produce two separate signals, namely relay output signal, u_r, 120 and periodic signal, U_k, 122 respectively. The two signals u_r, u_k are summed at a summing point 118 to produce the process input signal, u, 124, such that u = u_r + u_k.

For example, if the relay input signal 126 is e(t) = asinωt, after processing by the preload relay functional block 102, the amplitude (denoted by U₁) of the fundamental frequency of the process input signal, u, 124 will be increased by a value of Ka. Hence, if initially the input value of U₁ is 4μ/π, where μ is the amplitude of the ideal relay waveform of the relay functional block 116, the output value of U₁ will be 4μ/π + Ka. The amplitudes of the higher harmonics inherent in the processed input signal, u, 124 remain essentially unchanged. In this example, The describing function, N(a), of the preload relay functional block 102 is given by:

N(a)=[(4μ/πa) + K]

With reference to a Nyquist plot of the negative inverse of N(a), while the fundamental frequency has been increased, the Nyquist plot of the negative inverse of N(a) continues to lie on the negative real axis of the Nyquist plot albeit with a termination point at -1/K on the negative real axis. If an intersection occurs between the locus of -1/N(a) and the Nyquist curve of the process transfer function, G_p, and oscillation of the signals of the system is sustained, the critical frequency, ω_c, and the amplitude of the oscillation, K₀, can still be estimated as:

ω_c = CD₀SC . and Ko=(4μ/πa)+K,

where ω_osc is the oscillation frequency of the relay input signal 126.

For an intersection to occur under the describing function analysis above, it is necessary that, K<Kc

In the example embodiment, the gain K is fixed at 20%-30% of μ, that is

K = αμ,

where α is a value in the range of 0.2 to 0.3.

Figure 2 illustrates the results of the example embodiment of the present invention when α is in the range of 0.2 to 0.3. The results show the limit cycles at different values of α. Graph 200 contains a curve 204 representing the process input signal 124 when α = 0, a curve 206 representing the process input signal 124 when α = 0.2 and a curve 208 representing the process input signal 124 when α = 0.3. Graph 210 contains a curve 212 representing the output signal 106 when α = 0, a curve 214 representing the output signal 106 when α = 0.2 and a curve 216 representing the output signal 106 when α = 0.3.

From the graphs 200, 210 it can be seen that increasing α also increases the overall amplitude of the limit cycle oscillation. If it is desired that the amplitude be maintained at the same level, this can be achieved by increasing μ, since it is the relative amplitude of K to ω_c that is of interest in the example embodiment. From the graphs 200, 210, it can be seen that with the addition of the gain in the example embodiment, the periodic output signal becomes more sinusoidal (compare curves 212, 214, and 216), indicating that the resultant signal is now less distorted by other harmonics. A larger α means a higher gain, and therefore better accuracy, in the example embodiment, with a limit to α due to a constraint on the amplitude of oscillation allowable.

When the preload relay method of the example embodiment is used to tune a PID controller, information from the steady state oscillations are extracted using known techniques, and used for the tuning. The tuned PID controller is then connected in the usual closed-loop feedback way, i.e., the process output is fedback to the controller input and the controller output goes to the process input. The tuning duration is directly dependent on how fast the oscillations settle to the steady state. A shorter tuning duration is clearly desirable. The preload relay method of the example embodiment effectively provides a higher feedback gain for the oscillating frequency, which can be adjustable by a user. With a higher gain threshold at the oscillating frequency, the limit cycle can settle into the stationary state at a faster rate, thus enabling a faster tuning time when the setup is used for control tuning purposes.

Figure 3 shows two plots 306, 308 of output signal versus time (seconds). The process, function of the process is G_p = 4e^"001s/(s²+s+4). The first plot 306 shows the output signal versus Time (seconds) plot for applying the Preload Relay method of the example embodiment to tune the process. The second plot 308 shows the output signal versus Time (seconds) plot for applying the Conventional Relay method to tune the process. With the Preload Relay method of the example embodiment, the time for the system to settle to steady state oscillations 302 (based on less than 2% deviation in consecutive oscillation amplitudes) is 3.9s whereas the time for the system to settle to steady state oscillations 304 for the conventional relay method is 6.5s.

Figures 4 to 7 show comparative results of simulations of critical point estimation of known real processes, between using the preload relay method of the example embodiments and using the conventional relay feedback technique. The percentage errors (PE) (%) of the parameters of the critical point estimations, that is critical frequency, ω_C) of the oscillation of the processes and amplitude of the oscillation, K₀, at the critical frequency are calculated for comparison between the tests conducted. The value of α is fixed at α = 0.3 for the preload relay method of the example embodiment utilised for obtaining the data in Figures 4 to 7.

L in Figures 4 to 7 is a variable value representing the time delay of the respective process functions used. Setting L as different values will produce variants of different time delays of the process functions.

In Figure 4, for the Conventional Relay test, the PE values are listed in column 428 for the estimated K₀ values in column 416, and the PE values are listed in column 430 for the ω₀ values in column 418. For the Preload Relay test, the PE values are listed in column 432 for the estimated K₀ values in column 420 and the PE values are listed in column 434 for the ω_c values in column 422. The input process for the data shown in Figure 4 is given at numeral 402.

In columns 424 and 426 in Figure 4, the improvements in the PE values for K₀ and ω_c respectively for the Preload Relay test of the example embodiment are summarized, compared to the Conventional Relay method.

In Figure 5, for the Conventional Relay test, the PE values are listed in column 528 for the estimated K₀ values in column 516, and the PE values are listed in column 530 for the ω_c values in column 518. For the Preload Relay test, the PE values are listed in column 532 for the estimated K₀ values in column 520 and the PE values are listed in column 534 for the ω_c values in column 522. The input process for the data shown in Figure 5 is given at numeral 502.

In columns 524 and 526 in Figure 5, the improvements in the PE values for K₀ and ω_c respectively for the Preload Relay test of the example embodiment are summarized, compared to the Conventional Relay method.

In Figure 6, for the Conventional Relay test, the PE values are listed in column 628 for the estimated K₀ values in column 616, and the PE values are listed in column 630 for the ω₀ values in column 618. For the Preload Relay test, the PE values are listed in column 632 for the estimated K₀ values in column 620 and the PE values are listed in column 634 for the ω_c values in column 622. The input process for the data shown in Figure 6 is given at numeral 602.

In columns 624 and 626 in Figure 6, the improvements in the PE values for K₀ and ω₀ respectively for the Preload Relay test of the example embodiment are summarized, compared to the Conventional Relay method.

In Figure 7, for the Conventional Relay test, the PE values are listed in column 728 for the estimated K₀ values in column 716, and the PE values are listed in column 730 for the ω_c values in column 718. For the Preload Relay test, the PE values are listed in column 732 for the estimated K₀ values in column 720 and the PE values are listed in column 734 for the co_c values in column 722. The input process for the data shown in Figure 7 is given at numeral 702. In columns 724 and 726 in Figure 7, the improvements in the PE values for K₀ and ω₀ respectively for the Preload Relay test of the example embodiment are summarized, compared to the Conventional Relay method.

From Figures 4 to 7, it is apparent that the Preload Relay method of the example embodiment provides improvement in the accuracy of the estimated values of K₀ and ω_c, thus improving the critical point estimate, over the Conventional Relay method.

As the critical point estimate is improved, the example embodiment can lead to improved control performance when the critical point is used as the basis for direct tuning of a controller for a process, or for deriving a model to indirectly tune the controller. Utilising the example embodiment, better performance can be achieved with improved critical point estimation in terms of how close the user specifications can be met.

Figure 8 shows a comparison of results obtained for controller tuned utilising the Conventional Relay method and the Preload Relay method in an example

embodiment for a process function ₇ ?e^~5s . The desired values of a gain margin

and a phase margin of the process tested are 3 and 1.05 respectively. As can be seen in column 802, the Preload Relay method provides a PE improvement of 5.76% over the Conventional Relay in terms of gain margin, and a PE improvement of 4.66% over the Conventional Relay method in terms of phase margin. Also, there is a PE improvement with values of 7 and 0.04 in the combined gain and phase margin specifications. For the data presented in Figure 8, PID controllers for the process were tuned using the method described in astrom, KJ. , and T.Hagglund, "Automatic Tuning Of Simple Regulators With Specification On Face And Amplitude Margins Phases And Amplitude Margins", Automatica, Volume 20(5), Pages 645- 651, 1984 and Ho W.K., Hang, CC, Cro, LS., "Tuning of PDI controllers based on gain and face margin specifications", Automatica, Volume 31, Pages 497-502, 1995. In the case of unstable processes, for instance, when the process function is

7 ^e^~sL , the preload relay method in the example embodiment also yields

(1Os-I) improved estimation accuracy as evident in the results shown in Figure 9.

It is noted that no stable limit cycle oscillation result from the conventional relay method for long time delay cases when L = 8 and 9 respectively. With the example embodiment a stable limit cycle oscillation can still be obtained and a critical point can still be estimated.

For a process with long delay, there can be several intersection points between its Nyquist plot and the negative real axis of the complex plane. The critical point is usually defined as the first intersection point as the frequency increases. It is observed that the conventional relay method yields the critical point, but not the outermost point although the outermost point can be a more crucial point to consider for controller design. Figure 10 shows the results of the critical point and outermost

point of a process with process function, , tuned utilising the

Conventional Relay method and the Preload Relay method of the example embodiment. By appropriately adjusting the gain, in the example embodiment by using α=0.6, the outermost point can be obtained. This is not achievable utilising the Conventional Relay method. It is noted that the outermost intersection point is associated with a higher critical frequency K₀. This is due to the resonance in the frequency response of the process.

Figure 11 shows a flow chart 1100 illustrating a method of measuring a frequency response for tuning of a process controller, according to an example embodiment. At step 1102, a process functional block is provided having a transfer function representative of a process. At step 1104, a feedback signal is provided from an output of the process functional block and a feedback loop is formed around the process functional block. At step 1106, self-oscillation is induced based on a reference input signal and the feedback signal utilising a relay element. At step 1108, a fundamental frequency component of the self-oscillation is boosted relative to other harmonics in the self-oscillation. At step 1110, the frequency response at the output of the process functional block is measured. Example embodiments of the present invention may have the following features and advantages.

Apart from improved critical point estimation, example embodiments of the present invention may be applied to other applications, including robustness assessment, friction and valve nonlinearities modelling and compensation, etc, which involve self- oscillation and deriving parameters from the steady state oscillations in the frequency response at the output of the functional block under investigation. The arrangement of example embodiments can also increase the probability of sustaining an oscillation in some classes of processes, for example unstable processes.

If employed as an add-on feature to current process control systems that are already widely used in the industries, example embodiments of the present invention can yield significantly improved estimates of the critical point without incurring significant modification expenses. For example, in some current systems, only an additional gain associated with the preload relay functional block (102 in Figure 1) may need to be added.

Example embodiments of the present invention can serve to achieve consistent and improved estimation accuracy without having to add complexities to the process control systems that are already widely practised in the industries. Improved estimation accuracy can lead to improved control and assessment performance.

Other benefits of the example embodiments include an applicability to certain classes of processes when the current techniques employed by process control systems fail, shortened transient time to reach steady state oscillations for the process control system and providing versatility in estimating other intersection points of the frequency response of the process control system.

Example embodiments of the present invention may be employed in industrial applications, including, but not limited to, automatic controllers, e.g. PID controller or the like, in the petrol, petrol chemical, food and pharmaceutical, semiconductor, and general automation industries. Many modifications and other embodiments can be made to the system and its method by those skilled in the art having the understanding of the above described disclosure together with the drawings. Therefore, it is to be understood that the device and its utility is not to be limited to the above description contained herein only, and that possible modifications are to be included in the claims of the disclosure.

Claims

1. A method of measuring a process frequency response, the method comprising: providing a process functional block having a transfer function representative of a process; providing a feeback signal from an output of the process functional block and forming a feedback loop around the process functional block; inducing self-oscillation based on a reference input signal and the feedback signal utilising a relay element; boosting a fundamental frequency component of the self-oscillation relative to other harmonics in the self-oscillation; and measuring the frequency response at the output of the process functional block.

2. The method as claimed in claim 1, wherein the boosting of the fundamental frequency component is provided by providing a proportional gain to the fundamental frequency component.

3. The method as claimed in claim 2, wherein the method comprises utilising a preload relay functional block connected before the process functional block, the preload relay functional block comprising a parallel connection of a relay and a proportional gain element.

4. The method as claimed in any one of claims 1 to 3, further comprising determining one or more parameters from the frequency response at the output of the process functional block once the self-oscillation has reached a steady state.

5. The method as claimed claim 4, further comprising utilising the parameters for directly tuning a process controller for controlling the process.

6. The method as claimed claim 4, further comprising utilising the parameters for deriving a model to indirectly tune a process controller for controlling the process.

7. A system for measuring a process frequency response, the system comprising: a process functional block having a transfer function representative of a process; a feedback loop around the process functional block providing a feeback signal from an output of the process functional block; a relay element for inducing a self-oscillation of the system based on a reference input signal and the feedback signal; a gain element for boosting a fundamental frequency component of the self- oscillation relative to other harmonics in the self-oscillation; and an analyser element for measuring the frequency response at the output of the process functional block.

8. The system as claimed in claim 7, wherein gain element for the boosting of the fundamental frequency component provides a proportional gain to the fundamental frequency component of the self-oscillation.

9. The system as claimed in claim 8, wherein the gain element for boosting of the fundamental frequency component and the relay element are provided as a preload relay functional block connected before the process functional block, the preload relay functional block comprising a parallel connection of a relay and a proportional gain element.

10. The system as claimed in any one of claims 7 to 9, wherein the analyser elment is arranged for determining one or more parameters of the frequency response at the output of the process functional block once the self- oscillation has reached a steady state.