WO2013128214A1 - A method for auto-tuning of pid controllers and apparatus therefor - Google Patents

A method for auto-tuning of pid controllers and apparatus therefor Download PDF

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WO2013128214A1
WO2013128214A1 PCT/GR2012/000010 GR2012000010W WO2013128214A1 WO 2013128214 A1 WO2013128214 A1 WO 2013128214A1 GR 2012000010 W GR2012000010 W GR 2012000010W WO 2013128214 A1 WO2013128214 A1 WO 2013128214A1
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overshoot
controller
tuning
closed loop
parameters
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PCT/GR2012/000010
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WO2013128214A8 (en
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Nikolaos MARGARIS
Konstantinos Papadopoulos
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Aristole University Of Thessaloniki-Research Committee
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Publication of WO2013128214A8 publication Critical patent/WO2013128214A8/en

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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
    • G05B13/0205—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric not using a model or a simulator of the controlled system
    • G05B13/024—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric not using a model or a simulator of the controlled system in which a parameter or coefficient is automatically adjusted to optimise the performance

Definitions

  • the invention refers to a method for automatic tuning of various types PID controllers notably for Single Input Single Output closed loop control systems to be applied in any minimum or non minimum phase linear stable SISO process of some form that is met in numerous chemical, mechanical and electrical industrial applications, such as electrical drives.
  • the betragsoptimum design criterion is applied for the design of type-I closed loop control systems, i.e. systems able to track only step reference signals as stated by R.C. Dorf, R.H. Bishop, in "Modern Control Systems” pp. 386-387, Prentice Hall, 2004.
  • the symmetrical optimum design criterion is applied for the control of integrating processes, leading to type-II closed loop control systems, i.e. systems able to track faster than step reference signals such as ramp inputs.
  • a basic feature of the aforementioned design criteria is that both methods try to design a controller such that the final closed loop control system exhibits optimal output disturbance rejection d 0 (s) as set out by Oldenbourg, Sartorius above. This is achieved when the magnitude of the frequency response of the closed loop transfer function is rendered as close as possible to unity in the widest possible frequency range as taught therein.
  • the term optimal is to be understood as the final closed loop (SISO) control system exhibiting optimal rejection of the output disturbance d 0 (s).
  • the design via said betragsoptimum criterion leads always to a closed loop control system that exhibits a step and frequency response with specific shape. Indeed, it was found that despite the type of control applied to the process via the betragsoptimum design criterion, the shape of the step response of the final closed loop control system is preserved.
  • One of the step response features that are being preserved is the overshoot, which remains constant and equal to 4,47%.
  • the property of the shape preservation is also observed in the frequency domain. The transition from I to PID control increases the robustness of the final closed loop control system. The property of shape preservation appears to constitute a much attractive feature that drives effortlessly to the automatic tuning of PID type controllers.
  • controller's design via the betragsoptimum and the symmetrical optimum methods presents two critical drawbacks that lead to suboptimal results in terms of output disturbance rejection.
  • controller parameters zeros of the controller transfer function
  • exact pole-zero cancellation between process's poles and controller's zeros has to be achieved.
  • This assumption results in a suboptimal control law, since it restricts the controller parameters to be tuned only with the dominant time constants of the process.
  • both the betragsoptimum and symmetrical optimum methods determine the optimal values of the PID type controller via compensation between the process's poles and the controller's zeros.
  • exact pole-zero cancellation between the process poles and the controller's zeros has to be achieved so that both design methods are applied.
  • both the betragsoptimum and symmetrical optimum methods make use of the controllers that restrict their parameters to be tuned only with real zeros. This constraint leads to suboptimal results, since for the derivation of the optimal control law, only the dominant time constants of the process are considered.
  • controller parameters are determined analytically as a function of all plant parameters, and not as a function of the plant's dominant time constants. More specifically, for a clear presentation of the invention, some assumptions are made as to the linear time invariant process described by its transfer function.
  • the revised optimal controller parameters actually depend on all process parameters, and not only on the dominant time constants of the process, in contrast with the betragsoptimum design criterion.
  • the revision of the symmetrical optimum method leads to similar results concerning the revision of the betragsoptimum method.
  • the revision of the symmetrical optimum criterion determines the controller parameters as a function of all process parameters, and not as a function of the dominant time constants of the process. For that reason, the application of the revised control law to any given process leads to improved output d 0 (s) disturbance rejection compared to the symmetrical optimum method.
  • the problem is to find automatically the optimal values for parameters so that the final closed loop
  • the loop control system exhibits a determined shape of overshoot, in particular wherein the overshoot of the final closed loop control system remains substantially constant and equal to a predetermined value, in particular 4,47%.
  • the tuning procedure is remarkable in that it is automatic wherein it consists of the following steps including
  • Step 1 determination of the plant's dc gain k p ,
  • Step 2 determination of the time constant T ⁇ x _
  • Step 3 determination of the time constant ⁇
  • Step 4 determination of the time constant Tvx.
  • the integration time constant T h which results via the betragsoptimum method, preserves the shape of the system time responses. This is true in cases of PI and PID control law, and only if exact zero-pole cancellation between the process poles and the controller's zeros occurs. As a result, if the dominant plant time constants are known, the PID type controller parameters can be automatically tuned properly via the integration time constant, so that the overshoot of the step response reaches the limit of 4,47%.
  • said gain k p is determined from the step response of the plant at steady state, in particular wherein an estimation of the sum time constant T ⁇ p of the plant is derived from the step response in some way, more particularly wherein
  • t ss is the settling time of the process, wherein an auxiliary loop is then placed in the closed loop system for tuning said controller C x (s).
  • the operation of the auxiliary loop is the following: a series of small step variations of the reference input with alternating sign are imposed, so that the plant does not diverge far from its operating point, wherein during these variations, the overshoot, undershoot, is measured and is compared with the reference overshoot, respectively undershoot.
  • the controlled process is represented by G(s), which method is remarkable in that the tuning of said controller's parameters C x (s) is based on measuring the output's overshoot, wherein an overshoot reference ovs re / is adjusted at the output of the process with which the actual overshoot of the output is every time compared at every tuning step.
  • the absolute value of the reference overshoot is 0,0447, which method is remarkable in that the error is fed into a PI controller, which tunes the controller C x (s) in succession, so that the overshoot, respectively undershoot, of the closed loop step response converges to said predetermined value.
  • the absolute value of the reference overshoot is 0,0447, which method is remarkable in that the controller C x (s) is given the form where T ⁇ x , T m and T vx are time constants that are determined.
  • both T m & T vx are set 0, wherein a series of step variations is imposed in succession on the reference input and the time constant is tuned, so that the
  • overshoot, resp. undershoot is 4,47 %, for which
  • for determining the time constant is set 0 with the value of given, wherein a series of step variations of the reference input is imposed again in succession, and is tuned, so
  • the parasitic time constant is relatively large, wherein the procedure is
  • step 4 continued by attempting the abovementioned step 4, in particular wherein the parasitic time constant is sufficiently small, wherein PI control is retained.
  • I control is initially applied to the process according to Step 2, and the integration time constant is tuned properly so that the final closed loop control system exhibits overshoot equal to 4,47%.
  • the next step is to implement a mechanism able to estimate the desired overshoot responsible for tuning properly the PI controller parameters, wherein the same mechanism operates in the case of PID control for tuning also its parameters properly, wherein for implementing this specific mechanism an Adaptive-Network-based Fuzzy Inference System designated as ANFIS is used.
  • ANFIS Adaptive-Network-based Fuzzy Inference System
  • said controller has the form of
  • step open loop experiment at the controlled process so that the plant's dc gain and settling time are measured;
  • step 2' tuning of parameter T ⁇ ,
  • step 3' estimation of the desired overshoot reference for tuning the PI controller's parameters
  • step 4' estimation of the desired overshoot reference for tuning the PID controller's parameters, thereby yielding automatically the optimal values for parameters X x , Y x , T ⁇ x , so that the step response of the final closed loop control system exhibits the observed shape.
  • said Step 1 ' consists of an open loop experiment of the process carried out at the controlled process so that the plant's dc gain and steady state time are measured, in particular wherein the plant's dc gain k p (2), and the settling time of the plant's step response t ss are measured, in particular wherein
  • a max-min detector is adjusted at the output of the process that is responsible of detecting the maximum and minimal value of the step response of the closed loop control system during the time of tuning, and an overshoot reference ovs re f is adjusted at the output of the process with which, the actual overshoot of the output is every time compared at every tuning step whereby the tuning of the controller's parameters is based on measuring the output's overshoot.
  • said operation of the auxiliary loop is the following: a series of small step variations of the reference input with alternating sign is imposed, so that the plant does not diverge far from its operating point, wherein during these variations, the overshoot (undershoot) is measured and is compared with the reference overshoot (undershoot).
  • said control law, the absolute value of the reference overshoot varies in the range of values presented, the error is fed into a PI controller, which tunes the controller
  • the overshoot of the final closed loop control system remains constant and equal to said predetermined value, in particular 4,47%, parameter being tuned, yielding that the final closed loop control system exhibits overshoot equal to being said predetermined value.
  • said step 2' consists of tuning of parameter
  • Controller is initialized by setting
  • the process is conceived as a first order one.
  • the tuning of parameter follows the next steps. At every rise-fall of the step reference input during the series of small step variations, the actual overshoot (undershoot) of the closed loop control system is measured by the max-min detector and is compared with the overshoot reference value said predetermined value.
  • the tuning procedure keeps carrying on until an actual overshoot (undershoot) of said predetermined value is observed by the max-min detector. The moment that the max-min detector measures that the actual overshoot is equal to the reference overshoot, the tuning procedure is terminated.
  • said step 3' consists in that for tuning the PI controller's parameters, the desired overshoot reference is estimated which acts as a guide for tuning the C x (s) PI controller's 1 parameters, particularly wherein the estimation of the desired overshoot is carried out by said ANFIS 7 network, wherein the stored parameters of the previous step, enter the ANFIS network 7
  • the tuning of parameter X x is carried out as follows. Again a series of small step variations at the reference input with alternating sign are imposed, so that the plant does not diverge from its operating point. At every rise-fall of the step reference input, the actual overshoot (undershoot) of the closed loop control system is measured by the max-min detector and is compared with the overshoot reference , wherein
  • the tuning procedure is terminated.
  • the shape of the step response of the final closed loop control system is identical to the expected one observed during the design of the closed loop control system after the application of the proportional integral control law to any given process.
  • the known parameters are
  • said step 4' further consists of the estimation of the desired overshoot reference for tuning the PID controller's parameters.
  • optimal automatic tuning of the PI, PID type controller's parameters for SISO closed loop control systems is remarkable by a constancy of the shape of the step response of the closed loop control system despite the type of controller applied to the process.
  • said parameters of the PID type controller are automatically tuned according to a fixed relationship.
  • the fixed relationship yields to a closed loop control system exhibiting optimal disturbance rejection.
  • the reference overshoot is estimated by an adaptive network based fuzzy inference system.
  • a control system for use to automatically tune the PID controller parameters according to the invention, it comprises the structure of the PID type controller which is automatically tuned wherein the structure allows the PID type Controller parameters to be tuned either with real or conjugate complex values, on the one hand, and the reference overshoot responsible for the automatic tuning of the PID type controller parameter, on the other hand.
  • the adaptive network fuzzy inference system for estimating the reference overshoot for tuning the PI controller, resp. the PID controller, and/or the product of the controller zeros for tuning the PID controller.
  • Figure 1 is a block diagram showing a general structure of a closed loop control system.
  • Figure 2 shows the step response of the final closed loop control system after the application of an I, PI, resp. PID control to the process.
  • Figure 3 shows the frequency response of the final closed loop control system with I, PI and PID control.
  • Figure 4 shows the step response and output disturbance rejection of the type-II closed loop control system according to the symmetrical optimum criterion.
  • Figure 5 shows the frequency response of type-II closed loop control system.
  • Figure 6 shows a so-called closed loop system with a two degree of freedom controller.
  • Figure 7 shows the step response and output disturbance rejection for said two degree of freedom controller.
  • Figure 8 shows the closed loop control system during the automatic tuning of the PID type controller according to the invention.
  • Figure 9 shows a typical example of an open loop of the process.
  • Figure 10 shows the block diagram of the control system and tuning loop according to the invention, wherein the overshoot reference is equal to 4,47%.
  • Figure 11 represents a series of small step variations at the reference input with alternating sign being imposed, so that the plant does not diverge far from its operating point.
  • Figure 12 shows the block diagram of the control system and tuning loop, that includes the ANFIS network for estimating the desired reference overshoot, according to the invention.
  • Figures 13 and 14 show the snapshots from the tuning procedure, wherein the transition from I to PK) control leads to a faster closed loop control system, although the shape of the step response is preserved.
  • Figure 1 shows the general structure of the SISO closed loop control system, wherein G(s) is the plant transfer function, C(s) is the controller transfer function, r(s) is the reference signal, d a (s) and dj(s) are the input and disturbance signals respectively and are the noise signals at
  • the invention can be applied in any minimum or non-minimum phase linear stable SISO process of the form
  • a basic feature of the aforementioned betragsoptimum and the symmetrical optimum design criteria is that both methods try to design a controller such that the final closed loop control system exhibits optimal output disturbance rejection d 0 (s). This is achieved when the magnitude of the frequency response of the closed loop transfer function is rendered as close as possible to unity in the widest possible frequency range. In other words, if T(s) stands for the closed loop transfer function, then the magnitude of the frequency response of T(s) has to satisfy condition
  • controller's design via the betragsoptimum and the symmetrical optimum methods presents two critical drawbacks that lead to merely suboptimal results in terms of output disturbance rejection d 0 (s).
  • controller parameters zeros of the controller transfer function
  • exact pole-zero cancellation between process's poles and controller's zeros has to be achieved.
  • This assumption results in a suboptimal control law, since it restricts the controller parameters to be tuned only with the dominant time constants of the process.
  • pole-zero cancellation the attenuation of load disturbances may be poor if the cancelled poles are excited by disturbances and if they are slow compared to the dominant closed- loop poles.
  • FIG.3 shows the frequency response of the final closed loop control system, I PI PID control.
  • the design of PID type controllers via the symmetrical optimum design criterion reveals advantages and disadvantages that are similar with the betragsoptimum method. It is assumed again a SISO linear integrating process of the form
  • T m stands for the integrating time constant of the plant.
  • process eq. (53 ) can be considered of having the form
  • the step response of eq.69 is represented in Fig.4 showing that the control system's output exhibits an undesired overshoot of 43,4%.
  • the frequency response of type-II closed loop control system of eq. 69 represented in Figure justifies the great overshoot in the time domain since in the higher frequency region, an undesired maximum is also observed.
  • the reference signal is filtered by an external controller as shown in Fig.6
  • Fig.7 representing the step response and output disturbance rejection for said two degree of freedom controller.
  • both the betragsoptimum and symmetrical optimum methods determine the optimal values of the PED type controller via compensation between the process's poles and the controller's zeros. In other words, exact pole-zero cancellation between the process pole's and the controller's zeros has to be achieved so that both design methods are applied.
  • the current optimization method is getting improved by revising the drawbacks described above.
  • the current invention firstly introduces the PID controller of the form
  • controller parameters are determined analytically as a function of all plant parameters and not as a function of the plant's dominant time constants.
  • Table 3 shows the range of overshoot of the final closed loop control system after the application of the revised control law to any given process - Type II closed loop control systems.
  • the revision of the symmetrical optimum method leads to similar results presented previously concerning the revision of the betragsoptimum method.
  • the revision of the symmetrical optimum criterion determines the controller parameters as a function of all process parameters and not as a function of the dominant time constants of the process. For that reason, the application of the revised control law to any given process leads to improved output d 0 (s) disturbance rejection compared to the symmetrical optimum method.
  • the property of the shape conservation of the step response of the final closed loop control system still exists after the revision of the symmetrical optimum criterion, Table 3.
  • the integration time constant ⁇ which results via the betragsoptimum method, preserves the shape of the system time responses. This is true in cases of PI and PID control law and only if exact zero-pole cancellation between the process poles and the controller's zeros occurs. As a result, if the dominant plant time constants gets known, the PID type controller parameters can be automatically tuned properly via the integration time constant, so that the overshoot of the step response reaches the limit of 4,47 %.
  • the invention is implemented based on the betragsoptimum method for the implementation of the invention, reference is made to Fig. 8, showing a closed loop control system during the automatic tuning of the PID type controller, where C x (s) (1) stands for the PID type controller, the parameters of which are getting automatically tuned.
  • the closed loop control system exhibits the specific shape observed in terms of overshoot. According to the betragsoptimum design criterion, the overshoot of the final closed loop control system remains constant and equal to 4,47 %.
  • the automatic tuning procedure consists of the following steps.
  • a first step consists of the determination of the gain k p .
  • the gain k p is determined from the step response of the plant at steady state as shown in Fig.9 representing a typical example of an open loop of the process.
  • an estimation of the sum time constant T ⁇ p of the plant can be derived from the step response in various wa s. For example,
  • t ss is the settling time of the process.
  • an auxiliary loop is placed in the closed loop system of Fig. 8, as shown grey shaded in Fig. 10 representing a block diagram of the control system and tuning loop.
  • the purpose of this loop is the tuning of the controller C x (s).
  • the overshoot reference is equal to 4,47%.
  • C x (s) 1 stands for the controller whose parameters are getting automatically tuned.
  • the plant's dc gain is represented by k p 2 whereas the controlled process is represented by G(s) 3.
  • an overshoot reference ovs re / is adjusted at the output of the process with which the actual overshoot of the output is compared every time at every tuning step as shown in Fig. 11.
  • the operation of the auxiliary loop is thus the following: A series of small step variations of the reference input with alternating sign are imposed, so that the plant does not diverge far from its operating point, as represented in Fig. 11. During these variations, the overshoot, resp. undershoot is being measured and is compared with the reference overshoot, resp. undershoot. According to the preceding analysis, the absolute value of the reference overshoot is 0,0447. The error is fed into a PI controller 5, which tunes the controller C x (s) in succession, so that the overshoot, resp. undershoot of the closed loop step response converges to 4,47%.
  • the controller C x (s) is given the form where ⁇ & , T m and T vx are time constants that must be determined.
  • Step 2 then consists of the determination of the time constant T ⁇ x .
  • a series of step variations on the reference input is imposed in succession, and the time constant T ⁇ x is tuned so that the overshoot, resp. undershoot, is 4,47%. As shown above, this occurs when T ⁇ x ⁇ T ⁇ .
  • a further step 3 consists of the determination of the time constant T m .
  • Said step 4 consists of the determination of the time constant Tvx. Given the values of T ⁇ x and Tm, Tvx is tuned, so that the overshoot is again 4,47%, by imposing again a series of step variations on the reference input. As shown above, this occurs when Tvx ⁇ T P 2.
  • the optimal control law set out in Table 2 shows that the preservation of the shape of the final step response in the control loop, in terms of the overshoot, does not remain constant and equal to 4,47% but ranges in the region according to Table 2. For that reason, a mechanism able to estimate the desired overshoot has to be provided for enabling to automatically tune the controller parameters so that the final control loop exhibits a specific shape in terms of the overshoot.
  • the purpose of the invention is to initially apply I control to the process according to Step 2, and tune properly the integration time constant so that the final closed loop control system exhibits overshoot equal to 4,47%. Based on Table 2, the resulting closed loop control system is optimal according to the analysis presented above.
  • the next step is to implement a mechanism able to estimate the desired overshoot responsible for tuning properly the PI controller parameters.
  • the same mechanism has to operate in the case of PID control for tuning also its parameters properly.
  • ANFIS Adaptive-Network-based Fuzzy Inference System
  • the problem is to find automatically the optimal values for parameters so that the final
  • closed loop control system exhibits the specific shape observed in terms of overshoot.
  • the problem is to find automatically the optimal values for parameters so
  • the automatic tuning procedure consists of the following steps :
  • Step 1 consists of the Open loop experiment at the controlled process so that the plant's dc gain and steady state time are measured.
  • Fig. 12 shows a block diagram of the control system and tuning loop.
  • the tuning loop includes the ANFIS network for estimating the desired reference overshoot. Because of the fact that there is sufficiently little information about the process integral control is initially applied so
  • a max-min detector 6 responsible of detecting the maximum and minimal value of the step response of the closed loop control system during the time of tuning is adjusted at the output 8 of the process. Moreover, because of the fact that the tuning of the controller's parameters 1 is based on measuring the output's overshoot 8, an overshoot reference ovs ref is adjusted at the output of the process with which, the actual overshoot of the output will every time be compared at every tuning step shown in Fig. 11.
  • the operation of the auxiliary loop is the following.
  • a series of small step variations of the reference input with alternating sign is imposed, so that the plant does not diverge far from its operating point, as shown in Fig. 11.
  • the overshoot, resp. undershoot is being measured and is compared with the reference overshoot, resp. undershoot.
  • Table 1 the absolute value of the reference overshoot varies in the range of values presented in Table 2.
  • the error is fed into a PI controller 5, which tunes the
  • controller C x (s) 1 in succession, so that the overshoot , resp. undershoot of the closed loop step response converges to The time the controller parameters are considered to
  • parameter T ⁇ x is tuned so that the final closed loop control system exhibits overshoot equal to
  • Step 2 consists of the tuning of parameter Controller 95 is initialized by setting
  • Step 3 consists of the estimation of the desired overshoot reference for tuning the PI controller's parameters.
  • the desired overshoot reference acting as a guide for tuning the C x (s) PI controller's (1) parameters has to be estimated.
  • the estimation of the desired overshoot is carried out by an ANFIS (Adaptive Neurofuzzy Inference System) 7 network.
  • the stored parameters of the previous step enter the
  • the tuning of parameter X x is carried out as follows. Again a series of small step variations at the reference input with alternating sign are imposed, so that the plant does not diverge from its operating point. At every rise-fall of the step reference input, the actual overshoot, resp. undershoot of the closed loop control system is measured by the max-min detector and is compared with the overshoot reference .
  • parameter ⁇ is increased by the PI controller 5, as shown in Fig. 12.
  • the integral gain of the controller is automatically tuned according to eq.(99).
  • the tuning procedure keeps carrying on until an overshoot, resp. undershoot of is observed by the max-min detector. The moment the max-min detector measures that the actual overshoot is equal to the overshoot reference, the tuning procedure is terminated.
  • the shape of the step response of the final closed loop control system is identical to the expected one observed during the design of the closed loop control system after the application of the proportional integral control law to any given process Table 1.
  • the known parameters are
  • Step 4 consists of the estimation of the desired overshoot reference for tuning the PID controller's parameters. Since the optimal overshoot of the final closed loop control system ranges when the PID control law Table 1 is applied to any given process Table 2 (3% - 8,5%), the desired overshoot reference acting as a guide for tuning the PID controller's (I) parameters has to
  • the estimation of the desired overshoot is carried out by an ANFIS (Adaptive Neurofuzzy Inference System) 7 network.
  • ANFIS Adaptive Neurofuzzy Inference System
  • Controller C x (s) eq.93 takes the form
  • Controller C x (s) eq.93 is initialized with
  • X P1 is the value we found at the end of the previous step.
  • X x value is tuned the same way as described in Step 3.
  • Y x parameter is not related with X x through a straightforward expression, as visible in control law Table 1, every time X x is tuned, parameter Y x has to be estimated. This estimation is carried out through the ANFIS network 7. For that reason, if the actual overshoot, resp. undershoot is then parameter X x is decreased by the PI controller 5 and parameter
  • Y x is estimated through the ANFIS network, Fig. 12.
  • Y x we use is made of which act as in input at the ANFIS network.
  • X PI is the value found at
  • parameter Y x is estimated through the ANFIS network, Fig.12.
  • X P1 is the value found at the end of step 3 and X x is the output of the PI controller at every rise-fall of the step reference input.
  • the integral gain of the controller is automatically tuned through eq.(102).
  • the tuning procedure keeps carrying on until an overshoot (undershoot) of is observed by the max-min detector. The moment the max-min detector measures that

Abstract

A robust method and apparatus of optimum auto-tuning for PID type controllers is presented. The method can be applied in any linear single input single output stable process. The method automatically provides the correct parameters of the PID type controller so that the final closed loop control system exhibits optimal output disturbance rejection.

Description

A METHOD FOR AUTO-TUNING OF PID CONTROLLERS AND APPARATUS THEREFOR
Field of the invention
The invention refers to a method for automatic tuning of various types PID controllers notably for Single Input Single Output closed loop control systems to be applied in any minimum or non minimum phase linear stable SISO process of some form that is met in numerous chemical, mechanical and electrical industrial applications, such as electrical drives.
Background of the invention
Since the invention of PID control in 1910, there has been developed a vast number of methods and products towards automatic tuning of PID type controllers as stated by Ang K.H., Chong G. and Li Y., by PID Control System Analysis, Design, and Technology", published in IEEE Trans, on Control Systems Technology, vol.13, No.4, pp. 559-576, July 2005.
However, according to K.J. Astrom and T. Hagglund in "PID Controllers: theory, design and tuning", published in Instrument Society of America in 2005, the majority of these industrial products fail to operate efficiently. As a result, the tuning of regulators is still carried out manually and is set out only by experts, T. Hagglund and KJ Astrom, both in WO 83/00753 disclosing "A method and an apparatus in tuning a PID regulator", and in "Automatic tuning of simple regulators with Specifications on Phase and Amplitude Margins", Automatica, Vol.20, No.5, pp. 645-651, Pergamon Press in 1984.
This represents a serious limitation for said products. The current invention is focused for being confronted mainly with that problem. To this respect, one is based on the well known methods of tuning design criteria betragsoptimum, set out by Sartorius H. in his Dissertation of "Die zweckmassige Festlegung der frei wahlbaren Regelungskonstanten" I back to 1945 and by Oldenbourg R. C, Sartorius H. in" A uniform approach to the optimum adjustment of control loops", Trans, of the ASME, pp. 1265-1279 in 1954, on the one hand, and on the so-called symmetrical optimum design criteria, on the other hand. The latter are widely used by the German industry as taught by Frohr F., Orttenburger F. in "Introduction to electronic control engineering" , Siemens, in 1982; by Umland W. J., Safiuddin M. in "Magnitude and symmetrical optimum criterion for the design of linear control systems: What is it and how does it compare with the others ? ", IEEE Trans, on Industry Applications, vol.26, No.3, pp. 489-497, in 1990; by Lutz H, Wendt W in "Taschenbuch der Regelungstechnik" Verlag Harri Deutsch in 1998, and by Follinger O. in "Regelungstechnik", Hiithig in 1994.
The betragsoptimum design criterion is applied for the design of type-I closed loop control systems, i.e. systems able to track only step reference signals as stated by R.C. Dorf, R.H. Bishop, in "Modern Control Systems" pp. 386-387, Prentice Hall, 2004. The symmetrical optimum design criterion is applied for the control of integrating processes, leading to type-II closed loop control systems, i.e. systems able to track faster than step reference signals such as ramp inputs.
A basic feature of the aforementioned design criteria is that both methods try to design a controller such that the final closed loop control system exhibits optimal output disturbance rejection d0(s) as set out by Oldenbourg, Sartorius above. This is achieved when the magnitude of the frequency response of the closed loop transfer function is rendered as close as possible to unity in the widest possible frequency range as taught therein. The term optimal is to be understood as the final closed loop (SISO) control system exhibiting optimal rejection of the output disturbance d0(s).
The design via said betragsoptimum criterion leads always to a closed loop control system that exhibits a step and frequency response with specific shape. Indeed, it was found that despite the type of control applied to the process via the betragsoptimum design criterion, the shape of the step response of the final closed loop control system is preserved. One of the step response features that are being preserved is the overshoot, which remains constant and equal to 4,47%. In addition, the property of the shape preservation is also observed in the frequency domain. The transition from I to PID control increases the robustness of the final closed loop control system. The property of shape preservation appears to constitute a much attractive feature that drives effortlessly to the automatic tuning of PID type controllers.
However, the controller's design via the betragsoptimum and the symmetrical optimum methods presents two critical drawbacks that lead to suboptimal results in terms of output disturbance rejection. For determining the controller parameters zeros of the controller transfer function, exact pole-zero cancellation between process's poles and controller's zeros has to be achieved. This assumption results in a suboptimal control law, since it restricts the controller parameters to be tuned only with the dominant time constants of the process. Moreover, according to K.J. Astrom and T. Hagglund in "PID Controllers: theory, design and tuning", Instrument Society of America, 2005, when a control law leads to pole-zero cancellation, the attenuation of load disturbances may be poor if the cancelled poles are excited by disturbances and if they are slow compared to the dominant closed-loop poles.
The second drawback of both the betragsoptimum and the symmetrical optimum methods consists in that they restrict the PID controller parameters to be tuned only with real values, whereas the optimal ones are highly likely to be complex conjugate.
Said symmetrical optimum design criterion is further presented hereafter similarly again for integral I-control, wherein the final closed loop control system is unstable. For that reason, the analysis is proceeded by applying proportional integral ΡΙ-control, wherein the resulting closed loop control system is unstable again. The symmetrical optimum criterion assumes exact pole- zero cancellation between the process's poles and the controller's zeros for determining parameter T„. For that reason, the analysis is proceeded by applying proportional integral differential control PID-control.
The design of PID type controllers via the symmetrical optimum design criterion reveals similar advantages and disadvantages with the betragsoptimum method, as taught by Kessler C. in "Das Symmetrische Optimum", Regelungstechnik, No. 3 pp. 432-426, 1958. Advantages and disadvantages concerning both the betragsoptimum and symmetrical optimum design criteria can thus be summarized as follows.
Firstly, both the betragsoptimum and symmetrical optimum methods determine the optimal values of the PID type controller via compensation between the process's poles and the controller's zeros. In other words, exact pole-zero cancellation between the process poles and the controller's zeros has to be achieved so that both design methods are applied.
Secondly, both the betragsoptimum and symmetrical optimum methods make use of the controllers that restrict their parameters to be tuned only with real zeros. This constraint leads to suboptimal results, since for the derivation of the optimal control law, only the dominant time constants of the process are considered.
Thirdly, both methods have been tested in simple linear SISO processes, and not in more complex plants such as non minimum phase plants or plants exhibiting large time delay.
However, despite the above drawbacks, the attractive advantage that both the betragsoptimum and symmetrical optimum design methods exhibit, which is the preservation of the shape of the step response of the final closed loop control system, is exploited hereafter. Along with the aid of that important property, a systematic procedure that leads to the automatic tuning of PID type controllers' parameters is developed. Aim of the invention
In this respect, the current optimization approach is improved by revising said drawbacks, wherein a PID controller of some form is introduced according to the current invention, the parameters of which are freed to be tuned whether with real or complex conjugate values.
Besides, for determining said parameters, certain optimization conditions are applied straightforward to the final closed loop transfer function. As a result, no compensation between process poles and controller's zeros has to take place. For that reason, controller parameters are determined analytically as a function of all plant parameters, and not as a function of the plant's dominant time constants. More specifically, for a clear presentation of the invention, some assumptions are made as to the linear time invariant process described by its transfer function.
The revised optimal controller parameters actually depend on all process parameters, and not only on the dominant time constants of the process, in contrast with the betragsoptimum design criterion. The application of the control law to any given process, compared to the control law proved via the betragsoptimum method, reveals that the output d0(s) disturbance rejection is improved.
Moreover, the property of the preservation of the shape of the final closed loop step response still exists in the case of the optimal control law. The difference is that when said control law is applied to a SISO linear process, despite its complexity, minimum or non-minimum phase, plant with large time delay, then the overshoot does not remain constant and equal to 4,47%, as in said betragsoptimum method, but varies according to presented values.
Furthermore, the revision of the symmetrical optimum method leads to similar results concerning the revision of the betragsoptimum method. The revision of the symmetrical optimum criterion determines the controller parameters as a function of all process parameters, and not as a function of the dominant time constants of the process. For that reason, the application of the revised control law to any given process leads to improved output d0(s) disturbance rejection compared to the symmetrical optimum method. One also could find out that the property of the shape conservation of the step response of the final closed loop control system still exists after the revision of the symmetrical optimum criterion. For that reason, the property of the shape conservation of the step response of the final closed loop control system is exploited hereafter according to the invention, in order to develop a systematic procedure for the automatic tuning of notably PID type controllers.
The design of closed loop control systems via the betragsoptimum method revealed that the overshoot of the step response remains constant and equal to 4,47% despite the distribution of the plant time constants. The measure of the automatic tuning of the PID type controllers lies on the fact that despite the process complexity, a systematic tuning procedure may be developed so that the step response of the final closed loop control system preserves this specific shape in terms of exhibiting a constant overshoot.
For the implementation of the invention based on the betragsoptimum method, the problem is to find automatically the optimal values for parameters so that the final closed loop
Figure imgf000006_0002
control system exhibits the specific shape observed in terms of overshoot. According to the betragsoptimum design criterion, the overshoot of the final closed loop control system remains constant and equal to 4,47%. Summary of the invention
There is proposed for the purpose according to the present invention a method as defined in main claim 1, thereby including a method for self tuning of controllers of PID type notably, wherein said controller has the form of
Figure imgf000006_0001
where Cx(s) (1) stands for the PID type controller the parameters of which are automatically tuned, for which target values for parameters are computed so that the final closed
Figure imgf000006_0003
loop control system exhibits a determined shape of overshoot, in particular wherein the overshoot of the final closed loop control system remains substantially constant and equal to a predetermined value, in particular 4,47%. The tuning procedure is remarkable in that it is automatic wherein it consists of the following steps including
Step 1 : determination of the plant's dc gain kp,
Step 2: determination of the time constant T∑x_
Step 3: determination of the time constant Τηχ, and
Step 4: determination of the time constant Tvx.
The integration time constant Th which results via the betragsoptimum method, preserves the shape of the system time responses. This is true in cases of PI and PID control law, and only if exact zero-pole cancellation between the process poles and the controller's zeros occurs. As a result, if the dominant plant time constants are known, the PID type controller parameters can be automatically tuned properly via the integration time constant, so that the overshoot of the step response reaches the limit of 4,47%.
According to a particular embodiment of the method according to the invention, said gain kp is determined from the step response of the plant at steady state, in particular wherein an estimation of the sum time constant T∑p of the plant is derived from the step response in some way, more particularly wherein
Figure imgf000007_0001
where tss is the settling time of the process, wherein an auxiliary loop is then placed in the closed loop system for tuning said controller Cx(s).
According to a more particular embodiment of the method according to the invention, the operation of the auxiliary loop is the following: a series of small step variations of the reference input with alternating sign are imposed, so that the plant does not diverge far from its operating point, wherein during these variations, the overshoot, undershoot, is measured and is compared with the reference overshoot, respectively undershoot.
According to an even more particular embodiment of the method according to the invention, the controlled process is represented by G(s), which method is remarkable in that the tuning of said controller's parameters Cx(s) is based on measuring the output's overshoot, wherein an overshoot reference ovsre/ is adjusted at the output of the process with which the actual overshoot of the output is every time compared at every tuning step.
According to a still more particular embodiment of the method according to the invention, the absolute value of the reference overshoot is 0,0447, which method is remarkable in that the error is fed into a PI controller, which tunes the controller Cx(s) in succession, so that the overshoot, respectively undershoot, of the closed loop step response converges to said predetermined value.
According to a yet more particular embodiment of the method according to the invention, the absolute value of the reference overshoot is 0,0447, which method is remarkable in that the controller Cx(s) is given the form
Figure imgf000007_0002
where T∑x, Tm and Tvx are time constants that are determined.
According to a further particular embodiment of the method according to the invention, for determining the time constant T∑x , both Tm & Tvx are set 0, wherein a series of step variations is imposed in succession on the reference input and the time constant is tuned, so that the
Figure imgf000008_0009
overshoot, resp. undershoot is 4,47 %, for which
Figure imgf000008_0006
According to an even further particular embodiment of the method according to the invention, for determining the time constant
Figure imgf000008_0007
is set 0 with the value of
Figure imgf000008_0008
given, wherein a series of step variations of the reference input is imposed again in succession, and is tuned, so
Figure imgf000008_0011
that the overshoot, resp. undershoot becomes again 4,47%, for which
Figure imgf000008_0005
According to a still further particular embodiment of the method according to the invention, the parasitic time constant is relatively large, wherein the procedure is
Figure imgf000008_0004
continued by attempting the abovementioned step 4, in particular wherein the parasitic time constant is sufficiently small, wherein PI control is retained.
According to a yet further particular embodiment of the method according to the invention, for determining the time constant
Figure imgf000008_0003
is tuned, the values of
Figure imgf000008_0010
being given, so that the overshoot is again 4,47%, by imposing again a series of step variations on the reference input, for which
Figure imgf000008_0002
According to a yet further particular preferred embodiment of the method according to the invention, I control is initially applied to the process according to Step 2, and the integration time constant is tuned properly so that the final closed loop control system exhibits overshoot equal to 4,47%.
According to a yet particular preferred embodiment of the method according to the inventtion, the next step is to implement a mechanism able to estimate the desired overshoot responsible for tuning properly the PI controller parameters, wherein the same mechanism operates in the case of PID control for tuning also its parameters properly, wherein for implementing this specific mechanism an Adaptive-Network-based Fuzzy Inference System designated as ANFIS is used.
According to a yet additional particular preferred embodiment of the method according to the invention for self tuning of controllers of PID type notably, said controller has the form of
Figure imgf000008_0001
for which target values for parameters
Figure imgf000008_0012
are calculated, so that the final closed loop control system exhibits a specific shape of overshoot, wherein the controller's integrating time constant is equal to
Figure imgf000008_0013
where depends explicitly on the process time constants , wherein
Figure imgf000009_0001
is remarkable in that in particular wherein the automatic tuning procedure consists of the following steps including
step : open loop experiment at the controlled process so that the plant's dc gain and settling time are measured;
step 2': tuning of parameter T∑ ,
step 3': estimation of the desired overshoot reference for tuning the PI controller's parameters, step 4': estimation of the desired overshoot reference for tuning the PID controller's parameters, thereby yielding automatically the optimal values for parameters Xx, Yx, T∑x, so that the step response of the final closed loop control system exhibits the observed shape.
According to a particular preferred embodiment thereof, said Step 1 ' consists of an open loop experiment of the process carried out at the controlled process so that the plant's dc gain and steady state time are measured, in particular wherein the plant's dc gain kp (2), and the settling time of the plant's step response tss are measured, in particular wherein
Figure imgf000009_0002
wherein, at the end of that step, the known parameters are , after which an auxiliary loop
Figure imgf000009_0003
is placed in the closed loop an integral control is initially applied, so that at least zero steady state position error is ensured at the process output, and both Xx = 0 Yx = 0 are set in eq. (3) yielding
Figure imgf000009_0004
after which, a max-min detector is adjusted at the output of the process that is responsible of detecting the maximum and minimal value of the step response of the closed loop control system during the time of tuning, and an overshoot reference ovsref is adjusted at the output of the process with which, the actual overshoot of the output is every time compared at every tuning step whereby the tuning of the controller's parameters is based on measuring the output's overshoot.
According to a more particular preferred embodiment of the method according to the invention, said operation of the auxiliary loop is the following: a series of small step variations of the reference input with alternating sign is imposed, so that the plant does not diverge far from its operating point, wherein during these variations, the overshoot (undershoot) is measured and is compared with the reference overshoot (undershoot). According to a still further particular preferred embodiment of the method according to the invention, said control law, the absolute value of the reference overshoot varies in the range of values presented, the error is fed into a PI controller, which tunes the controller
Figure imgf000010_0001
Cx(s) 1 in succession, so that the overshoot (undershoot) of the closed loop step response converges to The time The controller parameters are tuned optimally. After
Figure imgf000010_0004
Figure imgf000010_0005
the application of the integral control law, for any given plant, the overshoot of the final closed loop control system remains constant and equal to said predetermined value, in particular 4,47%, parameter being tuned, yielding that the final closed loop control system exhibits overshoot equal to being said predetermined value.
According to a yet further particular preferred embodiment of the method according to the invention, said step 2' consists of tuning of parameter
Figure imgf000010_0003
Controller is initialized by setting
Figure imgf000010_0002
being assumed that the process is conceived as a first order one. The tuning of parameter
Figure imgf000010_0013
follows the next steps. At every rise-fall of the step reference input during the series of small step variations, the actual overshoot (undershoot) of the closed loop control system is measured by the max-min detector and is compared with the overshoot reference value said predetermined value.
If the actual overshoot (undershoot) is
Figure imgf000010_0010
said predetermined value then parameter
Figure imgf000010_0009
is increased by the PI controller 2, whereas if the actual overshoot (undershoot) is
Figure imgf000010_0008
said predetermined value then parameter is decreased by the PI controller 2, wherein the tuning
Figure imgf000010_0011
procedure keeps carrying on until an actual overshoot (undershoot) of said predetermined value is observed by the max-min detector. The moment that the max-min detector measures that the actual overshoot is equal to the reference overshoot, the tuning procedure is terminated.
According to a more particularly preferred embodiment of the method according to the invention, said step 3' consists in that for tuning the PI controller's parameters, the desired overshoot reference is estimated which acts as a guide for tuning the Cx(s) PI controller's 1 parameters, particularly wherein the estimation of the desired overshoot is carried out by said ANFIS 7 network, wherein the stored parameters
Figure imgf000010_0007
of the previous step, enter the ANFIS network 7
Figure imgf000010_0006
which returns at its output, the desired overshoot reference {ovsrej according to
Figure imgf000011_0001
Controller Cx(s) eq. (3) becoming
Figure imgf000011_0002
the tuning of parameter Xx is carried out as follows. Again a series of small step variations at the reference input with alternating sign are imposed, so that the plant does not diverge from its operating point. At every rise-fall of the step reference input, the actual overshoot (undershoot) of the closed loop control system is measured by the max-min detector and is compared with the overshoot reference , wherein
Figure imgf000011_0005
if the actual overshoot (undershoot) is then parameter Xx is decreased by the PI
Figure imgf000011_0003
controller, whereas if the actual overshoot (undershoot) is then parameter Xx is
Figure imgf000011_0004
increased by the PI controller.
At each step of tuning parameter Xx the integral gain of the controller is automatically tuned according to eq. (9). The tuning procedure keeps carrying on until an overshoot (undershoot) of is observed by the max-min detector. The moment at which the max-min detector
Figure imgf000011_0007
measures that the actual overshoot is equal to the overshoot reference, the tuning procedure is terminated. At that point, the shape of the step response of the final closed loop control system is identical to the expected one observed during the design of the closed loop control system after the application of the proportional integral control law to any given process. At the end of that step the known parameters are
Figure imgf000011_0006
According to a still more particularly preferred embodiment of the method according to the invention, said step 4' further consists of the estimation of the desired overshoot reference for tuning the PID controller's parameters.
According to a yet particularly preferred embodiment of the method of the invention, optimal automatic tuning of the PI, PID type controller's parameters for SISO closed loop control systems, particularly according to the invention, is remarkable by a constancy of the shape of the step response of the closed loop control system despite the type of controller applied to the process.
According to a more particularly preferred embodiment of the method according to the invention, said parameters of the PID type controller are automatically tuned according to a fixed relationship. The fixed relationship yields to a closed loop control system exhibiting optimal disturbance rejection.
According to another particularly preferred embodiment according to the invention, the reference overshoot is estimated by an adaptive network based fuzzy inference system.
In a particular embodiment of a control system for use to automatically tune the PID controller parameters according to the invention, it comprises the structure of the PID type controller which is automatically tuned wherein the structure allows the PID type Controller parameters to be tuned either with real or conjugate complex values, on the one hand, and the reference overshoot responsible for the automatic tuning of the PID type controller parameter, on the other hand.
In addition, according to a further embodiment of the system of the invention, it allows to the adaptive network fuzzy inference system for estimating the reference overshoot for tuning the PI controller, resp. the PID controller, and/or the product of the controller zeros for tuning the PID controller.
Further features and properties of the method and device according to the invention will emerge from the following description in detail of some embodiments of the invention, which are illustrated with the aid of the attached drawings.
Brief description of the drawings
Figure 1 is a block diagram showing a general structure of a closed loop control system.
Figure 2 shows the step response of the final closed loop control system after the application of an I, PI, resp. PID control to the process.
Figure 3 shows the frequency response of the final closed loop control system with I, PI and PID control.
Figure 4 shows the step response and output disturbance rejection of the type-II closed loop control system according to the symmetrical optimum criterion.
Figure 5 shows the frequency response of type-II closed loop control system.
Figure 6 shows a so-called closed loop system with a two degree of freedom controller.
Figure 7 shows the step response and output disturbance rejection for said two degree of freedom controller.
Figure 8 shows the closed loop control system during the automatic tuning of the PID type controller according to the invention.
Figure 9 shows a typical example of an open loop of the process.
Figure 10 shows the block diagram of the control system and tuning loop according to the invention, wherein the overshoot reference is equal to 4,47%.
Figure 11 represents a series of small step variations at the reference input with alternating sign being imposed, so that the plant does not diverge far from its operating point.
Figure 12 shows the block diagram of the control system and tuning loop, that includes the ANFIS network for estimating the desired reference overshoot, according to the invention.
Figures 13 and 14 show the snapshots from the tuning procedure, wherein the transition from I to PK) control leads to a faster closed loop control system, although the shape of the step response is preserved.
Description
Figure 1 shows the general structure of the SISO closed loop control system, wherein G(s) is the plant transfer function, C(s) is the controller transfer function, r(s) is the reference signal, da(s) and dj(s) are the input and disturbance signals respectively and are the noise signals at
Figure imgf000013_0002
the reference input and process output respectively.
The invention can be applied in any minimum or non-minimum phase linear stable SISO process of the form
or
or
Figure imgf000013_0001
Such processes are met in numerous chemical, mechanical and electrical industrial applications, such as electrical drives.
A basic feature of the aforementioned betragsoptimum and the symmetrical optimum design criteria is that both methods try to design a controller such that the final closed loop control system exhibits optimal output disturbance rejection d0(s). This is achieved when the magnitude of the frequency response of the closed loop transfer function is rendered as close as possible to unity in the widest possible frequency range. In other words, if T(s) stands for the closed loop transfer function, then the magnitude of the frequency response of T(s) has to satisfy condition
Figure imgf000014_0001
Since T(s) has in general the form
Figure imgf000014_0005
by substituting into e .5 results in
Figure imgf000014_0009
Figure imgf000014_0002
The magnitude of eq.6 is defined by
Figure imgf000014_0008
Figure imgf000014_0003
where
Figure imgf000014_0004
and
Figure imgf000014_0006
Substituting equations eq. 8, eq.9 into eq.7 leads to
Figure imgf000014_0007
Therefore, expression eq.(4) is satisfied in the widest possible frequency range if
Figure imgf000015_0003
holds by expression (11) and according to eq.(8) and eq.(9), results in the following conditions
Figure imgf000015_0001
designated hereafter as the optimization conditions.
For a clear presentation of the betragsoptimum design criterion with integral control (I- control), it is assumed that the plant is described by the transfer function
Figure imgf000015_0004
where quantities are all small time constants. Under these circumstances, the
Figure imgf000015_0005
process is conceived as a first order one of the form
where
Figure imgf000015_0002
is the sum of the small parasitic time constants of the process. Since the plant transfer function has the form of eq. (18), for controlling the process, we will use integral action of the form
Figure imgf000015_0006
where is a small time constant standing for the parasitic unmodelled dynamics of the controller's electronic circuit. According to Fig.1, the plant transfer function takes the form of
where is equal to
Figure imgf000015_0007
By applying optimization conditions eq. (12) and eq. (13) into eq. (21), the resulting values for both the feedback gain and the integration time constant are given respectively by and
Figure imgf000016_0001
Substituting the control law eq.(23), eq.(24) into eq.(21), the optimal transfer function of the closed loop control system is derived as
Setting eq.(25) becomes
Figure imgf000016_0002
In case of proportional integral control (PI-control) and in case where is a large dominant
Figure imgf000016_0008
time constant, the process given in eq.(17) is approximated by
where
Figure imgf000016_0003
is the sum of the small parasitic time constants of the process. For controlling the process defined by eq.(28), PI control of the form
Figure imgf000016_0004
is applied, where is a small time constant standing for the parasitic unmodelled dynamics of
Figure imgf000016_0007
the controller's electronic circuit.
Based on Fig.l, the final closed loop transfer function is defined by
where
Figure imgf000016_0005
The optimization procedure according to the betragsoptimum design criterion moves on by setting
Figure imgf000016_0006
In other words, exact pole zero cancellation between the plant's dominant time constant and the controller's zero has to be achieved. In addition, the resulting closed loop transfer function takes the form
Figure imgf000017_0002
The application of optimization conditions eq.(12) and eq.(13) into eq.(34) drives again to determining the values of both the feedback gain and the controller's integration time constant respectively, which are finally given by
Figure imgf000017_0003
Substituting eq.35, eq.36 into eq.34 results in
Setting yields
Figure imgf000017_0004
from which we observe that equations eq.(27) and eq.(39) are the same.
In case of proportional integral differential control (PID-control) and if process defined by eq.(17) has two dominant time constants , then eq.(17) can be approximated by
Figure imgf000017_0006
where
Figure imgf000017_0005
stands for the sum of the small parasitic time constants of the process. For controlling the process defined by eq.(40), PID control of the form
Figure imgf000017_0007
is applied. Again, is a small time constant representing the parasitic unmodelled dynamics of
Figure imgf000017_0008
the controller's electronic circuit. The resulting transfer function of the closed loop control system is given by where
Figure imgf000017_0001
The optimization procedure according to the betragsoptimum design criterion moves on by assuming exact pole-zero cancellation between the process's poles and the controller's zeros. For that reason it is set
Figure imgf000018_0002
This yields by also substituting eq.(45), eq.(46) into eq.(43),
Figure imgf000018_0001
The application of optimization conditions eq.(12) and eq.(13) into eq.(47) leads to determining the optimal values for both the feedback gain and the controller's integration time constant respectively as
Figure imgf000018_0003
By substituting equations eq.(48) and eq.(49) into eq.(47) results in
By setting eq.(50) becomes
Figure imgf000018_0004
Comparing eq.(52) with eq.(27) and eq.(39) yields a closed loop transfer function of specific form in all three cases of I, PI and PUD control. The respective step response of the final closed loop control system after the application of said I, PI, resp. PID control to the process of each one of eq.(27), eq.(39) and eq.(52) is shown in Fig.2, wherein it is visible that the form of the response is preserved in all three cases in terms of the output's overshoot which remains constant and equal to 4,47%.
From Fig.2, it is thus stated that despite the type of control applied to the process via the betragsoptimum design criterion, the shape of the step response of the final closed loop control system is preserved. One of the step response features that are being preserved is the overshoot, which remains constant and equal to 4,47%. In addition, the property of the shape preservation is also observed in the frequency domain. Judging from the frequency response represented in Fig.3, it is observed that the transition from I to PID control increases the robustness of the final closed loop control system since in the lower frequency region it is apparent that
Figure imgf000018_0005
The above yields that the design via the betragsoptimum criterion always leads to a closed loop control system that exhibits a step and frequency response with specific shape. It is shown below that the property of shape preservation constitutes a much attractive feature that drives effortlessly to the automatic tuning of PID type controllers.
On the contrary, the controller's design via the betragsoptimum and the symmetrical optimum methods, presents two critical drawbacks that lead to merely suboptimal results in terms of output disturbance rejection d0(s). For determining the controller parameters (zeros of the controller transfer function), exact pole-zero cancellation between process's poles and controller's zeros has to be achieved. This assumption results in a suboptimal control law, since it restricts the controller parameters to be tuned only with the dominant time constants of the process. When a control law leads to pole-zero cancellation, the attenuation of load disturbances may be poor if the cancelled poles are excited by disturbances and if they are slow compared to the dominant closed- loop poles.
The second drawback of both the betragsoptimum and the symmetrical optimum methods is the fact that they restrict the PID controller parameters to be tuned only with real values, whereas the optimal ones, thereby not allowing complex conjugate values.
For the presentation of the symmetrical optimum method in Integral control (I-control), Fig.3 shows the frequency response of the final closed loop control system, I PI PID control. The design of PID type controllers via the symmetrical optimum design criterion reveals advantages and disadvantages that are similar with the betragsoptimum method. It is assumed again a SISO linear integrating process of the form
Figure imgf000019_0001
where Tm stands for the integrating time constant of the plant. In case where the quantities are small time constants, process eq. (53 ) can be considered of having the form
Figure imgf000019_0002
where
Figure imgf000019_0003
is the sum of the small parasitic time constants of the process. If I-control is applied to eq.(54) through eq.(20), it is easily proved that the final closed loop control system is unstable. For that reason the analysis is proceeded by applying Pi-control.
For proportional integral control (PI-control), in case where is the plant's dominant
Figure imgf000019_0004
time constant in eq.(53), the process is approximated by where
Figure imgf000020_0001
is the sum of the small parasitic time constants of the process. If PI control defined by eq.30 is applied to process eq.56, then it is proved that the resulting closed loop control system is again unstable. In similar fashion with the betragsoptimum criterion, the symmetrical optimum criterion assumes exact pole-zero cancellation between the process's poles and the controller's zeros for determining parameter Tn . For that reason, PID control given by
Figure imgf000020_0002
is applied to the process defined by eq.56.
In proportional integral differential control (PID-control) and for controlling the process defined by eq.56, the PID type regulator given by
Figure imgf000020_0003
is used according to the analysis presented above, where is a small time constant standing for
Figure imgf000020_0004
the parasitic unmodelled dynamics of the controller's electronic circuit. In that case, the resulting closed loop transfer function is given by
where
Figure imgf000020_0005
The optimization procedure according to the symmetrical optimum design criterion moves on by setting
Figure imgf000020_0006
In that, exact pole-zero cancellation is assumed to take place, between the process's pole and the controller's zero. The closed loop transfer function takes the form
Figure imgf000020_0007
The application of optimization conditions eq. 12, eq. 13, and eq. 14 to eq. 63 leads to the optimal PID control law which is finally defined by
Figure imgf000020_0008
Figure imgf000021_0001
By substituting eq.64, eq.65, eq.66 into eq.63 it is found that
Setting equation eq.67 becomes,
Figure imgf000021_0002
from which a specific form is observed with respect to the analysis presented in terms of the betragsoptimum method.
The step response of eq.69 is represented in Fig.4 showing that the control system's output exhibits an undesired overshoot of 43,4%. The frequency response of type-II closed loop control system of eq. 69 represented in Figure justifies the great overshoot in the time domain since in the higher frequency region, an undesired maximum is also observed. In order to overcome that obstacle, the reference signal is filtered by an external controller as shown in Fig.6
Figure imgf000021_0006
[Horowitz I.M., "Synthesis of Feedback Systems", London, Academic Press, 1963]. For that reason, the resulting control scheme is called a closed loop system with a two degree of freedom controller. If an external filter of the form
Figure imgf000021_0005
Figure imgf000021_0003
is chosen, the overshoot decreases from 43,4% to 8,1%, as shown in Fig.7 representing the step response and output disturbance rejection for said two degree of freedom controller.
Finally, the following advantages and disadvantages concerning both the betragsoptimum and symmetrical optimum design criteria set out above are summarized as follows.
At first, both the betragsoptimum and symmetrical optimum methods determine the optimal values of the PED type controller via compensation between the process's poles and the controller's zeros. In other words, exact pole-zero cancellation between the process pole's and the controller's zeros has to be achieved so that both design methods are applied.
In addition, both the betragsoptimum and symmetrical optimum methods make use of the controller of the form
Figure imgf000021_0004
In other words, both methods make use of controllers that restrict their parameters to be tuned only with real zeros. This constraint leads to merely suboptimal results since for the derivation of the optimal control law, only the dominant time constants of the process are considered.
At last, both methods have yet been tested in simple linear SISO processes, but not in more complex plants such as non minimum phase plants or plants exhibiting large time delay.
However, despite the above drawbacks, the attractive advantage is exploited that both the betragsoptimum and symmetrical optimum design methods exhibit, consisting in the preservation of the shape of the step response of the final closed loop control system. A systematic procedure that leads effortlessly to the automatic tuning of PID type controllers' parameters is developed hereafter along with the aid of that important property.
For that reason, the current optimization method is getting improved by revising the drawbacks described above. As a result, the current invention firstly introduces the PID controller of the form
Figure imgf000022_0001
the parameters of which, are freed to be tuned whether with real or complex conjugate values. Then, for determining parameters X , Y the optimization conditions defined under eq.(12) to (16), are applied straightforwardly to the final closed loop transfer function. As a result, no compensation between process poles and controller's zeros has to take place. For that reason, controller parameters are determined analytically as a function of all plant parameters and not as a function of the plant's dominant time constants.
In a more specifical embodiment of the invention, the linear time invariant process is assumed to be described by the following transfer function
Figure imgf000022_0002
For controlling the above process, the PID type controller defined by eq.(72) is applied. In that case and according to Fig.l, the transfer function of the closed loop control system is given by
where and
Setting
Figure imgf000022_0003
where c1 is a constant value, the polynomials
Figure imgf000023_0007
are rewritten by the following form )(
Figure imgf000023_0004
where
Figure imgf000023_0001
Setting
Figure imgf000023_0002
and
Figure imgf000023_0005
results in
Figure imgf000023_0003
and
Figure imgf000023_0006
The substitution of eq.(82), eq.(83) into eq.(74) and the straightforward application of the final closed loop transfer function optimization conditions eq.(12-15) into eq.(74), results in the optimal values of the PID type controller presented in Table 1 hereunder showing optimal values for the PID type controller parameters:
Figure imgf000024_0002
where
Figure imgf000024_0001
From the above, it appears that in contrast with the betragsoptimum design criterion, the revised optimal controller parameters depend on all process parameters and not only on the dominant time constants of the process. The application of the above control law according to Table 1 to any given process, compared to the control law proved via the betragsoptimum method, reveals that output d0(s) disturbance rejection is improved.
Moreover, the property of the preservation of the shape of the final closed loop step response still exists in the case of the optimal control law. The difference is that when the control law according to Table 1 is applied to a SISO linear process, despite its complexity, minimum or non-minimum phase, plant with large delay time, then the overshoot does not remain constant and equal to 4,47% (betragsoptimum method) but varies according to the values presented in Table 2 below showing the range of overshoot of the final closed loop control system after the application of the revised control law to any given process - Type I closed loop control systems.
Figure imgf000024_0003
Table 3 below shows the range of overshoot of the final closed loop control system after the application of the revised control law to any given process - Type II closed loop control systems.
Figure imgf000025_0001
Furthermore, the revision of the symmetrical optimum method leads to similar results presented previously concerning the revision of the betragsoptimum method. The revision of the symmetrical optimum criterion determines the controller parameters as a function of all process parameters and not as a function of the dominant time constants of the process. For that reason, the application of the revised control law to any given process leads to improved output d0(s) disturbance rejection compared to the symmetrical optimum method. Besides, the property of the shape conservation of the step response of the final closed loop control system still exists after the revision of the symmetrical optimum criterion, Table 3.
For that reason, the property of the shape conservation of the step response of the final closed loop control system is exploited in order to develop a systematic procedure for the automatic tuning of PID type controllers.
The design of closed loop control systems via the betragsoptimum method revealed that the overshoot of the step response remains constant and equal to 4,47% despite the distribution of the plant time constants. The purpose of the automatic tuning of the PID type controllers lies on the fact that despite the process complexity, a systematic tuning procedure can be developed so that the step response of the final closed loop control system preserves this specific shape in terms of exhibiting a constant overshoot.
From the analysis set out above, it is stated that the integration time constant Γ,, which results via the betragsoptimum method, preserves the shape of the system time responses. This is true in cases of PI and PID control law and only if exact zero-pole cancellation between the process poles and the controller's zeros occurs. As a result, if the dominant plant time constants gets known, the PID type controller parameters can be automatically tuned properly via the integration time constant, so that the overshoot of the step response reaches the limit of 4,47 %.
The invention is implemented based on the betragsoptimum method for the implementation of the invention, reference is made to Fig. 8, showing a closed loop control system during the automatic tuning of the PID type controller, where Cx(s) (1) stands for the PID type controller, the parameters of which are getting automatically tuned.
Controller has the form of
Figure imgf000026_0003
The point is to find automatically the optimal values for parameters so that the final
Figure imgf000026_0004
closed loop control system exhibits the specific shape observed in terms of overshoot. According to the betragsoptimum design criterion, the overshoot of the final closed loop control system remains constant and equal to 4,47 %. The automatic tuning procedure consists of the following steps.
A first step consists of the determination of the gain kp. The gain kp is determined from the step response of the plant at steady state as shown in Fig.9 representing a typical example of an open loop of the process. Moreover, an estimation of the sum time constant T∑p of the plant can be derived from the step response in various wa s. For example,
Figure imgf000026_0001
where tss is the settling time of the process. Then, an auxiliary loop is placed in the closed loop system of Fig. 8, as shown grey shaded in Fig. 10 representing a block diagram of the control system and tuning loop. The purpose of this loop is the tuning of the controller Cx(s). The overshoot reference is equal to 4,47%. In Fig.10, Cx(s) 1 stands for the controller whose parameters are getting automatically tuned. The plant's dc gain is represented by kp 2 whereas the controlled process is represented by G(s) 3. Moreover, because of the fact that the tuning of the controller's parameters Cx(s) 1 is based on measuring the output's overshoot 8, an overshoot reference ovsre/ is adjusted at the output of the process with which the actual overshoot of the output is compared every time at every tuning step as shown in Fig. 11.
The operation of the auxiliary loop is thus the following: A series of small step variations of the reference input with alternating sign are imposed, so that the plant does not diverge far from its operating point, as represented in Fig. 11. During these variations, the overshoot, resp. undershoot is being measured and is compared with the reference overshoot, resp. undershoot. According to the preceding analysis, the absolute value of the reference overshoot is 0,0447. The error is fed into a PI controller 5, which tunes the controller Cx(s) in succession, so that the overshoot, resp. undershoot of the closed loop step response converges to 4,47%.
According to the analysis presented "above, the controller Cx(s) is given the form
Figure imgf000026_0002
where Τ&, Tm and Tvx are time constants that must be determined.
Step 2 then consists of the determination of the time constant T∑x . Setting Tm = Tvx = 0 in eq. 90, a series of step variations on the reference input is imposed in succession, and the time constant T∑x is tuned so that the overshoot, resp. undershoot, is 4,47%. As shown above, this occurs when T∑x ~ T∑.
A further step 3 consists of the determination of the time constant Tm. With the value of T∑x given, Tvx = 0 is set in eq. 90. A series of step variations of the reference input is imposed again and Tm, is tuned so that the overshoot, resp. undershoot becomes again 4,47%. As shown above, this occurs when Tm ~ Tpl. If the parasitic time constant T∑1 = T∑x - Tm is relatively large, the procedure can be continued by attempting Step 4. If the parasitic time constant is sufficiently small, PI control is retained.
Said step 4 consists of the determination of the time constant Tvx. Given the values of T∑x and Tm, Tvx is tuned, so that the overshoot is again 4,47%, by imposing again a series of step variations on the reference input. As shown above, this occurs when Tvx ~ TP2.
The optimal control law set out in Table 2, shows that the preservation of the shape of the final step response in the control loop, in terms of the overshoot, does not remain constant and equal to 4,47% but ranges in the region according to Table 2. For that reason, a mechanism able to estimate the desired overshoot has to be provided for enabling to automatically tune the controller parameters so that the final control loop exhibits a specific shape in terms of the overshoot. The purpose of the invention is to initially apply I control to the process according to Step 2, and tune properly the integration time constant so that the final closed loop control system exhibits overshoot equal to 4,47%. Based on Table 2, the resulting closed loop control system is optimal according to the analysis presented above.
The next step is to implement a mechanism able to estimate the desired overshoot responsible for tuning properly the PI controller parameters. The same mechanism has to operate in the case of PID control for tuning also its parameters properly. For implementing this specific mechanism, a well known Adaptive-Network-based Fuzzy Inference System, ANFIS [Jang J. S. R., "ANFIS: Adaptive-Network-based Fuzzy Inference Systems", IEEE Trans, on Systems, Man, and Cybernetics, vol. 23, No.3, pp. 665-685, 1993] is used. Further details concerning the systematic tuning method operates are presented below.
For implementing the invention based on the optimal control law, reference is made to Fig. 8, where Cx(s) 1 stands for the PID type controller the parameters of which are getting automatically tuned. Controller has the form of
Figure imgf000028_0001
The problem is to find automatically the optimal values for parameters so that the final
Figure imgf000028_0004
closed loop control system exhibits the specific shape observed in terms of overshoot.
According to the control law Table 1, the controller's integrating time constant is equal to
Figure imgf000028_0003
where depends explicitly on the process time constants. Because of the fact that in reality, quantity is an unknown parameter of the process G(s) 3, eq.(91) takes the form
Figure imgf000028_0002
As a result, the problem is to find automatically the optimal values for parameters so
Figure imgf000028_0011
that the step response of the final closed loop control system exhibits the shape observed in Table 2. The automatic tuning procedure consists of the following steps :
Step 1 consists of the Open loop experiment at the controlled process so that the plant's dc gain and steady state time are measured.
An open loop experiment of the process is carried out. A typical example of an open loop experiment of the process is presented in Fig. 9 where the plant's dc gain kp 2, and the settling time of the plant's step response
Figure imgf000028_0010
can be easily measured. The feedback gain 4 has to satisfy condition
Figure imgf000028_0005
As a result, at the end of that step, the known parameters are
Figure imgf000028_0006
Then, an auxiliary loop as shown grey shaded in Fig. 12 is placed in the closed loop of Fig. 8. Fig. 12 shows a block diagram of the control system and tuning loop. The tuning loop includes the ANFIS network for estimating the desired reference overshoot. Because of the fact that there is sufficiently little information about the process integral control is initially applied so
Figure imgf000028_0007
that at least zero steady state position error is ensured at the process output.
For that reason, are set in eq.(93). As a result, eq.(93) becomes
Figure imgf000028_0008
Figure imgf000028_0009
Afterwards, a max-min detector 6 responsible of detecting the maximum and minimal value of the step response of the closed loop control system during the time of tuning is adjusted at the output 8 of the process. Moreover, because of the fact that the tuning of the controller's parameters 1 is based on measuring the output's overshoot 8, an overshoot reference ovsref is adjusted at the output of the process with which, the actual overshoot of the output will every time be compared at every tuning step shown in Fig. 11.
The operation of the auxiliary loop is the following. A series of small step variations of the reference input with alternating sign is imposed, so that the plant does not diverge far from its operating point, as shown in Fig. 11. During these variations, the overshoot, resp. undershoot is being measured and is compared with the reference overshoot, resp. undershoot. According to the control law Table 1, the absolute value of the reference overshoot varies in the range of values presented in Table 2. The error is fed into a PI controller 5, which tunes the
Figure imgf000029_0007
controller Cx(s) 1 in succession, so that the overshoot , resp. undershoot of the closed loop step response converges to
Figure imgf000029_0008
The time the controller parameters are considered to
Figure imgf000029_0006
have been tuned optimally.
Because of the fact that after the application of the integral control law Table 1, for any given plan, the overshoot of the final closed loop control system remains constant and equal to 4,47%, parameter T∑x is tuned so that the final closed loop control system exhibits overshoot equal to
Figure imgf000029_0005
Step 2 consists of the tuning of parameter Controller 95 is initialized by setting
Figure imgf000029_0009
Figure imgf000029_0001
¾
since the process is assumed to be conceived as a first order one. The tuning of parameter T∑x follows the next steps. At every rise-fall of the step reference input during the series of small step variations, the actual overshoot, resp. undershoot of the closed loop control system is measured by the max-min detector and is compared with the overshoot reference value
Figure imgf000029_0003
If the actual overshoot, resp. undershoot is then parameter T∑x is increased by the
Figure imgf000029_0002
PI controller (2), as visible from Fig.12, on the one hand, whereas if the actual overshoot (undershoot) is
Figure imgf000029_0004
then parameter T∑x is decreased by the PI controller (2), as visible from Fig.12, on the other hand.
The tuning procedure keeps carrying on until an actual overshoot, resp. undershoot of 4,47% is observed by the max-min detector. The moment the max-min detector measures that the actual overshoot is equal to the reference overshoot, the tuning procedure is terminated. At that point, the shape of the step response of the final closed loop control system is identical to the expected one observed during the design of the closed loop control system after the application of the integral control law to any given process. Step 3 consists of the estimation of the desired overshoot reference for tuning the PI controller's parameters. Since the optimal overshoot of the final closed loop control system ranges when PI control Table 1 law is applied to any given process 3%-8% Table 2, the desired overshoot reference acting as a guide for tuning the Cx(s) PI controller's (1) parameters has to be estimated. The estimation of the desired overshoot is carried out by an ANFIS (Adaptive Neurofuzzy Inference System) 7 network. The stored parameters of the previous step, enter the
Figure imgf000030_0008
ANFIS network 7
Figure imgf000030_0001
which returns at its output, the desired overshoot reference ovsre/ according to
Figure imgf000030_0002
Controller eq.(93) now takes the form
Figure imgf000030_0003
Since the desired overshoot reference is available, the tuning of parameter Xx is carried out as follows. Again a series of small step variations at the reference input with alternating sign are imposed, so that the plant does not diverge from its operating point. At every rise-fall of the step reference input, the actual overshoot, resp. undershoot of the closed loop control system is measured by the max-min detector and is compared with the overshoot reference .
Figure imgf000030_0005
If the actual overshoot, resp. undershoot is then parameter Xx is decreased by the
Figure imgf000030_0004
PI controller 5 as shown in Fig.12, whereas if the actual overshoot, resp. undershoot) is
Figure imgf000030_0006
then parameter^ is increased by the PI controller 5, as shown in Fig. 12.
Figure imgf000030_0007
At each step of tuning parameter Xx the integral gain of the controller is automatically tuned according to eq.(99). The tuning procedure keeps carrying on until an overshoot, resp. undershoot of is observed by the max-min detector. The moment the max-min detector measures that the actual overshoot is equal to the overshoot reference, the tuning procedure is terminated. At that point, the shape of the step response of the final closed loop control system is identical to the expected one observed during the design of the closed loop control system after the application of the proportional integral control law to any given process Table 1. At the end of that step, the known parameters are
Figure imgf000031_0001
Step 4 consists of the estimation of the desired overshoot reference for tuning the PID controller's parameters. Since the optimal overshoot of the final closed loop control system ranges when the PID control law Table 1 is applied to any given process Table 2 (3% - 8,5%), the desired overshoot reference acting as a guide for tuning the PID controller's (I) parameters has to
Figure imgf000031_0003
be estimated. Again a series of small step variations of reference input with alternating sign are imposed, so that the plant does not diverge from its operating point. At every rise-fall of the step reference input, the actual overshoot, resp. undershoot of the closed loop control system is measured by the max-min detector and is compared with the overshoot reference
Figure imgf000031_0002
The estimation of the desired overshoot is carried out by an ANFIS (Adaptive Neurofuzzy Inference System) 7 network. The stored parameters of the previous step,
Figure imgf000031_0004
enter the ANFIS network 7
Figure imgf000031_0005
which returns at its output, the desired overshoot reference ovsre/ according to
Controller Cx(s) eq.93 takes the form
Figure imgf000031_0006
Since the desired overshoot reference is available eq. lOl, the tuning of parameter Xx is carried out as follows. Controller Cx(s) eq.93 is initialized with
Figure imgf000031_0007
where XP1 is the value we found at the end of the previous step. Following the same line presented in Step 3, at every rise-fall of the step reference input and according to the actual overshoot measured at the output, Xx value is tuned the same way as described in Step 3. However, because of the fact that Yx parameter is not related with Xx through a straightforward expression, as visible in control law Table 1, every time Xx is tuned, parameter Yx has to be estimated. This estimation is carried out through the ANFIS network 7. For that reason, if the actual overshoot, resp. undershoot is then parameter Xx is decreased by the PI controller 5 and parameter
Figure imgf000031_0008
Yx is estimated through the ANFIS network, Fig. 12. For estimating parameter Yx we use is made of which act as in input at the ANFIS network. XPI is the value found at
Figure imgf000031_0009
the end of Step 3 and Xx is the output of the PI controller at every rise-fall of the step reference input. In similar fashion, if the actual overshoot, resp. undershoot is then parameter Xx is
Figure imgf000032_0001
increased by the PI controller 5 and parameter Yx is estimated through the ANFIS network, Fig.12. For estimating parameter Yx
Figure imgf000032_0002
which act as in input at the ANFIS network is used again. XP1 is the value found at the end of step 3 and Xx is the output of the PI controller at every rise-fall of the step reference input.
At each tuning step of parameter Xx, Yx, the integral gain of the controller is automatically tuned through eq.(102). The tuning procedure keeps carrying on until an overshoot (undershoot) of is observed by the max-min detector. The moment the max-min detector measures that
Figure imgf000032_0004
the actual overshoot is equal to the overshoot reference , the tuning procedure is
Figure imgf000032_0003
terminated.
Although the shape of the step response is preserved, the transition from I to PID control leads to a faster closed loop control system presents snapshots of the tuning process of the PID type controller. From there, it appears that the step response of the closed system is preserved. The transition from I to PID control leads to a closed loop control system with faster response, Fig.12.

Claims

1. Method for self tuning of controllers of PID type notably, wherein said controller has the form of
Figure imgf000033_0001
where (1) stands for the PID type controller the parameters of which are tuned, for which target values for parameters
Figure imgf000033_0002
are computed so that the final closed loop control system exhibits a determined shape of overshoot, in particular wherein the overshoot of the final closed loop control system remains substantially constant and equal to a predetermined value, in particular 4,47%, characterized in that the tuning procedure is automatic wherein it consists of the following steps:
Step 1 : determination of the plant's dc gain
Figure imgf000033_0003
Step 2: determination of the time constant
Figure imgf000033_0004
Step 3: determination of the time constant
Figure imgf000033_0005
Step 4: determination of the time constant
Figure imgf000033_0006
2. Method according to claim 1, characterized in that said gain is determined from the
Figure imgf000033_0011
step response of the plant at steady state, in particular wherein an estimation of the sum time constant of the plant is derived from the step response in a way, particularly wherein
Figure imgf000033_0009
where is the settling time of the process, wherein an auxiliary loop is then placed in the closed loop system for tuning said controller
Figure imgf000033_0010
3. Method according to the preceding claim 2, characterized in that the operation of the auxiliary loop is the following: a series of small step variations of the reference input with alternating sign are imposed, so that the plant does not diverge far from its operating point, wherein during these variations, the overshoot, undershoot, is measured and is compared with the reference overshoot, respectively undershoot.
4. Method according to claim 2 or 3, wherein the controlled process is represented by G(s) (3), characterized in that the tuning of said controller's parameters (1) is based on
Figure imgf000033_0008
measuring the output's overshoot (8), wherein an overshoot reference is adjusted at the
Figure imgf000033_0007
output of the process with which the actual overshoot of the output is every time compared at every tuning step.
5. Method according to one of the preceding claims, wherein the absolute value of the reference overshoot is 0,0447, characterized in that the error is fed into a PI controller (5), which tunes the controller Cx(s) in succession, so that the overshoot, respectively undershoot, of the closed loop step response converges to said predetermined value.
6. Method according to one of the preceding claims wherein the absolute value of the reference overshoot is 0,0447, characterized in that the controller Cx(s) is given the form
Figure imgf000034_0001
where are time constants that are determined.
Figure imgf000034_0003
7. Method according to one of the preceding claims, characterized in that for determining the time constant , both are set 0, wherein a series of step variations is imposed in
Figure imgf000034_0004
Figure imgf000034_0005
succession on the reference input and the time constant is tuned, so that the overshoot, resp.
Figure imgf000034_0006
undershoot is 4,47 %, for which
Figure imgf000034_0009
8. Method according to the preceding claim, characterized in that for determining the time constant Tm ,T„ is set 0 with the value of given, wherein a series of step variations of the
Figure imgf000034_0008
reference input is imposed again in succession, and Tm is tuned, so that the overshoot, resp. undershoot becomes again 4,47%, for which
Figure imgf000034_0007
9. Method according to one of the preceding claims, characterized in that the parasitic time constant is relatively large, wherein the procedure is continued by attempting
Figure imgf000034_0002
step 4, in particular wherein the parasitic time constant is sufficiently small, wherein PI control is retained.
10. Method according to one of the preceding claims wherein for determining the time constant is tuned, the values of being given, so that the overshoot is again
Figure imgf000034_0010
Figure imgf000034_0011
4,47%, by imposing again a series of step variations on the reference input, for which Tvx ~ Tp2.
11. Method according to one of the preceding claims, characterized in that I control is initially applied to the process according to Step 2, and the integration time constant is tuned properly so that the final closed loop control system exhibits overshoot equal to 4,47%.
12. Method according to one of the preceding claims, characterized in that the next step is to implement a mechanism able to estimate the desired overshoot responsible for tuning properly the PI controller parameters, wherein the same mechanism operates in the case of PID control for tuning also its parameters properly, wherein for implementing this specific mechanism an Adaptive-Network-based Fuzzy Inference System, ANFIS is used.
13. Method for self tuning of controllers of PID type notably, wherein said controller has the form of
Figure imgf000035_0001
for which target values for parameters Xx, Yx, Tb are calculated, so that the final closed loop control system exhibits a specific shape of overshoot, wherein the controller's integrating time constant is equal to
Figure imgf000035_0002
where T∑x depends explicitly on the process time constants , wherein
Figure imgf000035_0003
characterized in that the automatic tuning procedure comprises the following steps :
Step 1' consists of an open loop experiment at the controlled process so that the plant's dc gain and settling time are measured,
Step 2' consists of tuning parameter
Figure imgf000035_0004
Step 3' consists of the estimation of the desired overshoot reference for tuning the PI controller's parameters,
Step 4' consists of the estimation of the desired overshoot reference for tuning the PED controller's parameters,
thereby yielding automatically the optimal values for parameters so that the step
Figure imgf000035_0005
response of the final closed loop control system exhibits the observed shape.
14. Method according to the preceding claim, characterized in that said Step 1' consists of an open loop experiment of the process carried out at the controlled process so that the plant's dc gain and steady state time are measured, in particular wherein the plant's dc gain kp (2), and the settling time of the plant's step response are measured, in particular wherein
Figure imgf000035_0009
Figure imgf000035_0006
wherein, at the end of that step, the known parameters are , after which an auxiliary loop
Figure imgf000035_0007
is placed in the closed loop an integral control is initially applied, so that at least zero steady state position error is ensured at the process output, and both Xx - 0 Yx = 0 are set, yielding
Figure imgf000035_0008
after which, a max-min detector (6) is adjusted at the output (8) of the process that is responsible of detecting the maximum and minimal value of the step response of the closed loop control system during the time of tuning, and an overshoot reference is adjusted at the output of the
Figure imgf000036_0008
process with which, the actual overshoot of the output is every time compared at every tuning step whereby the tuning of the controller's parameters (1) is based on measuring the output's overshoot (8).
15. Method according to the preceding claim, characterized in that said the operation of the auxiliary loop is the following: a series of small step variations of the reference input with alternating sign is imposed, so that the plant does not diverge far from its operating point, wherein during these variations, the overshoot, resp. undershoot is measured and is compared with the reference overshoot, resp. undershoot.
16. Method according to the preceding claim, characterized in that according to the said control law, the absolute value of the reference overshoot varies in the range of values presented , the error
Figure imgf000036_0001
is fed into a PI controller (5), which tunes the controller Cx(s) (1) in succession, so that the overshoot, resp. undershoot of the closed loop step response converges to in that the time and that the controller parameters are tuned optimally, in
Figure imgf000036_0010
Figure imgf000036_0007
that after the application of the integral control law, for any given plant, the overshoot of the final closed loop control system remains constant and equal to said predetermined value, in particular 4,47%, parameter T∑x being tuned, yielding that the final closed loop control system exhibits overshoot equal to being said predetermined value.
Figure imgf000036_0009
17. Method according to one of the claims 13 to 16 in particular the preceding claim, characterized in that said Step 2' consists of tuning of parameter wherein said controller is
Figure imgf000036_0003
initialized by setting , being assumed that the process is conceived as a first order
Figure imgf000036_0002
one, wherein the tuning of parameter T∑x follows the next steps, wherein at every rise-fall of the step reference input during the series of small step variations, the actual overshoot, resp. undershoot of the closed loop control system is measured by the max-min detector and is compared with the overshoot, reference value said predetermined value, wherein if the
Figure imgf000036_0006
actual overshoot, resp. undershoot is said predetermined value then parameter T∑x is
Figure imgf000036_0005
increased by the PI controller (2), whereas if the actual overshoot (undershoot) is
Figure imgf000036_0004
said predetermined value then parameter T∑x is decreased by the PI controller (2), wherein the tuning procedure keeps carrying on until an actual overshoot, resp. undershoot of said predetermined value is observed by the max-min detector, wherein at the moment the max-min detector measures that the actual overshoot is equal to the reference overshoot, the tuning procedure is terminated.
18. Method according to one of the claims 13 to 17 in particular the preceding claim, characterized in that said Step 3' consists in that for tuning the PI controller's parameters the desired overshoot reference is estimated which acts as a guide for tuning the Cx(s) PI controller's (1) parameters, particularly wherein the estimation of the desired overshoot is carried out by said ANFIS (7) network, wherein the stored parameters of the previous step, enter the
Figure imgf000037_0005
ANFIS network (7)
Figure imgf000037_0002
which returns at its output, the desired overshoot reference according to
Figure imgf000037_0004
Figure imgf000037_0003
controller Cx(s) becoming
Figure imgf000037_0006
and the tuning of parameter Xx is carried out in that again a series of small step variations at the reference input with alternating sign are imposed, so that the plant does not diverge from its operating point, in that at every rise-fall of the step reference input, the actual overshoot resp. undershoot of the closed loop control system is measured by the max-min detector and is compared with the overshoot reference , wherein if the actual overshoot resp. undershoot is
Figure imgf000037_0007
then parameter Xx is decreased by the PI controller (5) whereas if the actual
Figure imgf000037_0008
overshoot resp. undershoot is then parameter is increased by the PI controller
Figure imgf000037_0009
Figure imgf000037_0010
(5),
at each step of tuning parameter Xx the integral gain of the controller is automatically tuned according to the latter equation, the tuning procedure keeps carrying on until an overshoot resp. undershoot of is observed by the max-min detector, wherein at the moment the max-min
Figure imgf000037_0011
detector measures . that the actual overshoot is equal to the overshoot reference, the tuning procedure is terminated, wherein at that point, the shape of the step response of the final closed loop control system is identical to the expected one observed during the design of the closed loop control system after the application of the proportional integral control law to any given process, and wherein at the end of that step the known parameters are
Figure imgf000037_0001
19. Method according to one of the claims 13 to 18 particularly the preceding claim, characterized in that said Step 4' consists of the estimation of the desired overshoot reference for tuning the PID controller's parameters.
20. A method of optimal automatic tuning of the PI, PID type controller's parameters for SISO closed loop control systems, in particular according to one of the preceding claims, characterized by a constancy of the shape of the step response of the closed loop control system despite the type of controller applied to the process.
21. Method according to the preceding claim, characterized in that said parameters of the PID type controller are automatically tuned according to a fixed relationship, which yields to a closed loop control system exhibiting optimal disturbance rejection.
22. Method wherein the reference overshoot is estimated by an adaptive network based fuzzy inference system.
23. A control system for use to automatically tune the PID controller parameters comprising
the structure of the PID type controller which is automatically tuned, which structure allows the PID type controller parameters to be tuned either with real or conjugate complex values, and
the reference overshoot responsible for the automatic tuning of the PID type controller parameter.
24. The adaptive network fuzzy inference system for estimating the reference overshoot for tuning the PI controller.
25. The adaptive network fuzzy inference system for estimating the reference overshoot for tuning the PID controller.
26. The adaptive network fuzzy inference system for estimating the product of the controller zeros for tuning the PID controller.
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