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Closed-loop Identification based PID Tuning without External Excitation *
Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress Proceedings of the 20th9-14, World The International International Federation of Congress Automatic Control Toulouse, France, July 2017 The Federation of Automatic Control Available online at www.sciencedirect.com The International of Automatic Control Toulouse, France, July Toulouse, France,Federation July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 3995–4000 Closed-loop Closed-loop Identification Identification based based PID PID Closed-loop Identification based PID⋆⋆ Tuning without External Excitation Tuning without without External External Excitation Excitation ⋆ Tuning
Yan Wengang, Zhu Yucai, Zhao Jun Yan Yan Wengang, Wengang, Zhu Zhu Yucai, Yucai, Zhao Zhao Jun Jun Yan Wengang, Zhu Yucai, Zhao Jun State Key Laboratory of Industrial Control Technology, College of State Laboratory of Control College State Key Key Laboratory of Industrial IndustrialZhejiang Control Technology, Technology, College of of Control Science and Engineering, University, Hangzhou, State Key Laboratory of IndustrialZhejiang Control Technology, College of Control Science and Engineering, University, Control Science and310027, Engineering, Zhejiang University, Hangzhou, Hangzhou, P.R.China (e-mail: Control Science and310027, Engineering, Zhejiang University, Hangzhou, P.R.China (e-mail: 310027, P.R.China (e-mail: [email protected];[email protected];[email protected];). 310027, P.R.China (e-mail: [email protected];[email protected];[email protected];). [email protected];[email protected];[email protected];). [email protected];[email protected];[email protected];). Abstract: Identification based PID tuning is studied in this work. A method is proposed Abstract: Identification based tuning this work. A is Abstract: Identification based PID tuning is is studied studied inThe thisbasic work.idea A method method is proposed proposed to achieve the informativity of PID the closed-loop system.in is shifting the PID Abstract: Identification based PID tuning is studied inThe thisbasic work.idea A method is proposed to achieve the informativity of the closed-loop system. is shifting the to achieve the informativity of the closed-loop system. The basic idea is shifting the PID PID parameters appropriately according to the amplitude of the feedback error. In this way, not to achieve the informativity of the to closed-loop system. Thefeedback basic idea is shifting the PID parameters appropriately according the of In not parameters appropriately according the amplitude amplitude of the the feedback error. In this this way, way, not only the closed-loop system can be to identified accurately without usingerror. test signals, but also parameters appropriately according to the amplitude of the feedback error. In this way, not only the closed-loop system can be identified accurately without using test signals, but also onlycontrol the closed-loop system canimproved be identified accurately testAn signals, but also the performance can be during the data without collectionusing period. internal model only the closed-loop system can be identified accurately without using test signals, but also the control performance can be improved during the data collection period. An internal model the control performance can beisimproved collection period. An internal model control (IMC) design method applied toduring tune the data PID parameters. The effectiveness of the the control performance can beis the data collection period. An internal model control (IMC) design method applied to tune PID The of control (IMC) design method isimproved applied toduring tune the PID parameters. parameters. The effectiveness effectiveness of the the proposed approach is illustrated by simulations. control (IMC) design method is by applied to tune the PID parameters. The effectiveness of the proposed approach is simulations. proposed approach is illustrated illustrated by simulations. proposed approach is illustrated by of simulations. © 2017, IFAC (International Federation Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: closed-loop system; PID; informativity; control performance; IMC tuning rules. Keywords: closed-loop closed-loop system; system; PID; PID; informativity; informativity; control control performance; performance; IMC IMC tuning tuning rules. rules. Keywords: Keywords: closed-loop system; PID; informativity; control performance; IMC tuning rules. 1. INTRODUCTION process model (Ljung,1999). It has also been shown that, 1. INTRODUCTION INTRODUCTION process model It been that, 1. process model (Ljung,1999). (Ljung,1999). It has has also also been shown shown that, if a reference signal is sufficiently persistently exciting, 1. INTRODUCTION process model (Ljung,1999). It has also been shown that, if a reference signal is sufficiently persistently exciting, if a reference signal is sufficiently persistently exciting, it is possible to identify the process from closedAlthough model predictive control (MPC) is becoming then if a reference signal is sufficiently persistently exciting, then it is possible to identify the process from closedAlthough model predictive controland (MPC) isof becoming becoming then data it is possible to identify process from quanticlosedAlthough model control (MPC) (Ljung,1999). More the recently, general more popular in predictive process control a lot is work for loop then it is possible to identify process from quanticlosedAlthough model predictive controland (MPC) isof becoming (Ljung,1999). More recently, general more popular in process control and lot work for loop data data (Ljung,1999). Moreforthe recently, general more popular control aa lot work tative informative conditions identification of quanticlosedMPC such as in Qinprocess and Badgwell (2003) andof Zhu et for al. loop loop data (Ljung,1999). More recently, general quantimore popular in process control and a lot of work for tative informative conditions for identification of MPC such as Qin and Badgwell (2003) and Zhu et al. tativesystem informative conditionsinforGevers identification of closedclosedMPC Qinpublished, and Badgwell Zhu al. loop were presented et al. (2008) and (2013) such has as been most (2003) control and loops areetstill informative conditionsin identification of closedMPC such Qinpublished, and Badgwell (2003) and Zhu etstill al. tative loop system were Gevers et (2008) and (2013) has as been published, most control loops are still loop system were presented presented informodels Gevers et al. al.two (2008) and (2013) has been control loops Gevers et al. (2010) for different using different PID controllers. PID tuning most is also part of the are pre-test system were presented in models Gevers using et al.two (2008) and (2013) has been PID published, control loops are still loop et (2010) for different PID controllers. PID tuning most isgood also part of of the pre-test Gevers et al. al.an (2010) for different different models using two different PID controllers. tuning is also part pre-test approaches: expectation-based and information matrixin a MPC project. Therefore, tuning ofthe PID loops Gevers Gevers et al. (2010) for different models using two different PID controllers. PID tuning is also part of the pre-test approaches: an expectation-based and information matrixin a MPC project. Therefore, good tuning of PID loops approaches: an expectation-based and information matrixin a MPC project. to Therefore, of PIDofloops approaches. is very important maintain good goodtuning performance the based an expectation-based and information matrixin very a MPC project. to Therefore, good of PIDof based is very important to maintain goodtuning performance ofloops the approaches: based approaches. approaches. is important maintain performance the overall process control system. good based approaches. is very important to maintain good performance of the Most identification based PID tuning methods use test overall process process control control system. system. overall based tuning test ˚ Most identification identification based PID PID test, tuning methods use test overall signals during the identification see,methods e.g., Zhuuse (2001). A str¨omprocess and H¨acontrol gglund system. (1995) contains a good summary Most Most identification based PID tuning methods use test ˚ signals during the identification test, see, e.g., Zhu (2001). A str¨ o m and H¨ a gglund (1995) contains a good summary ˚ signals during the identification test, see,operation, e.g., Zhu (2001). A str¨ o m and H¨ a gglund (1995) contains a good summary However, Test signals disturb process which of PID tuning rules, among which IMC-based PID tuning ˚ signals during the identification test, see, e.g., Zhu (2001). A str¨ o m and H¨ a gglund (1995) contains a good summary signals process operation, which of PID PID tuning rules, among which IMC-based PID tuning tuning However, However, Test signals disturb disturb process operation, which of which IMC-based PID is a cost Test in production unit. For reducing the cost of rules aretuning widelyrules, usedamong in process control. The methodology signals disturb process operation, which of PID tuning rules, among which IMC-based PID tuning However, in unit. For of rules are widelybased usedon in IMC, process control. The methodology is aa cost cost Test in production production unit. For reducing the cost of rules are widely used in process control. methodology identification, it would be ideal toreducing use no the testcost signal of PID tuning which was The firstly introduced is is a cost in production unit. For reducing the cost of rules are widely used in process control. The methodology identification, it would be ideal to use no test signal of PID tuning based on IMC, which was firstly introduced identification, it would be ideal no test signal of tuning based on IMC, which the identification test. In to thisuse paper, a method by PID Rivera et al. (1986), consists of was threefirstly steps:introduced factoring during identification, it would be ideal to use no test signal of PID tuning based on IMC, which was firstly introduced during the identification test. In this paper, a method by Rivera et al. (1986), consists of three steps: factoring during the identification test.using In this method by al. (1986), consists ofand three steps: factoring of closed-loop test without test paper, signalsa for PID the Rivera model, et defining the controller being implemented the identification test.using In this a for method by Rivera et al. (1986), consists ofand three steps: factoring during closed-loop test test signals PID the model, defining the practical controller and being implemented of closed-loop test without without using test paper, signals for PID the model, defining the controller being implemented tuning is proposed and studied. The idea follows the as PID controller. For use, tuning formulae for of closed-loop test without using The test idea signals for PID thePID model, definingFor the practical controlleruse, andtuning being implemented tuning is and follows as PID controller. For practical use, tuning formulae for of tuning is proposed proposed and studied. studied. The idea follows the as controller. formulae for early result on closed-loop identification and method isthe to typical process models was formed in tables by Chien and is proposed and studied. The and idea method follows is as PID process controller. For was practical use, tuningby formulae for tuning result on closed-loop to typical process models was formed in (2001) tables by Chien and and earlynonlinear result on PID closed-loop identification and method isthe to typical models formed tables Chien use rules toidentification obtain informativity without Fruehauf (1990). Furthermore, Zhuin extended the early early result on PID closed-loop identification and method is to typical process models was formed in tables by Chien and use nonlinear rules to obtain informativity without Fruehauf (1990). Furthermore, Zhu (2001) extended the use nonlinear PID rules to obtain informativity without Fruehauf (1990). Furthermore, Zhu (2001) extended the external excitation. By applying the approach, not only IMC tuning rules to nonlinear PID controller, and an error nonlinear PID rules to obtainthe informativity without Fruehauf (1990). Furthermore, Zhu (2001) extended the use external excitation. By applying approach, not only IMC tuning rules to nonlinear PID controller, and an made error external excitation. By can applying the approach, not only IMC tuning to nonlinear an error the closed-loop system be identified accurately but bound of therules linear model partPID was controller, estimated, and which external excitation. By applying the approach, not only IMC tuning rules to nonlinear PID controller, and an error the closed-loop system can be identified accurately but bound of the linear model part was estimated, which made the closed-loop system can be identified during accurately but bound of the linearpossible. model part was estimated, which made also the control performance is improved the test. the robust tuning the closed-loop system can be identified accurately but bound of the linear model part was estimated, which made also the control performance is improved during the test. the robust tuning possible. also the control is improved the test. the robust tuning possible. IMC-based PID performance tuning rules are applied toduring determine the the control performance is improved during the test. the robust tuning possible. Model identification using closed-loop test data is pre- also IMC-based PID tuning rules are applied to determine the IMC-based PID tuning rules are applied to determine the PID parameters after the model is identified. Model identification using closed-loop test data is prePID tuning rules are is applied to determine the Model identification closed-loop test data pre- IMC-based ferred for stable andusing safe process operation. Theis early PID parameters after the model identified. PID parameters after the model is identified. Model for identification closed-loop test data pre- PID parameters after the model is identified. ferred for stable and andusing safe process process operation. Theis early early ferred stable safe operation. The The rest of the paper is organised as follows: informativity work on closed-loop identification, Box and MacGregor ferred for stable and safe process operation. The early The restcontrolled of the the paper paper is organised organised as follows: follows: informativity work on closed-loop identification, Box and MacGregor rest of is as informativity work closed-loop identification, Box andproperty MacGregor for PID closed-loop system is studied in section (1974,on 1976), introduced the identifiability for The The restcontrolled of the paper is organised as follows: informativity work on closed-loop identification, Box andproperty MacGregor for PID closed-loop system is studied in (1974, 1976), introduced the identifiability property for for PID controlled closed-loop system is studied in section section (1974, 1976), introduced the identifiability for 2; the method to obtain informativity is discussed in processes operation in closed-loop operation. A similar, for PID controlled closed-loop system is studied in section (1974, 1976), introduced the identifiability property for 2; the method to obtain informativity is discussed in processes operation in closed-loop operation. A similar, 2; the method to obtain is discussed processes operation in closed-loop A similar, 3; IMC-based PID informativity tuning rules are presented in more general result was considered operation. in Gustavsson et al. section 2; the method to obtain informativity is discussed in processes operation in closed-loop operation. A similar, section 3; IMC-based PID tuning rules are presented more general result was considered in Gustavsson et al. 3; simulations IMC-based PID tuning are5;presented more was identifiability considered in has Gustavsson et al. section 4; are given in rules section section 6 in is (1977).general Later, result the term become inforsection 3; IMC-based PID tuning rules are presented in more general result was considered in Gustavsson et al. 4; (1977). Later, Later, the term termInidentifiability identifiability has become infor4; simulations simulations are are given given in in section section 5; 5; section section 66 is is (1977). the become the conclusion. mativity (Ljung,1999). a closed-loophas test, when inforthere section section 4; simulations are given in section 5; section 6 is (1977). Later, the termInidentifiability has become informativity (Ljung,1999). closed-loop test, whenat there the conclusion. conclusion. mativity (Ljung,1999). aa closed-loop test, when there is no external excitationIn but there is disturbance the the mativity (Ljung,1999). Inbut a closed-loop test, whenatthere is no external external excitation there is disturbance disturbance the the conclusion. is no excitation is the output, it is shown that ifbut the there controller is of higherat order is no external excitation but there is disturbance at the output, it is shown that if the controller is of higher order output, is shown that significant if the controller is of higher order than theitprocess or with nonlinearities, then the output, itprocess is shown that significant if the controller is of higher order than the process or with significant nonlinearities, then the than the or with nonlinearities, then the data is informative and it is possible to obtain accurate than the process or with nonlinearities, then the data is informative informative and significant it is is possible possible to obtain obtain accurate accurate data is and it to ⋆ Thisis work is supported by 973 Program of China (No. data informative and it is possible to obtain accurate ⋆
This supported by 973 of (No. ⋆ 2012CB720500) by National Foundation of China This work work is isand supported by Science 973 Program Program of China China (No.( ⋆ This work is supported by 973 Program of China (No.(( 2012CB720500) and by National Science Foundation of China No.61673343). 2012CB720500) and by National Science Foundation of China 2012CB720500) No.61673343). No.61673343). and by National Science Foundation of China ( No.61673343). Copyright © 2017 IFAC 4068 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017 IFAC 4068 Copyright ©under 2017 responsibility IFAC 4068Control. Peer review of International Federation of Automatic Copyright © 2017 IFAC 4068 10.1016/j.ifacol.2017.08.713
Proceedings of the 20th IFAC World Congress 3996 Yan Wengang et al. / IFAC PapersOnLine 50-1 (2017) 3995–4000 Toulouse, France, July 9-14, 2017
a Box-Jenkins model structure. Write the estimated model in the form: ˆ Cˆ B (6) y(t) = u(t) + ε(t) ˆ Aˆ D
The prediction error is given by: ˆ ˆ D B ε(t) = [y(t) − u(t)] Cˆ Aˆ
Fig. 1. Closed-loop process 2. INFORMATIVITY FOR PID CONTROLLED CLOSED-LOOP SYSTEMS
The asymptotic loss function associated with the PEM is given by:
2.1 Background information Given a closed-loop process as shown in Fig.1, the process is described as: C(q −1 ) B(q −1 ) u(t) + e(t) (1) y(t) = −1 A(q ) D(q −1 ) where A(q −1 ) = 1 + a1 q −1 + · · · + ana q −na , ana ̸= 0 B(q −1 ) = b1 q −1 + · · · + bnb q −nb , bnb ̸= 0 C(q −1 ) = 1 + c1 q −1 + · · · + cnc q −nc , cnc ̸= 0 D(q −1 ) = 1 + d1 q −1 + · · · + dnd q −nd , dnd ̸= 0
where S(q −1 ) = s0 + s1 q −1 + · · · + sns q −ns , sns ̸= 0 R(q −1 ) = 1 + r1 q −1 + · · · + rnr q −nr , rnr ̸= 0
where E(•) denotes the expectation. Note that ε(t) is a stationary process, then we have: [ ] [ ] T T V (θ) = E ε(t) ε(t) ≥ E e(t) e(t) (9) ε(t) =
(3)
(4)
nr and ns are the orders of R-polynomial and Spolynomial. Note that the feedback (3) does not include any external signal. For simplicity of presentation, the backward shift operator q −1 will be dropped in the following sections. From (1) and (3), the transfer function of the closed-loop system is readily found: ARC e(t) (5) y(t) = D(AR + BS) We make the following assumptions: A1 A2 A3 A4 A5
(8)
It follows easily from (3) and (7) that:
The system is controlled under the following feedback: S(q −1 ) y(t) R(q −1 )
N 1 ∑ T E[ε(t) ε(t)] N →∞ N t=1
V (θ) = lim
(2)
na , nb , nc and nd are the orders of corresponding polynomials. The process described in (1) is a Box-Jenkins model. Denote u(t) and y(t) as the input and output of the process; r(t) is the external reference signal. The disturbance signal e(t) is assumed to be a white noise signal. The notation q −1 means the backward shift operator, i.e. q −1 y(t) = y(t − 1). u(t) = −
(7)
ˆ + BS) ˆ D(AR y(t) ˆ Cˆ AR
(10)
Assume that the global minimum of the loss function can be achieved namely the equality of (9) is satisfied, it is required that: ε(t) = e(t)
(11)
Comparing (5) and (10), it can be seen that (11) is equivalent to: ˆ AR ˆ + BS) ˆ D( D(AR + BS) (12) = ˆ ˆ AC AC Result 1. Consider the system given by (1) with the controller (3) and the system (1) is identified by using a direct PEM approach. Assumptions A1-A5 are fulfilled and the global minimum of the loss function can be achieved. Assume further that AC and D(AR + BS) are coprime. Then the sufficient condition for informativity of the close-loop system is: max{nr − nb , ns − na } ≥ nd
(13)
The proof of Result 1 can be done in the same way as that in S¨oderstr¨om and Stoica (1989) except that the open loop process there is an autoregressive moving average model with exogenous input (ARMAX) model. A similar result can be seen in Gevers et al. (2008). To summarize, if there is no external exciting signal, the degree condition (13) implies that the controller order must be large enough to guarantee the informativity of the system.
There is no common factor to A, B, C and D. There is no common factor to R and S. A and C have all their zeros inside the unit circle. The closed loop system (5) is stable. The integers na , nb , nc and nd are known.
2.3 PID controller
2.2 Informativity condition for closed-loop system The system (1) is assumed to be identified by using a direct prediction error method (PEM) approach applied to
In this paper, we focus on the PID controller. In general, the discrete PID regulator can be written as: ∑ u(t) = Kp ef (t) + KI ef (t) + KD [ef (t) − ef (t − 1)](14)
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ů
3997
ů
Fig. 2. Closed-loop system with shifted regulator where ef (t) denotes the feedback error,i.e. ef (t) = r(t) − y(t). Substituting t for t − 1 in (14), we have: ∑ u(t−1)=Kp ef(t−1)+KI ef (t−1)+KD[ef(t−1)−ef (t−2)](15)
It follows from (14) and (15) that: u(t) =
(Kp + KI + KD )−(Kp +2KD )q −1 +KD q −2 ef (t)(16) 1 − q −1
Fig. 3. IMC structure Case 1: If the closed-loop system has a slow response, then the following shifting rule is used to speed up the closedloop response: 1 0 KD , if |ef (t)|> δ KP =MP1 KP0 , KI =MI1 KI0 , KD=MD 1 MP1 > 1, MI1 > 1, 1 > MD >0 (18) K = K 0 , K = K 0 , K = K 0 , otherwise P I D P I D
Case 2: If the closed-loop system has an oscillating reComparing (16) with (3), if there is no external input, we sponse, then the following shifting rule is used to slow down the closed-loop response: have: 2 0 KD , if |ef (t)|> δ KP =MP2 KP0 , KI =MI2 KI0 , KD=MD S(q −1 ) = (Kp + KI +KD ) − (Kp + 2KD )q −1 + KD q −2 2 2 2 (17) −1 −1 0 < M < 1, 0 < M < 1, 1 < MD (19) P I R(q ) = 1 − q K = K 0 , K = K 0 , K = K 0 , otherwise P I D P I D
It can be seen from (17) that the degrees nr and ns are respectively 1 and 2. When the order of the open-loop process is equal to or bigger than 2, the closed-loop system may not be identifiable as the sufficient informativity condition (13) is not fulfilled. To apply model-based PID tuning rules, the informativity problem must be solved. 3. ACHIEVING INFORMATIVITY IN CLOSED-LOOP TEST
According to S¨oderstr¨ om and Stoica (1989) and Ljung (1999), there are many ways to obtain informativity if the input is determined through a linear low-order feedback from the output, for example: (1) To add an additional (external) exciting signal to the input.
i , i = 1, 2 and δ are all positive numbers. where Mpi , MIi , MD i , i = 1, 2 should be The determination of Mpi , MIi , MD in the defined limits. Note that the defined limits in (18) and (19) just offer a direction for the selection of these parameters. Under the direction, these parameters are adjusted to make sure the control performance is better than before. The threshold δ is set in proportion to the standard deviation of the feedback error so that the regulator would shift from one to another.
In this approach, we aim to achieve the informativity and simultaneously improve the control performance. Note that the main goal of this section is to achieve the informativity and identify the model. As for the proof of the informativity for closed-loop system with nonlinear regulator, it can be seen in S¨oderstr¨om and Stoica (1989). 4. IMC-BASED PID TUNING RULES
(2) To add a time delay in the feedback regulator. (3) To use a nonlinear controller. The first method is used by control engineers. This method is costly as the use of external excitations will have a bad impact on control performance. Similarly, the control performance will degrade when adding delays in the controller. Therefore, the disadvantage of method (1) and (2) is that the informativity is obtained at the cost of control performance.
There are many model-based PID tuning rules, such as dominant pole placement, optimization by minimizing integral square error (ISE) or integral absolute error (IAE), and internal model control (IMC) tuning; see ˚ Astr¨om and H¨agglund (1995). Here we will use the IMC tuning rules introduced by Rivera et al. (1986). The block diagram for IMC structure is shown in Fig.3.
In this work, the third way is adopted to achieve the informativity in closed-loop identification, namely, to use a nonlinear controller. The nonlinearity of the controller is obtained by using a linear regulator that shifts between different settings which is shown in Fig.2.
The IMC-based PID tuning rules can be divided into three steps:
Specially, for a PID regulator, the following shifting rules are used.
ˆ p+ contains all the time delay and rightsuch that G ˆ −1 is stable and does not half plane zeros; consequently G p− involve predictors.
0 Denote KP0 , KI0 and KD as the existing PID parameters.
Step 1: Factor the model ˆp = G ˆ p+ G ˆ p− G
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Proceedings of the 20th IFAC World Congress 3998 Yan Wengang et al. / IFAC PapersOnLine 50-1 (2017) 3995–4000 Toulouse, France, July 9-14, 2017
Step 2: Define the IMC controller by
where F is a low-pass filter and given as 1 F = τs + 1
13
12
(22)
The parameter τ is time constant which determines the speed of the response. Step 3: IMC implemented as PID controller. The relationship between IMC controller Q and classical controller Gc can be deduced as: ˆ −1 G Q p− Gc = (23) = −1 ˆ ˆ p+ 1 − Gp Q F −G In this work, Gc is a PID controller and the tuning formulae for typical process models are available in tables; see, e.g., Chien and Fruehauf (1990). Therefore, when a process model is identified, it is straightforward to obtain PID parameters by following the 3 steps. For controller tuning, the user only needs to specify the time constant τ of the filter, or the desired speed of the closed-loop system. In general, a large time constant leads to a slow response and a more robust controller; a small time constant leads to a fast response, but a less robust controller. 5. SIMULATION In order to confirm the effectiveness of the approach proposed above, a second-order Box-Jenkins process is investigated. The process is given as y(t) =
0.1q −1 + 0.32q −2 1 + 0.23q −1 u(t) + e(t) (24) −1 −2 1 − 1.2q + 0.35q 1 − 0.9q −1
The simulation is divided into 3 periods. The control performance of the existing PID controller is tested using a step signal added to the reference signal in first period. Then, according to the control performance, the proper shifting rule of Section 3 will be applied to the closedloop system. Identification for the system is done using the input-output data of the second period. Based on the identified model, the IMC-based PID tuning rules are applied in the third period. In this period, in order to test the control performance of the newly-tuned PID controller, step signals are added at the reference signal r(t) and then at input u(t). In total 100 simulations are run and the performance of model identification and of the control tuning are verified in a statistic manner and compared with those when initial linear PID controller is used. Case 1 : The existing closed-loop system has a slow response. The initial PID parameters are set as 0 KP0 = 0.05, KI0 = 0.01, KD =0
(25)
11
10
9
8
200
400
600 800 Sample(seconds)
1000
1200
1400
Fig. 4. System output in first and second period, the green solid lines are the responses under the linear PID controller, the red dashed lines are those under shifted PID controller, only 10 of the 100 responses are plotted Fig.4 shows the system outputs in the first and second period of the simulation, where the green lines are the step responses of the linear PID controller, and the red dashed lines are those of the shifted PID controller. Only 10 of the 100 simulations are plotted in the figure in order to avoid a too messy picture. The first period lasts from 0 to 400s, and starting from 200s the corresponding shifted rules are applied. From this period, it can be seen that the initial closed-loop system has a slow response and after the shifted rules are applied, the dynamic performance is much better than that of the initial PID controlled system. Input-output data of the second period from 400s to 1400s are used for the identification of the open loop process. Table 1 contains the statistics of models identified using data from shifted PID and those from the linear PID; their model step responses are shown in Fig.5. Table 1. Result of identified parameters
The white noise signal e(t) has zero mean and a variance of 0.01.
and the corresponding shifting rule is given as { 0 , if |ef (t)|> 0.2 KP =6KP0 , KI =4KI0 , KD=KD 0 0 0 KP = KP , KI = KI , KD = KD , otherwise
Setpoint Intial PID control Shitfed PID control
(21)
Output
Q = G−1 p− F
Parameters True value a1 a2 b1 b2 c1 d1
-1.2 0.35 0.1 0.32 0.23 -0.9
Identified results under initial PID control -0.97±0.361 0.08±0.305 -0.35±42.216 1.41±39.016 0.15±0.200 -0.94±0.006
Identified results under shifted PID control -1.19±0.151 0.33±0.100 0.11±0.156 0.31±0.120 0.24±0.060 -0.89±0.020
The results of Table 1 and Fig.5 imply that the system under initial linear PID control is not identifiable, which is in agreement with the theory result that if the condition (13) is not fulfilled the informativity of the closed-loop system will not be guaranteed. Meanwhile, from Table 1 and Fig.5, it can be seen that the open loop process can be identified accurately when the shifted PID is used. As for the control performance in this period, the advantage of the shifted PID controller is qualified by ISE as shown in Fig.6. One can see that the shifted PID not only achieves informativity in closed-loop identification, but also has much higher control performance during the identification test. In the third period, the IMC-based PID tuning rule is used to obtain the new PID parameters and they are used in the closed-loop control. Models identified from the linear
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Step response
10
13
True process Identified models (Intial PID control) Identified models (Shitfed PID control)
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Setpoint Intial PID control Shitfed PID control
12
Output
Amplitude
6
4
11
10
2
9
0
−2 0
10
20 30 Time(seconds)
40
8
50
Fig. 5. Step response of the identified models, the green dashed lines are the step responses under the linear PID controller, the red dashed lines are those under shifted PID controller, only 10 of the 100 responses are plotted 90
ISE
600 800 Sample(seconds)
1000
1200
1400
Fig. 8. System output in first and second period, the green solid lines are the responses under the linear PID controller, the red dashed lines are those under shifted PID controller, only 10 of the 100 responses are plotted
Case 2 : The closed-loop has an oscillating response. The initial PID parameters are set as 0 KP0 = 1, KI0 = 0.3, KD =0
70 60
and the corresponding shifting rule is given as { 0 KP =0.25KP0 , KI =0.4KI0 , KD=KD , if |ef (t)|> 0.2 (26) 0 0 0 KP = KP , KI = KI , KD = KD , otherwise
50 40 30 0
20
40
Simulation
60
80
As shown in Fig.8, the closed-loop system under existing PID control has both over-shooting and oscillation. The dynamic performance is improved when the shifted PID controller is applied from 200s. Identification period lasts from 401s to 1400s. The estimated parameters and step response of the identified models are shown in Table 2 and Fig.9.
100
Fig. 6. ISE comparison in second period Output under IMC−based PID control 13
12
Output
400
the control performance of the newly-tuned PID controller using the shifted PID controller is very good.
Initial PID control Shifted PID control
80
200
Setpoint Output(initial PID) Output(shifted PID)
Table 2. Result of identified parameters Parameters True value
11
10
a1 a2 b1 b2 c1 d1
9
8
1450 1500 1550 1600 1650 1700 1750 1800 1850 1900 Sample(seconds)
Fig. 7. Simulation of the third period, the green dashed lines are the outputs of newly-tuned PID controlled system using the linear PID controller, the red solid lines are those under shifted PID controller, only 10 of the 100 responses are plotted PID data and from shifted PID data are used in the PID tuning and their performances are compared. Models from the data under linear PID control are poor, which result in many unstable simulations as shown in Fig.7. To check the setpoint tracking, a step signal is added to the reference signal r(t) from 1500s to 1600s; and to test the disturbance reduction of the closed-loop system, a step signal with amplitude 20% of u(t) in steady state, is added to the input u(t) starting from 1700s. From Fig.7, it is obvious that
-1.2 0.35 0.1 0.32 0.23 -0.9
Identified results under initial PID control -1.04±0.086 0.14±0.082 0.03±0.026 0.43±0.040 0.11±0.053 -0.96±0.003
Identified results under shifted PID control -1.20±0.128 0.35±0.126 0.11±0.031 0.31±0.122 0.23±0.070 -0.88±0.030
Similarly, it can be seen that by applying the shifted PID controller the closed-system can be identified accurately. The ISE comparison is shown in Fig.10, which indicates the shifted PID controller has a better control performance as well. In third period, the IMC-based PID tuning rules are applied. The tracking property and the disturbance reduction are tested the same way as Case 1. Good control performance of newly-tuned PID controller is shown in Fig.11. 6. CONCLUSIONS A method of closed-loop identification based PID tuning is developed where no external excitation is used in closedloop test. To achieve the informativity, a shifted PID
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Proceedings of the 20th IFAC World Congress 4000 Yan Wengang et al. / IFAC PapersOnLine 50-1 (2017) 3995–4000 Toulouse, France, July 9-14, 2017
mativity of data for model identification in normal closedloop operation. The method can be easily generalized for non PID controllers and for multivariable processes.
Step response
10
True process Identified models (Intial PID control) Identified models (Shitfed PID control)
8
REFERENCES
Amplitude
6
4
2
0
−2 0
10
20 30 Time(seconds)
40
50
Fig. 9. Step response of the identified models, the green dashed lines are the responses under the linear PID controller, the red dashed lines are those under shifted PID controller, only 10 of the 100 responses are plotted 70
Initial PID control Shifted PID control
65 60
ISE
55 50 45 40 35 0
20
40
Simulation
60
80
100
Fig. 10. ISE comparison in second period Output under IMC−based PID control
13
Setpoint Output(initial PID) Output(shifted PID)
12.5 12 11.5
Output
11 10.5 10 9.5 9 8.5 8
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1500
1550
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1650
1700
Sample(seconds)
1750
1800
1850
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˚ Astr¨om, K.J. and H¨agglund, T. (1995). PID controllers: Theory, design and tuning. Instrument Society of America Research Triangle Park Nc. Box, G. and MacGregor, J. (1974). The analysis of closed loop dynamic-stochastic systems. Technometrics, 16(3), 391–398. Box, G. and MacGregor, J. (1976). Parameter estimation with closed-loop operating data. Technometrics, 18(4), 371–380. Chien, I.L. and Fruehauf, P. (1990). Consider IMC tuning to improve controller performance. Chemical Engineering Progress, 86(10), 33–41. Gevers, M., Bazanella, A., and Miˇskovi´c, L. (2008). Informative data: How to get just sufficiently rich? In Proceedings of the IEEE Conference on Decision and Control, 1962–1967. Gevers, M., Bazanella, A.S., Bombois, X., and Miˇskovi´c, L. (2010). Identification and the information matrix: How to get just sufficiently rich? IEEE Transactions on Automatic Control, 54(12), 2828–2840. Gustavsson, I., Ljung, L., and S¨oderstr¨om, T. (1977). Survey paper: Identification of process in closed loop identifiability and accuracy aspects. Automatica, 13(1), 59–75. Ljung, L. (1999). System identification. Theory for the user. Prentice Hall PTR, Upper Saddle River, NJ, United States of America, 2nd edition. Qin, S.J. and Badgwell, T.A. (2003). A survey of industrial model predictive control technology. Control Engineering Practice, 11(7), 733–764. Rivera, D., Morari, M., and Skogestad, S. (1986). Internal model control: 4. pid controller design. Industrial & Engineering Chemistry Process Design & Development, 25(1), 252–265. S¨oderstr¨om, T. and Stoica, P. (1989). System identification. Prentice Hall, New York, NY, Unitied States of America. Zhu, Y. (2001). Multivariable System Identification for Processes Control. Elsevier Science, Oxford,London. Zhu, Y., Patwardhan, R., Wagner, S.B., and Zhao, J. (2013). Toward a low cost and high performance MPC: The role of system identification. Computers & Chemical Engineering, 51(14), 124–135.
Fig. 11. Simulation of the third period, the green dashed lines are the outputs of newly-tuned PID controlled system using the linear PID controller, the red solid lines are those under shifted PID controller, only 10 of the 100 responses are plotted regulator is applied. Using the approach, not only the closed-loop system can be identified accurately but also the control performance is improved during the test. IMCbased PID tuning rules are applied to determine the PID parameters using the identified model. The effectiveness of the method is confirmed by simulations. Based on the result of this study, we recommend that industrial PID controllers be implemented using the shifted PID controllers as proposed here and this will ensure the infor4073
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IFAC PapersOnLine 50-1 (2017) 3995–4000 Closed-loop Closed-loop Identification Identification based based PID PID Closed-loop Identification based PID⋆⋆ Tuning without External Excitation Tuning without without External External Excitation Excitation ⋆ Tuning
Yan Wengang, Zhu Yucai, Zhao Jun Yan Yan Wengang, Wengang, Zhu Zhu Yucai, Yucai, Zhao Zhao Jun Jun Yan Wengang, Zhu Yucai, Zhao Jun State Key Laboratory of Industrial Control Technology, College of State Laboratory of Control College State Key Key Laboratory of Industrial IndustrialZhejiang Control Technology, Technology, College of of Control Science and Engineering, University, Hangzhou, State Key Laboratory of IndustrialZhejiang Control Technology, College of Control Science and Engineering, University, Control Science and310027, Engineering, Zhejiang University, Hangzhou, Hangzhou, P.R.China (e-mail: Control Science and310027, Engineering, Zhejiang University, Hangzhou, P.R.China (e-mail: 310027, P.R.China (e-mail: [email protected];[email protected];[email protected];). 310027, P.R.China (e-mail: [email protected];[email protected];[email protected];). [email protected];[email protected];[email protected];). [email protected];[email protected];[email protected];). Abstract: Identification based PID tuning is studied in this work. A method is proposed Abstract: Identification based tuning this work. A is Abstract: Identification based PID tuning is is studied studied inThe thisbasic work.idea A method method is proposed proposed to achieve the informativity of PID the closed-loop system.in is shifting the PID Abstract: Identification based PID tuning is studied inThe thisbasic work.idea A method is proposed to achieve the informativity of the closed-loop system. is shifting the to achieve the informativity of the closed-loop system. The basic idea is shifting the PID PID parameters appropriately according to the amplitude of the feedback error. In this way, not to achieve the informativity of the to closed-loop system. Thefeedback basic idea is shifting the PID parameters appropriately according the of In not parameters appropriately according the amplitude amplitude of the the feedback error. In this this way, way, not only the closed-loop system can be to identified accurately without usingerror. test signals, but also parameters appropriately according to the amplitude of the feedback error. In this way, not only the closed-loop system can be identified accurately without using test signals, but also onlycontrol the closed-loop system canimproved be identified accurately testAn signals, but also the performance can be during the data without collectionusing period. internal model only the closed-loop system can be identified accurately without using test signals, but also the control performance can be improved during the data collection period. An internal model the control performance can beisimproved collection period. An internal model control (IMC) design method applied toduring tune the data PID parameters. The effectiveness of the the control performance can beis the data collection period. An internal model control (IMC) design method applied to tune PID The of control (IMC) design method isimproved applied toduring tune the PID parameters. parameters. The effectiveness effectiveness of the the proposed approach is illustrated by simulations. control (IMC) design method is by applied to tune the PID parameters. The effectiveness of the proposed approach is simulations. proposed approach is illustrated illustrated by simulations. proposed approach is illustrated by of simulations. © 2017, IFAC (International Federation Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: closed-loop system; PID; informativity; control performance; IMC tuning rules. Keywords: closed-loop closed-loop system; system; PID; PID; informativity; informativity; control control performance; performance; IMC IMC tuning tuning rules. rules. Keywords: Keywords: closed-loop system; PID; informativity; control performance; IMC tuning rules. 1. INTRODUCTION process model (Ljung,1999). It has also been shown that, 1. INTRODUCTION INTRODUCTION process model It been that, 1. process model (Ljung,1999). (Ljung,1999). It has has also also been shown shown that, if a reference signal is sufficiently persistently exciting, 1. INTRODUCTION process model (Ljung,1999). It has also been shown that, if a reference signal is sufficiently persistently exciting, if a reference signal is sufficiently persistently exciting, it is possible to identify the process from closedAlthough model predictive control (MPC) is becoming then if a reference signal is sufficiently persistently exciting, then it is possible to identify the process from closedAlthough model predictive controland (MPC) isof becoming becoming then data it is possible to identify process from quanticlosedAlthough model control (MPC) (Ljung,1999). More the recently, general more popular in predictive process control a lot is work for loop then it is possible to identify process from quanticlosedAlthough model predictive controland (MPC) isof becoming (Ljung,1999). More recently, general more popular in process control and lot work for loop data data (Ljung,1999). Moreforthe recently, general more popular control aa lot work tative informative conditions identification of quanticlosedMPC such as in Qinprocess and Badgwell (2003) andof Zhu et for al. loop loop data (Ljung,1999). More recently, general quantimore popular in process control and a lot of work for tative informative conditions for identification of MPC such as Qin and Badgwell (2003) and Zhu et al. tativesystem informative conditionsinforGevers identification of closedclosedMPC Qinpublished, and Badgwell Zhu al. loop were presented et al. (2008) and (2013) such has as been most (2003) control and loops areetstill informative conditionsin identification of closedMPC such Qinpublished, and Badgwell (2003) and Zhu etstill al. tative loop system were Gevers et (2008) and (2013) has as been published, most control loops are still loop system were presented presented informodels Gevers et al. al.two (2008) and (2013) has been control loops Gevers et al. (2010) for different using different PID controllers. PID tuning most is also part of the are pre-test system were presented in models Gevers using et al.two (2008) and (2013) has been PID published, control loops are still loop et (2010) for different PID controllers. PID tuning most isgood also part of of the pre-test Gevers et al. al.an (2010) for different different models using two different PID controllers. tuning is also part pre-test approaches: expectation-based and information matrixin a MPC project. Therefore, tuning ofthe PID loops Gevers Gevers et al. (2010) for different models using two different PID controllers. PID tuning is also part of the pre-test approaches: an expectation-based and information matrixin a MPC project. Therefore, good tuning of PID loops approaches: an expectation-based and information matrixin a MPC project. to Therefore, of PIDofloops approaches. is very important maintain good goodtuning performance the based an expectation-based and information matrixin very a MPC project. to Therefore, good of PIDof based is very important to maintain goodtuning performance ofloops the approaches: based approaches. approaches. is important maintain performance the overall process control system. good based approaches. is very important to maintain good performance of the Most identification based PID tuning methods use test overall process process control control system. system. overall based tuning test ˚ Most identification identification based PID PID test, tuning methods use test overall signals during the identification see,methods e.g., Zhuuse (2001). A str¨omprocess and H¨acontrol gglund system. (1995) contains a good summary Most Most identification based PID tuning methods use test ˚ signals during the identification test, see, e.g., Zhu (2001). A str¨ o m and H¨ a gglund (1995) contains a good summary ˚ signals during the identification test, see,operation, e.g., Zhu (2001). A str¨ o m and H¨ a gglund (1995) contains a good summary However, Test signals disturb process which of PID tuning rules, among which IMC-based PID tuning ˚ signals during the identification test, see, e.g., Zhu (2001). A str¨ o m and H¨ a gglund (1995) contains a good summary signals process operation, which of PID PID tuning rules, among which IMC-based PID tuning tuning However, However, Test signals disturb disturb process operation, which of which IMC-based PID is a cost Test in production unit. For reducing the cost of rules aretuning widelyrules, usedamong in process control. The methodology signals disturb process operation, which of PID tuning rules, among which IMC-based PID tuning However, in unit. For of rules are widelybased usedon in IMC, process control. The methodology is aa cost cost Test in production production unit. For reducing the cost of rules are widely used in process control. methodology identification, it would be ideal toreducing use no the testcost signal of PID tuning which was The firstly introduced is is a cost in production unit. For reducing the cost of rules are widely used in process control. The methodology identification, it would be ideal to use no test signal of PID tuning based on IMC, which was firstly introduced identification, it would be ideal no test signal of tuning based on IMC, which the identification test. In to thisuse paper, a method by PID Rivera et al. (1986), consists of was threefirstly steps:introduced factoring during identification, it would be ideal to use no test signal of PID tuning based on IMC, which was firstly introduced during the identification test. In this paper, a method by Rivera et al. (1986), consists of three steps: factoring during the identification test.using In this method by al. (1986), consists ofand three steps: factoring of closed-loop test without test paper, signalsa for PID the Rivera model, et defining the controller being implemented the identification test.using In this a for method by Rivera et al. (1986), consists ofand three steps: factoring during closed-loop test test signals PID the model, defining the practical controller and being implemented of closed-loop test without without using test paper, signals for PID the model, defining the controller being implemented tuning is proposed and studied. The idea follows the as PID controller. For use, tuning formulae for of closed-loop test without using The test idea signals for PID thePID model, definingFor the practical controlleruse, andtuning being implemented tuning is and follows as PID controller. For practical use, tuning formulae for of tuning is proposed proposed and studied. studied. The idea follows the as controller. formulae for early result on closed-loop identification and method isthe to typical process models was formed in tables by Chien and is proposed and studied. The and idea method follows is as PID process controller. For was practical use, tuningby formulae for tuning result on closed-loop to typical process models was formed in (2001) tables by Chien and and earlynonlinear result on PID closed-loop identification and method isthe to typical models formed tables Chien use rules toidentification obtain informativity without Fruehauf (1990). Furthermore, Zhuin extended the early early result on PID closed-loop identification and method is to typical process models was formed in tables by Chien and use nonlinear rules to obtain informativity without Fruehauf (1990). Furthermore, Zhu (2001) extended the use nonlinear PID rules to obtain informativity without Fruehauf (1990). Furthermore, Zhu (2001) extended the external excitation. By applying the approach, not only IMC tuning rules to nonlinear PID controller, and an error nonlinear PID rules to obtainthe informativity without Fruehauf (1990). Furthermore, Zhu (2001) extended the use external excitation. By applying approach, not only IMC tuning rules to nonlinear PID controller, and an made error external excitation. By can applying the approach, not only IMC tuning to nonlinear an error the closed-loop system be identified accurately but bound of therules linear model partPID was controller, estimated, and which external excitation. By applying the approach, not only IMC tuning rules to nonlinear PID controller, and an error the closed-loop system can be identified accurately but bound of the linear model part was estimated, which made the closed-loop system can be identified during accurately but bound of the linearpossible. model part was estimated, which made also the control performance is improved the test. the robust tuning the closed-loop system can be identified accurately but bound of the linear model part was estimated, which made also the control performance is improved during the test. the robust tuning possible. also the control is improved the test. the robust tuning possible. IMC-based PID performance tuning rules are applied toduring determine the the control performance is improved during the test. the robust tuning possible. Model identification using closed-loop test data is pre- also IMC-based PID tuning rules are applied to determine the IMC-based PID tuning rules are applied to determine the PID parameters after the model is identified. Model identification using closed-loop test data is prePID tuning rules are is applied to determine the Model identification closed-loop test data pre- IMC-based ferred for stable andusing safe process operation. Theis early PID parameters after the model identified. PID parameters after the model is identified. Model for identification closed-loop test data pre- PID parameters after the model is identified. ferred for stable and andusing safe process process operation. Theis early early ferred stable safe operation. The The rest of the paper is organised as follows: informativity work on closed-loop identification, Box and MacGregor ferred for stable and safe process operation. The early The restcontrolled of the the paper paper is organised organised as follows: follows: informativity work on closed-loop identification, Box and MacGregor rest of is as informativity work closed-loop identification, Box andproperty MacGregor for PID closed-loop system is studied in section (1974,on 1976), introduced the identifiability for The The restcontrolled of the paper is organised as follows: informativity work on closed-loop identification, Box andproperty MacGregor for PID closed-loop system is studied in (1974, 1976), introduced the identifiability property for for PID controlled closed-loop system is studied in section section (1974, 1976), introduced the identifiability for 2; the method to obtain informativity is discussed in processes operation in closed-loop operation. A similar, for PID controlled closed-loop system is studied in section (1974, 1976), introduced the identifiability property for 2; the method to obtain informativity is discussed in processes operation in closed-loop operation. A similar, 2; the method to obtain is discussed processes operation in closed-loop A similar, 3; IMC-based PID informativity tuning rules are presented in more general result was considered operation. in Gustavsson et al. section 2; the method to obtain informativity is discussed in processes operation in closed-loop operation. A similar, section 3; IMC-based PID tuning rules are presented more general result was considered in Gustavsson et al. 3; simulations IMC-based PID tuning are5;presented more was identifiability considered in has Gustavsson et al. section 4; are given in rules section section 6 in is (1977).general Later, result the term become inforsection 3; IMC-based PID tuning rules are presented in more general result was considered in Gustavsson et al. 4; (1977). Later, Later, the term termInidentifiability identifiability has become infor4; simulations simulations are are given given in in section section 5; 5; section section 66 is is (1977). the become the conclusion. mativity (Ljung,1999). a closed-loophas test, when inforthere section section 4; simulations are given in section 5; section 6 is (1977). Later, the termInidentifiability has become informativity (Ljung,1999). closed-loop test, whenat there the conclusion. conclusion. mativity (Ljung,1999). aa closed-loop test, when there is no external excitationIn but there is disturbance the the mativity (Ljung,1999). Inbut a closed-loop test, whenatthere is no external external excitation there is disturbance disturbance the the conclusion. is no excitation is the output, it is shown that ifbut the there controller is of higherat order is no external excitation but there is disturbance at the output, it is shown that if the controller is of higher order output, is shown that significant if the controller is of higher order than theitprocess or with nonlinearities, then the output, itprocess is shown that significant if the controller is of higher order than the process or with significant nonlinearities, then the than the or with nonlinearities, then the data is informative and it is possible to obtain accurate than the process or with nonlinearities, then the data is informative informative and significant it is is possible possible to obtain obtain accurate accurate data is and it to ⋆ Thisis work is supported by 973 Program of China (No. data informative and it is possible to obtain accurate ⋆
This supported by 973 of (No. ⋆ 2012CB720500) by National Foundation of China This work work is isand supported by Science 973 Program Program of China China (No.( ⋆ This work is supported by 973 Program of China (No.(( 2012CB720500) and by National Science Foundation of China No.61673343). 2012CB720500) and by National Science Foundation of China 2012CB720500) No.61673343). No.61673343). and by National Science Foundation of China ( No.61673343). Copyright © 2017 IFAC 4068 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017 IFAC 4068 Copyright ©under 2017 responsibility IFAC 4068Control. Peer review of International Federation of Automatic Copyright © 2017 IFAC 4068 10.1016/j.ifacol.2017.08.713
Proceedings of the 20th IFAC World Congress 3996 Yan Wengang et al. / IFAC PapersOnLine 50-1 (2017) 3995–4000 Toulouse, France, July 9-14, 2017
a Box-Jenkins model structure. Write the estimated model in the form: ˆ Cˆ B (6) y(t) = u(t) + ε(t) ˆ Aˆ D
The prediction error is given by: ˆ ˆ D B ε(t) = [y(t) − u(t)] Cˆ Aˆ
Fig. 1. Closed-loop process 2. INFORMATIVITY FOR PID CONTROLLED CLOSED-LOOP SYSTEMS
The asymptotic loss function associated with the PEM is given by:
2.1 Background information Given a closed-loop process as shown in Fig.1, the process is described as: C(q −1 ) B(q −1 ) u(t) + e(t) (1) y(t) = −1 A(q ) D(q −1 ) where A(q −1 ) = 1 + a1 q −1 + · · · + ana q −na , ana ̸= 0 B(q −1 ) = b1 q −1 + · · · + bnb q −nb , bnb ̸= 0 C(q −1 ) = 1 + c1 q −1 + · · · + cnc q −nc , cnc ̸= 0 D(q −1 ) = 1 + d1 q −1 + · · · + dnd q −nd , dnd ̸= 0
where S(q −1 ) = s0 + s1 q −1 + · · · + sns q −ns , sns ̸= 0 R(q −1 ) = 1 + r1 q −1 + · · · + rnr q −nr , rnr ̸= 0
where E(•) denotes the expectation. Note that ε(t) is a stationary process, then we have: [ ] [ ] T T V (θ) = E ε(t) ε(t) ≥ E e(t) e(t) (9) ε(t) =
(3)
(4)
nr and ns are the orders of R-polynomial and Spolynomial. Note that the feedback (3) does not include any external signal. For simplicity of presentation, the backward shift operator q −1 will be dropped in the following sections. From (1) and (3), the transfer function of the closed-loop system is readily found: ARC e(t) (5) y(t) = D(AR + BS) We make the following assumptions: A1 A2 A3 A4 A5
(8)
It follows easily from (3) and (7) that:
The system is controlled under the following feedback: S(q −1 ) y(t) R(q −1 )
N 1 ∑ T E[ε(t) ε(t)] N →∞ N t=1
V (θ) = lim
(2)
na , nb , nc and nd are the orders of corresponding polynomials. The process described in (1) is a Box-Jenkins model. Denote u(t) and y(t) as the input and output of the process; r(t) is the external reference signal. The disturbance signal e(t) is assumed to be a white noise signal. The notation q −1 means the backward shift operator, i.e. q −1 y(t) = y(t − 1). u(t) = −
(7)
ˆ + BS) ˆ D(AR y(t) ˆ Cˆ AR
(10)
Assume that the global minimum of the loss function can be achieved namely the equality of (9) is satisfied, it is required that: ε(t) = e(t)
(11)
Comparing (5) and (10), it can be seen that (11) is equivalent to: ˆ AR ˆ + BS) ˆ D( D(AR + BS) (12) = ˆ ˆ AC AC Result 1. Consider the system given by (1) with the controller (3) and the system (1) is identified by using a direct PEM approach. Assumptions A1-A5 are fulfilled and the global minimum of the loss function can be achieved. Assume further that AC and D(AR + BS) are coprime. Then the sufficient condition for informativity of the close-loop system is: max{nr − nb , ns − na } ≥ nd
(13)
The proof of Result 1 can be done in the same way as that in S¨oderstr¨om and Stoica (1989) except that the open loop process there is an autoregressive moving average model with exogenous input (ARMAX) model. A similar result can be seen in Gevers et al. (2008). To summarize, if there is no external exciting signal, the degree condition (13) implies that the controller order must be large enough to guarantee the informativity of the system.
There is no common factor to A, B, C and D. There is no common factor to R and S. A and C have all their zeros inside the unit circle. The closed loop system (5) is stable. The integers na , nb , nc and nd are known.
2.3 PID controller
2.2 Informativity condition for closed-loop system The system (1) is assumed to be identified by using a direct prediction error method (PEM) approach applied to
In this paper, we focus on the PID controller. In general, the discrete PID regulator can be written as: ∑ u(t) = Kp ef (t) + KI ef (t) + KD [ef (t) − ef (t − 1)](14)
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Proceedings of the 20th IFAC World Congress Yan Wengang et al. / IFAC PapersOnLine 50-1 (2017) 3995–4000 Toulouse, France, July 9-14, 2017
ů
3997
ů
Fig. 2. Closed-loop system with shifted regulator where ef (t) denotes the feedback error,i.e. ef (t) = r(t) − y(t). Substituting t for t − 1 in (14), we have: ∑ u(t−1)=Kp ef(t−1)+KI ef (t−1)+KD[ef(t−1)−ef (t−2)](15)
It follows from (14) and (15) that: u(t) =
(Kp + KI + KD )−(Kp +2KD )q −1 +KD q −2 ef (t)(16) 1 − q −1
Fig. 3. IMC structure Case 1: If the closed-loop system has a slow response, then the following shifting rule is used to speed up the closedloop response: 1 0 KD , if |ef (t)|> δ KP =MP1 KP0 , KI =MI1 KI0 , KD=MD 1 MP1 > 1, MI1 > 1, 1 > MD >0 (18) K = K 0 , K = K 0 , K = K 0 , otherwise P I D P I D
Case 2: If the closed-loop system has an oscillating reComparing (16) with (3), if there is no external input, we sponse, then the following shifting rule is used to slow down the closed-loop response: have: 2 0 KD , if |ef (t)|> δ KP =MP2 KP0 , KI =MI2 KI0 , KD=MD S(q −1 ) = (Kp + KI +KD ) − (Kp + 2KD )q −1 + KD q −2 2 2 2 (17) −1 −1 0 < M < 1, 0 < M < 1, 1 < MD (19) P I R(q ) = 1 − q K = K 0 , K = K 0 , K = K 0 , otherwise P I D P I D
It can be seen from (17) that the degrees nr and ns are respectively 1 and 2. When the order of the open-loop process is equal to or bigger than 2, the closed-loop system may not be identifiable as the sufficient informativity condition (13) is not fulfilled. To apply model-based PID tuning rules, the informativity problem must be solved. 3. ACHIEVING INFORMATIVITY IN CLOSED-LOOP TEST
According to S¨oderstr¨ om and Stoica (1989) and Ljung (1999), there are many ways to obtain informativity if the input is determined through a linear low-order feedback from the output, for example: (1) To add an additional (external) exciting signal to the input.
i , i = 1, 2 and δ are all positive numbers. where Mpi , MIi , MD i , i = 1, 2 should be The determination of Mpi , MIi , MD in the defined limits. Note that the defined limits in (18) and (19) just offer a direction for the selection of these parameters. Under the direction, these parameters are adjusted to make sure the control performance is better than before. The threshold δ is set in proportion to the standard deviation of the feedback error so that the regulator would shift from one to another.
In this approach, we aim to achieve the informativity and simultaneously improve the control performance. Note that the main goal of this section is to achieve the informativity and identify the model. As for the proof of the informativity for closed-loop system with nonlinear regulator, it can be seen in S¨oderstr¨om and Stoica (1989). 4. IMC-BASED PID TUNING RULES
(2) To add a time delay in the feedback regulator. (3) To use a nonlinear controller. The first method is used by control engineers. This method is costly as the use of external excitations will have a bad impact on control performance. Similarly, the control performance will degrade when adding delays in the controller. Therefore, the disadvantage of method (1) and (2) is that the informativity is obtained at the cost of control performance.
There are many model-based PID tuning rules, such as dominant pole placement, optimization by minimizing integral square error (ISE) or integral absolute error (IAE), and internal model control (IMC) tuning; see ˚ Astr¨om and H¨agglund (1995). Here we will use the IMC tuning rules introduced by Rivera et al. (1986). The block diagram for IMC structure is shown in Fig.3.
In this work, the third way is adopted to achieve the informativity in closed-loop identification, namely, to use a nonlinear controller. The nonlinearity of the controller is obtained by using a linear regulator that shifts between different settings which is shown in Fig.2.
The IMC-based PID tuning rules can be divided into three steps:
Specially, for a PID regulator, the following shifting rules are used.
ˆ p+ contains all the time delay and rightsuch that G ˆ −1 is stable and does not half plane zeros; consequently G p− involve predictors.
0 Denote KP0 , KI0 and KD as the existing PID parameters.
Step 1: Factor the model ˆp = G ˆ p+ G ˆ p− G
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Step 2: Define the IMC controller by
where F is a low-pass filter and given as 1 F = τs + 1
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The parameter τ is time constant which determines the speed of the response. Step 3: IMC implemented as PID controller. The relationship between IMC controller Q and classical controller Gc can be deduced as: ˆ −1 G Q p− Gc = (23) = −1 ˆ ˆ p+ 1 − Gp Q F −G In this work, Gc is a PID controller and the tuning formulae for typical process models are available in tables; see, e.g., Chien and Fruehauf (1990). Therefore, when a process model is identified, it is straightforward to obtain PID parameters by following the 3 steps. For controller tuning, the user only needs to specify the time constant τ of the filter, or the desired speed of the closed-loop system. In general, a large time constant leads to a slow response and a more robust controller; a small time constant leads to a fast response, but a less robust controller. 5. SIMULATION In order to confirm the effectiveness of the approach proposed above, a second-order Box-Jenkins process is investigated. The process is given as y(t) =
0.1q −1 + 0.32q −2 1 + 0.23q −1 u(t) + e(t) (24) −1 −2 1 − 1.2q + 0.35q 1 − 0.9q −1
The simulation is divided into 3 periods. The control performance of the existing PID controller is tested using a step signal added to the reference signal in first period. Then, according to the control performance, the proper shifting rule of Section 3 will be applied to the closedloop system. Identification for the system is done using the input-output data of the second period. Based on the identified model, the IMC-based PID tuning rules are applied in the third period. In this period, in order to test the control performance of the newly-tuned PID controller, step signals are added at the reference signal r(t) and then at input u(t). In total 100 simulations are run and the performance of model identification and of the control tuning are verified in a statistic manner and compared with those when initial linear PID controller is used. Case 1 : The existing closed-loop system has a slow response. The initial PID parameters are set as 0 KP0 = 0.05, KI0 = 0.01, KD =0
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Fig. 4. System output in first and second period, the green solid lines are the responses under the linear PID controller, the red dashed lines are those under shifted PID controller, only 10 of the 100 responses are plotted Fig.4 shows the system outputs in the first and second period of the simulation, where the green lines are the step responses of the linear PID controller, and the red dashed lines are those of the shifted PID controller. Only 10 of the 100 simulations are plotted in the figure in order to avoid a too messy picture. The first period lasts from 0 to 400s, and starting from 200s the corresponding shifted rules are applied. From this period, it can be seen that the initial closed-loop system has a slow response and after the shifted rules are applied, the dynamic performance is much better than that of the initial PID controlled system. Input-output data of the second period from 400s to 1400s are used for the identification of the open loop process. Table 1 contains the statistics of models identified using data from shifted PID and those from the linear PID; their model step responses are shown in Fig.5. Table 1. Result of identified parameters
The white noise signal e(t) has zero mean and a variance of 0.01.
and the corresponding shifting rule is given as { 0 , if |ef (t)|> 0.2 KP =6KP0 , KI =4KI0 , KD=KD 0 0 0 KP = KP , KI = KI , KD = KD , otherwise
Setpoint Intial PID control Shitfed PID control
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Output
Q = G−1 p− F
Parameters True value a1 a2 b1 b2 c1 d1
-1.2 0.35 0.1 0.32 0.23 -0.9
Identified results under initial PID control -0.97±0.361 0.08±0.305 -0.35±42.216 1.41±39.016 0.15±0.200 -0.94±0.006
Identified results under shifted PID control -1.19±0.151 0.33±0.100 0.11±0.156 0.31±0.120 0.24±0.060 -0.89±0.020
The results of Table 1 and Fig.5 imply that the system under initial linear PID control is not identifiable, which is in agreement with the theory result that if the condition (13) is not fulfilled the informativity of the closed-loop system will not be guaranteed. Meanwhile, from Table 1 and Fig.5, it can be seen that the open loop process can be identified accurately when the shifted PID is used. As for the control performance in this period, the advantage of the shifted PID controller is qualified by ISE as shown in Fig.6. One can see that the shifted PID not only achieves informativity in closed-loop identification, but also has much higher control performance during the identification test. In the third period, the IMC-based PID tuning rule is used to obtain the new PID parameters and they are used in the closed-loop control. Models identified from the linear
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Fig. 5. Step response of the identified models, the green dashed lines are the step responses under the linear PID controller, the red dashed lines are those under shifted PID controller, only 10 of the 100 responses are plotted 90
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Fig. 8. System output in first and second period, the green solid lines are the responses under the linear PID controller, the red dashed lines are those under shifted PID controller, only 10 of the 100 responses are plotted
Case 2 : The closed-loop has an oscillating response. The initial PID parameters are set as 0 KP0 = 1, KI0 = 0.3, KD =0
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and the corresponding shifting rule is given as { 0 KP =0.25KP0 , KI =0.4KI0 , KD=KD , if |ef (t)|> 0.2 (26) 0 0 0 KP = KP , KI = KI , KD = KD , otherwise
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As shown in Fig.8, the closed-loop system under existing PID control has both over-shooting and oscillation. The dynamic performance is improved when the shifted PID controller is applied from 200s. Identification period lasts from 401s to 1400s. The estimated parameters and step response of the identified models are shown in Table 2 and Fig.9.
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Fig. 6. ISE comparison in second period Output under IMC−based PID control 13
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the control performance of the newly-tuned PID controller using the shifted PID controller is very good.
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Table 2. Result of identified parameters Parameters True value
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Fig. 7. Simulation of the third period, the green dashed lines are the outputs of newly-tuned PID controlled system using the linear PID controller, the red solid lines are those under shifted PID controller, only 10 of the 100 responses are plotted PID data and from shifted PID data are used in the PID tuning and their performances are compared. Models from the data under linear PID control are poor, which result in many unstable simulations as shown in Fig.7. To check the setpoint tracking, a step signal is added to the reference signal r(t) from 1500s to 1600s; and to test the disturbance reduction of the closed-loop system, a step signal with amplitude 20% of u(t) in steady state, is added to the input u(t) starting from 1700s. From Fig.7, it is obvious that
-1.2 0.35 0.1 0.32 0.23 -0.9
Identified results under initial PID control -1.04±0.086 0.14±0.082 0.03±0.026 0.43±0.040 0.11±0.053 -0.96±0.003
Identified results under shifted PID control -1.20±0.128 0.35±0.126 0.11±0.031 0.31±0.122 0.23±0.070 -0.88±0.030
Similarly, it can be seen that by applying the shifted PID controller the closed-system can be identified accurately. The ISE comparison is shown in Fig.10, which indicates the shifted PID controller has a better control performance as well. In third period, the IMC-based PID tuning rules are applied. The tracking property and the disturbance reduction are tested the same way as Case 1. Good control performance of newly-tuned PID controller is shown in Fig.11. 6. CONCLUSIONS A method of closed-loop identification based PID tuning is developed where no external excitation is used in closedloop test. To achieve the informativity, a shifted PID
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mativity of data for model identification in normal closedloop operation. The method can be easily generalized for non PID controllers and for multivariable processes.
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REFERENCES
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˚ Astr¨om, K.J. and H¨agglund, T. (1995). PID controllers: Theory, design and tuning. Instrument Society of America Research Triangle Park Nc. Box, G. and MacGregor, J. (1974). The analysis of closed loop dynamic-stochastic systems. Technometrics, 16(3), 391–398. Box, G. and MacGregor, J. (1976). Parameter estimation with closed-loop operating data. Technometrics, 18(4), 371–380. Chien, I.L. and Fruehauf, P. (1990). Consider IMC tuning to improve controller performance. Chemical Engineering Progress, 86(10), 33–41. Gevers, M., Bazanella, A., and Miˇskovi´c, L. (2008). Informative data: How to get just sufficiently rich? In Proceedings of the IEEE Conference on Decision and Control, 1962–1967. Gevers, M., Bazanella, A.S., Bombois, X., and Miˇskovi´c, L. (2010). Identification and the information matrix: How to get just sufficiently rich? IEEE Transactions on Automatic Control, 54(12), 2828–2840. Gustavsson, I., Ljung, L., and S¨oderstr¨om, T. (1977). Survey paper: Identification of process in closed loop identifiability and accuracy aspects. Automatica, 13(1), 59–75. Ljung, L. (1999). System identification. Theory for the user. Prentice Hall PTR, Upper Saddle River, NJ, United States of America, 2nd edition. Qin, S.J. and Badgwell, T.A. (2003). A survey of industrial model predictive control technology. Control Engineering Practice, 11(7), 733–764. Rivera, D., Morari, M., and Skogestad, S. (1986). Internal model control: 4. pid controller design. Industrial & Engineering Chemistry Process Design & Development, 25(1), 252–265. S¨oderstr¨om, T. and Stoica, P. (1989). System identification. Prentice Hall, New York, NY, Unitied States of America. Zhu, Y. (2001). Multivariable System Identification for Processes Control. Elsevier Science, Oxford,London. Zhu, Y., Patwardhan, R., Wagner, S.B., and Zhao, J. (2013). Toward a low cost and high performance MPC: The role of system identification. Computers & Chemical Engineering, 51(14), 124–135.
Fig. 11. Simulation of the third period, the green dashed lines are the outputs of newly-tuned PID controlled system using the linear PID controller, the red solid lines are those under shifted PID controller, only 10 of the 100 responses are plotted regulator is applied. Using the approach, not only the closed-loop system can be identified accurately but also the control performance is improved during the test. IMCbased PID tuning rules are applied to determine the PID parameters using the identified model. The effectiveness of the method is confirmed by simulations. Based on the result of this study, we recommend that industrial PID controllers be implemented using the shifted PID controllers as proposed here and this will ensure the infor4073