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Single-loop controller design via IMC principles
Automatica 37 (2001) 2041}2048
Brief Paper
Single-loop controller design via IMC principles夽 Qing-Guo Wang*, C. C. Hang, Xue-Ping Yang Department of Electrical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore Received 24 January 2000; revised 16 January 2001; received in "nal form 15 June 2001
Abstract In this paper, a new internal model control (IMC)-based single-loop controller design is proposed. The model reduction technique is employed to "nd the best single-loop controller approximation to the IMC controller. Compared with the existing IMC-based methods, the proposed design is applicable to a wider range of processes, and yields a control system closer to the IMC counterpart. It can be made automatic for on-line tuning. The users have the option to choose between PID and high-order controllers to suit the applications better. It turns out that high-order controllers may be necessary to achieve high performance for essentially high-order processes. 2001 Elsevier Science Ltd. All rights reserved. Keywords: IMC; Single-loop control; PID control; Model reduction; On-line tuning
1. Introduction Among the single variable control structures, the vast majority of the controllers used in the process industry are of the PID type. Numerous methods for PID tuning have been reported in As stroK m and HaK gglund (1995), and references therein. Recently, great e!orts have been made to develop PID tuning strategies for more general processes (Barnes, Wang, & Cluett, 1993; Sung & Lee, 1996; Sung, Lee, Lee, & Yi, 1996; Datta, Ho, & Bhattacharyya, 2000). Each method was derived for its particular optimization objectives and plant model assumptions, and therefore perform well only in their own areas. It is a common experience that users are not certain which tuning method should be chosen to provide good control to a given process. It would hence be desirable to develop a design method that works universally with a high performance for general stable linear processes. The internal model control (IMC) has also been exploited to tune conventional PID controllers in singleloop con"gurations (Chien, 1988; Fruehauf, Chien, & Lauritsen, 1994; Lee, Lee, Park, & Brosilow, 1998). They have attracted the attention of industrial users because 夽 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Li Qiu under the direction of Editor Roberto Tempo. * Corresponding author. Tel.: #65-874-2282; fax: #65-779-1103. E-mail address: [email protected] (Q.-G. Wang).
there is only one user-de"ned tuning parameter, which is directly related to the closed-loop time constant or equivalently, the closed-loop bandwidth. A serious drawback of the existing IMC}PID methods is that they are only applicable to speci"c types of processes such as "rst-order plus dead time (FOPDT) and second-order plus dead time (SOPDT) processes (Chien, 1988; Ho, Hang, & Cao, 1995). In their works, either a "rst-order PadeH approximation, a "rst-order Taylor series expansion or a Maclaurin series expansion is applied to derive the tuning rules. This inevitably imposes some limitations on the applicability of the methods and the performance of the designed PID controller. For a process whose Nyquist curve exhibits a strange shape, especially around the cross-over frequency, a PID controller could be di$cult to shape it satisfactorily. What one can do is de-tune the PID controller by trading o! some performance to a su$cient extent to generate a stable closed-loop if the original process is stable. However, this necessarily results in a sluggish response and poor performance, which may not be desirable. We are thus motivated to develop a new method for IMCbased single-loop controller design, and consider, in addition to PID, a high-order controller as well for complex processes, where PID controllers are not adequate. In this paper, model reduction is employed to "nd the best single-loop controller approximation to the IMC controller. Compared with the existing IMC-based methods, the proposed design is applicable to a wider
0005-1098/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 1 ) 0 0 1 7 0 - 4
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range of processes, and yields a control system closer to the IMC counterpart. Furthermore, it is also simple and can be made automatic for on-line tuning. The users have the option to choose between PID and high-order controllers to suit applications better. It turns out that high-order controllers may be necessary to achieve high performance for essentially high-order processes. This paper is organized as follows. An overview of the proposed single loop controller design is presented in Section 2. The speci"c PID and high-order controller cases are detailed in Sections 3 and 4, respectively. In Section 5, stability of the proposed method is analyzed. Finally, some concluding remarks are drawn in Section 6.
2. Design methodology The schematic of the IMC system is depicted in Fig. 1, where G(s) is the given stable process to be controlled, GK (s) a model of the process and C(s) the IMC primary controller. The design procedure for IMC systems is well documented (Morari & Za"riou, 1989). The model is factorized as GK (s)"GK (s)GK (s), (1) > \ such that GK (s) contains all the dead time and right half > plane zeros of GK (s):
1! s G , Re( )'0, (2) GK (s)"e\*Q G > 1# s G G while GK (s) is stable and of minimum phase with no \ predictors. The primary controller takes the form C"GK \f, (3) \ where f is a user speci"ed low-pass "lter and usually chosen as
closed-loop response but the manipulated variable is moved more vigorously, while a larger provides a slower but smoother response. A larger is also less sensitive to model mismatches. In the process control practice, the closed-loop bandwidth can rarely ex ceed ten times of the open loop process bandwidth (Morari & Za"riou, 1989), i.e., )10 . Usually, the desired closed-loop bandwidth is chosen as " , 3[0.5,10]. Using (4), it can be readily seen that (2!1 P " ,
3[0.5,10]. (5) In the case of model uncertainty, should be increased su$ciently enough to meet the condition for which the system is robustly stable (Morari & Za"riou, 1989). In order to keep the action of the manipulated variable within bounds (;M ), it is also required that should satisfy, at least,
r *P , ;M
(6)
where "lim [GK \(s)/sP], and r is a positive set Q \ point change. For the case of no plant-model mismatch, the nominal closed-loop transfer function of the IMC system between the set point r and output y is
1 1! s G H"GK f" e\*Q. (7) > 1# s ( s#1)P G G The IMC system in Fig. 1 can be formally redrawn into the equivalent single-loop (SL) feedback system in Fig. 2, if the SL controller K is related to the IMC-controller C via
1 f (s, )" , (4) ( s#1)P where r is su$ciently large in order to guarantee that the IMC controller C is proper. Also, is the only tuning parameter to be selected by the user to achieve the appropriate compromise between performance and robustness and to keep the action of the manipulated variable within bounds. A smaller provides faster
C(s, ) . (8) K(s, )" 1!C(s, )GK (s) In Chien (1988), K is chosen as the PID type, and the value of is set according to (5) as if the single-loop PID controller could achieve the same performance as that of the more complex IMC controller. The dead time is approximated by either a "rst-order PadeH or a "rst-order Taylor series, while the PID controller parameters are obtained by matching the "rst few Markov coe$cients of (8) for the selected speci"c process models. The results are
Fig. 1. IMC control system.
Fig. 2. Single-loop control system.
Q.-G. Wang et al. / Automatica 37 (2001) 2041}2048
listed in Chien (1988). However, it is noted that the use of PadeH approximation or a "rst-order Taylor expansion introduces extra modeling errors. Furthermore, Chien's rules are only applicable to FOPDT and SOPDT processes. Other processes have to be reduced to such models, and such a reduction may have bad accuracy or even have no solution. This inevitably restricts the general applicability of the method and the performance of the resulting controller. The present work proposes a new IMC-based design methodology. It can yield the best single-loop controller approximation to the IMC controller regardless of the process order and characteristics. The resulting singleloop performance can be better guaranteed and well predicted from the IMC counterpart. Our design idea is very simple: given the equivalent single-loop controller K in (8), which may be unnecessary or too complicated to implement, apply a suitable model reduction to obtain the best approximation KK for K. If the user speci"es the type of KK (say, PID), then the model reduction algorithm will generate its parameters. If the approximation accuracy is satisfactory, then the design is completed; otherwise, the algorithm will adjust the IMC controller performance down until its single-loop approximation is satisfactory. On the other hand, if the user has no preference for controller structure, our algorithm starts with a PID type, and gradually increases the controller complexity such that the simplest approximation KK is attained with the guaranteed accuracy of K. This allows a uni"ed treatment of all cases and facilitates auto-tuning application. A crucial issue in IMC}SL controller design is to get a suitable value for which can lead to a good single loop controller approximation to the corresponding IMC one. Note the inherent di!erence of IMC and SL systems in their con"gurations (Figs. 1 and 2), where the former has the output prediction, while the latter does not. In fact, not all IMC systems can be approximated reasonably by single-loop systems (see the remark in the end of Section 4). The given by (5) is suitable for IMC systems, but it does not consider the performance limitations of single-loop feedback systems due to nonminimum phase zero and dead time. Such limitations are usually expressed by some integral relationships (Freudenberg & Looze, 1987). Recently, As stroK m (1999) proposed the following simple non-integral inequality for the gain cross-over frequency of the open-loop trans fer function GK K, where GK ( j )K( j , )"1
2043
where is the desired phase margin, and the selection
of re#ects the control system robustness to the pro cess uncertainty (As stroK m, 1999). A large is required
for a large uncertainty. Due to the lack of information on uncertainty size, a typical range for could be 303}803.
Our design objective is basically to achieve a non-oscillatory response as speci"ed by (7) and yet have the response as fast as possible. It approximately translates to a damping ratio of "0.7, and the empirical formula "100 (Franklin, Powell, & Workman, 1990) yields
an estimate of as "703 for "0.7. Our studies
suggest that "653 will usually be a good choice and
we will use this throughout the paper. With speci
"ed, we then "nd the smallest H which satis"es (9) and (10). In short, for single-loop controller design the tuning parameter in the "lter (4) should be, in general, chosen to meet (5), (6), (9) and (10), simultaneously. If the process is of minimum phase, (9) and (10) vanish, while (5) and (6) are in action. On the other hand, if the process has any non-minimum element, our study shows that the de rived from (6), (9) and (10) always appears in the range given in (5) so that (6), (9) and (10) would be enough to determine in this case. In the subsequent two sections, PID and general controllers will be considered, respectively.
3. PID controller Owing to its simple structure, the PID controller is the most widely used controller in the process industry, even though many advanced control algorithms have been introduced. Consider a PID controller in the form k K "k # #k s, .'" s
(11)
where k is the proportional gain, k the integral gain (units of time), and k the derivative gain (units of time). Our task is to "nd the three PID parameters, so as to match KK "K to K"C/(1!CGK ) as well as possible. .'" This objective can be realized by minimizing the loss function, K min JOmin K ( j )!K( j ), k , k , k '0 .'" G G ).'" ).'" G (12)
(9)
meets arg GK ( j )*!1803# !arg GK ( j )K( j , ), >
\ (10)
whose solution is obtained by the standard non-negative least square to give the optimal PID parameters as [kH kH kH]2" H. Our studies suggest that the fre quency range [ , ] in the optimal "tting (12) be + chosen as (0.1 , ) with the step of ( & ) , where is the desired closed-loop bandwidth.
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Once a PID controller is found, the following criterion should be used to validate the solution:
KK ( j)!K( j) ) , K( j)
E" max (13) SZ S
where is the user-speci"ed "tting error threshold. is speci"ed according to the desired degree of performance, or accuracy of the SL approximation to the IMC one. Usually, may be set as 3%. If (13) holds true, then the design is completed. On the other hand, if the given threshold cannot be met, one can always detune the PID controller by relaxing the IMC speci"cation, i.e., by increasing . In gen eral, E decreases as increases. It provides a simple way to select a minimum with respect to the speci"c accuracy threshold. In practice, however, it is inconvenient to draw such a curve. It is found that the decreasing rate, dE/d , is highly in#uenced by plant dead time ¸ and the right half plane (RHP) zeros \, which limit G the achievable bandwidth. is virtually una!ected by the presence of the "lter (Rivera, Morari, & Skogestad, 1986) until reaches an order of magnitude comparable to ¸ and , respectively. Hence, it is e!ective and e$cient G to choose the increment of in the PID detuning procedure as the maximum of ¸ and Re , i.e., G I>"I # I max(¸, Re ), (14) G where k represents the kth iteration, and is an adjustable factor re#ecting the approximation accuracy of the present iteration and is set at , and 1, when 3%(E)20%, 20%(E)100% and 100%(E, respectively. The iteration continues until the accuracy bound is ful"lled. Our detuning rule (14) for implicitly assumes that E would be su$ciently small when is large enough. In this connection, it would be interesting to see if E"0. Eq. (5) can be rewritten as lim O
P (2!1 . "
(15) When increases to in"nity, it is easy to see from (15) that lim "0, G( j) can be replaced by G(0), for O ) , and K becomes 1/(G (0)(( s#1)P!1)). For \ r"1, K"1/sG (0) is a pure integrator and can be \ realized precisely by a PID controller with no error. In general, the Nyquist curve of K( j) for 3(0, ) will approach a straight line, when tends to in"nity. Note that the Nyquist curve for the PID controller is always a vertical straight line, and can match that of K( j) as well as desired for PR. One thus expects E to converge to 0 as approaches in"nity. We now present some simulation examples to demonstrate our PID tuning algorithm and compare it with the original IMC and the PID tuning in Chien (1988). Chien
(1988) used the following PID form:
1 ¹ s " KI "K 1# # , .'" A ¹ s (¹ /N)s#1 ' " where the PID settings are given in Chien (1988). The ideal PID controller in (11) used for our algorithm development is not physically realizable and is thus replaced by k k s K "k # # . (16) .'" N s (k /N)s#1 In both cases, N is suggested to be chosen within [5, 20]. The simulations are done under the perfect model matching condition, i.e. GK "G (model mismatch will be considered in Section 5). To have a fair and comprehensive assessment of controller performance, most performance indices popularly used in process control are measured and they include both time domain ones such as overshoot in percentage(M ), rising time (from 10% to 90%) in seconds(t ), setting time (to 1%) in seconds(t ), and integral absolute-error (IAE" r!y dt where the up per limit R may be replaced by ¹, which is chosen su$ciently large so that e(t) for t'¹ is negligible); and the frequency domain E de"ned in (13). Simulation is done for three typical plants, and the results are tabulated in Table 1. The following case is used here to illustrate the proposed method. Example. Consider a high order and oscillatory process e\Q G" . (s#s#1)(s#0.6s#1) From (6), (9) and (10), one can readily "nd H "0.6687, which gives rise to 0.3962s 0.2139 # . K "0.2860# .'" (0.3962/N)s#1 s
(17)
This controller has the approximation error E"17.46%, which cannot ful"ll the accuracy threshold; and the closed-loop response is very poor, as shown in Fig. 3(a). Then is adjusted to " #0.25¸"0.6687# 0.5"1.1687 according to the proposed tuning rule (14). The new results in 0.1498 0.1229s K "0.1016# # . (18) .'" s (0.1229/N)s#1 The approximation error E of the proposed method has met the speci"ed approximation accuracy E)3%. The closed-loop responses are shown in Fig. 3(c). One observes that the di!erence between our SL system and the original IMC system now becomes invisible. It can be seen from the simulation study in Table 1 that the proposed method always yields a PID controller with much better approximation to the IMC counterpart than
6.6798 0 1.0000 5.77 0
10.00 5.77 0
10.00
0.39 5.1016
6.6798
4.6798 0 3.30 *
0
8.72
5.0012
4.6798 0.0045
5.4687 17.48
3.5488 8.72 3.30
2.67 * 7.4365
0
10.00
5.2860
2.8378 0
2.8388 2.66
4.8031 2.17
6.18
2.11
0
IMC
Proposed PID
0.1498 0.1229s 0.1016# # s (0.1229/N)s#1 e\Q G" (s#s#1)(s#0.6s#1)
1.1687
Chien
IMC
Proposed high-order
0.2139 0.3962s 0.2860# # s (0.3962/N)s#1 0.6687
Chien Proposed PID
Proposed PID
(!0.5s#1)e\Q G" (s#1)(2s#1)
0.8328
IMC
0.3569 0.9765s 1.1194# # s (0.9765/N)s#1
Chien
0.3111 0.8365s 1.3779 1# # s (0.8365/N)s#1
IMC
Proposed PID 0.3333
Chien
0.9339
8.46
6.1204
2.8728 26.33 1.54
0.73 0
14.0690
0.73 0
11.66
6.3779
0.8343 0 2.23
3.0184
0.8351 1.23 2.31
6.3945
1.0571 42.59
e\ Q G" s#1
Scheme Plant
Table 1 Summary of simulation results
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Chien's method, regardless of what is chosen. Our experience indicates that for FOPDT and SOPDT processes and slow closed-loop response requirement of
1.8044 4.15 11.7728
1.2029(1#1/s) 1.2050 0.1856s 1.4005# # s (0.1856/N)s#1
0.72
M (%) Controller
t
t
;
E (%)
IAE
Q.-G. Wang et al. / Automatica 37 (2001) 2041}2048
' (2!1/ P , both the proposed IMC-PID method and Chien's rules generate responses similar to the IMC counterpart. Especially, the proposed method can always achieve E(3%, and thus the closed-loop performance can be well predicted from the corresponding IMC system. But, when fast closed-loop response,
generally ' , i.e. ( (2!1/ P , is required, the proposed method shows signi"cant improvement over Chien's rules. The improvement is also evident for complex processes with slow responses. Moreover, under fast response requirement, ' , the PID controller derived from Chien's rules may cause large peaks in the manipulated variable, which is harmful to the system. It is however noticed for high-order processes with fast responses, that none of the above two IMC-PID methods is able to generate PID systems similar to IMC ones. It implies that the controller in the PID form is insu$cient to obtain the desired performance. In this case, a higher order of controller has to be considered for a better "tting and performance, which will be discussed in the next section.
4. High-order controller In this section, model reduction is employed to "nd the lowest high order of controller, which may achieve a speci"ed approximation accuracy. A number of methods for rational approximation are surveyed by Pintelon, Guillaume, Rolain, Schoukens and Van Hamme (1994). The recursive least-squares (RLS) algorithm is suitable for our application and is brie#y described as follows. The problem at hand is to "nd an nth-order rational function approximation: b sL#b sL\#2#b s#b L\ KK " L sL#a sL\#2#a s L\ with an integrator such that
(19)
+ JIO =( j )(KK ( j )!K( j )). (20) G G G G is minimized, where k denotes the index for the kth recursion in the iterative weighted linear least-squares method, and let =( j ) G = M " ( j )L#aI\( j )L\#2#aI\( j ) G L\ G G operate as a weighting function in the standard leastsquares problem which depends on the parameters generated in the last recursion. In this paper, = M is chosen as 1 and standard LS is applied in each iteration. On
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Fig. 3. (a) "0.6687 nominal performance; (b) "0.6687 with a gain uncertainty of 50%; (c) "1.1687 nominal performance; (d) "1.1687 with a gain uncertainty of 50%; Comparison of set-point responses for e\Q/((s#s#1)(s#0.6s#1)) ((! ) ! ) !) PID, (} } }) high-order controller, (**) IMC).
convergence, the resultant parameters will form one solution to (20). Like the LS algorithm, the frequency range in RLS is also chosen as (0.1 , ) with the step of ( & ) . In this range, RLS yields satisfactory "tting results in a frequency domain. The above algorithm deals with the problem of approximating a given, probably non-rational transfer function by a rational function. Error bounds for such an approximation have been investigated (Wahlberg & Ljung, 1992; Yan & Lam, 1999). Wahlberg and Ljung (1992) proposed the approach based on weighted leastsquares estimation, and provided hard frequencydomain transfer function error bounds. However, it is not easy to calculate such a bound, and the convergence of the estimation has not been addressed. In our work, we use a maximum likelihood index E to evaluate the approximation accuracy, and assume that the accuracy threshold can be achieved when the controller order is high enough. When a is chosen, we "rst "nd the PID controller with the standard least-squares method and evaluate the corresponding approximation error E in (13) as described
in the following section. If E cannot achieve the speci"ed approximation accuracy (usually 3%), we recommend a high-order controller in (19), and start from a controller order of 2 until the smallest integer n such that E) . 4.1. Tuning procedure Step 1: Find the smallest H from (6), (9) and (10), and let "H . Step 2: Find the PID controller from (12) and evaluate the corresponding approximation error E in (13). If E achieves the speci"ed approximation accuracy (usually 3%), then end the design. Step 3: Otherwise, we have two ways to solve this problem: if PID controller is desired, update by (14), and go to Step 2; else, go to Step 4. Step 4: Adopt the high-order controller in (19), start from a controller order of 2 until the smallest integer n is reached with E) . Example (continued). Reconsider e\Q G" (s#s#1)(s#0.6s#1)
Q.-G. Wang et al. / Automatica 37 (2001) 2041}2048
for which "0.6687 and a PID has been obtained there with E"17.46%. For a high-order controller, our procedure ends with
Note
from
(4)
2047
that
f ( j)
decays
quickly
* " (2!1/ P and (22) is likely to hold for
3.5488s#14.9135s#23.9669s#29.6333s#18.2712s#9.2024 KK " s#4.5140s#25.8787s#22.8686s#43.0211s with the "tting error E less than "3%. The closed-loop step responses are shown in Fig. 3(a), and their performance indices are also tabulated in Table 1. We can see that the new controller KK restores the IMC performance, while the previous PID controller in (17) is not capable of that under such a tight performance speci"cation. If is chosen to be smaller than the value suggested by (6), (9) and (10), this breaks the limitation of singleloop feedback systems, and then no single-loop controller solution with stability can be found for the corresponding IMC system. For instance, in the above example, choose 50% less than , i.e., "0.3344, we could not "nd a controller in form of (19) with E less than 3%, which implies that SL controllers are unlikely to achieve the performance tighter than that speci"ed by (6), (9) and (10). It is observed from our simulation study that usually, the approximation error magnitude of high-order controller obtained by the RLS is in the order of 10\ or less, while the controller order is less than 6, and the controller yields the closed-loop response quite similar to that of IMC loop provided that is set by (6), (9) and (10). The high-order controller provides signi"cant performance enhancement over PID for complex processes. The proposed method is a simple, e!ective, and e$cient way to design such high performance controllers.
From the results obtained so far, it is possible to state that the single-loop system with KK derived from the proposed method performs similar to the corresponding IMC loop. Thus, the stability of the resulting single-loop control system can be well related to that of the IMC system. In this section, we will consider both nominal stability (G"GK ) and robust stability (GOGK ). Assume that G"GK in the absence of model uncertainty, the nominal stability of IMC system automatically guarantees the stability of the feedback system in Fig. 2 with K determined from (8). But the proposed design makes a controller approximation KK , where KK (s)"K(s)(1# (s)). With the standard assumption that ) KK has the same number of unstable poles as K, the nominal single-loop feedback system is stable if and only if f ( j) ( j) (1. )
(21)
high frequencies. Assume that (22) is true for * , and the nominal closed-loop is thus stable if
" )
K( j)!KK ( j) )1, 3[0, ]. K( j)
(22)
(23)
In the proposed algorithm, the approximation accuracy has to meet (13), where is usually speci"ed as 3%. The resulting controller KK then satis"es (23) with a big margin and the nominal stability of the designed single-loop system is thus expected. Consider now the model uncertainty. Let the actual plant be G(s)"GK (s)(1# (s)), ) () and % ) ) ) (). It follows from the stability robustness % % theorem (Doyle, Wall, & Stein, 1982) and some algebra that the uncertain feedback system remains stable for all "diag , if ) % () f ( j)# () f ( j)#4 () () f ( j))1, ) % ) % ∀. (24)
As f ( j) decays quickly for * "( (2!1)/ P , then (24) is likely to hold for high frequencies. Thus, assume that (24) is true for * . In the proposed method, is made small, i.e., ())3%. Let ) ) "3%, then the robust stability of the closed-loop can ) be guaranteed by ())94.13%, 3[0, ]. %
5. Stability analysis
for
(25)
Example (continued). Reconsider e\Q G" (s#s#1)(s#0.6s#1) with the nominal " "1. When "0.6687, the proposed method yields a "fth-order controller in (21) and the nominal performance is shown in Fig. 3(a). It can be seen that the system is indeed nominally stable. In order to observe robustness, introduce a 50% perturbation in gain , giving "1.5;1"1.5. Fig. 3(b) shows the resulting performances, and indicates that the singleloop high-order controller K derived from the proposed method exhibits a similar robust performance as does the IMC loop. When "1.1687, the proposed method yields a PID controller in (18) and the nominal performance is shown in Fig. 3(c). We also introduce a 50% perturbation in
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Q.-G. Wang et al. / Automatica 37 (2001) 2041}2048
gain, giving "1.5;1"1.5. Fig. 3(d) shows the result ing performance, which is still stable, and more robust than that shown in Fig. 3(d) for "0.6687.
6. Conclusion A new IMC-based single-loop controller design has been proposed in this paper. Extensive simulations have been performed to show that the proposed IMC-based method is generally applicable regardless of the process model involved, and that it allows a uni"ed treatment of all linear stable processes. A guideline for setting the single tuning parameter has also been provided. Simu lation studies show that using the given guideline to select gives consistent and satisfactory performance over a large class of processes. The proposed single-loop controller design method also provides users with the option to achieve speci"ed closed-loop performance at the cost of controller complexity or retain simple PID controller with possible deterioration in the closed-loop performance.
References As stroK m, K. J. (1999). Limitations on control system performance. European Journal of Control, submitted for publication. As stroK m, K. J., & HaK gglund, T. (1995). PID controllers: Theory, design, and tuning. International Society for Measurement and Control, Research Triangle Park, NC. Barnes, T. J., Wang, L., & Cluett, W. R. (1993). A frequency domain design method for PID controllers. Proceedings of the American control conference, San Francisco (pp. 890}894). Chien, I. L. (1988). IMC-PID controller design-an extension. IFAC Proceedings Series, vol. 6 (pp. 147}152). Datta, A., Ho, M. T., & Bhattacharyya, S. P. (2000). Structure and synthesis of PID controllers. London, UK: Springer. Doyle, J., Wall, J., & Stein, G. (1982). Performance and robustness analysis for structured uncertainty. Proceedings of the IEEE conference on decision and control, Orlando, FL, vol. 2 (pp. 629}636). Franklin, G. F., Powell, J. D., & Workman, M. L. (1990). Digital control of dynamic systems. Reading, MA: Addison-Wesley. Freudenberg, J. S., & Looze, D. P. (1987). A sensitivity trade-o! for plants with time delay. IEEE Transactions on Automatic Control, AC-32, 99}104. Fruehauf, P. S., Chien, I. L., & Lauritsen, M. D. (1994). Simpli"ed IMC-PID tuning rules. ISA Transactions, 33(1), 43}59. Ho, W. K., Hang, C. C., & Cao, L. S. (1995). Tuning of PID controllers based on gain and phase margin speci"cations. Automatica, 31(3), 497}502. Lee, Y., Lee, M., Park, S., & Brosilow, C. (1998). PID controller tuning for desired closed-loop responses for SI/SO system. A.I.Ch.E. Journal, 44(1), 106}115. Morari, M., & Za"riou, E. (1989). Robust process control. Englewood Cli!s, NJ: Prentice-Hall. Pintelon, R., Guillaume, P., Rolain, Y., Schoukens, J., & Van Hamme, H. (1994). Parametric identi"cation of transfer functions in the frequency domain*a survey. IEEE Transactions on Automatic Control, 39(11), 2245}2260.
Rivera, D. E., Morari, M., & Skogestad, S. (1986). Internal model control. 4 PID controller design. Industrial and Engineering Chemistry, Process Design and Development, 25, 252}265. Sung, S. W., & Lee, I. (1996). Limitations and countermeasures of PID controllers. Industrial and Engineering Chemistry Research, 35, 2596}2610. Sung, S. W., Lee, I., Lee, J., & Yi, S. (1996). Automatic tuning of PID controller using second-order plus time delay model. Journal of Chemical Engineering of Japan, 29, 990}999. Wahlberg, B., & Ljung, L. (1992). Hard frequency-domain model error bounds from least-squares like identi"cation techniques. IEEE Transactions on Automatic Control, 37(7), 900}912. Yan, W. -Y., & Lam, J. (1999). An approximate approach to H optimal model reduction. IEEE Transactions on Automatic Control, 44(7), 1341}1358.
Qing-Guo Wang received B.Eng. in Chemical Engineering in 1982, M. Eng. in 1984 and Ph.D. in 1987, both in Industrial Automation from Zhejiang University of the People's Republic of China. Since 1992, he has been with the Department of Electrical Engineering of National University of Singapore, where he is currently an Associate Professor. He held the Alexandervon-Humboldt Research Fellowship of Germany with Duisburg University and Kassel University, from 1990 to 1992. His present research interests are mainly in systems theory, robust, adaptive and multivariable control and optimization with emphasis on their applications in process, chemical and environmental industries.
C.C. Hang graduated with a First Class Honours Degree in Electrical Engineering from the University of Singapore in 1970. He received the Ph.D. degree in Control Engineering from the University of Warwick, England, in 1973. From 1974 to 1977, he worked as a Computer and Systems Technologist in the Shell Eastern Petroleum Company (Singapore) and the Shell International Petroleum Company (The Netherlands). Since 1977, he has been with the National University of Singapore, serving in various positions such as the Vice-Dean of the Faculty of Engineering and Head of the Department of Electrical Engineering. From 1994 to 2000, he served as the Deputy Vice-Chancellor in charge of research. His major areas of research is adaptive control in which he has published two books, 230 international journal and conference papers and 6 patents. He was a Visiting Scientist in Yale University in 1983, and in Lund Institute of Technology in 1987 and 1992. From 1992 to 1999, he served as Principal Editor (Adaptive Control) of the Automatica Journal. He was elected a Fellow of IEEE in 1998, a Fellow of Third World Academy of Sciences in 1999 and a Foreign Member of the Royal Academy of Engineering, UK, in 2000.
Xue-Ping Yang received the Bachelor degree of Engineering in Automation and Bachelor degree of Economics in Enterprises Management in 1998, both from Tsinghua University, Beijing, People's Republic of China. She is currently working toward the Ph.D. degree in the Department of Electrical and Computer Engineering, National University of Singapore. Her present research interests are process identi"cation, model predictive control, PID auto-tuning and real-time control systems.
Brief Paper
Single-loop controller design via IMC principles夽 Qing-Guo Wang*, C. C. Hang, Xue-Ping Yang Department of Electrical Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore Received 24 January 2000; revised 16 January 2001; received in "nal form 15 June 2001
Abstract In this paper, a new internal model control (IMC)-based single-loop controller design is proposed. The model reduction technique is employed to "nd the best single-loop controller approximation to the IMC controller. Compared with the existing IMC-based methods, the proposed design is applicable to a wider range of processes, and yields a control system closer to the IMC counterpart. It can be made automatic for on-line tuning. The users have the option to choose between PID and high-order controllers to suit the applications better. It turns out that high-order controllers may be necessary to achieve high performance for essentially high-order processes. 2001 Elsevier Science Ltd. All rights reserved. Keywords: IMC; Single-loop control; PID control; Model reduction; On-line tuning
1. Introduction Among the single variable control structures, the vast majority of the controllers used in the process industry are of the PID type. Numerous methods for PID tuning have been reported in As stroK m and HaK gglund (1995), and references therein. Recently, great e!orts have been made to develop PID tuning strategies for more general processes (Barnes, Wang, & Cluett, 1993; Sung & Lee, 1996; Sung, Lee, Lee, & Yi, 1996; Datta, Ho, & Bhattacharyya, 2000). Each method was derived for its particular optimization objectives and plant model assumptions, and therefore perform well only in their own areas. It is a common experience that users are not certain which tuning method should be chosen to provide good control to a given process. It would hence be desirable to develop a design method that works universally with a high performance for general stable linear processes. The internal model control (IMC) has also been exploited to tune conventional PID controllers in singleloop con"gurations (Chien, 1988; Fruehauf, Chien, & Lauritsen, 1994; Lee, Lee, Park, & Brosilow, 1998). They have attracted the attention of industrial users because 夽 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Li Qiu under the direction of Editor Roberto Tempo. * Corresponding author. Tel.: #65-874-2282; fax: #65-779-1103. E-mail address: [email protected] (Q.-G. Wang).
there is only one user-de"ned tuning parameter, which is directly related to the closed-loop time constant or equivalently, the closed-loop bandwidth. A serious drawback of the existing IMC}PID methods is that they are only applicable to speci"c types of processes such as "rst-order plus dead time (FOPDT) and second-order plus dead time (SOPDT) processes (Chien, 1988; Ho, Hang, & Cao, 1995). In their works, either a "rst-order PadeH approximation, a "rst-order Taylor series expansion or a Maclaurin series expansion is applied to derive the tuning rules. This inevitably imposes some limitations on the applicability of the methods and the performance of the designed PID controller. For a process whose Nyquist curve exhibits a strange shape, especially around the cross-over frequency, a PID controller could be di$cult to shape it satisfactorily. What one can do is de-tune the PID controller by trading o! some performance to a su$cient extent to generate a stable closed-loop if the original process is stable. However, this necessarily results in a sluggish response and poor performance, which may not be desirable. We are thus motivated to develop a new method for IMCbased single-loop controller design, and consider, in addition to PID, a high-order controller as well for complex processes, where PID controllers are not adequate. In this paper, model reduction is employed to "nd the best single-loop controller approximation to the IMC controller. Compared with the existing IMC-based methods, the proposed design is applicable to a wider
0005-1098/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 1 ) 0 0 1 7 0 - 4
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Q.-G. Wang et al. / Automatica 37 (2001) 2041}2048
range of processes, and yields a control system closer to the IMC counterpart. Furthermore, it is also simple and can be made automatic for on-line tuning. The users have the option to choose between PID and high-order controllers to suit applications better. It turns out that high-order controllers may be necessary to achieve high performance for essentially high-order processes. This paper is organized as follows. An overview of the proposed single loop controller design is presented in Section 2. The speci"c PID and high-order controller cases are detailed in Sections 3 and 4, respectively. In Section 5, stability of the proposed method is analyzed. Finally, some concluding remarks are drawn in Section 6.
2. Design methodology The schematic of the IMC system is depicted in Fig. 1, where G(s) is the given stable process to be controlled, GK (s) a model of the process and C(s) the IMC primary controller. The design procedure for IMC systems is well documented (Morari & Za"riou, 1989). The model is factorized as GK (s)"GK (s)GK (s), (1) > \ such that GK (s) contains all the dead time and right half > plane zeros of GK (s):
1! s G , Re( )'0, (2) GK (s)"e\*Q G > 1# s G G while GK (s) is stable and of minimum phase with no \ predictors. The primary controller takes the form C"GK \f, (3) \ where f is a user speci"ed low-pass "lter and usually chosen as
closed-loop response but the manipulated variable is moved more vigorously, while a larger provides a slower but smoother response. A larger is also less sensitive to model mismatches. In the process control practice, the closed-loop bandwidth can rarely ex ceed ten times of the open loop process bandwidth (Morari & Za"riou, 1989), i.e., )10 . Usually, the desired closed-loop bandwidth is chosen as " , 3[0.5,10]. Using (4), it can be readily seen that (2!1 P " ,
3[0.5,10]. (5) In the case of model uncertainty, should be increased su$ciently enough to meet the condition for which the system is robustly stable (Morari & Za"riou, 1989). In order to keep the action of the manipulated variable within bounds (;M ), it is also required that should satisfy, at least,
r *P , ;M
(6)
where "lim [GK \(s)/sP], and r is a positive set Q \ point change. For the case of no plant-model mismatch, the nominal closed-loop transfer function of the IMC system between the set point r and output y is
1 1! s G H"GK f" e\*Q. (7) > 1# s ( s#1)P G G The IMC system in Fig. 1 can be formally redrawn into the equivalent single-loop (SL) feedback system in Fig. 2, if the SL controller K is related to the IMC-controller C via
1 f (s, )" , (4) ( s#1)P where r is su$ciently large in order to guarantee that the IMC controller C is proper. Also, is the only tuning parameter to be selected by the user to achieve the appropriate compromise between performance and robustness and to keep the action of the manipulated variable within bounds. A smaller provides faster
C(s, ) . (8) K(s, )" 1!C(s, )GK (s) In Chien (1988), K is chosen as the PID type, and the value of is set according to (5) as if the single-loop PID controller could achieve the same performance as that of the more complex IMC controller. The dead time is approximated by either a "rst-order PadeH or a "rst-order Taylor series, while the PID controller parameters are obtained by matching the "rst few Markov coe$cients of (8) for the selected speci"c process models. The results are
Fig. 1. IMC control system.
Fig. 2. Single-loop control system.
Q.-G. Wang et al. / Automatica 37 (2001) 2041}2048
listed in Chien (1988). However, it is noted that the use of PadeH approximation or a "rst-order Taylor expansion introduces extra modeling errors. Furthermore, Chien's rules are only applicable to FOPDT and SOPDT processes. Other processes have to be reduced to such models, and such a reduction may have bad accuracy or even have no solution. This inevitably restricts the general applicability of the method and the performance of the resulting controller. The present work proposes a new IMC-based design methodology. It can yield the best single-loop controller approximation to the IMC controller regardless of the process order and characteristics. The resulting singleloop performance can be better guaranteed and well predicted from the IMC counterpart. Our design idea is very simple: given the equivalent single-loop controller K in (8), which may be unnecessary or too complicated to implement, apply a suitable model reduction to obtain the best approximation KK for K. If the user speci"es the type of KK (say, PID), then the model reduction algorithm will generate its parameters. If the approximation accuracy is satisfactory, then the design is completed; otherwise, the algorithm will adjust the IMC controller performance down until its single-loop approximation is satisfactory. On the other hand, if the user has no preference for controller structure, our algorithm starts with a PID type, and gradually increases the controller complexity such that the simplest approximation KK is attained with the guaranteed accuracy of K. This allows a uni"ed treatment of all cases and facilitates auto-tuning application. A crucial issue in IMC}SL controller design is to get a suitable value for which can lead to a good single loop controller approximation to the corresponding IMC one. Note the inherent di!erence of IMC and SL systems in their con"gurations (Figs. 1 and 2), where the former has the output prediction, while the latter does not. In fact, not all IMC systems can be approximated reasonably by single-loop systems (see the remark in the end of Section 4). The given by (5) is suitable for IMC systems, but it does not consider the performance limitations of single-loop feedback systems due to nonminimum phase zero and dead time. Such limitations are usually expressed by some integral relationships (Freudenberg & Looze, 1987). Recently, As stroK m (1999) proposed the following simple non-integral inequality for the gain cross-over frequency of the open-loop trans fer function GK K, where GK ( j )K( j , )"1
2043
where is the desired phase margin, and the selection
of re#ects the control system robustness to the pro cess uncertainty (As stroK m, 1999). A large is required
for a large uncertainty. Due to the lack of information on uncertainty size, a typical range for could be 303}803.
Our design objective is basically to achieve a non-oscillatory response as speci"ed by (7) and yet have the response as fast as possible. It approximately translates to a damping ratio of "0.7, and the empirical formula "100 (Franklin, Powell, & Workman, 1990) yields
an estimate of as "703 for "0.7. Our studies
suggest that "653 will usually be a good choice and
we will use this throughout the paper. With speci
"ed, we then "nd the smallest H which satis"es (9) and (10). In short, for single-loop controller design the tuning parameter in the "lter (4) should be, in general, chosen to meet (5), (6), (9) and (10), simultaneously. If the process is of minimum phase, (9) and (10) vanish, while (5) and (6) are in action. On the other hand, if the process has any non-minimum element, our study shows that the de rived from (6), (9) and (10) always appears in the range given in (5) so that (6), (9) and (10) would be enough to determine in this case. In the subsequent two sections, PID and general controllers will be considered, respectively.
3. PID controller Owing to its simple structure, the PID controller is the most widely used controller in the process industry, even though many advanced control algorithms have been introduced. Consider a PID controller in the form k K "k # #k s, .'" s
(11)
where k is the proportional gain, k the integral gain (units of time), and k the derivative gain (units of time). Our task is to "nd the three PID parameters, so as to match KK "K to K"C/(1!CGK ) as well as possible. .'" This objective can be realized by minimizing the loss function, K min JOmin K ( j )!K( j ), k , k , k '0 .'" G G ).'" ).'" G (12)
(9)
meets arg GK ( j )*!1803# !arg GK ( j )K( j , ), >
\ (10)
whose solution is obtained by the standard non-negative least square to give the optimal PID parameters as [kH kH kH]2" H. Our studies suggest that the fre quency range [ , ] in the optimal "tting (12) be + chosen as (0.1 , ) with the step of ( & ) , where is the desired closed-loop bandwidth.
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Q.-G. Wang et al. / Automatica 37 (2001) 2041}2048
Once a PID controller is found, the following criterion should be used to validate the solution:
KK ( j)!K( j) ) , K( j)
E" max (13) SZ S
where is the user-speci"ed "tting error threshold. is speci"ed according to the desired degree of performance, or accuracy of the SL approximation to the IMC one. Usually, may be set as 3%. If (13) holds true, then the design is completed. On the other hand, if the given threshold cannot be met, one can always detune the PID controller by relaxing the IMC speci"cation, i.e., by increasing . In gen eral, E decreases as increases. It provides a simple way to select a minimum with respect to the speci"c accuracy threshold. In practice, however, it is inconvenient to draw such a curve. It is found that the decreasing rate, dE/d , is highly in#uenced by plant dead time ¸ and the right half plane (RHP) zeros \, which limit G the achievable bandwidth. is virtually una!ected by the presence of the "lter (Rivera, Morari, & Skogestad, 1986) until reaches an order of magnitude comparable to ¸ and , respectively. Hence, it is e!ective and e$cient G to choose the increment of in the PID detuning procedure as the maximum of ¸ and Re , i.e., G I>"I # I max(¸, Re ), (14) G where k represents the kth iteration, and is an adjustable factor re#ecting the approximation accuracy of the present iteration and is set at , and 1, when 3%(E)20%, 20%(E)100% and 100%(E, respectively. The iteration continues until the accuracy bound is ful"lled. Our detuning rule (14) for implicitly assumes that E would be su$ciently small when is large enough. In this connection, it would be interesting to see if E"0. Eq. (5) can be rewritten as lim O
P (2!1 . "
(15) When increases to in"nity, it is easy to see from (15) that lim "0, G( j) can be replaced by G(0), for O ) , and K becomes 1/(G (0)(( s#1)P!1)). For \ r"1, K"1/sG (0) is a pure integrator and can be \ realized precisely by a PID controller with no error. In general, the Nyquist curve of K( j) for 3(0, ) will approach a straight line, when tends to in"nity. Note that the Nyquist curve for the PID controller is always a vertical straight line, and can match that of K( j) as well as desired for PR. One thus expects E to converge to 0 as approaches in"nity. We now present some simulation examples to demonstrate our PID tuning algorithm and compare it with the original IMC and the PID tuning in Chien (1988). Chien
(1988) used the following PID form:
1 ¹ s " KI "K 1# # , .'" A ¹ s (¹ /N)s#1 ' " where the PID settings are given in Chien (1988). The ideal PID controller in (11) used for our algorithm development is not physically realizable and is thus replaced by k k s K "k # # . (16) .'" N s (k /N)s#1 In both cases, N is suggested to be chosen within [5, 20]. The simulations are done under the perfect model matching condition, i.e. GK "G (model mismatch will be considered in Section 5). To have a fair and comprehensive assessment of controller performance, most performance indices popularly used in process control are measured and they include both time domain ones such as overshoot in percentage(M ), rising time (from 10% to 90%) in seconds(t ), setting time (to 1%) in seconds(t ), and integral absolute-error (IAE" r!y dt where the up per limit R may be replaced by ¹, which is chosen su$ciently large so that e(t) for t'¹ is negligible); and the frequency domain E de"ned in (13). Simulation is done for three typical plants, and the results are tabulated in Table 1. The following case is used here to illustrate the proposed method. Example. Consider a high order and oscillatory process e\Q G" . (s#s#1)(s#0.6s#1) From (6), (9) and (10), one can readily "nd H "0.6687, which gives rise to 0.3962s 0.2139 # . K "0.2860# .'" (0.3962/N)s#1 s
(17)
This controller has the approximation error E"17.46%, which cannot ful"ll the accuracy threshold; and the closed-loop response is very poor, as shown in Fig. 3(a). Then is adjusted to " #0.25¸"0.6687# 0.5"1.1687 according to the proposed tuning rule (14). The new results in 0.1498 0.1229s K "0.1016# # . (18) .'" s (0.1229/N)s#1 The approximation error E of the proposed method has met the speci"ed approximation accuracy E)3%. The closed-loop responses are shown in Fig. 3(c). One observes that the di!erence between our SL system and the original IMC system now becomes invisible. It can be seen from the simulation study in Table 1 that the proposed method always yields a PID controller with much better approximation to the IMC counterpart than
6.6798 0 1.0000 5.77 0
10.00 5.77 0
10.00
0.39 5.1016
6.6798
4.6798 0 3.30 *
0
8.72
5.0012
4.6798 0.0045
5.4687 17.48
3.5488 8.72 3.30
2.67 * 7.4365
0
10.00
5.2860
2.8378 0
2.8388 2.66
4.8031 2.17
6.18
2.11
0
IMC
Proposed PID
0.1498 0.1229s 0.1016# # s (0.1229/N)s#1 e\Q G" (s#s#1)(s#0.6s#1)
1.1687
Chien
IMC
Proposed high-order
0.2139 0.3962s 0.2860# # s (0.3962/N)s#1 0.6687
Chien Proposed PID
Proposed PID
(!0.5s#1)e\Q G" (s#1)(2s#1)
0.8328
IMC
0.3569 0.9765s 1.1194# # s (0.9765/N)s#1
Chien
0.3111 0.8365s 1.3779 1# # s (0.8365/N)s#1
IMC
Proposed PID 0.3333
Chien
0.9339
8.46
6.1204
2.8728 26.33 1.54
0.73 0
14.0690
0.73 0
11.66
6.3779
0.8343 0 2.23
3.0184
0.8351 1.23 2.31
6.3945
1.0571 42.59
e\ Q G" s#1
Scheme Plant
Table 1 Summary of simulation results
2045
Chien's method, regardless of what is chosen. Our experience indicates that for FOPDT and SOPDT processes and slow closed-loop response requirement of
1.8044 4.15 11.7728
1.2029(1#1/s) 1.2050 0.1856s 1.4005# # s (0.1856/N)s#1
0.72
M (%) Controller
t
t
;
E (%)
IAE
Q.-G. Wang et al. / Automatica 37 (2001) 2041}2048
' (2!1/ P , both the proposed IMC-PID method and Chien's rules generate responses similar to the IMC counterpart. Especially, the proposed method can always achieve E(3%, and thus the closed-loop performance can be well predicted from the corresponding IMC system. But, when fast closed-loop response,
generally ' , i.e. ( (2!1/ P , is required, the proposed method shows signi"cant improvement over Chien's rules. The improvement is also evident for complex processes with slow responses. Moreover, under fast response requirement, ' , the PID controller derived from Chien's rules may cause large peaks in the manipulated variable, which is harmful to the system. It is however noticed for high-order processes with fast responses, that none of the above two IMC-PID methods is able to generate PID systems similar to IMC ones. It implies that the controller in the PID form is insu$cient to obtain the desired performance. In this case, a higher order of controller has to be considered for a better "tting and performance, which will be discussed in the next section.
4. High-order controller In this section, model reduction is employed to "nd the lowest high order of controller, which may achieve a speci"ed approximation accuracy. A number of methods for rational approximation are surveyed by Pintelon, Guillaume, Rolain, Schoukens and Van Hamme (1994). The recursive least-squares (RLS) algorithm is suitable for our application and is brie#y described as follows. The problem at hand is to "nd an nth-order rational function approximation: b sL#b sL\#2#b s#b L\ KK " L sL#a sL\#2#a s L\ with an integrator such that
(19)
+ JIO =( j )(KK ( j )!K( j )). (20) G G G G is minimized, where k denotes the index for the kth recursion in the iterative weighted linear least-squares method, and let =( j ) G = M " ( j )L#aI\( j )L\#2#aI\( j ) G L\ G G operate as a weighting function in the standard leastsquares problem which depends on the parameters generated in the last recursion. In this paper, = M is chosen as 1 and standard LS is applied in each iteration. On
2046
Q.-G. Wang et al. / Automatica 37 (2001) 2041}2048
Fig. 3. (a) "0.6687 nominal performance; (b) "0.6687 with a gain uncertainty of 50%; (c) "1.1687 nominal performance; (d) "1.1687 with a gain uncertainty of 50%; Comparison of set-point responses for e\Q/((s#s#1)(s#0.6s#1)) ((! ) ! ) !) PID, (} } }) high-order controller, (**) IMC).
convergence, the resultant parameters will form one solution to (20). Like the LS algorithm, the frequency range in RLS is also chosen as (0.1 , ) with the step of ( & ) . In this range, RLS yields satisfactory "tting results in a frequency domain. The above algorithm deals with the problem of approximating a given, probably non-rational transfer function by a rational function. Error bounds for such an approximation have been investigated (Wahlberg & Ljung, 1992; Yan & Lam, 1999). Wahlberg and Ljung (1992) proposed the approach based on weighted leastsquares estimation, and provided hard frequencydomain transfer function error bounds. However, it is not easy to calculate such a bound, and the convergence of the estimation has not been addressed. In our work, we use a maximum likelihood index E to evaluate the approximation accuracy, and assume that the accuracy threshold can be achieved when the controller order is high enough. When a is chosen, we "rst "nd the PID controller with the standard least-squares method and evaluate the corresponding approximation error E in (13) as described
in the following section. If E cannot achieve the speci"ed approximation accuracy (usually 3%), we recommend a high-order controller in (19), and start from a controller order of 2 until the smallest integer n such that E) . 4.1. Tuning procedure Step 1: Find the smallest H from (6), (9) and (10), and let "H . Step 2: Find the PID controller from (12) and evaluate the corresponding approximation error E in (13). If E achieves the speci"ed approximation accuracy (usually 3%), then end the design. Step 3: Otherwise, we have two ways to solve this problem: if PID controller is desired, update by (14), and go to Step 2; else, go to Step 4. Step 4: Adopt the high-order controller in (19), start from a controller order of 2 until the smallest integer n is reached with E) . Example (continued). Reconsider e\Q G" (s#s#1)(s#0.6s#1)
Q.-G. Wang et al. / Automatica 37 (2001) 2041}2048
for which "0.6687 and a PID has been obtained there with E"17.46%. For a high-order controller, our procedure ends with
Note
from
(4)
2047
that
f ( j)
decays
quickly
* " (2!1/ P and (22) is likely to hold for
3.5488s#14.9135s#23.9669s#29.6333s#18.2712s#9.2024 KK " s#4.5140s#25.8787s#22.8686s#43.0211s with the "tting error E less than "3%. The closed-loop step responses are shown in Fig. 3(a), and their performance indices are also tabulated in Table 1. We can see that the new controller KK restores the IMC performance, while the previous PID controller in (17) is not capable of that under such a tight performance speci"cation. If is chosen to be smaller than the value suggested by (6), (9) and (10), this breaks the limitation of singleloop feedback systems, and then no single-loop controller solution with stability can be found for the corresponding IMC system. For instance, in the above example, choose 50% less than , i.e., "0.3344, we could not "nd a controller in form of (19) with E less than 3%, which implies that SL controllers are unlikely to achieve the performance tighter than that speci"ed by (6), (9) and (10). It is observed from our simulation study that usually, the approximation error magnitude of high-order controller obtained by the RLS is in the order of 10\ or less, while the controller order is less than 6, and the controller yields the closed-loop response quite similar to that of IMC loop provided that is set by (6), (9) and (10). The high-order controller provides signi"cant performance enhancement over PID for complex processes. The proposed method is a simple, e!ective, and e$cient way to design such high performance controllers.
From the results obtained so far, it is possible to state that the single-loop system with KK derived from the proposed method performs similar to the corresponding IMC loop. Thus, the stability of the resulting single-loop control system can be well related to that of the IMC system. In this section, we will consider both nominal stability (G"GK ) and robust stability (GOGK ). Assume that G"GK in the absence of model uncertainty, the nominal stability of IMC system automatically guarantees the stability of the feedback system in Fig. 2 with K determined from (8). But the proposed design makes a controller approximation KK , where KK (s)"K(s)(1# (s)). With the standard assumption that ) KK has the same number of unstable poles as K, the nominal single-loop feedback system is stable if and only if f ( j) ( j) (1. )
(21)
high frequencies. Assume that (22) is true for * , and the nominal closed-loop is thus stable if
" )
K( j)!KK ( j) )1, 3[0, ]. K( j)
(22)
(23)
In the proposed algorithm, the approximation accuracy has to meet (13), where is usually speci"ed as 3%. The resulting controller KK then satis"es (23) with a big margin and the nominal stability of the designed single-loop system is thus expected. Consider now the model uncertainty. Let the actual plant be G(s)"GK (s)(1# (s)), ) () and % ) ) ) (). It follows from the stability robustness % % theorem (Doyle, Wall, & Stein, 1982) and some algebra that the uncertain feedback system remains stable for all "diag , if ) % () f ( j)# () f ( j)#4 () () f ( j))1, ) % ) % ∀. (24)
As f ( j) decays quickly for * "( (2!1)/ P , then (24) is likely to hold for high frequencies. Thus, assume that (24) is true for * . In the proposed method, is made small, i.e., ())3%. Let ) ) "3%, then the robust stability of the closed-loop can ) be guaranteed by ())94.13%, 3[0, ]. %
5. Stability analysis
for
(25)
Example (continued). Reconsider e\Q G" (s#s#1)(s#0.6s#1) with the nominal " "1. When "0.6687, the proposed method yields a "fth-order controller in (21) and the nominal performance is shown in Fig. 3(a). It can be seen that the system is indeed nominally stable. In order to observe robustness, introduce a 50% perturbation in gain , giving "1.5;1"1.5. Fig. 3(b) shows the resulting performances, and indicates that the singleloop high-order controller K derived from the proposed method exhibits a similar robust performance as does the IMC loop. When "1.1687, the proposed method yields a PID controller in (18) and the nominal performance is shown in Fig. 3(c). We also introduce a 50% perturbation in
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gain, giving "1.5;1"1.5. Fig. 3(d) shows the result ing performance, which is still stable, and more robust than that shown in Fig. 3(d) for "0.6687.
6. Conclusion A new IMC-based single-loop controller design has been proposed in this paper. Extensive simulations have been performed to show that the proposed IMC-based method is generally applicable regardless of the process model involved, and that it allows a uni"ed treatment of all linear stable processes. A guideline for setting the single tuning parameter has also been provided. Simu lation studies show that using the given guideline to select gives consistent and satisfactory performance over a large class of processes. The proposed single-loop controller design method also provides users with the option to achieve speci"ed closed-loop performance at the cost of controller complexity or retain simple PID controller with possible deterioration in the closed-loop performance.
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Qing-Guo Wang received B.Eng. in Chemical Engineering in 1982, M. Eng. in 1984 and Ph.D. in 1987, both in Industrial Automation from Zhejiang University of the People's Republic of China. Since 1992, he has been with the Department of Electrical Engineering of National University of Singapore, where he is currently an Associate Professor. He held the Alexandervon-Humboldt Research Fellowship of Germany with Duisburg University and Kassel University, from 1990 to 1992. His present research interests are mainly in systems theory, robust, adaptive and multivariable control and optimization with emphasis on their applications in process, chemical and environmental industries.
C.C. Hang graduated with a First Class Honours Degree in Electrical Engineering from the University of Singapore in 1970. He received the Ph.D. degree in Control Engineering from the University of Warwick, England, in 1973. From 1974 to 1977, he worked as a Computer and Systems Technologist in the Shell Eastern Petroleum Company (Singapore) and the Shell International Petroleum Company (The Netherlands). Since 1977, he has been with the National University of Singapore, serving in various positions such as the Vice-Dean of the Faculty of Engineering and Head of the Department of Electrical Engineering. From 1994 to 2000, he served as the Deputy Vice-Chancellor in charge of research. His major areas of research is adaptive control in which he has published two books, 230 international journal and conference papers and 6 patents. He was a Visiting Scientist in Yale University in 1983, and in Lund Institute of Technology in 1987 and 1992. From 1992 to 1999, he served as Principal Editor (Adaptive Control) of the Automatica Journal. He was elected a Fellow of IEEE in 1998, a Fellow of Third World Academy of Sciences in 1999 and a Foreign Member of the Royal Academy of Engineering, UK, in 2000.
Xue-Ping Yang received the Bachelor degree of Engineering in Automation and Bachelor degree of Economics in Enterprises Management in 1998, both from Tsinghua University, Beijing, People's Republic of China. She is currently working toward the Ph.D. degree in the Department of Electrical and Computer Engineering, National University of Singapore. Her present research interests are process identi"cation, model predictive control, PID auto-tuning and real-time control systems.