Assessment of Performance for Single Loop Control Systems

Aement of Performance for Single Loop Control Sytem Hiao-Ping Huang and Jyh-Cheng Jeng Department of Chemical Engineering National Taiwan Univerity Taipei 1617, Taiwan Abtract Aement of performance in tracking et-point change for ingle loop ytem i preented. In contrat to the work that ued tochatic performance, the current aement ue a determinitic performance meaure, i.e. the integration of abolute tracking error (abbr. IAE). A benchmark ytem that ha an open loop tranfer function (abbr. OLTF) compried of one LHP zero, one integrator, and dead time i etablihed. Thi benchmark ytem i ued to provide a goal of performance that the exiting ytem can practically achieve. For aeing an exiting control ytem, the performance of the ytem in term of IAE i computed and an index that indicate the extent of achievement toward thi benchmark ytem i computed. Evaluation of control baed on thi index can then be made. The model required in computing the aforementioned performance index can be obtained from an auto-tuning procedure. Keyword Single loop, Aement, Benchmark ytem, IAE performance, Auto-tuning Introduction Aement and monitoring of control ytem with tochatic performance ha been an active area of reearch for the lat decade(harri, 1989; Stanfelj et al., 1993; Harri et al., 1996; Qin, 1998; Harri et al., 1999; Leung and Romagnoli, 2). The development of reearch work have been focued on formulating the performance in term of variance of the ytem which are diturbed by tochatic input. A a reult, with few exception, for example: Leung and Romagnoli (2), controller are implemented with dicrete-time algorithm to puruit minimum variance. On the other hand, the reearche regarding aement for determinitic performance have been reported, lately. For uch aement, technical development have been focued on etimating maximum log module (L c,max ) of cloedloop, (Chiang and Yu, 1993; Ju and Chiu, 1997), frequency repone (Kendra and Cinar, 1997), proce characteritic(piovoo et al., 1992), rie time (Åtröm et al., 1992), and ettling time (Swanda and Seborg, 1999), etc. But, thoe work mentioned did not provided any implication regarding the bet performance that an exiting ytem can practically achieve. In thi paper, a benchmark ytem baed on an IAE meaure for an exiting control ytem of ingle loop i preented. Thi benchmark ytem i to provide a practical goal of performance that the exiting ytem can achieve. It ha a loop tranfer function compried of one LHP zero, one integrator, and dead time. An index baed on thi benchmark ytem i thu preented to evaluate the exiting ytem comparing with it control limit. Thi control limit i obtained baed on what ha been known about the proce in term of a model with pecific dynamic order. Model to be required for thi purpoe can be obtained from an ATV experiment Phone: 886-2-2363-8999 Fax: 886-2-2362-3935 Email: huanghpc@ccm.ntu.edu.tw Figure 1: The conventional feedback control ytem. with relay feedback. Developing Benchmark Sytem for Performance Aement According to the IMC deign principle, an IMC equivalent controller in a ingle loop of Figure 1 i given by: Ḡ c () = Ḡ 1 p, F () 1 Ḡp,+F () (1) Where, Ḡ p () deignate the model for the proce, and Ḡ p, and Ḡp,+ deignate the invertible and noninvertible part of Ḡp. The tranfer function F () i an IMC filter. For deign purpoe, the dynamic of a open loop proce can in general be repreented by a tranfer function either of firt-order-plu-dead-time (abbr. FOPDT) or of econd-order-plu-dead-time (abbr. SOPDT) of the following: FOPDT: SOPDT: Ḡ p () = k pe θ τ + 1 (2) Ḡ p () = k p(a + 1)e θ τ 2 2 + 2τζ + 1 ; a (3) 384

Aement of Performance for Single Loop Control Sytem 385 Notice that the SOPDT proce with RHP zero can alo be modeled with the one of Equation 3 by making ue of the following approximation. Let the RHP zero i given in the form of 1 β. Then, β + 1 e 2β (1 + β) (4) By uing G c in Equation 1 and G p in Equation 2 3, the reulting open loop tranfer function (abbr. OLTF) of the ytem become: G OL () = G c G p () = G p,+f () 1 G p,+ F () = e θ F () 1 e θ F () (5) Thu, if G c i implemented exactly with Equation 1, the performance of the ytem depend on the choice of F (). The limiting performance of uch a ytem can be obtained from the reult of Holt and Morari (1985) with a light modification to take into account the additional dead time. For conventional loop, G c i ued to be confined to have the following form: G c () = b m m + b m 1 m 1 + + b a n n + a n 1 n 1 + + a (6) To obtain G c () in the above form, e θ in the denominator of Equation 1 hould be replaced with a Pade approximation of proper order. After introducing the ame Pade approximation into the denominator of Equation 5, it i eay to ee that the OLTF become: G OL = H() e θ (7) where, H() i conidered a loop filter and it functional form varie with the choice of F () and the Pade approximation being ued. But, in general, H() conit of finite number of pole and zero. Due to the approximation that ha been made for yntheizing G c (), the performance limit et by the IMC will no longer be applicable to the ytem in Figure 1. Thu, the performance limit for a ingle loop ytem ha to be obtained from a minimization proce. The minimization proce tart with an H() of the following form: β + 1 H() = k o (8) γ + 1 In other word, with an OLTF of the following: β + 1 e θ G OL = k o γ + 1 (9) If the performance in term of IAE i ued, then the procedure i to find the value of k o and thoe of β and IAE Model or method θ Remark Benchmark ytem 1.38 ko =.76 θ ; β =.47θ Rovira(PI) 1.93 FOPDT; τ = θ Rovira(PID) 1.52 FOPDT; τ = θ Sung et al.(pid) 2.6 G p () = 1 (+1) 5 Sung et al.(pid) 2.22 e G p () = (9 2 +2.4+1)(+1) Swanda and Seborg 2. PI control Smith and Corripio (1997, pg. 325) Sung et al. (1996) Swanda and Seborg (1999) value for different con- Table 1: The minimum IAE θ trol ytem. γ that minimize the following integral: J IAE = Min [ko,β,γ] (1) where e(t) i given a the invere tranformation of e(), which i the tracking error of the ytem, i.e.: { } e(t) = L 1 1 R() (11) 1 + G OL Thi above optimization problem wa olved numerically by imulation. The reult turn out to be: where, G OL = k o(1 + β ) e θ (12) k o =.76 θ, β =.47θ, andγ = (13) Notice that G OL hould not have exce number of zero than pole. Thu, for procee of Equation 2 and 3, the H() in the form of Equation 8 i mot appropriate for developing the benchmark ytem. Thu, with the optimization reult given above, the benchmark ytem i elected a the one that ha OLTF of the following: G OL() =.76(1 +.47θ) θ e θ (14) The IAE value of thi benchmark ytem ubjected to an unit tep et-point change i found to be 1.38 θ. It i alo found that thi benchmark ytem ha a gain margin of 2.11, and a phae margin of 64.4 o. The ytem with uch margin value i conidered to have acceptable tability robutne. Baed on the FOPDT model of Equation 2, and the the SOPDT procee of Equation 3, the controller in the form of Equation 6 that yield the benchmark OLTF are given a follow:

386 Hiao-Ping Huang and Jyh-Cheng Jeng FOPDT: G c,1() = ( ).76 (τ + 1)(.47θ + 1) k pθ (15) SOPDT: G c,2() = ( ).76 (τ 2 2 + 2τζ + 1)(.47θ + 1) k pθ (a + 1) (16) Obviouly, thee controller are not phyically realizable. For controller to be realizable, one or two low pa filter with mall time contant have to be introduced omewhere in G c. The value of the reulting IAE will thu be degraded. But, the change i really too mall to be conidered. Thu, if G c () ha not been confined to the conventional PID controller, the minimum achievable IAE will be: IAE = 1.38θ (17) In Table 1, ome IAE value of everal ytem are given. Thee ytem include the benchmark one and ome other, which have optimal controller from different ource. It i to how none of thee other ytem from different ource ha IAE le than the benchmark one. Thi above equation can be adapted to apply to the cae where G p () ha 1 β a a factor in the numerator (i.e. a RHP zero). In thi cae, the minimum IAE become: IAE = 1.38 (θ + 2β) (18) A an example, Conider a ytem that ha open loop tranfer function of the following: G c G p () = k o(1 + a)(1.5)e (1 +.5) The minimum IAE of thi cloed loop ytem occur when k o =.4 and a =.21, and ha a value of 2.8, which i about 1.38 (1 + 2.5) (i.e. 2.76). Thu, aement for a ingle loop control ytem that ha G p () of Equation 2 or 3, will be targeting at the minimum IAE value of Equation 17 and 18. Aement for Control Baed on IAE In the previou ection, it ha been mentioned that the minimum achievable IAE value of a conventional feedback loop i 1.38 θ. In order to meaure the achievement of an exiting ytem toward thi achievable target for et-point tracking, the following index i defined: Φ = 1.38θ (19) Where, in the denominator, the IAE meaure of the exiting ytem for tracking tep et-point change i ued. Thi IAE meaure can be obtained from experiment or from prediction baed on a model for G p. The index Φ, Figure 2: The Φ value for FOPDT proce with Rovira PI and PID controller. alway le than one, i ued to repreent the ability of an exiting ytem in eliminating the tracking error. If Φ i cloe to one, it indicate that the ytem i near it performance limit. To illutrate aeing with thi preented index, firt, the control of FOPDT proce with PID controller i conidered. In order to be more incluive for the reult, the FOPDT model i normalized with dimenionle time unit. The normalized tranfer function for the FOPDT model i: Ḡ p () = k pe (2) τ + 1 where, τ = τ/θ. Then, for each τ, Rovira tuning formula(smith and Corripio, 1997, pg. 325) are ued to tune the PI or PID controller, which were claimed optimal for the IAE meaure. A hown in Figure 2, the value of Φ reulting from uch PI and PID control ytem are given. It i thu found that, thee optimal PID controller give value of Φ higher than.8. On the other hand, thoe for optimal PI controller, have value itting between.65 and.75. Thu, a far a the control for FOPDT procee i concerned, the PID controller i a good choice. Next, for dynamic ytem of SOPDT, a normalized G p with dimenionle time unit are alo conidered for illutration: Ḡ p () = k p(ā + 1)e ( τ 2 2 + 2 τζ + 1) (21) where, ā deignate a θ, and τ deignate τ θ. The normalized tranfer function how that τ and ζ can be ued to characterize the dynamic behavior of uch a proce. Thu, PID control ytem for G p with different τ and ζ are ued for illutration. For PID control of SOPDT procee, tuning rule of Sung et al. (1996) are ued to compute the value of Φ. The reult

Aement of Performance for Single Loop Control Sytem 387 Figure 3: The Φ value for SOPDT proce with PID controller given by Sung et al. are given in Figure 3, where Φ i plotted along the value of θ/τ in the range between.1 and 2. The damping factor ζ ha been ued a a parameter. The computed Φ for uch ytem indicate that uch PID controller are more cloe to the limit for well overdamped G p. In thi figure, it i alo oberved that the Φ value computed baed on the IAE from SOPDT model are lower than thoe from FOPDT model. Thi doe not imply, for control purpoe, an FOPDT model i uperior to the SOPDT one. Intead, it indicate that with more knowledge about the dynamic, the control performance would have more tringent limit, and, if achievable, the performance would be uperior to thoe with impler (uch a FOPDT) model. In general, a G p () of high order may have an FOPDT and an SOPDT model at the ame time a approximation of different accuracy. One may be quetioned which type of model to be ued. To jutify, let θ 1 and θ 2 deignate two apparent dead time in the FOPDT and SOPDT model, repectively. In general, θ 2 θ 1. Two value of Φ will be reulted: Then, we hall have: where, Φ 1 = 1.38θ 1 Φ 2 = 1.38θ 2 Φ 2 = (1 η) Φ 1 (22) η = θ 1 θ 2 (23) θ 1 The value of η implie the portion of apparent dead time, from thi FOPDT model, that can be reduced by the controller baed on SOPDT model. Thu, if the value of η i too high, that mean FOPDT model i not ufficient for deigning good control ytem. Aement with Relay Feedback Experiment The information needed for aeing the control will be a model of FOPDT or of SOPDT that ha an apparent dead time. To obtain thi model, a relay feedback ytem can be ued. The ue of relay feedback ha advantage on a few apect. The mot important one i that the control loop i till operated under cloed loop and, hence, i till under control. The relay feedback ytem i the ame one a ha been ued in the ATV tet of Åtröm and Hagglund (1984). The experiment conit of two tage. In the firt tage, the ytem i perturbed with a bia to the output of the relay and wait until the ytem to appear contant cycling at the output. At thi time, the cycling period (deignated a P ) and the amplitude (deignated a a) of the cycle are meaured to calculate the proce gain: k p = t+p t t+p t y(t)dt u(t)dt (24) Then, in the econd tage, the bia to the relay i et to zero and wait again until the ytem appear contant cycling again. The period a well a the amplitude of the cycle are meaured for etimating the other dynamic parameter. With thee data obtained on-line, parameter etimation are carried out. The procedure and the algorithm for thee etimation can be found elewhere(huang et al., 1996, 2). Thu, with the etimated model, we can calculate the predicted IAE for a et-point change by imulating the following with computer, i.e.: e() = 1 1 1 + G c ()Ḡp() (25) Prediction of IAE via thi identification procedure and imulation have been carried out over everal example procee of high order dynamic, and pretty cloe reult are obtained. In other word, the computation of Φ for aement can be performed with an ATV tet mentioned above. Concluion For a ingle loop control, a benchmark ytem that aim at minimuming IAE for et-point change ha been etablihed. Thi benchmark ytem ytem provide a performance limit for all feaible controller in the form of Equation 6. It ha an open loop tranfer function (abbr. OLTF) compried of one integrator, one imple lead, and dead time. For aeing an exiting ytem, an index for indicating the extent of achievement toward thi benchmark ytem i preented. Thi index i cloely aociated with the knowledge of dynamic being available, and the knowledge i uually contained in model of

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