A model-based sliding mode control methodology applied to the HDA-plant

Transcription

1 Journal of Proce Control 13 (2003) A model-baed liding mode control methodology applied to the HDA-plant Guido Herrmann, Sarah K. Spurgeon*, Chritopher Edward Department of Engineering, Control and Intrumentation Reearch Group, Univerity of Leiceter, Univerity Road, Leiceter, LE1 7RH, UK Abtract A liding mode control methodology uing output information i demontrated in application to the HDA-plant, a plant for production of benzene. Thi proce i a highly integrated, non-linear large cale proce with non-minimum phae and relative degree zero characteritic. The non-linear control law i deigned on the bai of a linear oberver-baed control ytem. The nonlinear control law ue the tate of the linear oberver. The performance in the liding mode i determined by a linear table ubmanifold of the linear cloed loop control ytem choen via a robut pole election cheme. The liding mode control i optimized to operate in a wide operating region. # 2002 Elevier Science Ltd. All right reerved. Keyword: Sliding-mode control H 1 -Control Chemical procee 1. Introduction The HDA-plant, a chemical plant for production of benzene, ha been of longtanding interet to chemical proce engineer [1 3] and control engineer [4 6]. Since thi non-linear, large cale and relative degree zero proce i highly integrated and non-minimum phae, it form an important tet bed for new control approache. The HDA-proce model wa initially implemented a a 68 tate model by Brognaux [4] and then extenively re-developed into a 270-tate ytem by Cao et al. [7] under SpeedUp. 1 The model ha been re-implemented under Apen Cutom Modeler 1 by the author for thi paper. The plant ha undergone a detailed analyi with repect to the available combination of ingle-input ingle-output (SISO) control cheme and the mot effective actuator [5,8,9]. However, control cheme have been generally limited to SISO-control method a een in Luyben et al. [6] for a different HDA-model imulation et-up. Hence, the application of modelbaed, multi-variable control a conidered here i novel. For the multivariable control, a linear mixed enitivity H 1 -control problem i poed. Enforcing pole * Correponding author. Tel.: fax: addree: gh17@un.engg.le.ac.uk (G. Herrmann), eon@le.ac.uk (S.K. Spurgeon), ce14@le.ac.uk (C. Edward). 1 SpeedUp and Apen Cutom Modeler are trademark of Apen Technology, Inc., Cambridge, USA. eparation for the linear controller and oberver ytem, a non-linear control i augmented uing the oberver tate. The non-linear controller improve robutne and performance via peudo-liding forcing the cloed loop tate into the vicinity of a table manifold of the linear cloed loop control. The article dicue firt the plant, problem of nonlinearity, the tracking problem and the control methodology. Finally, the model reduction, controller deign and imulation reult are preented. 2. The HDA-plant: a benzene producing chemical proce In the HDA-plant (Fig. 1), benzene i produced via hydrodealkylation (HDA) of toluene. The reaction taking place comprie an exothermic reaction and an equilibrium reaction. There are two input ubtance, toluene and hydrogen and three product ubtance, benzene, diphenyl and methane (Fig. 1). For baic operation, Brognaux [4] and alo Cao et al. [7] introduced PID controller to tabilize the ytem and to keep certain plant variable in a well-defined range. The remaining meaurable output value in Table 1 have to be controlled. A tracking controller i uppoed to follow production demand change while keeping the interaction with the other four meaurement a low a poible. Theoretical input/ouput controllability analyi applied to the HDA-plant by Cao and Roiter [8], Cao and Roiter [9] and Cao et al. [5] howed that ix out of /02/$ - ee front matter # 2002 Elevier Science Ltd. All right reerved. PII: S (02)

2 130 G. Herrmann et al. / Journal of Proce Control 13 (2003) Fig. 1. The benzene production proce, actuator=italic, meaurement=italic. 13 poible actuator are the mot effective for control (Table 3). Further, the external diturbance of Table 2 affect the cloed loop ytem performance. Chemical plant are often ubject to delay: Thee are to be expected within the HDA-plant for the product purity meaurement. Cao et al. [10] invetigated delay of up to 10 min. The HDA-plant model i highly non-linear. One characteritic of the repone of output meaurement of the open loop plant to mall poitive and negative tep change of the input i that they are not inverted to each other. Fig. 2 how that the non-linear fat dynamic eem to match the repone of the Apen- Cutom-Modeller-provided linearization, while low dynamic prove to be highly non-linear. Thi ha been Table 1 Meaurement available for control Meaurement Required range a Scaling 1. Flah inlet temperature (F) O 1 ¼ Production rate (lb mol/h) O 2 ¼ Product purity At leat (%) (mol-%) O 3 ¼ 1 4. Hydrogen to aromatic ratio 5. Flah outlet vapour preure lb mol=h lb mol=h 0:01 O 4 ¼ 1 0: (pia) O 5 ¼ 1 5 a Subtance quantity: 1 lb mol= mol, (lb mol)=poundmole: Temperature: T (F) 1.8T ( C)+32, (F) Fahrenheit preure: 1 pia=6.895 kpa, (pia) pound per quare inch (abolute preure). alo verified by invetigating the frequency repone of the linearization in the conidered operation envelope. The next ection will conider the neceary theoretical background for controller deign which i employed for the HDA-plant in connection with an order reduced linear model. 3. An H 1 -controller baed approach Suppoe a linear, time-invariant ytem i given by: x : ¼AxþBuþB d d x 2 R n B 2 R nm B d 2 R nm d y¼cx þ DuþD d d C 2 R pn t: p: A BB d C DD d where u 2 R m ðn 5 mþ are the available actuator, d the plant diturbance and y 2 R p the output meaurement. A mixed enitivity H 1 -control problem can be poed Table 2 Diturbance influencing the control Diturbance Nominal value Scaling 1. Pre-flah cooler: cooling effluent 59 (F) I 7 ¼ 9 temperature 2. Purge down tream preure 350 (pia) I 8 ¼ 50

3 G. Herrmann et al. / Journal of Proce Control 13 (2003) Table 3 and S 1 ðþ¼ I G ðþk 1 ðþw y ðþ A ufficient condition for thi i that T yh ½ Mot effective actuator for low control effort, high robutne and 0 r 0 Š Weight W y have good diturbance rejection been ued with pole at 0 which preclude the ue of the Actuator Unit a Scaling tandard H 1 -ynthei procedure of Zhou et al. [11]. lbmol=h Safonov pole hifting methodology ha thu been ued [12]. 1. Benzene column plitter: reflux ratio I lbmol=h 1 ¼ 0: Compreor hore power (hp) I 2 ¼ 64: A linear oberver baed cloed loop control ytem 3. Pre-flah cooler duty (BTU/h) I 3 = Ga feed flow (lb mol/h) I 4 = with auxiliary non-linear control input ignal 5. Toluene feed flow (lb mol/h) I 5 = Purge outlet valve opening I 6 =9.183 Conidering only the controlled plant input u and the a Power: 1 hp=745.71w, (hp)=hore power energy: 1 BTU= J, (BTU) Britih thermal unit. according to Fig. 3 chooing the enitivity weight W y a part of the controller and including integrator to achieve a teady tate error of 0. Further, the actuator i weighted by W uu and W ur, a filter for untructured uncertainty and a filter limiting the derivative of the actuator ignal. W p hape the cla of diturbance d. TheH 1 -optimization criterion minimize the H 1 normt yh ½ 0 r 0 Š 0 1 of the tranfer function T yh ½ 0 r 0 0 Š from the exogenou input T r T T to the controlled output y H ¼ y T Hy y T H T yt uu H ur (Fig. 3). Robutne with repect to untructured additive uncertainty, G þ W uu kk 1 < 1 of the nominal plant, G t: p: (A, B, C, D) i achieved if W uu K 1 W y S where K 1 i the deigned control meaured output y Hy from Fig. 3, the plant augmented with the H 1 -deign weight i given by G: t: p: G x: G ¼ A G B G xg ð1þ y Hy C G D G u and a linear controller for (1) with auxiliary input u NL and u X i 2 3 K t: x^: p: G ¼ A K B K B NL B X x^g u K 6 7 y K C K D K u NL 5 u X where the controller K 1 ða K B K C K D K Þ ha been deigned for u K ¼ y Hy and u ¼ y K o that x^g are oberver tate and at leat a weak pole eparation principle ([13], p. 178) applie to thi cloed loop ytem. The matrice B NL,B X and the input u X are determined later. Fig. 2. Repone to mall poitive/negative tep-change (1%) of the benzene column reflux ratio for non-linear plant and linearization (, nonlinear repone to po. tep..., non-linear repone to neg. tep.., linear repone to po. tep ----, linear repone to neg. tep).

4 132 G. Herrmann et al. / Journal of Proce Control 13 (2003) Fig. 3. Augmented plant for H 1 -optimization conidering additive uncertainty W uu ðkk 1 4 1Þ. For the cloed loop ytem invetigated here, aume the auxiliary ignal u NL (Fig. 4) i added to the linear controller output ignal y K, o that u ¼ u NL þ y K u K ¼ y Hy. For e G ¼ x G x^g, the cloed loop can be derived from a tranfer matrix formula for a feedback connection ([11], p. 66) x : G e : G ¼ A A 1 B 0 A 2 A 3 2 xg 6 e G B 1 B X 4 u NL u X ð2þ where B 1 ¼ B NL B K D G R 2 þ B G R 2 and R 2 =(I D K D G ) 1. If e G define the dynamic of an oberver error with at leat weak pole eparation, then generally ka 2 k ka 3 k where kk i the (induced) Euclidean norm. Hence, exact oberver/controller pole eparation applie if B def NL ¼ B K D G R 2 þ B G R 2 B def X ¼A 2 u def X ¼x^G and the matrix A 3 þa 2 ha table eigenvalue. It i readily undertood that the exogenou input ½ 0 r 0 Š 0 do Fig. 4. Cloed loop ytem with auxiliary input. not affect the actual cloed loop tability and thi permit them to be neglected when conidering tability. The introduction of integral action enure zero teady tate error for a given demand r=cont. Having etablihed a table linear cloed loop ytem with pole eparation of oberver and feedback, a liding-mode tate-feedback control can be contructed Derivation of a liding mode hyperplane from a table plant repreentation Sliding-mode control i a powerful, non-linear, robut control method which ha been uccefully applied to many practical ytem uch a furnace [14], car engine [15,16] or induction motor driven tram ytem ([16], p. 271). The approach of thi non-linear control method i to achieve a liding motion by forcing the cloed loop tate onto a table ub-manifold of the tate-pace, the liding-mode hyperplane. By auming that uncertainty, un-modeled non-linearitie and diturbance are matched, i.e. confined to the range of the actuator, a controller achieving a liding motion render the cloed loop ytem invariant to thee diturbance and introduce reduced order liding dynamic. For linear uncertain ytem, an extenive frame work of non-linear controller tructure ha been developed [16,17]. The mot common choice of a liding mode urface i a linear hyperplane [16]. Furthermore, problem of unmatched uncertainty can be uitably addreed for linear uncertain ytem [17]. The deign of a liding urface and a non-linear controller will be undertaken for the table ytem pair (A B) in (2), uing the linear oberver tate and an appropriate non-linear auxiliary input u NL.

5 G. Herrmann et al. / Journal of Proce Control 13 (2003) Auming B i full column rank, then a non-ingular U ¼ U 11 U 12 U linear tranformation [17] exit o that, diregarding U 21 U 11 2 R ðn m Þ ð N m Þ 22 the interaction with the oberver error and the exogenou input, (A B) can be repreented a: then S 1 ¼ U12 T S 2 ¼ U22 T ½ I N m0šv~ ¼ U 11 ^ 1V. Hence, T x : ¼AxþBu NL x 2 R N A2R NN B2R Nm exit only if U22 T or U 22 i invertible and ½I N m 0ŠV~ ha B¼ 0 only full rank if U 11 ha. Schur formula implie that an B 2 2 R mm u NL 2 R m ð3þ invertible U 11 i equivalent to an invertible U 22. Therefore, the exitence of T i equivalent to the full rank B 2 condition for ½I N m 0ŠV~. Now, the econd part of the where B 2 i full rank, A a table matrix. Define L to be a claim (5) ha to be hown. Note that diagonal matrix containing the eigenvalue of A S 1 S 2 V ~ ¼ 0 () S 1 S 2 ½ VC v 1 v 2 :::v r Š ¼ 0: AV ¼ V V 2 C NN fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} : ð4þ V The following Lemma, a theoretical extenion of an approach of Bhatti ([15], pp ), how that it i poible to find a tranformation realizing the canonical form required for liding mode control. Lemma 1. Suppoe there exit a choice of (N m) ditinct eigenvalue of A, which contain r 0 4 r 4 N m, real eigenvalue l 1 l 2 l 3... l r 2 R: A½v 1...v r Š ¼ ½v 1...v 2 Šdiagðl 1... l r Þ and N m r 2 pair of complex conjugate eigenvalue l rþ1 ¼ l rþ2...l N m 1 ¼ l N m 2 CnR AV C ¼ V C diag l rþ1 l rþ2... l N m with V def C ¼½v rþ1 þiv rþ2 v rþ1 iv rþ2... Š v k 2R N 14k4N m: Then there i a tranformation matrix T and a matrix with eigenvalue l 1... l r l rþ1... l N m uch that T ¼ I 0 S 1 S 2 TAT 1 ¼ A 12 ð5þ 0 S 2 2 R mm 2 R mm where ½S 1 S 2 Š ¼ ½0 I m ŠU T V~ ¼ U^ V " # ^ ¼ ^ 1 0 V~ def ¼½v 1...v r v rþ1... v N m Š and U 2 R NN V2R ðn m Þ ð N m Þ orthogonal and ^ 1 > 0 ^ 1 2 R ðn m Þ ð N m Þ diagonal reulting from a ingular value decompoition, if and only if ½I N m 0 ŠV~ ha full rank. Proof. For brevity, the proof i ketched. In Herrmann [18], the full proof i provided. Suppoe Aume without lo of generality h V ¼ V i ¼ ¼ diagðl 1 l 2 l 3... l N m Þ then oberving the tructure of TAT 1 ðtvþ ¼ ðtvþ and uing the fact that ½I N m 0ŠeV i invertible it follow that TAT 1 ha the tructure of (5) and ha the eigenvalue 1. Note that if the eigenvalue of A are all real and ditinct then there exit at leat one choice of ðn mþ eigenvalue o that there i a tranformation matrix T and a matrix A with the eigenvalue a choen. Thu, the tabilized ytem (3) can be tranformed for a uitable choice of eigenvalue into: z : 1ðÞ¼z t 1 ðþþa t 12 ðþ t : ðþ¼ t ðþþ t ðs 2 B 2 Þu NL ðþ t where the table matrice and and A 12 are defined in (5). By contruction, the hyperplane defining the liding urface i given by S¼ z 1 : ¼ 0 Suppoe the ub-tate canbeforcedtoremainonthe liding mode hyperplane in finite time then uncertainty and diturbance within the range pace of B 2 can be completely rejected and the control ytem dynamic are governed by the reduced order dynamic z : 1 ¼ z 1.Definea Lyapunov matrix P ¼ P T > 0 ðp 2 R mm Þ, where P þ T P ¼ I m and uppoe uncertainty and diturbance are in the range pace of B 2 and are parametric or contant bounded, then the non-linear control ignal ð u NL ¼ S 2B 2 Þ 1 P 1 z T 1 kpkþ T þ 2 NL ð6þ NL > can achieve liding, ¼ 0, for NL ¼ 0 in finite time and large enough 1 2 [17]. However in practice, the choice NL > 0 i ued to prevent chattering, high frequency

6 134 G. Herrmann et al. / Journal of Proce Control 13 (2003) witching of the non-linear control ([14] and [17], p. 15). Thi implie that ultimately peudo-liding motion, kk4 ð NL Þ, can be aured only where ðþ : R ) R i an increaing continuou function atifying ð0þ ¼ 0. The application of an oberver alo reduce chattering of the control: uch control tructure have been ued in practice by Bhatti [15] and have been reported by Utkin [16] a practically important. However, trict mathematical proof of robut tability for thi linear oberver/ non-linear controller et-up have not been hown o far. A quetion remain in the cae of the availability of everal choice for the liding mode plane a to which choice i preferable. For high order plant, the number N! of choice (max. ðn mþ!m! ) prevent teting of each poibility via imulation. A criterion involving performance or robutne, for aement of the liding mode plane prior to imulation, i preferable. It ha been argued by Edward and Spurgeon [17], uing idea of robut eigentructure aignment, if the condition number of the matrix of right hand eigenvector ½I N m 0ŠV of i minimized then the liding mode pole become inenitive to uncertaintie or perturbation of the ytem matrix A. The minimization of the condition number provide a robutne meaure for the ytem in liding motion, but doe not necearily imply good performance. It i therefore better to evaluate pole combination with condition number cloe to the optimum. A earch algorithm from Cao and Roiter [9] baed on the branch-and-bound technique can be ued. The algorithm i generic and i applicable to any combinatorial problem where N m element have to be choen with a minimal cot J(N m) from a bai et of N element. Each election of k ð0 4 k 4 N mþ, element i aociated with a cot J(k) which monotonically increae with k, J(k+1) 5 J(k), when adding a new element to the exiting et of k element. Hence, conider the following characteritic: & Remark 1. A matrix ½# 1 # 2...# kþ1 Š of ðk þ 1Þ ðð1 4 k < N mþ m 5 NÞ, linearly independent vector # ðþ 2 R ðn mþ ha exactly k+1 non-zero ingular value where the larget ingular value i 1 = ðkþ1þ and the mallet i ðkþ1þ= ðkþ1þ > 0. For the matrix ½# 1 # 2...# k Š of k vector the larget ingular value i 1=k > 0 and the mallet i k=k > 0. It follow 1=k = k=k 4 1= ðkþ1þ = ðkþ1þ= ðkþ1þ : For (k+1)=(n m) the relation 1= ðkþ1þ = ðkþ1þ= ðkþ1þ i the condition number of a quare matrix. Thu, the monotonicity of J(k)= 1=k = k=k in k can be eaily ued with the et of n available vector ½I N m 0ŠV with V from (4). However, the algorithm ha to conider that a election of vector V containing a complex valued vector ha to include the complex conjugate eigenvector. The branch-and-bound algorithm allow thi modification and it ha been ued for deign decribed in the next ection. 4. Controller deign for the HDA-plant The control deign take the two diturbance, ix actuator and the five output meaurement into account. Thu, a model linearization G HDA ðj! Þ of 270 tate, eight input and five output for the operation point of the nominal value of Table 1 and 2 (middle of required value range) and a production rate of 265 lb mol/h i conidered. For model order reduction and controller deign, input and output caling I c ¼ diagði 1 I 2... I 8 Þ, O c ¼ diagðo 1 O 2... O 5 Þ (Table 1 3) have been ued. From an evaluation of the Hankel ingular value of the balanced realization of O c G HDA ðjwþi c, a 10 20th order model wa een to be mot appropriate. Balanced truncation ha been preferred to achieve good high frequency model matching becaue of the uncertainty aociated with the linearization at low frequency. A 17th order model G~ HDA ha been obtained. The H 1 controller from Section 3 employ the derived order reduced model and the weight: W ur ðþ¼diag ð1:9! ur 2:3! ur! ur 0:1! ur! ur! ur ðþ Þ þ 1! ur ¼ þ W uu ðþ¼diag 0:0077! uu 0:009! uu 0:009! uu 0:01! uu 0:01! uu 0:01! uu! uu ¼ 450:93 þ 4: þ 3: þ 5: þ 2: þ 1: þ 2: W y ðþ¼diag 0:1 þ21:43 0:7143 þ 2:069 1:111 þ 2 0:9091 þ 4:444 þ 1:444 W p ðþ¼diag 0:5 0:01 0:01 0:5 þ0:01 þ 0:01 ð7þ where the deigner choen weight W ur ha been introduced to limit the aggreivene of the actuator ignal, while W y wa ued to achieve the required performance of a ettling time of le than 5 h. The diturbance have been modeled a rather low procee via W p. W uu model additive uncertainty. A the plant i highly nonlinear and highly integrated, a change of one parameter, uch a the operation point of the ga and toluene feed (Fig. 1), affect many other plant variable and ubequently the dynamic repone of the plant to a change of any actuator or diturbance. Hence, it i difficult to obtain a complete tranfer function model of the actual uncertainty due to the non-linearity in the operation area around the nominal et-point of G~ HDA. However, by changing each of the available eight input, ix actuator (Table 3) and two diturbance (Table 2), 1% from it nominal value one at a time, ixteen

7 G. Herrmann et al. / Journal of Proce Control 13 (2003) model linearization G j are obtained to evaluate the degree of uncertainty. Thi procedure keep the etpoint in reaonable proximity of the nominal et-point and doe not introduce an unnecearily high, puriou uncertainty which i introduced by teting all available combination. An envelope of max j G j G~ HDA ha been fitted with a tranfer function. The reulting tranfer function wa ued to contruct W uu. Safonov pole hifting methodology [12] i ued for the H 1 - deign, hifting the augmented plant pole by a value of p~= for deign. The choice of plant model order and the weight reult in a control ytem order of 48 tate and an additional five tate for the enitivity weight W y and integrator which are included in the cloed loop. The numerical H 1 -optimization with the choen deign weight give a cloed loop norm of T yh ½ 0 r 0 Š 1 4 2:3. For the pecified controller deign weight, the reulting controller ha the property that W uu K 1 W y S (Fig. 5b) for part of the frequency range hence, robutne to plant non-linearitie ha been acrificed for improved controller performance. Since W uu K 1 W y S 1 i contant in large part of the conidered frequency range, a further reduction of thi norm by H 1 -optimization i not poible without performance degradation. Thu, thi trade-off lead to an indirect reduction of the effective operation area of the cloed loop controller. Thi procedure i common in applied H 1 -control (e.g. [19]) a the primary interet i to achieve practically robut performance in the operation area of interet. The cloed loop performance and robutne of the linear control for the non-linear HDA-plant i improved by the introduction of a liding motion uing the linear oberver derived from the deigned H 1 -control. For thi reaon, the linear control i modified to introduce pole eparation (Section 3.1). Thi i found to be feaible ince A 2 from (2) i mall compared to A 3 ka 2 k ¼ 0: ka 3 k ¼ kak ¼ 2: and the matrix ða 3 þa 2 Þ i table. The liding mode plane i choen o that the condition number of ½I N m 0ŠV i mall. The pole combination with condition number value of yield good performance. The condition number of the eigenvector matrix for ome other pole choice can reach value of the order indicating poor robutne to unmatched uncertaintie. It Fig. 5. Frequency repone of H 1 -controller for linearized plant model.

8 136 G. Herrmann et al. / Journal of Proce Control 13 (2003) ha been found that other ad-hoc pole election cheme for the liding mode were le ucceful. The parameter for the non-linear control element from (6) have been choen with 1 ¼ ¼ 0, which allowed practically fat attainment of the peudo-liding mode ^ 1 while NL ¼ ha been choen to prevent chattering of the control and numerical inaccuracy of the imulation. Fig. 6. Production rate tracking repone of H 1 and linear oberver baed liding mode control Controller performance The linear H 1 -controller wa deigned o that the ettling time of the linear repone were le than 5 h. The non-linear imulation how fat rie time for the production rate (Fig. 6), but long ettling time. Thi reult from the fact that low frequency uncertainty i particularly high (Section 2) which only allow the deigner to obtain fat initial dynamic. Interaction with the remaining four output meaurement for the H 1 - controller i relatively low (Fig. 7). Sytematic tet with controller employing variou deign weight have hown that increaing the bandwidth of the enitivity function S 1 of the linear cloed loop H 1 -controller by adjuting the weight W y doe not reult in a fater repone with improved interaction of the non-linear imulated cloed loop ytem while a decreae of the required bandwidth would decreae rie time and ubequently ettling. Thi ha been particularly detected for the purity meaurement repone. Subequently, a liding mode controller i deigned baed on the H 1 - controller. Thi reduce the repective interaction and give fater ettling time for the four output meaurement from Fig. 7, while for the production rate the tracking repone characteritic are very imilar to the linear control. Thi ha been alo verified employing Fig. 7. Tracking error of linear oberver-baed control due to production demand (ee Fig. 6).

9 G. Herrmann et al. / Journal of Proce Control 13 (2003) Table 4 lb mol Mean abolute error evaluating the tracking error for everal demand change between 255 and 280 h in the interval ð0 h 54:5hÞ Control method Production tracking error Purity Hydr.-to-arom.-ratio Flah outlet vapour preure Flah inlet temperature H Sliding mode model i ued for control deign. Therefore, the linear 270-tate model retrieved via a oftware linearization tool had to be order-reduced. The balanced truncation methodology reducing the plant model to 17 tate ha been found to introduce minor model error compared to the linear 270-tate model. The large low frequency gain uncertainty of the HDA-proce affect the ettling time of the linear control and thi can be improved by introducing a non-linear liding-mode control. The nonlinear control i table for a wide operating region and i robut to meaurement delay. Fig. 8. Non-linear repone of the two H 1 -baed controller to a production rate ramp demand change during 0.5 and 1 h from 265 to 240 (, linear, liding mode controller). lb mol h 1 output data ampled at 3min and analyzed via the mean abolute error (Table 4) of the tracking error (y r): e j ¼ 1 X N y N j ðt i Þ r j ðt i Þ j ¼ ð8þ i¼1 The mean abolute error reduce by at leat 30% up to 90% for the four interaction output which i a ignificant improvement for general plant operation and product quality. The non-linear controller alo operate in a wider envelope than the H 1 -controller. The liding mode controller remain table for production rate lb mol demand below 240 h, while the H1 -controller lb mol become ocillatory for a demand of 240 h and untable below thi value (Fig. 8). The liding mode controller and the linear H 1 -controller are both robut to product purity meaurement delay of more than 20 min. The liding mode control generally how better cloed loop repone to mall cooling-effluent temperature diturbance than the H 1 - control, while for purge-down tream preure diturbance both controller have imilar performance. 5. Concluion An oberver baed liding-mode control cheme ha been preented and applied to the highly non-linear, benzene producing HDA-plant. The plant ha a particular highly non-linear teady tate characteritic. A linear Acknowledgement The author would like to acknowledge the upport for G. Herrmann from the European Commiion (TMR-grant, project number: FMBICT983463). Reference [1] J.M. Dougla, Conceptual Deign of Chemical Procee, McGraw Hill, [2] J.J. McKetta, W.A. Cunningham, Encyclopedia of Chemical Proceing and Deign, Marcel Dekker, New York, [3] R. Smith, Chemical Proce Deign, McGraw Hill, New York, [4] C. Brognaux, A Cae Study in Operability Analyi: The HDA plant, Mater Thei, Imperial College, London, Prince Conort Rd., London SW7 2BY, [5] Y. Cao, D. Roiter, D. Owen, Input election for diturbance rejection under manipulated variable contraint, Computer and Chemical Engineering 21 (SS) (1997) S403 S408. [6] W.L. Luyben, M.L. Luyben, B.D. Tyreu, Plantwide Proce Control, McGraw-Hill, New York, [7] Y. Cao, D. Roiter, D. Edward, J. Knechtel, D. Owen, Modelling iue for control tructure election in a chemical proce, Computer and Chemical Engineering 22 (SS) (1998) S411 S418. [8] Y. Cao, D. Roiter, An input pre-creening technique for control tructure election, Computer and Chemical Engineering 21 (6) (1997) [9] Cao, Y., Roiter, D., Globally optimal control tructure election uing the branch-and-bound method, in: Proceeding of the 5th IFAC Sympoium on Dynamic and Control of Proce Sytem, Corfu, Greece, [10] Y. Cao, D. Roiter, D. Owen, Controllability improvement uing econdary meaurement, in: Proceeding of the Conf. on Advanced Proce Control 5, [11] K. Zhou, J.C. Doyle, K. Glover, Robut and Optimal Control, Prentice Hall, NJ, [12] R.Y. Chiang, M.G. Safonov, Robut control toolbox, in Matlab Toolbox Uer Guide, Mathwork Inc., [13] P. Colaneri, J.C. Geromel, A. Locatelli, Control Theory and

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