Habituating Control for Nonsquare Nonlinear Processes

Transcription

1 Ind. Eng. Chem. Res. 1996, 35, Habituating Contol fo Nonsquae Nonlinea Pocesses Richad B. McLain, Michael J. Kutz, and Michael A. Henson* Depatment of Chemical Engineeing, Louisiana State Univesity, Baton Rouge, Louisiana Fancis J. Doyle III School of Chemical Engineeing, Pudue Univesity, West Lafayette, Indiana A contolle design stategy fo nonlinea systems with moe manipulated inputs than contolled outputs is poposed. The technique is called habituating contol as a esult of its similaity to contol schemes commonly used in biological systems. Nonlinea contol laws that povide inputoutput lineaization while simultaneously minimizing the cost of affecting contol ae deived. In the single-output case, the cost function employed diffes accoding to the elative degees of the two inputs. Local stability analysis shows that the esulting contolle can povide a simple solution to the singulaity and nonminimum phase poblems. An extension of the contolle design stategy fo multiple-output pocesses also is pesented. The poposed method is evaluated using nonlinea models of chemical and biochemical eaction systems. 1. Intoduction Feedback contol systems typically employ equal numbes of manipulated inputs and contolled outputs. In many applications, supeio pefomance and obustness can be achieved by intoducing additional input o output vaiables. A well-established example of this appoach is cascade contol, whee a second output measuement allows impoved distubance ejection using the existing manipulated input. Also widely studied is the intoduction of additional input vaiables to fom a nonsquae system with moe inputs than outputs. A vaiety of linea contolle design techniques have been poposed fo both the single-output (Henson et al., 1995; Popiel et al., 1986; Shinskey, 1978) and multiple-output (Medanic, 1993; Muske and Rawlings, 1993; Siljak, 198) cases. Significantly fewe esults ae available fo nonlinea systems with moe inputs than outputs. The design of input-output lineaizing contolles fo nonminimum phase nonlinea systems with a single output and two inputs has been investigated (Doyle et al., 199; Mc- Clamoch and Schumache, 1993). The fist input is used to achieve input-output lineaization, while the second input is used to stabilize the othewise unstable zeo dynamics. Nonlinea model pedictive contol (Bequette, 1991; Rawlings et al., 1994) povides a systematic means fo handling nonsquae nonlinea systems with multiple outputs. Howeve, this method has seveal disadvantages including lage computational equiements. In this pape, we popose an input-output lineaizing contol stategy fo nonsquae nonlinea pocesses with moe manipulated inputs than contolled outputs. The undelying pemise is that the contol objectives can be satisfied moe easily by utilizing additional input vaiables. Because the additional inputs povide exta degees of feedom, the nonlinea contolle is designed to povide input-output lineaization at the minimum cost. As explained below, the technique is called habituating contol due to its similaity to contol schemes utilized in biological systems. The emainde of the pape is oganized as follows. The biological motivation fo the habituating contol * To whom coespondence should be addessed. Phone: Fax: henson@nlc.che. lsu.edu. S (96)85- CCC: $1. stategy is discussed in section. Section 3 contains a detailed pesentation of the nonlinea contolle design technique fo the single-output case. An extension fo multiple-output pocesses is pesented in section 4. The poposed method is evaluated via two simulation examples in section 5. Finally, section 6 contains a summay and conclusions.. Biological Motivation The baoecepto eflex (baoeflex) is esponsible fo shot-tem egulation of ateial blood pessue (Kichheim, 1976; Sagawa, 1983). Blood pessue measuements ae povided by baoecepto neuons located within the ateial walls. The pessue measuements ae integated with othe cadioespiatoy signals in the lowe bain stem. The integated sensoy infomation is pocessed by two distinct neual contolles, the paasympathetic and sympathetic nevous systems. These contolles maintain blood pessue at the desied value by manipulating cadiac output and vascula esistance. The paasympathetic system affects only cadiac output, while the sympathetic system pimaily affects vascula esistance. As a esult, the effect of the paasympathetic system on ateial pessue is quite apid as compaed to that of the sympathetic system. Howeve, sustained vaiations in cadiac output ae physiologically expensive as compaed to long-tem vaiations in peipheal esistance. Theefoe, the paasympathetic system is used pefeentially duing tansient conditions, while the sympathetic system is pimaily esponsible fo steady-state contol. In conjunction with collaboatos fom DuPont, the thid autho has developed linea habituating contol stategies by evese engineeing this biological contol system (Henson et al., 1995). The technique is applicable to linea systems with a single contolled output and two manipulated inputs which diffe in tems of thei elative costs and dynamic effects. The pimay input, which is analogous to the vascula esistance, is used mainly fo steady-state contol. The seconday input is analogous to the cadiac output; it is used only duing tansients unde nomal opeating conditions. Contolle design techniques based on diect synthesis and model pedictive contol have been developed. Simulation esults demonstate the supeio pefomance that can be achieved with habituating contol 1996 Ameican Chemical Society

2 468 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 as compaed to conventional techniques that utilize a single manipulated input. In this pape, we popose a habituating contol stategy fo pocesses descibed by nonlinea pocess models. 3. Single-Output Pocesses Initially we conside the following class of nonlinea systems, x )f(x)+g 1 (x)u 1 +g (x)u (1) y ) h(x) whee x is an n-dimensional vecto of state vaiables, u 1 and u ae scala manipulated input vaiables, and y is a scala output vaiable. We assume that the state vecto is measued o estimated fom available measuements. The objective is to design nonlinea feedback contol laws fo u 1 and u such that input-output lineaization is achieved. The following notation (Henson and Sebog, 1991; Isidoi, 1989; Kavais and Kanto, 199) is useful. The Lie deivative of the scala field h(x) with espect to the vecto field f(x) is defined as: L f h(x) [ h(x) x ] T f(x) () Highe-ode Lie deivatives ae defined ecusively: L k f h(x) [ L k-1 T f h(x) x ] f(x) (3) The ith input u i has elative degee i at the point x if i is the smallest intege such that L gi L i -1 f h(x ) *. Utilizing both inputs to achieve input-output lineaization offes seveal potential advantages as compaed to the standad appoach of using a single input. As discussed below, singula points (Henson and Sebog, 199; Hischon and Davis, 1987) and unstable zeo dynamics (Doyle et al., 1996; Hause et al., 199a) may peclude exact lineaization using the pimay input u 1. We show that it may be possible to ovecome such obstuctions by intoducing a seconday input u. It is impotant to note that the lineaization objective does not yield unique contol laws; an additional objective must be specified to obtain a well-defined contol poblem. We design the contol laws to minimize a pefomance index which coesponds to the cost of affecting contol. The index diffes accoding to the elative degees of the two inputs Equal Relative Degees. Fist we assume that the two inputs have equal elative degees: 1 ). In this case, the th deivative of the output can be epesented as: y () ) L f h(x) + L g1 L f -1 h(x) u 1 + L g L f -1 h(x) u (4) Without loss of geneality, we use the fist input u 1 to achieve input-output lineaization: v - L f h(x) - L g L -1 f h(x) u u 1 ) L g1 L -1 f h(x) (5) Unde this contol law, the closed-loop system has a linea input-output map: y () ) v. Consequently, the new input v can be designed to place the closed-loop poles and to povide integal action fo offset-fee tacking, v )-R L -1 f h(x)-... -R L f h(x)+r 1 [y sp - h(x)] + t R [ysp - h(x)] dτ (6) whee y sp is the setpoint and R i ae contolle tuning paametes chosen such that the chaacteistic polynomial s +1 +R s R 1 s+r is Huwitz. Note that the input-output lineaizing contol law (5) is a function of the second input u. The objective is to design the contol law fo u such that the cost associated with affecting contol is minimized. It is impotant to note that this appoach is moe geneal, and moe biologically plausible (Schmidt et al., 1971), than linea contolle design techniques (Henson et al., 1995; Popiel et al., 1986; Shinskey, 1978) in which the moe expensive input is etuned to its esting value at steady state. We popose the following cost function: whee uj i and γ i ae the desied steady-state value and the cost of input u i, espectively. Note that the cost function penalizes instantaneous deviations of the inputs fom thei steady-state values. Minimizing I with espect to u yields whee I ) 1 γ 1(u 1 - uj 1 ) + 1 γ (u - uj ) (7) di du ) ) γ 1 (u 1 - uj 1 ) u 1 u + γ (u - uj ) (8) u 1 L g L -1 f h(x) )- u L g1 L -1 f h(x) Solving (5) and (8) fo u yields the following state feedback contol law: RL g L -1 f h(x) u ) [L g1 L -1 f h(x)] +R[L g L -1 f h(x)] [v - L fh(x)] + [L g1 L -1 f h(x)] uj -RL g1 L -1 f h(x)l g L -1 f h(x)uj 1 (1) [L g1 L -1 f h(x)] +R[L g L -1 f h(x)] (9) whee R)γ 1 /γ. In pactice, R may be employed as a tuning paamete that detemines the elative contibutions of the two inputs. By substituting (1) into (5), the following contol law fo u 1 is obtained: L g1 L -1 f h(x) u 1 ) [L g1 L -1 f h(x)] +R[L g L -1 f h(x)] [v - L fh(x)] + R[L g L -1 f h(x)] uj 1 - L g1 L -1 f h(x) L g L -1 f h(x) uj (11) [L g1 L -1 f h(x)] +R[L g L -1 f h(x)] It is insightful to examine the contol laws (1) and (11) fo limiting values of the tuning paamete R. In the limit as R f, u ) uj and the contol law fo u 1 becomes:

3 Ind. Eng. Chem. Res., Vol. 35, No. 11, By constuction, the manipulated inputs w 1 and w have the same elative degee ). The extended system can be used to design the nonlinea contol laws as descibed in the pevious section, whee the fom of the input v in (6) is modified accodingly. It is impotant to emembe that w 1 ) u (µ) 1 when analyzing the esulting contol laws. Based on the equal elative degee case, the following esults ae easily deived: 1. In the limit as R f, u ) uj and u 1 povides the lineaizing feedback.. In the limit as R f, u (µ) 1 ) uj (µ) 1 ). If u 1 () ) uj 1 and the system is initially at est, then u 1 ) uj 1 and u povides the lineaizing feedback. 3. At points whee L g1 L 1-1 f h(x) ), u (µ) 1 ) uj (µ) 1 ) and both inputs contibute to the lineaizing feedback. If µ ) 1, then u 1 is held constant on the singulaity manifold. 4. At points whee L g L -1 f h(x) ), u ) uj and u 1 povides the lineaizing feedback. 5. At steady state, u ) uj and u 1 maintains the output at the setpoint. By analogy to linea habituating contol (Henson et al., 1995), the final esult shows that u 1 and u can be identified as the pimay input and seconday input, espectively Local Stability. Next we pefom a local stability analysis of the closed-loop system obtained with the habituating contolle. Of paticula inteest is the case whee the zeo dynamics associated with one of the inputs is unstable. Standad input-output lineaization techniques based on a single input cannot be applied to such nonminimum phase systems. Below we show that nonlinea habituating contol can povide an effective method fo ovecoming the nonminimum phase poblem. The zeo dynamics associated with the input u 1 ae constucted as follows. Unde the lineaizing contol law (1), thee exists a nonlinea coodinate tansfou 1 ) 1 L L g1 L -1 f h(x) [v - L g L -1 f h(x) fh(x)] - L g1 L -1 f h(x) uj (1) z 1 ) z z µ-1 ) z µ (14) This coesponds to the case whee the cost associated with manipulating u is much highe than the cost of manipulating u 1. Consequently, u 1 povides the lineaizing feedback, while u is maintained at its steady-state value. In the limit as R f, u 1 ) uj 1 and the contol law fo u becomes: u ) 1 L L g L -1 f h(x) [v - L g1 L -1 f h(x) fh(x)] - L g L -1 f h(x) uj 1 (13) In this case, the cost of manipulating u 1 is much highe than the cost of manipulating u. Theefoe, u is active and u 1 is held at its steady-state value. The poposed contol stategy can povide a simple and effective means fo ovecoming the singulaity poblem. The point x is temed a singula point with espect to u i if L gi L -1 f h(x ) ), but L gi L -1 f h(x) * fo some points x in a neighbohood of x (Henson and Sebog, 199; Hischon and Davis, 1987). Standad input-output lineaizing contol laws based on a single input u i ae not well-defined on the singulaity manifold, which is defined as: M s ) {x: L gi L -1 f h(x) ) }. As an illustation, conside the lineaizing contol law (1) obtained when u is held at its steady-state value. Because the tem L g1 L -1 f h(x) appeas in the denominato, the contolle poduces unbounded values of u 1 on the singulaity manifold. Although design techniques fo systems with singulaities have been poposed, the esulting contol laws povide only appoximate lineaization and/o ae difficult to analyze (Castillo, 1991; Couch et al., 1991; Hause et al., 199b). By contast, singulaities ae handled easily with the habituating contol stategy. On the singulaity manifold whee L g1 L -1 f h(x) ), u 1 ) uj 1 and the contol law (13) with the last tem vanishing is obtained fo u.at points whee L g L -1 f h(x) ), u ) uj and the contol law fo u 1 is (1) with the last tem vanishing. Note that the contol laws ae not well-defined at points whee L g1 L -1 f h(x) ) L g L -1 f h(x) ). 3.. Diffeent Relative Degees. Now the contolle design pocedue is genealized to systems in which the two inputs have diffeent elative degees. Without loss of geneality, assume that the elative degee of the fist input is less than the elative degee of the second input: 1 <. When computing deivatives of the output, we assume that u appeas via the vecto field f(x) athe than g 1 (x). This simplifies the contolle design since it ensues that u will fist appea in the th deivative via the function L g L -1 f h(x). The fist step is to constuct an extended system (Henson and Sebog, 199; Nijmeije and van de Schaft, 199) in which the two manipulated inputs have the same elative degee. The extended system is obtained by intoducing µ) - 1 integatos into the u 1 input channel, z µ ) w 1 u 1 ) z 1 whee z i epesent contolle state vaiables and w 1 is a new manipulated input that eplaces u 1 in the contolle design. The extended system has the following state-space epesentation: [ x z 1 ] [f(x) + g1(x) z1 [ [g(x) z ) + ]w1 + ]u (15) z µ-1 z µ z µ 1 ] By defining x e ) [x T z T ] T and w ) u, the extended system can be ewitten as: y ) h(x) x e ) f e (x e ) + g 1e (x e ) w 1 + g e (x e ) w (16) y ) h e (x e )

4 47 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 mation [ξ T, η T ] ) Φ T (x) such that the nonlinea system (1) has the following nomal fom epesentation (Isidoi, 1989): We assume that the ank of the matix A(x) at the point x is geate than o equal to p. This is a necessay and sufficient condition fo achieving local input-output decoupling with static state feedback (Isidoi, 1989). Unde this assumption, the input vecto can be patiξ )Aξ+Bv η )q(ξ,η) (17) y ) Cξ The zeo dynamics ae defined as dynamics of the (n - 1 )-dimensional nonlinea subsystem when the vaiables of the 1 -dimensional linea subsystem ξ ) : η )q(,η) (18) Stability of the zeo dynamics is a necessay and sufficient condition fo the contol law (1) to yield local closed-loop stability (Bynes and Isidoi, 1988). The zeo dynamics associated with the input u can be deived similaly. Note that the intoduction of integatos does not affect the stability of the zeo dynamics (Isidoi, 1989). Stability analysis is based on the Jacobian appoximation of the extended nonlinea system (16): x e ) Ax e + b 1 w 1 + b w (19) y ) cx e The input-output behavio of the lineaized system can be epesented as, y(s) ) β n-1 s n- 1 + β n-1-1 sn β 1 s + β w s µ (s n +R n-1 s n-1 1 (s) R 1 s+r ) s n- βˆn- +βˆn- -1 sn βˆ 1s + βˆ w s n +R n-1 s n-1 (s) R 1 s+r N(s) s µ D(s) w(s)+nˆ(s) 1 D(s) w (s) () whee N(s)/D(s) and Nˆ (s)/d(s) ae (possibly nonminimal) ealizations of the tansfe functions y(s)/u 1 (s) and y(s)/u (s), espectively. It is impotant to note that the zeos of the tansfe function y(s)/u 1 (s) ae identically equal to the eigenvalues of the lineaized vesion of the zeo dynamics (18) (Isidoi, 1989). Similaly, the zeos of the tansfe function y(s)/u (s) equal the eigenvalues of the lineaized zeo dynamics associated with u. Theoem 1. If the chaacteistic polynomial β n-1 N(s) +Rβˆn- s µ Nˆ(s)) (1) associated with the lineaized zeo dynamics of the extended nonlinea system (16) is Huwitz, then the habituating contolle is locally stabilizing. The poof is pesented in the Appendix. If the two inputs have equal elative degees (µ ) ), Theoem 1 shows that the lineaized zeo dynamics of the single-input, single-output systems u 1 /y and u /y ae ecoveed in the limit as R f and R f, espectively. By contast, only the zeo dynamics of u 1 /y can be ecoveed when the elative degees ae diffeent (µ g 1). Coollay 1 follows diectly fom these obsevations. Coollay 1. Thee exists a tuning paamete R [, ) such that the habituating contolle is locally stabilizing if (i) the two inputs have equal elative degees and the lineaized zeo dynamics associated with eithe u 1 o u is stable o (ii) the two inputs have diffeent elative degees and the lineaized zeo dynamics associated with u 1 is stable. Fo systems that do not satisfy the conditions of Coollay 1, thee may exist values of R such that (1) is a Huwitz polynomial and the closed-loop system is locally stable. Coollay povides a necessay condition fo the existence of a stabilizing R. Coollay. If the two inputs have diffeent elative degees and the lineaized zeo dynamics associated with u 1 ae unstable, then the habituating contolle is locally stabilizing only if: (i) the lineaized zeo dynamics associated with u 1 and u do not have common eigenvalues with positive eal pat; and (ii) the lineaized zeo dynamics associated with u 1 has an even numbe of eigenvalues with positive eal pat. The fist condition follows diectly fom (1), while a poof fo the second condition is pesented in the Appendix. 4. Multiple-Output Pocesses Now we genealize the nonlinea habituating contol technique to multiple-output systems of the fom x )f(x)+g(x)u () y ) h(x) whee x is an n-dimensional vecto of state vaiables, u is an m-dimensional vecto of manipulated inputs, and y is a p-dimensional vecto of contolled outputs. We assume that the numbe of inputs is stictly geate than the numbe of outputs (m > p). The objective is to design state feedback contol laws such that the input-output esponse is both linea and decoupled and the cost of affecting contol is minimized. The ith output is said to have elative degee i at the point x if i is the smallest intege such that L gj L i -1 f h i (x ) * fo at least one j [1, m]. Theefoe, time deivatives of the outputs can be epesented as (Isidoi, 1989): ( [y1 1 )] ) [Lf 1 h 1 ) + ( y p p L p f h p (x)] L 1-1 f h 1 (x) L gm L 1-1 f h 1 (x) L g1 L p -1 f h p (x) L gm L p -1 f h p (x)]u [Lg1 b(x)+a(x)u(3)

5 tioned such that ( [y1 1 )] ) ) b(x) + A 1 (x) u 1 + A (x) u (4) ( y p p whee u 1 is a p-dimensional vecto, u is an (m - p)- dimensional vecto, A 1 (x) is a p p matix that is invetible at x, and A (x) isap m-pmatix. Note that the patitioning may not be unique. The inputoutput decoupling contol law is obtained by setting the output deivatives equal to an m-dimensional vecto of new inputs and solving the esulting equation fo u 1 : u 1 ) A 1-1 (x) [v-b(x)-a (x)u ] (5) Unde this contol law, the closed-loop system has a linea and decoupled input-output map: y i ( i ) ) v i. Consequently, the new input v i can be designed as in the single-output case. We use the additional manipulated inputs u to design a second state feedback contol law that minimizes the cost of affecting contol. The cost function utilized is I ) 1 (u 1 - uj 1 )T Γ 1 (u 1 - uj 1 ) + 1 (u - uj )T Γ (u - uj ) I 1 + I (6) whee uj 1 is the desied steady-state value of u 1 and Γ 1 is a diagonal matix with non-negative elements that coespond to the cost of manipulating the individual inputs. The vecto uj and matix Γ epesent analogous quantities fo u. Minimizing I with espect to u yields di ) I 1 u 1 + I ) du u 1 u u -(A -1 1 A ) T Γ 1 (u 1 - uj 1 ) + Γ (u - uj ) ) (7) whee the state dependence has been omitted fo simplicity. The state feedback contol laws esult fom simultaneous solution of the two sets of equations (5) and (7): [ A 1 A -(A -1 1 A ) T Γ 1 Γ ][ u 1 u ] ) [ v - b -(A 1-1 A ) T Γ 1 uj 1 + Γ uj ] (8) The habituating contolle exists if and only if these equations have a unique solution at x. A poof of the following theoem is pesented in the Appendix: Theoem. The habituating contolle exists if ank[a 1 (x )] ) p and eithe of the following conditions hold: (i) ank(γ ) ) m - p o (ii) ank(γ 1 ) ) p and ank[a -1 1 (x )A (x )] ) m - p. Unde these conditions, (8) can be solved to yield Ind. Eng. Chem. Res., Vol. 35, No. 11, u 1 ) [A (A -1 1 A ) -1 (A -1 1 A ) T Γ 1 A -1 1 ](v - b) + (A -1 1 A ) -1 (A -1 1 A ) T Γ 1 uj 1 - (A -1 1 A ) -1 Γ uj (9) u ) -1 (A -1 1 A ) T Γ 1 A -1 1 (v - b) - -1 (A -1 1 A ) T Γ 1 uj Γ u whee the matix Γ + (A -1 1 A ) T Γ 1 (A -1 1 A )isinvetible at x. It is inteesting to note that the nonlinea habituating contolle educes to the standad inputoutput decoupling contolle (Isidoi, 1989) when the cost associated with manipulating u is much highe than the cost of manipulating u 1 (Γ 1 ) ). In this case, the contol law (9) yields (5) with u ) uj. A moe complicated esult is obtained if the cost of manipulating u 1 is vey high as compaed to the cost of manipulating u (Γ ) ). Both u 1 and u ae needed to achieve inputoutput decoupling, in geneal, although only u is utilized if thee ae twice as many inputs as outputs (m ) p). Note that this condition is satisfied in the single-output case. The habituating contol technique can be advantageous fo pocesses that have singula decoupling matices with espect to the pimay inputs (Kavais and Sooush, 199). In this case, seconday inputs ae intoduced and the input vecto is patitioned such that the system is input-output decouplable. Then the weighting matices (Γ 1, Γ ) ae used to detemine the elative contibution of the individual inputs. This epesents a vey simple and effective appoach as compaed to standad input-output decoupling techniques, which do not utilize all the available inputs (Isidoi, 1989) o poduce a complicated dynamic contol law using just the pimay inputs (Nijmeije and Respondek, 1988). We cuently ae investigating the stability popeties of the habituating contolle in the multiple-output case. 5. Simulation Examples Chemical Reacto. Fist we apply the habituating contol stategy to a nonlinea chemical eacto. The pocess model descibes a evesible eaction A h B that occus in a constant-volume, stied-tank eacto (Economou et al., 1986), Ċ A ) q V (C Ai - C A ) - k 1 (T) C A + k (T) C B Ċ B ) q V (C Bi - C B ) + k 1 (T) C A - k (T) C B (3) Ṫ ) q V (T i - T) + (- H) FC p [k 1 (T) C A - k (T) C B ] whee k 1 (T) ) C 1 exp(-e 1 /RT) and k (T) ) C exp(-e / RT). Symbol definitions and nominal opeating conditions ae given in Table 1. Economou et al. (1986) have designed a nonlinea intenal model contolle fo this system using the feed tempeatue T i and effluent concentation C B as the manipulated input (u 1 ) and contolled output (y), espectively. Fo this choice of vaiables, the elative degee 1 ) and the lineaizing contol law is singula on the manifold: E 1 k 1 (T) C B ) E 1 k 1 (T) + E k (T) (31)

6 47 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 Table 1. Nominal Opeating Conditions fo Chemical Reacto symbol definition nominal value q inlet flow ate 1 L/s C Ai inlet concentation of A 1 mol/l C Bi inlet concentation of B mol/l T i inlet tempeatue 39.4 K V eacto volume 6 L C 1 peexponential facto fo s -1 fowad eaction C peexponential facto fo s -1 evese eaction E 1 activation enegy fo cal/mol fowad eaction E activation enegy fo cal/mol evese eaction - H heat of eaction 5 cal/mol F density 1 kg/l C p heat capacity 1 cal/kg K C A effluent concentation of A.6 mol/l C B effluent concentation of B.4 mol/l T eacto tempeatue K It is inteesting to note that the optimum convesion (C B /C Ai ).58) belongs to the singulaity manifold. Consequently, input-output lineaization should not be applied if T i is the only input vaiable. We conside intoducing an additional manipulated input to ovecome the singulaity poblem. Assume that opeational equiements dictate that the thoughput q emain constant. In this case, the inlet concentation C Ai may be chosen as the second input. Howeve, lage excusions of C Ai fom its nominal value may lead to inceased aw mateial costs. Theefoe, it is desiable to use both T i and C Ai as manipulated inputs. It is easy to show that C Ai has elative degee ) and the lineaized zeo dynamics associated with both inputs is stable fo all opeating points of inteest. We compae a nonlinea habituating contolle (NHC) that manipulates both T i and C Ai and an input-output lineaizing contolle (IOLC) that manipulates only T i. In both cases, the input v is designed to yield the closedloop chaacteistic polynomial (ɛs + 1) 3 ), whee ɛ ) 15 s. NHC is tuned with R)5 1-5, Th i ) 39.4 K, and Ch Ai ) 1 mol/l. The contolles ae compaed fo a setpoint change to the optimum convesion (whee C B ).58 g/l) in Figue 1. The contolles appea to yield the same output esponse. Howeve, IOLC poduces vey lage contol moves as the singulaity at the optimum is appoached. As a esult, the simulation fails completely at appoximately t ) 185 s. NHC geneates easonable T i changes by employing C Ai as an additional manipulated input. Note that only small vaiations in C Ai ae needed to avoid the singulaity. Figue 1 also shows that NHC can handle much lage setpoint changes (C B ).6 g/l) without equiing lage contol moves. In Figue, the contolles ae compaed fo a sudden change in the activation enegy of the second eaction (E ) cal/mol) while opeating at a constant setpoint. IOLC is unable to handle the distubance because the singulaity manifold is encounteed. As a esult, the simulation fails at t ) 31 s. NHC povides smooth distubance ejection and easonable contol moves. Note that a faste esponse could be obtained by etuning the contolle. In Figues 3 and 4, we utilize the following initial conditions: C A () ) C B () ).5 g/l, T() ) 44.9, T i () ) 4.4 K. Note that these values coespond to a steady state much close to the optimum convesion than the steady state in Table 1. NHC is etuned with Figue 1. IOLC and NHC fo setpoint changes (chemical eacto). Figue. IOLC and NHC fo a sudden change in the activation enegy E (chemical eacto). Th i ) 4.4 K to account fo the initial condition change. Figue 3 compaes NHC and IOLC fo a step distubance in the inlet flow ate (q ) 1.1 L/s). IOLC is unable to handle the distubance because a singulaity is encounteed, and the simulation fails at t ) 1 s. By contast, NHC povides effective distubance ejection by employing C Ai as an additional manipulated input. Figue 3 also shows that NHC can handle much

7 Ind. Eng. Chem. Res., Vol. 35, No. 11, Table. Nominal Opeating Conditions fo Biochemical Reacto symbol definition nominal value D dilution ate.4 h -1 S f feed substate concentation g/l Y X/S cell-mass yield.4 g/g R kinetic paamete. g/g β kinetic paamete. h -1 µ m maximum specific gowth ate.48 h -1 P m poduct satuation constant 5 g/l K m substate satuation constant 1. g/l K i substate inhibition constant g/l X biomass concentation 6.9 g/l S substate concentation 4.76 g/l P poduct concentation 43.9 g/l Biochemical Reacto. Now we apply the habituating contol stategy to the following biochemical eacto model (Henson and Sebog, 199): whee the gowth ate µ is Ẋ ) -DX + µx Ṡ ) D(S f - S) - 1 Y X/S µx (3) Ṗ ) -DP + (Rµ + β)x Figue 3. IOLC and NHC fo step distubances in the inlet flow ate (chemical eacto). Figue 4. Effect of R on NHC pefomance (chemical eacto). lage flow ate distubances (q ) 1.33 L/s). The effect of the NHC tuning paamete R on closed-loop pefomance fo a flow ate distubance (q ) 1.33 L/s) is shown in Figue 4. As expected, the contibution of the second input C Ai inceases as R is inceased. In this case, lage values of R yield impoved distubance ejection. µ ) µ m (1 - P/P m )S K m + S + S /K i (33) Symbol definitions and nominal opeating conditions ae given in Table. A common choice fo the manipulated input (u 1 ) and contolled output (y) ae the dilution ate D and the substate concentation S, espectively. These vaiables yield a well-defined elative degee 1 ) 1, but the lineaized zeo dynamics associated with u 1 ae unstable at the opeating point in Table. As a esult, stable input-output lineaization cannot be achieved using D as the only input vaiable. An intenally stable closed-loop system can be obtained by employing the feed substate concentation S f as an additional manipulated input. Howeve, lage vaiations in S f ae undesiable in some applications. We compae a nonlinea habituating contolle that manipulates both D and S f and an input-output lineaizing contolle that manipulates only D. The dilution ate is constained as e D e.1 h -1. To obtain the contol affine fom (1), the second input fo NHC design is defined as u ) DS f. It is easy to show that this input has elative degee ) 1 and stable lineaized zeo dynamics. Both contolles utilize an input v that is designed to yield the closed-loop chaacteistic polynomial (ɛs + 1) ), whee ɛ ) 3 h. NHC employs taget values that coespond to the opeating conditions in Table : Dh ).4 h -1, Sh f ) g/l. Theoem 1 can be used to detemine the ange of R values that yield a locally stable closed-loop system. As mentioned above, the lineaized zeo dynamics associated with D ae unstable when S f is not used. Howeve, the stability of zeo dynamics associated with D is changed damatically when S f is intoduced as an additional input. It can be shown that the lineaized zeo dynamics actually ae stable in this case. This seemingly anomalous esult is attibutable to the definition of the second input as u ) DS f. Consequently, Theoem 1 shows that NHC povides local stability when <R< ; we choose R).

8 474 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 Figue 5. IOLC and NHC fo stabilization at the nominal opeating point (biochemical eacto). Figue 6. eacto). IOLC and NHC fo setpoint changes (biochemical Figue 7. IOLC and NHC fo a sudden change in the cell-mass yield (biochemical eacto). Figue 5 compaes the two contolles fo opeation at the nominal opeating point in the pesence of a vey small initial condition eo. IOLC is unable to stabilize the system as a esult of the unstable zeo dynamics, and D satuates at the uppe constaint. When the constaint is emoved, D is inceased such that the system moves into the minimum phase egion. The new steady state coesponds to the desied substate concentation, but the poduct concentation (19.1 g/l) is much less than the value in Table. This behavio is undesiable in applications whee a low dilution ate and/o high poduct concentation ae desied. Figue 6 compaes the contolles fo positive (5.5 g/l) and negative (4 g/l) step changes in the substate setpoint. IOLC cannot handle eithe setpoint change. The positive setpoint change causes D to satuate at the uppe constaint, while the negative change causes satuation at the lowe constaint. When the constaints ae emoved, the positive change can be handled as the system moves into the minimum phase egion. Howeve, the closed-loop system is unstable fo the negative change. By contast, NHC povides effective tacking of both setpoint changes by utilizing S f as a second manipulated input. The contolles ae compaed fo a sudden change in the cell-mass yield (Y X/S ).45 g/g) in Figue 7. Due to the unstable zeo dynamics, IOLC cannot handle the distubance as D satuates at the lowe constaint. The system moves into the minimum phase egion when the constaint is emoved. NHC povides excellent distubance ejection by using both D and S f. Figue 8 shows the effect of the NHC tuning paamete R fo the same paamete change. The distubance ejection pefomance is compaable fo all thee values of R. Note that the utilization of both D and S f is inceased as R is deceased. This behavio can be explained by consideing the NHC cost function (7). In this example, a contol affine system is obtained by defining the second input as u ) DS f. Unexpected contol moves ae obseved because DS f is penalized athe than S f.

9 Ind. Eng. Chem. Res., Vol. 35, No. 11, degee. The associated tansfe function model () allows a convenient epesentation of the linea plant opeato in obsevability canonical state-space fom: z ) ( -R 1 -R 1 1 -R n- )z + 1 -R n--1 -R n- 1 -R n-1 Figue 8. effect of R on NHC pefomance (biochemical eacto). 6. Summay and Conclusions By emulating a contol stategy used in biological systems, we have developed a contolle design technique fo nonlinea pocesses with moe manipulated inputs than contolled outputs. The motivation fo habituating contol is that impoved closed-loop pefomance can be achieved if all the available inputs ae utilized. The nonlinea contolle povides inputoutput lineaization while simultaneously minimizing the cost of affecting contol. In the single-output case, we have shown that the poposed method can povide a simple means to ovecome the singulaity and nonminimum phase poblems. An extension of the contolle design stategy fo multiple-output pocesses also has been pesented. The habituating contol technique was successfully applied to nonlinea chemical and biochemical eacto models. Acknowledgment R.B.M., M.J.K., and M.A.H. acknowledge NSF suppot though a Caee Development Awad (CTS ). F.J.D. acknowledges NSF suppot though a Young Investigato Awad (CTS-95759). (β β 1 β n- Coesponding to this state-space model, we intoduce the following definition of Ã, b 1, b, and c : It is shown by Isidoi (1989) that one can constuct an invetible tansfomation, ( ξ1 ξ ξ 3 ) ( ) c z c Ãz c à z ) Tx ) (36) ξ c à -1 z η 1 z 1 η n- z n- y ) (... 1)z )u1 + (βˆ βˆ 1 βˆ n- )u (34) z )Ãz+b 1u 1 +b u (35) y ) c z Appendix Poof of Theoem 1. Fist we conside the Jacobian lineaization (19) of the extended nonlinea system assuming that the two inputs have equal elative which patitions the state vecto into obsevable (ξ) and unobsevable (η) vaiables. Intenal stability of the input-output lineaized system equies that the unobsevable dynamics ae stable. The unfoced subsystem of unobsevable state vaiables, also known as the zeo dynamics, is obtained by setting ξ ) :

10 +( 476 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 β 1 β 1 η )( + 1 β n-- 1 )η β ( n--1)u1 βˆ βˆ1 (37) βˆn-- βˆn--1)u The Jacobian lineaization of the habituating contolle (1)-(11) is given by: u 1 ) u ) β n- β n- β n- +Rβˆn- Rβˆ n- +Rβˆn- (v-z n- ) (38) (v-z n- ) When these elations ae substituted into (37), the following esult is obtained: η ) β ( β n- +Rβˆβˆn- - β n- +Rβˆn- 1 - β 1 β n- +Rβˆ1 βˆn- β n- +Rβˆn- )η 1 - β n-- β n- +Rβˆn-- βˆn- β n- +Rβˆn- 1 - β n--1 β n- +Rβˆn--1 βˆn- β n- +Rβˆn- (39) It is staightfowad to show that the stability of this subsystem is equivalent to the condition stated in the theoem; that is, the polynomial Rβˆ n- Nˆ (s) + β n- N(s) (4) is Huwitz. Now conside the case whee the two inputs have diffeent elative degees. The tansfe function model () can be manipulated such that the individual tansfe functions have a common denominato: y(s) ) N(s) w 1 (s) + sµ Nˆ (s) w (s) s µ D(s) Ñ 1 (s) w 1 (s) + Ñ (s) w (s) (41) D (s) This model has the same fom as that in the equal elative degee case. Recall that the final esult hinged upon the stability of a weighted vesion of the individual numeato polynomials. In this case the numeato polynomial fo w is modified as shown, and the esult in the theoem is obtained. Poof of Coollay. Let N(s) be the numeato polynomial associated with u 1 and Nˆ (s) be the numeato polynomial fo u : N(s) ) β n-1 s n- 1 + β n-1-1 sn β 1 s + β Nˆ (s) ) βˆ n- s n- + βˆ n- -1 sn βˆ 1s + βˆ We want to check the oots of the chaacteistic equation: β n-1 N(s) +Rβˆn- s - 1 Nˆ(s) M(s) By assumption, the input u 1 is nonminimum phase and has a lesse elative degee than u (i.e., N(s) contains ight-half plane oots and - 1 > ). We define two eal-valued functions ove the polynomials: Note the following: Thus, to pove the coollay, it is sufficient to show that β n-1 β e fo an odd numbe of ight-half plane (RHP) zeos since the fist citeion of the Routh-Huwitz test fails in this case. Fist note that β n-1 *. If β n-1 β ), then β ), which implies that M(s) has a oot at the oigin fo any value of the tuning paamete R and any numbe of additional closed RHP oots. This implies M(s) isnot Huwitz. Now assume β * and wite N(s) ) Q(s) P(s), whee Q(s) contains all the open left-half plane (LHP) oots of N(s) and P(s) contains all the closed ight-half plane (RHP) oots. Since Q(s) has only negative oots, the following must be tue: Futhe patition P(s) ) P c (s) P (s), whee P c (s) contains all the closed RHP complex oots and P (s) contains all the RHP eal oots. Fist we conside the contibutions fom P c (s), whee a i c i > and a i b i e. It is clea that the following holds: Now we conside the contibution of P (s), Rg {LC(p(x)): F[x] f R} coefficient of the lowest degee of x in p(x) {HC(p(x)): F[x] f R} coefficient of the highest degee of x in p(x) HC(M(s)) ) β n-1 LC(M(s)) ) β n-1 β +Rβ n- HC(Q(s)) LC(Q(s)) > > Rg P c (s) ) (a 1 s + b 1 s + c 1 )(a s + b s + c )... (a l s + b l s + c l ) HC(P c (s)) LC(P c (s)) ) (a 1 a... a l )(c 1 c... c l ) ) (a 1 c 1 )(a c )... (a l c l ) > P (s) ) (a 1 s + b 1 )(a s + b )... (a p s + b p )

11 whee a i, b i R ae such that a i b i <. If p is even and if an odd numbe of a i <, then an odd numbe of b i <. Similaly, if p is even and an even numbe of a i <, then an even numbe of b i <. This shows that HC(P (s)) < w LC(P (s)) < HC(P (s)) > w LC(P (s)) > which implies that HC(P (s)) LC(P (s)) > if thee is an even numbe of eal oots in P (s). If p is odd then, P (s) ) (γs + δ)(a p-1 s p a ) whee γδ < and a p-1 a >. Thus, if p is odd: HC(P (s)) LC(P (s)) ) (γa p-1 )(δa ) ) (γδ)(a p-1 a ) < We now can combine all the contibutions to N(s): β n-1 β ) HC(N(s)) LC(N(s)) ) HC(Q(s)) HC(P c (s)) HC(P (s)) LC(Q(s)) LC(P c (s)) LC(P (s)) ) [HC(Q(s)) LC(Q(s))][HC(P c (s)) LC(P c (s))][hc(p (s)) LC(P (s))] Thus, β n-1 β { < if N(s) contains an odd numbe of RHP oots > if N(s) contains an even numbe of RHP oots and the poof is complete. Poof of Theoem. The set of equations (8) has a unique solution if the m m matix [ A 1 (x) A (x) -(A -1 1 (x) A (x)) T Γ 1 Γ ] (4) has full ank m at x. Fo simplicity, we omit the state dependence of the matices. Because A 1 is invetible by assumption, the fist set of equations in (8) can be pemultiplied by A -1 1 to yield the matix: [ I p A -1 1 A -(A -1 1 A ) T (43) Γ 1 Γ ] This matix can be ewitten as follows afte a simple ow eduction opeation: [ I p A -1 1 A Γ + (A -1 1 A ) T Γ 1 (A -1 (44) 1 A ) ] Because of its block diagonal stuctue, this matix is full ank, and theefoe the equations (8) have a unique solution, if: ank[γ + (A -1 1 A ) T Γ 1 (A -1 1 A )] ) m - p (45) It is easy to show that this condition holds if eithe of the two assumptions in the theoem ae satisfied. Liteatue Cited Bequette, B. W. Nonlinea Contol of Chemical Pocesses: A Review. Ind. Eng. Chem. Res. 1991, 3, Bynes, C. I.; Isidoi, A. Local Stabilization of Minimum-Phase Nonlinea Systems. Syst. Contol Lett. 1988, 11, Castillo, B. Output Tacking Though Singula Points fo a Class of Nonlinea Systems. In Poc. Eu. Contol Conf Couch, P. 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