Approximate Feedback Control for Uncertain Nonlinear Systems

1420

Ind. Eng. Chem. Res. 1999, 38, 1420-1431

PROCESS DESIGN AND CONTROL Approximate Feedback Control for Uncertain Nonlinear Systems Wei Wu† Department of Chemical Engineering, Hwa Hsia College of Technology and Commerce, Taipei 235, Taiwan, Republic of China

Recent developments in geometrical linearization techniques cannot be directly applied to nonlinear chemical processes, such as those with unstable zeros. This paper deals with the problem of designing a robust tracking controller for uncertain nonlinear systems with a nonminimum phase. With the approximate linearization and parametric transformation algorithms, the actual output can converge to the desired trajectory with an arbitrary degree of accuracy. When the bounds of “lumped” uncertainties can be available, with the use of the easily estimating and feasible tuning scheme the overall system stability can be guaranteed. Inspired by the model-based control strategy, we introduce a minimum-phase, approximate model as an open-loop observer such that the observer-based controller has stable inverse and the asymptotic output regulation for uncertain, nonminimum-phase systems can be achieved. Finally, in the illustrative example, an adiabatic stirred-tank reactor using the Van de Vusse reaction, the present methodologies are verified, obtaining the expected results. 1. Introduction In recent years, the developments in feedback linearization control theory have given a new perspective to the problems associated with nonlinear control design. Feedback linearization control technique has been used successfully to address some practical control problems (Henson and Seborg, 1997). However, there are also a number of serious shortcomings and limitations associated with the feedback linearization control approach. In essence, the central idea of this approach is to transform algebraically a nonlinear process under stable nonlinear inversion condition into a linear one, and then linear control techniques can be applied. In other words, the input-output (I/O) linearization approach needs to satisfy the minimum-phase condition. However, for many practical chemical processes, such as a CSTR process with Van de Vusse reaction (Kravaris and Daoutidis, 1990; Wright and Kravaris, 1992), the intrinsic nonminimum phase property unfortunately resides. It is well recognized that an effective controller should be able to explicitly or implicitly generate a process inverse. When dealing with nonminimum-phase linear systems, an appropriate decomposition of the process model into a stable part and an unstable part is necessary and the controller must invert only the part with stable inverse (Morari and Zafiriou, 1989). In the nonlinear case, a controller that generates a process inverse is also the dominant idea in general synthesis methods for minimum-phase processes (Kravaris and Chung, 1987; Henson and Seborg, 1991). How to construct the appropriate control algorithm for processes with unstable zeros has recently been a popular issue. † Telephone: 886-2-2949-0418. Fax: E-mail: [email protected].

886-2-8668-8164.

During the past decade, some research groups have proposed the useful nonlinear control strategies for a class of nonminimum-phase processes. Kravaris and Daoutidis (1990) first proposed a first-order nonlinear all-pass to make the closed-loop system equivalent, but it is available only for the second-order system. Wright and Kravaris (1992) presented a minimum-phase output predictor to estimate a statically equivalent, minimumphase output on-line and control it to set point. Kravaris et al. (1994) introduced a Smith-type abstract operator that allows the reduction of the controller synthesis problem for nonminimum-phase processes to one for minimum-phase processes. One assumption central in all predictor-type approaches and necessary in their works is that the nonminimum-phase system is required to be open-loop stable. Allgo¨wer (1996) has recently proposed an approximate stable/antistable factorization for nonlinear processes with poorly behaved zero dynamics, e.g., an exothermic Van de Vusse reaction (Doyle et al., 1992). With respect to the approximate input-output linearization technique, Hauser et al. (1992) have proposed a novel method for the design of a feedback linearizing control law based on a minimumphase approximation to the actual system such that the closed-loop system has bounded state trajectory. However, this technique is only applicable for nonlinear systems with slight nonminimum-phase; it means that the corresponding unstable zeros via the Jacobian linearization are very close to the imaginary axis. Lian et al. (1993) have developed a composite control structure which consists of a fast and a slow feedback control for a class of singular perturbation plants with weak nonminimum-phase. Benvenuti et al. (1994) have proposed approximating a class of nonminimum-phase systems by minimum-phase systems. Their main idea is to modify the output of the nonlinear system with a transformation performed on the Jacobian linearization

10.1021/ie9806500 CCC: $18.00 © 1999 American Chemical Society Published on Web 03/11/1999

Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999 1421

assumptions, the mass and energy balances describing the dynamic behavior of the process take the form

Table 1. Nominal Parameters ∆H1 ) -5 kJ‚kmol-1 ∆H2 ) -15 kJ‚kmol-1 ∆H3 ) -20 kJ‚kmol-1 k10 ) 50 s-1 k20 ) 100 s-1 k30 ) 10 m3‚kmol-1‚s-1 R ) 8.314 kJ‚kmol-1‚K-1

E1 ) 15 kJ‚kmol-1 E2 ) 25 kJ‚kmol-1 E3 ) 15 kJ‚kmol-1 CAf ) 10 kmol‚m-3 Tf ) 300 K Vr ) 1 m3 F0cp ) 1 kJ‚m-3‚K-1

Vr

Vr

of the system. In the recent approaches, Gopalswamy and Hedrick (1993) utilized a new output function as well as a sliding control for stabilizing internal dynamics such that the unacceptable zero dynamics with respect to this new output become acceptable. Wu (1999) provided a stable inverse control technique for finding a bounded control and state trajectories such that the output tracking error between the actual output and thedesired trajectory can be minimized in the finite time interval. Despite the significant progress and growing interest in nonlinear, nonminimum-phase systems, the robust output tracking control design for nonlinear, nonminimum-phase systems with modeling errors and unmeasurable disturbances has rarely been addressed. In fact, for real processes where there exist inevitable uncertainties in those constructed models, the robust output tracking control design is an important subject. Inspired by the works of Hauser et al. (1992) and Gopalswamy and Hedrick (1993), we propose the approximate state/ output feedback controls for achieving output tracking control to ensure the stable control of a continuously stirred tank reactor (CSTR) with the Van de Vasse reaction and practical applications. The main remarkable result is that the system states are ultimately bounded and the output tracking errors will converge to a prescribed bounded set within the finite time. In addition, the complete theoretical results are given to prove these properties. The simulation results show that the proposed methodology has guaranteed a more reliable control strategy.

To address the control problem of nonlinear, nonminimum-phase processes, we first present a CSTR process with the Van de Vusse reaction that will be used in the main discussion. We then work through the details of input-output linearization for this CSTR process and present illustration results showing the qualitative behavior of the nonminimum-phase process. We consider an adiabatic CSTR (Doyle et al., 1992 (for the exothermic case)), where the series/parallel Van de Vusse reaction is taking place: k2

A 98 B 98 C k3

(1)

2A 98 D The rates of formation of A and B are

rA ) -k1(T)CA - k3(T)CA2 rB ) k1(T)CA - k2(T)CB

F0cp

dCB ) F(-CB) + Vr(k1(T)CA - k2(T)CB) dt

(2)

where ki(T) ) ki0 exp(-Ei/RT), i )1, 2, 3. Under standard

(3)

dT ) -∆H1k1(T)CA - ∆H2k2(T)CB dt ∆H3k3(T)CA2 +

F (T - T) Vr f

where F is the inlet flow rate of component A, Vr is the reactor volume, and CA and CB are the concentrations of A and B inside the reactor, respectively. The concentration of A in the feed stream is given by CAf. It is desired to maintain CB at a constant value by manipulating the dilution rate F/Vr. The system may be put into a standard state space form by letting

x1 ) CA x2 ) CB x3 ) T u)

F Vr

which results in

x˘ 1 ) f1(x) + g1(x)u ) -k1(x3)x1 - k3(x3)x12 + u(CAf - x1) x˘ 2 ) f2(x) + g2(x)u ) k1(x3)x1 - k2(x3)x2 - ux2

2. Input-Output Linearization for a CSTR

k1

dCA ) F(CAf - CA) + Vr(-k1(T)CA - k3(T)CA2) dt

(4)

x˘ 3 ) f3(x) + g3(x)u ) (F0cp)-1[-∆H1k1(x3)x1 - ∆H2k2(x3)x2 ∆H3k3(x3)x12 + u(Tf - x3)] The parameters of the reactor model are given in Table 1. Furthermore, we consider a stable equilibrium point as the initial condition of the model, for us ) 1.067 kJ‚s-1 there exists initials, x1s ) 0.202 kmol/m3, x2s ) 0.1 kmol/ m3 and x3s ) 494.467 K. To manipulate concentration CB from initials to track the desired concentration trajectory, variable x2 can be denoted as the controlled output and y ) h(x) ) x2. Proceeding in the geometrical linearization algorithm (Kravaris and Chung, 1987), we differentiate the output until the manipulated input appears. Since the system relative order is one, the static state feedback control can be directly obtained

u)

k1(x3)x1 - k2(x3)x2 - ν , y ) x2 * 0 x2

(5)

where ν is a new input. Then the closed-loop system is

1422 Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999

y˘ ) ν x˘ 1 ) -k1(x3)x1 - k3(x3)x12 +

(6a) CAf - x1

[k1(x3)x1 - k2(x3)y - ν]

{

x2

(6b)

x˘ 3 ) (F0cp)-1 -∆H1k1(x3)x1 - ∆H2k2(x3)y - ∆H3k3(x3) ×

} (6c)

T f - x3 y

x12 + [k1(x3)x1 - k2(x3)y - ν]

This linearizing control makes its input-output map linear, but the internal dynamics of x˘ 1 and x˘ 3 in parts band c of eq 6 are unobservable. To guarantee the internal stability of the system, parts b and c of eq 6 can be denoted as zero dynamics by setting v ) 0 and y ) x2s, and they are usually described as η˘ ) q(0,η). Furthermore, the use of local linearization at operating points of zero dynamics (parts b and c of eq 6) is shown as

η˘ )

(∂η∂q)

η)ηs

(η - ηs)

(7)

where ηs is an equilibrium point. Then, the zeros of closed-loop system in eq 6 can be determined by Laplace transformation of eq 7 and written as

-0.1s2 + 482.5s + 515.1 ) 0

(8)

Analysis of eq 8 has one positive root, so this closedloop model (6) has an RHP zero. Parts A and B of Figure 1 depict that the unobservable modes of x˘ 1 and x˘ 3 have unbounded behavior, respectively, and the corresponding input responses are also unbounded as shown in Figure 1C. Therefore, the conventional I/O linearization cannot be directly applied to this nonminimum-phase nonlinear system. From the above analysis and simulations, we think that the unbounded behavior in the system could be caused by the control of CB by manipulating the dilution rate F/Vr, which should produce changes in concentrations of the reactor and in heat generation producing internal instabilities. Besides, the true plant often contains modeling errors or unmeasurable disturbances; for example, the CSTR in the example may suffer from inlet flow rate perturbation. Therefore, how to improve this inappropriate control design for nonminimumphase systems as well as stabilize the uncertain system will be addressed as follows. 3. Problem Statement and Control Designs In this section, we consider the control of uncertain nonlinear systems of the form

Figure 1. Closed-loop profile of states and input, in the case of using the static state feedback: (A) Response of concentration of A (x1); (B) corresponding response of reactor temperature (x3); (C) corresponding control input.

x˘ ) f(x) + g(x)u + E(x,d)

x˘ ) f(x) + g(x)u

y ) h(x)

(9)

where x ∈ Rn, u, y ∈ R. f:Rn f Rn and g:Rn f Rn are unknow but smooth vector fields. h:Rn f R is a scalar function. d ∈ Rm represent the unmeasureable disturbance and the nonlinear function E:Rn × Rm f Rn which can be denoted as a “lumped” uncertainty. The nominal system, i.e. E(‚) ) 0 in eq 9, has nonminimumphase behavior and is shown as

y ) h(x)

(10)

In the following, we consider the operating condition to be located at an open neighborhood U of the equilibrium point. All statements that we make, such as the existence of certain diffeomorphism, can be assumed merely to hold in U. Next, we will precisely define and develop the tracking control problem of nonminimumphase systems.


3.1. Approximate State Feedback Control (ASFC). To introduce our design concepts clearly and completely, the technique of approximate linearization is first addressed. The method presented here can be viewed as a simple extension of the procedure proposed by Hauser et al. (1992) which uses the approximate I/O linearization to overcome the nonlinear system with slight nonminimum phase. Suppose that the relative order of the nominal system is one and define the following two sets of local diffeomorphisms, for all x ∈ Rn,

(ξT,ηT)T ) $(x) ) (ξ1 ) h(x), η1(x), ..., ηn-1(x))T

(11)

and

Lgh(x) ) ω(x) LgLfh(x) ) ω(x)λ1(x) l

(18)

F-2

LgLf

h(x) ) ω(x)λF-2(x)

LgLfF-1h(x) ) β(ω,x) where F > r ) 1. The notations λ1(x), ..., λF-2(x), and β(ω,x) are smooth nonlinear functions. The approximate relative order F of the system can be determined by the nonlinear function β(ω,x) * ω(x)λF-1(x). Under the above expression, the approximate transformation can be defined as

(ξT,η˜ T)T ) (h(x),Lfh(x),...,LfF-1h(x),η˜ 1(x),...,η˜ n-F(x))T

(ξ˜ T, η˜ T)T ) $ ˜ (x)

(19)

) (ξ˜ 1 ) h(x),ξ˜ 2 ) Lfh(x),η˜ 1(x), ..., η˜ n-2(x))T (12)

with Lgη˜ i(x) ) 0, i ) 1, 2, ..., n - F. Then the approximate linearized system can be shown as

ξ˜˙ 1 ) ξ˜ 2 + ω(x)u

with

Lgηi(x) ) 0, i ) 1, 2, ..., n - 1

(13)

and

Lgη˜ i(x) ) 0, i ) 1, 2, ..., n - 2

(14)

Via the coordinate in eq 11, the nominal system (eq10) in normal form is shown as

ξ˙ 1 ) Lfh(x) + Lgh(x)u η˘ ) q(ξ, η)

(15)

where q(ξ,η) ) ((∂η/∂x)f(x))o$ ˜ -1(ξ,η). The approximate model via the coordinate in eq 12 can be written as

ξ˜˙ 1 ) ξ˜ 2 ξ˜˙ 2 ) Lf h(x) + LgLfh(x)u 2

ξ˜˙ F-1 ) ξ˜ F + ω(x)λF-2(x)u

(20)

ξ˜˙ F ) LfFh(x) + LgLfF-1h(x)u η˜˙ ) q˜ (ξ˜ ,η˜ ) LgLfF-1h(x)

Moreover, for ) β(ω,x) * 0, the approximate state feedback control can be expressed as

u ) Ψ(x,v) )

-LfFh(x) + v LgLfF-1h(x)

(21)

where v is the new control input. If y is required to track the desired trajectory yd, then we can choose v as

v ) y(F) d - R1(h(x) - yd) - R2(Lfh(x) - yd) - ... ) (22) RF(LfF-1h(x) - y(F-1) d

(16)

η˜˙ ) q˜ (ξ˜ ,η˜ ) where q˜ (ξ˜ ,η˜ ) ) (∂η˜ /∂x)f(x))oω ˜ -1(ξ˜ ,η˜ ). Since the nominal model has nonminimum-phase behavior, the zero dynamics, q(0,η), in eq 15 is unstable. Let us define a nonlinear function ω(x) ) Lgh(x), the approximate linearized system through the specified coordinate in eq 12 can be described as

ξ˜˙ 1 ) ξ˜ 2 + ω(x)u ξ˜˙ 2 ) Lf2h(x) + LgLfh(x)u

l

(17)

η˜˙ ) q˜ (ξ˜ ,η˜ ) If the new zero dynamics, q˜ (0,η˜ ), in eq 17 is stable and a value of |ω(x)| is sufficiently small, then we can think of this nominal system as having a slight nonminimumphase. The preceding discussion may be generalized to the case whose difference in relative order between the nominal model in eq 15 and the approximate model in eq 16 is greater than one. Let

where y(i) d represents the derivatives of the desired and smooth signal. The parameters Ri > 0 should be chosen such that

sF + RFsF-1 + ... + R1

(23)

is a Hurwitz polynomial. Therefore, the new input v as a stabilizing feedback for the linearized system in eq 20. Substituting eqs 21 and 22 into eq 20 and using the following definitions

e1 ) h(x) - yd e2 ) Lfh(x) - y˘ d l

(24)

eF ) LfF-1h(x) - y(F-1) d we can write the closed-loop nominal system as

η˜ ) q˜ (ξ˜ ,η˜ )

(25b)

[]


[ ][ e˘ 1

· · · e˘ F-1 e˘ F

0 · · ) · 0

1 · · · 0

· · · · · · · · ·

-R1 -R2 · · ·

0 · · · 1 -RF

][ ] e1

1

λ1(x) · · · · + ω(x) · u · eF-1 λF-2(x) eF 0

(25a)

Remark 1. In the review work of Hauser et al. (1992), the given parameter ‘” is replaced here by the scalar function, ωx. If it is sufficiently small and can be reduced to a constant parameter, the system can be viewed as a slight nonminimum-phase system. However, most chemical processes, such as the illustrated example, have poorly behaved internal dynamics. Therefore, the novel parametrized feedback control design aims to stabilize this approximate linearized system. In the following theorem, the bounded and stable output regulation of the nominal system (eq 10) would be established via the previous control algorithm and assumptions. Theorem 1. Suppose that the desired trajectory yd and its r derivatives are bounded, for Bd > 0, T ||(yd,y˘ d,...,y(F) d ) || e Bd

(26)

approximation is continuous as well as vanishing at operating points. Remark 3. According to this approximate feedback (eq 21) and with the assumptions of Theorem 1 being satisfied, the output tracking is bounded but a small output tracking error cannot be guaranteed. It is dueto the existence of the value of ω(x), which may be large. However, from the theoretical stability analysis in the Appendix, the magnitude of |ω(x)| must be small enough to satisfy eq A7. Therefore, we think that its magnitude can be reduced by choosing the appropriate nonlinear function, h ˜ (x), as the “pseudo” output function. Consider a scalar function, h ˜ (x), if the one-to-one map between the new output function, h ˜ (x) and the actual output function, h(x), exists such that the following inequality is satisfied

|Lgh ˜ ≡ω ˜ (x)| < |ω(x)|

(31)

and the value of F introduced in eq 18 is still constant. Moreover, the modified state feedback can be written as F ˜ (x))-1(Y(F) u)Ψ ˜ (x,Yd) ) (LgLfF-1h d - Lf h(x) F

the zero dynamics of the approximate linearized system (20) are locally exponentially stable, and the nonlinear function, λˆ (x,u) ) (u,λ1(x)u,...,λF-2(x)u,0)T, satisfies the local Lipschitz condition. Then, the system states of eq 25 will be bounded and the output tracking error is

|e1| ≡ |y - yd| e κ |ω(x)|

(27)

for some κ < ∞ and x ∈ U. The detailed proof is shown in the Appendix. Remark 2. It is clear that the first constraint in Theorem 1 is sufficient to ensure boundedness of the tracking error, so that the following inequality holds

||ξ˜ || e ||e|| + Bd

(28)

Second, the zero dynamics are locally exponentially stable. It is a typical constraint stated in the literature. In fact, owing to the local exponential stability of the zero dynamics, the converse Lyapunov theorem (Khalil, 1996) implies the existence of a Lyapunov function, Vo(η˜ ), for the zero dynamics, η˜ ) q˜ (0,η˜ ), satisfying

for LgLfF-1h ˜ (x) * 0 and Yd ) h ˜ (yd). Moreover, a similar result of Theorem 1, |e1| e κ|ω ˜ (x)|, can be directly obtained. Note that the tracking error can be improved such that the output offset would be smaller than that in the previous design (eq 21). If the system with uncertainty is considered, the bounded output tracking performance could become poor due to the lack of robust feedback design involved in the above design. 3.2. Parametrized State Feedback Control (PSFC). In this subsection, we think that the system has both nonminimum-phase behavior and unmatched uncertainty; this mathematical model has been described in eq 9. Since the nominal model can be approximated through the previous transformation, the uncertain system is then of the form

ξ˜˙ 1 ) ξ˜ 2 + ω(x)u + LEh(x) ξ˜˙ F-1 ) ξ˜ F + ω(x)λF-2(x)u + LELfF-2h(x)

(29)

∂V0 || e a4||η˜ || ∂η˜

||

(33) ξ˜˙ F ) LfFh(x) + LgLfF-1h(x)u + LELfF-1h(x) η˜˙ ) q˜ (ξ˜ ,η˜ ) +

for some positive constants ai, i ) 1, 2, 3, 4. The third constraint used in Theorem 1 is the Lipschitz condition, whereby λˆ (x,u) satisfies the inequality

||λˆ (x,u) - λˆ (xs,u)|| e L||x - xs||

(32)

l

a1||η˜ ||2 e V0(η˜ ) e a2||η˜ ||2 ∂V0 q˜ (0,η˜ ) e - a3||η˜ ||2 ∂η˜

Ri(Lfi-1h ˜ (x) - Y(i-1) )) ∑ d i)1

(30)

where L is a Lipschitz constant. Equation 30 represents the point that the perturbation in eq 25 caused by

∂η˜ E(x,d) ∂x

y ) ξ˜ 1 Since the approximate errors and transformed uncertainties in eq 33 can destabilize the overall system., we need an effective method to compensate for these perturbations. First, we introduce a parametric coordinate transformation where 0 < e 1. Then eq 33 can


||∆q˜ (x,d)|| e γˆ 4||ξ˜ || + γˆ 5

eˆ 1 ) e1 ) h(x) - yd l F-2

eˆ F-1 )

F-2

eF-1 )

(34)

(Lf

F-2

h(x) -

y(F-2) ) d

eˆ F ) F-1eF ) F-1(LfF-1h(x) - y(F-1) ) d be written as

eˆ˙ ) Aceˆ + B[F(LfFh(x) + LgLfF-1h(x)u - y(F) d ) + F

Rieˆ i] + ∆E ˆ (x,u) ∑ i)1 η˜˙ ) q˜ (ξ˜ ,η˜ ) + ∆q˜ (x,d)

(35)

e1 ) Ceˆ

[

]

where B ) [0,...,0,1]T, ∈ RF×1, C ) [1,0,...,0] ∈ R1×F, and

ω(x)u + LEh(x) l ∈ RF×1; ∆E ˆ (x,u) ) F-2 (ω(x)λF-2(x)u + LELfF-2h(x)) F-1(LELfF-1h(x)) ∂η˜ ∆q˜ (x,d) ) E(x,d) (36) ∂x Furthermore, we can choose a parametrized state feedback control law F u ) Φ(x,yd,) ) (LgLfF-1h(x))-1(y(F) d - Lf h(x) F

-F(

Rieˆ i)) ∑ i)1

(37)

such that eq 35 can be expressed as a closed-loop system as follows.

ˆ (x,u) eˆ˙ ) Aceˆ + ∆E

(38a)

η˜˙ ) q˜ (ξ˜ ,η˜ ) + ∆q˜ (x,d)

(38b)

Remark 4. Under the parametric transformation in eq 34, the approximate feedback linearizable system (eq 33) can be further transformed into an -coupling feedback linearizable system (eq 38). If the parameter is sufficiently small, this system dynamics can be separated into two parts, including the fast dynamics eˆ and the slow dynamics η˜ , then this transformed system is a classical singularly perturbed system. Since the controller in eq 37 has a small , the tracking error dynamic (eq 38a) can become so fast as to neglect the dynamic behavior of perturbation. Note that under appropriate assumptions the elastic tuning parameter played can directly attenuate the effect of the nonlinearity, ∆E ˆ (‚). More precisely, the tuning direction of would rely on the dynamic behavior of nonlinearities ∆E ˆ (‚) and ∆q˜(‚) in eq 38 and also the process constraints. If the transformed nonlinear functions ∆E ˆ (‚) and ∆q˜ (‚) are to be smooth enough, the bounded-input bounded-state (BIBS) property of closed-loop systems is necessary (Henson and Seborg, 1991). These are stated in the following assumption. Assumption 1. There exist four positive constants γˆ i, i ) 1, ..., 5, such that, for all x ∈ Rn

ˆ (x,ub)|| e γˆ 1||ξ˜ || + γˆ 2||η˜ || + γˆ 3 ||2Ps∆E

(39a)

(39b)

where Ps > 0 is a solution of eq A3 and ub represents the feasible range for controller action, i.e. |u| e ub. Remark 5. These inequalities in eq 39 aim to describe the bounds of those nonlinearities in eq 38. They provide 2-fold information: (1) The coefficients of eq 39 can be estimated, such that the feasible ranges of tuning parameters can be established. (2) The nonzero coefficients of γˆ 3 and γˆ 4 show that the perturbation is nonvanishing at operating points (Khalil, 1996; Lian et al., 1993). Furthermore, the extension of Theorem 1 is obtained with the properties as summarized in the following theorem. Theorem 2. Suppose that the desired output yd and its r derivatives are bounded, the zero dynamics of eq 38, η˜˙ ) q˜ (0,η˜ ), is locally exponentially stable, and Assumption 1 holds. Then, for ∈ [min,max], where min,max > 0, the system states of eq 38 will be bounded and the output tracking error is

|e1| e κˆ , t > 0

(40)

for some κˆ < ∞. This detailed proof is shown in the Appendix. Remark 6. Notice that the result of Theorem 2 depends only on the adjustable parameter with above assumptions being satisfied. If an arbitrary small is employed, the perfect tracking can result in an arbitrary degree of accuracy, that is limf0|e1| ) 0. Although perfect control ( f 0) is impossible, it is clear that the stable output regulation of uncertain nonminimumphase systems can be achieved for the parameter ∈ [min,max]. Besides, this control scheme could be potentially denoted as a new type of robust control scheme due to the structure of plant uncertainty being not exactly known and only with the aid of the tuning control scheme for stabilization. From a theoretical approach and practical interest, the control action must be bounded as well as f min being permitted, so that the output trajectory could converge to the bounded set and the output offset cannot be completely eliminated. From the above analysis and design, the output tracking control problem for uncertain nonminimumphase systems can be simplified as the control of an -coupling feedback linearization. However, the process constraints, such as states cannot be completely measured, need to be overcome in practical design. 3.3. Approximate Output Feedback Control (AOFC). The output feedback design for nonminimumphase systems would be addressed from an application point of view. Inspired by the stable model-based control technique for nonlinear processes (Henson and Seborg, 1991; Kravaris et al., 1994), we first choose a stable and minimum-phase model

x˜˙ ) f0(x˜ ) + g0(x˜ )u

(41)

Via the previous approximation and transformation in (42) eqs 18 and 19, this model can be transformed into

η˜˙ ) q˜ (ξ˜ ,η˜ )


ξ˜˙ 1 ) ξ˜ 2 l ξ˜˙ F-1 ) ξ˜ F ξ˜˙ F ) LfF0h(x˜ ) + Lg0LfF-1 h(x˜ )u 0 If the model in eq 41 is denoted as an open-loop observer, then the approximate output feedback controlcan be written as

h(x˜ ))-1(R1-Fed u)Φ ˜ (x˜ ,ed,) ) (Lg0LfF-1 0

Figure 2. Approximate output feedback control scheme.

F

Ri-F+i-1Lf i-1h(x˜ )) ∑ i)1

Lg0LfF0h(x˜ ) -

0

where ed ) yd - y + h(x˜ ). Moreover, with the above design, the transformed internal model can be further written as

ξ˜ 2(x) ) k1(x3)x1 - k2(x3)x2

(46)

η˜ (x) ) (CAf - x1)/(Tf - x3) The CSTR model (eq 4) can be transformed into

ξ˜˙ 1 ) ξ˜ 2

ξ˜˙ 1(x) ) ξ˜ 2(x) + ω(x)u

l ξ˜˙ F-1 ) ξ˜ F

ξ˜ 1(x) ) x2

(43)

(44)

ξ˜˙ F ) -R1-Fξ˜ 1 - ... - RF-1ξ˜ F + R1-Fed

ξ˜˙ 2(x) ) Lf2h(x) + LgLfh(x)u η˜˙ (x) )

η˜˙ ) q˜ (ξ˜ ,η˜ )

(47)

CAf - x1 -1 (x˘ 1) + (x˘ 3) Tf - x3 (Tf - x3)2

Since the internal dynamics of eq 44 is assumed stable by the previously approximate linearization algorithm and the controller with the Hurwitz coefficients (Ri), the closed-loop system of eq 44 is asymptotically stable and the asymptotic output regulation can be achieved, that is limtf∞ y ) yd. Besides, if we choose the similar transformation for eq 34,

where h(x) ) x2 and ω(x) ) -x2. To determine the stability of internal dynamics of eq 47, a positive function V0(η˜ ) ) η˜ 2 is selected with its differentiation V˙ 0(η˜ ) ) 2η˜ η˜ ; we obtain V˙ 0(η˜ ) < 0 and the asymptotical stability of zero dynamic. Since LgLfh(x) * ω(x)λ1(x), by eqs 21 and 22 the approximate state feedback control can be written as

ξˆ 1 ) h(x˜ )

u ) -(LgLfh(x))-1[Lf2h(x) + R1(ξ˜ 1 - ysp) + R2ξ˜ 2] (48)

l F-2

ξˆ F-1 )

Lf0F-2h(x˜ )

(45)

ξˆ F ) F-1Lf0F-1h(x˜ ) then eq 44 can also be reduced to an -coupling feedback linearizable model. It is well-known that the adjustable parameter is also played for output performance enhancement. Consequently, the present control structure as shown in Figure 2 is a typical nonlinear internal model control strategy for nonminimum-phase systems. From the above state and output feedback control designs, the overall system can be stabilized and the perturbation from approximate errors and uncertainties can be attenuated by virtue of the same parametric transformation. As for the small value of introduced in the above control laws, it can be denoted as the highgain feedback approach (Khalil, 1994). For a practical implementation, the high-gain feedback may result in peaking states of systems or destroy the closed-loop stability due to input constraints. Therefore, we must be careful in adjusting the parameter () by following the result of Theorem 2 which will be addressed in the following demonstration. 4. Simulation Results 4.1. Implementation of ASFC. Following the above mappings in eq 19, we choose

where ysp represents a constant step change. Although the control scheme in eq 21 can follow a desired and time-varying signal, many of the process control strategies prefer steady-state operation. To demonstrate this design, Figure 3A depicts that this CSTR can be stabilized for ysp ) 0.2 and ysp ) 0.5 if (R1,R2) ) (1,2) is fixed, and the corresponding control action is shown in Figure 3B. Since this approximate feedback may give rise to the large offset of output tracking, the tuning parameters (R1,R2) are required for detuning. In addition, this figure also shows that the output offset depends on the magnitude of |-x2|. Thus, we can draw a conclusion on how to reduce the magnitude of |ω| via the new output function, if we select h ˜ (x) ) (1/3)x23. With the following inequality

˜ ) -x23| < |Lgh ) -x2|, for 0 e x2 < 1 |Lgh

(49)

being satisfied, the modified state feedback control from eq 32 can be written as

˜ (x))-1[Lf2h ˜ (x) + R1(h ˜ (x) - Yd) + R2Lfh ˜ (x)] u ) -(LgLfh (50) with Yd ) (1/3)ysp3. To demonstrate this design, Figure 4A depicts that the output performance of the modified design (eq 50) is better than that of the previous design (eq 48), but the corresponding control action is large as


Figure 3. Closed-loop profiles for setpoint tracking, in the case of using the approximate state feedback: (A) output response of concentration of B (x2); (B) corresponding control input.

Figure 4. Closed-loop profiles for setpoint tracking, in the case of using the modified state feedback: (A) output response of concentration of B (x2) (B) corresponding control input.

shown in Figure 4B. Therefore, the control law in eq 50 can provide a larger controller action than that in eq 48. Since the true process would contain modeling errors or unmeasurable disturbances, we can suppose that the illustrated CSTR has an inlet flow rate perturbation and its perturbation is described as F(1 + δ)/Vr, for |δ| e 1. Under the same operating point and having a positive input perturbation (δ ) 2%), Figure 5A shows that the closed-loop system cannot be stabilized by the above designs of eqs 48 and 50. The corresponding control motions also become unbounded as shown in Figure 5B. From the simulation results, it is noted that the approximate feedback design can stabilize the nonminimum-phase process, but it lacks robustness. 4.2. Implementation of PSFC. To analyze the CSTR example with inlet perturbation, the transformed nonlinearities from eq 36 can be written as

as

[

∆E ˆ (x,u) ) - δux2 δu{k1(x3)(CAf - x1) + k2(x3)x2 + [k1(x3)E1x1 - k2(x3)E2x2](F0cpRx32)-1(Tf - x3)}

]

(53)

ˆ (x,u)/∂x|(xs,us), A2 ) 2Ps∆E ˆ (xs,us), A3 where A1 ) 2Ps∂∆E s s s s ) ∂∆q˜ (x,u)/∂x|(x ,u ), and A4 ) ∆q˜ (x ,u ). With the nominal parameters in Table 1, we obtain

γˆ 1 ) 130.37|δ|; γˆ 2 ) 593.90|δ|; γˆ 3 ) γˆ 4 ) γˆ 5 ) 0 Evidently, if the flowrate perturbation (δ) is measured, then all coefficients in Assumption 1 can be calculated. Moreover, the developed parametrized state feedback from eq 37 can be written as

u ) -(LgLfh(x))-1[Lf2h(x) + R1-2(ξ˜ 1 - ysp) + R2-1ξ˜ 2] (54)

(51)

In the same way, the control law of eq 50 can also be parametrized and shown as

u ) -(LgLfh ˜ (x))-1[Lf2h ˜ (x) + R1-2(h ˜ (x) - Yd) + ˜ (x)] (55) R2-1Lfh

and

∆q˜ (x,u) ) δu(x1 - CAf)/(Tf - x3)[1 - (F0cp)-1]

γˆ i ) ||Ai|| ) [λmax(AiTAi)]1/2 i ) 1, ..., 5

(52)

Under Assumption 1 and the known initial operating point (xs,us), the coefficients of eq 39 can be calculated

If (R1,R2) ) 1,2) and in eq 54, these feedback controls in eqs 54 and 55 can obviously be reduced and are equivalent to those in the previous designs (eqs 48 and 50). For the nominal process in computer simulation,


Figure 5. Closed-loop profiles for effect of inlet flowrate perturbation, in the case of using the approximate (dashed-line) and modified (solid-line) state feedback control: (A) output response of concentration of B (x2); (B) corresponding control input.

Figure 6A shows that these control feedback with ) 0.1 provide the bounded tracking performance and the performance, can obviously be improved better than that of the previous design as shown in Figures 3A and 4A. However, the corresponding control motions shown in Figure 6B will become fast and large. Moreover, to stabilize the process with inlet perturbation and determine the appropriate region of tuning parameter , we first obtain a coefficient γˆ 1 () 130.37δ*) g γ1() 130.37|δ|) where δ* g |δ| > 0. If δ* ) 0.02 is selected, then by eq A15, max e 1/(4γˆ 1) ) 0.1 is obtained. For the process with 2% inlet flowrate perturbation and the same operating condition and control parameters, Figure 7 shows that the parametrized feedback controls in eqs 54 and 55 with ) max can stabilize this CSTR with bounded inlet perturbation. It is interesting to inform the design of eq 54 that it can provide a smaller offset than that in the control design (55). However, it is a counter result to the output tracking performance as shown in Figures 4A and 6A. It is owing to the perturbation only appear in the space of control input and the modified state feedback in eq 50 has been denoted as a high-gain approach, such that the control action by eq 55 can be enlarged more than that in the other design. 4.3. Implementation of AOFC. Since part of the knowledge of states cannot be measured in practice, we

Figure 6. Closed-loop profiles for setpoint tracking, in the case using the parametrized approximate (dashed-line) and modified (solid-line) state feedback controls: (A) output response of concentration of B (x2); (B) corresponding control input.

Figure 7. Closed-loop profiles for effect of inlet flowrate perturbation, in the case of using the parametrized approximate (dashedline) and modified (solid-line) state feedback controls.

first choose a stable nonlinear model by setting ω(x) ) 0, yielding

x˜˙ 1 ) -k1(x˜ 3)x˜ 1 - k3(x˜ 3)x˜ 12 + u(CAf - x˜ 1) x˜˙ 2 ) k1(x˜ 3)x˜ 1 - k2(x˜ 3)x˜ 2

(56)

x˜˙ 3 ) (F0cp)-1[-∆H1k1(x˜ 3)x˜ 1 ∆H2k2(x˜ 3)x˜ 2 - ∆H3k3(x˜ 3)x˜ 12 + u(Tf - x˜ 3)] as an open-loop observer. According to the internal model control scheme, the approximate output feedback


ing the good regulation performance. While a negative perturbation is introduced, the corresponding Figure 9B shows that the output regulation can also be asymptotically achieved. 5. Conclusion

Figure 8. Closed-loop profiles for setpoint tracking, in the case of using the approximate output feedback control.

In this paper, the approximate state and output feedback control associated with parametric transformation for uncertain nonminimum-phase systems, such as an adiabatic stirred-tank reactor with the Van de Vusse reaction, have been developed. With complete theoretical description and detailed proofs, the uncertain nonminimum-phase nonlinear system can be successfully transformed into an -coupling feedback linearizable system, and the feasible region of the single controller parameter can be determined through the underlying Lyapunov stability and the available bounds of uncertainties. Since the appropriate tuning region for one parameter can be directly estimated and it involves the practical control problem, as discussed in the example, we believe that the developed controller can work well, is robust against uncertainty, and can be applied to a general class of nonminimum-phase systems. Acknowledgment The author would like to thank Prof. Y. S. Chou for being a great source of inspiration and also acknowledges the National Taiwan University of Science and Technology for its partial financial support. Nomenclature

Figure 9. Closed-loop profiles for effect of inlet flowrate perturbation, in the case of using the approximate output feedback control: (A) output response of concentration of B (x2) for δ ) 10%; for (B) output response of concentration of B (x2) for δ ) -10%.

A ) reactant B ) a desired product C, D ) undesired products CA ) concentration of species A in reactor, kmol‚m-3 CAf ) reactant concentration in the feed, kmol‚m-3 CB ) concentration of species B in reactor, kmol‚m-3 cp ) heat capacity of reactor, kJ‚kg-1‚K-1 Ei ) activation energy for reactions, kJ‚mmol-1 F ) volumetric feed rate, m3‚s-1 k1,k2,k3 ) rate constants for reactions k10,k20,k30 ) frequency factor for reactions r ) relative order of controlled y with respect to u R ) ideal gas constant, kJ‚mol-1‚K-1 T ) reactor temperature, K Tf ) feed temperature, K u ) control input v ) control input of linearized model Vr ) tank volume, m3 yd ) reference trajectory ysp ) set point

control from eq 43 will be

Greek Letters

u ) -(LgLfh(x˜ ))-1[Lf2h(x˜ ) - R1-2(ysp - y) +

Ri ) controller tuning parameters ) controller tuning parameters ∆Hi ) heat of reaction i, kJ‚kmol-1 F0 ) density of reactor, kmol‚m-3

R2-1Lfh(x˜ )] (57) To demonstrate this design, Figure 8 depicts that the output of the nominal CSTR can be asymptotically achieved, with the small ()0.1) employed obtaining good tracking performance if (R1,R2) ) (1,2) is fixed a priori. To test output regulation for this CSTR with a bounded perturbation, Figure 9A depicts that the output regulation can be asymptotically achieved for a positive perturbation, with the small ()0.1) employed obtain-

Math Symbols λmax(‚) ) the maximum eigenvalue of a matrix λmin(‚) ) the minimum eigenvalue of a matrix o ) composition of functions ≡ ) is defined Lf0h ) h(x) n Lfkh ) LfLfk-1h ) ∑j)1 (∂Lfk-1h/∂xj)fj(x), k ) 1, 2, ...

1430 Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999 ) Lie derivative of the scalar function h(x) with respect to the field f(x)

we have

V˙ e -

Appendix Proof of Theorem 1. Consider that the system (25) can be expressed as

e˘ ) Ace + ω(x)λˆ (x,u)

(A1)

η˜˙ ) q˜ (ξ˜ ,η˜ )

Thus, it implies that ||η˜ || and ||e|| are bounded and there exists ||η˜ || e Bc, for some Bc > 0. Using the differentiating Ve ) eTPse, we have

V˙ e e -||e||2 + |ω(x)|||e||[l1(||e|| + Bd) + l2||η˜ ||]

where Ac ∈ RF×F have been defined in eq 25a. Since the zero dynamics are assumed to be exponentially stable, a Lyapunov function for the system (eq A1) is chosen as

V(e,η˜ ) ) eT Pse + µV0(η˜ )

(A2)

2 (a4L ˜ Bd)2 ||e||2 µa3||η˜ || + (|ω(x)|l1Bd)2 + µ 4 2 a3 (A8)

(

e-1-

| |)

1 1 - - ω(x) l1 ||e||2 + 4 4 |ω(x)|2(l2Bc)2 + |ω(x)|2(l1Bd)2 (A9)

where Ps is a symmetric positive definite matrix satisfying the Lyapunov equation

Using the inequality, λmin(Ps)||e||2 e Ve e λmax(Ps)||e||2 and eq A7, the derivative of Ve by eq A9 can be written as

AcTPs + PsAc ) -I

V˙ e e -reVe + |ω(x)|2R2e

(A3)

and µ is a positive constant to be determined later. Under the third constraint of Theorem 1, the nonlinear function q˜ (ξ˜ ,η˜ ) is also locally Lipschitz and can be written as

||q˜ (ξ˜ ,η˜ ) - q˜ (0,η˜ )|| e L ˜ ||ξ˜ ||

(A4)

where γe ) 1/4λmax(Ps) and Re ) Then it implies that

[

Ve(t) e Ve(0) -

(A10)

x(l2Bc)2 + (l1Bd)2.

]

|ω(x)|2 Re2 |ω(x)|2 Re2 exp(-ret) + re re

On te basis of coordinate transformation, the required inequality can be written as

˜ -1(ξ˜ ,η˜ )|| e l1||ξ˜ || + l2||η˜ || ||2Psλˆ (x,u) o $

(A5)

(A11) Under the appropriate initial value Ve(0), the state e will converge to the ball

where L ˜ , l1 and l2 are Lipschitz constants. Taking the derivative of V(e,η˜ ) along the trajectories of eq A1, we find

∂V0 q˜ (ξ˜ ,η˜ ) ∂η˜

||e|| e κ|ω(x)| )

(

)

4λmax(Ps) 2 R λmin(Ps) e

1/2

|ω(x)|

(A12)

Proof of Theorem 2. To obtain the boundedness result, we proceed with the following Lyapunov function

V˙ e -||e||2 + 2 ω(x)|eT Psλˆ (x,u) + µ

V ˆ (eˆ ,η˜ ,) ) V1(eˆ ,) + µV0(η˜ )

e -||e||2 + |ω(x)|||e||(l1(||e|| + Bd) + l2||η˜ ||) +

(A13)

2

(

e-

(

˜ ||η˜ ||(||e|| + Bd)) µ(-a3||η˜ || + a4L

| | )

||e|| - ω(x) l1Bd 2

2

+ (|ω(x)|l1Bd)2 -

|| ||)

||e|| ˜ ) η˜ - (|ω(x)|l2 + µa4L 2

(

µa4L ˜ )2||η˜ ||2 - µa3

2

+ (|ω(x)|l2 +

)

˜ Bd ||η˜ || a4L 2 a3

2

2

+µ

( | |)

(a4L ˜ Bd) a3

3 1 - ω(x) l1 ||e||2 - µa3||η˜ ||2 2 4 1 3 e - - ω(x) l1 ||e||2 - µa3 - (|ω(x)|l2 + 2 4 (a4L ˜ Bd)2 µa4L ˜ )2 ||η˜ ||2 + (|ω(x)|l1Bd)2 + µ (A6) a3

( | |) )

Define µ* ) a3/4(l2 + satisfying

(

R4L ˜ )2,

then for all µ e µ* and

( )

1 |ω(x)| e min µ, 4l1

where V1(eˆ ,) ) eˆ TPseˆ , for all > 0 and µ > 0. Under the same assumption of Theorem 1 and inequalities of eq 39, differentiating V ˆ (‚) along the trajectory of eq 38, yielding

˜ + V ˆ˙ e -(1 - γˆ 1)||eˆ ||2 + (γˆ 1Bd + γˆ 3)||eˆ || + (µa4(L ˜ + γˆ 4) + γˆ 5)||η˜ || γˆ 5) + γˆ 2)||eˆ ||||η˜ || + µa4(Bd(L µa3||η˜ ||2

(21 - γˆ )||eˆ|| -[43µa - (µa (L˜ + γˆ ) + ˜ + γˆ ) + γˆ ) ]||η˜ || + (γˆ B + γˆ ) + µa a (B (L 2

e-

1

2

2

2

3

4

2

2

1

d

3

4

-1 2 3 4

d

4

2

γˆ 5) (A14) So we can define γ/1 ) 1/4γ1 and γ/2 ) 3a3/8(γˆ 2 + a4(L ˜ + γˆ 4))2. Then, for all

µ e γ/2 and e max ) min(µ,γ/1) (A7) we have

(A15)


||eˆ ||2 µa3 V ˆ˙ e ||η˜ ||2 + 2(γˆ 1Bd + γˆ 3)2 + 4 4 2 ˜ + γˆ 4) + γˆ 5)2 (A16) µa-1 3 a4 (Bd(L Thus, V ˆ˙ g 0 whenever ||η˜ || or ||eˆ || is large, which implies that ||η˜ || and ||eˆ || must be bounded. Therefore, there exists ||η|| e Bc, for some Bc > 0. Using V1(eˆ ,) and its derivative, we obtain

V˙ 1 e -||eˆ ||2 + ||eˆ ||(γˆ 1||eˆ || + γˆ 2||η˜ || + γˆ 1Bd + γˆ 3)

(

e-1-

1 1 - - γˆ 1 ||eˆ ||2 + 2(γˆ 1Bd + γˆ 3)2 + 4 4 (γˆ 2Bc)2

)

e -γˆ eV1 + 2D20 where D0 )

[

(A17)

x(γˆ 1Bd+γˆ 3)2+(γˆ 2Bc)2. This it implies that

]

2D20 2D20 exp(-γet) + V1(t) e V1(0) γe γe

(A18)

Under the appropriate initial value V1(0), the state eˆ will converge to the ball

x

4λmax(Ps) 2 D , t>0 λmin(Ps) 0

|e1| e ||eˆ || e κˆ )

(A19)

Since |u| e ub, eq 37 can be described as F |u| ) |y(F) d - Lf0h(x) F

Rieˆ i)||(Lg LfF-1h(x))-1| e ub ∑ i)1

-F(

0

0

(A20)

Under the bounded states and bounded reference signals of eq 38, if we select Rmax ) max{R1,...,RF}, then eq A20 can be written as

-FRmax||eˆ || e

ub F-1 |LgoLf0 h(x))-1|

F + |y(F) d | + |Lf0h(x)| )

obtained

g min g [u/max/(Rmaxκˆ )]1/F-1

(A22)

Literature Cited Allgo¨wer, F.; Approximate Input-output Linearization of Nonminimum Phase Nonlinear Systems. Technical report 1996; Swiss Federal Institute of Technology: Berne, Switzerland, 1996. Benvenuti, L.; Di Benedetto, M. D.; Grizzle, J. W. Approximate Output Tracking for Nonlinear Nonminimum Phase Systems with an Application to Flight Control. Int. J. Robust Nonlinear Control 1994, 4, 397-414. Doyle, F., III; Allgo¨wer, F.; Oliveira, S.; Gilles, E.; Morari, M. On Nonlinear Systems with Poorly Behaved Zero Dynamics. Proceedings of the American Control Conference 1992, Chicago, IL, 1992; pp 2571-2573. Gopalswamy, S.; Hedrick, J. K. Tracking Nonlinear Nonminimum Phase Systems Using Sliding Control. Int. J. Control 1993, 57, 1141-1158. Hauser, J.; Sastry, S.; Meyer, G. Nonlinear Control Design for Slightly Nonminimum Phase Systems: Application to V/STOL Aircraft. Automatica 1992, 28, 665-679. Henson, M. A.; Seborg, D. E.; An Internal Model Control Strategy for Nonlinear Systems. AIChE J. 1991, 37, 1065-1080. Henson, M. A.; Seborg, D. E. Nonlinear Process Control; Prentice Hall: Englewood Cliffs, NJ, 1997. Khalil, H. K. Nonlinear Systems, 2nd ed.; Macmillan: Pub. Co., New York, 1996. Khalil, H. K. Robust Servomechanism Output Feedback Controllers for Feedback Linearizable Systems. Automatica 1994, 30, 1587-1599. Kravaris, C.; Chung, C. B. Nonlinear State Feedback Synthesis by Global Input/Output Linearization. AIChE J. 1987, 34, 592603. Kravaris, C.; Daoutidis, P. Nonlinear State Feedback Control of Second-order Nonminimum-phase Nonlinear Systems. Comput. Chem. Eng. 1990, 14, 439-449. Kravaris, C.; Daoutidis, P.; Wright, R. A. Output Feedback Control of Nonminimum-phase Nonlinear Processes. Chem. Eng. Sci. 1994, 49, 2107-2122. Lian, K.-Y.; Fu, L.-C.; Liao, T.-L. Robust Output Tracking for Nonlinear Systems with Weakly Nonminimum Phase. Int. J. Control 1993, 58, 301-316. Morari, M.; Zafiriou, E. Robust Process Control; Prentice-Hall: Englewood Cliffs, NJ, 1989. Wright, R. A.; Kravaris, C. Nonminimum-phase Compensation for Nonlinear Processes. AIChE J. 1992, 38, 26-40. Wu, W. Stable Inverse Control for Nonminimum-phase Nonlinear Processes. J. Proc. Control 1999, 9, 171-183.

u/max (A21)

Received for review October 13, 1998 Revised manuscript received January 11, 1999 Accepted January 12, 1999

Therefore, the lower bound of tuning parameter is

IE9806500

Approximate Feedback Control for Uncertain Nonlinear Systems

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