Ind. Eng. Chem. Res. 1995,34, 2383-2392
2383
PROCESS DESIGN AND CONTROL Nonlinear Control of a CSTR Disturbance Rejection Using Sliding Mode Control Maria C. Colantonio PWIQUI-UNS-CONICET, 12 de Octubre 1842, 8000 Bahia Blanca, Argentina
Alfred0 C. Desages CIC-Departamento de Ingenieria Elictrica, UNS, Avenida Alem 1253, 8000 Bahia Blanca, Argentina
Jose k Romagnoli Department of Chemical Engineering, The University of Sydney, Sydney 2006 NSW, Australia
Ahmet Palazoglu* Department of Chemical Engineering and Materials Science, University of California, Davis, California 95616
This paper addresses the control of a continuous stirred tank reactor (CSTR) by feedback linearization and a second-order sliding mode (SOSM)construction around an unstable equilibrium point. The problem formulation features the use of the reactor temperature, as opposed to the concentration, as the controlled output. The system in the controllability canonical normal form exhibits stable zero dynamics. It is shown how SOSM can be applied to estimate on-line unmeasured disturbances in the inlet concentration, with which the nonlinear controller fully decouples the output from disturbances achieving asymptotic tracking. 1. Introduction
Many chemical processes exhibit strong nonlinear dynamic behavior, as in the familiar example of a pH neutralization process. Linear controllers designed as a first resort to maintain stability and performance requirements for these systems yield suboptimal results due to the dominance of the process nonlinearities, and often times fail as the process strays away from the linear regime. This typically necessitates the use of nonlinear compensators, but the design of such control systems is nontrivial. In the past decade, feedback linearization techniques have attracted the attention of many control engineers because they allow the application of well-established linear control analysis and design methods t o nonlinear processes. A number of techniques t o solve nonlinear process control problems using feedback linearization have been presented. The reader is referred to two recent review papers on this subject for a complete account of this area of research (Kravaris and Kantor, 1990a,b; Kravaris and Arkun, 1991). As discussed by Kravaris and Arkun (1991),a possible nonlinear control strategy involves the modification of the system dynamics such that a pole assignment problem can be formulated. This is referred to as “linearizationof the dynamics”and proceeds by finding a coordinate transformation for the states t o yield a linear input-state behavior. Another approach is the “linearization of the input/output (UO)behavior”, and this achieves essentially a partial linearization of the dynamics as seen from the input-output standpoint. ~
* To whom all correspondence should be addressed.
The YO linearization approach presents an advantage if one is specifically interested in the behavior of measured outputs, providing a convenient framework for relating design parameters to performance specifications. Naturally, the choice of the system outputs is dictated by practical considerations. In some cases, the nonlinear system cannot be totally linearized (Kravaris and Arkun, 1991) with feedback (in the input-state sense) using the available output set. In this case, a part of the transformed system remains nonlinear. Dynamic properties of this part, called “zero dynamics”, have a crucial role in the design of the controller and can preclude satisfactory process response. In the presence of disturbances and/or parameter uncertainties, the above strategy may fail t o be robust. This prompts the control engineer to modify the feedback control law such that the process output remains within specifications (Isidori, 1989). Among the alternatives, sliding mode control (SMC)has been suggested (Ferndndez and Hedrick, 1987) as a complementary “outer loop” feedback (Sira-Ramirez,1989) to robustify the control structure. However, we show here that if certain disturbances and states are unmeasured, disturbance rejection may not be achieved, although the output may be still bounded. In order t o solve this problem, we propose the use of a second-order-slidingmode (SOSM) technique (Elmali and Olgag, 1992; Chiacchiarini et al., 1995) in a nonlinear estimation framework. The resulting controller is shown to be robust under unmeasured disturbances, and the characteristic SMC chattering behavior of sliding mode control is not exhibited. This paper is concerned with the control of a continuous stirred tank reactor (CSTR) using temperature
0 1995 American Chemical Society Q888-5885l95l2634-2383~~9.QQl~
2384 Ind. Eng. Chem. Res., Vol. 34, No. 7, 1995 measurements instead of concentration, due to practical limitations associated with composition measurements. The controller must maintain the temperature below an allowable maximum while achieving disturbance rejection, in the presence of measured and unmeasured disturbances in the inlet stream. On the basis of an early work of Daoutidis and Kravaris (1989), we shall also classify the disturbances and analyze their effects on the feedback system. In section 2, some basic aspects of I/O linearization and SMC are reviewed. Two nonlinear control strategies to robustify the controller are presented where both make use of the combination of YO linearization and SMC. Section 3 contains the application of the above strategies to the CSTR. A n estimation algorithm derived from the use of the SOSM over the estimation error is presented. The conclusions are presented in section 4. 2. Robust Nonlinear Control Strategies 2.1. Preliminaries. We shall briefly describe the main aspects of the YO linearization approach to introduce the basic concepts. For more on the background and details of the methodology, the reader is referred to papers by Kravaris and Kantor (1990a,b)and Kravaris and Arkun (1991). We shall consider a nonlinear system of the general form x = fix)
+ g(x)u + zu(x)d y = h(x)
(1)
-
E R", u E R, y E R, d E Rm;A*): R" R", go): R", and wC): Rm R" are Cwvector fields, and he): R" R is a Cmfunction. We shall define L,kh(x)as the Kth Lie derivative of the scalar field h(x)with respect
--
where x
R"
-
to the vector field A x ) . Definition 1: It is said that system 1 has a local relative degree at a point x o if (i) L&h(x) = 0 for all x in the vicinity of xo and for allk r, the effect of the disturbance is less direct than the control input. All the information from disturbance di is available from the system states, and thus no action is needed to decouple y(t) from di. Class If ei = r, the disturbance and the control input affect the output of the system in the same manner, and feedforward action is necessary for achieving decoupling. Class 6! If ei < r, then the disturbance affects the output more directly than the control input. Some sort of "anticipative action" must be taken in order to achieve disturbance rejection. It is desired that the output y(t) track some reference signal. From the coordinate transformation (Isidori, 19891, the control input u is found to be
-Lpz(x) U =
+v
L&;-'h(x)
+ d(x,d)
(4)
where 6(x,d) is a function which decouples the output from measured disturbances and v is an external input that can be designed to assign a specific set of eigenvalues. Thus, this external input has the form:
The decoupling function d(x,d)is obtained from the main result of Daoutidis and Kravaris (1989) which is summarized in the following theorem. Theorem: For the nonlinear system described by (11, let ei be the relative degree given by definition 2 and A,@, and Gthe classes of disturbances outlined above. Then, the function d(x,d) that renders y(t) independent of the dL'sis given by
r 2,
ti = V(Z),
?-
=v
+ 1 Ii In
Y =z1
(2)
with z = [h(x)L&(x>Lfh(x)...L'-'h(x) q 1 ( ~ ) . . . q , - ~ h ( x ) l ~ (Isidori, 1989). It can be seen &hatif r < n, the system Proof: Please refer to the Appendix. Remark Note that if any disturbance is a function appears to be decomposed into a linear subsystem with of time, the decoupling function may be dependent on dimension r , which is responsible for the I/O behavior, and a nonlinear subsystem of dimension n - r whose its derivatives. For the sake of simplicity, we have not
Ind. Eng. Chem. Res., Vol. 34, No. 7, 1995 2385 included this feature in the above notation. Note also that, in the literature, the notation d(x) is generally used although x is not the only argument. The only drawback of the control strategy proposed above is that one can only guarantee local robustness conditions. Note that neither modeling errors nor parameter uncertainty is considered at this stage. It is assumed that the nonlinear transformation is exactly k n o w n and that all states are available. It is argued that the same method yields more globally robust results under disturbances and parameter uncertainty if combined with the SMC approach (Fernbdez and Hedrick, 1987). We shall focus on the solution of the I/O linearization problem in the presence of unmeasured disturbances and states. In the following, we present two control strategies which essentially combine this method with the SMC approach leading to more robust results. 2.2. Strategy 1: YO Linearization and Sliding Mode Control. Sliding mode control (SMC) is a technique derived from a more general approach k n o w n as the variable structure control (VSC) that has been originally studied by Utkin (1977, 1992). The method is particularly attractive due t o its ability to deal with nonlinear systems, time-varying systems, as well as uncertainties and disturbances in a direct manner. Moreover, it may be possible to obtain robust control systems. In VSC systems, the control can modify its structure, that is, travel among members of a set of possible continuous functions of the state. The design problem is to select the parameters of each structure and to define the traveling logic. This complexity is overshadowed by the possibility of combining useful properties of each structure. Moreover, a variable structure system can have properties that are not present in the individual structures. This defines the approach referred to as the sliding mode control. Such control systems are well-established by the works of Utkin (1977, 19921, Hung et al. (1993), and Fernandez and Hedrick (1987). The first step in SMC is to define a sliding surface S(t)that is generally linear and stable, with the choice of the sliding surface being problem dependent. The goal is to reach this surface and to remain (to slide) on it. For nonlinear systems in normal form, S(t)is chosen as a differential operator acting on some error function:
where I represents the bandwidth of error dynamics, determining the performance of the system on the sliding surface. When the system is outside the sliding surface S(t), one needs t o define the condition under which the system moves toward and reaches the surface. This condition, k n o w n as the attractiveness equation or reaching condition, can be chosen to directly specifythe dynamics of the switching function. The reaching law method gives the suitable condition for achieving this objective (Hung et al., 1993). Let the reaching condition be defined, for example, as
which is referred t o as a constant rate law. Then, one can argue that the system will reach the surface S(t) = 0 within a finite time and "slide" along it, implying that e(t) approaches zero with a bandwidth of 2. The
existence of sliding regimes is guaranteed for systems having relative degree 1 (Sira-Ramirez, 1989). When one is interested in combining SMC with the I/O linearization, the surface is defined as a stable linear operator of order r - 1: *-1
(9) k=O
From eqs 7 and 8, the modified control law is obtained as
Equation 10 establishes the relationship between I/O linearization and the sliding mode approach when the error dynamics on the sliding surface is chosen to be linear and time-invariant. The classical sliding mode exhibits chattering in the control signal, which is highly undesirable in most practical applications. Methods for reducing or suppresing chattering are available (Hung et al., 1993).For instance, the parameters 3 and a in eq 8 can be chosen in a manner to smooth the control signal. 2.3. Strategy 2: YO linearization and SecondOrder Sliding Modes (SOSM). The second method is studied previously by Elmali and Olga$ (1992) and Chiacchiarini et al. (1995). Suppose that a sliding surface S is defined and that it must be reached with zero velocity. Then, the derivative S must also be zero on the surface. To asymptotically reach the point (S,S) = (O,O), let us propose the following candidate Lyapunov function:
L = ~ s ~ + v S a,q>O ~,
(11)
To achieve our .goal, it is sufficient to ensure the negativeness of L,i.e., 2(as
+ VS) c 0
(12)
The point (S,$) = (0,O)can be reached within a finite time with prespecified dynamics by modifylng the reaching condition (12): 2(as
+ lyS>I-7
sgn
s
(13)
The sliding mode so defined is called second-order sliding mode (SOSM). This strategy will be used in this paper, within the context of a nonlinear state observer, to obtain the values of unmeasured states needed t o compute the I/O linearization control law (4). In essence, the sliding surface will be set equal to the estimation error. From condition 13 and using the state-space desmiption of the system, one can obtain an expression for the unknown entry (parameter uncertainty or unmeasured disturbance) which will be considered as an input to the model. A simple on-line estimation algorithm can be derived, where the asymptotic convergence is guaranteed by the design of the SOSM. As a result, all the states will be exactly k n o w n and the nonlinear feedback control law (4) will ef-
2386 Ind. Eng. Chem. Res., Vol. 34,No. 7, 1995 Table 1. Dimensionless Parameters for the CSTR Model activation energy y = EIRTo adiabatic temperature rise B = (-AE&AfdecpTfo Da = ko exp(-yV)lFo Damkohler number heat transfer coefficient P =MecPo dimensionless time t =t'(Fm dimensionless composition X 1 = ( C A f o - cA)/CAfo x z = (T - TfoYTfo dimensionless temperature dimensionless control input u = (Tc - TJTfo di = (Tf- Tfo)/Tfo feed temperature disturbance feed composition disturbance dz = (CM - CAfo)lCAfa ~~~
~
fectively linearize the perturbed system from the L/O standpoint. A very important feature of this approach is that the input derived from the SOSM does not affect the real plant. Hence, chattering would eventually affect the model but not the system. In the following section, we shall apply the above strategies to the case of a CSTR. It should be noted that estimation by SOSM is favored in this paper, first, to present a novel application of this new methodology and, second, because it is very easy t o design. Furthermore, when compared with other estimation methods, one of the distinct advantages is the explicit consideration of stability during the design phase and the lack of nonlinear differential expressions for estimation algorithms as this becomes critical for real-time applications. 3. Nonlinear Control of the CSTR
We shall consider the dimensionless modeling equations for a CSTR where an exothermic, first-order, irreversible reaction takes place that was previously studied by Ray (1981) and Calvet and Arkun (1988): *l = -xl
k2 = -x2
+ Da(1 - x l ) exp(kd;;+
+ BDa(1 - x , ) exp
1) - d2
1) -
dl and d2 are disturbances in the inlet temperature and concentration, respectively. For the original nomenclature, the reader is referred to Ray (1991). The disturbances are commonly assumed as steps. The dimensionless parameters for this system are defined in Table 1. For the parameter set, B = 8, a = 0.3, y = 20, Da = 0.078, and xzC = 0, the reactor exhibits multiple steady states as shown below:
(xl,xz)A= (0.144, 0.886)
stable equilibrium point: unstable equilibrium point:
( ~ 1 q z ) B= (0.4472,
stable equilibrium point:
2.7517)
( x 1 ~ 2 )=c(0.7646, 4.705)
Our goal is to design a controller such that the process output exhibits satisfactory set-point tracking and disturbance rejection characteristics. The control of the CSTR by feedback linearization considering concentration as the available output was previously investigated by Calvet and Arkun (1988). We focus on the temperature as the system output y = h(x)= x2
(15)
~~
Figure 1. I/O linearization control structure.
for two reasons. First, reliable concentration measurements are typically unavailable in industrial environments, and second, there is a restriction in the maximum admissible temperature in order to avoid secondary reactions to occur. These constraints lead us to select the unstable equilibrium point for a safe operation under control. For the chosen output, we have r = 1, el = 1,and QZ =2. Thus, the control law that linearizes the system from the I/O viewpoint and decouples the output from measurable disturbances is
Replacing the Lie derivatives and taking into account the compensation for initial conditions due t o the existence of zero dynamics, eq 16 yields
u = [x2 - BDa(1 - xl)exp(xd((xdy)
+ 1))+/3x2 -
The zero dynamics is observed to be minimum phase; hence the internal stability condition is verified and this approach can be applied. To implement the compensator given in eq 17, one needs information on the system states. More specifically, the I/O linearizing control law (17) requires that the concentration must be known. Since we have already eliminated the possibility of concentration measurements, this problem will be solved by resorting to a model-based control structure (Figure 1). The model of the CSTR will be used like an observer to supply the concentration measurements for the control law. Note that the disturbance d2 is unmeasured. Initially, we will consider this disturbance in the plant P but not in the model M. Besides, as the output is known, we will make use of this information in M. The input signal t o the system ( u ) is extracted from the I/O linearized model, so that the input to both P and M is the same. Hence, bounds over the input affect the system and the model in a natural way. It is well-known that if all states are available and the disturbances are measured, in the absence of modeling errors and parameter uncertainty, the I/O linearization approach works well. For the sake of comparison with the case of perturbed systems, Figure 2 depicts the stabilization of the closed-loop system around the equilibrium point B using the feedback law (171, when both disturbances are zero. If the measurable disturbance d l occurs, only the magnitude of the control law is modified and the output remains the same due to the decoupling. During start-up, the temperature must follow a predetermined trajectory toward the steady-state. In general, this trajectory has the form
Ind. Eng. Chem. Res., Vol. 34, No. 7,1995 2387 3
I
U
-0.5
00
-
-0.5
-I
0
.
5
15
IO
5
20
25
35
30
40
I
-2 -1 I
0
-I f
0
IO
IO
20
20
30
30
40
40
50
50
60
60
70
70
80
80
90
90
I
100
I
100
Figure 3. (a) Tracking during start-up. (b) Input control law.
where kl and k2 depend on practical restrictions and xpS is the desired steady-state value. The responses obtained for different values of kz, in the absence of disturbances, are shown in Figure 3a. The corresponding inputs are depicted in Figure 3b. More abrupt control laws are needed for faster heating, so there is a compromise between the constraints over the input and the heating speed. Now, we shall consider a 5% step disturbance in the feed concentration (d2). The system strays away from equilibrium at first, but after the disturbance has disappeared, it returns to the steady state due to the controller action (Figure 4a). Daoutidis and Kravaris (1989) argue that measurement of this disturbance is not needed, a t least if all the states are available. However, in this particular application, the information
0
IO
20
30
40
50
60
70
80
100
90 I
Figure 4. Response in the presence of the unmeasured disturbance dz.
on dz is not completely present in the states because the concentration is not measured. Consequently, decoupling of this unmeasured disturbance is not possible without a good estimation of the concentration. The situation is more complicated if d2 has a longer duration because the system goes to another steady state (Figure 4b). The control structure of Figure 1 does not consider the differences between the system and its model, although the latter has some information through the output. An interesting alternative could be the application of the internal model control (IMC) structure (Morari and Zafiriou, 1989), but the results that we obtained were not acceptable, primarily due t o the fact that IMC, as applied here, is a one-degree-of-freedom controller. It is important to note that when all states are available, the IMC strategy has been applied with satisfactory results (Calvet and Arkun, 1988; Henson and Seborg, 1991). This problem is quite akin to the case of modeling uncertainty. As mentioned in section 2, modellplant mismatch is a significant handicap in the application of feedback linearization control methods because the control laws are obtained from a model. Hence, if the model is not good enough, the control law does not linearize the system in the desired sense. However, it is expected that strategy 1(presented in section 2.2.), will generally yield more robust results. 3.1. Robust Control by Strategy 1. The goal is to make the outputy(t) follow the reference inpUtYR(t) even in the presence of dz. Then, following the aforementioned SMC procedure and with r = 1, the sliding surface is
2388 Ind. Eng. Chem. Res., Vol. 34, No. 7, 1995
a
31
I
2.5 -
x2
2.5
~
21.5
2-
-
1.5
~
0'5c 1-
-1
0
U
,
-0.5
b
10
0
20
30
40
60
50
70
0
70
80
4
r-
2.5
-
x2
21.5
-
I0.5
i
-1.5
A'-
10
20
30
40
I 60
50 t
Figure 5. (a) I/O linearization and SMC for dz = 0.02, a = 1, 7 = 0.5. (b) I/O linearization and SMC for d2 = 0.01, a = 1, 7 = 0.5.
U
-2
IO
0
20
30
40
60
50
Figure 6. (a) I/O linearization and SMC for dz = 0.02, a = 0.52, 7 = 0.55. (b) I/O linearization and SMC for dz = 0.01, a = 0.52, 7 = 0.55.
Applying the reaching law given by eq 8, the control input results in
Figure 5 shows the results obtained for different values of dz, a = 1,and q = 0.5. The control law is not smooth for all the values of the disturbance. For a 0.52 and q = 0.55, the chattering is supressed (Figure 6). The output remains in the neighborhood of the desired value, thus maintaining stability and indicating a more robust result. However, the disturbance dz is not completely rejected, due to the fact that the model is not correct. Hence, the computed value of the concentration, which affects the SMC through the reaching law, remains wrong. Consequently, the derivative of the switching surface is also found with some uncertainty. Another issue is that, even after adjusting the parameters a and 7 in a judicious manner, the system may still be affected by chattering. Further, the more we reduce the chattering, the greater becomes the difference between y ( t ) and y ~ ( t ) To . determine the optimum values of these parameters in order t o minimize the distance t o the sliding surface with a smooth control law is a nontrivial task. The main conclusion from the above results is that safe operation and good disturbance rejection will be only achieved by improving the concentration observer.
The stability and robustness properties gained by the addition of the SMC loop to the UO linearizing controller led us to consider the possibility of applying sliding modes in order to estimate the unmeasured disturbance. The underlying idea is to use the difference between the system and its model as a sliding surface to design an estimator for dz and thus obtain the correct value of concentration. As a result, the control law given by eq 20 will actually linearize the system from the UO standpoint, achieving complete disturbance rejection. In the following section, we outline a procedure for the estimation of dz using a SOSM strategy.
3.2. Robust Control by Strategy 2: SOSM and Nonlinear Estimation. We shall consider the following system of equations for the plant and its model: kl = -xl
h1= -m,
+ D d l - x,)exp( W YXZ> + 1)- dz
+ Da(1 - m,) exp((xz/y)+ 1)- dzm XZ
Ind. Eng. Chem. Res., Vol. 34, No. 7, 1995 2389
BDaE(d2, - d 2 ) (23)
+
where E = exp(xz/((xz/y) 1)). Then, the reaching law (13)results in Figure 7. Control structure with state estimation through SOSM.
Ys = x 2
n
Y m = m2
(21)
where dzmis the value to be estimated. When dzm f dz, the states x and m will be different. Considering the fact that x2 and m2 are the only available outputs, we define the sliding surface:
S = x 2 - m2
where XI is obtained from (21) and f 2 is evaluated by finite differences. Equation 25 allows us t o construct an iterative procedure to estimate d2. Algorithm. Step 1: At time t, solve the equations which corresponds to the model in (21)and evaluate (25). If Ad2 t 0, then initialize d2m= Adz. Step 2: Evaluate (21) using the control law (17). Step 3: For k = 1-p, compute (25) p-. d z m ( k ) = Ad2(k) + d2m(k - 1). If Ad&) f 0 go to step 2, else e d2m = &m(K). STOP.
(22)
and construct a SOSM like (11). From (19),we have
S = BDa(ml - x1)E
0.09 b 0.09-
1
0.080.070.06
~
0.05
~
0.04 0.03 0.03 ~
I
0.01 0 .010.02
A
0.5 ~
50
0'
d
1
0
50
100
150
200
250
300
350
400
I50 150
Zbo ZOO
250
3bo 300
350 350
4bO
400
500
4SO 450 t
31
0.5
-0.6
IbO 100
-
m,
I
500
450 t
Figure 8. (a) Reactor states. (b) Estimation of d2. (c) Control law. (d) Model states.
x)
2390 Ind. Eng. Chem. Res., Vol. 34, No. 7, 1995
-
Note that as d2m approaches dz, Ad, 0. In this context, dz, is used like an input to the model with the objective of reaching the surface and the sliding mode. This input contains a switching term which does not affect the physical system, thus, will not result in chattering. Now, the SMC is applied over the estimation error. The CSTR will be controlled only by the YO linearization law (17). The resulting control structure is shown in Figure 7. Note that the estimation procedure takes some time. The updating of d2, can be performed in a continuous or a discrete form. In the following, we shall analyze the main results of the application of the above procedure. Suppose that the system follows a reference trajectory that goes exponentially to the point xzS. After reaching this steady state at some time t (for example, t = 2501, a 5% step disturbance in the inlet concentration is introduced. Figure 8a depicts the state trajectories. The estimated disturbance dzm is shown in Figure 8b. Note that the input is smooth (Figure 8c) but exhibits an overshoot. This is due to the fact that we did not consider the first derivative of Y R ( t ) in the simulation, in order to illustrate the effect of the initial conditions of the zero dynamics (Isidori, 1989), if they are not set a priori. In Figure 8d, the model states are shown. The difference between the system and the model outputs is the result of updating dam in a discrete manner by the above algorithm. It can be shown that if dzm is updated continuously, both outputs become equal. Since we are interested in the system output, which is at the reference value, we decided to update the estimation in a discrete manner to minimize the simulation effort. We shall now consider a 10% step disturbance, with finite duration, in the inlet concentration (Figure 9a). The comparison between the system and the observer states is depicted in Figure 9b,c. Once again the disturbance is quickly estimated and rejected. Note that there is a difference between the system temperature and the model temperature. It can be proven by simulations that this is due to the fact that the value of the disturbance was updated in a discrete form. If we consider updating dam continuously, both values would again be equal, as in the previous case.
I
-0.04 50
la)
150
200
250
300
350
400
450
sa)
4. Conclusions
The method of I/O feedback linearization in combination with SMC is studied t o solve a control problem in the presence of unmeasured disturbances and states. Strategy 1, based on the work by FernAndez and Hedrick (1987), yields better results than the I/O linearization alone. However, the controlled output is not completely decoupled from the unmeasured disturbance. The need for the exact value of the unmeasured state to compute the control law for the sake of complete decoupling led us to design a nonlinear observer with zero error. We proposed the use of SOSM to design the nonlinear estimator as strategy 2. We demonstrated how to make the estimation error exactly zero with this approach. Despite using SMC, the system was not affected by undesirable chattering. The main result of this paper is that the use of nonlinear estimation with SOSM over the error allows us t o apply nonlinear control geometric methods, even if it is not possible to have measurements of all the system states. The controller so designed presents good disturbance rejection and performance characteristics. Future developments will be related to the application of the methodology in discrete time. Although the
I
Figure 9. (a)Finite time disturbance estimation. (b) Comparison of actual and observed concentration. (c) Comparison of actual and observed temperature.
algorithm has been implemented using real-time application sofiware, sampling times and delays in the application of the control law have not been explicitly considered. Further, it would be critical to study the performance of the algorithm in the presence of constraints.
Acknowledgment A Cooperative research grant by NSF (INT-9215832) to A.P. and by DITAC (92/3171) t o J.A.R. is gratefully acknowledged. Appendix. Proof of Theorem Based on the earlier results on disturbance decoupling by Isidori (1989), a proof is constructed by differentiat-
Ind. Eng. Chem. Res., Vol. 34,No. 7, 1995 2391 ing the output successively. The proof is presented in three parts according to the classes of disturbances. (a)Class A @i > r). In this case, one can verify that L&p(x) = 0, at least for i = 0, 1, ..., r - 1. Then, the disturbance di will not appear in the first r equations of the normal form, which will be identical to the corresponding ones of the system in the controllability canonical form. The remaining equations, corresponding t o the zero dynamics, will be dependent on di. Suppose that the following feedback control law is chosen:
+
S = f(x) g(x)
4x1 + g(x)B(x)u + (&XI
&x)+ w(x))d,
y = h(x)
In this case, the first r - 1equations of the normal form are identical to those obtained for the unperturbed system. But
y'"(t) = L;+.tec,+S"fi(x)+ Lg&+J;-lh(xM, where
Lgd+&;-lh(x) Then, the feedback system is described by
+ L&;-lh(x)
d(X)
+ L&;-lh(x)
Then, for y ( t ) to be independent from d,, and for the input u to be such that (r) y ( t )= 2, = u
the decoupling function must be selected as
2,
d(x)= -
=u
q = q ( 4 + m)d, where it can be seen that the output (state 21) is completely decoupled from the disturbance d,. Then, the condition e; > r is necessary for the output to be independent of di with the UO linearizing control law u. We shall also prove that this is a necessary condition. Let u = a(x) B(x)u be such that y ( t ) is independent from di. The original closed loop system is
+
+ &x)a(x)+ &x)p(x)u + w(x)d,
S = f(x)
L&h(X) L&;-'h (x)
and d(x,d) = d(x)d, has the structure given by the theorem for d, E @. (c)Class Gdisturbances (pi < r). We shall consider the control law
+
u = a(x) B(x)u + d(x)d, because these disturbances will appear in the gi-order derivative of the output. We know that
Y'l)(t)= Z 2 ( t ) = Lf+,,h(X)
+ d(x)d,L&X) + L&(xkl, = L&x)
y = h(x)
We shall take u ( t ) for simplicity. Then,
y'l'(t>= z,(t) = Lf+,,h(X)
+ L&(x)d,
will be independent from di only if Lwh(x)= 0. Assuming that this is satisfied, we have
if e, > 1. Then
~ ' ~ '=( z3(t) t ) = Lf+,,h(X)
LJfh(X)d, If
e, > 2, the last equation becomes
y"'(t) = ~ 3 ( t= ) L,2+gah(x)+ L&f+gah(xMi Once again, LwLf+gah(X)must be zero. Repeating the calculations, y'"(t) = L;+,,h(X)
y'2'(t) = z,(t) = L,2h(x)
continuing up to y'@J(t) = L,,&y'h(x)
+ LJ;;;,h(x)di
where LwL;;iah(x) must be zero. Then, we can conclude that LwL;&h(x) = 0 a t least for i = 0, 1, ..., r 1,that is, ei > r. In conclusion, the control law u does not contain any explicit term t o reject class A disturbances, since all the information about them will be present in the state vector. (b) Class Li? Disturbances @i = r). We shall consider
which includes a feedforward action for di. The original system under control results in
+ G(x)~,L&&(x)+
+ d(x)d,L&fP'-lh(x)+ L&B-'h(x)d,
and considering that r >
e,, we have
y'@z)(t)= Lqlh(x)
+
+ L&ylh(x>d,
If r > ei 1,the calculation is repeated up to the rth derivative of the output, which results in the following expression:
2392 Ind. Eng. Chem. Res., Vol. 34, No. 7, 1995
Then,
The output is decoupled from class G disturbance di by means of
Finally, considering the results of all parts together, eq 6 is obtained. Q.E.D.
Literature Cited Calvet, J. P.; Arkun, Y. Feedforward and Feedback Linearization of Nonlinear Systems and Its Implementation Using Internal Model Control (IMC). Ind. Eng. Chem. Res. 1988,27, 18221831. Chiacchiarini, H.; Desages, A. C.; Romagnoli, J. A.; Palazoglu, A. Variable Structure Control with a Second-Order Sliding Condition: Application to a Steam Generator. Automatica 1995, in press.
Daoutidis, P.; Kravaris, C. Synthesis of FeedforwardState Feedback Controllers for Nonlinear Processes. AIChE J . 1989,35, 1602. Elmali, H.;Olga$, N. Robust Output Tracking Control of Nonlinear MIMO Systems via Sliding- Mode Technique. Automatica 1992, 45,145--151. Fernhndez. B.: Hedrick. K. Control of Multivariable Nonlinear Systems by the Sliding Mode Method. Int. J. Control 1987,46, 1019-1040. Henson, M. A.; Seborg, D. E. An Internal Model Control Strategy for Nonlinear Systems. AIChE J. 1991,37,1065-1081. Hung, J. Y.; Gao, W.; Hung, J. C. Variable Structure Control: a Survey. IEEE Trans. Ind. Electronics 1993,40,2-22. Isidori, A. Nonlinear Control Systems, An Introduction; Springer-Verlag: New York, 1989. Kravaris, C.; Kantor, J. C. Geometric Methods for Nonlinear Process Control: 1. Background. Ind. Eng. Chem. Res. 1990a, 29,2295-2310. Kravaris, C.; Kantor, J. C. Geometric Methods for Nonlinear Process Control: 2. Synthesis Methods. Ind. Eng. Chem. Res. 1990b,29,2310-2323. Kravaris, C.; Arkun, Y. Geometric Nonlinear Control-An Overview. In CPC-N; Ray, W. H., Arkun, Y., Eds.; AIChE Publications: New York, 1991; pp 477-515. Morari, M.; Zafiriou, E. Robust Process Control; Prentice Hall: New York, 1989. Ray, W. H. Advanced Process Control; McGraw Hill: New York, 1981. Sira-Ramirez, H. Sliding Regimes in General Nonlinear Systems: A Relative Degree Approach. Int. J. Control 1989,50,14871506.
Received for review October 12, 1994 Revised manuscript received April 11, 1995 Accepted April 21, 1995@ IE9405910 Abstract published in Advance A C S Abstracts, June 1, 1995. @
