Ind. Eng. Chem. Res. 2002, 41, 3179-3185
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PROCESS DESIGN AND CONTROL Sliding Mode Compensation for Windup and Direction of Control Problems in Two-Input-Two-Output Proportional-Integral Controllers Ricardo J. Mantz* and Herna´ n De Battista Laboratorio de Electro´ nica Industrial Control e Instrumentacio´ n, Universidad Nacional de La Plata, CC91, 1900 La Plata, Argentina
The paper deals with the problems of windup and change in the direction of control in multiinput-multi-output processes with actuator constraints. A sliding mode control is proposed to solve these problems for the case of two-input-two-output proportional-integral-controlled systems. The control strategy is based on the conditioning of the filtered references vector through a discontinuous action. 1. Introduction The control input of all industrial processes is subject to physical constraints. Because of the presence of saturation, the real plant input may substantially differ from the controller output. As a consequence, the performance of the process may seriously deteriorate. The major problems of multi-input-multi-output (MIMO) processes are (a) the controller windup and (b) the socalled problem of directional change in the control.1 As happens with single-input-single-output (SISO) systems, saturation leads MIMO processes to perform as an open loop. Therefore, unstable and poorly damped modes of the system may be excited, driving the control signal far away from the control limits. As a result, large overshoots, long settling times, and even instability may appear in the system response. This problem, known as reset windup (RW), has been extensively studied in the last few years.2,3 Besides, the required closed-loop performance of MIMO processes strongly depends on their decoupling degree. The introduction of multiple saturation affects the coupling between the controller and the plant. Then, the direction and the frequency of the control signals vector necessary to achieve the intended degree of decoupling are changed, thus degrading the transient response in addition to the windup problem.1,3,4 The main contribution of the present paper is to introduce notions of variable structure system theory and the associated sliding regimes to overcome these problems. The analysis is limited to proportionalintegral (PI)-MIMO controllers. The reason is that, as is well-known, PI controllers are by far the most widely used ones in process control. For simplicity, the twoinput-two-output case is discussed. The following section briefly reviews the basic concepts on sliding mode (SM) control. In the section after the next, the windup and process decoupling problems are revisited. Then, the main results of the paper are * To whom all correspondence should be addressed. Email:
[email protected]. Tel/Fax: +54 221 425 9306.
presented and validated through a pair of examples. Finally, conclusions are summarized. 2. Basic Concepts of SM Control A variable structure system comprises a set of continuous subsystems with a switching logic that is a function of the system state. A particular operation is achieved when switching occurs at a very high frequency, ideally infinite, constraining the system state to a subspace, named the sliding surface. This kind of operation is called SM and has many attractive properties. It is robust to parameter uncertainties and external disturbances, it reduces the order of the sliding dynamics which becomes dependent on the designer-chosen sliding surface, and it is easy to implement.5-9 Because of the interesting features of the SM, a large number of papers presenting practical applications of SM control have been reported in the last few years. For instance, Kantor,10 Kawata et al.,11 Sira-Ramı´rez,12 Hanczyc and Palazoglu,13,14 and Camacho et al.15 have discussed the application of SM to chemical process control. Consider the following dynamical system: m
x3 ) Ax +
biwi ) Ax + Bw ∑ i)1
(1)
where x ∈ R n is the system state and w ∈ R m is the control vector. Matrixes A and B (and its column vectors bi) are of consistent dimensions. The variable structure control law is defined componentwise as
wi )
{
wi+ if si(x) > 0 wi- if si(x) < 0
i ) 1, ..., m
(2)
according to the sign of the scalar switching functions si(x) ) Ri - kiTx. The sliding surface S is defined as the intersection of the m so-called individual sliding surfaces Si defined by
10.1021/ie010198b CCC: $22.00 © 2002 American Chemical Society Published on Web 05/30/2002
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Si ) {x ∈ R n: si(x) ) 0}
(3)
Thus, m Si ) {x ∈ R n: s(x) ) R - KTx ) 0} (4) S ) ∩i)1
with s(x) being the vector of switching functions, i.e., s(x) ) [s1(x), s2(x), ..., sm(x)]]T. Besides, R ) [R1, R2, ..., Rm]T is the SM reference vector. Finally, the columns of the matrix K are the feedback gain vectors ki, i.e., K ) [k1, k2, ..., km]. A sliding motion locally exists on a particular individual sliding surface Sj if, as a result of the switching logic (2), the following reaching condition is satisfied:
{
s˘ j < 0 if sj > 0 s˘ j > 0 if sj < 0
(5)
In fact, once the individual surface Sj is reached, the control action wj switches (ideally at infinite frequency), constraining the state trajectory to Sj. If the individual sliding motion converges toward the intersection surface S, where all controllers induce sliding motions on their individual surfaces, then the combination of individual sliding motions results in a collective sliding regime. The ideal sliding dynamics can be found by the equivalent control method.16 The equivalent control is defined as the fictitious smooth control action that forces the system state to remain on the sliding surface S. Its expression is obtained from the sliding invariance condition
{
s)0 s3 ) 0
(6)
weq(x) ) -[(KTB)-1KTAx]|x∈S
(7)
and results in
Note that weq(x) is well-defined; i.e., it exists and is unique if and only if the transversality condition det(KTB) * 0 holds. The following existence condition for SM existence can be written in terms of the equivalent control:5,16
of the controller, leading to undesirable transient behavior, with large overshoots and long settling times. This problem is known as integral windup or RW. Although it was originally addressed for proportionalintegral-derivative (PID) controllers, RW also occurs with controllers having unstable or slow dynamics.17 To avoid this problem, a two-step design of the controller is usually developed.2,18 This approach consists of designing the controller, ignoring the physical limitations, and then adding the windup compensation known as anti-reset-windup (ARW). The additional compensation must satisfy the following specifications: (a) stability, (b) action when the limitation is active only, (c) graceful degradation with respect to the unrestricted control system. One of the pioneering ARW two-step techniques was proposed by Fertik and Ross.19 It is known as “back calculation” and addresses the windup problem in systems comprising PID controllers. A large number of contributions have been subsequently reported on this topic for SISO systems. Two other techniques worth mentioning are the “conditioning technique” formulated by Hanus et al.,18 which is based on the concept of “realizable reference”, and one based on observers that was contributed by Åstro¨m.18,20 Another outstanding contribution was presented by Kothare and coauthors.2 They proposed a unified framework for the design of ARW algorithms, which allows parametrization of all of the known techniques as a function of two parameters. Among many other techniques, concepts of internal model control,21 linear matrix inequalities,22,23 model predictive control,24 and linear parameter varying25,26 have been recently exploited to find solutions to RW problems in MIMO systems. MIMO processes undergoing actuator saturation may present another type of undesirable behavior. In fact, saturation acts on each actuator independently of the others, thus changing the direction of the process input vector with respect to the controller output. Considering that the gain of a MIMO process is a function of the input direction, a further degradation of the system performance occurs.1,27,28 For instance, if the controller output vector takes the value
min {wi+, wi-} < wieq < max {wi+, wi-}, i ) 1, ..., m (8) When w in eq 1 is replaced by that in eq 7, the ideal sliding dynamics yields
x3 ) A ˜ x ) [I - B(KTB)-1KT]Ax
u)
3. Windup and Directional Change in the Control Control systems are subject to restrictions on the process itself (for instance, in the form of temperature, volume, and pressure constraints) as well as on the actuators (saturation in the amplitude and in the rate of change). When saturation occurs, the control system performs as an open loop. In fact, the process input is inconsistent with the internal state of the controller. Consequently, the evolution of the system state becomes independent
1 3
(10)
and the actuator output variables (i.e., the process input variables) are bounded by (1, then
(9)
Notice that eq 9 is redundant. In fact, the SM dynamics is of order n - m because it is constrained to S.
[]
u ˆ )
[
] []
sat(u1) 1 ) sat(u2) 1
(11)
Hence, the resulting direction of the process input vector u ˆ differs from the controller output vector u. When the saturation error (u - u ˆ ) is in the direction of a high gain of the process, the difference among the process output variables produced by u and u ˆ will be maximal. On the other hand, when the saturation error is aligned with a low gain direction of the plant, the effect on the output will be less important. Observation. Because multiple saturation acts independently at each actuator, it has a diagonal structure. It is well-known from the theory of robust control for linear systems that certain combinations of plants
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Figure 1. PI controller with SM compensation.
Figure 2. Pair of block diagrams of the same SM compensating loop for windup and directionality of control.
and controllers yield serious performance degradation at the presence of diagonal uncertainties in the input. To avoid this directionality problem when saturation occurs, it is possible to resort to the attenuation of the control vector. Thus, deterioration of the system response will only be caused by the error in the amplitude of the input vector but not by the error in its direction. Another possibility is to modify the reference signal in order to avoid any change in the direction of the control. It must be remarked that the problems of windup and directionality of control in MIMO systems are closely related. In fact, if the directional changes in the control are avoided, then the windup problem is also avoided because the internal variables of the controller are now consistent with its output. On the other hand, solving the windup problem does not necessary mean that the control direction is preserved or, in other words, that the coupling degree is not altered.1
troller, the actuator saturation, and the set-point filter F. The SM compensating loop of the filter state can also be identified. The output of the SM controller s(x) drives the switches with the aim of conditioning the filtered reference. It is important to remark that the problem of chattering, commonly considered a serious drawback of variable structure control, can be ignored in the current application. In fact, as Figure 1 shows, the sliding regime is confined to the low-power side of the system. Consequently, fast electronic devices can be used for switching. The following statements are assumed to be true: 1. P is open-loop stable. 2. The controller ignoring saturation (i.e., the first step of the design) is properly designed. 3. The rate of change of the process output is finite even in the presence of disturbances. 4. The directionality compensation dynamics is much faster than the main loop dynamics. 5. The electronic switches are fast enough and are activated only when saturation takes place, i.e., uˆ i ) ui w wi ) 0. Under these assumptions, a pair of compensating loops such as the one shown in Figure 2a is proposed for the two-input-two-output PI controller. First-order filters f1 and f2 of the set points r1 and r2 are considered, with eigenvalues λf1 and λf2, respectively. Therefore, the state vector of the controller with the proposed compensation has six variables:
x ) [x11 x12 x21 x22 xf1 xf2 ]T
{
Four of them are the PI-MIMO state variables, and the others correspond to the set-point filters. The state equations are
kp11 (-λf1xf1 - y1) T11 kp12 x˘ 12 ) (-λf2xf2 - y2) T12 kp21 (-λf1xf1 - y1) x˘ 21 ) T21 kp22 (-λf2xf2 - y2) x˘ 22 ) T22 x˘ f1 ) λf1xf1 + r1 + w1 x˘ f2 ) λf2xf2 + r2 + w2
x˘ 11 )
4. Correction of the Windup and Control Direction by SM In this section, the use of SM control concepts for the correction of windup and control directionality for twoinput-two-output PI controllers is suggested. Although the basic ideas can be exploited to solve the problem for generic PI-MIMO controllers, the extension of the analysis is out of the scope of this paper. Based on concepts introduced by Walgama et al.,4 a compensation method involving the addition of a reference filter with state adaptation is developed. It is noteworthy that the filter has not of itself the aim of avoiding saturation. In fact, this simple solution would lead to an extremely conservative design. On the contrary, the real compensation action lies on the conditioning of the filter state variables as a function of the restrictions. In the present work, the state adaptation is governed by a sliding regime in such a way that the filtered reference does not produce actuator saturation, thus avoiding both windup and directionality problems. Figure 1 sketches the proposed control. It shows the blocks representing the process P, the PI-MIMO con-
(12)
(13)
where r ) [r1, r2]T is the set-point vector, y ) [y1, y2]T is the vector of output variables, Tij represents the integral times of the PI-MIMO controller, and kpij depicts the proportional gains. It is assumed in the following that kp11 * 0, kp22 * 0, and ∆ ) kp11kp22 - kp21kp12 * 0. Furthermore, the outputs of the controller (u ) [u1, ˆ ) [uˆ 1, uˆ 2]T) are given by u2]T) and actuators (u
{
u1 ) x11 + kp11(-λf1xf1 - y1) + x12 + kp12(-λf2xf2 - y2)
u2 ) x21 + kp21(-λf1xf1 - y1) + x22 + kp22(-λf2xf2 - y2) (14)
and
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{
uˆ i ) ui if |ui| e sat(ui) uˆ i ) sat(ui) if |ui| > sat(ui)
(15)
It is convenient to express the estate equations in the matrix form
{
x3 ) Ax + Brr + Byy + Bww u ) KxTx + KyTy
{
{
s1(x) ) sat(u1) - k1xTx - k1yTy ) 0 s˘ 1(x) ) -k1xTx3 - k1yTy3 ) 0
(16)
where the matrixes A, Br ) [b1r, b2r], By ) [b1y, b2y], Bw ) [b1w, b2w], Kx ) [k1x, k2x], and Ky ) [k1y, k2y] can be readily identified from eqs 13 and 14. The blocks of the compensating systems (Figure 2a) are rearranged in Figure 2b to put in evidence the objective of the proposed SM control. The switching action associated with the proposed SM compensation is
wi ) wi+ if si > 0 if si ) 0 i ) 1, 2 wi ) 0 wi ) wi if si < 0
output tends to saturation. On the other hand, the second inequality is a condition on w1- that should be verified to avoid saturation and hence windup. During the sliding regime, the following invariance condition is therefore satisfied:
From eq 20, the equivalent control signal yields
w1,eq ) -(k1xTb1w)-1(k1xTAx + k1xTb1rr1 + k1xTb1yy1 - k1yTy3 ) (21) which is well defined because k1xTb1w * 0. When w1 in eq 16 is replaced by eq 21 and w2 ) 0, the ideal sliding dynamics of the PI controller with SM compensation can be derived:
˜ r1r + B ˜ y1y + B ˜ y1y3 x3 ) A ˜ 1x + B
(17)
s1(x) ) uˆ 1 - u1 s2(x) ) uˆ 2 - u2
A ˜ 1 ) (I - b1w(k1xTb1w)-1k1xT)A (18)
Note that the system (16) has a well-defined relative degree of [1, 1]. Hence, the proposed switching action may lead to the existence of sliding regimes. In fact, if wi+ and wi- are chosen properly, one of the following operation modes is established: 1. The controller output is within the limits of the actuators. Then, s1(x) ) s2(x) ) 0 and hence w1 ) w2 ) 0. 2. The actuator j reaches its saturation limit, i.e., uˆ j ) sat(uj). In this case, sat(uj) is the reference for the SM compensating loop. Then, the switching action enforces a sliding motion on the individual sliding surface Sj ) {x ∈ R n: sat(uj) - uj ) 0}. During this operating mode, the reference signal rfj is conditioned to avoid saturation of the controller output uj. 3. Both actuators reach their saturation limits. Hence, a collective sliding motion is established on the intersection surface S ) S1 ∩ S2. Thus, both controller output variables are prevented from exceeding their saturation limits by means of the conditioning of the filtered references rf1 and rf2. The cases of individual and collective sliding motions are separately analyzed. In particular, conditions for the stability of the SM compensating loops are obtained. Case 1. Let us consider first the case when only one actuator reaches its saturation limit. That is, one of the individual sliding surfaces is reached, for instance, S1 ) {x ∈ R n: s1 ) sat(u1) - u1 ) 0}. Then, w1 switches according to s1(x), whereas w2 ) 0. A sliding regime is established on S1 provided the reaching condition (5) is satisfied in the vicinity of S1, i.e.
{
(22)
In particular, the matrix A ˜ 1 is given by
where
{
(20)
u˘ 1 ) k1xT(Ax + b1rr1 + b1yy1) - k1yTy3 > 0 if u1 < sat(u1) u˘ 1 ) k1xT(Ax + b1rr1 + b1yy1) - k1yTy3 + k1xTb1ww1- < 0 if u1 < sat(u1) (19)
The first inequality reflects the fact that the controller
(23)
The eigenvalues vector of A ˜ 1 is given by
eig(A ˜ 1) ) [0 -1/T11 0 0 0 λf2 ]
(24)
One of the eigenvalues at the origin has no physical significance. In fact, it is associated with the order reduction during the sliding regime and the consequent redundancy of A ˜ 1 in describing the sliding dynamics. Besides, as a result of the ARW compensation, the pole of PI11 at the origin is shifted to its zero at -1/T11. On the other hand, the other integrators of the PI controller are not in the SM compensation loop; hence, their poles remain at the origin. Similarly, the pole of filter f2 (which is not in the SM loop) is also an eigenvalue of A ˜ 1. On the contrary, the pole of f1 has been removed because of the aforementioned order reduction. Case 2. When both actuators reach their saturation limits, a collective SM is established on the sliding surface S. The equation defining S is of the form
s(x) ) R - KxTx - KyTy ) 0
(25)
where
[ ] [
R sat(u1) R ) R1 ) sat(u2) 2
]
(26)
From the invariance condition (6), the equivalent control vector is obtained:
weq ) -(KxTBw)-1(KxTAx - KxTBrr - KxTByy - KyTy3 ) (27) where det(KxTBw * 0). Then, the ideal sliding dynamics is of the form in eq 22, where the matrix A ˜ is given by
A ˜ ) (I - Bw(KxTBw)-1KxT)A
(28)
Matrix A ˜ also presents four poles at the origin. There are now two of them associated with the overdimension of A ˜ . In fact, during the SM both prefilter poles are
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removed, reducing the dynamics to a fourth order. The other poles at the origin are related to the PI controller. Finally, A ˜ has two other eigenvalues which are lefthand-side poles provided
{
(
)
(
)
kp12kp21 1 1 1 1 1 1 + + + >0 T11 T22 ∆ T11 T22 T12 T21 kp12kp21 1 1 1 + >0 T11T22 ∆ T11T22 T12T21 (29)
(
)
Note that when the PI controller is decoupled, i.e., kp12 ) kp21 ) 0, these conditions are always satisfied. Remark. On the one hand, it is evident from eq 19 and Figure 2 that the controller output is not able to change abruptly, even despite the “disturbances” r and y. This is due to the presence of the filters f1 and f2 and to assumption 3 (i.e., the rate of change of the process output is bounded). The controller output u is therefore continuous, and the system cannot fall suddenly in saturation. On the other hand, it is clear from eqs 21 and 27 that the equivalent control is bounded. Then, with a proper selection of wi+ and wi-, the existence condition (8) of sliding motions on S1 (or S) is guaranteed [see also the reaching condition (19)]. This means that once the actuator reaches its saturation limits the sliding regime is established (i.e., the invariance condition is verified). For the previous reasons, the system never operates in saturation. The last one is an important advantage with respect to conventional ARW algorithms. Although conventional algorithms reduce significantly the saturation periods, they cannot completely avoid saturation. Therefore, the directionality of the control is lost during these periods; i.e., conventional windup correction reduces but cannot avoid control direction problems. Contrarily, if the proposed SM windup compensation is used, the system never operates in saturation and the decoupling degree is therefore not altered. Effectively, the filtered reference is continuously adjusted so that the controller output is within its saturation limits. Consequently, both windup and directionality problems are overcome. 5. Examples The features of the proposed SM compensation for windup and directionality of the control are corroborated through a pair of examples. Example 1. Let us consider first the Wood/Berry binary distillation column process:29,30
[
] [ ]
12.8e-s -18.9e-3s 3.8e-8s + 1 21s + 1 U + 1 D(s) ˆ (s) + 14.9s -3s Y(s) ) 16.7s -7s -19.4e-3s 6.6e 4.9e 10.9s + 1 14.4s + 1 13.2s + 1 (30) The following multiloop PI controller K has been designed in ref 30:
K(s) )
[
(
0.699 1 + 0
1 8.42s
)
0
(
-0.0895 1 +
1 9.52s
)
]
(31)
The response of the closed-loop system to a step change in the set point r1 is shown in Figure 3. The step
Figure 3. Step response without (solid) and with (dashed) restrictions in the actuators: (a and b) output variables y1 and y2, respectively; (c) controller output u1 and process input uˆ 1; (d) controller output u2 ≡ uˆ 2.
Figure 4. Step response with restrictions in the actuators and SM compensation (solid): (a and b) output variables y1 and y2, respectively; (c and d) controller output variables u1 ≡ uˆ 1 and u2 ≡ uˆ 2, respectively. Idem without SM compensation (dashed).
response of the system with unbounded actuators is depicted in solid lines, whereas the response with actuator constraints is shown in dashed lines. In the last case, a saturation nonlinearity uˆ i ∈ [-0.2, 0.2] is taken into consideration. It can be observed that saturation occurs on uˆ 1 and that a large overshoot appears on the controlled variable y1, revealing a typical windup behavior. Figure 4 shows the step response of the closed-loop system with bounded actuators and the proposed SM compensation. A pair of first-order reference filters with λf1 ) λf2 ) - 1 are included in the compensating loop. Then, the state of filter f1 is continuously conditioned in order to avoid saturation of uˆ 1. The amplitudes of the control signals are w1+ ) -w1- ) 1. For the sake of comparison, the response without ARW compensation is repeated with dashed lines. Note that the overshoot in the controlled variable is eliminated. Besides, it is convenient to remark that the controller and actuator outputs coincide at any time. That is, the actuator never saturates during the step response. Example 2. This problem has been proposed in ref 28 and then used as a benchmark example to test other
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Figure 5. (a) Process outputs and (b) process inputs of the system with unbounded actuators. (c) Process outputs and (d) controller outputs (- - -) and process inputs (s) of the system with restrictions in the actuators.
Figure 6. System with restrictions in the actuators and SM compensation: (a) process outputs, (b) controller outputs.
control solutions (see, for instance, refs 21 and 31). The dynamics of the process under consideration is given by
P(s) )
[
4 -5 1 110s + 1 -3 4
]
(32)
A controller designed to obtain a decoupled linear performance has been developed in ref 28:
K(s) )
[ ]
10s +1 4 5 3 4 s
(33)
Figure 5a depicts the time evolution of the system output assuming unbounded actuators when the reference vector suddenly changes from zero to
r)
[ ] 0.6 0.8
(34)
whereas Figure 5b shows the process input variables. On the other hand, Figure 5c displays the response of the system excited by the same reference signals but now considering saturation in the actuators uˆ i ∈ [-15, 15]. The corresponding controller output (u1, u2) and process input variables (uˆ 1, uˆ 2) are drawn in Figure 5d with dashed and solid lines, respectively. Large overshoots and long settling times can be observed in the transient response (Figure 5c). As is claimed in ref 28, windup is a relatively minor problem in this example. Then, to accomplish a satisfactory system behavior, the problem of change in the direction of control should be overcome. The proposed SM compensation has been applied to the controller (33) in order to solve the problem of control directionality. First-order filters (with eigenvalues in λf1 ) λf2 ) -1) for the reference signals are included in the controller. A sliding regime is established (when necessary) for the conditioning of their state variables. The amplitudes of the control signals are wi+ ) -wi- ) 1. The effectiveness of the proposed SM strategy is corroborated in Figure 6. In fact, the overshoots and the large settling times in the system response are eliminated (Figure 6a). Besides, Figure 6b shows that the controller outputs do not surpass their saturation limits.
Figure 7. Process outputs with prefilters and restrictions in the actuators but without SM compensation.
Remark. Obviously, prefilters can be used to smooth the reference signal, thus helping to maintain the actuator output within its saturation limits. However, conservative design of these filters should be avoided. In this paper, the set-point filters are not designed to produce by themselves a substantial improvement in the system performance. This is verified in Figure 7 that shows the step response of the system with the set-point filters but without SM compensation. In fact, it is observed that this response is not appreciably better than the one shown in Figure 5c. It could be noticed that a satisfactory response (similar to the one obtained in this paper) can also be accomplished for this example using the method described in ref 28. Nevertheless, it is worth mentioning that the main contribution of the paper is to introduce the basic ideas on SM control to develop new ARW algorithms. The method proposed in this paper presents some promising features. For instance, it incorporates the attractive properties of SM, it is very simple and easy to implement, it introduces no nonlinearities into the loop, and it completely avoids actuator saturation, thus eliminating windup and directionality problems. 6. Conclusions A solution to integral windup and directionality of control problems for two-input-two-output PI-controlled systems has been developed in the context of variable structure control. One of the main goals of the proposed SM strategy is the simplicity the control objective renders into the sliding surface design. Each reference signal is continuously adjusted (when necessary) by an SM compensating loop such that actuator saturation is
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completely avoided. Then, both windup and directionality of control problems are overcome. It is important to remark that the chattering problem usually associated with variable structure control is negligible in the current application. The reason is that the sliding regime is confined to the low-power side of the system. The discontinuous control can therefore be implemented with fast electronic devices. Acknowledgment Financial support from UNLP, CONICET, and ANPCyT is gratefully acknowledged. R.J.M. is a member of CICpBA. H.D. is with CONICET. Literature Cited (1) Walgama, K.; Sternby, J. Conditioning Technique for Multiinput-Multioutput Processes with Input Saturation. IEE Proc. D 1993, 140, 231-241. (2) Kothare, M.; Campo, P.; Morari, M.; Nett, K. A Unified Framework for the Study of Anti-Windup Design. Automatica 1994, 30, 1869-1883. (3) Peng, Y.; Vrancic, D.; Hanus, R.; Weller, S. Anti-Windup Design for Multivariable Controllers. Automatica 1998, 34, 15591565. (4) Walgama, K.; Ronnback, S.; Sternby, J. Generalization of Conditioning Technique for Anti-Windup Compensators. IEE Proc. D 1992, 139, 109-118. (5) Bu¨hler, H. Re´ glage par Mode de Glissement; Presses polytechniques romandes: Lausanne, Switzerland, 1986. (6) Edwards, C.; Spurgeon, S. Sliding Mode Control: Theory and Applications; Taylor & Francis: London, 1998. (7) Hung, J. Y.; Gao, W.; Hung, J. C. Variable Structure Control: A Survey. IEEE Trans. Ind. Electron. 1993, 40, 2-22. (8) Sira-Ramı´rez, H. Differential Geometric Methods in Variable Structure Control. Int. J. Control 1988, 48, 1359-1390. (9) Utkin, V. Sliding Mode Control in Electromechanical Systems; Taylor & Francis: London, 1999. (10) Kantor, J. Nonlinear Sliding Mode Controller and Objective Function for Surge Tanks. Int. J. Control 1989, 50, 20252047. (11) Kawata, S.; Yamamoto, A.; Masubuchi, M.; Okabe, N.; Sakata, K. Modeling and Sliding Mode Temperature Control of a Semi-Batch Polymerization Reactor Implemented at the Mixer. Proc. 11th IFAC World Congr. 1990, 6, 112-117. (12) Sira-Ramı´rez, H. On the Dynamical Discontinuous Feedback Strategies in the Regulation of Nonlinear Chemical Processes. IEEE Trans. Control Syst. Technol. 1994, 2, 11-21. (13) Hanczyc, E.; Palazoglu, A. Sliding Mode Control of Nonlinear Distributed Parameter Chemical Processes. Ind. Eng. Chem. Res. 1995, 34, 557-566. (14) Hanczyc, E.; Palazoglu, A. Sliding Mode Control of a
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Received for review March 1, 2001 Revised manuscript received February 27, 2002 Accepted April 5, 2002 IE010198B
