2.30

Sliding Mode Control in Process Industry O. CAMACHO

(2005)

Sliding mode control (SMC) is a powerful robust nonlinear control (see Section 2.19) technique that has been intensively 1,2 developed during the last 35 years. Sliding mode control has, for many years, been accredited as one of the key approaches for the systematic design of robust controllers for complex nonlinear dynamic systems operating under conditions of uncertainty. The most important advantage of sliding mode controllers is their innate insensitivity to parameter variations and disturbances once in the sliding mode, thereby 1 eliminating the necessity of exact modeling. Another distinguishing feature is their capability of order reduction, which enables the simplification of design and the decoupling of the system. With these advantages, sliding mode control is a promising area of study for both theoretical and applicationoriented problems. The potential of control resources may be used to the fullest extent within the framework of nonlinear control methods since the actuator limitations and other performance specifications may be included in the design procedure. The scope of SMC studies includes varied problems such as math2–5 ematical methods, design principles, and applications. The SMC design is composed of two steps. In the first step, a custom-made surface should be designed. While on the sliding surface, the plant’s dynamics is restricted to the equations of the surface and is robust to match plant uncertainties and external disturbances. In the second step, a feedback control law should be designed to provide convergence of a system’s trajectory to the sliding surface; thus, the sliding surface should be obtained in a finite time. The system’s motion on the sliding 1,2,5 Perfect tracking can be surface is called the sliding mode. 7 achieved at the price of control chattering. Chattering is a highfrequency oscillation around the desired equilibrium point. It is undesirable in practice because it involves high control activity and can excite high-frequency dynamics ignored in the mod1–7 eling of the system. The aim of this section is to present the overall concept of sliding mode control. Some references to industrial applications in electromechanical systems are discussed, and a practical approach, using a reduced-order model of chemical processes, for single-input single-output (SISO) systems and multiple-input multiple-output (MIMO) systems is also presented. Furthermore, a proposed SMC implementation of a PID algorithm is also suggested.

DESIGN PROCEDURE The idea behind SMC is to define a surface along which the 1 process can slide to its desired final value. The structure of the controller is intentionally altered in accordance with a prescribed control law as its state crosses the surface. First Step The first step in SMC is to define the sliding surface, S(t), which represents a desired global behavior, such as stability and tracking performance. The S(t) selected in this work, pre6 sented by Slotine and Li, is a proportional, integral-differential equation acting on the tracking-error expression: d  S (t ) =  + λ   dt 

n

∫ e(t)dt t

2.30(1)

0

where e(t) is the tracking error, that is, the difference between the reference value or set point, r(t), and the output measurement, x(t), namely e(t) = r(t) – x(t). λ is a tuning parameter that helps to define S(t). This term is selected by the designer and determines the performance of the system on the sliding surface. n is the system order. The control objective is to ensure that the controlled variable is equal to its reference value at all times, meaning that e(t) and its derivatives must be zero. Once the reference value is reached, Equation 2.30(1) indicates that S(t) has reached a constant value, meaning that e(t) is zero at all times; it is desired to set dS (t ) =0 dt

2.30(2) 1,2

Equation 2.30(2) is called the sliding condition. Second Step

As a second step, once the sliding surface has been selected, attention must be turned to designing the control law that

351 © 2006 by Béla Lipták

352

Control Theory

drives the controlled variable to its reference value and satisfies Equation 2.30(2). The SMC control law, U(t), consists of two additive parts: a continuous part, UC(t), and a discontinuous part, UD(t). That is

Process output (Fraction TO)

1.5 1 0.5

U (t ) = UC (t ) + U D (t )

2.30(3) 0

0

10

20

The continuous part is given by

40

50

60

70

80

90

100

80

90

100

80

90

100

80

90

100

Controller output (Fraction CO)

1.5

UC (t ) = f ( x (t ), r (t ))

30

1

2.30(4)

0.5

where f ( x (t ), r (t )) is a function of the controlled variable and the reference value. The continuous part of the controller is obtained by combining the process model and sliding condition, Equation 2.30(2). The discontinuous part is nonlinear and represents the switching element of the control law. The discontinuous part of the controller is discontinuous across the sliding surface. Mainly, UD(t) is designed based on a relay-like function, i.e., UD(t) = KD sign S(t), because it allows for changes between the structures with a hypothetical infinitely fast speed. In practice, however, it is impossible to achieve the high switching control because of the presence of finite time delays for control computations or limitations of the physical actuators, thus causing chattering around of 1–7 the sliding surface. The aggressiveness to reach the sliding surface depends on the control gain (i.e., KD), but if the controller is too aggressive it can collaborate with the chattering. To reduce the chattering, one approach is to reattach the relay-like func7 tion by a saturation or sigma function, which can be written as follows: S (t ) |S (t )| + δ

0

10

20

30

40

50

60

70

Process output (Fraction TO)

1.5 1 0.5 0

0

10

20

30

40

50

60

70

Controller output (Fraction CO)

1 0.5 0 –0.5

0

10

20

30

40

50

60

70

FIG. 2.30a Chattering reduction using a saturation function. (a: δ = 0, b: δ = 1.5).

2.30(5)

where KD is the tuning parameter responsible for the reaching mode. δ is a tuning parameter used to reduce the chattering problem. Figure 2.30a shows the effect of δ variations on the process output. Figure 2.30b shows the effect of KD variations on the system trajectory on a phase plane, which illustrates how the rate of the trajectory moves to reach its final value from an initial state to a final state. For example, when KD = K3 is used, the error rate moves faster, reducing the error until the final value is reached, than when KD = K1 or K2 are used to tune the controller. Therefore, KD represents a gain, which provides the aggressivity of the controller. Figure 2.30c confirms the previous conclusion in the time domain. In summary, the control law usually results in fast motion to bring the state onto the sliding surface, UD(t), and a slower motion to proceed until a desired state UC(t), is reached. See Figure 2.30d.

© 2006 by Béla Lipták

–1

0.2

K1 K2 K3

0.1 Initial value

Final value

0

td/ed

U D (t ) = K D

0 –0.5

–0.1 –0.2 –0.3 –0.4 –0.5 –0.4 –0.2

0

0.2

0.4

0.6

0.8

1

e (t)

FIG. 2.30b Graphical interpretation for KD variations (K1 < K2 < K3 ).

1.2

2.30 Sliding Mode Control in Process Industry

speed control, servo systems applications, power systems, power supply switching, robotics, gas turbines, and turbojet engines, as well as in aircraft systems and anti-skid automotive systems. Good references for electromechanical system appli8 cations can be found in Utkin, Guldner, and Shi and Edwards 9 10 and Spurgeon. A special issue of ASME is also recommended for readers interested in applications in this control field.

Process output (Fraction TO)

1.5 1 0.5 0

0

10

20

30

40

50

60

70

80

90

100

Chemical Systems

Controller output (Fraction CO)

2

Even though SMC has been widely investigated for a variety of processes, as mentioned earlier, to the best of our knowledge none of the previous works have resulted in a general approach for process industry. Since industrial chemical processes modeled using first principles tend to be of higher order and complexity, the traditional SMC procedures present disadvantages in their application. An efficient alternative could be the use of empirical modeling methods. Empirical methods use low-order linear models; usually, first-order plus deadtime (FOPDT) models are adequate for process control analysis and design. This kind of model is able to adequately represent the dynamics of many industrial processes, especially chemical processes, over a 11,12 and is easily obtained from the popular range of frequencies, reaction curve (Section 2.22). Although reduced order models cause uncertainties, which arise from the imperfect knowledge of the model, and nonlinear effects of the process contribute to performance degradation in the controllers, a sliding mode controller (SMCr) could be designed on the basis that the robustness of the controller will compensate for modeling errors that arise from the linearization of the nonlinear model of the process. Thus, if a general reduced-order model of the process can be obtained and chattering can be reduced, this control strategy can

1.5 1 0.5 0 –0.5

0

10

20

30

40

50

60

70

80

90

100

FIG. 2.30c System responses to a set point change for different KD values .---: KD = 0.25. —: KD = 0.44 — .—:. KD = 3.54.

APPLICATIONS Although a large number of reports talk about SMC applications, few process companies have reported their results publicly. Therefore, the information presented here will be limited to the contents of the reports found in the literature. Two types of applications will be described, first to electromechanical systems and second, the applications for chemical processes. Electromechanical Systems Sliding mode control has most been often used in the electrical control of machines, including the control of induction motors,

de(t)/dt UD (t) = KD sign (S(t))

Reaching Mode

Sliding Surface S(t) = 0

Chattering!! UD(t) = KD

S(t) S(t) + d

e(t)

U(t) = UC (t) + UD(t)

UC (t)

Sliding Mode

dS(t) =0 dt

Substituting system dynamic equations

FIG. 2.30d Sliding mode control components.

© 2006 by Béla Lipták

353

Sliding Surface

  d S(t) =  + l   dt

nt

∫0 e(t)dt

354

Control Theory

become one of the most important discoveries in the field of process control. SMC for SISO Chemical Process Control Here a general equation of the SMCr for SISO systems is presented. First, it is discussed in connection with self-regulating processes. After that, the subjects of inverse response, integrating, and a robust dead time compensation are covered. Self-Regulating Processes SMCr is a type of model-based control (MBC) (Section 2.13), and the order of the model is proportional to the numbers of tuning parameters. The controller equation was derived by using an FOPDT process model, 11,12 (see Figure 2.30e). which is the recommended approach The controller obtained significantly simplifies the application of SMC to industrial processes having this type of dynamics. 13,14 Based on the references, the complete SMCr is   t τ   X (t ) S (t ) U (t ) =  0   + λ0e(t )  + K D |S (t )| +δ  K   t0τ 

2.30(6a)

model; second, the action of the controller is considered (in the sliding surface equation), by including the term sign(K). Note that sign(K) only depends on the static gain; therefore, it never switches. For industrial applications, Equation 2.30(6b) 15 represents a PID algorithm. To complete the controller, the following tuning equations 13,14 can be considered: •

For the continuous part of the controller and the sliding surface,

λ1 =

t0 + τ –1 [=] [time] t0τ

2.30(7a)

λ12 –2 [=] [time] 4

2.30(7b)

λ0 ≤ •

For the discontinuous part of the controller,

KD =

0.51  τ    | K |  t0 

0.76

[=] [fraction CO]

2.30(7c)

δ = 0.68 + 0.12(|K | K D λ1) [=] [fraction TO/time]

with

2.30(7d)

 dX (t ) S (t ) = sign( K )  − + λ1e(t ) + λ0 dt 

 e(t )dt  0 

∫

t

2.30(6b)

Figure 2.30f shows a schematic of the controller. These equations present advantages from the process control point of view. First, they have a fixed structure depending on the λ parameters and the characteristic parameters of the FOPDT

e(t)

+ R(t)

SMCr

U(t)

Valve

Process

– X(t)

C(t)

Equations 2.30(7c) and 2.30(7d) are used when the signals from the transmitter and controller are in fractions (0 to 1). Sometimes, the control signals are scaled in percentages, so the signals are in 0-to-100% ranges. In these cases, the values of KD and δ are multiplied by 100. Inverse Response Systems When the initial direction of the process response is contrary to the direction of the final steady state, the process is called a nonminimum phase or inverse response process. This characteristic is only related to the input/output behavior of a process. Several chemical processes can have this kind of response, such as drum boilers, reactors, and reboilers. This kind of system can be described by the following transfer function,

Transmitter

G1 (s) = SMCr

2.30(8)

BIAS

It can be approximated by

e(t) X(t)

UC (t) S(t)

UD(t)

FIG. 2.30e Schematic of a sliding mode controller.

© 2006 by Béla Lipták

K (1 − τ 1s) (τ 2s + 1) ⋅ (τ 3s + 1)L(τ n s + 1)

+

+ +

U(t)

G1 (s) ≅ K

e − to s τs +1

2.30(9)

This means that the step response of the nonminimum phase system (Equation 2.30[8]) can be approximated by an

2.30 Sliding Mode Control in Process Industry

355

Reaction Curve Method output 0.63 Y Y 0.28 Y t1 t2 Hot water input

Heat exchanger

Cold oil input

First Order Plus Dead Time FOPDT

Hot oil output

Cold water output

Tank with continuous stirring

time

G(s) =

K −tos e ts + 1

Input flow

Input flow 1 Output flow

Chemical process

Mixing tank Input flow 2 Mixture

FIG. 2.30f Empirical model to design Sliding mode controllers for chemical processes.

FOPDT model, such as the one in Equation 2.30(9) (see Section 2.22). Therefore, the design procedure for systems 14,16 represented using FOPDT models can be applied. The resulting SMCr is similar to the ones given by Equation 2.30(6a) and 2.30(6b). The equations for the first tuning 16 estimates of the controller parameters were obtained and are listed in Table 2.30k. Integrating Systems In general, the following model can approximate integrating systems with dead time: G (s ) = K

e − to′s s

2.30(10)

The identification of the characteristic parameters of this model (K, τ, and to) is included in Section 2.22. Using Equation 2.30(10) and the procedure described previously, the 17 controller obtained is

which is summarized in Figure 2.30g. Tuning settings for integrating processes are listed in Table 2.30k. Robust Dead Time Compensator Dead time is present in many industrial processes, where it creates a serious control problem. The performance of feedback control systems will drop drastically as the ratio of dead time to time constant increases. The Smith predictor (SP— see Section 2.27) or dead time compensator (DTC), as it is also known, has many weak points, including possible instability and poor performance, if the model is inaccurate, and poor response to dis18 turbances. In addition, the original structure of the SP cannot reject constant load disturbance for processes with integration. 14 Camacho and Smith showed the design of an SMCr from an FOPDT. The design of SMCrs for dead time processes

+

e(t)

R(t) –

U (t ) =

λoto′ S (t ) e(t ) + K D |S (t )| + δ K

2.30(11)

Equation 2.30(11) does not have the term that includes the deviation of the controlled variable; it suggests a modification,

© 2006 by Béla Lipták

SMCr

U(t)

Valve

Integrating C(t) process

X(t)

Transmitter

FIG. 2.30g Modified sliding mode controller for integrating systems.

356

Control Theory

R(t) +

U(t)

SMCr

–

X(t)

Process

–

S(t) K Gm−(s) =ts + 1

e−tos

xm(t)–

+

em(t)

FIG. 2.30h Robust dead time compensator scheme.

13,14

which can be solved using requires some assumptions, the internal model approach. It was shown that by merging these two control techniques, a robust one (SMC) and a predictive one (IMC), the resulting controller will provide 19 the benefits of both. Thus, the mixing of SMC and predictive structures can produce better controllers than the original 20 ones. The robust dead time compensator is designed in two steps: First, an invertible model of the process is chosen, and from it, the sliding surface is designed. Second, the reaching condition is satisfied, and the robustness is guaranteed. Figure 2.30h shows the proposed scheme. The nonlinear process has been modeled as an FOPDT. This model can be separated into two parts: Gm+ = e − tos Gm− =

2.30(12)

K τs +1

2.30(13)

Since Gm− (s) eliminates the dead time term from the model, this simplification facilitates the SMC design. The controller obtained is as follows:  S (t ) = sign(k ) em− (t ) + λ  u(t ) =



∫ e(t)dt  t

2.30(14)

0

  τ  x m− (t ) s + λ [ R(t ) − x m (t ) − em (t )] + K D 144424443  K τ |s| +δ e (t )   2.30(15)

em− (t ) is the error between the reference, R(t), and the model output without deadtime, x m− (t ), x(t) is the controlled variable,

© 2006 by Béla Lipták

and em(t) is the error between the process output and the complete model output, also known as modeling error. It is observed that this sliding surface includes a prediction term, em− (t ), which gives a faster response to reach the set point than the original one, when processes present an elevated controllability relationship (to /τ). The tuning parameters for this controller are included in Table 2.30k. SMC for MIMO Chemical Process Control In this section, a summary is provided of the SMCr modification that should be applied when the chemical processes 10 are multivariable (see Section 2.18). The objective is to extend the applicability of the SMCrs, which were designed for SISO processes, to multivariable systems. Industrial processes are by nature multivariable. Control of multivariable and nonlinear processes is not an easy task because of the interaction among variables (see Section 2.12) and because of the presence of dead time. The interaction results from the high probability that a manipulated variable can affect several controlled variables. This interaction among the variables degrades control quality. The proposed approach discussed below assumes that the total control system consists of multiple SMCrs acting in addition to a decoupling system (Section 2.12). The amount of interaction between the variables can be determined by the steady-state gains Kij, using the relative 11,12 gain array (Section 2.25) or Bristol Array method, which allows obtaining a dimensionless interaction index µij. In systems with strong interaction, a common practice is to use 11,12 decoupling of the interacting loops. Once the system is decoupled, it is possible to use a developed SMCr based on an FOPDT model. The original SMCr controller has a set of tuning equations for SISO-type processes. The idea in this part is to consider the effect of the multivariable characteristic parameters and the interaction index in the controller tuning equations.

2.30 Sliding Mode Control in Process Industry

357

0 GP′ 22 (s) ... 0

0   M1 (s)    0   M 2 (s )   ...     GPnn ′ (s)   M N (s) 

... ... ... ...

Process

 C1 (s)  GP′ 11 (s)    C2 (s )  =  0  ...   ...    Cn (s)   0

I/O Subsystem

Using the concepts of decoupling, a multivariable control system can be represented as:

2.30(16)

SMCR IMPLEMENTATION OF PID ALGORITHM Digital PID algorithms are widely used in distributed control systems (DCS), in programmable logic controllers (PLC), and in remote terminal unit (RTU) applications. A common PID algorithm is as follows:  1 MV(t ) = Kc e(t ) + τ I 

∫ e(t)dt − τ t

0

D

d CV(t )   dt 

2.30(17)

where MV(t) is the manipulated variable, τI is the integral time, τD is the derivative term, and CV(t) is the controlled

© 2006 by Béla Lipták

FIG. 2.30i Implementation of PID algorithm in an industrial controller.

variable. Note that the derivative term is calculated using the measured controlled variable, not the error. In this respect, Equations 2.30(6b) and 2.30(17) are similar. The discussion here will focus on implementing the SMCr 15 based on the PID algorithms, using the presently existing control devices and plant operator mentality. The PLC-based implementation of a discrete algorithm for a PID is shown in Figure 2.30f. The term X(t) in Equation 2.30(6b) is the same as the term CV(t) in 2.30(17). The next step is to tune the PID based on the SMCr tuning equations. From Equation 2.30(6b) and Equation 2.30(17), the tuning parameters of the PID algorithm can be converted into the tuning parameters of the sliding surface S(t), as follows: Kc = λ1

2.30(18)

τ I = λ1/λO

2.30(19)

τ D = (λ1 )−1

2.30(20)

With the above conversions, the PID algorithm can be expressed in terms of the sliding surface term S(t). The SMCr is divided in two parts—the continuous part, which is an algebraic function of the process variables and the set point, and the discontinuous part, which is also an algebraic function of S(t), which is calculated from the PID algorithm that exists on the PLCs or RTUs used in industry. Figures 2.30i and 2.30j show the implementation of this approach, where the output of the PID, which was an input

I/O Subsystem

Discussion of the Results Due to space limitations, detailed results are not included here, but they can be seen for the 14 minimum-phase case in Camacho and Smith, for the inverse response case in Reference 16 and for the integrating systems case in Reference 17. A new control scheme, mixing the internal model approach with the SMC concept for processes with high dead time– to–time constant ratios was tested in Reference 19. Finally, the multivariable systems case can be seen in the ASME special 10 issue. Summarizing the results, the same SMCr designed for minimum phase systems can be used with good results for a wide class of nonlinear chemical processes. These SMCrs gave robust performance on processes with dead time and in applications with modeling errors and disturbances.

PID

Industrial controller

Process

where Ci(s) represents the i-controlled variable; Mj (s) is the j-manipulated variable; and Gpij(s) is an FOPDT transfer function relating the i-controlled and j-manipulated variables, respectively. Once the system is decoupled, it is possible to 13 use SMCr based on an FOPDT model. The parameters tojj , τ jj , and K jj needed to calculate the initial tuning settings of the controller are obtained from the open-loop step responses (see Section 2.22), for the j-j pairing of the manipulated and controlled variable pairs after decoupling. In comparison with the original SMCr tuning values, only the KDj tuning parameter has changed to include the interaction index. This provides the necessary aggressiveness to reach the sliding surface in multivariable applications (Table 2.30k).

Process Data

PID Process Data

SMC algorithm

Industrial controller

FIG. 2.30j Implementation of SMCr using an industrial PID controller.

358

Control Theory

TABLE 2.30k Summary of the Tuning Equations Used for SMCrs Self-Regulating

Inverse Response

Integrating

λ1

t0 + τ t0 τ

t0 + τ t0 τ

1 t0

λ0

≤

KD

0.51  τ j    | K |  toj 

δ

λ12 4

≤

0 ,76

0.68 + 0.12(KKD λ 1)

λ12 8

0.064  τ    K  t0 

λ≤

 λ1   4  0.76

0.68 + 0.12 (KKD λ 1)

Three steps are required to implement the total algorithm. In each iteration, the term S(t) is calculated from the PID output. The continuous and discontinuous parts of the controller algorithm are easily programmable algebraic equations. This procedure is based on the PID algorithm and already exists in most industrial controllers. The sequence of statements summarizing the proposed algorithm is: a1 = t0*tau/K do forever input x et = x - ref y = PID(et) st = sign(k)*y ut = (x/K + a1*lambda0*et) + Kd*(st/(abs(st)+delta) End Table 2.30k provides a summary of the tuning equations for the SMCrs. The parameters (t0, τ, and K) that are required to calculate the initial tuning settings of the controller are obtained from the open-loop step response (Section 2.22).

t0 jj τ jj ≤

0.76

KD ≥

0.8  τ  | K |  t0 

0.76

0.68 + 0.12(KKD λ)

µij

λij2 4

0.51  τ jj    | K jj |  t0 jj 

0.76

0.68 + 0.12(KijK Dij λ jj)

This section has described SMC and provided some examples of industrial applications in electromechanical systems and a practical approach for using a reduced-order model in chemical process applications. SISO and MIMO systems were also discussed. The SMCr can be implemented from a PID algorithm, where the PID represents the sliding surface, and the rest of the controller is built using algebraic blocks. This represents an advantage from a process control engineer’s point of view because the controller is of a fixed structure. This controller, having adjustable parameters, can be easily implemented in DCS. Finally, the approach proposed for chemical industry applications has the simplicity of PID controllers and the robustness of SMCrs, and the procedure used can be easily extended to electrical and mechanical systems if reduced-order models can be obtained for those processes.

References 1. 2.

3.

4.

CONCLUSIONS In the past, SMC has been an important theoretical research field. Today, it is becoming a good source of solutions to real-world problems. Its potential is limited only by the imagination of the people working in process control. Therefore, the prospects for the application for SMC in the processing industries are immense.

tojj + τ jj

1 τ + t0

—

0.68 + 0.12(KKD λ1)

SMCr Implementation Methodology

Multivariable

2

0.64  1    K  t0 

to the process (manipulated variable), became an input to the discontinuous part of the SMCr.

© 2006 by Béla Lipták

Dead-Time Compensator

5.

6. 7.

Utkin, V. I., “Variable Structure Systems with Sliding Modes,” Transactions of IEEE on Automatic Control, AC–22, pp. 212–222, 1977. Hung, J. Y., Gao, W., and Hung, J. C., “Variable Structure Control: A Survey,” IEEE Transactions on Industrial Electronics, 40(1), pp. 2–21, 1993. Sira-Ramirez, H., and Llanes-Santiago, O., “Dynamical Discontinuous Feedback Strategies in the Regulation of Nonlinear Chemical Processes,” IEEE Transactions on Control Systems Technology, 2(1), pp. 11–21, 1994. Colantino, M. C., Desages, A. C., Romagnoli, J. A., and Palazoglu, A., “Nonlinear Control of a CSTR: Disturbance Rejection Using Sliding Mode Control,” Industrial & Engineering Chemistry Research, 34, pp. 2383–2392, 1995. Young, K. D., Utkin V. I., and Özgumer, Ü., “A Control Engineer’s Guide to Sliding Mode Control,” IEEE Transactions on Control Systems Technology, 7(3), pp. 328–342, 1999. Slotine, J. J., and Li, W., Applied Nonlinear Control, Englewood Cliffs, NJ: Prentice Hall, 1991. Zinober, A. S. I., Variable Structure and Liapunov Control, London: Springer-Verlag, 1994.

2.30 Sliding Mode Control in Process Industry

8. 9. 10.

11. 12. 13.

14.

15.

16.

17.

Utkin, V., Guldner, J., and Shi, J. X., Sliding Mode Control in ElectroMechanical Systems, London: Taylor and Frencis, 1999. Edwards, C., and Spurgeon, S., Sliding Mode Control: Theory and Applications, London: Taylor and Francis, 1999. Camacho, O., and Rojas, R., “A General Approach of Sliding Mode Control for Nonlinear Chemical Processes,” Transactions of the ASME Journal of Dynamic Systems, Measurements, and Control, 122(4), pp. 650–656, 2000. Smith, C. A., and Corripio, A. B., Principles and Practice of Automatic Process Control, New York: John Wiley & Sons, 1997. Marlin, T. E., Process Control, New York: McGraw-Hill, 1995. Camacho, O., “A New Approach to Design and Tune Sliding Mode Controllers for Chemical Processes,” Ph.D. Dissertation, University of South Florida, Tampa, 1996. Camacho, O., and Smith, C., “Sliding Mode Control: An Approach to Regulate Nonlinear Chemical Processes,” ISA Transactions, 39(2), pp. 205–218, 2000. Camacho, O., Smith, C., and Chacón, E., “Toward an Implementation of Sliding Mode Control to Chemical Processes,” Proceedings of ISIE ’97, Guimaraes-Portugal, pp. 1101–1105, 1997. Camacho, O., Rojas, R., and Garcia, W., “Variable Structure Control Applied to Chemical Processes with Inverse Response,” ISA Transactions, 38, pp. 55–72, 1999. Camacho, O., García, W., and Rojas, R., “Sliding Mode Control: A Robust Approach to Integrating Systems with Deadtime,” Proceedings of IEEE-ICCDCS, Margarita, Venezuela, 1998.

© 2006 by Béla Lipták

18. 19.

20.

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