Some long time delay sliding mode control approaches

ISA Transactions 46 (2007) 95–101 www.elsevier.com/locate/isatrans

Some long time delay sliding mode control approaches Oscar Camacho ∗ , Rub´en Rojas, Winston Garc´ıa-Gab´ın Grupo de Investigaci´on en Nuevas Estrategias de Control Aplicadas, Escuela de Ingenier´ıa El´ectrica, Facultad de Ingenier´ıa, Universidad de Los Andes, La Hechicera M´erida 5101, Venezuela Received 7 December 2005; accepted 28 June 2006 Available online 9 January 2007

Abstract This paper presents a combined approach of predictive structures with sliding mode control (SMC). Control schemes have been proposed looking for performance and robustness improvement. These structures were designed for processes that can be approximated either by a first order plus time delay or an integral first order plus time delay model broadly used on chemical processes. The proposed schemes were tested for performance and robustness against set point changes and disturbances as compared with classical approaches. c 2006, ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Chemical processes; Predictive structures; Sliding mode control; Time delay

1. Introduction The presence of time delays in many industrial processes is a well-recognized problem. Time lag, transportation lag, time delay and dead time are common phenomena in industrial processes. Time delay can be produced by measurement lag, analysis and computation time, communication lag or the transport time required for a fluid to flow through a pipe. The achievable performance of typical feedback control systems can decline if a process has a relatively large time delay compared to the dominant time constant [1]. Predictive structures and sliding mode controllers have been used to solve such problems. Primarily, internal model control (IMC) and the Smith predictor (SP) are the most popular predictive structures used for time delay compensation [2,3]. Furthermore, when the process presents an integral behavior the original structures cannot be used since a constant load disturbance results in a steady-state error [4]. To overcome this obstacle different approaches have been proposed [4,5]. Simulation studies have shown that the set point and load disturbances are either very oscillatory or highly damped when the process has a large time delay [6]. To deal with this additional problem new structures were proposed, decoupling the disturbance from the set point response [5,7,8]. In general, these approaches have some

problems: they are sensitive to modeling errors, since the design requires the use of a process model, which can be difficult to obtain in practice. Modeling errors are unavoidable and they result in a mismatch between the model and the actual plant. Thus, the controllers designed using particular models may perform quite differently when they are implemented on the actual process. On the other hand, SMC has been used to design controllers based on its strength for dealing with model–plant mismatches [9]. Even though this controller has proved to be robust against modeling errors and disturbances, its overall performance was too sluggish. The aim of this paper is to summarize an approach that combines simple predictive structures with sliding modes. In that sense, three different controllers are presented to show the performance of this approach: an internal model based sliding mode controller (IM-SMCr), a time delay sliding mode controller (TDSMCr), and a Smith predictor based sliding mode controller (SPSMCr). So, the paper is organized as follows. Section 2 shows the background necessary for developing these controllers. Section 3 gives a brief description of the proposed controller design procedure. Section 4 presents some computer simulation results. Finally, some conclusions are offered. 2. Background 2.1. Models of the processes

∗ Corresponding author. Tel.: +58 274 2402903; fax: +58 274 2402903.

E-mail address: [email protected] (O. Camacho).

Nonlinear high order models describe most processes in industry. It is well known that a simplified model of a nonlinear

c 2006, ISA. Published by Elsevier Ltd. All rights reserved. 0019-0578/$ - see front matter doi:10.1016/j.isatra.2006.06.002

96

O. Camacho et al. / ISA Transactions 46 (2007) 95–101

Fig. 2. Smith predictor scheme. Fig. 1. Internal model control.

high order model can be used to design a controller. In chemical processes the reaction curve is an often-used method for identifying dynamic models [10]. It is simple to use, and provides adequate models for many applications. The curve is obtained by introducing a step change in the controller output and recording the transmitter output. That curve allows model parameter calculation. A first order plus time delay (FOPTD) model, Eq. (1), is able to adequately represent the dynamics of many chemical processes over a range of frequencies [11]: Km Y (s) = e−t0m s U (s) τm s + 1

Fig. 3. Graphical interpretation of SMC.

(1)

where K m , τm and t0m are the static gain, time constant and the time delay of the model, respectively. For processes presenting an integrating behavior, an integrating first order plus time delay (IFOPTD) process model, Eq. (2), must be considered: Y (s) Km = e−t0m s . U (s) s(τm s + 1)

(2)

In both equations, Y (s) is the Laplace transform of the controlled variable (transmitter output), and U (s) is the Laplace transform of the manipulated variable (controller output). Both Y (s) and U (s) are deviation variables.

2.2.2. Smith predictor structure The Smith predictor structure is shown in Fig. 2. In it, y(t) is the process output, r (t) is the set point or reference input, G− m (s) is the invertible part of the process model, ym (t) is the process model output and em (t) is the output modeling error. So, the SP incorporates a model of the process, and thus it is able to predict its output. This allows the controller to be designed as though the system is delay free, therefore retaining the simple tuning features of PID controllers [12,13]. The closed-loop transfer function of the system, coming from the previous figure, can be written as G c (s)G p (s) Y (s) = − U (s) 1 + G c (s)G m (s) + G c (s)[G p (s) − G m (s)]

(4)

2.2.1. Internal model structure The internal model structure is shown in Fig. 1. The idea behind this scheme is firstly to obtain a model of the process, and then to decompose it into two parts, a directly invertible + term G − m (s), and another, noninvertible term G m (s). Thus, the model can be represented in the following way:

where G c (s), G p (s) and G m (s) are the controller, process and process model transfer functions, respectively. If the model and the process match, the characteristic equation will not contain a time delay, because G p (s) and G m (s) will cancel. Therefore, in this case, the characteristic equation involves only the expression 1 + G c (s)G − m (s), which allows an aggressive adjustment of the manipulated variable. Obviously, the true process is never known exactly, and therefore the performance should decrease.

− G m (s) = G + m (s)G m (s).

2.3. Sliding mode control

2.2. Predictive structures

(3)

The noninvertible part has an inverse that is not causal or is unstable, such as time delay or unstable poles. On the other hand, the invertible component is causal and stable, which allows one to design a realizable controller [12,13]. Therefore, the IMC procedure eliminates all elements in the process model that can produce an unrealizable controller. Thus, the design of the controller takes into consideration only the invertible one.

Sliding mode control is a technique derived from variable structure control (VSC) which was originally studied by Utkin [14]. A controller designed using the SMC method is particularly appealing due to its ability to deal with nonlinear systems and time-varying systems, showing a robust behavior [15]. The idea behind SMC is to define a surface along which the process output can slide to its desired final value. Fig. 3

97

O. Camacho et al. / ISA Transactions 46 (2007) 95–101

depicts the SMC objective. The structure of the controller is intentionally altered as its state crosses the surface in accordance with a prescribed control law. Thus, the first step in SMC is to define the sliding surface, S(t), which represents a desired global behavior, like stability and tracking performance. The sliding surfaces used to derive the different control schemes presented in this work are based on an integral–differential equation acting on the tracking-error expression [15]:   Z de(t) S(t) = sign(K m ) f e(t), e(t)dt, , λ, n (5) dt where e(t) is the tracking error, that is, the difference between the reference value or set point, r (t), and the output measurement, y(t), namely e(t) = r (t) − y(t). λ is a tuning parameter, which 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 model’s order, and therefore once its value is included in the sliding surface it does not appear explicitly. The sign(K m ) function was included in the sliding surface equation to guarantee the appropriate action of the controller [9]. Note that sign(K m ) only depends on the static gain of the plant model; for that reason it never switches, giving just a positive or negative sign multiplying f (•). The control objective is to ensure that the controlled variable be equal to its reference value at all times, meaning that e(t) and its derivatives must be zero. Once the reference value is reached, S(t) has reached a constant value; it is desired to set dS(t) = 0. dt

U D (t) = K D

S(t) |S(t)| + δ

(9)

where K D is the tuning parameter responsible for the reaching mode. δ is a tuning parameter used to reduce the chattering problem. In summary, the control law usually results in a fast motion to bring the state onto the sliding surface, and a slower motion to proceed until a desired state is reached. 3. Controller design

(7)

The continuous part is given by UC (t) = f (y(t), r (t))

dynamics ignored in the modeling of the system [15]. The aggressiveness for reaching the sliding surface depends on the control gain (i.e. α), but if the controller is too aggressive it can collaborate with the chattering. To reduce the chattering, one approach is to replace the relay-like function by a saturation or sigma function, which can be written as follows:

(6)

Once the sliding surface has been selected, attention must be turned to designing the control law that drives the controlled variable to its reference value and satisfies Eq. (6). The control law, U (t), consists of two additive parts: a continuous part, UC (t), and a discontinuous part, U D (t). That is U (t) = UC (t) + U D (t).

Fig. 4. Proposed scheme of IM based SMC.

(8)

where f (y(t), r (t)) is a function of the controlled variable and the reference value. The discontinuous part is nonlinear and represents the switching element of the control law. This part of the controller is discontinuous across the sliding surface. Mainly, U D (t) is designed on the basis of a relay-like function (i.e. U D (t) = αsign(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 the sliding surface [14,15]. Chattering is a high frequency oscillation around the desired equilibrium point. It is undesirable in practice, because it involves high control activity and can excite high frequency

In this section the syntheses of the sliding mode controllers using the different predictive structures are presented. The controllers are developed for processes that can be approximated by FOPTD and IFOPTD models. 3.1. Internal model based sliding mode controller The design of a sliding mode controller (SMCr) from an FOPTD model was described by Camacho and Smith [11] but their approach requires some assumptions and approximations to deal with the time delay term. The internal model based sliding mode controller (IM-SMCr) approach takes advantage of choosing the invertible part of the model process to design the controller. Fig. 4 shows the proposed scheme. The nonlinear process was modeled as an FOPTD. As was suggested in Section 2.2.1, the model can be separated into two parts: −t0m s G+ m =e

(10)

Km G− m = τm s + 1

(11)

where G − m (s) eliminates the time delay term from the model; this simplification facilitates the SMC design. Let us propose the following sliding surface: Z t − S(t) = em (t) + λ e(t)dt (12) 0

98

O. Camacho et al. / ISA Transactions 46 (2007) 95–101

where − em (t) = r (t) − ym− (t)

(13)

e(t) = r (t) − y(t)

(14)

where ym− (t) is the model output without time delay. Now, following the procedure described in [11], the SMCr is obtained:   τm ym− (t) S(t) U (t) = + λe(t) + K D (15) Km τm |S(t)| + δ   Z t − e(t)dt . (16) S(t) = sign(K m ) em (t) + λ 0

To complete the SMCr design, it is necessary to establish a set of tuning equations. For the tuning equations as first estimates, using the Nelder–Mead searching algorithm [16,17], the following equations were obtained: λ≤ KD

1 τm + t0m   0.8 τm 0.76 ≥ |K m | t0m

δ = 0.68 + 0.12|K m |K D λ.

(17)

Fig. 5. Smith predictor based sliding mode controller.

where S(t) is the following sliding surface: S(t) = K S · sign(K m ) · e1 (t)

and the K S parameter is responsible for the controller aggressiveness, K S > 0. To complete the controller design, it is necessary to have a set of tuning equations. The tunings for first estimate values were determined using time-domain performance methods [5], resulting in the following equations: 0.72 = |K m |



τm t0m

0.76

(18)

KD

(19)

δ = 0.68 + 0.12|K m |K D

These equations have a fixed structure depending on the λ parameter and the characteristic parameters of the FOPTD model which is an advantage from the process control tuning point of view. For an industrial application, Eq. (16) can be implemented by a PI algorithm [18].

(22)

K S = 15K P

(23) t0m + τm t0m τm

(24) (25)

where K P can be calculated as the controller gain of a PID controller. Thus, the value chosen for this parameter depends on the tuning equations used, giving a more aggressive controller if the Ziegler–Nichols are chosen instead of Dahlin equations [2].

3.2. Time delay sliding mode controller The time delay sliding mode controller (TDSMCr) was developed using an approach similar to that presented in the previous subsection. In this case the Smith predictor structure was used and a different controller was obtained [12,13]. When the nonlinear process is modeled as an FOPTD, the delay free part, G − m (s), can be used to design the controller. This simplifies the procedure of obtaining a conventional SMCr without delay compensation. Then, the following proportional sliding surface was proposed: S(t) = K S · e1 (t)

(20)

where K S is a design gain and e1 (t) is the SP-like error (e1 (t) = r (t) − (ym− (t) + em (t))) that it is reduced to the difference between the reference, r (t), and the free delay model output, ym− (t), when a perfect model matching is considered, i.e. em (t) = 0. Thus, the process time delay is not considered, shortening the controller design. Although previous considerations are not so real, it is assumed that the controller robustness will compensate for this. The TDSMCr obtained is given by the following equation:   S(t) τm dr (t) ym− (t) + KD U (t) = + (21) Km dt τm |S(t)| + δ

3.3. The Smith predictor based sliding mode controller for integrating processes The Smith predictor based sliding mode controller (SPSMCr) presented uses the standard SP architecture while the controller is a sliding mode controller (SMC). The block diagram of the proposed scheme is shown in Fig. 5. It is well known that the original SP controller is not effective for integral processes because it cannot reject a constant load disturbance. Many modified SP for integral processes with different structures have been proposed in the literature to remove the steady-state error produced by a constant load disturbance. Tian and Gao [20], and Matauˇsek and Mici´c [8] added derivative action, G d (s), to their proposed TDC to overcome this problem. This was also considered in our approach. G d (s) = K O (τd s + 1).

(26)

To develop the SPSMCr, an integrating first order plus time delay (IFOPTD) process model was considered. The model transfer function without time delay can be written as G− m (s) =

Km . s(τm s + 1)

(27)

Again assuming perfect model matching, em (t) = 0. Thus, the time delay part is not considered in the SMC design. Then,

99

O. Camacho et al. / ISA Transactions 46 (2007) 95–101

the complete SPSMCr can be represented as   1 dym− (t) − U (t) = (1 − τm λ1 ) + τm λ0 em (t) Km dt + KD

S(t) |S(t)| + δ

(28)

with   Z t dy − (t) − − em (t)dt (29) S(t) = sign(K m ) − m + λ 1 em (t) + λ0 dt 0 − − where ym− (t) is the G − m output and em (t) = r (t) − ym (t). Eqs. (28) and (29) define the SPSMCr. In this case the tuning equations for first estimate values were determined from a systematic set of simulations as a function of the controllability relationship, CR = τtm0 . The following values provide satisfactory system performance and robustness against modeling errors [19]:  4 −1   if CR ≤ 4  τ [=][time] m λ1 = (30)  1.5   [=][time]−1 if CR ≥ 4 τm

λ21 [=][time]−2 8   0.75 t0 −0.76 = [=][fraction CO] |K m | τm

λ0 =

(31)

KD

(32)

δ = 2[0.68 + 0.12(|K m |K D λ1 )][=][fraction TO/time].

0.7239 K m (τm + t0m ) τd = 0.4(τm + t0m ).

Table 1 Controller tuning parameters IM-SMC

Values

SMC

Values

IMC

Values

λ KD δ

0.10 0.90 0.69

λ0 λ1 KD δ

0.19 0.87 0.21 0.70

τm Km τf

1.5 1.0 1.8

For this system a FOPTD model was obtained by using the reaction curve procedure [3],

(33)

A proportional-derivative controller given in Eq. (26) is used in the proposed SPSMCr. The parameters of the G d (s) controller for enabling the load disturbance rejection were found as recommended by Matauˇsek and Mici´c [8]: K0 =

Fig. 6. Process response for set point and disturbance changes.

(34) (35)

4. Simulation results To illustrate the performance of the proposed controllers a high order model with long time delay of a system is given, for each case. The different controllers were tested when set point and disturbances changes were applied to the process. Finally, IM-SMCr was compared against IMC and SMC; TDSMCr was compared against a dead time compensator (DTC), and for this comparison an SP was used as a DTC; and SPSMCr was compared against a predictive controller structure proposed by Matauˇsek and Mici´c (MM99) [8].

G m (s) =

e−5.68s . (1.3s + 1)

(37)

In Table 1 are shown the tuning values for each controller. These values were kept constant in all simulations. The tuning values for the SMCr were obtained as given in Camacho [9,17] and those for the IMCr were obtained as given in Marlin [2], where τm is the model time constant, K m is the model static gain, and τ f is the robustness filter time constant (see Fig. 1). Figs. 6 and 7 shows the process and controller outputs when a set point change was introduced at time t = 0 units of time (UT) and a disturbance change was applied at time t = 100 UT. It is observed that the IM-SMC and IMC controllers presented very close behavior, while the SMC controller presented a slower and oscillatory response, with a higher overshoot. 4.2. Time delay sliding mode controller In this case a fourth order system with long time delay and a controllability ratio above 1 was used (t0m /τm ≈ 8.22): e−10s . (s + 1)(0.5s + 1)(0.25s + 1)(0.125s + 1)

(38)

4.1. Internal model based sliding mode controller

G(s) =

In this case a fourth order system with time delay and a controllability ratio above 1 was used (t0m /τm ≈ 4.38):

For this system a FOPTD model was obtained by using the reaction curve procedure,

G(s) =

e−5s . (s + 1)(0.5s + 1)(0.25s + 1)(0.125s + 1)

(36)

G m (s) =

e−10.68s . (1.3s + 1)

(39)

100

O. Camacho et al. / ISA Transactions 46 (2007) 95–101

Fig. 7. Controller output when set point and disturbance changes were applied.

Fig. 9. Process response for disturbance change applied to G(s).

Fig. 8. Process response for set point change applied to G(s).

Fig. 10. Responses of the proposed SPSMCr and MM99 for G(s). Nominal case.

Table 2 Tuning parameters

4.3. The Smith predictor based sliding mode controller for integrating processes

DTC

G(s)

DTSMC

G(s)

Kc Ti Td

0.07 1.5 5.25

Ks KD δ

1.07 0.164 0.695

In Table 2 the tuning values for each controller are shown. These values were kept constant in all simulations. The tuning values for the PID adjustment in the DTC were obtained using the Dahlin equations [3] and for the TDSMC, using Eqs. (23)–(25). The system with both controllers, DTC and DTSMC, was tested against set point changes and disturbances. Fig. 8 shows the system responses when a set point change of 10% was simulated; both responses were overdamped, but the DTSMC produces a faster response than that given by the DTC. Fig. 9 depicts the responses for the same system when a disturbance of 5% is simulated at time t = 400 UT, similar to the set point change result; the DTSMC presents a better performance than the DTC.

In this case a fourth order system with long time delay and a controllability ratio above 1 was used (t0 /τ ≈ 11): G(s) =

e−20s s(s + 1)(0.5s + 1)(0.2s + 1)(0.1s + 1)

(40)

and the corresponding IFOPTD model was G m (s) =

e−20.64s . s(1.28s + 1)

(41)

For MM99, the equivalent time constant, Te , is set to 2.4 UT as in [8] to improve robustness. The rest of the parameters of the MM99 and SPSMCr are given in Table 3. A unit step input was introduced at time t = 0 UT and a −10% load disturbance at time t = 70 UT. Fig. 10 shows the system response when both control schemes were used. Fig. 11 shows the effect of a 20% time delay modeling error on the system performance. The MM99 scheme becomes unstable.

O. Camacho et al. / ISA Transactions 46 (2007) 95–101

Fig. 11. Responses of the proposed SPSMCr and MM99 for G(s). 20% error in t0 . Table 3 Controller tuning parameters for process G(s) MM99 Ko

Tr

Kr

2.4

0.417

0.027

Td

λ1

λ0

SPSMCr KD δ

Ko

Td

10.56

1.172

0.172

0.09

0.033

8.77

0.139

5. Conclusions A combined approach of predictive structures with sliding mode control was presented. This control schemes showed the benefits for dealing with long time delays using the predictive structure plus the robustness of the sliding mode theory. The proposed structures work well for processes that can be approximated either by a first order plus time delay or an integral first order plus time delay model, broadly used on chemical processes. The proposed schemes showed a better performance and robustness against set point changes and disturbances when they were compared with classical approaches. The use of simple predictive structures and the provision of tuning equations make implementation easy and give a good starting point for the adjustment. References [1] Tan KK, Lee TH, Leu FM. Predictive PI versus Smith control for deadtime compensation. ISA Transactions 2001;40:17–29. [2] Marlin TE. Process control. New York: McGraw-Hill; 1995. [3] Smith C, Corripio A. Principles and practice of automatic process control. New York: John Wiley & Sons, Inc.; 1997. [4] Watanabe K, Ito M. A process-model control for linear systems with delay. IEEE Transactions on Automatic Control 1981;26(6):1261–9. [5] Zhang WD, Sun YX. Modified Smith predictor for controlling integrator/time delay processes. Industrial Engineering Chemistry Research 1996;35:2769–72. [6] Astrom KJ, Hang CC, Lim BC. A new Smith predictor for controlling a process with an integrator and long dead-time. IEEE Transactions on Automatic Control. 1994;39(2):343–5. [7] Matauˇsek MR, Mici´c AD. A modified Smith predictor for controlling a process with an integrator and long dead-time. IEEE Transactions on Automatic Control 1996;41(8):1199–203.

101

[8] Matauˇsek MR, Mici´c AD. On the modified Smith predictor for controlling a process with an integrator and long dead-time. IEEE Transactions on Automatic Control 1999;44(8):1603–6. [9] Camacho O, Smith CA. Sliding mode control: An approach to regulate nonlinear chemical process. ISA Transactions 2000;39:205–18. [10] Camacho O, Rojas R. A general sliding mode controller for nonlinear chemical processes. Transactions of ASME 2000;122:650–5. [11] Camacho O, Smith C, Moreno W. Development of an internal model sliding mode controller. Industrial Engineering Chemistry Research 2003; 41:568–73. [12] Camacho O. A predictive approach-based sliding mode control. In: Proceedings of 15th IFAC triennial world congress B’02; 2002. [13] Camacho O, Rojas R. An approach of sliding mode control for dead time systems. WSEAS Transactions on Circuits and Systems 2004;4:789–93. [14] Utkin VI. Variable structure systems with sliding modes. IEEE Transactions on Automatic Control 1997;AC-(22):212–22. [15] Slotine JJ, Li W. Applied nonlinear control. New Jersey: Prentice-Hall; 1991. [16] Himmelblau DM. Applied nonlinear programming. New York: McGrawHill; 1972. [17] Camacho O. A new approach to design and tune Sliding Mode Controller for Chemical Process. Ph.D. dissertation, University of South Florida; 1996. [18] Camacho O, Smith C, Chac´on E. Toward an implementation of sliding mode control to chemical processes. In: Proceedings of ISIE’97. 1997. p. 1101–5. [19] Camacho O, De la Cruz F. Smith predictor-based sliding mode controller for integrating processes with elevated deadtime. ISA Transactions 2004; 43:55–72. [20] Tian Y, Gao F. Control of integrator processes with dominant time delay. Industrial Engineering Chemistry Research 1999;38:2979–83. Oscar Camacho received an Electrical Engineering and M.S. in Control Engineering degrees from Universidad de Los Andes (ULA), M´erida, Venezuela, in 1984 and 1992, respectively, and an M.E. and Ph.D. in Chemical Engineering at University of South Florida (USF), Tampa, Florida, in 1994 and 1996, respectively. He has held teaching and research positions at ULA, PDVSA, and USF. His current research interest includes sliding mode control, dead time compensation, and fault detection systems. He is the author of more than 60 publications in journals and conference proceedings. Rub´en Rojas received a Systems Engineering degree from Universidad de Los Andes (ULA), M´erida, Venezuela, in 1986, and an M.Sc. and Ph.D. in Biomedical Engineering at University of lows (UI), Iowa City, Iowa, in 1994 and 1997, respectively. He has held teaching and research positions at ULA. His current research interest includes sliding mode control, dead time compensation, mathematical modeling, and identification. He is the author of more than 50 publications in journals and conference proceedings. Winston Garc´ıa-Gab´ın received an Electrical Engineering, and M.S. in Automation and Instrumentation degrees from Universidad de Los Andes, M´erida, Venezuela, in 1994 and 1998, respectively, and a Ph.D. in Industrial Engineering at University of Seville, Spain, in 2002. He has held teaching and research positions at the Electrical Engineering School at Universidad de Los Andes. His current research interests include sliding mode control, model predictive control, and time delay systems. He is the author of more than 50 publications in journals and conference proceedings. He has been in consultancy work, as well as lecturing in short courses for companies.