Smith Predictor Based-Sliding Mode Controller for Integrating Process with Elevated Deadtime

Transcription

1 Sith Predictor Based-Sliding Mode Controller for Integrating Process with Elevated Deadtie Oscar Caacho, a, * Francisco De la Cruz b a Postgrado en Autoatización e Instruentación. Grupo en Nuevas Estrategias de Control Aplicado. Universidad de los Andes, Mérida 5. Venezuela b Departaento de Ingeniería Electrónica. Vice Rectorado Barquisieto Universidad Nac. Exp. Politécnica Antonio José de Sucre, Barquisieto 3. Venezuela *Corresponding author. Tel: ; fax: ; e-ail address: ocaacho@ula.ve

2 Sith predictor based - sliding ode controller for integrating processes with elevated deadtie ABSTRACT An approach to control integrating processes with elevated deadtie using a Sith Predictor Sliding Mode Controller is presented. A PID sliding surface and an integrating first order plus deadtie odel have been used to synthesize the controller. Since the perforance of existing controllers with a Sith Predictor decrease in the presence of odeling errors, this article presents a siple approach to cobining the Sith Predictor with the Sliding Mode concept, which is a proven siple and robust procedure. The proposed schee has a set of tuning equations as a function of the characteristic paraeters of the odel. For ipleentation of our proposed approach, coputer based industrial controllers that execute PID algoriths can be used. The perforance and robustness of the proposed controller are copared with the Mataušek-Micić schee for linear systes using siulations. Keywords: Sliding Mode Control, Sith Predictor, integrating process, deadtie. INTRODUCTION The presence of tie delays in any industrial processes is a well-recognized proble. The achievable perforance of typical feedback control systes can decline if a process has a relatively large tie delay copared to the doinant tie constant []. In the case of integral processes, deadtie copensation is necessary in order to enhance the perforance of the control syste. O Dwyer [][3] considered a wide variety of ethods for the copensation of processes with tie delay, in both the continuous tie and discrete tie doains. The copensators discussed are:. PID controllers and its variations.. Lead, lag or lead-lag controllers. 3. The Sith predictor [4] and its variations.

3 3 4. Direct synthesis ethods, which are typically based on designing the controller to eet a required output specification; pole placeent controllers are an exaple. 5. Optial controller design ethods, which ay be based on a iniu variance or linear quadratic control strategy. 6. Predictive controllers. 7. Other copensation strategies for processes with tie delays, including fuzzy ipleentation, neural networks and expert systes. The wide spectru of ethods covered, and the dependence of the choice of copensator ethod on the application, eans that an overall conclusion as to the best ethod to use is not appropriate. Viewing the copensation proble fro a variety of perspectives, appear to present the Sith predictor as the optial (or a coponent of the optial) controller for doinant delay processes []. In those works, the copensation of integrating process with dead tie is only achieved using PID controllers, Sith predictors or its variations. Others approaches has not been reported for this kind of systes. The Sith Predictor (SP), or deadtie copensator (DTC) as is also known, has any weak points, including possible instability and poor perforance under odeling errors, and poor response to disturbances []. In addition, the original structure of SP cannot reject constant load disturbance for processes with integration [5]. To overcoe this obstacle any variations of SP has been proposed. Sliding Mode Control (SMC) is a robust and siple procedure to develop controllers for linear and nonlinear processes [6]. The design of a Sliding Mode Controller (SMCr) depends on the process odel and the nuber of tuning paraeters is proportional to the odel order. Caacho and Sith [7] developed a siple and practical ethod for the design of a SMCr based on a siplified odel of the actual process. The SP perfors well for eliinating deadtie and the SMCr is a proven robust controller. It is desired cobining the in a single control structure that preserves the good qualities of both techniques and iprove the bad qualities of SP. A robust controller for integral

4 4 processes with deadtie, the Sith Predictor Sliding Mode Controller (SPSMCr), will be developed. This paper is organized as follows: Section introduces the basic concepts of the sliding ode controller, the SP architecture and a review of the odified SP for integrating processes that have previously been proposed. Section 3 covers the developent of the SPSMCr based on an integrating first-order process odel with deadtie. Section 4 shows the SPSMCr ipleentation using a PID algorith of a coputer based industrial controller. Section 5 provides siulation results to illustrate the approach and to copare it with the perforance and robustness of a previous schee. Finally, soe conclusions are presented in section 6.. BASIC CONCEPTS.. Sliding Mode Control (SMC). Sliding Mode Control is a technique derived fro variable structure (VSC) which was originally studied by [8]. A controller designed using the SMC ethod is particularly appealing due to its ability to deal with nonlinear systes and tie-varying systes [9]. The robustness to the uncertainties becoes an iportant aspect in designing any control systes. The idea behind SMC is to define a surface along which the process can slide to its desired final value. Figure 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, for instance stability and tracking perforance. Figure The S(t) selected in this work, presented by [9], is an integral-differential equation acting on the tracking-error expression: de() t St () = f et (), etdt (),, λ, n () dt

5 5 where e(t) is the tracking error, that is, the difference between the reference value or set point, r(t), and the output easureent, x(t), naely e(t) = r(t) - x(t). λ is a tuning paraeter, which helps to define S(t). This ter is selected by the designer, and deterines the perforance of the syste on the sliding surface. n is the syste order. The objective of control is to ensure that the controlled variable be equal to its reference value at all ties, eaning that e(t) and its derivatives ust be zero. Once the reference value is reached, Eq. () indicates that S(t) reaches a constant value, eaning that e(t) is zero at all ties; it is desired to set ds( t) = dt () Once the sliding surface has been selected, attention ust be turned to design the control law that drives the controlled variable to its reference value and satisfies Eq. (). The SMC control law, M(t), consists of two additive parts: a continuous part, U C (t), and a discontinuous part, U D (t). That is The continuous part is given by M ( t) = U ( t) U ( t) (3) C C D U () t = f( x(), t r()) t (4) where f ( x( t), r( t)) is a function of the controlled variable and the reference value. The discontinuous part is nonlinear and represents the switching eleent of the control law. This part of the controller is discontinuous across the sliding surface. Mainly, U D (t) is designed based on a relay-like function (i.e. U D (t) = α sign((s(t))), because it allows for changes between the structures with a hypothetical infinitely fast speed. In practice, however, it is ipossible to achieve the high switching control because of the presence of finite tie delays for control coputations or liitations of the physical actuators causing chattering around of the sliding surface [8][9]. Chattering is a high-frequency oscillation around the desired equilibriu point. It is undesirable in practice, because it involves high control activity and can excite high-frequency dynaics ignored in the odeling of the

6 6 syste [9][]. The aggressiveness to reach 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 siga function, which can be written as follows: U () D() t St = K D St () δ (5) where K D is the tuning paraeter responsible for the reaching ode. δ is a tuning paraeter used to reduce the chattering proble. Figure shows the effect of δ variations in the shape of saturation function. Figure 3 shows the effect of K D variations on the syste trajectory on the phase plane fro an initial state to a final state Figure Figure 3 In suary, the control law usually results in a fast otion to bring the state onto the sliding surface, and a slower otion to proceed until a desired state is reached... Sith Predictor. Basic Concepts and Previous Schees for Integrating Processes. As stated before, SP is a popular schee for deadtie copensation. Figure 4 shows the architecture of the SP. The process transfer function is G ( s) p t s = G( s) e which is assued to consist of a rational stable transfer function G(s) and a deadtie, t. A odel of the process without deadtie, G (s), is used to predict the effect of the control action on the process output and to increase the perforance of the syste. The difference between the output of the process and the odel is fed back in order to correct odeling errors and load disturbances. Figure 4 If there is no process/odel isatch, G (s) = G(s) and t = t, then the odeling error e (t) = y(t) -y (t) =. Since the deadtie is separated fro the odel, and e =, the feedback only consists of the odel without delay. Therefore, the deadtie is isolated and

7 7 copensated [], and thus, for controller design purposes, it can be ignored. However, the SP cannot be used with its original structure to control processes with integration since a constant load disturbance results in a steady-state error [5]. To overcoe this obstacle any variations of SP has been proposed. Watanave and Ito [5] proposed a odification of the SP as shown in Figure 5. The syste can reject a load disturbance if the tie delay of the process is exactly known. Otherwise, there will be a sall steady-state error []. Siulations studies have shown that the setpoint and load disturbances are either very oscillatory or highly daped when the process has a large dead tie [3]. Figure 5 Aströ et al. [3] proposed a new SP structure, as shown in Figure 6, where the disturbance response is decoupled fro the setpoint response. The controller has four adjustable paraeters but a systeatic tuning ethod was not given. The authors ts considered only the restricted integrating process Gs () = e / s. Figure 6 Zhang and Sun [] iproved the results of Aströ et al. retaining the separation nature of load response fro the setpoint response. The structure of the odified SP is shown in Figure 7. M(s) was recoended as sm () s ( θt λt) s T M() s =, M () s = ts st M ( s) e ( s ) Ts o o o λ The Eq. (6) contains a positive feedback loop that is a potential instability source, resulting in liited robustness [5]. Zhan and Sun rely on a general guideline to tune K r and λ rather than a systeatic approach [6]. The proposed schee does not provide uch better perforance than Aströ s []. When the process is described by a high order odel, the controller is a derivative or a derivative with lag and a ore coplex M(s) will result. Figure 7 (6)

8 8 Mataušek and Micić [7][8] proposed a odified SP and gave a siple controller tuning ethod. Their schee, given in Figure 8, has a controller to reove the load disturbance although the setpoint and load responses cannot be decoupled. In a first paper [7], F(s) is a proportional controller, K o, obtaining a siple structure with three adjustable paraeters. However, for high values of the dead tie, the schee has poor disturbance rejection [9] and becoes significantly oscillatory with tie delay deviation [5]. Figure 8 A second Mataušek and Micić s odified SP [8], where F(s) is a lead-lag copensator, provides considerably faster load disturbance rejection but requires a trial-an-error procedure of the ain controller gain in a tradeoff between closed loop syste perforance and stability/robustness. This schee will be used to copare the proposed controller. The Tian and Gao s control schee [5] has the sae structure as Astro s SP and a paraeters tuning as in [8]. The structure of this SP includes four controllers, as shown in Figure 9. Figure 9 A local proportional feedback K o is introduced to prestabilize the integrator process. The introduction of G o (s) eliinates the effect of K o on setpoint tracking. To copensate the phase lag caused by integrator and tie delay a proportional-derivative controller G c (s) is suggested, where: G K K o p tos o( s) = e ; Gc ( s) K c ( Td s) s = (7) This structure behaves slightly better than the latest Mataušek and Micić s schee (setpoint ITAE index reduction <.5%) [5], despite the added structural coplexity and their tuning strategy can be applied to only pure integrator plus tie delay odels [6]. 3. THE SMITH PREDICTOR BASED SLIDING MODE CONTROLLER FOR INTEGRATING PROCESSES

9 9 The Sith Predictor based Sliding Mode Controller (SPSMCr) proposed in this paper uses the standard SP architecture while the controller is a sliding ode controller (SMC). The block diagra of the proposed schee is shown in Figure. As entioned, the original structure of SP is ineffective for integrator processes because it cannot reject a constant load disturbance [5] and proportional-derivative controller G d (s) is used for load disturbance rejection, where G d (s) = K o (T d s ) (8) This controller will be discussed later. Figure To develop SPSMCr, an integrating first order plus deadtie (IFOPDT) process odel given in Eq. (9) is considered ts K Gp () s = G() s e = e s( τ s ) ts where K is the process gain, t is the process deadtie and τ is the process tie constant. If there is no process/odel isatch, the odel transfer function without deadtie is X () s K G () s = = M() s s( τ s ) where K is the odel gain and τ is the odel tie constant. (9) () Since, the deadtie ter has been isolated using a SP structure, we can ignore it in the SMC design. Then, transforing Eq. () into differential equation for τ d X() t dx() t K () M t dt dt = () and d X t = K () M t dt τ () dx () t dt In this case, we use an IFOPDT process and n =. Then, fro Eq. (), we selected as S(t) an integral-differential equation acting on the tracking error expression represented by ()

10 de() t t St () = λet () λ etdt () dt (3) Equation (3) represents a PID surface. The paraeters λ and λ can be chosen independently and, in our case, they were select to obtain an overdaped response. Fro Eq. () ds() t d e() t de() t = λ λet () = (4) dt dt dt but e ( t ) = R ( t ) X ( t ) and substituting into the abode equation gives d X () t dx () t et () = (5) dt dt dt d R() t dr() t λ λ λ dt Caacho [] has shown that the derivates of the reference value can be discarded without any effect on the control perforance. Thus Substituting Eq. () into Eq. (6) d X() t dx() t λ λ Thus, the continuous part of the controller is dt = et () (6) dt dx () t dx () () () t λ λ et = K M t dt dt τ dx() t M () t = ( τ λ ) τ λ e() t C K dt Then, the coplete SPSMCr can be represented as dx() t St ( ) Mt () = ( τ λ ) τ λ et () K K dt S t δ D () (7) (8) (9) with dx() t t St () = signk ( ) λet () λ etdt () dt () The function sign(k), in Eq. (), is included in the sliding surface equation to guarantee the appropriate action of the controller for the given syste. Note that sign(k) only depends on the static gain of the plant; therefore it never switches [6].

11 Equations (9) and () define the controller equations to be used in the SPSMCr, which can be siplified by setting λ [ ][ ] = = tie () τ Furtherore, to assure that the sliding surface behaves as a critical or overdaped syste, λ should be λ λ [ = ][ tie] () 4 Nuerous siulations showed that the values of λ and λ are a function of controllability t relationship, CR =, and the following values provide satisfactory syste perforance τ and robustness with tie delay deviation: λ 4 τ [ ][ tie] = if CR 4 =.5 = if CR τ [ ][ tie] 4 (3) λ λ = [ = ][ tie] (4) 8 The paraeters δ and K D has a relationship with syste speed, overshot and chattering. Based on previous approaches [6][][4] where the Nelder-Mead searching algorith were used, the tuning paraeters of the controller discontinuous part are K.75 t.76 D = = K τ [ ][ fractionco] [ K K λ ][ ][ fraction TO tie] (5) δ =.68.( ) = / (6) D 3.. Disturbance rejection As entioned, any odified SP with different structures have been proposed in the literature for integrator processes to reove the steady-state error produced by a constant

12 load disturbance. Tian and Gao [5], and Mataušek and Micić [8] added derivative action to their proposed DTC to overcoe this proble. A proportional-derivative controller G d (s), given in Eq. (8), is used in the proposed SPSMCr to enable the load disturbance rejection. The paraeters of this controller, using α =.4, Φ p = 64º =.7 rad, as recoended in [8], are: K o π Φp = = π K ( τ t ) ( α) Φp α.739 K τ t ( ) (7) T d = α (τ t o ) =.4 (τ t o ) (8) 4. SPSMCr IMPLEMENTATION USING A PID ALGORITHM. Coputer based industrial controllers that execute PID algoriths (Prograable Logic Controllers and Reote Terinal Units) can be used for SPSMCr ipleentation. The ore coon PID algorith is based on the following equation: t dcv ( t) MV ( t) = Kc e( t) e( t) dt τ D (9) τ i dt Where MV(t) is the anipulated variable, τ i is the integral tie, τ D is the derivative ter and CV(t) is the controlled variable. Note that the derivative ter is calculated using the easured controlled variable, not the error. Figure shows the block diagra that ipleents an algorith for a PID. Figure. As can be observed, Eq. (9) and Eq. () present siilitude. The ter X (t) in Eq () is the sae CV(t) in Eq. (9). Therefore, to represent the sliding surface S(t), the next step is to tune the PID based on the SPSMCr tuning equations as follows: K c = λ τ = λ / λ i τ = / λ D (3)

13 3 With these equations the PID algorith can be changed, representing S(t). Then, to achieve the ipleentation, the sliding surface value S(t) is calculated fro the PID output. Because the continuous and discontinuous parts of the controller are algebraic equations, they are prograable easily. The block diagra that ipleents an algorith for the SPSMCr is shown in Figure. Figure 5. SIMULATION RESULTS To illustrate the SPSMCr perforance, two exaples are given. In exaple, the syste response to different paraeters adjustent is presented to show the relationship with the syste perforance. In exaple, the response of an integrating syste with a long deadtie is shown and the results are copared to those obtained using the proposed controller in [8]. Exaple Let us consider a process with IFOPDT transfer function G (s) G ( s) e s(3s ) 6s = (3) with CR = t /τ =. Using Eq. (3)-(6), the paraeters of SPSMCr are: 4 4 λ = = =.333 ; τ 3 K λ λ = = t.75 6 D = = = K τ [ K K λ ] [ ] δ = *.68.( ) = *.68.()(.443)(.333) =.5 D K o = = =.89 K τ t ( ) 3 ( 6) T d =.4 (τ t o ) =.4 (9) = 3.6

14 4 Figure 3 and Figure 4 show the process output to a setpoint change when λ and λ variations were introduced. In these figures, the dotted lines correspond to responses using the values obtained with Eq. (3)-(4). The saller values of λ increase speed and overshoot. The saller values of λ increase speed response and decrease settling tie. Nuerous siulations showed that the larger speed, the saller syste robustness. Figure 3 Figure 4 Figure 5 shows the syste responses to a setpoint change when reductions of δ value were introduced. Figure 5a corresponds to δ value obtained with Eq. (5). Figure 5b corresponds to δ =. The saller δ increases speed and chattering. Figure 5 Figure 6 shows the syste responses to a setpoint change when reductions of K D value were introduced. In this figure, the solid lines correspond to responses to K D value using Eq. (6). A larger K D increase speed and overshot syste and the saller K D values increase settling tie. Figure 6 Exaple Let us consider a process with a transfer function G (s), which has been investigated in [5], [7], [8] and [9] G ( s) The corresponding IFOPDT odel is with CR = 6.5. e s( s )(.5s )(.s )(.s ) = s (3) G ( s) e s(.8s ).64s = (33) In this exaple, the SPSMCr is copared to the controller proposed in [8] and denoted by MM99. For MM99, the equivalent tie constant, T e, is set to.4 as in [8] to iprove robustness. The reainder paraeters of MM99 and SPSMCr are given in Table. A unit

15 5 step set point is introduced at tie t = and a load disturbance d =. is introduced at tie t = 7. When the odel is exact, Figure 7a shows the response of both control schees. Figure 7b shows the effect of a % deadtie odeling error on syste perforance. The MM99 schee becoes unstable. 6. CONCLUSIONS Table Figure 7 In this paper, a robust control schee for integrating systes with deadtie using a Sith Predictor Sliding Mode Controller (SPSMCr) has been presented. Sliding Mode Control iproves robustness and stability to the new schee while Sith Predictor isolates and copensates the deadtie. This new ethod allows for a robust controller while copensating for the deadtie within the process. The robustness of the steady-state perforance is iproved by the sliding ode with % odeling errors. This controller cobines the siplicity of the SP architecture ipleentation and the robustness of a sliding ode controller. Besides, the disturbance rejection for integrator processes of the original structure of SP has been iproved adding a proportional-derivative controller originally proposed by Mataušek and Micić [8]. The new proposed controller has a siple tuning procedure if an IFOPDT odel is available. REFERENCES [] Tan, K. K., Lee, T. H., and Leu, F. M., Predictive PI versus Sith control for deadtie copensation, ISA Transactions,, 4, 7-9. [] O Dwyer, A., The estiation and copensation of processes with tie delays. Ph. D. Thesis. Dublin City University, Ireland [3] O Dwyer, A., A survey of techniques for the estiation and copensation of processes with tie delay. Technical Report Nuber: AOD..3. Dublin Institute of Technology, Ireland.. [4] Sith, O. J. M., Close Control Loops with Dead Tie. Che. Eng. Progress. 957, 53, 7-9

16 6 [5] Watanabe, K., and Ito, M., A process-odel control for linear systes with delay. IEEE Tran. on Autoatic Control, 98. 6(6), [6] Caacho, O., Rojas, R., and García, W. M., Variable Structure Control Applied to Cheical Processes with Inverse Response. ISA Transactions, 999, 38, [7] Caacho, O. and Sith, C. A., Sliding Mode Control: An approach to regulate nonlinear cheical process. ISA Transactions,, 39, 5-8 [8] Utkin, V. I., Variable structure systes with sliding odes. Transactions of IEEE on Autoatic Control. 997, AC-(), -. [9] Slotine, J. J., and Li, W., Applied Nonlinear Control. Prentice-Hall, New Jersey, 99. [] Caacho, O. and Rojas, R., A general Sliding Mode Controller for Nonlinear Cheical Processes. Transactions of ASME,,, [] Caacho, O., A new approach to design and tune Sliding Mode Controller for Cheical Process. Ph. D. Dissertation, University of South Florida, 996. [] Zhang, W. D. and Sun, Y. X., Modified Sith Predictor for Controlling Integrator/Tie Delay Processes. Ind. Eng. Che. Res., 996, 35, [3] Astro, K. J., Hang, C. C. and Li, B. C. A New Sith Predictor for Controlling a Process with an Integrator and Long Dead-Tie. IEEE Trans. on Autoatic Control. 994, Vol. 39, No., [4] Caacho, O., Rojas, R., García, W., and Álvarez, A., Sliding Mode Control: A Robust Approach to Integrating Systes whit Dead Tie. Proc. Second IEEE Int. Caracas Conf on Devices, Circuits and Systes, ICCDSC-98, Margarita, Venezuela, 998. [5] Tian, Y. and Gao, F., Control of Integrator Processes with Doinant Tie Delay. Ind. Eng. Che. Res., 999, 38, [6] Kwak, H. J., Sung S. W., and Lee, I., Modified Sith Predictors for Integrating Processes: Coparisons and proposition. Ind. Eng. Che. Res., 4, [7] Mataušek, M. R. and Micić, A. D., A Modified Sith Predictor for Controlling a Process with an Integrator and Long Dead-Tie. IEEE Trans. on Autoatic Control, 996, 4(8), 99-3.

17 7 [8] Mataušek, M. R. and Micić, A. D., On the Modified Sith Predictor for Controlling a Process with an Integrator and Long Dead-Tie. IEEE Tran. on Autoatic Control, 999, 44(8), [9] Noray-Rico, J. E. and Caacho, E. F. Robust Tuning of Dead-Tie Copensators for Processes with an Integrator and Long Dead-Tie. IEEE Trans. on Autoatic Control, 996, 44(8),

18 8 Figure. Graphical interpretation of SMC. List of captions Figure. Chattering reduction using a saturation function (a: δ = ; b: δ =.; c: δ =.; d: δ =.). Figure 3. Graphical interpretation for K D variations. Figure 4. Sith Predictor schee. Figure 5. The odified SP proposed by Watanabe and Ito Figure 6. The SP structure proposed by Astro et al. Figure 7. The structure of Zhan and Sun s SP. Figure 8. Modified SP proponed by Mataušek and Micić. Figure 9. Tian and Gao s control schee. Figure. Sith predictor based Sliding Mode Controller. Figure. Ipleentation of PID algorith in an industrial controller. Figure. Ipleentation of SPSMCr using an industrial controller. Figure 3. Syste response to a setpoint change for different λ values: --- : λ =.33. : λ =.67. : λ =.33. Figure 4. Syste response to a setpoint change for different λ values: --- : λ =.. : λ =.44. : λ =.56. Figure 5. Syste responses using SPSMCr for G (s): (a) δ =.5 and (b) δ =. Figure 6. Syste response to a setpoint change for different K D values: : K D = : K D =.5. : K D = Figure 7. Responses of the proposed SPSMCr and MM99 for G (s). (a) Noinal case, (b) % error in t. Table. Controller tuning paraeters for process G (s).

19 9 e (t) Desired final value e(t) Reaching Mode Sliding Mode Sliding Surface Figure. Graphical interpretation of SMC..5.5 a b c d Figure. Chattering reduction using a saturation function (a: δ = ; b: δ =.; c: δ =.; d: δ =.).

20 .. K K K3 Final value Initial value -. de/dt e(t) Figure 3. Graphical interpretation for K D variations. K<K<K3 r Controller ts G( s) e y G (s) x t e s y Figure 4. Sith Predictor schee.

21 r G c (s) d ts e s y G () s e s ts Figure 5. The odified SP proposed by Watanabe and Ito r d y s ts k e s e ts M(s) Figure 6. The SP structure proposed by Astro et al. r d y e Ts K r ts Ts e ts M(s) Figure 7. The structure of Zhan and Sun s SP

22 r K r F(s) d K p /s ts G( s) e e t s y Figure 8. Modified SP proponed by Mataušek and Micić. r K r G o (s) d G( s) e t s y K p /s G c (s) K o t s e Figure 9. Tian and Gao s control schee. r SMC G d (s) d G (s) ts G( s) e x t s e y y Figure. Sith predictor based Sliding Mode Controller.

23 3 PROCESS I/O Subsyste Process Data PID Industrial controller Figure. Ipleentation of PID algorith in an industrial controller. PROCESS I/O Subsyste Process Data PID SPSMC algorith Industrial controller Figure. Ipleentation of SPSMCr using an industrial controller.

24 4.5 Process output, (Fraction TO) Controller output, (Fraction CO) Figure 3. Syste response to a setpoint change for different λ values: --- : λ =.33. : λ =.67. : λ =.33.

25 5.5 Process output, (Fraction TO) Controller output, (Fraction CO) Figure 4. Syste response to a setpoint change for different λ values: --- : λ =.. : λ =.44. : λ =.56.

26 6.5 Process output, (Fraction TO) Controller output, (Fraction CO) (a).5 Process output, (Fraction TO) Controller output, (Fraction CO) (b) Figure 5. Syste responses using SPSMCr for G (s): (a) δ =.5 and (b) δ =.

27 7.5 Process output, (Fraction TO) Controller output, (Fraction CO) Figure 6. Syste response to a setpoint change for different K D values: : K D = : K D =.5. : K D = 3.54.

28 8.5 SPSMCr MM99 Process output, y(t), (Fraction TO) Tie (Unit of tie) (a) 5 4 SPSMCr MM99 Process output, y(t), (Fraction TO) Tie (Unit of tie) (b) Figure 7. Responses of the proposed SPSMCr and MM99 for G (s). (a) Noinal case, (b) % error in t.

29 9 Table Controller tuning paraeters for process G (s). MM99 SPSMCr T r K r K o T d λ λ K D δ K o T d