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Sliding-Mode Control of Stable Processes Ibrahim Kaya Dicle UniVersity, Faculty of Engineering and Architecture, Department of Electrical and Electronics Engineering, 21280, Diyarbakır, Turkey
The robustness of a controller to parameter variations or modeling errors is an important issue in process control. Sliding-mode control (SMC) is well-known for its property of being insensitive to parameter variations, modeling errors, and external disturbances. Hence, an appropriately designed SMC can be used to overcome the aforementioned weaknesses, which are unavoidable in practice. In this paper, a new sliding surface with four parameters has been introduced to achieve a satisfactory closed-loop system performance. Simulation examples are provided to show the value of the proposed SMC. Comparisons with an existing SMC method are also given to illustrate the superiority of the proposed SMC. 1. Introduction In practice, controller designs are usually performed based on an approximate model of the actual process. Generally, a first-order plus dead time (FOPDT) model is adequate to represent open-loop stable processes. However, there may be discrepancies between the actual plant and the obtained FOPDT model for the use in control. Parameter variations, modeling of higher and complex plant transfer functions by the approximate FOPDT model, or unmodeled dynamics may be the reasons behind these discrepancies. Therefore, ensuring a satisfactory closed-loop system performance in practice, despite such plant and model mismatches is an important task. Sliding-mode control (SMC)1,2 is well-known for its robustness to modeling errors, as well as its insensitivity to parameter variations and disturbances. This property of the SMC is the reason it has been used in many successful practical applications, such as use in robotics3,4 and electric drives.5,6 There are two phases in the SMC, namely, the reaching phase and the sliding phase. In the reaching phase, the system state is derived onto a specified and user chosen surface, which is called sliding surface, in a finite time. Once in the sliding mode, the system dynamics are strictly determined by the dynamics of the sliding surface and, therefore, the closed-loop system becomes insensitive to parameter changes and disturbances.2 However, no such insensitivity to parameter variations and disturbances can be possessed during the reaching phase. Therefore, to ensure a good closed-loop system response, the control system should be designed in such a way that the initial reaching phase is as short as possible.2 The robustness property of the SMC to modeling errors and parameter changes, which are unavoidable in practice, can be used to design robust process control systems. Recently, the use of SMC to control chemical processes has gained great attraction. Camacho and Smith7 proposed a SMC based on the FOPDT model for controlling open-loop stable chemical processes. Camacho et al.8 gave the use of the SMC in the internal model control. Rojas et al.9 extended the use of SMC to control open-loop unstable processes. Camacho and De La Cruz10 presented Smith-predictor-based SMC for integrating processes. Cheng and Peng11 have also given the design of a SMC system for chemical processes. * To whom correspondence should be addressed. Tel.: +90 412 2488030, ext 3580. Fax: +90 412 2488405. E-mail address:
[email protected].
In this paper, a new sliding surface is introduced to achieve reasonably satisfactory closed-loop responses in controlling open-loop stable processes. The design procedure is based on the use of the FOPDT model. For the modeling, a relay feedback test12,13 is conducted to identify parameters of the FOPDT model exactly, assuming no measurement errors. Extensive simulation examples are provided, and comparisons are given to illustrate the advantages and superiority of the proposed SMC approach over the one suggested by Camacho and Smith.7 The reason to choose the SMC that was suggested by Camacho and Smith7 for comparison is that it is the best candidate, because it is also proposed for controlling stable processes and has a similar structure to the one suggested in this paper. The SMC method proposed in this paper has several advantages over the one suggested by Camacho and Smith.7 First, they use a step test to identify the unknown parameters of the FOPDT model. Because the step test is an open-loop test, it is very sensitive to external disturbances. As stated previously, a relay feedback test13,14 is performed to identify parameters of the FOPDT model exactly. The proposed relay feedback method considers the effect of static load disturbances in the derived expressions so that the identification procedure results in exact estimates, even under static load disturbances. Second, the open-loop step test cannot be performed for the modeling of underdamped processes. On the other hand, the relay feedback test can also be performed for modeling underdamped processes with the FOPDT transfer function. Third, the SMC suggested by Camacho and Smith7 usually results in unsatisfactory closed-loop system performances in the sense of large overshoots and long settling times. All the aforementioned shortcomings have been eliminated in this paper, to achieve a better closed-loop performance. 2. Model Parameter Estimation Identification based on relay feedback control, as given in Figure 1, has become an accepted practical procedure. There are several reasons behind the success of the relay feedback identification method. First, relay feedback control, as normally used, gives important information about the process frequency response at the critical gain and frequency, which are the essential data that are required for controller design. Second, the relay feedback identification method is performed under closed-loop control. If appropriate values of the relay parameters are chosen, the process may be kept in the linear region, where the frequency response is of interest. Third, the relay feedback
10.1021/ie0607806 CCC: $37.00 © 2007 American Chemical Society Published on Web 12/14/2006
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phase (with s(t) * 0). In correspondence with the two phases, two types of control lawsthat is, the continuous control uc(t) and discontinuous control ud(t)scan be derived separately.15 Hence,
u(t) ) uc(t) + ud(t)
Figure 1. Relay feedback control under static load disturbances.
method eliminates the need for a careful choice of frequency, which is the case for traditional methods of process identification, because an appropriate signal can be generated automatically. Finally, the method is so simple that operators understand how it works. In this paper, it is assumed that the process dynamics can be modeled effectively by an FOPDT transfer function given by
G(s) )
C(s) Ke-Ls ) U(s) (Ts + 1)
(1)
Note that the FOPDT model is only used to simplify calculations, and the actual process may be a higher-order process or a process with complex poles as well. In this paper, it is assumed that the dead time is relatively small, because a first-order Taylor series approximation is used for the dead time to obtain the control law in the next section. For larger dead times, a different control strategy, such as that given by Camacho et al.,8 must be used, which is not within the scope of this paper. Expressions to give exact solutions for a limit cycle to occur for the FOPDT transfer function under a relay-controlled feedback loop with static load disturbances have been derived by Kaya12 or Kaya and Atherton.13 The details of parameter estimation are not given here; interested readers can refer to Kaya12 and Kaya and Atherton13 to determine the unknown parameters of the FOPDT plant transfer function. However, the required equations for the FOPDT plant transfer function to identify its unknown parameterssnamely, K, T and Lsare given in the Appendix for convenience. 3. Sliding-Mode Control of Stable FOPDT Processes The idea behind the SMC is to derive the system state trajectory onto a specified surface along which the process can slide to its desired final value. Hence, the first step in the SMC is to define the sliding surface s(t), which is expected to respond in the desired control specifications and performance. In the literature, the sliding surfaces are usually selected to have the form of a proportional integral (PI) or proportional integral derivative (PID) controller. However, it is well-known that PI and PID controllers have structural limitations and may not give satisfactory responses for processes with large time constants or poorly located complex poles.14 On the other hand, PI-PD controllers are shown to result in much better closed-loop responses in the aforementioned cases.14 Hence, in this paper, the following sliding surface s(t), which is the control signal of a PI-PD controller, is proposed.
s(t) ) k1e(t) + k2
∫0t e(t) dt - k3c(t) - k4
dc(t) dt
(2)
In eq 2, e(t) is the tracking error, that is, the difference between the set point value r(t) and the system output c(t). The terms k1, k2, k3, and k4 are the tuning parameters used to define the sliding surface s(t) and, hence, helps to determine the performance of the system in the sliding mode. The process of SMC can be divided into two phases: namely, the sliding phase (with s(t) ) 0 and s˘ (t) ) 0) and the reaching
(3)
The continuous control law corresponds to the condition to maintain s˘ (t) ) 0.15 Taking the derivative of eq 2 gives
s˘ (t) ) k1
de(t) dc(t) d2c(t) + k2e(t) - k3 - k4 2 ) 0 dt dt dt
(4)
In the literature, different approximations are used for the time delay term, such as first-order Pade´ or first-order Taylor series approximations. However, the most widely used one is the first-order Taylor series approximation. Camacho and Smith7 have shown that a first-order Pade´ approximation gives better solutions than a first-order Taylor series approximation when the time delay is small. However, if the time delay is relatively larger, first-order Taylor series approximation gives better solutions than a first-order Pade´ approximation. Therefore, in this paper, a first-order Taylor series approximation is used for the time delay. Equation 1 then can be rearranged as follows:
G(s) )
C(s) K ) U(s) (Ts + 1)(Ls + 1)
(5)
In differential equation form, the equation reads
dc(t) d2c(t) TL 2 + (T + L) + c(t) ) Ku(t) dt dt
(6)
The tracking error is the difference between the set point variable, r(t), and the controlled variable, c(t); that is,
e(t) ) r(t) - c(t)
(7)
Solving eq 6 for the second derivative of the controlled variable, d2c(t)/dt, in conjunction with eq 7, and replacing the expressions into eq 4, the following equation is obtained:
s˘ (t) ) k1
(
)
k4 dr(t) + k2r(t) + - k2 c(t) + dt TL k4(T + L) dc(t) Kk4 - k1 - k3 u(t) ) 0 (8) TL dt TL
[
]
Solving for u(t), which corresponds to the continuous control law, gives
uc(t) )
{[
]
(
)
k4 dc(t) TL k4(T + L) - k1 - k3 + - k2 c(t) + Kk4 TL dt TL dr(t) + k2r(t) dt
k1
}
(9)
Extensive simulation examples show that the derivative of the set-point variable r(t) can be eliminated without affecting the closed-loop system performance. This is also stated by Camacho and Smith.7 Furthermore, to simplify, the continuous control law
k 1 + k3 )
k4(T + L) TL
(10)
is selected. Hence, the final continuous control law is given by
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uc(t) )
[(
)
TL k4 - k2 c(t) + k2r(t) Kk4 TL
]
(11)
It is necessary to find the four parameters of the sliding surface. Because, in the sliding mode, the dynamics of the closed-loop system is solely determined by the sliding surface, the condition Uc(s)G(s) ) C(s) must be hold. Hence, using eq 5 in conjunction with eq 11, the following closed-loop transfer function is achieved:
k2TL/k4 C(s) ) 2 R(s) TLs + (T + L)s + k2TL/k4
(12)
Dividing both the numerator and denominator by k2TL/k4, the closed-loop system transfer function during the sliding mode becomes
C(s) 1 ) R(s) (k4/k2)s2 + [k4(T + L)/(k2TL)]s + 1
(13)
To force the closed-loop system to have an overdamped response, it can easily be shown that the following condition must be satisfied:
k4 TL 2 g4 k2 T+L
(
)
(14)
The following procedure can be followed to determine the four parameters of the proposed sliding surface. For a selected value of k2, k4 is observed to satisfy the inequality given in eq 14 and for a selected value of k1, k3 is calculated from eq 10. The larger k4 value to satisfy the inequality given in eq 14 is the more overdamped system, that is, the more-sluggish closedloop response. The designed method that was suggested by Camacho and Smith7 has sliding surface parameters, namely, λ1 and λ0, which are the parameters corresponding to proportional and integral terms in their sliding surface. These two parameters, respectively, can be used as initial values of k1 and k2, which are the parameters that correspond to the proportional and integral terms in the proposed sliding surface. The discontinuous control law must be included to account for the presence of modeling errors and external disturbances. It is discontinuous across the sliding surface s(t), which leads to a serious and undesirable phenomenon, namely chattering. To avoid this phenomenon, the discontinuous control law must be suitably smoothed. One possible solution2 is to use
s(t) ud(t) ) KD |s(t)| + δ
(15)
which was used by Camacho and Smith.7 In eq 15, KD and δ are the tuning parameters that are responsible for the reaching mode and reducing the chattering phenomenon. In the SMC, the initial reaching phase must be as short as possible (Edwards and Spurgeon, 1998) to ensure the insensitivity of the closed loop system to parameter variations and disturbances. Thus, KD and δ should be kept as large as possible. However, it has been observed from extensive simulations that using larger KD and δ values in eq 15 may result in an oscillatory response. Although there are several other discontinuous control laws, it has been reported16,17 that the following hyperbolic tangent function allows us to use larger k and Ω values, the parameters
Figure 2. Closed-loop responses to set-point tracking and disturbance rejection for example 1.
responsible for the reaching mode, and reducing the chattering phenomenon, respectively, similar to KD and δ.
ud(t) ) k tanh
( ) s(t) Ω
(16)
Therefore, this function is chosen for the discontinuous control. It is necessary to determine the parameters k and Ω to complete the design. Extensive simulations have shown that using k ) KD and Ω ) 1.5δ, as the initial values, gives quite satisfactory closed-loop responses. Throughout the paper, these values will be used for the discontinuous control law. Therefore, the overall control law is given by eq 3, with uc(t) and ud(t), respectively, given by eqs 11 and 16. 4. Simulation Examples In this section, simulation examples are provided to illustrate the use of the proposed SMC approach for controlling stable processes. The identification method given by Kaya12 has been used for all transfer functions in the examples; however, because it gives essentially exact results on simulation data, the estimated plant transfer functions are only given for original plants of higher order. The required expressions to identify parameters of a stable FOPDT transfer function are given in the Appendix for the integrity. Results obtained from the proposed SMC method have been compared with results obtained from the SMC method suggested by Camacho and Smith,7 which is the best candidate for comparison. Although fine-tuning can be performed for both the proposed SMC and that suggested by Camacho and Smith,7 initial tuning parameters of the sliding surfaces are used for both methods to illustrate the superiority of the proposed SMC. Example 1. Consider the FOPDT plant transfer function of G(s) ) e-s/(2s + 1). The initial tuning parameters used by Camacho and Smith7 are λ1 ) (L + T)/TL ) 1.500, λ0 ) (L + T)2/[4(TL)2] ) 0.563 for the continuous control law. The discontinuous control law has the initial tuning parameters of KD ) (0.51/K)(T/L)0.76 ) 0.864, δ ) 0.68 + 0.12|K|KDλ1 ) 0.836. Using k2 ) λ0 ) 0.563, as suggested in section 3, k4 g 1.001 (from eq 14). Thus, k4 is selected to be 1.500. Substituting this value of k4 and k1 ) λ1 ) 1.5 into eq 10 gives k3 ) 0.750. The discontinuous control tuning parameters are k ) KD ) 0.864 and Ω ) 1.5δ ) 1.254. Figure 2 gives closed-loop responses to a unity set-point change and step disturbance with magnitude -0.5 introduced at time t ) 50 s. The proposed SMC gives
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Figure 5. Control efforts for example 1. Figure 3. Closed-loop responses under measurement noises for example 1.
Figure 6. Responses under assumed parameter variations for example 1. Figure 4. Variation of sliding surfaces (s(t)) during control for example 1.
less overshoot than that suggested by Camacho and Smith.7 The settling time is similar for both approaches. The disturbance rejection of the proposed SMC is slightly faster than that suggested by Camacho and Smith.7 In practice, signals may contain measurement noises. Responses for band-limited white noise with a maximum power of 10-3 are given in Figure 3. Figure 4 gives the variations of sliding surfaces for both the proposed SMC and that suggested by Camacho and Smith.7 Control magnitudes for both approaches are compared in Figure 5, which illustrates that a smaller initial control effort is required for the proposed SMC. Figure 5 also shows that the control action for the proposed method is quite fast and, for example, not all valves can respond as fast as the controller demands. In this case, a larger k4 value should be selected to achieve a slower control action, as stated in section 4. Because parameter changes can, generally, be encountered in practice, -10% and +100% changes in parameters of the FOPDT plant transfer function are assumed and simulations for both the proposed SMC and that suggested by Camacho and Smith7 are repeated with the existing tuning parameters. The assumed parameter changes are the amounts used by Camacho and Smith,7 and it is assumed that the changes occur in all parameters with the same amount, that is, -10% or +100% variations in K, T, and L. The latter amount of change in the parameters is unreasonable; however, the intent is to judge the performance of controllers.7 The closed-loop responses for both
cases and SMC approaches are illustrated in Figure 6. Again, the figure shows that the closed-loop performance of the proposed SMC over that suggested by Camacho and Smith7 is evident. Example 2. In this example, the FOPDT model transfer function of G(s) ) 1.6e-3s/(13s + 1), which has relatively a large time constant and a time delay (when compared with the first example), has been considered. The FOPDT model transfer function was obtained from a step response test applied to a chemical reactor.7 Using the aforementioned FOPDT model, Camacho and Smith7 obtained the following tuning parameters: λ1 ) 0.410 and λ0 ) 0.0421 for the continuous control law. The discontinuous control law was observed to have the following initial tuning parameters: KD ) 0.960 and δ ) 0.760. Using k2 ) λ0 ) 0.0421 as suggested in section 3, the inequality k4 g 1.001must be satisfied (from eq 14). Thus, the k4 value is selected to be 2.000. Substituting this value of k4 and k1 ) λ1 ) 0.410 into eq 10 gives k3 ) 0.411. The discontinuous control tuning parameters are k ) KD ) 0.960 and Ω ) 1.5δ ) 1.140. Responses to a unity set-point change and step disturbance with a magnitude of -0.5 introduced at time t ) 100 s are given in Figure 7. As is observed from the figure, in comparison to the first example, the proposed SMC is now resulting in not only significantly less overshoot but also a shorter settling time. On the other hand, the rise time has slightly degraded. The disturbance rejection of the proposed SMC is also now much faster than that suggested by Camacho and Smith.7 Again, to
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Figure 7. Closed-loop responses to set-point tracking and disturbance rejection for example 2.
Figure 10. Control efforts for example 2.
Figure 11. Responses under assumed parameter variations for example 2. Figure 8. Closed-loop responses under measurement noises for example 2.
Figure 9. Variation of sliding surfaces (s(t)) during control for example 2.
illustrate the effect of measurement noises, responses for bandlimited white noise with a maximum power of 10-3 are given in Figure 8. Variations of sliding surfaces are provided in Figure 9, which illustrates a much faster reaching mode for the proposed SMC. Figure 10 shows the control efforts for both the proposed SMC and that suggested by Camacho and Smith.7 It is observed that
much less initial control effort is requested by the proposed SMC. Responses under assumed parameter changes of -10% and +100% are given in Figure 11. The figure clearly shows that better closed-loop responses are achieved with the proposed SMC. Example 3. This example is given to illustrate the advantages of the proposed relay feedback identification over the step response test suggested by Camacho and Smith.7 It will also be shown that the closed-loop system performance deteriorates, because of poor estimation of the FOPDT model when the openloop step test is used. Consider the high-order plant transfer function G(s) ) 1/(s + 1)5. Table 1 gives the identified FOPDT models obtained from the proposed relay feedback test and the open-loop step test suggested by Camacho and Smith,7 under assumed static load disturbances. The relay parameters in the simulations were selected to be h1 ) 1, h2 ) -0.8, and ∆ ) 0. The table shows that the open-loop test that was suggested by Camacho and Smith7 gives the same time constant and the time delay for different static load disturbance values but different gains, which is expected. To demonstrate how good the obtained models are, Nyquist plots for the actual plant transfer function and the models obtained from the proposed relay test and step test suggested by Camacho and Smith7 are given in Figure 12. As is distinguished from the figure, the Nyquist plots, obtained from the models based on the proposed relay feedback identification under different static load disturbances, are almost the same, so that they cannot be observed for different cases.
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Figure 12. Nyquist plots for the actual plant transfer function and models obtained from the relay test (left) and step test (right). Table 1. First-Order Plus Dead Time (FOPDT) Models Obtained under Different Static Load Disturbances proposted relay feedback test
open loop step test
d)0
d ) 0.1
d ) 0.3
d )0
d ) 0.1
d ) 0.3
e-2.632s/(4.400s + 1)
e-2.573s/(4.300s + 1)
e-2.393s/(4.054s + 1)
e-2.596s/(2.840s + 1)
1.10e-2.596s/(2.840s + 1)
1.30e-2.596s/(2.840s + 1)
Table 2. Tuning Parameters for Example 2 proposed SMC d)0
d ) 0.1
k1 ) 0.607 k2 ) 0.0922 k3 ) 0.607 k4 ) 2.000 k ) 0.754 Ω ) 1.102
k1 ) 0.621 k2 ) 0.0965 k3 ) 0.621 k4 ) 2.000 k ) 0.754 Ω ) 1.104
SMC suggested by Camacho and Smith d ) 0.3
d)0
d ) 0.1
d ) 0.3
k1 ) 0.665 λ1 ) 0.737 λ1 ) 0.737 λ1 ) 0.737 k2 ) 0.111 λ0 ) 0.136 λ0 ) 0.136 λ0 ) 0.136 k3 ) 0.664 KD ) 0.546 KD ) 0.496 KD ) 0.420 k4 ) 2.000 δ ) 0.728 δ ) 0.728 δ ) 0.728 k ) 0.761 Ω ) 1.111
On the other hand, the figure shows that there are quite large differences between the models obtained from the step test, performed under different static load disturbances. Also, it is quite straightforward to predict from the figure that larger errors in the modeling can be expected for larger static load disturbance magnitudes. As a consequence, it can be expected that the closed-loop system performance will also be affected greatly. Table 2 gives the tuning parameters for the proposed SMC and that suggested by Camacho and Smith.7 The tuning parameters for each case, which are related to different static load disturbance magnitudes, are obtained using the corresponding FOPDT model. The tuning parameters for the SMC of Camacho and Smith7 are the same for all cases, except KD, because they are dependent on the time constant and time delay of the FOPDT model, which are the same for their modeling approach. Figures 13 and 14 show the closed-loop responses to a unity set-point change and disturbance with a magnitude of -0.5 introduced at time t ) 50 s for the proposed SMC and that suggested by Camacho and Smith,7 respectively. Figure 13 illustrates that the closed-loop system gives very similar responses for the proposed SMC and proposed relay feedback parameter estimation of the FOPDT model. In contrast, the SMC and the open-loop step test suggested by Camacho and Smith7 results in a deteriorating closed-loop system performance, especially for disturbance rejection capability, as the static load disturbance magnitude gets larger. Furthermore, comparing Figures 13 and 14, the superior closed-loop performance of the proposed SMC over that recommended by Camacho and Smith7 is apparent. The SMC suggested by Camacho and Smith7 gives steady-state errors, in response to disturbance rejections.
Example 4. This example is given to show another shortcoming of the SMC approach that was suggested by Camacho and Smith.7 Consider a high-order process transfer function with complex poles, G(s) ) e-s/(s + 1)(s2 + s + 1). Because the
Figure 13. Responses for the proposed SMC using models obtained from relay feedback tests under different load disturbances.
Figure 14. Responses for the SMC suggested by Camacho and Smith,7 using models obtained from open-loop step tests under different load disturbances.
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heights when the relay is on and off, respectively. ∆ is the hysteresis in the relay, d the static load disturbance magnitude, and ω the limit cycle frequency. The variables amax and amin are, respectively, the maximum and minimum of limit cycle amplitude. λ is given by ωT.
K
K
Figure 15. Closed-loop response (upper) and control effort (lower) for example 4.
open-loop process transfer function has complex poles, the openloop test used by Camacho and Smith7 cannot be used for parameter estimation of the FOPDT model. Conducting the proposed relay feedback test, with relay parameters of h1 ) 1, h2 ) -0.8, and ∆ ) 0.2, the limit cycle parameters were measured to be ω ) 1.100, amax ) 0.789, amin ) -0.583 and ∆t1 ) 2.543. The disturbance magnitude in the simulation was put at d ) 0.1. Using these limit cycle parameters in eqs A1A4 in the Appendix, in conjunction with eqs A5 and A6 in the Appendix, the FOPDT model was determined to be G(s) ) e-1.761s/(1.214s + 1). Selecting k2 ) 0.500, then eq 14 requires that k4 g 1.033. Hence, k4 ) 2.000 is chosen. Constraining k1 ) 1.500, eq 10 gives k3 ) 1.283. Using the expressions for KD and δ from Camacho and Smith,7 one obtains KD ) 0.384 and δ ) 0.744. Hence, k ) KD ) 0.384 and Ω ) 1.5δ ) 1.116 are found for the discontinuous control law. The closed-loop responses to a unity set-point change and a disturbance with a magnitude of -0.1 introduced at time t ) 50 s, and the control effort during the control are given in Figure 15.
(
)
{
Appendix Specific expressions used to estimate unknown parameters of the FOPDT plant transfer function in the relay feedback control under static load disturbances are provided here, for the sake of convenience. (To see the procedure for obtaining these equations, one can refer to Kaya12 and Kaya and Atherton13). In the expressions, ∆t1 and ∆t2 are the positive and negative pulse durations of the relay output, and h1 and h2 are the relay
}
)
(ω∆t1 - 2π) πeL/T[e(-ω∆t1+2π)/λ - 1] + ) 2 e2π/λ - 1 G(0)(h1∆t1 + h2∆t2) π dG(0) - ∆ + (A2) h 1 - h2 P
(
amin ) dG(0) +
)
G(0)(h1∆t1 + h2∆t2) + P (h1 - h2)K -ω∆t1 π(e∆t1/T - 1) + 2π/λ (A3) π 2 e -1
(
)
G(0)(h1∆t1 + h2∆t2) + P (h1 - h2)K -ω∆t1 πe2π/λ(1 - e-∆t1/T) (A4) + π 2 (e2π/λ - 1)
amax ) dG(0) +
(
)
Using these four equations (that is, eqs A1, A2, A3, and A4), determination of the three unknown parameters of the FOPDT plant transfer function and the disturbance magnitude is possible. However, because the equations are nonlinear, initial guesses are required. Thus, to reduce the number of unknowns and make the solution easier to find, Fourier analysis can be used to identify the steady-state gain (K) and disturbance magnitude (d). It is assumed that the steady-state gain can be calculated from
∫t t+P c(t) dt K ) G(0) ) t+P ∫t u(t) dt
5. Conclusions The paper has introduced a new sliding surface with four tuning parameters to achieve a better closed-loop performance in process control systems. A systematic guideline, co-operating the results from Camacho and Smith7(2000), has been provided to find the four tuning parameters of the proposed sliding-mode control (SMC). The FOPDT model is used to derive the continuous control law of the proposed SMC. The parameters of the FOPDT model are determined from a single relay feedback control under static load disturbances, using exact expressions for a limit to occur. The FOPDT model is only used to simplify the calculations, and the actual process may be a higher-order process or a process with complex poles as well. Extensive simulation examples and comparisons are given to illustrate the value and advantages of the proposed SMC.
(
- ω∆t1 πeL/T(e∆t1/T - 1) -π ) dG(0) + ∆ + + 2 h1 - h2 e2π/λ - 1 G(0)(h1∆t1 + h2∆t2) (A1) P
(A5)
where c(t) and u(t) are the plant output and input, respectively, and P is the period of the limit cycle. After steady-state operation occurs, the disturbance magnitude can be calculated from
d)
∫t t+P c(t) dt -
1 G(0)P
h1∆t1 + h2∆t2 P
(A6)
Therefore, with the plant transfer function gain K and disturbance magnitude d being determined, respectively, from eqs A5 and A6, the time constant T can be calculated from eq A3, if the amin value is measured, or eq A4, if the amax value is measured. Because the measurements generally are not free of error, both amin and amax can be measured and the time constant T can be calculated from eqs A3 and A4 separately and then the average value of these two results can be used as a final value to ensure better accuracy. Finally, with K and T being known, the dead time L can be computed using either eq A1 or A2. Again, to ensure better accuracy, the average of these two equations can be used for the time delay estimation. Literature Cited (1) Utkin, V. I. Variable structure systems with sliding modes. IEEE Trans. Autom. Control 1977, 22, 212-222.
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ReceiVed for reView June 19, 2006 ReVised manuscript receiVed October 30, 2006 Accepted November 8, 2006 IE0607806
