A two-degree-of-freedom control structure is proposed for a class of unstable processes with time delay based on modified Smith predictor control; the superior performance of disturbance rejection and good robust stability are gained for the system. The set-point tracking controller is designed using the direct synthesis method; the IMC-PID controller for disturbance rejection is designed based on the internal mode control design principle. The controller for set-point response and the controller for disturbance rejection can be adjusted and optimized independently. Meanwhile, the two controllers are designed in the form of PID, which is convenient for engineering application. Finally, simulation examples demonstrate the validity of the proposed control scheme.
1. Introduction
Unstable processes are well known to be difficult to control especially when there exists pure time delay. A time delay is introduced into the transfer function description of such system due to the measurement delay or an actuator delay [1–5]. A lot of academic research had been devoted to developing effective control strategies for such processes. Generally, PI or PID controllers are designed using a unity feedback control structure for these systems. Two-degree-of-freedom methods based on PID control are the most common methods [6–11]. Internal mode control and Smith predictor (SP) control are regarded as the most effective methods for process control and most widely used in industry but cannot be used directly for unstable process with time delay. Owing to the standard SP control structure which is in essence equivalent to the internal model control structure for time delay processes, a number of control schemes based on modified Smith predictor had been developed in recent years [12–18]. By using the Smith predictor, Rao and Chidambaram [19] proposed a two-degree-of-freedom control scheme for unstable systems with time delay; in the scheme three controllers were used to improve the system performance. Liu et al. [20] proposed a modified form of Smith predictor in a two-degree-of-freedom control scheme, which demonstrated the remarkable improvement of regulatory capacity for both of reference input tracking and load disturbance rejection. García and Albertos [21] proposed a scheme that is equivalent to the Smith predictor but able to cope with any kind of systems; the results showed a substantial improvement in the performance/robustness tradeoff as well as in the tuning process. Vijayan and Panda [22] proposed a double-feedback loop method which was used to achieve stability and better performance of the process. By comparison, few papers [23, 24] developed discrete-time domain control methods for advanced regulation of unstable processes. Besides, nonlinear control schemes were presented to deal with integrating and unstable processes with time delay [25]. Dey et al. [26] proposed an autotuning proportional-derivative control scheme; the proportional and derivative gains were adjusted using a nonlinear gain updating factor to achieve an overall improved performance.
Evaporation processes are common in the food industry. It is the stage in which the water contained in a juice is eliminated in order to obtain a juice with a higher concentration. The dynamics of the evaporation can be represented with integrating first order plus time delay process [27]. The control of such system is difficult because of the limitations imposed by the integrator and the time delay on the system performance and stability.
Disturbance rejection is much more important than set-point tracking for many process control applications. But the methods proposed previously for the disturbance rejection have not gained much popularity; what is more, it is difficult to be carried out in process industries. The objective of the present study is to develop a practicable method to obtain enhanced disturbance rejection performance and perfect set-point tracking performance. So a two-degree-of-freedom control scheme based on modified Smith predictor shown in Figure 1 is proposed. The set-point tracking controller K1(s) and disturbance rejection controller K2(s) are designed in the form of PID. The scheme can lead to substantial control performance improvement, especially for the disturbance rejection. The analysis has been carried out for the two typical transfer function models: P(s)=ke-θs/s(Ts-1) and P(s)=ke-θs/s(Ts+1).
Modified Smith control structure.
In Figure 1P0(s) is the transfer function of the process model without the time delay, that is, P(s)=P0(s)e-θs, K1(s) is used for set-point tracking, K2(s) is used for disturbance rejection,F(s) is the set-point filter, r(s) is the set point, y(s) is the process output, anddi(s) anddo(s)are the disturbances before and after process, respectively. As can be seen, the performance of set point and load disturbance rejection response are decoupled completely and can be monotonically tuned to meet a good performance by controller K1(s) and K2(s), respectively.
From Figure 1, the transfer function from y(s) to r(s) can be determined in the form of
(1)Hr(s)=y(s)r(s)=P(s)K1(s)1+P0(s)K1(s)1+K2(s)P0(s)e-θs1+K2(s)P(s).
In the nominal case, that is P(s)=P0(s)e-θs, the set-point tracking transfer function can be simplified as
(2)Hr(s)=y(s)r(s)=P(s)K1(s)1+P0(s)K1(s).
Obviously, there is no dead-time element in the characteristic equation of the nominal set-point tracking transfer function; K1(s) can be obtained if the transfer function is determined:
(3)K1(s)=Hrd(s)1-Hrd(s)1P0(s).
Considering the implementation and system performance, the desired set-point tracking transfer function is proposed:
(4)Hrd(s)=y(s)r(s)=a2s2+a1s+1(λs+1)3,
whereλis the adjustable parameter; as for the unstable process type P(s)=ke-θs/s(Ts-1), the controller can be derived from (3) and (4):
(5)K1(s)=s(Ts-1)(a2s2+a1s+1)k[(λs+1)3-(a2s2+a1s+1)].
Because of simple structure and better control performance than the direct-action tuner, the ability of PID controllers to meet most of the control objectives has led to their widespread acceptance in the control industry. As we know, distributed control system is widely used in process industry. PID module is the basic and the most used module in the distributed control system; over 90% control points were designed in PID form [28]. To obtain a realizable controller, K1(s)should be realized in discrete form or approximated by a rational transfer function. So K1(s)can be expressed as
(6)K1(s)=(Ts-1)(a2s2+a1s+1)k[λ3s2+(3λ2-a2)s+(3λ-a1)].
According to the model transform method [29], order a1=4λ, a2=6λ2+1. A PID controller with first order lag filter can be approximated
(7)K1(s)=k1(1+1τi1s+τd1s)1αs+1.
After approximate comparison, we can obtain the parameters of PID, k1=a1/k is proportional gain, τi1=a1 is integral gain, τd1=a2/a1 is derivative gain, and α=λ4/Tis the filter parameter.
As can be seen from (4), numerator of the transfer function will result in undesired system overshoot. So in order to improve the set-point tracking performance and reduce the overshoot, a set-point filer is designed F(s)=1/(a2s2+a1s+1).
Analogously, as for the process type P(s)=ke-θs/s(Ts+1), K1(s) can be obtained from (3) and (4):
(8)K1(s)=(Ts+1)(a2s2+a1s+1)k[λ3s2+(3λ2-a2)s+(3λ-a1)].
Using the similar method, we obtain a1=3λ, a2=3λ2-λ3/T. The K1(s) can be derived in the form of PID as follows:
(9)K1(s)=k1(1+1τi1s+τd1s),
where k1=3T/kλ2 is proportional gain, τi1=3λ is integral gain, and τd1=λ(1-λ/3T) is derivative gain.
2.2. Disturbance Rejection Controller K2(s)
In the proposed control structure shown in Figure 1, the load disturbance transfer functions are given by
(10)Hdi(s)=y(s)di(s)=P(s)1+K2(s)P(s),Hdo(s)=y(s)do(s)=11+K2(s)P(s).
At the same time, we can obtain the closed-loop complementary sensitivity function between the process input and output for the load disturbance rejection as
(11)T(s)=K2(s)P(s)1+K2(s)P(s).
Here, K2(s) is designed using the method of unit feedback based on internal mode control theory [30]:
(12)K2(s)P(s)1+K2(s)P(s)=P(s)C(s),
where C(s) is the internal mode controller, P(s)=P-(s)P+(s), C(s)=P--1(s)f(s), in which f(s) is the filter, P-(s) contains the invertible portion of the model, and P+(s) contains all the noninvertible portion. The invertible portions are the part of the model with stable poles and unstable poles. The noninvertible portions are the portion of the model with right half plane zeros and time delays. In order to ensure that the system is internally stable, the filter is designed as
(13)f(s)=∑i=1mbisi+1(λ′s+1)n,
where λ′ is an adjustable parameter which controls the tradeoff between the performance and robustness, determined to cancel the unstable and integrating poles of P(s). m is the number of unstable and integrating poles. n is selected to be large enough to make the internal mode controller proper; bi is determined by 1-P(s)C(s)|s=z1,…,zm=0, where z1,…,zm are the unstable and integrating poles.
As for the unstable process type P(s)=ke-θs/s(Ts-1), it can be transformed as P(s)=k′e-θs/(T′s-1)(Ts-1); time constantT′is selected to be large enough. The filter is designed as
(14)f(s)=b2s2+b1s+1(λ′s+1)4.
Correspondingly by using (12) and (14), the controller K2(s) can be obtained as
(15)K2(s)=(T′s-1)(Ts-1)(b2s2+b1s+1)k′[(λ′s+1)4-e-θs(b2s2+b1s+1)],
where b1and b2 are determined by the two constraints
(16)lims→1/THd0(s)=0,lims→1/T′Hd0(s)=0,thatis,lims→1/T[1-b2s2+b1s+1(λs+1)4e-θs]=0,lims→1/T′[1-b2s2+b1s+1(λs+1)4e-θs]=0.
Following a simple calculation, we obtain
(17)b1=(T′2(λ′T′+1)4eθ/T′-T2(λ′T+1)4eθ/Tss+T2-T′2(λ′T′+1)4)×(T′-T)-1,b2=T′2[(λ′T′+1)4eθ/T′-1]-b1T′.
The dead time e-θs in (15) is approximated using Pade expansion:
(18)e-θs=1-θs/21+θs/2.
Then substituting (18) into (15) obtains the controller K2(s) as
(19)K2(s)=b2s2+b1s+1η×(T′s-1)(Ts-1)(1+θs/2)1+l1s+l2s2+l3s3+l4s4,
where η=4λ′-b1+θ, l1=(6λ′2+2λ′θ+b1θ/2-b2)/η, l2=(4λ′3+3λ′2θ+b2θ2/2)/η, l3=(λ′4+2λ′3θ)/η, and l4=λ′4θ2/2η. Since the resulting controller does not have a standard PID controller form, a procedure is employed to produce a PID controller cascade with a first order lead-lag filter:
(20)K2(s)=k2(1+1τi2s+τd2s)1+α′s1+βs.
As can be seen from (19), the first term (b2s2+b1s+1)/η can be interpreted as the standard PID controller in the form of k2(1+(1/τi2s)+τd2s). The later term can be interpreted as a first order lead-lag filter (1+α′s)/(1+βs), α′=0.5θ; β can be obtained by
(21)dds(νs2+βs+1)|s=0=dds[1+l1s+l2s2+l3s3+l4s4(T′s-1)(Ts-1)]|s=0.
Since the νs2 term has little impact on the overall control performance, it can be ignored. Calculating by using (19) and (21), we can obtain the parameters for the controller K2(s). k2=b1/k′(4λ′+θ-b1) is proportional gain, τi2=b1 is integral gain, and τd2=b2/b1 is derivative gain. α′=0.5θ, β=((b1θ/2-b2+2λ′θ+6λ′2)/(θ+4λ′-b1))+T+T′, α and β are parameters of the lead-lag filter.
As for the process type P(s)=ke-θs/s(Ts+1), it can be transformed as P(s)=-k′e-θs/[(T′s-1)(-Ts-1)], and the executable controller K2(s) can be obtained in the form of PID by using (15), (18), and (20). On the basis of simulation study on integrating first order plus time delay processes, the use of 0.1β instead of β is suitable and βis about 0.2–1.2 generally.
3. System Robust Stability Analysis
A control system is robust if it is insensitive to differences between the actual system and the model of the system which was used to design the controller. These differences are referred to as model mismatch or simply model uncertainty [31]. A study of robustness analysis is an important task because no mathematical model of a system will be a perfect representation of the actual system. Small-gain theorem is ∥lmT(s)∥∞<1; it expresses the robustly stable condition of a control system, where lm(s) is the bound on the process multiplicative uncertainty and T(s) is the closed-loop complementary sensitivity function [32].
For the process type P(s)=ke-θs/s(Ts-1), if there are uncertainties that exist in all three parameters, that is,
(22)P′(s)=(k+Δk)e-(θ+Δθ)ss(T2s-1)(ΔTs+1).
The bound on the process multiplicative uncertainty lm(s) can be obtained as
(23)lm(s)=|P′(s)-P(s)P(s)|=(1+(Δk/k))e-Δθs(ΔTs+1)-1.
Then the tuning parameters should be selected in such a way that
(24)∥b2s2+b1s+1(λ′s+1)4∥∞<1∥((1+(Δk/k))e-Δθs/(ΔTs+1))-1∥∞.
If the uncertainty exists in the time delay, the tuning parameters should be selected in a way that
(25)∥b2s2+b1s+1(λ′s+1)4∥∞<1|e-jΔθw-1|.
At the same time, in order to compromise the nominal performance with the robust stability of the closed-loop for the load disturbance rejection, the following constraint is required to meet [31]
(26)|lm(s)T(s)|+|ω(s)(1-T(s))|<1,
where ω(s) is a weight function of the closed-loop sensitivity function S=1-T(s), which usually can be chosen as 1/s for the step change of the load disturbance. It indicates that tuning the adjustable parameter λ′ aims at the tradeoff between the nominal performance of the closed-loop and its robust stability. That is to say, decreasing λ′improves the disturbance rejection performance of the closed-loop but decays its robust stability in the presence of the process uncertainty. On the contrary, increasing λ′ tends to strengthen the robust stability of the closed-loop but degrades its disturbance rejection performance.
4. Simulation Example 1.
Consider the unstable process studied by Liu et al. [20], P(s)=e-0.2s/s(s-1).
In the proposed method, take λ=3.3θ=0.66, the parameters of controllers are obtained, k1=2.64, τi1=2.64, τd1=1.367, α=0.19, a1=2.64, and a2=3.61. As for the controller K2(s), transform the process as P(s)=100e-0.2s/(100s-1)(s-1), take λ′=2θ=0.4, and obtain
(27)K2(s)=3.02(1+11.79s+1.06s)0.1s+10.008s+1.
The method proposed in [20] is better than others which can be seen in this simulation effects, so only compare with the method in [20]. By adding a unit step change to the set-point input at t=0, an inverse unit step change of load disturbance to process output at t=20, the simulation results are obtained as shown in Figure 2.
Nominal system responses for Example 1.
Now suppose that there exists 30% increment for estimating the process time delay and 30% reduction for the time constant of the process model; then the perturbed system responses are provided in Figure 3.
Perturbed system responses for Example 1.
It can be seen from the simulation results that the proposed method gives better performances for the unstable process.
Example 2.
Consider a process studied by Liu and Gao [15], P(s)=0.1e-5s/s(5s+1).
In the proposed method, as for controllersK1(s), take λ=0.8θ=4, and obtain k1=9.375, τi1=12, τd1=2.933, a1=12, and a2=35.2. As for controller K2(s), transform the process as P(s)=-10e-5s/(100s-1)(-5s-1), take λ′=0.8θ=4, and obtain the controller
(28)K2(s)=2.2(1+122s+3.864s)2.5s+10.8s+1.
Add a unit step change to the set-point input at t=0 and an inverse step change of load disturbance to the process output at t=200. The simulation results are obtained as shown in Figure 4, and the corresponding control action responses are shown in Figure 5. It can be observed from the figure that the control action response of the proposed method shows smooth variation compared to that of Liu and Gao [15].
Nominal system responses for Example 2.
Nominal system control signal for Example 2.
Now suppose that there exist 10% error for estimating the process time delay and the time constant of the process model, such as both of them are actually 10% larger. The perturbed system responses are provided in Figure 6, and the corresponding control action responses are shown in Figure 7. It can be seen that the proposed control action is smooth and the performance of disturbance rejection is better than that of Liu and Gao [15].
Perturbed system responses for Example 2.
Perturbed system control signal for Example 2.
5. Conclusion
In order to improve the system performance of disturbance rejection, a modified Smith predictor scheme has been proposed based on a two-degree-of-freedom control structure. In the proposed control structure, both of the set-point response and the load disturbance response can be tuned separately by the set-point tracking controller and the disturbance estimator, respectively. The most advantage of the method is that the designed system has good performance of disturbance rejection as well as performance of set-point tracking. The two controllers are all designed in the form of PID, and they are simple and easy to be used in process industry. Comparisons with the previous methods demonstrate a clear advantage of the proposed method in both nominal and robust performances in disturbance rejection.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research is funded by the Natural Science Foundation of Shandong Province (Grant no. ZR2012CQ026) and the Science and Technology Funds of Shandong Education Department (Grant no. J11LD16).
ChienI.-L.PengS. C.LiuJ. H.Simple control method for integrating processes with long deadtime200212339140410.1016/S0959-1524(01)00040-32-s2.0-0036532959HuzmezanM.GoughW. A.DumontG. A.KovacS.Time delay integrating systems: a challenge for process control industries. A practical solution200210101153116110.1016/S0967-0661(02)00060-62-s2.0-0036808955DongH.WangZ.GaoH.Distributed H∞ filtering for a class of markovian jump nonlinear time-delay systems over lossy sensor networks201360104665467210.1109/TIE.2012.22135532-s2.0-84878121569Normey-RicoJ. E.BordonsC.CamachoE. F.2007Berlin, GermanySpingerWangZ.DongH.ShenB.GaoH.Finite-horizon H∞ filtering with missing measurements and quantization effects20135871707171810.1109/TAC.2013.2241492MR30728552-s2.0-84879974725VisioliA.Optimal tuning of PID controllers for integral and unstable processes2001148218018410.1049/ip-cta:200101972-s2.0-0035269268HwangC.HwangJ.Stabilisation of first-order plus dead-time unstable processes using PID controllers20041511899410.1049/ip-cta:200400352-s2.0-1542685431PandaR. C.Synthesis of PID controller for unstable and integrating processes200964122807281610.1016/j.ces.2009.02.0512-s2.0-67349245152LeeY.LeeJ.ParkS.PID controller tuning for integrating and unstable processes with time delay200055173481349310.1016/S0009-2509(00)00005-12-s2.0-0034035657Seshagiri RaoA.RaoV. S.ChidambaramM.Direct synthesis-based controller design for integrating processes with time delay20093461385610.1016/j.jfranklin.2008.06.004MR24937062-s2.0-58149503670PadhanD. G.MajhiS.A new control scheme for PID load frequency controller of single-area and multi-area power systems201352224225110.1016/j.isatra.2012.10.0032-s2.0-84875235031KayaI.Autotuning of a new PI-PD smith predictor based on time domain specifications200342455957510.1016/S0019-0578(07)60006-82-s2.0-0242368927MajhiS.AthertonD. P.Obtaining controller parameters for a new Smith predictor using autotuning200036111651165810.1016/S0005-1098(00)00085-6MR18317212-s2.0-0034326216Normey-RicoJ. E.CamachoE. F.Dead-time compensators: a survey200816440742810.1016/j.conengprac.2007.05.0062-s2.0-38549098249LiuT.GaoF. R.Enhanced IMC design of load disturbance rejection for integrating and unstable processes with slow dynamics201150223924810.1016/j.isatra.2010.11.0042-s2.0-79952439734CongE.-D.HuM.-H.TuS.-T.XuanF.-Z.ShaoH.-H.A novel double loop control model design for chemical unstable processes201453249750710.1016/j.isatra.2013.11.0032-s2.0-84897630993Normey-RicoJ. E.GarciaP.GonzalezA.Robust stability analysis of filtered Smith predictor for time-varying delay processes201222101975198410.1016/j.jprocont.2012.08.0122-s2.0-84866007809NandongJ.ZangZ.High-performance multi-scale control scheme for stable, integrating and unstable time-delay processes201323101333134310.1016/j.jprocont.2013.08.0072-s2.0-84885146585RaoA. S.ChidambaramM.Analytical design of modified Smith predictor in a two-degrees-of-freedom control scheme for second order unstable processes with time delay200847440741910.1016/j.isatra.2008.06.0052-s2.0-50249138289LiuT.ZhangW.GuD.Analytical design of two-degree-of-freedom control scheme for open-loop unstable processes with time delay200515555957210.1016/j.jprocont.2004.10.0042-s2.0-13444267364GarcíaP.AlbertosP.Robust tuning of a generalized predictor-based controller for integrating and unstable systems with long time-delay20132381205121610.1016/j.jprocont.2013.07.0082-s2.0-84882288229VijayanV.PandaR. C.Design of PID controllers in double feedback loops for SISO systems with set-point filters201251451452110.1016/j.isatra.2012.03.0032-s2.0-84861642657GarcíaP.AlbertosP.HägglundT.Control of unstable non-minimum-phase delayed systems200616101099111110.1016/j.jprocont.2006.06.0072-s2.0-33750333989AlbertosP.GarcíaP.Robust control design for long time-delay systems200919101640164810.1016/j.jprocont.2009.05.0062-s2.0-71849113974MorenoJ. C.GuzmánJ. L.Normey-RicoJ. E.BañosA.BerenguelM.A combined FSP and reset control approach to improve the set-point tracking task of dead-time processes201321435135910.1016/j.conengprac.2012.12.0022-s2.0-84873717352DeyC.MudiR. K.SimhachalamD.A simple nonlinear PD controller for integrating processes201453116217210.1016/j.isatra.2013.09.0112-s2.0-84892372534Normey-RicoJ. E.CamachoE. F.Unified approach for robust dead-time compensator design201121710801091AstromK. J.HagglundH.19952ndResearch Triangle Park, NC, USAInstrument Society of AmericaOgunnaikeB. A.RayW. H.1994New York, NY, USAOxford UniversityShamsuzzohaM.LeeM.Enhanced performance for two-degree-of-freedom control scheme for second order unstable processes with time delayProceedings of the International Conference on Control, Automation and Systems (ICCAS '07)October 2007Seoul, Republic of Korea24024510.1109/ICCAS.2007.44069152-s2.0-48349131481MorariM.ZafiriouE.1989Englewood Cliffs, NJ, USAPrentice HallDoyleJ. C.FrancisB. A.TannenbaumA. R.1992New York, NY, USAMacmillan Publishing CompanyMR1200235