Abstract

A decentralized model predictive controller applicable for some systems which exhibit different dynamic characteristics in different channels was presented in this paper. These systems can be regarded as combinations of a fast model and a slow model, the response speeds of which are in two-time scale. Because most practical models used for control are obtained in the form of transfer function matrix by plant tests, a singular perturbation method was firstly used to separate the original transfer function matrix into two models in two-time scale. Then a decentralized model predictive controller was designed based on the two models derived from the original system. And the stability of the control method was proved. Simulations showed that the method was effective.

1. Introduction

Applications of singular perturbation in control theory can be traced back to 1970s [1]. When there are both fast and slow dynamics in a system, the perturbation conception can be adopted to simplify the system. Phillips first combined the optimal control theory with a singular perturbation system and presented a two-stage design of linear feedback control [2]. Later, nonlinear singular perturbation systems [3], large scale systems [4], and high-gain feedback systems [5] were investigated with optimal control theory, too. Recently, process control researchers took notice of the mechanical reasons for two-time scale characteristics of systems. And the existence of fast and slow dynamics was studied in several previous papers. Yi and Luyben analyzed the dynamic characteristic of coupled reactor/column systems in a series of papers and illuminated the mechanisms of fast and slow dynamics in some special systems [68]. Contou-Carrere and Daoutidis focused on the coexistence of fast and slow dynamics in integrated process networks [9]. They explained that large flow rates brought in fast dynamic and led to a time scales separation of dynamics. They also designed a precompensator for a distillation model. Sequentially, Kumar and Daoutidis further analyzed the dynamics of process with material and energy recycle, and introduced a controller design framework consisting of properly coordinated controllers in fast and slow time scales [10]. Vora and Daoutidis introduced a nonstandard form singular perturbation method and analyzed the existence of fast and slow dynamics in a nonlinear system [11]. Kumar and Christofides dealt with two-time scale chemical processes and modeled them by using nonlinear ordinary differential equations with large parameters of the form 1/𝜀 [12]. They obtained a standard singularly perturbed representation. All those methods focused on process thermodynamic models for which took much work to get precise data.

In actual processes, model predictive control (MPC) was regarded as “the only advanced control methodology which has made a significant impact on industrial control engineering” [13]. And MPC has no limit of the model form. The input-output model which can be obtained easily by identification is usually used. However, all the methods mentioned above is from the point of optimal control which is an offline method. In addition, all the above methods regard the steady state output of fast dynamics as the input or known quantity of slow dynamics but pay little attention on the influence of slow dynamics on fast dynamics. MPC adopts online rolling optimization and uses feedback to correct prediction. The challenge of applying MPC in singular perturbation system becomes the compromise between the control interval and the predictive horizon. Fast dynamic needs small control interval, while slow dynamic needs large predictive horizon. There are few papers about two-time scale MPC. Buescher and Baum introduced a two-time scale approach to nonlinear model predictive control, and used a “gapping” method to smooth the control quality [14]. But they did not consider the dynamic characteristics of the model in their control algorithm which may be not suitable for special systems.

Therefore, we focused on designing an MPC for systems with two-time scale characteristic in this paper. First, we introduced the background of this field. Then we described the two-time scale decomposition of a transfer function matrix with different dynamics in different channels. At part 3, we presented a kind of two-time scale decentralized MPC algorithm step by step and proved its stability. At last, we gave several simulations to test the validity of the two-time scale decentralized MPC algorithm.

2. Two-Time Scale Decomposition of a Transfer Function Matrix

In some systems, the dynamics varies with different channels. And the response speeds of those different channels vary so much even in different time scales. The characteristics can be got from the transfer function matrix intuitively. We simply took a two-in-two-out first-order transfer function matrix, for example and gave the following definition.

Definition 1. A Two-in-two-out first-order transfer function matrix 𝐺(𝑠) with two-time scale characteristic is presented below, where 𝐺(𝑠)=𝐺11(𝑠)𝐺12(𝑠)𝐺21(𝑠)𝐺22(𝑠)𝐺11(𝑠)=𝑎11𝑏11𝑠+1𝑒𝜏11𝑠,𝐺12(𝑠)=𝑎12𝑏12𝑠+1𝑒𝜏12𝑠,𝐺21(𝑠)=𝑎21𝑏21𝑠+1𝑒𝜏21𝑠,𝐺22(𝑠)=𝑎22𝑏22𝑠+1𝑒𝜏22𝑠,(1) when 𝑎11, 𝑎12, 𝑎21, 𝑎22 are in the same order, 𝑏11,𝑏12𝑏21,𝑏22, 𝜏11,𝜏12,𝜏21,𝜏22 are in the same order.
Considering the system
𝑌=𝐺(𝑠)𝑈,(2) the response speed of output 𝑦1 is much faster than that of output 𝑦2. Then a traditional central controller cannot satisfy the demands of 𝑦1 and 𝑦2 simultaneously. For example, for the fast channel a very short control interval is needed to provide enough dynamic characteristics, while for the slow channel a very large predictive horizon is needed to ensure the stability of the controller. Thus a normal model predictive controller cannot satisfy both demands. Therefore we designed a decentralized controller based on a two-time scale method.
Some papers [9, 11, 15] introduced a singular perturbation method to obtain two-time scale models which are based on state space model. In order to transform the transfer function matrix into a two-time scale form, the transfer function matrix form should be transferred into a state space form. But the delay terms cannot be expressed in a state space form. Thus we firstly took a transfer function matrix without delay to illustrate the two-time scale decomposition, and then discussed the situation with delay terms.

2.1. Without Delay Terms

Based on the method mentioned in literature[16], system 𝐺(𝑠)=𝐺11(s)𝐺12(s)𝐺21(s)𝐺22(s),(3) where 𝐺11(s)=𝑎11/𝑏11𝑠+1, 𝐺12(s)=𝑎12/𝑏12𝑠+1, 𝐺21(s)=𝑎21/𝑏21𝑠+1, 𝐺22(s)=𝑎22/𝑏22𝑠+1, 𝑎11, 𝑎12, 𝑎21, 𝑎22 are in the same order, 𝑏11,𝑏12𝑏21,𝑏22, can have this form

̇𝑋=𝐴𝑋+𝐵𝑈,𝑌=𝐶𝑋,(4) where 𝐴=diag𝐴11,𝐴12,𝐴21,𝐴22,𝐵=𝐵11𝐵12𝐵21𝐵22,𝐶=𝐶11𝐶12𝐶21𝐶22;(5)(𝐴𝑖𝑗,𝐵𝑖𝑗,𝐶𝑖𝑗) is a one-dimension state space form of 𝐺𝑖𝑗(s):

𝑏11,𝑏12𝑏21,𝑏22||𝐴11||,||𝐴12||||𝐴21||,||𝐴22||.(6)

So we can rewrite (4) in this form:

𝜀̇𝑥1𝜀̇𝑥2̇𝑥3̇𝑥4=𝜀𝐴110𝜀𝐴12𝐴210𝐴22𝑥1𝑥2𝑥3𝑥4+𝜀𝐵11𝜀𝐵12𝐵21𝐵22𝑢1𝑢2,𝑦1𝑦2=𝐶11𝐶12𝐶21𝐶22𝑥1𝑥2𝑥3𝑥4,(7) where 𝜀 is a very small positive constant.

This form can be regarded as

̇𝑋=𝐴11𝑋+𝐴12𝑍+𝐵1𝑈,𝜀̇𝑍=𝐴21𝑋+𝐴22𝑍+𝐵2𝑈,𝑌=𝐶1𝑋+𝐶2𝑍+𝐷𝑈,(8) where 𝐴11=𝐴2100𝐴22,𝐴12=0000,𝐴21=0000,𝐴22=𝜀𝐴1100𝜀𝐴12,𝐵1=𝐵2100𝐵22,𝐵2=𝜀𝐵1100𝜀𝐵12,𝐶1=00𝐶21𝐶22,𝐶2=𝐶11𝐶1200,𝐷=0000.(9) And the system in this form was discussed in several literatures [9, 15, 17], too. We got the slow model

𝑑𝑋𝑠𝑑𝑡=𝐴𝑠𝑋𝑠+𝐵𝑠𝑈,𝑌=𝐶𝑠𝑋𝑠+𝐷𝑠𝑈,(10) and the fast model

𝑑𝑍𝑓𝑑𝜏=𝐴𝑓𝑍𝑓+𝐵𝑓𝑈,𝑌=𝐶𝑓𝑍𝑓+𝐷𝑓𝑈,(11) where 𝐴𝑠=𝐴11𝐴12𝐴122𝐴21,𝐵𝑠=𝐵1𝐴12𝐴122𝐵2,𝐶𝑠=𝐶1𝐶2𝐴122𝐴21,𝐷𝑠=𝐷𝐶2𝐴122𝐵2,𝐴𝑓=𝐴22,𝐵𝑓=𝐵2,𝐶𝑓=𝐶2,𝐷𝑓=𝐷.(12)

We denoted the transfer function of the slow model 𝐺𝑠(𝑠) by

𝐺𝑠(𝑠)=𝐶𝑠𝑠𝐼𝐴𝑠1𝐵𝑠+𝐷𝑠=𝑎11𝑎12𝐺21(𝑠)𝐺22(𝑠)(13) and the transfer function of the fast model 𝐺𝑓(𝑠) in 𝜏 time scale by

𝐺𝑓(𝑠)=𝐶𝑓𝑠𝐼𝐴𝑓1𝐵𝑓+𝐷𝑓=𝐶2𝑠𝐼𝐴221𝐵2+𝐷=𝐶2𝑠𝐼𝐴221𝐵2=𝜀𝑎11𝑏11𝑠+𝜀𝜀𝑎12𝑏12𝑠+𝜀00,(14) and the fast model in t-time scale is

𝐺𝑟(𝑠)=𝐺𝑓(𝜀𝑠)=𝑎11𝑏11𝑠+1𝑎12𝑏12𝑠+100.(15)

𝐺𝑓(𝑠), 𝐺𝑟(𝑠), and 𝐺𝑠(𝑠) are all descriptions from different points of the real system 𝐺(𝑠). And they are all two-in-two-out systems but have less state variables than the real system 𝐺(𝑠). If we choose a long enough sample interval for 𝐺(𝑠) and 𝐺𝑠(𝑠), the sample values can be similar, because the fast responses turn to steady states that can be regarded as constants in a very short time. If we choose a very short sample interval for 𝐺(𝑠) and 𝐺𝑟(𝑠) in a short enough period of time, the sample values can be similar, too. Because in such a short period of time the slow channel response is so slow that can be regarded as zero. But the output of the slow channel is not zero. Therefore, we modified the fast model in t-time scale when we designed a controller based on this fast model. Literature [18] proved the relationship between 𝐺𝑠(𝑠) and 𝐺𝑓(𝑠), and also the transfer function of the original system 𝐺(𝑠):

lim𝑠𝐺𝑠(𝑠)=𝐷𝑠=lim𝑠0𝐺𝑓(𝑠),(16)𝐺(𝑠)=𝐺𝑠(𝑠)+𝐺𝑓(𝜀𝑠)𝐷𝑠+𝑂(𝜀).(17)

Equation (16) denotes that the initial value of the slow model equals to the final value of the fast model. And (17) denotes that the original system can be regarded as a sum of the slow model, the fast model, and a very little item 𝑂(𝜀). Let

𝑂(𝜀)=00𝑂21(𝜀)𝑂22(𝜀),(18) and let

𝐺𝑡(𝑠)=𝐺𝑟(𝑠)+𝑂(𝜀)=𝑎11𝑏11𝑠+1𝑎12𝑏12𝑠+1𝑂21(𝜀)𝑂22(𝜀)=𝐺11(𝑠)𝐺12(𝑠)𝑂21(𝜀)𝑂22(𝜀).(19)

We had expressions of the fast model 𝐺𝑡(𝑠) and the slow model 𝐺𝑠(𝑠) about the model 𝐺(𝑠) without delay terms in t-time scale. Next we would consider the model with delay terms.

2.2. With Delay Terms

We took system (2), where 𝐺(𝑠) shows two-time scale characteristic, as an example. Let

𝑋11(𝑠)=𝐺11(𝑠)𝑈1(𝑠)=𝐺11(𝑠)𝑈1(𝑠)𝑒𝜏11𝑠,𝑋12(𝑠)=𝐺12(𝑠)𝑈2(𝑠)=𝐺12(𝑠)𝑈2(𝑠)𝑒𝜏12𝑠,𝑋21(𝑠)=𝐺21(𝑠)𝑈1(𝑠)=𝐺21(𝑠)𝑈1(𝑠)𝑒𝜏21𝑠,𝑋22(𝑠)=𝐺22(𝑠)𝑈2(𝑠)=𝐺22(𝑠)𝑈2(𝑠)𝑒𝜏22𝑠,(20)

Here 𝐺𝑖𝑗(𝑠), 𝑖,𝑗=1,2, denotes the transition process, and 𝑒𝜏𝑖𝑗𝑠, 𝑖,𝑗=1,2, denotes the delay time. When 𝜏11,𝜏12,𝜏21,and𝜏22 are in the same order, 𝐺𝑖𝑗(s), 𝑖,𝑗=1,2, reflects the main dynamic characteristic. We can get the fast model in t-time scale

𝐺𝑡𝑑(𝑠)=𝐺11(𝑠)𝑒𝜏11𝑠𝐺12(𝑠)𝑒𝜏12𝑠𝑂21(𝜀)𝑂22(𝜀).(21) and the slow model in t-time scale

𝐺𝑠𝑑(𝑠)=𝑎11𝑎12𝐺21(𝑠)𝑒𝜏21𝑠𝐺22(𝑠)𝑒𝜏22𝑠(22)

Then we can design a decentralized controller based on characteristics of the fast model 𝐺𝑡(𝑠)(𝐺𝑡𝑑(𝑠)) and the slow model 𝐺𝑠(𝑠)(𝐺𝑠𝑑(𝑠)).

3. Two-Time Scale Decentralized MPC

MPC is the only advanced control methodology which has made a significant impact on industrial control engineering [13]. And MPC is based on a predicted model. The MPC algorithm can be regarded as a combination of three parts: model prediction, roll optimization, and feedback rectification [19].

For the two-in-two-out system mentioned above with two-time scale characteristic, we designed a decentralized controller based on different time scales. We took the model without delay (𝐺(𝑠)) to illustrate the algorithm. The fast model provided abundant fast dynamic information to ensure the control quality. And the control interval was determined by the fast model. The slow model provided prediction horizon long enough to ensure the controller’s stability. In order to illustrate the algorithm, we defined that 𝑃𝑠 is the prediction horizon, 𝑇𝑠 is the sampling interval based on the slow model, 𝑃𝑓 is the prediction horizon, 𝑇𝑓 is the sampling interval based on fast model, and 𝑀 is the manipulate horizon. 𝑎𝑖,𝑗(𝑡) is the step response of 𝑦𝑖 from 𝑢𝑗 at 𝑇𝑓 sample interval. We got the model vector 𝑎𝑖,𝑗=[𝑎𝑖,𝑗(1)𝑎𝑖,𝑗(𝑁)]𝑇, 𝑖=1,2, 𝑗=1,2, and 𝑁 is a number large enough to fully reflect the fast and the slow part of the model. The two-time scale decentralized DMC algorithm was shown as follows.

Step 1. Model prediction based on slow model 𝐺𝑠(𝑠).
On the t-time scale, the fast dynamic achieved a steady state. The model can be fully expressed by information not so necessary as 𝑎𝑖,𝑗. Let 𝑎𝑠,𝑗=[𝑎𝑠,𝑗(1)𝑎𝑠,𝑗(𝑃𝑠)]𝑇, 𝑗=1,2, be the slow model vector, where 𝑎𝑠,𝑗(𝑖)=[𝑎2,𝑗((𝑖1)(𝑇𝑠/𝑇𝑓)+1)𝑎2,𝑗((𝑖1)(𝑇𝑠/𝑇𝑓)+𝑀)], 𝑖=1,,𝑃𝑠. The slow predict model is ̃𝑦𝑠,𝑃𝑀(𝑝)=̃𝑦𝑠,𝑃0(𝑝)+𝐴𝑠Δ𝑢,𝑗=1,2,(23) where, ̃𝑦𝑠,𝑃𝑀(𝑝)𝑅𝑃𝑠 is the predict value of future output and ̃𝑦𝑠,𝑃0(𝑝)𝑅𝑃𝑠 is the prime predict value of future output: 𝐴𝑠=𝑎𝑠,𝑗(1)0𝑎𝑠,𝑗(𝑀)𝑎𝑠,𝑗(1)𝑎𝑠,𝑗𝑃𝑠𝑎𝑠,𝑗𝑃𝑠𝑀+1.(24)

Step 2. Feedback Correction based on slow model 𝐺𝑠(𝑠).
Let error vector be 𝑒𝑠(𝑝+1)=𝑦2(𝑝+1)̃𝑦𝑠,𝑃𝑀(𝑝+1𝑝),(25) and we can get ̃𝑦𝑠,cor(𝑝+1)=̃𝑦𝑠,𝑃𝑀(𝑝)+𝐻𝑠𝑒𝑠(𝑝+1),(26) where 𝐻𝑠=𝑠(1)𝑠𝑃𝑠,(27) Like in Step 4, at 𝑝+1 time point, the time origin changes from 𝑝 to 𝑝+1 time point, then the elements of vector ̃𝑦𝑟,cor(𝑝+1) should be moved, and the operation can be expressed by ̃𝑦𝑠,𝑁0(𝑝+1)=𝑆𝑠,0̃𝑦𝑠,cor(𝑝+1),(28) where 𝑆𝑠,0=01001001.(29)

Step 3. Model prediction based on fast model 𝐺𝑡(𝑠).
On the 𝜏-time scale, the slow dynamic can be regarded as 0 which means that the slow dynamic changes very little. The model also can be fully expressed by information not so necessary as 𝑎𝑖,𝑗. Let 𝑎1,𝑗=[𝑎1,𝑗(1)𝑎1,𝑗(𝑁𝑟)]𝑇, 𝑗=1,2. The fast predict model is ̃𝑦𝑟,𝑃𝑀(𝑘)=̃𝑦𝑟,𝑃0(𝑘)+𝐴𝑟Δ𝑢,𝑗=1,2,(30) where 𝐴𝑟=𝑎𝑟,𝑗(1)0𝑎𝑟,𝑗(𝑀)𝑎𝑟,𝑗(1)𝑎𝑟,𝑗𝑃𝑓𝑎𝑟,𝑗𝑃𝑓𝑀+1.(31)

Step 4. Feedback Correction based on fast model 𝐺𝑡(𝑠).
The input 𝑢(𝑘) was applied to the plant 𝐺(s) at each 𝑘 time point, and (22) gave the predictive output ̃𝑦𝑟,𝑃𝑀(𝑘+1𝑘). Let error vector be 𝑒𝑟(𝑘+1)=𝑦1(𝑘+1)̃𝑦𝑟,𝑃𝑀(𝑘+1𝑘),(32) where 𝑦1(𝑘+1) is the sample value. We used this error vector to modify the infection by some unsure factors, and we gave the error a weight vector to modify the prediction of output: ̃𝑦𝑟,cor(𝑘+1)=̃𝑦𝑟,𝑃𝑀(𝑘)+𝐻𝑟𝑒𝑟(𝑘+1),(33) where 𝐻𝑟=[𝑟(1)𝑟(𝑃𝑓)]. At 𝑘+1 time point, the time origin changes from 𝑘 to 𝑘+1 time point, then the elements of vector ̃𝑦𝑟,cor(𝑘+1) should be moved, and the operation can be expressed by ̃𝑦𝑟,𝑃0(𝑘+1)=𝑆𝑟,0̃𝑦𝑟,cor(𝑘+1),(34) where 𝑆𝑟,0=01001001.(35)

Step 5. Rolling horizon optimization based on fast model 𝐺𝑡(𝑠).
Let objective function be min𝐽(𝑘)=𝜔1(𝑘)̃𝑦𝑟,𝑃𝑀(𝑘)2𝑄1+𝜔2(𝑝)̃𝑦𝑠,𝑃𝑀(𝑝)2𝑄2+Δ𝑢(𝑘)2𝑅s.t.Δ𝑢minΔ𝑢Δ𝑢max,𝑢min𝑢𝑢max,(36) where 𝜔𝑖(𝑘), 𝑖=1,2, are the reference value: 𝑄1=diag𝑞1(1)𝑞1𝑃𝑓,𝑄2=diag𝑞2(1)𝑞2𝑃𝑠,𝑅=blockdiag𝑅1,𝑅2,𝑅𝑖=diag𝑟𝑖(1)𝑟𝑖𝑀𝑓,𝑖=1,2.(37) Without constraint, we can get the manipulated variables: Δ𝑢(𝑘)=𝐿𝐴𝑇𝑟𝑄1𝐴𝑟+𝐴𝑇𝑠𝑄2𝐴𝑠+𝑅1×𝐴𝑇𝑟𝑄1𝜔1(𝑘)̃𝑦𝑟,𝑃0(𝑘)+𝐴𝑇𝑠𝑄2𝜔2(𝑝)̃𝑦𝑠,𝑃0(𝑝),(38) where 𝐿=100𝑀00100𝑀.(39) Step 3 to step 5 form the inner circulate, and in every 𝑇𝑓 time interval fast controller calculates a manipulate variable. Step 1 and Step 2 form the exterior circulate, and the slow controller provides the predictive value of the slow output to correct the fast controller every 𝑛𝑇𝑓 (n is a positive integer) time interval. So the framework of the control system was showed as shown in Figure 1.

4. Stability of the Two-Time Scale Decentralized MPC

We introduced the algorithm step by step in the above sections. And we would like to discuss the stability of the controller in this section. First we put forward a sufficient condition of the controller with one inner circulate.

Theorem 1. A two-time scale decentralized MPC with the control parameters 𝑇𝑓, 𝑇𝑠, and 𝑃𝑠 and one inner circulate is stable under the sufficient condition of a standard MPC, which has a control interval 𝑇𝑓 and a predictive horizon 𝑃𝑠(𝑇𝑠/𝑇𝑓).

Proof. Let the prediction horizon 𝑃𝑠(𝑇𝑠/𝑇𝑓), be the manipulate horizon, let 𝑀 be the objective function of the standard MPC be min𝐽(𝑘)=𝜔(𝑘)̃𝑦𝑃𝑀(𝑘)2𝑄+Δ𝑢(𝑘)2𝑅s.t.Δ𝑢minΔ𝑢Δ𝑢max,𝑢min𝑢𝑢max,(40)𝐽(𝑘)=𝜔(𝑘)̃𝑦𝑃𝑀(𝑘)2𝑄+Δ𝑢(𝑘)2𝑅=𝜔1(𝑘)̃𝑦1,𝑃𝑀(𝑘)2𝑄1+𝜔2(𝑝)̃𝑦2,𝑃𝑀(𝑝)2𝑄2+Δ𝑢(𝑘)2𝑅=𝑃𝑠(𝑇𝑠/𝑇𝑓)𝑖=1𝑞1𝜔1(𝑘)̃𝑦1,𝑃𝑀(𝑖+𝑘𝑘)2+𝑃𝑠(𝑇𝑠/𝑇𝑓)𝑖=1𝑞2𝜔2(𝑘)̃𝑦1,𝑃𝑀(𝑖+𝑘𝑘)2+𝑀𝑖=1𝑟Δ𝑢(𝑘+𝑖𝑘)2.(41) Because the response speed of 𝑦1 is very fast, the predictive value ̃𝑦1,𝑃𝑀(𝑖+𝑘𝑘) can be a fixed number when 𝑖 is larger than a certain number (𝑃𝑓): 𝑃𝑠(𝑇𝑠/𝑇𝑓)𝑖=1𝑞1𝜔1(𝑘)̃𝑦1,𝑃𝑀(𝑖+𝑘𝑘)2=𝑃𝑓𝑖=1𝑞1𝜔1(𝑘)̃𝑦1,𝑃𝑀(𝑖+𝑘𝑘)2+Const.(42)
The response speed of 𝑦2 is very slow, so between a short time interval (𝑇𝑠) the predictive values ̃𝑦2,𝑃𝑀(𝑖+𝑘𝑘), 𝐶𝑖𝐶+(𝑇𝑠/𝑇𝑓) (𝐶 is a positive number), can be linear correlation:
𝐶+(𝑇𝑠/𝑇𝑓)𝑖=𝐶𝑞2𝜔2(𝑘)̃𝑦1,𝑃𝑀(𝑖+𝑘𝑘)2=𝑞2𝜔2(𝑘)̃𝑦1,𝑃𝑀(𝐶+𝑘𝑘)2,(43)𝑃𝑠(𝑇𝑠/𝑇𝑓)𝑖=1𝑞2𝜔2(𝑘)̃𝑦1,𝑃𝑀(𝑖+𝑘𝑘)2=𝑃𝑠𝑗=1𝑞2𝜔2(𝑘)̃𝑦1,𝑃𝑀(𝑗+𝑘𝑘).(44) Combining (38), (40), (42), and (44), we got 𝐽(𝑘)=𝐽(𝑘)+const.(45)
So min𝐽(𝑘) and min𝐽(𝑘) had same answers, and the sufficient condition was proved.

When the inner circulate is large than one, that is, 𝑛>1, the algorithm can maintain its stability if the slow dynamic changes little in the time interval of the inner circulate. But it is hard to find an upper limit for 𝑛, because it is determined by the characteristic of the slow dynamic which can be quite different in different systems.

5. Case Study

A Model with Delay
We considered a two-in-two-out system. Two streams flow into a reactor, and 𝑢1 and 𝑢2 are the flow rates. The liquid level 𝑦1 and the temperature 𝑦2 are two controlled variables. 𝑦10=50cm and 𝑦20=295K are the initial stable states. The linear model of the system is 𝑦1𝑦2=𝐺(𝑠)𝑢1𝑢2,(46) where 𝐺(𝑠)=𝐾11𝑇11𝑠+1𝑒𝜏11𝑠𝐾12𝑇12𝑠+1𝑒𝜏12𝑠𝐾21𝑇21𝑠+1𝑒𝜏21𝑠𝐾22𝑇22𝑠+1𝑒𝜏22𝑠,(47)
The response of output 𝑦1 is much faster than that of 𝑦2. If a standard DMC controller is applied on this system, the sample interval is determined by the fast response and should be very small, and the predictive horizon is determined by the slow response and should be very large. In such a small interval, it is difficult to calculate the optimal manipulate variables, and the control quality may be bad. If we compromise the sample time interval of different channels, we can get the following control effect to track step signals. Liquid level 𝑦1 is set as 51 cm, and temperature is set as 296 K. Considering the uncertainty of the model, we chose the plant that each parameter above has 20% uncertainty to carry through the simulation. The plant is
𝑃(𝑠)=𝐾11+0.2𝐾11𝜀𝑒𝜏11+0.2𝜏11𝜀𝑠𝑇11+0.2𝑇11𝜀𝑠+1𝐾12+0.2𝐾12𝜀𝑒𝜏12+0.2𝜏12𝜀𝑠𝑇12+0.2𝑇12𝜀𝑠+1𝐾21+0.2𝐾21𝜀𝑒𝜏21+0.2𝜏21𝜀𝑠𝑇21+0.2𝑇21𝜀𝑠+1𝐾22+0.2𝐾22𝜀𝑒𝜏22+0.2𝜏22𝜀𝑠𝑇22+0.2𝑇22𝜀𝑠+1,(48) where 1<𝜀<1 is a random number. (1)𝑇=0.2, 𝑃=50.(2)𝑇=10, 𝑃=50.When the control interval is short, as seen from Figures 2 and 3, the maximum overshot of fast channel is too big and the response speed of slow channel is too slow. The reason is that the predictive horizon is not long enough, the prediction model is not fully used, and the feedback correction plays an important role in this set point tracking process. When the control interval is long, as seen from Figures 4 and 5, the slow channel shows good control quality, but the respond speed of fast channel is still a little slow. A long control interval means a low control frequency and the fast channel achieves its set point by several control steps, so the fast channel response is a little slow. And the fast channel can achieve a temporary stable state, so the response curve may have a stair shape. The compromise methods are not so perfect. Therefore, we designed a decentralized DMC controller, and let 𝑇𝑓=0.2, 𝑇𝑠=20, 𝑛=5, 𝑃𝑠=50, and 𝑃𝑓=50. We used this decentralized controller to track step signals. As shown in Figures 6 and 7, both fast channel and slow channel showed very good control quality. This method combined fast dynamic information and stable state information. The model information was fully used. From the simulation, we found that the decentralized method showed better control quality than the two compromised method.

A Continuously Stirred Tank Reactor (CSTR) Model
Chen studied the nonlinearity of a CSTR and modeled the CSTR by the following nonlinear equations [20]: 𝑑𝐶𝐴𝑑𝑡=̇𝑉𝑉𝑅𝐶𝐴0𝐶𝐴𝑘1(𝑇)𝐶𝐴𝑘3(𝑇)𝐶2𝐴,𝑑𝐶𝐵𝑑𝑡=̇𝑉𝑉𝑅𝐶𝐵+𝑘1(𝑇)𝐶𝐴𝑘2(𝑇)𝐶𝐵,𝑑𝑇𝑑𝑡=̇𝑉𝑉𝑅𝑇0𝑇1𝜌𝐶𝑝𝑘1(𝑇)𝐶𝐴Δ𝐻𝑅𝐴𝐵+𝑘2(𝑇)𝐶𝐵Δ𝐻𝑅𝐵𝐶+𝑘3(𝑇)𝐶2𝐴Δ𝐻𝑅𝐴𝐷+𝑘𝑤𝐴𝑅𝜌𝐶𝑝𝑉𝑅𝑇𝐾𝑇,𝑑𝑇𝐾𝑑𝑡=1𝑚𝐾𝐶𝑝𝐾𝑄𝐾+𝑘𝑤𝐴𝑅𝑇𝑇𝐾.(49)
We chose 𝑢1=̇𝑉/𝑉𝑅 and 𝑢2=𝑄𝐾 as the manipulate variables, and 𝐶𝐴 and 𝑇𝐾 as the controlled variables. Parameter values were given in Table 2.
We chose a steady state and identified the input-output model to design a model predictive controller.
The input-output model is
𝐶𝐴𝑇𝑘=2.868𝑠+42.8503.633𝑠+24.940.04559𝑠+14.24𝑢1𝑢2.(50)
The respond speed of 𝑇𝑘 is a little faster than that of 𝐶𝐴. And the sample frequency of the two kinds of sensors cannot be same due to the limits of the sensors. The temperature sensor can only be sampled in high frequency, while the concentration sensor can only be sampled in low frequency. In order to maintain high control frequency, soft-sensing methods are often used in standard MPC. Due to computational errors and other errors, the soft-sensing method is not a perfect way. The decentralized method presented in this paper can also deal with different sample frequencies in different channels. We compared 0.1 step tracing effect of the standard MPC and that of the decentralized method. Let the sample interval of the temperature sensor be 0.1 minute, and let the sample interval of the concentration sensor be 1 minute.
(1)Perfect soft-sensing, 𝑇=0.1 minute, 𝑃=50.(2)Without soft-sensing 𝑇=1 minute, 𝑃=50.(3)Decentralized MPC 𝑇𝑓=0.1minute, 𝑇𝑠=2minutes, 𝑛=10.
For the systems with little dynamic differences in different channels, if the controlled variables can provide sufficient reliable information in high frequency, the standard MPC can be applied and good control quality can be achieved (Figures 8 and 9). If the controlled variables can only provide reliable information in low frequency, the tracking speed turns slow (Figures 10 and 11). Although the dynamic characteristics of the two channels have a few differences, the decentralized MPC can also achieve good control quality (Figures 12 and 13) because the decentralized method takes full advantage of the reliable information of different channels.

6. Conclusion

In this article, we focused on a kind of special system and designed a decentralized model predictive controller for it. This kind of system has different dynamics in different channels and exhibits two-time scale. A centralized MPC controller cannot satisfy the fast and the slow channels simultaneously. We used singular perturbation method to get the fast and the slow model from the original system. In actual processes, input-output models that can be obtained easily by identification were usually used to describe the real system. We demonstrated the singular perturbation method applying in transfer function matrix. Then we presented a decentralized model predictive controller based on the fast and the slow model and provided a sufficient condition for the algorithm stability when 𝑛=1. Finally, the decentralized model predictive control algorithm was applied in two examples by simulation, and the validity of the control algorithm was tested. The simulation results proved that the two-time scale MPC is superior to the traditional MPC when the system had two-time scale characteristic.

The algorithm is based on the idea of fully using the information of the system. For the systems with two-time scale characteristics, the fast and slow channels are controlled, respectively, in the decentralized algorithm. This algorithm makes best use of the transition information of the fast channels and the slow channels and reduces the computation burden, which provides short control interval and increases the response speed. For those systems without two-time scale characteristics, this algorithm also works well. MPC has intensively been applied in the industrial process. The two-time scale MPC algorithm which is presented in this paper extends the applying scopes of MPC.

Acknowledgments

The authors gratefully acknowledge the financial support of 863 Program of China (no. 2007AA041402), National Key Scientific and Technical Project of China (no. 2007BAF22B05) and National Science Foundation of China (no. 60804023).