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Two-Degree-of-Freedom Controller Design for an Ill-Conditioned Process Using H2 Decoupling Control Wei Zhang,† Xing He,† Shihe Chen,‡ and Weidong Zhang*,† †

Department of Automation, Shanghai Jiao Tong University, and Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai 200240, P. R. China ‡ Guangdong Electric Power Research Institute, Guangzhou 510080, P. R China S Supporting Information *

ABSTRACT: In this paper, a two-degree-of-freedom (TDOF) controller is designed for an ill-conditioned process based on the H2 decoupling control method. The ill-conditioned process considered stems from the benchmark problem formulated in the IEEE Conference on Decision and Control (CDC). The goal of this paper is to show that lower order controllers can be reached with respect to the given plant and the corresponding design specifications. The orders of the designed TDOF controllers are 2 and 1, respectively, which is much lower than the previously developed methods. The additional benefit is that the new design procedure is simpler than the developed methods as well. In the proposed design procedure, no weight function needs to be chosen and the controller is given in an analytical form. Simulation shows that the designed controllers satisfy all design specifications of the CDC problem. problem in refs 12−15. Yaniv et al.12 proposed a quantitative feedback theory (QFT) approach for the ill-conditioned plant. No iterative steps were needed to find the feedback controller, while the prefilter required five iterations. Hoyle et al.13 designed TDOF controllers based on H∞ control, where the feedback controller and the prefilter were designed in a single step in an H∞ optimization framework. Limebeer et al.14 designed TDOF controllers for the CDC problem based on the loop-shaping design procedure, where uncertainties were modeled as H∞-bounded perturbations in the normalized coprime factors of the plant. Different from ref 14, Lundstrom et al.15 designed a TDOF controller using μ synthesis for the CDC problem, where uncertainty is modeled as linear fractional uncertainty and performance is specified as in a standard H∞ control problem. Nevertheless, the controllers given in these TODF methods are still high. In practice, high-order controllers are difficult to implement and may cause numerical problems. Thus, it is of great significance to explore the design method of lower order controllers. The goal of this paper is to design lower order controllers for the CDC benchmark problem by introducing the H2 design method of Zhang et al.16 To design controllers for the CDC problem, we first employ the H2 design method to obtain a controller that can decouple the closed system. As the response of the closed system can be tuned separately, the design complexity of the CDC benchmark problem is reduced. Second, TDOF controllers based on H2 decoupling control are designed to avoid exceeding overshoot. Since the TDOF control structure can isolate the disturbance from the reference, the controller of the reference loop can be used to improve the

1. INTRODUCTION In a control area, the condition number of a plant is defined as the ratio between its maximal and minimal singular values. The plant with a high condition number is said to be ill conditioned. The ill-conditioned plant has a strong directionality. Such a plant is potentially extremely sensitive to model uncertainty, which makes satisfactory control difficult to achieve.1−4 One class of ill-conditioned plant that has received extensive treatment is the high-purity distillation column. To compare different controller design methods for ill-conditioned plants, Limebeer5 formulated a benchmark problem in the IEEE Conference on Decision and Control (CDC) based on the work of Skogestad, Morari, and Doyle.6,7 The ill-conditioned plant in this problem is based on a real distillation column, and the design requirement is typical in similar systems. The problem includes many possibly conflicting requirements and thus is extremely difficult. Even though the problem has been presented for two decades, it can still be regarded as a challenge for testing controller design methods. The distillation problem in ref 6 and variants of this problem, like the CDC problem in ref 5, have been studied by many researchers using different control structures and design methods. Some papers studied the one-degree-of-freedom (ODOF) controller. Zhou and Kimura8 investigated how to use the robust stability degree assignment method in the controller design for an ill-conditioned distillation column. Semino et al.9 designed a double-filter internal model control (IMC) controller for ill-conditioned distillation columns. Razzaghi and Shahraki10,11 showed that acceptable closedloop performance could be achieved for an ill-conditioned highpurity distillation column by use of the structured singular value. The benefit of this method is its reliance on a simple filter structure depending on fewer parameters. Since the order of the ODOF controller is usually high, TDOF controllers are designed for the CDC benchmark © 2012 American Chemical Society

Received: Revised: Accepted: Published: 14752

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reference response independently. In this way, all design specifications of the CDC problem are satisfied. Compared with previously developed methods, the benefit of the proposed procedure is its simplicity. In the design procedure, no weight function needs to be chosen and the controller is given in an analytical form.

2. PROBLEM STATEMENT The distillation column described in ref 5 is used to separate the input feed into its light and heavy components. The light component is concentrated in the distillate, while the heavy component is concentrated in the bottom product. The manipulated variables are the reflux flow and boil-up flow. The measurements are the product components. The resulting model is ill conditioned, which can be expressed as G̃ (s) =

1 75s +

−θ s 0 ⎤ ⎡ 0.878 −0.864 ⎤⎡⎢ k1e 1 ⎥ ⎢ ⎥ 1 ⎣1.082 −1.096 ⎦⎢⎣ 0 k 2e−θ2s ⎥⎦

Figure 1. Schematic of the IMC control structure.

Figure 2. Unity feedback control loop.

(1)

ki ∈ [0.81.2], θi ∈ [0.01.0]

This is equivalent to a gain uncertainty of ±20% and a delay up to 1 min in each input channel. The set of possible plants defined by eq 1 is denoted as Π in the sequel. The aim is to design a controller which meets the following quantitative robust stability and robust performance specifications denoted by S1−S3 for every G̃ (s)∈Π: S1, closed-loop stability; S2, for a unit step demand in channel 1 at t = 0 the plant outputs γ1(tracking) and γ2 (interaction) should satisfy: • γ1(t) ≥ 0.9 for all t ≥ 30 min; • γ1(t) ≤ 1.1 for all t; • 0.99 ≤ γ1(∞) ≤ 1.01; • γ2(t) ≤ 0.5 for all t; • −0.01 ≤ γ2(∞) ≤ 0.01. Corresponding requirements hold for a unit step demand in channel 2. S 3: To avoid controllers with unrealistic gains and bandwidths, the gain of the transfer matrix between output disturbances and plant inputs should be limited to about 50 dB, and the unity-gain, crossover frequency of the largest singular value should be below 150 rad/min. In this paper, we will provide a new design method in the framework of TDOF decoupling control, the basis of which is the H2 analytical decoupling controller developed in ref 16. While many solutions have been developed, there are two reasons we use this newly developed method to deal with the CDC problem. First, the controller designed with the H2 design method is analytical. This feature can be used to reduce the complexity of the design procedure. Second, the controller order is closely related to the plant’s order. This feature provides a simple way to estimate the complexity of the solution in studying the CDC benchmark problem. As it is shown in later sections, the proposed method provides a simple design procedure and results. Such a design method can also be applied to other similar control systems.

Figure 3. Schematic of the TDOF structure.

3. TDOF H2 DECOUPLING CONTROL 3.1. H2 Decoupling Control. The H2 analytical decoupling control method is first introduced in this section, and a TDOF design method based on it is developed in the next section. Consider the IMC structure in Figure 1, where G(s) is a square plant of dimension n, Gm(s) is the model, and Q(s) is the

Figure 4. Flowchart of the procedure of the TDOF control method.

IMC controller. The IMC structure can be related to the unity feedback loop (see Figure 2) through 14753

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the maximum time delay that can be separated is 2. In the following element

(2)

In the case that the model is exact, that is, G(s) = Gm(s), the sensitivity transfer matrix (i.e., the transfer matrix from set point r(s) to the error e(s)) is given by S(s ) = I − G (s )Q (s )

⎛ e − 5s e − 3s e 2s 1 ⎞ 2s + =⎜ + ⎟e 2s + 1 3s + 1 ⎝ 2s + 1 3s + 1 ⎠

the maximum time delay that can be separated is −2. According to ref 16, the plant G(s) can be factorized as

(3)

and the complementary transfer matrix (i.e., the transfer matrix from set point r(s) to the system output y(s)) is given by T (s ) = G (s )Q (s )

G(s) = G D(s)G N(s)GMP(s)

(4)

where GD(s) is the time delay part, GD(s) and GN(s) are diagonal, GA(s) = GD(s)GN(s) is the all-pass portion of the plant, and GMP(s) is the MP part of G(s). Definition 3.1.16 Let θi be the maximum prediction of the ith column of G−1(s), that is θi = maxj θji, j = 1, 2, ..., n. The factorization for the time delay part is

Let W1(s) and W2(s) be the performance weighting functions. The performance index of the MIMO IMC design is min W2(s)S(s)W1(s)

2

(5)

If the plant is rational it can be factorized into a stable all-pass portion GA(s) and a minimum phase (MP) portion GM(s) such that G(s) = GA (s)GM(s)

G D(s) = diag{e−θ1s , ···, e−θns}

Example 3.2. Assume that

(6)

⎡* *e 4s *⎤ ⎢ ⎥ G−1(s) = ⎢* *e−3s *⎥ ⎢ ⎥ ⎣* *e6s *⎦

G−1 M (s)

where GA(s) and are stable and GA*(s)GA(s) = I. Here the asterisk (*) denotes the complex conjugate transpose of a matrix. Morari and Zafiriou3 have shown that, for a unit step input, the optimal controller is given by

Here the asterisk (*) denotes an arbitrary entry. The time delays of the elements in the second column are 4, −3, and 6. Then θ2 = 6. Assuming that zj (j = 1, 2, ..., rz) is a right half plane zero of G(s), it is an unstable pole of G−1(s). GN(s) can be constructed as follows. Definition 3.2.16 Let ki be the largest multiplicity of the unstable pole zj in the ith column of G−1(s)GD(s). The factorization for the right half plane zeros is

−1 Q opt(s) = GM (s )

The IMC design provides an excellent result, but the design procedure is only developed for plants without time delay. When there are time delays in the plant, these time delays must be expanded by rational approximations. Furthermore, the result of the IMC design is not decoupled in the general case. Aimed at designing a decoupled IMC controller for plants with time delays, Zhang et al.16 gave a modified inner−outer factorization for plants with time delays. The procedure of the factorization is given as follows. Assume that the plant is expressed as ⎡G (s)e−θ11s ··· G (s)e−θ1ns ⎤ 1n ⎢ 11 ⎥ ⎢ ⎥ ⋮ ⋱ ⋮ G (s ) = ⎢ ⎥ ⎢⎣ Gn1(s)e−θn1s ··· Gnn(s)e−θnns ⎥⎦

k1j ⎧ rz ⎛ − s + zj ⎞ ⎪ ⎟⎟ , ···, G N(s) = diag ⎨∏ ⎜⎜ ⎪ j = 1 ⎝ s + z*j ⎠ ⎩

⎛ −s + z ⎞knj ⎫ ⎪ j ⎟⎟ ⎬ ∏ ⎜⎜ * s + zj ⎠ ⎪ j=1 ⎝ ⎭ rz

In particular, for MP plants GN(s) = I. It is easy to verify that GA*(s)GA(s) = I and GA(0) = I. This definition is illustrated in the following example. Example 3.3. Assume that the second column of G−1(s) GD(s) involves only one unstable pole, that is, z1 = 1. The multiplicities of its elements are given below

(7)

where Gij(s) (i, j = 1, 2, ..., n) are scalar rational transfer functions and θij > 0 are time delays. Let the inverse of the plant be

⎡ ⎤ * *⎥ ⎢* * 4 (s − 1) ⎢ ⎥ ⎢ ⎥ * *⎥ G−1(s)G D(s) = ⎢* * 3 (s − 1) ⎢ ⎥ ⎢ ⎥ * ⎢* * ⎥ * ⎢⎣ (s − 1)6 ⎥⎦

n1 ⎤ ⎡ 11 −θ11s ··· Gn1(s)e−θ s ⎥ ⎢ G (s)e ⎥ G−1(s) = ⎢ ··· ⋱ ··· ⎢ ⎥ 1n nn ⎢⎣G1n(s)e−θ s ··· Gnn(s)e−θ s ⎥⎦ ji

where Gji (s)e−θ s (j = 1, 2, ..., n) are the elements of G−1(s) and θji are the real numbers denoting the maximum time delays that can be separated from each element. It can be negative, meaning it is a prediction. The following example is used to illustrate how to compute θji. In particular, for rational plants GD(s) = I. This definition is illustrated in the following example. Example 3.1. Consider the following element

The multiplicities of this RHP pole in the elements of the second column are 4, 3, and 6, respectively. Then k2 = 6. The idea of the H2 optimal control is to find a controller that stabilizes the system and minimizes the performance index defined in eq 5. To design the controller, the two weighting functions must be determined first. Instead of choosing complex weighting functions, for step inputs or inputs similar to steps, Zhang et al.16 gave a reasonable but simple way to determine the weighting functions. The weighting functions can be chosen as W1(s) = s−1I and W2(s) = I. Therefore, the

⎛ e −s e − 3s e − 2s 1 ⎞ − 2s ⎟e + =⎜ + 2s + 1 3s + 1 ⎝ 2s + 1 3s + 1 ⎠ 14754

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limited. In this case, it is impossible to use complicated methods. The design method introduced in this section provides a lowcomplexity choice. It can be seen that in the design procedure no weight function needs to be chosen. The controller order is directly related to the plant order. Since the design procedure is analytical, it is very easy to design a controller. The design procedure of the TDOF design approach is summarized in the flowchart shown in Figure 4.

performance index of the H2 optimal control can be rewritten as min || S(s)W1(s)||2

(9)

Once the factorization has been finished, the optimal IMC decoupling controller for eq 9can be obtained by ref 16 −1 Q opt(s) = GMP (s )

(10)

Since the controller is generally improper, a filter J(s) should be introduced to make the controller proper. Besides, the filter should make the closed-loop system internally stable and satisfy the asymptotic tracking condition. For step inputs and stable plants, the structure of the filter can be chosen as

4. APPLICATION OF TDOF H2 DECOUPLING CONTROL In this section, TDOF H2 decoupling controllers are designed for the model in eq 1 with respect to S1−S3 in the CDC benchmark problem.

⎡ ⎤ 1 ⎢ ⎥ n1 λ + ( s 1) ⎢ 1 ⎥ ⎥ ⋱ J (s ) = ⎢ ⎢ ⎥ 1 ⎢ ⎥ ⎢⎣ (λns + 1)nn ⎥⎦

Table 1. Orders of the Designed Controllers in Different References different design methods controller orders

where λi (i = 1, 2, ..., n) is a positive real constant and defined as performance degree and ni (i = 1, 2, ..., n) is the smallest relative degree of the ith column of Qopt(s) for strictly proper plants. Consequently, the H2 analytical decoupling controller can be derived as

Q (s) = Q opt(s)J(s) C(s) = Q (s)[I − G(s)Q (s)]

Gm(s) =

(13)

3.2. TDOF Control Based on H2 Decoupling Control. In this paper, the TDOF control structure shown in Figure 3 is chosen to satisfy S1−S3 in the CDC benchmark problem. Here, C1(s) is the controller of the disturbance loop and C2(s) is the controller of the reference loop. Here, we will design both of the two TDOF controllers in the framework of ref 16. The controller of the disturbance loop is just the controller of the unity feedback loop −1

C1(s) = Q (s)[I − G(s)Q (s)]

22

2, 1

1 ⎡ 0.878 −0.864 ⎤⎡ e−0.5s 0 ⎤ ⎥ ⎢ ⎥⎢ 75s + 1 ⎣1.082 −1.096 ⎦⎢⎣ 0 e−0.5s ⎥⎦

(18)

(19)

where ⎡ e−0.5s ⎤ ⎥, G D(s) = ⎢ ⎢⎣ e−0.5s ⎥⎦

(14)

⎡1 0⎤ , and G N (s ) = ⎢ ⎣ 0 1 ⎥⎦

(15)

GMP(s) =

1 75s +

⎡ 0.878 −0.864 ⎤ ⎥ ⎢ 1 ⎣1.082 −1.096 ⎦

One can readily obtain the optimal IMC controller as follows

(16) −1 (s ) = Q opt(s) = GMP

Introducing a diagonal filter J2(s) to the optimal controller C2(s) = J1−1(s)G N−1(s)J2 (s)

5, 5

Gm(s) = G D(s)G N(s)GMP(s)

Regard T(s) as a new plant and C2(s) as the controller in the next step design. The controller of the reference loop can be obtained as follows: C2opt(s) = J1−1(s)G N−1(s)

4, 4

In other words, we choose the “center” of Π (k1 = k2 = 1 and θ1 = θ2 = 0.5) as the nominal plant. Consequently, the nominal plant can be factorized as follows

with Q(s) = Qopt(s)J1(s). The closed-loop transfer matrix of the corresponding unity feedback loop is T (s) = G(s)Q (s) = G D(s)G N(s)J1(s)

38

Limebeer14 Lundstrom15 proposed

In order to design the TDOF H2 analytical decoupling controllers, the nominal plant should be chosen first. The nominal model used for design is

(12) −1

Zhou8 Yaniv12

75s + 1 ⎡−1.096 0.864 ⎤ ⎢ ⎥ −0.0741 ⎣−1.082 0.878 ⎦

(20)

Since the plant is stable, the filter can be chosen as

(17)

⎡ 1 ⎢ λ1s + 1 J1(s) = ⎢⎢ 1 ⎢ λ 2s + ⎣

J2(s) has a similar structure to J1(s). Roughly speaking, there are two kinds of design objectives in developing controller design methods: improving the performance or reducing the design complexity. In many applications, the problem of reducing the design complexity is even more important than the problem of improving the performance. For example, in an ultrasupercritical power plant there are more than 100 loops. The time for configuration and tuning is very

⎤ ⎥ ⎥ ⎥ ⎥ 1⎦

(21)

Introducing the filter to Qopt(s), the suboptimal proper controller Q(s) can be obtained as follows 14755

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Table 2. Control Performance in Channel 1 for Designed TDOF Controllers with Gain Uncertainty and a First-Order Taylor Expansion Approximation of a 1 min Delay gain uncertainties

set-point tracking

interaction

step channel

k1

k2

t = 30 min

max

t = 60 min

max

t = 60 min

1 1 1 1 1 1 1 1 1 1

0.8 0.8 0.8 0.8 0.9 1.1 1.2 1.2 1.2 1.2

0.8 1.0 1.1 1.2 1.1 0.9 0.8 0.9 1.0 1.2

0.9488 0.9348 0.9298 0.9256 0.9377 0.9605 0.9726 0.9648 0.9586 0.9494

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.9974 0.9968 0.9965 0.9963 0.9969 0.9980 0.9986 0.9982 0.9979 0.9975

0.0045 −0.2898 −0.3964 −0.4858 −0.2370 0.2364 0.4864 0.3259 0.1970 0.0034

−0.0000 −0.0009 −0.0012 −0.0015 −0.0007 0.0007 0.0015 0.0010 0.0006 −0.0000

C2(s) =J1−1(s)G N−1(s)J2 (s)

Q (s) = Q opt(s)J1(s) ⎡ 1 0 ⎢ 75s + 1 ⎡−1.096 0.864 ⎤⎢ λ1s + 1 = ⎢ ⎥ 1 −0.0741 ⎣−1.082 0.878 ⎦⎢ ⎢ 0 λ 2s + ⎣

⎤ ⎥ ⎥ ⎥ ⎥ 1⎦

⎡ λ1s + 1 ⎤ ⎥ =⎢ ⎢⎣ λ 2s + 1⎥⎦ ⎡ 1 ⎢ γ1s + 1 ⎢ ×⎢ ⎢ ⎢⎣ ⎡ λ1s + 1 ⎤ ⎢ ⎥ ⎢ γ1s + 1 ⎥ =⎢ ⎥ λ 2s + 1 ⎥ ⎢ ⎢⎣ γ2s + 1 ⎥⎦

(22)

Then the controller of the disturbance loop in the TDOF structure can be derived C1(s) =Q (s)[I − G(s)Q (s)]−1 =

75s + 1 ⎡− 1.096 0.864 ⎤ ⎢ ⎥ − 0.0741 ⎣− 1.082 0.878 ⎦ ⎡ 1 0 ⎢ −0.5s ⎢ (λ1s + 1) − e ×⎢ 1 ⎢ 0 (λ 2s + 1) − e−0.5s ⎣

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

(25)

It can be seen that the controller has two nonzero elements, and both of the orders are 1. From the design procedure of the TDOF analytical H2 decoupling controller it can be seen that the controllers are designed analytically and the orders of the two controllers are low. In the two controllers, two performance degrees need to be determined. Zhang et al.16 gave a very simple engineering tuning rule for tuning: Increase the performance degrees monotonically until the required response is obtained. Using this rule one can easily obtain the performance degrees satisfying S1−S3. To illustrate the benefit of the proposed method, we compare the order of the proposed controller with those designed by other methods. From Table 1, we can see that the orders of the proposed TDOF H2 decoupling controllers are much lower than those of other design methods.

(23)

Remark 1: Many design methods transform the nominal model into a rational plant by approximation first and then design the controller based on the approximate model. Instead of this, the controller here is directly based on the nominal plant with time delays and an exact solution is obtained. The obtained controller is of infinite dimension. A lower order controller can be achieved by expanding the controller with the rational approximation in the form of the first order lag, which is given as follows C1(s) ≈

1 γ2s + 1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥⎦

75s + 1 ⎡− 1.096 0.864 ⎤ ⎢ ⎥ − 0.0741 ⎣− 1.082 0.878 ⎦ ⎡ ⎤ 0.5s + 1 0 ⎢ ⎥ 2 λ + λ + 0.5 s ( 0.5) s 1 1 ⎢ ⎥ ×⎢ ⎥ 0.5s + 1 ⎢ ⎥ 0 2 0.5λ 2s + (λ 2 + 0.5)s ⎦ ⎣

5. SIMULATIONS To illustrate that the designed controllers satisfy S1−S3, 10 different gain combinations with 1 min delays are chosen, which are listed in Table 2. Increasing the performance degrees from small to large, we can easily determine that λ1 = λ2 = 6.4 and γ1 = γ2 = 600 for the required response. Figure 5 gives the closed-loop step responses of the plants with 10 different gain combinations with 1 min delays chosen in Table 2. Figure 5a gives the responses for channel 1, and Figure 5b gives the responses for channel 2. It can be seen that the closed-loop system for every chosen plant is stable, which implies that S1 is satisfied. Tables 2 and 3 summarize the

(24)

It is seen that the controller involves two nonzero elements, and both of the orders are 2. Now let us consider the design of the controller for the reference loop in the TDOF structure. Since the closed-loop system is decoupled, it is easy to design this controller. On the basis of eq 17 in the last section, the designed controller is 14756

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Table 4. Four Extreme Uncertainty Combinations with 1 min Delay k1

k2

0.8 0.8 1.2 1.2

0.8 1.2 0.8 1.2

Figure 6. Maximum and minimum singular values of D(s) for TDOF H2 decoupling controllers of four chosen plants.

compute a large set of step responses corresponding to θ1 and θ2 in the range of 0 ≤ θ1, θ2 ≤ 1 min and k1 and k2 in the range of 0.8 ≤ k1, k2 ≤ 1.2. Due to limited space, these curves are not given, but they are similar to the results in Figure 5. On the basis of the TDOF control structure in this paper, the transfer matrix between output disturbances and plant inputs can be calculated by D(s) = C1(s)[I + G̃(s)C1(s)]−1

According to S3, the maximum peak of the largest singular value of D(s) for every G̃ (s) ∈ Π should be less than 50 dB. In order to test S3, four extreme uncertainty combinations with 1 min delays listed in Table 4 are chosen. Figure 6 gives the maximum and minimum singular values of D(s) for the designed TDOF controllers of the four chosen plants. The maximum peak of the

Figure 5. Closed-loop responses for the designed controller with plant−model mismatch in Table 3.

simulation results in channels 1 and 2, respectively. From Tables 2 and 3, it can be seen that the designed TDOF controllers satisfy S2. As a complete simulation study, we

Table 3. Control Performance in Channel 2 for Designed TDOF Controllers with Gain Uncertainty and a First-Order Taylor Expansion Approximation of a 1 min Delay gain uncertainties

set-point tracking

interaction

step channel

k1

k2

t = 30 min

max

t = 60 min

max

t = 60 min

2 2 2 2 2 2 2 2 2 2

0.8 0.8 0.8 0.8 0.9 1.1 1.2 1.2 1.2 1.2

0.8 1.0 1.1 1.2 1.1 0.9 0.8 0.9 1.0 1.2

0.9487 0.9631 0.9683 0.9726 0.9605 0.9378 0.9256 0.9336 0.9400 0.9494

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.9974 0.9982 0.9984 0.9986 0.9980 0.9969 0.9963 0.9967 0.9970 0.9975

0.0009 0.1854 0.2535 0.3105 0.1515 −0.1517 −0.3115 −0.2091 −0.1263 0.0124

−0.0000 0.0006 0.0008 0.0009 0.0005 −0.0005 −0.0009 −0.0006 −0.0004 −0.0000

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work is partly supported by the National Science Foundation of China under Grant Nos. 11072144, 61025016, and 61034008.

■ Figure 7. Maximum and minimum singular values of G(s)C1(s)C2(s) for TDOF H2 decoupling controllers of four chosen plants.

largest singular value for the chosen four plants is 312. It is less than 316 (that is, 50 dB). The unity-gain, crossover frequency is the frequency at which the largest singular value of the open loop transfer function matrix between set points and plant outputs is equal to 1. Consider the control structure in this study, the open loop transfer function matrix between references and plant outputs is G̃ (s)C1(s)C2(s), where G̃ (s) is the plant and C1(s), C2(s) are the designed controllers. To satisfy the unity-gain, crossover frequency of the largest singular value should be below 150 rad/min; the frequency corresponding to σmax[G̃ (s)C1(s)C2(s)] = 1 for every G̃ (s) ∈ Π should be below 150 rad/min. Figure 7 gives the maximum and minimum singular values of G̃ (s)C1(s)C2(s) for the chosen four extreme uncertainty combinations with 1 min delay plants. The unity-gain, crossover frequency of the largest singular value is 3.114 rad/ min, which is less than 150 rad/min. Therefore, the designed TDOF controllers satisfy S3. From the simulation results above, it can be seen that the designed TDOF H2 decoupling controllers satisfy S1−S3 formulated in the CDC benchmark problem.

6. CONCLUSIONS In this paper, a well-known benchmark problem is studied in which the plant is an ill-conditioned distillation column. For this challenging problem the design requirement is achieved with a very simple design procedure. Compared with previously developed methods, the proposed procedure has two important features: First, no weighting functions need to be chosen. Second, the designed controllers are analytical. One finds that the orders of the designed TDOF controllers are 2 and 1, respectively, which is much lower than other developed methods. This design method can also be applied to other similar control systems.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

Tables S1−S4, and Figures S1−S7. This material is available free of charge via the Internet at http://pubs.acs.org. 14758

dx.doi.org/10.1021/ie2028848 | Ind. Eng. Chem. Res. 2012, 51, 14752−14758