Article pubs.acs.org/IECR
Partial Decoupling Control for Disturbance Rejection for Multivariable Systems with Time Delays Q. B. Jin and Q. Liu* Institute of Automation, Beijing University of Chemical Technology, Beisanhuan East Road 15, Chaoyang District, Beijing 100029, People’s Republic of China ABSTRACT: Motivated by the fact that the partial decoupling control sometimes provides better disturbance rejection than decoupling control and decentralized control, we focus on the design of the partial decoupler in this paper. The goal of the partial decoupler here is to remove the interactions of the decoupled loops and to retain the interactions of the loops that are not decoupled. We first analyze the effects of the disturbance on the outputs with and without interactions among loops. An appropriate decoupled matrix to achieve the above goal then is proposed. Based on this decoupled matrix, the proposed partial decoupler is derived. Also based on the decoupled matrix, the equivalent model of each loop is easily obtained. An optimization problem, which is to maximize the integral gain with certain robustness levels, is established to design the proportional-integral/ proportional-integral-derivative (PI/PID) controller for each loop. Simulation studies show the effectiveness of the proposed partial decoupler.
1. INTRODUCTION Most encountered chemical processes have a multivariable nature. Although several advanced control schemes can handle the interactions among loops for multi-input/multi-output (MIMO) processes, the traditional proportional-integral/ proportional-integral-derivative (PI/PID) controller is still widely used, because of its simple implementation. Compared to the controller design for single-input/single-output (SISO) processes, the controller design for MIMO processes is difficult, because of the interactions among loops. The two common control schemes for MIMO processes are full decoupling control (FDC) and decentralized control. FDC has been welladdressed in the literature, including the ideal decoupling,1,2 the simplified decoupling,1,3 and the inverted decoupling.1,4 FDC can effectively remove the interactions, and, in this case, the well-established methods for SISO processes can be employed to design the controllers independently. The key procedure for FDC is to design the decoupler, which is complexity for highdimensional systems.5 As for the decentralized control, it is simpler and more easily understood than FDC. The key point in this case still is how to handle the interactions. The existing methods include the detuning method,6 the sequential loop closing (SLC) method,7 the optimization method,8 and the autotuning method.9 Recently, with the assumption of “perfect control”, the equivalent open loop transfer function (EOTF) has been proposed to realize the independent design of the decentralized controllers.10 The similar idea also can be found in ref 11. Besides the FDC and the decentralized control, there is another control scheme: partial decoupling control (PDC). PDC means some loops in a multivariable system are decoupled and the others are not. The reason for using PDC is that it may provide better disturbance rejection, which is the primary concern for process control, than FDC and the decentralized control in some cases.12 Naturally, two problems arise for PDC: (1) how to determine whether one loop needs to be decoupled or not; and (2) how to design the partial © 2014 American Chemical Society
decoupler and the controllers. To determine whether a loop needs to be decoupled or not, the designer should first evaluate the effect of the interactions on the disturbance response. The relative disturbance gain (RDG)13,14 and the relative load gain (RLG)15−17 are the two common indices used to measure the performance for disturbance rejection for different control schemes. This issue has been addressed well in the literature, although some limitations still exist. In this paper, much attention is paid to the design of the partial decoupler and the controllers. Different from FDC, where the interactions are fully removed for all the loops, PDC should remove the interactions of the loops decoupled and retain the interactions of the loops that are not decoupled. Retaining the interactions of the nondecoupled loops means that the dynamics of these loops with partial decoupling should be similar to that without decoupling. However, the general form of the partial decoupler satisfying the above goal is not found. In refs 12 and 18, based on the evaluation of the disturbance rejection capability, the one-way decoupling for two-input/two-output (TITO) processes was proposed. The authors in ref 17 reported the generalized decoupling control for MIMO systems, which satisfies the basic requirements of PDC. In their work, the interactions of the decoupled loops were removed and those of the nondecoupled loops were still present. However, the resulting dynamics of the nondecoupled loops was different from those without decoupling, and the resulting decoupled models are conservative. Another issue is how to design the controllers if the partial decoupler is determined, especially for the loops that have not been decoupled. The authors of ref 19 stated that this issue remains unsolved. In this study, we try to give a solution Received: Revised: Accepted: Published: 15932
April 22, 2014 September 12, 2014 September 22, 2014 September 22, 2014 dx.doi.org/10.1021/ie502785c | Ind. Eng. Chem. Res. 2014, 53, 15932−15945
Industrial & Engineering Chemistry Research
Article
to the two problems above. To meet the requirements of PDC, we separate the loops from the MIMO system with the equivalent controlled transfer function. Based on the equivalent SISO loops, a desired decoupled transfer function matrix is introduced and the proposed partial decoupler is derived from the decoupled transfer function matrix. The static and dynamical forms of the decoupler are analyzed based on our previous work.5 For the decoupled loops, the controllers are designed using the well-established tuning rules of PI/PID controllers for SISO systems. With the partial decoupler, the loops that have not been decoupled have dynamics similar to that of decentralized control. Because of this fact, the controllers for the loops not decoupled can be designed as decentralized controllers. In ref 10, the decentralized controllers are independently designed under the assumption of “perfect control”. Based on the idea of independent design, in this paper, an optimization problem is established to obtain the optimal parameters of the PI/PID controllers for disturbance rejection. It is beneficial that the controllers for the loops that have not been decoupled can also be independently designed as the decentralized control. The effectiveness of the proposed partial decoupling control is demonstrated, along with two simulation examples. This paper is organized as follows. Section 2 discusses the FDC and decentralized control, and the existing indices to determine whether one loop needing to be decoupled or not are derived. The partial decoupler is proposed in section 3 based on the analyses in section 2. In section 4, details about the controller design is given. The proposed method is tested with two examples in section 5. Finally, we draw the conclusions in section 6.
Figure 1. Decentralized control system.
where t is the time. For simplicity, the term “(s)” will be dropped in the equations that follow. To study the effect of the disturbance on the outputs with the interactions, we separate the loops from the MIMO control systems. From eq 1, the following relation is derived: G−1Y = U + G−1LD
Assume that the ith loop is open and the subsystem without the ith loop, denoted by Σ, is closed. The relationship of Σ between inputs and outputs is expressed as follows, based on eq 1: Y −i = G−iiC−i(I + G−iiC−i)−1R −i
(3)
where I is the identity matrix, G−ii and C−i are the transfer function matrix G with the ith column and row removed, respectively, and R−i and Y−i are the corresponding set-points and outputs of the subsystem Σ, respectively. Under the perfect control conditions,15 the outputs of the subsystem Σ are equal to the inputs at all times, i.e., G−iiC −i(I + G−iiC−1)−1 = I The perfect control is achieved at steady state if the process transfer function is stable and the controller includes the integral term. Generally speaking, this assumption holds at low frequencies. The more detail about the perfect control can be found in ref 15. With perfect control, the relationship of the ith loop based on eq 2 is derived as
2. DECENTRALIZED CONTROL AND FULL DECOUPLING CONTROL (FDC) In this section, we discuss the two common control strategies for MIMO plants: decentralized control and FDC. We focus on how to separate the single loops from the MIMO systems and study the effect of the disturbance on the output performance. Then, we derive the two indices to evaluate the disturbance rejection. This section is the base of the design of the partial decoupler. 2.1. Decentralized Control. Decentralized control is a widely used control strategy for MIMO processes, since it is easily implemented and understood. The decentralized control system is shown in Figure 1 and is described as follows: ⎧ Y(s) = G(s)U(s) + L(s)D(s) ⎪ ⎪ U(s) = C(s)(R(s) − Y(s)) ⎪ ⎪ T ⎨ Y(s) = [y1(s), y2 (s), ···, yn (s)] ⎪ ⎪ R(s) = [r1(s), r2(s), ···, rn(s)]T ⎪ ⎪ U(s) = [u (s), u (s), ···, u (s)]T ⎩ 1 2 n
(2)
[G−1]ii yi = ui + [G−1L]i D ⇒ yi =
[G−1L]i 1 u + D i [G−1]ii [G−1]ii (4)
Figure 2. Equivalent single-input/single-output (SISO) loop for the ith loop with perfect control.
Hence, the ith loop can be considered as an independent loop with perfect control, which is shown in Figure 2. It can be seen from eq 4 that the effect of the disturbance on the output is directly related to the inverse of the process transfer function matrix with perfect control. Next, we discuss the corresponding relationship in the ideal full decoupling control. 2.2. FDC. The FDC is the effective strategy to remove the interactions among loops. For this control strategy, the process
(1)
where G(s), C(s), and L(s) are the process transfer function matrix, the decentralized controller matrix, and the disturbance transfer function matrix, respectively. The disturbance signal is denoted by D(s) in the frequency domain; and the vectors of the setpoints, the outputs, and the manipulated variables are donated by R(s), Y(s), and U(s), respectively. The above variables in the time domain are donated by r(t), y(t), and u(t), 15933
dx.doi.org/10.1021/ie502785c | Ind. Eng. Chem. Res. 2014, 53, 15932−15945
Industrial & Engineering Chemistry Research
Article
transfer function matrix is modified as G̅ = GQ = diag{g11, g22, ..., gnn}, where Q is the ideal decoupler. In this case, the ith output is expressed as follows, based on eq 4: yi =
Relative Load Gain (RLG). RLG directly measures the gain of the disturbance to the outputs. The definition of this index can be found in refs 15−17. For the ith loop, the gain between disturbance and output with interactions is expressed as follows from eq 4:
[G̅ −1L]i 1 u + D i [G̅ −1]ii [G̅ −1]ii
= giiui + liD
∂yi ∂D
(5)
The independent SISO loop of the ith loop can be drawn as Figure 3.
=
[G−1(0)L(0)]i [G−1(0)]ii
(8)
Similarly, the disturbance gain for FDC is given by ∂yi ∂D
= li(0)
(9)
Therefore, RLG of the ith loop is computed as λi =
2.3. Indices for Evaluating the Disturbance Rejection. The interactions among loops have not always had a negative effect on the disturbance rejection. If the interactions degrade the disturbance rejection, removing the interactions is required; otherwise, a good control system needs to be designed to take advantage of the interactions. The PDC is proposed to achieve these goals if some interactions need to be removed but the others need to be retained. Therefore, the first step is to evaluate the effect of the disturbance on the outputs with and without interactions, and determine whether one loop needs to be decoupled or not. Two indices are derived here to measure the disturbance rejection for different control schemes. Relative Disturbance Gain (RDG). RDG was first proposed by Stanley for TITO systems,14 which was defined as the ratio of the required manipulated variable changes for FDC and for the decentralized control. Assume that the disturbance is a unit step change, and the required manipulated variable at steady state for decentralized control to reject the disturbance is determined by the following equation, based on eq 4:
⎡ G11 G12 ⎤ needing to be decoupled ⎥ G=⎢ ⎣G21 G22 ⎦ not needing to be decoupled
[G (0)L(0)]i 1 ui(0) + D(0) [G (0)]ii [G−1(0)]ii −1
(6)
In the case of FDC, the required manipulated variable is computed as ui (0) = li(0)/gii(0). Hence, the value of RDG for the ith loop is given by [G−1(0)L(0)]i li(0)/gii(0)
(11)
where G11 ∈ Rm×(n−m), G12 ∈ Rm×(n−m), G21 ∈ R(n−m)×m, and G22 ∈ R(n−m)×(n−m). With PDC, the process transfer function matrix is modified as T = GP, where T is the decoupled transfer function matrix and P is the partial decoupler. The goal of the partial decoupler is to remove the interactions for the first m loops but retain the interactions for the remaining n−m loops. Note that the decoupled matrix T determines whether the above goal can be achieved or not. Therefore, we first study which decoupled matrix meets the requirements of PDC. Consider decentralized control. It can be found from eq 4 that the interactions and the disturbance effect are directly related to the inverse of G with perfect control condition. The inverse of G, in terms of eq 11, is given by
⇒ ui(0)
βi =
(10)
3. THE PARTIAL DECOUPLER Assume that there are m (m ≤ n) loops needing to be decoupled and n−m loops not needing to be decoupled for a n × n process. The process transfer function can be described as the following block form:
−1
= −[G−1(0)L(0)]i
[G−1(0)]ii li(0)
Note that, if λi < 1, the equivalent disturbance gain with the interactions is lower than that without interactions. In this case, the decentralized control is better than FDC for this loop. Otherwise, if λi < 1, it suggested that this loop needs to be decoupled to provide a good disturbance rejection. Both RDG and RLG reflect the effect of the disturbance on the outputs with and without interactions. In fact, the two indices only reflect the disturbance effect at steady state and fail to measure the dynamical information on the disturbance effect. Motivated by this limitation, some dynamic indices13 were proposed. Even so, this problem is still not solved perfectly, since the full dynamics information on the disturbance effect is controller-dependent. In this study, we still use these two indices to select the structure of the partial decoupler and focus on the design of the partial decoupler and the controller.
Figure 3. Equivalent SISO loop for the ith loop with ideal decoupling control.
0=
[G−1(0)L(0)]i
(7)
The value of RDG indicates whether the interaction is favorable or not, in terms of disturbance rejection. That is, if βi < 1, the interactions suppress the disturbance rejection and this loop does not need to be decoupled; if βi < 1, the interactions amplify the disturbance rejection and this loop needs to be decoupled.
⎡G −1 + G −1G X−1G G −1 − G −1G X−1⎤ 11 11 12 21 11 11 12 ⎥ G−1 = ⎢ −1 −1 −1 ⎢ ⎥ X G G X − ⎣ ⎦ 21 11 15934
(12)
dx.doi.org/10.1021/ie502785c | Ind. Eng. Chem. Res. 2014, 53, 15932−15945
Industrial & Engineering Chemistry Research
Article
where X = G22 − G21G11−1G12. Therefore, the equivalent relationship of the loops not needing to be decoupled can be expressed as follows, based on eq 4: yi =
1 [X−1](i − m)(i − m)
ui +
{[−X−1G21G11−1 X−1]L}i − m [X−1](i − m)(i − m)
(i = m + 1, m + 2 , ..., n)
D (13)
In the case of FDC, the inverse of the decoupled matrix G̅ is given by G̅
−1
⎡G̅ −1 0 ⎤ 11 ⎥ =⎢ ⎢⎣ 0 G̅ 22−1⎥⎦
(14) Figure 4. Partial decoupling control (PDC) system.
where G̅ 11 = diag{g11 , g22, ..., gmm} and G̅ 22 = diag{g(m+1)(m+1), g(m+2)(m+2), ..., gnn}. For the decoupled loops, the outputs are not affected by the interactions and are described as yi =
1 ui + liD [G̅ 11−1]i
g12 ⎤ ⎡ ⎢1 − ⎥ g11 ⎥ P=⎢ ⎢⎣ 0 1 ⎥⎦
(i = 1, 2 , ..., m) (15)
Note that the proposed partial decoupler is frequencydependent and directly related to the inverse of G11. However, if two or more loops need to be decoupled, the inverse of G11 is highly complex. In addition, because of the presence of the time delays, the inverse of G11 may be unfeasible in some cases. Motivated by the two facts, here, we briefly introduce how to obtain the feasible P. The partial decoupler at a given frequency ωk can be expressed as follows:
Note that eq 4 is a general form, which is also suitable for PDC. That is, in the case of PDC, the closed-loop performance also is determined by the inverse of the decoupled transfer function matrix T. Hence, to achieve the goal of PDC is to select an appropriate matrix T such that eqs 15 and 13 hold. In this regard, the inverse of T is set as follows here: ⎡ G̅ −1 0 ⎤ 11 ⎥ T −1 = ⎢ ⎢⎣−X−1G G−1 X−1⎥⎦ 21 11
⎡G−1(jω )G̅ (jω ) −G−1(jω )G (jω )⎤ k 11 k 11 k 12 k ⎥ P(jωk) = ⎢ 11 ⎣ ⎦ 0 I
⎡ ⎤ G̅ 11−1 0 ⎥ =⎢ ⎢⎣−(G − G G−1G )−1G G−1 (G − G G−1G )−1⎥⎦ 22 21 11 12 21 11 22 21 11 12
(k = 0, 1, 2 ,..., N )
(16)
where the first m loops of T−1 are the same as the loops of G̅ −1 and the remaining n−m loops are the same as those of G̅ −1. Moreover, T is given by ⎡ G̅ 11 ⎤ 0 ⎥ T = (T−1)−1 = ⎢ −1 −1 ⎢⎣G21G11 G̅ 11 (G22 − G21G11 G12)⎥⎦
(19)
(20)
where j = −1 and N is large enough to produce sufficient points to describe the partial decoupler. Specially, if ωk = 0, P(0) denotes the partial decoupler at steady state, namely, the static partial decoupler, which is given by 2
⎡G−1(0)G̅ (0) −G−1(0)G (0)⎤ 11 11 12 ⎥ P(0) = ⎢ 11 ⎣ ⎦ 0 I
(17)
Then, the proposed partial decoupler is determined as follows:
(21)
Based on eq 20, one element of the partial decoupler, denoted by pij, is described by the following Nyquist set:
⎤ ⎡ G11 G12 ⎤ G̅ 11 0 ⎥ ⎥ ⎢ P = G−1T =⎢ − − 1 1 ⎣G21 G22 ⎦ ⎢⎣G21G11 G̅ 11 (G22 − G21G11 G12)⎥⎦ −1⎡
{pij (jω0), pij (jω1), ···, pij (jωN )}
⎡G−1G̅ − G−1G ⎤ 11 12 ⎥ =⎢ 11 11 ⎣ 0 ⎦ I
= {[P(jω0)]ij , [P(jω1)]ij , ···, [P(jωN )]ij }
(22)
Therefore, the element of the partial decoupler can be fitted into a desired transfer function form by the complex curve fitting technique. The detail about complex curve fitting can be found in ref 20 and easily implemented by the Matlab function invfreqs. This method can approximate the decoupler by solving the n2 curve-fitting problem, instead of computing the exact partial decoupler first. In this regard, this method is suitable for high dimensional systems. In addition, because of the time delays, the inverse of G11 may contain advance terms, and in this case, the right half poles may exist when the decoupler is reduced into low-order form,
(18)
It can be seen that the proposed partial decoupler is a general decoupler. If G11 = G, the partial decoupling control results in FDC. To understand the difference between the partial decoupling control and the dull decoupling, we plot the block diagram of this control schemes for MIMO systems in Figure 4. Some reports reported the one-way decoupling for TITO processes. In the case of TITO processes, the partial decoupler is given as follows, based on eq 18: 15935
dx.doi.org/10.1021/ie502785c | Ind. Eng. Chem. Res. 2014, 53, 15932−15945
Industrial & Engineering Chemistry Research
Article
⎧ ⎡ y ⎤ ⎡ g G ⎤⎡ u ⎤ ⎡ l 0 ⎤ i i* i ⎪ ⎢ i ⎥ = ⎢ ii ⎥⎢ ⎥D ⎥+⎢ ⎪ U ⎢ Y ⎢ ⎥ ⎣ ⎦ ⎣ 0 L−i ⎥⎦ ⎪ ⎣ −i ⎦ ⎣G*i G−ii ⎦ −i ⎨ ⎪ ⎡ ui ⎤ ⎡ ci 0 ⎤⎡ ri − yi ⎤ ⎪⎢ ⎥⎢ ⎥ ⎥=⎢ ⎪ ⎣ U − i ⎦ ⎣ 0 C − i ⎦⎣ R − i − Y − i ⎦ ⎩
leading to an unfeasible partial decoupler. The partial decoupler is modified as follows to overcome this limitation: ⎡G−1G̅ ′ −G−1G′ ⎤ 11 11 11 12 ⎥ P′ = PZ = ⎢ ⎢⎣ 0 Z 2 ⎥⎦
(23)
)where G̅ 11 ′ = G̅ 11Z1 and G̅ 12 ′ = G12Z2. Z1 and Z2 are defined as follows: Z1 = diag{z11, z12 , ..., z1m},
where Gi* denotes the ith row of G with gii removed, G*i denotes the ith column of G with gii removed, and L−i denotes the L with li removed. Another form of the equivalent model of the ith loop can be derived as follows, based on eq 29:
Z 2 = diag{z 21, z 22 , ···, z 2(n − m)} (24)
where z1a =
1 e−L1as T1as + 1
and
z 2b =
1 e−L2bs T2bs + 1
(a = 1, 2, ..., n; b = 1, 2, ..., n − m)
gii′ =
(26)
(27)
gii′ = gii = [G̅ 1]ii
(i = 1, 2 , ..., m)
(32)
Equations 31 and 32 describe the equivalent models of the nondecoupled loops and decoupled loops, respectively. If G12 and G̅ 11 are modified as G12 ′ and G̅ 11 ′ , respectively, to provide a feasible partial decoupler, the corresponding equivalent models are computed based on G′ and G̅ ′11. So far, we obtain the controlled models to design the controllers. Based on these models, the equivalent SISO feedback control system for each loop can be redrawn as Figure 5, where
4. CONTROLLER DESIGN The controller design for PDC includes the design of the controllers for the decoupled loops and nondecoupled loops. The controller design for the decoupled loops is easier than that for the nondecoupled loops, since the interactions are removed for decoupled loops and the well-established design methods for SISO loops can be directly employed to design these controllers. However, the interactions among nondecoupled loops make design of the corresponding controller difficult. Our goal in this study is to realize the independent design for all of the loops. Motivated by this goal, we first consider the equivalent models, which are the controlled models, of the nondecoupled loops. Based on eq 4, the equivalent model of the ith loop under the assumption of “perfect control” is given by 1 [G−1]ii
= gii − Gi *G−−ii1[G−iiC−i(I + G−iiC−i)−1]G i *
gii′ ≈ gii − Gi *G−ii−1G i (31) * It can be proved that eqs 28 and 31 are equivalent. To achieve the goal of independent design of the controllers, we use eq 31 as the controlled model of the ith loop with the interactions. Although eq 31 describes the equivalent model for decentralized control, it can be used to design the controllers for the nondecoupled loops in PDC. The reason for this is because the proposed PDC does not change the interactions of the original loops after decoupling. This property is an advantage of the proposed partial decoupling scheme. The controlled models of the decoupled loops are easily determined as
Note that the gains of the elements of the compensator are unity, which means that values of RLG or RDG of G and G′ are the same. That is, the effects of the interactions on the disturbance rejection are unchanged if RLG or RDG is used to evaluate these effects.
gii′ ≈
∂ui
Equation 30 gives the exact form of the equivalent model of the ith loop with the interactions. It can be found that the equivalent model is dependent on the controllers of other loops, because of the interactions. This fact results that the decentralized controllers are not be designed independently. However, in the case of “perfect control”, namely, G−iiC−i(I + G−iiC−i)−1 = I, eq 30 is approximated as
That is, the loops that need to be decoupled are decoupled as G̅ ′11. Moreover, the interactions of the nondecoupled loops are the same as the loops of the following modified process:
⎡ G11 G′12 ⎤ ⎥ G′ = ⎢ ⎢⎣G21 G′22 ⎥⎦
∂yi
(30)
(25)
The reason for unfeasible elements of the decoupler is that the elements have advance phases. The function of the compensator is to make the advance phases lag behind. The detail about selecting Z1 and Z2 can be found in our previous work.5 After compensating, the decoupled matrix becomes ⎡ G̅ ′11 ⎤ 0 ⎥ T′ = GP′⎢ −1 −1 ⎢⎣G21G11 G̅ ′11 G′22 − G21G11 G̅ ′12 ⎥⎦
(29)
⎛1⎞ ci = kP , i + kI , i⎜ ⎟ + kD , is ⎝s⎠
(28)
On the other hand, the output of the ith loop can also be described as follows:
Figure 5. Equivalent SISO control system for the ith loop with partial decoupling. 15936
dx.doi.org/10.1021/ie502785c | Ind. Eng. Chem. Res. 2014, 53, 15932−15945
Industrial & Engineering Chemistry Research
Article
and g′ii and l′i are expressed as ⎧ g ′ = g = [G̅ 1]ii (i = 1, 2 , ..., m) ⎪ ii ii ⎨ ⎪ g ′ ≈ g − Gi *G−−1iiG i (i = m + 1 , ..., n) ⎩ ii ii *
The advantages of using the MIGO method here are as follows: (i) The objective function is simple and only related to the integral terms. (ii) The design method is not dependent on the exact controlled model; it is only dependent on the frequency response of the equivalent model, which is suitable for the nondecoupled loops, where the exact model cannot be easily obtained. Consider the ith equivalent loops, and the frequency response of the equivalent model of this loops is defined as
(33)
and ⎧ li (i = 1, 2 , ..., m) ⎪ ⎪ li′ = ⎨ gii ⎪ ⎪[G−1L] (i = m + 1 , ..., n) ⎩ i
(34)
Γi ≜ {gii′(jωz) = α(ωz) + jβ(ωz)|z = 1, 2 , ..., M }
Our primary concern for controller design in this study is the disturbance rejection. Assume that the disturbance D is a unitstep signal, and the transfer function from disturbance to the output at low frequency can be approximated as gii′li′ 1 + cigii′
≈
1 kI , i
where M is large enough to capture the main dynamic of the equivalent model g′ii. Based on the frequency response (eq 39) and the definition of Ms, the sensitivity function of the ith equivalent loop is expressed by the following equation: ⎡ ⎛ kI , i ⎞⎤ 1 + (α + jβ)⎢kP , i − j⎜ − kD , iωz⎟⎥ ⎢⎣ ⎝ ωz ⎠⎥⎦
(35)
The integrated error (IE) of this control system for disturbance response is given by |IE| =
∫0
∞
e(t ) dt =
li′(0) kI , i
z = 1, 2, ..., M
⎧ ⎫ ′ (jωz) ci(jωz)giiii ⎪ ⎪ ⎬ M t = max⎨ ⎪ ⎪ ′ (jωz) ⎭ ⎩ 1 + ci(jωz)giiii
z = 1, 2, ..., M
1 ≥0 γ2 (40)
(α 2 + β 2)kP2, i + 2αkP , i + (α 2 + β 2)A2 − 2βA +
γ2 − 1 ≥0 γ2
(z = 1, 2 , ..., M )
(41)
Obviously, eq 41 is a quadratic equation, indicating that the range of kP,i for a given ωz with fixed kI,i and kD,i, denoted by ϕz, can be easily determined by the two roots of this quadratic equation. In this case, for a given kI,i and kD,i, the available kP,i is the intersection of the following sets: kP , i ∈ ΦS = ϕ1 ∩ ϕ2 ∩ ... ∩ ϕM
(42)
Similarly, the complementary sensitivity function is expressed as the following equation based on the definition given by eq 37: (α 2 + β 2)kP2, i + 2α − 2β
η2 kP , i + (α 2 + β 2)A2 η2 − 1
η2 η2 A+ 2 ≥0 η −1 η −1 2
(z = 1, 2 , ..., M ) (43)
(37)
Equation 43 is also a quadratic inequality. Hence, for a given ωz, and with fixed kI,i and kD,i, the range of kP,i, denoted by ψz, can be easily obtained by solving this quadratic inequality. Once kD,i is fixed, for a given integral gain kP,i, the available kP,i is the intersection of ψz(z = 1, 2, ..., M):
where M is a sufficiently large constant such that {g′iiii(jω1), giiii ′ (jω2), ···, giiii ′ (jωM)} can capture the main dynamic character of gii′. The typical values of Ms and Mt are 1.2−2.0 and 1.0−1.5, respectively.21 Hence, the optimization problem can be established as follows to design the PID parameters for the ith loop:
kP , i ∈ ΨT = ψ1 ∩ ψ2 ∩ ... ∩ ψM
(44)
Another constraint of the proposed optimization problem is to ensure that the system is stable. The stability region of the parameters provides an available region to search the optimal integral term satisfying robust specifications. However, to compute the stability region of the controllers for the MIMO systems is very complicated. For simplicity, here, we compute the stability region for gii instead of the region for g′ii. Tan proposed a simple way to compute the stability region of PI/PID controller for classical process with time delays.23
(38)
such that
(1.2 ≤ γ ≤ 2.0) (1.0 ≤ η ≤ 1.5)
−
Let A = (−(kI,i/ωz) + kD,iωz), and eq 40 is rewritten as follows: (36)
⎧ ⎫ ⎪ ⎪ 1 ⎬ Ms = max⎨ ⎪ ⎪ ⎩ 1 + ci(jωz)gii′(jωz) ⎭
⎧ stability ⎪ ⎪ ⎨ Ms ≤ γ ⎪ ⎪M ≤ η ⎩ t
2
(z = 1, 2 , ..., M )
where e is the control error. It can be found that the criterion IE is close to the widely used performance specification integrated absolute error (IAE) if the disturbance response is welldamped. In this regard, IE can be used to evaluate the output performance, and, in this case, the controller design is converted to maximize the integral gain kI,i. The method of tuning the PID parameters for disturbance rejection by maximizing integral gain is called MIGO,21,22 as proposed by Aström. Another issue that must be considered in the optimization procedures is the robustness, which is captured by maximum sensitivity (Ms) and complementary sensitivity (Mt) functions:
⎛ ⎞ 1 ⎟ min⎜⎜ ⎟ ⎝ kI , i ⎠
(39)
(38) 15937
dx.doi.org/10.1021/ie502785c | Ind. Eng. Chem. Res. 2014, 53, 15932−15945
Industrial & Engineering Chemistry Research
Article
According to Tan’s method, the stability region with a fixed kD,i for a given model gii can be determined by the following loci on the (kP,i,kI,i) plane: ⎧ kP , i = 0 ⎪ ⎪ α′ ⎪ kP , i = − 2 ω>0 ⎨ α′ + β′ 2 ⎪ β′ω ⎪ 2 ω>0 ⎪ kI , i = kD , iω − α′ 2 + β′ 2 ⎩
the proposed method is suitable for the controller design here, since the exact low-order model cannot be easily obtained for the nondecoupled loops.
5. SIMULATION STUDIES Two examples here are given to demonstrate the effectiveness of the proposed partial decoupling scheme. The output response is evaluated by the integrated absolute error (IAE), which is defined as (45)
IAE =
where gii(ω) = α′(ω) + jβ′(ω). Based on the analyses above, we propose the following algorithm to search the optimal integral term. The basic idea of the proposed optimization method is to maximize the integral term by computing the range of kP,i with a given kD,i. This optimization method, which is used in our previous paper,24 is described in Chart 1.
∫0
∞
|e(t )| dt
(46)
In addition, the robustness stability is another important issue that must be considered in control system design, since the models cannot capture the entire dynamics information on the processes. Two common uncertainties are considered here: the output uncertainty and the input uncertainty, which are described by G(s)[I + Δ0(s)] and G(s)[I + ΔI(s)], respectively. If these two uncertainties occur, the control system is stable if the following conditions hold:15
Chart 1. Proposed Optimization Procedures of Designing PID Controllers
|| ΔI (jω)||
