Article pubs.acs.org/IECR
Decoupling Proportional−Integral−Derivative Controller Design for Multivariable Processes 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, PR China ABSTRACT: In this article, the decoupling proportional−integral−derivative (PID) controller is designed for multi-input multioutput (MIMO) processes with time delays. Three basic issues on the decoupling control are restudied here: the computation of the decoupler, the selection of the decoupled matrix, and the design of the controller. Considering the difficulty of computing the inverse-based dynamic decoupler, especially for nonsquare processes, here, we introduce a new concept of nyquist set to describe the dynamic of the decoupler. Then, based on the nyquist set, the decoupler is approximated as a low order transfer function matrix though the complex-curve fitting technique. In addition, some rules are proposed to reselect the decoupled matrix if it cannot produce a feasible decoupler. With the proposed decoupler, a model reduction technique and a new robust internal model control (IMC) principle are employed to design the decentralized PI controller. The decoupler and the decentralized PI controller are implemented by a centralized PID controller, which is the desired decoupling controller. Simulation studies demonstrate the effectiveness and merits of the presented methodology.
1. INTRODUCTION Most industrial processes are multivariable in nature. Because of the interactions that exist among the input and output variables, controller design for multi-input multi-output (MIMO) systems is very difficult. When the interactions are modest, we always take the MIMO systems as single-input− single-output (SISO) systems and use the well-established proportional−integral−derivative (PID) tuning rules for SISO systems, such as SIMC1 and AMIGO2 to control them. However, in order to provide a high product quality, the interactions must be considered, especially for the processes with strong interactions. Unfortunately, good controller design methods for SISO processes can hardly be extended to MIMO processes directly. Over the last few decades, many works have been done to extend the classical controller design methods to MIMO systems. A MIMO process can be controlled by a simple decentralized controller, where the MIMO system is divided into several SISO loops.3,4 Because of the interactions, these loops cannot be tuned independently. Many methods have been developed to deal with the interactions among loops, including the detuning method,5 the optimization method,6,7 and the relay autotuning method.8 The biggest advantage of these control strategies is that they can be easily implemented and understood. Nevertheless, the drawback is that the multiloop controllers may not provide good performance if the interactions are strong.9,10 The decoupling control is one of the effective control strategies to deal with the interactions. The typical decoupling methods are the static,11,12 ideal,13,14 and simplified10,15−17 decoupling. The decoupler is obtained according to the equation D(s) = G−1(s)T(s), where D(s), G(s), and T(s) are the decoupler, the process transfer function matrix, and the desired decoupled matrix. The static decoupling deal with the interactions at steady-state, and the decoupler is obtained as follows: D(s) = G−1(0)T(0). The ideal decoupling and the © 2013 American Chemical Society
simplified decoupling are the dynamical decoupling. The difference between them is that the decoupled matrix for the ideal decoupling is selected as the diagonal elements of G(s), whereas the decoupled matrix for the simplified decoupling is designed as a special form to set n elements of the decoupler as unity. The aim of the simplified decoupling is to produce a simple decoupler, but the computation of the decoupled matrix, the elements of which are equivalent to the effective open-loop transfer functions (EOTFs),18 is complex. Some papers have reported another decoupling scheme called as inverted decoupling.15,19,20 In fact, the inverted decoupling is another implementation of the ideal decoupling, where the resulting decoupled matrix is equivalent to that of the ideal decoupling. Two problems occur in the decoupling control. One is the computation of the inverse-based decoupler; the other is the selection of the decoupled matrix. The complexity of the dynamical inverse or pseudoinverse of a transfer function matrix with time delays brings about a big challenge for the implementation of the inverse-based decoupler. Some papers have addressed this issue. For the normalized relative gain array (NRGA)-based decoupling method termed as normalized decoupling control,21,22 the inverse of the process model matrix has been approximated as the transposition of the open loop effective transfer function matrix under the assumption of the “perfect control”. Nevertheless, because of the assumption, much dynamic of G−1(s) has been neglected and the resulting decoupling performance is poor (see section 4). In addition, the existing schemes of selecting the decoupled matrix may cause some unfeasible elements of the decoupler due to the existences of the advance terms in the decoupler. Received: Revised: Accepted: Published: 765
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Figure 1. Decoupling control system for n × n processes.
For two-input-two-output (TITO) processes, Nordfeldt et al.23 has focused on the selection of the decoupled matrix to produce a feasible decoupler. However, the proposed is based on the analyses of the adjoint matrix of the process transfer function matrix. As a result, it is suitable for TITO processes and difficult to be applied to high dimensional systems or nonsquare systems. Zhang and Gao proposed a nonzero-pole-cancellation decoupling24 to provide always feasible decoupler. For the nonzero-pole-cancelation, the decoupler, which is not inversebased, is selected as the adjoint of process transfer function matrix. The limitation of this method is that the decoupled matrix is fixed and conservative, which leads to a slow closedloop response (see section 4). With the decoupler, a decentralized controller matrix can be designed using the well-established PID tuning rules for SISO processes. Another decoupling control is to use a full matrix controller called centralized controller. In fact, the centralized controller can be treated as the combination of the decoupler and the decentralized controller. Wang et al.25 developed a centralized controller for TITO processes based on the IMC principle. Liu et al.26,27 proposed an analytical decoupling control strategy and developed a centralized controller for MIMO systems. With the centralized controller, the resulting control system has a simple structure. However, the resulting controller is high order and cannot be implemented by the dominated PID form. For example, Liu’s controller is implemented by a unity feedback structure.27 This work attempts to solve the problems mentioned above. First, in order to avoid computing the inverse or the pseudoinverse of the transfer function matrix, the dynamic decoupler at low frequencies is described by a defined nyquist set, and a model fitting technique is employed to approximate the decoupler. In the proposed procedures, the nyquist set is computed instead of computing the inverse or the pseudoinverse of the transfer function matrix. On the basis of this method, the dynamic decoupling control is easily implemented for the high dimensional square systems and for the nonsquare systems. Second, taking into account the feasibility of the decoupler if some elements of the decoupler are unfeasible, we propose four rules to reselect a decoupled matrix to produce a simple but feasible decoupler on the base of the analyses of the effect of the decoupled matrix on the decoupler. The issue on the selection of the decoupled matrix is the one of the contributions of this work. On the basis of the presented decoupler, a decentralized PI controller is designed independently in line with the proposed robust IMC (internal model control) principle. Moreover, the proposed decoupler and the decentralized controller are implemented by a centralized PID controller, namely the desired decoupling PID controller. The examples
show the design procedures and demonstrate the effectiveness of the proposed method not only for the square systems but also for the nonsquare systems. This paper is organized as follows. Section 2 gives the methodology including the concept of the decoupling control, the proposed method of computing the decoupler, and the selection of the decoupled matrix. In section 3, the performance index and robust stability for MIMO systems are introduced. Simulation results are discussed in section 4. Finally, section 5 summarizes this work and draws the final conclusions.
2. METHODOLOGY 2.1. General Decoupling Control. In this section, the general decoupling control for MIMO processes is detailed. For simplicity, we introduce the basic control scheme for square processes. The scheme can be extended to nonsquare processes. Consider the general transfer function matrix for a n × n process with time delays expressed as follows: ⎡ g11 g12 ⎢ ⎢ g21 g22 G (s ) = ⎢ ⎢⋮ ⋮ ⎢⎣ gn1 gn2
... g1n ⎤ ⎥ ... g2n ⎥ ⋱ ⋮ ⎥⎥ ... gnn ⎥⎦
(1)
where gij = gi̅ j(s)e−θij (i, j = 1, 2, ..., n), gi̅ j(s) denotes the physically proper, stable, and delay-free transfer function and θij denotes the time delay. For a MIMO process with time delays expressed as eq 1, the typical decoupling control is depicted by Figure 1, where D(s) is the decoupler and C(s) is the decentralized controller. In order to remove the interactions among loops, the decoupler D(s) is designed such that the decoupled matrix T(s) is a diagonal matrix, namely, T (s) = G(s)D(s)
(2)
Then, the decoupler is given by
D(s) = G−1(s)T (s)
(3)
−1
where G (s) is the inverse of the process transfer function matrix. If the decoupling control scheme is extended to the nonsquare processes, G−1(s) denotes the pseudoinverse of the processes transfer function matrix. The interactions among loops are removed by the decoupler from eq 2. Therefore, the decentralized controller C(s) can be designed independently using the well-established controller design methods for SISO processes. In fact, the decentralized controller and the decoupler can be implemented by a centralized controller denoted by C̅ (s) depicted in Figure 1, where C̅ (s) = D(s)C(s). However, because of the complexity of the 766
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where j2 = −1, Dk(jωk) = G−1(jωk)T(jωk). The constant s is large enough to produce sufficient points to capture the main property of the decoupler at low frequencies. In addition, to obtain a low order decoupler element, the selection of ωs must guarantee that the normalized frequency responses of the elements are in quadrant 1 or 4 on the complex plane (see section 2.3). Dk(jωk) represents the required decoupler at a given frequency ωk. The dynamic of each element of the decoupler is reflected by the following nyquist points:
decoupler, the centralized controller always is not the conventional PID form in many decoupling control methods. The goal of this study is to develop a centralized PID controller for MIMO processes. Hence, the elements of the decoupler and the decentralized controller have the following form: dij = kijdij̅ = kij
Aij s + 1 Bij s + 1
,
i , j = 1, 2, ..., n , Bij > 0 (4)
⎛ 1 ⎞ cii(s) = kcii⎜1 + ⎟, Tciis ⎠ ⎝
{dij(jω0), dij(jω1), ..., dij(jωk), ..., dij(jωs)}
i = 1, 2, ... n (5)
≈ {[D0(0)]ij , [D1(jω1)]ij , ..., [Ds(jωs)]ij },
where d̅ij is the normalized transfer function. On the basis of eqs 4 and 5, the elements of the centralized controller are described as follows: ⎛ 1 ⎞⎟⎛⎜ Aij s + 1 ⎞⎟ cij̅ = kijkcjj⎜⎜1 + , Tjjs ⎟⎠⎜⎝ Bij s + 1 ⎟⎠ ⎝
i , j = 1, 2, ... n
These points can be fitted into the model 4 though a complexcurve fitting technique.28,29 Then, the corresponding element of the decoupler is obtained. On the basis of the Levi’s complexcurve fitting method,28 this problem can be expressed as the following optimization problem:
i , j = 1, 2, ..., n (6)
s
Equation 6 can be represented as a standard PID form described as KC(1 + 1/(TIs) + TDs)(1/(TFs + 1)), where ⎧ Tjj + Bij ⎪ K C = kijkcjj Tjj ⎪ ⎪ ⎪TI = Tjj + Bij ⎨ TjjBij ⎪ ⎪TD = Tjj + Bij ⎪ ⎪ ⎩TF = Aij
min
A ij , Bij
∑ Wt(ωk)|[Dk(jωk)]ij (Bij(jωk) + 1) k=0
− kij(Aij (jωk) + 1)|2
(11)
where Wt(ωk) is the weighting function to minimize the relative error criterion. The problem is solved by the iterative leastsquares method, and the weighting function is selected as Wt(ωk) = 1/[Dk(jωk)]ij(B0ijjωk) + 1),29 where B0ij is the optimal constant of the previous iteration. Then, the decoupler is obtained by solving n2 optimization problems. It can be seen from the procedures above the computation of the inverse or pseudoinverse of the process transfer function is omitted in the procedures. The decoupler is described by the defined nyquist set. Without doubts, obtaining the nyquist set is easier than obtaining the inverse or pseudoinverse of the transfer function for high dimensional square processes and for square processes. In this regard, the advantage of this method is manifested in the application of the decoupling control for the high dimensional systems and the nonsquare systems. 2.3. Design of the Decoupled Matrix. The decoupled matrix not only has a direct influence on the closed-loop performance but also determines the decoupler. Hence, the selection of the decoupled matrix plays a significant role in the decoupling control. As stated in the Introduction, there are two common methods to select the decoupled matrix. The typical decoupled matrix is selected as follows:
(7)
2.2. Computation of the Decoupler. The decoupler can be determined by eq 3 if the decoupled matrix T(s) is specified. The key procedure is to compute the inverse or the pseudoinverse of the process transfer function matrix. In the conventional methods of computing the decoupler, the inverse or the pseudoinverse is computed first, and a model reduction technique is utilized to approximate the decoupler. Because of the complexity of the inverse or the pseudoinverse of a transfer function matrix, the computation procedures are tedious and not suitable for high dimensional square processes or nonsquare processes. In this study, a new nyquist set is introduced to describe the decoupler. According to eq 3, the decoupler at a given frequency can be expressed as follows: D(jω) = G−1(jω)T (jω)
(10)
(8)
T (s) = diag{g11, g22 , ..., gnn}
For the static decoupling, the decoupler is selected as D(s) = G−1(0)T(0), in which the resulting decoupled matrix is diagonal at steady-state. An alternative approximate decoupling is that the decoupled matrix is as diagonal as possible at a critical frequency ω0, where the decoupler is designed as D(s) = G−1(jω0)T(jω0). As stated by Skogested,11 the bandwidth frequency is a good selection for ω0 in that the effect on performance of reducing interaction is normally greatest at this frequency. In fact, we hope that the interactions can be removed at as many as possible frequencies. Define the nyquist set of the decoupler as follows:
(12)
This technique is called ideal decoupling introduced by Luyben.13 The advantage of the ideal decoupling is that the decoupled matrix is easily achieved. The other technique is called simplified decoupling, where the elements of the decoupled matrix is selected as the equivalent open loop transfer function. In the decoupling control, the elements of the decoupler always are reduced into low order models. For the existing decoupled matrix, however, the reduced decoupler may be unfeasible because of the advance terms. To overcome this limitation, the corresponding elements of the decoupler are set to unity in the existing methods.4,16 Although this method is simple, the resulting decoupling performance is degraded. Few papers have reported how to obtain a feasible decoupler by
{D0(0), D1(jω1), ..., Dk (jωk), ..., Ds(jωs)}, k ∈ {0, 1, ..., s} (9) 767
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• For area III, (Bij > 0, Aij < Bij) • For area IV, (Aij < Bij < 0) It is concluded that the resulting models for the areas II and III are stable. Nevertheless, the models for the areas I and IV are unstable because there is one pole in the right half plane for these models. That is, the nyquist points in the areas I and IV cannot be fitted into a feasible first order model. In this case, the only way to give a stable decoupler is to select a new decoupled matrix to drag the nyuist points in areas I and IV into area II or III. The decoupled matrix is modified as follows:
selecting a decoupled matrix. In this study, we give a new approach to design the decoupled matrix to achieve this goal. The decoupled matrix is first selected as eq 12 for simplicity. In this case, the decoupler is described as the nyquist set 9. For one element of the decoupler, dij, its normalized frequency response can be described as follows: {dij̅ (jω0), dij̅ (jω1), ..., dij̅ (jωk), ..., dij̅ (jωs)} ⎧ [Ds(jωs)]ij ⎫ [Dk (jωk)]ij [D1(jω1)]ij ⎪ ⎪ ⎬, , ..., , ..., ≈⎨ 1, ⎪ [D0(0)]ij ⎪ [D0(0)]ij [D0(0)]ij ⎩ ⎭ i , j = 1, 2, ..., n
T ′(s) = T (s)Q (s) = diag{g11q11 , g22q22 , ..., gnnqnn}
(13)
(15)
where Q(s) = diag{q11, q22, ..., qm} is defined as a compensator matrix. Then, the elements of the new decoupler are changed into
where d̅ij(jωk) = [Dk(jωk)]ij/[D0(0)]ij = Reij(ωk) +jImij(ωk). Note that each element of the set in eq 13 represents one point in the complex plane and the first element presents the critical point (1,0). The plots of these nyquist points are depicted in Figure 2. As shown in Figure 2, the nyquist curves start from
dij′ = dijqjj
(16)
In this study, the elements of the compensator matrix are elected as a first-order plus time delay model for simplicity: qii =
1 e−Liis , Tiis + 1
i = 1, 2, ..., n , Tii ≥ 0, Lii ≥ 0 (17)
Then, the normalized frequency response of the element of the new decoupler at a given frequency ωk, is expressed as follows: ⎡ ⎢ 1 dij̅ ′(jωk) = [Reij(ωk) + jImij(ωk)]⎢ e−Ljjs + T s 1 ⎢⎣ jj
⎤ ⎥ ⎥ ⎦ s = jωk ⎥
⎛ 1 = [Reij(ωk ) + jImij(ωk )]⎜⎜ 2 2 + 1 T ⎝ jj ω −j
⎞ ⎟[cos(Ljjω) − j sin(Ljjω)] 1 + Tjj 2ω 2 ⎟⎠ Tjjω
= Re′ij(ωk) + jIm′ij(ωk)
Figure 2. Four trends of the normalized nyquist points in eq 13.
(18)
where the critical point (1,0), and four trends occur along with the increase of the frequency. Our goal is to provide a low order decoupler at low frequencies. Hence, ωs must guarantee that the normalized frequency response in quadrant 1 or 4. The right-half complex plane can be divided into the following four areas corresponding to the four trends. • Area I: (0 < Reij(ω) < 1, Imij(ω) > 0) (0 ≤ ω ≤ ωs) • Area II: (Reij(ω) > 1, Imij(ω) > 0) (0 ≤ ω ≤ ωs) • Area III: (0 < Reij(ω) < 1, Imij(ω) < 0) (0 ≤ ω ≤ ωs) • Area IV: (Reij(ω) > 1, Imij(ω) < 0) (0 ≤ ω ≤ ωs) In this study, the nyquist points need to be fitted into a first order transfer function given by eq 4. The corresponding normalized first order transfer function for the normalized nyqusit points is expressed as follows: Aij s + 1 dij̅ = Bij s + 1 (14)
Re′ij(ωk) = {Reij(ωk)[cos(Ljjωk) − (Tjjωk)sin(Ljjωk)] + Imij(ωk)[sin(Ljjωk ) + (Tjjωk )cos(Ljjωk)]} /{1 + Tjj 2ωk 2} Im′ij(ωk) = {− Reij(ωk)[sin(Ljjωk ) + (Tjjωk )cos(Ljjωk)] + Imij(ωk)[cos(Ljjωk) − (Tjjωk)sin(Ljjωk)]} /{1 + Tjj 2ωk 2} (19)
In line with the conclusions above, the normalized nyquist points should be dragged into the stability areas (areas I or IV) though compensation if the origin points are in the instability areas (areas II or III), while the normalized nyquist points should be still in the stability areas if the origin points are in the area II or III. Therefore, the compensation elements are required to comply with the following rules: • Rule 1: if (Reij(ωk) < 1, Imij(ωk) > 0), then (Re′ij(ωk) > 1, Im′ij(ωk) > 0) or (Re′ij(ωk) < 1, Im′ij(ωk) < 0. • Rule 2: (Reij(ωk) > 1, Imij(ωk) > 0), then (Reij′ (ωk) > 1, Imij′ (ωk) > 0) or (Reij′ (ωk) < 1, Imij′ (ωk) < 0.
The parameters of model 14 also can be divided into four types corresponding to the four areas above. • For area I, (Bij < 0), (Aij > Bij) • For area II, (Aij > Bij > 0) 768
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• Rule 3: if (Reij(ωk) < 1, Imij(ωk) < 0), then (Re′ij(ωk) < 1, Im′ij(ωk) < 0). • Rule 4: if (Reij(ωk) > 1, Imij(ωk) < 0), then (Re′ij(ωk) < 1, Im′ij(ωk) < 0) where k = 1,2, .., s. The ranges of the parameters of qjj are determined by the four compensation rules. To obtain the ranges of the parameters of qjj needs to solve two transcendental inequalities. Nevertheless, solving these two inequalities is not an easy work since the sine and the cosine functions exist. In order to simplify the problem, the following assumptions are introduced at low frequencies: Tjj = Ljj ,
sin(Ljjωk) ≈ Ljjωk ,
response of the reduced model. The procedures are formulated as an optimization problem, the goal function of which is given by ⎡ kgii min ∑ ⎢ ⎢ Tii′(jωk) + 1 Tii′ k=1 ⎣ r
kgii Tii′s + 1
e−Lii′s
(26)
η = [kgii 2 − |gii′(jω1)|2 kgii 2 − |gii′(jω2)|2 ... kgii 2 − |gii′(jωr )|2 ]T
(27)
Then, the time constant Tii′ is estimated as follows: (ΦTΦ)−1ΦTη
Tii′ =
(21)
(28)
The time delay, similarly, is estimated by minimizing the difference between the phase angle of the real model and that of reduced model: ⎡ ⎛ ⎤2 ⎞ kgii −Lii′(jωk ) ⎢ min ∑ ∠⎜ e ⎟ − ∠gii′(jωk)⎥ ′ ⎢ ⎥⎦ Lii′ T ( j ) 1 ω + ⎝ ⎠ ⎣ ii k k=1 r
(29)
The corresponding least-squares problem is expressed as follows:
ψLii′ = ς
(30)
where Ψ = [ω1ω2 ... ωr ]T ⎡ −tan−1(T′ω ) − ∠g ′(jω ) ⎤ ii 1 1 ⎥ ii ⎢ ⎢ ⎥ −tan−1(Tii′ω2) − ∠gii′(jω2)⎥ ⎢ ς= ⎢ ⎥ ⋮ ⎢ ⎥ ⎢ −tan−1(T′ω ) − ∠g ′(jω ) ⎥ ii r r ⎣ ⎦ ii
(22)
(31)
The least-squares estimation gives Lii′ = (ΨTΨ)−1ΨTς
(32)
This reduction method is widely used and can be found in other papers, such as in ref 30. The other effective reduction methods can also be employed here. In line with the IMC principle, the feedback controller for the FOPTD model 23 is obtained as follows:
(23)
First, the steady state gain kgii of the approximation model is the same as that of the real model:
kgii = gii(0)
(25)
Φ = [ω1 |gii′(jω1)|2 ω2 |gii′(jω2)|2 ... ωr |gii′(jωr )|2 ]T
In this study, the IMC-PI tuning rules are applied to design the controller for each SISO control system. In order to produce a PI controller, the controlled process needs to be modeled by a first-order plus time delay (FOPTD) model expressed as follows:
gii′ ≈
⎥⎦
where
cos(Ljjωk) ≈ 1
The compensator matrix Q(s) can make the decoupler feasible. Unfortunately, the compensator element from eq 15 may slow the response of each decoupled element and degrade the closed-loop performance. The parameters of compensator element are as small as possible in terms of the closed-loop performance. On the other hand, a large value of Tjj may produce “better” decoupler elements. That is, the elements of the decoupler can be more exactly modeled by a low order transfer function with a larger Tjj. Hence, the concrete values of the parameters are designed by considering the trade-off between a faster response and an easier implementation. It may be argued that this procedure is complex. In fact, once the range is determined, the value of Tjj can be easily obtained according to the guidelines above (see the following examples). In this study, we focus on how to obtain the reliable interval of Tjj. 2.4. Decentralized PI Controller Design. With a decoupler, a multivariable control system is derived into n SISO control systems, where the controller is cii and the controlled plant is giiqii denoted by 1 e−Liis Tiis + 1
− |gii′(jωk)|
Φ(Tii′)2 = η
The desired compensator qjj, which makes the corresponding element dij feasible, can be determined under the assumptions above. In fact, it can be seen from eq 16 that the function of qjj is to make all the elements of the jth column of D(s) feasible. Therefore, if the desired qjj for dij is denoted by Uij, the resulting qjj is the intersection of the n corresponding sets, denoted by
gii′ = gii
2⎥
The optimization problem 25 can be rewritten as a leastsquares problem:
(20)
Tjj ∈ U1j ∩ U2j ... ∩ Uij ... ∩ Unj
⎤2
2
cii =
(24)
(Tii′s + 1) 1 kgii (λiis + 1) − e−Lii′s
(33)
This controller is implemented by a PI controller with the first Taylor series expansion of the time delay term, and the controller parameters are given by
The time constant T′ii of the reduced model is determined by minimizing the difference between the amplitude of the frequency response of the real model and that of the frequency 769
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Robust Stability. Numerous methods develop the controller based on the precise model of the process. In fact, it is not possible to obtain the exact model in practice. Hence, the resulting control system should be robust to the uncertainty of the process model. That is, the control system should be stable if some uncertainties exist in the process model. Two common uncertainties are considered in this study. One is the input uncertainty which is described as G(s)[I + ΔI(s)], where I is an identity matrix and ΔI(s) is the input perturbation model. In this case, the closed-loop system is stable if the following equation holds:11,32
Tii′ kgii(λii + Lii′)
kcii =
Tcii = Tii′
(34)
where λii is the IMC filter constant which is directly related to the closed-loop performance and robustness for a SISO control system. A good criterion to measure the robustness of a SISO system is the maximum sensitivity (Ms), which is defined as follows: 1 1 + ciigii′
Ms =
(35)
∞
∥ΔI(jω)∥
