Decentralized Unconditional Stability Conditions Based on the

Ind. Eng. Chem. Res. 2002, 41, 1569-1578

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Decentralized Unconditional Stability Conditions Based on the Passivity Theorem for Multi-loop Control Systems Wen Z. Zhang,† Jie Bao,*,† and Peter L. Lee‡ School of Chemical Engineering & Industrial Chemistry, The University of New South Wales, UNSW, Sydney, New South Wales 2052, Australia, and School of Engineering, Murdoch University, GPO Box S1400, Murdoch, Western Australia 6001, Australia

On the basis of the Passivity Theorem, this paper provides a new approach to stability analysis for multi-loop control systems. The destabilizing effect of interactions of a multi-input multioutput system is studied using the passivity index. The decentralized unconditional stability condition, which implies closed-loop stability of decentralized control systems under control loop failure, is derived. An easy-to-use stability test is performed on open-loop stable process transfer matrices, independent of controller designs. 1. Introduction Multi-loop control is a control strategy that uses multiple single-input single-output (SISO) controllers to control a multivariable plant. Because of its hardware simplicity and relative effortlessness to achieve failuretolerant design,1 multi-loop control is the most widely used strategy in industrial process control. Current multi-loop control design approaches can be classified into three categories: detuning methods,2,3 independent design methods,1 and sequential loop-closing methods.4,5 In any of these design procedures, loop interactions have to be taken into consideration, as they may have deteriorating effects on both control performance and closed-loop stability. It is also very desirable if the multiloop control system is decentralized unconditionally stable, that is, any subset of the control loops can be detuned independently to an arbitrary degree (decentralized detunable) or even turned off without endangering closed-loop stability. This will ensure failuretolerant control and make on-line fine-tuning significantly easier. Independent design methods are often preferred as they are systematic control synthesis procedures and can often achieve failure tolerance.1 In these methods, each control loop is designed on the basis of the paired transfer function while satisfying some stability constraints due to process interactions. Perhaps the most widely used decentralized stability conditions are those based on the µ-interaction measure.1,6 Derived from µ-stability criteria, these conditions require the modulus (gains) of each closed-loop subsystem to be constrained by the µ-interaction measure of the given multivariable process. As the µ-interaction measure does not contain phase information of the process, the above conditions can be conservative.7 Although extensions of µ-stability criteria have been proposed to utilize, explicitly or inexplicitly, the phase information of multi-variable processes,8,9 none of these techniques have been adopted in decentralized stability analysis, probably because of the enormous computational complexity required when

applied to high-dimensional processes. To reduce the conservativeness of the µ-interaction measure, Lee and Edgar7 proposed a set of additional phase conditions for decentralized stability, which can be used together with the gain conditions. The explicit phase-gain conditions can be extended such that they can be used for decentralized unconditional stability analysis.10 However, a large amount of computation and manual analysis is required to verify the phase-gain conditions, as closedloop subsystems need to satisfy either the phase condition or the gain condition at each frequency. While it is possible to design controllers for small-scale processes to satisfy the phase-gain conditions by trial and error, it is difficult to develop a systematic control synthesis method on the basis of these conditions. In this paper, new, easy-to-use conditions for decentralized unconditional stability are proposed using the concept of passive systems11 and the Passivity Theorem.12 Passive systems are special dissipative systems that have certain input-output properties.11 The concept of dissipativeness/passivity has been playing an important role in stability analysis of interconnected subsystems.11 If each subsystem possesses a certain type of dissipativeness, then the stability of the composite system can be deduced according to the interconnection topology.13 Moylan and Hill14 derived a general inputoutput stability result for large-scale systems that can be broken down into a number of dissipative SISO subsystems. In this paper, the passivity property of subsystems (the process and the controller) is studied, from which the stability of the closed-loop system can be derived. The destabilizing effect of interactions is measured by the passivity index of the multi-variable process. The passivity index comprises both phase and gain information and is easy to compute. The proposed conditions can deal with systems with unlimited gain, such as controllers with integral action, and thus it can be used directly to analyze decentralized controllability in an open-loop fashion. 2. Passive Systems and the Passivity Theorem

* To whom correspondence should be addressed. Tel.: +61 (2) 9385-6755. Fax: +61 (2) 9385-5966. E-mail: j.bao@ unsw.edu.au. † The University of New South Wales. ‡ Murdoch University.

Definition 2.1. Linear Passive Systems.11 A linear time-invariant system, Σ: T(s): ) C(sI - A)-1B + D, (T(s) is a n × n transfer function matrix), is passive if and only if T(s) is Positive Real (PR), or equivalently,

10.1021/ie001037v CCC: $22.00 © 2002 American Chemical Society Published on Web 02/13/2002

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Ind. Eng. Chem. Res., Vol. 41, No. 6, 2002

generalized diagonal dominance is not necessary for closed-loop stability. For example, the following system

G(s) )

[

(s +1 1) -a a (s +2 2)

]

(2)

with a large value of a will not be diagonal-dominant. However, it is easy to verify that G(s) is strictly passive and could be stabilized by any passive controller. This indicates that a passivity-based study could provide a new avenue for analysis of the destabilizing effects of interactions. Encouraged by the above facts, the passivity-based decentralized unconditional stability conditions are proposed in the next section.

Figure 1. Passivity Theorem.

Figure 2. Decentralized control.

(1) T(s) is analytic in Re(s) > 0. (2) T(jω) + T*(jω) g 0 for all frequency ω that jω is not a pole of T(s). If there are poles p1, p2, ..., pm of T(s) on the imaginary axis, they are nonrepeated and the residue matrix at the poles limsfpi (s - pi)T(s) (i ) 1, ..., n) is Hermitian and positive semidefinite. System Σ is said to be strictly passive or Strictly Positive Real (SPR)15 if (1) T(s) is analytic in Re(s) g 0 (2) T(jω) + T*(jω) > 0 for ω ∈ (-∞, +∞) and is extended strictly passive or Extended Strictly Positive Real (ESPR) if it is strictly passive and T(j∞) + T*(j∞) > 0. Passive systems are phase-bounded.16,17 Therefore, the small gain condition is not required for a closedloop system comprising two passive systems. Theorem 2.1. Passivity Theorem.12 Consider the closed-loop system of G1, G2 (as shown in Figure 1) with e2 ≡ 0 so that

u1 ) e1 - G2(u2) u2 ) G1(u1) e1 ∈ L2

(1)

with G1, G2: L2e f L2e. Assume that for any e1 ∈ L2 there are solutions u1, u2 ∈ L2e. If G1 is passive and G2 is strictly passive, then u2 ) G1(u1) ∈ L2, where L2e is the extended L2 space, which consists of all measurable signals f(t) such that its truncation fT(t) ∈ L2. For linear systems, the Passivity Theorem states that a feedback system comprised of a passive system and a strictly passive system is asymptotically stable. For the multi-loop control system as shown in Figure 2, if the multi-variable process system is strictly passive, then the closed-loop system is stable if the multi-loop controller K(s) is passive. A passive multi-loop controller K(s) remains passive even when one or more sub-loops fail or are detuned. As a result, the closed-loop system is decentralized unconditionally stable. By definition, systems with nonrepeated poles at the origin can be passive. Therefore, multi-loop PI and PID controllers with non-negative control gains are passive. Therefore, the Passivity Theorem can be directly used to analyze closed-loop stability involving integral action. Large loop interactions generally lead to control performance degradation and even instability in decentralized control of closed-loop systems. Most current stability conditions imply “generalized diagonal dominance”.6,18,19 The Passivity Theorem indicates that

3. Passivity-Based Stability Conditions Many chemical processes are not passive, and thus the passivity-based condition given in Theorem 2.1 cannot be directly used to analyze decentralized unconditional stability for those systems. In this section, the passivity-based condition has been extended to cope with both passive and nonpassive processes. Theorem 3.1. For a given stable nonpassive process with a transfer function matrix of G(s), there exists a diagonal, stable, and passive transfer function matrix W(s) ) w(s)I such that H(s) ) G(s) + W(s) is passive. Proof. Because both G(s) and W(s) are analytic in Re(s) g 0, so is H(s). Therefore, H(s) is passive if the minimum eigenvalue of [H(jω) + H*(jω)] is non-negative for any ω ∈ [-∞, +∞],

λmin(H(jω) + H*(jω)) ) λmin([G(jω) + G*(jω)] + [W(jω) + W*(jω)]) (3) Because both [G(jω) + G*(jω)] and [W(jω) + W*(jω)] are Hermitian, from the Weyl inequality,20 we have

λmin(H(jω) + H*(jω)) gλmin(G(jω) + G*(jω)) + λmin(W(jω) + W*(jω)) )λmin(G(jω) + G*(jω)) + 2{Re}(w(jω))

(4)

If w(s) is chosen to be “passive enough” such that

1 Re(w(jω)) g - λmin(G(jω) + G*(jω)) 2

(5)

for any ω ∈ [-∞, +∞], then H(s) ) G(s) + W(s) can be rendered passive. If Re(w(jω)) > -1/2λmin(G(jω) + G*(jω)), then H(s) will be strictly passive. 9 Therefore, the minimum feedforward required provides a measure of how far the process is from being passive and thus is defined as the passivity index.21 Definition 3.1. Passivity Index. For an n × n linear time-invariant stable system G(s), the passivity index is defined as a frequency-dependent function:22

1 ν(G(s),ω) } -λmin [G(jω) + G*(jω)] 2

(

)

(6)

Apparently, for a system G(s) with its passivity index ν(G(s),ω), if a stable and minimum phase transfer function w(s) is chosen such that

Ind. Eng. Chem. Res., Vol. 41, No. 6, 2002 1571

ν(w(s),ω) < -ν(G(s),ω)

(7)

then G(s) + w(s)I is strictly passive. The passivity index of a linear system comprises both phase and gain information about the system in question.17 This can be seen from a simple SISO example. Given a system T(s), its passivity index at frequency ω is given by

ν[T(s),ω]) -{Re}[T(jω)] ) -|T(jω)|cos(φ)

Figure 3. Loop shifting.

(8)

where

φ ) tan-1

(

)

Im(T(jω)) Re(T(jω))

(9)

It is possible to use the passivity index as a single index, rather than use both phase and gain margins, to study unconditional stability. To facilitate future stability analysis, we define G+(s) ) G(s)U, where U is a diagonal matrix with either 1 or -1 along the diagonal. The signs of U elements are determined such that the diagonal elements of G+(s) are positive at steady state, that is, G+ii(0) g 0, i ) 1, ..., n. We denote

K+(s) ) U-1K(s) ) UK(s)

(10)

G+(s) ) G(s)U

(11)

Theorem 3.2. For an interconnected system (as shown in Figure 2) comprising a stable subsystem G(s) and a decentralized controller K(s) ) diag{ki(s)}, i ) 1, ..., n, if a stable and minimum phase transfer function w(s) is chosen such that ν(w(s),ω) < -ν(G+(s),ω), then the closed-loop system will be decentralized unconditionally stable if for any loop i ) 1, ..., n, ki′(s) ) ki+(s)[1 - w(s)ki+(s)]-1 is passive, where ki+(s) ) Uiiki(s) and U ) diag{Uii}, i ) 1, ..., n. Proof. When loop shifting is used, a closed-loop system equal to that in Figure 2 is obtained, as shown in Figure 3, where

G′(s) ) G(s)U + w(s)I

(12)

K′(s) ) U-1K(s)[I - w(s)U-1K(s)]-1

(13)

and

From Theorem 2.1, if G′(s) is SPR and K′(s) is PR, the closed-loop system will be stable. Because K+(s) is diagonal, so is the subsystem K′(s). Then, K′(s) is passive if and only if its diagonal element ki′(s) is passive for each loop i ) 1, ..., n. In addition, when ki′(s) is passive, K′(s) will remain passive when its gain matrix is reduced to K′(s)E ) diag{ki′(s)i}, 0 e i e 1, i ) 1, ..., n. Therefore, the positive realness of ki′(s) ensures the decentralized unconditional stability of the closed-loop system. 9 Similar to the diagonal scaling treatment for calculating maximum stability gain margins,23 the conservativeness of the sufficient stability condition given in Theorem 3.2 could be reduced by using a constant, real, and positive-definite diagonal rescaling matrix (as matrix D shown in Figure 4). The closed-loop system in Figure 1 is stable if and only if the feedback system shown in Figure 4 is stable. Note that, for any diagonal system K+(s), K+(s) ) D-1K+(s)D. However, the passivity index of D-1G+(s)D

Figure 4. Rescaling.

can be significantly reduced by using an appropriate D matrix:

ν(D-1G+(s)D,ω) < ν(G+(s),ω)

(14)

The rescaling matrix D can be chosen such that the rescaled G+(s) at steady state is positive-real; that is, the following inequality is satisfied:

D-1G+(0)D + D[G+(0)]TD-1 > 0

(15)

Because D is nonsingular, we have

D{D-1G+(0)D + D[G+(0)]TD-1}D > 0

(16)

G+(0)DD + DD[G+(0)]T > 0

(17)

Define M ) DD; thus, M is a constant, real, and positive-definite diagonal matrix. Inequality (15) is equivalent to the following inequality:

G+(0)M + M[G+(0)]T > 0

(18)

Inequality (18) is a typical linear matrix inequality (LMI) problem and can be solved by using any semidefinite programming tools such as Matlab LMI Toolbox. The continuity of the transfer function G+(s) implies that inequality (18) holds not only at steady state but also for a certain frequency range [0,ω1]:

G+(jω)M + M[G+(jω)]T > 0, ∀ ω ∈ [0,ω1] (19) The following theorem can be derived directly from Theorem 3.2. Theorem 3.3. Given a stable and rational LTI MIMO process with its transfer function matrix G(s) ∈ Cn×n, if the rescaled passivity index of G+(s) ) G(s)U is bounded by

νs(G(s),ω) ) max{- minν(D-1G+(s)D,ω),} (20) D

then any multi-loop controller

K(s) ) diag{ki(s)}, i ) 1, ..., n

(21)

satisfying the following conditions will stabilize the closed-loop system and achieve decentralized unconditional stability:

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Interactions. The passivity index does not quantify interactions directly but it does indicate the relative stability margin for MIMO systems under decentralized control if passive controllers are used. For a process with its transfer function matrix given by

[

g11(s) g12(s) · · · g1n(s) g2n(s) g (s) g22(s) G(s) ) 21· ·· · ·· ·· · gn1(s) gn2(s) · · · gnn(s) Figure 5. Passivity index and controller gain.

{

}

+

ki (jω)

(1)

Re

g 0, 1 - νs(G(s),ω)ki+(jω) (22) ∀ ω ∈ R, i ) 1, ..., n

(2)

K(s) is analytic in Re(s) > 0

where is an arbitrarily small positive number and D is positive-definite and νs(ω) is a frequency-dependent real positive function, ki+(s) ) Uiiki(s) ) 0. The (i,i)th element in matrix U indicates whether the corresponding ith loop is direct acting or reverse acting.

Achievable Control Performance. The decentralized unconditional stability condition given in eq 22 implies

(a) ki+(s) is passive and

|

(b) ki

| |

(24)

The relative stability is affected by the phase and gain margin of diagonal subsystems g11(s), ..., gii(s), ..., gnn(s) and the interactions arising from the off-diagonal subsystems. The overall destabilizing effect is captured by the passivity index and thus makes the condition given in inequality (22) a simple but effective tool for decentralized unconditional stability analysis. Robustness. The passivity-based analysis can be extended to deal with processes with model-plant mismatch. Assume the “true” process is GT(s) ) G(s) + ∆(s), where ∆(s) is the uncertainty. The passivity index of the “true” process can be estimated as

ν(GT(s),ω)

Remarks and Discussions

+

]

|

1 1 (jω) e , 2νs(G(s),ω) 2νs(G(s),ω) ∀ ω ∈ R, i ) 1, ..., n (23)

At frequency ω, ki+(jω), which satisfies the above conditions, is confined within the disk on the s plane, as shown in Figure 5. The size of the disk changes with frequency, but the disks are tangential to the imaginary axis. The amplitude ratio of the multi-loop controller at frequency ω, |K(jω)|, is limited to |1/(νs(G(s),ω))|. For a given process, the passivity indices at different frequencies can be estimated, and the permissible bandwidth of passive decentralized controllers can be obtained, which indicates limits on the achievable performance of the passivity-based decentralized unconditionally stabilizing controllers. As the passivity index is the property of the process, independent of the controllers, the achievable performance can be estimated in the early stage of control system design, prior to controller synthesis. The passivity index can be used for choosing pairing schemes: the pairing scheme results in the transfer function matrix G(s). The pairing should be chosen such that inequality (19) holds for the largest frequency band and this will result in the best achievable control performance. Equation 23 can be used to determine whether integral action can be used for offset free control. If the rescaled passivity index of the process system at steady state is not greater than zero, then passive controllers with infinite steady-state gain (with integral action) are allowed. For a given stable process with its steady-state gain matrix G(0), if a rescaling matrix M can be found such that inequality (18) is satisfied, then this process is decentralized integral controllable (DIC).24

1 ) -λmin [∆(jω) + ∆*(jω) + G(jω) + G*(jω)] 2

(

)

1 e -λmin [∆(jω) + ∆*(jω)] (25) 2 1 λmin [G(jω) + G*(jω)] 2

(

)

(

)

) ν(G(s),ω) + ν(∆(s),ω) When the estimate of the upper bound of the passivity index of the uncertainty is added to the passivity index of the nominal plant, the decentralized unconditional stability condition in inequality (22) can be used in the presence of model-plant mismatch. Inherent Passivity. Recent studies have revealed the connection between passivity and thermodynamics.25 It is possible to analyze decentralized unconditional stability based on the inherent passivity of certain chemical processes. This is currently under investigation. Unstable Processes. The decentralized unconditional stability analysis tool proposed here can be applied to a class of unstable processes. If an unstable process can be stabilized by inner-loop static diagonal output feedback, the stabilized system can then be controlled by a decentralized unconditional stabilizing controller, which is designed on the basis of the stability conditions provided in this section. Because failure of the inner-loop stabilizing controllers may lead to closedloop instability, redundant control loops (sensors, actuators, and controllers) should be used and a fault detection mechanism employed. The choice of static diagonal output feedback (a decentralized proportional only controller) as the inner-loop controller can minimize the amount of redundant devices and therefore reduce the cost of a failure-tolerant control system. Passivity Index Bound. Scalar passivity index is implemented in proposed conditions. The conservativeness of the passivity index may be reduced by introducing a passivity index vector,

Ind. Eng. Chem. Res., Vol. 41, No. 6, 2002 1573 νs(G(s),ω) ) [νs,1(G(s),ω) νs,2(G(s),ω) ‚‚‚ νs,i(G(s),ω) ‚‚‚ νs,n(G(s),ω) ]

T

(26) which contains an individual passivity measure for each input-output pair (νs,i for loop i). The stability condition in Theorem 3.2 is still valid after a minor modifications eq 22 in Theorem 3.2 should be replaced by

Re

{

ki+(jω)

1 - νs,i(G(s),ω)ki+(jω)

}

decentralized unconditional stability of the closed-loop system as well as good performance, a controller tuning method is proposed to minimize the sensitivity function of each loop, subject to condition (22). For multi-loop PI controller synthesis, this tuning problem is converted into the following optimizations problem: Problem 4.1.

min (-γi)

kc,i+,τI,i

g 0,

such that

|

∀ ω ∈ R, i ) 1, ..., n (27)

Defining a real and diagonal matrix N ) diag{νs(G(s),ω)}, the passivity index vector of G(s) at frequency ω, νs(G(s),ω), can be calculated by solving the following optimization problem. Problem 3.1.

min Trace(N) N

subject to G(jω) + G*(jω) + 2N g 0

(28)

N>0

(29)

The complex matrix inequality constraint can be easily transformed into a real matrix inequality. The matrix N is constrained to be positive semidefinite and thus the minimum passivity index would be zero. Problem 3.1 is convex. However, the computational complexity of solving the passivity index vector is significantly higher than finding the scalar index. Numerical experiments showed that using the passivity index vector may be beneficial in certain cases but does not always lead to a less conservative result. In most cases, implementation of the passivity index vector may improve passivity bounds of certain loops while worsening the bounds of other loops. Use of frequency-dependent rescaling matrices will also lead to an improvement on the passivity bound. In the current treatment, the rescaling matrix D is obtained from the steady-state condition (as in inequality (18)), which may not be optimal for non-zero frequencies. An optimal diagonal and real rescaling matrix D may be found by solving the following optimization problem for each frequency. Problem 3.2.

min t D

such that D-1G+(jω)D + D[G+(jω)]*D-1 + 2tI > 0

(30)

D>0

(31)

This problem is, however, nonconvex and is very computationally complex. The approaches to solving frequency-dependent rescaling matrices are still under investigation. It is possible to approximate the constraint given in inequality (30) by a linear matrix inequality such that a suboptimal D matrix can be obtained. 4.Multi-loop PI Controller Design Multi-loop PI controllers can be designed based on the proposed stability conditions in section 3. To achieve

+

1 + Gii (jω)kc,i

|

γi 1 < 1 (32) [1 + 1/(τI,i × jω)] jω

+

and τI,i2 g

kc,i+νs(ω) [1 - kc,i+νs(ω)]ω2,

∀ ω ∈ R, i ) 1, ..., n (33)

A larger γi in inequality (32) implies a wider bandwidth of the control system. Condition (33) is the reduced form of condition (22) for PI-type controllers. Problem 4.1 can be solved using any nonlinear optimization procedures (e.g., Matlab Optimization Toolbox). The tuning procedure is as follows. Procedure 4.1. (1) Find matrix U and calculate G+(s). (2) Check the pairing. Examine the proposed pairing using the DIC condition (18). If it is not satisfied, another pairing scheme should be considered. If no pairs satisfy condition (18), then no decentralized detunable PI controllers can be obtained using the proposed method. In the case where more than one pair meets condition (18), choose the pairing with larger permissible frequency bandwidth ω1, satisfying inequality (19). (3) Use the M matrix obtained in step 2 to derive the rescaling matrix D ) M1/2. (4) Calculate the passivity indices ν(D-1G+(s)D,ω) for different frequency points. These frequency points form a set Ω ∈ [0,ωE], where ωE is the frequency which is high enough such that ν(D-1G+(s)D,ω) f 0 for ω > ωE. This results in a list of νs(ω) for ω ∈ Ω. For each loop of the controller (e.g., ith loop, i ) 1, ..., n), do step (5): (5) Solve Problem 4.1 and find the optimal controller parameters kc,i+ and τI,i. (6) Obtain the final controller settings: kc,i ) Uiikc,i+. This method is limited to open-loop stable systems in its current form. Less conservative passivity-based stability conditions and design methods are currently under investigation. 5. Illustrative Examples Example 1: Decentralized Control System Stability Analysis. Consider a 2 × 2 system with the following transfer function matrix under decentralized control:6,7

[

5 2.5e-5s 4s + 1 (2s + 1)(15s + 1) G(s) ) 1 -4e-6s 20s + 1 3s + 1

]

(34)

This system has only two possible pairing schemes: ((1,1), (2,2)) or ((1,2), (2,1)). The frequency-dependent

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Figure 6. Passivity index of the rescaled process (Example 1).

Figure 7. Passivity index of the rescaled process (Example 2).

passivity index was calculated and plotted in Figure 6 for the two schemes. It shows that the first pairing ((1,1)), ((2,2)) has a frequency band of [0, 2.11rad/min], where the passivity indices are negative (although its passivity index is very close to zero at its first peak at the frequency of 0.058 rad/min). The other pairing has a frequency band of [0, 0.058 rad/min] with negative passivity indices. From the guideline discussed above, the first pairing scheme was chosen for better performance. This example has also been tested against existing stability conditions.6,7 The condition based on Interaction Measure6 appears to be very conservative, as it even cannot conclude closed-loop stability with any pairing scheme when integral action is employed. The result from phase-gain condition7 is very close to those from the proposed passivity-based stability condition. How-

ever, the proposed stability condition requires much less computational effort than the phase-gain condition. In this example, the number of floating point calculations required to test stability for one frequency point is 1043 (proposed method) vs 13551 (phase-gain condition). In addition, the proposed condition also indicates unconditional stability. Example 2: Multi-loop PI Controller Synthesis. The process considered is a 4 × 4 distillation column, which was used by Luyben2 to demonstrate the BLT design method. It has the following transfer function matrix: The pairing scheme of (1,1), (2,2), (3,3), and (4,4) satisfies the DIC condition (18) and thus is used in control design. A multi-loop PI controller was obtained by following the tuning procedure given in section 4. The rescaled passivity indices of this process were


Figure 8. Step-change response of loop 1 (Example 2).


shown in Figure 7. For comparison purposes, PI controller parameters were also calculated using the BLT method.2 In the BLT design, two different “biggest log modulus” values (Lmax ) 4 and Lmax ) 8) were used. BLT

design with a smaller Lmax would result in a more robustly stable control system while use of a larger Lmax can lead to better performance with a smaller stability margin. The step-change response of each loop was

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illustrated in Figures 8 -11. The corresponding controller outputs with the proposed PI settings were shown in Figure 12. Simulation studies have been conducted to assess the performance of the proposed controller and the BLT controllers. The PI controller parameters and the control errors (ITAE) found using simulation were listed in Table 1. In terms of ITAE, the proposed approach achieves better performance than the BLT method with Lmax ) 4. It is noticed that, compared with the BLT controller designed using Lmax ) 8, the proposed controller performs better in Loop 1 and worse in other loops. This can be explained as follows: the proposed design method is based on the passivity index of the full process system, which reflects the overall stability constraint resulting from both the interactions and dynamics of the entire process system. The large time delay and the second-order dynamics in G11(s) lead to a

Table 1. PI Controller Setting PI controller

loop 1

loop 2

loop 3

loop 4

0.923 1.16 k BLT method c τI 61.7 13.2 (Lmax ) 8) ITAE 1.12 × 104 273

0.727 13.2 4224

0.393 0.495 k BLT method c τ 145 31 (Lmax ) 4) I ITAE 1.95 × 105 4400

0.31 0.927 31 93.8 1.06 × 105 9.12 × 104

proposed method

0.696 kc τI 38.2 ITAE 8131

0.692 37.2 2218

2.17 40 4051

0.692 0.717 37.2 47.3 2.18 × 104 2.10 × 104

large overall passivity index. This large passivity index imposes a severe constraint on the achievable performance for each loop. While this constraint is appropriate to the first control loop, it is too conservative for the rest of the loops, which themselves alone would have a smaller passivity index due to their smaller dead-time or/and simpler dynamics. Therefore, the passivity-based


Figure 12. Control action with respect to step change in each loop (Example 2).

approach results in controller settings that may sacrifice performance for the second, third, and fourth loop. Taking into account the fact that decentralized unconditional stability (DUS) is a very demanding requirement, the loss of performance in proposed control is acceptable. Simulation was also performed when one or more loops were arbitrarily detuned. The proposed controller demonstrated DUS. For this particular case, the BLT controller maintained closed-loop stability in our tests. However, as pointed out by Luyben,2 the BLT design does not guarantee DUS. 6. Conclusion A new analysis tool for decentralized unconditional stability is proposed in this paper. On the basis of the Passivity Theorem, this approach gives some new insight into system interactions and their destabilizing effect. For a given open-loop stable process, the proposed conditions also indicate achievable performance by any passivity-based decentralized unconditional stabilizing controllers and thus can be used to decide pairing schemes. The proposed method does not require manual analysis and can be easily implemented in systematic control synthesis. Case studies have shown that the new approach is easy to use and requires much less computational effort than existing methods. Acknowledgment This work was supported by the Australian Research Council (Grant A00104473). The authors wish to thank the anonymous reviewers for their useful comments and helpful suggestions regarding this work. Nomenclature G(0) ) steady-state gain matrix G* ) complex conjugate transpose of a complex matrix G A > 0 ) matrix A is positive definite A g 0 ) matrix A is positive semidefinite Re(c) ) real part of a complex number c Im(c) ) imaginary part of a complex number c

λmin(A) ) minimum eigenvalue of matrix A Trace(A) ) trace of matrix A

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Received for review December 4, 2000 Revised manuscript received November 2, 2001 Accepted November 27, 2001 IE001037V

Decentralized Unconditional Stability Conditions Based on the

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