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Passivity-Based Decentralized Failure-Tolerant Control Jie Bao,*,† Wen Z. Zhang,† and Peter. L. Lee‡ School of Chemical Engineering & Industrial Chemistry, The University of New South Wales UNSW, Sydney, New South Wales 2052, Australia, and Division of Engineering & Science, Curtin University of Technology, GPO Box U1987, Perth, Western Austrailia 6845, Australia
This paper provides a new approach to decentralized failure-tolerant control using passivitybased stability conditions. A controller design method is developed to achieve both decentralized unconditional stability (a transformed passivity condition) and user-specified H2 performance. The control synthesis problem is solved by using successive semidefinite programming. The proposed method is illustrated by a case study of decentralized control of a distillation column. 1. Introduction Decentralized control is a control strategy that uses multiloop single-input single-output (SISO) controllers to control multivariable plants. Decentralized control is the most widely used control strategy in the process industry because of its simplicity in controller design and implementation and its potential to achieve failuretolerant control.1,2 In process control applications, failures of control components such as actuators, sensors, or controllers are often encountered. These problems not only degrade the performance of control systems but also may induce instability. At present, most fault-tolerant control systems are built with redundant controllers (e.g., see Willsky,3 Siljak,4 and Yang et al.5). The backup control loop is employed once failure of the main controller is detected. However, control loop failure cannot be guaranteed to be detected swiftly and accurately. Sometimes the fault detection system itself could be a possible source of failure.6 This approach also requires a significant number of redundant control components, which may increase the system cost to an unacceptable level. For stable processes, it is possible to design decentralized control systems that maintain closed-loop stability when one or more subloops are arbitrarily detuned or switched off by sensor or actuator failures. Closed-loop stability under the above circumstances is called decentralized unconditional stability (DUS)7 or decentralized detunability.8 Decentralized unconditionally stabilizing controllers can achieve failure tolerance without using redundant control components. It is noted that even if the process itself is stable, the closed-loop system with a decentralized controller may not be unconditionally stable.9 The DUS test for decentralized IMC controllers was proposed by Hovd and Skogestad.8 Lee et al.10 presented a DUS condition for control systems whose diagonal closed-loop subsystems are first-order with a steady-state gain of 1. These conditions, however, are difficult to implement in systematic control synthesis. Direct synthesis methods for decentralized unconditional stabilizing controllers have not been reported. This paper presents an approach to decentralized failure-tolerant control design, using the concept of * 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. ‡ Curtin University of Technology.
passive systems. From the “passivity theorem”, a strictly passive multivariable plant can be stabilized by any decentralized passive controller. The decentralized passive controller remains passive when one or more of its subloops are arbitrarily detuned and thus decentralized unconditional stability is achieved. Therefore, the passivity theorem provides a new avenue for decentralized unconditionally stabilizing control. The concept of passivity was used to derive a new condition for decentralized integral controllability.11 Recent studies by the authors led to a set of easy-to-use passivity-based DUS conditions.12 In this paper, a control synthesis method is developed on the basis of the above conditions. The proposed method can be classified as an independent design method. Each control loop is designed such that the passivity-based stability condition is satisfied to ensure DUS and user-specified H2 performance is achieved. A new diagonal scaling technique is introduced to reduce the conservativeness of the above condition over the whole frequency horizon. The stability and performance constraints are cast into bilinear matrix inequalities. A computational approach to this design problem is developed using successive semidefinite programming (SSDP). This paper is organized as follows. The concept of passive systems and the passivity-based stability conditions are briefly introduced in section 2. The failure-tolerant control approach is described in section 3. The numerical solution to the controller design problem is developed in section 4. Section 5 presents the distillation column control case study. The paper concludes with discussion in section 6. 2. Passivity and Passivity-Based Conditions Definition 1. Linear Passive Systems.13 A linear time-invariant system Σ: T(s) :) C(sI - A)-1B + D (T(s) is an n × n transfer function matrix) is passive if and only if T(s) is positive real or, equivalently, (1) T(s) is analytic in Re(s) > 0 and (2) T(jω) + T*(jω) g 0 for all frequency ω such 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)14 if (1) T(s) is analytic in Re(s) g 0 and (2) T(jω) + T*(jω) > 0 for ω ∈ (-∞, +∞). For linear systems, the passivity theorem states that a system formed by the negative feedback of a passive
10.1021/ie0201314 CCC: $22.00 © 2002 American Chemical Society Published on Web 10/16/2002
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(s),ω), then the closed-loop system will be decentralized unconditionally stable if
K′(s) ) K+(s)[1 - wp(s)K+(s)]-1
is passive, where K+(s) ) U-1K(s) ) UK(s). Condition (4) implies that K+(s) is passive but it is a more stringent condition than the passivity of K+(s).
Figure 1. Control system diagram.
system and a feedforward strictly passive system is asymptotically stable. A passive decentralized controller unconditionally stabilizes a strictly passive process, as the passivity condition is still valid when one or more of its subloops are arbitrarily detuned. Such passive control systems include controllers with nonrepeated poles at s ) 0. Thus, integral actions can be incorporated into the control system to achieve offset free control. As the passivity theorem can deal with systems with unlimited gain explicitly, it can be used to study closedloop stability in an open-loop fashion by simply examining the passivity of its subsystems. The passivity-based stability analysis has been extended to nonpassive systems, which are typical of chemical processes. The extended stability conditions were obtained by using the passivity index and simple loop shifting. The passivity index was first introduced by Wen to measure how far a system of concern is from being passive.15 In this paper, a frequency-dependent passivity index is adopted. Definition 2. Passivity Index.16 For an n × n linear time-invariant stable system G(s), the passivity index at frequency ω is defined as
1 ν(G(s),ω) ∆ -λmin [G(jω) + G*(jω)] 2 )
(
)
(1)
The passivity index indicates how much feedforward at different frequencies is required to render a nonpassive system passive. It can be seen that, for a given system G(s) ∈ Cn×n, if a stable and strictly positive real transfer function wp(s) can be found such that
ν(wp(s),ω) < -ν(G(s),ω), ∀ ω ∈ R
3. Passivity-Based Decentralized Unconditionally Stabilizing Control A design method for “decentralized unconditionally stabilizing” (DUS) controllers is developed in this section. Given an n × n process G(s) (assuming 1-1/2-2/ ...n-n pairing), its passivity indices at different frequencies can be calculated. A decentralized controller K(s) ) diag{ki(s)} is designed such that the DUS condition given in Theorem 1 is satisfied and an H2 performance for each control loop is optimized. Problem 1. For the ith controller,
min{γi} ki(s)
subject to
k′i(s) ) uiki(s)[1 - wp(s)uiki(s)]-1 is passive, ∀ i ) 1...n (5) and
||Ti(Gii(s),ki(s))||2 < γi
(3)
where U ) diag{ui}, i ) 1...n, is a diagonal matrix with either 1 or -1 along the diagonal elements such that G+ii(0) g 0, i ) 1, ..., n. A passive controller K+(s) can be designed for G+(s). The final controller K(s) ) UK+(s) absorbs the sign changes. The following stability condition was derived by using simple loop shifting. Theorem 1.12 For an interconnected system (as shown in Figure 1) comprising a stable subsystem G(s) and a decentralized controller K(s) ) diag{ki(s)}, i ) 1, ..., n, if a stable and minimum phase scalar transfer function wp(s) is chosen such that ν(wp(s),ω) < -ν(G+-
(6)
where Gii(s) is the (i,i)th element of G(s), ui is the (i,i)th element of the diagonal “sign” matrix U, and Ti(Gii(s),ki(s)) is the ith closed-loop transfer function whose H2 norm needs to be minimized. Typical performance specifications concern control error and energy consumption, which are reflected by the following choice of Ti(Gii(s),ki(s)):
Ti(Gii(s),ki(s)) )
(2)
for any frequency ω, then G(s) + wp(s)I is strictly passive.12 Passive systems are phase-bounded. The phase of MIMO LTI passive systems lies in [-π/2, +π/2],17 as does each control loop in any decentralized (multiloop) passive control system. Therefore, diagonal elements of G(s) need to have positive steady-state gain so that passive controllers (which are reverse acting) can be used. Define
G+(s) ) G(s)U
(4)
[
wk,iki(s)[I + Gii(s)ki(s)]-1 ws,i(s)[I + Gii(s)ki(s)]-1
]
(7)
where ws,i(s) is the weighting function for the sensitivity function (for small control error) and wk,i is a constant weight to penalize the controller gain (indirectly). It is noted that eq 5 gives the same open-loop condition for each controller, depending on the passivity indices only. 3.1. Constraint on Control Performance. Equation 4 requires each controller loop, ki+(s), to be passive and the following condition to be satisfied:12
|k
+ i
(jω) +
| |
|
1 1 e , 2ν(wp(s),ω) 2ν(wp(s),ω) ∀ ω ∈ R, i ) 1, ..., n (8)
At frequency ws,i(s), the controller transfer function ki+(s), which satisfies eq 8, is confined within a disk centered at(-1/(2ν(wp(s),ω)],0) with a radius of 1/|2ν(wp(s),ω)| on the s-plane. The size of the disk varies with frequency. The amplitude ratio of the multiloop controller at frequency ω, |ki+(jw)|, is limited to 1/|ν(G(s),ω)|. This implies limits on the achievable performance of the passivity-based controllers: At a particular frequency,
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Because D(ω) is nonsingular, inequality (9) is equivalent to the following inequalities:
D(ω){D(ω)-1G+(jω)D(ω) + D(ω)[G+(jω)]*D(ω)-1}D(ω) + tD(ω)D(ω) > 0 (11) G+(jω)D(ω)D(ω) + D(ω)D(ω)[G+(jω)]* + tD(ω)D(ω) > 0 (12) Figure 2. Diagonal scaling.
Define M ) D(ω)D(ω), we have
the larger the passivity index is, the smaller the controller amplitude ratio is allowed. With the signs of the diagonal elements of G(s) adjusted using eq 3, the passivity indices of G+(s) at low frequencies are usually small. If the passivity index of a process at steady state is less than or equal to zero, the process is “decentralized integral controllable”11 and a decentralized controller with integral action can be used to achieve offset free control while maintaining DUS. In this case, there exits a low-frequency band [0,ωb], in which the passivity indices are no greater than zero. In this frequency band, arbitrarily large amplitude ratios of the controller are permissible. The larger the upper bound ωb is, the larger bandwidth a passivitybased controller can have and a faster response can be achieved. 3.2. Frequency-Dependent Diagonal Scaling of Passivity Indices. It has been found that the passivity index can be diagonally scaled to reduce the conservativeness of the passivity-based control. Define “D” as a diagonal positive definite matrix. The closed-loop system in Figure 1 is stable if and only if the feedback system shown in Figure 2 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 can be significantly reduced by using an appropriate D matrix. A scaling method using a constant matrix was developed to minimize the passivity index at steady state.12 In this paper, the scaling technique is extended by introducing a frequencydependent diagonal scaling matrix D(ω) such that the passivity indices at different frequencies can be reduced. For a given stable process with signs adjusted G+(s) ∈ Cn×n, the problem of diagonal scaling for a passivity index at frequency ω can be described as Problem 2. Problem 2. D
subject to +
+
D(ω) G (jω)D(ω) + D(ω)[G (jω)]*D(ω)
-1
+ tI > 0 (9)
and
D(ω) > 0
(13)
Define G+(jω) ) X(ω) + jY(ω), where both X(ω) and Y(ω) are real matrices. This leads to
(X(ω)M + MX(ω)T) + j(Y(ω)M - MY(ω)T) + tM > 0 (14) or equivalently
-(X(ω)M + MX(ω)T) - j(Y(ω)M - MY(ω)T) - tM < 0 (15) The above inequality holds if and only if
[
]
-X(ω)M - MX(ω)T Y(ω)M - MY(ω)T -Y(ω)M + MY(ω)T -X(ω)M - MX(ω)T t
[ ]
M 0 < 0 (16) 0 M
Therefore, Problem 2 can be converted into the following generalized eigenvalue problem with constraints described in real matrix inequalities. Problem 3.
min{t} M
subject to
[
] [ ]
-X(ω)M - MX(ω)T Y(ω)M - MY(ω)T M 0 0
(18)
For each frequency ω, a real matrix of M can be obtained by solving the above optimization problem, using any SDP solver (e.g., Matlab LMI Toolbox). Then the diagonally scaled passivity index can be calculated as
min{t}
-1
G+(jω)M + M[G+(jω)]* + tM > 0
(10)
where D(ω) ∈ Rn×n is a diagonal matrix and t is a real scalar variable. Problem 2 cannot be directly handled by “semidefinite programming” solvers because inequality (9) is nonlinear and complex. Inequality (9) can be converted into a real and linear matrix inequality by the following steps.
1 νs(G+(s),ω)∆-λmin [M-1/2G+(jω)M1/2 + 2 )
(
)
M1/2(G+(jω))*M-1/2] (19) With replacement of passivity index νs(G+(s),ω) in inequality (2) with the scaled index νs(G+(s),ω), a less conservative control design can be obtained. Figure 4 shows the passivity indices of the process in the illustrative example described later in this paper before and after diagonal scaling, from which it can be seen that the proposed scaling method can reduce the passivity indices significantly. 3.3. Choosing Weighting Function wp(s). To use the stability condition given in Theorem 1, a minimum
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where T′i(Gii(s),k′i(s)) gives the performance constraint as a function of k′i(s). For the equivalent performance specification as given in eq 7,
T′i(Gii(s),k′i(s)) )
[
]
wk,iuik′i(s)[I + Gii(s)uik′i(s) + wp(s)k′i(s)]-1 ws,i(s)[I + wp(s)k′i(s)][I + Gii(s)uik′i(s) + wp(s)k′i(s)]-1 (26)
Figure 3. Feedback system with loop shifting.
phase and stable scalar transfer function wp(s) needs to be chosen. Because this transfer function will be absorbed into the controllers, a simple and low-order wp(s) is desirable. If G+(s) is positive real at steady state, then wp(s) could be chosen to have the following form,
wp(s) )
ks(s + a) (s + b)(s + c)
(20)
where a, b, c, and k are positive and real parameters to be determined. If a passivity index profile
v ) [νs(G+(s),ω1) νs(G+(s),ω2) ‚‚‚ νs(G+(s),ωm)] is obtained, the parameters of wp(s) can be found by solving the following problem. Problem 4. m
min
∑(Re(wp(jωi)) - νs(G+(s),ωi))2
(21)
a,b,c,k i)1
subject to
Re(wp(jωi)) > νs(G+(s),ωi), ∀ i ) 1...m
(22)
The above problem can be solved using any nonlinear optimization solver, such as Matlab Optimization Toolbox. 3.4. Controller Synthesis. Problem 1 cannot be solved by existing H2/H∞/passivity multiobjective control synthesis approaches (e.g., ref 18) because conditions (5) and (8) are about two entirely different feedback systems. To simplify control design, we construct the DUS controller K(s) ) diag{ki(s)} (i ) 1...n) indirectly. From eq 4,
K(s) ) UK′(s)[I + wp(s)K′(s)]-1
(23)
Therefore, the feedback system in Figure 1 is equivalent to that shown in Figure 3. We can design the system K′(s), which is required to be passive, and then obtain the final controller using (23). The passive controller K′(s) ) diag{k′i(s)} (i ) 1...n) can be found by solving the following problem. Problem 5. For the ith controller,
min{γi} k′i(s)
subject to
k′i(s) is passive, ∀ i ) 1...n
(24)
and
|T′i(Gii(s),k′i(s))|2 < γi
(25)
A numerical solution to Problem 5 is presented in section 4. In summary, the DUS controller synthesis procedure can be described as follows: (1) For a given stable process G(s), adjust the signs of diagonal subsystems and derive G+(s) by using eq 3. (2) Calculate the diagonally scaled passivity indices of G+(s), νs(G+(s),ω), at a number of frequencies so that a profile of the passivity indices of the process can be produced. This is done by solving Problem 3 and using eq 19 at each frequency. (3) Choose a stable and minimum phase scalar transfer function wp(s) such that ν(wp(s), ω) < -νs(G+(s),ω), by solving Problem 4. (4) Derive the DUS controller by solving Problem 5. 3.5. Remarks. Fundamental Performance Limitations. The fundamental bandwidth limitations on decentralized control caused by interactions, RHP zeros, and time delays are reflected by the passivity indices. As explained in section 3.1, a large passivity index at a given frequency implies that the controlled variables may be poorly controlled at that frequency. Therefore, the proposed method does not offer a workaround to these fundamental bandwidth limitations. Pairing. Different paring schemes result in different transfer function matrices G(s), which normally have different passivity indices. As the passivity index implies a constraint on achievable performance, a pairing scheme should be chosen such that the resulting G(s) has small passivity indices at all frequencies concerned. If the process is decentralized integral controllable, the best pairing scheme should lead to a transfer function matrix G(s) which has the largest frequency band [0,ωb] such that νs(G(s),ω) e 0, ∀ ω ∈ [0,ωb]. State-Space Passivity Conditions. It is worthwhile to point out that the passivity condition can be presented in state space using the Positive Real Lemma.19 By using the multiplier theory, a generalized passivity condition can be derived by using transfer function multipliers.20,21 This condition can be verified without a frequency-by-frequency search. However, finding a general solution to such transfer function multipliers can be very difficult, as it is a feasibility problem with “bilinear matrix inequalities”. In addition, most chemical processes have time delays, which cannot be represented using finite dimensional state-space models. Therefore, the frequency domain condition developed in this paper would be more practical in applications of process control. 4. Numerical Solution to the Control Synthesis Problem Our research shows that the existing methods for H2/ passive controller synthesis often lead to very conservative designs and they are not applicable to many types of performance specifications, such as that in eq 7.22-24
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Therefore, a new approach is developed in this paper. The basic idea is to represent the H2 conditions in matrix inequalities and then solve the optimization problem using successive semidefinite programming (SSDP). For a given process transfer function matrix G(s) ∈ Cn×n, assume the following state-space representations:
Gii(s) :) (Agi, Bgi, Cgi, Dgi), i ) 1...n
(27)
wp(s) :) (Ap, Bp, Cp, Dp)
(28)
wsi(s) :) (Awi, Bwi, Cwi, Dwi), i ) 1...n
(29)
Consider the ith controller loop where the input to and output from Gii(s) are ui and yi, respectively. Assume controller k′i(s) to be designed is strictly proper and has a state-space representation:
k′i:
{
x˘ C ) AkixC + Bkiyi ui ) CkixC
]
(30)
The following augmented plant
[
]
{
x˘ ) Ax + B1w + B2u P11(s) P12(s) :) z ) C1x + D11w + D12u Pi(s) ) P21(s) P22(s) y ) C2x + D21w + D22u (31)
where L ) (ΠC2T + B1D21T)(D21D21T)-1 is the observer gain matrix and Π is the solution to the Riccati equation below:
ΠAT - C2T(D21D21T)-1D21B1T + A - B1D21T(D21D21T)-1C2Π ΠC2T(D21D21T)-1C2Π + B1B1T B1D21T(D21D21T)-1D21B1T ) 0 (35) As in the standard LQG/H2 framework, the controller structure in eq 33 is that it does not produce integral control. If an integral action is introduced in the performance weighting function wsi(s), eq 35 will not have a solution because the pole s ) 0 in wsi(s) is uncontrollable in the feedback system. This problem can be overcome by explicitly introducing integral action into the controller structure. For any passive controller k′i(s), k′i(s) + ksi/s is still passive when ksi g 0. Therefore, the inclusion of an integrator does not violate the stability condition. The structure for k′i(s) is now reformed as follows:
Aki )
[
ki(s) )
|(Fi(Pi(s)),k′i)||2 ) ||P11(s) + P12(s)k′i(s)[I P22(s)k′i(s)]-1P21(s)||2 ) |T′i(Gii(s),k′i(s))||2 (32)
[
] [] [ ]
Agi 0 0 0 Ap 0 , B1 ) 0 , A) 0 Bwi -BwiCgi 0 Awi Bgi B2 ) Bp , C1 ) [-DwiCgi 0 Cwi ], -BwiDgi D11 ) Dp, D12 ) -DpDgi, C2 ) [-Cgi -Cp 0 ], D21 ) I, D22 ) -Dgi - Dp
{
x˘ C ) AxC + B2ui + L(yi - C2xC - D22ui) ui ) -KgixC
]
(33)
In this approach, the assumptions of B1D21T ) 0 and D12TC1 ) 0, which are required in existing methods,22-24 are removed. As a result, H2 problems with any performance specifications, such as those in eq 7, can be solved. The controller has the following state-space representation,
Aki ) A + B2Kg - LC2 - LD22Kg Bki ) L Cki ) Kgi Dki ) 0
(34)
uik′i(s) 1 + wp(s)k′i(s)
(37)
will retain the integral action as long as wp(0) ) 0, which is the case for “decentralized integral controllable” processes. This leads to the desirable offset free control. If a plant/pairing is not decentralized integral controllable (DIC), it is not possible to design a DUS controller with integral action for it. In this case, the proposed control synthesis method will automatically produce a DUS controller without integral action. This can be explained as follows: Any non-DIC plant/pairing has a positive (rescaled) passivity index at the frequency of zero. Therefore, the weighting function wp(s) has to be chosen such that Re(wp(0)) > νs(G+(s),0) > 0, which does not possess a zero at s ) 0. The weighting function would have the following form:
4.1. Controller Structure. To keep the complexity of the control synthesis procedure to a manageable level, an ad hoc LQG control structure (an observer plus state feedback), similar to Geromel and Gapski,22 is adopted:
k′i:
[]
The final controller for the ith loop,
is constructed such that
where
]
A + B2Kgi - LC2 - LD22Kgi 0 L , Bki ) 1 0 0 (36) Cki ) [Kgi ksi ], Dki ) 0
wp(s) )
k(s + a)(s + b) (s + c)(s + d)
(38)
With such a weighting function wp(s), the final controller K(s) ) diag{ki(s)} (i ) 1...n) obtained using eq 37 does not have integral action. Assume Fl(Pi(s),k′i(s)) :) (Acl,Bcl,Ccl,Dcl). By using the “positive real lemma” and the property of H2 norm,25 the control synthesis problem can be cast into the following problem. Problem 6.
min {Tr(Q)}
Kgi,ksi P1,P2,Q
subject to
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[
(A + B2Kgi - LC2 - LD22Kgi)TP1 + P1(A + B2Kgi - LC2 - LD22Kgi) LTP1 - Kgi
[
ksi > 0
(40)
P1 > 0
(41)
]
AclTP2 + P2Acl P2Bcl 0 Ccl Q
(43)
where
[
]
B2ksi B2Kgi A Acl ) LC2 A + B2Kgi - LC2 LD22ksi , C2 D22Kgi D22ksi
]
P1L - KgiT e0 0
(39)
P20. The decision variables of the approximated problem are deviation variables. Details of the approximated problem (Problem 7) are given in the Appendix. When solving the approximated problem, the solution radii of the deviation variables need to be restricted. 4.3. Initial Solution. The iterative SSDP approach needs an initial solution such that all the constraints in Problem 6 are satisfied (although Tr(Q) does not need to be minimized). One obvious choice of the initial point is an arbitrary passive controller. With the assumption of Kgi ) LTP1, the following inequality gives a sufficient condition for inequality (39):
(A + B2LTP1 - LC2 - LD22LTP1)TP1 +
[ ]
B1 Bcl ) LD21 D21
P1(A + B2LTP1 - LC2 - LD22LTP1) < 0 (48) (44)
With left and right multiplying of P1-1, and defining W ) P1-1, the following linear matrix inequality (LMI) can be obtained:
W(A - LC2)T + (A - LC2)W + LB2T + B2LT -
Cdl ) [C1 D12Kgi D12ksi ], Dcl ) D11 This problem has four matrix decision variables P1, P2, Q, and Kgi and one scalar variable ksi. Inequalities (39)(41) imply passivity of k′i(s). Inequalities (42)-(44) are H2 conditions. The trace of matrix Q gives the upper bound of ||Fl(Pi(s),k′i)||22. Because Acl, Bcl, Ccl, and Dcl are functions of Kgi and ksi, inequalities (39) and (42) are “bilinear matrix inequalities” (BMIs). Therefore, conventional “semidefinite programming” (SDP) techniques cannot be used to solve Problem 6. In this section, a “successive semidefinite programming” (SSDP) approach, analogous to “successive linear programming” (SLP),26 is developed to solve this optimization problem iteratively. 4.2. Approximation of Bilinear Constraints. The proposed approach is an extension to the SSDP method for BMI problems proposed by Bao et al.27 Starting with a feasible solution, the problem is solved iteratively. The bilinear constraints are approximated by linear matrix inequalities in the neighborhood of estimated solutions and the linearized subproblem is solved by using semidefinite programming. The bilinear terms in inequalities (39) and (42) are approximated by the following simple linearization treatment: Assume X and Y are two independent decision matrix variables. Their product around (X0, Y0) is approximated by using the following equation:
LD22TLT - LD22LT < 0 (49)
(46)
Matrix variable W can be solved using any semidefinite programming (SDP) tool, such as Matlab LMI Toolbox. The state feedback gain matrix can be calculated as Kgi ) LTW-1. The initial value of ksi can be set to an arbitrary positive value (or can be set as 1). With Kgi and ksi fixed, and leaving P1, P2, and Q as decision variables, inequalities (39)-(43) are linear matrix inequalities. Problem 6 can be solved to obtain the optimal P1, P2, and Q by using any SDP algorithm. The results can be used as the initial point for SSDP iteration. 4.4. SSDP Procedure. The following procedure is implemented in designing each controller loop ki(s), i ) 1...n. Step 1. Find the initial set of solutions using the method described in section 4.3. They are named Kgi0, ksi0, P10, P20, and Q0, respectively. Step 2. Set the solution radius ) 0. It should be set to a small positive number so that the optimization solver in Step 3 will search in the small neighborhood of the initial values. Also set convergence tolerance ζ and the maximum number of iterations η. Step 3. Check whether the maximum number of iterations has been reached. If yes, terminate the SSDP procedure and jump to Step 6. Step 4. Solve Problem 7 (the problem with approximated constraints) with initial values δKgi ) 0, δksi ) 0, δP1 ) 0, δP2 ) 0, and δQ ) 0 and restrictions on the solution radii,
where δX ) X - X0 and δY ) Y - Y0. Both δX and δY are restricted by
||δKgi|| < , ||δksi|| < , ||δP1|| < , ||δP2|| < , and ||δQ|| < (50)
XY ) X0δY + δXY0 + δXδY + X0Y0 ≈ X0δY + δXY0 + X0Y0
||δX|| e , ||δY|| e
(45)
(47)
where is an arbitrary small positive number such that the solution region of eq 46 is not too far from that of eq 45. Define δKgi ) Kgi - Kgi0, δksi ) ksi - ksi0, δP1 ) P1 P10, and δP2 ) P2 - P20. By using eq 46, inequalities (39)-(43) can be approximated around Kgi0, ksi0, P10, and
and obtain δKgi, δksi, δP1, δP2, and δQ. Because zero initial values of decision variables satisfy all the constraints in Problem 7, there always exists a feasible solution. Step 5. Update Kgi ) Kgi0 + δKgi and ksi ) ksi0 + δksi. Fix Kgi and ksi, and solve Problem 6 with the linear matrix inequalities for P1, P2, and Q. This step provides the solution to the original BMI problem.
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[
[
(A + B2Kgi - LC2 - LD22Kgi)TP1 + P1(A + B2Kgi - LC2 - LD22Kgi) LTP1 - Kgi 2.22e-2.5s (36s + 1)(25s + 1)
-2.33e-5s (35s + 1)2 G(s) ) -1.06e-22s (17s + 1)2 -5.73e-2.5s (8s + 1)(50s + 1)
-2.94(7.9s + 1)e-0.05s (23.7s + 1)2 3.46e-1.01s 32s + 1 3.511e-13s (12s + 1)2 4.32(25s + 1)e-0.01s (50s + 1)(5s + 1)
(a) If no feasible solution is obtained, the solution radius in Step 4 is too large and the solution to the approximated problem does not satisfy the original nonlinear constraints. Reduce solution radius by replacing with 0.9 and go to Step 3. (b) If a feasible solution is obtained, there are two scenarios: (i) If Tr(Q) > Tr(Q0), the new solution is less optimal than the previous solution. Increase solution radius by replacing with 1.1 and go to Step 3. (ii) If Tr(Q) < Tr(Q0) with the new solution, a better result is obtained. Now check for convergence. If |Tr(Q) - Tr(Q0)| < ζ, then an acceptable solution is obtained and proceed to Step 6. Otherwise, replace the initial values for Step 3. Set Kgi0 ) Kgi, ksi0 ) ksi, P10 ) P1, P20 ) P2, and Q0 ) Q. Go to Step 3. Step 6. Calculate k′i(s) using eq 36 and the final controller ki(s) is computed using eq 37. Inequality (39) is negative semidefinite and thus may cause some numerical problems for some SDP solvers, such as Matlab LMI Toolbox, which is based on the interior-point method. One of the workarounds is to replace inequality (39) with the following strictly negative definite inequality (eq 51) where ξ is a very small positive number. The user tuning parameters in the SSDP procedures are initial solution radius 0, maximum number of iterations η, and the convergence tolerance ζ. It is noted that both the complexity and dimensionality of the iterative procedure are reasonably high. This complexity can be hidden from users as both the problem formulation and solution procedure proposed here are entirely mechanical and can be performed automatically by a computer. There are many efficient and reliable algorithms available for the execution of each LMI step (Steps 4 and 5). The original optimization Problem 6 is not convex, and as a result, the global optimum is hard to find.28 Thus, the proposed SSDP algorithm can only be guaranteed to find a local optimum. In our trial tests, we found the algorithm converged smoothly. Further issues of convergence and algorithm stability of the proposed SSDP approach are beyond the scope of this paper. Like many other independent designs, such as [8], [9], and [10], performance design in the proposed method is based on individual loops. It is noted that a good H2 norm for the individual loops does not guarantee a good
]
P1L - KgiT 0
(63)
Z31T Z32T Z33 Z43
]
Z41T Z42T 0
Z21T Z22 Z32 Z42
Z11 Z21 Z31 Z41
]
δP1TL - δKgiT + P10TL - Kgi0T e0 0
P2230 + δP223T P2330 + δP233 D12Ksi0 + D12δksi
P2120 + δP212 P2220 + δP222 P2230T + δP223T
(64)
]
C1T D12Kgi0TD12T + δKgiTD12T >0 Ksi0TD12T + δksiTD12T Q0 + δQ
]
P2130 + δP212 P2230 + δP223 > 0 P2330 + δP233
(65)
(66)
where
S ) P10E + δP1E + P10FKg0 + P10FδKgi + δP1FKgi0 + ETP10T + ETδP1T + Kgi0TFTP10T + δKgiTFTP10T + Kgi0TFTδP1T (67) E ) A - LC2
(68)
F ) B2 - LD22
(69)
Z11 ) P2110A + δP211A + P2120LC2 + δP212LC2 + P2130C2 + δP213C2 + ATP2110T + ATδP211T + C2TLTP2120T + C2TLTδP212T + C2TP2130T + C2TδP213T (70) Z21 ) Kgi0TB2TP2110T + δKgiTB2TP2110T + Kgi0TB2TδP211T + ATP2120T + ATδP212T + Kg0TB2TP2120T + δKgiTB2TP2120T + Kg0TB2TδP212T - C2TLTP2120T - C2TLTδP212T + P2220LC2 + δP222LC2 + P2120TA + δP212TA + P2230C2 + δP223C2 + Kgi0TD22TP2130T + δKgiTD22TP2130T + Kgi0TD22TδP213T (71)
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Z22 ) P2120TB2Kgi0 + P2120TB2δKgi + δP212TB2Kgi0 + P2220A + δP222A + P2220B2Kgi0 + P2220B2δKgi + δP222B2Kgi0 - P2220LC2 - δP222LC2 + P2230D22Kg0 + δP223D22Kgi0 + P2230D22δKg + Kgi0TB2TP2120 + δKgiTB2TP2120 + Kgi0TB2TδP212 + ATP2220T + ATδP222T + Kgi0TB2TP2220T + δKgiTB2TP2220T + Kgi0TB2TδP222T - C2TLTP2220T - C2TLTδP222T + Kgi0TD22TP2230T + δKgiTD22TP2230T + Kgi0TD22TδP223T (72) Z31 ) P2130TA + δP213TA + P2230TLC2 + δP223TLC2 + P2330C2 + δP233C2 + ksi0B2TP2110T + δksiB2TP2110T + ksi0B2TδP211 + ksi0D22TLTP2120 + δksiD22TLTP2120T + ksi0D22TLTδP212T + ksi0D22TP2130T + δksiD22TP2130T + ks0D22TδP213T (73) Z32 ) P2130TB2Kgi0 + δP213TB2Kgi0 + P2130TB2δKgi + P2230TA + δP223TA + P2230TB2Kgi0 + δP223TB2Kgi0 + P2230TB2δKgi - P2230TLC2 - δP223TLC2 + P2330D22Kgi0 + δP233D22Kgi0 + P2330D22δKgi + Ks0B2TP2120 + δksiB2TP2120 + ksi0B2TδP212 + ks0D22TLTP2220T + δksiD22TLTP2220T + Ks0D22TLTδP222T + ksi0D22TP2230T + δksiD22TP2230T + ksi0D22TδP223T (74) Z33 ) P2130TB2ksi0 + δP213TB2ksi0 + P2130TB2δksi + P2230TLD22ksi0 + δP223TLD22ksi0 + P2230TLD22δksi + P2330D22ksi0 + δP233D22ksi0 + P2330D22δksi + ksi0B2TP2130 + δksiB2TP2130 + ksi0B2TδP213 + ksi0D22TLTP2230 + δksiD22TLTP2230 + ksi0D22TLTδP223 + ksi0D22TP2330T + δksiD22TP2330T + ksi0D22TδP233T (75) Z41 ) B1TP2110T + B1TδP211T + D21TLTP2120T + D21TLTδP212T + D21TP2130T + D21TδP213T
(76)
Z42 ) B1TP2120 + B1TδP212 + D21TLTP2220T + D21TLTδP222T + D21TP2230T + D21TδP223T
(77)
Z43 ) B1TP2130 + B1TδP213 + D21TLTP2230 + D21TLTδP223 + D21TP2330T + D21TδP233T
(78)
Z44 ) -I
(79)
Nomenclature G(0) ) steady-state gain matrix G* ) complex conjugate transpose of a complex matrix G AT ) transpose of matrix A 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 Tr(A) ) trace of matrix A ||T(s)||2 ) H2 norm of transfer function matrix T(s)
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Received for review February 12, 2002 Revised manuscript received July 29, 2002 Accepted August 26, 2002 IE0201314
