Ind. Eng. Chem. Res. 1999, 38, 3407-3419
3407
Robust Multiloop PID Controller Design: A Successive Semidefinite Programming Approach J. Bao,† J. F. Forbes,*,‡ and P. J. McLellan§ School of Chemical Engineering and Industrial Chemistry, The University of New South Wales, Sydney, New South Wales, Australia 2052, Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G6, and Department of Chemical Engineering, Queen’s University, Kingston, Ontario, Canada, K7L 3N6
The problem of robust multiloop proportional-integral-derivative (PID) controller tuning for multivariable processes is addressed in this paper. The problem is formulated in the H∞ control framework, and the controller parameters are determined based on both user-specified performance and robust stability. The PID settings are computed by solving a nonlinear optimization problem with matrix inequality constraints, using a successive semidefinite programming approach. The proposed method is illustrated by a simple case study that investigates robust PID control of a distillation column. 1. Introduction Proportional-integral-derivative (PID) controllers have been used extensively in the process industries since they are simple and often effective and represent the basic building blocks available in many process control systems. Despite their wide spread use and considerable history, PID tuning is still an active area of research, both academic and industrial (e.g., Wang and Cluett,1 A° stro¨m et al.,2 and Hovd and Skogestad3,4). During the past five decades, a comprehensive PID tuning literature has developed. The first significant tuning method was proposed by Ziegler and Nichols.5 Analytical methods to obtain PID parameters based on simple first- or second-order transfer function models were developed by Rivera et al.6 and Gawthrop and Nomikos.7 For more complicated transfer functions or transfer matrices for multiinput-multioutput (MIMO) systems models, numerical search procedures that minimize different performance objective functions were also proposed (Radke and Isermann,8 Zhuang and Atherton,9 Vega et al.,10 A° stro¨m et al.2). A method for autotuning fully coupled multivariable PID controllers from decentralized relay feedback was presented by Wang et al.11 Since the process models used for controller design are often simplifications or approximations, it is essential that the PID tunings obtained by such methods should tolerate model-plant mismatch. Unfortunately, the above controller design methods do not deal with the robustness issue explicitly and in many design schemes, only control performance is optimized. An H∞ (robust) PID controller synthesis method was first presented by Grimble.12 A genetic algorithm was then proposed to determine PID controller tuning to achieve H∞ optimality by Chen et al.13 In both of these approaches, the H∞ norm of the weighted sensitivity function and complementary sensitivity function are * Author to whom correspondence should be addressed. Phone: (780) 492-0873. Fax: (780) 492-2881. E-mail:
[email protected]. † The University of New South Wales. ‡ University of Alberta. § Queen’s University.
minimized; however, these design approaches are limited to single-input-single-output (SISO) models. For control application engineers, multiloop PID controllers can be preferred to the multivariable approach. Many plants have older, or “legacy”, control systems that do not possess the capabilities to support the implementation of complex multivariable controllers. For these plants, a multiloop PID control scheme does not require purchase of additional control system hardware, as may be required to implement multivariable controllers. Multiloop designs also can have better failure tolerance than the multivariable approach; however, multiloop controllers may suffer from control performance loss and even the instability of closed-loop system when each individual loop is tuned by using SISO tuning methods. This results from the interactions among different loops, which a decentralized control structure cannot deal with.14 The problem of multiloop robust PID controller tuning for MIMO models is addressed in this paper. To minimize the performance loss due to the restriction of a decentralized controller structure, the controller parameters are computed based on the closed-loop system consisting of the full process model and the multiloop controller. This leads to a less conservative design compared to the decentralized control approaches, which use the diagonal subsystem as a design model and treat the known interactions as uncertainties (Morari and Zafiriou,15 and Samyudia et al.16). Hovd and Skogestad3,4 provide a sequential method for building a multiloop design by closing one loop at a time, as well as a good introduction to the area of decentralized controller design. To achieve robustness, the tuning problem posed in this paper is formulated in the H∞ control framework with constraints on the controller structure (i.e., fixedorder, PID, and decentralized structure). This approach also leads to a systematic and unified tool for multiloop PID tuning with user-specified performance and robust stability, which needs much less heuristic user interaction than many of the manual and time-consuming approaches mentioned above. A numerical optimization procedure is proposed to solve the structure-constrained H∞ problem within the framework of semidefinite programming (SDP), since the robust multiloop PID
10.1021/ie980746u CCC: $18.00 © 1999 American Chemical Society Published on Web 08/11/1999
3408 Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999
where
W1(s) ) w1(s)I ∈ Cm×m
(4)
T(s) ) [I + G(s)C(s)]-1 G(s)C(s)
(5)
and
Figure 1. PID controller diagram.
tuning problem can be framed in terms of matrix inequality constraints (Boyd et al.17). Due to the bilinear terms in the matrix inequalities, the optimization problem is nonconvex (Goh et al.18). A computational approach to this problem is developed based on successive semidefinite programming (SSDP). The contributions in this paper are two-fold. First, a formulation of and a practical algorithm for solving the multiloop PID controller design problem is presented. Second, the methods presented here can serve as a basis for analyzing and justifying the benefits of control system improvement projects. This paper begins by formulating the robust PID tuning problem in section 2. The solution to the tuning problem using semidefinite programming is described in section 3. Section 4 presents the distillation column control case study. The paper concludes with discussion in section 5. 2. Robust Multiloop PID Controller Tuning Problem In this section, the robust PID controller tuning problem is formulated in an H∞ control framework. The PID controllers are tuned to optimize controller performance subject to a robust stability constraint. The controller performance and robustness are measured by the H∞ norms of the sensitivity function and complementary sensitivity function, respectively. The norm conditions are converted into matrix inequalities, and the multiloop PID tuning problem is cast as an optimization problem, in SDP form. 2.1. Tuning Problem Formulation. Consider the feedback control diagram as shown in Figure 1. The m × m plant is modeled as a transfer function matrix G(s). The multiloop PID controller consists of a set of individual control loops, which can be arranged in matrix form as
C(s) ) diag{c1(s), c2(s), ..., ci(s), ..., cm(s)}
is the complementary sensitivity function. Assume Ks is the set of stabilizing decentralized controllers with PID structure. The problem in eq 6 maximizes H∞ nominal performance subject to the robust stability constraint given in equality 3
max γ
(6)
|W1(s)T(s)|∞ < 1
(7)
|γW2(s)S(s)|∞ < 1
(8)
S(s) ) [I + G(s)C(s)]-1
(9)
C(s)∈Ks
subject to
where
is the sensitivity function, which denotes the transfer function from setpoint r control error e or from disturbance -d to e, and W2(s) is the frequency-dependent weighting function, which penalizes the control error e (as shown in Figure 1). Therefore, a smaller H∞ norm of S(s) implies better nominal performance in terms of tracking precision and disturbance attenuation. Weighting function W1(s) is determined by the gain of uncertainty at different frequencies. Generally the gain of W1(s) is small at low frequencies and large at high frequencies, because models often describe well the steady-state and low-frequency behavior of processes but become inaccurate at high-frequencies.15 Weighting function W2(s) defines the frequencyweighted, user-specified performance, which is normally chosen such that the performance requirement is satisfied when
|W2(s)S(s)|∞ < 1
(10)
As stated previously, the multiloop PID controller C(s) is designed to optimize controller performance while the robust stability is guaranteed. From the Small-Gain Theorem,19 a sufficient condition for robust stability is given by
(i.e., γ ) 1). The optimal value of γ indicates the level of performance actually achieved relative to the performance specification. If γ > 1, the controller [C(s)] obtained from the solution of eq 6 will provide better performance than what is specified in eq 8; if γ < 1, the performance achieved is at a lower level compared to the specification in eq 8. A rule of thumb for choosing W2(s) is as follows. (1) The gain of W2(s) is high at low frequencies where performance is required and low at high frequencies where robustness is emphasized. (2) If the L2 gain of the frequency response of disturbance d on output y is desired to be attenuated to |d(jω1)|2/R at frequency ω1, then gain of W2(jω1) should be
|W1(s)T(s)|∞ < 1
σmin[W2(jω1)] > R
(1)
The model-plant mismatch is represented as multiplicative uncertainty ∆(s). Further, it is assumed that system ∆(s) is stable and its maximum singular value is bounded by
σmax[∆(jω)] < |w1(jω)| ∀ω ∈ R
(2)
(3)
(11)
Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999 3409
(3) For a desired settling time Ts, the crossover frequency ωc of W2(s) is chosen from the following empirical formula:20
4 9 e ωc e Ts Ts
(12)
Both W1(s) and W2(s) should be stable, minimum phase, and diagonal. The nominal performance and robustness compete with each other, thus the achievable performance is limited by the gain of the uncertainty. As a result, the crossover frequency ωc of W2(s) given in eq 10 must be sufficiently below the crossover frequency of W1(s), otherwise the robustness and performance requirements in eqs 3 and 8 cannot be satisfied simultaneously. Typical choices of the weighting functions are given in the simple case study in section 4. 2.2. A Matrix Inequality Approach The problem defined in section 2.1 is actually an H∞ controller synthesis problem with constraints on the controller structure. Although there are numerous solvers for the standard H∞ control problem in the robust control literature (Doyle et al.21 and Gahinet and Apkarian22), none of these approaches can deal with the controller structure constraints required here (Safonov et al.23). In this section, a direct matrix inequality approach is developed for this problem. Denoting
Tcl(s) )
[
W1(s)T(s) γW2(s)S(s)
]
(13)
the controller tuning problem can be formulated as the optimization problem:
min -γ
(14)
C(s)∈K
Figure 2. Linear fractional transformation.
All actuators have physical constraints. Actuator saturation may even destabilize the closed-loop system. Note that in eq 14 the controller gains are indirectly limited by eq 15. Therefore, when controller saturation is a concern, ensuring the weighting function W1(s) has a sufficiently large gain can limit the controller gain and avoid actuator saturation. Further, heuristics are normally built into PID loops to compensate for the effects of controller saturation (e.g., antireset wind-up). It is noted that when |Tcl(s)|∞ < (2)-1/2 the robust sensitivity performance is also obtained (Chiang24). In this case, the performance specification |γW2(s)S(s)|∞ < 1 is achieved for any multiplicative uncertainty ∆(s) bounded by eq 2. That is, the user-specified performance is guaranteed for the worst-case model-plant mismatch. To represent different controller specifications in a unified framework, the model G(s) is augmented into a two-port generalized plant (McFarlane and Glover25):
P(s) )
P11(s) P12(s) P21(s) P22(s)
]
(18)
As shown in Figure 2, the system from w to z is given as the lower linear fractional transformation (LFT):
F l[P(s),C(s)] } P11(s) + P12(s)C(s)[I - P22(s)C(s)]-1P21(s) (19) whenever
det[I - P22(s)C(s)] * 0
subject to
||
W1(s)T(s) |Tcl(s)|∞ ) γW2(s)S(s) Tcl(s) ∈ RH∞
||
∞
0 where
(25)
)
BTX
CT
where the matrix inequality represents a partial ordering and indicates negative definiteness (), or positive semidefiniteness (g). Assume the model is G(s) ) (Ag, Bg, Cg, Dg), the weighting functions are W(s) ) (Aw1, Bw1, Cw1, Dw1), W2(s) ) (Aw2, Bw2, Cw2, Dw2) and the multiloop PID controller to be designed is C(s) ) (Ak, Bk, Ck, Dk). Equation 14 is cast into the following minimization problem with constraints given by matrix inequalities:
Acl )
kri TDi 1TLi TLi
TDi k4,i ) kri TLi
]
[
ATX + XA XB
(24)
kri k2,i ) TIiTLi k3,i )
0
0 0 · · · · · · · · · [k2,m k3,m]
0
[
[
0 1 0 -k1,m
· · ·
0 1
0
0
Dk )
where
0
0
Bk )
Ck )
]
(23)
Ccl, Dcl). The details of deduction of (Acl, Bcl, Ccl, Dcl) from any augmented plant P are shown in Appendix B. The H∞ constraints can be converted into a matrix inequality by using the following lemma. Lemma 1. Bounded Real Lemma (Gahinet and Apkarian22). Consider a continuous-time transfer function T(s) of (not necessarily minimal) realization T(s) ) D + C(sI - A)-1 B. The following statements are equivalent. (1) |D + C(sI - A)-1 B|∞ < γ and T(s) is stable {Re[λi(A)] < 0} and (2) there exists a real symmetric positive definite solution X to the LMI:
[
Ag - BgDkCg 0
BgCk
-Bw2Cg
Aw2 0
Bw1Cg
0
Aw1 0
0
0
-BkCg
Bcl ) (27) Ccl )
[]
0 Ak
BgDk
(26)
The state-space representation above incorporates both the tuning parameters and the structure of multiloop PID controllers. The state-space realization of Tcl(s), which contains the realizations of the controller, the plant model, optimization variable γ, and the weighting functions explicitly, can then be obtained as (Acl, Bcl,
0
[
Bw2
(32)
0
Bk
0 Dw1Cg Dcl )
γCw2 0
[ ] γDw2 0
]
(31)
0
0
Cw1 0
]
(33)
(34)
Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999 3411
If Acl ∈ Rn×n and the control system has m loops, then m
Ak ) E0 +
xiEi ∑ i)1
the bilinear terms in eq 30, and X ) X0 and Y ) Y0 are feasible solutions to eq 29. Then, the bilinear terms can be approximated about X0 and Y0 as
(35) XY ) X0δY + δXY0 + X0Y0 + δXδY ≈ X0δY + δXY0 + X0Y0
2m
Ck ) F0 +
xi+mFi ∑ i)1
(36)
xi+3mGi ∑ i)1
(37)
|δX| e and |δY| e
(38)
where is an arbitrary small positive number such that the solution region of eq 44 is not too far from eq 43. The solution of eq 29 in the neighborhood of Ak0, Ck0, Dk0, and X0 can be found by solving the following approximated semidefinite programming problem:
n(n+1)/2
X ) H0 +
∑ i)1
xi+4mHi
(44)
where δX ) X - X0 and δY ) Y - Y0. Both δX and δY are restricted by
m
Dk ) G0 +
(43)
The realization of Tcl(s) ) (Acl, Bcl, Ccl, Dcl) is obtained under the assumption that the model is strictly proper, which greatly simplifies the optimization procedure. The matrix variables are Ak, Ck, Dk, and X, where Ak, Ck, and Dk contain the controller parameters to be tuned and X is a symmetric auxiliary matrix variable. Equations 35-37 impose controller structure constraints on the optimization problem. The constant matrices Hi [i ) 0, ..., n(n + 1)/2] are symmetric and constant matrices Ei, Gi, (i ) 0, ..., m) and Fi (i ) 0, ..., 2m) are dictated by the controller structure given in eq 23 (as shown in Appendix C). The stability condition for the closed-loop system of G(s) and C(s) is implied in eq 29. The decision variables are x1, ..., x4m+n(n+1)/2. From eqs 35 to 37 and 23, it can be seen that the first 4m decision variables correspond to the state-space controller variables k1i, k2i, k3i, and k4i:
k1i ) xi i ) 1...m
(39)
k2i ) xi+m i ) 1...m
(40)
k3i ) xi+2m i ) 1...m
(41)
k4i ) xi+3m i ) 1...m
(42)
The controller tuning parameters can be calculated from eqs 24 to 27. 3. Successive Semidefinite Programming Conventional SDP techniques cannot be used to solve eq 29 because the decision variables x ) [x1, ..., x4m+n(n+1)/2] in the matrices Ak, Ck, Dk, and X form bilinear terms in eq 30. In this section, a successive semidefinite programming approach, analogous to successive linear programming (SLP) or successive quadratic programming (SQP),27 is developed to solve this optimization problem iteratively. Thake et al.28 first proposed one approach to SSDP for controller approximation. This method is extended to solve the optimization problem with bilinear constraints. The nonlinear constraints are approximated by linear matrix inequalities in the neighborhood of estimated solutions, and the linearized subproblem is solved by using semidefinite programming. 3.1. Linear Approximation of the Constraints. Suppose X and Y are the matrix variables, which form
(45)
min -γ
(46)
x
subject to
[
δATclX0+ATcl0δX+ATcl0X0+ δXBcl0+X0δBcl+ δCTcl+CTcl0 δXAcl0+X0δAcl0 X0Bcl0 δBTclX0+BTcl0δX+BTcl0X0 δCcl+Ccl0
-I δDcl+Dcl0
δDTcl+DTcl0 -I
]
0
(48)
δXT ) δX
(49)
where δAcl and δBcl are functions of δAk, δCk, δDk and δCcl, and δDcl are functions of γ. Matrix variables δAk, δCk, δDk, and δX are functions of the decision variable x ) [x1, ..., x4m+n(n+1)/2]T, similar to eqs 35-38. Matrices Ak0, Ck0, Dk0, and X0 are constants. The detail of the linearized matrix eqs 47 and 48 constraints are shown in Appendix D. 3.2. Successive Semidefinite Programming Procedure. The successive semidefinite programming (SSDP) procedure proposed here is described as follows. (1) Choose initial values of Ak0, Ck0, and Dk0. Fix Ak ) Ak0, Ck ) Ck0, and Dk ) Dk0. Solve the resulting LMI problem (eq 29) and use the solution (X ˜ , γ˜ ) as initial ˜ and γ0 ) γ˜ . Set the initial solution radius value of X0 ) X as 0, the maximum number of iterations as n, convergence tolerance as ζ, and preset iteration counter k ) 0. (2) Solve eq 46 with the initial values Ak0, Ck0, Dk0, and X0, and γ0. Assume at kth iteration, the solutions δAk, δCk, δDk, δX, and γk are obtained with their radii restricted by
|δAk| e k |δCk| e k |δDk| e k |δX| e k
(50)
˜ k ) Ck0 + δCk, D ˜k ) (3) Compute A ˜ k ) Ak0 + δAk, C ˜ ) X0 + δX. Dk0 + δDk, and X (a) If A ˜ k, C ˜ k, D ˜ k, and X ˜ are feasible solutions to eq 29, (i) If |γk - γk-1| e ζ, then acceptable solution obtained and proceed to step 4.
3412 Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999
(ii) If |γk - γk-1| > ζ, then A ˜ k, C ˜ k, D ˜ k, and X ˜ are used as the reference values for the next approximated ˜ k, Ck0 ) C ˜ k, Dk0 ) D ˜ k, optimization problem. Let Ak0 ) A ˜ , and k+1 ) Rk (where R is user-specified tuning X0 ) X parameter, typically R ) 0.95), and go to step 2. (b) If A ˜ k, C ˜ k, D ˜ k, and X ˜ are not feasible solutions to eq 29, (i) If k > n, then eq 29 is infeasible in the neighborhood of (Ak0, Ck0, Dk0, and X0) and stop. (ii) If k e n, then reject these solutions and choose k+1 ) βk (where β is another user-specified tuning parameter, typically β ) 1.05) and go to step 2. (4) The state-space representation of the multiloop ˜ k, PID controller is obtained as C(s) ) diag{ci(s)} ) (A ˜ k, D ˜ k). The controller parameters in eq 22 are B ˜ k, C calculated by the following equations:
kri )
k3i + k4ik1i k1i
TLi ) 1/k1i
(51) (52) Figure 3. Distillation column.
k3i + k4ik1i TIi ) k2i
(53)
k4i k3i + k4ik1i
(54)
TDi )
Equation 46 is a SDP problem. It is convex and can be easily solved by Matlab LMI Toolbox. If eq 29 is infeasible in the neighborhood of (Ak0, Ck0, Dk0, X0), the optimization procedure should be restarted with a new initial point. The user tuning parameters in the SSDP procedures are initial solution radius 0, maximum number of iterations n, the convergence tolerance ζ, and the solution radius modifiers R and β. Although the solution radius is automatically adjusted during the iterations, the initial value of 0 should be chosen small enough such that the solution of each LMI step does not violate the nonlinear constraints in eq 29. It is noted that both the complexity and dimensionality may increase significantly due to the introduction of the auxiliary matrix variable X into the optimization problem given in eq 29 and δX in eq 46. This increased complexity can be hidden from the user as both the problem formulation and solution procedure proposed here are entirely mechanical and can be performed automatically by a computer. The proposed SSDP procedure has the advantage that standard, very efficient, and reliable algorithms are available for the execution of each LMI step. The optimization of eq 29 is not convex, and as a result, the global optimum may be hard to find.18 Thus, the proposed SSDP algorithm can only be guaranteed to find a local optimum. Further issues of convergence and algorithm stability of the proposed SSDP approach are beyond the scope of this paper. 4. Distillation Control Case Study A simple robust PID control case study is presented in this section to illustrate the proposed tuning method.
The system under consideration is a distillation column as shown in Figure 3. Tray temperatures are regulated, as inferential variables for composition, by manipulating liquid reflux and vapor boilup rates. The transfer function of the plant is given as follows [cf. Ogunnaike and Ray,29 (p 813)]:
[
]
G(s) )
-33.89 32.63 (98.02s + 1)(0.42s + 1) (99.6s + 1)(0.35s + 1) 34.84 -18.85 (75.43s + 1)(0.30s + 1) (110.5s + 1)(0.03s + 1) (55)
where the outputs 1 and 2 are temperatures of trays 21 and tray 7 and inputs 1 and 2 are the liquid reflux and vapor boilup, respectively. 4.1. Controller Design Specifications. From relative gain array (RGA) analysis, a suitable multiloop configuration consists of output 1 paired with input 1 and output 2 paired with input 2 (Thake et al.8). The multiloop PID controllers are designed to achieve the following robustness and performance specifications. (1) Robust stability. The closed-loop system including the plant and the controller is robustly stable with the following multiplicative uncertainty ∆(s):
σmax[∆(jω)]
0 δX34 + X340 δX44 + X440
(71)
Literature Cited (1) Wang, L.; Cluett, W. R. PID controller design for integrating processes. IEE Proc.-D, Control Theory Appl. 1997, 144 (5), 385392. (2) A° stro¨m, K. J.; Panaagopoulos, H.; Hagglund, T. Design of PI Controllers Based on Non-convex Optimization. Automatica 1998, 34 (5), 585-601. (3) Hovd, M.; Skogestad, S. Improved Independent Design of Robust Decentralized Controllers. J. Process Control 1993, 3 (1), 43-51. (4) Hovd, M.; Skogestad, S. Sequential Design of Decentralized Controllers. Automatica, 1994, 30 (10), 1601-1607. (5) Ziegler, J. G.; Nichols, N. B. Optimum Settings for Automatic Controllers. Trans. ASME 1942, 62, 759-768. (6) Rivera, D. E.; Morari, M.; Skogestad, S. Internal Model Control. 4. PID Control Design. Ind. Eng. Chem. Process Des. Dev. 1986, 25, 252-265. (7) Gawthrop, P. J.; Nomikos, P. E. Automatic Tuning of Commercial PID Controllers for Single-loop and Multi-loop Applications IEEE Conrol Syst. Mag. 1990, 10, 34-42. (8) Radke, F.; Isermann, R. A Parameter-adaptive PID Controller with Stepwise Parameter Optimization. Automatica 1987, 23, 449-457. (9) Zhuang, M.; Atherton, D. P. Automatic Tuning of Optimum PID Controllers. IEE Proc.-D, Control Theory Appl. 1993, 140, 216-224. (10) Vega, P.; Prada, C.; Alexiandre, V. Self-tuning Predictive PID Control. IEE Proc.-D 1991, 138, 303-311. (11) Wang, Q. G.; Zou, B.; Lee, T. H.; Qiang, B. Auto-tuning of Multivariable PID Controllers from Decentralized Relay Feedback. Automatica 1997, 33 (3), 319-330. (12) Grimble, M. J. H∞ Controllers with a PID Structure J. Dyn. Syst. Meas. Control 1990, 112, 325-336. (13) Chen, B. S.; Cheng, Y. M.; Lee, C. H. A Genetic Approach to Mixed H2/H∞ Optimal PID Control. IEEE Control Syst. 1995, 15 (5), 51-60.
(14) Grosdidier, P.; Morari, M. Interaction Measures for Systems Under Decentralised Control. Automatica 1996, 22 (3), 309319. (15) Morari, M.; Zafiriou, E. Robust Process Control; Prentice Hall: Englewood-Cliffs, NJ, 1989. (16) Samyudia, Y.; Green, M.; Lee, P. L.; Cameron, I. T. A New Approach to Decentralised Control Design. Chem. Eng. Sci. 1995, 50 (11), 1695-1706. (17) Boyd, S.; Ghaoui, L. E.; Feron, E.; Balakrishnan, V. Linear Matrix Inequalities in System and Control Theory; SIAM: Philidelphia, PA, 1994. (18) Goh, K. C.; Safonov, M. G.; Ly, J. H. Robust Synthesis via Bilinear Matrix Inequalities. Int. J. Robust Nonlinear Control 1996, 6 (9-10), 1079-1095. (19) Zames, G. On the Input and Output Stability of TimeVarying Nonlinear Feedback Systems. IEEE Trans. Autom. Control 1966, 11, 228-476. (20) Franklin, G. F.; Powell, J. D.; Naeini, A. E. Feedback Control of Dynamic Systems; Addision-Wesley: Reading, MA, 1994. (21) Doyle, J. C.; Glover, K.; Khargonekar, P.; Francis, B. State Space Solution to Standard H2 and H∞ Control Problems. IEEE Trans. Autom. Control 1989, 34 (8), 832-847. (22) Gahinet, P.; Apkarian, P. A. Linear Matrix Inequality Approach to H∞ Control, Int. J. Robust and Nonlinear Control 1994, 4, 421-448. (23) Safonov, M. G.; Goh, K. C.; Ly, J. H. Control System Synthesis via Bilinear Matrix Inequalities. Proc. Am. Control Conf. (Baltimore, MA) 1994. (24) Chiang, R. Y. Modern Robust Control Theory. Ph.D. Dissertation, University of South California, Los Angeles, CA, 1988. (25) McFarlane, D. C.; Glover, K. Robust Control Design using Normalized Coprime Factor Plant Descriptions (Lecture Notes in Control and Information Sciences); Springer-Verlag: Berlin, 1990. (26) Seborg, D. E.; Edgar, T. F.; Mellichamp, D. A. Process Dynamics and Control; John Wiley: New York, 1989. (27) Gill, P. E.; Murray, W.; Wright, M. H. Practical Optimization; Academic Press: New York, 1981. (28) Thake, A. J.; Forbes, J. F.; McLellan, P. J. Approximation of High-Dimensional Multivariable Controllers using SemiDefinite Programming. IFAC Int. Symp. Adv. Control Chem. Processes (Banff, Canada) 1997. (29) Ogunnaike, B. A.; Ray, W. H. Process Dynamics, Modeling and Control; Oxford University Press: New York, 1994.
Received for review November 30, 1998 Revised manuscript received June 4, 1999 Accepted June 7, 1999 IE980746U
