Article pubs.acs.org/IECR
Control Configuration Selection for Multivariable Nonlinear Systems Hamid Reza Shaker*,† and Mohammad Komareji‡ †
Department of Energy Technology, Aalborg University, Pontoppidanstræde 101, 9220 Aalborg, Denmark Singapore University of Science and Technology, 20 Dover Drive Singapore 138682
‡
ABSTRACT: Control configuration selection is the procedure of choosing the appropriate input and output pairs for the design of SISO (or block) controllers. This step is an important prerequisite for a successful industrial control strategy. In industrial practices, it is often the case that systems, which are needed to be controlled, are nonlinear, and linear models are insufficient to describe the behavior of the processes. The focus of this work is on the problem of control configuration selection for such systems. A gramian-based interaction measure for control configuration selection of MIMO nonlinear processes is described. In general, most of the results on the control configuration selection, which have been proposed so far, can only support linear systems. The proposed gramian-based interaction measure not only supports nonlinear processes but also can be used to propose a richer sparse or block diagonal controller structure. The interaction measure is used for control configuration selection of the nonlinear quadruple-tank process.
1. INTRODUCTION The technological world of today has been witnessing increased complexity due to the rapid development of the process plants and the manufacturing processes. The computational complexity, the reliability problems, and the restrictions in communication make the centralized control of such large-scale complex systems expensive and difficult. To cope with these problems, several decentralized control structures have been introduced and implemented over the last few decades.1 The decentralized controllers have several advantages, which make them popular in industry. The decentralized controllers are easy to understand for operators and easy to implement and to retune.1,2 The decentralized control systems design is a two-step procedure. The first step is the control configuration selection, which is the procedure of choosing the appropriate input and output pairs for the design of SISO (or block) controllers. The controller synthesis for each channel is the second step of the decentralized control. The focus of this work is on control configuration selection. This issue is a key problem in the design of the decentralized and distributed control systems, which directly affects the stability and the performance of the control systems. The interaction measures play an important role in the suitable pairing and the controller structure selection for the decentralized and the distributed control. Interaction measures make it possible to study input−output interactions and to partition a process into subsystems in order to reduce the coupling, to facilitate the control and to achieve a satisfactory performance. The interaction measures have received a lot of attention over the last few decades.2−4 There are two broad categories of interaction measures in the literature. The first category is the relative gain array (RGA) and its related indices5−10 and the second category is the family of the gramian-based interaction measures.11−14 The most well-known and commonly used interaction measure is the relative gain array (RGA), which was first proposed by Bristol.5 In the RGA, d.c. gain of the process is used for the construction of the channel interaction measure. © 2012 American Chemical Society
The RGA is not sensitive to delays and more importantly, it considers the process just in the particular frequency. The RGA has been studied by several other researchers.6,7 There are also other similar measures of interaction, which use the d.c. gain of the process, e.g., the NI (the Niederlinski index).8 The NI does not provide more information for control configuration selection compared to RGA. The RGA and the NI have been extended for input−output pairing of unstable MIMO systems.2 The relative interaction array (RIA) is an interaction measure, which is similar to RGA and it is based on considering the interaction as an unmodelled term at d.c. RIA, however, does not provide more information than the RGA about the channel interactions of the process. These indices use the model of the processes at zero frequency. The relative dynamic gain array (RDGA) was proposed for the first time by Witcher and McAvoy.7 The RDGA shows how the interaction varies over the frequency. The idea is further generalized by the generalized relative dynamic gains (GRDG).10 The second category of the interaction measures is the family of the gramian-based methods. A method from this category was first proposed by Conley and Salgado.11,12 In this category, the observability and the controllability gramians are used to form the Participation Matrix (PM). The elements of the PM encode the information of the channel interactions. PM is used for control configuration selection. The Hankel Interaction Index Array (HIIA) is a similar interaction measure.13 The gramian-based interaction measures have several advantages over the interaction measures in the RGA category. The gramian-based interaction measures take the whole frequency range into account rather than a single frequency. This family of the interaction measures provides more information on channel interactions and allows more complicated controller structures. Received: Revised: Accepted: Published: 8583
May 2, 2012 May 30, 2012 June 4, 2012 June 4, 2012 dx.doi.org/10.1021/ie301137k | Ind. Eng. Chem. Res. 2012, 51, 8583−8587
Industrial & Engineering Chemistry Research
Article
x(̇ t ) = Ax(t ) + Bu(t ), x(t ) ∈ n , u(t ) ∈ p ,
More details on the applications and the differences between two main categories of the interaction measures can be found in the literature.4,12−15 A potential weakness of the gramian based interaction measure is that the PM is sensitive to input and output scaling. However, methods to deal with this issue have already been presented.12 The results on the gramian-based interaction measures, which have been proposed so far, only support linear systems. However, in industrial practices, it is often the case that the system, which is needed to be controlled, is nonlinear and linear models are insufficient to describe the process. In this work, a gramian-based interaction measure is extended to support a nonlinear system. The proposed interaction measure is used for control configuration selection. The proposed gramian-based interaction measure not only supports nonlinear processes but also can be used to propose a richer sparse or block diagonal controller structure. The work is organized as follows. In the next section, we review the concept of the gramians for the linear systems as well as nonlinear systems. The interpretation of the controllability and observability, gramians is also discussed in this section. Section 3 presents how gramians can be used to quantify the channel interactions for nonlinear systems. The application of the proposed interaction measure in control configuration selection is explained in this section. In Section 4, the proposed interaction measure is used for the control configuration selection of the nonlinear quadruple-tank process. Section 5 concludes the work. The notation used in this work is as follows: M* denotes the transpose of matrix M if M ∈ n×m and complex conjugate transpose if M ∈ n×m. The standard notation >, ≥ ( 1/ p2, therefore their associated elementary subsystems are good candidates to be involved in the nominal model. However, the best paring for a decentralized controller can be obtained with (u1, y1), (u2, y3), and (u3, y2), which are associated with the following:
(12)
where, ψij =
(15)
∑ = ψ11 + ψ23 + ψ32 = 0.4307 trace[Wc , jWo , i] p p ∑i = 1 ∑j = 1 trace[Wc , jWo , i]
The structure of the nominal model is as follows: ⎡* 0 0⎤ ⎢ ⎥ struc(Πo) = ⎢ 0 0 * ⎥ ⎢⎣ 0 * 0 ⎥⎦
(13)
Note that 0 ≤ ψij < 1 and ∑i =p 1∑j =p 1ψij = 1. The participation matrix highlights the elementary subsystems, which are more important in the description of MIMO systems, and in this way, it shows the suitable pairing and the appropriate controller structure to select. For control configuration selection, the structure of the nominal system Πn needs to be obtained. The nominal model is a model, which is obtained by keeping some of the elementary subsystems of the actual MIMO process and ignoring the rest. For example, assume that one of the ordinary methods for pairing is used and a decentralized control is synthesized. If the inputs and outputs are relabeled, then one only needs to design p independent SISO controller loops, for elementary diagonal subsystems. In this case: ⎡* 0 0⎤ ⎢ ⎥ struc(Πo) = ⎢ 0 ⋱ 0 ⎥ ⎢⎣ 0 0 * ⎥⎦ p×p
A simple controller structure for selection is the structure of Π−1 o : struc(C) =
struc(Π−o 1)
⎡* 0 0⎤ ⎢ ⎥ = ⎢ 0 0 *⎥ ⎢⎣ 0 * 0 ⎥⎦
If it is practically possible to use more complicated control structures than completely decentralized control, y2 could be commanded from u2 and then we will have:
∑ = ψ11 + ψ12 + ψ23 + ψ32 = 0.599 The structure of the nominal model then will be as follows: ⎡* * 0⎤ ⎢ ⎥ struc(Πo) = ⎢ 0 0 * ⎥ ⎢⎣ 0 * 0 ⎥⎦
(14)
For the designed controller C, we have: 8585
dx.doi.org/10.1021/ie301137k | Ind. Eng. Chem. Res. 2012, 51, 8583−8587
Industrial & Engineering Chemistry Research
Article
A simple controller structure to select:
Table 1. Quadruple-Tank Plant Parameters
⎡* 0 *⎤ ⎢ ⎥ −1 struc(C) = struc(Π o ) = ⎢ 0 0 * ⎥ ⎢⎣ 0 * 0 ⎥⎦
In this case, the structure is partially decentralized.
4. CONTROL CONFIGURATION SELECTION FOR QUADRUPLE-TANK PLANT In this section, the proposed interaction measure is used for control configuration selection of a Quadruple-tank.4,23 The schematic of the Quadruple-tank is shown in Figure 2. The
parameters
value
A1, A3 [cm2] A2, A4 [cm2] a1, a3 [cm2] a2, a4 [cm2] g [cm/s2 ] k1[cm2/(V s)] k2 [cm2/(V s)] γ1 γ2
28 32 0.071 0.057 981 3.33 3.35 0.7 0.6
Therefore, the suggested decentralized structure for the nominal model is as follows: ⎡* 0⎤ struc(Πo) = ⎢ ⎥ ⎣ 0 *⎦
The suggested decentralized controller structure for this model is the structure of Π−1 o : ⎡* 0⎤ struc(C) = struc(Π−o 1) = ⎢ ⎥ ⎣ 0 *⎦
This participation matrix furthermore suggests a more complicated (partially decentralized) structure for the nominal systems:
⎡ * *⎤ struc(Πo) = ⎢ ⎥ ⎣ 0 *⎦ The suggested decentralized controller structure for this model is the structure of Π−1 o , which is as follows:
Figure 2. Schematic diagram of the quadruple-tank process.4,23
⎡ * *⎤ struc(C) = struc(Π−o 1) = ⎢ ⎥ ⎣ 0 *⎦
objective of the control is to control the two lower tanks levels, h1 and h2, using the two pumps. Input voltages to the pumps are v1 and v2. The equations which describe the plant are as follows:
5. CONCLUSIONS Control configuration selection for nonlinear systems has been addressed in this work. A general gramian-based interaction measure for the control configuration selection for such systems has been proposed. The proposed MIMO interaction measure is the extension of its gramian-based analogous counterpart, which was proposed for input−output pairing, as well as for the controller architecture selection for the processes with standard state-space. The proposed measure reveals more information about the ability of the channels to be controlled and to be observed and provides hints for the selection of richer controller structures such as triangular, sparse, and block diagonal.
γ k1 a dh1 a = − 1 2gh1 + 3 2gh3 + 1 v1 dt A1 A1 A1 γ k2 a dh2 a = − 2 2gh2 + 4 2gh4 + 2 v2 dt A2 A2 A2 (1 − γ2)k 2 dh3 a = − 3 2gh3 + v2 dt A3 A3 (1 − γ2)k1 dh4 a v1 = − 4 2gh4 + dt A4 A4
■
(16)
where Ai is the cross section of tank i, ai is the cross section of the outlet hole and hi is the water level. The inputs are chosen as u1 = v1, u2 = v2 and the outputs as y1 = h1, y2 = h2. The parameters of the Quadruple-tank plant which are used in our study are shown in Table 1. For these parameters, the empirical gramians are computed and the empirical gramian based participation matrix for control configuration selection will be as follows:
AUTHOR INFORMATION
Corresponding Author
*Tel: +45-9940-9282. Fax: +45-9815-1411. E-mail:
[email protected]. dk. Notes
The authors declare no competing financial interest.
■
REFERENCES
(1) Scattolini, R. Architectures for distributed and hierarchical model predictive controlA review. J. Proc. Control 2009, 19, 723−731. (2) Hovd, M.; Skogestad, S. Pairing criteria for decentralised control of unstable plants. Ind. Eng. Chem. Res. 1994, 33, 2134−2139.
⎡ 0.5218 0.4334 ⎤ Ψ=⎢ ⎥ ⎣ 0.0316 0.0132 ⎦ 8586
dx.doi.org/10.1021/ie301137k | Ind. Eng. Chem. Res. 2012, 51, 8583−8587
Industrial & Engineering Chemistry Research
Article
(3) Van de Wal, M.; De Jager, B. A review of methods for inputoutput selection. Automatica 2001, 37, 487−510. (4) Khaki-Sedigh, A.; Moaveni, B. Control Configuration Selection for Multivariable Plants. Lecture Notes in Control and Information Sciences; Springer: Berlin/Heidelberg, 2009. (5) Bristol, E. H. On a new measure of interaction for multivariable process control. Automatic control. IEEE Trans. 1966, 11, 133−134. (6) Skogestad, S.; Morari, M. Implications of large RGA elements on control performance. Ind. Eng. Chem. Res. 1987, 26, 2323−2330. (7) Witcher, M. F.; McAvoy, T. J. Interacting control systems: Steady-state and dynamic measurement of interaction. ISA Trans. 1977, 16, 35−41. (8) Niederlinski, A. A heuristic approach to the design of linear multivariable interacting control systems. Automatica 1971, 7, 691− 701. (9) Bristol, E. H.. Recent results on interaction in multivariable process control. AIChE 71st Annual Meeting; Miami, FL, 1978. (10) Gagnon, E.; Desbiens, A.; Pomerleau, A. Selection of pairing and constrained robust decentralized PI controllers. Proceedings of the American Control Conference; San Diego, CA, 1999. (11) Conley, A.; Salgado, M. E. Gramian based interaction measure. Proceedings of the 39th IEEE Conference on Decision & Control; Sydney, Australia, 2000. (12) Salgado, M. E.; Conley, A. MIMO interaction measure and controller structure selection. Int. J. Control 2004, 77, 367−383. (13) Wittenmark, B.; Salgado, M. E. Hankel-norm based interaction measure for input−output pairing. Proceedings of the 15th IFAC World Congress; Barcelona, Spain, 2000. (14) Halvarsson, B. Comparison of some Gramian based interaction measures. Proceedings of the IEEE Multi-conference on Systems and Control; San Antonio, Texas, 2008. (15) Samuelsson, P.; Halvarsson, B.; Carlsson, B. Interaction analysis and control structure selection in a wastewater treatment plant model. Control systems technology. IEEE Trans. 2005, 13, 955−964. (16) Anthoulas, A. C. (2005). Approximation of Large-Scale Dynamical Systems. Advances in Design and Control, Philadelphia SIAM (17) Gugercin, S.; Antoulas, A. C. A survey of model reduction by balanced truncation and some new results. Int. J. Control 2004, 77, 748−766. (18) Moore, B. Principal component analysis in linear systems: Controllability, observability, and model reduction. Automatic Control. IEEE Trans. 1981, 26, 17−32. (19) Lall, S., Marsden, J. E., Glavaski, S. Hankel-norm based interaction measure for input-output pairing. Proceedings of the 14th IFAC World Congress; Beijing, P. R. China, 1999. (20) Lall, S.; Marsden, J. E.; Glavaski, S. A subspace approach to balanced truncation for model reduction of nonlinear control systems. Int. J. Robust Nonlinear Control 2002, 12, 519−535. (21) Hahn, J.; Edgar, T. F. An improved method for nonlinear model reduction using balancing of empirical Gramians. Comput. Chem. Eng. 2002, 26, 1379−1397. (22) Hahn, J.; Edgar, T. F.; Marquardt, W. Controllability and observability covariance matrices for the analysis and order reduction of stable nonlinear systems. J. Proc. Control 2003, 13, 115−127. (23) Johansson, K. H. The quadruple-tank process: A multivariable laboratory process with an adjustable zero. Control systems technology. IEEE Trans. 2000, 8, 456−465.
8587
dx.doi.org/10.1021/ie301137k | Ind. Eng. Chem. Res. 2012, 51, 8583−8587
