Frequency-Interval Interaction Measure for Control Configuration Selection for Multivariable Processes Hamid Reza Shaker Abstract— In many applications one is interested in control and analysis within a bounded frequency-interval. For such applications, the input-output interactions within the desired frequency-interval need to be quantified rather than the interactions within the whole frequency range. However, the interaction measures, which have been proposed so far, either quantify the input-output interactions for a single frequency or the whole frequency range. Motivated by this, a new interaction measure is proposed in this paper. The proposed interaction measure uses frequency-interval gramians and it can be used for control configuration selection of MIMO processes. Compared with other counterparts, the proposed gramianbased interaction measure can encode more information on the input-output interactions over a desired frequency-interval. This interaction measure in addition to decentralized control can be used to propose a richer sparse or block diagonal controller structure for distributed and partially decentralized control of multivariable systems. The method is illustrated with the help of a numerical example.
I. I NTRODUCTION The technological world of today has been witnessing the 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, easy to implement and to re-tune [1], [2]. The decentralized control systems design is a two-step procedure. The controller configuration selection is the first main step and the controller synthesis for each channel is the second step. Control configuration selection is the procedure of choosing the appropriate input and output pairs for the design of decoupled (SISO or block) controllers for multivariable systems. The focus of this paper is on the controller configuration selection for decentralized and partially decentralized control systems. This is an important issue and it directly affects the stability and the performance of the control systems. The interaction measures play an important role in the suitable controller configuration selection. 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 H. R. Shaker is with the Department of Energy Technology, Aalborg University, Pontoppidanstrde 101, 9220 Aalborg, Denmark
[email protected]
satisfactory performance. There are two broad categories of interaction measures in the literature. The first category is the relative gain array (RGA) and its related indices [10][15] and the second category is the family of the gramianbased interaction measures [16]-[22]. The relative gain array (RGA) is the most well-known interaction measure. It was first proposed in [10]. RGA uses d.c. gain of the process to quantify the channel interactions. The RGA can also be computed for a particular frequency other than zero. However, RGA is not sensitive to delays. The RGA has been studied by several other researchers (see, e.g. [11],[12]). There are also other similar measures of interaction, which use dc gain of the process e. g. the NI (the Niederlinski index) [13]. The second category of the interaction measures is the family of the gramian based methods. A method from this category was first proposed in [16] and further in [17]. 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 on the channel interactions. PM is used for pairing and the controller structure selection. The Hankel Interaction Index Array (HIIA) is a similar interaction measure, which was proposed in [18]. 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 suggests more suitable pairing and allows more complicated controller structures. For more details on the applications and the differences between two main categories of the interaction measures, see [17] and [19]. It is clear from this short review that the interaction measures, which have been proposed so far, either show the input-output interactions for a single frequency or the whole frequency range. However, in many applications, one is interested in analysis and control of a system within a frequency-interval. In such cases, the interaction measure should focus on the frequency-interval of interest rather than the whole frequency range. In this paper a new interaction measure is proposed for linear multivariable systems. Since the proposed interaction measure uses frequency-interval gramians it encodes more information on interactions over the desired frequencyinterval. The proposed interaction measure is used for control configuration selection of linear MIMO systems. The proposed gramian-based interaction measure shows the input-output interactions for any desired frequency-interval, and can be used to propose a richer sparse or block diagonal
controller structure. The paper is organized as follows. Section II is dedicated to the frequency-interval gramians and their interpretation and computations. In Section III , the frequencyinterval interaction measure is proposed. This section also presents the application of the proposed interaction measure in the frequency-interval control configuration selection. The method is further illustrated with the help of a numerical example in Section IV and the results are discussed. Finally, the last section concludes the paper. The notation used in this paper is as follows: M ∗ denotes transpose of matrix if M ∈ Rn×m and complex conjugate transpose if M ∈ Cn×m . For the system Π , Struc(Π) = [πij ]p×p shows the structure of the system and it is a symbolic array where π = ∗, if there exist a subsystem in Π with input uj and output yi . Otherwise: π = 0 . The standard notation > , ≥ (< , ≤) is used to denote the positive (negative) definite and semidefinite ordering of matrices. II. F REQUENCY-I NTERVAL C ONTROLLABILITY AND O BSERVABILITY G RAMIANS , AND T HEIR I NTERPRETATION The notion of controllability and observability is very important in control theory. Many classical control problems have been studied using information on the controllability and observability. The gramians are matrices with the embedded controllability and observability information and therefore are very important in control theory. The controllability and observability gramians were first introduced in [23] and [24] and more recently in [25]. It is wellknown that the controllability gramian shows the level of controllability. Similarly, the observability gramian contains information of the level of observability for a system. An important application of these gramians is in balanced truncation which is a well-known method of model reduction. The frequency-interval balanced truncation is among the methods which improves the accuracy of the ordinary balanced truncation. The method was developed in particular for the applications in which one is interested to approximate a system in a given frequency-interval. This method was first proposed in [26] and has received a lot of attention over the last few decades [26]. In [26], it was suggested that the frequency-interval balanced realization can be obtained based on the frequency-interval gramians. The frequencyinterval gramians are enhanced gramians which compared to ordinary gramians contain more information on the level of controllability and observability within a chosen frequencyinterval. For a dynamic system with the minimal realization: G(s) := (A, B, C, D)
(1)
where G(s) is the transfer matrix with the state-space representation: x(t) ˙ = Ax(t) + Bu(t), x(t) ∈ Rn y(t) = Cx(t) + Du(t).
(2)
The ordinary controllability gramian P and the ordinary observability gramians Q are given by [25],[27]: Z ∞ ∗ P = eAτ BB ∗ eA τ dτ , (3) 0
Z
∞
Q=
∗
eA τ C ∗ CeAτ dτ .
(4)
0
For the frequency-interval [ω1 , ω2 ] , the frequency-interval gramians are defined as [26],[27]: P (ω1 , ω2 ) = P (ω2 ) − P (ω1 ), Q(ω1 , ω2 ) = Q(ω2 ) − Q(ω1 ),
(5)
where: 1 Q(ω) := 2π
1 P (ω) := 2π
Z
ω
−1
(−Ijθ − A∗ )
C ∗ C(Ijθ − A)
−1
−ω
Z
dθ, (6)
ω
−1
(Ijθ − A)
−1
BB ∗ (−Ijθ − A∗ )
dθ.
−ω
(7) These gramians are the solutions of the following Lyapunov equations [26],[27]: AP (ω1 , ω2 ) + P (ω1 , ω2 )A∗ + Wc (ω1 , ω2 ) = 0, A∗ Q(ω1 , ω2 ) + Q(ω1 , ω2 )A + Wo (ω1 , ω2 ) = 0.
(8)
where: 1 S(ω) := 2π
Z
ω
−1
(Ijθ − A)
dθ,
(9)
−ω
Wc (ω) = S(ω)BB ∗ + BB ∗ S ∗ (ω) , Wo (ω) = C ∗ CS(ω) + S ∗ (ω)C ∗ C ,
(10)
Wc (ω1 , ω2 ) = Wc (ω2 ) − Wc (ω1 ), Wo (ω1 , ω2 ) = Wo (ω2 ) − Wo (ω1 ).
(11)
III. F REQUENCY-I NTERVAL I NTERACTION M EASURE C ONTROL C ONFIGURATION S ELECTION FOR L INEAR M ULTIVARIABLE S YSTEMS
FOR
In this section, an interaction measure for the square continuous-time MIMO processes (Fig. 1) is built upon the notion of the frequency-interval gramians. The method can be easily generalized to discrete-time MIMO processes and further to non-square systems. The trace of the frequencyinterval cross gramian is used as a convenient basis to present the channel interaction and to select the most appropriate controller structure. For a square MIMO system with representation (1), we have: B = [b1 b2 · · · bm ] , C ∗ = [c1 c2 · · · cm ] .
(12)
A set of elementary SISO systems can be associated to the MIMO system, such that each SISO system has a single input uj (t) and single output yi (t) . The state-space representation of each elementary system is given by:
u1
y1
u2
y2
3
# um 1
# ym 1
um Fig. 1.
ym A MIMO system with input u ∈ Rm and output y ∈ Rm .
Πij :
x(t) ˙ = Ax(t) + bj uj (t), yi (t) = c∗i x(t),
(13)
Let Pj (Ω) and Qj (Ω) be the frequency-interval controllability gramian and the observability gramian for these elementary subsystems and let Ω = [ω1 , ω2 ]. These gramians are solutions to generalized Lyapunov equations and can be obtained for the elementary subsystems as described in the previous section. The information of the channel interaction which is obtained from the frequency-interval gramians of the elementary systems is encompassed into the so-called participation matrix (PM): Ψ(Ω) = [ψij ] ∈ Rm×m
(14)
trace [Pj (Ω)Qi (Ω)] ψij = P m P m trace [Pj (Ω)Qi (Ω)]
(15)
where:
i=1 j=1
Note that 0 ≤ ψij < 1 and
m P m P
tion 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 pairing and the controller structure 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 re-labeled, one only needs to design m independent SISO controller loops, for elementary diagonal subsystems. In this case: ∗ 0 0 Struc(Πo ) = 0 . . . 0 (16) 0
To pair inputs and outputs for decentralized control structure, we have to select one element per row and one element per column. ψ11 , ψ12 , ψ21 > 1/m2 , 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 (u P1 , y1 ), (u2 , y3 ), (u3 , y2 ) which are associated with: = ψ11 + ψ23 + ψ32 = 0.4307
ψij = 1. The participa-
i=1 j=1
0
The elements of the PM show which elementary subsystems are significant and should be considered in the nominal model. When ψij is small, the associated elementary subsystem to the pair (i, j) is either hard to control or hard to observe. This shows that, that subsystem does not have any significant effect in the actual input-output relation and could be ignored in the nominal model. When ψij is larger than 1/m2 , some states in the elementary system with output yj and input uj are easy to control and easy to observe and therefore Πij is a good candidate to be kept in the nominal system. The suitability of the pairing and the performance of the controller structure highly depend on how close the sum of the chosen ψij elements is to one. When the sum of the chosen ψij elements are close to one, the nominal and the actual model are close to each other and the error is not significant. The complexity of the selected controller structure depends on the number of the elements. In the completely decentralized control, which is the least complicated controller structure, the number of the chosen elements would be m . For example consider a 3 × 3 process model with PM: 0.1833 0.1685 0.0861 Ψ(Ω) = 0.1200 0.0445 0.1783 0.0639 0.0691 0.0863
∗
m×m
For the designed controller C, we have: ∗ 0 0 Struc(C) = 0 . . . 0 0 0 ∗ m×m
(17)
The structure of the nominal model is:
∗ Struc(Πo ) = 0 0
0 0 ∗
0 ∗ 0
A simple controller structure for selection is the structure of Π−1 o :
∗ 0 Struc(C) = Struc(Π−1 ) = o 0
0 0 ∗
0 ∗ 0
If practically it is possible to use more complicated control structures than completely decentralized control, y1 could be commanded from u2 and then we will have: P = ψ11 + ψ12 + ψ23 + ψ32 = 0.599 The structure of the nominal model then will be: ∗ ∗ 0 Struc(Πo ) = 0 0 ∗ 0 ∗ 0 A simple controller structure to select:
∗ 0 0 0 Struc(C) = Struc(Π−1 ) = o 0 ∗
∗ ∗ 0
In this case the structure is partially decentralized. IV. A N I LLUSTRATIVE EXAMPLE In this section, the proposed method for control configuration selection is illustrated with the help of a numerical example. Consider a multivariable system described by: x(t) ˙ = Ax(t) + Bu(t), (18) y(t) = Cx(t). where:
−13.1 A = −5.84 5.79 −1.21 0 B= 1.63 0.888 C= 0 0.325
5.79 −5.35 , −7.65 −0.303 0.294 , 0 −1.15 −1.07 −2.94 1.44 0 0
−5.84 −16.3 −5.35 0.489 1.03 0.727
The frequency-interval PM 0.127 Ψ(Ω) = 0.118 0.362
for Ω = [0, 1] is computed as: 0.0473 0.0348 0.0437 0.0323 0.135 0.0992
Therefore, the suggested decentralized structure for the nominal model is: 0 ∗ 0 Struc(Πo ) = 0 0 ∗ ∗ 0 0 The suggested decentralized controller structure for this model is the structure of Πo −1 : 0 0 ∗ Struc(C) = (Struc(Πo −1 ) = ∗ 0 0 0 ∗ 0 This participation matrix furthermore suggests a more complicated (partially decentralized) structure for the nominal systems: 0 ∗ 0 Struc(Πo ) = 0 0 ∗ ∗ ∗ 0 The suggested partially decentralized controller structure for this model is the structure of Πo −1 , which is: ∗ 0 ∗ Struc(C) = Struc(Πo −1 ) = ∗ 0 0 . 0 ∗ 0 V. C ONCLUSION In many applications one is interested in control and analysis within a bounded frequency-interval. For such applications, the interaction measure which is used for control configuration selection should focus on the frequency-interval of
interest rather than the whole frequency range. However, the interaction measures, which have been proposed so far, either show the input-output interactions for a single frequency or the whole frequency range. Motivated by this, a new interaction measure is proposed in this paper. The proposed interaction measure uses frequency-interval gramians and it can be used for control configuration selection of MIMO processes. Compared with other counterparts, the proposed gramian-based interaction measure can encode more information on the input-output interactions over a desired frequency-interval. This interaction measure in addition to decentralized control can be used to propose a richer sparse or block diagonal controller structure for distributed and partially decentralized control of multivariable systems. R EFERENCES [1] R. Scattolini, Architectures for distributed and hierarchical model predictive control - a review, Journal of Process Control,vol. 19, no. 5, pp. 723-731, 2009. [2] A. Khaki-Sedigh, and B. Moaveni, Control Configuration Selection for Multivariable Plants, Lecture Notes in Control and Information Sciences, Springer, 2009. [3] M. Van de Wal, and B. De Jager, A review of methods for input/output selection, Automatica, Vol. 37, No. 4, pp.487510, 2001. [4] D. Georges, The use of observability and controllability Gramians or functions for optimal sensor and actuator location in finite-dimensional systems, Proceedings of IEEE Conference on Decision and Control, vol. 4 , pp. 3319-3324,1995. [5] B. Marx, D. Koenig and D. Georges, Optimal sensor/actuator location for descriptor systems using Lyapunov-Like equations, Proceedings of IEEE Conference on Decision and Control, pp. 4541-4542, 2002. [6] B. Marx, D. Koenig, D. Georges, Optimal sensor and actuator location for descriptor systems using generalized grammian and balanced realization, Proceesdings of American Control Conference, pp. 27292734, 2004. [7] A.K. Singh and J. Hahn, Sensor Location for Stable Nonlinear Dynamic Systems: Multiple Sensor Case, Industrial Engineering Chemistry Research, 45, No. 10, pp. 3615-3623 , 2006. [8] A.K. Singh and J. Hahn, Determining Optimal Sensor Locations for State and Parameter Estimation for Stable Nonlinear Systems, Industrial Engineering Chemistry Research, 44, No. 15, pp. 56455659 , 2005. [9] H. R. Shaker and M. Tahavori, Optimal sensor and actuator location for unstable systems, Journal of Vibration and Control, 2012 doi:10.1177/1077546312451302. [10] E. H. Bristol, On a new measure of interaction for multivariable process control, IEEE Transactions on Automatic Control, 11, pp. 133134,1966. [11] S. Skogestad and M. Morari, Implications of large RGA elements on control performance, Industrial Engineering Chemistry Research, 26, pp.2323-2330, 1987. [12] M. F. Witcher, and T. J. McAvoy, Interacting control systems: Steadystate and dynamic measurement of interaction , ISA Transactions, 16,3, pp. 35-41,1977. [13] A. Niederlinski, A heuristic approach to the design of linear multivariable interacting control systems, Automatica, 7, pp. 691-701,1971. [14] E. H. Bristol, Recent results on interaction in multivariable process control. 71st AIChE Conference, 1978. [15] E. Gagnon, A. Desbiens, and A. Pomerleau , Selection of pairing and constrained robust decentralized PI controllers, Proceedings of American Control Conference, pp. 4343-4347,1999. [16] A. Conley and M. E. Salgado, Gramian based interaction measure, The Proceedings of 39th IEEE Conference on Decision and Control, pp. 5020-5022, 2000. [17] M. E. Salgado and A. Conley, MIMO interaction measure and controller structure selection, International Journal of Control, 77, pp. 367-383, 2004. [18] B. Wittenmark and M. E. Salgado, Hankel-norm based interaction measure for input-output pairing. , Proceedings of the 2002 IFAC World Congress, 2002.
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