TRIBUTE TO FOUNDERS: ROGER SARGENT. PROCESS SYSTEMS ENGINEERING
Experiment Design for Control-Relevant Identification of Partially Known Stable Multivariable Systems Shyam Panjwani and Michael Nikolaou Chemical & Biomolecular Engineering Dept., University of Houston, Houston, TX 77204-4004 DOI 10.1002/aic.15212 Published online March 21, 2016 in Wiley Online Library (wileyonlinelibrary.com)
Design of experiments for identification of control-relevant models is at the heart of robust controller design. In a number of prior publications, experiment designs have been developed that generate input/output data for efficient identification of models satisfying the integral controllability (IC) condition. The design of process inputs for such experiments is often, but not always, based on the concept of independent random rotated inputs, with appropriately proportioned amplitudes. However, prior publications do not account for models that may already be partially known before identification. In this work, this issue is addressed by developing a general experiment design framework for efficient identification of partially known models that must satisfy the IC condition. This framework produces optimal designs by solving appropriately formulated optimization problems, based on a number of rigorous theoretical results. Numerical simulaC 2016 American Institute of tions illustrate the proposed approach and potential future extensions are suggested. V Chemical Engineers AIChE J, 62: 2986–3001, 2016 Keywords: design of experiments, robust control, integral controllability, multivariable systems
Introduction A good mathematical model is useful for the design of a good automatic controller, whether the model is used explicitly or implicitly. However, mere proximity between the dynamic behavior of a mathematical model and that of the controlled system does not necessarily imply that the model can be used effectively with a certain controller design method. For control of multivariable systems, a simple classic example is a case in point: Consider a (stable) system with steady-state gain matrix (SSGM)1 " # 0:878 20:864 G5 (1) 1:082 21:096 with two alternative models " 0:870 ^ G15 1:092 and
" ^ 25 G
20:880
# (2)
21:096
1:054
20:691
1:298
20:877
# (3)
^ 2 are approxi^ 1 and G where the respective relative errors for G mately 1% and 20% compared to G. It turns out (see Appendix A for details) that G can be controlled robustly (i.e., retains closed-loop stability for a wide range of tunings) by a fully Correspondence concerning this article should be addressed to M. Nikolaou at
[email protected]. C 2016 American Institute of Chemical Engineers V
2986
decoupling controller that employs the far more inaccurate ^ 1 . In fact, closed-loop stability is ^ 2 , rather than G model G problematic for any tuning, when the far more accurate model ^ 1 is used. The underlying reason for this seeming paradox is G ^ 2 , along with G, satisfies the integral controllability that G (IC) condition2 ^ 21 Þ < 0 for all i 2Re½ki ðGG 21
(4) 21
^ Þ is an eigenvalue of GG ^ ) whereas G ^1 (where ki ðGG does not. Consequently, despite its much larger approximation ^ 2 is preferable to G ^ 1 for the design of the intended error, G multivariable controller. The preceding realization suggests that generating data for identification of models satisfying the IC condition through deliberately designed experiments requires an approach that departs from standard design of experiments (DOE) and takes IC explicitly into account. A fairly general mathematical framework for such design was developed by Darby and Nikolaou,3 inspired by pioneering ideas of Koung and MacGregor.4–6 While these investigations have provided valuable insight (as well as simple recipes, in some occasions, involving rotated inputs with appropriately proportioned amplitudes) for DOE that enables efficient identification of IC-compliant models, the focus of the investigations was entirely on identification of complete models, without assuming any partial prior knowledge. Yet, it is not uncommon to encounter situations where part of a multivariable model to be identified is already known.7–10 For instance, structural knowledge about a model may dictate that a number of elements in a transfer matrix are identically zero, or that certain entries should be trivially equal to each other, or that they add up to zero, and so on. One could reasonably anticipate that
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incorporation of such partial knowledge into DOE for identification of IC-compliant models would have distinct benefits. In this article, we show that this intuition is indeed correct. That is, explicitly incorporating partial knowledge in DOE for identification of IC-compliant models produces data from which such models can be identified a lot faster than from data generated through DOE that does not take partial knowledge into account. The proposed approach expands substantially on the mathematical framework presented by Darby and Nikolaou.3 Specifically, we present theoretical results and a numerical optimization framework that enables DOE for efficient identification of IC-compliant models for systems that are partially known at the outset. Through numerical simulations on industrial and literature models, we demonstrate that the proposed approach results in significant efficiency improvements over standard approaches. The rest of the article is structured as follows: In the next section, we first provide a brief background on IC and on DOE that is relevant for this work. In the next section, we present our main results. Numerical simulations in the subsequent section exemplify the main results. Finally, suggestions are made for future work.
Background Integral controllability The precise formulation of the result on IC proved by Garcıa and Morari,2 Theorem 2 is as follows: Assume that internal model control (IMC) with a diagonal (decoupling) fil12a is used to control an n3n stable, ter matrix FðzÞ5diag 12az 21 linear, time-invariant system with steady-state input-output behavior y5Gm, y; m 2 Rn ; G 2 Rn3n . Then, there exists an a 2 ½0; 1Þ such that the closed loop remains stable for all a ^ are ^ 21 (where G and G 2 ½a; 1Þ if and only if the matrix GG the actual and estimated steady-state gain matrices, respec ^ 21 in the right half of the comtively) has eigenvalues k GG plex plane, that is, satisfies the IC condition, Eq. 4. This result establishes the achievable robustness of decoupling multivariable controllers with integral action.
Uncertainty description Since G is not known exactly, Eq. 4 must be satisfied for all ^ is the outcome of G in an uncertainty set U. Assuming that G standard least-squares identification using data over Tf time steps, the uncertainty set U for G can be defined as the standard ellipsoidal uncertainty set 8 > >
> :
gT1 ⯗ gTn
9 > > =
3
7 7 2 Rn3n : gT 2^ g Ti MT Mðgi 2^ g i Þ c2 ; i51; . . .; n i 5 > > ;
(5) where the information matrix MT M results from the input matrix M 2 RTf 3n and c2 5s2 nF12c ðn; Tf 2nÞ r2noise v212c ðnÞ
(6)
for confidence level c, and for the F-distribution with ðn; Tf 2nÞ degrees of freedom or, approximately for Tf n for the chisquare distribution with n degrees of freedom.
Ensuring integral controllability Assessing whether Eq. 4 holds for all G in the set U defined in Eq. 5 is not trivial. Worse yet, the IC condition involves AIChE Journal
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eigenvalue inequalities whose “main weakness is that they consist of a coupling between the plant model and the true plant, which is highly cumbersome” for design of ICcompliant identification experiments, in that the plant inputs to be selected by DOE do not appear in that inequality.11 To remedy that problem, Darby and Nikolaou3 proved that the IC inequality in Eq. 4 is satisfied for all G in the set U defined in Eq. 5 if qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n X k^ u k k1 v^ Tk ðMT MÞ21 v^k < 1 (7) ^ c Ju 5 ^k r k51 where ^ G5
n X
^ k u^ k v^Tk r
(8)
k51
is the singular-value decomposition (SVD) of the steady-state gain matrix of the identified model. The advantage of Eq. 7, compared to Eq. 4, is that it does not include the uncertain matrix G directly, whereas the manipulated inputs appear directly and explicitly in the matrix MT M. Therefore, Eq. 7 can be used (as a constraint or objective) in DOE much more easily than Eq. 4, as was demonstrated by the theoretical and numerical results developed in Darby and Nikolaou.3 In fact, in some cases it leads to analytical results that are in full agreement with recipes that have appeared in literature, and in other cases it leads to novel such recipes of similar simplicity.3 Such recipes rely on the idea of rotated inputs, with appropriately proportioned amplitudes.5 Note also that, while the inequality in Eq. 7 is only sufficient for IC, numerical tests suggest that it is not overly conservative.12
Design of experiments for identification of models satisfying integral controllability To design an identification experiment numerically based on Eq. 7, a standard approximation of the information matrix MT M can first be considered in terms of the input covariance matrix, Cm , and duration of the experiment, Tf , as MT M ðTf 21ÞCm ; then, Cm can be parametrized in terms of a triangular matrix Q through a Cholesky factorization Cm 5QQT ; finally, Eq. 7 can be used either as a constraint, or the lefthand side of Eq. 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n X k^ u k k1 v^ Tk ðQQT Þ21 v^ k (9) b5 ^ r k k51 can be used as an objective to be minimized in an optimization problem that produces Q and from that the corresponding input covariance matrix Cm .
Adaptive design The inequality in Eq. 7 characterizes the optimal inputs, dic^ Eq. 8, tated by DOE, in terms of the SVD of the estimate G, which results from the identification experiment. However, because the identification experiment has not yet been con^ is not yet known at ducted, and consequently the estimate G the time of the design, optimal input design—rather than mere characterization—requires some sort of an adaptive approach. This was already proposed by Darby,13 was later modified by Kulkarni,14 and was found to work quite satisfactorily on a number of simulations.15 The general idea of the adaptive approach is as follows.
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1. Develop a preliminary model from input-output data using standard PRBS inputs for limited time. 2. Based on the available model, perform DOE that complies with IC (e.g., by minimizing b, subject to input and/or output constraints). 3. Implement the inputs determined in the above step for limited time, and collect input-output data, to update the model. 4. If the updated model does not satisfy Eq. 7 (therefore IC), go to step 2. Else, stop. Details can be found in Appendix B.
Mathematical Problem Formulation and Main Results In this section, we first formulate mathematically the problem at hand, and then explain how this formulation leads to a numerical solution. We consider that the steady-state gain matrix G of the system to be identified is partially known, in terms of linear equality constraints. A typical kind of such constraints arises when matrix elements are known a priori to be equal to zero, because pertinent inputs are known to have no effect on related outputs. Another kind arises from fundamental balance equations. Examples of each kind are discussed in the Case Studies section. From a mathematical viewpoint, linear equality constraints capturing partial process knowledge involve elements either from individual rows of G or from multiple rows of G. It turns out that each of these two classes of constraints needs to be handled differently mathematically. The linear constraints mentioned above are certainly just one form of partial knowledge available. Other forms of partial knowledge include linear inequality constraints, nonlinear constraints, constraints for the full dynamic model, and so on. While all such possibilities are worth examining—both from a theoretical and from a practical viewpoint—the possibility examined here, namely linear equality constraints on the steady-state gain matrix, is not uncommon and is of practical interest. At the same time, it is challenging enough to warrant investigation on its own right. Therefore, we quickly summarize linear-equality constrained least-squares next, and proceed to develop our main results afterwards.
Least-squares identification for partially known systems System Model. Consider an n3n multivariable system around a nominal steady state, and steady-state deviations of inputs and outputs m; y 2 Rn , respectively. Assume that the steady-state input-output behavior of the system is described by the equation yðt11Þ5GmðtÞ1eðtÞ
(10)
where G5½g ^ 1 . . . gn T 2 Rn3n ; t is discrete time; and eðtÞ is Gaussian white noise with zero mean and covariance matrix r2e I. Note that the sampling period has been assumed to be long enough, to eliminate transient dynamics in Eq. 10. This assumption is reasonable for most plants, which have a fairly smooth low-pass type of frequency response without pronounced resonance frequencies, and makes the problem manageable enough to allow for a realistic solution to be obtained with reasonable effort.4–6 The more general case of a full dynamic model will be examined in the future, along the lines of Darby and Nikolaou.16 2988
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Identification with Linear Equality Constraints for Each Individual Row of G. Assume that there is partial knowledge about G in terms of equality constraints for each row gTi of G, namely Hi gi 5hi ; i51; . . .; n
(11)
where Hi 2 Rni 3n , hi 2 Rni with ni equal to the number of equality constraints for that row. Then, the identification problem can be viewed as a collection of n multi-input-single output (MISO) problems. The least-squares estimate g^i of gi for the model of Eq. 10 subject to the equality constraints in Eq. 11 is17 21 21 g^i 5 MT M ðMT yi 1HTi ½Hi MT M HTi 21 21 ½hi 2Hi MT M MT yi Þ 21 21 5g^i; LS 1 MT M HTi ½Hi MT M HTi 21 ðhi 2Hi g^i; LS Þ (12) and the ellipsoidal uncertainty region of gi for confidence level c is ðgi 2g^ i ÞT Ai ðgi 2g^i Þ rA2 ; i51; . . .; n
(13)
M5½ mð0Þ mðT21Þ T
(14)
yi 5½ yi ð1Þ yi ðTÞ T ; i51; . . .; n 21 g^i; LS 5 MT M MT yi ; i51; . . .; n
(15)
where
(16)
is the unconstrained least-squares estimate of gi rA2 r2e v212c ðnÞ T Ai 5R1;i D21 1;i R1;i ;
i51; . . .; n
(17) (18)
and D1;i is the diagonal matrix of non-zero singular values and R1;i is the matrix of corresponding singular vectors in the SVD of the matrix T 21 T 21 M i Mi 5 ^ M M ðIn 2HTi ½Hi ðMT MÞ21 HTi 21 " #2 T 3 R1;i D1;i 0 (19) 21 4 5 Hi ðMT MÞ Þ5½ R1;i R2;i 0 0 RT2;i In terms of the input covariance matrix Cm and total experimentation time Tf , Eq. 19 can be rewritten as T 21
1 T 21 In 2HTi Hi C21 Hi C21 Mi Mi C21 m m Hi m Tf 21 1 5 ^ C21 ; i51; . . .; n Tf 21 m;i (20) Identification with Linear Equality Constraints Relating Multiple Rows of G. As there are constraints relating multiple rows of the matrix G, a different formulation of the constrained estimation problem is needed, to include the constraints in the least-squares problem. To accomplish this, one can vectorize G and proceed with its identification as a full multi-input-multi-output (MIMO) model, as follows: The input-output system described by Eq. 10 can be rewritten as
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yðt11Þ5WðtÞT vec GT 1eðtÞ
(21)
T
T 2 ^ g1 . . . gTn 2 Rn vec GT 5
(22)
where 2
T 6 mðtÞ 6 6 WðtÞT 56 0 6 4 .. .
0
..
..
0
.
0
3
.
mðtÞ
T
7 7 2 7 ^ n mðtÞT 2 Rn3n 75I 7 5
(23) with denoting the standard Kronecker product. Partial knowledge about G in terms of equality constraints relating multiple rows of the matrix G can be written as Kvec GT 5k (24) 2
where K 2 Rp3n , k 2 Rp , with p equal to the total number of linear equality constraints.
T ^ T Þ5 ^ g^ T1 . . . g^Tn of The estimate vecðG T least-squares vec G for the model of Eq. 21 subject to the equality constraints in Eq. 24 is17 T h i21 ^ 5 XT X 21 XT Y1KT K XT X 21 KT vec G h i 21 k2K XT X XT Y (25) T 21 T h T 21 T i21 T ^ 5vec G LS 1 X X K K X X K h T i ^ k2Kvec G LS
and the ellipsoidal uncertainty region of vec GT for confidence level c is T T T ^ ^ rB2 ; B vec GT 2vec G vec GT 2vec G (26) where
2
6 X56 4
3 2
Wð0ÞT ⯗ T
7 6 756 5 4
3
In mð0ÞT ⯗ T
7 7 2 RnTf 3n2 5
WðTf 21Þ In mðTf 21Þ T T ~ 2 RnTf y 5 y ð 1Þ yT T f 2 2 XT X5In MT M 2 Rn 3n T 21 T T ^ X y~ vec G LS 5 X X
(27) (28) (29) (30)
is the unconstrained least-squares estimate of vec GT rB2 r2e v212c ðn2 Þ
(31)
~ 1D ~ 21 R ~T B5R 1 1
(32)
~ 1 are the diagonal matrix of non-zero singular values ~ 1, R and D and matrix of corresponding singular vectors, respectively, obtained from SVD of the symmetric matrix h 21 21 i21 In2 2KT K XT X KT C5 ^ XT X "~ #2 ~ T 3 (33) R1 0 1 T 21
D 4 5 ~2 ~1 R K X X 5 R ~T 0 0 R 2
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with Ik the identity matrix of dimensions k3k. Equation 33 can be rewritten in terms of the input covariance matrix Cm and total experimentation time Tf as T 21 1 C21 C In In2 2KT K In C21 K In C21 m m K m Tf 21 (34)
Ensuring IC in identification of partially known models As already mentioned above, the purpose of DOE is to determine inputs that will generate data enabling the identifi^ detercation of an IC-compliant model, that is, a model G, mined by Eqs. 12 or 25, which satisfies Eq. 4 along with the real system G constrained in Eqs. 13 or 26, respectively. As has been argued already,3,11 directly relying on Eq. 4 results in an overly complicated DOE task. To circumvent that difficulty, in the following theorems we develop inequalities that guarantee satisfaction of Eq. 4 yet are much simpler in that they a. directly involve the inputs to be determined by DOE, and b. do not contain the uncertain matrix G. Consequently, these inequalities are much easier to use in DOE for identification of IC-compliant models, as will be explained afterwards. Theorem 1 Sufficient condition for IC of model identified as multiple, partially known MISO models ^ partially known through Eq. 11 and identified A model G, according to Eq. 12, satisfies IC for all potential G in the set D5fG5½g1 . . . gn T 2 Rn3n : ðgk 2g^k ÞT Ak ðgk 2g^k Þ rA2 ; 1 k ng
(35)
suggested by Eq. 13, where Ak and rA are as in Eqs. 18–20, if qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n X k^ u k k1 ^k < 1 v^Tk A21 (36) ^ rA Jc 5 k v ^ r k k51 Proof. Placing the matrix Ak in place of MT M in Theorem 1 in Darby and Nikolaou,3 the proof of this Theorem follows the exact same pattern, and is omitted for brevity. Note the similarity between the sets D in Eq. 35 and U in Eq. 5. Note also that, in contrast to Eq. 4, the above Eq. 36 in Theorem 1 directly involves plant inputs through the matrix Ak , which, as Eqs. 18 and 19 indicate, is a function of both the data matrix, M, and the constraint matrix, Hk . In addition, the uncertain matrix G, which is present in Eq. 4, has been eliminated in Eq. 36. The above Theorem 1 is applicable to those cases where individual rows of the plant steady-state gain matrix G are not related to each other through constraining equalities. However, as already argued, situations with constraints involving multiple rows of G are not uncommon. The subsequent Theorem 2, following the intermediate Lemma 1 in Appendix C, is developed for such situations. 䊏 Theorem 2 Sufficient condition for IC of model identified as a single, partially known MIMO model ^ partially known through Eq. 24 and identified A model G, according to Eq. 25, satisfies IC for all potential G in the set
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3. Implement the inputs determined in the above step for limited time, and collect input-output data, to update the model. 4. If the updated model does not satisfy Eq. 36 (therefore IC), go to step 2. Else, stop.
Table 1. Case1: Parameters Used in Simulation for Adaptive Designs Time steps of initial identification with PRBS inputs Standard deviation of output noise Length of time segment at the end ^ is updated and DOE repeated of which G Total number of identification steps
55 0:25 1 1000
T ^ TÞ ~ D5 G5½g1 . . . gn T 2 Rn3n : vec GT 2vecðG (37) T T 2 ^ B vec vec G 2vecðG Þ rB suggested by Eq. 26, where B and rB given in Eqs. 32–34, if 1 lmax ðB UÞ < 2 nrB 21
(38)
^ T GÞ ^ 21 , and lmax ðB21 UÞ is the largest where U5In ðG eigenvalue of the matrix B21 U. Proof. See Appendix D. Theorem 1 and Theorem 2 immediately suggest how DOE for identification of an IC-compliant model can be formulated as an optimization problem. The optimization resulting from Theorem 1 is not far from the formalism in Darby and Nikolaou.3 By contrast, Theorem 2 departs significantly from that formalism and is more challenging. Therefore, in the following we will first discuss briefly the DOE implications of Theorem 1, and will concentrate more on discussing the DOE implications of Theorem 2. 䊏
Design of experiments for IC-compliant model to be identified as multiple, partially known MISO models Theorem 1 suggests that DOE for identification of an ICcompliant model as in Eq. 10, with partial model knowledge captured in constraints as in Eq. 11, can be performed using Eq. 36, as follows. Find qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! n X k^ u k k1 opt ^k v^Tk A21 (39) min rA Cm 5arg k v ^k r Cm 5QQT k51 subject to variance constraints on individual outputs and inputs, such as var ðmi Þ5½Cm kk Mk 2 ; k51; . . .; n T
^ mG ^ Yk 2 ; k51; . . .; n var ðyi Þ5½GC kk
(40) (41)
where the input covariance matrix is parametrized through a Cholesky factorization as Cm 5QQT . Other kinds of constraints, in place of Eqs. 40 and 41 may also be considered, as discussed in.3 It should be noted, that Eq. 36 rigorously characterizes (rather than prescribes) the optimal inputs under the respective set of identification constraints, as it involves the elements u^ k ; ^ As already men^ k of the SVD of the identified model G. v^k ; r tioned above, an adaptive DOE can address this issue, as follows. 1. Develop a preliminary model from input-output data using standard PRBS inputs for limited time. 2. Based on the available model, perform DOE that complies with IC by minimizing Jc , Eq. 36, subject to input and/or output constraints. 2990
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Design of experiments for IC-compliant model to be identified as a single, partially known MIMO model Theorem 2 suggests that DOE for identification of an ICcompliant model as in Eq. 24, with partial model knowledge captured in constraints as in Eq. 21, can be carried out using Eq. 38, as follows: Find
opt 21 min T lmax ðB UÞ (42) Cm 5arg Cm 5QQ
subject to variance constraints on outputs and inputs, as in Eqs. 40 or 41. While the task in Eq. 42 is challenging, we explain below that minimization of an upper bound on lmax ðB21 UÞ is manageable in realistic situations. In fact the analysis that follows establishes that, in the absence of partial knowledge about the model, DOE based on rotated inputs emerges naturally as the result of minimizing that upper bound on lmax ðB21 UÞ, subject to various constraints on inputs and outputs. The corresponding results are summarized in Theorem 3 and Theorem 4, after the related context is set first. Design of Experiments for IC-Compliant Model to be Identified as a Single MIMO Model. In standard linear regression, absent any partial knowledge on G, the matrix B in Eq. 37 is B5I MT M
(43)
T
where M M in the information matrix containing the inputs that must be designed, and can be diagonalized as MT M5PKPT
(44)
Given this, it can be shown (Appendix E) that the DOE problem in Eq. 42 can be cast as the following optimization problem: Find ! 22 T ^ 24 (45) min min tr QK Q R K
T
^ P Q5V T
^ P and the diagwith respect to the orthonormal matrix Q5V onal matrix K, subject to pertinent input and output constraints. More specifically, such constraints can be ðTf 21Þvar ðmk Þ ½MT Mkk 5½PKPT kk ðTf 21ÞMk2 ; k51; . . .; n
(46)
^V ^ T PKPT V ^T ^T ðTf 21Þvar ðyk Þ ½YT Ykk 5½T kk 2 ðTf 21ÞYk ; k51; . . .; n
(47)
ðTf 21Þ
n X k51
ðTf 21Þ
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n X
var ðmk Þ tr½MT M5
n X
k2k ðTf 21ÞM2 (48)
k51
^ 2V ^ T PKPT V ^ ðTf 21ÞY 2 var ðyk Þ tr½YT Y5tr½R
k51
(49)
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Figure 1. Optimal inputs and outputs for 535 FCC unit as characterized by design D1. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
2
T
^ V ^ PKPT V ^ 1ð12vÞ tr½PKPT w2 v tr½R |fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} ðTf 21Þvar ðyÞ
(50)
ðTf 21Þvar ðmÞ
^5 ^ ^ 1; . . . ; r ^ n Þ and v; 12v are relative weights where T ^ Udiag ðr of the output and input variances, respectively, with 0 v 1. Analytical Solution to Eq. 45 in the Absence of Partial Model Knowledge: Rotated Inputs. To ensure that Eq. 45 poses a meaningful numerical optimization problem, it would be useful to examine the outcome of this optimization when it accepts an analytical solution. We show below (Theorem 3 and Theorem 4) that in the case of constraints on the total input or output variance, as in Eq. 50, an explicit solution to the minimi^ 24 Þ can be found, in terms of rotated zation of trðQK22 QT R inputs, as discussed in the Introduction section. Specifically, the minimization problem in Eq. 45 can proceed in two steps: a. by finding the optimal (orthonormal) P, given K, in the first step, and b. by finding the optimal K, in the second step. The first step is performed in Theorem 3 and the second step in Theorem 4 below.
^ ½ v^1 v^n 2 Rn3n orthonormal, K5diag V5 Given 2 k1 ; . . . ; k2n with diagonal entries indexed in increasing order as 0 < k‘1 < < k‘n
^ ^ n Þ with diagonal entries indexed in ^ 1; . . . ; r ðr and R5diag decreasing order as ^n > 0 ^1 > > r r
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(52)
24
^ Þ with respect to the orthothe minimum of trðQK22 QT R ^ 2 Rn3n is normal matrix P5VQ n X 1 ^ 24 5 (53) min tr QK22 QT R 4 4 ^ ^ r P5VQ k51 k k‘k and is obtained for ^ ½ v^‘1 Popt 5VP5
v^‘n 2 Rn3n ;
(54)
where P is a permutation matrix such that ½‘1 ; . . . ‘n 5½1; . . . nP:
(55) 䊏
Proof. See Appendix F. Theorem 3 Emergence of rotated inputs as optimal choice for identification of IC-compliant models
(51)
Corollary 1 Rotated inputs as optimal choice for identification of IC-compliant models
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Figure 2. Optimal inputs and outputs for 535 FCC unit as characterized by design D2. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
Under the conditions of Theorem 3, optimal inputs (in the 22 T ^ 24 sense of min P5VQ ^ trðQK Q R Þ) for the process to be identified have the form
n X 1 1 < 2 4 4 ^ k k‘k nrB k51 r
^ m5Vn
Proof. Immediate, from combination of Eqs. 53 in Theorem 3, (38) in Theorem 2, (42), and (45). 䊏
(56)
^ is a rotation (orthonormal) matrix in the SVD of where V Eq. 8, and var ðnk Þ5k2‘k
(57)
Theorem 4 Emergence of optimal scaling of rotated inputs for identification of IC-compliant models
Proof. Equations 44 and 54 imply min k‘k
^ 5NT N5PKPT 5diag k2 ; . . . ; k2 ^ T MT MV V ‘ ‘ 1 n |fflfflffl{zfflfflffl} |{z} NT
(58)
N
^ T m, rather than the original with the rotated inputs n5V inputs m, being uncorrelated, with covariance matrix Cn 2 2 T 1 1 䊏 Tf 21 N N5 Tf 21 diagðk‘1 ; . . . ; k‘n Þ
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n X 1 4 4 ^ k k‘k k51 r
(60)
subject to the constraint in Eq. 50 and the ordering of k‘k in Eq. 51, is obtained at k‘ k 5
w 2
^ k bk r
1=3 hXn i51
1 b 4=3 i1=2
(61)
i
^i r
2 1=2 r k 112v where bk 5 v^
Corollary 2 Sufficient condition for IC in process identification without partial knowledge IC is guaranteed for identification of a process as in Eq. 10 if
(59)
Proof. See Appendix G.
䊏
Corollary 3 Optimal scaling of rotated inputs for identification of IC-compliant models Under the conditions of Theorem 4
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Figure 3. Optimal inputs and outputs for 535 FCC unit as characterized by design D3. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 1=6 var nj k‘j v^ r k 1ð12vÞ^ r 4k rjk ðvÞ5 ^ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 5 1=6 var ðnk Þ k‘k v^ r 6 1ð12vÞ^ r4 j
(62)
j
Proof. Immediate from Eqs. 61 and 57. Note that for v51, that is, constraint only on total output variance through Eq. 50, the optimal ratio rjk ðv51Þ of rotated-input amplitudes expressed in Eq. 62 recovers the ^ k =^ r j suggested by Koung, MacGregor.6 However, for ratio r v50, that is, constraint only on total input variance through Eq. 50, the optimal ratio rjk ðv50Þ of rotated-input 2=3ampli^ k =^ rj . This tudes expressed in Eq. 62 turns out to be r 1=3 ^ k =^ result is related to the design rjk ðv50Þ5 r rj suggested 3 In fact, both designs, that is, rjk by Darby and Nikolaou. 1=3 2=3 ^ ^ k =^ ðv50Þ5 r k =^ rj and rjk ðv50Þ5 r rj , provide rigorous justification to the heuristic suggestion of using a ratio rjk 5e^ r k =^ r j with 0 < e < 1 at the beginning of an identification experiment (to account for uncertainty) proposed by 䊏 Bruwer and MacGregor.18
Case Studies The IC-optimal designs proposed in the previous section will be illustrated next with numerical simulations on two multivariable systems, namely a 535 industrial fluid catalytic AIChE Journal
September 2016 Vol. 62, No. 9
cracker (FCC) reactor/regenerator unit and a 232 multistage absorber unit, presented below. The reason for selecting the FCC unit is that its steady-state gain matrix has a number of entries known to be identically zero. This kind of partial system knowledge conforms with Eq. 11. Therefore, the corresponding theory, epitomized by Eq. 36, will be used. The reason for selecting the two-stage absorber unit is that entries of its steady-state gain matrix are related through linear constraints that conform with Eq. 24. Therefore, the corresponding theory, epitomized by Eq. 42, will be used. In addition to IC-optimal designs, D-optimal designs19 will be illustrated, for comparison. The reason for this comparison is that D-optimal designs maximize the determinant of the input covariance matrix, Cm Tf 121 MT M. By contrast, ICoptimal designs involve the quantities indicated in Eqs. 36 and 42, which are different functions of Cm . Four different designs will be illustrated, summarized as follows: a. Design D1 is the IC-optimal DOE approach based on minimization of Jc , Eq. 39 (or minimization of lmax ðB21 UÞ, Eq. 42), taking into account partial knowledge, Eq. 11 (or Eq. 24), and input-output variance constraints, for example, Eqs. 40, 41 (or any of 46–50).
Published on behalf of the AIChE
DOI 10.1002/aic
2993
Figure 4. Optimal inputs and outputs for 535 FCC unit as characterized by design D4. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
b. Design D2 is similar to D1, but without use of partial knowledge; that is, D2 minimizes Ju , Eq. 7, subject to input-output variance constraints, as above. c. Design D3 is D-optimal DOE19 with partial knowledge; that is, D3 minimizes the covariance of the parameter 21 estimator, diagððMT1 M1 Þ21 ; . . . ; MTn Mn Þ as in Eqs. 18–20, for least squares identification with partial knowledge as in Eq. 11 ! n X log det ðD1;i Þ (63) min T Cm 5QQ
i51
subject to input-output variance bounds, as above; or minimization of the covariance of the parameter estimator, C, as in Eqs. 32–34 for least squares identification with partial knowledge as in Eq. 24 ~ 1Þ (64) min T log det ðD Cm 5QQ
subject to input-output variance bounds, as above. d. Design D4 is similar to D3 but without use of partial knowledge; that is, D4 minimizes the covariance of the parameter estimator, Cm Tf 121 MT M min ð2log ðdetðCm ÞÞÞ
Cm 5QQT
subject to input-output variance constraints, as above. 2994
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(65)
For all four designs, identification is performed through constrained least-squares, namely minimization of the sum of the squared errors subject to the equality constraints, Eq. 11 (or Eq. 24), emanating from partial knowledge. Given that the IC-optimal designs D1 and D2 through Eqs. ^ the plant esti39 or 42, respectively, require knowledge of G, mate after the identification, two different kinds of designs D1 and D2 are performed in the simulations that follow: a. DOE assuming perfect knowledge of G, and replacing ^ by G in Eqs. 39 or 42 at the time of DOE. For G numerical simulations resulting from such DOE, satisfaction of IC is indirectly checked through Eqs. 36 or 38, as checking satisfaction of Eq. 4 is not trivial. While this design is obviously unrealistic from a practical viewpoint, it provides a clear characterization of the optimal inputs and illustrates what can be anticipated in a best-case, if unrealistic, scenario. b. Adaptive DOE, as outlined in Adaptive design of the ^ to Background section, using the latest estimate of G design optimal inputs for the next time-segment of ^ will be updated identification, at the end of which G and DOE repeated. After an initial period of identification with PRBS inputs, Eqs. 39 or 42 are adaptively ^ being the latest estiused at the time of DOE, with G mate. Satisfaction of IC is again checked through Eqs. 36 or (38). While this design confounds the IC-optimal design with the adaptation method used, it provides a
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AIChE Journal
Table 2. Characterization of Inputs and Outputs for Designs D1–D4 for 535 FCC Unit; Active Constraints are in Bold Design D1 D2 D3 D4
det ðCm Þ
det ðAi Þ, i51; . . .; 5
var ðmi Þ, i51; . . .; 5
var ðyi Þ, i51; . . .; 5
0:02 1:01 0:99 1:16
1:49; 0:60; 0:49; 0:01; 0:88 1:5; 0:37; 0:75; 0:69; 0:55 1:5; 0:53; 0:66; 0:75; 0:77 1:5; 0:39; 0:77; 0:80; 0:58
1:50; 0:65; 3:00; 1:16; 1:34 1:50; 0:42; 3:00; 1:50; 1:04 1:50; 0:55; 3:00; 1:50; 1:36 1:50; 0:41; 3:00; 1:50; 1:17
0:22; 0:35; 0:65; 0:17; 0:19 0:22; 0:35; 0:65; 0:09; 0:18 0:23; 0:35; 0:65; 0:14; 0:20 0:22; 0:35; 0:65; 0:13; 0:19
with emphasis on Eq. 39. The state-space model of the system can be found in Darby.13 The SSGM G1 of the system is 2 3 0:3866 0 0 0 0 6 7 6 0 20:6935 0 0 20:5805 7 6 7 6 7 7 0:1192 1:5461 0:5224 0 20:3667 G1 56 6 7 6 7 6 0 7 20:1313 20:1298 0:1058 20:2057 4 5 0:0631 20:2462 0 0 20:4435 (66)
clear illustration of how the proposed approach would work in practice. It is also clear that for very long identification experiments, this design will eventually converge toward producing inputs similar to those produced by design (a). The effect of adaptation on convergence warrants its own investigation, and is examined elsewhere.15 For numerical solution of the optimization problems in Doptimal designs D3 and D4, we use the semidefinite programming solver SeDuMi20 of YALMIP-MATLABV. For the ICoptimal designs D1 and D2, we use the nonlinear optimization function fmincon available in MATLABV, with initial guesses obtained from corresponding D-optimal designs. To avoid local minima, the Multistart algorithm available in MATLABV is also employed with fmincon. The computational time required for fmincon with Multistart was found to increase by a number approximately equal to the number of initial guesses considered by Multistart. However, no improvement in the solution was observed, suggesting small probability of convergence to local minima. R
R
with identically zero elements evident. This partial knowledge can be included in DOE in the form of linear constraints on the parameters, as shown in Eq. 11. All four experiment designs D1–D4 use the following variance constraints on individual inputs (mi ) and outputs (yi ) var ðy1 Þ 0:35 var ðm1 Þ 1:5
R
var ðy2 Þ 0:35
var ðm2 Þ 1:5
var ðy3 Þ 0:65
var ðm3 Þ 3:0
Fluid catalytic cracking reactor-regenerator
var ðy4 Þ 0:35
var ðm4 Þ 1:5
A 535 FCC reactor-regenerator unit identified from plant testing13 is used here to compare performances of designs D1–D4,
var ðy5 Þ 0:35
var ðm5 Þ 1:5
(67)
Table 3. Input and Output Correlations Matrices for Designs D1–D4 for 535 FCC Unit Design D1
2
Rm 1
6 6 6 6 6 6 6 6 4 D2
2
1
2 6 6 6 6 6 6 6 6 4
D4
2 6 6 6 6 6 6 6 6 4
20:87 1
1
6 6 6 6 6 6 6 6 4 D3
20:08 20:07
1
3
2
7 20:55 7 7 7 20:84 0:59 7 7 7 1 20:34 7 5
6 6 6 6 6 6 6 6 4
0:24
0:15
0:49
1 3 0:082 7 1 20:71 20:23 20:38 7 7 7 1 0:33 0:41 7 7 7 1 0:16 7 5 1 20:19 0:003 2331024 0:14
20:11
1
0:002
531028
1
631029 1
20:11 2131028 1
1 3 20:02 0:14 7 6 6 1 0:002 0:50 0:91 7 6 7 6 7 6 7 1 0:12 0:26 6 7 6 7 6 1 0:71 7 4 5 1 3 2 1 431029 20:16 20:04 0:08 7 6 6 1 2131028 0:50 0:89 7 7 6 7 6 6 1 0:37 0:33 7 7 6 7 6 7 6 1 0:77 5 4 1 0:001
3
2
7 20:43 7 7 7 0:50 7 7 7 131029 7 5
6 6 6 6 6 6 6 6 4
1 2731029
3
7 20:53 7 7 7 0:60 7 7 7 2331029 7 5 1
1
131029
20:71
29310210
1
25310210 1
0:08
1
0:01
231028 731029 1
1 3 20:02 0:13 7 0:51 0:91 7 7 7 0:13 0:26 7 7 7 1 0:73 7 5 1
1
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3
7 20:01 0:33 0:86 7 7 7 1 0:49 0:41 7 7 7 1 0:70 7 5
1
2
0:009
20:80
Ry 1 20:09 20:11 0:04 0:03
Published on behalf of the AIChE
DOI 10.1002/aic
2995
Figure 5. Adaptive DOE for 535 FCC unit: Identification time required for IC satisfaction when inputs are produced by designs D1–D4. The vertical axis is Jc , Eq. 11, for designs D1 and D3, and Ju , Eq. 7, for designs D2 and D4. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
Parameters used in the simulation for all adaptive designs are given in Table 1. The optimal inputs and outputs for 535 FCC unit as characterized by designs D1–D4 are shown in Figures 1–4, respectively.
Important observations from Figures 1–4, Tables 2 and 3 are as follows: For designs D2 and D4, Ak , k51; . . .; 5, are information matrices, according to Eq. 35. Larger value of detðAk Þ imply more accurate parameter estimation. Because the design D4 is D-optimal, detðAk Þ for that design is larger than detðAk Þ for design D2, which is IC-optimal. This is clearly in agreement with the fact that IC-optimal designs sacrifice some accuracy in parameter estimation to achieve IC satisfaction. All designs result in input or output pairs that may be from highly correlated to fairly uncorrelated. This is due to the fact that in the presence of input and/or output constraints, optimal experiments are obtained through constrained numerical optimization. These observations suggest that the frequently mentioned rule of thumb “opt for correlated inputs and uncorrelated outputs when identifying illconditioned systems” is not universally applicable. Similar findings were also stated by Darby and Nikolaou.16 Figure 5 shows comparison of identification time required for IC satisfaction (J < 1) for designs D1–D4. Designs D1 and D3, which use partial knowledge about the system, require fewer time steps than designs D2 and D4. This clearly establishes the usefulness of using partial knowledge in designing the experiments. Design D1, being an IC-optimal design, performs slightly better than D-optimal design D3. Figure 6 shows the convergence of elements of identified SSGM using given input/output constraints. Evidently, designs
Figure 6. Convergence of SSGM elements of 535 FCC unit for designs D1–D4. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
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AIChE Journal
Table 4. Case 2: Parameters Used in Simulation for Adaptive Designs Time steps of initial identification with PRBS inputs Standard deviation of output noise Length of time segment at the end of ^ is updated and DOE repeated which G Total number of identification steps
55 0:5 1 300
Figure 8. Optimal inputs and outputs for 232 twostage absorber unit as characterized by design D2. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
Figure 7. Optimal inputs and outputs for 232 twostage absorber unit as characterized by design D1. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
D1–D4 adaptively produce models which are close to the actual system G1 . However, IC-optimal designs D1 and D2 deliberately compromise the accuracy of some of the parameters, in exchange of IC satisfaction.
Case 2: Two-stage absorber unit A 232 two-stage absorber21 is used here to compare performances of designs D1–D4, with emphasis on Eq. 42. Input variables are solute concentrations in the liquid and vapor streams entering the absorber, and output variables are concentrations of solute in the liquid at each of the two stages. The SSGM G2 for this system is " # 0:2632 0:1053 (68) G2 5 0:1579 0:2632 Partial knowledge is available in terms of linear equality between the first and second row of the matrix G2 G2 ð1; 1Þ5G2 ð2; 2Þ
Optimal inputs and outputs for 232 two stage absorber unit as characterized by designs D1–D4 are shown in Figures 7– 10, respectively. As in the first case study, it can be observed here that detðBÞ (a measure of parameter estimation accuracy) for design D4 is larger than detðBÞ for design D2 (Table 5). This again establishes the fact that IC-optimal designs sacrifice some accuracy in parameter estimation to achieve IC satisfaction. Figure 11 compares the identification time required for IC satisfaction (J < 1) for designs D1–D4. Designs with partial knowledge (D1 and D3) lead to faster IC satisfaction than designs without partial knowledge (D2 and D4) as anticipated. The overlapping profiles for design D1 and D3 show close agreement between the IC-optimal design D1 and the Doptimal design D3. Figure 12 shows the convergence of the SSGM elements for the identified models. Despite nearly the same closeness of the matrix elements to the actual process over the adaptations, some models satisfy IC and some do not. This again illustrates the claim made in the Introduction about closeness of the model to the real system and control relevance.
Conclusion and Future Work A general mathematical framework was presented to design experiments for efficient identification of partially known models that are required to satisfy the IC condition. The mathematical framework relies on guaranteeing IC (Eq. 4) by satisfaction of simpler inequalities (Eqs. 36 or 38, depending on the nature of partial knowledge of the identified model), in
(69)
G2 ð1; 2Þ1G2 ð2; 1Þ5G2 ð2; 2Þ Input and output variance constraints are var ðyi Þ 0:1; var ðmi Þ 0:5; i51; 2
(70)
Parameters used in the simulation for all adaptive designs are given in Table 4. Table 5. Results for Designs D1–D4 for 232 Two-Stage Absorber. Active Constraints are in Bold Design det ðCm Þ det ðBÞ var ðmi Þ D1 D2 D3 D4
931025 0:22 131028 0:25
AIChE Journal
0:45 0:18 0:45 0:25
0:5; 0:5 0:5; 0:5 0:5; 0:5 0:5; 0:5
var ðyi Þ
qm1 ; m 2
0:07; 0:09 0:99 0:07; 0:09 20:35 0:07; 0:09 1:00 0:04; 0:05 2431029
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qy1 ; y2 1:00 1:00 1:00 0:79
Figure 9. Optimal inputs and outputs for 232 twostage absorber unit as characterized by design D3. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
Published on behalf of the AIChE
DOI 10.1002/aic
2997
Figure 10. Optimal inputs and outputs for 232 twostage absorber unit as characterized by design D4. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
which process inputs appear explicitly. This framework produces experiment designs either analytically, in simple cases, or through solution of an appropriately formulated constrained optimization problem, when input and/or output constraints are present. The proposed framework was illustrated with numerical simulations on two multivariable systems, namely a 535 industrial FCC reactor/regenerator unit and a 232 multistage absorber unit. These simulations showed agreement of the observed results with results in literature under similar conditions. They also demonstrated novel results, under conditions not examined in literature before. A number of items related to this work can be examined in the future, such as the following: D-optimal and IC-optimal designs produced inputs with different but not entirely dissimilar characteristics. What are the reasons underlying this outcome? In the present work, partial knowledge of the process to be identified is expressed in terms of equality constraints. Can these ideas be extended to inequality constraints, and would that be worthwhile from a practical viewpoint? The framework presented emphasizes steady-state behavior. Can it be extended to the DOE for identification of
Figure 11. Adaptive DOE for 232 two-stage absorber: Identification time required for IC satisfaction when inputs are produced by designs D1–D4. The vertical axis is Jc , Eq. 11, for designs D1 and D3, and Ju , Eq. 7, for designs D2 and D4. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
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Figure 12. Convergence of SSGM elements of 232 two-stage absorber for designs D1–D4. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]
partially known dynamic systems along the lines of Darby and Nikolaou?16
Acknowledgment The authors would like to thank Dr. Mark Darby for bringing to their attention the problem addressed in this article.
Literature Cited 1. Skogestad S, Morari M. Control configuration selection for distillation-columns. AIChE J. 1987;33(10):1620–1635. 2. Garcia CE, Morari M. Internal model control: 2. design procedure for multivariate systems. Ind Eng Chem Process Des Dev. 1985;24: 472–484. 3. Darby ML, Nikolaou M. Multivariable system identification for integral controllability. Automatica. 2009;45(10):2194–2204. 4. Koung CW, MacGregor JF. Geometric analysis of the global stability of linear inverse-based controllers for bivariate nonlinear processes. Ind Eng Chem Res. 1991;30(6):1171–1181. 5. Koung CW, MacGregor JF. Design of identification experiments for robust-control: a geometric approach for bivariate processes. Ind Eng Chem Res. 1993;32(8):1658–1666. 6. Koung CW, MacGregor JF. Identification for robust multivariable control: the design of experiments. Automatica. 1994;30(10):1541– 1554. 7. Johansen TA. Identification of non-linear systems using empirical data and prior knowledge: An optimization approach. Automatica. 1996;32(3):337–356. 8. Abonyi J, Babuska R, Verbruggen HB, Szeifert F. Incorporating prior knowledge in fuzzy model identification. Int J Syst Sci. 2000; 31(5):657–667. 9. Timmons WD, Chizeck HJ, Casas F, Chankong V, Katona PG. Parameter-constrained adaptive control. Ind Eng Chem Res. 1997; 36(11):4894–4905. 10. Kothare SL, Lu Y, Mandler JA, Inventors. Constrained system identification for incorporation of a priori knowledge. US patent US 2004/0181498 A12004. 11. Featherstone AP, Braatz RD. Integrated robust identification and control of large-scale processes. Ind Eng Chem Res. 1998;37(1):97– 106. 12. Panjwani S, Nikolaou M. Ensuring integral controllability for robust multivariable control; submitted, Comput Chem Eng. 2016. 13. Darby ML. Studies of Online Optimization Methods for Experimental Test Design and State Estimation. Houston: Chemical and Biomolecular Engineering, University of Houston, 2008. 14. Kulkarni P. Design of Experiments for Identification of Dynamic Models Satisfying the Condition of Integral Controllability [M.S. Thesis]. Houston: Electrical Engineering and Chemical & Biomolecular Engineering, University of Houston, 2012.
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15. Misra S, Nikolaou M. Adaptive design of experiments to identify dynamic models for integral controllability. submitted, Comput Chem Eng. 2016. 16. Darby ML, Nikolaou M. Identification test design for multivariable model-based control: an industrial perspective. Control Eng Pract. 2014;22:165–180. 17. Seber GAF, Lee AJ. Linear Regression Analysis, 2nd ed. Wiley, Hoboken, New Jersey, 2003. 18. Bruwer MJ, MacGregor JF. Robust multi-variable identification: optimal experimental design with constraints. J Process Contr. 2006; 16(6):581–600. 19. Mehra RK. Optimal input signals for parameter estimation in dynamic systems survey and new results. IEEE Trans Autom Contr. 1974;19(6):753–768. 20. Sturm JF. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim Method Softw. 1999;11-2(1–4):625– 653. 21. Amundson N. Mathematical Methods in Chemical Engineering. New Jersey: Prentice-Hall, 1966.
FðzÞðI2FðzÞÞ21 =f ðzÞ
(A1)
For a diagonal IMC filter FðzÞ5 12a z2a I, the closed-loop sensitivity function becomes
21 21 12a ^ ðI1Gf ðzÞCðzÞÞ 5 I1GG z21 21 ^ 21 Gðz21Þ1Gð12aÞ ^ 5ðz21ÞG 21
(A2)
with closed-loop characteristic equation a2 1a1 z1z2 50, where the parameters a1 ; a2 are
a2 50:021514:67a23:69a2 ; a1 54:7112:71a
(A3)
^ 1 , and for G 2
a2 520:041711:04a ; a1 50:83322:08a
a2 21 < 0
a1 2a2 21 < 0
Copt m 5 min b Cm
^ mG ^ T Yi 2 var ðyi Þ5½GC ii
(A6)
(A5)
(A7)
opt 3. Calculate Qopt by Cholesky decomposition of Copt m 5Q opt T opt ðQ Þ and design input m5Q z where z is a zeromean PRBS with unit covariance. 4. Add new input m in the set of input signals; perform ^ identification and update model G. 5. Check for satisfaction of IC sufficient condition (Eq. 9); stop if maximum number of iterations achieved, otherwise go to step-2 and repeat subsequent steps.
Appendix C: Proof of Lemma 1 The following Lemma 1 is proved first, to make it easier to follow the proof of the subsequent Theorem 2.
Lemma 1. Sufficient condition for IC of model identified as a single, partially known MIMO model ^ 2 Rn3n and a real plant G 2 Rn3n with For a model G ^ the IC condition (Eq. 4) is satisfied if uncertainty D5G2G, ^ Jc 5
(A4)
^ 2. for G For the roots of the characteristic equation to be inside the unit disk, the Jury stability criterion requires
2a1 2a2 21 < 0
^ from input-output data 1. Develop a preliminary model G using standard PRBS inputs for limited time. 2. Calculate Cm by solving below optimization problem
var ðmi Þ5½Cm ii Mi 2
^ ðzÞ, where f ðzÞ is a stable Given a transfer matrix model Gf transfer function with stable inverse, the IMC method can be used ^ ðzÞQðzÞÞ21 by to design a feedback controller CðzÞ5QðzÞðI2Gf 21 ^ selecting QðzÞ5G FðzÞ=f ðzÞ, which yields 21
Appendix B: Adaptive Design to Build an IC-Compliant Model
subject to variance constraints on individual outputs and inputs
Appendix A: Controller Design for Motivating Example in Introduction
^ CðzÞ5G
Figure 13 indicates that these inequalities cannot be satisfied ^1 simultaneously for any value of a in the interval ½0; 1Þ when G is used in the controller structure of Eq. A1, but they can be ^ 2 is used in easily satisfied for all a in the interval ½0; 1Þ when G that controller structure.
n X 1 ^T T kZ k vec D k2 < 1 ^ r k51 k
(A8)
where 2 6 6 T 6 ^ Z k 56 6 4
v^Tk 0 ..
.. .
..
0
0
.
3 .
0 v^Tk
7 7 2 7 T 75In v^k 2 Rn3n ; k51; . . .; n 7 5
(A9)
Figure 13. Satisfaction of the inequalities in Eq. A5, required by the Jury stability criterion for closed-loop stability, ^ 1 (left) and G ^ 2 (right) is used in the feedback controller, Eq. A1. when G
AIChE Journal
September 2016 Vol. 62, No. 9
Published on behalf of the AIChE
DOI 10.1002/aic
2999
^ T vec DT k 5kZ ^ T vec DT k v k ki2 5kZ kD^ v k u^ Tk ki2 5kD^ i2 2 k k
v^k , k51; . . .; n are the right singular vectors of the SVD of ^ G5
n X
rk u^ k v^ Tk
(A19)
(A10)
Proof. Equation 4 is equivalent to
where k ki2 is the induced 2-norm of a matrix. Consequently, Eq. A19 implies that the inequality in Eq. A17 is satisfied if n X 1 ^ T T (A20) Z k vec D < 1 ^ 2 r k51 k
h h i 21 i ^ 21 > 0 ^ ^ > 0 () 11Re k DG Re k G1D G
Appendix D: Proof of Theorem 2
k51
and
T ^ 5 vec DT 5vec GT 2vec G ^ dT1
...
dTn
T
(A11)
䊏 By the Cauchy-Schwarz inequality, we have that
(A12) which is satisfied if
^ 21 < 1 k DG
(A13)
By the matrix spectral radius theorem we have ^ 21 ^ 21 k DG DG i
n qffiffiffiffiffiffiffiffiffi X xTk xk
21
ki < 1
(A22)
U
vec DT
T
1 Uvec DT < n
The worst case of model uncertainty corresponds to the maximum in the left-hand side of the final inequality in Eq. A22, namely
T max vec DT U vec DT D2D |fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflffl{zfflfflfflfflffl}
(A23)
T T T 2 D5 D : vec D B vec D rB |fflfflfflfflffl{zfflfflfflfflffl} |fflfflfflfflffl{zfflfflfflfflffl}
(A24)
x
xT
(A17) with
6 7 6 7 6 7 6 d1 7 6 7 7 2 36 6 7 . T .. 6 7 ^ 0 v k 6 76 7 6 76 7 6 76 7 6 6 7 7 ^T T .. 56 0 ⯗ 6 7 7 5Z k vec D . 0 76 6 7 6 76 7 4 56 7 6 7 .. 6 7 . 0 v^Tk 6 7 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 6 7 T 6 7 ^ Zk 6 dn 7 6 7 4 5
xT
x
according to Eq. 26. Using the method of Lagrange multipliers (see below), the maximum of xT Ux in the above optimization problem of Eqs. A23 and A24 can be easily shown (see below) to be equal to lmax B21 U rB2 and to be attained for xmax being the eigenvector to corresponding the largest eigenvalue of the matrix B21 U, lmax B21 U P . ^k Finally, it is straightforward to show that U5 ^ nk51 r^12 Z T T k 21 ^ ^ ^ Z k 5In ðG GÞ , because
T
^ G ^ G
21
T ^ U ^S ^ U ^ S^ V ^T 5 V T
!21
5
n X 1 v^ v^T 2 k k ^ r k51 k
(A25)
and the mixed-product property of the Kronecker product, ðA BÞðC DÞ5ðACÞ ðBDÞ, implies
|fflfflffl{zfflfflffl} vecðDT Þ
3000
(A21)
n X T 1 ^ kZ ^ T vec DT < 1 () vec DT Z k 2 ^ r k51 k ! T T X n 1 1 T ^ Z ^ vec DT < () vec D Z 2 k k k51 r n ^k |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
(A15)
for any induced matrix norm k ki . Using the vec operator, the term D^ v k in the above inequality can be written as 2 2 3 3 dT1 v^Tk d1 6 6 7 7 6 6 7 7 7v^k 56 7 D^ v k 56 ⯗ ⯗ 6 6 7 7 4 4 5 5 dTn v^Tk dn 2 3
Therefore,
xTk xk
k51
n
which is satisfied if n X 1 kD^ v k u^ Tk ki < 1 ^ r k51 k
n X
^ T vec DT , Applied to the left-hand side of Eq. A20 with xk 5 ^ r^1k Z k Eq. A21 implies that the inequality in Eq. A20 is satisfied if
(A14)
for some induced matrix norm k ki . ^ Pn r ^ k v^Tk , Eq. A15 is equivalent to Using the SVD G5 k51 ^ k u ! 21 n n X X D^ v k u^ Tk T D < 1 (A16) ^ ^ ^ k uk vk r < 1 () ^ k i r i k51 k51
n
k51
for any induced matrix norm k ki . Therefore, Eq. A14 is satisfied if ^ kDG
!2
n n X 1 ^ ^T X 1 ðI v^ k Þ In v^Tk Z Z 5 2 k k 2 n ^ ^ k51 r k k51 r k n X 1 ðI I Þ v^ k v^Tk 5 2 n n ^k k51 r
(A26)
n X 1 T 5In v^k v^k ^ 2k k51 r T 21 ^ G ^ 5In G
(A18) DOI 10.1002/aic
Published on behalf of the AIChE
September 2016 Vol. 62, No. 9
AIChE Journal
Lagrange multiplier method: First, maxD2D xT Ux such that xT Bx rB2 is attained for
x
T
Bx5rB2
@L @q‘
dL 52ðU2gBÞx50 () B21 Fx5gx dxT
xT Fx5xT BB21 Fx5gxT Bx5grB2
Premultiplying Eq. A33 by qT‘ for ‘51; . . . ; n gives
qT‘
! n X ^ 24 R ð‘Þ 2g I q 50 ( ) bkk q2‘;k 50; ‘51; . . . ; n ‘‘ ‘ k4‘ k51 |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}
(A29)
(A34) ^ distinct (Eq. 52), the diagonal With singular values in R ^ 24 matrix Bð‘Þ 5 Rk4 2g‘‘ I can only have one entry equal to zero, ‘ for a corresponding choice of g‘‘ . Consequently, q‘ in Eq. A34 can only have one non-zero entry, corresponding to the diagonal zero entry of Bð‘Þ ; that is q‘ must be a unit vector ep‘ 5 ^ p‘ ; 0; . . . 0T with p‘ 6¼ 0. Substituting q‘ in Eq. A32 ½0; . . . 0;P n 1 yields k51 k4 r4 , where index ½‘1 ; . . . ‘n 5½1; . . . nP. By the ‘k k rearrangement inequality, it can be trivially shown that the miniPn 1 mum value of k51 k4 r4 is obtained when rk and k‘k are ‘ k matched in reverse order.k Therefore
Appendix E: Proof of Eq. 45 21
It is straightforward to show that the eigenvalues of lðB UÞ satisfy the equalities
21 T 21 ^ G ^ lðB21 UÞ5l MT M G 0 1 (A30) B ^T 21 T ^ ^ 22 ^ T 21 T ^ ^ 22 C 5l PK P V R V 5l@|ffl{zffl} P V R A; V P K |ffl{zffl}
Popt 5arg
P2R
^ opt 5VP ^ tr A21 H 5VQ (A35)
min n3n
; orthonormal
Appendix G: Proof of Theorem 4 ^ from Eq. 54 into Eq. 50 yields Substituting V n X
v^ r 2k 112v k2‘k w2 k51 |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}
QT
the last equality owing to the similarity of the matrices PK21 PT ^ T PK21 PT V ^ T ðPK21 PT V ^R ^ 22 V ^ T and V ^R ^ 22 5 V ^R ^ 22 VÞ ^ V ^ T. V 21 T ^ 22 Because QK Q R is positive definite, its largest eigen^ 22 Þ is equal to its spectral radius value lmax ðQK21 QT R ^ 22 Þ. A standard upper bound for the spectral radius qðQK21 QT R is the Frobenius norm of the corresponding matrix, which yields the following
^ lmax QK21 QT R
(A36)
b2k
Using the Lagrange multiplier method, minimization of the objective function (Eq. 60) subject to constraints as in Eq. A36 and ordering of k‘k in Eq. 51 proceeds as follows P P Define the Lagrangian as L5 nk51 r^ 41k4 1 ni50 li gi where
g0 5
^ 22 5q QK21 QT R
n X k51
k ‘k
v^ r 2k 112v
|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}
k2‘k 2w2
0
b2k
(A37)
g1 52k‘1 < 0
22
^ k kQK21 QT R F rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 22 ^ QK21 QT QK21 QT R ^ 22 5 tr R 5
(A33)
Bð‘Þ
which implies that g must be the largest of all eigenvalues of B21 U, lmax B21 U , x its corresponding and the eigenvector, maximum of xT Ux is xTmax Uxmax 5lmax B21 U rB2 .
22
‘ n X ^ 24 q‘ X R 2 g q 2 gi‘ qi 50 ‘k k k4‘ k51 i5‘
(A28)
which, in turn, implies that ðg; xÞ is an eigenvalue-eigenvector pair for the matrix B21 U. Now, at the optimum, the objective function will be
52
(A27)
because, if the optimum were attained for ðxopt ÞT Bðxopt Þ < rB2 , then scaling up xopt would trivially yield a larger value for xT Fx. Then, the Lagrangian is L5xT Fx2gðxT Bx2rB2 Þ, which implies that at the optimum
Q
T
gi 5k‘i21 2k‘i < 0; 2 i n (A31)
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ 24 tr QK22 QT R
Applying the Karush-Kuhn-Tucker conditions
@L 4 52 4 5 1l0 2k‘k b2k 1ðlk11 2lk Þ50 @k‘k ^ k k‘ r
(A38)
k
li gi ðk‘k Þ50; 0 i n
(A39)
li 0; 0 i n
(A40)
Appendix F: Proof of Theorem 3 By the properties of matrix trace it follows that
! n n qT R ^ 24 qj X X qk qTk ^ 24 j 22 T ^ 24 tr QK Q R (A32) 5tr 5 R 4 k4j j51 k51 kk Pn qTj R^ 24 qj Pn Pi TDefining the Lagrangian as L5 j51 k4j 2 i51 k51 gik qi qk 2dik and setting its partial derivatives with respect to q‘ equal to zero yields
AIChE Journal
September 2016 Vol. 62, No. 9
yields k‘k 5ðl
1=6
2
^ 4k b2k 0r
Solving
Þ
, g0 50, and li 50, for 1 i n. 1=6
k‘k 5ðl r^24 b2 Þ 0 k k
k‘k 5
with g0 50, yields the optimal solution
w 2
ð^ r k bk Þ
1=3
Pn
i51
1 bi 4=3 1=2
(A41)
^i r
Manuscript received Dec. 23, 2015, and revision received Feb. 5, 2016.
Published on behalf of the AIChE
DOI 10.1002/aic
3001
