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Journal of Process Control 20 (2010) 787–799
Contents lists available at ScienceDirect
Journal of Process Control
journa l homepage: www.e lsev ier .com/ locate / jprocont
tate estimation and nonlinear predictive control of autonomousybrid system using derivative free state estimators
. Prakasha, Sachin C. Patwardhanb, Sirish L. Shahc,∗
Department of Instrumentation Engineering, Madras Institute of Technology, Anna University, Chennai 600044, IndiaDepartment of Chemical Engineering, Indian Institute of Technology, Bombay, Powai, Mumbai 400076, IndiaDepartment of Chemical and Materials Engineering, University of Alberta, Edmonton T6G 2G6, Canada
r t i c l e i n f o
rticle history:eceived 15 July 2009eceived in revised form 3 February 2010ccepted 2 April 2010
a b s t r a c t
In this work, we develop a state estimation scheme for nonlinear autonomous hybrid systems, which aresubjected to stochastic state disturbances and measurement noise, using derivative free state estimators.In particular, we propose the use of ensemble Kalman filters (EnKF), which belong to the class of particlefilters, and unscented Kalman filters (UKF) to carry out estimation of state variables of autonomous hybridsystem. We then proceed to develop novel nonlinear model predictive control (NMPC) schemes using
eywords:nsemble Kalman filternscented Kalman filterutonomous hybrid systemonlinear model predictive controllertate estimation based NMPC
these derivative free estimators for better control of autonomous hybrid systems. A salient feature of theproposed NMPC schemes is that the future trajectory predictions are based on stochastic simulations,which explicitly account for the uncertainty in predictions arising from the uncertainties in the initialstate and the unmeasured disturbances. The efficacy of the proposed state estimation based controlscheme is demonstrated by conducting simulation studies on a benchmark three-tank hybrid system.Analysis of the simulation results reveals that EnKF and UKF based NMPC strategies is well suited foreffective control of nonlinear autonomous three-tank hybrid system.
. Introduction
Nonlinear model predictive control (NMPC) has become aajor research area over the last few decades. Unlike many other
dvanced techniques, NMPC has been successfully applied in manyrocess industries for controlling complex unit operations oper-ted in continuous or semi-batch mode. If we view these NMPCpplications from the viewpoint of model development, then theseormulations typically employ models represented by ordinaryifferential equations or differential-algebraic equations, which
nvolve continuous state variables. However, dynamic systems thatnvolve continuous and discrete state variables, broadly classifieds hybrid systems, are often encountered in engineering applica-ions. Thus, model predictive control of hybrid system has gainedncreased attention in recent years [1].
Many methodologies have been proposed to model hybrid sys-ems, to analyze their behaviour, and to design model based control
chemes that guarantee closed loop stability and performancepecifications. Among the several different approaches, the mixedogical dynamic (MLD) formalism [1], which is described by linearynamic equations with continuous, discrete and mixed variables,∗ Corresponding author.E-mail address: [email protected] (S.L. Shah).
959-1524/$ – see front matter © 2010 Elsevier Ltd. All rights reserved.oi:10.1016/j.jprocont.2010.04.001
© 2010 Elsevier Ltd. All rights reserved.
and linear inequality constraints, has been widely used in mod-elling hybrid system. Also, this framework facilitates developmentof model based predictive control for hybrid systems. It may benoted that, many approaches are now available in the literature forcontrol of hybrid systems based on linear hybrid models [2,4,5,26],but most of these approaches resort to using linear approximationof nonlinear system dynamics and they employ multi-parametricoptimization approach for controller synthesis. In comparison tothese approaches, not much work has been reported on the directuse of nonlinear hybrid models for control. Sonntag et al. [6] havedeveloped an NMPC scheme for nonlinear autonomous hybridsystem. By parameterizing discrete input signals as pulses, theyconvert the resulting MINLP problem into NLP with respect to con-tinuous variables.
Recently, Nandola and Bhartiya [7] have proposed a novelapproach for predictive control of nonlinear hybrid systems basedon multi-model approach. They propose to develop multiple locallinear models that are combined using Bayesian approach to gener-ate a control relevant nonlinear model for carrying out predictions.In another publication [8], they exploit structural features of the lin-
earized multi-model to evolve a computationally efficient schemebased on generalized outer approximation for solving the result-ing MINLP. An experimental validation of model predictive controlusing the identified model for a three-tank benchmark hybrid sys-tem has been reported by Nandola and Bhartiya [9].
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Most of the above mentioned control schemes are developednder deterministic settings and the effect of stochastic unmea-ured disturbances are not considered. In practice, however, mostf the systems are influenced by unmeasured disturbances. In addi-ion, measurements used for feedback are corrupted with noise. Asconsequence, the scheme used for state estimation and unmea-
ured disturbance prediction forms a key component of any NMPCmplementation. The conventional state observer based MPC for-
ulations make use of the Kalman filter (KF) (Ricker [22]) andxtended Kalman filter (EKF) [10] as a state estimator. For lin-ar systems, Kalman filters generate optimal estimates of staterom observations when uncertainties in state and measurementquations can be adequately modelled as Gaussian white noiserocesses. The Kalman filter has attracted widespread attention
n the engineering community because of the recursive nature ofts computational scheme. For nonlinear systems, the extendedalman filter (EKF) is a natural extension of the linear filter to
he nonlinear domain through local linearization. In EKF formu-ations, the state covariance propagation is carried out using Tayloreries expansion of the nonlinear state transition operator. This stepequires analytical computation of Jacobians at each time step. Thismplies that nonlinear function vectors appearing in state dynamicsnd state to output map should be smooth and at least once dif-erentiable. However, dynamic models for hybrid systems involveiscontinuities, which are introduced by switching of the discreteariables. As a consequence, EKF cannot be used for state estimationn autonomous hybrid systems as Jacobians cannot be computed foron-smooth functions. The use of moving horizon based state esti-ation for hybrid system has been reported in Ferrari Trecate et al.
11] and Bemporad et al. [3]. If a fixed arrival cost is used in the MHEormulation, then the need to compute Jacobians can be avoided.owever, the use of fixed arrival cost can result in sub-optimal statestimates. State estimation using robust hybrid observer and faultiagnosis for uncertain hybrid systems has been reported in Wangt al. [12]. With the exception of these few references, to the bestnowledge of the authors, the problem of estimating state variablesf a nonlinear hybrid system has hardly received any attention inhe process control literature.
Thus, state estimation and estimator based control of hybridystems poses a challenging problem. In recent years, a numberf derivative free nonlinear filtering techniques have been pro-osed in the literature [13]. For example, the unscented Kalmanlter (UKF) [14] or central difference Kalman filter (CDKF) [15]ave been proposed as an alternative to the EKF where the above
imitations have been overcome using the concept of sample statis-ics. Also, a new class of filtering technique, called particle filtering,an deal with state estimation problems arising from multi-modalnd non-Gaussian distributions [16]. A particle filter (PF) approx-mates multi-dimensional integration involved in the propagationnd update steps in the general Bayesian state estimation schemesing Monte Carlo sampling. The ensemble Kalman filter (EnKF),riginally proposed by Evensen [17], belong to the class of parti-le filters. In the EnKF formulation, similar to the EKF or UKF, thebserver gain is computed using the covariance matrix of the inno-ations and cross covariance matrix between the predicted statestimation errors and the innovations. However, the main differ-nce is that the covariance information is generated using Montearlo sampling without making any assumption on the nature ofnderlying distribution of state. In the context of developing statestimator based predictive control schemes for hybrid systems,hese derivative free estimation schemes appear to be promising
andidates.In this work, we develop a state estimation scheme for nonlin-ar autonomous hybrid systems, which are subjected to stochastictate disturbances and measurement noise, using the ensemblealman filter (EnKF) and unscented Kalman filter (UKF). We further
Control 20 (2010) 787–799
develop a novel derivative free estimator based nonlinear modelpredictive control schemes for optimal control of autonomoushybrid system [18]. The distinguishing feature of the proposedNMPC schemes is that the future trajectory predictions are based onstochastic simulations which explicitly account for the uncertaintyin predictions arising from the uncertainties in the initial state andthe unmeasured disturbances. The efficacy of the proposed stateestimation based control scheme is demonstrated by conductingsimulation studies on the benchmark three-tank hybrid system.
The organization of the paper is as follows. Section 2 discussesthe details of state estimation in an autonomous hybrid systemusing UKF and EnKF algorithms. The design of derivative free stateestimators based nonlinear model predictive control scheme for anautonomous hybrid system is given in Section 3. The process con-sidered for the illustrative simulation study is outlined in Section4. Simulation results are also presented in Section 4 followed byconcluding remarks in Section 5.
2. State estimation of autonomous hybrid systems
A particular class of hybrid systems, namely autonomous hybridsystems, is of interest in this work. These systems can be repre-sented by the following set of differential-algebraic equations:
x(k) = x(k − 1) +∫ kT
(k−1)T
F[x(�), u(k − 1), d + w(k − 1), z(�)]d� (1)
z(�) = G[x(�)] (2)
y(k) = H[x(k), v(k)] (3)
In the above process model, x(k) is the system state vector(x ∈ Rn), u(k) is known system input (u ∈ Rm), d ∈ Rp is the unknownsystem input, w(k) is the state noise (w ∈ Rp) with known distri-bution, y(k) is the measured state variable (y ∈ Rr) and v(k) is themeasurement noise (v(k) ∈ Rr) with known distribution. The indexk represents the sampling instant, F[·] and H[·] are the nonlin-ear process and measurement models respectively. In Eq. (2), zrepresents a (h × 1) discrete variables such that it can take onlyfinite integer values, such as {−1, 0, 1} depending on some events,which are functions of continuous state variables in the case ofan autonomous hybrid system. The function vector G[·] can beexpressed using a combination of logical operators, such as AND,OR, XOR, etc.
The first step in the development of any NMPC scheme is the for-mulation of a scheme for state estimation. In this work, we intend touse the derivative free nonlinear state estimation schemes, namely,the ensemble Kalman filter [19] and unscented Kalman filter [14]for estimation of state variables in an autonomous hybrid system.The major advantage of using EnKF and UKF is that they avoidexplicit computation of Jacobian matrices. The second order statis-tics necessary for estimation of observer gain in UKF and EnKF isgenerated using sample points (particles). The sample points strad-dle the discontinuity and, hence, can approximate the effect ofdiscontinuity [14]. In the following section we present the EnKFalgorithm and UKF algorithm for state estimation of an autonomoushybrid system in more detail.
2.1. Ensemble Kalman filter
The filter is initialized by drawing N particles {x(j)(0|0)} from
a suitable distribution. At each time step, N samples {w(j)(k −1), v(j)(k) : j = 1, . . . , N} for {w(k)} and {v(k)} are drawn randomlyusing the distributions of state noise and measurement noise.These sample points together with particles {x(j)(k − 1|k − 1) : j =1, . . . , N} are then propagated through the system dynamics to
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ompute a cloud of transformed sample points (particles) as fol-ows:
ˆ(i)(k|k − 1) = x(i)(k − 1|k − 1) +∫ kT
(k−1)T
F[x(�), u(k − 1),
¯ + w(i)(k − 1), G[x(�)]]d� i = 1, 2, . . . , N (4)
These particles are then used to estimate the sample mean andovariance terms as follows:
¯ (k|k − 1) = 1N
N∑i=1
x(i)(k|k − 1) (5)
¯ (k|k − 1) = 1N
N∑i=1
H[x(i)(k|k − 1), v(i)(k)] (6)
ε,e(k) = 1N − 1
N∑i=1
[ε(i)(k)][e(i)(k)]T
(7)
e,e(k) = 1N − 1
N∑i=1
[e(i)(k)][e(i)(k)]T
(8)
here,
(i)(k) = x(i)(k|k − 1) − x(k|k − 1) (9)
(i)(k) = H[x(i)(k|k − 1), v(i)(k)] − y(k|k − 1) (10)
The Kalman gain and cloud of updated samples (particles) arehen computed as follows:
(k) = Pε,e(k)[Pe,e(k)]−1 (11)
(i)(k|k − 1) = {y(k) − H[x(i)(k|k − 1), v(i)(k)]} (12)
ˆ(i)(k|k) = x(i)(k|k − 1) + L(k)� (i)(k|k − 1) (13)
(i)(k|k) = {y(k) − H[x(i)(k|k)]} (14)
The updated state estimate is computed as the mean of thepdated particles cloud, i.e.
ˆ(k|k) = 1N
N∑i=1
x(i)(k|k) (15)
ˆ(k|k) = G[x(k|k)] (16)
The accuracy of the estimates depends on the number of dataoints (N). Gillijns et al. [19] have indicated that ensemble sizeetween 50 and 100 suffices even for large dimensional systems.
.2. Unscented Kalman filter
In this subsection, we describe the steps involved in obtainingtate estimates of autonomous hybrid system using UKF. Considerhe augmented state vector (xa(k|k)) defined as follows
a(k|k)�
(x(k|k)w(k)v(k)
)(17)
The expected value of the augmented state at instant (k − 1) andhe covariance of the augmented state vector are as follows:
[xa(k − 1|k − 1)] =
⎛⎜⎝
x(k − 1|k − 1)
0p×1
0r×1
⎞⎟⎠ (18)
Control 20 (2010) 787–799 789
Pa(k − 1|k − 1) =
⎡⎢⎣
P(k − 1|k − 1) 0n×p 0n×r
0p×n Q 0
0r×n 0 R
⎤⎥⎦ (19)
A set of 2(n + r + p) + 1 deterministic sigma points �a togetherwith the associated weights {ˇ(i): i = 1, . . ., 2(n + r + p) + 1} are cho-sen symmetrically around xa(k − 1|k − 1) as follows:
�a = {xa(k − 1|k − 1), xa(k − 1|k − 1) + �(i)(k − 1), xa(k − 1|k − 1)
− �(i)(k − 1) : i = 1, 2, . . . (n + p + r)} (20)
where the vector �(i)(k − 1) represents the ith column of matrix√
(n + r + p + �)Pa(k − 1|k − 1). The associated weights are chosenas{
ˇ0= �
n + r + p + �, ˇi=
12(n + r + p + �)
: i = 1, 2, . . . (n + r + p)}
(21)
where � is a tuning parameter. For Gaussian distribution, thistuning parameter can be obtained from the following relation� = 3 − (n + r + p) [14]. The 2(n + r + p) + 1 sigma points have beenderived from the augmented state (xa(k − 1|k − 1)) and covarianceof the augmented state vector (Pa(k − 1|k − 1)), where (n + r + p)is the dimension of the augmented state. In the prediction step,the sigma points are propagated through the nonlinear differentialequations to obtain the predicted set of sigma points as
x(i)(k|k − 1) = x(i)(k − 1|k − 1) +∫ kT
(k−1)T
F[x(�), u(k − 1),
d + w(i)(k − 1), G[x(�)]]d� i = 0 : 2(n + p + r) (22)
The statistics of the nonlinearly transformed sigma points iscomputed as
x(k|k − 1) =2(n+p+r)∑
i=0
ˇix(i)(k|k − 1) (23)
where
y(k|k − 1) =2(n+p+r)∑
i=0
ˇiH[x(i)(k|k − 1), v(i)(k)] (24)
Pε,e(k) =2(n+p+r)∑
i=0
ˇi[ε(i)(k)][e(i)(k)]
T(25)
Pe,e(k) =2(n+p+r)∑
i=0
ˇi[e(i)(k)][e(i)(k)]
T(26)
where,
ε(i)(k) = x(i)(k|k − 1) − x(k|k − 1) (27)
e(i)(k) = H[
x(i)(k|k − 1), v(i)(k)]
− y(k|k − 1) (28)
It may be noted that w(i) and v(i) are the sampled process andmeasurement noise terms, respectively. The computation of theobserver gain L(k) is as given by Eq. (11) and the updated continuousand discrete state estimates are given as follows:
� (k|k − 1) = {y(k) − y(k|k − 1)} (29)
x(k|k) = x(k|k − 1) + L(k)� (k|k − 1) (30)
� (k|k) = {y(k) − H[x(k|k)]} (31)
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ˆ(k|k) = G[x(k|k)] (32)
Since in UKF formulation the sigma points are computed basedn covariance, the state error covariances are updated as follows:
P(k|k − 1) =2(n+p+r)∑
i=0
ˇi[ε(i)(k)][ε(i)(k)]
T
P(k|k) = P(k|k − 1) − L(k)Pe,e(k)L(k)T
(33)
Remark 1: If the state noise and the measurement noise aredditive, zero mean and Gaussian, then it is possibly to simplifyKF and EnKF computations. For example, in the UKF formulation,qs. (33) and (25) is modified as follows:
P(k|k − 1) =2n∑i=0
ˇi[ε(i)(k)][ε(i)(k)]
T + Q
Pe,e(k) =2n∑i=0
ˇi[e(i)(k)][e(i)(k)]
T + R
(34)
ithout requiring to draw samples for w(k) and v(k). Similar mod-fications can be carried out in the EnKF formulation.
Remark 2: The central difference filter (CDKF) [15], which haseen derived based on Sterling’s interpolation formula, is anotherxample belonging to this class of derivative free state estimatorsr sigma point filters. The main difference between UKF and CDKFs in the choice of sigma points and weights [20]. It may be notedhat any other appropriate sigma point filter can be used for statestimation of autonomous hybrid system.
. Nonlinear MPC formulation for an autonomous hybridystem
The first step in the development of NMPC formulation is gen-ration of model based future predictions. At this stage, it becomesecessary to incorporate measures for dealing with plant-modelismatch. There are two strategies available in the literature for
ealing with plant-model mismatch. They are as follows:
.1. State augmentation
Using this approach, artificial states, say �(k), equal to numberf outputs are introduced in the state dynamics as follows [10]:
x(k)=x(k−1)+∫ kT
(k−1)T
F[x(�), u(k−1), �(k−1), d+w(k − 1), z(�)]d�
�(k) = �(k − 1) + w�(k − 1)(35)
(�) = G[x(�)] (36)
(k) = H[x(k), v(k)] (37)
Here w�(k) represents zero mean Gaussian white noise processith covariance Q�. The state observer is designed for the aug-ented system by treating entries in Q� as tuning parameters. The
rtificially introduced state variables �(k), which are modelled asntegrated white noise, provide integral action to the estimator. Typ-cally �(k) is chosen as a bias term in the input or to capture drifts inome model parameters. For example, if �(k) is chosen to representhe bias in the manipulated inputs, then the state dynamics used
Control 20 (2010) 787–799
in estimation and predictions is modified as follows:
x(k) = x(k − 1) +∫ kT
(k−1)T
F[x(�), u(k − 1) + � ��(k − 1),
d + w(k − 1), z(�)]d� (38)
Here, �� is a (m × l) matrix consisting of ones and zeros thatintroduces bias in the selected subset of manipulated inputs. Itmay be noted that the number of artificial states or drifting modelparameters (l) cannot exceed the number of measurements (r). Themain difficulty with this approach is that the closed loop perfor-mance is a complex function of Q� and the values have to be selectedby trial and error [21]. In addition, state augmentation implies thata large number of particles have to be used in EnKF and UKF for-mulations, thereby increasing the computational burden.
3.2. Innovation based correction
Recently, Srinivasrao et al. [21] have proposed a plant-modelmismatch compensation strategy, which is similar to the schemedeveloped by Ricker [22]. By this approach, the future state x(k +j|k) and the future output y(k + j|k) predictions are carried out usingthe original un-augmented state space model. The predicted statesx(k + j|k) are corrected using integrated white noise signal
�e(k + j + 1|k) = �e(k + j|k)�e(k|k) = � f (k|k − 1)
(39)
where � f (k|k − 1) represents filtered values of model residual andare computed as follows:
� f (k|k − 1) = ˚e� f (k − 1|k − 2) + [I − ˚e]� (k|k − 1) (40)
Similarly, the predicted outputs y(k + j|k) are corrected assum-ing integrated white noise signals
�d(k + j + 1|k) = �d(k + j|k)�d(k|k) = � f (k|k)
(41)
where � f (k|k) represents filtered values of model residual com-puted as follows
� f (k|k) = ˚e� f (k − 1|k − 1) + [I − ˚e]� (k|k) (42)
Here, ˚d and ˚e are diagonal matrices of the form
˚d = diag[
�1 �2 · · · �n
]˚e = diag
[˛1 ˛2 · · · ˛n
] (43)
0 ≤ �i < 1 and 0 ≤ ˛i < 1 (44)
where �i and ˛i are tuning parameters. It may be noted that thesefilters are identical to that of filters in the feedback path of thenonlinear internal model control scheme [23] and provide extradegrees of freedom for achieving robustness against plant-modelmismatch. Srinivasrao et al. [21] have shown that it is easier to tunethese filters than choosing elements in Q� for the artificial states.Recently, Huang et al. [24] have established sufficient conditionsfor robust stability of the innovation feedback based NMPC formu-lation. In this work, we develop a nonlinear MPC scheme using thelatter approach.
3.3. Multi-step ahead future prediction using EnKF
Consider the problem of generating multi-step ahead predic-tions over time horizon [k + 1, k + Np], where Np represent theprediction horizon. The two sources of uncertainties that arisewhile performing such predictions are (a) uncertainty in initial stateat the beginning of the horizon and (b) unmeasured disturbances
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w(i)(k + j|k) : j = 1, . . . , Np} that may occur in future. To overcomehe difficulties arising from the uncertainties in the initial state,e propose to carry out predictions by propagating all particles
ver the prediction horizon. Thus, given future manipulated inputoves u(k|k) . . . u(k + 1|k) . . . u(k + Np − 1|k), the future state pre-
ictions are generated as follows:
ˆ(i)(k + j + 1|k) = x(i)(k + j|k) +∫ (k+j+1)T
(k+j)T
x(�), u(k + j|k),
+ w(i)(k + j|k), z(�)d� i = 1, 2, . . . , N and j = 0, 1, 2, . . . , Np − 1
(45)
The predicted mean of state and output trajectories are thenomputed as follows
ˆ(k + j + 1|k) = 1N
N∑i=1
[x(i)(k + j + 1|k) + L(k)�(i)e (k + j + 1|k)] (46)
ˆ(k + j + 1|k) = 1N
N∑i=1
[H[x(i)(k + j + 1|k)] + �(i)d
(k + j + 1|k)] (47)
These filtered signals are computed for each particle as follows:
�(i)e (k + j + 1|k) = �(i)
e (k + j|k)
�(i)e (k|k) = � (i)
f(k|k − 1)
(48)
�(i)d
(k + j + 1|k) = �(i)d
(k + j|k)
�(i)d
(k|k) = � (i)f
(k|k)(49)
(i)f
(k|k − 1) = ˚e� (i)f
(k − 1|k − 2) + [I − ˚e]� (i)(k|k − 1) (50)
(i)f
(k|k) = ˚d� (i)f
(k − 1|k − 1) + [I − ˚d]� (i)(k|k) (51)
It may be noted that � (i)f
(k|k − 1) and � (i)f
(k|k) are filtered values
f signals � (i)(k|k − 1) and � (i)(k|k), respectively, which are definedy Eqs. (12) and (14).
.4. Multi-step ahead future prediction using UKF
Multi-step ahead future predictions are computed somewhatifferently when UKF is used for state estimation. In the UKF for-ulation, new sigma points are generated at each prediction step
sing prediction error covariance matrix as follows:
(j)p = {x(k + j|k), x(k + j|k) + �
(i)(k + j|k), x(k + j|k)
− �(i)(k + j|k) : i = 1, 2, . . . , n + p} (52)
here the vector �(i)(k + j|k) represents the ith column of matrix
(n + p + �)Pa(k + j|k).
a(k + j|k) =[
P(k + j|k) 0n×q
0q×n Q
]
The prediction error covariance matrix required in the abovetep is updated as follows:
(i)(k + j + 1|k) = x(i)(k + j + 1|k) − x(k + j + 1|k) (53)
(k + j + 1|k) =2(n+p)∑
i=0
ˇi[ε(i)(k + j + 1|k)][ε(i)(k + j + 1|k)]
T(54)
Control 20 (2010) 787–799 791
where{ˇ0 = �
n + p + �, ˇi = 1
2(n + p + �): i = 1, 2, . . . , (n + p)
}The predicted mean is computed as follows:
x(i)(k + j + 1|k) = x(i)(k + j|k) +∫ (k+j+1)T
(k+j)T
F[x(�), u(k + j|k),
d + w(i)(k + j|k), G[x(�)]]d� i = 0, 1, . . . , 2(n + p)
and j = 0, 1, . . . , Np − 1 (55)
x(k + j + 1|k) =2(n+p)∑
i=0
ˇix(i)(k + j + 1|k) (56)
x(k + j + 1|k) = x(k + j + 1|k) + L(k)�e(k + j + 1|k) (57)
�e(k + j + 1|k) = �e(k + j|k)�e(k|k) = � f (k|k − 1)
(58)
� f (k|k − 1) = ˚e� f (k − 1|k − 2) + [I − ˚e] � (k|k − 1) (59)
y(k + j + 1|k) = H[x(k + j + 1|k)] + �d(k + j + 1|k) (60)
�d(k + j + 1|k) = �d(k + j|k)�d(k|k) = � f (k|k)
(61)
� f (k|k) = ˚e� f (k − 1|k − 1) + [I − ˚e] � (k|k) (62)
It may be noted that �f (k|k − 1) and �f (k|k) are filtered valuesof signals � (k|k − 1) and �(k|k), respectively, which are defined byEqs. (29) and (31). These filtered signals are computed as describedin the previous subsection.
Remark 3: The main difference between the UKF based and theEnKF based formulation arises due to the distributional assump-tions implicit in the propagation step. The sampling scheme inthe UKF formulation implicitly assumes that the probability dis-tribution of the state are represented by the multivariate Gaussiandistribution, which can turn out to be an oversimplification in com-plex nonlinear systems. On the other hand, the EnKF formulationdoes not make such distributional assumption while propagatingthe particles. Thus, the EnKF based formulation is applicable to amore general class of nonlinear systems.
Remark 4: If the state noise and the measurement noise areadditive, zero mean and Gaussian, then it is possibly to simplify UKFand EnKF based predictions as indicated in Remark 1. Moreover,there is no need to draw samples for state noise while carrying outpredictions in this case.
3.5. NMPC formulation
At any sampling instant k, the NMPC is formulated as aconstrained nonlinear optimization problem where the futuremanipulated inputs are determined by minimizing a suitable objec-tive function. It is possible to treat the objective function itself as astochastic quantity and formulate NMPC as minimization of expec-tation of this quantity. However, to simplify on-line computations,we propose to work with the expected value of predicted trajecto-ries in the NMPC formulation. Thus, the objective function used inour formulation is defined as follows:
J = u(k|k)min. . .u(k + Np − 1|k)
NP∑E(k + j|k)T WEE(k + j|k)
j = 1
+NC −1∑j=0
�u(k + j|k)T WU�u(k + j|k) (63)
7 rocess
u
x
u
�
w
E
�
wtisocstptm(holf
rcpdttch
3
niwsotrfs
•
92 J. Prakash et al. / Journal of P
Subject to prediction given by Eqs. (55)–(62) or Eqs. (45)–(51)
(k + j|k) = u(k + NC − 1) for j = Nc, Nc + 1, . . . , Np − 1 (64)
L ≤ x(k + j|k) ≤ xU(j = 1, . . . , NP) (65)
L ≤ u(k + j|k) ≤ uU(j = 0, . . . , Nc − 1) (66)
uL ≤ �u(k + j|k) ≤ �uU(j = 0, . . . , NC − 1) (67)
here
(k + j|k) = yr(k + j|k) − y(k + j|k) (68)
u(k + j|k) = u(k + j|k) − u(k + j − 1|k) (69)
here yr(k + j|k), in Eq. (68) represents future setpoint trajec-ory. It may be noted that the state and the output constraints aremposed on the expectations and not on individual particles (origma points). The cost function is minimized subject to constraintsn the state variables, output variables, manipulated variables andhange in manipulated variables. Eq. (64) together with Eq. (69)tates that no future control moves are planned beyond the con-rol horizon of Nc steps. The desired closed loop performance of theroposed NMPC scheme can be achieved by appropriately selectinghe prediction horizon Np, control horizon Nc, the error weighting
atrix (WE), input move weight matrix (WU) and filter matrices˚d, ˚e). Further, the NMPC scheme is implemented in a recedingorizon framework i.e. only the first move u(k|k) is implementedn the plant and the constrained optimization problem is reformu-ated at the next sampling instant based on the updated informationrom the plant.
Remark 5: It may be noted that, in the present work, we haveestricted ourselves to autonomous hybrid systems, i.e. the dis-rete states are functions of continuous process states alone. In theroposed estimation and future trajectory prediction schemes, theiscrete states are evaluated using predicted estimates of the con-inuous states as z(�) = G[x(�)]. Moreover, the decision variables inhe NMPC formulation are assumed to be continuous variables. As aonsequence, the proposed NMPC formulation for the autonomousybrid systems reduces to an NLP and not an MINLP.
.6. Comparison with EKF based NMPC formulation
Consider the problem of state estimation and prediction inonlinear systems that do not involve any discontinuities. If it
s desired to carry out predictions for general nonlinear systems,ith or without discontinuities in the state dynamics and in the
tochastic settings, it becomes necessary to consider two sourcesf uncertainty, namely the errors in the initial state estimate andhe unmeasured disturbances influencing the state dynamics. Witheference to the handling of these uncertainties in predictions, theollowing observations can be made regarding the conventionaltate estimator based NMPC formulation proposed in the literature:
The uncertainty in the future trajectory predictions {x(k + j|k) :j = 1, 2, . . . , Np}, which arises due to the uncertainty in initialstate, x(k|k), has been completely ignored. The future trajectorypredictions are carried out by assuming that the expected valueof a nonlinear function of a random variable can be approximatedby propagating the mean of the random variable through the non-linear function, i.e. E[g(x)] ≈ g[E(x)]. In fact, as no measurementsare available in the future while carrying out predictions, theuncertainty in x(k + j|k) increases with j and future predictionsbecome more and more inaccurate with the above mentioned
approximation. To overcome this problem, we propose to carryout the predictions by explicitly accounting for these uncertain-ties over the prediction horizon. In UKF based formulation this isachieved by propagating the covariance associated with predic-tion errors, Pa(k + j|k), over the future and drawing samples atControl 20 (2010) 787–799
jth future time point based on the propagated covariance infor-mation (see Eqs. (52)–(54)). In the EnKF based formulation, thisis achieved by propagating the particles over the entire horizon.
• Another difficulty with the conventional prediction scheme basedon the state estimator, such as EKF, is the simplistic assumptionthat the unmeasured disturbances in the state are linear and addi-tive. In practice, however, the unmeasured disturbances, suchas feed flow or concentration fluctuations, influence the statedynamics in a nonlinear manner. If it is desired to account forthis uncertainty in a systematic manner, then it becomes neces-sary to explicitly use the probability density function associatedwith the state noise while carrying out the predictions. In theproposed approach, this is achieved through the covariance prop-agation and through drawing samples for the uncertainty in thestate dynamics while carrying out the predictions.
Thus, even in the context of nonlinear systems whose dynamicsare governed by only the continuous states, the proposed approachaddresses the issue of uncertainties in the predictions, which islargely ignored in the conventional formulation. It may be notedthat, the need to address the issue of uncertainty in predictionsis also recognized in the recent work by Huang et al. (submittedfor publication). They propose to create multiple scenarios of thefuture trajectories by explicitly considering uncertainties in modelparameters.
4. Simulation studies
In this section, we present simulation studies to demonstratethe effectiveness of the proposed state estimation based nonlin-ear control on a benchmark system often used in hybrid systemliterature. The standard deviation of the estimation errors (E) ofcontinuous state variables, defined as SSEE =∑L
i=1[x(k) − x(k|k)]is used as a performance index to assess the performance of theproposed state estimation schemes. The performance of the NMPCscheme is assessed using conventional metric of integral squareerror (ISE).
4.1. Process description
The system considered for simulation study has three tanks [25],which are connected through the valves u1 to u7. The governingequations of the three-tank hybrid system are as follows (Fig. 1):
Adh1
dt= qmaxu1 − q2 − q3 − q6 (70)
Adh2
dt= q2 + q3 − q4 − q7 − q8 (71)
Adh3
dt= qmaxu5 + q4 + q7 (72)
q3 = k3sgn(h1 − h2)√
|h1 − h2|u3 (73)
q4 = k4sgn(h2 − h3)√
|h2 − h3|u4 (74)
q6 = k6
√h1 + hdu6 (75)
q8 = k8
√h2 + hz (76)
There are four possible values (modes of behaviour – [25]) forthe flow q2 (see Eq. (77)) which depends on the water levels h1 and
J. Prakash et al. / Journal of Process Control 20 (2010) 787–799 793
Fig. 1. Three-tank h
Table 1Three-tank hybrid system model parameters.
Parameter Value
qmax 2e−4k2 and k3 3e−4 6e−4k and k 6e−4 2e−4
h
z
d
z
4 6
k7 and k8 3e−4 2e−4hT and hd 0.3 0.02hz and A 0.02 0.0108
2 as follows (Table 1):
q2 = z1k2
√|(h1 − h2)|u2 {h1, h2 ≤ hT }
= z1k2
√(h1 − hT )u2 {(h1 � hT ) AND (h2 ≤ hT )}
= z1k2
√(h2 − hT )u2 {h1 ≤ hT ) AND (h2 � hT )}
= z1k2
√|(h1 − h2)|u2 {h1, h2 � hT }
(77)
1 =
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
0 {(h1 ≤ hT ) AND (h2 ≤ hT )} OR {h1 = h2}
+1
{[(h1 � hT ) AND (h2 ≤ hT )]OR
[[(h1 � hT ) AND (h2 � hT )] AND (h1 � h2)]
}
−1
{[(h1 ≤ hT ) AND (h2 � hT )]OR
[[(h1 � hT ) AND (h2 � hT )] AND (h1 ≺ h2)]
}
⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭(78)
There are four possible values for the flow q7 (see Eq. (79)) whichepends on the water levels h2 and h3 as follows:
q7 = z2k7
√|(h2 − h3)|u7 {h2, h3 ≤ hT }
= z2k7
√(h2 − hT )u7 {{(h2 � hT ) AND (h3 ≤ hT )}
= z2k7
√(h3 − hT )u7 {h2 ≤ hT ) AND (h3 � hT )}
= z2k7
√|(h2 − h3)|u7 {h2, h3 � hT }
(79)
=
⎧⎪⎪⎪⎪⎪⎨0 {(h2 ≤ hT ) AND (h3 ≤ hT )} OR {h1 = h2}
+1
{[(h2 � hT ) AND (h3 ≤ hT )]OR
[[(h � h ) AND (h � h )] AND (h � h )]
}⎫⎪⎪⎪⎪⎪⎬
2 ⎪⎪⎪⎪⎪⎩2 T 3 T 2 3
−1
{[(h2 ≤ hT ) AND (h3 � hT )] OR
[[(h2 � hT ) AND (h3 � hT )] AND (h2 ≺ h3)]
}⎪⎪⎪⎪⎪⎭(80)
ybrid system.
It may be noted that the flow q7 and the flow q2 depend onthe discrete variables z1 and z2 respectively. The discrete variablesz1 and z2 can take only finite integer values, such as {−1, 0, 1}depending on the level of fluids in the tanks. For example
• If the discrete variable (z1) value is zero, it implies that eitherthe water levels in the first tank and second tank are below thepredefined threshold value (hT) or the water levels in the firsttank and second tank are equal.
• If the discrete variable (z1) value is +1, it denotes that either thewater level in the first tank alone is above the threshold value orwater levels in the first tank and second tank are both above thethreshold value, with water level in the first tank greater thanwater level in the second tank causing flow in the pipe.
• If the discrete variable (z1) value is −1, it indicates that either thewater level in the second tank alone is above the threshold valueor water levels in the first tank and second tank are both abovethe threshold value, with water level in the second tank greaterthan water level in the first tank causing flow in the pipe.
It may be noted that existence, non-existence and direction offlow between tanks through the connection pipes placed at a heighthT from the bottom of the tanks can be obtained with the help of thediscrete variables. Further, the switching functions (Eqs. (78) and(80)) depend on the water level in the tanks and discrete variables.Eqs. (73)–(76) are valid for all the modes. The objective is to esti-mate the water levels in the tanks. Further the estimated values willbe used in designing MPC for the control of nonlinear three-tankhybrid system.
4.2. Simulation of closed loop control
In all the simulation runs, the process is simulated using thenonlinear first principles model ((70)–(72)). Only the level in Tank 1and level in Tank 3 are considered as measured variables. The covari-ance matrices of measurement noise and state noise are selectedas follows:
R =[
(0.0079)2 00 (0.0073)2
]and Q =
[(0.0111)2 0 0
0 (0.0098)2 00 0 (0.0104)2
]
We have assumed that x(0|0) = N [x(0|0), P(0|0)] , where
x(0|0) = [0.5086 0.4765 0.4876] and P(0/0) = Q. The controlproblem is to track levels h1 and h3 by manipulating valves u1 andu5. The NMPC schemes (EnKF based NMPC and UKF based NMPC)for the three-tank hybrid system have been developed with the pre-diction horizon of NP = 10 and control horizon of Nc = 3. The error
794 J. Prakash et al. / Journal of Process Control 20 (2010) 787–799
Table 2Regulatory control problem: comparison of estimator performances.
State estimation E (h1) E (h2) E (h3)
w
N
fuiun
4
tNscitta1tc
•••
svb
UKF 0.0213 0.0484 0.0130EnKF (N = 10) 0.0306 0.0548 0.0184EnKF (N = 25) 0.0242 0.0488 0.0141
eighting matrix and the controller weighting matrix used in the
MPC formulations are WE =[
1 00 1
]and WU =
[1 00 1
]. The
ollowing constraints on the manipulated inputs are imposed 0 ≤1 ≤ 1 and 0 ≤ u5 ≤ 1. In the UKF and EnKF based NMPC schemes,
dentical realization of state and measurement noises have beensed in all the simulation trials. Further, we have assumed the stateoise and the measurement noise to be additive.
.2.1. Regulatory performanceSimulation studies have been carried out to demonstrate the dis-
urbance rejection capability of the proposed UKF and EnKF basedMPC formulations. It may be noted that the three-tank hybrid
ystem is regulated around the setpoint h1 = h3 = hT = 0.3. A stephange in the valve position (from 0.2 to 1 in the case of u6) isntroduced at the 25th sampling instants and that value is main-ained up to the 75th sampling instant. Another step change inhe valve position (from 1 to 0.2 in the case of u6) is introducedt the 75th sampling instant and this value is maintained up to the50th sampling instants. A comparative study of the performance ofhe following NMPC formulations was carried out for the followingases:
UKF (with 7 sigma points).EnKF (with N = 10).EnKF (with N = 25).
Table 2 presents the comparison of state estimators based onum of the squares of the estimation errors. The estimates of stateariables generated by UKF and EnKF are shown in Figs. 2–5. It maye noted that, as a consequence of plant-model mismatch intro-
Fig. 3. Evolution of true and estimated value of discrete variables of thre
Fig. 2. Evolution of true and estimated states of three-tank hybrid system in thepresence of step change in the valve position u6 (UKF based NMPC). (a) Level inTank 1, (b) level in Tank 2, and (c) level in Tank 3.
duced by the sustained change in the unmeasured disturbance, theestimated states become biased (Figs. 2 and 3) subsequent to thestep changes. In fact, from Table 2, it is evident that, in terms ofstandard deviations, the effect of model-plant mismatch is morepronounced on EnKF than on UKF. In spite of the biases in theestimated states, both the NMPC schemes are able to reject thedisturbance and bring the process variables back to the respectivesetpoint values (refer to Figs. 6 and 7).
The closed loop performance indices (integral square error or ISEvalues) and average computation time per iteration (Matlab 7.7.0-R2008(b), Intel Core 2 Duo Processor-2.13 GHz) computed for ENKF-MPC and UKF-MPC schemes are reported in Table 3. Figs. 6 and 7
present the closed loop responses obtained using UKF-MPC andENKF-MPC (N = 10), respectively. It may be noted that the UKF basedNMPC, with only 7 sample points, performs better than EnKF basedNMPC with 10 particles.e-tank hybrid system with UKF based NMPC (regulatory response).
J. Prakash et al. / Journal of Process Control 20 (2010) 787–799 795
Fig. 4. Evolution of true and estimated states of three-tank hybrid system in thepresence of step change in the valve position u6 (EnKF Based NMPC). (a) Level inTank 1, (b) level in Tank 2, and (c) level in Tank 3.
Table 3Regulatory control problem: comparison of performances of UKF-NMPC and EnKF-NMPC schemes.
NMPC scheme ISE (h1) ISE (h3) Average computation timeper iteration (s)
Nttsis
UKF-NMPC 3.5869 1.5212 59.05EnKF-NMPC (N = 10) 3.5732 1.6199 84.18EnKF-NMPC (N = 25) 3.3928 1.4011 206.44
To achieve closed loop performance equivalent to that of UKF-MPC using EnKF based NMPC, it becomes necessary to increase
he ensemble size to 25, which increases the average computationime significantly. The samples points in the UKF formulation areelected deterministically in such as way that the entire probabil-ty density function of state is well approximated with only 2n + 1amples. The unscented Kalman filter (UKF) solves the state estima-
Fig. 5. Evolution of true and estimated value of discrete variables of three
Fig. 6. Regulatory response of three-tank system with UKF based NMPC. (a) Processoutput (h1), (b) process output (h3), and (c) controller outputs.
tion problem through an unscented transformation, in which thestate variable is approximated as a Gaussian random variable, usinga minimal set of sample points. These minimal sample points aredeterministically chosen, rather than random sampling (as in EnKF)and completely capture the mean and covariance accurately to atleast the 3rd order for any nonlinearity and the higher moments toat least 2nd order. On the other hand EnKF propagates the prob-ability distribution of the state using a randomly selected set ofrealizations (ensemble) through the nonlinear model. The neces-sary statistics are then obtained from the ensemble analysis. Inorder to completely capture the mean and covariance accurately,the number of particles required are large. Thus, when the probabil-
ity distribution of the state can be well approximated by Gaussiandensities, then UKF based NMPC formulation appears to be a bet-ter choice than the EnKF based NMPC formulation. However, whenthe probability distribution of the state are multi-modal and non--tank hybrid system with EnKF based NMPC (regulatory response).
796 J. Prakash et al. / Journal of Process Control 20 (2010) 787–799
Fig. 7. Regulatory response of three-tank system with EnKF based NMPC. (a) Processoutput (h1), (b) process output (h3), and (c) controller outputs.
Fs(
Gh
ttsaaFftooadtideEtn
Fig. 9. Regulatory response of three-tank system with EnKF based NMPC (withoutstate and measurement noise). (a) Process output (h1), (b) process output (h3), and(c) controller outputs.
Fig. 10. Servo response of three-tank hybrid system with UKF based NMPC. (a)Process output (h1), (b) process output (h3), and (c) controller outputs.
ig. 8. Regulatory response of three-tank system with EnKF based NMPC (withouttate and measurement noise). (a) Process output (h1), (b) process output (h3), andc) controller outputs.
aussian, the random sampling strategy in EnKF formulation canave an edge over the UKF based formulation.
To get a better insight into the ability of the proposed approacho achieve offset free control, simulations were carried out underhe hypothetical scenario where the state noise and the mea-urement noise are switched off. The EnKF and UKF, however,re formulated by assuming that the noise and the disturbancesre present. The resulting closed loop behaviour is shown inigs. 8 and 9. As evident from Fig. 8, the proposed UKF-NMPCormulation gives rise to a slight offset due to the determinis-ic sampling scheme used in UKF. The EnKF-NMPC formulation,n the other hand, does not lead to an offset. The controlledutput oscillates in the neighbourhood of the desired set pointsnd the offset is removed in the mean-square sense. This is airect consequence of the random sampling procedure used inhe EnKF based formulation. Since the NMPC objective functionn both the cases is formulated using estimated mean of the pre-icted particle/sigma point trajectories, the proposed approach is
xpected to eliminate offset only in the mean-square sense. FornKF based formulation, this holds even when the state noise andhe measurement noise in the plant are switched off as random-ess is part of the EnKF formulation. In the case of UKF-NMPC, thisFig. 11. Servo response of three-tank hybrid system with EnKF based NMPC. (a)Process output (h1), (b) process output (h3), and (c) controller outputs
J. Prakash et al. / Journal of Process Control 20 (2010) 787–799 797
FN
bp
4
aotNacsotsp
Fig. 13. Evolution of true and estimated states of three-tank hybrid system withEnKF based NMPC. (a) Level in Tank 1, (b) level in Tank 2, and (c) level in Tank 3.
Table 4Servo control problem: comparison of performances of UKF-NMPC and EnKF-NMPCschemes.
NMPC scheme ISE (h1) ISE (h3) Average computation timeper iteration (s)
process parameters k6 and k8 (refer Eqs. (75) and (76)) from 2e−4
ig. 12. Evolution of true and estimated states of three-tank system with UKF basedMPC. (a) Level in Tank 1, (b) level in Tank 2, and (c) level in Tank 3.
ehaviour is revealed only when the stochastic disturbances areresent.
.2.2. Servo responseTo assess the tracking capability of the proposed UKF-NMPC
nd EnKF-NMPC schemes, sequence of step changes in setpointsf h1 and h3 are introduced as shown in Figs. 10 and 11, respec-ively. From these figures, it can be observed that the proposedMPC schemes are able to maintain water levels of Tanks 1 and 3t desired setpoints. A comparison of the estimated and the trueontinuous state variables generated using UKF and EnKF is pre-ented in Figs. 12 and 13, respectively. The corresponding evolution
f true and estimated value of discrete variables (z1 and z2) of three-ank hybrid system with UKF-NMPC and EnKF-NMPC schemes arehown in Figs. 14 and 15, respectively. A comparison of closed looperformances based on ISE values is presented in Table 4.Fig. 14. Evolution of true and estimated value of discrete varia
UKF-NMPC 3.3278 3.7203 60.65EnKF-NMPC (N = 25) 4.8246 3.8264 219.2EnKF-NMPC (N = 40) 4.4599 3.2629 389.7
4.2.3. Performance in the presence of model parameter mismatchTo assess the performance of the proposed UKF-NMPC and
EnKF-NMPC schemes in the presence of plant-model mismatch,simulation studies were performed by deliberately changing the
to 1.0e−4 at the 25th sampling instants. The controller compu-tations, however, are based on the nominal model parameters. Itshould be noted that the three-tank hybrid system is regulatedaround the setpoint h1 = h3 = hT = 0.3. It can be observed that
bles of three-tank hybrid system with UKF based NMPC.
798 J. Prakash et al. / Journal of Process Control 20 (2010) 787–799
Fig. 15. Evolution of true and estimated value of discrete variables of three-tank hybrid system with EnKF based NMPC.
Table 5Parameter mismatch case: comparison of performances of UKF-NMPC and EnKF-NMPC schemes.
NMPC Scheme ISE (h1) ISE (h2)
UKF-NMPC 0.9231 1.0579
btmpsto
FN
EnKF-NMPC (N = 25) 0.9339 1.0276EnKF-NMPC (N = 40) 0.8647 0.9688
oth the controllers are able to maintain the process variables athe desired setpoint values even in this presence of plant-model
ismatch (see Figs. 16 and 17). The corresponding closed looperformance in terms of ISE values is reported in Table 5. Theimulation study again endorses the observation that increasing
he number of particles improves the closed loop performancebtained using EnKF-NMPC.ig. 16. Closed response in the presence of plant-model mismatch (UKF basedMPC).
Fig. 17. Closed response of three-tank hybrid system in the presence of plant-modelmismatch (EnKF based NMPC).
5. Concluding remarks
In this paper, we have proposed a state estimation scheme foran autonomous hybrid system using derivative free nonlinear stateestimators. Further, state estimation based nonlinear model pre-dictive control scheme for an autonomous hybrid system has beenproposed. The efficacy of the proposed state estimation and controlscheme has been demonstrated on the three-tank hybrid bench-mark system. From simulation studies on the three-tank hybridsystem, it can be concluded that the proposed nonlinear stateestimation based NMPC schemes has good setpoint tracking anddisturbance rejection capabilities. The UKF-NMPC formulation wasable to produce satisfactory servo and regulatory behaviour at rela-
tively lower computational cost. To achieve equivalent closed loopbehaviour, it was found necessary to increase number of particlesin EnKF-NMPC formulation. Thus, when the probability distribu-tion of the state can be well approximated by Gaussian densities,
rocess
tt
A
pRoms
R
[
[
[
[
[
[
[
[
[
[
[
[
[
[
[
J. Prakash et al. / Journal of P
he UKF based NMPC formulation appears to be a better choice thanhe EnKF based NMPC formulation.
cknowledgements
The authors would like to acknowledge financial sup-ort through the NSERC-MATRIKON-SUNCORE-ICORE Industrialesearch Chair program in Computer Process Control at Universityf Alberta. The first author would also like to thank the Depart-ent of Science and Technology, Government of India for providing
upport in the form of the BOYSCAST fellowship.
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