State estimation of stochastic non-linear hybrid dynamic system using an interacting multiple model algorithm

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State estimation of stochastic non-linear hybrid dynamic system using an interacting multiple model algorithm M. Elenchezhiyan, J. Prakash n Department of Instrumentation Engineering, Madras Institute of Technology Campus, Anna University, Chennai 600044, India

art ic l e i nf o

a b s t r a c t

Article history: Received 8 August 2014 Received in revised form 2 May 2015 Accepted 1 June 2015 This paper was recommended for publication by A.B. Rad.

In this work, state estimation schemes for non-linear hybrid dynamic systems subjected to stochastic state disturbances and random errors in measurements using interacting multiple-model (IMM) algorithms are formulated. In order to compute both discrete modes and continuous state estimates of a hybrid dynamic system either an IMM extended Kalman filter (IMM-EKF) or an IMM based derivative-free Kalman filters is proposed in this study. The efficacy of the proposed IMM based state estimation schemes is demonstrated by conducting Monte-Carlo simulation studies on the two-tank hybrid system and switched non-isothermal continuous stirred tank reactor system. Extensive simulation studies reveal that the proposed IMM based state estimation schemes are able to generate fairly accurate continuous state estimates and discrete modes. In the presence and absence of sensor bias, the simulation studies reveal that the proposed IMM unscented Kalman filter (IMM-UKF) based simultaneous state and parameter estimation scheme outperforms multiple-model UKF (MM-UKF) based simultaneous state and parameter estimation scheme. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Hybrid dynamic system State estimation Interacting multiple-model based Kalman filter algorithm and extended Kalman filter Unscented Kalman filter and ensemble Kalman filter

1. Introduction Dynamic systems that are described by an interaction between continuous dynamics and discrete modes are called hybrid dynamic systems [12,13]. In practice, hybrid dynamic systems are affected by random disturbances and measurements are corrupted with random noise. Thus, state estimation of stochastic hybrid dynamic system poses a challenging problem [15]. The Kalman update based non-linear state estimators such as the extended Kalman filter (EKF) [10,27], the unscented Kalman filter (UKF) [11] and the ensemble Kalman filter [22,26] and Particle filter [25] and their variants have been widely used to estimate the state variables of chemical processes such as Distillation column, continuous stirred tank reactors, Polymerization reactors etc. [20]. In the context of developing state estimator for hybrid dynamic systems, [17] suggested that the derivative-free state estimation schemes appear to be promising candidates. Excellent review articles on state and parameter estimation schemes have appeared recently in the process control literature [19,20]. The use of moving horizon approach based state estimation for hybrid system is reported in [2,8,9]. The design of a non-linear state feedback control system for a class of switched non-linear system has been addressed in [7]. It may be noted that [7] have designed a deterministic high gain observer for the estimation of state variables. n

Corresponding author. E-mail address: [email protected] (J. Prakash).

Robust state estimation and fault diagnosis for an uncertain hybrid dynamic system is reported in [23]. Recently, a state estimation scheme for a non-linear autonomous hybrid system, which is subjected to stochastic state disturbances and measurement noise, using the ensemble Kalman filter and unscented Kalman filter, has been proposed by [17]. Simultaneous estimation of both continuum and non-continuum state variables with an application to the distillation process using a moving horizon estimator is reported in [15,16]. Fault detection and monitoring scheme for a non-isothermal chemical reactor with uncertain mode transitions using an unknown input observer theory and results from Lyapunov stability theory is proposed recently by [24]. Many chemical processes are characterized by strong interactions between continuous dynamics and discrete events and are more appropriately modeled by hybrid systems [12,13]. On-line estimation of state variables of such systems is very important from the view point of fault detection and identification and model based control [4,6,14,24]. It may be noted that the interacting multiplemodel algorithm is a well-established state estimation technique and being widely used to estimate the continuous states and discrete modes in the area of target tracking applications [1,3]. The major contributions of the work are as follows: a state estimation scheme for stochastic non-linear hybrid system which has continuous dynamics and discrete modes modeled by a finite Markov chain using either IMM-EKF (or) IMM-UKF (or) IMM based ensemble Kalman filter (IMM-EnKF) is proposed. An IMM based simultaneous state and parameter estimation scheme to estimate the state variables and discrete modes of hybrid dynamic system in the presence of

http://dx.doi.org/10.1016/j.isatra.2015.06.005 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Elenchezhiyan M, Prakash J. State estimation of stochastic non-linear hybrid dynamic system using an interacting multiple model algorithm. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.005i

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sensor bias is also proposed. The efficacy of the proposed state estimation schemes is validated by carrying out Monte-Carlo simulation studies on the simulated models of non-linear two-tank hybrid system and switched non-isothermal continuous stirred tank reactor. Also, the performance of the IMM-UKF based simultaneous state and parameter estimation scheme has been validated with the experimental data and is found to be consistent with the simulation results. The organization of the paper is as follows: Section 2 describes the problem formulation and Section 3 deals with state estimation for hybrid dynamic system using IMM based state estimators. Extensive simulation studies have been reported in Section 4 and followed by concluding remarks in Section 5.

where λij j ðk  1j k  1Þ is mixing probability, Ωij is the assumed transition probability for the Markov chain according to which the system model switches from model ‘i’ to model ‘j’, μðiÞ ðk 1Þ is the mode probability at discrete time instant ‘k 1’. The input to each Kalman update based non-linear state estimator matched to model ‘j’ is obtained from an interaction of the ‘N’ Kalman update based non-linear state estimator which consists ðiÞ of the mixing of estimates x^ ðk  1j k  1Þ with the weights λij j ðk  1j k  1Þ, called the mixing probabilities xðjÞ ðk  1j k  1Þ ¼

N X

ðiÞ x^ ðk  1j k 1Þλij j ðk  1j k  1Þ;

j ¼ 1:::N

i¼1

ð5Þ 2. IMM based state estimation scheme for a hybrid dynamic system

The error covariance matrix is computed as ðjÞ

In this work, it is proposed to formulate the state estimation problem associated with the sub-class of hybrid dynamic system represented by the following equations: "Z # kT ðiÞ xðkÞ ¼ xðk  1Þ þ F ½xðτÞ; uðk  1Þdτ þ wðiÞ ðkÞ ð1Þ

P ðk  1j k  1Þ ¼

N X

λij j ðk  1j k  1Þ½PðiÞ ðk 1j k 1Þ

i¼1 ðiÞ

ðiÞ

þ ½x^ ðk  1j k  1Þ  xðjÞ ðk  1j k  1Þ½x^ ðk  1j k  1Þ  xðjÞ ðk  1j k  1ÞT ;

j ¼ 1:::N

ð6Þ

ðk  1ÞT

yðkÞ ¼ H½xðkÞ þ vðkÞ

ð2Þ n

In the above process model, xðkÞ is the system state vector ðx A R Þ, uðkÞ is the known system input ðu A Rm Þ, wðiÞ ðkÞ is the process noise (wðiÞ A Rn ) with known distribution, yðkÞ is the measured state variable (y A Rr ) and vðkÞ is the measurement noise (vðkÞ A Rr ) with known distribution. The index ‘k’ represents the sampling instant and the index ‘i’ A f1; 2::::Ng represents the discrete mode whose evolution is governed by the finite state Markov chain as shown below μðkÞ ¼ Ωμðk 1Þ

ð3Þ NN

N

is the mode transition matrix and μðkÞ A R is where Ω ¼ fΩij g A R the mode probability at time ‘k’. The non-linear process and measurement models for each mode F ðiÞ and H ðiÞ are assumed to be known in this work. In this work, it is assumed that the initial state vector follows multivariate normal distribution with known mean vector and covariance matrix and the noise sequences {w(i)(k)} and {v(k)} are assumed to be zero mean, normally distributed and mutually uncorrelated white noise sequences with known covariance matrices namely Q ðiÞ and R respectively. For each mode, we recommend the use of either an extended Kalman filter or an unscented Kalman filter or an ensemble Kalman filter to estimate the continuous state variables of the hybrid dynamic system. The steps involved in obtaining state estimates and discrete modes of hybrid dynamic system using the IMM approach are as follows. The IMM algorithm consists of ‘N’ interacting Kalman update based non-linear filters (EKF or UKF or EnKF algorithms as outlined in Appendix A, B, and C respectively) operating in parallel. However, it should be noted that in the IMM approach, at discrete time ‘k’ the state estimate is computed under each model using ‘N’ Kalman update based non-linear filters, with each non-linear Kalman filter using a different combination of the previous model-conditioned estimates (mixed initial condition [1]). The one cycle of the IMM algorithm [1] consists of the following steps:

(ii) Mode-matched filtering [1] The state estimate (xðjÞ ðk 1j k  1Þ) and the error covariance ðjÞ

matrix at previous time instant (P ðk  1j k  1Þ ) are used as input to each Kalman update based non-linear filter (refer to Appendix A, B, and C) matched to M ðjÞ which uses the current ðjÞ measurement yðkÞ to yield x^ ðkj kÞ and P ðjÞ ðkj kÞ. The likelihood functions corresponding to the ‘N’ Kalman update based non-linear state estimators are computed as follows:    h i1 exp  0:5 γðjÞ ðkj k  1ÞT VðjÞ ðkÞ γðjÞ ðkj k  1Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ΛðjÞ ðkÞ ¼ ð7Þ ð2πÞr j VðjÞ ðkÞj

In the above equation, γðjÞ ðkj k  1Þ, VðjÞ ðkÞ are the innovation and innovation covariance matrices respectively. (iii) Mode probability update [1] The mode probabilities are computed as follows: hP i N ðiÞ h i ΛðjÞ ðkÞ i ¼ 1 Ωij μ ðk 1Þ ðjÞ k ðjÞ hP i μ ðkÞ ¼ P M j Y ¼ P N N ðlÞ ðiÞ l ¼ 1 Λ ðkÞ i ¼ 1 Ωij μ ðk  1Þ j ¼ 1; 2; :::N

ð8Þ

(iv) State estimate and covariance combination [1] The updated state estimates and updated error covariance matrix are computed using the following equations:

^ kÞ ¼ xðkj

N X

ðjÞ x^ ðkj kÞμðjÞ ðkÞ

ð9Þ

j¼1

Pðkj kÞ ¼

N X

 h ih iT  ðjÞ ^ kÞ x^ ðjÞ ðkj kÞ  xðkj ^ kÞ μðjÞ ðkÞ P ðjÞ ðkj kÞ þ x^ ðkj kÞ  xðkj

j¼1

(i) Calculation of the mixing probabilities and mixed initial condition [1] The probability that the mode M ðiÞ was in effect at discrete time instant ‘k 1’ given that M ðjÞ is in effect at discrete time instant ‘k’ conditioned on Y k  1 is Ωij μðiÞ ðk  1Þ λij j ðk  1j k  1Þ ¼ PN ðiÞ i ¼ 1 Ωij μ ðk  1Þ

i; j ¼ 1; 2; :::N

ð4Þ

ð10Þ

3. Simulation studies The efficacy of the state estimation schemes namely IMM-EKF, IMM-UKF and IMM-EnKF has been validated on the two benchmark examples namely (i) two-tank hybrid system and (ii)

Please cite this article as: Elenchezhiyan M, Prakash J. State estimation of stochastic non-linear hybrid dynamic system using an interacting multiple model algorithm. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.005i

M. Elenchezhiyan, J. Prakash / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎ Table 1 Two-tank hybrid system model parameters.

V2

V1

Tank 1

V3

hv

V5

Tank 2

V4

switched non-isothermal continuous stirred tank reactor. The sum of square estimation errors (SSEE) is used as a performance index. Statistics of SSEE computed for each simulation run is used to assess the efficacy of the estimation scheme [17]. 3.1. Two-tank hybrid system [18] The two-tank hybrid system has four modes that depend on the relation between the water levels in both tanks ðh1 and h2 Þ. A schematic representation of the two-tank hybrid system is shown in Fig. 1. The model parameters of two-tank hybrid system are reported in Table 1. The governing equations of two-tank hybrid system for different modes are as follows: – MODE-1 ðh1 o hv Þ and ðh2 o hv Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi 9 1 Adh ¼ qmax V 1 k4 sgnðh1  h2 Þ j h1  h2 j V 4  k5 h1 V 5 = dt pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Adh ¼ qmax V 2 þk4 sgnðh1  h2 Þ j h1  h2 j V 4  k6 h2 V 6 ; dt ð11Þ – MODE-2 ðh1 Z hv Þ and ðh2 o hv Þ 9 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 > > ¼ qmax V 1  k4 sgnðh1  h2 Þ j h1  h2 j V 4 Adh > dt > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi > > =  k5 h1 V 5  k3 j h1  hv j V 3 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p ffiffiffiffiffi dh2 A dt ¼ qmax V 2 þk4 sgnðh1  h2 Þ j h1  h2 j V 4  k6 h2 V 6 > > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > ; þk j h h j V 1

v

3

ð12Þ – MODE-3 ðh1 o hv Þ and ðh2 Z hv Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 1 > ¼ q V k sgnðh  h Þ j h1  h2 j V 4 > Adh 1 4 1 2 max > dt > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi > > =  k5 h1 V 5 þ k3 j h2  hv j V 3 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi dh2 A dt ¼ qmax V 2 þk4 sgnðh1  h2 Þ j h1  h2 j V 4 > > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi > > ;  k6 h2 V 6  k3 j h2  hv j V 3

ð13Þ

– MODE-4 ðh1 Z hv Þ and ðh2 Z hv Þ 9 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 > > ¼ qmax V 1  k4 sgnðh1  h2 Þ j h1  h2 j V 4 Adh > dt pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > >  k5 h1 V 5  k3 sgnðh1 h2 Þ j h1  h2 j V 3 = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dh2 > A dt ¼ qmax V 2 þ k4 sgnðh1  h2 Þ j h1  h2 j V 4 > > pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > ;  k6 h2 V 6 þ k3 sgnðh1 h2 Þ j h1  h2 j V 3 >

Parameter

Value

qmax k3 and k4 k5 and k6 A

5e  5 3.6050e  5 4.0565e  5 0.0176625 (m2) 0

V min i V max i hmax

V6

Fig. 1. Two-tank hybrid system.

3

3

ð14Þ

The simulation studies have been carried out in the presence of additive Gaussian state noise and measurement noise. It is assumed that water level in second tank alone is available for

4.8960e  5 3.5653e  5

1 0.6

measurement. The objective is to generate an estimate of the water level in the first tank and a filtered estimate of water level in the second tank. The parameters associated with IMM-EKF, IMMUKF and IMM-EnKF algorithms are reported in Table 2. It should be noted that in the EKF algorithm reported in Appendix-A, the first moment (conditional mean) of prior distribu^ k  1Þ and posterior distribution xðkj ^ kÞ are suitably modified tion xðkj in order to satisfy the bound constraints. Also, in UKF algorithm (reported in Appendix-B), those sample/sigma points lying outside the bounds are clipped and then the first two moments of the prior ^ k  1Þ; Pðkj k  1Þ are computed. Redrawn sample distribution xðkj points lying outside the bounds are also clipped and being used in the computation of the innovation, innovation covariance matrix and Kalman gain. It may be noted that in EnKF algorithm (reported in Appendix-C) also, those particles lying outside the bounds were clipped in the prediction as well as update step. The above modification is needed for generating unbiased state estimates for the arbitrary choice of the initial error covariance matrix Pð0j 0Þ as well as process noise covariance matrices. The evolution of true and estimated continuous state variables generated by IMM-EKF, IMM-UKF and IMM-EnKF for variations in V 1 and V 2 (see Fig. 5) is shown in Fig. 2. From Fig. 2, it can be concluded that IMM-EKF, IMM-UKF and IMM-EnKF are able to generate fairly accurate filtered state estimates of the water level in tank-2 and a fairly accurate estimate of the water level in tank-1. The mode probability update of IMM-EKF, IMM-UKF and IMM-EnKF is shown in Fig. 3(a), (b) and (c). From Fig. 3(a)–(c), it can be concluded that the IMM-EKF, IMM-UKF and IMM-EnKF transition between modes as designed. However the performance of IMM-UKF is found to be better compared to IMM-EKF and IMM-EnKF. The evolution of the measured water level in the tank-2 and the filtered estimate of the water level in the tank-2 generated by IMM-EKF, IMM-UKF and IMM-EnKF during the interval ð1200 r k r 1500Þ are shown in Fig. 4(a), (b) and (c). From Fig. 4(a), (b) and (c), it can be inferred that the proposed state estimation schemes (IMM-EKF, IMM-UKF and IMM-EnKF) are able to efficiently suppress the measurement noise. The average SSEE (ASSEE) based on NT (¼ 25) trials is reported in Table 3. From Table 3 it can be inferred that the ASSEE was found to be less in IMM-UKF compared to IMM-EKF and IMM-EnKF. The evolution of state variables (True and Estimated) using IMM-UKF and MM-UKF is shown in Fig. 6 and the mode probability for both the schemes is also shown in Fig. 7. From Fig. 7, it can be inferred that MM based UKF scheme has got locked on to mode-1, and it took a very long time for the mode probabilities (weights) to reflect the change, resulting in the performance degradation of the state estimation scheme during the interval ð750 r k r 1050Þ as evident from Fig. 6. In order to assess the efficacy of the proposed state estimation scheme in the presence of sensor bias, simulation studies are carried out. In the presence of sensor bias, it was observed that there exist an offset between the estimated value and the true value of the process variables in the case of MM-UKF based simultaneous state and bias parameter estimation scheme, whereas IMM-UKF based simultaneous state and bias parameter estimation scheme generated an

Please cite this article as: Elenchezhiyan M, Prakash J. State estimation of stochastic non-linear hybrid dynamic system using an interacting multiple model algorithm. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.005i

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Table 2 Two-tank hybrid system—parameters associated with IMM-EKF, IMM-UKF and IMM-EnKF. Parameter

Value

Measurement noise covariance matrix R Process noise covariance Matrix Q(i)

1e  4 " 1e  8 2

Markov chain transition matrix Ωij

1e  8

0

0:99 6 0:0033 6 6 4 0:0033

# i ¼ 1; 2; 3; 4

0:0033 0:99

0:0033 0:0033

0:0033

0:99

0:0033 0:0033 0:0033 1, 0 and 0   10 0 i ¼ 1; 2; 3; 4 0 10  T 0:15 0:15 i ¼ 1  T 0:45 0:15 i ¼ 2  T 0:15 0:45 i ¼ 3  T 0:45 0:45 i ¼ 4  T 0:95 0:02 0:02 0:01

α; β and κ Initial error covariance P ðiÞ Initial state vector ðiÞ x^ ð0j 0Þ

Mode probability μð0j 0Þ   H xðkÞ ¼ CxðkÞ

True

0



IMM-EKF

IMM-UKF

IMM-EnKF

0

1

3 0:0033 0:0033 7 7 7 0:0033 5 0:99



Estimated

Measured

IMM-EKF

h2(m)

0.4

1

h (m)

0.4

0.35

0.2 1200

1300

1350

1400

1450

1500

Sampling Instants

0

200

400

600

800

1000

1200

1400

Sampling Instants IMM-EKF

IMM-UKF

IMM-EnKF

h2(m)

True

IMM-UKF

0.4

0.6 0.4

0.35 1200

1250

1300

1350

1400

1450

1500

Sampling Instants

2

h (m)

1250

0.2

IMM-EnKF

0

200

400

600

800

1000

1200

1400

h (m) 2

0.4

Fig. 2. Evolution of true and estimated states of two-tank hybrid system using IMMEKF, IMM-UKF and IMM-EnKF (a) water level in tank-1 and (b) water level in tank-2.

Mode Probability

IMM-EKF

1200

1300

1350

1400

1450

1500

Sampling Instants

Fig. 4. Evolution of filtered estimate of water level in tank-2 using IMM-EKF, IMMUKF and IMM-EnKF and measured water level in tank-2 ð1200 r k r 1500Þ.

1

0 400

600

800

1000

1200

0.7

1400

0.68

V

IMM-UKF

1

Sampling Instants

Mode Probability

1250

0.5

200

0.66

1

0.64

0.5

0.62 0.6

0 200

400

600 800 1000 Sampling Instants

1200

0

1400

500

1000

1500

1000

1500

Sampling Instants 0.25

IMM-EnKF

0.24 2

1 0.5

V

Mode Probability

0.35

0.23 0.22

0 200

400

600

800

1000

1200

1400

Sampling Instants Mode-1

Mode-2

0.21 0.2

Mode-3

Mode-4

Fig. 3. Mode probability update: IMM-EKF, IMM-UKF and IMM-EnKF.

0

500 Sampling Instants

Fig. 5. Evolution of V 1 and V 2 versus sampling instants.

Please cite this article as: Elenchezhiyan M, Prakash J. State estimation of stochastic non-linear hybrid dynamic system using an interacting multiple model algorithm. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.005i

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0.4 1

h (m)

Table 3 Two-tank hybrid system: performance comparison of state estimation schemes using IMM-EKF, IMM-UKF and IMM-EnKF.

5

State estimator

Water level (h1)

Water level (h2)

IMM-EKF IMM-EnKF (N¼ 10) IMM-UKF MM-UKF

1.0354 0.7476 0.2165 0.2529

0.0099 0.0142 0.0072 0.0430

0.2

True 200

400

600

IMM-UKF 800

1000

MM-UKF 1200

1400

Sampling Instants

2

h (m)

0.4 0.2 200

0.4

400

600

0.3 0.2 0.1 0

IMM-UKF 800

1000

MM-UKF 1200

1400

1200

1400

Sampling Instants

True 200

400

600

MM-UKF 800

1000

IMM-UKF 1200

1400

Sensor Bias

1

h (m)

True

True

IMM-UKF

MM-UKF

0.1 0.05 0

Sampling Instants

200

400

600

800

1000

Sampling Instants Fig. 8. Evolution of true and estimated states and sensor bias parameter of twotank hybrid system using IMM-UKF and MM-UKF (a) water level in tank-1, (b) water level in tank-2 and (c) sensor bias.

0.4

0.2

Mode-1

0

True 200

400

600

MM-UKF 800

1000

1200

1400

Sampling Instants Fig. 6. Evolution of true and estimated states of two-tank hybrid system using IMM-UKF and MM-UKF (a) water level in tank-1 and (b) water level in tank-2.

Mode Probability

Mode-1

Mode-2

Mode-3

Mode-3

Mode-4 IMM-UKF

1

0.5

0 200

Mode-4 IMM-UKF

400

600

800

1000

1200

1400

Sampling Instants

1

MM-UKF 0.5

0 200

400

600

800

1000

1200

1400

Sampling Instants

1 Mode-1

0.5

Mode-2

Mode-3

Mode-4

0

MM-UKF

Mode Probability

Mode-2

IMM-UKF

Mode Probability

0.1

Mode Probability

h2(m)

0.3

200

400

600

800

1000

1200

1400

Sampling Instants

1

Fig. 9. Sensor bias: mode probability update: (a) IMM-UKF and (b) MM-UKF 0.5

3.2. Switched non-isothermal continuous stirred tank reactor [5,24] 0 200

400

600

800

1000

1200

1400

Sampling Instants

Fig. 7. Mode probability update: MM-UKF and IMM-UKF.

unbiased state estimate as shown in Fig. 8. It may be noted that in order to estimate the state variables and discrete modes accurately in the presence of sensor bias, the state equations have been augmented with an additional differential equation of the form dβ ¼ 0, where β is dt the sensor bias parameter to be estimated. The mode probability update for the sensor bias case study is shown in Fig. 9. From Fig. 9 it can be inferred that in the presence of sensor bias also MM based UKF scheme has got locked on to mode-1, resulting in the performance degradation of the MM based simultaneous state estimation and parameter scheme during the interval ð750 r k r 1500Þ as evident from Fig. 9(b).

The schematic diagram of the switched non-isothermal continuous stirred tank reactor Fig. 10. Irreversible exothermic chemical reaction is taking place inside the reactor with A being the reactant species and B being the desired product. The governing mass and energy balance equations of the reactor for various modes are as follows: – MODE-1 In mode-1, the reactor is provided with feed inflow rate F1, molar concentration CA1 and temperature TA1 of the reactant species A  E C_ A ¼ FV1 ðC A1  C A Þ  k0 exp  RT C A   T_ ¼ F 1 ðT A1  T Þ þ  ΔHr k0 exp  E C A þ V

ρcp

RT

9 = Q1 ρcp V

;

ð15Þ

Please cite this article as: Elenchezhiyan M, Prakash J. State estimation of stochastic non-linear hybrid dynamic system using an interacting multiple model algorithm. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.005i

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Fig. 10. Switched non-isothermal continuous stirred tank reactor.

– MODE-2 In mode-2, the reactor is supplied with another feed with inflow rate as F2, molar concentration CA2 and temperature TA2 9  E = C_ A ¼ FV1 ðC A1  C A Þ þ FV2 ðC A2  C A Þ  k0 exp  RT C A   E ð16Þ Q2 r T_ ¼ FV1 ðT A1  TÞ þ FV2 ðT A2  T Þ þ ρcΔH k exp þ C 0 A ρcp V ; RT p – MODE-3 Third feed with reactant species A having inflow rate as F3, molar concentration CA3 and temperature TA3 has been added to the reactor in mode-3 9  E = C_ A ¼ FV1 ðC A1  C A Þþ FV2 ðC A2 C A Þ þ FV3 ðC A3  C A Þ k0 exp  RT C A   E Q3 r C k exp þ T_ ¼ FV1 ðT A1 TÞ þ FV2 ðT A2 T Þ þ FV3 ðT A3 T Þþ ρcΔH 0 A RT ρcp V ; p ð17Þ

The problem at hand is to generate an estimate of the reactor concentration as it was considered as an unmeasured variable in this study and a filtered estimate of reactor temperature using IMM based state estimation schemes. Based on the desired requirement of the yield, the mode transition is initiated and the reactor switches among three modes. The parameters associated with the switched non-isothermal continuous stirred tank reactor are shown in Table 4. The parameters associated with IMM-EKF,

Table 4 Switched non-isothermal continuous stirred tank reactor – process parameters. Parameter

Value

F1 F2 F3 CA1 CA2 CA3 TA1 TA2 TA3 T nom A1 Q nom 1 Q nom 2 Q nom 3 V R ΔHnom r k0 E ρ cp C sA1 C sA2 C sA3 Ts

4.998 m3/h 12.998 m3/h 16.998 m3/h 4.0 kmol/m3 4.5 kmol/m3 5.0 kmol/m3 295.0 K 320.0 K 340.0 K 300.0 K 0 kJ/h 187,768 kJ/h 367,978 kJ/h 1.0 m3 8.314 kJ/kmol K  5.0  104 kJ/kmol 3.0  106 h  1 5.0  104 kJ/kmol 1000.0 kg/m3 0.231 kJ/kg K 3.59 kmol/m3 4.23 kmol/m3 4.60 kmol/m3 388.57 K

Please cite this article as: Elenchezhiyan M, Prakash J. State estimation of stochastic non-linear hybrid dynamic system using an interacting multiple model algorithm. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.005i

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mode probability update of IMM-EKF, IMM-UKF and IMM-EnKF is shown in Fig. 12. The measured reactor temperature and the filtered estimate of reactor temperature generated by IMM-EKF, IMM-UKF and IMM-EnKF are shown in Fig. 13. From Fig. 13, it can be inferred that the proposed state estimation schemes (IMM-EKF, IMMUKF and IMM-EnKF) are able to suppress the measurement noise. The average sum of square of estimation errors based on NT ( ¼ 25) trials is reported in Table 6. From Table 6 it can be

IMM-UKF and IMM-EnKF state estimation schemes are also reported in Table 5. It is assumed that the state equations are affected by random disturbances and reactor temperature measurements are affected by measurement noise. The modifications that are suggested in Section 3.1 for the EKF, UKF and EnKF algorithms have been incorporated, while carrying out simulation studies in order to generate non-negative estimates of reactor concentration. The true and estimated continuous state variables generated by IMM-EKF, IMM-UKF and IMM-EnKF are shown in Fig. 11. From Fig. 11, it can be concluded that IMM-EKF, IMM-UKF and IMM-EnKF are able to generate an estimate of reactor concentration and a filtered estimate of reactor temperature with a reasonable accuracy. The

Mode-1

Mode-2

Mode-3

IMM-EKF

Mode

1 0.5 0 Table 5 Non-isothermal continuous stirred tank reactor parameters associated with IMMEKF, IMM-UKF and IMM-EnKF.

100

200

300

400

500

600

700

Sampling Instants IMM-UKF

Value

Measurement noise covariance matrix ðR Þ

1 "

Process noise covariance matrix ðQ ðiÞ Þ   Markov chain transition matrix Ωij

2

1

Mode

Parameter

#

ð0:005Þ2

0

0

ð0:05Þ2

i ¼ 1; 2; 3

100

ðiÞ

Initial state vector x^ ð0j 0Þ

Mode probability μð0j 0Þ

4.5

500

600

700

0.5 0 100

200

300

400

500

600

700

Sampling Instants Fig. 12. Non-isothermal continuous stirred tank reactor: mode probability update: IMM-EKF, IMM-UKF and IMM-EnKF.

Estimated

IMM-EnKF

4.5 CA

CA

400

IMM-EnKF

4.5

4

300

1

True

IMM-EKF

200

Sampling Instants

Mode



Initial error covariance P ðiÞ

CA

0

3

0:99 0:005 0:005 6 0:005 0:99 0:005 7 4 5 0:005 0:005 0:99 1, 0 and 0 " # ð0:005Þ2 0 i ¼ 1; 2; 3 2 0 ð0:05Þ  T 3:59 388:57 i ¼ 1  T 4:23 388:57 i ¼ 2  T 4:60 388:57 i ¼ 3  T 0:98 0:01 0:01

α ; β and κ

0.5

4

4

IMM-UKF

3.5

3.5

3.5

200 400 600 Sampling Instants

200 400 600 Sampling Instants

391

391

390

390

390

389

389

389

388

388

388

387 386 385

T

391

T

T

200 400 600 Sampling Instants

387 IMM-EKF

386 385

IMM-UKF

387 386

200 400 600

200 400 600

Sampling Instants

Sampling Instants

385

IMM-EnKF

200 400 600

Sampling Instants

Fig. 11. Evolution of true and estimated value of reactor concentration and reactor temperature using IMM-EKF, IMM-UKF and IMM-EnKF.

Please cite this article as: Elenchezhiyan M, Prakash J. State estimation of stochastic non-linear hybrid dynamic system using an interacting multiple model algorithm. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.005i

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8

Estimated

Measured

Mode-1

IMM-EKF

0.8

100

200

300

400

500

600

700

0.6

Mode

388 386

0.4 0.2

Sampling Instants

0

IMM-UKF

100

200

300

400

500

600

700

Sampling Instants

388

MM-UKF 1

386

100

200

300

400

500

600

700

Sampling Instants IMM-EnKF

0.8 0.6

Mode

T

390

T

IMM-UKF

Mode-3

1

390

T

Mode-2

0.4

390

0.2

388

0 100

386

100

200

300

400

500

600

200

300

Sampling Instants Fig. 13. Evolution of measured and filtered estimate of reactor temperature using IMM-EKF, IMM-UKF and IMM-EnKF.

State estimator

Reactor concentration (CA)

Reactor temperature (T)

IMM-EKF IMM-UKF IMM-EnKF

7.8147 7.7677 8.6405

52.8018 52.8747 177.8655

500

600

700

Fig. 15. Switched non-isothermal continuous stirred tank reactor: mode probability update: IMM-UKF and MM-UKF.

True

Estimated(IMM-UKF)

Estimated(MM-UKF)

A

4.5

C

Table 6 Switched non-isothermal continuous stirred tank reactor: performance comparison of state estimation schemes using IMM-EKF, IMM-UKF and IMM-EnKF.

400

Sampling Instants

700

4 3.5

100

200

300

400

500

600

700

500

600

700

500

600

700

Sampling Instants

T

390 388 386 384

100

200

300

400

Sampling Instants

Estimated(IMM-UKF)

Estimated(MM-UKF) Sensor Bias

True

4.5

4

2 1 0 -1

100

200

300

400

Sampling Instants

3.5

100

200

300

400

500

600

700

Fig. 16. Evolution of true and estimated value of reactor concentration and reactor temperature and sensor bias using IMM-UKF and MM-UKF.

391 390 389 388 387 386

100

200

300

400

500

600

700

Fig. 14. Evolution of true and estimated value of reactor concentration and reactor temperature using IMM-UKF and MM-UKF.

inferred that the average SSEE for the reactor concentration was found to be less in IMM-UKF compared to IMM-EKF and IMMEnKF.

The IMM-UKF and MM-UKF performances are shown in Fig. 14 and the mode probability for both the schemes is shown in Fig. 15. From Fig. 15, it can be inferred that MM based UKF scheme has got locked on to mode-1, resulting in the performance degradation of the state estimation scheme as evident from Fig. 14. That is, there exists an offset between the estimated value and the true value of the process variables in the case of MM-UKF. In the presence of sensor bias in reactor temperature, simulation studies reveal that there exist an offset between the estimated value and the true value of the process variables in the case of MM-UKF based simultaneous state and bias parameter estimation scheme, whereas IMM-UKF based simultaneous state and bias parameter estimation scheme generated an unbiased state estimate as shown in Fig. 16. From Fig. 17, it can be inferred that in the presence of sensor bias also MM based UKF

Please cite this article as: Elenchezhiyan M, Prakash J. State estimation of stochastic non-linear hybrid dynamic system using an interacting multiple model algorithm. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.005i

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Mode-1

Mode-2

Mode-3

IMM-UKF

IMM-UKF 1

1 0.8

0.8

0.6

0.6

Mode

Mode

9

0.4 0.2

Mode-1 Mode-2 Mode-3 Mode-4

0.4 0.2

0 100

200

300

400

500

600

0

700

1000

2000

Sampling Instants

3000

4000

5000

MM-UKF

MM-UKF 1

1

0.8

0.8

0.6

0.6

Mode

Mode

6000

Time (s)

0.4

Mode-1 Mode-2 Mode-3 Mode-4

0.4

0.2 0.2

0 100

200

300

400

500

600

0

700

1000

2000

3000

Sampling Instants

4000

5000

6000

Time (s)

Fig. 17. Switched non-isothermal continuous stirred tank reactor: mode probability update: IMM-UKF and MM-UKF.

Fig. 20. Mode probability update of IMM-UKF and MM-UKF based estimation schemes (experimental validation).

0.5 Estimated(MM-UKF) Measured Estimated(IMM-UKF)

0.3 0.2 0.1 0

0.4

h1(m)

h1(m)

0.4

2000

3000 4000 Time (s)

5000

6000

1000

2000

1000

2000

3000

4000

5000

6000

h2(m)

0.5

0.4 0.3

Estimated(MM-UKF) Measured Estimated(IMM-UKF)

0.2

Estimated

0

2000

3000 4000 Time (s)

5000

6000

Fig. 18. Evolution of measured and estimated value of water level in tank-1 and tank-2 using IMM-UKF and MM-UKF (experimental validation).

Sensor Bias(m)

1000

Measured(w/o bias)

3000

Measured

4000

5000

6000

4000

5000

6000

Time (s)

0.1 0

Measured

Estimated

0

1000

0.5

h2(m)

0.2

0.2 True

Estimated

0.1 0 1000

2000

3000 Time (s)

0.43 Measured 0.428

Fig. 21. Evolution of measured and estimated states and sensor bias parameter of two-tank hybrid experimental setup using IMM-UKF (a) water level in tank-1, (b) water level in tank-2 and (c) sensor bias.

0.426

h2(m)

0.424

scheme has got locked on to mode-1, resulting in the performance degradation of the MM based simultaneous state estimation and parameter scheme as evident from Fig. 16 during the time interval 250 r k r 750.

0.422 0.42 0.418

4. Experimental evaluation

0.416 0.414 4000

4500

5000

5500

6000

6500

Time (s) Fig. 19. Evolution of measured value of water level in tank-2 during the time interval 4000–6500 s.

It should be noted that from the extensive Monte-Carlo simulation studies, it was observed that IMM-UKF based state estimation scheme outperforms IMM-EKF based state estimation scheme on the simulated model of two-tank hybrid system as well as switched non-isothermal continuous stirred tank reactor. The experimental validation of the IMM-UKF based state

Please cite this article as: Elenchezhiyan M, Prakash J. State estimation of stochastic non-linear hybrid dynamic system using an interacting multiple model algorithm. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.005i

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IMM-UKF

1

Mode

0.8

Mode-1 Mode-2 Mode-3 Mode-4

0.6 0.4 0.2 0 1000

2000

3000

4000

5000

6000

Time (s) MM-UKF 1 Mode-1 Mode-2 Mode-3 Mode-4

Mode

0.8 0.6 0.4 0.2 0 1000

2000

3000

4000

5000

6000

Time (s) Fig. 22. Mode probability update of IMM-UKF and MM-UKF based estimation schemes in the presence of sensor bias (experimental validation).

0.5

h1(m)

Fig. 24. Hybrid tank experimental test setup.

Estimated 0

1000

2000

Estimated

Measured 3000

4000

Measured(w/o bias)

5000

6000

Measured

h2(m)

0.5

0

1000

2000

3000

4000

5000

6000

4000

5000

6000

Time (s)

bias(m)

0.2 True

Estimated

0.1 0

1000

2000

3000 Time (s)

Fig. 23. Evolution of measured and estimated states and sensor bias parameter of two-tank hybrid experimental setup using MM-UKF (a) water level in tank-1, (b) water level in tank-2 and (c) sensor bias.

estimation scheme has been carried out using the data obtained from the Laboratory scale interacting hybrid-tank system available at the Process Control Laboratory, Department of Instrumentation Engineering, MIT Campus, Anna University. It may be noted that each tank has a circular area of 177 cm2 and a height of 63 cm. The setup has two inputs (two metering pumps) which can be manipulated to control the water levels in the first and second tanks. The water level in the first and second tanks is measured with the help of the differential pressure transmitter (accuracy of the transmitter 7 0.25% of full range). However, for the IMM-UKF implementation, we have considered only the water level in the second tank as a measured variable. That is using the available measurement h2 the IMM-UKF based state estimation will generate a filtered estimate of water level in the second tank as well as an estimate of water level in the first tank.

Water is continuously pumped into the two tanks using metering pumps, which can accurately deliver water at any flow rate between 0 and 4 liters per minute. The metering pumps and the level transmitters are interfaced to the personal computer using an NI-USB data acquisition card (NI USB DAQ 6251). The data from the experimental setup (refer to Fig. 24) has been acquired using the Lab VIEW software package. The evolution of the measured value of the water level (h2) in the second tank for the time interval 4000–6500 s is reported in Fig. 19. From Fig. 19, it can be inferred that the acquired data is corrupted by measurement noise. The evolution of filtered estimate of the water level in the second tank and an estimate of water level in the first tank generated by IMM-UKF and MM-UKF are shown in Fig. 18. Since water level in both the tanks (first and second tanks) has been measured, we have superimposed the measured values with the estimated values generated by the IMM-UKF and MMUKF algorithms. From Fig. 18, it can be concluded that the IMMUKF is able to generate fairly accurate estimates of water level in first and second tank. The mode probability update of IMM-UKF and MM-UKF is shown in Fig. 20. From Fig. 20, it can be concluded that the IMM-UKF transitions between modes as designed. From Fig. 20, it can be inferred that MM based UKF scheme has got locked on to mode-2, and it took a very long time for the mode probabilities (weights) to reflect the change, resulting in the performance degradation of the state estimation scheme during the interval ð3200 r k r 5700Þ as evident from Fig. 18. It can be concluded that the performance of both IMMUKF and MM-UKF based state estimation schemes is found to be consistent with the simulation results. In the presence of sensor bias it was observed that there exist an offset between the estimated value and the measured value (without bias) of the process variables in the case of MM-UKF based simultaneous state and bias parameter estimation scheme (see Fig. 23), whereas IMM-UKF based simultaneous state and bias parameter estimation scheme generated fairly accurate state estimates as shown in Fig. 21. From Fig. 22(b) it can be inferred that in the presence of sensor bias also MM based UKF scheme has got

Please cite this article as: Elenchezhiyan M, Prakash J. State estimation of stochastic non-linear hybrid dynamic system using an interacting multiple model algorithm. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.005i

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Table 7 The recommendation for the choice of IMM based state estimator for hybrid system. Hybrid system

Conditional densities

Distribution of state noise and measurement Constraints on state noise variables

Choice of the state estimator for hybrid system

Nonlinear

Can be well approximated as Gaussian

Gaussian

Absent

Nonlinear Nonlinear

Non-Gaussian Can be well approximated as Gaussian

Non-Gaussian Gaussian

Absent Present

Nonlinear

Non-Gaussian

Non-Gaussian

Absent

IMM-EKF IMM-UKF IMM-EnKF IMM-PF with EnKF as proposal IMM-CEKF IMM-CUKF IMM-C-EnKF IMM-CPF with C-EnKF as proposal

locked on to mode-2, resulting in the performance degradation of the MM based simultaneous state estimation and parameter scheme during the interval ð3200r k r 6250Þ as evident from Fig. 23.

The measurement prediction, computation of innovation and covariance matrix of innovation are computed as follows:   yðkj k  1Þ ¼ H xðkj k  1Þ ðA4Þ γðjÞ ðkj k  1Þ ¼ yðkÞ yðkj k  1Þ

ðA5Þ

and

5. Concluding remarks In this paper, design and implementation of state estimation schemes for two-tank hybrid system and non-isothermal continuous reactor using an IMM-EKF algorithm, an IMM-UKF algorithm and IMM-EnKF algorithm have been reported. The efficacies of the proposed state estimation schemes have been demonstrated by carrying out Monte-Carlo simulation studies on the two-tank hybrid system as well as switched non-isothermal continuous stirred tank reactor. The simulation studies indicate that the proposed nonlinear state estimation schemes have good tracking capabilities. From the extensive simulation studies, it can be observed that IMM-EKF, IMM-UKF and IMM-EnKF have been able to generate fairly accurate estimates of continuous state variables and correctly detect discrete mode transitions. It can be concluded in the presence of and absence of sensor bias the proposed IMMUKF is able to generate fairly accurate state and sensor bias parameter estimates as compared to MM-UKF. The summary of recommendations for the choice of the IMM based state estimator for stochastic non-linear hybrid systems is reported in Table 7. The experimental studies conducted on the two-tank hybrid system corroborate completely the inference we have obtained through simulation studies and demonstrate the efficacy of the proposed state estimation scheme.

Appendix A. Extended Kalman filter The extended Kalman filter algorithm is as follows: The one step ahead predicted state estimates are obtained as follows: Z k   xðkj k 1Þ ¼ xðjÞ ðk  1j k  1Þ þ F ðjÞ xðτÞ; uðk  1Þ dτ ðA1Þ k1

ðjÞ

ðA2Þ

where ΦðkÞ ¼ exp½AðkÞnT and AðkÞ, is nothing but Jacobian matrices of partial derivatives of FðjÞ with respect to xðjÞ ðk  1j k 1Þ and is expressed as " # ∂F ðjÞ ðA3Þ AðkÞ ¼  ∂x  ðjÞ x ðk  1j k  1Þ;uðk  1Þ

ðA6Þ

where CðkÞ is the Jacobian matrix of partial derivatives of H with respect to xðkj k  1Þ   ∂H CðkÞ ¼ ðA7Þ ∂x ½xðkj k  1Þ;uðk  1Þ The Kalman gain is computed using the following equation:  1 KðkÞ ¼ Pðkj k  1ÞCðkÞT V ðkÞ

ðA8Þ

The updated state estimates are obtained using the following equation: ðjÞ x^ ðkj kÞ ¼ xðkj k  1Þ þ KðkÞγðjÞ ðkj k  1Þ

ðA9Þ

The covariance matrix of estimation errors in the updated state estimates is obtained as   P ðjÞ ðkj kÞ ¼ I  KðkÞCðkÞ Pðkj k  1Þ

ðA10Þ

Appendix B. Unscented Kalman Filter The unscented Kalman filter algorithm proposed by [11] is as follows: A set of 2nþ 1 sigma points, χ ðk  1j k  1; lÞ with the associated weights, WðlÞ are chosen symmetrically about xðjÞ ðk  1j k 1Þ as given below χ ðk  1j k  1; 0Þ ¼ xðjÞ ðk 1j k 1Þ; χ ðk  1j k  1; lÞ ¼ xðjÞ ðk  1j k  1Þ þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðjÞ ðn þ λÞP ðk  1j k  1Þ l

l ¼ 1; :::; n χ ðk  1j k  1; lÞ ¼ xðjÞ ðk  1j k  1Þ 

The covariance matrix of estimation errors in the predicted estimates is obtained as follows: Pðkj k  1Þ ¼ ΦðkÞP ðk 1j k  1ÞΦðkÞT þ Q ðjÞ

VðkÞ ¼ CðkÞPðkj k  1ÞCðkÞT þRðjÞ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðjÞ ðn þ λÞP ðk  1j k  1Þ ln

l ¼ n þ 1; :::; 2n λ ; W m ð0Þ ¼ nþλ λ þ ð1  α2 þ βÞ; λ ¼ α2 ðn þ κÞ  n W c ð0Þ ¼ nþλ 1 ; l ¼ 1; :::; 2n W c ðlÞ ¼ W m ðlÞ ¼ 2ðn þ λÞ where κ, α and β are the parameters associated with UKF [21]. In the prediction step, the sigma points are propagated through the nonlinear differential equations to obtain the predicted set of

Please cite this article as: Elenchezhiyan M, Prakash J. State estimation of stochastic non-linear hybrid dynamic system using an interacting multiple model algorithm. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.005i

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h i ðjÞ k  1Þ  N xðjÞ ðk  1j k  1Þ; P ðk  1j k  1Þ . These particles fxðlÞ ðk 

sigma points as Z χ ðkj k  1; lÞ ¼ χ ðk  1j k  1; lÞ þ

kT

ðk  1ÞT

  F ðjÞ χ ðτ; lÞ; uðk  1Þ dτ;

l ¼ 0; ::::; 2n

1j k  1Þ : l ¼ 1; …; Ng are then propagated through the system dynamics to compute transformed sample points as follows: ðB1Þ

The predicted state estimates xðkj k  1Þ is obtained from the predicted sigma points as 2n X

xðkj k 1Þ ¼

" xðlÞ ðkj k 1Þ ¼ xðlÞ ðk  1j k  1Þ þ

Z

kT

h i F ðjÞ xðτÞ; uðk  1Þ; d dτ

#

ðk  1ÞT

W m ðlÞχðkj k  1; lÞ

þ wðlÞ ðk  1Þ :

ðB2Þ

l ¼ 1; 2; :::; N

ðC1Þ

l¼0

The error covariance matrix Pðkj k 1Þ is obtained from the predicted sigma points as 2n X

Pðkj k  1Þ ¼

^ k  1Þgnfχðkj k 1; lÞ W c ðlÞfχðiÞ ðkj k  1; lÞ  xðkj

These particles are then used to estimate sample mean and covariance as follows: xðkj k  1Þ ¼

N 1X xðlÞ ðkj k  1Þ Nl¼1

ðC2Þ

yðkj k  1Þ ¼

N h i 1X H xðlÞ ðkj k  1Þ Nl¼1

ðC3Þ

l¼0

^ k  1ÞgT þ Q ðjÞ  xðkj

ðB3Þ

Sigma points are redrawn using the predicted state estimate as given below χ n ðkj k  1; 0Þ ¼ xðkj k  1Þ;

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi χ n ðkj k  1; lÞ ¼ xðkj k  1Þ þ ðn þ λÞPðkj k  1Þ l ¼ 1; :::; n l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n χ ðkj k  1; lÞ ¼ xðkj k  1Þ  ðn þ λÞPðkj k  1Þ l ¼ n þ 1; :::; 2n

Pε;e ðkÞ ¼

VðkÞ ¼

ln

Redrawn sigma points are then propagated through the nonlinear measurement equation to obtain the predicted measurement as yðkj k  1Þ ¼

2n X

  W m ðlÞnH χn ðkj k 1; lÞ

ðB4Þ

l¼0

The covariance matrix of the innovations VðkÞ and the cross covariance matrix between the predicted state estimate errors and innovations P xy ðkÞ are computed as VðkÞ ¼

2n X

N h ih iT 1 X εðlÞ ðkÞ eðlÞ ðkÞ N 1 l ¼ 1

N h ih iT 1 X eðlÞ ðkÞ eðlÞ ðkÞ N1 l ¼ 1

ðC5Þ

where εðlÞ ðkÞ ¼ xðlÞ ðkj k 1Þ  xðkj k  1Þ

ðC6Þ

h i eðlÞ ðkÞ ¼ H xðlÞ ðkj k  1Þ yðkj k 1Þ

ðC7Þ

The Kalman gain and cloud of updated samples (particles) are then computed as follows:  1 ðC8Þ LðkÞ ¼ Pε;e ðkÞ VðkÞ n h io ϒ ðlÞ ðkj k  1Þ ¼ yðkÞ þ vðlÞ ðkÞ  H xðlÞ ðkj k  1Þ

W c ðlÞfH½χn ðkj k 1; lÞ yðkj k  1ÞgnfH½χn ðkj k  1; lÞ

ðC4Þ

ðC9Þ

l¼0

 yðkj k  1ÞgT þ RðjÞ Pxy ðkÞ ¼

2L X

ðB5Þ

ðC10Þ

The updated state estimate is computed as follows: W c ðlÞfχn ðkj k 1; lÞ  xðkj k 1ÞgnfH½χn ðkj k  1; lÞ

l¼0

 yðkj k  1ÞgT

ðB6Þ

The innovation is computed as follows: ðjÞ

γ ðkj k  1Þ ¼ yðkÞ  yðkj k  1Þ

ðB7Þ

The updated state estimates are obtained using the following equation: x^ ðkj kÞ ¼ xðkj k 1Þ þ KðkÞγ ðkj k  1Þ ðjÞ

ðjÞ x^ ðkj kÞ ¼

N 1X xðlÞ ðkj kÞ Nl¼1

ðC11Þ

The covariance of the updated estimates can be computed as

The Kalman gain is computed using the following equation:  1 ðB8Þ KðkÞ ¼ P xy ðkÞ VðkÞ

ðjÞ

xðlÞ ðkj kÞ ¼ xðlÞ ðkj k  1Þ þ LðkÞϒ ðlÞ ðkj k 1Þ

ðB9Þ

The covariance matrix of estimation errors in the updated state estimates is obtained as  T P ðjÞ ðkj kÞ ¼ Pðkj k  1Þ  KðkÞVðkÞ KðkÞ ðB10Þ

Appendix C. Ensemble Kalman filter [17,22] At each time step, N samples fwðlÞ ðk  1Þ; vðlÞ ðkÞ : l ¼ 1; ::Ng for {wðkÞ} and {vðkÞ } are drawn randomly using the distributions of process noise and measurement noise of jth mode. N samples are drawn from the multivariate normal distribution; xðlÞ ðk  1j

PðjÞ ðkj kÞ ¼

N h ih iT 1 X ξðlÞ ðkÞ ξðlÞ ðkÞ N 1 l ¼ 1 ðjÞ

ξðlÞ ðkÞ ¼ xðlÞ ðkj kÞ  x^ ðkj kÞ γ ðjÞ ðkj k  1Þ ¼

N 1X ϒ ðlÞ Nl¼1

ðC12Þ

ðC13Þ

ðC14Þ

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Please cite this article as: Elenchezhiyan M, Prakash J. State estimation of stochastic non-linear hybrid dynamic system using an interacting multiple model algorithm. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.005i

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Please cite this article as: Elenchezhiyan M, Prakash J. State estimation of stochastic non-linear hybrid dynamic system using an interacting multiple model algorithm. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.06.005i