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State Estimation of a Non-linear Hybrid System Using an Interacting Multiple Model Algorith
8th IFAC Symposium on Advanced Control of Chemical Processes The International Federation of Automatic Control Singapore, July 10-13, 2012
State Estimation of a Non-linear Hybrid System Using an Interacting Multiple Model Algorithm J. Prakash*, M. Elenchezhiyan *, S.L. Shah**
*Department of Instrumentation Engineering, Madras Institute of Technology Campus, Anna University Chennai-44. India [email protected]. ** Department of Chemical and Materials Engineering, University of Alberta, Canada ([email protected])
Abstract: In this work, we formulate a state estimation scheme for a nonlinear hybrid system that is subjected to stochastic state disturbances and measurement noise using an interacting Multiple-Model Algorithm (IMM). In particular, we propose the use of an IMM Extended Kalman Filter (IMM-EKF) and an IMM Unscented Kalman filter (IMM-UKF), which belongs to the class of derivative free estimators to carry out estimation of state variables of hybrid system. The efficacy of the proposed state estimation schemes is demonstrated by conducting simulation studies on a two-tank hybrid system. Analysis of the simulation results reveals that the proposed state estimation schemes are able to generate fairly accurate filtered estimate of state variables.
noise, using the Ensemble Kalman filter and Unscented Kalman filter has been proposed by Prakash et al. (2010). Simultaneous estimation of both continuum and noncontinuum states with an application to the distillation process is reported recently in Olanrewaju et al. (2012) With the exception of these few references, to the best knowledge of the authors, the problem of estimating state variables of a nonlinear hybrid system subjected to state noise and measurement noise has hardly received any attention in the process control literature. In this work, we develop a state estimation scheme for nonlinear two-tank hybrid system, which are subjected to stochastic state disturbances and measurement noise, using the IMM algorithm. The organization of the paper is as follows. Section 2 discusses the details of state estimation for hybrid system using IMM-EKF algorithm and IMM-UKF algorithm. The process considered for the illustrative simulation study is outlined in section 3. Simulation results are also presented in section 3 followed by concluding remarks in section 4.
1. INTRODUCTION Dynamic systems that are described by an interaction between continuous and discrete dynamics are called hybrid systems. The switching events in hybrid systems can be classified into (i) Autonomous switching of dynamics, (ii) Autonomous state jumps (iii) Controlled switching of dynamics and (iv) Controlled state jumps [Lunze and Lamnabhi-Lagarrigue, 2009]. Many methodologies have been proposed to model hybrid systems, to analyze their behaviour, and to design model based control schemes that guarantee closed-loop stability and performance specifications [Lunze and Lamnabhi-Lagarrigue, 2009]. In practice, however, most of the systems are influenced by unmeasured disturbances. In addition, measurements used for feedback are corrupted with noise. Thus, state estimation of hybrid system poses a challenging problem (Prakash et al. 2010). The non-linear Kalman filters such as the Extended Kalman Filter (EKF) and the Unscented Kalman Filter (UKF) have attracted widespread attention in the engineering community because of the recursive nature of their computational scheme and their ability to deal with non-linear processes. Also, a new class of filtering techniques, called Particle Filtering, can deal with state estimation problems arising from multimodal and non-Gaussian distributions. A Particle Filter (PF) approximates multi-dimensional integration involved in the propagation and update steps in the general Bayesian state estimation scheme using Monte Carlo sampling (Arulampalam et al., 2002). In the context of developing state estimator for hybrid systems, these derivative free estimation schemes appear to be promising candidates (Prakash et al. 2010). The use of moving horizon based state estimation for hybrid system has been reported in Ferrari-Trecate et al. (2000, 2002) and Bemporad et al. (1999). Robust state estimation and fault diagnosis for uncertain hybrid systems has been reported in Wang et al. (2007). Recently, a state estimation scheme for nonlinear autonomous hybrid systems, which are subjected to stochastic state disturbances and measurement
978-3-902823-05-2/12/$20.00 © 2012 IFAC
2. STATE ESTIMATION OF AUTONOMOUS HYBRID SYSTEMS Two types of discrete events can occur in a hybrid system, namely external jumps and autonomous jumps. The autonomous jumps are functions of continuous states of the systems. In this work, it is proposed to formulate the state estimation problem associated with this sub-class of hybrid systems namely autonomous switching of dynamics. These systems can be represented by the following set of differential equations: kT x(k ) x(k 1) F z [x(), u(k 1)] d w(k ) (1) ( k 1)T y(k ) H [x(k )] v(k ) (2) In the above process model, x(k) is the system state vector (x R L ) , u(k) is the known system input (u R m ) ,
w(k ) is the state noise ( w R L ) with known distribution,
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8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
P(k | k ) I K(k )C(k ) P(k | k 1)
y(k) is the measured state variable ( y R r ) and v(k) is the measurement noise ( v(k) R ) with known distribution.
(10)
r
2.2 Unscented Kalman Filter
The index k represents the sampling instant, Fz . and H . are the nonlinear process and measurement models respectively. In equation (1), z Q represents discrete modes such that it can take only finite integer values. Q : 1, 2,...., r is the mode set and also assumed to be an unknown mode transition function. The continuous dynamics switches autonomously between r discrete modes depending on whether or not the state variables exceed the threshold level.
Over the last two decades, many approaches have been proposed in the literature, which overcome some limitations of the EKF. A class of filters, called sigma point filters or derivative free filters has emerged, which relies on the fact that estimation of the moments of a nonlinear function using samples is significantly easier (computationally less demanding) than computation of the derivatives of the nonlinear function. The most popular approach in this class is the Unscented Kalman filter as developed by Julier and Uhlmann (2004). A set of 2L+1 sigma points, (k 1| k 1, i) with the associated weights, W (i) are chosen symmetrically about Xˆ (k 1| k 1) as given below.
2.1 Extended Kalman Filter The basic idea of the EKF is to linearize F and H using a first order Taylor series expansion, and then apply the standard Kalman Filter. We have assumed in this work that the initial state x(0) and the sequence w(k) and v(k) are white, Gaussian and independent of each other. The well-known Extended Kalman Filter is as follows: The predicted mean of the estimate is computed as follows: k t
ˆ k | k 1) x( ˆ k 1| k 1) x(
χ (k 1| k 1,0) xˆ (k 1| k 1); χ (k 1| k 1, i) xˆ (k 1| k 1)
iL
W (0) ; L W c (0) (1 2 ); 2 ( L ) L L 1 W c (i ) W m (i ) ; i 1,..., 2 L 2( L ) Where is a secondary scaling parameter and is a factor determining the spread of sigma points around Xˆ (k 1| k 1) and is usually set between 1e-4 to 1 (Van der Merwe, 2003). The parameter is used to incorporate prior knowledge of distribution of x and for Gaussian distribution its optimum value is 2 (Van der Merwe, 2003). The 2L+1 sigma points have been derived from the state xˆ (k | k 1) and covariance of the state vector P(k 1 | k 1) , where L is the dimension of the state vector. In the prediction step, the sigma points are propagated through the nonlinear differential equations to obtain the predicted set of sigma points as
ˆ k 1| k 1) (3) xˆ (k 1)t x( The covariance matrix of estimation errors in the predicted state estimates is obtained as follows: (4) P(k | k 1) Φ(k )P(k 1| k 1)Φ(k )T Q Where, Φ( k) exp[A( k)* T] and A(k ), is nothing but matrix of partial derivatives of F with respect to F xˆ (k 1| k 1) and is expressed as A(k ) x xˆ ( k 1|k 1) It should be noted that the extended Kalman filter (EKF) computes covariance matrix using the linear propagation. The measurement prediction ( yˆ (k | k 1) ), computation of innovation ( γ(k ) ) and covariance matrix of innovation
( Pyy (k ) ) are computed as follows:
Pyy (k ) C(k ) P(k | k 1)C(k )T R
(7)
i
m
F x(), u(k 1) d
(5)
( L )P(k 1| k 1) i 1,..., L
χ (k 1| k 1, i) xˆ (k 1| k 1) ( L )P(k 1| k 1) i L 1,..., 2 L
( k 1) t
yˆ (k | k 1) H xˆ (k | k 1) γ(k ) y(k ) yˆ (k | k 1)
k t
χ (k | k 1, i) χ (k 1| k 1, i)
(6)
Fχ (, i), u(k 1) d ;
( k 1) t
(11)
i 0,....,2 L
The predicted state estimates xˆ (k | k 1) is obtained from the predicted sigma points as
Where C(k) is the measurement matrix of partial derivatives of H with respect to xˆ (k | k 1)
2L
xˆ (k | k 1) W m (i)χ (k | k 1, i) .
H C(k ) x xˆ ( k |k 1) The Kalman gain is computed using the following equation, (8) K(k ) P(k | k 1)C(k )T Pyy 1 (k )
(12)
i 0
The error covariance matrix P(k | k 1) is obtained from the predicted sigma points as 2L
P(k | k 1) W c (i){χ (k | k 1, i) xˆ (k | k 1)}*
The updated state estimates are obtained using the following equation, xˆ (k | k ) xˆ (k | k 1) K(k ) γ(k ) (9) The covariance matrix of estimation errors in the updated state estimates is obtained as
i 0
(13)
{χ (k | k 1, i) xˆ (k | k 1)}T Q Sigma points are redrawn using the predicted state estimate as given below χ (k | k 1,0) xˆ (k | k 1); 508
8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
(L )P(k | k 1) i 1,..., L χ (k | k 1, i) xˆ (k | k 1) ( L )P(k | k 1) i L 1,...,2L
χ (k | k 1, i) xˆ (k | k 1)
(mixed initial condition).The conditional probability density function of the state x is decomposed according to the total probability theorem as follows:
i
iL
r
2L
yˆ (k | k 1) W m (i) * H χ (k | k 1, i)
j 1
r
j 1
(14)
P x(k )| M j ( k ), y ( k ) , Y k 1
The covariance matrix of the innovations Pyy (k) and the cross covariance matrix between the predicted state estimate errors and innovations Pxy (k) are computed as Pyy (k ) [W (i){H[χ (k | k 1, i)] yˆ (k | k 1)}*
P y (k ) | M j (k ), x( k ) P y (k ) | M j (k ), Y k 1
i 0
r
P
(15)
i 1
T
{H[χ (k | k 1, i)] yˆ (k | k 1)} ] R 2L
Pxy (k ) W (i){χ (k | k 1, i ) xˆ (k | k 1)}* i 0
The one cycle of the IMM algorithm consists of the following Calculation of the mixing probabilities and mixed initial condition The probability that the mode M i was in effect at instant
(16)
{H[χ (k | k 1, i)] yˆ (k | k 1)}T The innovation are computed as follows (17) γ(k ) y(k ) yˆ (k | k 1) . The Kalman gain matrix K(k) can be determined as follows 1 yy
K(k ) Pxy (k )P (k )
x(k )| M j ( k ), M i ( k 1), xˆ i ( k 1| k 1), P i ( k 1| k 1) i | j (k 1| k 1)
c
P x( k )| M j ( k ), Y k 1
P x(k )| M j (k ),Y k 1
c
P[x( k )| M j ( k ) , y( k ) , Y k 1 ] j ( k )
i 0
2L
P x(k )| Y k P x( k )| M j ( k ) , Y k P M j ( k ) | Y k
Redrawn sigma points are then propagated through the nonlinear measurement equation to obtain the predicted measurement as
(k-1) given that M j is in effect at k conditioned on Y k-1 is i| j (k 1| k 1) P[ M i (k 1)| M j (k ), Y k 1 ]
(18)
The updated state estimates xˆ (k | k ) are then obtained using the linear update equation as in the Kalman filter using the equation, (19) xˆ (k | k ) xˆ (k | k 1) K(k )* γ(k ) The covariance matrix of error in the updated state estimates P(k | k ) is computed using the equation,
i | j (k 1| k 1)
Pij i (k 1) r
Pij i (k 1)
(21)
i, j 1, 2,...r
(22)
i 1
Where i|j k-1|k-1 is mixing probability,
(20) P(k | k ) P(k | k 1) K(k )*Pyy (k )KT (k ) The UKF does not approximate the nonlinear functions of system and measurement models as required by the EKF. Instead, the nonlinear functions are applied to sigma points to yield transformed samples, and the propagated mean and covariance are calculated from the transformed samples. Under the assumption that the prior probability density function is Gaussian, it has been shown that the UKF formulation results in approximations that are accurate up to the third moment for all types of nonlinearities. For nonGaussian uncertainties, approximations are accurate to at least the second-moment, with the accuracy of third and higher order moments determined by the choice of tuning parameters (Julier and Uhlmann, 2004). A major advantage of UKF is that it avoids explicit computation of the Jacobian matrices. 2.3 Interacting Multiple Model Estimator: IMM-UKF In this subsection, we describe the steps involved in obtaining state estimates of system using the IMM approach. The IMM algorithm consists of r interacting non-linear Kalman filters operating in parallel. However, it should be noted that in the IMM approach, at time k the state estimate is computed under each possible current model using ‘r’ non-linear Kalman filters, with each non-linear Kalman filter using a different combination of the previous model-conditioned estimates
P i, j
is the assumed
transition probability for the Markov chain according to which the system model switches from model i to model j, ( k 1) is the model probability. The input to the filter i matched to model j is obtained from an interaction of the ‘r’ non-linear Kalman filters, which consists of the mixing of estimates xˆ (i ) (k 1| k 1) with the weights i|j k-1|k-1 , called the mixing probabilities. r
xˆ 0 j (k 1| k 1) xˆ i (k 1| k 1) i | j (k 1| k 1) ; j 1...r j 1
The covariance matrix is computed as r
P 0 j (k 1| k 1) i | j (k 1| k 1)[ Pi (k 1| k 1) j 1
[ xˆ i (k 1| k 1) xˆ 0 j (k 1| k 1)] (24) [ xˆ i (k 1| k 1) xˆ 0 j (k 1| k 1)]T ] Mode-Matched Filtering The estimate and the covariance are used as input to the filter matched to M j , which uses the current measurement y(k) to
yield
xˆ j (k | k ) and P j (k | k ) .
The
likelihood
functions
corresponding to the 'r' non-linear Kalman filters are computed according to j (k ) P[ y(k )| M j (k ), Y k 1 ] P[ y(k )| M j (k ), xˆ 0 j (k 1| k 1),P 0 j (k 1| k 1)]
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8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
exp 0.5 γ j (k ) Pyyj (k ) γ j (k )T j (k ) (2) r | Pyyj (k ) |
The governing equations of two-tank hybrid system for different modes are as follows: MODE -1
1
(26)
dh1 qmaxV1 k4 sgn(h1 h2 ) | h1 h2 | V4 k5 h1V5 dt dh A 2 qmaxV2 k4 sgn(h1 h2 ) | h1 h2 | V4 k6 h2 V6 dt A
In the above equation, (k ) , P (k 1) are the innovation and innovation covariance matrix respectively. Mode Probability Update j
j yy
MODE -2 A
The model probabilities are updated as follows: j (k ) P[ M j (k )|Y k ] r (k ) Pij i (k 1) i 1 (27) j (k ) r j 1, 2,...r r j (k ) Pij i (k 1) j 1 i 1 State Estimate and Covariance Combination Combination of model conditioned estimates and covariance matrices is carried out using the following equations
A
j
dh1 (qmaxV1 k4 sgn(h1 h2 ) | h1 h2 | V4 dt k5 h1 V5 k3 | h2 hv | V3 ) dh2 A (qmaxV2 k4 sgn(h1 h2 ) | h1 h2 | V4 dt k6 h2 V6 k3 | h2 hv | V3
(28)
A
The two-tank hybrid system has four modes that depend on the relation between the water levels in both tanks. For each mode, we have a set of ordinary differential equations and the system transitions between modes are based on the state variables (h1 and h2). A schematic representation of the twotank hybrid system is given in Figure 1.
A
V5
V3
Tank 2
V6
V4
dh1 (qmaxV1 k4 sgn(h1 h2 ) | h1 h2 | V4 dt k5 h1 V5 k3 sgn(h1 h2 ) | h1 h2 | V3 ) dh2 (qmaxV2 k4 sgn(h1 h2 ) | h1 h2 | V4 dt k6 h2 V6 k3 sgn(h1 h2 ) | h1 h2 | V3
Parameter
Value
q max
5e-5
k 3 and k 4
3.6050e-5
4.8960e-5
k 5 and k 6
4.0565e-5
3.5653e-5
0.0176625(m2)
A Fig. 1. Two-tank Hybrid system
V i
h2(t)
1 2 3 4
< hv >= hv < hv >= hv
< hv < hv >= hv > =hv
0
max
1
The measurement noise covariance matrix is assumed as (0.01)2 R 0
Table 1. Discrete modes and their dependence on the continuous state variables h1(t)
min
V i
It should be noted that these transitions are autonomous since they depend on the continuous behaviour of the system as described by the threshold conditions (Refer to Table -1). The model parameters of two-tank hybrid system are reported in Table 2.
Mode (z)
(32)
Table 2. Two-tank hybrid system model parameters
V2
hv
(31)
MODE -4
3. SIMULATION STUDIES
Tank 1
(30)
dh2 qmaxV2 k4 sgn(h1 h2 ) | h1 h2 | V4 k6 h2 V6 dt k3 | h1 hv | V3 )
A
j 1
V1
dh1 (qmaxV1 k4 sgn(h1 h2 ) | h1 h2 | V4 dt k5 h1 V5 k3 | h1 hv | V3 )
MODE -3
r
xˆ (k | k ) xˆ j (k | k ) j (k )
(29)
(0.01)2 0
The initial error covariance matrices for each filter is assumed as 10 0 Pr (0 | 0) ; 0 10
r 1, 2,3, 4
The process noise covariance matrices for each filter is assumed as 0 1e 8 Qr ; 1e 8 0
510
r 1, 2,3, 4
8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
The initial state vector for each filter is assumed as
0.4 h1(m)
ˆ r (0 | 0) 0.15 0.1T r 1, 2,3, 4 X
The Markov chain transition matrix has been chosen as
0.3 0.2 True
Estimated
0.1 500
0.99 0.01/3 0.01/3 0.01/3 0.01/3 0.99 0.01/3 0.01/3 Pij 0.01/3 0.01/3 0.99 0.01/3 0.01/3 0.01/3 0.01/3 0.99
1000
1500
2000 2500 3000 Sampling Instants
3500
4000
4500
5000
h2(m)
0.4
The initial estimate model probability ( (0 | 0) ) is assumed
0.3 0.2 True
T
Estimated
0.1
as (0 | 0) 0.95 0.02 0.02 0.01 . , and associated with UKF are assumed as 1, 0 and 0 respectively. The objective is to generate filtered estimate of the water levels in the tanks. The simulation studies have been carried out in the presence of additive Gaussian state noise and measurement noise. The evolution of true and estimated continuous state variables generated by IMM-EKF and IMM-UKF for variations in V1 and V2 (See figure 8) is shown in Figures 2 and 3 respectively. From Figures 2 and 3, it can be concluded that IMM –EKF and IMM-UKF are able to generate fairly accurate filtered state estimates.
500
1000
1500
2000 2500 3000 Sampling Instants
3500
4000
4500
5000
Fig. 3. Evolution of true and estimated states of Two-tank hybrid system using IMM-UKF (a) Level in Tank1 (b) Level in Tank2
4. CONCLUDING REMARKS In this paper, we have proposed state estimation schemes for two-tank hybrid system using an IMM-EKF algorithm and an IMM-UKF algorithm. The efficacy of the proposed state estimation schemes has been demonstrated by carrying out simulation studies on the two-tank hybrid system. 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 the IMM-UKF is determined to be a good alternative for the IMM-EKF, because, UKF doesn’t approximate the nonlinear function and computation of the Jacobian matrices at each time step is not required.
The mode probability update of IMM-EKF and IMM-UKF are shown in Figures 4 and 5. From Figures 4 and 5, it can be concluded that the IMM-EKF and IMM-UKF transition between modes as designed. It is seen that the performance of IMM-EKF and IMM-UKF are found to be identical for this study case. The evolution of true, measured and estimated continuous state variables generated by IMM-EKF and IMMUKF during the interval (200 k 1500) is shown in Figures 6 and 7 respectively. From Figures 6 and 7, it can be inferred that the proposed state estimation schemes is able to efficiently suppress the measurement noise.
1.5 Mode-1
Mode-2
Mode-3
Mode-4
1
0.3
Mode
h1(m)
0.4 0.5
0.2 True
Estimated
0.1 500
1000
1500
2000 2500 3000 Sampling Instants
3500
4000
4500
0
5000
h2(m)
0.4
-0.5
0.3 0.2 True
0
500
1000
1500
2000 2500 3000 Sampling Instants
3500
4000
4500
5000
Fig. 4. Model Probability Update: IMM-EKF
Estimated
0.1 500
1000
1500
2000 2500 3000 Sampling Instants
3500
4000
4500
5000
ACKNOWLEDGEMENTS
The financial support through the NSERC – MATRIKON – SUNCORE - ICORE Industrial research chair program in CPC at University of Alberta is also gratefully acknowledged. The technical support rendered by Mr. N. Prabhakar, Distinguished Scientist, DRDL and Mr. Vamsi Krishna Reddy, Scientist-D, DRDL Hyderabad is gratefully acknowledged.
Fig. 2. Evolution of true and estimated states of Two-tank hybrid system using IMM-EKF (a) Level in Tank1 (b) Level in Tank2
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8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
Models, in Proceedings of the Workshop on Advances in Machine Learning, Montreal, Canada. 1.5 Mode-1
Mode-2
Mode-3
Wenhui Wang, D.H. Zhou, Zhengxi Li (2006). Robust state estimation and fault diagnosis for uncertain hybrid systems. Nonlinear analysis, Volume 65, pp. 2193-2215.
Mode-4
1
0.3 0.5
h1(m)
Mode
0.32
0.28 True
0.26 0
200
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600
Estimated
Measured
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1200
1400
0.3 0
500
1000
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2000 2500 3000 Sampling Instants
3500
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True
0.28
5000 h2(m)
-0.5
Fig. 5. Model Probability Update: IMM-UKF
Estimated
Measured
0.26 0.24 0.22 200
REFERENCES
400
600
800 1000 Sampling Instants
1200
1400
Figure 6: Evolution of true, estimated, measured states of Two-tank hybrid system using IMM-EKF (a) Level in Tank1 (b) Level in Tank2 (200 k 1500)
Bemporad A, D. Mignone and M. Morari (1999), Moving horizon estimation for hybrid systems and fault detection. Proceedings of American Control Conference, San Diego, CA, pp. 2471–2475.
0.32
Ferrari-Trecate G, D. Mignone and M. Morari (2000), Moving horizon estimation for hybrid systems. Proceedings of American Control Conference. Chicago, IL, 3 pp.1684– 1688.
h1(m)
0.3 0.28 True
0.26 200
Ferrari-Trecate G, D. Mignone, and M. Morari, (2002) Moving horizon estimation for hybrid systems.IEEE Transactions on Automatic Control, 47, pp.1663–1676.
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600
Estimated
Measured
800 1000 Sampling Instants
1200
1400
0.3 True
Estimated
Measured
h2(m)
0.28
Julier S.J. and J.K. Uhlmann (2004), Unscented filtering and nonlinear estimation, Proceedings of the IEEE, 92 (3), pp. 401–422.
0.26 0.24 0.22 200
Lunze, J. and Francoise Lamnabhi-Lagrarrigue (2009) Handbook of Hybrid Systems Control Theory, Tools, Applications, Cambridge University Press, New York.
400
600
800 1000 Sampling Instants
1200
1400
Figure 7: Evolution of true, estimated, measured states of Two-tank hybrid system using IMM-UKF (a) Level in Tank1 (b) Level in Tank2
Moshood J. Olanrewaju, Biao Huang, Artin Afacan (2012) Development of a Simultaneous Continuum and Noncontinuum State Estimator with Application on a Distillation Process, AIChE , 58, pp. 480–492.
0.75
V1
0.7 0.65 0.6
Prakash, J., Sachin C. Patwardhan and S. L. Shah (2010) State Estimation and Nonlinear Predictive Control of Autonomous Hybrid system using Derivative free State Estimators, Journal of Process Control, 20, pp.787–799.
0.55
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4000
4500
5000
500
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2000 2500 3000 Sampling Instants
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5000
0.25
V2
0.24
Sanjeev Arulampalam M., Simon Maskell, Neil Gordon, and Tim Clapp (2002), A tutorial on particle filters for online Nonlinear/Non-Gaussian Bayesian tracking, IEEE Transactions on Signal Processing, 50(2), pp. 174–188.
0.23 0.22 0.21 0.2
Figure 8: Evolution of V1 and V2 versus sampling Instants
Van der Merwe, R. and E Wan (2003). Sigma-Point Kalman Filters for Probabilistic Inference in Dynamic State-Space 512
State Estimation of a Non-linear Hybrid System Using an Interacting Multiple Model Algorithm J. Prakash*, M. Elenchezhiyan *, S.L. Shah**
*Department of Instrumentation Engineering, Madras Institute of Technology Campus, Anna University Chennai-44. India [email protected]. ** Department of Chemical and Materials Engineering, University of Alberta, Canada ([email protected])
Abstract: In this work, we formulate a state estimation scheme for a nonlinear hybrid system that is subjected to stochastic state disturbances and measurement noise using an interacting Multiple-Model Algorithm (IMM). In particular, we propose the use of an IMM Extended Kalman Filter (IMM-EKF) and an IMM Unscented Kalman filter (IMM-UKF), which belongs to the class of derivative free estimators to carry out estimation of state variables of hybrid system. The efficacy of the proposed state estimation schemes is demonstrated by conducting simulation studies on a two-tank hybrid system. Analysis of the simulation results reveals that the proposed state estimation schemes are able to generate fairly accurate filtered estimate of state variables.
noise, using the Ensemble Kalman filter and Unscented Kalman filter has been proposed by Prakash et al. (2010). Simultaneous estimation of both continuum and noncontinuum states with an application to the distillation process is reported recently in Olanrewaju et al. (2012) With the exception of these few references, to the best knowledge of the authors, the problem of estimating state variables of a nonlinear hybrid system subjected to state noise and measurement noise has hardly received any attention in the process control literature. In this work, we develop a state estimation scheme for nonlinear two-tank hybrid system, which are subjected to stochastic state disturbances and measurement noise, using the IMM algorithm. The organization of the paper is as follows. Section 2 discusses the details of state estimation for hybrid system using IMM-EKF algorithm and IMM-UKF algorithm. The process considered for the illustrative simulation study is outlined in section 3. Simulation results are also presented in section 3 followed by concluding remarks in section 4.
1. INTRODUCTION Dynamic systems that are described by an interaction between continuous and discrete dynamics are called hybrid systems. The switching events in hybrid systems can be classified into (i) Autonomous switching of dynamics, (ii) Autonomous state jumps (iii) Controlled switching of dynamics and (iv) Controlled state jumps [Lunze and Lamnabhi-Lagarrigue, 2009]. Many methodologies have been proposed to model hybrid systems, to analyze their behaviour, and to design model based control schemes that guarantee closed-loop stability and performance specifications [Lunze and Lamnabhi-Lagarrigue, 2009]. In practice, however, most of the systems are influenced by unmeasured disturbances. In addition, measurements used for feedback are corrupted with noise. Thus, state estimation of hybrid system poses a challenging problem (Prakash et al. 2010). The non-linear Kalman filters such as the Extended Kalman Filter (EKF) and the Unscented Kalman Filter (UKF) have attracted widespread attention in the engineering community because of the recursive nature of their computational scheme and their ability to deal with non-linear processes. Also, a new class of filtering techniques, called Particle Filtering, can deal with state estimation problems arising from multimodal and non-Gaussian distributions. A Particle Filter (PF) approximates multi-dimensional integration involved in the propagation and update steps in the general Bayesian state estimation scheme using Monte Carlo sampling (Arulampalam et al., 2002). In the context of developing state estimator for hybrid systems, these derivative free estimation schemes appear to be promising candidates (Prakash et al. 2010). The use of moving horizon based state estimation for hybrid system has been reported in Ferrari-Trecate et al. (2000, 2002) and Bemporad et al. (1999). Robust state estimation and fault diagnosis for uncertain hybrid systems has been reported in Wang et al. (2007). Recently, a state estimation scheme for nonlinear autonomous hybrid systems, which are subjected to stochastic state disturbances and measurement
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2. STATE ESTIMATION OF AUTONOMOUS HYBRID SYSTEMS Two types of discrete events can occur in a hybrid system, namely external jumps and autonomous jumps. The autonomous jumps are functions of continuous states of the systems. In this work, it is proposed to formulate the state estimation problem associated with this sub-class of hybrid systems namely autonomous switching of dynamics. These systems can be represented by the following set of differential equations: kT x(k ) x(k 1) F z [x(), u(k 1)] d w(k ) (1) ( k 1)T y(k ) H [x(k )] v(k ) (2) In the above process model, x(k) is the system state vector (x R L ) , u(k) is the known system input (u R m ) ,
w(k ) is the state noise ( w R L ) with known distribution,
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8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
P(k | k ) I K(k )C(k ) P(k | k 1)
y(k) is the measured state variable ( y R r ) and v(k) is the measurement noise ( v(k) R ) with known distribution.
(10)
r
2.2 Unscented Kalman Filter
The index k represents the sampling instant, Fz . and H . are the nonlinear process and measurement models respectively. In equation (1), z Q represents discrete modes such that it can take only finite integer values. Q : 1, 2,...., r is the mode set and also assumed to be an unknown mode transition function. The continuous dynamics switches autonomously between r discrete modes depending on whether or not the state variables exceed the threshold level.
Over the last two decades, many approaches have been proposed in the literature, which overcome some limitations of the EKF. A class of filters, called sigma point filters or derivative free filters has emerged, which relies on the fact that estimation of the moments of a nonlinear function using samples is significantly easier (computationally less demanding) than computation of the derivatives of the nonlinear function. The most popular approach in this class is the Unscented Kalman filter as developed by Julier and Uhlmann (2004). A set of 2L+1 sigma points, (k 1| k 1, i) with the associated weights, W (i) are chosen symmetrically about Xˆ (k 1| k 1) as given below.
2.1 Extended Kalman Filter The basic idea of the EKF is to linearize F and H using a first order Taylor series expansion, and then apply the standard Kalman Filter. We have assumed in this work that the initial state x(0) and the sequence w(k) and v(k) are white, Gaussian and independent of each other. The well-known Extended Kalman Filter is as follows: The predicted mean of the estimate is computed as follows: k t
ˆ k | k 1) x( ˆ k 1| k 1) x(
χ (k 1| k 1,0) xˆ (k 1| k 1); χ (k 1| k 1, i) xˆ (k 1| k 1)
iL
W (0) ; L W c (0) (1 2 ); 2 ( L ) L L 1 W c (i ) W m (i ) ; i 1,..., 2 L 2( L ) Where is a secondary scaling parameter and is a factor determining the spread of sigma points around Xˆ (k 1| k 1) and is usually set between 1e-4 to 1 (Van der Merwe, 2003). The parameter is used to incorporate prior knowledge of distribution of x and for Gaussian distribution its optimum value is 2 (Van der Merwe, 2003). The 2L+1 sigma points have been derived from the state xˆ (k | k 1) and covariance of the state vector P(k 1 | k 1) , where L is the dimension of the state vector. In the prediction step, the sigma points are propagated through the nonlinear differential equations to obtain the predicted set of sigma points as
ˆ k 1| k 1) (3) xˆ (k 1)t x( The covariance matrix of estimation errors in the predicted state estimates is obtained as follows: (4) P(k | k 1) Φ(k )P(k 1| k 1)Φ(k )T Q Where, Φ( k) exp[A( k)* T] and A(k ), is nothing but matrix of partial derivatives of F with respect to F xˆ (k 1| k 1) and is expressed as A(k ) x xˆ ( k 1|k 1) It should be noted that the extended Kalman filter (EKF) computes covariance matrix using the linear propagation. The measurement prediction ( yˆ (k | k 1) ), computation of innovation ( γ(k ) ) and covariance matrix of innovation
( Pyy (k ) ) are computed as follows:
Pyy (k ) C(k ) P(k | k 1)C(k )T R
(7)
i
m
F x(), u(k 1) d
(5)
( L )P(k 1| k 1) i 1,..., L
χ (k 1| k 1, i) xˆ (k 1| k 1) ( L )P(k 1| k 1) i L 1,..., 2 L
( k 1) t
yˆ (k | k 1) H xˆ (k | k 1) γ(k ) y(k ) yˆ (k | k 1)
k t
χ (k | k 1, i) χ (k 1| k 1, i)
(6)
Fχ (, i), u(k 1) d ;
( k 1) t
(11)
i 0,....,2 L
The predicted state estimates xˆ (k | k 1) is obtained from the predicted sigma points as
Where C(k) is the measurement matrix of partial derivatives of H with respect to xˆ (k | k 1)
2L
xˆ (k | k 1) W m (i)χ (k | k 1, i) .
H C(k ) x xˆ ( k |k 1) The Kalman gain is computed using the following equation, (8) K(k ) P(k | k 1)C(k )T Pyy 1 (k )
(12)
i 0
The error covariance matrix P(k | k 1) is obtained from the predicted sigma points as 2L
P(k | k 1) W c (i){χ (k | k 1, i) xˆ (k | k 1)}*
The updated state estimates are obtained using the following equation, xˆ (k | k ) xˆ (k | k 1) K(k ) γ(k ) (9) The covariance matrix of estimation errors in the updated state estimates is obtained as
i 0
(13)
{χ (k | k 1, i) xˆ (k | k 1)}T Q Sigma points are redrawn using the predicted state estimate as given below χ (k | k 1,0) xˆ (k | k 1); 508
8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
(L )P(k | k 1) i 1,..., L χ (k | k 1, i) xˆ (k | k 1) ( L )P(k | k 1) i L 1,...,2L
χ (k | k 1, i) xˆ (k | k 1)
(mixed initial condition).The conditional probability density function of the state x is decomposed according to the total probability theorem as follows:
i
iL
r
2L
yˆ (k | k 1) W m (i) * H χ (k | k 1, i)
j 1
r
j 1
(14)
P x(k )| M j ( k ), y ( k ) , Y k 1
The covariance matrix of the innovations Pyy (k) and the cross covariance matrix between the predicted state estimate errors and innovations Pxy (k) are computed as Pyy (k ) [W (i){H[χ (k | k 1, i)] yˆ (k | k 1)}*
P y (k ) | M j (k ), x( k ) P y (k ) | M j (k ), Y k 1
i 0
r
P
(15)
i 1
T
{H[χ (k | k 1, i)] yˆ (k | k 1)} ] R 2L
Pxy (k ) W (i){χ (k | k 1, i ) xˆ (k | k 1)}* i 0
The one cycle of the IMM algorithm consists of the following Calculation of the mixing probabilities and mixed initial condition The probability that the mode M i was in effect at instant
(16)
{H[χ (k | k 1, i)] yˆ (k | k 1)}T The innovation are computed as follows (17) γ(k ) y(k ) yˆ (k | k 1) . The Kalman gain matrix K(k) can be determined as follows 1 yy
K(k ) Pxy (k )P (k )
x(k )| M j ( k ), M i ( k 1), xˆ i ( k 1| k 1), P i ( k 1| k 1) i | j (k 1| k 1)
c
P x( k )| M j ( k ), Y k 1
P x(k )| M j (k ),Y k 1
c
P[x( k )| M j ( k ) , y( k ) , Y k 1 ] j ( k )
i 0
2L
P x(k )| Y k P x( k )| M j ( k ) , Y k P M j ( k ) | Y k
Redrawn sigma points are then propagated through the nonlinear measurement equation to obtain the predicted measurement as
(k-1) given that M j is in effect at k conditioned on Y k-1 is i| j (k 1| k 1) P[ M i (k 1)| M j (k ), Y k 1 ]
(18)
The updated state estimates xˆ (k | k ) are then obtained using the linear update equation as in the Kalman filter using the equation, (19) xˆ (k | k ) xˆ (k | k 1) K(k )* γ(k ) The covariance matrix of error in the updated state estimates P(k | k ) is computed using the equation,
i | j (k 1| k 1)
Pij i (k 1) r
Pij i (k 1)
(21)
i, j 1, 2,...r
(22)
i 1
Where i|j k-1|k-1 is mixing probability,
(20) P(k | k ) P(k | k 1) K(k )*Pyy (k )KT (k ) The UKF does not approximate the nonlinear functions of system and measurement models as required by the EKF. Instead, the nonlinear functions are applied to sigma points to yield transformed samples, and the propagated mean and covariance are calculated from the transformed samples. Under the assumption that the prior probability density function is Gaussian, it has been shown that the UKF formulation results in approximations that are accurate up to the third moment for all types of nonlinearities. For nonGaussian uncertainties, approximations are accurate to at least the second-moment, with the accuracy of third and higher order moments determined by the choice of tuning parameters (Julier and Uhlmann, 2004). A major advantage of UKF is that it avoids explicit computation of the Jacobian matrices. 2.3 Interacting Multiple Model Estimator: IMM-UKF In this subsection, we describe the steps involved in obtaining state estimates of system using the IMM approach. The IMM algorithm consists of r interacting non-linear Kalman filters operating in parallel. However, it should be noted that in the IMM approach, at time k the state estimate is computed under each possible current model using ‘r’ non-linear Kalman filters, with each non-linear Kalman filter using a different combination of the previous model-conditioned estimates
P i, j
is the assumed
transition probability for the Markov chain according to which the system model switches from model i to model j, ( k 1) is the model probability. The input to the filter i matched to model j is obtained from an interaction of the ‘r’ non-linear Kalman filters, which consists of the mixing of estimates xˆ (i ) (k 1| k 1) with the weights i|j k-1|k-1 , called the mixing probabilities. r
xˆ 0 j (k 1| k 1) xˆ i (k 1| k 1) i | j (k 1| k 1) ; j 1...r j 1
The covariance matrix is computed as r
P 0 j (k 1| k 1) i | j (k 1| k 1)[ Pi (k 1| k 1) j 1
[ xˆ i (k 1| k 1) xˆ 0 j (k 1| k 1)] (24) [ xˆ i (k 1| k 1) xˆ 0 j (k 1| k 1)]T ] Mode-Matched Filtering The estimate and the covariance are used as input to the filter matched to M j , which uses the current measurement y(k) to
yield
xˆ j (k | k ) and P j (k | k ) .
The
likelihood
functions
corresponding to the 'r' non-linear Kalman filters are computed according to j (k ) P[ y(k )| M j (k ), Y k 1 ] P[ y(k )| M j (k ), xˆ 0 j (k 1| k 1),P 0 j (k 1| k 1)]
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exp 0.5 γ j (k ) Pyyj (k ) γ j (k )T j (k ) (2) r | Pyyj (k ) |
The governing equations of two-tank hybrid system for different modes are as follows: MODE -1
1
(26)
dh1 qmaxV1 k4 sgn(h1 h2 ) | h1 h2 | V4 k5 h1V5 dt dh A 2 qmaxV2 k4 sgn(h1 h2 ) | h1 h2 | V4 k6 h2 V6 dt A
In the above equation, (k ) , P (k 1) are the innovation and innovation covariance matrix respectively. Mode Probability Update j
j yy
MODE -2 A
The model probabilities are updated as follows: j (k ) P[ M j (k )|Y k ] r (k ) Pij i (k 1) i 1 (27) j (k ) r j 1, 2,...r r j (k ) Pij i (k 1) j 1 i 1 State Estimate and Covariance Combination Combination of model conditioned estimates and covariance matrices is carried out using the following equations
A
j
dh1 (qmaxV1 k4 sgn(h1 h2 ) | h1 h2 | V4 dt k5 h1 V5 k3 | h2 hv | V3 ) dh2 A (qmaxV2 k4 sgn(h1 h2 ) | h1 h2 | V4 dt k6 h2 V6 k3 | h2 hv | V3
(28)
A
The two-tank hybrid system has four modes that depend on the relation between the water levels in both tanks. For each mode, we have a set of ordinary differential equations and the system transitions between modes are based on the state variables (h1 and h2). A schematic representation of the twotank hybrid system is given in Figure 1.
A
V5
V3
Tank 2
V6
V4
dh1 (qmaxV1 k4 sgn(h1 h2 ) | h1 h2 | V4 dt k5 h1 V5 k3 sgn(h1 h2 ) | h1 h2 | V3 ) dh2 (qmaxV2 k4 sgn(h1 h2 ) | h1 h2 | V4 dt k6 h2 V6 k3 sgn(h1 h2 ) | h1 h2 | V3
Parameter
Value
q max
5e-5
k 3 and k 4
3.6050e-5
4.8960e-5
k 5 and k 6
4.0565e-5
3.5653e-5
0.0176625(m2)
A Fig. 1. Two-tank Hybrid system
V i
h2(t)
1 2 3 4
< hv >= hv < hv >= hv
< hv < hv >= hv > =hv
0
max
1
The measurement noise covariance matrix is assumed as (0.01)2 R 0
Table 1. Discrete modes and their dependence on the continuous state variables h1(t)
min
V i
It should be noted that these transitions are autonomous since they depend on the continuous behaviour of the system as described by the threshold conditions (Refer to Table -1). The model parameters of two-tank hybrid system are reported in Table 2.
Mode (z)
(32)
Table 2. Two-tank hybrid system model parameters
V2
hv
(31)
MODE -4
3. SIMULATION STUDIES
Tank 1
(30)
dh2 qmaxV2 k4 sgn(h1 h2 ) | h1 h2 | V4 k6 h2 V6 dt k3 | h1 hv | V3 )
A
j 1
V1
dh1 (qmaxV1 k4 sgn(h1 h2 ) | h1 h2 | V4 dt k5 h1 V5 k3 | h1 hv | V3 )
MODE -3
r
xˆ (k | k ) xˆ j (k | k ) j (k )
(29)
(0.01)2 0
The initial error covariance matrices for each filter is assumed as 10 0 Pr (0 | 0) ; 0 10
r 1, 2,3, 4
The process noise covariance matrices for each filter is assumed as 0 1e 8 Qr ; 1e 8 0
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8th IFAC Symposium on Advanced Control of Chemical Processes Singapore, July 10-13, 2012
The initial state vector for each filter is assumed as
0.4 h1(m)
ˆ r (0 | 0) 0.15 0.1T r 1, 2,3, 4 X
The Markov chain transition matrix has been chosen as
0.3 0.2 True
Estimated
0.1 500
0.99 0.01/3 0.01/3 0.01/3 0.01/3 0.99 0.01/3 0.01/3 Pij 0.01/3 0.01/3 0.99 0.01/3 0.01/3 0.01/3 0.01/3 0.99
1000
1500
2000 2500 3000 Sampling Instants
3500
4000
4500
5000
h2(m)
0.4
The initial estimate model probability ( (0 | 0) ) is assumed
0.3 0.2 True
T
Estimated
0.1
as (0 | 0) 0.95 0.02 0.02 0.01 . , and associated with UKF are assumed as 1, 0 and 0 respectively. The objective is to generate filtered estimate of the water levels in the tanks. The simulation studies have been carried out in the presence of additive Gaussian state noise and measurement noise. The evolution of true and estimated continuous state variables generated by IMM-EKF and IMM-UKF for variations in V1 and V2 (See figure 8) is shown in Figures 2 and 3 respectively. From Figures 2 and 3, it can be concluded that IMM –EKF and IMM-UKF are able to generate fairly accurate filtered state estimates.
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3500
4000
4500
5000
Fig. 3. Evolution of true and estimated states of Two-tank hybrid system using IMM-UKF (a) Level in Tank1 (b) Level in Tank2
4. CONCLUDING REMARKS In this paper, we have proposed state estimation schemes for two-tank hybrid system using an IMM-EKF algorithm and an IMM-UKF algorithm. The efficacy of the proposed state estimation schemes has been demonstrated by carrying out simulation studies on the two-tank hybrid system. 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 the IMM-UKF is determined to be a good alternative for the IMM-EKF, because, UKF doesn’t approximate the nonlinear function and computation of the Jacobian matrices at each time step is not required.
The mode probability update of IMM-EKF and IMM-UKF are shown in Figures 4 and 5. From Figures 4 and 5, it can be concluded that the IMM-EKF and IMM-UKF transition between modes as designed. It is seen that the performance of IMM-EKF and IMM-UKF are found to be identical for this study case. The evolution of true, measured and estimated continuous state variables generated by IMM-EKF and IMMUKF during the interval (200 k 1500) is shown in Figures 6 and 7 respectively. From Figures 6 and 7, it can be inferred that the proposed state estimation schemes is able to efficiently suppress the measurement noise.
1.5 Mode-1
Mode-2
Mode-3
Mode-4
1
0.3
Mode
h1(m)
0.4 0.5
0.2 True
Estimated
0.1 500
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2000 2500 3000 Sampling Instants
3500
4000
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0
5000
h2(m)
0.4
-0.5
0.3 0.2 True
0
500
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1500
2000 2500 3000 Sampling Instants
3500
4000
4500
5000
Fig. 4. Model Probability Update: IMM-EKF
Estimated
0.1 500
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3500
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5000
ACKNOWLEDGEMENTS
The financial support through the NSERC – MATRIKON – SUNCORE - ICORE Industrial research chair program in CPC at University of Alberta is also gratefully acknowledged. The technical support rendered by Mr. N. Prabhakar, Distinguished Scientist, DRDL and Mr. Vamsi Krishna Reddy, Scientist-D, DRDL Hyderabad is gratefully acknowledged.
Fig. 2. Evolution of true and estimated states of Two-tank hybrid system using IMM-EKF (a) Level in Tank1 (b) Level in Tank2
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Models, in Proceedings of the Workshop on Advances in Machine Learning, Montreal, Canada. 1.5 Mode-1
Mode-2
Mode-3
Wenhui Wang, D.H. Zhou, Zhengxi Li (2006). Robust state estimation and fault diagnosis for uncertain hybrid systems. Nonlinear analysis, Volume 65, pp. 2193-2215.
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0.32
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Fig. 5. Model Probability Update: IMM-UKF
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800 1000 Sampling Instants
1200
1400
Figure 6: Evolution of true, estimated, measured states of Two-tank hybrid system using IMM-EKF (a) Level in Tank1 (b) Level in Tank2 (200 k 1500)
Bemporad A, D. Mignone and M. Morari (1999), Moving horizon estimation for hybrid systems and fault detection. Proceedings of American Control Conference, San Diego, CA, pp. 2471–2475.
0.32
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h1(m)
0.3 0.28 True
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Figure 7: Evolution of true, estimated, measured states of Two-tank hybrid system using IMM-UKF (a) Level in Tank1 (b) Level in Tank2
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0.75
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Prakash, J., Sachin C. Patwardhan and S. L. Shah (2010) State Estimation and Nonlinear Predictive Control of Autonomous Hybrid system using Derivative free State Estimators, Journal of Process Control, 20, pp.787–799.
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0.23 0.22 0.21 0.2
Figure 8: Evolution of V1 and V2 versus sampling Instants
Van der Merwe, R. and E Wan (2003). Sigma-Point Kalman Filters for Probabilistic Inference in Dynamic State-Space 512