Skip to main content
SpringerLink
Log in
Book cover

Nonlinear Filtering pp 29–57Cite as

Kalman Filter and Linear State Estimations

  • Chapter
  • First Online:

  • 779 Accesses

Abstract

As discussed in the earlier chapters, the system state contains vital information for control system analysis and design, but it is not always directly measurable. Control engineers will have to obtain the information of system states based on accessible information of the known system dynamics, inputs and outputs (measurements).

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    The Kalman’s original derivation did not use the Bayes’ rule and does not require the exploitation of any specific error distribution information. The Kalman filter is the minimum variance estimator if the noise is Gaussian, and it is the linear minimum variance estimator for linear systems with non-Gaussian noises (Kalman 1960; Simon 2006; Julier and Uhlmann 2004; Van Der Merwe 2004).

  2. 2.
    $$\begin{aligned} (\mathbf P _{k|k-1}\mathbf H _k^T\mathbf K _k^T)^T= & {} \mathbf K _k\mathbf H _k\mathbf P _{k|k-1} \\ Tr(\mathbf P _{k|k-1}\mathbf H _k^T\mathbf K _k^T)^T= & {} Tr(\mathbf K _k\mathbf H _k\mathbf P _{k|k-1}) \end{aligned}$$
  3. 3.

    For any matrix, M, and a symmetric matrix, N,

    $$\begin{aligned} \frac{\partial Tr({MNM^T})}{\partial M}= 2MN. \end{aligned}$$
  4. 4.

    \(\mathcal{H}_{\infty }\) filters minimises the worst-case energy gain from the noise input to the estimation error, which is equivalent to minimising the corresponding \(\mathcal{H}_{\infty }\) norm.

References

  • Banavar RN, Speyer JL (1991) A linear-quadratic game approach to estimation and smoothing. In: 1991 American control conference, pp 2818–2822. IEEE

    Google Scholar 

  • Bandyopadhyay B, Fulwani D (2009) High-performance tracking controller for discrete plant using nonlinear sliding surface. IEEE Trans Ind Electron 56(9):3628–3637

    Article  Google Scholar 

  • Bandyopadhyay B, Deepak F, Kim K-S (2009) Sliding mode control using novel sliding surfaces, vol 392. Springer, Berlin

    Google Scholar 

  • Basar T (1991) Optimum performance levels for minimax filters, predictors and smoothers. Syst Control Lett 16(5):309–317

    Article  MathSciNet  Google Scholar 

  • Biswas PP, Srivastava R, Ray S, Samanta AN (2009) Sliding mode control of quadruple tank process. Mechatronics 19(4):548–561

    Article  Google Scholar 

  • Chandra KPB (2013) Nonlinear state estimation algorithms and their applications. PhD thesis, University of Leicester

    Google Scholar 

  • Cheng G, Peng K (2007) Robust composite nonlinear feedback control with application to a servo positioning system. IEEE Trans Ind Electron 54(2):1132–1140

    Article  Google Scholar 

  • Csorba M (1998) Simultaneous localisation and map building. PhD thesis, University of Oxford

    Google Scholar 

  • Durrant-Whyte H (2001) Introduction to estimation and the kalman filter. Aust Cent Field Robot 28(3):65–94

    Google Scholar 

  • Einicke GA, White LB (1999) Robust extended kalman filtering. IEEE Trans Signal Process 47(9):2596–2599

    Article  Google Scholar 

  • Eustice RM, Singh H, Leonard JJ, Walter MR (2006) Visually mapping the RMS Titanic: conservative covariance estimates for slam information filters. Int J Robot Res 25(12):1223–1242

    Article  Google Scholar 

  • Grimble MJ, El Sayed A (1990) Solution of the \(h_\infty \) optimal linear filtering problem for discrete-time systems. IEEE Trans Acoust Speech Signal Process 38(7):1092–1104

    Google Scholar 

  • Imtiaz SA, Roy K, Huang B, Shah SL, Jampana P (2006) Estimation of states of nonlinear systems using a particle filter. In: 2006 IEEE international conference on industrial technology, ICIT 2006, pp 2432–2437. IEEE

    Google Scholar 

  • Johansson KH (2000) The quadruple-tank process: a multivariable laboratory process with an adjustable zero. IEEE Trans Control Syst Technol 8(3):456–465

    Article  Google Scholar 

  • Johnsen JK, Allöwer F (2007) Interconnection and damping assignment passivity-based control of a four-tank system. In: Lagrangian and hamiltonian methods for nonlinear control 2006, pp 111–122. Springer, Berlin

    Google Scholar 

  • Julier SJ, Uhlmann JK (2004) Unscented filtering and nonlinear estimation. Proc IEEE 92(3):401–422

    Article  Google Scholar 

  • Kailath T (1968) An innovations approach to least-squares estimation-part i: linear filtering in additive white noise. IEEE Trans Autom Control 13(6):646–655

    Article  Google Scholar 

  • Kailath T, Sayed AH, Hassibi B (2000) Linear estimation, vol 1. Prentice Hall, Upper Saddle River

    MATH  Google Scholar 

  • Kalman RE (1960) A new approach to linear filtering and prediction problems. J Basic Eng 82(1):35–45

    Article  Google Scholar 

  • Kolmogorov AN, Doyle WL, Selin I (1962) Interpolation and extrapolation of stationary random sequences. Bulletin of academic sciences, Mathematics series, USSR, vol 5, 1941. Translation by W. Doyle and J. Selin, RM-3090-PR, Rand Corporation, Santa Monica, California

    Google Scholar 

  • Mercangöz M, Doyle FJ (2007) Distributed model predictive control of an experimental four-tank system. J Process Control 17(3):297–308

    Article  Google Scholar 

  • Mutambara AG (1998) Decentralized estimation and control for multisensor systems. CRC press, Boca Raton

    MATH  Google Scholar 

  • Napgal K, Khargonekar P (1991) Filtering and smoothing in an h-infinity setting. IEEE Trans Autom Control 36:152–166

    Article  Google Scholar 

  • Okpanachi AV (2010) Developing advanced control strategies for a 4-tank laboratory process. Master’s thesis report

    Google Scholar 

  • Park P, Kailath T (1995) New square-root algorithms for kalman filtering. IEEE Trans Autom Control 40(5):895–899

    Article  MathSciNet  Google Scholar 

  • Raff T, Huber S, Nagy ZK, Allgower F (2006) Nonlinear model predictive control of a four tank system: an experimental stability study. In: 2006 IEEE international conference on control applications, CCA’06, pp 237–242. IEEE

    Google Scholar 

  • Said SH, M’Sahli F (2008) A set of observers design to a quadruple tank process. In: 2008 IEEE international conference on control applications, CCA 2008, pp 954–959. IEEE

    Google Scholar 

  • Shaked U (1990) \(h_\infty \)-minimum error state estimation of linear stationary processes. IEEE Trans Autom Control 35(5):554–558

    Article  MathSciNet  Google Scholar 

  • Shen X-M, Deng L (1997) Game theory approach to discrete \(h_\infty \) filter design. IEEE Trans Signal Process 45(4):1092–1095

    Article  Google Scholar 

  • Simon D (2006) Optimal state estimation: Kalman, H infinity, and nonlinear approaches. Wiley, New York

    Book  Google Scholar 

  • Utkin V, Guldner J, Shi J (2009) Sliding mode control in electro-mechanical systems, vol 34. CRC press, Boca Raton

    Book  Google Scholar 

  • Vadigepalli R, Gatzke EP, Doyle FJ (2001) Robust control of a multivariable experimental four-tank system. Ind Eng Chem Res 40(8):1916–1927

    Article  Google Scholar 

  • Van Der Merwe R (2004) Sigma-point kalman filters for probabilistic inference in dynamic state-space models

    Google Scholar 

  • Wiener N (1949) Extrapolation, interpolation, and smoothing of stationary time series, vol 7. MIT press, Cambridge

    MATH  Google Scholar 

  • Yaesh I, Shaked U (1992) Game theory approach to optimal linear state estimation and its relation to the minimum \(h_\infty \) norm estimation. IEEE Trans Autom Control 37(6):828–831

    Article  MathSciNet  Google Scholar 

  • Yang F, Wang Z, Lauria S, Liu X (2009) Mobile robot localization using robust extended h8 filtering. Proc Inst Mech Eng Part I J Syst Control Eng 223(8):1067–1080

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. GMR Institute of Technology, Rajam, Andhra Pradesh, India

    Kumar Pakki Bharani Chandra

  2. Department of Engineering, University of Leicester, Leicester, UK

    Da-Wei Gu

Authors
  1. Kumar Pakki Bharani Chandra
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Da-Wei Gu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Da-Wei Gu .

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Chandra, K.P.B., Gu, DW. (2019). Kalman Filter and Linear State Estimations. In: Nonlinear Filtering. Springer, Cham. https://doi.org/10.1007/978-3-030-01797-2_3

Download citation

Publish with us

Policies and ethics

Search

Navigation