Elsevier

Automatica

Volume 93, July 2018, Pages 435-443
Automatica

Brief paper
Set-membership approach and Kalman observer based on zonotopes for discrete-time descriptor systems

https://doi.org/10.1016/j.automatica.2018.03.082Get rights and content

Abstract

This paper proposes a set-membership state estimator and a zonotopic Kalman observer for discrete-time descriptor systems. Both approaches are developed in a set-based context considering system disturbances, measurement noise, and unknown inputs. This set-membership state estimation approach determines the set of consistent states with the model and measurements by constructing a parameterized intersection zonotope. Two methods to minimize the size of this intersection zonotope are provided: one inspired by Kalman filtering and the other based on solving an optimization problem involving a series of linear matrix inequalities. Additionally, we propose a zonotopic Kalman observer for discrete-time descriptor systems. Moreover, the relationship between both approaches is discussed. In particular, it is proved that the zonotopic Kalman observer in the current estimation type is equivalent to the set-membership approach. Finally, a numerical example is used to illustrate and compare the effectiveness of the proposed approaches.

Introduction

In many industrial applications involving distribution or collection networks (as e.g. water and electrical networks), mass and energy balance static equations must hold. A standard model including only the dynamical part (described by ordinary differential/difference equations) is not enough to represent system dynamics subject to static relations among system variables. Such systems, known as descriptor systems (also known as singular, implicit or differential–algebraic systems), are better represented by a set of differential and algebraic equations describing the generalized dynamic and static behaviors. In the literature, descriptor models have been considered to address a large amount of applications, such as water distribution networks (Wang, Puig, & Cembrano, 2017), chemical systems (Biegler, Campbell, & Mehrmann, 2012), electrical circuits Duan (2010), Riaza (2008), aircraft systems (Stevens, Lewis, & Johnson, 2016), biological systems (Zhang, Liu, & Zhang, 2012) as well as economic systems (Zhang, Lam, & Zhang, 1999). For monitoring purposes and for developing control strategies, state estimation is usually required. Some research works on state estimation for discrete-time descriptor systems have been carried out (see as e.g. Hsieh, 2013, Ishihara, Terra, & Bianco, 2010, where system states are estimated by using different versions of Kalman filtering).

Research on set-based state estimation has been quite active for the last decades, e.g. Alamo, Bravo, and Camacho (2005), Combastel (2015b), Jaulin, Kieffer, Didrit, and Walter (2001), Puig, Cuguero, and Quevedo (2001), Raïssi, Ramdani, and Candau (2004) and Thabet, Raïssi, Combastel, Efimov, and Zolghadri (2014) among others. In the literature, set-based state estimation approaches can be classified according to whether they follow a set-membership or an interval observer-based paradigm. A set-membership approach relies on over-bounding the uncertain estimated states considering unknown-but-bounded uncertainties (Schweppe, 1968). An interval observer-based approach bounds the set of estimated states by means of an observer structure in which the gain is designed assuming that uncertainties are modeled in a deterministic way (as e.g. using intervals for bounding them Efimov, Perruquetti, Raïssi, & Zolghadri, 2013) or in a stochastic way (as e.g. using the Kalman filtering Kalman (1960), Kalman and Bucy (1961)). From the application point of view, the set-based approaches are very popular in the fault diagnosis framework, e.g. Puig (2010), Xu, Puig, Ocampo-Martinez, Stoican, and Olaru (2014) and Wang, Wang, Puig, & Cembrano (2017).

Zonotopes are a special class of geometrical sets. The symmetry properties of zonotopes help toreduce the computational load of using them in an iterative way. Worst-case state estimation for dynamical systems using zonotopes is investigated in Puig et al. (2001). A state bounding observer based on zonotopes is introduced in Combastel (2003). The zonotopic observer in combination with Kalman filtering is addressed in Combastel (2015a), Combastel (2015b). Moreover, a set-membership approach based on zonotopes is proposed for dynamical systems in Alamo et al. (2005) and Alamo, Bravo, Redondo, and Camacho (2008).

The main contribution of this paper is to propose a set-membership state estimator and a zonotopic Kalman observer for discrete-time descriptor systems. Basically, three types of system uncertainties are considered: unknown inputs and unknown-but-bounded system disturbances and measurement noise. One limitation for the use of zonotopic approaches in real applications is that some system disturbances are unknown and it may not be possible to bound them in a predefined zonotope as a-prior knowledge. To overcome this problem, two classes of unknown system disturbances are considered: (i) bounded disturbances in a zonotope; (ii) unbounded disturbances, which are considered to be unknown inputs and can be decoupled in the observer design.

For the proposed set-membership approach, the consistent states with measurements are enclosed by a parameterized intersection zonotope. To reduce the size of the intersection zonotope, the FW-radius (Combastel, 2015a) and W-radius (Le, Stoica, Alamo, Camacho, & Dumur, 2013) criteria are considered. The FW-radius criterion is used through the Kalman filtering procedure while the W-radius criterion is taken into account via an optimization problem including linear matrix inequalities (LMIs). For the designed zonotopic Kalman observer, we present the explicit solution of the optimal Kalman gain based on the FW-radius criterion. Moreover, the relationship between the proposed set-membership approach and zonotopic Kalman observer is discussed.

The paper is organized as follows. The problem statement is expressed in Section 2. The set-membership approach for discrete-time descriptor systems is proposed in Section 3 and the zonotopic Kalman observer for discrete-time descriptor systems is designed in Section 4. The relationship between both approaches is discussed in Section 5. A numerical example is provided to illustrate the effectiveness of both approaches and comparison results are also shown in Section 6. Finally, conclusions are presented in Section 7.

An m-order zonotope ZRn (mn) is defined by a hypercube Bm=1,+1m affine projection with the center pRn and the generator matrix HRn×m as Z=p,H=p+Hz,zBm.

Denote the Minkowski sum as  and the linear image product as . The zonotope Z in (1) can also be defined by Z=pHBm. Besides, the following properties hold: p1,H1p2,H2=p1+p2,H1H2,Lp,H=Lp,LH,p,Hp,rs(H),where L is a matrix of appropriate dimension. p,rs(H) is called interval hull of the zonotope Z=p,H and rs(H) returns a diagonal matrix with diagonal elements of rs(H)i,i=j=1m|Hi,j| for i=1,,n.

For Z=p,H, the weighted zonotope reduction operator proposed in Combastel (2015b) is denoted by q,W(H) satisfying the inclusion property p,Hp,q,W(H), where qn specifies the maximum number of columns of q,W(H) and W is a weighting matrix of appropriate dimension.

For XRn×n, we use tr(X)=i=1nXii and rank(X) to denote the trace and the rank of X, and if X is non-singular, we use X1 to denote the inverse matrix of Xvec(X) denotes the vectorization of X. X0 denotes positive definiteness if the scalar xTXx is positive for arbitrary non-zero column vector x of real numbers. Similarly, X0 denotes positive semi-definiteness. If X is symmetric, we use  to denote a symmetric element in X. We use Im to denote an identity matrix of dimension m. For two matrices X and Y, the Kronecker product of these two matrices is denoted by XY.

Let X, A, B and C be matrices of appropriate dimensions. The following matrix calculus regarding the matrix trace holds: XtrAXTB=ATBT,XtrAXBXTC=BXTCA+BTXTATCT.

For HRn×m, with WRn×n and W=WT0, the weighted Frobenius norm of H is defined by HF,W=tr(HTWH) and HF=tr(HTH), obtained with W=In, is the non-weighted Frobenius norm. For hRn, the weighted and non-weighted 2-norms of h are denoted by h2,W=hTWh and h2=hTh obtained with W=In, respectively.

Section snippets

Problem statement

Consider the discrete-time descriptor linear system as Exk+1=Axk+Buk+Dωk+Dddk,yk=Cxk+Fυk,where xRnx denotes the vector of system states, uRnu denotes the vector of known inputs, dRnd denotes the vector of unknown inputs, yRny denotes the vector of measurement outputs, ERnx×nx, ARnx×nx, BRnx×nu, CRny×nx, DRnx×nw, DdRnx×nd and FRny×nv. Besides, the initial state x0 is given in the inclusion zonotope X0=p0,H0, where p0Rnx and H0Rnx×nx are the center and generator matrix of this

Set-membership approach for discrete-time descriptor systems

In this section, we propose a set-membership state estimation approach based on zonotopes for discrete-time descriptor system (4). This approach uses the structure of the parameterized intersection zonotope for implementing the measurement consistency test including unknown inputs. Some preliminary definitions are introduced as follows.

Definition 1 Uncertain State Set

Given the descriptor system (4) with x0p0,H0, ωW, kN, the uncertain state set X̄ is defined by X̄=xRnxExAX̄BuDddDW.

Zonotopic Kalman observer for discrete-time descriptor systems

In this section, we design a zonotopic Kalman observer for the descriptor system (4). Unlike the set-membership approach proposed in Section 3, this zonotopic observer structure is defined based on the Luenberger observer structure.

Relationship between two proposed approaches

Comparing the parameterized intersection zonotope structure proposed in Theorem 1 and the zonotopic observer structure proposed in Theorem 5, the intersection zonotope is formulated by considering the measurement output y+ to implement the system consistency test while the zonotopic observer includes measurement outputs y and y+.

To find the relationship between these two approaches, we also consider a current estimation-type zonotopic observer for the descriptor system (4) only containing the

Illustrative example

To illustrate the proposed state estimation approaches, a discrete-time descriptor system as defined in (4) is considered with E=100010000,A=0.5000.80.95010.51,B=100100,Dd=000.8,C=101110,D=0.10001.50000.6,F=0.5001.5.

The input signal u is given by u=0.5sin(t)+12cos(t), for t0,10π with 100 sampling steps. The system disturbances ω and measurement noise υ are random Gaussian white noise bounded in zonotopes: ωW=0,I3 and υ0,I2, kN. Since E, C and Dd satisfy the rank

Conclusion

In this paper, we have proposed a set-membership state estimation approach and a zonotopic Kalman observer for discrete-time descriptor systems subject to uncertainties and unknown inputs. In the proposed set-membership approach, we provide several methods for finding the correction matrix to characterize the intersection zonotope. In the proposed zonotopic Kalman observer, we propose the optimal Kalman observer gain in prediction estimation-type. Furthermore, we prove that the zonotopic Kalman

Acknowledgments

We would like to thank Prof. Teodoro Alamo for useful discussions about this work. We also acknowledge editors and anonymous reviewers for useful suggestions to help us improve this paper.

Ye Wang received his B.Sc. degree in Engineering from the Northeast Forestry University, Harbin, P.R. China in 2012, and M.Sc. degree in Automatic Control and Robotics from the Technical University of Catalonia (UPC), Barcelona, Spain, in 2014. Currently, he is a Ph.D. candidate at the UPC and a research fellow of Spanish National Research Council (CSIC) at the Institute of Robotics and Industrial Informatics (IRI), CSIC-UPC. He has participated in the European project EFFINET (ref.

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    Ye Wang received his B.Sc. degree in Engineering from the Northeast Forestry University, Harbin, P.R. China in 2012, and M.Sc. degree in Automatic Control and Robotics from the Technical University of Catalonia (UPC), Barcelona, Spain, in 2014. Currently, he is a Ph.D. candidate at the UPC and a research fellow of Spanish National Research Council (CSIC) at the Institute of Robotics and Industrial Informatics (IRI), CSIC-UPC. He has participated in the European project EFFINET (ref. FP7-ICT-2011-8-31855) and the Spanish projects ECOCIS (ref. DPI2013-48243-C2-1-R), HARCRICS (ref. DPI2014-58104-R) and DEOCS (ref. DPI2016-76493-C3-3-R). His current research interests include economic model predictive control, set-theoretic fault diagnosis and fault-tolerant control with application to water systems and smart grids.

    Vicenç Puig received his B.Sc./M.Sc. Degree in Telecommunications Engineering in 1993 and Ph.D. degree in Automatic Control, Vision and Robotics in 1999, both from the Technical University of Catalonia (UPC). He is a full professor at the Automatic Control Department of the UPC and a researcher at the Institute of Robotics and Industrial Informatics (IRI), CSIC-UPC. He is the chair of the Automatic Control Department and the head of the research group on Advanced Control Systems (SAC) at the UPC. He has developed important scientific contributions in the areas of fault diagnosis and fault tolerant control, using interval and linear-parameter-varying models using set-based approaches. He has participated in more than 20 European and national research projects in the last decade. He has also led many private contracts with several companies and has published more than 140 journal articles as well as over 400 contributions in international conference/workshop proceedings. He has supervised over 20 Ph.D. dissertations and over 50 Master’s theses/final projects. He is currently the vice-chair of the IFAC Safeprocess TC Committee 6.4 (2014–2018). He was the general chair of the 3rd IEEE Conference on Control and Fault-Tolerant Systems (SysTol 2016) and is the IPC chair of IFAC Safeprocess 2018.

    Gabriela Cembrano received her M.Sc. and Ph.D. degrees, in Industrial Engineering and Automatic Control, respectively, from the Technical University of Catalonia-BarcelonaTech. She is tenured researcher of the Spanish National Research Council (CSIC) at the Institute of Robotics and Industrial Computing (IRI). Since 2007, she is also a Scientific Advisor of CETaqua Water technology Centre. Her main research area is Control Engineering and its applications to water systems management. She has taken part in numerous fundamental and industrial research projects in this field since 1990. Most recently, she has been Scientific Director of EC project EFFINET Integrated Real-time Monitoring and Control of Drinking Water Networks and she is currently CSIC leader in project LIFE-EFFIDRAIN Efficient Integrated Real-time Control in Urban Drainage and Wastewater Treatment plants for Environmental Protection and in National Research Grant DPI2016-76493-C3-3-R Data-driven Monitoring, Diagnosis and Fault-tolerant Control of Cyber–physical Systems. She has published over 50 journal and conference papers in the field.

    This work was partially supported by the Spanish State Research Agency (AEI) and the European Regional Development Fund (ERFD) through the projects ECOCIS (ref. DPI2013-48243-C2-1-R), DEOCS (ref. DPI2016-76493-C3-3-R) and HARCRICS (ref. DPI2014-58104-R), and the FPI grant (ref. BES-2014-068319). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Brett Ninness under the direction of Editor Torsten Söderström.

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