×

zbMATH — the first resource for mathematics

Positive state-bounding observer for positive interval continuous-time systems with time delay. (English) Zbl 1274.93039
Summary: This paper is concerned with the problem of positive observer synthesis for positive systems with both interval parameter uncertainties and time delay. Conventional observers may no longer be applicable for such kind of systems due to the positivity constraint on the observers, and they only provide an estimate of the system state in an asymptotic way. A pair of positive observers with state-bounding feature is proposed to estimate the state of positive systems at all times in this paper. A necessary and sufficient condition for the existence of desired observers is first established, and the observer matrices can be obtained easily through the solutions of a set of linear matrix inequalities (LMIs). Then, to reduce the error signal between the system state and its estimates, an iterative LMI algorithm is developed to compute the optimized state-bounding observer matrices. Finally, a numerical example is presented to show the effectiveness and applicability of the theoretical results.

MSC:
93B07 Observability
93C15 Control/observation systems governed by ordinary differential equations
93C05 Linear systems in control theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Farina, Positive Linear Systems: Theory and Applications (2000) · Zbl 0988.93002
[2] Berman, Nonnegative Matrices in the Mathematical Sciences (1994) · Zbl 0815.15016
[3] Kaczorek, Positive 1D and 2D Systems (2002)
[4] Valcher, Controllability and reachability criteria for discrete time positive systems, International Journal of Control 65 (3) pp 511– (1996) · Zbl 0873.93009
[5] Bru, Canonical forms for positive discrete-time linear control systems, Linear Algebra and its Applications 310 (1-3) pp 49– (2000) · Zbl 0968.93016
[6] van den Hof, Positive linear observers for linear compartmental systems, SIAM Journal on Control and Optimization 36 (2) pp 590– (1998) · Zbl 0913.93006
[7] Benvenuti, A tutorial on the positive realization problem, IEEE Transactions on Automatic Control 49 (5) pp 651– (2004) · Zbl 1365.93001
[8] Kaczorek, Practical stability of positive fractional 2D linear systems, Multidimensional Systems and Signal Processing 21 (3) pp 231– (2010) · Zbl 1202.93045
[9] Anderson, Nonnegative realization of a linear system with nonnegative impulse response, IEEE Transactions on Circuits and Systems (I) 43 (2) pp 134– (1996)
[10] Benvenuti, Minimal positive realizations of transfer functions with positive real poles, IEEE Transactions on Circuits and Systems (I) 47 (9) pp 1370– (2000) · Zbl 1030.93009
[11] Benvenuti, Filtering through combination of positive filters, IEEE Transactions on Circuits and Systems (I) 46 (12) pp 1431– (1999)
[12] Germani, Representation of a class of MIMO systems via internally positive realization, European Journal of Control 16 (3) pp 291– (2010) · Zbl 1198.93059
[13] Boyd, Linear Matrix Inequalities in System and Control Theory (1994) · Zbl 0816.93004
[14] Ait Rami, Controller synthesis for positive linear systems with bounded controls, IEEE Transactions on Circuits and Systems (II) 54 (2) pp 151– (2007)
[15] Luenberger, An introduction to observers, IEEE Transactions on Automatic Control 16 (6) pp 596– (1971)
[16] Dautrebande N Bastin G Positive linear observers for positive linear systems
[17] Ait Rami M Cheng CH de Prada C Tight robust interval observers: an LP approach 2967 2972
[18] Ait Rami M Helmke U Tadeo F Positive observation problem for linear time-delay positive systems · Zbl 1245.93026
[19] Ait Rami M Helmke U Tadeo F Positive observation problem for linear time-lag positive systems · Zbl 1245.93026
[20] Back, Design of positive linear observers for positive linear systems via coordinate transformations and positive realizations, SIAM Journal on Control and Optimization 47 (1) pp 345– (2008) · Zbl 1157.93009
[21] Shu, Positive observers and dynamic output-feedback controllers for interval positive linear systems, IEEE Transactions on Circuits and Systems (I) 55 (10) pp 3209– (2008)
[22] Gao, New results on stability of discrete-time systems with time-varying state delay, IEEE Transactions on Automatic Control 52 (2) pp 328– (2007) · Zbl 1366.39011
[23] He, Augmented Lyapunov functional and delay-dependent stability criteria for neutral systems, International Journal of Robust and Nonlinear Control 15 (18) pp 923– (2005) · Zbl 1124.34049
[24] He, Delay-range-dependent stability for systems with time-varying delay, Automatica 43 (2) pp 371– (2007) · Zbl 1111.93073
[25] Xu, Further results on delay-dependent robust stability conditions of uncertain neutral systems, International Journal of Robust and Nonlinear Control 15 (5) pp 233– (2005) · Zbl 1078.93055
[26] Xu, A survey of linear matrix inequality techniques in stability analysis of delay systems, International Journal of Systems Science 39 (12) pp 1095– (2008) · Zbl 1156.93382
[27] Wang, Robust H observer design of linear state delayed systems with parametric uncertainty: the discrete-time case, Automatica 35 (6) pp 1161– (1999)
[28] Wang, Robust H observer design of linear time-delay systems with parametric uncertainty, Systems and Control Letters 42 (4) pp 303– (2001)
[29] Gao, Delay-dependent robust H and L2-L filtering for a class of uncertain nonlinear time-delay systems, IEEE Transactions on Automatic Control 48 (9) pp 1661– (2003)
[30] Lewis, Insensitivity of a class of nonlinear compartmental systems to the introduction of arbitrary time delays, IEEE Transactions on Circuits and Systems 27 (7) pp 604– (1980) · Zbl 0519.93026
[31] Haddad, Stability theory for nonnegative and compartmental dynamical systems with time delay, Systems and Control Letters 51 (5) pp 355– (2004) · Zbl 1157.34352
[32] Liu, Stability analysis of positive systems with bounded time-varying delays, IEEE Transactions on Circuits and Systems (II) 56 (7) pp 600– (2009)
[33] Liu, Constrained control of positive systems with delays, IEEE Transactions on Automatic Control 54 (7) pp 1596– (2009) · Zbl 1367.93280
[34] Lemesle, Hybrid bounded error observers for uncertain bioreactor models, Bioprocess and Biosystems Engineering 27 (5) pp 311– (2005)
[35] Moisan, Near optimal interval observers bundle for uncertain bioreactors, Automatica 45 (1) pp 291– (2009) · Zbl 1154.93321
[36] Jacquez, Qualitative theory of compartmental systems, SIAM Review 35 (1) pp 43– (1993) · Zbl 0776.92001
[37] El Ghaoui, Advances in Linear Matrix Inequality Methods in Control (2000)
[38] Cao, Static output feedback stabilization: an ILMI approach, Automatica 34 (12) pp 1641– (1998) · Zbl 0937.93039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.