# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Reduced-order $H_{\infty}$ filtering for singular systems. (English) Zbl 1120.93321
Summary: This paper solves the problem of reduced-order $H_{\infty }$ filtering for singular systems. The purpose is to design linear filters with a specified order lower than the given system such that the filtering error dynamic system is regular, impulse-free (or causal), stable, and satisfies a prescribed $H_{\infty }$ performance level. One major contribution of the present work is that necessary and sufficient conditions for the solvability of this problem are obtained for both continuous and discrete singular systems. These conditions are characterized in terms of linear matrix inequalities (LMIs) and a coupling non-convex rank constraint. Moreover, an explicit parametrization of all desired reduced-order filters is presented when these inequalities are feasible. In particular, when a static or zeroth-order $H_{\infty }$ filter is desired, it is shown that the $H_{\infty }$ filtering problem reduces to a convex LMI problem. All these results are expressed in terms of the original system matrices without decomposition, which makes the design procedure simple and directly. Last but not least, the results have generalized previous works on $H_{\infty }$ filtering for state-space systems. An illustrative example is given to demonstrate the effectiveness of the proposed approach.

##### MSC:
 93B51 Design techniques in systems theory 93C15 Control systems governed by ODE 93C41 Control problems with incomplete information 93C05 Linear control systems
Full Text:
##### References:
 [1] Anderson, B. D. O.; Moore, J. B.: Optimal filtering. (1979) · Zbl 0688.93058 [2] Bassong-Onana, A.; Darouach, M.: Optimal filtering for singular systems using orthogonal transformations. Control theory adv. Tech. 8, 731-742 (1992) [3] Bernstein, D. S.; Hyland, D. C.: The optimal projection equations for reduced-order state estimation. IEEE trans. Automat. control 30, 583-585 (1985) · Zbl 0555.93056 [4] Bettayeb, M.; Kavrano&gbreve, D.; Lu: Reduced order H$\infty$ filtering. Proceedings of the American control conference, 1884-1888 (June, 1994) [5] Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V.: Linear matrix inequalities in system and control theory. SIAM studies in applied mathematics. (1994) · Zbl 0816.93004 [6] Brown, R. G.; Hwang, P. Y. C.: Introduction to random signals and applied Kalman filtering. (1992) · Zbl 0759.93074 [7] Chung, R. C.; Bélanger, P. R.: Minimum-sensitivity filter for linear time-invariant stochastic systems with uncertain parameters. IEEE trans. Automat. control 21, 98-104 (1976) [8] Dai, L.: Singular control systems. (1989) · Zbl 0669.93034 [9] Darouach, M.; Zasadzinski, M.; Hayar, M.: Reduced-order observer design for descriptor systems with unknown inputs. IEEE trans. Automat. control 41, 1068-1072 (1996) · Zbl 0857.93016 [10] De Souza, C. E.; Palhares, R. M.; Peres, P. L. D.: Robust H$\infty$ filter design for uncertain linear systems with multiple time-varying state delays. IEEE trans. Signal process. 49, 569-576 (2001) [11] Fu, M.: Interpolation approach to H$\infty$ optimal estimation and its interconnection to loop transfer recovery. Systems and control lett. 17, 29-36 (1991) · Zbl 0733.93018 [12] Gahinet, P.; Apkarian, P.: A linear matrix inequality approach to H$\infty$ control. Int. J. Robust and nonlinear control 4, 421-448 (1994) · Zbl 0808.93024 [13] Gershon, E.; Limebeer, D. J. N.; Shaked, U.; Yaesh, I.: Robust H$\infty$ filtering of stationary continuous-time linear systems with stochastic uncertainties. IEEE trans. Automat. control 46, 1788-1793 (2001) · Zbl 1016.93067 [14] Grigoriadis, K. M.: Optimal H$\infty$ model reduction via linear matrix inequalities: continuous- and discrete-time cases. Systems and control lett. 26, 321-333 (1995) · Zbl 0877.93017 [15] Grigoriadis, K. M.; Watson, J. T.: Reduced-order H$\infty$ and L2-L$\infty$ filtering via linear matrix inequalities. IEEE trans. Aerospace electron systems 33, 1326-1338 (1997) [16] Iwasaki, T.; Skelton, R. E.: All controllers for the general H$\infty$ control problems: LMI existence conditions and state space formulas. Automatica 30, 1307-1317 (1994) · Zbl 0806.93017 [17] Lewis, F. L.: A survey of linear singular systems. Circuits, syst. Signal process. 5, 3-36 (1986) · Zbl 0613.93029 [18] Masubuchi, I.; Kamitane, Y.; Ohara, A.; Suda, N.: H$\infty$ control for descriptor systems: a matrix inequalities approach. Automatica 33, 669-673 (1997) · Zbl 0881.93024 [19] Nagpal, K. M.; Khargonekar, P. P.: Filtering and smoothing in an H$\infty$ setting. IEEE trans. Automat. control 36, 152-166 (1991) · Zbl 0758.93074 [20] Park, P. G.; Kailath, T.: H$\infty$ filtering via convex optimization. Internet J. Control 66, 15-22 (1997) · Zbl 0870.93041 [21] Rawson, J. L.; Hsu, C. S.; Rho, H.: Reduced order H$\infty$ filters for discrete linear systems. Proceedings of the 36th IEEE conference on decision and control, 3311-3316 (1997) [22] Shaked, U.: H$\infty$ minimum error state estimation of linear stationary processes. IEEE trans. Automat. control 35, 554-558 (1990) · Zbl 0706.93063 [23] Shaked, U.; Theodor, Y.: H$\infty$-optimal estimation: a tutorial. Proceedings of the 31st IEEE conference on decision and control, 2278-2286 (December, 1992) [24] Syrmos, V. L.: Observer design for descriptor systems with unmeasurable disturbances. Proceedings of 31st IEEE conference on decision and control, 981-982 (1992) [25] Xu, S.; Chen, T.: Reduced-order H$\infty$ filtering for stochastic systems. IEEE trans. Signal process. 50, 2998-3007 (2002) [26] Xu, S.; Lam, J.; Zhang, L.: Robust D-stability analysis for uncertain discrete singular systems with state delay. IEEE trans. Circuits syst. I 49, 551-555 (2002) [27] Xu, S.; Van Dooren, P.; Stefan, R.; Lam, J.: Robust stability and stabilization for singular systems with state delay and parameter uncertainty. IEEE trans. Automat. control 47, 1122-1128 (2002) [28] Yaesh, I.; Shaked, U.: Game theory approach to optimal linear state estimation and its relation to the minimum H$\infty$-norm estimation. IEEE trans. Automat. control 37, 828-831 (1992) · Zbl 0769.90087 [29] Yan, W. -Y.; Xie, L.: Optimal filter reduction via a double projection. IEEE trans. Signal process. 48, 896-900 (2000) · Zbl 1038.93011 [30] Zhang, H.; Xie, L.; Soh, Y. C.: Optimal recursive filtering, prediction, and smoothing for singular stochastic discrete-time systems. IEEE trans. Automat. control 44, 2154-2158 (1999) · Zbl 1136.93451 [31] Zhang, L.; Lam, J.; Xu, S.: On positive realness of descriptor systems. IEEE trans. Circuits syst. I 49, 401-407 (2002)