\(H_ \infty\) estimation for discrete-time linear uncertain systems. (English) Zbl 0754.93050
Given a prescribed level of noise attenuation \(\gamma>0\), the authors show sufficient conditions for designing an estimator \(\{x_ e,z_ e\}\) of the system with uncertainties
\[
\Delta: x_{k+1}=(A+\Delta_ 1(k))x_ k+Bw_ k, y_ k=(C+\Delta_ 2(k))x_ k+Dw_ k, z_ k=Lx_ k,
\]
\[ [\Delta_ 1,\Delta_ 2]^ T=[H_ 1,H_ 2]^ T F(k), F^ T(k)F(k)=I, \] such that the augmented of the system and the estimator is quadratically stable and \(\|{\mathcal G}\|<\gamma\) under zero initial conditions for \(x(k)\) and \(x_ e(k)\), where \(\|{\mathcal G}\|\) is the induced operator norm of the mapping \(\mathcal G\) from the noise, \(w\), to the estimation error, \(e=\{z(k)-z_ e(k)\}\), and the inequality holds for all the mentioned \(F(k)\). In the case when \(A\) is stable and invertible, \((C,A)\) is detectable and \(DD^ T>0\), a solution to such a robust \(H_ \infty\) estimation problem has been given in terms of two algebraic Riccati equations. It is similar to the authors’ results for continuous-time case [\(H_ \infty\) estimation for continuous-time linear uncertain systems, Proc. IFAC symp. on design methods of control systems (1991, to appear)]. The Riccati equation approach allows them to solve a \(H_ \infty\) estimation problem associated with a linear certain system with the parameters \(\gamma/\varepsilon\cdot H_ 1\), \(\gamma/\varepsilon\cdot H_ 2\), \(\varepsilon>0\), and then the main result is obtained by converting the robust \(H_ \infty\) estimation problem to solving a discrete-time co-spectral factorization with a suitable state-space realization and the standard \(H_ \infty\) estimation problem.
\[ [\Delta_ 1,\Delta_ 2]^ T=[H_ 1,H_ 2]^ T F(k), F^ T(k)F(k)=I, \] such that the augmented of the system and the estimator is quadratically stable and \(\|{\mathcal G}\|<\gamma\) under zero initial conditions for \(x(k)\) and \(x_ e(k)\), where \(\|{\mathcal G}\|\) is the induced operator norm of the mapping \(\mathcal G\) from the noise, \(w\), to the estimation error, \(e=\{z(k)-z_ e(k)\}\), and the inequality holds for all the mentioned \(F(k)\). In the case when \(A\) is stable and invertible, \((C,A)\) is detectable and \(DD^ T>0\), a solution to such a robust \(H_ \infty\) estimation problem has been given in terms of two algebraic Riccati equations. It is similar to the authors’ results for continuous-time case [\(H_ \infty\) estimation for continuous-time linear uncertain systems, Proc. IFAC symp. on design methods of control systems (1991, to appear)]. The Riccati equation approach allows them to solve a \(H_ \infty\) estimation problem associated with a linear certain system with the parameters \(\gamma/\varepsilon\cdot H_ 1\), \(\gamma/\varepsilon\cdot H_ 2\), \(\varepsilon>0\), and then the main result is obtained by converting the robust \(H_ \infty\) estimation problem to solving a discrete-time co-spectral factorization with a suitable state-space realization and the standard \(H_ \infty\) estimation problem.
Reviewer: T.N.Pham (Hanoi)
MSC:
| 93C55 | Discrete-time control/observation systems |
| 93E20 | Optimal stochastic control |
| 93E11 | Filtering in stochastic control theory |
| 93B35 | Sensitivity (robustness) |
Keywords:
uncertain discrete-time system; robust estimation; algebraic Riccati equations; \(H_ \infty\) estimationReferences:
| [1] | and , Optimal Filtering, Prentice Hall, Englewood Cliffs, NJ, (1979). |
| [2] | Barmish, J. Optimiz. Theory Appl. 46 pp 399– (1985) |
| [3] | Bernstein, Systems & Control Lett. 12 pp 9– (1989) |
| [4] | and , On the discrete-time bounded lemma with application in characterization of static state feedback H controllers, Tech. Report EE9097, Dept. Elec. & Comp. Eng., University of Newcastle, NSW, Australia, Oct. 1990, submitted for publication. |
| [5] | Doyle, IEEE Trans. Automat. Control AC-39 pp 831– (1989) |
| [6] | , and , On H filtering in multivariable discrete-time systems, Proc. IEEE Int. Symp. on Circuits and Systems, Helsinki, June (1988). |
| [7] | A course in H Control Theory, Springer-Verlag, New York (1987). · doi:10.1007/BFb0007371 |
| [8] | Interpolation approach to H optimal estimation, Systems & Control Lett., to appear. |
| [9] | , and , Quadratic stabilization and H control of discrete-time uncertain systems, Proc. the International Symp. Math. Theory of Networks and Systems, Kobe, Japan, June (1991). |
| [10] | , and , H estimation for continuous-time linear uncertain systems, Proc. IFAC Symp. on Design Methods of Control Systems, Zurich, Sept. 1991, to appear. |
| [11] | and , An algebraic approach to discrete-time H control problems, Proc. 1990 American Control Conf., San Diego, CA, May (1990). |
| [12] | H design of optimal linear filters, in Linear Circ., Syst. and Signal Processing: Theory and Application, and (Eds.), North-Holland, Amsterdam, pp. 533–540 (1988). |
| [13] | Grimble, IEEE Trans. Acoust. Speech ASSP-38 pp 1092– (1990) |
| [14] | , and , Discrete time H control, Proc. 28th IEEE Conf. Decision and Control, Tampa, FL, Dec. (1989). |
| [15] | Nagpal, IEEE Trans. Automat. Control AC-36 pp 152– (1991) |
| [16] | Shaked, IEEE Trans. Automat. Control AC-35 pp 554– (1990) |
| [17] | and , H state estimation for linear periodic systems, Proc. 29th IEEE Conf. Decision and Control, Honolulu, Hawaii, Dec. (1990). |
| [18] | and , Game theory approach to optimal linear estimation in the minimum H-norm sense, Proc. 28th IEEE Conf. Decision and Control, Tampa, FL, Dec. (1989). |
| [19] | and , A transfer function approach to the problems of discrete-time systems H optimal linear control and filtering, Tech. Report, Dept. Electronic Systems, Tel-Aviv University, 1990. |
| [20] | and , Minimum H-norm regulation of linear discrete-time systems and its relation to linear quadratic discrete games, Proc. 28th IEEE Conf. Decision and Control, Tampa, FL, Dec. (1989). |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
