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Systems, Structure and Control 2012 Part 5

Tham khảo tài liệu 'systems, structure and control 2012 part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Differential Neural Networks Observers development stability analysis and implementation 73 Figure 4. Estimation of x3 t 2 s Figure 5. Estimation of x3 t 20 s 74 Systems Structure and Control Figure 6. Estimation of x4 t 1 s Figure 7. Estimation of x4 t 5 s As it can be seen the projectional DNNO has significantly better quality in state estimation especially in the beginning of the process when negative values and over-estimation have been obtained by a non-projectional DNNO. Differential Neural Networks Observers development stability analysis and implementation 75 6. Conclusion and future work The complete convergence analysis for this class of adaptive observer is presented. Also the boundedness property of the adaptive weights in DNN was proven. Since the projection method leads to discontinuous trajectories in the estimated states a nonstandard Lyapunov - Krasovski functional is applied to derive the upper bound for estimation error in average sense which depends on the noise power output and dynamics disturbances and on an unmodelled dynamic. It is shown that the asymptotic stability is attained when both of these uncertainties are absent. The illustrative example confirms the advantages which the suggested observers have being compared with traditional ones. Appendix proof of Theorem 2 Evidently that ố í -ố í - h Lfìịt - t - h tVI n ti J a 2n t An 1A1n 2n t j K 1 nwiỉ K1f 2 n VII II n II II l 0 Aj if 2 T 1 2 1 2 f tị A- A f f f1ll1 t J Af 1 where s t x t - x t is the state estimation error at time t . Consider the next nonstandard Lyapunov-Krasovskii energetic function V t f i cotrlvv t W t 1 dT t - h t L -I where W tt W t - W. Since the problem under consideration contains uncertainties and external output disturbances we won t demonstrate that the time-derivative of this energetic function is strictly negative. Instead we will use it to obtain an upper bound for the averaged state estimation error. Taking time derivative of Lyapunov-Krasovski function and