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ANN-Based Control Systems
Published in Jitendra R. Raol, Ramakalyan Ayyagari, Control Systems, 2020
Jitendra R. Raol, Ramakalyan Ayyagari
Using the trace identity, we obtain the following () V˙=−eTQe−2eTPBΛεf(x)+2trace{ΔKxT(Γx−1K˙^x+xeTPBsgn(Λ))|Λ|}+2trace{ΔKrT(Γr−1K˙^r+reTPBsgn(Λ))|Λ|}+2trace{ΔΘT(ΓΘ−1Θ˙^+Φ(x)eTPBsgn(Λ))|Λ|}
Shared control of ship autopilots and human pilots for maritime autonomous surface ship in the presence of actuator anomalies
Published in Systems Science & Control Engineering, 2022
Minghao Ruan, Anqing Wang, Dan Wang
The tracking error dynamics has been given in (23). If the control deficiency signal , we construct the following Lyapunov function Taking the time derivative of the Lyapunov function in (24) along the system trajectories (12) and (23), it leads to According to the vector trace identity , which is valid for any two column vectors a and b, one has that Substituting (26) into (25) yields Substituting the adaptation laws (12) into (27) and using Property 1, one has that Consequently, , whenever the trajectory of e is outside the set Thus, one has that the tracking error e is ultimately bounded, and the boundedness of all the adaptive parameters is ensured by the operator(Lewis, 2010).
Riemann–Hilbert approach of the coupled nonisospectral Gross–Pitaevskii system and its multi-component generalization
Published in Applicable Analysis, 2021
Xiaotong Chen, Yi Zhang, Rusuo Ye
The organization of this paper is as follows. In Section 2, with the aid of the zero-curvature formulation and the trace identity, we analyze analytical properties of matrix eigenfunctions for an equivalent spectral problem of the coupled nonisospectral GP system, and then formulate a kind of Riemann–Hilbert problems associated with the newly introduced spectral problem. In Section 3, the N-soliton solutions of the coupled nonisospectral GP systems are calculated from a specific Riemann–Hilbert problem, which possesses the identity jump matrix on the real axis. In Section 4, we generalize the multi-component coupled nonisospectral GP system. Finally, the conclusion will be addressed in Section 5.