Some alternative derivations of Craig’s formula. (English) Zbl 1507.33002
Summary: In the performance of digital communication systems over fading channels, the error analysis is typically modelled using a Gaussian probability distribution. One function central to the analysis is what engineers routinely refer to as the (Gaussian) \(Q\)-function and is defined by \[Q(x)=\frac{1}{\sqrt{2\pi}}\int_x^\infty\exp(-\frac{u^2}{2})du.\] This is the canonical representation used for the function. In this paper a number of derivations of an important alternative representation for the \(Q\)-function known as Craig’s formula will be given.
MSC:
| 33B20 | Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals) |
References:
| [1] | Abid, H.; Toumi, A., Adaptive fuzzy sliding mode controller for a class of SISO nonlinear time-delay systems, Soft Comput, 20, 2, 649-659 (2016) · Zbl 1369.93326 · doi:10.1007/s00500-014-1529-9 |
| [2] | Asadinia, MS; Binazadeh, T., Finite-time stabilization of descriptor time-delay systems with one-sided Lipschitz nonlinearities: application to partial element equivalent circuit, Circuits Syst Signal Process, 38, 12, 5467-5487 (2019) · doi:10.1007/s00034-019-01129-7 |
| [3] | Bakefayat, AS; Tabrizi, MM, Lyapunov stabilization of the nonlinear control systems via the neural networks, Appl Soft Comput, 42, 459-471 (2016) · doi:10.1016/j.asoc.2015.07.012 |
| [4] | Baleghi, N.; Shafiei, M., Stability analysis and stabilization of a class of discrete-time nonlinear switched systems with time-delay and affine parametric uncertainty, J Vib Control, 25, 7, 1326-1340 (2019) · doi:10.1177/1077546318819737 |
| [5] | Binazadeh, T.; Bahmani, M., Design of robust controller for a class of uncertain discrete-time systems subject to actuator saturation, IEEE Trans Autom Control, 62, 3, 1505-1510 (2017) · Zbl 1366.93324 · doi:10.1109/TAC.2016.2580662 |
| [6] | Binazadeh, T.; Yousefi, M., Designing a cascade-control structure using fractional-order controllers: time-delay fractional-order proportional-derivative controller and fractional-order sliding-mode controller, J Eng Mech, 143, 7, 04017037 (2017) · doi:10.1061/(ASCE)EM.1943-7889.0001234 |
| [7] | Chen, M.; Sun, J., H∞ finite time control for discrete time-varying system with interval time-varying delay, J Frankl Inst, 355, 12, 5037-5057 (2018) · Zbl 1395.93343 · doi:10.1016/j.jfranklin.2018.05.031 |
| [8] | Dastaviz, A.; Binazadeh, T., Simultaneous stabilization of a collection of uncertain discrete-time systems with time-varying state-delay via discrete-time sliding mode control, J Vib Control, 25, 16, 2261-2273 (2019) · doi:10.1177/1077546319852485 |
| [9] | Dastaviz, A.; Binazadeh, T., Simultaneous stabilization for a collection of uncertain time-delay systems using sliding-mode output feedback control, Int J Control, 93, 9, 2135-2144 (2020) · Zbl 1453.93188 · doi:10.1080/00207179.2018.1543897 |
| [10] | Dower, PM; Zhang, H., A max-plus primal space fundamental solution for a class of differential Riccati equations, Math Control Signals Syst, 29, 3, 15 (2017) · Zbl 1373.93153 · doi:10.1007/s00498-017-0200-2 |
| [11] | Fei, Z.; Guan, C.; Li, Z.; Xu, Y., On finite frequency H∞ performance for discrete linear time delay systems, Int J Syst Sci, 48, 7, 1548-1555 (2017) · Zbl 1362.93047 · doi:10.1080/00207721.2016.1271917 |
| [12] | Fridman, E., Introduction to time-delay systems: analysis and control (2014), Berlin: Springer, Berlin · Zbl 1303.93005 · doi:10.1007/978-3-319-09393-2 |
| [13] | Gholami, H.; Binazadeh, T., Robust finite-time H∞ controller design for uncertain one-sided Lipschitz systems with time-delay and input amplitude constraints, Circuits Syst Signal Process, 38, 7, 1-21 (2019) · doi:10.1007/s00034-018-01018-5 |
| [14] | Gholami, H.; Binazadeh, T., Sliding-mode observer design and finite-time control of one-sided Lipschitz nonlinear systems with time-delay, Soft Comput, 23, 15, 6429-6440 (2019) · Zbl 1418.93042 · doi:10.1007/s00500-018-3297-4 |
| [15] | Gholami, H.; Shafiei, MH, Finite-time boundedness and stabilization of switched nonlinear systems using auxiliary matrices and average dwell time method, Trans Inst Meas Control, 42, 7, 1406-1416 (2020) · doi:10.1177/0142331219891609 |
| [16] | Jafari E, Binazadeh T (2020) Low-conservative robust composite nonlinear feedback control for singular time-delay systems. J Vib Control 1077546320953736 |
| [17] | Kang, W.; Zhong, S.; Shi, K.; Cheng, J., Finite-time stability for discrete-time system with time-varying delay and nonlinear perturbations, ISA Trans, 60, 67-73 (2016) · doi:10.1016/j.isatra.2015.11.006 |
| [18] | Khalil, HK, Nonlinear control (2014), Hoboken: Prentice-Hall, Hoboken |
| [19] | Kumar, R.; Srivastava, S.; Gupta, J., Lyapunov stability-based control and identification of nonlinear dynamical systems using adaptive dynamic programming, Soft Comput, 21, 15, 4465-4480 (2017) · Zbl 1387.93081 · doi:10.1007/s00500-017-2500-3 |
| [20] | Li, X.; Yang, X., Lyapunov stability analysis for nonlinear systems with state-dependent state delay, Automatica, 112, 108674 (2020) · Zbl 1430.93159 · doi:10.1016/j.automatica.2019.108674 |
| [21] | Liu, F.; Zhang, B.; Yang, Z., Lyapunov stability and numerical analysis of excursive instability for forced two-phase boiling flow in a horizontal channel, Appl Therm Eng, 159, 113664 (2019) · doi:10.1016/j.applthermaleng.2019.04.074 |
| [22] | Liu, H.; Shi, P.; Karimi, HR; Chadli, M., Finite-time stability and stabilisation for a class of nonlinear systems with time-varying delay, Int J Syst Sci, 47, 6, 1433-1444 (2016) · Zbl 1333.93157 · doi:10.1080/00207721.2014.932467 |
| [23] | Ma, Y.; Fu, L.; Jing, Y.; Zhang, Q., Finite-time H∞ control for a class of discrete-time switched singular time-delay systems subject to actuator saturation, Appl Math Comput, 261, 264-283 (2015) · Zbl 1410.93074 |
| [24] | Mazenc, F.; Malisoff, M.; Bhogaraju, INS, Sequential predictors for delay compensation for discrete time systems with time-varying delays, Automatica, 122, 109188 (2020) · Zbl 1451.93295 · doi:10.1016/j.automatica.2020.109188 |
| [25] | Ning, B.; Jin, J.; Zheng, J., Fixed-time consensus for multi-agent systems with discontinuous inherent dynamics over switching topology, Int J Syst Sci, 48, 10, 2023-2032 (2017) · Zbl 1371.93014 · doi:10.1080/00207721.2017.1308579 |
| [26] | Song, J.; He, S., Finite-time robust passive control for a class of uncertain Lipschitz nonlinear systems with time-delays, Neurocomputing, 159, 275-281 (2015) · doi:10.1016/j.neucom.2015.01.038 |
| [27] | Song, J.; He, S., Robust finite-time H∞ control for one-sided Lipschitz nonlinear systems via state feedback and output feedback, J Frankl Inst, 352, 8, 3250-3266 (2015) · Zbl 1395.93128 · doi:10.1016/j.jfranklin.2014.12.010 |
| [28] | Song, R.; Xiao, W.; Wei, Q.; Sun, C., Neural-network-based approach to finite-time optimal control for a class of unknown nonlinear systems, Soft Comput, 18, 8, 1645-1653 (2014) · Zbl 1326.49041 · doi:10.1007/s00500-013-1170-z |
| [29] | Stojanovic, SB, Robust finite-time stability of discrete time systems with interval time-varying delay and nonlinear perturbations, J Frankl Inst, 354, 11, 4549-4572 (2017) · Zbl 1380.93196 · doi:10.1016/j.jfranklin.2017.05.009 |
| [30] | Trang, TT; Phat, VN; Samir, A., Finite-time stabilization and h∞ control of nonlinear delay systems via output feedback, J Ind Manam Optim, 12, 1, 303-315 (2016) · Zbl 1317.93222 · doi:10.3934/jimo.2016.12.303 |
| [31] | Wang, Y.; Li, D., Adaptive synchronization of chaotic systems with time-varying delay via aperiodically intermittent control, Soft Comput, 24, 17, 12773-12780 (2020) · Zbl 1491.34053 · doi:10.1007/s00500-020-05161-7 |
| [32] | Yang, R.; Zhang, G.; Sun, L., Finite-time robust simultaneous stabilization of a set of nonlinear time-delay systems, J Robust Nonlinear Control, 30, 5, 1733-1753 (2020) · Zbl 1465.93192 · doi:10.1002/rnc.4848 |
| [33] | Yao, L.; Zhang, W.; Xie, X-J, Stability analysis of random nonlinear systems with time-varying delay and its application, Automatica, 117, 108994 (2020) · Zbl 1442.93046 · doi:10.1016/j.automatica.2020.108994 |
| [34] | Zhang, Y.; Jiang, T., Finite-time boundedness and chaos-like dynamics of a class of Markovian jump linear systems, J Frankl Inst, 357, 4, 2083-2098 (2020) · Zbl 1451.93415 · doi:10.1016/j.jfranklin.2019.11.050 |
| [35] | Zhang, Y.; Shi, Y.; Shi, P., Resilient and robust finite-time H∞ control for uncertain discrete-time jump nonlinear systems, Appl Math Model, 49, 612-629 (2017) · Zbl 1480.93247 · doi:10.1016/j.apm.2017.02.046 |
| [36] | Zhang, Z.; Zhang, Z.; Zhang, H.; Zheng, B.; Karimi, HR, Finite-time stability analysis and stabilization for linear discrete-time system with time-varying delay, J Frankl Inst, 351, 6, 3457-3476 (2014) · Zbl 1290.93147 · doi:10.1016/j.jfranklin.2014.02.008 |
| [37] | Zong, G.; Wang, R.; Zheng, W.; Hou, L., Finite-time H∞ control for discrete-time switched nonlinear systems with time delay, J Robust Nonlinear Control, 25, 6, 914-936 (2015) · Zbl 1309.93057 · doi:10.1002/rnc.3121 |
| [38] | Zuo, Z.; Li, H.; Wang, Y., New criterion for finite-time stability of linear discrete-time systems with time-varying delay, J Frankl Inst, 350, 9, 2745-2756 (2013) · Zbl 1287.93081 · doi:10.1016/j.jfranklin.2013.06.017 |
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.
