Jointly identification of the sample delay and the parameters of time-delay discrete-time ARMAX models are addressed in this article. Like any identified problem, the proposed approach is based on the solution of an optimization problem. It consists then of two main steps: the first is the problem formulation and the second is the problem resolution. The problem formulation defines the criterion to be minimized according to an unknown parameter vector consisting of the sample delay and the parameters with their corresponding observation vector. Since the sample delay is of integer type, the obtained optimization problem cannot be efficiently solved using a real optimization algorithm. To overcome this difficulty, the rounding properties are used to transform the obtained problem into a real optimization problem. The solution is then done using the least square approach. Consistency of the estimates with their convergence rates is established under the persistent excitation condition. Finally, some numerical simulation and experimental results are presented to illustrate the performance of the proposed method.
DingFLiuYJBaoB. Gradient based and least squares based iterative estimation algorithms for multi-input multi-output systems. Proc IMechE, Part I: J Systems and Control Engineering2012; 226(1): 43–55.
2.
PanJMaHZhangX, et al. Recursive coupled projection algorithms for multivariable output-error-like systems with coloured noises. IET Sig Process2020; 14(7): 455–466.
3.
PanJMaHZhangX, et al. Partially coupled gradient-based iterative algorithms for multi-variable output-error-like systems with autoregressive moving average noises. IET Contr Theo Appl2020; 14(7): 2613–2627.
4.
DingF. Coupled-least-squares identification for multivariable systems. IET Contr Theo Appl2013; 7(1): 68–79.
5.
MaHPanJ, et al. Partially-coupled least squares based iterative parameter estimation for multi-variable output-error-like autoregressive moving average systems. IET Contr Theo Appl2019; 13(18): 3040–3051.
6.
DingF. The innovation algorithms for multivariable state-space models. Int J Adapt Contr Sig Process2019; 33(11): 1601–1608.
7.
WangYJDingF. Novel data filtering based parameter identification for multiple-input multiple-output systems using the auxiliary model. Automatica2016; 71: 308–313.
8.
XuLDingFLuX, et al. Hierarchical multi-innovation generalised extended stochastic gradient methods for multivariable equation-error autoregressive moving average systems. IET Contr Theo Appl2020; 14(10): 1276–1286.
9.
WangYJDingFWuMH. Recursive parameter estimation algorithm for multivariate output-error systems. J Frank Inst2018; 355(12): 5163–5181.
10.
WanLJ. Decomposition-and gradient-based iterative identification algorithms for multivariable systems using the multi-innovation theory. Circ Syst Sig Process2019; 38(7): 2971–2991.
11.
DingF. Identification methods for Hammerstein nonlinear systems. Digit Sig Process2011; 21(2): 215–238.
12.
DingF. Hierarchical multi-innovation stochastic gradient algorithm for hammerstein nonlinear system modeling. Appl Math Model2013; 37(4): 1694–1704.
13.
DingF. Gradient-based and least-squares-based iterative algorithms for Hammerstein systems using the hierarchical identification principle. IET Contr Theo Appl2013; 7(2): 176–184.
14.
AtitallahABedouiSAbderrahimK. An optimal two stage identification algorithm for discrete Hammerstein time delay systems. IFAS: Paper Online2016; 49(10): 19–24.
15.
XuL. Auxiliary model multiinnovation stochastic gradient parameter estimation methods for nonlinear sandwich systems. Int J Robust Nonl Contr2021; 31(1): 148–165.
16.
XuL. Separable multi-innovation stochastic gradient estimation algorithm for the nonlinear dynamic responses of systems. Int J Adapt Contr Sig Process2020; 34(7): 937–954.
17.
FanYMLiuXM. Two-stage auxiliary model gradient-based iterative algorithm for the input nonlinear controlled autoregressive system with variable-gain nonlinearity. Int J Robust Nonl Contr2020; 30(14): 5492–5509.
18.
LiuXMFanYM. Maximum likelihood extended gradient-based estimation algorithms for the input nonlinear controlled autoregressive moving average system with variable-gain nonlinearity. Int J Robust Nonl Contr2020; 31(9): 4017–4036.
19.
JiYJiangXKWanLJ. Hierarchical least squares parameter estimation algorithm for two-input Hammerstein finite impulse response systems. J Frank Inst2020; 357(8): 5019–5032.
20.
JiYZhangCKangZ, et al. Parameter estimation for block-oriented nonlinear systems using the key term separation. Int J Robust Nonl Contr2020; 30(9): 3727–3752.
21.
JiYKangZ. Three-stage forgetting factor stochastic gradient parameter estimation methods for a class of nonlinear systems. Int J Robust Nonl Contr2021; 31(3): 871–987.
22.
XiaBZhaoXDe CallafonR, et al. Accurate lithium ion battery parameter estimation with continuous-time system identification methods. Appl Energ2016; 179: 426–436.
23.
BedouiSLtaiefMAbderrahimK. Using a recursive least square algorithm for identification of interconnected linear discrete-time delay multivariable systems. In: Proceedings of the 17th IEEE international conference on methods & models in automation & robotics (MMAR2012), Miedzyzdroje, 27–30 August2012. New York: IEEE.
24.
BedouiSLtaiefMAbderrahimK. New method for the systematic determination of the model’s base of time varying delay system. Int J Comput Appl2012; 46(1): 13–19.
25.
YangXLiuXLiZ. Multimodel approach to robust identification of multiple-input single-output nonlinear time-delay systems. IEEE T Ind Inform2020; 16(4): 2413–2422.
26.
LiuXYangX. Robust variational inference for LPV dual-rate systems with randomly delayed outputs. IEEE T Instr Meas2021; 70: 1–9.
27.
AtitallahABedouiSAbderrahimK. Multistage for identification of wiener time delay systems based on hierarchical gradient approach. Math Comput Model Dyn Syst2017; 23(2): 222–239.
28.
BedouiSLtaïefMAbderrahimK. Hierarchical gradient based identification of discrete-time delay systems. In: Proceedings of the 52nd IEEE conference on decision and control, Firenze, 10–13 December2013, pp.4565–4570. New York: IEEE.
29.
BedouiSLtaiefMAbderrahimK, et al. Representation and control of time delay system: multimodel approach. In: Proceedings of the Eighth international multi-conference on systems, signals & devices, Sousse, 22–25 March2011, pp.1–6. New York: IEEE.
30.
GarnierH. Direct continuous-time approaches to system identification: overview and benefits for practical applications. Eur J Contr2015; 24: 50–62.
31.
GawthropPJNihtilaMT. Identification of time delays using polynomial identification method. Syst Contr Lett1985; 5: 276–271.
32.
KurzHGoedeckeW. Digital parameter adaptive control of process with unknown dead time. Automatica1981; 17(1): 245–252.
33.
GaoWLiYCLiuGJ, et al. An adaptive fuzzy smith control of time-varying processes with dominant and variable delay. Proc Am Contr Conf2003; 1: 220–224.
34.
GaoWZhouMLLiYC, et al. An adaptive generalized predictive control of time varying delay system. In: Proceedings of the second world conference on machine learning and cybernetics, Xi’an, China, 26–29 August2004, pp.878–881. New York: IEEE.
35.
YangZSunZAndersenC. Nonlinear FOPDT model identification for the superheat dynamic in a refrigeration system. In: Proceedings of the 37th annual conference on IEEE industrial electronics society, Melbourne, VIC, Australia, 7–10 November2011. New York: IEEE.
36.
GuYZhuQNouriH. Bias compensation-based parameter and state estimation for a class of time-delay non-linear state-space models. IET Contr Theo Appl2020; 14(15): 2176–2185.
37.
GuYLiuJLiX, et al. State space model identification of multirate processes with time-delay using the expectation maximization. J Frank Inst2019; 356(3): 1623–1639.
38.
GuYChouYLiuJ, et al. Moving horizon estimation for multirate systems with time-varying time-delays. J Frank Inst2019; 356(4): 2325–2345.
39.
LiuXYangXXiongW. A robust global approach for LPV-FIR model identification with time-varying time delays. J Frank Inst2018; 355(15): 7401–7416.
40.
ZhengGPolyakovALevantA. Delay estimation via sliding mode for nonlinear time-delay systems. Automatica2018; 89: 266–273.
41.
BedouiSAbderrahimK. ARMAX time delay systems identification based on least square approach. In: Proceedings of the 17th IFAC symposium on system identification, Beijing, China, 19–21 October2015. New York: IEEE.
42.
BedouiSLtaiefMAbderrahimK. New results on discrete time delay systems. Int J Autom Comput: IJAC2012; 9(6): 570–577.
43.
BedouiSLtaiefMAbderrahimK. A new recursive algorithm for simultaneous identification of discrete time delay systems. In: Proceedings of the 16th IFAC symposium on system identification (SYSID 2012), Brussel, 11–13 July2012. IFAC Proceedings Volumes (IFAC Papers-OnLine).
44.
XuLSongGL. A recursive parameter estimation algorithm for modeling signals with multi-frequencies. Circ Syst Sig Process2020; 39(8): 4198–4224.
45.
GuoLChenHF. Identification and stochastic adaptive control (Systems & control: foundations & applications). Berlin: Springer, 1991.
46.
DingFLiuG. Hierarchical least squares identification for linear SISO systems with dual-rate sampled-data. IEEE T Autom Contr2011; 56(11): 2677–2683.
47.
HübnerRSchöbelA. When is rounding allowed? A new approach to integer nonlinear optimization. In: Proceedings of the ORP3 meeting, 13–17September2011, https://www.torrossa.com/en/resources/an/4657531
48.
BruceALGoelABernsteinDS. Convergence and consistency of recursive least squares with variable-rate forgetting. Automatica2020; 119: 109052.
49.
HuangDGuoL. Estimation of nonstationary ARMAX models based on the Hannan–Rissanen method. Ann Stat1999; 18: 1729–1756.
50.
ShiYDingFChenT. Performance analysis of estimation algorithms of nonstationary ARMA processes. IEEE T Sig Process2006; 54(3): 1041–1053.
51.
XiaoYDingFZhouY, et al. On consistency of recursive least squares identification algorithms for controlled auto-regression models. Appl Math Model2008; 32: 2207–2215.
52.
YinLGaoH. Moving horizon estimation for ARMAX processes with additive output noise. J Frank Inst2019; 356(4): 2090–2110.
53.
HuangLHjalmarssonH. A multi-time-scale generalization of recursive identification algorithm for ARMAX systems. IEEE T Autom Contr2015; 60(8): 2242–2247.
54.
YinLLiuSGaoH. Regularised estimation for ARMAX process with measurements subject to outliers. IET Contr Theo Appl2018; 12(7): 865–874.
55.
GuoL. Self-convergence of weighted least-squares with applications to stochastic adaptive control. IEEE T Autom Contr1996; 41(1): 79–89.
56.
ChenZWangYSunM, et al. Convergence and stability analysis of active disturbance rejection control for first-order nonlinear dynamic systems. Trans Inst Meas Contr2019; 41(7): 2064–2076.
57.
LandauIDZitoG. Digital control systems: design, identification and implementation. Berlin: Springer, 2006.
58.
Fong-ChweeTSirisenaHR. Self-tuning PID controllers for dead time processes. IEEE T Ind Electr1988; 35(1): 119–125.
