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Singularly perturbed control law for linear time-invariant delay SISO systems. (English) Zbl 1305.93095
Summary: This paper considers the problem of stabilizing a single-input single-output linear time-invariant delay system, where the parameters of the system are unknown. Using norm bounds of the unknown parameters, a low-order controller and reference model are proposed. The closed-loop system is a linear singularly perturbed retarded system with uniform asymptotic stability behavior. We calculate bounds \(\epsilon \in (0,\epsilon ^{*})\) as in the book of P. Kokotović et al. [Singular perturbation methods in control: analysis and design. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics (1999; Zbl 0989.93001)], such that the uniform asymptotic stability of the linear singularly perturbed delay system is guaranteed. We show how to design a control law such that the system dynamics is assigned by a Hurwitz polynomial with constant coefficients.

MSC:
93C23 Control/observation systems governed by functional-differential equations
93C70 Time-scale analysis and singular perturbations in control/observation systems
93C41 Control/observation systems with incomplete information
93D15 Stabilization of systems by feedback
93D20 Asymptotic stability in control theory
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[1] Ali, R., Amir, M.: Extreme properties go eigenvalues of Hermitian transformation and singular values of the sum and product of linear transformations. Duke Math. J. 23, 463–467 (1956) · Zbl 0071.01601 · doi:10.1215/S0012-7094-56-02343-2
[2] Bellman, R., Cooke, K.L.: Differential-Difference Equations. Academic Press, UK (1963) · Zbl 0105.06402
[3] Desoer, C.A., Vidyasagar, M.: Fedback Systems: Input-Output Properties. SIAM, Philadelphia (2009)
[4] Fridman, E.: Effects of small delay on stability of singularly perturbed systems. Automatica 38, 897–902 (2002) · Zbl 1014.93025 · doi:10.1016/S0005-1098(01)00265-5
[5] Fridman, E.: Robust sampled data $${\(\backslash\)cal {H}}_{\(\backslash\)infty }$$ H control of linear stability of singularly perturbed systems. IEEE Trans. Autom. Control 51, 470–475 (2006) · Zbl 1366.93154 · doi:10.1109/TAC.2005.864194
[6] Fridman, E.: Decoupling transformation of singularly perturbed systems with small delay and its application. Z. Angew. Math. Mech 76(2), 201–204 (1996) · Zbl 0886.34063
[7] Glizer, V.Y.: Singularly perturbed linear controlled systems with small delay: an overview of some recent results. IN: 47th IMACS World Congress, France (2005)
[8] Glizer, V.Y., Fridman, E.: $${\(\backslash\)cal {H}}_{\(\backslash\)infty }$$ H Control of linear singularly perturbed systems with small state delay. J. Math. Anal. Appl. 250, 49–85 (2000) · Zbl 0965.93049 · doi:10.1006/jmaa.2000.6955
[9] Góreki, H., Fuksa, S., Grabowski, P., Korytowsky, A.: Analysis and Synthesis of Time Delay Systems. Wiley, New York (1989)
[10] Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations, Springer, New York (1993) · Zbl 0787.34002
[11] Hardy, G.H.: A Course of Pure Mathematics. Cambridge University Press, UK (1975) · Zbl 0796.26002
[12] Kim, J.H.: Robust stability of linear systems with delay perturbation. IEEE Trans. Autom. Control 41, 1820–1822 (1996) · Zbl 0944.93025 · doi:10.1109/9.545749
[13] Kokotović, P.V., Khalil, H.K., O’Reilly, J.: Singular Perturbation Methods in Control: Analysis and Design. SIAM, Philadelphia (1999) · Zbl 0989.93001
[14] Liu, X., Shen, X., Zhang, Y.: Exponential stability of singularly perturbed systems with time delay. J. Appl. Anal. 82, 117–130 (2003) · Zbl 1044.34031 · doi:10.1080/0003681031000063775
[15] Pan, S.T., Hsiao, F.H., Teng, C.C.: Stability bound of multiple time delay singularly perturbed systems. Electron. Lett. 32, 1327–1328 (1996) · doi:10.1049/el:19960644
[16] Polderman, J.W., Willems, J.C.: Introduction to Mathematical System Theory. Text in Applied Mathematics. Springer, New York (1997) · Zbl 0940.93002
[17] Puga, S., Bonilla, M., Malabre, M.: Singularly perturbed implicit control law for linear time varying SISO system. In: 49th IEEE CDC, pp. 6870–6875 (2010)
[18] Puga, S., Bonilla, M., Malabre, M., Lozano, R.: Singularly perturbed implicit control law for linear time varying SISO systems. Int. J. Robust Nonlinear Control (2013). doi: 10.1002/rnc,3013 · Zbl 1291.93202
[19] Shampine, L.F., Thompson, S., Kierzenka, J.: Solving delay differential equations withdde23. http://www.radford.edu/thompson/webddes/tutorial.pdf (2002)
[20] Shao, Z.H.: Stability bounds of singularly perturbed delay systems. IEE Proc Control Theory Appl. 151, 585–588 (2004)
[21] Stewart, G.W.: Introduction to Matrix Computations. Academic Press, New York (1973) · Zbl 0302.65021
[22] William Luse, D.: Multivariable singularly stability perturbed feedback systems with time delay. IEEE Trans. Autom. Control AC 32, 990–994 (1987) · Zbl 0628.93053
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