LPV systems with parameter-varying time delays: Analysis and control. (English) Zbl 0969.93020

Wu, Fen; Grigoriadis, Karolos M.

Automatica 37, No. 2, 221-229 (2001).

The paper considers the system \[ \begin{aligned} \dot x(t) &= A(\rho(t))x(t)+ A_h(\rho(t))x (t-h(\rho(t)))+ B(\rho(t)) u(t),\\ y(t) &= C(\rho(t))x(t)+ C_h(\rho(t)) x(t- h(\rho(t)))+ D(\rho(t)) u(t)\end{aligned} \] with bounded nonincreasing delay. Also \(\rho(\cdot)\) is a function with bounded derivative. The authors develop a Lyapunov functional theory for linear stability of the equilibrium of the free system as well as for feedback stabilization.

Reviewer: Vladimir Răsvan (Craiova)

MSC:

93C23	Control/observation systems governed by functional-differential equations
93D20	Asymptotic stability in control theory
93D15	Stabilization of systems by feedback

Keywords:

time-varying linear system; delay; Lyapunov functional theory; linear stability; feedback stabilization

Software:

LMI toolbox

References:

[1]	Apkarian, P.; Adams, R. J., Advanced gain-scheduling techniques for uncertain systems, IEEE Transactions on Control Systems Technology, 6, 21-32 (1998)
[2]	Apkarian, P.; Gahinet, P., A convex characterization of gain-scheduled \(H∞\) controllers, IEEE Transactions on Automatic Control, 40, 853-864 (1995) · Zbl 0826.93028
[3]	Becker, G.; Packard, A. K., Robust performance of linear parametrically varying systems using parametrically-dependent linear feedback, Systems Control Letters, 23, 205-215 (1994) · Zbl 0815.93034
[4]	Boyd, S. P.; El Ghaoui, L.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in systems and control theory (1994), SIAM: SIAM Philadelphia, PA · Zbl 0816.93004
[5]	Dugard, L.; Verriest, E. I., Stability and control of time-delay systems (1998), Springer: Springer London · Zbl 0901.00019
[6]	Driver, R. D., Ordinary and delay differential equations (1977), Springer: Springer New York · Zbl 0374.34001
[7]	Gahinet, P.; Apkarian, P.; Chilali, M., Affine parameter-dependent Lyapunov functions and real parametric uncertainty, IEEE Transactions on Automatic Control, 41, 436-442 (1996) · Zbl 0854.93113
[8]	Gahinet, P.; Nemirovskii, A.; Laub, A. J.; Chilali, M., LMI control toolbox (1995), Mathworks: Mathworks Natick, MA
[9]	Skelton, R. E.; Iwasaki, T.; Grigoriadis, K. M., A unified algebraic approach to linear control design (1998), Taylor & Francis: Taylor & Francis London
[10]	Kolmanovskii, V. B.; Shaikhet, L. E., Control of systems with aftereffect, vol. 157 (1996), American Mathematical Society: American Mathematical Society Providence, RI
[11]	Mahmoud, M. S.; Al-Muthairi, N. F., Design of robust controllers for time-delay systems, IEEE Transactions on Automatic Control, 39, 995-999 (1994) · Zbl 0807.93049
[12]	Malek-Zavarei, M.; Jamshidi, M., Time delay systems: analysis, optimization and applications (1987), North-Holland: North-Holland Amsterdam · Zbl 0658.93001
[13]	Packard, A. K., Gain scheduling via linear fractional transformations, Systems Control Letters, 22, 79-92 (1994) · Zbl 0792.93043
[14]	Rugh, W. J., Analytical framework for gain scheduling, IEEE Control Systems Magazine, 11, 74-84 (1991)
[15]	Scherer, C. W., Mixed \(H_2/H_∞\) control for time-varying and linear parametrically varying systems, International Journal of Robust and Nonlinear Control, 6, 929-952 (1996) · Zbl 0861.93009
[16]	Shamma, J. S.; Athans, M., Analysis of nonlinear gain-scheduled control systems, IEEE Transactions on Automatic Control, 35, 898-907 (1990) · Zbl 0723.93022
[17]	Shamma, J. S.; Athans, M., Gain scheduling: Potential hazards and possible remedies, IEEE Control Systems Magazine, 12, 101-107 (1992)
[18]	Vandenberghe, L.; Boyd, S., A primal-dual potential reduction method for problems involving matrix inequalities, Mathematical Programming, 69, 205-236 (1994) · Zbl 0857.90104
[21]	Wu, F.; Yang, X. H.; Packard, A. K.; Becker, G., Induced \(L2\) norm control for LPV systems with bounded parameter variation rates, International Journal of Robust and Nonlinear Control, 6, 983-998 (1996) · Zbl 0863.93074