Uncertain Linear Singular Time-delay Systems With Delay-dependent H ¡Þ Variable Structure Control

Research in variable structure control (VSC) can be back to Soviet Union. Soviet scholars Utkin et al. began VSC research in 1950s. One of the most favorable advantages of variable structure control systems is its robustness confronted with change of the system dynamics, change of parameters and perturbation of external disturbances. When systems have matched uncertainties or matched disturbances and so on, sliding mode dynamics is invariable to change of parameters and disturbances. What makes VSC superior to bang-bang control is that the track of the sliding mode dynamics is confined to a lower order subspace and so its order of moving differential equation is also reduced. Advantages of VSC make it a hot point of control science. And scholars have achieved a lot in variable structure control of multivariable linear systems and nonlinear systems, for example [1]-[21] etc. However, since the reduced-order equivalent sliding mode dynamics restricted to the sliding surface falls under the sway of the uncertainties and it does not any more remain insensitive to the uncertainties in the case of the uncertain multivariable system with mismatched uncertainties, the design problem becomes difficult. Over the years, along with in-depth research, scholars have broadened their VSC research in systems with mismatched conditions. Recently, based on the linear matrix inequality (LMI) approach, Choi[1]-[4] has developed variable structure control design methods for a class of mismathched uncertain systems without delay. Yuanqin Xia(2003) et al. suggested a robust sliding mode control for uncertain time-delay systems with matched disturbances; a delay-independent sufficient condition for the existence of a linear sliding surface and quadratic stability of the sliding mode dynamics is given in terms of LMI. Because LMI can be solved very efficiently via various powerful LMI optimization algorithms, this method has been used widely in control design. Unfortunately, there are a few papers on variable structure control of multivariable singular systems. The reason is obvious: singular systems are very complex and their state track usually contains impulse which makes variable structure control of singular systems very difficult. Furthermore, few VSC research on uncertain singular systems with time-delay is considered. To the best of our knowledge, it seems that there are no previous results on delay-dependent H_∞ variable structure control. This has motivated our research.For a class of linear singular time-delay systems with uncertainties, we bring forward a new method to design a switching surface. The singular systems with mismatched norm-bounded uncertainties and mismatched norm-bounded external disturbances. Withthe help of LMI, we give a delay-dependent sufficient condition which ensure that the sliding mode dynamics restricted to the switching surface is not only regular, impulse free but internally stable with disturbance attenuation. A state feedback variable structure controller is given as well. We also give an explicit formula of such linear switching surface, together with a design example.In the second section, we discuss a uncertain singular time-delay system :Exit) = [A + AA]x(t) + [Ad + AAd)]x{t -T) + [B + AB]u{t) + [F + AF]w{t) z{t) = Cx(t) x{t) = if>(t), tG{-r,Q)where x(t) e Rn is the state, u{t) € Rm is the control input, z(t) €■ Rp is the output. w €. R' is the square-integrable external disturbance whose norm is bounded by the known scalar ￡, i.e., || w{t) ||< L Delay r is a known positive scalar. (t) € C([-r,0],i?n) is a compatible vector valued continuous function. The constant matrices E, A, Ad, B, C and F are of appropriate dimensions, rankB = m, rank E = p, 0 < p < n. Assume the uncertainties AA, A Ad, AB and AF are time-invariant matrices representing norm-bounded uncertainties and have the following form|| AA ||< Pl, || AAd \\< p2, || AJ5 ||< p3, II AF ||< p4where pi , i = 1, ..., 4, are all known positive constants. It gives a sliding surfaceQ, - {x : a(x) = SEx = 0}where 5 G Rmxn is to be design.Regarding the above system, in theorem 2 we give a delay-dependent sufficient condition, guaranteeing that the equivalent system on the sliding surface is regular, impulse free and internally stable with disturbance attenuation.have solutions N, Y and X > 0, M > 0, Q > 0, R > 0, Z > 0, ^ > 0, d2 > 0, A > 0. Where J, j and tt are described at the back.A sufficient condition given by theorem 2 can be deduced to solvability of LMIs. We can calculate LMIs by MATLAB software. Switching parameter matrix S can be obtained by S = LBTX~1.Theorem 3 gives the variable structure controller:u{t) = -wherep(x(t)) = -{e+\\So\\-\pi\\x(t)\\+p2\\x(t-r) A1+p3 I! S0Ax{t) + S0Adx{t - r) ||]and e is a positive scalar.This controller ensures that the system state trajectory can reach the sliding surface Q in finite time and stay on it.In section 3, we give an example to illustrate the validity of our algorithm.