This paper addresses the problem of the detemzil/ation of regiam (f s/abiliry for linear systems \\'ith delayed inputs and subject to inpul saturation through anti-windllp strategies. Differently of the most t/nti-windup techniques, where the desigl/ of the anti-windllP loop is introduced lI'ith the objectil'e of minimizing the pelfor-/11al/ce degradatiOI/. we are particular/)' interested in the synthesis ()f allti-windup gains in order to guarantee the stability of the c1osed·loop systemfor regio/lS of admissible initial states as large as possible, With Ihis aill/, due to the presence of delay in the input
lntroductionConsider the linear continuous-time delay system: J xU)=Ax(t)+Bu (t-r)
ly(t)=Cx(t)with the initial conditions (I) where x( t) e (R", u( t) e (Rm, y (t) e (RP are the state, the input and the measured output vectors, respectively. Matrices A, B and C are real constant matrices of appropriate dimensions.Considering system (I), we assume that an nc-order dynamic output stabilizing compensator described bywhere TJ(t) e (R"< is the controller state, y(t) is the controller input, and v(t) is the controller output, has been determined. In fact, the control signal to be injected in the system is a saturated one, that is, Journal of Dynamic Systems, Measurement, and Control where each component of sat (v(t» are defined for i= I, ... ,m as: sat(v(i) (t»=sign(v(i) (t»min(lv(i) (t)I,uo(i) where UO(i»O denotes the control bound of the ith input. In order to mitigate the undesirable effects of windup caused by input saturation, an antiwindup term Ec(sat(v(t»-v(t» can be added to the controller [I]. Thus, considering the dynamic controller and this anti-windup strategy, the c1osed-loop system reads:Let us now define the extended state vector
TJ(t)and the following matrices of appropriate dimensionsHence. the closed-Ioop system reads:In Eg. (5) where A is a diagonal matrix whose diagonal elements A(i.i) are simply denoted by À(i)' with O";;;À(i)< I, and the polyhedral set,,;;;Klilg";;;uo(i/I-À(i) ,i= I, ... ,m}.The augmented system (5) admits an augmented initial condition 'V Oe [-1',0] where cpç( O) is supposed to satisfy 1 1 cp.;llc";;; v, v>O. Note that the initial condition of the dynamic output controller CP.,,It is important to note that the c1osed-loop matrix A will be asymptotically stable if and only if matrices A and A c are asymptotically stable. Indeed, even if Ac has been chosen asymptotically stable, A may admit some unstable or not wished eigenvalues. In this case, we should study the stability of cJosed loop system (5) in a delay dependent context. System (5) is said globally asymptotically stable if for any initial condition satisfying 11 CP€llc";;; v with any finite v>O, the trajectories of system (5) converge asymptotically to the origino Similar to the case of delay-free (1'= O), the determination of a global stabilizing controller is possible only when some stability hypothesis are verified by the open-Ioop system (u(t)=O) [6]. When this hypothesis is not verified, it is only ...