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 ...

SUMMARYThis paper addresses the problem of controlling a linear system subject to actuator saturations and to L 2 -bounded disturbances. Linear matrix inequality (LMI) conditions are proposed to design a state feedback gain in order to satisfy the closed-loop input-to-state stability (ISS) and the closed-loop finite gain L 2 stability. By considering a quadratic candidate Lyapunov function, two particular tools are used to derive the LMI conditions: a modified sector condition, which encompasses the classical sector-nonlinearity condition considered in some previous works, and Finsler's Lemma, which allows to derive stabilization conditions which are adapted to treat multiple objective control optimization problems in a potentially less conservative framework.

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This paper addresses the problem of tracking periodic references for uncertain linear systems subject to control saturation. Accordingly to the internal model principle, a control loop containing the modes of both the references and additive disturbances is considered. From this structure, conditions in a "quasi" LMI form are proposed to simultaneously compute a stabilizing state feedback gain and an anti-windup gain. Provided that the references and disturbances belong to a certain admissible set, these gains guarantee that the trajectories of the closed-loop system starting in a certain invariant ellipsoidal set contract to the linearity region of the closed-loop system, ensuring therefore the perfect reference tracking. Based on these conditions, an optimization problem aiming at the maximization of the invariant set of admissible states and/or the maximization of the set of admissible references/disturbances is proposed. Numerical examples to illustrate the method are provided.

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