Delay-Dependent Anti-Windup Loops for Enlarging the Stability Region of Time Delay Systems With Saturating Inputs

Abstract

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 ...

Stabilization of Neutral Systems With Saturating Inputs

“…As will seen in the sequel, differently from the classical sector condition (used for instance in (Tarbouriech et al, 2003)), this condition will allow to obtain stability conditions directly in an LMI form. Theorem 1.…”

Anti-windup design with guaranteed regions of stability: an LMI-based approach

“…Following now a similar development to the one used in [16], but using the result stated in Lemma 1, the following Theorem can be stated. The proof of the Theorem 1 is omitted for reasons of place.…”

“…Furthermore, (22) is simpler than the equivalent one in [16]. The synthesis of E c aiming at the maximization of the set of admissible initial conditions can then be carried out by using the optimization criterion discussed in [16]. Taking the gain computed in [11] and applying the condition given in that paper in an analysis context it turns out that the maximal achievable 1 is 36.2187.…”

“…We consider now the case where the input is delayed by , i.e., the open-loop system reads _ x(t) = Ax(t) + Bu(t 0 ) y(t) = Cx(t) : (20) In this case, given a dynamic controller as in (2), the closed-loop system in the presence of input saturation can be rewritten as Problem 1 considering (21) has been recently addressed in [16]. In that paper, the stabilization conditions were stated by using a "classical" sector condition, as detailed in Remark 1.…”

“…Furthermore, (22) is simpler than the equivalent one in [16]. The synthesis of E c aiming at the maximization of the set of admissible initial conditions can then be carried out by using the optimization criterion discussed in [16].…”