An observer-based compensator for distributed delays

-This paper presents an algorithm for compensating delays that are distributed between the sensor(s), controller and actuator(s) within a control loop. This observer-based algorithm is specially suited to compensation of network-induced delays in integrated communication and control systems. The robustness of the algorithm relative to plant model uncertainties has been examined. 1. In troduct ion AN EFFICIENT means for realizing Integrated Control Systems is to interconnect the spatially distributed components by a computer communication network (Ray, 1987). Such a network introduces randomly varying delays in the control loop, which degrade the system dynamic performance and are a source of potential instability. This paper considers control systems with distributed delays that occur in both measurements and control inputs. The plant and controller dynamics at the sample time k are modeled as: xk+ t = A x k + BUk_at----plant dynamics with delayed control (1) Yk = Cxk s e n s o r measurement (2) wt, = Yk-,,2---'delay ed output (3) u k = ~e(Wk)----control function (4) where x ¢ R ~, u ~ R ~ and y a R" , and the matrices A, B and C, are of compatible dimensions; the finite non-negative integers A1 and A2 represent the number of delayed samples in control inputs and measurements, respectively. The control law u k is a linear function of the history W k := {w k, wk_ t . . . . } of the delayed measurements. The objective is to construct the control function ~ such that the effects of the delays on the control system performance are mitigated. A major motivation for considering the delayed control system described above is the recent interest in Integrated Communication and Control Systems (ICCS) (Ray, 1987, 1988; Ray and Phoha, 1989; Halevi and Ray, 1988; Ray and Halevi, 1988) for applications to diverse processes. Since the individual system components in ICCS are interconnected via a time-division-multiplexed network, the delays A1 and A2 in (1) and (3) arise because of network-induced delays between the controller and actuator, and the sensor and * Received 20 March 1987; revised 28 July 1989; received in final form 25 November 1989. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor P. G. Ferreira Guinares under the direction of Editor H. Kwakernaak. This work was supported in part by NASA Lewis Research Center Grant No. NAG 3-823. t Mechanical Engineering Department, Mississippi State University, Drawer ME, Mississippi State, MS 39762, U.S.A. , Mechanical Engineering Department, The Pennsylvania State University, University Park, PA, U.S.A. Author to whom all correspondence should be addressed. controller, respectively. Recently Ray and Halevi (1988; also Halevi and Ray, 1988) have reported analysis and design of ICCS (also described in Ray, 1987, 1988; Ray and Phoha, 1989) where the delays A1 and A2 are deterministically or randomly varying. Necessary and sut~cient conditions were obtained for the case of periodically varying delays. The problem of delay compensation under the non-periodic and random traffic has been discussed in Halevi and Ray (1988) and Ray and Halevi (1988), but no specific solution has been given. Several investigators have addressed the problems of delay compensation in closed loop control systems. An intuitive approach (Isermann, 1981) is to augment the system model to include delayed variables as additional states. Unfortunately, this renders some of the states uncontrollable even when the original system is completely controllable (Marianni and Nicoletti, 1973; Drouin et al., 1985). For the case of delayed control inputs, Pyndick (1972) proposed a predictor for the optimal state trajectory based on past control inputs. Zahr and Slivinsky (1974) considered the problem of controlling a computer-controlled system with measurement and computational delays. It was pointed out that the delays in multi-variable systems may result in: (1) an increase in the magnitudes of the transients and poor response during the inter-sampling time, (2) loss of decoupling between individual SISO control loops although decoupling may be restored for a stable process at the steady-state, and (3) a possible decrease in the stability margin. Their algorithm was verified by simulation but the use of an observer to estimate the unavailable states was not discussed. A significant amount of research work has been reported for observer and controller design (Drouin et al., 1985; Bhat and Kiovo, 1976; Fairman and Kumar, 1986) for processes with inherent constant delays that occur within the process to be controlled. In contrast to the system under consideration in (1)-(4), such processes are described as follows: d x ( t ) / d t = A x ( t ) + D x ( t h ) + G u ( t ) (5) y ( t ) = Cx ( t ) (6) where h is a constant. By setting G = 0 and D =BK, equation (5) reduces to a delayed state-variable-feedback system. The reported literature in delay compensation does not apparently address the problem of distributed delays in both the input and output variables, which is the case with ICCS. A methodology for compensation of distributed delays, as an essential step to ICCS design, is presented in this paper. The ICCS network can be designed on the assumption that the induced delays are bounded within a specified confidence interval. It has been established by several investigators that a stable controller designed on the basis of a constant delay which is equal to the supremum of the varying delay may not ensure the system stability (Halevi and Ray, 1988). The proposed delay compensator circumvents the detrimental effects of bounded network-induced delays by using a multi-step predictor. The key idea in the compensator design