Stability analysis of multiple time-delay systems and design of time-delay filters

Time-delay is a common and important fact seen in biological, chemical and physical processes or in engineering systems. Systems which has the dynamics dependent not just on its states at a certain time, but as well on its states at previous times, are called as time-delay systems. Time-delay determines the lag in time for a previous state value to effect the present state. Besides arising in natural dynamics of the systems, time-delays may be inserted in control design of some engineering systems to stabilize systems or to achieve relatively easy implementation and good performance. The major contributions of this dissertation are in two main directions: one in stability analysis of time-delay systems and one in utilization of delay phenomena for control purposes of flexible systems. In the first direction, a new method providing necessary and sufficient conditions to test delay-independent stability for general linear time invariant systems with constant delays is proposed. The method is utilized for systems with single delay, and more importantly for incommensurate multiple delay systems. The proposed method offers an approach to determine the exact boundaries of unknown parameters such as controller gains or system parameters ensuring delay-independent stability, as well as exhibiting an efficient test for real parameters. The technique is based on non-existence of unitary complex zeros of an auxiliary characteristic polynomial obtained via extended Kronecker summation. A special feature of the polynomial, i.e. self-inversive property, is proved and utilized to check its unitary zeros to determine delay-independent stability by an efficient zero location test. The methodology is executed employing simple algebraic operations and inspection of the number of sign variations in the obtained sequence. For single delay case, the procedure does not require parameter (or frequency) sweeping, equation solving and point-wise testing even for determination of the delay-independent stabilizing regions of unknown parameters. In case of systems with $p$ multiple delays, ($p-2$) agent parameters in the range $[0,2\pi]$ and one parameter in $[0,\pi]$ are swept to determine delay-independent stability without requirement of solving equations. A graphical projection approach for multiple delays is proposed in case unknown parameters exist. The complete delay-independent stability analysis of a second order PD-controlled system with single delay is presented executing the proposed method. Also, the method is applied to find the exact delay-independent stabilizing regions of unknown parameters of systems with two and three delays. Next to the contributions for delay-independent stability analysis, using the symmetry feature resulting from the self-inversive property of auxiliary characteristic polynomial and the efficiency of utilized zero location test, a well-known procedure (Cluster Treatment of Characteristic Roots) for delay-dependent stability analysis of multiple time-delay systems is improved. Actually, the computational load in numerical procedure is reduced by proposing an partial analytical approach to determine stable regions in delay parameter space. The achieved improvement is demonstrated by applying the new procedure to a system with three delays, which was analyzed before. Moreover, the proposed method is performed to obtain the stable regions in delay space to assure safe distance in the train following problem with multiple delays, where the delays arise up to data transfer in communication between the trains and the wayside control unit. In the second direction, as a delay based control technique, the design of time-delay filters to pre-compensate oscillatory modes of flexible systems is dealt with. In particular, we focus on the parametrization of well-known time-delay filters, which are referred in the literature as command smoothing profiles (smoothers) and input shapers, with distributed delays. These smoothers and shapers can readily be parametrized by analytical formulation if the oscillatory mode is undamped. However, the parametrization requires inherently the numerical solution of related equations (usually with an optimization procedure) for the damped oscillatory modes. To overcome this issue, we modify the structure of the filters by utilizing a straightforward complex domain transformation and a compulsory gain compensation, which enables the closed-form parametrization for the damped case. The reference and the system output signals settle without vibration preserving the time lengths of the pre-forms for the undamped case, when the modified filtering structures are applied to the reference. The transformation with the gain compensation is performed on Trapezoidal, S-curve and Trigonometric smoothers, Jerk Limited shaper, and Distributed-delay Zero Vibration shaper which was proposed recently. Then, we examine the obtained new types of smoothers and shapers with exponential distribution of delays in essence, in time and frequency domain. Consequently, the basic properties such as performance of the responses, spectrum distribution and robustness analysis, are confirmed and cross-compared in a case study example. In addition to these, the proposed new shaper with lumped and exponentially weighted delay distribution is considered in the feedback architecture with its inverse form, for vibration suppression of flexible systems controlled with magnitude saturated actuators. It is shown that such distributed-delay shapers with the inverse form in the feedback path have the capability of canceling the undesired vibration caused by saturation limit of actuators. The main idea is to treat the saturation effect as a disturbance on the control input of the actuator which can be canceled by the shapers in the feedback control loop. The theoretical results are compared with an existing solution and verified on a laboratory set-up.