A methodology to compute and design the delay margin (DM) of large-scale linear time-invariant systems is presented. Different from existing work, this methodology is scalable; does not impose restrictions on the system to be able to invoke simultaneous triangularization or simultaneous diagonalization; and sheds light on how the finite number of delay-free system eigenvalues can be effectively utilized to compute or design the DM.
KEYWORDSdelay margin, delay margin design, time-delay systems
INTRODUCTIONThe presence of time delays in feedback control systems has attracted tremendous interest in engineering, mathematics, and physics communities for over six decades. [1][2][3][4][5][6] The main reason for this is that delays can dramatically affect the dynamics of control systems. Notably, they can cause poor performance and even instability; and this must be properly dealt with either through controller design or through system design. Virtually, in all control applications, delays exist because it takes time to transmit information, sense information, formulate a proper decision, and execute such decisions. Widely studied problems include machine tool chatter, 7,8 teleoperation, 9,10 human reaction times in driving 11,12 and when operating an aircraft, 13,14 power networks, 15,16 population dynamics, 17,18 supply chain networks, 19,20 and gene regulatory networks. 21,22 In the pursuit of addressing the fundamental stability analysis and control design needs, one faces numerous challenges and opportunities. For example, the presence of delays makes the system infinite dimensional. In linear time-invariant (LTI) systems, this means that one has to deal with infinitely many poles, and thus, standard tools available for finite-dimensional problems are immediately ruled out as prospects to analyze stability and/or design controllers. Moreover, the presence of delay is not necessarily detrimental to the dynamics. An appropriate amount of delay can render improved disturbance rejection capabilities, 23 can increase stability margins, 24 can be deliberately introduced as part of a controller to enhance the performance of a closed-loop system, 25-28 or can provide stability. 29,30 One of the widely studied problems in the context of time-delay systems is the problem of computing the delay margin * (DM) of a system. An LTI system is stable for all delays from zero up to the DM ∈ [0, * ), which implies that when the delay value is equal to the DM = * , the system has at least one pole on the imaginary axis of the complex plane. 1 Many techniques have been developed to calculate the DM of LTI systems. Efforts in this direction possibly go back to 31 where the authors developed an approach to create "stability maps" of LTI systems with delay. These maps represent, on the plane of the delay and a parameter of interest, the boundaries along which the system can have imaginary poles ∓ j and which side of the boundaries indeed correspond to the stable operation of the system. Approaches to obtain similar maps have also been deve...
