General Stabilization Approach of the Low-Order Controller for Both SISO Systems and MIMO Systems with Multiple Time Delays

The stabilization of the system with multiple time delays is much more difficult than that of the rational system since the delay multiplicity complicates the system analysis. For an arbitrarily-given linear time-invariant (LTI) multiple time-delay plant, the stabilization problems of the PID controllers are solved in this paper. Necessary and sufficient conditions for the stability of general single-input and single-output (SISO) systems and multiple-input and multiple-output (MIMO) systems with multiple time delays are presented by considering the argument transformation of closed-loop characteristic function and real and imaginary roots of related quasipolynomial in a sufficiently large frequency range. Based on such the conditions, analytical algorithms are then proposed to compute the complete stabilizing sets of low-order controllers. For the MIMO system with multiple time delays, firstly, MIMO systems are decomposed into several independent single loop subsystems with multiple time delays by the method of equivalent transfer function (ETF). Then, the stabilizing PID controllers for all independent single-loop subsystems are presented on the basic of the stabilizing result of the PID controller for SISO systems with multiple time delays, from which the stabilizing regions of the PID controllers for MIMO system with multiple time delays can be derived. More importantly, the results accomplished here provide a strong theoretical basic and technical guidance for the low-order controller design and tuning of the system with multiple time delays. Numerical examples are provided to illustrate the validity of the proposed algorithms.