Honghai Wang

On τ-Decomposition Frequency-Sweeping Test for a Class of Time-Delay Systems. Part II: Multiple Roots Case

Abstract This paper addresses the stability analysis problem of a class of linear system with commensurate delays in a frequency-domain setting. In Part I of this paper, only the simple imaginary roots (SIRs) case is considered. In Part II of the paper, the case of multiple imaginary roots (MIRs) will be studied and general results on the issues of Puiseux series expansion, invariance, and instability persistence will be presented. By the approach proposed in this two-part paper, we can study the complete stability for the time-delay systems under consideration.

New result on PID controller design of LTI systems via dominant eigenvalue assignment

This note considers the problem of assigning the dominant eigenvalues of a linear time-invariant (LTI) system to the desired positions by using proportional-integral-derivative (PID) controllers. The procedure is based on first setting some rightmost eigenvalues of the system at the desired positions and then guaranteeing the dominance of those rightmost eigenvalues by using the generalization of the Hermite-Biehler Theorem. It is worth pointing out that this work aims to ascertain the gains of PID controllers in a straightforwardly computational way which plays an important role in practical applications.

Controller design for delay systems via eigenvalue assignment – on a new result in the distribution of quasi-polynomial roots

This paper considers the eigenvalue distribution of a linear time-invariant (LTI) system with time delays and its application to some controllers design for a delay plant via eigenvalue assignment. First, a new result on the root distribution for a class of quasi-polynomials is developed based on the extension of the Hermite–Biehler theorem. Then, such result is applied to proportional–integral (PI) controller parameter design for a first-order plant with time delay through pole placement. The complete region of PI gains can be obtained so that the rightmost eigenvalues in the infinite eigenspectrum of the closed-loop system with delay plant are assigned to desired positions in the complex plane. Furthermore, on the basis of the previous result, this paper also extended the PI control to the proportional–integral–derivative (PID) control. It is worth pointing out that this work aims to improve the performance of the closed-loop system on the premise of guaranteeing the stability.

On τ-Decomposition Frequency-Sweeping Test for a Class of Time-Delay Systems. Part I: Simple Imaginary Roots Case

Abstract This paper addresses the stability analysis problem of a class of linear system with commensurate delays in a frequency-domain setting. In Part I of this paper, only the simple imaginary roots (SIRs) case is considered. First, the characteristic quasipolynominal is reformulated into a factorization form. Then, it is found that (i) the crossing direction of imaginary roots (CDIRs) can be reflected by an appropriate frequency-sweeping test, (ii) the CDIRs are determined by a certain order of the imaginary roots by magnitude and finally, (iii) a time-delay system is ultimately unstable if there is at least one crossing imaginary root. To apply the properties (i)-(iii), an easily implemented frequency-sweeping method is proposed, by which the CDIRs can be obtained without any further computation. The method can be used for the complete stability of the system. In Part II of the paper, the MIR case will be discussed.

Proportional-derivative controllers for stabilisation of first-order processes with time delay

This paper considers the problems of determining the complete stabilising set of proportional-derivative controllers for a first-order process with time delay. First, by employing a version of the Hermite–Biehler theorem applicable to quasi-polynomials, a complete set of all stabilising proportional-derivative parameters for first-order processes with constant time delay are obtained. Next, we provide an approach to design a robust PD controller to stabilise a first-order process with uncertain time delay, which lies inside a known interval.

All stabilizing sets for proportional–integral controller of high-order delay processes

This article considers the problem of determining the complete stabilizing set of proportional–integral controllers, which is applied in a class of high-order processes with time delay in complex-frequency domain. First, we give a necessary condition for a proportional–integral controller to stabilize the process with certain constant delay using Descartes’ rule of signs. Then by employing the generalization of the Hermite–Biehler theorem applicable to quasi-polynomials, a complete set for all stabilizing proportional–integral parameters is derived to stabilize the open-loop stable and unstable systems. In the case of uncertain time delay, we provide a design approach to stabilize the related plant with a robust proportional–integral controller, where the unknown but constant time delay lies inside a known interval. Three examples illustrate the effectiveness of the proposed results.

New Result on Low-Order Controller Design for First-Order Delay Processes via Eigenvalue Assignment

Abstract This article developed a new result on the eigenvalue distribution for a certain class of time delay systems based on the extension of the Hermite-Biehler Theorem. Such result is applied to proportional-integral (PI) controller parameter design for a first-order plant with time delay via eigenvalue assignment. Using the method provided in this paper, one can assign the rightmost eigenvalues of the closed-loop system to desired positions in the complex plane. Further, on the basis of the previous result, this paper also extended the PI control to the proportional-integral-derivative (PID) case.

Nuclear norm subspace identification for continuous-time stochastic systems based on distribution theory method

A novel method for nuclear norm subspace identification of continuous-time stochastic systems based on distribution theory is proposed. The time-derivative problem of the system is solved by using random distribution theory, which is the key to obtain the input-output algebraic equation in the time-domain. Due to the fact that the system encounters the stochastic noise, we design a Kalman filter to achieve the state estimation and noise reduction. Nuclear norm minimization is constructed to optimize the system order in the process of subspace identification. Further, the optimization problem is solved by the alternating direction method of multipliers. Simulation results are provided to show the effectiveness of the proposed method.

PID controller tuning for neutral type systems with time delay via dominant eigenvalue assignment

This paper considers the problem of proportional-integral-derivative (PID) controller design for a closed-loop feedback system with time delay according some desired performance indexes. In this study, the characteristic equation of the closed-loop system is considered as that of a neutral type system with time delay. Besides, it is expected that the step response of the closed-loop systems has no overshoot. To achieve the objective of control, combining with Smith Predictor, we propose a new approach on the PID controller design for a neutral type delay system via dominant eigenvalue assignment. Such a method can make the performance of the system very close to the desired performance indexes for a standard first-order system. A numerical example is given to illustrate the effectiveness of the proposed approach.

New Results on Eigenvalue Distribution and Controller Design for Time Delay Systems

This paper considers the eigenvalue distribution of a linear time-invariant (LTI) system with commensurate time delays and its application to proportional-integral-derivative (PID) controller design for a delay plant via dominant eigenvalue assignment. A new result on the root distribution of a quasi-polynomial is first produced by applying part of Pontryagins conclusions. This result which gives a necessary and sufficient condition can be directly used to judge the number of the right-half plane eigenvalues of the characteristic equation of a time delay system. Based on the proposed result, necessary and sufficient conditions on dominant eigenvalue assignment for PID control of time delay systems are presented and an algorithm is then provided to determine the PID controller gains. The proposed approaches can assign some (one or two) eigenvalues to the desired positions and all the other eigenvalues to the left of a given line to guarantee the dominance of the assigned ones, which enables us to design the controller according to the desired performance indexes for a standard first-order system or a standard second-order system in addition to stability. The method is effective for the closed-loop characteristic equation being retarded type or neutral type. The controller gains to achieve the control objective can be characterized by a straightforward computation. Further, a result on degradation to proportional-integral (PI) control of time delay systems is given.