The τ Decomposition Method for PID Controllers of First Order Delayed Unstable Processes

Abstract

The stabilization of first‐order delayed (FOD) unstable processes with proportional–integral–derivative (PID) controllers is considered, and all the feasible PID controllers are determined. Different from the existing results which are based on the D‐partition technique and partitioning complex's real and imaginary parts, a novel procedure enlightened by the τ decomposition method is proposed to characterize the space of controller parameters. Generally speaking, the parameter space is mainly divided into four regions: delay independent stable region, delay independent unstable region, delay interval dependent stable region, and delay interval dependent unstable region. The depiction of the space partition boundaries and the determination of stable intervals become the main problems. In our work, these boundaries are related to the existence of the purely imaginary roots (PIRs) of the systems' characteristic equation, and the determination of stable intervals refers to the number and value of the PIRs. Thus, the key points are the number and calculation of PIRs. According to our discussion, analytical results on both topics are obtained in explicit form. Finally, vivid numerical simulations are given to illustrate the results.

All‐Stabilizing Proportional Controllers for First‐Order Bi‐Proper Systems with Time Delay: An Analytical Derivation

“…Furthermore, a possible future work is to extend these results for second-order biproper systems with time delay for different types of controllers such as P, PI, and lead/lag. 27, it is sufficient to prove that the numerator of this expression is negative for 0 < ω < ∞. Apparently for ω ≤ ffiffiffiffiffi pz p , the expression is negative.…”

“…In , an analytical characterization of all‐stabilizing PD controllers for first‐order unstable delay processes is given by employing the Hermite‐Biehler theorem for quasipolynomials. In , the controller parameter space of P, PI, PD, and PID controllers is successfully divided into intervals of delay‐independent (dependent) stability–instability regions. In , based on the Nyquist stability criterion, an algorithm is proposed to compute all‐stabilizing P controllers for general‐order time‐delay systems with strictly proper transfer functions, which can also be used for first‐order systems.…”

Synthesis of a linear controller for a control system with interval parameters on a base of D‐partition in vertices of a coefficient's polyhedron of characteristic polynomial

“…The main goal in the IDS design is to ensure the desired control quality. There are a lot of studies which deal with the methodology for parametric design of linear robust controllers based on the ICP [1–27]. One should single out root methods that differ from others by their simplicity and visualization [3–6,9,11,13,14,16,17,20,22–25,27].…”

“…To determine the linear controller parameters that ensure allowable root quality indices in the control system at any values of the ICP coefficients, it is proposed to use the D-partition method [2,4,5,7,8,10,12,14,18,20,23,25]; the interval extension of this methodology is carried out in terms of the ICP. The disadvantage of the proposed solution is in the necessity to apply the D-partition method for all 2 m vertices of the coefficient polyhedron (m is the number of the interval ICP coefficients), which is a rather laborious procedure.…”

Integral-proportional derivative controller tuning rules obtained from optimum responses for set-point tracking of unstable processes with time delay

“…There are enormous approaches for tuning PID controllers to control open-loop unstable processes. Some common approaches used to design PID controllers are as follows: D-partition-based design methods (Cai et al, 2016; Hwang and Hwang, 2004; Shafiei and Shenton, 1997), gain-phase margin–based design methods (Amini and Rahmani, 2021; Ho and Wang, 2003; Tang et al, 2007; Wang, 2014), relay feedback–based design methods (Majhi and Atherton, 1999; Vivek and Chidambaram, 2005; Wang, 2020), internal model control (IMC)–based controller design methods (Begum et al, 2016; Chu et al, 2018; Jin et al, 2014; Kumar and Sree, 2016; Panda, 2009), and direct synthesis–based design methods (Cho et al, 2014; Ravikishore et al, 2020; Siddiqui et al, 2020; Vanavil et al, 2015).…”