Elsevier

Knowledge-Based Systems

Volume 146, 15 April 2018, Pages 12-43
Knowledge-Based Systems

Performance analysis of robust stable PID controllers using dominant pole placement for SOPTD process models

https://doi.org/10.1016/j.knosys.2018.01.030Get rights and content

Highlights

  • Dominant pole placement tuning of PID controllers is proposed for SOPTD processes.

  • Analytical expressions are derived considering 3 closed loop non-dominant pole type.

  • Third order Pade approximation is used to handle the time delay terms.

  • Robust stable solution of PID controller are achieved using k-means clustering.

  • Different time and frequency domain performances are shown for 9 test-bench plants.

Abstract

This paper derives new formulations for designing dominant pole placement based proportional-integral-derivative (PID) controllers to handle second order processes with time delays (SOPTD). Previously, similar attempts have been made for pole placement in delay-free systems. The presence of the time delay term manifests itself as a higher order system with variable number of interlaced poles and zeros upon Pade approximation, which makes it difficult to achieve precise pole placement control. We here report the analytical expressions to constrain the closed loop dominant and non-dominant poles at the desired locations in the complex s-plane, using a third order Pade approximation for the delay term. However, invariance of the closed loop performance with different time delay approximation has also been verified using increasing order of Pade, representing a closed to reality higher order delay dynamics. The choice of the nature of non-dominant poles e.g. all being complex, real or a combination of them modifies the characteristic equation and influences the achievable stability regions. The effect of different types of non-dominant poles and the corresponding stability regions are obtained for nine test-bench processes indicating different levels of open-loop damping and lag to delay ratio. Next, we investigate which expression yields a wider stability region in the design parameter space by using Monte Carlo simulations while uniformly sampling a chosen design parameter space. The accepted data-points from the stabilizing region in the design parameter space can then be mapped on to the PID controller parameter space, relating these two sets of parameters. The widest stability region is then used to find out the most robust solution which are investigated using an unsupervised data clustering algorithm yielding the optimal centroid location of the arbitrary shaped stability regions. Various time and frequency domain control performance parameters are investigated next, as well as their deviations with uncertain process parameters, using thousands of Monte Carlo simulations, around the robust stable solution for each of the nine test-bench processes. We also report, PID controller tuning rules for the robust stable solutions using the test-bench processes while also providing computational complexity analysis of the algorithm and carry out hypothesis testing for the distribution of sampled data-points for different classes of process dynamics and non-dominant pole types.

Introduction

The dynamic behaviours of many industrial processes are affected and governed by significant amount of time delays in the control loops. The time delay is caused by the flow of information, energy and transport of physical variables, computer processing time etc. [40]. The introduction of the time delay makes the continuous time closed loop system to have an infinite order [7] upon exponential series expansion of the delay term (eLs) which is difficult to handle with a finite term controller [36]. To alleviate this problem, there have been several works to design controllers for such systems e.g. in [68]. It is well known that most of the controllers used in the process industries are of PID type [6], [68] due to its simplicity and ease of implementation, nice disturbance rejection, tracking performance etc. Amongst many other available approaches, the Internal Model Control (IMC) based tuning of PID controllers has been quite popular to handle First Order Plus Time

Delay (FOPTD) and Second Order Plus Time Delay (SOPTD) processes, as well as Integral Process with Dead Time (IPDT) [50], [56] because of its good robustness on uncertain plants. Another approach on the design of smith predictor augmented PID controller to handle time delay processes have been reported in [4] which yields improved tracking and load disturbance rejection performances. A modified methodology is proposed with combined Smith predictor and PID controller in [35] considering challenging higher order integrating plants with delays. However the main drawback of this method is that it cannot handle unstable process with delay [40], unless an additional observer is used [22]. To overcome these problems of complicated time delay processes, the model predictive control (MPC) has got attention by many researchers but there are only few results for time delay systems [19]. Initially the MPC was developed mainly to control slower processes as it requires large computational burden for prediction and optimisation-based control. MPC controller design for time delay systems is mostly an open area and there are only few results like [28]. Another important area is designing output feedback controller [55] as well as state feedback controller [38] for stabilizing time delayed systems which are gradually gaining increased attention. In the literature some control algorithms are proposed using linear matrix inequalities (LMIs) for time delayed systems e.g. [39] to enforce robustness and several closed loop performance measures. Despite having these results, traditional pole placement remained quite challenging for time delay systems because of its increased or even infinite order.

In this paper, we propose an analytical formulation for dominant pole placement tuning of PID controllers to handle SOPTD systems. This is due to the fact that in many process industries, the dynamical behaviours in a large variety of self-regulating processes can be modelled using the SOPTD template with the flexibility of showing both sluggish and oscillatory open loop dynamics as well as different lag to delay ratio or normalized dead time [41]. PID controllers are traditionally tuned by various means like time and frequency domain performance criteria or design specifications [11]. Amongst many available approaches, the dominant pole placement method has been quite promising as the designer can specify his demand of closed loop performance, as the equivalent second order system's damping ratio, time constant or natural frequency [62]. Amongst previous approaches, dominant pole placement based PID controller design for delay free second order systems have been addressed in [13], [52] whereas its time delay version has been extended in this paper and the method has been verified on several test-bench SOPTD plants.

A continuous pole placement method (controlling the rightmost root of the closed loop system and shift it to the left half of the s-plane in quasi-continuous way) has been proposed to design a semi-automated pole placement based state feedback controller for retarded [36] and neutral type [37] delay systems. By this method, the closed loop roots lying in the extreme right-hand side is shifted to the far possible left-hand side. However, the methodology does not allow direct pole placement for SOPTD system and only monitoring the real part of the roots. To overcome the above problem another methodology is proposed in [38] which combines direct pole placement and the minimization of the spectral abscissa for determining controller parameters in retarded time-delay systems. There are some other approaches on stability analysis of time delayed single input single output (SISO) systems to derive controller gains by computing the root locus. Using the characteristic equation which leads to a transcendental equation in the presence of delays which is also known as the quasi-polynomial, several methods have been proposed to construct the root locus which creates horizontal asymptotes [29], [34], [64]. Other root locus based stabilization methods are also reported to analyse state space models with input delay [20], state delay or both [58] by using the root locus.

The PID controller tuning by direct pole assignment is found to be a difficult approach for time delay systems, as the time delay in a process makes the closed loop system to have an infinite order. Therefore, to handle time delay systems, the direct pole placement in complex s-plane is not recommended as suggested in the pioneering work on dominant pole placement tuning in [62]. The methodology in [62] also suggested a Nyquist based design for frequency domain stabilization of the time delay systems using PID controllers. The main hurdle with the pole placement for delay systems has been the fact that the exponential delay term in Laplace domain (i.e. eLs), manifests itself as very high order transfer function upon approximations using Pade methods of a specified order [57]. Therefore, such a pole placement approach will need relocation of many open loop poles at a time with a compact finite term (three-term for PID) controller when the number of interlacing pole-zeroes, arising due to the approximation of the time delay term are too many to handle for a chosen order of approximation. In our derivations, the order and approximation method are considered to be fixed. In particular, we apply a third order Pade approach to approximate the time delay term in the process model. Therefore, it can be considered as a new area of research to get a clearer picture of the dominant pole placement design for time delay systems where the task is to handle many poles and zeros of the combined plant and delay with finite number of controller parameters. To demonstrate the methodology, SOPTD processes with various delay to lag ratio have been used which shows the strength of the algorithm for even quite challenging plants e.g. delay dominant systems with low open loop damping which are much harder to control using standard PID controller tuning methods.

Section snippets

Theoretical formulation

In this section, the dominant pole placement based PID controller design has been shown to handle SOPTD systems. The time delay term has been approximated using a third order Pade's approximation instead of considering it as transcendental term in the quasi-polynomial [57]. The co-efficient matching based pole placement method has been described previously in [6], [33] for the control of delay free systems. We apply here a similar co-efficient matching method to design dominant pole placement

Determining the robust stable solutions using centroid of the stability region

We here use the k-means clustering algorithm to determine the centroid of the stability region in the PID controller space which can have a complex shape in the 3D parameter space of the controller gains {Kp, Ki, Kd}. The stability regions are determined for nine classes of SOPTD plants and it is also checked that a single centroid represents the stability regions of a unimodal distribution in the controller parameter space. Otherwise if a multi-modal distribution is discovered, indicating more

Test-bench SOPTD processes for performance evaluation

For each of the nine classes of test-bench plants under investigation, we also tabulate which pole configuration and expression for the PID controller gains yield the largest stability region, as explored from the number of stabilizing solutions obtained from the Monte Carlo simulations on the chosen design parameter space. The percentage volume can be represented as the ratio of accepted stable solutions to the total number of uniformly distributed (105) random samples drawn from the chosen

Discussions

This paper proposes three types of dominant pole placement based PID controller design methods for processes with time delays using different criteria for the non-dominant poles i.e. all complex conjugate, all real and mixture of them. The assumption of different types of non-dominant poles are used to get many alternative candidate expressions for the PID controller gains, so that the designer can choose the most robust stable one amongst competing solutions, while also maintaining acceptable

Conclusion

In order to handle SOPTD processes, dominant pole placement based PID controller tuning has been considered with three different non-dominant pole types. The k-means clustering technique is used here to achieve the robust stable PID controller gains inside the stability region in the controller parameter space, obtained by random Monte Carlo sampling of the chosen pole placement parameter intervals. Different closed loop performance measures in both time and frequency domain have been analysed

Acknowledgement

KH acknowledges the support from the University Grants Commission (UGC), Govt. of India under its Basic Scientific Research (BSR) scheme.

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