Performance analysis of robust stable PID controllers using dominant pole placement for SOPTD process models
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 () 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. ), 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.
References (69)
- et al.
Fractional adaptive control for an automatic voltage regulator
ISA Trans.
(2013) - et al.
PID control in terms of robustness/performance and servo/regulator trade-offs: a unifying approach to balanced autotuning
J. Process Control
(2013) - et al.
PID autotuning for weighted servo/regulation control operation
J. Process Control
(2010) Advanced controller auto-tuning and its application in HVAC systems
Control Eng. Pract.
(2000)Design of a fractional order PID controller for hydraulic turbine regulating system using chaotic non-dominated sorting genetic algorithm II
Energy Convers. Manage.
(2014)- et al.
Improved model reduction and tuning of fractional-order PIλDμ controllers for analytical rule extraction with genetic programming
ISA Trans.
(2012) On the selection of tuning methodology of FOPID controllers for the control of higher order processes
ISA Trans.
(2011)- et al.
Multi-objective LQR with optimum weight selection to design FOPID controllers for delayed fractional order processes
ISA Trans.
(2015) - et al.
Pareto optimal robust design of fractional-order PID controllers for systems with probabilistic uncertainties
Mechatronics
(2012) - et al.
Design of PID-type controllers using multiobjective genetic algorithms
ISA Trans.
(2002)
New methods for process identification and design of feedback controller
Chem. Eng. Res. Des.
Continuous pole placement for delay equations
Automatica
An eigenvalue based approach for the stabilization of linear time-delay systems of neutral type
Automatica
Control design for time-delay systems based on quasi-direct pole placement
J. Process Control
Chaotic multi-objective optimization based design of fractional order PIλDμ controller in AVR system
Int. J. Electr. Power Energy Syst.
Frequency domain design of fractional order PID controller for AVR system using chaotic multi-objective optimization
Int. J. Electr. Power Energy Syst.
PID tuning rules for SOPDT systems: review and some new results
ISA Trans.
Design and performance analysis of PID controller for an automatic voltage regulator system using simplified particle swarm optimization
J. Franklin Inst.
A conformal mapping based fractional order approach for sub-optimal tuning of PID controllers with guaranteed dominant pole placement
Commun. Nonlinear Sci. Numer. Simul.
A novel optimal PID plus second order derivative controller for AVR system
Eng. Sci. Technol. Int. J.
Tuning of PID controller for an automatic regulator voltage system using chaotic optimization approach
Chaos Solitons Fractals
Guaranteed dominant pole placement with PID controllers
J. Process Control
PID auto-tuner based on sensitivity specification
Chem. Eng. Res. Des.
Fractional order PID control design for semi-active control of smart base-isolated structures: a multi-objective cuckoo search approach
ISA Trans.
Design of a fractional order PID controller for an AVR using particle swarm optimization
Control Eng. Pract.
Design of fractional order PID controller for automatic regulator voltage system based on multi-objective extremal optimization
Neurocomputing
CAS algorithm-based optimum design of PID controller in AVR system
Chaos Solitons Fractals
A new Smith predictor for controlling a process with an integrator and long dead-time
IEEE Trans. Autom. Control
Revisiting the Ziegler–Nichols step response method for PID control
J. Process Control
Computer-Controlled Systems: Theory and Design
Continuous firefly algorithm for optimal tuning of PID controller in AVR system
J. Electr. Eng.
PID controllers: recent tuning methods and design to specification
IEE Proc. Control Theory Appl.
Inverse optimal control formulation for guaranteed dominant pole placement with PI/PID controllers
Cited by (21)
Optimal super twisting sliding mode control strategy for performance improvement of islanded microgrids: Validation and real-time study
2024, International Journal of Electrical Power and Energy SystemsStabilizing region in dominant pole placement based discrete time PID control of delayed lead processes using random sampling
2022, Chaos, Solitons and FractalsCitation Excerpt :It is well-known that the control performance of a closed-loop system depends on its pole locations. Thus, dominant pole placement based PID controller design has got popularity to handle time delay systems [4,11], where the user can specify desired closed loop performance criteria. However, it is quite challenging to design PID controllers for time delay systems, since the presence of time delay makes the closed loop system to have infinite order transcendental known as the quasi-polynomial [3,12].
Analytical design of discrete PI–PR controllers via dominant pole assignment
2022, ISA TransactionsCitation Excerpt :It is known that pole assignment method is well-known technique providing a simple design procedure and the closed-loop performance is adjustable as desired. In order to deal with the difficulty of infeasibility issues that may arise in trying to locate all the poles for some (high order) systems, dominant pole assignment method can be used [6,23–28]. This method aims to place a few (usually two) so called dominant poles considering a given set of CLS performance specifications such as rise time, settling time, overshoot, etc.
Linear active disturbance rejection control for oscillatory systems with large time-delays
2021, Journal of the Franklin InstituteCitation Excerpt :So far, almost all tuning formulas for SOPDT underdamped systems are based on PID. For example, a method of dominant pole assignment of the proportional integral differential (PID) for the oscillatory SOPDT is proposed in Das et al. [8]; A fuzzy neural network control method for stabilizing underdamped plants is proposed in Shen [9]; A new control structure based on Posicast control and PID control is proposed for underdamped second-order system to achieve minimum overshoot in Oliveira and Vrančić [10]; A PID controller with cascaded delay filters is used for the control of SOPDT system in Nath et al. [11]; And a model free PID controller is designed to stabilize SOPDT system in Siddiqui et al. [12]. However, PID is an error feedback control method which will lead to some deviations in the output of the systems [13].
Genetic-Algorithm-Assisted Self-Scheduled Multidelay PIR Control: Experiments in a Car-Like Vehicle System
2024, IEEE Transactions on Cybernetics