A Conformal Mapping Based Fractional Order Approach for Sub-optimal Tuning of PID Controllers with Guaranteed Dominant Pole Placement

A novel conformal mapping based Fractional Order (FO) methodology is developed in this paper for tuning existing classical (Integer Order) Proportional Integral Derivative (PID) controllers especially for sluggish and oscillatory second order systems. The conventional pole placement tuning via Linear Quadratic Regulator (LQR) method is extended for open loop oscillatory systems as well. The locations of the open loop zeros of a fractional order PID (FOPID or PI{\lambda}D{\mu}) controller have been approximated in this paper vis-\`a-vis a LQR tuned conventional integer order PID controller, to achieve equivalent integer order PID control system. This approach eases the implementation of analog/digital realization of a FOPID controller with its integer order counterpart along with the advantages of fractional order controller preserved. It is shown here in the paper that decrease in the integro-differential operators of the FOPID/PI{\lambda}D{\mu} controller pushes the open loop zeros of the equivalent PID controller towards greater damping regions which gives a trajectory of the controller zeros and dominant closed loop poles. This trajectory is termed as "M-curve". This phenomena is used to design a two-stage tuning algorithm which reduces the existing PID controller's effort in a significant manner compared to that with a single stage LQR based pole placement method at a desired closed loop damping and frequency.Comment: 23 pages, 7 figures, in press; Communications in Nonlinear Science and Numerical Simulations, 201

DECOUPLING CONTROL OF TITO SYSTEM SUPPORTED BY DOMINANT POLE PLACEMENT METHOD

Appropriate approach to the nature of systems is a significant precondition for its successful control. An always actual issue of its mutual coupling is considered in this paper. A multivariable system with two-inputs and two-outputs (TITO) is in the focus here. The dominant pole placement method is used in trying to tune the PID controllers that should support the decoupling control. The aim is to determine parameters of the PID controllers which, in combination with decoupler, can obtain a good dynamical behavior of the system. Therefore, this kind of the centralized analytically obtained controller is used for object control. Another goal is to simplify the tuning procedure of PID controllers and enlarge the possibility for introducing the given approach into practice. But the research results indicate that the proposed procedure leads to the usage of P controllers because they enable the best performances for the considered object. Also, it is noticed that some differences from the usual rules in selection of the dominant poles gives better results. The research is supported by simulations and, therefore, the proposed method effectiveness, regarding the system behavior quality, is presented on several examples

Iterative Method for Tuning Multiloop PID Controllers Based on Single Loop Robustness Specifications in the Frequency Domain

Multiloop proportional-integral-derivative (PID) controllers are widely used for controlling multivariable processes due to their understandability, simplicity and other practical advantages. The main difficulty of the methodologies using this approach is the fact that the controllers of different loops interact each other. Thus, the knowledge of the controllers in the other loops is necessary for the evaluation of one loop. This work proposes an iterative design methodology of multiloop PID controllers for stable multivariable systems. The controllers in each step are tuned using single-input single-output (SISO) methods for the corresponding effective open loop process (EOP), which considers the interaction of the other loops closed with the controllers of the previous step. The methodology uses a frequency response matrix representation of the system to avoid process approximations in the case of elements with time delays or complicated EOPs. Consequently, different robustness margins on the frequency domain are proposed as specifications: phase margin, gain margin, phase and gain margin combination, sensitivity margin and linear margin. For each case, a PID tuning method is described and detailed for the iterative methodology. The proposals are exemplified with two simulations systems where the obtained performance is similar or better than that achieved by other authors

Root cause isolation of propagated oscillations in process plants

Persistent whole-plant disturbances can have an especially large impact on product quality and running costs. There is thus a motivation for the automated detection of a plant-wide disturbance and for the isolation of its sources. Oscillations increase variability and can prevent a plant from operating close to optimal constraints. They can also camouflage other behaviour that may need attention such as upsets due to external disturbances. A large petrochemical plant may have a 1000 or more control loops and indicators, so a key requirement of an industrial control engineer is for an automated means to detect and isolate the root cause of these oscillations so that maintenance effort can be directed efficiently. The propagation model that is proposed is represented by a log-ratio plot, which is shown to be ‘bell’ shaped in most industrial situations. Theoretical and practical issues are addressed to derive guidelines for determining the cut-off frequencies of the ‘bell’ from data sets requiring little knowledge of the plant schematic and controller settings. The alternative method for isolation is based on the bispectrum and makes explicit use of this model representation. A comparison is then made with other techniques. These techniques include nonlinear time series analysis tools like Correlation dimension and maximal Lyapunov Exponent and a new interpretation of the Spectral ICA method, which is proposed to accommodate our revised understanding of harmonic propagation. Both simulated and real plant data are used to test the proposed approaches. Results demonstrate and compare their ability to detect and isolate the root cause of whole plant oscillations. Being based on higher order statistics (HOS), the bispectrum also provides a means to detect nonlinearity when oscillatory measurement records exist in process systems. Its comparison with previous HOS based nonlinearity detection method is made and the bispectrum-based is preferred

Modern Design of Classical Controllers

Classical controller design emphasizes simple low-order controllers. These classical controllers include Proportional-Integral (PI), Proportional-Integral-Derivative (PID), and First Order. In modern control theory, it is customary to design high-order controllers based on models, even for simple plants. However, it was shown that such controllers are invariably fragile, and this led to a renewal of interest in classical design methods. In the present research, a modern approach to the design of classical controllers (by introducing a complete stabilizing set in the space of the design parameters) is described. When classical specifications such as gain margin, phase margin, bandwidth, and time-delay tolerance are imposed, the achievable performance can be easily determined graphically. The objective of this research is to determine the controller gains, contained in the stabilizing set, which satisfy desired performance specifications such as crossover frequency and closed-loop stability margins. The design procedure starts with the calculation of the stabilizing set using recent methods. Then, a simple parametrization produces ellipses and straight lines (for PI controller design) and cylinders and planes (for PID controller design) in the space of controller gains. Each set of ellipses/cylinders and straight lines/planes represents constant magnitude and constant phase loci for the controller. The main result is that the crossing points, which are the intersection of ellipses/cylinders and straight lines/planes, are selected such that they are contained in the stabilizing set of controllers. They provide the controller gains that we need to satisfy our desired robust performance, seen as desired gain margin, phase margin, gain crossover frequency, and time-delay tolerance in our system. Then, using these crossing points contained in the stabilizing set, a new plot with information about the achievable performance in terms of gain margin, phase margin, and gain crossover frequency is constructed. Each point of this achievable performance can be used to retrieve the controller’s gains, which are contained in the stabilizing set. This result provides the possibility to analyze the system’s achievable performance by exploring the stabilizing set and considering different desired configurations in the performance capabilities for the system using a PI or PID controller. This expands our possibilities when designing controllers by considering different classical controller’s configurations. This research considers the discrete-time and continuous-time linear time invariant systems and cases including First Order with time-delay in the system, and the extension to the controller design for multivariable systems. Finally, the design procedure is illustrated with different examples and real applications for all such cases

Dominant pole placement for multi-loop control systems

10.1016/S0005-1098(02)00009-2Automatica3871213-1220ATCA

Dominant pole placement for multi-loop control systems

Proceedings of the American Control Conference31965-1969PRAC