CASE STUDIES OF COEFFICIEANT DIAGRAM METHOD – PRACTICAL POLYNOMIAL DESIGN APPROACHES

Vibration suppression controls of two-inertia systems have been proposed by many researchers. However, for three-inertia systems, it is difficult to obtain the adequate gains of the PID controller, which suppress the vibration of the speed of the load side. The Coefficient Diagram Method (CDM) that can design the optimum controller’s gains for the 2-inertia systems has been proposed. In this paper, the control method of vibration of multi-inertia systems, such as 3-inertia and 4-inertia systems, based on the CDM is proposed. Especially, the coefficient diagram is used to determine the controller gains. The validity of the proposed method is shown by the computer simulation for both the 3-inertia system and the 4-inertia one. Furthermore experimental results is also presented for the 3-inertia system.

Coefficient diagram method is a controller design technique for linear time-invariant systems. This design procedure occurs into two different domains: an algebraic and a graphical. The former is closely paired to a conventional pole placement method and the latter consists on a diagram whose reading from the plotted curves leads to insights regarding closed-loop control system time response, stability and robustness. The controller structure has two degrees of freedom and the design process leads to both low overshoot closed-loop time response and good robustness performance regarding mismatches between the real system and the design model. This article presents an overview on this design method. In order to make more transparent the presented theoretical concepts, examples in Matlab ®code are provided. The included code illustrates both the algebraic and the graphical nature of the coefficient diagram design method.

Importance of Polynomial in Coefficient Diagram Method

A new robust control strategy coefficient diagram method (CDM), which is an algebraic approach applied to a polynomial loop in the parameter space, is studied in this paper. Here, a model-based CDM is developed and proposed for uncertain systems. A special diagram called a coefficient diagram, which is used as a vehicle to carry the necessary information and as the criteria of good performance, is designed and modified accordingly. The performance of this novel controller is illustrated through several examples, and numerical results are provided.

A new adaptive controller based on coefficient diagram method (CDM) as an active queue management (AQM) method is introduced. In the developed AQM (ACDM), the output feedback pole placement is implemented in an indirect model reference adaptive control (MRAC) scheme, in which the closed loop characteristic polynomial is determined by CDM and the system parameters are estimated using a modified recursive gradient method. This method preserves existing features of CDM controller in the control of networks with unknown or varying parameters. Simulation results illustrate the good performance of the controller

This paper proposes a novel load frequency control (LFC) approach formulated on an optimal structure of the coefficient diagram method (CDM) in a two-area thermal power system. As part of a realistic analysis, nonlinearities related to governor dead band (GDB) and generation rate constraint (GRC) have been considered. In this article, a hybrid CDM method is combined with the optimization of its mathematical equations to achieve an innovative controller. Furthermore, a new metaheuristic optimization technique called the water cycle algorithm (WCA) is used to determine the optimal coefficients of the CDM controller. For the purpose of demonstrating the validity of the proposed scheme, a wide range of uncertainties in the dynamic parameters of a nonlinear power system were tested. In addition, a comparative study is presented between the results obtained from a classical integral, CDM alone, optimized fuzzy, optimized PID, and the suggested controller. In this new approach to improved control, algebraic support provides a robust and responsive controller that can provide fast and stable dynamic responses and effectively overcome the detrimental effects of nonlinearities and uncertainties in the parameters of the power system.

In this work, a PI-P control strategy based on the coefficient diagram method (CDM) is proposed. The proposed CDM–PI-P control strategy is tested with the pH neutralization system. The system is modeled as first order plus time delay (FOPTD) transfer function using the experimental step test method. Closed loop simulations are carried out at two different critical operating points and the servo performance of the CDM–PI-P controller in terms of integral square error (ISE), integral absolute error (IAE), total variation (TV), and time domain specifications are reported. The results of the proposed control strategy are compared with other conventional PI control techniques.

Solution to a Benchmark Control Problem using the CDM Design Method

We consider the problem of designing P, PI and PID controllers for interval plants. It is known from the area of parametric robust control that the stabilization problem for an interval plant are reduced to the equivalent problem of stabilizing either a finite set of certain extremal plants. This paper shows that under special conditions imposed on controller parameters and the order of plant, the number of extremal plants required for stabilization on PID controllers are reduced. Thereafter, using Ho's method, it allows one to obtain all the stabilizing set of PID gains for a interval plant family. We assure that the coefficient diagram method (CDM) is very useful for designing a controller because the design in CDM is not only carried out directly in the parameter space but also easy to meet the time domain specification, as shown in a numerical example.

The aim of this paper is to determine which type of control loop is better for each particular type of standard voltage source inverter (VSI) load (static, dynamic, nonlinear rectifier resistive-capacitive RC load). A comparison of three different types of controllers for single-phase VSIs is presented. The first two are of the single input single output (SISO)/proportional-integral-derivative controller (PID) and coefficient diagram method (CDM) types, and the third is of the multi-input single output (MISO)/passivity-based control (PBC) type. The selections of the gains for SISO and MISO controllers are presented, including the problem in the choice of PBC controller gains caused by the imperfection of the pulse width modulation (PWM) modulator. For a standard nonlinear rectifier RC load, the new control quality factor (CQF) is defined to distinguish the properties of the controllers. The results show the superiority of the MISO-PBC controller for the RC load; however, for a linear dynamic load, the MISO-PBC performed worse than the SISO-PID or CDM. Therefore, the choice between either a simple and inexpensive SISO with one measurement trace or an expensive MISO with three measurement traces depends on the purpose of the VSI. The initial simulations and measurements of the experimental models are presented and discussed.

This work proposes the coefficient diagram method (CDM)-based two degree of freedom proportional integral (CDM-PI) controller tuning rules for stable and unstable first order plus time delay (FOPTD) processes and pure integrating processes with time delay (PIPTD). To derive the tuning rules, a general first order plus time delay (FOPTD) model, the first order Taylor denominator (TD) approximation technique and the pole allocation strategy named CDM is used. The tuning rules derived here are novel and they relate the controller parameters to the process model parameters directly. The performance of the CDM-PI controller utilising the proposed tuning rules is tested with numerical examples of stable, unstable and pure integrating processes with time delay models. The test results indicate that the proposed tuning rules yield promising results over the other PI controllers. Performance measures confirm the effectiveness of the proposed tuning method.

Load frequency control is a major issue in power systems operation.The performance of adaptive particle swarm optimization (PSO) , coefficient diagram method (CDM) and conventional integral (PI) controller are compared for an isolated single area non re-heated turbine microgrid with using both of electric vehicle (EVs) and heat pump water heaters (HPs) as controllable loads.The dynamic response of LFC is studied using MATLAB 2017a Simulink package to validate the effectiveness of the proposed scheme.The results indicated that the system with PSO technique has better convergence and robustness than CDM in face of uncertainties due to load perturbations.

This chapter, on practical nonlinear control design, includes a variety of open-loop and closed-loop control design techniques that not only do not avoid the nonlinearities of the model but sometimes utilize the nonlinearities as part of a successful control strategy. The chapter includes many MATLAB Simulink examples. One of the chapter purposes is to serve as a MATLAB Simulink tutorial. Nonlinearities in the process model are often the “heart and soul” of the process model. We tried to demonstrate it with many real-life examples—aircraft motion models, missile guidance, physiological models, and more. In all these examples, the nonlinearities play a central role. Dynamic simulations, if done carefully and systematically, can compensate for the lack of full control design experience and training. A simulator in the loop that performs multiple runs of a slow enough process can provide meaningful control insight that can be put into action in real time. Sliding mode control and Lyapunov Design are powerful control design techniques in their own right. These control design methods particularly shine in the presence of modeling uncertainties and poorly modeled disturbance signals. One may even apply these methods to linear process in order to obtain superior performance at certain circumstances. Due to time and space constraints, several subjects regrettably ended up being omitted from this chapter (awaiting possibly future editions of this book). These subjects include Model Linearization by means of MATLAB and Simulink and PID Control Auto-Tuning.

Time delays are inherent in modern motion-control and electric-drive loops due to sensing, filtering, sampling and computation, communication, and actuation scheduling. When such delays are only partially known, they can markedly reduce stability margins and narrow the admissible range of state-feedback gains, especially in high-bandwidth servo applications. This paper develops a design-oriented state-feedback framework for delay-affected plants based on the Manabe polynomial concept and the Coefficient Diagram Method (CDM). The plant is represented as a chain of integrators of order two to four with an effective input gain, and the feedback gain is synthesized for the nominal delay-free model by matching a standard Manabe/CDM characteristic polynomial using the classical CDM stability-index pattern. When an unmodeled input delay is present, the closed loop is governed by a delay-dependent characteristic equation. By introducing a normalized representation, the analysis yields explicit delay-stability limits that directly translate into a lower bound on the equivalent time constant used for tuning. The degradation of the phase margin and gain margin with increasing normalized delay is quantified as design charts, and a simple phase-margin-based inequality is proposed for selecting the tuning time constant, with gain-margin checks recommended as a verification step.

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