In this chapter, a Closed Loop Reaction (CRC) curvemethod for the identification of stable TITO system is discussed. The system under consideration is controlled by decentralized PI or PID controllers. The responses and interactions are modelled by the extension of Yuwana and Seborg (1982) method and the transfer function matrix for the closed loop system is obtained. Using the relation between the open loop and the closed loop transfer function matrices, the open loop transfer function matrix is obtained in Laplace (s) domain. The responses of the obtained and the actual transfer functions matrix with the original controller settings are compared. Better results are obtained if the parameters of the identified transfer function models are used as initial guess values for any optimization method and the transfer function model is identified. The higher order models are approximated to a FOPTD model. Since the method is proposed for stable systems,many of the higher order stable systems can be approximated to a FOPTD system.

Identification Method

Identification of Individual Responses

Consider a stable 2 × 2 transfer function matrix (GP ) as in Eq. (4.1a).

Let the controller settings be defined by the matrix given in Eq. (4.2).

The PI/PID controllers are selected so as to get a closed loop under-damped response. Initial controller settings are taken, based on the knowledge of the gain alone, using Davison method (Section 3.13).

Consider Fig. 4.1a. A step change of known magnitude is given to the set point yr1 and the other set point is kept unchanged. The main response obtained is y11 and the interaction response is y21. Similarly for Fig. 4.1b. A step change of the same magnitude is given as yr2 and the other set point is kept unchanged. Let the set point matrix be given as in Eq. (4.3).

The main response is denoted as y22, the interaction as y21, and the output matrix is given in Eq. (4.4).

The transfer function of the closed loop responses is assumed to be an under-damped SOPTD system. From the closed loop response curve for each case, the values yp1, yp2, ym1 and T are noted (Fig. 2.6). Using the formulas given by Yuwana and Seborg (1982), the value of τe and ζ are noted for each case. The closed loop time delay is noted from the responses.

In this chapter, a method is presented to identify the Steady State Gain Matrix (SSGM) of an unstable multivariable system under closed loop control. Effects of disturbances and measurement noise on the identification of SSGM are also studied. Davison (1976) method is modified to design single-stage multivariable PI controllers using only the multivariable gain matrix of the system. Since the overshoots in the responses are larger, a two-stage P–PI control system is proposed. Based on the SSGM, a simple proportional controller matrix is designed by the modified Davison (1976) method to stabilize the system. Based on the gain matrix of the stabilized system, diagonal PI controllers are designed.

Introduction

SISO unstable systems with time delay are difficult to control compared to stable systems. The performance specifications like overshoot, settling time for such systems are larger for unstable systems than that of the stable systems (Padmasree and Chidambaram, 2006). Methods of designing PI/PID controllers for scalar unstable systems are given by pole placement method, synthesis method, IMC method, gain and phase margin method, and optimization method (Padmasree and Chidambaram, 2006). Conventional PI controllers give a relatively large overshoot for unstable systems. Jacob and Chidambaram (1996) have proposed a two-stage design of controllers for unstable SISO systems. First, the unstable system is stabilized by a simple proportional controller. For the stabilized system, a PI controller is designed. It is shown that an improved performance is obtained. Even for the stable multivariable systems, the method of designing controllers is complicated due to the interactions among the loops. Simplemethods of designing centralized PI controllers for stable systems are reviewed by Tanttu and Lieslehto (1991) and Katebi (2012).

Govindakannan and Chidambaram (1997) have applied the method of Tanttu and Lieslehto (1991) to unstable MIMO processes. Decentralized PI controllers are also designed by the detuning method proposed by Luyben and Luyben (1997). The decentralized PI controllers do not stabilize the system if the unstable pole is present in each of the transfer functions of the system. Only centralized PI controllers stabilize such systems. Papastathopoulo and Luyben (1990) have proposed a method of calculating the steady state gain of stable multivariable systems by introducing a step change in one set point at a time.

6.1 Introduction to System-Level Modeling

In this introductory chapter, the general concepts and classification of dynamic systems in engineering are introduced; commonly used methods and computer software for modeling, simulation, and analysis of dynamic systems are previewed; and the scope of this book is outlined.

Contents

Mechanical systems are seen in machines, devices, equipment, and structures in a wide variety of engineering applications. Modeling of a mechanical system involves forces and motion about relevant objects, which can be either solids or fluids. Depending on specific concerns in an application, there are several types of mathematical models for mechanical systems at different levels of complexity, including lumped-parameter models, rigid-body models, and deformable- or flexible-body models. For instance, in studying the dynamic behaviors of an airplane, a lumped-parameter model may be good enough to describe its flight trajectory and a rigid-body model may be sufficient to study its three-dimensional motion in space. However, a flexible-body model is necessary to understand the vibration and stress in the airplane structure in performance evaluation and failure analysis.

This chapter assembles six problems of combined dynamic systems from engineering applications, namely, vibration analysis of a moving car, speed control of a coupled engine–propeller system (electromechanical system), modeling and analysis of a bimetallic strip thermometer (thermomechanical system), modeling and analysis of a resistive-heating element (electro-thermo-mechanical system), feedback control of a water purification system component (liquid-level system), and the working principles of sensors, electroacoustic, and piezoelectric devices. Each problem is presented in one section.

This chapter provides the reader with a brief refresher course on the mathematical apparatus crucial for modeling of dynamic systems. Sections 2.1 and 2.2 present basic concepts and terminology of vector and matrix algebra. Definitions and basic operations on complex numbers are introduced in Section 2.3. Section 2.4 is devoted to one of the important methods for solving differential equations – the Laplace transform. Sections 2.5 and 2.6 discuss the types of differential equations widely encountered in modeling of common dynamic systems and develop methods for solving these equations. Section 2.7 introduces the mathematical foundation for deriving transfer functions and creating block diagrams of various linear time-invariant dynamic systems. Section 2.8 presents a brief overview of solving differential equations numerically, with MATLAB and Wolfram Mathematica.

Fundamental Principles of Thermal Systems

Practically every modern engineered dynamic system has electrical components such as motors, sensors, controllers, or, at the very least, power sources. Therefore, an understanding of the physical processes occurring in the typical electrical circuits, and the ability to model behavior of an electrical subsystem are essential for anyone interested in dynamic systems.

Controls in engineering are efforts to change, design, or modify the behaviors of dynamic systems. Automatic control is control that involves only machines and devices, and that has no human intervention. Examples of automatic control are diverse, including room-temperature control, cruise control of cars, missile guidance, trajectory control of robots, control of appliances such as washing machines and refrigerators, and control of industrial processes like papermaking and steelmaking. In this chapter, for simplicity, an automatic control system is called a control system. Because the focus of this text is on the modeling and analysis of dynamic systems, only basic concepts of feedback control are introduced. For theories and methods about feedback control systems, one may refer to standard textbooks for control courses, including those listed at the end of the chapter.