Following our discussions in Ch. 5 on developing analytical and numerical solutions of system responses, Ch. 6 is focused on critical features of these solutions, known as , for systems with responses that do not tend to infinity when excited by a pulse input. The primary characteristics of these responses include the settling time, the percent overshoot, and the frequency of oscillations (i.e., if they exist), all of which can be obtained analytically, numerically, or experimentally using simple test inputs – such as steps or sinusoids. Once the response characteristics are established for a test input, the implications for more complex excitations can be understood by applying the concept of superposition (see Sec. 1.3) for linear systems.

Through our discussions in Chs. 5–10, we developed many tools and techniques and are now equipped to analyze, design, and simulate linear feedback control systems. Up to this point, we have focused heavily on idealized mathematical representations and their numerical simulation. To put this knowledge into practice, we must now address several practical considerations and non-idealities that arise when implementing feedback control systems. By doing so, we aim to fortify our intuition for the behavior of real-world dynamic systems, improve our ability to quickly develop feasible designs, and develop an understanding of the important challenges and choices that arise in the implementation of feedback control systems prior to moving forward with our discussions of advanced techniques in Part IV of this text.

In this chapter we will discuss techniques for developing mathematical models of mechanical systems. In doing so, we will take a direct approach to derive governing equations for planar rigid bodies and point masses by first reviewing the equations that describe the behavior of individual mechanical elements and later applying well-known interconnection laws to combine these elements into meaningful representations of physical dynamics. Through this approach, known as mechanical modeling, we will develop mathematical and physical models of mechanical systems with translational, rotational, levered, and geared elements.

Continuing our discussions on the methods used to analyze dynamic systems and the characteristics of their responses, we will now focus our efforts on a special kind of Laplace-domain representation known as theand will develop a set of tools and techniques relevant to transfer function analysis. By analyzing transfer functions we will be able to identify properties of dynamic systems – rather than properties of their responses – and, later, specify desirable system design characteristics in terms of transfer function properties. In turn, transfer functions will enable our eventual transition from analysis tasks to design tasks and will be pivotal in moving forward. Accordingly, we will begin by demonstrating how a linear system can be put into the form of a transfer function, and we will review some basic properties of these representations.

As we wrap up our introduction to control engineering, it is worth reflecting on the techniques covered thus far. Beginning in Ch. 8, the concept of feedback has been pivotal in designing closed-loop systems that achieve transient and steady-state performance for continuous-time plants such as the mechanical, electrical, thermal, fluidic, and other systems modeled in Chs. 1–4. Using the Laplace transform and linear-algebra-based state-space methods, we have reviewed numerous techniques for placing the poles of the closed-loop system, tracking reference inputs, and rejecting disturbances such as impulses, steps, ramps, and sine waves. These topics form the foundation of control engineering practice and are essential for understanding and applying advanced techniques. In fact, with some additional mathematics, we can extend many of these concepts and results to a broader range of dynamics and greatly improve the performance, robustness, and operating regimes of our control systems.

Thus far, we have discussed the dynamics and response characteristics of various physical systems. In doing so, we have developed many useful modeling and analysis techniques that help us to understand the behavior of first-, second-, and higher-order dynamic systems. One important application of these tools is to design systems that meet certain requirements or criteria in the responses of their output variables, such as the settling time and final value of a motor’s velocity in response to a step input.

The method suggested in the previous chapter for the identification of Multi Input Multi Output (MIMO) First Order Plus Time Delay (FOPTD) transfer function is extended to identify Critically Damped Second Order Plus Time Delay (CSOPTD) model parameters of higher order MIMO system. The closed loop transfer function is identified as a third order system. The normalized step response curve given by Clark (2005) is used to identify the main responses of the closed loop system as a third order transfer function model. From this model, the open loop model is identified by the Closed Loop Reaction Curve (CRC) method.

Identification of Multivariable Systems

A step input is given to the yr1 and the closed loop main response yc11 is obtained. The closed loop step response is assumed to be of third order of the form in Eq. (6.1).

For a step input, the transfer function will be of the form given in Eq. (6.2).

The normalized time τ is given by Eq. (6.3). Final steady state value is given in Eq. (6.5).

The shapes of the closed loop main responses are comparedwith the normalized curves given for the third order systems by Clark (2005) where the quadratic term has a damping ratio (ζ) and the most matched curve is selected. A plot is shown in Fig. 6.1. The values of the damping ratio (ζ) and β are noted. The value of ωn can be obtained from the normalized time τ given by Eq. (6.3). The value of the pole P is obtained from Eq. (6.4). The value of K is obtained from Eq. (6.5). Thus, the closed loop main responses are identified.

The interaction responses yc21 and yc12 are identified using the Yuwana and Seborg (1982) method since the method given by Clark (2005) does not have normalized plots for responses whose final steady state is 0. The closed loop interaction responses yc21 and yc12 are assumed of the form given in Eq. (6.6). To identify ζ and τe, Yuwana and Seborg (1982) method is used. The value of K is obtained from the Laplace inverse of Eq. (6.6) as shown in Fig. 6.1.

The equation for the transfer function of a closed loop response is given byMelo and Friedly (1992) as in equations Eq. (6.7) to Eq. (6.11).

This book, Identification and Classical Control of LinearMultivariable Systems, has been structured for an elective course for undergraduate students and for a core course for post graduate students of Chemical Engineering, Instrumentation and Control Engineering, and Electrical Engineering.

Many systems are described byMulti Input and Multi Output (MIMO) systems. To design controllers for such systems, we require the identification of a transfer function matrix of the system. If the system is mildly interactive, then we can design decentralized Proportional and Integral (PI) controllers based on diagonal transfer functions with appropriate detuning of the PI controllers. If the interaction is significant, then a centralized PI control system is to be designed. The design of the controller becomes complicated if the system is unstable in nature.

Classical Control Theory studies the physical systems and the control design in the frequency domain, while Modern Control Theory studies it in the time domain. Systems in the frequency domain are expressed in transfer functions (via Laplace Transforms), while the time-domain systems are described in state-space representations (a set of differential equations). In Chapter 1, a basic reviewis given of the Classical Control Theory and the Modern Control Theory.

In Chapter 2, basics of open loop and closed loop identification of transfer function models of Single Input and Single Output (SISO) systems are reviewed. An open loop method for identifying First Order Plus Time Delay (FOPTD) model and Critically Damped Second Order Plus Time Delay (CSOPTD) transfer function model is proposed. The closed loop reaction curve method and optimization method are discussed. A review of the techniques available for design of PI controllers for transfer function models for SISO systems is brought out.

In Chapter 3, the concept of relative gain array for the measure of interactions in a multivariable system is given. The need to detune the diagonal controllers’ settings is brought out and also the method of tuning decentralized PI controllers by relay auto tune method. Simple methods of designing centralized PI controllers and the analyses of robust stability and robust performances are discussed.

In Chapter 4, a Closed Loop Reaction Curve (CRC) method for the identification of a stableMIMO system is discussed.

In this chapter, the simple centralized controller tuning methods – Davison method, and Tanttu and Lieslehto method – are extended to non-square systems with the Right Half-Plane (RHP) zeros. The proposed methods are applied by simulation on two examples – coupled pilot plant distillation columns and a crude distillation unit. The performances of the square and non-square controllers are compared.

Genetic algorithm (GA) optimization technique is applied for tuning of centralized and decentralized controllers for linear non-square multivariable systems with RHP zeros. Using Ziegler–Nichols (ZN) method, all the loops are initially tuned independently. To obtain the range of controller parameters, a detuning factor is used and this range is used to improve the range of GA search. Simulation studies are applied on a non-square (3 input, 2 output) coupled distillation column. ISE values show that centralized controllers designed by the GA technique give good performance when compared to the decentralized controllers designed by the GA method and other analytical methods. The method of designing decentralized PI controllers for non-square systems by relay auto-tuning method is also proposed in this chapter.

Introduction

Processes with unequal number of inputs and outputs often arise in the chemical process industry. Such non-square systems may have either more outputs than inputs or more inputs than outputs. Non-square systems with more outputs than inputs are generally not desirable as all the outputs cannot be maintained at the set points since they are over specified. The control objective in this case is to minimize the sum of square errors of the outputs with the given (fewer) inputs. For these systems, robust performance (with no offset) is impossible to achieve due to the presence of an inevitable permanent offset that results in at least one of the outputs.

More frequently encountered in the chemical industry are non-square systems withmore inputs than outputs. Here, better control can be achieved by redesigning the controller, eliminating the steady state offsets. Examples of non-square systems are mixing tank process (Reeves and Arkun, 1989), 2 × 3 system, Shell standard control problem (Prett and Morari, 1987), 5 × 7 system, crude distillation unit (Muske et al., 1991), 4 × 5 system, and so on. A common approach towards the control of non-square processes is to first ‘square up’ or ‘square down’ the system through the addition or removal of appropriate inputs (manipulated variables) or outputs (controlled variables) to obtain a square system matrix.