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.

In this chapter, basics of open loop and closed loop identification methods of transfer function models of Single Input Single Output (SISO) systems are reviewed. Open loop identification methods are proposed to identify the parameters of First Order Plus Time Delay (FOPTD) model and Critically Damped Second Order Plus Time Delay (CSOPTD) model. The methods under closed loop identification include the reaction curve method and the optimization method. A review of these methods is given in this chapter. Methods of designing PI controllers for transfer function models for SISO systems are brought out.

Identification of SISO Systems

Process identification is the method of obtaining a model which can be used to predict the behaviour of the process output for a given process input. Transfer function model identification is important for the design of controllers. Identification can be carried out in an open loopmanner or in a closed loopmanner. Excellent books are available for process identification (Ljung, 1998; Soderstrom and Stoica, 1989;Wang et al., 2008; Sung et al., 2009). Pintelon and Schoukens (2001) have given an excellent description of the methods of system identification from the data represented in frequency domain. Clark (2005) plotted the normalized step response curve in time domain for second order systems and third order systems for different damping ratios. These graphs can be used for the estimation of model parameters. Keesman (2011) has given an excellent account of the method available for system identification based on the discrete time models.

Open Loop Identification of Stable Systems

In open loop identification of the system, for a given change in the input variable, the change in output response is noted. The block diagram of the open loop identification is shown in Fig. 2.1. Many works have been reported on open loop identification methods. Identification of a FOPTD model is adequate for design of Proportional Integral Derivative (PID) controllers. Sundaresan and Krishnaswamy (1978) (SK method) presented a reaction curve method for process identification of FOPTD models. Sundaresan et al. (1978) pointed out the error sensitivity involved in the inflection point in the graphical method and suggested a new method of parameter estimation for non-oscillatory, oscillatory, and systems with delay, which involves utilizing the first moment of the step response curve.

In this chapter, interactions in multivariable systems, interaction measure, and suitable variables pairing are described. The design methods of decentralized controllers using a detuning factor to reduce the interactions are discussed. The methods of designing decouplers followed by the methods of designing single loop controllers are also discussed. The use of relay tuning to design decentralized Proportional and Integral (PI) controllers is brought out. The methods of designing simple centralized PI controllers are reviewed. The performance and robust stability analysis of the closed loop system are discussed.–

Stable Square Systems

Consider Fig. 3.1, where the connection of inputs and outputs are shown for TITO multivariable systems. For the step change in u1, record the change in y1 (as y1 versus t); this is called response and the change on y2 is called interaction of u1 on y2. Similarly, for the step change in u2, the change in y2 is called the response and that on y1 is called interaction. The control of output y1 by changing u1 and control of output y2 by u2 by the two controllers as shown in Fig. 3.2 is called decentralized control system. Here we have Eq. (3.1). For n input and n output systems, there are n controllers for a decentralized control system. If the interactions are significant, then the centralized control system is required as shown in Fig. 3.3.

Here, [u1 u2] = GcE

Where,

is a full matrix. And the error vector is defined the same as earlier. For n input and n output systems, there are n×n controllerswhich are to be used for the centralized control system.

Decentralized control system is popular in industries due to the following reasons:

• (1) Decentralized controllers are easy to implement.

• (2) They are easy for the operator to understand.

• (3) The operator can easily retune the controllers to take into account changing process conditions.

• (4) Tolerance for failures in measuring devices or in the final control elements is more easily incorporated into the design of decentralized controllers.

• (5) The operator can easily bring the control system into service during process start up and can take it gradually out of service during shutdown.

In this chapter, the identification method for a First Order Plus Time Delay (FOPTD) model, discussed in Chapter 7, is extended to identify a Second Order Plus Time Delay (SOPTD) model of multivariable systems. This chapter discusses the need for step-up and step-down excitation in the set points to obtain the parameter convergence with significantly reduced computational time.

Introduction

It should be observed that the time domain identification method is easier to formulate and it is conceptually simple. Viswanathan et al. (2001) presented a method for the closed loop identification of FOPTD and SOPTD models by using genetic algorithm (GA) for TITO systems. The computational time for the identification of a SOPTD TITO system using the GA (Viswanathan et. al, 2001) ranges from 1 hour 55 minutes to 5 hours 28 minutes (for different controller settings and test timings). Further, the converged values from GA are not reliable. The converged values are refined by using a local optimizer (Gupta, 1995) (Broyden–Fletcher–Goldfarb–Shannon method). For the SOPTD model identification, Viswanathan et al. (2001) reported the success rate for 10 trials ranging from 25% to 100% to get the global minimum. There is no guarantee to get the exact transfer function model (global minimum) in a single trial. If we obtain the least IAE values also, some deviations are found among the time constant and the damping coefficient (similar to what we get in any model reduction technique). They provide success rate, in which howmany times, they obtained the global minima (exact model parameters) or local accepted minima are not clearly specified. It is to be inferred that the attainment of global minimum (in SOPTD model identification test) is difficult from the step response. The computational time given above is for one trial. The computational time is higher (about 20 to 50 hours) considering together all the 10 trials. Viswanathan et al. (2001) have presented the upper and lower bounds for the parameters in the GA arbitrarily for each of the simulation examples.

For the limit values of process gains, Viswanathan et al. (2001) used a method reported by Papastathopoulo and Luyben (1990). It requires the process input data also for obtaining the guess values of steady state gains.

Simulink Block for WB Distillation Column Process

For systems with significant interactions, a centralized control system is preferred over a decentralized control system. The closed loop identification of model parameters of FOPTD models of a multivariable system under the control of centralized PI control system is carried out by an optimizationmethod. Themethod of getting the initial guess values of the model parameters is given. The closed loop performances of the original system are compared with the closed loop of the identified model with the same centralized controllers.

Identification Method

The method used here is basically the one proposed in the previous chapter wherein the control system was a decentralized control system whereas in the present work a centralized PI control system is considered. Let G(s) and Gc(s) be the transfer function matrix (of size n × n) for the process and the centralized control system as given in Eq. (8.1) and Eq. (8.2).

Here i = 1, 2, … ,N and j = 1, 2, … ,N. The block diagram in Fig. 3.3 (Chapter 3) shows a centralized TITO multivariable system. The process transfer function models are given by FOPTD model as in Eq. (8.3).

With a suitable set of the controller parameters, the closed loop system gives a stable response. A unit step change is introduced in the set point yr1. The main response y11 and the interaction response y21 are obtained. Similarly, a unit step change is given separately to yr2 and the main response y22 and the interaction response y12 are obtained. Thus, we have Eq. (8.4).

The guess values for the model parameters can be obtained from the response matrix. Let us consider the proposedmethod for the calculation of the guess values for the centralized control system of a TITO system. The guess value for kpii is considered as 1/kcii. The guess values of the interaction gains (kp21 and kp12) are to be calculated. The method given in the previous chapter is suitably modified. The Laplace transforms of interaction response y21(s⋆) and y12(s⋆) are given by the expression by Rajapandiyan and Chidambaram (2012a) and Wang et al. (2008) as in Eq. (8.5) and Eq. (8.6)

In this chapter, some of the methods of designing multivariable control system are reviewed. These methods include Gain and Phase Margin (GPM) method, Internal Model Control (IMC) method, synthesis method, PI controllers with no proportional kick, analytical methods, method of inequalities, goal attainment method, Effective Transfer Function (ETF) method, and Model Reference Controller (MRC) design method.

Gain and Phase Margin Method

The gain and phase margins (GPMs) are typical loop specifications associated with the frequency response (Franklin et al., 1989). The GPMs have always served as important measures of robustness (Kaya, 2004, Lee, 2004). It is known from classical control that phasemargin is related to the damping of the system and can therefore also serve as a performance measure (Franklin et al., 1989). The controller design methods to satisfy GPM criteria are not new, and have been widely used (Ho et al., 1995; Hu et. al, 2011). Simple formulae are given by Maghade and Patre (2012) to design a PI/PID controller to meet user-defined Gain Margin (GM) and Phase Margin (PM) specifications.

Using definitions of GPM, the set of equations from Eq. (15.1) to Eq. (15.4) can be written.

where Amii and ϕmii are GM and PM respectively. Here ωgii and ωpiiare gain and phase crossover frequencies. The PI controller parameters are given for FOPTD model as in Eq. (15.5).

Where,

PID controller parameters with first order filter are as in Eq. (15.7) and Eq. (15.8).

It is recommended that a gain margin of 2 and phase margin of 45◦ can be used.

Internal Model Control Method

The concept of Internal Model Control (IMC) was introduced by Garcia & Morari (1982) for scalar systems, and further extended to multivariable (linear and non-linear) systems to design a decentralized control system by Garcia and Morari (1985a), Garcia and Morari (1985b), Ravera et al. (1986) and Economou et al. (1986). Garcia andMorari (1985a) extended themethod toMIMO discrete time systems.

A model predictive control law formulation is suggested by Garcia and Morari (1985b) for the computation of the IMC and the controller parameters can be calculated explicitly without inversion of the polynomia

The majority of existing techniques for identification are based on the frequency domain approach. For any optimization method, the selection of initial guess values plays an important role in computational time and convergence. In this chapter, a simple and generalized method for obtaining reasonable initial guess values for the First Order Plus Time delay (FOPTD) transfer function model parameters are discussed. A method to obtain the upper and lower bounds for the parameters to be used in the optimization routine is also presented. The method gives a quick and guaranteed convergence. The standard lsqnonlin routine is used for solving the optimization problem in Matlab. This method is applied to FOPTD and higher order transfer function models of multivariable systems.

Identification of Decentralized Controlled Systems

Identification Method

Consider an n-input and n-output multivariable system. G(s) and GC(s) are process transfer function matrix and decentralized controller matrix with compatible dimensions, expressed in Eq. (7.1) and Eq. (7.2).

The controller parameters can be chosen arbitrarily for the multivariable systems such that the closed loop system is stable with reasonable responses.

Consider a decentralized TITO multivariable system as shown in Fig. 3.2. The process transfer functionmodels are identified by FOPTDmodels.AFOPTDmodel is given in Eq. (7.3).

In this case, a known magnitude of step change is introduced in the set point yr1 with all the remaining set points unchanged and all other loops kept under closed loop operation. From the prescribed step change in the set point yr1, we obtain the main response y11 and interaction response y21. Similarly, the same magnitude of step change is introduced in set point yr2, and we obtain the main response y22 and interaction response y12. The response matrix of the TITO system can be expressed as Eq. (7.4).

The first column in the response matrix in Eq. (7.4) contains the responses (main and interaction) obtained by the step change in the set point yr1 in the first loop. The second column contains the responses (main and interaction) obtained by the step change in set point yr2 in the second loop. From these step responses, the initial guess values of the model parameters are obtained.

In any optimization method, the selection of initial guess values plays a vital role.