In this chapter, using a single asymmetric relay feedback test, a method proposed by Ramakrishnan and Chidambaram (2003) is reviewed to identify four parameters of a transfer function model. The proposed method is used to identify all the parameters in a second order plus time delay model (SOPTD). The parameters estimated are of adequate accuracy for designing suitable controllers. The estimated SOPTD model has a step response behavior matching with that of the actual process. The method can also be used for identifying open loop unstable transfer function models. For unstable systems, the closed loop step responses are compared. Simulation results are given for four case studies.

Introduction

Identification of dynamic transfer function models from experimental data is essential for model-based controller design. Often derivation of rigorous models is difficult due to the complex nature of chemical processes. Hence, system identification is a valuable tool to identify low order models, based on input–output data, for controller design.

As stated earlier in Chapter 1, Åström and Hägglund (1984) suggested the relay feedback test to generate sustained oscillations of the controlled variable and get the ultimate gain (Ku) and ultimate frequency (ωu) directly from the experiment. Since only process information Ku and ωu are available, the additional information such as steady-state gain or time delay should be known a priori in order to fit a typical transfer function model such as first order plus time delay (FOPTD).

C.1 Relay Tuning of Integrating Plus FOPTD Systems

The most used dynamic model for chemical engineering process is the FOPTD model. In many cases, time constant is very large; in some cases, dead time is the dominant one. In the literature, industrial examples of large time constants are reported by McNeill and Sacks (1969) for distillation and by Westerlund et al. (1980) for cement production. Some of the processes such as heating boilers, liquid storage tanks and batch chemical reactors are examples of integrating processes in industrial and chemical plants (Liu et al., 2005)

Zhang et al. (1999) proposed PID controller design method for integrating process with dead time and time constant. Kwak et al. (1997) proposed an online identification and autotuning method for integrating process. Kookos et al. (1999) proposed online PI controller tuning for integrating/dead time processes. Tian and Gao (1999) proposed a control scheme for integrating process with dominant time delay. The proposed scheme consist of a local proportional feedback to pre-stablize the process, a proportional controller for set-point tracking and a PD controller for load disturbance rejection. Huzmezan et al. (2002) designed a PID controller based on adaptive predictive control strategy to handle integrating type processes with large time constant. An industrial example of temperature control of a process that involves the heating and cooling of a batch reactor is studied.

C.2 Improved Analysis of Relay Tuning

The transfer function model for the integrating plus FOPTD system is given by Kp exp(-Ds)/[s(τs + 1)].

This work focuses attention on (i) incorporation of higher order harmonics in the analysis of relay tuning of controllers for a single loop controller, cascade control systems, single loop saturation relay test, single loop unstable FOPTD system and single loop stable SOPTD system; (ii) providing a simple method of designing P/PI controllers for cascade control schemes; (iii) estimation of model parameters of unstable FOPTD, stable SOPTD and unstable SOPTDZ systems using a single relay feedback test and (iv) application to multivariable systems.

Improved Autotune Identification Method

A method is suggested to formulate an additional equation so that the process gain can also be estimated using the conventional relay autotune method. This method avoids getting a negative time constant of an FOPTD model. For systems showing higher order harmonics in the response, a modification of the calculation for the model parameters of FOPTD model using the conventional relay feedback method is also proposed. This method does not assume the complete filtering of higher order harmonics. The method of calculation is also simple. The present method gives an improved value for the controller ultimate gain. The method gives a more accurate result [on ku and on the identified FOPTD model parameters] than that proposed by Luyben (1987) and Li et al. (1991). Simulation results show that the present method gives improved open loop as well as closed loop performances.

The proposed modification in the asymmetrical relay test gives improved values of the parameters of the FOPTD model.

A simple method is proposed to design PID controllers (i) for a series cascade control system and (ii) for a parallel cascade control system. The method is based on equating coefficients of the corresponding powers of q and q2 in the numerator to α1 and α2 times that in the denominator of the closed loop transfer function model for a servo problem. This method can be used only when the inner and outer loop transfer functions are known. If these transfer functions are not known, then an identification step needs to be carried out. The method is first applied to design a proportional (P) controller for the inner loop and then to design a proportional plus integral (PI) controller for the outer loop. Performances of the controllers are evaluated for the FOPTD models of the inner loop and outer loop.

Introduction

As stated earlier, cascade control is one of the most popular structures for process control. A cascade control system consists of a primary controller and a secondary controller (refer to Fig. 5.1). Cascade control scheme is used to improve the dynamics response of the closed loop system when the disturbance enters the inner loop or disturbances are present in the manipulated variable. The frequency response method (Edgar et al., 1982) is usually employed to design such controllers. The method involves trial and error graphical method. Krishnaswamy et al. (1990) proposed a tuning chart that predicts the primary controller settings by minimizing the ITAE criterion due to load disturbances on the secondary loop.

In this chapter, the method discussed in Chapter 10 on identifying second order plus time delay (SOPTD) transfer function models by asymmetric relay tuning method is extended to identify a multivariable system. The decentralized relay feedback method suggested by Wang et al. (1997) is applied for the m × m system and m relay tests are required for identifying the entire transfer function matrix. Although most of the processes can be adequately approximated by a first order plus time delay (FOPTD) model, some of the processes are under-damped and higher order processes can be better incorporated by an SOPTD than an FOPTD model. Certain higher order stable models when approximated to an FOPTD model give a negative time constant, hence, identifying a second order model is necessary. The proposed method (Ganesh and Chidambaram, 2005) is applied to a 2 × 2 transfer function matrix. Multivariable IMC controllers are designed for the identified model and the closed loop performances of the actual and identified models are compared.

Introduction

As stated earlier in Chapter 2, Srinivasan and Chidambaram (2004) proposed a method to analyze the conventional relay autotune data for estimating the three parameters of the FOPTD model using only one relay experiment. They proposed an additional equation, which along with the phase angle and amplitude criteria gives three parameters. Srinivasan and Chidambaram (2003) also proposed a modified asymmetrical relay feedback method to get improved estimates of the parameters of the FOPTD model.

Using a single symmetric relay feedback test, a method is proposed to identify all the three parameters of a first order plus time delay (FOPTD) unstable model. It is found by simulation that the relay autotune method gives -23% error in the calculation of ku, when D/τ = 0.6. In the present work, a method is proposed by incorporating the higher order harmonics to explain the error in the calculation of ku and to estimate all the parameters of an unstable FOPTD system. Two simulation results are given on unstable first and second order plus time delay transfer function models. The estimated values of the parameters of the unstable FOPTD model are compared with the methods of Majhi and Atherton (2000) and Thyagarajan and Yu (2003). PID controllers are designed for the identified model and for the actual system. The proposed method gives results close to that of the actual system. For a second order plus time delay system, the latter two methods fail to identify the FOPTD model. Simulation results are also given for a non-linear bioreactor system. The PID controller designed on the model identified by the present method gives a performance closer to that of the controller designed on the locally linearized model.

Introduction

As stated earlier, the relay feedback method has become very popular because it is time efficient as compared to the conventional method. The amplitude (a) and period of oscillation (pu) are noted from the sustained oscillation of the system output.

1. Consider a system given by y(s)/u(s) = 2/(I + 1)3. With a proportional controller (withgain k = 2), get the response in y for a set point change of 0.1 of the closed loop system. Similarly, with each value of kc = 2.5, 3.0, 3.5 and 4, get the response for a step change in the set point. Can you get a sustained oscillation in the output when kc = 4? From the period of oscillation, and kc,max value, design a suitable PID controller using the Ziegler–Nichols tuning formulae. With the designed controller, obtain the response in the output for a step change in the set point.

2. For the system given in problem 1, replace the proportional gain by an on–off symmetric relay (h = + 1 and − 1 height). No change in the set point is considered. Check whether you can get a sustained oscillation in the output. Note down the period of oscillation. Using Eq.(1.8) calculate the controller ultimate gain. Hence, calculate the PID settings using the Ziegler–Nichols method based on continuous cycling oscillation method. With the designed controller, obtain the response in the output for a step change in the set point.

3. Consider the system y(s)/u(s) = 2 exp(−2s)/[(8s +1)(s +1)]. Using a symmetric relay with h = +1 and −1, obtain the oscillation in the output and identify the model parameters of FOPTD as given in section 2.2.

In this chapter, a technique for identification of multivariable transfer function matrix proposed by Ganesh and Chidambaram (2003) is reviewed. They extended the method of Srinivasan and Chidambaram (2003) meant for asymmetric relay feedback for a scalar system to a multivariable system. The method identifies the FOPTD model for each element of the transfer function matrix model without any priori knowledge of the value of time delay or gain of the system. In this method, a decentralized relay feedback method suggested by Wang et al. (1997) is applied to the m × m system and ‘m’ relay feedback tests are required for identifying the entire transfer function matrix. The identification method is a closed loop relay feedback method and hence this method is less susceptible to disturbance and measurement noise and can be used for identifying an unstable system. The method is applied for various 2 × 2 transfer function matrices as Wood and Berry's methanol-water distillation column process (1973) and a multivariable system with higher order individual elements. IMC controller (Tanttu and Lieslehto, 1991) is designed for the identified model, and using this controller the closed loop performances of actual and identified models are compared.

Introduction

The transfer function model of the system is required for designing a suitable control system. The dynamic models of process are determined from the fundamental physical and chemical laws. But for the process that is already in operation, there is an alternative approach based on experimental dynamic data obtained from plant test.

This chapter reviews the methods proposed by Srinivasan and Chidambaram (2003; 2004) to accurately estimate the model parameters of a first order plus time delay (FOPTD) transfer function model using (1) the conventional relay autotune method and (2) asymmetric relay autotune method. Usually, the value of delay is assumed or noted from the initial portion of the response of the system. Whenever identifying a higher order dynamics system by an FOPTD model, this method wrongly identifies the time constant as negative (Li et al., 1991) due to an error in identifying the time delay, which is due to an error in the model structures. Using conventional relay autotune method, an additional equation is formulated to calculate accurately the parameters of the FOPTD model. Even when the actual system is FOPTD and the time delay to time constant ratio is larger, higher order harmonics cannot be neglected in the output response. Hence, there is a need to consider the higher order harmonics of the relay oscillations to get improved accurate values for the controller ultimate gain. For the asymmetric relay tuning method, analytical solutions are given for the evaluation of the model parameters.

Introduction

Luyben (1987) used the relay feedback method to identify the model parameters (kp, τ and D) of an FOPTD model. Using the controller ultimate gain and period of oscillation, two equations are formulated using the amplitude criterion and phase angle criterion.

Entrepreneur’s Diary

Like it or not, you often will be negotiating something in life, and always will be negotiating something in an entrepreneurial pursuit. The negotiation could be direct or indirect, obvious or subtle, but as an entrepreneur, you always will be negotiating.

As just one example, my colleague Michael (first author of this text) shared with me a very important lesson in this regard from one of the angel investors that was his first major outside investor, Peter. Peter was seventy-one years old when he first became involved in my friend’s commercial fish farming company. He was the master at just about everything to do with starting a new business. Peter had left a Big 4 accounting firm (probably was the Big 8 back then) to launch his entrepreneurial career, when he had four children and a wife to support at the time. Among other deal points, Peter negotiated with Michael to purchase a 20% equity stake in his company in exchange for Peter’s cash investment. Michael violated a soon-to-be-learned rule, one that Peter was about to teach him the hard way. Peter had originally agreed to invest in the company for 18%, but later simply said that he’d feel a lot better if my friend could “round up” this figure to 20%. It wasn’t necessary, but it would be “nice.” Wanting to get the deal done, given that the parties were so close, and concerned he might otherwise disappoint or aggravate Peter, Michael said, “OK.” Michael didn’t ask what he would get for bumping up Peter’s equity stake or say something such as, “I’ll do that for you if you do XYZ in return.” Peter later told Michael that this was part of the negotiation, not simply a casual request.

Entrepreneur’s Diary

You’re making great progress. You’ve got a board, you are about to hire two key employees, and you are just about ready to start executing your company business strategy. But, you need a company structure to embody these attributes. In fact, you can’t even open a bank account without an Employer’s Identification Number (EIN). This is no simple decision. In particular, your board (typically made up of “older” types) will probably steer you in the direction of a conventional commercial corporation, aka C-Corp. Why, because they are familiar with it. People just about always think what they are familiar with is best, else why would they have been doing it all these years? I believe that a limited liability corporation (LLC) is probably your best bet at this point of your company history for a variety of reasons we discuss in this chapter. I’m not a lawyer, and this is where you should seek legal advice once you think you know what you want to do. In fact, never go to a lawyer and ask them an open-ended question such as “What should I do?” They will just about always guide you in the most conservative direction legally, and this may not be best for your company.

Types of Ownership Structures

Before you can decide on an ownership structure for your business, you should learn a little bit about how each structure works. As a good reference on deciding which ownership structure is most suitable for your business, read “Choosing the Best Ownership Structure for Your Business” (go to www.nolo.com/lawcenter).

Entrepreneur’s Diary

It seems as if addition, subtraction, multiplication, and division should be fairly simple. It is, until you want to apply it to accounting and do your tax calculations! There are many ramifications resulting from your choice of corporate structure on how you end up paying taxes. For example, who in their right mind would choose to pay taxes twice on profits you make from your company? Well, if you choose a C-Corp for your company structure, that’s just what you’ll be doing! But sometimes a C-Corp is the best choice, and you might ask why. I think my accountant was probably the key professional with whom I interacted with during the early days of my start-up. I recommend seeking an accountant’s advice early, before you create your company structure.

Getting Started

At this point in the text, we’ve covered the basic steps of defining your business, developing a marketing strategy, and differentiating your business from the competition. In this chapter, we cover the creation and use of financial accounting statements in the typical business plan, including depreciation and taxation issues. We stress the importance of creating these financial statements from the top down versus the bottom up by basing the figures on details from the demand side of the equation, that is, sales and the costs of production.

Where Are We Going?

If you ask a group of engineers, most of them would probably tell you that they have thought about starting a business. Most did nothing. Some others did some research and preliminary planning. Some of these then invested some personal funds in creating the shell of a company in some legal format. A rare few took their companies forward to the point that they became operational businesses.

Most of us will see opportunities to start a business. This book is for those among us who are contemplating the start of a business and those that may have already taken that big first step. As teachers and successful entrepreneurs, our goal in writing this book is to help you the reader maximize your chances of entrepreneurial success. You will find guidance, instruction, and practical lessons that will assist you, the prospective entrepreneur, with your first steps toward realizing a dream.