The system with one or more right half plane transmission (RHPT) zero is generally called as a non-minimum phase system. In this chapter, two methods of designing multivariable controllers for the systems with RHPT zeros are compared (Reddy et al., 2006). The methods used are decoupled internal model controller (IMC) (Wang et al., 2002) and a simple tuning method (Davison, 1976). In decoupled IMC method (Wang et al., 2002), controller design procedure is developed with the help of a model reduction theorem. Davison (1976) proposed a simple tuning method, which is based on the inverse of the steady-state gain matrix [G(0)]. Previously the method was applied to multivariable minimum phase systems only. In the present work, Davison method is applied to systems with multivariable right half plane zero. The decoupled IMC method (Wang et al., 2002) involves some complex calculations in designing the controllers whereas the Davison method (1976) is simple to apply. Simulation results are given for the following 2 × 2 transfer function matrix of the systems: (i) four-tank system, a benchmark example of multivariable system (Johansson, 2000) (ii) system (Morari et al., 1987) with individual and multivariable transmission zero at s = 1 (iii) binary distillation column (Wang et al., 2002). The performance comparisons are given by the sum of IAE values based on response and interaction both for servo and regulatory problems. The simple tuning method gives an improved performance.

Identification of transfer function models of a system is required for an improved tuning of controllers. Several methods have been reported in the literature for identification of transfer function models with two, three and four parameters (pure delay system, first order plus time delay (FOPTD), second order plus time delay (SOPTD), etc.) using relay feedback approach. In this section, the basics of conventional relay feedback method and modifications in the original autotuning method are reviewed for single-input single-output systems. Excellent reviews on relay tuning methods are given by Yu (1999, 2006), Hang et al. (2002) and Wang et al. (2003). Methods of designing PI/PID controllers based on the transfer function models are also briefly reviewed.

Relay Feedback Method

Åström and Hägglund (1984) suggested the use of an ideal (on–off) relay (Fig. 1.1) to generate a sustained oscillation in the closed loop. For positive gain process, on–off relay is defined by u = umax if e ≥ 0, and u = umin, if e < 0. For negative gain processes, on–off relay is defined by u = umin, if e ≥ 0, and u = umax if e < 0. Amplitude (a) and period of oscillation (pu) are noted from the sustained oscillation. This is a closed loop method for identification of transfer function models. The method is based on the observation that when an open loop output lags the input by π radians, the closed loop system may oscillate (Fig. 1.2) with a period Pu.

The present chapter is concerned with simultaneous relay autotuning of cascade controllers. Traditionally, autotuning procedures for cascade controllers are applied using a sequential, one-loop-at-a-time method. Saraf et al. (2003) proposed a simultaneous method of the relay tuning of series cascade control system. In this method, they assumed the principal harmonic analysis of relay oscillations. This results in error in calculation of the ultimate gains. The higher order harmonics are to be considered in the analysis of simultaneous relay tuning of series and parallel cascade control systems. Using improved ultimate gains, the inner loop (PI) and outer loop (PID) controllers are designed by the Ziegler–Nichols(Z-N) tuning method. The improved performance of the control is compared with the principal harmonic analysis reported by Saraf et al. (2003)

Introduction

Hang et al. (1994) proposed a sequential relay autotuning of series cascade control of stable systems. In this method, the outer loop is open. The on–off relay is used in the inner loop and the value of ku is obtained from 4h/(πa0). The controllers in the inner loop are tuned using the Ziegler–Nichols tuning formulae. With the inner loop under PI control action, the relay test is repeated for the outer loop.

Simultaneous relay tuning method is required whenever the system is to be kept under a closed loop in order to reduce the effect of disturbances or when the open loop system is unstable.

To design proportional plus integral (PI)/PID controllers, the ultimate values of the controller gain (ku) and frequency of oscillation (ωu) should be known. The conventional Ziegler and Nichols continuous cycling method requires a large number of experiments to calculate these values. Åström and Hägglund (1984) suggested the use of ideal relay to generate closed loop oscillations. The ultimate gain and ultimate frequency can be found in a single-shot experiment. However, the method is still approximate because of the use of the principal harmonics approximation. Li et al. (1991) reported that for an open loop, stable first order plus time delay (FOPTD) system, an error of -18% to 27% is obtained in the calculation of ku. Yu (1999) suggested a saturation feedback test to get better results for the ultimate gain and frequency. However, Yu (1999) did not report any result for large values of delay-to-time constant ratio.

An SOPTD model can incorporate higher order process dynamics better than an FOPTD model. The controller designed on the basis of the SOPTD model gives a better closed loop response than the one designed on an FOPTD model. It is better to have an SOPTD model with equal time constants since only three parameters are to be identified. Li et al. (1991) showed that the conventional analysis of the relay autotune method for an SOPTD model with equal time constants gives -11% to 27% error in the calculation of ku.

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.