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.

The focus of this chapter is the control of spark timing. As discussed in Chapter 3, the spark is ignited in advance of TDC during the compression stroke. The exact timing can influence performance, fuel economy, emissions, and knock. As discussed in Chapter 1, advancing the spark timing can improve performance and reduce fuel consumption. However, advanced spark timing also can lead to engine knocking and potential engine damage. Spark-timing control can be used, for example, in idle-speed control (see Chapter 8) with throttle control. In this chapter, we focus on the occurrence of engine knock and the control of knock by adjustment of spark timing.

Knock Control

Knock occurs when an unburned part of the air–fuel mixture within the combustion chamber explodes prematurely. This is called knocking because it generates resonating gas-pressure oscillations, which are heard as a knocking sound. Knocking can lead to serious engine damage (Heywood 1989). Historically, a low-compression ratio or conservative spark timing was used to ensure that knocking did not occur; however, this approach sacrifices performance and fuel economy. Knock control can be used when a feedback sensor becomes available, which adjusts the spark timing based on a measured variable that indicates knock. Suitable measurements include the cylinder pressure (e.g., the 5- to 15-kHz region was found to be a good knock indicator), engine-block vibrations, light emission within the combustion chamber, and ion current through the gas mixture.