A simple method is given in this chapter to design multivariable Proportional and Integral (PI) and Proportional Integral and Derivative (PID) controllers for stable multivariable systems. The method needs only the Steady State Gain Matrix (SSGM). The method is based on the static decoupler design followed by the SISO PI/PID controllers design and combines the resulting decoupler and the diagonal PI(D) controllers as centralized controllers. The result of the present method is shown to be equivalent to the empirical method proposed by Davison (1976). Three simulation examples have been given in this chapter. The performance of the controllers has been compared with the reported centralized controller based on the multivariable transfer function matrix.

Introduction

Design of PI controllers for stable MIMO processes is difficult when compared to the SISO processes due to the interaction between the input–output variables. As discussed in Chapter 3, the MIMO processes can be controlled by decentralized or decoupled controllers or by centralized PI controllers. For mild interacting stable MIMO processes, design of decentralized PI controllers based on the diagonal processes (based on proper pairing) is carried outwith a suitable detuningmethod. The detuning step involves decreasing the controller gains by F, multiplying integral by F, and decreasing derivative times by F. Here, F is the detuning factor which may vary from 1.5 to 4, depending on the extent of interactions, dictated by the RGA. For systems with large interactions, the decoupler (D) is designed so as to make the MIMO processes into n SISO processes. The PI controllers (Gc) are designed for the resulting SISO processes. The overall control system is the combined decoupler and the diagonal controllers (DGc). We can also design, straight away, the centralized PID controllers. The SSGM of the multivariable system can be obtained more easily than identifying the dynamic model parameters and, hence, it is easier to tune the controllers. In this chapter, derivation of a simple method of designing multivariable PI controllers is explained, based on SSGM of the system.

Multivariable System

The stable multivariable system (Fig. 3.3 in Chapter 3) is assumed to be of the form given in Eq. (11.1).

For unstable non-square multivariable systems, the Davison method (Davison, 1976) is modified to design single-stage multivariable PI controllers using only the SSGM of the system. Since the overshoots in the closed loop responses are larger, a two-stage P–PI control system is proposed. Based on the SSGM, a simple proportional controller matrix is first designed by the modified Davison (1976) method to stabilize the system. Based on the gain matrix of the stabilized system, diagonal PI controllers are designed. The performance of the two-stage control system (inner loop centralized P controllers and outer loop diagonal PI controllers) is evaluated and compared with the single-stage multivariable control system. Simulation results on two examples show the effectiveness of the proposed methods for both the servo and regulatory problems. A method is presented to identify the SSGM of a non-square multivariable unstable system under closed loop control. Effects of disturbances and measurement noise on the identification of SSGM are also studied.

Introduction

Control of unstable systems with time delay is a challenging task when compared to the stable systems. Non-square process has a greater number of inputs than outputs or a greater number of outputs than the inputs. Non-square multivariable systems with a greater number of inputs than outputs arise in chemical industries. Examples include vacuum distillation column unit with five inputs and four outputs (Treiber, 1984), a mixing tank with three inputs and one output (Reeves and Arkun, 1989) and a distillation column with four measured impurities and three manipulated variables (Treiber, 1984).

Studies on decentralized control of stable non-square multivariable systems have been reported by (Loh and Chiu, 1997). Ganesh and Chidambaram (2010) have reported the design of centralized multivariable controllers by an optimization method such as the GA. Delay compensator for non-square systems have been reported by Rao and Chidambaram (2006b) and Chen and Ding (2011). Xu and Shin (2011) have presented the design of self-tuning fuzzy logic controller for stable non-square systems. Sarma and Chidambaram (2005) have extended the method of Davison and the method of Tanttu and Lieslehto (1991) to non-square stable systems. Sarma and Chidambaram (2005) have presented that these simple methods for controller design based on SSGM give a good performance for the non-square stable multivariable systems also.

In this chapter, a Closed Loop Reaction (CRC) curvemethod for the identification of stable TITO system is discussed. The system under consideration is controlled by decentralized PI or PID controllers. The responses and interactions are modelled by the extension of Yuwana and Seborg (1982) method and the transfer function matrix for the closed loop system is obtained. Using the relation between the open loop and the closed loop transfer function matrices, the open loop transfer function matrix is obtained in Laplace (s) domain. The responses of the obtained and the actual transfer functions matrix with the original controller settings are compared. Better results are obtained if the parameters of the identified transfer function models are used as initial guess values for any optimization method and the transfer function model is identified. The higher order models are approximated to a FOPTD model. Since the method is proposed for stable systems,many of the higher order stable systems can be approximated to a FOPTD system.

Identification Method

Identification of Individual Responses

Consider a stable 2 × 2 transfer function matrix (GP ) as in Eq. (4.1a).

Let the controller settings be defined by the matrix given in Eq. (4.2).

The PI/PID controllers are selected so as to get a closed loop under-damped response. Initial controller settings are taken, based on the knowledge of the gain alone, using Davison method (Section 3.13).

Consider Fig. 4.1a. A step change of known magnitude is given to the set point yr1 and the other set point is kept unchanged. The main response obtained is y11 and the interaction response is y21. Similarly for Fig. 4.1b. A step change of the same magnitude is given as yr2 and the other set point is kept unchanged. Let the set point matrix be given as in Eq. (4.3).

The main response is denoted as y22, the interaction as y21, and the output matrix is given in Eq. (4.4).

The transfer function of the closed loop responses is assumed to be an under-damped SOPTD system. From the closed loop response curve for each case, the values yp1, yp2, ym1 and T are noted (Fig. 2.6). Using the formulas given by Yuwana and Seborg (1982), the value of τe and ζ are noted for each case. The closed loop time delay is noted from the responses.

In this chapter, a method is presented to identify the Steady State Gain Matrix (SSGM) of an unstable multivariable system under closed loop control. Effects of disturbances and measurement noise on the identification of SSGM are also studied. Davison (1976) method is modified to design single-stage multivariable PI controllers using only the multivariable gain matrix of the system. Since the overshoots in the responses are larger, a two-stage P–PI control system is proposed. Based on the SSGM, a simple proportional controller matrix is designed by the modified Davison (1976) method to stabilize the system. Based on the gain matrix of the stabilized system, diagonal PI controllers are designed.

Introduction

SISO unstable systems with time delay are difficult to control compared to stable systems. The performance specifications like overshoot, settling time for such systems are larger for unstable systems than that of the stable systems (Padmasree and Chidambaram, 2006). Methods of designing PI/PID controllers for scalar unstable systems are given by pole placement method, synthesis method, IMC method, gain and phase margin method, and optimization method (Padmasree and Chidambaram, 2006). Conventional PI controllers give a relatively large overshoot for unstable systems. Jacob and Chidambaram (1996) have proposed a two-stage design of controllers for unstable SISO systems. First, the unstable system is stabilized by a simple proportional controller. For the stabilized system, a PI controller is designed. It is shown that an improved performance is obtained. Even for the stable multivariable systems, the method of designing controllers is complicated due to the interactions among the loops. Simplemethods of designing centralized PI controllers for stable systems are reviewed by Tanttu and Lieslehto (1991) and Katebi (2012).

Govindakannan and Chidambaram (1997) have applied the method of Tanttu and Lieslehto (1991) to unstable MIMO processes. Decentralized PI controllers are also designed by the detuning method proposed by Luyben and Luyben (1997). The decentralized PI controllers do not stabilize the system if the unstable pole is present in each of the transfer functions of the system. Only centralized PI controllers stabilize such systems. Papastathopoulo and Luyben (1990) have proposed a method of calculating the steady state gain of stable multivariable systems by introducing a step change in one set point at a time.