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.

Details of the Relay Auto-Tuning Method for TITO Systems

The method proposed by Palmor et al. (1995) concentrated on the identification of DCP (Desired Critical Point) along with the steady state gains. The usual method involves the open loop step or pulse response test. Palmor et al. (1995) recommend a closed loop method. The process output y(t) and manipulated variable u(t) are determined by two experiments conducted on the system with two different reference (that are not both 0 mean) signals along with the relays. The steady state gain of the system is then calculated as (Palmor et al., 1995):

where p1 and p2 (vectors) are calculated from the area under the corresponding waveform for one period duration in the responses and interactions in the output (Palmor et al., 1995). Similarly, q1 and q2 (vectors) are calculated from the area under the corresponding waveform for one period duration in themanipulated variables. GP (0) which is the SSGM of the system can be easily calculated as discussed in Chapter 11.

To identify DCP, relative importance is given to one loop over another using a weighting factor C, as given in Eq. (B.2).

K1c and K2,c are the critical gains of the controllers. Once the steady state gains and the resultant error amplitudes are known, the relay height ratio (h1 and h2) can be calculated as in Eq. (B.3).

Here a1 and a2 are the amplitude of output oscillations obtained and Cd indicates the desired quantity of C. None of these quantities in Eq. (B.3) is available initially to the auto tuner. The relay ratio may be calculated by a series of limit cycle experiments. The algorithm is repeated till the convergence criterion is satisfied |ϕ − ϕd| < ϵ. Once the optimum relay height is known, the DCP can be obtained by the relay feedback test.

From the Fourier series expansion, the amplitude of the relay response is assumed to be the result of the primary harmonic output. Therefore the ultimate gain can then be calculated (Yu, 2006; Astrom and Hagglund, 1995) from Eq. (B.4).

The Classical Control Theory studies the physical systems and the control design in the frequency domain, while the Modern Control Theory is 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 first order differential equations). State–space is commonly used to model Multiple Input and Multiple Output (MIMO) systems, like spacecrafts, aircrafts, automobiles, marine vessels, and so on. In this chapter, a basic review of the Classical Control Theory and the Modern Control Theory will be done.

Classic Control Theory vs Modern Control Theory

The Classical Control Theory and the Modern Control Theory are the two control theories available. The classical control method (introduced before 1950) considers (Seborg et al. 2006) root locus, and Bode, Nyquist, and Routh–Hurwitz methods, which make use of the transfer function model. Multiple Input and Multiple Output (MIMO) systems were considered one loop at a time. Modern control method refers to stat–space methods developed in the late 1950s and early 1960s. Here the systemmodels are considered in the time domain.Analysis and design are carried out in time domain by using computers and advanced numerical methods.

During 1960s, the system representation and analysis were carried out in state variable form (representation by n first-order differential equations). An excellent review of classical control of multivariable systems is given by Skogestad and Postlethwaite (2005). Excellent review on Modern control theory is given by Gopal (2014) and Ogata (2017).

Systemmodels can be developed by two distinct methods.Analyticalmodelling consists of a systematic application of basic physical laws to system components and the inter connection of these components. Experimental modelling, or modelling by synthesis, is the selection of mathematical relationships which seem to fit observed input–output data.

The computational effort in analysing and designing controllers by the Modern Control Theory is increased only marginally for higher order systems. To design the controllers, all the state variables are to be available for measurement. In the Classical Control Theory, the interactions among the feedback loops are to be evaluated and taken into account in designing the controllers. Given the system description in the state–space form, the derivation of the transfer function matrix model is unique, whereas given the transfer function matrix, the derivation of the state space model description is not unique (Gopal, 2014).

A technique is presented to identify a transfer function of an unstable SOPTD system with a 0. An optimization method is used to estimate the model parameters to match the closed loop responses. A method is given for the initial guess values of the model parameters (time delay, steady state gain and the values of poles, and 0). The method is applied to three simulation examples. The presented method gives good results. The measurement noise and disturbance effects on the model identification are also reported.

Introduction

Transfer function models are required to design PID controllers. Closed loop identificationmethod is not sensitive to disturbances and is essential to identify the transfer function models for unstable systems. Kavdia and Chidambaram (1996) and Srinivas and Chidambaram (1996) have extended the method proposed by Yuwana and Seborg (1982) for stable systems to identify an unstable FOPTDmodel. The method uses a proportional controller, and a closed loop response for a step change in the set point is used to identify the model. Harini and Chidambaram (2005) have extended the method to identify an unstable SOPTD model. Since only the proportional controller is used, an offset in the response is present and the method may not be employed in chemical plants. In addition, in case of certain parameter values of unstable FOPTD systems (for example, when the ratio of time delay to time constant is greater than 0.7), a proportional controller alone cannot stabilize the process. Ananth and Chidambaram (1999), Cheres (2006), and Padmasree and Chidambaram (2006) have proposed methods for identifying an unstable FOPTD model using a closed loop response of the system under the PID control mode. All the above methods have used analytically derived nonlinear algebraic equations relating an open loop and closed loop model parameters.

Pramod and Chidambaram (2000, 2001) have proposed an optimization method for identifying an unstable FOPTD model using the step response of a PID-controlled system's response. Padmasree and Chidambaram (2002) have extended the method for identifying an unstable FOPTD model with a 0.

For stable systems, there are several methods available for closed loop identification of SOPTD systems under PID control mode (Suganda et al., 1998; Cheres and Eydelzon, 2000; Sung et al., 2009; Dhanya Ram and Chidambaram, 2015).

In this chapter, a Closed Loop Reaction Curve (CRC) method is given for identifying analytically the Single Input Single Output (SISO) systems by Critically Damped Second Order Plus Time Delay (CSOPTD) process transfer function models from the closed loop step response. The material presented here will form the basis for the next chapter on identification of multivariable critically damped systems.

CSOPTD Systems

Introduction

The dynamics of many of the stable processes can be described by that of an over-damped second order system whereas that of the closed loop system is represented by a second order under-damped response. For the purpose of designing PID controllers, many processes are described by a stable second order critically damped transfer function model:

Since the critically damped SOPTD model has only three parameters, the identificationmay be easier than the four parameters required for the over-damped SOPTD models. The time delay is due to the measurement delay or the actuator delay and/or due to approximation of higher order systems by a simple SOPTD model. In this chapter, the method to identify a SOPTD model with equal time constants is discussed. A simple method of calculating the process steady state gain is applied.

Identification of Critically Damped SOPTD and FOPTD Systems

For the purpose of designing controllers, the process is assumed as given in Eq. (5.1). The form of equation in Eq. (5.2) for PID controller is used.

Where,

The closed loop transfer function model is derived as in Eq. (5.4).

For a given step change in the set point, the closed loop response is obtained as shown in Fig. 5.1. Chidambaram and Padmasree (2006) used the u versus time and y versus time profile to get the steady state gain of the system. Since an integral action is present in the controller, the output will reach the desired steady state and hence the input variable of the process will also reach a steady state. The ratio of change in y (output variable deviation) and change in u (manipulated variable deviation) at the steady state gives the steady state gain (kp).

In this chapter, a generalized technique is discussed to obtain the initial guess values for individual transfer function processes of the unstable Two Input Two Output (TITO) multivariable systems. To determine the lower and the upper bounds to be used in the optimization technique, a simple method is explained. Section 10.1 discusses the proposed design method to identify the model parameters of a TITO system under decentralized controller along with the analytical expressions to determine the initial guess values. The applicability of the method is demonstrated with two simulated unstable systems. The method is also extended to unstable TITO system under centralized controllers. Simulation examples show that the proposed method gives a quick convergence with less computational time. For solving the optimization problem, lsqnonlin routine in Matlab is used.

Identification of Systems with Decentralized PI Controllers

The decentralized multivariable system shown in Fig. 3.2 (Chapter 3) is considered. The main loop diagonal transfer function models are unstable FOPTD. The process transfer function matrix Gp(s) and the decentralized controller matrix Gc(s) are given as in Eq. (10.1).

The unstable system can be stabilized by the decentralized control scheme. The controller settings can be selected to obtain a reasonable stable process response. The present example focuses on the identification of the systems having the main loop diagonal transfer functions (g11 and g22) as unstable and the off diagonal transfer function (g12 and g21) as stable.

In general, the transfer function models used are expressed as in Eq. (10.2).

The set point yr1 is perturbed with the other loops closed and other set points unchanged. From this set point change, the main response y11 and the interaction y21 is obtained. Similarly, yr2 is perturbed to obtain the main response y22 and the interaction y12. The initial guess value plays an important role in the optimization technique. For the identification of the model, these response values are used to find the initial guess values for which a straightforward method is suggested. The initial guess values for time delay are considered to be the same as the corresponding closed loop time delay values and the time constant is considered as ts/8 where ts are the settling time of closed loop responses. The initial guess values for kp11 and kp22 are obtained from the relation given in Eq. (10.3) (Chidambaram, 1998).

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.