Controllability SISO systems

We use a simple liquid level controller to illustrate the concept of a classic feedback control system.1 In this example (Fig. 5.1), we monitor the liquid level in a vessel and use the information to adjust the opening of an effluent valve to keep the liquid level at some user-specified value (the set point or reference). In this case, the liquid level is both the measured variable and the controlled variable—they are the same in a single-input single-output (SISO) system. In this respect, the controlled variable is also the output variable of the SISO system. A system refers to the process which we need to control plus the controller and accompanying accessories such as sensors and actuators.2... 

We first establish the closed-loop transfer functions of a fairly general SISO system. After that, we ll walk through the diagram block by block to gather the thoughts that we must have in synthesizing and designing a control system. An important detail is the units of the physical properties. 

In a SISO system, we manipulate only one variable, so we must make a decision. Since our goal is to control the tank temperature, it would be much more sensible to manipulate the steam temperature TH instead of the inlet temperature. We can arrive at this decision with physical... 

After we have chosen the controlled and manipulated variables, the remaining ones are taken as load variables in a SISO system. 

We now return to the use of state space representation that was introduced in Chapter 4. As you may have guessed, we want to design control systems based on state space analysis. State feedback controller is very different from the classical PID controller. Our treatment remains introductory, and we will stay with linear or linearized SISO systems. Nevertheless, the topics here should enlighten( ) us as to what modem control is all about. 

There are many advanced strategies in classical control systems. Only a limited selection of examples is presented in this chapter. We start with cascade control, which is a simple introduction to a multiloop, but essentially SISO, system. We continue with feedforward and ratio control. The idea behind ratio control is simple, and it applies quite well to the furnace problem that we use as an illustration. Finally, we address a multiple-input multiple-output system using a simple blending problem as illustration, and use the problem to look into issues of interaction and decoupling. These techniques build on what we have learned in classical control theories. 

It is apparent from Eq. (10-22) that with interaction, the controller design of the MIMO system is different from a SISO system. One logical question is under what circumstances may we make use of SISO designs as an approximation Or in other words, can we tell if the interaction may be weak This takes us to the next two sections. 

After proper pairing of manipulated and controlled variables, we still have to design and tune the controllers. The simplest approach is to tune each loop individually and conservatively while the other loop is in manual mode. At a more sophisticated level, we may try to decouple the loops mathematically into two non-interacting SISO systems with which we can apply single loop tuning procedures. Several examples applicable to a 2 x 2 system are offered here. 

Skogestad and Morari recommend the use of uncertainty models for the design of robust controllers. The idea is easy to visualize for an SISO system. Suppose we have a process with the following openloop transfer function ... 

So the multiloop SISO diagonal controller remains an important structure. It is the base case against which the other structures should be compared. The procedure discussed in this chapter was developed to provide a workable, stable, simple SISO system with only a modest amount of engineering effort. The resulting diagonal controller can then serve as a realistic benchmark, against which the more complex multivariable controller structures can be compared. 

One of the major questions in multivariable control is how to tune controllers in a diagonal multiloop SISO system. If PI controllers are used, there are 2N tuning parameters to be selected. The gains and reset times must be specified so that the overall system is stable and gives acceptable load responses. Once a consistent and rational tuning procedure is available, the pairing problem can be attacked. 

In theory, the internal model control methods discussed for SISO systems in Chap. 11 can be extended to multivariable systems (see the paper by Garcia and Morari in lEC Process Design and Development, Vol. 24, 1985, p. 472). 

The minimum variance control for an SISO system finds the unrestricted minimum of the expected value of a quadratic objective function ... 

Various performance indices have been suggested [54, 53, 149, 20, 148] and several approaches have been proposed for estimating the performance index for SISO systems, including the normalized performance index approach [53], the three estimator approach [175[, and the filtering and correlation analysis (FCOR) approach [115[. A model free approach for linear quadratic CPM from closed-loop experiments that uses spectrum analysis of the input and output data has been suggested [136]. Implementation of SISO loop based CPM tools for refinery-wide control loop performance assessment has been reported [294]. 

Select the control configuration. In the case of multi SISO systems this step consists in the determining the pairing between controlled and manipulated variables. 

SISO system with only a modest amount of engineering effort. The resulting diagonal controller can then serve as a realistic benchmark, against which the more complex multivariable controller structures can be compared. 

Ackermanns formula improves the traditional pole assignment standard algorithm of SISO system. The open loop eigenpolynomial of system is not requisite. Ackermanns formula is used for designing the control law u, and the desired closed-loop pole is obtained from designed ideal pole distribution with differential transformation method. 

In the examples of controlled processes that we have been discussing, we have worked with systems handling one resource and with one target variable, i.e., single-input/single-output (SISO) systems. However, in actual practice, the process is affected by multiple input variables and has multiple target variables, i.e., a multiple-input/multiple-output (MIMO) system. 

Valasek M., Olgac N., 1995 b. Efficient pole placement technique for linear time-variant siso systems. lEE Proceedings - Control Theory and Applications, Vol. 142, No. 5, pp. 451-458. 

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