Tuning the controller

Try tuning the controller to obtain the best response to a change in feed concentration and feed rate. 

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. 

To minimize /, you balance the error between the setpoint and the predicted response against the size of the control moves. Equation 16.2 contains design parameters that can be used to tune the controller, that is, you vary the parameters until the desired shape of the response that tracks the setpoint trajectory is achieved (Seborg et al., 1989). The move suppression factor A penalizes large control moves, but the weighting factors wt allow the predicted errors to be weighted differently at each time step, if desired. Typically you select a value of m (number of control moves) that is smaller than the prediction horizon / , so the control variables are held constant over the remainder of the prediction horizon. 

The minimization of the quadratic performance index in Equation (16.2), subject to the constraints in Equations (16.5-16.7) and the step response model such as Equation (16.1), forms a standard quadratic programming (QP) problem, described in Chapter 8. If the quadratic terms in Equation (16.2) are replaced by linear terms, a linear programming program (LP) problem results that can also be solved using standard methods. The MPC formulation for SISO control problems described earlier can easily be extended to MIMO problems and to other types of models and objective functions (Lee et al., 1994). Tuning the controller is carried out by adjusting the following parameters ... 

Figure 5.3 shows results for a step change in the disturbance of 0.2 at time equal zero. An integration step size of 0.1 min is used. We will return to this simple system later in this book to discuss the selection of values for and r, that is, how we tune the controller. . 

In this chapter we will study control equipment, controller performance, controller tuning, and general control-systems design concepts. Some of the questions that wc will explore are how do we decide what kind of control valve to use what type of sensor can be used and what are some of the pitfalls that you should be aware of that can give faulty signals what type of controller should we select for a given application and how do we tune the controller. 

Water is heated by passing it through a steam-heated kettle (Fig. 7.78) at a mass flowrate w. The inlet and outlet temperatures of the water are 9, and 9 respectively. Steam condenses in the jacket of the kettle at a temperature 0, and a pressure f. It is intended to control the temperature of the water by placing a temperature sensor in the water in the kettle and using this measurement to manipulate the flow of steam to the kettle jacket. In order to tune the controller it is necessary to derive the transfer functions relating 0o to 0j, 0, and w. 

In tuning the controller we assume that three small lags rm exist in the loop, so the controller sees the total openloop transfer function ... 

Filters are likely to be more effective on fast processes and can complicate or limit the response of a PID controller. One way to compensate for this is to tune the controller with the filter inserted into the feedback loop. 

Based on current knowledge of the process and its disturbance characteristics, one may know or choose a reasonable difference equation structure for the controller algorithm. Starting with some assumed initial parameter values in the controller equation, the controller can be implemented on the process as shown. The control algorithm is coupled with an on-line recursive estimation algorithm which receives information on the inputs and outputs at each sampling interval and uses this to recursively estimate the optimal controller parameters on-line and to update the controller accordingly. The idea is to use the data collected from the on-line control manipulations to tune the controller directly. 

These nonlinearities are the main reason an operating margin must be considered when tuning the controller. If the loop is to be robust and operate in a stable manner over a wide range of conditions, conservative values of the tuning parameters must be chosen. Unfortunately, this results in poorer performance under most conditions. One technique to handle known nonlinearities is to provide tuning parameters that vary based on measured process conditions. 

Simultaneously satisfying each of these objectives is never possible therefore, tuning is a compromise. For example, tuning for minimum deviation from setpoint for normal disturbances is contrary to tuning the controller to remain stable for major disturbances. That is, if the controller is tuned for normal disturbances, the closed-loop system may go unstable when a major disturbance enters the process. On the other hand, if the controller is tuned for the largest possible disturbance, control performance is likely to be excessively sluggish for normal disturbance levels. 

Even at this early stage in designing your controller, keep in mind that some of its targets may be set by a nonlinear optimizer. It is useful to decide which CVs or MVs will have their targets set by an optimizer. This becomes important later, particularly when tuning the controller. 

Let us tune the controller using a phase margin equal to 30°. Then we have... 

The reader can easily show that a phase margin of 45° is enough to tune the controller in case 1 and provide the necessary safety factor for absorbing a 50% error in the dead time. The value of the proportional gain Kc for a 45° phase margin is found to be Kc = 5.05. Assume that there is an error in the time constant which has a true value of 0.25 instead of the assumed 0.5. Then the crossover frequency is found from the equation... 

Larger uncertainty in the parameters of a model (static gain, time constant, dead time) requires larger or smaller gain and phase margins for tuning the controller s parameters ... 

To suppress errors that persist for long times, the ITAE criterion will tune the controllers better because the presence of large t amplifies the effect of even small errors in the value of the integral. 

Figure 16.3 demonstrates, in a qualitative manner, the shape of the expected closed-loop responses. When we tune the controller parameters using ISE, IAE, or ITAE performance criteria, we should remember the following two points ... 

The process reaction curve for this system provides us with an experimental model of the overall process which we can use to tune the controller without requiring detailed knowledge of the dynamics for the reactor, heating jacket, thermocouple, and control valve. 

After introducing the necessary decouplers, can you tune the controllers of two loops separately so that the stability of the overall process is guaranteed (Hint Examine closely the closed-loop characteristic equations of two decoupled loops.)... 

Implement and tune the controllers of units, usually of PID type. 

Closed loop simulation. Here the first task consists of implementing and tuning the controllers. The use of prescribed local control structures, or setting perfect control for fast loops simplifies this task and preserves the plantwide character of the analysis. Here... 

A number of critical questions must be answered in developing a control system for a plant. What should be controlled What should be manipulated How should the controlled and manipulated variables be paired in a multivariable plant How do we tune the controllers ... 

---