Self tuning regulators

1. Self tuning regulators ( STR)

2. Introduction to Self Tuning Regulator (STR)

3. In STR, the controller parameters or the process parameters are estimated in real time. • These estimates are then used as if they are equal to the true parameters (i.e. the uncertainty in estimation is not considered). This is called the certainty equivalence principle. • In an STR, the parameters of the process are estimated online by the estimation techniques discussed • Later these estimated parameters are used to design the controller parameters. • Then these controller parameters are varied in the controller, ultimately resulting in control of the plant. • TYPES OF STR Indirect STR Direct STR

4. In an indirect STR, the parameters of the process are estimated by the available estimation technique, and these estimated parameters are used for design of the controller. This is a time consuming process. This approach is also referred to as explicit self tuning control. • In a direct STR, the model is first re-parameterized in terms of the controller parameters then the parameters are estimated by the available estimation technique. Hence the result of parameter estimation will now be directly the controller parameters, instead of process parameters. Thus a lot of computation time is saved. The only computation effort here will be to re-parameterize the model in terms of the controller parameters, which needs to be done only once, so lot of time is saved in this approach. This approach is also referred to as implicit self tuning control.

5. Deterministic Vs Stochastic Process A stochastic system is one which cannot be expressed precisely but can be expressed with the help of statistical data. A deterministic system on the other hand can be expressed precisely. Usually deterministic systems are encountered in case of servo systems. Stochastic systems are encountered in case of regulator systems. Self Tuning Regulators are available both for deterministic and stochastic systems.

6. Deterministic Indirect Self Tuning Regulators • Pole placement design a simple method of controller design. The idea of this method is to determine a controller that gives desired closed-loop poles. In addition it is required that the system follow the command signals in a specified manner

7. process model be described by

9. The key idea of the design method is to specify the desired closed loop characteristic polynomial AC The polynomials R and S can then be solved form equation (6). Equation (6) is called the Diophantine equation or Bezout identity or the Aryabhatta equation The Diophantine equation given by (6) can be used to determine only the polynomials R and S. To determine T, some other conditions need to be introduced. This requires that the response from the command signal to the output be described by the dynamics:

10. This model-following condition says that the response of the closed loop system to command signals is as specified by the model (7). Whether model-following can be achieved or not depends on the model, the system, and the command signal. If it is possible to make the error zero for all command signals, then perfect model-following is achieved. From (6), it can be deduced that there are cancellations of factors of BT and Ac

11. Now factorize the B polynomial

13. Minimum Degree Pole placement (MDPP) algorithm

16. Algorithm for Indirect Self-Tuning Regulator • obtained by combining the parameter estimation and controller design part. • An algorithm combining the Recursive Least Squares (RLS ) parameter estimation and Minimum Degree Pole Placement (MDPP) controller design is given below:

17. Deterministic Direct Self Tuning Regulators • In a direct STR, the model is first re-parameterized in terms of the controller parameters and then the parameters are estimated by the available estimation technique. • Hence the result of parameter estimation will now be directly the controller parameters, instead of process parameters. • Thus it can be seen that a lot of computation time is saved. • The only computation effort here will be to re-parameterize the model in terms of the controller parameters, which needs to be done only once, so lot of time is saved in this approach. • This approach is also referred to as implicit self tuning control.

18. Consider a process described by:

19. The equation (18) represents a process re-parameterized in terms of the coefficients B- ,R and S. • If the parameters in the process given by (18) are estimated, the control law is thus obtained directly without any design calculations. It should however be noted that the process given by • (18) is nonlinear in parameters because the right-hand side is multiplied by B-. • The difficulties caused can be avoided in the special case of minimum phase systems for which B- = b0= constant

20. Minimum phase systems

23. Direct self tuning regulator algorithm for minimum phase systems

24. Non-minimum phase systems

25. Direct self tuning regulator algorithm for non-minimum phase systems: