Performance Limitations of Some Industrial PID Controllers

Transcription

1 Performance Limitations of Some ndustrial P Controllers Flávio Faccin and Jorge O. Trierweiler * Chemical Engineering epartment Federal University of Rio Grande do Sul, Porto Alegre - RS, Brazil Abstract This paper intends to make a study about industrial P controllers, how difficult it is to adjust them, what parameterizations are most sensitive and show more performance limitations, and other important factors. At first, all P controller structures are converted to a common base: a two degree-of-freedom control configuration. After, the optimal parameters for some controller parameterizations are synthesized with a design method based on the frequency response approximation. n general, the parallel form shows better performance, robustness, and flexibility than the series. Keywords: industrial P controllers, performance limitations and frequency domain. 1. ntroduction Although exist several advanced control techniques (e.g., MPC, fuzzy controllers), well tuned P controllers still be satisfactory for most industrial control loops. Therefore, the P controller has continued to be the most widely used process control technique for many decades (Chen and Seborg, 00). Nevertheless, despite all P controllers are based on the same principles, they can differ considerably to each other concerning implementation aspects. n this way, the parameters calculated using usual design methods are not compatible with all industrial controllers. As pointed out by Goodwin et al. (001), caution must be exercised when applying P tuning rules, as there are a number of possible parameterizations. n this paper, to make easier to tune P controllers and possible the conversion between different parameterizations, it is used a two degree-of-freedom control configuration as a common base, which can represent all possible P parameterizations.. ndustrial P Controllers n Fig. 1a, a classical representation of a standard feedback control configuration is shown, where G, C, y and y SET are the plant model, the controller, the control variable and the setpoint, respectively. The P controller (C) can assume several different forms. The parallel (or non-interacting) and the series (or interacting) forms are given by the equations (1) and (). * Author to whom correspondence should be adressed: jorge@enq.ufrgs.br

2 y SET - C G y y SET C SP C P G - y C PV (a) (b) Fig 1: The standard (a) and two degree-of-freedom (b) feedback control configuration. 1 C KP 1 T syset Y T s 1 C KP 1 1Ts YSET Y T s (1) () ue to the facility for manual tuning of the derivative action and simplicity for implementing in analog controllers (since it is necessary only one amplifier circuit), the series form is wide used in several industrial controllers. However, the parallel form is more general, and can achieve better performance for some processes, especially oscillatory ones (Skogestad, 003). Additionally, the ideal forms (1) and () are only used for educational purposes. Practical implementations need to include a first order filter for the derivative part to reduce the strong control action at high frequency produced by measurement noise and suddenly setpoint changes. The setpoint problem is solved by applying only the derivative action to process variable. The filter time constant is usually 10 times lower than the derivative time constant (T )..1 The Common Base The main difference among industrial Ps occur in derivative action. P controllers are normally the same. ue to this fact, we have used a two degree-of-freedom control configuration that keeps the basic nucleus of a P controller (C P ). Two new blocs are added: the setpoint and process variable filters (respectively, C SP and C PV ). Fig. 1b shows how the blocks are connect to represent a feedback loop. t can be easily shown that all industrial P controllers can be converted to this common base. The C P block structure is fixed and given by (3) and C SP and C PV blocks are lead-lag filters easily implemented into control systems. Table 1 shows some typical industrial P parameterizations and the corresponding conversion to the common base form. The main advantage presented by the common base configuration is to divide a typical nonconvex optimization problem formulated in the standard control loop form into two convex problems, which can be solved sequentially as is shown in the next section. C P 1 KP 1 T s (3)

3 Table 1: Conversion to the common base form of typical P parameterizations. Type Parameterization Conversion Parallel 1 1 Ts 1 TT s T Ts1 C KP 1 YSET 1 Y CSP 1; CPV (P) Ts Ts 1 Ts TT s T Ts1 Series 1Ts 1Ts 1Ts Ts 1 C KP YSET Y CSP 1; CPV (S) Ts Ts 1Ts Ts 1 P-error 1 Ts 1 TT s T Ts1 C KP 1 YSET Y (Pe) CSP CPV Ts 1 Ts TT s T Ts1 bt s 1 CSP ; 1 1 Ts Ts 1 SA C KP b YSET 1 Y Ts Ts 1 Ts 1 TT s T Ts1 CPV TT s T T s1 3. Sequential terative Optimization Method (SOM) To design and convert the P parameters, it is formulated as an optimization problem in frequency domain, which consists of minimizing the difference between the closedloop transfer function (T) and the attainable closed-loop performance (T O ), i.e., 1 OF min T s T s P s O (4) n (4) the term (1/s) is used to emphasize that both response should be as close as possible for stepwise setpoint changing. The closed-loop response for the OF control configuration of Fig. 1b is given by Gs () CP scsp() s T() s. (5) 1 Gs ( ) C ( s) C ( s) P PV f controller blocks C P (s), C PV (s), and C SP (s) are simultaneously optimized, the optimization problem (4) is nonconvex. To overcome the nonconvexity, it is recommended to apply a sequential iterative procedure, where the P block is separately optimized to the other two blocks. The procedure starts optimizing C P (s) with C PV (s) = 1 and C SP (s) = 1, after that C PV (s), and C SP (s) are calculated considering C P (s) constant. With the converged C PV (s) and C SP (s) a new C P (s) is determined (step 1) and then with the new C P (s), the blocks C PV (s) and C SP (s) can be updated (step ), and thus for ahead until all 3 blocks completely converge. An important factor to make the convergence of (4) simple and fast is the attainable closed-loop performance. Typically, it is desired that a closed-loop response twice to four times faster than the open loop response, but the closed-loop response must be compatible with the plant model and include the desired closed loop behavior (e.g. to be stable, small settling time, small overshoot). For stepwise setpoint changing, closed loop response which minimizing the TAE criterion is the best choice.

4 Table : The Optimum Coefficients of T O (s) based on the TAE Criterion for a Step nput. s n s 1.4n sn s 1.75 s.15 s 3 3 n n n s.1 s 3.4 s.7 s n n n n s.8 s 5.0 s 5.5 s 3.4 s n n n n n s 3.5 s 6.6 s 8.6 s 7.45 s 3.95 s n n n n n n The corresponding analytical expressions for this kind of closed loop response were given by orf and Bishop (1998, p. 59). These functions have no zero, unit gain, and their coefficients are optimized based on the TAE criterion for a step input, with only one adjustable parameter (to choice the dynamic) for several model order. The denominators of this functions are shown in Table. Using this functions, it is just necessary to select the model order and to set the parameter n, which is inversely proportional to the settling time, to compose the final attainable performance function. 4. Case Studies n this section, the P parameterizations are compared considering the following four plant models obtained by varying the damping factor i : 1 G s exp 5 s with 0.3; 0.7; 1.0, and.0 i 100s 0 is1 i (6) The corresponding transfer functions are denoted by G 1, G, G 3 and G 4, respectively. Fig. shows the step response of these transfer functions with the specified closed loop performance (T O ), which was obtained considering a fourth order attainable performance with n = This T O was chosen because it is suitable and feasible to be achieved for all models. The P parameters (K P, T and T ) of the configurations P, S, and Pe are shown in Table 3, calculated using the SOM method presented in section 3. The different parameterization were compared considering TAE and AE performance criteria, gain margin (GM) and phase margin (PM) as robustness criteria (see e.g., Marlin (1995) definition of these indices). Table 3 shows these results for the process models G 1, G, G 3, and G 4 and the three controller parameterizations (P, S, and Pe). n this table, T is the number of iterations and OF-S1 and OF-S are the converged corresponding values of the objective function (4) calculated in the step 1 and of the SOM procedure, respectively. Table 3 shows clearly that when more overdamped is the model, easier is the controller tuning for all controller parameterizations and smaller are the difference between the controller parameters (K P, T, and T ) for the three studied controller parameterization. The reason for that is the derivative action, which becomes smaller making the configuration P, S, and Pe converge to the same P controller.

5 1.4 G T O G G G Time (s) Fig : Step response comparison for the models G 1, G, G 3, G 4, and for T O. On the other hand, when the damping factor is reduced (G 1 and G ) the difference between the P and S forms are very big. n fact, for system G 1 the S form could not even stabilize the closed-loop. These results are waited, since the series form cannot produce the complex zeros necessary to compensate the complex poles of G 1 and G. n general, the parallel form produces better performance and stable responses than the series form (smaller AE and TAE and higher GM and PM). The P-error parameterization often produces higher GM and good results. Based on the number of iterations (T), we can concluded that for the proposed method the series form is normally more difficult to converge and take more number of iterations than the parallel forms. Table 3: Optimal results obtained with the proposed method. Model K P T T T OF-S1 OF-S AE TAE GM PM G 1 - P G 1 - S not converged G 1 - Pe G - P G - S G - Pe G 3 - P G 3 - S G 3 - Pe G 4 - P G 4 - S G 4 - Pe

6 Table 4: Comparison results for the case studies. Case K P T T AE TAE GM PM OV A % B C % Aiming to analyze the sensitivity to parameter change, the P parameters obtained for the model G 3 were changed in relation to the original values of the P form in -0% (case A) and +0% (case B). The simulations results obtained with these parameters are shown in Table 4, which also include the similar results calculated for the S form (cases C and ). Based on Table 4 it can be concluded that the parallel form is much less sensitive to controller parameters mismatches, since it has better performance and robustness (higher GM and PM and smaller overshoot - OV) than series form. The better performance can be analyzed by the AE and TAE indices, which have varied less (both in absolute and relative way). t occurs because the series form is interacting, and the effects of small parameter variations are higher, in general. 5. Conclusions The results in this work indicate that the use of parallel form produces more stable loops and more performance responses with less sensitivity due to parameter variations than series form. Therefore, if it is possible it is recommended to use always the parallel form, although the series is the most used form in commercial controllers (Aström and Hägglund, 1995, p. 110). 6. References Aström, K.J. and T. Hägglund, 1995, P Controllers: Theory, esign, and Tuning, nstrument Society of America, Research Triangle Park. Chen,. and.e. Seborg, P/P Controller esign Based on irect Synthesis and isturbance Rejection, nd. Eng. Chem. Res., 41. orf, R.C. and R.H. Bishop, 1998, Modern Control Systems, Addison-Wesley, Menlo Park. Goodwin, G.C., S.F. Graebe and M.E. Salgado, 001, Control System esign, Prentice- Hall. Marlin, T.E., 1995, Process Control, McGraw-Hill, New York. Skogestad, S., 003, Simple Analytic Rules for Model Reduction and P Controller Tuning, Journal of Process Control, 13.