Fall 2003 BMI 226 / CS 426 AUTOMATIC SYNTHESIS OF IMPROVED TUNING RULES FOR PID CONTROLLERS

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1 Notes LL-1 AUTOMATIC SYNTHESIS OF IMPROVED TUNING RULES FOR PID CONTROLLERS

2 Notes LL-2 AUTOMATIC SYNTHESIS OF IMPROVED TUNING RULES FOR PID CONTROLLERS The PID controller was patented in 1939 by Albert Callender and Allan Stevenson of Imperial Chemical Limited of Northwich, England. The PID controller was an enormous improvement over previous manual and automatic methods for control. The quality of PID tuning rules is of considerable practical importance because a small percentage improvement in the operation of a plant can translate into large economic savings or other (e.g., environmental) benefits.

3 Notes LL-3 IMPROVED TUNING RULES CONTINUED In 1942, Ziegler and Nichols developed a set of mathematical rules for automatically selecting the parameter values associated with the proportional, integrative, and derivative blocks of a PID controller. The 1942 Ziegler-Nichols PID tuning rules do not require an analytic model of the plant. Instead, they are based on several parameters that provide a simple characterization of the to-be-controlled plant. These parameters have the practical advantage of being measurable for plants in the real world by means of relatively straightforward testing in the field. The Ziegler-Nichols rules have been in widespread use for tuning PID controllers since World War II.

4 Notes LL-4 IMPROVED TUNING RULES CONTINUED The question arises as to whether it is possible to improve upon the Ziegler-Nichols tuning rules. Åström and Hägglund answered that question in the affirmative in their 1995 book PID Controllers: Theory, Design, and Tuning in which they identified 4 families of plants that are representative for the dynamics of typical industrial processes.

5 Notes LL-5 ÅSTRÖM AND HÄGGLUND S 4 FAMILIES OF PLANTS Plants represented by transfer functions of the form -s e G( s) = (1+ st ) where T=0.1,, 10 n-lag plants G( s) 2, [A] 1 =, [B] n (1 + s) where n=3, 4, and 8 Plants represented by transfer functions of the form G( s) (1 + s)(1 + α s)(1 + α s)(1 + α s) =, [C] where α=0.2, 0.5, and 0.7 Plants represented by transfer functions of the form 1- αs G ( s) = 3, [D] (s+ 1) where α=0.1, 0.2, 0.5, 1.0, and 2.0

6 Notes LL-6 ÅSTRÖM AND HÄGGLUND S APPROACH Åström and Hägglund (1995) developed rules for automatically tuning PID controllers for the 16 plants from these 4 industrially representative families of plants. The PID tuning rules developed by Åström and Hägglund (like those of Ziegler and Nichols) are based on several parameters representing important overall characteristics of the plant that can be obtained by straightforward testing in the field. In one version of their method, Åström and Hägglund characterize a plant by two frequency-domain parameters: ultimate gain, K u (the minimum value of the gain that must be introduced into the feedback path to cause the system to oscillate) ultimate period, T u (the period of this lowest frequency oscillation)

7 Notes LL-7 ÅSTRÖM AND HÄGGLUND S APPROACH CONTINUED In another version of their method, Åström and Hägglund characterize a plant by two time-domain parameters time constant, T r dead time, L (the period before the plant output begins to respond significantly to a new reference signal). Åström and Hägglund (1995) describe a procedure for estimating these two timedomain parameters from the plant s response to a simple step input.

8 Notes LL-8 ÅSTRÖM AND HÄGGLUND S APPROACH CONTINUED Åström and Hägglund (1995) set out to develop new tuning rules to yield improved performance with respect to the multiple (often conflicting) issues associated with most practical control systems including setpoint response, disturbance rejection, sensor noise attenuation, and robustness in the face of plant model changes as expressed by the stability margin. Åström and Hägglund approached the challenge of improving on the 1942 Ziegler- Nichols tuning rules with a shrewd combination of mathematical analysis, domain-specific knowledge, rough-and-ready approximations, creative flair, and intuition sharpened over years of practical experience.

9 Notes LL-9 ÅSTRÖM AND HÄGGLUND S APPROACH CONTINUED Åström and Hägglund started by identifying industrially representative analytic plant models Next, they characterized each of the plants in their test bed in terms of parameters that are easily measured in the field. Then they applied the known analytic design technique of dominant pole design to the analytic plant models and recorded the resulting parameters for PID controllers They decided that functions of the form f(x) = + 0 * ax a x a e, where x=1/k u, were the appropriate form for the tuning rules. Finally, they fit approximating functions of the chosen form to the PID controller parameters produced by the dominant pole design technique.

10 Notes LL-10 ÅSTRÖM AND HÄGGLUND (1995) RESULT The tuning rules developed by Åström and Hägglund in their 1995 book outperform the 1942 Ziegler-Nichols tuning rules on all 16 industrially representative plants used by Åström and Hägglund. As Åström and Hägglund observe, [Our] new methods give substantial improvements in control performance while retaining much of the simplicity of the Ziegler-Nichols rules.

11 Notes LL-11 THE TOPOLOGY OF A PID CONTROLLER WITH NONZERO SETPOINT WEIGHTING OF THE REFERENCE SIGNAL IN THE PROPORTIONAL BLOCK 270 BUT NO SETPOINT WEIGHTING FOR THE DERIVATIVE BLOCK 280 Reference Signal 200 Plant Feedback Equation 1 - Equation Equation /s Control Variable Equation s

12 Notes LL-12 ÅSTRÖM AND HÄGGLUND (1995) TUNING RULES The PID tuning rules developed by Åström and Hägglund are expressed by 4 equations of the form f(x) = a + 0 * e, ax a x where x=1/k u. Equation 1 implements setpoint weighting 210 of the reference signal 200 (the setpoint) for the proportional (P) part Equation 1 specifies that the setpoint weighting, b, is given by b = * K u e K u. [1] Equation 2 specifies the gain, K p, for the proportional (P) block 230 of the controller K p = * * K u K u u K e. [2] 2

13 Notes LL-13 ÅSTRÖM AND HÄGGLUND (1995) TUNING RULES CONTINUED Equation 3 specifies the gain 250, K i, that is associated with the input to the integrative (I) block 260 K i = 0.72* K * e u 0.59* T * e u K u Ku K u Ku 2 Equation 4 specifies the gain 280, K d, that is associated with the input to the derivative (D) block 290 K d =. [3] Ku K 2 Ku K 2 u u 0.108* * * * K T e e. [4] u u

14 Notes LL-14 ÅSTRÖM AND HÄGGLUND (1995) TUNING RULES CONTINUED The transfer function for the Åström- Hägglund PID controller is K A = K ( * - ) + i p b R P ( R- P) + Kd * s* (-P), s where A is the output of the Åström- Hägglund controller, R is the reference signal, P is the plant output, and b, K p, K i, and K d are given by the four equations above Note that only P (as opposed to R P) appears in the derivative term of this transfer function because Åström and Hägglund apply a setpoint weighting of zero to the reference signal R in the derivative block.

15 Notes LL-15 IMPROVED TUNING RULES USING GP The goal here is to use genetic programming is used to discover PID tuning rules that improve upon the tuning rules developed by Åström and Hägglund in their 1995 book. The topology of the to-be-evolved controller is not open-ended here, but, instead, fixed as the PID controller topology. That is, we are not searching here for a parameterized topology

16 Notes LL-16 TEST BED OF 33 PLANTS Family Parameter value Ku Tu L Tr Runs in which the plant is used A T = A, P, 1, 2, 3 A T = A, P, 1, 2, 3 A T = A, P, 1, 2, 3 A T = A, P, 1, 2, 3 A T = P A T = P, 2, 3 A T = P A T = P A T = A, P, 1, 2, 3 B n = A, P, 1, 2, 3 B n = A, P, 1, 2, 3 B n = P, 2, 3 B n = P, 1, 2, 3 B n = P, 2, 3 B n = A, P, 1, 2, 3 B n = C α = C α = A, P, 1, 2, 3 C α = P C α = P C α = P C α = P, 2, 3 C α = P, 2, 3 C α = A, P, 1, 2, 3 C α = P, 2, 3 C α = A, P, 1, 2, 3 C α = D α = A, P, 1, 2, 3 D α = A, P, 1, 2, 3 D α = A, P, 1, 2, 3 D α = P, 2, 3 D α = A, P, 1, 2, 3 D α = A, P, 1, 2, 3

17 Notes LL-17 TEST BED OF 33 PLANTS 16 A plants used in Åström-Hägglund (1995) 30 P plants that are used to evolve the improved PID tuning rules 20 1 plants that are used to evolve the 1 st non-pid controller 24 2 / 3 plants used to evolve the 2 nd /3 rd non-pid controller

18 Notes LL ADDITIONAL PLANTS All 18 additional plants are members of Åström and Hägglund s families A, C, and D Family Parameter K u T u L T r value A A A A A A C C C C C C D D D D D D

19 Notes LL-19 PERFORMANCE OF THE ÅSTRÖM- HÄGGLUND PID CONTROLLER ON THE 33 PLANTS Plant Plant parameter value ITAE 1 ITAE 2 ITAE 3 ITAE 4 ITAE 5 ITAE 6 Stability margin A A A A A A A A A B B B B B B B C C C C C C C C C C C D D D D D D Sensor noise

20 Notes LL-20 PERFORMANCE OF THE ÅSTRÖM- HÄGGLUND PID CONTROLLER ON THE 18 ADDITIONAL PLANTS Plant Plant parameter value ITAE 1 ITAE 2 ITAE 3 ITAE 4 ITAE 5 ITAE 6 Stability margin A A A A A A C C C C C C D D D D D D Sensor noise

21 Notes LL-21 SEARCH FOR IMPROVED PID TUNING RULES The controller s topology is not subject to evolution, but instead, is prespecified to be the PID topology We are seeking four mathematical expressions (for K p, K i, K d, and b). Each of the expressions may contain free variables representing the plant s ultimate gain, K u, and the plant s ultimate period, T u. Note that there is no search for, or evolution of, the topology here. The problem is simply a search for four surfaces in threedimensional space.

22 Notes LL-22 SEARCH FOR IMPROVED PID TUNING RULES CONTINUED Initial experimentation quickly confirmed the fact that the Åström and Hägglund tuning rules are highly effective. Starting from scratch, genetic programming readily evolved threedimensional surfaces that closely match the Åström-Hägglund surfaces and that outperform the Åström-Hägglund tuning rules on average. However, these genetically evolved surfaces did not satisfy our goal, namely outperforming the Åström-Hägglund tuning rules for every plant in the test bed.

23 Notes LL-23 SEARCH FOR IMPROVED PID TUNING RULES CONTINUED Thus, we decided to approach the problem of discovering improved tuning rules by building on the known and highly effective Åström-Hägglund results. Specifically, we use genetic programming to evolve four mathematical expressions (containing the free variables K u and T u ) which, when added to the corresponding mathematical expressions developed by Åström and Hägglund, yield improved performance on every plant in the test bed. Although we anticipated that the resulting three-dimensional surfaces would be similar to the Åström-Hägglund surfaces (i.e., the magnitudes of the added numbers would be relatively small), this additive approach can, in fact, yield any surface (or, more precisely, any surface that can be represented by the generously large number of points that a single program tree may contain).

24 Notes LL-24 PREPARATORY STEPS FOR IMPROVED PID TUNING RULES PROGRAM ARCHITECTURE The architecture of each program tree in the population has four result-producing branches (one associated with each of the problem s four independent variables, namely K p, K i, K d, and b). TERMINAL SET The terminal set, T, for each of the four result-producing branches is T = {R, KU, TU}, where R denotes a perturbable numerical value between 5.0 and FUNCTION SET F = {+, -, *, %, REXP, RLOG, POW}. The two-argument POW function returns the value of its first argument to the power of the value of its second argument.

25 Notes LL-25 PREPARATORY STEPS FOR IMPROVED PID TUNING RULES CONTINUED FITNESS Fitness is measured by means of eight separate invocations of the SPICE simulator for each plant under consideration. When the aim is to automatically create a solution to a category of problems in the form of mathematical expressions containing free variable(s), the possibility of overfitting is especially salient. Overfitting occurs when an evolved solution performs well on the fitness cases that are incorporated in the fitness measure (i.e., in the training phase), but then performs poorly on previously unseen fitness cases.

26 Notes LL-26 PREPARATORY STEPS FOR IMPROVED PID TUNING RULES CONTINUED FITNESS CONTINUED In this problem, four presumptively nonlinear and complicated functions mathematical expressions (each incorporating two free variables) must be evolved. In applying genetic programming to this program, we were especially concerned about overfitting because there are only 16 plants in the test bed used by Åström and Hägglund. Moreover, there are only three plants in two of the four families. GP must usually consider a multiplicity of data points in order to discover even a linear relationship. A fortiori, a multiplicity of data points is required when the underlying function is actually nonlinear (as is usually the case in non-trivial problems).

27 Notes LL-27 PREPARATORY STEPS FOR IMPROVED PID TUNING RULES CONTINUED FITNESS CONTINUED Concerned about possible overfitting, we used the 30 plants marked P (in the table) so that the fitness measure entails 240 separate invocations of the SPICE simulator. The choice of 30 plants for this first run was governed primarily by considerations of computer time. Ideally, we would have used even more plants. We did not know in advance whether 30 plants would prove to be sufficient to enable genetic programming to unearth the complicated relationships inherent in this problem.

28 Notes LL-28 PREPARATORY STEPS FOR IMPROVED PID TUNING RULES CONTINUED CONTROL PARAMETERS The population size is 100,000

29 Notes LL-29 RESULTS IMPROVED PID TUNING RULES The best-of-run individual emerged in generation 76, so the fitness of approximately 7,700,000 individuals was evaluated during the run. The average time to evaluate the fitness of an individual (i.e., to perform the 240 SPICE simulations) was 51 seconds. It took hours (4.4 days) to produce the best-of-run individual from generation 76. The improved PID tuning rules are obtained by adding the genetically evolved adjustments below to the values of K p, K i, and K d, and b developed by Åström and Hägglund in their 1995 book.

30 Notes LL-30 RESULTS IMPROVED PID TUNING RULES CONTINUED The quantity, K p-adj, that is to be added to K p for the proportional part of the controller is K p-adj -6 = * T u *10. The quantity, K i-adj, that is to be added to K i for the integrative part is K K i-adj = * u T u The quantity, K d-adj, that is to be added to K d for the derivative part is log log K d-adj = ( ) ( K u T ) e u. The quantity, b adj, that is to be added to b for the setpoint weighting of the reference signal in the proportional block is K b adj = u K e u.

31 Notes LL-31 RESULTS IMPROVED PID TUNING RULES CONTINUED In other words, after adding, the final values (K p-final, K i-final, K d-final, and b final ) for the genetically evolved PID tuning rules are as follows: K 2 K p-final -6 = u 0.72* K * Ku u e * T u *10 K i-final = K u Ku * Ku * e K * + T K u * T * e Ku u u u K K d-final u K Ku K T = 0.108* K * * u * u ( u u Tu e e e ) K ( u ) log log K 2 b final = u 0.25* Ku K u e + K e u The authors refer to these genetically evolved PID tuning rules as the Keane-Koza- Streeter (KKS) PID tuning rules. The automatically designed controller is parameterized (i.e., general) in the sense that it contains free variables (K u and T u ) and thereby provides a solution to an entire category of problems (i.e., the control of all the plants in all the families).

32 Notes LL-32 RESULTS IMPROVED PID TUNING RULES CONTINUED Averaged over the 30 plants used in this run, the best-of-run tuning rules from generation 76 have 91.6% of the setpoint ITAE of the Åström-Hägglund tuning rules, 96.2% of the disturbance rejection ITAE of the Åström-Hägglund tuning rules, 99.5% of the reciprocal of minimum attenuation of the Åström-Hägglund tuning rules, and 98.6% of the maximum sensitivity, M s, of the Åström-Hägglund tuning rules.

33 Notes LL-33 IMPROVED PID TUNING RULES CONTINUED CROSS-VALIDATION Averaged over the 18 additional plants, the best-of-run tuning rules from generation 76 have 89.7% of the setpoint ITAE of the Åström-Hägglund tuning rules, 95.6% of the disturbance rejection ITAE of the Åström-Hägglund tuning rules, 99.5% of the reciprocal of minimum attenuation of the Åström-Hägglund tuning rules, and 98.5% of the maximum sensitivity, M s, of the Åström-Hägglund tuning rules. As can be seen, the results obtained for the 18 previously unseen additional plants are similar to those for the results for the plants used by the evolutionary process.

34 Notes LL-34 IMPROVED PID TUNING RULES CONTINUED COMPARISON TO ÅSTRÖM AND HÄGGLUND Averaged over the 16 plants used by Åström and Hägglund in their 1995 book, the best-of-run tuning rules from generation 76 have 90.5% of the setpoint ITAE of the Åström-Hägglund tuning rules, 96% of the disturbance rejection ITAE of the Åström-Hägglund tuning rules, 99.3% of the reciprocal of minimum attenuation of the Åström-Hägglund tuning rules, and 98.5% of the maximum sensitivity, M s, of the Åström-Hägglund tuning rules.

35 Notes LL-35 COMPARISON TO ÅSTRÖM- HÄGGLUND TUNING RULES THE GENETICALLY EVOLVED ADJUSTMENT, K P-ADJ, FOR THE PROPORTIONAL (P) PART The genetically evolved adjustment becomes as large as 5% only in the combination of circumstances when K u is at the low end of its range and T u is at the high end of its range.

36 Notes LL-36 COMPARISON TO ÅSTRÖM- HÄGGLUND TUNING RULES CONTINUED ADJUSTMENT, K I-ADJ, FOR THE INTEGRATIVE (I) PART BBB3862 The genetically evolved adjustment is about 5.5% for most values of K u, but drops to as little as 4% when K u is at the low end of its range.

37 Notes LL-37 COMPARISON CONTINUED ADJUSTMENT, K D-ADJ, FOR THE DERIVATIVE (D) PART As can be seen from the figure, the genetically evolved adjustment is near zero for most values of K u and T u,but becomes as large as 40% when both K u and T u are at the low ends of their ranges.

38 Notes LL-38 COMPARISON CONTINUED ADJUSTMENT, B ADJ, FOR THE SETPOINT WEIGHTING OF THE REFERENCE SIGNAL OF THE CONTROLLER S P BLOCK The genetically evolved adjustments range between 0% and about +80% and depend only on K u. It is near zero for most values of K u, but rises sharply when K u is at the low end of its range.

39 Notes LL-39

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