International Conference on Computer Systems and Technologies - CompSysTech’2009
LIMITATIONS OF MULTIVARIABLE CONTROLLER TUNING USING GENETIC ALGORITHMS Dr. Karl O. Jones and Wilfried Hengue Abstract: In recent years Evolutionary Computation has come of age, with Genetic Algorithms (GA) being possibly the most popular technique. A study is presented revealing the performance of a GA in determining the PID tuning parameters for a multivariable process, including decoupling controllers. The process used for this investigation is a distillation column which is a MIMO high-order, nonlinear system. The results indicate some limitations of using GAs for controller tuning when MIMO systems are involved. Keywords: Genetic algorithm, multivariable, PID, tuning.
INTRODUCTION The most common controller used in industry is the Proportional-Integrate-Derivative (PID) controller. In order for the PID controller to work properly it is essential that its parameters are tuned, and there are numerous methods that can be used, such as Zeigler-Nichols [1] or Cohen-Coon [2], both of which are widely used experimental approaches. While these tuning techniques are wholly suitable for single-input singleoutput (SISO) systems, their use becomes extremely problematic when a multi-input multioutput (MIMO) system is considered. A characteristic feature of MIMO systems is process interaction where each manipulated variable can affect each output variable. Control of multi-input multi-output (MIMO) processes can be achieved either by using a set of independent SISO controllers or by employing a centralized multivariable controller. Although centralized multivariable control has a number of benefits, multiple SISO controllers are often employed to control interacting multivariable processes because of their simplicity. The benefits of multiple SISO controllers over centralized control include fault tolerance, simplified design, operation flexibility and so on [3]. It should be noted that because of the single loop structure, a decentralized controller cannot remove process interactions [4]. The problem on interaction can be reduced with the inclusion of a decoupling element in the control system, although this further increases the complexity, for example an nxn process requires a total of n PID controllers and n decouplers. Simultaneously tuning a number of PID controllers can prove extremely difficult and very time consuming. One possible resolution to this manual tuning process is to utilise an automatic optimisation process. Traditional optimisation approaches can become trapped in local minima, however newer approaches such as Evolutionary Computation involve large search populations and thus have more opportunity to find a global optima. Evolutionary systems have been considered as an optimisation tool since the early 1950s, and until the 1960s, the field of evolutionary systems was working in parallel with research into Genetic Algorithm (GA). When the two areas began interacting, the field of evolutionary programming appeared, introducing concepts of evolution, selection and mutation. Holland [5] defined the concept of the GA as a metaphor of the Darwinian theory of evolution. GAs view learning in terms of competition amongst an evolving population. GAs have been applied to numerous problems, and are known as an efficient optimisation method that is commonly used in industry. GAs encode potential solutions to a problem on a simple chromosome-like data structure and apply recombination operators to these structures to preserve critical information. A GA implementation begins with a population of chromosomes, often random, which is then evaluated and reproductive opportunities allocated so that those chromosomes which represent a better solution to the problem are given more chance to “reproduce”. The suitability of a solution is usually defined with respect to the current -
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International Conference on Computer Systems and Technologies - CompSysTech’2009
population [6]. GA techniques have a solid theoretical foundation [5], based on the Schema Theorem [6]. BINARY DISTILLATION COLUMN The Binary Distillation Column separates a mixture of two components having different boiling points, with the process enhanced by making separation occur in stages within the column. One way to achieve this is by incorporating a liquid stream of high purity distillate (known as reflux) back into the column through a series of sieve trays. Although there are benefits to using reflux, the resultant process dynamics creates an extremely complex control system. The distillation system has two inputs, S and R, and two outputs, B and D (Figure 1), where S is the hot steam used to heat the column, B are the impurities being rejected to R which in turn are the impurities being returned to the column in order to make them pure, finally D is the pure liquid obtained from the column
Figure 1 Distillation Column The system transfer function has been modelled by Wood and Berry [7] as: 12.8e s y1(s ) 16.7s 1 7s y 2 (s ) 6.6e 10.9s 1
18.9e 3s 21.0s 1 u1(s ) 19.4e 3s u 2 (s ) 14.4s 1
(1)
where y1(s) and y2(s) are the overhead and bottom compositions of methanol, respectively; u1(s) is the reflux flow rate and u2(s) is the steam flow rate to the reboiler; d(s) is the feed flow rate, a disturbance variable. The time constants are in minutes. The process can be represented by the block diagram shown in Figure 2. Controllers D11 and D22 are the primary control for the system, while D12 and D21 are the decoupling controllers, r1 and r2 are the input references, u1 and u2 are the combined controller signals, and y1 and y2 are the outputs.
Figure 2 Distillation Column model with PID Controllers and GA for Tuning. -
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International Conference on Computer Systems and Technologies - CompSysTech’2009
GENETIC ALGORITHM OPERATION The Genetic Algorithm (GA) is a powerful and broadly applicable stochastic search and optimisation technique, and is probably the most widely recognised evolutionary computation technique. A GA is an iterative process, with an iteration known as a generation. Common practice is to terminate a GA after a specified number of generations and then examine the best chromosomes. If no satisfactory solution is determined, then the GA can be restarted. Table 1 Genetic Algorithm process 1. Randomly generate an initial population P(0) 2. Fitness evaluation F(i) for each individual i in the current population P(t) 3. Do a stochastic selection S(i) for each individual i in the population P(t) in order to have S(i) proportional to F(i) 4. Generate P(t+1) by probabilistically selecting individuals from P(t) to produce offspring via genetic operators The process repeats until the maximum number of generations is reached
Importance of the Fitness Function The fitness function measures the quality of the represented solution and it is always dependant on the problem under consideration. Seen as the most important parameter within any optimisation system and in particular in a GA, the fitness function determines how individuals are selected to provide for the next generation. As stated the fitness function depends on the problem under consideration, hence certain cost functions will not provide a good solution for some systems but for others, which have a different structure. A poorly selected fitness function will not provide an optimal solution. APPLICATION RESULTS The distillation process described by equation (1) was simulated within Simulink in MATLAB. The GA was executed using a population of 50 over 1000 generations. The GA was designed to determine 10 parameters which represented the PID parameters for D11 and D22 as well as the PI values for D12 and D21. The fitness function employed was based on the Integral Absolute Error (IAE) of each process output: = ( ∫| − |) + ( ∫| − |) (2) where Sp1 and Sp2 refers to the two set-point signals, and a and b are weighting factors. While most workers use a step input, this work uses a square wave input profile to ensure that the developed controller is capable of maintaining regulation for both positive and negative step changes. The first tests involved setting all the process time-delays to zero and removing the decoupling controllers. The resulting transient response (Figure 3) shows reasonable control for both outputs, and highlights that the GA is capable of finding suitable PID parameters. Next the GA was applied to the process with the delays included (no decoupling controllers). Figure 4 illustrates that the developed PID controllers can provide some form of control, however there is evidence of the time-delays reducing the effectiveness of the controllers. There is clear interaction between the two loops, however since there are no decoupling controllers this is only to be expected. The next test included both the process time-delays and the decoupling controllers (D12 and D21). In this, instance the GA determined controller parameters that produced an acceptable controlled performance (Figure 5), limited owing to loop interaction. It can be stated, that the decoupling controllers are only performing an adequate function. During -
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International Conference on Computer Systems and Technologies - CompSysTech’2009
another test the PID parameters for the four controllers produced a response that deteriorates as time progresses (Figure 6). This would suggest that having considered some 50,000 sets of PID values during its operation, the GA has not been able to determine a set that is optimal. DISCUSSION Throughout the literature there are numerous cases of genetic algorithms being able to determine the optimal set of parameters for a given problem. Furthermore, GAs are frequently recommended as the most appropriate tool for any optimisation problem. The results presented here suggest that there is a limitation to the capabilities of GAs. It is not clear where the problem lies: is it a limitation of the genetic algorithm as an optimisation methodology, or is the failing down to an inadequately designed fitness function? The latter might seem the most likely problem area, however the fitness function used in this work was designed to minimise the error between the loop set-points and outputs and it is clear from the result in Figure 6 that there is a significant level of error during a steady set-point period. CONCLUDING REMARKS This work has focused on how to tune a MIMO process. Classical methods like Zeigler-Nichols have been used widely within the industry for tuning controllers. The method is robust and widely used because of its online tuning but it is also described as old fashioned and an aggressive tuning which might create error. The aim of a control is not to create errors but to control them. It is not clear that the application of GAs to tuning a set of PID controllers on a MIMO system is wholly successful. The results presented for a system with no time-delays is quite successful, as is the application to a process with decoupling. However, the results for the complete distillation model (which involves decoupling controllers and time-delays on the process signals) indicate that the GA has not determined optimal values. The reasons behind this lack of success need further investigation: firstly, is the full process not appropriate for PID controllers? Secondly, would an alternative more complicated fitness function provide improved results?
[1] [2] [3]
[4] [5] [6]
[7]
REFERENCES Zeigler, J. G. and Nichols, N. B. 1943. Process lags in automatic-control circuits. Transactions of the ASME, volume 65, p. 433–444. Cohen, G. H. and Coon, G. A. 1953. Theoretical Consideration of Related Control. Transactions of the ASME, vol. 75, p. 827-834. Campo, P. and M. Morari. 1994. Achievable closed-loop properties of systems under decentralized control: conditions involving the steady-state gain. IEEE Transactions on Automatic Control. 39 (5), p. 932-943. Garelli, F., R.J. Mantz and H. De Battista. 2006. Limiting interactions in decentralized control of MIMO systems. Journal of Process Control. Vol. 16. p. 473-483. Holland, J. H. 1975. Adaptation in Natural and Artificial Systems. University of Michigan Press, USA. Whitley, D. 1993. A Genetic Algorithm Tutorial. Computer Science Department, Colorado State University Fort Collins, CO 80523, USA. Wood, R. K. and Berry, M. W. 1973. Terminal Composition Control of a Binary Distillation Column, Chemical Engineering Science, Vol. 28,p. 1707-1717.
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International Conference on Computer Systems and Technologies - CompSysTech’2009
ABOUT THE AUTHORS Dr. Karl O. Jones is a Principal Lecturer in the School of Engineering at Liverpool John Moores University, while Wilfried Hengue was an MSc student working under his supervision. Having gained his MSc, Wilfried now works in France. Phone: +44 151 231 2199, Е-mail:
[email protected]. RESULT FIGURES 1.2
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Figure 3 Process with no delays and no decoupling controllers.
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Figure 5 Process with delays and decoupling controllers included. 3
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Figure 6 Deteriorating controlled response (process with delays and decoupling).
Figure 4 Process with no delays (decoupling controllers included).
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