Design and Tuning of Parallel Distributed Compensation-based Fuzzy Logic Controller for Temperature

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A GA algorithm is used to objectively optimize the parameters of the time lag elements in the suggested Smodel structure. The optimal parameters computed are q_p=[K₁=13.71, K₂=12.2, K₃=8.3, T₁=57.3, T₂=96.5, T₃=49.1, T₄=30, T₅=11.6, T₆=15.4, T₇=2.5]. The GA optimization minimizes the integral squared relative error accepted as a fitness function

(1)

where E(t)=y_Smodel(t)-y_e(t) is the difference between the model output y_Smodel(t) from simulations and the real plant output from experiments y_e(t) – both as responses to one and the same pattern of input step changes u(t). The GA optimization reaches minimum of F_p=2.9. The pattern of input signal u(t) is selected according to the requirements to be rich in magnitudes and frequencies and to cover the whole range of the input signal, so that the Smodel can learn the real plant nonlinearity. The ready Sugeno plant model is validated comparing its response to the real plant experimental response for input signals, different from the used in modeling. The red line in Figure 1 shows the Smodel responses which are close to the experimental ones in blue from the real time operation of the plant. The parameters of the GA are: population size=20; number of generations=20; elite=2; crossover rate = 0.8 and method – single point; mutation operator –adapt feasible; fitness scaling – rank based; selection – roulette wheel; binary coding.

3. Design and GA Optimization of PID PDC-FLC

When the plants dynamics can be partitioned in a finite number of linear models, a TSK plant model with a small number of rules can be derived to allow a PDC-FLC design. The standard PDC-FLC representation is a single fuzzy block with linear controllers’ state space descriptions in the conclusions of the same number of rules with the same premise as in the TSK plant model.

Figure 2. Modified TSK plant model

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Figure 3. Modified PDC-FLC

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Here, first this PDC-FLC is equivalently substituted by a Sugeno-dynamic structure similar to the new plant Sugeno modification of TSK model, shown in Figure 2. The new structure is depicted in Figure 3. It contains the same Sugeno model block used in producing the degrees μ_iof belonging of the current temperature y to each of the three linearization zones, and j PID linear controllers with input the system error e. The controller output u is computed as weighted by μ_j sum of the individual PID controllers’ outputs u_j thus performing fuzzy blending and defuzzification.

The modified PDC-FLC shows a simple way for turning linear controllers into nonlinear fuzzy as well as easy embedding of the PDC-FLC into a PLC algorithm for industrial real time operation. Several linear PID controllers are smoothly mixing outputs when y is moving along different linearization zones, characterized by different linear plant models. The fuzzy blending is ensured by a simple Sugeno fuzzy model of j rules with singletons as outputs –available as a fuzzy block in most PLCs ^{[12, 13, 14, 15, 32, 33]}. This modified PDC-FLC facilitates the design – each of the PID controllers is tuned considering the mean linear plant model in the corresponding linearization zone without turning plant and PID controller description into a state space representation. Different tuning criteria and techniques can be selected from the linear control theory – robust stability, robust performance, engineering approaches, pole placement, desired performance indices, etc. In this research it is accepted that the PID block operates as PI algorithm.

The complexity of tuning of a number of linear controllers considering local linear plant model uncertainties can be decreased using a GA based optimization. The parameters to be searched are the parameters of the three PI controllers q_PI=[K_p1, T_i1; K_p2, T_i2; K_p3, T_i3].

The fitness function combines four objectives – minimization of: 1) mean absolute error (MAE) ; 2) mean control effort (MCE); 3) maximal relative output with respect to reference y_r and the range of the step changes in reference D_r - an estimate of the maximal overshoot σ= y/y_r -1; 4) relative with respect to D_r maximal deviation y_m from reference y_m:

(2)

where N is the total number of samples, collected from the experiment/simulation.

The fitness function is computed using (2) and the recorded values for the corresponding variables from simulation of the closed loop system with an initial set of parameters for the three PI controllers. The initial parameters are taken from the user defined bounds for the parameters, which are determined using some engineering tuning approach. The closed system model used in the simulation with the GA optimizer for tuning of the modified PDC-FLC parameters is shown in Figure 4, where a Limiter bounds the control signal to correspond to the laboratory set-up for operation of the dryer. The GA performs selection, crossover and mutation to build a new generation of offspring thus provides a new set of controller’s parameters. After a new simulation the fitness (2) is computed. This cycle continues till the end condition is met – the final number of generations of 20 is reached. The simulation is fast, more realistic, with full control on the experiment, without unplanned noise or disturbances, safe for the plant which ensures the best solution with respect to system stability and gradual parameter changes without restrictions on parameters and signals. It allows considering of various inputs – reference changes and disturbances with different magnitude and frequency that cover the realistic industrial environment impact range. However, a reliable plant model is required for the whole range of operation conditions.

Figure 4. Closed loop system of modified PID PDC-FLC and TSK plant model with GA optimizer

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The optimal PI controllers’ parameters that minimize (2) in the GA optimization are q_PI^opt=[K_p1=0.27, T_i1=99s; K_p2=0.36, T_i2=120s; K_p3=0.83, T_i3=200s]. A linear PI controller for comparison is tuned by GA optimization and its optimal parameters are K_p=0.25, T_i3=85s.

The stability of the designed global nonlinear system can be proven using the methodology, developed in ^{[2, 5]}.

4. Design and GA Optimization of Fuzzy Logic Supervisor

A further improvement of the closed system performance is possible by connecting a properly designed FLS. The FLS operates on-line in real time and can compensate for signals, difference in plan of experiments and changes of plant parameters which are not included in the sample used by the GA for optimization.

Here it is suggested that the FLS has an input the bounded control u^*_k-1 in previous time moment t_k-1 and output u_c which to provide a multiplicative correction of the PDC-FLC output u according to Figure 5. The correction is assumed to vary in the range [0.3, 1.7] not to influence system stability but also for the purpose of its tuning a scaling gain K_u is introduced. The input and output MFs of the FLS fuzzy block and the rules are selected standard and are shown also in Figure 4. The meaning of this embedding of a FLS is to keep a reasonable control action thus ensuring energy efficiency in control. The bigger the control – the longer the period when the dryer and the fan are connected to the nets supply, so the greater the consumed electrical energy for heating. Oscillations with great peaks of control action are not good for the heater durability and reliability, and also show energy inefficiency – too much heating followed by a necessary cooling.

The scaling gain is optimized by GA minimization of the fitness function (2) using a modified simulation model from Figure 4 with the connection of the FLS from Figure 5. Besides, the PDC-FLC parameters remain fixed to their optimal values and the tuning parameter is q_FLS=[K_u]. As a result the optimal FLS parameter computed is K_u=0.71 and the minimal value for F reached is F =1.

Figure 5. Fuzzy logic supervisor block diagram

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5. Simulations and Real Time Control of Temperature in a Laboratory Dryer

The designed PDC-FLC, consisting of a Sugeno model and three local PI linear controllers (PDC-3PI) with GA optimized parameters, and the GA optimized FLS are applied for the control of the air temperature in a laboratory dryer. The plan of the experiments is to prove first in simulations and then in real time control:

- expected improvement of closed loop system performance (settling time and overshoot reduction) and increased robustness to plant model uncertainty - similar step responses in different operation points despite the different plant parameters;

- improvements in the step responses of the system with the PDC-3PI with respect to the system with the linear PI controller, and improvements in the step responses of the system with the PDC-3PI with the FLS with respect to the system without the FLS;

- improvements in the control action u(t) (smooth, without peaks and oscillations) and in the energy efficiency, defined as ;

- closeness between the step responses in simulations and in the real time control;

- preservation of good performance for step inputs different from the used in the GA optimization.

Figure 6. Step responses of systems with PDC-3PI and linear PI controller from simulation and real time control

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Figure 7. Step responses of systems with PDC-3PI with FLS and without FLS from simulation and real time control

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The step responses of temperature in systems with PDC-3PI and linear PI controller from simulation and real time control are depicted in Figure 6. Simulation and real time results are close and lead to the same conclusion:

– the PDC-3PI system shows faster settling for high reference, where the plant has the smallest gain for the whole temperature range, which makes the settling time of all system step responses almost the same - an evidence for high robustness;

- the settling time of the PI control system step responses differ in the different operation points -increase with the increase of reference, so the this system is less robust to the plant parameter changes with the operation point;

- both systems have very good performance for small references, PDC-3PI system improves performance where the linear PI system is slow at high references.

The step responses of temperature of the PDC-3PI system and the PDC-3PI system with FLS from simulation and real time control are shown in Figure 7. The FLS does not lead to significant improvements in temperature responses since it is designed to reduce control effort and GA optimized to minimize a fitness function that unites several criteria – three related to system performance (MAE, σ, y_m), and one - to the control action (MCE). Both in simulations and in real time the two systems have close settling time – only for the last reference the FLS system has a slower response.

The improvements due to the FLS are observed in Figure 8, where the EEF and the control action take smaller values – the control is more economic and smooth (small peaks and oscillations) and the system more energy efficient.

The GA optimization of the controllers’ parameters is carried out for the developed plant model and the accepted fitness function and sample of step responses. The fact that simulation results are close to the obtained in real time control in all previous investigations gives grounds to test the three systems by simulation to a different pattern of step reference changes. The results lead to the same conclusions and this confirms their validity for all possible step inputs in the given range.

Figure 8. Energy efficiency (up) and control action (down) from real time control with linear PI controller and with PDC-3PI with and without FLS

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