Chapter 5 Nonlinear System Control Using Functional-Link-Based
5.3 Parameter Learning Phase
Ratnaweera et al. [61] stated that the lack of population diversity in PSO algorithms is understood to be a factor in their convergence on local optima.
Therefore, the addition of a mutation operator to PSO should enhance its global search capacity and thus improve its performance. There are mainly two types of mutation operators: one type is based on particle position [118] and the other type is based on particle velocity [117]. The former method is the most common technique, and the mutation operator we proposed in this research is also based on particle position.
In [116], Li, Yang, and Korejo modified the PSO by adding a mutation operator;
the mutation operator provides a chance to escape from local optima. They focused on determining which random generator of the mutation operator is good for improving the population. However, the timing of application of the mutation operator is the most important thing. If mutation operator is applied too early, when the particles are
not nearly convergent, the local search ability of PSO is destroyed. If the mutation operator is applied too late, the parameter learning algorithm will be very inefficient.
Hence, it is an important issue to consider when to apply mutation operator. In our study, we used the distances between each particle as a measure to determine whether the mutation operator needed to be applied or not, and the modified PSO we used is the so called distance-based mutation particle swarm optimization (DMPSO).
Comparing the basic PSO with DMPSO, a convergent detection unit used to detect the particle convergent status is introduced. If the particles are convergent, the mutation operator will be processed. Otherwise, the mutation operator will be skipped.
The convergent detection unit computes the average distance from every particle to the particle that has global best value using Eq. (5.4)
( ) 1
After the average distance is computed, the threshold Thconv is used to determine whether the particles are close enough or not according to Eq. (5.5). If all particles are close enough, meaning that all particles are converging to the same position, the mutation operator will be applied. Otherwise, the mutation operator will be skipped.
( )t Thconv
(5.5)
In this study, every particle has its own mutation probability. If the average distance is greater than Thconv, implying that the majority of particles are not convergent, the mutation probability is set to zero, meaning that every particle does
not mutate and the behavior of every particle is like a generic PSO. If the average distance is less than Thconv, meaning that all particles are converging to the same position, named Gbestt , the mutation probability ( MP ) of each particle is computed by Eq. (5.6). to 1 only when the i particle is successfully evolved at the th t iteration, meaning th that the local best fitness value is improved at the t iteration, and th progress t is ( ) the number of successful evolution particles at time step t .
The design of mutation probability is based on the ratio of improved population.
If the ratio of the improved population is higher, the mutation probability becomes smaller. Most particles are moving toward to the best value that they have currently found. The lower probability guarantees the direction of the moving group will not be destroyed by the mutation operator. On the other hand, if most particles do not improve their fitness value, the population is in the stable status. There are two possibilities: the first possibility is that the particles have converged to the global optimum (or near global optimum). The application of the mutation operator at the moment will not destroy the moving group, because the particles still remember the global optimum, and the mutated particles will move toward the global optimum in the next iteration. The second possibility is that the particles have converged to the local optimum, or in other words, they have fallen into a trap. The mutation operator provides a chance to escape from the trap. If some particles mutate and the new
position the particle reaches has a better fitness value than the local optima, the other particles that are trapped will fly to the new position in the next iteration according to the PSO, meaning that the trapped particles can escape from the local optimum.
5.4 Illustrative Examples
In this section, we demonstrate the performance of the proposed FLNFN model using DMPSO algorithm (FLNFN-DMPSO) for nonlinear system control. The FLNFN-DMPSO is adopted to design controllers in three simulations of nonlinear system control problems: multi-input multi-output (MIMO) plant control [114], control of the truck backing system [119], and a water bath temperature control system [120].
Example 1: Multi-Input Multi-Output Plant Control
In this example, the MIMO plants [114] to be controlled are described by the equations specified by the following 250 pieces of data;
1
Figure 5.2 plots the learning curves of the best performance of the FLNFN-DMPSO model for the affinity/fitness value, the CNFC-ISEL [121], the SEFC [122], and the Mamdani-type fuzzy system using symbiotic evolution algorithm (MFS-SE) [123], after the learning process of 600 generations. To demonstrate the performance of the proposed controller, Figure 5.3 plots the control results of the desired output (solid line) and the model output (dotted line) after the learning process of 600 generations, and Figure 5.4 shows the errors of the proposed method. Table 5.1 presents the best and averaged affinity/fitness values after 600 generations of training. The comparison indicates that the best and averaged affinity/fitness values of FLNFN-DMPSO are better than those of other methods.
0 100 200 300 400 500 600
Figure 5.2: Learning curves of best performance of the FLNFN-DMPSO, CNFC-ISEL, SEFC and MFS-SE in MIMO plant control.
0 50 100 150 200 250
Figure 5.3: Desired (solid line) and model (dotted line) output generated by FLNFN-DMPSO in MIMO plant control.
0 50 100 150 200 250
Figure 5.4: Errors of proposed FLNFN-DMPSO in MIMO plant control.
Table 5.1: Performance comparison of the FLNFN-DMPSO, FLNFN-PSO, CNFC-ISEL, SEFC and MFS-SE controllers for the MIMO plant.
Method Affinity/Fitness Value (Best) Affinity/Fitness Value (Avg.)
FLNFN-DMPSO 0.9898 0.9856
FLNFN-PSO 0.9506 0.9149
CNFC-ISEL [121] 0.9786 0.9721
SEFC [122] 0.9581 0.9553
MFS-SE [123] 0.8560 0.8503
Example 2: Control of Backing Up the Truck
Backing a truck into a loading dock is difficult. It is a nonlinear control problem for which no traditional control method exists [119]. Figure 5.5 shows the simulated truck and loading zone. The truck’s position is exactly determined by three state variables , x and y , where is the angle between the truck and the horizontal, and the coordinate pair ( , )x y specifies the position of the center of the rear of the truck in the plane. The steering angle of the truck is the controlled variable.
Positive values of represent clockwise rotations of the steering wheel and negative values represent counterclockwise rotations. The truck is placed at some initial position and is backed up while being steered by the controller. The objective of this control problem is to use backward only motion of the truck to make it arrive at the desired loading dock (xdesired,ydesired) at a right angle (desired . The truck 90 ) moves backward as the steering wheel moves through a fixed distance (df) in each step. The loading region is limited to the plane [0 100] [0 100] .
Figure 5.5: Diagram of simulated truck and loading zone.
The input and output variables of the FLNFN-DMPSO controller must be specified. The controller has two inputs: truck angle and cross position x. When the clearance between the truck and the loading dock is assumed to be sufficient, the y coordinate is not considered to be an input variable. The output of the controller is the steering angle . The ranges of the variables x, , and are as follows:
The equations of backward motion of the truck are
1
where l is the length of the truck. Equation (5.10) yields the next state from the present state.
Learning involves several attempts, each starting from an initial state and terminating when the desired state is reached; the FLNFN-DMPSO is thus trained.
The training process continues for 2000 generations. The affinity/fitness value of the
FLNFN-DMPSO is approximately 0.9637, and the learning curve of FLNFN-DMPSO is compared with those obtained using various existing models [121-123], as shown in Figure 5.6. Figure 5.7 plots the trajectories of the moving truck controlled by the FLNFN-DMPSO, starting at initial positions ( , , )x y (40, 20, 30 ) , (10, 20,150 ) ,
(70, 20, 30 ) and (80, 20,150 ) , after the training process has been terminated. The considered performance indices include the best affinity/fitness and the average affinity/fitness value. Table 5.2 compares the results. According to these results, the proposed FLNFN-DMPSO outperforms various existing models.
0 200 400 600 800 1000 1200 1400 1600 1800 2000 0.75
0.8 0.85 0.9 0.95 1
Generation
Affinity/Fitness Value FLNFN-DMPSO
CNFC-ISEL SEFC MFS-SE
Figure 5.6: Learning curves of best performance of the FLNFN-DMPSO, CNFC-ISEL, SEFC and MFS-SE in control of backing up the truck.
0 20 40 60 80 100
0 20 40 60 80 100
Figure 5.7: Trajectories of truck, starting at four initial positions under the control of the FLNFN-DMPSO after learning using training trajectories.
Table 5.2: Performance comparison of various controllers to control of backing up the truck.
Method Affinity/Fitness Value (Best) Affinity/Fitness Value (Avg.)
FLNFN-DMPSO 0.9637 0.9502
FLNFN-PSO 0.9423 0.9355
CNFC-ISEL [121] 0.9558 0.9511
SEFC [122] 0.9516 0.9451
MFS-SE [123] 0.9398 0.9332
Example 3: Control of Water Bath Temperature System
The goal of this example is to elucidate the control of the temperature of a water bath system according to
0 ( ) thermal capacity of the system and T is the equivalent thermal resistance between R the borders of the system and the surroundings.
T and R C are assumed to be essentially constant, and the system in Eq. (5.11) is rewritten in discrete-time form to some reasonable approximation. The system
0.5 ( ) 40 0
y (C ), which were obtained from a real water bath plant considered elsewhere [120]. The plant input ( )u k is limited between 0V and 5V where V represents the voltage unit. The sampling period is Ts 30 second.
Figure 5.8 presents a block diagram for the conventional training scheme. This block diagram has two phases – the training phase and the control phase. In the
training phase, the switches S1 and S2 are connected to nodes 1 and 2, respectively, to form a training loop. In this loop, training data with input vector
( ) [ p( 1) p( )]
I k y k y k and desired output ( )u k can be defined, where the input vector of the FLNFN controller is the same as that used in the general inverse modeling [124] training scheme. In the control phase, the switches S1 and S2 are connected to nodes 3 and 4, respectively, forming a control loop. In this loop, the control signal ˆ( )u k is generated according to the input vectors
( ) [ ref( 1) p( )]
Figure 5.8: Conventional training scheme.
A sequence of random input signals urd( )k limited between 0V and 5V is injected directly into the simulated system described in Eq. (5.12), using the training scheme for the FLNFN-DMPSO controller. The 120 training patterns are selected based on the input–outputs characteristics to cover the entire reference output. The temperature of the water is initially 25 C, and rises progressively when random input signals are injected.
This dissertation compares the FLNFN-DMPSO controller to the FLNFN controller [32], the proportional-integral-derivative (PID) controller [125], the
manually designed fuzzy controller [8], the FLNN [80], and the TSK-type neuro-fuzzy network (TSK-type NFN) [24]. Each of these controllers is applied to the water bath temperature control system. The performance measures include the set points regulation, the influence of impulse noise, large parameter variations in the system and the tracking capability of the controllers.
The first task is to control the simulated system to follow three set points
ref
Figure 5.9 presents the regulation performance of the FLNFN-DMPSO controller.
The regulation performance was also tested using the FLNFN controller, the PID controller, the fuzzy controller, the FLNN controller and the TSK-type NFN controller.
To test their regulation performance, a performance index, the sum of absolute error (SAE), is defined by simulated system, respectively. The SAE values of the FLNFN-DMPSO, the FLNFN controller, the PID controller, the fuzzy controller, the FLNN controller and the TKS-type NFN controller are 352.32, 352.84, 418.5, 401.5, 379.22 and 361.96, which values are given in the second column of Table 5.3. The proposed FLNFN-DMPSO controller has a much better SAE value of regulation performance than the other controllers.
The second set of simulations is performed to elucidate the noise rejection ability of the six controllers when some unknown impulse noise is imposed on the process.
One impulse noise value of 5C is added to the plant output at the 60th sampling instant. A set point of 50C is adopted in this set of simulations. For the
FLNFN-DMPSO controller, the same training scheme, training data and learning parameters were used as in the first set of simulations. Figure 5.10 presents the behaviors of the FLNFN-DMPSO controller under the influence of impulse noise.
The SAE values of the FLNFN-DMPSO controller, the FLNFN controller, the PID controller, the fuzzy controller, the FLNN controller and the TSK-type NFN controller are 270.29, 270.41, 311.5, 275.8, 324.51 and 274.75, which values are shown in the third column of Table 5.3. The FLNFN-DMPSO controller performs quite well. It recovers very quickly and steadily after the occurrence of the impulse noise.
One common characteristic of many industrial control processes is that their parameters tend to change in an unpredictable way. The value of 0.7 (u k is 2) added to the plant input after the 60th sample in the third set of simulations to test the robustness of the six controllers. A set point of 50C is adopted in this set of simulations. Figure 5.11 presents the behaviors of the FLNFN-DMPSO controller when the plant dynamics change. The SAE values of the FLNFN-DMPSO controller, the FLNFN controller, the PID controller, the fuzzy controller, the FLNN controller and the TSK-type NFN controller are 262.91, 263.35, 322.2, 273.5, 311.54 and 265.48, which values are shown in the fourth column of Table 5.3. The results present the favorable control and disturbance rejection capabilities of the trained FLNFN-DMPSO controller in the water bath system.
In the final set of simulations, the tracking capability of the FLNFN-DMPSO controller with respect to ramp-reference signals is studied. Define
ref
Figure 5.12 presents the tracking performance of the FLNFN-DMPSO controller.
The SAE values of the FLNFN-DMPSO controller, the FLNFN controller, the PID controller, the fuzzy controller, the FLNN controller and the TSK-type NFN controller are 42.45, 44.28, 100.6, 88.1, 98.43 and 54.28, which values are shown in the fifth column of Table 5.3. The results present the favorable control and tracking capabilities of the trained FLNFN-DMPSO controller in the water bath system. The aforementioned simulation results, presented in Table 5.3, demonstrate that the proposed FLNFN-DMPSO controller outperforms other controllers.
0 20 40 60 80 100 120
0 10 20 30 40 50 60 70 80
Control Input → 5V
Sampling Instant
Temperature(Degree C)
Water Bath Temperature Control (case 1)
Reference Signal Actual Signal
Figure 5.9: The regulation performance of the FLNFN-DMPSO controller for the water bath system.
0 20 40 60 80 100 120
Water Bath Temperature Control (case 2)
Reference Signal Actual Signal
Figure 5.10: The behavior of the FLNFN-DMPSO controller under impulse noise for the water bath system.
0 20 40 60 80 100 120
Water Bath Temperature Control (case 3)
Reference Signal Actual Signal
Figure 5.11: The behavior of the FLNFN-DMPSO controller when a change occurs in the water bath system dynamics.
0 20 40 60 80 100 120
Water Bath Temperature Control (case 4)
Reference Signal Actual Signal
Figure 5.12: The tracking performance of the FLNFN-DMPSO controller for the water bath system.
Table 5.3: Performance comparison of various controllers for the water bath temperature control system.
Effect of Change in Plant Dynamics
This chapter proposes an evolutionary neural fuzzy system, designed using
FLNFN model embedded with DMPSO algorithm. The proposed learning scheme consists of structure learning and parameter learning for the FLNFN model. The structure learning depends on the entropy measure to determine the number of fuzzy rules. The proposed DMPSO parameter learning method can adjust the shape of fuzzy rule’s membership function and the corresponding weighting of FLNN. The simulation results have shown the proposed FLNFN-DMPSO method has more chance of converging to the global optimum and yields better performance than other existing models under some circumstances.
Chapter 6
Comparisons and Discussions
PSO is an efficient tool for optimization and search problems. However, it is easy to be trapped into local optima due to its information sharing mechanism. Many researchers have worked on improving its performance in various ways, thereby deriving many interesting variants. This dissertation develops three novel learning algorithms embedded with particle swarm optimizer, named IPSO, BFPSO and DMPSO for the neuro-fuzzy systems.
6.1 Comparisons
In this section, skin color detection problem is performed to evaluate the performance of the proposed IPSO, BFPSO and DMPSO methods.
The skin color detection experimental results of the IPSO and BFPSO methods are given in Section 3.4 and Section 4.3, respectively. In the following subsection, the skin color detection problem is performed to assess the performance of the DMPSO approach in classification application.
6.1.1 Skin Color Detection Using DMPSO
The description of the system is the same as Section 3.4. We set three rules constituting a neuro-fuzzy classifier. In this example, the performance of the DMPSO method is compared with the PSO method [41]. First, the learning curves of DMPSO and PSO methods are shown in Figure 6.1. In Figure 6.1, we find that the performance of the proposed DMPSO method is superior to the PSO method.
Furthermore, the comparison items include the training and testing accuracy rates are tabulated in Table 6.1.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9
Generation
Affinity/Fitness Value DMPSO
PSO
Figure 6.1: The learning curves of PSO and DMPSO methods using the CIT database.
Table 6.1: Performance comparison with PSO and DMPSO methods from the CIT database (Training data: 6000; Generations: 2000)
Method DMPSO PSO
Average training accuracy rate 98.05% 96.77%
Average testing accuracy rate 87.26% 83.64%
The CIT facial database consists of complex backgrounds and diverse lighting.
Hence, from the comparison data listed in Table 6.1, the average of the test accuracy rate is 83.64% for the PSO method and 87.26% for the proposed DMPSO method.
This demonstrates that the CIT database is more complex and does not lead to a decrease in the accuracy rate. The proposed DMPSO method maintains a superior accuracy rate. The color images from the CIT facial database are shown in Figure 6.2.
The corresponding fitness maps generated by well-trained network using the proposed DMPSO method are shown in Figure 6.3. With proper selection of the threshold value, a well-trained network can generate binary outputs (1/0 for skin/non-skin) to detect a facial region. Figure 6.4 shows the masks generated by the proposed skin color classifier. Figure 6.5 shows that the proposed approach determines a facial region accurately.
Figure 6.2: Original face images from CIT repository.
Figure 6.3: Fitness maps generated by a well-trained FLNFN-DMPSO
Figure 6.4: Masks generated by a well-trained skin color classifier.
Figure 6.5: Results of skin color detection with YCbCr color space
6.1.2 Skin Color Detection Results Comparison with Different Approaches
In this subsection, the skin color detection experimental results of neuro-fuzzy classifier embedded with different parameter learning algorithms demonstrated. In this research, we select the FLNFN model as our neuro-fuzzy architecture to develop the skin color classifier. The aim of the skin color detection is to distinguish between skin and non-skin pixels based on the Y, Cb and Cr information. Table 6.2 summarized the average accuracy rates of testing and training data with different approaches.
Table 6.2: Performance comparison with various existing models from the CIT database (Training data: 6000; Generations: 2000)
Method No. of fuzzy rules Average accuracy rate (training data)
Average accuracy rate (testing data)
IPSO 4 93.32% 90.18%
IA [94] 4 88.1% 82.63%
4 79.05% 74.32%
PSO [41]
3 96.77% 83.64%
BFO 3 96.5% 82.39%
BFPSO 3 97.63% 85.82%
DMPSO 3 98.05% 87.26%
6.2 Discussions
In the IPSO and BFPSO approaches, we investigated hybridization by combining PSO with IA and BFO, respectively. In IPSO method, the major parameter learning
In the IPSO and BFPSO approaches, we investigated hybridization by combining PSO with IA and BFO, respectively. In IPSO method, the major parameter learning