Abstract
In this paper the optimal static and the full order anti-windup compensator (AWC) is designed using the improved algorithm of particle swarm optimization (PSO). The existing optimization techniques of AWC use the LMI and the nonlinear control concepts that require complex mathematics. For the optimization, an improved PSO algorithm is used. The PSO algorithm first explores the problem-space with constant non-zero inertia factor, until the fitness function saturates to a local or global minima. The solution obtained thus far is then further refined by rerunning the algorithm but with inertia factor set to zero. This slight improvement enables a better search for a solution within the already explored problem-space. The new optimized AWC is compared with the existing LMI-based AWC and it is observed that the optimized static and full order AWC give better IAE and ISE values as compare to the present conventional methods.
Introduction
Actuator saturation in the control system not only degrades the performance of the system but also incorporate the undesirable effects to the system that make the system unstable. The Anti-windup compensator is an additional controller which is activated onlywhen the system goes in saturation region and used to improve the closed loop response [1]. As AWC only take part when the system go into saturation region, so in the designing phase the linear controller is designed without considering the saturation effect and after that the AWC is designed.
Until now numerous efforts have been made for the designing of the AWC design of SISO and single loop MIMO stable systems. The AWC theory is not only well developed for the stable system as well as for the unstable system [2-5]. The AWC schemes have been developed for the cascaded [6] and multiple loop MIMO system and nonlinear systems [7, 8]. Nowadays the robust AWC designing is a hot area for the researchers [9, 10, 11].
Full order, low order and static AWC has been designed successfully based on the L2 norm reduction and optimize using linear matrix inequalities (LMI) [12]. But this L2 norm reduction and optimization using the LMI requires a lot of complex mathematics and concepts of the nonlinear control. Until now there is not much work about the optimization of the static, low order and full order AWC coefficients. In this paper we are optimizing the coefficients of the AWC using the particle swarm optimization (PSO).
Particle Swarm Optimization algorithm is an evolutionary algorithm which simulates the search of food by bird-flocks. It was first presented as an optimization tool by Kennedy and Eberhart (1995) [13]. Since then, many improvements have been suggested to the algorithm, with almost all having the objective of decreasing the time required to achieve the optimum solution. These include use of linearly decreasing inertia factor (w) [14], introduction of constriction factor (χ) [15], development of Binary PSO [16], Bare Bone PSO [17], and fully informed PSO (FIPS) [18]. In this paper, we have made use of modified PSO algorithm by Shi and Eberhart [14], with a small innovation which allowed better search in our case, to be described later. There are some improvements have been done in the PSO to solve the numerical optimization problem by modifying the velocity updating equation to balance the global search and local search abilities of PSO called NPSO [20]. Recently the PSO has been combined with the differential evolution and the neural network to improve the response and trajectory tracking [21, 22]. In this paper we have improved the PSO algorithm so that it search the problem space with non-zero inertia factor and then obtained solution is refined by putting the inertia factor equals to zero.
The outlined of the paper is as follow; in section-II the improvement in the PSO algorithm is discussed. In section-III the concept about the optimization of AWC using PSO is discussed in detail. In section-IV the simulation results showing the improvement in the optimal value of IAE and ISE are discussed.
The improved Particle Swarm Optimization (PSO) algorithm
The PSO algorithm begins by simulating a problem-space with dimensions (D) equal to the parameters to be tuned and bounds dictated by the user for each parameter. This problem-space is analogous to the environment in which the birds search for food. A fixed number of particles (NP) are initialized at random positions (xi) in this D-dimensional problem-space within the specified bounds. A cost-function is then defined, analogous to the distance of the bird from the food. The aim of the algorithm is to minimize the cost-function through iterative process. The velocity (vi) of each simulated particle is also initially set to zero.
The movement of the particles in the problem-space is dictated by three influences: inertia(vi), cognitive influence (pbest) and social influence (gbest).The directions of cognitive and social influences depend on the direction vectors of the bird's position from pbest and gbest respectively. Both, pbest and gbest are compared with the current cost on each iteration and updated if required. The relative dependence of particles' motion on these three influences is maintained by learning factors (c1 and c2) and inertia factor (w) [19]. Furthermore, randomness is maintained in the system by multiplying the learning factors by some random numbers between 0 and 1.
while (k<Kmax)
for(i=1:NP)
computecosti,(k) as a function of xi(k)
if (costi(k)<pbesti)
pbesti= costi(k)
end if
end for
gbest = minimum of all pbest values
for (i=1:NP)
vi(k+1) = w-vi(k) + c1-rand- (pbesti - xi(k)) + c2-rand- (gbest - xi(k))
xi(k+1) = xi(k) + vi(k+1)
end for
k =k+1
end while
The algorithm mentioned above is run for a fixed number of iterations Kmax, as defined by the user. However, it is quite possible that the best possible solution is still not achieved within these iterations. This can happen when either the algorithm gets stuck in a local minima or the solution does not converge to a minima of the problem. To perform a better search of the solution, the findings by Shi and Eberhart have been utilized. According to them, for large values of inertia factor(w), the algorithm tries to exploit new areas in the problem space. On the other hand, for small values of w, the algorithm further searches within the explored region for a better solution.
In view of these findings, the PSO algorithm is run in two stages: exploration stage and refinement stage. An overview of the modifications in the algorithm is given in the Figure-1.
untitled.jpg
Figure-1. Overview of modification in PSO algorithm
As shown in the figure, the usual PSO algorithm is first run for a particular large value of inertia factor (w0), until smallest cost-functionvalue (gbest)achieved thus far, becomes constant for a fair number of iterations (K0). This ends the exploration stage of the algorithm. It should be noted here that the w0 is not kept very large as they may cause divergence of the solution. After this stage, the algorithm is paused and inertia factor (w) is dropped from w0 to zero. This marks the beginning of refinement stage. The algorithm is then resumed and again allowed to run till the stage when the gbest again reaches a steady state, say after Ktotal iterations. The achievement of steady state for a particular stage is assumed only after at least 30 iterations have been run in that stage.
IMPLEMENTATION OF IMPROVED PSO ON AWC
In our case, Particle Swarm Optimizer is used to tune the parameters of the static and dynamic controllers to compensate the wind-up effect on the control system. The anti-windup compensator activated only when the system goes under saturation. The optimize gains and coefficients of the AWC are very important and there are different techniques to optimize the gains like L2 norm optimization using LMI. In the Figure-1 the structure of the anti-windup compensator is shown. The main task is to optimize the gains Theta-1 and Theta-2 for the better results. As in the PSO we require the error for the cost function. In our case the point of error (e) is shown in Figure-2 which is simply the error between reference (r) and the output (y).
1.jpg
Figure-2. Error coefficient from AWC scheme for PSO
In the Figure-2 G(s) is the transfer function of the plant and K(s) is the transfer function matrix of the controller, which has state-space realizations as follows:
(1)
(2)
For the analysis here, it is convenient to split the plant function as where is the transfer function from disturbance to the measured output y(t) and is the transfer function from the plant input to the output. Similarly, the controller is portioned as.
This optimization of the gains and coefficients can not only be applied on the static AWC but also on the dynamical anti-windup compensator. The details about the implementation of the improved PSO on AWC are discussed in the subsections.
Optimization of Static AWC
The static AWC are most commonly used anti-windup compensator in industry and practical application because of simple in optimization and the easy in implementation. The general block diagram of the static AWC is shown in Figure-3.
Figure-3. Block diagram of the static anti-windup compensator (AWC)
In this static AWC the main task is to optimize the θ1 and θ2 gains. In PSO algorithm, for optimizing the gains or coefficients, the limits and the signs of the gains must be well defined. As the gain θ1 is going to subtract from the loop so its value must be positive, so that it would bring the actuator value within the range of saturation by overall subtracting the magnitude from the loop. In the similar way as the gain θ2 is going to add in the feedback loop so it value must be negative so that it will also try to decrease the magnitude and will try to bring it between the saturation the limits.
Gain θ1: [0,∞)
Gain θ2: (-∞,0]
The optimization of the static AWC is very simple because we have to optimize only two gains and the details about the sign and the range of these gains have been discussed. The detailed example and comparison with the results from Ian Postelwaithe et. al [9] are discussed in the next section.
Optimization of Dynamic AWC
The optimization of the dynamical AWC is similar as the static AWC but instead of having the static gains it have the transfer functions at the place of gains. The overall structure and block diagram of the dynamical AWC is shown in Figure-4.
Figure-4. Block diagram of the dynamic anti-windup compensator (AWC)
In the Figure-3 the detail of the transfer functions M(s)-I and G2M(S) is given below [9].
(4)
In the state space representation of the matrices the F is a column matrix and its number or column are equal to the order of the system. If system is 2nd order then the dimensions of the F matrix will be 1-2, similarly if the system is 3rd order then the dimension of the F matrix will be 1-3.
As in the PSO algorithm we optimize the gains or coefficients, it would be more convenient if the M(s) and N(s) are described in the form of transfer function. From the state space representations of these matrices it is obvious that the denominator of these transfer functions will be same and the numerator will be different. For the 2nd order systemthe transfer functions in terms of tuned parameters can be written as:
In the above transfer function the p1 to p5 are the coefficients that have to be tuned for the optimal results.
In this paper we are dealing with the optimization of the stable plants, so the transfer function M(s) and N(s) must be stable as well for the overall stability of the system. So, coefficients p3 and p4 must be positive. Similarly for the system stability and to not go into the non-minimum phase system the p1, p2 and p5 must also be positive. So, the limits and the signs of the coefficients, that are necessary for the PSO algorithm, are defined as below:
Coefficient p1 : [0,+∞) Coefficient p2 : [0,+∞)
Coefficient p3 : [0,+∞) Coefficient p4 : [0,+∞)
Coefficient p5 : [0,+∞)
The dynamical AWC is normally preferred over the static AWC. Although it is little more complex to design but it gives the improved results as compare to the static AWC. The dynamical AWC also have the ability to provide the robustness against the perturbations. The detailed example and comparison with the results from Turner et. al. [9] are discussed in the next section and comparison and results are outlined.
Simulation Results
The improved PSO algorithm was implemented to both the static AWC and dynamic AWC. In both the cases, two different cost-functions were used: IAE (Integral Absolute Error) and ISE (Integral Square error), where error refers to the difference between the reference (r) and the output (y). The values of c1 and c2 are set to 1.2 and 1.6 respectively. The value of w0 is varied and the results of the PSO algorithm for these values are then compared to come up with the best possible solution. In this section both the static as well as the dynamic AWC are discussed. The detailed examples and the discussions are given as below:
Results for Static AWC
The plant and the controller choose for the simulation purposes are taken from the paper Turner et al [9]. The nominal plant is considered in this paper, which is given as follows:
(7)
The 2-D linear controller which was designed for the plant G(s) is described by the state space matrices
(8)
The static gains that are calculated using the LMI in [9] are as follows:
(9)
For the optimization of the gains using improved PSO algorithm we select the following range of the gains. We can select the wide range of the gains but the small gains give the more stability.
Gain θ1: [-1, 10]
Gain θ2: [0, 10]
The following table shows the results obtained after implementing the improved PSO on static AWC using IAE cost-function.The IAE value for solution by Turner et al [9] is 16.8592.
w0
K0
IAE value (stage 1)
Ktotal
IAE value (final)
0.6
43
16.7022
100
16.7015
0.7
74
16.7021
120
16.7019
0.8
33
16.7024
150
16.7016
0.9
30
16.7026
80
16.7018
1.0
42
16.7021
130
16.7022
Table-1. Optimization the gains of the static AWC for IAE
The parameter values which gave the best solution are:
(10)
The table-2 shows the results obtained after implementing the improved PSO on static AWC using ISE cost-function. The ISE value for solution by Turner et al [9] is 10.3886.
w0
K0
ISE value (stage 1)
Ktotal
ISE value (final)
0.6
66
10.1381
96
10.1381
0.7
68
16.1381
100
10.1380
0.8
39
10.1386
150
10.1381
0.9
32
10.1383
100
10.1382
1.0
32
10.1398
100
10.1380
Table-2. Optimization the gains of the static AWC for ISE
The parameter values which gave the best solution are:
(11)
From the simulations results of the static AWC it is clear that the value of the ISE and IAE using the improved PSO algorithm has been improved as compare to the Turner et. al [9]. The values of the IAE and ISE are more optimal and give more convergence.
Results for Dynamic AWC
For the simulation and comparison we are considering the same plant and controller as shown in Eq. (7) and Eq. (8). The value of the optimized F using LMI from Turner et. al. [9] is given below:
(12)
Based on this value of F, the calculated transfer functions of M(s)-I and N(s) are given in Eq. (13) and Eq. (14).
(13)
(14)
The IAE value for solution by Turner et al [9] is 15.7626.
As this system is 2nd order so the generalized form of the M(s)-I and N(s) are shown in Eq. (5-6). The range of the parameters for the optimization is given below.
Coefficient p1 : [0,100] Coefficient p2 : [0,100]
Coefficient p3 : [0,100] Coefficient p4 : [0,100]
Coefficient p5 : [0,100]
The results for dynamic AWC using IAE cost-function are given in Table-3. For the optimization purpose the different values of the w0 was selected.
w0
K0
IAE value (stage 1)
Ktotal
IAE value (final)
0.6
78
15.7034
108
15.7034
0.7
150
15.6933
180
15.6933
0.8
109
15.6923
150
15.6921
0.9
97
15.7071
100
15.7041
1.0
39
15.7425
74
15.7398
Table-3. Optimization the coefficents of the dynamic AWC for IAE
The parameter values which gave the best solution are:
p1= 84.1183; p2= 26.4177;
p3= 2.3473; p4= 1.9501;
p5= 41.5470
Based on these parameters the optimal value of IAE is 15.6921 and the resultant transfer function of M(s)-I and N(s) is shown in Eq. (15) and Eq. (16).
(15)
(16)
Similarly, the results for dynamic AWC using ISE cost-function are shown in Table-4. The ISE value for solution by Turner et al [9] is 8.7166. The optimized value of ISE based on improved PSO algorithm is 7.7297.
w0
K0
ISE value (stage 1)
Ktotal
ISE value (final)
0.6
72
8.6604
104
8.6599
0.7
100
8.6612
150
8.6597
0.8
74
8.6569
129
8.6562
0.9
100
7.8123
300
7.7297
1.0
82
8.6640
135
8.6631
Table-4. Optimization the coefficents of the dynamic AWC for ISE
The parameter values which gave the best solution are:
p1= 46.6578; p2= 0.0104
p3= 48.9349; p4= 14.0907
p5= 1.4146
Based on these parameters the resultant transfer function of M(s)-I and N(s) is shown in Eq. (17) and Eq. (18).
(17)
(18)
The optimal value of IAE and ISE by Turner et. al [9] using LMI are 15.7626 and 8.7166 respectively for the dynamic AWC. The values of IAE and ISE optimized by PSO are 15.6921 and 7.7297.
From these simulation results it is shown that the values of IAE and ISE are more optimized as compare to the previous results.
Conclusion
In this paper the optimization of the anti-windup compensator is considered. Proposed improved PSO algorithm provides the optimal IAE and ISE value as compare to the existing methodologies that are optimized using convex optimization using the linear matrix inequalities (LMI). First the PSO algorithm explores the problem-space with constant non-zero inertia factor, then further refined by rerunning the algorithm but with inertia factor set to zero. The comparison of the AWC optimization using PSO is compared with the existing techniques. The static and the dynamic AWC is considered for comparison and the optimization using PSO proves the optimality in both cases.
