ISA Transactions 53 (2014) 517–523
Contents lists available at ScienceDirect
ISA Transactions journal homepage: www.elsevier.com/locate/isatrans
Research Article
Controller design based on μ analysis and PSO algorithm Ali Lari, Alireza Khosravi n, Farshad Rajabi Faculty of Electrical and Computer Engineering, Babol University of Technology, Babol 47135-484, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 12 April 2013 Received in revised form 28 October 2013 Accepted 3 November 2013 Available online 7 December 2013 This paper was recommended for publication by Prof. A.B. Rad
In this paper an evolutionary algorithm is employed to address the controller design problem based on μ analysis. Conventional solutions to μ synthesis problem such as D–K iteration method often lead to high order, impractical controllers. In the proposed approach, a constrained optimization problem based on μ analysis is defined and then an evolutionary approach is employed to solve the optimization problem. The goal is to achieve a more practical controller with lower order. A benchmark system named two-tank system is considered to evaluate performance of the proposed approach. Simulation results show that the proposed controller performs more effective than high order H1 controller and has close responses to the high order D–K iteration controller as the common solution to μ synthesis problem. & 2013 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Robust control μ Synthesis problem PSO Structure-specified controller Two-tank system
1. Introduction Designing structure-specified controllers is one of the common ways to obtain controllers which are much easier to implement. Robust control is one of the areas in which designing controllers using structure-specified approach has been examined to simplify the controller structure. In the last decade, designing structurespecified controllers (especially in robust control) using evolutionary algorithms has attracted considerable attention [1–6]. In this approach, coefficients of a low order structure-specified controller are calculated by an evolutionary algorithm so that the resulting controller satisfies the robust design objectives. In the presence of uncertainties in models dynamic, μ synthesis problem is one of the most powerful robust controller design techniques. In μ synthesis problem, controller design is Carried out based on the structured singular values (μ) stability and performance analysis whereas in some other robust techniques like H1, singular values are the basis of controller design. The differences between these two approaches arise from definition of structured singular values and singular values. In fact, using structured singular values is the most effective and conservative technique to controller design in case there is a structured uncertainty in system modeling. D–K iteration is the most common method to solve μ synthesis problem. Unfortunately the resulting controllers in this method are often impractical, high order controllers. Moreover, the reduced
n
Corresponding author. Tel.: þ 98 47435484. E-mail addresses: [email protected] (A. Lari), [email protected] (A. Khosravi), [email protected] (F. Rajabi).
order controllers which are close to the main D–K iteration controller are usually not yet practical enough to be used instead of the main high order controller. In addition, one of the main drawbacks of D–K iteration is that it does not guarantee convergence to a global or even local minimum, which leads to nonoptimality of the resulting controller [7]. To design a more practical controller satisfying stability and performance criterions based on μ analysis and coping with the mentioned problem of the other solutions like D–K iteration method, this paper proposes an evolutionary approach to design a structure-specified μ synthesis controller. A rather similar approach on simple systems such as a DC motor system and a mass damper spring system has been examined by authors and promising results have been achieved [8]. In this paper the effectiveness of the proposed approach is verified through designing robust controllers for a benchmark MIMO system. In this approach, μ synthesis problem is solved as a constrained optimization problem in which the μ analysis is used to examine robust stability and performance criterions; robust stability as the constraint of the optimization problem and robust performance as the cost function which should be minimized. A simple improved particle swarm optimization (PSO) algorithm as an evolutionary algorithm is employed to find the required coefficients of a practical structure-specified controller such that minimizes the cost function and satisfies the constraint. PSO as a population-based algorithm can solve a variety of difficulties associated with optimization problems. Compared with GA, PSO takes less time for each function evaluation as it does not use many operators. Due to the simple concept, easy implementation and quick convergence, nowadays PSO has gained much attention and has wide applications in different fields [9–11].
0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.11.006
518
A. Lari et al. / ISA Transactions 53 (2014) 517–523
A multi input multi output (MIMO) benchmark system named two-tank system has been chosen to evaluate efficiency of the proposed method. Dynamic of this system is comprehensively discussed in [12–14]. Simulation results on the benchmark system show that the designed first order controller by the proposed approach has close responses to those of the high order D–K iteration controller and performs better than high order H1 controllers designed in [14]. The rest of this paper is organized as follow: Section 2 discusses the basic μ synthesis problem and the evolutionary algorithm. In Section 3, the proposed approach is illustrated step by step and in Section 4 description of the two-tank system is given. Simulation result is presented in Section 5 and finally Section 6 is dedicated to conclusion.
Δ
d
v
z
M
w
Fig. 2. Analysis framework.
ΔP
Δ
v
d 2. Preliminary Defining an appropriate cost function is the initial step to solve an optimization problem. In this section first, the requirements to define the optimization problem based on μ analysis is introduced, next the evolutionary algorithm to solve the problem is described. 2.1. Cost function In the cost function, the aim briefly is: “searching for the unknown parameters of a structure-specified controller with higher performance based on μ performance analysis, among controllers which satisfy stability condition based on μ analysis.” Therefore, to define the cost function mathematically, it is required to know the μ stability and performance criterions. 2.1.1. Robust stability based on μ analysis Fig. 1, shows a general framework used in μ analysis. In this figure, K is the controller, P is the nominal plant, Δ is the uncertainty set (perturbation matrix). Moreover, w denotes the exogenous input typically including command signals, disturbances, noises, etc.; z denotes the error output; v and d are the input and output signals of the dynamic uncertainties, respectively. Finally, u and v are the control input and the measurement signal, respectively. Using M(P,K) ¼Fl(P,K), Fig. 1 reduces to Fig. 2 (Fl(P;K) is the lower linear fractional transformation). In this figure, M can be partitioned as follows, # " M 11 M 12 v d ð1Þ ¼ M 21 M 22 z w |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} M
Regarding matrix M11, the stability condition based on μ analysis is defined as (2). jjM 11 ðsÞjjμ o 1
or
μΔ ðM 11 Þ o 1
ð2Þ
where μ is the structured singular value, jj:jjμ is μ norm and Δ is
w
M
z
Fig. 3. Structure for robust performance analysis.
uncertainty set. The definition of the structured singular value, μ, depends on the underlying block structure of perturbations which is defined as follows. Perturbation matrix Δ Cnn is defined as Eq. (3). Δ ¼ fdiag½δ1 I r1 ; :::; δs I rs ; Δ1 ; :::; ΔF : δi A C; Δj A Cmj mj g
ð3Þ
The block diagonal matrix Δ, includes two types of blocks: repeated scalar block (δi ) and full block (Δj ). In Eq. (3), S and F represent the number of repeated scalar blocks and the number of full blocks, respectively. For consistency among all the dimensions, Eq. (4) should be held. S
F
i¼1
j¼1
∑ r i þ ∑ mj ¼ n
ð4Þ
Structure singular value, μΔ ðMÞ, of a Matrix M A Cnn with respect to a block structure Δ A Δ is defined as Eq. (5). μΔ ðMÞ : ¼
1 min fsðΔÞ : Δ A Δ; detðI MΔÞ ¼ 0g
ð5Þ
Unless no Δ A Δ makes I MΔ singular, in which case μΔ ðMÞ : ¼ 0. 2.1.2. Robust performance based on μ analysis Defining an augmented block structure Δp as (6) in which Δf is a full block uncertainty matrix from z to w in Fig. 3, and Δ is the uncertainty matrix, the performance condition based on μ analysis is defined as (7). Δp A Δ : ¼ fdiag fΔ; Δf g : ΔA BΔ; jjΔf jj1 r 1g ; BΔ ¼ fΔ A Δ : s ðΔÞ r 1g ð6Þ jjMðsÞjjμ o 1
or
μΔp ðMÞ o1
ð7Þ
2.2. Evolutionary algorithm
Δ d
w
v z
P
u
y
K Fig. 1. General framework.
Particle swarm optimization (PSO) has been selected as a simple evolutionary algorithm to solve the optimization problem. Quick convergence is the main reason of using PSO in this paper because cost calculation in this case is time consuming. PSO takes less time for each function evaluation as it does not use many operators (such as mutation, crossover and selection operators used in GA). Particle swarm optimization (PSO) is originally attributed to Kennedy and Eberhart [15], based on the social behavior of collection of animals such as birds flocking. The swarm moves
A. Lari et al. / ISA Transactions 53 (2014) 517–523
randomly along the search space, remembering the best solution found by each particle and by the whole population. Regarding the information obtained by each particle and the swarm, position of the particles are updated by (8) and (9). V id ¼ wV id þ c1 r 1 ðP id X id Þ þ c2 r 2 ðP gd X id Þ
ð8Þ
X id ¼ X id þV id ; d ¼ 1; 2; …; N
ð9Þ
i ¼ 1; 2; …; S
where Xid represents the current position of the particle, Vid is the velocity that changes the particle's position, Pid is the best individual particle position, Pgd denotes the best swarm position, the parameters c1 and c2 are cognitive and social parameters, respectively; the parameters r1 and r2 are random numbers between 0 and 1. Finally, w is the inertia weight to balance the global and local search abilities. A large inertia weight facilitates global search while a small inertia weight facilitates local search. In an empirical study on PSO [16], Shi and Eberhart claimed that a linearly decreasing inertia weight could improve local search towards the end of a run, rather than using a constant value throughout. A decreasing function for the dynamic inertia weight can be devised in the following winitial wf inal w ¼ ðiter max iter cur Þ: ð10Þ þwf inal itermax where winitial and wfinal represent the initial and final inertia weights at the start of a given run, respectively; itermax is the maximum number of iterations in an offered run, and iter cur denotes the current iteration number at the present time step. However, due to the utilization of a linearly decreasing inertia weight, the global search ability at the end of the run may be inadequate. The PSO may fail to find the required optimal in cases the problems are too complicated. But to some extent, this can be overcome by employing a self-adapting strategy for adjusting the acceleration coefficients. Suganthan [17] applied the optimizing method that makes the two factors decrease linearly with the increase of iteration numbers, but the results were not as good as the fixed value 2 of c1 and c2. Ratnaweera [18] improved the convergence of particles to the global optima based on the way that makes c1 decrease and c2 increase linearly with the increase of iteration numbers, as they are given by the following equations: c1 ¼ c1s þ iter cur ðc1e c1s Þ=iter max c2 ¼ c2s þ iter cur ðc2e c2s Þ=iter max
ð11Þ
where c1s and c2s represent the initial values of c1 and c2, respectively; and c1e and c2e show the final values of c1 and c2, respectively.
3. The proposed approach
Step 2 Set the parameters of PSO algorithm. Step 3 Initialize a group of random particles in search space (Each particle includes unknown parameters of the structure specified controller). It should be checked that all of initial particles satisfy μ stability condition. Step 4 Calculate best individual position (Pid), best swarm position (Pgd) and the cost function for each particle regarding to the cost function introduced in (12). Step 5 Update PSO parameters c1, c2 and w according to the iteration number given in Eqs. (10) and (11). Step 6 Update the velocities and positions of each particle according to Eqs. (8) and (9). Step 7 If each new position verifies the stability constraint, set it as the new position. Else, set the last position as the new position. Step 8 If the number of iterations is lower than a predefined value, go to step 4, else, to step 9. Step 9 Consider Pgd as the coefficients of desired controller. If the obtained controller is not efficient enough, the order of structure-specified controller can be incremented. Using the proposed approach, also, makes it possible to embody time objectives in the cost function introduced in (12).
4. Case study: Two-tank system A benchmark problem called two-tank system has considered to evaluate efficiency of the proposed controller. A detailed description has given by [12–14]. The two-tank system is shown schematically in Fig. 4. This system consists of two water tanks in cascade, named tank 1 and tank 2 in the figure. Tank 1 is fed hot and cold water via controllable valve and tank 2 is fed by water from an exit at the bottom of tank 1. A constant level is maintained in tank 2 by means of an overflow. Tank 2 is also fed by a cold water bias stream which enables the tanks to have different steady-state temperatures [14]. The two-tank system, used in simulation, has two measured signals: t1, t2. Third measured signal is water level in tank 1 (h1), which is not controlled. This quantity is, however, important for assessment of controller performance. Also, the system has two input signals: fh and fc. These inputs are commands to hot flow (fhc) and cold flow (fcc) actuators, which are transformed to hot water flow fh and cold water flow fc [14]. Using the plant parameters and the selected operating point considered in [14], nominal model from the inputs fh, fc to the outputs t1, t2 can be written in two-dimensional transfer matrix
In this section it is described how robust controllers can be designed based on the new approach to solve μ synthesis problem. At the core of the proposed approach, the simple PSO algorithm, described in Section 2 is employed to solve μ synthesis problem as a constrained optimization problem shown in (12). As it can be seen in (12), stability condition based on μ analysis is the constraint and μ performance condition is the cost function which should be minimized. min jjMðsÞjjμ KðsÞ
such that : jjM 11 ðsÞjjμ o 1
ð12Þ
where M and M11 are defined in Eq. (1) and K(s) is the controller. Generally, the procedure of the proposed approach is summarized as follows: Step 1 Specify the structure of the controller.
519
Fig. 4. Schematic diagram of the two-tank system [14].
520
A. Lari et al. / ISA Transactions 53 (2014) 517–523
Fig. 5. Schematic representation of the perturbed, linear, two-tank model [14].
Fig. 7. Interconnection structure for m synthesis [14].
Fig. 6. Closed-loop interconnection [14].
as (13).
"
Pnom ðsÞ 2
P 11 ðsÞ
P 12 ðsÞ
P 21 ðsÞ
P 22 ðsÞ
0:0036s2
#
þ 0:0001s þ 7:815710
7
6 s3 þ 0:0491s2 þ 0:0007s þ 3:068410 6 ¼ 4 0:0109s2 0:0004s 2:344710 6 s3 þ 0:0491s2 þ 0:0007s þ 3:068410 6
6
0:0004s þ 4:664310 s3 þ 0:0491s2 þ 0:0007s þ 3:068410 6 0:156210 4 s þ 0:018710 4 s3 þ 0:0491s2 þ 0:0007s þ 3:068410 6
3 7 5
ð13Þ Differences from the nominal model are utilized by multiplicative perturbations at the outputs of measured quantities (Fig. 5). 4.1. Design scheme The problem, is tracking of set point commands for t1 and t2. Fig. 6, illustrates the desired closed-loop configuration. The controller has access to both the reference inputs and the temperature measurements. This configuration is less restrictive than the design which only uses the error between the temperature and the set points [14]. The layout of the interconnection structure for μ synthesis is shown in Fig. 7. It is desired to weight both the amplitude and the rate of the actuator in the case of the actuation weights. Using fhc as an example; the approach is to create an actuator model with fh and dfh/dt as outputs. These can then be separately weighted with constant weights [14]. Fig. 8 illustrates the form of such a weighted actuator model. The actuator bandwidth, denoted by BW in Fig. 8, is, BW ¼ 20 radians=s
Fig. 8. Model of the flow valve actuator including magnitude and rate weightings [14].
proposed controller is compared with high order, complicated D–K iteration controller (of order 22) and H1 controller (of order 15) designed in [14]. The proposed controller is also compared with the reduced order D–K controllers of orders 1 and 2. A simple controller containing elements of order 1, is selected as the structure-specified controller. This controller is shown in (15). 2 a11 a12 a13 a14 3 CðsÞ ¼ 4
s þ b11 a21 s þ b21
s þ b12 a22 s þ b22
s þ b13 a23 s þ b23
s þ b14 a24 s þ b24
5
ð15Þ
To calculate coefficients of the controller, parameters of PSO algorithm is adjusted as Table 2. Search space of PSO algorithm is also adjusted as (16). 1 o aij o 1 and 0 obij o1 for i ¼ 1; 2 and j ¼ 1; :::; 4
ð16Þ
Using these parameters the proposed approach leads to a controller as (17). 2 0:037 3 0:072 0:003 0:220 CðsÞ ¼ 4
s þ 0:200 0:242 s þ 0:136
s þ 0:039 0:015 s þ 0:491
s þ 0:089 0:143 s þ 0:254
s þ 0:227 0:113 s þ 0:489
5
ð17Þ
ð14Þ
In the described two-tank system, tank 1, supplies the majority of the water flowing into tank 2, which means that changes in t2 are dominated by changes in t1. It makes more sense to express the reference weighting in terms of t1 and t2–t1, denoted by Wt1cmd and Wtdiffcmd in Table 1. In order to compare with the controllers designed in [14], all the weighting functions have selected exactly the same as weighting functions used in [14]. Table 1 shows these functions. In this table h1ss ¼ 0.75 is steady state value of h1.
5. Simulation results In the following the two-tank system introduced in Section 4, is employed to evaluate the proposed approach. Then, the
5.1. A comparison between D–K iteration and structure-specified controllers D–K iteration controller and its reduced orders (using Hanklenorm approximation) are compared with the proposed structurespecified controller through μ plots in Figs. 9 and 10. Fig. 9, shows that all controllers have almost equally well stability margins. The μ plot in whole frequency range is lower than 0.2 and it means that robust stability margin against perturbations is well enough. Based on Fig. 10, the proposed controller has close behavior to the high order D–K iteration controller. Even in some frequencies from .03 to 0.2 rad/s, robust performance of the proposed controller is better than that of D–K iteration controller. In addition, regarding to Fig. 10, reduced order D–K iteration controllers have a poor robust performance;
A. Lari et al. / ISA Transactions 53 (2014) 517–523
521
Table 1 Weighting functions. Perturbation weights
Tracking error weights
0:5s W h1 ¼ 0:01 þ 0:25s þ1 20h1ss s W t1 ¼ 0:1 þ 0:2s þ 1 100s W t2 ¼ 0:1 þ s þ 21
100 W t1perf ¼ 400s þ1 50 W t2perf ¼ 800s þ 1
Sensor noise weights
Reference weights
W h1noise ¼ 0:01 W t1noise ¼ 0:03 W t2noise ¼ 0:03
W t1cmd ¼ 0:1 W tdif f cmd ¼ 0:01
Table 2 Parameters of PSO algorithm.
Actuator weights W hact ¼ 0:01 W hrate ¼ 50 W cact ¼ 0:01 W crate ¼ 50
0.8
Value
Parameter
Value
c1e c1s c2e c2s
0.5 1.5 1.5 0.5
Number of particles Number of iterations winitial wfinal
50 150 1 0.4
0.75 0.7
Measurements
Parameter
h1 t1 t2 t1ref t2ref
0.65 0.6 0.55
0.2
0.5 Structure - Specified D -K Reduced D- K of Order 2 Reduced D - K of Order 1
0.18
Mu upper bound
0.16
0.45 0
100
200
300
400
500
600
700
800
Time (sec)
0.14 0.12
Fig. 11. Response to reference signal for H1 controller.
0.1 0.08 0.06
0.8
0.04 0.75
0.02 -4
10
-3
10
-2
10
-1
10
0
10
0.7
1
Frequency (rad/sec) Fig. 9. Comparison of robust stability between the controllers.
Measurements
10
h1 t1 t2 t1ref t2ref
0.65 0.6 0.55 0.5
3.5 0.45 0
Mu upper bound
3
100
200
300
400
500
600
700
800
Time (sec) 2.5 Fig. 12. Response to reference signal for D–K iteration controller.
2 1.5
controllers which retain robust features like a high efficient but high order, impractical μ synthesis controller.
1 0.5 0 -4
10
-3
10
-2
10
-1
10
0
10
1
10
Frequency (rad/sec)
5.2. A comparison between D–K iteration, H1 and structurespecified controllers
Fig. 10. Comparison of robust performance between the controllers.
especially in low frequencies they cannot meet robust performance requirements. From Figs. 9 and 10, it can be concluded that the proposed approach to solve μ synthesis problem can lead to relatively simple
In this subsection, the controllers designed based on D–K iteration, H1 approach (designed in [14]) and the proposed approach are compared through time responses and worst case analysis. This is completely in accordance with the procedure considered in [14] to compare the designed controllers. Figs. 11–13 show time responses to ramp reference signal (sensor noise has also been applied).
522
A. Lari et al. / ISA Transactions 53 (2014) 517–523
0.8
0.9
0.75
0.8 0.7 h1 t1 t2 t1ref t2ref
0.65 0.6
Actuators
Measurements
0.7
fhc fcc
0.6 0.5
0.55
0.4
0.5
0.3
0.45 0
100
200
300
400
500
600
700
0.2
800
0
100
200
Time (sec)
300
400
500
600
700
800
Time: seconds
Fig. 13. Response to reference signal for the proposed controller.
Fig. 16. Actuator commands for the proposed controller.
2.5 0.9
Nominal
Magnitude
Actuators
0.7 0.6
Worst-case
2
fhc fcc
0.8
1.5
1
0.5 0.5 0.4 0
0.3
-4
-3
10
10
-2
10
-1
10
0
10
1
10
Frequency (red/sec)
0.2 0
100
200
300
400
500
600
700
800
Time: seconds
Fig. 17. Performance analysis for H1 controller.
Fig. 14. Actuator commands for H1 controller.
0.9 fhc fcc
0.8
Actuators
0.7 0.6 0.5 0.4 0.3 0.2 0
100
200
300
400
500
600
700
800
Time: seconds Fig. 15. Actuator commands for D–K iteration controller.
signals like H1 controller. That's why small fluctuations in actuator command of these controllers can be seen. In the following, the worst-case performance of the closed-loop systems are investigated in Figs. 17–19. Unlike Figs. 18 and 19, in Fig. 17, the worst-case performance of the closed-loop system is significantly worse than the nominal performance, which verifies that the H1 controller is not robust to modeling errors. Finally, variation of cost function versus iteration in PSO algorithm is depicted in Fig. 20. At the end, it must be noted that in this paper only first order structure-specified controller (15) has been examined and acceptable results has been obtained. It is obvious that higher order structure-specified controllers can lead to better results. But, using a higher order controller results in an increase in the number of unknown coefficients and a larger search space in which PSO algorithm may not be converged to a desired solution. Regarding to these issues, it seems that second order structure-specified controller is still a proper choice for the two-tank system if a more advanced evolutionary algorithm is used.
6. Conclusions Based on these figures, it is clear that the H1 controller has poor time response (steady-state error) with respect to D–K iteration and the proposed controllers. Figs. 14–16 show different actuator commands for the controllers in nominal closed-loop system. Based on these figures, D–K iteration and the proposed controllers cannot reject noise
In this paper a structure-specified μ synthesis controller has been designed. In this approach, μ synthesis problem is solved as an optimization problem using particle swarm optimization (PSO) as an evolutionary algorithm. Structure of controller in the proposed approach is selectable. Unlike the D–K iteration method,
A. Lari et al. / ISA Transactions 53 (2014) 517–523
results on a MIMO benchmark problem, a two-tank system, show that the proposed first order controller has close robust performance features to the high order D–K iteration controller and can give better results with respect to the high order H1 controller.
1.4
Nominal 1.2
Worst-case
Magnitude
1 0.8
References 0.6
[1] Ho S-J, Ho S-Y, Hung M-H, Shu L-S, Huang H-L. Designing structure-specified mixed H2/H1 optimal controllers using an intelligent genetic algorithm IGA. IEEE Trans Control Syst Technol 2005;13:1119–24. [2] Zamini M, Sadati N, Karimi Ghartemai M. Design of an H1 PID controller using particle swarm optimization. Int J Control Autom Syst 2009;7:273–80. [3] Chen B-S, Cheng Y-M. A structure-specified H1 optimal control design for practical applications: a generic approach. IEEE Trans Control Syst Technol 2005;13:1119–24. [4] Kaitwanidvilai S, Olarnthichachart P, Ngamroo I. PSO based automatic weight selection and fixed-structure robust loop shaping control for power system control applications,. Int J Innovative Comput Inf Control (IJICIC) 2011;7: 1549–63. [5] Herreros A, Baeyens E, Perán JR. Design of PID-type controllers using multiobjective genetic algorithms. ISA Trans 2002;41:457–72. [6] Shabani H, Vahidi B, Ebrahimpour M. A robust PID controller based on imperialist competitive algorithm for load-frequency control of power systems. ISA Trans 2013;52:88–95. [7] Stein G, Doyle J. Beyond singular values and loop shapes. AIAA J Guid Control 1991;14:5–16. [8] Lari A, Khosravi A, Karamnejad R. A new approach to design a simple structure μ synthesis controller. In: Application to an uncertain DC motor, seventh IFAC symposium on robust control design; 2012. 660–5. [9] Beghi A, Cecchinato L, Cosi G, Rampazzo M. A PSO-based algorithm for optimal multiple chiller systems operation. Appl Therm Eng 2012;32:31–40. [10] Pourjafari E, Mojallali H. Predictive control for voltage collapse avoidance using a modified discrete multi-valued PSO algorithm. ISA Trans 2011;50: 195–200. [11] Garg H. Performance analysis of complex repairable industrial systems using PSO and fuzzy confidence interval based methodology. ISA Trans 2013;52: 171–83. [12] Smith RS, Doyle J, Morari M, Skjellum A. A case study using mu. In: Laboratory process control problem, proceedings of the tenth IFAC World Congress. 8; 1987. 403–15. [13] Smith RS, Doyle J. Thetwotank experiment: a benchmark control problem. Proceedings American control conference 1988;3:403–15. [14] Balas GJ, Doyle JC, Glover K, Packard A, Smith R. μ-Analysis and synthesis toolbox for use with MATLAB. The MathWork, Inc; 2005 (3rd version). [15] Kennedy J, Eberhart.R. Particle swarm optimization. In: Proceedings of the IEEE international conference on neural networks; 1995. 1942–48. [16] Shi Y, Eberhart RC. Parameter selection in particle swarm optimization. In: Evolutionary programming VII 1998:. 591–600. [17] Suganthan PN. Particle swarm optimizer with neighborhood operator. In: Proceedings of IEEE international conference on evolutionary computation; 1999. 1958–62. [18] Ratnaweera A, Halgamuge SK, Watson HC. Self organizing hierarchical particle swarm optimizer with time-varying acceleration coefficients. IEEE Trans Evol Comput 2004;8:240–55.
0.4 0.2 0 -4
10
-3
-2
10
10
-1
0
10
10
1
10
Frequency (rad/sec) Fig. 18. Performance analysis for D–K iteration controller.
1.6
1.4
Magnitude
1.2
1
0.8
Nominal
0.6
Worst-case 0.4 -4
10
-3
10
-2
10
-1
0
10
10
1
10
Frequency (rad/sec) Fig. 19. Performance analysis for the proposed controller.
5.5 5 4.5 4
Cost
523
3.5 3 2.5 2 1.5 1 0
50
100
150
Iteration Fig. 20. Variation of cost function.
the proposed approach significantly improves in practical control viewpoint by simplifying the controller structure, reducing the controller order and retaining the robust performance. Simulation
Contents lists available at ScienceDirect
ISA Transactions journal homepage: www.elsevier.com/locate/isatrans
Research Article
Controller design based on μ analysis and PSO algorithm Ali Lari, Alireza Khosravi n, Farshad Rajabi Faculty of Electrical and Computer Engineering, Babol University of Technology, Babol 47135-484, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 12 April 2013 Received in revised form 28 October 2013 Accepted 3 November 2013 Available online 7 December 2013 This paper was recommended for publication by Prof. A.B. Rad
In this paper an evolutionary algorithm is employed to address the controller design problem based on μ analysis. Conventional solutions to μ synthesis problem such as D–K iteration method often lead to high order, impractical controllers. In the proposed approach, a constrained optimization problem based on μ analysis is defined and then an evolutionary approach is employed to solve the optimization problem. The goal is to achieve a more practical controller with lower order. A benchmark system named two-tank system is considered to evaluate performance of the proposed approach. Simulation results show that the proposed controller performs more effective than high order H1 controller and has close responses to the high order D–K iteration controller as the common solution to μ synthesis problem. & 2013 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Robust control μ Synthesis problem PSO Structure-specified controller Two-tank system
1. Introduction Designing structure-specified controllers is one of the common ways to obtain controllers which are much easier to implement. Robust control is one of the areas in which designing controllers using structure-specified approach has been examined to simplify the controller structure. In the last decade, designing structurespecified controllers (especially in robust control) using evolutionary algorithms has attracted considerable attention [1–6]. In this approach, coefficients of a low order structure-specified controller are calculated by an evolutionary algorithm so that the resulting controller satisfies the robust design objectives. In the presence of uncertainties in models dynamic, μ synthesis problem is one of the most powerful robust controller design techniques. In μ synthesis problem, controller design is Carried out based on the structured singular values (μ) stability and performance analysis whereas in some other robust techniques like H1, singular values are the basis of controller design. The differences between these two approaches arise from definition of structured singular values and singular values. In fact, using structured singular values is the most effective and conservative technique to controller design in case there is a structured uncertainty in system modeling. D–K iteration is the most common method to solve μ synthesis problem. Unfortunately the resulting controllers in this method are often impractical, high order controllers. Moreover, the reduced
n
Corresponding author. Tel.: þ 98 47435484. E-mail addresses: [email protected] (A. Lari), [email protected] (A. Khosravi), [email protected] (F. Rajabi).
order controllers which are close to the main D–K iteration controller are usually not yet practical enough to be used instead of the main high order controller. In addition, one of the main drawbacks of D–K iteration is that it does not guarantee convergence to a global or even local minimum, which leads to nonoptimality of the resulting controller [7]. To design a more practical controller satisfying stability and performance criterions based on μ analysis and coping with the mentioned problem of the other solutions like D–K iteration method, this paper proposes an evolutionary approach to design a structure-specified μ synthesis controller. A rather similar approach on simple systems such as a DC motor system and a mass damper spring system has been examined by authors and promising results have been achieved [8]. In this paper the effectiveness of the proposed approach is verified through designing robust controllers for a benchmark MIMO system. In this approach, μ synthesis problem is solved as a constrained optimization problem in which the μ analysis is used to examine robust stability and performance criterions; robust stability as the constraint of the optimization problem and robust performance as the cost function which should be minimized. A simple improved particle swarm optimization (PSO) algorithm as an evolutionary algorithm is employed to find the required coefficients of a practical structure-specified controller such that minimizes the cost function and satisfies the constraint. PSO as a population-based algorithm can solve a variety of difficulties associated with optimization problems. Compared with GA, PSO takes less time for each function evaluation as it does not use many operators. Due to the simple concept, easy implementation and quick convergence, nowadays PSO has gained much attention and has wide applications in different fields [9–11].
0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.11.006
518
A. Lari et al. / ISA Transactions 53 (2014) 517–523
A multi input multi output (MIMO) benchmark system named two-tank system has been chosen to evaluate efficiency of the proposed method. Dynamic of this system is comprehensively discussed in [12–14]. Simulation results on the benchmark system show that the designed first order controller by the proposed approach has close responses to those of the high order D–K iteration controller and performs better than high order H1 controllers designed in [14]. The rest of this paper is organized as follow: Section 2 discusses the basic μ synthesis problem and the evolutionary algorithm. In Section 3, the proposed approach is illustrated step by step and in Section 4 description of the two-tank system is given. Simulation result is presented in Section 5 and finally Section 6 is dedicated to conclusion.
Δ
d
v
z
M
w
Fig. 2. Analysis framework.
ΔP
Δ
v
d 2. Preliminary Defining an appropriate cost function is the initial step to solve an optimization problem. In this section first, the requirements to define the optimization problem based on μ analysis is introduced, next the evolutionary algorithm to solve the problem is described. 2.1. Cost function In the cost function, the aim briefly is: “searching for the unknown parameters of a structure-specified controller with higher performance based on μ performance analysis, among controllers which satisfy stability condition based on μ analysis.” Therefore, to define the cost function mathematically, it is required to know the μ stability and performance criterions. 2.1.1. Robust stability based on μ analysis Fig. 1, shows a general framework used in μ analysis. In this figure, K is the controller, P is the nominal plant, Δ is the uncertainty set (perturbation matrix). Moreover, w denotes the exogenous input typically including command signals, disturbances, noises, etc.; z denotes the error output; v and d are the input and output signals of the dynamic uncertainties, respectively. Finally, u and v are the control input and the measurement signal, respectively. Using M(P,K) ¼Fl(P,K), Fig. 1 reduces to Fig. 2 (Fl(P;K) is the lower linear fractional transformation). In this figure, M can be partitioned as follows, # " M 11 M 12 v d ð1Þ ¼ M 21 M 22 z w |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} M
Regarding matrix M11, the stability condition based on μ analysis is defined as (2). jjM 11 ðsÞjjμ o 1
or
μΔ ðM 11 Þ o 1
ð2Þ
where μ is the structured singular value, jj:jjμ is μ norm and Δ is
w
M
z
Fig. 3. Structure for robust performance analysis.
uncertainty set. The definition of the structured singular value, μ, depends on the underlying block structure of perturbations which is defined as follows. Perturbation matrix Δ Cnn is defined as Eq. (3). Δ ¼ fdiag½δ1 I r1 ; :::; δs I rs ; Δ1 ; :::; ΔF : δi A C; Δj A Cmj mj g
ð3Þ
The block diagonal matrix Δ, includes two types of blocks: repeated scalar block (δi ) and full block (Δj ). In Eq. (3), S and F represent the number of repeated scalar blocks and the number of full blocks, respectively. For consistency among all the dimensions, Eq. (4) should be held. S
F
i¼1
j¼1
∑ r i þ ∑ mj ¼ n
ð4Þ
Structure singular value, μΔ ðMÞ, of a Matrix M A Cnn with respect to a block structure Δ A Δ is defined as Eq. (5). μΔ ðMÞ : ¼
1 min fsðΔÞ : Δ A Δ; detðI MΔÞ ¼ 0g
ð5Þ
Unless no Δ A Δ makes I MΔ singular, in which case μΔ ðMÞ : ¼ 0. 2.1.2. Robust performance based on μ analysis Defining an augmented block structure Δp as (6) in which Δf is a full block uncertainty matrix from z to w in Fig. 3, and Δ is the uncertainty matrix, the performance condition based on μ analysis is defined as (7). Δp A Δ : ¼ fdiag fΔ; Δf g : ΔA BΔ; jjΔf jj1 r 1g ; BΔ ¼ fΔ A Δ : s ðΔÞ r 1g ð6Þ jjMðsÞjjμ o 1
or
μΔp ðMÞ o1
ð7Þ
2.2. Evolutionary algorithm
Δ d
w
v z
P
u
y
K Fig. 1. General framework.
Particle swarm optimization (PSO) has been selected as a simple evolutionary algorithm to solve the optimization problem. Quick convergence is the main reason of using PSO in this paper because cost calculation in this case is time consuming. PSO takes less time for each function evaluation as it does not use many operators (such as mutation, crossover and selection operators used in GA). Particle swarm optimization (PSO) is originally attributed to Kennedy and Eberhart [15], based on the social behavior of collection of animals such as birds flocking. The swarm moves
A. Lari et al. / ISA Transactions 53 (2014) 517–523
randomly along the search space, remembering the best solution found by each particle and by the whole population. Regarding the information obtained by each particle and the swarm, position of the particles are updated by (8) and (9). V id ¼ wV id þ c1 r 1 ðP id X id Þ þ c2 r 2 ðP gd X id Þ
ð8Þ
X id ¼ X id þV id ; d ¼ 1; 2; …; N
ð9Þ
i ¼ 1; 2; …; S
where Xid represents the current position of the particle, Vid is the velocity that changes the particle's position, Pid is the best individual particle position, Pgd denotes the best swarm position, the parameters c1 and c2 are cognitive and social parameters, respectively; the parameters r1 and r2 are random numbers between 0 and 1. Finally, w is the inertia weight to balance the global and local search abilities. A large inertia weight facilitates global search while a small inertia weight facilitates local search. In an empirical study on PSO [16], Shi and Eberhart claimed that a linearly decreasing inertia weight could improve local search towards the end of a run, rather than using a constant value throughout. A decreasing function for the dynamic inertia weight can be devised in the following winitial wf inal w ¼ ðiter max iter cur Þ: ð10Þ þwf inal itermax where winitial and wfinal represent the initial and final inertia weights at the start of a given run, respectively; itermax is the maximum number of iterations in an offered run, and iter cur denotes the current iteration number at the present time step. However, due to the utilization of a linearly decreasing inertia weight, the global search ability at the end of the run may be inadequate. The PSO may fail to find the required optimal in cases the problems are too complicated. But to some extent, this can be overcome by employing a self-adapting strategy for adjusting the acceleration coefficients. Suganthan [17] applied the optimizing method that makes the two factors decrease linearly with the increase of iteration numbers, but the results were not as good as the fixed value 2 of c1 and c2. Ratnaweera [18] improved the convergence of particles to the global optima based on the way that makes c1 decrease and c2 increase linearly with the increase of iteration numbers, as they are given by the following equations: c1 ¼ c1s þ iter cur ðc1e c1s Þ=iter max c2 ¼ c2s þ iter cur ðc2e c2s Þ=iter max
ð11Þ
where c1s and c2s represent the initial values of c1 and c2, respectively; and c1e and c2e show the final values of c1 and c2, respectively.
3. The proposed approach
Step 2 Set the parameters of PSO algorithm. Step 3 Initialize a group of random particles in search space (Each particle includes unknown parameters of the structure specified controller). It should be checked that all of initial particles satisfy μ stability condition. Step 4 Calculate best individual position (Pid), best swarm position (Pgd) and the cost function for each particle regarding to the cost function introduced in (12). Step 5 Update PSO parameters c1, c2 and w according to the iteration number given in Eqs. (10) and (11). Step 6 Update the velocities and positions of each particle according to Eqs. (8) and (9). Step 7 If each new position verifies the stability constraint, set it as the new position. Else, set the last position as the new position. Step 8 If the number of iterations is lower than a predefined value, go to step 4, else, to step 9. Step 9 Consider Pgd as the coefficients of desired controller. If the obtained controller is not efficient enough, the order of structure-specified controller can be incremented. Using the proposed approach, also, makes it possible to embody time objectives in the cost function introduced in (12).
4. Case study: Two-tank system A benchmark problem called two-tank system has considered to evaluate efficiency of the proposed controller. A detailed description has given by [12–14]. The two-tank system is shown schematically in Fig. 4. This system consists of two water tanks in cascade, named tank 1 and tank 2 in the figure. Tank 1 is fed hot and cold water via controllable valve and tank 2 is fed by water from an exit at the bottom of tank 1. A constant level is maintained in tank 2 by means of an overflow. Tank 2 is also fed by a cold water bias stream which enables the tanks to have different steady-state temperatures [14]. The two-tank system, used in simulation, has two measured signals: t1, t2. Third measured signal is water level in tank 1 (h1), which is not controlled. This quantity is, however, important for assessment of controller performance. Also, the system has two input signals: fh and fc. These inputs are commands to hot flow (fhc) and cold flow (fcc) actuators, which are transformed to hot water flow fh and cold water flow fc [14]. Using the plant parameters and the selected operating point considered in [14], nominal model from the inputs fh, fc to the outputs t1, t2 can be written in two-dimensional transfer matrix
In this section it is described how robust controllers can be designed based on the new approach to solve μ synthesis problem. At the core of the proposed approach, the simple PSO algorithm, described in Section 2 is employed to solve μ synthesis problem as a constrained optimization problem shown in (12). As it can be seen in (12), stability condition based on μ analysis is the constraint and μ performance condition is the cost function which should be minimized. min jjMðsÞjjμ KðsÞ
such that : jjM 11 ðsÞjjμ o 1
ð12Þ
where M and M11 are defined in Eq. (1) and K(s) is the controller. Generally, the procedure of the proposed approach is summarized as follows: Step 1 Specify the structure of the controller.
519
Fig. 4. Schematic diagram of the two-tank system [14].
520
A. Lari et al. / ISA Transactions 53 (2014) 517–523
Fig. 5. Schematic representation of the perturbed, linear, two-tank model [14].
Fig. 7. Interconnection structure for m synthesis [14].
Fig. 6. Closed-loop interconnection [14].
as (13).
"
Pnom ðsÞ 2
P 11 ðsÞ
P 12 ðsÞ
P 21 ðsÞ
P 22 ðsÞ
0:0036s2
#
þ 0:0001s þ 7:815710
7
6 s3 þ 0:0491s2 þ 0:0007s þ 3:068410 6 ¼ 4 0:0109s2 0:0004s 2:344710 6 s3 þ 0:0491s2 þ 0:0007s þ 3:068410 6
6
0:0004s þ 4:664310 s3 þ 0:0491s2 þ 0:0007s þ 3:068410 6 0:156210 4 s þ 0:018710 4 s3 þ 0:0491s2 þ 0:0007s þ 3:068410 6
3 7 5
ð13Þ Differences from the nominal model are utilized by multiplicative perturbations at the outputs of measured quantities (Fig. 5). 4.1. Design scheme The problem, is tracking of set point commands for t1 and t2. Fig. 6, illustrates the desired closed-loop configuration. The controller has access to both the reference inputs and the temperature measurements. This configuration is less restrictive than the design which only uses the error between the temperature and the set points [14]. The layout of the interconnection structure for μ synthesis is shown in Fig. 7. It is desired to weight both the amplitude and the rate of the actuator in the case of the actuation weights. Using fhc as an example; the approach is to create an actuator model with fh and dfh/dt as outputs. These can then be separately weighted with constant weights [14]. Fig. 8 illustrates the form of such a weighted actuator model. The actuator bandwidth, denoted by BW in Fig. 8, is, BW ¼ 20 radians=s
Fig. 8. Model of the flow valve actuator including magnitude and rate weightings [14].
proposed controller is compared with high order, complicated D–K iteration controller (of order 22) and H1 controller (of order 15) designed in [14]. The proposed controller is also compared with the reduced order D–K controllers of orders 1 and 2. A simple controller containing elements of order 1, is selected as the structure-specified controller. This controller is shown in (15). 2 a11 a12 a13 a14 3 CðsÞ ¼ 4
s þ b11 a21 s þ b21
s þ b12 a22 s þ b22
s þ b13 a23 s þ b23
s þ b14 a24 s þ b24
5
ð15Þ
To calculate coefficients of the controller, parameters of PSO algorithm is adjusted as Table 2. Search space of PSO algorithm is also adjusted as (16). 1 o aij o 1 and 0 obij o1 for i ¼ 1; 2 and j ¼ 1; :::; 4
ð16Þ
Using these parameters the proposed approach leads to a controller as (17). 2 0:037 3 0:072 0:003 0:220 CðsÞ ¼ 4
s þ 0:200 0:242 s þ 0:136
s þ 0:039 0:015 s þ 0:491
s þ 0:089 0:143 s þ 0:254
s þ 0:227 0:113 s þ 0:489
5
ð17Þ
ð14Þ
In the described two-tank system, tank 1, supplies the majority of the water flowing into tank 2, which means that changes in t2 are dominated by changes in t1. It makes more sense to express the reference weighting in terms of t1 and t2–t1, denoted by Wt1cmd and Wtdiffcmd in Table 1. In order to compare with the controllers designed in [14], all the weighting functions have selected exactly the same as weighting functions used in [14]. Table 1 shows these functions. In this table h1ss ¼ 0.75 is steady state value of h1.
5. Simulation results In the following the two-tank system introduced in Section 4, is employed to evaluate the proposed approach. Then, the
5.1. A comparison between D–K iteration and structure-specified controllers D–K iteration controller and its reduced orders (using Hanklenorm approximation) are compared with the proposed structurespecified controller through μ plots in Figs. 9 and 10. Fig. 9, shows that all controllers have almost equally well stability margins. The μ plot in whole frequency range is lower than 0.2 and it means that robust stability margin against perturbations is well enough. Based on Fig. 10, the proposed controller has close behavior to the high order D–K iteration controller. Even in some frequencies from .03 to 0.2 rad/s, robust performance of the proposed controller is better than that of D–K iteration controller. In addition, regarding to Fig. 10, reduced order D–K iteration controllers have a poor robust performance;
A. Lari et al. / ISA Transactions 53 (2014) 517–523
521
Table 1 Weighting functions. Perturbation weights
Tracking error weights
0:5s W h1 ¼ 0:01 þ 0:25s þ1 20h1ss s W t1 ¼ 0:1 þ 0:2s þ 1 100s W t2 ¼ 0:1 þ s þ 21
100 W t1perf ¼ 400s þ1 50 W t2perf ¼ 800s þ 1
Sensor noise weights
Reference weights
W h1noise ¼ 0:01 W t1noise ¼ 0:03 W t2noise ¼ 0:03
W t1cmd ¼ 0:1 W tdif f cmd ¼ 0:01
Table 2 Parameters of PSO algorithm.
Actuator weights W hact ¼ 0:01 W hrate ¼ 50 W cact ¼ 0:01 W crate ¼ 50
0.8
Value
Parameter
Value
c1e c1s c2e c2s
0.5 1.5 1.5 0.5
Number of particles Number of iterations winitial wfinal
50 150 1 0.4
0.75 0.7
Measurements
Parameter
h1 t1 t2 t1ref t2ref
0.65 0.6 0.55
0.2
0.5 Structure - Specified D -K Reduced D- K of Order 2 Reduced D - K of Order 1
0.18
Mu upper bound
0.16
0.45 0
100
200
300
400
500
600
700
800
Time (sec)
0.14 0.12
Fig. 11. Response to reference signal for H1 controller.
0.1 0.08 0.06
0.8
0.04 0.75
0.02 -4
10
-3
10
-2
10
-1
10
0
10
0.7
1
Frequency (rad/sec) Fig. 9. Comparison of robust stability between the controllers.
Measurements
10
h1 t1 t2 t1ref t2ref
0.65 0.6 0.55 0.5
3.5 0.45 0
Mu upper bound
3
100
200
300
400
500
600
700
800
Time (sec) 2.5 Fig. 12. Response to reference signal for D–K iteration controller.
2 1.5
controllers which retain robust features like a high efficient but high order, impractical μ synthesis controller.
1 0.5 0 -4
10
-3
10
-2
10
-1
10
0
10
1
10
Frequency (rad/sec)
5.2. A comparison between D–K iteration, H1 and structurespecified controllers
Fig. 10. Comparison of robust performance between the controllers.
especially in low frequencies they cannot meet robust performance requirements. From Figs. 9 and 10, it can be concluded that the proposed approach to solve μ synthesis problem can lead to relatively simple
In this subsection, the controllers designed based on D–K iteration, H1 approach (designed in [14]) and the proposed approach are compared through time responses and worst case analysis. This is completely in accordance with the procedure considered in [14] to compare the designed controllers. Figs. 11–13 show time responses to ramp reference signal (sensor noise has also been applied).
522
A. Lari et al. / ISA Transactions 53 (2014) 517–523
0.8
0.9
0.75
0.8 0.7 h1 t1 t2 t1ref t2ref
0.65 0.6
Actuators
Measurements
0.7
fhc fcc
0.6 0.5
0.55
0.4
0.5
0.3
0.45 0
100
200
300
400
500
600
700
0.2
800
0
100
200
Time (sec)
300
400
500
600
700
800
Time: seconds
Fig. 13. Response to reference signal for the proposed controller.
Fig. 16. Actuator commands for the proposed controller.
2.5 0.9
Nominal
Magnitude
Actuators
0.7 0.6
Worst-case
2
fhc fcc
0.8
1.5
1
0.5 0.5 0.4 0
0.3
-4
-3
10
10
-2
10
-1
10
0
10
1
10
Frequency (red/sec)
0.2 0
100
200
300
400
500
600
700
800
Time: seconds
Fig. 17. Performance analysis for H1 controller.
Fig. 14. Actuator commands for H1 controller.
0.9 fhc fcc
0.8
Actuators
0.7 0.6 0.5 0.4 0.3 0.2 0
100
200
300
400
500
600
700
800
Time: seconds Fig. 15. Actuator commands for D–K iteration controller.
signals like H1 controller. That's why small fluctuations in actuator command of these controllers can be seen. In the following, the worst-case performance of the closed-loop systems are investigated in Figs. 17–19. Unlike Figs. 18 and 19, in Fig. 17, the worst-case performance of the closed-loop system is significantly worse than the nominal performance, which verifies that the H1 controller is not robust to modeling errors. Finally, variation of cost function versus iteration in PSO algorithm is depicted in Fig. 20. At the end, it must be noted that in this paper only first order structure-specified controller (15) has been examined and acceptable results has been obtained. It is obvious that higher order structure-specified controllers can lead to better results. But, using a higher order controller results in an increase in the number of unknown coefficients and a larger search space in which PSO algorithm may not be converged to a desired solution. Regarding to these issues, it seems that second order structure-specified controller is still a proper choice for the two-tank system if a more advanced evolutionary algorithm is used.
6. Conclusions Based on these figures, it is clear that the H1 controller has poor time response (steady-state error) with respect to D–K iteration and the proposed controllers. Figs. 14–16 show different actuator commands for the controllers in nominal closed-loop system. Based on these figures, D–K iteration and the proposed controllers cannot reject noise
In this paper a structure-specified μ synthesis controller has been designed. In this approach, μ synthesis problem is solved as an optimization problem using particle swarm optimization (PSO) as an evolutionary algorithm. Structure of controller in the proposed approach is selectable. Unlike the D–K iteration method,
A. Lari et al. / ISA Transactions 53 (2014) 517–523
results on a MIMO benchmark problem, a two-tank system, show that the proposed first order controller has close robust performance features to the high order D–K iteration controller and can give better results with respect to the high order H1 controller.
1.4
Nominal 1.2
Worst-case
Magnitude
1 0.8
References 0.6
[1] Ho S-J, Ho S-Y, Hung M-H, Shu L-S, Huang H-L. Designing structure-specified mixed H2/H1 optimal controllers using an intelligent genetic algorithm IGA. IEEE Trans Control Syst Technol 2005;13:1119–24. [2] Zamini M, Sadati N, Karimi Ghartemai M. Design of an H1 PID controller using particle swarm optimization. Int J Control Autom Syst 2009;7:273–80. [3] Chen B-S, Cheng Y-M. A structure-specified H1 optimal control design for practical applications: a generic approach. IEEE Trans Control Syst Technol 2005;13:1119–24. [4] Kaitwanidvilai S, Olarnthichachart P, Ngamroo I. PSO based automatic weight selection and fixed-structure robust loop shaping control for power system control applications,. Int J Innovative Comput Inf Control (IJICIC) 2011;7: 1549–63. [5] Herreros A, Baeyens E, Perán JR. Design of PID-type controllers using multiobjective genetic algorithms. ISA Trans 2002;41:457–72. [6] Shabani H, Vahidi B, Ebrahimpour M. A robust PID controller based on imperialist competitive algorithm for load-frequency control of power systems. ISA Trans 2013;52:88–95. [7] Stein G, Doyle J. Beyond singular values and loop shapes. AIAA J Guid Control 1991;14:5–16. [8] Lari A, Khosravi A, Karamnejad R. A new approach to design a simple structure μ synthesis controller. In: Application to an uncertain DC motor, seventh IFAC symposium on robust control design; 2012. 660–5. [9] Beghi A, Cecchinato L, Cosi G, Rampazzo M. A PSO-based algorithm for optimal multiple chiller systems operation. Appl Therm Eng 2012;32:31–40. [10] Pourjafari E, Mojallali H. Predictive control for voltage collapse avoidance using a modified discrete multi-valued PSO algorithm. ISA Trans 2011;50: 195–200. [11] Garg H. Performance analysis of complex repairable industrial systems using PSO and fuzzy confidence interval based methodology. ISA Trans 2013;52: 171–83. [12] Smith RS, Doyle J, Morari M, Skjellum A. A case study using mu. In: Laboratory process control problem, proceedings of the tenth IFAC World Congress. 8; 1987. 403–15. [13] Smith RS, Doyle J. Thetwotank experiment: a benchmark control problem. Proceedings American control conference 1988;3:403–15. [14] Balas GJ, Doyle JC, Glover K, Packard A, Smith R. μ-Analysis and synthesis toolbox for use with MATLAB. The MathWork, Inc; 2005 (3rd version). [15] Kennedy J, Eberhart.R. Particle swarm optimization. In: Proceedings of the IEEE international conference on neural networks; 1995. 1942–48. [16] Shi Y, Eberhart RC. Parameter selection in particle swarm optimization. In: Evolutionary programming VII 1998:. 591–600. [17] Suganthan PN. Particle swarm optimizer with neighborhood operator. In: Proceedings of IEEE international conference on evolutionary computation; 1999. 1958–62. [18] Ratnaweera A, Halgamuge SK, Watson HC. Self organizing hierarchical particle swarm optimizer with time-varying acceleration coefficients. IEEE Trans Evol Comput 2004;8:240–55.
0.4 0.2 0 -4
10
-3
-2
10
10
-1
0
10
10
1
10
Frequency (rad/sec) Fig. 18. Performance analysis for D–K iteration controller.
1.6
1.4
Magnitude
1.2
1
0.8
Nominal
0.6
Worst-case 0.4 -4
10
-3
10
-2
10
-1
0
10
10
1
10
Frequency (rad/sec) Fig. 19. Performance analysis for the proposed controller.
5.5 5 4.5 4
Cost
523
3.5 3 2.5 2 1.5 1 0
50
100
150
Iteration Fig. 20. Variation of cost function.
the proposed approach significantly improves in practical control viewpoint by simplifying the controller structure, reducing the controller order and retaining the robust performance. Simulation
