International Journal of Control, Automation, and Systems (2009) 7(2):273-280 DOI 10.1007/s12555-009-0213-9
http://www.springer.com/12555
Design of an H∞ PID Controller Using Particle Swarm Optimization Majid Zamani, Nasser Sadati, and Masoud Karimi Ghartemani* Abstract: This paper proposes a novel method to designing an H ∞ PID controller with robust stability and disturbance attenuation. This method uses particle swarm optimization algorithm to minimize a cost function subject to H ∞ -norm to design robust performance PID controller. We propose two cost functions to design of a multiple-input, multiple-output (MIMO) and single-input, single-output (SISO) robust performance PID controller. We apply this method to a SISO flexible-link manipulator and a MIMO super maneuverable F18/HARV fighter aircraft system as two challenging examples to illustrate the design procedure and to verify performance of the proposed PID controller design methodology. It is shown with the MIMO super maneuverable F18/HARV fighter system that PSO performs well for parametric optimization functions and performance of the PSO-based method without prior domain knowledge is superior to those of existing GA-based and OSA-based methods for designing H ∞ PID controllers. Keywords: Genetic algorithm, H ∞ -optimal controller, particle swarm optimization, PID controller, simulated annealing.
1. INTRODUCTION It has been shown that H∞-based control [1-3] can systematically deal with various formats of model uncertainty. In the past, H∞-based control has been extensively applied in the area of flexible structure control [4-6] to design high performance controllers such that the closed-loop systems are robust to model uncertainty and disturbance. However, the order of the resulting controller is at least as high as the model order and even higher in the cases where the plant is augmented by dynamical scaling or weights in order to achieve the desired robustness or performance requirements. The high-order controllers may not be feasible for real-time implementation because of hardware and computational limitations. Unfortunately, the fixed-order H∞ controller design is computationally intractable [7] using those H∞-based control synthesis methods. Although the high-order controller can be approximated by a reduced-order controller, it is usually at the cost of closed-loop robustness and performance __________ Manuscript received June 27, 2007; revised March 29, 2008 and June 14, 2008; accepted September 4, 2008. Recommended by Editorial Board member Jietae Lee under the direction of Editor Young-Hoon Joo. This work was supported by the Iranian Telecommunication Research Center (ITRC) under Grant T50011629. Majid Zamani is with the School of Electrical Engineering, University of California, Los Angeles, U.S.A. (e-mail: zamani@ ee.ucla.edu). Nasser Sadati and Masoud Karimi Ghartemani are with the School of Electrical Engineering, Sharif University of Technology, Tehran, Iran (e-mails:
[email protected], karimig@ sharif.edu). * Corresponding author. © ICROS, KIEE and Springer 2009
degradation. Despite the advent of many sophisticated control theories and techniques, proportional-integral-derivative (PID) control is still one of the widely used control structures in industrial applications. The popularity of PID control is mainly due to its structural simplicity, demonstrated reliability, and broad applicability. With rigorous theoretical justification, recently several PID control synthesis methods [8-12] have been proposed. These results are applicable only to a given arbitrary SISO linear time-invariant plant. The PID stabilization problems were solved in [8-10]. By converting the H∞ design problem into simultaneous complex polynomial stabilization and using the complex PID stabilization results, [11] and [12] provided a linear-programmingbased characterization of all admissible H∞ PID controllers for a given plant. This characterization besides being computationally efficient revealed important structural properties of H∞ PID controllers. It was shown that for a fixed proportional gain, the set of admissible integral and derivative gains lie in a union of convex sets. Based on a frequency gridding approach [12], provided an alternative H∞ PID control synthesis technique. Beside the mentioned analytical methods, there are several numerical methods [13-16] which design a H∞ PID controller according to minimization of an H2 criterion such as integral of squared-error (ISE) for both SISO and MIMO systems. These methods not only satisfy the H∞-norm constraint to achieve the robust performance but also improve the step response of the system with minimizing the H2 criterion. In this paper, we propose a novel method which designs a H∞ PID controller according to minimization of an arbitrary criterion to achieve our desires using Particle Swarm
Majid Zamani, Nasser Sadati, and Masoud Karimi Ghartemani
274
Optimization (PSO). We use Particle swarm optimization (PSO) algorithm [17,18] to minimize the cost function as a powerful optimization method with high efficiency in comparison to other methods such as Genetic Algorithm (GA) [19] and Simulated Annealing (SA) [20] which have been used in [13-16]. Especially, PSO is very powerful than GA in applications with highly epistatic objective function (i.e., where the parameters being optimized are highly correlated) [21]. This technique combines social psychology principles in socio-cognition human agents and evolutionary computations. PSO has been motivated by behaviors of organisms such as fish schooling and bird flocking. Generally, PSO is characterized as a simple concept, easy to implement, and computationally efficient algorithm. PSO has a flexible and well-balanced mechanism to enhance global and local exploration abilities. Structure of the paper is as follows. An overview of the PSO algorithm is provided in Section 2. Section 3 presents an algorithm to numerically obtain a solution for the optimization problem. Illustrative examples are studied in Section 4 to show usefulness of the proposed method to improve existing designs. 2. PARTICLE SWARM OPTIMIZATION PSO is a population-based evolutionary algorithm that is developed from research on swarm such as fish schooling and bird flocking [17,18]. It has become one of the most powerful methods for solving optimization problems. The method is proved to be robust in solving problems featuring nonlinearity and nondifferentiability, multiple optima, and high dimensionality. The advantages of the PSO are its relative simplicity and stable convergence characteristic with good computational efficiency. The PSO consists of a swarm of particles moving in a D dimensional search space where a certain quality measure and fitness are being optimized. Each particle has a position represented by a position vector X i = ( xi1, xi 2 ,…, xiD ) and a velocity represented by a velocity vector Vi = (vi1, vi 2 ,…, viD ), which is clamped to a maximum velocity Vmax = (vmax1, vmax 2 ,…, vmaxD ). Each particle remembers its own best position so far in a vector Pi = ( pi1, pi 2 ,…, piD ), where i is the index of that particle. The best position vector among all the neighbors of a particle is then stored in the particle as a vector p g . The modified velocity and position of each particle can be manipulated according to the following equations: (t +1) (t ) vid = wvid + c1r1 ( pid − xid ) + c2 r2 ( pgd − xid ),
(1)
(t +1) (t ) (t +1) xid = xid + vid ,
(2)
d = 1,… , D ,
where w can be expressed by the inertia weights approach [22] and often decreases linearly from wmax
(of about 0.9) to wmin (of about 0.4) during a run. In general, the inertia weight w is set according to the following equation w = wmax −
wmax − wmin .iter itermax
(3)
where itermax represents the maximum number of iterations, and iter is the number of current iteration or generation. c1 and c2 are the acceleration constants which influence the convergence speed of each particle and are often set to 2.0 according to the past experiences [23] and r1, r2 are random numbers in the range of [0,1]. If Vmax is too small, the particles may not explore sufficiently beyond local solutions. In many experiences with PSO, Vmax is often set to the maximum dynamic range of the variables on each dimension vdmax = xdmax . 3. DESIGN OF H∞ PID CONTROLLER 3.1. Problem description Consider a control system as shown in Fig. 1 , where G ( s) is an ni -input and no -output actual plant which has some uncertainty, K(s) is PID controller as the following form
K ( s) = K P +
KP =
k P11 k P no 1
KI + K D s, s … k P1n
(4)
i
, … k Pn n
(5)
o i
where coefficients of controller have the same structure and is shown in (5). r(t) is the reference input, u (t ) is the control input, e(t ) is the tracking error, d (t ) is the external disturbance and y(t) is the output of the system. A PID controller K(s) is to be designed to achieve the following conditions: 1) the nominal closed loop system is asymptotically stable and 2) the robust stability performance satisfies the following inequality g1 ≤ 1 where:
W1T ( s) ∞ , g1 = W1K ( s ) S ( s)
for multiplicative uncertainty; ∞
, for additive uncertainty. (6)
Fig. 1. Controlled system with PID controller.
Design of an H∞ PID Controller Using Particle Swarm Optimization
where A ∞ := maxω σ ( A( jω)), W1 ( s ) is the bound of plant perturbation and 3) the disturbance attenuation performance satisfies the following inequality: g 2 = W2 ( s ) S ( s)
∞
≤ 1,
(7)
where W2 ( s ) is a stable weighting function matrix specified by designer. S ( s) and T ( s ) are the sensitivity and complementary sensitivity functions of the system with the following representations: S ( s) = ( I + K ( s )G ( s)) ,
(8)
T ( s ) = K ( s )G ( s )( I + K ( s )G ( s ))−1 ,
(9)
where G ( s ) is the nominal plant. Kwakernaak [27] suggests a balanced performance criterion to simultaneously satisfy (6) and (7) as g∞ = g12 + g 22 ≤ 1.
(10)
3.2. Cost function 1) SISO: There are several performance criteria for design of controllers such as integral of absolute error (IAE), the integral of squared-error (ISE) or integral of time-weighted-squared-error (ITSE). A disadvantage of the IAE and ISE criteria is that they may result in a response with a relatively small overshoot but a long settling time because they weigh all errors uniformly over time. Although the ITSE performance criterion can overcome this drawback, it cannot ensure desirable stability margin [14]. The IAE, ISE and ITSE performance criteria are defined as follows ∞
∞
0
0
IAE = ∫ | r (t ) − y (t ) | dt = ∫ | e(t ) | dt , ∞ 2 e (t )dt , 0
(11) (12)
∞
ITSE = ∫ te2 (t )dt .
(13)
0
In this paper, a new performance criterion containing the time domain and frequency domain indices is proposed. This performance criterion includes the overshoot M P , rise time tr , settling time ts , steady-state error ESS , the ( ITSE ), gain margin (GM ) and phase margin ( PM ). Control over PM and GM improves the stability margins. The ITSE must be computed numerically and the integral is evaluated up to T which is chosen sufficiently large so that e(t ) is negligible for t > T . Therefore, the proposed performance criterion is expressed as J (θ) = w1 g ∞ + w2 M P + w3tr + w4t s + w5 ESS T
+ w6 ∫ te2 (t )dt + 0
θ = [k P k I k D ].
(15)
The proposed performance criterion (14) comprises eight terms the significance of each is determined by a weight factor wi . The weight factors must properly be set in order to attain the desired specification. 2) MIMO: We use the ISE criterion for a MIMO system to facilitate comparison with other methods which exist. ∞
J (θ) = g∞ + ∫ eT (t )e(t )dt ,
(16)
0
−1
ISE = ∫
275
w7 w + 8 , PM GM
where θ is the controller parameters as following
(14)
where θ is the controller parameters as following
θ = [k P11 … k P1n k P21 … k P2n … k Dn 1 … k Dn i
i
o
o ni
]T , (17)
e(t ) = r (t ) − y (t ) is the tracking error and mentioned integral is its H 2 -norm. An approach using penalty function [24] is employed to solve the stability problem as following equation F (θ) = J (θ) + P (θ),
(18)
where P(θ) is defined as following
P , if θ is unstable; P (θ) = 1 else. 0,
(19)
If the parameter θ does not satisfy the stability then θ is an unstable parameter and it is penalized with a very large positive constant P1. Automatically, θ does not survive the evolutionary process. Otherwise, the parameter θ is feasible and is not penalized. 3.3. Design of H∞ PID controller using PSO The H∞ PID controller design problem is to find the optimal θ from search space that minimizes the objective function F (θ) in (18). The proposed PSObased method for finding a solution to the H∞ PID controller design problem is described as follows. 1) Randomly, initialize the individuals of the population including searching points and velocities in the search space. 2) For each initial individual θi of the population, i = 1, ..., n, where n denotes the population size of PSO, calculate the values of the evaluation function in (18). 3) Compare each individual’s evaluation value with its personal best Pi . The best evaluation value among the Pi is denoted as pg . 4) Modify the member velocity of each individual θi according to (1) where the value of w is set by (3). 5) Modify the member position of each individual θi according to (2). 6) If the number of iterations reaches the maximum, then go to Step 7, otherwise, go to Step 2.
Majid Zamani, Nasser Sadati, and Masoud Karimi Ghartemani
276
Fig. 3. Schematic diagram of the experimental setup. Table 1. Parameters of the identified model. n6
-14340.4953
d9
1
n5
0.4446×107
d8
486.7
n4
0.5697×109
d7
69317.7
d6
0.1616×108
d5
0.1062×1010
d4
0.6167×1011
d3
0.2624×1013
d2
0.3595×1014
d1
0.142×1015
d0
0
-0.1908×10
11
n2
-0.9354×10
12
n1
0.6919×1013
n3
n0
Fig. 2. Flowchart of the mentioned method. 7) The latest pg is the optimal controller parameter. The flowchart of the algorithm is shown in Fig. 2. 4. ILLUSTRATIVE EXAMPLE In this section, a SISO and a MIMO examples are given to illustrate the proposed design procedures. Example 1: An experimental single-link flexible manipulator is shown by the schematic diagram in Fig. 3. The flexible link is a rectangular stainless-steel bar with 34.5cm length, 3cm width, and 0.045cm thickness. The link is coupled to a permanent magnet DC motor by a hub. The transmitter of an ultrasonic sensor is mounted at the free end tip of the link and its receiver is fixed on top of the hub. This ultrasonic sensor is used to measure the tip deflection of the link. The transmitter of the ultrasonic sensor also acts as a payload. An optical encoder with resolution 1000 pulses/rev attached to the shaft of the DC motor is used to measure the angular position of the shaft. A nominal ninth-order model for this system identified as [26]:
G( s) =
n6 s6 +…+ n1s + n0
d9 s9 + …+ d1s + d0
,
0.2839×10
15
where the values of the parameters of the identified model are given in Table 1. The error between the identified model and the actual system is considered as an additive uncertainty. Hence, the set of models is assumed to be of the perturbed form: G ( s) = G ( s ) + W1 ( s )∆( s ),
where ∆( s) is any stable and proper transfer function satisfying
the
H∞-norm
bound
∆( s)
∞
< 1.
The
additive uncertainty weight W1 ( s ) is chosen to be [26] W1 ( s ) = 0.11
( 1 s +1)( 1 s +1)2 11
720
( 1 s +1)3
.
70
The performance weighting function W2 ( s ) is chosen as the approximation of the inverse sensitivity function of an acceptable closed-loop system to achieve the following step-response performance specifications: • Settling time of 6 seconds for the system output to achieve 95% of its final value. • Overshoot of less than 20%. Thus the resulting expression for W2 ( s ) is [26] W2 ( s ) =
(0.7217 s +1.4874)2 ( s +1)2
.
We use a PID controller to achieve the H∞-optimal design objective in (18). For the current study, we choose the weight factor wi in (14) as following: w1 = 10,
Design of an H∞ PID Controller Using Particle Swarm Optimization
277
Table 2. Comparison of the H∞ PID and conventional H∞ controller for Example 1. Type of Controller
KP
KI
KD
M P (%)
tr
ts
ESS
PM
GM
g∞ (θ )
Proposed PID Conventional H∞
0.854 -
0 -
0.0284 -
0.183 0.7
0.838 0.67
1.42 1.18
0 0.3
70.68 100.21
10.631 7.5
0.85166 0.667
proposed in [13,15,28], the same MIMO optimal control design problem is treated using the proposed algorithm. Consider the design problem of a longitudinal control system of the super maneuverable F18/HARV fighter aircraft in horizontal flight at an altitude of 15000(ft) with Mach number 0.24, airspeed VT = 238.7 (ft/s), attack angle α = 25 and pitch angle β = 25. The trim value of the path angle is β − α = 0 and the trim pitch rate is γ = 0( /s ). The longitudinal dynamics of the system can be described as
Fig. 4. Step response of the system of Example 1 with both PID and H∞ controllers. w2 = w3 = w4 = w5 = w6 = 1, w6 = 200 and w7 = 10. The following parameters are used for carrying out the H∞ PID controller design using PSO: • The members of each individual in PSO algorithm are θ = [ K I K P K D ]. • Population size =30. • Inertia weight factor w is set as (3), where wmax = 0.9 and wmin = 0.4. • The limit of change in velocity is set to maximum dynamic range of the variables on each dimension. • Acceleration constants c1 = 2 and c2 = 2. • Maximum iteration is set to 1000. • T in (14) is set to 100s. • P1 in (19) is set to 1010. The lower bounds of the three controller parameters are zero and their upper bounds are set to K Pmax = K Imax = K Dmax = 100. We performed 10 trials for the proposed method. The best solutions are summarized in Table 2 in comparison with conventional H∞ controller which has an order of fifteen. Response of the system to a unit step command is shown in Fig. 4 for the conventional H∞ controller and the proposed H∞ PID controller. The conventional H∞ controller has lower H∞ -norm but the proposed PID controller exhibits more desirable step response characteristics. This means that the proposed controller presents a simplified structure with improved step response at the cost of degrading the robust stability and disturbance attenuation characteristics. The obtained controller, nevertheless, is still H∞ -optimal due to existence of g∞ in (14). Example 2: To facilitate comparison with the GAbased, Improved GA-based and OSA-based methods
x = Ax + Bu y = Cx 0 −36.16 −0.075 −24.05 −0.0009 −0.1959 0.9896 0 A= , −0.0002 −0.1454 −0.1677 0 0 1 0 0 1 0 0 C = 0 −1 0 0 0 0 −0.023 −0.0002 B= −0.0067 0
0 1 1 0
−0.0729 −0.0001 −0.0004 −0.0007 −0.012 0 0 0.0393 0
−0.0411 0.16 −0.0003 −0.0003 −0.0006 0.0007 0.0005 (20) 0 0 0
where x = [VT , α, γ, β]T and u = [uTV , u AS , uSS , uLE , uTE , uT ]T where uTV , u AS , uSS , u LE , uTE and uT are the perturbations in symmetric thrust vectoring vane deflection, symmetric aileron deflection, symmetric stabilator deflection, symmetric leading edge flap deflection, symmetric trailing edge flap deflection, and throttle position, respectively. Note that the rank of B is three. It is important to remove the redundancy in the control inputs. By employing the pseudocontrol technique [29], the six control inputs (uTV , u AS , uSS , uLE , uTE and uT ) are transformed to three linearly independent inputs. Therefore, the system can be rewritten as x = Ax + Bv v,
where Bv and v are given in
Majid Zamani, Nasser Sadati, and Masoud Karimi Ghartemani
278
1 0 Bv = 0 0
0 0 1 0 , 0 1 0 0 0 −0.0729 −0.023 v = −0.0002 −0.0001 −0.0004 −0.0067 −0.0007 −0.012
(a)
(b)
(c)
(d)
0.0393 −0.0411 0.16 0 −0.0003 −0.0003 u. (21) −0.0006 0.0007 0.0005
Suppose the reference input is r (t ) = [0,1 − e −3t , 1 − e −6t ]T and the system is encountering the external
disturbance
d (t ) = 0.01e−0.2t cos(3162.3t )[1,1,1]T . The
multiplicative uncertainty weight W1 ( s ) is W1 ( s ) =
0.0125s 2 + 1.2025s + 1.25 s 2 + 20s + 100
Fig. 5. Outputs of systems using various controllers. (a) GA-based PI controller [15]. (b) Improved GAbased PI controller [28]. (c) OSA-based PI controller [13]. (d) PSO-based PI controller.
I 3×3 .
To attenuate disturbance, the stable weighting function W2 ( s ) is chosen as W2 ( s ) =
0.25s + 0.025 s 2 + 0.4s + 10000000
Controller
I 3×3 .
A PI type structure kP 11 k P21 kP K v ( s ) = 31
k P12 k P22 k P32
kI k P13 11 k P23 s + k I 21 k P33 k I31 s
k I12 k I 22 k I32
k I13 k I 23 k I33
is considered to carry out the proposed design algorithm. The PSO parameters are similar to previous example but the members of each individual are θ = [k P11 , k P12 , k P13 , k P21 , k P22 , k P23 , k P31 , k P32 , k P33 , k I11 , k I12 , k I13 , k I21 , k I 22 , k I 23 , k I31 , k I32 , k I33 ]T ,
the population size=15 and the maximum iteration is set to 100. Inspired from practical requirements, all parameters are assumed to be in the following range: −20000 ≤ θi ≤ 20000, for i = 1, 2,… ,18.
We performed 10 trials and the best PI controller is obtained using 100 iterations, 1500 function evaluations as follows: −50 2097.12 1300 K ( s ) = −1100 −1100 30 −15350 8750 19900
Table 3. Performance comparison for various controllers related to Example 3. J
g∞
Neval
GA-based PI [15]
NA
0.8194
18000
Improved GA-based PI [28]
0.1114
0.7682
4500
OSA-based PI [13]
0.0374
0.6299
3781
0.00004 0.62654
1500
PSO-based PI Conventional H∞
+
19550 14450 3102.47
-
0.5603
-
2976.10 −3800 −19900 −14389.17 −19777.5 13914.2 . S
Outputs of the system using the proposed controller are shown in Fig. 5(d). Outputs of the systems using the GA-based [15], Improved GA-based [28] and OSAbased [13] for the same system are also as shown in Figs. 5(a), (b) and (c). The OSA-based method uses the nonoverlapping group size N = 13, cooling rate CR = 0.95, initial temperature T0 = 80 and N stop = 3, and a stopping criterion of δ = 9 × 10−3. The GAbased method uses the population size N pop = 100, crossover rate pc = 0.9, mutation rate pm = 0.2 and 200 generations. The improved GA-based method uses N pop = 100, pc = 0.9, pm = 0.02 and 50 generations. More comparisons among various controllers are given in Table 3. The N eval is the number of function evaluations. The proposed algorithm uses minimal
Design of an H∞ PID Controller Using Particle Swarm Optimization
[5]
(a)
(b) [6]
[7]
(c) Fig. 6. Detailed tracking response of the first 0.1-sec period. (a) Improved GA-based PI controller [28]. (b) OSA-based PI controller [13]. (c) PSObased PI controller. number of evaluations. The conventional H∞ designed using γ -iteration algorithm has the lower H∞-norm but does not guarantee a desirable step response. Moreover, it has an order of 16 which shows a highly complex structure comparing to the simple PI structure. A detailed picture of the tracking responses of the first 0.1-s period in Fig. 6 reveals that the PSO-based PI controller has smaller oscillations than the improved GA-based one [28] and OSA-based one [13].
[8]
[9]
[10]
[11]
[12] 5. CONCLUSIONS In this paper, we propose a method of designing H∞ PID controller with minimizing a desired cost function. The high performance and validity of the proposed method are demonstrated by a MIMO super maneuverable F18/HARV fighter aircraft system, a SISO flexible-link manipulator system. It is shown empirically that the performance of the proposed method is much superior to those of existing GA-based, OSA-based and H∞-based methods. The PSO-based method can be widely used for designing high-performance optimal controllers. [1]
[2]
[3]
[4]
REFERENCES M. J. Balas, “Feedback control of flexible systems,” IEEE Trans. on Automat. Contr., vol. 23, no. 4, pp. 673-679, Aug. 1978. J. C. Doyle, K. Glover, P. Khargonekar, and B. A. Francis, “State space solutions to standard H2 and H∞ control problems,” IEEE Trans. on Automat. Contr., vol. AC-34, pp. 831-847, Aug. 1989. G. J. Balas, J. C. Doyle, K. Glover, A. Packard, and R. Smith, µ-Analysis and Synthesis ToolBox, Mathworks, Natick, MA, 1993. R. S. Smith, C. C. Chu, and J. L. Fanson “The
[13]
[14]
[15]
[16]
[17]
[18]
279
design of H∞ controllers for an experimental noncollocated flexible structure problem,” IEEE Trans. on Control Systems Technology, vol. 2, no. 2, pp. 101-109, Mar. 1994. G. J. Balas and J. C. Doyle, “Robustness and performance trade-offs in control design for flexible structures,” IEEE Trans. on Control Systems Technology, vol. 2, no. 4, pp. 352-361, Dec. 1994. I. N. Kar, T. Miyakura, and K. Seto, “Bending and torsional vibration control of a flexible plate structure using H∞ -based robust control law,” IEEE Trans. on Control Systems Technology, vol. 8, no. 3, pp. 545-553, May 2000. V. D. Blondel and J. N. Tsitsiklis, “A survey of computational complexity results in systems and control,” Automatica, vol. 36, pp. 1249-1274, Sep. 2000. G. J. Silva, A. Datta, and S. P. Bhattacharyya, “New results on the synthesis of PID controllers,” IEEE Trans. on Automat. Contr., vol. 47, no. 2, pp. 241252, Feb. 2002. H. Xu, A. Datta, and S. P. Bhattacharyya, “Computation of all stabilizing PID gains for digital control systems,” IEEE Trans. on Automat. Contr., vol. 46, no. 4, pp. 647-652, Apr. 2001. L. H. Keel, J. I. Rego, and S. P. Bhattacharyya, “A new approach to digital PID controller design,” IEEE Trans. on Automat. Contr., vol. 48, no. 4, pp. 687-692, Apr. 2003. M. T. Ho and C. Y. Lin, “PID controller design for robust performance,” IEEE Trans. on Automat. Contr., vol. 48, no. 8, pp. 1404-1409, Aug. 2003. F. Blanchini, A. Lepschy, S. Miani, and U. Viaro “Characterization of PID and Lead/Lag compensators satisfying given H∞ specifications,” IEEE Trans. on Automat. Contr., vol. 49, no. 5, pp. 736-740, May 2004. S. J. Ho, S. Y. Ho and L. S. Shu, “OSA: orthogonal simulated annealing algorithm and its application to designing mixed H2/H∞ optimal controllers,” IEEE Trans. Sys. Man and Cyber., vol. 34, no. 5, pp. 588600, Sep. 2004. R. A. Krohling and J. P. Rey, “Design of optimal disturbance rejection PID controllers using genetic algorithm,” IEEE Trans. Evol. Comp., vol. 5, no. 1, pp. 78-82, Feb. 2001. B. S. Chen and Y. M. Cheng, “A structure-specified H∞ optimal control design for practical applications: a genetic approach,” IEEE Trans. Control Sys. Tech., vol. 6, no. 6, pp. 707-718, Nov. 1998. B. S. Chen, Y. M. Cheng, and C. H. Lee, “A genetic approach to mixed optimal PID control,” IEEE Control Sys. Mag., vol. 15, pp. 51-60, Oct. 1995. J. Kennedy, “The particle swarm: social adaptation of knowledge,” Proc. IEEE Int. Conf. Evolutionary Comput., Indianapolis, IN, pp. 303-308, 1997. J. J. Liang, A. K. Qin, P. N. Suganthan, and S. Baskar, “Comprehensive learning particle swarm
280
[19]
[20]
[21]
[22]
[23]
[24] [25]
[26]
[27]
[28]
[29]
Majid Zamani, Nasser Sadati, and Masoud Karimi Ghartemani
optimizer for global optimization of multimodal functions,” IEEE Trans. Evol. Comp., vol. 10, no. 3, pp. 281-295, June 2006. O. Chao and L. Weixing, “Comparison between PSO and GA for parameters optimization of PID controller,” Proc. of International Conference on Mechatronics and Automation, pp. 2471-2475, June 2006. N. Sadati, M. Zamani, and H. Mahdavian, “Hybrid particle swarm-based-simulated annealing optimization techniques,” Proc. of the IEEE Inter. Conf. on Indus. Elect., Paris, France, Nov. 2006. D. B. Fogel, Evolutionary Computation Toward a New Philosophy of Machine Intelligence, IEEE, New York, 1995. Y. Shi and R. Eberhart, “A modified particle swarm optimizer,” Proc. IEEE Int. Conf. on Evolutionary Computation, pp. 69-73, 1998. R. Eberhart and Y. Shi, “Particle swarm optimization: Developments, applications and resources,” Proc. IEEE Int. Conf. on Evolutionary Computation, pp. 81-86, 2001. M. Avriel, Nonlinear Programming: Theory and Algorithms, Wiley, New York, 1979. K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control, Prentice Hall, Upper Saddle River, NJ, 1996. M. T. Ho and Y. W. Tu, “PID controller design for a flexible-link manipulator,” Proc. IEEE Conf. on Decision and Control, Spain, pp. 6841-6846, Dec. 2005. H. Kwakernaak, “Minimax frequency domain performance and robustness optimization of feedback systems,” IEEE Trans. Automat. Contr., vol. 30, pp. 994-1004, 1985. I. Kitsios, T. Pimenides, and P. Groumpos, “A genetic algorithm for designing H∞ structured specified controllers,” Proc. IEEE Int. Conf. Contr. Applicat., Mexico , pp. 1196-1201, 2001. P. Voulgaris and L. Valavani, “High performance H and H designs for supermaneuverable F18/harv fighter aircraft,” AIAA J. Guid. Cont. Dyn., vol. 14, no. 1, pp. 157-165, 1991.
Majid Zamani received the B.Sc. and M.Sc. degrees in Electrical Engineering in 2005 and 2007 from Isfahan University of Technology, and Sharif University of Technology, Iran, respectively. Currently, He is a Ph.D. student in Electrical Engineer-ing Department of University of California, Los Angeles, U.S.A.
Nasser Sadati was born in Iran in 1960. He received the B.S. degree from Oklahoma State University, Stillwater, in 1982, and the M.S. and Ph.D. degrees from Cleveland State University, Cleveland, OH, USA, in 1985 and 1989, respectively, all in Electrical Engineering. From 1986 to 1987, he was with the NASA Lewis Research Center, Cleveland, to study the albedo effects on space station solar array. In 1989, he conducted postgraduate research at Case Western Reserve University, Cleveland, OH. Since 1990, he has been with the Sharif University of Technology, Tehran, Iran, where he is currently a Full Professor in the Department of Electrical Engineering, the Head of Control Group, and the Director of the Intelligent Systems Laboratory and the CoDirector of Robotics and Machine Vision Laboratory. He was the first to introduce the subject of fuzzy logic and intelligent control as course work in the universities engineering program in Iran. He has published two books in Persian and over 200 technical papers in peer-reviewed journals and conference proceedings, and is currently working on two more books in English (Intelligent Control of Large-Scale Systems) and Persian (Neural Networks). His research interests include intelligent control and soft computing, large-scale systems, robotics and pattern recognition. Dr. Sadati was the recipient of the Academic Excellence Award for 1998-1999 from the Sharif University of Technology. He is a Founding Member of the Iranian Journal of Fuzzy Systems (IJFS). He is the Founder and Chairman of the First Symposiums on Fuzzy Logic, and Intelligent Control and Soft Computing in Iran. He is the editorial board members of International Journal of Advances in Fuzzy Mathematics (AFM) and the Journal of Iranian Association of Electrical and Electronics Engineers (IAEEE). He also has served as the CoChair of the First International Conference on Intelligent and Cognitive Systems (ICICS’96). Dr. Sadati is a Founding Member of the Center of Excellence in Power System Management and Control (CEPSMC), Sharif University of Technology, Tehran, Iran and the Foreign Member of the Institute of Control, Robotics, and Systems (ICROS), Korea. Masoud Karimi Ghartemani received the B.Sc. and M.Sc. in Electrical Engineering in 1993 and 1995 from Isfahan University of Technology, Iran, where he continued to work as a Teaching and Research Assistant until 1998. He received the Ph.D. degree in Electrical Engineering from University of Toronto in 2004. He was a Research Associate and a Post-doctoral Researcher in the Department of Electrical and Computer Engineering of the University of Toronto from 1998 to 2001 and from 2004 to 2005, respectively. He joined Sharif University of Technology, Tehran, Iran, in 2005 as a Faculty Member. His research topics include nonlinear and optimal control, novel control and signal processing techniques/algorithms for control and protection of modern power systems, power electronics, power system stability and control, and power quality.
