JOURNAL OF LOW FREQUENCY NOISE, VIBRATION AND ACTIVE CONTROL
Pages 123 – 150
Active vibration control of flexible plate structures with distributed disturbances S. Julai1 and M. O. Tokhi2 1Department of Mechanical Engineering, University of Malaya, Kuala Lumpur, Malaysia 2Department of Automatic Control and Systems Engineering, The University of Sheffield, United Kingdom Email:
[email protected] Received 8th March 2011
ABSTRACT This paper presents the development of an active vibration control (AVC) system with distributed disturbances using genetic algorithms, particle swarm optimization, and ant colony optimization. The approaches are realized with multiple-input multiple-output and multiple-input single-output control configurations in a flexible plate structure. A simulation environment characterizing a thin, square plate, with all edges clamped, is developed using the finite difference method as a platform for test and verification of the developed control approaches. Simulations are carried out with random disturbance signal. The control design comprises a direct minimization of the error (observed) signal by allowing a collective determination of detection and secondary source locations together with controller parameters. The algorithms are formulated with a fitness function based on the mean square of the observed vibration level. In this manner, knowledge of the input/output characterization of the system is not required for design of the controller. The performance of the system is assessed and analyzed both in the time and frequency domains and it is demonstrated that significant vibration reduction is achieved with the proposed schemes. Keywords: Active vibration control, genetic algorithms, particle swarm optimization, ant colony optimization, flexible plate structure.
1. INTRODUCTION Interest has increased in the active control of vibration in mechanically flexible systems. These in aerospace and aircraft structures include space-based radar antennal and solar panels, space robotics, propeller and aircraft fuselage and wings, in electromechanical systems that include turbo generator shafts, gas turbine rotors and electric transformer cores, and in civil engineering applications include skyscrapers and bridges [1-6]. Such structures may be damaged or become ineffective under the influence of undesired vibrational loads they constantly experience. Due to these problems, an effective control mechanism is required to attenuate the vibration in order to preserve structural integrity of such systems. The conventional form of external passive damping to change the dynamic characteristics of the structure is not preferred as the addition of a damper adds to the overall system weight, which is undesirable and makes the system less transportable especially for space applications. This has led to extensive research into active control techniques where the disturbance to be cancelled or the properties of the controlled system vary with time. Active control solutions are known to achieve good vibration suppression performance in comparison to passive control [7].
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Active vibration control of flexible plate structures with distributed disturbances Active vibration control (AVC) consists of artificially generating cancelling sources to interfere destructively with the unwanted source and thus result in a reduction in the level of vibration (disturbance) at desired location(s). This unwanted source can be either from one excitation source or a number of excitation sources that can be modelled as a set of point sources distributed around the surface of the structure. To suppress the vibration from one excitation source, a single detector is generally sufficient to obtain the required signal information needed to generate the cancelling signal. In the case where the vibration is from distributed sources, a multiple set of detectors is required to obtain sufficient signal information in either single-output or multiple-output control configuration. This will lead to the realization of a multiple-input single-output (MISO) or multiple-input multipleoutput (MIMO) control system [8-9]. AVC is realized with actuators, sensors and electronic control to reduce vibration without necessarily adding damping to the system. Due to the broadband nature of the disturbances, it is required that the control strategy realizes suitable frequencydependent characteristics so that cancellation over a broad range of frequencies is achieved [8]. Remarkable advances in smart materials, such as piezoelectric transducers which are used extensively as distributed sensors and actuators, and computing technology have lead AVC to provide cost-effective solutions to most sound- and vibration-control problems [10]. However, careful consideration must be taken in positioning the sensors and actuators to ensure good control [11]. This means that the locations of these transducers have significant influence on the performance of the control system as well as the controlled response. Many techniques have been reported to find optimal locations of sensors and actuators. Methods for optimal placement of sensors and actuators have been investigated by many researchers [12-15]. All the research works have shown the importance of optimal placement of sensor and actuator for achieving a significant level of vibration suppression. The aforementioned researches have utilized GA and PSO in finding optimal locations of sensors and actuators for vibration control. However, the use of ant colony optimization (ACO) has not been reported in this context. GA was introduced by John Holland [16] based on the principle of the Charles Darwinian Theory of evolution to natural biology. It is a search procedure based on the mechanics of natural selection and natural genetics. A possible solution to an optimization problem can be encoded as an individual (or a chromosome), and later ranked by comparison to a particular fitness function. New individuals are created using genetic operators such as mutation and crossover that produce the genetic composition of a population (finite number of individuals). PSO was developed by Eberhart and Kennedy [17] in 1995, inspired by the social behaviour of bird flocking or fish schooling. It is a population based stochastic optimization technique with a set of randomly generated potential solutions (initial swarm) referred to as particles that propagate in the search space towards the optimal solution. Each particle adjusts its position in the search space according to its own flying experience and the flying experience of other particles. ACO was developed by Dorigo and his colleagues in 1991, inspired by the cooperative search behaviour of real ants where the problem of interest is to mimic almost blind ants to establish shortest paths from their nest to food source and back, through local message exchange via the deposition of pheromone trails [18]. Traditional optimization methods emphasize accurate and exact computation, but may fail on achieving the global optimum. GA, PSO, and ACO on the other hand, have appeared as promising algorithms for handling optimization in complex realworld problems, and are widely used as design tools and problem solvers in optimization problems due to their versatility and ability to optimize in complex multimodal search spaces with nondifferentiable cost functions. These algorithms are nature inspired with population based stochastic search and have been very popular in the field of computational intelligence. Solving an optimization problem using stochastic search can often out-perform classical methods of optimization
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S. Julai and M. O. Tokhi when applied to difficult real-world problems. They offer better chances to achieve the global optimum since they do not use gradient information and are useful in solving problems where such information is unavailable or very costly to obtain [19]. Classical methods based on gradient information, on the other hand, emphasize accurate and exact computation, but may fail on achieving the global optimum. In this work, a multi-input-AVC system is developed using a non-model based approach where the controller is designed directly based on minimization of the observed signal with GA, PSO, and ACO, by allowing a combined determination of detection and secondary source locations together with controller parameters. The goal of this paper is to propose an approach for vibration reduction of flexible plate structures by using GA-AVC, PSO-AVC, and ACO-AVC strategies based on minimization of the observed signal. This approach is used to determine the controller parameters in such a way that the value of the objective function assigned is minimized. The approach does not require knowledge of the input/output characterization of the system for controller design. The performances of optimization techniques are presented in terms of convergence rate, computational time, and spectral attenuation achieved at the dominant modes of vibration of the plate. The rest of the paper is structured as follows: Section 2 presents classical governing dynamic equations of a thin rectangular plate and the corresponding numerical simulation algorithm based on finite difference (FD) method. The GA, PSO, and ACO algorithms are described in Section 3. The AVC strategies are presented in Section 4. Section 5 presents performance results of the AVC approaches within the flexible plate simulation environment. The results are analyzed and discussed from the time- and frequency-domain perspectives. The paper is concluded in Section 6. 2. THE FLEXIBLE PLATE STRUCTURE Flexible structures offer several advantages including fast response, low energy consumption, reduced mass and low cost, in comparison to their rigid and bulky counterparts. Besides the advantages these structures offer, a major constraint to take into consideration is the system vibration arising from structural flexibility. Accordingly, control of flexible systems has been a challenge for researchers and engineers [20]. Vibration suppression is focused more on flexible structures due to their potential widespread use as they are capable of being operated at high speeds and handling of larger payloads with the same actuator capabilities [22]. Dynamic modeling and simulation of a flexible plate structure using the FD method has been reported in [23], where a flat, square plate with all edges clamped has been considered. The classical dynamic equation of a thin rectangular plate is described with a partial differential equation (PDE) formulation as [24, 25]:
∂w ∂w ∂w ρ ∂w q +2 + + = ∂x D ∂t D ∂x ∂ y ∂ y 4
4
4
2
4
2
2
4
(1)
2
where w is the lateral deflection, r is the mass density per unit area, q=q(x, y) is the transverse external force at point (x, y) on the plate and has dimensions of force per unit area, ∂w/∂t2 is the lateral acceleration, D=[EH3/12(1-v)] is the flexural rigidity with υ representing the Poisson ratio, h the thickness of the plate, and E the Young’s modulus. A simulation algorithm characterizing the dynamic behaviour of the plate is developed through discretisation of the PDE. The plate can be divided into n × m sections, along the x and y axes as x = i∆x and y = j∆y. Using central difference approximations for the first-, second-, third- and fourth-order derivatives of the response, a linear relation for the deflection of each section (mesh) can be developed using FD approximations, as
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Active vibration control of flexible plate structures with distributed disturbances
D∆t 20 w ρ∆ 2
w
i , j ,k +1
=−
4
xy
(
i , j ,k
+ 2 w i + 2 , j ,k
+w
i , j + 2 ,k
i−1, j +1,k
+w
i − 2 , j ,k
+w
i+1, j ,k
+w
i+1, j +1,k
+w
+ 8 w
+w
i , j +1,k
+w
i , j − 2 ,k
+w
i−1, j ,k
+w
i−1, j −1,k
i+1, j −1,k
) + 2w
+w
i , j −1,k
+ (2)
∆t q 2
i , j ,k
−w
i , j ,k −1
+
i, j
ρ
where the x-axis is represented with the reference index i, y-axis with the reference index j, and time, t represented with a reference index k. Here, Wi,j,k+1 is the deflection of nodal point (xi, yj) of the plate at time step k+1. For the case of all edges clamped, the deflection is always zero along the edges, and the tangent of the deflection at the edge is equal to zero, e.g. at y = a a, wy=a = ∂w/∂yy=a = 0. This condition needs to be satisfied at every nodal point along the clamped edge within the FD formulation. The plate is divided into 20 × 20 sections with a sampling time of 0.0016 sec. An aluminium type plate with boundary condition of all edges clamped and specifications given in Table I is considered. The simulation algorithm developed is used as a platform for test and performance evaluation of AVC approaches developed in this work.
Table 1 Parameters of the flexible plate structure. Parameter
Value
Length, L
1.0 m
Width, W
1.0 m
Thickness, T
m
Moment of inertia, I
m
Mass density per area, Young s Modulus, E Simulation time, t
4 seconds
Poisson s ratio, υ
0.3
3. OPTIMIZATION TECHNIQUES Traditional optimization methods based on gradient information emphasize accurate and exact computation, but may fail in achieving the global optimum. Optimization techniques such GA, PSO, and ACO offer better chances to achieve the global optimum since they do not use gradient information and are very useful in solving problems where such information is unavailable or very costly to obtain [19]. Although the algorithms are population-based, but due to different mechanisms, they have different features and operators. 3.1. Real-coded genetic algorithm Genetic algorithms are search and optimization techniques that are motivated by the principle of natural evolution. In traditional GA, all the variables of interest must first be encoded as binary digits (genes) forming a string (chromosome).This representation is known as binary-coded genetic algorithm (BCGA). Each chromosome (individual) has an associated fitness which is the value of the objective function for that set of parameters and contains sub-strings or genes as units that contribute in different ways to the overall fitness of the individual. BCGA
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S. Julai and M. O. Tokhi is found to be a robust search technique avoiding local optima, but the major drawback is the difficulty faced when it is applied to problems with large search space and requiring high precision. Large string lengths result in more precise solution but lead to increase in computational cost [26]. To overcome the difficulties related to binary representation, a floating-point representation of parameters as chromosomes known as real-coded genetic algorithm (RCGA) is used. All genes in a chromosome used in RCGA are real numbers. The use of this floating point representation outperforms binary representations in real-valued optimization problems because they are more consistent, precise, and lead to faster convergence [27]. In RCGA, the length of chromosomes becomes shorter than those with the equivalent binary representation. This implies that computer programming for such algorithms can be easily performed. The tuning mechanism for mutation and crossover operations is also performed using floating point numbers instead of long strings of zeros and ones. In RCGA, three basic operations are used: selection, crossover, and mutation. (i) Selection Selection is an important aspect of evolutionary computation. It dictates what member of the current population affects the next population. More fit individuals are generally given a higher chance to participate in the recombination process. Stochastic universal sampling (SUS) provides zero bias and minimum spread. The individuals are mapped to contiguous segments of a line, such that each individual’s segment is equal in size to its fitness exactly as in roulette-wheel selection. Here equally spaced pointers are placed over the line as many as there are individuals to be selected [28]. After the selection, the parent chromosomes are combined and mutated to form the offspring chromosomes. (ii) Crossover Let x = (x1, ..., xn) and y = (y1, ..., yn) be the parent strings, where the generic xi and yi are real variables. In extended intermediate recombination
z = x +α (y − x ) k
k
k
k
(3)
k
where ak is chosen uniform randomly in the interval . Intermediate recombination is capable of producing any point within a hypercube slightly larger than that defined by the parents. Fig. 1 (a) shows the possible area of offspring after intermediate recombination. (ii) Mutation This function takes a vector containing the real representation of the individuals in the current population, mutates the individuals with probability pm and returns the resulting population. The mutation operator is able to generate most points in the hypercube defined by the variables of the individual and the range of the mutation, as in Fig. 1 (b).
area of possible offspring variable 2
possible offspring
after mutation
parents variable 1 (a)
Fig. 1.
before mutation
variable 2
variable 1 (b)
(a) Possible areas of the offspring after intermediate recombination, and (b) effect of mutation.
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Active vibration control of flexible plate structures with distributed disturbances 3.2 . Particle swarm optimization Particle swarm optimization shares many similarities with evolutionary computation techniques such as GA. It is inspired by the ability of flocks of birds, schools of fish, and herds of animals to adapt to their environment, find rich sources of food by implementing an information sharing approach, and it possesses the properties of easy implementation and fast convergence [17, 29]. The algorithm has been widely applied to continuous and discrete optimization problems and has received great attention in systems and control engineering, automatic recognition, radio systems, etc. Starting with a randomly initialized population, each particle in the PSO flies through the d-dimensional problem space and remembers the best position it has seen. The particles evaluate their positions relative to a global fitness during each iteration and use their memorized best positions to adjust their own velocities and subsequent positions. In this way, the particles tend to fly towards better and better searching areas through the search process. In a PSO algorithm, the position vector and the velocity vector of the i-th particle in a d-dimensional search space can be represented as Xi = (xi1, xi2, ..., xid) and Vi = (vi1, vi2, ..., vid), respectively. The best position of each particle (which corresponds to the best fitness value obtained by that particle at time t) is denoted as Pi = (pi1, pi2, ..., pid) , and the fittest particle found so far at time t as Pg = (pg1, pg2, ..., pgd) . Then the new velocities and positions of the particles for the next fitness evaluation are calculated using the following equations:
v (t + 1) = ω × v (t ) + c × rand(⋅) × ( p − x (t )) id
id
1
id
id
(4)
) + c × Rand(⋅) × ( p − x (t )) 2
gd
id
x (t + 1) = x (t ) + v (t + 1) id
id
(5)
id
where ω is the inertia weight, c1 is the cognition factor, c2 is the social factor, and rand (·) and Rand (·) are two separately generated uniformly distributed random numbers in the range [0, 1] [30]. Abd. Latiff and Tokhi, [31] have proposed a strategy to guarantee fast convergence as an improvement to the concept of time-varying acceleration coefficients (TVAC), developed by Ratnaweera et al., [32], to effectively control the global search and convergence to the global best solution. They have established that by continuously modifying the value of inertia weight, superior results can be achieved as compared to the case when the inertia weight is fixed. For fast convergence purposes, the particles have to know their location and relative distance from each other in exploring the search space. To do so, the spread factor (SF) has been introduced, which measures the distribution of particles in the search space as well as the precision and accuracy of the particles with respect to the global optimum. Precision (or spread of particles in the search space) refers to the maximum distance between particles in the best and worst positions while accuracy (or deviation of the particles) refers to the distance of average particle position from the global best particle. The value of spread factor varies from the maximum range of the search space down to the desired convergence precision and can be calculated as
spread + deviation SF = 0.5 × range − range id
id
id
max
(6)
min
where spreadid = xmax id – xmin id and deviationid = ∑(xid – gbestid)/N, with xmax id and xmin id representing maximum and minimum values of the ith particle’s position, N is the number of particles, and rangemax and rangemin are maximum and minimum
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S. Julai and M. O. Tokhi range of the specified variables. This factor is used to modify the inertia weight,
ω = exp( − iter / ( SF × max _iter )) with
(7)
c = 2 × (1 − iter / max _iter ) 1
(8)
c =2 2
where c1 is linearly reduced to zero from its initial value of 2, and c2 is maintained at 2 to ensure all particles are pulled towards global optimum. 3.3. Ant colony optimization The basic ACO algorithm actually fits a discrete problem only and is not suitable for solving continuous optimization problems such as linear or non-linear programming. Applying this algorithm to continuous domains was not straightforward where the main issue is how to model a continuous nest neighbourhood with a discrete structure. As the design problem can always be formulated as optimization problem in continuous design space, the ACO algorithm applied in any field should be modified accordingly [33 – 34]. Continuous ACO was proposed by Wang and Wu, [35] and Quan and Chao, [36]. This algorithm is referred to as ACO1 throughout this work. Here, the optimization problem is solved by a cooperation of artificial ant colony by exchanging information via pheromone deposited on graph edges. The essence of ACO1 is composed of two parts: 1. A state transition rule used by an ant to determine the next destination of a complete tour, and 2. A pheromone updating rule in allocating a greater amount of pheromone to shorten the tour. Let the vector X = [x1, x2, ..., xn] be the parameters to be optimized, where n represents the total number of parameters in the AVC system, along with upper and lower bounds to be xi∈D(xi) = [xi_low, xi_up] with i = 1,2, ..., n. . The definition field D(xi) is divided into M subspaces, and the middle of each subspace defines a node. A single artificial ant k (k = 1, 2, ..., Nant) where Nant is the maximum number of ants, would choose to move from one node to the other, in the total of P nodes in each D(xi). The length of each sub-space hi can be expressed by
h=
x
i
i _ up
−x
i _ low
(9)
M
For each level which has P nodes on it, there are M × n nodes in total. A modification has been made to this algorithm by the author to the state vector of the ant k that completed its tour, as shown in Fig. 2, with travel index [i8, i7, i6, ...,i4]. This index depends on the entries of cumulative probability (CP) from the probability Pij of the ant k to move to the ith node on the jth level. For example, if M = 10, CP = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0] and the generated random number lies between 0.7 and 0.8, the first travel index, i8, is chosen as 8 (eighth column of the CP). This process is done until all travel indexes are found. The values of the parameters X, held by ant k, are as
[ x , x , x …, x ] = [ x 1
2
3
n
1_ low
+i ×h, x
+ i × h ,…, x 6
3
8
n_low
1
2 _ low
+ i × h ,x 7
2
3_ low
(10)
+i ×h] 4
n
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Active vibration control of flexible plate structures with distributed disturbances The state transition rule of the ant k is expressed as
P =τ ij
n
ij
∑τ
(11)
ij
i=1
where Pij is the probability of the ant k to move to the ith node on the jth level, and τij is the amount of the pheromone at the node. When all the ants have finished their tours, the pheromone is updated by using
τ = (1 − ρ )τ + Q / f ij
ij
(12) best
where 0 < ρ < 1 is a pheromone decay parameter, Q is the quantity of pheromone laid by an ant per iteration cycle, τ0 is a constant for the initial value of τij (for initialization τij on the RHS is set to be τ0), and is the objective function value given by the best ant of each searching period.
Fig. 2.
State space graph for ACO1.
4. ACTIVE VIBRATION CONTROL DESIGN WITH DISTRIBUTED SOURCES OF DISTURBANCE 4.1. MIMO-AVC A schematic diagram of the geometric arrangement of a MIMO feedforward AVC structure considered in this study is shown in Fig. 3 (a). The disturbance signals, UD emitted by n primary point sources are detected by a set of n detectors. These detection signals, UM are processed by the controller and fed to a set of k secondary sources, UC. The aim of the controller design is to minimize the observed signals Y, where Y =[y1, y2, ..., yk], via UC by generating anti-phase control signals to counteract the vibration produced by UD. This can be achieved through optimization of the assigned objective function using GA, PSO or ACO to obtain the best value of the controller parameters. Fig. 3 (b) shows a block diagram of the MIMO-AVC system [8], where E is an n × n matrix representing transfer characteristics of the propagation paths between the primary sources and the detectors, F is a k × n matrix representing transfer characteristics of the paths between the secondary sources and the detectors, G is an n × k matrix representing transfer characteristics of the paths between the primary sources and the observers, H is a k × k matrix representing transfer characteristics of the acoustic paths between the secondary sources and the observers, M is an n × n diagonal matrix representing transfer characteristics of the detectors, L is a k × k matrix representing transfer characteristics of the secondary sources, UD is a 1 × n matrix representing the primary signals at the source points, P0 is a 1 × k matrix representing the primary signals at the observation points, UC is a 1 × k matrix representing the secondary signals at the source points, S0 is a 1 × k matrix representing the secondary signals at the 130
JOURNAL OF LOW FREQUENCY NOISE, VIBRATION AND ACTIVE CONTROL
S. Julai and M. O. Tokhi observation points, UM is a 1 × n matrix representing the detected signals, and O is a 1 × k matrix representing the combined primary and secondary signals at the observation points.
Optimization
Y
C UC
UM
Secondary sources
Detectors
UD
Observed signals
Primary sources
(a)
G F P0 + E
+ UM
•
+ M
L
C
UD
+ •
H UC
S0
O
Optimization (b)
Fig. 3.
AVC structure for flexible plate: (a) schematic diagram, and (b) block diagram.
To suppress the vibration at the observation points, it requires the observed primary and secondary signals at each observation point to be equal in magnitudes and have a phase difference of 180˚:
S = −P 0
(13)
0
Using the block diagram in Fig. 3(b), P0 and S0 can be expressed as
P = PG , S = PEMCL[I − FMCL] H −1
0
0
(14)
where I is the identity matrix. The required controller transfer function, C, can be obtained by substituting equ. (14) into equ. (13), and simplifying as:
C = M ∆ GH L −1
−1
−1
−1
(15)
where ∆ is an n × n matrix given by
∆ = GH F − E −1
(16)
From equ. (15), the matrix dimension for C can be obtained as n × k, given by
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Active vibration control of flexible plate structures with distributed disturbances
C=
c c c c c c c
11
c
c
12
21
22
n1
1k
2k
n2
(17)
nk
where cim(i = 1, 2, ..., n; m = 1, 2, ..., k) represents the controller transfer function along the secondary path from detector i to secondary source m. This controller can be realized in MIMO form as shown in Fig. 4. In this work, the number of primary sources and secondary sources are set to n = 2 and k = 2, as presented in Fig. 5 (a). The controller transfer function in equ. (17) then becomes
C
c = c
MIMO
11
c
12
21
c
22
(18)
where each transfer function is realized in a linear parametric form, as p
q
U (t ) = ∑ b ⋅ U (t − j ) − ∑ a ⋅ U (t − i ) C 11
j 11
M1
j 11
j =0
i=1
p
q
C 11
U (t ) = ∑ b ⋅ U (t − j ) − ∑ a ⋅ U (t − i ) C 12
j 12
M1
j 12
j =0
i=1
p
q
C 12
U (t ) = ∑ b ⋅ U (t − j ) − ∑ a ⋅ U (t − i ) C 21
j 21
M2
j 21
j =0
i=1
p
q
C 21
U (t ) = ∑ b ⋅ U (t − j ) − ∑ a ⋅ U (t − i ) C 22
j 22
M2
j 22
j =0
C 22
(19)
(20)
(21)
(22)
i=1
where ai, bj are the controller parameters and q ≥ p represents the order of the controller. The number of parameters to be estimated is (q + p + 1) × q × p. Randomly selected controller parameters, i.e. a1,...,aq and b0, ..., bp are identified for different, arbitrarily chosen orders to fit to the system. The stability of the obtained controller must be ensured. In the discrete-time case, pole-zero diagram of the corresponding controller transfer function provides a simple and effective means of assessing its stability. Using, for example equ. (19), the transfer function of the controller can be formed as
b + b z + b z + ... + b z −1
C (z) =
011
−2
111
211
−p
p 11
1 + a z + a z + a z + ... + a z −1
11
−2
111
211
−3
311
−q
(23)
q 11
Optimization techniques are carried out to achieve vibration reduction by feeding the observed signal Y to the controller C via optimization routines. The optimization process is based on minimizing the mean-squared error (MSE) of the observed signal, Y. This is formulated as:
f
error ( MIMO )
=
(
)
1 ∑ Y (i) + Y (i) + ... + Y (i) ; subject to –1 ≤ ai ≤ 1 and –1 ≤ bj ≤ 1 (24) S S
i=1
1
2
k
2
where S represents the number samples. 132
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S. Julai and M. O. Tokhi 4.2. MISO-AVC Unlike MIMO-AVC system as mentioned above, the MISO-AVC system has more than one input but only one output. The MISO-AVC structure for the flexible plate with two primary sources is illustrated in Fig. 5 (b). The controller is designed to minimize the deflection at observation point 1 via UC by generating anti-phase control signal to counteract the vibration produced by UD. For this case the number of primary sources and secondary source are set to n = 2 and k = 1. The controller transfer function in equ. (17) then becomes
C
MISO
c = c
11
21
(25)
where each of the transfer functions is realized in a linear parametric form as p
q
U (t ) = ∑ b ⋅ U (t − j ) − ∑ a ⋅ U (t − i ) C 11
j 11
M1
j 11
j =0
(26)
Ci 1
i=1
p
q
U (t ) = ∑ b ⋅ U (t − j ) − ∑ a ⋅ U (t − i ) C 21
j 21
M2
j 21
j =0
(27)
C 21
i=1
The number of parameters to be estimated here is (q + p + 1) × n. The transfer function of controller can be formed from equ. (26) and (27). The performance of the MISO-AVC system can be assessed with the mean-squared error (MSE) of the observed signal, Y. The objective function is thus used is
f
error ( MISO )
=
(
)
1 ∑ Y (i) ; subject to –1 ≤ ai ≤ 1 and –1 ≤ bj ≤ 1 S S
i=1
2
UM1 C11
C12
C1k
+
+
(28)
Observation point-1
+
UM2
+ C21
C22
C2k
+
Observation point-2
+
UMn Cn1
Cn2
Cnk
+
+
Observation point-k
+
Fig. 4.
MIMO controller.
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Active vibration control of flexible plate structures with distributed disturbances
UM1 C11
UC11 +
C12
Secondary signal 1
UM1
UC11
C11
UC21 +
+
UC12 UC22 +
UM2 C21
C22
+ (a)
Fig. 5.
Secondary signal 2
Secondary signal 1
+
UM2 C21
UC21 (b)
AVC with multiple-input: (a) multiple-output, and (b) single-output.
4.3. System geometry The effectiveness of AVC for controlling the vibration of a flexible structure is directly affected by the geometrical arrangement of the system components: detection, observation and secondary source points. A typical implementation comprises a sensor, to detect the vibration mode to be suppressed, and control electronics driving actuators to generate cancelling sources to interfere destructively with the unwanted source from primary point. To control the vibration on the structure requires that the actuators are optimally situated. In this work, two mechanisms of geometrical arrangement for detection and secondary source points are investigated, variable geometry and fixed geometry. For both mechanisms, points of primary and observation for both cases are kept at specific mesh-points. 4.3.1. Variable geometry In this mechanism, the locations for the detection and secondary source points as well as the parameters of the controller are determined by the global search technique of GA, PSO, and ACO. Since the algorithms use stochastic random searches, there are a few possibilities that need to be considered. In MIMO- and MISO-AVC cases, the algorithms may converge to primary and observation points for the detection and secondary source points, or to the same point for detection point 1 and secondary source 2, for example. To ensure such situations are avoided, penalty has been added to the algorithms. On the other hand, when the location of detection point 1 is the same as secondary point 1 or detection point 2 is the same as secondary point 2 (MIMO case), or secondary source 1 is the same as detection point 1 or detection point 2 (MISO case), such a situation is allowable. This is known as collocation, where the advantage is reduction of the required space to install actuator and sensor in mechanical design of the system [37]. Given that the plate is divided into 20 × 20 sections and the deflection along the boundaries is zero, the range of coordinates for the detection and secondary source points must be within 1 and 19. Here, about possible detection and secondary source locations are to be explored to achieve the highest degree of effectiveness to produce the lowest value of the objective function. However, after several investigations, it was found that points located in the smaller range [4, 16] have produced more vibration attenuation than the range [1,19]. For this reason, the execution of the RCGA-AVC, PSO-SF-AVC, and ACO1-AVC of the flexible plate are used to find these points in the range of [4, 16], as shown in the white area of the plate section in Fig. 6. The optimization procedures are described as below: (i) RCGA-AVC step 1 Initialize parameters of the flexible plate and location of disturbance points: P1 = (7, 7) is primary point 1, P2 = (10, 6) is primary point 2,and observation points; O1 = (9, 14) is observation point 1 (MISO and MIMO cases), and O2 = (13,8) is observation point 2 (MIMO case only). step 2 Initialize range of variables for optimal location of detection and secondary source points, in the range , and range of variables for controller parameters, in the range [-1, 1]. step 3 Generate random population of X chromosomes where the initial chromosome for ith individual is represented as below: 134
JOURNAL OF LOW FREQUENCY NOISE, VIBRATION AND ACTIVE CONTROL
S. Julai and M. O. Tokhi MISO case:
X = ( d , d , s , s , d , d , a ,…, a i
i (1)
i(2)
i ( 3)
i(4)
i ( 5)
i(6)
i ( q × k +6 )
i(7)
,b
i ( q × k +7 )
,…, b
i (( p + q )× k +8 )
)
(29)
where the first six variables are coordinates for detection point 1, D1 = (di1, di2) and secondary source point 1, S1 = (si3, si4), and detection point 2, D2 = (di5, di6). The remaining variables are the controller parameters to form transfer functions as in equ. (26) and (27). MIMO case:
X = ( d , d , s , s , d , d , s , s , a ,…, a i
i (1)
i(2)
i ( 3)
i(4)
i ( 5)
b
i (( p + q )× k +11)
i(6)
i(7)
i(8)
,…, b
i (( p + q )× k × n+12 )
i (( p + q )× k +10 )
i(9)
, (30)
)
where the first eight variables are coordinates for detection point 1, D1 = (di1, di2), secondary source point 1, S1 = (si3, si4), and detection point 2, D2 = (di5, di6), and secondary source point 2, S1 = (si3, si4), and detection point 2, S2 = (si7, si8). The remaining variables are the controller parameters to form transfer functions as in equ. (19) – (22). step 4
Add penalty to Xiif if any of the following conditions occur:
( d , d ) + penalty, D = ( d , d ), i1
i2
1
i1
i2
if ( d , d ) = P or ( d , d ) = P or i1
i2
i1
1
i2
2
( d , d ) = O or ( d , d ) = O or i1
i2
i1
1
i2
2
( d , d ) = D or ( d , d ) = S i1
i2
i1
2
i2
(31) 2
otherwise
( s , s ) + penalty, if ( s , s ) = P or ( s , s ) = P or ( s , s ) = O or ( s , s ) = O or S = [( s , s ) = S or ( s , s ) = D ] otherwise ( s , s ), i3
i4
i3
i3
1
i4
i4
i3
i3
1
i3
1
i4
2
MIMO case only
i3
step 5 step 6
i4
i4
i3
2
2
i4
2
(32)
i4
Evaluate the fitness value, i.e. MSE of the observed signal, for each chromosome in the population according to equ. (24) or (28). Create a new population by repeating the following steps until the new population is complete: 6.1. For selection operator, select two parent chromosomes from a population using SUS algorithm, according to their fitness. Parents that produced the smallest MSE value are considered as the best fit individuals. 6.2. For crossover operator, cross over the parents to form a new offspring (children) according to equ. (3). If no crossover was performed, offspring is an exact copy of parents. 6.3. For mutation operator, mutate the new offspring to alter the genes of some of the children, according to the probability of mutation, pm.
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Active vibration control of flexible plate structures with distributed disturbances step 7 step 8 step 9
step 10 step 11
Add penalty to the offspring as in step 4. Evaluate the fitness values of all offspring. Reinsert the offspring into the original population using fitness-basedreinsertion, with the new offspring replacing the least fit member of the original population. If the specified maximum number of generation is not reached repeat step 6 until convergence, else, end the algorithm. Output the values of the decision variables for final evaluation of the AVC system. End of procedure.
(ii) PSO-SF-AVC step 1-2 Same as in RCGA-AVC procedure. step 3 Initialize the population of particles with random positions and velocities, and maximum iteration, where the initial position vector for ith particle is represented as in step 3 in RCGA-AVC procedure. step 4 Same as RCGA-AVC procedure. step 5 Evaluate the fitness value, i.e. MSE of the observed signal, for each particle of the population according to equ. (24) or (28). The best particle that produces the smallest fitness value with its position will be stored as global best position. step 6 Calculate spread, deviation, SF, ω, c1, and, c2 as in equ. (6), (7), and (8), for all the variables and particles. step 7 Update the position and velocity of particle according to equ. (4) and (5). step 8 Add penalty to the updated position as in step 4. step 9 Evaluate the fitness values of all particles and determine the best particle for the current iteration. If the MSE observed signal is smaller than the MSE value of the global best position, then update the global best position and its MSE value with the position and MSE value of the current best particle. step 10-11 Same as RCGA-AVC procedure (iii) ACO1-AVC step 1-2 Same as RCGA-AVC procedures. step 3 Initialize randomly the tours of ants within the predefined range of variables, initial pheromone, initial probability, and maximum iteration, where the initial tour vector for kth ant is represented as in step 3 in RCGA-AVC procedure. step 4 Same as RCGA-AVC procedure. step 5 Evaluate the initial fitness value, i.e. MSE of the observed signal, for each ant of the colony according to equ. (24) or (28). The best ant that produces the smallest fitness value with its vector will be stored as the best vector. step 6 Evaluate the MSE value of every ant using the solution vector constructed from equ. (10). Add penalty to the updated position as in step 4. If the MSE value of the observed signal is smaller than the MSE value of the best vector, then update the best vector and its fitness value. Repeat until all ants finish the tour. step 7 Update the pheromone and calculate the probability of the ant k to move to the ith node on the jth level as in equ. (11) and (12), for all the variables and ants. The probability constructed here will be used in the next iteration. step 8-9 Same as RCGA-AVC procedures. 4.3.2. Fixed geometry For the case of fixed geometry, the locations of detection and secondary source points are fixed. This is done after variable-geometry-mechanism has been carried
136
JOURNAL OF LOW FREQUENCY NOISE, VIBRATION AND ACTIVE CONTROL
S. Julai and M. O. Tokhi out for all the algorithms for both MISO and MIMO cases. Since all the algorithms are stochastic search, the solutions obtained are different from one another, giving more than one possible location for the search points. Therefore, in order to choose the best locations, the levels of attenuation using different algorithms are compared and the searched points giving the maximum mean of the attenuation for the first five dominant vibration modes are used. Here, the optimization process is carried out only on the parameters of the controller so as to minimize the objective function defined in equ. (24) for the MIMO case, or equ. (28) for the MISO case.
Fig. 6.
Predefined region of arrangement for systems components.
5. IMPLEMENTATION AND RESULTS This section presents an assessment of the performances of the MISO- and MIMOAVC systems with RCGA, PSO-SF, and ACO1 design approaches. The random disturbance signal was applied to the plate at t = 0.2 to t = 0.5 sec. For the detection and secondary source points within the predefined range, it was observed that the first five dominant modes of the plate were at 10.35 rad/sec, 33.76 rad/sec, 64.94 rad/sec, 81.88 rad/sec, and 99.37 rad/sec. The results presented in this section are assessed on a comparative basis in terms of spectral density attenuation for both cases of variable-geometry-AVC and fixed-geometry-AVC with RCGA-, PSO-SF-, and ACO1-optimization techniques. 5.1. Simulation results for MISO-AVC 5.1.1. Variable-geometry A simulation environment was developed using m-file in MATLAB software. The simulation was carried out to find the optimal locations of detection (4 variables) and secondary source (2 variables) points together with controller parameters (8 variables). The best (minimum) MSE levels obtained using RCGA-AVC, PSO-SFAVC, and ACO1-AVC are as shown in Fig. 7 and Table II. Since the algorithms are stochastic in nature, the solutions found may vary at each run. Stability of the algorithm was carried by running the program 10 times with the corresponding best parameters for each algorithm. The results tabulated in Table II are based on statistical performance measures, i.e. minimum, maximum, mean, and standard deviation of the objection function. The smallest MSE value was obtained using ACO1-AVC. It is noted that RCGA achieved consistent MSE values over multiple runs, followed by ACO1 and PSO-SF. It is observed from Fig. 7 that the MSE convergence for all algorithms was large for iterations less than 5, and rapid convergence was achieved in the first 10 iterations, where the algorithms tried to locate the optimum region for detection and secondary source points. However, no or minor improvement was made after 30 iterations. As shown in Fig. 8, the computational time for RCGA was significantly lower than PSO-SF and ACO1 mechanisms.
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Active vibration control of flexible plate structures with distributed disturbances Fig. 9 shows the performance results of RCGA-AVC, PSO-SF-AVC, and ACO1AVC. The best controller order that produced the smallest MSE was 2 for all algorithms. All algorithms were run with 20 individuals/particles/ants and maximum generation/iteration of 40. For RCGA-AVC, the specific parameters used were generation gap of 0.80, crossover rate of 0.67, and mutation rate of 0.0714. For ACO1-AVC, the initial parameters used were τ0 0.01, ρ = 0.99, M = 10 and Q = 10. It is noted that all algorithms significantly reduced vibration at the first mode with minor reduction or reinforcements at the remaining modes. The amounts of cancellation/reinforcement with the corresponding best location obtained with the optimization techniques are summarized in Table III. It is noted that ACO1-AVC has achieved the highest mean attenuation followed by RCGA-AVC, and PSO-SFAVC. Therefore, the best locations for detection and secondary source points were found with ACO1-AVC at D1(11, 7), D2(6, 10), and S1(7, 6). These three locations obtained were nearer to the primary source 1 and primary source 2, within the range [6, 11]. The distance between the primary source and the detection point was thus shorter, enabling the sensor to detect most of the dynamic characteristics of the disturbance signal for the controller. The shorter distance between primary and secondary source results in shorter travelling path between cancelling and unwanted signals, thus enhancing the vibration suppression. Thus, in this case, the physical extent of vibration cancellation around the observation point is higher when the detection and secondary source points are located close to the primary source.
Table II MSE value for 10 trials for variable-geometry MISO-AVC. Algorithm
MSE value Minimum
Maximum
Mean
Standard deviation
RCGA
2.93 10
14
6.68 10
14
4.98 10
14
9.26 10
15
PSO-SF
4.18 10
14
7.02 10
13
1.50 10
13
2.02 10
13
ACO1
1.95 10
14
1.44 10
13
5.41 10
14
3.57 10
14
-8
8
x 10
MISO-RCGA-AVC MISO-PSO-SF-AVC MISO-ACO1-AVC
6
-14
MSE
x 10 6
4
4 2
2 20
0 0
Fig. 7.
40 40
Time (min)
MISO-RCGA-AVC MISO-PSO-SF-AVC MISO-ACO1-AVC
100
50
0 0
138
30
20 30 No. of iterations
Convergence graph for variable-geometry MISO-AVC
150
Fig. 8.
10
10
20 30 No. of iterations
40
Computational time for MISO-AVC with RCGA, PSO-SF, and ACO1.
JOURNAL OF LOW FREQUENCY NOISE, VIBRATION AND ACTIVE CONTROL
S. Julai and M. O. Tokhi Table III Spectral attenuation achieved at resonance modes with variable-geometry MISO-AVC. Algorithm
Position
Mode 1
Spectral attenuation (dB) Mode 2 Mode 3 Mode 4
Mode 5
Mean (dB) 5.65
RCGA
D1 (13, 6), D2 (8, 9) , S1 (7, 6)
22.63
2.93
-2.80
4.93
0.55
PSO-SF
D1 (9, 8), D2 (10, 10) S1(7, 9)
24.23
0.87
-4.84
0.28
-1.54
3.80
ACO1
D1 (11, 7), D2 (6, 10) , S1(7, 6)
34.57
4.54
-3.56
2.51
1.05
7.82
Note: Negative value indicates spectral reinforcement.
uncontrolled RCGA-controlled PSO-SF-controlled ACO1-controlled
-6
x 10
1 0.5 0 -0.5
1
2 Time (sec)
3
4
(a) Time-domain at observation point
Fig. 9.
uncontrolled RCGA-controlled PSO-SF-controlled ACO1-controlled
-80 -90 -100 -110 -120
-1 0
-70
Magnitude (dB)
Deflection (m)
1.5
-60
-130 0
50 100 Frequency (rad/sec)
150
(b) Spectral density at observation point
Performance of variable-geometry MISO-AVC.
5.1.2. Fixed-geometry The best location obtained for detection and secondary source points as mentioned in section 5.1.1, defined as fixed-geometry, are used here where the algorithms will only find the controller parameters (8 variables) to minimize the objective function. The best (minimum) MSE levels obtained using RCGA-AVC, PSO-SF-AVC, and ACO1-AVC are as shown in Fig. 10 and Table IV. The simulation was run 10 times for each algorithm to check its stability in searching the controller parameters. From the results presented in Table IV, it is noted that the minimum MSE values was obtained by PSO-SF-AVC. It is noted from the values of maximum, mean, and standard deviation that the algorithms yielded consistent MSE value over multiple runs. The standard deviation values were significantly lower than these in variable-geometry MISO-AVC, indicating that they are clustered closely around the mean value, i.e. the algorithms are stable in finding the controller parameters. The level of vibration reduction achieved at the observation points is demonstrated with the time-domain responses and spectral densities of the system responses before and after cancellation of RCGA-, PSO-SF-, and ACO1-AVC performances, as shown in Fig. 11. The deflection throughout the plate is further shown with the 3-dimensional plots of system response before and after control at t = 0.5 sec. It is noted that significant reduction in plate vibration was achieved. The spectral attenuations achieved are summarized in Table V. All algorithms achieved spectral attenuation at the first, second, third, and fifth modes. At the fourth mode, in contrast, minor reinforcement occurred. The mean spectral attenuations obtained with RCGA-, PSO-SF-, and ACO1-AVC were 8.65 dB, 9.93 dB, and 8.85 dB, respectively. All controllers were stable as revealed by the pole-zero plots in Fig. 12 where all the poles were inside the unit circle. Furthermore, the controlled signal in time domain at both observation points was at consistent level from 0.3 sec to 4 sec. The discrete transfer functions of the controllers at a sampling time of 0.0016 sec thus obtained are
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Active vibration control of flexible plate structures with distributed disturbances
C
C
C
RCGA
−0.199 z + 0.179 0.011z − 0.171 (33) C = (z) = C = z + 0.301z + 0.102 z + 0.129 z − 0.251 11
PSO-SF
ACO1
21
2
2
−0.016 z + 0.091 (34) 0.019 z − 0.244 C = (z) = C = z − 0.107 z + 0.049 z + 0.124 z + 0.233 11
21
2
2
−0.021z − 0.141 (35) 0.046 z − 0.103 C = (z) = C = z + 0.211z + 0.183 z + 0.188 z − 0.214 11
21
2
2
Table IV MSE value for 10 trials for fixed-geometry MISO-AVC. Algorithm
MSE value Minimum
1.90 10
14
PSO-SF ACO1
RCGA
Maximum
Mean
2.11 10
14
1.79 10
14
1.88 10
14
Standard deviation
1.99 10
14
6.09 10
16
2.21 10
14
1.95 10
14
1.23 10
15
2.15 10
14
1.97 10
14
7.73 10
16
Table V Spectral attenuation achieved at resonance modes with fixed-geometry MISO-AVC. Algorithm
Spectral attenuation (dB) Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Mean (dB) 8.65
RCGA
29.13
4.63
8.64
-0.70
1.57
PSO-SF
32.36
4.35
14.81
-2.74
0.89
9.93
ACO1
31.63
4.70
6.38
-0.01
1.57
8.85
Note: Negative value indicates spectral reinforcement.
-13
2
MISO-RCGA-AVC MISO-PSO-SF-AVC MISO-ACO1-AVC
x 10
1.5
-14
MSE
2.2
x 10
2
1
1.8 10
0.5
0
Fig. 10.
140
10
20
20 30 No. of iterations
30
40
40
Convergence graph for fixed-geometry MISO-AVC.
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uncontrolled RCGA-controlled PSO-SF-controlled ACO1-controlled
-70
Magnitude (dB)
Deflection (m)
1.5 1 0.5 0 -0.5
uncontrolled RCGA controlled PSO-SF-controlled ACO1-controlled
-60
-80 -90 -100 -110
0
1
2 Time (sec)
3
-130 0
4
50 100 Frequency (rad/sec)
(a) Time-domain at observation point -7
0
10
150
y (j)
-7
5 0 20
y (j)
5
10
15
x (i)
(d) System response after cancellation for RCGA-AVC
15
20
x (i)
-7
10 5 0 20
10
10
x 10
Deflection (m)
10
5
(c) System response before cancellation
x 10
Deflection (m)
Deflection (m)
5
(b) Spectral density at observation point
x 10
Fig. 12.
10
20
-120
-1
Fig. 11.
-7
x 10
Deflection (m)
-6
x 10
10 5 0 20
20 10 y (j)
5
10
15
20
x (i)
(e) System response after cancellation for PSO-SF-AVC
10 y (j)
5
10
15
20
x (i)
(f) System response after cancellation for ACO1-AVC
Performance of fixed-geometry MISO-AVC.
(a) Pole zero plot C(z)11 for RCGA
(b) Pole zero plot C(z)21 for RCGA
(c) Pole zero plot C(z)11 for PSO-SF
(d) Pole zero plot C(z)21 for PSO-SF
(e) Pole zero plot C(z)11 for ACO1
(f) Pole zero plot C(z)21 for ACO1
Pole-zero plots of the fixed-geometry MISO-AVC controllers.
5.2. Simulation results for MIMO-AVC The control structure in this case comprises two excitation points (P1 and P2), two detection points (D1 and D2), two secondary source points (S1 and S2) and the vibration is observed at two observation points (O1 and O2). For the detection and secondary source points within the predefined range, it was observed that the first five dominant modes of the plate were at 10.35 rad/sec, 33.76 rad/sec, 64.94 rad/sec, 81.88 rad/sec, and 99.37 rad/sec. The same parameters for the algorithms as in section 5.1.1 were used 5.2.1. Variable-geometry The simulation was carried out to find the optimal locations of detection (4 variables) and secondary source (4 variables) points together with controller parameters (16 variables). The best (minimum) MSE levels obtained using RCGA-AVC, PSO-SFAVC, and ACO1-AVC are shown in Fig. 13 and Table VI. The minimum MSE value was obtained using ACO1-AVC. From the maximum, mean, and standard deviation values, it is noted that the algorithms have performed consistently at similar levels. A similar behaviour as with variable-geometry MISO-AVC can be observed where RCGA achieved consistent MSE values over multiple runs, followed by ACO1 and
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Active vibration control of flexible plate structures with distributed disturbances PSO-SF. The MSE values for all algorithms were large number for iterations less than 5, and rapid convergence was made in the first 10 iterations. For most algorithms, no or minor improvement was made after 30 iterations. In terms of the computational times, RCGA has performed faster than ACO1 and PSO-SF. Fig. 14 shows the performance results of RCGA-AVC, PSO-SF-AVC, and ACO1-AVC. The best controller order that produced the smallest MSE was 2 for all algorithms. The results of attenuation/reinforcement achieved with the corresponding best locations of the detection and secondary source points are listed in Table VII. It is observed that ACO1-AVC has achieved the highest mean attenuation followed by RCGA-AVC, and PSO-SF-AVC. As summarized in Table VII, the maximum mean attenuation achieved was achieved using ACO1-AVC with best locations at D1(7, 10), D2(11, 7), S1(6, 7), and S2(8, 8). All the four points were located within the range , nearer to the primary source 1 and primary source 2, which was similar to variable-geometry-MISO-AVC case. It can be concluded that the short distance between the primary source and the detection point enables the sensor to detect most of the dynamic characteristics of the disturbance signal, while short distance between primary and secondary source results in shorter travelling path between cancelling and unwanted signals, thus enhancing the vibration suppression. Thus, in this case, the physical extent of vibration cancellation around the observation point is higher when the detection and secondary source points are located closer to the primary source. Again, the physical extent of vibration cancellation around the observation point is higher when the detection and secondary source points are located closer to the primary source.
Table VI MSE value for 10 trials for variable-geometry MIMO-AVC. MSE value
Algorithm
Minimum
RCGA PSO-SF ACO1
Maximum
Mean
Standard deviation
9.49 10
14
5.41 10 13
1.87 10 13
1.51 10 13
1.02 10
13
1.80 10
12
6.20 10
13
6.62 10 13
6.29 10
14
6.32 10
13
2.57 10
13
1.91 10 13
Table VII Spectral attenuation achieved at resonance modes with variable-geometry MIMO-AVC. Position
D1 (10, 9), D2 (8, 11) S1 (7, 8), S2 (9, 7)
Algorithm (observation point) RCGA (1)
Mode 1 26.19
D1 (11, 8), D2 (10,9) S1 (8, 6), S2 (7, 11) D1 (7, 10), D2 (11, 7) S1 (6, 7), S2 (8, 8)
Spectral attenuation (dB) Mode 2 Mode 3 Mode 4 8.64 -0.57 1.45
Mode 5 -2.41
Mean (dB) 6.66
RCGA (2)
26.14
4.94
3.88
1.62
-3.05
6.71
PSO-SF (1)
23.65
5.26
-9.32
-1.27
-4.45
2.77
PSO-SF (2)
23.52
1.91
-5.12
-3.17
-5.00
2.43
ACO1 (1)
32.13
18.00
-4.47
5.83
-0.47
10.20
ACO1 (2)
32.96
15.69
-1.38
3.20
-0.97
9.90
Note: Negative value indicates spectral reinforcement.
-7
x 10
MIMO-RCGA-AVC MIMO-PSO-SF-AVC MIMO-ACO1-AVC
1
MSE
0.8
-13
x 10
0.6
1.4 1.2 1 0.8 0.6
0.4 0.2 0 0
Fig. 13. 142
20
10
30
20 30 No. of iterations
40
40
Convergence graph for variable-geometry MIMO-AVC.
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-6
uncontrolled RCGA-controlled PSO-SF-controlled ACO1-controlled
x 10
0.5 0 -0.5
0
1
2 Time (sec)
3
-90 -100 -110
-130 0
4
(a) Time-domain at observation point-1 x 10
1.5
50 100 Frequency (rad/sec)
150
(b) Spectral density at observation point-1
uncontrolled RCGA-controlled PSO-SF-controlled ACO1-controlled
-6
-60
1
Magnitude (dB)
Deflection (m)
-80
-120
-1
2
uncontrolled RCGA-controlled PSO-SF-controlled ACO1-controlled
-70
1
Magnitude (dB)
Deflection (m)
1.5
-60
0.5 0 -0.5
uncontrolled RCGA-controlled PSO-SF-controlled ACO1-controlled
-80
-100
-120
-1 -1.5 0
1
2 Time (sec)
3
(c) Time-domain at observation point-2
Fig. 14.
-140 0
4
50 100 Frequency (rad/sec)
150
(d) Spectral density at observation point-2
Performance of variable-geometry MIMO-AVC.
5.2.2. Fixed-geometry The best locations obtained for detection and secondary source points in 5.2.1, defined as fixed-geometry, are used here where the algorithms will only find the controller parameters that minimize the objective function. The convergence graphs thus obtained are shown in Fig. 15 and listed in Table VIII. It is noted that the minimum MSE values was obtained by ACO1-AVC. From the values of maximum, mean, and standard deviation for the algorithms, it is noted that when the locations of all points are fixed, the algorithms yield consistent MSE value over multiple runs. The standard deviation values are significantly lower those that in variablegeometry MIMO-AVC, indicating that they are clustered closely around the mean value, i.e. the algorithms are stable in finding the controller parameters. The system performance results with RCGA-, PSO-SF-, and ACO1-AVC are shown in Fig. 16 and summarized in Table IX. The level of vibration reduction achieved at the observation points is demonstrated with the time-domain responses and spectral densities of the system responses before and after cancellation. The deflection throughout the plate is further shown with the 3-dimensional plots of system response before and after control. Vibration at most of the dominant modes was reduced at both observation points, except a minor reinforcement at the third mode at the observation point 1 with RCGA-AVC. The mean attenuations achieved were 9.19 dB and 9.17 dB, at observation point 1 and point 2, respectively. For PSOSF-AVC, it is noted that there were a few minor reinforcements at both points. The mean spectral attenuations achieved at observation point 1 and observation point 2 were 10.91 dB and 9.63 dB, respectively. For ACO1-AVC, the mean spectral attenuations achieved at observation point 1 and observation point 2 were 10.35 dB and 10.83 dB, respectively. There were few minor reinforcements at both points. The controlled signal in time domain at both observation points has shown stable vibration amplitude from 0.3 sec to 4 sec. The discrete transfer functions of the controllers at a sampling time of 0.0016 sec thus obtained are given below, which are stable as all the poles inside the unit circle, as shown in Fig. 17.
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C
RCGA
−0.792 z − 0.793 0.044 z − 0.434 C = C = z − 0.272 z − 0.379 z − 0.595z − 0.343 (z) = 0.478 z − 0.354 C = −0.839 z + 0.169 C = z − 0.701z − 0.131 z + 0.358 z + 0.996 11
2
12
2
21
2
22
2
−0.301z + 0.099 C = z + 0.268 z + 0.292 (z) = C = −0.301z + 0.299 z + 0.045z − 0.267 11
C
PSO-SF
2
21
C
ACO1
2
C = 12
(36)
−0.301z + 0.256 z + 0.295z − 0.301 2
−0.299 z − 0.298 C = z + 0.207 z + 0.167 22
(37)
2
−0.271z + 0.121 0.059 z − 0.159 C = C = z + 0.065z − 0.024 z + 0.115z − 0.101 (z) = −0.165z − 0.070 C = −0.119 z − 0.014 C = z + 0.216 z − 0.276 z + 0.059 z + 0.167 11
2
12
2
21
2
22
2
(38)
Table VIII MSE value for 10 trials for fixed-geometry MIMO-AVC. MSE value
Algorithm RCGA PSO-SF ACO1
Minimum
Maximum
Mean
6.17 10 14
7.05 10 14
6.68 10 14
Standard deviation
3.30 10 15
6.67 10
14
7.43 10
14
6.97 10
14
2.10 10 14
6.15 10
14
7.12 10
14
6.63 10
14
4.12 10 14
Table IX Spectral attenuation achieved at resonance modes with fixed-geometry MIMO-AVC. Spectral attenuation (dB)
Algorithm (observation point)
Mode 1
Mode 2
Mode 3
RCGA (1)
32.08
12.57
-1.59
2.11
0.79
9.19
RCGA (2)
33.17
9.08
2.04
1.28
0.29
9.17
PSO-SF (1)
37.70
15.41
-3.05
5.23
-0.74
10.91
PSO-SF (2)
38.01
12.36
-0.19
-0.82
-1.21
9.63
ACO1 (1)
35.73
14.61
-5.09
7.19
-0.71
10.35
ACO1 (2)
36.85
12.24
-1.76
8.08
-1.24
10.83
Mode 4
Mode 5
Note: Negative value indicates spectral reinforcement.
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Mean (dB)
S. Julai and M. O. Tokhi
MIMO-RCGA-AVC MIMO-PSO-SF-AVC MIMO-ACO1-AVC
-12
6
x 10
5
MSE
4
-14
x 10 7 6.8 6.6 6.4 6.2
3 2 1
20
0 0
Fig. 15.
10
30
40
20 30 No. of iterations
40
Convergence graph for fixed-geometry MIMO-AVC.
uncontrolled RCGA-controlled PSO-SF-controlled ACO1-controlled
-6
x 10
Magnitude (dB)
Deflection (m)
1.5
uncontrolled RCGA-controlled PSO-SF-controlled ACO1-controlled
-60
1 0.5 0 -0.5
-80
-100
-120
-1 0
1
2 Time (sec)
3
-140 0
4
(a) Time-domain at observation point-1 uncontrolled RCGA-controlled PSO-SF-controlled ACO1-controlled
2
-60
Magnitude (dB)
Deflection (m)
1.5 1 0.5 0 -0.5
-80
-7
x 10
-100
10 5 0
-120
20
-1 0
1
2 Time (s)
3
50 100 Frequency (rad/sec)
150
y (j)
(d) Spectral density at observation point-2
-7
20
5 0 20
10 y (j)
5
10
15
x (i)
(b) System response after cancellation for RCGA-AVC
15
20
x 10
10
Deflection (m)
Deflection (m)
0
10 x (i)
-7
x 10
5
5
(a) System response before cancellation
-7
x 10 10
Fig. 16.
10
-140 0
4
(c) Time-domain at observation point-2
Deflection (m)
150
(b) Spectral density at observation point-1 uncontrolled RCGA-controlled PSO-SF-controlled ACO1-controlled
Deflection (m)
-6
x 10
50 100 Frequency (rad/sec)
10 5 0 20
20 10 y (j)
5
10
15
x (i)
(c) System response after cancellation for PSO-SF-AVC
20 10 y (j)
5
10
15
20
x (i)
(d) System response after cancellation for ACO1-AVC
Performance of fixed-geometry-MIMO-AVC.
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Fig. 17.
(a) Pole-zero plot C(z)11_RCGA
(b) Pole-zero plot C(z)12_RCGA
(c) Pole-zero plot C(z)21_RCGA
(e) Pole-zero plot C(z)11_PSOSF
(f) Pole-zero plot C(z)12_PSOSF
(g) Pole-zero plot C(z)21_PSOSF
(i) Pole-zero plot C(z)11_ACO1
(j) Pole-zero plot C(z)12_ACO1
(k) Pole-zero plot C(z)21_ACO1
Pole-zero plots of the fixed-geometry MIMO-AVC controllers.
5.3. Comparative assessment of system performances From the results presented in the previous sections, ACO1-AVC has successfully achieved the highest mean spectral attenuation both in variable-geometry-MISOand -MIMO-AVC. For fixed-geometry case, PSO-SF-AVC has achieved the highest mean spectral attenuation with MISO-AVC, while ACO1-AVC has performed well with MIMO-AVC. Thus, ACO1 has performed well at finding optimal controller parameters, outperformed RCGA and PSO-SF for suppressing the vibration of the flexible plate. The best location for detection and secondary source points were found to be within the x- and y-section of on the flexible plate. This indicates that the vibration can be controlled better if detection and secondary source(s) points are located nearer to the primary source. As presented in Tables II and VI, the standard deviations of the MSE value for variable-geometry cases were relatively higher than those in fixed-geometry cases shown in Table IV and VIII. In variable-geometry, MISO-AVC case utilized 14 design variables (three sets of (x, y) coordinates for detection and secondary source points and 8 controller parameters) while in MIMO-AVC case, 24 design variables (four sets of (x, y) coordinates for detection and secondary source points and 16 controller parameters) were utilized. In contrast, MISO- and MIMO-AVC fixed geometry cases dealt with fewer design variables, i.e. 8 for MISO- and 16 for MIMO-AVC, resulting in consistent value of MSE and relatively low standard deviation. The large number of design variables has contributed to the high standard deviation in variable-geometry. Also, the effect of the location of detection and secondary source points has significant influence on the performance of the controlled system. In terms of level spectral attenuation achieved as listed in Tables III and V, variable-geometry-AVC systems have shown significant vibration reduction with all the systems at the first dominant mode for all the disturbances. As presented in Tables V and IX, the vibration reduction obtained with fixed-geometry-AVC
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S. Julai and M. O. Tokhi systems at this mode was considerably higher than those in variable-geometry cases. This is followed by minor attenuation or reinforcement at the other modes which has insignificant effect to the overall reduction in vibration level. Considering that most of the vibration energy is in the first dominant mode, attenuation obtained within 22.63 – 34.57 dB with variable-geometry MISO-AVC, 29.13 – 32.36 dB with fixedgeometry MISO-AVC, 23.52 – 32.96 dB with variable-geometry MIMO-AVC, and 32.08 – 38.01 dB with fixed-geometry MIMO-AVC system, at this mode reduced the system vibration considerably. It is noted that the performance of the system with the MIMO controller was significantly superior to MISO controller. The improvement in mean spectral attenuation by utilizing a multiple set of cancelling sources, i.e. MIMO case, for RCGA-AVC, PSO-SF-AVC, and ACO1-AVC, at observation point 1 was 6.24%, 9.87%, and 16.95%. These improvements have shown that the utilization of multiple set of cancelling sources enhances the performance of the system in terms of vibration reduction. 6. CONCLUDING REMARKS In this paper, the development of an AVC strategy with GA, PSO, and ACO techniques with multiple inputs has been presented. The controllers have been designed based on direct optimization of the location of the detector and secondary source, and the controller parameters based on minimizing the MSE level of the error (observed) signal. The approach does not require knowledge of the dynamic characterization of the plant for controller adaptation. The performances of GA, PSO, and ACO based AVC systems have been assessed within the simulation environment of a flexible plate system. It has been demonstrated that all the developed control systems perform successfully in suppressing the vibration of the system. The utilization of multiple set of cancelling sources in MIMO case has been shown to enhance the performance of the system in terms of vibration reduction compared to MISO case. Overall, the results indicate that GA, PSO, and ACO algorithms can be used effectively for optimal placement of system components and optimization of controllers for vibration suppression in flexible structures. ACKNOWLEDGMENTS S. Julai acknowledges the financial support of the Ministry of Higher Education Malaysia and University of Malaya, Kuala Lumpur, Malaysia. REFERENCES [1] Hu Q. (2009) A composite control scheme for attitude maneuvering and elastic mode stabilization of flexible spacecraft with measurable output feedback. Journal of Aerospace Science and Technology, 13 (2-3), 81-91. [2]
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