Swarm Control Designs Applied to a Micro-Electro-Mechanical Gyroscope System (MEMS) Fábio Roberto Chavarette1,3, José Manoel Balthazar2, Ivan Rizzo Guilherme2, and Orlando Saraiva do Nascimento Junior3 1
Faculty of Engineering, UNESP – Univ Estadual Paulista, DM, Avenida Brasil, 56, 15385-000, Ilha Solteira, SP, Brazil 2 Geoscience and Exact Science Institute, UNESP – Univ Estadual Paulista, DEMAC, PO BOX 178, 13500-230, Rio Claro, SP, Brazil 3 Ometto Herminio University Center at Araras – UNIARARAS, Engineering Center, Av. Dr. Maximiliano Baruto, 500, Jd. Universitário, 13607-339, Araras, SP, Brazil
[email protected], {jmbaltha,ivan}@rc.unesp.br,
[email protected]
Abstract. This paper analyzes the non-linear dynamics of a MEMS Gyroscope system, modeled with a proof mass constrained to move in a plane with two resonant modes, which are nominally orthogonal. The two modes are ideally coupled only by the rotation of the gyro about the plane’s normal vector. We demonstrated that this model has an unstable behavior. Control problems consist of attempts to stabilize a system to an equilibrium point, a periodic orbit, or more general, about a given reference trajectory. We also developed a particle swarm optimization technique for reducing the oscillatory movement of the nonlinear system to a periodic orbit. Keywords: Particle Swarm Optimization, MEMS Gyroscope, Evolutionary Algorithms.
1 Introduction The field of micro machining is forcing a profound redefinition of the nature and attributes of electronic devices. The technology of micro electro mechanical systems (MEMS) has found numerous applications in recent years, for example the electromechanically filters, biological and chemical sensing, force sensing and scanning probe microscopes [1-4]. This technology allows motion to be incorporated into the function of micro scale devices. However, the design of such mechanical systems may be quite challenging due to nonlinear effects that may strongly affect the dynamics. Microscopic gyroscopes [5, 6] are helping enable an emerging technology called electronic stability control. The resulting system helps prevent accidents by automatically activating brakes on out-of-control vehicles. The technology may be particularly useful for vehicles with a higher center of gravity, which makes them prone to rolling. N. García-Pedrajas et al. (Eds.): IEA/AIE 2010, Part II, LNAI 6097, pp. 308–317, 2010. © Springer-Verlag Berlin Heidelberg 2010
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Electronic stability control is available in luxury vehicles, but sensors made from quartz were too expensive for widespread installation. Innovations in MEMS gyroscope technology make these systems more affordable. Control problems consist of attempts to stabilize an unstable system to equilibrium point, a periodic orbit, or more general, about a given reference trajectory. In the last years, a significant interest in control of the nonlinear systems, exhibiting unstable behavior, has been observed and many of the techniques discussed in the literature [7-9]. Among strategies of control with feedback the most popular is OGY (Ott-GrebogiYork) method [7]. This method uses the Poincaré map of the system. Recently, a methodology, based on the application of the Lyapunov-Floquet transformation, was proposed by Sinha et al. [8] in order to solve this kind of problem. This method allows directing the chaotic motion to any desired periodic orbit or to a fixed point. It is based on linearization of the equations, which described the error between the actual and desired trajectories. Another one technique was proposed by Rafikov and Balthazar in [9], where the Dynamic Programming was used to solve the formulated optimal control problems. Several techniques that can be applied to a wide range of problems. Different from numerical approach, mentioned above, there are other algorithm based approach. In this sense, here we proposed the algorithm based approach, that used particle swarm optimization (PSO) algorithms. The PSO algorithms is a population based stochastic optimization technique developed by Kennedy and Eberhart in 1995 [10]. Using PSO algorithms allows directing the chaotic motion to any desired periodic orbit or to a fixed point. In this work, we proposed and develop a PSO based optimization algorithms for control the unstable movement of MEMS gyroscope. The paper is outlined as follows. In Section are showed the concepts related with the nonlinear model to the MEMS gyroscope. In Section 3, is shown the application of the Particle Swarm Optimization algorithm in the MEM gyroscope. In Section 4, the proposed control swarm approach are presented. In Section 5, we do some concluding remarks of this work. In section 6, we list the main bibliographic references used.
2 MEMS Gyroscope Model The technology of micro electro mechanical systems (MEMS) has found numerous applications in recent years, for example, the MEMS gyroscope ( Fig. 1). Here, we consider a mechanical model and the derivation of governing equations done by [5] for the MEMS gyroscope, commonly function on the coupling of two linear resonant modes via the Coriolis force. The micro gyroscope device consists itself of a perforated proof mass constrained to move in the plane by a suspension of micro beams. It is forced along one axis, the so-called drive axis, by a set of non-interdigitated comb drives, and its motion along the other axis, is detected by a set of parallel plate capacitors. The governing equations of motion of MEMS gyroscope were obtained by [5] and they are:
[
] [
]
mx + cx + k11 + r1V a2 (1 + cos(2 wt )) x + k 31 + r3V a2 (1 + cos(2 wt )) x 3 − 2Ωy = 0 my + cy + k12 y + k 32 y + 2Ωx = 0 3
(1)
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Fig. 1. Micrograph of the micro gyroscope [5]
And by using the non-dimensionless variables
rV 2 k L2 ⎫ y x m c , qs = , τ = t , εζ = , ελ1 = 1 a , εν 3 = 31 ⎪ L L k11 ⎪ k11 k11 mk11 ⎬ r3 L2Va2 k12 k 32 L2 m ⎪ , εγ = 2Ω ελ3 = , εδ = − 1, εξ = ⎪ k11 k 11 k11 k11 ⎭ qd =
(2)
We will obtain:
qd′′ + 2εζq ′d + [1 + ελ1 (1 + cos(2 wt ))]q d +
[εv 3 + ελ3 (1 + cos(2wt ))]qd3 − εγqs′ = 0 q′s′ + 2εζq′s + (1 + εδ )q s + εξq s3 + εγqd′ = 0
(3)
By applying the method of averaging in (3) and rewriting the equations of the dynamical system, in state form, the governing equations may be written as being[5]: x1 =
ε 8
x2 = − x3 = x4 =
[4 x2γ cos(x3 − x4 ) + x1 (−8ζ + (2λ1 + λ3 x12 ) sin(2 x1 ))], ε 2
ε
8 x1
ε 8 x2
[x1γ cos(x3 − x4 ) + 2ζx2 ] [−4 x2γsin(x3 − x4 ) + x1 ( 4λ1 − 8σ + 3(v3 + λ3 ) x12 + 2(λ1 + λ3 x12 ) cos(2 x3 ))],
(4)
[ −4 x1γsin(x3 − x4 ) + x2 (4δ − 8σ + 3ξx22 )]
Here the parameter x1 is the amplitude of oscillation the drive axis, and x2 is the amplitude of oscillation along the sensing axis. The variables x3 and x4 are the phases of oscillation for the two axes.
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Fig. 2. Dynamical behavior of the time history: x1 and x2
Fig. 3. Phase Portrait
In the Figure 2-5 is showed the dynamics behavior of the adopted dynamics model, by using numerical values, for the chosen parameters ε=0.001; λ1=1; λ3=2; ζ=0.1; γ=56; ξ=0.1; v3=0.01; δ=-0.01; σ=δ/2; x1=9; x2=9; x3=9 and x4=9. In the Figure 2 is showed the dynamics behavior of time history for the x1 and x2. In the Figure 3 shown the phase portrait for x1 and x2. In the Figure 4 shows the diagram of the stability for x1 (with the region’s control applied it is illustrate).
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Fig. 4. Stability Diagram for x1 with the region’s control applied
3 Swarm Control Design The particle swarm optimization (PSO), this technique is a population based stochastic optimization technique developed by Kennedy and Eberhart in 1995 [10]. A particle swarm optimization algorithm consists of a number of individuals refining their knowledge of the given search space. In each iteration, the particle swarm optimization algorithm refines its search by attracting the particles to positions with good solutions, considering the best solution until the moment and the best solution of the iteration. The particle swarm optimization technique has ever since turned out to be a competitor in the field of numerical optimization. The PSO approach to nonlinear and control has been observed and discussed in the literature [11-13]. Here, we propose a method for control of unstable systems using the Particle Swarm Optimization with optimization techniques. The method, is used for control the unstable movement of MEMS Gyroscope to stabilize the system to period orbit. The proposed method formulates the nonlinear system identification as an optimization problem in parameter space and then particle swarm optimization are used in the optimization process to find the estimation values of the parameters. 3.1 Particle Swam Optimization The Particle Swarm Optimization (PSO) algorithm, introduced by Kennedy and Eberhart [10], is a computational simulation of social and biological inspired algorithm. PSO consists of a algorithm with low computational cost and information sharing innate to the social behavior of the composing individuals. These individuals, also called particles, flow through the multidimensional search space looking for feasible
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solutions of the problem. The position of each particle in this search space represents a possible solution whose feasibility is evaluated using an objective function. The PSO algorithms refines in each iteration, its search by attracting the particles G to positions with good solutions, using the best solution ( p i ) found by the particle in G
last iteration and the best solution found so far considering all the particles ( pg ).
G
In each iteration, a particle i having position xi have its velocity following way:
G vi updated in the
G G G G G G G G vi = Χ (wvi + ϕ 1i ( pi − xi ) + ϕ 2i ( p g − xi ))
(5)
where X is know as the constriction coefficient described in [14], w is the inertia G G weight, pi is best solution found by the particle in last iteration and the p g best
G
G
solution found so far considering all the particles, and ϕ 1 and ϕ 2 are random values different for each particle and for each dimension. The position of each particle is updated during the execution of iteration. This is done by adding the velocity vector to the 1 position vector, i.e.,
G G G xi = xi + vi
(6)
Setting for the velocity parameters determine the performance of the particle swarm optimization to a large extent. This process is repeated until the desired result is obtained or a certain number of iterations is reached or even if the solution possibility is discarded.
Fig. 5. PSO-Controlled and non-controlled time history x1
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Fig. 6. PSO - Controlled and non-controlled time history x2
Fig. 7. Phase portrait: PSO controlled and non-controlled
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4 Control Swarm Approach The proposed algorithm, showed above, formulates the nonlinear system identification as an optimization problem in parameter space, and then adaptive particle swarm optimizations are used in the optimization process to find the estimation values of the parameters. The algorithm is used for control the behavior unstable of the nonlinear dynamics model (4). The goal of this control synthesis is find the estimation values of the parameters, to drive the orbit of the system to a periodic orbit. We apply the Particle Swarm Optimization algorithm, presented in earlier section for the MEMS Gyroscope (4), to reduce the unstable behavior of this nonlinear system to a period orbit. The Fig. 5, 6 and 7 showed the behavior controlled and uncontrolled of the system (4). In comparing, non-controlled system (see Fig. 2 and 3) with of numerical results of PSO (Fig. 5-7) we can verify that control orbit generated by PSO approach has small diameter. Algorithm of the Particle Swarm Optimization Create and initialize an nx-dimensional swarm, S, through the system of equations (4) and shown in the projection of the phase space (figure 3). S.xi i is used to denote the position of particle i in swarm S. S.yi i is used to denote the best position of particle i in swarm S. S. yˆ , is used to denote the global best position of particle i in swarm S. repeat for each particle i = 1,….,S.ns do // set the personal best position if f(S.xi) < f(S.yi) then S.yi = S.xi; end // set the global best position if
ˆ
f (S.yi) < f(S. y ) then
ˆ
S. y = S.yi; end end for each particle i=1,…,S.ns do update the velocity using equation (5); update the position using equation (6); end until stopping condition is true;
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5 Conclusion In this work, a dynamics of the MEMS gyroscope, proposed by [5, 6] it is investigated. We applied the particle swarm optimization technique applied to control MEMS Gyroscope. This control allows reduction of the oscillatory movement of the system to a desired period orbit. In comparing of numerical results of PSO with the non controlled system (Fig. 2 and 3) we can verify that control orbit generated by PSO approach (Fig. 5-7) has small diameter. The particle swarm optimization technique presents a computational algorithm motivated by a social analogy. The algorithm control allowed reducing the oscillatory movement of the nonlinear systems to a period orbit. The Fig 5-7, illustrate the effectiveness of the control algorithm to these problem.
Acknowledgments The first author thanks all the support of the Fundação Hermino Ometto and program of postdoctoral from the State University of São Paulo at Rio Claro. The second author thanks Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) and Conselho Nacional de Pesquisas (CNPq) for a financial supports.
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