LiLiandFaouziGhribDepartmentofCivilandEnvironmental

,单击此处编辑母版标题样式,单击此编辑母版文本样式,第二级,第三级,第四级,第五级,*,Li Li and Faouzi Ghrib,Department of Civil and Environmental Engineering,University of Windsor,Modal IdentificationUsing a Genetic and Nelder-Mead Approach,1,Modal identification means the determination of the modal parameters of structures from vibration measurements. The modal parameters are natural frequencies, mode shapes and damping ratios of each mode.,Mo,dal parameters are important because they describe the inherent dynamic properties of structure. They are the,eigenvalues,and eigenvectors of dynamic equations.,What is Modal Identification?,2,These modal parameters can serve as input to Finite Element model updating (such as: the minimum rank perturbation and sensitivity based model update), and proceed to subsequent steps like damage identification and health monitoring.,Why Modal Identification?,3,Equation solving approaches,Minimization approaches (have strong relation with optimization problem),Correlation approaches,Subspace approaches (based on the state space innovation model),According to Ljung (1999), a simple classification of identification techniques is:,4,Parameter identification as an optimization approach,However, among all these approaches, the most widely used in civil engineering modal identification are the correlation and subspace approaches, not the seemly straightforward minimization (optimization) approaches.,5,Parameter identification as an optimization approach,The Prediction Error Method (PEM),The Bayesian maximum a posteriori (MAP) estimation,Direct time domain least-mean square approach,6,Parameter identification as an optimization approach,1. The Prediction Error Method (PEM):,This method uses an ARMAX model structure (Autoregressive Moving Average with exogenous excitation) of the form.,7,Parameter identification as an optimization approach,1. The Prediction Error Method (PEM):,The parameter estimation of the ARMAX model structure is by solving a minimization problem of the residual function.,The minimization is a nonlinear problem due to the nonlinear dependency of the model residual upon the model parameter vector.,8,Parameter identification as an optimization approach,2. The Bayesian maximum a posteriori (MAP) estimation:,In the Bayesian approach, the modal parameters are assumed to be random variables whose particular realization we must estimate.,In this approach, we attempt to minimize the Bayesian mean-square error (BMSE) defined as,9,Parameter identification as an optimization approach,2. The Bayesian maximum a posteriori (MAP) estimation:,The estimator that minimizes the BMSE is difficult to get, thus a suboptimal option is adopted that maximizes the conditional mean of the parameter,given observations Y,10,Time-domain nonlinear least square problem,The equation of motion for a damped dynamic system using finite element method can be formulated as:,Here M,C,K ,R,nn,denote the mass, damping and stiffness matrix, whereas,g(t,) denotes the load vector.,11,Time-domain nonlinear least square problem,The modal matrix = ,k, is solution of K =,M,with respect to the,eigenvalues,. The modal transformation ,T,. transforms the mass and the stiffness matrix into diagonal matrices,12,Time-domain nonlinear least square problem,the objective function to be minimized is:,Where,t,j,is the sampling time, and,is the vector of parameters to be identified. M is the number of observations, usually this is a very large number.,13,Time-domain nonlinear least square problem,the objective function can be written as:,The vector R=(r,1, ,r,M,) is called the residual.,We see that:,The necessary conditions for optimality require that,14,Parameter identification as an optimization approach,All these minimizations are nonlinear problems due to the nonlinear dependency of the model residual upon the modal parameter vector.,15,Non-convex and non-differentiable objective functions in high-dimensional spaces.,The analytical gradient and Hessian are not available.,Many local minimums.,The measured outputs are always contaminated by noise!,Thus make it to possess many local minimums.,Parameter identification as an optimization approach,16,Due to these difficulties, global convergence methods must be employed.,Two methods has been hybridized to solve the optimization problem, the,genetic algorithm,and the,Nelder-Mead,method.,Nelder-Mead simplex is a kind of,direct,method; that means, it does not require the evaluation of derivatives.,Solving the NLS problem,17,In modal identification application, the Nelder-Mead is much better than Levenberg-Marquardt when the initial guess is not very close to the exact solution, and the noise level is high.,This is partly due to the difficulty in computing gradients; the Levenberg-Marquardt requires the gradients, but Nelder-Mead does not.,And the calculation of gradient in the modal identification is not easy.,Solving the NLS problem,18,Using fourth-order,Runge-Kutta,method to integrate the systems response under a impulsive load. The simulated system is treated as the “true system”, and the simulated acceleration outputs are taken as the measurements. Noise is added to simulated outputs to mimic the measurement noise which is unavoidable in real engineering.,In most civil engineering structures, acceleration is the only response that can be measured well. Accelerometers are more accurate and cheaper than other measurement-meters.,Example-1: a SDOF spring-mass system,19,The exact value of stiffness to the “true system” is 10, and damping ratio is 0.05. The mass is taken as 1, there is no loss of generality here, since we can always transform the mass of a system to be unity using modal transform.,There are only two parameters to be estimated, the stiffness and the damping ratio. Below is the identification results using,Nelder,-Mead method.,The iteration of,Nelder,-Mead terminates when the diameter of the simplex is smaller than the tolerance. The tolerance is set to be 0.001.,Example-1: a SDOF spring-mass system,20,Initial guess= 8 9 11; 0.02 0.03 0.06,tol = 0.001, 50% noise,21,Example-1: a SDOF spring-mass system,22,Example-1: a SDOF spring-mass system,Observations:,The initial guess is crucial in the performance of nonlinear LS identification.,With a good initial guess, this method still performs well even under a large portion of noise.,Damping are more susceptible to noise. If noise is large, damping estimates are poor, but stiffness estimate can be good.,Initial guess of damping is more crucial if the noise level is high.,23,Example-2: a MDOF truss structure,An electrical transmission tower,24,Example-2: a MDOF truss structure,An electrical transmission tower,The exact frequency and damping of first mode,:,Frequency =,17.366, damping ratio =,0.03,a bad initial guess: 50 51 52 ; 0.03 0.04 0.03,Identified:,237.3501, 1.8068,a good initial guess: 15 17 18 ; 0.03 0.04 0.03,Identified:,17.3655, 0.0300,25,Example-2: a MDOF truss structure,An electrical transmission tower,The convergence to a local minimum and the global minimum,26,we can identify several several modes at the same time by using a larger simplex.,Provide good initial guesses:,16.0000 17.0000 18.0000 15.0000 15.5000,50.0000 51.0000 52.0000 51.5000 53.0000,0.0300 0.0400 0.0300 0.0400 0.0340,0.0500 0.0450 0.0510 0.0440 0.0460,Example-2: a MDOF truss structure,An electrical transmission tower,27,The identified first two modal properties:,17.3490 17.3490 17.3490 17.3489 17.3490,51.7733 51.7732 51.7731 51.7730 51.7730,0.0303 0.0303 0.0303 0.0303 0.0303,0.5856 0.5854 0.5854 0.5855 0.5855,The exact solutions:,17.366,0.03,50.213, 0.05,Example-2: a MDOF truss structure,An electrical transmission tower,28,How can we get the good,Initial guess!,Initial guess!,Initial guess!,Now the goal is:,29,Use a “global method” to localize a promising area likely to contain a global minimum; it is necessary to well explore the whole search domain.,When a promising area is detected, a “local convergence method” must be used to exploit this area and obtain the optimum as accurately and quickly as possible.,Problem,the global optimization of multi-minima functions,30,The ideal of combine global method and local convergence method is not new.,Jer,-Nan,Juang,et al proposed an OKID-LS approach. Where they start nonlinear LS iterations from OKID (observer/,Kalman,identification) algorithm results. OKID is very effective in practice and it is one state-of-the-art method in system identification of modal structures; however it gives poor, usually overestimate or underestimate damping ratio in lightly damped structures.,Using LS as post-processor of OKID,Juang,got good damping results.,Problem,the global optimization of multi-minima functions,31,Damping is always the difficult part in identification of modal properties, and we see the nonlinear Least-Square provides one promising approach to fine-tuning it.,LS- to optimize the estimate of damping,32,GA is a paradigm that mimic the Darwinian theory of natural selection.,As natural selection works solely,By and for the good of each being,All corporeal and mental endowments,Will tend to progress toward perfection.,Charles Darwin, Origin of Species,the global optimization of multi-minima functions, using Genetic Algorithm,33,Basic steps of genetic algorithm (Davis, 1991),Step1: establish a base population of chromosomes.,Step2: determine the fitness value of each chromosome.,Step3: create new chromosomes by mating the current generation. (reproduction, crossover, mutation),Step4: delete old members of the population.,Goto step2, continue until the predetermined condition is achieved.,34,GA is efficient to explore a wide search space and detect a promising valley, it is a self-start method, no need for good initial guess, it is also robust against noise; but it is slow in fine-tuning, takes too much time to search for the bottom of this valley.,Slow convergence of GAs before providing an accurate solution is a well-known drawback, closely related to their lack of exploiting any local information.,the global optimization of multi-minima functions, using Genetic Algorithm,35,Various combinations of GA and some hill-climbing algorithm have been proposed in the literature.,the global optimization of multi-minima functions, using Genetic Algorithm and a local-convergence algorithm,36,Assume parameters (chromosomes) - simulate response,- calculate the fitness.,Binary encoding of the modal parameters (number of chromosome bits controls the resolution of identified parameters).,The fitness function is simply the inverse of the least mean square error of output accelerations.,Genetic algorithm,For modal identifications,37,Two kinds of combinations:,Sequential combination and nested combinations.,For,nested combinations, the GA loop forms the outer loop, and the LS forms an inner loop; the LS modify the offspring to seek better solutions with a specified number of iterations.,For,sequential combinations, simply use LS to do fine-tuning after GA.,the global optimization of multi-minima functions, using Genetic Algorithm and a local-convergence algorithm,38,Example-1: a SDOF spring-mass system,Use Genetic Algorithm to solve this problem:,The initial population is randomly distributed between a pre-specified lower bound and upper bound of the parameters to be identified.,The range for the initial population of k is chosen as 1100,The range for the initial population of c is chosen as 0.010.20,They are very wide ranges, we can safely assume the true values must be within this range.,A 20% level Gaussian white noise is added to the measurement.,39,Use Genetic Algorithm to solve example - 1,Evolution parameters:,population size = 50,crossover rate = 0.8,mutation rate = 0.15,after 100 generations, the b,est member:,k(1) = 10.306 c(2) = 0.067 fitness = 298.011,k(1) = 10.306 c(2) = 0.051 fitness = 299.078,k(1) = 10.306 c(2) = 0.049 fitness = 299.096,Exact values (k=10, c=0.05),As we can see, this is a very good initial guess for Nelder-Mead to start with.,40,Solving modal identification problems as a nonlinear Least Square optimization problem is a promising direction.,Global convergent methods, such as the GA algorithm, can be used to search an good initial guess for local convergent methods to do fine-tuning.,With good initial iterations, the,Nelder,-Mead direct method works well in modal identification, at least for simple structural models;,Even the damping ratios can be accurately estimated. (this is especially remarkable and useful),Conclusions,41,The competence of minimization modal identification approaches rely on the advance of optimization techniques.,Conclusions,42,Rachid Chelouah, Patrick Siarry,Genetic and NelderMead algorithms hybridized for a more accurate global optimization of continuous multiminima functions,European Journal of Operational Research, 148(2003) 335348.,Z. Michalewicz,Genetic Algorithms + Data Structures = Evolution Programs, Springer-Verlag, Heidelberg, 1996.,C.G. Koh, Y.F. Chen, C.Y. Liaw,A hybrid computational strategy for identification of structural parameters,computers and structures, 81 (2003) 107117.,Jer-Nan Juang,Optimized system identification, research report, NASA/TM-1999-209711,Joanna Iwaniec, Tadeusz Uhl, the application of the nonlinear least squares frequency domain method to estimation of the modal model parameters, 2002,References,43,Thank you！,44,