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Parameter identification problem description for three van der Pol-Duffing oscillators: (a-c) single-well, (d-f) double-well, and (g-i) double-hump. (a,d,g): experimental data used for parameter identification (x 1e (t)); (b,e,h): shape of objective function over β (J(β)) for four values of α and with µ = 0.1; (c,f,i): basins of attraction with µ = 0.1. In panels (c,f,i), initial parameter guesses in the dark-red regions lead to local minima when a simple gradient-based optimizer is used; points in the light-green regions lead to the global minimum, indicated with a "target" symbol.
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Dynamic models of physical systems often contain parameters that must be estimated from experimental data. In this work, we consider the identification of parameters in nonlinear mechanical systems given noisy measurements of only some states. The resulting nonlinear optimization problem can be solved efficiently with a gradient-based optimizer, bu...
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... generate synthetic experimental data by numerically integrating Equation (7) for 20 s [22] using a fourth-order Runge-Kutta method and storing only x 1e (t), as shown in Figure 1a. The shape of the objective function is shown in Figure 1b as a function of β, for four values of α and with µ = 0.1. ...
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... generate synthetic experimental data by numerically integrating Equation (7) for 20 s [22] using a fourth-order Runge-Kutta method and storing only x 1e (t), as shown in Figure 1a. The shape of the objective function is shown in Figure 1b as a function of β, for four values of α and with µ = 0.1. There are clearly many peaks and valleys in J(β) and, thus, there are many local minima in the objective function J( ˆ p). ...
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... are clearly many peaks and valleys in J(β) and, thus, there are many local minima in the objective function J( ˆ p). We show the basins of attraction in Figure 1c and confirm that a simple gradient-based optimizer is very likely to converge to a local minimum. In panels (c,f,i), initial parameter guesses in the dark-red regions lead to local minima when a simple gradient-based optimizer is used; points in the light-green regions lead to the global minimum, indicated with a "target" symbol. ...
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... now repeat the process for double-well and double-hump VDPD oscillators; the experimental data, objective function shape, and basins of attraction for these systems are shown in Figure 1d-i. The double-hump oscillator has a highly unstable response, so we use a simulation duration of 2.5 s in this case. ...
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... 0.1581] (maximum relative error of 58%) and [1.4754, −0.4767, 0.0975] (maximum relative error of 5%), respectively. Notice the relatively small basin of attraction in Figure 1i, which suggests that the parameter identification problem is relatively challenging in the double-hump case. Even when the GA and PSO strategies are successful, they are substantially more computationally expensive than the proposed PD controller approach. ...
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... of the identified parameters differ somewhat from the corresponding experimental values. However, when the identified parameters are used in the original mathematical model (Equations (18) and (19)), the response is in close agreement with the "experimental" response ( Figure 10). Thus, the identified parameter values nevertheless produce an accurate overall mathematical model. ...
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... the identified parameter values nevertheless produce an accurate overall mathematical model. We again compare the performance of the PD controller with that of GA and PSO strategies [36][37][38], and we again observe that the PD controller obtains a lower objective function value in fewer function evaluations ( Figure 11). The comparison between the PD controller and stochastic optimization strategies is summarized in Table 9. ...
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... Usually, for each specific identification problem (see examples of modeling pollutants moving in the river corridor [4], parameter identification in nonlinear mechanical systems [5], identification of conductivity coefficient in heat equation [6][7][8]), a separate special study is carried out, including goal setting, mathematical formalization of the problem, its study, creating a numerical model, preparing a computer program, solving a numerical problem and studying the results, including error estimation, etc. ...
This work is aimed at numerical studies of inverse problems of experiment processing (identification of unknown parameters of mathematical models from experimental data) based on the balanced identification technology. Such problems are inverse in their nature and often turn out to be ill-posed. To solve them, various regularization methods are used, which differ in regularizing additions and methods for choosing the values of the regularization parameters. Balanced identification technology uses the cross-validation root-mean-square error to select the values of the regularization parameters. Its minimization leads to an optimally balanced solution, and the obtained value is used as a quantitative criterion for the correspondence of the model and the regularization method to the data. The approach is illustrated by the problem of identifying the heat-conduction coefficient on temperature. A mixed one-dimensional nonlinear heat conduction problem was chosen as a model. The one-dimensional problem was chosen based on the convenience of the graphical presentation of the results. The experimental data are synthetic data obtained on the basis of a known exact solution with added random errors. In total, nine problems (some original) were considered, differing in data sets and criteria for choosing solutions. This is the first time such a comprehensive study with error analysis has been carried out. Various estimates of the modeling errors are given and show a good agreement with the characteristics of the synthetic data errors. The effectiveness of the technology is confirmed by comparing numerical solutions with exact ones.
... Focusing on frequency-variant dynamics, the curve fitting method can be applied in the low-and high-frequency bands using two different sets of parameters [16,17]. Regardless of whether the mathematical model is linear or nonlinear, curve fitting can achieve good agreement with the experimental results [18]. ...
Active hydraulic mounts (AHMs) provide an effective solution for refining ride comfort noise and vibration in passenger cars. AHMs with inertia tracks, decoupler membranes, and oscillating coil actuators (AHM-IT-DM-OCAs) have been extensively studied owing to their compact structures and strong damping characteristics in the low-frequency band. This study focuses on the full parameter identification based on the distinct features of external dynamics, which can be used to obtain an accurate and reliable estimate of the transfer functions required for active control algorithms. A lumped parameter model was established for the AHM-IT-DM-OCAs, and the analytical frequency bands were defined by the two resonance frequencies of the fluid channel and actuator mover. Methods for nonlinear model simplification were proposed in different bands and verified theoretically. Based on the simplified models, the distinct features of the active and passive dynamics are successively revealed, which include three resonances and seven horizontal segments. Subsequently, a series of experimental studies on the distinct features were carried out, which agreed well with the theoretical results. However, owing to the limitations of the test equipment and fixture modalities, only the distinct features of one fixed point, two resonance peaks, and three horizontal segments can be used for parameter identification. Based on the validated distinct features, a procedure for full parameter identification is proposed, and all six key parameters are identified. The obtained results showed good consistency and rationality, indicating that this approach can be used for the transfer function estimation of the primary and secondary paths of the AHM-IT-DM-OCAs.
... Artificial neural networks and adaptive neuro-fuzzy inference systems are the two techniques utilized in [26] to identify the unknown parameters in dynamic systems. Researchers in [27] utilized a morphing parameter and a PID controller to convert a parameter identification problem into a convex optimization problem, which can be solved using a gradient-based optimization algorithm. The research work presented in [28] introduces a variational method to identify the parameters in a 2D diffusion equation, subjected to boundary conditions of the Neumann type. ...
... In order to ensure that the dynamic equation, derived in Section III, is correct, MATLAB Simulink/Simscape environment is utilized. To do so, the mechanical system of the robot is simulated in Simscape, as depicted in Fig. 3; moreover, the dynamic equations of the robot, (27), is represented in the state-space and simulated in the Simulink environment. Then, equal random white noise torques are given to both these systems as an input, and the output joint angles are compared afterward. ...
... The outcomes from this section verify the dynamic model, derived in Section III, both qualitatively and quantitatively. (27) are going to be identified in this section, namely, M (mass of the EE) and F v (frictional coefficient of the winches). M is needed to be identified, because it is subject to change by adding or removing an object to or from the EE, such as a camera to be held by the EE. ...
... Optimization algorithms, least squares, time series, statistical methods, spectral analysis, Volterra series, wavelets, and orthogonal functions have been used for development of parametric identification techniques [1][2][3]. In [4], optimization techniques, combined with classic control theory, have been introduced for system parameters' identification, and a special application for parameters' identification for active vibration absorption schemes is reported in [5] where the offline modal analysis are implemented in the presence of noise. A model for linear nonviscous damping and a time-domain method for the identification of this parameter is proposed in [6]. ...
A novel algebraic scheme for parameters’ identification of a class of nonlinear vibrating mechanical systems is introduced. A nonlinearity index based on the Hilbert transformation is applied as an effective criterion to determine whether the system is dominantly linear or nonlinear for a specific operating condition. The online algebraic identification is then performed to compute parameters of mass and damping, as well as linear and nonlinear stiffness. The proposed algebraic parametric identification techniques are based on operational calculus of Mikusiński and differential algebra. In addition, we propose the combination of the introduced algebraic approach with signals approximation via orthogonal functions to get a suitable technique to be applied in embedded systems, as a digital signals’ processing routine based on matrix operations. A satisfactory dynamic performance of the proposed approach is proved and validated by experimental case studies to estimate significant parameters on the mechanical systems. The presented online identification approach can be extended to estimate parameters for a wide class of nonlinear oscillating electric systems that can be mathematically modelled by the Duffing equation.
... Particle swarm optimization is the best algorithm to tune the parameters of control techniques [29,[32][33][34]. Particle swarm optimization is optimizing the multiobjective function against the different values of the tuning parameters. ...
This research work is focused on the nonlinear modeling and control of a hydrostatic thrust bearing. In the proposed work, a mathematical model is formulated for a hydrostatic thrust bearing system that includes the effects of uncertainties, unmodelled dynamics, and nonlinearities. Depending on the type of inputs, the mathematical model is divided into three subsystems. Each subsystem has the same output, i.e., fluid film thickness with different types of input, i.e.,
viscosity, supply pressure, and recess pressure. An extended state observer is proposed to estimate the unavailable states. A backstepping control technique is presented to achieve the desired tracking performance and stabilize the closed-loop dynamics. The proposed control technique is based on the Lyapunov stability theorem. Moreover, particle swarm optimization is used to search for the best tuning parameters for the backstepping controller and extended state observer. The effectiveness of the proposed method is verified using numerical simulations
... Particle swarm optimization is the best algorithm to tune the parameters of control techniques [29,[32][33][34]. Particle swarm optimization is optimizing the multiobjective function against the different values of the tuning parameters. ...
This research work is focused on the nonlinear modeling and control of a hydrostatic thrust bearing. In the proposed work, a mathematical model is formulated for hydrostatic thrust bearing system which includes the effects of uncertainties, un modelled dynamics and nonlinearities. Depending on the type of inputs, the mathematical model is divided into three subsystems. Each subsystem has the same output i.e. fluid film thickness with different type of inputs i.e. viscosity, supply pressure and recess pressure. An extended state observer is proposed to estimate the unavailable states. A backstepping control technique is presented to achieve the desired tracking performance and to stabilize the closed loop dynamics. The proposed control technique is based on the Lyapunov stability theorem. Moreover particle swarm optimization is used to search the best tuning parameters for the backstepping controller and extended state observer. The effectiveness of the proposed method is verified using numerical simulations
... The tuning of parameters of controller with the help of swarm optimization is gaining popularity in the field of automation applications [37,38]. In order to tune the parameters of PID surface-based sliding mode controllers, a six-dimensional vector is introduced for k , k , k , λ, φ, k . ...
Research in the field of tribo-mechatronics has been gaining popularity in recent decades. The objective of the current research is to improve static/dynamics characteristics of hydrostatic bearings. Hydrostatic bearings always work in harsh environmental conditions that effect their performance, and which may even result in their failure. The current research proposes a mathematical model-based system for hydrostatic bearings that helps to improve its static/dynamic characteristics under varying conditions of performance-influencing variables such as temperature, spindle speed, external load, and clearance gap. To achieve these objectives, the capillary restrictors are replaced with servo valves, and a mathematical model is developed along with robust control design systems. The control system consists of feedforward and feedback control techniques that have not been applied before for hydrostatic bearings in the published literature. The feedforward control tries to remove a disturbance before it enters the system while feedback control achieves the objective of disturbance rejection and improves steady-state characteristics. The feedforward control is a trajectory-based controller and the feedback controller is a sliding mode controller with a PID sliding surface. The particle swarm optimization algorithm is used to tune the 6-dimensional vector of the tuning parameters with multi-objective performance criteria. Numerical investigations have been carried out to check the performance of the proposed system under varying conditions of viscosity, clearance gap, external load and the spindle speed. The comparison of our results with the published literature shows the effectiveness of the proposed system.
... The end-to-end delay predictability required by a real-time distributed system has been achieved by assigning proper intermediate deadlines of individual real-time tasks executed on each node [1,[5][6][7][8][9][10][11][12][13][14][15][16][17]. However, although existing studies for the intermediate deadline assignment have succeeded to utilize mathematical theories (including convex optimization [18][19][20]) of distributed computation/control for intermediate deadline assignment, they have assumed that every task operates in a cooperative manner, which does not always hold in real-world situations. In addition, existing studies have not addressed how to exploit a trade-off between end-to-end delay fairness among real-time tasks and performance for minimizing total end-to-end delays of the entire real-time distributed system. ...
... In this section, we show how the cooperative distributed framework in Section 3.1 and the non-cooperative one in Section 3.2 operate with the utility function designed in Section 4. To this end, we target a typical real-time distributed system shown in Figure 2. The system consists of a set of 9 nodes N = {N a , N b , N c , N d , N e , N f , N g , N h , N i } and a set of 6 tasks τ = {τ 1 , τ 2 , τ 3 , τ 4 , τ 5 , τ 6 }. Each task executes on three different nodes sequentially; for example, τ 1 is executed on N a , N b and N c in a sequential manner, and τ 4 is executed on N a , N d and N g in a sequential manner, as shown in Figure 2. We set C (1,a) , C (1,b) , C (1,c) , C (2,d) , C (2,e) , C (2, f ) , C (3,g) , C (3,h) , C (3,i) , C (4,a) , C (4,d) , C (4,g) , C (5,b) , C (5,e) , C (5,h) , C (6,c) , C (6, f ) and C (6,i) to 10,10,10,15,15,15,20,20,20,10,10,10,15,15,15,20,20 and 20, respectively. We set the period of each task is 40. ...
... In this section, we show how the cooperative distributed framework in Section 3.1 and the non-cooperative one in Section 3.2 operate with the utility function designed in Section 4. To this end, we target a typical real-time distributed system shown in Figure 2. The system consists of a set of 9 nodes N = {N a , N b , N c , N d , N e , N f , N g , N h , N i } and a set of 6 tasks τ = {τ 1 , τ 2 , τ 3 , τ 4 , τ 5 , τ 6 }. Each task executes on three different nodes sequentially; for example, τ 1 is executed on N a , N b and N c in a sequential manner, and τ 4 is executed on N a , N d and N g in a sequential manner, as shown in Figure 2. We set C (1,a) , C (1,b) , C (1,c) , C (2,d) , C (2,e) , C (2, f ) , C (3,g) , C (3,h) , C (3,i) , C (4,a) , C (4,d) , C (4,g) , C (5,b) , C (5,e) , C (5,h) , C (6,c) , C (6, f ) and C (6,i) to 10,10,10,15,15,15,20,20,20,10,10,10,15,15,15,20,20 and 20, respectively. We set the period of each task is 40. ...
In real-time distributed systems, it is important to provide offline guarantee for an upper-bound of each real-time task’s end-to-end delay, which has been achieved by assigning proper intermediate deadlines of individual real-time tasks at each node. Although existing studies have succeeded to utilize mathematical theories of distributed computation/control for intermediate deadline assignment, they have assumed that every task operates in a cooperative manner, which does not always hold for real-worlds. In addition, existing studies have not addressed how to exploit a trade-off between end-to-end delay fairness among real-time tasks and performance for minimizing aggregate end-to-end delays. In this paper, we recapitulate an existing cooperative distributed framework, and propose a non-cooperate distributed framework that can operate even with selfish tasks, each of which is only interested in minimizing its own end-to-end delay regardless of achieving the system goal. We then propose how to design utility functions that allow the real-time distributed system to exploit the trade-off. Finally, we demonstrate the validity of the cooperative and non-cooperative frameworks along with the designed utility functions, via simulations.
In this paper, we consider a problem of parametric identification of a piece-wise linear mechanical system described by ordinary differential equations. We reconstruct the phase space of the investigated system from accelerometer data and perform parameter identification using iteratively reweighted least squares. Two key features of our study are as follows. First, we use a differentiated governing equation containing acceleration and velocity as the main independent variables instead of the conventional governing equation in velocity and position. Second, we modify the iteratively reweighted least squares method by including an auxiliary reclassification step into it. The application of this method allows us to improve the identification accuracy through the elimination of classification errors needed for parameter estimation of piece-wise linear differential equations. Simulation of the Duffing-like chaotic mechanical system and experimental study of an aluminum beam with asymmetric joint show that the proposed approach is more accurate than state-of-the-art solutions.
