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Journal articles on the topic 'Nonlinear PDE`s'
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1
Khodja, Brahim. "A nonexistence result for a nonlinear PDE with Robin condition." International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–12. http://dx.doi.org/10.1155/ijmms/2006/62601.
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Under the assumptionλ>0andfverifyingf(x,y,0)=0inD,2F(x,y,u)−uf(x,y,u)≥0,u≠0, and ifΩ=R×D, we show the convexity of functionE(t)=∬D|u(t,x,y)|2dxdy, whereu:Ω→ℝis a solution of problemλ(∂2u/∂t2)−(∂/∂x)(p(x,y)(∂u/∂x))−(∂/∂y)(q(x,y)(∂u/∂y))+f(x,y,u)=0 in Ω,u+ε(∂u/∂n)=0 on ∂Ω, considered inH2(Ω)∩L∞(Ω),p,q:D¯→ℝare two nonnull functions onD,εis a positive real number, andD=]a1,b1[×]a2,b2[,(F(x,y,s)=∫0sf(x,y,t)dt).
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Geldhauser, Carina, and Enrico Valdinoci. "Optimizing the Fractional Power in a Model with Stochastic PDE Constraints." Advanced Nonlinear Studies 18, no. 4 (November 1, 2018): 649–69. http://dx.doi.org/10.1515/ans-2018-2031.
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AbstractWe study an optimization problem with SPDE constraints, which has the peculiarity that the control parameter s is the s-th power of the diffusion operator in the state equation. Well-posedness of the state equation and differentiability properties with respect to the fractional parameter s are established. We show that under certain conditions on the noise, optimality conditions for the control problem can be established.
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Ji, Binxin, Xiangxing Tao, and Yanting Ji. "Barrier Option Pricing in the Sub-Mixed Fractional Brownian Motion with Jump Environment." Fractal and Fractional 6, no. 5 (April 29, 2022): 244. http://dx.doi.org/10.3390/fractalfract6050244.
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This paper investigates the pricing formula for barrier options where the underlying asset is driven by the sub-mixed fractional Brownian motion with jump. By applying the corresponding Ito^’s formula, the B-S type PDE is derived by a self-financing strategy. Furthermore, the explicit pricing formula for barrier options is obtained through converting the PDE to the Cauchy problem. Numerical experiments are conducted to test the impact of the barrier price, the Hurst index, the jump intensity and the volatility on the value of barrier option respectively.
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KNESSL, CHARLES. "Asymptotic analysis of the American call option with dividends." European Journal of Applied Mathematics 13, no. 6 (December 2002): 587–616. http://dx.doi.org/10.1017/s0956792502004898.
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We consider an American call option and let C(S, T0) be the price of an option corresponding to asset price S at some time T0 prior to the expiration time TF . We analyze C(S, T0) in various asymptotic limits. These include situations where the interest and dividend rates are large or small, compared to the volatility of the asset. We also analyze the optimal exercise boundary for the option. We use perturbation methods to analyze either the PDE that C(S, T0) satisfies, or a nonlinear integral equation that is satisfied by the optimal exercise boundary.
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Philippin, G. A., and A. Safoui. "Some applications of the maximum principle to a variety of fully nonlinear elliptic PDE?s." Zeitschrift f�r Angewandte Mathematik und Physik (ZAMP) 54, no. 5 (September 1, 2003): 739–55. http://dx.doi.org/10.1007/s00033-003-3200-7.
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Bonazebi-Yindoula, Joseph. "Laplace-SBA Method for Solving Nonlinear Coupled Burger's Equations." European Journal of Pure and Applied Mathematics 14, no. 3 (August 5, 2021): 842–62. http://dx.doi.org/10.29020/nybg.ejpam.v14i3.3932.
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Burger’s equations, an extension of fluid dynamics equations, are typically solved by several numerical methods. In this article, the laplace-Somé Blaise Abbo method is used to solve nonlinear Burger equations. This method is based on the combination of the laplace transform and the SBA method. After reminders of the laplace transform, the basic principles of the SBA method are described. The process of calculating the Laplace-SBA algorithm for determining the exact solution of a linear or nonlinear partial derivative equation is shown. Thus, three examplesof PDE are solved by this method, which all lead to exact solutions. Our results suggest that this method can be extended to other more complex PDEs.
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Cavalcanti, Marcelo M., Valéria N. Domingos Cavalcanti, Irena Lasiecka, and Claudete M. Webler. "Intrinsic decay rates for the energy of a nonlinear viscoelastic equation modeling the vibrations of thin rods with variable density." Advances in Nonlinear Analysis 6, no. 2 (May 1, 2017): 121–45. http://dx.doi.org/10.1515/anona-2016-0027.
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AbstractWe consider the long-time behavior of a nonlinear PDE with a memory term which can be recast in the abstract form$\frac{d}{dt}\rho(u_{t})+Au_{tt}+\gamma A^{\theta}u_{t}+Au-\int_{0}^{t}g(s)Au(t% -s)=0,$where A is a self-adjoint, positive definite operator acting on a Hilbert space H, ${\rho(s)}$ is a continuous, monotone increasing function, and the relaxation kernel ${g(s)}$ is a continuous, decreasing function in ${L_{1}(\mathbb{R}_{+})}$ with ${g(0)>0}$. Of particular interest is the case when ${A=-\Delta}$ with appropriate boundary conditions and ${\rho(s)=|s|^{\rho}s}$. This model arises in the context of solid mechanics accounting for variable density of the material. While finite energy solutions of the underlying PDE solutions exhibit exponential decay rates when strong damping is active (${\gamma>0,\theta=1}$), this uniform decay is no longer valid (by spectral analysis arguments) for dynamics subjected to frictional damping only, say, ${\theta=0}$ and ${g=0}$. In the absence of mechanical damping (${\gamma=0}$), the linearized version of the model reduces to a Volterra equation generated by bounded generators and, hence, it is exponentially stable for exponentially decaying kernels. The aim of the paper is to study intrinsic decays for the energy of the nonlinear model accounting for large classes of relaxation kernels described by the inequality ${g^{\prime}+H(g)\leq 0}$ with H convex and subject to the assumptions specified in (1.13) (a general framework introduced first in [1] in the context of linear second-order evolution equations with memory). In the context of frictional damping, such a framework was introduced earlier in [15], where it was shown that the decay rates of second-order evolution equations with frictional damping can be described by solutions of an ODE driven by a suitable convex function H which captures the behavior at the origin of the dissipation. The present paper extends this analysis to nonlinear equations with viscoelasticity. It is shown that the decay rates of the energy are intrinsically described by the solution of the dissipative ODE${S_{t}+c_{1}H(c_{2}S)=0}$with given intrinsic constants ${c_{1},c_{2}>0}$. The results obtained are sharp and they improve (by introducing a novel methodology) previous results in the literature (see [20, 19, 21, 6]) with respect to (i) the criticality of the nonlinear exponent ρ and (ii) the generality of the relaxation kernel.
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Bridges, Thomas J. "Canonical multi-symplectic structure on the total exterior algebra bundle." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 462, no. 2069 (February 8, 2006): 1531–51. http://dx.doi.org/10.1098/rspa.2005.1629.
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The aim of this paper is to construct multi-symplectic structures starting with the geometry of an oriented Riemannian manifold, independent of a Lagrangian or a particular partial differential equation (PDE). The principal observation is that on an n -dimensional orientable manifold M there is a canonical quadratic form Θ associated with the total exterior algebra bundle on M . On the fibre, which has dimension 2 n , the form Θ can be locally decomposed into n classical symplectic structures. When concatenated, these n -symplectic structures define a partial differential operator, J ∂ , which turns out to be a Dirac operator with multi-symplectic structure. The operator J ∂ generalizes the product operator J (d/d t ) in classical symplectic geometry, and M is a generalization of the base manifold (i.e. time) in classical Hamiltonian dynamics. The structure generated by Θ provides a natural setting for analysing a class of covariant nonlinear gradient elliptic operators. The operator J ∂ is elliptic, and the generalization of Hamiltonian systems, J ∂ Z =∇ S ( Z ), for a section Z of the total exterior algebra bundle, is also an elliptic PDE. The inverse problem—find S ( Z ) for a given elliptic PDE—is shown to be related to a variant of the Legendre transform on k -forms. The theory is developed for flat base manifolds, but the constructions are coordinate free and generalize to Riemannian manifolds with non-trivial curvature. Some applications and implications of the theory are also discussed.
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Wei, Hongbo, Xuerong Cui, Yucheng Zhang, and Jingyao Zhang. "$ H_\infty $ deployment of nonlinear multi-agent systems with Markov switching topologies over a finite-time interval based on T–S fuzzy PDE control." AIMS Mathematics 9, no. 2 (2024): 4076–97. http://dx.doi.org/10.3934/math.2024199.
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<abstract><p>The deployment of multi-agent systems (MASs) is widely used in the fields of unmanned agricultural machineries, unmanned aerial vehicles, intelligent transportation, etc. To make up for the defect that the existing PDE-based results are overly idealistic in terms of system models and control strategies, we study the PDE-based deployment of clustered nonlinear first-order and second-order MASs over a finite-time interval (FTI). By designing special communication protocols, the collective dynamics of numerous agents are modeled by simple fist-order and second-order PDEs. Two practical factors, external disturbance and Markov switching topology, are considered in this paper to better match actual situations. Besides, T–S fuzzy technology is used to approximate the unknown nonlinearity of MASs. Then, by using boundary control scheme with collocated measurements, two theorems are obtained to ensure the finite-time $ H_\infty $ deployment of first-order and second-order agents, respectively. Finally, numerical examples are provided to illustrate the effectiveness of the proposed approaches.</p></abstract>
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Daoues, Adel, Amani Hammami, and Kamel Saoudi. "Multiple positive solutions for a nonlocal PDE with critical Sobolev-Hardy and singular nonlinearities via perturbation method." Fractional Calculus and Applied Analysis 23, no. 3 (June 25, 2020): 837–60. http://dx.doi.org/10.1515/fca-2020-0042.
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AbstractIn this paper we investigate the following nonlocal problem with singular term and critical Hardy-Sobolev exponent$$\begin{array}{} ({\rm P}) \left\{ \begin{array}{ll} (-\Delta)^s u = \displaystyle{\frac{\lambda}{u^\gamma}+\frac{|u|^{2_\alpha^*-2}u}{|x|^\alpha}} \ \ \text{ in } \ \ \Omega, \\ u >0 \ \ \text{ in } \ \ \Omega, \quad u = 0 \ \ \text{ in } \ \ \mathbb{R}^{N}\setminus \Omega, \end{array} \right. \end{array}$$where Ω ⊂ ℝN is an open bounded domain with Lipschitz boundary, 0 < s < 1, λ > 0 is a parameter, 0 < α < 2s < N, 0 < γ < 1 < 2 < $\begin{array}{} \displaystyle 2_s^* \end{array}$, where $\begin{array}{} \displaystyle 2_s^* = \frac{2N}{N-2s} ~\text{and}~ 2_\alpha^* = \frac{2(N-\alpha)}{N-2s} \end{array}$ are the fractional critical Sobolev and Hardy Sobolev exponents respectively. The fractional Laplacian (–Δ)s with s ∈ (0, 1) is the nonlinear nonlocal operator defined on smooth functions by$$\begin{array}{} \displaystyle (-\Delta)^s u(x)=-\frac{1}{2} \displaystyle\int_{\mathbb{R}^N} \frac{u(x+y)+u(x-y)-2u(x)}{|y|^{N+2s}}{\rm d }y, \;\; \text{ for all }\, x \in \mathbb{R}^N. \end{array}$$By combining variational and approximation methods, we provide the existence of two positive solutions to the problem (P).
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Katzourakis, Nikos, and Tristan Pryer. "Second-order L∞ variational problems and the ∞-polylaplacian." Advances in Calculus of Variations 13, no. 2 (April 1, 2020): 115–40. http://dx.doi.org/10.1515/acv-2016-0052.
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AbstractIn this paper we initiate the study of second-order variational problems in {L^{\infty}}, seeking to minimise the {L^{\infty}} norm of a function of the hessian. We also derive and study the respective PDE arising as the analogue of the Euler–Lagrange equation. Given {\mathrm{H}\in C^{1}(\mathbb{R}^{n\times n}_{s})}, for the functional\mathrm{E}_{\infty}(u,\mathcal{O})=\|\mathrm{H}(\mathrm{D}^{2}u)\|_{L^{\infty}% (\mathcal{O})},\quad u\in W^{2,\infty}(\Omega),\mathcal{O}\subseteq\Omega,{}the associated equation is the fully nonlinear third-order PDE\mathrm{A}^{2}_{\infty}u:=(\mathrm{H}_{X}(\mathrm{D}^{2}u))^{\otimes 3}:(% \mathrm{D}^{3}u)^{\otimes 2}=0.{}Special cases arise when {\mathrm{H}} is the Euclidean length of either the full hessian or of the Laplacian, leading to the {\infty}-polylaplacian and the {\infty}-bilaplacian respectively. We establish several results for (1) and (2), including existence of minimisers, of absolute minimisers and of “critical point” generalised solutions, proving also variational characterisations and uniqueness. We also construct explicit generalised solutions and perform numerical experiments.
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Polly, James B., and J. M. McDonough. "Application of the Poor Man's Navier-Stokes Equations to Real-Time Control of Fluid Flow." Journal of Applied Mathematics 2012 (2012): 1–18. http://dx.doi.org/10.1155/2012/746752.
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Control of fluid flow is an important, underutilized process possessing potential benefits ranging from avoidance of separation and stall on aircraft wings to reduction of friction in oil and gas pipelines to mitigation of noise from wind turbines. But the Navier-Stokes (N.-S.) equations, whose solutions describe such flows, consist of a system of time-dependent, multidimensional, nonlinear partial differential equations (PDEs) which cannot be solved in real time using current computing hardware. The poor man's Navier-Stokes (PMNS) equations comprise a discrete dynamical system that is algebraic—hence, easily (and rapidly) solved—and yet which retains many (possibly all) of the temporal behaviors of the PDE N.-S. system at specific spatial locations. Herein, we outline derivation of these equations and discuss their basic properties. We consider application of these equations to the control problem by adding a control force. We examine the range of behaviors that can be achieved by changing this control force and, in particular, consider controllability of this (nonlinear) systemvianumerical experiments. Moreover, we observe that the derivation leading to the PMNS equations is very general and may be applied to a wide variety of problems governed by PDEs and (possibly) time-delay ordinary differential equations such as, for example, models of machining processes.
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Saushin, Aleksandr S., Gennady M. Mikheev, Viatcheslav V. Vanyukov, and Yuri P. Svirko. "The Surface Photogalvanic and Photon Drag Effects in Ag/Pd Metal-Semiconductor Nanocomposite." Nanomaterials 11, no. 11 (October 25, 2021): 2827. http://dx.doi.org/10.3390/nano11112827.
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We performed the investigation of the polarization-sensitive photocurrent generated in silver-palladium metal-semiconductor nanocomposite films under irradiation with nanosecond laser pulses at the wavelength of 2600 nm. It is shown that in both the transverse and the longitudinal configuration, the surface photogalvanic (SPGE) and photon drag effects (PDE) contribute to the observed photocurrent. However, the temporal profile of the transverse photocurrent pulse is monopolar at any polarization and angle of incidence, while the temporal profile of the longitudinal photocurrent pulse depends on the polarization of the excitation beam. Specifically, the irradiation of the film with the s-polarized excitation beam produces a monopolar photoresponse, while at p-polarized excitation, the photoresponse is bipolar, having a short front and long tail. Obtained experimental results are in agreement with the developed phenomenological theory, which describes transverse and longitudinal photocurrents due to SPGE and PDE in terms of relevant second-order nonlinear susceptibilities and allows us to obtain their dependences on the angle of incidence and polarization of the excitation laser beam. The pronounced dependence of the photocurrent on the angle of incidence and polarization of the excitation beam opens avenues toward the development of polarization- and position-sensitive detectors for industrial and space applications.
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Zhang, Sumei, Hongquan Yong, and Haiyang Xiao. "Option Pricing with Fractional Stochastic Volatilities and Jumps." Fractal and Fractional 7, no. 9 (September 11, 2023): 680. http://dx.doi.org/10.3390/fractalfract7090680.
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Empirical studies suggest that asset price fluctuations exhibit “long memory”, “volatility smile”, “volatility clustering” and asset prices present “jump”. To fit the above empirical characteristics of the market, this paper proposes a fractional stochastic volatility jump-diffusion model by combining two fractional stochastic volatilities with mixed-exponential jumps. The characteristic function of the log-return is expressed in terms of the solution of two-dimensional fractional Riccati equations of which closed-form solution does not exist. To obtain the explicit characteristic function, we approximate the pricing model by a semimartingale and convert fractional Riccati equations into a classic PDE. By the multi-dimensional Feynman-Kac theorem and the affine structure of the approximate model, we obtain the solution of the PDE with which the explicit characteristic function and its cumulants are derived. Based on the derived characteristic function and Fourier cosine series expansion, we obtain approximate European options prices. By differential evolution algorithm, we calibrate our approximate model and its two nested models to S&P 500 index options and obtain optimal parameter estimates of these models. Numerical results demonstrate the pricing method is fast and accurate. Empirical results demonstrate our approximate model fits the market best among the three models.
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Mascia, C., P. Moschetta, and C. Simeoni. "Phase transitions of biological phenotypes by means of a prototypical PDE model." Communications in Applied and Industrial Mathematics 11, no. 1 (January 1, 2020): 1–17. http://dx.doi.org/10.2478/caim-2020-0001.
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AbstractThe basic investigation is the existence and the (numerical) observability of propagating fronts in the framework of the so-called Epithelial-to-Mesenchymal Transition and its reverse Mesenchymal-to-Epithelial Transition, which are known to play a crucial role in tumor development. To this aim, we propose a simplified one-dimensional hyperbolic-parabolic PDE model composed of two equations, one for the representative of the epithelial phenotype, and the second describing the mesenchymal phenotype. The system involves two positive constants, the relaxation time and a measure of invasiveness, moreover an essential feature is the presence of a nonlinear reaction function, typically assumed to be S-shaped. An identity characterizing the speed of propagation of the fronts is proven, together with numerical evidence of the existence of traveling waves. The latter is obtained by discretizing the system by means of an implicit-explicit finite difference scheme, then the algorithm is validated by checking the capability of the so-called LeVeque–Yee formula to reproduce the value of the speed furnished by the above cited identity. Once such justification has been achieved, we concentrate on numerical experiments relative to Riemann initial data connecting two stable stationary states of the underlying ODE model. In particular, we detect an explicit transition threshold separating regression regimes from invasive ones, which depends on critical values of the invasiveness parameter. Finally, we perform an extensive sensitivity analysis with respect to the system parameters, exhibiting a subtle dependence for those close to the threshold values, and we postulate some conjectures on the propagating fronts.
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Said, Hamid. "An analytical mechanics approach to the first law of thermodynamics and construction of a variational hierarchy." Theoretical and Applied Mechanics, no. 00 (2020): 11. http://dx.doi.org/10.2298/tam200315011s.
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A simple procedure is presented for the study of the conservation of energy equation with dissipation in continuum mechanics in 1D. This procedure is used to transform this nonlinear evolution-diffusion equation into a hyperbolic PDE; specifically, a second-order quasi-linear wave equation. An immediate implication of this procedure is the formation of a least action principle for the balance of energy with dissipation. The corresponding action functional enables us to establish a complete analytic mechanics for thermomechanical systems: a Lagrangian-Hamiltonian theory, integrals of motion, bracket formalism, and Noether?s theorem. Furthermore, we apply our procedure iteratively and produce an infinite sequence of interlocked variational principles, a variational hierarchy, where at each level or iteration the full implication of the least action principle can be shown again.
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Hwang, Feng-Nan. "Three-dimensional trajectory optimization for multi-stage launch vehicle mission using a full-space quasi-Lagrange–Newton method." ANZIAM Journal 60 (August 30, 2019): C172—C186. http://dx.doi.org/10.21914/anziamj.v60i0.14067.
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Many aerospace industrial applications require robust and efficient numerical solutions of large sparse nonlinear constrained parameter optimization problems arising from optimal trajectory problems. A three-dimensional multistage launcher problem is taken as a numerical example for studying the performance and applicability of the full-space Lagrange–Newton–Krylov method. The typical optimal trajectory, control history and other important physical qualities are presented, and the efficiency of the algorithm is also investigated.
References J. T. Betts. Practical methods for optimal control and estimation using nonlinear programming. Advances in Design and Control. SIAM, 2nd edition, 2010. doi:10.1137/1.9780898718577. R. T. Marler and J. S. Arora. Survey of multi-objective optimization methods for engineering. Struct. Multidiscip. Opt., 26(6):369395, 2004. doi:10.1007/s00158-003-0368-6. W. Roh and Y. Kim. Trajectory optimization for a multi-stage launch vehicle using time finite element and direct collocation methods. Eng. Opt., 34:1532, 2002. doi:10.1080/03052150210912. G. D. Silveira and V. Carrara. A six degrees-of-freedom flight dynamics simulation tool of launch vehicles. J. Aero. Tech. Manag., 7:231239, 2015. doi:10.5028/jatm.v7i2.433. H.-H. Wang, Y.-S. Lo, F.-T. Hwang, and F.-N. Hwang. A full-space quasi-LagrangeNewtonKrylov algorithm for trajectory optimization problems. Electron. T. Numer. Anal., 49:103125, 2018. doi:10.1553/etna_vol49s103. H. Yang, F.-N. Hwang, and X.-C. Cai. Nonlinear preconditioning techniques for full-space Lagrange-Newton solution of PDE-constrained optimization problems. SIAM J. Sci. Comput., 38:A2756A2778, 2016. doi:10.1137/15M104075X.
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Cheng, Bin, and Zisis N. Sakellaris. "Near-resonance approximation of rotating Navier–Stokes equations." Nonlinearity 36, no. 4 (February 28, 2023): 2074–127. http://dx.doi.org/10.1088/1361-6544/acb7c5.
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Abstract We formalise the concept of near resonance for the rotating Navier–Stokes equations, based on which we propose a novel way to approximate the original partial differential equation (PDE). The spatial domain is a three-dimensional flat torus of arbitrary aspect ratios. We prove that the family of proposed PDEs are globally well-posed for any rotation rate and initial datum of any size in any H s space with s ⩾ 0 . Such approximations retain many more 3-mode interactions, and are thus more accurate, than the conventional exact-resonance approach. Our approach is free from any limiting argument that requires physical parameters to tend to zero or infinity, and is free from any use of small divisors (so that all estimates depend smoothly on the torus’s aspect ratios). The key estimate hinges on the counting of integer solutions of Diophantine inequalities rather than Diophantine equations. Using a range of novel ideas, we handle rigorously and optimally challenges arising from the non-trivial irrational functions in these inequalities. The main results and ingredients of the proofs can form part of the mathematical foundation of a non-asymptotic approach to nonlinear, oscillatory dynamics in real-world applications.
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Nichols, James. "Coarse reduced model selection for nonlinear state estimation." ANZIAM Journal 62 (February 7, 2022): C192—C207. http://dx.doi.org/10.21914/anziamj.v62.16169.
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State estimation is the task of approximately reconstructing a solu- tion u of a parametric partial differential equation when the parameter vector y is unknown and the only information is m linear measurements of u. Cohen et al. [arXiv:2009.02687, Nov. 2020] proposed a method to use a family of linear reduced spaces as a generalised nonlinear reduced model for state estimation. A computable surrogate distance is used to evaluate which linear estimate lies closest to a true solution of the pde problem. In this article we propose a strategy of coarse computation of the surrogate distance while maintaining a fine mesh reduced model, as the computational cost of the surrogate distance is large relative to the reduced modelling task. We demonstrate numerically that the error induced by the coarse distance is dominated by other approximation errors. References P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova, and P. Wojtaszczyk. Convergence rates for geedy algorithms in reduced basis methods. SIAM J. Math. Anal. 43.3 (2011), pp. 1457–1472. doi: 10.1137/100795772. P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova, and P. Wojtaszczyk. Data assimilation in reduced modeling. SIAM/ASA J. Uncert. Quant. 5.1 (2017), pp. 1–29. doi: 10.1137/15M1025384 A. Cohen, W. Dahmen, O. Mula, and J. Nichols. Nonlinear reduced models for state and parameter estimation. arXiv:2009.02687 [cs, math] (2020). url: http://arxiv.org/abs/2009.02687 (visited on 01/07/2021) A. Cohen, D. Wolfgang, R. DeVore, and J. Nichols. Reduced basis greedy selection using random training sets. ESAIM: Math. Model. Num. Anal. 54 (2020), pp. 1509–1524. doi: 10.1051/m2an/2020004. J. S. Hesthaven, G. Rozza, and B. Stamm. Certified reduced basis methods for parametrized partial differential equations. SpringerBriefs in Mathematics. Springer, 2016. doi: 10.1007/978-3-319-22470-1. Y. Maday, A. T. Patera, J. D. Penn, and M. Yano. A parameterized-background data-weak approach to variational data assimilation: formulation, analysis, and application to acoustics. Int. J. Num. Meth. Eng. 102.5 (2015), pp. 933–965. doi: 10.1002/nme.4747.
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Drin, Yaroslav, Iryna Drin, and Svetlana Drin. "THE NONLOCAL PROBLEM FOR FRACTAL DIFFUSION EQUATION." Journal of Automation and Information sciences 1 (January 1, 2022): 47–55. http://dx.doi.org/10.34229/1028-0979-2022-1-5.
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Over the past few decades, the theory of pseudodifferential operators (PDO) and equations with such operators (PDE) has been intensively developed. The authors of a new direction in the theory of PDE, which they called parabolic PDE with non-smooth homogeneous symbols (PPDE), are Yaroslav Drin and Samuil Eidelman. In the early 1970s, they constructed an example of the Cauchy problem for a modified heat equation containing, instead of the Laplace operator, PDO, which is its square root. Such a PDO has a homogeneous symbol |σ|, which is not smooth at the origin. The fundamental solution of the Cauchy problem (FSCP) for such an equation is an exact power function. For the heat equation, FSCP is an exact exponential function. The Laplace operator can be interpreted as a PDO with a smooth homogeneous symbol |σ|^2, σ ∈ Rn. A generalization of the heat equation is PPDE containing PDO with homogeneous non-smooth symbols. They have an important application in the theory of random processes, in particular, in the construction of discontinuous Markov processes with generators of integro-differential operators, which are related to PDO; in the modern theory of fractals, which has recently been rapidly developing. If the PDO symbol does not depend on spatial coordinates, then the Cauchy problem for PPDE is correctly solvable in the space of distribution-type generalized functions. In this case, the solution is written as a convolution of the FSCP with an initial generalized function. These results belong to a number of domestic and foreign mathematicians, in particular S. Eidelman and Y. Drin (who were the first to define PPDO with non-smooth symbols and began the study of the Cauchy problem for the corresponding PPDE), M. Fedoruk, A. Kochubey, V. Gorodetsky, V . Litovchenko and others. For certain new classes of PPDE, the correct solvability of the Cauchy problem in the space of Hölder functions has been proved, classical FSCP have been constructed, and exact estimates of their power-law derivatives have been obtained [1–4]. Of fundamental importance is the interpretation of PDO proposed by A. Kochubey in terms of hypersingular integrals (HSI). At the same time, the HSI symbol is constructed from the known PDO symbol and vice versa [6]. The theory of HSI, which significantly extend the class of PDO, was developed by S. Samko [7]. We extends this concept to matrix HSI [5]. Generalizations of the Cauchy problem are non-local multipoint problems with respect to the time variable and the problem with argument deviation. Here we prove the solvability of a nonlocal problem using the method of steps. We consider an evolutionary nonlinear equation with a regularized fractal fractional derivative α ∈ (0, 1] with respect to the time variable and a general elliptic operator with variable coefficients with respect to the second-order spatial variable. Such equations describe fractal properties in real processes characterized by turbulence, in hydrology, ecology, geophysics, environment pollution, economics and finance.
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Kiessling, Michael K. H. "Correction to: On the Quasi-Linear Elliptic PDE $${-\nabla\cdot({\nabla}{u}/ \sqrt{1-|{\nabla}{u}|^2}) = 4\pi\sum_k a_k \delta_{s_k}}$$ - ∇ · ( ∇ u / 1 - | ∇ u | 2 ) = 4 π ∑ k a k δ s k in Physics and Geometry." Communications in Mathematical Physics 364, no. 2 (October 11, 2018): 825–33. http://dx.doi.org/10.1007/s00220-018-3261-2.
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Duan, Ruirui, Junmin Li, Yanni Zhang, Ying Yang, and Guopei Chen. "Stability analysis and H∞ control of discrete T–S fuzzy hyperbolic systems." International Journal of Applied Mathematics and Computer Science 26, no. 1 (March 1, 2016): 133–45. http://dx.doi.org/10.1515/amcs-2016-0009.
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Abstract This paper focuses on the problem of constraint control for a class of discrete-time nonlinear systems. Firstly, a new discrete T–S fuzzy hyperbolic model is proposed to represent a class of discrete-time nonlinear systems. By means of the parallel distributed compensation (PDC) method, a novel asymptotic stabilizing control law with the “soft” constraint property is designed. The main advantage is that the proposed control method may achieve a small control amplitude. Secondly, for an uncertain discrete T–S fuzzy hyperbolic system with external disturbances, by the proposed control method, the robust stability and H∞ performance are developed by using a Lyapunov function, and some sufficient conditions are established through seeking feasible solutions of some linear matrix inequalities (LMIs) to obtain several positive diagonally dominant (PDD) matrices. Finally, the validity and feasibility of the proposed schemes are demonstrated by a numerical example and a Van de Vusse one, and some comparisons of the discrete T–S fuzzy hyperbolic model with the discrete T–S fuzzy linear one are also given to illustrate the advantage of our approach.
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Xu, Li, and Fei Liu. "Fuzzy Based Model Predictive Control for Uncertain Nonlinear System." Advanced Materials Research 383-390 (November 2011): 2404–10. http://dx.doi.org/10.4028/www.scientific.net/amr.383-390.2404.
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In this paper, a model predictive control (MPC) scheme is investigated for uncertain nonlinear system with time delay and input constraint. First, the Takagi-Sugeno (T-S) fuzzy model is used to approximate the dynamics of nonlinear processes and the parallel distributed compensation (PDC) controllers which are parameter dependent and mirror the structure of the T-S plant model are proposed. Then a novel feedback PDC predictive controller obtained from the linear matrix inequality (LMI) solutions which can guarantee the stability of the closed-loop overall fuzzy system is put forward. Finally, a numerical example is provided to demonstrate the effectiveness and feasibility of the proposed method.
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Hsiao, Feng-Hsiag, and Wei-Ling Chiang. "Application of Fuzzy H∞ Control via T–S Fuzzy Models for Nonlinear Time-Delay Systems." International Journal on Artificial Intelligence Tools 12, no. 02 (June 2003): 117–37. http://dx.doi.org/10.1142/s0218213003001174.
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This paper deals with the problem of stability analysis and stabilization via Takagi-Sugeno (T-S) fuzzy models for nonlinear time-delay systems. First, Takagi-Sugeno (T-S) fuzzy models and some stability results are recalled. To design fuzzy controllers, nonlinear time-delay systems are represented by Takagi-Sugeno fuzzy models. The concept of parallel-distributed compensation (PDC) is employed to determine structures of fuzzy controllers from the T-S fuzzy models. LMI-based design problems are defined and employed to find feedback gains of fuzzy controller and common positive definite matrices P satisfying stability a delay-dependent stability criterion derived in terms of Lyapunov direct method. Based on the control scheme and this criterion, a fuzzy controller is then designed via the technique of PDC to stabilize the nonlinear time-delay system and the H∞ control performance is achieved in the mean time. Finally, the proposed controller design method is demonstrated through numerical simulations on the chaotic and resonant systems.
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CHEN, CHENG-WU, WEI-LING CHIANG, CHUNG-HUNG TSAI, CHEN-YUAN CHEN, and MORRIS H. L. WANG. "FUZZY LYAPUNOV METHOD FOR STABILITY CONDITIONS OF NONLINEAR SYSTEMS." International Journal on Artificial Intelligence Tools 15, no. 02 (April 2006): 163–71. http://dx.doi.org/10.1142/s0218213006002618.
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This paper proposes a fuzzy Lyapunov method for stability analysis of nonlinear systems represented by Tagagi-Sugeno (T-S) fuzzy model. The fuzzy Lyapunov function is defined in fuzzy blending quadratic Lyapunov functions. Based on fuzzy Lyapunov functions, some stability conditions are derived to ensure nonlinear systems are asymptotic stable. By using parallel distributed compensation (PDC) scheme, we design a nonlinear fuzzy controller for the nonlinear system. This control problem will be reformulated into linear matrix inequalities (LMI) problem.
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U'Ren, A. B., K. Banaszek, and I. A. Walmsley. "Photon engineering for quantum information processing." Quantum Information and Computation 3, special (October 2003): 480–502. http://dx.doi.org/10.26421/qic3.s-3.
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We study distinguishing information in the context of quantum interference involving more than one parametric downconversion (PDC) source and in the context of generating polarization-entangled photon pairs based on PDC. We arrive at specific design criteria for two-photon sources so that when used as part of complex optical systems, such as photon-based quantum information processing schemes, distinguishing information between the photons is eliminated guaranteeing high visibility interference. We propose practical techniques which lead to suitably engineered two-photon states that can be realistically implemented with available technology. Finally, we study an implementation of the nonlinear-sign shift (NS) logic gate with PDC sources and show the effect of distinguishing information on the performance of the gate.
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Pokorný, Miroslav, Tomáš Dočekal, and Danica Rosinová. "Nonlinear modelling and optimal control via Takagi-Sugeno fuzzy techniques: A quadrotor stabilization." Journal of Electrical Engineering 71, no. 1 (February 1, 2020): 1–10. http://dx.doi.org/10.2478/jee-2020-0001.
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AbstractUsing the principles of Takagi-Sugeno fuzzy modelling allows the integration of flexible fuzzy approaches and rigorous mathematical tools of linear system theory into one common framework. The rule-based T-S fuzzy model splits a nonlinear system into several linear subsystems. Parallel Distributed Compensation (PDC) controller synthesis uses these T-S fuzzy model rules. The resulting fuzzy controller is nonlinear, based on fuzzy aggregation of state controllers of individual linear subsystems. The system is optimized by the linear quadratic control (LQC) method, its stability is analysed using the Lyapunov method. Stability conditions are guaranteed by a system of linear matrix inequalities (LMIs) formulated and solved for the closed loop system with the proposed PDC controller. The additional GA optimization procedure is introduced, and a new type of its fitness function is proposed to improve the closed-loop system performance.
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Zhang, Liang, Xiaoheng Chang, and Hamid Reza Karimi. "Fuzzy Modeling and Control for a Class of Inverted Pendulum System." Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/936868.
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Focusing on the issue of nonlinear stability control system about the single-stage inverted pendulum, the T-S fuzzy model is employed. Firstly, linear approximation method would be applied into fuzzy model for the single-stage inverted pendulum. At the same time, for some nonlinear terms which could not be dealt with via linear approximation method, this paper will adopt fan range method into fuzzy model. After the T-S fuzzy model, the PDC technology is utilized to design the fuzzy controller secondly. Numerical simulation results, obtained by Matlab, demonstrate the well-controlled effectiveness based on the proposed method for the model of T-S fuzzy system and fuzzy controller.
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Rajesh, R., and M. R. Kaimal. "T–S fuzzy model with nonlinear consequence and PDC controller for a class of nonlinear control systems." Applied Soft Computing 7, no. 3 (June 2007): 772–82. http://dx.doi.org/10.1016/j.asoc.2006.01.014.
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Liu, Xiangjie, and Mengyue Wang. "Nonlinear Fuzzy Model Predictive Control for a PWR Nuclear Power Plant." Mathematical Problems in Engineering 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/908526.
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Reliable power and temperature control in pressurized water reactor (PWR) nuclear power plant is necessary to guarantee high efficiency and plant safety. Since the nuclear plants are quite nonlinear, the paper presents nonlinear fuzzy model predictive control (MPC), by incorporating the realistic constraints, to realize the plant optimization. T-S fuzzy modeling on nuclear power plant is utilized to approximate the nonlinear plant, based on which the nonlinear MPC controller is devised via parallel distributed compensation (PDC) scheme in order to solve the nonlinear constraint optimization problem. Improved performance compared to the traditional PID controller for a TMI-type PWR is obtained in the simulation.
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Seddik, Taieb, Debidatta Behera, Mohammed Batouche, Walid Ouerghui, Houda Ben Abdallah, Ram Krishna Sarkar, Mostafa M. Salah, Ahmed Shaker, and Sanat Kumar Mukherjee. "Electronic Properties, Linear and Nonlinear Performance of KAgCh (Ch = S, Se) Compounds: A First-Principles Study." Crystals 13, no. 5 (April 25, 2023): 726. http://dx.doi.org/10.3390/cryst13050726.
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In the current study, the peculiar nonlinear optical (NLO) properties of KAgCh (Ch = S, Se) and their structural, electronic, and thermodynamic properties are computed utilizing the FP-LAPW (full-potential linearized augmented plane wave) approach as embedded in Wein2K code. The Perdew–Burke–Ernzerh of generalized gradient approximation (PBE-GGA) was considered for the structural optimization. The computed bandgaps are found to be 2.57 and 2.39 eV for KAgS and KAgSe, respectively. Besides the structural and electronic properties, we also computed the refractive indices n(ω), surface energy loss function (SELF), and nonlinear optical susceptibilities. The estimated refractive indices, energy band gap, and their frequency dependence for the investigated KAgCh (Ch = S, Se) compounds, along with the NLO coefficients, are found to be in good agreement with the earlier reports. These current findings suggest that KAgCh (Ch = S, Se) can be recommended for nonlinear optical applications in the near-infrared spectrum.
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Fang, Yunmei, Shitao Wang, Juntao Fei, and Mingang Hua. "Adaptive Control of MEMS Gyroscope Based on T-S Fuzzy Model." Discrete Dynamics in Nature and Society 2015 (2015): 1–10. http://dx.doi.org/10.1155/2015/419643.
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A multi-input multioutput (MIMO) Takagi-Sugeno (T-S) fuzzy model is built on the basis of a nonlinear model of MEMS gyroscope. A reference model is adjusted so that a local linear state feedback controller could be designed for each T-S fuzzy submodel based on a parallel distributed compensation (PDC) method. A parameter estimation scheme for updating the parameters of the T-S fuzzy models is designed and analyzed based on the Lyapunov theory. A new adaptive law can be selected to be the former adaptive law plus a nonnegative in variable to guarantee that the derivative of the Lyapunov function is smaller than zero. The controller output is implemented on the nonlinear model and T-S fuzzy model, respectively, for the purpose of comparison. Numerical simulations are investigated to verify the effectiveness of the proposed control scheme and the correctness of the T-S fuzzy model.
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Hori, Tsuyoshi, and Kazuo Tanaka. "State Feedback Stabilization in Nonlinear Time-Delay Systems." Journal of Advanced Computational Intelligence and Intelligent Informatics 6, no. 3 (October 20, 2002): 109–14. http://dx.doi.org/10.20965/jaciii.2002.p0109.
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In this paper, a class of nonlinear time-delay systems based on the Takagi-Sugeno (T-S) fuzzy model is defined. We investigate the delay-independent stability of this model. A model-based fuzzy stabilization design utilizing the concept of parallel distributed compensation (PDC) is employed. The main idea of the controller design is to derive each control rule to compensate each rule of a fuzzy system. Moreover, the problem of H∞ of this class of nonlinear time-delay systems is considered. The associated control synthesis problems are formulated as linear matrix inequality (LMI) problems.
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Chak, Yew-Chung, and Renuganth Varatharajoo. "A parallel distributed compensation approach to fuzzy control of spacecraft combined attitude and sun tracking." International Journal of Engineering & Technology 7, no. 4.13 (October 9, 2018): 28. http://dx.doi.org/10.14419/ijet.v7i4.13.21324.
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A spacecraft combined attitude and sun tracking system (CASTS) is a synergized system in which solar array drive assemblies are used as sun trackers and simultaneously as attitude control actuators. This paper, a continuous research on CASTS, addresses its attitude control problem. The kinematics and dynamics equations of a rigid spacecraft attitude motion is inherently nonlinear. In the attitude regulation problem, the attitude motion can be treated as a simple linear system for a constrained range of operating conditions, but it has impacts on the accuracy of the linear model when a model-based controller is implemented. Naturally, this is a compromise between simplicity and accuracy that all design engineers have to face. In this paper, we present a systematic approach to improve the accuracy while preserving the model as linear as possible, by deriving a quasi-linear approximation of a nonlinear spacecraft attitude motion. The quasi-linear approximation is based on the framework of the Takagi–Sugeno (T–S) fuzzy model. If a spacecraft can be modeled in the form of a rule-based T-S fuzzy system that acts as an interpolator between linear state-space systems, an approach called parallel distributed control (PDC) can be used to stabilize the attitude motion. The design philosophy of PDC is to create a simple fuzzy controller, where each rule’s consequent is a control law designed to stabilize the linear system in the corresponding consequent of the spacecraft T-S fuzzy system. Numerical results validate that the attitude and sun-tracking performances are achievable using the proposed PDC strategy.
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Muhammad, Mustapha, Amir A. Bature, Umar Zangina, Salinda Buyamin, Anita Ahmad, and Mohamad A. Shamsudin. "Velocity control of a two-wheeled inverted pendulum mobile robot: a fuzzy model-based approach." Bulletin of Electrical Engineering and Informatics 8, no. 3 (September 1, 2019): 808–17. http://dx.doi.org/10.11591/eei.v8i3.1594.
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This paper presents the design of a fuzzy tracking controller for balancing and velocity control of a Two-Wheeled Inverted Pendulum (TWIP) mobile robot based on its Takagi-Sugino (T-S) fuzzy model, fuzzy Lyapunov function and non-parallel distributed compensation (non-PDC) control law. The T-S fuzzy model of the TWIP mobile robot was developed from its nonlinear dynamical equations of motion. Stabilization conditions in a form of linear matrix inequalities (LMIs) were derived based on the T-S fuzzy model of the TWIP mobile robot, a fuzzy Lyapunov function and a non-PDC control law. Based on the derived stabilization conditions and the T-S fuzzy model of the TWIP mobile robot, a state feedback velocity tracking controller was then proposed for the TWIP mobile robot. The balancing and velocity tracking performance of the proposed controller was investigated via simulations. The simulation result shows the effectiveness of the proposed control scheme.
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Hu, Songlin, Dong Yue, Min Shi, and Xiangpeng Xie. "Discrete-Time Event-Triggered Control of Nonlinear Wireless Networked Control Systems." Abstract and Applied Analysis 2014 (2014): 1–14. http://dx.doi.org/10.1155/2014/860438.
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This paper investigates the problem of stabilization of nonlinear discrete-time networked control systems (NCSs) with event-triggering communication scheme in the presence of signal transmission delay. A Takagi-Sugeno (T-S) fuzzy model and parallel-distributed compensation (PDC) scheme are first employed to design a nonlinear fuzzy event-triggered controller for the stabilization of nonlinear discrete-time NCSs. The idea of the event-triggering communication scheme (i.e., a soft computation algorithm) under consideration is that the current sensor data is transmitted only when the current sensor data and the previously transmitted one satisfy a certain state-dependent trigger condition. By taking the signal transmission delay into consideration and using delay system approach, a T-S fuzzy delay system model is established to describe the nonlinear discrete-time NCSs with event-triggering communication scheme. Attention is focused on the design of fuzzy event-triggered controller which ensures asymptotic stability of the closed-loop fuzzy systems. Linear matrix inequality- (LMI-) based conditions are formulated for the existence of admissible fuzzy event-triggered controller. If these conditions are feasible, a desired fuzzy event-triggered controller can be readily constructed. A nonlinear mass-spring-damper mechanical system is presented to demonstrate the effectiveness of the proposed method.
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Ku, Cheung-Chieh, Yun-Chen Yeh, Yann-Hong Lin, and Yu-Yen Hsieh. "Fuzzy Static Output Control of T–S Fuzzy Stochastic Systems via Line Integral Lyapunov Function." Processes 9, no. 4 (April 15, 2021): 697. http://dx.doi.org/10.3390/pr9040697.
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Considering some unmeasurable states, a fuzzy static output control problem of nonlinear stochastic systems is discussed in this paper. Based on a modelling approach, a Takagi–Sugeno (T–S) fuzzy system, constructed by a family of stochastic differential equations and membership functions, is applied to represent nonlinear stochastic systems. Parallel distributed compensation (PDC) technology is used to construct the static output controller. A line-integral Lyapunov function (LILF) is used to derive some sufficient conditions for guaranteeing the asymptotical stability in the mean square. From the LILF, a potential conservatism produced by the derivative of the membership function is eliminated to increase the relaxation of sufficient conditions. Furthermore, those conditions are transferred into linear matrix inequality (LMI) form via projection lemma. According to the convex optimization algorithm, the feasible solutions are directly obtained to establish the static output fuzzy controller. Finally, a numerical example is applied to demonstrate the effectiveness and usefulness of the proposed design method.
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Shao, Xuejuan, Jinggang Zhang, and Xueliang Zhang. "Takagi-Sugeno Fuzzy Modeling and PSO-Based Robust LQR Anti-Swing Control for Overhead Crane." Mathematical Problems in Engineering 2019 (April 4, 2019): 1–14. http://dx.doi.org/10.1155/2019/4596782.
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The dynamic model of overhead crane is highly nonlinear and uncertain. In this paper, Takagi-Sugeno (T-S) fuzzy modeling and PSO-based robust linear quadratic regulator (LQR) are proposed for anti-swing and positioning control of the system. First, on the basis of sector nonlinear theory, the two T-S fuzzy models are established by using the virtual control variables and approximate method. Then, considering the uncertainty of the model, robust LQR controllers with parallel distributed compensation (PDC) structure are designed. The feedback gain matrices are obtained by transforming the stability and robustness of the system into linear matrix inequalities (LMIs) problem. In addition, particle swarm optimization (PSO) algorithm is used to overcome the blindness of LQR weight matrix selection in the design process. The proposed control methods are simple, feasible, and robust. Finally, the numeral simulations are carried out to prove the effectiveness of the methods.
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Huang, Ping-Tzan, Chein-Chung Sun, Cheung-Chieh Ku, and Yun-Chen Yeh. "Passive Stabilization of Static Output Feedback of Disturbed Nonlinear Stochastic System." Mathematics 11, no. 21 (October 26, 2023): 4435. http://dx.doi.org/10.3390/math11214435.
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This paper investigates the Static Output (SO) control issue of the disturbed nonlinear stochastic system, which achieves passivity. Through the application of fuzzy sets and the stochastic differential equation, a Takagi–Sugeno (T-S) fuzzy model with the terms of multiplicative noise and external disturbance can be constructed to describe the considered systems. Furthermore, the Parallel Distributed Compensation (PDC) concept is used to design a fuzzy controller exhibiting an SO feedback scheme structure. To attenuate the effect of external disturbance, the PDC-based SO fuzzy controller is designed to exhibit passivity. During the derivation of some sufficient conditions, a line-integral Lyapunov function is utilized to avoid the conservative term produced using the derivative membership function. Using converting technologies, a stability criterion belonging to Linear Matrix Inequality (LMI) forms is proposed such that the derived conditions are convex hull problems and are solved through an optimization algorithm. Then, the proposed criterion is used to discuss the problem of SO controller design of ship fin stabilizing systems with added disturbance and noise.
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Félix-Herrán, L. C., Driss Mehdi, José de Jesús Rodríguez-Ortiz, Victor H. Benitez, Ricardo A. Ramirez‐Mendoza, and Rogelio Soto. "Disturbance Rejection in a One-Half Semiactive Vehicle Suspension by means of a Fuzzy-H∞ Controller." Shock and Vibration 2019 (July 31, 2019): 1–14. http://dx.doi.org/10.1155/2019/4532635.
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A fuzzy-H∞ control, improved with weighting functions, has been designed and applied to a novel model of a one-half semiactive lateral vehicle (OHSLV) suspension. The herein contribution resides in the development and computation of an H∞ controller with parallel distributed compensation (PDC) designed from a highly nonlinear system modelled via the Takagi–Sugeno (T-S) fuzzy approach. A fuzzy-H∞ controller is synthesized for an OHSLV T-S fuzzy model of a suspension with two magnetorheological (MR) dampers including actuators’ nonlinear dynamics. The realism of results has been improved by considering the MR damper’s behaviours (viscoplasticity, hysteresis, and saturation) and the handling of the phase angle of the sinusoidal disturbance, not included in other reported work. Time-domain tests remark transient time achievements, whereas precise performance criterion indices in the frequency domain are employed to assess the generated outcomes. The proposed solution complies with all performance criteria compared with a benchmark passive average suspension that fails in satisfying most of the performance criteria.
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HSIAO, FENG-HSIAG, WEI-LING CHIANG, CHENG-WU CHEN, SHENG-DONG XU, and SHIH-LIN WU. "APPLICATION AND ROBUSTNESS DESIGN OF FUZZY CONTROLLER FOR RESONANT AND CHAOTIC SYSTEMS WITH EXTERNAL DISTURBANCE." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 13, no. 03 (June 2005): 281–95. http://dx.doi.org/10.1142/s0218488505003461.
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A robustness design of fuzzy control via model-based approach is proposed in this paper to overcome the effect of approximation error between nonlinear system and Takagi-Sugeno (T-S) fuzzy model. T-S fuzz model is used to model the resonant and chaotic systems and the parallel distributed compensation (PDC) is employed to determine structures of fuzzy controllers. Linear matrix inequality (LMI) based design problems are utilized to find common definite matrices P and feedback gains K satisfying stability conditions derived in terms of Lyapunov direct method. Finally, the effectiveness and the feasibility of the proposed controller design method is demonstrated through numerical simulations on the chaotic and resonant systems.
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Abdelmalki, Fouad, and Najat Ouaaline. "T-S Fuzzy observer and controller of Doubly-Fed Induction Generator." International Journal of Power Electronics and Drive Systems (IJPEDS) 7, no. 3 (September 1, 2016): 617. http://dx.doi.org/10.11591/ijpeds.v7.i3.pp617-624.
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This paper aims to ensure a stability and observability of doubly fed induction generator DFIG of a wind turbine based on the approach of fuzzy control type T-S PDC (Parallel Distributed Compensation) which determines the control laws by return state and fuzzy observers. First, the fuzzy TS model is used to precisely represent a nonlinear model of DFIG proposed and adopted in this work. Then, the stability analysis is based on the quadratic Lyapunov function to determine the gains that ensure the stability conditions. The fuzzy observer of DFIG is built to estimate non-measurable state vectors and the estimated states converging to the actual statements. The gains of observatory and of stability are obtained by solving a set of linear matrix inequality (LMI). Finally, numerical simulations are performed to verify the theoretical results and demonstrate satisfactory performance.
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Ghorbel, Chekib, Amira Tiga, and Naceur Benhadj Braiek. "Proportional PDC Design-Based Robust Stabilization and Tracking Control Strategies for Uncertain and Disturbed T-S Model." Complexity 2020 (March 9, 2020): 1–9. http://dx.doi.org/10.1155/2020/8910132.
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This paper presents a proportional parallel distributed compensation (PPDC) design to the robust stabilization and tracking control of the nonlinear dynamic system, which is described by the uncertain and perturbed Takagi–Sugeno (T-S) fuzzy model. The proposed PPDC control design can greatly reduce the number of adjustable parameters involved in the original PDC and separate them from the feedback gain. Furthermore, the process of finding the common quadratic Lyapunov matrix is simplified. Then, the global asymptotic stability with decay rate and disturbance attenuation of the closed-loop T-S model affected by uncertainties and external disturbances are discussed using the H∞ synthesis and linear matrix inequality (LMI) tools. Finally, to illustrate the merit of our purpose, numerical simulation studies of stabilizing and tracking an inverted pendulum system are presented.
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You, Guodong, Tao Xu, Honglin Su, Xiaoxin Hou, Xue Wang, Chengxin Fang, and Jisheng Li. "Fault-Tolerant Control of Doubly-Fed Wind Turbine Generation Systems under Sensor Fault Conditions." Energies 12, no. 17 (August 22, 2019): 3239. http://dx.doi.org/10.3390/en12173239.
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This paper studies the fault-tolerant control problem of uncertain doubly-fed wind turbine generation systems with sensor faults. Considering the uncertainty of the system, a fault-tolerant control strategy based on a T-S fuzzy observer is proposed. The fuzzy observer is established based on the T-S fuzzy model of the uncertain nonlinear system. According to the comparison and analysis of residual between the state estimation of the fuzzy observer output and the measured value of the real sensor, a fault detection and isolation (FDI) based on T-S fuzzy observer is designed. Then by using a Parallel Distributed Compensation (PDC) method we design the robust fuzzy controller. Finally, the necessary and sufficient conditions for the stability of the closed-loop system are proved by quoting Lyapunov stability theory. The simulation results verify the effectiveness of the proposed control method.
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Gritli, Wafa, Hajer Gharsallaoui, Mohamed Benrejeb, and Pierre Borne. "Electronic Throttle Valve Takagi-Sugeno Fuzzy Control Based on Nonlinear Unknown Input Observers." International Journal of Computers Communications & Control 13, no. 5 (September 29, 2018): 808–23. http://dx.doi.org/10.15837/ijccc.2018.5.3281.
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This paper deals with the synthesis of a new fuzzy controller applied to Electronic Throttle Valve (ETV) affected by an unknown input in order to enhance the rapidity and accuracy of trajectory tracking performance. Firstly, the Takagi-Sugeno (T-S) fuzzy model is employed to approximate this nonlinear system. Secondly, a novel Nonlinear Unknown Input Observer (NUIO)-based controller is designed by the use of the concept of Parallel Distributed Compensation (PDC). Then, based on Lyapunov method, asymptotic stability conditions of the error dynamics are given by solving Linear Matrix Inequalities (LMIs). Finally, the effectiveness of the proposed control strategy in terms of tracking trajectory and in the presence of perturbations is verified in comparison with a control strategy based on Unknown Input Observers (UIO) of the ETV described by a switched system for Pulse-Width-Modulated (PWM) reference signal.
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Borodin, M. A. "Simulating the Profile of a Radio Altimeter Echo Signal." Journal of the Russian Universities. Radioelectronics 25, no. 4 (September 29, 2022): 52–62. http://dx.doi.org/10.32603/1993-8985-2022-25-4-52-62.
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Introduction. Radio altimeters are used for remote monitoring of the Earth and sea surface based on the signal power profile, i.e. the dependence of echo signal power vs time, averaged over a set of sounding measurements. Calculation of the power profile in the presence of sea waves requires information about the probability distribution density (PDD) of sea surface applicates. The existing applicate PDD models (linear, Gram-Charlier, combined model) are characterized by certain drawbacks leading to errors in assessing sea surface parameters based on the signal power profile. In this work, the echo signal of a radio altimeter is simulated using the nonlinear Creamer model for calculating sea surface applicates.Aim. To test the validity of the non-linear Creamer model for obtaining the PDD of sea surface applicates in the problem of simulating the power profile of a radio altimeter echo signal while providing an unambiguous relationship between the statistical parameters of PDD and wind speed.Materials and methods. A two-dimensional sea surface of finite dimensions with an Elfohaily spatial spectrum was simulated in the MATLAB environment based on both linear and non-linear Creamer models.Results. Under small wind speed values of up to 3 m/s, the PDD of sea surface applicates formed by the non-linear Creamer model obey the Gauss law. For the non-linear model, an increase in the duration of an radio altimeter echo signal along with an increase in the wind speed was shown to be 5.47 % slower on average compared to that for the linear model. The time shift of the leading edge of an echo signal for the nonlinear model relative to the linear model due to the PDD transformation may reach 70 ns, which is equivalent to 10.5 m. The conducted study confirmed the validity of the non-linear Creamer model for obtaining the PDD of sea surface applicates when simulating the power profile of a radio altimeter echo signal.Conclusion. The non-linear Creamer model can be recommended for simulating an echo signal power profile for use as input data when estimating the parameters of a radio altimeter echo signal at the stage of post-processing.
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Yang, Li, Fuzhao Yang, Weitao Sheng, Kun Zhou, and Tianmin Huang. "Fuzzy Chaos Control of Fractional Order D-PMSG for Wind Turbine with Uncertain Parameters by State Feedback Design." Energies 14, no. 21 (November 5, 2021): 7369. http://dx.doi.org/10.3390/en14217369.
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To research the chaotic motion problem of the direct-drive permanent magnet synchronous generator (D-PMSG) for a wind turbine with uncertain parameters and fractional order characteristics, a control strategy established upon fuzzy state feedback is proposed. Firstly, according to the working mechanism of D-PMSG, the Lorenz nonlinear mathematical model is established by affine transformation and time transformation. Secondly, fractional order nonlinear systems (FONSs) are transformed into linear sub-model by Takagi–Sugeno (T-S) fuzzy model. Then, the fuzzy state feedback controller is designed through Parallel Distributed Compensation (PDC) control principle to suppress the chaotic motion. By applying the fractional Lyapunov stability theory (FLST), the sufficient conditions for Mittag–Leffler stability are formulated in the format of linear matrix inequalities (LMIs). Finally, the control performance and effectiveness of the proposed controller are demonstrated through numerical simulations, and the chaotic motions in D-PMSG can be eliminated quickly.
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Rahmawaty, Made. "Modeling, Simulation, and Stabilization of Two Wheels Inverted Pendulum Robot Using Hybrid Fuzzy Control." Indonesian Journal of electronics, electromedical engineering, and medical informatics 3, no. 3 (August 27, 2021): 93–98. http://dx.doi.org/10.35882/ijeeemi.v3i3.2.
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Two wheels inverted pendulum robot has the same characteristics as inverted pendulum, which are unstable and nonlinear. Nonlinear systems can often be linearized by approximating them by a linear system obtained by expanding the nonlinear solution in a series, and then linear techniques can be used. Fuzzy logic control is the famous nonlinear controller that has been used by researchers to analyze the performance of a system due to the easiness to understand the nature of the controller. This research discusses about two wheels inverted pendulum robot design using hybrid fuzzy control. There are two types of fuzzy control, namely Fuzzy Balanced Standing Control (FBSC) to maintain stability and Fuzzy Traveling and Position Control (FTPC) to maintain position. Based on Takagi-Sugeno (T-S) fuzzy model on two wheels inverted pendulum robot, FBSC control used Parallel Distributed Compensation (PDC) with pole placement technic. Based on two wheels inverted pendulum robot movement characteristics, FTPC was designed using Mamdani Fuzzy architecture. FTPC control is used to help FBSC to maintain robot stability and to adjust to the desired position. Simulation result shows that controller for two wheels inverted pendulum robot can stabilize pendulum angle in 0 radian and close to the desired position
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Liu, Chuang, Jinxia Wu, and Weidong Yang. "Robust $ {H}_{\infty} $ output feedback finite-time control for interval type-2 fuzzy systems with actuator saturation." AIMS Mathematics 7, no. 3 (2022): 4614–35. http://dx.doi.org/10.3934/math.2022257.
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<abstract><p>The finite-time $ {H_\infty } $ performance of the interval type-2 Takagi-Sugeno fuzzy system (IT2 T-S) in presence of immeasurable states and input saturation is investigated. At first, an observer associated with IT2 T-S states is considered to address the problem of immeasurable states. After that, the input saturation is described based on the polyhedron model, and accordingly, a robust $ {H_\infty } $ observer-based finite-time controller is proposed via non-PDC algorithm. Then, on the basis of the Lyapunov function method and LMIs theory, the sufficient conditions for the finite time stability of fuzzy systems are derived. At last, the feasibility of the designed algorithm is verified by two examples of the nonlinear mass-spring-damping system and tunnel diode circuit system, respectively.</p></abstract>
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Qiao, Liang, and Zhaomin Lv. "Finite-Time Admissibility and Controller Design for T-S Fuzzy Stochastic Singular Systems with Distinct Differential Term Matrices." Complexity 2021 (July 21, 2021): 1–12. http://dx.doi.org/10.1155/2021/2858980.
Full textAbstract:
The finite-time admissibility analysis and controller design issues for extended T-S fuzzy stochastic singular systems (FSSSs) with distinct differential term matrices and Brownian parameter perturbations are discussed. When differential term matrices are allowed to be distinct in fuzzy rules, such fuzzy models can describe a wide class of nonlinear stochastic systems. Using fuzzy Lyapunov function (FLF), a new and relaxed sufficient condition is proposed via strict linear matrix inequalities (LMIs). Different from the existing stability conditions by FLF, the derivative bounds of fuzzy membership functions are not required in this condition. Based on admissibility analysis results, a design method for parallel distribution compensation (PDC) controller of FSSSs is given to guarantee the finite-time admissibility of the closed-loop system. Finally, the feasibility and effectiveness of the proposed methods in this article are illustrated with three examples.