Optimization Of Linear Systems Research Articles

Bayes Synthesis of Linear Nonstationary Stochastic Systems by Wavelet Canonical Expansions

This article is devoted to analysis and optimization problems of stochastic systems based on wavelet canonical expansions. Basic new results: (i) for general Bayes criteria, a method of synthesized methodological support and a software tool for nonstationary normal (Gaussian) linear observable stochastic systems by Haar wavelet canonical expansions are presented; (ii) a method of synthesis of a linear optimal observable system for criterion of the maximal probability that a signal will not exceed a particular value in absolute magnitude is given. Applications: wavelet model building of essentially nonstationary stochastic processes and parameters calibration.

Influence of Power Take-Off Modelling on the Far-Field Effects of Wave Energy Converter Farms

The study of the potential impact of wave energy converter (WEC) farms on the surrounding wave field at long distances from the WEC farm location (also know as “far field” effects) has been a topic of great interest in the past decade. Typically, “far-field” effects have been studied using phase average or phase resolving numerical models using a parametrization of the WEC power absorption using wave transmission coefficients. Most recent studies have focused on using coupled models between a wave-structure interaction solver and a wave-propagation model, which offer a more complex and accurate representation of the WEC hydrodynamics and PTO behaviour. The difference in the results between the two aforementioned approaches has not been studied yet, nor how different ways of modelling the PTO system can affect wave propagation in the lee of the WEC farm. The Coastal Engineering Research Group of Ghent University has developed both a parameterized model using the sponge layer technique in the mild slope wave propagation model MILDwave and a coupled model MILDwave-NEMOH (NEMOH is a boundary element method-based wave-structure interaction solver), for studying the “far-field” effects of WEC farms. The objective of the present study is to perform a comparison between both numerical approaches in terms of performance for obtaining the “far-field” effects of two WEC farms. Results are given for a series of regular wave conditions, demonstrating a better accuracy of the MILDwave-NEMOH coupled model in obtaining the wave disturbance coefficient (Kd) values around the considered WEC farms. Subsequently, the analysis is extended to study the influence of the PTO system modelling technique on the “far-field” effects by considering: (i) a linear optimal, (ii) a linear sub-optimal and (iii) a non-linear hydraulic PTO system. It is shown that modelling a linear optimal PTO system can lead to an unrealistic overestimation of the WEC motions than can heavily affect the wave height at a large distance in the lee of the WEC farm. On the contrary, modelling of a sub-optimal PTO system and of a hydraulic PTO system leads to a similar, yet reduced impact on the “far-field” effects on wave height. The comparison of the PTO systems’ modelling technique shows that when using coupled models, it is necessary to carefully model the WEC hydrodynamics and PTO behaviour as they can introduce substantial inaccuracies into the WECs’ motions and the WEC farm “far-field” effects.

The Multi-step Transformation Control and Exponentially Stability for Linear Discrete-time Systems with Additional Control Delay

Background: Few results about the controller of linear discrete-time system with control delay are reported. Methods: By means of the multi-step transformation with memory, the linear discrete-time systems with additional control delay can be transformed to the equivalent linear discrete-time systems without control delay, and the dimension of the transformed system is not increased. By designing the optimal controller of the finite horizon optimal controller of linear discrete time systems without time delay, the controller of linear discrete time systems with additional control delay can be obtained. At the same time, by designing the optimal controller of the infinite horizon optimal controller of linear discrete time systems without time delay, the controller of linear discrete time systems with additional control delay can be obtained as well. Results: The corresponding finite horizon optimal controller has proved to render the closed-loop system exponentially stable. And the corresponding infinite horizon optimal controller has proved to render the closed-loop system exponentially stable when the open-loop system is either controllable or stabilizable. Finally, two examples are used to verify the theoretical results of this paper. Conclusion: The controller design and the exponentially stability of discrete-time linear system with additional state delay will be investigated in the future.

Distributed Model Predictive Control and Optimization for Linear Systems With Global Constraints and Time-Varying Communication

In the article, we study the distributed model predictive control (DMPC) problem for a network of linear discrete-time systems, where the system dynamics are decoupled, the system constraints are coupled, and the communication networks are described by time-varying directed graphs. A novel distributed optimization algorithm called the push-sum dual gradient (PSDG) algorithm is proposed to solve the dual problem of the DMPC optimization problem in a fully distributed way. We prove that the sequences of the primal, and dual variables converge to their optimal values. Furthermore, to solve the implementation issues, stopping criteria are designed to allow early termination of the PSDG Algorithm, and the gossip-based push-sum algorithm is proposed to check the stopping criteria in a distributed manner. It is shown that the optimization problem is iteratively feasible, and the closed-loop system is exponentially stable. Finally, the effectiveness of the proposed DMPC approach is verified via an example.

(1912-5594) Optimal control of linear fuzzy time-variant controlled systems

In this paper, we study linear fuzzy time-variant optimal control systems using the generalized differentiability concept and we present the general form of optimal controls and states. Some examples are provided to illustrate our results.

Geometric Programming for Optimal Positive Linear Systems

This paper studies the parameter tuning problem of positive linear systems for optimizing their stability properties. We specifically show that, under certain regularity assumptions on the parametrization, the problem of finding the minimum-cost parameters that achieve a given requirement on a system norm reduces to a \emph{geometric program}, which in turn can be exactly and efficiently solved by convex optimization. The flexibility of geometric programming allows the state, input, and output matrices of the system to simultaneously depend on the parameters to be tuned. The class of system norms under consideration includes the $H^2$ norm, $H^\infty$ norm, Hankel norm, and Schatten $p$-norm. Also, the parameter tuning problem for ensuring the robust stability of the system under structural uncertainties is shown to be solved by geometric programming. The proposed optimization framework is further extended to delayed positive linear systems, where it is shown that the parameter tunning problem jointly constrained by the exponential decay rate, the $\mathcal L^1$-gain, and the $\mathcal L^\infty$-gain can be solved by convex optimization. The assumption on the system parametrization is stated in terms of posynomial functions, which form a broad class of functions and thus allow us to deal with various interesting positive linear systems arising from, for example, dynamical buffer networks and epidemic spreading processes. We present numerical examples to illustrate the effectiveness of the proposed optimization framework.

Effect of the PTO Damping Force on the 2D Oscillating Buoy Wave Energy Converter Integrated into a Pile-Restrained Floating Breakwater

The aim of the study is to evaluate the parametric effect of a power take-off (PTO) system on an integration system of an oscillating buoy wave energy device with a pile-restrained floating breakwater. To consider the viscous effect, the study is carried out with three conditions using OpenFOAM, i.e., the linear optimal PTO system, the fixed PTO damping coefficient, and the Coulomb PTO system. The results show that the maximum wave energy capture ratio (CWR) of the device can merely reach 0.35 in the present work, when considering the viscous effect. This result is smaller than that obtained from potential flow. In addition, the linear optimal PTO coefficient under viscous flow should be larger than the theoretical optimal coefficient in terms of the potential flow. A comparison of the Coulomb damping model and the linear damping model also shows that the Coulomb damping model has a good performance in terms of both wave energy capture efficiency and coastal protection. Hence, the effect of the PTO system on such an integration system can be an essential factor to achieve an optimal design.

Analysis of the faster-than-Nyquist optimal linear multicarrier system

Faster-than-Nyquist signalization enables a better spectral efficiency at the expense of an increased computational complexity. Regarding multicarrier communications, previous work mainly relied on the study of non-linear systems exploiting coding and/or equalization techniques, with no particular optimization of the linear part of the system. In this article, we analyze the performance of the optimal linear multicarrier system when used together with non-linear receiving structures (iterative decoding and direct feedback equalization), or in a standalone fashion. We also investigate the limits of the normality assumption of the interference, used for implementing such non-linear systems. The use of this optimal linear system leads to a closed-form expression of the bit-error probability that can be used to predict the performance and help the design of coded systems. Our work also highlights the great performance/complexity trade-off offered by decision feedback equalization in a faster-than-Nyquist context.

Stabilization and optimization of linear systems via pathwise state-feedback impulsive control

In this work, we address the stabilization and optimization issues for a class of linear systems in terms of pathwise state-feedback impulsive control. The pathwise state-feedback impulsive mechanism is shown universal in that any asymptotically stabilizable system admits such impulsive control law making the system stable. To improve the system performance, we establish some conditions the optimal impulse instants should satisfy using variational method, and design gradient-based algorithms to search for optimal impulse instants. A schematic algorithm is further designed to compute a series of parameters and construct a new and better impulsive control law. Finally, a numerical example shows the efficiency of the proposed approach.

Linear system construction of multilateration based on error propagation estimation

Iterative localization algorithms are critical part in the control of mobile autonomous robots because they feed fundamental position information to the robots. In a harsh unknown environment, the estimation of environmental noise is hardly obtained during the movement of the robots. It means that the state-of-the-art methods, which increase localization accuracy using error management, are unsuitable. In this paper, we deduced an upper bound of the localization error without knowing the precise model of environment noise when the anchor nodes have position errors. Utilizing the minimum upper bound, we can construct an optimal localization linear system of iterative localization algorithms based on least square. An algorithm of generating localization linear system is proposed by using the minimum upper bound. The algorithm reduces the impact of the shortage of environmental information on localization error propagation. Our simulation results show that the algorithm is insensitive to noise and can improve the localization accuracy by constructing a proper localization linear system with a high probability.

Optimal expansions of multivariable ARX processes on Laguerre bases via the Newton–Raphson method

SummaryIn this article, an optimal linear MIMO system approximation by using discrete‐time MIMO autoregressive with exogenous input (ARX) model is proposed. Each polynomial function of the MIMO ARX model associated with the inputs and with the outputs is expanded on independent Laguerre orthonormal basis. The resulting model is entitled MIMO ARX–Laguerre model. The optimal approximation of which is ensured once the poles characterizing each Laguerre orthonormal basis are set to their optimal values. In this paper, a new method to estimate, from input/output measurements, the optimal Laguerre poles of the MIMO ARX–Laguerre model is proposed. The method consists in applying the Newton–Raphson's iterative technique in which the gradient and the Hessian are expressed analytically. The proposed algorithm is tested on a numerical example and on a benchmark system. Simulation results show the effectiveness of the proposed optimal modeling method. Copyright © 2015 John Wiley & Sons, Ltd.

Nested Iteration and First-Order Systems Least Squares for a Two-Fluid Electromagnetic Darwin Model

In this paper, a two-fluid plasma (TFP) model is presented. The model couples the conservation of momentum and conservation of number density of both ions and electrons to Maxwell's equations. A Darwin approximation of Maxwell is used to eliminate spurious light waves from the model. After scaling and modification, the TFP-Darwin model yields a nonlinear, first-order system of equations whose Frechet derivative is shown to be uniformly $\mathcal{H}^1$-elliptic in a neighborhood of the exact solution. This system is addressed numerically by nested iteration (NI) and a first-order system least squares discretization. An important goal of NI is to produce an approximation that is within the basin of attraction for Newton's method on a relatively coarse mesh and, thus, on all subsequent meshes. $\mathcal{H}^1$ ellipticity yields optimal finite element performance and linear systems amenable to solution with algebraic multigrid. Numerical tests demonstrate the efficacy of this approach, yielding an approximate...

Solving Optimal Control Linear Systems by Using New Third kind Chebyshev Wavelets Operational Matrix of Derivative

In this paper, a new third kind Chebyshev wavelets operational matrix of derivative is presented, then the operational matrix of derivative is applied for solving optimal control problems using, third kind Chebyshev wavelets expansions. The proposed method consists of reducing the linear system of optimal control problem into a system of algebraic equations, by expanding the state variables, as a series in terms of third kind Chebyshev wavelets with unknown coefficients. Example to illustrate the effectiveness of the method has been presented.

Solving Optimal Control Linear Systems by Using New Third kind Chebyshev Wavelets Operational Matrix of Derivative

In this paper, a new third kind Chebyshev wavelets operational matrix of derivative is presented, then the operational matrix of derivative is applied for solving optimal control problems using, third kind Chebyshev wavelets expansions. The proposed method consists of reducing the linear system of optimal control problem into a system of algebraic equations, by expanding the state variables, as a series in terms of third kind Chebyshev wavelets with unknown coefficients. Example to illustrate the effectiveness of the method has been presented.

Architecture and Structure of Robust PID Controllers

In previous papers by the use of the architecture generating property of the optimal linear quadratic tracking theory those problems that give the PI, PD and PID controllers had been presented, thus avoiding heuristics and giving a systematic approach to explanation for their excellent performance. This approach has been used also in derivation of the family of generalized PImDn-1controllers. In this paper by the combined use of the optimal linear quadratic tracking and system sensitivity theories a problem is formulated and solutions are shown that give a family of robust PID controllers. Examples of the control architectures and structures for first and second order systems are presented.

Optimal Linear Multihop System for DF Relaying in a Poisson Field of Interferers

This paper provides a strategy for multihop transmission with decode-and-forward (DF) in a Poisson field of interferers. We analyze the optimal linear multihop system such as optimal resource allocation, optimal placement of the relay nodes, and optimal number of hops for multihop DF relaying which minimize their outage probability.

Quasi-Chebyshev accelerated iteration methods based on optimization for linear systems

In this paper, we present a quasi-Chebyshev accelerated iteration method for solving a system of linear equations. Compared with the Chebyshev semi-iterative method, the main difference is that the parameter ω is not obtained by a Chebyshev polynomial but by optimization models. We prove that the quasi-Chebyshev accelerated iteration method is unconditionally convergent if the original iteration method is convergent, and also discuss the convergence rate. Finally, three numerical examples indicate that our method is more efficient than the Chebyshev semi-iterative method.

A new approach for the optimal fuzzy linear time invariant controlled system with fuzzy coefficients

In this article, a new technique for solving the linear time invariant dynamical system with fuzzy parameters is proposed. This approach is used to solve a fuzzy controlled system by using α-cuts and presentation of numbers in a more compact form by moving to the field of complex numbers. By considering a new feature of the Hamiltonian function and using the Pontryagin maximum principle, the solution of the fuzzy optimal controlled system is considered.

A revisit to inverse optimality of linear systems

In this article, we revisit the problem of inverse optimality for linear systems. By applying certain explicit formulae for coprime matrix fraction descriptions (CMFD) of linear systems, we propose a necessary and sufficient condition for a given stable state feedback law to be optimal for some quadratic performance index. Compared to existing results in the literature, the proposed condition is simpler to check and interpret. Moreover, it reduces the redundancy in the solutions of the associated algebraic Riccati equation (ARE). As a direct application of the proposed results, we consider the problem of inverse optimality of observer-based state feedback. To be specific, for the case where the state is not fully known, we consider the inverse optimality problem of an observer-based state feedback for the closed-loop system augmented by an observer. More precisely, it is shown that the observer-based state feedback is inverse optimal for the closed-loop system with some general forms of cost functions, only if the original state feedback is inverse optimal for the original system with certain cost functions, irrespective of the choice of the observer. This coincides with existing results in the literature. Some other applications of the proposed results are also discussed. We also illustrate the proposed results through an example.

Optimization of linear systems with controllable coefficients

Consideration was given to the characteristic properties of the linear differential equations with controllable coefficients, their dynamic invariants and constraints on the reachability domain that are important for the problems of control, properties of the Pontryagin extremals of this class, and numerical methods of control optimization.