Approximation of nonlinear systems having outputs

Suppose we have a nonlinear system with output $$\begin{gathered} \dot x = f(x) + g(x)u, \hfill \\ y = h(x), \hfill \\ \end{gathered}$$ an open setO of state spaceR ⁿ , and a positive integerk. We find conditions onf,g, andh so that for eachx ₀ ε O there is ann-dimensional affine linear system, which depends onx ₀ but not onu, having the property that the output time responses (starting at the statex ₀ of the original nonlinear system and this approximating linear system agree through orderk for any admissable controlu. Several possible applications of our results are examined.     

Extended Robust \mathcal{H}_{2} and \mathcal{H}_{\infty} Filter Design for Discrete Time-Invariant Linear Systems with Polytopic Uncertainty

This paper is concerned with the problem of robust $\mathcal{H}_{2}$ and $\mathcal{H}_{\infty}$ filter design for discrete-time linear time-invariant systems with polytopic parameter uncertainties. Less conservative robust $\mathcal{H}_{2}$ and $\mathcal{H}_{\infty}$ filter design procedures are proposed in terms of single-parameter minimization problems with linear matrix inequality constraints. To this end, we generalize the filter structures available in the literature to date in such a way that the filter’s next state is built by summing the filter’s states over several samples from the past to the present. For stability of the filtering error system, the homogeneous polynomial parameter-dependent Lyapunov functions are employed. Finally, illustrative examples are given to demonstrate the merits of the proposed methods.     

Some results concerning the abstract degenerate nonlinear equationD t Mu(t) +Lu(t)=f(t,Ku(t))

We apply some abstract existence results, proved in Sections 1 and 2 of the paper, to some cases of the following problem (P), formally stated as (P) $$\left\{ \begin{gathered} D_t \left( {Mu\left( t \right)} \right) + Lu\left( t \right) = f\left( {t,Ku\left( t \right)} \right) 0< t \leqslant \tau \hfill \\ Mu\left( t \right) \to u_0 as t \to 0 + \hfill \\ \end{gathered} \right.$$ whereK,L,M are linear and/is a nonlinear term. The unknown functionu is required to lie in various functional spaces.     

Multiple time scales for nonlinear systems

In this paper we extend the results on the multiple time-scale structure for linear autonomous systems of the form $$\dot x = A( \in )x$$ (cf. Coderchet al. [1]) to nonlinear autonomous systems. Our main result is in obtaining conditions under which the linearized system and the nonlinear system around an equilibrium point have the same time-scale structure.     

A Rate \text{ M }_{\mathrm{T}} Full Diversity STF Block Coded 4\times 4 MIMO-OFDM System with Reduced Complexity

Multiple input multiple output (MIMO) communication systems with orthogonal frequency division multiplexing (OFDM) has a great role to play for 4G broadband wireless communications. In this paper, a space time frequency (STF) code is presented with reduced decoder complexity and to achieve code rate $\text{ M }_\mathrm{T}$ with full diversity of $\text{ M }_{\mathrm{T}} \text{ M }_{\mathrm{R}} \text{ N }_{\mathrm{b}}$ L i.e., product of number of transmit antennas ( $\text{ M }_\mathrm{T}$ ), receive antennas $(\text{ M }_{\mathrm{R}})$ , fading blocks $(\text{ N }_{\mathrm{b}})$ and channel taps (L). The maximum achievable diversity with high rate of STF block coded MIMO-OFDM is analyzed and verified by simulation results. The decoder complexity is resolved by employing several approaches like maximum likelihood (ML), sphere decoder (SD) and array processing. The performance of STF code is compared with existing layered algebraic STF code in terms of decoder complexity and bit error rate (BER). Further, the closed form expressions for BER performance of STFBC MIMO-OFDM systems are derived and evaluated for frequency selective block fading channels with MPSK constellations.     

WCDMA Multiclass Downlink Capacity and Interference Statistics of Cigar-Shaped Microcells in Highways

In this paper, the multiclass downlink capacity and the interference statistics of the sectors of a cigar-shaped microcells using wideband code-division multiple-access with soft handover mode are analyzed. The two-slope propagation model with log-normal shadowing is used in the analysis where a model of 8 cigar-shaped microcells is utilized. The performance of the downlink is studied for different [sector range R, standard deviation of the shadowing ( $\sigma _{1}$ and $\sigma _{2})$ and propagation exponents ( $\text{ s}_{1}$ and $\text{ s}_{2})$ ]. It is found that increasing the sector range from 500 to 1,000 m will increase the sector downlink capacity. Also, it is found that increasing the value of the propagation parameters ( $\sigma _{1}$ and $\sigma _{2})$ will reduce the downlink sector capacity. It is noticed that, the effect of changing the propagation exponent $\text{ s}_{1}$ is null while increasing the propagation exponent $\text{ s}_{2}$ will increase the downlink capacity.     

Decentralized \mathcal{L}_{2}–\mathcal{L}_{\infty} Filtering for Interconnected Markovian Jump Systems with Delays

This paper deals with the problem of decentralized $\mathcal{L}_{2}$ – $\mathcal{L}_{\infty}$ filtering for a class of interconnected (or large-scale) Markovian jump systems with constant time delays. The purpose is to present delay-dependent conditions for the existence of mode-dependent decentralized filters, which guarantees that the filtering error system is stochastically stable with a prescribed $\mathcal{L}_{2}$ – $\mathcal{L}_{\infty}$ disturbance attenuation level. Such a purpose is achieved by using a mode-dependent centralized Lyapunov functional together with the so-called Jensen’s inequality. The obtained synthesis conditions are expressed in terms of linear matrix inequalities (LMIs), which leads to a convex design method for the concerned filters. An example including numerical and simulation results is provided finally to illustrate the effectiveness of the proposed design method.     

Influence of a strong longitudinal static electric field on free carrier faraday effect in an n-type InSb at room temperature at submillimeter wave frequencies

The expression for free carrier Faraday rotation θ and for ellipticity Δ, as the function of the applied parallel static electric field \(\mathop {E_0 }\limits_ \to \) and static magnetic field \(\mathop {B_0 }\limits_ \to \) for a given value of wave angular frequency and electron concentration N₀, are obtained and theoretically analyzed with the aid of one-dimensional linearized wave theory and Kane's non-parabolic isotropic dispersion law. It is shown that the maximum Faraday rotation occurs near the cyclotron resonance condition, which can be expressed as \(\chi \omega = \omega _{ce} \) , where \(\chi = 1{1 \mathord{\left/ {\vphantom {1 {\sqrt {1 - ({{v_0 } \mathord{\left/ {\vphantom {{v_0 } {v_c }}} \right. \kern-0em} {v_c }})^2 } }}} \right. \kern-0em} {\sqrt {1 - ({{v_0 } \mathord{\left/ {\vphantom {{v_0 } {v_c }}} \right. \kern-0em} {v_c }})^2 } }}\) , \(v_c = \sqrt {{{\varepsilon _g } \mathord{\left/ {\vphantom {{\varepsilon _g } {2m}}} \right. \kern-0em} {2m}}} *\) , and \(\omega _{ce} = ({{eB_0 } \mathord{\left/ {\vphantom {{eB_0 } {m*}}} \right. \kern-0em} {m*}})\) . Here m^* and e denote the effective mass and charge of electron, respectively. ?_g is the forbidden bandgap of semiconductor. v₀ is the carrier drift velocity, which is a non-linear function of E₀ in high field condition. A possibility of a simple way of determining the non-linear “v₀ vs E₀” characteristics of semiconductors by the measurement of Faraday rotation is also discussed.     

A novel image segmentation model with an edge weighting function

A variational model for image segmentation consists of a data term and a regularization term. Usually, the data term is chosen as squared $\text{ L }_{2}$ norm, and the regularization term is determined by the prior assumption. In this paper, we present a novel model in the framework of MAP (maximum a posteriori). A new iteratively reweighted $\text{ L }_{2}$ norm is used in the data term, which shares the advantages of $\text{ L }_{2}$ and mixed $\text{ L }_{21}$ norm. An edge weighting function is addressed in the regularization term, which enforces the ability to reduce the outlier effects and preserve edges. An improved region-based graph cuts algorithm is proposed to solve this model efficiently. Numerical experiments show our method can get better segmentation results, especially in terms of removing outliers and preserving edges.     

Soft X-ray Absorption and Photoemission Spectroscopy Study of Cobalt-Based Thermoelectric Oxides: \hbox{Ca}_3\hbox{Co}_4\hbox{O}_9, \hbox{Ca}_3\hbox{Co}_2\hbox{O}_6, and \hbox{Bi}_{2}\hbox{Sr}_{2}\hbox{Co}_2\hbox{O}_{y}

The electronic structures of Co-based potential thermoelectric (TE) oxides, including $\hbox{Ca}_3\hbox{Co}_4\hbox{O}_9$ and $\hbox{Bi}_{2}\hbox{Sr}_{2}\hbox{Co}_2\hbox{O}_{y}$ (y = 8 + δ) single crystals and polycrystalline $\hbox{Ca}_3\hbox{Co}_2\hbox{O}_6$ , have been investigated by employing soft x-ray absorption spectroscopy (XAS) and photoemission spectroscopy (PES). Co 2p XAS measurements show that Co ions are nearly trivalent ( $\hbox{Co}^{3+}$ ) in all of these Co-based TE oxides with a small mixture of $\hbox{Co}^{4+}$ ions in $\hbox{Bi}_{2}\hbox{Sr}_{2}\hbox{Co}_2\hbox{O}_{y}$ . Valence-band PES and O 1s XAS measurements show that the occupied Co 3d states are located at the top of the valence bands and that the lowest unoccupied states have the primarily Co 3d character, respectively. These findings suggest the importance of the Co 3d electronic structures in determining TE properties of these Co-based oxides.     

A Novel Haar Wavelet-Based BPSK OFDM System Robust to Spectral Null Channels and with Reduced PAPR

Orthogonal frequency-division multiplexing (OFDM) has lately gained a great deal of attention and is considered as a strong candidate for many next-generation wireless communication systems. However, OFDM is very sensitive to nonlinear effects due to the high peak-to-average power ratio (PAPR) owned by the transmitted signals and does not show robustness to spectral null channels. This paper proposes a novel BPSK OFDM system based on Haar wavelet transformation. The Haar wavelet transformation operates decomposition over the data symbol sequence after binary-to-complex mapping shows that half of the data symbols are zeros and the rest are either ${\sqrt{2}}$ or ${-\sqrt{2}}$ . Then, we have the PAPR reduced by ${10\log_{10} 2\approx 3}$ ?dB at most, compared with the conventional OFDM system. We also propose a novel decoding algorithm for the proposed OFDM system to show robustness to spectral null channels, and derive the bit error rate (BER) performance in theory from unbalanced QPSK modulation. Finally, we compare BER performance of our proposed OFDM with the conventional OFDM over different channels to show the excellent performance of our proposed OFDM system.     

Structure theorems for nonlinear systems

One of the main results is a proposition to the effect that under some typically mild conditions finite sums of the form $$\sum\limits_\ell {K_\ell \sigma } \left[ {\sum\limits_m {\eta _{\ell m} Q_m (\cdot) + \rho _\ell } } \right]$$ are dense in an important sense in the set of shift-invariant approximately-finite-memory mapsG(·) that take a certain type of subsetU ofR intoR, whereR is the set of real-valued functions defined onR ⁿ orZ ⁿ . Here theQ _m (·) are linear, σ is any element of a certain set of nonlinear maps fromR toR, and the κ_?, ρ_?, and η_?m are real constants. Approximate representations comprising only affine elements and lattice nonlinearities are also presented.     

Bandgap and lattice constant of GaInAsP as a function of alloy composition

Published data for the composition dependence of the room-temperature bandgap (E_g) and lattice constant (a_o) in the pseudobinary Ga_yIn_1-yAs, Ga_yIn_1-yP, GaAs_xP_l-x, and InAs_xP_l-x systems have been used to derive the following equations for the quaternary Ga_yIn_l-yAs_x P_l-x, alloys: $$\begin{gathered} a_o ({\AA}) = 5.87 + 0.18x - 0.42y + 0.02xy \hfill \\ E_g (eV) = 1.35 - x + 1.4y - 0.33xy - (0.758 - 0.28x)y(1 - y) \hfill \\ - (0.101 + 0.109y) x(1 - x). \hfill \\ \end{gathered} $$ Available experimental data are in excellent agreement with these equations.     

Application of photonic crystal ring resonator nonlinear response for full-optical tunable add–drop filtering

In this paper, we investigate the application of Kerr-like nonlinear photonic crystal (PhC) ring resonator (PCRR) for realizing a tunable full-optical add–drop filter. We used silicon (Si) nano-crystal as the nonlinear material in pillar-based square lattice of a 2DPhC. The nonlinear section of PCRR is studied under three different scenarios: (1) first only the inner rods of PCRR are made of nonlinear materials, (2) only outer rods of PCRR have nonlinear response, and (3) both of inner and outer rods are made of nonlinear material. The simulation results indicate that optical power required to switch the state of PCRR from turn-on to turn-off, for the nonlinearity applied to inner PCRR, is at least \(2000\, \hbox {mW}{/}\upmu \hbox {m}^{2}\) and, for the nonlinearity applied to outer PCRR, is at least \(3000\, \hbox {mW}{/}\upmu \hbox {m}^{2}\) which corresponds to refractive index change of \(\Delta n_\mathrm{NL }= 0.085\) and \(\Delta n_\mathrm{NL }= 0.15\), respectively. For nonlinear tuning of add–drop filter, the minimum power required to 1 nm redshift the center operating wavelength \((\lambda _{0} = 1550\, \hbox {nm})\) for the inner PCRR scenario is \(125\, \hbox {mW}{/}\upmu \hbox {m}^{2}\) (refractive index change of \(\Delta n_\mathrm{NL}= 0.005)\). Maximum allowed refractive index change for inner and outer scenarios before switch goes to saturation is \(\Delta n_\mathrm{NL }= 0.04\) (maximum tune-ability 8 nm) and \(\Delta n_\mathrm{NL }= 0.012\) (maximum tune-ability of 24 nm), respectively. Performance of add–drop filter is replicated by means of finite-difference time-domain method, and simulations displayed an ultra-compact size device with ultra-fast tune-ability speed.     

Effect of the stoichiometric composition of the Si(111)\sqrt {21} × \sqrt {21} -(Au, Ag) surface phase on substrate conductivity

The conductivity of a silicon substrate with a Si(111) $\sqrt {21} $ × $\sqrt {21} $ -(Au, Ag) surface phase is studied. It is found that the surface conductivity of such a substrate varies depending on the ratio of the amounts of gold and silver in the given structure. An analysis of the behavior of the Si(111) $\sqrt {21} $ × $\sqrt {21} $ -(Au, Ag) surface conductivity during silver adsorption indicates the effect of a space-charge layer in the surface region of the substrate on the measurement results.     

Thiele’s continued fractions in digital implementation of noninteger differintegrators

A rational approximation is the preliminary step of all the indirect methods for implementing digital fractional differintegrators s ^??, with ${\nu \in \mathbb{R}, 0<|\nu| <1 }$ , and where ${s \in \mathbb{C}}$ . This paper employs the convergents of two Thiele??s continued fractions as rational approximations of s ^??. In a second step, it uses known s-to-z transformation rules to obtain a rational, stable, and minimum-phase z-transfer function, with zeros interlacing poles. The paper concludes with a comparative analysis of the quality of the proposed approximations in dependence of the used s-to-z transformations and of the sampling period.     

Approximating linearizations for nonlinear systems

Given a nonlinear control system $$\dot x(t) = f(x(t)) + \sum\limits_{i = 1}^m {u_i (t)g_i (x(t))}$$ on ? ⁿ and a pointx ₀ in ? ⁿ , we want to approximate the system nearx ₀ by a linear system. Of course, one approach is to use the usual Taylor series linearization. However, the controllability properties of both the nonlinear and linear systems depend on certain Lie brackets of the vector field under consideration. This suggests that we should construct a linear approximation based on Lie bracket matching atx ₀. In general, the linearizations based on the Taylor method and the Lie bracket approach are different. However, under certain mild assumptions, we show that there is a coordinate system for ? ⁿ nearx ₀ in which these two types of linearizations agree. We indicate the importance of this agreement by examining the time responses of the nonlinear system and its linear approximation and comparing the lower-order kernels in Volterra expansions of each.     

δ-Bit serial binary addition with linear threshold networks

In this paper we investigate δ-bit serial addition in the context of feed-forward linear threshold gate based networks. We show that twon-bit operands can be added in $2\left\lceil {\sqrt n } \right\rceil $ overall delay with a feed-forward network constructed with $\left\lceil {\sqrt n } \right\rceil + 1$ linear threshold gates and $\frac{1}{2}\left( {5\left\lceil {\sqrt n } \right\rceil ^2 + 9\left\lceil {\sqrt n } \right\rceil } \right) + 2$ latches. The maximum weight value is $2^{\left\lceil {\sqrt n } \right\rceil } $ and the maximum fan-in is $3\left\lceil {\sqrt n } \right\rceil + 1$ . We also investigate the implications our scheme have to the performance and the cost under small weights and small fan-in requirements. We deduce that if the weight values are to be limited by a constantW, twon-bit operands can be added in $\left[ {\log W} \right] + \tfrac{n}{{\left[ {\log W} \right]}}$ overall delay with a feed-forward network that has the implementation cost [logW]+1, in terms of linear threshold gates, $\tfrac{1}{2}(5[\log W]^2 + 9[\log W]) + 2$ in terms of latches and a maximum fan-in of 3[logW]+1. We also prove that, if the fan-in values are to be limited by a constantF+1, twon-bit operands can be added in $[\tfrac{F}{3}] + \tfrac{n}{{[\tfrac{F}{3}]}}$ overall delay with a feed-forward network that has the implementation cost $[\tfrac{F}{3}] + 1$ , in terms of linear threshold gates, $\tfrac{1}{2}(5[\tfrac{F}{3}]^2 + 9[\tfrac{F}{3}]) + 2$ in terms of latches, and a maximum weight value of $2^{[\tfrac{F}{3}]} $ . An asymptotic bound of $O(\tfrac{n}{{\log n}})$ is derived for the addition overall delay in the case that the weight values have to be linearly bounded, i.e., in the order ofO(n). The implementation cost in this case is in the order ofO(logn), in terms of linear threshold gates, and in the order ofO(log² n), in terms of latches. The maximum fan-in is in the order ofO(logn). Finally, a partition technique, that substantially reduces the overall cost of the implementation for all the schemes in terms of delay, latches, weights, and fan-in with some few additional threshold gates, is also presented.     

Four-Dimensional Gallant–Lambert–Vanstone Scalar Multiplication

The GLV method of Gallant, Lambert, and Vanstone (CRYPTO 2001) computes any multiple kP of a point P of prime order n lying on an elliptic curve with a low-degree endomorphism Φ (called GLV curve) over $\mathbb{F}_{p}$ as $$kP = k_1P + k_2\varPhi(P) \quad\text{with } \max \bigl\{ |k_1|,|k_2| \bigr\} \leq C_1\sqrt{n} $$ for some explicit constant C ₁>0. Recently, Galbraith, Lin, and Scott (EUROCRYPT 2009) extended this method to all curves over $\mathbb{F}_{p^{2}}$ which are twists of curves defined over $\mathbb{F}_{p}$ . We show in this work how to merge the two approaches in order to get, for twists of any GLV curve over $\mathbb{F}_{p^{2}}$ , a four-dimensional decomposition together with fast endomorphisms Φ,Ψ over $\mathbb{F}_{p^{2}}$ acting on the group generated by a point P of prime order n, resulting in a proven decomposition for any scalar k∈[1,n] given by $$kP=k_1P+ k_2\varPhi(P)+ k_3\varPsi(P) + k_4\varPsi\varPhi(P) \quad \text{with } \max_i \bigl(|k_i| \bigr)< C_2\, n^{1/4} $$ for some explicit C ₂>0. Remarkably, taking the best C ₁,C ₂, we obtain C ₂/C ₁<412, independently of the curve, ensuring in theory an almost constant relative speedup. In practice, our experiments reveal that the use of the merged GLV–GLS approach supports a scalar multiplication that runs up to 1.5 times faster than the original GLV method. We then improve this performance even further by exploiting the Twisted Edwards model and show that curves originally slower may become extremely efficient on this model. In addition, we analyze the performance of the method on a multicore setting and describe how to efficiently protect GLV-based scalar multiplication against several side-channel attacks. Our implementations improve the state-of-the-art performance of scalar multiplication on elliptic curves over large prime characteristic fields for a variety of scenarios including side-channel protected and unprotected cases with sequential and multicore execution.     

\mathcal{H}_{\infty} Filtering for Short-Time Switched Discrete-Time Linear Systems

In this paper, the $\mathcal{H}_{\infty}$ filtering problem for a class of short-time switched discrete-time linear systems is investigated. For such systems, switching always occurs in some short interval. Since the error state may attain large unacceptable values in short-time switching intervals, besides the asymptotic stability of error dynamics, the boundedness of error state is also significant for short-time switched systems. Thus the designed filter is composed of two parts: asymptotic filter, based upon the existing results, ensures the asymptotic stability of the system during normal, relatively long interval, and finite-time filter ensures system to be finite-time bounded during the short interval of switching, which is the main concern in this paper. By introducing the concept of finite-time boundedness, the proposed filter is formulated as a set of sub-filters ensuring the error dynamics $\mathcal{H}_{\infty}$ finite-time bounded in the short switching interval. Finally, a numerical example is provided to illustrate the effectiveness of this approach.