Receivers use Equalisers to counter the ill effects of channel. Ideally, it offers the inverse response of the channel. Designing equalizers for **Linear** **time** **invariant** channels is relatively easier [8]. Designing of equalizers for MC channels is not a simple task. Adaptive filtering has a huge role in signal processing applications [ 9]. Earlier, **linear** adaptive equalizers were frequently used for its low cost implementation and simplicity. These equalizers are more suitable when channel properties are known before hand [9]. In **time** varying channels, the performance of **linear** equalizer is poor. Non **linear** signal processing techniques such as neural **networks** are employed in Bayesian equalizers [10]. Fuzzy filters are explored to understand their efficacy as non **linear** filters in equalization [8,9].

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and synthesizes problem of discrete continues system. Various algebraic stability test algorithms have been proposed for multi dimension system. However, they required huge amount of computation **time** for all. The stability investigation associated with multidimensional digital filters used in the areas like seismology needs to be considered. Other applications arise in obtaining reliability properties of impedances of **networks** and transmission lines which represent multidimensional continuous systems in the form of multidimensional continuous filters. Foremost among them is the stability of two and multi dimension system which find the application in the process of bio medical, sonar and radar data. The study of multi dimension discrete shift **invariant** system has received considerable attention among the researches. The stability problem is an important issue in the design and analyses of multi dimension **linear** discrete system. A huge amount of research was dedicated to developing technique for multi dimension system. Jury proposed the stability test for multi dimension system. It should be motivated by practical applications. It is done mainly by engineers in the mathematical literate. The stability analysis of mul- tidimensional digital system is much more complex than single dimensional system. Result for the general case of any multidimensional realization is still warranted which will the topic of future investigation. The Jury’s inner wise determinant method of stability test is useful for numerical testing of stability and for the design of **linear** **time** **invariant** discrete systems. At the result of this paper multi-dimensional discrete system is converted into single dimensional discrete system. The necessary condition is that the roots of the characteristic equation should lie within the left half of the s-plane is verified. Then the characteristic equation forms the (n + 1) * ( n + 1) matrix and the inner determinant is calculated with the help of inner Jury’s concept and the sufficient condition is the inner determinant value is positive should be verified. This shows that the system is stable.

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These control systems also occur in medical fields. For example in pumps that intravenously administer medication to a patient’s bloodstream. An artificial pancreas is an example of a positive control system since insulin can only be added to a patient’s body, and cannot be extracted via the pump. Many other control systems can be listed which can be categorized as positive control. For example economic systems with nonnegative investements, or in electrical **networks** with diode elements (which have low resistance in one direction and high, ideally infinite, resistance in the other). Also the classical cruise control is an example of positive control since the control system only operates the throttle and not the brakes. 1 The control systems as described in the above examples are commonly represented by a system of differential equations. Stabilization of such systems has been studied extensively in the field of control theory. Consider the control system of n coupled differential equations with p control inputs given by

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As foreshadowed in Chapter 1, periodic controllers used in conjunction with FDLTI plants can offer a new dimension of flexibility in the design process. To recap, they have been used to achieve equivalent state feedback without observers, pole assignment, zero assignment, gain margin improvement, strong and simultaneous stabilisation and the removal of decentralised fixed modes in decentralised control. One of these results which is both theoretically inter esting and practically significant corresponds to the problem of gain margin improvement. The advantage of periodic controllers over LTI controllers in im proving the gain margin for a nonminimum-phase FDLTI plant seems to have been first indicated in [44]. Nevertheless, the result of [44] is only relevant to SISO discrete-**time** FDLTI bicausal plants with periodic discrete-**time** dy namic compensators. Some years later, a similar gain margin result for SISO continuous-**time** FDLTI plants with periodic continuous-**time** dynamic compen sators of a particular form was reported in [49]. At this juncture, it is important to mention that these two results do not involve the implementation of a digital

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under study where part of the delayed dynamics is deleted. Some suﬃcient condi- tions for the system to be g.u.e.s. dependent on delay are also obtained by using the same mathematical outlines. Extensions are given for the case when the system is forced by impulsive inputs and also by considering the closed-loop stabilization of **time**-delay systems of the given class. The paper is organized as follows. Section 2 deals with the class of homogeneous delayed systems under study and with the def- inition of the auxiliary system. Section 3 is devoted to the main uniform stability re- sult and the related ones for some particular auxiliary systems of interest. Some of those systems are deﬁned by considering only delay-free dynamics or either point- delayed, distributed-delayed, or even Volterra-type delayed dynamics together with a delay-free dynamics. Section 4 extends the above results to the presence of impulsive forcing functions. Section 5 is devoted to the stabilization of closed-loop systems of the given class under **linear** state or output-feedback controllers which can include de- lays. Some simple examples are discussed in Section 6 and, ﬁnally, conclusions end the paper.

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of imperfect electronic components, whose values are specified to a limit tolerance (e.g. resistor values often have a tolerance of ±5%) and which may also change with temperature and drift with **time**. As the order of an analog filter increases, and thus its component count, the effect of variable component errors is greatly magnified. In digital filters, the coefficient values are stored in computer memory, making them far more stable and predictable. Because the coefficients of digital filters are definite, they can be used to achieve much more complex and selective designs – specifically with digital filters, one can achieve a lower passband ripple, faster transition, and higher stopband attenuation than is practical with analog filters. Even if the design could be achieved using analog filters, the engineering cost of designing an equivalent digital filter would likely be much lower. Furthermore, one can readily modify the coefficients of a digital filter to make an adaptive filteror a user-controllable parametric filter. While these techniques are possible in an analog filter, they are again considerably more difficult. Digital filters can be used in the design of finite impulse response filters. Analog filters do not have the same capability, because finite impulse response filters require delay elements. Digital filters rely less on analog circuitry, potentially allowing for a better signal-to-noise ratio. A digital filter will introduce noise to a signal during analog low pass filtering, analog to digital conversion, digital to analog conversion and may introduce digital noise due to quantization. With analog filters, every component is a source of thermal noise (such as Johnson noise), so as the filter complexity grows, so does the noise. However, digital filters do introduce a higher fundamental latency to the

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Theorem 1.1 — Let F be a field with characteristic different from two. Let J : α → α J be a fixed non-trivial involutory automorphism on F. Let V be a vector space over F of dimension at least 2. Let T : V → V be an invertible **linear** map. Then V admits a T -**invariant** non-degenerate J-hermitian, resp. J -skew-hermitian form if and only if an elementary divisor of T o is either self-dual, or its dual

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The logic µ(U) is the fixpoint extension of the “Until”-only fragment of **linear**-**time** temporal logic. It also happens to be the stutter-**invariant** fragment of **linear**-**time** µ-calculus µ( ♦ ). We provide complete axiomatizations of µ(U) on the class of finite words and on the class of ω-words. We introduce for this end another logic, which we call µ(♦ Γ ), and which is a variation of µ(♦) where the Next **time** operator

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A novel impedance spectroscopy technique has been developed for high speed single biological particle analysis. A microfluidic cytometer is used to measure the impedance of single micrometre sized latex particles at high speed across a range of frequencies. The setup uses a technique based on maximum length sequence (MLS) analysis, where the **time**-dependent response of the system is measured in the **time** domain and transformed into the impulse response using fast M-sequence transform (FMT). Finally fast Fourier transform (FFT) is applied to the impulse response to give the transfer-function of the system in the frequency domain. It is demonstrated that the MLS technique can give multi-frequency (broad-band) measurement in a short **time** period (ms). The impedance spectra of polystyrene micro-beads are measured at 512 evenly distributed frequencies over a range from 976.5625 Hz to 500 kHz. The spectral information for each bead is obtained in approximately 1 ms. Good agreement is shown between the MLS data and both circuit simulations and conventional AC single frequency measurements.

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Some classes of 2D/nD **linear** systems share strong structural links with, in particular, standard (1D) **linear** systems, e.g. so-called differential and discrete **linear** repetitive processes (see, for exam- ple, [8]) where the common structure assertion arises from structural similarities between the state space models which describe the underlying dynamics. This immediately suggests that nD systems can be studied by direction extension of existing/emerging 1D systems theory. Experience has shown, however, that there are a great many problems in generalizing 1D systems theory to the nD case. Some of these problem are fundamentally algebraic in nature, e.g. the distinction in the nD case between factor primeness, minor primeness, and zero primeness, or the lack of an Euclidean algorithm.

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characterization of this stiffness is needed. The current industrial method for static stiffness determination has several downsides, amongst others its **time** consuming set-up preparation. Principle objective of this paper is to study the effect of vibrations on the material properties of structures, especially automobile Body-in-White (BIW) by using finite element analysis tool. A Body-in-White (BIW) is the automobile designing (or manufacturing) stage where the car body is formed by assembled metal sheets, and the main components as chassis, power train, doors, etc. are not still mounted. By determining the natural frequencies of the automobile body, the probability of failure are dramatically reduced and life is substantially increased. Vibration generated from various sources (engine, road surface, tires, exhaust, etc.) should be considered in the design of a car body. These vibrations travel through transfer systems to the steering wheel, seats and other areas where it is detected by the passengers of the vehicle. Transmission routes must be studied and efforts made to keep transfer systems from amplifying vibration and to absorb it instead.

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A modified PI-PD Smith predictor, which leads to significant improvements in the control of processes with large **time** constants or an integrator or unstable plant transfer functions plus long dead-**time** for reference inputs and disturbance rejections was proposed by Kaya in 2003 [22]. Skogestad developed analytic rules for PID controller tuning that are simple and still result in a good closed-loop behavior [23]. The starting point has been the IMC-PID tuning rules that have achieved widespread industrial acceptance. The rule for integral term has been modified to improve disturbance rejection for integrating processes. Furthermore, rather than deriving separate rules for each transfer function model, there is a single tuning rule for FOPDT or SOPDT model. The only drawback of the method is that the model order reduction is required for higher order systems.

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bounds for model classes with finite covering numbers (a sufficient condition for which is finite VC-dimension of a related class) but additionally require that Y is compact and ` is bounded. Early work in signal processing (Modha and Masry, 1998) proposes predictors based on sequences of parametric models of increasing memory which minimize a complex- ity regularized least squares criterion and establish that these predictors deliver the same statistical performance as oracle predictors. Steinwart and Christmann (2009) prove an or- acle inequality regularized ERM algorithms when observations are α-mixing which are close to the optimal i.i.d. rates. Mohri and Rostamizadeh (2009) give results using Rademacher complexity which are both tighter than those using VC-dimension or covering numbers as well as being computable from the data in many cases. Mohri and Rostamizadeh (2010) and Agarwal and Duchi (2013) consider another family of bounds for φ-mixing and β-mixing sequences when the predictors are algorithmically stable. Many classes of common machine learning algorithms are amenable to either Rademacher or algorithmic-stability bounds: Kernel-regularized methods, support-vector machines, relative-entropy based regulariza- tion, and kernel ridge regression among others. However, methods common to **time**-series such as AR models, ARIMA models, ARCH and GARCH models (Engle, 1982, 2001), state- space models, and other Box-Jenkins type predictors are not because they are not explicitly regularized, the loss functions are not bounded, and the predictions can depend on more than simply a fixed dimensional past. McDonald et al. (2011b) shows that stationarity alone can be used to impose a kernel-type regularization on an AR model, and hence, following the results of Mohri and Rostamizadeh (2009), is amenable to Rademacher complexity for a bounded loss function.

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The paper is organized as follows: The remaining part of section 1 defines the notation used in the paper and section 2 reviews the main definitions and properies of strong stability in the continuous- **time** case. Section 3 defines the notion of strong stability in discrete-**time**, develops numerous necessary and sufficient conditions for three refined strong stability notions and establishes connections between strong stability in the continuous and discrete domains via the bilinear transformation. Section 4 deals with numerous properties of strongly stable systems, establishes the invariance of strong stability under orthogonal transformations and characterizes the class of discrete systems for which strong and asymptotic stability are identical or approximately equivalent notions. Connections between strong stability and skewness of eigen-frame of the state-matrix are also developed in this section, and the Schur form is used for defining parameter-dependent conditions for strong stability. Section 4 also examines examines strong stability for systems subjected to arbitrary coordinate and balancing transformations. Section 5 poses and solves three variants of the strong stabilization problem under state feedback, output injection and output feedback, using easily verifiable necessary and sufficient conditions and gives a complete parametrization of the family of all optimal solutions in each case. The notation used in the paper is standard and is summarized here for convenience. N , R and C denote the sets of natural, real and complex numbers, respectively. The set of complex numbers with negative real part is denoted by C − and is referred to as the open-left-half-plane. The set of complex

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In this paper, a controller design method based on QFT is used for a two-link rigid-flexible manipulator system. To control the manipulator’s end point position for accurate tracking of a desired trajectory, motion of the manipulator is divided to two rigid and flexible parts. To control the rigid part of the motion, nonlinear equations of the motion of a two link rigid manipulator is replaced by a family of uncertain **linear** **time**-**invariant** equivalent systems using Rafeeyan-Sobhani’s method [12] which results in two decoupled transfer functions established in the Laplace domain; one is from input torque of the first motor to the first output hub angle and the other is from input torque of the second motor to the second output hub angle. To control the flexible part of motion, the third equation of the motion of the two- link rigid-flexible manipulator (elastic deflection equation) is treated in the same approach (RS method) and another transfer function is obtained. This transfer function is from input voltage to output tip-deflection of the second link. Since the equations of motion are decoupled by the RS method, the three input-three output robot system is reduced to three uncertain and **linear** single input- single output systems. Then the QFT method (a known robust control method) is used for each of these SISO systems. Thus, three controllers and prefilters are synthesized independently using the QFT method and a diagonal matrix is developed as the system controller. This proposed method can be extended to manipulators with more flexible links. 2. Dynamic Modeling

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Before defining a “mixed” small gain and passivity **time** domain property for nonlinear systems, which then leads us to the main business of the chapter of pro viding the associated input-output feedback interconnection stability result, let us derive a **time** domain analog of the LTI “mixed” small gain and passivity frequency domain property. This will help to motivate the definition of the “mixed” property for nonlinear systems. We must first establish some preliminary mathematics and notation (some of which has been repeated from Chapter 2 for convenience). The field of real numbers is denoted by R. Suppose that X and y are real inner product spaces. The inner product of X is denoted by (•, •) : X x X —» M. A norm for each element of X is defined to be ||/ ||^ = ( / ,/ ) • An important property of inner prod uct spaces is the so-called Cauchy-Schwarz inequality; that is \(f,g)\ < ll/HaHMU V/, g £ X. Suppose that 7i and /C are Hilbert spaces. For a bounded **linear** operator H : 7i —* /C, the Hilbert adjoint : /C —> H of H is defined by (Hh, k) — (h, H~h) for all h £ PC and k £ 1C.

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as a digital filter, an important consideration is the stability of the system. From the control view point stability analysis of a 2-D model is also of interest, since a variety of distributed systems, such as **time**-delay systems, **linear** multi pass processes and systems governed by certain types of partial differential equations, fit quite naturally in the framework of 2-D system theory. All these initial studies of the stability were carried in fre- quency domain (Z domain). More recently, however, the introduction of state-space models for 2-D systems al- lowed the investigation of the stability in the state-space approach. This is of particular interest due to the recent development of design technique using state-space models. The stability test of 2-D recursive digital filter is mostly numerical computation. The problem of the stable region of filter coefficients has been involved. The stability test is carried out by the new stability test theorem we presented in frequency domain which has the limitation of Jury’s and can be used in both conditions of **linear** and non-**linear** systems. Further Jury 1971 had presented the positive inner wise and positive definite symmetric matrices for stability of the system. In this method the formulations of symmetric matrices were very complicated and this criterion was rarely used by en- gineers for high order system. In this present paper a simple and direct scheme is proposed to find stability of **linear** **time** **invariant** discrete systems compared to the Jury (1971) method. The proposed scheme accounts all the coefficient of the unity shifted unit circle one dimensional equivalent characteristic equation in order to form the matrix followed by applying left shifting and right shifting principle to form X and Y matrix respectively. The single square matrix H = X + Y has been constructed from X and Y matrix with respected to Jury’s pro- posal. For an absolute stable system, the H value needs to be positive inner-wise which was identified for all its determinants that started from the centre elements and proceeding outwards the need to be positive. Furthermore, one more necessary condition is proposed along with Jury’s condition for stability.

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Abstract- Convolution and de convolution algorithms play a key role in digital processing applications. They involve many multiplication and division steps and consume a lot of processing **time**. As such, they play a vital role in determining the performance of the digital signal processor. Convolution and de convolution implemented with Vedic mathematics proved fast as compared to those using conventional methods of multiplication and division. This paper presents a novel VHDL implementation of convolution and de convolution algorithm with multiplier using radix-256 booth encoding to reduce the partial product rows by eight fold and carry propagate free redundant binary addition for adding the partial products, thus, contributing to higher speed. The design had been implemented for 32 bit signed and unsigned sequences. The delay was reduced by 18.27%. The entire design was implemented in Xilinx ISE 14.7 targeted towards SPORTON 3E.

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program. This is because each of the three constrains can be checked by testing if a trigono- metric polynomial is non-negative. A simple inner approximation can be constructed by requiring the constraints to hold on an a finite grid of size O(n). One can check that this provides a tight, polyhedral approximation to the set B α , following an argument similar to Appendix C of Bhaskar et al Bhaskar et al. (2013). Projection to this polyhedral takes at most O(n 3.5 ) **time** by **linear** programming and potentially can be made faster by using fast Fourier transform. See Section F for more detailed discussion on why projection on a polytope suffices. Furthermore, sometimes we can replace the constraint by an ` 1 or ` 2 -

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Received: 25 February 2011 – Revised: 23 May 2011 – Accepted: 26 July 2011 – Published: 17 October 2011 Abstract. Model order reduction appears to be beneficial for the synthesis and simulation of compliant mech- anisms due to computational costs. Model order reduction is an established method in many technical fields for the approximation of large-scale **linear** **time**-**invariant** dynamical systems described by ordinary di ff erential equations. Based on system theory, underlying representations of the dynamical system are introduced from which the general reduced order model is derived by projection. During the last years, numerous new pro- cedures were published and investigated appropriate to simulation, optimization and control. Singular value decomposition, condensation-based and Krylov subspace methods representing three order reduction methods are reviewed and their advantages and disadvantages are outlined in this paper. The convenience of apply- ing model order reduction in compliant mechanisms is quoted. Moreover, the requested attributes for order reduction as a future research direction meeting the characteristics of compliant mechanisms are commented.

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