Lumped model of a non-linear distributed system by finite element method

The finite element method has been used to find an approximate lumped parameter model of a non-linear distributed parameter system. A one dimensional non-linear dispersion system is considered. The space domain is divided into a finite set of k elements. Each element, has n nodes. Within each element the concentration is represented by C(x,t)^(e) = [N][C] ^T where [N] = [n₁(x),n₂(x), [tdot] n_n(x)] and [C] = [C₁(t),C₂(t), [tdot] C_n(t)]. By using Galerkin's criterion a set of (k × n ? n+ 1) first order differential equations are obtained for C_i(t). These equations are solved by an iterative method. The concepts are illustrated by an example taking five three-node elements in the space domain. The results are compared with those obtained by a finite difference method. It is shown that the finite element method can be used effectively in modelling of a distributed system by a lumped system.     

A Simple Robust Sliding-Mode Fuzzy-Logic Controller of the Diagonal Type

This paper derives and analyzes a new robust fuzzy-logic sliding-mode controller of the diagonal type, which does not need the prior design of the rule base. The basic objective of the controller is to keep the system on the sliding surface so as to ensure the asympotic stability of the closed-loop system. The control law consists of two rules: (i) IF sign(e(t)(t)) < 0 THEN maintain the control action, and (ii) IF sign(e(t)(t)) > 0 THEN change the control action, where e(t) = x(t) – x^d(t) is the system state error, and the control action can be either an increase or decrease of the control signal, which is realized through the use of fuzzy rules. The proposed controller, which does not need the prior knowledge of the system model and the prior design of the membership functions" shape, was tested, by simulation, on linear and nonlinear systems. The performance was in all cases excellent (very fast trajectory tracking, no chattering) . Of course, as in traditional control, there was a trade-off between the rise-time and the overshoot of the system response.     

Generalized isocline method of plotting phase-plane trajectories

By considering simultaneously theN-x(or theN-dot{x}), whereN=ddot{x}/dxand thedot{x}-xplanes, second order non-linear autonomous systems (not easily amendable to the existing methods) described by the differential equationddot{x}=F(x, dot{x})can be studied. Changes in system behavior due to changes in initial conditions, nonlinear dampling and restoring forces can easily be studied. Simple and general as it is, the method presented here is believed to be novel.     

Stabilization and Destabilization of Nonlinear Differential Equations by Noise

This paper considers the stabilization and destabilization by a Brownian noise perturbation that preserves the equilibrium of the ordinary differential equation x'(t) = f(x(t)). In an extension of earlier work, we lift the restriction that f obeys a global linear bound, and show that when f is locally Lipschitz, a function g can always be found so that the noise perturbation g(X(t)) dB(t) either stabilizes an unstable equilibrium, or destabilizes a stable equilibrium. When the equilibrium of the deterministic equation is nonhyperbolic, we show that a nonhyperbolic perturbation suffices to change the stability properties of the solution.     

Recursive pointwise design for nonlinear systems

This paper presents a recursive pointwise design (RPD) method for a class of nonlinear systems represented by x(t)=f(x(t))+g(x(t))u(t). A main feature of the RPD method is to recursively design a stable controller by using pointwise information of a system. The design philosophy is that f(x(t)) and g(x(t)) can be approximated as constant vectors in very small local state spaces. Based on the design philosophy, we numerically determine constant control inputs in very small local state spaces by solving linear matrix inequalities (LMIs) derived in this paper. The designed controller switches to another constant control input when the states move to another local state space. Although the design philosophy is simple and natural, the controller does not always guarantee the stability of the original nonlinear system x(t)=f(x(t))+g(x(t))u(t). Therefore, this paper gives ideas of compensating the approximation caused by the design philosophy. After addressing outline of the pointwise design, we provide design conditions that exactly guarantee the stability of the original system. The controller design conditions require to give the maximum and minimum values of elements in the functions f(x(t)) and g(x(t)) in each local state - space. Therefore, we also present design conditions for unknown cases of the maximum and minimum values. Furthermore, we construct an effective design procedure using the pointwise design. A feature of the design procedure is to subdivide only infeasible regions and to solve LMIs again only for the subdivided infeasible regions. The recursive procedure saves effort to design a controller. A design example demonstrates the utility of the RPD method.     

Some properties of cyclic codes†

This paper will investigate the properties of cyclic codes under the transformation,$ It is discovered that for all the non-shortened cyclic codes (binary or (/-nary), if (g(x), (x?1)) =1 then the set of all the codewords is invariant under the transformation f_b. If (g(x), Lx?1)) = (x; ? 1), then the set of all the code words is generated by g(x)/(x ? 1). Next this paper shows a systematic method for the construction of orthogonal cyclic codes in which all the code vectors are orthogonal to each other even to itself     

Global transformations of nonlinear systems

Recent results have established necessary and sufficient conditions for a nonlinear system of the formdot{x}(t) = f(x(t))-u(t)g(x(t)). withf(0) = 0, to be locally equivalent in a neighborhood of the origin in Rⁿto a controllable linear system. We combine these results with several versions of the global inverse function theorem to prove sufficient conditions for the transformation of a nonlinear system to a linear system. In doing so we introduce a technique for constructing a transformation under the assumptions that{gldot[fdotg],...,(ad^{n-1}fldotg)}span ann-dimensional space and that{gldot[fldot g],...,(ad^{n-2}fldotg)}is an involutive set.     

Small-amplitude limit cycles of Liénard systems

This paper is concerned with the study of second order differential equations of Liénard type: (A) $$\ddot x + f(x)\dot x + g(x) = 0$$ where f and g are polynomials. The equation (A) can also be written as a system of the form (B) $$\dot x = y - F(x),\dot y = - g(x),$$ , where \(F(x) = \mathop \smallint \limits_0^x f(\xi )d\xi \) . The results described here are mainly concerned with small amplitude limit cycles; that is, limit cycles which may be bifurcated from the origin on perturbation of the coefficients of F and g. The problem is to estimate the maximum number of limit cycles which various classes of systems of the form (B) can have; this is a special case of the second part of Hilbert’s sixteenth problem. Most of the calculations have been carried out on a computer using the REDUCE symbolic manipulation package.     

On the boundedness properties of a perturbed nonlinear system

It is shown that certain asymptotic equivalence hypotheses on the equationsu(t) = F(t, u(t))+G(t, u(t)) andv(t) = F(t, v(t)) imply that uniform boundedness in the second equation induces eventual uniform boundedness in the first. Also, under these hypotheses, a characterization is given of the unbounded solutions of the first equation.     

On a Second Derivative Test due to Qi

Qi [12] has given a theorem which guarantees the existence and uniqueness of a zero x* of a function f : Rn Rn in a bounded closed rectangular convex set [x] Rn under more general sufficient conditions than those described by Moore [9] and has defined an operator (the so-called second-derivative operator) together with a test, involving the second-derivative operator, for the existence but not the uniqueness of a zero x* of f in [x].The present paper has three purposes: (i) to establish sufficient conditions for the uniqueness of x* involving the second-derivative operator; (ii) to show that under the hypotheses for the convergence of sequences generated from Newton's method given in [15] a set [x] exists which satisfies the sufficient conditions in (i); (iii) to show how the second-derivative operator can be used in a manner similar to that which has been done with the Krawczyk operator in [4].     

Exact control of a system of diffusion equations

It is shown how a vector control function f(x,t) = (f_i(x,t)) can be found that will drive the solution of the linear diffusion system from an arbitrary initial condition u(x,0) = φ(x) = (φ_i(x), i = 1,2,...,n, to an arbitrary final condition u(x,T) = ψ(x) = (ψ_i(x)), i = 1,2,...,n, in arbitrary time T.     

Predictors of whole-body vibration exposure experienced by highway transport truck operators

Whole-body-vibration (WBV) exposure levels experienced by transport truck operators were investigated to determine whether operator's exposure exceeded the 1997 International Standards Organization (ISO) 2631-1 WBV guidelines. A second purpose of the study was to determine which truck characteristics predicted the levels of WBV exposures experienced. The predictor variables selected based on previous literature and our transportation consultant group included road condition, truck type, driver experience, truck mileage and seat type. Tests were conducted on four major highways with 5 min random samples taken every 30 min of travel at speeds greater than or equal to 80 km/h (i.e. highway driving). Results indicated operators were not on average at increased risk of adverse health effects from daily exposures when compared to the ISO WBV guidelines. Significant regression models predicting the frequency-weighted RMS accelerations for the x (F((5,97)) = 8.63, p < 0.01), y (F((5,97)) = 7.74, p < 0.01), z (F((5,61)) = 9.83, p < 0.01) axes and the vector sum of the orthogonal axes (F((5,61)) = 13.89, p < 0.01) were observed. Road condition was a significant predictor (p < 0.01) of the frequency-weighted RMS accelerations for all three axes and the vector sum of the axes, as was truck type (p < 0.01) for the z-axis and vector sum. Future research should explore the effects of seasonal driving, larger vehicle age differences, greater variety of seating and suspension systems and team driving situations.     

Stabilization of uncertain systems via linear control

This note considers the problem of stabilizing a linear dynamical system (Σ) whose state equation includes a time-varying uncertain parameter vectorq(cdot). Given the dynamicsdot{x}(t)=A(q(t))x(t)+ B(q(t))u(t)and a bounding setQfor the valuesq(t), the objective is to choose a control lawu(t)=p(x(t))guaranteeing uniform asymptotic stability for all admissible variations ofq(cdot). Our results differ from previous work in one fundamental way; that is, we show that when working with linear controllers, it is possible to dispense with all assumptions onB(cdot)which have been made by previous authors (e.g., see [1]-[9]). This elimination of hypotheses onB(cdot)is accomplished roughly as follows: the system(Sigma) {underline {underline Delta}} (A(q), B(q))is shown to be equivalent to another system(Sigma^{+}) {underline {underline Delta}} (A^{+}(q), B^{+})as far as stabilization is concerned. SinceB^{+}is a constant matrix (independent ofq), the desired result is readily obtained.     

Elliptic ball stability criterion for discrete-time systems

The elliptic ball criterion is proved for the discrete-time non-linear feedback control equation p(6) x+ BM(t)q(B)x = 0, in which θxlpar;t) = x( t + 1). It is a geometrical condition for stability and instability which reduces to the well-known circle criterion in. the special case when the matrix M(t) is a scalar. For a certain class of equations the elliptic ball criterion is shown to be both necessary and sufficient for the existence of a special kind of quadratic Lyapunov function. The analogy with differential equations is found to fail in one respect     

A transformation of non-linear dynamical systems with a single singular singularly perturbed differential equation†

Singularly perturbed state differential equations of the form [xdot] = f(x, z, t, ?), x(t₀, ?) = x₀(?); μ(?)? = g(x, z, t, ?), z(t₀ ?) = z₀(?) with lim μ(?) = 0; ?, μ > 0 are considered, where the nominal equation 0 = g(x, z, t, 0)^{? → ∞} does not have to be solvable for z. A fairly general transformation of the above system into a form [xdot]^* = f ^*(x^*, z, t; z(1),...,z^(d?1), ? ); μ^*(?)z^(d)= g^*(x^*. z(⁰),...z^(d?1), t; ?), with dim x^* = dim x ?(d ? 1), d ? 1 is proposed. The transformed system stands a better chance of being analysed by existing methods (especially by those proposed by Hoppensteadt (1971) and Hoppensteadt and Mi ranker (1976)) than the original singular singularly perturbed form. Informative examples are presented.     

Converging multistep methods for initial value problems involving multivalued maps

Every consistent and strongly stable multistep method of stepnumberk yields a solution, of the setvalued initial value problem \(\dot y \in F(t,y),y(t_0 ) = y_0 \) . The setF(t, z) is assumed to be nonvoid, convex and closed. Upper semicontinuity of F with respect to both variables is not required everywhere. If the initial value problem is uniquely solvable, the solutions of the multistep method will converge to the solution of the continuous problem. These results carry over to functional differential equations \(\dot y \in F(t,M_t y)\) of Volterra type and to discontinuous problems \(\dot y(t) = f(t,M_t y)\) in the sense of A.F. Filippov. A difference method is applied to the discontinuous delay equation \(\ddot x(t) + 2D\dot x(t) + \omega ^2 x(t) = = - \operatorname{sgn} (x(t - \tau ) + \dot x(t - \tau ))\) . In the limit τ→0 we obtain results for the problem \(\ddot x + 2D\dot x + \omega ^2 x = = - \operatorname{sgn} (x + \dot x)\) which cannot be solved classically everywhere.     

Theory and performance of a subroutine for solving Volterra Integral Equations

The author considers Volterra Integral Equations of either of the two forms $$u(x) = f(x) + \int\limits_a^x {k(x - t)g(u(t))dt, a \leqslant } x \leqslant b,$$ wheref, k, andg are continuous andg satisfies a local Lipschitz condition, or $$u(x) = f(x) + \int\limits_a^x {\sum\limits_{j = 1}^m {c_j (x)g_j (t,u(t))dt} ,} $$ wheref,c _j , andg _j ,j=1,2,...,m, are continuous and eachg _j satisfies a local Lipschitz condition in its second variable. It is shown that in each case the respective integral equation can be solved by conversion to a system of ordinary differential equations which can be solved by referring to a described FORTRAN subroutine. Subroutine VE1. In the case of the convolution equation, it is shown how VE1 converts the equation via a Chebyshev expansion, and a theorem is proved, and implemented in VE1, wherein the solution error due to truncation of the expansion can be simultaneously computed at the discretion of the user. Some performance data are supplied and a comparison with other standard schemes is made. Detailed performance data and a program listing are available from the author.     

State-space for non-linear algebraically constrained dynamical systems

Non-linear state differential equations x = f(x, u) with algebraic constraints g(x, u, e) = 0, e = e (t), which describe possibly singular systems, are considered. The derivation of equivalent unconstrained state differential equations x* = f* (x*, e, ?,...), {x* }c:{x} ≥ {x} with the ‘ output’ equations u = h*(x*, u*, e, ?,…) and x equals; h* * (x*, e, ?…) is studied. Instead of an extension of the linear matrix-oriented singular system theory, the non-linear system inversion ideas are found to be easily applicable, to preserve much of the original system structure, and to give insight into the possibly distributional behaviour of and the possible incompatibilities in the system.     

A Quadratic Lower Bound for Three-Query Linear Locally Decodable Codes over Any Field

A linear (q,δ,,m(n))-locally decodable code (LDC) C : Fn → Fm(n) is a linear transformation from the vector space Fn to the space Fm(n) for which each message symbol xi can be recovered with probability at least 1/(|F|) +∈ from C(x) by a randomized algorithm that queries only q positions of C(x),even if up to δm(n) positions of C(x) are corrupted.In a recent work of Dvir,the author shows that lower bounds for linear LDCs can imply lower bounds for arithmetic circuits.He suggests that proving lower bounds for LDCs over the complex or real field is a good starting point for approaching one of his conjectures.Our main result is an m(n) = Ω (n2) lower bound for linear 3-query LDCs over any,possibly infinite,field.The constant in the Ω (·) depends only on ε and δ.This is the first lower bound better than the trivial m(n) = Ω (n) for arbitrary fields and more than two queries.