Unstable Roots Research Articles

Further results on stability of [formula omitted

Our aim in this communication is to improve the existing results concerning the stability of xdot(t) = Ax(t) + Bx(t−τ). It is shown that the unstable characteristic roots could be more accurately located in a bounded and restrictive region of the complex plane. An iterative scheme is presented for numerical implementation of the results. Numerical computations are presented for illustration and comparison with previous results.

Fuel slosh in skewed tanks

Previous treatments of fuel slosh coupling with aircraft dynamics are extended to the case in which fuel tanks are skewed to the direction of flight, as in swept wing tanks. General and linearized equations of motion are developed and example calculations are made based on the Boeing 747-100 airplane. Slosh appears to be possible for swept wings only over a limited pitch attitude range, dependent on wing dihedral. For the example airplane at landing approach attitudes, coupling cannot happen, because partial inboard main wing tank fuel is fixed by gravity at the inboard tank ends. The gravity fixity is removed for an assumed lower dihedral angle. Then, slosh coupling with the Dutch roll mode of motion is small, but an unstable real root develops, similar to the spiral mode. Fuel slosh coupling to the Dutch roll mode is even smaller for centerline (unskewed) tanks. Tank locations forward of the aircraft's center of gravity are destabilizing.

Time-step limits for stable solutions of the ice-sheet equation

Various spatial discretizations for the ice sheet are compared for accuracy against analytical solutions in one and two dimensions. The computational efficiency of various iterated and non-iterated marching schemes is compared.The stability properties of different marching schemes, with and without iterations on the non-linear equations, are compared. Newton–Raphson techniques permit the largest time steps. A new technique, which is based on the fact that the dynamics of unstable iterated maps contain information about where the unstable root lies, is shown to improve substantially the performance of Picard iteration at a negligible computational cost.

Time-step limits for stable solutions of the ice-sheet equation

Various spatial discretizations for the ice sheet are compared for accuracy against analytical solutions in one and two dimensions. The computational efficiency of various iterated and non-iterated marching schemes is compared. The stability properties of different marching schemes, with and without iterations on the non-linear equations, are compared. Newton–Raphson techniques permit the largest time steps. A new technique, which is based on the fact that the dynamics of unstable iterated maps contain information about where the unstable root lies, is shown to improve substantially the performance of Picard iteration at a negligible computational cost.

AN ANALYSIS OF THE IMPACT OF FINITE HORIZONS ON MACROECONOMIC CONTROL

THIS PAPER is concerned with the practical application of optimal control techniques to the macroeconomic policy design problem using empirically based econometric models which embody forward-looking behaviour. In the context of UK models, such work can be divided into two broad approaches. One involves the application of well known non-linear optimal control algorithms directly to non-linear rational expectations models, as in Holly and Zarrop (1983) and Westaway and Wren-Lewis (1993).1 In the other, linearised versions of non-linear models are derived, allowing the more tractable properties of linear rational expectations models to be exploited, as in Christodoulakis et al. (1991) and Weale et al. (1989). Surprisingly, relatively little work has been done to reconcile these two approaches, in particular to compare the policy implications which might emerge from applying the different techniques to the same problem. In this paper we explain why these differences are important by emphasising the role of the finite horizon. We show that this can seriously distort the optimal control outcome. Typically, macroeconomic models are large and non-linear; for example the National Institute model of the UK economy on which the empirical analysis in this paper is based contains just over 350 variables with the behavioural content embodied in around half of these (see NIESR 1991). Inevitably, this forward-looking model can only be solved using numerical techniques, such as the extended Gauss-Seidel method discussed by Fisher and Hughes Hallet (1988). These solution techniques can be computationally expensive especially when the dominant unstable root of the model associated with this forwardlooking behaviour is close to the unit circle. By contrast, linear models, or linearised versions of non-linear models, can be solved at much less computational expense, even for rational expectations models (using, for example, the method suggested by Blanchard and Kahn 1980). When considering the policy design problem, an important implication of having to solve models by numerical methods is the need to replace analytic infinite horizon solutions by finite horizon approximation. The significance arises in two ways. First, in the presence of forward-looking expectations, any finite horizon model solution requires the imposition of a terminal condition

Testing for an unstable root in conditional and structural error correction models

This paper proposes a class of Wald tests for the hypothesis of an unstable root in conditional error correction models. Both single-equation models and simultaneous equations models in structural form are considered. The asymptotic distribution of the test statistics under the null hypothesis is derived in terms of a vector Brownian motion process, and critical values are obtained via Monte Carlo simulation. In an empirical model of the demand for money and the rate of inflation in the UK, the tests reject the instability hypothesis.

Variscan extension in the Pyrenees

Evidence for Variscan extension in the Pyrenees is summarized and used with other geological and geochronological data to reconstruct the Variscan extensional geometry and kinematics at a crustal scale. A regionally consistent pattern of ESE‐WNW trending synmetamorphic stretching directions indicates Variscan extension roughly parallel to the present alpine trend of the belt. The spatial distribution of Variscan low pressure ‐ high temperature metamorphic domes is interpreted as being possibly due to the activity of few moderately dipping, crustal‐scale shear zones allowing heterogeneous ductile extension of the Variscan middle and lower crust. Extension occurred synchronous with uplift and emergence of large parts of the crust and deposition, during the Upper Carboniferous and Permian, of up to 2500 m of continental sediments and intercalated basaltic‐andesitic volcanics in fault‐bounded extensional halfgrabens. The extensional structures and related metamorphism postdate a Namurian to early Westphalian stage of crustal thickening, characterized by thrusting and steep folding. It is suggested that Variscan extension in the Pyrenees was driven by the gravitational potential energy of an initially thickened lithosphere, enhanced by detachment of a cold and gravitationally unstable lithospheric root, which caused progressive uplift, tectonic denudation, and a highly anomalous thermal structure of the Variscan crust leading to extensive lower crustal melting and granodioritic intrusion during the Early Permian.

Further results on stability of [formula omitted

In this paper, it is shown that the restricted region where unstable characteristic roots of time-delay systems X ̇ (t) = AX(t) + BX(t − τ) exist could be more accurately estimated, and then by applying this key observation an improved delay-dependent stability criterion is proposed. A simple example is worked out to illustrate the result.

Saddlepath solutions for multivariate linear rational expectations models

Linear rational expectations models which include `forward-looking' variables typically imply equations of motion with unstable roots. This paper derives a closed form expression for saddlepath restrictions which purge unstable roots from the model's reduced form. It also shows that, for the class of models considered, the reduced form may be written as a finite-order autoregression. An example is provided along with Monte Carlo experiments which demonstrate the gains in estimation accuracy which obtain when the results of this paper are used to concentrate the likelihood function of the model's parameters.

Time-periodic control of a multi-blade helicopter

Abstract : The flap-lag equations of motion of an isolated rotor blade and those for a rigid helicopter containing four blades free to flap and lag are derived. Control techniques are developed which stabilize both systems for a variety of flight conditions. Floquet theory is used to investigate the stability of a rotor blade's flap-lag motion. A modal control technique, based on Floquet theory, is used to eliminate the blade's instabilities using collective and cyclic pitch control mechanisms. The technique shifts the unstable roots to desired locations while leaving the other roots unaltered. The control, developed for a single design point, is shown to significantly reduce or eliminate regions of flap-lag instabilities for a variety of off-design conditions. Both scalar and vector control are successfully used to stabilize the blade's motion. Coupling the flap-lag equations of motion of four rotor blades to a rigid airframe alters the flap, lg, and airframes roots. The airframe roots are stabilized using a combination of the body's pitch attitude and pitch rate feedback to the main rotor's longitudinal cyclic pitch. The modal control technique is used to eliminate multiple blade instabilities by first controlling a pair of unstable roots at a specific design point. The resulting closed loop system is a new linear system which periodic coefficients. Another modal controller is designed for this new system to shift a second pair of unstable roots to desired locations. This process is repeated until all instabilities are eliminated. Numerical inaccuracies, however, become noticeable when modal control is used more than once.

Stabilising discrete polynomials using spectral factorisation

A simple technique for stabilising discrete polynomials is presented. The technique does not require the roots of the polynomial to be computed but instead uses the familiar spectral factorisation method. If the polynomial has unstable roots, they are automatically reflected back inside the unit circle. Stable polynomials remain unaffected.

Robust stability of polynomials with multilinear parameter dependence

The problem is studied of testing for stability a class of real polynomials in which the coefficients depend on a number of variable parameters in a multilinear way. We show that the testing for real unstable roots can be achieved by examining the stability of a finite number of corner polynomials (obtained by setting parameters at their extreme values), while checking for unstable complex roots normally involves examining the real solutions of up to m+1 simultaneous polynomial equations, where m is the number of parameters. When m=2, this is an especially simple task.

An alternative proof of Kharitonov's theorem

An alternative proof is presented of Kharitonov's theorem for real polynomials. The proof shows that if an unstable root exists in the interval family, then another unstable root must also show up in what is called the Kharitonov plane, which is delimited by the four Kharitonov polynomials. This fact is proved by using a simple lemma dealing with convex combinations of polynomials. Then a well-known result is utilized to prove that when the four Kharitonov polynomials are stable, the Kharitonov plane must also be stable, and this contradiction proves the theorem.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

PROPERTIES OF PREDICTORS FOR MULTIVARIATE AUTOREGRESSIVE MODELS WITH ESTIMATED PARAMETERS

Abstract. The k‐dimensional pth‐order autoregressive processes {Yt} that are either stationary or have one unstable or explosive root are considered. The properties of the s‐periods‐ahead predictor Ŷn+s, obtained by replacing the unknown parameters in the expression for the best linear predictor YTn+s by their least‐squares estimators, is shown to be asymptotically equivalent to the optimal predictor except in the explosive case. An expression for the mean squared error of Ŷn+s is derived through terms of order n‐1 for normal stationary processes when the parameters are estimated from the realization to be predicted. In addition, small‐sample properties of Ŷn+s are investigated.

On Long-Run Solutions to Dynamic Econometric Equations Under Rational Expectations

Models of this type abound in the RE literature and include, inter alia, the permanent income model of consumption (Hall, I 978; Sargent, I 978 a), models of factor demand (Sargent, I978 b), portfolio balance during hyperinflations (Sargent, I977), foreign exchange rate determination (Mussa, I976, I982), the term structure of interest rates (Shiller, I972; Modigliani and Shiller, I973), and the demand for money (Cuthbertson and Taylor, i987) . The general formulation and estimation of such models is discussed in Hansen and Sargent (i 980). In particular, this type of equation arises naturally in models displaying the familiar RE 'saddle path' properties where the unstable root is 'solved forward' (Sargent, I979). If we define X zt= (I-A) E Ai xt+ i=o (3) may be written: Yt = KYtt1 +te, t +et. ()

On Asymptotic Prediction Problems for Multivariate Autoregressive Models in the Unstable Nonexplosive Case

This paper considers the prediction problems of a k-dimensional, pth order autoregressive process with unstable but non-explosive roots and dependent error variables. The estimated predictor has been shown to be asymptotically equivalent to the optimal predictor. An expression for the meansquare error of the estimated predictor has also been derived .

A note on non-uniqueness and the role of policy

In general, models embodying rational expectations must have unstable roots in order that a unique non-explosive solution exists. Peel 1981 has identified a role for the authorities in ensuring uniqueness by judicious choice of their feedback policy parameter to influence the eigenvalues of the model. This may mean, however, a loss of degree of freedom in stabilizing the economy if the stability condition acts as a binding constraint on the policy parameter. This paper offers an alternative way in which the authorities can formulate policy and determine a well-defined solution but which leaves stabilization policy unconstrained.

On stabilization by state feedback for neutral differential difference equations

In this paper we consider a class of feedback operators for a linear differential difference equation of the neutral type. We show that if a system has unstable neutral root chains, any stabilizing feedback operator in this class must contain a derivative of the delayed state component. Under the assumption of exact controllability, we construct a stabilizing feedback operator.

On convergence of least-squares identifiers of autoregressive models having stable and unstable roots

A proof is presented for establishing the convergence of least-squares (LS) identification algorithms when applied to autoregressive (AR) time-series models where some or all poles my be unstable, i.e., outside the unit circle in the complex <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</tex> -plane. The only assumption on the time-series model is that its residual or driving sequence is a zero-mean uncorrelated (white noise) sequence with finite second moment which is second-moment-ergodic (SME). In cases where the SME condition cannot be established, the resulting identified parameters will relate to a model driven by an SME process which is the LS approximation to the actual process whose identification was sought.

A Necessary and Sufficient Condition for GYROLITE Stability

The attitude stability problem of a GYROLITE spacecraft is investigated by using modern control tools. It is shown that the spacecraft may have a stable attitude even in presence of unstable roots. The only requirement to guarantee this stability is that the sum of all the roots of the system characteristic equation must be negative.