Auxiliary State Variables Research Articles

High-Gain Disturbance Observer-Based Backstepping Control With Output Tracking Error Constraint for Electro-Hydraulic Systems

We propose a high-gain disturbance observer (HGDOB)-based backstepping control with position tracking error constraint for electro-hydraulic systems to improve the position tracking performance in the presence of disturbances. The HGDOB is designed to estimate the disturbances that include the friction, the load force, and the parameter uncertainties. Auxiliary state variables are proposed to avoid amplification of the measurement noise in the HGDOB. To compensate for the disturbances while guaranteeing tolerance of the position tracking error, the backstepping controller is proposed by using the barrier Lyapunov function. As a result, the proposed method satisfies the output constraint and improves the position tracking performance in the presence of disturbances. Its performance is validated via simulations and experiments.

Sensor Fault Estimation Filter Design for Discrete-time Linear Time-varying Systems

This paper proposes a sensor fault diagnosis method for a class of discrete-time linear time-varying (LTV) systems. In this paper, the considered system is firstly formulated as a descriptor system representation by considering the sensor faults as auxiliary state variables. Based on the descriptor system model, a fault estimation filter which can simultaneously estimate the state and the sensor fault magnitudes is designed via a minimum-variance principle. Then, a fault diagnosis scheme is presented by using a bank of the proposed fault estimation filters. The novelty of this paper lies in developing a sensor fault diagnosis method for discrete LTV systems without any assumption on the dynamic of fault. Another advantage of the proposed method is its ability to detect, isolate and estimate sensor faults in the presence of process noise and measurement noise. Simulation results are given to illustrate the effectiveness of the proposed method.

An inverse-model approach to multivariable norm optimal iterative learning control with auxiliary optimisation

Motivated by the commonly encountered problem in which tracking is only required at selected intermediate points within the time interval, a general optimisation-based iterative learning control (ILC) algorithm is derived that ensures convergence of tracking errors to zero whilst simultaneously minimising a specified quadratic objective function of the input signals and chosen auxiliary (state) variables. In practice, the proposed solutions enable a repeated tracking task to be accurately completed whilst simultaneously reducing undesirable effects such as payload spillage, vibration tendencies and actuator wear. The theory is developed using the well-known norm optimal ILC (NOILC) framework, using general linear, functional operators between real Hilbert spaces. Solutions are derived using feedforward action, convergence is proved and robustness bounds are presented using both norm bounds and positivity conditions. Algorithms are specified for both continuous and discrete-time state-space representations, with the latter including application to multi-rate sampled systems. Experimental results using a robotic manipulator confirm the practical utility of the algorithms and the closeness with which observed results match theoretical predictions.

Multivariable norm optimal iterative learning control with auxiliary optimisation

The paper describes a substantial extension of norm optimal iterative learning control (NOILC) that permits tracking of a class of finite dimensional reference signals whilst simultaneously converging to the solution of a constrained quadratic optimisation problem. The theory is presented in a general functional analytical framework using operators between chosen real Hilbert spaces. This is applied to solve problems in continuous time where tracking is only required at selected intermediate points of the time interval but, simultaneously, the solution is required to minimise a specified quadratic objective function of the input signals and chosen auxiliary (state) variables. Applications to the discrete time case, including the case of multi-rate sampling, are also summarised. The algorithms are motivated by practical need and provide a methodology for reducing undesirable effects such as payload spillage, vibration tendencies and actuator wear whilst maintaining the desired tracking accuracy necessary for task completion. Solutions in terms of NOILC methodologies involving both feedforward and feedback components offer the possibilities of greater robustness than purely feedforward actions. Results describing the inherent robustness of the feedforward implementation are presented and the work is illustrated by experimental results from a robotic manipulator.

Design of robust fault detection observer for Takagi-Sugeno models using the descriptor approach

This paper deals with the design of a robust fault detection observer for a Takagi-Sugeno (T- S) fuzzy model affected by sensor and actuator faults and unknown bounded disturbances simulta- neously. An observer based on the technique of descriptor systems is studied. Indeed, by considering faults as auxiliary state variables, both states and faults are estimated simultaneously. In order to guar- antee the best robustness to disturbances and sensitivity to faults, the developed observer combine the H-/H∞ performances. Then, based on Lyapunov method, asymptotic stability conditions are given to design the observer parameters. In order to get convenient and reliable faults estimator in computations, an iterative linear matrix inequality (LMI) algorithm is developed. This algorithm, solved easily using existing numerical tools, allows to minimize influences of disturbances and maximize the ones of faults. Finally, a numerical example is proposed to illustrate the effectiveness of the result.

A New Optimal Control Formulation to Ensure the Stability of NMPC Systems

AbstractNonlinear model predictive control (NMPC) has been considered as a promising control algorithm, since a rigorous model can be explicitly employed and state variable restrictions ensured by formulating inequality constraints. However, due to the nonlinear problem formulation, it is difficult to analyze the stability properties of NMPC systems. In this study we propose a new formulation of the optimal control problem to ensure the stability of NMPC systems. Auxiliary state variables and linear state equations will be introduced and heir eigenvalues optimized in the dynamic optimization framework. System state variables will be constrained by these auxiliary variables so that they will conform to the stability properties of the auxiliary variables. Features of this stabilization approach are analyzed. Results of case studies indicate satisfactory effectiveness of the proposed NMPC.

When is discretion superior to timeless perspective policymaking?

The monetary policy literature assumes increasingly that policy is formulated according to the timeless perspective (Woodford, 1999a). However, by treating appropriately the auxiliary state variables that characterize the timeless perspective equilibrium when evaluating policy performance, this paper shows that discretionary policymaking can be superior to timeless perspective policymaking and identifies model features that make this outcome more likely. Using standard New Keynesian DSGE models, discretion is found to dominate timeless perspective policymaking when the price/wage Phillips curves are relatively flat, due, perhaps, to firm-specific capital (or labor) and/or Kimball (1995) aggregation in combination with nominal rigidities. These results suggest that studies applying the timeless perspective might also usefully compare its performance to discretion, paying careful attention to how policy performance is evaluated.

When is Discretion Superior to Timeless Perspective Policymaking?

The monetary policy literature assumes increasingly that policies are formulated according to the timeless perspective (Woodford, 1999a). However, treating the auxiliary state variables that characterize the timeless perspective equilibrium appropriately when evaluating policy performance, this paper shows that discretionary policymaking can be superior to timeless perspective policymaking and identifies model features that make this outcome more likely. Using standard New Keynesian DSGE models, discretion is found to dominate timeless perspective policymaking when the price/wage Phillips curves are relatively flat, due, perhaps, to firm-specific capital (or labor) and/or Kimball (1995) aggregation in combination with nominal rigidities. These results suggest that studies applying the timeless perspective might also usefully compare its performance to discretion, paying careful attention to how policy performance is evaluated.

Sequential Inference for Nonlinear Models using Slice Variables

This paper develops particle-based methods for sequential inference in nonlinear models. Sequential inference is notoriously difficult in nonlinear state space models. To overcome this, we use auxiliary state variables to slice out nonlinearities where appropriate. This induces a Fixed-dimension conditional sufficient statistics and, given these, we adapt existing particle learning algorithms to update posterior beliefs about states and parameters. We provide three illustrations. First, a dynamic exponential model with Gaussian errors. Second, a stochastic growth model with nonlinear state evolution and t-distributed errors. Finally, a bivariate radar tracking problem which was originally analyzed in the nonlinear Monte Carlo Filtering literature. In all cases, we illustrate the efficiency of our methodology.

Identical phase oscillators with global sinusoidal coupling evolve by Möbius group action

Systems of N identical phase oscillators with global sinusoidal coupling are known to display low-dimensional dynamics. Although this phenomenon was first observed about 20 years ago, its underlying cause has remained a puzzle. Here we expose the structure working behind the scenes of these systems by proving that the governing equations are generated by the action of the Möbius group, a three-parameter subgroup of fractional linear transformations that map the unit disk to itself. When there are no auxiliary state variables, the group action partitions the N-dimensional state space into three-dimensional invariant manifolds (the group orbits). The N−3 constants of motion associated with this foliation are the N−3 functionally independent cross ratios of the oscillator phases. No further reduction is possible, in general; numerical experiments on models of Josephson junction arrays suggest that the invariant manifolds often contain three-dimensional regions of neutrally stable chaos.

On Pricing Derivatives in the Presence of Auxiliary State Variables

Binomial, trinomial, and other lattice-based option pricing procedures are extremely useful in dealing with early exercise because solution by backward induction is possible. As one rolls back through the tree, the option value at each node depends on the current price for the underlying and the paths that might be followed going forward. But for many kinds of derivatives, the value at a given point in time depends on both the current asset price and the price path that it has followed up to the current point. Derivatives on interest rates in the Heath-Jarrow-Morton (HJM) model are one example. Another is option pricing when the underlying volatility varies over time according to a GARCH model. The most successful technique for adapting a lattice model to a path-dependent process has been to carry along a set of auxiliary variables that summarize (approximately) the necessary path information at each node. These techniques work, but they are typically very time consuming. In this article, Lin and Ritchken present a way to greatly increase the efficiency of this procedure. Some auxiliary path information must still be carried through the tree, but much less is needed. They demonstrate the sharp increase in performance this makes possible in an HJM application and for an underlying that follows a GARCH process. <b>TOPICS:</b>Options, simulations

A spatial domain decomposition method for parabolic optimal control problems

We present a non-overlapping spatial domain decomposition method for the solution of linear–quadratic parabolic optimal control problems. The spatial domain is decomposed into non-overlapping subdomains. The original parabolic optimal control problem is decomposed into smaller problems posed on space–time cylinder subdomains with auxiliary state and adjoint variables imposed as Dirichlet boundary conditions on the space–time interface boundary. The subdomain problems are coupled through Robin transmission conditions. This leads to a Schur complement equation in which the unknowns are the auxiliary state adjoint variables on the space-time interface boundary. The Schur complement operator is the sum of space–time subdomain Schur complement operators. The application of these subdomain Schur complement operators is equivalent to the solution of an subdomain parabolic optimal control problem. The subdomain Schur complement operators are shown to be invertible and the application of their inverses is equivalent to the solution of a related subdomain parabolic optimal control problem. We introduce a new family of Neumann–Neumann type preconditioners for the Schur complement system including several different coarse grid corrections. We compare the numerical performance of our preconditioners with an alternative approach recently introduced by Benamou.

A Simple Induction Approach and an Efficient Trinomial Lattice for Multi-State Variable Interest Rate Derivatives Models

This paper presents an alternative approach for interest rate lattice construction in the Ritchken and Sankarasubramanian (1995) framework. The proposed method applies a parsimonious induction technique to represent the distribution of auxiliary state variables and value interest rate derivatives. In contrast to other approaches, this technique requires no numerical interpolations, approximations and iterative procedures for pricing interest rate options using a simple backward induction and, therefore, provides significant computational advantages and flexibility with respect to existing implementations. Also, the proposed trinomial interest rate lattice specification provides for a further reduction in computational costs with additional flexibility. The results of this work can be extended to a class of derivatives pricing models with path dependent state variables and generalized to multi-factor models.

Methods of fluid–structure coupling in frequency and time domains using linearized aerodynamics for turbomachinery

Two methods of fluid–structure coupling for turbomachinery are presented, the first one in the frequency domain and the second in both frequency and time domains. In both methods, the structure and the fluid are assumed to have circumferential cyclic symmetric properties and the unsteady aerodynamic forces are assumed to be linear in terms of the structural displacements. The motion equation of the reference sector in the travelling wave coordinates is projected on the complex eigenmodes for each phase number. The generalized unsteady aerodynamic forces are computed by solving the Euler equations and by assuming the structural motion to be harmonic with a constant phase angle between two adjacent sectors. In the frequency domain, the complex, nonlinear eigenvalue problem for the aeroelastic stability analysis is solved iteratively either by the double scanning method or by using Karpel's minimum state smoothing of the aerodynamic coefficient matrix. In the time domain, Karpel's smoothing method is used to obtain an approximation of the generalized unsteady aerodynamic forces by means of auxiliary state variables. These coupling methods are tested on a compressor blade row and the good agreement obtained between their results and those of the direct coupling method shows that the proposed numerical methods, already used in aircraft applications, are adapted to turbomachinery.

Reduced-Order Dynamic Aeroelastic Model Development and Integration with Nonlinear Simulation

Piloted and batch simulations of the aeroservoelastic response are essential tools in the development of advanced flight control systems. In these simulations the number of differential equations must be sufficiently large to yield the required accuracy, yet small enough to enable real-time evaluations of the aircraft flying qualities, The challenge of these conflicting demands is reinforced by nonlinearities in the quasi-steady equations of motion and by the complex characteristies of the oscillatory forces. Our solution to the problem is based on a unique formulation that eliminates the need for auxiliary state variables to represent the unsteady aerodynamics. We also address transformations from the mean flight path axes to a body axes coordinate system and describe how the structural dynamie equations of motion are integrated with the quasi-steady, nonlinear, six-degree-of-freedom plant model. The unified model, which accurately preserves the roots of the dynamic aeroelastic system, includes provisions for control surface inputs and atmospheric turbulence.

Roundoff error reduction in state-space digital filters of cascade type

Some problems with realization of digital filters are caused by roundoff errors due to the word-length restriction. Effects of these errors can be reduced if the processing range for large signals is chosen so as to avoid overflow. For filters built according to the state-space expressions, one uses scaling which is aimed at the state variables. However, since no constraints apply to the output value computation, it results in a non-efficient use of the dynamic range and an increase in the relative error. On the other hand, in a cascade-type structure, the coefficient matrix of the entire system is computed from the coefficient matrices of all sections. Such a computing procedure leads to an increased number of multipliers and, therefore, increased roundoff error. To solve this problem, the authors have proposed a method based on scaling aimed at the output value computation which enables efficient use of the dynamic range. In addition, the realization was based on the original coefficients of the sections in a cascade structure, which allowed reducing the number of multipliers. The proposed computing procedure offers significant time savings. This realization method was demonstrated by introducing auxiliary state variables. In particular, for the cascade structure, significant reduction of the roundoff error was achieved. As a numerical example, we computed the roundoff error for a low-pass elliptic filter. © 1997 Scripta Technica, Inc. Electron Comm Jpn Pt 3, 80(10): 87–96, 1997

An extension of the runge-kutta method for nth order initial value problems

A Runge-Kutta algorithm for an nth order initial value problem (IVP) is developed by means of simple modifications of the algorithm used for the first-order IVP. Auxiliary state variables are used to convert the nth order IVP into a system of first-order IVPs. Consequently, the corresponding computer program for the nth order is developed by modification of the program used for the first-order IVP in a suitable fashion. It is shown that the method is straightforward and its computer program is extremely compact. For illustration, the program is applied to a fourth-order IVP involving nonlinear terms.

Crop growth models without hormones.

For the simulation of organ formation and assimilate partitioning, information is required on the current level of activities like CO2 assimilation and the growth of various organs, as well as state variables such as leaf and root wt., N content and carbohydrate reserves and exogenous variables like radiation and temp. This information may be retained in auxiliary state variables by considering the dynamic equilibrium between growth of roots and shoots. Auxiliary state variables are not tangible quantities but mathematical artefacts of the simulation program; it is speculated that in real plants similar information may be retained and transferred by the hormonal system. A hormonal system is a communication system and such systems may be analysed either in terms of means (of the hardware used) or in terms of purpose (of the messages transferred). In dynamic models of crop growth, interest should be focused on the latter. Wheat, maize and ryegrass are used as examples. (Abstract retrieved from CAB Abstracts by CABI’s permission)

Application of stochastic optimal control to a hydrofoil boat problem

In this paper the solution of a stochastic optimal control problem described by linear equations of motion and a nonquadratic performance index is presented. The theory is then applied to the dynamics of a single-foil and a hydrofoil boat flying on rough water. The random disturbances caused by sea waves are represented as the response of an auxiliary system to a white noise input. The control objective is formulated as an integral performance index containing a quadratic acceleration term and a nonquadratic term of the submergence deviation of the foil from calm water submergence. The stochastic version of the maximum principle is used in the formulation of a feedback control law. The Riccati equations and the feedback gains associated with a nonquadratic performance index are non-linear functions of the state and auxiliary state variables. These equations are integrated forward with the state equations for the steady-state solution of the problem. The controller for a nonquadratic performance index contains computing elements which perform the integration of the Riccati equations to generate the instantaneous values of the feedback gains. The effect of a nonquadratic penalty on the submergence deviation and the effect of a nonquadratic control penalty on the response of the system are investigated. A comparison between an optimal nonlinear control law and a suboptimal linear control law is presented.