Closed-loop eigenstructure assignment by state feedback in multivariable linear systems with slow and fast modes

In this paper, results are presented which facilitate the complete exploitation of state feedback in the assignment of the entire closed-loop eigenstructure of multivariable linear systems. These results provide a method for the direct computation of the state-feedback matrix which assigns prescribed Jordan canonical forms, eigenvectors, and generalized eigenvectors to the plant matrices of closed-loop systems. This method is illustrated by assigning the entire closed-loop eigenstructure of a third-order two-input discrete-time system in such a way that the resulting closed-loop system exhibits finite settling time behaviour.

An interactor plays a basic role in the design problems of multivariable control systems, such as the model matching control, decoupling control, adaptive control, and disturbance decoupling control. For example, it is well known that a linear multivariable system is decouplable by the static state feedback if and only if it has a diagonal interactor. But, in general, the interactor is a lower left triangular matrix, which makes the design problems of multivariable control systems complicated. From this point of view, the precompensator such that the total system has a diagonal interactor is important for multivariable control systems to simplify their design procedure. This paper concerns the systematic design method of the precompensator for a diagonal interactor using Markov parameters of the plant. It will also be proved that this compensator is always obtained by the proposed design algorithm. Furthermore, this argument will be a basic idea when this technique is applied to the nonlinear multivariable systems. © 2008 Wiley Periodicals, Inc. Electr Eng Jpn, 163(1): 65–72, 2008; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.20599

A systematic design method of assigning a required time-scale structure to a given multivariable system is developed. The time-scale structure assignment is done by high-gain feedback of either state or output. The method decomposes a given multivariable system into several single-input, single-output (SISO) subsystems each of which is designed separately. A standard method of design for SISO system is developed, and this forms a building block for multivariable system design. The state-feedback design depends only on the stage sequence of the given system, while the output feedback design depends only on the set of integers which specifies the order of infinite zeros. In this way the design is made robust with respect to certain parameter uncertainties. >

It is shown that the asymptotic values of the invariant zeros and associated zero-directions of multivariable systems with ‘slow’ and ‘ fast ’ modes can be readily computed by considering separately the system matrices associated with the ‘ slow ’ and ‘ fast ’ sub-systems of such systems. The general results are illustrated by determining both the ‘ slow ’ and ‘ fast ’ invariant zeros and associated zero-directions of a third-order system with ‘ slow ’ and ‘ fast ’ modes.

Aims. We present an investigation into how the cooling of the background plasma influences the propagation of slow and fast MHD wave modes supported by an unbounded, homogenous plasma. Previous investigations have suggested that the cooling of the plasma and a reduction in density could lead to the damping of fast magneto-acoustic oscillations. We aim to investigate whether cooling of the background plasma at a constant density may be responsible for the damping of slow and fast modes. Methods. The plasma is assumed homogeneous and the background temperature (pressure) is decreasing with time. The temperature change is assumed to be due to optically thin radiation. A special case of the radiative function is chosen to allow an analytical assessment of the effects of cooling on magneto-acoustic MHD modes and ensures the temperature evolution of the background plasma due to this radiation also matches the observed cooling profile of coronal loops. Results. A time-dependent dispersion relation is obtained on the slow timescale of cooling and full time-dependent solutions are found. Leading order equations for the amplitude of the waves are obtained and solved analytically for the slow and fast MHD modes. The cooling of the plasma is found to cause the frequency of the magneto-acoustic modes to decrease with time. The slow modes are found to experience a greater change in frequency than the fast modes. More importantly, the radiative losses also provide a significant damping of the slow mode and a small damping of the component of the fast mode perpendicular to the magnetic field. The damping of the slow mode is found to be strong within typical lifetimes of oscillations observed in coronal structures. Cooling could have important consequences and needs to be assessed when trying to determine what mechanism is responsible for the observed damping of coronal oscillations.

Many real-world multivariable systems need to be modeled to capture the interconnected behavior of their physical variables and to understand how uncertainty in actuators and sensors affects the system dynamics. In system identification, some estimation algorithms are formulated as multivariate data problems by assuming symmetric noise distributions, yielding deterministic system models. Nevertheless, modern multivariable systems must incorporate the uncertainty behavior as a part of the system model structure, taking advantage of asymmetric distributions to model the uncertainty. This paper addresses the uncertainty modeling and identification of a class of multivariable linear dynamic systems, adopting a Stochastic Embedding approach. We consider a nominal system model and a Gaussian mixture distributed error-model driven by an exogenous input signal. The error-model parameters are treated as latent variables and a Maximum Likelihood algorithm that functions by marginalizing the latent variables is obtained. An Expectation-Maximization algorithm that jointly uses the measurements from multiple independent experiments is developed, yielding closed-form expressions for the Gaussian mixture estimators and the noise variance. Numerical simulations demonstrate that our approach yields accurate estimates of both the multivariable nominal system model parameters and the noise variance, even when the error-model non-Gaussian distribution does not correspond to a Gaussian mixture model.

ABSTRACTAdiabatic and nonadiabatic dynamics in classical systems composed of fast and slow modes are formulated. Theoretical consequences from both adiabatic and nonadiabatic dynamics are discussed. In particular, we aim to identify a possible physical origin for drastic change of the slow dynamics to be induced by instability that happens to emerge in the fast mode. The Haken slaving principle and other theories like the adiabatic elimination method focus on the order formation led by the growth of the slow modes, with slaving the fast modes so as to faithfully follow the system dynamics. In this context, the fast modes are quite often regarded as ‘noise’ or a factor of small fluctuation. However, in molecular nonadiabatic dynamics, due to the breakdown of the Born–Oppenheimer approximation, the roles of the fast mode (due to electron dynamics) is far from that of ‘noise’, and sometimes bring about a qualitative bifurcation of the dynamics of the slow subsystem, which we usually refer to as nonadiabatic transition. We show the similar sudden change can be caused by the fast mode in classical mechanics too. The similarity and difference between quantum and classical nonadiabatic dynamics will be discussed.

A systematic design method for assigning a required time-scale structure to a given multivariable system is developed. The time-scale structure assignment is carried out using high-gain feedback of either the state or the output. The method developed decomposes a given multivariable system into several single-input single-output subsystems, each of which is designed separately. A standard method of design for single-input single-output systems is developed, and this forms a building block for the multivariable system design. The state feedback design depends only on the stage sequence of the given system, while the output feedback design depends only on the set of integers that specifies the order of infinite zeros. In this way, our design is made robust with respect to certain parameter uncertainties. High-gain parameter(s) can be adjusted on line, and this provides a tool for the fine tuning of the desired performance characteristics of a control system.

Disturbance rejection in multivariable systems by state feedback controller

This paper generalizes a recently reported method [1] of closed-loop eigenstructure assignment via state feedback in a linear multivariable system (with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> states and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</tex> control inputs). By introducing a lemma on the differentiation of determinants, the class of assignable eigenvectors and generalized eigenvectors associated with the assigned eigenvalues is explicitly described by a complete set of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n r</tex> -dimensional free parameter vectors. This parametric characterization conveniently organizes the nonuniqueness of the solution of the eigenvalue-assignment problem and thereby provides an efficient means of further modifying the system dynamic response. A numerical example is worked out to demonstrate the feasibility of the method.

Abstract. We test a simple technique based on breeding to separate fast and slow unstable modes in coupled systems with different time scales of evolution and variable amplitudes. The technique takes advantage of the earlier saturation of error growth rate of the fastest mode and of the lower value of the saturation amplitude of perturbation of either the fast or the slow modes. These properties of the coupled system allow a physically-based selection of the rescaling time interval and the amplitude of initial perturbations in the "breeding" of unstable modes (Toth and Kalnay, 1993, 1996, 1997; Aurell et al., 1997; Boffetta et al., 1998) to isolate the desired mode. We perform tests in coupled models composed of fast and slow versions of the Lorenz (1963) model with different strengths of coupling. As examples we present first a coupled system which we denote "weather with convection", with a slow, large amplitude model coupled with a fast, small amplitude model, second an "ENSO" system with a "tropical atmosphere" strongly coupled with a "tropical ocean", and finally a triply coupled system denoted "tropical-extratropical" in which a fast model (representing the "extratropical atmosphere") is loosely coupled to the "ENSO" system. We find that it is always possible to isolate the fast modes by taking the limit of small amplitudes and short rescaling intervals, in which case, as expected, the results are the same as the local Lyapunov growth obtained with the linear tangent model. In contrast, slow modes cannot be isolated with either Lyapunov or Singular vectors, since the linear tangent and adjoint models are dominated by the fast modes. Breeding is successful in isolating slow modes if rescaling intervals and amplitudes are chosen from physically appropriate scales.

We present several extensions of the classical Bode integral relations to multivariable systems. The main contributions are an extended gain-phase integral formula and an extended sensitivity integral relation. These results are useful for analyzing fundamental design limitations that arise in linear multivariable feedback systems. In particular, they provide important insights into feedback properties which are unique to multivariable systems and which cannot be inferred from the previously known integral relations applicable to single-input/single-output systems. Both results show that design limitations in multivariable systems depend strongly upon certain directionality properties.

S-54-2 - The integration of intrinsic properties and synaptic inputs in locus coeruleus

This paper is concerned with the nonlinear static state feedback stabilization of time-invariant linear multivariable continuous-time systems with friction driven backlash in the control inputs. A time-invariant static nonlinear state feedback compensator is proposed, easily obtainable solving an algebraic Riccati equation. The global asymptotic stability of the resulting closed-loop nonlinear system is proved using the Lyapunov's direct method and the La Salle's invariance principle. A simulation study shows that the proposed nonlinear state feedback compensator achieves an overall stabilization performance significantly better than the known backlash inverse compensator.

Robust eigenstructure assignment via dynamical compensators