Second-order Linear Systems Research Articles

Assessing Manual Pursuit Tracking in Parkinson’s Disease Via Linear Dynamical Systems

Quantitative assessment of motor performance is important for diseases of motor control, such as Parkinson's disease (PD). Manual tracking tasks are well suited for motor assessment, as they can be performed concomitantly with brain mapping techniques. Here we propose utilizing second-order linear dynamical systems to assess manual pursuit tracking performance. With the desired trajectory as the input, and the subject's actual motor response as the output, a linear model characterized by natural frequency and damping ratio is identified for each subject. We applied this method to 10 PD subjects (on and off L: -dopa medication) and 10 normal subjects performing a multi-frequency sinusoidal tracking task. Model parameters were more sensitive than overall tracking errors in determining significant differences between groups. The effect of L: -dopa medication was to reduce the damping ratio and make the range in natural frequency across individuals approach that of normal subjects. We interpret the changes in damping ratio and natural frequency as possibly related to suppression of compensatory cerebellar activity and/or improvement of motor program selection, and the changes in natural frequency as an implicit strategy to maintain settling time in the face of reduce damping ratio. Our results suggest that simple linear dynamical system models can be a powerful method to assess tracking performance in Parkinson's disease because of the additional insight they can provide into neurological processes.

The Transformation of Second-Order Linear Systems into Independent Equations

The class of second-order linear dynamical systems is considered. A method and algorithm are presented to transform any system with n degrees of freedom into n independent second-order equations. The conversion utilizes a real, invertible but nonlinear mapping and is applicable to practically every linear system. Two examples from earthquake engineering are provided to indicate the utility of this approach.

Existence of solutions for a class of second-order sublinear and linear Hamiltonian systems with impulsive effects

Existence of solutions for a class of second-order sublinear and linear Hamiltonian systems with impulsive effects

The reducibility of second-order linear time-varying homogeneous systems

Problems of the theory of the reduction of first- and second-order homogeneous time-varying systems are briefly described. Problems of the motion of a gyrohorizon compass and of the periodic motion of a rotor, attached to a flexible shaft are considered as interesting examples.

Stabilization of the second-order linear time-invariant control systems by a delayed feedback

We consider a static stabilization problem for a two-dimensional linear time-invariant control system with a delayed feedback. We obtain the necessary and sufficient conditions for the stabilizability of the system under consideration. The theorems proved in this paper show that such a delayed feedback approach is efficient in stabilizing the second-order linear systems.

Dynamics and contouring control of a 3-DoF parallel kinematics machine

In machining applications, instead of tracking error (the difference between the actual position and the desired position), contouring error (the minimum distance from the actual position to the desired trajectory) characterizes product quality. In this paper, we propose a generalized moving task coordinate frames based contouring control for parallel kinematics machines, whose dynamics is in general coupled and strongly nonlinear. The Orthopod, a 3 degree-of-freedom purely translational parallel kinematics machine, is introduced as a control plant. The Lagrange-D’Alembert formulation is used to model the system dynamics. The developed dynamic model in Cartesian space is transformed and parametrized by tangential error, normal error, and binormal error in moving task coordinate frames. The contouring error is then approximated by the normal error and the binormal error, which is the projection of tracking error to the normal plane at the desired position. By employing the structural properties of the transformed dynamics, a special feedback linearization, the computed torque control is applied. It leads to a stabilization problem for a second-order linear time-invariant system. Coulumb plus viscous friction model is used to compensate friction effects. Friction parameters are identified by least-squares approach. For comparison purpose, the tracking error based computed torque control is also carried out. Experiments demonstrate that the proposed control scheme not only leads to improved contouring accuracy, but also produces smaller and smoother control input torques, which may contribute to smaller vibration.

Second-order numerator-dynamics systems: Effects of initial discontinuities

Linear second-order systems with numerator-dynamics are considered and their transient responses are investigated for discontinuous inputs. Characteristic parameters for second-order systems with numerator dynamics are identified. The solution profiles of these systems depend upon the value of one such parameter relative to that of the others on the number line. Effects of initial jump discontinuities of the inputs on the correctness of the solutions are also treated. A methodology for analysis of discontinuities is presented. Initial discontinuities of the calculated response either get accounted for in the input function or the initial condition of the input function for all but the following case. For the impulse response alone, that too among inherent numerator-dynamics systems only, an initial di scontinuity cannot be accounted for in the input. The value of that discontinuity is thus required for correct solutions of these cases. The results are general and extendable to different inputs and higher order systems.

On dynamic reconstruction of input in the second-order linear systems

Consideration was given to the problem of dynamic reconstruction of input in the second-order linear systems. Procedures for its real-time solution were specified. They feature dynamicity, which means that the current values of the approximated input are generated in real time, and stability, which means that for a sufficiently small measurement error approximation is arbitrarily precise.

Ion Channel Biosensors—Part II: Dynamic Modeling, Analysis, and Statistical Signal Processing

This paper deals with the dynamic modeling, analysis, and statistical signal processing of the ion channel switch biosensor. The electrical dynamics are described by a second-order linear system. The chemical kinetics of the biosensor response to analyte concentration in the reaction-rate-limited regime are modeled by a two-timescale nonlinear system of differential equations. Also, the analyte concentration in the mass-transport-influenced regime is modeled by a partial differential equation subject to a mixture of Neumann and Dirichlet boundary conditions. By using the theory of singular perturbation, we analyze the model so as to predict the performance of the biosensor in transient and steady-state regimes. Finally, we outline the use of statistical signal processing algorithms that exploit the biosensor dynamics to classify analyte concentration.

Isometric force generated by locust skeletal muscle: responses to single stimuli

A mathematical model of the locust hind leg extensor muscle is described. The model accounts for the force response of the muscle to well-separated input stimuli under isometric conditions. Experimental data was collected by stimulating the extensor muscle and measuring the force generated at the tibia. In developing a model it was assumed that the response to a single isolated stimulus was linear. A linear model was found to fit well to the response to an isolated stimulus. No assumptions were made about the model order and models of various order were fitted to data in the frequency domain, using a least squares fit. The stimulus can be approximated as an impulse, with the response to each stimulus well described by a linear second-order system. Using a third-order model provided a better fit to data, but the improvement in fit was marginal and the model uses one extra parameter. A fourth-order model, which is often used to describe the behaviour of isometric muscle was found to overfit the data. Using a second-order model provides a simpler way of describing the behaviour of an isometric twitch.

Proportional damping approximation using the energy gain and simultaneous perturbation stochastic approximation

The design of vector second-order linear systems for accurate proportional damping approximation is addressed. For this purpose an error system is defined using the difference between the generalized coordinates of the non-proportionally damped system and its proportionally damped approximation in modal space. The accuracy of the approximation is characterized using the energy gain of the error system and the design problem is formulated as selecting parameters of the non-proportionally damped system to ensure that this gain is sufficiently small. An efficient algorithm that combines linear matrix inequalities and simultaneous perturbation stochastic approximation is developed to solve the problem and examples of its application to tensegrity structures design are presented.

Inverse Dynamics Modeling and Nonlinear Controller Design of the Similar Model of a Forging Manipulator

Forging manipulator is one of the most important equipment in the automation of forging operation,and plays an important role in improving productivity and ensuring manufacturing quality. To improve the forging manipulator's performance,a similar model of a forging manipulator is designed to facilitate experiments. The inverse dynamics of this similar model is obtained by using the principal of virtual work;and to decrease the negative effect of the nonlinearity of the mechanism dynamics,the dynamic feedforward control method is adopted,distributing driving forces,which are needed for the tongs carrier of the manipulator to generate desired motions,to each hydraulic cylinder. The tracking performance of the tongs carrier is chosen to be guaranteed by improving the position response performance of each hydraulic cylinder. So,by using feedback linearization method and considering the nonlinearities of hydraulic systems,a hydraulic cylinder system is transformed into a second-order linear system which can be easily adjusted by setting four parameters to get desired damp ratio and natural frequency and to improve the response dynamics. Simulation results finally attest the validity of the proposed method.

On a Mixed Problem for a Constant Coefficient Second-Order System

The paper is devoted to the study of an initial boundary value problem for a linear second-order differential system with constant coefficients. The first part of the paper is concerned with the existence of the solution to a boundary value problem for the second-order differential system, in the strip , where is a suitable positive number. The result is proved by means of the same procedure followed in a previous paper to study the related initial value problem. Subsequently, we consider a mixed problem for the second-order constant coefficient system, where the space variable varies in and the time-variable belongs to the bounded interval , with sufficiently small in order that the operator satisfies suitable energy estimates. We obtain by superposition the existence of a solution , by studying two related mixed problems, whose solutions exist due to the results proved for the Cauchy problem in a previous paper and for the boundary value problem in the first part of this paper.

Balanced Truncation of Linear Second-Order Systems: A Hamiltonian Approach

We present a formal procedure for structure-preserving model reduction of linear second-order and Hamiltonian control problems that appear in a variety of physical contexts, e.g., vibromechanical systems or electrical circuit design. Typical balanced truncation methods that project onto the subspace of the largest Hankel singular values fail to preserve the problem's physical structure and may suffer from lack of stability. In this paper, we adopt the framework of generalized Hamiltonian systems that covers the class of relevant problems and that allows for a generalization of balanced truncation to second-order problems. It turns out that the Hamiltonian structure, stability, and passivity are preserved if the truncation is done by imposing a holonomic constraint on the system rather than standard Galerkin projection.

Coupled equations for reverse time migration in transversely isotropic media

Reverse time migration (RTM) images reflectors by using time-extrapolation modeling codes to synthesize source and receiver wavefields in the subsurface. Asymptotic analysis of wave propagation in transversely isotropic (TI) media yields a dispersion relation describing coupled P- and SV-wave modes. This dispersion relation can be converted into a fourth-order scalar partial differential equation (PDE). Increased computational efficiency can be achieved using equivalent coupled second-order PDEs. Analysis of the corresponding dispersion relations as matrix eigenvalue systems allows one to characterize all possible coupled linear second-order systems equivalent to a given linear fourth-order PDE and to determine which ones yield optimally efficient finite-difference implementations. Setting the shear velocity along the axis of symmetry to zero yields a simpler approximate TI wave equation that is more efficient to implement. This simpler approximation, however, can become unstable for some plausible combinations of anisotropic parameters. The same eigensystem analysis can be applied using finite vertical shear velocity to obtain solutions that avoid these instability problems.

Integral representations of solutions to second-order linear elliptic systems with an interior supersingular point

Integral representations of solutions to second-order linear elliptic systems with an interior supersingular point

A Dynamic Causal Model of the Coupling Between Pulse Stimulation and Neural Activity

We present a dynamic causal model that can explain context-dependent changes in neural responses, in the rat barrel cortex, to an electrical whisker stimulation at different frequencies. Neural responses were measured in terms of local field potentials. These were converted into current source density (CSD) data, and the time series of the CSD sink was extracted to provide a time series response train. The model structure consists of three layers (approximating the responses from the brain stem to the thalamus and then the barrel cortex), and the latter two layers contain nonlinearly coupled modules of linear second-order dynamic systems. The interaction of these modules forms a nonlinear regulatory system that determines the temporal structure of the neural response amplitude for the thalamic and cortical layers. The model is based on the measured population dynamics of neurons rather than the dynamics of a single neuron and was evaluated against CSD data from experiments with varying stimulation frequency (1-40 Hz), random pulse trains, and awake and anesthetized animals. The model parameters obtained by optimization for different physiological conditions (anesthetized or awake) were significantly different. Following Friston, Mechelli, Turner, and Price (2000), this work is part of a formal mathematical system currently being developed (Zheng et al., 2005) that links stimulation to the blood oxygen level dependent (BOLD) functional magnetic resonance imaging (fMRI) signal through neural activity and hemodynamic variables. The importance of the model described here is that it can be used to invert the hemodynamic measurements of changes in blood flow to estimate the underlying neural activity.

Organizing syllables into groups—Evidence from F 0 and duration patterns in Mandarin

In this study we investigated grouping-related F 0 patterns in Mandarin by examining the effect of syllable position in a group while controlling for tone, speaking mode, number of syllables in a group, and group position in a sentence. We analyzed syllable duration, F 0 displacement, ratio of peak velocity to F 0 displacement ( v p/ d ratio) and shape of F 0 velocity profile (parameter C) in sequences of Rising, Falling and High tones. Results showed that syllable duration had the most consistent grouping-related patterns. In a short phrase of 1–4 syllables, duration is longest in the final position, second longest in the initial position, and shortest in the medial positions. In Rising and Falling tone sequences, syllable duration was positively related to F 0 displacement, but negatively related to v p/ d ratio. Sequences consisting of only the High tone, however, showed no duration-matching F 0 variations. Modeling simulations with a second-order linear system showed that duration variations alone could generate F 0 displacement and v p/ d ratio variations comparable to those in actual data. We interpret the results as evidence that grouping is encoded directly by syllable duration, while the corresponding variations in F 0 displacement, v p/ d ratio and velocity profile are the consequences of duration control.

Stabilization for a class of second-order switched systems with saturation constrains

This paper proposes a global stabilization method for a class of planar switched systems with input saturation constrains by using a state feedback and switching strategy. This method is proved to be effective by analyzing the characteristics of the trajectory of second-order linear systems with input saturation.

Vibration control of base-isolated structures using mixed H2/H∞ output-feedback control

A mixed H2/ H∞ output-feedback control design methodology for vibration reduction of base-isolated building structures modelled in the form of second-order linear systems is presented. Sufficient conditions for the design of a desired control are given in terms of linear matrix inequalities. A controller that guarantees asymptotic stability and a mixed H2/ H∞ performance for the closed-loop system of the structure is developed, based on a Lyapunov function. The performance of the controller is evaluated by means of simulations in MATLAB/Simulink.