scholarly journals Transformation of Uncertain Linear Systems with Real Eigenvalues into Cooperative Form: The Case of Constant and Time-Varying Bounded Parameters

Algorithms ◽  
2021 ◽  
Vol 14 (3) ◽  
pp. 85
Author(s):  
Andreas Rauh ◽  
Julia Kersten
Keyword(s):  
Linear Systems ◽  
Initial Conditions ◽  
A Priori ◽  
Real Life ◽  
Reachability Analysis ◽  
Interval Methods ◽  
Space Representation ◽  
Uncertain Parameters ◽  
Time Varying ◽  
Similarity Transformations

Continuous-time linear systems with uncertain parameters are widely used for modeling real-life processes. The uncertain parameters, contained in the system and input matrices, can be constant or time-varying. In the latter case, they may represent state dependencies of these matrices. Assuming bounded uncertainties, interval methods become applicable for a verified reachability analysis, for feasibility analysis of feedback controllers, or for the design of robust set-valued state estimators. The evaluation of these system models becomes computationally efficient after a transformation into a cooperative state-space representation, where the dynamics satisfy certain monotonicity properties with respect to the initial conditions. To obtain such representations, similarity transformations are required which are not trivial to find for sufficiently wide a-priori bounds of the uncertain parameters. This paper deals with the derivation and algorithmic comparison of two different transformation techniques for which their applicability to processes with constant and time-varying parameters has to be distinguished. An interval-based reachability analysis of the states of a simple electric step-down converter concludes this paper.

Amplitudes Decay in Different Kinds of Nonlinear Oscillators

2015 ◽  
Vol 137 (3) ◽  
Author(s):  
Mohammad A. Al-Shudeifat
Keyword(s):  
Nonlinear Oscillator ◽  
Initial Conditions ◽  
A Priori ◽  
Nonlinear Oscillators ◽  
Transformation Method ◽  
Equation Of Motion ◽  
Time Varying ◽  
Initial Displacement ◽  
Restoring Force ◽  
First Order

A transformation is employed to obtain expressions for the decay of the displacement, the velocity, and the energy for various forms of nonlinear oscillators. The equation of motion of the nonlinear oscillator is transformed into a first-order decay term plus an energy term, where this transformed equation can be decoupled into a set of two analytically solvable equations. The decoupled equations can be solved for the decay formulas. Unlike other methods in the literature, this transformation method is directly applied to the equation of motion, and an approximate solution is not required to be known a priori. The method is first applied to a purely nonlinear oscillator with a non-negative, real-power restoring force to obtain the decay formulas. These decay formulas are found to behave similarly to those of a linear oscillator. In addition, these formulas are employed to obtain an accurate formula for the frequency decay. Based on this result, the exact frequency formula given in the literature for this oscillator is generalized by substituting the initial values of the envelopes for the actual initial conditions. By this modification, the formulas for the initial and time-varying frequencies become valid for any combination of the initial displacement and velocity. Furthermore, a generalized nonlinear oscillator for which the transformation is always valid is introduced. From this generalized oscillator, the proposed transformation is applied to analyze various types of oscillators.

Guaranteed Asymptotic Output Stability for Systems With Constant Disturbances

1987 ◽  
Vol 109 (2) ◽  
pp. 186-189 ◽  
Author(s):  
W. E. Schmitendorf ◽  
B. R. Barmish
Keyword(s):  
Feedback Control ◽  
Linear Systems ◽  
State Feedback ◽  
Initial Conditions ◽  
State Feedback Control ◽  
Control Law ◽  
Uncertain Parameters ◽  
Output Stability ◽  
Linear State ◽  
Parameter Values

For a class of linear systems in which there are uncertain parameters in the system and input matrices, as well as constant additive disturbances, a linear state feedback control law is derived. The only information available about the uncertain parameters is the bounding sets in which they lie. The design guarantees that the specified output approaches zero for all possible parameter values and for all initial conditions. Two examples illustrate the application of the theory.

scholarly journals A Systematic Review and Qualitative Synthesis Resulting in a Typology of Elementary Classroom Movement Integration Interventions

2020 ◽  
Vol 6 (1) ◽  
Author(s):  
Spyridoula Vazou ◽  
Collin A. Webster ◽  
Gregory Stewart ◽  
Priscila Candal ◽  
Cate A. Egan ◽  
...  
Keyword(s):  
Physical Activity ◽  
Teacher Collaboration ◽  
A Priori ◽  
Real Life ◽  
Dose Intensity ◽  
Routine Practice ◽  
Qualitative Synthesis ◽  
Wide Range ◽  
Meta Analyses ◽  
Movement Integration

Abstract Background/Objective Movement integration (MI) involves infusing physical activity into normal classroom time. A wide range of MI interventions have succeeded in increasing children’s participation in physical activity. However, no previous research has attempted to unpack the various MI intervention approaches. Therefore, this study aimed to systematically review, qualitatively analyze, and develop a typology of MI interventions conducted in primary/elementary school settings. Subjects/Methods Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) guidelines were followed to identify published MI interventions. Irrelevant records were removed first by title, then by abstract, and finally by full texts of articles, resulting in 72 studies being retained for qualitative analysis. A deductive approach, using previous MI research as an a priori analytic framework, alongside inductive techniques were used to analyze the data. Results Four types of MI interventions were identified and labeled based on their design: student-driven, teacher-driven, researcher-teacher collaboration, and researcher-driven. Each type was further refined based on the MI strategies (movement breaks, active lessons, other: opening activity, transitions, reward, awareness), the level of intrapersonal and institutional support (training, resources), and the delivery (dose, intensity, type, fidelity). Nearly half of the interventions were researcher-driven, which may undermine the sustainability of MI as a routine practice by teachers in schools. An imbalance is evident on the MI strategies, with transitions, opening and awareness activities, and rewards being limitedly studied. Delivery should be further examined with a strong focus on reporting fidelity. Conclusions There are distinct approaches that are most often employed to promote the use of MI and these approaches may often lack a minimum standard for reporting MI intervention details. This typology may be useful to effectively translate the evidence into practice in real-life settings to better understand and study MI interventions.

scholarly journals The VIT Transform Approach to Discrete-Time Signals and Linear Time-Varying Systems

Eng ◽  
2021 ◽  
Vol 2 (1) ◽  
pp. 99-125
Author(s):  
Edward W. Kamen
Keyword(s):  
Discrete Time ◽  
Initial Conditions ◽  
Linear Time ◽  
Order Term ◽  
Initial Time ◽  
Time Varying ◽  
Varying Coefficients ◽  
Time Signals ◽  
Time Varying Systems ◽  
Time Varying Coefficients

A transform approach based on a variable initial time (VIT) formulation is developed for discrete-time signals and linear time-varying discrete-time systems or digital filters. The VIT transform is a formal power series in z−1, which converts functions given by linear time-varying difference equations into left polynomial fractions with variable coefficients, and with initial conditions incorporated into the framework. It is shown that the transform satisfies a number of properties that are analogous to those of the ordinary z-transform, and that it is possible to do scaling of z−i by time functions, which results in left-fraction forms for the transform of a large class of functions including sinusoids with general time-varying amplitudes and frequencies. Using the extended right Euclidean algorithm in a skew polynomial ring with time-varying coefficients, it is shown that a sum of left polynomial fractions can be written as a single fraction, which results in linear time-varying recursions for the inverse transform of the combined fraction. The extraction of a first-order term from a given polynomial fraction is carried out in terms of the evaluation of zi at time functions. In the application to linear time-varying systems, it is proved that the VIT transform of the system output is equal to the product of the VIT transform of the input and the VIT transform of the unit-pulse response function. For systems given by a time-varying moving average or an autoregressive model, the transform framework is used to determine the steady-state output response resulting from various signal inputs such as the step and cosine functions.

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