scholarly journals Exponential Stabilization of Linear Time-Varying Differential Equations with Uncertain Coefficients by Linear Stationary Feedback

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 853
Author(s):  
Vasilii Zaitsev ◽  
Inna Kim
Keyword(s):  
Differential Equation ◽  
Decay Rate ◽  
Linear Time ◽  
Sufficient Conditions ◽  
Exponential Stabilization ◽  
Time Varying ◽  
Static State ◽  
Linear Time Invariant ◽  
Linear Differential ◽  
Time Invariant

We consider a control system defined by a linear time-varying differential equation of n-th order with uncertain bounded coefficients. The problem of exponential stabilization of the system with an arbitrary given decay rate by linear static state or output feedback with constant gain coefficients is studied. We prove that every system is exponentially stabilizable with any pregiven decay rate by linear time-invariant static state feedback. The proof is based on the Levin’s theorem on sufficient conditions for absolute non-oscillatory stability of solutions to a linear differential equation. We obtain sufficient conditions of exponential stabilization with any pregiven decay rate for a linear differential equation with uncertain bounded coefficients by linear time-invariant static output feedback. Illustrative examples are considered.

scholarly journals Continuous LTI Input–Output Stable Systems on $${L^{p}(\mathbb {R})}$$ and $${\mathscr {D'}_{L^{p}}(\mathbb {R})}$$ Associated with Differential Equations: Existence, Invertibility Conditions and Inversion

2021 ◽  
Author(s):  
M. Ciampa
Keyword(s):  
Differential Equation ◽  
Linear Time ◽  
Sufficient Conditions ◽  
Inverse System ◽  
Continuous Linear ◽  
Input Output ◽  
Single Input Single Output ◽  
Linear Time Invariant ◽  
Linear Differential ◽  
Time Invariant

AbstractA usual problem in analog signal processing is to ascertain the existence of a continuous single-input single-output linear time-invariant input–output stable system associated with a linear differential equation, i.e., of a continuous system such that, for every input signal in a given space of signals, yields an output, in the same space, which verifies the equation with known term the input, and to ascertain the existence of its inverse system. In this paper, we consider, as space of signals, the usual Banach space of $${L^{p}}$$ L p functions, or the space of distributions spanned by $${L^{p}}$$ L p functions and by their distributional derivatives, of any order (input spaces which include signals with not necessarily left-bounded support), we give a systematic theoretical analysis of the existence, uniqueness and invertibility of continuous linear time-invariant input–output stable systems (both causal and non-causal ones) associated with the differential equation and, in case of invertibility, we characterize the continuous inverse system. We also give necessary and sufficient conditions for causality. As an application, we consider the problem of finding a suitable almost inverse of a causal continuous linear time-invariant input–output stable non-invertible system, defined on the space of finite-energy functions, associated with a simple differential equation.

scholarly journals Consensus of Noisy Multiagent Systems with Markovian Switching Topologies and Time-Varying Delays

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Multiagent Systems ◽  
Linear Time ◽  
Design Method ◽  
Sufficient Conditions ◽  
Time Varying ◽  
Linear Time Invariant ◽  
Graph Properties ◽  
Linear Time Invariant Systems ◽  
Necessary And Sufficient ◽  
Time Invariant

Stochastic multiagent systems have attracted much attention during the past few decades. This paper concerns the continuous-time consensus of a network of agents under directed switching communication topologies governed by a time-homogeneous Markovian process. The agent dynamics are described by linear time-invariant systems, with random noises as well as time-varying delays. Two types of network-induced delays are considered, namely, delays affecting only the output of the agents’ neighbors and delays affecting both the agents’ own output and the output of their neighbors. We present necessary and sufficient consensus conditions for these two classes of multiagent systems, respectively. The design method of consensus gains allows for decoupling the design problem from the graph properties. Numerical simulations are implemented to test the effectiveness of our obtained results as well as the tightness of necessary/sufficient conditions.

Zeno-free adaptive event-triggered control design

2018 ◽  
Vol 41 (8) ◽  
pp. 2328-2337 ◽  
Author(s):  
Hassan Adloo ◽  
Mohammad Hossein Shafiei
Keyword(s):  
Degrees Of Freedom ◽  
Linear Time ◽  
Sufficient Conditions ◽  
Performance Criterion ◽  
Linear Time Invariant ◽  
Event Triggering ◽  
Linear Time Invariant Systems ◽  
System States ◽  
Time Invariant ◽  
Event Triggered

This paper presents a new general framework for adaptive event-triggered control strategy to extend average inter-event interval, while maintaining the performance of the system. The proposed event-triggering mechanism is acquired from input to state stability conditions, which is defined in terms of system states as well as an adaptation parameter. Under the Lipschitz assumption, a positive lower bound on sampling durations is also established that is essential to restrain the Zeno behavior. Applying the proposed method to linear time-invariant systems, leads to sufficient conditions to guarantee asymptotic stability in the form of matrix inequalities. Moreover, it is shown that there exist more degrees of freedom to improve the performance criterion from theoretical aspects. Finally, in order to show capability of the proposed method and its better performance compared with some recent works, numerical simulations are presented.

scholarly journals FE Analysis of Communications Systems for Drive-Thru Restaurants in a Business Dispute Over Specifications and Design Process

2021 ◽  
Vol 38 (1) ◽  
Author(s):  
Robert Peruzzi
Keyword(s):  
Communication System ◽  
Communication Systems ◽  
Linear Time ◽  
Forensic Analysis ◽  
Time Varying ◽  
Audio Quality ◽  
Linear Time Invariant ◽  
Linear Time Invariant Systems ◽  
Quick Service ◽  
Time Invariant

Forensic analysis in this case involves the design of a communication system intended for use in Quick Service Restaurant (QSR) drive-thru lanes. This paper provides an overview of QSR communication system components and operation and introduces communication systems and channels. This paper provides an overview of non-linear, time-varying system design as contrasted with linear, time-invariant systems and discusses best design practices. It also provides the details of how audio quality was defined and compared for two potentially competing systems. Conclusions include that one of the systems was clearly inferior to the other — mainly due to not following design techniques that were available at the time of the project.

Existence of Solutions of Riccati Differential Equations

2012 ◽  
Vol 134 (3) ◽  
Author(s):  
Sinan Kilicaslan ◽  
Stephen P. Banks
Keyword(s):  
Differential Equation ◽  
Nonlinear Systems ◽  
Existence Of Solutions ◽  
Linear Time ◽  
Sufficient Conditions ◽  
Necessary Condition ◽  
Time Varying ◽  
Riccati Differential Equation ◽  
Riccati Differential Equations ◽  
Time Varying Systems

A necessary condition for the existence of the solution of the Riccati differential equation for both linear, time varying systems and nonlinear systems is introduced. First, a necessary condition for the existence of the solution of the Riccati differential equation for linear, time varying systems is proposed. Then, the sufficient conditions to satisfy the necessary condition are given. After that, the existence of the solution of the Riccati differential equation is generalized for nonlinear systems.

Linear Approximations for Swing Leg Motion During Gait

1984 ◽  
Vol 106 (2) ◽  
pp. 137-143 ◽  
Author(s):  
W. H. Lee ◽  
J. M. Mansour
Keyword(s):  
Linear Systems ◽  
System Analysis ◽  
Linear Time ◽  
Time Varying ◽  
Two Dimensional ◽  
State Variables ◽  
Linear Time Invariant ◽  
Linear Time Invariant Systems ◽  
Leg Motion ◽  
Time Invariant

The applicability of a linear systems analysis of two-dimensional swing leg motion was investigated. Two different linear systems were developed. A linear time-varying system was developed by linearizing the nonlinear equations describing swing leg motion about a set of nominal system and control trajectories. Linear time invariant systems were developed by linearizing about three different fixed limb positions. Simulations of swing leg motion were performed with each of these linear systems. These simulations were compared to previously performed nonlinear simulations of two-dimensional swing leg motion and the actual subject motion. Additionally, a linear system analysis was used to gain some insight into the interdependency of the state variables and controls. It was shown that the linear time varying approximation yielded an accurate representation of limb motion for the thigh and shank but with diminished accuracy for the foot. In contrast, all the linear time invariant systems, if used to simulate more than a quarter of the swing phase, yielded generally inaccurate results for thigh shank and foot motion.

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