Cauchy formula for the time-varying linear systems with caputo derivative

AbstractIn the paper, existence, uniqueness and a Cauchy formula for the solution to a time-varying linear system containing fractional Caputo derivative is obtained. This formula shows that nonnegativity of the data of the system implies nonnegativity of the solution. In the context of a strenghthening of this result, an example illustrating the absence (in the case of Caputo derivative) of the standard relation “monotonicity of function - sign of derivative”.

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On the Design of Finite Time Settling Control for Linear Systems

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.2897356 ◽

1992 ◽

Vol 114 (3) ◽

pp. 359-368 ◽

Cited By ~ 2

Author(s):

S. Choura

Keyword(s):

Linear System ◽

Linear Systems ◽

Finite Time ◽

Linear Time ◽

Minimum Energy ◽

Time Varying ◽

Parameter Uncertainties ◽

Final State ◽

Variable Function ◽

Time Varying Systems

The design of controllers combining feedback and feedforward for the finite time settling control of linear systems, including linear time-varying systems, is considered. The feedforward part transfers the initial state of a linear system to a desired final state in finite time, and the feedback part reduces the effects of uncertainties and disturbances on the system performance. Two methods for determining the feedforward part, without requiring the knowledge of the explicit state solutions, are proposed. In the first method, a numerical procedure for approximating combined controls that drive linear time-varying systems to their final state in finite time is given. The feedforward part is a variable function of time and is selected based on a set of necessary conditions, such as magnitude constraints. In the second method, an analytical procedure for constructing combined controls for linear time-invariant systems is presented, where the feedforward part is accurately determined and it is of the minimum energy control type. It is shown that both methods facilitate the design of the feedforward part of combined controllers for the finite time settling of linear systems. The robustness of driving a linear system to its desired state in finite time is analyzed for three types of uncertainties. The robustness analysis suggests a modification of the feedforward control law to assure the robustness of the control strategy to parameter uncertainties for arbitrary final times.

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Pole Placement for Single-Input Linear System by Proportional-Derivative State Feedback

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.4028713 ◽

2014 ◽

Vol 137 (4) ◽

Cited By ~ 4

Author(s):

Taha H. S. Abdelaziz

Keyword(s):

Linear System ◽

Linear Systems ◽

Degrees Of Freedom ◽

State Feedback ◽

Pole Placement ◽

Feedback Gain ◽

Time Varying ◽

Placement Problem ◽

Single Input ◽

Time Invariant

This paper deals with the direct solution of the pole placement problem for single-input linear systems using proportional-derivative (PD) state feedback. This problem is always solvable for any controllable system. The explicit parametric expressions for the feedback gain controllers are derived which describe the available degrees of freedom offered by PD state feedback. These freedoms are utilized to obtain closed-loop systems with small gains. Its derivation is based on the transformation of linear system into control canonical form by a special coordinate transformation. The solving procedure results into a formula similar to Ackermann’s one. In the present work, both time-invariant and time-varying linear systems are treated. The effectiveness of the proposed method is demonstrated by the simulation examples of both time-invariant and time-varying systems.

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Finite-Time Stabilization of Uncertain Switched Positive Linear Systems with Time-Varying Delays

Discrete Dynamics in Nature and Society ◽

10.1155/2015/954782 ◽

2015 ◽

Vol 2015 ◽

pp. 1-9 ◽

Cited By ~ 2

Author(s):

Tianjian Yu ◽

Yanke Zhong ◽

Tefang Chen ◽

Chunyang Chen

Keyword(s):

Linear System ◽

Linear Systems ◽

Finite Time ◽

State Feedback ◽

Positive System ◽

Time Varying ◽

Polytopic Uncertainties ◽

Finite Time Stabilization ◽

Finite Time Boundedness ◽

Definition Of

This paper is concerned with finite-time stabilization (FTS) analysis for a class of uncertain switched positive linear systems with time-varying delays. First, a new definition of finite-time boundedness (FTB) is introduced for switched positive system. This definition can simplify FTS analysis. Taking interval and polytopic uncertainties into account, a robust state feedback controller is built such that the switched positive linear system is finite-time bounded. Finally, an example is employed to illustrate the validities of obtained results.

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