Sufficient Conditions for Minimum-Phase Second-Order Linear Systems

1995 ◽  
Vol 1 (2) ◽  
pp. 183-199 ◽  
Author(s):  
Jong-Lick Lin ◽  
Jer-Nan Juang
Keyword(s):  
Linear System ◽  
Linear Systems ◽  
Common Knowledge ◽  
Sufficient Conditions ◽  
Second Order ◽  
Geometrical Approach ◽  
Minimum Phase ◽  
Sensors And Actuators ◽  
Influence Matrix ◽  
Order Linear

Several sufficient conditions are derived for a second-order linear system to be minimum phase. It is commonly known that a second-order linear system with collocated sensors and actuators is minimum phase. In general, this common knowledge is correct mathematically when the input influence matrix is the transpose of the output influence matrix. In this paper, we extend this knowledge to the case with noncollocated actuators and sensors. First, a conventional approach is used to prove some sufficient conditions of sensor and actuator locations for a system to be minimum phase. Second, a geometrical approach is introduced to discuss the sufficient conditions, and then used to derive other more useful sufficient conditions for a minimum-phase second-order linear system. Many illustrative examples are provided for the readers to better understand the theories developed in this paper.

The Stability of a Second Order Linear Periodic System

1969 ◽  
Vol 91 (2) ◽  
pp. 207-210 ◽  
Author(s):  
E. J. Davison
Keyword(s):  
Linear System ◽  
Periodic System ◽  
Sufficient Conditions ◽  
Second Order ◽  
Necessary And Sufficient Conditions ◽  
Order Linear ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Linear Periodic System

Necessary and sufficient conditions are obtained for the stability of the following second order linear system: x˙=θ(t)x,θ(t)=θt+∑i=1lTi and θ(t) =A1,0<t<T1=A2,T1<t<T1+T2⋮=Al,∑i=1l−1Ti<t<∑i=1lTi in terms of the eigenvalues and elements of the matrices Ai, i = 1, 2…l. The conditions become very simple for the case that l = 2. An example of a pendulum with a vibrating support is included.

Minimum-Phase Robustness for Second-Order Linear Systems

1999 ◽  
Vol 22 (2) ◽  
pp. 229-234 ◽  
Author(s):  
Jong-Lick Lin ◽  
Shin-Ju Chen ◽  
Jer-Nan Juang
Keyword(s):  
Linear Systems ◽  
Second Order ◽  
Minimum Phase ◽  
Order Linear

ACCELERATION OF CONVERGENCE OF WAVEFORM RELAXATION ITERATIONS FOR SECOND ORDER LINEAR SYSTEM

1996 ◽  
Vol 20 (1) ◽  
pp. 87-103
Author(s):  
Z. Jackiewicz ◽  
J. Knap
Keyword(s):  
Linear System ◽  
Differential Equations ◽  
Linear Systems ◽  
Second Order ◽  
Waveform Relaxation ◽  
Relaxation Techniques ◽  
Speed Of Convergence ◽  
Acceleration Of Convergence ◽  
Order Linear ◽  
Systems Of Differential Equations

The solution of second order linear systems of differential equations by waveform relaxation techniques is investigated. It is demonstrated that overlapping of components of the system improves significantly the speed of convergence of the resulting waveform relaxation iterations.

Robust model reference control for second-order descriptor linear systems subject to parameter uncertainties

2021 ◽  
pp. 014233122110450
Author(s):  
Guang-Tai Tian ◽  
Guang-Ren Duan
Keyword(s):  
Linear Systems ◽  
Tracking Error ◽  
Sufficient Conditions ◽  
Second Order ◽  
Parameter Uncertainties ◽  
Robust Controller ◽  
Model Reference ◽  
Robust Model ◽  
Robust Compensation ◽  
Descriptor Linear Systems

This paper is devoted to designing the robust model reference controller for uncertain second-order descriptor linear systems subject to parameter uncertainties. The parameter uncertainties are assumed to be norm-bounded. The design of a robust controller can be divided into two separate problems: a robust stabilization problem and a robust compensation problem. Based on the solution of generalized Sylvester matrix equations, we obtain some sufficient conditions to guarantee the complete parameterization of the robust controller. The parametric forms are expressed by a group of parameter vectors which reveal the degrees of freedom existing in the design of the compensator and can be utilized to solve the robust compensation problem. In order to reduce the effect of parameter uncertainties on the tracking error vector, the robust compensation problem is converted into a convex optimization problem with a set of linear matrix equation constraints. A simulation example is provided to illustrate the effectiveness of the proposed technique.

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