Formulation of A State Equation Including Fractional-Order State Vectors

2008 ◽  
Vol 3 (2) ◽  
Author(s):  
Masaharu Kuroda
Keyword(s):  
Fractional Calculus ◽  
Control Theory ◽  
Fractional Order ◽  
State Equation ◽  
The State ◽  
Space Representation ◽  
Fractional Dynamics ◽  
Modern Control Theory ◽  
Order State ◽  
Modern Control

In recent years, applications of fractional calculus have flourished in various science and engineering fields. Particularly in engineering, control engineering appears to be expanding aggressively in their applications. Exemplary are the CRONE controller and the PIλDμ controller, which is categorizable into applications of fractional calculus in classical control theory. A state equation can be called as the foundation of modern control theory. However, the relationship between fractional derivatives and the state equation has not been examined sufficiently. Consequently, a systematic procedure referred to every researcher on the fractional-calculus side or control-theory side has not yet been established. For this study, therefore, involvement of fractional-order derivatives into a state equation is demonstrated here for ready comprehension by researchers. First, the procedures are explained generally; then the technique to incorporate a fractional-order state vector into a conventional state equation is given as an example of the applications. The state-space representation in this study is useful not only to model a controlled system with fractional dynamics but also for design and implementation of a controller to control fractional-order states. After introducing the basic parts, the benefits of modern control theory including robust control theories, such as H∞ and μ-analysis and synthesis in their integrities, can be applied to this fractional-order state equation.

Formulation of a State Equation Including Fractional-Order State-Vectors

2007 ◽  
Author(s):  
Masaharu Kuroda
Keyword(s):  
Fractional Calculus ◽  
Control Theory ◽  
Fractional Order ◽  
State Equation ◽  
The State ◽  
Space Representation ◽  
Fractional Dynamics ◽  
Modern Control Theory ◽  
Order State ◽  
Modern Control

In recent years, applications of fractional calculus have flourished in various science and engineering fields. Particularly in engineering, control engineering appears to be expanding aggressively in its applications. Exemplary are the CRONE controller and the PIλDμ controller, which is categorizable into applications of fractional calculus in classical control theory. A state equation can be called the foundation of modern control theory. However, the relationship between fractional derivatives and the state equation has not been examined sufficiently. Consequently, a systematic procedure referred to by every researcher on the fractional-calculus side or control-theory side has not yet been established. For this study, therefore, involvement of fractional-order derivatives into a state equation is demonstrated here for ready comprehension by researchers. First, the procedures are explained generally; then the technique to incorporate the fractional-order state-vector into a conventional state equation is given as an example of the applications. The state-space representation in this study is useful not only for modeling a controlled system with fractional dynamics, but also for design and implementation of a controller to control fractional-order states. After we complete installation of the basic parts, we can apply the benefits of modern control theory, including robust control theories such as H-infinity and μ-analysis and synthesis in their integrities, to this fractional-order state-equation.

Modern control theory

2001 ◽  
pp. 23-70
Author(s):  
Branislav Kisačanin ◽  
Gyan C. Agarwal
Keyword(s):  
Control Theory ◽  
Modern Control Theory ◽  
Modern Control

Carathe´odory Controllability Criterion for Nonlinear Dynamical Systems

1978 ◽  
Vol 100 (3) ◽  
pp. 209-213 ◽  
Author(s):  
G. Langholz ◽  
M. Sokolov
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Control Theory ◽  
Linear Systems ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical ◽  
Modern Control Theory ◽  
Open Question ◽  
Modern Control

The question of whether a system is controllable or not is of prime importance in modern control theory and has been actively researched in recent years. While it is a solved problem for linear systems, it is still an open question when dealing with bilinear and nonlinear systems. In this paper, a controllability criterion is established based on a theorem by Carathe´odory. By associating a given dynamical system with a certain Pfaffian equation, it is argued that the system is controllable (uncontrollable) if its associated Pfaffian form is nonintegrable (integrable).

Modern Control Theory and the Limits of Criminal Justice

2019 ◽  
Author(s):  
Michael Gottfredson ◽  
Travis Hirschi
Keyword(s):  
Criminal Justice ◽  
Control Theory ◽  
Problem Behaviors ◽  
Self Control ◽  
Social Control Theory ◽  
Modern Control Theory ◽  
Criminal Sanctions ◽  
Course Outcomes ◽  
Modern Control

Modern Control Theory and the Limits of Criminal Justice updates and extends the authors’ classic general theory of crime (sometimes referred to as “self-control theory”). In Part I, contemporary evidence about the theory is summarized. Research from criminology, psychology, economics, education, and public health substantially supports the lifelong influence of self control as a significant cause of problem behaviors, including delinquency and crime, substance abuse, school problems, many forms of accidents, employment instability, and many poor health outcomes. Contemporary evidence is supportive of the theory’s focus on early socialization for creation of higher levels of self control and other dimensions of the theory, including the roles of self control, age and the generality or versatility of problem behaviors, as well as the connections between self control and later teen and adult problem behaviors. The book provides methodological assessments of research on the theory, contrasting the control theory perspective with other developmental perspectives in criminology. The role of opportunity, the relationship between self and social control theory, and the role of motivation are addressed. In Part II, control theory is taken to be a valid theory and is used to explore the role of criminal sanctions, especially policing and prisons, and policies about immigration, as methods to impact crime. Modern control theory provides an explanation for the general lack of effectiveness of formal, state sanctions on crime and instead provides substantial justification for prevention of delinquency and crime by a focus on childhood. The theory effectively demonstrates the limits of criminal sanctions and the connection between higher levels of self control and positive life-course outcomes.

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