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The presentation centers on four classes of systems that can be treated as linear systems over a ring. These are: (1) discrete-time systems over a ring of scalars such as the integers; (2) continuous-time systems containing time delays; (3) large-scale discrete-time systems; and (4) time-varying discrete-time systems.
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Part of the Presidential Policy Directive 21 (PPD-21) (PPD 2013) mandate includes evaluating safety, security, and safeguards (or nonproliferation) mechanisms traditionally implemented within the nuclear reactors, materials, and waste sector of critical infrastructure—including a complex, dynamic set of risks and threats within an all-hazards approach. In response, research out of Sandia National Laboratories (Sandia) explores the ability of systems theory principles (hierarchy and emergence) and complex systems engineering concepts (multidomain interdependence) to better understand and address these risks and threats. Herein, this Sandia research explores the safety, safeguards, and security risks of three different nuclear sector-related activities—spent nuclear fuel transportation, small modular reactors, and portable nuclear power reactors—to investigate the complex and dynamic risk related to the PPD-21-mandated all-hazards approach. This research showed that a systems-theoretic approach can better identify inter-dependencies, conflicts, gaps, and leverage points across traditional safety, security, and safeguards hazard mitigation strategies in the nuclear reactors, materials, and waste sector. Resulting from this, mitigation strategies from applying systems theoretic principles and complex systems engineering concepts can be (1) designed to better capture interdependencies, (2) implemented to better align with real-world operational uncertainties, and (3) evaluated as a systems-level whole to better identify, characterize, and manage PPD-21's all hazards strategies.
System theory - matrices, feedback control system, network synthesis, set theory, stability, shift registers, coding, theorem proving, polynomial roots, channels, and signal flow graphs
Dynamical systems which contain several identical subsystems occur in a variety of applications ranging from command and control systems and discretization of partial differential equations, to the stability augmentation of pairs of helicopters lifting a large mass. Linear models for such systems display certain obvious symmetries. In this paper, we discuss how these symmetries can be incorporated into a mathematical model that utilizes the modern theory of algebraic systems. Such systems are inherently related to the representation theory of algebras over fields. We will show that any control scheme which respects the dynamical structure either implicitly or explicitly uses the underlying algebra.
A system theory approach to thermospheric modeling is developed, based upon a linearization method which is capable of preserving nonlinear features of a dynamical system. The method is tested using a large, nonlinear, time-varying system, namely the thermospheric general circulation model (TGCM) of the National Center for Atmospheric Research. In the linearized version an equivalent system, defined for one of the desired TGCM output variables, is characterized by a set of response functions that is constructed from corresponding quasi-steady state and unit sample response functions. The linearized version of the system runs on a personal computer and produces an approximation of the desired TGCM output field height profile at a given geographic location.
Concept of state in system theory defined as sets of ordered pairs of time functions
The conceptual system and methodology of living systems theory appear to be of value to research on life in isolated environments. A space station, which must provide suitable conditions for human life in a stressful environment that meets none of the basic needs of life, is an extreme example of such isolation. A space station would include living systems at levels of individual human beings, groups of people engaged in a variety of activities, and the entire space crew as an organization. It could also carry living systems of other species, such as other animals and plants. Using the subsystem analysis of living systems theory, planners of a station, either in space or on a celestial body, would make sure that all the requirements for survival at all these levels had been considered. Attention would be given not only to the necessary matter and energy, but also the essential information flows that integrate and control living systems. Many variables for each subsystem could be monitored and kept in steady states. Use of living systems process analysis of the five flows of matter energy and information would assure that all members of the crew received what they needed.
Discussion of certain problems in system theory which have been or might be solved using some basic concepts from differential geometry. The problems considered involve differential equations, controllability, optimal control, qualitative behavior, stochastic processes, and bilinear systems. The main goal is to extend the essentials of linear theory to some nonlinear classes of problems.
Optimal control system theory applied to manual control systems - adaptive control - mathematical models
Dynamical systems theory has been used to study nonlinear aircraft dynamics. A six degree of freedom model that neglects gravity has been analyzed. The aerodynamic model, supplied by NASA, is for a generic swept wing fighter and includes nonlinearities as functions of the angle of attack. A continuation method was used to calculate the steady states of the aircraft, and bifurcations of these steady states, as functions of the control deflections. Bifurcations were used to predict jump phenomena and the onset of periodic motion for roll coupling instabilities and high angle of attack maneuvers. The predictions were verified with numerical simulations.
Linear dynamic systems governed by first-order linear differential equations in state variables, emphasizing time reversal and differential controllability
The problem of estimating parameters of dynamic systems is addressed in order to present the theoretical basis of system identification and parameter estimation in a manner that is complete and rigorous, yet understandable with minimal prerequisites. Maximum likelihood and related estimators are highlighted. The approach used requires familiarity with calculus, linear algebra, and probability, but does not require knowledge of stochastic processes or functional analysis. The treatment emphasizes unification of the various areas in estimation in dynamic systems is treated as a direct outgrowth of the static system theory. Topics covered include basic concepts and definitions; numerical optimization methods; probability; statistical estimators; estimation in static systems; stochastic processes; state estimation in dynamic systems; output error, filter error, and equation error methods of parameter estimation in dynamic systems, and the accuracy of the estimates.
Differential equations and their relationships to dynamic systems and control theory, with bibliographies
This paper explores some basic concepts of Biochemical Systems Theory (BST) and Metabolic Control Analysis (MCA), two frameworks developed to understand the behavior of biochemical networks. Initially introduced by Savageau, BST focuses on system stability and employs power laws in modeling biochemical systems. On the other hand, MCA, pioneered by authors such as Kacser and Burns and Heinrich and Rapoport, emphasizes linearization of the governing equations and describes relationships (known as theorems) between different measures. Despite apparent differences, both frameworks are shown to be equivalent in many respects. Through a simple example of a linear chain, the paper demonstrates how BST and MCA yield identical results when analyzing steady-state behavior and logarithmic gains within biochemical pathways. This comparative analysis highlights the interchangeability of concepts such as kinetic orders, elasticities and other logarithmic gains.
Dynamical systems theory has been used to study the nonlinear dynamics of the F-14. An eight degree of freedom model that does not include the control system present in operational F-14s has been analyzed. The aerodynamic model, supplied by NASA, includes nonlinearities as functions of the angles of attack and sideslip, the rotation rate, and the elevator deflection. A continuation method has been used to calculate the steady states of the F-14 as continuous functions of the control surface deflections. Bifurcations of these steady states have been used to predict the onset of wing rock, spiral divergence, and jump phenomena which cause the aircraft to enter a spin. A simple feedback control system was designed to eliminate the wing rock and spiral divergence instabilities. The predictions were verified with numerical simulations.
The state of stochastic system and control theory as it impacts restructurable control issues is addressed. The multivariable characteristics of the control problem are addressed. The failure detection/identification problem is discussed as a multi-hypothesis testing problem. Control strategy reconfiguration, static multivariable controls, static failure hypothesis testing, dynamic multivariable controls, fault-tolerant control theory, dynamic hypothesis testing, generalized likelihood ratio (GLR) methods, and adaptive control are discussed.
A methodology for modeling nonlinear unsteady aerodynamic responses, for subsequent use in aeroservoelastic analysis and design, using the Volterra-Wiener theory of nonlinear systems is presented. The methodology is extended to predict nonlinear unsteady aerodynamic responses of arbitrary frequency. The Volterra-Wiener theory uses multidimensional convolution integrals to predict the response of nonlinear systems to arbitrary inputs. The CAP-TSD (Computational Aeroelasticity Program - Transonic Small Disturbance) code is used to generate linear and nonlinear unit impulse responses that correspond to each of the integrals for a rectangular wing with a NACA 0012 section with pitch and plunge degrees of freedom. The computed kernels then are used to predict linear and nonlinear unsteady aerodynamic responses via convolution and compared to responses obtained using the CAP-TSD code directly. The results indicate that the approach can be used to predict linear unsteady aerodynamic responses exactly for any input amplitude or frequency at a significant cost savings. Convolution of the nonlinear terms results in nonlinear unsteady aerodynamic responses that compare reasonably well with those computed using the CAP-TSD code directly but at significant computational cost savings.