Engineering topics

Siljak, D. D.

Publications and source records attributed to Siljak, D. D..

At least 37 records · Page 2

NASA NTRS1974

Exponential stability of large-scale discrete systems

The concept of vector Liapunov functions is used to obtain conditions for the exponential stability of large-scale discrete systems which can be decomposed into a number of interconnected subsystems with the same stability property. Both the structurally invariant composite systems and the large-scale systems under structural perturbations are considered. Connective absolute stability of a large-scale system composed of the interconnected Lur'e-type subsystems is defined and resolved in this context, resulting in a computationally and conceptually attractive alternative to a straightforward stability analysis of the system by frequency-domain criteria.

Grujic, L. T.↗

NASA NTRS1974

Decomposition-aggregation stability analysis of the spinning Skylab

Stability of an 11-th order linear model of the spinning Skylab is determined by the decomposition-aggregation method based upon the comparison principle and vector Liapunov functions. To reduce the inherent conservativeness of the method an optimization problem is formulated and resolved producing the optimum comparison system. The system provides the best estimate of the stability region of the important structural parameter - asymmetry in the boom settings.

Cuk, S. M.↗

NASA NTRS1974

Connective stability of nonlinear matrix systems

Consideration of stability under structural perturbations of free dynamic systems described by the differential equation dx/dt = A(t,x)x, where the matrix A(t,x) has time-varying nonlinear elements. The concept of 'connective stability' is introduced to study the structural properties of competitive-cooperative nonlinear matrix systems. It is shown that stability reliability in such systems is high and that they remain stable despite time-varying (including 'on-off') interaction among individual agents present in the system. The results obtained can be used to study stability aspects of mathematical models arising in as diverse fields as economics, biology, arms races, and transistor circuits.

Siljak, D. D.↗

NASA NTRS1973

Asymptotic stability and instability of large-scale systems

The purpose of this paper is to develop new methods for constructing vector Lyapunov functions and broaden the application of Lyapunov's theory to stability analysis of large-scale dynamic systems. The application, so far limited by the assumption that the large-scale systems are composed of exponentially stable subsystems, is extended via the general concept of comparison functions to systems which can be decomposed into asymptotically stable subsystems. Asymptotic stability of the composite system is tested by a simple algebraic criterion. By redefining interconnection functions among the subsystems according to interconnection matrices, the same mathematical machinery can be used to determine connective asymptotic stability of large-scale systems under arbitrary structural perturbations.

Grujic, L. T.↗

NASA NTRS1973

Decomposition-aggregation stability analysis

This report presents the development and description of the decomposition aggregation approach to stability investigations of high dimension mathematical models of dynamic systems. The high dimension vector differential equation describing a large dynamic system is decomposed into a number of lower dimension vector differential equations which represent interconnected subsystems. Then a method is described by which the stability properties of each subsystem are aggregated into a single vector Liapunov function, representing the aggregate system model, consisting of subsystem Liapunov functions as components. A linear vector differential inequality is then formed in terms of the vector Liapunov function. The matrix of the model, which reflects the stability properties of the subsystems and the nature of their interconnections, is analyzed to conclude over-all system stability characteristics. The technique is applied in detail to investigate the stability characteristics of a dynamic model of a hypothetical spinning Skylab.

Siljak, D. D.↗

NASA NTRS1973

On stability of discrete composite systems.

Conditions are developed under which exponential stability of a composite discrete system is implied by exponential stability of its subsystems and the nature of their interactions. Stability of the system is determined by testing positive definiteness property of a real symmetric matrix the dimension of which is equal to the number of subsystems.

Grujic, L. T.↗

NASA NTRS1973

Algebraic criteria for positive realness relative to the unit circle.

A definition is presented of the circle positive realness of real rational functions relative to the unit circle in the complex variable plane. The problem of testing this kind of positive reality is reduced to the algebraic problem of determining the distribution of zeros of a real polynomial with respect to and on the unit circle. Such reformulation of the problem avoids the search for explicit information about imaginary poles of rational functions. The stated algebraic problem is solved by applying the polynomial criteria of Marden (1966) and Jury (1964), and a completely recursive algorithm for circle positive realness is obtained.

Siljak, D. D.↗

NASA NTRS1973

On stability of large-scale systems under structural perturbations.

The concept of connective stability of large-scale systems is widened to include arbitrary structural perturbations. Conditions for this broader kind of connective stability are derived by developing 'connective' versions of both the linear and the nonlinear comparison principles.

Siljak, D. D.↗

NASA NTRS1973

On stability of discrete composite systems

Grujic, L. T.↗

NASA NTRS1973

Algebraic criteria for positive realness relative to the unit circle.

A purely algebraic algorithm is developed for testing positive real character of real rational functions and matrices relative to the unit circle in the complex plane. Since the algorithm is entirely recursive and is performed in finite number of steps, it is suitable for machine computations.

Siljak, D. D.↗

NASA NTRS1973

On stability of large-scale systems under structural perturbations.

The concept of connective stability of large-scale systems is widened to include arbitrary structural perturbations. Conditions for this broader kind of connective stability are derived by developing connective versions of both the linear and the nonlinear comparison principle.

Siljak, D. D.↗

NASA NTRS1972

Stability of large-scale systems under structural perturbations.

A large-scale system is considered as a system constituted of subsystems which may be connected or disconnected from each other during operation. A new concept of connective stability is introduced by which a large-scale system is regarded as stable if it remains stable (in the sense of Lyapunov) under structural perturbations produced by the on-off participation of the subsystems. Algebraic conditions are developed that guarantee exponential connective stability of large-scale systems which may be composed of linear and nonlinear time-varying subsystems coupled by linear or nonlinear connections.

Siljak, D. D.↗

NASA NTRS1972

Singular perturbation of absolute stability.

It was previously shown (author, 1969) that the regions of absolute stability in the parameter space can be determined when the parameters appear on the right-hand side of the system equations, i.e., the regular case. Here, the effect on absolute stability of a small parameter attached to higher derivatives in the equations (the singular case) is studied. The Lur'e-Postnikov class of nonlinear systems is considered.

Siljak, D. D.↗

NASA NTRS1972

Absolute stability analysis of attitude control systems for large boosters.

A method for performing absolute stability analyses of attitude control systems for large launch vehicles is presented. Absolute stability of these systems is shown in a finite region of the state space. The regions are computed by using the Lur'e-Postnikov Liapunov function. This function is chosen to provide additional information about the exponential property of absolute stability. Significant advantages of the method proposed in this paper are: it is independent of the order of the system; algebraic operations involved in the computations are relatively simple and convenient for machine implementation; and the obtained results are valid not only for a particular nonlinearity but also for an entire class of nonlinear characteristics that satisfy certain general conditions. A system model representing the Saturn V launch vehicle is used to illustrate the method.

Seltzer, S. M.↗

NASA NTRS1972

Stability of large-scale systems

A survey is presented of the results obtained in a stability study of large scale systems based upon the comparison principle and vector Liapunov function.

Siljak, D. D.↗

NASA NTRS1972

Stability of large-scale systems under structural perturbations

A concept of connective stability is introduced by which a large-scale system is regarded as stable if it remains stable (in the sense of Liapunov) under structural perturbations produced by the on-off participation of the subsystems. Algebraic conditions are developed that guarantee exponential connective stability of large scale systems which may be composed of linear and nonlinear time varying subsystems coupled by linear or nonlinear connections.

Siljak, D. D.↗

NASA NTRS1972

Singular perturbation of absolute stability.

The influence of a small parameter at the higher derivatives in the differential equations describing nonlinear systems of the Lur'e-Postnikov class on absolute stability in the parameter space is investigated. The conditions leading to singular perturbations of absolute stability are examined.

Siljak, D. D.↗

NASA NTRS1972

Stability of large-scale systems with stable and unstable subsystems.

The purpose of this paper is to develop new methods for constructing vector Liapunov functions and broaden the application of Liapunov's theory to stability analysis of large-scale dynamic systems. The application, so far limited by the assumption that the large-scale systems are composed of exponentially stable subsystems, is extended via the general concept of comparison functions to systems which can be decomposed into asymptotically stable subsystems. Asymptotic stability of the composite system is tested by a simple algebraic criterion. With minor technical adjustments, the same criterion can be used to determine connective asymptotic stability of large-scale systems subject to structural perturbations. By redefining the constraints imposed on the interconnections among the subsystems, the considered class of systems is broadened in an essential way to include composite systems with unstable subsystems. In this way, the theory is brought substantially closer to reality since stability of all subsystems is no longer a necessary assumption in establishing stability of the overall composite system.

Grujic, Lj. T.↗