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.↗
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Stability regions of large-scale systems.
Using an aggregated comparison system, methods are developed for estimating regions of asymptotic stability for large-scale systems composed of interconnected, exponentially stable subsystems; in general some of these subsystems are not globally stable. Three different forms of Liapunov functions are shown to be suitable for describing the stability behavior of the aggregated system, but one form in particular (heretofore not used in this context) is found to be the most natural. Using this function, some general results are obtained.
Weissenberger, S.↗
Stability of large-scale systems.
The purpose of this paper is to present the results obtained in stability study of large-scale systems based upon the comparison principle and vector Liapunov functions. The exposition is essentially self-contained, with emphasis on recent innovations which utilize explicit information about the system structure. This provides a natural foundation for the stability theory of dynamic systems under structural perturbations.
Siljak, D. D.↗
Stability regions of large-scale systems
Using an aggregated comparison system, methods are developed for estimating regions of asymptotic stability for large-scale systems composed of interconnected, exponentially stable subsystems; in general, some of these subsystems are not globally stable. Three different forms of Liapunov functions are shown to be suitable for describing the stability behavior of the aggregated system, but one form in particular, heretofore not used in this context, is found to be the most natural. Using this function, some general results are obtained.
Weissenberger, S.↗
Model-reference adaptive control system design technique
This paper considers the model-reference adaptive control problem which has received considerable attention in the literature in the last few years. An adaptive control scheme is proposed which has terms in the Liapunov function used in the design procedure which are not included in previously proposed schemes. The relationship of this new scheme to existing schemes is shown by considering the root-loci of the linearized error equations between plant and model. Finally, a second order example is given which illustrates the difference between the two previously proposed model-reference adaptive methods and the one proposed in this paper.
Sutherlin, D. W.↗
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.↗
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.↗
Application of the comparison principle to analysis of nonlinear systems
A comparison principle based on a Kamke theorem and Lipschitz conditions is presented along with its possible applications and modifications. It is shown that the comparison lemma can be used in the study of such areas as classical stability theory, higher order trajectory derivatives, Liapunov functions, boundary value problems, approximate dynamic systems, linear and nonlinear systems, and bifurcation analysis.
Gunderson, R. W.↗
Large-scale systems: Complexity, stability, reliability
After showing that a complex dynamic system with a competitive structure has highly reliable stability, a class of noncompetitive dynamic systems for which competitive models can be constructed is defined. It is shown that such a construction is possible in the context of the hierarchic stability analysis. The scheme is based on the comparison principle and vector Liapunov functions.
Siljak, D. D.↗
Comparison theorems, numerical integration, and satellite orbits
A converse theorem using Liapunov functions is used to obtain upper error bounds in optimal estimates of perturbed two body problems.
Stokes, A.↗
Large-scale systems - Stability, complexity, reliability
This paper establishes the connective stability concept in the framework of the comparison principle and vector Liapunov functions, as a natural setting for resolving the complexity vs reliability problem in the control of large-scale dynamic systems. The central result is the following: a stable complex system when composed as a competitive structure of the interconnected stable subsystems remains stable despite the on-off participation of the subsystems. This result is important in that it can be used efficiently to synthesize reliable complex systems by multilevel feedback control.
Siljak, D. D.↗
Global analysis of a buck regulator
Sufficient conditions for global stability of a buck regulator using a discrete control law are found. The method of paired systems and Liapunov functions are used to establish global stability and to study the convergence of the regulator. A heuristic argument is given that the optimal switching curves associated with the paired continuous systems approximate the optimal switching curves of the discrete systems.
Edwards, D. B.↗
Stability of sequences generated by nonlinear differential systems
A local stability analysis is presented for both the analytic and numerical solutions of the initial value problem for a system of ordinary differential equations. It is shown that, using a proper choice of Liapunov function, a connected region of stable initial values of both the analytic solution and the one-leg k-step numerical solution can be approximated. Attention is given to the example of the two-dimensional problem involving the stability of the longitudinal equations of motion of a gliding jet aircraft.
Brown, R. L.↗
Theory of design using nonlinear transformations
This paper is presenting an overview of the theory of transformations from nonlinear systems to linear systems. Topics covered include (1) necessary and sufficient conditions for transformations to exist, (2) a method of constructing transformations (3) robustness in design (based on transformations theory) and Liapunov functions, (4) estimation theory, and (5) the relationship between transformation theory and 'nonlinear zeros'. Application of these results to automatic flight control is presented in another paper at this session.
Su, R.↗
Gyro motion boundedness under uncertain vehicle spin and acceleration
Using a quadratic Liapunov function, a condition is obtained for the boundedness of the motion of the gyro mounted in a vehicle which has a time-varying uncertain angular acceleration and deceleration omega-x(t) about the output axis, and is spinning with uncertain angular velocity omega-z(t) about the spin axis of the gyro. A region of ultimate boundedness in the theta-theta plane is obtained which the motion of the gyro eventually enters and in which it remains after a finite interval of time for any uncertain omega-x(t) and omega-z(t). The gyro motion is shown to be bounded if the uncertainty in the spin velocity does not exceed a certain threshold value. This condition disappears if omega-z(t) = 0.
Singh, S. N.↗
Reduced conservatism in stability robustness bounds by state transformation
This note addresses the issue of 'conservatism' in the time domain stability robustness bounds obtained by the Liapunov approach. A state transformation is employed to improve the upper bounds on the linear time-varying perturbation of an asymptotically stable linear time-invariant system for robust stability. This improvement is due to the variance of the conservatism of the Liapunov approach with respect to the basis of the vector space in which the Liapunov function is constructed. Improved bounds are obtained, using a transformation, on elemental and vector norms of perturbations (i.e., structured perturbations) as well as on a matrix norm of perturbations (i.e., unstructured perturbations). For the case of a diagonal transformation, an algorithm is proposed to find the 'optimal' transformation. Several examples are presented to illustrate the proposed analysis.
Yedavalli, R. K.↗
Attitude control of an object commonly held by multiple robot arms - A Lyapunov approach
Multiple robot arms moving a commonly held object can be viewed as complex actuators whose purpose is to provide net forces and moments to the object. These forces and moments can be used to control the orientation, or attitude, of the object via the Euler equation describing attitude evolution in response to applied moments at the mass center. In contrast to the common approach that feedback-linearizes the attitude dynamics to a double integrator form with respect to some three-parameter local representation of orientation, the authors control the object using a globally nonsingular representation. Using an energy-motivated Liapunov function, globally stable control of attitude is shown.
Kreutz, Kenneth↗
Lyapunov-based control designs for flexible-link manipulators
A feedback controller for the stabilization of closed-loop systems is proposed which is based on the Liapunov stability criterion. A feedback control law is first generated for the linear portion of the system equation using linear control theory. A feedback control is then designed for the nonlinear portion of the system equation by making negative the time derivative of a positive definite Liapunov function.
Juang, Jer-Nan↗