Stabilization Of Unstable Systems Research Articles

Analyzing Controllability of Bilinear Systems on Symmetric Groups: Mapping Lie Brackets to Permutations

Bilinear systems emerge in a wide variety of fields as natural models for dynamical systems ranging from robotics to quantum dots. Analyzing controllability of such systems is of fundamental and practical importance, for example, for the design of optimal control laws, stabilization of unstable systems, and minimal realization of input-output relations. Tools from Lie theory have been adopted to establish controllability conditions for bilinear systems, and the most notable development was the Lie algebra rank condition (LARC). However, the application of the LARC may be computationally expensive for high-dimensional systems. In this paper, we present an alternative and effective algebraic approach to investigate controllability of bilinear systems. The central idea is to map Lie bracket operations of the vector fields governing the system dynamics to permutation multiplications on a symmetric group, so that controllability and controllable submanifolds can be characterized by permutation cycles. The method is further applicable to characterize controllability of systems defined on undirected graphs, such as multi-agent systems with controlled couplings between agents and Markov chains with tunable transition rates between states, which in turn reveals a graph representation of controllability through the graph connectivity.

Stabilization problems for a system with output feedback (review)

Algorithms for solving the problem of design of static output feedback controllers for stationary linear systems with continuous and discrete time are reviewed. The inverse problem is considered. The algorithms of synthesis of output feedback controllers are generalized to the case of a periodic discrete-time system. To solve such problems, it might be more natural to use an approach based on multi-criterion optimization. It is also shown that these algorithms can be used for the optimal stabilization of unstable systems with delay. In this connection, the parameters of a controller with given structure for a controlled unstable scalar system with delay are optimized. To this end, the system is first approximated by a system without delay, with the exponent approximated by a fractionally rational function. Since the structure of the controller is given, the quality of approximation is estimated as the difference (in the space of controller coefficients) between the stability domains of the original and approximating systems. At the next stage, the gain coefficients of the controller for the reduced system are optimized. The efficiency of the thus synthesized controller is assessed through mathematical modeling of a system with delay whose feedback loop is defined by the gain coefficients found. The approach is illustrated by stabilizing an inverted simple pendulum with a proportional–derivative controller with delay. The problem of synthesis of a robust controller for this example is considered. Some examples of designing a robust controller, including for a third-order system in which the delay rather than some parameter is uncertain are presented

A new approach to the theory of the stability of linear canonical systems of differential equations with periodic coefficients

The main element of the proposed approach to constructing a theory of the stability of canonical systems is an index function (defined below), which contains all the necessary information on the system. The fundamental results of existing theory, in particular, the necessary and sufficient condition for strong stability, are expressed in new terms. The corresponding proofs only use simple mathematical means; moreover, they are much shorter than existing proofs. A number of new assertions are established, in particular, a simple sufficient condition for strong stability is obtained, which essentially generalizes the well-known Yakubovich theorem [1] of the directed width of the stability regions, and the necessary and sufficient condition for their directed convexity is obtained. Using them, certain non-local qualitative results on the regions of stability of parametric oscillations of canonical systems are established (which enable, in particular, the existing practice of constructing stability regions in accordance with their boundaries to be justified), and the conditions for high-frequency parametric stabilization of unstable systems are obtained.

Almost Sure Stability of Linear Stochastic Systems with Poisson Process Coefficients

We consider the problem of determining the sample path stability of a class of linear stochastic differential equations with point process coefficients. Necessary and sufficient conditions are obtained which are similar in spirit to those derived by Khas’minskii and Pinsky for diffusion processes. The conditions are based on the deep theorems of Furstenburg on the asymptotic behavior of products of random matrices. Estimates on the probabilities of large deviations for stable processes are also given, together with a result on the stabilization of unstable systems by feedback controls.