Minimal Invariant Set Research Articles

Receding-Horizon Control of LTI Systems with Quantized Inputs

This paper deals with the stabilization problem for a particular class of hybrid systems, namely discrete-time linear systems subject to a uniform (a priori fixed) quantization of the control set. Results of our previous work on the subject provided a description of minimal (in a specific sense) invariant sets that could be rendered maximally attractive under any quantized feedback strategy. In this paper, we consider the design of stabilizing laws that optimize a given cost index on the state and input evolution on a finite, receding horizon. Application of Model Predictive Control techniques for the solution of similar hybrid control problems through Mixed Logical Dynamical reformulations can provide a stabilizing control law, provided that the feasibility hypotheses are met. In this paper, we discuss precisely what are the shortest horizon length and the minimal invariant terminal set for which it can be guaranteed a stabilizing MPC scheme. The simulation of the application of the control scheme to a practical quantized control problem is finally reported.

More on minimal invariant sets for nonexpansive mappings

Minimal invariant sets for nonexpansive mappings share some singular geometrical properties. Here we present some seemingly unknown ones.

Systems with persistent disturbances: predictive control with restricted constraints

This paper addresses predictive state regulation of linear discrete-time systems subject to persistent bounded disturbances and to state and/or control constraints. It is well known that the joint presence of constraints and disturbances can drive a predictive controller to infeasibility and instability. Here it is shown how robustness against persistent bounded disturbances can be enforced by inserting in the predictive controller suitable constraint restrictions. The robust predictive controller obtained in this way guarantees, for all admissible disturbances, constraint fulfillment and asymptotic state regulation, i.e. convergence of the state to a minimal robust invariant set, provided that the initial state is feasible. Simulation results demonstrate the effectiveness of the proposed approach.

Minimal Invariant Sets of Discrete Dynamic Systems with a Stable Linear Part

Minimal Invariant Sets of Discrete Dynamic Systems with a Stable Linear Part

Analysis and synthesis of optimal and robustly optimal control systems under bounded disturbances

Abstract Analysis of dynamic (closed-loop control) systems under bounded disturbances by calculating minimal invariant acta is discussed. It is suggested to use the radius of A minimal invariant set as a particular numerical measure of the stability degree of a system and apply such a criterion for optimal linear control synthesis by means of minimizing this radius. It has been proven that the radius of a minimal invariant set reaches its minimum at a nilpotent parameter matrix of a system. The solution to the optimal control synthesis problem can be found, therefore, by calculating the linear feedback parameters providing the nilpotent matrix of the closed-loop system. In the case of scalar controls, the problem has got an analytical solution

Properties of minimal invariant sets for nonexpansive mappings

Properties of minimal invariant sets for nonexpansive mappings

Analysis of Some Classes of Nonlinear Discrete Systems Under Bounded Disturbances

The constructive solution of the calculating minimal invariant sets problem for some classes of autonomous and nonautonomous discrete systems has been obtained. For the nonautonomous discrete systems the problem is solved on the assumption that only set-valued estimates for additive bounded disturbances are available. The class of systems with bounded nonlinear parts is considered. This class contains systems with saturation, relay systems having a dead zone and others. An example illustrates the proposed method.

Minimal invariant sets of dynamic systems with bounded disturbances

Minimal invariant sets of dynamic systems with bounded disturbances

Limit periodic functions, adding machines, and solenoids

We prove that a stable adding machine invariant set for a homeomorphism of the plane is the limit of periodic points and also that a stable solenoid minimal invariant set for a three dimensional flow is the limit of periodic orbits. We give an example to show that a similar result is false in higher dimensions.

Operator Semigroups, Flows, and Their Invariant Sets

Let B( H) be the bounded operators on a Hubert space H. An operator semi-group Σ is an absolutely convex, unital subsemigroup of the ball of B( H). Such a Σ induces a flow on H 1, the unit ball of H, namely ( s, x) → sx. Let l(Σ) be the lattice of closed invariant sets of this flow. Both the weak and norm topologies on H 1 induce Hausdorff metrics on the hyperspace of closed convex subsets of H 1. We show that various subsets of I(Σ) are closed subspaces of these hyperspaces. There exist nontrivial operator semigroups which have the same invariant sets as the ball of B( H); i.e., the "transitive semigroup problem" has a negative solution. However, we can characterize the ball of B( H) in terms of certain transitivity properties. We obtain an analogue of Lomonosov′s Theorem for operator semigroups which are transitive on the unit sphere of H. Let A be a weakly closed unital subalgebra of B( H), and let A 1 be its unit ball. We show that the existence of "minimal invariant sets" for A yields hyperinvariant subspaces. We characterize various operator algebras for which A 1 is strongly precompact including von Neumann algebras, triangular algebras, CSL algebras, and the standard function algebras acting by multiplication on a L 2 space. This yields structural results for certain algebras generated by a subnormal operator.

Controllability and observability indices for linear time-invariant systems: A new approach

This paper is devoted to an investigation of the controllability and observability indices for linear time-invariant discrete-time systems. These indices are here introduced at an intrinsic level in the context of the behavioral approach to systems theory. For the class of the complete controllable systems it is proven that the controllability indices can be interpreted as a minimal and complete set of invariants with respect to a very natural systems equivalence. Autonomous phenomena are also discussed.

$\mathrm{SO}(2)$ invariants of a set of $2 \times 2$ matrices.

We give an alternative proofof a result due to Sibirskii on the polynomial invariants ofSO(2, C) (or SO (2, R)) acting on M(2, C) (or M(2, R)) by conjugation. We show that the invariants are given in terms of traces and Pfaffians, and we find a minimal basis which is a minimal complete set of invariants in the real case.

Invariant sets of arcs in network flow problems

This paper considers sets of arcs A = A + ∪ A − in a network which have the property that for every max flow, the sum of the flows on the arcs A + minus the sum of the flows on the arcs A − is equal to a fixed constant. We show that minimal invariant sets always arise from cuts (not necessarily min cuts) in a natural way. These results are applied to matrices with given row and column sums to generalize a result of Brualdi and Ross.