Subset Of State Space Research Articles

Approximate bang-bang control assisted rapid switching feedback stabilization for stochastic qubit systems

For stochastic qubit systems under measurement feedback, an approximate bang-bang control assisted rapid switching control scheme with fewer switching times between the two subsets of state space is proposed in this paper. An eigenstate for single-qubit systems and any GHZ entangled state for multi-qubit systems are prepared successfully. In our control scheme, the state space is partitioned into two parts: a subset S1 containing the target state and its complementary set S2. On each subset, a Lyapunov function is selected and the corresponding control law is designed, where the control laws in S1 are of approximate bang-bang form and therefore can speed up the control process and the control laws in S2 ensure the strictly monotonic descent of the corresponding Lyapunov function and therefore can reduce the switching times. In particular, for multi-qubit systems with degenerate measured observables, we use two control channels to distinguish the target state from other isospectral eigenstates. By properly constructing the control Hamiltonians and using stochastic Lyapunov stability theory, we prove that the switching control laws in this paper render the system trajectory to switch at most twice on the boundary of S1 and S2 and to converge to the target state contained in S1 rapidly. Numerical simulations verify the effectiveness and rapidity of the proposed rapid control scheme.

Sliding sector control using new equivalent sector control

ABSTRACTThis paper presents a variable structure control design method for linear and nonlinear continuous-time systems with a special type of sliding sector using a new equivalent sector control. The PR-sliding sector is the subset of state space in which the derivative of the square P-norm of the state, , is less than or equal to with a positive semi-definite matrix where is observable. The new equivalent sector control provides the derivative of being negative. The design method of and is proposed for the general linear single input system by using the Riccati equation. Under the proposed equivalent sector control, the state trajectory may jump at the particular uncontrollable states when , and the control is set zero. If becomes positive, the state is outside of the sector, and then the state shall be transferred to the inside of the sector by the discontinuous nonlinear control input. Numerical simulations show that the proposed method gives satisfactory results.

Finite time sliding sector control for spacecraft atmospheric entry guidance

The paper presents a novel variable structure control with finite time sliding sector for spacecraft atmospheric entry guidance. The finite time convergence of tracking error in the presence of system uncertainty can be guaranteed. In contrast with the normally used notion of asymptotic stability in conventional sliding sector, a finite time sliding sector for the entry guidance is designed as a subset of state space in which the Lyapunov function candidate satisfies the finite time stability condition. Then, the finite time sliding sector controller is designed to guarantee the tracking error to converge to a small region of zero. The chattering existing on the boundary of the sliding sector is further reduced by introducing inner sector and outer sector which are the subsets of the proposed sector. Finally, numerical simulation results are given to demonstrate the effectiveness of the proposed method.

Finite time sliding sector guidance law with acceleration saturation constraint

A novel guidance law utilising variable structure control with finite time sliding sector is proposed. In contrast with the normally used notion of asymptotic stability in conventional sliding sector, a finite time sliding sector is defined as a subset of state space in which the Lyapunov function candidate satisfies the finite time stability condition. For the guidance system, the analytical solution of a state-dependent differential Riccati equation is achieved to determine the finite time sliding sector. Then, based on the finite time sliding sector, a sliding sector guidance law is designed by considering the target acceleration as a bounded uncertainty. Furthermore, the finite time sliding sector guidance law is even extended to the case where missile acceleration saturation constraint is considered. Simulation results show that the new guidance law is highly effective.

Guidance Law Design via Variable Structure Control with Finite Time Sliding Sector

A new guidance law utilizing variable structure control with finite time sliding sector is proposed. First, a finite time sliding sector is defined. The finite time sliding sector is a subset of state space in which the Lyapunov function candidate satisfies the finite time stability condition, in contrast to the commonly used notion of asymptotic stability in conventional sliding sector. Then, based on the finite time sliding sector, a sliding sector control law is designed to move the system state in to the sector in finite time. The target acceleration is considered as an uncertainty. The proposed sliding sector guidance law is derived by supposing the target acceleration upper bound can be estimated as a priori. Simulation results show that the new guidance law is highly effective.

Complete synchronization of chaotic atmospheric models by connecting only a subset of state space

Abstract. Connected chaotic systems can, under some circumstances, synchronize their states with an exchange of matter and energy between the systems. This is the case for toy models like the Lorenz 63, and more complex models. In this study we perform synchronization experiments with two connected quasi-geostrophic (QG) models of the atmosphere with 1449 degrees of freedom. The purpose is to determine whether connecting only a subset of the model state space can still lead to complete synchronization (CS). In addition, we evaluated whether empirical orthogonal functions (EOF) form efficient basis functions for synchronization in order to limit the number of connections. In this paper, we show that only the intermediate spectral wavenumbers (5–12) need to be connected in order to achieve CS. In addition, the minimum connection timescale needed for CS is 7.3 days. Both the connection subset and the connection timescale, or strength, are consistent with the time and spatial scales of the baroclinic instabilities in the model. This is in line with the fact that the baroclinic instabilities are the largest source of divergence between the two connected models. Using the Lorenz 63 model, we show that EOFs are nearly optimal basis functions for synchronization. The QG model results show that the minimum number of EOFs that need to be connected for CS is a factor of three smaller than when connecting the original state variables.

An Intuitionistic Model of Single Electron Interference

The double slit experiment for a massive scalar particle is described using intuitionistic logic with quantum real numbers as the numerical values of the particle's position and momentum. The model assigns physical reality to single quantum particles. Its truth values are given open subsets of state space interpreted as the ontological conditions of a particle. Each condition determines quantum real number values for all the particle's attributes. Questions, unanswerable in the standard theories, concerning the behaviour of single particles in the experiment are answered.

H∞ ALMOST DISTURBANCE DECOUPLING FOR MIMO BILINEAR SYSTEMS

The H∞ almost disturbance decoupling problem is considered. In this paper, a nonlinear design is proposed to find a state feedback controller for bilinear systems. The closed-loop system is internally stable and achieves disturbance attenuation in nonlinear H∞ sense. We defined a special form of Lyapunov function, which is constructed in terms of one or a set of positive definite constant matrices. If, except of the origin of system, the corresponding polynomial of the positive definite matrix (or several polynomials relevant to the positive definite constant matrices) has (have) no zero on a given subset of state space, then we can construct a controller to solve our problem. It is found that the controller structure could be complicated, but is feasible in computation and may require optimization technique to search the solution. We consider both SIMO and MIMO cases with illustrated examples.

ハイブリッドシステムの可到達性による電圧不安定性の解析

Analysis of voltage instability of power systems is of basic importance for prevention of cascading outages and large blackouts. We propose to use reachability analysis of hybrid automaton for voltage instability analysis of power systems. Hybrid automaton is a well-known mathematical model of hybrid systems. Reachability analysis is performed by computing forward or backward reach sets for unsafe sets of a hybrid automaton. The automaton can represent the interaction between continuous voltage oscillation and discrete control of power equipments. The unsafe sets are also subsets of state space in which system voltages show unacceptable levels such as low or high values. For single machine-load bus (SMLB) system with controlled route switching, we show that computation of backward reach sets of hybrid automaton makes it possible to determine voltage instability in the SMLB system. The obtained result implies that voltage instability leading to cascading outages can be detected before its occurrence.

Local characterization of invariant sets of an autonomous differential inclusion

An application in robotics motivates us to characterize the evolution of a subset in state space due to a compact neighborhood of an arbitrary dynamical system—an instance of a differential inclusion. Earlier results of Blagodat·skikh and Filippov (1986) and Butkovskii (1982) characterize the boundary of theattainable set and theforward projection operator of a state. Our first result is a local characterization of the boundary of the forward projection ofa compact regular subset of the state space.

On practical stability with the settling time via vector norms

This paper gives a workable method for practical stability study (with settling time) of nonlinear time-varying disturbed systems. Sufficient conditions are obtained by the use of vector norms, which allow estimation of initial, final, admissible subsets of state space, related to chosen settling time and final time. A numerical example is given to illustrate results.

Analysis of a counting process associated with a semi-Markov process: number of entries into a subset of state space

Let N(t) be a finite semi-Markov process on 𝒩 and let X(t) be the associated age process. Of interest is the counting process M(t) for transitions of the semi-Markov process from a subset G of 𝒩 to another subset B where 𝒩 = B ∪ G and B ∩ G = ∅. By studying the trivariate process Y(t) =[N(t), M(t), X(t)] in its state space, new transform results are derived. By taking M(t) as a marginal process of Y(t), the Laplace transform generating function of M(t) is then obtained. Furthermore, this result is recaptured in the context of first-passage times of the semi-Markov process, providing a simple probabilistic interpretation. The asymptotic behavior of the moments of M(t) as t → ∞ is also discussed. In particular, an asymptotic expansion for E[M(t)] and the limit for Var [M(t)]/t as t → ∞ are given explicitly.

Analysis of a counting process associated with a semi-Markov process: number of entries into a subset of state space

Let N(t) be a finite semi-Markov process on 𝒩 and let X(t) be the associated age process. Of interest is the counting process M(t) for transitions of the semi-Markov process from a subset G of 𝒩 to another subset B where 𝒩 = B ∪ G and B ∩ G = ∅. By studying the trivariate process Y(t) =[N(t), M(t), X(t)] in its state space, new transform results are derived. By taking M(t) as a marginal process of Y(t), the Laplace transform generating function of M(t) is then obtained. Furthermore, this result is recaptured in the context of first-passage times of the semi-Markov process, providing a simple probabilistic interpretation. The asymptotic behavior of the moments of M(t) as t → ∞ is also discussed. In particular, an asymptotic expansion for E[M(t)] and the limit for Var [M(t)]/t as t → ∞ are given explicitly.

Asymptotic independence and individual ratio limit theorems

Let X n be a discrete parameter Markov process on general state space with transition functions P n (x, A). Individual ratio limit theorems are results about convergence as n→∞ of the ratios $$\frac{{P^{n + k} (x,A)}}{{P^n (y,B)}}$$ (1) where k is a fixed integer, A and B are subsets of state space, and x and y are starting points. This paper investigates the relationship of individual ratio limit theorems to various forms of asymptotic independence. When the state space is denumerable it is shown that, subject to standard auxiliary conditions, the much studied Strong Ratio Limit Property is equivalent to a form of asymptotic independence related to mixing conditions of Rosenblatt and Blum, Hanson, and Koopmans.