Finite Number Of Switchings Research Articles

Secure State Estimation for Nonlinear Systems Under Sparse Attacks With Application to Robotic Manipulators

Secure state estimation against sparse injection attacks and disturbances is a challenging problem of both theoretical and practical importance, and existing results mainly focus on linear systems despite many practical systems being nonlinear. In this paper, a novel secure state estimation scheme is proposed for a class of nonlinear systems with application to robotic manipulators. A kind of high-gain K-filters is constructed to estimate the unmeasured states, which can attenuate the disturbances to an arbitrary level. Moreover, a monitoring function and a switching scheme are introduced, which successfully preclude attacked measurements after a finite number of switchings. With these efforts, the proposed estimation scheme steers the estimation error into a residual set which can be made arbitrarily small by properly choosing some design parameters, regardless of the disturbances and possibly unbounded attacks. Both simulation and experimental results on a robotic manipulator demonstrate the effectiveness of the proposed method.

Adaptive Control of Aircraft in Icing Using Regulation-Triggered Batch Least-Squares Identifier

Extensive effort in controller design of aircraft systems is invested ensuring safe, stable behavior in spite of large parametric uncertainties. Particular attention is placed on anomalous flight conditions harbored by the atmosphere, especially icing. This paper presents a regulation trigger-based adaptive controller to cope with the impact of ice on the aircraft equations of motion and to control the aircraft pitch to the commanded angle. For a pitch model of an aircraft system with the impact of icing, the design of a stabilizing certainty-equivalence controller using backstepping is first given, and it is succeeded by the introduction of a regulation-triggered batch least-squares identifier. A theorem is incorporated that guarantees that 1) pitch angle and pitch rate converge to the setpoint asymptotically, and 2) parameter estimates end after a finite number of switches. Finally, simulation results of an aircraft experiencing icing demonstrate the effectiveness of the identifier, with the trajectory of the iced system using the proposed identifier closely following that of the nominal system.

Observer design for Linear Hybrid Systems with unknown inputs and Petri-net discrete dynamics

This work investigates the observability analysis and the observer synthesis for Linear Hybrid Systems (LHS’s) considering both known and unknown inputs and constrained discrete dynamics. In this framework, none assumption is made over the unknown inputs (that can represent faults or disturbances), moreover, the discrete dynamics are represented by a Petri net, whose behaviour constraints the possible switchings. For this, the concept of eventual observability is considered as the possibility of uniquely determining both the discrete and the continuous states after a finite number of switchings. In this way, the information provided by the continuous and the discrete outputs of the LHS is combined to determine the discrete state after a finite number of switchings. Next, based on the knowledge of the visited discrete states, a continuous observer estimates the continuous state. Under this approach, neither the observability of the linear systems nor the observability of the underlying discrete event system are required for estimating the complete state of the LHS.

Switching-based Regulation of Uncertain Stable Linear Systems Affected by an Unknown Harmonic Disturbance

The problem of rejection of a sinusoidal disturbance with unknown frequency acting on an unknown single-input-single-output stable linear system is addressed in this paper. We present a new hybrid approach that does not require knowledge of the frequency response of the transfer function over the range of frequencies of interest. The proposed controller reposes upon a switching strategy within a family of linear controllers based on the adaptive feedforward / internal model control methodology. A dead-beat frequency estimation method is integrated in the controller. The method also accounts for the presence of bounded sensor noise as well as imprecise frequency estimation. It is shown that, within a finite number of switchings, the regulation error is ultimately bounded by a function of the norm of the noise that depends on the choice of the controller and the estimator gains.

Observability of Linear Hybrid Systems with unknown inputs and discrete dynamics modeled by Petri nets.

This work deals with the observability analysis for LHS’s considering both known and unknown inputs and constrained discrete dynamics, modeled by Petri nets. For this, the concept of eventual observability is recalled as the possibility of uniquely determining both the discrete and the continuous states after a finite number of switchings. In this way, the information provided by the continuous and the discrete outputs of the LHS can be combined to determine the discrete state after a finite number of switchings. Next, based on the knowledge of the visited locations, a continuous observer can estimate the continuous state. It is shown that under this approach the observability conditions are greatly relaxed with respect to other approaches in the literature, in particular, neither the observability of the linear systems nor the observability of the underlying discrete event system are required.

Markov inequality rule for switching among time optimal controllers in a multiple vehicle intercept problem

In this paper, a Markov inequality based switching rule is proposed to switch among numerically computed, time optimal controllers in a multiple vehicle intercept problem. Each controller is optimal for the intercept of a single vehicle, i.e., for the segment of the complete time varying multiple vehicle target set. The switching rule guarantees that after every switch the time to the target set is shorter with a certain predefined probability. Furthermore, the rule guarantees that the target set is reached after a finite number of switches and the rule scales well with the number of vehicles, i.e., the segments covering the target set. The problem and results are illustrated by a numerical example.

Discrete-time switching periodic adaptive control for time-varying parameters with unknown periodicity

ABSTRACTIn this paper, we address the problem of unknown periodicity for a class of discrete-time nonlinear parametric systems without assuming any growth conditions on the nonlinearities. The unknown periodicity hides in the parametric uncertainties, which is difficult to estimate with existing techniques. By incorporating a logic-based switching mechanism, we identify the period and bound of unknown parameter simultaneously. Lyapunov-based analysis is given to demonstrate that a finite number of switchings can guarantee the asymptotic tracking for the nonlinear parametric systems. The simulation result also shows the efficacy of the proposed switching periodic adaptive control approach.

Planar tilting maneuver of a spacecraft: Singular arcs in the minimum time problem and chattering

In this paper, we study the minimum time planar tilting maneuver of a spacecraft, from the theoretical as well as from the numerical point of view, with a particular focus on the chattering phenomenon. We prove that there exist optimal chattering arcs when a singular junction occurs. Our study is based on the Pontryagin Maximum Principle and on results by M.I. Zelikin and V.F. Borisov. We give sufficient conditions on the initial values under which the optimal solutions do not contain any singular arc, and are bang-bang with a finite number of switchings. Moreover, we implement sub-optimal strategies by replacing the chattering control with a fixed number of piecewise constant controls. Numerical simulations illustrate our results.

Optimal scheduling of multiple sensors in continuous time

This paper considers an optimal sensor scheduling problem in continuous time. In order to make the model more close to the practical problems, suppose that the following conditions are satisfied: only one sensor may be active at any one time; an admissible sensor schedule is a piecewise constant function with a finite number of switches; and each sensor either doesn’t operate or operates for a minimum non-negligible amount of time. However, the switching times are unknown, and the feasible region isn’t connected. Thus, it’s difficult to solve the problem by conventional optimization techniques. To overcome this difficulty, by combining a binary relaxation, a time-scaling transformation and an exact penalty function, an algorithm is developed for solving this problem. Numerical results show that the algorithm is effective.

OPTIMAL VACCINATION STRATEGIES FOR AN INFLUENZA EPIDEMIC MODEL

We present an optimal control model for influenza vaccination strategies in an open population. The model is based on an extended Kermack–McKendrick model with the vaccination rate being a measurable function. The objective of this optimal control model is to describe the vaccination strategies so that the total cost arising from vaccination and infections is minimized. We show that the optimal control is a non-singular bang-bang control which has a finite number of switchings. A scheme for the solution of the optimal control problem is formulated using the shooting method. We also carry out numerical simulations to illustrate the general results and to examine the effects of parameters on the optimal vaccination strategy. The simulations show that the ratio of the per capita treatment cost and per capita vaccination cost has a significant effect on the optimal strategy, while the vaccination rate of the newly recruited class turns out to have less effect.

Stability Switches in a Host–Pathogen Model as the Length of a Time Delay Increases

The destabilising effects of a time delay in mathematical models are well known. However, delays are not necessarily destabilising. In this paper, we explore an example of a biological system where a time delay can be both stabilising and destabilising. This example is a host–pathogen model, incorporating density-dependent prophylaxis (DDP). DDP describes when individual hosts invest more in immunity when population densities are high, due to the increased risk of infection in crowded conditions. In this system, as the delay length increases, there are a finite number of switches between stable and unstable behaviour. These stability switches are demonstrated and characterised using a combination of numerical methods and analysis.

The periodic solution of two-dimensional linear nonstationary systems and estimation of the attraction domain boundary in the problem of control of a wheeled robot

For the problem of control of the plane motion of a wheeled robot the estimate of the attraction domain is set up, which ensures the prescribed exponential speed of asymptotic stability and the absence of overcontrol. This statement of the problem is the generalization of the problems “on the attraction domain,” “on overcontrol,” and “on monotone damping,” which were stated by A.M. Letov in the work [1]. To define the attractive domain estimate, periodic trajectories are considered, which exist at the boundary of the stability domain of twodimensional linear nonstationary systems. Periodic solutions have a finite number of switchings over the period and describe the attraction domain boundary satisfying the preset geometric constraints.

Global exponential stability of switched systems

This paper proposes a method for the stability analysis of deterministic switched systems. Two motivational examples are introduced (nonholonomic system and constrained pendulum). The finite collection of models consists of nonlinear models, and a switching sequence is arbitrary. It is supposed that there is no jump in the state at switching instants, and there is no Zeno behavior, i.e., there is a finite number of switches on every bounded interval. For the analysis of deterministic switched systems, the multiple Lyapunov functions are used, and the global exponential stability is proved. The exponentially stable equilibrium of systems is relevant for practice because such systems are robust to perturbations.

Periodic Motions for Estimation of the Attraction Domain in the Wheeled Robot Stabilization Problem

AbstractExtremum properties of the two-dimensional linear time-varying (LTV) system are used in this paper to estimate boundary of the attraction domain in the problem of the wheeled robot control.The motion is supposed to be planar without a lateral slippage. The control goal is to drive the target point of the robot platform to the specified trajectory and to stabilize the motion along it. The trajectory consists of line segments and circular arcs. The current curvature of the trajectory of the target point is taken as control. The control must satisfy two-sided constraints. The attraction domain must be inscribed into certain region of the ‘distance to the trajectory – orientation’ phase space. Time-varying curvature of the target trajectory is considered as arbitrary varying function which takes values from the specified interval. The feedback linearization scheme is used for synthesis of the control law. The ‘saturation function’ is then used to take into account control constraints. The closed loop system takes form of the nonlinear system with parametric disturbances.The absolute stability approach is explored for stability analysis. Some nonlinearities take values from the interval. Other nonlinearities satisfy sector constraints. Along with the nonlinear time-varying system the uncertain linear time varying system is considered. Every solution of the nonlinear system is also solution of the time varying system for certain set of time-varying disturbances.To estimate the attraction domain of the nonlinear closed loop system, the Lyapunov function for the LTV system is constructed. A convex invariant function is known to exist at the boundary of the absolute stability region. For the second order case, the extremum trajectory, corresponding to the boundary of the absolute stability region, belongs to the level set of the invariant function. The periodic solution has finite number of switches on the period. It circumscribes the boundary of the attraction domain estimate. Two illustrative examples are considered.

Maximal Open-Loop Operation Under Integral Error Constraints

A linear time invariant system, subjected to bounded parametric uncertainties and time varying disturbances, is to be controlled in open loop for the maximum possible time period. The open loop operation is continued as long as the L 2 norm of the states is bounded by a pre-specified threshold, for every possible value of the uncertainty and disturbance. It is shown that the optimal open loop input that achieves the maximal open loop operation exists and that it can be approximated for computational purposes by a bang-bang input with a finite number of switches. Similarly, the worst case disturbance can also be computed using a bang-bang approximation.

Exponential stability of stochastic switched systems

This paper proposes a method for exponential m-stability analysis of stochastic switched systems. The models, in a finite set of models, are non-linear stochastic models. It is assumed that there is no jump in the state at switching instants and there is no Zeno behaviour, ie, there is finite number of switches on every bounded interval. The stochastic hybrid systems have wide applications: transmission control protocol flows with congestion avoidance and slow-start modes; estimation in distributed networked systems; air traffic control; process control and communication networks for control systems. For analysis of stochastic switched system the multiple Lyapunov functions are used and exponential m-stability is proved. From the main result of the paper: 1) the exponential m1 -stability of stochastic switched systems whereby m1 ∈ (0, m); 2) the stability in probability. The exponentially stable equilibrium of system is relevant for practice because such systems are robust to perturbations.

Variable Structure Model Reference Adaptive Control with Unknown High Frequency Gain Sign

A variable structure model reference adaptive control for plants with relative degree greater than one and unknown high frequency gain sign is proposed. A switching scheme is introduced based on a monitoring function designed for the first auxiliary error of the close loop system. It is shown that under the supervision of the monitoring function, the switching stops after at most a finite number of switchings and the tracking error converges to a residual set that can be made arbitrarily small by appropriately choosing some design parameters.

Optimal chattering modes in the problem of the control of a Timoshenko beam

The linear problem of the control in a plane of the motion of a Timoshenko beam, one end of which is clamped to a rotating disc is considered. The angular acceleration of the disc serves as the control. It is proved that, in the problem of the quenching of the first mode, the optimal control has an infinite number of switchings in a finite time interval (a chattering control). The construction of a suboptimal control with a finite number of switchings is described.

Optimal control for a Timoshenko beam

We consider a linear model of a rotating Timoshenko beam, which is clamped at one end to a disk the other being free. The motion of the beam is controlled by the angular acceleration of the disk. We study the minimization problem of mean square deviation of the Timoshenko beam from a given position. For the minimization problem of the first mode we prove that optimal control is the chattering control, i.e., it has an infinite number of switches in a finite time interval. We construct a suboptimal control with a finite number of switches. To cite this article: M.I. Zelikin, L.A. Manita, C. R. Mecanique 334 (2006).

On the Stability of a Certain Class of Linear Time-Varying Systems

This study firstly considers the exponential stability of unforced linear systems of slowly time-varying dynamics. Possible switchings of the system structure to unstable dynamics during certain finite time intervals are admitted. The maintenance of global exponential stability does not necessarily require at most a finite number of switchings in the dynamics while infinitely many switches can also lead to stability. The mechanism to achieve stability under infinitely many switches in the dynamics is to maintain the system in the stable region during time intervals of sufficient large length without switches provided that the system dynamics evolves at a sufficiently small rate with time. Special attention is paid to the robust tolerance for a class of state disturbances and to the case of time-varying matrix of dynamics that possess either piecewise constant or constant eigenvalues. The obtained results can be relevant for their use in stability issues for the cases of multimodel non- adaptive and adaptive control with improved transient performances.