State Of Nonlinear System Research Articles

Dynamic Regret Minimization for Control of Non-stationary Linear Dynamical Systems

We consider the problem of controlling a Linear Quadratic Regulator (LQR) system over a finite horizon T with fixed and known cost matrices Q,R, but unknown and non-stationary dynamics A_t, B_t. The sequence of dynamics matrices can be arbitrary, but with a total variation, V_T, assumed to be o(T) and unknown to the controller. Under the assumption that a sequence of stabilizing, but potentially sub-optimal controllers is available for all t, we present an algorithm that achieves the optimal dynamic regret of O(V_T^2/5 T^3/5 ). With piecewise constant dynamics, our algorithm achieves the optimal regret of O(sqrtST ) where S is the number of switches. The crux of our algorithm is an adaptive non-stationarity detection strategy, which builds on an approach recently developed for contextual Multi-armed Bandit problems. We also argue that non-adaptive forgetting (e.g., restarting or using sliding window learning with a static window size) may not be regret optimal for the LQR problem, even when the window size is optimally tuned with the knowledge of $V_T$. The main technical challenge in the analysis of our algorithm is to prove that the ordinary least squares (OLS) estimator has a small bias when the parameter to be estimated is non-stationary. Our analysis also highlights that the key motif driving the regret is that the LQR problem is in spirit a bandit problem with linear feedback and locally quadratic cost. This motif is more universal than the LQR problem itself, and therefore we believe our results should find wider application.

Routh reducibility and controllability of unstable mechanical systems

Routh reduction presents the minimum number of differential equations that uniquely describe the state of nonlinear mechanical systems where the state variables can be separated into essential ones and cyclic ones. This work extends Routh reducibility for a relevant set of controlled mechanical systems. A chain of theorems is presented for identifying the conditions when reduced order rank conditions can be applied for determining the Kalman controllability of Routh reducible mechanical systems where actuation takes place along the cyclic coordinates only, while some of the essential coordinates and their derivatives are observed. Four mechanical examples represent the advantages of using reduced rank conditions to check and/or to exclude linear controllability in such systems.

Barrier-function-based distributed predictive control for operational safety of nonlinear processes

This article focuses on the design of distributed model predictive control (DMPC) systems for nonlinear processes with input constraints using a Control Lyapunov-Barrier Function (CLBF) to achieve simultaneous closed-loop stability and process safety. Specifically, we first use a constrained CLBF to design explicit control laws for each subsystem and to characterize a set of initial conditions, starting from which the closed-loop states of the overall nonlinear system are guaranteed to converge to the operating steady-state under the CLBF-based control laws while avoiding unsafe regions in state space. We then propose the CLBF-based DMPC, and prove its feasibility and effectiveness in ensuring the stability and avoidance of unsafe regions under sample-and-hold implementation of DMPC control actions. The CLBF-based DMPC is applied to both sequential and iterative DMPC designs in the general sense, and a modification to the DMPC formulation is presented for special cases of systems where the coupling between subsystems is in a one-way cascading manner. The proposed CLBF-DMPC method is demonstrated via a nonlinear chemical process example consisting of two subsystems.

A new strategy of smooth variable structure filter to estimate unmeasured states in target tracking

In the field of target tracking, the Kalman filter is one of the most popular filters, but its robustness will decrease when target maneuvers. The novel smooth variable structure filter (SVSF) has attracted much attention for its robustness against disturbances and uncertainties. In linear systems, the existing methods of SVSF can estimate unmeasured states, which refer to the states that cannot be measured by sensors, but the estimation is with low accuracy. And there is little research on the estimation for unmeasured states by SVSF in nonlinear systems. Therefore, this paper has put forward a novel strategy to improve the estimation accuracy of unmeasured states in both linear and nonlinear systems. The proposed strategy uses the correlation of error covariance among state elements to calculate the value of unmeasured states. On the basis of the strategy deduced from linear system, one algorithm in linear system and two new extended algorithms in nonlinear systems are proposed. In simulations, different filter models are used by all filters to process Gaussian and non-Gaussian noise in linear and nonlinear systems. The comprehensive results prove that, the proposed linear algorithm has better performance than the existing methods of SVSF in linear systems. Two nonlinear algorithms also show good performance in the estimation for unmeasured states in nonlinear and non-Gaussian systems.

Formation of unmanned aircraft trajectory when flying around prohibited areas

Considered the problem of flying over restricted areas by an unmanned aerial vehicle (UAV), which have various shapes and restrictions, set on the basis of the international airspace classification system for aviation in accordance with the Chicago Convention and the recommended principles for the formation of forbidden zones, rules for creating a flight route along forbidden zones and actions in case of border violations of restricted areas. The problem of analytical synthesis of the control acceleration of an unmanned aerial vehicle (UAV) is solved during its flight along a route passing along the boundaries of the forbidden zone of a given shape, along a given trajectory, which consists of subsequent segments located at the same height relative to the earth’s surface, in a given coordinate system. The optimal control synthesis problem is solved as an analytical definition of the optimal control of a linear non-stationary system based on the quadratic quality functional. A mathematical model of UAV motion in the horizontal plane is proposed, in the form of a system of ordinary differential equations in the Cauchy form. A law for measuring the control acceleration of the UAV’s center of mass is obtained on the basis of specifying the minimized quality functional and the corresponding constraints, which is a feature of the considered method of solving the problem. The proposed quality functional takes into account the parameters of coordinates and speed of the UAV, which correspond to the given points in the airspace, which characterize the necessary trajectory for flying around the restricted area. The derived mathematical dependences make it possible to implement them on board a UAV and minimize energy costs when guiding a UAV moving through specified points in space. Computer modeling of the derived analytical results, mathematical dependencies representing the optimal trajectory of the UAV flight along the boundaries of the forbidden zone, as well as the corresponding processes of changing the control acceleration and speed of the UAV movement was carried out, which made it possible to draw conclusions about the efficiency of the proposed method and the feasibility of its further use as a basis. for the initial stage of the synthesis of the UAV control system.

Study of the degree of controllability of a spacecraft during flight along a reference trajectory in the atmosphere of Mars

The process of landing a spacecraft on Mars or another planet is studied. During the descent of the aircraft in the atmosphere, a test mathematical model of guidance is chosen. The criterion of the controllability degree is applied to determine the degree of controllability when moving along various trajectories. Mathematical modeling of the spacecraft guidance process during descent in the Martian atmosphere according to tracking scheme of the program trajectory by using a control algorithm with active disturbance compensation and a proportionalintegralderivative (PID) adjuster is considered. Based on the results of the simulation, it is possible to evaluate and compare the optimality of various reference trajectories by using a criterion that is used as an indicator to study the influence of trajectory parameters on the landing process from the point of view of controllability. Keywords spacecraft, degree of controllability criterion, linear non-stationary system, reference trajectory guidance, control with active disturbance compensation

Investigation of the controllability degree criterion for the state variables of linear dynamical systems

A new approach to solving the problem of the controllability degree determining for the state variables of linear dynamical systems is presented. A numerical criterion for the state variables controllability degree of linear nonstationary systems is developed. The criterion, built on the principle of control with minimizing energy consumption, takes into account the normalization of the system original matrix and allows carrying out a comparative analysis of the controllability degree for dynamic systems with different parameters. A control example of an inverted pendulum on a trolley by using the proposed criterion is given and further research directions are outlined. Keywords criterion of the controllability degree, linear non-stationary system, state variables

Method for hierarchical study of spacecraft descent controllability in the Martian atmosphere

The problem of determining the degree of controllability of a linear non-stationary system (LNS) is considered and a scalar method of hierarchical controllability study for the integration of a highly dynamic guidance system for an autonomous landing of spacecraft is proposed. A new criterion of the controllability degree (DOC) for the problem of optimal control with minimization of energy consumption by calculating the singular values and the condition number of the Gramian controllability matrix is developed. The proposed DOC criterion allows to measure the degree of controllability of the system and each state variable. In addition, the normalization and modification method has been applied in calculating the degree of controllab ility. Mathematical modeling of the spacecraft guidance process during descent in the Martian atmosphere according to the tracing scheme of the program trajectory using the control algorithm with active disturbance compensation (ADRC) and PID is presented. Based on the results of the simulation, it is possible to evaluate and compare the optimality of various reference trajectories using the DOC criterion. The DOC criterion can function as an indicator to investigate the effect of trajectory parameters on the landing process from a controllability point of view. Keywords criterion of controllability degree, linear unsteady system, spacecraft, atmospheric descent

Moving Horizon Estimator Design for a Nonlinear Diffusion-Reaction System with Sensor Dynamics

Moving horizon estimation (MHE) is applied to estimate the spatial-temporal evolution of the state of nonlinear diffusion-reaction system with linear sensor dynamics. A late-lumping approach is introduced, which exploits directly the partial differential equation (PDE) and the ordinary differential equation (ODE) of the sensor dynamics to solve the optimal estimation problem. This leads to the determination of an adjoint PDE-ODE system, whose solution yields a gradient that is used for the implementation of a line search approach to achieve fast convergence to the minimum of the cost functional. The initial state of the MHE herein serves as decision variable and is determined on a receding horizon. Simulation results for in-domain measurement illustrate the performance of the approach.

A Convex Duality Approach for Assigning Probability Distributions to the State of Nonlinear Stochastic Systems

In order to optimally assign a desired probability distribution to the state of a nonlinear stochastic system, a convex duality approach is proposed to arrive at the associated optimality conditions. For a general class of stochastic systems governed by controlled Itô differential equations and subject to constraints on the probability distribution of the state at a fixed terminal time, a measure theoretic formulation is presented and it is shown that the original problem is embedded in a convex linear program on the space of Radon measures and that the embedding is tight, i.e., the optimal solution of both the original and the convex relaxation problems are equal. By exploiting the duality relationship between the space of continuous functions and that of measures, the associated optimality conditions are identified in the form of Hamilton-Jacobi problems where the optimization objective, in addition to the value function evaluation at the initial conditions, includes an extra term which is the integral of the product of the value function at the terminal time and the desired probability distribution. Numerical examples are provided to illustrate the results.

Event-Triggered Safe Control for the Zero-Sum Game of Nonlinear Safety-Critical Systems With Input Saturation

In this paper, a novel adaptive dynamic programming (ADP)-based event-triggered safe control method is proposed to solve the zero-sum game problem of nonlinear safety-critical systems with safety constraints and input saturation. First, the barrier function-based system transformation, the zero-sum game problem with safety constraints and input saturation is transformed into an equivalent input saturation zero-sum game problem, so as to guarantee that the system does not violate the safety constraints. Furthermore, the non-quadratic utility function is introduced into the performance function to solve input saturation. Then, a critic neural network (NN) is constructed to approximate the optimal safety value function. Subsequently, a novel event-triggered scheme is developed to determine the update instant of the control law and the disturbance law. Therefore, the proposed ADP-based event-triggered safe control method can ensure that the states of nonlinear safety-critical systems satisfy the safety constraints, while greatly reducing the amount of calculation and saving communication resources. In addition, during the learning process, the concurrent learning is used to relax the persistence of excitation (PE) condition. According to the Lyapuov theory, it is proved that the weight estimation error of the critic neural network and the states are uniformly ultimately bounded (UUB), and the Zeno behavior is excluded. Finally, a simulation example verifies the effectiveness of the proposed method.

On Joint Reconstruction of State and Input-Output Injection Attacks for Nonlinear Systems

We address the problem of robust state reconstruction for discrete-time nonlinear systems when the actuators and sensors are injected with (potentially unbounded) attack signals. Exploiting redundancy in sensors and actuators and using a bank of unknown input observers (UIOs), we propose an observer-based estimator capable of providing asymptotic estimates of the system state and attack signals under the condition that the numbers of sensors and actuators under attack are sufficiently small. Using the proposed estimator, we provide methods for isolating the compromised actuators and sensors. Numerical examples are provided to demonstrate the effectiveness of our methods.

A New Adaptive Event-Triggered Scheme Enabling Inter-Execution Time Pre-Evaluation and Double-Side Communication Reduction

This letter validates adaptive event-triggered stabilization confronting the coupling of estimate signal and system states for nonlinear systems with intractable large uncertainties. Notably, in the context, the evaluation on the changing rate of relative error is hampered while unavoidable for inter-execution time analysis. By taking advantage of the structure feature of controller function, that is not exploited in previous works, a convergence-preserving adaptive event-triggering mechanism is proposed in this letter. The relative/absolute thresholds are delicately online adjusted to maintain the effectiveness of the adaptive mechanism, while explicit pre-evaluation is provided for the minimum inter-execution time. The new event-triggered scheme enables double-side communication reduction, unlike those in related works only matching one-side event-triggered execution since continuous sensor-to-controller communication is required for certain event real-time directly evaluating the controller signal.

STATISTICAL CLOSURE MODELING FOR REDUCED-ORDER MODELS OF STATIONARY SYSTEMS BY THE ROMES METHOD

This work proposes a technique for constructing a statistical closure model for reduced-order models (ROMs) applied to stationary systems modeled as parametrized systems of algebraic equations. The proposed technique extends the reduced-order-model error surrogates (ROMES) method [Drohmann, M. and Carlberg, K., SIAM/ASA J. Uncertainty Quantif., 3(1):116-145, 2015] to closure modeling. The original ROMES method applied Gaussian-process regression to construct a statistical model that maps cheaply computable error indicators (e.g., residual norm, dual-weighted residuals) to a random variable for either (1) the norm of the state error or (2) the error in a scalar-valued quantity of interest. This work proposes instead a way to generate a statistical model for the state error itself, by constructing statistical models for the generalized coordinates characterizing both the in-plane error (i.e., the error in the trial subspace) and a low-dimensional approximation of the out-of-plane error. The former can be considered a statistical closure model, as it quantifies the error in the ROM generalized coordinates. Because any quantity of interest can be computed as a functional of the state, the proposed approach enables any quantity-of-interest error to be statistically quantified a posteriori, as the state-error model can be propagated through the associated quantity-of-interest functional. Numerical experiments performed on both linear and nonlinear stationary systems illustrate the ability of the technique (1) to improve (expected) ROM prediction accuracy by an order of magnitude, (2) to statistically quantify the error in arbitrary quantities of interest, and (3) to realize a more cost-effective methodology for reducing the error than a ROM-only approach in the case of nonlinear systems.

Identification of deformable droplets from boundary measurements: the case of non-stationary Stokes problem

In this paper, we use asymptotic expansion of the velocity field to reconstruct small deformable droplets (i.e. their forms and locations) immersed in an incompressible Newtonian fluid. Here the fluid motion is assumed to be governed by the non-stationary linear Stokes system. Taking advantage of the smallness of the droplets, our asymptotic formula and identification methods extend those already derived for rigid inhomogeneity and for stationary Stokes system. Our derivations, based on dynamical boundary measurements, are rigorous and proved by involving the notion of viscous moment tensor VMT. The viability of our reconstruction approach is documented by numerical results.

Optimal path tracking control for intelligent four-wheel steering vehicles based on MPC and state estimation

Four-wheel steering (4WS) vehicles have better stability control and path following performance than front-wheel steering (FWS) vehicles. Aiming at this characteristic, a new four-wheel active steering control strategy is proposed. For intelligent vehicle path tracking and nonlinear vehicle system state estimation, a path tracking control algorithm based on the MPC algorithm is designed to analyze the stability of the vehicle and set up constraints to achieve accurate tracking of the reference path. Automotive dynamic control systems require information on system variables. For example, the sideslip angle of electric vehicles cannot be measured directly. Based on UKF theory and the information input of low-cost sensors on the vehicle, an estimator is designed to estimate vehicle sideslip angle and yaw rate. According to the difference between the estimated value and the ideal value of the vehicle state, the LQR optimal controller is designed to realize the optimal control of the front and rear steering. And compared with the dynamic simulation results of front-wheel steering, proportional control four-wheel steering and yaw rate feedback four-wheel steering. The experimental results of the simulation platform show that the path tracking 4WS state feedback optimal control method has good lateral control stability and path tracking accuracy.

Asymptotic linear - nonlinear duality, indeterminism and mathematical intelligence

The formalism of asymptotic duality structure with applications to late time asymptotic states of linear and nonlinear systems is presented. A scale invariant linear differential system is shown to admit higher derivative discontinuous, nonsmooth solutions under the action of asymptotic duality transformations. The relevant asymptotic quantities are self dual and finitely differentiable. The geometrical meaning and applications to higher order nonlinear systems are presented on the space of differentiable functions. A novel duality relationship between linear and nonlinear systems is exposed. A linear system is shown to extend self similarly over nonlinear, classically inaccessible scales thereby extending classical smooth solution space to a nonsmooth one. A nonlinear system, on the other hand, is shown to proliferate into a system of self similar linear equations that could be solved exactly to have a very efficient high precision computation of periodic orbits. An extension of self dual asymptotics to critically self dual case in the space of continuous functions reveals a connection to prime number theorem and the associated correction term respecting the Riemann hypothesis.

Adaptive decentralized Kalman filters with non-common states for nonlinear systems

This paper presents two fault tolerance methods for decentralized Kalman filter with non-common states (DKF-NCS) for nonlinear systems. The DKF-NCS is used in a sensor network, where states are not uniform across all nodes. To detect and isolate faulty measurements, the χ2 test has been quite useful, where faulty measurements are detected based on the innovation error and innovation error covariance matrix. However, the innovation error and innovation error covariance matrix are dependent on the predicted state vector and its error covariance along with measurement model. The χ2 distribution test fails, if the predicted state vector is not consistent with its predicted error covariance matrix. Also, due to processing of set of measurements independently, fault detection happens more often in decentralized estimation compared to centralized estimation. Therefore, discarding the valid measurements based on χ2 detector may impact the performance of decentralized estimators significantly. To overcome this problem, we propose two adaptive fault tolerance methods. The first method handles faulty measurements at the assimilation step by applying weighted correction of the information and information matrix. The second method modifies the measurement noise matrix based on a closed-form solution, if fault is detected. These methods are validated using 100 simulation runs for a tracking problem. Overall, the proposed methods are demonstrated to be superior compared to the χ2 test and an existing adaptive extended Kalman filter.

Decentralized robust state estimation of multimachine power systems

The problem of online estimating the states of nonlinear (NL) Multimachine Power Systems (MPSs) is addressed within a constructive framework that combines notions and tools from electrical engineering and nonlinear estimation theory. First, the standard NL centralized model is realized as a set of linear decentralized robustly observable models, with augmented states that capture nonlinearity, parameter error, and intermachine state interaction. Then, based on the observability property of the decentralized model, a robustly convergent linear Geometric (Luenberger-like with integral action) estimator with simple tuning is constructed. With respect to previous MPS estimation techniques, the novelties are: (i) from a theoretical perspective, the comprehensiveness of the methodology, with model design, solvability in terms of observability, and robust functioning criteria coupled with a simple tuning scheme, and (ii) from an industrial applicability viewpoint, an online computation load considerably smaller than the one of the NL Extended Kalman Filter (EKF), and a tuning scheme appreciably simpler than the one of the NL sliding mode perturbation observer (SMPO). The proposed design methodology is illustrated through numerical simulation with a representative case example.

Synthesis of Programmed Motion Based on Special Optimal Control

A method is proposed for the synthesis of a closed-loop system with controls that ensure the movement of an object with minimal deviations from a given trajectory of the output coordinate and its higher derivatives and a transition to this set. To solve the problem, the Pontryagin maximum principle is used to study special situations without analysis of auxiliary variables, supplemented by the apparatus of general position conditions for nonlinear systems in an extended coordinate space, taking into account the object, a functional that is nonlinear regarding deviations of the output coordinate and the explicit occurrence of time. The combined use of these methods allows us, firstly, to find special trajectories of coordinates that are higher derivatives of the output coordinate, and after excluding time, a special phase trajectory is found, which is a switching line for reaching the final state, a given programmed motion along which in a closed system is carried out by special control. Secondly, access to a special phase trajectory from the initial state is carried out for linear objects by relay control, and for nonlinear objects, under certain boundary conditions, relay control is supplemented by a special control of the speed problem. Examples of control of programmed motion with oscillatory and aperiodic processes of a given duration for linear and nonlinear objects are given. Taking into account the nature of equilibrium states, determined by the methods of the qualitative theory of differential equations, and restrictions on control and coordinates, topologies of trajectories are obtained for the implementation of a continuous special control or sliding mode. New algorithms and structures of control systems are obtained. The results are accompanied by modeling, illustrating the effectiveness of algorithms and structures of control systems according to the proposed synthesis method and confirming analytical materials. The results of the work can be used to control linear and nonlinear objects in mechatronics, robotics, thermal processes and other industries.