Second-order Linear Systems Research Articles

Eigenstructure assignment of constrained second-order linear systems: an optimised positively-invariant set control approach

Second-order systems constitute a mathematical model class used to represent a variety of physical processes to be controlled, such as vehicle damping systems, hydraulic pump flow, and robot manipulators. Due to the practical limitation inherent to physical systems (or their mathematical representation) and constraints on the input variables, the constrained control theory became an important research field. In this work, based on set-invariance theory, we propose an eigenstructure assignment optimisation using a state-feedback control law that guarantees the respect of state and control constraints for second-order systems. Positive invariance is the property of keeping the trajectory of the system states inside the set. This characteristic is incorporated into the analysis and design of the control law to ensure local stability and constraint fulfilment.

Context-dependence of deterministic and nondeterministic contributions to closed-loop steering control.

In natural circumstances, sensory systems operate in a closed loop with motor output, whereby actions shape subsequent sensory experiences. A prime example of this is the sensorimotor processing required to align one's direction of travel, or heading, with one's goal, a behavior we refer to as steering. In steering, motor outputs work to eliminate errors between the direction of heading and the goal, modifying subsequent errors in the process. The closed-loop nature of the behavior makes it challenging to determine how deterministic and nondeterministic processes contribute to behavior. We overcome this by applying a nonparametric, linear kernel-based analysis to behavioral data of monkeys steering through a virtual environment in two experimental contexts. In a given context, the results were consistent with previous work that described the transformation as a second-order linear system. Classically, the parameters of such second-order models are associated with physical properties of the limb such as viscosity and stiffness that are commonly assumed to be approximately constant. By contrast, we found that the fit kernels differed strongly across tasks in these and other parameters, suggesting context-dependent changes in neural and biomechanical processes. We additionally fit residuals to a simple noise model and found that the form of the noise was highly conserved across both contexts and animals. Strikingly, the fitted noise also closely matched that found previously in a human steering task. Altogether, this work presents a kernel-based analysis that characterizes the context-dependence of deterministic and non-deterministic components of a closed-loop sensorimotor task.

Adaptive output-feedback trajectory tracking for a class of uncertain second-order linear systems

Abstract In this work, we present a novel design of an explicit control strategy to solve the trajectory tracking problem for an uncertain second-order linear system with only partial availability of the state. The system parameters are unknown, and an additive constant disturbance is considered. Despite the apparent simplicity of this system, the solution requires an intricate combination of available methodologies. The proposed approach corresponds to a Lyapunov-based adaptive control strategy consisting of a reduced-order observer, a dynamic slave system and an adaptive controller with dynamic extension. Simulation results are provided to highlight the effectiveness of the proposed adaptive control strategy in accomplishing the regulation and trajectory tracking problems.

New computational methods using seventh-derivative type for the solution of first-order initial value problems

This study uses finite power series as the basis function and interpolation and collocation techniques to study a class of implicit block methods of a seventh-derivative type. Discrete schemes are implicit two-point block methods that are obtained by selecting collocation points carefully and unevenly in order to improve the stability of the methods through testing. Nevertheless, in contrast to other current numerical equations, these methods require seventh-derivative functions. The novel techniques are identified, examined, and shown to be A-stable and convergent. Newton Raphson’s approach is used to accomplish method implementation. Trials demonstrated the effectiveness and precision of the derived equations in terms of computational time and absolute errors on a variety of first-order initial value issues, such as first-order, second-order linear differential systems and SIR model. When compared to similar methods that are currently in the literature, the suggested methods produce better numerical results.

Event-triggered Synchronization for Second-order Linear Systems in Complex Dynamical Network With Time-varying Topology

Event-triggered Synchronization for Second-order Linear Systems in Complex Dynamical Network With Time-varying Topology

Finite-time consensus control of second-order multi-agent systems with input saturation constraint

A novel finite-time consensus technique is derived for second-order linear multi-agent systems with input saturation constraints. The hyperbolic tangent–based saturation function is presented for achieving input saturation while reducing the chattering issue of classical saturation functions. The leader-following consensus controller with a single saturation function is constructed to address the issue of saturation level partition between position feedback and velocity feedback. Theoretically, the property of finite-time consensus, bounded input, and smoothness of a closed-form system are proved by the Lyapunov theory in detail. Numerical simulations are conducted to exhibit the effectiveness of the proposed method fully.

Common Quadratic Lyapunov Functions for Sets of Second-Order Linear Systems: A Simple Graphical Criterion

Common Quadratic Lyapunov Functions for Sets of Second-Order Linear Systems: A Simple Graphical Criterion

Model reduction for second-order systems with inhomogeneous initial conditions

In this paper, we consider the problem of finding surrogate models for large-scale second-order linear time-invariant systems with inhomogeneous initial conditions. For this class of systems, the superposition principle allows us to decompose the system behavior into three independent components. The first behavior corresponds to the transfer between the input and output having zero initial conditions. In contrast, the other two correspond to the transfer between the initial position or the initial velocity and the output when no input is applied. Based on this superposition of systems, our goal is to propose model reduction schemes that allow to preserve the second-order structure in the surrogate models. To this aim, we introduce tailored second-order Gramians for each system component and compute them numerically, solving Lyapunov equations. As a consequence, two methodologies are proposed. The first one consists in reducing each of the components independently using a suitable balanced truncation procedure. The sum of these reduced systems provides an approximation of the original system. This methodology allows flexibility on the order of the reduced-order model. The second proposed methodology consists in extracting the dominant subspaces from the sum of Gramians to build the projection matrices leading to a surrogate model. Additionally, we discuss error bounds for the overall output approximation. Finally, the proposed methods are illustrated by means of benchmark problems.

Stochastic Galerkin method and port-Hamiltonian form for linear dynamical systems of second order

We investigate linear dynamical systems of second order. Uncertainty quantification is applied, where physical parameters are substituted by random variables. A stochastic Galerkin method yields a linear dynamical system of second order with high dimensionality. A structure-preserving model order reduction (MOR) produces a small linear dynamical system of second order again. We arrange an associated port-Hamiltonian (pH) formulation of first order for the second-order systems. Each pH system implies a Hamiltonian function describing an internal energy. We examine the properties of the Hamiltonian function for the stochastic Galerkin systems. We show numerical results using a test example, where both the stochastic Galerkin method and structure-preserving MOR are applied.

Robust String Stability and Safety of CTH Predictor-Feedback CACC

We establish robustness of string stability to delay uncertainty as well as positivity of spacing and speed states, for homogeneous vehicular platoons under predictor-feedback Cooperative Adaptive Cruise Control (CACC). Each individual vehicle’s dynamics are described by a second-order linear system with delayed desired acceleration, under acceleration information transmitted to the ego vehicle from a single, preceding vehicle. The nominal design (in the delay-free case) is a constant time-headway (CTH) policy and no restriction on the delay size, in relation with the desired time headway, is imposed. The proofs rely on combination of an input-output approach (on the frequency domain) and on deriving estimates on explicit, closed-loop solutions; under specific, sufficient conditions that are derived on initial conditions and parameters of the baseline, CTH controller. We illustrate in simulation and numerical examples, the guarantees of robust stability and string stability as well as of collisions avoidance, of CTH predictor-feedback CACC design. We also present extensions of our design and analysis approach to heterogeneous, third-order dynamic models of vehicles.

Fundamental response in the vibration control of buildings subject to seismic excitation with ATMD

The linear quadratic regulator for vibration systems subject to seismic excitations is discussed in his own physical newtonian space as a second-order linear differential system with matrix coefficients. The linear quadratic regulator leads to a fourth-order system and second-order transversality conditions. Those systems are studied with a matrix basis generated by a fundamental matrix solution.

Explicit commutativity conditions for second-order linear time-varying systems with non-zero initial conditions

Explicit commutativity conditions for second-order linear time-varying systems with non-zero initial conditions

Synthesis of quasi-optimal fast filters by the least square criterion

The subject of the article research is special signal processing methods based on the optimal discrete filtering theory. The goal is to increase the efficiency of model-based methods for processing information signals by reducing computational costs and increasing the speed of optimal discrete filtering algorithms. Applied methods: description of dynamic processes in terms of state space using elements of vector-matrix algebra, weighted least squares method, elements of Kalman's theory of optimal discrete filtering, basic concepts of the O'Reilly–Luenberger theory of functional observers, elements of probability theory, statistical modeling by the Monte Carlo method. Results: a new method for reducing computational costs is proposed, which uses the approximation of the Kalman filter transfer matrix time dependence by given piecewise linear functions according to the least squares criterion. The effectiveness of the method was evaluated on the example of a second-order dynamical system. On the basis of a comparative analysis, several acceptable variants of the considered approximation are proposed. The practical significance of the work lies in the further development of methods for the synthesis of quasi-optimal high-speed filters. The operability of the proposed modifications is confirmed by the example of a second-order linear dynamic system. The efficiency of the algorithms was evaluated by the statistical modeling method according to the criterion "accuracy-computational costs". It is shown that the total savings in the number of multiplication and addition operations can reach tens of times due to insignificant losses in the accuracy of the filtering process.

Parametric design method for partial eigenstructure assignment of second-order linear systems via observer-based state feedback

In this article, a parametric method of observer-based partial eigenstrunt in second-order linear time-invariant (LTI) systems is explored. First, an observer is constructed to approximate the full sate vector when the entire state vector is not available for measurement. Meanwhile, the separation principle of the first-order linear systems is extended to second-order linear systems. Second, by partitioning the system matrix of the open-loop system into two parts, the satisfactory and unsatisfactory eigenstructures are selected. On the premise of retaining the satisfactory part in the open-loop system, the unsatisfactory eigenstructure assignment problem can be converted into the solutions of a type of generalized Sylvester equations (GSE). Third, by introducing a group of arbitrary parameters, complete parametric expressions of the proportional plus derivative (PD) state feedback controller can be obtained. The main advantage of the proposed method is that it provides all the design degrees of freedom and reduces the design complexity of the controller. Finally, the correctness of the proposed method is verified by two examples and simulation results. Using the degrees of freedom of the parameters, the gain matrices are optimized through the optimization index provided by the optimization function, and better performance is obtained.

Analytical expression of motion profiles with elliptic jerk

AbstractThe paper discusses the analytical expressions of a motion profile characterized by elliptic jerk. This motion profile is obtained through a kinematic approach, defining the jerk profile and then obtaining acceleration, velocity, and displacement laws by successive integrations. A dimensionless formulation is adopted for the sake of generality. The main characteristics of the profile are analyzed, outlining the relationships between the profile parameters. A kinematic comparison with other motion laws is carried out: trapezoidal velocity, trapezoidal acceleration, cycloidal, sinusoidal jerk, and modified sinusoidal jerk. Then, the features of these motion profiles are evaluated in a dynamic case study, assessing the vibrations induced to a second-order linear system with different levels of damping. The results show that the proposed motion law provides a good compromise between different performance indexes (settling time, maximum absolute values of velocity and acceleration).

Relative Controllability and Ulam–Hyers Stability of the Second-Order Linear Time-Delay Systems

We introduce the delayed sine/cosine-type matrix function and use the Laplace transform method to obtain a closed form solution to IVP for a second-order time-delayed linear system with noncommutative matrices A and Ω. We also introduce a delay Gramian matrix and examine a relative controllability linear/semi-linear time delay system. We have obtained the necessary and sufficient condition for the relative controllability of the linear time-delayed second-order system. In addition, we have obtained sufficient conditions for the relative controllability of the semi-linear second-order time-delay system. Finally, we investigate the Ulam–Hyers stability of a second-order semi-linear time-delayed system.

Controllability and observability of discretized satellite magnetic attitude control system

<abstract><p>In this paper, two different discrete schemes of the second-order linear time-varying system represented by the linearized satellite magnetic attitude control motion equation are obtained by Euler method. Then, the controllability and observability conditions of a new discrete second-order linear time-varying system are proposed and the validity of these conditions is further verified by some numerical examples. Next, the theoretical results are applied to investigate the controllability and observability of the discretized satellite magnetic control system. Different periods $ \tau $ are chosen to investigate the effect on the controllability and observability of the resulting discrete system. The corresponding conclusions are obtained.</p></abstract>

Efficient Near-Optimal Control of Large-Size Second-Order Linear Time-Varying Systems

Efficient Near-Optimal Control of Large-Size Second-Order Linear Time-Varying Systems

Decomposition of Second-Order Discrete-Time Linear Time-Varying Systems into First-Order Commutative Pairs

Decomposition of Second-Order Discrete-Time Linear Time-Varying Systems into First-Order Commutative Pairs

Human Decision-Making Modeling and Cooperative Controller Design for Human–Agent Interaction Systems

In this article, the problem of human intervention and autonomous controller design for human–agent interaction systems is addressed. In particular, a human drift diffusion model is developed for second-order linear multiagent systems subject to unknown external disturbances. The proposed human drift diffusion model can model human decision-making behavior using multiple sources of human decision-making information. Accurate human intervention timing is obtained by setting varying thresholds for different decision-making information. In addition, a fixed-time sliding mode adaptive behavioural controller is developed to execute human decisions in the framework of the null-space based behavioral control method. The controller guarantees that agents can follow human commands given an arbitrary initial task error and can finish tasks in fixed time. A simulation under various scenarios shows that the proposed human drift diffusion model is able to provide appropriate human intervention and the proposed controller can guarantee human command fulfillment within fixed time.