Linear Time-varying Dynamics Research Articles

Feasibility analysis and saturation control for underactuated spacecraft formation reconfiguration in elliptic orbits

This work proposes a saturation control scheme for underactuated spacecraft formation reconfiguration in elliptic orbits without radial or along-track thrust. Firstly, the rank criterion method is applied to analyze the controllability and feasibility of formation reconfiguration by linearizing the linear time-varying dynamics to linear time-invariant ones. Based on the inherent coupling of the linear time-varying system, the underactuated error dynamics are presented for either underactuated case. Subsequently, the developed underactuated saturation controller can ensure that the time-varying system trajectory asymptotically converges to the specified configuration. The Lyapunov-based analysis presents the constraint conditions of controller parameters and the stable reconfiguration accuracy of the system states. Finally, numerical simulations for both underactuated scenarios are performed in the environment with J2 perturbation to verify the validity of the proposed underactuated control scheme.

Distributed decentralized receding horizon control for very large-scale networks with application to satellite mega-constellations

The implementation feasibility of control algorithms over very large-scale networks calls for hard constraints regarding communication, computational, and memory requirements. In this paper, the decentralized receding horizon control problem for very large-scale networks of dynamically decoupled systems with a common, possibly time-varying, control objective is addressed. Each system is assumed to be modeled by linear time-varying dynamics, which can be leveraged to approximate nonlinear systems about successive points of operation. A distributed and decentralized receding horizon control solution is put forward, which: (i) takes communication delays into account; (ii) allows local communication exclusively; and (iii) whose computational and memory requirements in each computational unit do not scale with the dimension of the network. The scalability of the proposed solution enables emerging very large-scale applications of swarm robotics and networked control. This approach is applied to the orbit control problem of low Earth orbit mega-constellations, featuring high-fidelity numerical simulations for the Starlink mega-constellation.

Computing dispersion diagrams and forced responses of arbitrarily varying waveguides

Periodic structures can be used for vibration control thanks to their vibration attenuation stop bands. The dispersion diagram obtained for the structural unit cell can be used to characterize such wave attenuation bands. Several designs using spatially varying unit cells have recently been reported in the literature. Spatially varying properties can also be used to investigate the effect of variability on finite structures. Therefore, various methodologies to model spatially varying structures have been recently developed. This work presents new numerical methodologies to compute (i) dispersion diagrams and (ii) forced responses of one-dimensional spatially varying periodic waveguides. The first method extends previous work using a state-space formulation to compute dispersion relations using a linear time-varying system dynamics approach. The methodology is applied to elementary rods, Saint-Venant shafts, Euler–Bernoulli beams, and Timoshenko beams, and the results are validated with discretized spectral element models and the plane wave expansion method. The proposed approach is shown to be more efficient in cases where the plane wave expansion method presents a slow convergence. The second method is a recursive dynamic condensation to compute forced responses using the dynamic stiffness matrix of the unit cell. The results and computational times of the proposed recursive condensation are compared with the direct assembly of a global dynamic stiffness and its standard dynamic condensation. The two proposed methods are shown to be more efficient and accurate in the computation of dispersion diagrams and forced responses of one-dimensional structures for which the elastodynamic equations are known.

Minimal regret state estimation of time-varying systems

Kalman and H∞ filters, the most popular paradigms for linear state estimation, are designed for very specific specific noise and disturbance patterns, which may not appear in practice. State observers based on the minimization of regret measures are a promising alternative, as they aim to adapt to recognizable patterns in the estimation error. In this paper, we show that the regret minimization problem for finite horizon estimation can be cast into a simple convex optimization problem. For this purpose, we first rewrite linear time-varying system dynamics using a novel system level synthesis parametrization for state estimation, capable of handling both disturbance and measurement noise. We then provide a tractable formulation for the minimization of regret based on semi-definite programming. Both contributions make the minimal regret observer design easily implementable in practice. Finally, numerical experiments show that the computed observer can significantly outperform both H2 and H∞ filters.

An adaptive observer for uncertain linear time-varying systems with unknown additive perturbations

In this paper we are interested in the problem of adaptive state observation of linear time-varying (LTV) systems where the system and the input matrices depend on unknown time-varying parameters. It is assumed that these parameters satisfy some known LTV dynamics, but with unknown initial conditions. Moreover, the state equation is perturbed by an additive signal generated from an exosystem with uncertain constant parameters. Our main contribution is to propose a globally convergent state observer that requires only a weak excitation assumption on the system, which is satisfied in the present context of regulation.

Adaptive Event-Triggering Consensus for Multi-Agent Systems with Linear Time-Varying Dynamics

Adaptive Event-Triggering Consensus for Multi-Agent Systems with Linear Time-Varying Dynamics

A fundamental solution concept for a family of state constrained regulator problems

A family of continuous-time regulator problems, subject to linear time-varying dynamics and convex state constraints, is parameterized with respect to a space of semiconcave terminal costs. By embedding the state constraints into an extended real-valued running cost, a family of relaxations is employed to develop a min-plus convolution-based fundamental solution concept for the family of regulator problems. A representation of the fundamental solution concept is provided in terms of solutions of associated differential Riccati and related equations, and possible avenues for efficient numerical computation are explored.

Optimal trajectory design accounting for the stabilization of linear time-varying error dynamics

This study is dedicated to the development of a direct optimal control-based algorithm for trajectory optimization problems that accounts for the closed-loop stability of the trajectory tracking error dynamics already during the optimization. Consequently, the trajectory is designed such that the Linear Time-Varying (LTV) dynamic system, describing the controller’s error dynamics, is stable, while additionally the desired optimality criterion is optimized and all enforced constraints on the trajectory are fulfilled. This is achieved by means of a Lyapunov stability analysis of the LTV dynamics within the optimization problem using a time-dependent, quadratic Lyapunov function candidate. Special care is taken with regard to ensuring the correct definiteness of the ensuing matrices within the Lyapunov stability analysis, specifically considering a numerically stable formulation of these in the numerical optimization. The developed algorithm is applied to a trajectory design problem for which the LTV system is part of the path-following error dynamics, which is required to be stable. The main benefit of the proposed scheme in this context is that the designed trajectory trades-off the required stability and robustness properties of the LTV dynamics with the optimality of the trajectory already at the design phase and thus, does not produce unstable optimal trajectories the system must follow in the real application.

Discrete-time distributed Kalman filter design for networks of interconnected systems with linear time-varying dynamics

This paper addresses the problem of designing distributed state estimation solutions for a network of interconnected systems modelled by linear time-varying (LTV) dynamics in a discrete-time framework. The problem is formulated as a classical optimal estimation problem, for the global system, subject to a given sparsity constraint on the filter gain, which reflects the distributed nature of the network. Two methods are presented, both of them able to compute a sequence of well-performing stabilising gains. Moreover, both methods are validated by resorting to simulations of: (i) a randomly generated synthetic LTV system; and (ii) a large-scale nonlinear network of interconnected tanks. One of the proposed methods relies on a computationally efficient solution, thus it is computed very rapidly. The other achieves better performance, but it is computationally more expensive and requires that a window of the future dynamics of the system is known. When implemented to a nonlinear network, approximated by an LTV system, the proposed methods are able to compute well-performing gains that stabilise the estimation error dynamics. Both algorithms are scalable, being adequate for implementation in large-scale networks.

Best Linear Time-Varying Approximation of a General Class of Nonlinear Time-Varying Systems

This article presents a method for estimating a linear time-varying approximation of a general class of nonlinear time-varying (NLTV) systems. It starts from noisy measurements of the response of the NLTV system to a special class of periodic excitation signals. These measurements are subject to measurement noise, process noise, and a trend. The proposed method is a two-step procedure. First, the disturbing noise variance is quantified. Next, using this knowledge, the linear time-varying dynamics are estimated together with the NLTV distortions. The latter are split into even and odd contributions. As a result, the signal-to-nonlinear-distortion ratio is quantified. It allows one to decide whether or not a linear approximation is justifiable for the application at hand. The two-step algorithm is fully automatic in the sense that the user only has to choose upper bounds on the number of basis functions used for modeling the response signal. The obtained linear time-varying approximation is the best in the sense that the difference between the actual nonlinear response and the response predicted by the linear approximation is uncorrelated with the input. Therefore, it is called the best linear time-varying approximation (BLTVA). Finally, the theory is validated on a simulation example and illustrated on two measurement examples: the crystallographic pitting corrosion of aluminum and copper electrorefining.

Control of a nonlinear flexure-jointed X-Y positioning stage using LTV-FIR command prefiltering for finite-time error cancellation

This paper presents a new command input prefilter that achieves exact residual vibration cancellation for a single degree-of-freedom oscillator with linear time-varying (LTV) dynamics. The prefilter has a time-varying finite impulse response (FIR) function that is constructed using a time-warping technique and can be convolved with any command input signal to achieve finite settling time. Numerical results confirm that a discrete-time implementation of the prefilter can suppress vibration more effectively than robust linear time-invariant prefilter designs. The LTV-FIR prefilter was applied to a flexure-based X-Y micro-positioning platform having nonlinear dynamics and cross-coupling between motion axes. A configuration-dependent LTV model was constructed by using polynomial interpolation of a set of experimentally identified LTI models. Prefiltering was applied separately to X and Y axis command variables and a cross-coupling compensation technique used to minimize dynamic interaction between the motion axes. The LTV-FIR prefilter is shown to be effective for settling-time reduction in point-to-point motions (reduced from >6 to ≈0.4 seconds). Moreover, improved motion performance could be achieved over a large operating space without the use of high-gain feedback control, which is inappropriate in this case due to noise excitation and the risk of instability.

Sampled-Data Consensus of Linear Time-Varying Multiagent Networks With Time-Varying Topologies.

The main purpose of this article is to investigate the consensus of linear multiagent networks with time-varying characteristics under sampled-data communications, where the time-varying characteristics include both time-varying topologies and the node's linear time-varying dynamics. By using the decoupling method, we prove that the sampled-data consensus problem of multiagent networks is equal to the stability problem of sampled-data systems. Then, the globally asymptotical consensus is investigated for multiagent networks with time-varying characteristics by virtue of the Lyapunov function method. It should be noted that when the Lyapunov function method is utilized to investigate the stability problem of control systems, it is always assumed that the derivative of the constructed Lyapunov function is not more than zero. This assumption is removed here and as a replacement, the average value of the derivative of the Lyapunov function in a period to be negative is needed.

On the Approximated Reachability of a Class of Time-Varying Nonlinear Dynamic Systems Based on Their Linearized Behavior about the Equilibria: Applications to Epidemic Models

This paper formulates the properties of point reachability and approximate point reachability of either a targeted state or output values in a general dynamic system which possess a linear time-varying dynamics with respect to a given reference nominal one and, eventually, an unknown structured nonlinear dynamics. Such a dynamics is upper-bounded by a function of the state and input. The results are obtained for the case when the time-invariant nominal dynamics is perfectly known while its time-varying deviations together with the nonlinear dynamics are not precisely known and also for the case when only the nonlinear dynamics is not precisely known. Either the controllability gramian of the nominal linearized system with constant linear parameterization or that of the current linearized system (which includes the time-varying linear dynamics) are assumed to be non-singular. Also, some further results are obtained for the case when the control input is eventually saturated and for the case when the controllability gramians of the linear parts are singular. Examples of the derived theoretical results for some epidemic models are also discussed.

On the Mixed-Integer Linear-Quadratic Optimal Control With Switching Cost

This letter concerns the optimal control of a continuous-time dynamical system via continuous and discrete-valued control variables, where the objective functional also accounts for state-independent switching costs. The class of mixed-integer optimal control problems (OCPs) is interpreted as a bilevel problem, involving both switching times optimization, for a given sequence of modes, and purely continuous optimal control. Additionally, an original nonconvex formulation for the switching costs is introduced, in terms of cardinality, inspired by sparse optimization and compressed sensing techniques. We then adopt proximal algorithms to solve the resulting bilevel OCP with composite objective function. An efficient routine for evaluating the proximal operator is developed. Two examples are numerically solved via a proximal gradient method, discussed and compared with the literature. Although this letter focuses on switched linear time-varying dynamics and quadratic cost functionals with a specific formulation of the switching costs, the proposed approach may also apply to more general mixed-integer OCPs.

On the stability of inverse dynamics control of flexible-joint parallel manipulators in the presence of modeling error and disturbances

Inverse dynamics control is considered for flexible-joint parallel manipulators in order to obtain a good trajectory tracking performance in the case of modeling error and disturbances. It is known that, in the absence of modeling error and disturbance, inverse dynamics control leads to linear fourth-order error dynamics, which is asymptotically stable if the feedback gains are chosen to make the real part of the eigenvalues of the system negative. However, when there are modeling errors and disturbances, a linear time-varying error dynamics is obtained whose stability is not assured only by keeping the real parts of the frozen-time eigenvalues of the system negative. In this paper, the stability of such systems is investigated and it is proved that the linear time-varying system can be rendered stable by selecting the feedback gains such that the variation of the system becomes sufficiently slow. To illustrate the performance of the control method, deployment motion of a 3-RPR planar parallel manipulator subject to impact is simulated. For the impact model, the impulse-momentum and the coefficient of restitution equations for the system are derived.

Reset observer design for time-varying dynamics: Application to WIG crafts

In this article, a novel reset state estimator for time-varying dynamics is proposed. A first order reset law is used in the structure of observer to improve the performance of the state estimation response which decreases the settling time and overshoot of estimation. The sufficient conditions for stability of this observer based on systems with impulsive effect is addressed which is the main contribution of this study. This observer is applied to a WIG (Wing-In-Ground) craft in the presence of wave and gust which can be modeled by a linear time-varying dynamics. Finally, the five states for longitudinal dynamics of the WIG craft based on a full-order reset observer with time-varying parameters are estimated. The sufficient stability conditions of WIG state observer are stated and simulation results show the superior performance of the reset law.

FDI in Cyber Physical Systems: A Distributed Zonotopic and Gaussian Kalman Filter with Bit-level Reduction

The problem addressed is that of robust Fault Detection and Isolation (FDI) in Cyber Physical Systems (CPS) modeled by interconnected linear time-varying (LTV) dynamics in a noisy environment with both bounded disturbances and random noises. A Distributed Zonotopic and Gaussian Kalman Filter (DZG-KF) framework is proposed where each network node implements a local estimator using Set-membership (Zonotopic) and Probabilistic (Gaussian) Mergers (SPM/ZGM) to estimate its state. Each node communicates its own state information only to its neighbors in the network. In addition to the usual zonotope reduction, a bit-level reduction is introduced for the first time: its aim is to reduce the required communication channel capacity (bit rate) by approximating sets through quantization by the sender and reconstruction by the receiver. In order to deal with potential packet losses, the negotiation of data reconciliation strategies between agents is discussed. Fault detection and a first-level isolation emerge from agents local tests. Numerical simulations show the efficiency of the proposed scheme.

Performance comparison of distributed state estimation algorithms for power systems

A newly proposed distributed dynamic state estimation algorithm based on the maximum a posteriori (MAP) technique is generalised and studied for power systems. The system model involves linear time-varying load dynamics and nonlinear measurements. The main contribution of this paper is to compare the performance and feasibility of this distributed algorithm with several existing distributed state estimation algorithms in the literature. Simulations are tested on the IEEE 39-bus and 118-bus systems under various operating conditions. The results show that this distributed algorithm performs better than distributed quasi-steady state estimation algorithms which do not use the load dynamic model. The results also show that the performance of this distributed method is very close to that by the centralized state estimation method. The merits of this algorithm over the centralized method lie in its low computational complexity and low communication load. Hence, the analysis supports the efficiency and benefits of the distributed algorithm in applications to large-scale power systems.

Comparison of Several Filtering Methods for Linear Multi-agent Systems with Local Unknown Parametric Couplings

In this chapter, several filtering methods for multi-agent systems with coupling uncertainties are investigated. The considered multi-agent system is composed of many agents, each of which evolves with a discrete-time stochastic linear time-varying dynamics, and each agent will be locally influenced by its neighbor agents. So the states evolution of each agent is not only related with its previous states but also related with its neighbors’ previous states. Every agent can only observe its own measurements (outputs) and its neighbor agents’ outputs, while the states are invisible to any agent because of communication limitations existing in the considered multi-agent system. We suppose that each agent has a priori knowledge on the structure of itself and its neighbors, that is to say, the coefficient matrices occurred in the individual dynamic system are available for use by the agent itself and those agents who have interactions with the agent. Because of the information constraints, without knowing the coupling gains of the local interactions, it is not easy for each agent to estimate its states by traditional Kalman filter or other state observers. For the wide applications of multi-agent systems and the existence of coupling uncertainties in many practical applications, the problem of decentralized filtering of multi-agent systems is more important; this chapter introduces one general framework of multi-agent systems for convenience of filtering studies. For the considered coupled linear discrete-time multi-agent system with uncertain linear local couplings, based on the key idea of state augmentation and the certainty equivalence principle borrowed from the area of adaptive control, we introduce several filtering methods to resolve the fundamental problem considered in this chapter. By conducting extensive simulations, the consuming time and estimation errors of every method are compared for one typical example, which suggests which method is more precise and fast.

Nonlinear tension observers for web machines

Precise regulation of span tension in web-handling machines is critical to ensure product quality. Elevated tension causes web fracture during operation; depressed or highly variable tension leads to irregular and unacceptable spool. Web-handling machines typically employ costly load cell sensors to measure span tension. This work investigates the use of nonlinear estimators or observers to provide cost-effective span tension estimates from system measurements. After presenting the dynamic equations of a web-handling machine, their structure is exploited to design a reduced-order, nonlinear observer with linear time-varying error dynamics. Exponential stability of the error dynamics is demonstrated. Performance of the estimator is experimentally compared with two competing designs on a real web-handling machine.