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Safety for dynamical systems is often posed as an invariance constraint, requiring the system trajectory to remain in some safe subset of the state-space for all time. This note presents new tools for studying reachability and set invariance for nondeterministic systems subject to a disturbance input using the theory of mixed-monotone dynamical systems. The vector field of a mixed-monotone system is characterized as being decomposable into increasing and decreasing components, which allows the dynamics to be embedded in a higher dimensional embedding system. Even though the original system is nondeterministic due to the unknown disturbance input, the embedding system has no disturbance and a single simulation of the embedding system provides bounds for reachable sets of the original dynamics. In this article, we present an efficient method for identifying robustly forward invariant and attractive sets for mixed-monotone systems by studying equilibria and their stability properties of the corresponding embedding system. We show how this approach can be applied to either the backward-time dynamics or a set of linearly transformed dynamics to establish different robustly forward invariant sets for the original dynamics, and we show also how periodic solutions to the embedding system establish invariant regions for the original dynamics as well. The findings of this work are demonstrated through two numerical examples and two case studies, including a five-dimensional planar quadrotor system. [ABSTRACT FROM AUTHOR]

A novel adaptive method to achieve both path following and formation moving along desired orbits in the presence of a spatio-temporal flowfield is presented. The flowfield is a spatio-temporal general flow with unknown time-varying parameters. The so-called congelation of variables method is used to estimate the time-varying flow parameters, which do not have any restrictions on the rate of their variation. The asymptotic properties of the resulting adaptive system are studied in detail. Simulation results demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This article presents a method to synchronize a network of spatially distributed nonlinear oscillators that can only interact with each other intermittently and at very close range. This problem arises in applications where semipassive sensors drift along patterns of ambient geophysical flows that bring them close periodically and have to establish periodic rendezvous in order to efficiently exchange information or be retrieved. The problem is challenging because cooperative control action can only be applied over the short time window that agents are in rendezvous, and different subgroups that share some members meet asynchronously at different locations. In such cases, the ambient geophysical dynamics that drive the motion of the agents need to be directly incorporated into control design. This article presents a decentralized, intermittently activated, pairwise interacting control law for the agents, which, under certain conditions on overall network connectivity, brings the whole system into a steady state, where all agents synchronize their periodic rendezvous around configurations determined by the surrounding geophysical field. [ABSTRACT FROM AUTHOR]

A comprehensive theory for the stability boundaries and the stability regions of a general class of nonlinear discrete dynamical systems is developed in this article. This general class of systems is modeled by diffeomorphisms and admits as limit sets only fixed points and periodic orbits. Topological and dynamical characterizations of stability boundaries are developed. Necessary and sufficient conditions for fixed points and periodic orbits to lie on the stability boundary are derived. Numerical examples, including applications to associative neural networks illustrating the theoretical developments, are presented. [ABSTRACT FROM AUTHOR]

Nonlinear dynamical systems exhibiting complex structure in their limit sets, such as chaotic and closed orbits, do not admit energy functions. The theory of generalized energy functions, which may assume positive derivative in some bounded sets, appears as an alternative to study the asymptotic behavior of solutions of these systems. In this article, a generalized energy function and a complete characterization of the stability boundary and stability region are developed for a class of third-order dynamical systems. This class of systems appears in electrical power system models and has a class of quasi-gradient systems and second-order systems as particular cases. These systems may admit complex structure in their limit sets and do not admit an energy function that is general for the class. Numerical examples illustrate how generalized energy functions provide estimates of stability regions. [ABSTRACT FROM AUTHOR]

The proofs of the groundbreaking theorems of [1] rely on a lemma, which states that if the unstable manifold of a first hyperbolic closed orbit intersects transversely the stable manifold of a second (possibly the same) hyperbolic closed orbit, then the dimension of the unstable manifold of the first is strictly greater than the dimension of the unstable manifold of the second. However, we provide an example meeting the conditions of the lemma where the dimensions of the unstable manifolds are equal, thereby disproving the lemma. In particular, we present a hyperbolic closed orbit of a $C^\infty$ vector field over $\mathbb {R}^3$ whose stable and unstable manifolds have nonempty, transverse intersection. [ABSTRACT FROM AUTHOR]

Bilinear systems emerge in a wide variety of fields as natural models for dynamical systems ranging from robotics to quantum dots. Analyzing controllability of such systems is of fundamental and practical importance, for example, for the design of optimal control laws, stabilization of unstable systems, and minimal realization of input–output relations. Tools from the Lie theory have been adopted to establish controllability conditions for bilinear systems, and the most notable development was the Lie algebra rank condition (LARC). However, the application of the LARC may be computationally expensive for high-dimensional systems. In this article, we present an alternative and effective algebraic approach to investigate controllability of bilinear systems. The central idea is to map Lie bracket operations of the vector fields governing the system dynamics to permutation multiplications on a symmetric group, so that controllability and the controllable submanifold can be characterized by permutation orbits. The method is further shown to be applicable to characterize controllability of systems defined on undirected graphs, such as multiagent systems with controlled couplings between agents and Markov chains with tunable transition rates between states, which, in turn, reveals a graph representation of controllability through graph connectivity. [ABSTRACT FROM AUTHOR]

A conjunctive Boolean network (CBN) is a finite state dynamical system, whose variables take values from a binary set, and the value update rule for each variable is a Boolean function consisting only of logic and operations. We investigate the asymptotic behavior of CBNs by computing their periodic orbits. When the underlying digraph is strongly connected, the periodic orbits of the associated CBN have been completely understood, one-to-one corresponding to binary necklaces of a certain length given by the loop number of the graph. We characterize in the paper the periodic orbits of CBNs over arbitrary weakly connected digraphs. We establish, among other things, a new method to investigate their asymptotic behavior. Specifically, we introduce a graphical approach, termed system reduction, which turns the underlying digraph into a special weakly connected digraph, whose strongly connected components are all cycles. We show that the reduced system uniquely determines the asymptotic behavior of the original system. Moreover, we provide a constructive method for computing the periodic orbit of the reduced system, which the system will enter for a given, but arbitrary initial condition. [ABSTRACT FROM AUTHOR]

The Poincaré map is widely used to study the qualitative behavior of dynamical systems. For instance, it can be used to describe the existence of periodic solutions. The Poincaré map for dynamical systems with impulse effects (SIEs) was introduced in the last decade and mainly employed to study the existence of limit cycles (periodic gaits) for the locomotion of bipedal robots. We investigate sufficient conditions for the existence and uniqueness of Poincaré maps for dynamical SIEs evolving on a differentiable manifold. We apply the results to show the existence and uniqueness of Poincaré maps for systems with multiple domains. [ABSTRACT FROM AUTHOR]

We study the time optimal control problem for the evolution operator of an $n$ -level quantum system. For the considered models, the control couples all the energy levels to a given one and is assumed to be bounded in Euclidean norm. The resulting problem is a sub-Riemannian $K\hbox{--}P$ problem, (as introduced in articles by U. Boscain and by V. Jurdjevic), whose underlying symmetric space is $SU(n)/S(U(n-1) \times U(1))$. Following a method introduced by F. Albertini and D. D'Alessandro, we consider the action of $S(U(n-1) \times U(1))$ on $SU(n)$ as a conjugation $X \rightarrow KXK^{-1}$. This allows us to do a symmetry reduction and consider the problem on a quotient space. We give an explicit description of such a quotient space which has the structure of a stratified space. We prove several properties of sub-Riemannian problems with the given structure. We derive the explicit optimal control for the case of three level quantum systems where the desired operation is on the lowest two energy levels ($\Lambda$ -systems). We reduce the latter problem to an integer quadratic optimization problem with linear constraints, which we solve completely for a specific set of final data. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the relation between robustness of periodic orbits exhibited by systems with impulse effects and robustness of their corresponding Poincaré maps. In particular, we prove that input-to-state stability (ISS) of a periodic orbit under external excitation in both continuous and discrete time is equivalent to ISS of the corresponding zero-input fixed point of the associated forced Poincaré map. This result extends the classical Poincaré analysis for asymptotic stability of periodic solutions to establish orbital ISS of such solutions under external excitation. In our proof, we define the forced Poincaré map, and use it to construct ISS estimates for the periodic orbit in terms of ISS estimates of this map under mild assumptions on the input signals. As a consequence of the availability of these estimates, the equivalence between exponential stability (ES) of the fixed point of the zero-input (unforced) Poincaré map and the ES of the corresponding orbit is recovered. The results can be applied naturally to study the robustness of periodic orbits of continuous-time systems as well. Although our motivation for extending classical Poincaré analysis to address ISS stems from the need to design robust controllers for limit-cycle walking and running robots, the results are applicable to a much broader class of systems that exhibit periodic solutions. [ABSTRACT FROM AUTHOR]

Models of bipedal walking are hybrid, with continuous-time dynamics representing the swing phases and discrete-time dynamics representing the impact events. The feedback controllers for these systems can be two-level, including both continuous- and discrete-time (event-based) actions. This paper presents a systematic framework to design decentralized event-based controllers for robust stabilization of hybrid periodic orbits against possible disturbances in discrete-time phases. The properties of the Poincaré return map are investigated to study the orbital input-to-state stability for the closed-loop system with respect to disturbance inputs. An optimization problem involving bilinear matrix inequalities is then presented to design $\mathcal {H}_{2}$ - and $\mathcal {H}_{\infty }$ -optimal decentralized event-based controllers. The power of the proposed framework is finally demonstrated through designing a set of decentralized two-level controllers for underactuated walking of a three-dimensional autonomous bipedal robot with nine degrees of freedom and a decentralization scheme motivated by amputee locomotion with a transpelvic prosthetic leg. [ABSTRACT FROM AUTHOR]

Despite its potential use in several meaningful problems linear impulsive systems have not been extensively studied to account for dynamics in which the equilibria are out of the origin. In this work a novel characterization of the system equilibria and invariant regions—derived from the definition of two underlying discrete-time systems—is given, and based on this characterization, an unconstrained feedback control and a zone Model Predictive Control strategies are proposed. The controllers are tested in two drug administration problems: An intravenous bolus administration of Lithium ions and a nonlinear human immunodeficiency virus infection dynamics under Zidovudine treatment. [ABSTRACT FROM AUTHOR]

In this paper, we demonstrate that the dynamics of an $n$ -dimensional Lindblad control system can be separated into its inter- and intraorbit dynamics when there is fast controllability. This can be viewed as a control system on the simplex of density operator spectra, where projectors representing the eigenspaces are viewed as control variables. The local controllability properties of this control system can be analyzed when the control set of projectors is limited to a finite subset. In particular, there is a natural finite subset of $n!$ projector tuples that are effective for low-purity orbits. [ABSTRACT FROM PUBLISHER]

Synchronization and coordination of coupled oscillators are fundamental behaviors in complex dynamical systems. This paper considers the design of coupled harmonic oscillators to generate an orbitally stable limit cycle of prescribed oscillation profile. Based on the Floquet theory and averaging techniques, necessary and sufficient conditions are obtained for nonlinear coupling functions to achieve local exponential convergence to a desired orbit. Unlike globally convergent methods based on contraction analysis, the result applies to oscillators without flow invariance properties. Insights into coordination mechanisms are gained through interpretation of the coupling structure as a directed graph. The theory is illustrated by simple tutorial examples. [ABSTRACT FROM AUTHOR]

Recent research has established the importance of (strict) dissipativity for proving stability of economic MPC in the case of an optimal steady state. In many cases, though, steady-state operation is not economically optimal and periodic operation of the system yields a better performance. In this technical note, we propose ways of extending the notion of (strict) dissipativity for periodic systems. We prove that optimal $P$-periodic operation and MPC stability directly follow, similarly to the steady-state case, which can be seen as a special case of the proposed framework. Finally, we illustrate the theoretical results with several simple examples. [ABSTRACT FROM AUTHOR]

A new control law for elastic joint robots that allows to regulate an energy function of the system to a desired value is presented in this technical note. Being able to either remove energy from the system or inject into it, oscillations can be both damped out and induced. The proposed nonlinear dynamic state feedback controller forces the system to evolve on a submanifold of the configuration space. The reduced dynamics of the system and of the controller itself are similar to a single elastic joint, for which an asymptotically stable limit cycle is obtained regulating an energy function to a positive desired value. When the desired value of the energy function is chosen to be zero, then the asymptotically stable limit cycle reduces to an asymptotically stable equilibrium point. In this case the oscillations are damped out and the desired task-space configuration is reached. The design of the controller extensively uses the concept of conditional stability, so that the limit cycle can be designed for a lower dimensional dynamical system, although it will result to be a limit cycle for the whole system. [ABSTRACT FROM PUBLISHER]

This technical note shows that a barrier certificate exists for any safe dynamical system. Specifically, we prove converse barrier certificate theorems for a class of structurally stable dynamical systems. Other authors have developed a related result by assuming that the dynamical system has neither singular points nor closed orbits. In this technical note, we redefine the standard notion of safety to comply with dynamical systems with multiple singular elements. Hereafter, we prove the converse barrier certificate theorems and highlight the differences between our results and previous work by a number of illustrative examples. [ABSTRACT FROM AUTHOR]

We show that, near periodic orbits, a class of hybrid models can be reduced to or approximated by smooth continuous-time dynamical systems. Specifically, near an exponentially stable periodic orbit undergoing isolated transitions in a hybrid dynamical system, nearby executions generically contract superexponentially to a constant-dimensional subsystem. Under a non-degeneracy condition on the rank deficiency of the associated Poincaré map, the contraction occurs in finite time regardless of the stability properties of the orbit. Hybrid transitions may be removed from the resulting subsystem via a topological quotient that admits a smooth structure to yield an equivalent smooth dynamical system. We demonstrate reduction of a high-dimensional underactuated mechanical model for terrestrial locomotion, assess structural stability of deadbeat controllers for rhythmic locomotion and manipulation, and derive a normal form for the stability basin of a hybrid oscillator. These applications illustrate the utility of our theoretical results for synthesis and analysis of feedback control laws for rhythmic hybrid behavior. [ABSTRACT FROM PUBLISHER]

This paper studies symmetry in linear model predictive control (MPC). We define symmetry for model predictive control laws and for model predictive control problems. Properties of both MPC symmetries are studied by using a group theory formalism. We show how to efficiently compute MPC symmetries by transforming the search of MPC symmetry generators into a graph automorphism problem. MPC symmetries are then used to design model predictive control algorithms with reduced complexity. The effectiveness of the proposed approach is shown through a simple large-scale MPC problem whose explicit solution can only be found with the method presented in this paper. [ABSTRACT FROM AUTHOR]

The dynamical behaviors of an Euler discretized sliding mode control (SMC) systems based on equivalent control strategy are studied. A periodic-2 orbit in steady state is found for the switching function of the Euler discretized SMC systems. The time steps for the switching function to converge toward the periodic-2 orbit are obtained. When the discretized SMC system is stable, it further shows that the system states of the SMC systems will also enter into some periodic-2 orbits, and these periodic-2 orbits are characterized by explicit analytic expressions. Finally, simulation examples are given to illustrate the theoretical results. [ABSTRACT FROM PUBLISHER]

Marine Protected Areas (MPAs) are regions in the ocean or along coastlines where fishing is controlled to avoid the reduction or elimination of fish populations. A central question is where exactly to establish an MPA. We cast this as an optimal problem along a one-dimensional coast-line, where fish are assumed to move diffusively, and are subject to recruitment, natural death and harvesting through fishing. The functional being maximized is a weighted sum of the average fish density and the average fishing yield. It is shown that optimal controls exist, and that the location of the MPA is determined by two key model parameters, namely the size of the coast, and the weight of the average fish density in the functional. [ABSTRACT FROM PUBLISHER]

This paper presents a human-inspired control approach to bipedal robotic walking: utilizing human data and output functions that appear to be intrinsic to human walking in order to formally design controllers that provably result in stable robotic walking. Beginning with human walking data, outputs—or functions of the kinematics—are determined that result in a low-dimensional representation of human locomotion. These same outputs can be considered on a robot, and human-inspired control is used to drive the outputs of the robot to the outputs of the human. The main results of this paper are that, in the case of both under and full actuation, the parameters of this controller can be determined through a human-inspired optimization problem that provides the best fit of the human data while simultaneously provably guaranteeing stable robotic walking for which the initial condition can be computed in closed form. These formal results are demonstrated in simulation by considering two bipedal robots—an underactuated 2-D bipedal robot, AMBER, and fully actuated 3-D bipedal robot, NAO—for which stable robotic walking is automatically obtained using only human data. Moreover, in both cases, these simulated walking gaits are realized experimentally to obtain human-inspired bipedal walking on the actual robots. [ABSTRACT FROM PUBLISHER]

This paper addresses the problem of exponentially stabilizing periodic orbits in a special class of hybrid models—systems with impulse effects—through control Lyapunov functions. The periodic orbit is assumed to lie in a C^1 submanifold Z that is contained in the zero set of an output function and is invariant under both the continuous and discrete dynamics; the associated restriction dynamics are termed the hybrid zero dynamics. The orbit is furthermore assumed to be exponentially stable within the hybrid zero dynamics. Prior results on the stabilization of such periodic orbits with respect to the full-order dynamics of the system with impulse effects have relied on input-output linearization of the dynamics transverse to the zero dynamics manifold. The principal result of this paper demonstrates that a variant of control Lyapunov functions that enforce rapid exponential convergence to the zero dynamics surface, Z, can be used to achieve exponential stability of the periodic orbit in the full-order dynamics, thereby significantly extending the class of stabilizing controllers. The main result is illustrated on a hybrid model of a bipedal walking robot through simulations and is utilized to experimentally achieve bipedal locomotion via control Lyapunov functions. [ABSTRACT FROM PUBLISHER]

The main goal of this technical note is to show that the class of systems described by a planar differential equation having a rational proper Lyapunov function has asymptotically stable sets which are either locally asymptotically stable equilibrium points, stable algebraic limit cycles or asymptotically stable algebraic graphics. The use of the Zubov equation is then an adapted tool to investigate the study of an upper bound on the number of stable limit cycles and asymptotically stable graphics and their relative positions for this class of systems. [ABSTRACT FROM PUBLISHER]

A hybrid mathematical model of endocrine regulation obtained by augmenting the classical continuous Smith model with a pulse-modulated feedback to describe episodic (pulsatile) secretion is considered. Conditions for existence and local orbital stability of periodical solutions with m impulses in the least period (m-cycles) are derived. An important implication of the performed analysis is that the nonlinear dynamics of the pulse-modulated system and not the delay itself cause the sustained closed-loop oscillations. Furthermore, simulation and bifurcation analysis indicate that increasing the time delay in the system in hand typically, but not always, leads to less complex dynamic pattern in the closed-loop system by giving rise to stable cycles of lower periodicity. [ABSTRACT FROM AUTHOR]

This technical brief investigates virtual holonomic constraints for Euler–Lagrange systems with n degrees-of-freedom and n-1 controls. In our framework, a virtual holonomic constraint is a relation specifying n-1 configuration variables in terms of a single angular configuration variable. The enforcement by feedback of such a constraint induces a desired repetitive behavior in the system. We give conditions under which a virtual holonomic constraint is feasible, i.e, it can be made invariant by feedback, and it is stabilizable. We provide sufficient conditions under which the dynamics on the constraint manifold correspond to an Euler– Lagrange system. These ideas are applied to the problem of swinging up an underactuated pendulum while guaranteeing that the second link does not fall over. [ABSTRACT FROM AUTHOR]

Knowledge about to what extent quantum dynamical systems can be steered by coherent controls is indispensable for future developments in quantum technology. The purpose of this paper is to analyze such controllability aspects for finite dimensional bilinear quantum control systems. We use a unified approach based on Lie-algebraic methods from nonlinear control theory to revisit known and to establish new results for closed and open quantum systems. In particular, we provide a simplified characterization of different notions of controllability for closed quantum systems described by the Liouville–von Neumann equation. We derive new necessary and sufficient conditions for accessibility of open quantum systems modelled by the Lindblad–Kossakowski master equation. To this end, we exploit a well-studied topic of differential geometry, namely the classification of all matrix Lie-groups which act transitively on the Grassmann manifold or the punctured Euclidean space. For the special case of coupled spin-1/2 systems, we obtain a remarkably simple characterization of accessibility. These accessibility results correct and refine previous statements in the quantum control literature. [ABSTRACT FROM AUTHOR]

The existing characterization of stability regions was developed under the assumption that limit sets on the stability boundary are exclusively composed of hyperbolic equilibrium points and closed orbits. The characterizations derived in this technical note are a generalization of existing results in the theory of stability regions. A characterization of the stability boundary of general autonomous nonlinear dynamical systems is developed under the assumption that limit sets on the stability boundary are composed of a countable number of disjoint and indecomposable components, which can be equilibrium points, closed orbits, quasi-periodic solutions and even chaotic invariant sets. [ABSTRACT FROM AUTHOR]

This technical note establishes several versions of invariance principles for describing the eventual dynamical behaviors of discrete dynamical systems. Instead of the requirement of the so-called Lyapunov functions in the classical LaSalle invariance principle, some more relaxed conditions are imported. The established invariance principles thus can be applied to a more general class of discrete dynamical systems for classifying their orbits into two categories based on the eventual dynamical behaviors, and the proposed classification scheme is suitable for theoretically and numerically estimating the local or global attractors produced by the discrete dynamical systems. The practical usefulness of the analytical results is verified by systematically investigating several representative discrete systems. [ABSTRACT FROM AUTHOR]

The paper presents a parameter-dependent Lyapunov approach for the control of nonstationary linear-parameter varying systems. The work is motivated by the challenges encountered in controlling nonlinear systems about pre-specified aggressive trajectories, specifically ones which eventually settle into periodic orbits. The control design method uses the \ell2-induced norm performance measure, and the analysis and synthesis results are given in terms of parameterized linear matrix inequalities. Two fast and easy-to-implement algorithms for online controller construction are also provided. [ABSTRACT FROM PUBLISHER]

This paper contributes to the understanding of snake robot locomotion by employing nonlinear system analysis tools for investigating fundamental properties of snake robot dynamics. The paper has five contributions: 1) a partially feedback linearized model of a planar snake robot influenced by viscous ground friction is developed. 2) A stabilizability analysis is presented proving that any asymptotically stabilizing control law for a planar snake robot to an equilibrium point must be time-varying. 3) A controllability analysis is presented proving that planar snake robots are not controllable when the viscous ground friction is isotropic, but that a snake robot becomes strongly accessible when the viscous ground friction is anisotropic. The analysis also shows that the snake robot does not satisfy sufficient conditions for small-time local controllability (STLC). 4) An analysis of snake locomotion is presented that easily explains how anisotropic viscous ground friction enables snake robots to locomote forward on a planar surface. The explanation is based on a simple mapping from link velocities normal to the direction of motion into propulsive forces in the direction of motion. 5) A controller for straight line path following control of snake robots is proposed and a Poincaré map is investigated to prove that the resulting state variables of the snake robot, except for the position in the forward direction, trace out an exponentially stable periodic orbit. [ABSTRACT FROM AUTHOR]

This note deals with the control of linear discrete-time systems with uncertain initial conditions. Specifically, we consider the problem where the initial condition is known to reside in a norm ball of some radius, and the input disturbance is constrained to satisfy an independent norm condition. The. paper focuses on eventually periodic systems; these include both finite horizon and periodic systems as special cases. The main theorem provides exact synthesis conditions for the existence of eventually periodic controllers which both stabilize and provide performance in closed-loop control systems. These conditions are given in terms of a finite-dimensional semidefinite programming problem. We also give a version of the main result for the special case of linear time-invariant systems with uncertain initial states, and conclude with an illustrative example. [ABSTRACT FROM AUTHOR]