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Cyclic motions are fundamental patterns in robotic applications, including industrial manipulation and robot locomotion. This article proposes an approach for the online modulation of cyclic motions in robotic applications. For this purpose, we present an integrated programmable central pattern generator (CPG) for the online generation of the reference joint trajectory of a robotic system out of a library of desired periodic motions. The reference trajectory is then followed by the lower level controller of the robot. The proposed CPG generates a smooth reference joint trajectory convergence to the desired one while preserving the position and velocity joint limits of the robot. The proposed CPG consists of one novel bounded output programmable oscillator. We design the programmable oscillator for encoding the desired multidimensional periodic trajectory as a stable limit cycle. We also use the state transformation method to ensure that the oscillator’s output and its first time-derivative preserve the joint position and velocity limits of the robot. With the help of Lyapunov-based arguments, we prove that the proposed CPG provides the global stability and convergence of the desired trajectory. The effectiveness of the proposed CPG for trajectory generation is shown in a passive rehabilitation scenario on the Kuka iiwa robot arm, and also in walking simulation on a seven-link bipedal robot. [ABSTRACT FROM AUTHOR]

This article addresses the problem of learning the optimal control policy for a nonlinear stochastic dynamical. This problem is subject to the “curse of dimensionality” associated with the dynamic programming method. This article proposes a novel decoupled data-based control (D2C) algorithm that addresses this problem using a decoupled, “open-loop–closed-loop,” approach. First, an open-loop deterministic trajectory optimization problem is solved using a black-box simulation model of the dynamical system. Then, closed-loop control is developed around this open-loop trajectory by linearization of the dynamics about this nominal trajectory. By virtue of linearization, a linear quadratic regulator based algorithm can be used for this closed-loop control. We show that the performance of D2C algorithm is approximately optimal. Moreover, simulation performance suggests a significant reduction in training time compared to other state-of-the-art algorithms. [ABSTRACT FROM AUTHOR]

We consider the framework of transfer-entropy-regularized Markov decision process (TERMDP) in which the weighted sum of the classical state-dependent cost and the transfer entropy from the state random process to the control input process is minimized. Although TERMDPs are generally formulated as nonconvex optimization problems, an analytical necessary optimality condition can be expressed as a finite set of nonlinear equations, based on which an iterative forward–backward computational procedure similar to the Arimoto–Blahut algorithm is developed. It is shown that every limit point of the sequence generated by the proposed algorithm is a stationary point of the TERMDP. Applications of TERMDPs are discussed in the context of networked control systems theory and nonequilibrium thermodynamics. The proposed algorithm is applied to an information-constrained maze navigation problem, whereby we study how the price of information qualitatively alters the optimal decision polices. [ABSTRACT FROM AUTHOR]

Many safety-critical systems, such as autonomous vehicles and service robots, must achieve high-level task specifications with performance guarantees. Much recent progress toward this goal has been made through an automatic controller synthesis from temporal logic specifications. Existing approaches, however, have been limited to relatively short and simple specifications. Furthermore, existing methods either consider some prior discretization of the state space, deal only with a convex fragment of temporal logic, or are not provably complete. We propose a scalable, provably complete algorithm that synthesizes continuous trajectories to satisfy nonconvex temporal logic over reals (RTL) specifications. We separate discrete task planning and continuous motion planning on-the-fly and harness highly efficient Boolean satisfiability and linear programming solvers to find dynamically feasible trajectories that satisfy nonconvex RTL specifications for high-dimensional systems. The proposed design algorithms are proven sound and complete, and simulation results demonstrate our approach's scalability. [ABSTRACT FROM AUTHOR]

A new approach for trajectory optimization of musculoskeletal dynamic models is introduced. The model combines rigid-body and muscle dynamics described with a Hill-type model driven by neural control inputs. The objective is to find input and state trajectories that are optimal with respect to a minimum-effort objective and meet constraints associated with musculoskeletal models. The measure of effort is given by the integral of pairwise average forces of the agonist-antagonist muscles. The concepts of flat parameterization of nonlinear systems and sum-of-squares optimization are combined to yield a method that eliminates the numerous set of dynamic constraints present in collocation methods. With terminal equilibrium, optimization reduces to a feasible linear program, and a recursive feasibility proof is given for more general polynomial optimization cases. The methods of the article can be used as a basis for fast, and efficient solvers for hierarchical, and receding-horizon control schemes. Two simulation examples are included to illustrate the proposed methods. [ABSTRACT FROM AUTHOR]

This article presents a computational framework to efficiently generate feasible and safe trajectories for multiple autonomous vehicle operations. We formulate the optimal motion planning problem as a continuous-time optimal control problem, and approximate its solutions in a discretized setting using Bernstein polynomials. The latter possess convenient properties that allow to efficiently compute and enforce constraints along the vehicles’ trajectories, such as maximum speed and angular rates, minimum distance between trajectories and between the vehicles and known obstacles, etc. Thus, the proposed method is particularly suitable for generating trajectories in real-time for safe operations in complex environments and multiple vehicle missions. We show, using a rigorous mathematical framework, that the solution to the discretized optimal motion planning problem converges to that of the continuous-time one. The advantages of the proposed method are investigated through numerical examples. [ABSTRACT FROM AUTHOR]

This paper addresses a one-dimensional optimal persistent monitoring problem using second-order agents. The goal is to control the movements of agents to minimize a performance metric associated with the environment (targets) over a finite time horizon. In contrast to earlier results limited to first-order dynamics for agents, we control their accelerations rather than velocities, thus leading to a better approximation of agent behavior in practice and to smoother trajectories. Bounds on both velocities and accelerations are also taken into consideration. Despite these added complications to agent dynamics, we derive a necessary condition for optimality and show that the optimal agent trajectories can be fully characterized by two parameter vectors. A gradient-based algorithm is proposed to optimize these parameters and yield a minimal performance metric. In addition, a collision avoidance algorithm is proposed to solve potential collision and boundary-crossing problems, thus extending the gradient-based algorithm solutions. Finally, simulation examples are included to demonstrate the effectiveness of our results. [ABSTRACT FROM AUTHOR]

We study a notion of estimation entropy for continuous-time nonlinear systems, formulated in terms of the number of system trajectories that approximate all other trajectories up to an exponentially decaying error. We also consider an alternative definition of estimation entropy, which uses approximating functions that are not necessarily trajectories of the system, and show that the two entropy notions are equivalent. We establish an upper bound on the estimation entropy in terms of the sum of the desired convergence rate and an upper bound on the matrix measure of the Jacobian, multiplied by the system dimension. A lower bound on the estimation entropy is developed as well. We then turn our attention to state estimation and model detection with quantized and sampled state measurements. We describe an iterative procedure that uses such measurements to generate state estimates that converge to the true state at the desired exponential rate. The average bit rate utilized by this procedure matches the derived upper bound on the estimation entropy, and no other algorithm of this type can perform the same estimation task with bit rates lower than the estimation entropy. Finally, we discuss an application of the estimation procedure in determining, from the quantized state measurements, which of two competing models of a dynamical system is the true model. We show that under a mild assumption of “exponential separation” of the candidate models, detection always happens in finite time. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the 1-periodic schedule of a class of time interval models under the form of a polyhedron which can describe Timed Event Graphs and P-time Event Graphs. Using the duality and Stiemke's theorem, the main contribution is the determination of conditions where the extremum cycle times are finite and characteristic of a class of models. [ABSTRACT FROM AUTHOR]

We study three existing issues associated with optimization of long-run average rewards of time-nonhomogeneous Markov processes in continuous time with continuous state spaces: 1) the underselectivity, i.e., the optimal policies do not depend on their actions in any finite time period; 2) its related issue, the bias optimality, i.e., policies that optimize both long-run average and transient total rewards, and 3) the effects of a nonsmooth point of a value function on performance optimization. These issues require considerations of the performance in the entire period with an infinite horizon, and therefore are not easily solvable by dynamic programming, which works backwards in time and takes a local view at a particular time instant. In this paper, we take a different approach called the relative optimization theory, which is based on a direct comparison of the performance measures of any two policies. We derive tight necessary and sufficient optimality conditions that take the underselectivity into consideration; we derive bias optimality conditions for both long-run average and transient rewards; and we show that the effect of a wide class of nonsmooth points, called semismooth points, of a value function on the long-run average performance is zero and can be ignored. [ABSTRACT FROM AUTHOR]

We propose a new dynamic extension of the thrust variable in the quadcopter dynamics that preserves the positive sign of the thrust. This extension not only eliminates the positive sign constraint on the thrust variable, but also leads to global chartwise feedback linearization of the quadcopter dynamics. For the latter, an atlas is first constructed on the entire state space of the quadcopter and then the dynamically extended quadcopter system is transformed to a 14-dimensional linear controllable system on each chart in the atlas. Based on the chartwise dynamic feedback linearization, a global tracking strategy is proposed for the quadcopter and its excellent performance is demonstrated with a simulation. [ABSTRACT FROM AUTHOR]

We study a distributed consensus-based stochastic gradient descent (SGD) algorithm and show that the rate of convergence involves the spectral properties of two matrices: The standard spectral gap of a weight matrix from the network topology and a new term depending on the spectral norm of the sample covariance matrix of the data. This data-dependent convergence rate shows that distributed SGD algorithms perform better on datasets with small spectral norm. Our analysis method also allows us to find data-dependent convergence rates as we limit the amount of communication. Spreading a fixed amount of data across more nodes slows convergence; for asymptotically growing datasets, we show that adding more machines can help when minimizing twice-differentiable losses. [ABSTRACT FROM AUTHOR]

We present necessary and sufficient optimality conditions for finite-time optimal control problems for a class of hybrid systems described by linear complementarity models. Although these optimal control problems are difficult in general due to the presence of complementarity constraints, we provide a set of structural assumptions ensuring that the tangent cone of the constraints possesses geometric regularity properties. These imply that the classical Karush–Kuhn–Tucker conditions of nonlinear programming theory are both necessary and sufficient for local optimality, which is not the case for general mathematical programs with complementarity constraints. We also present sufficient conditions for global optimality. We proceed to show that the dynamics of every continuous piecewise affine (PWA) system can be written as the optimizer of a mathematical program, which results in a linear complementarity model satisfying our structural assumptions. Hence, our stationarity results apply to a large class of hybrid systems with PWA dynamics. We present simulation results showing the substantial benefits possible from using a nonlinear programming approach to the optimal control problem with complementarity constraints instead of a more traditional mixed-integer formulation. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of synchronizing a group of identical linear time-invariant agents that exchange information through a communication network. The agents may only broadcast information at discrete-time instants and the decision to execute a broadcast is based on an event-triggered communication protocol. We prove that with the proposed control architecture the state of each agent converges to and remains in a neighborhood of a desired reference signal and the closed-loop system does not exhibit Zeno solutions. A self-triggered implementation of the proposed event-triggered communication protocol is also derived. [ABSTRACT FROM AUTHOR]

Hybrid optimal control problems are studied for a general class of hybrid systems, where autonomous and controlled state jumps are allowed at the switching instants, and in addition to terminal and running costs, switching between discrete states incurs costs. The statements of the Hybrid Minimum Principle and Hybrid Dynamic Programming are presented in this framework, and it is shown that under certain assumptions, the adjoint process in the Hybrid Minimum Principle and the gradient of the value function in Hybrid Dynamic Programming are governed by the same set of differential equations and have the same boundary conditions and hence are almost everywhere identical to each other along optimal trajectories. Analytic examples are provided to illustrate the results. [ABSTRACT FROM AUTHOR]

The robust trajectory tracking of fully actuated Lagrange systems is studied. Exogenous perturbations as well as parameter uncertainties are taken into account. A family of set-valued passivity-based controllers is proposed, including first-order sliding-mode schemes. The existence of solutions and the stability of the closed-loop system are established in continuous time. An implicit discretization approach is proposed and the well posedness and the stability of the closed-loop system are studied. Numerical simulations illustrate the effectiveness of the proposed discrete-time controller. [ABSTRACT FROM AUTHOR]

The technical note presents the design of a reference governor that manages directly the state and input references for constrained output regulation problems. In particular, for the class of so-called weakly input redundant linear systems, the proposed method leads to a taxonomy in terms of non-intrusive and intrusive reference governors, on the basis of the effect of reference manipulation on the tracking error. Non-intrusive references constitute a degree of freedom that one can freely select to meet constraint requirements; intrusive references are also assignable during transient, but shall be forced to vanish in steady state in order to achieve exact regulation. A simulation study illustrates the features of the proposed approach in support of the theoretical findings. [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]

Swarms, flocks, and other group formations can be found in nature in many organisms ranging from simple bacteria to mammals. Such collective and coordinated behavior is effective for avoiding predators and/or for increasing the chances of finding food (foraging). In this technical note we develop a mathematical model for analyzing the benefits of social foraging in a noisy environment. We identify conditions on the nutrient profile ensuring that local agent actions will lead to cohesive foraging. For convex, smooth nutrient profiles, we formalize the way in which swarming for social foraging is better at handling the effects of noise when compared to the average of individual foraging strategies. Under a swarming discipline, observational noise realizations that induce trajectories differing too much from the group average are likely to be discarded because of each individual's need to maintain cohesion. As a result, the group trajectories are less affected by noise. Simulation experiments indicate our theoretical results are robust to inter-agent communication constraints and non-convex nutrient profiles. These results suggest that swarming-like approaches for the control of networked agents may provide an additional level of robustness. [ABSTRACT FROM AUTHOR]

This paper addresses the stability and L\infty-gain analysis problem for continuous-time positive fractional-order delay systems with incommensurate orders between zero and one. A necessary and sufficient condition is firstly given to characterize the positivity of continuous-time fractional-order systems with bounded time-varying delays. Moreover, by exploiting the monotonic and asymptotic property of the constant delay system by virtue of the positivity, and comparing the trajectory of the time-varying delay system with that of the constant delay system, it is proved that the asymptotic stability of positive fractional-order systems is not sensitive to the magnitude of delays. In addition, it is shown that the L\infty-gain of a positive fractional-order system is independent of the magnitude of delays and fully determined by the system matrices. Finally, a numerical example is given to show the validity of the theoretical results. [ABSTRACT FROM PUBLISHER]

This technical note addresses a tradeoff in the tree based trajectory optimization algorithms for open-loop optimal control problem of rigid body system. In this technical note, linear prediction based uniform state sampling method (LPUSS), which relaxes the tradeoff between solutions' quality and computational efficiency, is proposed on the basis of the local linearity of motion equations. LPUSS has a lower calculation order than randomized kinodynamic planning (RKP) and rapid semi-optimal motion planning (RASMO). In the validation using multi degree-of-freedom (DOF) under actuated manipulator models, the solutions' quality and the computational efficiency of LPUSS were better than those of RKP and RASMO. LPUSS finished the optimization for the 6 DOF model within 40 minutes. This was the world's first success of the optimization of the swing up motion of a 6 DOF under actuated manipulator model. [ABSTRACT FROM PUBLISHER]

In this technical note, a new model predictive controller for tracking arbitrary periodic references is presented. The proposed controller is based on a single layer that unites dynamic trajectory planning and control. A design procedure to guarantee that the closed loop system converges asymptotically to the optimal admissible periodic trajectory while guaranteeing constraint satisfaction is provided. In addition, the constraints of the optimization problem solved by the controller do not depend on the reference, allowing for sudden changes in the reference without loosing feasibility. The properties of the proposed controller are demonstrated with a simulation example of a ball and plate system. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of constructing a finite bisimulation quotient for a discrete-time switched linear system in a bounded subset of its state space. Given a set of observations over polytopic subsets of the state space and a switched linear system with stable subsystems, the proposed algorithm generates the bisimulation quotient in a finite number of steps with the aid of sublevel sets of a polyhedral Lyapunov function. Starting from a sublevel set that includes the origin in its interior, the proposed algorithm iteratively constructs the bisimulation quotient for the region bounded by any larger sublevel set. We show how this bisimulation quotient can be used for synthesis of switching laws and verification with respect to specifications given as syntactically co-safe Linear Temporal Logic formulae over the observed polytopic subsets. [ABSTRACT FROM AUTHOR]

An important area of cyber-physical systems research is the development of smart ground transportation systems due to their potentially significant impact on safety, the economy, and the environment. We propose an approach based on model predictive control (MPC) for the development of provably collision free autonomous ground transportation systems, and present an autonomous intersection management framework. The MPC approach enables a vehicle to generate its own motion locally in time based on an optimization framework, incorporating constraints based on the states of other vehicles in the neighborhood, the speed limit of a road, the maximum values of acceleration and deceleration, etc. Safety and liveness of the traffic are however system-wide properties, not merely neighborhood properties, and the challenge is to augment this distributed optimization with coordination rules that guarantee overall system-wide safety as well as liveness of the traffic. We design two vehicle-to-vehicle (V2V) coordination rules, along with a vehicle-to-infrastructure rule, and establish the system-wide safety and liveness of the autonomous traffic based on each vehicle's MPC motion planner, operating in conjunction with an algorithm that orders vehicles according to their runtime properties. We also conduct a comparative simulation study of the throughput performance at an intersection of the above approach against another popular algorithm, the All-Way STOP. [ABSTRACT FROM AUTHOR]

We are interested in assigning a pre-specified number of nodes as leaders in order to minimize the mean-square deviation from consensus in stochastically forced networks. This problem arises in several applications including control of vehicular formations and localization in sensor networks. For networks with leaders subject to noise, we show that the Boolean constraints (which indicate whether a node is a leader) are the only source of nonconvexity. By relaxing these constraints to their convex hull we obtain a lower bound on the global optimal value. We also use a simple but efficient greedy algorithm to identify leaders and to compute an upper bound. For networks with leaders that perfectly follow their desired trajectories, we identify an additional source of nonconvexity in the form of a rank constraint. Removal of the rank constraint and relaxation of the Boolean constraints yields a semidefinite program for which we develop a customized algorithm well-suited for large networks. Several examples ranging from regular lattices to random graphs are provided to illustrate the effectiveness of the developed algorithms. [ABSTRACT FROM AUTHOR]

This paper deals with the observer problem for dynamical systems in a behavioral context. We are given a dynamical system together with a partition of the system variables into a set of known or measured variables and a set of unknown, to be estimated variables. The observer problem is to find a system that produces an estimate of the unknown variables on the basis of the known or measured variables. For a given plant and partition, we establish a characterization of all error behaviors that can be achieved by interconnecting the plant with some observer. The main result of this paper is a very general, behavioral formulation of an internal model principle for observers. We will show that a nonintrusive observer achieves a stable error behavior if and only if, in addition to a detectability condition on the observer, the observer behavior contains the anti-stabilizable part of the plant behavior. [ABSTRACT FROM AUTHOR]

This technical note studies quantitatively asymptotic growth behaviors of trajectories (AGBT) of nonlinear autonomous discrete dynamical system that has unbounded domain, non-Lipschitz continuous nonlinear operator, and stable or unstable equilibrium point. We explain how trajectory motion speed is quantitatively determined in the system, and study exact computation and sharp estimation of the smallest exponential bound of trajectories. We characterize exponential stability and asymptotic stability of the system from a new point of view, and provide a simple condition to distinguish them from each other. These results extend existing results that were obtained in some special cases of the system, and are helpful for quantitative analysis and understanding of AGBT of the system. [ABSTRACT FROM AUTHOR]

This paper gives a solution to the problem of verifying stability of networks consisting of integral input-to-state stable (iISS) subsystems. The iISS small-gain theorem developed recently has been restricted to interconnections of two subsystems. For large-scale systems, stability criteria relying only on gain-type information that were previously developed address only input-to-state stable (ISS) subsystems. To address the stability problem involving iISS subsystems interconnected in general structure, this paper shows how to construct Lyapunov functions of the network by means of a sum of nonlinearly rescaled individual Lyapunov functions of subsystems under an appropriate small-gain condition. [ABSTRACT FROM AUTHOR]

This technical note deals with a general class of discrete 2-D possibly nonlinear systems based on the Roesser model. We first motivate the introduction of Lyapunov type definitions of asymptotic and exponential stability. This will allow us to introduce and discuss several particularities that cannot be found in 1-D systems. Once this background has been carefully designed, we develop different Lyapunov theorems in order to check asymptotic and exponential stability of nonlinear 2-D systems. Finally we propose the first converse Lyapunov theorem in the case of exponential stability. [ABSTRACT FROM AUTHOR]

Event-triggered control and self-triggered control have been recently proposed as new implementation paradigms that reduce resource usage for control systems. In self-triggered control, the controller is augmented with the computation of the next time instant at which the feedback control law is to be recomputed. Since these execution instants are obtained as a function of the plant state, we effectively close the loop only when it is required to maintain the desired performance, thereby greatly reducing the resources required for control. In this paper we present a new technique for the computation of the execution instants by exploiting the concept of isochronous manifolds, also introduced in this paper. While our previous results showed how homogeneity can be used to compute the execution instants along some directions in the state space, the concept of isochrony allows us to compute the executions instants along every direction in the state space. Moreover, we also show in this paper how to homogenize smooth control systems thus making our results applicable to any smooth control system. The benefits of the proposed approach with respect to existing techniques are analyzed in two examples. [ABSTRACT FROM PUBLISHER]

This paper considers the problem of distributed control of a platoon of vehicles with nonlinear dynamics. We present distributed receding horizon control algorithms and derive sufficient conditions that guarantee asymptotic stability, leader-follower string stability, and predecessor-follower string stability, following a step speed change in the platoon. Vehicles compute their own control in parallel, and receive communicated position and velocity error trajectories from their immediate predecessor. Leader-follower string stability requires additional communication from the lead car at each update, in the form of a position error trajectory. Predecessor-follower string stability, as we define it, implies leader-follower string stability. Predecessor-follower string stability requires stricter constraints in the local optimal control problems than the leader-follower formulation, but communication from the lead car is required only once at initialization. Provided an initially feasible solution can be found, subsequent feasibility of the algorithms are guaranteed at every update. The theory is generalized for nonlinear decoupled dynamics, and is thus applicable to fleets of planes, robots, or boats, in addition to cars. A simple seven-car simulation examines parametric tradeoffs that affect stability and string stability. Analysis on platoon formation, heterogeneity and size (length) is also considered, resulting in intuitive tradeoffs between lead car and following car control flexibility. [ABSTRACT FROM AUTHOR]

This paper proposes a computational method for the feasibility check and design of discrete abstract models of nonlinear dynamical systems. First, it is shown that a given discrete-time dynamical system can be transformed into a finite automaton by embedding a quantizer into its state equation. Under this setting, a sufficient condition for approximate bisimulation in infinite steps of time between the concrete model and its discrete abstract model is derived. The condition takes the form of a set of linear inequalities and thus can be checked efficiently by a linear programming solver. Finally, the iterative refinement algorithm, which generates a discrete abstract model under a given error specification, is proposed. The algorithm is guaranteed to terminate in finite iterations. [ABSTRACT FROM AUTHOR]

We present algebraic conditions on the trajectory of a dynamical system to approximately describe a certain type of system robustness. The corresponding equations can be used as constraints in a robust optimization procedure to select a set of optimal design parameters for a dynamical system which is subject to fast disturbances. Robustness is ensured by requiring the disturbance parameters to stay sufficiently far away from critical manifolds in the disturbance parameter space, at which the system would lose stability. The closest distance to the critical manifolds is measured along their normal vectors. [ABSTRACT FROM AUTHOR]

We address the issue of minimal time optimal control of fed-batch reactor in presence of complex non monotonic kinetics. Several extremal paths with singular arcs can be locally optimal. We show how a regularization technique can help determining the optimal synthesis, based on a numerical approach. [ABSTRACT FROM AUTHOR]

Sufficiently accurate finite state models, also called symbolic models or discrete abstractions, allow one to apply fully automated methods, originally developed for purely discrete systems, to formally reason about continuous and hybrid systems and to design finite state controllers that provably enforce predefined specifications. We present a novel algorithm to compute such finite state models for nonlinear discrete-time and sampled systems which depends on quantizing the state space using polyhedral cells, embedding these cells into suitable supersets whose attainable sets are convex, and over-approximating attainable sets by intersections of supporting half-spaces. We prove a novel recursive description of these half-spaces and propose an iterative procedure to compute them efficiently. We also provide new sufficient conditions for the convexity of attainable sets which imply the existence of the aforementioned embeddings of quantizer cells. Our method yields highly accurate abstractions and applies to nonlinear systems under mild assumptions, which reduce to sufficient smoothness in the case of sampled systems. Its practicability in the design of discrete controllers for nonlinear continuous plants under state and control constraints is demonstrated by an example. [ABSTRACT FROM AUTHOR]

Lyapunov methods and density functions provide dual characterizations of the solutions of a nonlinear dynamic system. This work exploits the idea of combining both techniques, to yield stability results that are valid for almost all the solutions of the system. Based on the combination of Lyapunov and density functions, analysis methods are proposed for the derivation of almost input-to-state stability, and of almost global stability in nonlinear systems. The techniques are illustrated for an inertial attitude observer, where angular velocity readings are corrupted by non-idealities. [ABSTRACT FROM AUTHOR]

Linear parameter-varying (LPV) systems are usually described in either state-space or input–output form. When analyzing system equivalence between different representations it appears that the time-shifted versions of the scheduling signal (dynamic dependence) need to be taken into account. Therefore, representations used previously to define and specify LPV systems are not equal in terms of dynamics. In order to construct a parametrization-free description of LPV systems that overcomes these difficulties, a behavioral approach is introduced that serves as a basis for specifying system theoretic properties. LPV systems are defined as the collection of trajectories of system variables (like inputs and outputs) and scheduling variables. LPV kernel, input–output, and state-space system representations are introduced with appropriate equivalence transformations. [ABSTRACT FROM AUTHOR]

Stability is arguably one of the core concepts upon which our understanding of dynamical and control systems has been built. The related notion of incremental stability, however, has received much less attention until recently, when it was successfully used as a tool for the analysis and design of intrinsic observers, output regulation of nonlinear systems, frequency estimators, synchronization of coupled identical dynamical systems, symbolic models for nonlinear control systems, and bio-molecular systems. However, most of the existing controller design techniques provide controllers enforcing stability rather than incremental stability. Hence, there is a growing need to extend existing methods or develop new ones for the purpose of designing incrementally stabilizing controllers. In this technical note, we develop a backstepping design approach for incremental stability. [ABSTRACT FROM AUTHOR]

We consider a stochastic optimal control problem of a forward-backward system in which the control variable consists of two components: the continuous control and the impulse control. The domain of the control is assumed to be convex. Necessary optimality conditions of the Pontryagin maximum principle type are obtained for this stochastic optimal control problem. We also give additional conditions, under which the necessary optimality conditions turn out to be sufficient. [ABSTRACT FROM AUTHOR]

This works considers the problem of control of linear systems with input constraints. The notion of control Lyapunov functions (CLFs) is first generalized and relaxed to define a constrained control Lyapunov function (CCLF) and a constructive procedure for CCLF is subsequently presented. [ABSTRACT FROM AUTHOR]

The coordinated path following problem, steering a fleet of unicycles along a path while achieving a desired inter-vehicle formation, is investigated. The path is a paved road in the world or a marked curve on the floor of work area, for which each vehicle can measure the distance to the path and the heading error as well as the curvature of the path segment in its sensing range. Moreover, the vehicles can measure the arc distances of their neighbors that lie in their sensing range. A hybrid control approach is proposed to solve the problem. Depending on the posture with respect to the partly observed path, each vehicle either interacts with one of the others or works at the single-agent level. It is shown that the path following error of each vehicle is eventually reduced to zero and vehicles of every weakly connected component asymptotically converge to formations with equal arc distances. [ABSTRACT FROM AUTHOR]

This technical note develops a unified Lyapunov approach to analysis of self-induced oscillations and stability for systems with piecewise linear elements. For self-induced oscillation within a global or regional attractor, invariant level sets of a piecewise quadratic Lyapunov function are obtained to bound the attractor via linear matrix inequality based optimization. The analysis results for self-induced oscillations are easily adapted to global or regional stability analysis. [ABSTRACT FROM AUTHOR]

This note presents simple controllers for achieving asymptotic trajectory (set-point and time-varying) tracking under the conditions of restricted input in vertical takeoff and landing aircraft. Unlike previous studies that apply nonlinear combination of saturated functions for stabilizing the closed-loop system, the proposed controllers are based on smooth functions that can easily be implemented in real time applications. [ABSTRACT FROM AUTHOR]

A fully actuated system can execute any joint trajectory. However, if a system is under-actuated, not all joint trajectories are attainable. In recent years, the authors have actively pursued novel designs of underactuated robotic arms which are both controllable and feedback linearizable. These robots can perform point-to-point motions in the state space, but potentially can be designed to work with fewer actuators, hence with lower cost. With this same spirit, the technical note investigates the property of differential flatness for a class of planar under-actuated open-chain robots having a specific inertia distribution, but driven by only one or two actuators. This technical note addresses the following theoretical question: What placement of one or two actuators will make an n-DOF planar robot differentially flat if it is designed so f/mat its center of moss always lies at joint 2? [ABSTRACT FROM AUTHOR]

The flatness-based design of a feedforward tracking control is considered for the solution of the trajectory planning problem for a boundary controlled diffusion-convection-reaction system with spatially and temporally varying parameters defined on a 1 ⩽ m-dimensional parallelepipedon with the nonlinear input being restricted to a (m 1)-dimensional hyperplane. For this, an implicit state and input parametrization in terms of a basic output is determined via a Volterra-type integral equation with operator kernel. By recursively computing successive series coefficients, a series solution of the integral equation is obtained, whose absolute and uniform convergence is verified by restricting the system parameters and the basic output to a certain but broad Gevrey class. Hence, prescribing an admissible desired trajectory for the basic output directly yields the feedforward control by evaluating the input parametrization. This results in a systematic procedure for trajectory planning and feedforward control design for boundary controlled parabolic distributed-parameter systems defined on higher-dimensional domains. [ABSTRACT FROM AUTHOR]

We consider planar bilinear control systems with measurable controls. We show that any point in the reachable set can be reached by a "nice" control, specifically, a control that is a concatenation of a bang arc with either 1) a bang-bang control that is periodic after the third switch; or 2) a piecewise constant control with no more than two discontinuities. Under the additional assumption that the bilinear system is positive (or invariant for any proper cone), we show that the reachable set is spanned by a concatenation of a bang arc with either 1) a bang-bang control with no more than two discontinuities; or 2) a piecewise constant control with no more than two discontinuities. In particular, any point in the reachable set can be reached using a piecewise-constant control with no more than three discontinuities. Several known results on the stability of planar linear switched systems under arbitrary switching follow as corollaries of our result. We demonstrate this with an example. [ABSTRACT FROM AUTHOR]

We present a global solution for an optimal feedback controller of the underactuated Heisenberg system or nonholonomic integrator. Employing a recently developed technique based on generating functions appearing in the Hamilton-Jacobi theory, we circumvent a singularity caused by underactuation to develop a nonlinear optimal feedback control in an implicitly analytical form. The systematic procedure to deal with underactuation indicates that generating functions should be effective tools for solving general underactuated optimal control problems. [ABSTRACT FROM AUTHOR]

The problem of constructing globally convergent, reduced-order observers for general nonlinear systems is addressed. It is shown that an asymptotic estimate of the unknown states can be obtained by rendering attractive an appropriately selected (invariant) manifold in the extended state space. Current results on nonlinear observer design require that the nonlinearities appearing in the system equations are either linear functions of the unmeasured states or monotonic functions of a linear combination of the states. In this paper we relax these two assumptions by allowing for a wider class of nonlinearities to appear in the system equations. The proposed approach is applied on several examples including a perspective vision system and a general two-degrees-of-freedom mechanical system. [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]

In this paper, we consider a problem termed "path following". This differs from the common problem of reference tracking, in that here we can adjust the speed at which we traverse the reference trajectory. We are interested in ascertaining the degree to which we can track a given trajectory, and in characterizing the class of paths for which we can generate an appropriate temporal specification so that the path can be tracked arbitrarily well in an L2 sense. We give various bounds on the achievable performance, as well as tight results in special cases. In addition, we give a numerical procedure based on convex optimization for computing the achievable performance. The results demonstrate that there are situations where arbitrarily good L2 performance may be achieved even though the origin is not in the convex hull of the positive limit set of the path to be followed. [ABSTRACT FROM AUTHOR]