Showing papers in "IEEE Transactions on Automatic Control in 1998"
TL;DR: Bendixson's theorem is extended to the case of Lipschitz continuous vector fields, allowing limit cycle analysis of a class of "continuous switched" systems.
Abstract: We introduce some analysis tools for switched and hybrid systems. We first present work on stability analysis. We introduce multiple Lyapunov functions as a tool for analyzing Lyapunov stability and use iterated function systems theory as a tool for Lagrange stability. We also discuss the case where the switched systems are indexed by an arbitrary compact set. Finally, we extend Bendixson's theorem to the case of Lipschitz continuous vector fields, allowing limit cycle analysis of a class of "continuous switched" systems.
3,289 citations
TL;DR: A class of bounded continuous time-invariant finite-time stabilizing feedback laws is given for the double integrator because Lyapunov theory is used to prove finite- time convergence.
Abstract: A class of bounded continuous time-invariant finite-time stabilizing feedback laws is given for the double integrator. Lyapunov theory is used to prove finite-time convergence. For the rotational double integrator, these controllers are modified to obtain finite-time-stabilizing feedback that avoid "unwinding".
1,389 citations
TL;DR: This work introduces a mathematical model of hybrid systems as interacting collections of dynamical systems, evolving on continuous-variable state spaces and subject to continuous controls and discrete transitions, and develops a theory for synthesizing hybrid controllers for hybrid plants in all optimal control framework.
Abstract: We propose a very general framework that systematizes the notion of a hybrid system, combining differential equations and automata, governed by a hybrid controller that issues continuous-variable commands and makes logical decisions. We first identify the phenomena that arise in real-world hybrid systems. Then, we introduce a mathematical model of hybrid systems as interacting collections of dynamical systems, evolving on continuous-variable state spaces and subject to continuous controls and discrete transitions. The model captures the identified phenomena, subsumes previous models, yet retains enough structure to pose and solve meaningful control problems. We develop a theory for synthesizing hybrid controllers for hybrid plants in all optimal control framework. In particular, we demonstrate the existence of optimal (relaxed) and near-optimal (precise) controls and derive "generalized quasi-variational inequalities" that the associated value function satisfies. We summarize algorithms for solving these inequalities based on a generalized Bellman equation, impulse control, and linear programming.
1,363 citations
TL;DR: The search for a piecewise quadratic Lyapunov function is formulated as a convex optimization problem in terms of linear matrix inequalities and the relation to frequency domain methods such as the circle and Popov criteria is explained.
Abstract: This paper presents a computational approach to stability analysis of nonlinear and hybrid systems. The search for a piecewise quadratic Lyapunov function is formulated as a convex optimization problem in terms of linear matrix inequalities. The relation to frequency domain methods such as the circle and Popov criteria is explained. Several examples are included to demonstrate the flexibility and power of the approach.
1,186 citations
TL;DR: A new solution to the problem of chattering elimination in variable structure control (VSC) is presented, inspired by the classical bang-bang optimal control strategy, and extended to the case of nonlinear systems with uncertainties of more general types.
Abstract: Relying on the possibility of generating a second-order sliding motion by using, as control, the first derivative of the control signal instead of the actual control, a new solution to the problem of chattering elimination in variable structure control (VSC) is presented. Such a solution, inspired by the classical bang-bang optimal control strategy, is first depicted and expressed in terms of a control algorithm by introducing a suitable auxiliary problem involving a second-order uncertain system with unavailable velocity. Then, the applicability of the algorithm is extended, via suitable modifications, to the case of nonlinear systems with uncertainties of more general types. The proposed algorithm does not require the use of observers and differential inequalities and can be applied in practice by exploiting such commercial components as peak detectors or other approximated methods to evaluate the change of the sign of the derivative of the quantity accounting for the distance to the sliding manifold.
992 citations
TL;DR: The method models the aircraft and the manoeuvre as a hybrid control system and calculates the maximal set of safe initial conditions for each aircraft so that separation is assured in the presence of uncertainties in the actions of the other aircraft.
Abstract: Air traffic management (ATM) of the future allows for the possibility of free flight, in which aircraft choose their own optimal routes, altitudes, and velocities. The safe resolution of trajectory conflicts between aircraft is necessary to the success of such a distributed control system. In this paper, we present a method to synthesize provably safe conflict resolution manoeuvres. The method models the aircraft and the manoeuvre as a hybrid control system and calculates the maximal set of safe initial conditions for each aircraft so that separation is assured in the presence of uncertainties in the actions of the other aircraft. Examples of manoeuvres using both speed and heading changes are worked out in detail.
957 citations
TL;DR: The control schemes the authors discuss introduce the notion that feedback is present in the receding-horizon implementation of the control, which leads to improved performance, compared to standard model predictive control, and resolves the feasibility difficulties that arise with the min-max techniques.
Abstract: Min-max feedback formulations of model predictive control are discussed, both in the fixed and variable horizon contexts. The control schemes the authors discuss introduce, in the control optimization, the notion that feedback is present in the receding-horizon implementation of the control. This leads to improved performance, compared to standard model predictive control, and resolves the feasibility difficulties that arise with the min-max techniques that are documented in the literature. The stabilizing properties of the methods are discussed as well as some practical implementation details.
897 citations
TL;DR: Some fundamental insights into observer design for the class of Lipschitz nonlinear systems are presented and a systematic computational algorithm is presented for obtaining the observer gain matrix so as to achieve the objective of asymptotic stability.
Abstract: This paper presents some fundamental insights into observer design for the class of Lipschitz nonlinear systems. The stability of the nonlinear observer for such systems is not determined purely by the eigenvalues of the linear stability matrix. The correct necessary and sufficient conditions on the stability matrix that ensure asymptotic stability of the observer are presented. These conditions are then reformulated to obtain a sufficient condition for stability in terms of the eigenvalues and the eigenvectors of the linear stability matrix. The eigenvalues have to be located sufficiently far out into the left half-plane, and the eigenvectors also have to be sufficiently well-conditioned for ensuring asymptotic stability. Based on these results, a systematic computational algorithm is then presented for obtaining the observer gain matrix so as to achieve the objective of asymptotic stability.
859 citations
TL;DR: A model for hybrid dynamical systems is formulated which covers a very large class of systems and which is suitable for the qualitative analysis of such systems and several types of (Lyapunov-like) stability concepts for an invariant set are defined.
Abstract: We first formulate a model for hybrid dynamical systems which covers a very large class of systems and which is suitable for the qualitative analysis of such systems. Next, we introduce the notion of an invariant set for hybrid dynamical systems and we define several types of (Lyapunov-like) stability concepts for an invariant set. We then establish sufficient conditions for uniform stability, uniform asymptotic stability, exponential stability, and instability of an invariant set of hybrid dynamical systems. Under some mild additional assumptions, we also establish necessary conditions for some of the above stability types (converse theorems). In addition to the above, we also establish sufficient conditions for the uniform boundedness of the motions of hybrid dynamical systems (Lagrange stability). To demonstrate the applicability of the developed theory, we present specific examples of hybrid dynamical systems and we conduct a stability analysis of some of these examples.
821 citations
TL;DR: In this paper, a solution algorithm is presented, which requires solving a finite number of finite-dimensional positive definite quadratic programs and is guaranteed to terminate in finite time with a computational cost with a reasonable upper bound compared to the minimal cost for computing the optimal solution.
Abstract: The paper is a contribution to the theory of the infinite-horizon linear quadratic regulator (LQR) problem subject to inequality constraints on the inputs and states, extending an approach first proposed by Sznaier and Damborg (1987). A solution algorithm is presented, which requires solving a finite number of finite-dimensional positive definite quadratic programs. The constrained LQR outlined does not feature the undesirable mismatch between open-loop and closed-loop nominal system trajectories, which is present in the other popular forms of model predictive control (MPC) that can be implemented with a finite quadratic programming algorithm. The constrained LQR is shown to be both optimal and stabilizing. The solution algorithm is guaranteed to terminate in finite time with a computational cost that has a reasonable upper bound compared to the minimal cost for computing the optimal solution. Inherent to the approach is the removal of a tuning parameter, the control horizon, which is present in other MPC approaches and for which no reliable tuning guidelines are available. Two examples are presented that compare constrained LQR and two other popular forms of MPC. The examples demonstrate that constrained LQR achieves significantly better performance than the other forms of MPC on some plants, and the computational cost is not prohibitive for online implementation.
568 citations
TL;DR: A systematic procedure is developed for designing global adaptive control of a class of nonlinear systems that possesses a triangular structure and can be of arbitrary dynamic order.
Abstract: Without a priori knowledge of the signs of the parameters called control directions (since they represent effectively the direction of motion under any control), a systematic procedure is developed for designing global adaptive control of a class of nonlinear systems. The class of systems possesses a triangular structure and can be of arbitrary dynamic order. No growth restrictions are imposed.
TL;DR: The approach is based on conceptual tools of predictive control and consists of adding to a primal compensated nonlinear system a reference governor which online handles the reference to be tracked, taking into account the current value of the state in order to satisfy the prescribed constraints.
Abstract: This paper addresses the problem of satisfying pointwise-in-time input and/or state hard constraints in nonlinear control systems. The approach is based on conceptual tools of predictive control and consists of adding to a primal compensated nonlinear system a reference governor. This is a discrete-time device which online handles the reference to be tracked, taking into account the current value of the state in order to satisfy the prescribed constraints. The resulting hybrid system is proved to fulfil the constraints as well as stability and tracking requirements.
TL;DR: The result is used to show that input-to-state stabilizability for nonlinear finite-dimensional control systems is robust, in an appropriate sense, to small time delays at the input.
Abstract: A Razumikhin-type theorem that guarantees input-to-state stability for functional differential equations with disturbances is established using the nonlinear small-gain theorem. The result is used to show that input-to-state stabilizability for nonlinear finite-dimensional control systems is robust, in an appropriate sense, to small time delays at the input. Also, relaxed Razumikhin-type conditions guaranteeing global asymptotic stability for differential difference equations are given.
TL;DR: A new method for achieving robust stabilization is presented for a class of uncertain time-delay systems via linear memoryless state feedback control through linear matrix inequalities.
Abstract: This paper deals with the problem of robust stabilization for uncertain systems with multiple state delays. The parameter uncertainties are time-varying and unknown but are norm-bounded, and the delays are time-varying. A new method for achieving robust stabilization is presented for a class of uncertain time-delay systems via linear memoryless state feedback control. The results depend on the size of the delays and are given in terms of several linear matrix inequalities.
TL;DR: This work discusses the design of safe and efficient hybrid controllers for regulation of vehicles on an AHS and uses game theoretic techniques to deal with the multiagent and multiobjective nature of the problem.
Abstract: The objective of an automated highway system (AHS) is to increase the safety and throughput of the existing highway infrastructure by introducing traffic automation. AHS is an example of a large scale, multiagent complex dynamical system and is ideally suited for a hierarchical hybrid controller. We discuss the design of safe and efficient hybrid controllers for regulation of vehicles on an AHS. We use game theoretic techniques to deal with the multiagent and multiobjective nature of the problem. The result is a hybrid controller that by design guarantees safety, without the need for further verification. The calculations also provide an upper bound on the performance that can be expected in terms of throughput at various levels of centralization.
TL;DR: This paper establishes passivity for the system which describes the attitude motion of a rigid body in terms of minimal three-dimensional kinematic parameters and shows that linear asymptotically stabilizing controllers and control laws without angular velocity measurements follow naturally from these passivity properties.
Abstract: In this paper we establish passivity for the system which describes the attitude motion of a rigid body in terms of minimal three-dimensional kinematic parameters. In particular, we show that linear asymptotically stabilizing controllers and control laws without angular velocity measurements follow naturally from these passivity properties. The results of this paper extend similar results for the case of the (nonminimal) Euler parameters.
TL;DR: The authors present an iterative procedure for determining the supremal controllable, observable, and diagnosable sublanguage of the legal language and for obtaining the supervisor that synthesizes this language and provide both a controller that ensures diagnosability of the closed-loop system and a diagnoser for online failure diagnosis.
Abstract: The need for accurate and timely diagnosis of system failures and the advantages of automated diagnostic systems are well appreciated. However, diagnosability considerations are often not explicitly taken into account in the system design. In particular, design of the controller and that of the diagnostic subsystem are decoupled, and this may significantly affect the diagnosability properties of a system. The authors present an integrated approach to control and diagnosis. More specifically, they present an approach for the design of diagnosable systems by appropriate design of the system controller. This problem, which they refer to as the active diagnosis problem, is studied in the framework of discrete-event systems (DESs); it is based on prior and new results on the theory of diagnosis for DESs and on existing results in supervisory control under partial observations. They formulate the active diagnosis problem as a supervisory control problem where the legal language is an "appropriate" regular sublanguage of the regular language generated by the system. They present an iterative procedure for determining the supremal controllable, observable, and diagnosable sublanguage of the legal language and for obtaining the supervisor that synthesizes this language. This procedure provides both a controller that ensures diagnosability of the closed-loop system and a diagnoser for online failure diagnosis. The procedure can be implemented using finite-state machines and is guaranteed to converge in a finite number of iterations. The authors illustrate their approach using a simple pump-valve system.
TL;DR: It has been shown that the above robust H/sub /spl infin//-filtering problem can be solved in terms of differential Riccati inequalities with finite discrete jumps.
Abstract: The paper is concerned with the problem of robust H/sub /spl infin// filtering for a class of systems with parametric uncertainties and unknown time delays under sampled measurements. The parameter uncertainties considered are real time-varying and norm-bounded, appearing in the state equation. An approach has been proposed for the designing of H/sub /spl infin// filters, using sampled measurements, which would guarantee a prescribed H/sub /spl infin// performance in the continuous-time context, irrespective of the parameter uncertainties and unknown time delays. Both cases of finite and infinite horizon filtering are studied. It has been shown that the above robust H/sub /spl infin//-filtering problem can be solved in terms of differential Riccati inequalities with finite discrete jumps.
TL;DR: A related linear dynamic system (RLDS) approximation to the nonlinear system (NLS) is defined, and it is shown that the differences between the NLS and the RLDS can be modeled as stochastic variables with known properties.
Abstract: This paper studies the asymptotic behavior of nonparametric and parametric frequency domain identification methods to model linear dynamic systems in the presence of nonlinear distortions under some general conditions for random multisine excitations. In the first part, a related linear dynamic system (RLDS) approximation to the nonlinear system (NLS) is defined, and it is shown that the differences between the NLS and the RLDS can be modeled as stochastic variables with known properties. In the second part a parametric model for the RLDS is identified. Convergence in probability of this model to the RLDS is proven. A function of dependency is defined to detect and separate the presence of unmodeled dynamics and nonlinear distortions and to bound the bias error on the transfer function estimate.
TL;DR: It is shown that input-to-state stabilizability (as defined by Sontag,1989, 1995) is both necessary and sufficient for the solvability of a Hamilton-Jacobi-Isaacs equation associated with a meaningful differential game problem similar to, but more general than, the "nonlinear H/sub /spl infin//" problem.
Abstract: We show that input-to-state stabilizability (as defined by Sontag,1989, 1995) is both necessary and sufficient for the solvability of a Hamilton-Jacobi-Isaacs equation associated with a meaningful differential game problem similar to, but more general than, the "nonlinear H/sub /spl infin//" problem. The significance of the result stems from the fact that constructive solutions to the input-to-state stabilization problem are available (presented in the paper) and that, as shown here, inverse optimal controllers possess margins on input-to-state stability against a certain class of input unmodeled dynamics. Rather than completion of squares, the main tools in our analysis are Legendre-Fenchel transformations and the general form of Young's inequality.
TL;DR: It is shown that the discontinuity plane for sliding mode control may be found in an explicit form using Ackermann's formula.
Abstract: The sliding mode control methods are developed to design systems which have the desired dynamic behavior and are robust with respect to perturbations. It is shown that the discontinuity plane for sliding mode control may be found in an explicit form using Ackermann's formula. Two design procedures are derived. First, static controllers are designed to enforce sliding modes with the desired dynamic properties after a finite-time interval. Then, dynamic controllers are designed that exhibit the desired dynamic properties during the entire control process.
TL;DR: The authors develop a systematic procedure for obtaining robust adaptive controllers that achieve asymptotic tracking and disturbance attenuation for a class of nonlinear systems which are described in the parametric strict-feedback form and are subject to additional exogenous disturbance inputs.
Abstract: The authors develop a systematic procedure for obtaining robust adaptive controllers that achieve asymptotic tracking and disturbance attenuation for a class of nonlinear systems which are described in the parametric strict-feedback form and are subject to additional exogenous disturbance inputs. Their approach to adaptive control is performance-based, where the objective for the controller design is not only to find an adaptive controller, but also to construct an appropriate cost functional, compatible with desired asymptotic tracking and disturbance attenuation specifications, with respect to which the adaptive controller is "worst case optimal". Three main issues of the paper are: the backstepping methodology, worst case identification schemes, and singular perturbations analysis. Closed-form expressions have been obtained for an adaptive controller and the corresponding value function. A numerical example involving a third-order system is given.
TL;DR: In this paper, a complementarity framework is described for the modeling of certain classes of mixed continuous/discrete dynamical systems, and the main theoretical results in this paper are concerned with uniqueness of smooth continuations; the solution of this problem requires the construction of a map from the continuous state to the discrete state.
Abstract: A complementarity framework is described for the modeling of certain classes of mixed continuous/discrete dynamical systems. The use of such a framework is well known for mechanical systems with inequality constraints, but we give a more general formulation which also applies, for instance, to switching control systems. The main theoretical results in the paper are concerned with uniqueness of smooth continuations; the solution of this problem requires the construction of a map from the continuous state to the discrete state. A crucial technical tool is the so-called linear complementarity problem from mathematical programming, and we introduce various generalizations of this problem.
TL;DR: This paper considers the use of the simultaneous perturbation stochastic approximation algorithm which requires only system measurements and it is shown that this algorithm can greatly enhance the efficiency over more standard stoChastic approximation algorithms based on finite-difference gradient approximations.
Abstract: Consider the problem of developing a controller for general (nonlinear and stochastic) systems where the equations governing the system are unknown. Using discrete-time measurement, this paper presents an approach for estimating a controller without building or assuming a model for the system. Such an approach has potential advantages in accommodating complex systems with possibly time-varying dynamics. The controller is constructed through use of a function approximator, such as a neural network or polynomial. This paper considers the use of the simultaneous perturbation stochastic approximation algorithm which requires only system measurements. A convergence result for stochastic approximation algorithms with time-varying objective functions and feedback is established. It is shown that this algorithm can greatly enhance the efficiency over more standard stochastic approximation algorithms based on finite-difference gradient approximations.
TL;DR: A receding horizon control scheme for nonlinear time-varying systems is proposed which is based on a finite-horizon optimization problem with a terminal state penalty and ensures exponential stability of the equilibrium.
Abstract: A receding horizon control scheme for nonlinear time-varying systems is proposed which is based on a finite-horizon optimization problem with a terminal state penalty. The penalty is equal to the cost that would be incurred over an infinite horizon by applying a (locally stabilizing) linear control law to the nonlinear system. Assuming only stabilizability of the linearized system around the desired equilibrium, the new scheme ensures exponential stability of the equilibrium. As the length of the optimization horizon goes from zero to infinity, the domain of attraction moves from the basin of attraction of the linear controller toward the basin of attraction of the infinite-horizon nonlinear controller. Stability robustness in the face of system perturbations is also established.
TL;DR: It is shown that the well-posedness conditions provided can be used to give a less conservative, yet computable bound on the real structured singular value, as illustrated by numerical examples.
Abstract: This paper establishes a framework for robust stability analysis of linear time-invariant uncertain systems, The uncertainty is assumed to belong to an arbitrary subset of complex matrices. The concept used here is well-posedness of feedback systems, leading to necessary and sufficient conditions for robust stability. Based on this concept, some insights into exact robust stability conditions are given, In particular, frequency domain and state-space conditions for well-posedness are provided in terms of Hermitian-form inequalities. It is shown that these inequalities can be interpreted as small-gain conditions with a generalized class of scalings given by linear fractional transformations (LFT). Using the LFT-scaled small-gain condition in the state-space setting, the "duality" is established between the H/sub /spl infin// norm condition with frequency-dependent scalings and the parameter-dependent Lyapunov condition. Connections to the existing results, including the structured singular value and the integral quadratic constraints, are also discussed. Finally, we show that our well-posedness conditions can be used to give a less conservative, yet computable bound on the real structured singular value. This result is illustrated by numerical examples.
TL;DR: A general framework for model-based fault detection and diagnosis of a class of incipient faults is developed and an automated fault diagnosis architecture using nonlinear online approximators with an adaptation scheme is designed and analyzed.
Abstract: Detection of incipient (slowly developing) faults is crucial in automated maintenance problems where early detection of worn equipment is required. In this paper, a general framework for model-based fault detection and diagnosis of a class of incipient faults is developed. The changes in the system dynamics due to the fault are modeled as nonlinear functions of the state and input variables, while the time profile of the failure is assumed to be exponentially developing. An automated fault diagnosis architecture using nonlinear online approximators with an adaptation scheme is designed and analyzed. A simulation example of a simple nonlinear mass-spring system is used to illustrate the results.
TL;DR: Using Lyapunov's direct method and LaSalle's invariance principle, a class of robot regulators consisting of a linear proportional-derivative feedback plus an integral action of a nonlinear function of position errors are characterized.
Abstract: Deals with the position control of robot manipulators. Proposed is a simple class of robot regulators consisting of a linear proportional-derivative (PD) feedback plus an integral action of a nonlinear function of position errors. By using Lyapunov's direct method and LaSalle's invariance principle, the authors characterize a class of such nonlinear functions, and they provide explicit conditions on the regulator gains to ensure global asymptotic stability. These regulators offer an attractive alternative to global regulation compared with the well-known partially model-based PD control with gravity compensation and PD control with desired gravity compensation.
TL;DR: This paper provides a new sufficient condition for the solvability of the above stabilizing output feedback control problem and discusses a simple procedure for the determination of a stabilizingoutput feedback gain assuring good suboptimal performance with respect to a given quadratic index.
Abstract: The main objective of this paper is to solve the following stabilizing output feedback control problem: given matrices (A; B/sub 2/; C/sub 2/) with appropriate dimensions, find (if one exists) a static output feedback gain L such that the closed-loop matrix A-B/sub 2/LC/sub 2/ is asymptotically stable. It is known that the existence of L is equivalent to the existence of a positive definite matrix belonging to a convex set such that its inverse belongs to another convex set. Conditions are provided for the convergence of an algorithm which decomposes the determination of the aforementioned matrix in a sequence of convex programs. Hence, this paper provides a new sufficient (but not necessary) condition for the solvability of the above stabilizing output feedback control problem. As a natural extension, we also discuss a simple procedure for the determination of a stabilizing output feedback gain assuring good suboptimal performance with respect to a given quadratic index. Some examples borrowed from the literature are solved to illustrate the theoretical results.