Abstract: An efficient Newton-type scheme for the approximate on-line solution of optimization problems as they occur in optimal feedback control is presented. The scheme allows a fast reaction to disturbances by delivering approximations of the exact optimal feedback control which are iteratively refined during the runtime of the controlled process. The contractivity of this real-time iteration scheme is proven, and a bound on the loss of optimality---compared with the theoretical optimal solution---is given. The robustness and excellent real-time performance of the method is demonstrated in a numerical experiment, the control of an unstable system, namely, an airborne kite that shall fly loops.

Abstract: In this paper, we introduce an approach for the efficient solution of motion-planning problems for time-invariant dynamical control systems with symmetries, such as mobile robots and autonomous vehicles, under a variety of differential and algebraic constraints on the state and on the control inputs. Motion plans are described as the concatenation of a number of well-defined motion primitives, selected from a finite library. Rules for the concatenation of primitives are given in the form of a regular language, defined through a finite-state machine called a Maneuver Automaton. We analyze the reachability properties of the language, and present algorithms for the solution of a class of motion-planning problems. In particular, it is shown that the solution of steering problems for nonlinear dynamical systems with symmetries and invariant constraints can be reduced to the solution of a sequence of kinematic inversion problems. A detailed example of the application of the proposed approach to motion planning for a small aerobatic helicopter is presented.

Abstract: The attitude tracking control problem of a rigid spacecraft with external disturbances and an uncertain inertia matrix is addressed using the adaptive control method. The adaptive control laws proposed in this paper are optimal with respect to a family of cost functionals. This is achieved by the inverse optimality approach, without solving the associated Hamilton-Jacobi-Isaacs partial differential (HJIPD) equation directly. The design of the optimal adaptive controllers is separated into two stages by means of integrator backstepping, and a control Lyapunov argument is constructed to show that the inverse optimal adaptive controllers achieve H/sub /spl infin// disturbance attenuation with respect to external disturbances and global asymptotic convergence of tracking errors to zero for disturbances with bounded energy. The convergence of adaptive parameters is also analyzed in terms of invariant manifold. Numerical simulations illustrate the performance of the proposed control algorithms.

Abstract: In this note, a new class of hybrid impulsive and switching models is introduced and their asymptotic stability properties are investigated. Using switched Lyapunov functions, some new general criteria for exponential stability and asymptotic stability with arbitrary and conditioned impulsive switching are established. In addition, a new hybrid impulsive and switching control strategy for nonlinear systems is developed. A typical example, the unified chaotic system, is given to illustrate the theoretical results.

Abstract: This note is concerned with the problem of stabilizing a nonlinear continuous-time system by using sampled encoded measurements of the state. We demonstrate that global asymptotic stabilization is possible if a suitable relationship holds between the number of values taken by the encoder, the sampling period, and a system parameter, provided that a feedback law achieving input-to-state stability with respect to measurement errors can be found. The issue of relaxing the latter condition is also discussed.

Abstract: In this work, a predictive control framework is proposed for the constrained stabilization of switched nonlinear systems that transit between their constituent modes at prescribed switching times. The main idea is to design a Lyapunov-based predictive controller for each constituent mode in which the switched system operates and incorporate constraints in the predictive controller design which upon satisfaction ensure that the prescribed transitions between the modes occur in a way that guarantees stability of the switched closed-loop system. This is achieved as follows: For each constituent mode, a Lyapunov-based model predictive controller (MPC) is designed, and an analytic bounded controller, using the same Lyapunov function, is used to explicitly characterize a set of initial conditions for which the MPC, irrespective of the controller parameters, is guaranteed to be feasible, and hence stabilizing. Then, constraints are incorporated in the MPC design which, upon satisfaction, ensure that: 1) the state of the closed-loop system, at the time of the transition, resides in the stability region of the mode that the system is switched into, and 2) the Lyapunov function for each mode is nonincreasing wherever the mode is reactivated, thereby guaranteeing stability. The proposed control method is demonstrated through application to a chemical process example.

Abstract: We present a constructive tool for generation and orbital stabilization of periodic solutions for underactuated nonlinear systems. Our method can be applied to any mechanical system with a number of independent actuators smaller than the number of degrees of freedom by one. The synthesized feedback control law is nonlinear and time-dependent. It is derived from a feedback structure that explicitly uses the general or full integral of the systems zero dynamics. The control law generates a periodic solution and makes it exponentially orbitally stable.

Abstract: In this note, we investigate the relationship between nonlinear control and passive walking in bipedal locomotion for the general case of an n degree-of-freedom biped in three dimensional space. We introduce the notion of controlled symmetry to capture the effect of the control input on the invariance of the system Lagrangian under group action. We then show the existence of a controlled symmetry for general bipeds under the action of SO(3) taking into account not only the kinetic energy but also the potential energy and impact dynamics. We use this result to show the existence of a nonlinear control law that reproduces so-called passive gaits independent of the particular ground slope. Our contribution in this note is two-fold. First, our result contains the first rigorous proof of the existence of so-called passivity mimicking control laws that explicitly accounts for the impact dynamics. Second, whereas previous papers have studied only planar bipeds with and without knees, our result is completely general. Our results can be viewed as direct extensions of several previous results, such as passivity-based control, virtual gravity, and virtual passive dynamic walking from the planar case to general n-degrees-of-freedom (DOF) robots in three-dimensional space.

Abstract: We present stability results for unconstrained discrete-time nonlinear systems controlled using finite-horizon model predictive control (MPC) algorithms that do not require the terminal cost to be a local control Lyapunov function. The two key assumptions we make are that the value function is bounded by a K/sub /spl infin// function of a state measure related to the distance of the state to the target set and that this measure is detectable from the stage cost. We show that these assumptions are sufficient to guarantee closed-loop asymptotic stability that is semiglobal and practical in the horizon length and robust to small perturbations. If the assumptions hold with linear (or locally linear) K/sub /spl infin// functions, then the stability will be global (or semiglobal) for long enough horizon lengths. In the global case, we give an explicit formula for a sufficiently long horizon length. We relate the upper bound assumption to exponential and asymptotic controllability. Using terminal and stage costs that are controllable to zero with respect to a state measure, we can guarantee the required upper bound, but we also require that the state measure be detectable from the stage cost to ensure stability. While such costs and state measures may not be easy to construct in general, we explore a class of systems, called homogeneous systems, for which it is straightforward to choose them. In fact, we show for homogeneous systems that the associated K/sub /spl infin// functions are linear, thereby guaranteeing global asymptotic stability. We discuss two examples found elsewhere in the MPC literature, including the discrete-time nonholonomic integrator, to demonstrate our methods. For these systems, we give a new result: They can be globally asymptotically stabilized by a finite-horizon MPC algorithm that has guaranteed robustness. We also show that stable linear systems with control constraints can be globally exponentially stabilized using finite-horizon MPC without requiring the terminal cost to be a global control Lyapunov function.

Abstract: Nonlinear control is an effective method for making two identical chaotic systems or two different chaotic systems be synchronized. However, this method assumes that the Lyapunov function of error dynamic (e) of synchronization is always formed as V (e) = 1/2eTe. In this paper, modification based on Lyapunov stability theory to design a controller is proposed in order to overcome this limitation. The method has been applied successfully to make two identical new systems and two different chaotic systems (new system and Lorenz system) globally asymptotically synchronized. Since the Lyapunov exponents are not required for the calculation, this method is effective and convenient to synchronize two identical systems and two different chaotic systems. Numerical simulations are also given to validate the proposed synchronization approach.

Abstract: A direct adaptive state-feedback controller is proposed for highly nonlinear systems. We consider uncertain or ill-defined nonaffine nonlinear systems and employ a neural network (NN) with flexible structure, i.e., an online variation of the number of neurons. The NN approximates and adaptively cancels an unknown plant nonlinearity. A control law and adaptive laws for the weights in the hidden layer and output layer of the NN are established so that the whole closed-loop system is stable in the sense of Lyapunov. Moreover, the tracking error is guaranteed to be uniformly asymptotically stable (UAS) rather than uniformly ultimately bounded (UUB) with the aid of an additional robustifying control term. The proposed control algorithm is relatively simple and requires no restrictive conditions on the design constants for the stability. The efficiency of the proposed scheme is shown through the simulation of a simple nonaffine nonlinear system.

Abstract: In this paper, a novel systematic design method, namely homogeneous domination approach, is developed for the global output feedback stabilization of nonlinear systems. The nonlinearities of the systems considered in this paper are neither linearly growing nor Lipschitz in immeasurable states, which make the most of existing methods inapplicable to solve the problem. By utilizing the homogeneous domination approach, a global output feedback stabilizer is explicitly constructed in two steps: i) we first design for the nominal linear system a unique homogeneous output feedback controller whose construction is genuinely nonlinear, rather than linear as used in the literature; ii) then we scale the homogeneous observer and controller with an appropriate choice of gain to render the nonlinear system globally asymptotically stable. The homogeneous domination approach not only enables us to completely remove the linear growth condition, which has been the common assumption for global output feedback stabilization, but also provides us a new perspective to deal with the output feedback control problem for nonlinear systems.

Abstract: In this note, the problem of robust output feedback control for a class of nonlinear time delayed systems is considered. The systems considered are in strict-feedback form. State observer is first designed, then based on the observed states the controller is designed via backstepping method. Both the designed observer and controller are independent of the time delays. Based on Lyapunov stability theory, we prove that the constructed controller can render the closed-loop system asymptotically stable. Simulation results further verify the effectiveness of the proposed approach.

Abstract: In this paper, an observer-based direct adaptive fuzzy-neural control scheme is presented for nonaffine nonlinear systems in the presence of unknown structure of nonlinearities. A direct adaptive fuzzy-neural controller and a class of generalized nonlinear systems, which are called nonaffine nonlinear systems, are instead of the indirect one and affine nonlinear systems given by Leu et al. By using implicit function theorem and Taylor series expansion, the observer-based control law and the weight update law of the fuzzy-neural controller are derived for the nonaffine nonlinear systems. Based on strictly-positive-real (SPR) Lyapunov theory, the stability of the closed-loop system can be verified. Moreover, the overall adaptive scheme guarantees that all signals involved are bounded and the output of the closed-loop system will asymptotically track the desired output trajectory. To demonstrate the effectiveness of the proposed method, simulation results are illustrated in this paper.

Abstract: In this paper, an analysis of the energy transfer properties of non-linear systems in the frequency domain is studied based on a new concept known as non-linear output frequency response functions (NOFRFs) The new concept allows the analysis to be implemented in a manner similar to the analysis of linear systems in the frequency domain, and provides great insight into the mechanisms which dominate the non-linear behaviour The new analysis is also helpful for the design of non-linear systems in the frequency domain

Abstract: These notes gather the material presented at a minicourse of 6 hrs at the conference mentioned above. The material we present here is not original and has been published in different papers. The adequate references are provided in the Bibliography. The general topic of study is Lyapunov stability of nonlinear time-varying cascaded systems. Roughly speaking these are systems in “open loop” as illustrated in the figure below.

Abstract: This paper considers probabilistic robust control of nonlinear uncertain systems. A combination of stochastic robustness and dynamic inversion is proposed for general systems that have a feedback-linearizable nominal system. In this paper, the stochastic robust nonlinear control approach is applied to a highly nonlinear complex aircraft model, the high-incidence research model (HIRM). The model addresses a high-angle-of-attack enhanced manual control problem. The aim of the flight control system is to give good handling qualities across the specified flight envelope without the use of gain scheduling and also to provide robustness to modeling uncertainties. The proposed stochastic robust nonlinear control explores the direct design of nonlinear flight control logic. Therefore, the final design accounts for all significant nonlinearities in the aircraft's high-fidelity simulation model. The controller parameters are designed to minimize the probability of violating design specifications, which provides the design with good robustness in stability and performance subject to modeling uncertainties. The present design compares favorably with earlier controllers that were generated for a benchmark design competition.

Abstract: This note presents a new method to design adaptive feedback controller for nonlinear time-delay systems. First, by using the LaSalle-Razumikhin theorem, a sufficient condition is derived that ensures the convergence of a part of the solution with stability for a class of functional differential equations. Then, using this condition, the adaptive stabilization problem is solved for nonlinear time-delay systems. Moreover, it is shown that for a class of nonlinear time-delay systems with triangular structure the adaptive stabilizing controller can be obtained by recursively constructing the Lyapunov-Razumikhin function. It will be shown that the provided recursive design approach, which is obviously motivated by the typical backstepping method, is not a trivial extension of the existing design method.

Abstract: This note considers the stabilization problem of an underactuated surface vessel. Three global smooth time-varying control laws are proposed with the aid of different techniques. The first proposed control law makes the state of the closed-loop system asymptotically converge to zero, while the second and the third control laws make the state of the closed-loop system globally exponentially converge to zero. Moreover, the exponential convergence rate of the state of the closed-loop system can be arbitrarily assigned with the third control law. Simulation results show that the proposed control laws are effective.

Abstract: This paper investigates the problem of robust output feedback stabilization for a family of uncertain nonlinear systems with uncontrollable/unobservable linearization. To achieve global robust stabilization via smooth output feedback, we introduce a rescaling transformation with an appropriate dilation, which turns out to be very effective in dealing with uncertainty of the system. Using this rescaling technique combined with the nonseparation principle based design method, we develop a robust output feedback control scheme for uncertain nonlinear systems in the p-normal form, under a homogeneous growth condition. The construction of smooth state feedback controllers and homogeneous observers uses only the knowledge of the bounding homogeneous system rather than the uncertain system itself. The robust output feedback design approach is then extended to a class of uncertain cascade systems beyond a strict-triangular structure. Examples are provided to illustrate the results of the paper.

Abstract: Models for deterministic continuous-time nonlinear systems typically take the form of ordinary differential equations. To utilize these models in practice invariably requires discretization. In this paper, we show how an approximate sampled-data model can be obtained for deterministic nonlinear systems such that the local truncation error between the output of this model and the true system is of order /spl Delta//sup r+1/, where /spl Delta/ is the sampling period and r is the system relative degree. The resulting model includes extra zero dynamics which have no counterpart in the underlying continuous-time system. The ideas presented here generalize well-known results for the linear case. We also explore the implications of these results in nonlinear system identification.