Abstract: In this paper, we propose a new algorithm for adaptive backstepping control of non-linear uncertain systems. Current backstepping algorithms require repeated differentiations of the modelled non-linearities. The addition of n first order low pass filters allows the algorithms to be implemented without differentiating any model non-linearities. The uncertainties are assumed to be linear in the unknown constant parameters. The combined adaptive backstepping/first order filter system is proven to be semi-globally stable for sufficiently fast filters by a singular perturbation approach.

Abstract: Dynamic positioning (DP) systems for ships are usually designed under the assumption that the kinematic equations be linearized about a constant yaw angle such that linear and gain scheduling techniques can be applied. This paper proposes a globally exponentially stable (GES) nonlinear control where this assumption is removed. A nonlinear observer is included in the design such that only position measurements are required. GES is proven by applying the backstepping design methodology and Lyapunov stability theory. The control law is simulated on two thruster-controlled ships.

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.

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.

Abstract: In this paper two non-linear control laws for ships are derived by using a non-linear ship model which includes the hydrodynamic effects due to time-varying speed and wave frequency. The non-linear ship model does not have the structural properties of symmetric inertia and positive damping at high speed, which are common assumptions, when designing non-linear ship tracking control systems. Backstepping control designs are used to circumvent the problem of a non-symmetrical inertia matrix. The first case is ship tracking with full actuation in surge, sway and yaw while the second case discusses an emergency situation where a ship is supposed to track a time-varying trajectory by using the bow thruster and one of the main propellers as actuation devices only. The application of speed- and wave-dependent models for ship control is a first step towards the design of a new generation of non-linear ship control systems that can operate under time-varying speed and wave frequency conditions. Previous work has often relied on the assumption of constant speed, and practical solutions have been based on switching between different control schemes at different speeds. Slowly varying environmental disturbances, such as ocean current, second-order wave-induced disturbances and wind forces, are compensated for by using adaptive backstepping. Global uniform asymptotic stability (GUAS) is proven for the fully actuated case by using a recent theorem for GUAS when backstepping with integral action, whereas asymptotic tracking of position and velocity are shown for the degraded control case. Computer simulations illustrate the performance and the robustness of the control laws for an offshore supply vessel. © 1998 John Wiley & Sons, Ltd.

Abstract: We consider parametrically uncertain systems satisfying a matching condition. A standard Lyapunov adaptive design for stabilization is modified to ensure a monotonically decreasing parametric error. This modified controller is used to learn a control which forces the state to reach a neighbourhood of the origin at a time t>0 in a finite number of learning passes of finite length. A performance analysis is completed. The iterative learning tracking problem is also considered. Corresponding results can also be obtained for strict feedback systems via adaptive backstepping, and this is briefly sketched.

Abstract: We study robustness of the adaptive backstepping design with tuning functions for linear systems. Under assumptions on unmodeled dynamics and disturbances equal to those for certainty equivalence schemes, we address an adaptive scheme not based on the certainty equivalence principle. In the process of redesign for robustness we employ only leakage in the estimator; we do not employ normalization, neither static nor dynamic. A fundamental difference between the tuning functions design and the certainty equivalence designs is that the controller in the former is inherently nonlinear, while in the latter it is nonlinear only in the parameter estimate. As a result, achievable robustness results for the tuning functions scheme are not global but regional, with a region of attraction inversely proportional to the "size" of the unmodeled dynamics. The tracking error is proportional to the size of the uncertainties.

Abstract: We propose a simple sliding mode based controller for nonlinear systems with mismatched uncertainties. The design methodology is similar to backstepping and multiple surface control method but with the inclusion of sliding mode filters for estimating the derivatives of the plant output.

Abstract: The results to date of a collaborative research project with BHP Steel in Port Kembla, Australia, are described. The project is concerned with control of strip tension and looper angle in BHP's hot strip steel rolling mill (the finishing mill). The new controllers have been successfully implemented on the process and now handle all the production at the mill. The paper describes a model of the process, similar to that reported in the work of others. The development of a full nonlinear controller is traced, based on a recursive nonlinear method (cf., backstepping). Insights are drawn into possible system structures, particularly in the output feedback case when tension measurements may not be available. Alternative controller schemes are examined, including a speculative design which contains filters, similar to those obtained from output feedback designs of linear systems, together with nonlinear operators which invite comparisons with variable structure designs. Simulation results, which provided the justification for the final implementation, together with results from actual production records are presented.

Abstract: We design a controller for a class of underactuated nonlinear systems. First, we try to find an appropriate global change of coordinates to transform the dynamics of the system into a desired form which consists of a lower order nonlinear subsystem plus a chain of integrators. Then, we find a control Lyapunov function (CLF) and its associated control law for the lower order subsystem. Finally, using a backstepping procedure we derive the control Lyapunov function and the controller for the whole system. The obtained controller renders the origin semiglobally asymptotically stable. As an special case, we demonstrate this procedure for the Acrobot example which is a two-link planar robot with a single actuator at the elbow.

Abstract: This paper presents a robust, adaptive, nonlinear controller for a class of magnetic-levitation systems, which includes active-magnetic bearings. The controller is analytically and experimentally shown to be superior to a classical linear control system in stability, control effort, step-response performance, robustness to parameter variations, and force-disturbance rejection performance, Using an adaptive backstepping approach, a Lyapunov function is generated along with an adaptive control law such that the nonlinear, closed-loop, continuous system is shown to guarantee stability of the equilibrium and convergence of the parameter estimates to constant values. The control system error coordinates are proven to be bounded in the presence of a bounded force disturbance input. The novelty of this controller is that it is digitally implemented using Euler integrators with anti-windup limits, it is single-input-single-output requiring only a measurement of the position of the levitating object, and it is designed to adaptively estimate not only the uncertain model parameters, but also the constant forces applied to the levitating object in order to ensure robustness to force disturbances. The experimental study was conducted on a single-axis magnetic-levitation device. The controller is shown to be applicable to active-magnetic bearings, under specific conditions, as well as any magnetic-levitation system that can be represented in output-feedback form.

Abstract: This paper discusses the development of a nonlinear controller design methodology and its application to the automated longitudinal control of automotive vehicles. The method is called the "multiple sliding surface" method and is closely related to sliding mode control, input/output linearization and integrator backstepping. The method was developed for a class of systems, typical of automotive control systems, where the uncertainties are "mismatched" and where many of the equations contain sparse, experimentally obtained maps. The error bounds on these maps are often unknown and their sparseness makes them difficult to differentiate. The developed method does not require any derivatives and has guaranteed semi-global stability. This paper summarizes the development of the method and applies it to design a combined brake/throttle controller for precision vehicle following. This controller was implemented on the California PATH vehicles in DEMO'97, an automated highway technology demonstration that occurred in San Diego, California in August of 1997. Some experimental data that shows the performance of the longitudinal controller will be presented in this paper.

Abstract: In the original paper, Fossen and Grovlen (ibid., vol.6, p.121-8, 1998) proposed the use of an observer-based backstepping method for the dynamic positioning of ships. This note points out that the observer design used does not cover unstable ship dynamics and suggests a remedy for an extended class of ships. The proof for the nonlinear observer used in the design in the above-mentioned papers only applies to ships with stable sway-yaw dynamics. An example concerning thruster assisted mooring of a tanker is considered. It does not fulfil the needed stability properties, so an extension to this case is highly motivated. We propose a method to extend the results, under a detectability condition. This condition implies stable surge dynamics, which is a natural assumption for ships.

Abstract: In this paper we propose a nonlinear Lyapunov-based controller to stabilize the famous cart-pole system. The novelty is in the use of a two-loop cascade controller. The inner loop uses an adaptive nonlinear controller, obtained by the backstepping recursive approach. It ensures the stabilization and the convergence towards zero of the angle tracking error and the (unknown) rod length estimation error. The reference signal to be tracked by the angle is generated by the outer loop linear controller. The controlled part of the system (rod angle) has thus a quasi-linear dynamics, which can be modeled by a double-integration with a variable gain. An indirect MRA controller is used to compensate the outer loop (cart position).

Abstract: Neural network (NN) controllers for the robust backstepping control of robotic systems in both continuous and discrete-time are presented. Control input is selected to achieve tracking performance for unknown nonlinear systems. Tuning methods are derived for the NN based on the delta rule. Novel weight tuning algorithms for the NN are obtained that are similar to /spl epsiv/-modification in the case of continuous-time adaptive control. Uniform ultimate boundedness of the tracking error and the weight estimates are presented without using the persistency of excitation (PE) condition. Certainty equivalence is not used and a regression matrix is not computed. No learning phase is needed for the NN and initialization of the network weights is straightforward.

Abstract: A multi-input control design procedure for uncertain non-linear systems expressible in multi-input parametric-strict feedback form with suitable structural features of the uncertain control matrix is presented in this paper. The proposed procedure, based on the well-known backstepping design technique, exploits the possibility of extending, under suitable assumptions, to multi-input uncertain systems, a second order sliding mode control approach recently developed, thus reducing the computational load, as well as increasing robustness, with respect to the purely backstepping design.

Abstract: Neural network (NN) controllers for the robust back stepping control of robotic systems in both continuous and discrete-time are presented. Control action is employed to achieve tracking performance for unknown nonlinear system. Tuning methods are derived for the NN based on delta rule. Novel weight tuning algorithms for the NN are obtained that are similar to e-modification in the case of continuous-time adaptive control. Uniform ultimate boundedness of the tracking error and the weight estimates are presented without using the persistency of excitation (PE) condition. Certainty equivalence is not used and regression matrix is not computed. No learning phase is needed for the NN and initialization of the network weights is straightforward. Simulation results justify the theoretical conclusions.

Abstract: This paper presents new adaptive robust control strategies suitable for permanent magnet (PM) synchronous motors. By using the backstepping approach, both the adaptive and robust controllers are appropriately designed to ensure global stability. Taking into account the variation ranges of system parameters, the adaptive robust methods are further developed to achieve better tracking performance. It is shown that the synthesized adaptive robust control schemes developed here can retain advantages of both adaptive and robust control schemes and overcome their shortcomings. To avoid discontinuous control laws which may cause problems in both theoretical and practical aspects, a continuous adaptive robust control method associated with a mu - modification scheme is also proposed to guarantee both the uniform boundedness of the system and suitably designated tracking precision. The paper includes simulation studies demonstrating the performance of the proposed control schemes.

Abstract: Develops a systematic methodology for the control of a class of feedback linearizable systems and applies it to an electrohydraulic servosystem. The class of systems to be dealt with are those that are single input and can be put in strict feedback form. Additionally, it is assumed that the relative degree of these systems is greater than one and the zero dynamics are stable. For a system with relative degree, r, a series of r system errors are defined. An input-output feedback linearization method is then used to control each error through the use of synthetic inputs. Subsequently, the entire closed loop system is analyzed for stability. The approach is conceptually similar to previously developed integrator backstepping methodologies. However, unlike some previous investigations which have relied exclusively on a Lyapunov analysis, this work presents a stability analysis including a passivity formulation. There are two main advantages of the proposed approach. One practical advantage is that the resulting controller leads to synthetic inputs which are decoupled in a certain sense. This leads to a compartmentalization of modeling efforts associated with the controller development. A second advantage of this method is that the system model need not be differentiated in the control formulation. A class of modeling error is introduced and compensated for. The resulting control is able to guarantee specified boundary layer tracking. Finally, the approach is implemented on a hydraulic cylinder governed by an electronically controlled servovalve.

Abstract: We design a controller for the beam-and-ball system that renders the origin semiglobally asymptotically stable (semi-GAS). First, we view the problem as designing a controller for a second-order subsystem of the system. Then, using a new backstepping procedure we reconstruct the control for the whole system. As our main result, we developed a new version of backstepping procedure for systems that do not have a triangular structure.

Abstract: : This paper examines the use of adaptive backstepping for multi-axis control of a high performance aircraft. The control law is demonstrated on a 6 Degree-of-Freedom simulation with nonlinear aerodynamic and engine models, actuator models with saturation, and turbulence. Simulation results are demonstrated for large pitch-roll maneuvers, and for maneuvers with failure of the right stabilator. There are substantial differences between the control law design and simulation models, which are used to demonstrate some robustness aspects of this control law. Actuator saturation is shown to be a considerable problem for this type of controller. However, the flexibility of the backstepping design provides opportunities for improvement. In particular, the Lyapunov function is modified so that the growth of integrated error and the rate of change of parameter growth are both reduced when the surface commands are growing at a rate that will likely saturate the actuators. In addition, the deadzone technique from robust linear adaptive control is applied to improve robustness to turbulence.

Abstract: Presents the application of backstepping feedback design technique to the speed control of a switched reluctance motor (SRM). Using the backstepping method, both feedback laws and Lyapunov based designs are applied to the controller design. The mathematical model for the SRM takes magnetic saturation into account. The controller takes phase currents, rotor position, rotor speed, and reference speed as inputs, and calculates the voltage required to maintain the motor speed close to the reference speed. The turn-on and conduction angles are continuously controlled to improve the system performance. An experimentally verified Saber model is used for simulation. A conventional PI controller is used for comparison. Simulation results confirm reduction in torque ripples, improved transient and steady state performance, and robustness of the controller.