Backstepping Procedure Research Articles

Robust numerical solution of the reservoir routing equation

The robustness of numerical methods for the solution of the reservoir routing equation is evaluated. The methods considered in this study are: (1) the Laurenson–Pilgrim method, (2) the fourth-order Runge–Kutta method, and (3) the fixed order Cash–Karp method. Method (1) is unable to handle nonmonotonic outflow rating curves. Method (2) is found to fail under critical conditions occurring, especially at the end of inflow recession limbs, when large time steps (greater than 12min in this application) are used. Method (3) is computationally intensive and it does not solve the limitations of method (2). The limitations of method (2) can be efficiently overcome by reducing the time step in the critical phases of the simulation so as to ensure that water level remains inside the domains of the storage function and the outflow rating curve. The incorporation of a simple backstepping procedure implementing this control into the method (2) yields a robust and accurate reservoir routing method that can be safely used in distributed time-continuous catchment models.

Output‐feedback stabilisation control for a class of under‐actuated mechanical systems

Output-feedback control of general underactuated mechanical systems is currently considered a major open problem. This study is focused on the output-feedback stabilisation control problems for a special class of underactuated mechanical systems, which appear in robotics and aerospace applications. For the synthesis of controller, first, the considered underactuated mechanical system is explicitly transformed into two cascade connected subsystems, and then an auxiliary filter-based virtual stabilisation controller is developed to locally asymptotically stabilise the first subsystem. Further, the designed virtual controller is again involved into the second subsystem using backstepping procedure to construct the actual control law, in which a series of auxiliary time-varying first-order low-pass filters are also implemented to avoid using the derivative of the system non-linear functions. Moreover, in the second step, finite-time observer technique is utilised to precisely reconstruct the immeasurable states to achieve the finite-time stabilisation control in the sense of output feedback. Lyapunov analysis shows the local asymptotic stability of the closed-loop system through the cascade system stability criteria. Simulation results are presented by using two benchmark non-linear underactuated mechanical systems to demonstrate the feasibility and the effectiveness of the proposed controller.

A sum of squares approach to backstepping controller synthesis for piecewise affine and polynomial systems

SUMMARY This paper develops a backstepping controller synthesis methodology for piecewise polynomial (PWP) systems in strict form. The main contribution of the paper is to formulate sufficient conditions for controller design for PWP systems in strict form as a sum of squares feasibility problem under the assumption that an initial control Lyapunov function exists to start the iterative backstepping procedure. This problem can then be translated into a convex SDP problem and solved by available software packages. The controller synthesis problem for PWP systems in strict feedback form is divided into two cases. The first case consists of the construction of a sum of squares polynomial control Lyapunov function for PWP systems with discontinuous vector fields. The second case addresses the construction of a PWP control Lyapunov function for PWP systems with continuous vector fields. One major advantage of the proposed method is the fact that it can handle systems with discontinuous vector fields and sliding modes. The new synthesis method is applied to several numerical examples. One of these examples offers the first convex optimization solution to piecewise affine (PWA) control of a benchmark circuit system addressed before in the literature using non-convex PWA control solutions. Copyright © 2013 John Wiley & Sons, Ltd.

Robust adaptive control for a class of cascaded nonlinear systems with applications to space interception

SUMMARYThis paper proposes a robust H ∞ ‐based adaptive backstepping control scheme for the output stabilization of a special class of cascaded nonlinear systems. This kind of systems possess the feature that the first sub‐equation is a linear perturbed system, whereas the rest ones perform a general semi‐strict feedback form. Different from the conventional backstepping design approach, the special cascaded structure ensures to introduce the H ∞ technique to the backstepping procedure such that both the robust performance and the robust stability can be simultaneously guaranteed. Within the Lyapunov framework, the proposed control scheme is proved to guarantee (i) the uniformly ultimate boundedness of the system signals with a bound that can be made arbitrarily small by suitably choosing control parameters; (ii) asymptotic output stabilization as long as the uncertain nonlinearities and external disturbances vanish; and (iii) ‐performance of the closed‐loop system. A space interception scenario is utilized to demonstrate the effectiveness of the proposed control scheme. Copyright © 2013 John Wiley & Sons, Ltd.

Adaptive Exponential Stabilization of Chained Form Systems with Application to Mobile Robot

This paper investigates the adaptive stabilization problem for chained form systems with strong nonlinear drifts. A parameter separation technique is introduced to transform the nonlinear parameterized system into a linear-like parameterized system. Then, using the parameter separation, input-state-scaling technique and integrator backstepping procedure, an adaptive state feedback controller is obtained. The adaptive control based switching strategy is proposed to eliminate the phenomenon of uncontrollability. Global exponential regulation of the closed-loop system and the boundedness of other signals are guaranteed. The proposed scheme can be applied to the adaptive control of mobile robot with small angle measurement error. Simulation results demonstrate the validity of the presented algorithm.

Observer-Based Robust Tracking Control for a Class of Switched Nonlinear Cascade Systems

This paper is devoted to robust output feedback tracking control design for a class of switched nonlinear cascade systems. The main goal is to ensure the global input-to-state stable (ISS) property of the tracking error nonlinear dynamics with respect to the unknown structural system uncertainties and external disturbances. First, a nonlinear observer is constructed through state transformation to reconstruct the unavailable states, where only one parameter should be determined. Then, by virtue of the nonlinear sliding mode control (SMC), a discontinuous nonlinear output feedback controller is designed using a backstepping like design procedure to ensure the ISS property. Finally, an example is provided to show the effectiveness of the proposed approach.

Adaptive neural decentralized control for strict feedback nonlinear interconnected systems via backstepping

An approximation based adaptive neural decentralized output tracking control scheme for a class of large-scale unknown nonlinear systems with strict-feedback interconnected subsystems with unknown nonlinear interconnections is developed in this paper. Within this scheme, radial basis function RBF neural networks are used to approximate the unknown nonlinear functions of the subsystems. An adaptive neural controller is designed based on the recursive backstepping procedure and the minimal learning parameter technique. The proposed decentralized control scheme has the following features. First, the controller singularity problem in some of the existing adaptive control schemes with feedback linearization is avoided. Second, the numbers of adaptive parameters required for each subsystem are not more than the order of this subsystem. Lyapunov stability method is used to prove that the proposed adaptive neural control scheme guarantees that all signals in the closed-loop system are uniformly ultimately bounded, while tracking errors converge to a small neighborhood of the origin. The simulation example of a two-spring interconnected inverted pendulum is presented to verify the effectiveness of the proposed scheme.

Micro-Helicopter for Long-Distance Missions: Description and Attitude Stabilization

This paper presents the development of a micro coaxial helicopter (MCR UAV) whose main characteristic is that it should be carried by an air shuttle transporter and then released in a desired place far away from the launching site, to develop surveillance missions in hover flight. A real-time embedded system is built in order to validate the proposed aerodynamic prototype, and a classic control law based on a classical backstepping procedure for the dynamic system is implemented to test this vehicle in autonomous flight. Finally, simulation and practical results are presented for hover flight.

Immersion- and invariance-based adaptive missile control using filtered signals

The article presents a new non-certainty-equivalent adaptive (NCEA) longitudinal autopilot for the control of a missile based on the immersion and invariance theory. The interest here is to control the angle of attack of the missile in the presence of large parametric uncertainties. For the derivation of the control law, a backstepping design procedure is used. At each step of the design, certain filtered signals are generated for the synthesis of a stabilizing control signal and a parameter estimator. Using Lyapunov stability analysis, it is shown that in the closed-loop system, trajectory control of the angle of attack is accomplished, and the trajectories of the system are attracted to certain manifold in the space of state variables and parameter errors. For stability in the closed-loop system, an explicit analytical relation involving the controller gains is obtained. It may be pointed out that recently an adaptive autopilot based on the immersion and inversion theory has been designed, but it has stringent requirements because for its synthesis, the derivatives of the Mach number and angle of attack must be known, and a large number of parameters must be updated. The derived control system of this article is synthesized using only the state variables, and its identifier is of lower order. A traditional certainty-equivalent adaptive autopilot is also presented for comparison. Simulation results are obtained which show that the designed NCEA control system can accomplish angle of attack control despite large parametric uncertainties; and it can give better tracking performance than the traditional controller.

Semiglobal Stabilization for Nonholonomic Mobile Robots Based on Dynamic Feedback With Inputs Saturation

This paper investigates the semiglobal stabilization problem for nonholonomic mobile robots based on dynamic feedback with inputs saturation. A bounded, continuous, time-varying controller is presented such that the closed-loop system is semiglobally asymptotically stable. The systematic strategy combines finite-time control technique with the virtual-controller-tracked method, which is similar to the back-stepping procedure. First, the bound-constrained smooth controller is presented for the kinematic model. Second, the dynamic feedback controller is designed to make the generalized velocity converge to the prespecified kinematic (virtual) controller in a finite time. Furthermore, the rigorous proof is given for the stability analysis of the closed-loop system. In the mean time, the position and torque inputs of robots are proved to be bounded at any time. Finally, the simulation results show the effectiveness of the proposed control approach.

Adaptive neural network dynamic surface control for perturbed nonlinear time-delay systems

This paper proposes an adaptive neural network control method for a class of perturbed strict-feedback nonlinear systems with unknown time delays. Radial basis function neural networks are used to approximate unknown intermediate control signals. By constructing appropriate Lyapunov-Krasovskii functionals, the unknown time delay terms have been compensated. Dynamic surface control technique is used to overcome the problem of "explosion of complexity" in backstepping design procedure. In addition, the semiglobal uniform ultimate boundedness of all the signals in the closed-loop system is proved. A main advantage of the proposed controller is that both problems of "curse of dimensionality" and "explosion of complexity" are avoided simultaneously. Finally, simulation results are presented to demonstrate the effectiveness of the approach.

Hybrid Control for an Overhead Crane

One of the common industrial structures that are used widely in many harbors and factories are overhead crane. Overhead cranes are usually operated manually or by some conventional control methods. In this study, a new combination of nonlinear backstepping scheme with fuzzy system is presented to control both position regulation and anti-swing control to avoid collision with other equipments. This study uses two different fuzzy systems: off-line and on-line to consider different concept. The backstepping design procedure consists of two steps. In all steps to design backstepping controller, Lyapunov stability theorem is used to stability analysis and control design problems. Furthermore, the stability of the overhead crane with nonlinear fuzzy backstepping controller is discussed. Simulation results illustrated the validity of the proposed control algorithm. Key words: On-line fuzzy design, off-line fuzzy design, backstepping method, over head crane.

Cascade Controller Including Backstepping for Hydraulic-Mechanical Systems

Development of a cascade controller structure including adaptive backstepping for a nonlinear hydraulic-mechanical system is considered in this paper where a dynamic friction (LuGre) model is included to obtain the necessary accuracy. The paper compares the performance of two variants of an adaptive backstepping tracking controller with earlier results. The new control architecture is analysed and enhanced tracking performance is demonstrated when including the extended friction model. The complexity of the backstepping procedure is significantly reduced due to the cascade structure. Hence, the proposed control structure is better suited to real-time implementation.

An Efficient Implementation of Backstepping Observers for Time-Varying Parabolic PDEs

AbstractA new computationally efficient implementation of the backstepping-based state observer is considered for a 1-dimensional diffusion-reaction system with spatially and time varying parameters. For this, the implicit formulation for the integral kernel, which results from the application of the backstepping procedure is discretized in space and approximated by means of the composite trapezoidal rule. This results in a system of first-order ordinary differential equations governing the pointwise explicit-in-time evolution of the backstepping-kernel. The resulting system of ODEs is numerically solved by applying the backward Euler method, which drastically reduces the computing time compared to the usually applied method of successive approximation. The impact of the proposed method on the computing time and on the observer error convergence is evaluated by means of numerical simulations.

Position and Yaw Angle Control for a Four Rotor Mini Helicopter Based on a Geometric Approach

In this paper, we present a position and yaw angle controller for a four-rotor mini helicopter, which is designed via the backstepping methodology. In order to avoid a singularity problem caused by using of Euler angles, only quaternion is employed to represent the attitude of the airframe. Although the use of quaternion makes it difficult to obtain a desired attitude in a step of the back-stepping procedure, a geometrical approach gives an effective solution for the problem. It is shown in some numerical simulations that the proposed controller can achieve global tracking to demanded position and yaw angle simultaneously.

Dynamic visual servoing of a small scale autonomous helicopter in uncalibrated environments

This paper presents a novel adaptive controller for image-based visual servoing of a small autonomous helicopter to cope with uncalibrated camera parameters and unknown 3D geometry of the feature points. The controller is based on the back-stepping technique, but its design has two new features. First, it incorporates the visual feedback into the last step of the backstepping procedure, while existing backsteppingbased methods employ the visual feedback at the early steps. Second, the controller maps the image errors onto the actuator space via a depth-independent interaction matrix to avoid estimation the depths of the feature points. The new design method makes it possible to linearly parameterize the closed-loop dynamics by the unknown camera parameters and coordinates of the feature points in the 3D space so that an adaptive algorithm can be developed to estimate the unknown parameters and coordinates on-line. Two potential functions are introduced in the controller to guarantee convergence of the image errors and to avoid trivial solutions of the estimated parameters. The Lyapunov method is used to prove the asymptotic stability of the proposed controller based on the nonlinear dynamics of the helicopter. Simulations have been also conducted to demonstrate the performance of the proposed method.

On the backstepping design procedure for multiple input nonlinear systems

SUMMARYIn a recent paper, we developed a structural decomposition for multiple input multiple output nonlinear systems that are affine in control but otherwise general. In this paper, we exploit the structural properties of such a decomposition in the application of the backstepping design technique on multiple input nonlinear systems. In particular, this decomposition simplifies the conventional backstepping design and motivates new backstepping design procedures that are able to stabilize systems on which the conventional backstepping procedure is not applicable. Copyright © 2011 John Wiley & Sons, Ltd.

An adaptive fuzzy design for fault-tolerant control of MIMO nonlinear uncertain systems

This paper presents a novel control method for accommodating actuator faults in a class of multiple-input multiple-output (MIMO) nonlinear uncertain systems. The designed control scheme can tolerate both the time-varying lock-in-place and loss of effectiveness actuator faults. In each subsystem of the considered MIMO system, the controller is obtained from a backstepping procedure; an adaptive fuzzy approximator with minimal learning parameterization is employed to approximate the package of unknown nonlinear functions in each design step. Additional control effort is taken to deal with the approximation error and external disturbance together. It is proven that the closed-loop stability and desired tracking performance can be guaranteed by the proposed control scheme. An example is used to show the effectiveness of the designed controller.

Backstepping design for cascade systems with relaxed assumption on Lyapunov functions

This study proposes a novel backstepping technique for non-linear cascade systems whose driven subsystems have a feedforward structure and include higher order terms. A small control is first assigned to stabilise the driven subsystem, and a simple backstepping procedure is then followed. When dealing with the driven subsystem, we adopt an approach of using a small control to attenuate the higher order term, rather than the usual method in which the higher order term is tackled under the guidance of a single Lyapunov function. The stability analysis is carried out using some boundedness information to explicitly compute the higher order term, and the global asymptotical stability of the whole closed-loop system is obtained using the ‘converging-input bounded-state’ criterion. This is in sharp contrast with previous designs where (involved) Lyapunov functions are utilised in both the control design and stability analysis. As applications, global stabilisation designs are presented for several classical mechanical systems including the inertia wheel pendulum, the translational oscillator with rotating actuator and the cart–pole system.

A novel adaptive control approach for nonlinear strict-feedback systems using nonlinearly parameterised fuzzy approximators

In this article, the problem of output tracking of perturbed nonlinear strict-feedback systems is addressed and a novel adaptive fuzzy control scheme is proposed. The considered systems are with unknown nonlinearities, so an adaptive fuzzy approximation approach is embedded into a backstepping procedure to get the proposed controller. However, unlike the exiting results, approximators used in this article are not linearly parameterised. Using nonlinearly parameterised adaptive fuzzy approximators, the controller can be obtained without the restriction that fuzzy basis functions of the approximators must be well defined. By managing to adapt the norm of on-line parameter vectors in the control design, the computation burden is largely reduced. The proposed controller can guarantee the stability and desired tracking performance of the closed-loop system. An example is included to demonstrate the effectiveness of the control scheme.