A globally exponentially stable speed observer for a class of mechanical systems: experimental and simulation comparison with high-gain and sliding mode designs

It is shown in the paper that the problem of speed observation for mechanical systems that are partially linearisable via coordinate changes admits a very simple and robust (exponentially stable) solution with a Luenberger-like observer. This result should be contrasted with the very complicated observers based on immersion and invariance reported in the literature. A second contribution of the paper is to compare, via realistic simulations and highly detailed experiments, the performance of the proposed observer with well-known high-gain and sliding mode observers. In particular, to show that – due to their high sensitivity to noise, that is unavoidable in mechanical systems applications – the performance of the two latter designs is well below par.     

On mechanical control systems with nonholonomic constraints and symmetries

This paper presents a computationally efficient method for deriving coordinate representations for the equations of motion and the affine connection describing a class of Lagrangian systems. We consider mechanical systems endowed with symmetries and subject to nonholonomic constraints and external forces. The method is demonstrated on two robotic locomotion mechanisms known as the snakeboard and the roller racer. The resulting coordinate representations are compact and lead to straightforward proofs of various controllability results.     

A simple global asymptotic convergent observer for uncertain mechanical systems

A simple nonlinear observer is proposed for a class of uncertain nonlinear multiple-input–multiple-output mechanical systems whose dynamics are first-order differentiable. The proposed observer is constructed without any detailed model knowledge of the system and the observer gains are easily chosen. Another interesting features of the proposed observer include that it is more immune to noise and is high robust against parameters variations, and thus it is readily implemented. Lyapunov's direct method is employed to prove global asymptotic convergent observation. Extensive simulations are presented to illustrate the effectiveness and improved performance of the proposed observer.     

A constructive speed observer design for general Euler–Lagrange systems

Astolfi, Ortega, and Venkatraman (2010) recently proved the existence of a globally exponentially convergent speed observer for general Euler–Lagrange systems. Key to their result, is a function defined by certain integrals which cannot be solved a priori, and may not have explicit analytic solutions. In this paper, this obstacle to a constructive design is removed and equations that solve the speed observer problem are given in closed form. The design is further simplified by removing up to one third of the observer states used in Astolfi et al. (2010). With the significant reduction in complexity, the new observer is easily applied to estimate the angular velocities in a Furuta pendulum example.     

A discontinuous output feedback controller and velocity observer for nonlinear mechanical systems

In this paper, we develop a new discontinuous output feedback tracking controller for a class of uncertain, nonlinear, multi-input/multi-output, mechanical systems whose dynamics are first-order differentiable. A novel filter design and Lyapunov-type stability analysis are used to prove semi-global asymptotic tracking. As a by-product of the proposed framework, we also present the design of a new simple, discontinuous velocity observer that ensures global asymptotic velocity observation.     

Full-order observer design for a class of port-Hamiltonian systems

We consider a special class of port-Hamiltonian systems for which we propose a design methodology for constructing globally exponentially stable full-order observers using a passivity based approach. The essential idea is to make the augmented system consisting of the plant and the observer dynamics to become strictly passive with respect to an invariant manifold defined on the extended state-space, on which the state estimation error is zero. We first introduce the concept of passivity of a system with respect to a manifold by defining a new input and output on the extended state-space and then perform a partial state feedback passivation which leads to the construction of the observer. We then illustrate this observer design procedure on two physical examples, the magnetic levitation system and the inverted pendulum on the cart system.     

Velocity observers for linear mechanical systems subject to single non-smooth impacts

In this paper, we consider the problem of the velocity estimation from position measurement for linear mechanical systems (with multiple degrees of freedom) subject to single non-smooth impacts, both elastic and inelastic (i.e., with coefficient of restitution e=1 and e(0,1), respectively). Through a simple example, it is shown that a classical Luenberger observer is not able to reproduce instantaneously the jumps in the mass velocities, since it recovers the error induced by such jumps only asymptotically: an infinite sequence of impacts can prevent the estimation error to asymptotically go to zero. A new observer structure is proposed for linear mechanical systems subject to single unilateral constraints, that guarantees that the corresponding error dynamics are exponentially stable, also in presence of an infinite sequence of non-smooth impacts. The observer that we propose switches at the impact times, that can be recognized by position measurements only. To validate the proposed observer, both simulation and experimental tests have been carried out and are briefly reported, pointing out the drawbacks and advantages of the observer.     

Exponentially convergent control laws for nonholonomic systems in power form

This paper introduces a method for constructing exponentially convergent control laws for n-dimensional nonholonomic systems in power form. The methodology is based on the construction of a series of nested invariant manifolds for the closed-loop system under a linear control law. A recursive algorithm is presented which uses these manifolds to construct a three-dimensional system in power form. It is shown that the feedback controller for the original system is the one for this three-dimensional system with a proper choice of the gains.     

多变量非线性系统的间接模糊输出反馈自适应控制

针对一类多输入多输出非线性不确定系统，提出一种基于观测器的模糊间接自适应控制方法，并基于李亚普诺夫函数方法，导出了输出反馈控制律以及参数的自适应律，证明了整个控制方案不但能保证闭环系统稳定，而且取得了良好的跟踪控制性能。     

The extended Luenberger observer for nonlinear systems

For nonlinear single-input single-output systems , the relationships for a state transformation into the nonlinear observer canonical form are developed. It is possible to dimension a nonlinear observer by an eigenvalue assignment without solving the nonlinear partial differential equations for the transformation, if the transformed nonlinearities are linearized about the reconstructed state. With reference to the extended Kalman filter algorithm, this nonlinear observer design is called the extended Luenberger observer.     

Interval observer design for consistency checks of nonlinear continuous-time systems

This paper deals with fault detection for nonlinear continuous-time systems. A procedure based on interval analysis is proposed to build a guaranteed qLPV (quasi-Linear Parameter-Varying) approximation of the nonlinear model. The interval qLPV approximation makes it possible to derive two point observers which estimate respectively the lower and the upper bound of the state vector using cooperativity theory. A set guaranteed to contain the actual value of the residual is then designed. The modelling uncertainties and measurement errors are taken into account at the design stage. The proposed methodology is illustrated through numerical simulations.     

一类非线性系统有限时间流形吸引的浸入与不变控制

本文对一类非线性系统,提出了一种设计渐近稳定控制律的有效方法.其中,通过更新系统浸入与不变流形理论的应用方法,流形的吸引坐标可以在有限时间内收敛到平衡点.为了得到闭环系统的稳定性,增广系统的各个信号被证明是有界的.本文得出的一个重要成果是流形吸引有限时间的计算方法.此外,在施加了有限时间流形吸引控制器之后,流形对外部有界未知扰动具有不敏感性.最后利用车摆系统来论述所提出的控制方法的设计步骤,以及通过仿真验证控制器的性能.     

Orbital stabilization of nonlinear systems via the immersion and invariance technique

Immersion and invariance is a technique for the design of stabilizing and adaptive controllers and state observers for nonlinear systems. In all these applications the problem considered is the stabilization of equilibrium points. Motivated by some modern applications, we show that the technique can also be used to solve the problem of orbital stabilization, where the final objective is to generate periodic solutions that are attractive. The feasibility of our result is illustrated by means of some classical mechanical engineering and power electronics examples.     

Further result on a dynamic recurrent neural-network-based adaptive observer for a class of nonlinear systems

In Kim et al. [(1997) A dynamic recurrent neural-network-based adaptive observer for a class of nonlinear systems. Automatica 33(8), 1539–1543], authors present an excellent neural network (NN) observer for a class of nonlinear systems. However, the output error equation in their paper is strictly positive real (SPR) which is restrictive assumption for nonlinear systems. In this note, by introducing a vector b₀ and Lyapunov equation, the observer design is obtained without requiring the SPR condition. Thus, our observer can be applied to a wider class of systems.     

Adaptive dynamic surface control using disturbance observer for nonlinear systems with input saturation and output constraints

An adaptive dynamic surface control (DSC) approach using fuzzy approximation and nonlinear disturbance observer (NDO) for uncertain nonlinear systems in the presence of input saturation, output constraint and unknown external disturbances is proposed in this paper. The issue of input saturation is addressed by introducing a lower bound assumption on the approximation function of saturation. The output constraint is handled by introducing an appropriate barried Lyapunov function. The nonlinear disturbance observer (NDO) is employed to estimate the unknown unmatched disturbances. It is manifested that the ultimately bounded convergence of all the variables in the closed-loop system is guaranteed and the tracking error can be made farely small by tuning the design parameters. Finally, two simulation examples illustrate the effectiveness and feasibility of the proposed approach.     

Neural network compensation control for mechanical systems with disturbances

Two novel compensation schemes based on accelerometer measurements to attenuate the effect of external vibrations on mechanical systems are proposed in this paper. The first compensation algorithm exploits the neural network as the feedback-feedforward compensator whereas the second is the neural network feedforward compensator. Each compensation strategy includes a feedback controller and a neural network compensator with the help of a sensor to detect external vibrations. The feedback controller is employed to guarantee the stability of the mechanical systems, while the neural network is used to provide the required compensation input for trajectory tracking. Dynamics knowledge of the plant, disturbances and the sensor is not required. The stability of the proposed schemes is analyzed by the Lyapunov criterion. Simulation results show that the proposed controllers perform well for a hard disk drive system and a two-link manipulator.     

Tracking control design for nonholonomic mechanical systems with affine constraints

The trajectory tracking control is considered for nonholonomic mechanical systems with affine constraints and dynamic friction. A new state transformation is proposed to deal with affine constraints, and then an integral feedback compensation strategy is used to identify the dynamic friction. The proposed controller ensures that the output tracking errors converge to zero as t →∞.As an application, a detailed example is presented to illustrate the effectiveness of the control scheme.     

A constructive solution for stabilization via immersion and invariance: The cart and pendulum system

Immersion and Invariance (I&I) is the method to design asymptotically stabilizing control laws for nonlinear systems that was proposed in [Astolfi, A., & Ortega, R. (2003). Immersion and invariance: A new tool for stabilization and adaptive control of nonlinear systems. IEEE Transactions on Automatic Control, 48, 590-606]. The key steps of I&I are (i) the definition of a target dynamics, whose order is strictly smaller than the order of the system to be controlled; (ii) the construction of an invariant manifold such that the restriction of the system dynamics to this manifold coincides with the target dynamics; (iii) the design of a control law that renders the manifold attractive and ensures that all signals are bounded. The second step requires the solution of a partial differential equation (PDE) that may be difficult to obtain. In this short note we use the classical cart and pendulum system to show that by interlacing the first and second steps, and invoking physical considerations, it is possible to obviate the solution of the PDE. To underscore the generality of the proposed variation of I&I, we show that it is also applicable to a class of n-dimensional systems that contain, as a particular case, the cart and pendulum system.     

Velocity observers for non-linear mechanical systems subject to non-smooth impacts

This paper is concerned with the estimation of the velocity variables (when the position variables are the measured outputs) for non-linear mechanical systems subject to non-smooth impacts, both elastic and inelastic (i.e., with coefficient of restitution e=1 and e(0,1), respectively). A reduced-order observer is proposed, which guarantees that the corresponding error system, despite the possible presence of an infinite sequence of non-smooth impacts, is locally exponentially stable. An estimate of the basin of attraction is also given.