Interval observers for discrete‐time systems

Interval observers are constructed for discrete‐time systems. First, time‐invariant interval observers are proposed for a family of nonlinear systems. Second, it is shown that, for any time‐invariant exponentially stable discrete‐time linear system with additive disturbances, time‐varying exponentially stable discrete‐time interval observers can be constructed. The latter result relies on the design of time‐varying changes of coordinates, which transform a linear system into a nonnegative one. Copyright © 2013 John Wiley & Sons, Ltd.     

非线性系统研究动态与展望

本文对近几年非线性控制系统的进展情况作了综合报导。文章共分三个部分:第一部分强调:什么是反馈意义下的非线性系统;第二部分介绍了近几年非线性控制系统研究中较集中的八个方面的问题;第三部对非线性控制理论今后主要研究动向作了预测。     

Discrete-time observers for singularly perturbed continuous-timesystems

The use of discrete-time observers for nonlinear singularly perturbed continuous-time systems is explored. The observer design approach is based on inversion of state-to-measurement maps. The two-time-scale nature of the system is exploited by constructing separate reduced-order observers for the approximate slow and fast subsystems, using multirate measurements and computation. The reduced-order observers are compared to a full-order observer designed for the same system. The comparison shows that although the reduced-order observers exhibit some approximation error, they also result in reduced computational requirements if the inversion is performed on-line, reduced memory requirements if the inversion is implemented by look-up table and reduced stiffness. The trade-off between accuracy and implementation requirements always favors the reduced-order approach for systems with significant separation of time scales. Numerical examples are provided, including a case-study of an observer-based control system for permanent-magnet synchronous motors     

Multirate modeling and control design for switched-mode powerconverters

A class of pulse-width modulated (PWM) systems that includes many switched-mode power converters is considered. A new multirate low-frequency sampling PWM scheme is introduced that not only provides the possibility for new types of operation, but also unifies the switching logic corresponding to established PWM schemes. An exact nonlinear discrete-time model is used to develop multirate nonlinear controllers and observers based on map-inversion concepts. This nonlinear design methodology is widely applicable, requiring only that the continuous-time small-signal average model be controllable and observable     

Application of nonlinear time–scaling for robust controller design of reaction systems

Even though the basic mechanisms of operation of reaction systems are relatively simple the dynamical models obtained from first principles are complex and contain highly uncertain terms. To develop reliable model‐based controllers it is therefore necessary to simplify the system dynamics preserving the features which are essential for control. Towards this end, co‐ordinate transformations identifying the states which are dependent/independent of the reactions and flows have been reported in the literature. This has allowed, for instance, the design of observers which are insensitive to the (usually unknown) reaction functions. The main contribution of this paper is to show the utility of nonlinear state‐dependent time‐scaling to simplify the system dynamics, and consequently the controller design. In particular, we show that with time‐scaling and an input transformation we can reveal the existence of attractive invariant manifolds, which allows us to reduce the dimension of the system. As an application we study the well‐known fourth order baker's yeast fed‐batch fermentation process model, whose essential dynamics is captured by a planar system perturbed by an exponentially decaying term. We then exploit this particular structure to design, with reduced control authority, a nonlinear asymptotically stabilizing control law which is robust with respect to the reaction function. Copyright © 2001 John Wiley & Sons, Ltd.     

Nonlinear observers for spacecraft attitude estimation in case of yaw angle measurement absence

This paper provides a brief presentation and a useful comparison between two nonlinear observers Extended Kalman Filter (EKF) and sliding mode observer (SMO). Both can be used for moderate-accuracy attitude determination systems for Low Earth Orbit (LEO) Earth-pointing spacecraft (s/c), which is typically achieved using Gyroscopes, Earth, and Sun sensors for attitude sensing. The use of these observers provides a substitute for the yaw data in case of the s/c eclipse periods or limited field of views. The nonlinear observability for this system is analytically investigated via the calculation of Lie derivatives to check the possibility of the system states estimation. The performances of both observers are presented, the SMO stability is proved and the SMO enhanced estimates are shown by simulation.     

Control of nonlinear, continuous, dynamic systems via finite elements, sensitivity analysis, and optimization

A general methodology to design open loop controllers for nonlinear, dynamic, continuous systems is presented and applied to control a single flexible link (SFL). In this application, the partial differential equations that describe the beam system are first analyzed via the finite element method (FEM) and Newmark integration method. Two open loop control inputs to achieve specified system performance criteria are then computed by posing and solving inverse dynamics problems. These analyses use nonlinear programming (NLP) algorithms and analytical gradients that are computed by the direct sensitivity method. The open loop control is verified experimentally. Closed loop controller synthesis for linear time invariant (LTI) and linear time varying systems (LTV) is relatively well understood. To apply this knowledge base to the control of the SFL, the nonlinear finite element plant model is linearized and recast in standard state space form.     

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.     

Output regulation of uncertain nonlinear systems with nonlinear exosystems

An adaptive control algorithm is proposed for output regulation of uncertain nonlinear systems in output feedback form under disturbances generated from nonlinear exosystems. A new nonlinear internal model is proposed to generate the desired input term for suppression of the disturbances. The proposed internal model design is based on boundedness of the disturbance, high gain design and saturation. It is capable to tackle disturbances in any specified initial conditions. Some uncertainties in the systems are allowed, provided that they do not affect the desired feedforward control term, and they are tackled by using nonlinear dominant functions and an adaptive control coefficient. The proposed control algorithm ensures the global convergence of the state variables to the invariant manifold, which implies that the measurement or the tracking error approaches to zero asymptotically.     

Closed loop observers bundle for uncertain biotechnological models

We propose nonlinear observers for a class of biotechnological processes. These observers are an extension of the open loop asymptotic observers (observers with unknown inputs) devoted to biotechnological systems for which some parts of the model are unknown. We take benefit of the additional outputs which are (nonlinear) functions of the state to design a closed loop observer. The global convergence of these nonlinear observers is proven. We use these observers to design interval based observers which predict guaranteed intervals in which the state is lying. We run simultaneously a broad set of interval observers and we select the best ones. The method is illustrated with a model describing the bioconversion of a substrate using micro-organisms in a bioreactor.     

Nonlinear observer for simultaneous states and unknown parameter estimation

This paper presents the design of an observer for the simultaneous estimation of states and unknown parameter for a class of nonlinear systems whose nonlinearity satisfies a bounded Jacobian condition. The paper presents two alternate observers based on the structure of the system. The conditions for the existence of these observers can be expressed as a linear matrix inequality and solved using standard solvers. The case of time-varying parameter and multiple unknown parameter have also been investigated. The use of the developed methodology is demonstrated through illustrative examples.     

Controllability of a class of underactuated mechanical systems with symmetry

In this paper we develop results based on geometric mechanics to study the controllability of a class of controlled under-actuated left invariant mechanical systems on Lie groups. We exploit the invariance of the controlled nonlinear dynamics to the group action (symmetry) to derive a set of reduced dynamics for the system. We first present sufficient conditions for the controllability of the reduced dynamics. We prove conditions (boundedness of coadjoint orbits and existence of a radially unbounded Lyapunov function) under which the drift vector field (of the reduced system) is weakly positively Poisson stable (WPPS). The WPPS nature of the drift vector field along with the Lie algebra rank condition is used to show controllability of the reduced system. We then extend these results to the unreduced dynamics, considering separately the cases when the symmetry group is compact and noncompact. In the noncompact case, under further assumptions of equilibrium controllability, sufficient conditions for controllability of the unreduced dynamics are derived. Jetpuck (hovercraft) and underwater vehicles are used as mechanical systems to motivate our work and illustrate our theoretical results.     

Generalised observer design for dissipative Lipschitz nonlinear systems in the presence of measurement noise

This paper presents two novel observer concepts. First, it develops a globally exponentially stable nonlinear observer for noise-free dissipative nonlinear systems. Second, for a dissipative nonlinear system with measurement noise, the paper develops an observer to guarantee a desired performance, namely an upper limit on the ratio of the square of the weighted L2 norm of the error to the square of the weighted L2 norm of the measurement noise. The necessary and sufficient conditions for both observers are reformulated as algebraic Riccati equations (AREs) so that standard solvers can be utilised. In addition, the paper presents necessary and sufficient conditions to be satisfied by the nonlinear system in order to ensure that the ARE (and hence the observer design problem) has a solution. The use of the methodology developed in this paper is demonstrated through illustrative examples. In literature, there is no previous observer for dissipative system that provides both necessary and sufficient conditions. Results for noisy system either rely on linearising the system about state trajectory (requiring initial estimates to be close to the actual states) or are for specialised systems that cannot be extended to dissipative systems.     

Practical stabilization of driftless systems on Lie groups: the transverse function approach

A general control design approach for the stabilization of controllable driftless nonlinear systems on finite dimensional Lie groups is presented. The approach is based on the concept of bounded transverse functions, the existence of which is equivalent to the system's controllability. Its outcome is the practical stabilization of any trajectory, i.e., not necessarily a solution of the control system, in the state-space. The possibility of applying the approach to an arbitrary controllable smooth driftless system follows in turn from the fact that any controllable homogeneous approximation of this system can be lifted (via a dynamic extension) to a system on a Lie group. Illustrative examples are given.     

Stabilization of a synchronous generator with a controllable series capacitor via immersion and invariance

The immersion and invariance (I&I) technique (it IEEE Trans. Automat. Control 2003; 48 (4):590–606) is a recently proposed control methodology for stabilizing nonlinear systems. Here we apply this philosophy to stabilize a single machine infinite bus (SMIB) system using a controllable series capacitor (CSC). The synchronous generator is modeled using the second‐order swing equation and a first‐order model is used for the CSC. The control objective here is to immerse a desired second‐order dynamic model into the higher order system manifold and design an asymptotically stabilizing control law. Copyright © 2011 John Wiley & Sons, Ltd.     

非线性广义系统的变结构控制设计

从线性定常广义系统出发，研究非线性广义系统的变结构控制设计问题。其主要思想为：选取一具有指定性能的线性定常广义系统作为参考模型，根据所控系统与参考模型的误差方程设计变结构控制，使系统的状态（输出）向参考模型的状态（输出）逼近，由参考模型的性能即得所研究的非线性广义系统所希望的性能。仿真例子验证了所建立方法的有效性。     

A New Algorithm to Design Minimal Multi‐Functional Observers for Linear Systems

Designing minimum possible order (minimal) observers for multi‐input multi‐output (MIMO) linear systems have always been an interesting subject. In this paper, a new methodology to design minimal multi‐functional observers for linear time invariant (LTI) systems is proposed. The approach is applicable, and it also helps in regulating the convergence rate of the observed functions. It is assumed that the system is functional observable or functional detectable, which is less conservative than assuming the observability or detectability of the system. To satisfy the minimality of the observer, a recursive algorithm is provided that increases the order of the observer by appending the minimum required auxiliary functions to the desired functions that are going to be estimated. The algorithm increases the number of functions such that the necessary and sufficient conditions for the existence of a functional observer are satisfied. Moreover, a new methodology to solve the observer design interconnected equations is elaborated. Our new algorithm has advantages with regard to the other available methods in designing minimal order functional observers. Specifically, it is compared with the most common schemes, which are transformation based. Using numerical examples it is shown that under special circumstances, the conventional methods have some drawbacks. The problem partly lies in the lack of sufficient numerical degrees of freedom proposed by the conventional methods. It is shown that our proposed algorithm can resolve this issue. A recursive algorithm is also proposed to summarize the observer design procedure. Several numerical examples and simulation results illustrate the efficacy, superiority and different aspects of the theoretical findings.     

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.     

Observers for Takagi-Sugeno fuzzy systems

We focus on the analysis and design of two different sliding mode observers for dynamic Takagi-Sugeno (TS) fuzzy systems. A nonlinear system of this class is composed of multiple affine local linear models that are smoothly interpolated by weighting functions resulting from a fuzzy partitioning of the state space of a given nonlinear system subject to observation. The Takagi-Sugeno fuzzy system is then an accurate approximation of the original nonlinear system. Our approach to the analysis and design of observers for Takagi-Sugeno fuzzy systems is based on extending sliding mode observer schemes to the case of interpolated multiple local affine linear models. Thus, our main contribution is nonlinear observer analysis and design methods that can effectively deal with model/plant mismatches. Furthermore, we consider the difficult case when the weighting functions in the Takagi-Sugeno fuzzy system depend on the estimated state.