Estimation and Control of Nonlinear Dynamical Systems

拟议研究计划的总体目标是：(I)发展和分析能够利用微分不变性的能力的线性和非线性系统的代数和递归状态和参数估计方法。代数估计是指必须使用有限时间间隔内的观测值来产生估计的情况。利用系统的任何已知的微分不变量的想法是有吸引力的，因为不变量携带独立于系统输出测量噪声的附加信息。开发和评估的方法将包括：(A)新版本的自适应核卡尔曼滤波器，其中递归估计将受到约束，以保持现有的微分不变性；(B)设计基于不变性的移动水平最小能量自适应滤波器，其表现出比扩展卡尔曼滤波器更快和更可靠的收敛特性；(C)设计对于未知的加性有色测量噪声具有鲁棒性的轨迹跟踪器。(Ii)开发受重力作用的空间运动链机械系统的新型在线非线性全局镇定控制器。这类系统的例子是人体姿势的机器人模型和多连杆垂直机械臂。研究方法和创新：所提出的估计器将利用作者最初提出的系统微分不变性的前向-后向核积分表示。对于用系统特征方程表示微分不变性的齐次线性时不变系统以及任意阶时变和变参数系统，导出了积分核的显式公式。对于被外部输入强迫的系统，核的显式表达也是可用的。最重要的是，积分系统表示的核产生了可以作为精确系统输出微分器的时域积分变换。文献中报道的现有代数估计方法对噪声敏感，在长时间间隔上使用时需要重新初始化。长期的研究目标包括：(A)一种系统的方法来构造多项式和有理系统中出现的非线性微分不变量的积分表示，这些系统在控制中是仿射的，并且具有平坦的输出，其微分产生系统的状态空间和参数；(B)在一般的非线性系统中构造可以计算和用于在线估计和滤波算法的近似微分不变量；深入分析所提出的估计方法的计算效率。应用的重要性：提出的高度自适应的非线性估计方法有望使许多应用受益，包括与目标跟踪和监视系统相关的应用。

The General Objectives of the Proposed Research Program are : (i) Development and analysis of algebraic and recursive state and parameter estimation methods for linear and nonlinear systems capable of exploiting the power of differential invariance. Algebraic estimation refers to the situation when the estimates must be produced using observations within a finite time interval. The idea of making use of any known differential invariants of the system is attractive because invariants carry additional information that is independent of system output measurement noise. The methods developed and evaluated will include: (a) novel versions of adaptive kernel Kalman filters in which the recursive estimates will be constrained to conserve the existing differential invariance; (b) design of invariance -based moving- horizon minimum- energy adaptive filters that exhibit accelerated and more reliable convergence properties than the extended Kalman filter; (c) design of trajectory trackers that are robust with respect to unknown additive coloured measurement noise. (ii) Development of novel on-line nonlinear globally stabilizing controllers for spatial kinematic chain mechanical systems that are subject to gravity. Examples of such systems are robotic models of the human posture and multi-link vertical robotic arms. Research Approach and Originality : The proposed estimators will exploit a forward--backward- kernel integral representation of system differential invariance originally proposed by the author. Explicit formulae for the integral kernels have been derived for homogeneous linear time invariant systems as well as time--varying and parameter--varying systems of arbitrary orders where the differential invariance was represented by the system characteristic equation. Explicit expressions of kernels are also available for systems forced by exogenous inputs. Most importantly, the kernels of the integral system representation give rise to time-domain integral transforms that can serve as exact system-output differentiators. Existing algebraic estimation methods reported in the literature are noise- sensitive and require re--initialization when used on long time intervals. Long Term Research Goals Include: (a) A systematic approach to the construction of integral representations of nonlinear differential invariants arising in polynomial and rational systems which are affine in control and which are equipped with flat outputs whose differentials generate the state space and parameters of the system; (b) Construction of approximate differential invariants in general nonlinear systems that can be computed and employed in on-line estimation and filtering algorithms; in depth analysis of the computational efficiency of the proposed estimation methods. The Importance for Applications: The proposed highly adaptive nonlinear estimation methods are expected to benefit many applications including those related to target tracking & surveillance systems.