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现代控制理论Modern Control Theory思维导图模板大纲
概念conception
原理:用状态空间模型,通过分析系统内部状态的变化,实现对多变量系统的最优控制 Principle:using a state-space model to achieve optimal control of multivariate systems by analyzing the changes in their internal states.
应用:航空航天、工业过程控制Applications: Aerospace, Industrial Process Control
控制步骤:建模、系统辨识、信号处理、综合控制输入Control Steps: Modeling, System Identification, Signal Processing, Synthesize Control Inputs
线性系统的能控性与能观性Controllability and Observability of Linear Systems
能控性controllability
定义:存在一个控制输入u(t),在有限时间内,能将系统从任意初始状态驱动到任意目标状态。Definition: There exists a control input u(t) that can drive the system from any initial state to any target state within a finite time interval.
判别:系统完全能控的充分必要条件是能控性矩阵满秩。Criterion: The necessary and sufficient condition is that its controllability matrix has full rank.
能观性:Observablity
定义:在有限时间内,通过观测系统的输出 y(t) 能够唯一地确定系统的初始状态 x(t₀)。Definition:The initial state x(t₀) of a system can be uniquely determined by observing the system's output y(t) within a finite time interval.
判别:系统完全能观的充要条件是能观性矩阵满秩。The necessary and sufficient condition is that the observability matrix has full rank.
李雅普诺夫方法Lyapunov's Method
李雅普诺夫第一法Lyapunov's First Method
定义:通过线性化非线性系统并分析其特征值,判断系统在平衡点附近的局部稳定性。Definition: Determine the local stability of a system near an equilibrium point by linearizing the nonlinear system and analyzing the eigenvalues of the resulting linear approximation.
线性系统的应用:所有的特征值都有负实部则系统渐进稳定 Application in Linear Systems:A system is asymptotically stable if all eigenvalues of its system matrix have negative real parts.
特点:计算简单Feature: Computationally simple
李雅普诺夫第二法Lyapunov's Second Method
定义:通过构造一个李雅普诺夫函数V(X),并通过分析该函数及其导数的正负性,直接判断系统的稳定性。Definition: determines the stability of a system directly by constructing a Lyapunov function V(X) and analyzing the sign definiteness of the function and its time derivative.
应用:计算该函数导数,如果导数小于0则系统渐近稳定。如果导数小于等于0则系统稳定。如果导数大于0则系统不稳定Application: Compute the time derivative of this function. If the derivative is strictly less than 0, the system is asymptotically stable. If the derivative is less than or equal to 0, the system is stable. If the derivative is greater than 0, the system is unstable.