1、线性系统理论与设计Linear system theory and designLinear system theory and designThe first group leader:Q.y zhang partner:D.b ge;T.h jiang;Y.j xu;Y.g xu第1章：线性系统的状态空间描述1.1 基本概念 1 状态变量：能完全描述系统行为的最小变量组中的每一个变量。2 状态向量：以状态空间为元所组成的向量。3状态空间：以状态变量 为坐标轴构成的n维空间。nxxx,21Chapter 1:State-space description of linear systems1
2、.1 Basic concept1 State Variables:A set of the least variables which can fully determine the state of system is called state variables.2 State Vector:Based on state space of vector.3 State Space:n dimensional space is made up of state variables for axis .nxxx,211.2 状态空间描述 状态方程：描述状态变量与输入信号之间关系的一阶微分方程
3、组。(1.2.1)输出方程：描述系统的输出量与状态变量、输入信号之间关系的代数方程。(1.2.2)()()(tButAxtx)()()(tDutCxty1.2 Description of the state space Equation of state：A first-order differential equations which describes the relati-onship between state variable and the input signal.(1.2.1)equation of output:A mathematical expression whic
4、h describes the relation between output rate and state variable and input signal.(1.2.2)()()(tButAxtx)()()(tDutCxty 1.3系统状态的线性变换 选择不同的状态变量组，可有不同形式的状态方程与输出方程 对状态变量的不同选取其实是状态向量的一种线性变换.（P是非奇异的）(1.3.1)cxybuAxx.xcyubxAx.xpx21npppP 1.3 Linear transformation of the system state Choose a different set of st
5、ate variables can have different forms of state equation and output equation.Different choices of state variables is actually a linear transformation of the state vector.(1.3.1)cxybuAxx.xcyubxAx.xpx21npppP第2章：线性系统的运动分析2.1 状态空间表达式的解 1.线性定常齐次方程的解：(2.1.1)2.状态转移矩阵：(2.1.2)0()!121()()()(22.xtAktAAtItxtAxt
6、xtk0()0()A t ttte Chapter 2:Linear Motion Systems Analysis2.1 Solution of state space expression1.The solution of linear time invariant homogeneous state equation (2.1.1)2.State transition matrix:(2.1.2)0()!121()()()(22.xtAktAAtItxtAxtxtk0()0()A t ttte2.2状态转移矩阵的计算 1.直接级数展开 (2.2.1)2.拉氏变换法 (2.2.2)3.待定
7、系数法（卡莱-汉密尔顿定理）(2.2.3)2 20111()2!nAtk kk kKteIAtA tA tA tkk 1-()AtteLSIA1（）110010nAtnknkkeat Iat Aat Aat A2.2 Calculate the state transition matrix1 Direct series expansion.(2.2.1)2 Mehod of Laplace transform.(2.2.2)3 Undetermined coefficient method(Carlisle-Hamilton Theorem).(2.2.3)2 20111()2!nAtk k
8、k kKteIAtA tA tA tkk 1-()AtteLSIA1（）110010nAtnknkkeat Iat Aat Aat A2.3 线性定常非齐次方程的解 (2.3.1)解释：此解有两部分组成 第一部分为起始状态的转移项 第二部分为控制作用下的受控项 满足系统的叠加原理)()-()(0txtttxdButtxtt)()()(0dButtXtttxtt)()()()()(000对初始状态的响应对输入作用的响应2.3 The solution of linear nonhomogeneous state equation.(2.3.1)Explanation:This solution
9、has two parts The first part is the initial state of the transfer of items.The second part is controlled under control action.dButtXtttxtt)()()()()(000)()-()(0txtttxdButtxtt)()()(0对初始状态的响应对输入作用的响应第三章 李雅普诺夫稳定性3.1 用二维空间来说明李雅普诺夫定义下的稳定性 1.稳定 渐进稳定 不稳定李亚普诺夫第二法:通过李氏函数 及其对时间的一次导数的定号性就可给出系统平衡状态稳定的信息 Chapter
10、3:Lyapunov stebility3.1 With the stability of a two-dimensional space is defined under Lyapunov stable Asymptotically stable Unstable Lyapunov Second Method:By the time Lees function and its first derivative of a given number of stable equilibrium system can give information.3.2.Lyapunov stability c
11、riterion Lyapunov second method not only for linear systems,but also nonlinear systems can also give information on a wide range of stability.unstablexnotvandvtheoremunstablevtheoremLsistablevtheoremstableallyasymptoticxnotvandvtheoremstableallyasymptoticvTheoremtxvxtxfxe),0(005,04.,03),0(002,010),(
12、0),(.，2.李亚普诺夫稳定性判据李亚普诺夫 第二法不仅对于线性系统，对于非线性系统也能给出大范围内稳定性的信息不稳定稳定，非渐进稳定渐进稳定渐进稳定在某一邻域负定在某一邻域正定，：定理负半定，在某个正定，：定理不恒为时，时，负定，正定，：定理负定正定，：定理vvxvvvttvxvvvvxtxfxe403000210),(03.2 Stability analysis of linear time-invariant systemsTheorem：System state equation isFor any real symmetric positive definite matrix Q
13、,the existence of real symmetric positive definite matrix is P，andThe systemis globally asymptotically stable at X=0.Energy function of the system(Lyapunov function)isXAXTA P PAQ()TVXXP X3.2 线性定常连续性系统的稳定性分析定理：设线性状态方程为 （A是非奇异的）对任意正定实对称矩阵Q，存在正定实对称矩阵P,使得则系统在X=0处的平衡状态是大范围渐进稳定的。系统的能量函数为 XAXTA P PAQ()TVXX
14、P XChapter IV:Controllability and Observability of the system4.1 Controllability and Observability Controllability:Controllability is to check the status of each component can be controlled by u(t),reflecting the impact on the system capacity.Observability:The output y can determine the status of X,
15、reflecting the system possible state is determined by the system output.第4章 系统的能控性与能观性4.1 能控性与能观性能控性：检查每一状态分量能否被u(t)控制，反映控制作用对系统的影响能力；能观性：由输出y能否判断状态X，反映由系统输出量确定系统状态的可能性。4.2系统的能控性与能观性的判据 已知线性定常系统的状态方程为：线性定常系统为完全能控的充要条件是 (4.2.1)线性定常系统为完全能观的充要条件是 (4.2.2)0)0(0txxBuAxx，nBABAABBrankn12nCACACrankTTTTT1-n.4
16、.2 The criterion of system controllability and observability State linear time-invariant systems of equations is Controllability criterion The necessary and sufficient condition of that linear time-invariant systems is fully controlled (4.2.1)Observability criterion The necessary and sufficient cond
17、ition of that linear time-invariant systems is fully observed (4.2.2)0)0(0txxBuAxx，nBABAABBrankn12nCACACrankTTTTT1-n.4.34.3对偶原理对偶原理线性系统 互为对偶系统。若 能控，则 能观测；若 能观测，则 能控。状态完全能控的充要条件 对于系统 的nnr 矩阵的秩，要求为 (4.3.1)对于系统 的nnm矩阵的秩，要求为 (4.3.2)状态完全能观的充要条件对于系统 的nnm矩阵的秩，要求为 (4.3.3)对于系统 的nnr 矩阵的秩，要求为 (4.3.4),(),(21TTT
18、BCACBA1212nBAABBrankrankQnk1nCACACrankrankQTnTTTk1nCACACrankrankQTnTTTg1nBAABBrankrankQng112214.3 Principle of dualityLinear Systems are mutually dual system，If can control，can be observed;If can observe,can control.Necessary and sufficient conditions of state fully controlled:For the system of rank
19、 n nr matrix requirement is (4.3.1)For the system of rank n nm matrix requirement is (4.3.2)Necessary and sufficient conditions state completely observableFor the system of rank n nm matrix requirement is (4.3.3)For the system of rank n nr matrix requirement is (4.3.4),(),(21TTTBCACBA1212nBAABBrankr
20、ankQnk112nCACACrankrankQTnTTTk1nCACACrankrankQTnTTTk1nBAABBrankrankQnk112Chapter 5:State feedback and state observer design5.1 State feedback (5.1.1)CxyBvAxx CxyBvxBKAx)(第5章 状态反馈与状态观测器设计5.1状态反馈状态反馈：状态反馈：将系统状态作为构成系统输入的一部分。能 改善系统性能。(5.1.1)CxyBvxBKAx)(CxyBvAxx 5.2 State Observer (5.2.1)CxyBuAxx xCyHyBu
21、xHCAx)(5.2 状态观测器：(5.2.1)CxyBuAxx CxyHyBuxHCAx)(5.3 State observer pole configurationFor single-input,single-output system,given arbitrary poles,for real or complex conjugate.In the n given pole polynomial rootObserver characteristic polynomialNote:State feedback pole freely configurable is system co
22、mpletely controllable conditions.The necessary and sufficient condition of all the poles of observer status can be any configured is the original system completely observable.*1*111()()nnninnifssssa sasa)()(det*sfHCAsI5.3 状态观测器的极点配置对于单输入、单输出系统，给定任意个极点，为实数或共轭复数。以这n个给定极点为根的多项式为观测器的特征多项式注：状态反馈极点可任意配置的条件是系统完全能控。可任意配置状态观测器全部极点的充要条件是原系统完全 可观。*1*111()()nnninnifssssa sasa)()(det*sfHCAsI
