哈工大现代控制理论CHP

1、1 2-1 线性定常齐次状态方程的解 2-2 矩阵指数函数 2-3 线性定常系统非齐次状态方程的解 2-4 线性时变系统的解 2-5 离散时间系统状态方程的解 2-6 连续时间状态空间表达式的离散化21.时变系统状态方程的特点2.线性时变齐次矩阵微分方程的解3.状态转移矩阵(t,t0)的基本性质4.线性时变系统非齐次状态方程式的解5.状态转移矩阵的计算3考察标量时变系统)()()(txtadttdxdttatxtdx)()()(datxtxtt0)()(ln)(ln0)()(0)(0txetxdatt)()(exp()(00txdatxtt)(exp(),(00datttt)(),()(00t

2、xtttx4考察矢量时变系统)()()(txtAtx)()(exp()(00txdAtxtt)()()()(00tAdAdAtAtttt？5)()(exp()(00txdAtxtt)()()()(00tAdAdAtAtttt证明：)()(exp()()()(exp0000txdAtAtxdAdtdtttt)(exp()()(exp00dAtAdAdtdttttdAdAdAdAtttttttt0000)()(! 21)(1)(exp)()(21)()(21)()(exp000tAdAdAtAtAdAdtdttttttdAtAtAdAtAtttt00)()()()(exp)(6对于齐次矩阵微分方程

3、)(| )(;)(00txtxxtAxttItt),(00),()(),(00tttAtt)(),()(00txtttx其解为:7)(),()()(),(0000txtttAtxttdtdItt),(00),()(),(00tttAtt)(),()(00txtttx证明：0000( )( , ) ( )x tt t x t8性质一),(),(),(011202tttttt)(),()(0011txtttx证明:)(),()(0022txtttx)(),(),()(),()(001121122txtttttxtttx所以),(),(),(011202tttttt9性质二Itt),(Itt),(0

4、010性质三),(),(001ttttItttttt),(),(),(00证明:所以Itttttt),(),(),(0000),(),(001tttt11线性时变系统非齐次状态方程为:)()()()()(tutBtxtAtxduBttxtttxtt)()(),()(),()(000其解为:且A(t)和B(t)的元素在时间区间t0tt2内分段连续12)(),()(),()(000txtttxtttxu证明:)()()()()(),()()(),(000tutBtxtAtxtttxtxttuu)()(),(00txtxttu)()()()()(),()()(0tutBtxtAtxtttxtAu)(

5、)(),()(01tutBtttxu)()(),(0tutBtt),()(),(00tttAtt的转移两部分组成。和控制作用激励的状态的转移看成由初始状态，将根据线性系统叠加原理)()()(0txtxtxu)()()()()(tutBtxtAtx)()()()()(),()()(),()(000tutBtxtAtxtttxtxtttAuu13)()()(),()(),()(00000txduBttxtttxutt)()(),()(01tutBtttxu)()(),(0tutBtt)()()(),()(000txduBttxuttu)(),()()(),()(),(00000txttduBttx

6、ttutt= 0)(),()(),()(000txtttxtttxu)()(),(00txtxttu14duBttxtttxtt)()(),()(),()(000零输入状态转移零状态状态转移)()()()()(tutBtxtAtxduBttCtxtttCtxtCtytt)()(),()()(),()()()()(000零输入响应零状态响应15Axx)()(exp)(00txttAtx定常系统)(exp)(00txAdtxtt时变系统)()()(txtAtx)()(00txtt)(),()(00txtttx)()()()(00tAdAdAtAtttt)(exp(),(00dAtttt16)()(

7、)()(00tAdAdAtAtttt0122100110000001000000)()()()()()(),(dddAAAddAAdAIttttttttttt皮亚诺-贝克(Peano-Baker)级数171.递推法2.z变换法18)()() 1(kHukGxkx)0(| )(0 xkxk)()0()(101jHuGxGkxkjjkk) 1()0()(10jkHuGxGkxkjjk19kGk )()() 1(kGkI )0(hhkhhhkhk111)()()()()(1kk)()(hkGhk20)()() 1(kHukGxkx)()()0()(zHUzGXzxzzX)()0()()(zHUzxz

8、XGzI)()()0()()(11zHUGzIzxGzIzX)()() 0()()(1111zHUGzILzxGzILkx)()0()(101jHuGxGkxkjjkkzGsILGk11)()()()(11101zHUGzILjHuGkjjk211.离散化方法2.近似离散化3.线性时变系统的离散化22DuCxyBuAxx)()()()()()()() 1(kDukCxkykuTHkxTGkxATeTG)(TAtdtBeTH0)(, 2 , 1 , 0, kkTt香农(Shannon)定理TktkTkTutu) 1()()(常数，23ATeTG)(TkkTTkABdeTH) 1() 1()(证明

9、：输出方程是状态矢量和控制矢量的某种线性组合，离散化之后，组合关系并不改变，故C和D保持不变dBuetxetxtttAttA)()()(00)(0)(TktkTt) 1(,0常数)()(kTutu)()() 1() 1() 1(kTuBdekTxeTkxTkkTTkAATTkt) 1(令dtd0)()(TAttBdeTHTAtdtBe024)()() 1() 1(kTTBukTxTATkx当采样周期T为系统最小时间常数的1/10时ITATG)(TBTH)(min101TT 25utDxtCyhTxutBxtAx)()()(;)()(初始条件为)()()()()()()()()() 1(kTuk

10、TDkTxkTCkTykTukTHkTxkTGTkxkTtkTtTkkTtDkTDtCkTCdBTkkTHkTTkkTG| )()(| )()()(,) 1()(,) 1()() 1(duBttxtttxtt)()(),()(),()(000IkTTAkTG)()()()(kTTBkTH近似化26IhThThTkTkTGhTTk),(),()(),)1(属性271.线性定常连续系统非齐次状态方程的解分为零输入的状态转移和零状态的状态转移；系统的输出响应由零输入响应和零状态响应两部分组成。2.线性定常连续系统齐次状态方程的解可表示为x(t)=(t-t0)x(t0)3.状态转移矩阵包含了系统运动的全部信息，它可以完全表征系统的动态特性4.线性时变系统非齐次状态方程的解在形式上类似与线性定常系统，即 duBttxtttxtt)()(),()(),()(0005.离散系统状态方程可采用递推法和z变换法来求解6.从系