09现代控制理论chp3-2

1、第三章线性控制系统的能控性和能观性a3-1 能控性的定义a3-2 线性定常系统的能控性判别a3-3 线性连续定常系统的能观性3-4 离散时间系统的能控性与能观性3-5 时变系统的能控性与能观性3-6 能控性与能观性的对偶关系3-7 状态空间表达式的能控标准型与能观标准型3-8 线性系统的结构分解3-9 传递函数矩阵的实现问题3-10 传递函数中零极点对消与状态能控性和能观性之间的关系13-4 离散时间系统的能控性与能观性1. 能控性矩阵M2. 能观性矩阵N23-4-1 能控性矩阵M单输入的n阶线性定常系统 n1 nn n1 x(k +1) = Gx(k) + hu(k)状态能控u(i) 11

2、x(k)0x(l)i = k, k + 1,L,l - 1系统状态完全能控33-4-1 能控矩阵M x(k +1) = Gx(k) + hu(k)M = hGhLGn-2hGn-1h系统能控的充要条件 rank(M ) = n4证明： x(k +1) = Gx(k) + hu(k)k -1x(k ) = Gk x(0) + Gk - j-1hu( j)j=0n-1- Gn x(0) = Gn- j-1hu( j)j=0- Gn x(0) = Gn-1hu(0) + Gn-2hu(1) +L+ Ghu(n - 2) + hu(n - 1)u(n - 1) n = -Gx(0)Mu(n - 2)u

3、(0)u(1)hLGhGn-2hGn-1 h53-4-2 能观矩阵N x(k +1) = Gx(k) + hu(k)1n-CGMCGN = GNT= CTGTCTL(Gn-1 )T CT y(k) = Cx(k)系统能观的充要条件rank ( NT ) = n rank(N ) = n6y(0)C x1(0)y(1) = CGx2 (0) y(n -1)MMMCGn-1nx(0)证明：x(k +1) = Gx(k) y(k) = Cx(k)x(k ) = Gk x(0) y(k ) = CGk x(0)y(0) = Cx(0)y(1) = CGx(0)y(n - 1) = CGn-1x(0)7

4、3-5 时变系统的能控性与能观性1. 能控性判别2. 能观性判别3. 连续时变系统可控性和可观性判别法和连续定常系统的判别法之间的关系8线性连续时变系统的能控性定义线性连续时变系统 x& = A(t)x + B(t)u状态能控系统状态完全能控u(t)x(t0 )x(t f)t0,tf状态x(t)的转移,与t0时刻的选取有关93-5-1 能控性判别有关线性时变系统能控性的几点说明(1) 定义中的允许控制u(t),在数学上要求其元在t0,tf区间绝对平方可积0L, rj2udt 0)充分条件系统在0，tf上完全能控192 t x1 + 0u0x102 x& x&1 = 0例3-9试判别下列系统的能

5、控性 方法一解： （1）求状态转移矩阵F(0,t) = 1+0 0t t+ 1 0 0t t +L= - 1 t 2 1200t d2! t d0201020 （2）计算能控性判别阵t f 1- 1 t2 010Wc (0,t f) = 02110- 1 t21dt012 1 t 4- 1 t 2 1 t5- 1 t3 =t f 42 20f6f 0-1 t 22dt1= 1t-3t6ff（3）判别Wc(0,tf)是否为非奇异6ft145fcdetW (0,t) =当t f 0,detWc (0,t f ) 0,所以系统在0，t是能控的212 t x1 + 0u0x102 x& x&1 = 0

6、例3-9 方法二解： B = B = 011 B= - A(t)B(t) + B&(t) = -0t 0= - t211001 0 Q (t) = B (t)B (t)= 0- tdet Q (t) = t10c12ct 0rankQc (t) = n = 2223-5-2 能观性判别有关线性时变系统能观性的几点说明(1) 时间区间t0,tf的大小和初始时刻t0的选择有关 (2) 不能观测状态的数学表达式C(t)F(t,t0 )x(t0 ) 0，t t0 ,t f 23 有关线性时变系统能观性的几点说明C(t)F(t,t0 )x(t0 ) 0 (3) 线性非奇异变换不改变系统的能观测性 若x(

7、t0)是不能观测的状态，它必满足x = Px , = CPCC(t) = C(t)P-1C(t)P(t)-1F(t,t)Px (t) 000C(t)F(t,t0 )x (t0 ) 024 有关线性时变系统能观性的几点说明(4) 如果x(t0) 是不能观测的，则ax(t0) 也是不能观测的（ a是任意非零实数）C(t)F(t,t0 )x(t0 ) 0C(t)F(t,t0 )ax(t0 ) 025 有关线性时变系统能观性的几点说明(5) 如果x01和x02都是不能观测的，则x01+x02也是不能观的C(t)F(t,t0 )x01 0C(t)F(t,t0 )x02 0C(t)F(t,t0 )( x0

8、1 + x02 ) 026 有关线性时变系统能观性的几点说明(6) 系统的不能观测状态状态构成状态空间中的一个子空间，称为不能观子空间，记为x0x + 1u12 0x&2 0 x1 11 = x&1 2 x 1 x1 y = 1y(t) =x1(t) +x2 (t)= F(t,t0 )x1(t0 ) + F(t,t0 )x2 (t0 )若x1(t) = -x2 (t)y(t) 0x10x1 = -x2x2不能观子空间27 线性时变系统的能观性判别x& = A(t)x + B(t)u y = C(t)x系统在t0,tf上状态完全能观测的充分必要条件是格拉姆矩阵00F(t,t)dt(t)C(t)0

9、00fTTF(t,t)CfttW (t ,t) = 非奇异28 证明: t f= F+ tx(t)(t,t0 )x(t0 )0F(t,t )B(t )u(t )dtt f0y(t) = C(t)F(t,t0 )x(t0 ) + C(t)tF(t,t )B(t )u(t )dtx(t) = F(t,t0 )x(t0 )y(t) = C(t)F(t,t0 )x(t0 )0000FT (t,t )CT (t) y(t) = FT (t,t )CT (t)C(t)F(t,t )x(t )00(t,t )dtx(t )(t)C(t)F0t00t0TTF(t,t )Cft(t) y(t)dt =TTF(t

10、,t )Cft= W0 (t0 ,t f )x(t0 )29 线性时变系统的能观性的一个实用判据x& = A(t)x + B(t)uy = C(t)x A(t),C(t)的元对时间t分别是(n-2)和(n-1)次连续可微的，记C1(t) = C(t)Ci (t) = -Ci-1(t)A(t) + C&i-1(t),i = 2,3,L,nnC (t)M2C (t)R(t) C1(t)系统在0，tf上完全能观rankQc (t f ) = n,(t f 0)充分条件303-5-3 连续时变系统可控性和可观性和连续定常系统的判别法之间的关系 定理：矩阵 H (t0 ,t)= h1(t0 ,t)h2

11、(t0 ,t)Lhn (t0 ,t)的列向量hi (t0 ,t)(i= 1,2,L, n)线性无关的充要条件是00t0ftG = HT (t ,t)H (t ,t)dt 非奇异31Wc (t0 ,t f ) =f F(tt00t00,t)B(t)BT (t)FT (t,t)dt=t ft0BT (t)FT (t,t)T BT (t)FT (t,t)dt00BT (t)FT (t ,t)的列矢量线性无关Wc (t0 ,t f )非奇异=320A(t -t )F(t0 - t) = e在定常系统中Wc (t0 ,t f)非奇异0BT (t)FT (t ,t)的列矢量线性无关= eA(t0 -t )

12、 B的行矢量线性无关= rankM = nn-1 1Mb bAn-1BLABBj0jb(t- t)A B = n-1j =0B =0A(t -t )e b033Wo (t0 ,t f ) =f FT (tt0t00,t)CT (t)C(t)F(t,t)dt=t ft0C(t)F(t,t0)T C(t)F(t,t)dt0C(t)F(t, t0 )的列矢量线性无关Wo (t0 ,t f )非奇异=343-6 能控性与能观性的对偶关系1. 线性系统的对偶关系2. 对偶原理3. 时变系统的对偶原理353-6-1线性系统的对偶关系x&1 = A1x1 + B1u1 y1 = C1x1x&2 = A2 x

13、2 + B2u2 y2 = C2 x2 n1 nn nr r1 12C= BT12B= CT12A= AT 对偶 n1 nn nm m1 m1 mnr1rn 122y= BT x12122x&= AT x+ CTu363-6-1线性系统的对偶关系x&2 = A2 x2 + B2u2 y2 = C2 x2 n1 nn nr r1 对偶 n1 nn nm r1 x&1 = A1x1 + B1u1 y1 = C1x1 m1 B1u1+ mn x&1x1+y1y2 m1 C2x2 rn B2x&2u2+C1A11BTA21CT1AT3738x1+ux2+y+xn+lnl2l1cnc2c1y = 11L

14、1xn c n M M l 2 uc x + c1 0 0 LLOLM0 0 M2l0l1 0x& = x1+u+ +x2+y+xn+lnl2c2l1cnc1cn xLc2y = c1 M1n M l x + u M110 0 LLOLM0 02l0l1 0x& = 对偶1111W (s) = C (sI - A )-1 Bx&1 = A1x1 + B1u1 y1 = C1x1W2 (s)= C2(sI- A2)-1 BW1ux(s) = (sI - A )B = -11W2xy(x)T21= BT (sI -W1xy(s) = C (sI - A )= -111W2ux(x)T1= BT (

15、sI -T )-1CTA1111A )-1T CT= W1(s)TsI - A2= sI -TsI - A3-6-1线性系统的对偶关系x&2 = A2 x2 + B2u2 y2 = C2 x2A=11393-6-2对偶原理S1 =（A1，B1，C1）S2 =（A2，B2，C2）能观性=能控性=能控性能观性40M= BA BLAn-1B222222= CTATCTL( AT)n-1CT 11111N=T1NT=CTATCTL( AT)n-1CT 22= B22A BL22An-1B 11111= M1413-6-3时变系统的对偶原理S1 = (A1(t)，B1(t)，C1(t)S2 = (A2

16、(t)，B2 (t)，C2 (t)对偶12C (t) = BT (t)12B (t) = CT (t)12A (t) = - AT (t)0102FT (t ,t) = F (t,t)42 证明x&2= A2 (t)x2+ B2 (t)u21122= - AT (t)x+ CT (t)u02102F&(t,t ) = -AT (t)F (t,t )由F2 (t0 ,t0 ) = I两边转置得FT (t,t0 ) = I222000亦即FT (t ,t)FT (t,t ) = I两边对时间求导有 d FT (t ,t)FT (t,t)= 00120FT (t ,t) = F (t,t )dt202043两边对时间求导有 d FT (t ,t)FT (t,t)= 0dt20202于是F&T (t,t)FT (t,t) + FT (t,t)F&T (t,t) = 0222200000即F&T (t,t) = -FT (t,t)F&T (t,t)FT (t,t)222000由两边转置得02102F&(t,t ) = -AT (t)F (t,t )F& T (t,t200) = -FT (t,t)A1(t)2F& T (t ,t) = FT (t ,t)FT (t,t )A (t)FT (t,t)202020120= A (t)FT (t,t)1200102FT (t ,t) =