华科电气自动控制理论m22-2

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1、Theorderofthenumeratorequalstotheorderofthedenominator.(分子分母阶次相等)nn−1bs+bs+...bs+bnn−110G(s)=nn−1s+as+...as+an−110n−1(b−ab)s+...+(b−ab)s+(b−ab)n−1n−1n11n00nG(s)=b+nnn−1s+as+...as+an−110n−1βs+...+βs+βn−110G(s)=b+nnn−1s+as+...as+an−110n−1βs+...+βs+βn−110G(s)=b+nnn−1s+as+...as+a

2、n−110⎧X=AX+bu⎨⎩y=CX+du⎡010...0⎤⎡0⎤⎢⎥⎢⎥001...00⎢⎥⎢⎥A=⎢#⎥,b=⎢#⎥⎢⎥⎢⎥⎢000...1⎥⎢0⎥⎢⎣−a0−a1−a2...−an−1⎥⎦⎢⎣1⎥⎦c=[]ββ…β01n−1d=bnStatevariablesSignalflowgraphofcanonicalcontrollabilityform(可控标准形状态变量图)bnβn-1βn-2β1xxn−1…xn111/s11/s11/sβu0y-axnxn−1x2x1n-1-an-2-a1-a0说明：•可控标准形和可观测标准形的各

3、矩阵之间的关系如下：对偶关系⎡010...0⎤⎡0⎤⎢⎥⎢⎥001...00⎢⎥⎢⎥A=⎢#⎥,b=⎢#⎥TccA=A⎢000...1⎥⎢0⎥co⎢⎥⎢⎥T⎢⎣−a0−a1−a2...−an−1⎥⎦⎢⎣1⎥⎦b=ccoc=[]bb…bc01n−1T⎡00...0−a⎤⎡b⎤c=b00co⎢⎥⎢⎥10...0−ab⎢1⎥⎢1⎥A=⎢01...0−a⎥,b=⎢#⎥•下标c表示可控标准形o2o⎢⎥⎢⎥#•下标o表示可观测标准形⎢⎥⎢⎥•T为转置符号⎢⎣00...1−an−1⎥⎦⎢⎣bn−1⎥⎦c=[]00...1oExample2：2s+3G(s)=3

4、2s+2s+s+5•Determinethestate-spacerepresentationincanonicalcontrollabilityformandcanonicalobservabilityform.(列写可控标准形和可观测标准形)⎡010⎤⎡0⎤⎢⎥⎢⎥x(t)=001x(t)+0u(t)⎢⎥⎢⎥⎢⎣−5−1−2⎥⎦⎢⎣1⎥⎦y=[]320x(t)⎡00−5⎤⎡3⎤⎢⎥⎢⎥x(t)=10−1x(t)+2u(t)⎢⎥⎢⎥⎢⎣01−2⎥⎦⎢⎣0⎥⎦y=[]001x(t)22.4State-SpaceModelfromagenera

5、lblockdiagramofsystemEx.3u1yu1y⎧x=−ax+us⎨s+a⎩y=xa22uωnyuωyEx.4n22s+2ζωs+ωs(s+2ξω)nnn21uωyns(S+2ξωn)uxxy21211ωns-s+2ζωnEx.5k1k2k3y(s)u(s)Ts1Ts+1T3s+21-k4xxxx33x2x121k1k1k1y(s)u(s)123TsTsTs12-3--11TT12k4xxxx33x2x121k1k1k1y(s)u(s)123TsTsTs12-3--11TT12k4⎡k⎤3k00x=3x⎢⎥⎡⎤12T

6、T3⎢3⎥⎢0⎥1k⎢1k2⎥⎢⎥2x=0−x+0ux2=−x2+x3⎢TT⎥⎢k⎥TT22122⎢⎥⎢⎥kk11kkk⎢−140−⎥⎢⎣T1⎥⎦141x=−x−x+u3T3T1T⎢⎣T1T1⎥⎦111y=xy=[]100x1Ex.6IftheT.F.withzeross+zk11y(s)u(s)s+psss++aa-z−pkx21x1y(s)u(s)s+pss+a-x3z−pkx21x1y(s)u(s)s+pss+a-x31x=x12sx=−ax+xs+a112kksx=−kx+kx+kux=x+(u−x)213231sssx=(p−z)

7、x−px+(z−p)u313z−px=(u−x)31y=xs+p1y=x1⎡−a10⎤⎡0⎤⎢⎥⎢⎥x=−k0kx+ku⎢⎥⎢⎥⎢⎣p−z0−p⎥⎦⎢⎣z−p⎥⎦y=[]100xu2x22x1yEx.7s+3s(s+1)sx32x1=(x2−x3)sx3=2x2−3x3s(s+1)sx=−2x−3x+2u2122x2=(u−x1)sx1=x3s+3y=xx=sx1310010⎡⎤⎡⎤y=x1⎢⎥⎢⎥x=−2−30x+2u⎢⎥⎢⎥⎢⎣02−3⎥⎦⎢⎣0⎥⎦y=[]100x22.5TransferFunctionfromState-SpaceMo

8、delFromthedifferentialequation(stateequation):x=Ax+BuWeassumeinitia