第五六讲(预习版)

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Discussion
Strict proper G(s) may have non-proper Smith-McMillan form The reason is that unimodular matrix may introduce poles at s=∞.
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Discussion
Smith-McMillan form is obtained by reducing the elements of
λ1(s) d(s) 1 O Λ(s) = λ r (s) d(s) d(s) 0
s (s + 1)2 (s + 2)2 G(s) = −s 2 (s + 2) s 2 (s + 2) −s (s + 2)2
Rank G(s)=2
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U(s)G(s)V( s) 1 = 2 (s + 1) s 0 (s + 1)2 (s + 2)2 −s 1 2 (s + 2) 0 2 s (s + 2) s 2 2 1 (s 1) − + (s + 2) −s 0 1 2 (s + 2)
s (s + 1)2 (s + 2)2 = 0
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Discussion
• Obtain irreducible MFD through Smith-McMillan form of G(s) Let M(s)=U(s)G(s)V(s)=Ε(s)ΨR-1(s) where
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Contents
• Smith-McMillan form of transfer function • Poles and zeros of MIMO transfer function • Nullspace; minimal polynomial bases and Kronecker indices
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Contents
• Smith-McMillan form of transfer function • Poles and zeros of MIMO transfer function • Nullspace; minimal polynomial bases and Kronecker indices
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频域方法 Frequency domain method
• 数学预备知识 Math preliminary • 传递函数矩阵的矩阵分式描述 Matrix fraction description (MFD) for transfer function matrix • 传递函数矩阵的结构性质 Structure property of transfer function matrix • 传递函数矩阵的状态空间实现 State space realization for transfer function matrix • 传递函数矩阵的多项式矩阵描述 Polynomial matrix description (PMD) for transfer function matrix • 频域分析与反馈综合 Analysis and feedback synthesis in frequency domain
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Discussion
• From λi(s)|λi+1(s) for i=1,…,r-1 we know that εi(s)/ψi(s)d(s)|εi+1(s)/ψi+1(s)d(s). This implies that εi(s)|εi+1(s) and ψi+1(s)|ψi(s). • Smith-McMillan form is unique but transformation matrices are not unique .
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频域方法 Frequency domain method
• 数学预备知识 Math preliminary • 传递函数矩阵的矩阵分式描述 Matrix fraction description (MFD) for transfer function matrix • 传递函数矩阵的结构性质 Structure property of transfer function matrix • 传递函数矩阵的状态空间实现 State space realization for transfer function matrix • 传递函数矩阵的多项式矩阵描述 Polynomial matrix description (PMD) for transfer function matrix • 频域分析与反馈综合 Analysis and feedback synthesis in frequency domain
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Intuitive explanation of poles
• For a strict proper G(s), let (A,B,C) be a minimum realization. z0 is a pole of G(s). Then the mode exp(z0t) can be generated at the output by an initial state without the application of any input. If z0 is not a pole, then this is not possible.
{D(s),N(s)} are coprime. N(s)D-1(s) is an irreducible MFD.
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Discussion
• Smith-McMillan form provides the theoretical basis for defining poles and zeros of MIMO LTI systems.
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Contents
• Smith-McMillan form of transfer function • Poles and zeros of MIMO transfer function • Nullspace; minimal polynomial bases and Kronecker indices
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Structure property of transfer function matrix
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The basic question is:
• For a SISO system, we know the zeros and poles of its transfer function g(s)=q(s)/p(s) are roots of its numerator q(s) and denominator p(s) respectively, what are the zeros and poles of transfer function G(s) for an MIMO system?
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Intuitive explanation of zeros
• For a strict proper G(s), let (A,B,C) is a minimum realization. z0 is a zero of G(s). When initial state x0 and input u0 are selected s.t. Cx0=0 and (z0I-A)x0=-Bu0, then the mode of exp(z0t) in system response to input u(t)=u0exp(z0t) is y(t)≡0 for t≥0, i.e., input of such form will be blocked by the system.
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Proof. (i) G(s)=N(s)D-1(s) since U-1(s)Ε(s)ΨR-1(s)V-1(s)=U-1(s)M(s)V-1(s). (ii) Since {εi(s),ψi(s)} are coprime, Ε(s) and ΨR(s) are right coprime. Thus there exists a unimodular matrix U(s)’ s.t. ΨR (s) R(s) U(s)' = Ε(s) 0 R(s) is also unimodular.
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Importance of poles and zeros
• Poles and zeros reflect structural nature of systems • System dynamic is mainly decided by poles and zeros • System synthesis can be reduced to pole assignment