Topic #21 16.31 Feedback Control MIMO Systems Singular Value Decomposition Multivariable Frequency Response Plots Copyright 2001 by Jonathan How
Topic #21 16.31 Feedback Control MIMO Systems • Singular Value Decomposition • Multivariable Frequency Response Plots Copyright 2001 by Jonathan How. 1
Fal!2001 16.3121-1 Multivariable Frequency Response In the MIMO case, the system G(s)is described by a p m trans fer function matrix (TFM) Still have that =C(sI-A)B+D But G(s)A, B, C, D MUCH less obvious than in SISO case Discussion of poles and zeros of mimo systems also much more complicated e In siso case we use the bode plot to develop a measure of the system“size Given z=Gu, where G(jw)=G (jw) lejo(u) Then w=lejwit+)applied to G(jw)lejo(u)yields lullGGwn)lej(ant++o(w1)=I Amplification and phase shift of the input signal obvious in the SISO case ● MIMO extension? Is the response of the system large or small? /s0 010
Fall 2001 16.31 21—1 Multivariable Frequency Response • In the MIMO case, the system G(s) is described by a p×m transfer function matrix (TFM) — Still have that G(s) = C(sI − A) −1B + D — But G(s) → A, B, C, D MUCH less obvious than in SISO case. — Discussion of poles and zeros of MIMO systems also much more complicated. • In SISO case we use the Bode plot to develop a measure of the system “size”. — Given z = Gw, where G(jω) = |G(jω)|ejφ(w) — Then w = |w|ej(ω1t+ψ) applied to |G(jω)|ejφ(w) yields |w||G(jω1)|ej(ω1t+ψ+φ(ω1)) = |z|ej(ω1t+ψo) ≡ z — Amplification and phase shift of the input signal obvious in the SISO case. • MIMO extension? — Is the response of the system large or small? G(s) = ∙ 103/s 0 0 10−3/s ¸
Fal!2001 16.3121-2 For MIMO systems, cannot just plot all of the Gi; elements of G Ignores the coupling that might exist between them So not enlightening Basic MIMo frequency response Restrict all inputs to be at the same frequency Determine how the system responds at that frequenc See how this response changes with frequency So inputs are w= wceJut, where Wc Ec Then we get z=Gs0,→2= zceu and zo∈C We need only analyze ac=G(jw)c As in the siso case, we need a way to establish if the system re- sponse is large or small How much amplification we can get with a bounded input Consider 2c=G(w wc and set lwcll2=whC 1. What can we say about the - Answer depends on w and on the direction of the input wc Best found using singular values
Fall 2001 16.31 21—2 • For MIMO systems, cannot just plot all of the Gij elements of G — Ignores the coupling that might exist between them. — So not enlightening. • Basic MIMO frequency response: — Restrict all inputs to be at the same frequency — Determine how the system responds at that frequency — See how this response changes with frequency • So inputs are w = wcejωt , where wc ∈ Cm — Then we get z = G(s)|s=jω w, ⇒ z = zcejωt and zc ∈ Cp — We need only analyze zc = G(jω)wc • As in the SISO case, we need a way to establish if the system response is large or small. — How much amplification we can get with a bounded input. • Consider zc = G(jω)wc and set kwck2 = pwc Hwc ≤ 1. What can we say about the kzck2? — Answer depends on ω and on the direction of the input wc — Best found using singular values
Fal!2001 16.3121-3 Singular value decomposition Must perform the SVD of the matrix G(s)at each frequency s=ju G(ju)∈CpmU∈C∑∈RmV∈Cnxm G=U∑Vh UHU=IUUH=I VHV=I VVH=I and 2 is diagonal Diagonal elements ok 20 of 2 are the singular values of G 入(GG)oro i(GGH the positive ones are the same from both formulas The columns of the matrices U and V(; and v;)are the asso- ciated eigenvectors G Gu GGHa o2u 0U If the rank(G)=r< min(p, m), then k>0,k=1,,r Ok=0,k=r+1,., min(p, m An SVD gives a very detailed description of how a ma trix(the system G) acts on a vector (the input w) at a particular frequency
Fall 2001 16.31 21—3 Singular Value Decomposition • Must perform the SVD of the matrix G(s) at each frequency s = jω G(jω) ∈ Cp×m U ∈ Cp×p Σ ∈ Rp×m V ∈ Cm×m G = UΣV H — U HU = I, UU H = I, V HV = I, V V H = I, and Σ is diagonal. — Diagonal elements σk ≥ 0 of Σ are the singular values of G. σi = q λi(GHG) or σi = q λi(GGH) the positive ones are the same from both formulas. — The columns of the matrices U and V (ui and vj) are the associated eigenvectors GHGvj = σj 2 vj GGHui = σi 2 ui Gvi = σiui • If the rank(G) = r ≤ min(p, m), then — σk > 0, k = 1,..., r — σk = 0, k = r + 1,..., min(p, m) • An SVD gives a very detailed description of how a matrix (the system G) acts on a vector (the input w) at a particular frequency
Fal!2001 16.3121-4 So how can we use this result? Fix the size wcl2= 1 of the input, and see how large we can make the output Since we are working at a single frequency, we just analyze the relation ze=Guc,Gu≡G(s=ju) Define the maximum and minimum amplifications as 0≡maNx.‖zcll g≡min.‖ll Then we have that (let q=min(p, m)) qp≥mtal 0p<m“wide Can use O and g to determine the possible amplification and atten uation of the input signals Since G(s) changes with frequency, so will o and g
Fall 2001 16.31 21—4 • So how can we use this result? — Fix the size kwck2 = 1 of the input, and see how large we can make the output. — Since we are working at a single frequency, we just analyze the relation zc = Gwwc, Gw ≡ G(s = jω) • Define the maximum and minimum amplifications as: σ ≡ max kzck2 kwck2=1 σ ≡ min kzck2 kwck2=1 • Then we have that (let q = min(p, m)) σ = ½ σ1 σq p ≥ m “tall” σ = 0 p < m “wide” • Can use σ and σ to determine the possible amplification and attenuation of the input signals. • Since G(s) changes with frequency, so will σ and σ