Singular Value Decomposition Analysis


 Clifton Smith
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1 Singular Value Decomposition Analysis Singular Value Decomposition Analysis Introduction Introduce a linear algebra tool: singular values of a matrix Motivation Why do we need singular values in MIMO control designs? Definition and properties of singular values Singular value decomposition (SVD) provides directional information 376_069 Multivariable feedback control V3 1 of 36
2 Singular Value Decomposition Analysis SISO sinusoidal steadystate (I) u(s) g(s) y(s) Assume: g(s) strictly stable Sinusoidal steadystate u(t) = ue jωt y(t) = ye jωt y = g(jω)u Note g(jω) is a complex scalar g(jω) = g(jω) e jφ(ω) Magnitude: g(jω) = g*(jω)g(jω) Phase: φ(ω) = arctan Im( g(j ω)) Re( g (j ω )) Above plotted in Bode plot defines frequency response of SISO plant g(s) g(jω) defines plant gain at frequency ω 376_069 Multivariable feedback control V3 2 of 36
3 Singular Value Decomposition Analysis SISO sinusoidal steadystate (II) Input with complex amplitude u(t) = ue jωt u = u e jψ u(t) = u ej(ωt + ψ) Interpretation Re{u(t)} = u cos(ωt + ψ) Im{u(t)} = u sin(ωt + ψ) Complex steadystate output y(t) = y e jωt y = g(jω) u = g(jω) e jφ(ω) u e jψ = g(jω) u ej(φ(ω) + ψ) y = g(jω) u Re{y(t)} = y cos(ωt + ψ + φ(ω)) Im{y(t)} = y sin(ωt + ψ + φ(ω)) 376_069 Multivariable feedback control V3 3 of 36
4 Singular Value Decomposition Analysis SISO Bode plot information Provides graphical "summary" of plant gain at different frequencies Concepts of "small" and "large" gain are clear g(jω) >> 1 g(jω) << 1 large gain small gain 376_069 Multivariable feedback control V3 4 of 36
5 Singular Value Decomposition Analysis MIMO sinusoidal steadystate u(s) g(s) y(s) Assume: G(s) strictly stable Sinusoidal inputs generate at steadystate sinusoidal outputs Sinusoidal steadystate u(t) = ue jωt ;u C m y(t) = ye jωt ;y C p y = G(jω)u G(jω) : p x m complex matrix Need notion of size of G(jω) vs. frequency want visualize MIMO gain on Bode plot 376_069 Multivariable feedback control V3 5 of 36
6 Singular Value Decomposition Analysis Issues Deal with complex vectors and complex matrices. How to quantify "large" and "small" Impact of directions y(s) = G(s) u(s) u(t) = ue jωt ;u C m y(t) = ye jωt ;y C p y = G(jω)u Direction and size of u and Plant properties at frequency ω yield Direction and size of y Singular value decomposition (SVD) provides the "tool" for analysis 376_069 Multivariable feedback control V3 6 of 36
7 Singular Value Decomposition Analysis Notes on complex vectors x is a complex vector; x C n x = x 1 x 2: x n x i = a i + jb i ; i = 1, 2,..., n Magnitude x = x 2 x H = [x* 1 x* 2... x* n ] x 2 = x H x Example 1 j + x = 2 3j ;x H = [1  j 2 + j3] x H x = (1  j)(1 + j) + (2 + j3)(2  j3) = 15 x 2 = _069 Multivariable feedback control V3 7 of 36
8 Singular Value Decomposition Analysis Notes on complex matrices A is a n x m matrix with complexvalued elements: a ik = α ik + jβ ik Notation A H : complexconjugate transpose of A A H is a m x n matrix Note: A H A m x m matrix AA H n x n matrix Fact: AA H and A H A have real nonnegative eigenvalues Example 1 1+ j A = j 2+ j A H 1 = j 1 j 2 j det(λι  A H A) = λ 29λ + 5 ;λ 1 = 8.41 ;λ 2 = 0.59 det(λι  A H A) = det(λι  AA H ) In this example λ i [A H A] = λ i [AA H ] > 0 376_069 Multivariable feedback control V3 8 of 36
9 Singular Value Decomposition Analysis Definition of singular values A is a n x m complex matrix Suppose : rank(a) = k Notation σ i (A): singular value of A Definition: The strictly positive square roots of the nonzero eigenvalues of A H A ( and AA H equivalently), are the singular values of A σ i (A) = λ i [A H A] = λ i [AA H ] > 0 i = 1, 2,..., k 376_069 Multivariable feedback control V3 9 of 36
10 Singular Value Decomposition Analysis The singular value decomposition A is a n x m complex matrix: rank(a) = k σ 1 σ 2... σ k 0 : are singular values of A Σ = σ σ σ k Σ n x m real matrix SVD A = U Σ V H A = U H Σ V U : n x n unitary matrix (U H = U 1 ) V : m x m unitary matrix (V H = V 1 ) Column vectors of U and V are orthonormal 376_069 Multivariable feedback control V3 10 of 36
11 Singular Value Decomposition Analysis More on SVD SVD A = U Σ V H A = U H Σ V U : n x n unitary matrix (U H = U 1 ) U = [u 1 u 2... u n ] ;u H i u j = δ ij u i : Left singular vectors of A (is right eigenvector of AA H associated with λ i [AA H ] ) V : m x m unitary matrix (V H = V 1 ) V = [v 1 v 2... v m ] ;v H i v j = δ ij v i : Right singular vectors of A (is right eigenvector of A H A associated with λ i [A H A] ) 376_069 Multivariable feedback control V3 11 of 36
12 Singular Value Decomposition Analysis Geometric interpretation A complex n x n matrix A 1 exists λ i (A) 0 Consider linear transformation y = Ax ;x, y R n x A y Euclidean norm x 2 = x'x ; y 2 = y'y Spectral norm of matrix A A 2 = max x 0 Ax 2 x 2 = max Ax x = Singular value relations σ max ( A) = max x 2 =1 Ax 2 = A 2 σ min ( A) = min x 2 =1 Ax 2 = 1 A _069 Multivariable feedback control V3 12 of 36
13 Singular Value Decomposition Analysis Graphical visualization Real case : y = Ax ; y, x R n, A R nxn ; n=2 SVD A = U Σ V H Σ = σ 1 = σ max 0 0 σ 2 =σ min U = [u 1 u 2 ] V = [v 1 v 2 ] Vector OD = v 1 ; Vector OD" = u 1 Vector OE = v 2 ; Vector OE" = u 2 Length of vector OD' = σ max = σ 1 Length of vector OE' = σ min = σ 2 376_069 Multivariable feedback control V3 13 of 36
14 Singular Value Decomposition Analysis MIMO frequency response u(t) = ue jωt G(s) y(t) = ye jωt Restrict u to unit (complex) sphere, i.e. u 2 = 1 u i (t) is complex sinusoid, e.g. u i (t) = u i e jψ ie jωt Re{u i (t)} = u i cos(ωt + ψ i ) Output response y(s) = G(s)u(s) Singular values σ max (G(jω)) = max G(j ω)u u = = y max (ω) 2 σ min (G(jω)) = min G(j ω)u u = = y min (ω) 2 Maximum and minimum singular values of G(jω) define max and min amplification of unit sinusoidal input at frequency ω 376_069 Multivariable feedback control V3 14 of 36
15 Singular Value Decomposition Analysis Bode plot visualization 376_069 Multivariable feedback control V3 15 of 36
16 Singular Value Decomposition Analysis Discussion The concept of singular values will be heavily exploited in analysis and design of MIMO feedback systems Correct interpretation of singular value plot hinges on units of physical variables (scaling) Singular value results assume "roundness" (convexity) of input signal space u(t) = ue jωt G(s) y(t) = ye jωt Input Space Output Space 376_069 Multivariable feedback control V3 16 of 36
17 Feedback Performance Specifications in the Frequency Domain Feedback Performance Specifications in the Frequency Domain Introduction Use singular values to establish nature of mimo performance specs in frequency domain Performance attributes  command following  disturbance rejection  insensitivity to sensor noise Stabilityrobustness to be addressed later 376_069 Multivariable feedback control V3 17 of 36
18 Feedback Performance Specifications in the Frequency Domain Fundamental relations True tracking error: e(s) = r(s)  y(s) Loop TFM: L(s) L(s) = G(s)K(s) Sensitivity TFM: S(s) S(s) = [Ι + L(s)] 1 Closedloop TFM: T(s) T(s) = [Ι + L(s)] 1 L(s) Performance equation e(s) = S(s)[r(s)  d(s)] + T(s)n(s) Constraint: T(s) + S(s) = Ι 376_069 Multivariable feedback control V3 18 of 36
19 Feedback Performance Specifications in the Frequency Domain Command following (I) Sinusoidal command yields sinusoidal error r(t) = re jωt e(t) = ee jωt Relation: e = S(jω)r e 2 σ max [S(jω)] r 2 Ω r range of frequencies where command input has energy Prescription for good command following make σ max [S(jω)] << 1 for all ω Ω r 376_069 Multivariable feedback control V3 19 of 36
20 Feedback Performance Specifications in the Frequency Domain Command following (II) Interpretation for unit command sinusoid r(t) = re jωt ; r 2 = 1 Worst error at frequency ω e(t) = ee jωt e 2 = σ max [S(jω)] Attained when r points along right singular vector associated with σ max Best error at frequency ω e 2 = σ min [S(jω)] Attained when r points along right singular vector associated with σ min In general σ min [S(jω)] e 2 σ max [S(jω)] 376_069 Multivariable feedback control V3 20 of 36
21 Feedback Performance Specifications in the Frequency Domain Command following (III) Objective: express prescription for good command following in terms of loop TFM L(s) = G(s)K(s) Singular value facts σ max [A 1 ] = σ min 1 [ A] σ min [A]  1 σ min [Ι + A] σ min [A] + 1 Recall: Need σ max [S(jω)] << 1 ;ω Ω r But S(jω) = [Ι + L(jω)] 1 σ max [S(jω)] = 1 << 1 σ min I +L(jω ) σ min [Ι + L(jω)] >> 1 ;ω Ω r But σ min [Ι + L(jω)] < σ min [L(jω)] + 1 Need σ min [L(jω)] >> 1 ;ω Ω r For good command following make σ min [G(s)K(s)] >> 1 for all ω Ω r 376_069 Multivariable feedback control V3 21 of 36
22 Feedback Performance Specifications in the Frequency Domain Disturbance rejection (I) Sinusoidal disturbance yields sinusoidal error d(t) = de jωt e(t) = ee jωt Relation e = S(jω)d e 2 σ max [S(jω)] d 2 Ω d range of frequencies where disturbance inputs has energy Prescription for good disturbance rejection make σ max [S(jω)] << 1 for all ω Ω d or make σ min [G(s)K(s)] >> 1 for all ω Ω d 376_069 Multivariable feedback control V3 22 of 36
23 Feedback Performance Specifications in the Frequency Domain Disturbance rejection (II) Interpretation for unit disturbance sinusoid d(t) = de jωt ; d 2 = 1 Worst error at frequency ω e 2 = σ max [S(jω)] Attained when d points along right singular vector associated with σ max Best error at frequency ω e 2 = σ min [S(jω)] Attained when d points along right singular vector associated with σ min In general σ min [S(jω)] e 2 σ max [S(jω)] 376_069 Multivariable feedback control V3 23 of 36
24 Feedback Performance Specifications in the Frequency Domain Quantitative relations Loop TFM: L(jω) = G(jω)K(jω) Sensitivity TFM: S(jω) = [Ι + L(jω)] 1 Closedloop TFM: T(jω) = [Ι + L(jω)] 1 L(jω) Ω p = Ω r Ω d Key relations Let 0 < δ << 1 If σ max [S(jω)] δ 1 ; all ω Ω p Then 1 << 1 δ δ σ min [L(jω)] and ; all ω Ω p 1  δ σ min [T(jω)] σ max [T(jω)] 1 + δ ; all ω Ω p T(jω) Ι  Proofs: not in this course 376_069 Multivariable feedback control V3 24 of 36
25 Feedback Performance Specifications in the Frequency Domain Comment Good command following and good disturbance rejection served by similar requirements ω Ω p = Ω r Ω d Large loop gain σ min [L(jω)] >> 1 Small sensitivity σ max [S(jω)] << 1 Flat closedloop response σ min [T(jω)] σ max [T(jω)] 1 376_069 Multivariable feedback control V3 25 of 36
26 Feedback Performance Specifications in the Frequency Domain Insensitivity to sensor noise Sinusoidal noise yields sinusoidal error n(t) = ne jωt e(t) = ee jωt Relation e = T(jω)n e 2 σ max [T(jω)] n 2 Ω n range of frequencies where noise has significant energy Prescription for good sensor noise rejection make σ max [T(jω)]<<1 for all ω Ω n 376_069 Multivariable feedback control V3 26 of 36
27 Feedback Performance Specifications in the Frequency Domain Conflict with performance Let 0 < γ << 1 Suppose that Then σ max [T(jω)] γ for all ω Ω n σ min [L(jω)] σ max [L(jω)] γ 1γ γ ω Ω n and low loop gain for all ω Ω n γ σ min [S(jω)] σ max [S(jω)] large sensitivity for all ω Ω n Bad command following and disturbance rejection in frequency range ω Ω n Consequence of constraint S(s) + T(s) = Ι 376_069 Multivariable feedback control V3 27 of 36
28 Feedback Performance Specifications in the Frequency Domain Design implications Need wide frequency separation between sets Ω p = Ω r Ω d and Ω n Cannot do good command following and disturbance rejection with noisy sensors that make low frequency errors (drift, bias, etc.) Stabilityrobustness to unmodelled highfrequency dynamics, faraway nonminimum phase zeros, and neglected small timedelays impact design in the same way as region Ω n 376_069 Multivariable feedback control V3 28 of 36
29 Feedback Performance Specifications in the Frequency Domain 376_069 Multivariable feedback control V3 29 of 36
30 Directional Information in Singular Value Plots Directional Information in Singular Value Plots Introduction Plots of min and max singular values vs. Frequency provide valuable insight into frequency domain properties of mimo systems Singular value decomposition provides directional information left singular vectors right singular vectors Need to understand and exploit directional information 376_069 Multivariable feedback control V3 30 of 36
31 Directional Information in Singular Value Plots SVD and linear equations y = Ax x A y SVD A = U Σ V H y = U Σ V H x Suppose : x = v i (right singular vector) y = U Σ V H v i Note (since v H i v j = δ ij ) V H v i = 0 : 1 : 0 (1 in row i) ; ΣV H v i = 0 : σ i : 0 (σ i in row i) then y i = σ i u i (u i left singular vector) Visualization 376_069 Multivariable feedback control V3 31 of 36
32 Directional Information in Singular Value Plots SVD directional information (I) Maximum singular value σ 1 (A) = σ max (A)  Associated max right singular vector v max = v 1 ; v max 2 = 1  Associated max left singular vector u max = u 1 ; u max 2 = 1 Max amplification direction Let If Then y = Ax x = v max y = σ max u max y 2 = σ max (A) ;max amplification Visualization y Vmax = x Umax 376_069 Multivariable feedback control V3 32 of 36
33 Directional Information in Singular Value Plots SVD directional information (II) Minimum singular value: Rank(A)= m σ m (A) = σ min (A)  Associated min right singular vector v min = v m ; v min 2 = 1  Associated min left singular vector u min = u m ; u min 2 = 1 Min amplification direction Let If Then y = Ax x = v min y = σ min u min y 2 = σ min (A) ;min amplification Visualization Vmin = x Umin y 376_069 Multivariable feedback control V3 33 of 36
34 Directional Information in Singular Value Plots Utilizing SVD directional information System: y(s) = G(s)u(s) ;G(s) m x m matrix Pick ω, calculate G(jω), do SVD G(jω) = U(jω) Σ(jω) V H (jω) Maximum direction analysis  Find σ max (ω), v max (ω), u max (ω)  Write [v max (ω)] i = a i e jψ i  Write [u max (ω)] i = b i e jφ i  Apply input u i (t) = a i sin(ωt + ψ i ) with u(t) = (u 1 (t); ;u i (t); u m (t)) then y i (t) = σ max b i sin(ωt + φ i ) Minimum directional analysis similar Complex directions of right and left singular vectors correspond to sinusoidal vectors with relative phaseshift among their elements 376_069 Multivariable feedback control V3 34 of 36
35 Directional Information in Singular Value Plots DC gain matrix analysis SVDbased direction analysis easiest at DC because plant is real (ω = 0) Plant model (strictly stable) dx(t)/dt = Ax(t) + Bu(t) y(t) = Cx(t) G(s) = C(sΙ  A) 1 B Plant DC gain matrix s = jω = 0 G(0) =  CA 1 B At steadystate u(t) = u = real constant vector y(t ) = y = real constant vector y = G(0)u or y =  CA 1 Bu 376_069 Multivariable feedback control V3 35 of 36
36 Directional Information in Singular Value Plots Steadystate analysis SVD at G(0) SVD at DC y = G(0)u G(0) =  CA 1 B = real G(0) = UΣV H (U, Σ, V = real) Max amplification direction If u = v max ; u 2 = 1 Then y = σ max u max Min amplification direction If u = v min ; u 2 = 1 Then y = σ min u min Above provides valuable insight upon MIMO plant characteristics at DC 376_069 Multivariable feedback control V3 36 of 36
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