Collocated versus non-collocated control [H04Q7]
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1 Collocated versus non-collocated control [H04Q7] Jan Swevers September
2 Contents Some concepts of structural dynamics Collocated versus non-collocated control Summary This lecture is based on parts taken from Vibration control of active structures (2nd edition) by A. Preumont, Kluwer Academic Publishers
3 Some concepts of structural dynamics Equation of motion of a lumped parameter system (i.e. discrete system) Vibration modes Modal decomposition Transfer function of collocated systems
4 Equation of motion of a lumped parameter system Consider the following system: Equations of motion: M m m ẍ 1 ẍ 2 ẍ 3 + c c 0 c 2c c 0 c c ẋ 1 ẋ 2 ẋ 3 + k k 0 k 2k k 0 k k x 1 x 2 x 3 = f 0 0
5 The general form of the equation of motion for a (non-gyroscopic) discrete flexible structure with a finite number of DOF s: Mẍ + Cẋ + Kx = f M, C, and K: mass, damping and stiffness matrices (symmetric and semi positive definite)
6 Vibration modes Free response of the undamped system : Mẍ + Kx = 0 If one tries a solution of the form x = φ i e jω it, φ i and ω i must satisfy the following eigenvalue problem: Properties: (K ω 2 i M)φ i = 0 the eigenvalues ω 2 i are real and non negative ω i are the natural frequencies φ i are the corresponding modes
7 Define the matrix of all mode shapes as: [ ] Φ = φ 1 φ 2... φ n Due to orthogonality conditions: Φ T MΦ = diag(µ i ) Φ T KΦ = diag(µ i ω 2 i ) µ i are the modal masses (modes can be scaled such that the µ i s are equal to one).
8 Modal decomposition Let us perform a change of variables from the physical coordinates x to modal coordinates z: x = Φz yielding: MΦ z + CΦż + KΦz = f Left multiplying by Φ T, and using orthogonality relationships: diag(µ i ) z + Φ T CΦż + diag(µ i ω 2 i )z = Φ T f If matrix Φ T CΦ is diagonal, the damping is said to be normal: the modal damping is defined as: Φ T CΦ = diag(2ξ i µ i ω i )
9 In case of modal damping, the modal equations are decoupled: with z + 2ξΩż + (Ω 2 )z = µ 1 Φ T f ξ = diag(ξ i ), Ω = diag(ω i ), µ = diag(µ i ) Frequency domain interpretation: dynamic flexibility matrix or yielding: or X = [ ω 2 M + jωc + K ] 1 F = G(ω)F { } 1 Z = diag µ i (ωi 2 Φ T F ω2 + 2jξ i ω i ω) { } 1 X = ΦZ = Φdiag µ i (ωi 2 Φ T F ω2 + 2jξ i ω i ω) G(ω) = with φ i the i-th column of φ. n i=1 φ i φ T i µ i (ω 2 i ω2 + 2jξ i ω i ω)
10 If the structure has NO rigid body modes, we can evaluate the system for ω = 0: n G(0) = K 1 φ = i φ T i µ i ωi 2 i=1 If we consider only frequencies ω < ω b << ω m, we get: G(ω) m i=1 m i=1 φ i φ T n i µ i (ωi 2 ω2 + 2jξ i ω i ω) + m+1 φ i φ T i µ i ω 2 i φ i φ T i µ i (ω 2 i ω2 + 2jξ i ω i ω) + K 1 m 1 φ i φ T i µ i ω 2 i The static contribution of the high frequency modes to the flexibility matrix is called the residual mode, denoted by R.
11 Structure with r rigid bodies (ω i = 0, for i = 1,...,r): G(ω) r i=1 T φ i φ m i µ i ω 2 + i=r+1 φ i φ T i µ i (ω 2 i ω2 + 2jξ i ω i ω) + R with R = n m+1 φ i φ T i µ i ω 2 i
12 Transfer function of collocated systems What is a collocated system? Dynamics are described by the diagonal elements of the dynamic flexibility matrix (for undamped system): G kk (ω) = r i=1 φ 2 i (k) m µ i ω 2 + i=r+1 φ 2 i (k) µ i (ω 2 i ω2 ) + R kk G kk is real and : dg kk (ω 2 ) dω 2 0 φ i (k) is the k-th element of the i-th column of matrix φ i.
13 In contrast to the resonance frequencies, the anti-resonance frequencies do depend on the actuator location. There will be just one anti-resonance between two consecutive resonances. For non-collocated actuator - sensor systems, the numerators of the various terms in the modal expansion of G kl (ω) become φ i (k)φ i (l): they can be positive or negative, such that the above properties are lost.
14 Pole/zero pattern of a structure with collocated actuator and sensor (a) undamped, (b) lightly damped Nyquist and Bode diagram of a lightly damped collocated system
15 each flexible mode introduces a circle in the Nyquist diagram more or less centered on the imaginary axis which is intersected at ω = ω i and ω = ω oi.
16 Collocated versus non-collocated control Collocated actuator and sensor pairs for lightly damped flexible structures leads to alternating poles and zeros near the imaginary axis. SISO control systems based on this: very robust (root locus techniques) This property does not hold for non-collocated control: root locus may experience severe alterations for small parameter changes, e.g. pole-zero flipping
17 Pole-zero flipping Root locus: locus of the solutions s of the closed-loop characteristic equation: when g goes from 0 to 1 + gg(s)h(s) = 0 G(s)H(s) = k m i=1 (s z i) n i=1 (s p i) Any point P on the locus is such that: m φ i i=1 n ψ i = l 360 i=1 with φ i and ψ i the angles of the vectors joining the zeros z i and poles p i to P respectively.
18 Departure angles from poles and arrival angles at zeros before and after zero-pole flipping: Pole-zero flipping may occur in two different ways: There are compensator zeros near to system poles (notch filter): if the actual poles of the system are different from those assumed in the compensator design, a pole-zero flipping may occur. Some actuator/sensor configurations may produce flipping within the system alone, for small parameter changes. This is not possible if the actuator and sensor are collocated.
19 Collocated control G 1 (s) = Y (s) F(s) = s 2 + 2ξω 0 s + ω 2 0 Ms 2 (s 2 + (1 + µ)(2ξω 0 s + ω 2 0 )) G 2 (s) = D(s) F(s) = 2ξω 0 s + ω 2 0 Ms 2 (s 2 + (1 + µ)(2ξω 0 s + ω 2 0 )) ω 2 0 Ms 2 (s 2 + (1 + µ)(2ξω 0 s + ω 2 0 )) with ω 2 0 = k m, µ = m M, 2ξω 0 = b m
20 The approximation for G 2 (s) is valid for low damping ξ << 1: the far away zero will not influence the closed-loop response. When the mass ratio µ is small, numerator and denominator of G 1 (s) are almost equal (except for Ms 2 ): pole-zero cancellation
21 Consider a collocated lead compensator: H(s) = g Ts + 1, (α < 1) αts + 1
22 The lead compensator always increases the damping of the flexible mode For several flexible modes: always as many flexible pole-pairs as flexible zero-pairs pole-zero excess remains 2: angles of asymptotes remains 90 deg lead compensator increases the damping of all flexible modes, but especially those having their natural frequency between the pole and the zero of the compensator
23 Consider the same lead compensator, applied in non-collocated control configuration The pole-zero excess is 3: the flexible modes are heading towards the asymptotes at ±60 deg in the right half plane
24 Even with small bandwidth, the gain margin is extremely small.
25 Pole-zero flipping in the structure Consider the first system: M m m ẍ 1 ẍ 2 ẍ 3 + c c 0 c 2c c 0 c c ẋ 1 ẋ 2 ẋ 3 + k k 0 k 2k k 0 k k x 1 x 2 x 3 = f 0 0 In Laplace form (with c = 0): Ms 2 + k k 0 k ms 2 + 2k k 0 k ms 2 + k X 1 X 2 X 3 = F 0 0
26 This yields: G 1 (s) = X 1(s) F(s) = s 4 + 3ω 2 0s 2 + ω 4 0 Ms 2 (s 4 + (3 + µ)ω 2 0 s2 + (1 + 2µ)ω 4 0 ) G 2 (s) = X 2(s) F(s) = ω 2 0(s 2 + ω 2 0) Ms 2 (s 4 + (3 + µ)ω 2 0 s2 + (1 + 2µ)ω 4 0 ) G 3 (s) = X 3(s) F(s) = ω 4 0 Ms 2 (s 4 + (3 + µ)ω 2 0 s2 + (1 + 2µ)ω 4 0 ) with ω 2 0 = k/m and µ = m/m.
27 Properties Poles: p 2 ω0 2 = (3 + µ) ± 5 2µ + µ 2 2 G 1 has two pairs of zeros, independent of the mass ratio For µ = 0, the poles and zeros of G 1 cancel each other. For µ > 0, the poles and zeros alternate on the imaginary axis (collocated system). G 2 has one pair of zeros, G 3 has no zeros.
28 Evolution of the poles with the mass ratio µ: The zeros do not depend on µ. There is a pole-zero flipping in G 2 at µ = 1. G 2 (s) ω 2 0(s 2 + ω 2 0) Ms 2 (s 2 + ω 2 0 (1+µ 2 ))(s2 + ω 2 0 (5+µ 2 ))
29
30 Effect on Bode plot : A pole-zero flipping near the imaginary axis produces a phase change of 360. m i=1 G(jω)H(jω) = k (jω z i) n i=1 (jω p i) The phase of GH for a specific value of jω is given by: m i=1 φ i n i=1 ψ i, with
31 Relation to the mode shapes Evolution of the zeros of a simple supported beam with a point force actuator at 0.1l
32 As the sensor moves away from the actuator, the zeros migrate along the imaginary axis. When the sensor reaches 0.2l, which is the nodal point of mode 5, the fourth zero becomes identical to jω 5. The third zero crosses jω 4 at 0.25l, node of mode 4. Etc...
33 Relation to the mode shapes: non-minimum phase systems Evolution of the zeros of a beam when the sensor moves away from the actuator
34 While the imaginary zeros migrate along the imaginary axis, every pair of zeros that disappears at infinity, reappears symmetrically at infinity on the real axis and moves towards the origin. Right-half plane zeros are non-minimum phase zeros. Non-minimum phase zeros do not cause difficulties if they lie well outside the desired bandwidth of the closed-loop system. They put severe restrictions on the control system if they interfere with the bandwidth.
35 Summary Difference between collocated and non-collocated systems. Influence on stability and robustness properties, pole/zero location. These properties play a role in feedback and feedforward control design.
36 Assignment 1: Collocated versus Non-collocated control and effect of pole-zero flipping Control of three mass-system of slide 3 Data: M = 1kg, ω 0 = = 1rad/s, c = 0.005Ns/m. Input: Force on first mass k m Task: Design a lead-compensator with phase-margin larger than 30 o, and maximal bandwidth for following systems: System 1: µ = m M System 2: µ = m M System 3: µ = m M System 4: µ = m M = 1.2 and Output: position of mass 1 (collocated) = 1.2 and Output: position of mass 2 (non-collocated) = 0.5 and Output: position of mass 1 (collocated) = 0.5 and Output: position of mass 2 (non-collocated) Discuss the results based on bode-diagram and root-locus plot.
37 Assignment 2: Pole-zero flipping in simply supported beam Reproduce figure of slide 30 for the following beam: l = 1m, b = 0.02m, h = 0.005m, ρ = 7500kg/m 3, E = N/m 2 Position of actuator: x a = 0.1m Useful formulas: Moment of inertia: I = bh3 12 Eigenfrequencies: ωn 2 = (nπ) 4 EI ml 4 Mode shapes: Φ n (x) = sin ( nπx l Frequency response function: G(s) = i=1 and x s the position of the sensor Discuss the results ) Φ i (x a )Φ i (x s ) µ(s 2 +ω 2 i ) with µ = ml 2 Hint: Calculate zeros of G(s), based on approximation with finite number of modes
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