Thesis 6
The detailed dynamic analysis of the sampled-data sliding oscillator, where the viscous damp- ing effects are also considered, results in the stability chart shown in the figure below. The corresponding dynamic model has the characteristic multipliers
z1,2 = 1
2(ε(1−P)−P θ+ 2)± 1 2
√
δ with ε= e−θ−1, where
δ=P2(e−2θ−2e−θ+ 1 + 2θε+θ2) +P(2θε−2(e−2θ−2e−θ+ 1)) + e−2θ−2e−θ+ 1.
0 2 4 6 8
Dimensionless sampling time - θ
Dimensionlessproportionalgain-P
0.0 0.5 1.0 1.5
2.0 θ≈3.721
P≈1.102 δ=0
0 2 4 6 8
Dimensionless sampling time -θ
Dimensionlessproportionalgain-P
0 2 4 6 8
Stability region without dry friction Extension of stability region
by dry friction
σ/x0
δ=0
Im
Re i
1 z
b2=0
Im Re i
1 z
b1=0
I
II III
Re -1
iIm 1
z
b0=0
In these expressions, the dimensionless proportional gain P =mkp/b2 and the dimensionless sampling time θ =bts/m, where m is the generalized mass, b is the generalized viscous damping coefficient, kp is the proportional control gain, and ts is the sampling time.
In this case, based on the detailed algebraic analysis of the characteristic equation of the model, the corresponding stable domain of control parameters is determined, and it is illustrated in the plane of the dimensionless sampling time θ and the dimensionless propor- tional gain P in the left panel of the figure below. In the left panel of this figure the colored regions represent the different stable domains of control parameters, where the characteristic multipliers z1,2 have the following properties:
• In region I, z1,2 ∈R, and 0<Re(z1,2)<1. The system shows first-order dynamics; no vibrations will develop, and at the boundary of stability saddle-node bifurcation occurs.
• In region II, z1,2 ∈C with Im(z1,2)6= 0. The system has similar oscillations as an under-damped second-order system, and at the boundary of stability Neimark-Sacker bifurcation occurs.
• In region III, z1,2 ∈R, and −1<Re(z1)<0 or −1<Re(z2)<0. The control force alternates, and at the boundary of stability period-doubling bifurcation occurs.
In the left stability chart above, the root separation curve is δ = 0, and the
different type of stability boundaries are defined by
b0 = 0 ⇐⇒ P = 2 eθ+ 1 eθ(θ−2) +θ+ 2 , b1 = 0 ⇐⇒ P = eθ−1
−θ+eθ−1, b2 = 0 ⇐⇒ P = 0.
The right panel shows that Coulomb friction stabilizes the motion at two different types of stability boundaries where Neimark-Sacker and period-doubling bifurcations take place. In both cases, the stability limit of the extended stable domain is at the critical dimensionless proportional control gain
Pcr= σ x0 coth
π 2
ln(ρ) ϑ
,
where ρ = abs (z1,2) and tan(ϑ) = arg(z1). In these expressions, σ = mfC/b2 where fC is the magnitude of the generalized dry friction force, and x0 is the initial position/perturbation with v0 = 0.
• Related publications: [BCs7, BCs8, BCs9, BCs10]
Appendix
Theorem and proof of the matrix sum determinant identity
Theorem: The determinant of the sum of two real element, two-by-two matrices can be expressed in the form
det(A+B) = detA+ detB+ tr(Aadj(B)). (A.1) Proof: By expanding the determinant of matrix sum, the left-hand-side of Eq. (A.1) becomes det(A+B) = (a11+b11)(a22+b22)−(a12+b12)(a21+b21), (A.2) and the right-hand-side is:
detA+ detB+ tr(Aadj(B)) = a11a22−a12a21+b11b22−b12b21+ +a11b22+b11a22−a12b21−b12a21.
(A.3)
Because of the left-hand-side is equal to the right-hand-side the statement in Eq. (A.1) is true.
Properties:
tr(Aadj(B)) = tr(Badj(A)) = tr(adj(B)A) = tr(adj(A)B) (A.4) It can be proven as the same way, then it was in case of the Theorem.
Application: This theorem can be applied during the derivation of the characteristic poly- nomial with the following substitutions A =−zI and B =W. It follows, the characteristic polynomial is
det(−zI) + detW+ tr(−zIadj(A)) =z2−ztrW+ detW, (A.5) because of
tr(−zIadj(A)) =−tr(Iadj(A)) = tr(adj(A)) = tr(A). (A.6)
Applying the Moebius transformation z= (w+ 1)/(w−1), the characteristic equation is reformulated, to the form b2w2+b1w+b0 = 0. In which, the coefficient of the monomials of the characteristic polynomial can be given in matrix-array form
b2
b1 b0
=
1 −1 1
2 0 −2
1 1 1
1 trW detW
=
1−trW+ detW 2−2 detW 1 + trW+ detW
. (A.7)
If the new coefficients are equal to zero, it follows
b2 b1 b0
=
0 0 0
⇔
trW−detW detW
−trW−detW
=
z1+z2−z1z2 z1+z2
−(z1+z2)−z1z2
=
1 1 1
. (A.8)
Based on this, it can be seen the followings: if b2 = 0, thenz1,2 = 1; ifb0 = 0, thenz1,2 =−1 and if b1 = 0, then z1,2 ∈C with Im(z1,2)6= 0 and abs (z1,2) = 1.
Equation (2.33), neglecting the effect of Coulomb friction, i.e.,fC = 0, and using that the control force in the form fu=Rzj, then
zj+1 =Adzj +BdRzj = (Ad+BdR)zj =Wzj, (A.9) where
Ad =
1 ts 0 1
, Bd =
t2s/(2m) ts/m
and RT =
−kp 0
. (A.10)
Therefore, applying the Theorem,
detW= detAd+ det (BdR) + tr(BdRadj(Ad)) = detAd+ tr(BdRadj(Ad)), (A.11) because of det(BdR) is always zero. Using the matrices in Eq. (A.10),
detAd = 1 and tr(BdRadj(Ad)) =kpt2s/(2m). (A.12) It follows
detW = 1 +t2s/(2m)kp = 1 +p/2. (A.13) Furthermore,
trW= trAd+ tr(BdR) = 2−kpt2s/(2m) = 2−p/2 (A.14) If these expressions are substituted back into the Eq. (A.8), it follows
b2 b1 b0
=
0 0 0
⇔
trW−detW detW
−trW−detW
=
1−p 1 +p/2
−3
=
1 1 1
⇒
b2
b1
⇔
p= 0 p= 0
. (A.15)
It follows,b0 never equals to zero, therefore there is no limit with period-doubling bifurcation [76]. Furthermore, if p= 0 then b1 =b2 = 0, therefore in this casez1,2 = 1.
List of Symbols
Here below, the most important symbols and notations are summarized.
Roman letters
A,W, W,c Wf state matrix
b viscous damping coefficient
bR effective viscous damping coefficient due to the armature resistance bC effective viscous damping coefficient due to the Coulomb friction C+,C−, fC+, fC− magnitude of the generalized Coulomb friction force
Cn, c1,c1 constant
e induced voltage, back electro-motive force fu generalized driving force
f0+, f0− boundary of the sticking/sliding region Fs sliding vector field
F1, F2 vector field
I identity matrix
J,JR second moment of inertia
Km DC motor torque constant
Ke DC motor electric constant
kL effective torsion stiffness due to the armature inductance kt torsion stiffness due to the gravity
kp proportional control gain
L armature inductance
m generalized mass
N(V) describing function
P1, P2 Jacobian of the Poincar´e map
R armature resistance
ts sampling time
u control force
u0 input voltage
xj, vj state variable x, y,z, zj state vector
z shifting operator
Greek letters
δu, δmax duty-cycle ζ damping ratio
θ dimensionless sampling time
τC magnitude of the Coulomb friction torque τm electric torque of DC motor
τu control torque τfr friction torque
τL torque due to the external load ϕd desired angular position
ϕ angular position
φ, Φ amplitude decay function Φ1, Φ2 trajectory
ω angular velocity
ωn natural angular frequency
˙
ω angular acceleration
¨
ω angular jerk
4.1 Introductory model (left) and asymmetric Coulomb friction model (right) . . 49 4.2 Phase portrait of the piecewise smooth system . . . 51 4.3 The ts−kp stability chart with x(0) = 0 and ˙x(0) = v0 . . . 58 5.1 Dimensionless stability chart with viscous damping . . . 63 5.2 Dimensionless stability chart considering viscous damping and dry friction . 65 5.3 Stability chart considering viscous damping and dry friction . . . 66 5.4 Simulation results at points A-C . . . 67 5.5 Simulation results at points D-F . . . 68
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List of Related Publications
[BCs1] Cs. Budai and L. L. Kov´acs. Limitations caused by sampling and quantization in position control of a single axis robot. In Proceedings of the XV. International PhD Workshop, pages 466–471, Wisla, Poland, October 2013.
[BCs2] Cs. Budai and L. L. Kov´acs. Robotok poz´ıci´o szab´alyoz´as´anak k´ıs´erleti sta- bilit´asvizsg´alata. In Proceedings of the Tavaszi Sz´el Konferencia, pages 11–20, De- brecen, Hungary, March 21–23, 2014.
[BCs3] G. Gyebr´oszki, G. Csern´ak, and Cs. Budai. Experimental investigation of micro- chaos. In Proceedings of the 8th European Nonlinear Dynamics Conference, pages 1–6, Vienna, Austria, July 6–11, 2014. Paper ID: 476.
[BCs4] Cs. Budai and L. L. Kov´acs. Friction effects on stability of a digitally controlled pendulum. Periodica Polytechnica Mechanical Engineering, 59(4):176–181, 2015.
[BCs5] Cs. Budai,L. L. Kov´acs, and J. K¨ovecses. Analysis of the effect of coulomb friction on haptic systems dynamics. In Proeedings of the International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Charlotte, NC, USA, August 21–24, 2016. no. IDETC2016-59961.
[BCs6] Cs. Budai, L. L. Kov´acs, J. K¨ovecses, and G. St´ep´an. Effect of dry friction on vibrations of sampled-data mechatronic systems. Nonlinear Dynamics, 88(1):349–
361, 2016.
[BCs7] Cs. Budaiand B. Szil´agyi. About a few elegant formulae in the matrix calculus. In Proceedings of the Matematik´at, Fizik´at ´es Informatik´at Oktat´ok XXXVII. Orsz´agos Konferenci´aja, pages 25–30, Miskolc, Hungary, August 26–28, 2013.