In order to analyze the stability of nonlinear systems passivity based techniques are often used in the literature [66, 67]. These analyze the controlled mechanical system from the energetic point of view by looking at the balance of the energy being generated due to sampling and the energy dissipated by the dissipative effects such as friction.
According to reference [57,58], the stability condition of the sampled-data sliding oscillator model, described in Eq. (3.1), can be given in the form
− kpts
2 + fC
vmax >0, (3.15)
where vmax denotes the maximum magnitude velocity during the motion. When the motion is initiated from zero position x(0) =x0 = 0 with positive initial velocity ˙x(0) =v0 >0, then the maximum velocity becomes vmax=v0. With this, the passivity-based stability condition of Eq. (3.1) can be given the form
−kpts 2 + fC
v0 >0 ⇐⇒ kp < 2fC v0
1
ts. (3.16)
The passivity-based stability condition in Eq. (3.16) is compared to the stability chart obtained via discrete numerical simulation (see Sec. 2.3) is presented in Fig. 3.1. For the calculations, the parameters collected in Tab. 3.1 were used.
Table 3.1: Parameters for calculations
m 0.1 kg x0 0 m
fC 1 N v0 0.5 m/s
In Fig. 3.1, the green dotted domain represents the stable operating region and the dashed blue line corresponds to the passivity condition in Eq. (3.16). It is easy to see that the passivity-based stability condition gives a conservative approximation of the exact stability limit.
Previous passivity-based condition
x(0) = 0
˙
x(0) =v0
0 2 4 6 8 10
Sampling time - ts [ms]
0 1000 2000 3000 4000 5000
Proportionalgain-kp[N/m]
Exact numerical simulation
Figure 3.1: The ts−kp stability chart withx(0) = 0 and ˙x(0) =v0
3.2.1 Effective viscous damping
Apart of passivity-based techniques, the method of describing functions can also be useful to model the non-linearities in a quasi-linear form [68, 69] and to make the dynamic analysis of the system possible by using conventional analysis methods in classical control theory.
Using the method of describing function (DF) analysis, the effect of dry friction can be approximated. The application of this method leads to an effective, frequency and amplitude dependent transfer function in general, called the describing function [68, 69].
Focusing on the approximation of the effect of dry friction, the equation of motion is considered in the form
m¨x(t) = u(t)−fCsgn ( ˙x(t)), (3.17) where u denotes the control force. This system is represented as a form of a block-diagram at the top of Fig. 3.2.
When in this system, self-sustained vibration occurs, its position can be given in the form of a harmonic function x(t) = Asin(ωt). Similarly, the velocity changes according to the harmonic function ˙x(t) =Aωcos(ωt) with amplitude V =Aω. Then, the Coulomb friction force becomes f(ωt) = fCsgn (cos(ωt)) which is a square-wave periodic function
x(t)
˙ x(t)
¨ x(t) R
fCsgn( ˙x(t))
1 m
u(t) R
˙
x(t)=Vcos(ωt)
u(s) x(s)
N(V)
1 s 1
ms
=⇒
L DF
Figure 3.2: Block diagram of the system f(ωt) =f(ωt+ 2kπ) with k ∈Z, where f(ωt) can be given as
f(ωt) =
fC, if −π2 < ωt < π2
−fC, if π2 < ωt < 3π2
. (3.18)
This 2π periodic square-wave function f(ωt) can be expanded into a Fourier series as f(ωt) = a0+
∞
X
n=1
(ancos(nωt) +bnsin(nωt)). (3.19)
Taking into account that f(ωt) is an even function bn = 0 for all n ≥ 0, and, as there is no offset along the vertical axis, a0 = 0. The remaining unknown coefficients are given by
an = 1 π
π/2
Z
−π/2
fCcos(nωt) dωt + 1 π
3π/2
Z
π/2
−fCcos(nωt) dωt= 4fC π
sin(nπ/2)
n . (3.20)
Neglecting the harmonics, i.e., the cases whenn ≥2, the fundamental friction force com- ponent is
fI(ωt) = 4fC
π cos(ωt). (3.21)
Recalling the input velocity ˙x(t) =V cos(ωt), the describing function becomes N(V) = 4fC
V π. (3.22)
This is the same result as the one presented in [68, 69]. The corresponding block-diagram is
shown in the lower part of Fig. 3.2.
Based on this figure, the transfer function between the position x(s) and the control torque u(s) is
G(s) = x(s)
u(s) = 1
ms2+N(V)s (3.23)
Applying the inverse-Laplace transformation, the equation of motion of the system becomes m¨x(t) + 4fC
V πx(t) =˙ u(t). (3.24)
As it can be seen, the describing function N(V) operates as a velocity dependent effective viscous damping coefficient bC.
When the motion is initiated with the following conditions x(0) =x0 = 0 and ˙x(0) = v0 >0, then the amplitude of the velocity function becomesV =v0. As it can be seen from Eq. (3.24), the effect of Coulomb friction can be modeled by an effective viscous damping term which depends also on the initial velocity. The corresponding effective viscous damping coefficient is
bC = 4fC
πv0. (3.25)
In order to understand the effect of viscous damping on the stability of the sampled- data damped oscillator, using the control law of the discrete-time proportional controller u=−kpx(tj), the following model is investigated
m¨x(t) +bx(t) =˙ −kpx(tj), t∈[tj, tj +ts), (3.26) where b=bC denotes the viscous damping coefficient.
3.2.2 Passivity condition with effective viscous damping
Introducing the notation for the sampled position data x(tj) = xj, the energy balance equa- tion for the same sampling period can be written as
m
tj+1
Z
tj
¨
x(τ) ˙x(τ) dτ +b
tj+1
Z
tj
˙
x2(τ) dτ =−kpxj tj+1
Z
tj
˙
x(τ) dτ. (3.27)
In this equation the first and third integrals can be calculated in closed form, which gives
Tj+1− Tj =−b
tj+1
Z
tj
˙
x2(τ) dτ −kpxj(xj+1−xj), (3.28)
where T denotes the kinetic energy.
Using the identity (xj+1+xj)2 = x2j+1+ 2xjxj+1 +x2j, the term kp(xjxj+1−x2j) can be rewritten as
kp xjxj+1−x2j
=kp x2j+1 2 − x2j
2 −(xj+1−xj)2 2
!
=Uj+1− Uj −kp(xj+1−xj)2
2 , (3.29) where U is the potential energy which represents the stored energy in a fictitious physical spring realized by the discrete-time proportional controller, while the second term with kp on the right hand side of Eq. (3.30) represents the energy instilling effect of the zero-order- hold [70]. The accumulation of this extra energy can lead to passivity violations and make the control unstable. Then the substitution of the above identity into the energy balance equation yields
Tj+1− Tj+Uj+1− Uj =−b
tj+1
Z
tj
˙
x2(τ) dτ +kp(xj+1−xj)2
2 . (3.30)
Using the following approximation of the velocity
˙
x(t)≈ xj+1−xj
tj+1−tj = xj+1−xj
ts , (3.31)
the remaining integral in Eq. (3.30) can be approximately determined as
tj+1
Z
tj
˙
x2(τ) dτ ≈
tj+1
Z
tj
(xj+1−xj)2
t2s dτ = (xj+1−xj)2
ts . (3.32)
It results that the energy balance equation in Eq. (3.30) is approximated as follows Tj+1− Tj+Uj+1− Uj =
kp 2 − b
ts
(xj+1−xj)2. (3.33)
The system becomes dissipative, if the total mechanical energy is decreasing. Thus, the left-hand-side of the equation above have to be negative and therefore the following inequality have to be satisfied
0≥ Tj+1− Tj +Uj+1− Uj = kp
2 − b ts
(xj+1−xj)2. (3.34) Obviously, the term (xj+1−xj)2 is always positive, therefore the total mechanical energy decreases when
0≥ kp 2 − b
ts ⇔ b ≥ kpts
2 . (3.35)
As it can be seen, it implies the same results that was presented in [66] with (B = 0) derived by classical control theory or in [58] by using the Cauchy-Schwarz inequality.
By plugging this effective dampingbC, into the passivity condition, a new passivity-based stability criterion can be given in the form
b > kpts
2 ⇐⇒
b=bC
kp< 8fC
πv0 1
ts (3.36)
which turns out to be the same as Eq. (3.16) but scaled with 4/π.
The describing function model is averaging the effect of Coulomb friction, while the passivity considerations take the worst-case scenario. In the present case, the combination of these techniques leads to almost the exact stability boundary. This is illustrated in Fig. 3.3 where the solid blue line corresponds to this new passivity-based stability condition.