Optimal Time Delay Selection

The Embedding Theorem states that the topological space of the dynamic system can be recon-structed using ν^∗. Such ν^∗, constructed with τ^∗and d^∗, comprises the essential dynamics of the system.

Based on this information, it is hypothesized that the utilization of ν^∗as control input would provide a DDC with sufficient information about the system’s dynamics, therefore enabling a good estimation of the control function. Taking the SDOF from Eq. (3.1) with the following control rule:

u(t) = −g₁x(t) − g₂x(t − τ) (3.33)

The upcoming subsection derives the optimal fixed time delay computed using the analytical so-lution of an SDOF system subjected to a harmonic excitation and controlled with a fixed time delay.

Also, the stability bounds on τ are determined. The subsequent subsection verifies the hypothesis that ν^∗could be used effectively as control input by comparing the analytical solution given by information theory to the analytical solution derived from the SDOF system.

3.4.1.1 Optimal time delay - SDOF analytical solution

Using the transformations ρ_g1= g₁/k and ρ_g2= g₂/k into Eq.(3.1) gives:

¨

x(t) + 2ξ ωnx(t) + ω˙ _n²x(t) = −ρg1ω_n²x(t) − ρg2ω_n²x(t − τ) + f (t)/m (3.34) where ωn=p

k/m and ξ = _2mω^c

n are the natural frequency and fundamental damping ratio of system, respectively.

The response of the system represented in Eq. (3.34) subjected to a harmonic forcing f (t) of the type

f(t) = ˆfsin(Ωt) (3.35)

where ˆf and Ω the are the magnitude and frequency of the excitation, respectively, can be expressed in the form

Solving for A and B in Eq. (3.37) and substituting back in Eq. (3.36) yields a transfer function H= |max(x(t)) · k/ ˆf|

H= 1

p[1 − ρ²+ ρg1+ ρg2cos(2πρτ)]²+ [ρg2sin(2πρτ) − 2ξ ρ]² (3.38)

Fig. 3.11is the plot of H versus ρ for various values of ρ_τ, where ρ = Ω/ωn and ρ_τ = τ/T . The figure is obtained using ρg1= 2 and ρg2= −1, and ρτ ≤ 0.25, as any additional delay would provide redundant information in terms of topology of the phase-space for a harmonic excitation. Results show that increasing ρτ reduces H, until a critical frequency ratio ρcr is reached. An expression for ρcr can be obtained by substituting appropriate values for ρτ in Eq.3.38

ρcr= ξ + q

ξ²+ ρ_g1+ 1 (3.39)

Figure 3.11: H function with ρg1= 2, ρ_g2= −1, and ρ_τ = [0, 0.05, 0.15, 0.25]

Before concluding on the optimal time delay, a stability analysis is conducted in what follows in order to provide bounds on ρ_τ, ρ_g1, and ρ_g2.

Stability Analysis

For the stability analysis, the homogeneous solution for x(t) in Eq. (3.36) is expressed in the form x(t) = ˆxe^{λ t}, where ˆxis an amplitude, yielding the characteristic equation

λ²+ 2ξ ωnλ + ω_n²+ ρ_g1ω_n²+ ρ_g2ω_n²e^−τλ = 0 (3.40) The exponential term can be expressed by the power series

e^−τλ = 1 − τλ +1

2(τλ )²−1

6(τλ )³+ ... (3.41)

Neglecting the higher order terms and retaining the first two terms only, equation (3.40) becomes

The system is stable if λR< 0, which gives an expression for ρ_g2

ρ_g2< 2ξ /(ωnτ ) (3.44)

Furthermore, if λ has two real numbers as roots, the imaginary part vanishes and λ becomes

λ = −ξ ωn+1

2ρ_g2ω_n²τ ±1 2ωn

q

(2ξ − ρ_g2ωnτ )²− 4ρ_g1− 4ρ_g2− 4 (3.45) The maximum root of λ needs to be negative for λ < 0, yielding

1 + ρg1+ ρ_g2> 0 (3.46)

Eqs. (3.44) and (3.46) are stability criterions for control gains g₁and g₂. In addition, the exponential term in (3.40) can be further expanded to study the stability of τ in terms of ρ_g1and ρ_g2

e^−τλ =1 −¹₂τ λ 1 +¹₂τ λ

+ O[(τλ )³] (3.47)

which yields a third degree polynomial in λ

τ λ³+ (2 + 2ξ ωnτ )λ²+ (4ξ ωn+ ω_n²τ + ρg1ω_n²τ − ρg2ω_n²τ )λ + 2ω_n²+ 2ρg1ω_n²+ 2ρg2ω_n²= 0 (3.48) Fig. 3.12 is a stability plot using Eq. (3.48) under various feedback coefficients (ρ_g1= 1 and ρg2= {−0.1, −0.2, −0.3, −0.4, −0.5}) applied to an SDOF with a period T = 2 sec with a fundamental damping ratio ξ = 2% (typical for a civil structure). Values for ρg1and ρg2were selected to meet the

stability conditions from (3.44) and (3.46). Fig. 3.12shows a bound on time delay for stability. This maximum time delay corresponds to λR= 0 or λ = λIiwhen the path shifts from the left half-plane to the right half-plane with varying ρg2. Substituting for λ in (3.40) leads to

− λ_I²+ ω_n²+ ρ_g1ω_n²+ ρ_g2ω_n²cos(τλI) + (2ξ ωnλI− ρ_g2ω_n²sin(τλI))i = 0 (3.49)

Figure 3.12: Stability condition of an SDOF system with a period of 2 sec and damping ratio ξ = 2%

Eq. (3.49) is satisfied when the real and imaginary terms vanish:

−λ_I²+ ω_n²+ ρ_g1ω_n²+ ρ_g2ω_n²cos(τλI) = 0 2ξ ωnλI− ρg2ω_n²sin(τλI) = 0

(3.50)

giving

λ_I⁴+ (4ξ²ω_n²− 2ω_n²− 2ρg1ω_n²)λ_I²+ (ω_n²+ ρg1ω_n²)²− ρ_g2² ω_n⁴= 0 (3.51) Eq. (3.51) can be used to obtain a condition for stability independent of time delay. Such stability is guaranteed if the system stay in the stable region and will not move cross to the unstable region for any τ. This occurs when the solution for λ has no imaginary parts:

ρ_g2² − 4ξ²ρg1< 4ξ²− 4ξ⁴ (3.52)

Figure 3.13: Stability of time delay dependent with ρ_g1and ρ_g2

The roots of Eq.(3.51) are

λI= ± r

ω_n²+ ρ_g1ω_n²− 2ξ²ω_n²±1 2

q

16ξ⁴ω_n⁴− 16ξ²ω_n⁴− 16ξ²ρ_g1ω_n⁴+ 4ρ_g2²ω_n⁴ (3.53) Eq. (3.51) can be used to obtain a condition for stability independent of time delay. Such stability is guaranteed if the solution for λ has no imaginary part or λI has complex roots. This occurs when

ρ_g2² − 4ξ²ρ_g1< 4ξ²− 4ξ⁴ (3.54) In the delay dependent region, only two positive values in Eq. (3.53) need to be considered since the maximum time delay correspond to ±λIi. The maximum allowable time delay can be obtained by solving the first part of Eq. (3.50) for τ in terms of λIand the minimum positive value is the maximum allowable time delay τ|_max, giving

τ |max= 1 λI

cos⁻¹ −ρ_g1ω_n²− ω_n²+ λ_I² ρ_g2ω_n²



(3.55)

The value for τ|maxis plotted in Fig.3.14for Tn= 2 s ,ξ = 2% under various values of ρg1and ρg2

that meet the stability condition for control gains. As ρg1increases, τ|maxdecreases, and varying ρg2

will influence τ|max.

Figure 3.14: Maximum time delay τ|maxas a function of ρg1and ρg2for ξ = 2%.

In summary, the study on stability led to three conditions:

1. ρ_g2< 2ξ /(ωnτ ) 2. 1 + ρ_g1+ ρ_g2> 0 3. τ|max=_λ¹

Icos⁻¹

_−ρ

g1ω_n²−ω_n²+λ_I² ρg2ω_n²



Combining these conditions with results obtained from the H function (Fig. 3.11), one obtains an optimal time delay τ^∗as a function of ρ. Fig.3.15plots the optimal time delay ratio ρτ^∗ for ρg1= 2 and ρ_g2= −1. The optimal time delay ratio is governed by the stability limit (red line; ρ_τ^∗ = τ|_max/T ) until it reaches the value obtained from optimal mitigation performance (black dashed line, ρ_τ^∗ = τ^∗/T ).

Once the excitation ratio ρ is higher than the critical frequency ratio ρcr= ξ +p

ξ²+ ρ_g1+ 1, no time delay (blue dashed-dotted line, ρ_τ^∗ = 0) provides an optimal performance.

0 0.5 1 1.5 2 2.5 3

Figure 3.15: Optimal time delay ratio ρ_τ under different frequency input with ρ_g1= 2 and ρ_g2= −1.

3.4.1.2 Optimal time delay - information theory

There exists an analytical MI test solution for τ^∗ for a harmonic signal, as derived in Reference [88] for a discretized signal. Briefly, consider two signals f₁(t) and f₂(t) that consist of two sinusoidal functions with a phase shift angle φ ∈ [0, 2π].

f₁= ˆf₁sin(θ )

where N is the length of the discretized signals, J(p_α) is the discrete entropy and p_α is the discrete probability for a particular value α in f1. The discrete entropy J(p_α) and probability p_αare given by

J(p_α) = −p_αlog₂(p_α) − (1 − p_α) log₂(1 − p_α)

Since N and ˆf₂are constants, the first local minima of MI( f1, f2) occurs when the discrete entropy J(p_α) reaches its maximum value J(p_α) = 1. It leads to the probability p_α= 1/2 and the optimal phase shift φopt

φopt= ±π

2 (3.60)

This is equivalent to a quarter of the excitation period 2π, or

τ^∗= 0.25T (3.61)

This solution is in agreement with the optimal value obtained from the analytical solution of the equation of motion presented in the last subsection.