On stability of SDOF systems with delayed position and velocity feedback

Abstract

In this paper, we study the stability problem of a single-degree-of freedom system with delayed position and velocity feedback, which can be found in the modeling of several engineering problems in control and vibration. For this class of delay system, we derive some necessary and sufficient conditions for delay-dependent exponential stability. These conditions determine a region in the feedback gains space for assuring the delay-dependent exponential stability. We show how the results can be applied to control and delayed resonator (DR) problems.

Keywords

single-degree-of freedom system delayed position and velocity feedback delay-dependent stability control delayed resonator

1. Introduction and problem formulation

Consider a single-degree-of freedom (SDOF) system with delayed position and velocity feedback of the form

m \ddot{x} (t) + c \dot{x} (t) + k x (t) = - g_{1} x (t - h) - g_{2} \dot{x} (t - h),

(1)

where m, c, k respectively denote the mass, damping, and stiffness, g₁ and g₂ are the feedback gains and h > 0 denotes the time delay. Since a SDOF system is the basic linear model for describing single mechanical systems, then (1) is widely used to model a number of engineering applications in control and vibration. Such applications can be divided into two different kinds depending on the source of the time delay.

The first application kind concerns with control applications, where proportional-derivative feedbacks are used for controlling the dynamics of SDOF systems under the presence of unavoidable time delays in the feedback loop due to the time involved in sensors, filters, actuators, and transmission of data and signals. An interesting application of this kind is the vibration suppression by active control of civil engineering structures, where the delay effect in stability and performance of the active control is one of the most important problems, see Udwadia (1991); Udwadia and Phohomsiri (2006); Agrawal and Yang (1997); Liu et al. (2010); Tran and Liu (2017); and Tang et al. (2018). Applications in mechanical systems can be found in Chen et al. (2008) for explaining the formation mechanism of squalling noise in automobile brakes, in Palkovics and Venhovens (1992) for modeling controlled wheel suspension, and in Ngouabo et al. (2021) for controlling micro-electro-mechanical systems resonators systems.

The second application kind concerns with the intentional introduction of the time delay as a control parameter for stabilization or performance improvement of the SDOF system. For instance, delayed position feedback was used to stabilize an inverted pendulum in Atay (1999), and position plus delayed position feedback was used to stabilize linear undamped systems in Liu and Hu (2008). An interesting application of this kind is the delayed resonator (DR). The main idea behind the DR is the introduction of a time delay in the feedback to achieve that the SDOF system has stable oscillatory solutions, which it becomes an ideal vibration absorber. The DR was introduced in Olgac and Holm-Hansen (1994) by using delayed position feedback and, since its introduction, several modifications have been proposed during the last decades. Thus, one finds DR by using delayed position, velocity, or acceleration. We refer the reader to the recent paper Vyhlidal et al. (2019), where a complete review along with a thorough analysis of the existing DR configurations are presented. An interesting different DR configuration with a delayed velocity and non-delayed position feedback is considered in Eris et al. (2018). In the DR context, the system (1) represents a DR by using delayed position and velocity feedback, which, to the best of our knowledge, it has not been considered in the literature.

A practical problem from the control application perspective is to select appropriate gains g₁ and g₂ for assuring the stability of the system (1) under the presence of a given delay h. On the other hand, a practical problem from the DR application point of view is to determine appropriate feedback gains g₁ and g₂, and delay h such that the system (1) has stable oscillatory solutions.

There is a number of results for the stability of linear time-invariant delay systems that can be used to address the above two problems, see, for instance, the recent book Gu et al. (2003) for a description of stability methods in frequency and time-domain approaches. Based on the general stability methods, one can find several works addressing the special problem of characterizing proportional-integral-derivative (PID) stabilizing controllers for linear delay systems, see, for instance, Silva et al. (2005); Hohenbichler (2009); Li et al. (2021). We refer the reader to the very recent paper Li et al. (2021) and the references therein for a review of some existing results concerning the PID stabilization of linear delay systems. However, the application of most of such methods demands the knowledge of all coefficients and delay values of the system. As a consequence, most of the existing methods can be used to verify the stability of a given system (1), but not necessarily allow us to explicitly determine a set of feedback gains and delay values for assuring neither the stability nor the existence of stable oscillatory solutions of the system (1).

There are two works particularly dedicated to the investigation of the stability of (1). The first one is Agrawal et al. (1993), where for given coefficients and feedback gains such that the system (1) is stable for delay equal to zero, an expression for determining the critical delay value for stability is given. Most interesting for our contribution is the second work Hu and Wang (1998). There, the stability region in the feedback gains space for delay-independent stability is given. It is important to mention that the same result is obtained in the recent paper Alikoc and Ergenc (2017) by the application of a new polynomial based method for delay-independent stability of general linear time-invariant systems with constant delays. By using some intuitive arguments, it was also shown in Hu and Wang (1998) that the characterization of the stability region in the feedback gains space for delay-dependent stability is a very complicated problem. In fact, to the best of our knowledge, the problem of analytically determining the complete stability region in the feedback gains space for delay-dependent stability is far to be completely solved, in spite of the apparent simplicity of the delay system (1). To illustrate this and motivates our present contribution, let us rewrite the system (1) as follows:

\ddot{x} (t) + 2 ω_{n} ζ \dot{x} (t) + ω_{n}^{2} x (t) + b_{1} x (t - h) + b_{2} \dot{x} (t - h) = 0,

(2)

where

ω_{n} = \sqrt{\frac{k}{m}}

is the natural frequency,

ζ = \frac{c}{2 m ω_{n}}

is the damping ratio,

b_{1} = \frac{g_{1}}{m}

and

b_{2} = \frac{g_{2}}{m}

. It is well-known that the system (2) is exponentially stable if and only the characteristic function (quasipolynomial)

f (s) = s^{2} + 2 ω_{n} ζ s + ω_{n}^{2} + b_{1} e^{- h s} + b_{2} s e^{- h s,}

(3)

has all zeros with negative real parts. For the subsequent stability analysis is convenient to consider the following quasipolynomial

q (s) = s^{2} + d a s + a^{2} + b_{1} e^{- h s} + b_{2} s e^{- h s},

(4)

where a, d, h are positive real. Clearly, the quasipolynomial (3) is a particular case of the quasipolynomial (4) when

a^{2} = ω_{n}^{2}

and d = 2ζ.

By following the main ideas of the procedure suggested by Neimark (1949), let us firstly observe that s = 0 is a zero of q(s) if and only if b₁ = −a². Now, let us assume that q(s) has a pure imaginary zero $s = i ω, ω \in ℝ_{+}$ . By separating the imaginary and real parts of q(iω), and then solving for b₁ and b₂ one arrives at the following parametrization

\begin{array}{l} b_{1} (ω) & = a d ω \sin (ω h) + (ω^{2} - a^{2}) \cos (ω h), \end{array}

(5)

\begin{array}{l} b_{2} (ω) & = \frac{1}{ω} ((ω^{2} - a^{2}) \sin (ω h) - a d ω \cos (ω h)) . \end{array}

(6)

The parametrization (5)–(6) defines a continuous curve, that begins at the point

(b_{1} (0), b_{2} (0)) = (- a^{2}, - a^{2} h - a d)

, in the parameter space (b₁, b₂) when ω varies from 0 to + ∞. This curve has a very complicated form depending on the parameters and delay, see Figures 1 and 2.

As it can be seen from Figures 1 and 2, the corresponding curves divide the plane (b₁, b₂) into an infinite set of connected open regions. For any pair (b₁, b₂) belonging to any of these regions, the quasipolynomial q(s) has exactly the same number of zeros with positive real part, and the stability region is the one where q(s) has no zeros with positive real part, see Neimark (1949). Since for (b₁, b₂) = (0,0) the quasipolynomial q(s) becomes a stable polynomial, then it follows that the region containing the origin of the plane (b₁, b₂) is the corresponding stability region, see shadowed regions in Figures 1 and 2. Therefore, by using the above procedure, one easily can determine the corresponding stability region in the feedback gains space (b₁, b₂) for any given values of parameters and delay. However, this numerical computation of the stability region cannot allow us to control the form and size of the stability region, which is important in the applications. More precisely, from the control application point of view, it is desirable to determine stability regions that provide a wide range of feedback gains to the designer. On the other hand, from the DR application point of view, it is desirable to determine stability regions, whose boundary is determined by a wide range of values of ω, as it will be shown later on. Thus, of the two stability regions in Figures 1 and 2, the one in Figure 1 should be preferred for the applications since it provides wider ranges of feedback gains and frequencies than the stability region in Figure 2. Based on the above, the following problem arises: to find conditions under which the stability region on the feedback gains space has a wide range of gains for the control application and a wide range of frequencies for the DR application. In this paper, we aim at contributing to a such complicated problem by providing a stability region in the feedback gain space for some parameters of the SDOF system.

2. Stability analysis

The following Lemma will play an important role in the main stability result.

Lemma 1

For any given $a, d, h \in ℝ_{+}$ , the function

g (ω) = \frac{d a \sin (ω h) + ω \cos (ω h)}{\cos (ω h) - 1},

(7)

has a local maximum at

ω_{k}^{*} \in (\frac{(2 k + 1) π}{h}, \frac{(2 k + 2) π}{h})

, where

ω_{k}^{*}, k = 0,1,2, \dots

, are the solutions of the equation m(ω) = n(ω), for

ω \in (\frac{2 k π}{h}, \frac{(2 k + 2) π}{h})

, with

m (ω) = \frac{\sin (ω h)}{\cos (ω h) - 1} and n (ω) = \frac{a d h - \cos (ω h)}{ω h} .

(8)

Now, if

1 \leq d \leq \frac{3 π}{2 a h},

(9)

then

ω_{k}^{*} \in (\frac{(2 k + 1) π}{h}, \frac{(4 k + 3) π}{2 h})

, and

g (ω_{k}^{*}) \geq \frac{a^{2}}{ω_{k}^{*}}

for k = 0, 1, 2, ….

Proof

Clearly, the function g(ω) given by (7) and its derivative

\frac{d g (ω)}{d ω} = (\frac{h ω}{\cos (ω h) - 1}) (m (ω) - n (ω)),

where

m (ω)

and

n (ω)

are given by (8), are well-defined for

ω \neq \frac{2 k π}{h}, k = 0,1,2, \dots

. Then,

\frac{d g (ω)}{d ω} = 0

if and only if

m (ω) = n (ω)

for some

ω \in (\frac{2 k π}{h}, \frac{2 (k + 2) π}{h}), k = 0,1,2, \dots

. The function

m (ω)

satisfies that

m (ω) \to + \infty

when

ω \to \frac{2 k π}{h} - 0, m (ω) = - \infty

when

ω \to \frac{2 k π}{h} + 0

, and

m (\frac{(2 k + 1) π}{h}) = 0

, for k = 0, 1, 2, …., see Figure 3. In order to analyze the behavior of n(ω), we consider the following three cases:

1. adh > 1. In this case, we have that adh − cos(ωh) > 1 − cos(ωh) > 0, which implies that n(ω) > 0 for all $ω \in ℝ_{+}$ . Additionally, we have that n(ω) → +∞ when ω → +0, see Figure 3.

2. adh = 1. In this case, we have that adh − cos(ωh) = 0 for $ω = \frac{2 k}{π}$ and adh − cos(ωh) > 0 for $ω \neq \frac{2 k}{π}$ , which implies that n(ω) > 0 for all $ω \neq \frac{2 k}{π}$ and n(ω) = 0 for $ω = \frac{2 k}{π}, k = 0,1,2, \dots$ ., see Figure 3.

3. adh < 1. In this case, we have that n(ω) → −∞ when ω → +0 and there exist $ω_{k} \in (\frac{2 k π}{h}, \frac{(2 k + 2) π}{h})$ such that n(ω_k) = 0, k = 0, 1, 2, …. Therefore, in this case, the function n(ω) is oscillatory.

From the above, it follows that in all the cases there exist $ω_{k}^{*} \in (\frac{(2 k + 1) π}{h}, \frac{(2 k + 2) π}{h})$ such that $m (ω_{k}^{*}) = n (ω_{k}^{*})$ , which implies that $\frac{d g (ω_{k}^{*})}{d ω_{k}^{*}} = 0$ for k = 0, 1, 2, …., see Figure 3.

By taking into account the behavior of the functions m(ω) and n(ω), it is easy to see that $\frac{d g (ω)}{d ω} > 0$ for $ω \in (\frac{(2 k + 1) π}{h}, ω_{k}^{*})$ and $\frac{d g (ω)}{d ω} < 0$ for $ω \in (ω_{k}^{*}, \frac{(2 k + 2) π}{h})$ , which imply that g(ω) has a local maximum at $ω_{k}^{*}, k = 0,1,2, \dots$ Now, observe that $m (\frac{(4 k + 3) π}{h}) = 1, k = 0,1,2, \dots$ On the other hand,

n (\frac{(4 k + 3) π}{h}) = \frac{2 a d h}{(4 k + 3) π} \leq n (\frac{3 π}{2 h}) = \frac{2 a d h}{3 π} .

Thus, if the condition (9) holds, then

n (\frac{(4 k + 3) π}{h}) \leq 1

, k = 0, 1, 2, … From this and the fact that

m (ω)

increases while that n(ω) decreases when

ω \in (\frac{(2 k + 1) π}{h}, \frac{(2 k + 2) π}{h}),

it follows that

ω_{k}^{*} < \frac{(4 k + 3) π}{2 h}, k = 0,1,2, \dots

Direct calculations from

m (ω_{k}^{*}) = n (ω_{k}^{*})

yield at

\cos (ω_{k}^{*} h) - 1 = \frac{ω_{k}^{*} h \sin (ω_{k}^{*} h)}{d a h - \cos (ω_{k}^{*} h)} .

By using this equation, one can arrive at the expression

g (ω_{k}^{*}) = \frac{d^{2} a^{2}}{ω_{k}^{*}} + p (ω_{k}^{*})

, where

p (ω_{k}^{*}) = \frac{\cos (ω_{k}^{*} h) (a d (ω_{k}^{*} h - \sin (ω_{k}^{*} h)) - ω_{k}^{*} h \cos (ω_{k}^{*} h))}{ω_{k}^{*} h \sin (ω_{k}^{*} h)} .

Since

ω_{k}^{*} h - \sin (ω_{k}^{*} h) > 0, \cos (ω_{k}^{*} h) < 0

and

\sin (ω_{k}^{*} h) < 0

for any

ω_{k}^{*} \in (\frac{(2 k + 1) π}{h}, \frac{(4 k + 3) π}{h})

, then

p (ω_{k}^{*}) > 0

, which implies

g (ω_{k}^{*}) \geq \frac{d^{2} a^{2}}{ω_{k}^{*}} \geq \frac{a^{2}}{ω_{k}^{*}}, k = 0,1,2, \dots .,

and ends the proof.

The following is the main result of the paper.

Theorem 2. Consider $h, a, d \in ℝ_{+}$ satisfying the following inequalities

1 \leq d \leq \frac{3 π}{2 a h} and d > {(\sqrt{2} + 1)}^{2} a h .

(10)

Then, all zeros of the quasipolynomial q(s), given by (4), have strictly negative real parts if and only if the pair

(b_{1}, b_{2})

belongs to the region Γ, see Figure 4, whose boundary in the space

(b_{1}, b_{2})

is given by

\begin{array}{l} \partial Γ = & {(b_{1} (ω), b_{2} (ω)) : ω \in [0, \hat{ω}]} \cup \\ {(b_{1}, b_{2}) = (- a^{2}, b_{2}) : b_{2} \in (b_{1} (0), b_{2} (\hat{ω}))}, \end{array}

(11)

where

b_{1} (ω)

and

b_{2} (ω)

are given by the parametrization (5)–(6), with

\hat{ω}

being the solution of the equation

\frac{a^{2}}{ω} = \frac{a d \sin (ω h) + ω \cos (ω h)}{\cos (ω h) - 1}, ω \in (0, \frac{π}{h}) .

(12)

Proof

In order to determine a stability region with a wide range for the parameters b₁ and b₂, one needs to assure that the curve, determined by the parametrization (5)–(6), does not intersect itself. Thus, since the curve starts at the vertical line b₁ = −a², then let us search for solutions of the equation $b_{1} (ω) = - a^{2}$ , which is equivalent to finding solutions of the equation r(ω) = g(ω), where g(ω) is given by (7) in Lemma 1 and $r (ω) = \frac{a^{2}}{ω}$ . Under the condition $1 \leq d \leq \frac{3 π}{2 a h}$ , Lemma 1 implies that there exist $ω_{k}^{*} \in (\frac{(2 k + 1) π}{h}, \frac{(4 k + 3) π}{h}) such that g (ω_{k}^{*}) \geq r (ω_{k}^{*})$ for k = 0,1,2, … Furthermore, since $g (ω_{k}^{*})$ is a local maximum of g(ω), for $ω \in (\frac{(2 k + 1) π}{h}, \frac{(2 k + 2) π}{h})$ , and r(ω) is a continuous decreasing and strictly positive function of ω, then there exist ${\hat{ω}}_{k, 0}$ and ${\hat{ω}}_{k, 1}$ satisfying

\frac{(4 k + 1) π}{h} < {\hat{ω}}_{k, 0} < \frac{(2 k + 1) π}{h} < ω_{k}^{*} < {\hat{ω}}_{k, 1} < \frac{(2 k + 2) π}{h},

such that

r ({\hat{ω}}_{k, j}) = g ({\hat{ω}}_{k, j})

for k = 0, 1, 2, … and j = 0, 1, see Figure 5.

In other words, the continuous curve determined by the parametrization (5)–(6) intersects the vertical line b₁ = −a² an infinite number of times. At these intersections, one has the following values for b₂(ω):

b_{2} ({\hat{ω}}_{k, j}) = \frac{- a (a \sin ({\hat{ω}}_{k, j} h) + d {\hat{ω}}_{k, j})}{{\hat{ω}}_{k, j} \cos ({\hat{ω}}_{k, j} h)} .

(13)

By taking into account that

{\hat{ω}}_{k, 0} \in (\frac{(4 k + 1) π}{h}, \frac{(2 k + 1) π}{h})

and

{\hat{ω}}_{k, 0} \to \frac{(4 k + 1) π}{h}

when k increases, one can arrive at the conclusion that

b_{2} ({\hat{ω}}_{k + 1,0}) > b_{2} ({\hat{ω}}_{k, 0}) > 0

for k = 0, 1, 2, …Now, from the condition

1 \leq d \leq \frac{3 π}{2 a h}

, one has that the following inequalities hold:

\begin{array}{l} g (\frac{3 π}{2 h} + \frac{2 k π}{h}) = a d & \geq \frac{2 a^{2} h}{3 π} = r (\frac{3 π}{2 h}) \\ \geq r (\frac{3 π}{2 h} + \frac{2 k π}{h}) = \frac{2 a^{2} h}{(4 k + 3) π}, \end{array}

r (\frac{7 π}{4 h} + \frac{2 k π}{h}) = \frac{4 a^{2} h}{(8 k + 7) π} > 0,

g (\frac{7 π}{4 h} + \frac{2 k π}{h}) = (\sqrt{2} + 1) (a d - (\frac{7 π}{4 h} + \frac{2 k π}{h})) < 0.

From these inequalities and the fact that g(ω) and r(ω) are decreasing for $ω \in (\frac{(2 k + 1) π}{h}, \frac{(2 k + 2) π}{h})$ , one gets that ${\hat{ω}}_{k, 1} \in (\frac{3 π}{2 h} + \frac{2 k π}{h}, \frac{7 π}{4 h} + \frac{2 k π}{h}), k = 0,1,2, \dots$ , which implies that $\cos ({\hat{ω}}_{k, 1} h) \in (0, \frac{1}{\sqrt{2}}), k = 0,1,2, \dots$ Then, the following inequalities hold:

\begin{array}{l} a^{2} h (1 + \cos ({\hat{ω}}_{k, 1} h)) & < a^{2} h (1 + \sqrt{2}), \end{array}

(14)

\begin{array}{l} a d (1 - \frac{1}{\sqrt{2}}) & < a d (1 - \cos ({\hat{ω}}_{k, 1} h)) . \end{array}

(15)

The condition

d > {(\sqrt{2} + 1)}^{2} a h

, which is equivalent to

a^{2} h (1 + \sqrt{2}) < d a (1 - \frac{1}{\sqrt{2}})

, implies from (14) and (15) that

\frac{1}{\cos ({\hat{ω}}_{k, 1} h)} (a^{2} h - d a) < - a^{2} h - d a .

From this inequality and

\frac{(- a^{2} h) \sin ({\hat{ω}}_{k, 1} h)}{({\hat{ω}}_{k, 1} h) \cos ({\hat{ω}}_{k, 1} h)} < \frac{a^{2} h}{\cos ({\hat{ω}}_{k, 1} h),},

one arrives at

b_{2} ({\hat{ω}}_{k, 1}) = \frac{- a (a \sin ({\hat{ω}}_{k, 1} h) + c {\hat{ω}}_{k, 1})}{{\hat{ω}}_{k, 1} \cos ({\hat{ω}}_{k, 1} h)} < - a^{2} h - d a = b_{2} (0) .

Furthermore, from the fact that ${\hat{ω}}_{k, 1} \to \frac{3 π}{2 h} + \frac{2 π}{h}$ when k increases one has $b_{2} ({\hat{ω}}_{k + 1,1}) < b_{2} ({\hat{ω}}_{k, 1})$ for all k = 0, 1, 2, …

Summarizing the above we conclude that, under the condition (10), the curve determined by the parametrization (5)–(6) is an increasing spiral, and therefore, it does not intersect itself as it is showed in Figure 6.

As it can be seen from Figure 6, the curve and the vertical line b = −a² divide the space (b₁, b₂) into an infinite set of connected open region Ψ_j, j = 0, 1, … Since the region Ψ₀ contains the origin of the plane (b₁, b₂), i.e., (b₁, b₂) = (0, 0) ∈ Ψ₀, then the region Γ = Ψ₀ is the desired stability region. The boundary of the stability region, ∂Γ, is given by the curve determined by the parametrization (5)–(6) for $ω \in (0, {\hat{ω}}_{0,0})$ and the segment $(b_{1}, b_{2}) = (- a^{2}, b_{2})$ with $b_{2} \in (b_{1} (0), b_{2} (\hat{ω_{0,0}}))$ as explicitly given by (11), which ends the proof.□

Theorem 2 characterizes the delay-dependent stability region in the feedback gains space for all parameters satisfying the condition (10). The condition (10) is equivalent to

0 < h < \min {\frac{3 π}{2 a d}, \frac{d}{{(\sqrt{2} + 1)}^{2} a}} and 1 \leq d .

(16)

Note that 1 ≤ d becomes

\frac{1}{2} \leq ζ

in terms of the damping ratio. Then, this condition imposes a certain restriction on the class of SDOF systems for which reference Theorem 2 can be applied. Nevertheless, it is sufficiently general for considering a subclass of underdamped systems, those with

\frac{1}{2} \leq ζ < 1

, critical damped system where ζ = 1, as well as the overdamped systems for which ζ > 1. Additionally, condition (16) gives an upper bound for the delay value, which can be conservative to the critical delay value for system exponential stability.

It is important to mention that these restrictions on the delay value and damping ratio are naturally obtained when proving in a formal way that the curve determined by the parametrization (5)–(6) is an increasing spiral, which does not intersect itself leading to stability regions in the feedback gains space of a particular form that give wide ranges of gains for the control application and wide ranges of frequencies for the DR applications.

In the following two sections, we show how the results in Theorem 2 can be applied to the control and DR problems.

3. Control application

As we mentioned in the introduction, the practical control problem consists in selecting appropriate gains g₁ and g₂ assuring the stability of (1) under the presence of an unavoidable delay h > 0. Note that for h = 0, the system (1) is exponentially stable if and only if

b_{1} > - a^{2} and b_{2} > - a d .

(17)

Several works, as for instance, Agrawal et al. (1993), addressed the control problem by firstly selecting the feedback gains to satisfy the condition (17) and then focused in calculating the critical delay value for which the stability is preserved. Our approach is different since reference Theorem 2 provides a way for selecting the feedback gains for any delay value h > 0 satisfying the condition (16). As a consequence, there are some cases with h > 0 such that the stability of (1) is assured and condition (17) does not hold. For instance, consider that m = 1 kg, k = 4 N/m and c = 2 N/m. Then, we obtain ω_n = 2 rad/s and ζ = 0.5 that yields at a = 2 and d = 1. From (16) we obtain that

h \leq \min {0.0858, 2.3562} .

(18)

Then, cite Theorem 2 allows us to determine the complete characterization of the delay-dependent stability region in the feedback gains space for any h > 0 satisfying (18). For h = 0.071 we obtain the stability region Γ showed in Figure 7. The condition (17) determines the region D in Figure 7. As it can be seen from Figure 7, there exists a region M ⊂ Γ, where for any pair (b₁, b₂) ∈ M we have that the system (1) is exponentially stable, but the condition (17) is not satisfied, see Figure 8 for a close up of this region M.

4. DR application

As we mentioned in the introduction, to the best of our knowledge, this is the first time when a DR by delayed position and velocity feedback is proposed. For this DR configuration, the problem consists in determining appropriate feedback gains g₁ and g₂, and delay h > 0 such that the system (1) has stable oscillatory solutions.

It follows from the proof of Theorem 2 that if (b₁, b₂) ∈ ∂Γ, then there exists $ω \in [0, \hat{ω}]$ , where $\hat{ω}$ is the solution of the equation (12), such that the quasipolynomial q(s) has a pair of pure imaginary zeros s_1,2 = ±iω, and the other (infinite number) zeros are located in the left open side of the complex plane. This particular location of the zeros of q(s) implies that the system (1) has stable oscillatory solutions at the frequency ω.

Based on the above, the following algorithm for designing a DR by delayed position and velocity feedback is proposed:

1. Given m, k and c, compute $a = ω_{n} = \sqrt{\frac{k}{m}}$ and $d = 2 ζ = \frac{c}{m ω_{n}}$ .

2. Select a delay h > 0 satisfying the inequality

h < \min {\frac{3 π}{2 a d}, \frac{d}{{(\sqrt{2} + 1)}^{2} a}} .

(19)

3. Compute $\hat{ω}$ as the numerical solution of the equation

\frac{a^{2}}{ω} = \frac{a d \sin (ω h) + ω \cos (ω h)}{\cos (ω h) - 1}, for ω \in (0, \frac{π}{h}) .

(20)

4. Select a desired frequency of oscillation $ω_{d} \in (0, \hat{ω})$ .

5. Compute the values $b_{1} = b_{1} (ω_{d})$ and $b_{2} = b_{2} (ω_{d})$ by means of the following expressions:

\begin{matrix} b_{1} (ω_{d}) & = a d ω_{d} \sin (ω_{d} h) + (ω_{d}^{2} - a^{2}) \cos (ω_{d} h), \\ b_{2} (ω_{d}) & = \frac{1}{ω_{d}} ((ω_{d}^{2} - a^{2}) \sin (ω_{d} h) - a d ω_{d} \cos (ω_{d} h)) . \end{matrix}

(21)

6. Finally, compute the controller’s gains g₁ = b₁m and g₂ = b₂m.

It is important to note that the algorithm for designing the DR by delayed position and velocity works for ζ ≥ 0.5 and for frequencies $ω_{d} \in (0, \hat{ω})$ .

In order to compare with the existing DR configurations, let us revise the conclusions stated in Vyhlidal et al. (2019) for lumped DR configurations, which are the following: (a) The acceleration feedback works for ω_d > 0.8ω_n and it does not work for ζ > 0.71, (b) The velocity feedback works for all ζ > 0, but $ω_{d} \in (0.3 ω_{n}, ω_{n})$ , and (c) The position feedback works for all ω_d > 0 when ζ > 0.71, and for all damping ratio when ω_d > 0.63ω_n.

To numerically illustrate the proposed DR design by delayed position and velocity, let consider the same numerical values used in the numerical example in section 3. By selecting h = 0.071, the condition (19) holds. Then, by numerically solving the equation (20), one gets $\hat{ω} \approx 23.4347$ rad/s. So, the frequency ω_d can be selected in the interval $(0, \hat{ω}) = (0,23.4347)$ . Let us select ω _d = 0.5 rad/s. Then, direct calculations from equation (21) yield at the values for the gains g₁ = b₁ = −3.7121 N/m and g₂ = b₂ = −2.2649 N/m. In Figure 9, we plot the corresponding stable oscillatory solution of the system (1).

It is important to mention that for this numerical example, none of the lumped DR configurations in Vyhlidal et al. (2019) can assign the frequency ω _d = 0.5 rad/s. On the other hand, it appears that the DR proposed design in Eris et al. (2018) with delayed velocity and non-delayed position can assign the frequency ω _d = 0.5 rad/s under an appropriate optimization of a free parameter associated with the non-delayed position feedback

Now, it is well-known that one of the main advantages of the DR is that the frequency of oscillation can be changed by means of the selection of the feedback gains, and this is also the case of the DR by delayed position and velocity feedback proposed here. Nevertheless, in contrast with the DR configurations proposed in Eris et al. (2018) and Vyhlidal et al. (2019), where both the gains and the delay value need to be changed to assign a different oscillatory frequency, it appears that the DR configuration by delayed position and velocity feedback can assign a different frequency (in the allowed range) by only changing the gains g₁ and g₂, but not the delay value. This special characteristic can simplify the DR hardware implementation since a change of the sampling period is not required when changing the desired oscillatory frequency.

To illustrate this, let us consider ω _d = 5 rad/s, which belongs to the permissible interval of frequencies (0, 23.4347). For this frequency, we obtain the gain values g₁ = b₁ = 23.1665 N/m and g₂ = b₂ = −0.4154 N/m. In Figure 10, we plot the corresponding stable oscillatory solution of the system (1).

5. Conclusions

A stability analysis of SDOF systems with delayed position and velocity feedback is performed. The stability analysis is motivated from control and DR applications. We determined conditions under the system parameters for which the stability region in the feedback gains space has a particular form that provides a wide range of gains for the control application and a wide range of frequencies for the DR application. The stability region is delay-dependent, and therefore, it is less conservative than the delay-independent stability regions presented in the literature. We showed how the results can be applied to control and DR problems. In particular, for the first time in the literature, an algorithm for designing a DR by delayed position and velocity feedback is presented. The problem of preserving the same particular form of the stability regions in the feedback gains space under less conservative conditions on the system parameters than the ones obtained in the present contribution is an interesting issue that deserves future research. Extensions of this work to different delays for both velocity and position feedback also deserve further study

Footnotes

Acknowledgments

L. Hernández-Villa thanks to Consejo Nacional de Ciencia y Tecnología (CONACYT) for the PhD scholarship. The authors wish to thank the associate editor and anonymous reviewers for their careful reading and very useful comments.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Daniel Melchor-Aguilar

References

Agrawal

Fujino

Bhartia

(1993) Instability due to time delay and its compensation in active control of structures. Earthquake Engineering and Structural Dynamics 22(3): 211–224.

Agrawal

Yang

(1997) Effect of fixed time delay on stability and performance of actively controlled civil engineering structures. Earthquake Engineering and Structural Dynamics 26(11): 1169–1185.

Alikoc

Ergenc

(2017) A polynomial method for stability analysis of LTI systems independent of delays. SIAM Journal on Control and Optimization 55(4): 2661–2683.

Atay

(1999) Balancing the inverted pendulum using position feedback. Applied Mathematical Letters 12(5): 51–56.

Chen

Liu

Jin

, et al. (2008) Stability analysis of a squealing vibration model with time delay. Journal of Sound and Vibration 311(1–2): 516–536.

Eris

Alikoc

Ergenc

(2018) A new delayed resonator design approach for extended operable frequency range. Journal of Vibration and Acoustic 140(4): 041003.

Kharitonov

Chen

(2003) Stability of Time-Delay Systems. Basel: Birkhauser.

Hohenbichler

(2009) All stabilizing PID controllers for time-delay systems. Automatica 45: 2678–2684.

Wang

(1998) Stability analysis of damped systems with two time delays in state feedback. Journal of Sound and Vibration 214(2): 213–225.

10.

Niculescu

Chen

J X

, et al. (2021) Characterizing PID controllers for linear time-delay systems: a parameter-space approach. IEEE Transactions on Automatic Control 66(10): 4499–4513.

11.

Liu

(2008) Stabilization of linear undamped systems via position and delayed position feedback. Journal of Sound and Vibration 312(3): 509–525.

12.

Liu

Lie

Dai

, et al. (2010) A single-degree-of-freedom system using a time-delayed control algorithm: principle and experimental study. Journal of Vibration and Control 17(12): 1840–1850.

13.

Neimark

(1949) D-subdivisions and spaces of quasiploynomials. Priklasnaya Matematika I Mekhanika 66(10): 349–380.

14.

Ngouabo

Tuwa

Noubissie

, et al. (2021) Nonlinear analysis of electrostatic micro-electro-mechanical systems resonators subject to delayed proportional-derivative controller. Journal of Vibration and Control 27(1–2): 220–233.

15.

Olgac

Holm-Hansen

(1994) A novel active vibration absortion technique: delayed resonator. Journal of Sound and Vibration 176(1): 93–104.

16.

Palkovics

Venhovens

(1992) Investigation on stability and possible chaotic motions in the controlled wheel suspension system. Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility 21(1): 269–296.

17.

Silva

Datta

Bhattacharyya

(2005) PID Controllers for Time-Delay Systems. Boston, Mass: Birkhauser.

18.

Tang

Dietza

, et al. (2018) The performance of delay-compensation in real-time dynamic substructuring. Journal of Vibration and Control 24(21): 5019–5029.

19.

Tran

Liu

(2017) Time-delay control of a switchable stiffness system. Journal of Vibration and Control 23(15): 2375–2390.

20.

Udwadia

(1991) Noncollocated point control of nondispersive continuous systems using time delays. Applied Mathematics and Computation 42(1): 23–63.

21.

Udwadia

Phohomsiri

(2006) Active control of structures using time delayed positive feedback proportional control designs. Structure Control and Health Monitoring 13(1): 536–552.

22.

Vyhlidal

Pilbauer

Alikoc

, et al. (2019) Analysis and design aspects of delayed resonator absorber with position, velocity or acceleration feedback. Journal of Sound and Vibration 459: 114831.