Stabilization Domains for Second Order Delay Systems

In this paper, an analytic condition is given for determining delayed positive feedback controller for stabilizing an oscillatory system. The <inline-formula> <tex-math notation="LaTeX">$\tau $ </tex-math></inline-formula>-decomposition and <inline-formula> <tex-math notation="LaTeX">$D$ </tex-math></inline-formula>-decomposition methods are employed in deriving this condition. The obtained results are then used to stabilize second order delay systems by a proportional controller. Under-damped and over-damped systems are treated separately, where the Smith-predictor structure is used in the over-damped case to obtain the stability conditions. Illustrative examples are given to show the effectiveness of the proposed approach.


I. INTRODUCTION
It was shown in [1] that delayed positive feedback can stabilize oscillatory systems. It was also proven that a single integrator is stabilizable by a single delay and that a chain of integrators can be stabilized by multiple delays [2]. The continuing interest in stabilizing systems by delay can be explained and motivated by the form of the controller which is simple and easily implementable, as it consists usually of delayed versions of the output multiplied by constants. In [3], the authors presented a delay based control strategy, called the integral retarded controller, to control the velocity of a DC servomotor. The stabilization problem of a vibration system by delayed state difference feedback was investigated in [4]. In [5], stabilization of longitudinal vibrations along an elastic beam with delayed feedback is analyzed. Delayed feedback was also employed to stabilize nonlinear systems. In [6], stability of a class of generalized gyroscope systems under delayed feedback was investigated. In [7], the authors studied the stabilization problem of an unstable highly nonlinear hybrid stochastic differential delay The associate editor coordinating the review of this manuscript and approving it for publication was Xiaojie Su. equation by delay feedback control. In [8], delayed positive feedback was used in the study and analysis of coupled genetic oscillators.
Although stabilizing by delayed feedback has advantages such as ease of implementation and its simplicity, the introduction of delay comes with difficulties in analyzing stability of the closed loop system, which is challenging even in the case of linear time invariant systems. The difficulties are mainly due to the resulting infinite dimensional eigenvalue problem caused by the delay. In fact, studying stability of delay systems continue to be one of the active areas of research [9]- [10], as time delay appears in a natural way in many mathematical models of engineering and physical systems. It is well known that the delay term complicates the analysis of these systems, deteriorates the performance and may lead to instability. Stability results for time delay systems can be achieved using a time domain approach, where Lyapunov theorem and its extensions are exploited. Usually, this approach results in solving linear matrix inequalities LMI's. Aggregation methods were employed to get sufficient stability conditions, see [11]- [12]and the references therein.
Frequency domain methods are also used in stabilizing time delay systems. Many researchers used a parametric approach to study this problem by concentrating on concluding stability or instability of the system with respect to a single or multiple system parameters. The main approaches can be divided into two main streams. Those studying system's stability with respect to delay value, known as τ -decomposition methods [13], where it is focused on the partition of the delays parameter space into countably many regions having the same number of unstable poles [14]. The second approach, studies stability with respect to system parameters, not the delay, and it is known as the D-decomposition method. See [15]- [16] and the references therein for a detailed explanation and applications of the D-decomposition method. Both approaches rely on the continuity property of the roots of the quasi-polynomial with respect to delay and system parameters. The D-decomposition method was largely used in robust stability problems and design of low fixed order controllers for time delay systems, as it permits to decompose the controller's parameter space and determine stability boundaries using a set of parametric equations. These equations explicitly depend on the control parameters, which results in finding the stability regions and can help in obtaining analytical expressions for tuning the controllers. In this line of research, the D-decomposition method was used in [15] to determine stabilizing PID controllers for dead time systems; and in [17], the set of all stabilizing first order controllers were parameterized. Other methods were also used, such as the extensions of the Hermite-Biehler theorem, to find the set of all stabilizing PI and PID controllers for first-order systems with time delay in [18]- [19], respectively.
The main contribution of the paper is the joint use of the D-decomposition and τ -decomposition methods to find analytic conditions on the proportional and delay terms of a delayed positive controller to guarantee stability of the closed loop of an oscillatory system. Stabilization of oscillatory systems finds applications in many fields such as robotics and flexible structures [1]. A similar result was derived in [1] using the Nyquist criterion. Another advantage of the proposed method is its direct application in stabilizing second order delay systems by proportional controllers. It is also shown how to use this approach with a Smith predictor to stabilize second order time delay system. Moreover, the obtained results are applied to the consensus problem of multi-agent network system [20]- [22]. Finally, determining stabilizing regions for second order delay system can be considered as first step in obtaining optimal low order controllers by optimizing other performance indices using genetic algorithms or particle swarm optimization techniques.
The paper is organized as follows. In section II, we determine the stability domains in the space of the couple ( k, τ ) representing the parameters of a controller applied to an oscillating second order system. Next, we generalize the obtained results to a delayed second order system, where under-damped systems and over-damped systems are treated separately. Section IV is devoted to illustrative examples. Finally, section V concludes the paper and draws some future alternatives.

II. SPECIAL SECOND ORDER SYSTEM
In this section, we consider determining the stabilizing regions in the (k, τ ) space of a delayed positive feedback controller with a positive feedback k and a time delay represented by τ > 0, applied to the system First, the admissible ranges of the positive feedback gain k are found by applying the D-decomposition method. Then, k is fixed within this range and the τ −decomposition method is used to determine the stabilizing delays τ . By sweeping over the values of k, the complete set of stabilizing controllers can be obtained. The novelty in the developed method resides in the fact that the ranges of stabilizing (k, τ ) values, for second order oscillatory system, are determined analytically. The closed-loop transfer function of Figure 1 is given by, Analytic determination of stability conditions for the closed loop system is a challenging task as a result of the obtained quasi-polynomial: That is why we will give a solution by dividing the original problem into two sub-problems.

A. DETERMINING ADMISSIBLE VALUES OF k
The D-decomposition technique relies on the continuity property of the roots of the quasi-polynomial (4) with respect to the coefficients. A necessary and sufficient condition for instability of a stable quasi-polynomial is the crossing of the imaginary axis by at least one of the roots. Suppose that τ is fixed, the previously mentioned result can be used to determine stabilizing intervals of k by partitioning the real line k as explained in the sequel. The determination of D−decomposition borders is obtained by setting s = jw in (4), which yields VOLUME 9, 2021 the real part and imaginary part of (5) are equated to zero, to obtain: which gives the following equations Two cases are considered: • Case 1: For even values of m, i.e. m = 2l, l = 0, 1, 2, . . ., we have • Case 2: For odd values of m, i.e. m = 2l + 1, l = 0, 1, 2, . . ., we have Equations (8) and (9) are written as follows: The resolution of this system of equations gives and Using (11) and (12), the quasi-polynomial (4) has roots on the imaginary axis when Note here that the above expression found for k is independent of τ . Taking account of the fact that k > 0, the admissible intervals as a function of l are obtained as follows: and where l = l = 0, 1, 2, . . .

B. COMPUTING STABILIZING VALUES OF τ
Once the admissible values of the gain k are determined, the conditions on the delay τ should be found. The τ -decomposition method is applied to obtain τ -values, which ensure stabilization of the system. Given the characteristic function, where the polynomials d(s) and n(s) are coprime and degree(n(s)) > degree(d(s)). The τ −decomposition technique allows specifying the τ values so that D(s, τ ) is stable. We notice that if s = jw is a root of D(s, τ ), then its complex conjugates = −jw is also a root, which gives the following system of equations: The elimination of the exponential term presented in (17) allows obtaining: However, when w i verifies W (w 2 i ) = 0 then s = jw i and s = −jw i denote the system characteristic roots for delays τ validating the equation below: The τ -decomposition technique is based on the algorithm written below: 1) Specify the delay free system roots (for τ = 0), which provide hints about the delay τ impact on the evolution of the roots. 2) Compute the polynomial W (w 2 ). 3) Calculate the real positive roots of W (w 2 ) (represented by w 2 i for i ≥ 1). In case there is no positive real solution for W , then system stability is the same for all delay values. Moreover, it is important to determine the motion of the roots located near jw i as τ increases. Thus, the sense of variation of W (.) should be examined according to w 2 . Obviously, the sense of variation of the functions Re(s, Thus, the τ −decomposition technique relies on the Re(s, τ ) variation sense in the points jw i neighborhood. In fact, when dRe(s,τ ) dτ ≤ 0 or dRe(s,τ ) dτ ≥ 0, then, for all τ 1 ≥ τ 2 we get respectively Re(s, τ 1 ) ≤ Re(s, τ 2 ) or Re(s, τ 1 ) ≥ Re(s, τ 2 ). This result shows that roots move to the right-half or to the left-half complex plane according to the change of the delay value. This procedure allows determining the values of τ constituting the upper and lower-bounds of the system stability interval. We observe, from Figure 2, the effect of varying the time delay in the neighborhood of the crossing points. At these points, the delay bounds can be determined. In fact, for an increasing function Re(s, τ ), as delay τ increases in the neighborhood of the points jw l , Re(s, τ ) will also augment as shown in Figure 2. As a consequence, the roots will migrate to the instability region. Hence, the upper bound of delay ensuring stability is obtained by increasing τ until the first pole of the system is located at the jω axis. In a similar manner, if Re(s, τ ) is a decreasing function of τ , the lower stability bound is calculated.
Applying this algorithm to the quasi-polynomial defined by (4), we get: 1) First compute zeros of the system without delay (τ = 0). The system characteristic zeros provided by s = ±j √ α − k, satisfying the following condition: Obviously, the zeros are placed on the imaginary axis.

2) As
we obtain 3) Solving W (w 2 ) = 0 and considering the condition (21), the following roots are determined: In this case, Re(s, .) is a decreasing function of τ . Therefore, increasing the value of τ will result in crossing the imaginary axis from right to left. Applying (19), we obtain: Thus, To maintain the roots in the left-half complex plane, the following relation should be satisfied Similar arguments are applied in the case of the roots ±j √ w 2 . The function Re(s, .) monotony test is obtained applying the equation written below: Roots cross the imaginary axis from left to right for the following values of τ : Which gives the following condition Thus, the system stability is achieved for: Now, taking into account (14), (15), (21) and (27), we get the following conditions on the values of k and τ that guarantee stability of the closed loop system defined by (3): for l = 0, 1, 2 . . ..

III. GENERAL SECOND ORDER DELAY SYSTEMS
In this section, taking advantage of the obtained result (28), we develop a method of stabilizing second order delay systems. Consider a prototype second order system with delay given by where ζ > 0 is the damping ratio, w n is the natural frequency and k 2 a constant, to be stabilized by a proportional controller having the form In this case, the transfer function of the closed-loop system of Figure 1 is given by  and the characteristic equation of (31) is as follows (32) Now, we impose that the poles of the closed loop system are located to the left of the damping factor ζ w n by the change of variable µ = s + ζ w n , we get It is clear that equation (33) has the same form of (4) where α = w 2 n (1 − ζ 2 ), k = k 1 w 2 n . The case ζ = 0 corresponds to an oscillatory system and we can directly apply the results of the previous section. Two cases, namely the under-damped and over-damped case, will be studied.

A. UNDER-DAMPED CASE
In this case, we assume that the open loop system is under-damped ( ζ < 1), the stability regions of couple (τ, k 1 ) is provided applying the technique presented in Section II. Therefore, by using the τ -decomposition method, the stabilizing delay values are given by the following condition and the D-decomposition method permits the determination of k 1 stabilizing gain values given by For w n = 1 and ζ = 0.707, the stability regions are given in Figure 3. Figure 4 and Figure 5 show the stability region in the (k 1 , τ, ζ ) space for l = 1 and l = 1, 2, . . . , 6, respectively.

B. OVER-DAMPED CASE
Consider the over-damped (ζ > 1) case. We can just choose k 1 so that the polynomial D(s, 0) is Hurwitz stable and the roots of W (w 2 ) are all negative. However, the difficulty in satisfying these two constraints comes from the fact that we imposed the use of a proportional controller C(s) = k 1 e −ζ w n . This may not always be feasible to achieve by the use of one   parameter k 1 . To solve this problem, Smith's predictor is a key. Assuming that C 1 (s) = k 3 , the transfer function of the dashed block part in Figure 6 is given by = k 2 k 3 w 2 n s 2 + 2ζ w n s + w 2 n + k 2 k 3 w 2 n e −τ s .
By an appropriate choice of controller C 1 (s) the system can become an oscillatory type system. In fact, by choosing C 1 (s) = k 3 = ζ 2 k 2 , and the same variable change µ = s + ζ w n we get D(s) = s 2 + 2ζ w n s + w 2 n + ζ 2 w 2 n = µ 2 + w 2 n .
In the other words, the choice of   as a delayed positive feedback, the transfer function of the overall system presented in Figure 6 becomes and the characteristic equation of (39) is as follows Therefore, we obtain in the same way, the relations defining the domain of (k 4 , τ 1 ) stabilizing values, given by the following inequalities and 0 < k 4 ≤ (4l + 1) (8l 2 + 4l + 1) 1 ζ 2 , l = 0, 1, 2, . . .

IV. ILLUSTRATIVE EXAMPLES
In this section, we present four illustrative examples. In the first example, we consider the stabilization of an oscillatory system. In the second example, a DC motor modeled by a second order under-damped system, is treated. In the third example, the proposed method is applied to a TCP/AQM system, modeled by a second order over-damped delay system, in congestion control. A fourth example is given to show the application of the proposed approach to second order multi-agent system with delayed positive feedback controller. Example 1: Consider stabilizing an oscillatory system with α = 1. Figure 9 gives the stability regions for l = 0, 1, . . . , 6 and Figure 10 illustrates the stabilizing regions in the (k, τ, α) plane using condition (28). The state responses are shown in Figures 11 and 12. When supplied by voltage on the stator and on the rotor, a motor usually exerts a torque T that acts on the mechanical structure, which is characterized by the rotor inertia J and the viscous friction coefficient B. The equation that links the torque and the angular velocity w is given below where K t is the torque constant and i is the armature current. The velocity of the motor can be used to get the position θ as follows (44) VOLUME 9, 2021  Substituting the (43) into (44), gives Therefore, the transfer function of the DC motor is given by Example 2: The mechanical representation of a DC motor is shown in Figure 13. Now, we stabilize the DC motor by a positive delayed feedback controller as shown in Figure 14. The closed loop transfer where G(s) is given by with µ = s + a 2 , w 2 n = bk 1 − a 2 4 and k = bk 2 . Choosing the same parameters given in [3], namely a = 5 and b = 53, we obtain an under-damped second order system.   Therefore, by using the τ -decomposition method, the stabilizing delay values are given by the following condition and the D-decomposition method permits the determination of k 2 stabilizing gain values expressed as Stability domains in the (k 2 , τ 1 ) plane are given in Figures 15, 16 and 17 for different values of l and k 1 . where b = k t J and a = B J . Example 3: In [24], the dynamic model of transmission control protocol TCP flows is introduced by applying the fluid flow model without taking into accounts low beginning and timeout mechanism. Using this system together with delays in the network, an active queue management AQM is built.  This model is depicted using the non-linear differential equation written below: whereẆ (t) andq(t) are W (t) and q(t) time-derivatives, respectively. W (t) represents the TCP window size, q(t) denotes the queue length in the router, p(t) denotes the probability packet marking/dropping ( p(t) ∈ [0 1]), R(t) denotes the round-trip time, C(t) designates the capacity of the link, N the number of nodes and T p refers to the delay of propagation. Equation (51) shows the progression of the TCP source congestion window w(t). p(t) signal represents the source loss rate, while delay τ corresponds to the RTT observed by the source. In system (51), (W , q) is the state of the system that should be controlled; p denotes the input. Thus, the equilibrium point (W 0 , q 0 , p 0 ) is given by resolv- After linearizing the nonlinear system [24], we obtain the following transfer function  In this case, we have the following values: w n = 1 The transfer function of the applied proportional controller to the closed-loop AQM system is given by C(s) = k 1 e −R 0 ζ w n . Since ζ > 1, we find values of k 1 such that all roots of D(s, 0) are situated in the left half of the complex plane. In addition, W (w 2 ) must have no positive roots. These two conditions lead to the determination of the values of k 1 ensuring stability of the system Using the proposed method presented in the previous section, we can draw the domains of (k 1 , τ 1 ) stabilizing values of the delayed positive feedback controller defined by (38). These domains are given in Figures 18, 19, 20, 21 and 22. Preliminary results for this example were given in [25]. Example 4: In this example we consider the consensus problem of a multi-agent network system. A second order linear dynamic model for each agent i is given below:   where x i ∈ R, is the position state, v i ∈ R is the velocity state of i th agent. The consensus protocol with delayed positive feedback can be written as   Similar to [22], we obtain its characteristic equation as follows: where f i (s, τ ), i = 2, 3, . . . , n are quasi-polynomials given by where λ i , i = 2, 3, . . . , n are the non-zero eigenvalues of the Laplacian matrix of the graph. Then, using (28), we obtain the following result   with τ 1 2lπ , τ 3 2lπ with λ max = max 2≤i≤n {λ i } and λ min = min 2≤i≤n {λ i }.  For simulation, we consider a network with the following topology: Its Laplacian matrix is given by which leads to: λ min = 4, λ max = 6. The stability domains for a = 3, b = 0, g 1 = 0, l = 0, 1, . . . , 5 are given in Figure 24. The state responses are shown in Figures 25 and IV. For a = 0,

V. CONCLUSION
Applying both τ -decomposition and D-decomposition techniques, stability of an oscillating second order systems with delayed positive feedback is examined. This method allows localizing stability domains of the parameters of a simple delayed positive feedback. Based on the obtained results, an extension was formulated to a delayed second order system. Four illustrative examples were given. In the first example, a delayed positive feedback for stabilizing an oscillatory system is given. In the second example, we stabilized an open loop under damped DC motor. In the third example, a controller was designed for AQM routers to address the congestion situations and minimize their impact. It consists of a simple positive feedback controller combined with Smith predictor. In the fourth example, the consensus problem of a second order multi-agent system with delayed positive feedback, was treated.