STABILITY ANALYSIS OF DAMPED SDOF SYSTEMS WITH TWO TIME DELAYS IN STATE FEEDBACK

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1 Journal of Sound and Vibration (1998) 214(2), Article No. sv STABILITY ANALYSIS OF DAMPED SDOF SYSTEMS WITH TWO TIME DELAYS IN STATE FEEDBACK H. Y. HU ANDZ. H. WANG Institute of Vibration Engineering Research, Nanjing University of Aeronautics and Astronautics, 2116, Nanjing, People s Republic of China (Received 6 November 1996, and in final form 17 December 1997) This paper presents the stability analysis of linear, damped SDOF vibration control systems with two time delays, one in the displacement feedback and the other in the velocity feedback. First, a sufficient and necessary algebraic criterion is proved to check the system stability independent of time delays. According to this criterion, all possible combinations of the feedback gains that guarantee the delay-independent stability are given. Then, the effect of the feedback gains on the system stability is discussed when the time delays are finite. The most dangerous case is found when the time delay in the displacement feedback is much longer than that in the velocity feedback Academic Press 1. INTRODUCTION The active control for attenuating the excessive vibration of mechanical systems has become more and more popular in engineering. One of the limits to the performance and application of this technique is the unavoidable time delays in controllers and actuators, particularly in hydraulic actuators for active suspensions of vehicles, active tendons of tall buildings, etc. These time delays, albeit very short, deteriorate the control performance or even cause the instability of the system, because the actuators may input energy at the exact moment when the controlled system does not need it. Consider a SDOF system with delayed state feedback, the motion of which yields the following dimensionless difference-differential equation x (t)+2x (t)+x(t)=f(t)+ x(t 1 )+ ẋ(t 2 ), (1) where is the damping ratio as usual, f(t) is the excitation, and are the feedback gains, 1 and 2 are the time delays in the paths of displacement feedback and velocity feedback, respectively. This system serves as a simple, but widely used model for active suspensions and active tendons in vibration control [1, 2]. To check the asymptotic stability of the steady state motion x(t), it is usual to study the variational equation of equation (1) which governs the small variation x(t) near x(t) x (t)+2x (t)+x(t)= x(t 1 )+ x (t 2 ). (2) Substituting the candidate solution x(t)=a e t into equation (2) yields a characteristic equation D(, 1, 2 ) exp ( 1 ) exp ( 2 )=. (3) 22 46X/98/ $3./ 1998 Academic Press

2 214 H. Y. HU AND Z. H. WANG Given the time delays 1 and 2, the steady state motion of system (1) is asymptotically stable if and only if all the roots of equation (3) have negative real parts. When system (1) is asymptotically stable for arbitrary time delays, it is said to be delay-independent stable. If there is no time delay in the state feedback, i.e., 1 = 2 =, equation (3) becomes a quadratic equation in D(,,)= 2 +(2 ) +(1 )=. (4) One can readily write out the stability condition 1, 2. (5) Except for this trivial case, equation (3) is transcendental. It is impossible to check the system stability by solving equation (3) for the infinite number of roots. Thus, it is not an easy task to give simple stability criteria for the retarded differential equation (1). In general, the criteria of delay-independent stability are simpler than those of delay-dependent stability. Hence, they have received much attention over the past decades [3, 4]. As the simplest case, the systems with single time delay have been intensively studied and the delay-independent stability criteria in terms of pure mathematical parameters have been given in [3, 5]. Yet, fewer successful studies have been made for the systems with multiple time delays and no practical stability criterion has been available. For instance, the sufficient condition given in [6] requires very tedious exponential matrix computation. The concise criteria proposed in [7] are not applicable to vibrating systems. In practice, the stability independent of time delays may be excessively restrictive. As a matter of fact, most controlled systems are stable only for bounded time delays. Hsu [8] first suggested several numerical methods to study the relationship of a single time delay and a system parameter when the system undergoes instability. Stepan [9] proposed a set of stability criteria for undamped systems. Chen [1] gave a number of algorithms to compute the maximal delay interval, wherein a controlled system is stable. However, if the system is damped or both delayed displacement and delayed velocity are taken as feedback, no simple criterion has been available for stability test. A practical problem in controller design is how to select appropriate feedback gains and such that the controlled system is stable if there exist time delays in the controller and actuators. Sometimes, the feedback gains and may have already been designed according to a control strategy, say, LQG, but the time delays in controller and the actuators were not taken into account in the previous design. One may want to know whether the controlled system is stable and robust with respect to the variation of the feedback gains. However, the publications mentioned above dealt with the stability criteria in terms of pure mathematical parameters, rather than the feedback gains. The aim of this paper is to find practical criteria of delay-independent stability for a damped vibrating system (1) with two time delays in state feedback. In section 2, a sufficient and necesssary algebraic condition of delay-independent stability is derived. Then, an equivalent condition in terms of feedback gains and is discussed in section 3 and the region of delay-independent stability in the plane of (, ) is given. In section 4, the stability of the systems with two finite time delays is analyzed. Finally, some concluding remarks are made in section 5.

3 STABILITY OF SYSTEMS WITH TWO DELAYS STABILITY CRITERION INDEPENDENT OF DELAYS The stability of system (1) is independent of time delays if equation (3) has no root with non-negative real part and D(i, 1, 2 ) = has no real root for any given 1 and 2. It is obvious that the critical condition D(i, 1, 2 ) = results in namely i = exp (i 2 i 1 )+i, (6) (1 2 ) 2 +(2) 2 = 1 +( ) 2 +2 sin ( 2 1 ). (7) Conversely, if, 1 and 2 satisfy equation (7), thre exists a non-negative number R such that and or and i = exp ( i)[ exp (i( 2 1 ))+i ], i = exp ( i){ +i exp [i( 1 2 )]}. This leads to D(i, 2 + 1, )= or D(i,, ) =. Thus, for given time delays 1 and 2, D(i, 1, 2 ) = has no real root if and only if equation (7) has no real root. Moreover, equation (7) can be rewritten as (8a) (8b) 4 +( ) =2 sin ( 2 1 ). (9) As both sides of equation (9) are even functions in variable, it is sufficient hereafter to study the case of only EQUAL TIME DELAYS When 1 = 2 =, equation (9) becomes where Equation (1) has four roots in the form 4 + p 2 + q =, (1) p , q 1 1. (11) 1,2 = 1 2( p p 2 4q), 3,4 = 1 2( p p 2 4q). (12) q = p 2 /4 q a c b o p d d Figure 1. Region of delay-independent stability in the (p, q) plane when 1 = 2.

4 216 H. Y. HU AND Z. H. WANG The number of real roots depends on the combination of p and q as follows. (a) p 2 4q: none of the roots is real; (b) p 2 4q, p, q : none of the roots is real; (c) p 2 4q, p, q : all of the roots are real; (d) p 2 4q, q : two roots 2 and 4 are real. These cases are shown in the (p, q) plane in Figure 1, where the shaded region represents the parameter combinations that guarantee the system stability independent of time delays. From Figure 1, the conditions (a) and (b) can be expressed in a simpler form p, q, or p, p 2 4q. (13) 2.2. UNEQUAL TIME DELAYS In the case of 1 2, one defines three functions where g() 4 + p 2 + q r sin ( 2 1 ), g () 4 + p 2 + q + r, g () 4 + p 2 + q r, (14) It is easy to prove that for all and r =2, r = r. (15) g () g() g (), (16) g () = g() = g () = q. (17) As 1 and 2 take arbitrary non-negative values, one can choose suitable 1 and 2 such that g()=g () org()=g () for any given. Thus, g() has no root in (, +) if g () for all org () for all. However, the second case is obviously impossible. Now, one focuses on the condition of g () for all. Consider the derivative of g () with respect to By letting g '()=4 3 +2p + r. (18) =(p/6) 3 +(r/8) 2, (19) one has three cases according to the solution of a cubic algebraic equation [11]. (a) If, g '() = has one real root and a pair of conjugate complex roots where 1 = u + v, 2 = u 1 + v 2, 3 = u 2 + v 1. (2) u = 3 r/8 +, v = 3 r/8, 1 = (b) If =, g '() = has three real roots 1 + i3 1 i3, 2 2 =. 2 (21) 1 =2 3 r/8, 2 = 3 = 3 r/8. (22)

5 (c) If, g'() = has three real roots STABILITY OF SYSTEMS WITH TWO DELAYS =2 p 6 cos 3, 2 =2 p 6 cos , 3 =2 p 6 cos , (23) where cos =6r/8p p/6,. (24) Noting r, one can find that only the root 1 in these cases is positive so that g'()=,, for (, 1 ) for ( 1,+). (25) This implies that g( 1 ) is the minimum of g() in [, +]. Hence, g() holds for all provided that g( 1 ). If this is the case, g() holds for all, 1, 2. In summary, the delay-independent stability criterion can be stated as follows. Criterion 1 The system (1) is delay-independent stable for any time delays if and only if either p, q, g ( 1 ) or p, p 2 4q, g ( 1 ) (26) holds, where 3 r/8 + + r/8 3, 1 = 2 (p/6) cos /3, for. (27) for For a given system, the stability test based on this criterion is an easy task including very simple algebraic computation. Table 1 shows 4 numerical examples. TABLE 1 Delay-independent stability of four illustrative examples Examples System parameters Stability tests Conclusions 2.1 = 1 p = 1 963, q = 99 Delay-independent = 1, = 5 p 2 4q = 186, stable g( 1) = = 1 p = 1 963, q = 91 Not = 3, = 5 p 2 4q = 2114, delay-independent g( 1) = 824 stable 2.3 = 5 p = 1 25, q = 75 Delay-independent = 5, = 5 p 2 4q = 1 436, stable g ( 1) = = 5 p = 1 9, q = 64 Delay-independent = 6, = 3 p 2 4q = 1 372, stable g ( 1) = 1345

6 218 H. Y. HU AND Z. H. WANG 3. DELAY-INDEPENDENT STABILITY CRITERION IN TERMS OF FEEDBACK GAINS In this section, all possible combinations of feedback gains that guarantee the delay-independent stability of system (1) are determined. Let D denote the region of those combinations in the (, ) plane, and D denote the region where the system with two equal time delays is delay-independent stable. For simplicity, these regions are referred to as the regions of delay-independent stability. It is obvious that (, )DD if EQUAL TIME DELAYS By substituting equation (12) into equation (13), one has or namely, or p = , q =1 1, (28a) p = , p 2 4q =( ) 2 4(1 1) ; (28b) 1 1, , (29a) , 4 1 +( ) 2 4. (29b) The second inequality in equation (29a) and the first inequality in equation (29b) are a pair of contradictory bounds for 2, holding for different damping. To gain insight in the second inequality in equation (29b), one can find all intersections of the curve with the axes of and. They are P(, ) 4 1 +( ) 2 4= (3) =, = 2, 2 2 1; =, = (31a, 31b) From equation (31a), it is easy to see that this curve, like an ellipse, has a pair of intersections on the and axes, if and only if the system is underdamped in the usual sense. Once the system is overdamped, the curve has no intersection on the -axis, but four intersections on the -axis. In fact, two separated ellipses appear in this case. By imposing P(, )/ =4 ( ) =, (32) 2ζ 1 2ζ 1 2ζ 1 ζ 2 < ζ < 1/ 2 2ζ 1 1 4ζ 2 2 2ζ 1 2ζ 2ζ 1/ 2 < ζ < 1 ζ > 1 Figure 2. Regions of delay-independent stability in the (, ) plane for different damping ratios when 1 = 2. (a) = 5; (b) = 1 ; (c) =

7 STABILITY OF SYSTEMS WITH TWO DELAYS 219 one obtains the extrema of on these two ellipses = 1. (33) On the basis of the above analysis, one can determine the criterion for delay-independent stability as the following and then plot the corresponding region of delay-independent stability as shown in Figure 2. (a) If 1/2, D is surrounded by an ellipse, i.e., 4 1 +( ) 2 4. (34) (b)if1/2 1, the boundary of D consists of two arcs of an ellipse and two sides of a rectangular, i.e., 1 1, , 4 1 +( ) 2 4, (35) (c) If 1, the boundary of D is composed of two sides of a rectangular and two arcs from two ellipses governed by equation (35), too. No matter what case happens, D is bounded in the rectangle {(, ) 1, 2}. So, it is the damping that makes the delay-independent stability possible UNEQUAL TIME DELAYS If 1 2, the boundary of D yields g ( 1 ) =. However, it is almost impossible to solve g ( 1 ) = for the explicit expression of the boundary. Thus, a qualitative analysis of region D will be made. Since p, q, r, p 2 4q, and g ( 1 ) are even functions in the variables and, the region D should be symmetric with respect to both and axes. In what follows, the study is confined to the first quadrant of (, ) plane. For and, one can readily verify that p/ =, q/ = 2, (/ )(4q p 2 )= 8, p/ = 2, q/ =, (/ )(4q p 2 )=4p, (p ) (36) g ( 1 )=g '( 1 ) 1 p q r + 1 s 1 = 2(s 1 s ), 1 g ( 1 )=g '( 1 ) 1 p q r + 2 s 1 = 2 2 s 1 ( + 1 ). (37) 2 These inequalities imply that if a given system with, and is delay-independent stable, so is the system with, and or with,,and. Example 3.1. As known from example 2.1, the system with = 1, = 1 and = 5 is delay-independent stable. According to the analysis above, the system with = 1, = 5 and 1, or with = 1, = 1 and 5 is delay-independent stable, too. Moreover, equation (37) leads to d /d g ( 1 )== (/ )g ( 1 )/(/ )g ( 1 )= 1/ 1. (38) Hence, the boundary defined by g ( 1 ) = in the first quadrant of (, ) plane is a simple curve. Along the boundary, decreases with the increase of. According to the above analysis of D, one obtains another stability criterion independent of time delays.

8 22 H. Y. HU AND Z. H. WANG 1. (a) 2. (b) (c) Figure 3. Regions of delay-independent stability in the first quadrant of the (, ) plane for different damping ratios when 1 =/ 2. Criterion 2 Let U = {(, ) a 1, a 2 }. It is sufficient to check (a 1, a 2 )D in order to make sure UD. Example 3.2. Consider a system with = 5. From Criterion 2 and Example 2.4, it is obvious that [ 6, 6] [ 3, 3]D. This rectangular can be broadened in D by further numerical tries. For instance, one can first fix = 6 and choose a larger, say = 5. A direct computation gives p = 1 25, q = 64, p 2 4q = 9975 and g ( 1 ) = 629. This means that the system is not delay-independent stable and such a is too large. Thus, one can choose a less larger, say = 4 as the second try. In this case, p = 1 16, q = 64, p 2 4q = 1 214, g ( 1 ) = 421. Hence, a larger rectangular [ 6, 6] [ 4, 4]D is obtained. For a given system, the region of delay-independent stability can be determined by a simple and short subroutine. The typical regions of delay-independent stability for underdamped, critically damped and over-damped systems are shown in Figure 3, where the grey regions and the dark regions correspond to the cases of equal time delays and unequal time delays respectively.

9 STABILITY OF SYSTEMS WITH TWO DELAYS STABILITY FOR FINITE TIME DELAYS As analyzed in section 3, the region of delay-independent stability is always bounded in the rectangular {(, ) 1, 2}. Since most mechanical systems are lightly damped, it is too restrictive for them to be delay-independent stable. In practice, the stability of many controlled systems is only required for bounded time delays, especially for short time delays in a bounded interval. Now, the system stability when two finite time delays are given is studied. By separating the real part and imaginary part of the critical condition D(i, 1, 2 )=, one has Re [D(i, 1, 2 )] 1 2 cos 1 sin 2 =, Im [D(i, 1, 2 )] 2 + sin 1 cos 2 =. (39) Solving equation (39) for and yields = (1 2 )cos 2 2 sin 2 cos [( 1 2 )], = (1 2 ) sin 1 +2 cos 1. (4) cos [( 1 2 )] In the plane of (, ), equation (4) gives the transition set where at least one characteristic root of equation (3) changes the sign of the real part. As both and are even functions in frequency, the transition set will be discussed in the semi-infinite interval [, +) EQUAL TIME DELAYS If 1 = 2 =, equation (4) becomes = [(1 2 )cos 2 sin ], = [(1 2 ) sin +2 cos ]/. (41) Thus, the transition set is a continuous curve C in the plane of (, ) when the parameter varies in [, +). It is easy to find from equation (41) that the curve C starts from the point A = (1, 2 + ) in the plane of (, ). As shown in Figure 4, the curve C becomes complicated and intersects itself if the time delay is long enough. 8 4 (a) A 4 (b) A (c) A 6 (d) A Figure 4. Transition sets in the (, ) plane for various equal delays ( = 5). values: (a) 1 5; (b) 2 ; (c) 4 ; (d) 6.

10 H. Y. HU AND Z. H. WANG τ m ζ Figure 5. Critical time delay versus damping ratio. From the viewpoint of an engineer, one may want to know the critical time delay m, with which the curve C just intersects itself at (1, 2,+) in(, ) plane. According to equation (41), such a time delay yields (1 2 )cos m 2 sin m =1, (1 2 ) sin m +2 cos m =(2 + m ). This condition can be written as (42) tan ( m + )=(2 + m ), = tan 1 (2/(1 2 )). (43) In addition, the self-intersection of curve C at (1, 2 + ) implies that equation (1) has two different positive roots when = 1 and =2 + m. From equations (11) and (12), these two roots are 1 =, 2 = p = 2 m +4 m + 2. (44) Substituting in equation (43) with 2 yields tan ( m 2 m +4 m +2+)=(2 + m ) 2 m +4 m +2, = tan m +4 m +2 1 ( 2 m +4 m +2). (45) 2ζ 2ζ + δ 2ζ + τ D δ 1 1 C τ C δ Figure 6. Stability region D 1 in the (, ) plane for equal time delays = 2 =.

11 STABILITY OF SYSTEMS WITH TWO DELAYS 223 C δ D δ 2 C τ C δ 2ζ 2ζ + δ 1 2ζ + τ C τ D δ k s C 1 δ Figure 7. Transition sets C and C in the (, ) plane for equal time delays. C τ C δ Solving equation (45) numerically for the minimal positive root, one obtains the critical time delay for given damping ratio. As shown in Figure 5, the critical time delay decreases with increase of the damping ratio. Given a time delay m, the transition set C shown in Figure 6 does not intersect itself. This curve, together with the lines of = 1 and =2, surrounds a shaded region denoted by D 1 in Figure 6. According to (, )D 1, all combinations of (, )D 1 guarantee the system stability for equal time delays 1 = 2 =. This region will be called the stability region hereafter for short. It should be emphasized that D 1 in Figure 6 is the unique stability region where the feedback gains guarantee the system stability for any equal time delays 1 = 2 =. What follows is an intuitive proof of this assertion. By differentiating equation (41) with respect to, one obtains the tangent of C at the starting location d = d = ((1 2 )/) cos +(( 2 1)/ 2 2 2) sin 2(1 + ) cos +( 2 2) sin = + O(). (46) If, the curves C and C intersect each other near (, c) as shown in Figure 6. According to the analysis in section 2, only the parameter combination (, ) under the condition p 2 4q, p, q enables equation (1) to have two positive roots. The third inequality here implies that C and C intersect each other only in the region of 1. Thus, C cannot enter into the stability region D 1 if 1. As a result, the combination of feedback gains (, )D 1 ensures that the system is stable for any equal time delays 1 = 2 =. Then, it is easy to see that the system is not stable for any if the combination of feedback gains (, ) falls into other shaded regions, say D 2 in Figure 7, where the roots of equation (3) seem to have the negative real parts again with variation of (, ). In fact, even for a very short time delay, the corresponding spiral C will enter D 2 as long as the frequency is high enough. This implies that the system will undergo an instability if the disturbance x(t) involves any harmonic components of sufficiently high frequency. As a result, the assertion made by Palkovics and Venhovens in [1] that there exist other possible stability regions in (, ) plane is not true.

12 224 H. Y. HU AND Z. H. WANG (a) (b) c c k k (c) (d) c c k k Figure 8. Transition sets and stability regions in the (, ) plane for unequal time delays. (a) 1 2 and small ; (b) 1 2 and large ; (c) 1 2 and small ; (d) 1 2 and large UNEQUAL TIME DELAYS In the case of 1 2, let = 1 2. It is obvious that both and in equation (4) will approach infinity when n/2, n =1,3,5,...Thus, the transition set given by equation (4) in this case is no longer a continuous curve. It consists of an infinite number of curves defined by the parametric equation of in the intervals [, /2), (/2, 3/2), etc. As analyzed in the previous subsection, the boundary of the stability region is a small part of the transition set corresponding to the lower frequency. So, one focuses on the transition set in the frequency range [, /2). Consider the case of 1 2 first. The transition set is a curve starting from (1, )in(, ) plane and approaching infinity when ( 2 1 )/2. It is easy to find that the tangent of the asymptotic line reads (1 2 ) sin 1 +2 cos 1 [(1 2 )cos( 1 + /2) 2 sin ( 1 + /2)] = 1/ = /2( 1 2 ). Obviously, if is small, the curve will spiral one or more rounds and go to infinity in the second quadrant or the fourth quadrant. Otherwise, it goes to infinity in the fourth quadrant. These two cases are shown in Figures 8a and 8b respectively. If 2 1, one can similarly find the tangent of the asymptotic line (1 2 ) sin ( 2 + /2) + 2 cos ( 2 + /2) = 1/ = /2( [(1 2 )cos 2 2 sin 2 ] 1 2 ) when ( 1 2 )/2. The transition set will approach infinity in the first quadrant or the third quadrant if is small. Otherwise, it goes to infinity in the third quadrant. These two cases are shown in Figures 8c and 8d respectively. Noting the stability conditions (5) for a system without time delays in feedback, one can determine the stability region shown as the shaded one in Figure 8. Figure 8d shows that the stability region shrinks to a very small size if 2 1. This is the most dangerous case and should be avoided in practice. (47) (48)

13 STABILITY OF SYSTEMS WITH TWO DELAYS CONCLUSIONS (a) It is found in this paper that there exists a sufficient and necessary algebraic criterion of the delay-independent stability for linear, SDOF vibrating systems with two arbitrary time delays in displacement feedback and velocity feedback respectively. With the help of this criterion, it is quite straightforward to check the stability of a given system. (b) The feedback gains and, with which the system is delay-independent stable, form a region D in the plane of (, ). The region D is bounded in the rectangular {(, ) 1, 2} and is symmetric with respect to both the - and -axes. When the two time delays are identical, the boundary of the region can be determined analytically. Otherwise, it is a simple curve without explicit expression. An arbitrary point (a 1, a 2 ) in region D can ensure that the rectangle U = {(, ) a 1, a 2 } falls into D. This property enables one to compute region D very efficiently. (c) Given short time delays 1 and 2, the feedback gains and that guarantee the system stability fall into a simple region D in the plane of (, ). In most cases, the boundary of region D consists of a spiral and two straight segments defined by = 1 and =2 respectively. However, if 2 1, the boundary becomes complicated and the region shrinks to a very small size. ACKNOWLEDGMENTS This work was supported in part by the National Natural Science Foundation of China under the Grant of The first author wishes to thank Professors E. H. Dowell and L. N. Virgin for hosting his visit to the Department of Mechanical Engineering and Material Sciences, Duke University, USA and for their helpful comments on the manuscript. REFERENCES 1. L. PALKOVICS andp. J. TH. VENHOVENS 1992 Vehicle System Dynamic1, Investigation on stability and possible chaotic motions in the controlled wheel suspension system. 2. L. ZHANG 1993 Journal of Engineering Mechanic19, Stability of active-tendon structural control with time delay. 3. Y. X. QIN 1989 Stability of Dynamic Systems with Delays. Beijing: The Science Press. 4. K. GOPALSAMY 1992 Stability and Oscillations in Delay Differential Equations of Population Dynamics. Dordrecht: Kluwer Academic Publishers. 5. T. MORI and H. KOKAME 1989 IEEE Transactions on Automatic Control 34, Stability of x (t)=ax(t)+bx(t ). 6. L. WANG and M. Q. WANG 1993, In Dynamic Systems (S. T. Liao et al., editors). Singapore: World Scientific On Lyapunov functionals in stability theory of systems with time lag. 7. R. J. WANG and W. J. WANG 1996 IEEE Transactions on Circuits and Systems I, 43, stability analysis of perturbed systems with multiple non-commensurate time delays. 8. C. S. HSU 197 Journal of Applied Mechanics 37, Application of the -decomposition method to dynamical systems subject to the retarded follower forces. 9. G. STEPAN 1989 Retarded Dynamical Systems: Stability and Characteristic Functions. England: Longman Scientific and Technical. 1. J. CHEN 1995 IEEE Transactions on Automatic Control 4, On computing the maximal delay intervals for stability of linear delay systems. 11. H. Z. WANG 1985 Handbook of Modern Mathematics for Engineers, 1, Wuhan: Press of Huazhong University of Science and Technology.