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Necessary conditions for the stability of one delay systems: a Lyapunov matrix approach*
Proceedings of the 10-th IFAC Workshop on Time Delay Systems The International Federation of Automatic Control Northeastern University, Boston, USA. June 22-24, 2012
Necessary conditions for the stability of one delay systems: a Lyapunov matrix approach S. Mondié ∗ , C. Cuvas ∗
∗
, A. Ramírez ∗ , A. Egorov ∗∗
Departamento de Control Automático , Cinvestav, IPN, México D.F., (e-mails: smondie,ccuvas,[email protected].) ∗∗ Saint-Petersburg State University, Saint-Petersburg, Russia, (e-mail: [email protected])
Abstract: Necessary conditions for the exponential stability of one delay linear systems expressed in terms of the Lyapunov matrix of the system are proved. The effectiveness of the proposed conditions is shown in illustrative examples. Keywords: Delay linear systems, stability, Lyapunov matrix, Lyapunov-Krasovskii approach. 1. INTRODUCTION
of delayed output feedback, or in the proportional delayed control of systems studied respectively in Mendez-Barrios et al. (2008) and Villafuerte and Mondié (2010), is a strong motivation for looking for better suited conditions for this case.
The approach of Lyapunov-Krasovskii functionals for the stability analysis of time delay systems, developed in Krasovskii (1956), provides fundamental results concerning the existence of the functional when the system is stable. For the linear case, the form of the functional which appeared first in Repin (1965) and Datko (1972), was established in the contributions of Infante and Castelan (1978), Huang (1989), where a cubic lower bound was found, and Louisell (1998), where substantial advances were presented. During the past decade, in Kharitonov and Zhabko (2003), Kharitonov (2006) fundamental concepts were clarified and extended to the cases of distributed and neutral type delay systems: the Lyapunov matrix function is obtained as the solution of the dynamic, symmetric and algebraic properties that play the role of the Lyapunov equation; the existence and uniqueness of its solution provided a spectrum Lyapunov like condition is satisfied is established in Kharitonov and Plischke (2006). This lead to the presentation of a functional, named of complete type, having a quadratic lower bound when the system is exponentially stable.
In this contribution, we analyze the stability of linear time delay systems of the form () = 0 () + 1 ( − ) (1) × where 0 1 ∈ R ≥ 0 is the delay and the initial condition is () = () − ≤ ≤ 0 ∈ PC[− 0] The organization of the paper is as follows: stability results on Lyapunov Krasovskii functionals with prescribed derivative for one delay systems, are recalled in section 2. The main contribution of the paper, necessary conditions that depend on the Lyapunov matrix, is proved in section 3. In section 4, nontrivial illustrative examples and comparison with known exact stability regions and with the conditions obtained in Mondié and Egorov (2011) show the reduction of conservatism, specially in cases where 1 is singular. The contribution ends with some concluding remarks. Notation: the Euclidian norm for vectors is denoted kk. For a given initial condition () in the set of piecewise functions defined on the interval [− 0] PC([− 0] R ) () = {( + ) ∈ [− 0)} denotes the restriction of the solution ( ) of system (1) on the interval [ − ) When the initial condition is not crucial, the argument is omitted. The set of piecewise functions is equipped with the norm kk = sup k()k
Having in mind the delay free case, where a stability test in terms of the positivity of the Lyapunov matrix through the Sylvester criteria is available, the above approach was employed in the past few years to prove a number of stability results. Conditions establishing, for the one delay scalar equation, the coincidence of the well known stability region with Lyapunov function conditions were recently obtained in Mondié (2012). Instability conditions were determined in Mondié et al. (2011) for retarded, neutral type and distributed delay systems. Finally, necessary conditions in terms of the Lyapunov matrix are reported in Mondié and Egorov (2011) for the case of single delay systems. These conditions are significantly conservative when matrix 1 is singular, as shown in the examples of section 4. The fact that this situation often arise in control, as in the problem
∈[−0]
×
For a symmetric matrix ∈ , the notation ( ≥ ) means that − is positive definite (positive semidefinite). 2. LYAPUNOV FRAMEWORK System (1) is said to be exponentially stable if there exist constants ≥ 1 and 0 such that for every initial
Supported by Conacyt, Mexico grant 61076. 978-3-902823-04-5/12/$20.00 © 2012 IFAC
13
10.3182/20120622-3-US-4021.00022
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
function ∈ PC([− 0] R ) the solution ( ) satisfies the following inequality: k( )k ≤ − kk The stability of a system can be analyzed with the help of a Lyapunov-Krasovskii functional.
( ) = − ()( − )() − ( − )( − )
≤ −min ( − ) k()k2 and the result follows from the choice 0 It is straightforward to see that Z 0 2 ()() ≤ max () kk
The initial results on the basic functional 0 ( ) with prescribed time derivative − () (), introduced in Huang (1989), are summarized below. Lemma 1. Given a positive definite matrix the functional 0 ( ) whose time derivative along the trajectories of system (1) is equal to − () () is of the form
−
Consider now the functional ( ) = ( ) − k()k2 where ( ) is given in (7). Then
0 () = (0) (0)(0) R0 +2 (0) − ( + )1 () R0 R0 + − (2 ) − 1 (2 − 1 )1 (1 )1 2
(2)
( ) = − ( ) = − ()( − )() − ( − )( − )
Then, if the system (1) is exponentially stable, the matrix function () ∈ [− ], is the unique solution of the dynamic equation (3) 0 () = ()0 + ( − )1 ≥ 0 with boundary conditions, called symmetric and algebraic properties, (−) = () ≥ 0 (4) 0 (0) + (0)0 + 1 () + ()1 = −
−2 () [0 () + 1 ( − )]
It follows that
¸ ∙ £ ¤ () ( ) = () ( − ) L() ( − )
where
(5)
L() =
It was shown in Huang (1989), Hale (1977), that this functional admits a local cubic lower bound: Theorem 2. If the system (1) is exponentially stable, then for any 0 there is a constant 0 such that () > k(0)k3 kkh 6 (6)
Here, we consider a particular case of the additional term introduced in Kharitonov and Zhabko (2003). In this case the functional is not of complete type, yet it satisfies a quadratic lower bound. The proof, which follows closely the one there presented, is included for completeness. Theorem 3. Let the delay system (1) be exponentially stable and let a positive definite matrix be given Then, for any 0 the functional () = 0 () + ˜() (7) where 0 is the basic functional introduced in Lemma 1 and ˜ is defined as Z 0 ˜() = ()() (8)
(12)
→∞
→∞
from the definition (11). Finally, as the system is stable, R∞ 0 () = 0 ( ) ( ) ≥ 0 hence 1 (0) = 0
Next, we remind some properties of the Lyapunov matrix. Lemma 4. (Kharitonov and Plischke, 2006) The matrix () ∈ [− 0] is infinitely many times differentiable on (− 0). Its first derivative has a jump discontinuity at = 0. Lemma 5. (Kharitonov and Zhabko, 2003) The first derivative of the Lyapunov matrix satisfies 0 () = ()0 + ( − )1 ≥ 0 (13)
−
2
¸ ∙ ¸ 0 + 0 1 − 0 + 0 1 0
Because of the stability assumption on system (1), lim ( ) = 0 hence − () ≤ 0 and the result follows
is such that there exist positive scalars () 1 () and 2 () such that 2 (9) ( ) ≤ −() k()k 2
∙
Let 1 be the first positive value for which the determinant of the matrix pencil L() vanishes (1 is a real number because the involved matrices are real and symmetric). As 0 the matrix L(0) is positive definite. It follows that for all ∈ [0 1 ) the matrix pencil L() is still positive definite hence ( ) = − ( ) ≤ 0 Integrating this inequality from zero to infinity yields Z ∞ Z ∞ ( ) ( ) ≤ 0 = − 0 0 therefore lim ( ) − () ≤ 0
Moreover, examples showing that the functional (2) does not admit a quadratic lower bound were given in Kharitonov and Zhabko (2003), leading to the presentation of a functional named complete that satisfies one.
1 () k(0)k ≤ () ≤ 2 () kk In addition, 1 (0) = 0
(11)
(10)
0 () = −0 () − 1 ( + )
Proof. Lemma 1 implies that the time derivative of the functional (7) along the system trajectories satisfies
0
(14)
Proof. The expression (13) follows straightforwardly from (3) and (4). The symmetric property implies that 14
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
− 0 (− ) = [ 0 ( )] 0
is such that
(15)
= [ ( )0 + ( − )1 ]
and
0 (− ) = −0 ( ) − 1 ( − ) Using again the symmetric property, 0 (− ) = −0 (− ) − 1 ( − ) and (14) is obtained by setting = −
()0 − £ +() ()0 + (
where () is the Dirac delta function.
(0) 0
∈ [0 ]
(22)
⎧ ⎨ 0 ¯ () = 0 ⎩
() =
+ 0 ()0 − 1 ()1 − )1 + 0 () + 1 ( +
(21)
Proof. We start with the functional (2) and we consider the special initial function
00
0 0 ()
∈ [0 ]
with ( ) the solution of (3), (4) and (5).
Furthermore, we prove the following equality. Lemma 6. The Lyapunov matrix satisfies for ∈ [− ] 0
K( ) ≥ 0
¤ ) (16)
∈ [− − ) ∈ [− 0− ) =0
∈
∈ (0 )
(23)
The first term of 0 () reduces to (¯ ) = ¯ (0) (0)¯ (0) = (0)
Proof. The expressions (13) and (14) can be summarized as £ ¤ 0 () = ( ≥ 0) ()0 + ( − )1 £ ¤ +(1 − ( ≥ 0)) −0 () − 1 ( + ) where ( ≥ 0) denotes the Heaviside step function whose value is zero for negative arguments and one for arguments greater or equal to zero. The second derivative is h £ ¤0 i 00 () = ( ≥ 0) 0 ()0 − ( − ) 1 £ ¤ +(1 − ( ≥ 0)) −0 0 () − 1 0 ( + ) £ ¤ +() ()0 + ( − )1 + 0 () + 1 ( + ) (17) Notice that
(24)
The simple integral term in 0 () can be written as () = 2 (0) =2
Z
−
−
+2
Z
Z
0
( + ) 1 ()
−
()1 ( + )(0)
0
()1 ( + )(0)
−
Substituting the initial function (23) yields
(¯ ) = 2
£ ¤0 0 ()0 − ( − ) 1 £ ¤ = 0 ()0 − ( − )0 + ()1 1
Z
− −
+2
Z
0 1 ( + )
0
0 1 ( + )
−
= 0 ()0 − 0 0 () + 0 ()0 − 1 ()1 (18) and that
using (14) and integration by parts results in
−0 0 () − 1 0 ( + ) £ ¤ = −0 0 () − 1 ( + )0 + (−)1
(¯ ) = −2 −0 (− ) + 2 −0 (−)
−2 (0) + 2 −0 (− )
= −0 0 () + 0 ()0 + 0 ()0 − 1 ()1(19) Finally, (16) follows by substituting (18) and (19) into (17).
(25)
The double integral, is decomposed as 1 () = R0 R − (1 ) − 1 (1 − 2 )1 (2 )2 1 − R − R0 + − (1 ) − 1 (1 − 2 )1 (2 )2 1 R0 R0 + − (1 ) − 1 (1 − 2 )1 (2 )2 1 R − R − + − (1 ) − 1 (1 − 2 )1 (2 )2 1
3. NECESSARY CONDITIONS Necessary conditions for the exponential stability of system (1), holding as well in the case 1 is singular are presented. They are obtained by substituting into the functional a set of initial functions of exponential form, and using the results of the previous sections allowing the elimination of matrix 1 Integration by parts, and the quadratic lower bound of Theorem 3 are also substantial elements of the proof. Lemma 7. If the delay system (1) is stable then the matrix ⎛ ⎞ (0) ( ) () (0) ( − ) ⎠ K( ) = ⎝ ( ) (20) () ( − ) (0)
Substituting the initial function (23) yields
1 (¯ ) = R0 R − − 0 1 − 1 (1 − 2 )1 0 2 2 1 R − R 0 + − 0 1 − 1 (1 − 2 )1 0 2 2 1 R R0 0 + − 0 1 − 1 (1 − 2 )1 0 2 2 1 R − R − + − 0 1 − 1 (1 − 2 )1 0 2 2 1
Appropriate subsbtitution of (16) and integration by parts leads to 15
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
© 1 (¯ ) = ( )−0 − ()−0 − 0
−
− 0
(0)−0 +
is negative, therefore, in view of the quadratic lower bound (10), if the system (1) is stable it necessary holds that ª © Ψ H ( )K( )H( ) Ψ ≥ 0 for ∈ [0 ] The vectors are arbitrary and the exponential terms in Ψ are non singular, hence H ( )K( )H( ) ≥ 0 for ∈ [0 ] Finally, as the orthogonal transformation H( ) is nonsingular for all the condition (21) follows. In addition, for the special case = 0 = 0, the condition (22) is satisfied.
o
( − )−0
n + −0 (− ) − −0 (0)−0
o −−0 (−) + −0 (− + )−0 © + (0) − ( )−0 o −−0 (− ) + −0 (0)−0 n + −0 (0)−0 − −0 ( − )−0 o −−0 ( − )−0 + −0 (0)−0
4. ILLUSTRATIVE EXAMPLES In this section, some examples illustrate the fact that the new conditions improve previously obtained results. In each case, the imaginary axis crossing loci of the characteristic quasipolynomial of the system are determined using the well known D-subdivision techniques, see Neimark (1949). In each example, these boundaries are depicted, along with points in the parametric space for which the necessary conditions reported in Mondié and Egorov (2011) or the new conditions (21-22) hold. Notice that in all the cases below, the matrix 1 is singular. Example 8. The characteristic equation of a second order system in closed loop with an output delayed proportional integral feedback controller is given by the quasipolynomial 2 + + + ( − 1)( + )− = 0 introduced in Mendez-Barrios et al. (2008). If the stability is analyzed, the problem reduces to the stability of the closed loop system (1) with ⎞ ⎛ 0 1 0 0 0 1 ⎠; 0 = ⎝ 3 2 2 − + −3 +2 − 3 − ⎛ ⎞ 0 0 0 0 ¡ 0 0 ⎠ 1 = ⎝ ¢ 2 ( + 1) − + (2 + 1) − − Here is the desired exponential decay and are the controller parameters. The closed loop quasipolynomial of this system is
(26)
and ) = − 2 (¯ −
Z
Z
−
0 0
− 0
0 0
(27)
−
Next, observe that substituting the initial function (23) into ˜ gives Z − ˜(¯ ) = 0 0 −
+
Z
0
0 0
(28)
−
Now, adding the terms (24), (25), (26) and (27) corresponding to 0 with (28) corresponding to ˜ and rearranging yields ) + ˜(¯ ) (¯ ) = 0 (¯ where
= Φ M( )Φ + 2 (¯ ) + ˜(¯ )
M( ) = ⎛ (0) ( ) () (0) ( ) ⎞ (0) ( − ) ( ) (0) ⎟ ⎜ ( ) ⎟ ⎜ ⎜ () ( − ) (0) () ( − ) ⎟ ⎝ (0) ( ) () (0) ( ) ⎠ (0) ( − ) ( ) (0) ( ) and ³ ´ Φ = −0 −0 − − −0
() = ( − )3 + ( − )2 + ( − ) £ ¤ + ( − )2 − + ( − )( − ) −(−) The exact stability domain is delimited by the parametric equations that result from the solution of the system µ ¶ = () () where () = µ 2 2 ( + − ) cos() − (2 + ) sin() 2 −(2 + 1) cos() + ( 2 − 2 −¶) sin() −( + 1) cos() + sin() cos() + ( + 1) sin() µ 3 ¶ − 2 + − 3 2 + 2 () = 3 − (3 2 − 2 + ) On Figure 1, the parameter numerical values are = 01 = 1 = −04 and = 3
The above can be written as ª © (¯ ) = Ψ H ( )K( )H( ) Ψ + 2 (¯ ) + ˜(¯ ) with ⎛ ⎞ ⎞ ⎛ 0 −0 Ψ = ⎝ −0 ⎠ H( ) = ⎝ 0 −0 (− ) ⎠ −0 0 0 and K( ) is defined in (20). As 0 , the quadratic term Z − 2 (¯ ) + ˜(¯ ) = 0 ( − )0 −
+
Z
0
−
0 ( − )0 16
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
Example 10. Let us consider the -stabilization of a second order system via proportional delayed output feedback. The closed loop system is in the form (1) with ¶ µ 0 1 0 = ; 2 −( + 2 − 2 + ) −2 ( − ) µ ¶ 0 0 1 = 0
0
−0.5
−1 k
i
−1.5
−2
−2.5 −0.5
0
0.5
kp
1
1.5
Here and are the frequency and the damping coefficient of the second order system, is the desired exponential decay and are the controller parameters. The closed loop quasipolynomial of this system is
2
Fig. 1. Example 8, conditions (21-22) or conditions in Mondié and Egorov (2011).
() = 2 + 2 ( − ) ¡ ¢ + 2 + 2 − 2 + − − The exact stability domain is delimited by the boundaries described by the parametric equations () = −2 ( − ) sin ( () ) ¡ 2 ¢ 2 2 −1 − + + − 2 + () = (1) cot (−2 ( − )) + = 0 1 3 4 2 2 and by = ( + −2+ ) where 0 ≤ ≤ 05. On Figures 4 and 5, the numerical values for the fixed parameters are = 2 = 31 = 176 = 00128 and = 2257
Example 9. The following is an example of the few convex directions 1 for systems the form (1), see Kharitonov et al. (2003). Here ¶ ¶ µ µ −1 1 0 −1 ; 1 = (29) 0 = −4 −1 0 0
We analyze the quasipolynomial () = det( − 0 − 1 ) = 2 + 2 + 5 − 4− The exact stability domain is described by the segment = 54 and the parametric equations q () = (5 − )2 + 4 2 4 ¶ µ q 2 −1 2 () = sin −2 (5 − ) + 4 + 2
for ∈ [0 ] = 0 1 .. For two dimensional systems, one can observe on Figure 3 that 1 defined in (29) is a convex direction, in other words, when increases, no stability reversal occurs.
60 50 40 30 kr 20
15
10 0 −10
10
−20 h
−30 0
0.05
0.1
0.15
0.2 h
0.25
0.3
0.35
0.4
5
0 0.95
Fig. 4. Example 10, conditions in Mondié and Egorov (2011). 1
1.05
1.1
μ
1.15
1.2
1.25
1.3
60
Fig. 2. Example 9, conditions in Mondié and Egorov (2011).
50 40 30 k 20 r
15
10 0 −10
10
−20 h
−30 0
0.05
0.1
0.15
0.2 h
0.25
0.3
0.35
0.4
5
Fig. 5. Example 10, conditions (21-22) 0 0.95
1
1.05
1.1
μ
1.15
1.2
1.25
1.3
A few observations concerning the above examples, where 1 is singular, are in order: As the number of unstable roots in a given region in the space of parameters
Fig. 3. Example 9, conditions (21-22) 17
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
REFERENCES
is constant, see Neimark (1949), the necessary stability conditions must hold in the whole region. Nevertheless, a natural query is how close these conditions are to possible sufficient conditions. In Example 8, although 1 is singular, the shaded regions coincide with the proved stability region in Mendez-Barrios et al. (2008), for conditions (2122) and conditions in Mondié and Egorov (2011). In Example 9, it appears that the conditions (21-22), which match the stability region, outperform the conditions in Mondié and Egorov (2011). Finally, in Example 10, the conditions (21-22) improve the upper estimate of the stability region obtained with conditions in Mondié and Egorov (2011), yet the shaded region does not match the known stability region, hence we conclude that these conditions are not sufficient. Further improvement may be achieved by employing the more complex initial condition with 1 2 3 4 ∈ ⎧ 0 ∈ [− − 2 ) ⎪ ⎨ 0 4 3 ∈ [− 2 − 1 ) ˜ () = 1 2 ∈ (0 ) 0 ∈ [− 1 0− ) ⎪ ⎩ 2 =0 1 The steps of the proof presented in this contribution lead to the new necessary conditions (21) and K4 ( ) ≥ 0 1 ∈ [0 ] 2 ∈ [0 ]
with
(30)
⎛
⎞ (0) ( 1 ) ( 2 ) () ⎜ ( 1 ) (0) ( 2 − 1 ) ( − 1 ) ⎟ ⎟ K4 ( ) = ⎜ ⎝ ( 2 ) ( 2 − 1 ) (0) ( − 2 ) ⎠ (0) () ( − 1 ) ( − 2 ) For these conditions, the estimate of the stability regions for Example 10 is still improved as shown on Figure 6, but they are not possibly sufficient either. 60 50 40 30 kr 20 10 0 −10 −20 −30 0
0.05
0.1
0.15
0.2 h
0.25
0.3
0.35
0.4
Fig. 6. Example 10, conditions (30-22)
5. CONCLUDING REMARKS In this contribution, we show that the necessary stability conditions (21-22) for linear one delay systems with 1 nonsingular expressed in terms of the Lyapunov matrix of the system are also valid in the case of singular matrix 1 . It is also shown with an example that the obtained conditions are not sufficient. Nevertheless, these necessary conditions allow the detection of candidate stability regions when the imaginary axis crossing boundaries are known. 18
Datko, R. (1972). An algorithm for computing Liapunov functionals for some differential difference equations. Ordinary Differential Equations. Academic Press, New York, 387-398. Hale, J.K. (1977). Theory of functional differential equations, springer-Verlag, New-York. Huang, W. (1989). Generalization of Liapunov’s theorem in a linear delay system, J. of math. anal. and appl., 142, 83-94. Infante, E.F. and W.B. Castelan. (1978). A Liapunov functional for a matrix difference-differential equation, J. of Dif. Equ., 29, 439—451. Kharitonov, V., Mondié, S., Santos, J. (2003). Matrix Convex directions for time delay systems, Int. J. of Robust and Nonlinear Contr., 13:14, 1259-1335. Kharitonov, V.L. and A.P. Zhabko. (2003). LyapunovKrasovskii approach for robust stability of time delay systems, Automatica, 39, 15-20. Kharitonov, V.L. (2006). Lyapunov matrices for a class of time delay systems, Syst. & Contr. Lett., 55, 610-617. Kharitonov, V.L. and E. Plischke. (2006). Lyapunov matrices for time-delay systems, Syst. & Contr. Lett., 55, 697-706. Krasovskii, N.N. (1956). On the application of the second method of Lyapunov for equations with time delays, Prikl. Mat. Mekh., 20, 315-327. Louisell, J. (1998). Numerics of the stability exponent and eigenvalues abscissas of a matrix delay equation, Stability and Control of Time Delay Systems, Lecture notes in Control and Information Sciences 228, SpringerVerlag, New-York. Mendez-Barrios, C., Niculescu S-I., Morarescu C-I. and Gu K. (2008). On the fragility of PI controllers for time delay SISO Systems, 16th Mediterranean Conference on Control and Automation, France. Mondié, S., Egorov A. (2011). Some necessary conditions for the exponential stability of one delay systems, 8th International Conference on Electrical Engineering, Computing Science and Automatic Control, Merida, Mexico, 103-108. Mondié, S., Ochoa G. and B. Ochoa. (2011). Instability conditions for linear time delay systems: a Lyapunov matrix function approach, Int. J. of Contr., 84:10), 1601—1611. Mondié, S. (2012). Exact stability region of the single delay scalar equation: a time domain criteria, accepted in IMA J. of Math Contr. and Inf.. Neimark, J. (1949). D-subdivisions and spaces of quasipolynomials , Prikl. Mat. Meh., 13, 349-380. Repin, M. Yu. (1965). Quadratic Lyapunov functionals for systems with delay, Prik. Mat. Meh., 29, 564-566. Villafuerte, R., Mondié S., and R. Garrido (2012). Tuning the leading roots of a second order DC servomotor with proportional retarded control, accepted in IEEE Trans. on Contr. Syst. Tech..
Necessary conditions for the stability of one delay systems: a Lyapunov matrix approach S. Mondié ∗ , C. Cuvas ∗
∗
, A. Ramírez ∗ , A. Egorov ∗∗
Departamento de Control Automático , Cinvestav, IPN, México D.F., (e-mails: smondie,ccuvas,[email protected].) ∗∗ Saint-Petersburg State University, Saint-Petersburg, Russia, (e-mail: [email protected])
Abstract: Necessary conditions for the exponential stability of one delay linear systems expressed in terms of the Lyapunov matrix of the system are proved. The effectiveness of the proposed conditions is shown in illustrative examples. Keywords: Delay linear systems, stability, Lyapunov matrix, Lyapunov-Krasovskii approach. 1. INTRODUCTION
of delayed output feedback, or in the proportional delayed control of systems studied respectively in Mendez-Barrios et al. (2008) and Villafuerte and Mondié (2010), is a strong motivation for looking for better suited conditions for this case.
The approach of Lyapunov-Krasovskii functionals for the stability analysis of time delay systems, developed in Krasovskii (1956), provides fundamental results concerning the existence of the functional when the system is stable. For the linear case, the form of the functional which appeared first in Repin (1965) and Datko (1972), was established in the contributions of Infante and Castelan (1978), Huang (1989), where a cubic lower bound was found, and Louisell (1998), where substantial advances were presented. During the past decade, in Kharitonov and Zhabko (2003), Kharitonov (2006) fundamental concepts were clarified and extended to the cases of distributed and neutral type delay systems: the Lyapunov matrix function is obtained as the solution of the dynamic, symmetric and algebraic properties that play the role of the Lyapunov equation; the existence and uniqueness of its solution provided a spectrum Lyapunov like condition is satisfied is established in Kharitonov and Plischke (2006). This lead to the presentation of a functional, named of complete type, having a quadratic lower bound when the system is exponentially stable.
In this contribution, we analyze the stability of linear time delay systems of the form () = 0 () + 1 ( − ) (1) × where 0 1 ∈ R ≥ 0 is the delay and the initial condition is () = () − ≤ ≤ 0 ∈ PC[− 0] The organization of the paper is as follows: stability results on Lyapunov Krasovskii functionals with prescribed derivative for one delay systems, are recalled in section 2. The main contribution of the paper, necessary conditions that depend on the Lyapunov matrix, is proved in section 3. In section 4, nontrivial illustrative examples and comparison with known exact stability regions and with the conditions obtained in Mondié and Egorov (2011) show the reduction of conservatism, specially in cases where 1 is singular. The contribution ends with some concluding remarks. Notation: the Euclidian norm for vectors is denoted kk. For a given initial condition () in the set of piecewise functions defined on the interval [− 0] PC([− 0] R ) () = {( + ) ∈ [− 0)} denotes the restriction of the solution ( ) of system (1) on the interval [ − ) When the initial condition is not crucial, the argument is omitted. The set of piecewise functions is equipped with the norm kk = sup k()k
Having in mind the delay free case, where a stability test in terms of the positivity of the Lyapunov matrix through the Sylvester criteria is available, the above approach was employed in the past few years to prove a number of stability results. Conditions establishing, for the one delay scalar equation, the coincidence of the well known stability region with Lyapunov function conditions were recently obtained in Mondié (2012). Instability conditions were determined in Mondié et al. (2011) for retarded, neutral type and distributed delay systems. Finally, necessary conditions in terms of the Lyapunov matrix are reported in Mondié and Egorov (2011) for the case of single delay systems. These conditions are significantly conservative when matrix 1 is singular, as shown in the examples of section 4. The fact that this situation often arise in control, as in the problem
∈[−0]
×
For a symmetric matrix ∈ , the notation ( ≥ ) means that − is positive definite (positive semidefinite). 2. LYAPUNOV FRAMEWORK System (1) is said to be exponentially stable if there exist constants ≥ 1 and 0 such that for every initial
Supported by Conacyt, Mexico grant 61076. 978-3-902823-04-5/12/$20.00 © 2012 IFAC
13
10.3182/20120622-3-US-4021.00022
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
function ∈ PC([− 0] R ) the solution ( ) satisfies the following inequality: k( )k ≤ − kk The stability of a system can be analyzed with the help of a Lyapunov-Krasovskii functional.
( ) = − ()( − )() − ( − )( − )
≤ −min ( − ) k()k2 and the result follows from the choice 0 It is straightforward to see that Z 0 2 ()() ≤ max () kk
The initial results on the basic functional 0 ( ) with prescribed time derivative − () (), introduced in Huang (1989), are summarized below. Lemma 1. Given a positive definite matrix the functional 0 ( ) whose time derivative along the trajectories of system (1) is equal to − () () is of the form
−
Consider now the functional ( ) = ( ) − k()k2 where ( ) is given in (7). Then
0 () = (0) (0)(0) R0 +2 (0) − ( + )1 () R0 R0 + − (2 ) − 1 (2 − 1 )1 (1 )1 2
(2)
( ) = − ( ) = − ()( − )() − ( − )( − )
Then, if the system (1) is exponentially stable, the matrix function () ∈ [− ], is the unique solution of the dynamic equation (3) 0 () = ()0 + ( − )1 ≥ 0 with boundary conditions, called symmetric and algebraic properties, (−) = () ≥ 0 (4) 0 (0) + (0)0 + 1 () + ()1 = −
−2 () [0 () + 1 ( − )]
It follows that
¸ ∙ £ ¤ () ( ) = () ( − ) L() ( − )
where
(5)
L() =
It was shown in Huang (1989), Hale (1977), that this functional admits a local cubic lower bound: Theorem 2. If the system (1) is exponentially stable, then for any 0 there is a constant 0 such that () > k(0)k3 kkh 6 (6)
Here, we consider a particular case of the additional term introduced in Kharitonov and Zhabko (2003). In this case the functional is not of complete type, yet it satisfies a quadratic lower bound. The proof, which follows closely the one there presented, is included for completeness. Theorem 3. Let the delay system (1) be exponentially stable and let a positive definite matrix be given Then, for any 0 the functional () = 0 () + ˜() (7) where 0 is the basic functional introduced in Lemma 1 and ˜ is defined as Z 0 ˜() = ()() (8)
(12)
→∞
→∞
from the definition (11). Finally, as the system is stable, R∞ 0 () = 0 ( ) ( ) ≥ 0 hence 1 (0) = 0
Next, we remind some properties of the Lyapunov matrix. Lemma 4. (Kharitonov and Plischke, 2006) The matrix () ∈ [− 0] is infinitely many times differentiable on (− 0). Its first derivative has a jump discontinuity at = 0. Lemma 5. (Kharitonov and Zhabko, 2003) The first derivative of the Lyapunov matrix satisfies 0 () = ()0 + ( − )1 ≥ 0 (13)
−
2
¸ ∙ ¸ 0 + 0 1 − 0 + 0 1 0
Because of the stability assumption on system (1), lim ( ) = 0 hence − () ≤ 0 and the result follows
is such that there exist positive scalars () 1 () and 2 () such that 2 (9) ( ) ≤ −() k()k 2
∙
Let 1 be the first positive value for which the determinant of the matrix pencil L() vanishes (1 is a real number because the involved matrices are real and symmetric). As 0 the matrix L(0) is positive definite. It follows that for all ∈ [0 1 ) the matrix pencil L() is still positive definite hence ( ) = − ( ) ≤ 0 Integrating this inequality from zero to infinity yields Z ∞ Z ∞ ( ) ( ) ≤ 0 = − 0 0 therefore lim ( ) − () ≤ 0
Moreover, examples showing that the functional (2) does not admit a quadratic lower bound were given in Kharitonov and Zhabko (2003), leading to the presentation of a functional named complete that satisfies one.
1 () k(0)k ≤ () ≤ 2 () kk In addition, 1 (0) = 0
(11)
(10)
0 () = −0 () − 1 ( + )
Proof. Lemma 1 implies that the time derivative of the functional (7) along the system trajectories satisfies
0
(14)
Proof. The expression (13) follows straightforwardly from (3) and (4). The symmetric property implies that 14
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
− 0 (− ) = [ 0 ( )] 0
is such that
(15)
= [ ( )0 + ( − )1 ]
and
0 (− ) = −0 ( ) − 1 ( − ) Using again the symmetric property, 0 (− ) = −0 (− ) − 1 ( − ) and (14) is obtained by setting = −
()0 − £ +() ()0 + (
where () is the Dirac delta function.
(0) 0
∈ [0 ]
(22)
⎧ ⎨ 0 ¯ () = 0 ⎩
() =
+ 0 ()0 − 1 ()1 − )1 + 0 () + 1 ( +
(21)
Proof. We start with the functional (2) and we consider the special initial function
00
0 0 ()
∈ [0 ]
with ( ) the solution of (3), (4) and (5).
Furthermore, we prove the following equality. Lemma 6. The Lyapunov matrix satisfies for ∈ [− ] 0
K( ) ≥ 0
¤ ) (16)
∈ [− − ) ∈ [− 0− ) =0
∈
∈ (0 )
(23)
The first term of 0 () reduces to (¯ ) = ¯ (0) (0)¯ (0) = (0)
Proof. The expressions (13) and (14) can be summarized as £ ¤ 0 () = ( ≥ 0) ()0 + ( − )1 £ ¤ +(1 − ( ≥ 0)) −0 () − 1 ( + ) where ( ≥ 0) denotes the Heaviside step function whose value is zero for negative arguments and one for arguments greater or equal to zero. The second derivative is h £ ¤0 i 00 () = ( ≥ 0) 0 ()0 − ( − ) 1 £ ¤ +(1 − ( ≥ 0)) −0 0 () − 1 0 ( + ) £ ¤ +() ()0 + ( − )1 + 0 () + 1 ( + ) (17) Notice that
(24)
The simple integral term in 0 () can be written as () = 2 (0) =2
Z
−
−
+2
Z
Z
0
( + ) 1 ()
−
()1 ( + )(0)
0
()1 ( + )(0)
−
Substituting the initial function (23) yields
(¯ ) = 2
£ ¤0 0 ()0 − ( − ) 1 £ ¤ = 0 ()0 − ( − )0 + ()1 1
Z
− −
+2
Z
0 1 ( + )
0
0 1 ( + )
−
= 0 ()0 − 0 0 () + 0 ()0 − 1 ()1 (18) and that
using (14) and integration by parts results in
−0 0 () − 1 0 ( + ) £ ¤ = −0 0 () − 1 ( + )0 + (−)1
(¯ ) = −2 −0 (− ) + 2 −0 (−)
−2 (0) + 2 −0 (− )
= −0 0 () + 0 ()0 + 0 ()0 − 1 ()1(19) Finally, (16) follows by substituting (18) and (19) into (17).
(25)
The double integral, is decomposed as 1 () = R0 R − (1 ) − 1 (1 − 2 )1 (2 )2 1 − R − R0 + − (1 ) − 1 (1 − 2 )1 (2 )2 1 R0 R0 + − (1 ) − 1 (1 − 2 )1 (2 )2 1 R − R − + − (1 ) − 1 (1 − 2 )1 (2 )2 1
3. NECESSARY CONDITIONS Necessary conditions for the exponential stability of system (1), holding as well in the case 1 is singular are presented. They are obtained by substituting into the functional a set of initial functions of exponential form, and using the results of the previous sections allowing the elimination of matrix 1 Integration by parts, and the quadratic lower bound of Theorem 3 are also substantial elements of the proof. Lemma 7. If the delay system (1) is stable then the matrix ⎛ ⎞ (0) ( ) () (0) ( − ) ⎠ K( ) = ⎝ ( ) (20) () ( − ) (0)
Substituting the initial function (23) yields
1 (¯ ) = R0 R − − 0 1 − 1 (1 − 2 )1 0 2 2 1 R − R 0 + − 0 1 − 1 (1 − 2 )1 0 2 2 1 R R0 0 + − 0 1 − 1 (1 − 2 )1 0 2 2 1 R − R − + − 0 1 − 1 (1 − 2 )1 0 2 2 1
Appropriate subsbtitution of (16) and integration by parts leads to 15
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
© 1 (¯ ) = ( )−0 − ()−0 − 0
−
− 0
(0)−0 +
is negative, therefore, in view of the quadratic lower bound (10), if the system (1) is stable it necessary holds that ª © Ψ H ( )K( )H( ) Ψ ≥ 0 for ∈ [0 ] The vectors are arbitrary and the exponential terms in Ψ are non singular, hence H ( )K( )H( ) ≥ 0 for ∈ [0 ] Finally, as the orthogonal transformation H( ) is nonsingular for all the condition (21) follows. In addition, for the special case = 0 = 0, the condition (22) is satisfied.
o
( − )−0
n + −0 (− ) − −0 (0)−0
o −−0 (−) + −0 (− + )−0 © + (0) − ( )−0 o −−0 (− ) + −0 (0)−0 n + −0 (0)−0 − −0 ( − )−0 o −−0 ( − )−0 + −0 (0)−0
4. ILLUSTRATIVE EXAMPLES In this section, some examples illustrate the fact that the new conditions improve previously obtained results. In each case, the imaginary axis crossing loci of the characteristic quasipolynomial of the system are determined using the well known D-subdivision techniques, see Neimark (1949). In each example, these boundaries are depicted, along with points in the parametric space for which the necessary conditions reported in Mondié and Egorov (2011) or the new conditions (21-22) hold. Notice that in all the cases below, the matrix 1 is singular. Example 8. The characteristic equation of a second order system in closed loop with an output delayed proportional integral feedback controller is given by the quasipolynomial 2 + + + ( − 1)( + )− = 0 introduced in Mendez-Barrios et al. (2008). If the stability is analyzed, the problem reduces to the stability of the closed loop system (1) with ⎞ ⎛ 0 1 0 0 0 1 ⎠; 0 = ⎝ 3 2 2 − + −3 +2 − 3 − ⎛ ⎞ 0 0 0 0 ¡ 0 0 ⎠ 1 = ⎝ ¢ 2 ( + 1) − + (2 + 1) − − Here is the desired exponential decay and are the controller parameters. The closed loop quasipolynomial of this system is
(26)
and ) = − 2 (¯ −
Z
Z
−
0 0
− 0
0 0
(27)
−
Next, observe that substituting the initial function (23) into ˜ gives Z − ˜(¯ ) = 0 0 −
+
Z
0
0 0
(28)
−
Now, adding the terms (24), (25), (26) and (27) corresponding to 0 with (28) corresponding to ˜ and rearranging yields ) + ˜(¯ ) (¯ ) = 0 (¯ where
= Φ M( )Φ + 2 (¯ ) + ˜(¯ )
M( ) = ⎛ (0) ( ) () (0) ( ) ⎞ (0) ( − ) ( ) (0) ⎟ ⎜ ( ) ⎟ ⎜ ⎜ () ( − ) (0) () ( − ) ⎟ ⎝ (0) ( ) () (0) ( ) ⎠ (0) ( − ) ( ) (0) ( ) and ³ ´ Φ = −0 −0 − − −0
() = ( − )3 + ( − )2 + ( − ) £ ¤ + ( − )2 − + ( − )( − ) −(−) The exact stability domain is delimited by the parametric equations that result from the solution of the system µ ¶ = () () where () = µ 2 2 ( + − ) cos() − (2 + ) sin() 2 −(2 + 1) cos() + ( 2 − 2 −¶) sin() −( + 1) cos() + sin() cos() + ( + 1) sin() µ 3 ¶ − 2 + − 3 2 + 2 () = 3 − (3 2 − 2 + ) On Figure 1, the parameter numerical values are = 01 = 1 = −04 and = 3
The above can be written as ª © (¯ ) = Ψ H ( )K( )H( ) Ψ + 2 (¯ ) + ˜(¯ ) with ⎛ ⎞ ⎞ ⎛ 0 −0 Ψ = ⎝ −0 ⎠ H( ) = ⎝ 0 −0 (− ) ⎠ −0 0 0 and K( ) is defined in (20). As 0 , the quadratic term Z − 2 (¯ ) + ˜(¯ ) = 0 ( − )0 −
+
Z
0
−
0 ( − )0 16
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
Example 10. Let us consider the -stabilization of a second order system via proportional delayed output feedback. The closed loop system is in the form (1) with ¶ µ 0 1 0 = ; 2 −( + 2 − 2 + ) −2 ( − ) µ ¶ 0 0 1 = 0
0
−0.5
−1 k
i
−1.5
−2
−2.5 −0.5
0
0.5
kp
1
1.5
Here and are the frequency and the damping coefficient of the second order system, is the desired exponential decay and are the controller parameters. The closed loop quasipolynomial of this system is
2
Fig. 1. Example 8, conditions (21-22) or conditions in Mondié and Egorov (2011).
() = 2 + 2 ( − ) ¡ ¢ + 2 + 2 − 2 + − − The exact stability domain is delimited by the boundaries described by the parametric equations () = −2 ( − ) sin ( () ) ¡ 2 ¢ 2 2 −1 − + + − 2 + () = (1) cot (−2 ( − )) + = 0 1 3 4 2 2 and by = ( + −2+ ) where 0 ≤ ≤ 05. On Figures 4 and 5, the numerical values for the fixed parameters are = 2 = 31 = 176 = 00128 and = 2257
Example 9. The following is an example of the few convex directions 1 for systems the form (1), see Kharitonov et al. (2003). Here ¶ ¶ µ µ −1 1 0 −1 ; 1 = (29) 0 = −4 −1 0 0
We analyze the quasipolynomial () = det( − 0 − 1 ) = 2 + 2 + 5 − 4− The exact stability domain is described by the segment = 54 and the parametric equations q () = (5 − )2 + 4 2 4 ¶ µ q 2 −1 2 () = sin −2 (5 − ) + 4 + 2
for ∈ [0 ] = 0 1 .. For two dimensional systems, one can observe on Figure 3 that 1 defined in (29) is a convex direction, in other words, when increases, no stability reversal occurs.
60 50 40 30 kr 20
15
10 0 −10
10
−20 h
−30 0
0.05
0.1
0.15
0.2 h
0.25
0.3
0.35
0.4
5
0 0.95
Fig. 4. Example 10, conditions in Mondié and Egorov (2011). 1
1.05
1.1
μ
1.15
1.2
1.25
1.3
60
Fig. 2. Example 9, conditions in Mondié and Egorov (2011).
50 40 30 k 20 r
15
10 0 −10
10
−20 h
−30 0
0.05
0.1
0.15
0.2 h
0.25
0.3
0.35
0.4
5
Fig. 5. Example 10, conditions (21-22) 0 0.95
1
1.05
1.1
μ
1.15
1.2
1.25
1.3
A few observations concerning the above examples, where 1 is singular, are in order: As the number of unstable roots in a given region in the space of parameters
Fig. 3. Example 9, conditions (21-22) 17
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
REFERENCES
is constant, see Neimark (1949), the necessary stability conditions must hold in the whole region. Nevertheless, a natural query is how close these conditions are to possible sufficient conditions. In Example 8, although 1 is singular, the shaded regions coincide with the proved stability region in Mendez-Barrios et al. (2008), for conditions (2122) and conditions in Mondié and Egorov (2011). In Example 9, it appears that the conditions (21-22), which match the stability region, outperform the conditions in Mondié and Egorov (2011). Finally, in Example 10, the conditions (21-22) improve the upper estimate of the stability region obtained with conditions in Mondié and Egorov (2011), yet the shaded region does not match the known stability region, hence we conclude that these conditions are not sufficient. Further improvement may be achieved by employing the more complex initial condition with 1 2 3 4 ∈ ⎧ 0 ∈ [− − 2 ) ⎪ ⎨ 0 4 3 ∈ [− 2 − 1 ) ˜ () = 1 2 ∈ (0 ) 0 ∈ [− 1 0− ) ⎪ ⎩ 2 =0 1 The steps of the proof presented in this contribution lead to the new necessary conditions (21) and K4 ( ) ≥ 0 1 ∈ [0 ] 2 ∈ [0 ]
with
(30)
⎛
⎞ (0) ( 1 ) ( 2 ) () ⎜ ( 1 ) (0) ( 2 − 1 ) ( − 1 ) ⎟ ⎟ K4 ( ) = ⎜ ⎝ ( 2 ) ( 2 − 1 ) (0) ( − 2 ) ⎠ (0) () ( − 1 ) ( − 2 ) For these conditions, the estimate of the stability regions for Example 10 is still improved as shown on Figure 6, but they are not possibly sufficient either. 60 50 40 30 kr 20 10 0 −10 −20 −30 0
0.05
0.1
0.15
0.2 h
0.25
0.3
0.35
0.4
Fig. 6. Example 10, conditions (30-22)
5. CONCLUDING REMARKS In this contribution, we show that the necessary stability conditions (21-22) for linear one delay systems with 1 nonsingular expressed in terms of the Lyapunov matrix of the system are also valid in the case of singular matrix 1 . It is also shown with an example that the obtained conditions are not sufficient. Nevertheless, these necessary conditions allow the detection of candidate stability regions when the imaginary axis crossing boundaries are known. 18
Datko, R. (1972). An algorithm for computing Liapunov functionals for some differential difference equations. Ordinary Differential Equations. Academic Press, New York, 387-398. Hale, J.K. (1977). Theory of functional differential equations, springer-Verlag, New-York. Huang, W. (1989). Generalization of Liapunov’s theorem in a linear delay system, J. of math. anal. and appl., 142, 83-94. Infante, E.F. and W.B. Castelan. (1978). A Liapunov functional for a matrix difference-differential equation, J. of Dif. Equ., 29, 439—451. Kharitonov, V., Mondié, S., Santos, J. (2003). Matrix Convex directions for time delay systems, Int. J. of Robust and Nonlinear Contr., 13:14, 1259-1335. Kharitonov, V.L. and A.P. Zhabko. (2003). LyapunovKrasovskii approach for robust stability of time delay systems, Automatica, 39, 15-20. Kharitonov, V.L. (2006). Lyapunov matrices for a class of time delay systems, Syst. & Contr. Lett., 55, 610-617. Kharitonov, V.L. and E. Plischke. (2006). Lyapunov matrices for time-delay systems, Syst. & Contr. Lett., 55, 697-706. Krasovskii, N.N. (1956). On the application of the second method of Lyapunov for equations with time delays, Prikl. Mat. Mekh., 20, 315-327. Louisell, J. (1998). Numerics of the stability exponent and eigenvalues abscissas of a matrix delay equation, Stability and Control of Time Delay Systems, Lecture notes in Control and Information Sciences 228, SpringerVerlag, New-York. Mendez-Barrios, C., Niculescu S-I., Morarescu C-I. and Gu K. (2008). On the fragility of PI controllers for time delay SISO Systems, 16th Mediterranean Conference on Control and Automation, France. Mondié, S., Egorov A. (2011). Some necessary conditions for the exponential stability of one delay systems, 8th International Conference on Electrical Engineering, Computing Science and Automatic Control, Merida, Mexico, 103-108. Mondié, S., Ochoa G. and B. Ochoa. (2011). Instability conditions for linear time delay systems: a Lyapunov matrix function approach, Int. J. of Contr., 84:10), 1601—1611. Mondié, S. (2012). Exact stability region of the single delay scalar equation: a time domain criteria, accepted in IMA J. of Math Contr. and Inf.. Neimark, J. (1949). D-subdivisions and spaces of quasipolynomials , Prikl. Mat. Meh., 13, 349-380. Repin, M. Yu. (1965). Quadratic Lyapunov functionals for systems with delay, Prik. Mat. Meh., 29, 564-566. Villafuerte, R., Mondié S., and R. Garrido (2012). Tuning the leading roots of a second order DC servomotor with proportional retarded control, accepted in IEEE Trans. on Contr. Syst. Tech..