New Results to Verify the Robust Stability Property of Interval Plants with Time-Delay1

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

New Results to Verify the Robust Stability Property of Interval Plants with Time-Delay ? Gerardo Romero ∗ Pedro Zamora ∗ Ivan Diaz ∗∗ Irma Perez ∗ David Lara ∗ ∗ These authors are with the Electronics Department at U.A.M.Reynosa-Rodhe, UAT, Carretera Reynosa San Fernando S/N cruce con canal Rodhe, Colonia Arco Iris, Reynosa, Tamaulipas M´exico, C.P. 88779. Tel: +52 (899) 921-33-00 Ext: 8313; Fax: + 52 (899) 921-33-01; E-mail: [email protected] ∗∗ This author is with the Texas A & M University, College Station, Texas, USA, E-mail: ivan diaz [email protected].

Abstract: This paper presents a sufficient condition to verify the robust stability property of a class of time delay systems which are described as an interval plant with uncertain time delay. The main result is obtained on the basis of some polynomials that can be easily computed using Kharitonov’s polynomials. This contribution considers the delay with interval uncertainty, since in most of the research reports only fixed delay is considered. The present work bases its results on the value set characterization for interval plants with uncertain time delay along with the use of the zero exclusion principle to compute the robust stability property of the close loop system. Keywords: Robust stability, interval plants, quasipolynomials, time-delay systems, value set. 1. INTRODUCTION

e.g, Lyapunov [1992]. In Su et al. [1989] the method of Razumikhin is used to guarantee stability. In Pepe et al. [2005], Xu and Lam [2005], Xu et al. [2006], and Zhu and Cheng [2005] apply Lyapunov and LMI techniques to verify the robust stability property of time-delay systems. b) Frequency domain analysis of stability: among the different forms that exist we find that of Mori and Kokame [1989], which presents a result to compute the margin of the delay on the basis of the measurement of µ of a previously defined matrix, that in this case is formed by the matrices of the dynamical system; this result uses the properties of the measurement µ of a matrix to guarantee that the roots of the characteristic equation be all contained in the left semiplane of the complex plane; in Tissir and Hmamed [1994] a similar analysis is presented. Many other important results related to the analysis in the frequency domain were presented in Barmish at al. [1992], Barmish and Zhi [1989], Boese [1992], Fu et al. [1989], Kharitonov and Zhabko [1994], Kogan and Leizarowitz [1995]. c) Design: the stabilization problem also has been boarded with great interest, for example, Manitius and Olbrot [1979] proposes a state feedback to assign a finite amount of poles of a dynamic differential-difference system; Niculescu et al. [1994] presents conditions for a static state feedback type control law, to be able to stabilize a system like the represented one by a time-delay differential equation with variant disturbance and considering saturation in the control law; some applications are presented in Gao and Wang [2003]. In Kojima and Ishijima [1994] they use the control technique H∞ in order to guarantee the

Time delay systems are interesting to many scientific researchers from different areas; this is due to the fact that this type of systems have multiple applications in industrial processes such as: thermal processes, electric power systems, biologic processes and many more, see Malek et al. [1987], Gorecki et al. [1989], Bellman and Cooke [1963], Hale and Verduyn-Lunel [1993]. One of the most important qualitative properties to consider in the analysis of time delay systems is the stability property that is obtained from the mathematical representation of a physical process. It is well known that the mathematical representation of a physical process does not accurately portraits its dynamic behavior, this in turn implies that the stability property can not be precisely obtained; this problem has been addressed by including dynamic uncertainty Green et al. [1995] or parametric uncertainty Ackermann [1993], Barmish [1994] in the mathematical model; the stability property that considers uncertainty in the mathematical model is defined as robust stability, this property will be discussed in the present paper. There exist different forms to establish the problem of control for systems of this type: a) Time domain analysis of stability: there exist many papers that analyze this type of systems on the basis of the Lyapunov methods ? The author Gerardo Romero is grateful to the Fondo Mixto de Fomento a la Investigacion Cientifica y Tecnologica - Gobierno del Estado de Tamaulipas, the financial support provided under grant No. 108166.

978-3-902823-04-5/12/$20.00 © 2012 IFAC

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10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012

stability of a system with input time delay, when a state feedback is applied to it.

∀ q ∈ Q; r ∈ R; τ ∈ [0, τmax ] We are interested in analyze the robust stability of control systems that are represented by the next block diagram: The property of robust stability is determined in terms of

This paper will present a sufficient condition to verify the robust stability property of interval plants with uncertain time delay; this result is based on the verification of some properties of some adequately selected polynomials. This paper is organized as follows: in section 2 the problem statement will be presented; section 3 includes some preliminary results; the main result will be presented in section 4, a numerical example in section 5 and finally, the conclusions in section 6. 2. PROBLEM STATEMENT The analysis presented in this paper is performed for interval plants with time delay that are defined as follows: Definition 1. An interval plant is a transfer function with parametric uncertainty that has the following structure: Pm − + i n(s, q) i=1 [qi , qi ]s g(s, q, r) = = (1) P n−1 n d(s, r) s + i=1 [ri− , ri+ ]si ∀ q ∈ Q; r ∈ R

Fig. 1. Closed-loop system for the interval plant with timedelay. the following characteristic equation: p(s, q, r, e−τ s ) = d(s, r) + n(s, q)e−τ s

(5)

where m < n, i.e. g(s, q, r) is a set of strictly proper rational functions, Q and R are sets that represent the parametric uncertainty and are defined as follows: n o T R ≡ r = [ r1 · · · rn−1 ] : ri− ≤ ri ≤ ri+ (2) n o T Q ≡ q = [ q1 · · · qm ] : qi− ≤ qi ≤ qi+

These kind of functions are defined as quasipolynomials. It is clear that the former characteristic equation (5) represents an infinite amount of characteristics equations that have to be considered to verify the robust stability property; this family will be defined as follows:  Pτ ≡ p(s, q, r, e−τ s ) : q ∈ Q; r ∈ R; τ ∈ [0, τmax ] (6)

n1 (s) = q0− + q1− s + q2+ s2 + q3+ s3 + . . .

where C+ is the set of complex numbers with real part greater than or equal to zero. From the equation (5) it can be seen that the robust stability property of the dynamic system is a property very difficult to verify due to the fact that this equation represents an infinite number of equations. The objective of this work is to present more simple results to verify the robust stability property of time delay systems as the ones shown in figure 1.

The robust asymptotic stability property is guaranteed if and only if the following equation is satisfied: p(s, q, r, e−τ s ) 6= 0 ∀ s ∈ C+ (7)

These type of sets are known as boxes by the way they are defined; the name of interval plants is used because the coefficients of the transfer function are uncertain values that belong to a closed interval. It is clear that interval plants represent an infinite number of transfer functions; however, the result of this paper will be expressed in terms of eight important elements of such interval plant, these elements are known as Kharitonov’s polynomials of the numerator and denominator of the interval plant which are defined as follows, see Kharitonov [1979]:

n2 (s) = q0+ n3 (s) = q0+ n4 (s) = q0− d1 (s) = r0− d2 (s) = r0+ d3 (s) = r0+ d4 (s) = r0−

+ q1− s + q2− s2 + q1+ s + q2− s2 + q1+ s + q2+ s2 + r1− s + r2+ s2 + r1− s + r2− s2 + r1+ s + r2− s2 + r1+ s + r2+ s2

+ q3+ s3 + q3− s3 + q3− s3 + r3+ s3 + r3+ s3 + r3− s3 + r3− s3

(3)

+...

3. PRELIMINARY RESULTS

+... The result presented in this paper is based on the value set characterization of the family of characteristic equations Pτ that is defined as follows:

+... +... +...

Definition 3. The value set of Pτ , noted by Vτ (ω), is the graph in the complex plane of p(s, q, r, e−τ s ) when s = jω is substituted; this is:   p(s, q, r, e−τ s ) : q ∈ Q; r ∈ R; Vτ (ω) = (8) τ ∈ [0, τmax ]; ω ∈ <

+...

+... by adding a time delay to the interval plant it is obtained the type of structure of the type of systems that will be analyzed in this paper, these are defined next:

It is clear that the value set of Pτ is a set of complex numbers plotted on the complex plane when values to qi , ri , ω and τ are assigned inside the defined boundaries, for example for a characteristic equation as presented in (5), but with the time-delay equal to zero, has a value set as the one shown in the figure below:

Definition 2 : An interval plant with time delay is a time delay system that has the following structure: g(s, q, r, e−τ s ) =

n(s, q) −τ s e d(s, r)

(4) 8

10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012

Lemma 2. Let V1 (ω0 ) and V2 (ω0 ) be two disaligned value sets. Then, the sum V1 (ω0 ) + V2 (ω0 ) is an octagon as it is shown in the Fig. 5:

Fig. 2. Value set when the time-delay is equal to zero. where the vertices are the Kharitonov’s polynomials defined in (3), for details see Dasgupta [1988]. Being able to characterize the value set Vτ (ω) is of great relevance because by applying additional results it is possible to determine the robust stability property. This characterization will be presented through the following results: Definition 4. Consider two rectangular value sets (as presented in Fig. 2) V1 (ω) and V2 (ω) for a fixed ω ∈ <, then V1 (ω) and V2 (ω) are said to be aligned for some ω0 ∈ < if, θ1 = θ2 + k Π2 for (k = 0, 1, 2, ...); where the angles θ1 and θ2 are obtained as shown in Fig. 3:

Fig. 5. Sum of two disaligned value sets. As can be seen clearly in the characteristic equation (5) there is a sum of two value sets, in which the second term is being affected by the time delay factor e−jωτ , which it does rotate the value set and then disaligned with respect to the other value set, depending on the term ωτ . Then, using the previous lemmas it is possible to obtain a characterization of the value set Vτ (ω) as follows, see Romero and Collado [1995]: Lemma 3. The value set Vτ (ω) is composed, for each value of ω, of octagons that change their shape with respect of the time delay τ . The vertices of each octagon have the following coordinates in the complex plane: vi+1 = di+1 (jω) + nk (jω)e−jωτ vi+5 = di+1 (jω) + nh (jω)e

(9)

−jωτ

where:

Fig. 3. Angles θ1 and θ2 of value sets.

i = 0, 1, 2, 3 Then, two value sets are aligned if they have the same inclination angle. The other hand, are said to be disaligned. With this definition is possible to present the following results: Lemma 1. Let V1 (ω0 ) and V2 (ω0 ) be two aligned value sets. Then, the sum V1 (ω0 ) + V2 (ω0 ) is a rectangle as it is shown in the Fig. 4:

k = (γ + i) mod4 + 1 h = (γ + i + 1) mod4 + 1  π  0 2nπ ≤ ωτ < + 2nπ   2  π    1 + 2nπ ≤ ωτ < π + 2nπ 2 γ≡ 3π  π + 2nπ ≤ ωτ < + 2nπ   2  2   3π  3 + 2nπ ≤ ωτ < 2π + 2nπ 2 n = 0, 1, 2, . . . the term (x)mod4 represents the well known entire module base four function as shown in the following examples: (2)mod4 = 2; (4)mod4 = 0; (7)mod4 = 3. This result was coded in MATLAB language for obtaining the corresponding figure of a fictitious Vτ (ω) for a fixed arbitrary ω, as shown in Fig. 6. It is worth to mention that although the value set may appear as describing a circle, it is not. From the value set definition it can be clearly observed that it represents all the values that the family Pτ can take when s = jω, this is, if the complex plane origin is contained in the

Fig. 4. Sum of two aligned value sets. 9

10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012

the centers of the corresponding value sets of the denominator, numerator, and the octagon which is the sum of value sets, as shown in Fig. 7.

9 8

14

6

12

5

10

4

8 Im

Im

7

3 2 −3

−2

−1

0

Re

1

2

3

Co(s)

Cd(s)

6

Cn(s)

4

4

2

Fig. 6. Value set for a fictitious Vτ (ω) and a fixed ω.

0 0

value set Vτ (ω) this means that Pτ has roots on the imaginary axis jω for some values of ω ∈ <, this in turn provokes instability in the time delay system. From the above mentioned it can be seen that the value set can be a tool to aid in verifying the robust stability property. An important question is how from a sweep over jω it can be determined that Pτ does not have roots on the right half plane which was defined as C+ . The answer to this question is found in the following result known as the zero exclusion principle, see Barmish [1994]. Lemma 4. Consider the characteristic equation (5), also called quasipolynomials. Suppose that (5) has at least one stable member. Then the robust stability property of the control system shown in Fig. 1 is guaranteed if and only if: 0∈ / Vτ (ω) ∀ω ≥ 0 (10)

1

2

3

4

5

Re

6

7

8

9

10

Fig. 7. Centers of the value sets corresponding. From the definition of co (s) it is possible to visualize that the center of the octagons that compose the value set Vτ (ω) takes the shape of arcs of circumferences centered in the polynomial cd (jω), these arcs have angles and radius equal to ωτ and | cn (jω) |, respectively. This property is important in the construction of Vbτ (ω) and will be defined next: Definition 5. The value set Vbτ (ω) is the graph in the complex plane that describes a set of solid disc segments with radius equal to r(jω) and centered at cd (jω) as it is shown in Fig. 8:

From previous lemma, the robust stability problem is transformed into a problem where it only needs verify the value set plot just avoids the zero of the complex plane and especially we are looking for a simple formula to help us verify the stability property of an interval plant with delay from its value set. 4. MAIN RESULT In this section the main result will be presented which consists in finding a set Vbτ (ω) that is more simple to analyze and that completely contains the value set Vτ (ω) in such a way that sufficient conditions of robust stability for the quasipolynomials family previously defined as Pτ can be determined based on the value set Vbτ (ω). The results of this paper are supported by the definitions of the following polynomials:

Fig. 8. Value set Vbτ (ω).

where re (jω) is as follows:

4

1X cd (s) = di (s) 4 i=1

(11) re (jω) =

4

cn (s) =

1X ni (s) 4 i=1

co (s) = cd (s) + cn (s)e−τ s

(12)

| d4 (jω) − d2 (jω) | + | n4 (jω) − n2 (jω) | 2

The previous Fig. 8 shows that the value set Vτ (ω) is completely contained in Vbτ (ω) and this is a relevant result in order to achieve a more simple test to verify the robust stability property of the time delay system shown in Fig. 1. Before presenting the next result it is essential to introduce the following definitions:

(13)

where di (s) and ni (s) are the Kharitonov’s polynomials as defined in (3). The definitions previously made, represent 10

10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012

Definition 6. The set Ωc is defined as follows: Ωc ≡ {ω ∈ (0, ∞) :| cd (jω) |= r(jω)}

10

(14)

Definition 7. The value of maximum delay is defined as: τ = min {τ1 , τ2 , . . . , τn } (15)

8 6

π + φmin (jωi ) − arg(cd (jωi )) ωi (i = 1, 2, ..., n) ∀ ωi ∈ Ωc

τi =

Im

where n represents the number of elements that the set Ωc has and the values of τi are attained by using the next formula:

4

(16)

2 0 −0.5

the angle φmin (jωi ) is computed using Kharitonov’s polynomials, as shown in the figure below Fig. 9. If the set Ωc is the empty set, then τ takes an infinity value (τ = ∞).

0

0.5

1

1.5

Re

2

2.5

3

3.5

4

Fig. 10. Value set for characteristic equation (18) . Additionally, in this paper will be considered uncertainty in the coefficients of the characteristic equation: p(s, e−τ s ) = [0.8, 1.2]s2 + [3.2, 4.8]s + [3, 4.5] − [.2, .3]e

(19)

−τ s

τ ∈ [0, 5] using the program coded in Matlab gives the value set presented in Fig. 11: 10 8

Fig. 9. Angle φmin (jωi ). Im

6

Theorem 5. Pτ is robustly stable if τmax < τ .

4

It is worth to mention that the prior result offers only sufficient conditions due to the strict enclosure of Vτ (ω) by Vbτ (ω), however, this result is less conservative than many others that utilize some other techniques for their analysis.

2 0 −3

5. NUMERICAL EXAMPLE

−2

−1

0

1 Re

2

3

4

5

Fig. 11. Value set for characteristic equation (19).

Example 1. The results presented in this paper will be applied to an example considered in previous publications.

using the result presented in the previous section, we determined that 19 is robustly stable.

Consider the time-delay systems in Hertz et al. [1995] in the following form:     −2 0 0 0.5 x(t) ˙ = x(t) + x(t − τ ) (17) 0.5 −2 0 0

6. CONCLUSIONS This paper presented a simple method used to obtain sufficient conditions of robust stability for a class of time delay system which is described by interval plants with time delay. These results may be considered as conditions of robust stability dependent of delay where the property is verified based in two polynomials adequately defined on the basis of Kharitonov’s polynomials of the interval plant. A recommendation for future research is obtaining conditions to verify the robust stability property considering another kind of uncertainty structure, such as the polynomial structure case. Also, an interesting

with the following characteristic equation: p(s, e−τ s ) = s2 + 4s + 3.75 − .25e−τ s (18) the example considered in Hertz et al. [1995] has no uncertainty in its parameters and there is only verifying the maximum delay to preserve the stability property. For this case, we can see that the set Ωc is an empty set and thus the τmax = ∞. The corresponding value set for τ = 20 to equation (18) is shown in Fig. 10: 11

10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012

problem is to obtain a new control law that improves the delay margin on the time delay systems, compared to that of previous results.

T. Mori, H. Kokame. Stability of x(t) ˙ = Ax(t)+Bx(t−τ ). IEEE Trans. on Automatic Control, Vol 34, No 4, pp 460-462, 1989. S.I. Niculescu, J.M. Dion, L. Dugard. Robust Stabilization for Uncertain Time-Delay Systems Containing Saturating Actuators. Proceedings of the 33rd Conference on Decison and Control, Lake Buena Vista Fl, pp. 431-432, December, 1994. P. Pepe, Z.P. Jiang. A Lyapunov-Krasovskii Methodology for ISS of Time-Dalay Systems. Proceeding of the 44th IEEEConference on Decision and Control, and European Control Conference, pp. 5782-5787, Seville Spain, December 12-15, 2005. G. Romero, J. Collado. Robust Stability of Interval Plants with Perturbed Time Delay. Proceedings of the 1995 American Control Conference, pp. 326-327, June 1995. T.J. Su, Y.Y. Sun, T.S. Kuo. Robust Stabilization of Observer based Linear Constrained Uncertain Time-Delay Systems. Control Theory and Advanced Technology, Vol 5, No 2, pp. 205-213, 1989. E. Tissir, A. Hmamed. Stability Test of Interval Time Delay Systems. Systems & Control Letters 23, pp. 263270, 1994. S. Xu, J. Lam. Improved Delay-Dependent Stability Criteria for Time-Delay Systems. IEEE Trans. on Automatic Control, Vol. 50, No 3, pp. 384-387, 2005. S. Xu, J. Lam, M. Zhong. New Exponential Estimates for Time-Delay Systems. IEEE Transactions on Automatic Control, Vol. 51, No 9, pp. 1501-1505, 2006. S. Zhu, Z. Cheng. Delay-dependent Robust Resilient Guaranteed Cost Control for Uncertain Singular TimeDelay Systems. Proceedings of the 2005 American Control Conference, pp. 2833-2838, Portland Oregon, USA, June 8-10, 2005.

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