Stabilization of LPV Positive Systems
53rd IEEE Conference on Decision and Control December 15-17, 2014. Los Angeles, California, USA
Stabilization of LPV Positive Systems M. Ait Rami, B. Boulkroune, A. Hajjaji and O. Pag`es
Abstract— This paper considers the stabilization issue for continuous-time linear parameter varying (LPV) positive systems. The time varying parameters are known and are modeled as belonging to the simplex set. The proposed stabilization approach relies on a parameter dependent Lyapunov function combined with a subtle choice of a slack variable that is not necessary diagonal. In fact, due to the positivity constraint on the closed-loop system the slack variable is chosen to be a Metzler matrix. Indeed, the particular case when the slack matrix is diagonal may work but the resulting stabilization conditions can be conservative. This fact is illustrated by a comparison example.
I. INTRODUCTION The main objective of the paper is to provide a flexible control design approach for continuous-time linear parameter varying (LPV) positive systems. A system is referred to be positive if it involves states that only take nonnegative values (see [9], [25], [16], [21] for general references). Such kind of system for which the states take physical nonnegative values, can be encountered in a wide range of applications such as in industrial processes involving chemical reactors, in water storage systems and in atmospheric pollution models. Other variety of systems exhibiting the positivity constraint on their states can be found in management science, economics, biology, medicine, etc. General LPV systems have been treated extensively in the literature. For a comprehensive overview on LPV sytems see the survey paper [27]. To the best of our knowledge, LPV positive systems are not yet treated in the literature. Only positive systems with parametric uncertainties have been considered in [1], [15]. In contrast, LTI and switched positive systems are well-reported and well-developed in the literature, without been exhaustive, see for instance, [22], [2], [3], [29], [26], [5], [12], [7], [19], [18], [11] in stability and control context and, [20], [4], [24], [6] in estimation context. This paper treats the synthesis of gain scheduled control laws that stabilize and maintain the positivity of a given continuous-time LPV positive system for which the time varying parameter are assumed to be known
and are modeled as belonging to the simplex set. For this purpose, two LMI-based procedures are provided. Also, an explanation on how to incorporate extra constraints on the gain scheduled control laws is presented. The proposed approach is based on a parameter dependent Lyapunov function combined with decoupling technique between the Lyapunov variables and the controller variables. This procedure leads to easy computational conditions. The decoupling technique has been proved to be efficient and has been used in different contexts [23], [13], [28], [14], [17]. This technique uses extra slack variables which leads to less conservative synthesis conditions. Due to the positivity constraint on the closed-loop system, the synthesis problem is not possible without a suitable choice of a special slack variable. Of course, one can use a slack variable as a diagonal matrix. However, this choice leads to conservative stabilization conditions since it provide a few degree of freedoms. Instead, we propose a subtle choice of slack variables that are Metzler matrices. Our numerical results show the highly advantage of the use of such slack variables. This paper is organized as follows. The description and characterization of positive LPV systems is presented in Section 2. Section 3 is devoted to the synthesis of gain scheduled control laws that stabilize and maintain the positivity of a given LPV system. Two numerical control designs are given. Also, it is shown how to add extra constraints on the gain scheduled control laws. In section 4, a numerical comparison of the proposed numerical procedures is provided. Concluding remarks are presented in the last section. Notations: M 0 denotes the transpose of the real matrix M. For a real matrix M (resp. a vector v), M ≥ 0 (resp. v ≥ 0) means that its components are nonnegative: Mi j ≥ 0 (resp. vi ≥ 0). M ≤ means −M ≥ 0. For symmetric real matrix M, M 0 (resp. M ≺ 0 ), means that M is definite positive (resp. negative). II. Positive LPV systems Here, our main objective is to recall some facts on positive systems and to provide a relatively concise characterization of an LPV positive system described by x(t) ˙ = A(θ )x(t)
(1a)
p
The author are with MIS Laboratory, UFR sciences, University of Picardie Jules Verne, 33 Rue st Leu 80000 Amiens France, e-mail : [email protected], [email protected],
with A(θ ) =
[email protected], [email protected]
θ is an exogenous parameter that can be time dependent
978-1-4673-6088-3/14/$31.00 ©2014 IEEE
∑ θ j A j , where x ∈ ℜn is the state vector and
j=1
4772
such that p
∑ θ j (t) = 1,
θ j (t) ≥ 0, ∀t ≥ 0, j = 1, . . . , p.
(2)
(3) is positive. Also, by duality we have that system (1) is asymptotically stable if and only if its dual system (3) is asymptotically stable.
j=1
Throughout this paper the free system is assumed to satisfy a positivity constraint on its states as follows. Definition 1: System (1) is said to be positive if for any nonnegative initial conditions x(0) ≥ 0, the corresponding trajectory is nonnegative: x(t) ≥ 0 for all t ≥ 0. An inherent property from the positivity of system (1) is related to Metzlerian matrices. Definition 2: A matrix M is called a Metzler matrix if its off-diagonal elements are nonnegative: Mi j ≥ 0, i 6= j.
III. Control design In this section, we treat the synthesis problem for LPV positive systems by means of gain scheduled control laws. The proposed approach is based on a parameter-dependent Lyapunov function combined with a suitable choice of slack matrix variables. In fact, it will be shown that Metzler matrix slack variables play a key role in ensuring positivity and stability of the closed-loop system. In addition, they can be useful for taking into account additional constraints on the stabilizing control laws. Consider the following LPV system
Remark 1: M is Metzler matrix is equivalent to the fact that there exists a diagonal matrix D such that M + D is positive: M + D ≥ 0.
x˙ = A(θ )x+B(θ )u p
Definition 3: A real matrix M is called a positive matrix if all its elements are nonnegative: Mi j ≥ 0. Also, M is called negative matrix if −M is positive. For a more complete treatment for Metzler and positive matrices and their properties see e.g. [10], [8] and [16]. The following result provides an useful property of Metzler matrices that are Hurwitz (see for instance [10]). Lemma 1: Let M be a Metzler matrix, then the following statements are equivalent (i) M is Hurwitz. (ii) M −1 ≤ 0.
with A(θ ) =
(4)
p
∑ θ j A j and B(θ ) = ∑ θ j B j , where x ∈ ℜn
j=1
j=1
is the state vector, u ∈ ℜm is the control input vector and θ is an exogenous parameter that can be time dependent with bounded derivatives such as p
∑ θi (t) = 1, θi (t) ≥ 0,
(5)
i=1
θ˙i (t) ≤ βi ,
i = 1, · · · , p
(6)
where βi are assumed to be known bounds on the derivatives of the parameters θi . Our objective is to design a gain scheduling control law of the form p
u(t) =
The following result can be viewed as an extension of the classical positivity result for LTI systems [25]. Lemma 2: System (1) is positive for any parameter function θ taking values in the simplex set, if and only if the matrices A1 , ..., A p are Metzler matrices. Now, consider the following dual system z˙(t) = A0 (θ )z(t)
(3)
(7)
∑ θ jKjx
j=1
that stabilizes system (4) and maintains the positivity of the closed-loop system. For this purpose, we provide the following result. Theorem 1: There exists a state-feedback of the form (7) that stabilizes and maintains the positivity of system (4), if there exist matrices P1 , . . . , Pp , Y1 . . . ,Yp , a diagonal matrix D and a scalar α such that the following conditions hold:
p
with A0 (θ ) =
∑ θ j A0j
and which depends on the same
Mii ≺ 0,
j=1
Pi 0
Mi j + M ji ≺ 0,
exogenous parameter vector θ as for system (1).
for i = 1, · · · , p
(8a)
for i < j = 1, · · · , p
(8b)
Ai D + BiY j + αI ≤ 0, In the sequel, we will exploit the two common properties connected to positivity and stability of system (1) and its dual. In fact, by the result of Lemma 2 we have that system (1) is positive if and only if its dual system
with Mi j =
4773
for i, j = 1, · · · , p (8c)
p
∑ βk Pk − DA0i − Ai D −Y j0 B0i − BiY j k=1
Pi + D − DA0i −Y j0 Bi
Pi + D − Ai D − BiY j . 2D
p
If so, a stabilizing gain K(θ ) =
∑ θ j K j that maintains
j=1
the closed-loop system positive is given by Ki = Yi D−1 , i = 1, . . . , p. Proof: The proof is based on duality. We shall exploit the fact that with the same control law, the positivity and the stability of system (4) is equivalent to the positivity and the stability of the dual closed-loop system given by z˙ = A0cl (θ )z (9) p
where Acl = A(θ ) + B(θ )K(θ ), with K(θ ) =
∑ θ jKj.
where
p
with Y j = K j D. (16) can be rewritten as p
V = z0 P(θ )z
(10)
∑ θ j Pj . Note that since all Pj are positive
j=1
The conditions of the theorem ensure that V˙ ≺ 0 which is implied by:
Mi j + M ji ≺ 0,
for i = 1, · · · , p
(18)
for i < j = 1, · · · , p
(19)
In order to show that the closed-loop system is positive, it suffices to ensure that each of its modes is a Metzler matrix. That is Ai + Bi K j is Metzler for all i, j. Since D ≺ 0 and D is diagonal, we obtain from condition (8c): Ai D + BiY j + αI ≤ 0, that (Ai D + BiY j + αI)D−1 ≥ 0.
definite matrices we have that P(θ ) > 0. The time derivative of V is given by: V˙ = z0 Pθ˙ z + z˙0 P(θ )z + z0 P(θ )˙z
(17)
i< j
i=1 j=1
Mii ≺ 0,
p
where P(θ ) =
p
p
V˙ ≤ ∑ ∑ θi2 ξ 0 Mii ξ + ∑ θi θ j ξ 0 (Mi j + M ji )ξ
j=1
Now, we shall first prove the stability of the dual system (9) by considering the following parameterdependent Lyapunov function:
Pi + D − Ai D − BiY j 2D
∑ βk Pk − DA0i − Ai D −Y j0 B0i − BiY j Mi j = k=1 Pi + D − DA0i −Y j0 Bi
(11)
Thus, we have Ai + Bi K j + αD−1 ≥ 0 which means by Remark 1 that Ai + Bi K j is a Metzler matrix.
p
with Pθ˙ =
∑ θ˙ j Pj .
j=1
In order to decouple the Lyapunov matrices from the system matrices, slack matrix variables are introduced as follows: 2[˙z0 D + z0 D] z˙−A0cl (θ )z = 0 (12) where D is a diagonal matrix. From (11) and (12), it follows that: V˙ = z0 Pθ˙ z + z˙0 P(θ )z + z0 P(θ )˙z
Note that the previous result involves a diagonal slack variable which provides a few degrees of freedom. A less conservative stabilization conditions can be established by using a Metzler matrix as a slack variable instead of a diagonal one. This is stated in the following result. Theorem 2: Let γ > 0 be a given a state-feedback that stabilizes and itivity of system (4), if there exist Y1 . . . ,Yp , a matrix G and a positive the following conditions hold
+ 2[˙z0 D + z0 D] z˙−A0cl (θ )z (13) Since θ˙ j ≤ β j ,
Mii ≺ 0,
j = 1, · · · , p, then V˙ satisfies:
Pi 0
Mi j + M ji ≺ 0, 0
p
+ 2[˙z D + z D] z˙−A0cl (θ )z (14)
0
for i < j = 1, · · · , p
(20b)
0
for i, j = 1, · · · , p (20c)
with
(20d)
p
0 0 0 0 ∑ β j Pj − GAi − Ai G −Y j Bi − BiY j Ξi j = j=1 Pi + G − GA0i −Y j0 B0i
By substituting Acl by its expression, we obtain: p
V˙ ≤ z0 ( ∑ β j Pj )z + z˙0 P(θ )z + z0 P(θ )˙z
Pi + G0 − Ai G0 − BiY j . G + G0 p
If so, a stabilizing gain K(θ ) =
j=1
+ 2[˙z0 D + z0 D] z˙−(A(θ ) + B(θ )K(θ ))0 z (15)
∑ θ j K j that maintains
j=1
the closed-loop system positive is given by
By substituting A(θ ), B(θ ) and P(θ ) by their ), K(θ z , we have expressions and taking ξ = z˙ p
(20a)
G + αI ≥ 0
j=1
0
for i = 1, · · · , p
Ai G + BiY j + γG ≤ 0,
V˙ ≤ z0 ( ∑ β j Pj )z + z˙0 P(θ )z + z0 P(θ )˙z
scalar. There exist maintains the posmatrices P1 , . . . , Pp , scalar α such that
Ki = Yi G−1 , i = 1, . . . , p.
p
V˙ ≤ ∑ ∑ θi θ j ξ 0 Mi j ξ i=1 j=1
(16)
Proof: By following the same line of arguments in the previous result, we have that V = z0 P(θ )z with
4774
p
P(θ ) =
∑ θ j Pj
IV. Numerical results
is a Lyapunov function. Thus, the
j=1
p
state-feedback control u =
∑ θ jKjx
with Ki = Yi (G0 )−1
j=1
is stabilizing. In order to complete the proof, we only need to show that the closed-loop system is positive or equivalently that each of its modes is a Metzler matrix. Let G be any matrix satisfying condition (20a) and (20b) which implies that the block matrix G + G0 satisfies G+G0 ≺ 0. In addition, conditions (20d) is equivalent to the fact that G is a Metzler matrix. Then, according to Lemma 1 the inverse of G is a negative matrix (G−1 ≤ 0). Due to this fact, we have that (Ai G0 + BiY j + γG0 )(G0 )−1 ≥ 0,
for i, j = 1, · · · , p.
Since by condition (20c), the matrix Ai G0 + BYi + γG0 is negative. Hence, it holds for each mode (Ai G0 + BiY j + γG0 )(G0 )−1 = Ai + Bi K j + γI ≥ 0,
Consider system (1) involving constant parameters a and b such that 0 3 1 1 A1 = , A2 = (21) a + 1 −1 b + 3 −1 2a 2−b B1 = , B2 = (22) −1 1 Based on Lemma 2, one can easily see that the free system is positive in the range : −1 ≤ a and −3 ≤ b since with these values A1 and A2 are Metzler matrices. The stabilizable region of the proposed approaches was investigated for various values of a and b. Figure 1 shows the stabilizable region (in blue color) of the parameterdependent Lyapunov function approach whit G diagonal for β j = 0.99, j = 1, · · · , 2. The stabilization result is significantly improved by using a general Metzler matrix variable G instead of a diagonal matrix as illustrated in figure 2. We can see that the stabilizable region is expanded by using Theorem 2.
which in turn shows that every mode of the closed-loop system is a Metzler matrix. Then, by Lemma 2 we can p
conclude that the state-feedback control u =
∑ θ jKjx
j=1
b
with Ki = Yi (G0 )−1 maintains the positivity of the closedloop system and the proof is complete. Our approach is flexible due to the fact that it can incorporate constraints on the stabilizing control laws. For instance, if a control law is needed to be nonnegative: u(t) ≥ 0, ∀t ≥ 0, such control can be computed by combining conditions (20a)-(20d) with the additional constraints: Yi ≤ 0,
a Fig. 1. The stabilizable region of the parameter-dependent Lyapunov function approach whit G diagonal for β j = 0.99, j = 1, · · · , 2.
i = 1, . . . , p.
In fact, due to the nonnegativity of the states, the p
nonnegativity of a control law of the form u =
∑ θ jKjx
j=1
is insured by the fact that the gains Ki must be positive matrices: Ki ≥ 0. Indeed, this condition can be recovered from b
Yi ≤ 0 ⇒ Yi (G0 )−1 = Ki ≥ 0, as we have previously shown that we can use a matrix G with negative inverse. In the same spirit, one can take into account prescibed upper and lower bounds on the gains of the control law such as: K ≤ Ki ≤ K. For this purpose, it suffices to add to conditions (20a)-(20d) the following extra constraints: KG0 ≤ Yi ≤ KG0 ,
a Fig. 2. The stabilizable region of the parameter-dependent Lyapunov function approach with a Metzler G for β j = 0.99, j = 1, · · · , 2.
i = 1, . . . , p. 4775
V. CONCLUSIONS We have treated the stabilization problem for timevarying continuous-time positive systems by using a new Lyapunov-based technique that involves Metzler slack variables. Due to the positivity constraint on the closedloop system this kind of slack variable are suitable. This choice of slack variables leads to an adequate stabilization result which can outperform the case when diagonal slack variables are involved. Our approach has been illustrated by a comparison example. ACKNOWLEDGMENTS This work was produced in the framework of CEREEV (Combustion Engine for Range-Extended Electric Vehicle), a European territorial cooperation project part-funded by the European Regional Development Fund (ERDF) through the INTERREG IVA France (Channel)-England program, and the research department of the Picardie, France. References [1] M. Ait Rami, and F.Tadeo, Controller Synthesis for positive Linear Systems with Bounded controls. IEEE Trans. Syst. And Circuits, 54, pp. 151–155, 2007. [2] M. Ait Rami. Stability and stabilization of positive systems with time varying delays. Lecture notes in control and information sciences, Springer, 389, pp. 205–215, 2009. [3] M. Ait Rami. Solvability of static output-feedback stabilization of positive LTI systems. Systems & control letters, 60, pp. 704–708, 2011. [4] M.Ait Rami, F. Tadeo, and U. Helmke. Positive observers for linear positive systems and their implications. International Journal of Control, 84, pp. 416–725, 2011. [5] M. AitRami, and D. Napp. Characterization and Stability of Autonomous Positive Descriptor Systems. IEEE Trans. Automat. Control, 57(10), pp. 2668–2673, 2012. [6] M. Ait Rami, M. Sch¨ onlein, and J. Jordan. Estimation of linear positive systems with unknown time-varying delays. Europeran Journal of control, 19: pp 179–187, 2013. [7] M. Ait Rami and D. Napp. Positivity of discrete singular systems and their stability: An LP-based approach. Automatica, 50: pp 84–91, 2014. [8] R. Bellman. Introduction to Matrix Analysis. SIAM, Philadelphia, 1997. [9] A. Berman, M. Neumann, and R. J. Stern, Nonnegative Matrices in Dynamic Systems. New York: Wiley, 1989. [10] A. Berman and R.J. Plemmons. Nonnegative Matrices in the Mathematical Sciences (1979), Academic Press, San Diego, CA, reprinted by SIAM, Philadelphia, 1994. [11] F. Blanchini, P. Colaneri and M.E. Valcher, Is stabilization of switched positive linear systems equivalent to the existence of an Hurwitz convex combination of the system matrices? IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), 2011. [12] C. Briat. Robust stability and stabilization of uncertain linear positive systems via integral linear constraints and L1-gains characterizations, International Journal of Robust and Nonlinear Control, 23, pp. 1932–1954, 2013. [13] M.C. de Oliveira, and R.E. Skelton. Stability tests for constrained linear systems. Lecture notes in control and information sciences. Perspectives in robust control, vol. 268, Berlin Springer, 2001. [14] Y. Ebihara and T. Hagiwara, A dilated LMI approach to robust performance analysis of linear time-invariant uncertain systems. Automatica, vol. 41 (11), pp. 1933–1941, 2005. [15] Y. Ebihara, D. Peaucelle, D.Arzelier. LMI approach to linear positive system analysis and synthesis. Systems & Control Letters, 63, pp. 50–6, 2014.
[16] L. Farina, and S. Rinaldi. Positive Linear Systems: Theory and Applications. John Wiley & Sons, Inc., New York, 2000. [17] Y. Feng, M.Yagoubi, P. Chevrel, Dilated LMI Characterizations for Linear Time-Invariant Singular Systems, International Journal of Control, 83, pp. 2276–2284, 2010. [18] E. Fornasini, M.E. Valcher, Linear copositive Lyapunov functions for continuous-time positive switched systems, IEEE Transactions on Automatic Control, 55 (8), pp. 1933–1937, 2010. [19] L. Gurvits, R. Shorten, O. Mason, On the stability of switched positive linear systems, IEEE Transactions on Automatic Control, 52 (6), pp. 1099–1103, 2007. [20] M. Hardin, and J.H. van Schuppen. Observers for linear positive systems. Linear Positive Algebra and its Applications, 425, 571–607, 2007. [21] T. Kaczorek. Positive 1D and 2D Systems. Springer, London, 2001. [22] F. Knorn, O. Mason, R. Shorten. On linear co-positive Lyapunov functions for sets of linear positive systems. Automatica, 45, pp. 1943–1947, 2009. [23] D. Peaucelle, D. Arzelier, O. Bachelier, J. Bernussou. A new robust D-stability condition for real convex polytopic uncertainty. Systems & Control Letters, 40(1), pp. 21–30, 2000. [24] P. Li and J. Lam. Positive state-bounding observer for positive interval continuous-time systems with time delay. International Journal of Robust and Nonlinear Control. 22, pp. 1244– 1257, 2011. [25] D.G. Luenberger. Introduction to Dynamic Systems: Theory, Models, and Applications. John Wiley & Sons, Inc., New York, 1979. [26] A. Rantzer, Distributed control of positive systems. IEEE Conference on Decision and Control, pp. 6608–6611, 2011. [27] W. Rugh, J. Shamma. Research on gain scheduling. Automatica, 36, 1401–1425, 2000. [28] F. O. Souza, R. M. Palhares and P. Ya Ekel, Asymptotic stability analysis in uncertain multi-delayed state neural networks via Lyapunov-Krasovskii theory. Mathematical and Computer Modelling, vol. 45, pp. 1350–1362, 2007. [29] T. Tanaka, C. Langbort, The real bounded lemme for internally positive systems and H-infinity structured static statefeedback, IEEE Trans. Automat. Control, 56, pp. 2218–2223, 2011.
4776
Stabilization of LPV Positive Systems M. Ait Rami, B. Boulkroune, A. Hajjaji and O. Pag`es
Abstract— This paper considers the stabilization issue for continuous-time linear parameter varying (LPV) positive systems. The time varying parameters are known and are modeled as belonging to the simplex set. The proposed stabilization approach relies on a parameter dependent Lyapunov function combined with a subtle choice of a slack variable that is not necessary diagonal. In fact, due to the positivity constraint on the closed-loop system the slack variable is chosen to be a Metzler matrix. Indeed, the particular case when the slack matrix is diagonal may work but the resulting stabilization conditions can be conservative. This fact is illustrated by a comparison example.
I. INTRODUCTION The main objective of the paper is to provide a flexible control design approach for continuous-time linear parameter varying (LPV) positive systems. A system is referred to be positive if it involves states that only take nonnegative values (see [9], [25], [16], [21] for general references). Such kind of system for which the states take physical nonnegative values, can be encountered in a wide range of applications such as in industrial processes involving chemical reactors, in water storage systems and in atmospheric pollution models. Other variety of systems exhibiting the positivity constraint on their states can be found in management science, economics, biology, medicine, etc. General LPV systems have been treated extensively in the literature. For a comprehensive overview on LPV sytems see the survey paper [27]. To the best of our knowledge, LPV positive systems are not yet treated in the literature. Only positive systems with parametric uncertainties have been considered in [1], [15]. In contrast, LTI and switched positive systems are well-reported and well-developed in the literature, without been exhaustive, see for instance, [22], [2], [3], [29], [26], [5], [12], [7], [19], [18], [11] in stability and control context and, [20], [4], [24], [6] in estimation context. This paper treats the synthesis of gain scheduled control laws that stabilize and maintain the positivity of a given continuous-time LPV positive system for which the time varying parameter are assumed to be known
and are modeled as belonging to the simplex set. For this purpose, two LMI-based procedures are provided. Also, an explanation on how to incorporate extra constraints on the gain scheduled control laws is presented. The proposed approach is based on a parameter dependent Lyapunov function combined with decoupling technique between the Lyapunov variables and the controller variables. This procedure leads to easy computational conditions. The decoupling technique has been proved to be efficient and has been used in different contexts [23], [13], [28], [14], [17]. This technique uses extra slack variables which leads to less conservative synthesis conditions. Due to the positivity constraint on the closed-loop system, the synthesis problem is not possible without a suitable choice of a special slack variable. Of course, one can use a slack variable as a diagonal matrix. However, this choice leads to conservative stabilization conditions since it provide a few degree of freedoms. Instead, we propose a subtle choice of slack variables that are Metzler matrices. Our numerical results show the highly advantage of the use of such slack variables. This paper is organized as follows. The description and characterization of positive LPV systems is presented in Section 2. Section 3 is devoted to the synthesis of gain scheduled control laws that stabilize and maintain the positivity of a given LPV system. Two numerical control designs are given. Also, it is shown how to add extra constraints on the gain scheduled control laws. In section 4, a numerical comparison of the proposed numerical procedures is provided. Concluding remarks are presented in the last section. Notations: M 0 denotes the transpose of the real matrix M. For a real matrix M (resp. a vector v), M ≥ 0 (resp. v ≥ 0) means that its components are nonnegative: Mi j ≥ 0 (resp. vi ≥ 0). M ≤ means −M ≥ 0. For symmetric real matrix M, M 0 (resp. M ≺ 0 ), means that M is definite positive (resp. negative). II. Positive LPV systems Here, our main objective is to recall some facts on positive systems and to provide a relatively concise characterization of an LPV positive system described by x(t) ˙ = A(θ )x(t)
(1a)
p
The author are with MIS Laboratory, UFR sciences, University of Picardie Jules Verne, 33 Rue st Leu 80000 Amiens France, e-mail : [email protected], [email protected],
with A(θ ) =
[email protected], [email protected]
θ is an exogenous parameter that can be time dependent
978-1-4673-6088-3/14/$31.00 ©2014 IEEE
∑ θ j A j , where x ∈ ℜn is the state vector and
j=1
4772
such that p
∑ θ j (t) = 1,
θ j (t) ≥ 0, ∀t ≥ 0, j = 1, . . . , p.
(2)
(3) is positive. Also, by duality we have that system (1) is asymptotically stable if and only if its dual system (3) is asymptotically stable.
j=1
Throughout this paper the free system is assumed to satisfy a positivity constraint on its states as follows. Definition 1: System (1) is said to be positive if for any nonnegative initial conditions x(0) ≥ 0, the corresponding trajectory is nonnegative: x(t) ≥ 0 for all t ≥ 0. An inherent property from the positivity of system (1) is related to Metzlerian matrices. Definition 2: A matrix M is called a Metzler matrix if its off-diagonal elements are nonnegative: Mi j ≥ 0, i 6= j.
III. Control design In this section, we treat the synthesis problem for LPV positive systems by means of gain scheduled control laws. The proposed approach is based on a parameter-dependent Lyapunov function combined with a suitable choice of slack matrix variables. In fact, it will be shown that Metzler matrix slack variables play a key role in ensuring positivity and stability of the closed-loop system. In addition, they can be useful for taking into account additional constraints on the stabilizing control laws. Consider the following LPV system
Remark 1: M is Metzler matrix is equivalent to the fact that there exists a diagonal matrix D such that M + D is positive: M + D ≥ 0.
x˙ = A(θ )x+B(θ )u p
Definition 3: A real matrix M is called a positive matrix if all its elements are nonnegative: Mi j ≥ 0. Also, M is called negative matrix if −M is positive. For a more complete treatment for Metzler and positive matrices and their properties see e.g. [10], [8] and [16]. The following result provides an useful property of Metzler matrices that are Hurwitz (see for instance [10]). Lemma 1: Let M be a Metzler matrix, then the following statements are equivalent (i) M is Hurwitz. (ii) M −1 ≤ 0.
with A(θ ) =
(4)
p
∑ θ j A j and B(θ ) = ∑ θ j B j , where x ∈ ℜn
j=1
j=1
is the state vector, u ∈ ℜm is the control input vector and θ is an exogenous parameter that can be time dependent with bounded derivatives such as p
∑ θi (t) = 1, θi (t) ≥ 0,
(5)
i=1
θ˙i (t) ≤ βi ,
i = 1, · · · , p
(6)
where βi are assumed to be known bounds on the derivatives of the parameters θi . Our objective is to design a gain scheduling control law of the form p
u(t) =
The following result can be viewed as an extension of the classical positivity result for LTI systems [25]. Lemma 2: System (1) is positive for any parameter function θ taking values in the simplex set, if and only if the matrices A1 , ..., A p are Metzler matrices. Now, consider the following dual system z˙(t) = A0 (θ )z(t)
(3)
(7)
∑ θ jKjx
j=1
that stabilizes system (4) and maintains the positivity of the closed-loop system. For this purpose, we provide the following result. Theorem 1: There exists a state-feedback of the form (7) that stabilizes and maintains the positivity of system (4), if there exist matrices P1 , . . . , Pp , Y1 . . . ,Yp , a diagonal matrix D and a scalar α such that the following conditions hold:
p
with A0 (θ ) =
∑ θ j A0j
and which depends on the same
Mii ≺ 0,
j=1
Pi 0
Mi j + M ji ≺ 0,
exogenous parameter vector θ as for system (1).
for i = 1, · · · , p
(8a)
for i < j = 1, · · · , p
(8b)
Ai D + BiY j + αI ≤ 0, In the sequel, we will exploit the two common properties connected to positivity and stability of system (1) and its dual. In fact, by the result of Lemma 2 we have that system (1) is positive if and only if its dual system
with Mi j =
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for i, j = 1, · · · , p (8c)
p
∑ βk Pk − DA0i − Ai D −Y j0 B0i − BiY j k=1
Pi + D − DA0i −Y j0 Bi
Pi + D − Ai D − BiY j . 2D
p
If so, a stabilizing gain K(θ ) =
∑ θ j K j that maintains
j=1
the closed-loop system positive is given by Ki = Yi D−1 , i = 1, . . . , p. Proof: The proof is based on duality. We shall exploit the fact that with the same control law, the positivity and the stability of system (4) is equivalent to the positivity and the stability of the dual closed-loop system given by z˙ = A0cl (θ )z (9) p
where Acl = A(θ ) + B(θ )K(θ ), with K(θ ) =
∑ θ jKj.
where
p
with Y j = K j D. (16) can be rewritten as p
V = z0 P(θ )z
(10)
∑ θ j Pj . Note that since all Pj are positive
j=1
The conditions of the theorem ensure that V˙ ≺ 0 which is implied by:
Mi j + M ji ≺ 0,
for i = 1, · · · , p
(18)
for i < j = 1, · · · , p
(19)
In order to show that the closed-loop system is positive, it suffices to ensure that each of its modes is a Metzler matrix. That is Ai + Bi K j is Metzler for all i, j. Since D ≺ 0 and D is diagonal, we obtain from condition (8c): Ai D + BiY j + αI ≤ 0, that (Ai D + BiY j + αI)D−1 ≥ 0.
definite matrices we have that P(θ ) > 0. The time derivative of V is given by: V˙ = z0 Pθ˙ z + z˙0 P(θ )z + z0 P(θ )˙z
(17)
i< j
i=1 j=1
Mii ≺ 0,
p
where P(θ ) =
p
p
V˙ ≤ ∑ ∑ θi2 ξ 0 Mii ξ + ∑ θi θ j ξ 0 (Mi j + M ji )ξ
j=1
Now, we shall first prove the stability of the dual system (9) by considering the following parameterdependent Lyapunov function:
Pi + D − Ai D − BiY j 2D
∑ βk Pk − DA0i − Ai D −Y j0 B0i − BiY j Mi j = k=1 Pi + D − DA0i −Y j0 Bi
(11)
Thus, we have Ai + Bi K j + αD−1 ≥ 0 which means by Remark 1 that Ai + Bi K j is a Metzler matrix.
p
with Pθ˙ =
∑ θ˙ j Pj .
j=1
In order to decouple the Lyapunov matrices from the system matrices, slack matrix variables are introduced as follows: 2[˙z0 D + z0 D] z˙−A0cl (θ )z = 0 (12) where D is a diagonal matrix. From (11) and (12), it follows that: V˙ = z0 Pθ˙ z + z˙0 P(θ )z + z0 P(θ )˙z
Note that the previous result involves a diagonal slack variable which provides a few degrees of freedom. A less conservative stabilization conditions can be established by using a Metzler matrix as a slack variable instead of a diagonal one. This is stated in the following result. Theorem 2: Let γ > 0 be a given a state-feedback that stabilizes and itivity of system (4), if there exist Y1 . . . ,Yp , a matrix G and a positive the following conditions hold
+ 2[˙z0 D + z0 D] z˙−A0cl (θ )z (13) Since θ˙ j ≤ β j ,
Mii ≺ 0,
j = 1, · · · , p, then V˙ satisfies:
Pi 0
Mi j + M ji ≺ 0, 0
p
+ 2[˙z D + z D] z˙−A0cl (θ )z (14)
0
for i < j = 1, · · · , p
(20b)
0
for i, j = 1, · · · , p (20c)
with
(20d)
p
0 0 0 0 ∑ β j Pj − GAi − Ai G −Y j Bi − BiY j Ξi j = j=1 Pi + G − GA0i −Y j0 B0i
By substituting Acl by its expression, we obtain: p
V˙ ≤ z0 ( ∑ β j Pj )z + z˙0 P(θ )z + z0 P(θ )˙z
Pi + G0 − Ai G0 − BiY j . G + G0 p
If so, a stabilizing gain K(θ ) =
j=1
+ 2[˙z0 D + z0 D] z˙−(A(θ ) + B(θ )K(θ ))0 z (15)
∑ θ j K j that maintains
j=1
the closed-loop system positive is given by
By substituting A(θ ), B(θ ) and P(θ ) by their ), K(θ z , we have expressions and taking ξ = z˙ p
(20a)
G + αI ≥ 0
j=1
0
for i = 1, · · · , p
Ai G + BiY j + γG ≤ 0,
V˙ ≤ z0 ( ∑ β j Pj )z + z˙0 P(θ )z + z0 P(θ )˙z
scalar. There exist maintains the posmatrices P1 , . . . , Pp , scalar α such that
Ki = Yi G−1 , i = 1, . . . , p.
p
V˙ ≤ ∑ ∑ θi θ j ξ 0 Mi j ξ i=1 j=1
(16)
Proof: By following the same line of arguments in the previous result, we have that V = z0 P(θ )z with
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p
P(θ ) =
∑ θ j Pj
IV. Numerical results
is a Lyapunov function. Thus, the
j=1
p
state-feedback control u =
∑ θ jKjx
with Ki = Yi (G0 )−1
j=1
is stabilizing. In order to complete the proof, we only need to show that the closed-loop system is positive or equivalently that each of its modes is a Metzler matrix. Let G be any matrix satisfying condition (20a) and (20b) which implies that the block matrix G + G0 satisfies G+G0 ≺ 0. In addition, conditions (20d) is equivalent to the fact that G is a Metzler matrix. Then, according to Lemma 1 the inverse of G is a negative matrix (G−1 ≤ 0). Due to this fact, we have that (Ai G0 + BiY j + γG0 )(G0 )−1 ≥ 0,
for i, j = 1, · · · , p.
Since by condition (20c), the matrix Ai G0 + BYi + γG0 is negative. Hence, it holds for each mode (Ai G0 + BiY j + γG0 )(G0 )−1 = Ai + Bi K j + γI ≥ 0,
Consider system (1) involving constant parameters a and b such that 0 3 1 1 A1 = , A2 = (21) a + 1 −1 b + 3 −1 2a 2−b B1 = , B2 = (22) −1 1 Based on Lemma 2, one can easily see that the free system is positive in the range : −1 ≤ a and −3 ≤ b since with these values A1 and A2 are Metzler matrices. The stabilizable region of the proposed approaches was investigated for various values of a and b. Figure 1 shows the stabilizable region (in blue color) of the parameterdependent Lyapunov function approach whit G diagonal for β j = 0.99, j = 1, · · · , 2. The stabilization result is significantly improved by using a general Metzler matrix variable G instead of a diagonal matrix as illustrated in figure 2. We can see that the stabilizable region is expanded by using Theorem 2.
which in turn shows that every mode of the closed-loop system is a Metzler matrix. Then, by Lemma 2 we can p
conclude that the state-feedback control u =
∑ θ jKjx
j=1
b
with Ki = Yi (G0 )−1 maintains the positivity of the closedloop system and the proof is complete. Our approach is flexible due to the fact that it can incorporate constraints on the stabilizing control laws. For instance, if a control law is needed to be nonnegative: u(t) ≥ 0, ∀t ≥ 0, such control can be computed by combining conditions (20a)-(20d) with the additional constraints: Yi ≤ 0,
a Fig. 1. The stabilizable region of the parameter-dependent Lyapunov function approach whit G diagonal for β j = 0.99, j = 1, · · · , 2.
i = 1, . . . , p.
In fact, due to the nonnegativity of the states, the p
nonnegativity of a control law of the form u =
∑ θ jKjx
j=1
is insured by the fact that the gains Ki must be positive matrices: Ki ≥ 0. Indeed, this condition can be recovered from b
Yi ≤ 0 ⇒ Yi (G0 )−1 = Ki ≥ 0, as we have previously shown that we can use a matrix G with negative inverse. In the same spirit, one can take into account prescibed upper and lower bounds on the gains of the control law such as: K ≤ Ki ≤ K. For this purpose, it suffices to add to conditions (20a)-(20d) the following extra constraints: KG0 ≤ Yi ≤ KG0 ,
a Fig. 2. The stabilizable region of the parameter-dependent Lyapunov function approach with a Metzler G for β j = 0.99, j = 1, · · · , 2.
i = 1, . . . , p. 4775
V. CONCLUSIONS We have treated the stabilization problem for timevarying continuous-time positive systems by using a new Lyapunov-based technique that involves Metzler slack variables. Due to the positivity constraint on the closedloop system this kind of slack variable are suitable. This choice of slack variables leads to an adequate stabilization result which can outperform the case when diagonal slack variables are involved. Our approach has been illustrated by a comparison example. ACKNOWLEDGMENTS This work was produced in the framework of CEREEV (Combustion Engine for Range-Extended Electric Vehicle), a European territorial cooperation project part-funded by the European Regional Development Fund (ERDF) through the INTERREG IVA France (Channel)-England program, and the research department of the Picardie, France. References [1] M. Ait Rami, and F.Tadeo, Controller Synthesis for positive Linear Systems with Bounded controls. IEEE Trans. Syst. And Circuits, 54, pp. 151–155, 2007. [2] M. Ait Rami. Stability and stabilization of positive systems with time varying delays. Lecture notes in control and information sciences, Springer, 389, pp. 205–215, 2009. [3] M. Ait Rami. Solvability of static output-feedback stabilization of positive LTI systems. Systems & control letters, 60, pp. 704–708, 2011. [4] M.Ait Rami, F. Tadeo, and U. Helmke. Positive observers for linear positive systems and their implications. International Journal of Control, 84, pp. 416–725, 2011. [5] M. AitRami, and D. Napp. Characterization and Stability of Autonomous Positive Descriptor Systems. IEEE Trans. Automat. Control, 57(10), pp. 2668–2673, 2012. [6] M. Ait Rami, M. Sch¨ onlein, and J. Jordan. Estimation of linear positive systems with unknown time-varying delays. Europeran Journal of control, 19: pp 179–187, 2013. [7] M. Ait Rami and D. Napp. Positivity of discrete singular systems and their stability: An LP-based approach. Automatica, 50: pp 84–91, 2014. [8] R. Bellman. Introduction to Matrix Analysis. SIAM, Philadelphia, 1997. [9] A. Berman, M. Neumann, and R. J. Stern, Nonnegative Matrices in Dynamic Systems. New York: Wiley, 1989. [10] A. Berman and R.J. Plemmons. Nonnegative Matrices in the Mathematical Sciences (1979), Academic Press, San Diego, CA, reprinted by SIAM, Philadelphia, 1994. [11] F. Blanchini, P. Colaneri and M.E. Valcher, Is stabilization of switched positive linear systems equivalent to the existence of an Hurwitz convex combination of the system matrices? IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), 2011. [12] C. Briat. Robust stability and stabilization of uncertain linear positive systems via integral linear constraints and L1-gains characterizations, International Journal of Robust and Nonlinear Control, 23, pp. 1932–1954, 2013. [13] M.C. de Oliveira, and R.E. Skelton. Stability tests for constrained linear systems. Lecture notes in control and information sciences. Perspectives in robust control, vol. 268, Berlin Springer, 2001. [14] Y. Ebihara and T. Hagiwara, A dilated LMI approach to robust performance analysis of linear time-invariant uncertain systems. Automatica, vol. 41 (11), pp. 1933–1941, 2005. [15] Y. Ebihara, D. Peaucelle, D.Arzelier. LMI approach to linear positive system analysis and synthesis. Systems & Control Letters, 63, pp. 50–6, 2014.
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