Robust Stabilization of Constrained Uncertain Continuous-time Fractional Positive Systems Fouad Mesquine a , Abdelaziz Hmamed b , Mohamed Benhayoun a , Abdellah Benzaouia a , Fernando Tadeo c a
LAEPT, URAC 28, Cadi Ayyad University, Faculty of Sciences Semlalia, BP 2390, Marrakech, Morocco b
LESSI, University Mohamed Ben Abdellah, Faculty of Sciences Dhar El Mehraz, BP 1796, Fes,
c
Dpto. Ingenieria de Sistemas y Automatica, Universidad de Valladolid, 47005 Valladolid, Spain
Abstract The robust stabilization problem is solved for uncertain fractional linear continuous-time systems having bounded asymmetric control, with additional condition of non negativity of the states. The synthesis of state-feedback controllers is obtained by giving conditions in terms of Linear Programs. An illustrative example is provided to show the usefulness of the results. Key words: Fractional systems, positive systems, stabilization, bounded controls, linear programming.
1
Introduction
In the literature, systems with non negative states are called positive systems (see [9], [17] for general references). These systems appear in many practical problems, when the states represent physical quantities that have an intrinsically constant sign (Absolute temperatures, levels, heights, concentrations, etc). Based on Gersgorin’s theorem, a sufficient condition is provided in [16] and formulated as a quadratic programming problem. Recently, in [11] a necessary and sufficient Linear Matrix Inequality (LMI) condition is proposed for the stabilization of positive linear systems. In comparison with these previous papers, the works of [1],[2], [13], [14] and [15] provide new treatments for the stabilization of positive linear systems where all the proposed conditions are necessary and sufficient, and expressed in terms of Linear Programming (LP). It was ? This paper was not presented at any IFAC meeting. Corresponding author F. Mesquine Tel. +212 5 24 434649. Fax +212 5 24 43 74 10. Email addresses:
[email protected] (Fouad Mesquine), hmamed
[email protected] (Abdelaziz Hmamed),
[email protected] (Mohamed Benhayoun),
[email protected] (Abdellah Benzaouia),
[email protected] (Fernando Tadeo).
Preprint submitted to Automatica
shown in these works that the constraints on the control can be handled easily. It is well known that this topic is of continuing interest and attracts researchers [3], [4], [5], [23], [6], [7] and references therein. On the other hand, the first introduction of fractional derivative was proposed by Liouville and Riemann at the XIX century as pointed out in [26]. Background works on mathematical fractional calculus can be found in [29], [26], [25] and [28]. The first work on control of fractional systems was given in [28]. Recent works on stability and robust stability were studied extensively in [18], [22], [19], [10] and [27]. Almost these investigations use algebraic stability conditions. However, few papers introduce the use of Lyapunov concept to deal with the problem of analysis stability of fractional systems as in [21]. Especially, [30] presents an interesting discussion about the meaning of Lyapunov function concept for fractional systems. Besides, a similar condition of stability as ordinary systems is obtained for a quadratic Lyapunv function candidate. New development using the methodology below and based on Lyapunov equation can be found in [8]. This paper proposes to extend the results established for uncertain linear systems to the uncertain fractional systems. A new approach for the stabilization problem of MIMO positive fractional linear systems by state feedback control presented in [8] is extended to uncertain
12 March 2014
and α ∈ R is the order of the fractional derivative. The Gamma function is defined by: Z ∞ Γ(z) = e−t tz−1 dt, z ∈ R. (3)
fractional case. Asymmetric bounds on the control and the state are taken into account while maintaining positivness of the pseudostate and ensuring asymptotic stability. The sysnthesis problem is solved by using linear programming approach. Robustness of the controllers in presence of both uncertainties of interval and polytopic forms is guaranteed.
0
The Caputo fractional derivative is defined as follows: The remainder of the paper is structured as follows: Section 2 is devoted to the problem statement. Some preliminaries are also recalled in Section 3. Section 4 deals with the main results concerning robustness of the controllers in presence of asymmetric constraints on the control and the state. Besides, example of an uncertain system is treated to illustrate the proposed method. Concluding remarks ends the paper in section 5. 1.1
c
Z 0
t
dx dt (τ ) dτ (t − τ )α
(4)
Note that the Caputo and Riemann-Liouville fractional derivatives are linked as follows: Dα x(t) = c Dα x(t) +
Notation and definitions
• Rn+ denotes the non-negative orthant of the ndimensional real space Rn . • M T denotes the transpose of the real matrix M . • A matrix M ∈ Rn×n is called a Metzler matrix if its off-diagonal elements are nonnegative. That is, for M = {mij }ni,j=1 , M is Metzler if mij ≥ 0 when i 6= j. • A matrix M (or a vector) is said to be nonnegative if all its components are nonnegative (by notation M ≥ 0). It is said to be positive if all its components are positive (M > 0).
t−α xo Γ(1 − α)
(5)
The problem we are addressing below is to use conditions on matrices A ∈ Rn×n and B ∈ Rn×m given in [8], such that there exists a state feedback matrix K ∈ Rm×n satisfying: • The fractional system is positive in closed-loop. • The fractional closed-loop system is asymptotically stable (limt→∞ x(t) = 0) while satisfying asymmetric control constraints. 3
2
1 Γ(1 − α) 0 < α ≤ 1,
Dα x(t) =
Problem statement
Preliminaries
Consider first the following continuous-time fractional linear autonomous system, without uncertainties:
Consider the uncertain continuous-time commensurate fractional order linear system:
α D x(t) = Ax(t) 0 0, matrices A+ , A− , B + , B − find a positive vector x ¯ that boubds the states trajectories (x(t) ≤ x ¯) and a state feedback matrix K giving bounded control ¯ such that the following uncerlaw −u ≤ u = Kx(t) ≤ u tain closed-loop system is positive and asymptotically stable: Dα x(t) = (A + BK)x(t) 0 < α ≤1 (14) x(0) = xo ≥ 0 −u ≤ Kx(t) ≤ u ¯
The problem we are dealing with is to design the controller which ensures asymptotic stability while maintaining the state positive even the open-loop system is
Now, we state the main result of this section which consists of a linear programming method for characterizing controllers K that ensures asymptotic stability while non negativness of the trajectory is guaranteed and the control is restricted to evolve in a prescribed region.
Consider the continuous-time fractional linear system given by (1). If one uses a state feedback control u(t) = Kx(t) the uncertain closed-loop system becomes Dα x(t) = (A + BK)x(t) 0 0, yi ≥ 0, (LP 2) zi ≥ 0, i = 1, . . . , n (15) Pn i=1 yi ≤ u ¯, Pn z ≤ u, i=1 i a− x − + ij ¯j + bi yj − bi zj ≥ 0, i 6= j = 1, . . . , n.
3
x1
2
1
0
Then, the closed-loop system (14) is positive and asymptotically stable for any initial condition 0 < xo ≤ x ¯, under the state-feedback bounded control law −u ≤ u = Kx ≤ u ¯, with
0
5
10
15 t (s)
20
25
30
0
5
10
15 t (s)
20
25
30
1 0.8 x2
0.6 0.4 0.2
K=
[¯ x−1 1 (y1
− z1 ) . . .
x ¯−1 n (yn
0
− zn )].
Proof 1 Let A− ≤ A ≤ A+ and B − ≤ B ≤ B + . Since x ¯ > 0, yi > one can write: P A− x ¯ ≤ Pn0, zi > 0, P n n + − + A¯ x ≤ A x ¯ , B y ≤ B y ≤ B i i i=1 i=1 i=1 Pn Pn Pn yi − and −B + i=1 zi ≤ −B i=1Pzi ≤ −BP i=1 zi . n n y − zi ) ≤ Summing up,P leads to A¯ x + B( i i=1 i=1 Pn n + A+ x ¯ + BP B − i=1 zi . Consequently, i=1 yi − P n n A+ x ¯ + B + i=1 yi − B − i=1 zi < 0 is sufficient to realize the first inequality of Theorem 3.3. Using the same reasonning, one can has: aij x ¯j + bi (yj − zj ) ≥ a− ¯j + ij x − + − − + bi yj − bi zj , which implies that aij x ¯j + bi yj − bi zj ≥ 0 for i 6= j is sufficient to realize the last inequality of Theorem 3.3. 2 4.2
Fig. 1. State Trajectories for the uncertain fractional system 1 0.8 x2
0.6 0.4 0.2 0
0
0.5
1
1.5 x1
2
2.5
3
0
5
10
15 t (t)
20
25
30
5 4
u
3 2 1 0
Fig. 2. Trajectory and Control evolution
Example: Positive Interval Bounded System
−1
0.6 0.58
a11
Consider an uncertain system described by (14), where
a12
−1.05 −1.1
A− =
−1.2 0.5
#
" ,
A+ =
−1
0.6
#
−1.2
0.52 0
5
10
15 t(s)
20
25
0.5
30
−0.2
−0.2 −0.6 −0.3 −0.7 # " # 0.4 0.5 B− = , B+ = . 0.2 0.3
0
5
10
15 t(s)
20
25
30
0
5
10
15 t(s)
20
25
30
−0.6
a21
−0.22
"
−0.24
a22
"
0.56 0.54
−1.15
−0.65
−0.26 −0.28 −0.3
0
5
10
15 t(s)
20
25
30
0.5
¯ = 10, α = 0.5. Solving the LP2, leads with u = 4, u to a feasible solution given by : x ¯T = [3.6588 1.6098] , and y1 = 6.2075, y2 = 1.2234, z1 = 0.3691, z2 = 2.2928. The robust feedback gain is given by h i K = 1.5957 −0.6643 .
0.48 b1
0.46 0.44 0.42 0.4
0
5
10
15 t (s)
20
25
30
0
5
10
15 t (s)
20
25
30
0.35
b2
0.3
0.25
0.2
It can be seen that the closed-loop system is positive, that the states are bounded by x ¯ and the control is al-
Fig. 3. System parameters evolution
4
4.3
4.4
Polytopic uncertainties
Consider an uncertain system described by (14), where
Consider the constrained uncertain fractional continuoustime system given by (1) with the following uncertainties: A=
PN
s=1 δs As ;
δs ≥ 0;
B=
PN
Example: Positive Bounded Polytopic System
" A1 =
PN
s=1 δs
s=1 δs Bs ;
A2 =
= 1.
A3 =
The aim here is to address the following problem: Given u, u ¯ > 0, matrices Ai , Bi , i = 1, . . . N, find a positive vector x ¯ and a state feedback matrix K giving bounded control law −u ≤ u = Kx(t) ≤ u ¯ such that the following uncertain closed-loop system is positive and asymptotically stable: Dα x(t) = (A + BK)x(t) 0 < α ≤1 x(0) = xo ≥ 0 −u ≤ Kx(t) ≤ u ¯
#
−0.9 0.6
−1.2 0.4 −0.3 −0.8
" , B1 =
#
"
#
# ,
0.4
# ,
0.3 "
, B3 =
0.5 0.3
, B2 =
−0.3 −0.7 "
where As and Bs are known vertices.
0.6
−0.2 −0.6 "
(16)
−1
0.4
# ,
0.5
with u = 10, u ¯ = 4, α = 0.4. Solving the LP2, leads to a feasible solution given by : x ¯T = [2.99 6.0941] , and y1 = 3.3054, y2 = 0.4311, z1 = 0.2108, z2 = 6.1686. The robust feedback gain is given by h i K = 1.035 −0.9415 .
(17)
It can be seen that the closed-loop system is positive, that the states are bounded by x ¯ and the control is always restricted between −10 and 4. For five different randomly chosen dynamics of the system each one applied during 6s, it can be seen in Figure 4 that the trajectories converge to 0 in the positive orthant. The corresponding control is shown in Figure 5 where it can be seen that the imposed bounds on u are fulfilled. The random variation of the parameters of A and B is shown in 6. The simulation was achieved, from xo = [2 4]T , with a time of 30s with a step of 0.1s and k = 400 in the exponential Mittag-Leffler expression (7).
Now, we state an important result: Theorem 4.2 For system (17), consider the following LP problem in the variables x ¯ = [¯ x1 . . . x ¯n ]T ∈ Rn , y1 , . . . , yn ∈ Rm and z1 , . . . , zn ∈ Rm : Pn Pn As x ¯ + Bs ( i=1 yi − i=1 zi ) < 0, x ¯ > 0, yi > 0, (LP 3) zi > 0, (18) i = 1, . . . , n, s = 1, . . . , N, Pn i=1 yi ≤ u ¯, Pn z ≤ u, i=1 i (a ) x s ij ¯j + (bs )i (yj − zj ) ≥ 0, i 6= j = 1, . . . , n. then, the closed-loop system (17) is positive and asymptotically stable for any initial condition 0 < xo ≤ x ¯, under the state-feedback bounded control law −u ≤ u = Kx ≤ u ¯, with K = [¯ x−1 ¯−1 n (yn − zn )]. 1 (y1 − z1 ) . . . x
5
Conclusions
In this paper, the robust stabilization problem for fractional linear continuous-time systems with bounded control is addressed, while imposing non negativeness of the states. The synthesis of robust state-feedback controllers is obtained by expressing the required conditions in terms of simple Linear Programing Problems for two kinds of uncertainties, namely the polytopic and the interval ones. Finally, two numerical examples are provided to illustrate the usefulness of the obtained results and their feasibility and simplicity for both kind of uncertainties. Acknowledgements This work has been funded by Morocco-Spain AECI research project AP/034911/11 and mcyt-CICYT project DP I2010 − 21589 − C05 − 05.
Proof 2 The proof is obvious using the convex arguments to retrieve the conditions of Theorem 3.3. 2
5
[8] A. Benzaouia, A. Hmamed, F. Mesquine, M. Benhayoun and F. Tadeo, Stabilization of Continuous-time Fractional Positive Systems by using Lyapunov function. IEEE Trans. Aut. Control, To appear, 2014.
4
x2
3 2 1 0
0
0.2
0.4
0.6
0.8
1 x1
1.2
1.4
1.6
1.8
[9] A. Berman, M. Neumann and R J. Stern, Nonnegative Matrices in Dynamic Systems. New York: Wiley, 1989.
2
1
[10] M. Efe, Fractional Order Systems in Industrial AutomationA Survey, IEEE Trans. Ind. Inf., Vol. 7, No. 4, 2011.
u
0
−1
−2
0
5
10
15 t(s)
20
25
[11] H. Gao, J. Lam, C. Wang and S. Xu, Control for stability and positivity: equivalent conditions and computation. IEEE Transactions on Circuits and Systems-II, 52, pp. 540 – 544, 2005.
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Fig. 4. State Trajectories for the uncertain fractional system
[12] A. Hmamed, F. Mesquine, A. Benzaouia, M. Benhayoun and F. Tadeo, Continuous-time Fractional Positive Systems with Bounded States, 52th CDC, Florence, Italy, pp. 2127 – 2132, 2013.
2
x1
1.5 1 0.5 0
0
5
10
15 t(s)
20
25
[13] A. Hmamed, A. Benzaouia, F. Tadeo and M. Ait Rami, Positive Stabilization of Discrete-Time Systems with Unknown Delay and Bounded Controls. European Control Conference, Kos, Greece, 2-5 July, 2007.
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4
x2
3
[14] A. Hmamed, A. Benzaouia, F. Tadeo and M. Ait Rami, Memoryless Control of Delayed Continuous-time Systems within the Nonnegative Orthant. IFAC World Congress, Korea, 2008.
2 1 0
0
5
10
15 t(s)
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[15] A. Hmamed, M. Ait Rami, A. Benzaouia and F. Tadeo Stabilization under constrained states and controls of positive systems with time delays European J. Control. Vol.18, no. 2, pp. 182 – 190, 2012.
Fig. 5. Trajectory and Control evolution
a11
−0.8
−1
−1.2
0
50
100
150 −1
t(10
200
250
[16] T. Kaczorek, Stabilization of positive linear systems by state feedback, Pomiary, Automatyka, Kontrola, pp. 2 – 5, 1999.
300
s)
[17] T. Kaczorek, Positive 1D and 2D Systems, Springer-Verlag, 2002.
b1
0.44
0.42
0.4
0
50
100
150
200
250
[18] T. Kaczorek, Fractional positive continuous-time linear systems and their reachability, in Int. J. App. Math. Comput. Sci., Vol. 18, N 2, pp. 223 – 228, 2008.
300
t(10−1 s)
Fig. 6. System parameters evolution
[19] T. Kaczorek, Necessary and sufficient stability conditions of fractional positive continuous-time linear systems, in Acta Mechanica et Automatica, Vol. 5, N 2, pp. 52 – 54, 2011.
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