t; t0
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Fig. 6. Tube E for Example 2.
3) Example 3: Consider now the time-varying system The solution with dynamic matrix and solution given respectively by [15, p. 149]
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[5] A. Ilchmann, D. H. Owens, and D. Prätzel-Wolters, “Sufficient conditions for stability of linear time-varying systems,” Syst. Control Lett., vol. 9, no. 2, pp. 157–163, Feb. 1987. [6] F. Amato, G. Celentano, and F. Garofalo, “New sufficient conditions for the stability of slowly varying linear systems,” IEEE Trans. Autom. Control, vol. 38, no. 9, pp. 1409–1411, Sep. 1993. [7] D. Guo and W. J. Rugh, “A stability result for linear parameter-varying systems,” Syst. Control Lett., vol. 24, no. 1, pp. 1–5, Jan. 1995. [8] W. Hahn, Stability of Motion. Berlin, Germany: Springer-Verlag, 1967. [9] H. D’Angelo, Linear Time-Varying Systems: Analysis and Synthesis. Boston, MA: Allyn and Bacon, 1970. [10] P. J. Antsaklis and A. N. Michel, Linear Systems. New York: McGraw-Hill, 1997. [11] P. Mullhaupt, D. Buccieri, and D. Bonvin, “A numerical sufficiency test for the asymptotic stability of linear time-varying systems,” Automatica, vol. 43, no. 4, pp. 631–638, Apr. 2007. [12] M. Wu, “A note on stability of linear time-varying systems,” IEEE Trans. Autom. Control, vol. AC-19, no. 1, pp. 162–162, Apr. 1974. [13] J. J. Zhu, “PD-spectral theory for multivariable linear time-varying systems,” in Proc. 36th IEEE Conf. Decision Contr., San Diego, CA, Dec. 1997, pp. 3908–3913. [14] G. Garcia, P. L. D. Peres, and S. Tarbouriech, “Necessary and sufficient numerical conditions for asymptotic stability of linear time-varying systems,” in Proc. 47th IEEE Conf. Decision Contr., Cancun, Mexico, Dec. 2008, pp. 5146–5151. [15] H. K. Khalil, Nonlinear Systems, 3rd ed. Upper Saddle River, NJ: Prentice-Hall, 2002.
A(t) = 0 1 +1 t ; x(t; t0 ; x0 ) = 11++tt0 x0 and thus the system is asymptotically stable but not uniformly asymptotically stable. In fact, Theorem 4 provides
Finite-Time Stability of Linear Time-Varying Systems: Analysis and Controller Design
X (t) = (1(1++tt0))2
Francesco Amato, Marco Ariola, Senior Member, IEEE, and Carlo Cosentino
2
as the solution of (4) and, consequently, limt (t) = limt (1 + t0 )=(1 + t) = 0, indicating that the system is asymptotically stable. On the other hand, applying the results from Theorem 5 one has
!1
Y (t; t0 ) = (1(1++tt0))2
2
!1
t 0 t0 ))
exp(2 (
as the solution of differential (9). Therefore, 8 > 0, we have limt Y (t; t0 ) ! 1 confirming the fact that the system is not uniformly asymptotically stable.
!1
Abstract—The note deals with the finite-time analysis and design problems for continuous-time, time-varying linear systems. Necessary and sufficient conditions and a sufficient condition for finite-time stability are devised. Moreover, sufficient conditions for the solvability of both the state and the output feedback problems are stated. Such results require the feasibility of optimization problems involving Differential Linear Matrix Inequalities. Some numerical examples illustrate the effectiveness of the proposed approach. Index Terms—Dynamic output feedback, finite-time stability, linear time-varying systems.
V. CONCLUSION Necessary and sufficient conditions for asymptotic stability and for uniform asymptotic stability of linear continuous time-varying systems have been given. The conditions are based on the numerical solution of a linear differential Lyapunov equation, associated to a tube in the state space that confines all the trajectories of the system. For periodic systems, it suffices to integrate the differential Lyapunov equation over one single period. Extensions of the proposed conditions to cope with control design problems, particularly in the case of periodic systems, are being investigated by the authors.
REFERENCES [1] H. H. Rosenbrook, “The stability of linear time-dependent control systems,” Int. J. Electr. Control, vol. 15, no. 1, pp. 73–80, Jul. 1963. [2] C. Desoer, “Slowly varying system x_ = A(t)x,” IEEE Trans. Autom. Control, vol. AC-14, no. 6, pp. 780–781, Dec. 1969. [3] G. Kreisselmeier, “An approach to stable indirect adaptive control,” Automatica, vol. 21, no. 4, pp. 425–431, Jul. 1985. [4] J. M. Krause and K. S. P. Kumar, “An alternate stability analysis framework for adaptive control,” Syst. Control Lett., vol. 7, no. 1, pp. 19–24, Feb. 1986.
I. INTRODUCTION The concept of finite-time stability (FTS) dates back to the 1960s, when this idea was introduced in the control literature [1], [2]. A system is said to be finite-time stable if, given a bound on the initial condition, its state does not exceed a certain threshold during a specified time interval. Recently, significant contributions have been given in this field, especially in the case of linear systems (see for instance [3]–[6]). Manuscript received November 14, 2007; revised March 04, 2009, July 05, 2009, and October 16, 2009. First published February 02, 2010; current version published April 02, 2010. Recommended by Associate Editor P. Colaneri. F. Amato and C. Cosentino are with the School of Computer and Biomedical Engineering, Magna Græcia University, Campus Salvatore Venuta, 88100 Catanzaro, Italy (e-mail: [email protected]; [email protected]). M. Ariola is with the Dipartimento per le Tecnologie, Università degli Studi di Napoli Parthenope, Centro Direzionale di Napoli, 80143 Napoli, Italy (e-mail: [email protected]). Color versions of one or more of the figures in this technical note are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2010.2041680
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Fig. 1. Open-loop free evolution from Example 1.
It is important to point out that FTS and Lyapunov asymptotic stability (LAS) are independent concepts; indeed, a system can be FTS but not LAS, and vice versa. While LAS deals with the behavior of a system within a sufficiently long (in principle infinite) time interval, FTS is a more practical concept, useful to study the behavior of the system within a finite (possibly short) interval, and therefore it finds application whenever it is desired that the state variables do not exceed a given threshold (for example to avoid saturations or the excitation of nonlinear dynamics) during the transients. The FTS dealt with in this note should not be confused with the FTS concept adopted in some other papers such as for instance the seminal paper [7], and the papers [8], [9]. In these works, the authors focus on the Lyapunov stability analysis of nonlinear systems whose trajectories converge to an equilibrium point in finite time and on the characterization of the associated settling-time. Thus, it should be remarked that the latter definition of FTS is unrelated to the one given in [1] and [2] and considered in the present note. A sufficient condition for FTS, requiring the solution of a linear matrix inequalities (LMIs)-based feasibility problem [10], has been provided in [3]. Extending the concept of FTS, which applies to open-loop systems, the finite-time stabilization problem concerns the design of a linear controller ensuring the FTS of the closed-loop system. Sufficient conditions for finite-time stabilization have been provided in [3] for the state feedback case. In the recent paper [11] the dynamic output feedback problem has been converted into a LMIs-based optimization problem. In [3] and [11], the authors deal with linear time-invariant systems and make use of an approach based on Lyapunov arguments which may render the technique rather conservative. Moreover, the drawback of the approach proposed in [11] is that the controller design is performed in two phases: the first step is devoted to design a static state feedback controller, then a state observer which tries to retain the properties guaranteed by the state feedback controller is synthesized. Therefore, this approach looks at a subset of the class of admissible dynamic output feedback controllers. Differently from previous works, the approach proposed in this note is reminiscent of optimal control techniques [12] and improves the existing literature in several ways. The first result of the technical note (Theorem 1) is a necessary and sufficient condition for FTS, which involves the computation of the state transition matrix; therefore the application of such condition is straightforward in the time-invariant case. We also provide in Theorem 1 a sufficient condition for FTS which requires the existence of a feasible solution of a certain differential linear matrix inequality (DLMI) [13]; such condition is more suitable for time-varying systems. Indeed, a similar approach has been also ex-
ploited in [6] to derive sufficient FTS analysis conditions for hybrid systems with finite state jumps. An important issue is the evaluation of the degree of conservatism of the sufficient condition, since it is the starting point for the derivation of the design conditions; this analysis is done by comparing the necessary and sufficient condition and the sufficient condition provided in Theorem 1 on 1000 randomly generated linear systems. With respect to the approach of [3], Theorem 1 provided in this note exhibits two advantages: 1) it enables to deal with time-varying systems and 2) even in the time-invariant case, both the necessary and sufficient condition and the sufficient condition for FTS turn out to be less conservative. Afterwards, we consider the output feedback design problem, which is solved making use of the proposed approach, ending up with DLMIs. The controller, differently from [11], is computed in one shot and therefore represents a more general solution. Throughout the note, a nontrivial example concerning the finite-time stabilization of a fourth-order system illustrates the way the proposed technique can be applied. The note is organized as follows: in Section II we give some preliminary definitions and we state the problems that we want to solve. Section III is devoted to the analysis conditions for FTS, whereas in Section IV, we present a sufficient condition for the design of an output feedback controller guaranteeing FTS. II. PROBLEM STATEMENT The following definitions deal with various finite-time control problems. All matrices and vectors are assumed to be of compatible dimensions; all time-varying matrices, unless otherwise specified, are assumed to be bounded and piecewise continuous functions of time. Definition 1 (Finite-Time Stability [3]): Given a positive scalar T , a positive definite matrix R, and a positive definite matrix-valued function 0(1), defined over [0; T ], such that 0(0) < R, the time-varying linear system
x_ (t) = A(t)x(t); x(0) = x0
(1)
where A(1) is a bounded piecewise continuous matrix-valued function, is said to be finite-time stable (FTS) with respect to (T; R; 0(1)), if
xT0 Rx0 1 ) x(t)T 0(t)x(t) < 1
t 2 [0; T ]:
8
(2)
Remark 1: The assumption 0(0) < R guarantees that the closed ellipsoid fx0 : xT0 Rx0 1g is a subset of the open ellipsoid fx0 : xT 0(0)x < 1g; this in turn guarantees well posedness of Definition 1. 0
0
Remark 2: LAS and FTS are independent concepts. Obviously, a system which is FTS wrt some (T; R; 0(1)) may be not LAS; conversely, a LAS system could be not FTS if, during the transients, its state exceeds the prescribed bounds. Example 1: Consider a LTI system in the form (1) with
0:0884 0:8984 01:0667 01:1599
0:4026 0:0276 2:8289 0:1808
0:5733 0:9771 A= : (3) 1:0680 01:2306 0 ), with This system is LAS but is not FTS with respect to (T; R; T = 10, R = I and 0 = diag(1=1:52 ; 1=102 ; 1=102 ; 1=102 ). Such 0
0:5796 0:3622 02:4719 00:0779 0
0
0
statement can be readily verified by means of numerical simulations. x(t), starting from the Consider for example the behavior of xT (t)0 initial condition
x0 = (00:2632 0:7616
0
0:0772 0:5871)T
x(t) exceeds the value depicted in Fig. 1. It is readily seen that x(t)T 0 0 < 1. of 1, whereas xT0 Rx
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Note that FTS refers to open-loop systems; the next problem fits FTS into the design framework. Poblem 1 (Finite-Time Stabilization via Output Feedback): Consider the time-varying linear system
x_ (t) = A(t)x(t) + B (t)u(t) x(0) = x0 y(t) = C (t)x(t)
(4a) (4b)
where u(t) is the control input and y (t) is the output. Then, given a positive scalar T , two positive definite matrices R and RK , and two positive definite symmetric matrix functions 0(1) and 0K (1), defined over [0; T ], such that 0(0) < R, 0K (0) < RK , find a dynamic output feedback controller in the form
x_ c (t) = AK (t)xc (t) + BK (t)y(t) u(t) = CK (t)xc (t) + DK (t)y(t)
(5a) (5b)
where xc (t) has the same dimension of x(t), such that the closed-loop system obtained by the connection of (4) and (5) is FTS with respect to
T; blockdiag(R; RK ); blockdiag (0(1); 0K (1))) :
(
III. ANALYSIS A. Main Results The following theorem gives three necessary and sufficient conditions for FTS of system (1) and one sufficient condition. Theorem 1: The following statements are equivalent. i) System (1) is FTS with respect to (T; R; 0(1)). ii) For all t 2 [0; T ]
t; 0)T 0(t)8(t; 0) < R
8(
where 8(t; 0) is the state transition matrix of system (1). iii) For all t 2 [0; T ] there exist a (sufficiently small) scalar such that the following differential matrix equation with terminal and initial conditions
P ) + A( )T P ( ) + P ( )A( ) + 2 I = 0 2 [0; t] _(
P (t) = 0(t) P (0) < R
(6a)
P (t) 0(t) P (0) < R
(6c)
(7a) (7b) (7c)
admits a piecewise continuously differentiable symmetric solution P (1). Moreover, the following condition is sufficient for FTS. v) The DLMI with terminal and initial conditions
P_ (t) + A(t)T P (t) + P (t)A(t) < 0 t 2 [0; T ] P (t) 0(t); t 2 [0; T ] P (0) < R
Remark 3: Condition ii) may be hard to apply in the time-varying case as it requires the computation of the state transition matrix; in the same way, condition iv) requires to check infinitely many DLMIs (one for each t 2 [0; T ]) and therefore it is not useful from a practical point of view. Conversely, condition v) requires to solve just one DLMI over the whole interval [0; T ] and can be solved by efficient numerical algorithms, however it is only sufficient; its degree of conservatism will be evaluated later, in Section III-B, by comparison with the necessary and sufficient condition ii). Condition v) will be the starting point for the solution of the design problem in the state feedback case. Finally, note that condition iii) is not useful per se, but rather it will turn useful in the proof of the theorem. Remark 4: When system (1) is time-invariant, it makes sense to compare the conditions in Theorem 1 with the sufficient condition for FTS provided in [3]. In this case, given a constant weight matrix 0, condition ii) reads
AT t)0 exp(At) < R;
exp(
(8a) (8b)
x(t)T 0(t)x(t) = xT0 8(t; 0)T 0(t)8(t; 0) x0 < xT0 Rx0 1:
Therefore, system (1) is FTS i) =) ii) By contradiction. Let us assume that for some t, x
T )8(t; 0)x xT Rx: 0) 0(t xT 8(t;
Now let x0 that
=
x, where is such that xT0 Rx0
= 1
(9)
. Then (9) implies
T )8(t; 0)x0 1: 0) 0(t xT0 8(t; Therefore
T )8(t; 0)x0 1 0) 0(t x(t)T 0(t)x(t) = xT0 8(t; which contradicts the initial assumption that system (1) be FTS. i) =) iii) Let us assume that system (1) is FTS. Then by continuity arguments, letting z (t) = x(t) for a sufficiently small , for all
t
;T]
2 [0
xT0 Rx0
1 )
2 x(t)T 0(t)x(t) + kz k[0 ;t] < 1
(10)
Let P (1) be the unique ([14, pp. 470-471]) solution of
P_ ( ) + A( )T P ( ) + P ( )A( ) = 0 2 I; P (t) = 0(t)
; t]
2 [0
(11a) (11b)
and assume, by contradiction, that for some x xT P (0)x xT Rx:
(8c)
admits a piecewise continuously differentiable symmetric solution P (1). Before proving the Theorem, some comments are in order.
t 2 [0; T ]
which is obviously less conservative than the condition provided in [3], which is only sufficient, while ii) is also necessary. Concerning condition v), roughly speaking it can be seen as a generalization of the main result in [3], since it optimizes over the set of the piecewise continuously differentiable matrix valued functions P (1)’s rather than looking to constant P ’s; therefore the approach proposed in this work is clearly less conservative than the methodology proposed in [3]. Again, a quantitative comparison between the two conditions is done in Section III-B. Proof: ii) =) i) Let xT0 Rx0 1. Then
(6b)
admits a piecewise continuously differentiable symmetric solution P (1). iv) For all t 2 [0; T ], the DLMI with terminal and initial conditions
P_ ( ) + A( )T P ( ) + P ( )A( ) < 0 2 [0; t]
1005
Now let x0
=
x, where is such that xT0 Rx0
(12) = 1
. Then (12) implies
xT0 P (0)x0 1:
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(13)
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From (11a), we obtain that
d d
T
x( ) P ( )x( )
= 02 x( )T x( ):
(14)
Integrating (14) from 0 to t, we have T
x(t) P (t)x(t)
0 xT0 P (0)x0 = 02 kxk[02
;t]
:
Therefore, from (11b) and (13) x(t)
T
0(t)x(t) = x(t)T P (t)x(t) = xT0 P (0)x0 0 2 kxk2[0;t] 1 0 kzk[02 ;t]
which contradicts (10) iii) =) iv ) It is straightforward to recognize that every solution P (1) to (6) is also solution to (7). iv ) =) i) Let t 2 [0; T ], 2 [0; t], and define V ( ; x) =
xT P ( )x. Then (7a) implies that V_ ( ; x) is negative definite along the trajectories of system (7a). Now let xT0 Rx0 1; then we have
x(t)
v)
T
0(t)x(t) (t)T P (t)x(t) in view of (7b) xT0 P (0)x0 since V ( ; x) is decreasing in [0; t] T < x0 Rx0 in view of (7c) 1:
=) iv) It is straightforward to check that a matrix function
1) satisfying conditions (8) also satisfies (7).
P(
one can reduce the conservatism at will, at the expense of a greater computational burden. By imposing a piecewise affine structure on P (1), condition (8) can be cast as an LMIs feasibility problem, with the initial and final values of P (1) in each subinterval being the optimization variables; such problem has been implemented and solved through the LMI control toolbox [15]. The percentage difference between the exact value of max , obtained by (15), and its estimates obtained via condition v) of Theorem 1 and [3, Lemma 6], has been computed for the 1000 randomly generated systems. The average “distance” from necessity is about 9.6% when applying the former condition and about 44.5% for the latter one. We can conclude that, at least for low order systems, condition v) of Theorem 1 introduces a slight conservatism in the analysis; at the same time there is a significant improvement with respect to the sufficient condition derived from [3]. IV. CONTROLLER DESIGN For the sake of brevity, only the sufficient (operative) condition for the existence of a controller will be given below; by following the same arguments of the proof of Theorem 1 it is possible to state the corresponding necessary and sufficient condition for the controller existence, which however requires the solution of infinitely many differential inequalities and therefore is not useful from a practical point of view (see also Remark 3). The following is the main result of the section. Theorem 2: Problem 1 is solvable if there exist piecewise continuously differentiable symmetric matrix-valued functions Q(1), S (1), a nonsingular matrix-valued function N (1) and matrix-valued func^K (1), C^K (1) and DK (1) such that (the time argument is tions A^K (1), B omitted for brevity)
B. Evaluation of the Sufficient Condition in Theorem 1 The goal of this section is, on one hand, to quantify the degree of conservatism introduced by condition v) with respect to the necessary and sufficient condition ii), and from the other hand to compare condition v) with the FTS sufficient condition derived from [3] when LTI systems are considered. To this end 1000 random LTI systems (500 of the second order and 500 of the third order) have been generated, in the form (1); condition ii) of Theorem 1 has been applied to each system to determine the largest
, say max , such that the system is FTS wrt (2; I; max I ); this is equivalent to computing, for each system
max
:=
1
maxt2[0;2] kexp(At)k2
:
Q
T 912 T 913 T 914
Fig. 6. Tube E for Example 2.
3) Example 3: Consider now the time-varying system The solution with dynamic matrix and solution given respectively by [15, p. 149]
1003
[5] A. Ilchmann, D. H. Owens, and D. Prätzel-Wolters, “Sufficient conditions for stability of linear time-varying systems,” Syst. Control Lett., vol. 9, no. 2, pp. 157–163, Feb. 1987. [6] F. Amato, G. Celentano, and F. Garofalo, “New sufficient conditions for the stability of slowly varying linear systems,” IEEE Trans. Autom. Control, vol. 38, no. 9, pp. 1409–1411, Sep. 1993. [7] D. Guo and W. J. Rugh, “A stability result for linear parameter-varying systems,” Syst. Control Lett., vol. 24, no. 1, pp. 1–5, Jan. 1995. [8] W. Hahn, Stability of Motion. Berlin, Germany: Springer-Verlag, 1967. [9] H. D’Angelo, Linear Time-Varying Systems: Analysis and Synthesis. Boston, MA: Allyn and Bacon, 1970. [10] P. J. Antsaklis and A. N. Michel, Linear Systems. New York: McGraw-Hill, 1997. [11] P. Mullhaupt, D. Buccieri, and D. Bonvin, “A numerical sufficiency test for the asymptotic stability of linear time-varying systems,” Automatica, vol. 43, no. 4, pp. 631–638, Apr. 2007. [12] M. Wu, “A note on stability of linear time-varying systems,” IEEE Trans. Autom. Control, vol. AC-19, no. 1, pp. 162–162, Apr. 1974. [13] J. J. Zhu, “PD-spectral theory for multivariable linear time-varying systems,” in Proc. 36th IEEE Conf. Decision Contr., San Diego, CA, Dec. 1997, pp. 3908–3913. [14] G. Garcia, P. L. D. Peres, and S. Tarbouriech, “Necessary and sufficient numerical conditions for asymptotic stability of linear time-varying systems,” in Proc. 47th IEEE Conf. Decision Contr., Cancun, Mexico, Dec. 2008, pp. 5146–5151. [15] H. K. Khalil, Nonlinear Systems, 3rd ed. Upper Saddle River, NJ: Prentice-Hall, 2002.
A(t) = 0 1 +1 t ; x(t; t0 ; x0 ) = 11++tt0 x0 and thus the system is asymptotically stable but not uniformly asymptotically stable. In fact, Theorem 4 provides
Finite-Time Stability of Linear Time-Varying Systems: Analysis and Controller Design
X (t) = (1(1++tt0))2
Francesco Amato, Marco Ariola, Senior Member, IEEE, and Carlo Cosentino
2
as the solution of (4) and, consequently, limt (t) = limt (1 + t0 )=(1 + t) = 0, indicating that the system is asymptotically stable. On the other hand, applying the results from Theorem 5 one has
!1
Y (t; t0 ) = (1(1++tt0))2
2
!1
t 0 t0 ))
exp(2 (
as the solution of differential (9). Therefore, 8 > 0, we have limt Y (t; t0 ) ! 1 confirming the fact that the system is not uniformly asymptotically stable.
!1
Abstract—The note deals with the finite-time analysis and design problems for continuous-time, time-varying linear systems. Necessary and sufficient conditions and a sufficient condition for finite-time stability are devised. Moreover, sufficient conditions for the solvability of both the state and the output feedback problems are stated. Such results require the feasibility of optimization problems involving Differential Linear Matrix Inequalities. Some numerical examples illustrate the effectiveness of the proposed approach. Index Terms—Dynamic output feedback, finite-time stability, linear time-varying systems.
V. CONCLUSION Necessary and sufficient conditions for asymptotic stability and for uniform asymptotic stability of linear continuous time-varying systems have been given. The conditions are based on the numerical solution of a linear differential Lyapunov equation, associated to a tube in the state space that confines all the trajectories of the system. For periodic systems, it suffices to integrate the differential Lyapunov equation over one single period. Extensions of the proposed conditions to cope with control design problems, particularly in the case of periodic systems, are being investigated by the authors.
REFERENCES [1] H. H. Rosenbrook, “The stability of linear time-dependent control systems,” Int. J. Electr. Control, vol. 15, no. 1, pp. 73–80, Jul. 1963. [2] C. Desoer, “Slowly varying system x_ = A(t)x,” IEEE Trans. Autom. Control, vol. AC-14, no. 6, pp. 780–781, Dec. 1969. [3] G. Kreisselmeier, “An approach to stable indirect adaptive control,” Automatica, vol. 21, no. 4, pp. 425–431, Jul. 1985. [4] J. M. Krause and K. S. P. Kumar, “An alternate stability analysis framework for adaptive control,” Syst. Control Lett., vol. 7, no. 1, pp. 19–24, Feb. 1986.
I. INTRODUCTION The concept of finite-time stability (FTS) dates back to the 1960s, when this idea was introduced in the control literature [1], [2]. A system is said to be finite-time stable if, given a bound on the initial condition, its state does not exceed a certain threshold during a specified time interval. Recently, significant contributions have been given in this field, especially in the case of linear systems (see for instance [3]–[6]). Manuscript received November 14, 2007; revised March 04, 2009, July 05, 2009, and October 16, 2009. First published February 02, 2010; current version published April 02, 2010. Recommended by Associate Editor P. Colaneri. F. Amato and C. Cosentino are with the School of Computer and Biomedical Engineering, Magna Græcia University, Campus Salvatore Venuta, 88100 Catanzaro, Italy (e-mail: [email protected]; [email protected]). M. Ariola is with the Dipartimento per le Tecnologie, Università degli Studi di Napoli Parthenope, Centro Direzionale di Napoli, 80143 Napoli, Italy (e-mail: [email protected]). Color versions of one or more of the figures in this technical note are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2010.2041680
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Fig. 1. Open-loop free evolution from Example 1.
It is important to point out that FTS and Lyapunov asymptotic stability (LAS) are independent concepts; indeed, a system can be FTS but not LAS, and vice versa. While LAS deals with the behavior of a system within a sufficiently long (in principle infinite) time interval, FTS is a more practical concept, useful to study the behavior of the system within a finite (possibly short) interval, and therefore it finds application whenever it is desired that the state variables do not exceed a given threshold (for example to avoid saturations or the excitation of nonlinear dynamics) during the transients. The FTS dealt with in this note should not be confused with the FTS concept adopted in some other papers such as for instance the seminal paper [7], and the papers [8], [9]. In these works, the authors focus on the Lyapunov stability analysis of nonlinear systems whose trajectories converge to an equilibrium point in finite time and on the characterization of the associated settling-time. Thus, it should be remarked that the latter definition of FTS is unrelated to the one given in [1] and [2] and considered in the present note. A sufficient condition for FTS, requiring the solution of a linear matrix inequalities (LMIs)-based feasibility problem [10], has been provided in [3]. Extending the concept of FTS, which applies to open-loop systems, the finite-time stabilization problem concerns the design of a linear controller ensuring the FTS of the closed-loop system. Sufficient conditions for finite-time stabilization have been provided in [3] for the state feedback case. In the recent paper [11] the dynamic output feedback problem has been converted into a LMIs-based optimization problem. In [3] and [11], the authors deal with linear time-invariant systems and make use of an approach based on Lyapunov arguments which may render the technique rather conservative. Moreover, the drawback of the approach proposed in [11] is that the controller design is performed in two phases: the first step is devoted to design a static state feedback controller, then a state observer which tries to retain the properties guaranteed by the state feedback controller is synthesized. Therefore, this approach looks at a subset of the class of admissible dynamic output feedback controllers. Differently from previous works, the approach proposed in this note is reminiscent of optimal control techniques [12] and improves the existing literature in several ways. The first result of the technical note (Theorem 1) is a necessary and sufficient condition for FTS, which involves the computation of the state transition matrix; therefore the application of such condition is straightforward in the time-invariant case. We also provide in Theorem 1 a sufficient condition for FTS which requires the existence of a feasible solution of a certain differential linear matrix inequality (DLMI) [13]; such condition is more suitable for time-varying systems. Indeed, a similar approach has been also ex-
ploited in [6] to derive sufficient FTS analysis conditions for hybrid systems with finite state jumps. An important issue is the evaluation of the degree of conservatism of the sufficient condition, since it is the starting point for the derivation of the design conditions; this analysis is done by comparing the necessary and sufficient condition and the sufficient condition provided in Theorem 1 on 1000 randomly generated linear systems. With respect to the approach of [3], Theorem 1 provided in this note exhibits two advantages: 1) it enables to deal with time-varying systems and 2) even in the time-invariant case, both the necessary and sufficient condition and the sufficient condition for FTS turn out to be less conservative. Afterwards, we consider the output feedback design problem, which is solved making use of the proposed approach, ending up with DLMIs. The controller, differently from [11], is computed in one shot and therefore represents a more general solution. Throughout the note, a nontrivial example concerning the finite-time stabilization of a fourth-order system illustrates the way the proposed technique can be applied. The note is organized as follows: in Section II we give some preliminary definitions and we state the problems that we want to solve. Section III is devoted to the analysis conditions for FTS, whereas in Section IV, we present a sufficient condition for the design of an output feedback controller guaranteeing FTS. II. PROBLEM STATEMENT The following definitions deal with various finite-time control problems. All matrices and vectors are assumed to be of compatible dimensions; all time-varying matrices, unless otherwise specified, are assumed to be bounded and piecewise continuous functions of time. Definition 1 (Finite-Time Stability [3]): Given a positive scalar T , a positive definite matrix R, and a positive definite matrix-valued function 0(1), defined over [0; T ], such that 0(0) < R, the time-varying linear system
x_ (t) = A(t)x(t); x(0) = x0
(1)
where A(1) is a bounded piecewise continuous matrix-valued function, is said to be finite-time stable (FTS) with respect to (T; R; 0(1)), if
xT0 Rx0 1 ) x(t)T 0(t)x(t) < 1
t 2 [0; T ]:
8
(2)
Remark 1: The assumption 0(0) < R guarantees that the closed ellipsoid fx0 : xT0 Rx0 1g is a subset of the open ellipsoid fx0 : xT 0(0)x < 1g; this in turn guarantees well posedness of Definition 1. 0
0
Remark 2: LAS and FTS are independent concepts. Obviously, a system which is FTS wrt some (T; R; 0(1)) may be not LAS; conversely, a LAS system could be not FTS if, during the transients, its state exceeds the prescribed bounds. Example 1: Consider a LTI system in the form (1) with
0:0884 0:8984 01:0667 01:1599
0:4026 0:0276 2:8289 0:1808
0:5733 0:9771 A= : (3) 1:0680 01:2306 0 ), with This system is LAS but is not FTS with respect to (T; R; T = 10, R = I and 0 = diag(1=1:52 ; 1=102 ; 1=102 ; 1=102 ). Such 0
0:5796 0:3622 02:4719 00:0779 0
0
0
statement can be readily verified by means of numerical simulations. x(t), starting from the Consider for example the behavior of xT (t)0 initial condition
x0 = (00:2632 0:7616
0
0:0772 0:5871)T
x(t) exceeds the value depicted in Fig. 1. It is readily seen that x(t)T 0 0 < 1. of 1, whereas xT0 Rx
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Note that FTS refers to open-loop systems; the next problem fits FTS into the design framework. Poblem 1 (Finite-Time Stabilization via Output Feedback): Consider the time-varying linear system
x_ (t) = A(t)x(t) + B (t)u(t) x(0) = x0 y(t) = C (t)x(t)
(4a) (4b)
where u(t) is the control input and y (t) is the output. Then, given a positive scalar T , two positive definite matrices R and RK , and two positive definite symmetric matrix functions 0(1) and 0K (1), defined over [0; T ], such that 0(0) < R, 0K (0) < RK , find a dynamic output feedback controller in the form
x_ c (t) = AK (t)xc (t) + BK (t)y(t) u(t) = CK (t)xc (t) + DK (t)y(t)
(5a) (5b)
where xc (t) has the same dimension of x(t), such that the closed-loop system obtained by the connection of (4) and (5) is FTS with respect to
T; blockdiag(R; RK ); blockdiag (0(1); 0K (1))) :
(
III. ANALYSIS A. Main Results The following theorem gives three necessary and sufficient conditions for FTS of system (1) and one sufficient condition. Theorem 1: The following statements are equivalent. i) System (1) is FTS with respect to (T; R; 0(1)). ii) For all t 2 [0; T ]
t; 0)T 0(t)8(t; 0) < R
8(
where 8(t; 0) is the state transition matrix of system (1). iii) For all t 2 [0; T ] there exist a (sufficiently small) scalar such that the following differential matrix equation with terminal and initial conditions
P ) + A( )T P ( ) + P ( )A( ) + 2 I = 0 2 [0; t] _(
P (t) = 0(t) P (0) < R
(6a)
P (t) 0(t) P (0) < R
(6c)
(7a) (7b) (7c)
admits a piecewise continuously differentiable symmetric solution P (1). Moreover, the following condition is sufficient for FTS. v) The DLMI with terminal and initial conditions
P_ (t) + A(t)T P (t) + P (t)A(t) < 0 t 2 [0; T ] P (t) 0(t); t 2 [0; T ] P (0) < R
Remark 3: Condition ii) may be hard to apply in the time-varying case as it requires the computation of the state transition matrix; in the same way, condition iv) requires to check infinitely many DLMIs (one for each t 2 [0; T ]) and therefore it is not useful from a practical point of view. Conversely, condition v) requires to solve just one DLMI over the whole interval [0; T ] and can be solved by efficient numerical algorithms, however it is only sufficient; its degree of conservatism will be evaluated later, in Section III-B, by comparison with the necessary and sufficient condition ii). Condition v) will be the starting point for the solution of the design problem in the state feedback case. Finally, note that condition iii) is not useful per se, but rather it will turn useful in the proof of the theorem. Remark 4: When system (1) is time-invariant, it makes sense to compare the conditions in Theorem 1 with the sufficient condition for FTS provided in [3]. In this case, given a constant weight matrix 0, condition ii) reads
AT t)0 exp(At) < R;
exp(
(8a) (8b)
x(t)T 0(t)x(t) = xT0 8(t; 0)T 0(t)8(t; 0) x0 < xT0 Rx0 1:
Therefore, system (1) is FTS i) =) ii) By contradiction. Let us assume that for some t, x
T )8(t; 0)x xT Rx: 0) 0(t xT 8(t;
Now let x0 that
=
x, where is such that xT0 Rx0
= 1
(9)
. Then (9) implies
T )8(t; 0)x0 1: 0) 0(t xT0 8(t; Therefore
T )8(t; 0)x0 1 0) 0(t x(t)T 0(t)x(t) = xT0 8(t; which contradicts the initial assumption that system (1) be FTS. i) =) iii) Let us assume that system (1) is FTS. Then by continuity arguments, letting z (t) = x(t) for a sufficiently small , for all
t
;T]
2 [0
xT0 Rx0
1 )
2 x(t)T 0(t)x(t) + kz k[0 ;t] < 1
(10)
Let P (1) be the unique ([14, pp. 470-471]) solution of
P_ ( ) + A( )T P ( ) + P ( )A( ) = 0 2 I; P (t) = 0(t)
; t]
2 [0
(11a) (11b)
and assume, by contradiction, that for some x xT P (0)x xT Rx:
(8c)
admits a piecewise continuously differentiable symmetric solution P (1). Before proving the Theorem, some comments are in order.
t 2 [0; T ]
which is obviously less conservative than the condition provided in [3], which is only sufficient, while ii) is also necessary. Concerning condition v), roughly speaking it can be seen as a generalization of the main result in [3], since it optimizes over the set of the piecewise continuously differentiable matrix valued functions P (1)’s rather than looking to constant P ’s; therefore the approach proposed in this work is clearly less conservative than the methodology proposed in [3]. Again, a quantitative comparison between the two conditions is done in Section III-B. Proof: ii) =) i) Let xT0 Rx0 1. Then
(6b)
admits a piecewise continuously differentiable symmetric solution P (1). iv) For all t 2 [0; T ], the DLMI with terminal and initial conditions
P_ ( ) + A( )T P ( ) + P ( )A( ) < 0 2 [0; t]
1005
Now let x0
=
x, where is such that xT0 Rx0
(12) = 1
. Then (12) implies
xT0 P (0)x0 1:
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(13)
1006
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From (11a), we obtain that
d d
T
x( ) P ( )x( )
= 02 x( )T x( ):
(14)
Integrating (14) from 0 to t, we have T
x(t) P (t)x(t)
0 xT0 P (0)x0 = 02 kxk[02
;t]
:
Therefore, from (11b) and (13) x(t)
T
0(t)x(t) = x(t)T P (t)x(t) = xT0 P (0)x0 0 2 kxk2[0;t] 1 0 kzk[02 ;t]
which contradicts (10) iii) =) iv ) It is straightforward to recognize that every solution P (1) to (6) is also solution to (7). iv ) =) i) Let t 2 [0; T ], 2 [0; t], and define V ( ; x) =
xT P ( )x. Then (7a) implies that V_ ( ; x) is negative definite along the trajectories of system (7a). Now let xT0 Rx0 1; then we have
x(t)
v)
T
0(t)x(t) (t)T P (t)x(t) in view of (7b) xT0 P (0)x0 since V ( ; x) is decreasing in [0; t] T < x0 Rx0 in view of (7c) 1:
=) iv) It is straightforward to check that a matrix function
1) satisfying conditions (8) also satisfies (7).
P(
one can reduce the conservatism at will, at the expense of a greater computational burden. By imposing a piecewise affine structure on P (1), condition (8) can be cast as an LMIs feasibility problem, with the initial and final values of P (1) in each subinterval being the optimization variables; such problem has been implemented and solved through the LMI control toolbox [15]. The percentage difference between the exact value of max , obtained by (15), and its estimates obtained via condition v) of Theorem 1 and [3, Lemma 6], has been computed for the 1000 randomly generated systems. The average “distance” from necessity is about 9.6% when applying the former condition and about 44.5% for the latter one. We can conclude that, at least for low order systems, condition v) of Theorem 1 introduces a slight conservatism in the analysis; at the same time there is a significant improvement with respect to the sufficient condition derived from [3]. IV. CONTROLLER DESIGN For the sake of brevity, only the sufficient (operative) condition for the existence of a controller will be given below; by following the same arguments of the proof of Theorem 1 it is possible to state the corresponding necessary and sufficient condition for the controller existence, which however requires the solution of infinitely many differential inequalities and therefore is not useful from a practical point of view (see also Remark 3). The following is the main result of the section. Theorem 2: Problem 1 is solvable if there exist piecewise continuously differentiable symmetric matrix-valued functions Q(1), S (1), a nonsingular matrix-valued function N (1) and matrix-valued func^K (1), C^K (1) and DK (1) such that (the time argument is tions A^K (1), B omitted for brevity)
B. Evaluation of the Sufficient Condition in Theorem 1 The goal of this section is, on one hand, to quantify the degree of conservatism introduced by condition v) with respect to the necessary and sufficient condition ii), and from the other hand to compare condition v) with the FTS sufficient condition derived from [3] when LTI systems are considered. To this end 1000 random LTI systems (500 of the second order and 500 of the third order) have been generated, in the form (1); condition ii) of Theorem 1 has been applied to each system to determine the largest
, say max , such that the system is FTS wrt (2; I; max I ); this is equivalent to computing, for each system
max
:=
1
maxt2[0;2] kexp(At)k2
:
Q
T 912 T 913 T 914
