Stability analysis and stabilization of systems presenting nested ...
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 8, AUGUST 2006
Stability Analysis and Stabilization of Systems Presenting Nested Saturations S. Tarbouriech, C. Prieur, and J. M. Gomes da Silva, Jr.
Abstract—This note addresses the problems of stability analysis and stabilization of systems presenting nested saturations. Depending on the open-loop stability assumption, the global stability analysis and stabilization problems are considered. In the (local) analysis problem, the objective is the determination of estimates of the basin of attraction of the system. Considering the stabilization problem, the goal is to design a set of gains in order to enlarge the basin of attraction of the closed-loop system. Based on the modelling of the system presenting nested saturations as a linear system with dead-zone nested nonlinearities and the use of a generalized sector condition, linear matrix inequality (LMI) stability conditions are formulated. From these conditions, convex optimization strategies are proposed to solve both problems. Index Terms—Linear matrix inequality (LMI), nested saturations, stability regions, stabilization.
I. INTRODUCTION The stability and stabilization of systems presenting saturations are problems that have been studied by many authors with different objectives in the last decade. This interest comes mainly from the fact that, due to physical, safety or technological constraints, control actuators cannot provide unlimited amplitude signals neither unlimited speed of reaction. This means that control systems are in general subject to amplitude and dynamics actuator saturations. In this case, the negligence of both amplitude and dynamics actuator limitations in control systems can be source of undesirable and even catastrophic behaviors, such as instability [3]. In this context, an important class of systems to be studied consists in systems presenting nested saturations. In particular, such structure appears when we deal with nonlinear actuators and sensors. For instance, it is common in aerospace control systems (e.g., launcher and aircraft control) that actuators are both limited in amplitude and rate (dynamics) (see, for example, [1], [8], and [10]). Furthermore, the presence of both sensor and actuator amplitude limitations can lead to a closed-loop system presenting nested saturations. This will be the case, for instance, in linear systems controlled by dynamic output feedback controllers in the presence saturating sensors and actuators. On the other hand, analysis and design methodologies for systems presenting nested saturations can be useful to address stability issues of more general classes of nonlinear systems. For instance, the use of nested saturations becomes very interesting when one uses forwarding techniques for cascade systems with linear part [15], [17], [18]. In this note, two complementary problems are addressed: The determination of stability regions and the design of stabilizing gains for systems presenting nested saturations. The objective in the analysis stability problem consists in determining an estimate of the basin of attraction of the system presenting generic nested saturations. Regarding the
Manuscript received June 2, 2004; revised October 10, 2005 and April 11, 2006. Recommended by Associate Editor M. Kothare. The work of J. M. Gomes da Silva, Jr. was supported by CNPq, Brazil. S. Tarbouriech and C. Prieur are with LAAS-CNRS, 31077 Toulouse cedex 4, France (e-mail: tarbour,[email protected]). J. M. Gomes da Silva Jr. is with the Department of Electrical Engineering, UFRGS, 90035-190 Porto Alegre-RS, Brazil (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2006.878743
stabilization problem, the objective is then to design a set of gains in order to maximize the basin of attraction of the closed-loop saturated system. Based on the modelling of the system presenting generic nested saturations as a linear system with dead-zone nested nonlinearities and the use of a generalized sector condition, linear matrix inequality (LMI) stability conditions are formulated. Although the objectives of the note are similar to the ones recently proposed in [1], considering the modelling of the saturated systems by a polytopic differential inclusion, it should be pointed out that we do not consider particular assumptions on the structure and the dimensions of matrices involved in the description of the system. Furthermore, our results allow also to address global stability issues, which is not considered in [1]. This note provides therefore an alternative solution for the problems formulated and addressed in [1] and, doing so, provides also some generalizations. The note is organized as follows. Section II describes the system under consideration and the problems we intend to address. In Section III, theoretical conditions for solving both the analysis of stability as well as the control design problems are presented. Section IV deals with convex optimization issues. A discussion concerning the numerical complexity with respect to our results and those proposed in the literature is quickly presented in the end of this section. Some illustrative examples are presented in Section V. Section VI summarizes the main contributions of the work and points to some open problems and future research directions. Notations: For any vector x 2 0. Thus, once v and w are elements of S (v0 ), we can conclude that '(v(i) )T(i;i) ('(v(i) ) + w(i) ) 0, 8T(i;i) > 0, 8i = 1; . . . ; m, whence follows (6). Remark 1: Particular formulations of Lemma 1 can be found in [7] (concerning the case of systems with a single saturation function, i.e., p = 1) and in [16] (concerning systems presenting both amplitude and dynamics restricted actuators). It should be pointed out that sector condition (6) is more generic than the classical one (see, for instance, [9] and [11]) given as (v )
0 T [ (v) + v] 0;
0
< 1:
(7)
Note that in our case we can consider w 6= v . This fact, differently from condition (7), allows to formulate conditions directly in LMI form. The following proposition provides theoretical sufficient conditions to solve Problem 1. Proposition 1: If there exist a symmetric positive definite matrix W , matrices Zjj , j = 1; . . . ; p, Yjl , j = 2; . . . ; p, l = 1; . . . ; p 0 1, j 6= l,
1 (x) = sat1 (Cx) 0 Cx 2 (x) = sat2 ((A1 + B1 C )x + B1 1 (x)) 0 [(A1 + B1 C )x + B1 1 (x)] 3 (x) = sat3 ((A2 + B2 (A1 + B1 C )) x + B2 2 (x) + B2 B1 1 (x)) 0 [(A2 + B2 (A1 + B1 C )) x + B2 2 (x) + B2 B1 1 (x)] .. .
p (x) = satp ((Ap01 + Bp01 (Ap02 + Bp02 (Ap03 + 1 1 1 + B2 (A1 + B1 C )))) x +Bp01 p01 (x) + Bp01 Bp02 p02 (x) + 1 1 1 + Bp01 Bp02 1 1 1 B1 1 (x)) 0 [(Ap01 + Bp01 (Ap02 + Bp02 (Ap03 + 1 1 1 + B2 (A1 + B1 C )))) x +Bp01 p01 (x) + Bp01 Bp02 p02 (x) + 1 1 1 + Bp01 Bp02 1 1 1 B1 1 (x)] Authorized licensed use limited to: LAAS. Downloaded on June 14,2010 at 09:32:17 UTC from IEEE Xplore. Restrictions apply.
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 8, AUGUST 2006
j > l, and diagonal positive matrices Sj , j = 1; . . . ; p, of appropriate dimensions satisfying1
W 0p + p W 0 S1 B1 1 1 1 Bp0 01 Bp0 0 Z11 S20 B20 1 1 1 Bp0 01 Bp0 0 Z22 .. .
Sp0 Bp0 0 Zpp
? ? 111 ? 02S ? 111 ? 0 Y 0 2S 1 1 1 ? .. .. . . 111 ? 0 Y 0 Y 1 1 1 0 2S 1
21
p1
0 W W C(0i) 0 Z11( i) 2 ? u1(i)
0;
< 0 (8)
2
p2
that the set E (W 01 ; 1), with P = W 01 , is included in S (u1 ) [2]. We prove now, by induction, that relations (9) and (10) imply that set E (W 01 ; 1) is included in \jp=1 S (uj ). First note that, from Lemma 1, if E (W 01 ; 1) S (u1 ), it follows that 10 T1 ( 1 + E11 x) 0, 8x 2 E (W 01 ; 1). Hence, provided that E (W 01 ; 1) S (u1 ), E (W 01 ; 1) will be contained in S (u1 ) \ S (u2 ) if
x 0 1
p
i = 1; . . . ; m1
P 0 0 0
0 u1
2 2(i)
0 01( i) 0 21( i)
01(i) 21(i)
x 0 1 8i = 1 ; . . . ; m 1 8x; 1 such that 2 10 T1 ( 1 + E11 x) 0
2
(9)
and (10), as shown at the bottom of the page. Then, the set E (W 01 ; 1) = fx 2 l. The satisfaction of relation (9) implies then 1The
8i = 1 ; . . . ; m : 1
T = T1 ; v = Cx; w = E11 x; v0 = u1 : • Second, in the case ' with
(11)
0;
i = 1; . . . ; mj ; j = 2; . . . ; p:
Sj 01 Bj0 01(i) 0 Yjj0 01(i) uj2(i)
Authorized licensed use limited to: LAAS. Downloaded on June 14,2010 at 09:32:17 UTC from IEEE Xplore. Restrictions apply.
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 8, AUGUST 2006
E (W 00 ; 1), the satisfaction of relations (9) and (10) implies that set E (W ; 1) is included in \ S (u ). Hence, the nonlinearities , 1
1
p j =1
j
1367
2E
0 P + P p x + 2 x0 P B p p p 0 0 +2x P Bp Bp01 p01 + 1 1 1 + 2x P Bp Bp01 1 1 1 B1 1 :
1
1
1
+ E11 x)
0 2 0 T ( 2
2
2
+ E22 x + E21 1 )
0111
02 0 Tp ( p + Epp x + Epp01 p01 + Epp02 p02 + . . . + Ep1 1 ): p
0
Considering = [x0 10 20 . . . p0 ] , the previous inequality can be rewritten as V_ 0 M with the equation shown at the bottom of the page. By recalling that W = P 01 , Sj = Tj01 , j = 1; . . . ; p, and by pre- and postmultiplying the matrix aforementioned M defined by W ? ?
.. .
?
0
S1 ?
.. .
0 0
S2
.. .
0 0 0 .. .
0 0 0 , it follows that, if relation (8) is satisfied, .. .
? ? ? Sp 01 ; 1), x = 0. Hence, one can conclude (W one has V_ (x) < 0, x that (W 01 ; 1) is a contractive set along the trajectories of system (1)
E
m
m
such that
8 2E
6
and, thus, it is a region where the asymptotic stability of system (1) is ensured, which concludes the proof. B. Global Asymptotic Stability Analysis Proposition 1 presents a local stability condition for the nested saturated system (1). On the other hand, in the case where Ap is Hurwitz the global asymptotic stability of the system can be checked. In this case, following the results based on integral quadratic constraints (IQCs) stated in [4], [12] for systems presenting repeated saturations,3 different nondiagonal multipliers T can be used as follows. Lemma 2: The nonlinearity '(v ) = satv (v ) 0 v satisfies the following inequality:
0
'(v ) T ('(v ) + v )
0
(14)
3By “repeated” saturations the authors in that papers mean the case where (sat(v )) = sat(v ) = sign((v ) min(v ; jv j) with v =u ; 8i = 1; . . . ; m appear in an additive way. It should be highlighted that these “repeated” saturations do not appear “nested” as in our case.
M=
0p P + P p B10 1 1 1 Bp0 01 Bp0 P 0 T1 E11 B20 1 1 1 Bp0 01 Bp0 P 0 T2 E22 B30 1 1 1 Bp0 01 Bp0 P 0 T3 E33 Bp0 P
.. .
0T
p
Epp
6
v0(j ) T(i;j ) ;
i = 1; . . . ; m:
(15)
Proof: Recalling that '(v ) = ['(v(1) ) . . . '(v(m) )]0 it follows that m
0
' v(i)
' (v ) T (' (v ) + v ) =
T(i;i) ' v(i) + v(i)
i=1
Since (9) and (10) are satisfied, sector conditions (6) hold 8 j , j = 8x 2 E (W 01 ; 1). Hence, 8x 2 E (W 01 ; 1) it follows that
V_ 0 2 0 T (
2< 2
j =1;i=j
m
+
1; . . . ; p ,
V_
and any matrix T
m
T(i;i) v0(i)
system (4) reads
0
m
j
j = 1; . . . ; p, associated to the appropriate v and w defined previously, 01 ; 1). (W satisfy sector conditions in the form of (6) for all x Consider now the quadratic Lyapunov function V (x) = x0 P x, with P = P 0 > 0. The time-derivative of V (x) along the trajectories of
V_ = x
2
0 In this case, '(v(i) ) + v(i) = 0v0(i) . Hence if (15) is satisfied one has m
T(i;j ) ('(v(j ) ) + [T(i;i) ('(v(i) ) + v(i) ) + i6=j;j =1 m T(i;j ) 0 v v(j ) )] v0(i) T(i;i) + i6=j;j =1 0(j ) and, therefore '(v(i) )[T(i;i) ('(v(i) ) + v(i) ) + m T(i;j ) ('(v(j ) )+v(j ) )] 0. i6=j;j =1 < 0 In this case, '(v(i) ) + • Case 2: '(v(i) ) v(i) = v0(i) . Hence if (15) is satisfied one has m T(i;j ) ('(v(j ) ) + [T(i;i) ('(v(i) ) + v(i) ) + i6=j;j =1 m v0(i) T(i;i) v T 0 v(j ) )] 0( j ) (i;j ) i6=j;j =1 and, therefore '(v(i) )[T(i;i) ('(v(i) ) + v(i) ) + m T(i;j ) ('(v(j ) )+v(j ) )] 0. i6=j;j =1 • Case 3: '(v(i) ) = 0 One has '(v(i) )[T(i;i) ('(v(i) ) + v(i) ) + m T(i;j ) ('(v(j ) )+v(j ) )] = 0. i6=j;j =1
j
0
j
j
0
j
Thus, from the three previous cases and from (16), we can conclude that '(v )0 T ('(v ) + v ) 0. Hence, by using Lemma 2, a condition for the global asymptotic stability can be stated. Proposition 2: If there exist a matrix P = P 0 > 0 and matrices Tk = Tk0 > 0, k = 1; . . . ; p, of appropriate dimensions, satisfying (17) and (18), as shown at the bottom of the next page. Then, the origin of system (1) is globally asymptotically stable. Proof: It follows the same steps of Proposition 1, considering w = v . In this case, we should consider E11 = C , E22 = A1 + B1 C; . . . ; Epp = (Ap01 + Bp01 (Ap02 +Bp02 (Ap03 + + B2 (A1 + B1 C )))) = = B1 , Epp01 = p01 , E21 Bp01 ; . . . ; Ep1 = Bp01 Bp02 . . . B2 B1 . It follows that the sector conditions (14), applied p-times to the nonlinearities defined in (2) are n . satisfied x Remark 2: If the control bounds are normalized to v0(i) = 1; i = m T(i;j ) , which cor1; . . . ; m, condition (15) reads T(i;i) j =1;j 6=i
111
8 2
l , Gk , k = 0; . . . ; p 0 1, and diagonal positive matrices Sj , j = 1; . . . ; p, of appropriate dimensions satisfying
0 L+ L 0 0 0 S1 B1 . . . Bp01 Bp 0 Z11 0 0 0 S2 B2 . . . Bp01 Bp 0 Z22 0
Sp Bp
?
02 0 0
Zpp
0
W
0 0
G0(i)
?
2
02
.. .
0 0
p
S2
Z11(i)
;
02
i
0
(20)
= 1; . . . ; m1
?
.. .
.. .
uj (1) Sj (1;1) > uj (2)
Sj (1;2)
uj (2) Sj (2;2) > uj (1)
Sj (1;2)
01 . . . B2 0 (A1 + B1 C )0 T2 0B10 T2
P B p Bp
02
?
... ... ...
T2
.. .
.. .
?
?
m
uk(j )
6
Tk(i;j )
;
=
uj (1)
Sj (2;1)
:
Only in this particular case it is therefore possible to find matrices Tj = 01 , j = 1; . . . ; p, satisfying (17) and (18). Sj
(21)
T1
?
?
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 8, AUGUST 2006
Stability Analysis and Stabilization of Systems Presenting Nested Saturations S. Tarbouriech, C. Prieur, and J. M. Gomes da Silva, Jr.
Abstract—This note addresses the problems of stability analysis and stabilization of systems presenting nested saturations. Depending on the open-loop stability assumption, the global stability analysis and stabilization problems are considered. In the (local) analysis problem, the objective is the determination of estimates of the basin of attraction of the system. Considering the stabilization problem, the goal is to design a set of gains in order to enlarge the basin of attraction of the closed-loop system. Based on the modelling of the system presenting nested saturations as a linear system with dead-zone nested nonlinearities and the use of a generalized sector condition, linear matrix inequality (LMI) stability conditions are formulated. From these conditions, convex optimization strategies are proposed to solve both problems. Index Terms—Linear matrix inequality (LMI), nested saturations, stability regions, stabilization.
I. INTRODUCTION The stability and stabilization of systems presenting saturations are problems that have been studied by many authors with different objectives in the last decade. This interest comes mainly from the fact that, due to physical, safety or technological constraints, control actuators cannot provide unlimited amplitude signals neither unlimited speed of reaction. This means that control systems are in general subject to amplitude and dynamics actuator saturations. In this case, the negligence of both amplitude and dynamics actuator limitations in control systems can be source of undesirable and even catastrophic behaviors, such as instability [3]. In this context, an important class of systems to be studied consists in systems presenting nested saturations. In particular, such structure appears when we deal with nonlinear actuators and sensors. For instance, it is common in aerospace control systems (e.g., launcher and aircraft control) that actuators are both limited in amplitude and rate (dynamics) (see, for example, [1], [8], and [10]). Furthermore, the presence of both sensor and actuator amplitude limitations can lead to a closed-loop system presenting nested saturations. This will be the case, for instance, in linear systems controlled by dynamic output feedback controllers in the presence saturating sensors and actuators. On the other hand, analysis and design methodologies for systems presenting nested saturations can be useful to address stability issues of more general classes of nonlinear systems. For instance, the use of nested saturations becomes very interesting when one uses forwarding techniques for cascade systems with linear part [15], [17], [18]. In this note, two complementary problems are addressed: The determination of stability regions and the design of stabilizing gains for systems presenting nested saturations. The objective in the analysis stability problem consists in determining an estimate of the basin of attraction of the system presenting generic nested saturations. Regarding the
Manuscript received June 2, 2004; revised October 10, 2005 and April 11, 2006. Recommended by Associate Editor M. Kothare. The work of J. M. Gomes da Silva, Jr. was supported by CNPq, Brazil. S. Tarbouriech and C. Prieur are with LAAS-CNRS, 31077 Toulouse cedex 4, France (e-mail: tarbour,[email protected]). J. M. Gomes da Silva Jr. is with the Department of Electrical Engineering, UFRGS, 90035-190 Porto Alegre-RS, Brazil (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2006.878743
stabilization problem, the objective is then to design a set of gains in order to maximize the basin of attraction of the closed-loop saturated system. Based on the modelling of the system presenting generic nested saturations as a linear system with dead-zone nested nonlinearities and the use of a generalized sector condition, linear matrix inequality (LMI) stability conditions are formulated. Although the objectives of the note are similar to the ones recently proposed in [1], considering the modelling of the saturated systems by a polytopic differential inclusion, it should be pointed out that we do not consider particular assumptions on the structure and the dimensions of matrices involved in the description of the system. Furthermore, our results allow also to address global stability issues, which is not considered in [1]. This note provides therefore an alternative solution for the problems formulated and addressed in [1] and, doing so, provides also some generalizations. The note is organized as follows. Section II describes the system under consideration and the problems we intend to address. In Section III, theoretical conditions for solving both the analysis of stability as well as the control design problems are presented. Section IV deals with convex optimization issues. A discussion concerning the numerical complexity with respect to our results and those proposed in the literature is quickly presented in the end of this section. Some illustrative examples are presented in Section V. Section VI summarizes the main contributions of the work and points to some open problems and future research directions. Notations: For any vector x 2 0. Thus, once v and w are elements of S (v0 ), we can conclude that '(v(i) )T(i;i) ('(v(i) ) + w(i) ) 0, 8T(i;i) > 0, 8i = 1; . . . ; m, whence follows (6). Remark 1: Particular formulations of Lemma 1 can be found in [7] (concerning the case of systems with a single saturation function, i.e., p = 1) and in [16] (concerning systems presenting both amplitude and dynamics restricted actuators). It should be pointed out that sector condition (6) is more generic than the classical one (see, for instance, [9] and [11]) given as (v )
0 T [ (v) + v] 0;
0
< 1:
(7)
Note that in our case we can consider w 6= v . This fact, differently from condition (7), allows to formulate conditions directly in LMI form. The following proposition provides theoretical sufficient conditions to solve Problem 1. Proposition 1: If there exist a symmetric positive definite matrix W , matrices Zjj , j = 1; . . . ; p, Yjl , j = 2; . . . ; p, l = 1; . . . ; p 0 1, j 6= l,
1 (x) = sat1 (Cx) 0 Cx 2 (x) = sat2 ((A1 + B1 C )x + B1 1 (x)) 0 [(A1 + B1 C )x + B1 1 (x)] 3 (x) = sat3 ((A2 + B2 (A1 + B1 C )) x + B2 2 (x) + B2 B1 1 (x)) 0 [(A2 + B2 (A1 + B1 C )) x + B2 2 (x) + B2 B1 1 (x)] .. .
p (x) = satp ((Ap01 + Bp01 (Ap02 + Bp02 (Ap03 + 1 1 1 + B2 (A1 + B1 C )))) x +Bp01 p01 (x) + Bp01 Bp02 p02 (x) + 1 1 1 + Bp01 Bp02 1 1 1 B1 1 (x)) 0 [(Ap01 + Bp01 (Ap02 + Bp02 (Ap03 + 1 1 1 + B2 (A1 + B1 C )))) x +Bp01 p01 (x) + Bp01 Bp02 p02 (x) + 1 1 1 + Bp01 Bp02 1 1 1 B1 1 (x)] Authorized licensed use limited to: LAAS. Downloaded on June 14,2010 at 09:32:17 UTC from IEEE Xplore. Restrictions apply.
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j > l, and diagonal positive matrices Sj , j = 1; . . . ; p, of appropriate dimensions satisfying1
W 0p + p W 0 S1 B1 1 1 1 Bp0 01 Bp0 0 Z11 S20 B20 1 1 1 Bp0 01 Bp0 0 Z22 .. .
Sp0 Bp0 0 Zpp
? ? 111 ? 02S ? 111 ? 0 Y 0 2S 1 1 1 ? .. .. . . 111 ? 0 Y 0 Y 1 1 1 0 2S 1
21
p1
0 W W C(0i) 0 Z11( i) 2 ? u1(i)
0;
< 0 (8)
2
p2
that the set E (W 01 ; 1), with P = W 01 , is included in S (u1 ) [2]. We prove now, by induction, that relations (9) and (10) imply that set E (W 01 ; 1) is included in \jp=1 S (uj ). First note that, from Lemma 1, if E (W 01 ; 1) S (u1 ), it follows that 10 T1 ( 1 + E11 x) 0, 8x 2 E (W 01 ; 1). Hence, provided that E (W 01 ; 1) S (u1 ), E (W 01 ; 1) will be contained in S (u1 ) \ S (u2 ) if
x 0 1
p
i = 1; . . . ; m1
P 0 0 0
0 u1
2 2(i)
0 01( i) 0 21( i)
01(i) 21(i)
x 0 1 8i = 1 ; . . . ; m 1 8x; 1 such that 2 10 T1 ( 1 + E11 x) 0
2
(9)
and (10), as shown at the bottom of the page. Then, the set E (W 01 ; 1) = fx 2 l. The satisfaction of relation (9) implies then 1The
8i = 1 ; . . . ; m : 1
T = T1 ; v = Cx; w = E11 x; v0 = u1 : • Second, in the case ' with
(11)
0;
i = 1; . . . ; mj ; j = 2; . . . ; p:
Sj 01 Bj0 01(i) 0 Yjj0 01(i) uj2(i)
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E (W 00 ; 1), the satisfaction of relations (9) and (10) implies that set E (W ; 1) is included in \ S (u ). Hence, the nonlinearities , 1
1
p j =1
j
1367
2E
0 P + P p x + 2 x0 P B p p p 0 0 +2x P Bp Bp01 p01 + 1 1 1 + 2x P Bp Bp01 1 1 1 B1 1 :
1
1
1
+ E11 x)
0 2 0 T ( 2
2
2
+ E22 x + E21 1 )
0111
02 0 Tp ( p + Epp x + Epp01 p01 + Epp02 p02 + . . . + Ep1 1 ): p
0
Considering = [x0 10 20 . . . p0 ] , the previous inequality can be rewritten as V_ 0 M with the equation shown at the bottom of the page. By recalling that W = P 01 , Sj = Tj01 , j = 1; . . . ; p, and by pre- and postmultiplying the matrix aforementioned M defined by W ? ?
.. .
?
0
S1 ?
.. .
0 0
S2
.. .
0 0 0 .. .
0 0 0 , it follows that, if relation (8) is satisfied, .. .
? ? ? Sp 01 ; 1), x = 0. Hence, one can conclude (W one has V_ (x) < 0, x that (W 01 ; 1) is a contractive set along the trajectories of system (1)
E
m
m
such that
8 2E
6
and, thus, it is a region where the asymptotic stability of system (1) is ensured, which concludes the proof. B. Global Asymptotic Stability Analysis Proposition 1 presents a local stability condition for the nested saturated system (1). On the other hand, in the case where Ap is Hurwitz the global asymptotic stability of the system can be checked. In this case, following the results based on integral quadratic constraints (IQCs) stated in [4], [12] for systems presenting repeated saturations,3 different nondiagonal multipliers T can be used as follows. Lemma 2: The nonlinearity '(v ) = satv (v ) 0 v satisfies the following inequality:
0
'(v ) T ('(v ) + v )
0
(14)
3By “repeated” saturations the authors in that papers mean the case where (sat(v )) = sat(v ) = sign((v ) min(v ; jv j) with v =u ; 8i = 1; . . . ; m appear in an additive way. It should be highlighted that these “repeated” saturations do not appear “nested” as in our case.
M=
0p P + P p B10 1 1 1 Bp0 01 Bp0 P 0 T1 E11 B20 1 1 1 Bp0 01 Bp0 P 0 T2 E22 B30 1 1 1 Bp0 01 Bp0 P 0 T3 E33 Bp0 P
.. .
0T
p
Epp
6
v0(j ) T(i;j ) ;
i = 1; . . . ; m:
(15)
Proof: Recalling that '(v ) = ['(v(1) ) . . . '(v(m) )]0 it follows that m
0
' v(i)
' (v ) T (' (v ) + v ) =
T(i;i) ' v(i) + v(i)
i=1
Since (9) and (10) are satisfied, sector conditions (6) hold 8 j , j = 8x 2 E (W 01 ; 1). Hence, 8x 2 E (W 01 ; 1) it follows that
V_ 0 2 0 T (
2< 2
j =1;i=j
m
+
1; . . . ; p ,
V_
and any matrix T
m
T(i;i) v0(i)
system (4) reads
0
m
j
j = 1; . . . ; p, associated to the appropriate v and w defined previously, 01 ; 1). (W satisfy sector conditions in the form of (6) for all x Consider now the quadratic Lyapunov function V (x) = x0 P x, with P = P 0 > 0. The time-derivative of V (x) along the trajectories of
V_ = x
2
0 In this case, '(v(i) ) + v(i) = 0v0(i) . Hence if (15) is satisfied one has m
T(i;j ) ('(v(j ) ) + [T(i;i) ('(v(i) ) + v(i) ) + i6=j;j =1 m T(i;j ) 0 v v(j ) )] v0(i) T(i;i) + i6=j;j =1 0(j ) and, therefore '(v(i) )[T(i;i) ('(v(i) ) + v(i) ) + m T(i;j ) ('(v(j ) )+v(j ) )] 0. i6=j;j =1 < 0 In this case, '(v(i) ) + • Case 2: '(v(i) ) v(i) = v0(i) . Hence if (15) is satisfied one has m T(i;j ) ('(v(j ) ) + [T(i;i) ('(v(i) ) + v(i) ) + i6=j;j =1 m v0(i) T(i;i) v T 0 v(j ) )] 0( j ) (i;j ) i6=j;j =1 and, therefore '(v(i) )[T(i;i) ('(v(i) ) + v(i) ) + m T(i;j ) ('(v(j ) )+v(j ) )] 0. i6=j;j =1 • Case 3: '(v(i) ) = 0 One has '(v(i) )[T(i;i) ('(v(i) ) + v(i) ) + m T(i;j ) ('(v(j ) )+v(j ) )] = 0. i6=j;j =1
j
0
j
j
0
j
Thus, from the three previous cases and from (16), we can conclude that '(v )0 T ('(v ) + v ) 0. Hence, by using Lemma 2, a condition for the global asymptotic stability can be stated. Proposition 2: If there exist a matrix P = P 0 > 0 and matrices Tk = Tk0 > 0, k = 1; . . . ; p, of appropriate dimensions, satisfying (17) and (18), as shown at the bottom of the next page. Then, the origin of system (1) is globally asymptotically stable. Proof: It follows the same steps of Proposition 1, considering w = v . In this case, we should consider E11 = C , E22 = A1 + B1 C; . . . ; Epp = (Ap01 + Bp01 (Ap02 +Bp02 (Ap03 + + B2 (A1 + B1 C )))) = = B1 , Epp01 = p01 , E21 Bp01 ; . . . ; Ep1 = Bp01 Bp02 . . . B2 B1 . It follows that the sector conditions (14), applied p-times to the nonlinearities defined in (2) are n . satisfied x Remark 2: If the control bounds are normalized to v0(i) = 1; i = m T(i;j ) , which cor1; . . . ; m, condition (15) reads T(i;i) j =1;j 6=i
111
8 2
l , Gk , k = 0; . . . ; p 0 1, and diagonal positive matrices Sj , j = 1; . . . ; p, of appropriate dimensions satisfying
0 L+ L 0 0 0 S1 B1 . . . Bp01 Bp 0 Z11 0 0 0 S2 B2 . . . Bp01 Bp 0 Z22 0
Sp Bp
?
02 0 0
Zpp
0
W
0 0
G0(i)
?
2
02
.. .
0 0
p
S2
Z11(i)
;
02
i
0
(20)
= 1; . . . ; m1
?
.. .
.. .
uj (1) Sj (1;1) > uj (2)
Sj (1;2)
uj (2) Sj (2;2) > uj (1)
Sj (1;2)
01 . . . B2 0 (A1 + B1 C )0 T2 0B10 T2
P B p Bp
02
?
... ... ...
T2
.. .
.. .
?
?
m
uk(j )
6
Tk(i;j )
;
=
uj (1)
Sj (2;1)
:
Only in this particular case it is therefore possible to find matrices Tj = 01 , j = 1; . . . ; p, satisfying (17) and (18). Sj
(21)
T1
?
?
