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Constrained output feedbacks for singularly perturbed imperfectly known nonlinear systems
Journal of the Franklin Institute 336 (1999) 449 — 472
Constrained output feedbacks for singularly perturbed imperfectly known nonlinear systems H.S. Binning, D.P. Goodall* Control Theory and Applications Centre, Coventry University, Priory Street, Coventry CV1 5FB, UK Received 9 March 1998; accepted 21 July 1998
Abstract Global attractors are investigated for a class of imperfectly known, singularly perturbed, nonlinear systems subject to control constraints. The uncertain systems are modelled as nonlinear perturbations to a known nonlinear idealized system. The model is represented by two time-scale systems involving a scalar singular perturbation parameter, which reduces to a system of lower order when the singular perturbation parameter is set to zero. A class of constrained static output feedback controllers is developed which guarantees global attractivity of a compact set, containing the state origin, for all values of the singular perturbation parameter less than some threshold value. 1999 The Franklin Institute. Published by Elsevier Science Ltd. Keywords: Control constraints; Global uniform attractor; Output feedback controls; Singularly perturbed systems; Uncertain systems
1. Introduction Often, in many systems, problems due to the coexistence of slow and fast dynamics in the plant to be controlled can arise. These problems can be addressed utilizing singular perturbation theory (for more details see [1] or, for a differential-geometric approach, see [2, 3]). For dynamical systems with the above behaviour, the plant may be modelled as a set of n differential equations, representing the slow subsystem, and
* Corresponding author. Tel.:#44 (0) 1203 838730; Fax:#44 (0) 1203 838080; E-mail: D. Goodall@ coventry.ac.uk 0016-0032/99/$ — See front matter 1999 The Franklin Institute. Published by Elsevier Science Ltd. PII: S 00 1 6-0 0 32 ( 9 8) 0 0 03 8 - 6
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l differential equations modelling the fast dynamics which are dependent upon a small positive real (perturbation) parameter. Loosely speaking, the singular perturbation parameter can be interpreted as some type of estimate of the ‘‘ratio’’ between the slow and fast dynamics. It is assumed that the set of n#l differential equations degenerates to the set of n differential equations, known as the reduced-order system, when the perturbation parameter is set to zero. Many stabilizing controllers for singularly perturbed systems have been constructed utilizing Lyapunov’s second method (see e.g., [4]). For singularly perturbed systems with uncertainty, design of deterministic stabilizing controls have been investigated using a Lyapunov-based approach in [5—13] to name but a few. The main objective of this paper is to design, using a deterministic approach, a class of robust output feedback controls for singularly perturbed uncertain systems, subject to control constraints, in order to achieve a stability property relating to existence of compact sets, containing the origin, which are global uniform attractors for the reduced-order and full-order systems. The stability analysis is similar to that used in [10] but, here, the reduced-order system is based on a nonlinear affine control system. For the problem of synthesizing a class of output feedbacks for the unconstrained problem, see [6]. A preliminary version of this paper was presented in [5].
2. Full-order singularly perturbed system Consider a singularly perturbed uncertain dynamical system consisting of coupled subsystems with the following structure: f (x)#F (x) y#G (x)u#h (t, x, y, u, k), xR (t)"a(t, x(t), y(t), u(t), k) " (1a) A(t) [ f (x)#y#G (x) u]#h (t, x, y, u, k), (1b) kyR (t)"b(t, x(t), y(t), u(t), k) " z(t)"r(x(t), y(t)),
(1c)
where (x(t), y(t))31L;1l is the state of the system, u(t)31K is the control input, [0,R) denotes the singular perz(t)31L is the output, 1)m)l, n, and k31> " turbation parameter, assumed to be ‘‘small’’. The dimension of the output space coincides with the dimension of the slow subsystem and the full-order system is defined on 1L>l. It is assumed that the vector fields f 3C(1L) and f 3C(1l) are known and satisfy f (0)"0, f (0)"0, F (x)3L(1l , 1L) (the set of all continuous linear maps from 1l into 1L), G (x)3L(1K, 1L) and G (x)3L(1K, 1l) are known. The uncertainty in the system is represented by A(t), a Lebesgue measurable matrixvalued function, and the vector fields h , h , which are smooth, that is h 3C(1L) and h 3C(1l). Remarks. The structure specified in Eqs. (1a)—(1c) admits a large nontrivial class of uncertain systems. For example, the uncertain system (discussed in [10]) xR (t)"Ax(t)#[B#*B(t)] y (t)#d(t, x(t)), x31L,
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with actuator dynamics kyR (t)"[C #*C (t)] (y (t)!u(t)),
y , u31K,
and sensor dynamics kyR (t)"[C #*C (t)] (y (t)!x(t)), y 31L, and output z(t)"y (t), where *B(t), *C (t), *C (t) and d(t, x(t)) model uncertainty in the system, can be identified with Eqs. (1a)—(1c). In this case, the model without uncertainty is essentially linear in nature. An example of a nonlinear model, that represents a nonlinear flexible mechanical system, is xR (t)"x (t), x (t), x (t)31 xR (t)"(1#i(t)) [k sin(x (t))!I (k y (t)#k y (t))]!d(t) A ? ? @ kyR (t)"y (t), y (t), y (t)31 kyR (t)"(1#i(t)) [I\I (K x (t)#K x (t)#k sin(x (t))) ? @ A !I (k y (t)#k y (t))!u(t)!k(I !I ) qN (t, x (t))], @ ? @ @ ? with input u(t)31 and output z(t)"[x (t) x (t)]2, where i(t), d(t) and qN (t, x (t)) model uncertainty in the system. This model can also be identified with Eqs. (1a)—(1c). Furthermore, a nonlinear electromechanical system considered in [8] is a candidate for the structure described in Eqs. (1a)—(1c). It is assumed that the uncertainty in the second subsystem (1b) is characterized by the following hypotheses: For a linear map ¸, the notation #¸# denotes +max p(¸2¸),, where p( ) ) denotes spectrum. Assumption 1. (a) For all t, there exists , 3[0, #P #\) and A 3L(1l, 1l) such that A(t)"A #AK (t),
#AK (t)#), ,
A is known and p(A )L"\, where P '0 is the unique, symmetric solution of the Lyapunov equation P A #A2P #I"O.
(2)
(b) For all (t, x, y, u), h (t, x, y, u, 0)"0. 3. Reduced-order uncertain system Setting k"0, system (1a)—(1b) degenerates to the set of n differential equations xR (t)"f (x(t))#F (x(t)) y(t)#G (x(t)) u(t)#h (t, x(t), y(t), u(t), 0)
(3)
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subject to the constraint f (x)#y#G (x) u"0.
(4)
The degenerate system (3)—(4) and (1c) are equivalent to xR (t)"fI (x(t))#GI (x(t)) u(t)#hI (t, x(t), u(t)),
(5a)
z(t)"r(x(t), (x(t), u(t))),
(5b)
f (x)!F (x) f (x), GI (x) " G (x)!F (x)G (x), (x, u) " !f (x)!G (x) u where fI (x) " and hI (t, x, u) " h (t, x, (x, u), u, 0), and is known as the reduced-order system. The function is determined uniquely in view of Assumption 1(a). The Euclidean inner product (on 1L or 1l as appropriate) and the induced norm are denoted by 1 ) , )2 and # ) #, respectively. Let ¸ v : 1NP1 denote the Lie derivative of D a scalar field x>v(x)31 along the vector field f3C(1L). In particular, let v : 1x, P x2, then (¸ v ) (x)"1P x, f (x)2. The reduced-order system is x>v (x) " D characterized by the prescribed triple ( fI , GI , hI ) for which the following hypotheses are assumed to hold: Assumption 2. (a) There exists e(x)3C(1), which is strictly increasing as #x#PR, with e(0)"0 and e(x)'0 for all xO0, and a symmetric positive definite matrix P 31L;L such that (i) (¸ I v ) (x))!e(x), for all x. D (0, R), such that (ii) There exists real constants c , c 31>, with 1> " K c " (¸ v ) (x) ")+v (x),)c +e(x),, EG G
where g (x) denotes the ith column of the matrix GI (x). G (b) There exists d , d 31> , satisfying d #d '0, such that # fI (x)#)d #d #x#. (c) There exists unknown vector fields p3C(1K) and q3C(1K) such that hI (t, x, u)"p(t, x, u)#GI (x)q(t, x, u), ∀(t, x, u), and there exist real constants a , a , a , b , b , b 31> , b (1, such that, for all (t, x, u), (i) #p(t, x, u)#)a #x##a #a K "u ", where u denotes the ith component G G G of u. (ii) "q (t, x, u)")b #x##b #b " u ", where q denotes the ith component of q. G G G Remarks. (a) Assumption 2(a)(i) ensures that the state origin of the drift system xR "fI (x) is a global uniform attractor (see Definition 4.1, below).
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(b) Analogous to the characterization of uncertainty for linear systems (see, e.g., [14]), the vector fields p, q are said to represent the matched and unmatched components, respectively, of the uncertainty in the nonlinear reduced-order system.
4. Class of constrained output feedback controls It is supposed that the constraints " u (t) ")o , ∀t*t , i"1, 2, 2 , m, (6) G G where o '0 are prescribed, are now imposed on the control inputs of system G (1a)—(1c). It is desired that a class of output feedback controls, satisfying the prescribed constraints, be designed so that (a) there exists a compact set, containing the origin, which is a global uniform attractor for the reduced-order system (5a)—(5b) in the sense of Definition 4.1 (given below), and (b) the full-order system has the same stability property in the presence of singular perturbations for all k3(0, k*), where k* is some computable real constant. The design procedure is such that the output feedback control functions depend on a design parameter e. Definition 4.1. Let !L1N be a compact set satisfying !5+0," and let ! (r) N denote the open unit ball of radius r centred at the origin in 1N. The set ! is a global uniform attractor for the system fQ (t)"Z(t, f(t)), f(t)31N
(7)
if (a) for each (t , f)31;1N, there exists a local solution f: [t , t )P1N satisfying Eq. (7) a.e. with f(t )"f and every such local solution can be extended into a solution on [t ,R); (b) for each r31>, there exists R(r)31> such that, for every t 31, f(t)3! (R(r)) for N all t*t on every solution f : [t ,R)P1N with f(t )3! (r); N (c) for each d31>, there exists D(d)31> such that, for every t 31, f(t)3!#! (d) N for all t*t on every solution f : [t ,R)P1N with f (t )3!#! (D(d)); N (d) for each d, r31>, there exists ¹(d, r)31> such that, for every t 31, f(t)3!#! (r) for all t't #¹(d, r) on every solution f: [t ,R)P1N of Eq. (7) N with f(t )3! (d). N The synthesis of an output feedback controller z>w(z) follows the design methodology of [6, 10]. Note that in the absence of uncertainty for the reduced order system and if r(x, y)"x, for all (x, y)31L>l, then the output feedback law !(¸ v ) (z(t)) renders the zero-state of system (5a)—(5b) as a global uniform u(t) " EG
i.e., an absolutely continuous function. +s #s : s 3S , s 3S ,L1N. For sets S , S L1N, S #S "
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attractor. Define x>cJ (x)"[cJ (x) cJ (x)2 cJ (x)]2, where K !bM \ [ cJ #b cJ [cJ " (¸ v ) (x) "#e ]\] (¸ v ) (x), cJ (x) " G G EG G G EG (1!b )'0, and cJ , c , e 31> are design parameters. bM " G G
(8)
Remark. Each cJ (x) consists of two parts and incorporates the matched uncertainty G parameter b . This term is used as a switch to activate the second part when b O0. Thus, when b "0 the second part is not activated and, in this case, Eq. (8) reduces to the simplified form: cJ (x)"!b\c(¸ v ) (x). G EG Let wJ (x), a saturating version of cJ (x), be defined by o s(o\ cJ (x)), wJ (x) " G G G G where s : 1P1 is defined by
1, h>s(h) " h,
(9)
h'1, " h ")1,
!1, h(!1.
r(x, I (x)): It is assumed that the function x>rJ (x) "
I (x) " (x, wJ (x)) has a well-defined inverse.
1LP1L,
where
Assumption 3. The function x>rJ \(x) exists and is defined uniquely for all x. The proposed class of constrained output feedback controls z>w(z)" [w (z) w (z)2 w (z)]2 is such that each component has the following structure K " w (z) wJ (rJ \(z)). (10) G G By the definition of w(z), " w (z) ")o and, for the reduced-order system, the output G G (5b) is satisfied by z(t)"rJ (x(t)). Therefore, for the reduced-order system, the constrained output feedback controls can be expressed in terms of the state vector for the reduced-order system as u(t)"w(z(t))"wJ (rJ \(z(t)))"wJ (x(t)). 5. Boundary-layer system The full-order system can be expressed as xR (t)"aJ (t, x(t), y(t), z(t), k), kyR (t)"bI (t, x(t), y(t), z(t), k), z(t)"r(x(t), y(t)), where aJ (t, x, y, z, k)"a(t, x, y, w(z), k) and bI (t, x, y, z, k)"b(t, x, y, w(z), k).
(11)
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Introducing the rescaled time variable q"(t!t*)/k, where t*31 is fixed, and the x(t*#kq), yN (q) " y(t*#kq)#f (x(t*))#G (x(t*))wJ (x(t*)), vector fields xN (q) " zN (q) " z(t*#kq) yields the system xN (q)"kaJ (t*#kq, xN (q), yN (q)# (x(t*), wJ (x(t*))), zN (q), k), yN (q)"bI (t*#kq, xN (q), yN (q)# (x(t*), wJ (x(t*))), zN (q), k), zN (q)"r(xN (q), yN (q)# (x(t*), wJ (x(t*)))), where ( ) denotes differentiation with respect to q. By construction, bI (t*, xN , I (xN ), zN , 0)" 0 and, therefore, any point (xN , I (xN ), zN , w(zN ))3S, !f (x)!G (x) u,, S"+(x, y, z, u)31L;1l;1L;1K: y" (x, u) " is an equilibrium point of the system at k"0. The behaviour of the system at k"0 is characterized by the system yN (q)"bI (t*, x(t*), yN (q)# (x(t*), wJ (x(t*))), zN (q), 0)"A(t*) yN (q),
(12)
which is known as the boundary-layer system. Assumption 1(a) guarantees that Eq. (12) is asymptotically stable.
6. Stability properties of the full and reduced order systems 6.1. Reduced-order system With output feedback u(t)"w(z(t))"wJ (x(t)) and wJ defined by Eq. (9), the reducedorder system (5a)—(5b) can be expressed as K xR (t)"fI (x(t))# [q (t, x(t), wJ (x(t)))#wJ (x(t))]g (x(t))#p(t, x(t), wJ (x(t))), G G G G
(13a)
z(t)"rJ (x(t)).
(13b)
To ensure that the constrained controller can cope with the uncertainty in the system, the following feasibility assumption is required. Assumption 4. The control constraints satisfy o 'bM \b for all i. G If Assumption 4 holds, then there exist parameters cJ *1 satisfying G b cJ (bM o . (14) G G The parameters cJ are to be identified with those introduced in Eq. (8). G The desired stability property is obtained by using essentially the same approach as that of [5, 6]. Examining the behaviour of the time derivative of the function
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v : 1LP1> along solutions to (13a)—(13b), one obtains, using standard Lyapunov analysis, the following result. Define
c\!(2c\b +p (P ),\ !2a p (P) , s" (15) p (P )
min [bM o !b cJ ], oL " (16) G G G K e , cJ " K cJ , o " K o , and let E(r) denote the set +x31L: v (x))r,. e" G G G G G G The following theorem may be deduced. Theorem 6.1. Suppose Assumptions 1—4 hold. If s'0 and the design parameters, c , cJ , G satisfy cJ 71 for all i, c '0 and, if a #a O0, G c oL s , (17) (a #a o)+2p (P ),
then, with constrained output feedback control u"w(z) (given by Eqs. (8)—(10)), the compact set E(r ), where C r " r\ [r #(r#4r b e] , C r "s!c\a bM \c +2p (P ), and r "(a #a bM \b cJ ) +2p (P ), ,
is a global uniform attractor for the uncertain system (13a)—(13b), with x(t )"x. c (c* "
Proof. See Appendix A.1. ) Remarks. (a) It is advantageous to choose c , cJ and e as small as possible with the G G result that the size of E(r ) can be reduced, but only in the sense that E(r ) will tend to C C the limiting set E(r*), where r*"s\(a #a bM \b ) +2p (P ),.
(b) If a "a "0, then E((b es\) is a global uniform attractor and, therefore by tuning the parameters e , the size of this set can be made as small as desired. G (c) If a "b "0, then, as a consequence of Eq. (21) in the proof of Theorem 6.1, +0, is a global uniform attractor. 6.2. Full-order system For this final stage it is shown that, using the constrained output feedback (10), the full-order uncertain system (1a)—(1c) has a global uniform attractor. The methodology follows that of [10] (see, also, [5, 11]). Consider the Lyapunov function candidate v : I v (x)#m v (x, y), where m 31>, a real constant 1L;1lP1> defined by v (x, y) " I I I dependent upon the singular perturbation parameter, is to be specified and 1y! I (x), P (y! I (x))2. Along solutions to Eqs. (1a)—(1c) and for v (x, y) "
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almost all t, d v (x(t), y(t))"» (t, x(t), y(t)), I dt I where 1( v ) (x), a(t, x, y, w(r(x, y)), k)2 » (t, x, y) " I #m +1( v ) (x, y), a(t, x, y, w(r(x, y)), k)2 I V #k\1( v ) (x, y), b(t, x, y, w(r(x, y)), k)2,. W Assumption 5. (a) For all x, there exist k , k 31> such that #F (x)#)k and #G (x)#)k . (b) There exists a continuous function j: 1>P1>, with j(0)"0, such that, for all (t, x, y, u)31;1L;1l;1K and k31>, #h (t, x, y, u, k)!h (t, x, y, u, 0)#)j(k). (c) There exist k , k 31> such that, for all y , y 31l and (t, x)31;1L, #h (t, x, y , w(r(x, y )), k)!h (t, x, y , w(r(x, y )), k) #)k #k #y !y #. Under Assumptions 1—5 and utilizing Eqs. (6) and (21), one may deduce the following lemma. Lemma 6.1. Suppose Assumptions 1—5 hold. For all (t, x, y)31;1L;1l and k31>, 1( v ) (x), a(t, x, y, w(r(x, y)), k)2)!r v (x)#(i #k(k)) +v (x),#b e #i +v (x),+v (x, y), , where
r #k +2p (P ),, if z"r(x, y) is independent of y or G (x)"O,
i " r #2oc\#(2ok k #k ) +2p (P ),, otherwise,
j(k)+2p (P ),. i "2(k #k ) +p (P )/p (P ),, and k>k(k) "
Proof. See Appendix A.2. ) Remark. If the output function z"r(x, y) is independent of y, then rJ (x)"r(x) and, hence, w(z)"wJ (rJ \(r(x)))"wJ (x). In this case, i "r #k +2p (P ), and, it is shown later, see Corollary 6.1, that, under an additional assumption, the asymptotic behaviour of the output controlled full-order system tends to that of the controlled reduced-order system. Assumpiton 6. The Fre& chet derivative of I satisfies #(D I ) (x)#,
K #(D I ) (x) g (x)# )k , k 31>, for all x. G G
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Remark. Assumptions 6, 5(a), 2(a) and (b) are quite severe restrictions on the nonlinearity of the full-order system. Lemma 6.2. Assuming the conditions stated in Assumptions 2(b), (c), 5 and 6 hold, then, for all (t, x, y)31;1L;1l and k31>, 1( v ) (x, y), a(t, x, y, w(r(x, y), k)2)(i +v (x),#i #k k(k)) +v (x, y), V #i v (x, y), where i "2k (d #a #mb ) +p (P )/p (P ),, i "k (d #a #mb #k #o(1#a #b #2k k )) +2p (P ),,
i "2k (k #k )+p (P )/p (P ),. Proof. See Appendix A.3. ) Assumption 7. There exist known scalars j , j , j 31> such that, for all k31> and (t, x, y)31;1L;1l , #h (t, x, y, w(r(x, y)), k)#)k[j #y! I (x)##j #x##j ]. Lemma 6.3. ºnder Assumptions 1(a), 5(a) and 7 and for all (t, x, y, k)3 1;1L;1l;1>, 1( v ) (x, y), b(t, x, y, w(r(x, y)), k)2)ki +v (x), +v (x, y), W #(ki #i )+v (x, y),#(ki !i ) v (x, y), where i "2j +p (P )/p (P ),,
i "j +2p (P ),,
0, if z is independent of y or G (x)"O, i" 2ok (#P A ##, #P #)+2/p (P ),, otherwise,
i "2j +p (P )/p (P ),,
i "(1!2, # P #) +p (P ),\.
Proof. See Appendix A.4. ) Let m "n(k/k*), where n, k*31>, and select real h31>. Then, since i '0, I
r i [r (i #i )#i (i #i )]\, i (i #i )O0, k*" r i [r (i #i )#h]\, otherwise,
(18)
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where
i (i #i )\, i h\, n" 4h(i #i )\, h,
i (i #i )O0, i O0, i #i "0, i "0, i #i O0, i "0, i #i "0 ,
has the property that the full-order uncertain system has the desired behaviour, provided k(k*. This type of structure for k* was first presented in [10, 11]. However, the value of k*, given in Eq. (18), is not unique, that is there are other possible threshold values, and the threshold estimate (18) may be rather conservative. Remarks. The formula that is provided to compute a threshold value of k, namely (18), often produces a very conservative value and examples have suggested that the value may differ from the true threshold value by a factor of at least 10. Other examples have suggested that some systems may be sensitive to the value of k in the sense that for a certain value of k there exists a compact attractor, but the system becomes unstable when k is changed by a relatively small amount. Theorem 6.2. Suppose Assumptions 1—7 are satisfied and k3(0, k*) is fixed. If s'0, then, with u(t)"w(z(t)) (defined in Eqs. (8) and (9)), c '0, cJ *1 for all i and, if G a #a O0, c satisfies Eq. (17), the compact set +(x, y)31L;1l : v (x, y))r,, W" I I I where v \ [gN #+gN #4lN b e,], r " I I I I I l " #M\ #\ min[1, m\] and g "([i #k(k)]#m [i #i #k\i I I I I I #k k(k)]), is a global uniform attractor for the full-order uncertain system (1a)—(1c) subject to the initial condition (x(t ), y(t ))"(x, y). Proof. See Appendix A.5. ) Under additional assumptions, the asymptotic behaviour of the solution of the controlled full-order system can be shown to tend to that of the controlled reducedorder system, as seen in the following corollary. Corollary 6.1. Suppose the conditions stated in ¹heorem 6.2 hold, then the set E(r ) is I a global uniform attractor under the dynamics of system (1a)—(1c). Moreover, if
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z"r(x, y) is independent of y and k "0, then lim d(E(r ), E(r ))"0, I C I where d denotes a Hausdorff metric. Proof. See Appendix A.6. )
7. Illustrative example Consider the controlled singularly perturbed uncertain nonlinear system
2#x(t) #y (t)#h (t, x(t), y(t), u(t), k), (19a) 1#x(t) kyR (t)"(!1#aL (t)) (y (t)!u(t)), (19b) kyR (t)"(!3#a' (t)) (y (t)!20x(t))#h (t, x(t), y(t), u(t), k), (19c) with initial conditions (x, y, y)"(3,!2, 5) and output z(t)"y (t)#y (t), where x(t), z(t)31, y(t)"[y y ]231. The system can be identified with (1a)—(1c) where (2#x) f (x)"!x , F (x)"[1 0], G (x)"0, 1#x xR (t)"!x(t)
0 ! !1#a' (t) 0 , A(t)" f (x)" , G " . !20x 0 0 !3#a' (t) It is assumed that the control constraint " u ")2 is imposed on the system. In order to perform a simulation on this system, it is supposed that a' (t)" (1#3e\R ) and a' (t)" sin(2t). Given 0 !1 0 , then P " satisfies Eq. (2) A " 0 !3 0 and, since
AK (t)"
1 1!2e\R 0 , 9 0 sin(2t)
it follows that , " . Hence, Assumption 1(a) is satisfied as , 3[0, #P #\)" [0, 1). For simulation purposes the following uncertainty functions are selected: h (t, x, y, u, k)" x! y cos(3t)! u sin(t)#sin(k) e\I, h (t, x, y, u, k)"k cos(t) [2(y !u)#5x#e\R sin(x)]. Thus, Assumption 1(b) is valid and Assumption 5 holds with k "1, k " , k ", k " , and j(k)"" sin(k)e\I ".
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The reduced-order equation is given by fI (x)"!x
5 1 (2#x) , GI (x)" , hI (t, x, u)" x! u [cos(3t)#sin(t)]. 1#x 4 8
Let P "1, then (¸ I v ) (x)"!x((2#x)/1#x) and (¸ v ) (x)" x. Therefore, EG D Assumption 2 holds with e(x)"x((2#x)/1#x), c "(2, c " , d "1 and d ". Since p(t, x, u)"0 for all (t, x, u), then a "a "a "0 and b " , b "0, b ", which validates Assumption 2(c). Since bM " and b "0, the control for the reduced-order system is wJ (x)" 2s(0.5cJ (x)), where cJ (x)"! cN x. Thus, wJ (x) s(!c x)
I (x)" (x, wJ (x))" " 20x 20x
and hence Assumption 6 holds with k "((cN /9)#400. Now rL (x)"r(x, I (x))"s(! cN x)#20x 20x!1, x'N , A N " (20!A ) x, !N )x)N , A A 20x#1, x(!N . A Selecting 20!(cN /3)'0, that is cN (60, Assumption 3 holds with
rL \(x)"
x!1 , 20 3x , 60!cN x#1 , 20
60 x(! #1, cN
" x ")
60 !1 , cN
60 x' !1. cN
Now consider Assumption 7. Note that " h (t, x, y, w(r(x, y)), k) ")k " 2(y !w(r(x, y)))#5x#e\R sin(x) " )k " 2(y !20x)#45x!2w(r(x, y))#e\R sin(x) " )k+2 " y !20x "#45 " x "#5,. But
y !s(! AN x) y !20x and so " y !20x ")#y! I (x)#. y! I (x)"
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Hence, Assumption 7 is satisfied with j "2, j "45 and j "5. A simple calculation shows that s" satisfies the stability criterion stated in Theorem 6.2. The constrained output feedback control for the system (19a)—(19c)has the following form: u(t)"w(y (t)#y (t))"2s(! cL r' \(y (t)#y (t))). Selecting cN "30 for simulation, u"2s(!10rL \(y #y )) 2, y #y (!1, " !2(y #y ), " y #y ")1, !2, y #y '1. The simulation results are shown in Figs 1—4. Figure 1 illustrates the dynamics of the full-order open-loop system, that is with no control, and shows the dynamics of the full-order closed-loop system when k"0.01. In addition, Figs. 2—4 show the state trajectories when the singular perturbation parameter has the value k"0.1, k"0.45 and k"0.46, respectively. Examination of Fig. 4 appears to indicate that the value of k has exceeded its threshold value, whilst Figs. 2 and 3 suggest that the threshold value is greater than 0.1. The theory predicts that k*+0.001, which is a conservative estimate for the threshold value, and a global attractor exists for the full-order system when k(k*.
Appendix A A.1. Proof of Theorem 6.1 For almost all t and along all solutions to Eqs. (13a) and (13b), K vR (x(t))"(¸ I v ) (x(t))# [q (t, x(t), wJ (x(t)))#wJ (x(t))] (¸ v ) (x(t)) D G G EG G #1P x(t), p(t, x(t), wJ (x(t)))2. Consider the case " w (x) "'o . G G Then, in view of Eq. (9), " wJ (x) ""o and wJ (x) (¸ v ) (x)"!o " (¸ v ) (x) ". Since G G G EG G EG #x#)2v (x)/p (P ) and #P x#)2p (P ) v (x)
and utilizing Assumptions 2(a) (i) and 2(c), K vR (x(t)))!e(x(t))#(2b +p (P ),\ +v (x(t)), " (¸ v ) (x(t)) " EG G p (P )
#2a v (x(t))#(a #a o) +2p (P ),+v (x(t)), p (P )
K ! [bM o !b ] " (¸ v ) (x(t)) ". G EG G
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Fig. 1. State histories for the open-loop system (left) and closed-loop system (right), including control history, with k"0.01.
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Fig. 2. State and control histories for the closed-loop system with k"0.1.
Fig. 3. State and control histories for the closed-loop system with k"0.45.
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Fig. 4. State and control histories for the closed-loop system with k"0.46.
In view of Eq. (15) and Assumptions (2a) (ii) and 5, vR (x(t)))!sv (x(t))#(a #a o ) +2p ( P ),+v ((x),. (A.1)
Invoking LaSalle’s invariance principle, all solutions of Eq. (A.1) tend to the largest invariant subset of +x31L: +v (x),)(a #a o) +2p (P ),/s,. S"
A subset of 1L is now determined for which the unconstrained feedback controller satisfies the prescribed constraints. As a consequence of Eq. (8) and since " cJ (x) "(bM \(c " (¸ v ) (x) "#b cJ ), G EG G then " cJ (x) ")o if " (¸ v ) (x) ")c \(bM o !b cJ ). The set G G EG G G 7 +x31L: " (¸ v ) (x) ")c \(bM oJ !b cJ ), R* " EG G G G is known as a constrained control region (see [15, 16]) and if the states of the system (13a)—(13b) lie in this region, the unconstrained controller cJ , defined in (8), satisfies the control constraints, namely wJ (x)"cJ (x) for all x3R*.
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min [(bM oJ (!b cJ )], As a consequence of Assumption 2(a) (ii) and defining oL " G G G S* " +x31L: +v (x),)c cN \oL ,LR* and, thus, S* is a compact constrained controlled region. Suppose a #a O0, then the parameter cN , which satisfies Eq. (17), is designed so that SLS*. Alternatively, if a "a "0, then S*LS"+0,. Hence, if x31L!S*, all solutions eventually enter S*, which is positively invariant, in finite time. Now suppose x3S*. Then, using Eq. (9),
b cJ G " (¸ v ) (x) ", wJ (x) (¸ v ) (x)"!bM \ cN # EG G EG cJ " (¸ v ) (x) "#e G EG G b cJ G " (¸ v )(x) ". " wJ (x) ""bM \ c # EG G cJ " (¸ v ) (x) "#e G EG G Utilizing Assumption 2, a straightforward calculation shows that
vR (x(t)))!(s!c\ a bM \cN +2p (P ),) v (x(t))
#(a #a bM \b cJ ) +2p (P ),+v (x(t)),
K K !cN "(¸ v ) (x(t)) "!b (cJ !1) (¸ v ) (x(t))#b e EG G EG G G )!(s!c\ a bM \cN +2p (P ),) v (x(t))
M \b #(a #a b cJ ) +2p (P ),+v (x(t)),#b e.
M \c Suppose that a #a O0. It is now shown that s'c\a b N +2p (P ),. Note
that, since cN (c*, oL sa bM \ . c\a bM \cN +2p (P ),(
a #a o By definition of oL , oL )bM o!b cJ and so a (o!bM \b cJ ) s s!c\a bM \cN +2p (P ),'s!
a #a o M \b (a #a b cJ ) s " a #a o *0. Thus, with r "s!c\a bM \cN +2p (P ),'0 and r "(a #a bM \b cJ ) +2p (P ),,
vR (x(t)))!r v (x(t))#r +v (x(t)),#b e (A.2) (0 for (t, x(t)),1;E(r ). (A.3) C
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In view of Eq. (A.2) and (A.3), every local solution evolves within the compact set +x31L: v (x) )v (x), 6 E(r ) and, hence, can be continued indefinitely. Property (b) C of Definition 4.1 holds with
[ #P\ # #P #]r, if r'(2 #P #\r , C (2#P\ #r , if r)(2 #P #\r . C C Moreover, property (c) of Definition 4.1 holds with D(d)"[ #P\ # #P #]\ d. Let ! be any compact set such that E(r ) is contained in an open set which is Z contained in !. Then, there exists a closed set E(rN ), with rN 'r such that E(rN )L!. Z Since, for each rJ *rN ('r ), C J P [r !r +v ,!b e]\ dv (R, 0)I(rJ , rN ) " PN property (d) of Definition 4.1 can be verified to hold with R(r)"
¹(d, r)"
I (( #P #r, rN ), 0
r'(2#P #\rN , r)(2#P #\rN .
This concludes the proof. A.2. Proof of Lemma 6.1 A straightforward calculation shows that f (x)#F (x)y#G (x) w(r(x, y))"fI (x)#GI (x) wJ (x)#F (x)(y! I (x)) #(GI (x)#F (x) G (x)) [w(r(x, y)) !w(r(x, I (x)))] and, therefore, for all (t, x, y)31;1L;1l and k31>, 1( v )(x), a(t, x, y, w(r(x, y)), k)2"1P x, fI (x)#GI (x)wJ (x) #h (t, x, I (x), w(r(x, I(x))), 0)2 #1P x, h (t, x, I (x), w(r(x, I (x))), k) !h (t, x, I (x), w(r(x, I (x))), 0)2 #1P x, h (t, x, y, w(r(x, y)), k) !h (t, x, I (x), w(r(x, I (x))), k)2 #1P x, GI (x) [w(r(x, y))!w(r(x, I (x)))]2 #1P x, F (x) G (x) [w(r(x, y))!w(r(x, I (x)))]2 #1P xF (x) (y! I (x))2. (A.4)
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Note that if G (x)"GI (x)#F (x) G (x)"O or z"r(x, y) is independent of y, then the 4th and 5th terms on the r.h.s of Eq. (A.4) are identically zero. In view of Assumption 5(a) and since #[w(r(x, y))!w(r(x, I (x)))]#)2o, 1P x, F (x) G (x) [w(r(x, y))!w(r(x, I (x)))]2#1P x, F (x)(y! I (x))2 p (P )
+v (x),+v (x, y),#2k k o +2p (P ),+v (x),. )2k
p (P )
Invoking Assumption 2(a) (ii),
K 1P x, GI (x) [w(r(x, y))!w(r(x, I (x)))]2" [w (r(x, y))!w (r(x, I (x)))] (¸ v ) (x) G G EG G )2oc\ +v (x),. As a consequence of Assumptions 5(a) and (b), 1P x, h (t, x, I (x), w(r(x, I (x))), k)!h (t, x, I (x), w(r(x, I (x))), 0)2 )j(k) +2p (P ),+v (x),
and 1P x, h (t, x, y, w(r(x, y)), k)!h (t, x, I (x), w(r(x, I (x))), k)2 p (P )
+v (x),+v (x, y),#k +p (P ),+v (x),. )2k p (P )
Thus, utilizing Eq. (A.2), the result follows.
A.3. Proof of Lemma 6.2 Since h (t, x, I (x), wJ (x), 0)"p(t, x, wJ (x))#GI (x) q(t, x, wJ (x)) and v (x, y)" V !(D I )2(x) P (y! I (x)), then, for all (t, x, y, k)31;1L;1l;1> , 1 v (x, y), a(t, x, y, w(r(x, y)), k)2)#(D I )(x)# #P (y! I (x))# [ # fI (x)# V ##p(t, x, wJ (x)###h (t, x, y, w(r(x, y)), k) !h (t, x, I (x), w(r(x, I (x))), k)# ##F (x)# #y! I (x) # ##h (t, x, I (x), w(r(x, I (x))), k) !h (t, x, I (x), w(r(x, I (x))), 0)# ##F (x)##G (x)##w(r(x, y))!w(r(x, I (x)))#] K ##P (y! I (x))# " w (r(x, y)) G G #q (t, x, wJ (x)) " #(D I ) (x)g (x)#. G G
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Utilizing Assumptions 2(b) and (c), # fI (x)###p(t, x, wJ (x))#)d #a #a oN #(d #a )#x#, " w (r(x, y))#q (t, x, wJ (x)) ")b #o (1#b )#b # x #. G G G As a consequence of Assumptions 5 and 6, 1 v (x, y), a(t, x, y, w(r(x, y)), k2)k #P [y! I (x)] # +(d #a #mb ) #x# V #d #a #mb #k #o(1#a #b I #2k k )#(k #k ) #y! (x)##j(k),. The statement of the lemma now follows in view of the results: #x#)+2/p (P ),+v (x),, #y! I (x)#)+2/p (P ),+v (x, y),,
I and #P [y! (x)]#)+2p (P ),+v (x, y),.
(A.5)
A.4. Proof of Lemma 6.3 Since v (x, y)" 1y! I (x) P [y! I (x)]2, 1( v ) (x, y), b(t, x, y, w(r(x, y)), k)2"1P [y! I (x)], A(t) [y! I (x)]2 W #P [y! I (x)], A(t) G (x) (w(r(x, y)) I !w(r(x, (x))))2 #1P [y! I (x)] h (t, x, y, w(r(x, y)), k)2. Utilizing Assumptions 1(a) and 5(a), 1P [y! I (x)], A(t) [y! I (x)]2)! (1!2i #P #) #y! I (x)#, 1P [y! I (x)], A(t) G (x) (w(r(x, y))!w(r(x, I (x))))2)2ok [#P A # #, #P #] #y! I (x)#. However, note that if G (x)"O or z"r(x, y) is independent of y, then 1P [y! I (x)], A(t)G (x) (w(r(x, y))!w(r(x, I (x))))2"0. Finally, as a consequence of Assumption 7, 1P [y! I (x)], h (t, x, y, w(r(x, y)), k)2)2kj [p (P )/p (P )]v (x, y) #2kj [p (P )/p (P )] +v (x, y), ;+v (x), #kj [2p (P )]+v (x, y),
and, in view of Eq. (A.5) and since #y! I (x)#*+2/p (P ) v (x, y),, the result then
follows.
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A.5. Proof of Theorem 6.2 As a consequence of Lemmas 6.1—6.3,
+v (x), +v (x), , M I +v (x, y), +v (x, y), #(i #k(k)) +v (x),#m (i #i #k\i I #k k(k)) +v (x, y),#b e,
» (t, x, y))! I
(26)
where
! (i #m (i #i )) r I . M" I ! (i #m (i #i )) m (k\i !i !i ) I I With k* defined as in Eq. (18), det(M )'0 for all k3(0, k*) if r '0 (see [11, 17]) and, I therefore, M is positive definite if s'0. Thus, if s'0 and k(k*, I » (t, x, y))!l v (x, y)#gN +v (x, y),#b e, I I I I I where l "# M\ #\ min[1, m\], I I I gN "([i #k(k)]#m [i #i #k\i #k k(k)]). I I Since 1 v (x, y)" I 2
x x , PI I y! I (x) y! I (x)
,
where
O P PI " , I O mP I standard arguments can be employed, analogous to those used in the proof of Theorem 6.1 (given in Appendix A.1), to show that properties (a)—(d) of Definition 4.1 hold. A.6. Proof of Corollary 6.1 Let i "i #i #k\i #k k(k). In view of Eq. (26), I » (t, x, y))!lN v (x, y)#(i #k(k)) +v (x),#m i +v (x, y),#b e I I I I I "!l v (x)#(i #k(k)) +v (x),!l [(m v (x, y)) I I I !iN l \ ]# m l \i #b e I I I I I )!l v (x)#(i #k(k)) +v (x),# m l \i #b e. I I I I
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This implies E(r' ) is an attractor for system (1a)—(1c), where I lN \ [i #k(k)#(g #4l b e], r' " I I I I and, since r' )r , E(r ) is a global attractor. I I I If z"r(x, y) is independent of y and k "0, then i "r and i "0. Some straightforward calculations show that lim #M\ #\"r , lim l "lim #M\#\"r and lim g "r . I I I I I I I I Thus, lim r "r , I C I where r is defined in Theorem 6.1. This concludes the proof. C References [1] P.V. Kokotovic´, H.K. Khalil, J. O’Reilly, Singular Perturbation Methods in Control: Analysis and Design, Academic Press, New York, 1986. [2] A. Isidori, Nonlinear Control Systems, Springer, Berlin, Germany, 1989. [3] R. Marino, P.V. Kokotovic´, A geometric approach to nonlinear singularly perturbed control systems, Automatica 24 (1988) 31—41. [4] A. Saberi, H. Khalil, Quadratic-type Lyapunov functions for singularly perturbed systems, IEEE Trans. Automat. Control AC-29 (1984) 542—550. [5] H.S. Binning, D.P. Goodall, Output feedback controls for a class of imperfectly known singularly perturbed nonlinear systems subject to control constraints, Proc. 4th IEEE Mediterranean Symp. on New Directions in Control and Automation, Maleme, Krete, Greece, 1996, pp. 53—58. [6] H.S. Binning, D.P. Goodall, Output feedback controls for a class of singularly perturbed uncertain nonlinear systems, Automat. Remote Control 59 (7) (1997) 81—97. [7] M. Corless, Asymptotic stability of singularly perturbed systems that have marginally stable boundary-layer systems, Dyn. Control 1 (1991) 95—108. [8] M. Corless, Robustness of a class of feedback-controlled uncertain nonlinear systems in the presence of singular perturbations, Proc. American Control Conf., Minneapolis, USA, 1987, pp. 1584—1589. [9] M. Corless, F. Garofalo, L. Glielmo, New results on composite control of singularly perturbed uncertain linear systems, Automatica 29 (1993) 387—400. [10] M. Corless, G. Leitmann, E.P. Ryan, Control of uncertain systems with neglected dynamics, in: A.S.I. Zinober (Ed.), Deterministic Control of Uncertain Systems, Peter Peregrinus, London, 1990, pp. 252—268. [11] M. Corless, E.P. Ryan, Robust feedback control of singularly perturbed uncertain dynamical systems, Dyn. Stability Systems 6 (1991) 107—121. [12] F. Garofalo, G. Leitmann, Nonlinear composite control of a class of nominally linear singularly perturbed uncertain systems, in: A.S.I. Zinober (Ed.), Deterministic Control of Uncertain Systems, Peter Peregrinus, London, 1990, pp. 269—288. [13] G. Leitmann, E.P. Ryan, A. Steinberg, Feedback control of uncertain systems: robustness with respect to neglected actuator and sensor dynamics, Int. J. Control 43 (1986) 1243—1256. [14] B.R. Barmish, G. Leitmann, On ultimate boundedness control of uncertain systems in the absence of matching assumptions, IEEE Trans. Automat. Control AC-27 (1982) 153—158.
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[15] D.P. Goodall, J. Wang, Practical stabilization of uncertain affine control systems with control constraints, Proc. IEEE Workshop on Robust Control via Variable Structure and Lyapunov Techniques, Benevento, Italy, 1994, pp. 56—63. [16] A.G. Soldatos, M. Corless, Stabilizing uncertain systems with bounded control, Dyn. Control 1 (1991) 227—238. [17] H.S. Binning, Robust feedback control of singularly perturbed uncertain nonlinear systems, Ph.D. Thesis, Coventry University, Coventry, UK, 1998. [18] G. Ambrosino, G. Celentano, F. Garofalo, Robust model tracking control for a class of nonlinear plants, IEEE Trans. Automat. Control AC-30 (1985) 275—279. [19] D.P. Goodall, Asymptotic stabilization of a class of uncertain composite systems, Dyn. Control 4 (1994) 311—326.
Constrained output feedbacks for singularly perturbed imperfectly known nonlinear systems H.S. Binning, D.P. Goodall* Control Theory and Applications Centre, Coventry University, Priory Street, Coventry CV1 5FB, UK Received 9 March 1998; accepted 21 July 1998
Abstract Global attractors are investigated for a class of imperfectly known, singularly perturbed, nonlinear systems subject to control constraints. The uncertain systems are modelled as nonlinear perturbations to a known nonlinear idealized system. The model is represented by two time-scale systems involving a scalar singular perturbation parameter, which reduces to a system of lower order when the singular perturbation parameter is set to zero. A class of constrained static output feedback controllers is developed which guarantees global attractivity of a compact set, containing the state origin, for all values of the singular perturbation parameter less than some threshold value. 1999 The Franklin Institute. Published by Elsevier Science Ltd. Keywords: Control constraints; Global uniform attractor; Output feedback controls; Singularly perturbed systems; Uncertain systems
1. Introduction Often, in many systems, problems due to the coexistence of slow and fast dynamics in the plant to be controlled can arise. These problems can be addressed utilizing singular perturbation theory (for more details see [1] or, for a differential-geometric approach, see [2, 3]). For dynamical systems with the above behaviour, the plant may be modelled as a set of n differential equations, representing the slow subsystem, and
* Corresponding author. Tel.:#44 (0) 1203 838730; Fax:#44 (0) 1203 838080; E-mail: D. Goodall@ coventry.ac.uk 0016-0032/99/$ — See front matter 1999 The Franklin Institute. Published by Elsevier Science Ltd. PII: S 00 1 6-0 0 32 ( 9 8) 0 0 03 8 - 6
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l differential equations modelling the fast dynamics which are dependent upon a small positive real (perturbation) parameter. Loosely speaking, the singular perturbation parameter can be interpreted as some type of estimate of the ‘‘ratio’’ between the slow and fast dynamics. It is assumed that the set of n#l differential equations degenerates to the set of n differential equations, known as the reduced-order system, when the perturbation parameter is set to zero. Many stabilizing controllers for singularly perturbed systems have been constructed utilizing Lyapunov’s second method (see e.g., [4]). For singularly perturbed systems with uncertainty, design of deterministic stabilizing controls have been investigated using a Lyapunov-based approach in [5—13] to name but a few. The main objective of this paper is to design, using a deterministic approach, a class of robust output feedback controls for singularly perturbed uncertain systems, subject to control constraints, in order to achieve a stability property relating to existence of compact sets, containing the origin, which are global uniform attractors for the reduced-order and full-order systems. The stability analysis is similar to that used in [10] but, here, the reduced-order system is based on a nonlinear affine control system. For the problem of synthesizing a class of output feedbacks for the unconstrained problem, see [6]. A preliminary version of this paper was presented in [5].
2. Full-order singularly perturbed system Consider a singularly perturbed uncertain dynamical system consisting of coupled subsystems with the following structure: f (x)#F (x) y#G (x)u#h (t, x, y, u, k), xR (t)"a(t, x(t), y(t), u(t), k) " (1a) A(t) [ f (x)#y#G (x) u]#h (t, x, y, u, k), (1b) kyR (t)"b(t, x(t), y(t), u(t), k) " z(t)"r(x(t), y(t)),
(1c)
where (x(t), y(t))31L;1l is the state of the system, u(t)31K is the control input, [0,R) denotes the singular perz(t)31L is the output, 1)m)l, n, and k31> " turbation parameter, assumed to be ‘‘small’’. The dimension of the output space coincides with the dimension of the slow subsystem and the full-order system is defined on 1L>l. It is assumed that the vector fields f 3C(1L) and f 3C(1l) are known and satisfy f (0)"0, f (0)"0, F (x)3L(1l , 1L) (the set of all continuous linear maps from 1l into 1L), G (x)3L(1K, 1L) and G (x)3L(1K, 1l) are known. The uncertainty in the system is represented by A(t), a Lebesgue measurable matrixvalued function, and the vector fields h , h , which are smooth, that is h 3C(1L) and h 3C(1l). Remarks. The structure specified in Eqs. (1a)—(1c) admits a large nontrivial class of uncertain systems. For example, the uncertain system (discussed in [10]) xR (t)"Ax(t)#[B#*B(t)] y (t)#d(t, x(t)), x31L,
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with actuator dynamics kyR (t)"[C #*C (t)] (y (t)!u(t)),
y , u31K,
and sensor dynamics kyR (t)"[C #*C (t)] (y (t)!x(t)), y 31L, and output z(t)"y (t), where *B(t), *C (t), *C (t) and d(t, x(t)) model uncertainty in the system, can be identified with Eqs. (1a)—(1c). In this case, the model without uncertainty is essentially linear in nature. An example of a nonlinear model, that represents a nonlinear flexible mechanical system, is xR (t)"x (t), x (t), x (t)31 xR (t)"(1#i(t)) [k sin(x (t))!I (k y (t)#k y (t))]!d(t) A ? ? @ kyR (t)"y (t), y (t), y (t)31 kyR (t)"(1#i(t)) [I\I (K x (t)#K x (t)#k sin(x (t))) ? @ A !I (k y (t)#k y (t))!u(t)!k(I !I ) qN (t, x (t))], @ ? @ @ ? with input u(t)31 and output z(t)"[x (t) x (t)]2, where i(t), d(t) and qN (t, x (t)) model uncertainty in the system. This model can also be identified with Eqs. (1a)—(1c). Furthermore, a nonlinear electromechanical system considered in [8] is a candidate for the structure described in Eqs. (1a)—(1c). It is assumed that the uncertainty in the second subsystem (1b) is characterized by the following hypotheses: For a linear map ¸, the notation #¸# denotes +max p(¸2¸),, where p( ) ) denotes spectrum. Assumption 1. (a) For all t, there exists , 3[0, #P #\) and A 3L(1l, 1l) such that A(t)"A #AK (t),
#AK (t)#), ,
A is known and p(A )L"\, where P '0 is the unique, symmetric solution of the Lyapunov equation P A #A2P #I"O.
(2)
(b) For all (t, x, y, u), h (t, x, y, u, 0)"0. 3. Reduced-order uncertain system Setting k"0, system (1a)—(1b) degenerates to the set of n differential equations xR (t)"f (x(t))#F (x(t)) y(t)#G (x(t)) u(t)#h (t, x(t), y(t), u(t), 0)
(3)
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subject to the constraint f (x)#y#G (x) u"0.
(4)
The degenerate system (3)—(4) and (1c) are equivalent to xR (t)"fI (x(t))#GI (x(t)) u(t)#hI (t, x(t), u(t)),
(5a)
z(t)"r(x(t), (x(t), u(t))),
(5b)
f (x)!F (x) f (x), GI (x) " G (x)!F (x)G (x), (x, u) " !f (x)!G (x) u where fI (x) " and hI (t, x, u) " h (t, x, (x, u), u, 0), and is known as the reduced-order system. The function is determined uniquely in view of Assumption 1(a). The Euclidean inner product (on 1L or 1l as appropriate) and the induced norm are denoted by 1 ) , )2 and # ) #, respectively. Let ¸ v : 1NP1 denote the Lie derivative of D a scalar field x>v(x)31 along the vector field f3C(1L). In particular, let v : 1x, P x2, then (¸ v ) (x)"1P x, f (x)2. The reduced-order system is x>v (x) " D characterized by the prescribed triple ( fI , GI , hI ) for which the following hypotheses are assumed to hold: Assumption 2. (a) There exists e(x)3C(1), which is strictly increasing as #x#PR, with e(0)"0 and e(x)'0 for all xO0, and a symmetric positive definite matrix P 31L;L such that (i) (¸ I v ) (x))!e(x), for all x. D (0, R), such that (ii) There exists real constants c , c 31>, with 1> " K c " (¸ v ) (x) ")+v (x),)c +e(x),, EG G
where g (x) denotes the ith column of the matrix GI (x). G (b) There exists d , d 31> , satisfying d #d '0, such that # fI (x)#)d #d #x#. (c) There exists unknown vector fields p3C(1K) and q3C(1K) such that hI (t, x, u)"p(t, x, u)#GI (x)q(t, x, u), ∀(t, x, u), and there exist real constants a , a , a , b , b , b 31> , b (1, such that, for all (t, x, u), (i) #p(t, x, u)#)a #x##a #a K "u ", where u denotes the ith component G G G of u. (ii) "q (t, x, u)")b #x##b #b " u ", where q denotes the ith component of q. G G G Remarks. (a) Assumption 2(a)(i) ensures that the state origin of the drift system xR "fI (x) is a global uniform attractor (see Definition 4.1, below).
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(b) Analogous to the characterization of uncertainty for linear systems (see, e.g., [14]), the vector fields p, q are said to represent the matched and unmatched components, respectively, of the uncertainty in the nonlinear reduced-order system.
4. Class of constrained output feedback controls It is supposed that the constraints " u (t) ")o , ∀t*t , i"1, 2, 2 , m, (6) G G where o '0 are prescribed, are now imposed on the control inputs of system G (1a)—(1c). It is desired that a class of output feedback controls, satisfying the prescribed constraints, be designed so that (a) there exists a compact set, containing the origin, which is a global uniform attractor for the reduced-order system (5a)—(5b) in the sense of Definition 4.1 (given below), and (b) the full-order system has the same stability property in the presence of singular perturbations for all k3(0, k*), where k* is some computable real constant. The design procedure is such that the output feedback control functions depend on a design parameter e. Definition 4.1. Let !L1N be a compact set satisfying !5+0," and let ! (r) N denote the open unit ball of radius r centred at the origin in 1N. The set ! is a global uniform attractor for the system fQ (t)"Z(t, f(t)), f(t)31N
(7)
if (a) for each (t , f)31;1N, there exists a local solution f: [t , t )P1N satisfying Eq. (7) a.e. with f(t )"f and every such local solution can be extended into a solution on [t ,R); (b) for each r31>, there exists R(r)31> such that, for every t 31, f(t)3! (R(r)) for N all t*t on every solution f : [t ,R)P1N with f(t )3! (r); N (c) for each d31>, there exists D(d)31> such that, for every t 31, f(t)3!#! (d) N for all t*t on every solution f : [t ,R)P1N with f (t )3!#! (D(d)); N (d) for each d, r31>, there exists ¹(d, r)31> such that, for every t 31, f(t)3!#! (r) for all t't #¹(d, r) on every solution f: [t ,R)P1N of Eq. (7) N with f(t )3! (d). N The synthesis of an output feedback controller z>w(z) follows the design methodology of [6, 10]. Note that in the absence of uncertainty for the reduced order system and if r(x, y)"x, for all (x, y)31L>l, then the output feedback law !(¸ v ) (z(t)) renders the zero-state of system (5a)—(5b) as a global uniform u(t) " EG
i.e., an absolutely continuous function. +s #s : s 3S , s 3S ,L1N. For sets S , S L1N, S #S "
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attractor. Define x>cJ (x)"[cJ (x) cJ (x)2 cJ (x)]2, where K !bM \ [ cJ #b cJ [cJ " (¸ v ) (x) "#e ]\] (¸ v ) (x), cJ (x) " G G EG G G EG (1!b )'0, and cJ , c , e 31> are design parameters. bM " G G
(8)
Remark. Each cJ (x) consists of two parts and incorporates the matched uncertainty G parameter b . This term is used as a switch to activate the second part when b O0. Thus, when b "0 the second part is not activated and, in this case, Eq. (8) reduces to the simplified form: cJ (x)"!b\c(¸ v ) (x). G EG Let wJ (x), a saturating version of cJ (x), be defined by o s(o\ cJ (x)), wJ (x) " G G G G where s : 1P1 is defined by
1, h>s(h) " h,
(9)
h'1, " h ")1,
!1, h(!1.
r(x, I (x)): It is assumed that the function x>rJ (x) "
I (x) " (x, wJ (x)) has a well-defined inverse.
1LP1L,
where
Assumption 3. The function x>rJ \(x) exists and is defined uniquely for all x. The proposed class of constrained output feedback controls z>w(z)" [w (z) w (z)2 w (z)]2 is such that each component has the following structure K " w (z) wJ (rJ \(z)). (10) G G By the definition of w(z), " w (z) ")o and, for the reduced-order system, the output G G (5b) is satisfied by z(t)"rJ (x(t)). Therefore, for the reduced-order system, the constrained output feedback controls can be expressed in terms of the state vector for the reduced-order system as u(t)"w(z(t))"wJ (rJ \(z(t)))"wJ (x(t)). 5. Boundary-layer system The full-order system can be expressed as xR (t)"aJ (t, x(t), y(t), z(t), k), kyR (t)"bI (t, x(t), y(t), z(t), k), z(t)"r(x(t), y(t)), where aJ (t, x, y, z, k)"a(t, x, y, w(z), k) and bI (t, x, y, z, k)"b(t, x, y, w(z), k).
(11)
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Introducing the rescaled time variable q"(t!t*)/k, where t*31 is fixed, and the x(t*#kq), yN (q) " y(t*#kq)#f (x(t*))#G (x(t*))wJ (x(t*)), vector fields xN (q) " zN (q) " z(t*#kq) yields the system xN (q)"kaJ (t*#kq, xN (q), yN (q)# (x(t*), wJ (x(t*))), zN (q), k), yN (q)"bI (t*#kq, xN (q), yN (q)# (x(t*), wJ (x(t*))), zN (q), k), zN (q)"r(xN (q), yN (q)# (x(t*), wJ (x(t*)))), where ( ) denotes differentiation with respect to q. By construction, bI (t*, xN , I (xN ), zN , 0)" 0 and, therefore, any point (xN , I (xN ), zN , w(zN ))3S, !f (x)!G (x) u,, S"+(x, y, z, u)31L;1l;1L;1K: y" (x, u) " is an equilibrium point of the system at k"0. The behaviour of the system at k"0 is characterized by the system yN (q)"bI (t*, x(t*), yN (q)# (x(t*), wJ (x(t*))), zN (q), 0)"A(t*) yN (q),
(12)
which is known as the boundary-layer system. Assumption 1(a) guarantees that Eq. (12) is asymptotically stable.
6. Stability properties of the full and reduced order systems 6.1. Reduced-order system With output feedback u(t)"w(z(t))"wJ (x(t)) and wJ defined by Eq. (9), the reducedorder system (5a)—(5b) can be expressed as K xR (t)"fI (x(t))# [q (t, x(t), wJ (x(t)))#wJ (x(t))]g (x(t))#p(t, x(t), wJ (x(t))), G G G G
(13a)
z(t)"rJ (x(t)).
(13b)
To ensure that the constrained controller can cope with the uncertainty in the system, the following feasibility assumption is required. Assumption 4. The control constraints satisfy o 'bM \b for all i. G If Assumption 4 holds, then there exist parameters cJ *1 satisfying G b cJ (bM o . (14) G G The parameters cJ are to be identified with those introduced in Eq. (8). G The desired stability property is obtained by using essentially the same approach as that of [5, 6]. Examining the behaviour of the time derivative of the function
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v : 1LP1> along solutions to (13a)—(13b), one obtains, using standard Lyapunov analysis, the following result. Define
c\!(2c\b +p (P ),\ !2a p (P) , s" (15) p (P )
min [bM o !b cJ ], oL " (16) G G G K e , cJ " K cJ , o " K o , and let E(r) denote the set +x31L: v (x))r,. e" G G G G G G The following theorem may be deduced. Theorem 6.1. Suppose Assumptions 1—4 hold. If s'0 and the design parameters, c , cJ , G satisfy cJ 71 for all i, c '0 and, if a #a O0, G c oL s , (17) (a #a o)+2p (P ),
then, with constrained output feedback control u"w(z) (given by Eqs. (8)—(10)), the compact set E(r ), where C r " r\ [r #(r#4r b e] , C r "s!c\a bM \c +2p (P ), and r "(a #a bM \b cJ ) +2p (P ), ,
is a global uniform attractor for the uncertain system (13a)—(13b), with x(t )"x. c (c* "
Proof. See Appendix A.1. ) Remarks. (a) It is advantageous to choose c , cJ and e as small as possible with the G G result that the size of E(r ) can be reduced, but only in the sense that E(r ) will tend to C C the limiting set E(r*), where r*"s\(a #a bM \b ) +2p (P ),.
(b) If a "a "0, then E((b es\) is a global uniform attractor and, therefore by tuning the parameters e , the size of this set can be made as small as desired. G (c) If a "b "0, then, as a consequence of Eq. (21) in the proof of Theorem 6.1, +0, is a global uniform attractor. 6.2. Full-order system For this final stage it is shown that, using the constrained output feedback (10), the full-order uncertain system (1a)—(1c) has a global uniform attractor. The methodology follows that of [10] (see, also, [5, 11]). Consider the Lyapunov function candidate v : I v (x)#m v (x, y), where m 31>, a real constant 1L;1lP1> defined by v (x, y) " I I I dependent upon the singular perturbation parameter, is to be specified and 1y! I (x), P (y! I (x))2. Along solutions to Eqs. (1a)—(1c) and for v (x, y) "
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almost all t, d v (x(t), y(t))"» (t, x(t), y(t)), I dt I where 1( v ) (x), a(t, x, y, w(r(x, y)), k)2 » (t, x, y) " I #m +1( v ) (x, y), a(t, x, y, w(r(x, y)), k)2 I V #k\1( v ) (x, y), b(t, x, y, w(r(x, y)), k)2,. W Assumption 5. (a) For all x, there exist k , k 31> such that #F (x)#)k and #G (x)#)k . (b) There exists a continuous function j: 1>P1>, with j(0)"0, such that, for all (t, x, y, u)31;1L;1l;1K and k31>, #h (t, x, y, u, k)!h (t, x, y, u, 0)#)j(k). (c) There exist k , k 31> such that, for all y , y 31l and (t, x)31;1L, #h (t, x, y , w(r(x, y )), k)!h (t, x, y , w(r(x, y )), k) #)k #k #y !y #. Under Assumptions 1—5 and utilizing Eqs. (6) and (21), one may deduce the following lemma. Lemma 6.1. Suppose Assumptions 1—5 hold. For all (t, x, y)31;1L;1l and k31>, 1( v ) (x), a(t, x, y, w(r(x, y)), k)2)!r v (x)#(i #k(k)) +v (x),#b e #i +v (x),+v (x, y), , where
r #k +2p (P ),, if z"r(x, y) is independent of y or G (x)"O,
i " r #2oc\#(2ok k #k ) +2p (P ),, otherwise,
j(k)+2p (P ),. i "2(k #k ) +p (P )/p (P ),, and k>k(k) "
Proof. See Appendix A.2. ) Remark. If the output function z"r(x, y) is independent of y, then rJ (x)"r(x) and, hence, w(z)"wJ (rJ \(r(x)))"wJ (x). In this case, i "r #k +2p (P ), and, it is shown later, see Corollary 6.1, that, under an additional assumption, the asymptotic behaviour of the output controlled full-order system tends to that of the controlled reduced-order system. Assumpiton 6. The Fre& chet derivative of I satisfies #(D I ) (x)#,
K #(D I ) (x) g (x)# )k , k 31>, for all x. G G
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Remark. Assumptions 6, 5(a), 2(a) and (b) are quite severe restrictions on the nonlinearity of the full-order system. Lemma 6.2. Assuming the conditions stated in Assumptions 2(b), (c), 5 and 6 hold, then, for all (t, x, y)31;1L;1l and k31>, 1( v ) (x, y), a(t, x, y, w(r(x, y), k)2)(i +v (x),#i #k k(k)) +v (x, y), V #i v (x, y), where i "2k (d #a #mb ) +p (P )/p (P ),, i "k (d #a #mb #k #o(1#a #b #2k k )) +2p (P ),,
i "2k (k #k )+p (P )/p (P ),. Proof. See Appendix A.3. ) Assumption 7. There exist known scalars j , j , j 31> such that, for all k31> and (t, x, y)31;1L;1l , #h (t, x, y, w(r(x, y)), k)#)k[j #y! I (x)##j #x##j ]. Lemma 6.3. ºnder Assumptions 1(a), 5(a) and 7 and for all (t, x, y, k)3 1;1L;1l;1>, 1( v ) (x, y), b(t, x, y, w(r(x, y)), k)2)ki +v (x), +v (x, y), W #(ki #i )+v (x, y),#(ki !i ) v (x, y), where i "2j +p (P )/p (P ),,
i "j +2p (P ),,
0, if z is independent of y or G (x)"O, i" 2ok (#P A ##, #P #)+2/p (P ),, otherwise,
i "2j +p (P )/p (P ),,
i "(1!2, # P #) +p (P ),\.
Proof. See Appendix A.4. ) Let m "n(k/k*), where n, k*31>, and select real h31>. Then, since i '0, I
r i [r (i #i )#i (i #i )]\, i (i #i )O0, k*" r i [r (i #i )#h]\, otherwise,
(18)
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where
i (i #i )\, i h\, n" 4h(i #i )\, h,
i (i #i )O0, i O0, i #i "0, i "0, i #i O0, i "0, i #i "0 ,
has the property that the full-order uncertain system has the desired behaviour, provided k(k*. This type of structure for k* was first presented in [10, 11]. However, the value of k*, given in Eq. (18), is not unique, that is there are other possible threshold values, and the threshold estimate (18) may be rather conservative. Remarks. The formula that is provided to compute a threshold value of k, namely (18), often produces a very conservative value and examples have suggested that the value may differ from the true threshold value by a factor of at least 10. Other examples have suggested that some systems may be sensitive to the value of k in the sense that for a certain value of k there exists a compact attractor, but the system becomes unstable when k is changed by a relatively small amount. Theorem 6.2. Suppose Assumptions 1—7 are satisfied and k3(0, k*) is fixed. If s'0, then, with u(t)"w(z(t)) (defined in Eqs. (8) and (9)), c '0, cJ *1 for all i and, if G a #a O0, c satisfies Eq. (17), the compact set +(x, y)31L;1l : v (x, y))r,, W" I I I where v \ [gN #+gN #4lN b e,], r " I I I I I l " #M\ #\ min[1, m\] and g "([i #k(k)]#m [i #i #k\i I I I I I #k k(k)]), is a global uniform attractor for the full-order uncertain system (1a)—(1c) subject to the initial condition (x(t ), y(t ))"(x, y). Proof. See Appendix A.5. ) Under additional assumptions, the asymptotic behaviour of the solution of the controlled full-order system can be shown to tend to that of the controlled reducedorder system, as seen in the following corollary. Corollary 6.1. Suppose the conditions stated in ¹heorem 6.2 hold, then the set E(r ) is I a global uniform attractor under the dynamics of system (1a)—(1c). Moreover, if
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z"r(x, y) is independent of y and k "0, then lim d(E(r ), E(r ))"0, I C I where d denotes a Hausdorff metric. Proof. See Appendix A.6. )
7. Illustrative example Consider the controlled singularly perturbed uncertain nonlinear system
2#x(t) #y (t)#h (t, x(t), y(t), u(t), k), (19a) 1#x(t) kyR (t)"(!1#aL (t)) (y (t)!u(t)), (19b) kyR (t)"(!3#a' (t)) (y (t)!20x(t))#h (t, x(t), y(t), u(t), k), (19c) with initial conditions (x, y, y)"(3,!2, 5) and output z(t)"y (t)#y (t), where x(t), z(t)31, y(t)"[y y ]231. The system can be identified with (1a)—(1c) where (2#x) f (x)"!x , F (x)"[1 0], G (x)"0, 1#x xR (t)"!x(t)
0 ! !1#a' (t) 0 , A(t)" f (x)" , G " . !20x 0 0 !3#a' (t) It is assumed that the control constraint " u ")2 is imposed on the system. In order to perform a simulation on this system, it is supposed that a' (t)" (1#3e\R ) and a' (t)" sin(2t). Given 0 !1 0 , then P " satisfies Eq. (2) A " 0 !3 0 and, since
AK (t)"
1 1!2e\R 0 , 9 0 sin(2t)
it follows that , " . Hence, Assumption 1(a) is satisfied as , 3[0, #P #\)" [0, 1). For simulation purposes the following uncertainty functions are selected: h (t, x, y, u, k)" x! y cos(3t)! u sin(t)#sin(k) e\I, h (t, x, y, u, k)"k cos(t) [2(y !u)#5x#e\R sin(x)]. Thus, Assumption 1(b) is valid and Assumption 5 holds with k "1, k " , k ", k " , and j(k)"" sin(k)e\I ".
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The reduced-order equation is given by fI (x)"!x
5 1 (2#x) , GI (x)" , hI (t, x, u)" x! u [cos(3t)#sin(t)]. 1#x 4 8
Let P "1, then (¸ I v ) (x)"!x((2#x)/1#x) and (¸ v ) (x)" x. Therefore, EG D Assumption 2 holds with e(x)"x((2#x)/1#x), c "(2, c " , d "1 and d ". Since p(t, x, u)"0 for all (t, x, u), then a "a "a "0 and b " , b "0, b ", which validates Assumption 2(c). Since bM " and b "0, the control for the reduced-order system is wJ (x)" 2s(0.5cJ (x)), where cJ (x)"! cN x. Thus, wJ (x) s(!c x)
I (x)" (x, wJ (x))" " 20x 20x
and hence Assumption 6 holds with k "((cN /9)#400. Now rL (x)"r(x, I (x))"s(! cN x)#20x 20x!1, x'N , A N " (20!A ) x, !N )x)N , A A 20x#1, x(!N . A Selecting 20!(cN /3)'0, that is cN (60, Assumption 3 holds with
rL \(x)"
x!1 , 20 3x , 60!cN x#1 , 20
60 x(! #1, cN
" x ")
60 !1 , cN
60 x' !1. cN
Now consider Assumption 7. Note that " h (t, x, y, w(r(x, y)), k) ")k " 2(y !w(r(x, y)))#5x#e\R sin(x) " )k " 2(y !20x)#45x!2w(r(x, y))#e\R sin(x) " )k+2 " y !20x "#45 " x "#5,. But
y !s(! AN x) y !20x and so " y !20x ")#y! I (x)#. y! I (x)"
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Hence, Assumption 7 is satisfied with j "2, j "45 and j "5. A simple calculation shows that s" satisfies the stability criterion stated in Theorem 6.2. The constrained output feedback control for the system (19a)—(19c)has the following form: u(t)"w(y (t)#y (t))"2s(! cL r' \(y (t)#y (t))). Selecting cN "30 for simulation, u"2s(!10rL \(y #y )) 2, y #y (!1, " !2(y #y ), " y #y ")1, !2, y #y '1. The simulation results are shown in Figs 1—4. Figure 1 illustrates the dynamics of the full-order open-loop system, that is with no control, and shows the dynamics of the full-order closed-loop system when k"0.01. In addition, Figs. 2—4 show the state trajectories when the singular perturbation parameter has the value k"0.1, k"0.45 and k"0.46, respectively. Examination of Fig. 4 appears to indicate that the value of k has exceeded its threshold value, whilst Figs. 2 and 3 suggest that the threshold value is greater than 0.1. The theory predicts that k*+0.001, which is a conservative estimate for the threshold value, and a global attractor exists for the full-order system when k(k*.
Appendix A A.1. Proof of Theorem 6.1 For almost all t and along all solutions to Eqs. (13a) and (13b), K vR (x(t))"(¸ I v ) (x(t))# [q (t, x(t), wJ (x(t)))#wJ (x(t))] (¸ v ) (x(t)) D G G EG G #1P x(t), p(t, x(t), wJ (x(t)))2. Consider the case " w (x) "'o . G G Then, in view of Eq. (9), " wJ (x) ""o and wJ (x) (¸ v ) (x)"!o " (¸ v ) (x) ". Since G G G EG G EG #x#)2v (x)/p (P ) and #P x#)2p (P ) v (x)
and utilizing Assumptions 2(a) (i) and 2(c), K vR (x(t)))!e(x(t))#(2b +p (P ),\ +v (x(t)), " (¸ v ) (x(t)) " EG G p (P )
#2a v (x(t))#(a #a o) +2p (P ),+v (x(t)), p (P )
K ! [bM o !b ] " (¸ v ) (x(t)) ". G EG G
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Fig. 1. State histories for the open-loop system (left) and closed-loop system (right), including control history, with k"0.01.
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Fig. 2. State and control histories for the closed-loop system with k"0.1.
Fig. 3. State and control histories for the closed-loop system with k"0.45.
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Fig. 4. State and control histories for the closed-loop system with k"0.46.
In view of Eq. (15) and Assumptions (2a) (ii) and 5, vR (x(t)))!sv (x(t))#(a #a o ) +2p ( P ),+v ((x),. (A.1)
Invoking LaSalle’s invariance principle, all solutions of Eq. (A.1) tend to the largest invariant subset of +x31L: +v (x),)(a #a o) +2p (P ),/s,. S"
A subset of 1L is now determined for which the unconstrained feedback controller satisfies the prescribed constraints. As a consequence of Eq. (8) and since " cJ (x) "(bM \(c " (¸ v ) (x) "#b cJ ), G EG G then " cJ (x) ")o if " (¸ v ) (x) ")c \(bM o !b cJ ). The set G G EG G G 7 +x31L: " (¸ v ) (x) ")c \(bM oJ !b cJ ), R* " EG G G G is known as a constrained control region (see [15, 16]) and if the states of the system (13a)—(13b) lie in this region, the unconstrained controller cJ , defined in (8), satisfies the control constraints, namely wJ (x)"cJ (x) for all x3R*.
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min [(bM oJ (!b cJ )], As a consequence of Assumption 2(a) (ii) and defining oL " G G G S* " +x31L: +v (x),)c cN \oL ,LR* and, thus, S* is a compact constrained controlled region. Suppose a #a O0, then the parameter cN , which satisfies Eq. (17), is designed so that SLS*. Alternatively, if a "a "0, then S*LS"+0,. Hence, if x31L!S*, all solutions eventually enter S*, which is positively invariant, in finite time. Now suppose x3S*. Then, using Eq. (9),
b cJ G " (¸ v ) (x) ", wJ (x) (¸ v ) (x)"!bM \ cN # EG G EG cJ " (¸ v ) (x) "#e G EG G b cJ G " (¸ v )(x) ". " wJ (x) ""bM \ c # EG G cJ " (¸ v ) (x) "#e G EG G Utilizing Assumption 2, a straightforward calculation shows that
vR (x(t)))!(s!c\ a bM \cN +2p (P ),) v (x(t))
#(a #a bM \b cJ ) +2p (P ),+v (x(t)),
K K !cN "(¸ v ) (x(t)) "!b (cJ !1) (¸ v ) (x(t))#b e EG G EG G G )!(s!c\ a bM \cN +2p (P ),) v (x(t))
M \b #(a #a b cJ ) +2p (P ),+v (x(t)),#b e.
M \c Suppose that a #a O0. It is now shown that s'c\a b N +2p (P ),. Note
that, since cN (c*, oL sa bM \ . c\a bM \cN +2p (P ),(
a #a o By definition of oL , oL )bM o!b cJ and so a (o!bM \b cJ ) s s!c\a bM \cN +2p (P ),'s!
a #a o M \b (a #a b cJ ) s " a #a o *0. Thus, with r "s!c\a bM \cN +2p (P ),'0 and r "(a #a bM \b cJ ) +2p (P ),,
vR (x(t)))!r v (x(t))#r +v (x(t)),#b e (A.2) (0 for (t, x(t)),1;E(r ). (A.3) C
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In view of Eq. (A.2) and (A.3), every local solution evolves within the compact set +x31L: v (x) )v (x), 6 E(r ) and, hence, can be continued indefinitely. Property (b) C of Definition 4.1 holds with
[ #P\ # #P #]r, if r'(2 #P #\r , C (2#P\ #r , if r)(2 #P #\r . C C Moreover, property (c) of Definition 4.1 holds with D(d)"[ #P\ # #P #]\ d. Let ! be any compact set such that E(r ) is contained in an open set which is Z contained in !. Then, there exists a closed set E(rN ), with rN 'r such that E(rN )L!. Z Since, for each rJ *rN ('r ), C J P [r !r +v ,!b e]\ dv (R, 0)I(rJ , rN ) " PN property (d) of Definition 4.1 can be verified to hold with R(r)"
¹(d, r)"
I (( #P #r, rN ), 0
r'(2#P #\rN , r)(2#P #\rN .
This concludes the proof. A.2. Proof of Lemma 6.1 A straightforward calculation shows that f (x)#F (x)y#G (x) w(r(x, y))"fI (x)#GI (x) wJ (x)#F (x)(y! I (x)) #(GI (x)#F (x) G (x)) [w(r(x, y)) !w(r(x, I (x)))] and, therefore, for all (t, x, y)31;1L;1l and k31>, 1( v )(x), a(t, x, y, w(r(x, y)), k)2"1P x, fI (x)#GI (x)wJ (x) #h (t, x, I (x), w(r(x, I(x))), 0)2 #1P x, h (t, x, I (x), w(r(x, I (x))), k) !h (t, x, I (x), w(r(x, I (x))), 0)2 #1P x, h (t, x, y, w(r(x, y)), k) !h (t, x, I (x), w(r(x, I (x))), k)2 #1P x, GI (x) [w(r(x, y))!w(r(x, I (x)))]2 #1P x, F (x) G (x) [w(r(x, y))!w(r(x, I (x)))]2 #1P xF (x) (y! I (x))2. (A.4)
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Note that if G (x)"GI (x)#F (x) G (x)"O or z"r(x, y) is independent of y, then the 4th and 5th terms on the r.h.s of Eq. (A.4) are identically zero. In view of Assumption 5(a) and since #[w(r(x, y))!w(r(x, I (x)))]#)2o, 1P x, F (x) G (x) [w(r(x, y))!w(r(x, I (x)))]2#1P x, F (x)(y! I (x))2 p (P )
+v (x),+v (x, y),#2k k o +2p (P ),+v (x),. )2k
p (P )
Invoking Assumption 2(a) (ii),
K 1P x, GI (x) [w(r(x, y))!w(r(x, I (x)))]2" [w (r(x, y))!w (r(x, I (x)))] (¸ v ) (x) G G EG G )2oc\ +v (x),. As a consequence of Assumptions 5(a) and (b), 1P x, h (t, x, I (x), w(r(x, I (x))), k)!h (t, x, I (x), w(r(x, I (x))), 0)2 )j(k) +2p (P ),+v (x),
and 1P x, h (t, x, y, w(r(x, y)), k)!h (t, x, I (x), w(r(x, I (x))), k)2 p (P )
+v (x),+v (x, y),#k +p (P ),+v (x),. )2k p (P )
Thus, utilizing Eq. (A.2), the result follows.
A.3. Proof of Lemma 6.2 Since h (t, x, I (x), wJ (x), 0)"p(t, x, wJ (x))#GI (x) q(t, x, wJ (x)) and v (x, y)" V !(D I )2(x) P (y! I (x)), then, for all (t, x, y, k)31;1L;1l;1> , 1 v (x, y), a(t, x, y, w(r(x, y)), k)2)#(D I )(x)# #P (y! I (x))# [ # fI (x)# V ##p(t, x, wJ (x)###h (t, x, y, w(r(x, y)), k) !h (t, x, I (x), w(r(x, I (x))), k)# ##F (x)# #y! I (x) # ##h (t, x, I (x), w(r(x, I (x))), k) !h (t, x, I (x), w(r(x, I (x))), 0)# ##F (x)##G (x)##w(r(x, y))!w(r(x, I (x)))#] K ##P (y! I (x))# " w (r(x, y)) G G #q (t, x, wJ (x)) " #(D I ) (x)g (x)#. G G
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Utilizing Assumptions 2(b) and (c), # fI (x)###p(t, x, wJ (x))#)d #a #a oN #(d #a )#x#, " w (r(x, y))#q (t, x, wJ (x)) ")b #o (1#b )#b # x #. G G G As a consequence of Assumptions 5 and 6, 1 v (x, y), a(t, x, y, w(r(x, y)), k2)k #P [y! I (x)] # +(d #a #mb ) #x# V #d #a #mb #k #o(1#a #b I #2k k )#(k #k ) #y! (x)##j(k),. The statement of the lemma now follows in view of the results: #x#)+2/p (P ),+v (x),, #y! I (x)#)+2/p (P ),+v (x, y),,
I and #P [y! (x)]#)+2p (P ),+v (x, y),.
(A.5)
A.4. Proof of Lemma 6.3 Since v (x, y)" 1y! I (x) P [y! I (x)]2, 1( v ) (x, y), b(t, x, y, w(r(x, y)), k)2"1P [y! I (x)], A(t) [y! I (x)]2 W #P [y! I (x)], A(t) G (x) (w(r(x, y)) I !w(r(x, (x))))2 #1P [y! I (x)] h (t, x, y, w(r(x, y)), k)2. Utilizing Assumptions 1(a) and 5(a), 1P [y! I (x)], A(t) [y! I (x)]2)! (1!2i #P #) #y! I (x)#, 1P [y! I (x)], A(t) G (x) (w(r(x, y))!w(r(x, I (x))))2)2ok [#P A # #, #P #] #y! I (x)#. However, note that if G (x)"O or z"r(x, y) is independent of y, then 1P [y! I (x)], A(t)G (x) (w(r(x, y))!w(r(x, I (x))))2"0. Finally, as a consequence of Assumption 7, 1P [y! I (x)], h (t, x, y, w(r(x, y)), k)2)2kj [p (P )/p (P )]v (x, y) #2kj [p (P )/p (P )] +v (x, y), ;+v (x), #kj [2p (P )]+v (x, y),
and, in view of Eq. (A.5) and since #y! I (x)#*+2/p (P ) v (x, y),, the result then
follows.
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A.5. Proof of Theorem 6.2 As a consequence of Lemmas 6.1—6.3,
+v (x), +v (x), , M I +v (x, y), +v (x, y), #(i #k(k)) +v (x),#m (i #i #k\i I #k k(k)) +v (x, y),#b e,
» (t, x, y))! I
(26)
where
! (i #m (i #i )) r I . M" I ! (i #m (i #i )) m (k\i !i !i ) I I With k* defined as in Eq. (18), det(M )'0 for all k3(0, k*) if r '0 (see [11, 17]) and, I therefore, M is positive definite if s'0. Thus, if s'0 and k(k*, I » (t, x, y))!l v (x, y)#gN +v (x, y),#b e, I I I I I where l "# M\ #\ min[1, m\], I I I gN "([i #k(k)]#m [i #i #k\i #k k(k)]). I I Since 1 v (x, y)" I 2
x x , PI I y! I (x) y! I (x)
,
where
O P PI " , I O mP I standard arguments can be employed, analogous to those used in the proof of Theorem 6.1 (given in Appendix A.1), to show that properties (a)—(d) of Definition 4.1 hold. A.6. Proof of Corollary 6.1 Let i "i #i #k\i #k k(k). In view of Eq. (26), I » (t, x, y))!lN v (x, y)#(i #k(k)) +v (x),#m i +v (x, y),#b e I I I I I "!l v (x)#(i #k(k)) +v (x),!l [(m v (x, y)) I I I !iN l \ ]# m l \i #b e I I I I I )!l v (x)#(i #k(k)) +v (x),# m l \i #b e. I I I I
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This implies E(r' ) is an attractor for system (1a)—(1c), where I lN \ [i #k(k)#(g #4l b e], r' " I I I I and, since r' )r , E(r ) is a global attractor. I I I If z"r(x, y) is independent of y and k "0, then i "r and i "0. Some straightforward calculations show that lim #M\ #\"r , lim l "lim #M\#\"r and lim g "r . I I I I I I I I Thus, lim r "r , I C I where r is defined in Theorem 6.1. This concludes the proof. C References [1] P.V. Kokotovic´, H.K. Khalil, J. O’Reilly, Singular Perturbation Methods in Control: Analysis and Design, Academic Press, New York, 1986. [2] A. Isidori, Nonlinear Control Systems, Springer, Berlin, Germany, 1989. [3] R. Marino, P.V. Kokotovic´, A geometric approach to nonlinear singularly perturbed control systems, Automatica 24 (1988) 31—41. [4] A. Saberi, H. Khalil, Quadratic-type Lyapunov functions for singularly perturbed systems, IEEE Trans. Automat. Control AC-29 (1984) 542—550. [5] H.S. Binning, D.P. Goodall, Output feedback controls for a class of imperfectly known singularly perturbed nonlinear systems subject to control constraints, Proc. 4th IEEE Mediterranean Symp. on New Directions in Control and Automation, Maleme, Krete, Greece, 1996, pp. 53—58. [6] H.S. Binning, D.P. Goodall, Output feedback controls for a class of singularly perturbed uncertain nonlinear systems, Automat. Remote Control 59 (7) (1997) 81—97. [7] M. Corless, Asymptotic stability of singularly perturbed systems that have marginally stable boundary-layer systems, Dyn. Control 1 (1991) 95—108. [8] M. Corless, Robustness of a class of feedback-controlled uncertain nonlinear systems in the presence of singular perturbations, Proc. American Control Conf., Minneapolis, USA, 1987, pp. 1584—1589. [9] M. Corless, F. Garofalo, L. Glielmo, New results on composite control of singularly perturbed uncertain linear systems, Automatica 29 (1993) 387—400. [10] M. Corless, G. Leitmann, E.P. Ryan, Control of uncertain systems with neglected dynamics, in: A.S.I. Zinober (Ed.), Deterministic Control of Uncertain Systems, Peter Peregrinus, London, 1990, pp. 252—268. [11] M. Corless, E.P. Ryan, Robust feedback control of singularly perturbed uncertain dynamical systems, Dyn. Stability Systems 6 (1991) 107—121. [12] F. Garofalo, G. Leitmann, Nonlinear composite control of a class of nominally linear singularly perturbed uncertain systems, in: A.S.I. Zinober (Ed.), Deterministic Control of Uncertain Systems, Peter Peregrinus, London, 1990, pp. 269—288. [13] G. Leitmann, E.P. Ryan, A. Steinberg, Feedback control of uncertain systems: robustness with respect to neglected actuator and sensor dynamics, Int. J. Control 43 (1986) 1243—1256. [14] B.R. Barmish, G. Leitmann, On ultimate boundedness control of uncertain systems in the absence of matching assumptions, IEEE Trans. Automat. Control AC-27 (1982) 153—158.
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[15] D.P. Goodall, J. Wang, Practical stabilization of uncertain affine control systems with control constraints, Proc. IEEE Workshop on Robust Control via Variable Structure and Lyapunov Techniques, Benevento, Italy, 1994, pp. 56—63. [16] A.G. Soldatos, M. Corless, Stabilizing uncertain systems with bounded control, Dyn. Control 1 (1991) 227—238. [17] H.S. Binning, Robust feedback control of singularly perturbed uncertain nonlinear systems, Ph.D. Thesis, Coventry University, Coventry, UK, 1998. [18] G. Ambrosino, G. Celentano, F. Garofalo, Robust model tracking control for a class of nonlinear plants, IEEE Trans. Automat. Control AC-30 (1985) 275—279. [19] D.P. Goodall, Asymptotic stabilization of a class of uncertain composite systems, Dyn. Control 4 (1994) 311—326.