IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 9, SEPTEMBER 2005
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Output Zeroing of MIMO Plants in the Presence of Actuator and Sensor Uncertain Hysteresis Nonlinearities Gianfranco Parlangeli and Maria Letizia Corradini Fig. 1. Block scheme of a plant driven by the actuator. Abstract—This note tackles the output zeroing problem of a square multiple-input–multiple-output plant containing uncertain nonsmooth hysteresis-like nonlinearities in actuator and sensor devices simultaneously. To this purpose, a robust sliding mode controller, based on the output of a reduced order observer, has been designed using it, such control law ensuring both the asymptotical zeroing of the sensing nonlinearities output and the boundedness of the unavailable states. Theoretical results have been validated by simulation. Index Terms—Hysteresis, nonsmooth nonlinearities, reduced-order observer, robust control, sliding-mode control.
I. INTRODUCTION The development of control techniques able to compensate for the presence of nondifferentiable nonlinearities is, at present, a challenging issue in control theory, mostly in view of the practical relevance of the problem. Indeed, traditional approaches tend to ignore those nonlinearities in control design for their nondifferentiable and nonmemoryless character, this resulting in severe deterioration of system performance which can even lead to instability. It cannot be denied, in fact, that a wide range of physical systems and devices (i.e., motor servo systems, mechanical actuators, electronic relay circuits) show nonsmooth behaviors, memory effects and/or time delays. In other words, nonsmooth nonlinearities such as hysteresis, backlash, dead-zone, are always present in real control plants, due to both physical imperfections and inherent characteristics of the controlled system. A key point in addressing such a challenge is the definition of a suitable model, to be used for control design, describing the nonlinear behavior. A number of different models have been developed for modeling the complex behavior shown by hysteresis, most of all heavy and hardly suitable for use in controller design [1], [2]. Incidentally, it is worth realizing that, as recalled in [3], hysteresis modeling is itself still a research topic [2]. Perhaps the most familiar hysteresis model is the piecewise linear model proposed in [4] and [5], which probably owes its popularity to its successful application in designing adaptive control techniques. Stemming from such model, following the lines of recently published results [6], [7], an “extended” hysteresis model has been here adopted, which contains deadzone and backlash as particular cases. Uncertain but bounded model parameters have been considered. Although physical evidence shows that nonsmooth nonlinearities are usually simultaneously present both in actuators and in sensors, nonlinearities affecting the actuator only have been more frequently addressed in the literature. In the adaptive framework, a well-known approach consists in building inverses of actuator deadzones, backlash, and hysteresis with unknown parameters [5], [8]–[11] in the controller, to the purpose of cancelling their effects. It is worth noting, however, that when such adaptive inverses are used for control, the effect of hysteresis may not be completely cancelled [4]. Variable structure control (VSC) techniques have been used as well [12]–[14].
Manuscript received July 30, 2004; revised May 3, 2005. Recommended by Associate Editor J. Berg. G. Parlangeli is with the Dipartimento di Ingegneria dell’Innovazione, Università di Lecce, 73100 Lecce, Italy. M. L. Corradini is with the Dipartimento di Matematica e Informatica, Università di Camerino, 62032 Camerino, Italy. Digital Object Identifier 10.1109/TAC.2005.854629
In the framework of tracking, and still addressing nonlinearities affecting the actuator only, [15] proposed to model the actuator nonlinearity as an upper semicontinuous set-valued map, and ensured the asymptotic convergence of the output within a known interval. In [16], asymptotic output tracking of any constant reference signal is guaranteed, at the price of stronger hypotheses on disturbances model (globally Lipschitz continuity is required). Less attention has been given, in the literature, to nonsmooth nonlinearities affecting plant output sensors also. Experimental evidence, however, shows that the performance of control systems is severely affected, as in the case of actuator nonlinearities [17]. In this framework, results based on adaptive control are available [18], [11], [5] as well as compensation techniques by disturbance observers [19]. Trying to fill this gap, this note addresses the simultaneous presence of uncertain hysteresis-like nonsmooth nonlinearities both in actuators and in sensors. Exploiting the well known robustness features and the discontinuous character of sliding mode control [20], a robust controller is here proposed, able to ensure both the asymptotical zeroing of the sensing nonlinearities output and the asymptotic boundedness of the unavailable states. Previous results have been extended to deal with multiple-input–multiple-output (MIMO) square plants, and the proposed controller has been validated by simulation using the model discussed in [21]. It may be worthwhile to note that model inversion is not required, to avoid the possible amplification of additive measurement disturbances which may result from inversion of the output nonlinearity. II. SYSTEM MODEL AND PROBLEM STATEMENT Consider a square system described by the equations
2
x_ = Ax + B u y = C x u = fu [ v ] y m = f y [y ]
(1) (2) (3)
2 2
n m where x(t) IR is the state vector at time t; u(t) IR is the n2n n2m is the state matrix, B IR is the system input vector, A IR input distribution matrix, y(t) IRm is the system output vector, C m2n IR is the output distribution matrix. As shown in Fig. 1, the plant (1) is considered driven by the input variable v of an actuator block preceding it. Each plant input ui is not directly available for control, since it is considered to be the output of an actuator affected by a nonsmooth m IR . This nonlinearity described by the vector function fu : IRm function connects each actuator input vi (actually available for control) with the corresponding system input ui through m hysteresis-like scalar blocks. Measurements are also affected by sensor nonsmooth m IR , connonlinearities described by the vector function fy : IRm necting each output ym i (actually available for measurement) with the corresponding system output yi through m nonlinear hysteresis-like scalar blocks. The characteristics of each hysteresis-like nonlinearity are shown in Fig. 2. The complete analytical description of the hysteresis is here omitted for sake of brevity, but details can be found in [7]. Hysteresis characteristics are completely described by ten paramt; l; r; b; s; i and four intersections eters, namely six slopes mj
0018-9286/$20.00 © 2005 IEEE
2
2
!
!
2f
g
2
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Fig. 2. Sketch of the nominal hysteresis and uncertainty region.
with the u 0 v axes cj 2 ft; l; r; bg. The values of these parameters are assumed uncertain. Remark 2.1: Note that backlash and dead zone are special cases of the adopted model. Backlash can be obtained with mt ; mb ; mr ml and sufficiently large ct and cb , while dead zone is obtained setting ct cb mt mb . CB 6 Assumption 2.1: Following [22], it is assumed that: i) , and ii) the invariant zeros of A; B; C are in 0 . This assumption has been shown [22] to be a necessary condition for an output feedback stable sliding motion to exist on the hyperplane S N C for a linear system A; B; C . Analogous assumptions are introduced also in paper pursuing different approaches [23], [15]. Assumption 2.2: The system initial state is assumed uncertain within a bounded set. Assumption 2.3: Coefficients describing the hysteresis nonlinmj earity are uncertain with bounded uncertainties, i.e., mj cj c j j cj j mj j mj j mj ; j 2 ft; l; r; b; s; ig; cj cj ; jj 2 ft; l; r; bg. Hysteresis loop slopes mj j 2 ft; l; r; bg are assumed nonnegative, whilst ms and mi are assumed strictly positive. As a consequence of Assumption 2.3, the outputs of a device affected by hysteresis has bounded uncertainties, too. It follows that: i) ! and u ! such that ui there exist functions fu u i vi ; ui vi ju i vi j 8vi 2 ; 8i ; ; m, and fu vi ! and functions ii) there exist strictly increasing functions fy y ! such that yi fy01 ym y i ymi ; y i ym i jy i ymi j 8ym 2 ; 8i ; ; m. By Assumption 2.3, it is always possible to find a value of the available input v such that any initial point lying inside the hysteresis loop is forced toward the boundary of the hysteresis loop, even in presence of uncertainties. The problem here addressed, provided that Assumptions 2.1–2.3 are satisfied, is finding an output feedback controller guaranteeing the vanishing of the measured output of the system (1) in the presence of uncertain hysteresis nonlinearities in actuators (2) and sensors (3).
0
=
=0
= =
= (
0
(
=0 )
=
det ( ) =
= ( )
)
= ^ + = ^ +1 ; 1 ;
1 ;1
^ : IR IR : IR IR = ^ ( )+ ( ) ( ) ( ) IR = 1 . . . ^ : IR IR ^ : IR IR = ( )+ ( ) ( ) ( ) IR = 1 . . .
III. PRELIMINARIES Some preliminary steps will be now performed making resort to the reduced order observer theory. As is well known, C being full row rank, there exists a linear state transformation such that the system (1) can
_ =[ = [ ] _= A ) +(B + B )u
_ ] = [ A A ][ ] + [ B ]u = _ A A B + 2 IR _ + _ = (A + A ) + (A + (A + A )
1 1 11 12 1 ;y be transformed as: y y 21 22 2 (n0m)2m Ly; L 1 , the dynamics O I . Defining L 21 1 Ly of is described by: 12 11 L 21 Ly , L 2 . By adding and subtracting L 22 y 11 1
it is straightforward to show that the best estimate of , based on meaA11 LA21 A12 LA22 0 surements availability, is A11 L 0 LA21L fy01 m B1 LB2 fu . Hence, the dynamics 0 are of the estimation error e
^_ = ( + )^ + ( ) ^ (y )+( + ) ^ (v) := ^
+
_ = (A + LA )e + (A + LA 0 A L 0 L A L ) y (y m ) + ( B + L B ) u (v ) (4) =: A(L)e + Bu (L)u (v) + By (L)y (ym ) where the pair (A ; A ) is observable in view of Assumption 2.1, and e
11
21
21
21
12 1
22
11
2
11
the eigenvalues of the error dynamical matrix can be properly chosen. Moreover, in view of Assumption 2.2, the initial state estimation error ev 0 is partially known: ev0 ev0 0 , with 0 unknown but bounded k 0 k 0 . The estimation error evolution is given by e t 'y t 'u t where e(A +LA )t e
=^ +
()
+ ( )+ ( )
( ) :=
'y t
( ) :=
'u t
t 0
0
t
=
A(t0) e y y
B (ym ()) d
e
A(t0)
Bu u (v()) d
(5)
(y )
are the forced responses generated by the uncertain terms y m and , respectively. Note that the design matrix L affects also the influence of (5) in the error dynamics (4). Therefore, it can be assumed to choose L, among the solutions providing a stable behavior of the error e , as that producing the smallest influence of u and y m over e . t e t , the first n 0 m components of the Being t t 0 Ly t state vector of the system evolve according to 1 t e t 0Ly m t . Define the sliding surface: t 0Lfy01 m t ; the achievement of a sliding motion on m m m guarantees of course the vanishing of the measured outputs. u
(v )
( ) = ^( ) + ( ) ( ^( ) ^ (y ( ))+ ( ) (y ( )) s (y ) = y = 0
(v ) (y ) ) ( )= ( ) ( )= y =0
IV. CASE I: HYSTERESIS AFFECTING SENSORS ONLY The simpler case of hysteresis nonlinearities affecting sensors only will be considered here. Note that the case with actuator-only hysteresis has been addressed recently by the same authors [7]. However, some short remarks will be included here to discuss this issue. Theorem 4.1: It is given the system (1) containing uncertain hysteresis nonlinearities in sensors (3). Under Assumptions 2.1–2.3, there
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 9, SEPTEMBER 2005
exist computable vector functions lowing control law:
6(t) and 9(t) such that the fol-
u(t) = 0B201 ( 6(t) + 9(t))
(6)
with arbitrary := diagf i g; i = 1; . . . ; m; i 1 ensures the achievement of a sliding motion on the surface ym = 0, hence the zeroing of the measured output. Moreover, the state vector remains always bounded, and asymptotically it holds
k (t)k ky (0)k 1 A 0B B A e 1 1
(
A 0 B B0 A
)t
12
0
1
2
1
dt:
22
(7)
The proof is reported in the Appendix. V. CASE II: HYSTERESIS AFFECTING BOTH ACTUATORS AND SENSORS In the case of hysteresis affecting both actuators and sensors the difference with respect to case I is twofold. First of all, it is necessary to take the system input uncertainty into account in the estimate of . This aspect can be easily managed introducing the extra term A21 'u (t) in y_ , due to the uncertainty on u in the estimate of [according to (5)]
y_ = A21 ^ + A21 e(A
A
e + A21 'y (t) + A21 'u (t) + (A22 0 A21 L) f^y01(ym ) + y (ym ) + B2 u: +L
)t
(8)
and are redefined as: (t) = A 'y (t) + A 'u (t)+A e A LA t +(A 0A L)y (ym ); (t) = My (t)+ Mu (t)+ M ( )+3y (ym (t)) where 3 is a matrix such that 3ij = jA 0 A Ljij and j(A 'u (t))i j < Mu (v); i = 1; . . . ; m, then y_ can be expressed again as y_ = 9(t) + (t) + B u and a sliding motion on ym = 0 would be still achieved def
Furthermore, if 21
21
+
(
0
22
)
22
21
def
21
0
21
21
2
driving the system by the control law (6). The control input u, however, cannot be directly applied to the system when uncertain actuator nonlinearities are also present. The definitions of = i (t)sgn(ym ) previously 6 def = diagfi g; i = 1; . . . ; m; ; i def introduced while proving Theorem 4.1, will be here used. A. Solution for Single-Input–Single-Output (SISO) Systems
In this section, the addressed control problem will be solved considering the case of SISO systems of the form (1), making reference to previous works [24], [7]. In order to concisely state the main result for the SISO case, some definitions will be given in the following. They formalize some useful relationships between the sliding surface, system dynamics and actuator characteristics. Note that B2 is a real number in the SISO case, with B2 6= 0 due to Assumption 2.1. Without loss of generality, it will be assumed B2 > 0. Let
(t) + B cj + B mj vej j (t) := B (m^ j 0 mj ) (t) + B (m^ k + mk )ck + B mk c^k + ck 0 vek k (t) := B (m^ k 0 mk ) ( )
2
2
2
2
2
( )
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Theorem 5.1: For the SISO system described by (1) and (2) under Assumptions 2.1, 2.3, the achievement of a sliding motion on ym (t) = 0, hence measured output stabilization and state boundedness according to (7) in Theorem 4.1, is guaranteed by the following control law:
v=
ve(i) + 2i i (t); ve(b) + 2b b (t); ve(r) + 2r r (t); ve(s) + 2s s (t);
> max vrmax ; i (t) + ve(i) (b) if v 2 > max vrmax ; b (t) + ve (r ) if v 3 > max vrmax ; r (t) + ve otherwise; if ym < 0 ve(s) 0 2s s (t); if v3 < min vlmin ; ve(s) 0 s (t) ve(t) 0 2t t (t); if v1 < min vlmin ; ve(t) 0 t (t) ve(l) 0 2l l (t); if v4 < min vlmin ; ve(l) 0 l (t) ve(i) 0 2i i (t); otherwise; if ym > 0 if v 4
(10)
where v j ; vj ; j = 1; 3; 4 are suitable extremal points in the hysteresis (j ) characteristics which can be easily found in Fig. 2, and with ve ; j = i; r; t; s; b; l chosen according to (9). Moreover, 2j ; j = i; r; t; s; b; l are selected as if ym < 0 (i) (i) max v 0 r max 1; i (t)ve < 2i < v40i (tv)e (b) (b) (b) max max 1; vr b0(t)ve ; v40b(tv)e < 2b < v20b(tv)e (r ) (r ) (r ) max max 1; vr r0(t)ve ; v20r (vt)e < 2r < v30r (tv)e (s) (s) max 2s > max 1; vr s0(t)ve ; v30s (tv)e if ym > 0 (s) (s) min v e 0 vl < 2s < ve 0 v3 max 1; s (t) s (t) (t) (t) (t) min max 1; ve 0t (tv)l ; vet(0t)v3 < 2t < vet(0t)v1 (l) (l) (l) min max 1; ve 0l (tv)l ; vel (0t)v1 < 2l < vel (0t)v4 (i) (i) min 2i > max 1; ve 0i (tv)l ; vei (0t)v4 : The proof of this theorem is omitted for sake of brevity. It follows the lines of the proof of the main result in [7]. Remark 5.1: Expression (7) clearly shows that the state vector asymptotically approaches a bounded neighborhood of the origin. If the sensors do not produce errors when y = 0, that is y (0) = 0, asymptotical state stabilization results.
2
for j = b; t; s; i; k = r; l, where ve ; j = b; t; s; i; r; l are suitable functions built using quantities of the nominal system: (j )
ve(j ) := 0 c^j m ^j
0 B9(m^t)j ; j = s; i; b; t vej := c^j 0 9(t) ; j = l; r: B m^ j 2
( )
2
(9)
B. Solution for MIMO Systems When MIMO plants are considered, the presence of hysteresis in each input channel is such that m nonlinearities are combined together by B2 (the case of B2 diagonal is a straightforward). For this reason, the following extra Assumption 5.1 will be now introduced before proposing a solution to the control problem considered in the general MIMO case. Indeed, it is not particularly restrictive, since it
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is reasonable to assume that, in practice, uncertainties cannot increase indefinitely. Assumption 5.1: Uncertainty functions ui vi are ultimately bounded, i.e., there exist u , such that u i vi u 8ui ; i ; ; m. fu 2 m jB2 i u < > 0 Lemma 5.1: The solutions set F mg is convex. i t 0 t i ; i The following proposition shows that a rectangular subset of arbitrary width can always be found in the feasible solution set F . The proof is reported in the Appendix. 1 ; ; m in +m , there Proposition 5.1: Given the vector fu 2 m u 1 2 A 1 0 always exists the rectangular set P 1 ; A1 1 ; ; um 2 Am 0 m ; Am m g such that P F . This proposition basically states that a feasible solution of our control problem can be sought treating each input channel independently of each other. This sort of “decoupling” will be used in the following Theorem, which ensures the existence of a measured output stabilizing controller v. A procedure for its explicit determination is given in the proof (see Appendix). Theorem 5.2: Under the Assumptions 2.1, 2.2, 5.1, there exists a computable input vector v ensuring the achievement of a sliding mo, hence, the zeroing of the measured output. tion on the surface ym The state vector remains always bounded, and asymptotically (7) holds. , (7) Remark 5.2: For actuator-only hysteresis, when y shows that the state vector vanishes asymptotically. This result is novel with respect to other control strategies presented in the literature. For example, in [15] convergence is guaranteed only to a bounded set whose diameter cannot be reduced to zero (though no knowledge of the plant dynamics is required to achieve this result), while the adaptive techniques of [4] and [5] merely ensure global boundedness of the state variables.
1
1 ... ( ) (9( )) = 1. . .
+ ] ...
=
=
[
( ) = ( ) 1 ; IR ( )
= [ . . . ] IR IR : [ + ]
~ =0
(0) = 0
VI. SIMULATION RESULTS In order to validate previous theoretical results, the proposed control approach has been applied by simulation on a system described by the linear model proposed by Tao in [21] representing the linearized lateral dynamic model of a Boeing 747 airplane. In such a system, not reported here for the sake of brevity, the control input u represents the rudder servo and the output y (the yaw rate) has been assumed not directly accessible, being u fu v and ym fy y . The actuator hysteresis ; ml : ; ms parameters have been chosen as follows: mr : ; mi : ; cr ; ct : ; mt : ; mb ; cl 0 ; cb 0 while for the sensor nonlinearity mr ; ml ; ms mt : ; mi mb cr ct : ; cl cb 0 : .A 25% variation was applied to coefficients cr ; ct ; cl and cb (assuming a 35% maximum uncertainty allowed), and a 15% variation to slopes ; has mr ; ml ; mt ; mt ; ms ; mi . A boundary layer of width been used in simulations. Results have been reported in Fig. 3 which, showing the observed and actual state x1 , confirms the expected boundedness of the estimation error, in the presence of uncertainties in the hysteresis parameters.
07 ^ ^ = ^ = ^ ^
requires that yi < , i.e., that yi A21 A21 e(A +LA )t e A21 'y t A22 0A21 L fy01 ym y ym B2u < . The previous expression, however, contains some uncertain terms associated to the presence of y ym , which can be bounded by known quantities in the worst case. Consider the term A21 'y t , and let A21 i denote the ith row of the matrix A21 . A bound can be easily computed as: j A21'y t i j 0t k A21 i e(A +LA )(t0) A12 LA22 0 A11 L 0 LA21L kky ym kd My ym . Analogously, it can be easily proved that j A21 e(A +LA )t e i j can be bounded by n n (A +LA )t jkj 0 def M0 t ; i ; ; m k=1 j =1 jA21 jik je where M0 t exponentially converges to zero. Denoting the exactly t def A21 A21 e(A +LA )t e known part of yi by 0 1 A22 0 A21 L fy ym ; the unknown but bounded part of yi by t def A21 'y t A21 e(A +LA )t A22 0 A21 L y ym and the bounding functions for the uncertain terms of yi by t def My t M 0 0 y ym t ( being a majA22 0 A21 Ljij ), the sliding conditrix such that ij tion for the ith component can be summarized as y i ; ; m. It follows that, t t B2 u i < ; i being B2 a square nonsingular matrix, the control input has to t t B2 u i < and a possible fulfill the inequality B201 w with wi 0 i i t ym 0 i t choice is u with i > , which proves (6) for ym > by setting def f i g; i ; ; m; i ym . Anali t and (6) follows. ogous arguments hold for the case ym < Finally, as the surface is reached, the reduced system satisfies 0, and the dynamics of t can be rewritten as ym 0 ; ym 1 A11 0 B1 B201 A21 1 A12 0 B1B201 A22 y . is bounded by y , Recalling that the forcing term y it follows that the system state is asymptotically bounded as k 1k k 01 e(A 0B B A )t A12 0 B1 B201 A22 y kdt hence (7) is proved. Proof of Proposition 5.1: A constructive approach will be followed. The set F of all solution of (6) can be equivalently parameB201 i w where: wi 0 i i ym 0 i ; i terized as ui ; ; m; i > . If a rectangular subset of solutions in F with arbitrary width exists, the equality (6) must hold for each ui 2 Ai 0 ; ; m with i a priori fixed. Being the set F i ; A i i ; i convex by Lemma 5.1, it is sufficient to check that for each vertex of i > such that ui Ai 6 i satisfies (6). Define P there 0exist f i g; i 2 . Of course, it is alA B2 1 0 ; 6 b2 1 6 such that: i i ways possible to find some i 2
( )+(
(
_
( ))
()
(
( ) =
( ) =
0
_ = ^+ + )( ^ ( )+ ( ))+ 0 ( ) () ( ) ( ) ( + ) ( ( )) =: ( ) ) ( = ( ) = 1 ...
_ 9( ) = )^ ( ) ( )+
^+
+(
( ) + ( ) + 3 ( ( )) 3 3 = 0 = 1 ... (9( ) + ( ) + ) = () = () ^ = 2 ^ = 25 ^ = ( ( ) + 9( ) + ) 0 = 0 25 ^ = 0 5 ^ = 0 2 ^ = 1 ^ = 2 ^ = 1 = ( )sgn( = =
2 ^ =5 ^ =8 ^ = 1 0 07 ^ = ^ = ^ = ^ = 05 ^ = ^ = 05 6 = diag = 1 ... = ( )sgn( ^ ^ ^ ^ 0 = 0 05 ^ ^ ^ ^ () = _ = _ = ( ) +( (0) ( APPENDIX
= 0 _ = [( )_] 0 =( ) = ) ) = 1 ... 1 1 ... _ + ] = 1 ... )_ 0 ( ) = 1 = ( 6 9) = diag ( ) _ 0 = 1. . . 0 IR
Proof of Theorem 4.1: The sliding condition on ym Ty requires that s ym s ym ym m ymT @ ym =@ y y < where @=@y ym @=@yi ym ; i ; ; m. Due to Assumption 2.3 of strict positiveness, the signs of ym and yi coincide, hence one can rewrite the sliding condition as m y where the term ym @=@yi ym i=1 m @ym =@yi yi < has known sign. A sufficient condition is the imposition on m separate m. Consider the i-th inequalities: ym @ym =@yi yi < 8 i component and assume ym > . In this case the sliding condition
( )_ ( ) = ) = diag((
(
_
Fig. 3. Simulation results: estimated states, actual states and estimation error of x .
(
^ + _ ) ( ) _
( _)
=
) 9() )
)( (0)) (0) ) (0)
sgn( ) 9 = [ IR
=
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 9, SEPTEMBER 2005
1 1 16 b2
IR
...
m i 2 for a possible combination of 6; b2 ; ; b2 being the elements of the ith row of the matrix B2 . Denote by ir the
i associated to an arbitrary combination of 6. Whichever this com ir can always be found. Put bination is, a i such that i i 0 0 i and consider the generic vertex ua of P . One can write i ir 0 i , B2 i uar B2 i A 6 i 0 i r ir ; ; m satisfy i i 0 0 i i 0. where i ; i Hence, it is possible to set i ir 0k, with k > , showing that each vertex of P belongs to F , i.e., P F . Proof of Theorem 5.2: Given u , Proposition 5.1 shows that it is possible to find suitable mi ; Mi such that the set: A fmi ui ; ; m is contained in F . In order Mi Mi 0 mi > u g; i to determine a control law vi satisfying the sliding-mode condition on ym , we seek m controllers vi such that mi fu vi Mi ; i ; ; m or equivalently, considering the worst case: mi u i vi f u v i M i 0 u i v i ; i ; ; m. By Assumption 5.1, and by the proper choice of the distance between mi and Mi , according to the set A, the interval mi u i vi ; Mi 0 u i vi is always nonempty. Choose ui arbitrarily in the set ui 2 mi u , and u ; Mi 0 calculate vi as vi fu01 ui . This choice necessarily ensures that fu vi 2 P for each value of u i vi , i.e., fu vi 2 F . Belonging the above solution to the feasible solution set, it follows that y t y t < holds, the zeroing of ym following. The proof of the boundedness of the state vector is analogous to that reported in Theorem 4.1.
max( ) 1 ) = 6 9 + 6 = ( + )6 9 ( ) =( )( + 1 + = 1 = 1 ... 1 + = 1 = ; = 1 ... 21 ( ) =0 = 1 ... + ( ) ^( ) ( ) = 1 ... [ + ( ) ( )] ~ 1 ] ~ [ +1 ~ ~ = ^ (~ ) (~ ) ( ) (~ ) ( ) _( ) 0
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[16] A. Ilchmann and E. Ryan, “On tracking and disturbance rejection by adaptive control,” Syst. Control Lett., vol. 52, pp. 137–147, 2004. [17] S. Ahmad, “Analysis of robot drive train errors, their static effects and their compensations,” IEEE J. Robot. Automat., vol. 4, no. 2, pp. 117–128, Apr. 1988. [18] G. Tao and P. Kokotovic, “Adaptive control of systems with unknown output backlash,” IEEE Trans. Autom. Control, vol. 40, no. 2, pp. 326–330, Feb. 1995. [19] S. Shahruz, “Performance enhancement of a class of nonlinear systems by disturbance observers,” IEEE/ASME Trans. Mechatron., vol. 5, no. 3, pp. 319–323, May 2000. [20] V. I. Utkin, Sliding Modes in Control Optimization. New York: Springer-Verlag, 1992. [21] G. Tao, S. Chen, and S. Joshi, “An adaptive actuator failure compensation controller using output feedback,” IEEE Trans. Autom. Control, vol. 47, no. 3, pp. 506–511, Mar. 2002. [22] C. Edwards and S. Spurgeon, “Sliding mode stabilization of uncertain systems using only output information,” Int. J. Control, vol. 59, no. 5, pp. 1211–1229, 1995. [23] F. Allgower, J. Ashman, and A. Ilchmann, “High gain adaptive -tracking for nonlinear systems,” Automatica, vol. 33, pp. 881–888, 1997. [24] M. L. Corradini and G. Parlangeli, “Robust stabilization of nonlinear uncertain plants with hysteresis in the actuator: A sliding mode approach,” in Proc. 2002 IEEE Int. Conf. on Systems, Man and Cybernetics (SMC2002), vol. 3, 2002.
REFERENCES [1] I. Mayergoyz, The Preisach Model for Hysteresis. New York: Springer-Verlag, 1991. [2] J. Macki, P. Nistri, and P. Zecca, “Mathematical models for hysteresis,” SIAM Rev., vol. 35, pp. 94–123, 1993. [3] C. Su, Y. Stepanenko, J. Svoboda, and T. Leung, “Robust adaptive control of class of nonlinear systems with unknown backlah-like hysteresis,” IEEE Trans. Autom. Control, vol. 45, no. 12, pp. 2427–2432, Dec. 2000. [4] G. Tao and P. Kokotovic, “Adaptive control of plants with unknown hysteresis,” IEEE Trans. Autom. Control, vol. 40, no. 2, pp. 200–212, Feb. 1995. [5] , Adaptive Control of Systems with Actuator and Sensor Nonlinearities. New York: Wiley, 1996. [6] M. L. Corradini and G. Parlangeli, “Dynamic output feedback variable structure control for the output stabilization of mimo uncertain plants with actuator and sensor hysteresis nonlinearities,” in Proc. IFAC Symp. Robust Control Design 2003 (ROCOND03), 2003. [7] M. L. Corradini, G. Orlando, and G. Parlangeli, “A vsc approach for the robust stabilization of nonlinear plants with uncertain nonsmooth actuator nonlinearities—A unified framework,” IEEE Trans. Autom. Control, vol. 49, no. 5, pp. 807–813, May 2004. [8] M. Grundelius and D. Angeli, “Adaptive control of systems with backlash acting on the input,” in Proc. 35th IEEE Conf. Decision and Control, vol. 4, 1996, pp. 4689–4694. [9] G. Tao and P. Kokotovic, “Adaptive control of plants with unknown dead-zones,” IEEE Trans. Autom. Control, vol. 39, no. 1, pp. 59–68, Jan. 1994. [10] , “Continuous-time adaptive control of systems with unknown backlash,” IEEE Trans. Autom. Control, vol. 40, no. 6, pp. 1083–1087, Jun. 1995. [11] G. Tao, “Adaptive control of partially known system,” IEEE Trans. Autom. Control, vol. 40, no. 10, pp. 1813–1818, Oct. 1995. [12] A. Azenha and J. Machado, “Variable structure control of robots with nonlinear friction and backlash at the joints,” in Proc. 1996 IEEE Int. Conf. Robotics and Automation, vol. 1, 1996, pp. 366–371. [13] M. L. Corradini and G. Orlando, “Robust stabilization of nonlinear uncertain plants with backlash or dead zone in the actuator,” IEEE Trans. Control Sys. Technol., vol. 10, no. 1, pp. 158–166, Jan. 2002. [14] H. Kwatny, M. Mattice, and C. Teolis, “Variable structure control of systems with uncertain nonlinear friction,” Automatica, vol. 38, pp. 1251–1256, 2002. [15] E. Ryan, “A nonlinear universal servomechanism,” IEEE Trans. Autom. Control, vol. 39, no. 4, pp. 753–7601, Apr. 1994.
Reduced-Order Observers for Linear Neutral Delay Systems M. Darouach Abstract—The problem of reduced-order observers design for linear neutral delay systems is investigated. Sufficient conditions for the existence of these observers are given. First, the general case is presented, then under more restrictive conditions a simple design method is developed for some special cases. Based on linear matrix inequality (LMI) and linear matrix equality (LME) formulation, independent of delays stability criteria are derived. A systematic design method of these observers is presented. Index Terms—Existence conditions, linear matrix equality (LME), linear matrix inequality (LMI), neutral systems, reduced-order observer, stability independent of delay.
I. INTRODUCTION Time delays are generally encountered in various types of systems, such as chemical processes, thermal processes, long transmission lines in pneumatic and hydraulic systems. Over the past decades, considerable attention has been undertaken on the stability and control of dynamic neutral delay systems; see [1]–[3], and references therein. A neutral delay system is depending not only on state delays but also on the derivatives of the delayed state. Neutral delay systems are used in the modeling of fluctuations of voltage and current in transmission lines, in the population dynamic, in vibration analysis of elastic systems, and in chemical engineering; see e.g., [1], [2], and [4]. The stabilization and control of these systems are often realized with the assumption that the Manuscript received April 4, 2004; revised April 10, 2005. Recommended by Associate Editor L. E. Holloway. The author is with CRAN-CNRS (UMR 7039), Université Henri Poincaré Nancy I, 54400 Cosnes et Romain, France (e-mail:
[email protected]). Digital Object Identifier 10.1109/TAC.2005.854630
0018-9286/$20.00 © 2005 IEEE
