Output Feedback Control of a Class of Integrator Nonlinear Systems in Presence of Actuator Saturation

11th IFAC International Workshop on Adaptation and Learning in Control and Signal Processing July 3-5, 2013. Caen, France

ThS5T1.4

Output Feedback Control of a Class of Integrator Nonlinear Systems in Presence of Actuator Saturation A. Mahjoub*, F. Giri**, F.Z. Chaoui**, N. Derbel* * University of Sfax, Sfax, Tunisia (e-mail: [email protected]) ** Normandie Université, UNICAEN, GREYC, F-14032 Caen, France (e-mail:[email protected])

Abstract. A new control strategy is presented to deal with the problem of controlling a class of integrator second-order nonlinear systems in presence of input saturation constraint. A saturated controller is designed using the backstepping technique and the closed-loop control system is analyzed using Lyapunov and input-output stability tools. It is formally shown that the controllers meet all its control objectives. Accordingly, the closed-loop system global stability is actually ensured and output-reference tracking is perfectly achieved in presence of slowly varying reference signals.

1. INTRODUCTION

closed stability region (Sussmann et al., 1994; Lin, 1998; Giri et al., 2012).

Signal limitations are unavoidable in control systems due to technological limitations. The latter are generally introduced in systems to ensure operation safety (e.g. limits of valve opening, stops in mechanical systems, voltage limits and saturations in power amplifiers…). In some control problems, limitations are deliberately introduced, in form of input or state constraints, in order to meet performance requirements (e.g. keeping a system around an operation point so that linear approximation can hold). The problem of controlling linear systems via saturating actuators has been extensively studied especially over the last fifteen years. This activity is reported in hundred of journal and conference papers as well as several monographs including (Glattfelder and Schaufelberger, 2003; Hu and Lin, 2001; Saberi et al., 1999; Tarbouriech et al., 2011). Roughly, two main approaches have been followed to design stabilizing regulators: the anti-windup compensator synthesis and the direct control design. In the first approach, a predefined controller ensuring satisfactory control performances in the absence of actuator saturation is supposed to be available. Then, an additional (static or dynamic) compensator is designed to minimize the adverse effect of actuator saturation on closed-loop performances. The direct control design problem is one where the input constraint is accounted for at the controller design stage. In turn, this is dealt with following two main approaches. The first one consists in enforcing the control signal to never violate the allowed limits, using either low-gain linear regulators (e.g. Dai et al., 2009; Glattfelder and Schaufelberger, 2003; Hu and Lin, 2001; Saberi et al., 1999; Stoorvogel and Saberi, 1999) or nested-saturation regulators (e.g. Bateman, 2003; Zhou et al., 2011). The second approach is one that allows the control signal to saturate. Then, feedback gets broken during time intervals of control saturation. It turns out that, closed-loop stabilization is only possible for systems with all poles in the

Controlling nonlinear systems in presence of actuator saturation is a far more challenging problem. Signal boundedness results have been achieved based on neural network- or fuzzy logical-based system approximation (e.g. Cao and Lin, 2003; Gao and Selmic, 2006). Model-based control schemes of the anti-windup type have been proposed in (Valluri and Soroush, 2003; Yoon et al., 2008). There, closed-loop local asymptotic stabilization is achieved for state-feedback linearizable systems. Global asymptotic stabilization of feedforward nonlinear systems has been achieved, using state-feedback nested-saturation controllers in (Jankovic et al., 1996; Mazenc and Praly, 1994). A common shortcoming of these works is that the proposed controllers are state-feedback, necessitating the measurements of all state variables, and the controller objective was limited to asymptotic stabilization, no tracking performances were achieved. In this paper, we seek the development of an output-feedback controller which, in addition, possesses some reference tracking capability. This control problem is considered for a class of nonlinear systems consisting of a series connection including an output integrator and a cascade of stable first order nonlinear subsystems (Fig. 1). It turns out that the controlled system is BIBO unstable. The control objective is to stabilize the system and to enforce its output to perfectly track any slowly varying reference signal, whatever the initial conditions. A saturated nonlinear controller is designed using a backstepping like technique. It is formally shown using Lyaponuv and input-output stability tools that, the proposed controller meets its objectives. This theoretical result is illustrated by simulation. The paper is organized as follows: the control problem is formulated in Section 2; the controller design and analysis are dealt with in Section 3; simulation results are presented in Section 4.

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11th IFAC ALCOSP July 3-5, 2013. Caen, France

u

1st order NL subsystem

1xst2 order NL subsystem

xn

x2

1 s

Proof. Part 1. Consider the Lyapunov function V0 = x 22 / 2 . Its derivative along the (1b) is the following: V
0 = x 2 x
2 = u x 2 − x 2 f ( x 2 ) ≤ u x 2 − k1 x 22 (using (2a))

y

(

≤ x 2 u M − k1 x 2 Fig. 1. Class of controlled systems

that x 2 ≤ u M / k 1 . Proof of Part 2. The subsystem (1b) assumes the feedback representation of Fig. 1. Applying the loop transformation gain (e.g. Vidyasagar, 2002), the system of Fig. 1 is given the equivalent form of Fig. 2 with: k +k λ= 1 2 (4) 2 φ ( x ) = f ( x ) − λx (5) The new feedback of Fig. 2 enjoys two key properties: 1 (i) The linear part, with transfer function , is s+λ asymptotically stable and so L p -stable, whatever p ∈ [1 ∞]

We are interested in controlling systems consisting of a cascade of nonlinear subsystem and an output integrator connected altogether in series as in Fig. 1. For space limitation, the forthcoming control development is illustrated for the second-order case. Analytically, the system is described by the following couple of equations: x
1 = x2 (1a)

x
2 = u − f ( x2 )

(1b)

where y = x1 and the control input u is subject to the following actuator constraint:

u (t ) ≤ uM

(1c)

(see e.g. Vidyasagar, 2002). (ii) The nonlinear part is centred, specifically: k − k k − k  φ ∈ S 1 2 2 1
(6) 2   2 Then, it follows, applying the small gain theorem , that the feedback of Fig. 2 is L1 -stable, if γ a γ b < 1 where γ a and

for some known constant u M > 0 . In (1b), the function f is piecewise continuous satisfying the sectoricity property,

k1 x 22 ≤ x 2 f ( x 2 ) , for all x2 ,

(2a)

for some real constants 0 < k1 < ∞ . denoted (e.g. Vidyasagar, 2002): f ∈ S [k1 ∞ )

That is commonly

γ b denote, respectively, the L1 -gains of the linear and nonlinear blocs in the feedback system of Fig. 2. It is well known that γ a = 1 / λ because the involved linear system is a

(2b)

It is checked (see Proposition 1, hereafter) that condition (2b) entails L1 / L∞ stability of the subsystem (1b). This will make it possible to achieve suitable control objectives for the whole system (1a-b) despite the saturation constraint (1c) and the fact that the subsystem (1a) is integrator. Specifically, the aim is to design a controller able to make the output system y = x1 track as closely as possible the reference

k 2 − k1 , due 2 k − k1 <1 to (2a). Then, the stability condition writes 2 2λ which holds implies, due to (4). From Fig. 2, one gets the following bounding: γa 1 x2 1 ≤ u = u . 1 − γ a γ b 1 k1 1 first order. Also, it is easily checked that γ b =

trajectory y * generated as follows:

1 yr (1 + Ts ) 2

(3a)

This completes the proof of Proposition 1

where T > 0 is chosen by the designer and yr is any bounded piecewise-constant satisfying: y
r ∈ L1 (3b)

+

u

f

Proposition 1. Consider the subsystem (1b) subject to (2b). Then one has the following properties: The

state

variable

x2

is

bounded,

Fig. 1. Feedback representation of (1b) u

specifically

x 2 ≤ u M / k1 . Then, there is a constant 0 < k 2 < ∞ such

1

≤k

1 s+λ

x2

φ

2) The subsystem (1b) is L1 -stable (with gain k 1−1 ). Consequently, x 2

+

-

that: x 2 f ( x 2 ) ≤ k 2 x 22 i.e. f ∈ S [k1 k 2 ] . −1 1

x2

1 s

-

The control problem description is completed with the following proposition that emphasizes stability properties of the subsystem (1b).

1)

(using (1c)

This show that V
0 < 0 whenever x 2 > u M / k1 . It turns out

2. CONTROL PROBLEM STATEMENT

y* =

)

u 1 , whatever u ∈ L1

Fig. 2. Equivalent feedback representation of Fig 1

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11th IFAC ALCOSP July 3-5, 2013. Caen, France

3.

γ 1 (θ )  −1 −1 T (10c) 
= θ ∆θ S C ( ) γ θ 2   Introduce the tracking error: e1 = xˆ1 − y * (11a) From (10a), it is readily seen that e1 undergoes the following equation: e
1 = xˆ 2 − γ 1 (θ ) ~ x1 − y
* (11b) This suggests the following stabilizing function: α 1 = y
* + γ 1 (θ ) ~x1 − c1 e1 (12a) with c1 > 0 is a design parameter. Introduce the second error, e2 = xˆ 2 − α 1 (12b) It follows from (12b) that (11b) rewrites as follows: e
1 = e2 − c1e1 (13a) It follows from (10b) that, e2 undergoes the following equation: e
2 = u − f ( xˆ 2 ) − γ 2 (θ ) ~ x1 − α
1 (13b)

CONTROLLER DESIGN AND ANALYSIS

3.1 State Observer Design In the system (1a-b), only the state x1 is measurable. To estimate the state x 2 , a high gain observer is used (e.g. Besançon, 2007). To this end, the system is first given the following compact form:

x
= Ax + ϕ (u ) + ψ ( x) ;

y = CT x

(7a)

with C T = [1 0] and

 0  0 1  0  A=
; ϕ (u ) = 
; ψ ( x ) = 
0 0 u      f ( x 2 ) The high gain observer writes as follows: x
ˆ = Axˆ + ϕ (u ) + ψ ( xˆ ) − θ∆−1 S −1C T (Cxˆ − y ) θ

(7b)

(8)

with,

Now, let us consider the following Lyapunov function candidate:

1 0  ∆θ = 
, 0 1 / θ 

e12 + e22 2 The time differentiation of V , along the trajectories defined by (12a-b), gives: V
= e1 e
1 + e 2 e
2 = e (e − c e ) + e (u − f ( xˆ ) − γ (θ ) ~ x − α
) V =

where θ > 0 is a design parameter and S is the unique solution of the algebraic Lyapunov equation:

S + AT S + SA − C T C = 0 The solution of (9) is symmetric positive definite and satisfies:

1

 C 1  2  S −1C T =  22
= 
C 2  1  It readily follows that the state estimation error ~ x = xˆ − x undergoes the following error equation: ~ x
= A~ x + (ψ ( xˆ ) − ψ ( x)) − θ ∆−θ1 S −1C T (Cxˆ − y )

2

1 1

2

2

2

1

1

This suggests the following control law: u = f ( xˆ 2 ) + γ 2 (θ ) ~ x1 + α
1 − c 2 e 2 (14) Substituting the right side of (14) to u in (13) yields: V
= −c1 e12 − c 2 e 22 + e1e2 (15)
It is readily seen that V can be made negative definite in the variables (e1 , e2 ) by letting c1c 2 > 1 / 2 . The point with the control law (14) is that, the resulting control action magnitude u can go beyond the allowed value u M . Therefore, the following saturated version of (14) is considered:

(9)

Proposition 2. The observer (8) when applied to the system (7a-b) is globally convergent i.e. there exists a real θ 0 > 0 , such that for all θ > θ 0 , there exists µ1 , µ 2 > 0 so that: ~ x (t ) ≤ µ θ 2 e − µ 2t ~ x (0) , whatever xˆ (0) ∈ R 2

def

u = sat (v) = min( v , u M ) sgn(v) v = f ( xˆ ) + γ (θ ) ~ x + α
− c e

1

2

The proof of this result can be found in many places e.g. (Besançon, 2007). The key conditions are presently satisfied especially the fact that the involved nonlinearities are Lipschitz. Presently, f ( x 2 ) is piecewise continuous by

2

1

1

2 2

(16a) (16b)

2.2 Controller Analysis Adding v − v in the right side of (13b) yields, using (16b):

assumption and x2 is bounded by Proposition 1. It turns out

e
2 = u − v − c2 e2

that f ( x 2 ) and ψ (x) are Lipschitz.

Equations (13a) and (17) are given the following input-output forms1:

3.2 Control Law Design

1 e2 s + c1 1 e2 = (u − v) s + c2 Substituting the right side of (19) to e2 in (18) yields: e1 =

The control design is dealt with using the backstepping technique on the basis of the observed model (8) which, in view of (7b), rewrites as follows:

x
ˆ1 = xˆ 2 − γ 1 (θ ) ~ x1
xˆ = u − f ( xˆ ) − γ (θ ) ~ x1 2 2 2 where

(10a)

(17)

(18) (19)

(10b) 1

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To avoid multiplication of notations, the same notation is used, throughout the paper, for a signal x and its Laplace transform. The distinction is made clear by the context.

11th IFAC ALCOSP July 3-5, 2013. Caen, France

1 (u − v) (20) ( s + c1 )( s + c2 ) On the other hand, using (12a), (13a) and (16b) one gets: v = c12 e1 − (c1 + c 2 )e 2 + f ( xˆ 2 ) +
y
* + (γ (θ ) − γ 2 (θ )) ~ x + γ (θ ) ~ x (21a) e1 =

2

1

1

1

=

(25) ˆ using (24). Substituting the right side of (25) to f ( x 2 ) in (21b) gives: 1

2

This, together with (18) and (19), yields: (c + c 2 ) s + c1 c 2 v= 1 (v − u ) + f ( xˆ 2 ) +
y
* ( s + c1 )( s + c 2 ) + (γ (θ ) − γ 2 (θ )) ~ x + γ (θ ) ~ x

( c1 + c 2 − λ ) s + c1 c 2 (v − u ) + (λ − λ ) xˆ 2 + λ y
* +
y
* ( s + c1 )(s + c 2 ) + (γ 2 (θ ) − γ 12 (θ ) + γ 1 (θ )) ~ x1 + γ 1 (θ ) ~ x2 (26) Using (3a), one has: (λ + s ) s * (λ + s ) * (27) λ y
* +
y
* = (λ + s ) sy * = y
y = 2 (1 + Ts ) (1 + Ts ) 2

Theorem 1. Assume that 0 < k 2 < 3k1 and let the design parameters (c1 , c 2 , θ ) be chosen such that c1c 2 > 1 / 2 ,

(λ + s ) is strictly proper and T > 0 , this transfer (1 + Ts ) 2 function is L1 -stable. Then, it follows from (27), using (3b) that: λ y
* +
y
* ∈ L1 (28a) ~ Also, as x is exponentially vanishing (by Proposition 2), As

k1 + k 2 ≤ c 2 and θ > θ 0 (where θ 0 is as in Proposition 2) 2 Then, one has the following properties: 1) All variables are bounded. 2) The control signals are small in the sense that v ∈ L1 and u ∈ L1 3) The tracking error is small in the sense that e1 belongs to L1 and vanishes asymptotically c1 ≤

one has: (γ 2 (θ ) − γ 12 (θ ) + γ 1 (θ )) ~ x1 + γ 1 (θ ) ~ x 2 ∈ L1

(c1 + c2 − λ ) s + c1c2 is not ( s + c1 )( s + c2 ) larger than 1. This is readily seen in the limit cases where c1 = λ and c2 = λ as then the above transfer function boils L1 -gain of the transfer function

f ( x 2 ) is bounded by (2a). Proposition 2, the same properties applies to xˆ and f ( xˆ ) Since c c > 1 / 2 , V
turns out to be

(28b)

On the other hand, as c1 ≤ λ ≤ c 2 , it can be checked that the

Proof. Part 1. The applied control signal u is bounded by (16a). Then, x 2 is bounded by Proposition 1 (Part 1) and

2

1

v=

(21b) 2 1 1 1 2 Equations (19-21) will now be based upon to establish the controller performances.

2

−λs (v − u ) + (λ − λ ) xˆ 2 ( s + c1 )( s + c 2 ) + λ y
* + γ (θ ) ~ x

1 2

negative definite in the variables (e1 , e2 ) . This

c2 and (s + c2 )

down to the first order transfer functions,

immediately implies that (e1 , e2 ) are both bounded. Then xˆ1

c1 , respectively. The corresponding impulse responses ( s + c1 )

is bounded by (11a) implying that x1 is also bounded, by Proposition 2. Then, it follows from (21a) that v is in turn bounded, using Proposition 2. Part 1 is established.

are respectively g 2 (t ) = c 2 e − c2t and g 1 (t ) = c1e − c1t (t ≥ 0) .

Proof of Part 2. Introduce the following auxiliary variable; def f ( x ˆ 2 (t )) λ (t ) = (22) xˆ 2 (t )

The L1 -gains of the corresponding transfer functions are both equal to 1 as (see e.g. Vidyasagar, 2002): g1 1 = g 2 1 = 1 . Then, taking the L1 norms of both sides of

It follows from (2a) that:

(26) yields:

v 1 ≤ v − u 1 + (λ − λ ) xˆ 2

k1 ≤ λ (t ) ≤ k 2 , for all t (23) From (12b) one has and (13a), one has xˆ 2 = e 2 + α 1 which, together with (12a), yields: xˆ 2 = e 2 − c1 e1 + y
* + γ 1 (θ ) ~ x1 This yields, using (19 and (20): −s xˆ 2 = (v − u ) + y
* + γ 1 (θ ) ~ x1 (24) ( s + c1 )( s + c 2 )

+ λ y
* +
y
*

≤ v 1 − u 1 + max λ (t ) − λ xˆ 2 t ≥0

1

1

+ λ y
* +
y
*

+ (γ 2 (θ ) − γ (θ ) + γ 1 (θ )) ~ x1 + γ 1 (θ ) ~ x2 2 1

where we have used the fact that v − u

1

1

1

(29)

= v 1 − u 1 . This is

a consequence of the fact that sgn(u ) = sgn(v) which is a known property of the saturation function (16a). Substracting λ from both sides of (23) gives: k1 − λ ≤ λ (t ) − λ ≤ k 2 − λ , for all t implying, k − k1 max λ (t ) − λ ≤ 2 (30) t ≥0 2

Then, (22) implies:

f ( xˆ 2 ) = λxˆ 2 = λ xˆ 2 + (λ − λ ) xˆ 2 with λ =

1

k1 + k 2 2

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11th IFAC ALCOSP July 3-5, 2013. Caen, France

Recall that, by Proposition 1 one has x 2 yields: xˆ 2 1 = ~ x2 − x 1 ≤ ~ x2

1

1

≤ k1−1 u

1

illustrated by Fig. 3. It is seen that the tracking quality is quite satisfactory despite the saturation constraint (1c). To better appreciate this result, the design parameters are now given different values not satisfying the conditions of Theorem 1. Specifically, the values c1 = 12 and c 2 = 0.01 are considered (while all other parameters are kept unchanged). The resulting control performances are illustrated by Fig. 4. Clearly, the control performances have much deteriorated compared to the preceding case.

which

+ x 1 ≤ k 1−1 u 1 + ~ x2 1 .

This together with (30) and (28) implies, due to (29):

 k2 − k1  * y* + 1 −

u 1 ≤ λ y
+

1 2 k  1 (γ 2 (θ ) − γ 12 (θ ) + γ 1 (θ )) x1 + γ 1 (θ ) x2 or equivalently,  2k 1 u 1 ≤  3k 1 − k 2

1

1 y



λ y
* +
y
* 1 x1 + γ 1 (θ ) ~ x2 + (γ 2 (θ ) − γ 12 (θ ) + γ 1 (θ )) ~

0.8

(

0.6

1

0.4

)

(31)

0.2

where we have used the fact that 0 < k 2 < 3k1 . This shows

0

that u ∈ L1 , using (28a-b). Now, let us show that:

v≤

y*

vM u uM

-0.2 -0.4

(32)

-0.6

where v M = max v(t ) which is finite because the signal v is

-0.8

bounded by Part 1. First, if v(t ) ≤ u M one has, by (16a),

-1

t ≥0

u (t ) = v(t ) and so (32) holds since v M > u M . If v(t ) > u M , v ∈ L1 because u ∈ L1 . It turns out that v − u ∈ L1 because L1 is a vector space. Part 3. Then, it follows from (20) that e1 ∈ L1 because 1 is L1 -stable. Furthermore, using Part 1 of ( s + c1 )( s + c 2 ) this theorem, it follows from (11b) that e
1 ∈ L∞ . This,

30

40

50

60

70

80

90

100

20

30

40

50

60

70

80

90

100

v

2 1

u

0 -1 -2 -3 -4

together with the fact that e1 ∈ L1 , implies that e1 vanishes asymptotically, by Barbalat's lemma. This ends the proof of Theorem 1

-5

0

10

Fig. 3. Control performances with design parameters satisfying the conditions of Theorem 1. Top: system output y (t ) (solid) and reference signal y *(t ) (dashed). Bottom:

4. SIMULATIONS The simulated system is a second-order of the form (1a-b) with f ( x 2 ) = x 2 + x 23 + x 25 . The system input is subject to the

computed control v(t ) (solid) and applied control action

u (t ) (dashed).

saturation constraint (1c) with uM = 2 . It is readily seen that x 2 f ( x 2 ) ≥ x 22 i.e. (2a) is satisfied with k1 = 1 . By

20

2 2

Proposition 1, x 2 ≤ u M / k1 and x 2 f ( x 2 ) ≤ k 2 x . Presently,

v

15

4

  uM 

= 21 . The reference signal is

+  k1

generated by y* =

20

3

inequality (32) does hold. This implies that

u k 2 = 1 + M  k1

10

4

one has u (t ) = u M sgn(v(t )) and again (32) holds. Hence

2

0

5

10 5 u

1

yr with yr a square signal with (1 + s )2 amplitude 0.9 and period 25. The saturated controller (1a-b) is implemented with c1 = 0.2 and c2 = 15 . These values meet

0 -5 -10

the two requirements in Theorem 1 i.e. c1c 2 > 1 / 2 and

-15

k + k2 c1 ≤ 1 ≤ c 2 . The resulting control performances are 2

-20

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0

5

10

15

20

25

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35

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11th IFAC ALCOSP July 3-5, 2013. Caen, France

reference inputs and disturbances. Int. J. Control, vol. 85, pp. 1694-1707. Glattfelder A.H. and W. Schaufelberger (2003). Control systems with input and output constraints. Springer, Berlin. Hu T. and, Z. Lin (2001). Control systems with actuator saturation: analysis and design. Birkhäuser Boston. Jankovic M, Sepulchre R, Kokotovic P V. Constructive Lyapunov stabilization of nonlinear cascade systems. IEEE Transactions on Automatic Control, 1996, vol. 41(12), pp. 1723−1735 Lin Z. (1998). Global control of linear systems with saturating actuators. Automatica, vol. 34, 897–905. Mazenc F, Praly L. (1996). Adding integrations, saturated controls, and stabilization for feedforward systems. IEEE TAC, vol. 41(11), pp. 1559−1578. Saberi A., A.A. Stoorvogel and, P. Sannuti (1999). Control of linear systems with regulation and input constraints, Springer. Stoorvogel A. A. and A. Saberi (Eds.) (1999). Special issue: Control problems with constraints. Int. Journal of Robust and Nonlinear Control, vol. 9, pp.583–734. Sussman H.J., E.D. Sontag, and Y. Yang (1994). A general result on the stabilization of linear systems using bounded controls. IEEE Trans. Autom. Contr., vol. 39, pp.2411-2425. Tarbouriech S., G. Garcia, J.M. Gomes da Silva Jr, I. Queinnec (2011). Stability and Stabilization of Linear Systems with Saturating Actuators. Springer. Valluri S. and M. Soroush (2003), ‘A nonlinear controller design method for process with actuator saturation nonlinearities’. International Journal of Control, vol. 76, pp. 698-716. Vidyasagar M (2002). Nonlinear Systems Analysis. 2d edition, SIAM, PA, USA. Yoon S.S., J.K. Park and T.W. Yoon (2008), ‘Dynamic antiwindup scheme for feedback linearizable nonlinear control’. Automatica, vol. 44, pp.3176-3180. Zhou B., W.X. Zheng, G.-R. Duan (2011). An improved treatment of saturation nonlinearity with its application to control of systems subject to nested saturation, Automatica, vol. 47, 306–315.

1.5

1 y 0.5 y* 0

-0.5

-1

-1.5

0

5

10

15

20

25

30

35

40

45

50

Fig. 4. Control performances with design parameters not satisfying the conditions of Theorem 1. Top: system output y (t ) (solid) and reference signal y *(t ) (dashed). Bottom: computed control

v(t ) (solid) and applied control action

u (t ) (dashed).

5. CONCLUSIONS In this paper, a new control approach is developed to deal with global output reference tracking for nonlinear systems of the form (1a-b) in presence of actuator saturation. The system is BIBO unstable as it includes an integrator. The control design and analysis of Section 3 combines Lyapunov and input-output stability tools. The presented control design generalizes to n -th order systems of the form:

x
1 = x2 ; x
i = x i +1 − f i ( x i ) i = 2  n − 1 x
n = u − f ( x n ) . REFERENCES Bateman A. and Z. Lin (2003). An analysis and design method for linear systems under nested saturation. Systems & Control Letters, vol. 48, pp. 41-52. Besançon G. (Ed.) (2007). Nonlinear observers and applications. Springer. Cao Y. Y. and Z. L. Lin, (2003). ‘Robust stability analysis and fuzzy-scheduling control for nonlinear systems subject to actuator saturation," IEEE Transactions on Fuzzy Systems, vol. 11, pp. 57-67. Dai D., T. Hu, A. R. Teel and L. Zaccarian (2009). Output feedback design for saturated linear plants using deadzone loop. Automatica, vol. 45(9) 2917-2924. Gao W. Z. and R. R. Selmic, (2006). ‘Neural network control of a class of nonlinear systems with actuator saturation’, IEEE Transactions on Neural Networks, vol. 17, no. 1, pp. 147-156. Giri F., F.Z. Chaoui, A. Chater, D. Ghani (2012). Tracking performances of saturated model-reference controllers in presence of not-necessarily constraint-compatible

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