Global Practical Stabilization of Planar Linear Systems in the ...
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[7] E. G. Strangas, H. K. Khalil, B. A. Oliwi, L. Laubnger, and J. M. Miller, “A robust torque controller for induction motors without rotor position sensor: Analysis and experimental results,” IEEE Trans. Energy Conversion, vol. 14, no. 12, pp. 1448–1458, Dec. 1999. [8] W. Leonhard, Control of Electrical Drives, 3rd ed. Berlin, Germany: Springer-Verlag, 2001. [9] M. Ghanes, J. Deleon, and A. Glumineau, “Validation of an interconnected high gain observer for sensorless induction motor on low frequencies benchmark: Application to an experimental set-up,” Proc. Inst. Elect. Eng. Control Theory Appl., vol. 152, pp. 371–378, Jul. 2005. [10] V. A. Bondarko and A. T. Zaremba, “Speed and flux estimation for an induction motor without position sensor,” in Proc. Amer. Control Conf., San Diego, CA, 1999, pp. 3890–3894. [11] A. T. Zaremba and S. V. Semonov, “Speed and rotor resistance estimation for torque control of an induction motor,” in Proc. Amer. Control Conf., Chicago, IL, 2000, pp. 605–608. [12] A. Pavlov and A. Zaremba, “Adaptive observers for sensorless control of an induction motor,” in Proc. Amer. Control Conf., Arlington, VA, 2001, pp. 1557–1562. [13] D. Nesic´, I. M. Y. Mareels, S. T. Glad, and M. Jirstrand, “Software for control system analysis and design: Symbol manipulation,” in Encyclopedia of Electrical Engineering, J. Webster, Ed. New York: Wiley, 2001 [Online]. Available: http://www.interscience.wiley.com:83/eeee/. [14] M. Diop and M. Fliess, “On nonlinear observability,” in Proceedings 1st European Control Conference. Paris, France: Hermés, 1991, pp. 152–157. [15] M. Diop and M. Fliess, “Nonlinear observability, identifiability and persistent trajectories,” in Proc. 36th Conf. Decision and Control, Brighton, U.K., 1991, pp. 714–719. [16] M. Fliess and H. Síra-Ramirez, “Control via state estimation of some nonlinear systems,” in Proc. Symp. Nonlinear Control Systems (NOLCOS-2004), Stuttgart, Germany, Sep. 2004. [17] M. Fliess, C. Join, and H. Síra-Ramirez, “Complex continuous nonlinear systems: Their black box identification and their control,” in Proc. 14th IFAC Symp. System Identification (SISYD 2006), Newcastle, Austalia, 2006. [18] S. Ibarra-Rojas, J. Moreno, and G. Espinosa-Pérez, “Global observability analysis of sensorless induction motors,” Automatica, vol. 40, pp. 1079–1085, 2004. [19] M. Li, J. Chiasson, M. Bodson, and L. M. Tolbert, “Observability of speed in an induction motor from stator currents and voltages,” in Proc. IEEE Conf. Decision and Control, Seville, Spain, Dec. 2005, pp. 3438–3443. [20] J. Chiasson, Modeling and High-Performance Control of Electric Machines. New York: Wiley, 2005. [21] S. Diop, J. W. Grizzle, P. E. Moraal, and A. Stefanopoulou, “Interpolation and numerical differentiation algorithms for observer design,” in Proc. Amer. Control Conf., 1994, pp. 1329–1333. [22] S. Diop, J. W. Grizzle, and F. Chaplais, “On numerical differentiation algorithms for nonlinear estimation,” in Proc. IEEE Conf. Decision and Control, Sydney, Australia, 2000, pp. 1133–1138. [23] J. Reger, H. S. Ramirez, and M. Fliess, “On non-asymptotic observation of nonlinear systems,” in Proc. 44th IEEE Conf. Decision and Control, Seville, Spain, Dec. 2005, pp. 4219–4224. [24] I.-J. Ha and S.-H. Lee, “An online identification method for both stator and rotor resistances of induction motors without rotational transducers,” IEEE Trans. Ind. Electron., vol. 47, no. 8, pp. 842–853, Aug. 2000. [25] Opal-RT Technologies RT-LAB [Online]. Available: http://www. opal-rt.com.
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Global Practical Stabilization of Planar Linear Systems in the Presence of Actuator Saturation and Input Additive Disturbance Haijun Fang and Zongli Lin
Abstract—In this note, we revisit the problem of global practical stabilization for planar linear systems subject to actuator saturation and input additive disturbances. A parameterized linear state feedback law is designed such that, by tuning the value of the parameter, all trajectories of the closed-loop system converge to an arbitrarily small neighborhood of the origin in a finite time and remain in there. Index Terms—Actuator saturation, disturbance rejection, practical stabilization.
I. INTRODUCTION Disturbances are common in control systems. There is vast literature that addresses the problem of disturbance rejection for linear systems subject to actuator saturation[3]–[13]. On this topic, two lines of research have been pursued. In the first line, disturbances are assumed to be in the Lp space and controllers are constructed to result in Lp state trajectories, possibly with a small Lp gain from the disturbance to the state[1], [2], [6], [8], [9]. In particular, it is established that Lp gain from the disturbance to the state can be made arbitrarily small by linear state feedback if the Lp disturbances are also bounded in magnitude[8]. Such boundedness assumption on the disturbances can be removed if nonlinear feedback control is allowed [9]. The other line of research focuses on disturbances that are magnitude bounded and may be persistent, such as constant disturbances and sinusoidal disturbances. The objective here is to design controllers such that the closed-loop trajectories enter and remain in an a-priori given arbitrarily small neighborhood of the origin in a finite time. Such a design objective is usually called practical stabilization. If all trajectories are required of the aforementioned behavior, the design objective is called global practical stabilization. If only trajectories starting from a prespecified, but arbitrarily large, bounded set of the state space, then the design objective is called semiglobal practical stabilization. Two pieces of work along this line are Lin [10] and Saberi et al. [13]. In particular, [13] established that semiglobal practical stabilization for a linear system subject to actuator saturation and input additive disturbances can be achieved as long as the open loop system is not exponentially unstable. For the same class of systems, Lin[10] constructed nonlinear feedback laws that achieve global practical stabilization. In this note, we revisit the problem of practical stabilization for planar linear systems subject to actuator saturation and input additive disturbances
R
x_ = Ax + B(u + d)
R
(1)
R
where x 2 2 is the state, u 2 is the input, d 2 is the disturbance, and is a standard saturation function with unity saturation level, i.e.,
(u) = sign(u) min(1; juj): Manuscript received October 6, 2005; revised December 2, 2005. Recommended by Associate Editor D. Angeli. This work was supported in part by the National Science Foundation under Grant CMS-0324329. H. Fang and Z. Lin are with the Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 229044743 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAC.2006.878777 0018-9286/$20.00 © 2006 IEEE
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Fig. 1. Trajectory of the system (11) in .
We assume that (A; B) is controllable, and both eigenvalues of the matrix A have nonpositive real parts. Also assumed is that the disturbance is bounded, i.e., jdj D , for some arbitrarily large positive number D . Under the previous assumptions on the matrices A and B , it is without loss of generality to assume that these two matrices are in one of the following forms: A=
0
1
0
0 0
or B =
0
0
1
0a
or
P =
2 (0; 1], D > 0 and I2 2 R222 is an identity matrix. Denote
p1
p2
p2
p3
to obtain
. Let A =
p p1 =
p
1
0a
or
where
0 0
1
0a 0b
0
p3 =
1
0
0
where a and b are positive numbers. These forms of the matrix pair (A; B) correspond to four different scenarios for the system (1). The objective of this note is to show that the system (1) can be globally practically stabilized by linear state feedback. This result will thus complement the results of [10] and [13]. The remainder of this note is organized as follows. In Section II, some preliminaries are established. The problem of global practical stabilization by linear state feedback is addressed in Section III. Simulation results are shown in Section IV to indicate the effectiveness of the proposed design method. Finally, concluding remarks are made in Section V.
0
01
For A = obtain
2
0
and B =
0
2
2
:
(5)
; > 0,
. We solve ARE (4)
p p2 =
(1 + D) 2(1 + D) + 1
1
we solve ARE (4) to
2 (1 + D)2 + 22 (1 + D)2
p1 =
2
(1 + D)
2 (1 + D)2 + 22
1
0 2 (1 + D)2 + 2
2 2 (1 + D)2 + 22 22
p2 =
p3 =
1
0
and B =
2(1 + D) + 1
(2) (3)
1
0
(1 + D)
(6)
0 (1 + D)
(7)
2 (1 + D)2 + 22
2
2
0 2 (1 + D)2 + 2 : (8)
II. PRELIMINARIES
The following lemma, adapted from [3], places saturated control inside a convex hull of two controls. Lemma 1: Let u; v 2 with jv j 1. Let E1 = 0; E2 = 1. Then
sat(u)
Consider the following algebraic Riccati equation (ARE): T PA + A P
0 (1 +2 D) P BBT P = 0 (1 +1 D) I2
(4)
R
(u)
2 cofu; vg = cofE u + E 0v; i = 1; 2g i
i
(9)
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where co denotes the convex hull and Ei0 = 1 0 Ei . For a given positive–definite matrix P 2 222 and a positive scalar , denote
R
(
" P;
) := fx 2 R2 ; xT P x g: III. MAIN RESULT
In this section, we will address the problem of global practical stabilization for (1). The proposed control law is linear state feedback of the form
u
= 0 1 B T P x
(10)
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Symmetrically, any trajectory in 0 is also a parabola. Then, we have the following facts. Fact 1: Any trajectory starting from the point xs on the line 1=B T P x = 1 + D, xs1 > 01, enters the interior of 0 ; any trajectory starting from the point xs on the line 1=B T P x = 1 + D , + + xs1 < 01, enters and will remain inside for a finite time T before it reaches the line 1=B P x = 1 + D again. Fact 2: Any trajectory starting from the point xs on the line 1=B T P x = 01 0 D, xs1 < 1, enters the interior of 0 ; any trajectory starting from the point xs on the line 1=B T P x = 01 0 D , with xs1 > 1, enters 0 and will remain inside 0 for a finite time before it reaches the line 1=B T P x = 01 0 D again. We choose the Lyapunov function V (x) = xT P x. According to (5) and (15), we have that, for any xs on the line 1=B T P x = 1 + D and with xs1 < 01
( ) 0 V (x ) p = 0x 1 0 2; 0x 2 + 2 2(1 + D) + 1
V xe
where 2 (0; 1] and P is a positive–definite matrix that will be determined according to the form of (A; B ) as given in (2)–(3). Theorem 1: Consider (1) with the matrices A and B defined in (2)-(3) and under the state feedback law (10). For any given arbitrarily small set 0 2 containing the origin in its interior, and any positive number D , there exists an 3 2 (0; 1] such that, for any 2 (0; 3 ], all trajectories of the closed-loop system will enter 0 in a finite time and remain in there. Proof: First, we consider the system
s
s
s
p 2 0x 1 0 2; 0x 2 + 2 2(1 + D) + 1 0 xT P x p = 4 1; 0 2(1 + D) + 1 P T p 2 x 1 + 1; x 2 0 2(1 + D) + 1 : s
R
_ = 00 10
x
+ 01 sat(u + d):
(11)
In this case, we choose the matrix P to be the solution to (4). Define
s
s
+ : =
x
0 : =
x
0 : =
x
2 R2 : 1 BT P x > 1 + D
0 x0 1
xs2
p 0 ( 2(1 + D) + 1) = 0x 1 0 1: s
( ) 0 V (x ) p 1; 0 2(1 + D) + 1 = p 4 2(1 + D) + 1
V xe
:
s
p
2
For an x 2 + , u + d < 01, and (11) reduces to
0 1 x_ = 0 0
2(1 + D) + 1
2(1 + D) + 1(x 1 + 1); 0(x 1 + 1) p 2+ 1 = p4(x 1 + 1) : 2 (1 + D ) 2(1 + D) + 1 s
:
(12)
1 x2 + x = 2 2 1
=
xs2
xe
=
(13)
xe1
sat 1 B T P x 0 d 2 co
Ei
xe2
:
= 0x 1 0 2 p x 2 = 0x 2 + 2 2(1 + D) + 1: s
e
s
1 BTP x 0 d
+E 0 (1 +1 D) B T P x; i 2 [1; 2] i
and, consequently (14)
It follows from (13) and (14) xe1
T
s
Recalling that xs1 < 01, we have V (xe ) < V (xs ). Similarly, we can also establish that V (xe ) < V (xs ) for any trajectory starting from a point xs on the line 1=B T P x = 01 0 D and with xs1 > 1. Now, for any x 2 0 , j1=(1 + D)B T P xj < 1. By Lemma 1, we have
where is a positive constant. The starting point xs and the ending point xe of the curve xs xe are on the line 1=B TP x = 1 + D . Let
xs
P
s
Through the analysis of the trajectory, we can easily verify that the system trajectory in + is a parabola (see Fig. 1) for a trajectory xs xe that starts from a point xs on 1=B T P x = 1 + D; xs1 < 01)
xs1
= 1 + D, we have
Hence
2 R2 : 1 BT P x > 01 0 D 2 R2 : 1 BT P x 1 + D
s
Noting that xs is on the line 1=B T P x
p
T
s
s
x
P
_ ( ) = xT P
V x
+ (15)
Ax
max 2[1 2] i
Ax
;
0 Bsat 1 BT P x 0 d
0 Bsat 1 BT P x 0 d T
x P
Ax
T
Px
0 BE 1 BT P x 0 d i
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Fig. 2.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 7, JULY 2006
, "(P; ) and S
.
Fig. 3. Trajectory of the system (19) in .
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0BE 0 (D1+ 1) B T P x i
+
Ax
0 BE 1 BT P x 0 d i
0BE 0 (D1+ 1) BT P x
T
i
xT
PA
Px
+ AT P 0 (1 +2 D) P BB T P
x
0 (D2D+ 1) xT P BE BT P x + 2xT P BE d i
i
0 (1 +1 D) xT x + 2D(D + 1):
(16)
Define
d :=
x
2 R2 : kxk22 22 D(1 + D)2
:
(17)
Clearly there exists a > 0 such that "(P; ) is the smallest ellipsoid that satisfies 0 \ d 0 \ "(P; ) (see Fig. 2). Hence, by (16), we have V_ (x) < 0, 8x 2 0 n d , which implies _V (x(t)) < 0, 8x 2 0 n "(P; ). Since D is given, d , and hence (P; ), shrink towards the origin as decreases to zero. Denote the set enclosed by arcs AD and BC and lines AB and CD as SABCD (see Fig. 2). Clearly, there exists an 3 2 (0; 1], such that, for all 2 (0; 3 ], SABCD is inside the unit circle and X0 . Then, by Fact 1, any trajectory starting from a point on the line AB enters 0 . Similarly, by Fact 2, any trajectory starting from a point on the line CD enters 0 . In summary, any trajectory starting from outside of SABCD will enter SABCD in a finite time and remain there. This completes the proof for the system (11). Next, we consider the system
_ = 00a 10
x
x
+ 01 sat(u + d)
(18)
where a > 0. By time scaling and change of coordinates, the system (18) with any a > 0 can be transformed into the following system:
_ = 001 10
x
x
+ 0 sat(u + d)
(19)
where = 1=a > 0. Hence, without loss of generality, we will consider (19) under the feedback (10) with P given by (6)–(8). It will be shown that, for sufficiently small > 0, all trajectories of the closed-loop system will enter X0 in a finite time and remain there. As with the system (11), we consider the three cases: x 2 + , x 2 0
and x 2 0 . By analysis of system trajectories, it is straightforward to verify that any system trajectory in + is an arc (see the curve xs xe in Fig. 3),
(x1 + )2 + x22 = where is a positive constant. Let xs be the starting point and xe the ending point. We have xs
+ xe = 2
0 2
2 p3 ((1 + D ) + p2 )
Fig. 4. State response of the closed-loop system (11) with (left plot) = 0:1 and with (right plot) = 0:01.
:
(20)
Symmetrically, any trajectory in 0 is also an arc. We have the following facts. Fact 3: Any trajectory starting from the point xs on the line 1=B T P x = 1 + D, xs1 > 0=2, enters the interior of 0 ; Any trajectory starting from the point xs on the line 1=B T P x = 1 + D , + + xs1 < 0=2, enters and will remain inside of for a finite time T before it reaches the line 1=B P x = 1 + D again. Fact 4: Any trajectory starting from the point xs on the line 1=B T P x = 01 0 D, xs1 < =2, enters the interior of 0 ; Any trajectory starting from the point xs on the line 1=B T P x = 01 0 D , 0 0 xs1 > =2, enters and will remain inside for a finite time T before it reaches the line 1=B P x = 01 0 D again. Choose the Lyapunov function V (x) = xT P x. In what follows, we will show that V (xe ) < V (xs ), where xs and xe are on the line 1=B T P x = 1 + D and xs1 < 0=2. Let xs
=
0 2
( (1 + D) + 2 p2 ) 0 0 pp
p3
;
0
> :
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Fig. 5. State response of the closed-loop system (18) with (left plot) = 0:1 and with (right plot) = 0:01.
We have
( ) 0 V (xs ) = 0 2 p1 p3p 0 p2 2
V xe
3
0, > 0, p1 p3 0 p22 > 0, and p3 > 0. Similarly, for xs and xe on the line 1=B T P = 01 0 D and xs1 > =2, V (xe ) < V (xs ) holds too. Now, for any x 2 0 , the same derivation of (16) holds and
1 xT x + 2D(D + 1) V_ (x) 0 (1 + D )
Fig. 6. State response of the closed-loop system (23) with (left plot) = 0:1 and with (right plot) = 0:01.
that SABCD is inside the set X0 and the circle with the radius =2. By Facts 3, 4, and (22), it can be concluded that any trajectory starting from outside of the set SABCD will enter the set SABCD and remain there. The proof for (19) is thus completed. We next consider the system
_ = 00 01a
x
(21)
which implies that
_( ) 0
8x 2 0 n d
+ 01 sat(u + d)
(23)
where a > 0. The control law is of the form (10) with P
V x
1 + D =) ax + x > (1 + D): 1 2
Then
_ = x2 (26) _ = 0ax2 0 1 which implies that ax_ 1 + x_ 2 = 01. Thus, ax1 + x2 decreases at a constant rate as time goes by, which implies that ax1 + x2 will be less than (1 + D) and any trajectory will enter 0 in a finite time. Symmetrically, any trajectory starting from inside 0 will enter 0 in a finite time. Hence, 0 is an invariant set. For x 2 0 , j1=B T P xj < 1 + D. Choose the Lyapunov function V (x) = xT P x, where P is x1 x2
defined in (24). Using Lemma 1 and following similar steps in deriving (16), we have
0 1 x + 0 sat(0 1 B T P x + d) 0 0a 1 T + 00 01a x + 01 sat(0 1 B T P x + d) P x 2 0 2 xT a0 10 x + 2D(D + 1): (27)
_ ( ) = xT P
V x
Define
2
d1 =: fx 2 R2 ; xT a0 10
x
2 D(D + 1)g:
Clearly, there exists an ellipsoid "(P; ); > 0, such that "(P; ) is the smallest ellipsoid and d1 \ 0 "(P; ) \ 0 . Therefore, V_ (x) < 0; 8x 2 0 n "(P; ), which implies that the set "(P; ) \ 0 is an invariant set. Any trajectory starting from outside of the set "(P; ) \
0 will enter this set in a finite time and remain there. By tuning the value of the parameter towards zero, the size of the set d1 , and hence 0 "(P; ) \ will shrink towards the origin. The result of Theorem 1 for the system (23) then follows. Finally, we consider the system
_ = 00a 01b
x
where the matrix [
0
x
+ 01 sat(u + d)
(28)
1
Fig. 7. State response of the closed-loop system (28) with (left plot) = 0:1 and with (right plot) = 0:01.
Similarly, (30) holds for x in deriving (16) that
0a 0b ] has two eigenvalues in the open left plane.
P
0
0a
1 + 0 1 T P = 0I : 2 0b 0a 0b
= xT P x. For x 0 T x P [ ] > (D + 1) and sat(u + d) = 01, then we have 1
_ ( ) = xT
V x
P
0 2x T P
0
0a 0 1
1 + 0 1 TP 0b 0a 0b 0
< :
V x
(29)
2 + ,
Choose the Lyapunov function V (x)
(30)
P
0
2 =: f 2 R2 k k2 2 2 ( + 1)g 0
\ ( )\
( )\ _( ) 0 ( )\
(31)
x ; x 2 D D . Choose an ellipsoid Denote d 2 0 " P; 0 . Then, by (30), for " P; ; > such that d 0 , V x < . In summary, any trajectory of any x outside " P; 0 in a finite time and remain there. the system (28) will enter " P; 0 By tuning the parameter towards zero, the size of the set " P;
(
x
1 + 0 1 TP x 0a 0b 0a 0b T 0 2 xT P 01 01 P x + 2D(D + 1) 0 1 xT x + 2D(D + 1):
_ ( ) xT
The feedback law is of form (10) with the matrix P being the solution to the following equality:
2 0 . For x 2 0 , it follows similar steps
)
(
)\
will shrink towards the origin. This completes the entire proof for the system (28) and the entire proof of the theorem. Q.E.D.
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IV. AN EXAMPLE Example: We consider four cases of the system (1). In the simulation, we have set both a and b in (2) to be 5, and chosen x(0) = [
10 10
Error Feedback and a Servomechanism Problem for Uncertain Nonlinear Systems
],
Hao Lei and Wei Lin
d = 10 sin t. Simulation results of the closed-loop system are shown
in Figs. 4–7. As seen in the simulation results, it is clear that practical stabilization is indeed achieved. V. CONCLUSION In this note, we revisited the problem of practical stabilization of linear planar systems subject to input saturation and input additive disturbance. A linear state feedback law was proposed. By tuning the parameter of the control law, the effect of the disturbance to the system can be reduced so that any trajectory of the closed-loop system will converge to an arbitrarily small neighborhood of the origin.
L
Index Terms—Error feedback, internal model, robust output regulation, uncertain nonlinear systems.
REFERENCES
stabilization of the double integrator subject to [1] Y. Chitour, “On the input saturation,” ESAIM COCV, vol. 6, pp. 291–331, 2001. [2] Y. Chitour, W. Liu, and E. Sontag, “On the continuity and incrementalgain properties of certain saturated linear feedback loops,” Int. J. Robust Nonlinear Control, vol. 5, pp. 413–440, 1995. [3] T. Hu and Z. Lin, Control Systems With Actuator Saturation: Analysis and Design. Boston, MA: Birkhäuser, 2001. [4] ——, “Practical stabilization on the null controllable region of exponentially unstable linear systems subject to actuator saturation nonlinearities and disturbance,” Int. J. Robust Nonlinear Control, vol. 11, no. 6, pp. 555–588, 2001. [5] H. Hindi and S. Boyd, “Analysis of linear systems with saturation using convex optimization,” in Proc. 37th IEEE Conf. Decision Control, 1998, pp. 903–908. [6] W. Liu, Y. Chitour, and E. Sontag, “On finite gain stabilizability of linear systems subject to input saturation,” SIAM J. Control Optim., vol. 34, pp. 1190–1219, 1996. [7] Z. Lin, Low Gain Feedback. London, U.K.: Springer-Verlag, 1998, vol. 240, Lecture Notes in Control and Information Sciences. [8] Z. Lin, A. Saberi, and A. R. Teel, “Almost disturbance decoupling with internal stability for linear systems subject to input saturation—state feedback case,” Automatica, vol. 32, no. 4, pp. 619–624, 1996. -almost disturbance decoupling with internal stability for [9] Z. Lin, “ linear systems subject to input saturation,” IEEE Trans. Autom. Control, vol. 42, no. 7, pp. 992–995, Jul. 1997. [10] Z. Lin, “Global control of linear systems with saturating actuators,” Automatica, vol. 34, pp. 897–905, 1998. [11] T. Nguyen and F. Jabbari, “Output feedback controllers for disturbance attenuation with bounded inputs,” in Proc. 36th IEEE Conf. Decision Control, 1997, pp. 177–182. [12] C. Paim, S. Tarbouriech, J. M. G. da Silva Jr., and E. B. Castelan, “Control design for linear systems with saturating actuators and L -bounded disturbances,” in Proc. 41st IEEE Conf. Decision Control, 2002, pp. 4148–4153. [13] A. Saberi, Z. Lin, and A. R. Teel, “Control of linear systems with saturating actuators,” IEEE Trans. Autom. Control, vol. 41, no. 3, pp. 368–378, Mar. 1996.
H
Abstract—The robust servomechanism problem, also known as the problem of robust output regulation, is considered for a family of uncertain triangular systems. In contrast to the previous work where the uncertain vector field of the controlled plant was assumed to be bounded by a linear growth of the unmeasurable states multiplied by a polynomial function of the system output, we show that the polynomial growth condition can be relaxed and robust output regulation by error feedback is still possible, as long as the uncertain system is dominated by a continuous function of the system output multiplied by a linear growth of the unmeasurable states.
I. INTRODUCTION AND DISCUSSION We consider the robust servomechanism problem, also known as the problem of robust output regulation by error feedback, for the uncertain feedback linearizable system x_ 1 = x2 + f1 (x1 ; !; )
.. . x_ n = u + fn (x1 ; . . . ; xn ; !; ) e1 = x 1
0 q(!; )
! _ = S!
(1)
where x = (x1 ; . . . ; xn )T , u 2 R are the system state and input, respectively, ! 2 Rs is the exosignal that represents a reference (respectively, disturbance) signal to be tracked (respectively, rejected). The exosystem !_ = S! is neutrally stable, i.e., all the eigenvalues of S are simple and lie on the imaginary axis. The parameter 2 Rp is an unknown vector ranging over a known compact set U . The functions fi ( 1 ); i = 1; . . . ; n and q ( 1 ) are smooth functions of their arguments. The error e1 is the only measurable signal that can be used in feedback design. The problem of global robust output regulation by error feedback is to find a smooth dynamic compensator of the form _ =
(; e1 )
u = (; e1 )
(2)
such that i) all the states (x(t); (t)) of the closed-loop system (1)–(2) are well-defined and uniformly bounded; e1 (t) = 0; 8 2 U , and 8(x(0); ! (0)) 2 Rn 2 W 0 , ii) limt where W 0 is an arbitrary but fixed known compact subset of Rs containing the origin.
!1
Manuscript received March 30, 2005; revised October 6, 2005 and April 8, 2006. Recommended by Associate Editor F. Bullo. This work was supported in part by the National Science Foundation under Grants DMS-0203387 and ECS-0400413, and in part by the Air Force Research Laboratory under Grant FA8651-05-C-0110. The authors are with the Department of Electrical Engineering and Computer Science, Case Western Reserve University, Cleveland, OH 44106 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2006.878778 0018-9286/$20.00 © 2006 IEEE
[7] E. G. Strangas, H. K. Khalil, B. A. Oliwi, L. Laubnger, and J. M. Miller, “A robust torque controller for induction motors without rotor position sensor: Analysis and experimental results,” IEEE Trans. Energy Conversion, vol. 14, no. 12, pp. 1448–1458, Dec. 1999. [8] W. Leonhard, Control of Electrical Drives, 3rd ed. Berlin, Germany: Springer-Verlag, 2001. [9] M. Ghanes, J. Deleon, and A. Glumineau, “Validation of an interconnected high gain observer for sensorless induction motor on low frequencies benchmark: Application to an experimental set-up,” Proc. Inst. Elect. Eng. Control Theory Appl., vol. 152, pp. 371–378, Jul. 2005. [10] V. A. Bondarko and A. T. Zaremba, “Speed and flux estimation for an induction motor without position sensor,” in Proc. Amer. Control Conf., San Diego, CA, 1999, pp. 3890–3894. [11] A. T. Zaremba and S. V. Semonov, “Speed and rotor resistance estimation for torque control of an induction motor,” in Proc. Amer. Control Conf., Chicago, IL, 2000, pp. 605–608. [12] A. Pavlov and A. Zaremba, “Adaptive observers for sensorless control of an induction motor,” in Proc. Amer. Control Conf., Arlington, VA, 2001, pp. 1557–1562. [13] D. Nesic´, I. M. Y. Mareels, S. T. Glad, and M. Jirstrand, “Software for control system analysis and design: Symbol manipulation,” in Encyclopedia of Electrical Engineering, J. Webster, Ed. New York: Wiley, 2001 [Online]. Available: http://www.interscience.wiley.com:83/eeee/. [14] M. Diop and M. Fliess, “On nonlinear observability,” in Proceedings 1st European Control Conference. Paris, France: Hermés, 1991, pp. 152–157. [15] M. Diop and M. Fliess, “Nonlinear observability, identifiability and persistent trajectories,” in Proc. 36th Conf. Decision and Control, Brighton, U.K., 1991, pp. 714–719. [16] M. Fliess and H. Síra-Ramirez, “Control via state estimation of some nonlinear systems,” in Proc. Symp. Nonlinear Control Systems (NOLCOS-2004), Stuttgart, Germany, Sep. 2004. [17] M. Fliess, C. Join, and H. Síra-Ramirez, “Complex continuous nonlinear systems: Their black box identification and their control,” in Proc. 14th IFAC Symp. System Identification (SISYD 2006), Newcastle, Austalia, 2006. [18] S. Ibarra-Rojas, J. Moreno, and G. Espinosa-Pérez, “Global observability analysis of sensorless induction motors,” Automatica, vol. 40, pp. 1079–1085, 2004. [19] M. Li, J. Chiasson, M. Bodson, and L. M. Tolbert, “Observability of speed in an induction motor from stator currents and voltages,” in Proc. IEEE Conf. Decision and Control, Seville, Spain, Dec. 2005, pp. 3438–3443. [20] J. Chiasson, Modeling and High-Performance Control of Electric Machines. New York: Wiley, 2005. [21] S. Diop, J. W. Grizzle, P. E. Moraal, and A. Stefanopoulou, “Interpolation and numerical differentiation algorithms for observer design,” in Proc. Amer. Control Conf., 1994, pp. 1329–1333. [22] S. Diop, J. W. Grizzle, and F. Chaplais, “On numerical differentiation algorithms for nonlinear estimation,” in Proc. IEEE Conf. Decision and Control, Sydney, Australia, 2000, pp. 1133–1138. [23] J. Reger, H. S. Ramirez, and M. Fliess, “On non-asymptotic observation of nonlinear systems,” in Proc. 44th IEEE Conf. Decision and Control, Seville, Spain, Dec. 2005, pp. 4219–4224. [24] I.-J. Ha and S.-H. Lee, “An online identification method for both stator and rotor resistances of induction motors without rotational transducers,” IEEE Trans. Ind. Electron., vol. 47, no. 8, pp. 842–853, Aug. 2000. [25] Opal-RT Technologies RT-LAB [Online]. Available: http://www. opal-rt.com.
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Global Practical Stabilization of Planar Linear Systems in the Presence of Actuator Saturation and Input Additive Disturbance Haijun Fang and Zongli Lin
Abstract—In this note, we revisit the problem of global practical stabilization for planar linear systems subject to actuator saturation and input additive disturbances. A parameterized linear state feedback law is designed such that, by tuning the value of the parameter, all trajectories of the closed-loop system converge to an arbitrarily small neighborhood of the origin in a finite time and remain in there. Index Terms—Actuator saturation, disturbance rejection, practical stabilization.
I. INTRODUCTION Disturbances are common in control systems. There is vast literature that addresses the problem of disturbance rejection for linear systems subject to actuator saturation[3]–[13]. On this topic, two lines of research have been pursued. In the first line, disturbances are assumed to be in the Lp space and controllers are constructed to result in Lp state trajectories, possibly with a small Lp gain from the disturbance to the state[1], [2], [6], [8], [9]. In particular, it is established that Lp gain from the disturbance to the state can be made arbitrarily small by linear state feedback if the Lp disturbances are also bounded in magnitude[8]. Such boundedness assumption on the disturbances can be removed if nonlinear feedback control is allowed [9]. The other line of research focuses on disturbances that are magnitude bounded and may be persistent, such as constant disturbances and sinusoidal disturbances. The objective here is to design controllers such that the closed-loop trajectories enter and remain in an a-priori given arbitrarily small neighborhood of the origin in a finite time. Such a design objective is usually called practical stabilization. If all trajectories are required of the aforementioned behavior, the design objective is called global practical stabilization. If only trajectories starting from a prespecified, but arbitrarily large, bounded set of the state space, then the design objective is called semiglobal practical stabilization. Two pieces of work along this line are Lin [10] and Saberi et al. [13]. In particular, [13] established that semiglobal practical stabilization for a linear system subject to actuator saturation and input additive disturbances can be achieved as long as the open loop system is not exponentially unstable. For the same class of systems, Lin[10] constructed nonlinear feedback laws that achieve global practical stabilization. In this note, we revisit the problem of practical stabilization for planar linear systems subject to actuator saturation and input additive disturbances
R
x_ = Ax + B(u + d)
R
(1)
R
where x 2 2 is the state, u 2 is the input, d 2 is the disturbance, and is a standard saturation function with unity saturation level, i.e.,
(u) = sign(u) min(1; juj): Manuscript received October 6, 2005; revised December 2, 2005. Recommended by Associate Editor D. Angeli. This work was supported in part by the National Science Foundation under Grant CMS-0324329. H. Fang and Z. Lin are with the Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 229044743 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAC.2006.878777 0018-9286/$20.00 © 2006 IEEE
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Fig. 1. Trajectory of the system (11) in .
We assume that (A; B) is controllable, and both eigenvalues of the matrix A have nonpositive real parts. Also assumed is that the disturbance is bounded, i.e., jdj D , for some arbitrarily large positive number D . Under the previous assumptions on the matrices A and B , it is without loss of generality to assume that these two matrices are in one of the following forms: A=
0
1
0
0 0
or B =
0
0
1
0a
or
P =
2 (0; 1], D > 0 and I2 2 R222 is an identity matrix. Denote
p1
p2
p2
p3
to obtain
. Let A =
p p1 =
p
1
0a
or
where
0 0
1
0a 0b
0
p3 =
1
0
0
where a and b are positive numbers. These forms of the matrix pair (A; B) correspond to four different scenarios for the system (1). The objective of this note is to show that the system (1) can be globally practically stabilized by linear state feedback. This result will thus complement the results of [10] and [13]. The remainder of this note is organized as follows. In Section II, some preliminaries are established. The problem of global practical stabilization by linear state feedback is addressed in Section III. Simulation results are shown in Section IV to indicate the effectiveness of the proposed design method. Finally, concluding remarks are made in Section V.
0
01
For A = obtain
2
0
and B =
0
2
2
:
(5)
; > 0,
. We solve ARE (4)
p p2 =
(1 + D) 2(1 + D) + 1
1
we solve ARE (4) to
2 (1 + D)2 + 22 (1 + D)2
p1 =
2
(1 + D)
2 (1 + D)2 + 22
1
0 2 (1 + D)2 + 2
2 2 (1 + D)2 + 22 22
p2 =
p3 =
1
0
and B =
2(1 + D) + 1
(2) (3)
1
0
(1 + D)
(6)
0 (1 + D)
(7)
2 (1 + D)2 + 22
2
2
0 2 (1 + D)2 + 2 : (8)
II. PRELIMINARIES
The following lemma, adapted from [3], places saturated control inside a convex hull of two controls. Lemma 1: Let u; v 2 with jv j 1. Let E1 = 0; E2 = 1. Then
sat(u)
Consider the following algebraic Riccati equation (ARE): T PA + A P
0 (1 +2 D) P BBT P = 0 (1 +1 D) I2
(4)
R
(u)
2 cofu; vg = cofE u + E 0v; i = 1; 2g i
i
(9)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 7, JULY 2006
where co denotes the convex hull and Ei0 = 1 0 Ei . For a given positive–definite matrix P 2 222 and a positive scalar , denote
R
(
" P;
) := fx 2 R2 ; xT P x g: III. MAIN RESULT
In this section, we will address the problem of global practical stabilization for (1). The proposed control law is linear state feedback of the form
u
= 0 1 B T P x
(10)
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Symmetrically, any trajectory in 0 is also a parabola. Then, we have the following facts. Fact 1: Any trajectory starting from the point xs on the line 1=B T P x = 1 + D, xs1 > 01, enters the interior of 0 ; any trajectory starting from the point xs on the line 1=B T P x = 1 + D , + + xs1 < 01, enters and will remain inside for a finite time T before it reaches the line 1=B P x = 1 + D again. Fact 2: Any trajectory starting from the point xs on the line 1=B T P x = 01 0 D, xs1 < 1, enters the interior of 0 ; any trajectory starting from the point xs on the line 1=B T P x = 01 0 D , with xs1 > 1, enters 0 and will remain inside 0 for a finite time before it reaches the line 1=B T P x = 01 0 D again. We choose the Lyapunov function V (x) = xT P x. According to (5) and (15), we have that, for any xs on the line 1=B T P x = 1 + D and with xs1 < 01
( ) 0 V (x ) p = 0x 1 0 2; 0x 2 + 2 2(1 + D) + 1
V xe
where 2 (0; 1] and P is a positive–definite matrix that will be determined according to the form of (A; B ) as given in (2)–(3). Theorem 1: Consider (1) with the matrices A and B defined in (2)-(3) and under the state feedback law (10). For any given arbitrarily small set 0 2 containing the origin in its interior, and any positive number D , there exists an 3 2 (0; 1] such that, for any 2 (0; 3 ], all trajectories of the closed-loop system will enter 0 in a finite time and remain in there. Proof: First, we consider the system
s
s
s
p 2 0x 1 0 2; 0x 2 + 2 2(1 + D) + 1 0 xT P x p = 4 1; 0 2(1 + D) + 1 P T p 2 x 1 + 1; x 2 0 2(1 + D) + 1 : s
R
_ = 00 10
x
+ 01 sat(u + d):
(11)
In this case, we choose the matrix P to be the solution to (4). Define
s
s
+ : =
x
0 : =
x
0 : =
x
2 R2 : 1 BT P x > 1 + D
0 x0 1
xs2
p 0 ( 2(1 + D) + 1) = 0x 1 0 1: s
( ) 0 V (x ) p 1; 0 2(1 + D) + 1 = p 4 2(1 + D) + 1
V xe
:
s
p
2
For an x 2 + , u + d < 01, and (11) reduces to
0 1 x_ = 0 0
2(1 + D) + 1
2(1 + D) + 1(x 1 + 1); 0(x 1 + 1) p 2+ 1 = p4(x 1 + 1) : 2 (1 + D ) 2(1 + D) + 1 s
:
(12)
1 x2 + x = 2 2 1
=
xs2
xe
=
(13)
xe1
sat 1 B T P x 0 d 2 co
Ei
xe2
:
= 0x 1 0 2 p x 2 = 0x 2 + 2 2(1 + D) + 1: s
e
s
1 BTP x 0 d
+E 0 (1 +1 D) B T P x; i 2 [1; 2] i
and, consequently (14)
It follows from (13) and (14) xe1
T
s
Recalling that xs1 < 01, we have V (xe ) < V (xs ). Similarly, we can also establish that V (xe ) < V (xs ) for any trajectory starting from a point xs on the line 1=B T P x = 01 0 D and with xs1 > 1. Now, for any x 2 0 , j1=(1 + D)B T P xj < 1. By Lemma 1, we have
where is a positive constant. The starting point xs and the ending point xe of the curve xs xe are on the line 1=B TP x = 1 + D . Let
xs
P
s
Through the analysis of the trajectory, we can easily verify that the system trajectory in + is a parabola (see Fig. 1) for a trajectory xs xe that starts from a point xs on 1=B T P x = 1 + D; xs1 < 01)
xs1
= 1 + D, we have
Hence
2 R2 : 1 BT P x > 01 0 D 2 R2 : 1 BT P x 1 + D
s
Noting that xs is on the line 1=B T P x
p
T
s
s
x
P
_ ( ) = xT P
V x
+ (15)
Ax
max 2[1 2] i
Ax
;
0 Bsat 1 BT P x 0 d
0 Bsat 1 BT P x 0 d T
x P
Ax
T
Px
0 BE 1 BT P x 0 d i
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Fig. 2.
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, "(P; ) and S
.
Fig. 3. Trajectory of the system (19) in .
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0BE 0 (D1+ 1) B T P x i
+
Ax
0 BE 1 BT P x 0 d i
0BE 0 (D1+ 1) BT P x
T
i
xT
PA
Px
+ AT P 0 (1 +2 D) P BB T P
x
0 (D2D+ 1) xT P BE BT P x + 2xT P BE d i
i
0 (1 +1 D) xT x + 2D(D + 1):
(16)
Define
d :=
x
2 R2 : kxk22 22 D(1 + D)2
:
(17)
Clearly there exists a > 0 such that "(P; ) is the smallest ellipsoid that satisfies 0 \ d 0 \ "(P; ) (see Fig. 2). Hence, by (16), we have V_ (x) < 0, 8x 2 0 n d , which implies _V (x(t)) < 0, 8x 2 0 n "(P; ). Since D is given, d , and hence (P; ), shrink towards the origin as decreases to zero. Denote the set enclosed by arcs AD and BC and lines AB and CD as SABCD (see Fig. 2). Clearly, there exists an 3 2 (0; 1], such that, for all 2 (0; 3 ], SABCD is inside the unit circle and X0 . Then, by Fact 1, any trajectory starting from a point on the line AB enters 0 . Similarly, by Fact 2, any trajectory starting from a point on the line CD enters 0 . In summary, any trajectory starting from outside of SABCD will enter SABCD in a finite time and remain there. This completes the proof for the system (11). Next, we consider the system
_ = 00a 10
x
x
+ 01 sat(u + d)
(18)
where a > 0. By time scaling and change of coordinates, the system (18) with any a > 0 can be transformed into the following system:
_ = 001 10
x
x
+ 0 sat(u + d)
(19)
where = 1=a > 0. Hence, without loss of generality, we will consider (19) under the feedback (10) with P given by (6)–(8). It will be shown that, for sufficiently small > 0, all trajectories of the closed-loop system will enter X0 in a finite time and remain there. As with the system (11), we consider the three cases: x 2 + , x 2 0
and x 2 0 . By analysis of system trajectories, it is straightforward to verify that any system trajectory in + is an arc (see the curve xs xe in Fig. 3),
(x1 + )2 + x22 = where is a positive constant. Let xs be the starting point and xe the ending point. We have xs
+ xe = 2
0 2
2 p3 ((1 + D ) + p2 )
Fig. 4. State response of the closed-loop system (11) with (left plot) = 0:1 and with (right plot) = 0:01.
:
(20)
Symmetrically, any trajectory in 0 is also an arc. We have the following facts. Fact 3: Any trajectory starting from the point xs on the line 1=B T P x = 1 + D, xs1 > 0=2, enters the interior of 0 ; Any trajectory starting from the point xs on the line 1=B T P x = 1 + D , + + xs1 < 0=2, enters and will remain inside of for a finite time T before it reaches the line 1=B P x = 1 + D again. Fact 4: Any trajectory starting from the point xs on the line 1=B T P x = 01 0 D, xs1 < =2, enters the interior of 0 ; Any trajectory starting from the point xs on the line 1=B T P x = 01 0 D , 0 0 xs1 > =2, enters and will remain inside for a finite time T before it reaches the line 1=B P x = 01 0 D again. Choose the Lyapunov function V (x) = xT P x. In what follows, we will show that V (xe ) < V (xs ), where xs and xe are on the line 1=B T P x = 1 + D and xs1 < 0=2. Let xs
=
0 2
( (1 + D) + 2 p2 ) 0 0 pp
p3
;
0
> :
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Fig. 5. State response of the closed-loop system (18) with (left plot) = 0:1 and with (right plot) = 0:01.
We have
( ) 0 V (xs ) = 0 2 p1 p3p 0 p2 2
V xe
3
0, > 0, p1 p3 0 p22 > 0, and p3 > 0. Similarly, for xs and xe on the line 1=B T P = 01 0 D and xs1 > =2, V (xe ) < V (xs ) holds too. Now, for any x 2 0 , the same derivation of (16) holds and
1 xT x + 2D(D + 1) V_ (x) 0 (1 + D )
Fig. 6. State response of the closed-loop system (23) with (left plot) = 0:1 and with (right plot) = 0:01.
that SABCD is inside the set X0 and the circle with the radius =2. By Facts 3, 4, and (22), it can be concluded that any trajectory starting from outside of the set SABCD will enter the set SABCD and remain there. The proof for (19) is thus completed. We next consider the system
_ = 00 01a
x
(21)
which implies that
_( ) 0
8x 2 0 n d
+ 01 sat(u + d)
(23)
where a > 0. The control law is of the form (10) with P
V x
1 + D =) ax + x > (1 + D): 1 2
Then
_ = x2 (26) _ = 0ax2 0 1 which implies that ax_ 1 + x_ 2 = 01. Thus, ax1 + x2 decreases at a constant rate as time goes by, which implies that ax1 + x2 will be less than (1 + D) and any trajectory will enter 0 in a finite time. Symmetrically, any trajectory starting from inside 0 will enter 0 in a finite time. Hence, 0 is an invariant set. For x 2 0 , j1=B T P xj < 1 + D. Choose the Lyapunov function V (x) = xT P x, where P is x1 x2
defined in (24). Using Lemma 1 and following similar steps in deriving (16), we have
0 1 x + 0 sat(0 1 B T P x + d) 0 0a 1 T + 00 01a x + 01 sat(0 1 B T P x + d) P x 2 0 2 xT a0 10 x + 2D(D + 1): (27)
_ ( ) = xT P
V x
Define
2
d1 =: fx 2 R2 ; xT a0 10
x
2 D(D + 1)g:
Clearly, there exists an ellipsoid "(P; ); > 0, such that "(P; ) is the smallest ellipsoid and d1 \ 0 "(P; ) \ 0 . Therefore, V_ (x) < 0; 8x 2 0 n "(P; ), which implies that the set "(P; ) \ 0 is an invariant set. Any trajectory starting from outside of the set "(P; ) \
0 will enter this set in a finite time and remain there. By tuning the value of the parameter towards zero, the size of the set d1 , and hence 0 "(P; ) \ will shrink towards the origin. The result of Theorem 1 for the system (23) then follows. Finally, we consider the system
_ = 00a 01b
x
where the matrix [
0
x
+ 01 sat(u + d)
(28)
1
Fig. 7. State response of the closed-loop system (28) with (left plot) = 0:1 and with (right plot) = 0:01.
Similarly, (30) holds for x in deriving (16) that
0a 0b ] has two eigenvalues in the open left plane.
P
0
0a
1 + 0 1 T P = 0I : 2 0b 0a 0b
= xT P x. For x 0 T x P [ ] > (D + 1) and sat(u + d) = 01, then we have 1
_ ( ) = xT
V x
P
0 2x T P
0
0a 0 1
1 + 0 1 TP 0b 0a 0b 0
< :
V x
(29)
2 + ,
Choose the Lyapunov function V (x)
(30)
P
0
2 =: f 2 R2 k k2 2 2 ( + 1)g 0
\ ( )\
( )\ _( ) 0 ( )\
(31)
x ; x 2 D D . Choose an ellipsoid Denote d 2 0 " P; 0 . Then, by (30), for " P; ; > such that d 0 , V x < . In summary, any trajectory of any x outside " P; 0 in a finite time and remain there. the system (28) will enter " P; 0 By tuning the parameter towards zero, the size of the set " P;
(
x
1 + 0 1 TP x 0a 0b 0a 0b T 0 2 xT P 01 01 P x + 2D(D + 1) 0 1 xT x + 2D(D + 1):
_ ( ) xT
The feedback law is of form (10) with the matrix P being the solution to the following equality:
2 0 . For x 2 0 , it follows similar steps
)
(
)\
will shrink towards the origin. This completes the entire proof for the system (28) and the entire proof of the theorem. Q.E.D.
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IV. AN EXAMPLE Example: We consider four cases of the system (1). In the simulation, we have set both a and b in (2) to be 5, and chosen x(0) = [
10 10
Error Feedback and a Servomechanism Problem for Uncertain Nonlinear Systems
],
Hao Lei and Wei Lin
d = 10 sin t. Simulation results of the closed-loop system are shown
in Figs. 4–7. As seen in the simulation results, it is clear that practical stabilization is indeed achieved. V. CONCLUSION In this note, we revisited the problem of practical stabilization of linear planar systems subject to input saturation and input additive disturbance. A linear state feedback law was proposed. By tuning the parameter of the control law, the effect of the disturbance to the system can be reduced so that any trajectory of the closed-loop system will converge to an arbitrarily small neighborhood of the origin.
L
Index Terms—Error feedback, internal model, robust output regulation, uncertain nonlinear systems.
REFERENCES
stabilization of the double integrator subject to [1] Y. Chitour, “On the input saturation,” ESAIM COCV, vol. 6, pp. 291–331, 2001. [2] Y. Chitour, W. Liu, and E. Sontag, “On the continuity and incrementalgain properties of certain saturated linear feedback loops,” Int. J. Robust Nonlinear Control, vol. 5, pp. 413–440, 1995. [3] T. Hu and Z. Lin, Control Systems With Actuator Saturation: Analysis and Design. Boston, MA: Birkhäuser, 2001. [4] ——, “Practical stabilization on the null controllable region of exponentially unstable linear systems subject to actuator saturation nonlinearities and disturbance,” Int. J. Robust Nonlinear Control, vol. 11, no. 6, pp. 555–588, 2001. [5] H. Hindi and S. Boyd, “Analysis of linear systems with saturation using convex optimization,” in Proc. 37th IEEE Conf. Decision Control, 1998, pp. 903–908. [6] W. Liu, Y. Chitour, and E. Sontag, “On finite gain stabilizability of linear systems subject to input saturation,” SIAM J. Control Optim., vol. 34, pp. 1190–1219, 1996. [7] Z. Lin, Low Gain Feedback. London, U.K.: Springer-Verlag, 1998, vol. 240, Lecture Notes in Control and Information Sciences. [8] Z. Lin, A. Saberi, and A. R. Teel, “Almost disturbance decoupling with internal stability for linear systems subject to input saturation—state feedback case,” Automatica, vol. 32, no. 4, pp. 619–624, 1996. -almost disturbance decoupling with internal stability for [9] Z. Lin, “ linear systems subject to input saturation,” IEEE Trans. Autom. Control, vol. 42, no. 7, pp. 992–995, Jul. 1997. [10] Z. Lin, “Global control of linear systems with saturating actuators,” Automatica, vol. 34, pp. 897–905, 1998. [11] T. Nguyen and F. Jabbari, “Output feedback controllers for disturbance attenuation with bounded inputs,” in Proc. 36th IEEE Conf. Decision Control, 1997, pp. 177–182. [12] C. Paim, S. Tarbouriech, J. M. G. da Silva Jr., and E. B. Castelan, “Control design for linear systems with saturating actuators and L -bounded disturbances,” in Proc. 41st IEEE Conf. Decision Control, 2002, pp. 4148–4153. [13] A. Saberi, Z. Lin, and A. R. Teel, “Control of linear systems with saturating actuators,” IEEE Trans. Autom. Control, vol. 41, no. 3, pp. 368–378, Mar. 1996.
H
Abstract—The robust servomechanism problem, also known as the problem of robust output regulation, is considered for a family of uncertain triangular systems. In contrast to the previous work where the uncertain vector field of the controlled plant was assumed to be bounded by a linear growth of the unmeasurable states multiplied by a polynomial function of the system output, we show that the polynomial growth condition can be relaxed and robust output regulation by error feedback is still possible, as long as the uncertain system is dominated by a continuous function of the system output multiplied by a linear growth of the unmeasurable states.
I. INTRODUCTION AND DISCUSSION We consider the robust servomechanism problem, also known as the problem of robust output regulation by error feedback, for the uncertain feedback linearizable system x_ 1 = x2 + f1 (x1 ; !; )
.. . x_ n = u + fn (x1 ; . . . ; xn ; !; ) e1 = x 1
0 q(!; )
! _ = S!
(1)
where x = (x1 ; . . . ; xn )T , u 2 R are the system state and input, respectively, ! 2 Rs is the exosignal that represents a reference (respectively, disturbance) signal to be tracked (respectively, rejected). The exosystem !_ = S! is neutrally stable, i.e., all the eigenvalues of S are simple and lie on the imaginary axis. The parameter 2 Rp is an unknown vector ranging over a known compact set U . The functions fi ( 1 ); i = 1; . . . ; n and q ( 1 ) are smooth functions of their arguments. The error e1 is the only measurable signal that can be used in feedback design. The problem of global robust output regulation by error feedback is to find a smooth dynamic compensator of the form _ =
(; e1 )
u = (; e1 )
(2)
such that i) all the states (x(t); (t)) of the closed-loop system (1)–(2) are well-defined and uniformly bounded; e1 (t) = 0; 8 2 U , and 8(x(0); ! (0)) 2 Rn 2 W 0 , ii) limt where W 0 is an arbitrary but fixed known compact subset of Rs containing the origin.
!1
Manuscript received March 30, 2005; revised October 6, 2005 and April 8, 2006. Recommended by Associate Editor F. Bullo. This work was supported in part by the National Science Foundation under Grants DMS-0203387 and ECS-0400413, and in part by the Air Force Research Laboratory under Grant FA8651-05-C-0110. The authors are with the Department of Electrical Engineering and Computer Science, Case Western Reserve University, Cleveland, OH 44106 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2006.878778 0018-9286/$20.00 © 2006 IEEE
