IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 2, FEBRUARY 2005
Global Exponential Stabilization of a Class of Nonlinear
Systems by Output Feedback
Ho-Lim Choi and Jong-Tae Lim
Abstract—This work extends the existing output feedback stabilization
schemes for the systems in a “perturbed chain-of-integrator” form. In particular, we further relax the triangular-type conditions imposed on the perturbed terms and analyze the robust property of the linear output feedback
control law using the newly proposed condition.
Index Terms—Global exponential stabilization, output feedback control.
255
Assumption 1 is a newly imposed condition on the perturbed term
(t; x; u). For comparison, we present two conditions which are special
cases of (2). First, the condition in [9] is as follows.
Triangularity Condition: For i = 1; . . . ; n, there exists a constant
c 0 such that
ji (t; x; u)j c(jx j + 1 1 1 + jxi j):
1
Note that the similar triangular-type condition is also imposed in [11].
Second, we consider a class of feedforward systems [13] with the
following condition.
Feedforward Condition: For i = 1; . . . ; n 0 2, there exists a constant c 0 such that
ji (t; x; u)j c(jxi j + 1 1 1 + jxn j)
I. INTRODUCTION
+2
The problem of global stabilization of nonlinear systems by output
feedback has received much attention. As investigated in [5], some
extra growth conditions on the unmeasurable states of the system are
usually necessary for the global stabilization of nonlinear systems via
output feedback. This restriction is recently relaxed such that growth
condition is no longer necessary in the case of planar systems [7], [8].
In this paper, the nonlinear system under consideration is a “perturbed
chain-of-integrator” form as treated in [9]–[13]. In [9], a backstepping-like design procedure on the observer error dynamics is introduced where global stabilization is achieved by a linear output feedback controller under the triangularity condition.
The main contribution of this note is the global stabilization of nonlinear systems by a linear output feedback under a generalized condition. We show that the existing conditions such as the triangularity
condition or the feedforward condition are special cases of our new
condition. Moreover, the observers for nonlinear systems are usually
studied as high-gain observers [1], [2], [9]. In our work, we show that
the “gain” of the observer needs to be properly tuned from the high-gain
to the low-gain depending on the nature of perturbed terms. Throughout
this note, the Euclidean two-norm is used. Otherwise, it will be specifically denoted by subscript.
II. PRELIMINARIES
We consider a class of single-input–single-output nonlinear systems
given by
x_ = Ax + Bu + (t; x; u)
y = Cx
(1)
where x 2 Rn is the state, u 2 R and y 2 R are the input
and the output of the system, respectively. Also, (A; B ) is the
Brunovsky canonical pair, C = [1; 0; . . . ; 0], and (t; x; u) =
T
n
[1 (t; x; u); . . . ; n (t; x; u)] . The mappings i (t; x; u) : R 2 R 2
R ! R; i = 1; . . . ; n, are continuous and satisfy the following
assumption.
Assumption 1: There exists a function () 0 such that for > 0
n
i=1
i01
ji (t; x; u)j ()
n
i=1
i01
jx i j :
(2)
(4)
with n01 (t; x; u) = n (t; x; u) = 0.
Note that this feedforward condition covers some class of feedforward systems treated in [3] and [4].
Remark 1: It can be easily verified that if triangularity condition (3)
holds, then (2) holds with () = c(1 + + 1 1 1 + n01 ). Similarly, if
feedforward condition (4) holds then (2) holds with () = c(02 +
1 1 1 + 0(n01) ).
We note that () of Assumption 1 characterizes the nonlinearities
of the system (1). In the following, we provide three examples showing
three particular types of ().
Example A: (Increasing-type) Consider the system in [9, Ex. 2.4]
x_ 1
x_ 2
y
x2 + x1 sin x22
2=3 1=3
= u + x1 x2
= x1 :
=
(5)
In view of Assumption 1, we have (1 + )jx1 j + jx2 j ()fjx1 j +
jx2 jg. Thus, we have () = maxf1 + ; 1g = 1 + .
Example B: (Fast decreasing-type) Consider (1) with n =
3; 1 (t; x; u) = (x3 =8) sin u; 2 (t; x; u) = 3 (t; x; u) = 0. In view
of Assumption 1, we have () = 1=(82 ).
Example C: (Slow decreasing-type) Consider (1) with n =
2; 1 (t; x; u) = (x1 x2 )=(1 + 50jx1 j); 2 (t; x; u) = 0. This
system does not satisfy either the triangularity condition or the
feedforward condition. In view of Assumption 1, we can check that
() = 0:0201 .
III. GLOBAL EXPONENTIAL STABILIZATION BY OUTPUT FEEDBACK
The linear output feedback controller is given by
u = K ()^
x
_
x^ = Ax
^ + Bu 0 L()(y 0 C x
^)
(6)
(7)
n
=
[(k1 = ); . . . ; (kn =)] and L()
=
where K ()
n T
[(l1 =); . . . ; (ln = )] with > 0.
First, we select K = [k1 ; . . . ; kn ] and L = [l1 ; . . . ; ln ]T such that
AK A + BK and AL A + LC are Hurwitz.
^i ; 1
i
n, from (1)
Stability Analysis: Defining ei = xi x
and (7)
0
e_ = AL ()e + (t; x; u) where AL ()e = A + L()C:
Manuscript received June 3, 2003; revised February 8, 2004 and July 20,
2004. Recommended by Associate Editor W. Kang. This work was supported
by the Korea Research Foundation under Grant KRF-2002-041-D00363.
The authors are with the Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, Taejon 305-701, Korea (e-mail:
[email protected]).
Digital Object Identifier 10.1109/TAC.2004.841886
(3)
(8)
From (1) and (6), the closed-loop system becomes
x_ = AK ()x + (t; x; u) 0 BK ()e where AK () = A + BK ():
(9)
Denote E () = diag[1; ; . . . ; n01 ]. Since AK and AL are Hurwitz, we have the following equalities. For j = K; L; Aj () =
0018-9286/$20.00 © 2005 IEEE
256
Fig. 1.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 2, FEBRUARY 2005
(a) Plot of det T () versus . (b) State trajectories with x (0) =
05; x (0) = 2; x^ (0) = x^ (0) = 0.
E ()01 Aj E (); ATj ()Pj () + Pj ()Aj () = 001 E ()2 , and
Pj () = E ()Pj E () where ATj Pj + Pj Aj = 0I .
First, we set Vo (e) = eT PL ()e. Then, from (8), Assumption 1, and
the previous equalities, we have
p
V_ o (e) 001 kE ()ek2 + 2 n ()kPL k kE ()ekkE ()xk: (10)
Second, we set Vc (x) = xT PK ()x. Note that E ()BK ()e =
01 BKE ()e. Then, from (9), Assumption 1, and the previous
equalities, we obtain
p
V_ c (x) 0f01 0 2 nkPK k 1 ()gkE ()xk2
+ 201 kPK k kK kkE ()xkkE ()ek: (11)
Now, we set V (e; x) = dL Vo (e) + dK Vc (x); dL > 0; dK > 0 for
the augmented system (8) and (9). Then, using (10) and (11), we have
V_ (e; x) 06()T T ()6() where
6() = kkEE(())xekk ;
dL 01
3
T () =
3 dK f01 0 2pnkPK k 1 ()g
p
(12)
with 3 = 0dK 01 kPK kkK k 0 dL n ()kPL k.
The matrix T () > 0 when det T () = dK dL 01 f01 0 1 ()g0
fdK 01 2 + dL 3 (p)g2 > 0 where 1 = 2pnkPK k; 2 =
kPK kkK k, and 3 = nkPL k.
The result is summarized in the following theorem.
Theorem 1: Suppose that: a) Assumption 1 holds, b) there exist
K; L, and such that AK and AL are Hurwitz, and det T () > 0
where T () is defined in (12). Then, with a linear output feedback control scheme (6) and (7), the origin of (1) is globally exponentially stable.
Corollary 1: Suppose that either the triangularity condition (3) or
the feedforward condition (4) holds. Then, with the linear output feedback scheme (6) and (7), the origin of the closed-loop system (1) is
globally exponentially stable.
Proof: We only need to show that the condition b) of Theorem 1
is satisfied. Obviously, each AK and AL can be made by Hurwitz by K
and L, respectively. As follows from Remark 1, under the triangularity
condition, we choose dK = to have det T () > 0 for 0 < < 3 .
Similarly, under the feedforward condition, we choose dK = 01 to
have det T () > 0 for 3 < < 1. The weighting factor dL is used
to avoid the extreme conservative 3 for each case.
Thus, each system in Examples A and B can be globally exponentially stabilized by the proposed scheme. Now, consider the next example for other case.
Example 1: We reconsider Example C. In this case, the output
feedback schemes in [9]–[12] are not applicable. First, we let
K = [02:25; 03:00] and L = [04; 04]T . With () = 0:0201 , we
choose dK = 2 and dL = 1. Then, there exists a range of such as
31 < < 32 to satisfy det T () > 0. We choose = 0:15. The plot
of det T () with respect to and control results are shown in Fig. 1.
IV. CONCLUSION
We have presented a new result on global stabilization of a class
of nonlinear systems by a linear output feedback control scheme. We
have shown that a larger class of nonlinear systems can be globally
stabilized by a linear output feedback than claimed in existing results.
In particular, the proposed method covers the systems of [9] and [11],
some class of feedworward systems, and plus other systems (illustrated
in Example C).
REFERENCES
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[3] W. Lin and C. Qian, “New results on global stabilization of feedforward
systems via small feedback,” in Proc. 37th IEEE Conf. Decision Control,
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[4] W. Lin and C. Qian, “Synthesis of upper-triangular nonliear systems
with marginally unstable free dynamics using state-dependent saruration,” Int. J. Control, vol. 72, no. 12, pp. 1078–1086, 1999.
[5] F. Mazenc, L. Praly, and W. P. Dayawansa, “Global stabilization by
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[6] L. Praly, “Asymptotic stabilization via output feedback for lower triangular systems with output dependent incremental rate,” in Proc. 40th
Conf. Decision Control, Orlando, FL, Dec. 2001, pp. 3808–3813.
[7] C. Qian and W. Lin, “Nonsmooth output feedback stabilization and
tracking of a class of nonlinear systems,” in Proc. 42nd IEEE Conf.
Decision Control, Maui, HI, 2003, pp. 43–48.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 2, FEBRUARY 2005
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Constrained MPC Algorithm for Uncertain Time-Varying
Systems With State-Delay
Seung Cheol Jeong and PooGyeon Park
Abstract—In this note, we present a model predictive control (MPC) algorithm for uncertain time-varying systems with input constraints and statedelay. Uncertainty is assumed to be polytopic, and delay is assumed to be
unknown but with a known upper bound. For a memoryless state-feedback
MPC law, we define an optimization problem that minimizes a cost function and relaxes it to two other optimization problems by finding an upper
bound of the cost function. One is solvable and the other is not. We prove
equivalence and feasibilities of the two optimization problems under a certain assumption on the weighting matrix. Based on these properties and optimality, we show that feasible MPC from the optimization problems stabilizes the closed-loop system. Then, we present an improved MPC algorithm
that includes relaxation procedures of the assumption on the weighting matrix and stabilizes the closed-loop system. Finally, a numerical example illustrates the performance of the proposed algorithm.
Index Terms—Closed-loop stability, input constraints, model predictive
control (MPC), state delay, uncertainty.
I. INTRODUCTION
Model predictive control (MPC), also known as receding horizon
control, has received much attention in control societies for its ability to
handle both constraints and time-varying behaviors as well as its good
tracking performance [1]–[10]. Moreover, it has been recognized as a
successful control strategy in industrial fields, especially in chemical
process control such as petrochemical, pulp, and paper control. The
basic concept of MPC is to solve an open-loop constrained optimization
problem at each time instant and implement only the first control move
of the solution. This procedure is repeated at the next time instant.
It is well known that parameter uncertainties and time-delays cannot
be avoided in practice, especially, in many chemical processes where
the MPC has been mainly applied. Since the parameter uncertainties
Manuscript received November 21, 2003; revised April 6, 2004, October 4,
2004, and October 5, 2004. Recommended by Associate Editor L. Magni. This
work was supported in part by the Ministry of Education of Korea through its
BK21 Program.
The authors are with the Division of Electrical and Computer Engineering,
Pohang University of Science and Technology (POSTETH), Pohang 790-784,
Korea (e-mail: [email protected]; [email protected]).
Digital Object Identifier 10.1109/TAC.2004.841920
257
and time-delays are frequently the main cause of performance degradation and instability, there has been an increasing interest in the robust control of uncertain time-delay systems in the control literature
[11]–[16].
However, only a few MPC algorithms [6], [17] have been published
that handle time-delayed systems explicitly. In [6], after designing
the novel robust constrained MPC for uncertain systems, the authors
argue that the control scheme can be extended to a delayed system
in a straightforward manner. This is true when the delay indices
are known. However, when the delay indices are unknown, it is not
straightforward and not easy to show the feasibility of the on-line
optimization problem and guarantee the closed-loop stability, which is
the main topic of this note. In [17], a simple receding horizon control
is suggested for continuous-time systems with state-delays, where a
reduction technique is used so that an optimal problem for state-delay
systems can be transformed to an optimal problem for delay-free
ordinary systems. They propose a set of linear matrix inequality (LMI)
conditions so as to check the closed-loop stability and a numerical
algorithm to compute the eigenvalues of general distributed delay
systems. However, as the author admits, the guaranteed stability of the
suggested control is not known. Moreover, neither model uncertainty
nor constraints are considered in the note, hence the application of the
controller is limited.
In this note, we present an MPC algorithm for uncertain time-varying
systems with input constraints and state-delay. The uncertainty is assumed polytopic and the delay is assumed unknown but with a known
upper bound. After defining an optimization problem that minimizes a
cost function at each time instant, we relax the problem into two other
optimization problems that minimize upper bounds of the cost function.
One is solvable and the other is not. We prove the equivalence and feasibility of the two optimization problems under a certain assumption
on the weighting matrix. Based on these properties and optimality, we
show that the MPC obtained through the solvable optimization problem
stabilizes the closed-loop system. Then, we present an improved MPC
algorithm that includes relaxation procedures of the assumption on the
weighting matrix.
The note is organized as follows. Section II states target systems, assumptions and the associated problem. Section III supplies a stabilizing
MPC algorithm for uncertain delayed systems with constraints. Section IV illustrates the performance of the proposed controller through
an example. Finally, in Section V, we make some concluding remarks.
Notation: Notations in this note are fairly standard. n and n2m
denote the n-dimensional Euclidean space and the set of all n 2 m
matrices, respectively. The notation X Y and X > Y (where X and
Y are symmetric matrices) means that X 0 Y is positive semidefinite
and positive definite, respectively. Inequality between vectors means
componentwise inequality. Finally, kxkW denotes xT W x.
II. PROBLEM STATEMENTS
Consider the following discrete-time uncertain time-varying
systems:
x(k + 1) = A(k)x(k) + A(k)x(k 0 d) + B (k)u(k);
x(k) = (k); k 2 [0d3 ; 0]
(1)
subject to input constraints
0u u(k) u; u 0; for all k 2 [0; 1)
(2)
where x(k) 2 n is the state, u(k) 2 m is the control and (k) 2 n
is the initial condition. d is an unknown constant integer representing
the number of delay units in the state, but being assumed 0 d d3
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