Observers for systems with nonlinearities satisfying an incremental quadratic inequality

Abstract

We consider the problem of designing observers to asymptotically estimate the state of a system whose non-linear time-varying terms satisfy an incremental quadratic inequality that is parameterized by a set of multiplier matrices. Observer design is reduced to solving linear matrix inequalities for the observer gain matrices. The proposed observers guarantee exponential convergence of the state estimation error to zero. In addition to considering a larger class of nonlinearities than previously considered, this paper unifies some earlier results in the literature. The results are illustrated by application to a model of an underwater vehicle.

Full text

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Observers for A Class of Systems with Nonlinearities Satisfying an
Incremental Quadratic Inequality
Ahmet Behget Agikmeqe and Martin Corless
Abstruct- We consider the problem of state estimation for
nonlinear time-varying systems whose nonlinearities satisfy
an incremental quadratic inequality. Observers are presented
which guarantee that the state estimation error exponentially
converges to zero. Observer design is reduced to solving linear
matrix inequalities for the observer gain matrices. Results are
illustrated by application to a simple model of an underwater
vehicle.
I. INTRODUCTION
A fundamental problem in system analysis and control
design is that of determining the state of a system from
its measured output. Many solutions to this problem use
an asymptotic observer (or state estimator) that produces
an estimate of the system state which asymptotically ap-
proaches the system state. Typical observers for linear
systems consist of a copy of the system dynamics along with
a linear correction term based on the output error, that is,
the difference between the measured output and its estimate
based on the estimated state [12], [6].
References [3] and [4] consider a class of nonlinear
systems whose nonlinearities have bounded derivatives.
They present asymptotic observers which involve a copy of
the system dynamics and two correction terms based on the
output error; one term is the usual linear correction term,
while the other term (called the nonlinear injection term)
enters into the nonlinear element in the observer. Some
other results on observers for nonlinear systems can also
be found in [ 101 and [ 131.
In this paper, we consider nonlinear time-varying systems
whose nonlinearities satisfy an incremental quadratic in-
equality. This inequality is characterized by a set 9d of mul-
tiplier matrices. The nonlinearities considered include many
of the common nonlinearities including those considered in
[3] and [4]. Section V exhibits some of the nonlinearities
under consideration along with their multiplier matrices.
For these systems, we present observers whose structure is
inspired by [3] and [4]. These observers are characterized by
two gain matrices: the gain matrix L for the linear correction
term and the gain matrix L, for the nonlinear injection
term. Initially, we consider L, fixed (for example zero) and
convert the problem of determining L into that of solving
linear matrix inequalities. Such inequalities can be readily
treated using the LMI toolbox in MATLAB [8].
Research was conducted while first author was at Purdue University
A. B AqikmeSe is with Guidance and Control Analysis Group, Jet
Propulsion Laboratory, California Institute of Technology, Pasadena, CA
91109,USA. behcet@jpl . nasa. gov
M. Corless is with the School of Aeronautics and Astronautics, Purdue
University, W. Lafayette, IN 47907, USA. corless@purdue. edu
We also consider the problem of simultaneously comput-
ing L and L,. By imposing a specific condition on the set
M of multiplier matrices describing the nonlinearities, we
also convert the problem of simultaneously determining L
and L, into that of solving linear matrix inequalities.
All of our results are based on analysis on the state esti-
mation error dynamics using quadratic Lyapunov functions.
To illustrate our results, we apply the proposed observers
to estimate the state of a simple model of an underwater
vehicle from [l 11.
11. SYSTEM DESCRIPTION
We consider nonlinearhime-varying systems described by
X = Ax+Bu+B,p(t,x,u)
y = Cx+Du+Dpp(t,x,u)
(1)
where x(t) E IR" is the state, u(t) E IRm is the control
input, y(t) E I R ' is the measured output and t E IR is the
time variable. All the nonlinearhime-varying elements in
the system are lumped into the term p(t,x,u) E R'p. We
suppose that
p(t,x,u) = y~(t,z) where
z = Cqx+Dqu (2)
and y~ is a piecewise continuous function o f t and a con-
tinuous function of z E IR'q. The matrices A , B, B,, C, D, D,
and C,,D, are constant and of appropriate dimensions.
Our characterization of w is based on a set 9d of sym-
metric matrices which we refer to as multiplier matrices.
Specifically, for all M E 94, the following incremental
quadratic inequality holds for all t E IR and zl ,z2 E IR'q:
( q(t,z2)-q(tA) )iM( dt,z2)-q(t,zl) ) > 0
yJ(t72) - y J ( t , Z l )
y J ( t 7 2 ) -yJ(t,z1)
(3)
q(t,z) = z+DqpyJ(t,z) '
(4)
where
Basically, the constant matrix D , and 94 provide a char-
acterization of y~ in a incremental sense. Section V exhibits
some of the nonlinearities under consideration along with
their multiplier matrices.
111. OBSERVERS
We propose the following observers to provide an esti-
mate f of the state x of the system described by (1):
f = A f +Bu + BPj? + L@ - y )
p = Cf+DU+DPj?
(5)

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where
fi = v(t, .?+L,,@-y))
and i= Cq2+Dqu.
(6)
Here, L is the gain for the linear correction term and L, is
the gain for the nonlinear injection term, which was also
introduced in [4]. The nonlinear injection term results in
additional flexibility in the design. As an example of the
usefulness of this term, suppose z = y. Then, L, = -I yields
,?+L,@ -y) = z and fi = v(t,z); hence we have an exact
copy of the nonlinearity in the observer.
In the observer description above, we have
fi = v(t, t+L,(Ci+Du-y) + L,Dpfi).
When LnDp # 0, this is an implicit equation for fi. So, we
assume that there is a continuous function Q such that for
all t and q, the equation
fi = ~ ( t ,
q +LnDpfi)
(7)
is uniquely solved by fi = Q(t,q). Then, fi is uniquely given
by
fi = $(t,i+L,(Ci+D~-y)).
The next section provides some sufficient conditions which
guarantee the existence of Q. Introducing the the state
estimation error,
e : = x - x ,
(8)
(9)
we have
j
v(t,z+ (Cq +LnC)e+LnDp(fi-p))
= $(t,z+ (Cq +L,C)e-LnDpp)
(10)
e = (A + LC)e+ (B, + LD,)Sp(t,e)
(1 1)
&p(t,e) = $ (t,z(t) + (cq
It follows from (1 0) that Sp satisfies
+LnC)e-LnDp\V(t,Z(t))) - ~ ( t , z ( t ) ) .
and the error dynamics are described by
where
Sp = v(t,z+ (Cq +LnC)e+L,DpGp) - v(t,z).
Let
Wt,e) = (Cq+LnC)e+ (Dqp +LnDp)Sp(t,e)
Then, using (3) with z1 = z and 22 = z + (C + LnCq)e +
LnDpSp we obtain that for all t and e,
The following result yields conditions for observer gains
which result in exponentially decaying estimation errors.
Theorem 1: Consider a system described by (1)-(2) and
satisfying (3) with a set M of matrices. Suppose that there
exist matrices P = PT > 0, L, L, and M E M satisfying
matrix inequality (1 3) (on the next page) for some a > 0.
Also suppose that there is a continuous function $ such that
fi = $(t,q) uniquely solves equation (7). Consider any input
u(.) and initial condition x(t0) =no such that system (1)
has a solution for all t 2 to. Then, the state estimation error
e = P-x corresponding to observer (5) decays exponentially
to zero with rate a.
The error dynamics given in (11) can be
described by
Proof
e= A,e+B,Sp(t,e)
6q = C,e+D,Sp(t,e),
where
(14)
A,:=A+LC,
C, := Cq +L,C,
B, := Bp i-
D, := Dqp +L,Dp ,
LD, ,
and Sp satisfies inequality (12) for all t, e, and M E M.
Inequality (13) now simplifies to
Pre- and post-multiplying both sides of the above inequality
by [eT 8pT] and its transpose and using condition (12) we
obtain that
eTP(A,e+B,Gp(t,e)) 5 -aeTPe for all t, e.
This shows that the error dynamics are quadratically stable
about zero with rate a; (see [7], [2] and/or [SI for a
definition of quadratic stability). This implies that the error
decays exponentially to zero with rate a.
The following corollary yields an observer design proce-
dure for a given L,.
Corollary 1: Consider a system described by (1)-(2) and
satisfying (3) with a set 94 of matrices. For a given L,,
suppose that there exist matrices P = PT > 0, R and M E M
such that the following matrix inequality holds:
PA +ATP+ RC+CTRT + 2 d PB, + RD,
B,TP+D,TR~
where C, and D, are given in (14) and let
L = P - ~ R .
(16)
Also suppose that there is a continuous function $ such
that fi = $(t,q) solves equation (7). Consider any input u(.)
and initial condition x(t0) = xo such that system (1) has a
solution for all t 2 to. Then the state estimation error for
the observer (5) decays exponentially to zero with a rate a.
Remark I: Note that, for a fixed a and L,, inequality
(15) is an LMI (linear matrix inequality) in the variables P,
R, and M.
A. On the Existence of a Solution to Equation (7)
As mentioned in the previous section, if LnDp # 0, then
we need to be able to solve equation (7) for j to implement
the observer. This equation defines an implicit relation
for j ? in terms o f t and q. Here q = I+Ln(C2+Du-y).
The following lemma provides a sufficient condition which

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guarantees that, for each t and q, equation (7) has a solution
j? = Cp(t,q), where Cp is continuous. Since the proof of this
lemma is rather long, it is not presented here. See [l] for
the proof.
Lemma I: Suppose that w satisfies (3) for all t, z1,z2 and
ME M. Given L,, suppose there are matrices M E ! M and
Q and a scalar p > 0 such that
Then, there is a continuous function Cp such that j? = Cp(t,q)
solves equation (7).
Reinark 2: When LnDp # 0, Lemma 1 suggests that we
can design an observer for a given L, by simultaneously
solving the LMIs (15) and (17) for P, M, R, Q and p. Then
L = P-'R. Consequently we obtain a well defined observer
to estimate the states, because equation (7) has a continuous
solution.
Iv. A CONDITION FOR SIMULTANEOUS DESIGN OF L
AND L, VIA LMIS
The previous section contains an observer design pro-
cedure where the observer gain L is designed for a fixed
L,. However, the simultaneous design of L and L, is not
addressed. The following condition, which is satisfied by
many common nonlinearities (see Section V), allows for
the simultaneous design of L and L,.
Condition I : There exist a nonsingular matrix T and a
set of matrix pairs (X, Y ) with Y E IRmp Xmp such that
XT = X > O,YT = Y 2 0 and the matrix
is in M. In addition, f i 2 + &DqP is nonsingular where
and f i 2 E IRmp x m p .
A. A Transformation
Suppose Condition 1 holds and note that
T ( ) = ( 51z+r12P )
fi1z + r22p
where
r12=q2+T11Dqp,,
and
r22=T22+fi1Dqp.
and q = z+ D,p. Now introduce the transformed nonlinear
term p defined by
:= T21z+ r22p.
(20)
Since by assumption, r 2 2 is nonsingular, we have
p = -rG1fi1Z+r;.p;
(21)
hence TI 1 z + r12p = Z + bqpF where
z=cz, z = ~ , ~ - r ~ ~ r ; ~ f i ~ ,
bqP=rl2r;I.
(22)
We now show that that E is invertible. Note that
(F,: :::)=( F ; : TI2
Since the two matrices on the righthandside of the second
equality are invertible, the matrix on the lefthandside of
the first equality is invertible. Since r 2 2 is assumed to be
invertible, by using the matrix inversion lemma [9], [14], the
first matrix above is invertible if and only if the following
Schur complement of the matrix is invertible:
f i z + f i ~ D ~ , ) = ~ (
+ TIID,,
0 I
D,P I )
T~~ -rI2r;;fi1
=x.
This implies that C is invertible. Consequently, z = C - ' ?
and
@(t,.,.) = q(t,z) := fi1~-1~+r22~(t,~-1~).
(23)
Letting
4"(t,Z) =Z+bqpi$(t,2)
we obtain that
1,
T ( W 7 2 ) -w(t,z1) >=(
Hence satisfaction of inequality (3) by w implies that the
transformed nonlinear hction q satisfies
W,&)
- V(t,,fl)
B(t,52)-B(t,&)
q(t,z2) -q(t,z1)
Now, using the transformed term 9, system (1) is described
by
x = Ax+Bu+B,ip(t,Z)
y = Cx+Butbp.jr(t,z)
z = Cqx+bqu
(25)
where @ satisfies (24) and

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B. Observer for the Transformed System
Based upon the previous section, we propose the follow-
ing observers for the transformed system (25):
i = Ai+Bu+BpB'+L(j-y)
p = &+Bu+BpB'
fi = \Tr(t, h+Ln(j-y))
(27)
h = eqi+Bqu
In the observer description we have,
@ = P(t, f+L,(&+Du-y)
So, when Lnbp # 0, we again assume that there is a
continuous fimction @ such that for all t and q, the equation
+L,bpB').
rj = \Tr(t, 1 1 +Lnbprj)
(28)
is uniquely solved by j = $(t,q). Then,
@ = $(t,h+L,(&+bu-y)).
(29)
Now, we can present the main result of this section, which
is a corollary to Theorem 1.
Corollary 2: Consider a system described by (1)-(2) and
satisfying (3) with a set %f of matrices which satisfy
Condition 1. Suppose that there exist matrices P = PT > 0,
R1, R2 and (X, Y ) E N which satisfy (30) (on the next page)
and let
L=P-'R1,
Also suppose that there is a continuous function @ such that
@ = @(t,q)
solves equation (28). Consider any input u(.) and
initial condition x(t0) = xo such that system (1) has a well
defined solution for all t 2 to. Then, the state estimation
error, e = P - x, decays exponentially to zero with a rate of
a.
L,=X-'R2.
(31)
Proof: Substitute (31) into inequality (30) and apply
a Schur complement result [5] to obtain
where e, := Eq + L,C and 8, := 8, +LnDP. The result now
follows by applying Theorem 1 to the transformed system.
w
Remark 3: Note that, for a fixed a , inequality (30) is an
LMI (linear matrix inequality) in the variables F', R1, R2, X
and Y.
When Lnbp # 0, the following corollary of Lemma 1
presents an LMI which guarantees a continuous solution to
equation (28) for @ (see [l] for a proof).
Corollary 3: Suppose that \Tr satisfies (24) for all t,zI ,z2
and (X,
(X,
-Q-QT+pI
Q
CiRT
Y ) E N. Given L,, suppose that there are matrices
Y ) E N, R2, Q and and a scalar p > 0 such that,
1 0 and R2=XLn.
(32)
R2bp Xbqp -X
Then there is a continuous function @ such that @ = @(t,q)
satisfies (28).
Remark 4: When LnBp # 0, Corollary 3 tells that we
can design the observer gains L and L, by simultaneously
solving LMI's (30) and (32) for P, R1, R2,X, Y, Q and p.
Then L=P-'Rl andLn=X-'R2.
v. SOME CLASSES OF NONLINEARITIES
In this section, we discuss some typical nonlinearities
satisfying (3). We also present additional conditions under
which these nonlinearities satisfy Condition 1.
SATISFYING (3)
A. Incrementally Sector Bounded Nonlinearities
Here we consider nonlinearities which, for all tl , t2, and
z1, z2, satisfy
(6y~-Kl?iq)'X(K2?iq-6y~) 2 0 for all X E X , (33)
where
6yr:=Ul(t2,z2)--(tl,ZI),
6q:=6z+Dqp6yr, 6z:=z2-z1, (34)
X is a set of symmetric positive definite matrices and
K1 , K2 are fixed matrices. Here, without loss of generality,
we assume that the set X is invariant under multiplication
by a positive number. It readily follows from (33) that a
set M of multiplier matrices for the nonlinearities under
consideration is given by
To satisfy Condition 1, suppose that there exists a pos-
itive scalar o such that SI - os2 is nonsingular where
SI :=. K2,Dqp -I and S2 := I - KIDqp. One can verify by
substitution that the following equality holds
where
-&K2-fiK1
K2 + oK~
--&I+&I
-I-OI
T = (
Here r 2 2 = SI - os2 is nonsingular. Therefore, Condition 1
is satisfied with the matrix T defined above and
When q and p are scalars, one can always choose a
positive scalar 0 such that SI -os2 is nonzero. To prove
this claim, note that if S1 - os2 is zero for all o > 0 then,
SI = S2 = 0. In this case, KI = K2 = 1 /Dqp and 6y1= K6q
where K := K1 = K2. Using 6q = 6z+Dqp6~
we have 6z = (1 - DqpK)6q = 0. However, 6z should be
arbitrary; hence we cannot have S1 = S2 = 0. Consequently,
Condition 1 is always satisfied by M in the scalar case.
As a specific example of a nonlinearity under consider-
ation, consider a globally Lipschitz nonlinearity which sat-
isfies 1)6y~ll 5 yll6qll for some y> 0. In this case, inequality
(33) holds with K1 = -yI, K2 = yI and X = {W : h > 0).
and 6y~= K6q,

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B. Incrementally Positive Real Nonlinearities
This class of nonlinearities is described by a set X of
symmetric positive definite matrices X such that for all tl , t2
and a,z2,
6qTX6yr 2 o
for all x E X,
(35)
where Sg and 6y are as defined in (34). It is clear from (35)
that, without loss of generality, we can assume that the set X
of matrices is invariant under multiplication by a positive
scalar. Note that nondecreasing nonlineanties satisfy (35)
with X = {W : h > 0). It readily follows from (35) that a
set M of multiplier matrices for the nonlinearities under
consideration is given by
M = { ( ; ;)
: ,EX>
To satisfy Condition 1 choose any scalar o > 0 such that
D, - GI is nonsingular. Then, we can readily show that
Consequently, if we let
then r 2 2 = D , - ol is nonsingular and Condition 1 holds.
C. Nonlinearities with Polytopic Parameterizations
These nonlinearities are assumed to satisfy
6yr = Q(s)6q
(36)
where 6yr and 6q are defined in (34), s := (tl,t2,zI ,z2) and
n(s) E Co { Q 1 , . . . , QV)
for all s ,
that is, for any s, n(s) = xi=lhkRk, where hk 2 0 and
h k = 1.
Since 6yr = Q(s)6q, a symmetric matrix M satisfies (3)
if
Let
where partitioning is in accordance with (6q,6yr). Then the
above inequalities can be expressed as
Consider now those matrices M which satisfy M22 5 0.
When Dqp = 0, we need only consider this case. With M22 5
0, the above inequalities can be shown to be equivalent to:
MI1 +M12nk+n[& +n[M22& 2 0 for k = 1,. . . , V .
(37)
Thus, the set M of symmetric matrices M which satisfy
( A k ) ' M ( A k ) 2 0 fork=l, ..., v andM2250,
(38)
is a set of multipliers matrices.
The above set of multiplier matrices does not necessarily
satisfy Condition 1 with a single transformation T. To
obtain a set satisfying Condition 1, choose any nonsingular
matrix T and consider multiplier matrices of the form given
in (18) where X and Y are symmetric matrices with X > 0
and Y 2 0. A matrix Mof this structure satisfies inequalities
(37) if and only if X and Y satisfy
TAXTl2 -T&YT22
5 0.
(39)
Then, provided 5 2 + EIDqp is invertible, Condition 1 is
satisfied with
N = { ( X , Y ) : X T = X > O andYT=YLO satisfy (39)).
Once T is chosen, (39) is a set of linear matrix inequal-
ities in X and Y. However, the choice of T to yield a large
subset of multipliers in some sense is not clear. Therefore, T
is treated as a design parameter at this point. For example,
the simple choice of T = I satisfies Condition 1 with N
defined by
N = {(X,Y) : XT =X> 0 and YT = Y 2 0 satisfy (40))
where
x - f l i Y f i k 2 0
fork= 1, ..., V .
(40)
VI. AN EXAMPLE: UNDERWATER
In this section we consider a simple model of an underwa-
ter vehicle dynamics. This example is taken from [ 1 13 where
a similar objective of designing observers is considered in
a different framework. A simplified model of the vehicle is
given by
VEHICLE
$1
= -36i)&i1+u
= illill- W 2 1 + 2 I ,
$2
where $1 is propeller angle, $2 is vehicle position and u is
the torque input to the propeller. It is assumed that we can
only measure $1 and $2. The angular velocity 41 of propeller
and the speed $2 of the vehicle will be estimated using an