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Observer Design for Lipschitz Nonlinear Systems
Observer design for Lipschitz systems was first considered by Thau in [22] where he obtained a sufficient condition to ensure the asymptotic stability of the observer. Thau’s condition is a very useful analysis tool but does not address the fundamental design problem. Encouraged by Thau’s result, several authors studied the observer design problem for Lipschitz systems. In [25], Zak considered the use of Lyapunov functions and the Bellman–Gronwall lemma for this design problem, with application in feedback stabilization. In [14] and [15], Raghavan formulated an iterative procedure to tackle this design problem based on solving an algebraic Riccati equation. Raghavan’s technique was later extended by Garg and Hedrick, [5], to study fault detection in Lipschitz systems. Unfortunately, Raghavan’s algorithm often fails to succeed even when the matrices (A; C) satisfy the usual observability assumptions. Moreover, it does not provide insight into what conditions must be satisfied by the observer gain to ensure stability. A rather complete solution of these problems was presented by Rajamani in [16] and [17]. Rajamani obtained a sufficient condition which is directly related to the observer matrix and that ensures asymptotic stability of the observer. He formulated a design procedure based on the use of a gradient-based optimization method. Rajamani also discussed the equivalence between the stability condition and the minimization of the H1 norm of a system in the standard form. However, he pointed out that the design is not solvable as a standard H1 optimization problem since the regularity assumptions required in the H1 framework are not satisfied. In this note, we show that the condition in [17] is related to a modified H1 problem satisfying all of the regularity assumptions. Based on this result, we propose a new observer design for Lipschitz nonlinear systems. The observer synthesis is carried out using H1 optimization and can, therefore, be done using commercially available software packages. Our approach is also less restrictive than the existing design techniques since it is directly related to the stability condition. Our formulation employs the input–output observer framework introduced in [8], [9] in which the static gain used in the classical observers is replaced with a dynamical filter. This approach has been further considered by the authors in [12] to study robustness in the context of the so-called unknown input observer (UIO) framework. Although the structure of the observer used in this note and the unknown input observer of [12] are very different, the underlying ideas in both articles are very similar. Very recently, the authors applied the results of their paper to the sensor fault detection problem, [13]. It is shown in [13] that the additional dynamics provided by the observer filter can have a significant impact on the resulting ability to detect faults with a certain frequency pattern. The rest of the note is organized as follows: Section II introduces some background results and notations. In Section III, we introduce our dynamic generalization of previously used Lipschitz observers and provide the sufficient condition for observer stability which is a generalization of the condition in [17]. We then present our main result, where we formulate the observer design for Lipschitz nonlinear systems as a regular H1 problem proving that its solution is necessary and sufficient for observer stability, and finally we present a parameterization of all possible convergent observers. Simulation results are shown in Section IV and conclusions are drawn in Section V.
A. M. Pertew, H. J. Marquez, and Q. Zhao
Abstract—The problem of observer design for Lipschitz nonlinear systems is considered. A new dynamic framework which is a generalization of previously used Lipschitz observers is introduced and the generalized sufficient condition that ensures asymptotic convergence of the state estimates is optimal conpresented. The equivalence between this condition and an trol problem which satisfies the standard regularity assumptions in optimization theory is shown and a parameterization of all possible observers is also presented. A design procedure which is less restrictive than the existing design approaches is proposed, and a simulation example is given to illustrate the observer design.
H
Index Terms—Dynamic observers,
H
H
synthesis, Lipschitz systems.
I. INTRODUCTION Nonlinear state observer design has been an area of constant research for the last three decades and, despite important progress, many outstanding problems still remain unsolved. Nonlinear coordinate changes, normal forms, output injection [10], [2]; high gain techniques [3]; adaptive observers [23]; numerical techniques [11]; smooth [24] and nonsmooth techniques [7] are some of the most notable concepts. However, most of the available techniques usually deal with special forms of nonlinearities, or give restrictive conditions to guarantee transformation into simpler forms, or develop assumptions that only guarantee local convergence, and the need of better design techniques still attracts researchers worldwide to work in this field. A class of nonlinear systems that has seen much attention in the literature and for which many Lyapunov-like design techniques have been proposed is the class of Lipschitz nonlinear systems of the form
x(t) _ = Ax(t) + 0(y; u; t) + 8(x; u; t) y(t) = Cx(t); A 2 n2n ; C 2 p2n
(1) (2)
and where the function 8(x; u; t) satisfies a uniform Lipschitz condition globally in x, i.e.,
k 8(x1 ; u; t) 0 8(x2 ; u; t) k k x1 0 x2 k (3) 8u 2 m ; t 2 and 8x1 ; x2 2 n . Here, 2 + is referred to as the
Lipschitz constant and is independent of x; u and t. Lipschitz systems constitute a very important class. Any nonlinear system x_ = f(x; u) can be expressed in the form of (1), at least locally, if f(x; u) is continuously differentiable with respect to x. Many nonlinearities are locally Lipschitz. Examples include trigonometric nonlinearities occurring in robotics, nonlinearities which are square or cubic in nature, etc. The function 8 can also be considered as a perturbation affecting the system [20]. Manuscript received December 14, 2003; revised June 3, 2005, October 7, 2005, February 1, 2006, and March 15, 2006. Recommended by Associate Editor A. Astolfi. This work was supported by the Natural Sciences and Engineering Council of Canada (NSERC). A. M. Pertew is with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada, and also with the Computer and Systems Engineering Department, University of Alexandria, Alexandria, Egypt (e-mail:
[email protected]). H. J. Marquez and Q. Zhao are with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada (e-mail:
[email protected];
[email protected]) Digital Object Identifier 10.1109/TAC.2006.878784
II. BACKGROUND RESULTS AND NOTATION In this section we summarize some preliminary results on observer design for systems of the form (1)–(3) and where the pair (A; C) is detectable. In all the literature available for this class of nonlinear systems, the observer proposed falls in the class of Luenberger-like observers, namely
0018-9286/$20.00 © 2006 IEEE
x^_ = A^x + 0(y; u; t) + 8(^x; u; t) + L(y 0 y^); L 2 n2p y^ = C^x:
(4) (5)
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The observer error dynamics is then given by
e_ = (A 0 LC)e + 8(x; u; t) 0 8(^x; u; t)
(6)
^. Thau was the first to introduce a sufficient condition where e = x 0 x for the asymptotic stability of the error in (6). His result was as follows. Theorem 1: [22] If the gain L is chosen such that < (min (Q)=2max (P )) with the Lyapunov equation (A 0 LC)T P + P (A 0 LC) = 0Q, then the estimation error in (6) is asymptotically stable. Theorem 1 provides a very important sufficient condition for the existence of an observer, but does not consider the design. Raghavan proposed an algorithm based on the following theorem. Theorem 2: [15] If there exists an " > 0 such that the algebraic Riccati equation (ARE) in (7) has a symmetric positive–definite solution P , then the observer gain L = (1=2")PC T stabilizes the error dynamics in (6) for all 8 with a Lipschitz constant AP + P AT + P 2 I 0 1 C T C P + I + "I = 0: "
(7)
Based on this result, Raghavan proposed an iterative binary search procedure over ", to obtain the observer gain. However, given a particular system of the form (1)–(3) with a specific Lipschitz constant 3 , this procedure may fail even if the pair of matrices (A; C) is observable. Moreover, Theorem 2 provides no insight into what conditions the matrix (A 0 LC) must satisfy to ensure observer stability. The answer to this puzzle was provided by Rajamani in the following theorem. Theorem 3: [17] The observer gain L stabilizes the error dynamics in (6) for all 8 with a Lipschitz constant if L is chosen so as to ensure that (A 0 LC) is stable and such that
min min (A 0 LC 0 j!I) > :
!2
(8)
Compared with Theorem 1, the beauty of this result is that it presents the condition for observer stability as a condition on the observer matrix itself. Besides, from the design prespective, it can be related to the H1 theory by rewriting (8) as [17]
k [sI 0 (A 0 LC)]01 k1< 1
(9)
where the left-hand side of (9) is equivalent to the H1 norm of the transfer function between ! and in the following so-called standard form (10)–(11), with the variables in (12):
z_ = [ A ]z + [ In
0In ] !
I 0 0n ! = n z+ n C 0pn 0pn ' ! = ~ = 8(x; u; t) 0 8(^x; u; t); = e = x 0 x^ = L(y 0 y^) and ' = y 0 y^:
(10) (11)
(12)
This can also be represented by Fig. 1 where the plant G has the state–space representation in (13) with the matrices in (10)–(12) and where the controller K is the static observer gain L
A ^ G(s) = C1 C2
B1 B2 D11 D12 : D21 D22
(13)
Therefore, focusing on the design problem, several design approaches have been proposed to satisfy the stability condition in (8). In [17], Rajamani considered the use of a gradient based optimization method to continuously change the locations of the closed-loop eigenvalues to minimize a performance index related to (8). The special case of A being Hurwitz has also been considered in [18], [1] introducing an analytical solution when a certain sufficient condition on the distance to unobservability of the pair (A; C) is satisfied. It can also be seen that the condition in Theorem 2 is a special form of (8). More recently, the result in [17] has been used to design reduced-order observers in [27]. However, all these design approaches are restrictive in the sense that they only provide sufficient conditions to satisfy (8). Moreover, they only consider the use of the Luenberger structure in (4)–(5) and, hence, put another restriction on the observer gain order which may prevent the existence of a solution to the H1 problem in (9). In this note, we focus on the design problem associated with the stability condition in (8). We first generalize this condition to a more general dynamic framework making use of the input–output observer framework introduced in [8], where an observer gain is seen as a filter designed so that the error dynamics has some desirable frequency domain characteristics. We then prove that the new condition is equivalent to a standard H1 problem satisfying all the regularity assumptions [unlike (10)–(11)]. Based on these results, we present a systematic design procedure (which is less restrictive than the existing design approaches) to compute the observer gain within the H1 framework. We also present a parameterization of all possible observer gains in this case. The following definitions and notation will be used throughout this note. Definition 1: (L2 space) The space L2 consists of all Lebesque measurable functions u : + ! q , having a finite L2 norm kukL , where 1 2 ku k L 0 k u(t) k dt, with ku(t)k as the Euclidean norm of the vector u(t). Consider now a system G : L2 ! L2 . We will represent by (G) the L2 gain of G defined by (G) = supu (kGukL =kukL ). It is well known that, for a linear system G : L2 ! L2 with a transfer matrix ^ (G) is equivalent to the H-infinity norm of G(s) ^ defined as G(s); ^ ^ k1 sup!2 max (G(j!)) , where max follows: (G) k G(s) ^ represents the maximum singular value of G(j!) . The matrices In ; 0n and 0nm represent the identity matrix of order n, the zero square matrix of order n and the zero n by m matrix, respectively. The symbol T^yu represents the transfer matrix from input u to output y . The partitioned matrix G
=
A C
B D
(when used as an operator from u to y , i.e.,
y = Gu) represents the state–space representation (_ = A+Bu; y = ^ = C(sI 0 C + Du) and in that case the transfer matrix is G(s) 0 1 A) B + D . We will also use the standard notation in the literature on H1 control [26] by using dom(Ric) to denote the domain that consists of all Hamiltonian matrices H with two properties, namely, H has no eigenvalues on the imaginary axis and the two related spectral subspaces are complementary. We will also use Ric(H) to denote the unique solution to the ARE associated with H in this case. III. NEW H1 DESIGN PROCEDURE A. Generalization to Dynamic Framework In this note, following the approach in [8] and [9], we make use of dynamical observers of the form
x^_ (t) = A^x(t) + 0(y; u; t) + 8(^x; u; t) + (t) y^(t) = C^x(t)
(14) (15)
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 7, JULY 2006
() () _ = AL + BL (y 0 y^); = CL + DL (y 0 y^);
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where t is obtained by applying a dynamical compensator K of order k (“k ” being arbitrary) on the output estimation error. In other words t is given from
AL 2 CL 2
2 ; BL 2 n2k ; DL 2 k k
2 n2p :
k p
(16) (17)
Fig. 1. Standard setup.
We will also write
K=
AL CL
BL DL
(18)
to represent the compensator in (16)–(17). It is straightforward to see that this observer structure reduces to the usual observer in (4)–(5) when
K
0k = 0nk
0kp L
. The additional dynamics brings addi-
tional degrees of freedom in the design, something that will be exploited in the proposed H1 procedure. In this section, we generalize Theorem 3 to the dynamic framework as follows. First, note that the observer error dynamics in (6) is now given by
e_ = Ae + 8(x; u; t) 0 8(^x; u; t) 0
(19)
which can also be represented by the setup in Fig. 1 where G has the state–space representation in (13) with the same matrices defined in (10)–(11) and with the same variables in (12) except for which is now given by
= = K (y 0 y^)
^
(20)
We denote by T! the transfer function between ! and for this setup. The following theorem is then the generalization of Theorem 3. Theorem 4: Given the Lipschitz system of (1)–(2), the state x of the observer (14)–(18) globally asymptotically converges to the system state x for all 1; 1; 1 satisfying (3) with a Lipschitz constant if the dynamic observer gain K is chosen such that
^
8(
)
sup max [T^! (j!)] < 1 : 2
(21)
!
Proof: The proof is similar to the sufficiency part of [12, Th. 3]. It follows by noting that T! can be represented as:
^
T^! = T^e ~ =
Fig. 2. Feedback interconnection.
A 0 DL C BL C In
0CL AL
In
0kn 0n
(22)
0nk and it satisfies (T^e ~ ) =k T^e ~ k1 < (1= ) according to (21). It also follows that the estimation error e can be represented by the feedback interconnection in Fig. 2 where 1 is the static nonlinear time-varying ^(t); u(t); t) 0 operator defined as follows: 1(t) : e ! ~ = 8(e + x 8(^x(t); u(t); t). It should be noted that, regardless of the presence of an exact expression for 1(t); (1) is true due to the Lipschitz condition in (3). A staightforward dissipativity argument can then be used (see [12] for more details) to show that T^e ~ and 1 are dissipative with the supply rates !1 = 0eT e+ (T^e ~ )2 ~T ~ and !2 = 0 ~T ~+ 2 eT e, respectively. Denoting S1 and S2 as their corresponding storage functions, it follows that S1 + aS2 ; a > 0 is a Lyapunov function for
( ^ ) (1) 1
the system and, since Te ~ < , the system is asymptotically stable and e ! as t ! 1: Corollary 1: Under the conditions of Theorem 4, if condition (3) holds locally, then local asymptotic convergence of the observer is guaranteed (and in this case the observer (14)–(18) is a local one, i.e., it is local in “x” and in the estimation error “e”). In the following sections, we present our main results by focusing on the design problem associated with the stability condition in (21). We first prove that this condition (and the one in (8) as a special case) is actually equivalent to a standard H1 problem satisfying all the regularity assumptions. We then propose a systematic design procedure to compute the observer gain within the H1 framework. This design has the advantage of being directly related to the stability condition, and hence avoids much of the restrictions associated with the designs in [15], [17], [18], and [1]. We finally present the parameterization of all possible observers in this case.
0
1
B. Regularization of the H1 Problem As mentioned earlier, the stability condition in (21) can be represented by the H1 norm of the setup in Fig. 1 where G has the state–space representation in (13) with the matrices defined in (10), (11). However, this H1 problem does not satisfy all the regularity T T assumptions in the H1 framework (notice that D12 D12 and D21 D21 are both singular). Although the LMI approach in [6], or the techniques in [19] and [21] can be used to solve this singular problem, here we focus on the Riccati approach in [4] by showing that the problem is actually equivalent to the so-called “Simplified H1 problem” defined in [4], [26]. This helps to directly relate the stability condition to two Riccati equations [instead of the one in (7)] and lays the ground to a systematic design procedure which is less restrictive than the existing design appoaches. This also has the advantage of classifying the set of all possible observer gains by using the standard parameterization of H1 controllers in [4], [26]. To this end, we adopt the following standard regularization procedure, which was also considered in [12]: to the output By adding a “weighted” disturbance term d t > (2), and using the same observer defined by (14)–(18), it can be seen that the standard form in (10) and (11) has now the form
( )(
z_ = [ A ] z + [ [ In '
=
0np ] 0In ]
! d
0n 0pn
! d
In
0n z C 0n 0np + 0n 0np [ 0pn Ip ]
0)
In
(23)
(24)
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Fig. 3. Parametrization of all observer gains.
which can still be represented by the setup in Fig. 1, by redefining the defined as: ! [ ! d(t) ]T matrices in (13) and by replacing ! by ! T and by defined as: = [ ] ; ( > 0). It follows that the standard form in (23) and (24) satisfy the conditions of the so-called “Simplified H1 problem” [4], [26] if and only if (A; C) is detectable, which does not impose any new design restrictions on the observer design. We now show the equivalence between the original problem and this “Simplified H1 problem” as follows: Assume T1 as the setup in Fig. 1 associated with (10)–(11) and T2 as the one associated with (23)–(24) where both use the controller K in (18). And let T^1 (s) and T^2 (s) be their corresponding transfer matrices. The following lemma demonstrates certain equivalence relationships among these two setups (the proof is similar to that of [12, Lemmas 2 and 3], hence omitted). Lemma 1: Given the same stabilizing controller K for the setups T1 and T2 , then k T^1 (s) k1 < if and only if 9 > 0; > 0 such that k T^2(s) k1 < . C. Parameterization of All Observer Gains We now present our main result in the form of a theorem showing that the observer gain K needed to stabilize the observer error dynamics according to Theorem 4 must solve a “Simplified H1 control problem” according to the definition used in [26]. To this end, we define the Lipschitz observer design problem as follows. Definition 2: (Lipschitz observer design problem) Given > 0 and > 0, find S , the set of admissible controllers K satisfying k T^! k1 < (1= ) for the setup in Fig. 1 with G having the state space representation in (13) along with the matrices in (23)–(24). Defining the following two Hamiltonian matrices associated with this problem:
A 2 In 0 1 In N1 = 0 In 0 AT T 2 A In 0 1 C T C J1 = 0A 0 In
available software packages or the analytical results in [26]) and any K 2 S is a candidate observer gain that stabilizes the error dynamics. (=2); ( =2). If or < r , a threshold Step 3) Set value, then stop; otherwise go to Step 2). Remarks: • This design procedure is less restrictive than the designs introduced in [15], [17], [18], and [1], since it is directly related to the stability condition through the result in Theorem 5. This will be numerically demonstrated in Section IV. • If the H1 problem can not be solved due to its infeasibility or due to the software limitations, one can decrease the Lipschitz constant . The word stop in Step 3) can then be replaced by: decrease and go to Step 1). The algorithm is then guaranteed to work as ! 0. However, the region of convergence is decreased when reducing . The choice of the threshold in Step 3) is also important to avoid numerical instability of the used software. • Design of the H1 observer can also be done by including appropriate weightings to emphasize the performance requirements of the observer over specific frequency ranges. • If some states are not affected by nonlinearities (i.e., if some entries of the Lipschitz function 8 are zeros), the corresponding 1’s of the Identity matrix in the matrix B1 of the setup (23)–(24) can be replaced by zeros. As long as (A; B1 ) is controllable, the regularity assumptions can be satisfied, and the observer design is still equivalent to a “Simplified H1 problem.” IV. SIMULATION RESULTS We consider the following example from [15] to show the advantages of using the proposed H1 design procedure. This is an example of a second-order system of the form (1)–(2) with A=
(25)
the main result is then summarized as follows. Theorem 5: There exists a dynamic observer gain K for the observer (14)–(18) [or a static gain L for (4)–(5)] that stabilizes the error dynamics according to Theorem 4 (or Theorem 3) if and only if 9; > 0 such that 1) N1 2 dom(Ric) and X1 = Ric(N1 ) > 0; 2) J1 2 dom(Ric) and Y1 = Ric(J1 ) > 0; 3) (X1 Y1 ) < (1= 2 ) (where (:) is the spectral radius). 1 Proof: A direct result of Theorem 4 and Lemma 1. Moreover, by using the result in [4], the set of all observer gains K can be represented by the set of all transfer matrices from y to u in Fig. 3: where A^1 = A + ( 2 0 (1= 2 ))X1 0 (1=2 )(In 0 2 Y1 X1 )01 Y1 C T C , and Q is such that kQk1 < (1= ). D. Design Procedure Following the approach in [15] and [12], the following iterative “binary search” procedure is then proposed to evaluate the observer gain. Design steps: Step 1) Set ; > 0. Step 2) Test solvability of the problem in Definition 2. If test fails then go to Step 3); otherwise, solve the problem (using
02 3 ; 0 = 3 1
0 0 u; 8 = and C = [ 0 1 ]. The Lipschitz constant is 1 k sin(x1 ) = jkj. The system is open-loop unstable and therefore the design
techniques in [18] and [1] cannot be used to design an observer of the form (4)–(5). In [15], the case of = 1:5 was considered and, although the design algorithm in Theorem 3 fails if the original system matrices are used, the static gain L was obtained after using a state transformation as L = [ 2 4 ]T . Case study 1: We here show the improvement that can be achieved by using a dynamic observer, in terms of the maximum achievable in (21). Using a static gain L = [ a b ]T in (22), we have T^! =
A C
B D
02 3 0 a ; B = C = I2 , and D = 02 . 3 10b x1 x 3 It then follows, [26], that kT^! k1 < iff 9X = > 0 such x3 x2 , with A
X A + AT X B T X C
=
X B 0 I2 D
C T that D T < 0. The term X A + AT X in 0 I2 04x1 + 6x3 y1 + 3x2 , where the previous matrix inequality is y1 + 3x2 y2 y1 = (3 0 a)x1 0 (b + 1)x3 ; y2 = (6 0 2a)x3 + (2 0 2b)x2 . We now use convex optimization as follows: By minimizing subject to the
previous linear matrix inequalities (LMIs), using the Matlab command
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In the simulations, the system initial condition x(0) = [ 1:6 2 ]T , and the objective is to stabilize the equilibrium point at the origin by using an observer-based state feedback control law. To this end, the control law u = [ 11 13]x, based on linearization, is proposed and Fig. 4(a), (b) compares the system performance for three different cases: i) pure state feedback, ii) observer-based state feedback with the static gain L = [ 2 4 ]T , and iii) observer-based state feedback with the dynamic ^(0) = [ 0 0 ]T . The gain in (27). The observers initial conditions is x figures show the improvement in the state convergence rate when the dynamic observer is used. It is also important to say that the case of unknown parameters in the A matrix was considered in [12] by using the unknown input observer scheme. V. CONCLUSION
1
A new H observer design for Lipschitz nonlinear systems is proposed. It is first shown that the classical “Luenberger-like” observers are special cases of a more general dynamic framework, one that shows promise given the additional degrees of freedom. The equivalence between the observer design problem and a standard H control problem that satisfies all of the regularity assumptions is shown. A systematic design procedure which is less restrictive than the existing design approaches and that can be carried out using commercially available software is presented.
1
ACKNOWLEDGMENT The authors would like to thank the Associate Editor A. Astolfi and the anonymous referees for helpful suggestions.
REFERENCES
Fig. 4. (a) Response of state
H
x . (b) Response of state x .
“mincx,” the optimal = 0:2774 which corresponds to = 3:6055. Therefore, the maximum possible over all static observers satisfying the stability condition in (21) is = 3:6055. Using the H design algorithm of Section III-D for = 25, a dynamic observer gain of order 2 can be obtained at = = 0:015625 as follows:
1
064:114 04:803 ; BL = 4:564 0:256 0146:361 87:373 62 :114 3:239 0 CL = 3:239 70:79 & DL = 0 :
AL =
(26)
This shows the advantage of using a dynamic observer gain in this case. Case study 2: In this case, we consider the case of = 1:5, comparing the dynamic observer with the static observer in [15], in a state feedback application. Using the H design algorithm of Section III.D (starting with = = 2), and without using any state transformations, a candidate dynamic gain for the observer (14)–(18) is obtained after four iterations (at = = 0:125) as
1
08:644 0:1753 6:8861 CL = 2:9276
AL =
08:5916 8:7669 021:5898 ; BL = 13:1211 2:9276 0 9:8137 & DL = 0
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H
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