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Automatica 40 (2004) 1931–1938 www.elsevier.com/locate/automatica Brief paper A framework for nonlinear sampled-data observer design via approximate discrete-timemodels and emulation� Murat Arcaka,∗, Dragan Nešic´b aDepartment of Electrical, Computer and Systems Engineering, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY 12180, USA bDepartment of Electrical and Electronic Engineering, The University of Melbourne, Parkville 3052, Victoria, Australia Received 12 January 2004; received in revised form 20 May 2004; accepted 2 June 2004 Available online 13 August 2004 Abstract We study observer design for sampled-data nonlinear systems using two approaches: (i) the observer is designed via an approximate discrete-time model of the plant; (ii) the observer is designed based on the continuous-time plant model and then discretized for sampled- data implementation (emulation). We investigate under what conditions, and in what sense, these designs achieve convergence for the unknown exact discrete-time model. We present examples which show that designs that violate our conditions may indeed lead to instability when implemented on the exact model. � 2004 Elsevier Ltd. All rights reserved. Keywords: Observers; Sampled-data systems; Nonlinear systems 1. Introduction The main drawback of the existing sampled-data observer theory is that the availability of exact discrete-time models is assumed, which is usually unrealistic. A more practical approach pursued in this paper is to employ approximate discrete-time models for design, and to study how robust such approximate designs would be when implemented on the exact model. This approach may be advantageous even when an exact model is available, because the very few con- structive tools for nonlinear observer design may be applica- ble only to an approximate model which preserves structural properties of the continuous-time model. � This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Jessy W. Grizzle under the direction of Editor Hassan Khalil. The work of both authors was supported by the Australian Research Council under the Discovery Grants Scheme while the first author was visiting The University of Melbourne. The work of the first author was also supported in part by NSF, under grant ECS-0238268. ∗ Corresponding author. Tel.: +1-518-276-6535; fax: +1-518-276-6261. E-mail addresses: arcakm@rpi.edu (M. Arcak), d.nesic@ee.mu.oz.au (D. Nešic´). 0005-1098/$ - see front matter � 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2004.06.004 Rather than develop a specific observer design method, our primary objective in this paper is to lay out a gen- eral framework which prescribes crucial properties that such methods must achieve. The study in this paper parallels re- cent results in Nešic´, Teel, and Kokotovic´ (1999), and Nešic´ and Teel (2004), for sampled-data control design based on approximate discrete-time models. However, observer de- sign problems pose additional difficulties which cannot be addressed with results from control design. We also em- phasize that our goal in this paper is to study observer convergence properties independently of a feedback design task. Results on output-feedback stabilization of nonlinear sampled-data systems include, among others, Dabroom and Khalil (2001) which studies discrete-time implementation of high-gain observers, and Laila, Nešic´, and Teel (2002), and Nešic´ and Teel (2004), which address dynamic sampled- data controllers, with observer-based control as a special case. Most results on discrete-time observer design in the literature rely on an exact model. Several of them, such as Boutayeb and Darouach (2000), present examples where the models are obtained from Euler approximations, but do not discuss the effect of the exact-approximate mismatch. 1932 M. Arcak, D. Nešic´ / Automatica 40 (2004) 1931–1938 For a class of state-affine systems, Nadri and Hammouri (2003) pursue a mixed continuous- and discrete-time design which circumvents the need for an explicit exact model with the help of continuous-time updates. The paper is organized as follows: Section 2 gives the pre- liminary definitions and the problem formulation. Section 3 presents examples of non-robust designs, which give clues of why observers based on approximate models may give rise to instability for the exact model. Following these clues, Section 4 derives a set of sufficient conditions for the ap- proximate design, which ensure its robustness when applied to the exact model. Section 5 presents analogous conditions for the emulation design. 2. Preliminaries and problem statement We consider the system x˙ = f (x, u), y = h(x), (1) where x ∈ Rn, u ∈ Rm, y ∈ Rp, and f (x, u) is locally Lipschitz. Given a sampling period T > 0, we assume that the control u is constant during sampling intervals [kT , (k+ 1)T ) and that the output y is measured at sampling instants kT; that is, y(k) := y(kT ). The family of exact discrete-time models of (1) is x(k + 1)= FeT (x(k), u(k)), y(k)= h(x(k)), (2) where FeT (x, u) is the solution of (1) at time T starting at x, with the constant input u. This model is well-defined when the continuous model (1) does not exhibit finite escape time. When there is finite escape time, (2) is valid on compact sets which can be rendered arbitrarily large by reducing T. To compute (2) we need a closed-form solution to the initial value problem x˙ = f (x, u(k)), x0 = x(k) (3) over one sampling interval [kT , (k+ 1)T ), which is impos- sible to obtain in general. It is realistic, however, to assume that a family of approximate discrete-time models is avail- able x(k + 1)= Fa T, �(x(k), u(k)), y(k)= h(x(k)). (4) This family is parameterized by the sampling period T, and a “modelling parameter” � which will be used to refine the approximate model when T is fixed. It can be interpreted as the integration period in numerical schemes for solving differential equations. The case where �= T is of separate interest because several approximations of this type (such as Euler approximation) preserve the structure and types of nonlinearities of the continuous-time system and, hence, may be preferable to the designer. When �= T we use the short-hand notation FaT (x, u) :=FaT,T (x, u). (5) We use this notation for other functions in the sequel; that is, whenever we drop the � subscript, we refer to the situation where �= T . For the linear system x˙=Ax, the forward Euler numerical scheme x(t + �)= (I + �A)x(t) can be used to generate an approximate model by dividing the sampling period T into N integration periods �=T/N , and by applying (I+�A) for each integration period; that is, Fa T, �=(I+�A)T/�x.As � → 0, this Fa T, � converges to the exact model F e T = exp(AT )x. If �= T , then we obtain FaT = (I + TA)x. Throughout the paper we assume that the approximate model (4) is consistent with the exact model, as defined in Nešic´ et al. (1999), and Nešic´ and Teel (2004), with motiva- tion from numerical analysis literature; see e.g. Stuart and Humphries (1996). Definition 1. (a) When � = T the family FaT (x, u) is said to be consistent with FeT (x, u) if for each compact set � ⊂ Rn×Rm, there exists a class-K function �(·) and a constant T0> 0 such that, for all (x, u) ∈ � and all T ∈ (0, T0], |FeT (x, u)− FaT (x, u)|�T �(T ). (6) (b) When � is independent of T, Fa T,�(x, u) is said to be consistent with FeT (x, u) if, for each compact set � ⊂ Rn×Rm, there exists a class-K function �(·) and a constant T0> 0, and for each fixed T ∈ (0, T0] there exists �0 ∈ (0, T ] such that, for all (x, u) ∈ � and � ∈ (0, �0], |FeT (x, u)− FaT,�(x, u)|�T �(�). (7) It is not necessary to know the exact model FeT (x, u) to verify the consistency property. Verifiable conditions to check (6) and (7) are given in Nešic´ et al. (1999) and Nešic´ and Teel (2004). For the approximate model (4), we design a family of observers (depending on T and �) of the form xˆ(k + 1)= Fa T, �(xˆ(k), u(k))+ �T, �(xˆ(k), y(k), u(k)), (8) and analyze under what conditions, and in what sense, this design guarantees convergence when applied to the exact model (2). Due to the mismatch of the exact and approximate models, the observer error system is now driven by the plant trajectories x(t) and controls u(t), which act as disturbance inputs.When these inputs are bounded, we want the observer to guarantee semiglobal practical convergence, as defined next. Definition 2. (a) When � = T we say that observer (8) is semiglobal practical in T, if there exists a class-KL func- tion �(·, ·) such that for any D>d > 0 and compact sets X ⊂ Rn,U ⊂ Rm, we can find a T ∗> 0 with the property that for all T ∈ (0, T ∗], |xˆ(0)− x(0)|�D, and x(k) ∈ X, u(k) ∈ U, ∀k�0 (9) M. Arcak, D. Nešic´ / Automatica 40 (2004) 1931–1938 1933 imply |xˆ(k)− x(k)|��(|xˆ(0)− x(0)|, kT )+ d. (10) (b) When � is independent of T we say that observer (8) is semiglobal in T and practical in �, if there exists a class- KL function �(·, ·) such that for any given real number D> 0, and compact sets X ⊂ Rn,U ⊂ Rm, we can find a T ∗> 0, and for any T ∈ (0, T ∗] and d ∈ (0,D), we can find �∗> 0 such that for all � ∈ (0, �∗], (9) implies (10). (c) We say that observer (8) is semiglobal in T and practical in T and �, if there exists a class-KL func- tion �(·, ·) such that for any D>d1> 0, and compact sets X ⊂ Rn,U ⊂ Rm, we can find a T ∗> 0, and for any T ∈ (0, T ∗] and d2 ∈ (0,D − d1), we can find �∗> 0 such that for all � ∈ (0, �∗], (9) implies |xˆ(k)− x(k)|��(|xˆ(0)− x(0)|, kT )+ d1 + d2. (11) Unlike Definition 2(b) where we can arbitrarily reduce the residual observer error d in (10) by decreasing �, in Definition 2(c) we can only reduce d2 with �, while d1 is dictated by the sampling period T. As we shall see in Section 5, this situation arises in emulation design where, decreasing � can reduce the residual observer error, but cannot eliminate it completely if T is held constant. 3. Examples of non-robust designs Example 1. Consider the quadruple chain of integrators x˙1 = x2, x˙2 = x3, x˙3 = x4, x˙4 = u, (12) where the output y = x1 is sampled at times t = kT , k ∈ Z. For the Euler approximate model x(k + 1)= AaT x(k)+ BaT u(k), y(k)= Cx(k), (13) where AaT =   1 T 0 0 0 1 T 0 0 0 1 T 0 0 0 1   , BaT =   0 0 0 T   , C = (1 0 0 0 ) we design a family of dead-beat observers xˆ(k + 1)= AaT xˆ(k)+ BaT u(k)− LT [y(k)− Cxˆ(k)], (14) in which the injection matrix LT = ( −4 − 6 T − 4 T 2 − 1 T 3 )′ places the eigenvalues of AaT + LT C at the origin for all T > 0. However, for the exact model, AeT =   1 T T 2 2! T 3 3! 0 1 T T 22! 0 0 1 T 0 0 0 1   , BeT =   T 4 4! T 3 3! T 2 2! T   the eigenvalues of AeT + LT C are, for all T, −0.5897± 1.6865i; 0.5897± 0.1334i, in which the first two are outside the unit circle. This means that, in the observer error e := xˆ − x dynamics e(k+1)=(AaT+LT C)e(k)+(AaT−AeT )x(k)+ (BaT−BeT )u, (15) the �∞-gain from x to e is �xe�1 for all values of T. (Oth- erwise, substituting x =−e (�ex = 1) and u= 0 in (15), we would infer from the small-gain property �ex�xe < 1 that the resulting system e(k + 1)= (AeT + LT C)e(k) is asymptot- ically stable, which contradicts our computation of eigen- values above.) Because the �∞-gain cannot be reduced ar- bitrarily by reducing T, we cannot assign d arbitrarily small in (10) even when x(k) is bounded. Thus, the approximate design (14) does not guarantee practical convergence. The reason why this dead-beat design is non-robust for the exact model is because, when T is reduced, it attempts to achieve faster convergence at the cost of larger overshoots. The combination of this “peaking” in the transients, and the mismatch between the exact and approximate discrete- time models, leads to instability of AeT + LT C. In the next example, the approximate design is non-peaking, but the convergence rate is slower for smaller T, which again leads to instability of AeT + LT C. Example 2. We consider the system x˙1 = x1 + x2, x˙2 = x2 + u, y = x1, (16) and, again, design a Luenberger observer based on the Euler approximation AaT = ( 1+ T T 0 1+ T ) . The injection matrix LT = (−2T − 2T 3 − T (1+ T 2)2)′ places both eigenvalues of AaT + LT C at 1 − T 3, which is inside the unit circle. However, for the exact model AeT = ( eT T eT 0 eT ) the eigenvalues of AeT + LT C are complex and located at {eT − T (1+ T 2)} ± {T (1+ T 2) √ eT − 1}i, which are outside the unit circle for all T > 0. 1934 M. Arcak, D. Nešic´ / Automatica 40 (2004) 1931–1938 4. Observer design via approximate discrete-time models In the examples of Section 3, either the overshoot or the convergence rate of the approximate design is not uniform in the sampling period T. We now derive conditions which exclude such designs, and guarantee semiglobal practical convergence for the exact model. For our analysis we first note from (2) and (8) that the observer error e := xˆ − x satisfies e(k + 1)=Fa T, �(xˆ(k), u(k))+ �T, �(xˆ(k), y(k), u(k)) −FeT (x(k), u(k)). (17) Adding and subtracting the approximate model Fa T, �(x(k), u(k)), we rewrite (17) as e(k + 1)=ET, �(e(k), x(k), u(k))+ FaT, �(x(k), u(k)) −FeT (x(k), u(k)), (18) where ET, �(e, x, u) :=FaT, �(xˆ, u)+ �T, �(xˆ, y, u) −Fa T, �(x, u) (19) represents the nominal observer error dynamics for the ap- proximate design, and Fa T, �(x(k), u(k)) − FeT (x(k), u(k)) is the mismatch between the approximate and exact plant models. We first study the case �=T , and prove semiglobal prac- tical convergence in T under conditions (i)–(iii) in Theorem 1 below. In particular, condition (iii) guarantees that we can find T-independent estimates for the overshoot and conver- gence rate in the approximate design and, thus, rules out the non-robust designs of the previous section. Theorem 1. (� = T ). Observer (8) is semiglobal practical in T as in Definition 2(a) if the following conditions hold: (i) �= T . (ii) FaT is consistent with FeT as in Definition 1(a). (iii) There exists a family of Lyapunov functions VT (x, xˆ), class-K∞ functions �1(·), �2(·), �3(·), �0(·), and nonde- creasing functions �0(·), �1(·), �2(·),with the following prop- erty: For any compact sets X ⊂ Rn, Xˆ ⊂ Rn, U ⊂ Rm, there exist constants T ∗> 0 and M> 0, such that, for all x, x1, x2 ∈ X, xˆ ∈ Xˆ, u ∈ U, and T ∈ (0, T ∗], |VT (x1, xˆ)− VT (x2, xˆ)|�M|x1 − x2|, (20) �1(|e|)�VT (x, xˆ)��2(|e|), (21) VT (F a T (x, u), F a T (xˆ, u)+ �T (xˆ, y, u))− VT (x, xˆ) T �− �3(|e|)+ �0(T )[�0(|e|)+ �1(|x|)+ �2(|u|)]. (22) The proof of Theorem 1 relies on the following proposi- tion, which is proved in Arcak and Nešic´ (2004). Proposition 1. Suppose all conditions of Theorem 1 hold. Then, for any quadruple of strictly positive numbers (�x,�xˆ ,�u, �), there exists T ∗> 0 such that for all x, xˆ, u and T satisfying |x|��x , |xˆ|��xˆ , |u|��u, T ∈ (0, T ∗], VT (F e T (x, u), F a T (xˆ, u)+ �T (xˆ, y, u))− VT (x, xˆ) T �− �3(|e|)+ �. (23) Proof of Theorem 1. We let X and U be as in Def- inition 2(a) and claim that, given any pair of numbers 0 0 such that, for all T ∈ (0, T ∗], r�VT (x(k), xˆ(k))�R implies VT (x(k + 1), xˆ(k + 1))− VT (x(k), xˆ(k)) T �− 1 2 �3(|e(k)|). (24) To see this, let (�x,�xˆ ,�u, �) be numbers such that �x� sup x∈X |x| �u� sup u∈U |u| �xˆ� sup x∈X |x| + �−11 (R) ��1 2 �3(� −1 2 (r)). (25) It then follows that |x(k)|��x , |u(k)|��u, and, from (49) and VT (x(k), xˆ(k))�R, |xˆ(k)|=|x(k)+ e(k)|�|x(k)| + |e(k)| � |x(k)| + �−11 (VT )� sup x∈X |x| + �−11 (R)��xˆ . (26) Thus, if we choose T ∗ as in Proposition 1, then we guarantee, for all T ∈ (0, T ∗], VT (x(k + 1), xˆ(k + 1))− VT (x(k), xˆ(k)) T �− �3(|e(k)|)+ �. (27) Moreover, VT (x(k), xˆ(k))�r and (21) together imply |e(k)|��−12 (r) and, hence, (24) follows from the choice of � in (25). Having proven (24) we next note from (27) that VT (x(k), xˆ(k))�r ⇒ VT (x(k + 1), xˆ(k + 1))�r + �T . (28) This means that, if we choose T such that r + �T d > 0 as in Definition 2(a), we can select R = �2(D), r = 12�1(d), and T ∗ small enough that �T ∗� 12�1(d), and verify from (30) that observer(8) is semiglobal practical in T. � Theorem 1 established semiglobal practical convergence by reducing the sampling period T. When T is fixed and can- not be reduced, it is still possible to achieve practical conver- gence by, instead, refining the accuracy of the approximate models with the parameter �. Theorem 2 (� independent of T). Observer (8) is semiglobal in T and practical in � as in Definition 2(b) if the following conditions hold: (i) � can be adjusted independently of T. (ii) Fa T, �(x, u) is consistent with the exact model F e T (x, u) as in Definition 1(b). (iii) There exists a family of Lyapunov functions VT, �(x, xˆ), class-K∞ functions �1(·), �2(·), �3(·), �0(·), and nondecreasing functions �0(·), �1(·), �2(·), with the following property: For any compact sets X ⊂ Rn, Xˆ ⊂ Rn, U ⊂ Rm, there exists a constant T ∗> 0, and for any fixed T ∈ (0, T ∗] there exists �∗> 0, and for any ε1> 0 there exists c > 0, such that, for all x, x1, x2 ∈ X, xˆ ∈ Xˆ, u ∈ U, and � ∈ (0, �∗], |x1 − x2|�c ⇒ |VT, �(x1, xˆ)− VT, �(x2, xˆ)|�ε1, (32) �1(|e|)�VT, �(x, xˆ)��2(|e|), (33) VT, �(F a T , �(x, u), F a T , �(xˆ, u)+ �T, �(xˆ, y, u))− VT, �(x, xˆ) T �− �3(|e|)+ �0(�)[�0(|e|)+ �1(|x|)+ �2(|u|)]. (34) We will use the following proposition, proved in Arcak and Nešic´ (2004). Proposition 2. Suppose that all conditions of Theorem 2 hold. Then, for any triple of strictly positive real numbers (�x,�xˆ ,�u) there exists T ∗> 0 such that, for any fixed T ∈ (0, T ∗] and �> 0, there exists �∗> 0 such that |x|��x , |xˆ|��xˆ , |u|��u and � ∈ (0, �∗] imply: VT, �(F e T , �(x, u), F a T , �(xˆ, u)+ �T, �(xˆ, y, u))− VT, �(x, xˆ) T �− �3(|e|)+ �. (35) Proof of Theorem 2. We first let �x , �xˆ and �u be as in (25), and determine T ∗ from Proposition 2. Next, we fix T ∈ (0, T ∗], and choose �> 0 to satisfy both (25) and �T� 12�1(d). (36) Finally, using Proposition 2 and arguments similar to those in the proof of Theorem 1, we can find �∗> 0 such that for all � ∈ (0, �∗] the estimate (30) holds. Note that, unlike Theorem 1 where we tune T to reduce r + �T in (30), here we ensure r+�T��1(d) by further restricting the choice of � by (36). Using the resulting �∗ obtained from Proposition 2, we conclude from (30) that observer (8) is semiglobal in T and practical in � as in Definition 2(b). � Example 3. Theorems 1 and 2 are also applicable to reduced-order observers when e is interpreted as the differ- ence between the unmeasured components of x, and their observer estimates. We now design such a reduced-order observer for the Duffing oscillator x˙1 = x2, x˙2 =−x1 − x31 , (37) from sampled measurements of its output y = x1. For our first design we use the Euler approximation with �= T x1(k + 1) = x1(k)+ T x2(k), x2(k + 1) = x2(k)+ T (−x1(k)− x1(k)3). (38) Observer design for this model is straightforward because the nonlinearity depends only on the output y=x1. Defining the new variable := x2 − y, which is governed by (k + 1)= (1− T ) (k)+ T [−2y(k)− y(k)3], (39) we employ the observer ˆ(k + 1)= (1− T ) ˆ(k)+ T [−2y(k)− y(k)3], (40) xˆ2 = ˆ+ y, (41) and note that the error variable e2= xˆ2−x2= ˆ− satisfies e(k + 1)= (1− T )e(k). (42) The assumptions of Theorem 1 hold because the Lyapunov function VT