49th IEEE Conference on Decision and Control December 15-17, 2010 Hilton Atlanta Hotel, Atlanta, GA, USA
A Robust Adaptive Observer for Nonlinear Systems with Unknown Inputs and Disturbances Habib Dimassi, Antonio Lor´ıa, Safya Belghith
Abstract— We present a simple robust adaptive observer for systems with unknown inputs. The class of systems includes partially linear systems with differentiable non-linearities provided that the solutions are bounded (the bound is unknown). Parametric uncertainty and additive disturbances are considered where the parameters are piece-wise constant. It is also assumed that measurable outputs are corrupted by noise. We show under certain necessary and sufficient conditions convergence of the estimation errors.
I. I NTRODUCTION
The problem of unknown input observer design consists in designing a state estimator for systems driven by unknown inputs to be reconstructed and perturbations to be rejected. That is, the estimator must meet certain sufficient and necessary structural conditions such that one is able to decouple the effect of unknown inputs. Since the pioneering work of [1] observers for (linear) systems subject to unknown inputs and perturbations are of great interest in various contexts. The problem is relevant for instance in fault detection -cf. [2]; in synchronization of chaotic systems with application to information decoding –cf. [3], [4]. A large majority of articles on unknown-input –observers applies to linear systems: time invariant –cf. [5]; time-varying -cf. [6] in continuous, or in discrete time –cf. [7]. The results of [6] apply to systems with input-dependent nonlinear dynamics yet, the analysis is carried out along trajectories by regarding the system as linear time-varying; geometric necessary and sufficient conditions for highgain observers are given. A few recent works focus on systems with nonlinearities (in the unmeasurable states). For instance [8] relies on the ability to transform the nonlinear system into a triangular canonical form to design a slidingmode observer. The work [9] is primarily addressed to synchronization of chaotic oscillators and relies on the assumption that the nonlinearity is globally Lipschitz. In this paper we propose a robust adaptive observer for partially nonlinear systems with disturbances and measurement noise. Nonlinearities are assumed to be once H Dimassi and A. Loria are with CNRS, LSSSupelec, 3 Rue Joliot-Curie, 91192 Gif sur Yvette, France,
[email protected] [email protected] S Belghith is with Ecole Nationale d’Ing´enieurs, Tunis, Tunisia
978-1-4244-7746-3/10/$26.00 ©2010 IEEE
continuously differentiable (we do not assume global Lipschitz as is common in the literature). However, we assume that the system operates in a regime such that state trajectories are bounded. Adaptation is used to estimate online such bound which is used to show estimation error convergence. The rest of the note is organized as follows. In the following section, we formulate the estimation problem precisely. Our main results are presented in Section III. We conclude with some remarks in Section V. Notation. The following notation is used throughout this paper. |·| denotes the absolute value for scalars, the Euclidean norm for vectors, and the induced norm for matrices. We use M + for the generalized inverse of a matrix M . The smallest and largest eigenvalues of M are denoted by λmin (M ) and λmax (M ) respectively. In particular, we may define |M | := [λmax (M ⊤ M )]1/2 . II. C ONTEXT AND P ROBLEM S TATEMENT
We consider the problem of observer design for nonlinear systems
x˙ = Ax + Bf0 (x) + Bg0 (x)m(t) + F d(t) (1a) y = Cx + Gd(t) (1b) where x ∈ Rn is the state vector, y ∈ Rp is a measurable output, f0 : Rn → Rs and g0 (x) : Rn → Rs×q are once continuously differentiable; A ∈ Rn×n , B ∈ Rn×s , C ∈ Rp×n , F ∈ Rn×r and G ∈ Rp×r are constant matrices. The functions m : R≥0 → Rq and d : R≥0 → Rr are assumed to be (component-wise) piece-wise constant and measurable respectively. The former represents a vector of unknown parameters and the latter are external disturbances to be rejected. The model (1) covers a wide number of natural scenarios as for instance that of systems with neglected dynamics and measurement noise. It is assumed that there exists Km > 0 such that
sup |m(t)| ≤ KM
(2)
t≥0
Standing assumption. We assume that the solutions x(·) to (1) are forward complete and uniformly bounded. Assuming boundedness of solutions is restrictive in general especially in the context of certaintyequivalence output feedback control. Yet, it is a common
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assumption in the literature of observer design and it holds for many physical systems such as oscillators1 . More particularly, chaotic oscillators. The assumption is made for technical reasons, namely to establish global results. Problem statement. It is required to design an adaptive (dynamic) observer for (1) such that for any r > 0
lim |x(t) − x ˆ(t)| = 0
t→∞
∀ x(0), xˆ(0) ∈ Br × Rn (3)
where Br := {x ∈ Rn : |x| < r}. In particular, it is required to reject the perturbation d which affects both the system dynamics and the measured output. For further development we introduce a transformation some times called “Lipschitz extension” –cf. [11]. The purpose is to rewrite the system dynamics using globally-Lipschitz functions. To that end, let ωi > 0 be arbitrarily given for each i ∈ {1, . . . n}. Define the compact Ω ⊂ Rn
Ω := {x ∈ Rn : |xi | ≤ ωi } and define the saturation function σ : Rn → Ω where each component σi of σ is given by ( xi if |xi | ≤ ωi i ∈ {1, . . . n} ⇒ σi (x) := ωi > 0 sgn(ωi ) |ωi | otherwise
III. M AIN R ESULT A. State Estimation and Disturbance Rejection
We show next that the following adaptive observer achieves the objective (3) in presence of external disturbances, parametric uncertainty (piece-wise constant) and measurement noise. The observer is given by the equations
z˙ = N z + Jy + Hf (ˆ x) + Hg(ˆ x)m ˆ 1ˆ ˆ) + βHM (T y − C1 x 2 x ˆ = z − Ey
Then, for all x ∈ Ω and t ∈ R≥0 consider the system
x˙ = Ax + Bf (x) + Bg(x)m(t) + F d(t) (4a) y = Cx + Gd(t) (4b)
The observer design problem under the standing assumption boils down to designing an observer for (4) that ensures (3). Indeed, it suffices to define Ω as the smallest set containing the solutions of (1) for all t. In general, Ω depends on r i.e., on the ball of initial states Br defined previously, however for certain physical systems such as chaotic oscillators, Ω is independent of r as it may be chosen as an arbitrary compact containing the attractor.
(5b)
and the adaptation laws
m ˆ˙ = δg(ˆ x)T M (T y − C1 x ˆ) ˙ˆ 2 β = γ |M (T y − C1 x ˆ)|
(6) (7)
The constants δ and γ are positive real numbers imposed by the designer. The matrices N , E , J , H , T , M and C1 are constant, of appropriate dimensions and are left to be determined. They are required to satisfy the following conditions
EG P A − N P − JC P F − N EG − JG H P TG TC
Define further f : Rn → Rs and g : Rn → Rs×q for each x ∈ Rn as f (x) := f0 ◦σ(x) and g(x) := g0 ◦σ(x). The following observations are in order (the proof of the second is provided in the Appendix): Fact 1: 1) for all x ∈ Ω we have f (x) = f0 (x) and g(x) = g0 (x); 2) For each Ω there exist positive reals Kf and Kg such that for any a, b ∈ Rn and w ∈ Rs ⊤ w [f (a) − f (b)] ≤ Kf |a − b| |w| ⊤ w [g(a) − g(b)] ≤ Kg |a − b| |w| .
(5a)
= = = = = = =
0 0 0 PB I + EC 0 C1
(8) (9) (10) (11) (12) (13) (14)
Furthermore, we require that the triple (N, H, C1 ) satisfy the stability conditions
N T Q + QN < 0 H T Q = M C1
(15) (16)
with Q = Q⊤ > 0. Proposition 1: Under the Standing Assumption and conditions (8)–(16) the expression (3) holds for the solutions x(·) of (1) and the observer trajectories x ˆ(·) of (5). This, for all initial states x(0) such that x(·) are bounded and for all initial states x ˆ(0) ∈ Rn . Proof of Proposition 1: We start by writing the estimation error dynamics in a convenient form. Then, we use standard Lyapunov arguments to show convergence to zero of the state estimation errors. To that end define the error variables e := x − x ˆ and m ˜ := m − m ˆ . Hence
1 The
reader shall not understand systems with periodic solutions. See [10] for several broad definitions of oscillators.
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e = x − z + Ey ⇐ (5b) = x − z + ECx + EGd ⇐ (1b) = Px − z ⇐ (8), (12)
Differentiating on both sides of the latter, we obtain
where we have used (16). Next, let condition (15) generate a positive real constant η such that N ⊤ Q + QN ≤ e˙ = P Ax + P Bf (x) + P Bg(x)m(t) + P F d(t) − N z −2η . Using Fact 1.2 with w⊤ = e⊤ QH , (19) and (2) we − Jy − Hf (ˆ x) − Hg(ˆ x)m(t) ˆ obtain 1ˆ V˙ 1 ≤ −2η |e|2 + 2 |M C1 e| [Kf + Kg Km ] |e| ˆ) . − βHM (T y − C1 x 2 2 ⊤ ˜ m(t) ˙ . −β |M C1 e|2 − m Observe that in view of (4b), (13) and (14) we have T y− δ C1 x ˆ = C1 e. Using this, (4b) again and z = x − e + Let β := [Kf + Kg Km ]2 /η then, using the triangle E(Cx + Gd) we obtain inequality on 2 |M C1 e| [Kf + Kg Km ] |e|, we finally obtain e˙ = (P A − JC)x − JGd(t) − N x + N e − N ECx − N EGd(t) + P Bf (x) + P Bg(x)m(t) ˜ ≤ −η |e|2 − 2 m ˜ ⊤ m(t) ˙ V˙ 1 (e, m, ˜ β) δ 1ˆ + P F d(t) − Hf (ˆ x) − Hg(ˆ x)m ˆ − βHM C1 e . which holds for all e such that x ∈ Ω. That is, for any 2 compact Ω and for all initial conditions x◦ ∈ Rn such Next, we regroup terms and observe from (10) that the that the solutions x(·) remain in the set Ω and for almost factors of d equal to zero hence, all t we have 1ˆ 2 e˙ = (P A − JC − N EC)x − N x ˆ − βHM C1 e ˜ V˙ 1 (e(t), m(t), ˜ β(t)) ≤ −η |e(t)|2 − m(t) ˜ ⊤ m(t) ˙ 2 δ + P B[f (x) + g(x)m(t)] − H[f (ˆ x) + g(ˆ x)m] ˆ . which, using the assumption that m(t) ˙ = 0 almost everywhere, implies that From (12) it follows that the Equation (9) is equivalent to P A − N EC − JC = N therefore, using (11), we ˜ V˙ 1 (e(t), m(t), ˜ β(t)) ≤ −η |e(t)|2 a.e. (22) obtain Integrating on both sides of the latter from 0 to ∞ we ˜ are bounded for all t and e˙ = N e + H[f (x) − f (ˆ x)] + H[g(x)m(t) − g(ˆ x)m] ˆ obtain that e(t), m(t) ˜ and β(t) 1ˆ moreover, e(t) is square integrable. Next, we make the (17) − βHM C1 e . following substitutions in (18): 2 Let β > 0 be a constant to be determined and define β˜ := β − βˆ. Adding Hg(ˆ x)m(t) to both sides of (17) we obtain
e˙ = N e + H[f (x) − f (ˆ x)] + H[g(x) − g(ˆ x)]m(t) 1 1 m ˜ x) HM C1 e ˜ − βHM C1 e + Hg(ˆ β 2 2 (18) where, according to the adaptation laws (6) and (7) m ˜ and β˜ are solutions to
m ˜˙ = −δg(ˆ x)T M C1 e a.e. ˙ β˜ = −γ |M C1 e|2 .
(19) (20)
Next, consider the positive definite and radially unbounded function 1 2 1 ˜2 ˜ m) V1 (e, β, ˜ := e⊤ Qe + m ˜ + β . (21) δ 2γ Its total derivative along the trajectories of (18)–(20) yields
V˙ 1 = e⊤ [N ⊤ Q + QN ]e + 2e⊤ QH[f (x) − f (ˆ x)] ⊤
⊤
− e Q[HM C1 e]β + 2e QH[g(x) − g(ˆ x)]m(t) 2 ⊤ ˜ [δg(ˆ x)⊤ M C1 e − m(t)] ˙ + 2e⊤ QH mg(ˆ ˜ x) − m δ
x = x(t) x ˆ = x(t) − e(t) e˙ = e(t) ˙ ˜ ˜ m ˜ = m(t) ˜ β = β(t) to conclude that e(t) ˙ is also uniformly bounded for all t (since m(t) is bounded). By [12, Lemma A.5] we conclude that (3) holds. B. Implementation
We have showed that the observer (5) achieves the estimation goal under conditions (8)–(16). In this section we show in detail how these conditions may be verified. In particular, we present a procedure to determine the design matrices involved in (8)–(16). Consider first Equations (13) and (14). They may be re-written as C 0 I T T = C1 0 0 (23) 0 G −I which is of the form
XR1 = R2
(24)
in which given R1 and R2 , it is required to find X . After [13] Equation (24) is solvable if and only if R Rank 1 = Rank[R1 ] (25) R2
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and the solution is given, for any Z , by
X = R2 R1+ − Z(I − R1 R1+ )
(26)
where R+ denotes the generalized inverse of R i.e., such that2 RR+ R = R. Thus, T may be obtained from (26) with R2 = [C1 0 0], C 0 I X = [T T ], R1 = . (27) 0 G −I Consider next, Equations (9) and (10). Given an arbitrary matrix K let J = −N E − K . Hence, Equations (9) and (10) become respectively,
P A + KC = N KG = −P F .
(28) (29)
The necessary and sufficient condition for (29) to be solvable for K is that G Rank = Rank[G] (30) −P F and, according to (26) its general solution is
K = −P F G+ − L(I − GG+ )
(31)
where G+ is a generalized inverse of G and L is an arbitrary matrix of appropriate dimensions. Using (31) in (28), we obtain
N = A1 − LC1
(32)
A1 = P A − P F G+ C
(33)
C1 = (I − GG+ )C
(34)
where
After [14], the necessary and sufficient conditions for the solvability of (15) and (16) are
Rank
The solution to the latter yields the minimum ρ = 0 and Q, M such that L = −Q−1 W satisfies (15), (16) with N = A1 − LC1 . ‘ We have showed that the adaptive observer (5), (7) and (6) achieve estimation error convergence for a fairly general class of nonlinear systems. The necessary and sufficient conditions are given by the rank conditions (25), (30), (35) and (36) which impose structural properties on the system. The second assumption, conservative in a general sense but not in the context of chaotic systems is boundedness of solutions. That is, such assumptions and only these restrict the class of systems for which the adaptive observer presented below applies. In such case, the following is a simple algorithm to compute the design matrices. Tuning Procedure To summarize, we propose the following procedure to determine the parameters of the unknown-input adaptive observer (5): Step 1: Compute C1 using equation (34) Step 2: Compute T using Equations (26) and (27) Step 3: If p = n, we choose E = T . Else, if p < n, we T choose E = 0 Step 4: Compute H and P using respectively equations (11 ) and (12 ) Step 5: Compute A1 using equation (33 ) Step 6: Find matrices M , L and Q by solving the convex optimization problem, presented in the above analysis. We can also apply the detailed method proposed in [14], to determine M , L and Q Step 7: Compute K and N using respectively equations (31) and (32) Step 8: Compute J = −N E − K .
Rank[C1 H] = Rank[H] (35) A1 − λI H = n + Rank[H] (36) C1 0
for each complex number λ such that Re(λ) ≥ 0. To solve (15), (16) with N = A1 − LC1 we consider the following convex optimization problem –cf. [14]: to minimize ρ subject to
Q > 0 (37) ⊤ ⊤ QA1 + + W C1 + C1 W < 0 (38) ⊤ ρI H Q − M C1 ≤ 0 . (39) ρI QH − C1⊤ M ⊤ A⊤ 1Q
IV. A N EXAMPLE OF M ASTER -S LAVE S YNCHRONIZATION
Consider a R¨ossler-like chaotic system with measurement noise and disturbances. The objective is to design a slave system that synchronizes with the R¨ossler system. The slave system is defined by the robust adaptive observer. The master system is
2 A practical method to compute R+ , is to find a singular value decomposition of R such that R = U SV ⊤ where U and V are unitary matrices and e.g., S := [S1 0] non-negative and of the same dimension than R, with ˜ ⊤ where S˜ := [S −1 0]⊤ . S1 square and diagonal. Hence, R+ = V SU 1 + By definition, R = 0 if R = 0.
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x˙ 1 x˙ 2 x˙ 3 y1 y2
= = = = =
−(x2 + x3 ) + d(t) x1 + ax2 + d(t) b + x3 (x1 − c) + m(t)x3 + d(t) x1 + 2d(t) x3 + d(t)
(40a) (40b) (40c) (40d) (40e)
−0.4 H = 0.8 , 1
which is of the form (1) with g0 (x) = x3 , 0 −1 −1 0 A = 1 0.398 0 , B = 0 , f0 (x) = 2 + x1 x3 , 0 −4 0 1 1 1 0 0 2 C= , G= , F = 1 . 0 0 1 1 1
−0.32 −1.2 0.24 A1 = 0.44 0.798 −3.08 , −0.4 0 −4.2 C1 =
0.2 0 −0.4 −0.4 0 0.8
,
5.1283 4.9395 −2.6505 6.5315 −3.2496 , Q = 4.9395 −2.6505 −3.2496 3.0395
19.9972 −39.9943 L = −40.0010 80.0020 −50.0030 100.0059
attractor of original chaotic system attractor of perturbed chaotic system
M=
12 10
−0.75 1.5
,
−20.3172 39.8343 −80.282 K = 39.441 49.603 −100.2059
3
7 X
1.2 0 −0.4 P = −0.4 1 0.8 , 0 0 1
The parameters are set to a = 0.398, b = 2 and c = 4 so the master system exhibits chaotic behavior, even in the presence of disturbances. The perturbation function d consists in an uniformly distributed noise (random) signal generated between lower and upper bounds respectively equal to 0 and 0.4. We assume that the unknown input to be estimated m(t), is a digital pulse train emulated by m(t) = 0.5sgn(sin(0.2t)). For comparison, Figure 1 depicts the attractors of systems (40) and an unperturbed R¨ossler system; the initial conditions are set to x0 = [0.2, −0.4, −0.2]⊤ for both systems.
4 1 −2 4
2
0
−2
−4
−6
X2
−8 −6
−3
0
3
6
−20.3172 −1.2 40.2343 N = 40.4410 0.798 −83.082 , 49.603 0 −104.2059
8
23.9006 −47.0012 95.82 J = −47.21 −59.5235 120.0471
X
1
Fig. 1. Attractors of an input-free chaotic R¨ossler system and the perturbed R¨ossler system (40)
R3
Let Ω := {x ∈ : kxi k ≤ ωi ∀ i ∈ {1 . . . 3} }. Note from Figure 1 that Ω strictly contains the systems’ attractors. Hence, the saturation levels are set to with ω1 = ω2 = ω3 = 12 and we define f (x) := f0 (σ(x)), g(x) = g0 (σ(x)). Clearly, f (x) = f0 (x) and g(x) = g0 (x) for all x ∈ Ω. Hence, the transmitter (40) takes the form (4) with f (x) = 2 + σ1 (x)σ3 (x), g(x) = σ3 (x) . The slave system is given by the observer equations (5) and the adaptation laws (6) and (7). To find explicit numeric values of the observer matrices we follow the tuning procedure previously described. We obtain 0.2 −0.4 0.2 −0.4 T = , E = −0.4 0.8 , −0.4 0.8 0 0
The initial conditions for the slave system set to x ˆ0 = ⊤ [−0.12, 0.24, 0] and m ˆ is initialized at m ˆ 0 = 0. 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 0
20
40
60
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100 Time[s]
120
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Fig. 2. Synchronization error e1 = x1 − x ˆ1 for the adaptive robust observer (5) in the absence of noise and disturbance
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1
chaotic oscillators. A natural application of our results is synchronization of chaotic systems with the motivation of secured transmission of information.
sent message recovered message
0.5 0
R EFERENCES
−0.5 −1 0
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120
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Fig. 3. The transmitted message m and the recovered message m ˆ by the adaptive observer (5) in the absence of noise and disturbances
The simulation results are showed in Figures 2 and 3. A second run of simulations was carried out in the presence of noise and disturbances; the results are showed in Figures 4 and 5 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 0
20
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100 Time[s]
120
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Fig. 4. Synchronization error e1 = x1 − x ˆ1 using observer ((5)) in presence of noise and disturbances
5 4
sent message recovered message
3 2
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1 0 −1 −2 −3 −4 0
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Fig. 5. The transmitted message m and the recovered message m ˆ by applying observer (5) in the presence of noise and disturbances
V. C ONCLUDING REMARKS
We have presented a robust adaptive observer for partially linear systems affected by additive disturbances and driven by unknown inputs. The results rely on structural necessary and sufficient conditions which nevertheless, hold for a wide class of nonlinear systems such as 2607
