Adaptive Observers With Persistency of Excitation for Synchronization of Chaotic Systems

Abstract

We address the problem of master-slave synchronization of chaotic systems under parameter uncertainty and with partial measurements. Our approach is based on observer-design theory hence, we view the master dynamics as a system of differential equations with a state and a measurable output and we design an observer (tantamount to the slave system) which reconstructs the dynamic behavior of the master. The main technical condition that we impose is persistency of excitation (PE), a property well studied in the adaptive control literature. In the case of unknown parameters and partial measurements we show that synchronization is achievable in a practical sense, that is, with "small" error. We also illustrate our methods on particular examples of chaotic oscillators such as the Lorenz and the Lü oscillators. Theoretical proofs are provided based on recent results on stability theory for time-varying systems.

Lag Quasi-Synchronization of Coupled Delayed Systems With Parameter Mismatch

“…The trajectory of the error system (4) converges exponentially to the set (8) i.e., the master system (1) and the slave system (2) achieve lag quasi-synchronized with an error level if there exist a nonsingular matrix and a matrix measure such that (9) Proof: Choose a Lyapunov function as (10) where is a non-singular matrix. Taking the upper-right Dini derivative of with respect to along the solution of (4) yields (11) Recalling Assumption 1, we have (12) From (11) and (12), we obtain (13) Let . From (9), we have that .…”

Rigid Body Attitude Control With Delayed Attitude Measurement

“…As a result, based on the trace property, λ(N) M 2 ≤ tr(M N M) = tr(M M N) [21], we have φ(t, t 0 ) 2 ≤ 9x 0 (λ(X)) −1 e −(t −t 0 ) . As proved in [22], under Assumption 1 and if x 0 ≥ β, X (t) is lower bounded as X (t) ≥ βρe −T I , which implies λ(X) ≥ βρe −T , and thus φ(t, t 0 ) 2 ≤ 9x 0 β −1 (e T /ρ)e −(t −t 0 ) .…”

Global estimation of rigid-body attitude/position using a single landmark and biased velocity measurements

“…Proof of Theorem 2: Item 1 : As proved in [19] a lower bound for the solution of matrix differential equationṖ (t) = 2C(t) ⊤ C(t) − ρP (t) − P (t)A(t) − A(t) ⊤ P (t) with initial condition P (t 0 ) = P 0 > 0 and persistency of excitation condition ∫ t+T t Φ A (t, t 0 ) ⊤ C(τ ) ⊤ C(τ )Φ A (t, t 0 ) dτ ≥ µI with positive numbers µ and T is σ(P ) ≥ µe −ρT , ∀t ≥ t 0 + T . For equation (12) it is evident that A(t) = 0, so Φ A (t, t 0 ) = I and the persistency of excitation condition becomes ∫ t+T t C(τ ) ⊤ C(τ ) dτ ≥ µT I which is the condition (8).…”