By Edward Ott, Tim Sauer, James A. Yorke

Brings jointly contemporary advances within the interpretive and useful functions of chaos, which carry nice promise for extensive applicability in the course of the actual sciences and engineering. DLC: Chaotic habit in platforms.

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**Extra info for Coping with Chaos: Analysis of Chaotic Data and Exploitation of Chaotic Systems**

**Example text**

As in Sect. 9 requires that there exists a symmetric matrix H such that Pz , Ac and Cc satisfy ⎡ ⎣ Pz − ⎤ H F + G K GCc ⎦ (F + G K )T Pz T (GCc ) T Pz 0 0, eiT H ei ≤ 1, i = 1, . . n C . 11 again apply. Using Ez as the constraint set in the online optimization in place of Z reduces the region of attraction of the MPC law. However, to compensate for this effect it is possible to design the prediction system parameters Ac and Cc so as to maximize the projection of Ez onto the x-subspace. 59b). 9 Optimized Prediction Dynamics 49 the case considered in Sect.

This uses an input–output model to express the vector of output predictions as an affine function of the vector of predicted inputs ⎤ ⎤ ⎡ y1|k Δu 0|k ⎥ ⎥ ⎢ ⎢ f .. yk = ⎣ ... ⎦ = C G Δuk + yk , Δuk = ⎣ ⎦ . y N |k Δu Nu −1|k ⎡ Here Nu denotes an input prediction horizon which is chosen to be less than or equal to the prediction horizon N . The matrix C G is the block striped (Toeplitz) lower 52 2 MPC with No Model Uncertainty triangular matrix comprising the coefficients of the system step response, C G Δuk f denotes the predicted forced response at time k, and yk denotes the free response at time k due to non-zero initial conditions.

27b). The stage cost (namely the part of the cost incurred at each prediction time step) has the general form x 2 Q + u 2 R = x 2 Q + Kx + c = z 2 , Qˆ 2 R = x T (Q + K T R K )x + c T E T R Ec Q + K T RK K T RE . 1, W is the (positive-definite) solution of the Lyapunov equation W = T W ˆ + Q. 34) The special structure of and Qˆ in this Lyapunov equation implies that its solution also has a specific structure, as we describe next. 27b) can be written as ⎡ J (xk , ck ) = xkT Wx xk + ckT Wc ck B T Wx B + R 0 ··· 0 TW B + R ··· ⎢ 0 B 0 x ⎢ Wc = ⎢ ..