Takagi Sugeno (TS) fuzzy model is used to model the chaotic dynamic system, while the uncertainties are decomposed such that the uncertain chaotic system can be rewritten as a set of local linear models with an additional disturbed input.
In the proposed method, a controller is designed in terms of the optimal linear model representation of the nominal system around each operating point of the trajectory, while the uncertainties are decomposed such that the uncertain nonlinear system can be rewritten as a set of local linear models with disturbed inputs.
In line with the above, the joint power allocation and CM problem (P1) in the two-tier femtocell/macrocell system can be rewritten into the following way summating over all connected users j ∈ S c of each cell c ∈ C in the system.
Assuming that (S t)+N t)=1), this equation system can be rewritten as left { textstylebegin{array}l} N t)[gamma S t)^{2}-gamma m S t)+beta_{2}]=0, -N t)[gamma S t)^{2}-gamma m S t)+beta_{2}]=0.
The Liénard type equation (1), for representing non-scalar systems, can be rewritten in a state space representation by considering (x t)), (dot{x}(t)) as state variables (x_{1}(t),x_{2}(t)in X) (X, being an adequate Banach space), leading to dot{x}_{1}(t)=x_{2}(t); qquad dot{x}_{2}(t)=-F_{0}bigl(x_{1}(t) bigr)x_{2}(t -G_{0}bigl(x_{1}(t -G_{).
Proof Substituting controller (3.9) into system (3.6), the error system (3.6) can be rewritten as D ∗ α e = ( B + K ) e. (3.10).
Let begin{aligned}& x t) (z)=x z,t), & y(t) (z)=y z,t), & fbigl t,x t bigr) (z)=fbigl z,t,x z,t bigr), & hbigl t,x t bigr) (z)=hbigl z,t,x z,t bigr). end{aligned} Then the fractional hybrid dynamic system (5.1) can be rewritten into the abstract fractional evolution system (4.1).
Next, we consider the case that the system (1) without Markovian jumping parameters and (h(t)=h), then the system (1) can be rewritten as begin{aligned} &dot{x}(t) = -Cx t-sigma)+Ax t-h)+Ax t-ht,x(t- sigma),x(t-h)bigr),quad t>0, &x(s)=phi(s), quad sinbigl[-max{sigma,h},0bigr].
The system (4.17) can be rewritten as (4.25).
Thus, the system (1.2) can be rewritten as (2.14).
Then, system (2.1) can be rewritten as (2.4).
