It also gives that the system parameter vector θ can be identified by the updating laws (4) when the lag synchronization is achieved.
According to the LaSalle invariance principle of differential equation, the orbits asymptotically converge to the set E. It shows that the system parameter vector θ can be identified by the updating laws (6) when the lag synchronization is achieved.
Consider the following uncertain dynamical system: dot{x}(t)=fbigl(x t),thetabigr triangleq f_{1}bigl(x t) bigr)+f_{2}bigl(x t bigr theta, (1) where (xin R^{n}) is the state vector, (thetain R^{m}) is an unknown system parameter vector.
The system parameter vector θ consists of positive rate coefficients, Michaelis Menten parameters or degradation and synthesis rates.
Given system (3.2), find a combined parameter vector ((boldsymbol{delta},boldsymbol{beta},T inLambda timesThetatimes 0,hat{T})) to minimize the objective functional (3.5) and meanwhile satisfy (3.3) and (3.4).
Thus, we can change the problem (({P_{0}})) into the following problem: (({P_{1}})): Given system (3.2), find a combined parameter vector ((boldsymbol{delta},boldsymbol{beta},T inLambda timesThetatimes 0,hat{T})) to minimize the objective functional (3.5) and meanwhile satisfy (3.3) and (3.4).
After building an augmented system state by concatenating the parameter vector and the state vector, the joint estimation of states and parameters reduces to filtering of the augmented state vector which makes SMC methods directly applicable to the problem.
We introduce four relative efficiencies to define the efficiency of estimator in two linear regression equations system with identical parameter vectors, also we give the lower and upper bounds of the four relative efficiencies.
where θ 1 ≥ ⋯ ≥ θ p is the ordered eigenvalues of X 1 ′ Q 1 X 1, η 1 ≥ ⋯ ≥ η p is the ordered eigenvalues of X 2 ′ Q 2 X 2. In this article, we have introduced four relative efficiencies in two linear regression equations system with identical parameter vectors, and we have also given the lower and upper bounds for the four relative efficiencies.
For example, Zhan and Yeung modeled a molecular pathway with the following ODEs [ 34]: (1) x ˙ (t ) = f (x (t ), u (t ), θ ), x (t 0 ) = x 0, y (t ) = g (x (t ) ) + η (t ), where x ∈ R n is the state vector of the system, θ ∈ R k is a parameter vector, u(t) ∈ R p is the system's input, y ∈ R m is the measured data, η(t) ~ N 0, σ) is the Gaussian white noise, and x0 denotes the initial state.
We show that if the system matrices depend affinely on the parameter vector, whose bounding set is a compact polyhedron, then this problem requires the solution of a finite number of eigenvalue problems associated with the vertices of such a polyhedron.
