Kalman Filtering, Theory and Practice Using Matlab 4.8-4.8.5 Wang Hongmei 20087123.

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Presentation transcript:

Kalman Filtering, Theory and Practice Using Matlab Wang Hongmei

Content 4.8 Matrix riccati differential equation Transformation to a linear equation Time-invariant problem Scalar time-invariant problem Parametric dependence of the scalar time-invariant solution Convergence issues

4.8.1 Transformation to a linear equation(1/3) Matrix Fractions Linearization by fraction decomposition Derivation Hamiltonian matrix Boundary constraints

4.8.1 Transformation to a linear equation(2/3) Matrix Fractions Linearization by fraction decomposition Derivation Fraction decomposition Matrix riccati differential equation numeratordenominator

4.8.1 Transformation to a linear equation(3/3) Derivation Hamiltonian Matrix Boundary constraints:

4.8.2 Time-invariant problem Boundary constraints:

4.8.3 Scalar time-invariant(1/7) Linearizing the differential equation Fundamental solution of the linear time- invariant differential equation General solution of scalar time-invariant riccati equation Singular values of denominator Boundary values

4.8.3 Scalar time-invariant(2/7) Linearizing the differential equation Matrix riccati differential equation Linearized equation

4.8.3 Scalar time-invariant(3/7) Fundamental solution of the linear time-invariant differential equation General solution Characteristic vectors of diagonalized

4.8.3 Scalar time-invariant(4/7) Fundamental solution of the linear time-invariant differential equation Solution of linearized system:

4.8.3 Scalar time-invariant(5/7) General solution of scalar time-invariant riccati equation Previous results:

4.8.3 Scalar time-invariant(6/7) Singular values of denominator

4.8.3 Scalar time-invariant(7/7)

4.8.4 Parametric dependence of the scalar time-invariant solution(1/6) Decay time constant (steady-state solution) Asymptotic and steady-state solutions Dependence on initial conditions Convergent and divergent solutions Convergent and divergent regions P(0)

4.8.4 Parametric dependence of the scalar time-invariant solution(2/6) Decay time constant

4.8.4 Parametric dependence of the scalar time-invariant solution(3/6) Asymptotic and steady-state solutions Corresponding steady-state differential equation (algebraic Riccati equation)

4.8.4 Parametric dependence of the scalar time-invariant solution(4/6) Dependence on initial conditions The initial conditions are parameterized by P(0) Two solutions: Nonnegative: stable initial conditions sufficiently near to it converge to it asymptotically Nonpositive: unstable infinitesimal perturbation nonpostive steady-state solution and converge nonnegative steady-state solution cause diverge to

4.8.4 Parametric dependence of the scalar time-invariant solution(5/6) Convergent and divergent solutions

4.8.4 Parametric dependence of the scalar time-invariant solution(6/6) Convergent and divergent regions, P=-1 Denominator=0

4.8.5 Convergence issues(1/2)

4.8.5 Convergence issues(2/2) Even unstable dynamic systems have convergent Riccati equations

Thank you!