Kalman Filtering, Theory and Practice Using Matlab

4.8-4.8.5

Wang Hongmei

20087123

Content

4.8 Matrix riccati differential equation

4.8.1 Transformation to a linear equation 4.8.2 Time-invariant problem 4.8.3 Scalar time-invariant problem 4.8.4 Parametric dependence of the scalar time-invariant solution 4.8.5 Convergence issues

4.8.1 Transformation to a linear equation(1/3)

Matrix FractionsLinearization by fraction decompositionDerivationHamiltonian matrixBoundary constraints

4.8.1 Transformation to a linear equation(2/3)

Matrix Fractions

Linearization by fraction decomposition Derivation

Fraction decomposition

Matrix riccati differential equation

numerator denominator

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 equationFundamental solution of the linear time-inv

ariant differential equationGeneral solution of scalar time-invariant ric

cati equationSingular values of denominator Boundary values

4.8.3 Scalar time-invariant(2/7)

Linearizing the differential equationMatrix 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 different

ial equation

Solution of linearized system:

4.8.3 Scalar time-invariant(5/7)General solution of scalar time-invariant ric

cati equationPrevious results:

4.8.3 Scalar time-invariant(6/7)

0( ) 0pD t

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 solutionsDependence on initial conditionsConvergent and divergent solutionsConvergent 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 asym

ptotically 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!