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!