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14
Riccati Equations
and their Solution
14.1 Introduction......................................................14-1
14.2 OptimalControlandFiltering:Motivation....14-2
14.3 Riccati Differential Equation............................14-3
14.4 Riccati Algebraic Equation...............................14-6
General Solutions • Symmetric Solutions
• Definite Solutions
14.5 Limiting Behavior of Solutions......................14-13
14.6 OptimalControlandFiltering:
Application......................................................14-15
14.7 NumericalSolution.........................................14-19
Invariant Subspace Method • MatrixSignFunction
Vladimír Kuceraˇ Iteration • Concluding Remarks
CzechTechnical University and Institute of Acknowledgments.....................................................14-21
Information Theory and Automation References ..................................................................14-21
14.1 Introduction
Anordinarydifferential equation of the form
x˙(t) +f (t)x(t)−b(t)x2(t)+c(t) = 0 (14.1)
is knownasa Riccati equation, deriving its name from Jacopo Francesco, Count Riccati (1676–1754) [1],
whostudiedaparticularcaseofthisequationfrom1719to1724.
For several reasons, a differential equation of the form of Equation 14.1, and generalizations thereof
comprise a highly significant class of nonlinear ordinary differential equations. First, they are intimately
related to ordinary linear homogeneous differential equations of the second order. Second, the solutions
of Equation 14.1 possess a very particular structure in that the general solution is a fractional linear
functionintheconstantofintegration.Inapplications,Riccatidifferentialequationsappearintheclassical
problemsofthecalculusofvariationsandintheassociateddisciplines of optimal control and filtering.
Thematrix Riccati differential equation refers to the equation
˙
X(t)+X(t)A(t)−D(t)X(t)−X(t)B(t)X(t)+C(t)=0 (14.2)
defined on the vector space of real m×n matrices. Here, A,B,C, and D are real matrix functions of
the appropriate dimensions. Of particular interest are the matrix Riccati equations that arise in optimal
control and filtering problems and that enjoy certain symmetry properties. This chapter is concerned
14-1
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14-2 Control System Advanced Methods
with these symmetric matrix Riccati differential equations and concentrates on the following four major
topics:
• Basic properties of the solutions
• Existence and properties of constant solutions
• Asymptoticbehaviorofthesolutions
• MethodsforthenumericalsolutionoftheRiccatiequations
14.2 Optimal Control and Filtering: Motivation
Thefollowingproblemsofoptimalcontrolandfilteringareofgreatengineeringimportanceandserveto
motivate our study of the Riccati equations.
Alinear-quadratic optimal control problem consists of the following. Given a linear system
x˙(t) = Fx(t)+Gu(t), x(t0) = c, y(t) = Hx(t), (14.3)
where x is the n-vector state, u is the q-vector control input, y is the p-vector of regulated variables, and
F,G,H areconstantrealmatricesoftheappropriatedimensions.Oneseekstodetermineacontrolinput
function u over some fixed time interval [t1,t2] such that a given quadratic cost functional of the form
t
2 ′ ′ ′
η(t ,t ,T) = [y (t)y(t)+u (t)u(t)]dt +x (t )Tx(t ), (14.4)
1 2 2 2
t
1
with T being a constant real symmetric (T = T′) and nonnegative definite (T ≥ 0) matrix, is afforded a
minimumintheclassofallsolutionsofEquation14.3,foranyinitialstatec.
Auniqueoptimalcontrolexistsforallfinite t2−t1 >0andhastheform
u(t) =−G′P(t,t2,T)x(t),
whereP(t,t2,T)isthesolutionofthematrixRiccatidifferential equation
˙ ′ ′ ′
−P(t)=P(t)F+F P(t)−P(t)GG P(t)+H H (14.5)
subject to the terminal condition
P(t2) = T.
Theoptimalcontrolisalinearstatefeedback,whichgivesrisetotheclosed-loopsystem
x˙(t) =[F −GG′P(t,t2,T)]x(t)
andyields the minimumcost
∗ ′
η (t1,t2,T) = c P(t1,t2,T)c. (14.6)
AGaussian optimal filtering problem consists of the following. Given the p-vector random process z
modeledbytheequations
x˙(t) = Fx(t)+Gv(t), (14.7)
z(t) = Hx(t)+w(t),
where x is the n-vector state and v,w are independent Gaussian white random processes (respectively,
q-vector and p-vector) with zero means and identity covariance matrices. The matrices F,G, and H are
constant real ones of the appropriate dimensions.
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Riccati Equations and their Solution 14-3
Given known values of z over some fixed time interval [t1,t2] and assuming that x(t1) is a Gaussian
randomvector,independentofvandw,withzeromeanandcovariancematrixS,oneseekstodetermine
anestimatexˆ(t2)ofx(t2) such that the variance
′ ′
σ(S,t ,t ) = Ef [x(t )−ˆx(t )][x(t )−ˆx(t )] f (14.8)
1 2 2 2 2 2
of the error encountered in estimating any real-valued linear function f of x(t2) is minimized.
Auniqueoptimalestimateexistsforallfinitet −t >0andisgeneratedbyalinearsystemoftheform
2 1
˙ ′
xˆ(t) = Fxˆ(t)+Q(S,t1,t)H e(t), xˆ(t0) = 0, e(t) = z(t)−Hxˆ(t),
whereQ(S,t1,t)isthesolutionofthematrixRiccatidifferential equation
˙ ′ ′ ′
Q(t)=Q(t)F +FQ(t)−Q(t)H HQ(t)+GG (14.9)
subject to the initial condition
Q(t1) =S.
Theminimumerrorvarianceisgivenby
σ∗(S,t ,t ) = f′Q(S,t ,t )f. (14.10)
1 2 1 2
Equations14.5and14.9arespecialcasesofthematrixRiccatidifferentialEquation14.2inthatA,B,C,
andDareconstantrealn×nmatricessuchthat
B=B′, C=C′, D=−A′.
Therefore, symmetric solutions X(t) are obtained in the optimal control and filtering problems.
WeobservethatthecontrolEquation14.5issolvedbackwardintime,whilethefilteringEquation14.9
is solved forward in time. We also observe that the two equations are dual to each other in the sense that
P(t,t2,T) = Q(S,t1,t)
onreplacing F,G,H,T,andt2−t inEquation14.5respectively,byF′,H′,G′,S,andt−t1 or,viceversa,
onreplacingF,G,H,S,andt−t1 inEquation14.9respectively,byF′,H′,G′,T,andt2−t.Thismakesit
possible to dispense with both cases by considering only one prototype equation.
14.3 Riccati Differential Equation
This section is concerned with the basic properties of the prototype matrix Riccati differential equation
˙ ′
X(t)+X(t)A+AX(t)−X(t)BX(t)+C=0, (14.11)
where A,B, and C are constant real n×n matrices with B and C being symmetric and nonnegative
definite, ′ ′
B=B, B≥0 and C=C, C≥0. (14.12)
By definition, a solution of Equation 14.11 is a real n×n matrix function X(t) that is absolutely
continuous and satisfies Equation 14.11 for t on an interval on the real line R.
Generally, solutions of Riccati differential equations exist only locally. There is a phenomenon called
finite escape time: the equation
x˙(t) = x2(t)+1
π π
has a solution x(t) = tant in the interval (− ,0) that cannot be extended to include the point t =− .
2 2
However,Equation14.11withthesign-definite coefficients as shown in Equation14.12 does have global
solutions.
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14-4 Control System Advanced Methods
Let X(t,t2,T) denote the solution of Equation 14.11 that passes through a constant real n×n matrix
T at time t . We shall assume that
2
T =T′ and T≥0. (14.13)
ThenthesolutionexistsoneveryfinitesubintervalofR,issymmetric,nonnegativedefiniteandenjoys
certain monotone properties.
Theorem14.1:
Under the assumptions of Equations 14.12 and 14.13 Equation 14.11 has a unique solution X(t, t ,T)
2
satisfying
X(t,t2,T) =X′(t,t2,T), X(t,t2,T) ≥0
for every T and every finite t,t2, such that t ≥ t2.
ThiscanmosteasilybeseenbyassociatingEquation14.11withtheoptimalcontrolproblemdescribed
in Equations 14.3 through 14.6. Indeed, using Equation 14.12, one can write B = GG′ and C = H′H for
somereal matrices G and H. The quadratic cost functional η of Equation 14.4 exists and is nonnegative
for every T satisfying Equation 14.13 and for every finite t2 −t. Using Equation 14.6, the quadratic form
c′X(t,t ,T)c can be interpreted as a particular value of η for every real vector c.
2
AfurtherconsequenceofEquations14.4and14.6follows.
Theorem14.2:
For every finite t1,t2 and τ1,τ2 such that t1 ≤ τ1 ≤ τ2 ≤ t2,
X(t1,τ1,0)≤X(t1,τ2,0)
X(τ2,t2,0)≤X(τ1,t2,0)
andforeveryT ≤T ,
1 2
X(t ,t ,T ) ≤ X(t ,t ,T ).
1 2 1 1 2 2
Thus, the solution of Equation 14.11 passing through T = 0 does not decrease as the length of the
intervalincreases,andthesolutionpassingthroughalargerT dominatesthatpassingthroughasmallerT.
TheRiccati Equation 14.11 is related in a very particular manner with linear Hamiltonian systems of
differential equations.
Theorem14.3:
Let
Φ Φ
Φ(t,t2) = 11 12
Φ Φ
21 22
be the fundamental matrix solution of the linear Hamiltonian matrix differential system
˙
U(t) = A −B U(t)
˙ ′
V(t) −C −A V(t)
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