In this paper some properties of the set
KR, whose members are the optimal feedback gains in linear-quadratic regulator problem, are described, and the technique of constructing subsets of
KR is given.
Controllable and stable plant as
x=Ax+bu is treated. The main results are as follows:
(1) Let
Qd=diag(
q1, q2, …, qn) be the standard form of weighting matrix
Q≥0 in parformance index, then
q1∼qn are the coefficients of a nonnegative polynomial.
(2) The set
K0L={P0b} is the largest convex cone in
KR, where
P0 satisfies the Lyapunov equation
A'P0+P0A+Q0=0, for any
Q0≥0.
(3)
P0b is obtained by solving
n simultaneous linear equations.
(4) The sets
KiL={
ki-1+Pib}(i=1, 2, …) are nondecreasing subsets in
KR, i.e.
Ki-1L⊂ KiL⊂KR, and its limit as
i tends to infinity equals
KR, Where
ki-1 is any member of
Ki-1L and
Pi satisfies the Lyapunov equation (
A-bki-1')'Pi+Pi(A-bki-1')+Qi=0, for any
Qi≥0.
(5) Let
R(ρ) be the solution of Riccati equation.
A'R(ρ)+R(ρ)A-R(ρ)bb'R(ρ)+ρQ=0, where ρ is scalar parameter of weighting matrix. Then
dR(ρ)/dρ=P(ρ) satisfies the Lyapunov equation
A(ρ)'P(ρ)+P(ρ)A(ρ)+Q=0, Where
A(ρ)=A-bb'R(ρ). And optimal feedback gain
R(ρ)b is obtained as the solution of a nonlinear differential equation.
Since various design specifications such as desired closed loop poles and constraints on state variables are treated directly in gain space, the subsets of
KR can be utilized to design the optimal regulator with those specifications and constraints.
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