On the Set of Optimal Feedback Gains in Linear-Quadratic Regulator Problem

Abstract

In this paper some properties of the set K_R, whose members are the optimal feedback gains in linear-quadratic regulator problem, are described, and the technique of constructing subsets of K_R is given.
Controllable and stable plant as x=Ax+bu is treated. The main results are as follows:
(1) Let Q_d=diag(q₁, q₂, …, q_n) be the standard form of weighting matrix Q≥0 in parformance index, then q₁∼q_n are the coefficients of a nonnegative polynomial.
(2) The set K⁰_L={P₀b} is the largest convex cone in K_R, 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 Kⁱ_L={k^i-1+P_ib}(i=1, 2, …) are nondecreasing subsets in K_R, i.e. K^i-1_L⊂ Kⁱ_L⊂K_R, and its limit as i tends to infinity equals K_R, Where k^i-1 is any member of K^i-1_L and P_i satisfies the Lyapunov equation (A-bk^i-1')'P_i+P_i(A-bk^i-1')+Q_i=0, for any Q_i≥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 K_R can be utilized to design the optimal regulator with those specifications and constraints.