翻訳お願いします。 The assumption that it is controllable guarantees that we can freely design a feedback gain to assign closed-loop poles to a state-feedback control system at any places in the s plane. We also assume that D = 0 for simplicity. If all components of the state are available, we design a state-feedback gain to asymptotically stabilize the plant at the origin (Itt). It is called the regulator problem The closed-loop system becomes i (t) = (A - BF)(t). We explain two widely used design methods for (8.16) below. 8.3.1 Pole-Placement Method Let the feedback gain in (8.16) be If the plant is given in the controllable canonical form (8.7), then a simple calculation shows that and the characteristic equation of the close-loop system, Φ(s) = |sI - (A-BF)| = 0, is On the other hand, if we choose the poles of the closed-loop system to be A comparison between (8.19) and (8.20) yields the state-feedback gain fi = a; -Фі, i = 0,1,2,..• ,n - 1.（8.21） If the system is not described in the controllable canonical form, then performing a state transformation (8.10) using yields the controllable canonical form. Note that Mc in (8.22) is the controllability matrix (8.14). 8.3.2 Linear-Quadratic Regulator The most widely used method of designing a feedback gain is the optimal control.Minimizing a quadratic performance index {x*(t)Qa(t) + u(t) Ru(t)} dt（8.23） gives the optimal state-feedback gain F = R-'BT P,(8.24) where P is the positive-definite symmetric solution of an algebraic Riccati equation Q and R in (8.23) are weighting matrices. They are usually chosen to be positive-definite matrices (Q > 0 and R > 0). However, if we choose Q to be semi-positive definite (Q ≥ 0), then we have to ensure that (Q'/2, A) is observablet. We choose a large entry on the main diagonal of Q to suppress the corresponding component of the state and use the same strategy tosuppress control input. The solution (8.24) is called the linear-quadratic regulator (LQR)
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