複選題

以下第二十四到第二十八題為多選題, 各題相互獨立。

24.以下關於回授控制機制的敘述, 何者為真?
(A) 可讓不穩定系統變穩定
(B) 可讓穩定系統變不穩定
(C) 可減少系統抵禦外來干擾的能力
(D) 可改變系統零點的位置
(E) 可增加系統的相位 邊界

25.以下關於 PD 控制器的敘述, 何者為真? (A) 通常會增加穩定度邊界 (B) 適合抑制高頻雜訊 (C) 通常會降低步階響應之上升時間 (D) 通常可增加欠阻尼二階系統之阻尼係數 (E) 是一種相 位超前補償器

26.以下關於 PI 控制器的敘述, 何者為真? (A) 通常會增加穩定度邊界 (B) 不可以消除或改善步階響應之穩態誤差 (C) 是一種相位落後補償器 (D) 不適合抑制低頻雜訊 (E) 可用來降低步階響 應之最大超越量

27.以下敘述何者為真? (A) 相對階數為 1 的穩定系統都有無窮大的增益邊界 (B) 相對階數為 2 的極小相位穩定系統有無窮大的增益邊界 (C) 任何穩定的極小相位系統都有無窮大的增益邊界 (D) 任何穩定系統都有可能有無窮大的相位邊界 (E) 回授系統中只要有訊號傳輸延遲, 系統就不可能有無窮大的的相位邊界

28.以下關於線性非時變狀態變數系統 (=Ax(t)+Bu(t), y(t)=Cx(t)+Du(t)) 的敘述, 何者為真? (A) u 到 y 之轉移函數必為 proper (B) 若該系統為可控制 (controllable), 則 u 到 y 之轉移函數的階數必然與狀態變數之個數相同 (C) 若該系統為可觀測 (observable), 則 u 到 y 之轉移函數的階數必然沒有不穩定的零點 (D) 系統的可控制性 (controllability) 無法由回授機制來改變 (E) 系統的可控制性 (controllability) 與可觀測性 (observability) 之間沒有任何必然的關聯

1. When using the Gaussian elimination to solve a linear equation Ax = b, elementary row operations (or multiplication by elementary matrices) are applied to the augmented matrix [A b]. Actually, there are many places in linear algebra where such a technique plays its role； e.g., the null space of A can be determined by setting b = 0. Which of the following cannot be determined by applying such a technique? (A) the range of A (B) the inverse of A (if it is nonsingular) (C) the QR factorization of A (D) the LU factorization of A (if it is square) (E) the determinant of A (if it is square).

2. The linear combination of a set of vectors is an essential element in linear algebra. We say a set V is invariant under linear combination if the implication "∀n ∈ and any set of vectors {v₁, ..., } ⊂ V ⇒ the set of all linear combinations {c₁v₁ + ... + } ⊂ V" holds. And we say a mapping L defined on a set X is invariant under linear combination if the form of linear combination is unchanged under L, or more precisely the statement ", and any set of vectors {x₁, ..., } ⊂ X, the identity L(c₁x₁ + ... + ) = c₁L(x₁) + ... + holds" is true. Which one of the following statements related to linear combination is false (A)Let S be a subset of a vector space V. Then S is a subspace of V if S is invariant under linear combination.  (B) Let {V₁, ..., } be a set of k subspaces of a vector space W and denote span{V₁, ..., } as the set of all linear combinations of the form c₁v₁ + ... + with each vᵢ chosen freely from . Then span{V₁, ..., } is also a subspace of W with dim(span{V₁, ..., }) = dim(V₁) + ... + dim()  (C) Let A and B be two matrices and denote C := AB. Then each column of C is a linear combination of all columns of A, and so rank≤ rank is implied. (D) A mapping L between two vector spaces is a linear transformation if and only if it is invariant under linear combination. (E) Let (V, (. ,. )ᵥ) be an inner product space. Then (. , .)ᵥ is invariant under linear combination at either one of its two arguments.

3. Consider the system represented by the differential equation ẍ+ẍ+kx. Which of the following is true? (A) the system is critically damped when k = 1 (B) the system is underdamped when k = 1/2 (C) the system is overdamped when k = 0.3 (D) the damping ratio of the system is increased when increasing k (E) none of the above

4. Consider the differential equation ẍ + bẋ + kx = cos(ωt) and its sinusoidal solution . Which of the following is true? (A) for a fixed b, the amplitude of is maximized when k = ω² (B) for a fixed b, the maximal amplitude of is 1/b (C) for fixed b and k, the amplitude of always grows as ω increases (D) the period of is 2πω (E) the period of depends on b and k.

5. Consider the differential equation tẋ + x = cos(ωt). Which of the following is true? (A) the general solution is sin(ωt) + C, C is a constant. (B) the particular (forced) solution is periodic with period ω (C) the solution is NOT bounded when t grows to infinity (D) If x(π/ω) = 1, then the solution is (sin(ωt) + π)/(ωt) (E) none of the above

複選題6. Consider a system whose dynamics is governed by a linear constant-coefficient ODE. Suppose the impulse response of this system is . Which of the following is true? (A) the system is order (B) the characteristic equation of the system has a pair of complex roots (C) the unit step response has oscillatory behavior (D) the unit step response does NOT converge to a constant value (E) the unit step response is equal to /3

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