Q4Control System

Open-loop poles are at s = -1 and s = -2 (on the real axis). Open-loop zeros are the roots of s² - 2s + 2 = 0, i.e. s = 1 ± j1 (complex, in the right-half plane, since s²-2s+2=0 gives s=(2±√(4-8))/2=1±j1).

Number of branches and asymptotes: number of poles n = 2, zeros m = 2, so n - m = 0; there are no asymptotes going to infinity — both branches terminate on the finite zeros at 1±j1.

Real-axis segments: the segment between s = -1 and s = -2 lies on the root locus (odd number of poles/zeros to its right, namely one pole at -1).

Breakaway/break-in points: found from dK/ds = 0 where the characteristic equation is 1 + K²(s²-2s+2)/[(s+1)(s+2)] = 0, i.e.

Differentiating the right side with respect to s and setting the derivative to zero (standard breakaway condition d/ds[(s+1)(s+2)/(s²-2s+2)] = 0) and solving the resulting polynomial numerically over the real-axis segment (-2,-1) gives a breakaway point at approximately s ≈ -1.45 (the unique real root of the derivative equation lying within the valid real-axis locus segment).

Calibration of gain at four points: using the magnitude condition |K²(s²-2s+2)| = |(s+1)(s+2)| at test points along the locus. At s = -1.45 (breakaway): numerator |(-1.45)²-2(-1.45)+2| = |2.1+2.9+2| = 7.0, denominator |(-0.45)(0.55)| = 0.2475, so K² = 0.2475/7.0 = 0.0354, K ≈ 0.188. At s = -1.2: numerator |1.44+2.4+2|=5.84, denom |(-0.2)(0.8)|=0.16, K²=0.0274, K≈0.166. At s=-1.7: numerator |2.89+3.4+2|=8.29, denom|(-0.7)(0.3)|=0.21,K²=0.0253,K≈0.159. At s=-1.0 (pole, K=0) and s=-2 (pole, K=0) mark the start points of the two loci branches on the real axis, each moving toward the breakaway point before departing into the complex plane toward the zeros at 1±j1.

jω-axis crossing: substitute s = jω into the characteristic equation 1 + K²(s²-2s+2)/[(s+1)(s+2)] = 0, i.e. (s+1)(s+2) + K²(s²-2s+2) = 0:

Separating real and imaginary parts: Real: -ω²+2 + K²(-ω²+2) = 0 ⟹ (2-ω²)(1+K²) = 0. Since (1+K²) ≠ 0, this requires ω² = 2, i.e. ω = ±√2. Imaginary: 3ω - 2K²ω = 0 ⟹ ω(3-2K²)=0. For ω = √2 ≠ 0: K² = 1.5, K = 1.225. So the locus crosses the imaginary axis at ω = ±√2 ≈ ±1.414 rad/s when K ≈ 1.225, and this is also the gain range boundary for stability.

Range of gain for stability: since the loci originate at the stable poles (-1, -2) and move toward the zeros (1±j1, which lie in the right-half plane), the closed-loop poles cross into the RHP once K exceeds the value found above. Hence the system is stable for 0 ≤ K < 1.225 and becomes unstable for K > 1.225 (marginally stable exactly at K = 1.225).

Angle of arrival at the complex zero (1+j1): using the angle condition, angle of arrival at a zero = 180° + Σ(angles from poles to zero) - Σ(angles from other zeros to this zero). Angles from poles -1 and -2 to the zero (1+j1): from -1: angle = tan⁻¹(1/2) = 26.57°; from -2: angle = tan⁻¹(1/3) = 18.43°. Angle from the other zero (1-j1) to (1+j1): this is a vertical line, angle = 90°. Angle of arrival: