• Notas
• 7mo Semestre
• Control de Sistemas Mecatrónicos

Ejercicio en clase Ubicación de polos

Ejercicio 1

1)

\[p_{LC} = (s + 6)^3 = s^3 + 18s^2 + 108s + 216\]

2)

\[\begin{aligned} p_{LC} &= det(sI - (A - Bk))\\ &= s^3 + s^2(k_2 + k_3 - 16) + s(85 + 2k_1 - 10k_2 -12k_3) + 23k_2 - 12 k_1 + 37k_3 - 150\\ \end{aligned}\]

3)

\[\begin{cases} k_2 + k_3 = 18 + 16\\ 2k_1 - 10k_2 - 12k_3 = 108 - 85\\ -12k_1 + 23k_2 + 37k_3 = 216+150\\ \end{cases}\]

Resolviendo el sistema:

\[k = [1062.5\ \ -847\ \ 881]\]

4)

Comando

Resultado

Ejercicio 2

1)

\[p_{LA} = \det(sI - A) = s^3 - 6s^2 + 9s - 4\]

2)

\[P^{-1} = [B\ \ AB\ \ A^2B]\begin{bmatrix} 1&-6&9\\ 0&1&-6\\ 0&0&1\\ \end{bmatrix} = \left[\begin{array}{ccc} 5 & 85 & -15\\ 20 & 35 & -55\\ 20 & -210 & 115 \end{array}\right]\] \[P = \left[\begin{array}{ccc} \frac{301}{10125} & \frac{53}{2025} & \frac{166}{10125}\\ \frac{136}{10125} & -\frac{7}{2025} & \frac{1}{10125}\\ \frac{196}{10125} & -\frac{22}{2025} & \frac{61}{10125} \end{array}\right]\]

3)

\[p_{LC} = (s + 5 - 2j)(s + 5 + 2j)(s + 4) = s^3 + 14s^2 + 69s + 116\]

4)

\[\bar{k} = [14 + 6\ \ 69 - 9\ \ 116 + 4] = [20\ \ 60\ \ 120]\]

5)

\[k = \bar{k}P = [3.7235\ \ -0.9877\ \ 1.0568]\]

6)

Comando

Resultado

Ejercicio 3

1)

\[p_{LC} = (s + 3 - 2j)(s + 3 + 2j)(s + 8) = s^3 + 14s^2 + 61s + 104\]

2)

\[\begin{aligned} p_{LC} &= det(sI - (A - Bk))\\ &= (^3 + k3s^2 + (2k2 - k1 - k3 - 2)s + 3k1 - 15 \end{aligned}\]

3)

\[\begin{cases} k_3 = 14\\ 2k2 - k1 - k3 - 2= 61\\ 3k_1 - 15 = 104\\ \end{cases}\]

Resolviendo el sistema:

\[k = [\dfrac{119}{3}\ \ \dfrac{175}{3}\ \ 14]\]

4)

Comando

Resultado

Ejercicio 4

1)

\[p_{LA} = \det(sI - A) = s^6 - 32s^4 + 128s^2\]

2)

\[\begin{aligned} P^{-1} &= [B\ \ AB\ \ A^2B\ \ A^3B\ \ A^4B\ \ A^5B]\left[\begin{array}{cccccc} 1 & 0 & -32 & 0 & 128 & 0\\ 0 & 1 & 0 & -32 & 0 & 128\\ 0 & 0 & 1 & 0 & -32 & 0\\ 0 & 0 & 0 & 1 & 0 & -32\\ 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right]\\ &= \left[\begin{array}{cccccc} 0 & -1 & 0 & 16 & 0 & 0\\ -1 & 0 & 16 & 0 & 0 & 0\\ 0 & 0 & 0 & 16 & 0 & 0\\ 0 & 0 & 16 & 0 & 0 & 0\\ 0 & 1 & 0 & -32 & 0 & 128\\ 1 & 0 & -32 & 0 & 128 & 0 \end{array}\right] \end{aligned}\] \[P = \left[\begin{array}{cccccc} 0 & -1 & 0 & 1 & 0 & 0\\ -1 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & \frac{1}{16} & 0 & 0\\ 0 & 0 & \frac{1}{16} & 0 & 0 & 0\\ 0 & \frac{1}{128} & 0 & \frac{1}{128} & 0 & \frac{1}{128}\\ \frac{1}{128} & 0 & \frac{1}{128} & 0 & \frac{1}{128} & 0 \end{array}\right]\]

3)

\[\begin{aligned} p_{LC} &= (s + 3 - 2j)(s + 3 + 2j)(s + 5 - 4j)(s + 5 + 4j)(s + 4)(s + 6)\\ &= s^6 + 26*s^5 + 298*s^4 + 1900*s^3 + 7029*s^2 + 14354*s + 12792\\ \end{aligned}\]

4)

\[\bar{k} = [\tilde{a}_1 - a_1\ \ \ldots\ \ \tilde{a}_6 - a_6] = [26 \ \ 330 1900\ \ 6901 \ \ 14354 \ \ 12792]\]

5)

\[k = \bar{k}P = [-230.0625\ \ 86.1406\ \ 861.2500\ \ 256.8906\ \ 99.9375 112.1406]\]

6)

Comando

Resultado