L = −kp sign(894)8q1:3 - ka (1±8q38q1:3) T L= -kp8q1:3-kaw (7.7) where kp and ka are positive scalar gains. Substituting Eq. (7.7) into Eq. (7.1b) gives the closed-loop system governed by Eq. (7.5) and i = -J¹ ([wx]Jw+kp8q1:3+kaw) The only equilibrium point is [8q3 ] = 0. (7.8) Stability is proven using Lyapunov's direct method. Reference [50] uses a difference between the actual and command quaternions in the definition of the candidate Lyapunov function. Here, a multiplicative approach is employed, which leads to the same result as in [50] but is more intuitive in terms of the quaternion error kinematics. Define the following candidate Lyapunov function: WTJw + kp8q38913+ ½ kp (1-894)² ≥ 0 (7.9) V = == Note that V0 when = 0 and 8q = = Iq, which is the equilibrium point. The time derivative of V is given by V = ½w³ Jw + kp8q|:38ġ1:3 − kp (1 − 894)8ġ4 (7.10) Substituting Eqs. (7.6) and (7.8) into Eq. (7.10) gives V == = - ———½ (w² 8q1:3) [kp + kp8q4 − k p(1 + 894)] — —½±ka ww 0 - (7.11) Thus, the closed-loop system is stable since V ≤0. Asymptotic stability can be proven using LaSalle's theorem (see Sect. 12.2.2). The equality in Eq. (7.11) is given when = 0, where 8q1:3 can be anything. We must check that the system cannot remain in a state where V = 0 while 8q1:30. Equation (7.11) guarantees that lim, ∞ = 0. The closed-loop dynamics in Eq. (7.8) shows that this asymptotic condition can only be achieved
L = −kp sign(894)8q1:3 - ka (1±8q38q1:3) T L= -kp8q1:3-kaw (7.7) where kp and ka are positive scalar gains. Substituting Eq. (7.7) into Eq. (7.1b) gives the closed-loop system governed by Eq. (7.5) and i = -J¹ ([wx]Jw+kp8q1:3+kaw) The only equilibrium point is [8q3 ] = 0. (7.8) Stability is proven using Lyapunov's direct method. Reference [50] uses a difference between the actual and command quaternions in the definition of the candidate Lyapunov function. Here, a multiplicative approach is employed, which leads to the same result as in [50] but is more intuitive in terms of the quaternion error kinematics. Define the following candidate Lyapunov function: WTJw + kp8q38913+ ½ kp (1-894)² ≥ 0 (7.9) V = == Note that V0 when = 0 and 8q = = Iq, which is the equilibrium point. The time derivative of V is given by V = ½w³ Jw + kp8q|:38ġ1:3 − kp (1 − 894)8ġ4 (7.10) Substituting Eqs. (7.6) and (7.8) into Eq. (7.10) gives V == = - ———½ (w² 8q1:3) [kp + kp8q4 − k p(1 + 894)] — —½±ka ww 0 - (7.11) Thus, the closed-loop system is stable since V ≤0. Asymptotic stability can be proven using LaSalle's theorem (see Sect. 12.2.2). The equality in Eq. (7.11) is given when = 0, where 8q1:3 can be anything. We must check that the system cannot remain in a state where V = 0 while 8q1:30. Equation (7.11) guarantees that lim, ∞ = 0. The closed-loop dynamics in Eq. (7.8) shows that this asymptotic condition can only be achieved
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
Related questions
Question
The control law in Eqn. 7.7 has the candidate Lyapunov function as Eqn. 7.9. What would be the Lyapunov function for control law equation in the second image?
Transcribed Image Text:L = −kp sign(894)8q1:3 - ka (1±8q38q1:3)
T
![L= -kp8q1:3-kaw
(7.7)
where kp and ka are positive scalar gains. Substituting Eq. (7.7) into Eq. (7.1b) gives
the closed-loop system governed by Eq. (7.5) and
i = -J¹ ([wx]Jw+kp8q1:3+kaw)
The only equilibrium point is [8q3 ] = 0.
(7.8)
Stability is proven using Lyapunov's direct method. Reference [50] uses a
difference between the actual and command quaternions in the definition of the
candidate Lyapunov function. Here, a multiplicative approach is employed, which
leads to the same result as in [50] but is more intuitive in terms of the quaternion
error kinematics. Define the following candidate Lyapunov function:
WTJw +
kp8q38913+ ½ kp (1-894)² ≥ 0 (7.9)
V = ==
Note that V0 when
= 0 and 8q
=
=
Iq, which is the equilibrium point. The
time derivative of V is given by
V = ½w³ Jw + kp8q|:38ġ1:3 − kp (1 − 894)8ġ4
(7.10)
Substituting Eqs. (7.6) and (7.8) into Eq. (7.10) gives
V
==
=
-
———½ (w² 8q1:3) [kp + kp8q4 − k p(1 + 894)] — —½±ka ww
0
-
(7.11)
Thus, the closed-loop system is stable since V ≤0.
Asymptotic stability can be proven using LaSalle's theorem (see Sect. 12.2.2).
The equality in Eq. (7.11) is given when = 0, where 8q1:3 can be anything.
We must check that the system cannot remain in a state where V = 0 while
8q1:30. Equation (7.11) guarantees that lim, ∞ = 0. The closed-loop
dynamics in Eq. (7.8) shows that this asymptotic condition can only be achieved](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad0d55fe-d83b-4711-86a1-cee8ecea510f%2F75d0317f-9b81-43e0-9375-0284872b6d0f%2Fxr5xh0b_processed.png&w=3840&q=75)
Transcribed Image Text:L= -kp8q1:3-kaw
(7.7)
where kp and ka are positive scalar gains. Substituting Eq. (7.7) into Eq. (7.1b) gives
the closed-loop system governed by Eq. (7.5) and
i = -J¹ ([wx]Jw+kp8q1:3+kaw)
The only equilibrium point is [8q3 ] = 0.
(7.8)
Stability is proven using Lyapunov's direct method. Reference [50] uses a
difference between the actual and command quaternions in the definition of the
candidate Lyapunov function. Here, a multiplicative approach is employed, which
leads to the same result as in [50] but is more intuitive in terms of the quaternion
error kinematics. Define the following candidate Lyapunov function:
WTJw +
kp8q38913+ ½ kp (1-894)² ≥ 0 (7.9)
V = ==
Note that V0 when
= 0 and 8q
=
=
Iq, which is the equilibrium point. The
time derivative of V is given by
V = ½w³ Jw + kp8q|:38ġ1:3 − kp (1 − 894)8ġ4
(7.10)
Substituting Eqs. (7.6) and (7.8) into Eq. (7.10) gives
V
==
=
-
———½ (w² 8q1:3) [kp + kp8q4 − k p(1 + 894)] — —½±ka ww
0
-
(7.11)
Thus, the closed-loop system is stable since V ≤0.
Asymptotic stability can be proven using LaSalle's theorem (see Sect. 12.2.2).
The equality in Eq. (7.11) is given when = 0, where 8q1:3 can be anything.
We must check that the system cannot remain in a state where V = 0 while
8q1:30. Equation (7.11) guarantees that lim, ∞ = 0. The closed-loop
dynamics in Eq. (7.8) shows that this asymptotic condition can only be achieved
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps
Recommended textbooks for you
Elements Of Electromagnetics
Mechanical Engineering
ISBN:
9780190698614
Author:
Sadiku, Matthew N. O.
Publisher:
Oxford University Press
Mechanics of Materials (10th Edition)
Mechanical Engineering
ISBN:
9780134319650
Author:
Russell C. Hibbeler
Publisher:
PEARSON
Thermodynamics: An Engineering Approach
Mechanical Engineering
ISBN:
9781259822674
Author:
Yunus A. Cengel Dr., Michael A. Boles
Publisher:
McGraw-Hill Education
Elements Of Electromagnetics
Mechanical Engineering
ISBN:
9780190698614
Author:
Sadiku, Matthew N. O.
Publisher:
Oxford University Press
Mechanics of Materials (10th Edition)
Mechanical Engineering
ISBN:
9780134319650
Author:
Russell C. Hibbeler
Publisher:
PEARSON
Thermodynamics: An Engineering Approach
Mechanical Engineering
ISBN:
9781259822674
Author:
Yunus A. Cengel Dr., Michael A. Boles
Publisher:
McGraw-Hill Education
Control Systems Engineering
Mechanical Engineering
ISBN:
9781118170519
Author:
Norman S. Nise
Publisher:
WILEY
Mechanics of Materials (MindTap Course List)
Mechanical Engineering
ISBN:
9781337093347
Author:
Barry J. Goodno, James M. Gere
Publisher:
Cengage Learning
Engineering Mechanics: Statics
Mechanical Engineering
ISBN:
9781118807330
Author:
James L. Meriam, L. G. Kraige, J. N. Bolton
Publisher:
WILEY