2.2 (Local linearization around a trajectory). A single-wheel cart (unicycle) mov- ing on the plane with linear velocity v and angular velocity @ can be modeled by the nonlinear system Px = v cose, Py = v sin 0, ė = w, (2.11) where (px, py) denote the Cartesian coordinates of the wheel and its orientation. Regard this as a system with input u := [v_ @]' = R². (a) Construct a state-space model for this system with state x = X1 X2 := X3 Px cose + (py - 1) sin -Px sin0 + (py - 1) cos Ө and output y := [x₁ x₂] € R². = (b) Compute a local linearization for this system around the equilibrium point xºq = 0, ueq = 0. (c) Show that w (t) = v(t) = 1, px(t) = sint, py(t) = 1 − cost, 0(t) = t, \t ≥ 0 is a solution to the system. (d) Show that a local linearization of the system around this trajectory results in an LTI system.
2.2 (Local linearization around a trajectory). A single-wheel cart (unicycle) mov- ing on the plane with linear velocity v and angular velocity @ can be modeled by the nonlinear system Px = v cose, Py = v sin 0, ė = w, (2.11) where (px, py) denote the Cartesian coordinates of the wheel and its orientation. Regard this as a system with input u := [v_ @]' = R². (a) Construct a state-space model for this system with state x = X1 X2 := X3 Px cose + (py - 1) sin -Px sin0 + (py - 1) cos Ө and output y := [x₁ x₂] € R². = (b) Compute a local linearization for this system around the equilibrium point xºq = 0, ueq = 0. (c) Show that w (t) = v(t) = 1, px(t) = sint, py(t) = 1 − cost, 0(t) = t, \t ≥ 0 is a solution to the system. (d) Show that a local linearization of the system around this trajectory results in an LTI system.
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Publisher:Robert L. Boylestad
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It is a problem from Linear Systems Theory , chapter 2, Linearization
i wanna detailed process for solving this problem. line by line
![2.2 (Local linearization around a trajectory). A single-wheel cart (unicycle) mov-
ing on the plane with linear velocity v and angular velocity can be modeled by the
nonlinear system
Px = v cos 0,
Py = v sin 0,
ė = w,
(2.11)
where (px, py) denote the Cartesian coordinates of the wheel and its orientation.
Regard this as a system with input u := [v]' € R².
(a) Construct a state-space model for this system with state
X =
X2 :=
X3.
Px cos 0 + (py - 1) sin
-Px sin 0 + (py - 1) cos 0
0
and output y := [x₁ x₂]' € R².
(b) Compute a local linearization for this system around the equilibrium point.xeq;
0, uºq = 0.
=
(c) Show that w (t) = v(t) = 1, px(t) = sint, py(t) = 1 − cost, 0(t) = 1,\t ≥ 0
is a solution to the system.
(d) Show that a local linearization of the system around this trajectory results in an
LTI system.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd0b5e2ea-caee-4914-ba9c-d8a6a9a4be5e%2F286d46df-f340-409f-b78c-8f5fccdc844e%2Fdkskqe_processed.png&w=3840&q=75)
Transcribed Image Text:2.2 (Local linearization around a trajectory). A single-wheel cart (unicycle) mov-
ing on the plane with linear velocity v and angular velocity can be modeled by the
nonlinear system
Px = v cos 0,
Py = v sin 0,
ė = w,
(2.11)
where (px, py) denote the Cartesian coordinates of the wheel and its orientation.
Regard this as a system with input u := [v]' € R².
(a) Construct a state-space model for this system with state
X =
X2 :=
X3.
Px cos 0 + (py - 1) sin
-Px sin 0 + (py - 1) cos 0
0
and output y := [x₁ x₂]' € R².
(b) Compute a local linearization for this system around the equilibrium point.xeq;
0, uºq = 0.
=
(c) Show that w (t) = v(t) = 1, px(t) = sint, py(t) = 1 − cost, 0(t) = 1,\t ≥ 0
is a solution to the system.
(d) Show that a local linearization of the system around this trajectory results in an
LTI system.
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VIEWStep 2: Construct a state-space model
VIEWStep 3: Compute a local linearization for this system around the equilibrium point.
VIEWStep 4: Show that the given condition is a solution to the system
VIEWStep 5: Show that computed local linearization of the system around this trajectory results in an LTI system
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