State Space Representation of a 1D Point Mass Floating in Space and Actuated by Two Lateral Thrusters

[djulzz1982]

Homework Statement

. A point-mass not subjected to gravity is "floating" in space.
. The point-mass has a mass m, and has two lateral thrusters, opposite one another.
. The first thruster generates a max force T1, the second thruster generates a max force T2.
. For the sake of problem formulation, the point-mass can move along the x direction.
. Assume no friction (the point-mass is in space).
1) Write the equations of motion (EOM) for the 1D point-mass
2) Convert the derived EOS(s) to a state space representation
3) If feasible, design an Linear Quadratic Regulator (LQR), that drives the point-mass from x(0) = 0 to x(t) = 4

Relevant Equations

According to Newton's 2nd Law, Sum of forces = m . a, so
(T1-T2) = m a, where:
- T1 is the maximum force generated by the first thruster
- T2 is the maximum force generated by the second thruster
- a is the point-mass' acceleration, so a = d^2(x) / d^2(t).

(u₁*T₁) + (u₂*T₂) = m (x dot dot), [1.1]

---

where (x dot dot) is the 2nd derivative of the point-mass position with respect to time, u₁ is the control input for the 1st thruster, u₂ is the control input for the second thruster.
Rearranging equation 1.1 yields

---

(x dot dot) = (T₁/m)*u₁+ (T2/m)*u₂ [1.2]

---

In order to design an LQR controller for such system, its state space formulation is required, as doing so allows to determine if the system is controllable.
The following states are thus defined

---

- x₁ = x dot ↔ x dot dot = x₁ dot
- x₂ = x ↔ x₂ dot = x₁ [1.3]

---

Rearranging equation [1.2] with the state space variables defined in [1.3] yields:
- x₁ dot = (T₁/m)*u₁ + (T₂/m)*u₂
- x₂ dot = x₁ [1.4]

---

The state space vector is thus x = [x₁ x₂]^T

 

[kuruman]

The state space vector is thus x = [x₁ x₂]^T

And your question is $\dots$?

 

[DaveE]

m (x dot dot)

Kind of hard to read. You'll get better response from us if you use LaTex. It's not hard to learn for basic stuff. There's a nice guide available at the link below your post window.

Also, it would be good to reserve "*" for inner products (or dot products) when vectors might be involved. There's an icon above the post window that looks like a little greek temple that you can use to insert some symbols, like "⋅"; or, better yet LaTex.

For example $(u_1T_1) + (u_2T_2) =m \ddot {x}$
Or $(\vec u_1 \cdot \vec T_1) + (\vec u_2 \cdot \vec T_2) =m \ddot {x}$
Or $(u_1 \vec T_1) + ( u_2 \vec T_2) =m \ddot { \vec x}$

No worries, just some suggestions.

 

[djulzz1982]

Homework Statement: . A point-mass not subjected to gravity is "floating" in space.
. The point-mass has a mass m, and has two lateral thrusters, opposite one another.
. The first thruster generates a max force T1, the second thruster generates a max force T2.
. For the sake of problem formulation, the point-mass can move along the x direction.
. Assume no friction (the point-mass is in space).
1) Write the equations of motion (EOM) for the 1D point-mass
2) Convert the derived EOS(s) to a state space representation
3) If feasible, design an Linear Quadratic Regulator (LQR), that drives the point-mass from x(0) = 0 to x(t) = 4
Relevant Equations: According to Newton's 2nd Law, Sum of forces = m . a, so
(T1-T2) = m a, where:
- T1 is the maximum force generated by the first thruster
- T2 is the maximum force generated by the second thruster
- a is the point-mass' acceleration, so a = d^2(x) / d^2(t).

(u₁*T₁) + (u₂*T₂) = m (x dot dot), [1.1]

---

where (x dot dot) is the 2nd derivative of the point-mass position with respect to time, u₁ is the control input for the 1st thruster, u₂ is the control input for the second thruster.
Rearranging equation 1.1 yields

---

(x dot dot) = (T₁/m)*u₁+ (T2/m)*u₂ [1.2]

---

In order to design an LQR controller for such system, its state space formulation is required, as doing so allows to determine if the system is controllable.
The following states are thus defined

---

- x₁ = x dot ↔ x dot dot = x₁ dot
- x₂ = x ↔ x₂ dot = x₁ [1.3]

---

Rearranging equation [1.2] with the state space variables defined in [1.3] yields:
- x₁ dot = (T₁/m)*u₁ + (T₂/m)*u₂
- x₂ dot = x₁ [1.4]

---

The state space vector is thus x = [x₁ x₂]^T

$$\underline{x} =\begin{bmatrix}
\dot{x_{1}} \\
\dot{x_{2}}
\end{bmatrix}
=
\begin{bmatrix}
0 & 0 \\
1 & 0
\end{bmatrix}
$$

Homework Statement: . A point-mass not subjected to gravity is "floating" in space.
. The point-mass has a mass m, and has two lateral thrusters, opposite one another.
. The first thruster generates a max force T1, the second thruster generates a max force T2.
. For the sake of problem formulation, the point-mass can move along the x direction.
. Assume no friction (the point-mass is in space).
1) Write the equations of motion (EOM) for the 1D point-mass
2) Convert the derived EOS(s) to a state space representation
3) If feasible, design an Linear Quadratic Regulator (LQR), that drives the point-mass from x(0) = 0 to x(t) = 4
Relevant Equations: According to Newton's 2nd Law, Sum of forces = m . a, so
(T1-T2) = m a, where:
- T1 is the maximum force generated by the first thruster
- T2 is the maximum force generated by the second thruster
- a is the point-mass' acceleration, so a = d^2(x) / d^2(t).

(u₁*T₁) + (u₂*T₂) = m (x dot dot), [1.1]

---

where (x dot dot) is the 2nd derivative of the point-mass position with respect to time, u₁ is the control input for the 1st thruster, u₂ is the control input for the second thruster.
Rearranging equation 1.1 yields

---

(x dot dot) = (T₁/m)*u₁+ (T2/m)*u₂ [1.2]

---

In order to design an LQR controller for such system, its state space formulation is required, as doing so allows to determine if the system is controllable.
The following states are thus defined

---

- x₁ = x dot ↔ x dot dot = x₁ dot
- x₂ = x ↔ x₂ dot = x₁ [1.3]

---

Rearranging equation [1.2] with the state space variables defined in [1.3] yields:
- x₁ dot = (T₁/m)*u₁ + (T₂/m)*u₂
- x₂ dot = x₁ [1.4]

---

The state space vector is thus x = [x₁ x₂]^T

Homework Statement: . A point-mass not subjected to gravity is "floating" in space.
. The point-mass has a mass m, and has two lateral thrusters, opposite one another.
. The first thruster generates a max force T1, the second thruster generates a max force T2.
. For the sake of problem formulation, the point-mass can move along the x direction.
. Assume no friction (the point-mass is in space).
1) Write the equations of motion (EOM) for the 1D point-mass
2) Convert the derived EOS(s) to a state space representation
3) If feasible, design an Linear Quadratic Regulator (LQR), that drives the point-mass from x(0) = 0 to x(t) = 4
Relevant Equations: According to Newton's 2nd Law, Sum of forces = m . a, so
(T1-T2) = m a, where:
- T1 is the maximum force generated by the first thruster
- T2 is the maximum force generated by the second thruster
- a is the point-mass' acceleration, so a = d^2(x) / d^2(t).

(u₁*T₁) + (u₂*T₂) = m (x dot dot), [1.1]

---

where (x dot dot) is the 2nd derivative of the point-mass position with respect to time, u₁ is the control input for the 1st thruster, u₂ is the control input for the second thruster.
Rearranging equation 1.1 yields

---

(x dot dot) = (T₁/m)*u₁+ (T2/m)*u₂ [1.2]

---

In order to design an LQR controller for such system, its state space formulation is required, as doing so allows to determine if the system is controllable.
The following states are thus defined

---

- x₁ = x dot ↔ x dot dot = x₁ dot
- x₂ = x ↔ x₂ dot = x₁ [1.3]

---

Rearranging equation [1.2] with the state space variables defined in [1.3] yields:
- x₁ dot = (T₁/m)*u₁ + (T₂/m)*u₂
- x₂ dot = x₁ [1.4]

---

The state space vector is thus x = [x₁ x₂]^T

 

[djulzz1982]

I wrote the potential solution to the problem, showing that with the given problem statement, LQR, PID, or any other control approach will not allow the system to be controlled.
Please check out the attached PDF, and please provide feedback if you can.

 

[djulzz1982]

And your question is $\dots$?

I posted the answer to the questions:
1) what is the state space formulation
2) is the system controllable
all in the attached PDF.
Regards

 

[DaveE]

I posted the answer to the questions:
1) what is the state space formulation
2) is the system controllable
all in the attached PDF.

So then why would we want to go get it and read it? Is it just another example of a solved HW problem (hint - it's your HW problem, not ours), or is there something particularly interesting about it?

 

[djulzz1982]

So then why would we want to go get it and read it? Is it just another example of a solved HW problem (hint - it's your HW problem, not ours), or is there something particularly interesting about it?

I am not sure of my answer, that's why I posted it. You are welcome to ignore it. And by solved, I solved it (hint).