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

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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
 
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  • #2
djulzz1982 said:
The state space vector is thus x = [x1 x2]T
And your question is ##\dots##?
 
  • #3
djulzz1982 said:
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.
 
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djulzz1982 said:
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
$$\underline{x} =\begin{bmatrix}
\dot{x_{1}} \\
\dot{x_{2}}
\end{bmatrix}
=
\begin{bmatrix}
0 & 0 \\
1 & 0
\end{bmatrix}
$$
djulzz1982 said:
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 said:
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
 
  • #5
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.
 

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  • #6
kuruman said:
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
 
  • #7
djulzz1982 said:
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?
 
  • #8
DaveE said:
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).
 

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

What is the state space representation of a 1D point mass floating in space?

The state space representation of a 1D point mass floating in space involves defining the state variables, typically position and velocity. For a point mass \(m\) with position \(x\) and velocity \(v\), the state vector can be defined as \(\mathbf{x} = [x, v]^T\). The state space equations can then be written as:\[\mathbf{\dot{x}} = \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u}\]where \(\mathbf{A}\) is the system matrix, \(\mathbf{B}\) is the input matrix, and \(\mathbf{u}\) is the input vector (thruster forces). For a 1D point mass, \(\mathbf{A} = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}\) and \(\mathbf{B} = \begin{bmatrix} 0 \\ \frac{1}{m} \end{bmatrix}\).

How do lateral thrusters affect the state space model of the system?

Lateral thrusters provide forces that can change the velocity of the point mass. If there are two lateral thrusters, their forces can be represented as \(u_1\) and \(u_2\). The net force \(u\) acting on the point mass is the sum of these forces. In the state space model, the input vector \(\mathbf{u}\) becomes \([u_1 + u_2]\), and the state space equation incorporating these thrusters is:\[\mathbf{\dot{x}} = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}\mathbf{x} + \begin{bmatrix} 0 \\ \frac{1}{m} \end{bmatrix}(u_1 + u_2)\]

What are the assumptions made in the state space representation of this system?

The primary assumptions in this state space representation include:1. The point mass is in a vacuum, so there is no air resistance or friction.2. The mass \(m\) is constant and does not change over time.3. The forces from the lateral thrusters are applied instantaneously and can be directly controlled.4. The system is linear, meaning the superposition principle applies.5. The thrusters provide forces only in the lateral direction, and there are no other external forces acting on the point mass.

How can we control the point mass using the lateral thrusters in the state space framework?

Control of the

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