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).
 

1. How is the state space representation of a 1D point mass floating in space derived?

The state space representation of a 1D point mass floating in space is derived by defining the state variables, input variables, and dynamics of the system. The state variables typically include the position and velocity of the point mass, while the input variables represent the forces applied by the lateral thrusters. The dynamics equations are then used to describe how the state variables change over time in response to the inputs.

2. What are the advantages of using state space representation for modeling a 1D point mass system?

State space representation allows for a compact and systematic way to model the behavior of a 1D point mass system. It provides a clear separation between the state variables and inputs, making it easier to analyze and control the system. Additionally, state space representation can easily be extended to higher dimensions and more complex systems.

3. How are the lateral thrusters actuated in the state space representation?

In the state space representation, the lateral thrusters are typically modeled as input variables that directly affect the dynamics of the system. By applying forces through the thrusters, the position and velocity of the point mass can be controlled and manipulated. The thrusters can be actuated individually or together to achieve different motion trajectories.

4. Can the state space representation of a 1D point mass system be used for control purposes?

Yes, the state space representation of a 1D point mass system is commonly used for control purposes. By designing appropriate control laws based on the state variables and inputs, the behavior of the system can be regulated and stabilized. Control algorithms such as PID controllers or state feedback controllers can be implemented to achieve desired performance objectives.

5. How can the state space representation of a 1D point mass system be simulated or implemented in practice?

The state space representation of a 1D point mass system can be simulated using numerical integration techniques such as Euler's method or Runge-Kutta methods. These simulations can be performed in software tools like MATLAB or Python. In practice, the state space representation can also be implemented in real-time control systems using microcontrollers or digital signal processors to actuate the lateral thrusters and stabilize the system.

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