State Space Representation

Oct 13, 2025 by wbrum.org 68 views

Introduction

In control theory, a linear state space representation is a fundamental concept used to model and analyze complex systems. It provides a mathematical framework for describing the behavior of a system in terms of its state variables and the inputs applied to it. In this article, we will derive a linear state space representation for a cart with 2 weights and a force applied, which is a classic problem in control theory.

Problem Formulation

The system consists of a cart with two weights, one on each side, and a force applied to the cart. The goal is to derive a linear state space representation for this system, which will allow us to analyze its behavior and design control strategies.

Equations of Motion

To derive the state space representation, we need to start with the equations of motion for the system. The forces acting on the cart are:

The force applied to the cart, denoted by $F$
The weight of the cart, denoted by $m_1g$
The weight of the first weight, denoted by $m_2g$
The weight of the second weight, denoted by $m_3g$

The equations of motion for the cart can be written as:

\begin{aligned} m_1 \ddot{x} &= F - m_1g - m_2g - m_3g \\ m_2 \ddot{y} &= m_2g \\ m_3 \ddot{z} &= m_3g \end{aligned}

where $x$ , $y$ , and $z$ are the positions of the cart, the first weight, and the second weight, respectively.

State Space Representation

To derive the state space representation, we need to define the state variables and the inputs. Let's define the state variables as:

$x_1 = x$
$x_2 = \dot{x}$
$x_3 = y$
$x_4 = \dot{y}$
$x_5 = z$
$x_6 = \dot{z}$

The inputs are the force applied to the cart, denoted by $F$ .

The state space representation can be written as:

\begin{aligned} \dot{x}_1 &= x_2 \\ \dot{x}_2 &= \frac{1}{m_1} (F - m_1g - m_2g - m_3g) \\ \dot{x}_3 &= x_4 \\ \dot{x}_4 &= \frac{1}{m_2} m_2g \\ \dot{x}_5 &= x_6 \\ \dot{x}_6 &= \frac{1}{m_3} m_3g \end{aligned}

Linearization

To obtain a linear state space representation, we need to linearize the system around an operating point. Let's assume that the operating point is at $x_1 = 0$ , $x_2 = 0$ , $x_3 = 0$ , $x_4 = 0$ , $x_5 = 0$ , and $x_6 = 0$ .

The linearized state space representation can be written as:

\begin{aligned} \dot{x}_1 &= x_2 \\ \dot{x}_2 &= \frac{1}{m_1} F \\ \dot{x}_3 &= x_4 \\ \dot{x}_4 &= 0 \\ \dot{x}_5 &= x_6 \\ \dot{x}_6 &= 0 \end{aligned}

Matrix Form

The linear state space representation can be written in matrix form as:

\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \\ \dot{x}_3 \\ \dot{x}_4 \\ \dot{x}_5 \\ \dot{x}_6 \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5 \\ x_6 \end{bmatrix} + \begin{bmatrix} 0 \\ \frac{1}{m_1} F \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}

Conclusion

In this article, we have derived a linear state space representation for a cart with 2 weights and a force applied. The state space representation is a fundamental concept in control theory, and it provides a mathematical framework for describing the behavior of a system in terms of its state variables and the inputs applied to it. The linear state space representation can be used to analyze the behavior of the system and design control strategies.

References

[1] Ogata, K. (2010). Modern Control Engineering. Prentice Hall.
[2] Franklin, G. F., Powell, J. D., & Emami-Naeini, A. (2014). Feedback Control of Dynamic Systems. Pearson Education.

Appendix

The following is a list of the variables used in this article:

$x$ : position of the cart
$y$ : position of the first weight
$z$ : position of the second weight
$F$ : force applied to the cart
$m_1$ : mass of the cart
$m_2$ : mass of the first weight
$m_3$ : mass of the second weight
$g$ : acceleration due to gravity
$x_1$ : position of the cart
$x_2$ : velocity of the cart
$x_3$ : position of the first weight
$x_4$ : velocity of the first weight
$x_5$ : position of the second weight
$x_6$ : velocity of the second weight
Q&A: Deriving a Linear State Space Representation for a Cart with 2 Weights ====================================================================

Q: What is the purpose of deriving a linear state space representation for a cart with 2 weights?

A: The purpose of deriving a linear state space representation for a cart with 2 weights is to provide a mathematical framework for describing the behavior of the system in terms of its state variables and the inputs applied to it. This allows us to analyze the behavior of the system and design control strategies.

Q: What are the state variables in the linear state space representation?

A: The state variables in the linear state space representation are:

$x_1 = x$ : position of the cart
$x_2 = \dot{x}$ : velocity of the cart
$x_3 = y$ : position of the first weight
$x_4 = \dot{y}$ : velocity of the first weight
$x_5 = z$ : position of the second weight
$x_6 = \dot{z}$ : velocity of the second weight

Q: What are the inputs in the linear state space representation?

A: The inputs in the linear state space representation are the force applied to the cart, denoted by $F$ .

Q: How do we linearize the system around an operating point?

A: To linearize the system around an operating point, we need to assume that the operating point is at $x_1 = 0$ , $x_2 = 0$ , $x_3 = 0$ , $x_4 = 0$ , $x_5 = 0$ , and $x_6 = 0$ . We then use the Taylor series expansion to approximate the system around this operating point.

Q: What is the matrix form of the linear state space representation?

A: The matrix form of the linear state space representation is:

\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \\ \dot{x}_3 \\ \dot{x}_4 \\ \dot{x}_5 \\ \dot{x}_6 \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5 \\ x_6 \end{bmatrix} + \begin{bmatrix} 0 \\ \frac{1}{m_1} F \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}

Q: What are the advantages of using a linear state space representation?

A: The advantages of using a linear state space representation are:

It provides a mathematical framework for describing the behavior of the system in terms of its state variables and the inputs applied to it.
It allows us to analyze the behavior of the system and design control strategies.
It is a fundamental concept in control theory and is widely used in many applications.

Q: What are the limitations of using a linear state space representation?

A: The limitations of using a linear state space representation are:

It assumes that the system is linear, which may not be the case in reality.
It assumes that the system is time-invariant, which may not be the case in reality.
It may not be able to capture the nonlinear behavior of the system.

Q: How do we design control strategies using a linear state space representation?

A: To design control strategies using a linear state space representation, we need to use control theory techniques such as pole placement, optimal control, and model predictive control. These techniques allow us to design control strategies that can stabilize the system, track a reference signal, and optimize a performance criterion.

Q: What are some common applications of linear state space representations?

A: Some common applications of linear state space representations include:

Control of mechanical systems such as robots and manipulators.
Control of electrical systems such as power systems and motor drives.
Control of chemical systems such as chemical reactors and distillation columns.
Control of biological systems such as population dynamics and epidemiology.

Q: What are some common tools used to analyze and design linear state space representations?

A: Some common tools used to analyze and design linear state space representations include:

MATLAB and Simulink.
Mathematica and Wolfram Language.
Python and NumPy and SciPy.
C++ and C# and Java.

Q: What are some common challenges in deriving and analyzing linear state space representations?

A: Some common challenges in deriving and analyzing linear state space representations include:

Linearization of nonlinear systems.
Time-varying systems.
Uncertain systems.
Nonlinear control systems.

Q: What are some common techniques used to overcome these challenges?

A: Some common techniques used to overcome these challenges include:

Nonlinear control techniques such as feedback linearization and sliding mode control.
Time-varying control techniques such as adaptive control and robust control.
Uncertain control techniques such as robust control and stochastic control.
Nonlinear control systems such as neural networks and fuzzy logic.