Help constructing state-space model of a system

Taking the derivative with respect to tdxdt = [v*cos(theta); v*sin(theta); omega]; % Simplifying the system dynamicsdydx = [px*cos(theta)+(py-1)*sin(theta); -px*sin(theta)+(py-1)*cos(theta)]; % Output equationA = jacobian(dxdt,x); % Jacobian matrix of dxdt with respect to xB = jacobian(dxdt,[v;omega]); % Jacobian matrix of dxdt with respect to uC = jacobian(dydx,x); % Jacobian matrix of dydx with respect to xD = jacobian(dydx,[v;omega]); % Jacobian matrix of dydx with
  • #1
tehipwn
16
0

Homework Statement



A single wheel cat moving on the plane with linear velocity v angular velocity [tex]\omega[/tex] can be modeled by the nonlinear system:

dpx/dt = v*cos([tex]\theta[/tex])

dpy/dt = v*sin([tex]\theta[/tex])

d[tex]\theta[/tex]/dt = [tex]\omega[/tex]

where (px,py) denote the cartesian coordinates of the wheel and [tex]\theta[/tex] its orientation. The system has input u=[v [tex]\omega[/tex]] '.

Construct a state-space model for this system with state

*note: These are both 3x1 matrixes.

[x1] = [[px*cos([tex]\theta[/tex]) + (py-1)*sin([tex]\theta[/tex])]
[x2] = [-px*sin([tex]\theta[/tex]) + (py-1)*cos([tex]\theta[/tex])]
[x3] = [[tex]\theta[/tex]]]

and output y = [x1 x2] '


Homework Equations



dpx/dt = v*cos([tex]\theta[/tex])

dpy/dt = v*sin([tex]\theta[/tex])

d[tex]\theta[/tex]/dt = [tex]\omega[/tex]


The Attempt at a Solution



I don't have much. I'm pretty sure to put the given state into a state-space model I will take the derivative of both sides of the system with respect to t. If that's the correct method, I guess I'm not sure of how to take the derivative of the right side with respect to t since I know the velocity and acceleration functions are functions of time, but the stated system doesn't explicitly show the variable t. Any ideas anyone?
 
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  • #2
I have came up with the following MATLAB code, let me know if it looks corrects if anyone is checking this thread:

syms v theta omega t

pxdt = v*cos(theta); % System dynamics
pydt = v*sin(theta);
thetadt = omega;

px = int(pxdt,t);
py = int(pydt,t);
theta = int(omega,t);
% Given system state:
x = [px*cos(theta)+(py-1)*sin(theta); -px*sin(theta)+(py-1)*cos(theta); theta];
% State-Space Model: Part (a)
dxdt = diff(x,t)
 

1. What is a state-space model?

A state-space model is a mathematical representation of a dynamic system that describes how the system's state changes over time. It consists of a set of differential equations that describe the relationship between the system's inputs, outputs, and internal states.

2. How do you construct a state-space model?

To construct a state-space model, you first need to identify the system's inputs, outputs, and internal states. Then, you can use mathematical techniques such as differential equations or transfer functions to describe the relationships between these variables. Finally, you can use software tools to simulate and analyze the behavior of the model.

3. What are the advantages of using a state-space model?

State-space models offer several advantages over other modeling approaches. They can accurately capture the dynamic behavior of complex systems, they are suitable for both linear and nonlinear systems, and they can easily incorporate uncertainties and disturbances into the model. Additionally, state-space models are flexible and can be easily modified and expanded as needed.

4. What are some real-world applications of state-space models?

State-space models are widely used in a variety of fields, including engineering, physics, economics, and biology. They are used to model and control complex systems such as aircraft, robots, chemical processes, and financial markets. State-space models are also commonly used in forecasting, system identification, and optimization problems.

5. What are some tips for constructing an accurate state-space model?

To construct an accurate state-space model, it is important to have a good understanding of the system and its behavior. This includes identifying the relevant variables, determining the appropriate modeling approach, and validating the model with data. It is also important to consider the model's limitations and potential sources of error, and to continuously refine and improve the model as needed.

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