I am working on a project that involves modeling a specific nonlinear dynamic system in Xcos. The core of the system is defined by the step response function:
y(t) = y(0) + \frac{Po}{f} \cdot A \cdot (1-\exp(-\frac{t}{\tau}))^\beta
In this equation, \beta, A, and \tau are constants, while Po and f are inputs to the system, and y(t) is the output. My goal is to simulate the dynamic behavior of this system for any step input, particularly focusing on accurately capturing the nonlinear characteristics defined by the power \beta and the interaction between Po and f.
Current Solution in Xcos:
To tackle this challenge, I’ve constructed an Xcos model that attempts to faithfully represent the dynamics of the described system. Here’s an overview of my model’s construction:
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Input Scaling: I’ve employed TK Scale blocks to adjust the values of Po and f dynamically, providing a way to simulate different input conditions.
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Step Input and Conditional Logic: To generate a step input that only activates when Po is positive, I’ve used an EXPRESSION block with the logic ((u1>0)*u2), where u1 represents Po and u2 is a constant 1, simulating the step signal.
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Transfer Function Modeling: The linear part of the system, (1-\exp(-\frac{t}{\tau})), is modeled using a CLR block configured as \frac{1}{\tau s + 1}, representing the transfer function H(s).
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Nonlinear Exponentiation: The output from the CLR block is fed into a POWBLK_f, which raises the signal to the power of \beta, adding the necessary nonlinearity to the model.
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Multiplication and Inversion: This signal is then multiplied by Po and an inverted f, followed by multiplication by A, to simulate the \frac{Po}{f} \cdot A part of the step response function.
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Summation with Initial Condition: Finally, the resulting signal is summed with y(0) to complete the model, aligning with the specified step response function.
Challenges and Considerations:
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The use of an EXPRESSION block for conditional step input generation, while effective, raises concerns about model complexity and readability. I am looking for alternative approaches that might simplify the implementation without compromising the dynamic response accuracy.
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Ensuring the model’s behavior aligns with the theoretical step response function, especially under varying input conditions and the nonlinear dynamics introduced by the power \beta.
Seeking Insights:
I am reaching out to the Scilab community for insights, suggestions, or validation of the approach I’ve taken. My experience with modeling such systems in Xcos is limited, and I would greatly appreciate any advice, especially on simplifying the model or alternative methods to accurately capture the system’s nonlinear dynamics. Are there more efficient ways to simulate the conditional step input and the interaction between Po and f while maintaining the system’s nonlinear characteristics?
Thank you in advance for your time and expertise!
Set Context
Te = 18
Pmax = 27000
Pomax = 135
rho = 998
c = 4187
tau=0.24
beta=60
f = 8
Po = 135
A = 60000*Pmax/(Pomax*rho*c)
Set Parameters
Final integration time 3.0E2
Solver kind Sundials/CVODE - ADAMS - FUNCTIONAL
XCOS Model
- Superblock
- ZCOS File
Stepresponse_Model.zcos (7.8 KB)
- ** Scope Output **
