Homework 1: Linear Muscle Mechanics Is Shown In Figure Below
Homework1 A Linear Muscle Mechanics Is Shown In Figure Below Here
Homework 1-) A linear muscle mechanics is shown in figure below. Here, the elastic element, Cp, is placed in parallel to the viscous damping element R and the contractile element, and the entire parallel combination is placed in series with the elastic element, Cs, and the lumped representation of the muscle mass, m. Derive an expression for the transfer function relating the extension of the muscle, x, to an applied force, F. Convert this transfer function description into the equivalent statespace model.
2-) Figure below displays the equivalent circuit of a short length of squid axon according to the Hodgkin-Huxley model of neuronal electrical activity. The elements shown as circles represent voltage sources that correspond to the Nemst potentials for sodium, potassium, and chloride ions. The resistances are inversely proportional to the corresponding membrane conductances for these three types of ions, while C represents membrane capacitance. Derive the Hodgkin-Huxley equation, i.e., the differential equation that relates the net current flowing through the membrane, I, to the applied voltage across the membrane, V.
3-) Figure below shows a physiological model of system. For this system, drive differential equation of the system. Show the block diagram of the system. Determine the transfer function. Determine the impulse response. Determine Vo(t) for t>0; if input voltage Vi=e-3tu(t).
4-) Figure below shows the block diagram of a sophisticated biomedical device for regulating the dosage of anesthetic gases being delivered to a patient during surgery. Note that the plant and controller are themselves feedback control systems. (a) Derive an expression for the open-loop gain of the overall control system. (b) Derive an expression for the closed-loop gain of the overall control system. (c) If GI = 1, G2 = 2, HI = 1, and H2 = 2, what is the loop-gain of the overall system?
Paper For Above instruction
The given set of problems highlights essential aspects of biomedical and neural systems modeling, involving both mechanical dynamics and electrical circuit analysis. This detailed exploration integrates system identification, transfer function derivation, state-space modeling, neuronal current equations, and feedback control analysis. The comprehensive approach provides profound insights into physiological modeling, essential for biomedical engineering applications.
Derivation of the Muscle Mechanical Transfer Function and State-Space Model
The first task involves modeling a linear muscle with combined passive elastic and viscous elements along with a contractile component, all working in tandem with a mass element. Considering the physical arrangement, the forces and displacements can be described through Newton’s laws and component constitutive equations.
Let x be the extension of the muscle, and F be the applied force. The system components include:
- Viscous damping R, proportional to velocity: \( F_R = R \cdot \dot{x} \)
- Parallel elastic element Cp, proportional to displacement: \( F_{Cp} = Cp \cdot x \)
- Contractile element—allows for an active force component, which can be modeled as a function of muscle activation state (assumed here as \(F_{act}\)).
- Series elastic element Cs, which also acts in the force equilibrium.
- Mass m, which contributes inertia: \( F_{mass} = m \ddot{x} \)
Applying Newton’s second law:
\[
m \ddot{x} + R \dot{x} + (Cp + Cs) x = F
\]
Transforming into the Laplace domain gives the transfer function from force \(F(s)\) to displacement \(X(s)\):
\[
X(s) / F(s) = \frac{1}{m s^2 + R s + (Cp + Cs)}
\]
The states can be defined as:
- State 1: \( x \), the displacement
- State 2: \( v = \dot{x} \), the velocity
State-space equations become:
\[
\dot{x} = v
\]
\[
\dot{v} = \frac{1}{m} [F - R v - (Cp + Cs) x]
\]
\end{pre>
Compact matrix form:
\[
\dot{\mathbf{x}}(t) = A \mathbf{x}(t) + B F(t)
\]
\[
\mathbf{y}(t) = C \mathbf{x}(t) + D F(t)
\]
\]
where
\[
A = \begin{bmatrix} 0 & 1 \\ -\frac{(Cp + Cs)}{m} & -\frac{R}{m} \end{bmatrix},
\quad B = \begin{bmatrix} 0 \\ \frac{1}{m} \end{bmatrix},
\quad C = \begin{bmatrix} 1 & 0 \end{bmatrix},
\quad D = 0
\]
Hodgkin-Huxley Equation Derivation
The Hodgkin-Huxley model describes the electrical behavior across the neuron membrane by balancing capacitive and ionic currents. The total current \(I\) flowing through the membrane can be expressed in terms of the membrane capacitance \(C\) and ionic currents:
\[
I = C \frac{dV}{dt} + I_{Na} + I_{K} + I_{Cl}
\]
Where:
- \(I_{Na} = g_{Na} m^3 h (V - V_{Na})\)
- \(I_{K} = g_{K} n^4 (V - V_{K})\)
- \(I_{Cl} = g_{Cl} (V - V_{Cl})\)
Here, \(g_{Na}\), \(g_{K}\), and \(g_{Cl}\) are maximal conductances, and \(V_{Na}\), \(V_{K}\), \(V_{Cl}\) are reversal potentials. The gating variables \(m, h, n\) follow voltage-dependent kinetics, typically expressed as first-order differential equations.
Substituting these, the core differential equation becomes:
\[
C \frac{dV}{dt} = -g_{Na} m^3 h (V - V_{Na}) - g_{K} n^4 (V - V_{K}) - g_{Cl} (V - V_{Cl}) + I
\]
Rearranged for \(I\):
\[
I = C \frac{dV}{dt} + g_{Na} m^3 h (V - V_{Na}) + g_{K} n^4 (V - V_{K}) + g_{Cl} (V - V_{Cl})
\]
This underlying differential equation describes neuronal membrane response based on ionic dynamics, which are coupled via gating variable kinetics.
Physiological System Differential Equation, Block Diagram, Transfer Function, and Response
Assuming the physiological model is linear and described by differential equations, the typical approach involves defining input \(V_i(t)\), output \(V_o(t)\), and state variables. For example, if the system's dynamics follow:
\[
\dot{x} = A x + B V_i(t)
\]
\[
V_o(t) = C x + D V_i(t)
\]
The block diagram synthesizes these relations, with transfer function \(G(s) = C (sI - A)^{-1} B + D\). To determine the transfer function, one Laplace transforms the differential equations:
\[
G(s) = C (sI - A)^{-1} B + D
\]
For the specified input \(V_i(t) = e^{-3t} u(t)\), the impulse response is obtained through inverse Laplace transform of \(G(s)\), multiplied by the Laplace transform of \(V_i(t)\):
\[
V_i(s) = \frac{1}{s + 3}
\]
\[
V_o(s) = G(s) \times V_i(s)
\]
\[
V_o(t) = \mathcal{L}^{-1} \{ V_o(s) \}
\]
This describes the output response \(V_o(t)\) for \(t > 0\), given the exponential input.
Analysis of the Biomedical Control System: Open Loop, Closed Loop Gain, and Loop Gain
The system configuration involves cascaded control loops. The open-loop transfer function \(G_{OL}(s)\) is the product of plant \(G_P(s)\) and controller \(G_C(s)\). Since both are feedback systems, the overall open-loop gain is:
\[
G_{OL}(s) = G_{plant}(s) \times G_{controller}(s)
\]
The closed-loop transfer function (system gain) is given as:
\[
G_{cl}(s) = \frac{G_{OL}(s)}{1 + G_{OL}(s) H(s)}
\]
\end{pre>
Given the specific gains \(G_I = 1\), \(G_2=2\), \(H_1=1\), \(H_2=2\), the overall loop gain (product of forward and feedback paths at the point of summation) is:
\[
L(s) = G_{I} \times G_{2} \times H_{1} \times H_{2} = 1 \times 2 \times 1 \times 2 = 4
\]
This quantifies the net loop amplification, critical for stability analysis and control design.
Conclusion
These derivations exemplify fundamental biomedical systems modeling techniques, including mechanical transfer functions, electrical neural models, and control feedback analysis. Mastery of these methods enables the development of advanced medical devices and understanding of physiological responses, forming the backbone of biomedical engineering innovations.
References
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