please could you draw a block diagram which represents the system in picture Q8The dynamic equation of a DC motor can be expressed as:dw(t) + bw(t) = K₁i(t)dtwhere omega(t) is the angular velocity of the motor, J is the moment of inertia of the rotor, b is thedamping coefficient of the motor, Kt is the torque constant of the motor, and i(t) is the current flowingthrough the motor.Now using Kirchhoff's voltage law to the circuit of the motor, we can obtain the follow the equationbelow:di(t)V(t) = Ri(t) + L + K₂w(t)dtwhere V(t) is the voltage applied to the motor, R is the resistance of the motor, L is the inductance ofthe motor, and Ke is the back-EMF constant of the motor.By substituting Kt i(t) from the mechanical equation into the electrical equation, we can obtain a singleequation that relates the input voltage V(t) to the output angular velocity omega(t):V(t) - Kew (t)Rdw (t)+ bw(t) = K₂(dtdw (t)dtSimplifying this equation, we get:- + bw(t) = K / V (t) -RKtKe-w (t)RThis equation can be written in the standard form of a second-order differential equation:d² w(t) bdw(t), K+Ke+w (t) +dt² J dt JRKt-V(t)JRExplanation:The equation governing the behavior of a DC motor establishes a connection between the inputvoltage and the output angular velocity, and is a second-order differential equation. The equation isobtained by applying Kirchhoff's voltage law to the motor's electrical circuit and taking intoconsideration its mechanical properties. Writing the equation in the standard form of a second-orderdifferential equation facilitates the analysis and control of the motor. Applying the Laplace transform tothe differential equation and simplifying the expression leads to the transfer function of the motor withthe PID controller. This transfer function mathematically links the Laplace transform of the inputvoltage to the Laplace transform of the output angular velocity, making it a valuable tool for designingand evaluating control systems for the motor. To regulate the motor's output, we can employ a PIDcontroller. The transfer function of the PID controller is given by:K₁G(s) = K₂ + + KasSwhere Kp, Ki, and Kd are the proportional, integral, and derivative gains of the controller, respectively.The transfer function of the motor with the PID controller can be obtained by substituting s withfrac{d}{dt} in the above equation and multiplying both sides by s:KtKes²w (t) + sbw (t) + -w(t) = K₂V (t)Rsb⇒ (s² + b + Keke) w(t) = KV (t)JRThe transfer function of the motor with the PID controller is then given by:G(s)K₂K₂ +S² +We can simplify this expression by multiplying the numerator and denominator by JRs²:G(s) =KtKt+ sbJRJRs²(KJRS + KJR + KaJRs³)G(s)K₁JRsb, KK₂+J JR-(K₂ + + Kas)K₁STaking the Laplace transform of the input voltage VExplanation:The transfer function of the motor with the PID controller is given by:KtJRS²+K₁sb KK K₂ +S+J JR+ Kas)In this system, there are various parameters of the motor denoted by Kt, Ke, R, L, J, and b. Thecontroller has three gains: proportional gain (Kp), integral gain (Ki), and derivative gain (Kd). The inputvoltage V(t) has a Laplace transform V(s), which is a function of the complex frequency variable s. Thetransfer function of the motor, when used with the PID controller, relates V(s) to the Laplacetransform of the output angular velocity omega(s). To express this transfer function as a rationalfunction of s, we apply the Laplace transform to the differential equation of the motor with the PIDcontroller. This transfer function can then be analyzed to determine system stability and performance,and to design the PID controller gains to meet specific performance requirements.

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The transfer function is given as…