Given The Following DC Permanent Magnet Motor Parameters ✓ Solved

Given the following DC permanent magnet motor parameters, de

Given the following DC permanent magnet motor parameters, design a cascaded controller to control the motor speed that meets the specified criteria. Parameters: V/rad/s = 0.0772, Nm/A = 0.0772, L = 0.8 mH, B = 0.0003 Nm/(rad/s), T_friction = 0.0756 Nm, TL = 0.1 Nm, supply voltage 42 V, speed/armature limits: current controller output limit 1 V, armature voltage limit 42 V, switching frequency 20 kHz, no-load speed 4000 rpm, no-load current 5 A.

Design steps: 1) Set up the Simulink model of the PM DC motor. 2) Design a torque (current) inner-loop controller. 3) Design a speed outer-loop controller. 4) Simulate with MATLAB/Simulink and verify against the specifications. Criteria: overshoot

Project report should include: description of cascaded controller design; block-by-block model description; final simulation results (speed reference and real, armature current reference and real, armature voltage input); conclusions.

Paper For Above Instructions

Introduction

The problem calls for a cascaded control structure to regulate the speed of a brushed DC motor with a permanent magnet rotor. A cascaded (inner torque/current and outer speed) arrangement decouples electrical dynamics from mechanical dynamics, enabling tighter regulation of motor torque while the outer loop governs speed. This separation helps address parameter uncertainties and external disturbances (load torque) while respecting actuator limits. A principled design follows a modeling framework, inner loop stabilization, outer loop tuning, and verification via simulation. Throughout this paper, we adopt standard PM DC motor modeling conventions and apply PI control laws for both loops, with practical saturation considerations to reflect hardware constraints.

System Modeling (PM DC Motor)

The PM DC motor can be described by a canonical electromechanical model consisting of electrical and mechanical dynamics. Let i_a denote armature current, ω the motor speed (rad/s), v_a the applied armature voltage, R the armature resistance, L the armature inductance, K_e the back-EMF constant (V·s/rad), and K_t the torque constant (Nm/A). The electrical equation is L (di_a/dt) + R i_a + K_e ω = v_a, and the mechanical equation is J (dω/dt) + B ω = K_t i_a − T_L − T_friction, where J is the rotor inertia, B is viscous friction, T_L is load torque, and T_friction models frictional losses. For a PM DC motor, K_e and K_t are related through units, and the given parameters (K_e = K_t = 0.0772) reflect the motor’s electromechanical coupling. In the cascaded controller, the inner loop regulates i_a (and hence torque T_m = K_t i_a) and the outer loop commands speed ω based on a reference ω_r.

Inner-Loop Controller Design (Torque/Current Control)

The inner loop treats the electrical submodel as the plant to be commanded by the current/torque controller. With the mechanical dynamics relatively slower than the electrical dynamics, a proportional-integral (PI) controller is commonly employed to achieve a desired current response and to reject steady disturbances. The inner-loop transfer from voltage command to armature current is shaped by selecting a crossover frequency fc,i well below the switching frequency (e.g., one to two orders lower). The PI gains (Kp,i and Ki,i) are chosen to meet a target phase margin and bandwidth, ensuring sufficient torque tracking with acceptable current ripple. Saturation at the limited current/voltage bounds (e.g., current controller output limited to 1 V, armature voltage limited to 42 V) must be accounted for in the control design and subsequent simulations.

Outer-Loop Controller Design (Speed Control)

The outer loop regulates speed by commanding a torque (current) reference based on speed error eω = ω_r − ω. This loop uses a PI controller to achieve the desired speed response, with attention to the interaction with the inner loop. The outer-loop crossover frequency fc,s is typically lower than fc,i to preserve the cascade’s decoupling and to provide robust performance in the presence of load steps. The resulting current reference i_a,ref is then fed to the inner loop. Practical limits include a 6 A limit (5 × 1.2 A) on the speed-controller output current, a 1 V limit on the current controller’s voltage command, and a 42 V limit on the resulting armature voltage. These limits constrain the achievable rise time and overshoot and must be accounted for in the tuning process.

Design Methodology and Gains Tuning

A practical tuning procedure follows standard control design steps. Step 1: choose fc,i as a frequency lower than the switching frequency by at least one order of magnitude to ensure adequate dwell time for discrete-time implementation. Step 2: compute PI gains for the inner loop to achieve the desired crossover and phase margin (commonly 45–70 degrees). Step 3: select fc,s (often one order lower than fc,i) and compute the outer-loop PI gains to achieve the desired closed-loop speed response while maintaining the inner loop’s stability margin. Step 4: validate through time-domain simulations including the specified test scenarios. If overshoot is excessive, reduce fc,i and/or fc,s; if response time is too slow, increase them, respecting actuation limits and the motor’s physical bandwidth. The hints for gain calculation often include using the electrical time constant τ_e = L/R to set Ki,i ≈ 1/τ_e and Kp,i ≈ R / K_t, then computing outer-loop gains based on the desired speed closed-loop characteristics (rise time, settling time, and damping).

Simulation Plan and Test Scenarios

Simulations should implement a PM DC motor model with the specified parameters and the cascaded PI controllers. Scenarios to validate design requirements include: (1) initial speed and load set to zero at t = 0; (2) a step load applied at t = 1 s; (3) a step speed command from 0 to 200 rad/s at t = 2 s; (4) a step speed command from 200 to 400 rad/s at t = 4 s. Output quantities of interest are the motor speed ω(t) (reference vs real), armature current i_a(t) (reference vs real), and the motor armature voltage v_a(t). The results should demonstrate an overshoot of less than or equal to 20% in the speed response, zero steady-state error in the steady-state regime, and a well-defined rise time within the physical limits of the actuators.

Discussion and Conclusions

Implementing a cascaded PI control framework for PM DC motor speed control provides decoupled and robust performance against load disturbances and parameter variations. The inner current loop ensures fast torque tracking and limits current, while the outer speed loop imposes the desired speed profile with acceptable overshoot and rise time. Careful attention to actuator saturation and the motor’s electrical and mechanical time constants is essential. The resulting control strategy is scalable to different motor parameters and can be extended to include feedforward terms, anti-windup schemes, and adaptive gains to further improve robustness in real hardware experiments.

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