DC motor speed controller design

This example will show how to model a DC motor's speed using different approaches in Collimator. The three approaches involve simulating diagrams from a differential equation, a transfer function, and a state variable model. Then, we will design and simulate a PID controller for the DC motor speed.

DC variable speed motor overview

A DC motor combines mechanical and electrical subsystems. The electrical part is an armature coil with a resistance and inductance to drive a field causing motion. The mechanical part can be modeled as an inertial rotor with friction. The DC motor is conceptually modeled as in Figure 1 with the parameters given in Table 1.

Table 1 DC motor model parameters:

\begin{array} {|l|l|}\hline Symbol & Description & Value & Unit \\ \hline R & \text{Shunt Resistance} & 4 & \Omega \\ \hline L & \text{Coil inductance} & .25 & H \\ \hline K & \text{Motor Electromotive/torque constant} & .05 & N.m/A \\ \hline j & \text{Rotor intertia} & .02 & Kg.m2 \\ \hline b & \text{Friction constant} & 0.1 & N.m.s \\ \hline \end{array}

DC variable speed motor model

The DC motor dynamics are given as the following system of differential equations:

$$L\frac{di}{dt} + Ri + K\frac{dϕ}{dt} = u$$

$$J\frac{d^2ϕ}{dt^2} +b \frac{dϕ}{dt} = Ki$$

Applying the Laplace transform for the above equations to get the domain equivalent as:

$$(Ls+R)I(s)+K\dotΦ(s) = U(s)$$

$$(Js+b) \dotΦ (s) = KI(s)$$

Then, both equations are combined by eliminating the current variable to get the open-loop transfer function as:

$$\frac{\dotΦ(s)}{U(s)} = \frac{K}{LJs^2+ (RJ +Lb)s + (Rb + K^2 )}$$

The transfer function can also be converted into an equivalent state space form as:

Equation 6

$$\frac{d}{dt}\begin{bmatrix}Φ\\\dotΦ\\i\end{bmatrix} = \begin{bmatrix}0 & 1 & 0\\0&-\frac{b}{J} &\frac{K}{J}\\0 & -\frac{K}{L} &-\frac{R}{L}\end{bmatrix}\begin{bmatrix}Φ\\\dotΦ\\i\end{bmatrix}+\begin{bmatrix}0\\0\\\frac{1}{L}\end{bmatrix}u $$

Equation 7

$$y=\begin{bmatrix}0 & 1 & 0\end{bmatrix}\begin{bmatrix}Φ\\\dotΦ\\i\end{bmatrix}$$

Simulating Models

Differential Equations Model

Substituting the parameter values in Table 1 within Equations $L\frac{di}{dt} + Ri + K\frac{dϕ}{dt} = u$ and, $J\frac{d^2ϕ}{dt^2}+b\frac{dϕ}{dt} = Ki$ , the DC motor speed model as a system of differential equations is described as:

$$0.25\frac{di}{dt} + 4i+ 0.05\frac{dϕ}{dt} = u$$

$$0.02\frac{d^2ϕ}{dt^2} + 0.1 \frac{dϕ}{dt} = 0.05i$$

The dataflow of the system of differential equations can be formulated as a simulation diagram in Collimator's model editor as shown in Figure 2.

**Figure 2** DC motor differential equations model

The parameters of the models are defined in Figure 3.

After running the simulation, the step response is visualized as in Figure 4.

**Figure 4** Step response for DC motor differential equations model

Transfer Function Model

The obtained transfer function for the DC motor is defined as:

$$\frac{\dotΦ(s)}{U(s)} =\frac{0.05}{0.005s^2+ 0.105s + 0.4025}$$

The transfer function model is implemented as in Figure 5 with the transfer function block is configured as in Figure 6. The visualization of the step response of the transfer function model is shown in Figure 7.

**Figure 5** DC motor transfer function model

**Figure 6** Configuration of the transfer function block

**Figure 7** Step response for DC motor transfer function model

Simulating State Space

By substituting the DC motor model parameters with the state space model of Equations shown in Equation 6 and Equation 7, the following models are obtained:

From Equation 6:

$$\frac{d}{dt}\begin{bmatrix}Φ\\\dotΦ\\i\end{bmatrix} = \begin{bmatrix}0 & 1 & 0\\0& -5 & 2.5\\0 & -0.2 & -16\end{bmatrix}\begin{bmatrix}Φ\\\dotΦ\\i\end{bmatrix}+\begin{bmatrix}0\\0\\4\end{bmatrix}u $$

From Equation 7:

$$y=\begin{bmatrix}0 & 1 & 0\end{bmatrix}\begin{bmatrix}Φ\\\dotΦ\\i\end{bmatrix}$$

The state space model is implemented in Figure 8 and the state space block is configured in Figure 9. The visualization of the step response of the state space model is shown in Figure 10.

**Figure 9** Configuration of the state space block

**Figure 10** Step response for DC motor state space model

DC motor speed controller design

The ideal parallel form the PID controller is defined as:

$$u(t)= K_pe(t)+K_i ∫_0^te(τ)dτ+K_d\frac{d}{dt}e(t)$$

The PID controller simulation model is implemented through Collimator's model editor as shown in Figure 11.

**Figure 11** PID controller for DC motor speed

First, we try an initial PID tuning as $K_p=K_i=K_d=1$. The step response of the closed-loop DC motor speed with the initial PID design is illustrated in Figure 12.

**Figure 12** Step response for PID with $K_p=K_i=K_d=1$

The initial PID tuning yields a very slow step response. Therefore, we increase the proportional gain to $K_p=80$ Then, the step response is plotted in Figure 13. The new PID controller has much faster response. However, there is a significant steady-state error about 9%.

To eliminate the steady-state error, we increase the integral gain to $ K_i= 50.$ This yields the step response in Figure 14. Now, the steady-state error is eliminated.