Engine Speed Control

This example uses a simple engine speed control application as illustration. The control system consists of a single PID loop and the PID controller gains must be tuned to adequately respond to step changes in the desired speed. Specifically, we want the response to settle in less than 5 seconds with little or no overshoot.

Figure 1: Engine Speed Control Loop

We use the following fourth-order model of the engine dynamics.

load rctExamples Engine
bode(Engine), grid

Specifying the Tunable Elements

We need to tune the four PID gains to achieve the desired performance. Use the tunablePID class to parameterize the PID controller.

PID0 = tunablePID('SpeedController','pid')

PID0 = 
  Tunable continuous-time PID controller "SpeedController" with formula:

             1            s    
  Kp + Ki * --- + Kd * --------
             s          Tf*s+1 

  and tunable parameters Kp, Ki, Kd, Tf.

Type "pid(PID0)" to see the current value and "get(PID0)" to see all properties.

Building a Tunable Model of the Feedback Loop

looptune tunes the generic SISO or MIMO feedback loop of Figure 2. This feedback loop models the interaction between the plant and the controller. Note that this is a positive feedback interconnection.

Figure 2: Generic Feedback Loop

For the speed control loop, the plant $$G$$ is the engine model and the controller $$C$$ consists of the PID and the prefilter $$F$$.

Figure 3: Feedback Loop for Engine Speed Control

To use looptune, create models for $$G$$ and $$C$$ in Figure 3. Assign names to the inputs and outputs of each model to specify the feedback paths between plant and controller. Note that the controller $$C$$ has two inputs: the speed reference "r" and the speed measurement "speed".

F = tf(10,[1 10]);  % prefilter

G = Engine;
G.InputName = 'throttle';
G.OutputName = 'speed';

C0 = PID0 * [F , -1];
C0.InputName = {'r','speed'};
C0.OutputName = 'throttle';

Here C0 is a generalized state-space model (genss) that depends on the tunable PID block PID0.

Tuning the Controller Parameters

You can now use looptune to tune the PID gains subject to a simple control bandwidth requirement. To achieve the 5-second settling time, the gain crossover frequency of the open-loop response should be approximately 1 rad/s. Given this basic requirement, looptune automatically shapes the open-loop response to provide integral action, high-frequency roll-off, and adequate stability margins. Note that you could specify additional requirements to further constrain the design, see "Decoupling Controller for a Distillation Column" for an example.

wc = 1;  % target gain crossover frequency 

[~,C,~,Info] = looptune(G,C0,wc);

Final: Peak gain = 0.965, Iterations = 5
Achieved target gain value TargetGain=1.

The final value is less than 1, indicating that the desired bandwidth was achieved with adequate roll-off and stability margins. looptune returns the tuned controller C. Use getBlockValue to retrieve the tuned value of the PID block.

PIDT = getBlockValue(C,'SpeedController')

PIDT =
 
             1            s    
  Kp + Ki * --- + Kd * --------
             s          Tf*s+1 

  with Kp = 0.00111, Ki = 0.00253, Kd = 0.000359, Tf = 1
 
Name: SpeedController
Continuous-time PIDF controller in parallel form.

Validating Results

Use loopview to validate the design and visualize the loop shaping requirements implicitly enforced by looptune.

clf, loopview(G,C,Info)

Next plot the closed-loop response to a step command in engine speed. The tuned response satisfies our requirements.

T = connect(G,C,'r','speed');  % closed-loop transfer from r to speed
clf, step(T)