Time-Domain Goals

For time-domain tuning goals, the tuning-goal plot is a time-domain plot of the relevant system response. The following plot, adapted from the example MIMO Control of Diesel Engine, shows a typical tuning-goal plot for a time-domain disturbance-rejection goal. The dashed lines represent the worst acceptable step response specified in the tuning goal. The solid line shows the corresponding response of the tuned system.

Frequency-Domain Goals

The plots for frequency-domain tuning goals show the target response and the tuned response in the frequency domain. The following plot, adapted from the example Fixed-Structure Autopilot for a Passenger Jet, shows a plot for a gain goal (TuningGoal.Gain at the command line). This tuning goal limits the gain between a specified input and output to a frequency-dependent profile. In the plot, the dashed line shows the gain profile specified in the tuning goal. If the tuned system response (solid line) enters the shaded region, the tuning goal is violated. In this case, the tuning goal is satisfied at all frequencies.

Margin Goals

For information about interpreting tuning-goal plots for stability-margin goals, see Stability Margins in Control System Tuning.

Continuous-Time Gain Profile for Discrete-Time Tuning

When you tune a discrete-time control system, you can specify frequency-dependent tuning goals using discrete-time or continuous-time transfer functions. If you use a continuous-time transfer function, the tuning algorithm discretizes the transfer function before tuning. For instance, suppose that you specify a tuning goal as follows.

W = zpk([],[0 -150 -150],1125000);
Req = TuningGoal.MaxLoopGain('Xloc',W);

Suppose further that you use the tuning goal with systune to tune a discrete-time genss model or slTuner interface. CL is the resulting tuned control system. To examine the result, generate a tuning-goal plot.

viewGoal(Req,CL)

The plot shows W, the continuous-time maximum loop gain that you specified, as a dashed line. The shaded region shows the discretized version of W that systune uses for tuning. The discretized maximum loop gain cuts off at the Nyquist frequency corresponding to the sample time of CL. Near that cutoff, the shaded region diverges from the dashed line.

The plot highlights that sometimes it is preferable to specify tuning goals for discrete-time tuning using discrete-time gain profiles. In particular, specifying a discrete-time profile gives you more control over the behavior of the gain profile near the Nyquist frequency.

Modifications for Numeric Stability

When you use a tuning goal with a frequency-dependent specification, the tuning algorithm uses a frequency-weighting function to compute the normalized value of the tuning goal. This weighting function is derived from the gain profile that you specify. For numeric tractability, weighting functions must be stable and proper. For numeric stability, their dynamics must be in the same frequency range as the control system dynamics. For these reasons, the software might adjust the specified gain profile to eliminate undesirable low-frequency or high-frequency dynamics or asymptotes. The process of modifying the tuning goal for better numeric conditioning is called regularization.

For example, consider the following tracking goal.

R1 = TuningGoal.Tracking('r','y',tf([1 0 0],[1 2 1]));
viewGoal(R1)

Here, the control bandwidth is about 1 rad/s and the gain profile has two zeros at s = 0, which become unstable poles in the weighting function (see TuningGoal.Tracking for details). The regularization moves these zeros to about 0.01 rad/s, and the maximum tracking error levels off at about 10^–3 (0.1%). If you need better tracking accuracy, you can explicitly specify the cutoff frequency in the error profile.

R2 = TuningGoal.Tracking('r','y',tf([1 0 5e-8],[1 2 1]));
viewGoal(R2)
ylim([1e-4,10])

However, for numeric safety, the regularized weighting function always levels off at very low and very high frequencies, regardless of the specified gain profile.

Access the Regularized Functions

When you are working at the command line, you can obtain the regularized gain profile using the getWeight or getWeights commands. For details, see the reference pages for the individual tuning goals for which the tuning algorithm performs regularization:

In Control System Tuner, you cannot view the regularized weighting functions directly. Instead, use the tuning-goal commands to generate an equivalent tuning goal, and use getWeight or getWeights to access the regularized functions.