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Loop Shape Goal


Shape open-loop response of feedback loops when using Control System Tuner.


Loop Shape Goal specifies a target gain profile (gain as a function of frequency) of an open-loop response. Loop Shape Goal constrains the open-loop, point-to-point response (L) at a specified location in your control system.

When you tune a control system, the target open-loop gain profile is converted into constraints on the inverse sensitivity function inv(S)  = (I + L) and the complementary sensitivity function T = 1–S. These constraints are illustrated for a representative tuned system in the following figure.

Where L is much greater than 1, a minimum gain constraint on inv(S) (green shaded region) is equivalent to a minimum gain constraint on L. Similarly, where L is much smaller than 1, a maximum gain constraint on T (red shaded region) is equivalent to a maximum gain constraint on L. The gap between these two constraints is twice the crossover tolerance, which specifies the frequency band where the loop gain can cross 0 dB.

For multi-input, multi-output (MIMO) control systems, values in the gain profile greater than 1 are interpreted as minimum performance requirements. Such values are lower bounds on the smallest singular value of the open-loop response. Gain profile values less than one are interpreted as minimum roll-off requirements, which are upper bounds on the largest singular value of the open-loop response. For more information about singular values, see sigma.

Use Loop Shape Goal when the loop shape near crossover is simple or well understood (such as integral action). To specify only high gain or low gain constraints in certain frequency bands, use Minimum Loop Gain Goal or Maximum Loop Gain Goal. When you do so, the software determines the best loop shape near crossover.


In the Tuning tab of Control System Tuner, select New Goal > Target shape for open-loop response to create a Loop Shape Goal.

Command-Line Equivalent

When tuning control systems at the command line, use TuningGoal.LoopShape to specify a loop-shape goal.

Open-Loop Response Selection


  • Shape open-loop response at the following locations


  • Compute response with the following loops open




Desired Loop Shape

Use this section of the dialog box to specify the target loop shape.

  • Pure integrator wc/s

    Check to specify a pure integrator and crossover frequency for the target loop shape. For example, to specify an integral gain profile with crossover frequency 10 rad/s, enter 10 in the Crossover frequency wc text box.

  • Other gain profile

    Check to specify the target loop shape as a function of frequency. Enter a SISO numeric LTI model whose magnitude represents the desired gain profile. For example, you can specify a smooth transfer function (tf, zpk, or ss model). Alternatively, you can sketch a piecewise target loop shape using an frd model. When you do so, the software automatically maps the profile to a smooth transfer function that approximates the desired loop shape. For example, to specify a target loop shape of 100 (40 dB) below 0.1 rad/s, rolling off at a rate of –20 dB/decade at higher frequencies, enter frd([100 100 10],[0 1e-1 1]).

    If you are tuning in discrete time, you can specify the loop shape as a discrete-time model with the same sample time that you are using for tuning. If you specify the loop shape in continuous time, the tuning software discretizes it. Specifying the loop shape in discrete time gives you more control over the loop shape near the Nyquist frequency.


Use this section of the dialog box to specify additional characteristics of the loop shape goal.

  • Enforce loop shape within

    Specify the tolerance in the location of the crossover frequency, in decades. For example, to allow gain crossovers within half a decade on either side of the target crossover frequency, enter 0.5. Increase the crossover tolerance to increase the ability of the tuning algorithm to enforce the target loop shape for all loops in a MIMO control system.

  • Enforce goal in frequency range


  • Stabilize closed loop system


  • Equalize loop interactions


  • Apply goal to




Evaluating Tuning Goals


For Loop Shape Goal, f(x) is given by:


S = D–1[I – L(s,x)]–1D is the scaled sensitivity function.

L(s,x) is the open-loop response being shaped.

D is an automatically-computed loop scaling factor. (If Equalize loop interactions is set to Off, then D = I.)

T = S – I is the complementary sensitivity function.

WS and WT are frequency weighting functions derived from the specified loop shape. The gains of these functions roughly match your specified loop shape and its inverse, respectively, for values ranging from –20 dB to 60 dB. For numerical reasons, the weighting functions level off outside this range, unless the specified gain profile changes slope outside this range. Because poles of WS or WT close to s = 0 or s = Inf might lead to poor numeric conditioning for tuning, it is not recommended to specify loop shapes with very low-frequency or very high-frequency dynamics. For more information about regularization and its effects, see Visualize Tuning Goals.

Implicit Constraints


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