Minimum loop gain constraint for control system tuning
TuningGoal.MinLoopGain object to enforce
a minimum loop gain in a particular frequency band. Use this tuning
goal with control system tuning commands such as
This tuning goal imposes a minimum gain on the open-loop frequency response (L) at a specified location in your control system. You specify the minimum open-loop gain as a function of frequency (a minimum gain profile). For MIMO feedback loops, the specified gain profile is interpreted as a lower bound on the smallest singular value of L.
When you tune a control system, the minimum gain profile is converted to a minimum gain constraint on the inverse of the sensitivity function, inv(S) = (I + L).
The following figure shows a typical specified minimum gain profile (dashed line) and a resulting tuned loop gain, L (blue line). The shaded region represents gain profile values that are forbidden by this tuning goal. The figure shows that when L is much larger than 1, imposing a minimum gain on inv(S) is a good proxy for a minimum open-loop gain.
only low-gain or high-gain constraints in certain frequency bands.
When you use these tuning goals,
the best loop shape near crossover. When the loop shape near crossover
is simple or well understood (such as integral action), you can use
specify that target loop shape.
a tuning goal for boosting the gain of a SISO or MIMO feedback loop.
The tuning goal specifies that the open-loop frequency response (L)
measured at the specified locations exceeds the minimum gain profile
Req = TuningGoal.MinLoopGain(
You can specify the minimum gain profile as a smooth transfer
function or sketch a piecewise error profile using an
Only gain values greater than 1 are enforced.
For MIMO feedback loops, the specified gain profile is interpreted as a lower bound on the smallest singular value of L.
a minimum gain profile of the form
Req = TuningGoal.MinLoopGain(
loopgain = K/s (integral
action). The software chooses
K such that the gain
gmin at the specified frequency,
Location at which the maximum open-loop gain is constrained, specified as a character vector or cell array of character vectors that identify one or more locations in the control system to tune. What loop-opening locations are available depends on what kind of system you are tuning:
Minimum open-loop gain as a function of frequency.
You can specify
loopgain = frd([100 100 10],[0 1e-1 1]);
When you use an
Only gain values larger than 1 are enforced. For multi-input,
multi-output (MIMO) feedback loops, the gain profile is interpreted
as a lower bound on the smallest singular value of
If you are tuning in discrete time (that is, using a
Frequency of minimum gain
Use this argument to specify a minimum gain profile of the form
Value of minimum gain occurring at
Use this argument to specify a minimum gain profile of the form
Minimum open-loop gain as a function of frequency, specified
as a SISO
The software automatically maps the input argument
Frequency band in which tuning goal is enforced, specified as
a row vector of the form
Req.Focus = [1,100];
Stability requirement on closed-loop dynamics, specified as
Default: 1 (
Toggle for automatically scaling loop signals, specified as
In multi-loop or MIMO control systems, the feedback channels
are automatically rescaled to equalize the off-diagonal terms in the
open-loop transfer function (loop interaction terms). Set
Location at which minimum loop gain is constrained, specified
as a cell array of character vectors that identify one or more analysis
points in the control system to tune. For example, if
The value of the
Models to which the tuning goal applies, specified as a vector of indices.
Req.Models = 2:4;
Feedback loops to open when evaluating the tuning goal, specified as a cell array of character vectors that identify loop-opening locations. The tuning goal is evaluated against the open-loop configuration created by opening feedback loops at the locations you identify.
If you are using the tuning goal to tune a Simulink model
of a control system, then
If you are using the tuning goal to tune a generalized state-space
For example, if
Name of the tuning goal, specified as a character vector.
For example, if
Req.Name = 'LoopReq';
Create a tuning goal that boosts the open-loop gain of a feedback loop to at least a specified profile.
Suppose that you are tuning a control system that has a loop-opening location identified by
PILoop. Specify that the open-loop gain measured at that location exceeds a minimum gain of 10 (20 dB) below 0.1 rad/s, rolling off at a rate of -20 dB/dec at higher frequencies. Use an
frd model to sketch this gain profile.
loopgain = frd([10 10 0.1],[0 1e-1 10]); Req = TuningGoal.MinLoopGain('PILoop',loopgain);
The software converts
loopgain into a smooth function of frequency that approximates the piecewise-specified gain profile. Display the tuning goal using
The dashed line shows the specified the gain profile. The shaded region indicates where the tuning goal is violated, except that gain values less than 1 are not enforced. Therefore, this tuning goal only specifies a minimum gain at frequencies below 1 rad/s.
You can use
Req as an input to
systune when tuning the control system. Then use viewGoal
(Req,T) to compare the tuned loop gain to the minimum gain specified in the tuning goal, where
T represents the tuned control system.
Create a tuning goal that specifies a minimum loop gain profile of the form L = K / s. The gain profile attains the value of -20 dB (0.01) at 100 rad/s.
Req = TuningGoal.MinLoopGain('X',100,0.01); viewGoal(Req)
viewGoal confirms that the tuning goal is correctly specified. You can use this tuning goal to tune a control system that has a loop-opening location identified as
'X'. Since loop gain values less than 1 are ignored, this tuning goal specifies minimum gain only below 1 rad/s, with no restriction on loop gain at higher frequency.
Although the specified gain profile (dashed line) is a pure integrator, for numeric reasons, the gain profile enforced during tuning levels off at very low frequencies, as described in Algorithms. To see the regularized gain profile, expand the axes of the tuning-goal plot.
The shaded region reflects the modified gain profile.
Examine a minimum loop gain tuning goal against the tuned loop gain. A minimum loop gain tuning goal is converted to a constraint on the gain of the sensitivity function at the location specified in the tuning goal.
To see this relationship between the minimum loop gain and the sensitivity function, tune the following closed-loop system with analysis points at
X2. The control system has tunable PID controllers
Create a model of the control system.
G2 = zpk(,-2,3); G1 = zpk(,[-1 -1 -1],10); C20 = tunablePID('C2','pi'); C10 = tunablePID('C1','pid'); X1 = AnalysisPoint('X1'); X2 = AnalysisPoint('X2'); InnerLoop = feedback(X2*G2*C20,1); CL0 = feedback(G1*InnerLoop*C10,X1); CL0.InputName = 'r'; CL0.OutputName = 'y';
Specify some tuning goals, including a minimum loop gain. Tune the control system to these requirements.
Rtrack = TuningGoal.Tracking('r','y',10,0.01); Rreject = TuningGoal.Gain('X2','y',0.1); Rgain = TuningGoal.MinLoopGain('X2',100,10000); Rgain.Openings = 'X1'; opts = systuneOptions('RandomStart',2); rng('default'); % for reproducibility [CL,fSoft] = systune(CL0,[Rtrack,Rreject,Rgain]);
Final: Soft = 2.75, Hard = -Inf, Iterations = 79
TuningGoal.MinLoopGain goal against the corresponding tuned response.
The plot shows the achieved loop gain for the loop at
X2 (blue line). The plot also shows the inverse of the achieved sensitivity function,
S, at the location
X2 (green line). The inverse sensitivity function at this location is given by
inv(S) = I+L. Here,
L is the open-loop point-to-point loop transfer measured at
The minimum loop gain goal
Rgain is constraint on
inv(S), represented in the plot by the green shaded region. The constraint on
inv(S) can be thought of as a minimum gain constraint on
L that applies where the gain of
L (or the smallest singular value of
L, for MIMO loops) is greater than 1.
Create requirements that specify a minimum loop gain of 20 dB (100) at 50 rad/s and a maximum loop gain of -20 dB (0.01) at 1000 rad/s on the inner loop of the following control system.
Create the maximum and minimum loop gain requirements.
RMinGain = TuningGoal.MinLoopGain('X2',50,100); RMaxGain = TuningGoal.MaxLoopGain('X2',1000,0.01);
Configure the requirements to apply to the loop gain of the inner loop measured with the outer loop open.
RMinGain.Openings = 'X1'; RMaxGain.Openings = 'X1';
Req.Openings tells the tuning algorithm to enforce the requirements with a loop open at the specified location. With the outer loop open, the requirements apply only to the inner loop.
By default, tuning using
TuningGoal.MaxLoopGain imposes a stability requirement as well as the minimum or maximum loop gain. Practically, in some control systems it is not possible to achieve a stable inner loop. In that case, remove the stability requirement for the inner loop by setting the
Stabilize property to
RMinGain.Stabilize = false; RMaxGain.Stabilize = false;
When you tune using either of these requirements, the tuning algorithm still imposes a stability requirement on the overall tuned control system, but not on the inner loop alone.
This tuning goal imposes an implicit stability constraint on
the closed-loop sensitivity function measured at
evaluated with loops opened at the points identified in
The dynamics affected by this implicit constraint are the stabilized
dynamics for this tuning goal. The
systuneOptions control the bounds on these
implicitly constrained dynamics. If the optimization fails to meet
the default bounds, or if the default bounds conflict with other requirements,
systuneOptions to change
When you tune a control system using a
the software converts the tuning goal into a normalized scalar value
f(x). Here, x is
the vector of free (tunable) parameters in the control system. The
software then adjusts the parameter values to minimize f(x)
or to drive f(x) below 1 if
the tuning goal is a hard constraint.
is given by:
Here, D is a diagonal scaling (for MIMO loops). S is
the sensitivity function at
Location. WS is
a frequency-weighting function derived from the minimum loop gain
MinGain. The gain of this function roughly
MaxGain for values ranging from –20
dB to 60 dB. For numerical reasons, the weighting function levels
off outside this range, unless the specified gain profile changes
slope outside this range. This adjustment is called regularization.
Because poles of WS close
to s = 0 or s =
lead to poor numeric conditioning of the
problem, it is not recommended to specify gain profiles with very
low-frequency or very high-frequency dynamics.
To obtain WS, use:
WS = getWeight(Req,Ts)
Req is the tuning goal, and
the sample time at which you are tuning (
Ts = 0 for
continuous time). For more information about regularization and its
effects, see Visualize Tuning Goals.
Although S is a closed-loop transfer function, driving f(x) < 1 is equivalent to enforcing a lower bound on the open-loop transfer function, L, in a frequency band where the gain of L is greater than 1. To see why, note that S = 1/(1 + L). For SISO loops, when |L| >> 1, |S | ≈ 1/|L|. Therefore, enforcing the open-loop minimum gain requirement, |L| > |WS|, is roughly equivalent to enforcing |WsS| < 1. For MIMO loops, similar reasoning applies, with ||S|| ≈ 1/σmin(L), where σmin is the smallest singular value.
For an example illustrating the constraint on S, see Minimum Loop Gain as Constraint on Sensitivity Function.