Minimum loop gain constraint for control system tuning
TuningGoal.MinLoopGain object to enforce
a minimum loop gain in a particular frequency band. Use this requirement
with control system tuning commands such as
This requirement 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 green region represents gain profile values that are forbidden by this requirement. 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 requirements,
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 requirement for boosting the gain of a SISO or MIMO feedback
loop. The requirement specifies that the open-loop frequency response
(L) measured at the specified locations exceeds
the minimum gain profile specified by
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 minimum open-loop gain is constrained, specified as a string or cell array of strings. These strings identify one or more loop-opening 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
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 requirement 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 string or cell array of strings. These strings identify one or more loop-opening locations in the control system to tune.
The value of the
Models to which the tuning requirement applies, specified as a vector of indices.
Req.Models = 2:4;
Feedback loops to open when evaluating the requirement, specified as a cell array of strings that identify loop-opening locations. The tuning requirement is evaluated against the open-loop configuration created by opening feedback loops at the locations you identify.
If you are using the requirement to tune a Simulink model
of a control system, then
If you are using the requirement to tune a generalized state-space
Name of the requirement object, specified as a string.
For example, if
Req.Name = 'LoopReq';
When you tune a control system using a
to specify a tuning requirement, the software converts the requirement
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 requirement is a hard constraint.
TuningGoal.MinLoopGain requirement, f(x)
is given by:
WS is the minimum
loop gain profile,
MaxGain. D is
a diagonal scaling (for MIMO loops). S is the sensitivity
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.
Create a requirement that boosts the open-loop gain of a feedback loop to greater than 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 exceed 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 requirement. Display the requirement using
The green region indicates where the requirement is violated, except that gain values less than 1 are not enforced. Therefore, this requirement 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.
Create a requirement 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); viewSpec(Req)
viewSpec confirms that the requirement is correctly specified. You can use this requirement to tune a control system that has a loop-opening location identified as
'X'. Since loop gain values less than 1 are ignored, this requirement specifies minimum gain only below 1 rad/s, with no restriction on loop gain at higher frequency.
Examine a minimum loop gain requirement against the tuned loop gain. A minimum loop gain requirement is converted to a constraint on the gain of the sensitivity function at the requirement location.
To see this relationship between the requirement 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 = ltiblock.pid('C2','pi'); C10 = ltiblock.pid('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 requirements, including a minimum loop gain requirement. 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'; [CL,fSoft] = systune(CL0,[Rtrack,Rreject,Rgain]);
Final: Soft = 1.07, Hard = -Inf, Iterations = 82
TuningGoal.MinLoopGain requirement 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 requirement
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 a requirement that specifies a minimum loop gain of 20 dB (100) at 50 rad/s on the inner loop of the following control system.
Req = TuningGoal.MinLoopGain('X2',50,100);
Configure the requirement to apply to the loop gain of the inner loop measured with the outer loop open.
Req.Openings = 'X2';
Req.Openings tells the tuning algorithm
to enforce this requirement with loops open at the specified location.
By default, tuning using
a stability requirement as well as the minimum loop gain requirement.
Practically, in some control systems it is not possible to achieve
a stable inner loop. When this occurs, remove the stability requirement
for the inner loop by setting the
Req.Stabilize = false;
The tuning algorithm still imposes a stability requirement on the overall tuned control system, but not on the inner loop alone.