Robust controller design using µ-synthesis
[k,clp,bnd] = dksyn(p,nmeas,ncont) [k,clp,bnd] = dksyn(p,nmeas,ncont,opt) [k,clp,bnd,dkinfo] = dksyn(p,nmeas,ncont,...) [k,clp,bnd,dkinfo] = dksyn(p,nmeas,ncont,prevdkinfo,opt) [...] = dksyn(p)
[k,clp,bnd] = dksyn(p,nmeas,ncont) synthesizes
a robust controller
k for the uncertain open-loop
p via the D-K or D-G-K algorithm for
p is an uncertain state-space
nmeas outputs and
p are assumed to be the measurement and control
k is the controller,
the closed-loop model and
bnd is the robust closed-loop
bnd are related as follows:
clp = lft(p,k); bnd1 = dksynperf(clp); bnd = 1/bnd1.LowerBound;
[k,clp,bnd] = dksyn(p,nmeas,ncont,opt) specifies
opt for the D-K or D-K-G
[k,clp,bnd,dkinfo] = dksyn(p,nmeas,ncont,...) returns
a log of the algorithm execution in
dkinfo. dkinfo is
an N-by-1 cell array where N is the total number
of iterations performed. The
ith cell contains
a structure with the following fields:
Robust performance bound on the closed-loop system (
Left D-scale, an
Right D-scale, an
Offset G-scale, an
Right G-scale, an
Center G-scale, an
Upper and lower µ bounds, an
Structure returned from
[k,clp,bnd,dkinfo] = dksyn(p,nmeas,ncont,prevdkinfo,opt)
allows you to use information from a previous dksyn iteration. prevdkinfo is
a structure from a previous attempt at designing a robust controller
prevdkinfo is used
dksyn starting iteration is not 1 (
= 1) to determine the correct D-scalings to initiate the
[...] = dksyn(p) takes
uss object that has two-input/two-output partitioning
as defined by
The following statements create a robust performance control design for an unstable, uncertain single-input/single-output plant model. The nominal plant model, G, is an unstable first order system .
G = tf(1,[1 -1]);
The model itself is uncertain. At low frequency, below 2 rad/s,
it can vary up to 25% from its nominal value. Around 2 rad/s the percentage
variation starts to increase and reaches 400% at approximately 32
rad/s. The percentage model uncertainty is represented by the weight
corresponds to the frequency variation of the model uncertainty and
the uncertain LTI dynamic object
Wu = 0.25*tf([1/2 1],[1/32 1]); InputUnc = ultidyn('InputUnc',[1 1]);
The uncertain plant model
the model of the physical system to be controlled.
Gpert = G*(1+InputUnc*Wu);
The robust stability objective is to synthesize a stabilizing
LTI controller for all the plant models parameterized by the uncertain
Gpert. The performance objective is
defined as a weighted sensitivity minimization problem. The control
interconnection structure is shown in the following figure.
The sensitivity function, S, is defined as
P is the plant model
K is the controller. A weighted sensitivity
minimization problem selects a weight
corresponds to the inverse of the desired sensitivity
function of the closed-loop system as a function of frequency. Hence
the product of the sensitivity weight
Wp and actual
closed-loop sensitivity function is less than 1 across all frequencies.
The sensitivity weight
Wp has a gain of 100 at
low frequency, begins to decrease at 0.006 rad/s, and reaches a minimum
magnitude of 0.25 after 2.4 rad/s.
Wp = tf([1/4 0.6],[1 0.006]);
The defined sensitivity weight
that the desired disturbance rejection should be at least 100:1 disturbance
rejection at DC, rise slowly between 0.006 and 2.4 rad/s, and allow
the disturbance rejection to increase above the open-loop level, 0.25,
at high frequency.
When the plant model is uncertain, the closed-loop performance
objective is to achieve the desired sensitivity function for all plant
models defined by the uncertain plant model,
The performance objective for an uncertain system is a robust performance
objective. A block diagram of this uncertain closed-loop system illustrating
the performance objective (closed-loop transfer function from d→e)
From the definition of the robust performance control objective,
the weighted, uncertain control design interconnection model, which
includes the robustness and performance objectives, can be constructed
and is denoted by
P. The robustness and performance
weights are selected such that if the robust performance structure
bnd, of the closed-loop uncertain
clp, is less than 1 then the performance
objectives have been achieved for all the plant models in the model
You can form the uncertain transfer matrix
[e; y] using the following commands.
P = [Wp; 1 ]*[1 Gpert]; [K,clp,bnd] = dksyn(P,1,1); bnd
bnd = 0.6806
K achieves a robust performance µ value
bnd of about 0.68. Therefore you have achieved the robust
performance objectives for the given problem.
You can use the
robgain command to analyze
the closed-loop robust performance of
[rpmarg,rpmargunc] = robgain(clp,1);
There are two shortcomings of the D-K iteration control design procedure:
Calculation of the structured singular value µΔ(·) is approximated by its upper bound. This is not a serious problem because the value of µ and its upper bound are often close.
The D-K iteration is not guaranteed to converge to a global, or even local minimum. This is a serious problem, and represents the biggest limitation of the design procedure.
In spite of these drawbacks, the D-K iteration control design technique appears to work well on many engineering problems. It has been applied to a number of real-world applications with success. These applications include vibration suppression for flexible structures, flight control, chemical process control problems, and acoustic reverberation suppression in enclosures.
Control of Spring-Mass-Damper Using Mixed mu-Synthesis
dksyn synthesizes a robust
controller via D-K iteration. The D-K iteration procedure is an approximation
to µ-synthesis control design. The objective of µ-synthesis
is to minimize the structure singular value µ of the corresponding
robust performance problem associated with the uncertain system
The uncertain system
p is an open-loop interconnection
containing known components including the nominal plant model, uncertain
ucomplex, and unmodeled LTI dynamics,
and performance and uncertainty weighting functions. You use weighting
functions to include magnitude and frequency shaping information in
the optimization. The control objective is to synthesize a stabilizing
k that minimizes the robust performance
µ value, which corresponds to
The D-K iteration procedure involves a sequence of minimizations, first over the controller variable K (holding the D variable associated with the scaled µ upper bound fixed), and then over the D variable (holding the controller K variable fixed). The D-K iteration procedure is not guaranteed to converge to the minimum µ value, but often works well in practice.
dksyn automates the D-K iteration procedure
and the options object
dksynOptions allows you
to customize its behavior. Internally, the algorithm works with the
generalized scaled plant model
P, which is extracted
uss object using the command
The following is a list of what occurs during a single, complete step of the D-K iteration.
(In the first iteration, this step is skipped.) The µ calculation (from the previous step) provides a frequency-dependent scaling matrix, Df. The fitting procedure fits these scalings with rational, stable transfer function matrices. After fitting, plots of
are shown for comparison.
(In the first iteration, this step is skipped.) The rational is absorbed into the open-loop
interconnection for the next controller synthesis. Using either the
previous frequency-dependent D's or the just-fit
rational , an estimate of an appropriate
the H∞ norm
is made. This is simply a conservative value of the scaled closed-loop H∞
using the most recent controller and either a frequency sweep (using
the frequency-dependent D's) or a state-space calculation
(with the rational D's).
(The first iteration begins at this
point.) A controller is designed using H∞ synthesis
on the scaled open-loop interconnection. If you set the
following information is displayed:
The progress of the γ-iteration is displayed.
The singular values of the closed-loop frequency response are plotted.
You are given the option to change the frequency range. If you change it, all relevant frequency responses are automatically recomputed.
You are given the option to rerun the H∞ synthesis
with a set of modified parameters if you set the
is convenient if, for instance, the bisection tolerance was too large,
maximum gamma value was too small.
The structured singular value of the closed-loop system is calculated and plotted.
An iteration summary is displayed, showing all the controller order, as well as the peak value of µ of the closed-loop frequency responses.
The choice of stopping or performing another iteration is given.
Subsequent iterations proceed along the same lines without the need to reenter the iteration number. A summary at the end of each iteration is updated to reflect data from all previous iterations. This often provides valuable information about the progress of the robust controller synthesis procedure.
AutoIter field in
that you interactively fit the D-scales each iteration.
During step 2 of the D-K iteration procedure, you are prompted to
enter your choice of options for fitting the D-scaling
data. You press return after, the following is a list of your options.
Enter Choice (return for list): Choices: nd Move to Next D-scaling nb Move to Next D-Block i Increment Fit Order d Decrement Fit Order apf Auto-PreFit mx 3 Change Max-Order to 3 at 1.01 Change Auto-Prefit Tol to 1.01 0 Fit with zeroth order 2 Fit with second order n Fit with n'th order e Exit with Current Fittings s See Status
you to move from one D-scale data to another.
to the next scaling, whereas
nb moves to the next
scaling block. For scalar D-scalings, these are
identical operations, but for problems with full D-scalings,
(perturbations of the form δI) they are different.
In the (1,2) subplot window, the title displays the D-scaling
block number, the row/column of the scaling that is currently being
fitted, and the order of the current fit (with
data when no fit exists).
You can increment or decrement the order of the current
fit (by 1) using
fits each D-scaling data. The default maximum state
order of individual D-scaling is 5. The
allows you to change the maximum D-scaling state
order used in the automatic prefitting routine.
be a positive, nonzero integer.
at allows you to
define how close the rational, scaled µ upper bound is to approximate
the actual µ upper bound in a norm sense. Setting
require an exact fit of the D-scale data, and is
not allowed. Allowable values for
at are greater
than 1. This setting plays a role (mildly unpredictable, unfortunately)
in determining where in the (D,K)
space the D-K iteration converges.
Entering a positive integer at the prompt will fit the current D-scale data with that state order rational transfer function.
e exits the D-scale
fitting to continue the D-K iteration.
s displays a status
of the current and fits.
 Balas, G.J., and J.C. Doyle, “Robust control of flexible modes in the controller crossover region,” AIAA Journal of Guidance, Dynamics and Control, Vol. 17, no. 2, March-April, 1994, p. 370-377.
 Balas, G.J., A.K. Packard, and J.T. Harduvel, “Application of µ-synthesis techniques to momentum management and attitude control of the space station,” AIAA Guidance, Navigation and Control Conference, New Orleans, August 1991.
 Doyle, J.C., K. Lenz, and A. Packard, “Design examples using µ-synthesis: Space shuttle lateral axis FCS during reentry,” NATO ASI Series, Modelling, Robustness, and Sensitivity Reduction in Control Systems, vol. 34, Springer-Verlag, Berlin 1987.
 Packard, A., J. Doyle, and G. Balas, “Linear, multivariable robust control with a µ perspective,” ASME Journal of Dynamic Systems, Measurement and Control, 50th Anniversary Issue, Vol. 115, no. 2b, June 1993, p. 310-319.
 Stein, G., and J. Doyle, “Beyond singular values and loopshapes,” AIAA Journal of Guidance and Control, Vol. 14, No. 1, January, 1991, p. 5-16.