mech550f_l15_musynthesis1

7/16/2019 MECH550F_L15_MuSynthesis1

http://slidepdf.com/reader/full/mech550fl15musynthesis1 1/23

2009/10 T1 MECH550F : Multivariable Feedback Control 1

MECH550F: MultivariableMECH550F: Multivariable

Feedback ControlFeedback Control

Dr. Ryozo NagamuneDr. Ryozo Nagamune

Department of Mechanical EngineeringDepartment of Mechanical EngineeringUniversity of British ColumbiaUniversity of British Columbia

Lecture 15Lecture 15

μ--synthesissynthesis




Review and todayReview and today’’s topics topic

In the previous lectures, weIn the previous lectures, we

have studiedhave studied structured singular valuestructured singular value μμ..

μμ--analysisanalysis of a feedback system.of a feedback system.(A controller K is given.)(A controller K is given.)

In todayIn today’’s lecture, we will studys lecture, we will study

μμ--synthesissynthesis of a feedbackof a feedbacksystem. (K is to be designed.)system. (K is to be designed.)




μμ--analysis (review)analysis (review)

UBUB

LBLB

Robustly stable caseRobustly stable case

UBUB

LBLB

Not robustly stable caseNot robustly stable case




Upper bound of Upper bound of μμ (review)(review)

Note: D andNote: D and ΔΔ

commute.commute.

For a complex matrix MFor a complex matrix M




μμ--synthesis problemsynthesis problem

Design a nominally stabilizing K Design a nominally stabilizing K

solving a minimization problemsolving a minimization problem

(Minimizing μ

upper bound over frequencies.)

(Minimizing scaled H∞ norm.)




Scaled HScaled H∞∞

minimizationminimization

If the minimized value is less than 1, then theIf the minimized value is less than 1, then theobtained K is a robust stabilizing controller.obtained K is a robust stabilizing controller.

The cost function is The cost function is nonconvexnonconvex w.r.tw.r.t. D & K.. D & K.

If we fix D, it is (convex!)If we fix D, it is (convex!) HH∞∞ optimal controlleroptimal controller

designdesign.. If we fix K, it is (convex!)If we fix K, it is (convex!) μμ--analysisanalysis..

We use theWe use the DD--K iterationK iteration to find a local optimum.to find a local optimum.




DD--K iteration (K iteration (dksyn.mdksyn.m))

Controller designController design

Controller analysisController analysis

Scaling designScaling design

Initialization of DInitialization of D

K K --design (with fixed D)design (with fixed D)

DD--design (with fixed K)design (with fixed K)

OK?OK? endendyesyes

nono




Initialization of Initialization of D(sD(s))

One standard choice of One standard choice of D(sD(s) is) is

Different initial DDifferent initial D

’’s will result ins will result in

different final results, due todifferent final results, due tononconvexitynonconvexity of the optimizationof the optimizationproblem.problem.

InitializationInitialization

K K --designdesign

DD--designdesign

OK?OK? endendyesyes

nono




Design of Design of K(sK(s))

For the fixedFor the fixed D(sD(s), design scaled), design scaled

((sub)optimalsub)optimal HH∞∞ controllercontroller

hinfsyn.mhinfsyn.m


K K --designdesign

DD--designdesign

OK?OK? endendyesyes

nono




Analysis of Analysis of K(sK(s))

Analyze the designedAnalyze the designed K(sK(s) by upper) by upper

bound of bound of μμ--value.value. In other words, forIn other words, for griddedgridded frequencyfrequency

ωω, compute (with, compute (with mussv.mmussv.m))

Check if it is satisfactory.Check if it is satisfactory.


K K --designdesign

DD--designdesign

OK?OK? endendyesyes

nono




Design of Design of D(sD(s))

For givenFor given DDωω, find, find D(sD(s)) s.ts.t. the. the

magnitude of magnitude of D(jD(jωω) is close to) is close to DDωω,,andand

User has freedom to selectUser has freedom to select deg(Ddeg(D))(fitting accuracy(fitting accuracy vsvs complexity).complexity).

deg(K deg(K )=deg(N)+2deg(D))=deg(N)+2deg(D)


K K --designdesign

DD--designdesign

OK?OK? endendyesyes

nono




Remarks on DRemarks on D--K iterationK iteration

DD--K iteration can find only a local optimum, andK iteration can find only a local optimum, and

there is no guarantee to find a global optimum.there is no guarantee to find a global optimum. Practical experience suggests that the methodPractical experience suggests that the method

works well in most cases.works well in most cases.

Final controller andFinal controller and μμ--value depend on:value depend on:

Initial choice of Initial choice of D(sD(s))

Order of Order of D(sD(s) to fit) to fit DDωω Numerical accuracy in optimization etc.Numerical accuracy in optimization etc.




Example (in Help of Example (in Help of dksyn.mdksyn.m))

Design K Design K s.ts.t..




Example: Bode plot of Example: Bode plot of inv(Wpinv(Wp))

10-4

10-3

10-2

10-1

100

101

102

-40

-30

-20

-10

0

10

20

M a g n i t u d e ( d B )

Bode Diagram

Frequency (rad/sec)




Example: Extracting K Example: Extracting K




Example:Example: MatlabMatlab codecode

ControllerController Optimized costOptimized cost

ClosedClosed--loop systemloop system

# of y# of y

# of u# of u




Example: Info of DExample: Info of D--K iterationK iteration

ControllerController

Optimized costOptimized cost

μμ--boundsbounds

DD--scalingscaling

11stst iterationiteration

22ndnd iterationiteration




10-4

10-2

100

102

104

-50

-40

-30

-20

-10

0

10

20


Bode Diagram

Frequency (rad/sec)

Example: Bode plotsExample: Bode plots

Samples of SSamples of S




10-4

10-2

100

102

104

-60

-50

-40

-30

-20

-10

0

10


Bode Diagram

Frequency (rad/sec)

Example: Redesign with 2WpExample: Redesign with 2Wp





10-4

10-2

100

102

104

-60

-50

-40

-30

-20

-10

0

10


Bode Diagram

Frequency (rad/sec)

Example: Redesign with 3WpExample: Redesign with 3Wp





SummarySummary

μ-synthesis for structured uncertainty

D-K iteration

If you want to design a controller by step-by-step

D-K iteration using dksyn.m, useoptions = dkitopt('AutoIter','off');

D-K iteration often generates very high-order

controllers. Apply model reduction to designedcontrollers if necessary.

Announcement: No lecture on November 5.




Last homework (Due Nov 10, 5pm)Last homework (Due Nov 10, 5pm)

In Lecture 14, we assumed that we have two PIDIn Lecture 14, we assumed that we have two PID

controllers, and analyzed their robustness.controllers, and analyzed their robustness. In this homework, you are required to design a robustIn this homework, you are required to design a robust

controller.controller.

The design specifications are (cf. Lec.14, slide 19) The design specifications are (cf. Lec.14, slide 19) The worst case sensitivity gain should be at most 8dB. The worst case sensitivity gain should be at most 8dB.

The bandwidth (the frequency that |S| passes 0dB) should be as The bandwidth (the frequency that |S| passes 0dB) should be aslarge as possible.large as possible.

Verify that your controller satisfies the specs by samplingVerify that your controller satisfies the specs by samplinguncertain plant.uncertain plant.

Submit only (readable!) mSubmit only (readable!) m

--file.file.




AppendixAppendix

In RP problem, what doesIn RP problem, what does

it mean if it mean if for allfor all ωω??

It meansIt means……..

mech550f_l15_musynthesis1

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