MECH550F: Multivariable Feedback Control Review and today's ...

MECH550F: Multivariable 

Feedback Control 

Lecture 15 

μ-synthesis 

Dr. Ryozo Nagamune 

Department of Mechanical Engineering 

University of British Columbia 

Review and today’s s topic 

• In the previous lectures, we 

have studied 

• structured singular value μ. 

• μ-analysis 

of a feedback system. 

(A controller K is given.) 

• In today’s s lecture, we will study 

μ-synthesis 

of a feedback 

system. (K is to be designed.) 

2009/10 T1 MECH550F : Multivariable Feedback Control 1 


μ-analysis (review) 

Upper bound of μ (review) 

• For a complex matrix M 

Robustly stable case 

Not robustly stable case 

UB 

LB 

UB 

LB 

Note: D and Δ commute. 



1

μ-synthesis problem 

Scaled H∞H 

minimization 

• Design a nominally stabilizing K 

solving a minimization problem 

(Minimizing μ upper bound over frequencies.) 

(Minimizing scaled H∞ norm.) 

• If the minimized value is less than 1, then the 

obtained K is a robust stabilizing controller. 

• The cost function is nonconvex w.r.t. . D & K. 

• If we fix D, it is (convex!) H∞ optimal controller 

design. 

• If we fix K, it is (convex!) μ-analysis. 

• We use the D-K K iteration to find a local optimum. 



D-K K iteration (dksyn.m( 

dksyn.m) 

Initialization of D(s) 

Initialization of D 

Initialization 

K-design 

Controller design 

Controller analysis 

Scaling design 

K-design (with fixed D) 

OK? 

no 

yes 

D-design (with fixed K) 

end 

• One standard choice of D(s) ) is 

• Different initial D’s D s will result in 

different final results, due to 

nonconvexity of the optimization 

problem. 

OK? 

end 

no 

yes 

D-design 



2

Design of K(s) 


K-design 

Analysis of K(s) 


K-design 

• For the fixed D(s), design scaled 

(sub)optimal 

H∞ controller 

OK? 

end 

no 

yes 

D-design 

• Analyze the designed K(s) ) by upper 

bound of μ-value. 

• In other words, for gridded frequency 

ω, , compute (with mussv.m) 

OK? 

end 

no 

yes 

D-design 

hinfsyn.m 

• Check if it is satisfactory. 



Design of D(s) 

Remarks on D-K D K iteration 

• For given Dω, , find D(s) s.t. . the 

magnitude of D(jω) ) is close to Dω, 

and 

• User has freedom to select deg(D) 

(fitting accuracy vs complexity). 

• deg(K)=deg(N)+2deg(D) 


K-design 

OK? 

end 

no 

yes 

D-design 

• D-K K iteration can find only a local optimum, and 

there is no guarantee to find a global optimum. 

• Practical experience suggests that the method 

works well in most cases. 

• Final controller and μ-value depend on: 

• Initial choice of D(s) 

• Order of D(s) ) to fit Dω 

• Numerical accuracy in optimization etc. 



3

Example (in Help of dksyn.m) 

Example: Bode plot of inv(Wp) 

20 

Bode Diagram 

10 

0 

Magnitude (dB) 

-10 

-20 

-30 

Design K s.t. 


-40 

10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 

Frequency (rad/sec) 


Example: Extracting K 

Example: Matlab code 

# of y 

# of u 

Controller 

Closed-loop loop system 

Optimized cost 



4

Example: Info of D-K D K iteration 

Example: Bode plots 

1 st iteration 

Controller 

Optimized cost 

D-scaling 

μ-bounds 

20 

10 

0 

Bode Diagram 

2 nd iteration 


-10 

-20 

-30 

Samples of S 

-40 

-50 

10 -4 10 -2 10 0 10 2 10 4 




Example: Redesign with 2Wp 

Example: Redesign with 3Wp 

10 

Bode Diagram 

10 

Bode Diagram 

0 

0 

-10 

-10 


-20 

-30 

-40 

Samples of S 


-20 

-30 

-40 

Samples of S 

-50 

-50 

-60 

10 -4 10 -2 10 0 10 2 10 4 



-60 

10 -4 10 -2 10 0 10 2 10 4 



5

Summary 

Last homework (Due Nov 10, 5pm) 

• 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, use 

options = dkitopt('AutoIter','off'); 

• D-K iteration often generates very high-order 

controllers. Apply model reduction to designed 

controllers if necessary. 

• Announcement: No lecture on November 5. 


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

controllers, and analyzed their robustness. 

• In this homework, you are required to design a robust 

controller. 

• The design specifications are (cf. Lec.14, slide 19) 

• The worst case sensitivity gain should be at most 8dB. 

• The bandwidth (the frequency that |S| passes 0dB) should be as 

large as possible. 

• Verify that your controller satisfies the specs by sampling 

uncertain plant. 

• Submit only (readable!) m-file. m 


Appendix 

• In RP problem, what does 

it mean if 

for all ω? 

• It means…. 


6

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