MECH550F: Multivariable Feedback Control Today's topics Why ...

MECH550F: Multivariable 

Feedback Control 

Lecture 2 

Internal stability of feedback systems 

Robust controller design process 

(Review) 

Reference 

Controller 

Actuator 

Input 

Sensor 

Disturbance 

Plant 

Output 

Dr. Ryozo Nagamune 

Department of Mechanical Engineering 

University of British Columbia 

4. Implementation 

Controller 

3. Design 

Uncertain model 

2. Analysis 

1. Modeling 

2009/10 T1 MECH550F : Multivariable Feedback Control 1 


Today’s s topics 

Why feedback? 

• Nominal stability of the feedback system 

C(s) 

P(s) 

• Why feedback? 

• What have you learned so far about nominal stability? 

• Well-posedness 

of the feedback system 

• Internal stability of the feedback system 

• Stabilizing controllers 

• To achieve system performance under 

uncertainties 

• Modeling error 

• System variations 

• Nonlinearities 

• Unknown signals 

• Stabilization of an unstable system 



1

Issue in stabilization 

by series connection 

• Consider three stable open-loop systems: 

• Pole/zero cancellation in unstable region by series 

connection is VERY BAD! 

• Unstable system MUST be stabilized by feedback!!! 


Issue in stabilization 

by series connection (cont’d) 

• Why is unstable cancellation by series 

connection not allowed? 

• Plant model is never exact! (and hence cancellation 

will not occur in real world.) 

• Neither nonzero initial condition nor input disturbance 

is allowed. (very fragile) 

• Some internal signal may go unbounded, even if 

output is bounded. 

• Internal stability (that we define later) can detect 

such cancellation! 



Nyquist stability criterion (Review) 


C(s) 

P(s) 







• P: # OL poles in open RHP 

(given) 

• N: # of clockwise 

encirclement of -11 by 

Nyquist plot of OL transfer 

function L(s) ) (counted by 

using Nyquist plot of L(s)) 

15 

10 

5 

0 

-5 

-10 

-15 

-5 0 5 10 15 



2

Observer-based based control (review) 

Comments 

Plant 

Observer 

Closed-loop loop A-matrixA 

• Nyquist stability criterion 

• Graphical method to decide NS 

of feedback (FB) system in s- 

domain. 

• An assumption: No hidden 

unstable pole/zero cancellation. 

• Algebraic method to decide 

NS of FB system in s-domain, 

even with hidden unstable 

pole/zero cancellation? 

C(s) 

P(s) 




Some terminologies 


• Real rational transfer function 

YES 

NO 

C(s) 

P(s) 

• Proper transfer function 







• Strictly proper transfer function 

• Biproper: : Inverse is also proper 

• Real rational proper transfer matrix 



3

A feedback structure 

Well-posedness 

posedness: : Definition 

P1 & P2: P : real rational proper 

• A feedback system is called well-posed 

if all closed-loop loop transfer matrices are 

• well-defined (inverses exist) and 

• proper. 

(The argument “s” is omitted.) 


Remark: Any meaningful feedback system has 

to be well-posed! 


Physical meanings 

Examples 

• If not well-defined, for an external signal, the 

internal signal is not uniquely determined. 

• If not proper, for high frequency signals, the gain 

goes to infinity. (Physically unrealizable!) 

• Roughly speaking, not-well 

well-posed feedback 

system does not make sense. 

Ex.1 

Ex.2 



4

Examples (cont’d) 


posedness: : Condition 

Ex.3 

• The feedback system 

is well-posed 

if and only if 

Remark: 

Non-proper! 

because 




condition in SS 

Examples (cont’d) 

Notation: 

• The feedback system is well-posed if and only if 

Ex.4 

Feedback system is NOT well-posed! 

• A sufficient condition (We often use this!) 

If one of P1(s) P (s) or P2(s) P 

is strictly proper (i.e. D1=0 D 

or 

D2=0), =0), the feedback system is well-posed. 

Ex.5 

Feedback system is well-posed! 



5


Internal stability: Definition 


C(s) 

P(s) 








• Assume that the CL 

system is well-posed. 

• The CL system is called 

internally stable if 

All the poles are in the open LHP. 


Example 

Remarks 

CL system is well-posed! Why? 

CL system is NOT internally stable! 

• Internal stability is a basic requirement for a 

practical feedback system. 

• Internal stability guarantees that all signals in the 

CL system are bounded, provided that the 

exogenous signals (w) are bounded. 

• Exogenous signals (w) are models of, for 

example, initial conditions, noises, and modeling 

errors. 

• Roughly speaking, not-internally 

internally-stable feedback 

system does not work fine. 



6


Stabilizing controllers for a stable plant 


C(s) 

P(s) 

• All stabilizing negative FB controllers C(s) ) for a 

stable plant P(s) ) is parameterized as 







• Q(s) ) is any stable proper rational transfer function 

matrix, i.e., 

• Youla parameterization (Q-parameterization) 

• A class of all the stabilizing controllers (even for an 

unstable plant: omitted in the lecture) 



Example 

Comments 

• Plant 

• All the stabilizing controllers 

• How to select Q? 

• Ex. If we want C to have an integrator, 

• Advantages of Youla parameterization 

• For a stable plant, the structure of the controller is 

identical to the one for (well-studied classical!) 

Internal Model Control (IMC). 

• Search over all stabilizing controllers is replaced by a 

search over stable Q(s). 

• All CL transfer functions are linear (or affine) in Q, 

simplifying optimization problems. 



7

Internal model control (IMC) 

Summary 

Q(s) 

Real plant 

P(s) 

• Feedback only the new information. 

• Design of Q(s) ) is relatively easy. (Proper and 

stable approximation of the inverse of P(s)) 



• Internal stability in s-domains 


• Announcements 

• No lecture on Sept 22 (Tue). 

• Today’s s HW is due Sept 22 (Tue) 5pm. 


Homework 

Homework (cont’d) 

• HW1: Prove 

• HW2: Check the well-posedness 

posedness, , as well as internal 

stability (if well-posed), for the following feedback 

systems. If unstable, stabilize the feedback system by 

modifying P2 P (P1 is fixed). 

• 

• 

• HW3: For MIMO feedback system (stable P(s)), 

verify that the following transfer functions are 

affine in Q(s) ) by using Youla parameterization. 

• From r to e 

• From r to u 

• From r to y 

• From d to e 

• From d to u 

• From d to y 

C(s) 

P(s) 



8

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