chapter 4: review of chapter 3 - unipi.itpeople.unipi.it/.../256/2019/10/4_csi_robustness.pdf ·...

Chapter 4: Review of Chapter 3

Measures of Vector Spaces, Signals, and Systems Norms of vectors and matrices Singular values decomposition and its use in the measure of signal amplification through a

matrix Energy and power of a dynamic system via H2 and H∞ norms, and their computation

Algebra of a 2x2 System Standard feedback realization, incorporation of weight functions for performance

requisites Well posed feedback loop: requirements in the frequency and time domains Internal stability of a feedback loop: requirements in the frequency and time domains;

definition of a stabilizing controller

Frequency Shaping in MIMO systems Performance requirements in terms of loop transfer, sensitivity, and complementary

sensitivity for MIMO systems

Reference Material Zhou, K., Essentials of Robust Control, Prentice Hall 1998, CH. 1‐6 Skogestad, S., Postlehwaite, I., Multivariable Feedback Control, Wiley 2010, Ch.

1‐5, App. A Mackenroth, U., Robust Control Systems, Springer 2004, Ch. 1‐7, App. A

Chapter 4: Robustness

General Robustness Problem

Review of SISO Uncertainty Characterization and Robustness

Robustness in MIMO Systems•Multivariable Nyquist Criterion•Small Gain Theorem

Shaping with Uncertainty •Case of unstructured uncertainty

Robustness of Optimal Controllers•Robustness of LQR, KBF, and LQG

Uncertainty in a 2‐Block Structure

Chapter 4: General Robustness Problem


The concept of robustness in the design of control systems deals, in a formal way, with the presence of uncertainties coming from a variety of sources

4. Good performance and stability in the presence of model errors, parameter variations, and nonlinearities

G(s)s+3(s+1+a)(s+2+b)K(s)

G’(s)

Robust Control

G(s)s+3(s+1)(s+2)K(s)

The main objectives for feedback control are, in general, 4:

1. Disturbance rejection;2. Improved internal stability;3. Reduction of tracking errors;

Nominal Control


• Neglected dynamics (sensors, actuators);• Truncated and unknown dynamics (model reduction, high frequency poorly known modes);• Sensitivity (to parameter variations);• Neglected nonlinearities;• Noise (internal and external), and sensor and actuator errors.

• The main sources of uncertainty in control design arise from:

1. The control engineer is required to design a system that, in addition to providing satisfactory nominal performance, is also robust to errors such those described above.

2. In order to perform this task, the uncertainty set must be • defined,• adequately modeled,• and its bounds selected, in order to determine the worst case, and design a

robust controller for it.

“Robustness of the controller is as good as the model of the uncertainty”.


The problem of robustness in MIMO systems developed in the late 1970’s and early 1980’s, with the idea of bringing frequency domain design methods back as primary tool, as opposed to time domain optimal control based techniques.

One reason for this was the highly sensitive properties of newly developed LQG controllers.

• Kalman’s contribution to stability margins of quadratic optimal compensators

• Rosenbrook’s work on Inverse Nyquist Array (1974)• Horowitz’s QFT (1970)• Doyle’s, Stein’s work on singular values and loop transfer

recovery (1981)• Zames’, Francis’ work on H theory (1981, 1984)• Doyle’s, Glover’s work on structured singular value (1988)• Boyd’s work on linear matrix inequalities, and convex

optimization (1990)• Ackermann’s work on parameter space robustness (1988).

Tools for evaluating robustness properties were missing and developed independently. We recall:


Nomenclature:• Uncertainty: analytical structure used to represent mathematically the

error(s) between model (or nominal plant) and real system (or more accurate model)

• Error: actual difference (due to physical approximation) between model and real system (or more accurate model).

• In the case of SISO systems, the most common approach (leading to the definition of stability margins) is to represent the uncertainty as a scalar, frequency ‐ dependent causal complex function described by its magnitude and phase.

• This uncertainty is called unstructured uncertainty, because the error in the model is not physically known in general, but is characterized analytically by stable complex rational function (bounded in magnitude and phase).

• Example: Consider a SISO unity feedback model, with an open loop transfer function g(j), and a ‘true’ loop transfer function ( )g jw


• Main Assumption: ( )g s ( )g sandhas the same number of encirclements about the critical point (‐1, 0). The errorbetween the two is their difference as a frequency dependent function for

have the same number of unstable poles, so their Nyquist diagram

[0, )w Î ¥

• The uncertainty present when using the model g(s), is due (in general) to two characterizations of the error:

1. additive error (absolute)2. multiplicative error (relative)


Additive Error: ( ) ( ) ( )e j g j g jw w w= -

( ) ( )

1( ) 1 ( )

( )

p e j

d g jS j

w w

w ww

=

= + =

• From Nyquist’s diagram, we can identify the error and return difference functions as:

( ) ( ), [0, )p dw w w< " Î ¥

• Asymptotic Stability of the ‘true’ closed loop system requires:


( ) ( ) ( )( ) 1

( ) ( )g j g j g j

e jg j g jw w w

ww w

-= = -

• Multiplicative Error: The error between model and true plant is measured in % of the model

[ ]( ) 1 ( ) ( )g j e j g jw w w= +

• The true plant transfer function is:

• Error and return difference functions are alsomeasured in % of the model:

( )( ) ( ) 1

( )g j

p e jg j

ww w

w= = -

11 ( )( ) 1 ( )

( )g j

d g jg j

ww w

w-+

= = +

• Asymptotic Stability of the true closed loop system still requires: ( ) ( )p dw w<

• Recall that for SISO systems input / output loop openings are the same


• Unstructured uncertainty is directly related to the classical notion of stability margins. Since stability margins are a “relative” measure of stability, additive and multiplicative errors must be represented in a (bounded) multiplicative uncertainty form.

• Multiplicative Uncertainty due to Additive Error

( )L jw

( ) ( ) ( ) ( ) ( )A

g j g j L j g j e jw w w w w= = + 1( ) 1 ( ) ( )A

L j e j g jw w w-= +

• For closed loop stability: ( ) ( ) ( ) 1 ( )Ap e j d g jw w w w= < = +

( )[ ( ) 1] ( ) ( ) 1 1 ( )g j L j g j L j g jw w w w w- £ - < +

1 1( ) 1 1 ( ) ( )CL

L j g j g jw w w- -- < + =

• Substituting the expression for the error

1( )

( ) 1CLg j

L jw

w<

-• with:


1( )

( ) 1CLg j

L jw

w<

-• Upper Bound on

Complementary Sensitivity

• Multiplicative Uncertainty due to Multiplicative Error

( ) ( ) ( ) 1 ( ) ( )M

g j g j L j e j g jw w w w wé ù= = +ê úë û ( ) ( ) 1M

e j L jw w= -• that yields:

• Substituting the expression for the error 1 1( ) 1 1 ( )CL

L j g j gw w- -- < + =


1 1( ) 1 1 ( )CL

L j g j gw w- -- < + =

• Lower Bound on the inverse of Complementary Sensitivity

1( ) ( )

1A Me s e ss

= =+

• Let us assume an error of the form:

• Example: Consider a second order model of a system, given by:2

1( )

1g s

s s=

+ +


21 2 2

( ) 1 ( ) ( )1A A

s sL s e s g s

s- + +

= + =+

2( ) 1 ( )

1M M

sL s e s

s+

= + =+

• The unstructured multiplicative uncertainty has the forms:

• Nominal System Response always stable


Alternate view for unstructured multiplicative uncertainty

( ) ( )( )

( )M

g j g je j

g jw w

ww

-=

( ) 1 ( ) ( )M

g j e j g jw w wé ù= +ê úë û

( ) ( ) ( ) ( ) , ( ) 1M M M

e j w j w jw w w w w£ = ⋅D D £

• How can we transform a parametric uncertainty into an equivalent unstructured uncertainty?


Example: given an uncertain system with delay

( ) ( )0.25, ( ) 2.5,

( )M

g j g je j

g jw w

w w ww

-< = <

( )1

skg s e

sq

t-=

+

• The uncertainty is due to variations of the three parameters k, , and : 2 , , 3k t q£ £

• Selected model transfer function:2.5

( )1 2.5 1

kg s

s st= =

+ +

1 0.2( ) , 4

12.5

M

Tsw j T

Ts

w+

= =+

22 1

2

1.6 1( ) ( )

1.4 1M M

s sw j w j

s sw w

+ +=

+ +

( )1

skg s e

sq

t-=

+

( )1

kg s

st=

+

( ) ( ) ( ) , ( ) 1M M

e j w jw w w w£ ⋅D D £

Chapter 4: Robustness in MIMO Systems

• The analysis of the influence of unstructured uncertainty on the performance and stability of a MIMO system requires the use of H2‐norms, H∞‐norms or singular values of appropriate loop transfer matrices.

Road to Nyquist Criterion for MIMO Systems

• Consider a generic multiplicative uncertainty. The relationship between uncertainty and modeling errors is given by:

[ ]

[ ]

( ) ( ) ( )

( ) ( )

A

M

E j L j I G j

E j L j I

w w w

w w

ì = -ïïïíï = -ïïî

• Relative stability analysis of the system using the MIMO extension of Nyquist criterion is not as direct as in the SISO counterpart. In particular, the following considerations hold:

• Directionality is relevant (input or output uncertainty)

• Necessary and sufficient conditions for closed loop stability are maintained.

• Capability of deriving stability margins is not necessarily maintained.

• Known Graphical tools are not easily available.


• Example:12

1 0 1

0 1 0 1

bì é ù é ùï -ï ê ú ê úï = +ï ê ú ê ú-í ê ú ê úë û ë ûïïï =ïî

x x u

y x

• The open loop system is asymptotically stable with poles in (-1, -1)

12

1

11 1( ) ( )

10

1

b

s sG s sI A B

s

-

é ùê úê ú+ += - = ê úê úê úê ú+ë û

• The subsystems are coupled via the b12 parameter

• Consider a full state unity feedback with Gain Matrix K = I

1 1 1

2 2 2

( ) ( ) ( ) cc

c

u u xs s s

u u x

ìï = -ï= - íï = -ïîu u y • The LoopTransfer Matrix is:

1( ) ( ) ( )OL

G s sI A B G sK -= - =


• The closed loop system becomes:

12 122 1

0 2 0 1 c

b bé ù é ù-ê ú ê ú= +ê ú ê ú-ê ú ê úë û ë ûx x u

• The closed loop system is asymptotically stable with poles in (-2, -2), which do not depend on b12.

1221 1( )

20

1

s bs sI G s

ss

é ù+ê úê ú+ ++ = ê ú+ê úê ú+ë û

• The return difference matrix is:

What happens if we apply the Nyquist criterion as known from SISO classical control ?


2

2 3( ) 1 det( ( ))

( 1)

ss I G s

s

+F = - + + =

+

Option 1: Nyquist diagram of the return difference matrix

• The Nyquist diagram is of (s) is 1. From the diagram, the point (‐1+j0) is not encircled, therefore the closed loop stability is guaranteed by SISO Nyquist.

2. Assuming the validity of SISO stability margins, in this case we would have:

GM ,

PM

13

106

denoting good robustnesscharacteristics

Option 2: Consider opening one loop at the time, and verify relative stability.


• If we open the first loop we have the following block diagram (The second loop is the same due to symmetry)

• We have then 1 1

1

( ) 1( )

( ) 1OL

y sG s

u s s= =

+

• with Nyquist diagram

1,

180

GM

PM

ì é ùï = - ¥ï ê úï ë ûíï @ ïïî

• Conclusions: From both analyses, the system is very robust in terms of stability margins. In addition, the closed loop eigenvalues do not depend on the interconnection parameter b12.

• Let us now introduce a cross feed disturbance from u1 and u2, which depends on b12. The closed loop system will become unstable even for an arbitrarily small b12!


• Open Loop System:

1

12

2

41 0

0 1 15

b

bì é ùï -ï é ù ê ú-ï ê úï ê ú= +ï ê úï ê ú-í ê ú ê úë ûï ê úï ë ûïï =ïïî

-x x u

y x

• Closed Loop System always unstable!

12 12

12 12

4 5

5 52 1

b b

b b

ì é ù é ùï - -ï ê ú ê úïï ê ú ê ú= +ïï ê ú ê ú- -í ê ú ê úï ê ú ê úï ë û ë ûïï =ïïî

cx x u

y x

• Closed loop poles 1 5

2s

=


G(s)yr e u

K(s)

1( ) ( )[ ( )]CL

G s L s I L s -= +

MIMO Nyquist Criterion (Rosenbrok, 1974, see Lavretski – Wise:, ‘Robust and Adaptive Control’, Ch. 5, p. 107 ):

Given the unity feedback MIMO system in the figure below:

• Denote with L(s) = G(s)K(s) the Loop transfer matrix , and L(s)[I + L(s)]-1 the closed loop transfer matrix (complementary sensitivity), then the following holds:

• The closed loop system is asymptotically stable if and only if

(0,det( ( )), )R OL

N I L s D P+ = - ( 1, 1 det( ( )), )R OL

N I L s D P- - + + =-or

• The number of clockwise encirclements of -1 by the det[I + L(s)] is equal to –POL, which is the number of unstable open loop poles over the Nyquist contour DR.

Comment: Proof and use is analytical (not graphical), based on properties of determinants


• Recall previous results about internal stability in a 2 block structure, for a well posed system

( )22( ) ( ) ( ), ( )

zwT s T s G s K s

( ) ( )( ) ( )

1 1 1

22 221 1

22 2222 22

22

( ) i o

i o

I K I KG K I G K

I G I K

S K

G I G

ST s

G G SK S

- - -

- -

é ùé ù- - -ê úê ú= = =ê úê ú ê ú- -ê úë û ê

é ùê úê

úë ûú- ê úë û

Conclusion No. 1: If the plant and controller structures are both stabilizable and detectable, the feedback system in Figure 2 is internally stable if T(s) or (Si, So, KSo, G22Si ) are stable matrices.

Internal Stability and Nyquist Criterion


Conclusion No. 2: The output Sensitivity matrix So(s) is stable if and only if the determinant (s) has all the transmission zeros ( its roots) in the open LHP. Where (s) is given by:

22( ) : det ( ) ( ) det ( )

os I G s K s I L sé ù é ùF = - = +ê ú ê úë û ë û

MIMO Nyquist Criterion Statement revisited 1 (Mackenroth Th 6.5.4. p. 170):

1. Let the realization Lo(s) be minimal and suppose that I + Do is invertible. 2. Moreover, assume that I + Lo has no transmission zero that is also a pole of this transfer

matrix. Denote by no the number of poles of Lo.

Then So is stable if and only if the origin does not lie on the Nyquist plot Cp and the Nyquist plot encircles the origin no times in the mathematical positive sense.

The closed loop system from figure 2 is internally stable if and only if it is well‐posed, condition (i) of theorem 5.7 is satisfied, and the Nyquist plot of (j) = det[I-G22(j)K(j)]-1 for encircles the origin nc + np times in the counter ‐ clockwise direction.

w-¥ £ £ ¥

MIMO Nyquist Criterion revisited (Zhou Th 5.8, p. 126):

Chapter 4: MIMO Shaping with Uncertainty

Robustness Analysis and Synthesis with respect to unstructured multiplicative Uncertainty (Historical perspective)

• Statement of the general Robust Control Problem:

• Given a System described by a transfer matrix ( )G s

( )G s• Given a Model described by the transfer matrix

• The system and its model have the same number of unstable poles

{ }( );G s G G -D "D Î D Î D =• Given an Uncertainty, which is bounded, stable, and unstructured

( ) ( ) ( )G s I G s G sé ù= +Dê úë û• Given the error between plant and model such that :

• Design a controller K(s) such that:

( ) 1I GK GK

-+1. The nominal closed loop system is stable ( Nominal Stability NS)

2. The Performance of the nominal closed loop system are satisfied ( Nominal Performance NP)

( ) 1I GK GK

-+ 3. The real closed loop system is stable in the presence of the assumed

uncertainty set G(s) ( Robust Stability RS)

4. The Performance of the real closed loop system are satisfied in the presence of he assumed uncertainty set ( Robust Performance RP)


( ) ( ) ( ) ( ) ( )G j I G j G j L j G jw w w w wé ù= +D =ê úë û

• Note: we consider the case of input multiplicative unstructured uncertainty. Other cases can be handled in a similar fashion (see later)

• The first requirement (NS) is met, for example, by direct application of MIMO Nyquistor the use of Mackenroth theorem 6.5.1.

• The second requirement (NP) is met, for example, by applying frequency shapingdesign or any other design method (LQR, etc.)

• The third and fourth requirements need additional analysis.

• Define a rational transfer matrix G(s, ) with coefficients functions of a parameter [0, 1], such that:

( , ) ( ) for 0

( , ) ( ) for 1

G s G s

G s G s

e e

e e

= =

= =


Robust Stability (RS)

• Since both system and model have the same number of unstable poles, the following holds:

• Rewrite in terms of singular values:

( )minmin

( ) ( , ) ( ) ( )

0

G s G s I L s G s

I GK I I L GKs e

e e

s

ì é ùï = = +ï ê úé ù+ + >ê ú

ï ë ûí é ùï + =ï ê úëï ë ûûî

• Replace with:


• From the property: min min minAB A Bs s sé ù é ù é ù³ê ú ê ú ê úë û ë û ë û We have:

1min min

( ) 0 GK I L GKs e s-é ù é ù+ + >ê úê ú ë ûë û

• From the property: min max min0A B A Bs s sé ù é ù é ù> + >ê ú ê ú ê úë û ë û ë û

• the top expression is satisfied for: 1min max max

( ) I GK L Ls s e e s-é ù é ù é ù+ > =ê ú ê úê ú ë û ë ûë û

( )1min

( ) 0 GK I L GKs e-é ù+ + >ê úë û

• Collecting GK and assuming (GK)-1 exists, yields:

• Conditions?


( ) 1 11

minmax

1 1

( ) LI GKe

ss- -

->

é ù é ù+ ê úê ú ë ûê úë û

• Use the identity: min 1max

1A

As

s -é ù =ê úë û é ù

ê úë ûTo obtain:

( ) ( ) ( )1 1 11 1( ) ( )( )I GK I GK GK GK I GK

- - -- -+ = + = +• From the identities:

( ) 11min max

1L GK I GKs s

e

-- é ùé ù > +ê úê úë û ê úë û• Rewrite:

• Consider multiplicative stable uncertainty complex matrix L(j ) bounded by some frequency dependent function lm():

max( ) ( )

mL j ls w wé ù <ê úë û


• Choose lm() such that:

( ) 1

max 0

1 1,

( )m

GK I GKl

s w we w

-é ù> + " >ê úê úë û

• The Robust Stability Requirement over the entire frequency range becomes:

( ) 1

max 0

2max 0

2

1 1,

( )

max

m

x

GK I GKl

AxA

x

s w we w

s

-

¹

ìï é ùï > + " >ê úïï ê úë ûïïíïï é ù =ï ê úë ûïïïî

(**)

• Conservative Approach: consider the worst case uncertainty ( = 1):


• From (**), the closed loop gain must decrease as the uncertainty increases.

• Since the uncertainty increases with frequency, and since at high frequency the following approximation holds:

( ) 1

max max

0

GK I GK GKs s

w w

-ì é ùï é ùï + »ê ú ê úï ë ûê úë ûíï ³ïïî

• We can define the following frequency shaping boundaries for Robust Stability with respect to unstructured input multiplicative uncertainties (loop broken at the output)

• Bandwidth Upper Limit in the presence of uncertainty


( ) ( )1

min max( ) ( ) ( ) ( ) , 0,

mI G j K j l L js w w w s w w

-é ù é ù+ > ³ " Î ¥ê ú ê úë ûê úë û

• NOTE: A similar procedure must be performedfor loop breaking at the input, usingK(s)G(s) as loop gain.

• The robustness stability characteristics over the entire frequency range can be expressed more easily. Consider (**) in the worst case uncertainty ( = 1):

• Using the identity: ( ) ( ) 11 1I G I G I-- -+ + + =

• The robust stability requirement over the entire frequency range becomes:

( ) 1

max

1( )

m

GK I GKl

sw

-é ù> +ê úê úë û

inverting, yields ( ) 1

max

1( )

ml

GK I GKw

s-

>é ù

+ê úê úë û


Robust Performance (RP)

• Recall that performance of a closed loop system are known in terms of transient response accuracy, speed, overshoot; steady state error, and disturbance rejection over appropriate frequency ranges

max min(( ) 1 o )r ( ) 1S j I KG jjs w s wwé ù é ù << + >>ê ú ê úë û ë û

( ) ( ) ( ) 0s s s= - »e r y

• Good tracking of a command requires error reduction :

• Select a design function p() , we can impose the requirement:

min( ) ( ) ( )I G j K j ps w w wé ù+ ³ê úë û

min( ) ( )I I L GK ps e wé ù+ + ³ê úë û

• Introducing the multiplicative uncertainty by substituting ( , ) ( )G s G se

and max( ) ( )

mL j ls w wé ù <ê úë û


min 1

( ),

1 ( )m

pGK

lw

s w ww

é ù ³ <ê úë û -

• The above inequality gives an analytical constraint for robust tracking, anddisturbance rejection requirements, within the appropriate frequency range.

• Combining previous robust stability and performance constraints yields:


• Frequency Shaping now differs from the nominal case due to the presence of a bounded unstructured uncertainty, which must be accounted for

max( ) ( ); ( )

mL j l L js w w wé ù < Î Dê úë û

Summary:


Example:

5 ( )

( 5)( 1)

TseG s

s s

-

=+ -

1( )

1G s

s=

-

• Given the system:

• Consider the model:

• The difference between real plant and model is a relative error characterized by an input multiplicative uncertainty

( ) ( )G s I L sD = + 5 ( ) ( ) 1

5

TseL s G s I

s

-

= D - = -+

0.05/2 0.025 0.9997,

/2 0.025 0.9997Ts ss T s

e es T s

- -- - += =

+ +

• using a Pade’ approximation to the first term, with:


2

5( 0.025 0.9997)( )

(0.025 0.9997)( 5)0.125 4.9984

0.025 1.1246 4.9984

sG s

s ss

s s

- +D = =

+ +- +

=+ +

max

2

2

( ) ( ) ( ) 1

0.125 4.9984 0.025 1.1246 4.9984

0.025 1.1246 4.9984 ( 49.984)

( 39.988)( 5)

ml s L s G s

s s s

s ss s

s s

s é ù= = D - =ê úë û- + - - -

=+ +

+=

+ +

with crossover frequency7 rad/secCRw »

10-1 100 101 102 103-40

-30

-20

-10

0

10

pulsazione (rad/sec)

ampi

ezza

(dB

)

incertezza massima (lm)

• We wish to design a controller such that:• the closed loop system is asymptotically stable• the steady state error to a unit step is zero• the closed loop system is robust to the assumed uncertainty.


• Using the model of the system 1

( )1

G ss

=-

we must design a controller that:

1. stabilizes the system2. makes the loop TF of type 13. produces a crossover frequency of less than 7 rad/sec4. satisfies the robustness requirements, that is:

( )

min 1

max

1

min

( ), 0.1 rad/sec

1 ( )1

,( )

( ),

m

CRm

m

pKG

l

KGl

I KG l

ws w w

w

s w ww

s w w-

ìïï é ù > < @ï ê úë ûï -ïïïïï é ù < >í ê úë ûïïïï é ùï + > "ê úïï ê úë ûïïî

• Select a PI controller of the form ( ) IP

KK s K

s= +

( ) ( ) ( 1)

IP

P

KK s

KK s G s

s s

æ ö÷ç ÷+ç ÷ç ÷çè ø=

-


• Step Response of the controlled Model

( )2

( ) ( )

( )1 ( ) ( ) 1

IP

PCL

P I

KK s

KK s G sG s

K s G s s K s K

æ ö÷ç ÷+ç ÷ç ÷çè ø= =

+ + - +

• The closed loop system is:

4, 3I PK K= = 1

2

-1.0000 + 1.7321

-1.0000 - 1.7321

p j

p j

==

0I

K >• For closed loop stability select: 1P

K >

0 2 4 6 8 100

0.5

1

1.5

tempo (sec)

ampi

ezza

risposta al gradino


• Robustness requirements ( ) 1/p sw =

10-2

10-1

100

101

102

103

-20

-10

0

10

20

30

40

50

Sin

gula

r V

alu

es (d

B)

p/(1 - lm

)min

(KG)

1st robus tness test

Frequenc y (rad/sec )

10-2

10-1

100

101

102

103

-60

-40

-20

0

20

40

60

80

Sin

gula

r V

alu

es (d

B)

1/lm

max(KG)

2nd robustness test


10-2

10-1

100

101

102

103

-60

-40

-20

0

20

40

60

Sin

gula

r Va

lues

(dB

)

lmmin

(I + (KG) -1)

3rd robus tness test


( ) 1

min( ),

mI KG ls w w

-é ù+ > "ê úê úë û • From the figure the two graphs are very

close at about = 2 rad/sec, so additional tuning may be necessary


The previous results describe the stability robustness conditions for a particular location of bounded unstructured uncertainty in the control loop. In the early 1980’s theorems were developed (Lehtomaki, Sandell, Doyle, IEEE‐TR‐AC‐2/1981 ) to derive similar conditions as function of the uncertainty location.

• Nomenclature: Consider a unity feedback loop with a full state feeback structure. Let the nominal model and the perturbed system be given respectively by:


Computational Details (properties of determinants):

2. Use the identity:1. Determinant of closed loop characteristic polynomial

3. Substitute in 1:

Chapter 4: MIMO Stability Margins

Stability Margins Implications

• Basic Idea: if the deformation of the Nyquist diagram from the model to the system is achieved without changing the number of encirclements, then there is no instability due to the perturbation.

• Note: the Nyquist Criterion does not require any information on the location of the unstructured uncertainty in the loop, and it provides (formally) information about absolute stability.


Computational details

Theorem 3

AD

( )CL

sFGiven the structure in the figure, the closed loop Perturbed system is stable, that is hasNO CRHP zeros if:

A. Points 1, 2, 3 of Theorem 2 are satisfiedB. The following holds:

min max( ) ( )

AI G ss sé ù+ > Dê úë û

( , ) [0,1]R

s De" Î ´

Sketch of Proof: we are interested in verifying if and when the return difference matrix for the perturbed system becomes singular , that is:

det ( , ) 0, ( , ) [0,1]

( , ) (1 ) ( ) ( )R

I G s s D

G s G s G s

e ee e e

ì é ùï + = " Î ´ï ê úï ë ûíï = - +ïïî

RD

Theorem 4


1min max

( ) ( )R

I G ss s-é ù+ > Dê úë û( , ) [0,1]

Rs De" Î ´

Given the structure in the figure, the closed loop Perturbed system is stable, that is hasNO CRHP zeros if:

( )CL

sF


• For the multiplicative error characterization:

( , ) ( ) ( ) ( )R

I G s I G s s G se e+ = + + D

• But I + G(s) is not singular, i.e. det[I + G(s)] ≠ 0 by assumption, therefore we must find:

min0

( ); ( ) ( )R

A B

A I G s B s G s

se

ì é ùï + =ï ê úï ë ûíï = + = Dïïî

• Assume (A + B) to be singular, then (A + B) is rank deficient. This implies:

{ }2

0, 1 ( ) 0 ( )A B A B$ ¹ = + = Î +n n n N

• Equivalently:

2 2A B B A =n =- n n n

• Using singular values:

min max2 2( ) ( )A B Bs s£ A = £n n

• Which yields:

min maxdet( ) 0 ( ) ( )A B A Bs s+ = £


• In conclusion:

min max( ) ( ) det( ) 0A B A Bs s> + ¹

min max( ) ( )

AI G ss sé ù+ > Dê úë û • This proves Theorem 3

Similarly we can write1 1( , ) ( ) ( ) ( ) ; det ( ) 0

RI G s G s I G s s G se e- -é ù é ù+ = + + D ¹ê úê ú ë ûë û

1min max

det ( , ) 0 ( ) ( )R

I G s I G se s s e-é ùé ù+ ¹ + > Dê ú ê úë û ë û

• Therefore we have the implication:

1min max max

( ) ( ) ( )R R

I G ss e s s-é ù+ > D > Dê úë û • This proves Theorem 4


min max

1min max

( ) ( )

( ) ( )

A

R

I G s

I G s

s s

s s-

ìï é ù+ > Dï ê úï ë ûïïíïï é ùï + > Dï ê úë ûïî

• Input/Output Additive direct

• Input/Output Multiplicative direct

Summary: The above are the sufficient conditions described by theorems 3 and 4. This result is conservative in the sense that:

1. It considers the worst case = 12. It considers full complex uncertainty (unstructured)

1 1

1 1

( ) ( ) ( )

( ) ( ) ( ) ( )

INVERSEA

INVEI

RNP

SER

s G s G s

s G s G s G s

- -

- -

D = -é ùD = -ê úë û

• Similarly to the previous derivation:

11

1

( , ) ( ) ( )

( , ) ( ) 1 ( )

A

R

G s s G s

G s G s s

e e

e e

--

-

é ù= D +ê úë ûé ù= + Dê úë û

For additive and multiplicative inverse errors:

11 1( , ) (1 ) ( ) ( )G s G s G se e e

-- -é ù= - +ê úë û

• from which


1( )= + ( )

RL s I s

-é ùDê úë û

• The inverse uncertainty in its multiplicative form is then:

• This yields the robustness conditions for inverse perturbations expressed in multiplicative form:

1min max

( ) ( )R

I G ss s-é ù+ > Dê úë û

Theorem 5


( , ) [0,1]R

s De" Î ´


( )CL

sF

min max( ) ( )


Theorem 6


( )CL

sF


( , ) [0,1]R

s De" Î ´

RD


Depending on the uncertainty type (additive, multiplicative) and location, robustness theorems lead to the following inequalities:

min max( ) ( )


1min max

( ) ( )M

I G ss s-é ù+ > Dê úë û

Question: Can we extend the concept of Stability Margins to the MIMO Case?

• Recall the SISO case: NO changes in the Nyquist diagram encirclement of (‐1, 0) can occur if:

1

1

1 ( ) ( ) 1

1 ( ) ( ) 1M M

A A

g s L s

g s L s

-

-

ìï + > - = Dïïíï + > - = Dïïî


• We can extend the constraints to the MIMO case by introducing the singular values of the appropriate transfer function matrices:

1min max max

1min max max

M M

A A

I G L I

I G L I

s s s

s s s

-

-

ì é ù é ù é ùï + > - = Dï ê ú ê úê úï ë û ë ûë ûí é ùé ù é ùï + > - = Dï ê ú ê úê úë û ë ûë ûïî• The uncertainty must appear simultaneously in all channels either as gain perturbation or

phase perturbation (this is a conservative result due to the use of singular values as measure of size of a matrix).


1. Assume the uncertainty present as gain perturbation from nominal: { }( ) ( )i

L j diag l jw w=

• Assume the error to be bound by some value : { }1min

min I Gw

s a-é ù+ =ê úë û

• From the robust stability requirement:

1min max max

( )

( ) 1M

i i i

I G L I s

s L I l

s s s

s s

-ì é ù é ù é ùï + > - = Dï ê ú ê úê úï ë û ë ûë ûí é ù é ùï D = - = -ï ê ú ê úë û ë ûïî

• If the largest |li – 1| is less than , then we have the following result:

1 1ila a- £ £ + • Which guarantees a gain margin interval: [ 1 – , 1 + a ]


2. Assume the uncertainty present as phase perturbation from nominal:

{ }( ) ( )i

L j diag l jw w=

( )1 1ij

il e f w b- = - £• This can be rewritten as:

• Using Euler formulas:

( ) 1 cos ( ) 1 sin ( )ij

i ie jf w f w f w bé ù é ù- = - + £ê ú ê úë û ë û

• Which guarantees a phase margin interval: 12 sin2b-

æ ö÷ç ÷ ç ÷ç ÷çè ø


Note # 1: from the previous corollaries, the best minimum singular value is obtained when = 1. This yields:

1

min

1min

11 , 6 ,

2

1 0,2 , 6

I G

I G

I G GM dB

I G GM dB

s

s -

+

-

+

ì é ùïï é ù é ùê ú+ = = +¥ = - +¥ï ê ú ê úï ë û ë ûê úí ë ûïï é ù é ù é ù+ = = = -¥ +ï ê ú ê úê ú ë û ë ûï ë ûî 1

1 0 0

1 0 0

12 sin 60 , 60

2

12 sin 60 , 60

2

I G

I G

PM

PM -

-+

-

+

ì é ùæ öïï ÷ç é ùê ú÷ï = = - +ç ÷ ê úï ê úç ë û÷çï è øê úï ë ûí é ùæ öï ÷ï ç é ùê ú÷= = - +ï ç ÷ ê úï ê úç ë û÷çè øï ê úë ûïî

Note # 2: Stability Margins are a conservative measure of robustness. They may fail in the case of ill‐conditioned systems:


Example: Consider the linear‐time‐invariant (LTI) short period pitch‐plane dynamics of an unmanned aircraft

• The system has 1 input and 2 outputs (SIMO). The autopilot (controller) for this plant contains proportional ‐ plus‐integral control elements in the inner rate loop closure and outer acceleration loop closure, given by:


• Add a second order actuator model and consider a flight condition corresponding to unstable dynamics

0.6

113 / secn

rad

zw

ìï =ïíï =ïî

2

2 22n

n ns s

w

xw w+ +

C


• Rewrite the model in state space format:

• Rewrite the controller in state space format (y is the model state vector, r is the commanded Acceleration AZC ):

0.0015

0.32

2.0

6.0

az

q

q

z

K

K

a

a

=

= --

==

• Using the following numerical values:


• Build the complete state space model:

• Build the complete closed loop system:

• NOTE: the matrix Z must be invertible for the feedback problem to be well‐posed!

1

0( ) 1 0.0005 0.32 1

0c pI D D

é ùé ù ê ú- = - - =ê ú ê úë û ê úë û


• In summary:

• Closed Loop Time Histories

• Exercise: Derive the loop transfer function matrices at input to the plant Li(s) and at the output of the plant Lo(s) respectively, in literal form:

1

1

( ) ( )

( ) ( )

i Li Li Li Li

o Lo Lo Lo Lo

L s C s sI A B D

L s C s sI A B D

-

-

ìï é ùï = - +ê úï ë ûíï é ù= - +ï ê úï ë ûî


• Consider the loop broken at the input and compute the classical stability margins:

Gain Margin

Phase Margin

0

8.8

50

GM dB

PM

ìï »ïïíï »ïïî


• Multivariable Gain and Phase Margins:

min( )I Ls + 1

min( )I Ls -+

• The minimum values can be computed:

1

min

1min

1 0.6379,2.3127 3.9,7.28

1 0.2695,1.7305 11.4, 4.7I L

I L

I L GM dB

I L GM dB

s

s -

+-

+

ì é ù é ù é ùï + = = = -ï ê ú ê ú ê úï ë û ë û ë ûí é ù é ù é ùï + = = = -ï ê ú ê úê ú ë û ë ûë ûïî

( )( )1

1 0

1 0

2 sin 0.5676 32.97

2 sin 0.7305 42.84I L

I L

PM

PM -

-+

-

+

ì é ù é ùï = = ï ê ú ê úï ë ûë ûí é ù é ùï = = ï ê ú ê úë ûë ûïî


08.8 , 50GM dB PM» »

• Note that the stability margins computed via singular values are more conservative than those using classical control.

• Consider the loop broken at the and compute the sensitivity and complementary sensitivity functions

1

1

( ) ( )

( ) ( ) ( )

o

o o

S s I L s

T s L s I L s

-

-

ìï é ùï = +ê úï ë ûíï é ù= +ï ê úï ë ûî

{ }{ }

1

max

1

max

( ) sup ( )

( ) sup ( ) ( )

o

o o

S s I L s

T s L s I L s

s

s

-

¥

-

¥

ìï é ùï = +ê úï ë ûïíï é ùï = +ê úï ë ûïî


{ }{ }

1

max

1

max

( ) sup ( ) 1.0257

( ) sup ( ) ( ) 1.0734

o

o o

S s I L s

T s L s I L s

s

s

-

¥

-

¥

ìï é ùï = + =ê úï ë ûïíï é ùï = + =ê úï ë ûïî

• Comments: • The sensitivity peak 1.0257 is very small peak indicating good stability margins at the

plant output.• The complementary sensitivity represents the acceleration closed‐loop transfer

function. The sup at 1.0734 is a measure of the peak resonance in the acceleration loop. This is also small value also indicating good margins in this loop.

• It is important to verify both input and output breaking points for stability margins

• In some multivariable systems, the margins at the plant input will be adequate, but at the plant output they could be low. It is always prudent to check margins at all loop break points to make sure no sensitivity problems exist.


• The Nyquist plots for both loops are shown next, indicating good stability margins:

max( )K ss é ùê úë û

• Controller Frequency Response

• The proportional – Integral behavior is evident from the figure

• If Noise rejection is not satisfactory, additional poles are necessary (low pass filtering), however in this case stability margins may decrease due to the increased phase lag.

Chapter 4: Stability Margins of LQR, KBF, LQG

• Consider the general unity feedback structure of a LQR controller:

( )K s 1( )sI A B--1 2e(s)

u(s)

x(s)r(s)±

y(s)C

2( ) 1I L s+ ³

• We can prove the Kalman’s Inequality for the Kalman Filter as well, since it is a feedback loop itself:

1( ) ( )f

L s C sI A K-= - -


( ) 1( )

LQG C C F FK s K sI A BK K C K

-= - - + +

0

0ˆF F

F c F

A K C D K

K C A BK K

é ù é ù é ù é ù- -ê ú ê ú ê ú ê ú= = +ê ú ê ú ê ú ê ú-ê ú ê ú ê ú ê úë ûë û ë û ë û

e wq q

vx

Stability Margins for the LQG compensator


Linear Optimal Control and Estimation play a fundamental role in the design of compensators for Multivariable linear systems. Their Robustness properties (in terms of stability margins) are therefore important to evaluate.

• Consider the general unity feedback structure of a LQR controller:

( ) 1

1

( )

( ) 0Tc

G s C sI A B

K s K R B P P

-

-

ìïï = -ïíï = = ³ïïî

( )K s 1( )sI A B--1 2e(s)

u(s)

x(s)r(s)±

y(s)C

0 0

T T T T T

c

J R dt C C R dt

K

¥ ¥

é ù é ù= + = +ê ú ê úë û ë û

-

ò òy y u u x x u u

u = x

( )( )

1

1

( )

( )

c

c

L s K sI A B

I K sI A B I L s

-

-

ìïï = -ïíï + - = +ïïî


• Recall the Single Input Case. From the Riccati equation we obtain the Kalman Inequality, leading to Guaranteed Gain and Phase Margins.

11 ( ) 1 1 ( ) 1c

K sI A b L s-+ - ³ + ³

• Kalman’s inequality can be extended to the MIMO case:

1 1 1 1( ) ( ) [ ( ) ] [ ( ) ]T T T T Tc c

R B sI A Q sI A B I B sI A K R I K sI A B R- - - -+ - - - = + - - + - ³

• Perform Spectral Factorization ( ) ( ) ( )Ts W s W sF = -

1 1

1

( ) ( ) ( )

( ) [ ( ) ]

T T

c

s R B sI A Q sI A B

W s R I K sI A B R

- -

-

F = + - - -

= + - ³

• Substituting:1 1[ ( ) ] [ ( ) ]

[ ( ) ] [ ( ) ]

Tc c

T

I RK sI A B R I RK sI A B R I

W s R W s R I

- -+ - + - ³

- - ³


1( ) ( )c

R I

L s K sI A B

r-

=

= -

• Let:

( )min max1

max

1[ ( )] 1 [ ] 1

( )I L s S

I L ss s

s-

+ = ³ £é ù

+ê úê ú

ë û

• Which is satisfied if and only if:

• Kalman’s Inequality can be written as:

[ ( )] [ ( )]TI L s I L s I+ - + ³

• From the definition of MIMO stability margins, LQR gives at the input of the plant:

• This holds also for any diagonal input weight matrix R (see Athans 1978):

0 01, 60 60

2GM PM

é ù é ùê ú= ¥ = - +ê úê ú ë ûë û


• Consider the general structure of a Kalman Filter Estimator:

0 0

ˆ ˆ ˆ( )

ˆ ˆ( )F

A B K C

t

= + + -=

x x u y xx x

0 0 0

( ) 0,

( ) 0,

( ) ,

t W

t V

t Q

ì é ùï ï ê úë ûïï é ùí ê úë ûïï é ùï ê úï ë ûî

w

v

x h

A B

C

ìï = + +ïíï = +ïî

x x u wy x v

{ } { }ˆ ˆ( ) ( )T TJ E Z E Z= = - -e e x x x x

1 0T TAQ QA W QC V CQ-+ + - =1TFK QC V -=

• We can prove the Kalman’s Inequality for the Kalman Filter as well, since it is a feedback loop itself:

1( ) ( )f

L s C sI A K-= - -


• Using the Filter Riccati equation:• If the measurement noise power spectral density

is diagonal, then at the input to the Kalman gain matrix Kf there will be an infinite Gain Margin, a gain reduction margin of 0.5 and a minimum Phase Margin of ± 60 degrees.

• Consequently, for a single‐output plant, the Nyquist diagram of the open‐loop filter transfer function will lie outside the unit circle with center at ‐1.

Stability Margins for the LQG compensator


• The use of a state estimator and/or observer can make the system arbitrarily sensitive, and all Robustness considerations are lost. The reason for this is that the LQG compensator operates on a system, which is not the internal Kalman filter model.

• The sensitivity of LQG with respect to loop uncertainty was proved Doyle (1981) with a simple SISO example:

1 1 0 1

0 1 1 1

1 0

u w

y n

ì é ù é ù é ùïï ê ú ê ú ê úï = + +ï ê ú ê ú ê úïí ê ú ê ú ê úë û ë û ë ûïï é ùï = +ê úï ë ûïî

x x

x

( )( )0,

0, 1

w

n

sìï =ïïíï =ïïî

• Choose the following weights:

0

1 1, 1

1 1Q q R

é ùê ú= =ê úê úë û

01 T

c fq K Ks= = =

• Consider now the complete system with an arbitrary controller gain m and a unity filter gain:

ˆ ...c

cCL

f c f

A BK

A BKA

K C A BK K

m

C

m

= - +é ù-ê ú = ê ú- -ê úë û

x x x 1 1 0 0

0 1

0 1 1

0 1

CL

mf mfA

d d

d d f f

é ùê úê ú- -ê ú = ê ú-ê úê ú- - -ê úë û


• The characteristic Polynomial is given by:

4 3 2(.) (.) (.)

4 2( 1) 1 (1 )d f m df m df

l l l

l

+ + +é ù é ù+ - + - + + -ê ú ê úë û ë û

4 2( 1) 0

1 (1 ) 0

d f m df

m df

ìé ùï + - + - >ïê úïë ûíé ùï + - >ê úïë ûïî

• From Algebra:

• From which we can infer that for large enough d and f, df > d + f, and the closed loop system becomes unstable for arbitrarily small values of m.

Chapter 4: Uncertainty in a 2x2 Block Structure

In the previous chapter lower linear fractional transformation was used to represent a nominal closed loop system in a convenient 2 – Block structure

d1 v1

d2v2

y

u

P22(s)

K(s)

22

22

o

i

L P K

L KP

ìï = -ïíï = -ïî

( )( )

1

1o o

i i

S I L

S I L

-

-

ìïï = +ïíï = +ïïî

{ }11 12 2 12 2 1-1

1( -+ ) = ( )I P KP P K P T s

In the following, we will use the upper linear fractional transformation to incorporate uncertainty.

• Advantages: Computer mechanization, account for unstructured (full complex) as well as structured (real diagonal) uncertainties.


Upper LFT representation

122 21 1 1 21 2 2

( ) ( )N N N TN sI -é ù+ D =ê úë û- D22

22

o

i

L N K

L KN

ìï = -ïíï = -ïî

( )( )

1

1o o

i i

S I L

S I L

-

-

ìïï = +ïíï = +ïïî

d1 v1

d2v2

y

u

(s)

N11(s)

111

( )I N -- D• Existence of the inverse necessary for well – posed closed loop system as seen by the uncertainty


• How does it work?

• Example: Consider the standard feedback loop for the controlled system, where G(s) is an uncertain LTI system approximated within 5% by the model G0(s) :

0 2

1( )

0.01 1G s

s s=

+ +

0( ) ( ) ( )G s G s I sé ù= +Dê úë û

• Therefore we can write

( ) 0.05, [0, )jw wD £ " Î ¥

1. (s) is a multiplicative error, modelled with any stable LTI system having a gain less than 0.05.

1

1

( ) ( ), ( ) 1, [0, )

0.05

j W

W

w w w w¥

D = ⋅ £ " Î ¥

=

D D• Or, using a weighting function


2. In addition, a structured uncertainty is present as a 10% fluctuating feedback gain about the nominal value k0

0 0( ) ( ), ( ) 0.1k t k t t kd d= + £

Unstructured multiplicative

Structured parametric

• Objective: Combine all uncertainties in a single block


1

2

p yp

p e

é ù é ùê ú ê ú= =ê ú ê úê ú ê úë û ë û

1. Introduce fictitious input in order to isolate the uncertainties and .

1 1

2 2

q pq

q p

dé ù é ùê ú ê ú= =ê ú ê úê ú Dê úë û ë û

2. Draw the original block diagram including fictitious p and q vectors

01

0 0

2 0 1

( )( )

1 ( )

( ) ( )

G sG s

k G s

G s k G s

ìïï =ïï +íïï =ïïî

11 12

21 22

( )p M q M r p q

y M q M r y rM s

ì é ù é ùï = +ï ê ú ê ú =í ê ú ê úï = + ê ú ê úïî ë û ë û

• Collect the system and uncertainty representation, with:


• We obtain:

2 2

112 2

12

21 1 1

22 1

1 1( )

1

1( )

1

( )

( )

G GM s

G G

M s

M s G G

M s G

é ù- -ê ú= ê ú- -ê úë ûé ùê ú= ê úê úë ûé ù= ê úë û

=

M(s)


• Note: Return Difference Matrix. Non singularity required from Nyquist Criterion.


Weighting Matrices can be used to shape desired design requirements as well as uncertainty.

Recall earlier results for robustness stability with shaping

( ) ( )1

min max( ) ( ) ( ) , 0,I G j K j L js w w s w w

-é ù é ù+ > " Î ¥ê ú ê úë ûê úë û

det det ( , ) ( ) 0; [0, )

as 0 1

I GK I G j K j

G(j ) G(j )

w e w we w w

é ù é ù+ = + ¹ " Î ¥ê ú ê úë û ë û

For SISO systems min max

( ) ( ) ( )G s G s G ss sé ù é ù= =ê ú ê úë û ë û

With L(j) stable, bounded, minimum phase uncertainty

( ) ( ) ( )

( ) ( ) ( )

( ) [ ( ) ] ( )

OL

OL

I I

L s G s K s

L s G s K s

G s I s G sw

ìï =ïïï =íï + Dïï =ïî

( ) ( )( ) [ ( ) ( ); ( ) 1,] ( ) ( )OL I I OL IOL I I

L s s LL s I s sK s jG s w ww w= + D D £+ D = "

G


From SISO Nyquist, worst case stability of closed loop real system GP(s) (assuming for simplicity a stable model)

ILw

( ) ( ) 1 ( ) ;I OL OL

s L s L sw w< + "( ) ( )

1;1 ( )

Is L s

L s

ww< "

+

1( ) ( ) 1 ( ) ;

( )II

s T s T ss

w ww

< < "

For a MIMO system, the above requirement becomes a normed weighted limit on the complementary sensitivity matrix. From the general definition of H∞ norm:

max( ) sup ( )G j G j

ww s w

¥ ÎÂ

é ù= ê úë û ( ) ( ) 1;I

s T sw w¥< "

• A similar result can be obtained using Sensitivity and for other locations of uncertainty

1

1

( ) [ ( ) ] ( )

( ) [ ( )]

1;

iI iI

OL

iI

G s I s G s

S j I L s

w

w

w

-

-

ìï = + Dïïïï = +íïïïD £ "ïïî

( ) ( ) 1 ( ) ( ) 1iI iI

s S s s S sw w¥

< <

ILw


• More general weighted representation of unstructured uncertainty

W1(s) and W2(s) are used to model the uncertainty, and later on incorporated in the controller analysis and synthesis processes.

1

is fully unknown but stable and bounded in magnitude:

• The block isolated from unstructured uncertainty is given by a full complex matrix

1

1 2( ) ( )G s I W W G s

-é ù= +ê úë ûD1

1 2( ) ( )G s G s I W W

-é ù= +ê D úë û

1 2( ) ( )G s I W W G sé ù= +ê úë ûD

1 2( ) ( )G s G s I W Wé ù= + Dê úë û

1 2( ) ( )G s G s W W= + D


• Example:

( ) ( )a a

W s s³ D ( ) ( )m m

W s s³ D


• 2 – Block structure built for direct incorporation of structured uncertainty.

( )1

o mx a x udd

d

ìï = - + +ïïíï <ïïî

• Example 1:

1/s-

oa

d md

o ma dd+

1/s-

1/s-

oa

m

M

u x

wp zp

o p

p m

x a x w u

z xd

ìï = - - +ïíï =ïî


1 2

( )( )( )

kG s

s sw w=

+ +

• Example 2: Consider a second order system with gain and corner frequency uncertainties

1, 1,2,3i

id £ =

• We can isolate the 3 uncertainties, as before, creating a fictitious input‐output pair for each uncertainty

1 11

2 22

3 3

1

2

0 0

0 0

0 0

0 1

p m

p m

p m

zx

z ux

z

xy

x

dd

d

é ù é ù é ùê ú ê ú ê úé ùê ú ê ú ê úê ú= +ê ú ê ú ê úê úê ú ê ú ê úê úë ûê ú ê ú ê úë û ë û ë û

é ùé ù ê ú= ê ú ê úë û ê úë û

11 10 1 0

22 20 2

3

0 1 1 0

0 0 0 0 1

p

p

p

wx x k

u wx x

w

ww

é ùê úé ù é ù é ù é ù é ù- - ê úê ú ê ú ê ú ê ú ê ú= + + ê úê ú ê ú ê ú ê ú ê ú- - ê úê ú ê ú ê ú ê ú ê úë û ë û ë û ë û ë û ê úë û

1 1 1

2 2 2

3 3 3

0 0

0 0

0 0

p p

p p

p p

w z

w z

w z

dd

d

é ù é ù é ùê ú ê ú ê úê ú ê ú ê ú=ê ú ê ú ê úê ú ê ú ê úê ú ê ú ê úë û ë û ë û

And the 3 constant gain uncertainty Block:


M

• The block isolated from the model has a definite structure (diagonal in this case) , therefore represents a structured uncertainty

• im constitute the weights


Uncertainty using state space representation: A B

C D

ìï = +ïíï = +ïî

x x uy x u

• Assume as an example the uncertainty be additive with respect to the system matrices

• Linear dependence on uncertain parameters

• In order to separate the uncertainty, we introduce a set of fictitious inputs and outputs


, 1,...pi iz pi

w z i kd= =

( )2 2 12 21 22, , , ,B C D D D• Construct So that the uncertainty enters as a feedback law:

M

M(s)


1 2

( )( )( )

kG s

s sw w=

+ +

1, 1,2, 3i

id £ =

• Recall Example 2:


Exercise: Inverted Pendulum Example

• Neglect DC motor dynamics at the hinge, and compute the transfer function between armature current and cart position:


Summary• The basic idea is to isolate uncertainty (structured and unstructured) in a separate block, to be connected in

feedback to the plant, via additional input – output pair.


Statement: Given an uncertain 2‐Block structure with stable uncertainty transfer matrix (s), furthermore let the uncertainty bounded in magnitude : stability robustness requires finding the largest , such that the feedback system below is internally stable.

Road to the Small Gain Theorem

Chapter 4: Small Gain Theorem

• Recall the Internal Stability Requirements for MIMO Systems:

• Definition 1 : An upper Fractional transformation (LFT) F (M, U) is said to be well posed if (I –M11U) -1 exists

𝑀 𝑀 𝛥 𝑰 𝑴𝟏𝟏𝜟 𝟏𝑀 𝑇 𝑠

• Definition 2 : An upper Fractional transformation (LFT) F (M, U) is said to be well posed if (I –M11U) has a proper and stable inverse


Small Gain Theorem (Zhou, p. 212): Let a nominal system M(s) be internally stable. Moreover, let > 0 (also in the presence of stable weights); then :

• The perturbed system is well‐posed and Internally stable for all stable perturbations for which:

• The Small Gain Theorem theorem has several versions, and it is an original extension of the MIMO Nyquist Criterion

{ }-1( ) = + ( - )zw zw zu yu yw

T s P P K I P K P g¥ ¥

£

• Question: What is the relationship between M(s) and P(s)?

• Iff when = 1


Small Gain Theorem (Lavretsky, 2013, p. 134):

• Given a block diagonal, bounded, stable uncertainty matrix (s), and a stable nominal system M(s), robust stability requires the return difference matrix to be nonsingular from Nyquist.

mindet( ) 0 ( ) 0I M I Ms-D ¹ -D >

maxmin maxmax max

xma

( ) (1

( ) ( )( )

) ( )IM

M Ms ss

s s sD > D ⋅> D > • Sufficient condition due to conservatism of inequality in red (*)

Small Gain Theorem (Mackenroth, 2004, p. 388):

• Given G22(s) RH∞ , K(s) RH∞ , and additionally then the feedback system is well – posed and internally stable

( ) 1o

L s¥<

(*) The Small Gain theorem gives a necessary and sufficient condition for unstructured uncertainty, and only a sufficient condition for structured uncertainty.


• Example: Consider a SISO model given by:

0

10( )

1( ) 1

1( )

2

G ss

s

K s

¥

ìïï =ïï -ïïï D £íïïïï =ïïïî

• Nominal Plant

• Input additive uncertainty

• Controller

101s -

( )sD

( )K s

wPzP

• Compute the transfer function from wP to zP

110

1( 1)( ) 2( )

1 ( ) ( ) 4

sK sM s

K s G s s

- --= =

+ +

• The nominal system is asymptotically stable

• The maximum singular value of M11(s) is:2

max 11 11 2

1 1( ) ( )

2 16M j M j

ws w w

w

+é ù = =ê úë û +

• The H∞ is given by:

2

11 max 2

1 1 1( ) sup

2 216M j

w

ww s

w¥

é ù+ê ú= =ê úê ú+ë û


• Example: Consider a loosely coupled Antenna Control problem

1 0.1 20%(0.1)P =

• Azimuth actuator and flexibility model are system’s uncertainties

Chapter 4: Summary

Preview of Approach to robust stability and robust performance using SISO systems ( Skogestad, Multivariable Feedback Control, Chapter 7.)

• A control system is robust if it is insensitive to differences between the actual system and the model of the system which was used to design the controller. These differences are referred to as model/plant mismatch or simply model uncertainty. The key idea in the H∞ robust control paradigm is to check whether the design specifications are satisfied even for the “worst‐case” uncertainty.

• The approach can then be described as follows:

1. Determine the uncertainty set: find a mathematical representation of the model uncertainty (“clarify what we know about the system and what we don’t know”).

2. Check Robust stability (RS): determine whether the system remains stable for all plants in the uncertainty set.

3. Check Robust performance (RP): if RS is satisfied, determine whether the performance specifications are met for all plants in the uncertainty set.

Chapter 4: Summary

• It should also be appreciated that model uncertainty is not the only concern when it comes to robustness. Other considerations include sensor and actuator failures, physical constraints, changes in control objectives, the opening and closing of loops, etc. Furthermore, if a control design is based on an optimization, then robustness problems may also be caused by the mathematical objective function not properly describing the real control problem. Also, the numerical design algorithms themselves may not be robust.

• Remark: Another strategy for dealing with model uncertainty is to approximate its effect on the feedback system by adding fictitious disturbances or noise. For example, this is the only way of handling model uncertainty within the so‐called LQG approach to optimal control. Is this an acceptable strategy? In general, the answer is no. This is easily illustrated for linear systems where the addition of disturbances does not affect system stability, whereas model uncertainty combined with feedback may easily create instability.

Chapter 4: Summary

• Uncertainty in the plant model may have several origins:

Chapter 4: Summary

• How do we combine all the uncertainties in a mathematically tractable set?

• Structured Uncertainty

• Unstructured Uncertainty

• Unstructured Uncertainty• Structured Uncertainty

Chapter 4: Summary

• Uncertainty approximation as a frequency dependent complex function (or matrix in MIMO settings disc (conservative)

Chapter 4: Summary

• Disc shape representation of a complex additive uncertainty

• A(s) is any stable transfer function, at each frequency no larger than one. At each frequency then GP(s) generates a disc‐shaped region with radius |wA(j)| centered at G(s).

• wA(j) can be thought as a weight introduced to normalize the perturbation to be less than one at each frequency. A typical selection is to be stable and minimum phase.

Question:What happens in the case of a MIMO system?

1. Mathematically nothing except for

2. Graphically we have no equivalent

( ) 1A

jw¥

D £

Chapter 4: Summary

• Disc shape representation of a complex multiplicative uncertainty

• Note that for SISO systems both uncertainties are equivalent as long as:

• However, multiplicative (relative) weights are often preferred because their numerical value is more informative. At frequencies where |wI(j)| > 1 the uncertainty exceeds 100% and the Nyquist curve may pass through the origin.

Chapter 4: Summary

How do we generate the weights w(s) so that we have a single lumped complex perturbation A or I ?

Chapter 4: Summary

• Example:

Chapter 4: Summary

( ) ( )a a

W s s³ D

( ) ( )m m

W s s³ D

Chapter 4: Summary

• We have seen that uncertainties may occur in different parts of the feedback loop, depending on the type of perturbation

• We know how to combine the uncertainties in a single block set

SISO Robust Stability

Chapter 4: Summary

• Consider a system with an input multiplicative perturbation from nominal

Chapter 4: Summary

• Graphical Derivation (valid for SISO systems)

Chapter 4: Summary

• Algebraic Derivation

• Since LP and L are both stable (as an example), then

Chapter 4: Summary

Using ZN tuning By trial and error

Chapter 4: Summary

• Problem: for the example on the previous page (Skogestad 2005, Example 7.6):• Verify, the results,• Plot the step input responses of the nominal system for both cases• Plot the step responses of the perturbed system for both cases

M – structure derivation of RS‐condition. This derivation is a preview of a general analysis presented later.

Chapter 4: Summary

Chapter 4: Summary

• Apply Nyquist Criterion to the loop transfer function M instead of LP .

Chapter 4: Summary

• The M – structure provides a very general way of handling robust stability, and is essentially a clever application of the small gain theorem.

• Consider a system with an inverse input multiplicative perturbation from nominal

Chapter 4: Summary

• Assume for simplicity that the loop transfer function Lp is stable, and assume stability of the nominal closed‐loop system. Robust stability is then guaranteed if encirclements by Lp(j) of the point ‐1 are avoided, and since Lp is in a norm‐bounded set we have

Chapter 4: Summary

SISO Robust Performance

• From nominal performance analysis, we know that the sensitivity function S is a very good indicator both for SISO and MIMO systems.

• The main advantage of considering S is that because we ideally want S small, it is sufficient to consider just its magnitude.

|𝒘𝑷 𝒋𝝎 |

Chapter 4: Summary

• For robust performancewe require that the nominal performance condition to be satisfied for all possible plants, that is, including the worst‐case uncertainty.

Chapter 4: Summary

1 1( ) ( ) 1,

1 1P P PP P

w j S j SL G K

w w < = =+ +

1 1 [1 ( ) ] 1,P P I I I

w L w j GKw< + = D D+ £+

1 1P I I I

w L w L L w L< + + D -+

• the worst‐case (maximum) is obtained at each frequency by selecting |I|=1 such that the terms (1 + L) and wI(j) (which are complex numbers) point in opposite directions.

Chapter 4: Summary

1P I

w w L L+ < + { }max 1P I

RP w S w Tw

+ <

Chapter 4: Summary

1. The Robust Performance requirement is similar to vector H∞computation:

{ }max 1P I

RP w S w Tw

+ <

For example, for a vector with m elements

Chapter 4: Summary

2. The Robust Performance condition can be used to derive bounds on the loop transfer function shape |L|

{ }max 1P I

RP w S w Tw

+ <

• The first condition is most useful at low frequencies where generally |wI|<1 and |wP|>1, and |L| needs to be large for tight performance.

• Conversely, the second condition is most useful at high frequencies where generally |wI|>1 (more than 100% uncertainty)and |wP|<1, and |L| needs to be small.

Chapter 4: Summary

3. The Robust Performance condition is related to the structured singular value(SSV) For a given matrix NRP (see later), we can define:

o SSV for the present problem

max1P

PSP PS

I

w Sw S

w Tm = =

-o Skewed‐SSV for the present problem, worst‐case weighted

sensitivity

Relationship between NP, RS, and RP

• Consider a nominally stable (NS) system with multiplicative uncertainty:

Chapter 4: Summary

• From this we see that a prerequisite for RP is that we satisfy NP and RS

• Robust performance may be viewed as a special case of robust stability (with multiple perturbations).

• Consider the problem of RP in the presence of an input multiplicative uncertainty

2 2 2( ) , 1w jw D D £

Chapter 4: Summary

• Now consider the RS with combined multiplicative and inverse multiplicative uncertainty:

2 2 2( ) , 1w jw D D £

1 1 1( ) , 1w jw D D £

• Since we use the H∞ norm to define both uncertainty and performance and since the weights are the same in both cases, the test for RP for the first system is the same as the test for RS for the second system.

Chapter 4: Summary

• Recall the RP condition

• Now derive the RS condition for the case where LP is stable (second diagram). RS is equivalent to avoiding encirclements of the critical point ‐1 by the Nyquist plot of LP.

Chapter 4: Summary

•

Conclusion: The RP of a system with input multiplicative uncertainty wPPcan be tested by adding a fictitious uncertainty wII and evaluating the RS of the resulting system

Chapter 4: Summary

Graphically 1:Want RP Introduce fictitious unstructured S

Check RS

Chapter 4: Summary

Graphically 2:

Chapter 4: Summary

Robust Stability

Robust Performance

Chapter 4: Summary

Implementation of 2 block format uses RedHeffer star product

chapter 4: review of chapter 3 - unipi.itpeople.unipi.it/.../256/2019/10/4_csi_robustness.pdf ·...

Documents