Download - ME675c5_IOStability_t0
-
7/29/2019 ME675c5_IOStability_t0
1/50
I/O Stability Lecture Notes by B.Yao
1
Input-Output (I/O) Stabil ity
-Stability of a System
Outline:
Introduction
White Box and Black Box Approaches Input-Output Description
Formalization of the Input-Output View
Signals and Signal Spaces
The Notions of Gain and Phase The Notion of Passivity The BIBO Stability Concept
I/O Stability Theorems
The Small Gain Theorem The Passivity Theorem Positive Real Functions and Kalman-Yakubovich Lemma
For LTI System
-
7/29/2019 ME675c5_IOStability_t0
2/50
I/O Stability Lecture Notes by B.Yao
2
1.Introduction
Systems can be modeled from two points of view: the internal (or state-space)
approach and the external (or input-output) approach. The state-space approach takesthe white box view of a system, and is based on a detailed description of the inner
structure of the system. In the input-output approach, a system is considered to be a
black box that transforms inputs to outputs. As a consequence, the system is modeled
as an operator. The operator can be represented by either a verbal description of the
input-output relationship, or a table look-up, or an abstract mathematical mapping that
maps an input signal (or a function) in the input signal space (or a functional space) to
an output signal (or a function) in the output signal space.
2. Formalization of I/O View2.1 Signals and Signal Spaces
2.1.1Single-Input Single-Output (SISO) Systems
In SISO systems, the input and output signals can be described by real valued timefunctions, e.g., an input signal is denoted by a time-function u inR, i.e.,
( ),u u t t R
Note that to specify an input signal u, it is necessary to give its value at any time, not
values at a point or an interval.
-
7/29/2019 ME675c5_IOStability_t0
3/50
I/O Stability Lecture Notes by B.Yao
3
The input and output signal spaces can be described by normed spaces; certain norms
have to be defined to measure the size of a signal. For example,
The super norm orL-norm
0
sup ( )t
u u t
(I.1)
Application examples of using the super norm are the problem of finding
maximum absolute tracking error and the problem of dealing with control
saturation.
2L -norm
1
22
2 0( )u u t dt
(I.2)
If we are interested in the energy of a signal, 2L -norm is a suitable norm tochoose.
pL -norm
1
0( )
p p
pu u t dt , 1 p (I.3)
-
7/29/2019 ME675c5_IOStability_t0
4/50
I/O Stability Lecture Notes by B.Yao
4
Corresponding to different norms, different norm spaces are defined:
L-space
the set of all bounded functions, i.e., the set of all functions with afiniteLnorm.
: for some 0L x x M M (I.4)
pL -space
the set of all functions with a finite pL -norm., i.e.,
1
0: or ( ) for some 0
p p
p pL x x M x t dt M M
(I.5)
In general, these normed spaces may be too restrictive and may not include certain
physically meaningful signals. For example, the 2L -space does not include the
common sinusoid signals (why?). To be practically meaningful, the following
extended space is defined to avoid this restrictiveness:
-
7/29/2019 ME675c5_IOStability_t0
5/50
I/O Stability Lecture Notes by B.Yao
5
Definition [Extended Space]
IfXis a normed linear subspace ofY, then the extended space eX is the set
: for any fixed 0e TX x Y x X T , (I.6)
where Tx represents the truncation ofx at Tdefined as
( ) 0( )
0T
x t t Tx t
t T
(I.7)
The extended 1L space is denoted as 1eL , the extended 2L space as 2eL , .
Example:
2 2( ) sin but( ) but
e
e
u t t u L u Ly t t y L y L
-
7/29/2019 ME675c5_IOStability_t0
6/50
I/O Stability Lecture Notes by B.Yao
6
2.1.2Multi-Input-Multi-Output (MIMO) Systems
Multiple inputs and multiple outputs can be represented by vectors of time functions.
For example, m inputs can be represented by a m-dimension of time functions u, i.e.,
: 0 mu R with
1
( )
( )
( )
m
m
u t
u t R
u t
(I.8)
Norms corresponding to (I.1)-(I.3) are defined as
The super norm
0
sup ( )t
u u t
(I.9)
where, for avoiding confusion, ( )u t is used to represent the norm of ( )u t inmR space1.
1The super-norm of ( )u t in mR is ( ) max ( )ii
u t u t
.
-
7/29/2019 ME675c5_IOStability_t0
7/50
I/O Stability Lecture Notes by B.Yao
7
The 2L -norm
22 20 0
( ) ( ) ( )Tu u t dt u t u t dt (I.10)
The pL -norm
1
0( )
p p
pu u t dt
, 1 p (I.11)
where ( )u t represents the norm of ( )u t in mR space2.
The corresponding norm spaces are represented bym
L , 2m
L andmpL respectively. The
extended spaces arem
eL , 2m
eL andmpeL respectively.
2 Thep-norm of ( )u t in mR is
1
1
( ) ( )m pp
ipi
u t u t
.
-
7/29/2019 ME675c5_IOStability_t0
8/50
I/O Stability Lecture Notes by B.Yao
8
2.1.3 Useful Math Facts
The following lemmas state some useful facts about pL -spaces.
Lemma I.1 [Page 144 of REF1] (Holders Inequality)
If , 1,p q , and1 1
1p q
, then, pf L , qg L imply that 1fg L , and
1 p qfg f g (I.12)i.e.,
1 1
0 0 0( ) ( ) ( ) ( )
p qp qf t g t dt f t dt g t dt
(I.13)
Note:
When 2p q , the Holders inequality becomes the Cauchy-Schwartz inequality,
i.e.,
1 1
2 22 2
0 0 0( ) ( ) ( ) ( )f t g t dt f t dt g t dt
(I.14)
-
7/29/2019 ME675c5_IOStability_t0
9/50
I/O Stability Lecture Notes by B.Yao
9
An extension of the Lemma I.1 to peL space can be stated as
Lemma I.2
If , 1,p q , and1 1 1p q
, then, pef L , qeg L imply that 1efg L , and
1
T Tp qT
fg f g , 0T (I.15)
or
1 1
0 0 0( ) ( ) ( ) ( )
T T Tp qp qf t g t dt f t dt g t dt 0T (I.16)
Lemma I.3 [Page 144 of REF1] (Minkowski Inequality)
For 1,p , , pf g L imply that pf g L , and
p p pf g f g (I.17)
I/O St bilit L t N t b B Y
-
7/29/2019 ME675c5_IOStability_t0
10/50
I/O Stability Lecture Notes by B.Yao
10
Note:
(a) A Banach space is a normed space which is complete in the metric defined by itsnorm; this means that every Cauchy sequence is required to converge. It can be
shown that pL -space is a Banach space. [Pages 76-82 ofREF1]
(b) Power Signals: [REF5]
The average power of a signaluover a time span Tis2
0
1( )
Tu t dt
T (I.18)
The signal uwill be called a power signal if the limit of (I.18) exists asT , andthen the square root of the average power will be denoted bypow( )u :
12
2
0
1pow( ) lim ( )
T
Tu u t dt
T
(I.19)
I/O Stability Lecture Notes by B Yao
-
7/29/2019 ME675c5_IOStability_t0
11/50
I/O Stability Lecture Notes by B.Yao
11
(c) The sizes of common pL spaces are graphically illustrated below [REF5]
Note:
1 2L L L
Example: (verify by yourself)
1( )
1f t
t
2( )f t L L , but 1( )f t L , 1( ) ef t L
0 if 0
1( ) 0 1
0 1
t
f t tt
t
1( )f t L , but ( )f t L , 2( )f t L
2L
1L
pow
L
I/O Stability Lecture Notes by B Yao
-
7/29/2019 ME675c5_IOStability_t0
12/50
I/O Stability Lecture Notes by B.Yao
12
2.2 Gain of A System
2.2.1Definitions
From an I/O point of view, a system is simply an operator Ssuch that maps a inputsignal u into an output signaly. Thus, it can be represented by
: ie oeS X X with y Su (I.20)
where the extended norm space ieX represents the input signal space (i.e., ieu X ) and
oeX is the output signal space. The gain of a linear system can thus be introduced
through the induced norm of a linear operator defined below:
Definition [Gain of A Linear System]
The gain ( )S of a linear system S is defined as
, 0( ) supeu X u
Su
S u (I.21)
where u is the input signal to the system. In other words, the gain ( )S is the
smallest value such that
( )Su S u , eu X (I.22)
I/O Stability Lecture Notes by B.Yao
-
7/29/2019 ME675c5_IOStability_t0
13/50
I/O Stability Lecture Notes by B.Yao
13
For a linear system described by a matrix m nijA a R , the gain ( )S is theinduced norm of the matrix given by
maxi ijj a (row sum) forsuper norm,
1maxj iji a (column sum) for1-norm, and
2T
M MA A A (maximum singular value) for2-norm.
Nonlinear systems may have static offsets (i.e., 0Su when 0u ). To accommodatethis particular phenomenon, a bias term is added in the above definition of gains:
Definition[Page 147 of REF1] [Gain of A Nonlinear System]
The gain ( )S of a nonlinear system Sis defined to be the smallest value such that
( )Su S u , eu X (I.23)
for some non-negative constant.
I/O Stability Lecture Notes by B.Yao
-
7/29/2019 ME675c5_IOStability_t0
14/50
y y
14
Definition[Page 197 of REF7] [Finite Gain Input-Output Stability]
A nonlinear system S is called finite-gainLp input-output stable if the gain ( )S
defined in (I.23) is bounded (or finite), in which theLp
norm is used for input and
output signals.
Note:
Whenp=, the above finite gainL stability results in bounded-input bounded-
output(BIBO) stability. The converse is in general not true. For example, the staticmappingy=u2 is BIBO stable but does not have a finite gain.
The above definitions are valid for all systems including acausal systems. Physical
dynamic systems are always causal, which is captured by the following definition
which further simplifies the analysis:
Definition [Page 146 of REF1] [Causal Nonlinear Operators]
A mapping : n mpe peS L L is said to be causal if for every input u and every T>0, itfollows that
, 0, nT peT TSu Su T u L
In other words, the truncation of the output to u(.) up to time Tis the same as theoutput to uT(.) up to time T.
I/O Stability Lecture Notes by B.Yao
-
7/29/2019 ME675c5_IOStability_t0
15/50
15
2.2.2Linear Time Invariant Systems
An LTI system with a TF matrix G(s) can be described in input/output forms by
( ) ( ) ( )Y s G s U s in s-domain
0( ) ( * )( ) ( ) ( )
ty t g u t g t u d in time-domain (I.24)
where 1( ) ( )g t L G s is the matrix of unit impulse responses. The unit impulseresponse matrix of a strictly proper stable ( )G s has the following properties:
(a) ( ) , 0tmg t g e t for some constants 0mg and 0
(b) ( ) (0 ) ( ) , 0tmdg t g t g e t for some constants 0mdg , where
( )g t represents the induced norm of the matrix ( )
m m
g t R
.
I/O Stability Lecture Notes by B.Yao
-
7/29/2019 ME675c5_IOStability_t0
16/50
16
Lemma I.4 For 1,p (note that p includes 1 and ), if 1g L and pu L ,then
1
*p p p
y g u g u (I.28)
Proof:
1,p , let q be1 1
1p q
. Then,
0
1 1
0 0
( ) * ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
t
t tp q
y t g u t g t u d
g t u d g t u g t d
Applying Holders inequality (Lemma I.2):11
0 0
11
1 0
( ) ( ) ( ) ( )
( ) ( )
q qt tppq
t ppq
y t g t u d g t d
g g t u d
Thus, 0,T
I/O Stability Lecture Notes by B.Yao
-
7/29/2019 ME675c5_IOStability_t0
17/50
17
1
1 0 0
1 0 0
1 0
1 0
1 0 0
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
pp T t pp p
qT p
p T t pq
p T T pq
p T T pq
p T pq
g
y g g t u d dt
g g t u d dt
g g t u dt d
g g t dt u d
g g t dt u d
1 10( )
p q T p p pq
pg u d g u
So,
1 1
p p p
p p p py g u y g u
which proves the lemma for 1,p . With the same change of order of theintegration as above, the case for 1p can be proved easily. Forp we can take
u out of the integral and the result follows. Q.E.D.
I/O Stability Lecture Notes by B.Yao
-
7/29/2019 ME675c5_IOStability_t0
18/50
18
THEOREM I.1 [Page 59 of REF2]
Let ( ) m mG s R be a strictly proper rational transfer function matrix of s.
(a) The system is stable (i.e. G(s) is analytic in Re[ ] 0s or has all poles in LHP)iff 1( )g t L .
(b)Suppose that the system is asymptotically stable (or exponentially stable), then,
(i) 1 1m m mu L y L L , 1my L , y is absolutely continuous and
( ) 0y t as t ;
(ii) 2 2m m m
u L y L L , 2m
y L , y is continuous, and ( ) 0y t as
t ;(iii)
m mu L y L ,my L , y is uniformly continuous;
(iv)mu L and if ( )u t u as t , then, in addition to (iii),
( ) (0)y t G u as t , and the convergence is exponential;
(v) For (1, )p , ,m mp pu L y y L .
I/O Stability Lecture Notes by B.Yao
-
7/29/2019 ME675c5_IOStability_t0
19/50
19
Sketch of Proof:
As G(s) is strictly proper rational TF, 1( ) and ( )g t g t L . Thus, from Lemma I.4,
[1, ],p m m
p pu L y L ,
and
m
py L .
As 1m
y L , ( ) ( )y t y . Since 1m
y L , ( ) 0y . Also, the fact that 1m
u L andmg L leads to
my L . Thus, (i) is true as the absolutely continuous of ( )y t can beproved by definition.
The fact that 2,m
y y L means that 1T
y y L .Noting2 2
02 ( ) (0)
t Ty ydt y t y ,
1Ty y L leads to the results that
2( ) my t L and ( )y t is continuous and converges
as t . Since 2m
y L , it must be ( ) 0 asy t t , which proves (ii).
THEOREM I.2
If G(s) is strictly proper and stable, then, the LTI system is finite gain pL input-
output stable for [1, ]p with1
( )S g . #
Proof:Follows directly from Theorem I.1 (a) and Lemma I.4. Q.E.D.
I/O Stability Lecture Notes by B.Yao
-
7/29/2019 ME675c5_IOStability_t0
20/50
20
We introduce two norms forG(s):
-Norm
( ) : sup ( )G s G j
(I.25)
2-Norm1
22
2 1( ) ( )2G s G j d
(I.26)
Note that ifG(s) is stable, then by Parsevals theorem,
1 1
2 22 2
2 20
1( ) ( ) ( )
2G s G j d g t dt g
THEOREM I.3
Assume that G(s) is stable and strictly proper. Then, its typical input and output
relationship can be summarized by the following two tables.
I/O Stability Lecture Notes by B.Yao
-
7/29/2019 ME675c5_IOStability_t0
21/50
21
Table I.1: Output Norms and Pow For Two Typical Inputs
( ) ( )u t t ( ) sin( )u t t
2y
2( )G s
y
g
( )G j
pow( )y 0 1
( )2
G j
Table I.2: System Gains ( )S
2u u
pow( )u
2y ( )G s
y
2
( )G s 1
g
pow( )y 0 ( )G s
( )G s
I/O Stability Lecture Notes by B.Yao
-
7/29/2019 ME675c5_IOStability_t0
22/50
22
2.3 The Notion of Phase
The phase angle between two vectors 1v and 2v innR space is
1 21 2
1 2 1 22 2 2 2
cosT v vv v
v v v v (I.29)
Similarly, we can define an inner product on the signal space 2mL as
1 2 1 20( ) ( )Tx x x t x t dt
1 2 2, mx x L (I.30)
It is obvious that the induced norm by the inner product is the 2L -norm defined early.
The phase ( )u of the system Sfor a given input u may thus be defined as
2 2 2 2
cos ( )y u Su u
uu y u Su
(I.31)
Note that unlike the gain ( )S , the above definition of phase depends on the input
signal.
I/O Stability Lecture Notes by B.Yao
-
7/29/2019 ME675c5_IOStability_t0
23/50
23
2.4 Passivity
The notion of passivity is introduced as an abstract formulation of the idea of energy
dissipation. It is observed that a physical system consisting of passive elements only
(e.g. resistors, capacitors, and inductors in electrical systems, mass, spring, and
damper in mechanical systems) can only store and dissipate energy. As a result, the
output of the system to any input will follow (or conform to) the input since the
system does not have active energy to fight against the input. This phenomenon ismathematically captured by the requirement that the inner product of the input signal u
and the output signaly is non-negative, i.e.,
0y u 2eu L (I.32)
which is equivalent to saying that the phase ( )u in (I.31) is between ,2 2
, i.e.,
( )2 2
u 2eu L (I.33)
The notion of passivity can thus be formally defined as follows
I/O Stability Lecture Notes by B.Yao
D fi iti [P 154 f REF1] [P i it S t ]
-
7/29/2019 ME675c5_IOStability_t0
24/50
24
Definition [Page 154 of REF1] [Passivity Systems]
A system with input u and outputy ispassive if there exists a constant 0 independent of the control input u(t), t>0, such that
y u 2eu L (I.34)
In addition, the system is input strictly passive (ISP) if there exists 0 independent ofu such that
2
2y u u 2eu L (I.35)
andoutput strictly passive (OSP) if there exists 0 independent ofu such that2
2y u y 2eu L (I.36)
Note:
In the above definition, a bias term is added to the inequality (I.32) to
represent the effect of non-zero initial conditions (or non-zero initial energy).
Note:
The passivity condition (I.34) is equivalent to
T Ty u 0T 2eu L or
I/O Stability Lecture Notes by B.Yao
T
-
7/29/2019 ME675c5_IOStability_t0
25/50
25
0( ) ( )
Ty t u t dt 0T 2eu L
Similarly, (I.35) and (I.36) are equivalent to2
2T T Ty u u 0T 2eu L and
2
2T T Ty u y 0T 2eu L
respectively since the physical system is causal.
Example I.1:A Mass System is Passive but neither ISP nor OSP
Consider a mechanical system consisting of a pure mass m shown below
The input to the system is the applied force Fand the output is the velocity v of the
mass. Then, 2eu L
2 2 2 2
0 0
1 1 1 1( ) ( ) ( ) ( ) ( ) (0) (0)
02 2 2 2
T T
T T
Ty u v t F t dt v t mv t dt mv mv T mv mv
which indicates that the system is passive. To see that the system is neither ISP nor
OSP, choose
1
ms
u=F y=v
I/O Stability Lecture Notes by B.Yao
( ) ( ) sinu t F t t
-
7/29/2019 ME675c5_IOStability_t0
26/50
26
( ) ( ) sinu t F t t
0
1 1( ) ( ) (0) ( ) (0) (1 cos )
ty t v t v F d v t
m m
From (I.40), 0T , T Ty u is bounded. Since2 2
2 2T Tu F and
2 2
2 2T Ty v asT , there does not exist an 0 such that (I.35) or(I.36) is satisfied. Thus, the system is neither ISP nor OSP. The physical explanation
is that an inertia element only stores energy but does not dissipate energy.
Example I.2:A Damper is Passive, ISP and OSP
A damper is described by:
( ) ( ) ( ) ( )y t F t bv t bu t Thus
2 22
2 20 0
1( ) ( ) ( )
T T
T T T T y u y t u t dt b u t dt b u y
b
which indicates that the system is ISP and OSP. The physical explanation is that a
damper dissipates energy.
Lemma I.5 [Passivity & Lyapunov Formulation]
Consider a system with input u and output y. Suppose that there exists a positivesemi-definite function ( , ) 0V x t such that
u=v(t) y=F(t)
b
I/O Stability Lecture Notes by B.Yao
-
7/29/2019 ME675c5_IOStability_t0
27/50
27
( ) ( ) ( )TV y t u t g t (I.41)
Then, the system is passive if 0g . In addition, the system is
(i) ISP if2
1 2g u , 2eu L , for some 0
(ii) OSP if2
1 2g y , u for some 0 .
Note:
(I.41) is an abstraction of the energy-conservation equation of the form
Stored Energy (kinetic potential)
External Power Input Internal Power Generation
d
dt
Thus, the system with the form (I.41) is normally referred to as a system in a powerform.
I/O Stability Lecture Notes by B.Yao
Proof:
-
7/29/2019 ME675c5_IOStability_t0
28/50
28
Proof:
From (I.41):
0 0
0
( ) ( ) (0) ( )
(0) ( )
T T
T T
T
y u V g t dt V T V g t dt
V g t dt
Thus, if ( ) 0g t , the system is passive. If2
1 2
g u , then the system is ISP.
If2
1 2g y , then the system is OSP. Q.E.D.
I/O Stability Lecture Notes by B.Yao
-
7/29/2019 ME675c5_IOStability_t0
29/50
29
2.5 Input-Output Stability
From an I/O point of view, a system is stable if a bounded input will result in a
bounded output, which is described by the following formal definition:
Definition [Page 197 of REF7] [Input-OutputL Stability]
A system :m qe eS L L is I/OL stable if there exists a class Kfunction , defined
on [0, ) , and a non-negative constant such that
T LT LSu u 0T meu L (I.44)
whereL
represents one of the norms defined for a function or a signal.
It is finite-gainL stable if there exist non-negative constants and independent
ofu such that
T LT LSu u 0T meu L (I.45)
I/O Stability Lecture Notes by B.Yao
-
7/29/2019 ME675c5_IOStability_t0
30/50
30
3.I/O Stability Theorems
Having defined the notions of BIBO stability and passivity, some useful criteria can
be developed to judge the I/O stability and passivity of a complex system based on the
I/O stability and passivity of its subsystems. Specially, consider the following
feedback system Swhich is an inter-connection of two subsystems 1Sand 2S :
Fig. I.1 Feedback Connection of 1Sand 2S
We would like to judge the I/O stability ofSbased on the I/O stability of 1S and 2S .
- 1Seu
22
S + n2u
I/O Stability Lecture Notes by B.Yao
THEOREM I.4 [The Small Gain Theorem (SGT)]
-
7/29/2019 ME675c5_IOStability_t0
31/50
31
THEOREM I.4 [The Small Gain Theorem (SGT)]
Consider the system shown in Fig. I.1. Suppose that 1S and 2S are finite-gain pL
I/O stable, i.e., there exist1
,2
0 and constants1
and2
such that 0T ,
1 1 1T Tp pT py S e e pee L
2 2 2 2 2 2T Tp pT py S u u 2 peu L (I.46)
if
1 2 1 (I.47)
then the closed-loop system is finite-gain pL stable with the gain from the input u
to the output y less than 1 1 21 . In fact,
1 1 2 1 2 11 2
1
1T T Tp p p
y u n
(I.48)
Note:
(a) The small gain theorem (SGT) is useful when we analyze the robustness of acontrol design to unmodeled dynamics; the stable nominal closed-loop system can
be considered as the subsystem1
S and the unmodeled dynamics can be
represented by the subsystem 2S .
I/O Stability Lecture Notes by B.Yao
-
7/29/2019 ME675c5_IOStability_t0
32/50
32
(b) The SGT is in general quite conservative since the condition (I.47) is quitestringent. For example, suppose that 1Sand 2S are stable LTI systems with TF
1( )G s and 2 ( )G s respectively. Then, their 2L gains are 1 1( )G s and2 2 ( )G s respectively. In order to satisfy the condition (I.43), it is necessary
that the Nyquist plot of the open-loop TF 1 2( ) ( )G s G s lies within the unit circle.
This is quite conservative compare to the Nyquist Stability Criterion which only
requires that the Nyquist plot does not encircle the 1,0 point.
Proof:
Note that2e u y 2T T Te u y
2u y n 2T T Tu y n (I.49)Substituting (I.49) into (I.46), we have
1 2 1
1 1 2 2 2 1
1 1 2 1 2 1
T T T
T T p
T T T
y u y
u u
u y n
(I.50)
which leads to (I.48) if (I.47) is satisfied. Q.E.D.
I/O Stability Lecture Notes by B.Yao
3.2. Passivity Theorems
-
7/29/2019 ME675c5_IOStability_t0
33/50
33
y
3.2.1 General Passivity Theorems
Lemma I.6 [Parallel Connection]Consider the parallel connection of the subsystem 1S and 2S shown below
System S
Fig. I.2 Parallel Connection of 1S and 2S
Assume that 1Sand 2S are passive. Then, the resulting system S is passive. In
addition,
(i) if any one of 1S and 2S is ISP, then S is ISP.
(ii) if both1S
and2S
are OSP, then S is OSP. Proof:
1S
2S
u y
2y
1y
+
I/O Stability Lecture Notes by B.Yao
Noting that
-
7/29/2019 ME675c5_IOStability_t0
34/50
34
g
1 2T T T T T T y u y u y u the proof of the lemma is straightforward. Q.E.D.
Lemma I.7 [Feedback Connection]
Consider the feedback connection of the subsystem1
S and2
S in Fig. I.1 with n=0.
If 1S and 2S are passive, then S is passive. In addition,
(i) if 1S is OSP or 2S is ISP, then S is OSP.
(ii) if 1S is ISP and 2S is OSP, then S is ISP
Note:
If the two subsystems 1S and 2S are in the power form (I.41) with the associated
energy function and the dissipative function being 1 1,V g and 2 2,V g respectively, then Sin either the parallel connection in Fig. I.2 or the feedback
connection Fig. I.1 can be described in the power form with 1 2V V V and
1 2g g g .
I/O Stability Lecture Notes by B.Yao
Proof:
-
7/29/2019 ME675c5_IOStability_t0
35/50
35
Noting that
2 2T T T T T T T T T y u y e y y e y y (I.51)the proof of the passivity ofSand OSP ofS in (i) are obvious. The following is to
prove (ii).
Assume that 1S is ISP and 2S is OSP. Then, there exist 1 0 , 2 0 , 1 , and
2 such that2
1 12T T Ty e e
2
2 2 2 22T T Ty y y (I.52)
Let 1 2min , , and 1 2 0 . From (I.51),
22 2
2 22 2 2 2
2 2
2 2 2
2
2 2
T T T T T T
T T T
y u e y e y
e y u
This shows that Sis ISP. Q.E.D.
I/O Stability Lecture Notes by B.Yao
THEOREM I.5 [Passivity Theorem]
-
7/29/2019 ME675c5_IOStability_t0
36/50
36
If a system S is OSP, then, the system S is I/O finite-gain 2L stable.
Proof:
Since Sis OSP, there exists 0 and 0 such that
22T T T T T
y u Su u y 0T (I.53)
Noting2 2T T T T
y u y u , we have
2
2 2 20T T Ty y u (I.54)
which leads to
2
2 2 22
4
2
T T T
T
u u uy
u (I.55)
which proves the theorem. Q.E.D.
I/O Stability Lecture Notes by B.Yao
3 2 2 Passivity of LTI Systems
-
7/29/2019 ME675c5_IOStability_t0
37/50
37
3.2.2 Passivity of LTI Systems
An important practical feature of the passivity formulation is that it is easy to
characterize passive LTI systems in terms of the positive realness of its TF. Thisallows linear blocks to be straightforwardly incorporated or added in a nonlinear
control problem formulated in terms of passive maps
Definition [Positive Real (PR) and Strictly Positive Real (SPR)]
A rational transfer function G(s) with real coefficients is positive real (PR) if
Re ( ) 0G s Re( ) 0s
and is strictly positive real (SPR) if ( )G s is positive real for some positive.
THEOREM I.6 [Page 127 of REF3] [Conditions For PR]
A rational TF G(s) with real coefficients is PR iff
(i) G(s) has no poles in the right half plane (RHP).(ii) If G(s) has poles on thej-axis, they are simple poles with positive residues.
(iii) For all for which s j is not a pole of G(s), one has Re ( ) 0G j
I/O Stability Lecture Notes by B.Yao
THEOREM I.7 [Page 127 of REF3] [Condition for SPR]
-
7/29/2019 ME675c5_IOStability_t0
38/50
38
Let( )
( )
( )
N sG s
D s
be a rational TF with real coefficients andn be the relative
degree of G(s) (i.e., deg ( ) deg ( )n D s N s ) with 1n . Then G(s) is SPR
iff
(i) G(s) has no poles in the closed RHP (i.e., all poles are in LHP only).
(ii) Re ( ) 0G j , , .
(iii) (a) When 1n , 2lim Re ( ) 0G j .
(b) When 1n ,( )
lim 0G j
j
.
I/O Stability Lecture Notes by B.Yao
Example:
-
7/29/2019 ME675c5_IOStability_t0
39/50
39
(1)1
( )G s
s
, ors is PR but not SPR.
(2)1
( )G ss
, ors , 0 is SPR.
(3)3
( )( 1)( 2)
sG s
s s
is PR but not SPR since it violates (iii) (a) of
Theorem I.7. In fact, you can verify that
2
2 2
6Re ( ) 0
2 9G j
,
but 0 ,
2 2
2 22 2 2
3 2 3
Re ( ) 02 3 3 2
G j
,
for large . Thus 0 , ( )G s will not be PR, andG(s) is not SPR.
I/O Stability Lecture Notes by B.Yao
Corollary I.1
1
-
7/29/2019 ME675c5_IOStability_t0
40/50
40
(i) G(s) is PR (SPR) iff 1( )G s
is PR (SPR).
(ii) If G(s) is PR, then, the Nyquist plot of ( )G j lies entirely in the closed RHP.
Equivalently, the phase shift of the system in response to sinusoidal inputs is
always within 90o .
(iii) If G(s) is PR, the zeros and poles of G(s) lie in closed LHP. If G(s) is SPR, thezeros and poles of G(s) lie in LHP (i.e., G(s) is asymptotic stable and strictly
minimum phase).
(iv) If 1n , then G(s) is not PR. In other words, if G(s) is PR, then 1n .(v) (Parallel Connection) If 1( )G s and 2 ( )G s are PR (SPR), so is
1 2
( ) ( ) ( )G s G s G s .
(vi) (Feedback Connection) If 1( )G s and 2( )G s are PR (SPR), so is1
1 2
( )( )
1 ( ) ( )
G sG s
G s G s
.
I/O Stability Lecture Notes by B.Yao
Example:
-
7/29/2019 ME675c5_IOStability_t0
41/50
41
2
1( )
sG s
s as b
is not PR.
2
1( )
sG s
s s b
is not PR.
2
1( )G s
s as b
is not PR.
2
1
( )s
G s s as b
is SPR for 1, 0a b , as
2
22 2 2
( 1)Re ( ) 0,
b aG j
b a
.
I/O Stability Lecture Notes by B.Yao
THEOREM I.8 [PR and Passivity]
-
7/29/2019 ME675c5_IOStability_t0
42/50
42
Assume that ( )G s is stable (i.e., all poles in LHP). Then,
(i) The system G(s) is passive iff G(s) is PR.
(ii) The system G(s) is ISP iff there exists 0 such that
Re ( ) 0G j , (I.57)
(iii) The system G(s) is OSP iff there exists 0 such that
2
Re ( ) ( )G j G j , (I.58)
Proof:
Since G(s) is stable, it follows from Parsevals theorem that
I/O Stability Lecture Notes by B.Yao
0
1( ) ( ) ( ) ( )
2y u y t u t dt Y j U j d
-
7/29/2019 ME675c5_IOStability_t0
43/50
43
0
2 2
0
( ) ( ) ( ) ( )2
1
( ) ( ) ( )2
1 1( ) ( ) Re ( ) ( )
2
y y j j
G j U j U j d
G j U j d G j U j d
(I.59)
where YandU are Laplace transforms ofy andu, respectively.
(i) Assume that G(s) is PR, then, Re ( ) 0,G j . From (I.59), it is clear that
0y u and the system is passive.
Assume that G(s) is passive, then 0 independent ofu, s.t.
,y u u (TI.8.1)
Suppose G(s) is not PR, i.e., 0 such that 0Re ( ) 0G j .By continuity, 1 2 s.t. 0 1 2, and Re ( ) 0G j c for some
0c and 1 2, . Let a control input 1( ) ( )u t L U j
be
I/O Stability Lecture Notes by B.Yao
1 2 2 1, ,( )
kU j
-
7/29/2019 ME675c5_IOStability_t0
44/50
44
( )0 else
U j
Then, for this input, the output satisfies (I.59):
2 2
1 1
22
22 1
1Re ( ) Re ( )
( )
ky u G j k d G j d
k c
Thus, ifkis chosen such that
2
2 1( )k
c
then y u and a contradiction is obtained. This proves that G(s) is PR.
(ii)Notice that G(s) is ISP iff 0 and 0 , s.t.2 2
2 0
1( ) ,y u u U j d u
From (I.59), the above is equivalent to
20
1Re ( ) ( ) ,G j U j d u
Following the same argument as in the proof (i), the above is true iff Re ( )G j
I/O Stability Lecture Notes by B.Yao
which proves the theorem.
-
7/29/2019 ME675c5_IOStability_t0
45/50
45
(iii)The same proof as in (ii) by noting
2 2 2 2
2 0 0
1 1( ) ( ) ( )y Y j d G j U j d
#
I/O Stability Lecture Notes by B.Yao
Example:
2 2
21 1s
-
7/29/2019 ME675c5_IOStability_t0
46/50
46
2
2 22
2 22
2 2
2 22
22
22
1 1( ) , ( ) 1 2
( 1) 1
1 1( ) 1 4
1
1
1Re ( ) 0
1
sG s G j j
s
G j
G j
G(s) is PR but not SPR (why?)since there are a pair of zeros on jaxis at 1,2z j . Obviously, for 1 ,
Re ( ) 0G j and thus by Theorem I.8, G(s) is not ISP.
Since
2
Re ( ) ( )G j G j , from (iii) of Theorem I.8, G(s) is OSP.
The following example shows that in general there is no direct connection between
SPR and ISP (or OSP).
I/O Stability Lecture Notes by B.Yao
Example:
-
7/29/2019 ME675c5_IOStability_t0
47/50
47
1( ) 2G s s is SPR, and it is also ISP, but not OSP.
2
1( )
s 2G s
is SPR, and it is OSP, but not ISP.
2
3 21( )1
sG ss
is PR, but not SPR, and it is OSP but not ISP.
However, if G(s) is proper, then G(s) being SPR implies OSP. The proof is omitted.
I/O Stability Lecture Notes by B.Yao
The following theorems make the connections between PR concepts and Lyapunov
stability formulation in state space.
-
7/29/2019 ME675c5_IOStability_t0
48/50
48
y p
THEOREM I.9 [PR and Lyapunov -- Kalman-Yakubovich (KY) Lemma]
[Page 240 of REF7]
Let1( ) ( )G s C sI A B D be a p p transfer matrix where (A,B) is
controllable and (A,C) is observable. Then, G(s) is positive real iff there exist
matrices 0TP P , , andL Wsuch thatT T
A P PA L L (I.60)T T
B P C W L (I.61)T T
W W D D (I.62)
THEOREM I.10 [SPR & Lyapunov--Kalman-Yakubovich-Popov (KYP) Lemma]
[Page 240 of REF7]
Let1( ) ( )G s C sI A B D be a p p transfer matrix where (A,B) is
controllable and (A,C) is observable. Then, G(s) is strictly positive real iff there
exist matrices 0TP P , ,L andW, and a positive constant such thatT T
A P PA L L P (I.63)T T
B P C W L (I.64)T T
W W D D (I.65)
I/O Stability Lecture Notes by B.Yao
In many applications of SPR concepts to adaptive systems, the TF G(s) involves stable
pole-zero cancellations, which implies that the system associated with (A,B,C) is
-
7/29/2019 ME675c5_IOStability_t0
49/50
49
p p y
uncontrollable or unobservable, and the above KY or KYP lemma cannot be applied
since they require the controllability of the pair(A,B). In these situations, the following
MKY lemma should be used instead.
THOREM I.11 [Page 129 of REF3] [Meyer-Kalman-Yakubovich (MKY) Lemma]
Given a stable matrix A, vectors B, C and a scalar 0d . If1( ) ( )G s d C sI A B
is SPR, then, for any given s.p.d. matrix L>0, there exists a scalar 0 , a vector qand an s.p.d. matrix P such that
T TA P PA qq L
and
2T TB P C d q (I.65)
Note:
The above PR and SPR concepts for a TF G(s) can be extended to MIMO systems
with a TF matrix G(s). For details, see [REF7].
I/O Stability Lecture Notes by B.Yao
References
-
7/29/2019 ME675c5_IOStability_t0
50/50
50
[REF1] Shankar Sastry (1999),Nonlinear Systems: Analysis, Stability, and Control,
Springer-Verlag, New York, Inc.
[REF2] C. A. Desoer and M. Vidyasagar (1975), Feedback Systems: Input-Output
Properties, Academic Press, New York, Inc.
[REF3] Ioannou, P.A., and Sun, Jing (1996), Robust Adaptive Control, Prentice-Hall.
[REF4] Slotine, J.J.E. and Li, Weiping (1991),Applied Nonlinear Control, Prentice-
Hall.
[REF5] J. C. Doyle, B. A. Francis, and A. R. Tannenbaum (1992), Feedback Control
Theory, Macmillan Publishing Co.
[REF6] AstromK. J. and Wittenmark B. (1995) Adaptive Control, Second edition,
Addison-Wesley.
[REF7] Khalil, H. K. (2002), Nonlinear Systems, Third edition, Prentice-Hall.