Negative Imaginary Systems Theory in the Robust Control of Highly Resonant Flexible Structures

0.11 MB·English

by  Ian R. Petersen

Negative Imaginary Systems Theory in the Robust Control of Highly
Resonant Flexible Structures
Ian R. Petersen
Abstract—This paper covers recent developments in the of the modes of a flexible structure. Furthermore, this con-
theory of negative imaginary systems and their application to trolleris robustagainst uncertaintyin the modalfrequencies
the control of highly resonant flexible structures. The theory
as well as unmodelled plant dynamics. We consider the
of negative imaginary systems arose out of a desire to unify a
theoryofnegative-imaginarysystemstorevealtherobustness
number of classical methods for the control of lightly damped
structures with collocated force actuators and position sensors properties of MIMO positive-position feedback controllers
includingpositivepositionfeedbackandintegralforcefeedback. and related types of controllers for flexible structures; e.g.,
The key result is a stability result which shows why these see [1], [7]–[15].
methods are guaranteed to yield robust closed loop stability in
3 The negative-imaginaryproperty of linear systems can be
thefaceofunmodelledspilloverdynamics.Relatedresultstobe
1 extended to nonlinear systems through the notion of coun-
presented connect the theory of negative imaginary systems to
0 positive real systems theory and a negative imaginary lemma terclockwise input-output dynamics [16]–[18]. It is shown
2
has been established which is analogous to the positive real in [19] for the single-input, single-output(SISO) linear case

n lemma. The paper also presents recent controller synthesis that the results of [16], [17] guarantee the stability of a
a results based on the theory of negative imaginary systems. positive-position feedback control system in the presence
J
I. INTRODUCTION of unmodelled dynamics and parameter uncertainties that
6
maintain the negative-imaginaryproperty of the plant.
1 The problem of vibration control for flexible structures
arises in a variety of applications such as large space struc-
A. Flexible Structure Modeling
]
tures, flexible dynamics in air vehicles, nano-positioning,
Y
the control of atomic force microscopes and the control of In modeling an undamped flexible structure with collo-
S
optical interfermetric sensing systems; e.g., see [1], [2]. A cated force actuators and position sensors, modal analysis
.
s particularproblemin vibrationcontrolforflexiblestructures can be applied to the relevant partial differential equation
c
[ isthatunmodelled(spillover)dynamicscanseverelydegrade leading to the transfer function matrix
the control system performance or lead to instability. Also, ∞
1 1
uncertainties in mode frequencies and damping can lead P(s)= ψ ψT,
2v to similar problems of poor control system performance or Xi=1 s2+κis+ωi2 i i
5 instability; e.g., see [3], [4].
where, for all i, κ >0, ω >0, and ψ is an m×1 vector.
5 An approach to the control of flexible structures which i i i
Therefore, the Hermitian-imaginary part
3 addresses these robustness issues is positive-position feed-
.
1 back, which involves the use of collocated force actuators 1 ∗
ℑ [P(ω)]=− (P(ω)−P (ω))
0 and position sensors [5], [6]. Roughly speaking a positive H 2
3 position feedback controller consists of a transfer function
1 of the frequency response matrix P(ω) satisfies
matrix which has the same form as a finite mode model of
:
v a flexible structure. This approach was originally proposed ∞ κ
Xi apsroaancheaslteirnnathtievecotnotrpoalsosifvliatyrgebasspeadcevesltoruccittyurefese.dTbhaeckoraigp-- ℑH[P(ω)]=−ωXi=1 (ωi2−ω2)i2+ω2κ2iψiψiT ≤0
r
a inal papers on positive position feedback included stability for all ω ≥ 0. That is, the frequency response matrix
proofs based on Routh-Hurwitz and Nyquist techniques but has a negative-semidefinite Hermitian-imaginarypart for all
the robustness properties were never fully quantified. The ω ≥0. We thusreferto the transferfunctionmatrix P(s) as
research presented in this paper aims to develop a general negative imaginary. Any flexible structure with collocated
systems theory framework for positive position control and force actuators and position sensors will have a negative
related controlmethodsforflexible structures. An important imaginary transfer function matrix.
special case of positive-position feedback involves the use
of piezoelectric actuators and sensors. A positive-position II. NEGATIVEIMAGINARYTRANSFERFUNCTION
feedbackcontrollercan bedesignedto increasethedamping MATRICES
ResearchsupportedbytheAustralianResearch Council (ARC) In this section, we present the formal definition of a
Ian R. Petersen is with the School of Engineering and Infor- negativeimaginarytransferfunctionmatrix.We also present
mation Technology, University of New South Wales at the Aus-
a number of extensions to this notion, which have appeared
tralian Defence Force Academy, Canberra ACT 2600, Australia.
[email protected] in the literature.
Definition 1 ( [20]–[23]): A square real-rational proper imaginary (SSNI) systems by imposing the additional con-
transfer function matrix R(s) is termed negative imaginary ditions that
(NI) if
∗
lim jω(R(jω)−R(jω) )>0
1) R(s) has no poles at the origin and in ℜ[s]>0; ω→∞
2) j[R(jω)−R∗(jω)]≥0forallω ∈(0,∞)exceptvalues and
1
of ω where jω is a pole of R(s); limj (R(jω)−R(jω)∗)>0.
3) If jω , ω ∈ (0,∞), is a pole of R(s), it is at most a ω→0 ω
0 0
simplepole,andtheresiduematrixK0 ,lims→jω0(s− This extension is useful in characterizing the strict negative
jω )jR(s) is positive semidefinite Hermitian. imaginary property using LMIs and in controller synthesis.
0
IntheSISOcase,atransferfunctionisnegativeimaginaryif The following lemmas state some useful properties of
and only if it has no poles in open right half plane (ORHP) negative imaginary transfer function matrices.
or at the origin and its phase is in the interval [−π, 0] at Lemma 2 ( [21], [23]): A square real-rational proper
all frequencies. Consequently, the Nyquist plot of a SISO transfer function matrix R(s) is negative imaginary if and
negative-imaginarytransferfunctionliesbelowtherealaxis; only if R(∞) = RT(∞) and Rˆ(s) , R(s) − R(∞) is
see Figure 1. negative imaginary.
Lemma 3 ( [15]): If the m×m transfer function matrix
Im R(s) is NI and the m×m transferfunction matrix R (s) is
s
NI, then the transfer function matrix
R (s)=R(s)+R (s) (1)
T s
is NI. Also, if in addition R (s) is SNI, then R (s) is SNI.
s T
Lemma 4 ( [15]): Consider the NI transfer function ma-
trices R(s) and R (s), and suppose that the positive-
s
feedback interconnection of this transfer functions is inter-
nally stable. Then the corresponding 2m×2m closed-loop
Re
transfer function matrix
R(s)(I −R (s)R(s))−1
R(ω) T(s) = (cid:20) R (s)(I −R(s)sR (s))−1R(s)
s s
R(s)(I −R (s)R(s))−1R (s)
R (s)(I−s R(s)R (s))−s1 (cid:21) (2)
s s
is NI. Furthermore, if in addition, either R(s) or R (s) is
Fig.1. Nyquistplotofanegative-imaginary system. s
SNI, then T(s) is SNI.
Remark 1: Note that this definition has been recently We can also can also extend the above definitions of NI
extendedto allow for a simple pole at the origin;[24], [25]. systems to the finite bandwidth case.
However, in this case, this leads to a slightly different form Definition 3 ( [28]): A square real-rational proper trans-
of the stability condition.Also, in [26],a lossless version of fer function matrix R(s) is said to be finite frequency
this definition is given along with a corresponding negative negative imaginary (FFNI) with bandwidth ω¯ if it satisfies
imaginary lemma. the following conditions:
The following lemma from [21], [23] describes the rela- 1) R(s) has no poles at the origin and in the open right-
tionshipbetweennegativeimaginarytransferfunctionmatri- half of the complex plane;
ces and positive real transfer function matrices. 2) j[R(jω)−R∗(jω)]≥0 for all ω ∈Ω, where
Lemma 1 ( [21], [23]): Given a square real-rational Ω={ω ∈R:0<ω ≤ω¯, jω is not a pole of R(s)};
strictly proper transfer function matrix R(s). Then R(s) is 3) EverypoleofR(s)onjΩ¯,ifany,issimpleandthecor-
negative imaginary if and only if respondingresiduematrixof jR(s)ispositivesemidef-
1) R(s) has no poles at the origin; inite Hermitian, where Ω¯ ={ω ∈R:0<ω ≤ω¯};
2) F(s),sR(s) is positive real. 4) R(∞)=RT(∞).
The following definition defines the strict version of the Based on the above definition, we have the following
negative imaginary property. properties.
Definition 2 (See [20], [21].): A square real-rational Lemma 5 ( [28]): If R(s) is an FFNI transfer function
proper transfer function matrix R(s) is termed strictly matrixwithbandwidthω¯,thenthefollowingpropertieshold:
negative imaginary (SNI) if 1) j[R(jω)−R∗(jω)]≤0 for all ω ∈Ω , where
1
1) R(s) has no poles in ℜ[s]≥0; Ω1 ={ω ∈R:−ω¯ ≤ω <0, jω is not a pole of R(s)}.
2) j[R(jω)−R∗(jω)]>0 for ω ∈(0,∞). 2) EverypoleofR(s)injΩ¯1,ifany,issimpleandthecor-
Remark 2: Inthepaper[27],thestrictnegativeimaginary respondingresiduematrixofjR(s)isnegativesemidef-
property is extended to a notion of strongly strict negative inite Hermitian, where Ω¯1 ={ω ∈R:−ω¯ ≤ω <0}.
Note that the stability result given in [29], uses a notion Then, A is Hurwitz and R(s) is Strongly Strict Negative
similartotheFFNIpropertydefinedabovealongwithasmall Imaginary if and only if D =DT and there exists a matrix
gain property to establish robust stability. Y =YT >0 such that
III. SOME NEGATIVEIMAGINARY LEMMAS AY +YAT <0 and B =−AYCT. (4)
The following Theorem provides a state space condition The followingtheoremfrom[28]givesa finite bandwidth
for a system to be negative imaginary. version of the Negative Imaginary Lemma.
Theorem 1 (Negative Imaginary Lemma, [20]–[23].): Theorem 5 (FFNI Lemma [28].):
Let (A,B,C,D) be a minimal state-space realization of an Consider a real-rational proper
m×m real-rational proper transfer function matrix R(s), transfer function matrix R(s) with a minimal state-
where A ∈ Rn×n, B ∈ Rn×m, C ∈ Rm×n, D ∈ Rm×m. space realization (A,B,C,D). Suppose all poles of R(s)
Then R(s) is negative imaginary if and only if are in the closed left-half of the complex plane, and the
1) det(A)6=0, D =DT, and poles on the imaginary axis, if any, are simple. Let a
2) there exists a matrix Y = YT > 0, Y ∈ Rn×n, such positive scalar ω¯ be given. Also, suppose that if A has
eigenvalues jω (i ∈ {1,...,q}) such that 0 < ω ≤ ω¯,
that i i
the residue of (sI − A)−1 at s = jw is given by
i
AY +YAT ≤0, and B+AYCT =0. Φi , lims→jωi(s − jωi)(sI − A)−1. Then the following
The following alternative form of the negative imaginary statements are equivalent:
lemma also holds in the case that the system has a simple
1) The transfer function matrix R(s) is FFNI with band-
pole at the origin.
width ω¯.
Theorem 2 (Modified NI Lemma, [22]–[25].): Let
2) det(A) 6= 0, D = DT, and the transfer function
(A,B,C,D) be a minimal realization of R(s) ∈ Rm×m,
matrix F(s) with a minimal state-space realization
where A ∈ Rn×n, B ∈ Rn×m, C ∈ Rm×n, D ∈ Rm×m,
(A,B,CA,CB) is FFPR with bandwidth ω¯.
m ≤ n. Then R(s) is negative-imaginary if and only if 3) det(A)6=0, D =DT, and CAΦ B =(CAΦ B)∗ ≥0
D =DT and there exist matrices P =PT ≥0, P ∈Rn×n, for all i ∈ {1,...,q} if A has aniy eigenvaluies on jΩ¯
W ∈ Rm×m, and L ∈ Rm×n such that the following LMI whereΩ¯ ={ω ∈R:0<ω ≤ω¯}.Also,thereexistreal
is satisfied:
symmetric matrices P = PT and Q = QT ≥ 0 such
PA+ATP PB−ATCT that
(cid:20) BTP −CA −(CB+BTCT) (cid:21)
PA+ATP −ATQA+ω¯2Q≤0, (5)
−LTL −LTW
=(cid:20) −WTL −WTW (cid:21)≤0. (3) C+BTA−TP =0, (6)
Remark 3: Note that a version of this negativeimaginary QA−1B =0. (7)
lemma for descriptor systems is presented in [30].
The followingtheoremsprovidecorrespondingconditions 4) det(A)6=0, D =DT, and CAΦiB =(CAΦiB)∗ ≥0
for strict negative imaginary properties. for all i ∈ {1,...,q} if A has any eigenvalues on jΩ¯
Theorem 3 (Weak SNI Lemma, [22], [23].): Let whereΩ¯ ={ω ∈R:0<ω ≤ω¯}.Also,thereexistreal
(A,B,C,D) be a minimal state-space realization of symmetric matrices Y = YT and X = XT ≥ 0 such
anm×m real-rationalpropertransferfunctionmatrix R(s), that
where A ∈ Rn×n, B ∈ Rn×m, C ∈ Rm×n, D ∈ Rm×m.
AY +YAT −AXAT +ω¯2X ≤0, (8)
Then R(s) is strictly negative imaginary if and only if
B+AYCT =0, (9)
1) A is Hurwitz, D =DT, rank(B)=rank(C)=m;
2) There exists a matrix Y = YT > 0, Y ∈ Rn×n, such CX =0. (10)
that Inadditiontotheabovenegativeimaginarylemmas,which
involve LMI conditions to test if a given system is negative
AY +YAT ≤0, and B+AYCT =0;
imaginaryorstrictlynegativeimaginary,thefollowingresult
3) The transfer function matrix M(s) ∼ uses an eigenvalue test of a corresponding Hamiltonian
A B matrix to test if a given system is NI.
has full column rank at s = jω
(cid:20) LY−1A−1 0 (cid:21) Theorem 6 ( [32]): Consider a transfer function matrix
for any ω ∈ (0,∞). Here LTL = −AY −YAT. That R(s)withaminimalrealization(A,B,C,D).Also,suppose
is, rank(M(jω))=m for any ω ∈(0,∞). AisaHurwitzmatrix,Q=CB+BTCT >0,andD =DT.
The following theorem provides a purely LMI approach Then, R(s) is NI if and only if the following conditions are
to test the strong SNI property. satisfied:
Theorem 4: [Strong SNI Lemma, [27], [31].] Given a 1. The Hamiltonian matrix,
square transfer function matrix R(s) ∈ Rm×m with a
sBtate∈-spRacne×mre,alCizat∈ionR(mA×,nB,aCn,dDD), w∈heRrem×Am.∈SuRpnp×osne, N =(cid:20)−−AAT+CBTQQ−−11CCAA AT −BAQT−C1TBQT−1BT(cid:21)
R(s)+R(−s)T hasnormalrankmand(C,A)isobservable. hasnopureimaginaryeigenvalueswithoddmultiplicity.
2. j(ωi+ωi+1)(R(jη )−R(jη )∗) has only positive real The above theorem can also be extended to give a suffi-
2 i i
eigenvalues for all i ∈ {1,k − 1}, where ω ∈ cient conditionforclosed loop stability in the case in which
i
Ω, and Ω = {ω ,ω ,...,ω } is the set of fre- the NI system has a simple pole at the origin.
1 2 k
quencies listed in strictly increasing order, such that Theorem 9 ( [24], [25]): Consider a strictly proper neg-
det[F(jωi)+F(jωi)∗] = 0. Here, F(s) = sR˜(s) ative imaginary transfer function matrix R(s) with minimal
where R˜(s)=R(s)−D. state space realization (A,B,C,0), which may contain a
Remark 4: This approachhas been extended in the paper simple pole at the origin. Also, consider a strictly negative
[33] to consider the problem of perturbing a non-NI system imaginary transfer function matrix R (s). Then the closed-
s
model in order to obtain an NI system model. loop positive feedback interconnection between R(s) and
Also, the paper [32] gives the following result for testing R (s) is internally stable if R (s) < 0 and the matrix
s s
the NI property of SISO systems. A+BR (0)C is not singular.
s
Assumption 1: Let A ∈ Rn×n,B ∈ Rn×1,C ∈
R1×n,D ∈ R, and suppose the transfer function G(s) = V. NEGATIVE-IMAGINARYFEEDBACK CONTROLLERS
C(sI −A)−1B +D has all of its poles and zeros in the
In this section, we present a number of different classes
closed left half of the complex plane excluding poles at the
of commonly used negative imaginary controllers; see also
origin. Also, any pole on the imaginary axis is assumed to
[15].
be simple.
Theorem 7 ( [32]): The SISO transfer function G(s) =
A. Positive-Position Feedback Controllers
C(sI−A)−1B+DwithCB >0,andsatisfyingAssumption
1 is NI if and only if the following conditions are satisfied: In the SISO case, a positive-position feedback controller
1. The matrix (I− 1 BC)A2 has no eigenvalues of odd (e.g.,see [5],[6],[12],[38]–[44])is a controllerof the form
CB
(algebraic) multiplicity on the open negative real axis. M
k
2. All residues of F(s)=s(G(s)−D) corresponding to C(s)= i , (11)
s2+2ζ ω s+ω2
poles on the imaginary axis are positive. Xi=1 i i i
IV. STABILITY OFINTERCONNECTIONS OFNEGATIVE where ωi >0, ζi >0, and ki >0 for i=1,2,...,M. Such
IMAGINARYSYSTEMS controllers are always SNI.
Thefollowingtheorem,whichisthemainstabilityresultin
B. Resonant Controllers
the theory of negative imaginary systems, gives a necessary
and sufficientconditionforthe stability of the positivefeed- Resonant controllers are proper but not strictly proper
back interconnection (see Figure 2) between an NI system SISO SNI controllers of the form
and an SNI system.
M −k s2
i
C(s)= , (12)
s2+2ζ ω s+ω2
w1 u1 y1 Xi=1 i i i
R(s)
+ + where ωi > 0, ζi > 0, and ki > 0 for i = 1,2,...,M; e.g,
see [8], [45]. The controller(12) can be implementedas the
positive-position feedback controller (11) using an accelera-
tion sensor rather than a position sensor. Alternatively, (12)
can be implemented as the positive-real feedback controller
y2 Rs(s) u2 + + w2 M k s
C¯(s)= i ,
s2+2ζ ω s+ω2
Xi=1 i i i
Fig.2. Apositive feedback interconnection. where ω > 0, ζ > 0, and k > 0 for i = 1,2,...,M,
i i i
using a velocity sensor rather than a position sensor. Such
Theorem 8 ( [20]–[23]): Given a negative imaginary controllers are always SNI.
transfer function matrix R(s) and a strictly negative imag-
inary transfer function matrix Rs(s) that also satisfy C. Integral Resonant Controllers
R(∞)R (∞) = 0 and R (∞) ≥ 0. Then the positive
s s MIMO transfer function matrices of the form
feedback interconnection [R(s),R (s)] is internally stable
s
if and only if λmax(R(0)Rs(0))<1. C(s)=[sI+ΓΦ]−1Γ
Remark 5: Note that in the papers [34], [35], the robust-
ness properties obtained by this theorem are compared with are SNI. Here, Γ is a positive-definite matrix and Φ is a
the robustness properties which are obtained from the µ positive-definite matrix. The use of a controller of this form
analysisapproach.Also,inthepapers[36],[37],thisstability when applied to a flexible structure with force actuatorsand
robustness result is extended to allow for additional norm position sensors is referred to as integral resonant control,
bounded uncertain parameters. or integral force control [1], [9], [10], [46]–[51].
VI. STATE-FEEDBACKCONTROLLERSYNTHESIS VII. CONCLUSIONS
Consider the feedback control system in Figure 3 in the We have surveyed some recent results in the area of
case that full state feedback is available. In this case, the negative imaginary systems theory. This results characterize
negativeimaginarylemmacan beused to synthesizea state- theclassofnegativeimaginarysystemsandgivemethodsfor
feedback control law such that the resulting closed-loop testingifa givensystemisnegativeimaginary.Theseresults
system is NI. Indeed, suppose the uncertain system shown alsoshowthattheclassofNIsystemsyieldsarobuststability
in Figure 3 is described by the state equations analysis result, which broadly speaking can be captured by
saying that if one system is negative imaginary and the
x˙ =Ax+B w+B u, (13) other system is strictly negative imaginary, then a necessary
1 2
and sufficient condition for internal stability of the positive-
z =C x, (14)
1
feedback interconnection of the two systems is that the dc
w =∆(s)z, (15)
loopgainislessthanunity.Thisresultprovidesaframework
forthe analysis of robuststability of lightly dampedflexible
where the uncertainty transfer function matrix ∆(s) is as-
structures with unmodelled dynamics.
sumed to be SNI with |λmax(∆(0))| ≤ 1 and ∆(∞) ≥ 0.
Applying the state-feedback control law u=Kx yields the
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