Practical Stabilization for Uncertain
Pseudo-Linear and Pseudo-Quadratic
MIMO Systems
Laura Celentano
Università degli Studi di Napoli Federico II, Dipartimento di Informatica e Sistemistica, Napoli, Italy Email: laura.celentano@unina.it
Received December 4, 2012; revised January 6, 2013; accepted January 14, 2013
ABSTRACT
In this paper the problem of practical stabilization for a significant class of MIMO uncertain pseudo-linear and pseudo- quadratic systems, with additional bounded nonlinearities and/or bounded disturbances, is considered. By using the concept of majorant system, via Lyapunov approach, new fundamental theorems, from which derive explicit formulas to design state feedback control laws, with a possible imperfect compensation of nonlinearities and disturbances, are stated. These results guarantee a specified convergence velocity of the linearized system of the majorant system and a desired steady-state output for generic uncertainties and/or generic bounded nonlinearities and/or bounded disturbances.
Keywords: Practical Stabilization; Linear and Nonlinear Uncertain Systems; Pseudo-Quadratic MIMO Uncertain
System; Lyapunov Approach
1. Introduction
The problem of practical stability and stabilization for linear and nonlinear systems subject to disturbances and parametric uncertainties together with an efficientrobust control has been in the past [1-10] one of the most re- search topic and nowadays remains actual and significant [11-21].
Indeed there exist many controlled or not systems lin-ear but with uncertain parameters, uncertain pseudo-lin- ear and with bounded coefficients, uncertain pseudo- quadratic and with bounded coefficients, having a boun- ded additional term, for which not always there exists an equilibrium state.
Regarding this, consider:
• the mechanical systems with not viscous friction and/ or with revolute joints (e.g. robots),
• the electrical and/or electro-mechanical systems with ferromagnetic devices,
• many chemical, ecological, meteorological, biological and medical systems,
with possible disturbances and reference signals which are non standard (not polynomial or cisoidal).
For the above significant systems, it is important to design a control law such that, in a finite time interval, the state evolution, for all the initial conditions belonging to a specified compact set, is bounded and such that the
evolution of the output (also the error signal) converges, with assigned minimum velocity, to chosen maximum values, that are bounded and not necessarily null.
In this paper a systematic method, in a more general framework with respect to the ones proposed in literature (see e.g. [1-5,8-10,13-18]), for the analysis and for the practical stabilization of a significant class of MIMO uncertain pseudo-linear and pseudo-quadratic systems, with additional bounded nonlinearity and/or bounded disturbances, is considered. In detail, by using the con- cept of majorant system, via Lyapunov approach, new fundamental theorems, from which derive explicit for- mulas and efficient algorithms to design state feedback control laws, with a possible imperfect compensation of nonlinearities and disturbances, are stated. These results guarantee a specified convergence velocity of the lin- earized system of the majorant system and a desired steady-state output for generic uncertainties and/or ge- neric bounded nonlinearities and/or bounded distur- bances (see also [19-21]). Finally two significant exam- ples of application, well showing the utility and the effi- ciency of the proposed results, are reported.
2. Problem Formulation and Preliminary
Results
tivariable systems ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 2 T 1,11 1,1 1 1 1, 1 1, T ,11 ,1 1 1 , 1 , m m m m y F y F y F y y F F y F F y y Gu y F F y F F y y ν ν ν ν ν ν ν ν νν ν ν ν ν νν − − − − − − = + + + + + , d + (1.1) where: ( )i di
( )
d ,i 0,1, , , y = y t t i= ν y∈Rm)
is the out- put, is the control input,models possible external sig- nals and/or particular nonlinearities of the system, , with P a compact set of
r u∈R
(
, , ,y y( )1, m d t ν − p ∈R R p∈Pμ, is the vector of the uncertain parameters, ( )1 and
)
m m(
, , i F y yν − ( )1 , yν − p ∈R × )
, m m p ∈R ×(
, ,Fk ij, y are bounded matrices, con-
tinuous with respect to their arguments,
( )
(
, , 1)
m rG y yν − ∈R × is a matrix continuous with
re-spect to its arguments and with rank m.
Now suppose that there exist the nominal values Fˆi,
, ˆ k ij F , ˆd of Fi , Fk ij, , d such that Fi = −Fi Fˆi , , k ij k ij , , ˆ k ij F F
(
, ,y F ∈ ϒ = − ( )ν −1 are functions multilinear with respect to and with respect to bounded function
where
p∈℘
g y
)
,{
p p: p−,p+}
⊇ ⊂P Rμ℘= ∈
and ϒ =
{
g g: ∈g−,g+}
⊂Rη are hyper-rectangles,and such that d= −d dˆ ≤δ
, n R ν , where δ is a constant. Pose: ( ) ( ) 1 1 1 1 ,11 ,1 ,11 ,1 , 1 , , 1 , , ˆ ˆ ˆ ˆ , , ˆ ˆ ˆ , ˆ ˆ ˆ , , 1, 2, , , n m j j j j j j j j j j n n j j n m y y x R F F F F y F F F F F F F F F F F F R j m ν ν ν ν ν νν ν νν ν × − − × = ⋅ ′ = ∈ = ∈ ′′= ′′= ′′ ′′∈ = (1.2)
and denote with F2, 1i F2, 2i F2,iν Rm n
×
∈
.
, , the matrices whose m rows are respec- tively the i-th rows of the m matrices
1, 2, ,
i= n
j F′′
Then, by controlling the system (1.1) with the follow- ing state feedback control law with a partial compensa-
tion
[
]
( )
T 1 † 1 1 T 1 † T T ˆ ˆ ˆ , ˆ , m x F u G K K K x F x x d x F G G GG ν ν − − ′′ ′ = − + + + ′′ = (1.3) it is easy to verify that the closed-loop system turns out to be(
)
[
]
1 1 2 2 1 1 1 2, 1 2, 2 2, 3 2, 1 2, 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 v v v n i i i i i i n i i i I I 0 x x I F K F K F K F K x x d F F F F I A x x A x Bd y I ν ν ν ν − − − − = = = − − − − + + = + + =
x=Cx. (1.4) To develop a practical stabilization method for the system (1.4), in a more systematic and general framework, which allows to calculate in a simple manner a control law that guarantees a specified convergence velocity, the following notations, definitions and preliminary results are provided.{
}
{
T T , , , , , , : , : ˆ , P I P r P P r P P r P r x x Px x x x x S x x r C x x r C C = = = = ≤ = = ⊇}
, (1.5) where is a symmetric and positive definitematrix and
n n
P∈R×
)
(
p d. . CˆP r, is a compact set.Definition 1. Give the system (1.4) and a sym- metric matrix . A first-order positive system
. . p d n n P∈R ×
(
,)
f ρ= ρ δ , υ η ρ=( )
, where( )
( )
P t x t ρ = and , d δ ≥ such that υ( )
t ≥ y t( )
0 is said to be majorant system of the system (1.4).Theorem 1. Consider the quadratic system
( )
2 2 1 2 2 1 1 2 0 , 0, , 0, 0 0, 0. ρ α ρ α ρ βδ α ρ α ρ α α α β ρ ρ δ = + + = + + < ≥ = ≥ ≥ (1.6) If 2 1 4 2 δ α< α β, it is (see Figure 1)1
ρ ρ0
2 ρ ρ ρ
Figure 1. Graphical representation of the system (1.6).
( )
1 0 lim , t→∞ρ t ≤ρ ∀ <ρ ρ2, (1.7) where ρ ρ ρ1, 2, 1<ρ2 2 2 1α ρ +α ρ+, are the roots of the algebraic
equation 0 . Moreover for the
practical convergence time t t
0 α = δ =0 0 5%
( )
5% ρ 5% = ρ isgiven by (see Figure 2)
0 20 5% 1 0 20 20 1 2 20 , 1 , ln 1 , l l t γτ τ α γ ρ ρ , ρ ρ ρ α α − = = − = − = − (1.8)
in which l is the time constant of the linearized of the
system (1.6) and τ ρ is the upper bound of the conver- 20 gence interval of for , i.e. of the system (1.6) in free evolution. ρ
(
)
t δ =0
Proof. The proof of (1.7) easily follows from Figure 1. Instead (1.8) easily derives by noting that the solution of (1.6) for d=0 is
( )
20 0 0 20 0 1 1 1 et l t τ ρ ρ ρ ρ ρ ρ = + − . (1.9)In Figure 2 the evolution of γ as a function of 20 0
ρ ρ is reported. By analyzing Figure 2, note that for 20 0 0.27
ρ ρ ≤ it is γ∈
[
3,3.3 ,]
i.e. 5% .Theorem 2. The solution of the Equation (1.6) with is 3 l t ≅ τ 2 1 4 0 2 0 α − α α > 1 2 2 0 1 1 2 0 2 , e 1 t a f f f ρ ρ ρ ρ ρ ρ ρ ρ ρ − − − = = − − , (1.10) where ρ ρ ρ1, 2, 1<ρ2 2 1 0 0 a ρ +aρ+a =
, are the roots of the equation .
2
Proof. From (1.6) it derives
2 1 0 1 2 1 2 2 2 2 d 1 1 1 d d , a a a a a t ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ = − − − − + + = (1.11)
from which (1.10) easily follows.
Lemma 1. If is a symmetric and p.d. matrix, is a symmetric matrix, continuous with respect to n n P∈R × n
( )
n n Q x ∈R ×x∈R , and g x
( )
∈Rn is continuous withrespect to x , then ∀ ≥ρ 0 it is:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 3 3.5 4 4.5 5 5.5 6 ρ0/ρ20 γ
Figure 2. Factor γ as a function of the normalized initial condition.
( )
(
( )
)
( )
(
)
, , , T 1 min 1 2 min ˆ min min min P P P x C x C x C x Q x x Q x P Q x P ρ ρ ρ λ ρ2 λ ρ − ∈ ∈ − ∈ ≥ ≥ (1.12)( )
( )
( )
( )
( )
, , , T T 1 T 1 ˆ max max max . P P P x C x C x C x g x g x P g x g x P g x ρ ρ ρ ρ ρ − ∈ ∈ − ∈ ≤ ≤ (1.13)Moreover, if Q x
( )
is linear with respect to x it is( )
(
( )
)
( )
(
)
, ,1 ,1 T 1 min 1 3 min ˆ min min min . P P P x C x C x C x Q x x Q x P Q x P ρ λ 3 ρ λ ρ − ∈ ∈ − ∈ ≥ ≥ (1.14)More in general, if Q x
( )
is pseudo-linear with respect to x with bounded coefficients, i.e. if( )
( )
,1 n i i i Q x Q x x = =
where Q xi
( )
are bounded, then( )
( )
( )
, ,1 ,1 T 1 1 3 min , 1 1 3 min ˆ , 1 min min min . P n P n P n i i x C i n i i x C z R i n i i x C z R i x Q x x x Q z x P Q z x P ρ λ ρ λ ρ ∈ = − ∈ ∈ = − ∈ ∈ = ≥ ≥
(1.15)Proof. Note that, if f x
( )
∈R nis a continuous func- tion with respect to x R∈ and X1⊂X2 are compact subsets of n, it is R
( )
( )
1 2 min min x X∈ f x ≥x X∈ f x ,( )
( )
1 2 max maxx X∈ f x ≤ x X∈ f x . Moreover, since P is p.d., there exists a symmetric nonsingular matrix such that
. Hence, by posing , it is
S
2
( )
( )
( )
( )
(
( )
(
)
( )
(
)
( )
(
)
, , , , , , , , , , T T , T 1 1 , 1 1 min , 1 1 1 min 1 2 min 1 2 min ˆ min min min min min min min , P P P I P I P P P P x C y C x C z C x C z C x C x C x C x C x Q x x y Q x y z S Q x S z S Q x S z z SS Q x S S Q x P Q x P ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ λ λ)
T 2 ρ λ ρ λ ρ ∈ ∈ ∈ − − ∈ ∈ − − ∈ ∈ − − − ∈ − ∈ − ∈ ≥ = = = = ≥ (1.16)and so (1.12) holds. Similarly
( )
( )
( )
( )
( )
( )
( )
( )
, , , , , , , , , T T , T 1 , 1 , T 1 T 1 ˆ max max max max max max , P P P I P I P P P x C y C x C z C x C z C x C x C x C x g x y g x z S g x z S g x g x P g x g x P g x ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ∈ ∈ ∈ − ∈ ∈ − ∈ ∈ − ∈ − ∈ ≤ = ≤ = ≤ (1.17)and hence (1.13). The inequalities (1.14) easily follow from the fact that, if Q x
( )
is linear with respect to x ,( )
x CP,( )
x CP,Q x Q x
ρ 1ρ
∈ = ∈ . The inequalities (1.15) an- alogously follow.
Remark 1. Clearly, if Q x
( )
and g x( )
are inde-pendent of x , (1.12) and (1.13) are valid with the equal sign. If Q x
( )
depends on x,( )
, T min P x C∈ ρx Q x x( )
is quite difficult to compute because, T P x C x Q x x ρ ∈ has in gen- eral different points of relative maximum, of relative minimum and of “inflection”; moreover, the second and the third member of (1.12) (of (1.14) or of (1.15)) allow an easier computation of a lower bound on
( )
, T P x C x Q x x ρ ∈ proportional to( )
2 3 ρ ρ , as it will be shown later on. A similar talking is valid if g x( )
de-pends on x .
Lemma 2. Consider a p.d. matrix and a ma- trix with rank m. If
n n
P∈R×
m n
C∈R ×
P
x ≤ then the smal- ρ
lest α such that v ≤ P, ere v=Cx ual
to
(
)
x α wh , is eq 1 T max CP C . α = λ −Proof. Since P is p.d., by posing , where is a sy
z=Sx
that
S
e mmetric nonsingular matrix such =S2, th n, in an equivalent way, the smallest α such that P
P
v ≤α x ≤αρ is also the sm llest a α such that the
matrix 2 1 T 1
I S C
α − −
values are non-n
CS− is positive sem efinite, i.e. all
its eigen egative. As
(
2 1 T 1)
2(
id)
1 1 T I−S C CS = − S C CS− − , λ α − − α λ(
1 T 1)
max S C CS .α = λ − − By taking into accou that, nt
it is
if F is a real m n× matrix with rank m , the matrix T
F F
nva
has n m− l eigenvalues and m positive ei- lues e al nulto the ones of T,
ge qu FF it follows that
(
CS S C1 1 T)
(
C T)
α = λ − − = λ 1 and, hence,
max max P C−
the proof.
3. Let be a symmetric p.d. matrix
w Lemma n n P∈R× 1 P− = h hen the
ith its inverse Pi aving unitary elements on the
main diagonal. T hyper-rectangle of n
R exter-
nally tangent to the hyper-ellipse
{
T}
x 2 ith : 1 n E= ∈x R Px=
is the hyper-quadrilateral (quadra
3
n= ) having as origin the centre and unitary
ide. Proof. I
ngle if n= , cube if
w half-s
t is easy to prove that the point of contact of the hyper-line orthogonal to the versor e ii, =1, 2, , , n
of n
R and tangent to the hyper-ellipse E is
T - column of i i i i i i i Pe p i th P e Pe = = . (1.18)
From this the proof easily follows.
l theorem is stated. The following significant and usefu
Theorem 3. Let { } 1 2 1 2 1 2 , , 1 2 , , , 0,1 π π , , πl l l i i i n n i i i l i i i A A R × ∈ =
∈ be a matrix multilinearly depending on the parameters
[
]
T{
l − +}
1 2
π π πl = ∈∏ = ∈π π R : π ≤ ≤π π
and let be a symmetric p.d. matrix. Then the (maxim n n P∈R× minimum um) of
(
−1)
(
−1)
min max π π QP QP λ λ ∈∏ ∈∏ , where(
T)
Q= − A P+PA of ∏., is attained in one of the vertices
. Note that for a constant it is
2l Proof π ,j j≠i, 0 πi 1, πi π , π .i i Q Q − − Q= + ∈ Moreover, g into T n x Qx, it turns out to by takin account that λ
(
QP−)
={
T}
1 min : 1 mi x∈x x Px= be that)
(
)
(
)
{
T}
(
1 min 0 1 π π ,π T 0 1 π π ,π , : 1 min π min π . i i i i i i i i x x x Px Q Q P x Q Q λ − + − + − ∈ ∈ ∈ = + = + x (1.19)Therefore, said π ,ˆ ˆi x the points of minimum of
(
)
(
)
{
T}
T 0 1 : 1 π ,i i x x x Px x x Q x ∈ = = +π , it is f Q(
)
(
)
(
)
(
)
{
}
1 min 0 1 π π ,π T T 0 0 π π ,π 0 1 0 1 0 1 π π ,π min π ˆ ˆ ˆ ˆ min π ˆ ˆ ˆ ˆ ˆ min π min π , π . i i i i i i i i i i i i i Q Q P x Q x x Q x c c c c c c λ − + − + − + − ∈ ∈ − + ∈ + = + = + = + + ˆi (1.20)From (1.20) the proof easily follows.
3.
result, concerning the analysis of sta-
m of the system (1.4) is
Main Result
Now the first main bility, can be stated.
Theorem 4. Give a symmetric p.d. matrix P∈Rn n× .
Then a majorant syste
2 1 2 ,v c , ρ α ρ α ρ= + +βδ = ρ (1.21) in which:
(
)
(
)
(
)
(
)
(
)
1 1 min 1 1 , T 1 1 1 1 min 2 1 2 , , T 2 2 2 T max 1 T max min , 2 min , 2 , , p g P p g p V g V n i i i x V p V g V i i i Q P Q A P PA Q x P Q A P PA B PB c CP C d λ α λ α β λ λ δ − ∈ ∈ − = ∈ ∈ ∈ − = − = − + = − = − + = = ≥
(1.22) where Vp and Vg andare the sets of vertices of the hyper- rectangle ℘ of the hyper-rectangle ϒ respec- tively, and V i the set of vertices of the hyper-rec- P1
tangle of n externally tangent to the hy r-ellipse
s
R pe
{
n: T 1}
x∈R x P = .
Proof. By choosing as “Lyapunov function” the quad-
x ratic form T 2 2 P V =x Px= x =ρ , for x belonging to a generic hyper-circumference
{
T 2}
, : P C ρ = x x Px=ρ , it is 2 , T T T 1 2 , , 1 max 2 P n i i x C ρ p t R x Q x x i Q x x x PBd . ρρ ∈ ∈℘ ∈ = ≤ − − +
(1.23) The proof easily follows from (1.23), Lemmas 1, Theorem 3.control law, follows from Theorem 4. 2 and The second main result, concerning the synthesis of the stabilizing Theorem 5. Let
( )
1 1 1 1 a p λ =βνaν+βν−aν−λ+ + β λa ν− +λν be th orth filter ofe characteristic polynomial of the low-pass Butterw order ν with cutoff frequency ω = . n a
If in (1.3) it is posed
[
Kν Kν 1]
ν ν − 1 1 1 1 , a I a I aI ν ν β β − β − K = (1.24)in which I is the identity matrix of order m, th
rant system of the system (1.4) with respect to the norm en a majo- P x , with P=T PTa 1 a, where
( )
{ }
{ }
1 V∗ − = 1 1 2 1 1 1 1 1 2 1 2 , 1 1 , 0 0 0 0 diag , 0 0 i j n i a P V I I I I I I V I I I I a I a I T a I I ν ν ν ν ν ν ν λ λ λ λ ν ν λ λ λ − − − − − − − = = = = (1.25) being ( ) π 2 1 2 ei k k ν ν λ = + − , k=1, 2, ,ν , the k-th root of for , is( )
a p λ a=1 2 1 2 1 1 , , a a a v aν ρ γ ρ γ ρ +βδ = − ρ (1.26) in which: = − + (
1)
(
)
min 1 1 T 1a m, 1 1 1 1 1 1 1 2 1 1 2 1 1 2 , , 2 in 0 0 0 0 0 0 0 0 0 p g a a a a p V g V a v v v Q P Q A P P A I I A I F F F F I I I I a aν ν aν ν aν ν λ γ β β β β − ∈ ∈ − − − − − − = − + = = − − − − (1.27) (
)
1 min 2 1 1 2 , , 1 T 2 2 1 1 2 2 1 1 floor 2, 1 2, 2 2, 1 2 min , 2 , 0 0 0 1 , 0 0 0 1 p g i n ai i i a p V g V z ai ai ai ai i m i i i Q z P Q A P P A A F F F a a a ν ν ν ν λ γ − = ∈ ∈ =± − − − − − = − = − + =
(1.28)( )
1 , 1.414, 2.236,3.696,6.236, , 2,3, 4,5, . P n n n β = = = (1.29)Proof. By making the change of variable , it is easy to prove that the system (1.4) becomes a
z=T x
(
)
1 2 1, , 10 0 . a ai i i n a y C z I aν z = − = = 1.30) , z=aA z+
A z z+Bd (Moreover, note that the Butterworth ei have unitary magnitudes; hence all the
elements of the matrix are unitary. From co , th genvalues λ k main diagonal 1 1 VV∗ =P− this
nsideration, from (1.30), from Lemmas 2, 3 and Theo- rem 4, the proof easily follows.
Theorem 6. For a→ ∞ e parameters γ and 1a 2a
γ of the majorant system (1.26) turn out to be
( )
(
)
1 ˆ1
lim co max Real
2 0.7071,0.5000,0.3827,0.3090, , a i a i 1 s π 2,3, 4,5, ν γ γ λ ν →∞ = = = − = ν = (1.31) − ( )
(
)
1 min 2 1 1 1 2 2 , , 1 T 2 2 1 1 2 2 2, , ˆ lim min , 2 0 0 0 . 0 0 0 0 0 p g i m n i i i m a p V g V z a i i i i i Q z P Q A P P A A F ν ν ν λ γ γ = − = − + ∈ ∈ =± →∞ − = = = − + =
(1.32) Proof. It is easy to verify that for the matrixis the limit of the eigenvectors m the matrix
a→ ∞
atrix of
V
1a
A (see (1.27)); hence it is that A1∞ = ΛV V−1,
T 1
1 1
A∞ =A∗∞=V∗−Λ∗ ∗V , where
{
}
{
}
{
1 1}
diag diag λ, ,λ , ,diag λν, ,λν
= . Therefore Λ
(
)
(
)
(
)
(
)
1 min 1 1 T 1 min 1 1 1 1 1 min 1 1 1 min max , Q P A PA P V V V V A VV V V V V λ λ λ λ λ − − ∞ ∞ ∗− ∗ ∗ ∗− − ∗ ∞ ∗− ∗ ∗ ∗− ∗ ∗ = − − = − Λ − = − Λ − Λ = −(
Λ + Λ)
(1.33) hence (1.31).Moreover it is easy to verify that
(
)
(
)
2 2, 1, 2, , 1 ; lim , 1 1, , . 0 0 0 0 0 ai a i i m 2 lim ai 0, a A 0 0 0 A i m m F ν ν ν ν →∞ = − = = − + →∞ = (1.34) From (1.28) and (1.34) the expression (1.32) easily follows.Remark 2. If m=1 it is easy to prove that
(
)
(
T 1)
2 max P1 e e P Pn n 1 1 2 γ =λ + − 2 ˆ2 2 2, lim a max , a F νν γ γ γ →∞ = = T 1.207,1.618, 2.348,3.618, , 2,3, 4,5, . n n e e ν = = (1.35)From Theorems 5 and 6 the following result de Theorem 7. Consider the system (1.4) with
rives. 1, 2, , K K Kν ram the tim system (1.26)
provided from (1.24). If the design pa- , t eter a is big enough, from a practical point of view
majoran e constant τ of the linearized of the l
is inverse proportional to a and it coincides with the maximum time constant of the linearized of the system (1.4). More in detail, if a is large enough it turns out to be
( )
(
)
(
)
1 1 1 ˆ 1 1 ,max real eig
l l a a a A a τ τ = γ ≅ − ≅ (1.36) 1 ˆl 1.414, 2.000, 2.613,3.236, , 2,3, 4,5, ; τ = ν =
moreover, if a is sufficiently large, for t large enough it is
( )
1 , 2.000, 4.472,9.657, 20.180, , 2,3, 4,5, , ˆ g y t a g νδ β ν γ ≤ ≅ = = (1.37) or, more in general, it is( )i
( )
, 0,1, , 1 i g y t i aν− δ ν ≤ = − .lows from (1.26), (1.31) and by noting that
1
0
(1.38) Proof. (1.36) easily fol
1 1 2 0 0 0 0 0 a a I 0 0 0 lim I A I I I I ν ν ν β β β β →∞ − − I = − − − − . (1.39)
The inequality (1.37) follows from (1.7), from th that if a is sufficiently large then
e fact 1 1 , a a β ρ δ γ ≅ from the se
king into acc .
x
4. Examples
The following examples show the utility and th in the previous sections. Consider the pseudo-linear uncertain cond of (1.26) and from (1.29) and (1.31).
(1.38) analogously follows by ta ount that
( )1
[
0 0 , ,]
( )[
0 0]
y = =y I x yν = I
(1.40)
e effi- ciency of the results stated
system
(
1 sin)
1 1 2 y= −p y− +p y y−p y+ +u d , (1.41) where p1=p2=1% 20%±(
, , , , ,y y y p p 1 2)
≤δ . By and 1 1 2 d t posing = and by of the system (1.41) controlled with the cont1 ˆ ˆ ˆ ˆ sin , 1, 0, 0 g = y p = p = g = d m 5, the majorant s applying Theore ystem rol law 2a y − turns out to be
(
) (
)
(
)
3 2 1 1 sin 2 1 u= −a y+ + y− a y+ 1 2 1 2.236 , . a a v a ρ= − γ ρ+ δ = ρFigure 3 the value of
(1.42) In γ for 1a a∈
[
1, 20]
at f
is re- ported. It is significant to note th or a≥ it is 6
1a 0.9517 γ ≥ , i.e. γ =1a 0.5 unless is 5%, in 0.4867; accord with Theorem 7. For a=10 it γ =1a hence
2.054 10 0.2054
l s
τ = = . Moreover, being γ = , it is 2a 0
1 4.849 a
ρ = and ρ = ∞ . Therefore, at “steady state”, 2
: , d d δ ∀ ≤ it is: 3 3 2 4.849 4. y ≤ δ 10 ≅ 472δ a , y ≤4.849δ a and 4.849 y ≤ δ a .
Example 2. Consider the system of Figure 4 d by the equation escribed
( )
( )
(
)
(
)
(
)
1 5 1 1 1 0 2 1000 p p p y 1000 p y = − − + − 3 3 5 2 1 1 2 1 4 1 2 4 1 2 2 4 1 2 1 2 0 0 0 1 2 1000 0 0 1 0 1000 1 0 1 0 1 , 1 1 1 1 1000 p p y p sg y y y p sg y p sg y y p sg y y y y p sg y y d u d − − + − + − + + − + + − +− (1.43) in which d1= − +5 δ δ1, 1 ≤1,d2 = − +5 δ δ2, 2 ≤bance actions due to several causes include the road, p1= p3=5% 20%± ,
1 are
the distur d the
slope of p2,p4∈
[ ]
0 2 , 5 1% 20 p = ±( )
% and(
)
( )
sg σ =2 π tan 1000a σ ≅sgn σ . By posing pˆ1=pˆ3 nd by a 5 = , a ppl 2 4 ˆ ˆ 0 p = p = , ying Theorem 5 ˆ 1 p = 5 the m , ajorant 1 2 ˆ ˆ 5 d =d = − 0 2 4 6 8 10 12 14 16 18 20 0 0.1 0.2 0.3 0.4 0.5 a γ1a Figure 3. γ1a as a function of a.system of (1.43), controlled by using the control law
(
)
2 1 2 2 2 5 1 0 1 0 5 2 5 1 1 1 1 5 1 0 1000 2 , 1 1 y y u y y a y ay − + = + +− − + (1.44) turns out to be 2 1a 0.0079 1.414 , γ ρ ρ δ 1 1 0 δ 1 2 max =2.236, . 1 1 a v a ρ δ δ = − + + ρ = − = .45)In Figure 5 the value of
(1 1a γ a∈
[ ]
1, 20 is re- 13.5 a≥ it forported. It is significant to no at for turns out to be te th 1a 0.6366 γ ≥ , i.e. γ =1a 0.7071 a= unless 10 it is 1a %, in For
accord with Theorem 7. 10 γ =0.6111; hence τ =l 0.1887s. Moreover it is ρ =1 0.5177, ρ = 2 773.0. Therefore, at “steady state”, ∀ , δ1 δ δ ≤ , 2: 1 1
2 1 δ ≤ , it is: 2 0.05177 2 2.236 , y ≤ ≅ × a 0.5177 2 2.236 . y ≤ ≅ a
Figure 6 shows the values of
( )
×
y t and of y t
( )
,obtained fo 1 2= , 1 p3 4 p5= , 1 δ1
( )
t and δ2( )
t square waves of amplitude 1 uency[
0.2 0.2 0]
0 0 0 0.2 0.2T. re lights that the proposed stabilization ho on ve, as it cr 1.
Th
met d is little c servati an be easily verified by simulating the stabilized system for severa di- tions and numerous values of the parameter
5.
r the roble of analysis d practical 5 p = , high p 0 5 = , T , x = 1 p = , and freq
[
]
2 Hz, x0 is figu= l initial con s.Conclusions
In this pape p m anFigure 4. Vehicles to be kept still at a distance ds.
0 0.8 2 4 6 8 10 12 14 16 18 20 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 γ a 1a Figure 5. γ1a as a function of a.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 [s] ||y|| ||dy/dt||
Figure 6. Possible time histories of y t and of
( )
y t .( )
stabilization of a significant class of MIMO nonlinea systems subject to parametric uncertainties, including linear and quadratic ones with an additional bounded nonlinearities and/or disturbances, has been approached By using the concept of majorant system and via Ly-apunov approach, new useful results, explicit form and efficient algorithms for designing state feedbacc rfect tion
onlinearities and disturbances, have been stated. Th
of Optimization r
. ulas k ontrol laws, with a possible impe compensa of
n ese
results have been proved that guarantee a specified con- vergence velocity of the linearized of the majorant sys- tem and a desired steady-state output for generic uncer- tainties and/or nonlinearities and/or bounded distur- bances.
The utility and the efficiency of the these results have been shown with two illustrative example.
The presented results can be used to establish further new useful theorems for the tracking of trajectories for relevant MIMO systems, like e.g. the robots.
In this direction the research of the author is going on.
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