Haddad W.M. Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach
Подождите немного. Документ загружается.
NonlinearBook10pt November 20, 2007
CONTENTS xi
Chapter 7. Input-Output St ability a nd Dissipativity 471
7.1 Introduction 471
7.2 Input-Output Stability 472
7.3 The Small Gain Theorem 476
7.4 Input-Output Dis sipativity Theory 478
7.5 Input-Output Operator Dissipativ ity Theory 481
7.6 Connections Between Input-Output Stab ility and Lyapunov
Stability 483
7.7 Induced Convolution Operator Norms of Linear Dynamical
Systems 485
7.8 Induced Convolution Operator Norms of Linear Dynamical
Systems and L
p
Stability 489
7.9 Finitely Computable Upper Bounds for |||G|||
(∞,p),(1,r)
499
7.10 Upper Bounds for L
1
Operator Norms 500
7.11 Problems 505
7.12 Notes and R eferences 509
Chapter 8. Optimal Nonlinear Feedback C ontrol 511
8.1 Introduction 511
8.2 Stability Analysis of Nonlinear Systems 512
8.3 Optimal Nonlinear Control 519
8.4 Inverse Optimal Control for Nonlinear Affine Systems 522
8.5 Gain, Sector, and Disk Margins of Nonlinear-Nonquadratic
Optimal Regulators 529
8.6 Inverse Optimality of Nonlinear Regulators 536
8.7 Linear-Quadratic Optimal Regulators 538
8.8 Stability M argins, Meaningful Inverse Optimality, and
Control Lyapunov Functions 540
8.9 Problems 547
8.10 Notes and R eferences 554
Chapter 9. Inverse Optimal Contr ol and Integrator Backstepping 557
9.1 Introduction 557
9.2 Cascade and Block Cascade Control Design 559
9.3 Optimal Integrator B ackstepping Controllers 566
9.4 Optimal Linear Block Backstepping Controllers 569
9.5 Optimal Nonlinear Block Backstepping Controllers 573
9.6 Rotating Stall and Surge Control for Axial Compression
Systems 581
9.7 Surge Control for Centrifugal Compressors 587
9.8 Problems 597
9.9 Notes and References 602
NonlinearBook10pt November 20, 2007
xii CONTENTS
Chapter 10. Disturbance Rejection Control for Nonlinear
Dynamical Systems 603
10.1 Introduction 603
10.2 Nonlinear Dissipative Dynamical Systems with Bounded
Disturbances 605
10.3 Specialization to Dissipative Systems with Quadratic Supply
Rates 608
10.4 A Riccati Equation Characterization for Mi x ed H
2
/L
1
Performance 615
10.5 Nonlinear-Nonquadratic Controllers for Systems with
Bounded Disturban ces 617
10.6 Optimal and Inverse Optimal Control for Affine Systems
with L
2
Disturbances 622
10.7 Stability Margins, Meaningful Inverse Optimali ty, a nd
Nonexpansive Control Lyapunov Functions 628
10.8 Nonlinear Controllers with Multi linear and Polynomial
Performance Criteria 633
10.9 Problems 637
10.10 Notes and References 646
Chapter 11. Robust Control for Nonlinear Uncertain Systems 649
11.1 Introduction 649
11.2 Robust Stability Analysis of Nonlinear Uncertain Systems 651
11.3 A Dissipative S y stems Perspective on Robust Stability 666
11.4 Robust Optimal Control for Nonlinear Uncertain Systems 669
11.5 Optimal and Inverse Opt imal Robust Control for Nonlinear
Uncertain Affine Systems 677
11.6 Nonlinear Guaranteed Cost Control 680
11.7 Stability Margins, Meaningful Inverse Optimali ty, a nd
Robust Control Lyapunov Functions 683
11.8 Robust Nonlinear Controllers with Polynomial Performance
Criteria 686
11.9 Robust Nonlinear Controllers with Multilinear Performance
Criteria 694
11.10 Problems 698
11.11 Notes and References 716
Chapter 12. Structured Parametric Uncertainty and Parameter-
Dependent Lyapunov Functions 719
12.1 Introduction 719
12.2 Linear Uncertain Systems and the S tructured Singular Value 722
12.3 Robust Stability Analysis of Nonlinear Uncertain Systems
via Parameter-Dependent Lyapunov Functions 731
NonlinearBook10pt November 20, 2007
CONTENTS xiii
12.4 Robust Optimal Control for Nonlinear Systems via Parameter-
Dependent Lyapunov Functions 743
12.5 Robust Control for Nonlinear Uncertain Affine Systems 748
12.6 Robust Nonlinear Controllers with Polynomial Performance
Criteria 750
12.7 Robust Nonlinear Controllers with Multilinear Performance
Criteria 754
12.8 Problems 755
12.9 Notes and R eferences 760
Chapter 13. Stability and Dissipativity Theory for Discrete-Time
Nonlinear Dynamical Systems 763
13.1 Introduction 763
13.2 Discrete-Time Lyapunov Stability Theory 763
13.3 Discrete-Time Invariant Set Stabili ty Theorems 768
13.4 Converse Lyapunov Theorems for Discrete-Time Systems 772
13.5 Partial Stability of Discrete-Time Nonlinear Dynamical
Systems 776
13.6 Stability Theory for Discrete-Time Nonlinear Time-Varying
Systems 783
13.7 Lagrange Stabili ty, Boundedness, and Ultimate B oundedness 786
13.8 Stability Theory via Vector Lyapunov Functions 792
13.9 Dissipative and Geometrically Dissipative Discrete-Time
Dynamical Systems 798
13.10 Extended Kalman-Yakubovich-Popov Conditions for
Discrete-Time Dynami cal Systems 807
13.11 Linearization of Dissipative Dynamical Systems 814
13.12 Positive Real and Bounded Real Discrete-Time Dynamical
Systems 816
13.13 Feedback Interconnections of Dissi pative Dynamical Sy stems 825
13.14 Stability Margins of Discrete Regulators 830
13.15 Control Lyapunov Functions for Discrete-Time Systems 832
13.16 Problems 834
13.17 Notes and References 843
Chapter 14. Discrete-Time Optimal Nonlinear Feedback Control 845
14.1 Introduction 845
14.2 Optimal Control a nd the Bellman Equation 846
14.3 Stability Analysis of Discrete-Time Nonlinear Systems 849
14.4 Optimal Discrete-Time Nonlinear Control 856
14.5 Inverse Optimal Control for Nonlinear Affine Systems 860
14.6 Gain, Sector, and Disk M argins of Dis crete-Time Optimal
Regulators 863
14.7 Linear-Quadratic Optimal Regulators 866
NonlinearBook10pt November 20, 2007
xiv CONTENTS
14.8 Stability Margins, Meaningful Inverse Optimali ty, a nd
Control Lyapunov Functions 868
14.9 Nonlinear Discrete-Time Dynamical Systems with B ounded
Disturbances 872
14.10 Specialization to Dissipa t ive Sy stems with Q u adratic Supply
Rates 876
14.11 A Riccati Equation Characterization for Mixed H
2
/ℓ
1
Per-
formance 882
14.12 R obust Stability A n alysis of Nonlinear Uncertain Discrete-
Time Systems 884
14.13 Problems 891
14.14 Notes and References 899
Bibliography 901
Index 939
NonlinearBook10pt November 20, 2007
Conventions and Notation
In a definition or when a word is defin ed in the text, the concept defined
is italicized. Italics in th e running text is also used for emphasis. The
definition of a word, p hrase, or symbol is to be understood as an “if and
only if” statement. Lower-case letters such as x denote vectors, upper-case
letters such as A denote matrices, upper-case script letters such as S denote
sets, and lower-case Greek letters su ch as α denote scalars; however, there
are a few exceptions to this convention. The notation S
1
⊂ S
2
means th at
S
1
is a proper subset of S
2
, whereas S
1
⊆ S
2
means that either S
1
is a
proper subset of S
2
or S
1
is equal to S
2
. Throughout the book we use two
basic types of mathematical statements, namely, existential and universal
statements. An existential statement has the form: there exists x ∈ X such
that a certain condition C is satisfied; whereas a universal statement has the
form: condition C holds for all x ∈ X. For universal statements we often
omit the words “for all” and write: condition C holds, x ∈ X. The notation
used in this book is fairly standard. The reader is urged to glance at the
notation below before starting to read the book.
Z set of integers
Z
+
, Z
+
, Z
−
, Z
−
set of nonnegative, positive, nonpositive, negative
integers
R set of real numbers
R
n×m
set of n ×m real matrices
R
n
R
n×1
(real column vectors)
R
+
, R
+
, R
−
, R
−
set of nonnegative, positive, nonpositive, negative
real numbers
C set of complex numbers
C
n×m
set of n ×m complex matrices
C
n
C
n×1
(complex column vectors)
C
+
, C
+
, C
−
, C
−
set of complex numbers with nonnegative,
positive, nonpositive, negative real parts
F, F
n
, F
n×m
R or C, R
n
or C
n
, R
n×m
or C
n×m
CLHP, OLHP closed, open left half plane
NonlinearBook10pt November 20, 2007
xvi CONVENTIONS AND NOTATION
CRHP, ORHP closed, open right half plane
√
−1
R imaginary numbers
Re z, Im z real part; imaginary part of a complex number z
△
= equals by definition
Ø empty set
{ }, { }
m
set, multiset
∪, ∩ union, intersection
∈, 6∈ is an element of, is not an element of
⊆, ⊂ is a subset of, is a proper s ubset of
→ approaches
0, 0
n×m
, 0
n
zero matrix, n ×m zero matrix, 0
n×n
I
n
, I n × n identity matrix
R(A), N(A) range space of A, null sp ace of A
x
i
, x
(i)
ith component of vector x ∈ R
n
A
(i,j)
(i, j) entry of A
col
i
(A), row
i
(A) ith column of A, ith row of A
diag[A
(1,1)
, . . . , A
(n,n)
] diagonal matrix
A
(1,1)
0
.
.
.
0 A
(n,n)
block−diag[A
1
, . . . , A
k
] block-diagonal matrix
A
1
0
.
.
.
0 A
k
,
A
i
∈ R
n
i
×m
i
, i = 1, . . . , k
A
T
transpose of A
¯
A complex conjugate of A
A
∗
¯
A
T
A
−1
inverse of A
A
†
Moore-Penrose generalized inverse of A
A
#
group generalized inverse of A
A
−T
, A
−∗
(A
T
)
−1
, (A
∗
)
−1
tr A trace of A
det A determinant of A
rank A rank of A
S
n
set of n ×n symmetric matrices
N
n
set of n ×n nonnegative-definite matrices
P
n
set of n ×n positive-definite matrices
A ≥≥ 0 (A >> 0) A
(i,j)
≥ 0 (A
(i,j)
> 0) for all i and j
A ≥≥ B (A >> B) A
(i,j)
≥ B
(i,j)
(A
(i,j)
> B
(i,j)
), where A and B
are matrices with identical dimensions
A ≥ 0 (A > 0) nonnegative (respectively, positive) definite
matrix; that is, symmetric matrix with
NonlinearBook10pt November 20, 2007
CONVENTIONS AND NOTATION xvii
nonnegative (respectively, positive) eigenvalues
A ≥ B A − B ∈ N
n
A > B A − B ∈ P
n
R
n
+
, R
n
+
{x ∈ R
n
: x >> 0}, {x ∈ R
n
: x ≥≥ 0}
⊗, ⊕ Kronecker product, Kronecker s um
x
[k]
x ⊗ ··· ⊗x (k times)
k
⊕ A A ⊕ A ⊕ ··· ⊕ A (k times)
N
(k,n)
{Ψ ∈ R
1×n
k
: Ψ x
[k]
≥ 0, x ∈ R
n
}
vec column-stacking operator
spec(A) spectrum of A including multiplicity
ρ(A) spectral radius of A
α(A) spectral abscissa of A
|α| absolute value of α
σ
i
(A) ith singular value of A
σ
min
(A), σ
max
(A) minimum, maximum singular value of A
k ·k, ||| · ||| vector or matrix norm, vector or matrix operator
norm
kxk
2
Euclidean norm of x (=
√
x
∗
x)
kxk
p
H¨older vector norms, [
P
n
i=1
|x
i
|
p
]
1/p
, 1 ≤ p < ∞
kxk
∞
max
i
|x
(i)
|
kAk
p
H¨older matrix norms,
h
P
m
i=1
P
n
j=1
|A
(i,j)
|
p
i
1/p
,
1 ≤ p < ∞
kAk
∞
max
i,j
|A
(i,j)
|
kAk
σp
[
P
r
i=1
σ
p
i
(A)]
1/p
, 1 ≤ p < ∞, r = rank A
kAk
σ∞
σ
max
(A)
kAk
s
spectral norm of A (= σ
max
(A))
kAk
F
Frobenius matrix norm of A (= (tr AA
∗
)
1/2
)
kAk
q,p
induced matrix norm
λ
i
(A) ith eigenvalue of A ∈ R
n×n
λ
min
(A), λ
max
(A) minimum, maximum eigenvalues of the
Hermitian matrix A
He A, Sh A
1
2
(A + A
∗
),
1
2
(A −A
∗
)
E
(i,j)
elementary matrix with unity in the (i, j)
entry and zeros elsewhere
log
e
logarithm with base e = 2.71828 ···
e
i
vector with unity in the ith compon ent and zeros
elsewhere
e [1, 1, . . . , 1]
T
L
p
Lebesgue space, 1 ≤ p ≤ ∞
L
2
space of square-integrable Lebesgue measurable
functions on [0, ∞)
L
∞
space of bounded L ebesgue measurable functions
NonlinearBook10pt November 20, 2007
xviii CONVENTIONS AND NOTATION
on [0, ∞)
|||f|||
p,q
{
R
∞
0
kf(t)k
p
q
dt}
1/p
, 1 ≤ p < ∞
|||f |||
∞,q
ess s up
t≥0
kf(t)k
q
hf, gi
R
∞
0
f
T
(t)g(t)dt
ℓ
p
sequence space, 1 ≤ p ≤ ∞
ℓ
2
space of square-summable sequences on Z
+
ℓ
∞
space of bounded sequences on Z
+
H
p
analytic function space
H
2
Hardy space of real-rational transfer function
matrices square-integrable on the imaginary
axis (unit disk) with analytic continuation
in the right half plane (outside the unit disk)
H
∞
Hardy space of real-rational transfer function
matrices bounded on the imaginary axis (unit
disk) with analytic continuation in
the right half plane (outside the unit disk)
ℜH
2
real-rational subspace of H
2
ℜH
∞
real-rational subspace of H
∞
[a, b] closed interval
(a, b) open interval
X × Y Cartesian p roduct of X and Y
f : X → Y function f with domain X and codomain Y
∂f
∂x
i
(x
0
) partial derivative of f with respect to x
i
at x
0
f
′
(x
0
) Fr´echet derivative of f at x
0
f
(k)
(x
0
) kth Fr´echet derivative of f at x
0
D
+
f(x
0
) upper right Dini derivative of f at x
0
D
+
f(x
0
) lower right Dini derivative of f at x
0
f
−1
(D) inverse image of the set D
f
2
◦ f
1
composition of two functions;
(f
2
◦f
1
)(·) = f
2
(f
1
(·))
L[z(t)] Laplace transform of z(·)
G(s) ∼
A B
C D
state space realization of transfer fun ction
G(s) = C(sI − A)
−1
B + D
G(s)
min
∼
A B
C D
minimal state sp ace realization of G(s)
B
ε
(α) {x ∈ R
n
: kx − αk < ε}
B
ε
[α] {x ∈ R
n
: kx − αk ≤ ε}
X\Y {x ∈ X : x 6∈ Y} for sets X and Y
∂S boundary of the set S
◦
S
interior of the set S
S closure of the set S
S
c
or S
∼
complement of the set S
NonlinearBook10pt November 20, 2007
CONVENTIONS AND NOTATION xix
inf infimum; greatest lower bound
sup supremum; least upper bound
lim in f
n→∞
f(x
n
) limit inferior of f(x
n
);
lim inf
n→∞
f(x
n
) = sup
n
inf
k≥n
f(x
k
)
lim sup
n→∞
f(x
n
) limit superior of f (x
n
);
lim sup
n→∞
f(x
n
) = inf
n
sup
k≥n
f(x
k
)
min, max minimum, maximum
C
0
continuous functions
C
r
functions with r-continuous derivatives
C
∞
infinitely differentiable functions
C[a, b] space of continuous functions
E expectation
a.e. almost everyw here
△ end of example
quod erat demonstrandum or end of proof
NonlinearBook10pt November 20, 2007