Chapter 1 Introduction
1.4 Organization of this Dissertation
This dissertation comprises five chapters. In Chapter 1, the introduction comprises motivation, related works, approach, and organization of this dissertation. In Chapter 2, foundations are described by providing concepts of Lyapunov stability and linear matrix inequality. In Chapter 3, LMI-based robust sliding control design methods are developed for different uncertain Takagi-Sugeno fuzzy models with
matched/mismatched parameter uncertainties and external disturbances which are bounded by known scalar valued functions and meantime we relaxed the restrictive assumption that each nominal local system model shares the same input channel, which is required in the traditional VSS-based fuzzy control design methods. Besides, a robust sliding control design method is also presented for the uncertain T-S time-delay model with mismatched parameter uncertainties and external disturbances. Finally, some examples are used to illustrate the effectiveness of the proposed methods for distinct uncertain T-S fuzzy models and to compare with the existing methods in each final subsection. In Chapter 4, LMI-based robust adaptive control design methods are proposed for distinct uncertain T-S fuzzy models which include matched/mismatched parameter uncertainties and unknown norm-bounded external disturbances. Moreover, a robust adaptive control design method is also proposed for the uncertain T-S time-delay model with mismatched parameter uncertainties and external disturbances. Finally, some examples are used to illustrate the effectiveness of the proposed methods for distinct uncertain T-S fuzzy models and to compare with the existing methods in each final subsection. In Chapter 5, the contributions are discussed and suggestions for future work are proposed.
8
Chapter 2 Foundations
In this chapter, the basic concepts that relate to the proposed control methods are introduced. The Lyapunov stability is discussed in the first section. Section 2.2 introduces the concept of linear matrix inequality (LMI).
2.1 Lyapunov Stability
Consider a general nonlinear system [47]
x&=A(x) (2.1) wherex∈Rnare the state variables and A:Rn →Rnis a nonlinear function. We assume that A is such that system (2.1) has a unique solution x(t) over [0,∞ for all initial ) conditions x(0) and that the solution depends continuously on x(0) . A vector
Rn
x0∈ is an equilibrium point of the system (2.1) ifA(x0)=0.
Without loss of generality, we can assume that x0 =0 is an equilibrium point of the system (2.1); that is, A(0)=0. Otherwise, we can perform a simple state transformation z= x−x0 to obtain a new state equation ~( ) ( )
x0
z A z A
z&= = + where
0 =0
z is an equilibrium point, that is, ~(0) ( ) 0.
0 =
=A x
A Clearly, the solution of the differential equation (2.1) shows that if x(0)=0, then x(t)=0, for all t>0 . However, this solution may or may not be stable.
Definition 2.1.1:
Stability: The equilibrium point x0 =0 of the system (2.1) is stable if for all ε >0, there exists a δ(ε) > 0 such that x(0) < δ(ε)⇒ x(t) < ε , ∀t≥0.
In other words, the equilibrium point x0 =0 is stable if arbitrarily small perturbations of the initial state x(0)=0 from the equilibrium point result in arbitrarily small perturbation of the corresponding state trajectoryx(t).
Definition 2.1.2:
Asymptotic Stability: The equilibrium point x0 =0 of the system (2.1) is asymptotically stable if it is stable and there exists some γ >0 such that if x(0) <γ , thenx(t)→0 as t→∞.
In other words, the equilibrium point x0 =0 is asymptotically stable if there exists a neighborhood of x0 =0 such that if the system starts in the neighborhood, then its trajectory converges to the equilibrium point x0 =0 as t→∞.
The equilibrium point x0 =0 of the system (2.1) is globally asymptotically stable if γ >0 can be arbitrarily large; that is, all trajectories converges to the equilibrium point 0x0 = .
Determining stability of a system may not be an easy task if the system is nonlinear. One approach often used to determine stability is that of Lyapunov.
Intuitively, the Lyapunov stability theorem can be explained as follows. Given a system with an equilibrium point x0 =0, let us define some suitable “energy” function of the system. The function must have the property that is zero at an equilibrium point x0 =0 and positive elsewhere. Assume further that the dynamic system is such that the energy of the system is monotonically decreasing with time and hence eventually reduces to zero. Then, the trajectories of the system have no other places to go but the origin.
Therefore, the system is asymptotically stable. This generalized energy function is called a Lyapunov function. If there exists a Lyapunov function, then we can prove the
10
asymptotic stability using the following Lyapunov stability theorem.
Theorem 2.1.1
The equilibrium point x0 =0 of the system (2.1) is asymptotically stable if there exists a Lyapunov function V :Rn →R such that V(x)>0 , x≠0 , V(x)=0 ,
=0
x ,V&(x)<0, and x≠0, 0V&(x)= ,x=0 is true in a neighborhood of x0 =0,
{
<γ}
= x x
N : for some γ >0. Proof:
We provide the following intuitive proof by contradiction. If the equilibrium point
0 =0
x of the system (2.1) is not asymptotically stable; that is, x(t)→0 as t→∞ is not true even if x(0) <γ for some γ >0, then V&(x)<−α for some α >0. Since
∫
= +∫
− = −+
=V x tV x d V x t d V x t
t x
V( ( )) ( (0)) 0 &( ) τ ( (0)) 0 α τ ( (0)) α .
For a sufficiently large t , 0V(x(t))< . This contradicts the assumption V(x(t))≥0. The key to proving stability of a system using the Lyapunov stability theorem is to construct a Lyapunov function. This construction must be done in a case-by-case basis.
There is no general method for the construction. The following example illustrates the application of the Lyapunov stability theorem.
Example 2.1.1
Let us consider the following system:
1 2
1 x 3x
x& = − , x&2 =−x23−2x1.
To prove it is asymptotically stable, let us consider the following Lyapunov function:
2 2 2
2 1
)
(x x x
V = + .
Clearly,V(x)>0, x≠0, V(x)=0, x=0.
On the other hand,
2 2 1
1 2
4 )
(x x x x x
V& = & + & =4x1(x2−3x1)+2x2(−x23−2x1)
2 1 4 2 2 1 2
1 12 2 4
4xx − x − x − xx
= =−12x12 −2x24.
Therefore,V&(x)<0, x≠0,V&(x)=0, x=0.
Finally, we can conclude that the system is asymptotically stable.
2.2 Linear Matrix Inequality
A linear matrix inequality (LMI) has the form [48]
0 )
(
1
0+ >
≡
∑
= m
i i iF x F
x
F (2.2)
where x∈Rmis the variable and the symmetric matrices Fi =FiT ∈Rn×n, i=0,...,m, are given. The inequality symbol in (2.2) means that F(x) is positive-definite, i.e.,
0 ) (x u >
F
uT for all nonzero u∈Rn. Thus, the LMI (2.2) is equivalent to a set of n polynomial inequalities inx, i.e., the leading principal minors of F(x) must be positive. We will also encounter nonstrict LMIs, which have the form
0 ) (x ≥
F . (2.3) The strict LMI (2.2) and the nonstrict LMI (2.3) are closely related.
The LMI (2.2) is a convex constraint onx, i.e., the set {x|F(x)>0}is convex.
Though the LMI (2.2) may seem to have a specialized form, it can represent a wide variety of convex constraints on x . In particular, linear inequalities, quadratic inequalities, matrix norm inequalities, and constraints that arise in control theory, such as Lyapunov and convex quadratic matrix inequalities, can all be cast in the form of an LMI.
Multiple LMIs F(1)(x)>0 ,⋅ ⋅⋅, F(p)(x)>0 can be expressed as the single LMI
12
0 )) ( , ), (
(F(1) x ⋅⋅⋅ F( ) x >
diag p . Therefore we will make no distinction between a set of LMIs and single LMI, i.e., “the LMIF(1)(x)>0,⋅ ⋅⋅, F(p)(x)>0” will mean “the LMI
0 )) ( , ), (
(F(1) x ⋅⋅⋅ F( ) x >
diag p ”.
When the matricesF are diagonal, the LMI i F(x)>0 is just a set of linear inequalities. Nonlinear (convex) inequalities are converted to LMI form using Schur complements. The basic idea is as follows: the LMI
) 0 ( ) (
) ( )
( >
x R x s
x S x Q
T (2.4) where Q(x)=Q(x)T, R(x)=R(x)T, and S(x)depend affinely on x, is equivalent to
0 ) (x >
R , 0Q(x)−S(x)R(x)−1S(x)T > . (2.5) In other words, the set of nonlinear inequalities (2.5) can be represented as the LMI (2.4).
As an example, the matrix norm constraint Z(x) <1, where Z(x)∈Rp×qand depends affinely on x, is represented as the LMI
) 0 (
) ( >
I x
Z
x Z I
T
Since Z <1 is equivalent to I −ZZT >0.
We will often encounter problems in which the variables are matrices, e.g., the Lyapunov inequality
<0 + PA P
AT (2.6) whereA∈Rn×nis given and P=PTis the variable. In this case we will not write out the LMI explicitly in the formF(x)>0, but instead make clear which matrices are the variables. The phrase “the LMIATP+ PA<0in P ” means that the matrix P is a variable. Of course, the Lyapunov inequality (2.6) is readily put in the form (2.2), as
follows. LetP1,⋅ ⋅⋅, P be a basis for symmetric m n×n matrices. Then take F0 =0 and A
P P A
Fi =− T i − i . Leaving LMIs in a condensed form such as (2.6), in addition to saving notation, may lead to more efficient computation.
As another related example, consider the quadratic matrix inequality
1 + <0
+
+PA PBR− B P Q P
AT T (2.7) where A , B , Q=QT , R=RT >0 are given matrices of appropriate sizes, andP=PTis the variable. Note that this is a quadratic matrix inequality in the variable P . It can be expressed as the linear matrix inequality
>0
− − −
R P
B
PB Q PA P A
T T
.
This representation also clearly shows that the quadratic matrix inequality (2.7) is convex in P , which is not obvious.
Finally, given an LMI F(x)>0, the corresponding LMI Problem (LMIP) is to find xfeassuch that F(xfeas)>0 or determine that the LMI is infeasible. Of course, this is a convex feasibility problem. We will say “solving the LMIF(x)>0” to mean solving the corresponding LMIP.
As an example of an LMIP, consider the “simultaneous Lyapunov stability problem”: We are givenAi∈Rn×n, i=1,⋅ ⋅⋅,L,and need to find P satisfying the LMI
>0
P , AiTP+ PAi <0, i=1,⋅ ⋅⋅,L or determine that no such P exists.
14
Chapter 3 LMI-Based Robust Sliding Control
In this chapter, LMI-based robust sliding control methods are developed for different uncertain Takagi-Sugeno fuzzy models/time-delay models. The introduction of this chapter is introduced in Section 3.1. In Section 3.2, a robust sliding control method is proposed for T-S fuzzy systems. Section 3.3 presents two kinds of robust sliding control methods for mismatched T-S fuzzy systems. A robust sliding control method is presented for mismatched T-S fuzzy time-delay systems in Section 3.4.
3.1 Introduction
Over the past two decades, fuzzy techniques have been widely and successfully exploited in nonlinear system modeling and control. The Takagi-Sugeno (T-S) model [5]
is a popular and convenient tool for handling complex nonlinear systems.
Correspondingly, the fuzzy feedback control design problem for a nonlinear system has been studied extensively by using T-S model where simple local linear models are combined to describe the global behavior of the nonlinear system [23-29]. In practice, the inevitable uncertainties may enter a nonlinear system model in a very complicated way. The uncertainty may include modeling errors, parameter variations, external disturbances, and fuzzy approximation errors. In such a situation, the fuzzy feedback control design methods of [23-29] may not work well anymore. To deal with the problem, some authors [30,31] have exploited the variable structure system (VSS) theory which has provided an effective means to design robust controllers for uncertain nonlinear systems where the uncertainties are bounded by known scalar valued functions.
In the VSS, the control design of the plant is intentionally changed by using a viable high-speed switching feedback control to obtain a desired system response, from which the VSS arises in finite time. The VSS drives the trajectory of the system onto a specified and user-design surface, which is called the sliding surface or the switching surface, and maintains the trajectory on this sliding surface for all subsequent time. The closed-loop response obtained from using a VSS control law comprises two distinct modes. The first is the reaching mode, also called nonsliding mode, in which the trajectory starting from anywhere on the state space is being driven towards the switching surface. The second is the sliding mode in which the trajectory asymptotically tends to the origin. The central feature of the VSS is the sliding mode on the sliding surface on which the system remains insensitive to internal parameter variations and external disturbance. In sliding mode, the order of the system dynamics is reduced. This enables simplification and decoupling design procedure [32-35].
However, all the VSS-based fuzzy control system design methods are based on the assumption that each nominal local system model shares the same input channel. This assumption is very restrictive and inadequate to modeling uncertainty/nonlinearity in various mechanical systems such as an inverted pendulum on a cart.
On the other hand, time-delay is often encountered in various industrial systems, such as the turbojet engine, electrical networks, nuclear reactor, rolling mill, and chemical process, etc. Recently, the feedback stabilization problem for uncertain time-delay systems is also a problem of interest because the existence of a delay is frequently a source of poor system performance or instability [41-43]. However, they are sensitive to the uncertainty, which directly affects the control systems.
In this chapter, we propose robust sliding control design methods for different uncertain T-S fuzzy models with matched/mismatched parameter uncertainties and
16
external disturbances which are bounded by known scalar valued functions. Each nominal local system model of the uncertain system under consideration may not share the same input channel. As the local controller, we use a sliding mode controller with a nonlinear switching feedback control term. We derive LMI conditions for existence of linear sliding surfaces guaranteeing asymptotic stability of the reduced order equivalent sliding mode dynamics, and we give an explicit formula of the switching surface parameter matrix in terms of the solution of the LMI existence conditions. The nonlinear switching feedback control term is also designed to drive the system trajectories so that a stable sliding motion is induced in finite time on the switching surface and the state converges to zero. Besides, a robust sliding control design method is also presented for the uncertain T-S time-delay model with mismatched parameter uncertainties and external disturbances. Finally, some examples are used to illustrate the effectiveness of the proposed methods for distinct uncertain T-S fuzzy models and to compare with the existing methods in each final subsection.
3.2 Robust Sliding Control for T-S Fuzzy Systems
In this section, system formulation for the uncertain T-S fuzzy model is described in Section 3.2.1. A robust sliding control method via LMI is proposed in Section 3.2.2.
Some examples are used to illustrate the effectiveness of the proposed methods and to compare with the existing methods in Section 3.2.3.
3.2.1 System Formulation
Consider the following uncertain T-S fuzzy model [49]:
[
( ) ( ) ( , )]
) ( )
(
1
x t h B t u B t x A t
x i i i
r
i
i + +
=
∑
=
θ
& β (3.1)
where x(t)∈Rnis the state, u(t)∈Rm is the control input, A ,i Bi are constant
matrices of appropriate dimensions, θ =[θ1,⋅⋅⋅,θs],θj(j =1,⋅⋅⋅,s) are the as the normalized weight of each IF-THEN rule and it satisfies that βi(θ)≥0,
∑
r=1β (θ)=1, uncertainties. We will assume that the followings are satisfied:A1: The n×m matrix B defined by =
∑
ri= Bi B 1r 1satisfies the rank constraint rank(B)=m, i.e., the matrix B has full column rankm. known positive integer.
The system (3.1) does not have to satisfy the restrictive assumption that all the input matrices of the local system models are in the same range space. It should be noted that the assumption A1 implies that rank(Bi)≤mand each nominal local system model may not share the same input channel. The assumption A2 with l=1 and
0 written as follows.
)]
18
[
( ), ,( )]
,2 1
1 B Br
B B
H = − ⋅ ⋅⋅ − G=
[
I,⋅ ⋅⋅,I]
T,[
(1 2 ( )) , ,(1 2 ( ))]
. )( diag 1 I I
F β = − β θ ⋅ ⋅⋅ − βr θ (3.3) It should be noted that the system (3.1) does not have to satisfy
2 ,
1 B Br
B = =⋅ ⋅⋅= which is used in almost all published results on VSS design methods including the VSS-based fuzzy control design methods of [33,34]. Hence the methods [30,31] cannot be applied to the above model (3.1). Since βi(θ)≥0 and
∑
i 1r= β(θ)=1,we can see that the following inequality always holds:. ) ( ) ( ) ( )
( F F F I
FT β β = β T β ≤ (3.4) Many examples in the literature and various mechanical systems such as motors and robots do not satisfy the restrictive assumptions that each nominal local system model shares the same input channel and they fall into the special cases of the above model [49].
3.2.2 Sliding Control Design via LMI
The Sliding Mode Control (SMC) design is decoupled into two independent tasks of lower dimensions: The first involves the design of m(n− )1 −dimensional switching surfaces for the sliding mode such that the reduced order sliding mode dynamics satisfies the design specifications such as stabilization, tracking, regulation, etc. The second is concerned with the selection of a switching feedback control for the reaching mode so that it can drive the system’s dynamics into the switching surface [33]. We first characterize linear sliding surfaces using LMIs.
Let us define the linear sliding surface as σ = xS = 0where Sis a m×nmatrix.
Referring to the previous results [33], [51], we can see that for the system (3.2) it is reasonable to find a sliding surface such that
P1
[
SB+SHF(β)G]
is nonsingular for any β satisfying βi(θ)≥0,i=1,⋅ ⋅⋅,r, and∑
r= =i 1βi(θ) 1.
P2 The reduced (n−m)order sliding mode dynamics restricted to the sliding surface
=0 x
S is asymptotically stable for all admissible uncertainties.
It should be noted that P1 is necessary for the existence of the unique equivalent control [33] and the assumption A1 is necessary for the nonsingularity of SB.
Define a transformation matrix and the associated vector vas M=[Λ(ΛTYΛ)−1,
From the equivalent control method [33], we can see that the equivalent control is given by ueq(t)= r1 ( )[I SHF( )G] 1SAix h(t,x).
Theorem 3.1 Let us consider the sliding mode dynamics (3.6). If there exist
matrices Y∈Rn×n, Λ∈Rn×(n−m)satisfying ,BTΛ =0,ΛTΛ=I scalarsc1∈R,c2∈R,η∈R, ),
min(BTB λ
κ = and ∗ represents blocks that are readily inferred by symmetry such that the following LMIs holds:
20
then, there exists a linear sliding surface parameter matrix Ssatisfying P1-P2 and the sliding surface
0 will guarantee that the sliding mode dynamics (3.6) is asymptotically stable.
Proof: By using Schur complement formula [48], we can easily show that in fact the
following LMIs are incorporated in the LMIs (3.7)-(3.9) ,
= is nonsingular and hence P1 holds .
By using the Schur complement formula, we can see that (3.8) and (3.11) imply ,
0<c1−1I <Y <c2I 0<c2−1I <Y−1 <c1I (3.14) and this leads to
T
Finally, by using the above inequalities (3.11) and (3.16), we can obtain
I
Now, we will show that Sof (3.10) guarantees P2. Using the matrix inversion lemma:
B
where A and B are compatible constant matrices such that (I+AB)is nonsingular, we can show that the sliding mode dynamics (3.6) is equivalent to
∑
=The sliding mode dynamics (3.18) is asymptotically stable if there exists a positive definite matrixP0∈R(n−m)×(n−m)such that the time derivative of the Lyapunov function
1 0
) 1
(t v Pv
Eg = T satisfies for some positive scalarτ
∑
=22 This and (3.19) imply that (3.18) is asymptotically stable if there exists a positive definite matrixP such that 0
0 where ∗ represents blocks that are readily inferred by symmetry.
Let z be i zi =[I −N(β)D0]−1N(β)Ci0ywherey∈R(n−m). Using (3.22) and (3.23), we can show that the Lyapunov inequality (3.21) is satisfied if the following inequality holds:
0
Using the Schur complement formula, we can rewrite the above inequality as
0
Let the positive definite matrix P be0 P0 =ΛTYΛ where Y is a solution to LMIs (3.7)-(3.9), which implies that the sliding mode dynamics (3.18) is asymptotically stable. Hence, the sliding mode dynamics (3.6) is asymptotically stable.
After the switching surface parameter matrix Sis designed so that the reduced )
(n−m order sliding mode dynamics has a desired response, the next step of the SMC design procedure is to design a switching feedback control law for the reaching mode such that the reachability condition is met. If the switching feedback control law satisfies the reachability condition, it drives the state trajectory to the switching surface
=0
= xS
σ and maintains it there for all subsequent time. With σ of (3.10), we design a sliding fuzzy control law guaranteeing that σ converges to zero. We will use the following nonlinear sliding switching feedback control law as the local controller.
Control rule i: IF θ is 1 µi1and ... and θsis µis, THEN inferred as the weighted average of the each local controller is given by
∑
= and we can establish the following theorem.
Theorem 3.2 Consider the closed-loop control system of the uncertain system (3.2) with control (3.26). Suppose that the LMIs (3.7)-(3.9) has a solution vector
24 )
, , ,
(Y c1 c2 η and the linear sliding surface is given by (3.10). Then the state converges to zero.
Proof: Since Theorem 3.1 implies that the linear sliding surface (3.10) guarantees
P1-P2, we only have to show that σ converges to zero. Define a Lyapunov function as Eg(t)=0.5σTσ.The time derivative of Eg(t)is E&g =σTσ&. From (3.2), (3.10), (3.26), SHF(β)G ≤ r SH =ω ,0≤ω<1, and A2, we obtain
) ( ) (
1
t x SAi
r
i i T
Tσ σ β θ
σ
∑
=
& = +σT[I +SHF(β)G][u+h(t,x)]
∑
=
+
≤ r
i
T i
T
i SAx t u
1
) ( )
(θ σ σ
β +{ω u +(1+ω)h(t,x)}σ .
This implies that (1 ) ( ) ( ) 0
1 1
2 − ≤
−
−
≤
∑ ∑
= =
r
i
r
i
i i i
i
E&g ω β θ χ σ β θ α σ which indicates that Eg ∈L2∩L∞,E&g ∈L∞. Finally, by using Barbalat’s lemma, we can conclude that σ converges to zero.
Remark 3.1 Theorem 3.1 and 3.2 can be summarized in the form of the following LMI-based design algorithm.
Step 1: Obtain =
∑
ri= Bi B 1r 1and
[
( ), ,( )]
2 1
1 B Br
B B
H = − ⋅⋅⋅ − for givenB . i
Step 2: Check that (Ai,B)is stabilization. If not, exit.
Step 3: Find a solution vector(Y,c1,c2,η) to LMI (3.7)-(3.9).
Step 4: Compute the sliding surface parameter matrixSby using the formula of (3.10).
Step 5: The controller is given by (3.26).
3.2.3 Numerical Examples
Example 3.1 Consider the following inverted pendulum on a cart [49]
2, and )d(t is related to external disturbances which may be caused by the frictional force.
),
ρ are known constants. To design the fuzzy controller (3.26), we must have a fuzzy 1
model. Here, we approximate the system (3.27) by the following two-rule fuzzy model.
Plant Rule 1: IF x1is about 0, THEN
26
The inverted pendulum on a cart (3.27) can be cast as (3.2) with data (3.28).
Because B1is not in the range space of B2,all existing VSS-based fuzzy control system design methods cannot be applied to the above system (3.28). Via LMI optimization with (3.28), we can obtain the sliding surfaceσ =Sx.
By setting ˆ(, ) sin , 5, 1, 2, 1, 1
we can obtain the following nonlinear controller:
Control Rule 1: IFx1is about 0, THEN
The final controller inferred as the weighted average of each local controller is given by .
To assure the effectiveness of our fuzzy controller, we apply the controller to the two-rule fuzzy model (3.28) with nonzero ).d(t We assume that
).
= ) 0
3(
x x4(0)=0. In Figure 3.1, Figure 3.2, and Figure 3.3, it should be noted that since it is impossible to switch the input u instantaneously, oscillations always occur in
x x4(0)=0. In Figure 3.1, Figure 3.2, and Figure 3.3, it should be noted that since it is impossible to switch the input u instantaneously, oscillations always occur in