Problem Statement

Consider a switched system consisting of two second-order subsystems, given by

( )x A

x= _σt (4.1)

where σ

( )

t :

[

0,∞

)

→

{ }

1,2 , x∈R² and both A1 and A2 are 2×2 constant non-Hurwitz matrices. The main purpose in this chapter is to determine the switching conditions for these two subsystems such that the overall system dynamics is stabilized.

Interestingly, (4.1) can be rewritten into a second-order homogeneous bilinear system, similar to the procedure in [37], controlled by a switching input as

Nx Ax

x= +u (4.2)

where A=0.5(A1+A2), N=0.5(A1−A2) and the switching input u∈{−1, 1}. Clearly, the stabilization problem in (4.1) is equivalent to designing the switching input u to stabilize (4.2). To deal with the stabilization problem in (4.2), first define λi and ci as the i-th eigenvaule and its corresponding left eigenvector of A+u0N, where u0 is a real number. Then, the bang-bang sliding mode control is proposed to stabilize the system (4.2) under the following assumptions:

Assumption 4.1.

There exists a u0 in the range of (−1, 1) such that the eigenvalues, λ1 and λ2, of A+u0N are stable and real.

Assumption 4.2.

There exists at least one left eigenvector ci corresponding to λi that satisfies rank([ci ;ciN])=2.

Similar to several literatures [32-34], the existence of stable matrices combination A+u0N is also required in this dissertation. Besides, λ1 and λ2 are further assumed to be real, as declared in Assumption 4.1, such that their left eigenvectors c1 and c2 are real and could be used as the coefficient vector of a sliding function [13,46]. As for Assumption 4.2, it is required to guarantee the existence of stable sliding motions, which will be explained later.

In next section, with both assumptions being satisfied, a switching controller based on the bang-bang sliding mode control will be proposed for the homogenous bilinear system (4.2). It will show that the switching input must be determined from two switching functions rather than a single one as the general sliding controls. The stability of the switched system will be proven and different switching behaviors resulting from these two switching functions will be clearly described.

4.2 Design Procedures of Bang-Bang Sliding Mode Control

Under Assumption 4.1, system (4.2) can be further rewritten as Nx

x A

x= _S +u_S (4.3)

where AS=A+u0N, uS=−u0+u and u∈{−1, 1}. Obviously, system (4.3) is still a homogeneous bilinear system but possesses a stable system matrix AS with real eigenvalues and a new input uS. Let λ be one of the eigenvalues of AS and c be the corresponding left eigenvectors, i.e.,

c

cA_S=λ (4.4)

Then, the sliding function s(x) can be defined as

( )

x =cx

s (4.5)

Note that c must satisfy the condition of rank([c;cN])=2. From (4.2)−(4.5), the derivative of s(x) with respective to time is

( ) (

^{x c A u}₀^N

)

^{x u}_S^{cNx s}

( )

^{x u}_S^cNx

s = + + =λ + ^(4.6)

Since u∈{−1, 1}, choose u=−sgn(ρ(x))·sgn(s(x)), where ρ(x)=cNx, and then uS will be

( )

(

x

)

sgn

(

s

( )

x

)

sgn u u u

u_S=− ₀+ =− ₀− ρ ⋅ ^(4.7)

From (4.5)−(4.7), we have

( ) ( )

^{x s x s}

( )

^x

[

^{u sgn}

( ( )

^x

)

^sgn

(

^s

( )

^x

) ] ( ) ( )

^{x s x}

s =λ ²+ − ₀− ρ ⋅ ⋅ρ ⋅

( )

^{x u}

( ) ( )

^{x s x sgn}

( ( )

^x

)

^sgn

(

^s

( )

^x

) ( ) ( )

^{x s x}

s − ⋅ ⋅ − ⋅ ⋅ ⋅

=λ ² ₀ ρ ρ ρ

( )

^x

[

u sgn

( ( )

^x

)

sgn

(

s

( )

^x

) ] ( ) ( )

^x s ^x

s − ⋅ ⋅ + ⋅ ⋅

=λ ² ₀ ρ 1 ρ

( )

^x

( ) ( )

^x s ^x

s − ⋅ ⋅

=λ ² α ρ

( ) ( )

x ⋅ s x

⋅

−

≤ α ρ ^(4.8)

where α=(1+u0·sgn(ρ(x))·sgn(s(x))) and 0<α<2. Note that there are two features different from general sliding mode controls. First, uS possesses two switching functions sgn(ρ(x)) and sgn(s(x)), i.e., uS will switch around two sets defined as Ωs={x|s(x)=0} and ΩN={x|ρ(x)=0}. According to Assumption 4.2, x(t) will not belong to Ωs and ΩN simultaneously, except the equilibrium point x=0, and thus the state space can be well separated into four regions as shown in Fig. 4.1, in which uS can be determined and it will switch when system trajectories pass through Ωs or ΩN.

Second, the equation (4.8) is similar to the RAS-condition (2.4), but the reaching rate in (4.8) is α multiplied with a state-dependent term |ρ(x)|. In general sliding mode controls, the system trajectories can be theoretically driven to the sliding mode only when the reaching rate is nonzero. However, in equation (4.8), |ρ(x)| may always be zero ( i.e., once the system trajectory reach ΩN, it will stay in ΩN thereafter) such that the reaching rate is zero. With this observation, the reaching modes related to the state-dependent term |ρ(x)| can be concluded into three types, as depicted in Fig. 4.2.

x

1

u _S =-u ₀ +1

u _S =-u ₀ -1 u _S =-u ₀ +1

u _S =-u ₀ -1

x

2

Fig. 4.1 Four regions in the state space separated by Ωs and Ω_N.

Ω

s

Ω

_N

For the reaching mode of type 1, without crossing ΩN, the system trajectory starts from p1 and then reaches Ωs at p2 in a finite time, which is the same as the reaching mode in general sliding controls. It is known that the equivalent control ueq can be obtained as

( )

0

0 S₌ = ⋅ = ⋅ =

= _{,u u eq} cNx _eq ρ x

s u u

s eq

(4.9) Since the system trajectory stays in Ωs and does not belong to ΩN, ueq will be zero, except the equilibrium point x=0. By substituting uS=ueq=0 into (4.3), the equivalent system dynamics becomes

x A

x= _S (4.10)

where AS is stable. Therefore, the system (4.3) is stabilized once its system trajectory is constrained in the sliding mode.

For the reaching mode of type 2, the system trajectory starts from q1 and passes through ΩN at q2 but not stay in ΩN. After that, it reaches Ωs at p3 in a finite time and then is stabilized with the same equivalent system dynamics as (4.10). Intuitively, when the system is switched from one subsystem to another subsystem through ρ(x)=0, the derivatives of ρ(x) with respective to time for both subsystems must have the same sign. As discussed in §2.2, the switching behaviors around ρ(x)=0 are of the refractive mode and can be mathematically represented by (2.10). In other words, the reaching mode of type 2 is guaranteed if

( )

x _A=A ⋅

( )

x _A=A >0 for

( )

x =0 and x≠0

2

1 ρ ρ

ρ i i (4.11)

which is equivalent to

( )

x _Ax

( )

x

( )

x x 0

x

A₁ ρ ⋅L ₂ ρ >0 for ρ =0 and ≠

L (4.12)

Note that _Axρ

( )

x

L 1 and _Axρ

( )

x

L 2 are the directional derivatives of ρ(x) with respective to the vector fields A1x and A2x. That means the projections of A1x and A2x onto the gradient of ρ(x) are in the same direction as shown in Fig. 4.3, in which ψ⁺

and ψ⁻ denote the regions of {x|cNA1x>0, cNA2x>0} and {x|cNA1x<0, cNA2x<0}, and they are bounded by two lines (L1: cNA1x=0 and L2: cNA2x=0). Clearly, if switching occurs inside the regions of ψ⁺ and ψ⁻ (i.e., (4.12) is satisfied), the system trajectories will pass through ΩN. Such reaching mode is classified into type 2.

As for the reaching mode of type 3, the system trajectory starts from r1 and hits ΩN

at r2. Different from that of type 2, this system trajectory is constrained in ΩN and cNx is always zero. From (4.6), the system dynamics is governed by

s

s=λ (4.13)

such that the system trajectory cannot reach Ωs in a finite time but exponentially approaches Ωs instead, i.e., this system trajectory will exponentially converge to the equilibrium point x=0, the intersection of Ωs and ΩN. Interestingly, the switching control (4.7) forces the system trajectories to switch along ΩN, which means the switching motion along ΩN is another sliding mode in this situation. Hence, the following inequality will be satisfied:

( )

x _{ρ( )}x→₀+ <0 and ρ

( )

x _{ρ( )}x→₀⁻ >0

ρ _(4.14)

and it can be further written as

( ) ( ) ( )

Obviously, (4.15) express another sliding condition along ΩN for s(x)>0 and s(x)<0.

From above, the bang-bang sliding mode control can theoretically stabilize the homogenous bilinear system (4.2). According to the switching input derived in (4.7), the switched system of (4.1) can be stabilized with the switching conditions as

( ) ( ) ( )

In this section, it shows that with Assumption 4.1 and Assumption 4.2 being satisfied, the bang-bang sliding mode control can be adopted to determine the

switching conditions for the switched systems consisting of two second-order unstable subsystems. In next section, it will be extended to another class of switched systems with model uncertainties.

x1

x₂

p1

q1

q₂ p2

q₃ Type 1

Type 2

x₁ x2

r₁ r₂

Type 3

(a) (b) Fig. 4.2 Three types of reaching modes.

(a) Reaching modes of type 1 and type 2. (b) Reaching mode of type 3.

x1

x2

( )

x =0 ρ

ψ

−

ψ

+

A1x A2x

( )x ρ

∇ A2x

( )x ρ

∇

A1x

L1

L2

L2

L1

( )

x =0 s

Fig. 4.3 Geometric representations of the directional derivatives for the reaching mode of type 2.

Ω

s

Ω

_N

Ω

s

Ω

_N

在文檔中 Bang-Bang順滑控制在切換式電源轉換器之設計 (頁 56-63)