Chapter 4 Bang-Bang Sliding Mode Control in Switched Systems
4.3 Stabilization of the Switched Systems with Model Uncertainties
As general sliding mode controls, the bang-bang sliding mode control is robust to model uncertainties in switched systems. Here, consider another class of switched systems containing model uncertainties, described by
( )x A ( )x A
x= σt +Δ σt (4.17)
where ΔAσ(t) are the model uncertainties and
( ) Δ ( )t ≤γ maxt σ
σ A , γ is a constant. With Assumption 4.1 and Assumption 4.2, (4.17) can be expressed as
( )x A Nx x
A
x= S +uS +Δ σt (4.18)
Obviously, the problem of designing a switching controller to stabilize (4.17) is changed into designing a robust switching controller for a homogenous bilinear system with state-dependent disturbance ΔAσ(t)x. Similar to the design procedures in
§4.2, the equivalent control in the sliding mode can be derived from
( ) 0
σ =
Δ +
⋅cNx c A t x
ueq (4.19)
Compared with (4.9), (4.19) contains cΔAσ(t)x and thus ueq may be affected by the model uncertainties. If cΔAσ(t)x is zero in the sliding mode, ueq will not be affected by the model uncertainties and is zero. However, in general cases, ueq should be expressed as
(
cNx)
c A( )t xueq =− −1⋅ Δ σ (4.20)
Note that the existence of (cNx)-1 is guaranteed in the sliding mode, except x=0.
Substituting (4.20) into (4.18), the equivalent system dynamics will be
(
cNx)
c A ( )x Nx A( )x xA
x= S − −1⋅ Δ σt ⋅ +Δ σt (4.21)
Assume that the norm of N is bounded by a constant β (i.e., N ≤β), and choose a symmetric positive-definite matrix P such that xTPx is a candidate of Lyapunov function, which is denoted as V. Then, the derivative of V with respective to time can be obtained as
( ) ( )
( )( )
where Q is a symmetric positive-definite matrix and φσ(t)(x)=cΔAσ(t)x. Besides, it is known that
( )
Q x 2 xTQx max( )
Q x 2min λ
λ ≤ ≤ (4.23)
where λmin(·) and λmax(·) denote the minimum eigenvalue and maximum eigenvalue of a matrix respectively. Further, for some xs in the sliding mode, define max|φσ(t)(xs)|
and |ρ(xs)| as lΔ and lN, and denote the ratio of lΔ to lΝ as l, which are shown in Fig. 4.4 in the geometric view.
x
1Substituting lN, lΔ and l into (4.22), it leads to
(
λ( )
Q −2β⋅ ⋅ P −2γ P)
x 2−
≤ l
V min (4.24)
Note that l is a constant value for all the xs in the sliding mode and it is finite since Assumption 4.2 is satisfied, i.e., |ρ(xs)| is nonzero, except xs=0. Thus, it can be concluded that V is a Lyapunov function if the following inequality is satisfied
( )
02 −β⋅ −γ >
λmin l P
Q (4.25)
and it yields x(t)Æ0 as tÆ∞.
From above, the stability of the switched system is guaranteed in the sliding mode if (4.25) is satisfied. However, it still needs to check the reaching condition for the switched system since the model uncertainties also affect the original reaching rate.
Similar to (4.8), the reaching condition for the switched system with model uncertainties is
( ) ( )
( )( )
(
⋅s x + ⋅ x − t x)
⋅s( )
x ≤− ⋅( ) ( )
x ⋅s x− λ α ρ φσ σ ρ (4.26)
where σ is some positive constant. Obviously, there are three state-dependent terms related to the reaching rate and the RAS-condition can be guaranteed only when (4.26) is satisfied for all x in R2, except x=0. While Assumption 4.2 is satisfied, cΔAi can be represented in the linear combination of c and cN as
cN c
A
cΔ i = ws ,i +wρ ,i (4.27)
where ws,i and wρ,i are the coefficients of i-th model uncertainty represented by vectors c and cN. With two model uncertainties in the switched system, the reaching rate can
be expressed as
(
λ − ws ,i)
⋅s( )
x +(
α− wρ ,i)
⋅ρ( )
x ,i=1 ,2 (4.28) Clearly, if ws,i and wρ,i are bounded by2 , 1
, < , i=
ws i λ (4.29)
and
2 , 1
, < , i=
wρ i α (4.30)
(4.26) is satisfied and the system trajectories will reach the sliding mode in a finite
time. Note that α takes two possible values of 1−u0 and 1+u0, and the minimum value of α is obtained as 1−|u0|, which is used to check (4.30). Besides, there may also exist the reaching mode of type 3 with the system dynamics described by (4.13). While (4.29) is satisfied, the system trajectory can still exponentially converge to the equilibrium point x=0.
From above, it shows that the proposed switching controller is robust to the model uncertainties if (4.25) and (4.26) (or, (4.29) and (4.30)) are both satisfied. In next section, numerical simulations will be given to demonstrate the effectiveness of the proposed bang-bang sliding mode control.
4.4 Numerical Simulation Results
In this section, several examples are given to show the cases discussed in §4.2 and
§4.3. First, three possible reaching modes under the bang-bang sliding mode control are illustrated in example 4.1 (with reaching modes of type 1 or type 2), and example 4.2 (with reaching modes of type 1 or type 3). Finally, a switched system with model uncertainties is considered in example 4.3.
Example 4.1:
Consider the second-order switched system given in [34], which consists of two unstable subsystems expressed as
⎥⎦
where the eigenvalues of A1 and A2 are {1.5+i2.1794, 1.5−i2.1794} and {0.3852,
−10.3852} respectively. First, rewrite (4.31) into a second-order homogenous bilinear system as
and choose u0 as 0.09. Then, Assumption 4.1 is satisfied since Ax+u0Nx has stable eigenvalues λ1=−1.0979 and λ2=−1.8171 with left eigenvectors c1=[0.9122 −0.4097]
and c2=[−0.5322 0.8466] correspondingly. Assumption 4.2 is also satisfied by checking rank[c1;c1N]=2. Therefore, the sliding function can be chosen as c1x and
from (4.16), the switching conditions of (4.31) can be obtained as
( ) ( ) ( )
Under the switching conditions (4.33), the simulation results of (4.31) are shown in Fig. 4.5 including four possible system trajectories. For the system trajectories starting from [0.5 0.5]T and [−0.5 −0.5]T, they reach the sliding mode Ωs in a finite time without crossing ΩN, which shows the reaching mode of type 1. As for the system trajectories starting from [−0.5 0.5]T and [0.5 −0.5]T, they present the reaching mode of type 2, passing through ΩN and then reaching the sliding mode Ωs. For all these system trajectories, once they are in the sliding mode Ωs, they will always stay in it and move toward the equilibrium point x=0.
Remark 4.1:
For the system trajectories with reaching mode of type 2, (4.12) must be satisfied.
This can be easily verified by substituting xN=[η −0.7η]T, where xN∈ΩN and η∈R, into cNA1x and cNA2x. For any nonzero η, we have
( )
2( )
1 2 2295536 2 01 ρ ⋅L ρ = ⋅ = . η >
LAx xN Ax xN cNAxN cNA xN (4.34)
which guarantees (4.12). Thus, for the system trajectories reaching ΩN, they will pass through ΩN and such reaching mode is classified into type 2 and only stable sliding motions exist in this switched system.
Example 4.2:
Consider a second-order switched system, consisting of two unstable subsystems given as
respectively. First, rewrite (4.35) into a second-order homogenous bilinear system as x
and choose u0 as −0.3. Then, Assumption 4.1 is satisfied since Ax+u0Nx has stable eigenvalues λ1=−0.1 and λ2=−0.9 with left eigenvectors c1=[−0.4472 −0.8944] and c2=[−0.6402 0.7682] correspondingly. Assumption 4.2 is also satisfied by checking rank[c1;c1N]=2. Therefore, the sliding function s can be chosen as c1x and from (4.16), the switching conditions of (4.35) can be obtained as
( ) ( ) ( )
Under the switching conditions (4.37), the simulation results of (4.35) are shown in Fig. 4.6 including four possible system trajectories. Different from example 4.1, the system trajectories may stay in Ωs or ΩN, depending on which sliding mode that the system trajectories reach first. For the system trajectories starting from [−0.5 0.5]T and [0.5 −0.5]T, they reach Ωs first, presenting the reaching mode of type 1. For the system trajectories starting from [0.2 0.8]T and [−0.2 −0.8]T, they reach ΩN first and then stay in it. Although the system trajectories cannot reach Ωs, they truly slide along ΩN and converge to x=0 exponentially as described in §4.3. Clearly, for all these system trajectories, they will stay in the sliding mode Ωs or ΩN, and move toward the equilibrium point x=0.
Remark 4.2:
For the system trajectories sliding along ΩN, the inequalities in (4.15) must be satisfied, which can be verified by the following process. First, under the condition of ρ(x) being around zero, assume s(x)=τ and ρ(x)=δ, where τ is a nonzero number and
Since Assumption 4.2 is satisfied, the square matrix in (4.38) is invertible and the solution of x can be obtained as
⎥⎦
This verifies (4.15) and ΩN is really a sliding mode in this example.
Example 4.3:
Consider the switched system given in example 4.1 again but containing the model uncertainties ΔAσ(t) with the values of
ΔA1= ⎥ a Q is given, a corresponding P can be obtained by Matlab software, e.g.,
⎥⎦ satisfied from the following calculation:
( )
which guarantees that the switched system is robust to the model uncertainties in the sliding mode. Next, check the reaching condition in (4.26) and the coefficients in (4.27) can be obtained as ws,1=−0.1635, wρ,1=−0.0168, ws,2=0.1546 and wρ,2=0.0143.
Obviously, (4.29) can be satisfied by checking:
2
Therefore, the reaching condition is also satisfied in the presence of model uncertainties. Then, the switching conditions in (4.37) can be used to stabilize the switched system and the system trajectories in the numerical simulation are shown in Fig. 4.7.
In this example, it shows that the bang-bang sliding mode control can stabilize the second-order switched system with model uncertainties if (4.25) and (4.26) are satisfied for the mode uncertainties.
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-Fig. 4.5 System trajectories of example 4.1: reaching modes of type 1 and type 2.
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Fig. 4.6 System trajectories of example 4.2: reaching modes of type 1 and type 3.
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Fig. 4.7 System trajectories of example 4.3.