Paper365: Day 1 Finite-Time Continuous Terminal Sliding Mode Control of Servo Motor Systems

Basic information

Title: Finite-Time Continuous Terminal Sliding Mode Control of Servo Motor Systems

Highlight authors: Xinghuo Yu, Zhenwei Cao

Publication: TIE, 2020

Contribution

• A novel full-order terminal sliding mode surface is proposed based on the bilimit homogeneous property, such that the sliding motion is finite-time stable independent of the system’s initial condition.
• A new continuous terminal sliding mode control algorithm is proposed to guarantee that the system states reach the sliding surface in finite-time.

PROBLEM FORMULATION

Considering nonlinear system:
x ˙ 1 ( t ) = x 2 ( t ) x ˙ 2 ( t ) = f ( x , t ) + d ( x , t ) + b u ( t ) (sys) \begin{array}{l} \dot{x}_{1}(t)=x_{2}(t) \\ \dot{x}_{2}(t)=f(x, t)+d(x, t)+b u(t) \end{array}\tag{sys} x˙1​(t)=x2​(t)x˙2​(t)=f(x,t)+d(x,t)+bu(t)​(sys)
Assumptions:

• ∣ d ( x , t ) ∣ ≤ l d |d(x,t)| \le l_d ∣d(x,t)∣≤ld​
• ∣ d ˙ ( x , t ) ∣ ≤ k d |\dot{d}(x,t)|\le k_d ∣d˙(x,t)∣≤kd​

FINITE-TIME CONTINUOUS SLIDING MODE CONTROLLER DESIGN

Basic knowledge and nations

Notions:
⌈ x ⌋ α = ∣ x ∣ α s i g n ( x ) , d ⌈ x ⌋ α d x = α ∣ x ∣ α − 1 , ⌈ x ⌋ 0 = s i g n ( x ) , ⌈ x ⌋ 2 = x ∣ x ∣ \lceil x \rfloor^\alpha = |x|^\alpha \mathrm{sign} (x),\quad \frac{d\lceil x \rfloor^\alpha}{dx} = \alpha|x|^{\alpha - 1},\quad \lceil x \rfloor ^0 = \mathrm{sign}(x),\quad \lceil{x}\rfloor^2 = x|x| ⌈x⌋α=∣x∣αsign(x),dxd⌈x⌋α​=α∣x∣α−1,⌈x⌋0=sign(x),⌈x⌋2=x∣x∣

Property 1: finite-time stability

Consider the scalar differential equation:
z ˙ = − λ ⌈ z ⌋ α − μ ⌈ z ⌋ γ \dot{z}=-\lambda\lceil z\rfloor^{\alpha}-\mu\lceil z\rfloor^{\gamma} z˙=−λ⌈z⌋α−μ⌈z⌋γ

with λ , μ > 0 , α > 1 \lambda,\mu >0,\alpha >1 λ,μ>0,α>1 and γ < 1 \gamma <1 γ<1. The equation admits a finite time T 0 T_0 T0​ uniform with respect to the initial condition z ( 0 ) z(0) z(0) and bounded by
T 0 ≤ T ∗ ( α , γ , λ , μ ) = 1 λ ( α − 1 ) + 1 μ ( 1 − γ ) T_{0} \leq T^{*}(\alpha, \gamma, \lambda, \mu)=\frac{1}{\lambda(\alpha-1)}+\frac{1}{\mu(1-\gamma)} T0​≤T∗(α,γ,λ,μ)=λ(α−1)1​+μ(1−γ)1​

Sliding Surface Design

Sliding surface is defined as:
s = x ¨ 1 + ( c 1 ⌈ x ˙ 1 ⌋ α 1 + c 2 ⌈ x ˙ 1 ⌋ α 2 + c 3 ⌈ x ˙ 1 ⌋ α 3 ) + ( k 1 ⌈ x 1 ⌋ β 1 + k 2 ⌈ x 1 ⌋ β 2 + k 3 ⌈ x 1 ⌋ β 3 ) = x ˙ 2 + ( c 1 ⌈ x 2 ⌋ α 1 + c 2 ⌈ x 2 ⌋ α 2 + c 3 ⌈ x 2 ⌋ α 3 ) + ( k 1 ⌈ x 1 ⌋ β 1 + k 2 ⌈ x 1 ⌋ β 2 + k 3 ⌈ x 1 ⌋ β 3 ) (SS) \begin{aligned} s=& \ddot{x}_{1}+\left(c_{1}\left\lceil\dot{x}_{1}\right\rfloor^{\alpha_{1}}+c_{2}\left\lceil\dot{x}_{1}\right\rfloor^{\alpha_{2}}+c_{3}\left\lceil\dot{x}_{1}\right\rfloor^{\alpha_{3}}\right) \\ &+\left(k_{1}\left\lceil x_{1}\right\rfloor^{\beta_{1}}+k_{2}\left\lceil x_{1}\right\rfloor^{\beta_{2}}+k_{3}\left\lceil x_{1}\right\rfloor^{\beta_{3}}\right) \\ =& \dot{x}_{2}+\left(c_{1}\left\lceil x_{2}\right\rfloor^{\alpha_{1}}+c_{2}\left\lceil x_{2}\right\rfloor^{\alpha_{2}}+c_{3}\left\lceil x_{2}\right\rfloor^{\alpha_{3}}\right) \\ &+\left(k_{1}\left\lceil x_{1}\right\rfloor^{\beta_{1}}+k_{2}\left\lceil x_{1}\right\rfloor^{\beta_{2}}+k_{3}\left\lceil x_{1}\right\rfloor^{\beta_{3}}\right) \end{aligned}\tag{SS} s==​x¨1​+(c1​⌈x˙1​⌋α1​+c2​⌈x˙1​⌋α2​+c3​⌈x˙1​⌋α3​)+(k1​⌈x1​⌋β1​+k2​⌈x1​⌋β2​+k3​⌈x1​⌋β3​)x˙2​+(c1​⌈x2​⌋α1​+c2​⌈x2​⌋α2​+c3​⌈x2​⌋α3​)+(k1​⌈x1​⌋β1​+k2​⌈x1​⌋β2​+k3​⌈x1​⌋β3​)​(SS)
Coefficients:

• c i > 0 , k i > 0 c_i >0, k_i >0 ci​>0,ki​>0
• α 1 = ν , α 2 = 1 , α 3 = 1 + 1 − ν 3 − 2 ν \alpha_{1}=\nu, \alpha_{2}=1, \alpha_{3}=1+\frac{1-\nu}{3-2 \nu} α1​=ν,α2​=1,α3​=1+3−2ν1−ν​
• β 1 = ν 2 − ν , β 2 = 1 , β 3 = 2 − ν 2 − ν , ν ∈ ( 0 , 1 ) \beta_{1}=\frac{\nu}{2-\nu}, \beta_{2}=1,\beta_3 = 2 - \frac{\nu}{2-\nu}, \nu \in (0,1) β1​=2−νν​,β2​=1,β3​=2−2−νν​,ν∈(0,1)

Once the system trajectories slide along the sliding surface, i.e., s = 0 s = 0 s=0, it yields:
x ˙ 1 ( t ) = x 2 ( t ) x ˙ 2 ( t ) = − ( c 1 ⌈ x 2 ⌋ α 1 + c 2 ⌈ x 2 ⌋ α 2 + c 3 ⌈ x 2 ⌋ α 3 ) − ( k 1 ⌈ x 1 ⌋ β 1 + k 2 ⌈ x 1 ⌋ β 2 + k 3 ⌈ x 1 ⌋ β 3 ) (SMM) \begin{aligned} \dot{x}_{1}(t)=& x_{2}(t) \\ \dot{x}_{2}(t)=&-\left(c_{1}\left\lceil x_{2}\right\rfloor^{\alpha_{1}}+c_{2}\left\lceil x_{2}\right\rfloor^{\alpha_{2}}+c_{3}\left\lceil x_{2}\right\rfloor^{\alpha_{3}}\right) \\ &-\left(k_{1}\left\lceil x_{1}\right\rfloor^{\beta_{1}}+k_{2}\left\lceil x_{1}\right\rfloor^{\beta_{2}}+k_{3}\left\lceil x_{1}\right\rfloor^{\beta_{3}}\right) \end{aligned}\tag{SMM} x˙1​(t)=x˙2​(t)=​x2​(t)−(c1​⌈x2​⌋α1​+c2​⌈x2​⌋α2​+c3​⌈x2​⌋α3​)−(k1​⌈x1​⌋β1​+k2​⌈x1​⌋β2​+k3​⌈x1​⌋β3​)​(SMM)

Theorem 1:

Consider the sliding mode motion ( S M M ) (SMM) (SMM) with the parameters c i , k i , α i , β i ( i = 1 , 2 , 3 ) c_i , k_i , α_i , β_i (i = 1, 2, 3) ci​,ki​,αi​,βi​(i=1,2,3). The finite-time stability of sliding mode motion to the origin is guaranteed. Furthermore, the convergence time is independent of initial values.

Continuous Sliding Mode Controller Design

Theorem 2:

For nonlinear system ( s y s ) (sys) (sys), the proposed sliding surface ( S S ) (SS) (SS) will be reached in finite-time, and the system trajectories converge to zero along the sliding surface ( S S ) (SS) (SS) within finite-time with the following control algorithm:
u = − b − 1 ( u e q + u n ) u e q = f ( x , t ) + ( c 1 ⌈ x 2 ⌋ α 1 + c 2 ⌈ x 2 ⌋ α 2 + c 3 ⌈ x 2 ⌋ α 3 ) + ( k 1 ⌈ x 1 ⌋ β 1 + k 2 ⌈ x 1 ⌋ β 2 + k 3 ⌈ x 1 ⌋ β 3 ) u ˙ n = ( k d + η ) sgn ⁡ ( s ) − λ ⌈ s ⌋ ξ − μ ⌈ s ⌋ ε \begin{aligned} u=&-b^{-1}\left(u_{e q}+u_{n}\right) \\ u_{e q}=& f(x, t)+\left(c_{1}\left\lceil x_{2}\right\rfloor^{\alpha_{1}}+c_{2}\left\lceil x_{2}\right\rfloor^{\alpha_{2}}+c_{3}\left\lceil x_{2}\right\rfloor^{\alpha_{3}}\right) \\ &+\left(k_{1}\left\lceil x_{1}\right\rfloor^{\beta_{1}}+k_{2}\left\lceil x_{1}\right\rfloor^{\beta_{2}}+k_{3}\left\lceil x_{1}\right\rfloor^{\beta_{3}}\right) \\ \dot{u}_{n}=&\left(k_{d}+\eta\right) \operatorname{sgn}(s)-\lambda\lceil s\rfloor^{\xi}-\mu\lceil s\rfloor^{\varepsilon} \end{aligned} u=ueq​=u˙n​=​−b−1(ueq​+un​)f(x,t)+(c1​⌈x2​⌋α1​+c2​⌈x2​⌋α2​+c3​⌈x2​⌋α3​)+(k1​⌈x1​⌋β1​+k2​⌈x1​⌋β2​+k3​⌈x1​⌋β3​)(kd​+η)sgn(s)−λ⌈s⌋ξ−μ⌈s⌋ε​
where k i , c i , α i , β i ( i = 1 , 2 , 3 ) k_i, c_i, α_i, β_i(i = 1,2,3) ki​,ci​,αi​,βi​(i=1,2,3) and k d k_d kd​ have been given previously; η > 0 η > 0 η>0 is an adjustable constant; λ , μ , ξ , ε λ, μ, ξ, ε λ,μ,ξ,ε are all positive constants. The upper bound of the time horizon is independent of initial condition.

The time upper bound of the reaching phase and the sliding phase can be obtained as:
T F X < T r ∗ ( λ , μ , ξ , ε ) + T s ∗ ( ϱ , d V ∞ , d V 0 , k s ∞ , k s 0 ) = 2 λ ( ξ − 1 ) + 2 μ ( 1 − ε ) + 1 ϱ ( d V ∞ k s ∞ + d V 0 ∣ k s 0 ∣ ) (FT) \begin{aligned} T_{F X} &<T_{r}^{*}(\lambda, \mu, \xi, \varepsilon)+T_{s}^{*}\left(\varrho, d_{V_{\infty}}, d_{V_{0}}, k_{s \infty}, k_{s_{0}}\right) \\ &=\frac{2}{\lambda(\xi-1)}+\frac{2}{\mu(1-\varepsilon)}+\frac{1}{\varrho}\left(\frac{d_{V_{\infty}}}{k_{s \infty}}+\frac{d_{V_{0}}}{\left|k_{s 0}\right|}\right) \end{aligned}\tag{FT} TFX​​<Tr∗​(λ,μ,ξ,ε)+Ts∗​(ϱ,dV∞​​,dV0​​,ks∞​,ks0​​)=λ(ξ−1)2​+μ(1−ε)2​+ϱ1​(ks∞​dV∞​​​+∣ks0​∣dV0​​​)​(FT)
Note that the time bound is independent of the initial conditions.

SIMULATION AND EXPERIMENTAL RESULTS

The responses of system state x 1 ( t ) x_1 (t) x1​(t) with different initial values

The continuous terminal sliding mode control input with four different initial conditions.

My finding

1. The chattering phenomenon is not alleviated well (which is obvious from the simulation figs).
2. The convergence rate for the system state is very slow.