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【论文推导】Tube-based MPC-辅助控制器设计

news 2025/10/16 11:59:17

引言

本文给出了Tube-based MPC-辅助系统控制器设计与稳定性证明的大致过程，以供自己回顾检查。

辅助系统控制器设计

首先建立状态误差方程：
$ϵ(k+1)=A~ϵ(k)+B~ϕη(k)+ω(k)(14)\epsilon(k + 1) = \tilde{A}\epsilon(k) + \tilde{B}\phi\eta(k) + \omega(k) \tag{14}$
其中， $ϵ(k)=χ^(k)−ξ^(k)\epsilon(k) = \hat{\chi}(k) - \hat{\xi}(k)$ 为实际系统与标称系统状态之间的误差。

定义切换函数为：
$C_{\text{e}} \epsilon(k) \tag{15}$
通过延迟估计的方式测量不确定扰动：
$ω^(k)=ϵ(k)−A~ϵ(k−1)−B~ϕη(k−1)=ω(k−1)(16)\hat{\omega}(k) = \epsilon(k) - \tilde{A}\epsilon(k - 1) - \tilde{B}\phi\eta(k - 1) = \omega(k - 1) \tag{16}$
设计滑膜趋近率为如下形式：
$\frac{\lambda}{\beta + \text{sgn}(s(k))} t \text{sig}^\alpha(s(k)) + C_{\text{e}} \sigma(k) \tag{17}$
其中， $σ(k)=ω(k)−ω(k−1)\sigma(k) = \omega(k) - \omega(k - 1)$ 为扰动误差； $sigα(s(k))=∣s(k)∣αsgn(s(k))\text{sig}^\alpha(s(k)) = |s(k)|^\alpha \text{sgn}(s(k))$ 。

将式15带入进式17中推导可得：
$s(k+1)=(1−qt)s(k)−λβ+sgn(s(k))tsigα(s(k))+Ceσ(k)Ce(k+1)=(1−qt)s(k)−λβ+sgn(s(k))tsigα(s(k))+Ceσ(k)Ce[A~(k)+B~ϕη(k)+ω(k)]=(1−qt)s(k)−λβ+sgn(s(k))tsigα(s(k))+Ceσ(k)CeB~ϕη(k)=(1−qt)s(k)−λβ+sgn(s(k))tsigα(s(k))+Ce[ω(k)−ω(k−1)−A~(k)−ω(k)]η(k)=(CeB~ϕ)−1{(1−qt)s(k)−λβ+sgn(s(k))tsigα(s(k))−Ce[ω(k−1)+A~(k)]}(18)\begin{aligned} s(k + 1) & = (1 - qt)s(k) - \frac{\lambda }{{\beta + {\rm{sgn}}(s(k))}}t{\rm{si}}{{\rm{g}}^\alpha }(s(k)) + {C_{\rm{e}}}\sigma (k)\\ {C_{\rm{e}}}(k + 1) & = (1 - qt)s(k) - \frac{\lambda }{{\beta + {\rm{sgn}}(s(k))}}t{\rm{si}}{{\rm{g}}^\alpha }(s(k)) + {C_{\rm{e}}}\sigma (k)\\ {C_{\rm{e}}}\left[ {\tilde A(k) + \tilde B\phi \eta (k) + \omega (k)} \right] & = (1 - qt)s(k) - \frac{\lambda }{{\beta + {\rm{sgn}}(s(k))}}t{\rm{si}}{{\rm{g}}^\alpha }(s(k)) + {C_{\rm{e}}}\sigma (k)\\ {C_{\rm{e}}}\tilde B\phi \eta (k) & = (1 - qt)s(k) - \frac{\lambda }{{\beta + {\rm{sgn}}(s(k))}}t{\rm{si}}{{\rm{g}}^\alpha }(s(k)) + {C_{\rm{e}}}\left[ {\omega (k) - \omega (k - 1) - \tilde A(k) - \omega (k)} \right]\\ \eta (k) & = {\left( {{C_{\rm{e}}}\tilde B\phi } \right)^{ - 1}}\left\{ {(1 - qt)s(k) - \frac{\lambda }{{\beta + {\rm{sgn}}(s(k))}}t{\rm{si}}{{\rm{g}}^\alpha }(s(k)) - {C_{\rm{e}}}\left[ {\omega (k - 1) + \tilde A(k)} \right]} \right\} \end{aligned} \tag{18}$

稳定性证明

证明需要包含两部分：

滑动模式曲面 s k ( )可以在 $k^*$ 步数内进入 qSM 区域 $Ω\Omega$ ，并无限期地保持在区域内；
$∣ s ∣$ 递减。

由式17建立Lyapunov方程： $V(k) = s^2(k)$

则有：

$ΔV(k)=V(k+1)−V(k)=s(k+1)2−s(k)2=−(qts(k)+λβ+sgn(s(k))tsigα(s(k))−Ceσ(k))×(2s(k)−qts(k)−λβ+sgn(s(k))tsigα(s(k))+Ceσ(k))(19)\begin{aligned} \Delta V(k) & = V(k + 1) - V(k)\\ & = s{(k + 1)^2} - s{(k)^2}\\ & = - \left( {qts(k) + \frac{\lambda }{{\beta + {\rm{sgn}}(s(k))}}t{\rm{si}}{{\rm{g}}^\alpha }(s(k)) - {C_{\rm{e}}}\sigma (k)} \right)\\ & \times \left( {2s(k) - qts(k) - \frac{\lambda }{{\beta + {\rm{sgn}}(s(k))}}t{\rm{si}}{{\rm{g}}^\alpha }(s(k)) + {C_{\rm{e}}}\sigma (k)} \right) \end{aligned} \tag{19}$
当滑膜面 $s (k)$ 在区域 $Ω\Omega$ 内是稳定的，这里主要考虑 $s (k)$ 在区域 $Ω\Omega$ 外的情况，

令 $\frac{\lambda}{\beta + \text{sgn}(s(k))} t \text{sig}^\alpha(s(k)) - C_{\text{e}} \sigma(k)$ 则原式19可重写为 $ΔV(k)=−A(k)(2s(k)−A(k))\Delta V(k) = - A(k)(2s(k) - A(k))$ ，此时就需要求解使得 $ΔV(k)<0\Delta V (k) \lt 0$ 即可，当$s(k) \gt \rho $
$λβ+1ts(k)α−Ceσ(k)≥λβ+1ts(k)α−∣Ceσ(k)∣≥λβ+1ts(k)α−σ∗(k)(20)\begin{aligned} \frac{\lambda }{{\beta + 1}}ts{(k)^\alpha } - {C_{\rm{e}}}\sigma (k) & \ge \frac{\lambda }{{\beta + 1}}ts{(k)^\alpha } - \left| {{C_{\rm{e}}}\sigma (k)} \right|\\ & \ge \frac{\lambda }{{\beta + 1}}ts{(k)^\alpha } - {\sigma ^*}(k) \end{aligned} \tag{20}$
其中 $∣Ceσ(k)∣≤σ∗|C_{\text{e}} \sigma(k)| \leq \sigma^*$ ，同时当$s(k) \gt \rho $时，可将$ qts(k) $忽略，其余的非线性项占主导，且有$ \text{sgn}(s(k)) = 1 $，$ \text{sig}^\alpha(s(k)) = s{(k)^\alpha }$。
$s(k)>ψ(α)[β+1λtσ∗(k)]1α>[β+1λtσ∗(k)]1α(21)\begin{aligned} s(k) & \gt \psi \left( \alpha \right){\left[ {\frac{{\beta + 1}}{{\lambda t}}{\sigma ^*}(k)} \right]^{\frac{1}{\alpha }}}\\ & \gt {\left[ {\frac{{\beta + 1}}{{\lambda t}}{\sigma ^*}(k)} \right]^{\frac{1}{\alpha }}} \end{aligned} \tag{21}$
其中， $∣Ceσ(k)∣≤σ∗|C_{\text{e}} \sigma(k)| \leq \sigma^*$ ，由此可得在式19中有
$\frac{\lambda}{\beta + 1} t s^\alpha(k) - C_{\text{e}} \sigma(k) \geq qts(k) + \sigma^* \psi^\alpha(\alpha) - |C_{\text{e}} \sigma(k)| \\ \geq qt \rho + \left[ \psi^\alpha(\alpha) - 1 \right] \sigma^* := \varrho \tag{22}$
令 $\frac{\lambda}{\beta + \text{sgn}(s(k))} t \text{sig}^\alpha(s(k))$ ，则 $s (k)$ 理论边界计算如下：
$(1−qt)s(k)−λβ+sgn(s(k))tsigα(s(k))>0s(k)>ψ(α)[λ(β+1)(1−qt)]11−α(23)\begin{aligned} &(1 - qt)s(k) - \frac{\lambda}{\beta + \text{sgn}(s(k))} t \text{si} \text{g}^{\alpha}(s(k)) > 0 \\ &s(k) > \psi(\alpha) \left[ \frac{\lambda}{(\beta + 1)(1 - qt)} \right]^{\frac{1}{1 - \alpha}} \end{aligned} \tag{23}$
如果直接使用理论边界，会因未考虑时滞引发的状态偏移，导致实际波动超出理论范围。故而在上述理论边界的基础上，添加放缩因子 $ψ(α)\psi(\alpha)$ ，使实际边界略大于理论边界，则实际边界为
$ρ=ψ(α)⋅max⁡{((β+sgn(s(k)))σ∗λt)1α,(λt(β+sgn(s(k)))(1−qt))11−α}(24)\rho = \psi(\alpha) \cdot \max\left\{ \left( \frac{(\beta + \text{sgn}(s(k))) \sigma^*}{\lambda t} \right)^{\frac{1}{\alpha}}, \left( \frac{\lambda t}{(\beta + \text{sgn}(s(k)))(1 - qt)} \right)^{\frac{1}{1 - \alpha}} \right\} \tag{24}$
其中， $ψ(α)=1+αα1−α−α11−α\psi(\alpha) = 1 + \alpha^{\frac{\alpha}{1 - \alpha}} - \alpha^{\frac{1}{1 - \alpha}}$ 。将式19、式21、式22联立可得：
$\frac{\lambda}{\beta + 1} t s^\alpha(k) + C_{\text{e}} \sigma(k) \geq qts(k) + \frac{\lambda}{\beta + 1} t s^\alpha(k) - |C_{\text{e}} \sigma(k)| \geq \varrho \tag{25}$
综上所述， $ΔV(k)≤−ϱ2\Delta V(k) \leq -\varrho^2$ 。针对 $-\rho$ 可以用相同的方法证明 $ΔV(k)≤−ϱ2\Delta V(k) \leq -\varrho^2$ 。

根据上述结论，
$1)^2 - s(k)^2 \leq -\varrho^2 \\ \begin{cases} s(1)^2 \leq s(0)^2 - \varrho^2 \\ s(2)^2 \leq s(0)^2 - 2\varrho^2 \\ \vdots \\ s(m)^2 \leq s(0)^2 - m\varrho^2 \end{cases} \tag{26}$
当 $s(m∗)2=ρ2s(m^*)^2 = \rho^2$ 时， $s (k)$ 恰好到达$\Omega $边界，该边界可表示为
$ρ=ψ(α)Φ=ψ(α)max⁡{((β+sgn(s(k)))σ∗λt)1α,(λt(β+sgn(s(k)))(1−qt))11−α}(27)\rho = \psi(\alpha) \Phi = \psi(\alpha) \max\left\{ \left( \frac{(\beta + \text{sgn}(s(k))) \sigma^*}{\lambda t} \right)^{\frac{1}{\alpha}}, \left( \frac{\lambda t}{(\beta + \text{sgn}(s(k)))(1 - qt)} \right)^{\frac{1}{1 - \alpha}} \right\} \tag{27}$
此时分两种情况：

当 $((β+sgn(s(k)))σ∗λt)1α>(λt(β+sgn(s(k)))(1−qt))11−α\left( \frac{(\beta + \text{sgn}(s(k))) \sigma^*}{\lambda t} \right)^{\frac{1}{\alpha}} \gt \left( \frac{\lambda t}{(\beta + \text{sgn}(s(k)))(1 - qt)} \right)^{\frac{1}{1 - \alpha}}$ 时，则有
${((β+sgn(s(k)))σ∗λt)1α≤(λt(β+sgn(s(k)))(1−qt))11−ασ∗≤λtβ+sgn(s(k))(λt(β+sgn(s(k)))(1−qt))α1−α{Φα≥(β+sgn(s(k)))σ∗λtσ∗≤λt(β+sgn(s(k)))Φα=λt(β+sgn(s(k)))(λt(β+sgn(s(k)))(1−qt))α1−α=(λt(β+sgn(s(k))))11−α(1(1−qt))α1−α=(λt(β+sgn(s(k)))(1−qt)α)11−α=(λt(1−qt)1−α(β+sgn(s(k)))(1−qt))11−α=(1−qt)Φ(28)\begin{array}{l} \left\{ \begin{array}{l} {\left( {\frac{{(\beta + {\rm{sgn}}(s(k))){\sigma ^*}}}{{\lambda t}}} \right)^{\frac{1}{\alpha }}} \le {\left( {\frac{{\lambda t}}{{(\beta + {\rm{sgn}}(s(k)))(1 - qt)}}} \right)^{\frac{1}{{1 - \alpha }}}}\\ {\sigma ^*} \le \frac{{\lambda t}}{{\beta + {\rm{sgn}}(s(k))}}{\left( {\frac{{\lambda t}}{{(\beta + {\rm{sgn}}(s(k)))(1 - qt)}}} \right)^{\frac{\alpha }{{1 - \alpha }}}} \end{array} \right.\\ \left\{ \begin{array}{l} {\Phi ^\alpha } \ge \frac{{(\beta + {\rm{sgn}}(s(k))){\sigma ^*}}}{{\lambda t}}\\ {\sigma ^*} \le \frac{{\lambda t}}{{(\beta + {\rm{sgn}}(s(k)))}}{\Phi ^\alpha } = \frac{{\lambda t}}{{(\beta + {\rm{sgn}}(s(k)))}}{\left( {\frac{{\lambda t}}{{(\beta + {\rm{sgn}}(s(k)))(1 - qt)}}} \right)^{\frac{\alpha }{{1 - \alpha }}}}\\ = {\left( {\frac{{\lambda t}}{{(\beta + {\rm{sgn}}(s(k)))}}} \right)^{\frac{1}{{1 - \alpha }}}}{\left( {\frac{1}{{(1 - qt)}}} \right)^{\frac{\alpha }{{1 - \alpha }}}}\\ = {\left( {\frac{{\lambda t}}{{(\beta + {\rm{sgn}}(s(k))){{(1 - qt)}^\alpha }}}} \right)^{\frac{1}{{1 - \alpha }}}}\\ = {\left( {\frac{{\lambda t{{(1 - qt)}^{1 - \alpha }}}}{{(\beta + {\rm{sgn}}(s(k)))(1 - qt)}}} \right)^{\frac{1}{{1 - \alpha }}}}\\ = (1 - qt)\Phi \end{array} \right. \end{array} \tag{28}$
当 $((β+sgn(s(k)))σ∗λt)1α<(λt(β+sgn(s(k)))(1−qt))11−α\left( \frac{(\beta + \text{sgn}(s(k))) \sigma^*}{\lambda t} \right)^{\frac{1}{\alpha}} \lt \left( \frac{\lambda t}{(\beta + \text{sgn}(s(k)))(1 - qt)} \right)^{\frac{1}{1 - \alpha}}$ 时，则有
$σ∗=λt(β+sgn(s(k)))Φασ∗1−α≥(λt(β+sgn(s(k))))1−α(λt(β+sgn(s(k)))(1−qt))α=λt(β+sgn(s(k)))(1−qt)ασ∗≤(1−qt)((β+sgn(s(k)))σ∗λt)1α=((1−qt)α(β+sgn(s(k)))σ∗λt)1α(29)\begin{array}{l} {\sigma ^*} = \frac{{\lambda t}}{{(\beta + {\rm{sgn}}(s(k)))}}{\Phi ^\alpha }{\rm{ }}\\ {\sigma ^*}^{1 - \alpha } \ge {\left( {\frac{{\lambda t}}{{(\beta + {\rm{sgn}}(s(k)))}}} \right)^{1 - \alpha }}{\left( {\frac{{\lambda t}}{{(\beta + {\rm{sgn}}(s(k)))(1 - qt)}}} \right)^\alpha } = \frac{{\lambda t}}{{(\beta + {\rm{sgn}}(s(k))){{(1 - qt)}^\alpha }}}\\ {\sigma ^*} \le (1 - qt){\left( {\frac{{(\beta + {\rm{sgn}}(s(k))){\sigma ^*}}}{{\lambda t}}} \right)^{\frac{1}{\alpha }}} = {\left( {\frac{{{{(1 - qt)}^\alpha }(\beta + {\rm{sgn}}(s(k))){\sigma ^*}}}{{\lambda t}}} \right)^{\frac{1}{\alpha }}} \end{array} \tag{29}$
综上所述 $σ∗≤(1−qt)Φ\sigma^* \leq (1 - qt)\Phi$ ， $s (k + 1)$ 可重新表述为
$qt)\theta\rho - \frac{\lambda}{\beta + \text{sgn}(s(k))} t \text{sig}^\alpha(\theta\rho) + C_e \sigma(k) \\ \leq (1 - qt)\psi(\alpha)\theta\Phi - \frac{\lambda}{\beta + \text{sgn}(s(k))} t \text{sig}^\alpha(\psi(\alpha)\theta)\Phi^\alpha + \sigma^* \tag{30}$
基于该关系，论证 $−ρ≤s(k+1)≤ρ-\rho \leq s(k + 1) \leq \rho$ ，同样分为两种情况

情况1： $\leq \rho$

若 $ψ(α)θ>0\psi (\alpha )\theta > 0$
$s.t.λβ+sgn(s(k))tΦα=σ∗s(k+1)≤(1−qt)ψ(α)θΦ−λβ+sgn(s(k))t(ψ(α)θ)αΦα+σ∗≤(1−qt)ψ(α)θΦ−(ψ(α)θ)ασ∗+σ∗(31)\begin{array}{l} s.t.\frac{\lambda }{{\beta + {\rm{sgn}}(s(k))}}t{\Phi ^\alpha } = {\sigma ^*}\\ s(k + 1) \le (1 - qt)\psi (\alpha )\theta \Phi - \frac{\lambda }{{\beta + {\rm{sgn}}(s(k))}}t{(\psi (\alpha )\theta )^\alpha }{\Phi ^\alpha } + {\sigma ^*}\\ \le (1 - qt)\psi (\alpha )\theta \Phi - {(\psi (\alpha )\theta )^\alpha }{\sigma ^*} + {\sigma ^*} \end{array} \tag{31}$
若 $ψ(α)θ>1\psi (\alpha )\theta > 1$
$s.t.{σ∗≤(1−qt)Φs(k+1)=θψ(α)Φρ=ψ(α)Φs(k+1)≤(1−qt)ψ(α)θΦ≤ψ(α)θΦ≤ψ(α)Φ=ρ(32)\begin{array}{l} s.t.\left\{ \begin{array}{l} {\sigma ^*} \le (1 - qt)\Phi \\ s(k + 1) = \theta \psi (\alpha )\Phi \\ \rho = \psi (\alpha )\Phi \end{array} \right.\\ s(k + 1) \le (1 - qt)\psi (\alpha )\theta \Phi \le \psi (\alpha )\theta \Phi \le \psi (\alpha )\Phi = \rho \end{array} \tag{32}$
若 $\le \psi (\alpha )\theta \le 1$
$s(k+1)=(1−qt)θρ−λβ+sgn(s(k))tsigα(θρ)+Ceσ(k)≤(1−qt)ψ(α)θΦ−λβ+sgn(s(k))tΦαsigα(ψ(α)θ)+σ∗≤(1−qt)ψ(α)θΦ−(1−qt)Φ(ψ(α)θ)α+σ∗≤(1−qt)ψ(α)θΦ−(ψ(α)θ)ασ∗+σ∗≤(1−qt)ψ(α)θΦ+[1−(ψ(α)θ)α]σ∗≤(1−qt)ψ(α)θΦ+(1−qt)Φ[1−(ψ(α)θ)α]=(1−qt)Φ[ψ(α)θ+1−(ψ(α)θ)α]≤(1−qt)Φ<ψ(α)Φ=ρ(33)\begin{aligned} s(k + 1) & = (1 - qt)\theta \rho - \frac{\lambda }{{\beta + {\rm{sgn}}(s(k))}}t{\rm{si}}{{\rm{g}}^\alpha }(\theta \rho ) + {C_e}\sigma (k)\\ & \le (1 - qt)\psi (\alpha )\theta \Phi - \frac{\lambda }{{\beta + {\rm{sgn}}(s(k))}}t{\Phi ^\alpha }{\rm{si}}{{\rm{g}}^\alpha }(\psi (\alpha )\theta ) + {\sigma ^*}\\ & \le (1 - qt)\psi (\alpha )\theta \Phi - (1 - qt)\Phi {\left( {\psi (\alpha )\theta } \right)^\alpha } + {\sigma ^*}\\ & \le (1 - qt)\psi (\alpha )\theta \Phi - {\left( {\psi (\alpha )\theta } \right)^\alpha }{\sigma ^*} + {\sigma ^*}\\ & \le (1 - qt)\psi (\alpha )\theta \Phi + \left[ {1 - {{\left( {\psi (\alpha )\theta } \right)}^\alpha }} \right]{\sigma ^*}\\ & \le (1 - qt)\psi (\alpha )\theta \Phi + (1 - qt)\Phi \left[ {1 - {{\left( {\psi (\alpha )\theta } \right)}^\alpha }} \right]\\ & = (1 - qt)\Phi \left[ {\psi (\alpha )\theta + 1 - {{\left( {\psi (\alpha )\theta } \right)}^\alpha }} \right]\\ & \le (1 - qt)\Phi \\ & < \psi (\alpha )\Phi \\ & = \rho \end{aligned} \tag{33}$
若 $ψ(α)θ≤0\psi (\alpha )\theta \le 0$
$\leq -(1 - qt)|\psi(\alpha)\theta|\Phi + |\psi(\alpha)\theta|^\alpha(1 - qt)\Phi + \sigma^* \tag{34}$
若 $ψ(α)θ≤−1\psi (\alpha )\theta \le -1$
$\leq \sigma^* \leq (1 - qt)\Phi < \psi(\alpha)\Phi = \rho \tag{35}$
若 $\le \psi (\alpha )\theta \le 0$
$s.t.{σ∗≤(1−qt)Φs(k+1)=θψ(α)Φρ=ψ(α)Φs(k+1)≤−(1−qt)∣ψ(α)θ∣Φ+∣ψ(α)θ∣ασ∗+σ∗≤−(1−qt)∣ψ(α)θ∣Φ+∣ψ(α)θ∣ασ∗+σ∗≤−(1−qt)∣ψ(α)θ∣Φ+∣ψ(α)θ∣α(1−qt)Φ+(1−qt)Φ=[1−∣ψ(α)θ∣+∣ψ(α)θ∣α](1−qt)Φ(36)\begin{array}{l} s.t.\left\{ \begin{array}{l} {\sigma ^*} \le (1 - qt)\Phi \\ s(k + 1) = \theta \psi (\alpha )\Phi \\ \rho = \psi (\alpha )\Phi \end{array} \right.\\ s(k + 1) \le - (1 - qt)\left| {\psi (\alpha )\theta } \right|\Phi + {\left| {\psi (\alpha )\theta } \right|^\alpha }{\sigma ^*} + {\sigma ^*} \le - (1 - qt)\left| {\psi (\alpha )\theta } \right|\Phi + {\left| {\psi (\alpha )\theta } \right|^\alpha }{\sigma ^*} + {\sigma ^*}\\ \le - (1 - qt)\left| {\psi (\alpha )\theta } \right|\Phi + {\left| {\psi (\alpha )\theta } \right|^\alpha }(1 - qt)\Phi + (1 - qt)\Phi \\ = \left[ {1 - \left| {\psi (\alpha )\theta } \right| + {{\left| {\psi (\alpha )\theta } \right|}^\alpha }} \right](1 - qt)\Phi \end{array} \tag{36}$
情况2： $\geq -\rho$ 则有
$|C_e \sigma(k)| \leq \sigma^* \\ s(k + 1) \geq (1 - qt)\psi(\alpha)\theta\Phi - \frac{\lambda}{\beta + \text{sgn}(s(k))} t \text{sig}^\alpha(\psi(\alpha)\theta)\Phi^\alpha - \sigma^* \tag{37}$
若 $ψ(α)θ>1\psi (\alpha )\theta > 1$
$\geq (1 - qt)\psi(\alpha)\theta\Phi - \frac{\lambda}{\beta + \text{sgn}(s(k))} t \text{sig}^\alpha(\psi(\alpha)\theta)\Phi^\alpha - \sigma^* \\ \geq - \sigma^* \geq -(1-qt) \Phi \geq - \rho \tag{38}$
若 $\le \psi (\alpha )\theta \le 1$
$s(k+1)≥(1−qt)ψ(α)θΦ−λβ+sgn(s(k))tsigα(ψ(α)θ)Φα−σ∗≥(1−qt)ψ(α)θΦ−(ψ(α)θ)ασ∗−σ∗≥(1−qt)ψ(α)θΦ−(1−qt)Φ(ψ(α)θ)α−(1−qt)Φσ∗=(1−qt)Φ(ψ(α)θ−(ψ(α)θ)α−1)(39)\begin{aligned} s(k + 1) & \ge (1 - qt)\psi (\alpha )\theta \Phi - \frac{\lambda }{{\beta + {\rm{sgn}}(s(k))}}t{\rm{si}}{{\rm{g}}^\alpha }(\psi (\alpha )\theta ){\Phi ^\alpha } - {\sigma ^*}\\ & \ge (1 - qt)\psi (\alpha )\theta \Phi - {(\psi (\alpha )\theta )^\alpha }{\sigma ^*} - {\sigma ^*}\\ & \ge (1 - qt)\psi (\alpha )\theta \Phi - (1 - qt)\Phi {(\psi (\alpha )\theta )^\alpha } - (1 - qt)\Phi {\sigma ^*}\\ & = (1 - qt)\Phi \left( {\psi (\alpha )\theta - {{(\psi (\alpha )\theta )}^\alpha } - 1} \right) \end{aligned} \tag{39}$
若 $\le \psi (\alpha )\theta \le 0$
$s(k+1)≥(1−qt)ψ(α)θΦ−λβ+sgn(s(k))tsigα(ψ(α)θ)Φα−σ∗≥−(1−qt)∣ψ(α)θ∣Φ+∣ψ(α)θ∣ασ∗−σ∗≥−(1−qt)∣ψ(α)θ∣Φ+(1−qt)Φ∣ψ(α)θ∣ασ∗−(1−qt)Φσ∗=(1−qt)Φ((ψ(α)θ)α−ψ(α)θ−1)≥−(1−qt)Φ≥−ρ(40)\begin{aligned} s(k + 1) & \ge (1 - qt)\psi (\alpha )\theta \Phi - \frac{\lambda }{{\beta + {\rm{sgn}}(s(k))}}t{\rm{si}}{{\rm{g}}^\alpha }(\psi (\alpha )\theta ){\Phi ^\alpha } - {\sigma ^*}\\ & \ge - (1 - qt)\left| {\psi (\alpha )\theta } \right|\Phi + {\left| {\psi (\alpha )\theta } \right|^\alpha }{\sigma ^*} - {\sigma ^*}\\ & \ge - (1 - qt)\left| {\psi (\alpha )\theta } \right|\Phi + (1 - qt)\Phi {\left| {\psi (\alpha )\theta } \right|^\alpha }{\sigma ^*} - (1 - qt)\Phi {\sigma ^*}\\ & = (1 - qt)\Phi \left( {{{(\psi (\alpha )\theta )}^\alpha } - \psi (\alpha )\theta - 1} \right)\\ & \ge - (1 - qt)\Phi \\ & \ge - \rho \end{aligned} \tag{40}$
若 $ψ(α)θ≤−1\psi (\alpha )\theta \le -1$
$s(k+1)≥(1−qt)ψ(α)θΦ−λβ+sgn(s(k))tsigα(ψ(α)θ)Φα−σ∗≥(1−qt)ψ(α)θΦ+λβ+sgn(s(k))t∣ψ(α)θ∣αΦα−σ∗≥(1−qt)Φψ(α)θ+∣ψ(α)θ∣ασ∗−σ∗≥(1−qt)Φψ(α)θ≥−ρ(41)\begin{aligned} s(k + 1) & \ge (1 - qt)\psi (\alpha )\theta \Phi - \frac{\lambda }{{\beta + {\rm{sgn}}(s(k))}}t{\rm{si}}{{\rm{g}}^\alpha }(\psi (\alpha )\theta ){\Phi ^\alpha } - {\sigma ^*}\\ & \ge (1 - qt)\psi (\alpha )\theta \Phi + \frac{\lambda }{{\beta + {\rm{sgn}}(s(k))}}t{\left| {\psi (\alpha )\theta } \right|^\alpha }{\Phi ^\alpha } - {\sigma ^*}\\ & \ge (1 - qt)\Phi \psi (\alpha )\theta + {\left| {\psi (\alpha )\theta } \right|^\alpha }{\sigma ^*} - {\sigma ^*}\\ & \ge (1 - qt)\Phi \psi (\alpha )\theta \\ & \ge - \rho \end{aligned} \tag{41}$
综上所述， $ρ<s(k+1)<ρ\rho \lt s(k+1) \lt \rho$ 。