Dynamic Optimization of Differential-algebraic Equations With Inequality Path Constraints
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摘要: 针对具有不等式路径约束的微分代数方程（Differential-algebraic equations，DAE）系统的动态优化问题，通常将DAE中的等式路径约束进行微分处理，或者将其转化为点约束或不等式约束进行求解.前者需要考虑初值条件的相容性或增加约束，在变量间耦合度较高的情况下这种转化求解方法是不可行的；后者将等式约束转化为其他类型的约束会增加约束条件，增加了求解难度.为了克服该缺点，本文提出了结合后向差分法对DAE直接处理来求解上述动态优化问题的方法.首先利用控制向量参数化方法将无限维的最优控制问题转化为有限维的最优控制问题，再利用分点离散法用有限个内点约束去代替原不等式路径约束，最后用序列二次规划（Sequential quadratic programming，SQP）法使得在有限步数的迭代下，得到满足用户指定的路径约束违反容忍度下的KKT（Karush Kuhn Tucker）最优点.理论上证明了该算法在有限步内收敛.最后将所提出的方法应用在具有不等式路径约束的微分代数方程系统中进行仿真，结果验证了该方法的有效性.Abstract: For dynamic optimization of differential algebraic systems with inequality path constraints, the equality constraint in differential-algebraic equations (DAE) is often differentiated or transformed into point constraints or inequality constraints to solve. As for the former, the compatibility of initial conditions needs to be checked or more constraints are added, making the original optimization problem infeasible in some cases. For the latter, the way that equality constraint is converted to other types of constraints increases the difficulty of solving. In order to overcome the above problems, a new method is proposed to solve the above problem directly based on the backward differentiation formula. Firstly, the control vector parameterization is used to convert the optimal control problem of infinite dimensions into a finite dimensional one. Then, a set of interior-time points by using the pointwise discretization method are used to replace the original inequality path constraints. Finally, the sequential quadratic programming (SQP) is used to locate the Karush Kuhn Tucker (KKT) points within finite-stepped iterations. Proof is provided for the finite convergence of the algorithm. The dynamic optimization algorithm is applied to the differential algebraic equation systems with inequality path constraint, and simulation studies are carried out to verify the effectiveness of the proposed method for the differential algebraic systems with inequality path constraints.
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$ H_{\infty} $控制理论主要研究抑制干扰和不确定性问题[1].在$ H_{\infty} $控制理论中, 传递函数(或系统)的$ H_{\infty} $范数是一项重要的性能指标, 用于度量扰动输入对系统输出的影响, 反映了闭环系统的抗扰能力.在$ H_{\infty} $控制理论研究中, 长期存在一个挑战性议题:是否能够直接给出关于$ H_{\infty} $范数的通用解析表达式, 进而避免针对线性矩阵不等式(Linear matrix inequality, LMI)约束条件的繁琐的$ H_{\infty} $范数近似寻优方案.
在20世纪80年代, $ H_{\infty} $控制理论的研究由频域转换到时域, 开启了基于状态空间方程描述的系统鲁棒性能研究[2].总的来说, $ H_{\infty} $性能时域分析面临的核心问题是如何选择适当的李雅普诺夫函数.具体表现为基于李雅普诺夫方程[3-4]或参数化Riccati不等式[5]均难以得到用于精确分析系统$ H_{\infty} $性能的最优李雅普诺夫函数, 因此在早期的研究中结果的保守性是难以避免的.
为精确求解$ H_{\infty} $范数, 有学者提出了有界实引理[6], 并将求解$ H_{\infty} $范数问题转化为时域状态空间的约束优化问题.基于有界实引理给出的LMI约束条件, $ H_{\infty} $范数能够被近似寻优[7-14].在LMI方法中, $ H_{\infty} $范数的寻优一般包含以下步骤:
1) 给出一个充分大的初始$ H_{\infty} $范数估计$ \mit\gamma $;
2) 解LMI问题;
3) 递减$ H_{\infty} $范数估计$ \mit\gamma $, 直到获得满足LMI条件的最小$ H_{\infty} $范数估计$ \mit\gamma $.
显然, 一旦最小$ H_{\infty} $范数估计得到, 则通过解LMI, 可以得到相应的近似最优李雅普诺夫函数.不难发现, LMI方法存在一定不足, 表现为:
1) 对于每一个给定的$ \mit\gamma $, LMI条件需要被重复求解, 直到找到最小的$ H_{\infty} $范数估计, 过程过于繁琐;
2) 这种试凑逼近方法无法揭示系统结构和参数对$ H_{\infty} $性能的影响, 在一定程度上限制了控制器精细设计的研究.
为了克服目前关于$ H_{\infty} $范数问题研究的不足, 一个可替换的方法是直接优化李雅普诺夫函数, 进而得到关于$ H_{\infty} $范数的通用解析表达式.目前, 针对系统具体性能, 难以找到李雅普诺夫函数设计的充要条件, 因此这方面的研究并不多见.事实上, 在分析系统具体性能时, 存在最优的李雅普诺夫函数, 并且这一最优李雅普诺夫函数与系统结构和参数存在内在关系[15].因此本文尝试寻找一种李雅普诺夫函数的直接优化途径, 进而实现$ H_{\infty} $性能的精确分析.
由于多数高阶系统在一定的条件下可以近似(或分解)为二阶系统来研究, 并且二阶系统的分析方法是分析高阶系统的基础[16], 因此为有效展现最优李雅普诺夫函数与系统结构和参数存在内在关系, 本文针对一类二阶系统的$ H_{\infty} $范数问题, 构造和优化李雅普诺夫函数, 进而得到$ H_{\infty} $范数的通用解析表达式.本文的研究避免了LMI方法中繁琐的近似寻优过程, 并展示了系统矩阵特征值的实部和虚部对$ H_{\infty} $性能的影响.本文结构如下:第1节分析$ H_{\infty} $范数问题; 第2节分析Riccati不等式中李雅普诺夫函数的选择对求解$ H_{\infty} $范数的影响; 第3节展现李雅普诺夫函数的直接优化方法, 并给出$ H_{\infty} $范数的通用解析表达式; 第4节给出算例, 验证李雅普诺夫函数直接优化方法的有效性.
1. 问题的提出
1.1 问题描述
系统描述为
$ \begin{align} \dot{\boldsymbol{ x}} = A {\boldsymbol{ x}}+ {\boldsymbol{ w}} \end{align} $
(1) 其中, $ {\boldsymbol{ x}} \in \textbf{R}^{2} $, $ A $为Hurwitz矩阵, $ A $的特征值为复数, $ {\boldsymbol{ w}} $为扰动输入, $ \|{\boldsymbol{ w}}\| \leq \delta $, $ \delta $为常数, $ \|{\boldsymbol{ w}}\| = (\Sigma^{2}_{i = 1}w^{2}_{i})^{\frac{1}{2}} $.
研究的问题是如何得到系统(1)的状态上界.在数学意义上, 这一问题可转化为关于输入–输出系统的$ H_{\infty} $范数问题, 其中系统描述为
$ \begin{align} \begin{cases} \dot{\boldsymbol{ x}} = A {\boldsymbol{ x}} + {\boldsymbol{ w}} \\ {\boldsymbol{ y}} = {\boldsymbol{ x}} \end{cases} \end{align} $
(2) 在$ H_{\infty} $控制理论中, 系统的$ H_{\infty} $范数定义为$ S $右半平面上解析的有理函数阵的最大奇异值.在标量函数中就是幅频特性的极大值, 代表了系统对峰值有界信号的传递特性.
1.2 LMI方法分析
令李雅普诺夫函数为$ V = {\boldsymbol{ x}}^{\rm T}P{\boldsymbol{ x}} $, $ \gamma $为系统(2)的$ H_{\infty} $范数, 即$ \mit\gamma = \|G\|_{\infty} $, 其中$ G(s) = (sI-A)^{-1} $为系统(2)的传递函数.根据有界实引理, 可得
$ \begin{align} \left[ \begin{array}{ccc} PA+A^{\rm{T}}P & P & I \\ P & -\gamma^{2} I & 0_{2\times 2} \\ I & 0_{2\times 2} & -I \\ \end{array} \right] < 0 \end{align} $
(3) LMI方法是寻找式(3)中$ \mit\gamma $的最小值$ \mit\gamma_{\rm{min}} $.由于李雅普诺夫函数$ V = {\boldsymbol{ x}}^{\rm T}P {\boldsymbol{ x}} $可以任意构造, 因此对于每一个给定的$ \mit\gamma $, 需要重复求解LMI, 以判断式(3)的存在性, 直到$ \mit\gamma_{\rm{min}} $被找到.显然, 在LMI方法中复杂的优化过程是不可避免的.事实上, $ \mit\gamma_{\rm{min}} $与最优的$ P $矩阵是一一对应的.如果能够直接给出最优的$ P $矩阵, 则$ \mit\gamma_{\rm{min}} $的表达式就能够得到, 进而避免LMI方法中复杂的优化过程.本文的工作是尝试提供一种新的途径来直接给出$ \mit\gamma_{\rm{min}} $的表达式.
2. $ \pmb H_{\boldsymbol{ \infty}} $范数分析
根据特征值和奇异值分解原理, 可以得到下面的特性.
特性1. 对于系统(2)中特征矩阵$ A $, 存在可逆矩阵$ T $, 满足
$ \begin{align} D = -TAT^{-1} = \left[ \begin{array}{cc} \lambda & \nu \\ -\nu & \lambda \\ \end{array} \right] \end{align} $
(4) 其中, $ T = \Theta_{T1} \times \text{diag}\{t_{1}, t_{2}\} \times \Theta_{T2} $, $ \Theta_{T1} $和$ \Theta_{T2} $为正交矩阵, $ t_{2} \geq t_{1} > 0 $, $ \lambda > 0 $, $ \nu > 0 $. $ \text{diag}\{t_{1}, t_{2}\} $表示对角元素为$ t_{1} $, $ t_{2} $的对角阵.
令$ \alpha = {t_{2}}/{t_{1}} \geq 1 $, $ {\boldsymbol{ y}} = \Theta_{T2} \times {\boldsymbol{ x}} $, $ {\boldsymbol{ {\Delta}}} = \Theta_{T2}\times{\boldsymbol{ w}} $.由式(2)和特性1, 得
$ \begin{align} \begin{cases} \dot{\boldsymbol{ y}} = E {\boldsymbol{ y}} + B {\boldsymbol{ {\Delta}}} \\ {\boldsymbol{ x}} = C {\boldsymbol{ y}} \end{cases} \end{align} $
(5) 其中, $ B = I $为单位阵, $ C = \Theta_{T2}^{-1} $, $ E = - \left[ {array}{cc} \lambda & \alpha \nu \\ -\frac{1}{\alpha}\nu & \lambda \\ {array} \right], $并且系统(2)和(5)具有相同的$ H_{\infty} $范数.
根据文献[5]中引理2.1, 可以得到下面的特性.
特性2. 对于系统(5), 存在正定矩阵$ X $, 满足Riccati不等式
$ \begin{align} E^{\rm T}X+XE+(1+\varepsilon)C^{\rm T}C+ \rho^{-2} XBB^{\rm T}X \leq 0 \end{align} $
(6) 其中, $ \gamma < \rho $, $ \gamma = \|G\|_{\infty} $为系统$ H_{\infty} $范数, $ \varepsilon $为趋于零的正数.
注1. 应用Riccati不等式一般会得到具有很强保守性的结果, 但这种保守性并不是Riccati不等式本身导致的.研究表明:基于李雅普诺夫函数的准确选择, 可以将特性2中Riccati不等式转化为等式, 进而精确给出$ H_{\infty} $范数.因此, 导致这种保守性的原因是:在应用Riccati不等式时, 目前尚没有有效的方法找到最优的李雅普诺夫函数.这正是本文研究李雅普诺夫函数构造(或优化)的动机.
令
$ \begin{align} \Upsilon = \, &K^{-1} \Theta \begin{bmatrix} \lambda & -\frac{1}{\alpha} \nu \\ \alpha \nu & \lambda \end{bmatrix}\Theta^{\rm T}\; + \nonumber \\&\Theta \begin{bmatrix} \lambda & \alpha \nu \\ -\frac{1}{\alpha} \nu & \lambda \\ \end{bmatrix} \Theta^{\rm T}K^{-1} - K^{-1}K^{-1} \end{align} $
(7) 其中, $ \alpha \geq 1 $,
$ \begin{align} K = \iota \left[ \begin{array}{cc} 1 & 0 \\ 0 & k \\ \end{array} \right], \;\;\;\; \Theta = \left[ \begin{array}{cc} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \\ \end{array} \right] \end{align} $
(8) $ \iota >0 $, $ k \geq 1 $, $ 0 \leq \theta \leq {\pi}/{4} $.
由式(8)构造的李雅普诺夫函数分解了"放缩"和"旋转"作用.这种功能的分解使李雅普诺夫函数的参数优化具有了可行性.
定理1. 对于系统(5), 系统$ H_{\infty} $范数$ \gamma $满足
$ \begin{align} \gamma < \rho_{\rm{min}} = \left[\sqrt{\lambda_{\rm{min}}(\Upsilon)} \right]^{-1} \end{align} $
(9) 其中, $ \lambda_{\rm{min}}(\Upsilon) $为矩阵$ \Upsilon $的最小特征值.
证明. 令$ X = \Theta^{\rm T} K \Theta $, 其中, $ K $和$ \Theta $由式(8)给出.根据特性2和式(7), 得
$ \begin{align} \rho^{-2} I \leq \Upsilon - \varepsilon K^{-1}K^{-1} \end{align} $
(10) 则$ \rho^{-2} \leq \lambda_{\rm{min}}(\Upsilon- \varepsilon K^{-1}K^{-1}) $, 由于$ \gamma < \rho $, 并且$ \varepsilon $为趋于零的正数, 则式(9)成立.
注2. 根据定理1, 可以优化李雅普诺夫函数的参数, 以最大化$ \lambda_{\rm{min}}(\Upsilon) $, 进而精确估计系统$ H_{\infty} $范数.因此, 定理1给出了一种新的途径以得到系统的$ H_{\infty} $范数.
3. 李雅普诺夫函数优化
考查式(7)给出的矩阵$ \Upsilon $.由式(7)和式(8), 可得
$ \begin{align} \Upsilon = \frac{1}{\iota} \left[ \begin{array}{cc} 2\lambda + \beta \nu - \frac{1}{\iota} & \frac{1}{k} \sigma \nu \\ \frac{1}{k} \sigma \nu & \frac{1}{k}(2 \lambda - \beta \nu) - \frac{1}{\iota k^{2}} \\ \end{array} \right] \end{align} $
(11) 其中,
$ \begin{align} \beta = &\ \left(\alpha-\frac{1}{\alpha}\right) \sin 2\theta \end{align} $
(12) $ \begin{align} \sigma = &\, \left[\alpha- (\alpha-\frac{1}{\alpha}) \sin^{2} \theta \right] -k \left[\frac{1}{\alpha} + (\alpha-\frac{1}{\alpha}) \sin^{2} \theta \right] = \\ &\ \frac{1}{2}(1-k)(\alpha+\frac{1}{\alpha}) +\frac{1}{2}(1+k) (\alpha-\frac{1}{\alpha}) \cos 2\theta \end{align} $
(13) 根据式(11), 以最大化$ \lambda_{\rm{min}}(\Upsilon) $为目标, 将给出一种李雅普诺夫函数的优化方法.
3.1 李雅普诺夫函数优化策略
令
$ \begin{align} \Upsilon_{1} = \Theta^{-1} \Upsilon \Theta, \; \; Y_{1} = X^{-1} \end{align} $
(14) 则由式(7)和$ X = \Theta^{\rm T}K\Theta $, 得
$ \begin{align} \Upsilon_{1} = EE^{\rm T}-(E+Y_{1})(E+Y_{1})^{\rm T} \end{align} $
(15) 令
$ \begin{align} &EE^{\rm T} = \Theta_{1}^{\rm T} \Lambda \Theta_{1}, \quad \Upsilon_{2} = \Theta_{1} \Upsilon_{1} \Theta_{1}^{\rm T} \end{align} $
(16) $ \begin{align} &E_{1} = \Theta_{1} E \Theta_{1}^{\rm T}, \qquad Y_{2} = \Theta_{1} Y_{1} \Theta_{1}^{\rm T} \end{align} $
(17) 其中, $ \Lambda = {\rm diag}\{\sigma_{1}, \sigma_{2}\} $, $ \sigma_{1} \geq \sigma_{2} $, 则
$ \begin{align} \Upsilon_{2} = \Lambda - (E_{1}+Y_{2})(E_{1}+Y_{2})^{\rm T} \end{align} $
(18) 令
$ \begin{align} E_{1} = E_{R}+E_{J}, \; \; Y_{3} = E_{R}+Y_{2} \end{align} $
(19) 其中, $ E_{R}^{\rm T} = E_{R} $, $ E_{J} = -E_{J}^{\rm T} $, 则
$ \begin{align} \Upsilon_{2} = \Lambda - (E_{J}+Y_{3})(E_{J}+Y_{3})^{\rm T} \end{align} $
(20) 令
$ \begin{align} Y_{3} = \left[ \begin{array}{cc} y_{1} & y_{3} \\ y_{3} & y_{2} \\ \end{array} \right], \; \; E_{J} = \left[ \begin{array}{cc} 0 & a \\ -a & 0 \\ \end{array} \right] \end{align} $
(21) 则根据$ \Lambda = \text{diag}\{\sigma_{1}, \sigma_{2}\} $, 有$ \sigma_{1} \geq \sigma_{2} $,
$ \begin{align} \Upsilon_{2} = & \left[ \begin{array}{cc} \sigma_{1}-(y_{3}+a)^{2}-y_{1}^{2} \\ -(y_{1}+y_{2})y_{3}-(y_{2}-y_{1})a \\ \end{array}\right.\\ &\qquad\qquad\qquad \left. \begin{array}{cc} & -(y_{1}+y_{2})y_{3}-(y_{2}-y_{1})a \\ & \sigma_{2} -(y_{3}-a)^{2}-y_{2}^{2} \\ \end{array} \right] \end{align} $
(22) 根据式(14), (16), (21), (22)和定理1, 存在$ Y_{3} $, 使$ \lambda_{\rm{min}}(\Upsilon_{2}) $ $ > $ $ 0 $, 即$ \Upsilon_{2} $正定.因此根据式(22), 为了最大化$ \Upsilon_{2} $的最小特征值, 应使下面两个条件成立.
1) $ (y_{1}+y_{2})y_{3}+ (y_{2}-y_{1})a = 0 $ (例如$ y_{2} = 0 $, $ y_{3} = a $; 或$ y_{1} = y_{2} = 0 $).
2) $ \Upsilon_{2} $的特征值相等(例如$ y_{1}^{2} = \sigma_{1}-\sigma_{2}-4a^{2} $; 或$ y_{3} $ $ = $ $ (\sigma_{1}-\sigma_{2})/{4a} $).
注意, $ \sqrt{\sigma_{2}} $为$ E $的最小奇异值, 因此$ \gamma \geq {1}/{\sqrt{\sigma_{2}}} $.令
$ \begin{align} \lambda_{1} = \frac{1}{\iota}\left( 2\lambda + \beta \nu - \frac{1}{\iota} \right), \; \; \lambda_{2} = \frac{1}{\iota}\left[ \frac{1}{k}(2 \lambda - \beta \nu) - \frac{1}{\iota k^{2}} \right] \end{align} $
(23) 基于以上分析, 并根据式(9), (11), (14), (16)和(23), 为了最大化$ \Upsilon $的最小特征值, 李雅普诺夫函数的优化策略设计为$ \sigma = 0 $和$ \lambda_{1} = \lambda_{2} $.
3.2 李雅普诺夫函数参数优化
基于所给李雅普诺夫函数优化策略, 进一步优化李雅普诺夫函数参数.
定理2. 对于系统(5), 系统$ H_{\infty} $范数$ \gamma $满足
$ \begin{align} \gamma < \rho(k, \iota) = \left[\min(\lambda_{1}, \lambda_{2}) \right]^{-\frac{1}{2}} \end{align} $
(24) 其中, $ \lambda_{1} $和$ \lambda_{2} $由式(23)给出, 式(23)中$ \beta $由下式给出.
$ \begin{align} \beta = \frac{2}{k+1}\sqrt{\left(k \alpha-\frac{1}{\alpha}\right)\left(\alpha- \frac{k}{\alpha}\right)} \end{align} $
(25) 证明. 考查式(11)给出的矩阵$ \Upsilon $.令$ \sigma = 0 $, 则
$ \begin{align} \cos 2\theta = \frac{(k-1)(\alpha+\frac{1}{\alpha})}{(k+1)(\alpha-\frac{1}{\alpha})} \end{align} $
(26) 因此根据式(11), (12), (23)和$ 0 \leq \theta \leq {\pi}/{4} $, 矩阵$ \Upsilon $的特征值为$ \lambda_{1} $和$ \lambda_{2} $, 其中$ \beta $由式(25)给出.根据定理1, 可得式(24).
注3. 基于李雅普诺夫函数参数矩阵$ \Theta $的优化策略, 定理2进一步给出系统$ H_{\infty} $范数的估计., 同时奠定了进一步优化李雅普诺夫函数参数$ k $和$ \iota $的基础.
定理3. 对于系统(5), 系统$ H_{\infty} $范数$ \gamma $满足
$ \begin{align} \gamma < \rho(k) = \begin{cases} \frac{1}{\lambda}, & \text{若}\; \alpha = 1\\ \left[ f(k)\right]^{-\frac{1}{2}}, & \text{若}\; \alpha >1 \end{cases} \end{align} $
(27) 其中,
$ \begin{align} f(k) = \frac{4k}{(k+1)^{2}} \left[ \lambda^{2} + \nu^{2} - \frac{k \nu^{2}}{(k-1)^{2}} \left(\alpha-\frac{1}{\alpha}\right)^{2} \right] \end{align} $
(28) 证明. 考查式(23)给出的矩阵$ \Upsilon $的特征值为$ \lambda_{1} $和$ \lambda_{2} $.令$ \lambda_{1} = \lambda_{2} $, 即
$ \begin{align} 2\lambda + \beta \nu - \frac{1}{\iota} = \frac{1}{k}(2 \lambda - \beta \nu) - \frac{1}{\iota k^{2}} \end{align} $
(29) 其中, $ \beta $由式(25)给出, $ \alpha \geq 1 $.
当$ \alpha > 1 $时, 由式(25)和式(29)可知$ k \neq 1 $, 并且得
$ \begin{align} \frac{1}{\iota} = \frac{2k \lambda}{k+1}+\frac{2k \nu}{k^{2}-1} \sqrt{\left(k \alpha- \frac{1}{\alpha}\right)\left(\alpha-\frac{k}{\alpha}\right)} \end{align} $
(30) 当$ \alpha = 1 $时, 由式(25)可知$ (k-1)^{2} \leq 0 $, 即$ k = 1 $.则根据式(23), (25), (29), $ \lambda_{1} = \lambda_{2} = \frac{1}{\iota} (2 \lambda-\frac{1}{\iota}) $.当$ \iota = \lambda $时, 得$ \max (\lambda_{1}) = \lambda^{2} $.
基于以上分析, 并根据定理2和式(23), (25), (29)以及(30), 可得结论.
注4. 通过给出李雅普诺夫函数参数$ \iota $的优化策略, 定理3进一步给出系统$ H_{\infty} $范数的估计.根据定理3, 可以直接优化李雅普诺夫函数参数$ k $, 进而得到系统$ H_{\infty} $范数的精确估计.
注5. 注意, 当$ \alpha > 1 $时, $ k \neq 1 $.因此定理3通过分别讨论$ \alpha > 1 $和$ \alpha = 1 $两种情况, 解决了$ f(k) $的奇异问题.
令
$ \begin{align} \kappa = k + \frac{1}{k} > 2 \end{align} $
(31) 则由式(28), 得
$ \begin{align} f(\kappa) = \frac{4(\lambda^{2} + \nu^{2})}{\kappa+2} - \frac{4\nu^{2}}{\kappa^{2}-4} \times \left(\alpha-\frac{1}{\alpha}\right)^{2} \end{align} $
(32) 定理4. 对于系统(5), 系统$ H_{\infty} $范数$ \gamma $满足
$ \begin{align} \gamma < \rho_{\text{opt}} = \begin{cases} \frac{1}{\lambda}, & \text{若}\; \alpha = 1\\ \frac{1}{2\lambda}\sqrt{\alpha^{2}+\frac{1}{\alpha^{2}}+2}, &\text{若}\; \kappa_{0} \geq \alpha^{2}+\frac{1}{\alpha^{2}}\\ \left[ f(\kappa_{0})\right]^{-\frac{1}{2}}, &\text{若}\; \kappa_{0} < \alpha^{2}+\frac{1}{\alpha^{2}} \end{cases} \end{align} $
(33) 其中
$ \begin{align} &f(\kappa_{0}) = \frac{4(\lambda^{2} + \nu^{2})}{\kappa_{0}+2} - \frac{4\nu^{2}}{\kappa_{0}^{2}-4} \times \left(\alpha-\frac{1}{\alpha}\right)^{2} \end{align} $
(34) $ \begin{align} &\kappa_{0} = 2 + \frac{\nu^{2} (\alpha-\frac{1}{\alpha})^{2}}{\lambda^{2} + \nu^{2}} \times \left[ 1+\sqrt{1+ \frac{4(\lambda^{2} + \nu^{2})}{\nu^{2} (\alpha-\frac{1}{\alpha})^{2}}} \right] \end{align} $
(35) 证明. 由式(32), 得
$ \begin{align} f'(\kappa) = \frac{{\rm d} f(\kappa)}{{\rm d} \kappa} = -\frac{4(\lambda^{2} + \nu^{2})}{(\kappa+2)^{2}} +\frac{8(\alpha-\frac{1}{\alpha})^{2} \nu^{2} \kappa}{(\kappa+2)^{2}(\kappa-2)^{2}} \end{align} $
(36) 令$ f'(\kappa) = 0 $, 即
$ \begin{align} \kappa^{2} - \left[ 4+ \frac{2(\alpha-\frac{1}{\alpha})^{2} \nu^{2}}{\lambda^{2} + \nu^{2}} \right] \kappa +4 = 0 \end{align} $
(37) 根据$ \kappa >2 $和式(35), 得$ \kappa = \kappa_{0} $.
根据式(35) $ \sim $ (37), 得
$ \begin{align} \lim \limits_{\varsigma \rightarrow 0} \frac{f'(\kappa_{0} + \varsigma)-f'(\kappa_{0})}{\varsigma} <0 \end{align} $
(38) 因此, 在$ 2 < \kappa < \infty $的条件下, $ \max f(\kappa) = f(\kappa_{0}) $, 如图 1 (a)和1 (b)所示.
注意, 定理2中李雅普诺夫函数参数矩阵$ \Theta $的优化策略为$ \sigma = 0 $, 则由式(13), 可得$ k \leq \alpha^{2} $.由于$ k >1 $, 因此根据式(31), 得
$ \begin{align} \Omega = \left\{ \kappa \in \textbf{R} | 2 < \kappa \leq \alpha^{2}+\frac{1}{\alpha^{2}} \right\} \end{align} $
(39) $ \begin{align} \max \limits_{k \in \Omega} f(\kappa) = \begin{cases} \frac{4\lambda^{2}}{\alpha^{2}+\frac{1}{\alpha^{2}}+2}, &\text{若}\; \kappa_{0} \geq \alpha^{2}+\frac{1}{\alpha^{2}}\\ f(\kappa_{0}), & \text{若}\; \kappa_{0} < \alpha^{2}+\frac{1}{\alpha^{2}} \end{cases} \end{align} $
(40) 因此由定理3可得结论.
注6. 通过对李雅普诺夫函数参数的直接优化, 定理4给出了系统$ H_{\infty} $范数上界的优化结果.应用定理4, 可以给出系统$ H_{\infty} $范数的精确估计.
注7. 不同于LMI方法, 本文提出的李雅普诺夫函数直接优化方法分析了李雅普诺夫函数的构造对系统性能分析的影响, 充分利用系统结构和参数以优化李雅普诺夫函数的设计.与LMI方法相比, 李雅普诺夫函数直接优化方法能够直接给出系统$ H_{\infty} $范数的精确结果, 进而避免了复杂的数值优化过程.因此本文的工作提供了一种新的途径以更为方便地分析系统动态性能.
4. 算例
考查系统
$ \begin{align} \dot{\boldsymbol{ x}} = -\left[ \begin{array}{cc} 1.25 & 1.25 \\ -1.25 & 2.75 \\ \end{array} \right]{\boldsymbol{ x}}+ {\boldsymbol{ w}} \end{align} $
(41) 其中, $ {\boldsymbol{ w}} $为扰动输入, $ \|{\boldsymbol{ w}}\| \leq 1 $, $ {\boldsymbol{ x}} $为状态输出.根据式(5), 得
$ \begin{align} \begin{cases} \dot{\boldsymbol{ y}} = - \left[ \begin{array}{cc} 2 & 2 \\ -0.5 & 2 \\ \end{array} \right] {\boldsymbol{ y}} + {\boldsymbol{ {\Delta}}} \\ {\boldsymbol{ x}} = \frac{\sqrt{2}}{2} \left[ \begin{array}{cc} 1 & -1 \\ 1 & 1 \\ \end{array} \right] {\boldsymbol{ y}} \end{cases} \end{align} $
(42) 因此, $ \lambda = 2 $, $ \nu = 1 $, $ \alpha = 2 $.
由式(34), 得$ \kappa_{0} = 3.8651< \alpha^{2}+\frac{1}{\alpha^{2}} = 4.25 $.则根据定理4, 得$ \gamma < \rho_{\text{opt}} = 0.622 $.因此$ \gamma \approx 0.622 $.应用MATLAB中$ H_{\infty} $范数求解函数hinfnorm (sys, 0.0000001)可得相同的结果.因此提出的李雅普诺夫函数直接优化方法能精确给出系统$ H_{\infty} $范数.
表 1进一步给出在不同参数条件下系统(5)的$ H_{\infty} $范数.表 1表明, 针对式(5)给出的具有不同参数的系统, 提出的李雅普诺夫函数直接优化方法都能精确给出系统$ H_{\infty} $范数.
表 1 $H_{\infty}$范数分析($\alpha = 2$)Table 1 $H_{\infty}$ norm analysis ($\alpha = 2$)$\lambda$ $\nu$ MATLAB 定理4 稳态误差$\|A^{-1}\|$ 状态上界 2 6 0.626 0.626 0.307 0.626 2 4 0.626 0.626 0.419 0.626 2 2 0.626 0.626 0.588 0.626 2 1.2 0.626 0.626 0.626 0.626 2 1 0.622 0.622 0.622 0.622 2 0 0.501 0.501 0.501 0.501 在$ \alpha $和系统特征值实部$ \lambda $确定(即$ \alpha = 2 $, $ \lambda = 2 $)的条件下, 表 1给出的结果表明, 随着系统特征值虚部$ \nu $变化, $ H_{\infty} $范数的变化具有一定规律性, 表现为:
1) 当$ \nu = \nu^{*} = 1.2 $ (即$ \kappa_{0} = \alpha^{2}+{1}/{\alpha^{2}} $)时, $ H_{\infty} $范数为$ \max \|A^{-1}\| $;
2) 当$ \nu < \nu^{*} $ (即$ \kappa_{0} < \alpha^{2}+{1}/{\alpha^{2}} $)时, $ H_{\infty} $范数与稳态指标$ \|A^{-1}\| $一致;
3) 当$ \nu > \nu^{*} $ (即$ \kappa_{0} > \alpha^{2}+{1}/{\alpha^{2}} $)时, $ H_{\infty} $范数为固定值(即$ H_{\infty} $范数的值与$ \nu $无关), 并且根据定理4, $ H_{\infty} $范数的表达式非常简洁.
由式(1), (3), (41), 得
$ \begin{align} \begin{bmatrix} -P \begin{bmatrix} 1.25 & 1.25 \\ -1.25 & 2.75 \\ \end{bmatrix} -\small{ \begin{bmatrix} 1.25 & -1.25 \\ 1.25 & 2.75 \\ \end{bmatrix}}P & P & I \\ P & -\gamma^{2} I & 0_{2\times 2} \\ I & 0_{2\times 2} & -I \end{bmatrix} < 0 \end{align} $
(43) 采用LMI方法求解$ H_{\infty} $范数的步骤为:
1) 选择足够大的$ \gamma $, 如$ \gamma = 10 $;
2) 应用MATLAB中LMI工具求解式(43), 可得$ P $存在;
3) 减小$ \gamma $取值, 如$ \gamma = 1 $, 应用LMI工具求解式(43), 可得$ P $存在;
4) 当$ \gamma = 0.622 $时, 应用LMI工具求解式(43), 可得$ P $存在;
5) 当$ \gamma = 0.621 $时, 应用LMI工具求解(43), 可得$ P $不存在.
基于以上步骤, LMI方法可给出$ H_{\infty} = 0.622 $.这一结果与定理4得到的结果一致, 如表 1所示.
事实上, LMI方法需要对$ \gamma $进行遍历寻找.当选$ \gamma $的间隔较大时, 保守的结果不可避免.与之相比, 本文的方法具有明显的优越性.
5. 结论
本文针对$ H_{\infty} $控制理论研究中难以精确求解系统$ H_{\infty} $范数的问题, 提出了一种李雅普诺夫函数的直接优化方法.通过优化Riccati不等式中的李雅普诺夫函数, 给出了$ H_{\infty} $范数的通用解析表达式, 进而提供了一个有效的途径以直接和精确求解系统$ H_{\infty} $范数.研究结果具有以下特点:
1) 与LMI方法相比, 本文所提方法避免了复杂的数值优化过程, 使求解系统$ H_{\infty} $范数简化.
2) 与早期关于李雅普诺夫方程和Riccati不等式的研究相比, 本文所提方法避免了由于李雅普诺夫函数选择的随意性导致的保守结果.
3) 本文所提方法能够展现系统矩阵特征值的实部和虚部对$ H_{\infty} $性能的影响, 为进一步精确(定量)控制系统$ H_{\infty} $性能提供借鉴.
在进一步的工作中, 将研究含有时滞及非线性项的系统.
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