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Data-Driven Stabilization of Input-Saturated Systems
输入饱和系统的数据驱动稳定

Valentina Breschi ⊖ ⊖ ^(⊖){ }^{\ominus}, Member, IEEE, Luca Zaccarian ® ® ^(®){ }^{\circledR}, Fellow, IEEE, and Simone Formentin ⊙ ⊙ ^(o.){ }^{\odot}, Member, IEEE
Valentina Breschi ⊖ ⊖ ^(⊖){ }^{\ominus} ，IEEE 会员，Luca Zaccarian ® ® ^(®){ }^{\circledR} ，IEEE 研究员，Simone Formentin ⊙ ⊙ ^(o.){ }^{\odot} ，IEEE 会员(Special Section on Data-Driven Analysis and Control)
（数据驱动分析与控制专题）

Abstract  抽象的

We provide a data-driven stabilization approach for input-saturated systems with formal Lyapunov guarantees. Through a generalized sector condition, we propose a convex design algorithm based on linear matrix inequalities for obtaining a regionally stabilizing data-driven static state-feedback gain. Regional, rather than global, properties allow us to address nonexponentially stable plants, thereby making our design broad in terms of applicability. Moreover, we discuss consistency issues and introduce practical tools to deal with measurement noise. Numerical simulations show the effectiveness of our approach and its sensitivity to the features of the dataset.
我们为具有正式 Lyapunov 保证的输入饱和系统提供数据驱动的稳定方法。通过广义扇区条件，我们提出了一种基于线性矩阵不等式的凸设计算法，以获得区域稳定的数据驱动静态状态反馈增益。区域性而非全局性的特性使我们能够解决非指数稳定的植物，从而使我们的设计在适用性方面具有广泛的范围。此外，我们讨论了一致性问题并介绍了处理噪声测量的实用工具。数值模拟证明了我们方法的有效性及其对数据集特征的敏感性。

Index Terms-Data-driven control, input saturation.
索引术语-数据驱动控制，输入饱和。

I. INTRODUCTION  一、引言

DATA-DRIVEN (DD) control is recently enjoying great success as an alternative to traditional model-based design. In particular, existing DD control techniques range from model-reference approaches, e.g., [5], [10], to the recently proposed optimal (e.g., [8], [9]) and predictive techniques [2], [6]. Nonetheless, few solutions have been devised to handle systems with input saturation apart from predictive strategies. Moreover, these alternatives to predictive control have mainly been tailored to the model-reference framework. Indeed, a VRFT-like strategy is proposed in [3], where saturation is accounted for by augmenting the cost of the optimization problem solved to design a fixed-structure (dynamic) controller. A similar principle is exploited in [4],
数据驱动（DD）控制作为传统基于模型的设计的替代方案，最近获得了巨大的成功。具体来说，现有的 DD 控制技术包括模型参考方法（例如 [5]、[10]）到最近提出的最优（例如 [8]、[9]）和预测技术 [2]、[6]。然而，除了预测策略之外，很少有解决方案可以处理输入饱和系统。此外，这些预测控制的替代方案主要针对模型参考框架进行了调整。事实上，[3] 中提出了一种类似 VRFT 的策略，其中通过增加解决优化问题的成本来设计固定结构（动态）控制器来解决饱和问题。 [4] 中也运用了类似的原则，
where a neural network controller is used to cope with input saturation. However, neither of these solutions provides stability guarantees. On the other hand, the MPC-based DD solution in [1] provides robust stability guarantees, but does not account for input saturation that would otherwise prevent global results to be achieved with general (possibly exponentially unstable) plants. In the model-based context, several tools are instead available, mostly stemming from generalized sector conditions leading to regional stability guarantees (with guaranteed estimates of the basin of attraction) with easily implementable feedbacks [14], [17] and less cumbersome online computations, as compared to MPC.
其中神经网络控制器用于应对输入饱和。然而，这两种解决方案都不能提供稳定性保证。另一方面，[1] 中基于 MPC 的 DD 解决方案提供了强大的稳定性保证，但没有考虑输入饱和，否则输入饱和会阻止一般（可能指数不稳定）工厂实现全局结果。在基于模型的背景下，有几种可用的工具，主要源于广义扇区条件，从而实现区域稳定性保证（具有吸引域的保证估计），与 MPC 相比，具有易于实现的反馈[14]，[17]和不太繁琐的在线计算。
In this note, we exploit generalized sector conditions to propose a novel DD solution for the design of regionally stabilizing controllers for input-saturated systems. To this end, we introduce a DD formulation of the model-based solutions in [14], starting from a behavioral description of linear systems [16] and its closed-loop counterpart [7]. Assuming the state to be fully accessible, the DD solution solely requires the explicit identification of the matrix B B BB linking the state evolution to the input. By working at the boundary between system identification and control design, we thus need to identify how the nonlinearity (i.e., the saturation) affects the system.
在本文中，我们利用广义扇区条件提出了一种新颖的 DD 解决方案，用于设计输入饱和系统的区域稳定控制器。为此，我们在 [14] 中引入了基于模型的解决方案的 DD 公式，从线性系统 [16] 及其闭环对应部分 [7] 的行为描述开始。假设状态完全可访问，DD 解决方案仅需要明确识别将状态演变与输入联系起来的矩阵 B B BB 。因此，通过在系统识别和控制设计之间的边界上工作，我们需要确定非线性（即饱和度）如何影响系统。
To handle noisy data, we propose to use an instrumental variable scheme [12]. A similar approach has been used for the design of DD predictive controllers [15]. With this well-known tool in system identification, we are able to recover an (asymptotically) unbiased description of the closed-loop system (in line with the one in [15]), which is ultimately exploited to design the regionally stabilizing feedback law.
为了处理噪声数据，我们建议使用工具变量方案[12]。类似的方法已用于DD预测控制器的设计[15]。利用这个著名的系统辨识工具，我们能够恢复闭环系统的（渐近）无偏描述（与 [15] 中的一致），最终利用该描述来设计区域稳定反馈律。
This letter is organized as follows. The problem statement is given in Section II. A first solution is given in Section III, in a deterministic setting. A design strategy with noisy data is then proposed in Section IV. Section V shows some numerical results, highlighting the potential of the proposed framework.
这封信的内容结构如下。问题陈述在第二部分中给出。第三部分在确定性设置中给出了第一个解决方案。然后在第四部分提出了一种针对噪声数据的设计策略。第五部分展示了一些数值结果，强调了所提出的框架的潜力。
Notation: For a full-row-rank matrix Y , Y † Y , Y † Y,Y^(†)Y, Y^{\dagger} denotes its right pseudo-inverse. For matrix A ∈ R n x × n x , He ( A ) = A + A ⊤ A ∈ R n x × n x , He ( A ) = A + A ⊤ A inR^(n_(x)xxn_(x)),He(A)=A+A^(TT)A \in \mathbb{R}^{n_{x} \times n_{x}}, \operatorname{He}(A)=A+A^{\top}. x + = A x x + = A x x^(+)=Axx^{+}=A x is a shorthand for x ( t + 1 ) = A x ( t ) x ( t + 1 ) = A x ( t ) x(t+1)=Ax(t)x(t+1)=A x(t). Given another sequence t ↦ w ( t ) ∈ R n w t ↦ w ( t ) ∈ R n w t|->w(t)inR^(n_(w))t \mapsto w(t) \in \mathbb{R}^{n_{w}}, for any 0 ≤ t 0 < t 1 ≤ T 0 ≤ t 0 < t 1 ≤ T 0 <= t_(0) < t_(1) <= T0 \leq t_{0}<t_{1} \leq T and 1 ≤ τ < t 1 1 ≤ τ < t 1 1 <= tau < t_(1)1 \leq \tau<t_{1}, we define the associated Hankel matrix as
符号：对于满行秩矩阵， Y , Y † Y , Y † Y,Y^(†)Y, Y^{\dagger} 表示其右伪逆。对于矩阵 A ∈ R n x × n x , He ( A ) = A + A ⊤ A ∈ R n x × n x , He ( A ) = A + A ⊤ A inR^(n_(x)xxn_(x)),He(A)=A+A^(TT)A \in \mathbb{R}^{n_{x} \times n_{x}}, \operatorname{He}(A)=A+A^{\top} 。 x + = A x x + = A x x^(+)=Axx^{+}=A x 是 x ( t + 1 ) = A x ( t ) x ( t + 1 ) = A x ( t ) x(t+1)=Ax(t)x(t+1)=A x(t) 的简写。给定另一个序列 t ↦ w ( t ) ∈ R n w t ↦ w ( t ) ∈ R n w t|->w(t)inR^(n_(w))t \mapsto w(t) \in \mathbb{R}^{n_{w}} ，对于任何 0 ≤ t 0 < t 1 ≤ T 0 ≤ t 0 < t 1 ≤ T 0 <= t_(0) < t_(1) <= T0 \leq t_{0}<t_{1} \leq T 和 1 ≤ τ < t 1 1 ≤ τ < t 1 1 <= tau < t_(1)1 \leq \tau<t_{1} ，我们将相关的 Hankel 矩阵定义为
W t 0 , τ , t 1 = [ w ( t 0 ) w ( t 0 + 1 ) ⋯ w ( t 1 − τ ) w ( t 0 + 1 ) w ( t 0 + 2 ) ⋯ w ( t 1 − τ + 1 ) ⋮ ⋮ ⋱ ⋮ w ( τ − 1 ) w ( τ ) ⋯ w ( t 1 − 1 ) ] W t 0 , τ , t 1 = w t 0 w t 0 + 1 ⋯ w t 1 − τ w t 0 + 1 w t 0 + 2 ⋯ w t 1 − τ + 1 ⋮ ⋮ ⋱ ⋮ w ( τ − 1 ) w ( τ ) ⋯ w t 1 − 1 W_(t_(0),tau,t_(1))=[[w(t_(0)),w(t_(0)+1),cdots,w(t_(1)-tau)],[w(t_(0)+1),w(t_(0)+2),cdots,w(t_(1)-tau+1)],[vdots,vdots,ddots,vdots],[w(tau-1),w(tau),cdots,w(t_(1)-1)]]W_{t_{0}, \tau, t_{1}}=\left[\begin{array}{cccc} w\left(t_{0}\right) & w\left(t_{0}+1\right) & \cdots & w\left(t_{1}-\tau\right) \\ w\left(t_{0}+1\right) & w\left(t_{0}+2\right) & \cdots & w\left(t_{1}-\tau+1\right) \\ \vdots & \vdots & \ddots & \vdots \\ w(\tau-1) & w(\tau) & \cdots & w\left(t_{1}-1\right) \end{array}\right]
while we denote the single row Hankel matrix as follows:
我们将单行 Hankel 矩阵表示如下：
W t 0 , t 1 = W t 0 , 1 , t 1 = [ w ( t 0 ) w ( t 0 + 1 ) ⋯ w ( t 1 ) ] W t 0 , t 1 = W t 0 , 1 , t 1 = w t 0 w t 0 + 1 ⋯ w t 1 W_(t_(0),t_(1))=W_(t_(0),1,t_(1))=[w(t_(0))w(t_(0)+1)cdots w(t_(1))]W_{t_{0}, t_{1}}=W_{t_{0}, 1, t_{1}}=\left[w\left(t_{0}\right) w\left(t_{0}+1\right) \cdots w\left(t_{1}\right)\right]
Fig. 1. Scheme of the (noisy) closed-loop system, where known/tunable blocks are depicted in white and the unknown plant is depicted in grey.
图 1. (噪声) 闭环系统方案，其中已知/可调块以白色表示，未知对象以灰色表示。
Definition 1 (Persistence of Excitation): A sequence w ( t ) ∈ R n w w ( t ) ∈ R n w w(t)inR^(n_(w))w(t) \in \mathbb{R}^{n_{w}} is persistently exciting of order τ τ tau\tau if the Hankel matrix W t 0 , τ , t 1 W t 0 , τ , t 1 W_(t_(0),tau,t_(1))W_{t_{0}, \tau, t_{1}} in (1) is full row rank, namely rank ( W t 0 , τ , t 1 ) = τ ⋅ n w rank W t 0 , τ , t 1 = τ ⋅ n w rank(W_(t_(0),tau,t_(1)))=tau*n_(w)\operatorname{rank}\left(W_{t_{0}, \tau, t_{1}}\right)=\tau \cdot n_{w}.
定义 1（激励持久性）：如果 (1) 中的 Hankel 矩阵 W t 0 , τ , t 1 W t 0 , τ , t 1 W_(t_(0),tau,t_(1))W_{t_{0}, \tau, t_{1}} 是满行秩，即 rank ( W t 0 , τ , t 1 ) = τ ⋅ n w rank W t 0 , τ , t 1 = τ ⋅ n w rank(W_(t_(0),tau,t_(1)))=tau*n_(w)\operatorname{rank}\left(W_{t_{0}, \tau, t_{1}}\right)=\tau \cdot n_{w} ，则序列 w ( t ) ∈ R n w w ( t ) ∈ R n w w(t)inR^(n_(w))w(t) \in \mathbb{R}^{n_{w}} 为 τ τ tau\tau 阶的持续激励。

II. Problem Formulation  二.问题表述

Consider a discrete-time, linear time-invariant (LTI) and controllable plant P P P\mathscr{P}, in cascade with an input saturation. Let this cascade be described by the difference equation
考虑一个离散时间、线性时不变 (LTI) 和可控对象 P P P\mathscr{P} ，与输入饱和级联。让这个级联用差分方程来描述
x + = A x + B v = A x + B sat ( u ) x + = A x + B v = A x + B sat ( u ) x^(+)=Ax+Bv=Ax+B sat(u)x^{+}=A x+B v=A x+B \operatorname{sat}(u)
where x ∈ R n x x ∈ R n x x inR^(n_(x))x \in \mathbb{R}^{n_{x}} is the state, u ∈ R n u u ∈ R n u u inR^(n_(u))u \in \mathbb{R}^{n_{u}} is the input to the saturation, and A ∈ R n x × n x A ∈ R n x × n x A inR^(n_(x)xxn_(x))A \in \mathbb{R}^{n_{x} \times n_{x}} and B ∈ R n x × n u B ∈ R n x × n u B inR^(n_(x)xxn_(u))B \in \mathbb{R}^{n_{x} \times n_{u}} are unknown matrices of suitable dimensions. The input nonlinearity sat : R n u → R n u R n u → R n u R^(n_(u))rarrR^(n_(u))\mathbb{R}^{n_{u}} \rightarrow \mathbb{R}^{n_{u}} is a known decentralized saturation, with components defined as
其中 x ∈ R n x x ∈ R n x x inR^(n_(x))x \in \mathbb{R}^{n_{x}} 是状态， u ∈ R n u u ∈ R n u u inR^(n_(u))u \in \mathbb{R}^{n_{u}} 是饱和度的输入， A ∈ R n x × n x A ∈ R n x × n x A inR^(n_(x)xxn_(x))A \in \mathbb{R}^{n_{x} \times n_{x}} 和 B ∈ R n x × n u B ∈ R n x × n u B inR^(n_(x)xxn_(u))B \in \mathbb{R}^{n_{x} \times n_{u}} 是适当维度的未知矩阵。输入非线性 sat : R n u → R n u R n u → R n u R^(n_(u))rarrR^(n_(u))\mathbb{R}^{n_{u}} \rightarrow \mathbb{R}^{n_{u}} 是已知的分散饱和度，其分量定义为
v i = sat i ( u i ) = max { − u ¯ i , n , min { u ¯ i , p , u i } } , i = 1 , … , n u , v i = sat i u i = max − u ¯ i , n , min u ¯ i , p , u i , i = 1 , … , n u , v_(i)=sat_(i)(u_(i))=max{- bar(u)_(i,n),min{ bar(u)_(i,p),u_(i)}},quad i=1,dots,n_(u),v_{i}=\operatorname{sat}_{i}\left(u_{i}\right)=\max \left\{-\bar{u}_{i, n}, \min \left\{\bar{u}_{i, p}, u_{i}\right\}\right\}, \quad i=1, \ldots, n_{u},
whose known lower and upper saturation bounds, gathered in vectors u ¯ n , u ¯ p ∈ R m u ¯ n , u ¯ p ∈ R m bar(u)_(n), bar(u)_(p)inR^(m)\bar{u}_{n}, \bar{u}_{p} \in \mathbb{R}^{m}, respectively, satisfy the componentwise inequalities u ¯ n ≥ u ¯ u ¯ n ≥ u ¯ bar(u)_(n) >= bar(u)\bar{u}_{n} \geq \bar{u} and u ¯ p ≥ u ¯ u ¯ p ≥ u ¯ bar(u)_(p) >= bar(u)\bar{u}_{p} \geq \bar{u} for some vector u ¯ = [ u ¯ 1 ⋯ u ¯ n u ] ⊤ ∈ R n u u ¯ = u ¯ 1      ⋯      u ¯ n u ⊤ ∈ R n u bar(u)=[[ bar(u)_(1),cdots, bar(u)_(n_(u))]]^(TT)inR^(n_(u))\bar{u}=\left[\begin{array}{lll}\bar{u}_{1} & \cdots & \bar{u}_{n_{u}}\end{array}\right]^{\top} \in \mathbb{R}^{n_{u}} having positive elements. Assume that we have access to noisy measurements of the state, i.e.,
其已知的饱和下限和上限分别集中在向量 u ¯ n , u ¯ p ∈ R m u ¯ n , u ¯ p ∈ R m bar(u)_(n), bar(u)_(p)inR^(m)\bar{u}_{n}, \bar{u}_{p} \in \mathbb{R}^{m} 中，对于某些具有正元素的向量 u ¯ = [ u ¯ 1 ⋯ u ¯ n u ] ⊤ ∈ R n u u ¯ = u ¯ 1      ⋯      u ¯ n u ⊤ ∈ R n u bar(u)=[[ bar(u)_(1),cdots, bar(u)_(n_(u))]]^(TT)inR^(n_(u))\bar{u}=\left[\begin{array}{lll}\bar{u}_{1} & \cdots & \bar{u}_{n_{u}}\end{array}\right]^{\top} \in \mathbb{R}^{n_{u}} ，满足分量不等式 u ¯ n ≥ u ¯ u ¯ n ≥ u ¯ bar(u)_(n) >= bar(u)\bar{u}_{n} \geq \bar{u} 和 u ¯ p ≥ u ¯ u ¯ p ≥ u ¯ bar(u)_(p) >= bar(u)\bar{u}_{p} \geq \bar{u} 。假设我们可以获得状态的噪声测量，即
x d = x + e x d = x + e x^(d)=x+ex^{d}=x+e
where x d ∈ R n x x d ∈ R n x x^(d)inR^(n_(x))x^{d} \in \mathbb{R}^{n_{x}} denotes the measured state, and e ∈ R n x e ∈ R n x e inR^(n_(x))e \in \mathbb{R}^{n_{x}} is a zero mean, white noise corrupting the true state x x xx, here assumed to be uncorrelated to the inputs.
其中 x d ∈ R n x x d ∈ R n x x^(d)inR^(n_(x))x^{d} \in \mathbb{R}^{n_{x}} 表示测量状态， e ∈ R n x e ∈ R n x e inR^(n_(x))e \in \mathbb{R}^{n_{x}} 是零均值，白噪声会破坏真实状态 x x xx ，这里假设与输入不相关。
Even though A A AA and B B BB in (3a) are unknown, assume that we can carry out experiments on P P P\mathscr{P} for data collection purposes. Specifically, suppose that we can feed the plant with a finite input sequence V T = { v d ( t ) } t = 0 T − 1 V T = v d ( t ) t = 0 T − 1 V_(T)={v^(d)(t)}_(t=0)^(T-1)\mathscr{V}_{T}=\left\{v^{d}(t)\right\}_{t=0}^{T-1}, saturated in accordance with (3b), that is persistently exciting of order n x + 1 n x + 1 n_(x)+1n_{x}+1 according to Definition 1, and that we can collect the corresponding (noisy) states X T = { x d ( t ) } t = 0 T X T = x d ( t ) t = 0 T X_(T)={x^(d)(t)}_(t=0)^(T)\mathscr{X}_{T}=\left\{x^{d}(t)\right\}_{t=0}^{T}. Given the dataset D T = D T = D_(T)=\mathscr{D}_{T}= { V T , X T } V T , X T {V_(T),X_(T)}\left\{\mathscr{V}_{T}, \mathscr{X}_{T}\right\}, our goal is propose a data-based solution alternative to existing (and viable) system identification+model-based approaches, to design a static state-feedback law, namely
尽管 (3a) 中的 A A AA 和 B B BB 未知，但假设我们可以对 P P P\mathscr{P} 进行实验以收集数据。具体来说，假设我们可以为植物提供一个有限输入序列 V T = { v d ( t ) } t = 0 T − 1 V T = v d ( t ) t = 0 T − 1 V_(T)={v^(d)(t)}_(t=0)^(T-1)\mathscr{V}_{T}=\left\{v^{d}(t)\right\}_{t=0}^{T-1} ，该序列根据 (3b) 饱和，根据定义 1，该序列持续激发阶为 n x + 1 n x + 1 n_(x)+1n_{x}+1 ，并且我们可以收集相应的（噪声）状态 X T = { x d ( t ) } t = 0 T X T = x d ( t ) t = 0 T X_(T)={x^(d)(t)}_(t=0)^(T)\mathscr{X}_{T}=\left\{x^{d}(t)\right\}_{t=0}^{T} 。给定数据集 D T = D T = D_(T)=\mathscr{D}_{T}= { V T , X T } V T , X T {V_(T),X_(T)}\left\{\mathscr{V}_{T}, \mathscr{X}_{T}\right\} ，我们的目标是提出一种基于数据的解决方案来替代现有（可行的）系统识别+基于模型的方法，设计一个静态状态反馈律，即
u = K x , K ∈ R n u × n x u = K x , K ∈ R n u × n x u=Kx,quad K inR^(n_(u)xxn_(x))u=K x, \quad K \in \mathbb{R}^{n_{u} \times n_{x}}
such that the origin of the closed loop in Fig. 1 is exponentially stable, while obtaining an (inner) estimate of the basin of attraction of the closed-loop system.
使得图 1 中的闭环原点是指数稳定的，同时获得闭环系统吸引盆的（内部）估计。
Remark 1 (On Experiment Design): For the rank condition in Definition 1 to be satisfied by the saturated input sequence V T V T V_(T)\mathscr{V}_{T}, the user-defined inputs U T = { u d ( t ) } t = 0 T U T = u d ( t ) t = 0 T U_(T)={u^(d)(t)}_(t=0)^(T)\mathscr{U}_{T}=\left\{u^{d}(t)\right\}_{t=0}^{T} from which V T V T V_(T)\mathscr{V}_{T} is originated ( v d ( t ) = sat ( u d ( t ) ) v d ( t ) = sat u d ( t ) (v^(d)(t)=sat(u^(d)(t)):}\left(v^{d}(t)=\operatorname{sat}\left(u^{d}(t)\right)\right. for all t ) t {:t)\left.t\right) should be chosen to avoid recurrent saturations as much as possible so that the input values are sufficiently rich to provide excitation.
注释 1（关于实验设计）：为了使饱和输入序列 V T V T V_(T)\mathscr{V}_{T} 满足定义 1 中的秩条件，对于所有 t ) t {:t)\left.t\right) ，应选择 V T V T V_(T)\mathscr{V}_{T} 所源自的用户定义输入 U T = { u d ( t ) } t = 0 T U T = u d ( t ) t = 0 T U_(T)={u^(d)(t)}_(t=0)^(T)\mathscr{U}_{T}=\left\{u^{d}(t)\right\}_{t=0}^{T} ( v d ( t ) = sat ( u d ( t ) ) v d ( t ) = sat u d ( t ) (v^(d)(t)=sat(u^(d)(t)):}\left(v^{d}(t)=\operatorname{sat}\left(u^{d}(t)\right)\right. ，以尽可能避免重复饱和，使得输入值足够丰富以提供激励。

III. Deterministic Data-Driven Design of
Regionally Stabilizing State-Feedback Laws
三确定性数据驱动设计 区域稳定状态反馈法

We initially focus on a noiseless setting, i.e., e = 0 e = 0 e=0e=0 in (4), to show how a regionally stabilizing static state feedback law can be designed from data, using the generalized sector condition in [14]. As in the linear case [7], initially considering this simplified framework allows us is obtain DD conditions that are equivalent to their model-based counterparts.
我们首先关注无噪声设置，即（4）中的 e = 0 e = 0 e=0e=0 ，以展示如何利用[14]中的广义扇区条件，根据数据设计区域稳定的静态反馈律。与线性情况 [7] 一样，最初考虑这个简化的框架使我们能够获得与基于模型的对应条件等价的 DD 条件。
At the core of our formulation lays the following foundational result, that stems directly from our setup and the fundamental lemma in [16].
我们公式的核心是以下基础结果，它直接源于我们的设置和[16]中的基本引理。
Lemma 1: Given the input sequence U T U T U_(T)\mathscr{U}_{T} in the available dataset D T D T D_(T)\mathscr{D}_{T}, assume that V T = { v d ( t ) } t = 0 T − 1 V T = v d ( t ) t = 0 T − 1 V_(T)={v^(d)(t)}_(t=0)^(T-1)\mathscr{V}_{T}=\left\{v^{d}(t)\right\}_{t=0}^{T-1}, is persistently exciting of order n x + 1 n x + 1 n_(x)+1n_{x}+1 and that the dataset length T T TT satisfies
引理 1：给定可用数据集 D T D T D_(T)\mathscr{D}_{T} 中的输入序列 U T U T U_(T)\mathscr{U}_{T} ，假设 V T = { v d ( t ) } t = 0 T − 1 V T = v d ( t ) t = 0 T − 1 V_(T)={v^(d)(t)}_(t=0)^(T-1)\mathscr{V}_{T}=\left\{v^{d}(t)\right\}_{t=0}^{T-1} 是持续激发的，阶数为 n x + 1 n x + 1 n_(x)+1n_{x}+1 ，并且数据集长度 T T TT 满足
T ≥ ( n u + 1 ) n x + n u T ≥ n u + 1 n x + n u T >= (n_(u)+1)n_(x)+n_(u)T \geq\left(n_{u}+1\right) n_{x}+n_{u}
(namely, it is sufficiently long). Then, the following rank condition holds:
（即足够长）。然后，以下排序条件成立：
rank ( [ V 0 , T − 1 d X 0 , T − 1 d ] ) = n u + n x rank V 0 , T − 1 d X 0 , T − 1 d = n u + n x rank([[V_(0,T-1)^(d)],[X_(0,T-1)^(d)]])=n_(u)+n_(x)\operatorname{rank}\left(\left[\begin{array}{l} V_{0, T-1}^{d} \\ X_{0, T-1}^{d} \end{array}\right]\right)=n_{u}+n_{x}

A. Characterizing the Closed-Loop Dynamics From Data
A. 从​​数据中表征闭环动力学

In a noise-free setting, the model-based representation (3a) is equivalent to the following DD characterization:
在无噪声设置中，基于模型的表示（3a）等同于以下 DD 表征：
x + = [ B A ] [ sat ( u ) x ] = X 1 , T d [ V 0 , T − 1 d X 0 , T − 1 d ] † [ sat ( u ) x ] x + = B      A sat ( u ) x = X 1 , T d V 0 , T − 1 d X 0 , T − 1 d † sat ( u ) x x^(+)=[[B,A]][[sat(u)],[x]]=X_(1,T)^(d)[[V_(0,T-1)^(d)],[X_(0,T-1)^(d)]]^(†)[[sat(u)],[x]]x^{+}=\left[\begin{array}{ll} B & A \end{array}\right]\left[\begin{array}{c} \operatorname{sat}(u) \\ x \end{array}\right]=X_{1, T}^{d}\left[\begin{array}{c} V_{0, T-1}^{d} \\ X_{0, T-1}^{d} \end{array}\right]^{\dagger}\left[\begin{array}{c} \operatorname{sat}(u) \\ x \end{array}\right]
with X 1 , T d = [ x d ( 1 ) ⋯ x d ( T ) ] X 1 , T d = x d ( 1 ) ⋯ x d ( T ) X_(1,T)^(d)=[x^(d)(1)cdotsx^(d)(T)]X_{1, T}^{d}=\left[x^{d}(1) \cdots x^{d}(T)\right], as stated next.
使用 X 1 , T d = [ x d ( 1 ) ⋯ x d ( T ) ] X 1 , T d = x d ( 1 ) ⋯ x d ( T ) X_(1,T)^(d)=[x^(d)(1)cdotsx^(d)(T)]X_{1, T}^{d}=\left[x^{d}(1) \cdots x^{d}(T)\right] ，如下所述。
Lemma 2 (DD Open-Loop With Saturation): Given the unknown system in (3a) and a set of sufficiently long (in the sense of (6)) noiseless data D T = { V T , X T } D T = V T , X T D_(T)={V_(T),X_(T)}\mathscr{D}_{T}=\left\{\mathscr{V}_{T}, \mathscr{X}_{T}\right\}, with V T V T V_(T)\mathscr{V}_{T} being persistently exciting of order n x + 1 n x + 1 n_(x)+1n_{x}+1, the dynamics in (3a) can be equivalently characterized from data as in (8).
引理 2（带饱和的 DD 开环）：给定 (3a) 中的未知系统和一组足够长（在 (6) 的意义上）的无噪声数据 D T = { V T , X T } D T = V T , X T D_(T)={V_(T),X_(T)}\mathscr{D}_{T}=\left\{\mathscr{V}_{T}, \mathscr{X}_{T}\right\} ，其中 V T V T V_(T)\mathscr{V}_{T} 为 n x + 1 n x + 1 n_(x)+1n_{x}+1 阶的持续激发，则 (3a) 中的动态可以用数据等效地表征，如 (8) 中所示。
Proof: As in [7, Sec. III.A], the proof straightforwardly follows from the definition
证明：与 [7, Sec. III.A] 中一样，证明直接来自定义
[ B A ] = argmin [ B A ] ‖ X 1 , T d − [ B A ] [ V 0 , T − 1 d X 0 , T − 1 d ] ‖ F = X 1 , T d [ V 0 , T − 1 d X 0 , T − 1 d ] † B      A = argmin B      A X 1 , T d − B      A V 0 , T − 1 d X 0 , T − 1 d F = X 1 , T d V 0 , T − 1 d X 0 , T − 1 d † [[B,A]]=argmin_([[B,A]])||X_(1,T)^(d)-[[B,A]][[V_(0,T-1)^(d)],[X_(0,T-1)^(d)]]||_(F)=X_(1,T)^(d)[[V_(0,T-1)^(d)],[X_(0,T-1)^(d)]]^(†)\left[\begin{array}{ll}B & A\end{array}\right]=\underset{\left[\begin{array}{ll}B & A\end{array}\right]}{\operatorname{argmin}}\left\|X_{1, T}^{d}-\left[\begin{array}{ll}B & A\end{array}\right]\left[\begin{array}{c}V_{0, T-1}^{d} \\ X_{0, T-1}^{d}\end{array}\right]\right\|_{F}=X_{1, T}^{d}\left[\begin{array}{c}V_{0, T-1}^{d} \\ X_{0, T-1}^{d}\end{array}\right]^{\dagger},
thanks to the rank property in (7).
这得益于（7）中的等级属性。
Introduce the dead-zone nonlinearity dz ( u ) = dz ( u ) = u − dz ( u ) = dz ( u ) = u − dz(u)=dz(u)=u-\mathrm{dz}(u)=\mathrm{dz}(u)=u- sat ( u ) sat ( u ) sat(u)\operatorname{sat}(u) induced by sat, with u u uu defined in (5). Starting from the identification-like result in Lemma 2, we provide the following data-based representation of the closed-loop (3), (5):
引入由 sat 引起的死区非线性 dz ( u ) = dz ( u ) = u − dz ( u ) = dz ( u ) = u − dz(u)=dz(u)=u-\mathrm{dz}(u)=\mathrm{dz}(u)=u- sat ( u ) sat ( u ) sat(u)\operatorname{sat}(u) ，其中 u u uu 定义在 (5) 中。从引理 2 中的类似识别的结果出发，我们给出闭环 (3)、(5) 的以下基于数据的表示：
x + = X 1 , T d G x x − X 1 , T d [ V 0 , T − 1 d X 0 , T − 1 d ] † [ dz ( K x ) 0 ] x + = X 1 , T d G x x − X 1 , T d V 0 , T − 1 d X 0 , T − 1 d † dz ( K x ) 0 x^(+)=X_(1,T)^(d)G^(x)x-X_(1,T)^(d)[[V_(0,T-1)^(d)],[X_(0,T-1)^(d)]]^(†)[[dz(Kx)],[0]]x^{+}=X_{1, T}^{d} G^{x} x-X_{1, T}^{d}\left[\begin{array}{c} V_{0, T-1}^{d} \\ X_{0, T-1}^{d} \end{array}\right]^{\dagger}\left[\begin{array}{c} \mathrm{dz}(K x) \\ 0 \end{array}\right]
with G x ∈ R ( T − 1 ) × n x G x ∈ R ( T − 1 ) × n x G^(x)inR^((T-1)xxn_(x))G^{x} \in \mathbb{R}^{(T-1) \times n_{x}} satisfying  其中 G x ∈ R ( T − 1 ) × n x G x ∈ R ( T − 1 ) × n x G^(x)inR^((T-1)xxn_(x))G^{x} \in \mathbb{R}^{(T-1) \times n_{x}} 满足
X 0 , T − 1 d G x = I X 0 , T − 1 d G x = I X_(0,T-1)^(d)G^(x)=IX_{0, T-1}^{d} G^{x}=I
This result can be formalized as follows.
该结果可以形式化如下。
Lemma 3 (DD Closed-Loop With Saturation): Given the unknown system in (3), and a sufficiently long (in the sense of (6)) set of noiseless data D T = { V T , X T } D T = V T , X T D_(T)={V_(T),X_(T)}\mathscr{D}_{T}=\left\{\mathscr{V}_{T}, \mathscr{X}_{T}\right\}, with V T V T V_(T)\mathscr{V}_{T} being persistently exciting of order n x + 1 n x + 1 n_(x)+1n_{x}+1, the closed loop (3), (5) is equivalent to (10), (11).
引理 3（带饱和的 DD 闭环）：给定 (3) 中的未知系统，以及一组足够长（在 (6) 的意义上）的无噪声数据 D T = { V T , X T } D T = V T , X T D_(T)={V_(T),X_(T)}\mathscr{D}_{T}=\left\{\mathscr{V}_{T}, \mathscr{X}_{T}\right\} ，其中 V T V T V_(T)\mathscr{V}_{T} 为阶数为 n x + 1 n x + 1 n_(x)+1n_{x}+1 的持续激励，则闭环 (3)、(5) 等价于 (10)、(11)。
Proof: Replacing sat ( u ) = sat ( K x ) = K x − dz ( K x ) sat ( u ) = sat ( K x ) = K x − dz ( K x ) sat(u)=sat(Kx)=Kx-dz(Kx)\operatorname{sat}(u)=\operatorname{sat}(K x)=K x-\mathrm{dz}(K x) and applying Lemma 2, the closed loop (8), (5) can be written as
证明：代入 sat ( u ) = sat ( K x ) = K x − dz ( K x ) sat ( u ) = sat ( K x ) = K x − dz ( K x ) sat(u)=sat(Kx)=Kx-dz(Kx)\operatorname{sat}(u)=\operatorname{sat}(K x)=K x-\mathrm{dz}(K x) ，应用引理2，闭环（8），（5）可以写成
x + = [ B A ] [ K x x ] − X 1 , T d [ V 0 , T − 1 d X 0 , T − 1 d ] † [ dz ( K x ) 0 ] = [ B A ] [ K I ] x − X 1 , T d [ V 0 , T − 1 d X 0 , T − 1 d ] † [ dz ( K x ) 0 ] . x + = B A K x x − X 1 , T d V 0 , T − 1 d X 0 , T − 1 d † dz ( K x ) 0 = B A K I x − X 1 , T d V 0 , T − 1 d X 0 , T − 1 d † dz ( K x ) 0 . {:[x^(+)=[[B,A]][[Kx],[x]]-X_(1,T)^(d)[[V_(0,T-1)^(d)],[X_(0,T-1)^(d)]]^(†)[[dz(Kx)],[0]]],[=[[B,A]][[K],[I]]x-X_(1,T)^(d)[[V_(0,T-1)^(d)],[X_(0,T-1)^(d)]]^(†)[[dz(Kx)],[0]].]:}\begin{aligned} x^{+} & =\left[\begin{array}{ll} B & A \end{array}\right]\left[\begin{array}{c} K x \\ x \end{array}\right]-X_{1, T}^{d}\left[\begin{array}{c} V_{0, T-1}^{d} \\ X_{0, T-1}^{d} \end{array}\right]^{\dagger}\left[\begin{array}{c} \mathrm{dz}(K x) \\ 0 \end{array}\right] \\ & =\left[\begin{array}{ll} B & A \end{array}\right]\left[\begin{array}{c} K \\ I \end{array}\right] x-X_{1, T}^{d}\left[\begin{array}{c} V_{0, T-1}^{d} \\ X_{0, T-1}^{d} \end{array}\right]^{\dagger}\left[\begin{array}{c} \mathrm{dz}(K x) \\ 0 \end{array}\right] . \end{aligned}
Proceeding as in [7], the rank condition in (7) enables introducing a new variable G x ∈ R ( T − 1 ) × n x G x ∈ R ( T − 1 ) × n x G^(x)inR^((T-1)xxn_(x))G^{x} \in \mathbb{R}^{(T-1) \times n_{x}}, satisfying
按照 [7] 的方法，(7) 中的秩条件可以引入一个新变量 G x ∈ R ( T − 1 ) × n x G x ∈ R ( T − 1 ) × n x G^(x)inR^((T-1)xxn_(x))G^{x} \in \mathbb{R}^{(T-1) \times n_{x}} ，满足
[ K I ] = [ V 0 , T − 1 d X 0 , T − 1 d ] G x , K I = V 0 , T − 1 d X 0 , T − 1 d G x , [[K],[I]]=[[V_(0,T-1)^(d)],[X_(0,T-1)^(d)]]G^(x),\left[\begin{array}{c} K \\ I \end{array}\right]=\left[\begin{array}{c} V_{0, T-1}^{d} \\ X_{0, T-1}^{d} \end{array}\right] G^{x},
whose existence is guaranteed by the Rouché-Capelli theorem. Substituting (13) into (12), the closed-loop representation (10) and the consistency characterization (11) follow.
其存在由 Rouché-Capelli 定理保证。将（13）代入（12），可得到闭环表示（10）和一致性表征（11）。
Note that, according to (13), the static state-feedback gain K K KK in (5) can be uniquely computed from G x G x G^(x)G^{x} as
请注意，根据 (13)，(5) 中的静态状态反馈增益 K K KK 可以根据 G x G x G^(x)G^{x} 唯一计算得出
K = V 0 , T − 1 d G x K = V 0 , T − 1 d G x K=V_(0,T-1)^(d)G^(x)K=V_{0, T-1}^{d} G^{x}
The data-based closed-loop representation in (10) can be then equivalently recast compactly as follows:
然后，可以将（10）中基于数据的闭环表示紧凑地等效重写如下：
x + = A cl d ( G x ) x − B cl d dz ( K x ) , x + = A cl d G x x − B cl d dz ( K x ) , x^(+)=A_(cl)^(d)(G^(x))x-B_(cl)^(d)dz(Kx),x^{+}=A_{\mathrm{cl}}^{d}\left(G^{x}\right) x-B_{\mathrm{cl}}^{d} \mathrm{dz}(K x),
where G x G x G^(x)G^{x} should satisfy the consistency condition (11) and
其中 G x G x G^(x)G^{x} 应满足一致性条件 (11)，并且
A cl d ( G x ) = X 1 , T d G x , B cl d = X 1 , T d [ V 0 , T − 1 d X 0 , T − 1 d ] † [ I 0 ] . A cl d G x = X 1 , T d G x , B cl d = X 1 , T d V 0 , T − 1 d X 0 , T − 1 d † I 0 . A_(cl)^(d)(G^(x))=X_(1,T)^(d)G^(x),quadB_(cl)^(d)=X_(1,T)^(d)[[V_(0,T-1)^(d)],[X_(0,T-1)^(d)]]^(†)[[I],[0]].A_{\mathrm{cl}}^{d}\left(G^{x}\right)=X_{1, T}^{d} G^{x}, \quad B_{\mathrm{cl}}^{d}=X_{1, T}^{d}\left[\begin{array}{c} V_{0, T-1}^{d} \\ X_{0, T-1}^{d} \end{array}\right]^{\dagger}\left[\begin{array}{c} I \\ 0 \end{array}\right] .
It is thus clearer from (15) that the state transition matrix A cl 1 d ( G x ) A cl 1 d G x A_(cl1)^(d)(G^(x))A_{\mathrm{cl} 1}^{d}\left(G^{x}\right) of the closed-loop depends on the data and on the variable G x G x G^(x)G^{x}, which plays the role of design parameter, while we need to exploit the estimate B cl d B cl d B_(cl)^(d)B_{\mathrm{cl}}^{d} of the matrix B B BB in (8) to characterize the impact of the nonlinearity on the closed loop.
因此从 (15) 可以清楚地看出，闭环的状态转移矩阵 A cl 1 d ( G x ) A cl 1 d G x A_(cl1)^(d)(G^(x))A_{\mathrm{cl} 1}^{d}\left(G^{x}\right) 取决于数据和变量 G x G x G^(x)G^{x} ，后者起着设计参数的作用，而我们需要利用 (8) 中的矩阵 B B BB 的估计值 B cl d B cl d B_(cl)^(d)B_{\mathrm{cl}}^{d} 来表征非线性对闭环的影响。

B. Regionally Stabilizing Static State Feedback
B. 区域稳定静态反馈

To derive a design algorithm for a regionally stabilizing K K KK in (5), let us recall the regional sector condition [14] for the dead-zone nonlinearity appearing in (15a), namely
为了推导 (5) 式中区域稳定 K K KK 的设计算法，我们回顾一下 (15a) 式中出现的死区非线性的区域扇区条件 [14]，即
dz ( H x ) = 0 ⟹ dz ( u ) ⊤ W ( u − dz ( u ) + H x ) ≥ 0 dz ( H x ) = 0 ⟹ dz ( u ) ⊤ W ( u − dz ( u ) + H x ) ≥ 0 dz(Hx)=0quad Longrightarrowquaddz(u)^(TT)W(u-dz(u)+Hx) >= 0\mathrm{dz}(H x)=0 \quad \Longrightarrow \quad \mathrm{dz}(u)^{\top} W(u-\mathrm{dz}(u)+H x) \geq 0
which holds for any H ∈ R n u × n x H ∈ R n u × n x H inR^(n_(u)xxn_(x))H \in \mathbb{R}^{n_{u} \times n_{x}}, any u ∈ R n u u ∈ R n u u inR^(n_(u))u \in \mathbb{R}^{n_{u}}, and any positive definite diagonal matrix W ∈ R n u × n u W ∈ R n u × n u W inR^(n_(u)xxn_(u))W \in \mathbb{R}^{n_{u} \times n_{u}}. Exploiting the quadratic Lyapunov function candidate
对于任何 H ∈ R n u × n x H ∈ R n u × n x H inR^(n_(u)xxn_(x))H \in \mathbb{R}^{n_{u} \times n_{x}} 、任何 u ∈ R n u u ∈ R n u u inR^(n_(u))u \in \mathbb{R}^{n_{u}} 以及任何正定对角矩阵 W ∈ R n u × n u W ∈ R n u × n u W inR^(n_(u)xxn_(u))W \in \mathbb{R}^{n_{u} \times n_{u}} 都成立。利用二次李雅普诺夫函数候选
V ( x ) = x ⊤ P x , V ( x ) = x ⊤ P x , V(x)=x^(TT)Px,V(x)=x^{\top} P x,
with P ∈ R n x × n x P ∈ R n x × n x P inR^(n_(x)xxn_(x))P \in \mathbb{R}^{n_{x} \times n_{x}} being symmetric and positive definite, we cast the DD design K K KK in (5) by solving the following linear matrix inequality (LMI) in the decision variables Q = Q ⊤ = P − 1 Q = Q ⊤ = P − 1 Q=Q^(TT)=P^(-1)Q=Q^{\top}=P^{-1}, M = W − 1 M = W − 1 M=W^(-1)M=W^{-1} diagonal, F = G x Q ∈ R ( T − 1 ) × n x F = G x Q ∈ R ( T − 1 ) × n x F=G^(x)Q inR^((T-1)xxn_(x))F=G^{x} Q \in \mathbb{R}^{(T-1) \times n_{x}} and N ∈ R n u × n x N ∈ R n u × n x N inR^(n_(u)xxn_(x))N \in \mathbb{R}^{n_{u} \times n_{x}} :
其中 P ∈ R n x × n x P ∈ R n x × n x P inR^(n_(x)xxn_(x))P \in \mathbb{R}^{n_{x} \times n_{x}} 是对称和正定的，我们通过求解决策变量 Q = Q ⊤ = P − 1 Q = Q ⊤ = P − 1 Q=Q^(TT)=P^(-1)Q=Q^{\top}=P^{-1} 、 M = W − 1 M = W − 1 M=W^(-1)M=W^{-1} 对角线、 F = G x Q ∈ R ( T − 1 ) × n x F = G x Q ∈ R ( T − 1 ) × n x F=G^(x)Q inR^((T-1)xxn_(x))F=G^{x} Q \in \mathbb{R}^{(T-1) \times n_{x}} 和 N ∈ R n u × n x N ∈ R n u × n x N inR^(n_(u)xxn_(x))N \in \mathbb{R}^{n_{u} \times n_{x}} 中的以下线性矩阵不等式 (LMI) 来构造 (5) 中的 DD 设计 K K KK ：
[ Q N j ⊤ N j u ¯ j 2 ] ≥ 0 , j = 1 , … , n u , He [ − Q 2 0 0 V 0 , T − 1 d F + N − M 0 X 1 , T d F − B cl d M − Q 2 ] < 0 , Q N j ⊤ N j u ¯ j 2 ≥ 0 , j = 1 , … , n u , He − Q 2 0 0 V 0 , T − 1 d F + N − M 0 X 1 , T d F − B cl d M − Q 2 < 0 , {:[{:[[Q,N_(j)^(TT)],[N_(j), bar(u)_(j)^(2)]] >= 0","quad j=1","dots","n_(u)",":}],[He[[-(Q)/(2),0,0],[V_(0,T-1)^(d)F+N,-M,0],[X_(1,T)^(d)F,-B_(cl)^(d)M,-(Q)/(2)]] < 0","]:}\begin{aligned} & {\left[\begin{array}{cc} Q & N_{j}^{\top} \\ N_{j} & \bar{u}_{j}^{2} \end{array}\right] \geq 0, \quad j=1, \ldots, n_{u},} \\ & \operatorname{He}\left[\begin{array}{ccc} -\frac{Q}{2} & 0 & 0 \\ V_{0, T-1}^{d} F+N & -M & 0 \\ X_{1, T}^{d} F & -B_{\mathrm{cl}}^{d} M & -\frac{Q}{2} \end{array}\right]<0, \end{aligned}
(where N j N j N_(j)N_{j} denotes the j j jj-th row of matrix N N NN and B cl d B cl  d B_("cl ")^(d)B_{\text {cl }}^{d} only depends on data, as defined in (15b)) together with the consistency condition
（其中 N j N j N_(j)N_{j} 表示矩阵 N N NN 的第 j j jj 行，而 B cl d B cl  d B_("cl ")^(d)B_{\text {cl }}^{d} 仅取决于数据，如 (15b) 中定义）以及一致性条件
X 0 , T − 1 d F = Q X 0 , T − 1 d F = Q X_(0,T-1)^(d)F=QX_{0, T-1}^{d} F=Q
The n u n u n_(u)n_{u} LMIs in (18a) have dimension n x + 1 n x + 1 n_(x)+1n_{x}+1 each, while LMI (18b) has dimension 2 n x + n u 2 n x + n u 2n_(x)+n_(u)2 n_{x}+n_{u}, thus depending on the order of the system and the dimension of the input. Overall, also the size of the decision vector depends on the system’s features (and not on the length of the dataset), due to Q , W Q , W Q,WQ, W and N N NN, amounting to n x ( n x + 1 ) 2 + n u + n u n x n x n x + 1 2 + n u + n u n x (n_(x)(n_(x)+1))/(2)+n_(u)+n_(u)n_(x)\frac{n_{x}\left(n_{x}+1\right)}{2}+n_{u}+n_{u} n_{x} variables. Only F F FF of size ( T − 1 ) n x ( T − 1 ) n x (T-1)n_(x)(T-1) n_{x} depends on the length of the dataset, eventually making the proposed approach computationally more demanding when a large dataset is available. This is the price to pay to avoid a full identification step. We state below our noise-free DD design result.
(18a) 中的 n u n u n_(u)n_{u} 个 LMI 的维度为 n x + 1 n x + 1 n_(x)+1n_{x}+1 ，而 LMI (18b) 的维度为 2 n x + n u 2 n x + n u 2n_(x)+n_(u)2 n_{x}+n_{u} ，因此取决于系统的阶数和输入的维度。总体而言，决策向量的大小还取决于系统的特征（而不是数据集的长度），因为 Q , W Q , W Q,WQ, W 和 N N NN 共计 n x ( n x + 1 ) 2 + n u + n u n x n x n x + 1 2 + n u + n u n x (n_(x)(n_(x)+1))/(2)+n_(u)+n_(u)n_(x)\frac{n_{x}\left(n_{x}+1\right)}{2}+n_{u}+n_{u} n_{x} 个变量。大小为 ( T − 1 ) n x ( T − 1 ) n x (T-1)n_(x)(T-1) n_{x} 的其中只有 F F FF 取决于数据集的长度，最终使得在有大型数据集可用时，所提出的方法在计算上要求更高。这是为了避免全面识别步骤而必须付出的代价。我们一直在遵循无噪音的 DD 设计结果。
Theorem 1 (Regionally stabilizing DD design): Given (3) and a noise free set of data D T D T D_(T)\mathscr{D}_{T} satisfying the conditions in (6) and (7), if there exist matrices Q = Q ⊤ , M Q = Q ⊤ , M Q=Q^(TT),MQ=Q^{\top}, M diagonal, F ∈ F ∈ F inF \in R ( T − 1 ) × n x R ( T − 1 ) × n x R^((T-1)xxn_(x))\mathbb{R}^{(T-1) \times n_{x}} and N ∈ R n u × n x N ∈ R n u × n x N inR^(n_(u)xxn_(x))N \in \mathbb{R}^{n_{u} \times n_{x}} satisfying (18), then the following gain selection in (5)
定理 1（区域稳定 DD 设计）：给定 (3) 和一组满足 (6) 和 (7) 条件的无噪声数据 D T D T D_(T)\mathscr{D}_{T} ，如果存在满足 (18) 的矩阵 Q = Q ⊤ , M Q = Q ⊤ , M Q=Q^(TT),MQ=Q^{\top}, M 对角线、 F ∈ F ∈ F inF \in R ( T − 1 ) × n x R ( T − 1 ) × n x R^((T-1)xxn_(x))\mathbb{R}^{(T-1) \times n_{x}} 和 N ∈ R n u × n x N ∈ R n u × n x N inR^(n_(u)xxn_(x))N \in \mathbb{R}^{n_{u} \times n_{x}} ，则 (5) 中的以下增益选择
K = V 0 , T − 1 d F Q − 1 K = V 0 , T − 1 d F Q − 1 K=V_(0,T-1)^(d)FQ^(-1)K=V_{0, T-1}^{d} F Q^{-1}
guarantees exponential stability of the origin for the closed loop (3), (5) with basin of attraction containing the set
保证闭环（3）、（5）的原点指数稳定性，其吸引域包含集合
E ( Q , 1 ) = { x ∈ R n x : x ⊤ Q − 1 x ≤ 1 } E ( Q , 1 ) = x ∈ R n x : x ⊤ Q − 1 x ≤ 1 E(Q,1)={x inR^(n_(x)):x^(TT)Q^(-1)x <= 1}\mathscr{E}(Q, 1)=\left\{x \in \mathbb{R}^{n_{x}}: x^{\top} Q^{-1} x \leq 1\right\}
of Lyapunov function V V VV in (17), with selection P = Q − 1 P = Q − 1 P=Q^(-1)P=Q^{-1}.
(17) 式中的 Lyapunov 函数 V V VV ，选择为 P = Q − 1 P = Q − 1 P=Q^(-1)P=Q^{-1} 。
Proof: Using the Lyapunov function candidate (17) with P = P ⊤ = Q − 1 P = P ⊤ = Q − 1 P=P^(TT)=Q^(-1)P=P^{\top}=Q^{-1}, which is positive definite due to the ( 1 , 1 ) ( 1 , 1 ) (1,1)(1,1) entry in (18b), we may study exponential stability of the origin by studying the sign of the increment
证明：使用 Lyapunov 函数候选 (17)，其中 P = P ⊤ = Q − 1 P = P ⊤ = Q − 1 P=P^(TT)=Q^(-1)P=P^{\top}=Q^{-1} ，由于 (18b) 中的 ( 1 , 1 ) ( 1 , 1 ) (1,1)(1,1) 项而具有正定义，我们可以通过研究增量的符号来研究原点的指数稳定性
Δ V ( x ) = ( x + ) ⊤ P x + − x ⊤ P x Δ V ( x ) = x + ⊤ P x + − x ⊤ P x Delta V(x)=(x^(+))^(TT)Px^(+)-x^(TT)Px\Delta V(x)=\left(x^{+}\right)^{\top} P x^{+}-x^{\top} P x
Starting from the solution Q , F , M , N Q , F , M , N Q,F,M,NQ, F, M, N of (18) assumed in the theorem statement, select G x = F Q − 1 G x = F Q − 1 G^(x)=FQ^(-1)G^{x}=F Q^{-1} so that, from (18c) the consistency condition (11) is satisfied. Due to the consistency condition and since (19) matches the assumed selection (14), apply Lemma 3 to characterize the increment in (21) by the following DD inequality, issued from (15),
从定理陈述中假设的 (18) 的解 Q , F , M , N Q , F , M , N Q,F,M,NQ, F, M, N 开始，选择 G x = F Q − 1 G x = F Q − 1 G^(x)=FQ^(-1)G^{x}=F Q^{-1} ，使得从 (18c) 满足一致性条件 (11)。由于一致性条件，且 (19) 与假设选择 (14) 相匹配，应用引理 3 来表征 (21) 中的增量，通过以下 DD 不等式 (由 (15) 得出)，
Δ V ( x ) = ( A cl d ( G x ) x − B cl d dz ( u ) ) ⊤ P ( A cl d ( G x ) x − B cl d dz ( u ) ) − x ⊤ P x = [ x dz ( u ) ] ⊤ ( [ ( A cl d ( G x ) ) ⊤ ( B cl d ) ⊤ ] P [ A cl d ( G x ) B cl d ] − [ P 0 0 0 ] ) [ x dz ( u ) ] . Δ V ( x ) = A cl d G x x − B cl d dz ( u ) ⊤ P A cl d G x x − B cl d dz ( u ) − x ⊤ P x = x dz ( u ) ⊤ A cl d G x ⊤ B cl d ⊤ P A cl d G x B cl d − P 0 0 0 x dz ( u ) . {:[Delta V(x)],[=(A_(cl)^(d)(G^(x))x-B_(cl)^(d)dz(u))^(TT)P(A_(cl)^(d)(G^(x))x-B_(cl)^(d)dz(u))-x^(TT)Px],[=[[x],[dz(u)]]^(TT)([[(A_(cl)^(d)(G^(x)))^(TT)],[(B_(cl)^(d))^(TT)]]P[A_(cl)^(d)(G^(x))B_(cl)^(d)]-[[P,0],[0,0]])[[x],[dz(u)]].]:}\begin{aligned} & \Delta V(x) \\ & =\left(A_{\mathrm{cl}}^{d}\left(G^{x}\right) x-B_{\mathrm{cl}}^{d} \mathrm{dz}(u)\right)^{\top} P\left(A_{\mathrm{cl}}^{d}\left(G^{x}\right) x-B_{\mathrm{cl}}^{d} \mathrm{dz}(u)\right)-x^{\top} P x \\ & =\left[\begin{array}{c} x \\ \mathrm{dz}(u) \end{array}\right]^{\top}\left(\left[\begin{array}{c} \left(A_{\mathrm{cl}}^{d}\left(G^{x}\right)\right)^{\top} \\ \left(B_{\mathrm{cl}}^{d}\right)^{\top} \end{array}\right] P\left[A_{\mathrm{cl}}^{d}\left(G^{x}\right) B_{\mathrm{cl}}^{d}\right]-\left[\begin{array}{cc} P & 0 \\ 0 & 0 \end{array}\right]\right)\left[\begin{array}{c} x \\ \mathrm{dz}(u) \end{array}\right] . \end{aligned}
Exploiting the sector condition (16) for any x x xx satisfying dz ( H x ) = 0 dz ( H x ) = 0 dz(Hx)=0\mathrm{dz}(H x)=0, we can upper bound
通过利用扇区条件 (16)，对于满足 dz ( H x ) = 0 dz ( H x ) = 0 dz(Hx)=0\mathrm{dz}(H x)=0 的任何 x x xx ，我们可以上界
Δ V ( x ) ≤ Δ V ( x ) + 2 dz ( u ) ⊤ W ( K x − dz ( u ) + H x ) = ξ ⊤ Ξ ξ Δ V ( x ) ≤ Δ V ( x ) + 2 dz ( u ) ⊤ W ( K x − dz ( u ) + H x ) = ξ ⊤ Ξ ξ Delta V(x) <= Delta V(x)+2dz(u)^(TT)W(Kx-dz(u)+Hx)=xi^(TT)Xi xi\Delta V(x) \leq \Delta V(x)+2 \mathrm{dz}(u)^{\top} W(K x-\mathrm{dz}(u)+H x)=\xi^{\top} \Xi \xi
where we denoted ξ = [ x dz ( u ) ] ξ = x dz ( u ) xi=[[x],[dz(u)]]\xi=\left[\begin{array}{c}x \\ \operatorname{dz}(u)\end{array}\right] and
其中我们表示 ξ = [ x dz ( u ) ] ξ = x dz ( u ) xi=[[x],[dz(u)]]\xi=\left[\begin{array}{c}x \\ \operatorname{dz}(u)\end{array}\right] 和
Ξ = He [ − P 2 0 0 W K + W H − W 0 X 1 , T d G x − B cl d − P − 1 2 ] Ξ = He − P 2 0 0 W K + W H − W 0 X 1 , T d G x − B cl d − P − 1 2 Xi=He[[-(P)/(2),0,0],[WK+WH,-W,0],[X_(1,T)^(d)G^(x),-B_(cl)^(d),-(P^(-1))/(2)]]\Xi=\operatorname{He}\left[\begin{array}{ccc} -\frac{P}{2} & 0 & 0 \\ W K+W H & -W & 0 \\ X_{1, T}^{d} G^{x} & -B_{\mathrm{cl}}^{d} & -\frac{P^{-1}}{2} \end{array}\right]
Based on the solution Q , F , M , N Q , F , M , N Q,F,M,NQ, F, M, N of (18), consider now the following selection of the variables appearing in (24),
基于 (18) 的解 Q , F , M , N Q , F , M , N Q,F,M,NQ, F, M, N ，现在考虑对 (24) 中出现的变量进行以下选择，
P = Q − 1 > 0 , G x = F Q − 1 , W = M − 1 > 0 diagonal, H = N Q − 1 , P = Q − 1 > 0 , G x = F Q − 1 , W = M − 1 > 0  diagonal,  H = N Q − 1 , {:[P=Q^(-1) > 0","quadG^(x)=FQ^(-1)","],[W=M^(-1) > 0" diagonal, "quad H=NQ^(-1)","]:}\begin{aligned} P & =Q^{-1}>0, \quad G^{x}=F Q^{-1}, \\ W & =M^{-1}>0 \text { diagonal, } \quad H=N Q^{-1}, \end{aligned}
where positive definiteness of P P PP follows from Q = Q ⊤ > 0 Q = Q ⊤ > 0 Q=Q^(TT) > 0Q=Q^{\top}>0 induced by the ( 1 , 1 ) ( 1 , 1 ) (1,1)(1,1) entry of ( 18 b ) ( 18 b ) (18 b)(18 b), and positive definiteness of W W WW is induced by the ( 2 , 2 ) ( 2 , 2 ) (2,2)(2,2) entry of ( 18 b ) ( 18 b ) (18 b)(18 b). Then, pre- and
其中 P P PP 的正定性由 ( 18 b ) ( 18 b ) (18 b)(18 b) 的 ( 1 , 1 ) ( 1 , 1 ) (1,1)(1,1) 项引起的 Q = Q ⊤ > 0 Q = Q ⊤ > 0 Q=Q^(TT) > 0Q=Q^{\top}>0 得出，而 W W WW 的正定性则由 ( 18 b ) ( 18 b ) (18 b)(18 b) 的 ( 2 , 2 ) ( 2 , 2 ) (2,2)(2,2) 项引起。然后，预
  1. Manuscript received 10 January 2023; revised 27 February 2023; accepted 21 March 2023. Date of publication 11 April 2023; date of current version 24 May 2023. This work was supported in part by the Italian Ministry of University and Research through the PRIN Projects “Data-Driven Learning of Constrained Control Systems” under Contract 2017J89ARP and “Design of Cooperative EnergyAware Aerial Platforms for Remote and Contact-Aware Operations” (DOCEAT) under Grant 2020RTWES4, and in part by ANR through “Hybrid and Networked Dynamical Systems” (HANDY) under Grant ANR-18-CE40-0010. Recommended by Senior Editor M. K. Camlibel. (Corresponding author: Simone Formentin.)
    稿件收到日期：2023 年 1 月 10 日； 2023 年 2 月 27 日修订；接受日期：2023 年 3 月 21 日。出版日期：2023 年 4 月 11 日；当前版本的日期为 2023 年 5 月 24 日。这项工作部分由意大利大学和研究部通过 PRIN 项目“受限控制系统的数据驱动学习”（合同 2017J89ARP）和“用于远程和接触感知操作的合作能量感知空中平台设计”（DOCEAT）（资助 2020RTWES4）资助，部分由 ANR 通过“混合和网络化动力系统”（HANDY）资助（资助 ANR-18-CE40-0010）。由高级编辑 MK Camlibel 推荐。 （通讯作者：西蒙尼·福门汀）
    Valentina Breschi is with the Department of Electrical Engineering, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands.
    Valentina Breschi 就职于荷兰埃因霍温理工大学电气工程系，邮编 5600 MB。
    Luca Zaccarian is with LAAS-CNRS, University of Toulouse, 31000 Toulouse, France, and also with the Dip. di Ing. Industriale, Universitá di Trento, 38122 Trento, Italy.
    Luca Zaccarian 是法国图卢兹大学 LAAS-CNRS（邮编 31000 图卢兹，法国）以及意大利特伦托大学工业工程文凭（邮编 38122 特伦托）的毕业生。
    Simone Formentin is with the Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, 20133 Milan, Italy (e-mail: simone.formentin @ polimi.it).
    Simone Formentin 就职于米兰理工大学电子、信息和生物工程系，地址：意大利米兰 20133（电子邮件：simone.formentin @ polimi.it ）。
    Digital Object Identifier 10.1109/LCSYS.2023.3266254
    数字对象标识符 10.1109/LCSYS.2023.3266254
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