随机线性混合系统的可观测性准则与估计器设计（篇 1）

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#随机线性混合系统

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Observability Criteria and Estimator Design for Stochastic Linear Hybrid Systems

随机线性混合系统的可观测性准则与估计器设计

Inseok Hwang, Hamsa Balakrishnan, Claire Tomlin
Hybrid Systems Laboratory
Department of Aeronautics and Astronautics
Stanford University, Stanford, CA 94305, U.S.A.
ishwang, hamsa, tomlin@stanford.edu

印石旭·黄（Inseok Hwang）、哈姆萨·巴拉科里什南（Hamsa Balakrishnan）、克莱尔·汤姆林（Claire Tomlin）
美国加利福尼亚州斯坦福市斯坦福大学航空航天系混合系统实验室，邮编：94305
电子邮箱：ishwang, hamsa, tomlin@stanford.edu

Abstract

摘要

A stochastic linear hybrid system is said to be observable if the hybrid state of the system is uniquely determined from the output. In this paper, we derive the conditions for the observability of stochastic linear hybrid systems by exploiting the information obtained from system noise characteristics. Having established the necessary criteria for observability, we study the effect of these conditions on estimator design, and also find bounds on the switching times of the system to achieve guaranteed estimator performance. We then apply these results to the estimation of a two-mode aircraft trajectory.
若能由输出唯一确定系统的混合状态，则称该随机线性混合系统具有可观测性。本文通过利用系统噪声特性所提供的信息，推导随机线性混合系统的可观测性条件。在建立可观测性的必要准则后，研究了这些准则对估计器设计的影响，并确定了系统切换时间的界，以保证估计器性能。最后，将所得结果应用于双模态飞行器轨迹的估计问题。

1 Introduction

引言

The tracking of aircraft trajectories is a problem that has been approached with some success using hybrid systems models [1]. Related problems of interest to us are the ability to estimate the hybrid states of such systems from their outputs, and also the design of estimators for such systems.
采用混合系统模型解决飞行器轨迹跟踪问题已取得一定成效（文献[1]）。本文关注的相关问题包括：如何由系统输出估计此类系统的混合状态，以及如何设计此类系统的估计器。

The problem of observability, namely, the ability to estimate or reconstruct the actual state of a system given its output, is well-known and has been studied extensively, both for continuous systems [2] as well as for discrete ones [3, 4]. More recently, several researchers have approached the problem of observability of hybrid systems. A practical problem that has received increasing research attention recently is the extension of these concepts to stochastic hybrid systems. In this paper, we address the issue of observability of a class of stochastic hybrid systems – systems where the continuous dynamics are affected by white Gaussian noise.
可观测性问题——即由系统输出估计或重构其真实状态的能力——是一个经典问题，目前已针对连续系统（文献[2]）和离散系统（文献[3, 4]）开展了大量研究。近年来，众多学者开始研究混合系统的可观测性问题。其中，将可观测性概念推广到随机混合系统的研究，是近年来受到越来越多关注的实际问题。本文旨在解决一类随机混合系统的可观测性问题，这类系统的连续动力学过程受白高斯噪声影响。

Alessandri and Coletta [5] proposed a Luenberger observer design methodology for deterministic linear hybrid systems, and proved that the error converges if the discrete state evolution is known. Balluchi et al. [6] developed a method of combining location observers for discrete state estimation with Luenberger observers for continuous state estimation for linear systems, such that they can guarantee the exponential convergence of the estimation error. Bemporad et al. [7] defined the concept of incremental observability of continuous-time linear hybrid systems, using the solutions of a mixed-integer linear program. Recently, Vidal et al. [8] derived observability conditions for linear hybrid systems with continuous-time continuous-state dynamics, given in the form of rank conditions similar to those for continuous-time linear system observability.
Alessandri 与 Coletta（文献[5]）提出了确定性线性混合系统的龙伯格（Luenberger）观测器设计方法，并证明：若离散状态的演化规律已知，则估计误差会收敛。Balluchi 等人（文献[6]）提出一种融合设计方法，将用于离散状态估计的位置观测器与用于连续状态估计的龙伯格观测器相结合，可保证线性系统估计误差的指数收敛。Bemporad 等人（文献[7]）利用混合整数线性规划的解，定义了连续时间线性混合系统的增量可观测性概念。近期，Vidal 等人（文献[8]）推导了连续时间、连续状态动力学线性混合系统的可观测性条件，其形式为秩条件，与连续时间线性系统可观测性的秩条件类似。

For stochastic systems, the definition of observability in its classical form, as proposed by Kalman for systems with no noise, fails; we therefore need to find a meaningful interpretation of observability for systems with random noise. Baram and Kailath [9] proposed the concept of estimability as a better criterion to gauge stochastic linear systems. While this is one way of approaching the problem, we try to extend the definition of observability to include stochastic hybrid systems.
对于随机系统，卡尔曼（Kalman）针对无噪声系统提出的经典可观测性定义不再适用；因此，需要为含随机噪声的系统寻找可观测性的有效定义。Baram 与 Kailath（文献[9]）提出了“可估计性”概念，将其作为衡量随机线性系统的更优准则。尽管这是解决该问题的一种思路，但本文尝试将可观测性定义推广至随机混合系统。

An important class of problems associated with applications in multi-target tracking [1] and speech recognition [10] pertains to the estimation of discrete-time Markov jump linear systems. Cost and do Val [11] analyzed such systems with finite Markov states and deterministic continuous dynamics, and derived the observability condition that the solution to the coupled Riccati equation associated with the quadratic control problem has a stabilizing solution. Vidal et al. [12] derived observability conditions for jump linear systems based on rank tests similar to those of deterministic linear hybrid systems.
在多目标跟踪（文献[1]）和语音识别（文献[10]）等应用中，一类重要问题涉及离散时间马尔可夫跳变线性系统的估计。Cost 与 do Val（文献[11]）分析了具有有限马尔可夫状态和确定性连续动力学的此类系统，推导得出其可观测性条件：与二次控制问题相关的耦合黎卡提方程的解需为镇定解。Vidal 等人（文献[12]）基于秩检验，推导了跳变线性系统的可观测性条件，该秩检验与确定性线性混合系统的秩检验类似。

The first part of this study is motivated by the results of Vidal et al. [12]. They proposed the notion of indistinguishability as: two initial states are indistinguishable if the corresponding outputs in free evolution are equal. This approach results in elegant rank tests for the observability of stochastic jump linear systems. Since in the design of estimators for aircraft tracking we have knowledge of not just the system dynamics, but also the noise covariances, we try to exploit this additional knowledge to improve our ability to differentiate between state trajectories. Since the output sequences of stochastic systems might be different from the same initial condition, we extend the notion of indistinguishability [12] for such systems, and based on our definition, we derive conditions for the observability of discrete-time stochastic linear hybrid systems. The latter part of this paper applies the approach of Balluchi et al. [6], so far used in the design of hybrid observers for deterministic hybrid systems with continuous-time state evolution, to discrete-time stochastic hybrid systems and estimator design.
本文前半部分的研究灵感来源于 Vidal 等人（文献[12]）的成果。他们提出“不可区分性”概念：若两个初始状态在自由演化过程中产生的输出相等，则这两个初始状态不可区分。该方法为随机跳变线性系统的可观测性提供了简洁的秩检验方法。在飞行器跟踪估计器设计中，我们不仅知晓系统动力学特性，还掌握噪声协方差信息；因此，本文尝试利用这一额外信息，提升区分状态轨迹的能力。由于随机系统在相同初始条件下可能产生不同输出序列，本文针对此类系统推广了“不可区分性”概念（文献[12]），并基于该定义推导离散时间随机线性混合系统的可观测性条件。本文后半部分将 Balluchi 等人（文献[6]）的方法——该方法此前用于设计具有连续时间状态演化的确定性混合系统的混合观测器——推广至离散时间随机混合系统及估计器设计中。

This paper is organized as follows: Section 2 presents the observability conditions of discrete-time stochastic jump linear systems. In Section 3, we obtain conditions on the system parameters that would guarantee the exponential convergence of hybrid estimators. Examples and conclusions are presented in Sections 4 and 5 respectively.
本文结构如下：第 2 节给出离散时间随机跳变线性系统的可观测性条件；第 3 节推导系统参数需满足的条件，以保证混合估计器的指数收敛；第 4 节给出示例，第 5 节给出结论。

2 Observability of discrete-time stochastic linear hybrid systems

离散时间随机线性混合系统的可观测性

In this section, inspired by Vidal et al. [12], we extend the concepts of indistinguishability, observability of the hybrid initial state, and discrete transition times as defined in [12] and derive more general observability conditions for discrete-time stochastic linear hybrid systems using the knowledge of noise covariances.
本节受 Vidal 等人（文献[12]）的启发，推广了文献[12] 中定义的“不可区分性”“混合初始状态可观测性”及“离散切换时间”等概念，并利用噪声协方差信息，推导离散时间随机线性混合系统更具一般性的可观测性条件。

We consider a discrete-time stochastic linear hybrid system
考虑如下离散时间随机线性混合系统：

$H:\left\{ \begin{array} {rcl} x_{k+1} & = & A(q_{k})x_{k} + w_{k}(q_{k}) \\ y_{k} & = & C(q_{k})x_{k} + v_{k}(q_{k}), \quad k \in \{0,1,\cdots\} \\ q_{k+1} & = & \delta(q_{k},\gamma_{k}) \end{array} \right. \tag{1}$

where $k$ is a non-negative integer ( $\in \mathbb{N}$ ); $x_{k} \in \mathbb{R}^{n}$ and $y_{k} \in \mathbb{R}^{p}$ are the continuous state and output variables, respectively; $q_{k} \in \{1,2,\cdots,N\}$ is the discrete state; $\gamma_k$ is a discrete control input; and $\delta(\cdot, \cdot)$ is a deterministic discrete transition relation that governs the evolution of the discrete state. We assume the event times at which discrete transitions occur are unknown. For $q_{k} \in \{1,2,\cdots,N\}$ , the system parameters $A(q_{k}) \in \mathbb{R}^{n \times n}$ and $C(q_{k}) \in \mathbb{R}^{p \times n}$ are real matrices. We assume that the initial state $x_{k_0}$ is an unknown, zero-mean white Gaussian random variable with covariance $E[x_{k_0}x_{k_0}^T] = \Pi_0$ ; the process noise $w_{k}(q_{k})$ and measurement noise $v_{k}(q_{k})$ are uncorrelated, zero-mean white Gaussian sequences with covariance matrices $E[w_{k}(q_{k})w_{k}(q_{k})^T] = \rho(q_{k})I$ and $E[v_{k}(q_{k})v_{k}(q_{k})^T] = \sigma(q_{k})I$ , respectively. These random sequences are assumed to be uncorrelated with the initial state, i.e., $E[x_{k_0}w_{k}(q_{k})^T] = E[x_{k_0}v_{k}(q_{k})^T] = 0$ . $I$ denotes the identity matrix. Since the state evolution of a hybrid system includes both continuous trajectories and discrete jumps, we define a “hybrid time trajectory” as follows:
其中， $k$ 为非负整数（ $\in \mathbb{N}$ ）； $x_{k} \in \mathbb{R}^{n}$ 和 $y_{k} \in \mathbb{R}^{p}$ 分别为连续状态变量和输出变量； $q_{k} \in \{1,2,\cdots,N\}$ 为离散状态； $\gamma_k$ 为离散控制输入； $\delta(\cdot,\cdot)$ 为确定性离散切换关系，用于描述离散状态的演化规律。假设离散切换发生的时刻未知。对于 $q_{k} \in \{1,2,\cdots,N\}$ ，系统参数矩阵 $A(q_{k}) \in \mathbb{R}^{n \times n}$ 和 $C(q_{k}) \in \mathbb{R}^{p \times n}$ 均为实矩阵。假设初始状态 $x_{k_0}$ 为未知零均值白高斯随机变量，其协方差为 $E[x_{k_0}x_{k_0}^T] = \Pi_0$ ；过程噪声 $w_k(q_k)$ 与测量噪声 $v_k(q_k)$ 为互不相关的零均值白高斯序列，其协方差矩阵分别为 $E[w_k(q_k)w_k(q_k)^T] = \rho(q_k)I$ 和 $E[v_k(q_k)v_k(q_k)^T] = \sigma(q_k)I$ 。假设这些随机序列与初始状态互不相关，即 $E[x_{k_0}w_k(q_k)^T] = E[x_{k_0}v_k(q_k)^T] = 0$ ； $I$ 表示单位矩阵。由于混合系统的状态演化既包含连续轨迹，又包含离散跳变，因此定义“混合时间轨迹”如下：

Definition 1 (Hybrid time trajectory) A hybrid time trajectory is a sequence of intervals
定义 1（混合时间轨迹）混合时间轨迹是一组区间序列：
$\left[k_{0}, k_{1}-1\right], \left[k_{1}, k_{2}-1\right], \cdots, \left[k_{i}, k_{i+1}-1\right], \cdots$

where $k_{i}(i \geq1)$ is the time at which the $i$ -th discrete state transition occurs.
其中， $k_i(i \geq1)$ 表示第 $i$ 次离散状态切换发生的时刻。

Before deriving the observability conditions, we review the definition of observability for discrete-time stochastic linear hybrid systems [12]:
在推导可观测性条件之前，首先回顾离散时间随机线性混合系统可观测性的定义（文献[12]）：

Definition 2 (Observability of discrete-time stochastic linear hybrid systems) A discrete-time linear hybrid system $H$ is observable on $k_{0}, k_{0}+K]$ if the hybrid state $q_{k}, x_{k})$ for $\in [k_{0}, k_{0}+K]$ is uniquely determined from the output sequence $Y_{K}=[y_{k_{0}}^{T} \cdots y_{k_{0}+K}^{T}]^{T}$ , where $\in \mathbb{N}$ .
定义 2（离散时间随机线性混合系统的可观测性）若对于所有 $\in [k_0, k_0+K]$ ，混合状态 $q_k, x_k)$ 可由输出序列 $Y_K=[y_{k_0}^T \cdots y_{k_0+K}^T]^T$ 唯一确定（其中 $\in \mathbb{N}$ ），则称离散时间线性混合系统 $H$ 在区间 $k_0, k_0+K]$ 上具有可观测性。

Vidal et al. [12] developed rank tests for the observability of stochastic jump linear systems of the form described by (Eq.(1)) using the notion of indistinguishability. Since we know the noise covariances as well as the system dynamics for a stochastic system, we use this additional knowledge to obtain a more general condition. Since the output sequences of stochastic systems could be different from the same initial condition, we extend the notion of indistinguishability [12] as follows:
Vidal 等人（文献[12]）利用“不可区分性”概念，针对式 (1) 所示形式的随机跳变线性系统，提出了可观测性的秩检验方法。对于随机系统，我们不仅知晓系统动力学特性，还掌握噪声协方差信息；因此，本文利用这一额外信息，得到更具一般性的可观测性条件。由于随机系统在相同初始条件下可能产生不同输出序列，本文针对此类系统推广“不可区分性”概念（文献[12]），定义如下：

Definition 3 (Indistinguishability of discrete-time stochastic linear hybrid systems) A discrete-time linear hybrid system $H$ is indistinguishable on $k_{0}, k_{0}+K]$ if there exist output sequences $y_{K}$ and $y_{K}'$ on $\in [k_{0}, k_{0}+K]$ starting from any two different hybrid states $q_{k_{0}}, x_{k_{0}})$ and $q_{k_{0}}', x_{k_{0}}')$ , whose covariances are equal.
定义 3（离散时间随机线性混合系统的不可区分性）若对于任意两个不同的混合初始状态 $q_{k_0}, x_{k_0})$ 和 $q_{k_0}', x_{k_0}')$ ，在区间 $\in [k_0, k_0+K]$ 上存在输出序列 $y_K$ 和 $y_K'$ ，且二者的协方差相等，则称离散时间线性混合系统 $H$ 在区间 $k_0, k_0+K]$ 上具有不可区分性。

2.1 Observability of the hybrid initial state

混合初始状态的可观测性

In this section, using a procedure similar to that in [12], we derive the conditions under which the hybrid initial state ( $q_{k_{0}}, x_{k_{0}})$ ) can be uniquely determined from the output sequence on $k_{0}, k_{1}-1]$ ( $k_{1}-1 \leq k_{0}+K$ ), i.e., before the first discrete transition occurs. We define $\kappa_{i}:=k_{i+1}-k_{i}(i \geq0)$ as the sojourn time, which denotes how long the system stays in a discrete state after the $i$ -th discrete transition. Based on Definition 2 and Definition 3, we get the following lemma:
本节采用与文献[12] 类似的方法，推导混合初始状态（ $q_{k_0}, x_{k_0})$ ）可由区间 $k_0, k_1-1]$ （满足 $k_1-1 \leq k_0+K$ ）上的输出序列唯一确定的条件，即第一次离散切换发生前的可观测性条件。定义 $\kappa_i := k_{i+1}-k_i(i \geq0)$ 为“停留时间”，表示第 $i$ 次离散切换后系统在该离散状态下停留的时长。基于定义 2 和定义 3，可得如下引理：

Lemma 1 The hybrid initial state of a discrete-time linear hybrid system $H$ is observable if and only if it is distinguishable.
引理 1 离散时间线性混合系统 $H$ 的混合初始状态具有可观测性，当且仅当该混合初始状态具有可区分性。

Proof: The proof follows directly from Definition 2 and Definition 3. □
证明：该证明可由定义 2 和定义 3 直接推出。□

In order to check if the hybrid initial state is indistinguishable, we have to compute the covariance of the output sequence $y_{\kappa_{0}}$ on $k_{0}, k_{1}-1]$ . The output sequence starting from the hybrid initial state $q_{k_{0}}, x_{k_{0}})$ on $k_{0}, k_{1}-1]$ is
为了判断混合初始状态是否具有不可区分性，需计算区间 $k_0, k_1-1]$ 上输出序列 $y_{\kappa_0}$ 的协方差。从混合初始状态 $q_{k_0}, x_{k_0})$ 出发，区间 $k_0, k_1-1]$ 上的输出序列为：

$Y_{\kappa_{0}}(q_{k_{0}})=O_{\kappa_{0}}(q_{k_{0}})x_{k_{0}}+T_{\kappa_{0}}(q_{k_{0}})W_{\kappa_{0}}(q_{k_{0}})+V_{\kappa_{0}}(q_{k_{0}}) \tag{2}$

where
其中，
$\begin{array}{ll} {W_{{\kappa _0}}}({q_{{k_0}}}) & = {[{w_{{k_0}}}{({q_{{k_0}}})^T},{w_{{k_0} + 1}}{({q_{{k_0}}})^T}, \cdots ,{w_{{k_1} - 1}}{({q_{{k_0}}})^T}]^T},{V_{{\kappa _0}}}({q_{{k_0}}})\\[1em] & = {[{v_{{k_0}}}{({q_{{k_0}}})^T},{v_{{k_0} + 1}}{({q_{{k_0}}})^T}, \cdots ,{v_{{k_1} - 1}}{({q_{{k_0}}})^T}]^T} , {O_{{\kappa _0}}}({q_{{k_0}}}) \in {^{p{\kappa _0} \times n}} \end{array}$

is the extended observability matrix for the linear system in Eq.(1) [12] and $T_{\kappa_{0}}(q_{k_{0}})$ is a Toeplitz matrix.
是式 (1) 所示线性系统的扩展可观测性矩阵（文献[12]）； $T_{\kappa_0}(q_{k_0})$ 是托普利茨（Toeplitz）矩阵。

If $rank[O_{\kappa_{0}}(q_{k_{0}})]=n$ , i.e., the linear system in discrete state $q_{k_{0}}$ is observable and $\kappa_{0} \geq n$ , then a least-squares solution (which we denote by $\hat{x}_{k_{0}}(q_{k_{0}})$ ) to Eq.(2) can be determined uniquely, where
若 $rank[O_{\kappa_0}(q_{k_0})]=n$ （即离散状态 $q_{k_0}$ 对应的线性系统具有可观测性）且 $\kappa_0 \geq n$ ，则式 (2) 的最小二乘解（记为 $\hat{x}_{k_0}(q_{k_0})$ ）可唯一确定，即：

$\hat{x}_{k_{0}}(q_{k_{0}})=O_{\kappa_{0}}^{\dagger}(q_{k_{0}})\left[Y_{\kappa_{0}}(q_{k_{0}})-T_{\kappa_{0}}(q_{k_{0}})W_{\kappa_{0}}(q_{k_{0}})-V_{\kappa_{0}}(q_{k_{0}})\right] \tag{3}$

and
其中，

$O_{\kappa_{0}}^{\dagger}(q_{k_{0}})=(O_{\kappa_{0}}^{T}(q_{k_{0}}) O_{\kappa_{0}}(q_{k_{0}}))^{-1} O_{\kappa_{0}}^{T}(q_{k_{0}})$ .
The last two terms on the right hand side of Eq.(3) represent the estimation error due to the process noise and the measurement noise. Similarly, the output sequence from another hybrid initial state $q_{k_{0}}', x_{k_{0}}')$ over $k_{0}, k_{1}-1]$ is
式 (3) 右端后两项表示由过程噪声和测量噪声引起的估计误差。类似地，从另一混合初始状态 $q_{k_0}', x_{k_0}')$ 出发，区间 $k_0, k_1-1]$ 上的输出序列为：

$Y_{\kappa_{0}}(q_{k_{0}}')=O_{\kappa_{0}}(q_{k_{0}}')x_{k_{0}}'+T_{\kappa_{0}}(q_{k_{0}}')W_{\kappa_{0}}(q_{k_{0}}')+V_{\kappa_{0}}(q_{k_{0}}') \tag{4}$

From Lemma 1, in order that the hybrid initial state of a discrete-time stochastic linear hybrid system be observable, it should be distinguishable, i.e., the covariances of $Y_{\kappa_{0}}(q_{k_{0}})$ and $Y_{\kappa_{0}}(q_{k_{0}}')$ satisfy:
由引理 1 可知，离散时间随机线性混合系统的混合初始状态要具有可观测性，需满足可区分性，即 $Y_{\kappa_0}(q_{k_0})$ 与 $Y_{\kappa_0}(q_{k_0}')$ 的协方差需满足：

$E[Y_{\kappa_{0}}(q_{k_{0}})Y_{\kappa_{0}}(q_{k_{0}})^{T}] \neq E[Y_{\kappa_{0}}(q_{k_{0}}')Y_{\kappa_{0}}(q_{k_{0}}')^{T}] \tag{5}$

where
其中：

$\begin{aligned} E[Y_{\kappa_{0}}(q_{k_{0}})Y_{\kappa_{0}}(q_{k_{0}})^T] &= O_{\kappa_{0}}(q_{k_{0}})\Pi_{0}O_{\kappa_{0}}^T(q_{k_{0}}) + T_{\kappa_{0}}(q_{k_{0}})\rho(q_{k_{0}})I T_{\kappa_{0}}^T(q_{k_{0}}) + \sigma(q_{k_{0}})I \\ E[Y_{\kappa_{0}}(q_{k_{0}}')Y_{\kappa_{0}}(q_{k_{0}}')^T] &= O_{\kappa_{0}}(q_{k_{0}}')\Pi_{0}O_{\kappa_{0}}^T(q_{k_{0}}') + T_{\kappa_{0}}(q_{k_{0}}')\rho(q_{k_{0}}')I T_{\kappa_{0}}^T(q_{k_{0}}') + \sigma(q_{k_{0}}')I \end{aligned} \tag{6}$
Then, the discrete initial state can be uniquely determined from the covariance of the output sequence and the continuous initial state can also be uniquely determined using Eq.(3). In order to reduce the required $\kappa_{0}$ for observability (the sojourn time in the discrete state $q_{k_{0}}$ required for observability of the hybrid initial state), we define $\tau(q_{k})$ as the minimum integer which satisfies $rank[O_{\tau}(q_{k})]=n$ ( $\forall q_{k} \in \{1,2, \cdots, N\}$ ), and $\bar{\tau}=max_{q_{k} \in \{1,\cdots,N\}} \tau(q_{k})$ (similar to the joint observability index used in [12]). Then, we have the following condition for the observability of the hybrid initial state:
此时，离散初始状态可由输出序列的协方差唯一确定，而连续初始状态可通过式 (3) 唯一确定。为减小可观测性所需的 $\kappa_0$ （即混合初始状态可观测性所需的、系统在离散状态 $q_{k_0}$ 中的停留时间），定义 $\tau(q_k)$ 为满足 $rank[O_{\tau}(q_k)]=n$ （对所有 $q_k \in \{1,2,\cdots,N\}$ ）的最小整数，并定义 $\bar{\tau}=max_{q_k \in \{1,\cdots,N\}} \tau(q_k)$ （与文献[12] 中使用的联合可观测性指数类似）。由此，混合初始状态的可观测性条件如下：

Lemma 2 (Observability of the hybrid initial state) If $A(q_{k}), C(q_{k}))$ are observable for each $q_{k} \in \{1, \cdots, N\}$ , and $\kappa_{0} \geq \bar{\tau}$ , the hybrid initial state $q_{k_{0}}, x_{k_{0}})$ is observable if and only if
引理 2（混合初始状态的可观测性）若对所有 $q_k \in \{1,\cdots,N\}$ ， $A(q_k), C(q_k))$ 均具有可观测性，且 $\kappa_0 \geq \bar{\tau}$ ，则混合初始状态 $q_{k_0}, x_{k_0})$ 具有可观测性，当且仅当

$E[Y_{\kappa_{0}}(q_{k_{0}})Y_{\kappa_{0}}(q_{k_{0}})^T] \neq E[Y_{\kappa_{0}}(q_{k_{0}}')Y_{\kappa_{0}}(q_{k_{0}}')^T]$

for all $q_{k_{0}} \neq q_{k_{0}}' \in \{1, \cdots, N\}$ .
对所有 $q_{k_0} \neq q_{k_0}' \in \{1,\cdots,N\}$ ，满足：

Proof: Since the linear system in each discrete state is observable and $\kappa_{0} \geq \bar{\tau}$ , the initial continuous state can be uniquely determined using Eq.(3) if the initial discrete state is identified.
证明：由于每个离散状态对应的线性系统均具有可观测性，且 $\kappa_0 \geq \bar{\tau}$ ，因此若能识别出初始离散状态，即可通过式 (3) 唯一确定初始连续状态。

(If) Since the covariances of the output sequences for each discrete state are distinct, the initial discrete state is uniquely determined by checking the covariance of the output sequence.
（充分性）由于每个离散状态对应的输出序列协方差互不相同，通过检验输出序列的协方差可唯一确定初始离散状态。

(Only if) The proof follows directly from Definition 3. □
（必要性）该证明可由定义 3 直接推出。□

We show through the following simple example how a noise-free unobservable discrete-time linear hybrid system may be rendered observable, if each discrete state has different measurement noise covariances.
下面通过一个简单示例说明：对于无噪声时不可观测的离散时间线性混合系统，若各离散状态具有不同的测量噪声协方差，则系统可变为可观测系统。

Example: Consider a discrete-time linear hybrid system with two discrete states $q = 1$ and $q = 2$ , whose dynamics are given by:
示例：考虑含两个离散状态（ $q = 1$ 和 $q = 2$ ）的离散时间线性混合系统，其动力学方程如下：

$\begin{cases} x_{k+1} = x_k & (q=1,2) \\ y_k = c_1 x_k + v_1 & (q=1) \\ y_k = c_2 x_k + v_2 & (q=2) \end{cases}$

where $c_{1} \neq 0$ , $c_{2} \neq 0$ , and $\in \mathbb{N}$ . The covariance of the initial state is $E[x_{0} x_{0}^{T}]=\Pi_{0} \in \mathbb{R}^{+}$ . $v_{1}$ and $v_{2}$ are uncorrelated, zero-mean white Gaussian sequences with covariances $E[v_{1} v_{1}^{T}]=\sigma_{1} \neq 0$ and $E[v_{2} v_{2}^{T}]=\sigma_{2} \neq 0$ respectively.
其中， $c_1 \neq 0$ 、 $c_2 \neq 0$ ，且 $\in \mathbb{N}$ 。初始状态的协方差为 $E[x_0x_0^T]=\Pi_0 \in \mathbb{R}^+$ 。 $v_1$ 和 $v_2$ 为互不相关的零均值白高斯序列，其协方差分别为 $E[v_1v_1^T]=\sigma_1 \neq 0$ 和 $E[v_2v_2^T]=\sigma_2 \neq 0$ 。

If $v_{1}=v_{2}=0$ (noise-free case), the hybrid system is unobservable because two different hybrid initial states $q_{1}, x_{0})$ and $(q_{2}, \frac{c_{1}}{c_{2}} x_{0})$ generate the same output sequences [8]. However, if $v_{1}$ and $v_{2}$ are not identically zero and have different covariances ( $\sigma_1 \neq \sigma_2$ ), then we can uniquely determine the hybrid initial state.
若 $v_1 = v_2 = 0$ （无噪声情况），则该混合系统不可观测。因为两个不同的混合初始状态 $q_1, x_0)$ 和 $(q_2, \frac{c_1}{c_2}x_0)$ 会产生相同的输出序列（文献[8]）。然而，若 $v_1$ 和 $v_2$ 均非零且协方差不同（ $\sigma_1 \neq \sigma_2$ ），则可唯一确定混合初始状态。

If we consider the case in which the actual hybrid initial state is $q_{1}, x_{0})$ , the output and its covariance are:
当实际混合初始状态为 $q_1, x_0)$ 时，输出及其协方差为：

$c_1 x_0 + v_1, \quad E[yy^T] = \Pi_0 c_1 c_1^T + \sigma_1 \tag{7}$

Next, if the actual hybrid initial state is $(q_{2}, \frac{c_1}{c_2}x_0)$ , the output and its covariance are:
接下来，当实际混合初始状态为 $(q_2, \frac{c_1}{c_2}x_0)$ 时，输出及其协方差为：

$c_{2}\left(\frac{c_{1}}{c_{2}}x_{0}\right)+v_{2}, \quad E[yy^{T}] = \Pi_{0}c_{1}c_{1}^{T}+\sigma_{2} \tag{8}$

Since $\sigma_{1} \neq \sigma_{2}$ , we can determine the discrete initial state uniquely. For instance, if the output comes from $q_{1}$ , then the estimate of the initial state is $\hat{x}_{0}=x_{0}+\frac{v_{1}}{c_{1}}$ ; if it comes from $q_{2}$ , the estimate is $\hat{x}_{0}=\frac{c_1}{c_2}x_0 + \frac{v_2}{c_2}$ .
由于 $\sigma_1 \neq \sigma_2$ ，可唯一确定离散初始状态。例如，若输出来自 $q_1$ ，则初始状态估计值为 $\hat{x}_0 = x_0 + \frac{v_1}{c_1}$ ；若输出来自 $q_2$ ，则初始状态估计值为 $\hat{x}_0 = \frac{c_1}{c_2}x_0 + \frac{v_2}{c_2}$ 。

2.2 Observability of the discrete transition times

离散切换时间的可观测性

Lemma 2 gives the condition for the hybrid initial state to be observable, over a time interval up to, but not including the first transition. In this section, we focus without loss of generality on deriving the conditions under which the first discrete transition time $k_{1}$ can be uniquely determined from the output sequence $Y_{K}$ on $k_{0}, k_{0}+K]$ ; the times of the ensuing transitions $k_{i}(i \in \{2, ...\})$ can be computed in the same way [12]. We define observability of the first discrete transition time as follows:
引理 2 给出了混合初始状态在“至第一次切换前（不含第一次切换）”区间内的可观测性条件。本节不失一般性，推导第一次离散切换时间 $k_1$ 可由区间 $k_0, k_0+K]$ 上的输出序列 $Y_K$ 唯一确定的条件；后续切换时间 $k_i(i \in \{2,\dots\})$ 可采用相同方法确定（文献[12]）。第一次离散切换时间的可观测性定义如下：

Definition 4 (Observability of the first discrete transition time) The first discrete transition time of a discrete-time linear hybrid system $H$ is observable on $k_{0}, k_{0}+K]$ if it can be determined uniquely from the output sequence $Y_{K}=[y_{k_{0}}^{T} \cdots y_{k_{0}+K}^{T}]^{T}$ .
定义 4（第一次离散切换时间的可观测性）若第一次离散切换时间可由输出序列 $Y_K=[y_{k_0}^T \cdots y_{k_0+K}^T]^T$ 唯一确定，则称离散时间线性混合系统 $H$ 的第一次离散切换时间在区间 $k_0, k_0+K]$ 上具有可观测性。

If there is a discrete transition at time $k_{1}$ (i.e., $q_{k_1} \neq q_{k_1-1}$ ), the output at time $k_{1}$ and its covariance are
若在时刻 $k_1$ 发生离散切换（即 $q_{k_1} \neq q_{k_1-1}$ ），则时刻 $k_1$ 的输出及其协方差为：

$\begin{aligned} y_{k_{1}} &= C(q_{k_{1}})A(q_{k_{0}})^{k_{1}-k_{0}}x_{k_{0}} + C(q_{k_{1}})F_{\kappa_{0}}(q_{k_{0}})W_{\kappa_{0}}(q_{k_{0}}) + v_{k_{1}}(q_{k_{1}}) \\ E[y_{k_{1}}y_{k_{1}}^{T}] &= C(q_{k_{1}})A(q_{k_{0}})^{k_{1}-k_{0}}\Pi_{0}(A(q_{k_{0}})^{k_{1}-k_{0}})^{T}C(q_{k_{1}})^{T} \\ &\quad + \rho(q_{k_{0}})C(q_{k_{1}})F_{\kappa_{0}}(q_{k_{0}})F_{\kappa_{0}}(q_{k_{0}})^{T}C(q_{k_{1}})^{T} + \sigma(q_{k_{1}})I \end{aligned} \tag{9}$

where $F_{\kappa_{0}}(q_{k_{0}})$ is a matrix related to the process noise accumulation over $k_0, k_1-1]$ . If there is no state transition at time $k_{1}$ (i.e., $q_{k_1} = q_{k_1-1}$ ), the output at time $k_{1}$ and its covariance are:
其中， $F_{\kappa_0}(q_{k_0})$ 是与区间 $k_0, k_1-1]$ 上过程噪声累积相关的矩阵。若在时刻 $k_1$ 未发生状态切换（即 $q_{k_1} = q_{k_1-1}$ ），则时刻 $k_1$ 的输出及其协方差为：

$\begin{aligned} y_{k_{1}} &= C(q_{k_{0}})A(q_{k_{0}})^{k_{1}-k_{0}}x_{k_{0}} + C(q_{k_{0}})F_{\kappa_{0}}(q_{k_{0}})W_{\kappa_{0}}(q_{k_{0}}) + v_{k_{1}}(q_{k_{0}}) \\ E[y_{k_{1}}y_{k_{1}}^{T}] &= C(q_{k_{0}})A(q_{k_{0}})^{k_{1}-k_{0}}\Pi_{0}(A(q_{k_{0}})^{k_{1}-k_{0}})^{T}C(q_{k_{0}})^{T} \\ &\quad + \rho(q_{k_{0}})C(q_{k_{0}})F_{\kappa_{0}}(q_{k_{0}})F_{\kappa_{0}}(q_{k_{0}})^{T}C(q_{k_{0}})^{T} + \sigma(q_{k_{0}})I \end{aligned} \tag{10}$
$$

In order that the transition at time $k_{1}$ be observable, the covariances of $y_{k_{1}}$ in Eq.(9) and Eq.(10) should be different. Thus, the observability condition of the first discrete transition time is:
要使时刻 $k_1$ 的切换具有可观测性，式 (9) 和式 (10) 中 $y_{k_1}$ 的协方差需互不相同。因此，第一次离散切换时间的可观测性条件如下：

Lemma 3 (Observability of the first discrete transition time) The first discrete transition time is observable if and only if
引理 3（第一次离散切换时间的可观测性）第一次离散切换时间具有可观测性，当且仅当

$\begin{aligned} &C(q_{k_{1}})A(q_{k_{0}})^{k_{1}-k_{0}}\Pi_{0}(A(q_{k_{0}})^{k_{1}-k_{0}})^T C(q_{k_{1}})^T + \rho(q_{k_{0}})C(q_{k_{1}})F_{\kappa_{0}}(q_{k_{0}})F_{\kappa_{0}}(q_{k_{0}})^T C(q_{k_{1}})^T + \sigma(q_{k_{1}})I \\ \neq &C(q_{k_{0}})A(q_{k_{0}})^{k_{1}-k_{0}}\Pi_{0}(A(q_{k_{0}})^{k_{1}-k_{0}})^T C(q_{k_{0}})^T + \rho(q_{k_{0}})C(q_{k_{0}})F_{\kappa_{0}}(q_{k_{0}})F_{\kappa_{0}}(q_{k_{0}})^T C(q_{k_{0}})^T + \sigma(q_{k_{0}})I \end{aligned}$

for all $q_{k} \neq q_{k}' \in \{1, \cdots, N\}$ .
对所有 $q_k \neq q_k' \in \{1,\cdots,N\}$ ，满足：

Proof: The proof follows by construction. □
证明：通过构造法可证。□

Therefore, from Lemma 2 and Lemma 3, the hybrid initial state and the first discrete transition time can be uniquely determined. The remaining state trajectories can be determined by repeating the procedure. For $k_{i}(i \geq1)$ , the $\hat{x}_{k_{i}}$ will be given from the initial state estimate [12]. Thus, we have the following observability condition:
综上，由引理 2 和引理 3 可唯一确定混合初始状态和第一次离散切换时间。重复该过程可确定剩余的状态轨迹。对于 $k_i(i \geq1)$ ， $\hat{x}_{k_i}$ 可由初始状态估计值得到（文献[12]）。因此，离散时间线性混合系统的可观测性条件如下：

Theorem 1 A discrete-time linear hybrid system $H$ is observable if and only if it satisfies Lemma 2 and Lemma 3.
定理 1 离散时间线性混合系统 $H$ 具有可观测性，当且仅当该系统满足引理 2 和引理 3 的条件。

Proof: The proof follows directly from Lemma 2 and Lemma 3. □
证明：该证明可由引理 2 和引理 3 直接推出。□

This test needs the operations of multiplication and addition of matrices which are system parameters and noise covariances; the computation is straightforward with computational complexity depending on data size.
该检验过程需对系统参数矩阵和噪声协方差矩阵进行乘法与加法运算；计算过程简单直接，计算复杂度取决于数据规模。