As with general linear systems, numerous concepts of stability have been defined for switched systems. In this paper, we use the definition of asymptotic stability for switched systems.

###
**Definition 1**

The switched system (10) is **asymptotically stable** if there exists some *δ*>0 such that ∥**x**^{∗}_{0}∥<*δ* implies ∀*k*,∥**x**^{∗}_{k}∥<*ε*\(\left (\!or \lim \limits _{k \rightarrow \infty } {\left \|{{\mathbf {x}^{*}}_{k}}\right \|} = 0\right)\) for all solutions **x**^{∗}_{k} of the system.

###
**Remark 1**

A switched system is marginally stable if it is neither asymptotically stable nor unstable.

Note that asymptotic stability gives a stronger condition for \(\lim \limits _{k \rightarrow \infty } {\left \|{{\mathbf {x}^{*}}_{k}}\right \|} < \infty \) since it not only requires convergence but requires convergence to the origin. The definition of marginal stability implies that the state trajectory is bounded but not necessarily convergent, which is equivalent to \(\lim \limits _{k \rightarrow \infty } {\left \|{{\mathbf {x}^{*}}_{k}}\right \|} < \infty \). Therefore, conditions for asymptotic stability are sufficient to guarantee \(\lim \limits _{k \rightarrow \infty } {\left \|{{\mathbf {x}^{*}}_{k}}\right \|} < \infty \). Also, because asymptotic stability is closely related to the stability of the corresponding autonomous system, it is typical to consider the stability of the autonomous system first. For the transformed switched system in (10), the corresponding autonomous system is:

$$ {\mathbf{x}^{*}}_{k} = {\mathbf{F}}_{{{q}}_{k}} {\mathbf{x}^{*}}_{k-1}. $$

(11)

Among the existing research works, there are primarily two approaches to address the stability of the autonomous switched system in (11). One approach involves solving the generalized/joint spectral radius (JSR) of a bounded set of matrices [39]. As proved in [40], testing whether the JSR of a bounded set of matrices is less than or equal to 1 is computationally undecidable. While the exact computation of JSR is Turing-undecidable in general, the approximation of JSR is an active area of research. The other approach is primarily built on the well-known Lyapunov theory. Specifically, it has been proved that the existence of a common quadratic Lyapunov function (CQLF) provides a sufficient condition for the asymptotic stability of the switched system in (11) which also implies the JSR of the bounded set of matrices is less than 1. Therefore, without dwelling on the approaches that involve approximations of JSR, our main results are built on Lyapunov theory. The analysis procedure is summarized in Fig. 2.

We use to denote the subsystem corresponding to mode *q*. The autonomous switched system (11) switches between for all *q*. The following lemma is introduced in [41].

###
**Lemma 1**

The switched system (11) is asymptotically stable under arbitrary switching signal if:

(i). \({\rho ({{\mathbf {F}}_{{{q}}}})} < 1, \forall {{q}} \in {\mathcal {Q}}\);

(ii). \(\exists \mathbf {P} = \mathbf {P}^{\prime } {\succ } 0, \quad {\mathbf {F}}^{\prime }_{{q}} \mathbf {P} {\mathbf {F}}_{{q}} - \mathbf {P} {\prec } 0\).

Condition (i) in Lemma 1 implies asymptotic stability of every subsystem and condition (ii) is the existence of common Lyapunov quadratic function (CQLF). Also, it is worth pointing out that the stability for each subsystem does not imply asymptotic stability of the switched system [42]. The converse does not always hold either. As discussed in [43], by choosing the switching signal carefully, the switched system can be made asymptotically stable even though the subsystem is not. In the following, we first study conditions such that

$$ {\rho({{\mathbf{F}}_{{q}}})} < 1, \forall {{q}} \in {\mathcal{Q}} $$

(12)

holds, i.e., each subsystem is asymptotically stable.

### 4.1 Stability of subsystem

By definition, **F**_{q} is composed of convex combination of matrices as:

$$\begin{array}{*{20}l} {\mathbf{F}}_{{{q}}} = \sum\limits_{{\mathsf{i}} = 1}^{d} {\lambda}_{{\mathsf{i}},{{q}}} {\boldsymbol{\Lambda}}_{{\mathsf{i}}} \end{array} $$

The task of checking spectral radius of summation of matrices is not trivial in general. If two matrices are commutable, i.e., **A****B**=**B****A**, then *ρ*(**A**+**B**)≤*ρ*(**A**)+*ρ*(**B**) [44]. If all the matrices are non-negative (element-wise), [45] proves that spectral radius is strictly convex. But all the mentioned results cannot be extended to general cases. Therefore, directly checking the spectral radius is not feasible. An alternative approach is built on Lyapunov theory which demonstrates the relationship between a quadratic Lyapunov function (QLF) and the spectral radius of system matrices.

###
**Lemma 2**

The following statements are equivalent:

(i) if there exists a positive definite matrix **P** such that \({\mathbf {F}}_{{q}}^{\prime } \mathbf {P} {\mathbf {F}}_{{q}} - \mathbf {P} {\prec } 0\);

(ii) *ρ*(**F**_{q})<1;

(iii) the subsystem is asymptotically stable.

We first illustrate a property related to the spectral radius of *Λ*_{i} in the following lemma.

###
**Lemma 3**

For a switched system defined in (1), if \(({\mathbf {A}}_{\mathsf {i}},{\mathbf {B}}_{\mathsf {i}} {\mathbf {Q}} {\mathbf {B}}^{\prime }_{\mathsf {i}})\) is controllable and (**C**_{i},**A**_{i}) is observable for all \({\mathsf {i}} \in {\mathcal {Q}}\), then \(\forall {\mathsf {i}} \in {\mathcal {Q}}\), *ρ*(*Λ*_{i})<1.

###
*Proof*

From the definition,

$${\boldsymbol{\Lambda}}_{{\mathsf{i}}} = {\mathbf{A}}_{{\mathsf{i}}} - {\mathbf{K}}_{{\mathsf{i}}}{\mathbf{C}}_{{\mathsf{i}}}{\mathbf{A}}_{{\mathsf{i}}} = ({\mathbf{I}} - {\mathbf{K}}_{{\mathsf{i}}}{\mathbf{C}}_{{\mathsf{i}}}){\mathbf{A}}_{{\mathsf{i}}}. $$

For any Kalman filter, the observer gain corresponding to mode i is defined as \(\mathbf {L}_{{\mathsf {i}}} = {\mathbf {A}}_{\mathsf {i}} {\mathbf {M}}_{\mathsf {i}} {\mathbf {C}}^{\prime }_{\mathsf {i}} {{\left ({\mathbf {C}}_{\mathsf {i}}{\mathbf {M}}_{\mathsf {i}} {\mathbf {C}}^{\prime }_{\mathsf {i}} + {\mathbf {R}}\right)}^{-1}}\). Here, **M**_{i} is the steady error covariance related to steady Kalman gain **K**_{i}. Given that \(\left ({\mathbf {A}}_{\mathsf {i}},{\mathbf {B}}_{\mathsf {i}} {\mathbf {Q}} {\mathbf {B}}^{\prime }_{\mathsf {i}}\right)\) is controllable and (**C**_{i},**A**_{i}) is observable for all \({\mathsf {i}} \in {\mathcal {Q}}\), the closed-loop dynamics **A**_{i}−**L**_{i}**C**_{i} is stable. That is,

$${\rho({{\mathbf{A}}_{\mathsf{i}} - \mathbf{L}_{{\mathsf{i}}} {\mathbf{C}}_{\mathsf{i}}})} < 1. $$

Rewrite it as:

$${\mathbf{A}}_{\mathsf{i}} - \mathbf{L}_{{\mathsf{i}}} {\mathbf{C}}_{\mathsf{i}} = {\mathbf{A}}_{\mathsf{i}} - {\mathbf{A}}_{\mathsf{i}} {\mathbf{K}}_{{\mathsf{i}}} {\mathbf{C}}_{\mathsf{i}} = {\mathbf{A}}_{\mathsf{i}} ({\mathbf{I}} - {\mathbf{K}}_{{\mathsf{i}}} {\mathbf{C}}_{\mathsf{i}}). $$

From commutativity property of spectral radius,

$${\rho({{\mathbf{A}}_{\mathsf{i}} - \mathbf{L}_{{\mathsf{i}}} {\mathbf{C}}_{\mathsf{i}}})} = {\rho({{\boldsymbol{\Lambda}}_{{\mathsf{i}}}})} < 1. $$

□

With the fact that all the matrices *Λ*_{i} are stable, we have the following theorem.

###
**Lemma 4**

If there is only one *λ*_{i,q}>0 for each \({{q}} \in {\mathcal {Q}}\), then the subsystem is asymptotically stable for all \({{q}} \in {\mathcal {Q}}\).

###
*Proof*

Let *k*_{q} be the index indicating the non-zero \({\lambda }_{k_{{q}}, {{q}}}\) for each \({{q}} \in {\mathcal {Q}}\); note that *k*_{q} also takes value in \({\mathcal {Q}}\). Based on the property of random variable **Ξ**_{t} discussed in Section 3, \({\lambda }_{k_{{q}}, {{q}}} = 1\). Therefore, we have

$${\mathbf{F}}_{{q}} = {\boldsymbol{\Lambda}}_{k_{{q}}}, \forall {{q}}. $$

From Lemma 3, it is straightforward to conclude that \({\rho ({{\mathbf {F}}_{{q}}})} = {\rho ({{\boldsymbol {\Lambda }}_{k_{{q}}}})} < 1, \forall {{q}} \in {\mathcal {Q}}\). According to Lemma 2, all the subsystems are asymptotically stable. □

Following the notation in proof of Lemma 4, we use *k*_{q} to denote the index indicating the non-zero \({\lambda }_{k_{{q}}, {{q}}}\) for each \({{q}} \in {\mathcal {Q}}\). Note that *k*_{q} is not necessarily equal to *q*. As *ρ*(*Λ*_{q})<1 for all *q*, even though the probability of mode mismatch between *q* and mode *k*_{q} is 1 (i.e., the mode mismatches always happen), all the subsystems are still stable. The physical interpretation behind the result seems inconsistent. However, this result is only related to the stability of the autonomous subsystem but not the complete switched system. In fact, if we take a close look at our system in (10), the choice of \({\lambda }_{k_{{q}},{{q}}}\) will have impact on the input matrix **G**_{q}. We will discuss this result in Section 4.3.

Lemma 4 gives a non-trivial condition such that the stability of each subsystem is guaranteed. However, the condition that only one *λ*_{i,q}>0 is not generally realistic since it eliminates the randomness associated with errors. The next theorem is built on the concept of CQLF and it is applicable for broader choices of *λ*_{i,q}.

###
**Theorem 2**

If for all \({\mathsf {i}} \in {\mathcal {Q}}\), *Λ*_{i} share a common quadratic Lyapunov function. That is, if there exists a positive definite matrix \(\mathbf {P} \in {\mathbb {R}}^{n \times n}\) such that

$$\begin{array}{*{20}l} {\boldsymbol{\Lambda}}^{\prime}_{\mathsf{i}} \mathbf{P} {\boldsymbol{\Lambda}}_{\mathsf{i}} - \mathbf{P} {\prec} 0, \forall {\mathsf{i}} \in {\mathcal{Q}}, \end{array} $$

(13)

then every subsystem \(\forall {{q}} \in {\mathcal {Q}}\) is asymptotically stable for all choices of *λ*_{i,q}.

###
*Proof*

$$\begin{array}{*{20}l} {\boldsymbol{\Lambda}}^{\prime}_{\mathsf{i}} \mathbf{P} {\boldsymbol{\Lambda}}_{\mathsf{i}} - \mathbf{P} {\prec} 0 \stackrel{(a)}{\Longleftrightarrow} \mathbf{P} - {\boldsymbol{\Lambda}}_{\mathsf{i}} \mathbf{P} {\boldsymbol{\Lambda}}^{\prime}_{\mathsf{i}} {\succ} 0 \stackrel{(b)}{\Longleftrightarrow} \left[\begin{array}{ll} \mathbf{P} & {\boldsymbol{\Lambda}}_{\mathsf{i}} \\ {\boldsymbol{\Lambda}}^{\prime}_{\mathsf{i}} & \mathbf{P}^{-1} \end{array}\right] {\succ} 0. \end{array} $$

(a) is due to the fact that **P** is positive definite and (b) is a result of Schur decomposition. According to Lemma 2, in order to prove is asymptotically stable for all *q*, we need to find if there exists some positive definite matrix **P**_{q} for each *q* such that \(\mathbf {P}_{{q}} - {\mathbf {F}}_{{q}} \mathbf {P}_{{q}} {\mathbf {F}}^{\prime }_{{q}} {\succ } 0\).

Since \(\mathbf {P} - {\boldsymbol {\Lambda }}_{\mathsf {i}} \mathbf {P} {\boldsymbol {\Lambda }}^{\prime }_{\mathsf {i}} {\succ } 0\), therefore, \(\mathbf {P} - {\lambda }_{{\mathsf {i}},{{q}}}^{2} {\boldsymbol {\Lambda }}_{\mathsf {i}} \mathbf {P} {\boldsymbol {\Lambda }}^{\prime }_{\mathsf {i}} {\succ } 0\) for 0≤*λ*_{i,q}≤1. For all \({{q}} \in {\mathcal {Q}}\), we have:

$$\begin{array}{*{20}l} \left[\begin{array}{ll} \mathbf{P} & {\lambda}_{{\mathsf{i}},{{q}}} {\boldsymbol{\Lambda}}_{{\mathsf{i}}} \\ {\lambda}_{{\mathsf{i}},{{q}}} {\boldsymbol{\Lambda}}^{\prime}_{{\mathsf{i}}} & {{\mathbf{P}}^{-1}} \end{array}\right]\! {\succ} 0 &\Longrightarrow \sum\limits_{{\mathsf{i}} = 1}^{d} \left[\begin{array}{ll} \mathbf{P} & {\lambda}_{{\mathsf{i}},{{q}}} {\boldsymbol{\Lambda}}_{{\mathsf{i}}} \\ {\lambda}_{{\mathsf{i}},{{q}}} {\boldsymbol{\Lambda}}^{\prime}_{{\mathsf{i}}} & {{\mathbf{P}}^{-1}} \end{array}\right] {\succ} 0 \\ & \Longrightarrow \left[\begin{array}{ll} \mathbf{P} & \sum\limits_{{\mathsf{i}} = 1}^{d} {\lambda}_{{\mathsf{i}},{{q}}} {\boldsymbol{\Lambda}}_{{\mathsf{i}}} \\ \sum\limits_{{\mathsf{i}} = 1}^{d} {\lambda}_{{\mathsf{i}},{{q}}} {\boldsymbol{\Lambda}}^{\prime}_{{\mathsf{i}}} & {{\mathbf{P}}^{-1}} \end{array}\right] {\succ} 0 \\ & \Longrightarrow \left[\!\begin{array}{ll} \mathbf{P} & {\mathbf{F}}_{{q}} \\ {\mathbf{F}}^{\prime}_{{q}} & {{\mathbf{P}}^{-1}} \end{array}\!\right] {\succ} 0 \Longrightarrow \mathbf{P} - {\mathbf{F}}_{{q}} \mathbf{P} {\mathbf{F}}^{\prime}_{{q}} {\succ} 0. \end{array} $$

By taking **P**_{q}=**P**, we proved that there exists positive definite matrix **P**_{q} for each *q* such that \(\mathbf {P}_{{q}} - {\mathbf {F}}_{{q}} \mathbf {P}_{{q}} {\mathbf {F}}^{\prime }_{{q}} {\succ } 0\). Therefore, every subsystem \(\forall {{q}} \in {\mathcal {Q}}\) is asymptotically stable for all choices of *λ*_{i,q}. □

As presented in Lemma 1, there are two conditions that can guarantee the stability of the autonomous switched system. Condition (i) is related to the stability of each subsystem and we have developed Lemma 4 and Theorem 2 determine *ρ*(**F**_{q})<1 for all \({{q}} \in {\mathcal {Q}}\). To complete the stability analysis for switched autonomous system in (11), we will study conditions such that constraint (ii) in Lemma 1 is satisfied in the following subsection.

### 4.2 Stability of switched autonomous systems

We have introduced the concept of CQLF in Lemma 1. For stability analysis and CQLF conditions, [46] provides an excellent survey on the progress that have been made in this research area. In general, determining algebraic conditions (on the subsystems’ state matrices) for the existence of a CQLF remains an open task. For switched system with only two modes, [47] derives a necessary and sufficient condition for the existence of a CQLF for a second-order (two dimensional) continuous-time switched system with two modes while a similar approach is proposed in [41] by considering a discrete-time system. Their approach is based on the stability of the matrix pencil constructed using the state matrices corresponding to the two modes. While the matrix pencil presents a different perspective on the CQLF existence problem, it also lacks an analytical solution.

In this work, the switched system in (11) contains unknown variable *λ*_{i,q} in the subsystem matrices **F**_{q}. Due to the unknown values in **F**_{q} and lack of algebraic solutions, we cannot directly solve the LMI conditions nor derive constraints on *λ*_{i,q} such that the existence of CQLF for **F**_{q} is guaranteed. In the following, we propose to establish a relationship between the existence of CQLF for *Λ*_{i} and **F**_{q} and then obtain conditions for stability of switched system (11) regardless of the choice of *λ*_{i,q}.

###
**Theorem 3**

If there exists a CQLF for \({\boldsymbol {\Lambda }}_{\mathsf {i}}, \forall {\mathsf {i}} \in {\mathcal {Q}}\), then there exists a CQLF for \({\mathbf {F}}_{{q}}, \forall {{q}} \in {\mathcal {Q}}\). As a consequence, the switched system (11) is asymptotically stable under arbitrary switching signal.

###
*Proof*

We will use the similar approach as shown in the proof of Theorem 2. If there exists a CQLF for *Λ*_{i}, we know that there exists a positive definite matrix \(\mathbf {P} \in {\mathbb {R}}^{n \times n}\) such that

$$\begin{array}{*{20}l} {\boldsymbol{\Lambda}}^{\prime}_{\mathsf{i}} \mathbf{P} {\boldsymbol{\Lambda}}_{\mathsf{i}} - \mathbf{P} {\prec} 0, \forall {\mathsf{i}} \in {\mathcal{Q}}. \end{array} $$

As a result of Theorem 2, for all \({{q}} \in {\mathcal {Q}}\), we have

$$\begin{array}{*{20}l} \sum\limits_{{\mathsf{i}} = 1}^{d} \!\left[\!\begin{array}{ll} \mathbf{P} & {\lambda}_{{\mathsf{i}},{{q}}} {\boldsymbol{\Lambda}}_{{\mathsf{i}}} \\ {\lambda}_{{\mathsf{i}},{{q}}} {\boldsymbol{\Lambda}}^{\prime}_{{\mathsf{i}}} & {{\mathbf{P}}^{-1}} \end{array}\!\right]\! {\succ} 0 \!\Longrightarrow\! \!\left[\!\begin{array}{ll} \mathbf{P} &\! {\mathbf{F}}_{{q}} \\ {\mathbf{F}}_{{q}} &\! {{\mathbf{P}}^{-1}} \end{array}\!\right]\! {\succ} 0\ \!\!\Longrightarrow\! {\mathbf{F}}^{\prime}_{{q}} \mathbf{P} {\mathbf{F}}_{{q}} - \mathbf{P} {\prec} 0. \end{array} $$

Therefore, there exists a CQLF for \({\mathbf {F}}_{{q}}, \forall {{q}} \in {\mathcal {Q}}\). From Lemma 1, the switched system (11) is asymptotically stable under arbitrary switching signal. □

The condition derived in Theorem 3 is only based on all the matrices *Λ*_{i} which can be determined given the system matrix. The LMI condition can be easily checked in practice via an LMI solver alleviating the lack of an analytical solution. As illustrated in Fig. 2, we have completed the discussion for the stability of autonomous switched system (11) thus far. In the following, we will consider stability of the complete transformed switched system (10) including the input term.

### 4.3 Bounded-input bounded-output (BIBO) stability

For the transformed switched system in (10), we introduce the notion of BIBO stability that has been defined in [48].

###
**Definition 2**

The system in (10) is **BIBO stable** if there exists a positive constant *η* such that for any essentially bounded input signal **u**, the continuous state **x**^{∗} satisfies

$$\begin{array}{*{20}l} \sup\limits_{k \geq 0} {\left\|{{\mathbf{x}^{*}}_{k}}\right\|} \leq \eta \sup\limits_{k \geq 0} {\left\|{{\mathbf{u}}_{k}}\right\|}. \end{array} $$

According to this definition, an input signal cannot be amplified by a factor greater than some finite constant *η* after passing through the system if the system is BIBO stable. It has been proven that if the corresponding autonomous switched system (11) is asymptotically stable, then the input-output system (10) is BIBO stable provided the input matrix **G**_{q} is uniformly bounded in time for all *q* [49]. This in fact is the case when the system switches between a finite family of matrices. In our transformed switched system, the input signal **u**_{k}=**x**_{k}, where **x**_{k} is the continuous state of original system (1). Therefore, depending on the stability of (1), **u**_{k} can be either bounded or unbounded. Therefore, we should consider two different scenarios based on the boundedness of **u**_{k} in the following discussions.

**Scenario 1: Original system in** (1) **is not asymptotically stable**

If the original system in (1) is unstable, then sup*k*≥0∥**u**_{k}∥= sup*k*≥0∥**x**_{k}∥=*∞*. Since **u**_{k} is an *n*-dimensional vector, when **u**_{k} is unbounded, at least one of the elements in the vector is unbounded. We refer to those elements as unstable components and these components are collected in the set \({\mathcal {I}}\):

$${\mathcal{I}} = \left\{ i: \enskip \sup \limits_{k \geq 0} {\mathbf{u}}^{[i]}_{k} = \infty \right\}. $$

For this situation, if the columns of **G**_{q} corresponding to those unstable components of **u**_{k} are 0, then the boundedness of sup*j*≥0,*q*∥**G**_{q}**u**_{j}∥ is guaranteed. The process of finding the stable region for each probability of mode mismatch error is summarized in Algorithm 2:

Generally, *λ*_{i,q}=1 for i=*q* should always be a solution of Algorithm 2 because of *Λ*_{i}=*Γ*_{i,q} for i=*q*. Furthermore, this condition along with the result of Lemma 4 indicate that *λ*_{i,q}=1 for i=*q* not only guarantees stability of subsystem but also BIBO stability of the switched system in (10). By definition, *λ*_{i,q} represents the probability that true mode is *q* while estimated mode is i. *λ*_{i,q}=1 for i=*q* meaning that there is no mode mismatch error. Therefore, the convergence of **x**^{∗}_{k} (i.e., the bias generated from mode-based Kalman filter) is reasonable. Besides the trivial solution, Algorithm 2 also gives a less conservative result. For those unstable components in the original SHS, if the difference of *Λ*_{i}−*Γ*_{i,q} at the column corresponding to the unstable components are all 0, the mode-based Kalman filter is still tolerant of the mode mismatch between i and *q*.

**Scenario 2: Original system in** (1) **is asymptotically stable**

If the original system in (1) is asymptotically stable, then the continuous state **x**_{k} (i.e., **u**_{k} in the transformed switched system) is bounded. Since linear transformations of a vector is a bounded operator in Euclidean space, for a bounded vector **u**, **G****u** is bounded. For this situation, we are interested in minimizing the upper bound of ∥**x**^{∗}_{k}∥. From the definition of BIBO stability, we can write

$$ {\left\|{{\mathbf{x}^{*}}_{k}}\right\|} \leq \eta \sup \limits_{k \geq 0,{{q}}} {\left\|{{\mathbf{G}}_{{q}} {\mathbf{u}}_{k}}\right\|} \stackrel{(a)}{\leq} \eta \max\limits_{{{q}}} {\left\|{{\mathbf{G}}_{{q}}}\right\|} \sup \limits_{j \geq 0} {\left\|{{\mathbf{u}}_{j}}\right\|}, $$

(14)

where *η* and sup*j*≥0∥**u**_{j}∥ are fixed constant for a given system and **G**_{q} is related to the unknown variable *λ*_{i,q}. The equality in (a) holds if and only if each row of **G**_{q} is linearly dependent of **u**_{k} for all *q*,*k*. In this framework, we seek to address the following questions:

(1) Given the probability of mode mismatch is \({\mathcal {P}}\), i.e., \(\sum _{\substack {{\mathsf {i}} = 1 \\ {\mathsf {i}} \neq {{q}}}}^{d} {\lambda }_{{\mathsf {i}},{{q}}} = {\mathcal {P}}, \forall {{q}}\), what is the lowest upper bound of ∥**x**^{∗}_{k}∥?

(2) Given a certain upper bound \({\mathcal {B}}\) of ∥**x**^{∗}_{k}∥, what is the largest tolerant region for mode mismatch probability \({\mathcal {P}}\) that will guarantee that \({\mathcal {B}}\) is achievable?

The following theorem is developed to answer the first question.

###
**Theorem 4**

Given the probability of mode mismatch \({\mathcal {P}} \neq 0\) and the original system in (1) is asymptotically stable, the lowest upper bound of ∥**x**^{∗}_{k}∥ that can be achieved is:

$$\begin{array}{*{20}l} {\left\|{{\mathbf{x}^{*}}_{k}}\right\|} \leq \eta \cdot {\mathcal{P}} \cdot \sup \limits_{j \geq 0} {\left\|{{\mathbf{u}}_{j}}\right\|} \cdot \max\limits_{{{q}}} \min \limits_{{\mathsf{i}}, {\mathsf{i}} \neq {{q}}} {\left\|{{\boldsymbol{\Lambda}}_{\mathsf{i}} - {\boldsymbol{\Gamma}}_{{\mathsf{i}},{{q}}}}\right\|}. \end{array} $$

###
*Proof*

From the definition of **G**_{q},

$$\begin{array}{*{20}l} {\left\|{{\mathbf{G}}_{{q}}}\right\|} &= {\left\|{\sum\limits_{{\mathsf{i}} = 1}^{d} {\lambda}_{{\mathsf{i}},{{q}}} ({\boldsymbol{\Lambda}}_{\mathsf{i}} - {\boldsymbol{\Gamma}}_{{\mathsf{i}},{{q}}})}\right\|} = {\left\|{\sum_{\substack{{\mathsf{i}} = 1 \\ {\mathsf{i}} \neq {{q}}}}^{d} {\lambda}_{{\mathsf{i}},{{q}}} ({\boldsymbol{\Lambda}}_{\mathsf{i}} - {\boldsymbol{\Gamma}}_{{\mathsf{i}},{{q}}})}\right\|} \\ &\leq \sum_{\substack{{\mathsf{i}} = 1 \\ {\mathsf{i}} \neq {{q}}}}^{d} {\lambda}_{{\mathsf{i}},{{q}}} {\left\|{{\boldsymbol{\Lambda}}_{\mathsf{i}} - {\boldsymbol{\Gamma}}_{{\mathsf{i}},{{q}}}}\right\|}. \end{array} $$

(15)

With the constraint that \(\sum _{\substack {{\mathsf {i}} = 1 \\ {\mathsf {i}} \neq {{q}}}}^{d} {\lambda }_{{\mathsf {i}},{{q}}} = {\mathcal {P}}\), we have:

$$\begin{array}{*{20}l} \min \limits_{{\lambda}_{{\mathsf{i}},{{q}}}} \sum_{\substack{{\mathsf{i}} = 1 \\ {\mathsf{i}} \neq {{q}}}}^{d} {\lambda}_{{\mathsf{i}},{{q}}} {\left\|{{\boldsymbol{\Lambda}}_{\mathsf{i}} - {\boldsymbol{\Gamma}}_{{\mathsf{i}},{{q}}}}\right\|} = {\mathcal{P}} \min \limits_{{\mathsf{i}}, {\mathsf{i}} \neq {{q}}} {\left\|{{\boldsymbol{\Lambda}}_{\mathsf{i}} - {\boldsymbol{\Gamma}}_{{\mathsf{i}},{{q}}}}\right\|}. \end{array} $$

(16)

From Eq. (14), we have the lowest bound of ∥**x**^{∗}_{k}∥ as a function of ∥**G**_{q}∥. Given the constraint on mode mismatch probability and results of (15) and (16), we get the lowest upper bound of ∥**x**^{∗}_{k}∥ that can be reached is:

$$\begin{array}{*{20}l} {\left\|{{\mathbf{x}^{*}}_{k}}\right\|} \leq \eta \cdot {\mathcal{P}} \cdot \sup \limits_{j \geq 0} {\left\|{{\mathbf{u}}_{j}}\right\|} \cdot \max\limits_{{{q}}} \min \limits_{{\mathsf{i}}, {\mathsf{i}} \neq {{q}}} {\left\|{{\boldsymbol{\Lambda}}_{\mathsf{i}} - {\boldsymbol{\Gamma}}_{{\mathsf{i}},{{q}}}}\right\|}. \end{array} $$

□

To assist in the analysis for the second question, we first define an auxiliary function \({\phi }: {\mathbb {R}}^{d-1} \rightarrow {\mathbb {R}}\) as:

$$\begin{array}{*{20}l} {\phi}({\pmb{\upsilon}}) = \max\limits_{{{q}}} {\left\|{\sum\limits_{i = 1}^{d-1} {\pmb{\upsilon}}^{[i]} \mathbf{S}_{i,{{q}}}}\right\|}, {\pmb{\upsilon}} \in {\mathbb{R}}^{d-1} \end{array} $$

where \(\mathbf {S}_{i,{{q}}} \in {\mathbb {R}}^{n \times n}\) is a series of known matrices for a given *q*. The following lemma illustrates the convexity of this function.

###
**Lemma 5**

*ϕ*(*υ*) is a convex function respect to *υ*.

###
*Proof*

In order prove that *ϕ*(*υ*) is a convex function respect to *υ*, we want to show that for all \({\pmb {\upsilon }}, {\pmb {\nu }} \in {\mathbb {R}}^{d-1}\), and *θ* with 0≤*θ*≤1, *ϕ*(*θ**υ*+(1−*θ*)*ν*)≤*θ**ϕ*(*υ*)+(1−*θ*)*ϕ*(*ν*). We have

$$\begin{array}{*{20}l} {\phi}(\theta {\pmb{\upsilon}} + (1-\theta) {\pmb{\nu}}) & = \max\limits_{{{q}}} {\left\|{\sum\limits_{i = 1}^{d-1} (\theta {\pmb{\upsilon}} + (1-\theta) {\pmb{\nu}})^{[i]} \mathbf{S}_{i,{{q}}}}\right\|} \\ & = \max\limits_{{{q}}} {\left\|{\theta \sum\limits_{i = 1}^{d-1} {\pmb{\upsilon}}^{[i]} \mathbf{S}_{i,{{q}}} + (1 - \theta) \sum\limits_{i = 1}^{d-1} {\pmb{\nu}}^{[i]} \mathbf{S}_{i,{{q}}}}\right\|} \\ & \leq \max\limits_{{{q}}} {\left\|{\theta \sum\limits_{i = 1}^{d-1} {\pmb{\upsilon}}^{[i]} \mathbf{S}_{i,{{q}}} + (1 - \theta) \sum\limits_{i = 1}^{d-1} {\pmb{\nu}}^{[i]} \mathbf{S}_{i,{{q}}}}\right\|} \\ & \leq \theta \max\limits_{{{q}}} {\left\|{\sum\limits_{i = 1}^{d-1} {\pmb{\upsilon}}^{[i]} \mathbf{S}_{i,{{q}}}}\right\|} + (1 - \theta) \max\limits_{{{q}}} {\left\|{\sum\limits_{i = 1}^{d-1} {\pmb{\nu}}^{[i]} \mathbf{S}_{i,{{q}}}}\right\|} \\ &= \theta {\phi}({\pmb{\upsilon}}) + (1-\theta) {\phi}({\pmb{\nu}}). \end{array} $$

Therefore *ϕ*(*υ*) is a convex function on *υ*. □

Recall that the second question is to derive the largest tolerant region for mode mismatch probability \({\mathcal {P}}\) such that an upper bound \({\mathcal {B}}\) of ∥**x**^{∗}_{k}∥ is achievable. In other words, we need to solve for *λ*_{i,q} such that \(\sum _{\substack {{\mathsf {i}} = 1 \\ {\mathsf {i}} \neq {{q}}}}^{d} {\lambda }_{{\mathsf {i}},{{q}}} = {\mathcal {P}}\) and \({\left \|{{\mathbf {x}^{*}}_{k}}\right \|} \leq {\mathcal {B}}\) holds. Based on Eq. (14), we have

$$\begin{array}{*{20}l} &{\left\|{{\mathbf{x}^{*}}_{k}}\right\|} \leq \eta \max\limits_{{{q}}} {\left\|{{\mathbf{G}}_{{q}}}\right\|} \sup \limits_{j \geq 0} {\left\|{{\mathbf{u}}_{j}}\right\|} \leq {\mathcal{B}} \\ \Longrightarrow &\max\limits_{{{q}}} {\left\|{{\mathbf{G}}_{{q}}}\right\|} \leq \frac{{\mathcal{B}}}{\eta \cdot \sup \limits_{j \geq 0} {\left\|{{\mathbf{u}}_{j}}\right\|}} \\ \Longrightarrow & \max\limits_{{{q}}} {\left\|{\sum_{\substack{{\mathsf{i}} = 1 \\ {\mathsf{i}} \neq {{q}}}}^{d} {\lambda}_{{\mathsf{i}},{{q}}} ({\boldsymbol{\Lambda}}_{\mathsf{i}} - {\boldsymbol{\Gamma}}_{{\mathsf{i}},{{q}}})}\right\|} \leq \frac{{\mathcal{B}}}{\eta \cdot \sup \limits_{j \geq 0} {\left\|{{\mathbf{u}}_{j}}\right\|}}. \end{array} $$

(17)

Use the auxiliary function and define \({\pmb {\lambda }} \in {\mathbb {R}}^{d-1}\) and **S**_{i,q}=*Λ*_{i}−*Γ*_{i,q}. We can write the left-hand side of (17) as:

$$\begin{array}{*{20}l} {\phi}({\pmb{\lambda}}) = \max\limits_{{{q}}} {\left\|{\sum\limits_{i = 1}^{d-1} {\pmb{\lambda}}^{[i]} \mathbf{S}_{i,{{q}}}}\right\|}. \end{array} $$

Since *ϕ*(*λ*) is convex in *λ*, a non-negative bound \({\mathcal {B}}\) is achievable by taking *λ*^{[i]}=0 for all *i*. To seek a *λ* such that

$$\begin{array}{*{20}l} {\phi}({\pmb{\lambda}}) \leq \frac{{\mathcal{B}}}{\eta \cdot \sup \limits_{j \geq 0} {\left\|{{\mathbf{u}}_{j}}\right\|}}, \end{array} $$

we will use triangle inequality to approximate *ϕ*(*λ*) and get a more conservative condition. Since

$$\begin{array}{*{20}l} {\phi}({\pmb{\lambda}}) \leq \max\limits_{{{q}}} \sum\limits_{i = 1}^{d-1} {\pmb{\lambda}}^{[i]} {\left\|{\mathbf{S}_{i,{{q}}}}\right\|}, \end{array} $$

with ∥**S**_{i,q}∥ is known for all *i* and *q*. The condition

$$\begin{array}{*{20}l} \max\limits_{{{q}}} \sum\limits_{i = 1}^{d-1} {\pmb{\lambda}}^{[i]} {\left\|{\mathbf{S}_{i,{{q}}}}\right\|} \leq \frac{{\mathcal{B}}}{\eta \cdot \sup \limits_{j \geq 0} {\left\|{{\mathbf{u}}_{j}}\right\|}} \end{array} $$

(18)

is a 1st degree polynomial inequality with *d*−1 variables, and this can provide a feasible region for each *λ*_{i,q} on the *d*−1 dimensions space.

The discussion of BIBO stability completes the convergent analysis of bias dynamics in a mode-based Kalman filter. Both stable and unstable original SHS have been taken into consideration. For an unstable system, we can still stabilize the bias dynamics by specifically choosing the probability *λ*_{i,q}. For an asymptotically stable system, we addressed two important questions regarding the minimization of the upper bound for the bias.