Observability-Blocking Controls for Double-Integrator and Higher Order Integrator Networks

Controllability and observability of dynamical networks have been extensively studied in the controls-engineering community [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. One key question that has found attention recently is how to design controls in a network to shape observability and controllability while maintaining network performance. Such design problems are particularly relevant in contexts where multiple stakeholders (including adversaries) have access to the network’s dynamics, and have conflicting objectives in modulating the dynamics. For example, increasing concern about cyber-attacks in the power grid has led to significant interest in designing wide-area control systems that prevent adversaries from estimating or manipulating the dynamics [11]. Similarly, controller designs for multi-vehicle systems require security from intruders probing a subset of vehicles to infer the system’s state [12]. In this regard, observability-blocking has found usefulness in protecting privacy in dynamical networks [13].

Based on this motivation, in our previous work we explored design of feedback controls in a linear synchronization network to block observability/controllablity at a remote set of nodes, while maintaining all the eigenvalues and most of the eigenvectors [14, 15, 16]. However, the synchronization model considered in our earlier works has a limitation that each node is associated with only a scalar state. In contrast, double-integrator networks (DIN) and higher-order integrator network dynamics, which are widely used to model and control high-fidelity multi-agent systems such as robotic motion control and tracking in unmanned vehicle networks, consider network nodes with multiple states [17, 18]. Specifically, DIN is extensively used as a network model for the formation control of unmanned vehicle networks, among many other applications [17, 12]. Therefore, in this work we focus on the design of observability blocking controls for DIN model. We also study generalization of our design for higher order integrator networks.

Our main focus of this study is to utilize the dynamics and topological structure of an integrator network to develop a design algorithm that can prevent observability of the measurements obtained from a given set of nodes. Our main contributions for this paper are the following:

1) An algorithm that blocks observability of a set of $m$ nodes in DIN is presented using state-feedback controls at another set of $q=m+2$ nodes. The algorithm builds upon the design from our earlier work [16] and maintains all the eigenvalues and most of the eigenvectors. Unlike previous work, the algorithm leverages a special property of eigenvectors in a DIN model to limit the actuation to $q=m+2$.

2) A sparser design scheme is then presented that exploits the topological structure of the DIN model. Similar to our previous work, we show that the number of actuation can be reduced by blocking observability at cutsets separating the actuation and measurement locations. However, the conditions required for this design in the context of DIN differ significantly from those established in our earlier work.

3) We then extend our results for $N$-th order integrator networks. The number of actuation nodes required for blocking observability remains the same as in the DIN case.

The paper is organized as follows: In section II, we formulate the observability-blocking controller design problem for DIN. We present our main results on observability-blocking controls in Section III for both DIN and $N$-th order integrator networks. Section IV is devoted to a numerical example, and in Section V we draw conclusions.

In this study, we first focus on a double-integrator network (DIN) model which will allow us develop design principles for higher order integrator network models. Specifically, here we consider a DIN with $n$ nodes labeled as $1,2,\cdots,n$. Each node $i$ is associated with two scalar states denoted as $s_{i}$ and $\dot{s}_{i}$. In the context of a vehicle network, $s_{i}$ and $\dot{s}_{i}$ correspond to the position and velocity of the vehicle $i$. The interactions between different nodes in the DIN are represented by a directed graph $\mathcal{G}(\mathcal{V},\mathcal{E})$, where $\mathcal{V}$ contains $n$ nodes/vertices and $\mathcal{E}\subset\mathcal{V}\times\mathcal{V}$ contains a collection of directed edges specified as an ordered pair of nodes. We assume the graph is strongly connected and use $\mathcal{N}(i)$ to denote the set of nodes such that $(j,i)\in\mathcal{E}$. In our DIN model, the node dynamics of each node $i$ is given by [19]:

$\displaystyle\ddot{s_{i}}$

$\displaystyle=$

$\displaystyle-\sum_{j\in\mathcal{N}(i)}{w_{ji}^{s}(s_{i}-s_{j}})-\sum_{j\in\mathcal{N}(i)}{w_{ji}^{\dot{s}}(\dot{s}_{i}-\dot{s}_{j}})+u_{i}$

Here, weights $w_{ji}^{s}\in\mathcal{W}^{s}$ and $w_{ji}^{\dot{s}}\in\mathcal{W}^{\dot{s}}$ quantify the influence of node $j$ on node $i$ through the edge $(j,i)\in\mathcal{E}$ and $u_{i}$ denotes the actuation applied to the node $i$. In our model, $w_{ji}^{s}$ and $w_{ji}^{\dot{s}}$ are positive and assumed to not be related for generality. As standard, in order to denote the dynamics conveniently, we consider two (asymmetric) Laplacian matrices $\mathbf{L}^{s}$ and $\mathbf{L}^{\dot{s}}$ on the graph $\mathcal{G}(\mathcal{V},\mathcal{E})$ for the edge-weight sets $\mathcal{W}^{s}$ and $\mathcal{W}^{\dot{s}}$. Specifically, $\mathbf{L}^{s}$ is an $n\times n$ matrix whose entries are as follows: each off-diagonal entry $L_{ij}^{s}$ is equal to $-w_{ij}$ if $(i,j)\in\mathcal{E}$ and otherwise is set to $0$; each diagonal entry $L^{s}_{ii}$ is equal to $-\sum_{j=1,j\neq i}^{n}L^{s}_{ij}$. We define $\mathbf{L}^{\dot{s}}$ similarly based on the edge weight set $\mathcal{W}^{\dot{s}}$. We further define the network state $\mathbf{x}(t)$ as $\mathbf{x}(t)=[s_{1}(t)~~s_{2}(t)~\cdots~s_{n}(t)~~\dot{s}_{1}(t)~~\dot{s}_{2}(t)~\cdots~\dot{s}_{n}(t)]^{T}$. We enhance this network model by capturing a set of nodes as actuation nodes where actuation can be provided by the network operator (i.e. $u_{i}\neq 0$ in eqn. (II)) and another distinct set of nodes as measurement nodes that can be accessed by an adversary. We assume there are $q$ actuation nodes and $m$ measurement nodes given by the set $\{r_{1},r_{2},\ldots,r_{q}\}$ and set $\{r^{\prime}_{1},r^{\prime}_{2},\ldots,r^{\prime}_{m}\}$, respectively. It is reasonable to assume that when an adversary finds access to a node $r^{\prime}_{i}$, where $i=1,2,\cdots,m$, it can measure both states $s_{r^{\prime}_{i}}$ and $\dot{s}_{r^{\prime}_{i}}$.

The dynamics of the DIN with actuation and measurement included are then given by the state space model:

	$\displaystyle\mathbf{\dot{x}}=$	$\displaystyle\begin{bmatrix}\mathbf{0}~~~~~~\mathbf{I}\\ -\mathbf{L}^{s}~~-\mathbf{L}^{\dot{s}}\end{bmatrix}\mathbf{x}+\begin{bmatrix}\mathbf{0}\\ \mathbf{\hat{B}}\end{bmatrix}\mathbf{u},$		(2a)
	$\displaystyle\mathbf{y}=$	$\displaystyle\begin{bmatrix}\mathbf{\hat{C}}~~\mathbf{0}\\ \mathbf{0}~~\mathbf{\hat{C}}\end{bmatrix}\mathbf{x}$		(2b)

where $\mathbf{\hat{B}}=[\mathbf{e}_{r_{1}}~\mathbf{e}_{r_{2}}\cdots~\mathbf{e}_{r_{q}}]$, $\mathbf{\hat{C}}=[\mathbf{e}_{r^{\prime}_{1}}~\mathbf{e}_{r^{\prime}_{2}}\cdots~\mathbf{e}_{r^{\prime}_{m}}]^{T}$ and $\mathbf{e}_{i}\in\mathbb{R}^{n}$ is a 0–1 indicator vector with $i$ entry equal to $1$. Note, in the above, $\mathbf{u}\in\mathbb{R}^{q}$ and $\mathbf{y}\in\mathbb{R}^{2m}$ denotes the inputs applied at the actuation nodes and measurements taken at the measurement nodes, respectively. Writing $\mathbf{A}=\begin{bmatrix}\mathbf{0}~~~~~~\mathbf{I}\\ -\mathbf{L}^{s}~~-\mathbf{L}^{\dot{s}}\end{bmatrix}$, $\mathbf{B}=\begin{bmatrix}\mathbf{0}\\ \mathbf{\hat{B}}\end{bmatrix}$ and $\mathbf{C}=\begin{bmatrix}\mathbf{\hat{C}}~~\mathbf{0}\\ \mathbf{0}~~\mathbf{\hat{C}}\end{bmatrix}$, we can write the dynamics in (2) as

	$\displaystyle\mathbf{\dot{x}}=$	$\displaystyle\mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{u},$		(3a)
	$\displaystyle\mathbf{y}=$	$\displaystyle\mathbf{C}\mathbf{x}$		(3b)

We assume throughout our development that the pair $(\mathbf{A},\mathbf{B})$ is controllable. In this study, we consider the design of linear feedback controllers at the actuation nodes, to block the observability of the network dynamics with respect to the measurements $\mathbf{y}$ obtained at the measurement nodes. As a nominal case, a static state feedback control scheme is considered, with the input at each node $r_{i}$, $i=1,2,\ldots,q$, specified as $u_{r_{i}}=\mathbf{k}_{r_{i}}^{T}\mathbf{x}$ where $\mathbf{k}_{r_{i}}$ is the control gain. Assembling the state feedback models for each actuation node and writing $\mathbf{F}=[\mathbf{k}_{r_{1}}~\mathbf{k}_{r_{2}}\cdots\mathbf{k}_{r_{q}}]^{T}$, we get

$$\mathbf{u}=\mathbf{F}\mathbf{x}$$

(4)

Therefore, the closed-loop dynamics for the feedback control become:

	$\displaystyle\mathbf{\dot{x}}=$	$\displaystyle(\mathbf{A}+\mathbf{B}\mathbf{F})~\mathbf{x},$		(5a)
	$\displaystyle\mathbf{y}=$	$\displaystyle\mathbf{C}\mathbf{x}.$		(5b)

Our focus here is to design the state-feedback controller, defined by the gain matrix $\mathbf{F}$, to enforce that the pair $(\mathbf{C},(\mathbf{A}+\mathbf{BF}))$ is unobservable, while maintaining as much of the open-loop eigenstructure as possible. Our main objective is to find sparse feedback controllers that only require a small number actuation by exploiting the dynamics and graph topology of the DIN. We also aim to generalize our result for $N$-th order integrator networks.

In this section, we present our results for the design of observability-blocking controls. First, we develop a design algorithm for DIN requiring $q=m+2$ actuation nodes. Then, we provide a sparser design that utilizes the topological structure of DIN to reduce the required number of actuation nodes. Finally, we extend our results to encompass $N$-th order integrator networks.

:

In this section, we present a numerical example that verifies our theoretical findings. We adopt the same example graph used in [16], but now considering DIN dynamics for the nodes. Specifically, an $11$ node network with an undirected graph $\mathcal{G}$ is used as an example, as shown in Fig. 2. The edge weights for $\mathcal{G}$ in the Laplacians $\mathbf{L}^{s}$ and $\mathbf{L}^{\dot{s}}$ are assigned random positive numbers. The measurement nodes are assumed to be $\{6,8,9,11\}$. We identify from Fig. 2 that node 5 is a potential single-node cutset that results in the graph’s measurement nodes coalesced into one partition. Thus, from Theorems 1 and 2, we should be able to build state feedback controllers at any three nodes from $\{1,2,3,4,10\}$ to block observability at node 5, and hence at the measurement nodes. However, since all the eigenvalues of $\mathbf{A}$ are real, $q=2$ is sufficient to block observability by using our algorithm. We choose nodes 1 and 10 for actuation.

To build the controller, we select one eigenvalue and its associated eigenvector of the matrix $\mathbf{A}$. Specifically we select $\lambda_{p}=-1.0099$ whose corresponding eigenvector is $\mathbf{v}_{p}=[0.0305,0.1342,-0.1163,-0.0948,0.0118,0.4642,-0.4413,$ $-0.0857,0.1786,-0.0594,-0.0218,-0.0308,-0.1355,$ $0.1174,0.0958,-0.0119,-0.4688,0.4457,0.0865,-0.1803,$ $0.0600,0.0220]^{T}$. Then we build state feedback controllers at nodes 1 and 10 according to our proposed algorithm to block observability at node 5. The control gains for controllers at nodes 1 and 10 are obtained as $[-0.2550,-1.1694,0.9500,0.5459,0.3783,-2.8561,2.1917,$ $-0.1588,-0.3110,0.3363,0.3482,-0.0535,-0.2354,$ $0.2040,0.1664,-0.0207,-0.8147,0.7745,0.1504,-0.3134,$ $0.1042,0.0382]^{T}$ and $[-0.1309,-0.6005,0.4878,0.2803,$ $0.1943,-1.4666,1.1254,-0.0816,-0.1597,0.1727,0.1788,$ $-0.0275,-0.1209,0.1048,0.0854,-0.0106,-0.4183,$ $0.3977,0.0772,-0.1609,0.0535,0.0196,]^{T}$respectively. After applying these controllers, the modified eigenvector $\mathbf{\hat{v}}_{p}$ is obtained as $[-0.0046,0.0033,-0.0021,0.0017,0,0,0,0,0,-0.0045,0,$ $0.0046,-0.0033,0.0022,-0.0017,0,0,0,0,0,0.0045,0]^{T}$. We see that the entries corresponding to node $5$ as well as all the nodes in the partition $\mathcal{V}_{2}$, i.e. the node set $\{6,7,8,9,11\}$, are zero. Thus, the DIN model has become unobservable at the measurement nodes, verifying our theorems.

A design algorithm has been developed for observability-blocking controls in a double integrator network (DIN). The topological structure of DIN has then been exploited to reduce the number of actuation nodes needed to block observability. We have generalized the observability-blocking design for $N$-th order integrator networks. A numerical example has been presented to verify our findings. One interesting future direction for this study includes developing design of observability-blocking controls for more complex networks, such as nonlinear dynamical networks.

We thank Prof. Sandip Roy at Texas A&M University for his valuable suggestions and insights regarding this work.

1. 1)

   Select one eigenvalue $\lambda_{p}$ of $\mathbf{A}$ and its associated eigenvector $\mathbf{v}_{p}$, where $p\in\{1,2,\ldots,2n\}$. If $\lambda_{p}$ is real, follow the steps under Sub-Algorithm 1 to obtain the observability-blocking controller. Otherwise follow the steps under Sub-Algorithm 2.

Sub-Algorithm 1:

1. 2)

   Compute a matrix $\mathbf{N}(\lambda_{p})\in\mathbb{C}^{(2n+q)\times q}$, whose columns are linearly independent and span the null space of $\mathbf{S}(\lambda_{p})=[(\mathbf{A}-\lambda_{p}\mathbf{I})~~\mathbf{B}]$. Then partition $\mathbf{N}(\lambda_{p})$ as $\mathbf{N}(\lambda_{p})=[\mathbf{N}_{1}(\lambda_{p})^{T}~\mathbf{N}_{2}(\lambda_{p})^{T}]^{T}$, where $\mathbf{N}_{1}(\lambda_{p})\in\mathbb{C}^{2n\times q}$ and $\mathbf{N}_{2}(\lambda_{p})\in\mathbb{C}^{q\times q}$. Therefore $\mathbf{N}_{1}(\lambda_{p})$ and $\mathbf{N}_{2}(\lambda_{p})$ satisfy:

   $\displaystyle[(\mathbf{A}-\lambda_{p}~\mathbf{I})~~\mathbf{B}]\left[\begin{array}[]{c}\mathbf{N}_{1}(\lambda_{p})\\ \mathbf{N}_{2}(\lambda_{p})\end{array}\right]=\mathbf{0}.$

   (17)

2. 3)

   Partition $\mathbf{N}_{1}(\lambda_{p})$ as $\mathbf{N}_{1}(\lambda_{p})=[\mathbf{N}_{3}(\lambda_{p})^{T}~\mathbf{N}_{4}(\lambda_{p})^{T}~\mathbf{N}_{5}(\lambda_{p})^{T}~\mathbf{N}_{6}(\lambda_{p})^{T}]^{T}$, where $\mathbf{N}_{3}(\lambda_{p})\in\mathbb{C}^{(n-m)\times q}$, $\mathbf{N}_{4}(\lambda_{p})\in\mathbb{C}^{m\times q}$, $\mathbf{N}_{5}(\lambda_{p})\in\mathbb{C}^{(n-m)\times q}$ and $\mathbf{N}_{6}(\lambda_{p})\in\mathbb{C}^{m\times q}$. Then find a vector $\mathbf{h}_{p}\neq\mathbf{0}$ which lies in the null space of $\mathbf{N}_{4}(\lambda_{p})$, i.e. which satisfies:

   $\displaystyle\mathbf{N}_{4}(\lambda_{p})~\mathbf{h}_{p}$

   $\displaystyle=$

   $\displaystyle\mathbf{0}.$

   (18)

3. 4)

   Compute the vectors $\hat{\mathbf{v}_{p}}$ and $\mathbf{z}_{p}$ as:

   	$\displaystyle\hat{\mathbf{v}}_{p}$	$\displaystyle=$	$\displaystyle\mathbf{N}_{1}(\lambda_{p})~\mathbf{h}_{p}$		(19)
   	$\displaystyle\mathbf{z}_{p}$	$\displaystyle=$	$\displaystyle\mathbf{N}_{2}(\lambda_{p})~\mathbf{h}_{p}.$		(20)

4. 5)

   Form the open-loop modal matrix $\mathbf{V}_{0}\in\mathbb{C}^{2n\times 2n}$ (i.e. $\mathbf{V}_{0}=[\mathbf{v}_{1}~\mathbf{v}_{2}~\cdots~\mathbf{v}_{2n}]$), and $q\times{2n}$ zero matrix $\mathbf{Z}_{0}$. Then check whether the $2n$ vectors in $\hat{V}_{0}=V_{0}\cup\{\hat{\mathbf{v}}_{p}\}\backslash\{\mathbf{v}_{p}\}$ are linearly independent, where $V_{0}$ is a vector set containing all the open-loop eigenvectors. If yes, then follow Step 6 below to construct the closed-loop modal matrix $\mathbf{V}$ and associated $\mathbf{Z}$ matrix. Otherwise, skip Step 6, and follow Steps 7 through 9 to construct $\mathbf{V}$ and $\mathbf{Z}$ matrices.

5. 6)

   Construct the matrix $\mathbf{V}\in\mathbb{C}^{2n\times 2n}$ from $\mathbf{V}_{0}$ by replacing the column containing $\mathbf{v}_{p}$ with $\hat{\mathbf{v}}_{p}$. Similarly, construct the matrix $\mathbf{Z}\in\mathbb{C}^{q\times 2n}$ from $\mathbf{Z}_{0}$ by replacing the corresponding column with $\mathbf{z}_{p}$. Therefore, $\mathbf{V}=[\mathbf{v}_{1}\cdots\mathbf{v}_{p-1}~\hat{\mathbf{v}}_{p}~\mathbf{v}_{p+1}\cdots\mathbf{v}_{2n}]$ and $\mathbf{Z}=[\mathbf{0}\cdots\mathbf{0}~\mathbf{z}_{p}~\mathbf{0}\cdots\mathbf{0}]$. Then jump to Step 10.

6. 7)

   Find the largest-cardinality subset $V_{1}$ of $\hat{V}_{0}$ such that $\hat{\mathbf{v}}_{p}\in V_{1}$ and $V_{1}$ is a self-conjugate set of linearly independent vectors.

7. 8)

   Find $\hat{\mathbf{v}}_{k}$ in the column space of $\mathbf{N}_{1}(\lambda_{k})$ for all $\mathbf{v}_{k}\in\hat{V}_{0}\backslash V_{1}$ such that the set $\hat{V}=V_{1}\cup\{\hat{\mathbf{v}}_{k}|\mathbf{v}_{k}\in\hat{V}_{0}\backslash V_{1}\}$ is a self-conjugate set of $2n$ linearly independent vectors. While doing so, maintain $\hat{\mathbf{v}}_{k_{2}}=\bar{\hat{\mathbf{v}}}_{k_{1}}$ whenever $\mathbf{v}_{k_{2}}=\bar{\mathbf{v}}_{k_{1}}$ and $\mathbf{v}_{k_{1}},\mathbf{v}_{k_{2}}\in\hat{V}_{0}\backslash V_{1}$. Next, find the corresponding $\mathbf{z}_{k}$ for all $\mathbf{v}_{k}\in\hat{V}_{0}\backslash V_{1}$ such that $\mathbf{z}_{k}=\mathbf{N}_{2}(\lambda_{k})~\mathbf{h}_{k}$, where $\mathbf{h}_{k}$ solves $\hat{\mathbf{v}}_{k}=\mathbf{N}_{1}(\lambda_{k})~\mathbf{h}_{k}$. Note that $\mathbf{z}_{k_{2}}=\bar{\mathbf{z}}_{k_{1}}$ whenever $\mathbf{v}_{k_{2}}=\bar{\mathbf{v}}_{k_{1}}$.

8. 9)

   Construct $\mathbf{V}$ from $\mathbf{V}_{0}$ by replacing the columns containing $\mathbf{v}_{p}$ and all $\mathbf{v}_{k}\in\hat{V}_{0}\backslash V_{1}$ with $\hat{\mathbf{v}}_{p}$ and corresponding $\hat{\mathbf{v}}_{k}$ respectively. In the same manner construct $\mathbf{Z}$ from $\mathbf{Z}_{0}$ by replacing the corresponding columns of $\mathbf{Z}_{0}$ with $\mathbf{z}_{p}$ and all $\mathbf{z}_{k}$ obtained in Step 8 respectively.

9. 10)

   Finally the gain matrix $\mathbf{F}$ for the observability-blocking controller is obtained by (21):

   $\displaystyle\mathbf{F}=\mathbf{Z}~\mathbf{V}^{-1}.$

   (21)

Sub-Algorithm 2:

1. 2-4)

   Steps 2 through 4 remain exactly the same as for Sub-Algorithm 1. Additionally in Step 4, it is necessary to obtain $\bar{\hat{\mathbf{v}}}_{p}$ and the associated $\bar{\mathbf{z}}_{p}$ by taking complex conjugates of $\hat{\mathbf{v}}_{p}$ and $\mathbf{z}_{p}$, respectively.

2. 5)

   Form the open-loop modal matrix $\mathbf{V}_{0}\in\mathbb{C}^{2n\times 2n}$ and $q\times 2n$ zero matrix $\mathbf{Z}_{0}$. Then check whether the $2n$ vectors in $\hat{V}_{0}=V_{0}\cup\{\hat{\mathbf{v}}_{p},\bar{\hat{\mathbf{v}}}_{p}\}\backslash\{\mathbf{v}_{p},\bar{\mathbf{v}}_{p}\}$ are linearly independent. If yes, follow Step 6; otherwise skip Step 6 and follow Steps 7 through 9 to find $\mathbf{V}$ and $\mathbf{Z}$.

3. 6)

   Construct the matrix $\mathbf{V}$ from $\mathbf{V}_{0}$ by replacing the columns having $\mathbf{v}_{p}$ and $\bar{\mathbf{v}}_{p}$ with $\hat{\mathbf{v}}_{p}$ and $\bar{\hat{\mathbf{v}}}_{p}$ respectively. Similarly, construct the matrix $\mathbf{Z}$ from $\mathbf{Z}_{0}$ by replacing the corresponding columns of $\mathbf{Z}_{0}$ with $\mathbf{z}_{p}$ and $\bar{\mathbf{z}}_{p}$ respectively. Then jump to Step 10.

4. 7)

   Find the largest-cardinality subset $V_{1}$ of $\hat{V}_{0}$ such that $\hat{\mathbf{v}}_{p}$, $\bar{\hat{\mathbf{v}}}_{p}\in V_{1}$ and $V_{1}$ is a self-conjugate set of linearly independent vectors.

5. 8)

   This step is the same as the Step 8 of Sub-Algorithm 1. Thus, find $\hat{\mathbf{v}}_{k}$ in the column space of $\mathbf{N}_{1}(\lambda_{k})$ for all $\mathbf{v}_{k}\in\hat{V}_{0}\backslash V_{1}$ such that the set $\hat{V}=V_{1}\cup\{\hat{\mathbf{v}}_{k}|\mathbf{v}_{k}\in\hat{V}_{0}\backslash V_{1}\}$ is a self-conjugate set of $2n$ linearly independent vectors, and find all the corresponding $\mathbf{z}_{k}$.

6. 9)

   Construct $\mathbf{V}$ from $\mathbf{V}_{0}$ by replacing the columns containing $\mathbf{v}_{p}$, $\bar{\mathbf{v}}_{p}$ and all $\mathbf{v}_{k}\in\hat{V}_{0}\backslash V_{1}$ with $\hat{\mathbf{v}}_{p}$, $\bar{\hat{\mathbf{v}}}_{p}$ and corresponding $\hat{\mathbf{v}}_{k}$ respectively. In the same manner, construct $\mathbf{Z}$ from $\mathbf{Z}_{0}$ by replacing the corresponding columns of $\mathbf{Z}_{0}$ with $\mathbf{z}_{p}$, $\bar{\mathbf{z}}_{p}$ and all $\mathbf{z}_{k}$ obtained in Step 8 respectively.

7. 10)

   Like Sub-Algorithm 1, compute the gain matrix $\mathbf{F}$ using (21).