7.2.1 System Description

The NCS Σ investigated in this section is the same as that of the previous chapters, which has also been adopted in [10–12]. This system is constituted from N linear time-invariant dynamic subsystems, whereas the dynamics of its ith subsystem ${\mathbf{\Sigma }}_i$ is described by the following state space model like representation:

$\left[\begin{array}{c}\hfill x(k+1,i)\hfill \\ \hfill z(k,i)\hfill \\ \hfill y(k,i)\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill A_{\mathbf{xx}}(i)\hfill & \hfill A_{\mathbf{xv}}(i)\hfill & \hfill B_{\mathbf{x}}(i)\hfill \\ \hfill A_{\mathbf{zx}}(i)\hfill & \hfill A_{\mathbf{zv}}(i)\hfill & \hfill B_{\mathbf{z}}(i)\hfill \\ \hfill C_{\mathbf{x}}(i)\hfill & \hfill C_{\mathbf{v}}(i)\hfill & \hfill D_{\mathbf{u}}(i)\hfill \end{array}\right]\left[\begin{array}{c}\hfill x(k,i)\hfill \\ \hfill v(k,i)\hfill \\ \hfill u(k,i)\hfill \end{array}\right],$

where $i=1,2,\cdots ,N$. Moreover, its subsystems are connected through

$v(k)=\mathrm{\Phi }z(k).$

As in the previous chapters, here $z(k)$ and $v(k)$ are respectively defined as $z(k)=\mathbf{col}\{z(k,i){|}_{i=1}^N\}$ and $v(k)=\mathbf{col}\{v(k,i){|}_{i=1}^N\}$. Moreover, k and i stand respectively for the temporal variable and the index number of a subsystem, $x(k,i)$ represents the state vector of the ith subsystem ${\mathbf{\Sigma }}_i$ at the time instant k, $z(k,i)$/$v(k,i)$ represent its output/input vector to/from other subsystems, and $y(k,i)$ and $u(k,i)$ represent respectively its output and input vectors. Once again, as in the previous chapters, to distinguish $z(k,i)$ and $v(k,i)$ respectively from $y(k,i)$ and $u(k,i)$, $z(k,i)$ and $v(k,i)$ are called internal output/input vectors, whereas $y(k,i)$ and $u(k,i)$ are called external output/input vectors.

Throughout this section, the dimensions of the vectors $x(k,i)$, $v(k,i)$, $u(k,i)$, $z(k,i)$, and $y(k,i)$, are assumed respectively to be $m_{\mathbf{x}i}$, $m_{\mathbf{v}i}$, $m_{\mathbf{u}i}$, $m_{\mathbf{z}i}$, and $m_{\mathbf{y}i}$. Moreover, we assume that every row of the matrix Φ has only one nonzero element, which is equal to one. As argued in the previous chapters, this assumption does not sacrifice any generality of the adopted system model. Note also that an approximate power-law degree distribution exists extensively in science and engineering systems, such as gene regulation networks, protein interaction networks, internet, electrical power systems, and so on. For these systems, interactions among subsystems are sparse, and the matrix Φ usually has a dimension significantly smaller than that of its state vector [10,12–14]. Under such a situation, results given in this section work well in general.

7.2.2 Stability of a Networked System

To develop a computationally attractive criterion for the stability of system Σ and for its robust stability against both parametric modeling errors and unmodeled dynamics, the following results are at first introduced, which are well known in system theories [1,15–17].

Note that well-posedness is essential for a system to work properly. In fact, a plant that does not satisfy this requirement is usually hard to be controlled, and/or its states are in general difficult to be estimated [1,4,12]. It is therefore assumed in the following discussions that the networked system Σ under investigation is well-posed, which can be explicitly expressed as a requirement that the associated matrix is invertible.

When each its subsystem is linear and time-invariant and the subsystem connection matrix is also time invariant, the networked system Σ itself is also linear and time invariant. From Lemma 7.1 it is clear that the requirement that the networked system Σ is stable can be equivalently expressed as that all the eigenvalues of its state transition matrix are smaller than 1 in magnitude. To make mathematical derivations more concise, we first define the following matrices: $A_{\mathbf{⁎}\mathrm{\#}}=\mathbf{diag}\{A_{\mathbf{⁎}\mathrm{\#}}(i){|}_{i=1}^N\}$, $B_{\mathbf{⁎}}=\mathbf{diag}\{B_{\mathbf{⁎}}(i){|}_{i=1}^N\}$, $C_{\mathbf{⁎}}=\mathbf{diag}\{C_{\mathbf{⁎}}(i){|}_{i=1}^N\}$, and $D_{\mathbf{u}}=\mathbf{diag}\{D_{\mathbf{u}}(i){|}_{i=1}^N\}$, where $\mathbf{⁎},\mathrm{\#}=\mathbf{x}$, v or z. Moreover, denote $\mathbf{col}\{u(k,i){|}_{i=1}^N\}$, $\mathbf{col}\{x(k,i){|}_{i=1}^N\}$, and $\mathbf{col}\{y(k,i){{|}_{i=1}}^N\}$ respectively by $u(k)$, $x(k)$, and $y(k)$. Then straightforward algebraic manipulations show that the well-posedness of the networked system Σ is equivalent to the regularity of the matrix $I-A_{\mathbf{zv}}\mathrm{\Phi }$, that is, this matrix is invertible. Moreover, when the networked system Σ is well-posed, its dynamics can be equivalently described by the following state space model:

$\left[\begin{array}{c}\hfill x(k+1)\hfill \\ \hfill y(k)\hfill \end{array}\right]=\{\left[\begin{array}{cc}\hfill A_{\mathbf{xx}}\hfill & \hfill B_{\mathbf{x}}\hfill \\ \hfill C_{\mathbf{x}}\hfill & \hfill D_{\mathbf{u}}\hfill \end{array}\right]+\left[\begin{array}{c}\hfill A_{\mathbf{xv}}\hfill \\ \hfill C_{\mathbf{v}}\hfill \end{array}\right]\mathrm{\Phi }{[I-A_{\mathbf{zv}}\mathrm{\Phi }]}^{-1}\left[A_{\mathbf{zx}}{B}_{\mathbf{z}}\right]\}\left[\begin{array}{c}\hfill x(k)\hfill \\ \hfill u(k)\hfill \end{array}\right].$

This expression gives a lumped state space model of the networked system Σ and is completely the same as that adopted in Lemma 7.1. This property can be understood more easily if we define the matrices A, B, C, and D respectively as

$\begin{array}{ccc}\hfill & \hfill \hfill & A=A_{\mathbf{xx}}+A_{\mathbf{xv}}\mathrm{\Phi }{(I-A_{\mathbf{zv}}\mathrm{\Phi })}^{-1}{A}_{\mathbf{zx}},B=B_{\mathbf{x}}+A_{\mathbf{xv}}\mathrm{\Phi }{(I-A_{\mathbf{zv}}\mathrm{\Phi })}^{-1}{B}_{\mathbf{z}},\hfill \\ \hfill & \hfill \hfill & C=C_{\mathbf{x}}+C_{\mathbf{v}}\mathrm{\Phi }{(I-A_{\mathbf{zv}}\mathrm{\Phi })}^{-1}{A}_{\mathbf{zx}},D=D_{\mathbf{u}}+C_{\mathbf{v}}\mathrm{\Phi }{(I-A_{\mathbf{zv}}\mathrm{\Phi })}^{-1}{B}_{\mathbf{z}}.\hfill \end{array}$

Clearly, all these matrices are time invariant. Moreover, the input–output relation of the networked system Σ has been rewritten completely in the same form as that in Lemma 7.1. As in the verification of controllability and observability of the networked system, this relation enables application of Lemma 7.1 to the stability and robust stability analysis of the networked system Σ. However, it is worth noting that a networked system usually has a great amount of subsystems, which means that the dimensions of the associated matrices in the lumped model are in general high. Hence, when a large-scale networked system is under investigation, it is usually not numerically feasible to straightforwardly apply the results of Lemma 7.1 to its analysis and synthesis. In addition, note that the inverse of the matrix $I-A_{\mathbf{zv}}\mathrm{\Phi }$ is required in the aforementioned expressions, and this inversion is in general not numerically stable when the dimension of this matrix is high and/or when this matrix is nearly singular. These imply that calculations of the parameter matrices A, B, C, and D of the lumped model itself might not be reliable. As discussed in Chapter 3, these difficulties happen also to verifications of the controllability and/or the observability of the networked system Σ.

To develop a computationally feasible and numerically reliable criterion for the stability of the networked system Σ, the following properties of the matrix A is first established.

On the basis of Eq. (7.5), it can be shown that the networked system Σ is stable, only when all the unstable modes of each of its subsystems are both controllable and observable. This conclusion can be established through a similarity transformation that divide the state space of each subsystem into a stable subspace and an unstable subspace, utilizing the relation of the following Eq. (7.11).

From Lemmas 7.1 and 7.2 it is clear that a necessary and sufficient condition for the networked system Σ to be stable is that, for every complex number λ satisfying $|\lambda |⩾1$, the matrix $I-\mathrm{\Phi }G(\lambda )$ is regular. However, it is worth noting that when the networked system under investigation is constructed from a large amount of subsystems, although the subsystem connection matrix Φ is usually sparse, its dimension is still usually high. Moreover, the associated inequality must be checked for each complex number λ satisfying $|\lambda |⩾1$. This is generally impossible through straightforward computations. By these observations it is safe to declare that some further efforts are still required to make the results of Lemma 7.2 applicable to a large-scale networked system.

To achieve these objectives, we recall the following property for the subsystem connection matrix Φ stated and proven in Chapter 3.

Let $m(i)$ stand for the number of subsystems that is directly affected by the ith element of the vector $z(k)$, $i=1,2,\dots ,M_{\mathbf{z}}$. Define the matrices $\mathrm{\Theta }(j)$, $j=1,2,\dots ,N$, and Θ respectively as $\mathrm{\Theta }(j)=\mathbf{diag}\{\sqrt{m(i)}{|}_{i=M_{\mathbf{z},j-1}+1}^{M_{\mathbf{z},j}}\}$ and $\mathrm{\Theta }=\mathbf{diag}\{\sqrt{m(i)}{|}_{i=1}^{M_{\mathbf{z}}}\}$. It has been proven in [11] and Chapter 3 that

${\mathrm{\Phi }}^T\mathrm{\Phi }={\mathrm{\Theta }}^2=\mathbf{diag}\left\{{{\mathrm{\Theta }}^2(j)|}_{j=1}^N\right\}.$

On the basis of these results, we derive a necessary and sufficient condition for the stability of the networked system Σ, which explicitly takes the structure of the networked system into account and is computationally attractive.

Compared with the available results, such as those of [2,3,6,7], an attractive characteristic of Theorem 7.1 is that there exist no restrictions on the model of the networked system Σ and the condition is both necessary and sufficient. On the other hand, note that the left-hand side of Eq. (7.12) depends linearly on the matrix P and that the structure of the networked system Θ, which is represented by the subsystem connection matrix Φ, is explicitly included in this equation. For a small size problem, feasibility of this matrix inequality can in general be easily verified using available linear matrix inequality (LMI) solvers. On the other hand, note that all the matrices $A_{\mathbf{⁎}\mathrm{\#}}$ with ⁎, $\mathrm{\#}$ = x, z are block diagonal and a networked system usually has a sparse structure, which is reflected by the subsystem connection matrix Φ. In addition, efficient methods have been developed for sparse semidefinite programming, such as those in [10,20,21]. It is expected that the above condition works well also for a moderate size problem. This expectation has been confirmed through some numerical simulations reported in [11].

From its proof it is clear that although the matrices $A_{\mathbf{⁎}\mathrm{\#}}$ with $\mathbf{⁎},\mathrm{\#}=\mathbf{x}$, v, or z are block diagonal, the matrix P is usually dense. In addition, when the square of the matrix Θ in inequality (7.12) is replaced by ${\mathrm{\Phi }}^T\mathrm{\Phi }$, the results of Theorem 7.1 become valid for an arbitrary subsystem connection matrix Φ. On the other hand, a necessary condition for the feasibility of this inequality obviously is

$P-A_{\mathbf{xx}}^{T}P{A}_{\mathbf{xx}}+A_{\mathbf{zx}}^T{\mathrm{\Theta }}^2{A}_{\mathbf{zx}}>0,$

which can be proved to be equivalent to the existence of positive definite matrices $P(i)$ for $i=1,2,\cdots ,N$ such that

$P(i)-A_{\mathbf{xx}}^T(i)P(i)A_{\mathbf{xx}}(i)+A_{\mathbf{zx}}^T(i){\mathrm{\Theta }}_i^2{A}_{\mathbf{zx}}(i)>0.$

The last inequality can be verified for each subsystem independently.

It is worth pointing out that from Lyapunov stability theory we can directly declare that system Σ is stable if and only if there exists a positive definite matrix P such that $P-A^{T}PA>0$, which is also a linear matrix inequality. However, as argued in [10,12] and the previous chapters, although the subsystem connection matrix Φ is generally sparse, the state transition matrix A is usually dense. This implies that computational complexity for feasibility verification of this equation is usually significantly greater than that of inequality (7.12), especially when the plant consists of a large amount of subsystems. Similar observations have also been reported in [10] for robust stability verification of system Σ with IQC described uncertainties.

When a system is of a very large-scale, numerical difficulties may still arise in verifying the condition of Theorem 7.1. To overcome these difficulties, one possibility is to find some matrices ${\mathrm{\Theta }}_{ij}^{[⁎]}=\mathbf{diag}\{{\mathrm{\Theta }}_{ij}^{[⁎]}(k){|}_{k=1}^N\}$ with $i,j=1,2$ and $⁎=l,h$ that have dimensions and partitions compatible with the matrix $A_{\mathbf{xx}}$ such that

$\left[\begin{array}{cc}\hfill {\mathrm{\Theta }}_{11}^{[l]}\hfill & \hfill {\mathrm{\Theta }}_{21}^{[l]T}\hfill \\ \hfill {\mathrm{\Theta }}_{21}^{[l]}\hfill & \hfill {\mathrm{\Theta }}_{22}^{[l]}\hfill \end{array}\right]⩽\left[\begin{array}{cc}\hfill 0\hfill & \hfill A_{\mathbf{zx}}^T{\mathrm{\Phi }}^T\hfill \\ \hfill \mathrm{\Phi }{A}_{\mathbf{zx}}\hfill & \hfill \mathrm{\Phi }{A}_{\mathbf{zv}}+A_{\mathbf{zv}}^T{\mathrm{\Phi }}^T\hfill \end{array}\right]⩽\left[\begin{array}{cc}\hfill {\mathrm{\Theta }}_{11}^{[h]}\hfill & \hfill {\mathrm{\Theta }}_{21}^{[h]T}\hfill \\ \hfill {\mathrm{\Theta }}_{21}^{[h]}\hfill & \hfill {\mathrm{\Theta }}_{22}^{[h]}\hfill \end{array}\right].$

With availability of these matrices, direct algebraic operations show that a necessary condition for the stability of system Σ is the existence of a PDM $P(i)$ for each $1⩽i⩽N$ such that

$\left[\begin{array}{cc}\hfill P(i)-A_{\mathbf{xx}}^T(i)P(i)A_{\mathbf{xx}}(i)+A_{\mathbf{zx}}^T(i){\mathrm{\Theta }}^2(i)A_{\mathbf{zx}}(i)-{\mathrm{\Theta }}_{11}^{[l]}(i)\hfill & \hfill {A_{\mathbf{zx}}}^T(i){\mathrm{\Theta }}^2(i)A_{\mathbf{zv}}(i)-A_{\mathbf{xx}}^T(i)P(i)A_{\mathbf{xv}}(i)-{\mathrm{\Theta }}_{21}^{[l]T}(i)\hfill \\ \hfill A_{\mathbf{zv}}^T(i){\mathrm{\Theta }}^2(i)A_{\mathbf{zx}}(i)-{A_{\mathbf{xv}}}^T(i)P(i)A_{\mathbf{xx}}(i)-{\mathrm{\Theta }}_{21}^{[l]}(i)\hfill & \hfill {A_{\mathbf{zv}}}^T(i){\mathrm{\Theta }}^2(i)A_{\mathbf{zv}}(i)-A_{\mathbf{xv}}^T(i)P(i)A_{\mathbf{xv}}(i)+I-{\mathrm{\Theta }}_{22}^{[l]}(i)\hfill \end{array}\right]>0.$

Moreover, when the superscript l is replaced by h, these inequalities become a sufficient condition. Clearly, these inequalities depend linearly on $P(i)$, and their dimensions are completely and independently determined by each individual system. These properties make them much more computationally attractive than Eq. (7.12). However, further efforts are required to find the matrices ${\mathrm{\Theta }}_{ij}^{[⁎]}(k)$ in Eq. (7.16) with $i,j=1,2$, $⁎=h,l$, and $k=1,2,\cdots ,N$ such that the associated inequalities are both tight and computationally attractive.

Based on the properties of the SCM Φ, another computationally attractive sufficient condition is derived for the stability of system Σ.

Note that ${\mathrm{\Theta }}_i{G}_i(\lambda )$ is completely determined by the SCM Φ and the parameters of the ith subsystem, and efficient methods exist for computing an upper bound of the $H_{\infty }$ norm of a TFM. Therefore, the condition of Theorem 7.2 can be easily verified, and its computational complexity increases only linearly with the increment of the subsystem number N. This means that for a large-scale networked system, the computation cost of Theorem 7.2 is in general significantly lower than that of Theorem 7.1. However, it is worth emphasizing that this condition is only sufficient and its conservativeness is still not clear. On the other hand, the numerical simulation results reported in [11] show that with the increment of the subsystem number and/or the magnitude of the subsystem matrix elements, conservativeness of Theorem 7.2 usually increases.

Note also that the finiteness of ${\Vert {\mathrm{\Theta }}_i{G}_i(\lambda )\Vert }_{\infty }$ implies the stability of the subsystem ${\mathbf{\Sigma }}_i$. Moreover, from the definition of the TFM $H_{\infty }$ norm we can see that ${\Vert {\mathrm{\Theta }}_i{G}_i(\lambda )\Vert }_{\infty }$ decreases monotonically with the decrement of any diagonal element of the matrix ${\mathrm{\Theta }}_i$. Theorem 7.2 therefore also makes it clear that for an NS consisting of stable subsystems, sparse connections are helpful in maintaining its stability, and subsystem interaction reduction can make an unstable system stable.

On the basis of the results given in [19], here we provide a linear matrix inequality-based condition for the verification of the $H_{\infty }$ norm-based condition.

Note that $\overline{\sigma }({\mathrm{\Theta }}_i{G}_i(\lambda ))<1$ is equivalent to

${\left[\begin{array}{c}\hfill {\mathrm{\Theta }}_i{G}_i(\lambda )\hfill \\ \hfill I\hfill \end{array}\right]}^H\left[\begin{array}{cc}\hfill -I\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill I\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\mathrm{\Theta }}_i{G}_i(\lambda )\hfill \\ \hfill I\hfill \end{array}\right]>0.$

It can be directly proved from the structure of $\mathbf{col}\{{\mathrm{\Theta }}_i{G}_i(\lambda ),I\}$ that the condition of Theorem 7.2 can be equivalently expressed as the following matrix inequality:

${[\star ]}^H\left[\begin{array}{cc}\hfill -A_{\mathbf{zx}}^T(i){\mathrm{\Theta }}_i^2{A}_{\mathbf{zx}}(i)\hfill & \hfill -A_{\mathbf{zx}}^T(i){\mathrm{\Theta }}_i^2{A}_{\mathbf{zv}}(i)\hfill \\ \hfill -A_{\mathbf{zv}}^T(i){\mathrm{\Theta }}_i^2{A}_{\mathbf{zx}}(i)\hfill & \hfill -A_{\mathbf{zv}}^T(i){\mathrm{\Theta }}_i^2{A}_{\mathbf{zv}}(i)+I\hfill \end{array}\right]\left[\begin{array}{c}\hfill {(\lambda I-A_{\mathbf{xx}}(i))}^{-1}{A}_{\mathbf{xv}}(i)\hfill \\ \hfill I\hfill \end{array}\right]>0$

for every complex number λ such that $|\lambda |⩾1$.

Recall that $|\lambda |⩾1$ is equivalent to

${\left[\begin{array}{c}\hfill \lambda \hfill \\ \hfill 1\hfill \end{array}\right]}^H\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill \end{array}\right]\left[\begin{array}{c}\hfill \lambda \hfill \\ \hfill 1\hfill \end{array}\right]⩾0.$

Based on the definition of the $H_{\infty }$ norm of a transfer function matrix and Theorems 1 and 3 of [19], we can directly claim that ${\Vert {\mathrm{\Theta }}_i{G}_i(\lambda )\Vert }_{\infty }<1$ is equivalent to the existence of a positive definite matrix $P(i)$ such that

$\begin{array}{cc}\hfill & {[\star ]}^T\left[\begin{array}{cc}\hfill P(i)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -P(i)\hfill \end{array}\right]\left[\begin{array}{cc}\hfill A_{\mathbf{xx}}(i)\hfill & \hfill A_{\mathbf{xv}}(i)\hfill \\ \hfill I\hfill & \hfill 0\hfill \end{array}\right]\hfill \\ \hfill & -\left[\begin{array}{cc}\hfill -A_{\mathbf{zx}}^T(i){\mathrm{\Theta }}_i^2{A}_{\mathbf{zx}}(i)\hfill & \hfill -A_{\mathbf{zx}}^T(i){\mathrm{\Theta }}_i^2{A}_{\mathbf{zv}}(i)\hfill \\ \hfill -A_{\mathbf{zv}}^T(i){\mathrm{\Theta }}_i^2{A}_{\mathbf{zx}}(i)\hfill & \hfill -A_{\mathbf{zv}}^T(i){\mathrm{\Theta }}_i^2{A}_{\mathbf{zv}}(i)+I\hfill \end{array}\right]<0,\hfill \end{array}$

which can be reexpressed as

$\left[\begin{array}{cc}\hfill P(i)-A_{\mathbf{xx}}^T(i)P(i)A_{\mathbf{xx}}(i)-A_{\mathbf{zx}}^T(i){\mathrm{\Theta }}_i^2{A}_{\mathbf{zx}}(i)\hfill & \hfill -A_{\mathbf{xx}}^T(i)P(i)A_{\mathbf{xv}}(i)-A_{\mathbf{zx}}^T(i){\mathrm{\Theta }}_i^2{A}_{\mathbf{zv}}(i)\hfill \\ \hfill -A_{\mathbf{xv}}^T(i)P(i)A_{\mathbf{xx}}(i)-A_{\mathbf{zv}}^T(i){\mathrm{\Theta }}_i^2{A}_{\mathbf{zx}}(i)\hfill & \hfill I-A_{\mathbf{xv}}^T(i)P(i)A_{\mathbf{xv}}(i)-A_{\mathbf{zv}}^T(i){\mathrm{\Theta }}_i^2{A}_{\mathbf{zv}}(i)\hfill \end{array}\right]>0.$

Clearly, the left-hand side of this matrix inequality depends linearly on the matrix $P(i)$, and its dimension is completely determined by that of the ith subsystem ${\mathbf{\Sigma }}_i$. Therefore, feasibility of this inequality can be effectively verified in general.

Note that a necessary condition for the feasibility of inequality (7.22) is the existence of a positive definite matrix $P(i)$ such that

$P(i)-A_{\mathbf{xx}}^T(i)P(i)A_{\mathbf{xx}}(i)-A_{\mathbf{zx}}^T(i){\mathrm{\Theta }}_i^2{A}_{\mathbf{zx}}(i)>0.$

This inequality further leads to

$P(i)-A_{\mathbf{xx}}^T(i)P(i)A_{\mathbf{xx}}(i)>0.$

According to the Lyapunov stability theory [1,2,4], this means that the matrix $A_{\mathbf{xx}}(i)$ is stable. In other words, a necessary condition for the feasibility of inequality (7.22) is that when every subsystem of system Σ is completely isolated from interactions with all the other subsystems of the plant, then all they are stable. Clearly, this is generally not required to guarantee the stability of the whole system, noting that a controllable and observable open loop unstable system can be stabilized by an appropriately designed feedback controller. From these aspects it is also clear that the associated condition is generally conservative.

7.2.3 Robust Stability of a Networked System

Modeling errors are unavoidable in practical engineering. A general requirement about a system property is that it is not sensitive to modeling errors, which is widely known as robustness of this property. This section investigates the robust stability of a networked system when there exist both parametric modeling errors and unmodeled dynamics in a state space model of each of its subsystems. To deal with this problem, an extra input vector and an extra output vector are introduced into the description of the dynamics of its ith subsystem ${\mathbf{\Sigma }}_i$ with the purpose of reflecting influences of modeling errors while the subsystem connections remain unchanged. More precisely, the state space model of Eq. (7.1) is modified into the following form:

$\left[\begin{array}{c}\hfill x(k+1,i)\hfill \\ \hfill w(k,i)\hfill \\ \hfill z(k,i)\hfill \\ \hfill y(k,i)\hfill \end{array}\right]=\left[\begin{array}{cccc}\hfill A_{\mathbf{xx}}(i)\hfill & \hfill A_{\mathbf{xd}}(i)\hfill & \hfill A_{\mathbf{xv}}(i)\hfill & \hfill B_{\mathbf{x}}(i)\hfill \\ \hfill A_{\mathbf{wx}}(i)\hfill & \hfill A_{\mathbf{wd}}(i)\hfill & \hfill A_{\mathbf{wv}}(i)\hfill & \hfill B_{\mathbf{w}}(i)\hfill \\ \hfill A_{\mathbf{zx}}(i)\hfill & \hfill A_{\mathbf{zd}}(i)\hfill & \hfill A_{\mathbf{zv}}(i)\hfill & \hfill B_{\mathbf{z}}(i)\hfill \\ \hfill C_{\mathbf{x}}(i)\hfill & \hfill C_{\mathbf{d}}(i)\hfill & \hfill C_{\mathbf{v}}(i)\hfill & \hfill D_{\mathbf{u}}(i)\hfill \end{array}\right]\left[\begin{array}{c}\hfill x(k,i)\hfill \\ \hfill d(k,i)\hfill \\ \hfill v(k,i)\hfill \\ \hfill u(k,i)\hfill \end{array}\right].$

In addition, the unmodeled dynamics in this subsystem and the parametric errors in the matrices $A_{\mathbf{⁎}\mathrm{\#}}(i)$ with $\mathbf{⁎},\mathrm{\#}=\mathbf{x},\mathbf{d},\mathbf{v},\mathbf{w},\mathbf{z}$ are respectively described by

$d(k,i)={\mathrm{\Delta }}_u(q,i)w(k,i),{\Vert {\mathrm{\Delta }}_u(q,i)\Vert }_{\infty }⩽1,$

$A(i)=A_0(i)+E_i{\mathrm{\Delta }}_p(i)F_i,\overline{\sigma }({\mathrm{\Delta }}_p(i))⩽1,$

$A(i)=\left[\begin{array}{ccc}\hfill A_{\mathbf{xx}}(i)\hfill & \hfill A_{\mathbf{xd}}(i)\hfill & \hfill A_{\mathbf{xv}}(i)\hfill \\ \hfill A_{\mathbf{wx}}(i)\hfill & \hfill A_{\mathbf{wd}}(i)\hfill & \hfill A_{\mathbf{wv}}(i)\hfill \\ \hfill A_{\mathbf{zx}}(i)\hfill & \hfill A_{\mathbf{zd}}(i)\hfill & \hfill A_{\mathbf{zv}}(i)\hfill \end{array}\right].$

Here q stands for the one-step forward shift operator, ${\mathrm{\Delta }}_u(q,i)$ and ${\mathrm{\Delta }}_p(i)$ have respectively a prescribed structure represented by ${\mathbf{\Delta }}_u(i)$ and ${\mathbf{\Delta }}_p(i)$. Moreover, $E_i$ and $F_i$ are known matrices with compatible dimensions. In this description, the matrices $E_i$ and $F_i$ are introduced to reflect how parametric errors influence subsystem parameters, whereas ${\mathrm{\Delta }}_u(q,i)$ and ${\mathrm{\Delta }}_p(i)$ denote respectively unmodeled dynamics and parametric errors existing in the state space model. This description is general enough to describe a large class of system dynamics. Moreover, a widely adopted structure for modeling errors is that they are block diagonal [1,22].

With this uncertainty model, the following results are obtained for the robust stability of the networked system Σ. Their proof is given in the appendix attached to this chapter.

Note that the condition of Theorem 7.3 can be verified independently for each subsystem. This means that the well-developed SSV analysis methods can be directly applied to its verification in general. Compared with the results of [10], Theorem 7.3 is valid for a wider class of uncertainty models and have a lower computational complexity. As a matter of fact, its computational complexity increases only linearly with the increment of the subsystem number N.

On the other hand, from Eq. (7.A.4) we can derive another SSV-based sufficient condition for the robust stability of system Σ simultaneously using all subsystem parameter matrices and the subsystem connection matrix. However, the corresponding matrix and the uncertainty structure generally have a high dimension, which is not appropriate to be applied to a large-scale networked system.

7.3.1 Modeling Errors Described by IQCs

In the description of modeling errors, a well-known but relatively abstract expression is the so-called integral quadratic constraints (IQCs) [23,24]. As its name indicates, in this description, both integration and a quadratic form are utilized. More precisely, let $\mathrm{\Pi }(\cdot ,t)$ be a bounded self-adjoint operator, and let $\mathrm{\Delta }(\cdot ,t)$ be a bounded and causal operator that maps a p-dimensional signal to a q-dimensional signal. If for each p-dimensional signal $v(t)$ with finite energy, that is, $v(t)\in {\mathcal{L}}_2^p$,

${\int }_0^{\infty }{\left[\begin{array}{c}\hfill v(t)\hfill \\ \hfill \mathrm{\Delta }(v(t),t)\hfill \end{array}\right]}^T\mathrm{\Pi }(\left[\begin{array}{c}\hfill v(t)\hfill \\ \hfill \mathrm{\Delta }(v(t),t)\hfill \end{array}\right],t)dt⩾0,$

then we say that the operator $\mathrm{\Delta }(\cdot ,t)$ satisfies the IQC defined by the operator $\mathrm{\Pi }(\cdot ,t)$, which is usually denoted by $\mathrm{\Delta }\in \mathcal{IQC}(\mathrm{\Pi }(\cdot ,t))$. This inequality is capable of describing many modeling errors adopted in system analysis and synthesis. For example, a $q\times p$-dimensional parametric modeling error Δ with $\overline{\sigma }(\mathrm{\Delta })<\gamma$ can be easily seen to satisfy

${\int }_0^{\infty }{\left[\begin{array}{c}\hfill v\hfill \\ \hfill \mathrm{\Delta }(v)\hfill \end{array}\right]}^T\left[\begin{array}{cc}\hfill I_p\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -{\gamma }^2{I}_q\hfill \end{array}\right]\left[\begin{array}{c}\hfill v\hfill \\ \hfill \mathrm{\Delta }(v)\hfill \end{array}\right]dt⩾0,\forall v(t)\in {\mathcal{L}}_2^p,$

whereas the $q\times p$-dimensional unmodeled dynamics $\mathrm{\Delta }(\cdot )$ with ${\Vert \mathrm{\Delta }\Vert }_{\infty }<\gamma$ can be straightforwardly proved to meet the IQC

${\int }_0^{\infty }{\left[\begin{array}{c}\hfill v\hfill \\ \hfill \mathrm{\Delta }(v)\hfill \end{array}\right]}^T\left[\begin{array}{cc}\hfill I_p\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -{\gamma }^2{I}_q\hfill \end{array}\right]\left[\begin{array}{c}\hfill v\hfill \\ \hfill \mathrm{\Delta }(v)\hfill \end{array}\right]dt⩾0,\forall v(t)\in {\mathcal{L}}_2^p.$

In addition to these, IQCs are also able to describe system properties like passivity, and combine several IQCs into a single IQC [25,26].

Utilization of IQCs in system analysis and synthesis can be traced back to the 1960s, in which they are extensively applied to the absolute stability analysis of a nonlinear system, whereas their applications to the analysis and synthesis of a system with modeling errors were started around the beginning of the 1990s [23–26]. A particularly important property of IQCs is that they have a clear frequency domain interpretation, which makes engineering intuition in system analysis and synthesis much clearer and easier to be understood. In particular, if the operator $\mathrm{\Pi }(\cdot ,t)$ of Eq. (7.27) is linear and time invariant and has a transfer function matrix representation, then the IQC of Eq. (7.27) can be equivalently rewritten as

${\int }_{-\infty }^{\infty }{\left[\begin{array}{c}\hfill v(j\omega )\hfill \\ \hfill \mathrm{\Delta }(v)(j\omega )\hfill \end{array}\right]}^H\mathrm{\Pi }(j\omega )\left[\begin{array}{c}\hfill v(j\omega )\hfill \\ \hfill \mathrm{\Delta }(v)(j\omega )\hfill \end{array}\right]d\omega ⩾0.$

In addition to this, an IQC constraint is also capable of describing nonlinear dynamic uncertainties, which makes the associated system analysis and synthesis methods able to cope with nonlinear modeling errors.

The following results are fundamental in robust stability analysis of a system with IQC described uncertainties [10,25]. They also play central roles in the derivations of a criterion for the robust stability of a networked system, which is the main objective of this section.