The Basic Idea

We will model real-world complex dynamical systems as the feedback interconnected block diagram shown in Figure 1.1. The mathematical representation of this system can be expressed as follows.

$$\begin{array}{l} z = \sum_{l=0}^{n}G_{0l}w_{l}+e_{0},\\v_{k}=\sum_{l=0}^{n}G_{kl}w_{l}+e_{k},\\ w_{k}=\Delta_{k}(v_{k}).\end{array}$$

Each block in the diagram represents a subsystem. The \(G_{kl}\) subsystems are stable linear time-invariant (LTI) transfer functions while the \(\Delta_{k}\) subsystems represents nonlinearities, time-varying coefficients, parametric uncertainties, and unmodelled dynamics. The exact mathematical description of (1.1) is in general very complicated due to the different nature of the various \(\Delta_{k}\) blocks. To make possible an accurate and rigorous analysis of stability and performance of such a complex system it is necessary to simplify the model. The basic idea behind the IQC framework is to replace each \(\Delta_{k}\) block and every external signal \(e_{k}\) with an IQC characterization. In this way we embed the real system model in (1.1) in an IQC relaxation which contains all possible solutions of the real system. Analysis of the relaxed model is much easier and can be done using convex optimization. This approximation is often the best available for analysis and it is in some cases also exact.

To discuss the IQC framework in some more detail we introduce the space of square integrable functions \(L_{2}[0,\ \infty)\) and the inner product on this space

$$ \langle v,w \rangle = \int_{0}^{\infty} v(t)^{T}w(t)dt = \int_{-\infty}^{\infty} \hat{v}(j\omega)^{*}\hat{w}(j\omega)d\omega$$

where \(\hat{v}\) and \(\hat{w}\) are the Fourier transforms of v and w, respectively. There are certain mathematical assumptions on the \(\Delta_{i}\) subsystems and on the interconnection in (1.1) that must be valid. The essential assumption is that all \(\Delta_{i}\) represents causal and bounded operators on \(L_{2}[0,\ \infty)\). For the exact meaning of this and other technical details below we refer to e.g.[2].

Figure 1.1: System under consideration

In the IQC framework, integral quadratic constraints are used to characterize external disturbances \(e_{k}\) and operators \(\Delta_{k}\). An operator \(\Delta\) is said to satisfy the IQC defined by \(\Pi_{\lambda}\) (notation: \(\Delta \in\) IQC(\(\Pi_{\lambda}\))), if for all input-output pairs \(w = \Delta(v)\), the integral inequality

$$ \left\langle \begin{bmatrix} v\\ w \end{bmatrix}, \Pi_{\lambda} \begin{bmatrix} v\\w \end{bmatrix} \right\rangle \geq 0$$

is valid. The operator \(\Pi_{\lambda}\) is often referred to as the multiplier of the IQC. Here, we consider multipliers which are linearly parameterized by \(\lambda\), where \(\lambda\) is a finite dimensional vector belonging to a convex cone \(\wedge\). The subscript \(\lambda\) is used to reflect this linear parameterization.

Similarly, for a given external disturbance e belonging to some set \(\Xi\), we say that e satisfies the integral quadratic constraint defined by a bounded, self-adjoint operator \(\Psi_{\lambda}\) on \(L_{2}[0,\ \infty )\) (and similar notation \(\Xi \in\) IQC(\(\Psi_{\lambda}\)) applies) if

$$ \langle e,\Psi_{\lambda}e\rangle \geq 0$$

for all \(e \in \Xi\) . This allows us to describe the spectral characteristics of the uncertainty.

To start with the IQC analysis, one assumes that the operators \(\Delta_{k}\) and the external disturbances \(e_{k}\) are known to satisfy integral quadratic constraints defined by \(\Pi_{k,\lambda}\) and \(\Psi_{k,\lambda}\), respectively. More comments on how to find IQCs for \(\Delta_{k}\) and \(e_{k}\), as well as the parameterization of IQCs, are given in the examples. The idea is then to replace these operators and sets by arbitrary signals that satisfy these IQCs. To illustrate this idea we let \(D_{k,\lambda} : \textbf{L}_{2}^{m_{k}}[0,\ \infty) \rightarrow P ( \textbf{L}_{2}^{m_{k}}[0,\ \infty ))\) be the set-valued function \(w_{k} \in D_{k,\lambda} (v_{k} )\), where

$$ D_{k,\lambda}(v_{k})=\left\{ w_{k} \in \textbf{L}_{2}^{m_{k}}[0,\ \infty) \left| \left\langle \begin{bmatrix} v_{k}\\ w_{k}\end{bmatrix}, \Pi_{k,\lambda} \begin{bmatrix} v_{k}\\ w_{k}\end{bmatrix} \right\rangle \geq 0 \right.\right\}$$

We also introduce the sets \(\Xi_{k,\lambda}\)

$$\Xi_{k,\lambda} = \left\{ e_{k} \in \textbf{L}_{2}^{l_{k}}[0,\ \infty) \left| \langle e, \Psi_{k,\lambda}e \rangle \geq 0 \right.\right\}$$

The essential idea of IQC analysis is to replace the operator \(\Delta_{k}\) by the set-valued function \(D_{k,\lambda}\) and the external disturbance e k by any signal from the set \(\Xi_{k,\lambda}\). More specifically, we now consider the system which can be described as in Figure 1.2. All possible solutions (in \(\textbf{L}_{2}[0,\ \infty)\)) of the original system are also valid solutions of the new system. This follows since, by definition,
every \(w_{k}=\Delta_{k}(v_{k})\) belongs to \(D_{k,\lambda}(v_{k})\)
every \(e_{k}\) belongs to \(\Xi_{k,\lambda}\)
The original system is now embedded in the IQC relaxation in Figure 1.2. It is sufficient to analyze the new system, and stability and worst-case performance of the new system implies the same properties hold for the original system.

Figure 1.2: IQC relaxation of the system in Figure 1.1

Now let \(G_{0}\) and \(G\) be the operators

$$ \begin{bmatrix} G_{0}\\\hline G \end{bmatrix} := \begin{bmatrix} G_{01} & \cdots & G_{0n}\\ G_{11} & \cdots & G_{1n}\\ \vdots & \ddots & \vdots\\ G_{n1} & \cdots & G_{nn} \end{bmatrix}$$

and let \(v^{T} = [ v_{1}^{T}~\cdots \ v_{n}^{T}],\ w^{T} = [w_{1}^{T} \ \cdots \ w_{n}^{T}],\ e^{T} = [e_{1}^{T}\ \cdots e_{n}^{T}]\), and \(e_{tot}^{T} = [e_{0}^{T}\ e^{T}]\). Also, define the following quadratic forms

$$ \sigma_{\Pi}(v,w)=\sum_{k=1}^{n}\left\langle \begin{bmatrix} v_{k}\\w_{k} \end{bmatrix},\Pi_{k,\lambda} \begin{bmatrix} v_{k}\\ w_{k} \end{bmatrix} \right\rangle$$

$$ \sigma_{\Psi}(e_{t})=\sum_{k=0}^{n}\langle e_{k},\Psi_{k,\lambda}e_{k}\rangle$$

Note that the quadratic forms \(\sigma_{\Pi}\) and \(\sigma_{\Psi}\) are linearly parameterized by \(\lambda\).

In the IQC\(\beta\) toolbox, checking stability is never the only objective of the analysis. Instead, stability is inferred from a robust performance problem. Typically, we consider the induced gain from some disturbance \(e_{0}\) to a particular performance output \(z\). If there exists a positive \(\gamma\) such that

$$ \int_{0}^{\infty} (|z|^{2}-\gamma |e_{0}|^{2})dt \leq 0$$

for all possible solutions of the IQC relaxation, then under some weak conditions the system is stable and the induced gain is \(\sqrt{\gamma}\). More generally, we can consider quadratic performance constraints of the following form

$$ \sigma_{\Theta}(z,e_{tot}) = \left\langle \begin{bmatrix} z\\ e_{tot}\end{bmatrix}, \Theta \begin{bmatrix} z\\e_{tot} \end{bmatrix} \right\rangle$$

where \(\Theta\) is a bounded, self-adjoint operator defined on \(\textbf{L}_{2}[0,\ \infty)\) space. What we would like to do is to check whether \(\sigma_{\Theta}(z,\ e) \leq 0\) for all solutions of the system when \(e_{tot} \in \Xi_{0,\lambda} \times \cdots \times \Xi_{n,\lambda}\). This can be carried out by solving a constrained feasibility problem
Find \(\lambda \in \wedge\), such that

$$ \sigma(G_{0}w+e_{0},Gw+e,e_{tot},w) \leq 0,\ \forall (e_{tot},w) \in \textbf{L}_{2}[0,\ \infty).$$

where \(\sigma(z,\ v,\ e_{tot},\ w) = \sigma_{\Theta}(z,\ e_{tot} ) + \sigma_{\Pi}(v,\ w) + \sigma_{\Psi}(e_{tot})\). This is a convex optimization problem that can be solved efficiently. Here, due to limitation of space, we will not discuss how the computational problem arises.