Integral Quadratic Constraints

August 10, 2009

1 Introduction

Integral Quadratic Constraints (IQC1) are inequalities used to describe (partially) pos-sible signal combinations within a given dynamical system. IQC offer a framework forabstracting ”challenging” (e.g. non-linear, time-varying, uncertain, or distributed) el-ements of dynamical system models to aid in rigorous analysis of robust stability andperformance (more specifically, to establish L2 gain bounds, passivity, and other systemproperties which can be expressed, exactly or approximately, in terms of generalized dis-sipativity). While the technique can be employed to prove general theorems, it is mostpowerful when used to derive optimization-based algorithms for certification of stabilityand robustness of specific feedback systems.

IQC can be viewed as implicit generalized dissipation inequalities with known quadra-tic supply rates and unspecified storage functions, not necessarily quadratic or sign defi-nite. Alternatively, they have frequency a domain interpretation as bounds on the degreeof harmonic distortion produced by a specific element of the complete model. The pastresearch in nonlinear systems and robust control can be harvested to extract rich IQCdescriptions of commonly used components of feedback systems. These IQC can then bere-used in a modular approach to system analysis.

The IQC framework is closely related to multiplier based passivity, upper boundingof structured singular values, quadratic relaxations in non-convex optimization (includ-ing the sums of squares approach to positivity of multivariable polynomials), and otherconstructive techniques for handling nonlinearity and uncertainty.

2 Getting Started With IQC

In this section we introduce basic elements of the IQC framework by having it appliedto the classical problem of global analysis of a feedback interconnection of a SISO LTI

1We will use ”IQC” for both singular and plural forms.

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system and a memoryless ”rate limited” nonlinearity. The overall model can be describedeither by the block diagram

G(s)

φ(·)

-

w y

or by the n-dimensional ordinary differential equation

x = Ax+Bw, w = φ(Cx), (2.1)

where A,B,C are given real matrices (det(sI − A) 6= 0 for Re(s) ≥ 0), G(s) = C(sI −A)−1B is the transfer function of the LTI subsystem, and φ : R 7→ R is such that φ(0) = 0and φ ∈ [0, 1]. The analysis objective is to establish global asymptotic stability of theequilibrium x = 0.

2.1 IQC Modeling

IQC analysis begins with IQC modeling, which includes (i) recognizing the exact modelS of the dynamical system under consideration; (ii) specifying the analysis objective asan IQC to be established for S; and (iii) finding a sufficiently rich family of IQC readilyknown to be satisfied on S.

To apply IQC analysis to system (2.1), let S be its behavioral model in terms of signalsw and x:

S = [w;x] : x = Ax+Bw, w = φ(Cx). (2.2)

2.1.1 Complete and Conditional IQC

For a set S = q of d-dimensional signals q = q(t), an IQC is defined by a quadraticform2 σ : Rd 7→ R (referred to as the supply rate of the IQC), and states existenceof a lower bound κ = κ(q(0)) for the integrals of σ(q(t)) over long intervals of time: acomplete IQC σ 0 on S means existence of a continuous function κ : Rd 7→ R+ suchthat κ(0) = 0 and ∫ T

0

σ(q(t))dt ≥ −κ(q(0)) (2.3)

2While it is possible to construct a similar theory utilizing general supply rates σ, requiring σ to bequadratic is important for practical feasibility of the approach: eventually, checking positive semidefinite-ness of linear combinations of different σ will be required to reach an analysis conclusion, and quadraticfunctions form the only generic class for which this can be done efficiently.

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for all q ∈ S and T ≥ 0, while a conditional IQC σ B 0 on S states that∫ ∞0

σ(q(t))dt ≥ −κ(q(0)) (2.4)

for all signals q ∈ S of finite energy.Informally, one can think of a complete IQC σ 0 as an implicit dissipation inequality

σ(q(t)) ≥ dV (xh(t))/dt where xh = xh(t) is the hidden state of the system, and V =V (xh) is an unknown non-negative storage function satisfying an unknown3 upper boundV (xh(0)) ≤ κ(q(0)). In a similar way, a conditional IQC σ B 0 corresponds to the casewhen the storage function has a not necessarily non-negative, but satisfies V (0) ≥ 0.

2.1.2 IQC as Analysis Objective and Background Information

IQC can be used to define an objective of system analysis. For example, since A is aHurwitz matrix, global asymptotic stability of the equilibrium x(0) = 0 in (2.1) is impliedby the energy bound ∫ T

0

|w(t)|2dt ≤ κ(x(0)), (2.5)

where κ : Rn 7→ R+ is continuous and such that κ(0) = 0. By definition, (2.5) is acomplete IQC σ∗ 0 on S, where

σ∗([w;x]) = −|w|2 (w ∈ R, x ∈ Rn). (2.6)

In the standard IQC analysis flow, the main effort in proving σ∗ 0 is given to findinga quadratic form σ0 such that σ0 B 0 and σ∗ ≥ σ0. This is accomplished by compilinga large set Λ0 = σ of quadratic forms for which the IQC σ B 0 are either trivial orreadily established, and then optimizing over the convex hull Λ of Λ0 to satisfy σ∗ ≥ σ forsome σ ∈ Λ. Thus, the IQC σ B 0 for σ ∈ Λ0 become elementary pieces of ”backgroundinformation” about S, and success of IQC analysis depends on one’s ability to generate aset Λ0 which is rich enough.

2.1.3 Sector IQC

Since φ(0) = 0 and φ ∈ [0, 1], φ(y) is between 0 and y for all y ∈ R, i.e. the points[y;w] ∈ R2 such that w = φ(y) lie in the sector w(y − w) ≥ 0.

- φ(·) - -

6

y(t)

w(t)

y w

3In most applications, it is possible to have explicit upper bounds κ(·), which can be useful in per-forming advanced analysis tasks.

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Hence w(t)[y(t) − w(t)] ≥ 0 for all t and all [w;x] ∈ S, y = Cx, which implies that thecomplete IQC σs 0 is satisfied on S for

σs([w;x]) = 2w(Cx− w) (w ∈ R, x ∈ Rn). (2.7)

The corresponding upper bound κ = κs can be chosen as κs ≡ 0.

2.1.4 Popov IQC

The sector IQC σs 0 does not reflect the time invariant nature of the relation betweeny = Cx and w: it is still satisfied when the equation w(t) = φ(y(t)) in the definition ofS is replaced by w(t) = α(t)−1φ(α(t)y(t)), where α = α(t) is an arbitrary time-varyingnon-zero coefficient. Two less obvious IQC can be derived from the observation that forevery [w;x] ∈ S and y = Cx we have

wy =d

dtψ(y), where ψ(h) =

∫ h

0

φ(τ)dτ.

Since, due to φ(0) = 0 and φ ∈ [0, 1], we have 0 ≤ ψ(y) ≤ y2/2 for all y ∈ R, the IQCσp 0 (with κ([w;x]) = |Cx|2 for w ∈ R, x ∈ Rn) and −σp B 0 (with κ ≡ 0) hold on Sfor

σp([w;x]) = 2xC(Ax+Bw) (w ∈ R, x ∈ Rn). (2.8)

- φ(·) - -

6

w(t)

w1/s-

y y y(t)

2.1.5 Pure Integrator IQC

Since, due to (2.1), Ax+Bw is the derivative of x, for every symmetric real matrix Q = Q′

the signal 2x′Q(Ax + Bw) is the derivative of x′Qx. Hence σI B 0 on S for all Q = Q′

and σI 0 on S for all Q = Q′ ≥ 0, where

σI([w;x]) = 2x′Q(Ax+Bw) (w ∈ R, x ∈ Rn), (2.9)

with κ([w;x]) = x′Qx.

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2.1.6 Zames-Falb IQC

To give a taste of less obvious IQC relations describing the memoryless nonlinearity ofsystem (2.1), consider the following IQC from a (much larger) family established by aclassical result by G. Zames and P. Falb (see the IQC Library section 4 for details).Define an extension Se of S by

Se = [w;x; ξ; ξ] : [w;x] ∈ S, ξ + 3ξ + 2ξ = w, ξ(0) = ξ(0) = 0.

It turns out that, due to the rate limit φ ∈ [0, 1], the complete IQC σz 0 is satisfied onSe, where

σz([w;x; ξ0; ξ1]) = 2(Cx− w)(w − 2ξ0) (w, ξ0, ξ1 ∈ R, x ∈ Rn), (2.10)

with κ ≡ 0.The IQC has an interesting interpretation in the frequency domain. Assume that

y = Cx has finite energy, and let v = y − w = y − φ(y). In general, the time domainrelation w(t)v(t) ≥ 0 does not imply the corresponding inequality w(jω)′v(jω) ≥ 0 forthe Fourier transforms (in general, w(jω)′v(jω) is a complex number anyway), thoughthe implication holds when φ(y) = δy is linear with δ ∈ [0, 1]. Also, due to the Parcevalidentity, the integral of Re[w′v] over all frequencies is always non-negative. The IQC σzB0claims that, due to the rate limit imposed on the memoryless nonlinearity φ, the integralof Re[w′Ge(jω)v], where Ge(s) = (s2 + 3s)/(s2 + 3s + 2), is also not negative. In otherwords, the IQC σz B 0 sets a limit on the amount of harmonic distortion which can beintroduced by the nonlinear transformation y 7→ φ(y).

While it can be shown that the IQC σz 0 indicates existence of a non-negativestorage function Vz = Vz(ξ, ξ) such that (Cx − w)(w − 2ξ) ≥ dVz/dt, no explicit formfor Vz is available4 – the proof is based on a converse storage function argument, and, assuch, is not very constructive. Despite its ”implicit” nature, the IQC σz 0, can be veryuseful in analysis of specific nonlinear systems.

Note that applying the ”extension” S 7→ Se does not remove any of the original IQC:the sector and Popov IQC σs 0, σp 0, −σp B 0 are still valid on Se5, the ”objective”IQC σ∗ 0 has same meaning on S and Se, and the family of ”pure integration” IQCσI B 0 generalizes to σIe B 0, where

σIe([w;x; ξ0; ξ1]) = 2

xξ0

ξ1

′Qe

Ax+Bwξ1

−2ξ0 − 3ξ1 + w

, Qe = Q′e. (2.11)

4To the best of author’s knowledge5Here we allow some abuse of notation, using the same identifier for a quadratic form σ : Rd 7→ R

and its ”extension” σ : Rd+2 7→ R defined by σ([w;x; ξ0; ξ1]) = σ([w;x]).

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2.1.7 Combining IQC

As long as linear operations are concerned, IQC can be handled as usual inequalities: ifσ1 B 0 and σ2 B 0 on S then c1σ1 + c2σ2 B 0 on S for arbitrary non-negative real numbersci. Similarly, if σ1 0 and σ2 0 then c1σ1 + c2σ2 0. This allows the user of the IQCframework to convert individual IQC satisfied for subsystems of a larger model into oneconvex set Λ = σ of quadratic forms σ defining valid conditional IQC σ B 0 on S. Inaddition, since many of the classical IQC are readily complete, a convex subset Λ+ ⊂ Λsuch that σ 0 for all σ ∈ Λ+ can be constructed as well.

In particular, for the behavioral model S defined by (2.1), the set Λ compiled so faris given by

Λ = csσs + cpσp + σI : cs ∈ R+, cp ∈ R, Q = Q′, (2.12)

where σs, σp, σI are defined in (2.7), (2.8), and (2.9). In contrast, the set Λ+ of recognizedcomplete IQC consists of σ = csσs + cpσp + σI with cs ≥ 0, cp ≥ 0, Q = Q′ ≥ 0. It isimportant to understand that, depending on the coefficient matrices A, B, C, there couldbe some σ ∈ Λ such that σ 0 on S but σ 6∈ Λ+: those are the quadratic forms definingvalid complete IQC which are not recognized as such.

Similarly, for the extended model Se,Λ = csσs + cpσp + σIe + czσz : cs, cz ∈ R+, cp ∈ R, Qe = Q′e, (2.13)

with σIe defined in (2.11), and the forms from Λ+ satisfy the additional constraints cp ≥ 0and Qe ≥ 0.

2.2 Feasibility Optimization and Post-Feasibility Analysis

Once the exact model S is defined, the objective IQC σ∗ is selected, and a convex setΛ = σ of valid IQC σ B 0 on S is compiled, the next step in the IQC framework isfeasibility analysis, understood as the search for σ0 ∈ Λ satisfying σ∗ ≥ σ0. In a typicalapplication, there is also a ”cost” parameter to be minimized (in the IQC model for (2.1),this could be γ, subject to γI ≥ |cp|C ′C + Q, bounding κ∗ in (2.5)), so there is a welldefined optimization criterion in addition to the feasibility task.

If there is no σ0 ∈ Λ such that σ∗ ≥ σ0, the IQC analysis fails, and the only option isto return to the IQC modeling step to find a larger set Λ. Sometimes it helps to consider(as in the Zames-Falb IQC example) replacing S with an extension Se of S: a cleverlydefined set Se = qe of signals of fixed dimension de > d such that for every q ∈ S thereis at least one signal g such that [q; g] ∈ Se.

Having σ∗ ≥ σ0 for some σ0 ∈ Λ means that σ∗ B 0, i.e. either σ∗ 0, as desired,or σ∗ B 0 but σ∗ 6 0. In most applications, the latter means that the system is stronglyunstable: subject to mild assumptions, the conditional IQC σ∗B0 on S is a ”certificate ofdichotomy”, i.e. it implies that S is either quite stable or very unstable, no middle ground.In most situations, general theorems of post-feasibility analysis provide easy criteria todecide which outcome actually takes place.

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2.2.1 Feasibility Optimization and Semidefinite Programming

When the set Λ is linearly parameterized by a vector variable which is in turn subjectto a linear matrix inequality (LMI) constraint, the search for σ0 ∈ Λ satisfying σ∗ ≥ σ0

becomes a semidefinite program.For example, in the analysis of (2.1) with Λ defined by (2.12), one possible semidefinite

program to solve is γ → min subject to γI ≥ ±cpC ′C +Q, cs ≥ 0, and[2cs − 1− 2cpCB −csC − cpCA−B′Q

−csC ′ − cpA′C ′ −QB −QA− A′Q

]≥ 0, (2.14)

where the decision variables are γ, cp, cs ∈ R and Q = Q′ ∈ Rn,n. While the linear matrixinequality in (2.14) looks quite cumbersome, the quadratic form notation is usually muchmore straightforward. For example, the functional version of (2.14) is

−|w|2 − 2csw(Cx− w)− 2cpwC(Ax+Bw)− 2x′Q(Ax+Bw) ≥ 0 ∀ w, x.Certain programming tools can be used to convert quadratic form notation into the formatof the standard semidefinite program solvers.

2.2.2 Feasibility Optimization and the KYP Lemma

While conversion to a semidefinite program, to be solved numerically, appears to be theonly practical approach to feasibility analysis of advanced IQC models, valuable theoret-ical insight can be gained by applying the following version of the classical KYP Lemma(sometimes also called positive real Lemma).

Theorem 2.1 For real matrices A,B of dimensions n-by-n and n-by-m respectively, anda Hermitian form σ : Cm+n 7→ R with real coefficients, let Ω be the set of ω ∈ R suchthat the matrix Aω = jωIn − A is invertible. For ω ∈ Ω let Lω = A−1

ω B. Consider thefollowing statements:

(a) σ([u;Lωu]) ≥ 0 for all u ∈ Cm and ω ∈ Ω;

(b) there exists ω ∈ Ω such that σ([u;Lωu]) > 0 for all u ∈ Cm, u 6= 0;

(c) quadratic form σ([u;x]) − 2x′Q(Ax + Bu) (where x ∈ Rn and u ∈ Rm) is positivesemidefinite for some real matrix Q = Q′.

Then

(i) (c) implies (a);

(ii) when the pair (A,B) is controllable, (a) implies (c);

(iii) when the pair (A,B) is CT stabilizable, (a) and (b) together imply (c).

Theorem 2.1 can be used to formulate frequency domain conditions of feasibility inIQC analysis when the ”pure integration” term 2x′Q(Ax+Bu) is present in the descriptionof Λ.

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2.2.3 The Circle Criterion

In the special case when only the pure integrator and sector IQC are used to representsystem (2.1), the feasibility analysis calls for finding cs ∈ R+ and Q = Q′ such that

−|w|2 − 2csw(Cx− w)− 2X ′Q(Ax+Bw) ≥ 0 ∀ w ∈ R, x ∈ Rn. (2.15)

Theorem 2.1 offers valuable insight into the feasibility of the associate semidefinite pro-gram: together with the identity CLω = G(jω), it allows one to conclude that Q = Q′

satisfying (2.15) exists for some cs ∈ R+ if and only if ρ < 1, where ρ is the minimalupper bound of Re[G(jω)] over ω ∈ R.

When ρ ≥ 1, the IQC analysis proves nothing: system (2.1) may be stable or unstabledepending on fine details not reflected in the IQC model used so far. When ρ < 1, wehave σ∗ ≥ σ0 where σ0 = csσs + σI for some cs > 0 and Q = Q′. Since σs 0 onS and σI 0 on S for Q ≥ 0, the desired complete IQC σ∗ 0 will be establishedas long as it is possible to guarantee that Q = Q′ is positive semidefinite. When Q isfound numerically by solving a semidefinite program, the inequality Q ≥ 0 can be checkeddirectly. Otherwise, checking that Q ≥ 0 becomes a typical post-feasibility analysis task.

Substituting w = 0 into the inequality in (2.15) yields QA + A′Q ≤ 0. Since A is aHurwitz matrix, which in turn implies Q ≥ 0, recovering the circle criterion: system(2.1) is globally asymptotically stable when Re[G(jω)] < 1 for all ω ∈ R.

2.2.4 The Popov Criterion

The case when the pure integrator, sector, and Popov IQC are used together to representsystem (2.1), the feasibility analysis calls for finding cs ∈ R+, cp ∈ R, and Q = Q′ suchthat

−|w|2−2csw(Cx−w)−2cpwC(Ax+Bw)−2x′Q(Ax+Bw) ≥ 0 ∀ w ∈ R, x ∈ Rn. (2.16)

Theorem 2.1 can be used to show that existence of cs ∈ R+, cp ∈ R, Q = Q′ ∈ Rn,n

satisfying (2.16) is equivalent to existence of p ∈ R such that ρp < 1, where ρp is theminimal upper bound of Re[(1 + jωp)G(jω)] over ω ∈ R (when ρp < 1, one can usecp = pcs with cs > 0 large enough).

Since the storage function for the IQC σs + pσp + σI B 0 is V (x) = x′Qx + 2pψ(Cx),the desired IQC σ∗ 0 is established when non-negativity of V is assured. Since (2.16)with w = 0 implies QA + A′Q ≤ 0, we know that Q ≥ 0 is positive semidefinite. Sinceψ(y) ≥ 0 for all y ∈ R, this implies V ≥ 0 when p ≥ 0.

The post-feasibility analysis becomes trickier in the case p ≤ 0. The upper boundψ(y) ≤ y2/2 can be used to show that V ≥ 0 whenever Q + cpC

′C ≥ 0. However, thisspecial treatment is not necessary, as, according to the general post-feasibility analysistheorems discussed in section 3.2, the inequality σ∗ ≥ csσs+cpσp+σI implies the completeIQC csσs+cpσp+σI 0 whenever cs ≥ 0 and A is a Hurwitz matrix. This establishes the

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classical Popov criterion: system (2.1) is globally asymptotically stable when thereexists p ∈ R such that Re[(1 + pjω)G(jω)] < 1 for all ω ∈ R ∪ ∞.

3 Theory of IQC Analysis

In general, IQC analysis follows the steps highlighted in section 2 (IQC modeling, feasi-bility optimization, post-feasibility analysis). This section gives a formal presentation ofthe framework.

3.1 IQC Modeling

An IQC model describes a system (understood as a set S of signals of fixed dimension d)and an analysis objective in terms of IQC, where each IQC is a property of S defined bya quadratic form σ on Rd.

3.1.1 Signals

A continuous time (CT) signal of dimension d is an element of the set Ld of all measurablelocally bounded functions q : [0,∞) 7→ Rd (we also use L for Ld with d = 1). Similarly,a discrete time (DT) signal of dimension d is an element of the set `d (or simply ` ford = 1) of all functions f : Z+ 7→ Rd. To simplify notation when making statements whichapply to both DT and CT cases, we use L as the ”variable” time domain designation, i.e.L = L or L = ` (of course, L has the same meaning throughout every single statementor definition).

We define p = q+ by p(t) = q(t+ 1) for q ∈ `d and by p = dq/dt for q ∈ Ld (in whichcase the constraint dq/dt ∈ Ld is imposed automatically). We use I[q] to denote the totalintegral (or sum) of q:

I[q] =

∫ ∞0

q(t)dt for q ∈ Ld, I[q] =∞∑t=0

q(t) for q ∈ `d

(I[q] is not defined for some q, and it is possible to have I[q] =∞ or I[q] = −∞). We use‖q‖ to denote the square root of the energy ‖q‖2 = I[|q|2] of signal q, and denote by Ld0the set of all finite energy signals in Ld. Given T ≥ 0 and q ∈ Ld, p = [q]T ∈ Ld0 denotesthe past history of q: p(t) = q(t) for t ≤ T , p(t) = 0 for t > T .

3.1.2 Quadratic Forms

We treat elements of Rd (or Cd) as d-by-1 column matrices. Accordingly, a quadraticform σ : Rd 7→ R is a function defined by σ(x) = x′Px where P is a real d-by-d matrix(the set of all such matrices to be denoted by Rd,d), and ′ is the operation of Hermitian

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conjugation (transposition for real matrices). Without loss of generality, P can be assumedto be symmetric, i.e. such that P = P ′.

The Hermitian extension σH : Cd 7→ R of σ is then the function defined by σ(x) =x′Px for all x ∈ Cd. We also use σ〈q〉 as a shortcut notation for I[σ(q)] when q ∈ Ld0. Inthe special case σ(X) = |X|2 (X ∈ Rd), we use ‖x‖2

T in place of σ〈[x]T 〉 for x ∈ Ld andT ≥ 0.

3.1.3 Systems

Mathematically, we view a general system as a behavioral model, i.e. simply a set Sof signals of given dimension. The elements of S are assumed to represent all possiblecombinations of all signals of interest (the ”outputs” may include components of thehidden state as well as auxiliary signals introduced specifically to simplify the analysis).Accordingly, a system is a subset of Ld, where L = L or L = ` and d is a positive integer.

We also consider the special case of operator models, which are functions ∆ : Lk 7→ Lmmapping signals to signals. An operator model ∆ can be described by a behavioral modeldefined as the graph G∆ = [v; ∆(v)] : v ∈ Lk of ∆. Operator model ∆ is calledcausal when [∆(v)]T is completely defined by [v]T for all T ≥ 0, i.e. [∆(v1)]T = [∆(v2)]Twhenever [v1]T = [v2]T . Operator model ∆ is called stable (in the ”bounded input energyimplies bounded output energy” sense) if ‖∆(v)‖ <∞ whenever ‖v‖ <∞.

In particular, a finite order linear time-invariant (LTI) operator defined by real ma-trices A,B,C,D is a causal operator model mapping v to y = Cx + Dv ∈ Lk where x isdefined by x+ = Ax + Bv and x(0) = 0. An LTI operator is completely defined by itstransfer matrix G(λ) = D+C(λI−A)−1B, an element of the set RLk,m of all real rationalk-by-m matrix functions of scalar complex argument λ which are proper (i.e. bounded as|λ| → ∞).

3.1.4 IQC

Let σ : Rd 7→ R be a quadratic form. We say that system S ⊂ Ld satisfies the conditionalIQC defined by σ (shortcut notation σB0 or σ(q)B0 when it is more convenient) if thereexists a continuous function κ : Rd 7→ R+, such that κ(0) = 0 and

σ〈q〉 ≥ −κ(q(0)) whenever q ∈ S, ‖q‖ <∞. (3.17)

We say that system S satisfies the complete IQC defined by σ (shortcut notation σ 0or σ(q) 0) if there exists a continuous function κ : Rd 7→ R+, such that κ(0) = 0 and

σ〈[q]T 〉 ≥ −κ(q(0)) whenever q ∈ S, T ≥ 0. (3.18)

3.1.5 Extended Systems and IQC

Introducing new signals into consideration (an action referred to as extension here) fre-quently helps to expose IQC relations in the original model. In general, the behavioral

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model S ⊂ Ld+N is called an extension of S0 ⊂ Ld if for every q0 ∈ S0 there exists g ∈ LNsuch that [q0; g] ∈ S.

Example 3.1 Consider the behavioral model S0 = [u3;u2] : u ∈ L (it is the graph ofthe memoryless nonlinear operator v 7→ v2/3). It can be shown that every quadratic formσ : R2 7→ R such that σB0 on S0 is positive semidefinite, i.e. S0 does not satisfy any non-trivialIQC. The extension S = [u3;u2;u; 1] : u ∈ L, however, satisfies a set of useful IQC σ 0,where

σ([x3;x2;x1;x0]) = c1(x0x2 − x21) + c2(x0x3 − x1x2) + c3(x1x3 − x2

2),

and the coefficients ci ∈ R are arbitrary.

3.1.6 Stable LTI Extensions and Frequency Weighted IQC

The situation in which an extension S of S0 is defined by a stable LTI operator E accordingto S = [q0;Eq0] : q0 ∈ S0 is of special importance in the IQC framework. Allowingsome abuse of notation, let E = E(s) also denote the transfer matrix of E. Assume thatσ B 0 on S for some quadratic form σ : Rd+N 7→ R.

Let Π = Pσ,E ∈ RLd,d be the transfer matrix defined (uniquely) by the identities

σH([u;E(λ)u]) = u′Π(λ)u, Π(λ) = Π(λ)′ ∀ u ∈ Cd, λ ∈ C0,

where C0 = s ∈ C : Re(s) = 0 in the CT case, and C0 = z ∈ C : |z| = 1 in theDT case. The IQC σ(q) B 0 on the extended system S can be interpreted as a frequencydomain weighted IQC on S0 (shortcut notation σH(E(λ)q0) B 0 or q′0Π(λ)q0 B 0, whereλ = s for CT systems, λ = z for DT systems). Indeed, due to the Parceval identity, σ〈q〉can be represented as the integral of q′0Πq0 over C0 for all q0 ∈ Ld0, q = [q0;Eq0], whereη denotes the Fourier transform of η ∈ Ld0. While Π is uniquely defined by E and σ, asingle Π corresponds to a variety of pairs (E, σ). All such ”realizations” are equivalent inthe framework of conditional IQC.

For example, in the special case of the Zames-Falb IQC in subsection 2.1.6, the IQCσzB0 on Se can be expressed as a frequency-weighted IQC (y−w)(w−H(s)w)B0, whereH(s) = 2/(s2 + 3s+ 2), relating signals y = Cx ∈ L0 and w = φ(y) ∈ L0.

3.2 Feasibility Optimization and Post-Feasibility Analysis

This section presents general theorems of post-feasibility analysis. It also describes theuse of the KYP lemma in deriving frequency domain conditions of IQC feasibility, asintroduced in the preamble to section 2.2.

3.2.1 KYP Lemma in Discrete Time

As mentioned before in subsection 2.1.5, for CT signals x ∈ Ln, u ∈ Lm, relation x+ =Ax+Bu implies σI([u;x])B0 for σI([u;x]) = 2x′Q(Ax+Bu), where Q = Q′ is an arbitrary

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symmetric n-by-n matrix. Moreover, σI([u;x]) 0 when Q ≥ 0 is positive semidefinite.Theorem 2.1 provides frequency domain conditions for the existence of Q = Q′ such thatσ ≥ σI , where σ is a given quadratic form.

For DT signals, x+ = Ax+Bu implies σ+([u;x]) B 0, where, for some Q = Q′,

σ+([u;x]) = (Ax+Bu)′Q(Ax+Bu)− x′Qx (u ∈ Rm, x ∈ Rn), (3.19)

with σ+([u;x]) 0 when Q ≥ 0.The following statement is a discrete time analog of Theorem 2.1.

Theorem 3.1 For real matrices A,B of dimensions n-by-n and n-by-m respectively, anda Hermitian form σ : Cm+n 7→ R with real coefficients, let Θ be the set of θ ∈ R suchthat the matrix Aθ = ejθIn − A is invertible. For θ ∈ Θ let Lθ = A−1

θ B. Consider thefollowing statements:

(a) σ([u;Lθu]) ≥ 0 for all u ∈ Cm and θ ∈ Θ;

(b) there exists θ ∈ Θ such that σ([u;Lθu]) > 0 for all u ∈ Cm, U 6= 0;

(c) quadratic form σ([u;x])+(Ax+Bu)′Q(Ax+Bu)−x′Qx (where x ∈ Rn and u ∈ Rm)is positive semidefinite for some real matrix Q = Q′.

Then

(i) (c) implies (a);

(ii) when the pair (A,B) is controllable, (a) implies (c);

(iii) when the pair (A,B) is DT stabilizable, (a) and (b) together imply (c).

Theorem 3.1 can be used to formulate frequency domain conditions of feasibility inIQC analysis of DT systems when the ”time delay” term (Ax+Bu)′Q(Ax+Bu)− x′Qxis present in the description of Λ.

3.2.2 Stable Operator Feedback Setup

An important class of IQC analysis scenarios is given by the stable operator feedback setupwhich calls for certifying the complete IQC r|f |2 − |e|2 0 (with r ≥ 0 being as smallas possible) in the feedback interconnection of a stable causal operator ∆ (input y ∈ Lk,output w ∈ Lm) and a stable LTI system M (input [f ;w] ∈ Ll+m, output [e; y] ∈ Ld+k,state x ∈ Ln), assuming the interconnection satisfies the IQC σδ([f ;w;x]) B 0.

-

-M

∆

-f e

w y

12

The IQC σδB0 is interpreted as one describing of ∆. While a typical IQC model will havemultiple IQC for every subsystem, the single IQC setup is general enough to fourmulateabstract statements relevant to feasibility optimization and post-feasibility analysis.

The quadratic form σδ : Rk+m+n 7→ R, as well as the real coefficient matricesA,Bi, Ci, Dij (where the pair (A, [B1, B2]) is controllable) in the equations

M :

eyx+

=

D11 D12 C1

D21 D22 C2

B1 B2 A

fwx

, x(0) ∈ X0 (3.20)

are given, while the feedback operator ∆ and the (non-empty) set of initial conditionsX0 ⊂ Rn are not expected to be known in detail.

The associated IQC model is given by the triplet (S, σ∗,Λ), where S = q is thebehavioral model consisting of all signals q = [f ;w;x], satisfying (3.20) and w = ∆(y) forsome e, y,

σ∗([f ;w;x]) = r|f |2 − |C1x+D11f +D12w|2 (3.21)

is the parameterized ”analysis objective” quadratic form, and

Λ = σ = cσδ + σQ : c ≥ 0, Q = Q′ ∈ Rn,n

is the set of quadratic forms σ for which the IQC σ B 0 on S is readily established,including σ = σQ([f ;w;x]) defined as 2x′Q(Ax+B1f +B2w) (”pure integrator” IQC) forCT models, or as (Ax+B1f +B2w)′Q(Ax+B1f +B2w)− x′Qx (”time delay” IQC) forDT models.

3.2.3 Operator Feedback Setup in the Frequency Domain

For i, j ∈ 1, 2 let Mij(λ) = Dij + CiA−1λ Bj, where Aλ = λIn − A, be the corresponding

blocks of the transfer matrix of M . Let Γ = Pσδ, A−1λ [B1, B2], where the operation

P = Pσ,E is defined in subsection 3.1.6. Since M is assumed to be stable, the IQCσδ B 0 on S can be expressed in the frequency domain format as [f ;w]′Γ(λ)[f ;w] B 0, aslong as X0 = 0.

The task of finding σ0 ∈ Λ such that σ∗ ≥ σ0 with r as small as possible can bedelegated to an appropriate optimization engine as a semidefinite program with decisionvariables c ∈ R+, Q = Q′ ∈ Rn,n, r ∈ R and objective r → min. Alternatively, onecan apply the KYP Lemma to show that Q = Q′ ∈ Rn,n, c ∈ R+, and r ∈ R satisfyingσ∗ ≥ cσδ + σQ can be found if and only if the condition

Γ22 ≤ −ε(Γ′12Γ12 +M ′12M12) ∀ λ ∈ C0, (3.22)

where Γ22 and Γ12 are the lower right and upper right corner blocks of Γ of dimensionsm-by-m and l-by-m respectively, is satisfied for some ε > 0.

13

3.2.4 Need for Post-Feasibility Analysis

In general, the existence of c ≥ 0 and Q = Q′ such that σ∗ ≥ cσδ +σQ does not imply thecomplete IQC σ∗ 0. This could be due to the fact that σδ B 0 but σδ 6 0, or becauseQ = Q′ is not positive semidefinite. On the other hand, it is possible to have σδ 6 0 andQ 6≥ 0 while σδ + σQ 0.

For example, consider the stable operator feedback setup from subsection 3.2.2 givenby state space equations x1 = −x1 + f , x2 = −x2 + w, e = y = x1 + x2 with X0 = 0:

1s+1 ∆e- - - -

6

f u wy

The conditional IQC σδ B 0, where σδ([f ;w;x1;x2]) = |x1|2 − 0.25|W |2 (or, equivalently,|y − w

s+1|2 − 0.25|w|2 B 0) is satisfied for ∆ = ∆0 as well as for ∆ = ∆1, where ∆0(y) ≡ 0

and ∆1(y) = 2y. Also, existence of Q = Q′, c ≥ 0, and r ∈ R such that σ∗ ≥ cσδ + σQ isguaranteed by the frequency domain condition (3.22), as in this case

Γ(s) =

[1

1−s2 0

0 −0.25

], Mij(s) =

1

1 + s.

Nevertheless, the complete IQC σ∗ 0 holds for r ≥ 1 when ∆ = ∆0, but the feedbackinterconnection is unstable, and the power gain from f to e equals infinity, when ∆ = ∆1.

3.2.5 Minimal Stability

Minimal stability is a simple condition6 to be imposed on σq = σ∗−cσδ, which guaranteesthat Q ≥ 0 whenever σq ≥ σQ. Then σQ 0 and σ∗ 0 follows as long as it is knownthat σδ 0.

In terms of the stable operator feedback setup from subsection 3.2.2, minimal stabilitymeans existence of real matrices K1, K2 such that AK = A + B1K1 + B2K2 is stable(i.e. Hurwitz in the CT case and Schur in the DT case), and σq(K1x,K2x, x) ≤ 0 for allx ∈ Rn. Indeed, since σq ≥ σQ, minimal stability implies QAK +A′KQ ≤ 0, hence Q ≥ 0.Such K1, K2 are usually easy to find when σδ has a simple structure, as in the classicaltheory of absolute stability.

For example, the derivation of the circle criterion in subsection 2.2.3 uses the sta-ble operator feedback setup with B1 = 0, B2 = B, ∆(y) = φ(y), and σδ([f ;w;x]) =2csw(Cx− w) satisfying σδ 0. It employs a minimal stability argument, with K2 = 0,to show that Q ≥ 0. The derivation of the Popov criterion in subsection 2.2.4 can use asimilar argument, with σδ([f ;w;x]) = 2csw(Cx − w) + 2cpwC(Ax + Bw), when cp ≥ 0(and hence 2cpwC(Ax+Bw) 0). When, cp < 0, σδ should be re-defined as

σδ([f ;w;x]) = 2csw(Cx− w) + 2cpwC(Ax+Bw)− 2cpx′C ′C(Ax+Bw).

6Invented and frequently used by V. Yakubovich

14

Since 2cpwC(Ax + Bw) − 2cpx′C ′C(Ax + Bw) 0 when cp ≤ 0, the minimal stability

argument can be used, with K2 = C being the natural selection. Establishing thatAK = A+BC is a Hurwitz matrix is the extra effort required in this case.

3.2.6 Completeness of IQC and Stability of Causal Operators

In this subsection, we consider a causal operator S with k-dimensional input f , m-dimensional output w, and graph GS = [f ;w] : w = S(f).

S- -f w

Assuming that a quadratic form σ∗ defines a valid conditional IQC σ∗B0 on GS, we explorethe relation between stability of S and the possibility of claiming that the complete IQCσ∗ 0 holds on GS as well. The analysis relies on σ∗([F ;W ]) being concave with respectto W ∈ Rm, which is common in applications (as, for example, in the analysis of thepower gain from f to w, where the IQC σ∗ 0 to be proven is defined by σ∗([F ;W ]) =r|F |2 − |W |2).

The first result states that, for a conditional IQC which is concave with respect to theoutput variable, the corresponding complete IQC can be derived from the assumptions ofcausality and stability.

Theorem 3.2 If S is causal and stable, σ∗B 0 on GS, and σ∗([0;W ]) ≤ 0 for all W ∈ Rm

then σ∗ 0 on GS.

The second result states that stability can be derived from a complete IQC which isstrictly concave with respect to the output variable.

Theorem 3.3 If S is causal, σ∗ 0 on GS, and there exists ε > 0 such that σ∗([0;W ]) ≤−ε|W |2 for all W ∈ Rm then S is stable.

3.2.7 Sensitivity of IQC to Small Perturbations

Let S1,S2 be two d-dimensional behavioral models. Define the (non-symmetric) relativedistance measure ρ = ρ(S1,S2) between S1 and S2 as the maximal lower bound of numbersr > 0 such that for every q2 ∈ S2 there exists q1 ∈ S1 satisfying the inequality ‖q2−q1‖T ≤r‖q2‖T for all T > 0.

The following statement claims a degree of semi-continuity of complete IQC withrespect to the distance measure ρ(·).

Theorem 3.4 For every quadratic form σ : Rd 7→ R and every ε > 0 there exists δ > 0such that σ(q) + ε|q|2 0 on S2 whenever σ 0 on S1 and ρ(S1,S2) < δ.

15

3.2.8 The Homotopy Argument

Combining Theorems 3.2, 3.3, and 3.4 yields the following homotopy argument of post-feasibility analysis, expressed by Theorem 3.5.

S(τ)- -f w

Theorem 3.5 Let S = S(τ) be a family of causal operator models with input f andoutput w, parameterized by τ ∈ [0, 1]. Assume that the dependence of S(τ) on τ ∈ [0, 1] iscontinuous, in the sense that for every δ > 0 there exists h > 0 such that ρ(GS(a),GS(b)) < δwhenever a ≤ b ≤ a + h. Then r|f |2 − |w|2 0 on GS(1) whenever r|f |2 − |w|2 0 onGS(0) and r|f |2 − |w|2 B 0 on GS(τ) for all τ ∈ [0, 1].

Application of the homotopy principle is facilitated by the following observation: ifsystems

S = [v;w;u; y] : w = ∆(v), y = G(u), S = [v;w;u; y] : w = ∆(v), y = G(u)

are interconnections of operators G,∆, G, ∆, as shown in the figure below, then

ρ(S1,S2) = maxρ(G∆,G∆), ρ(GG,GG).

G

∆

e- -

e ?

6

u y

w v

G

∆

ee

- -

?

6

u y

w v

3.2.9 The Minimax Argument

In this subsection, we consider the case when the behavioral model S has special structureexpressed by the block diagram in Figure 3.1, where ∆ : Lk 7→ Lm is a stable causaloperator, and M is defined by stable LTI state equations

x+ = Ax+B1w +B2v, x(0) ∈ X0, f = Cx+D1w +D2v.

Consider the following systems associated with the setup:

SLTI = [w; v;x] : w ∈ Lm0 , v ∈ Lk0, x+ = Ax+B1w +B2v, (3.23)

Sv = [v;x] : v ∈ Lk0, x+ = Ax+B2v, x(0) = 0, (3.24)

S = [w; v;x] : v ∈ Lk, w = ∆(v), x+ = Ax+B1w +B2v, x(0) ∈ X0. (3.25)

Let C+ = s ∈ C : Re(s) ≥ 0 in the CT case, and C+ = z ∈ C : |z| ≥ 1 in theDT case.

16

- -v w

∆

-M

-f

Figure 3.1: Minimax Post-Feasibility Analysis Setup

Theorem 3.6 Assume that real matrices A,B1, B2, C,D1, D2, quadratic form σ0, a setX0 ⊂ Rn, and stable causal operator ∆ : Lk 7→ Lm satisfy the following conditions:

(a) A is a stable (has no eigenvalues in C+);

(b) matrix D2 + C(λI − A)−1B2 is right invertible for all λ ∈ C+;

(c) supσ0〈[w; v;x]〉 : Cx+D1w +D2v = f <∞ for all f ∈ Lk0;

(d) σv B 0 on Sv for σv([v;x]) = σ0([0; v;x]);

(e) σw B 0 on SLTI where σw([w; v;x]) = cw|v|2 − σ0([w; v;x]) for some cw.

Then σ0 0 on S whenever σ0 B 0 on S.

4 IQC Library

This section provides a short list of IQC established for common nonlinear, time-varying,uncertain, and distributed elements of dynamical system models.

4.1 Harmonic Modulation

For a fixed ω0 > 0 let S0 = [v;w] : v ∈ L, w(t) = cos(ω0t)v(t) be the behavioral modeldescribing the harmonic modulation operator v(t) 7→ cos(ω0t)v(t):

- -

6

v(t) w(t)⊗

cos(ω0t)

It can be shown that S0 satisfies the frequency weighted IQC

[α(jω − jω0) + α(jω + jω0)]|v|2 − 2α(jω)|w|2 B 0

for every rational function α ∈ RL1,1 which is bounded and non-negative on the imaginaryaxis. The IQC limits the transfer of power between different parts of the signal spectrumin the transition from v to w, and can be proven by observing that the Fourier transformsv, w of finite energy [v;w] ∈ S0 satisfy

2|w(jω)|2 ≤ |v(jω + jω0)|2 + |v(jω − jω0)|2 ∀ ω ∈ R.

17

4.2 Multiplication with an Unknown Scalar

Let δ : R 7→ R be a scalar function. Let Sδ = [w; v] : v ∈ Ln, w(t) = δ(t) · v(t) bethe behavioral model describing the operation of multiplication with a unknown scalar:v(t) 7→ δ(t) · v(t).

4.2.1 IQC for Fast Time-Varying Scalar

Suppose δ satisfies |δ(t)| ≤ 1, ∀t. It is then straightforward to verify that Sδ satisfies thecomplete IQC σftvs 0, where

σftvs([W ;V ]) = V ′XV + V ′YW +W ′Y ′V −W ′XW,

where X = X ′ ≥ 0 and Y + Y ′ = 0 are matrices in Rn,n. Indeed, since w(t) = δ(t)v(t),∫ T

0

σftvs([w(t); v(t)])dt =

∫ T

0

(1− δ(t)2)v(t)′Xv(t)dt ≥ 0, ∀ T ≥ 0.

4.2.2 IQC for Slowly Time-Varying Scalar

Suppose δ satisfies |δ(t)| ≤ 1, and |δ(t)| ≤ d, ∀t. Obviously, Sδ still satisfies the completeIQC σftvs 0. Furthermore, the additional information on the rate of variation allowsmore IQC characterization to be derived based on extended models of Sδ.

The first example is a Popov IQC based on the extended model

Sδ,e1 = q = [w; v; f ] : [w; v] ∈ Sδ, v(t) = f(t), ∀t.

It can be shown that Sδ,e1 satisfies the conditional IQC σstvs1 B 0, where

σstvs1([W ;V ;F ]) = V ′ZV + 2W ′Y F,

and Y = Y ′, Z = Z ′ are n-by-n real matrices satisfying Z ≥ ±dY . To see this, note that,by integration by parts, we have for any q = [w; v; f ] ∈ L3m with ‖q‖ <∞∫ ∞

0

(v(t)′Zv(t) + 2w(t)′Y f(t))dt

= limt→∞

(δ(t)v(t)′Y v(t)− δ(0)v(0)′Y v(0)) +

∫ ∞0

v(t)′(Z − δ(t)Y )v(t)dt ≥ −κ(q(0))

where κ(q(0)) = max(v(0)′Y v(0),−v(0)′Y v(0)). The IQC becomes complete if we addthe term 2V ′Y F to σstvs1.

In the second example we use a swapping property to characterize the degree of non-commutativity between a time-varying parameter and a linear time-invariant system. Forthis purpose, we introduce the extended system

Sδ,e2 = q = [w; v;u;x1;x2;x3] : [w; v] ∈ Sδ, x1 = Ax1 +Bv, x2 = Ax2 +Bw,

x3 = Ax3 + u, u = δx1, x1(0) = x2(0) = x3(0) = 0,

18

where A is a Hurwitz matrix. Note that by definition the swapping relation

δ(t)x1(t)− x2(t) = x3(t), ∀ t

follows since the initial condition and the time derivative of both quantities are identical.The identities x2 + x3 = δx1 and u = δx1 gives rise to the following complete IQC

v′eX1ve−w′eX1we+v′eY1we+w

′eY′

1ve 0, d2x′1X2x1−u′X2u+x′1Y2u+u′Y ′2x1 0 (4.26)

where X1 = X1 ≥ 0, X2 = X2 ≥ 0, Y ′1 = −Y1, Y ′2 = −Y2 and ve = [x1; v] and we =[x2 + x3;w]. The IQC follows since we = δve and u = δx1, where δ(t) ∈ [−1, 1] andδ(t) ∈ [−d, d].

Note that the complete IQC σftvs 0 is a special case of (4.26); one obtains σftvs 0by setting parts of the free variables in X1, Y1 to zero.

4.2.3 IQC for Constant Scalar

Suppose δ satisfies |δ(t)| ≤ 1, and δ(t) = 0, ∀t. In other words, δ is a constant. In thiscase, in addition to the complete IQC σftvs 0, Sδ also satisfies the frequency weightedIQC σcs B 0, where

σHcs ([W ;V ]) = V ′X(jω)V −W ′X(jω)W + V ′Y (jω)W +W ′Y (jω)′V

for any rational function X ∈ RLn,n which is bounded and positive semidefinite, and anyrational function Y ∈ RLn,n which is bounded and skew Hermitian (i.e., Y (jω)+Y (jω)′ =0) on the imaginary axis. This IQC can be proven by observing that the Fourier transformsv, w of finite energy [w; v] ∈ Sδ satisfy

w(jω) = δ · v(jω) ∀ ω ∈ R.

The IQC σcs B 0 is readily verified by substituting the above relationship to the corre-sponding integral quadratic form.

4.3 Uncertain Time Delay

Let Dτ : Cm[−τ0, 0]→ Rm be defined as (Dτvt)(t) = v(t− τ(t)), where vt := v(t−ϕ), ϕ ∈[−τ0, 0], and τ : R → R is a scalar function. Let SD = [w; v] : v ∈ L,w(t) = (Dτvt)(t)be the behavioral model describing the time delay operation: v(t) 7→ v(t − τ(t)). Theinitial condition of v, v(θ), is a continuous function defined on [−τ0, 0].

19

4.3.1 IQC for Fast Time-Varying Delay

Suppose τ satisfies 0 ≤ τ(t) ≤ τ0, ∀t. In this case, consider the following extended modelof SD

SD,e1 = [w; v; f ] : [w; v] ∈ SD, v = f.

It can be shown that SD,e1 satisfies the complete IQC σftvd 0, where

σftvd([W ;V ;F ]) = α(τ 20 |F |2 − |V −W |2),

and α ≥ 0 is any non-negative real number. To prove this IQC, note that v(t) − w(t)

can be expressed as

∫ t

t−τ(t)

v(s)ds. Hence, by applying the Cauchy-Schwartz inequality,

we have

|v(t)− w(t)|2 ≤ τ0

∫ t

t−τ0|v(s)|2ds = τ0

∫ 0

−τ0|v(t+ s)|2ds

which in turn implies

‖v − w‖2T ≤

∫ T

0

(τ0

∫ 0

−τ0|v(t+ s)|2ds

)dt = τ0

∫ 0

−τ0

(∫ T

0

|v(t+ s)|2dt)ds

= τ0

∫ 0

−τ0

(∫ 0

s

|v(t)|2dt+

∫ T+s

0

|v(t)|2dt)ds

≤ κ(v(θ)) + τ 20

∫ T

0

|v(t)|2dt

where

κ(v(θ)) = τ0

∫ 0

−τ0

∫ 0

s

|v(t)|2dtds = τ0

∫ 0

−τ0

∫ t

−τ0|v(s)|2dsdt

Hence, the complete IQC σftvd 0 holds for SD,e1.

4.3.2 IQC for Slowly Time-Varying Delay

Suppose τ satisfies |τ(t)| ≤ d < 1, ∀t. In this case, SD satisfies the complete IQC σstvd 0,where

σstvd([W ;V ]) = α

(1

1− d|V |2 − |W |2

),

and α ≥ 0 is any non-negative real number. To derive this IQC, note that∫ T

0

|w(t)|2dt =

∫ T−τ(T )

−τ(0)

|v(s)|2 1

1− τ(t(s))ds ≤ 1

1− d

∫ T

−τ0|v(s)|2ds.

20

The first equality follows the change of variables s = t − τ(t), which is invertible since0 ≤ d < 1. This leads to the inequality∫ T

0

1

1− d|v(s)|2 − |w(s)|2ds ≥ −κ(v(θ)),

where κ(v(θ)) =τ0

1− dmax

θ∈[−τ0,0]|v(θ)|2. Hence, we have proven that the complete IQC

σstvd 0 is satisfied on SD.

4.3.3 IQC for Arbitrary Constant Time Delay

Suppose τ satisfies τ(t) = 0, ∀t; that is, parameter τ is a constant. Obviously, SD satisfiescomplete IQC σstvd 0, with d set to zero; i.e. σstvd([W ;V ]) = α (|V |2 − |W |2), whereα ≥ 0 is any non-negative real number.

The fact that τ is a constant allows frequency dependent weighting in the IQC. It canbe shown that SD satisfies frequency weighted IQC σcd,1 B 0, where

σHcd,1([W ;V ]) = α(jω)|V |2 − α(jω)|W |2

for every rational function α ∈ RL1,1 which is real-valued and bounded on the imaginaryaxis. This IQC can be proven by observing that the Fourier transforms v, w of finiteenergy [w; v] ∈ SD satisfy

|w(jω)|2 = |v(jω)|2 ∀ ω ∈ R.

4.3.4 IQC for Bounded Constant Time Delay

Suppose τ satisfies τ ∈ [0, τ0], τ(t) = 0, ∀t. In this case, consider the extended model

SD,e2 = [w; v; f ] : [w; v] ∈ SD, τ 20 f + aτ0f + bf = 2τ 2

0 v + cτ0v,

where c = 2√

12.5, b =√

50, and a =√

2b+ 6.5. It can be shown that SD,e2 satisfies thefrequency weighted IQC σcd,2 B 0, where

σHcd,2([W ;V ;F ]) = α(jω)|F |2 − α(jω)|W − V |2

for every rational function α ∈ RL1,1 which is bounded and non-negative on the imaginaryaxis. This IQC can be proven by observing that the Fourier transforms v, w, f of finiteenergy [w; v; f ] ∈ SD,e2 satisfy

f(jω) = H(jω)v(jω) ∀ ω ∈ R.

|w(jω)− v(jω)|2 = 4 sin(τω

2

)2

|v(jω)|2 ∀ ω ∈ R.

where H(s) =2(τ0s)

2 + cτ0s

(τ0s)2 + aτ0s+ b, and that under the assumption τ ∈ [0, τ0], |H(jω)| is

larger than or equal to 2| sin(τω2

)| for all ω ∈ R.

21

4.4 Simple IQC for Periodic Function

Let ∆H : Rm 7→ R2Nm be a collection of elementary harmonics

∆H(t) =

cos(k1ω0t)Isin(k1ω0t)I

...cos(kNω0t)sin(kNω0t)I

where ω0 > 0 and I = k1, . . . , kN where 0 < k1 < k2 < . . . < kN . Let

S∆H= [w; v] : v ∈ Lm;w(t) = ∆H(t)v(t)

be behavioral model representing multiplication with ∆H in the time domain. Suchoperation may arise in modelling of periodic systems. It can be shown that S∆H

satisfythe complete IQC σpf 0, where

σHpf ([W ;V ]) =

[VW

]′M

[VW

]where M = Re(U ′QHU),

U =

I 0 0 0 . . . 0 0 00 I iI 0 . . . 0 0 0

...0 0 0 0 . . . 0 I iI

,and QH is a real matrix of suitable dimension such that M11 ≥ 0, M22 ≤ 0, and[

A′HPAH − P A′PBH

B′HPAH B′HPBH

]+[CH DH

]′QH

[CH DH

]≥ 0

for some P = P ′ ∈ CnKN ,nKN . In the above statement, M11 ∈ Rn,n and M22 ∈ R2nN,2nN

are the submatrices of M defined by

M =

[M11 M12

M ′12 M22

],

and (AH , BH , CH , DH) is any controllable realization of

H(z) =1

(z + a)kN

Izk1I

...zkN I

.This IQC can be proven using the discrete time version of the KYP lemma and the factthat eikω0t = cos(kω0t) + i sin(kω0t).

22

4.5 LTI Unmodelled Dynamics

Let ∆ : Ln → Lm be a linear time-invariant bounded operator with the L2-induced gainbeing less than or equal to 1. Let Sud = [w; v] : v ∈ Ln, w = ∆v be the behavioralmodel describing the LTI transformation v 7→ ∆v. It can be shown that Sud satisfies thefrequency weighted IQC σud B 0, where

σHud([W ;V ]) = α(jω)|V |2 − α(jω)|W |2

for every rational function α ∈ RL1,1 which is bounded and non-negative on the imaginaryaxis. This IQC can be proven by observing that the Fourier transforms v, w of finite energy[w; v] ∈ Sud satisfy

w(jω) = ∆(jω)v(jω), ∀ω ∈ R,

where ∆(s) is the representation of ∆ in the Laplace domain. Since supω σ(

∆(jω))≤ 1,

where σ(M) denotes the maximal singular value of M , we conclude that

‖w(jω)‖2 ≤ ‖v(jω)‖2, ∀ω ∈ R,

and hence the IQC.

4.6 Polytope Uncertainties

Let ∆(t) be a Rm,n-valued function which satisfies ∆(t) ∈ ∆ := co∆1, · · · ,∆N, ∀ t,where coM denotes the convex hull of M, and ∆1 to ∆N are given real matrices ofdimension m-by-n. We assume 0 ∈ ∆. Let Spoly = [w; v] : v ∈ Ln, w(t) = ∆(t)v(t) bethe behavioral model describing multiplication in time domain by the uncertainty ∆(t).It can be shown that Spoly satisfies the complete IQC σpoly 0, where

σpoly([W ;V ]) = V ′ZV + V ′YW +W ′Y ′V −W ′XW

where Z = Z ′ ∈ Rn,n, Y ∈ Rn,m, and X = X ′ ∈ Rm,m satisfy X ≥ 0 and

Z + Y∆i + ∆′iY′ −∆′iX∆i ≥ 0, for i = 1, · · · , N.

This IQC can be proven by observing that w(t) can be expressed asN∑i=1

αi∆iv(t), for some

αi ≥ 0, αi + · · ·+ αN = 1. Since X ≥ 0, we have

w(t)′Xw(t) =

∥∥∥∥∥N∑i=1

αiX12 ∆iv(t)

∥∥∥∥∥2

≤N∑i=1

αiv(t)′∆′iX∆iv(t).

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Therefore,

v(t)′Zv(t) + v(t)′Y w(t) + w(t)′Y ′v(t)− w(t)′Xw(t) ≥N∑i=1

αiv(t)′ (Z + Y∆i + ∆′iY′ −∆′iX∆i) v(t) ≥ 0, ∀ t,

and hence the IQC.

5 Applications of IQC Analysis

The framework in the previous sections will here be applied to a number of examples ofvarying complexity.

5.1 System with Slowly Time-Varying Gain

Consider the linear system

x(t) = (A0 + δ(t)B0C0)x(t), x(0) = xi

where A0 ∈ Rn×n is Hurwitz, B0 ∈ Rn×m, C0 ∈ Rm×n and δ : R 7→ R satisfies |δ(t)| ≤ 1and |δ(t)| ≤ d, ∀t. The system can be represented as in equation (3.19) [Aug 8, version]with B1 = 0, B2 = B0, C1 = I, D11 = 0, D12 = 0, C2 = C0, D21 = I, D22 = 0, theexternal input f(t) = 0, and ∆(v) = δv.

The IQC for slowly time-varying systems in the previous section may be used inthe analysis of this system. For this purpose, let us introduce the extended dynamicsxe = Aexe +Be1v +Be2w +Be3ω, where

xe =

x1

x2

x3

, Ae =

A 0 00 A 00 0 A

, Be1 =

B00

, Be2 =

0B0

, Be3 =

00I

and an auxiliary input/output pair ψ = x1, ω = δψ. Based on the behavioral model

Se = q = [v;w;ψ;ω;x0;xe] ∈ Ln+4N+2m : x0 = A0x0 +B0w;

xe = Aexe +Be1v +Be2w +Be3ω; w = δv; ω = δψ

we may use the quadratic forms Λe defined as

σe([V ;W ; Ψ; Ω;X0;Xe] = V ′ZV + 2W ′Y (A0X0 +B0W )

+ V ′eX1Ve −W ′eX1We + V ′eY1We +W ′

eY′

1Ve + d2Ψ′X2Ψ− Ω′X2Ω + Ψ′Y2Ω + Ω′Y ′2Ψ

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where the matrices satisfy the conditions in the previous section. For the analysis weinclude the IQC for pure integration and use the set of quadratic forms Λ defined by

σ([V ;W ; Ψ; Ω;X0;Xe]) = σe(V ;W ; Ψ; Ω;X0;Xe)

+

[X0

Xe

]′Q

[A0X0 +B0W

AeXe +Be1V +Be2W +Be3Ω

]where σe ∈ Λe, Q = Q′. It follows that σ B 0 for any σ ∈ Λ.

The feasibility analysis reduces to a search for matrices Q = Q′, Y = Y ′, Z = Z ′,Z ≥ ±dY , and X1 = X ′1 ≥ 0, Y ′1 = −Y1, X2 = X ′2 ≥ 0, Y ′2 = −Y2 such that

σ∗([X]) ≥ σ([V ;W ; Ψ; Ω;X0;Xe]).

where σ∗[X]) = −|X|2. If we can prove σ∗ 0 on S then the system is stable in the sense

that∫ T

0|x(t)|2dt ≤ γ|x(0)|2, ∀T ≥ 0, for some γ > 0.

5.1.1 Post Feasibility Analysis

If the feasibility test was succesfull with Z = Y = 0 and Q ≥ 0 then σ∗ 0 and thesystem is proven stable. If Y 6= 0 or Q 6≥ 0 then σ∗ B 0. Then let S(τ) be defined as Sabove but with τδ and τ δ for τ ∈ [0, 1]. Clearly σB 0 for any σ ∈ Λ on the correspondingextended behavior model Se(τ). We conclude that σ∗ B 0 on S(τ), for τ ∈ [0, 1]. Clearly,the τ dependence is continuous and σ∗ 0 on S(0) (since A0 is Hurwitz). It follows fromTheorem 3.5 that σ∗ 0 on S(1) = S and the system is proven stable.

6 Appendix

This section contains proofs which are not included in the main text.

6.1 Homotopy Arguments

This section contains proofs of statements related to the homotopy argument in post-feasibility analysis.

6.1.1 Proof of Theorem 3.2

For every f ∈ Lm, w = S(f), q = [f ;w] and T ≥ 0 consider fT = [f ]T and wT = S(fT )(i.e. qT = [fT ;wT ] is obtained by ”switching input to zero” at time T ). Stability of Simplies that qT has finite energy, hence σ∗ B 0 implies σ∗〈qT 〉 ≥ 0. Since fT (t) = 0 fort > T , we have σ∗([fT (t);wT (t)]) ≤ 0 for t > T , hence σ∗〈[qT ]T 〉 ≥ σ∗〈qT 〉 ≥ 0. Bycausality of S, q(t) = qT (t) for t ≤ T , hence σ∗〈[q]T 〉 = σ∗〈[qT ]T 〉 ≥ 0.

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6.1.2 Proof of Theorem 3.3

There exist c > 0 such that σ∗([F ;W ]) ≤ c|F |2 − 0.5ε|W |2. Hence, for f ∈ Lk0, w = S(f)we have 0 ≤ σ∗〈[[f ;w]]T 〉 ≤ c‖f‖2

T − 0.5ε‖w‖2T for all T ≥ 0, which implies ‖w‖ <∞.

6.1.3 Proof of Theorem 3.4

The functionσ(F,W ) = σ(F +W )− σ(F )

is a quadratic form on Rd × Rd which is affine with respect to F , hence for every ε > 0there exists R > 0 such that

σ(F +W )− σ(F ) ≤ 0.5ε|F |2 +R|W |2 ∀ F,W ∈ Rd. (6.27)

Therefore, for δ = 0.5ε/R,

σ〈[q1〉T ] ≤ σ〈[q2]T 〉+ 0.5ε‖q2‖2T +R‖q1 − q2‖2

T ≤ σ〈[q2]T 〉+ ε‖q2‖2,

and therefore σ 0 on S1 implies σ(q) + ε|q|2 0 on S2.

6.1.4 Proof of Theorem 3.5

According to Theorem 3.4, applied to σ([F ;W ]) = r|F |2 − |W |2 and ε = 0.5, there existsδ > 0 such that (r + 0.5)|f |2 − 0.5|w|2 0 on GS(b) whenever r|f |2 − |w|2 0 on GS(a)

and ρ(GS(a),GS(b)) < δ. Since S(b) is causal, complete IQC (r + 0.5)|f |2 − 0.5|w|2 0implies, via Theorem 3.3, stability of S(b). Hence, Theorem 3.2 and conditional IQCr|f |2− |w|2 B 0 imply complete IQC r|f |2− |w|2 0 on GS(b). Therefore, when h is smallenough, r|f |2−|w|2 0 on GS(a) implies r|f |2−|w|2 0 on GS(b) for all b ∈ [a, a+h]∩[0, 1].Starting from a = 0, this proves the complete IQC r|f |2−|w|2 0 on GS(a) for all a ∈ [0, 1].

6.2 Proof of Theorem 3.6

The proof is based on the following version of the minimax theorem.

Theorem 6.1 Let V , W be a real pre-Hilbert space and a real vector space. Let σ :V ×W 7→ R be a quadratic functional satisfying the following conditions:

(i) there exists cv > 0 such that σ(v, w) ≤ cv(|v|2 + 1) for all v, w;

(ii) the function v 7→ σ(v, 0) is convex on V .

Then there exists vs ∈ V such that

limv→vs

supwσ(v, w) = sup

winfvσ(v, w).

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We begin by applying Theorem 6.1 to V = Lk0, W = Lm0 and σ(v, w) = Iσ0([w; v;x])where, for a fixed a ∈ Rn, x is defined by [w; v;x] ∈ SLTI and x(0) = a. Then assumption(e) implies (i), assumption (d) implies (ii), and assumption (c) implies infv σ(v, w) ≤ 0.Hence for every a ∈ Rn there exists vs ∈ Lk such that, subject to [w; v;x] ∈ SLTI ,

lim‖v−vs‖→0

supw∈Lm

0

Iσ0([w; v; a]) ≤ 0. (6.28)

For fixed q0 = [w0; v0;x0] ∈ S and T > 0 take vs ∈ Lk0 satisfying (6.28) with a = x0(T )and define v∗ ∈ Lk0 by

v∗(t) =

v0(t), t < T,vs(t), t ≥ T.

Let qi∞i=1 be the sequence (existence of which is guaranteed by assumption (b)) of finiteenergy signals qi = [wi; vi;xi] ∈ S such that [qi]T = [q0]T for all i, and ‖v∗ − vi‖ → 0as i → ∞. Since σ0 B 0 implies Iσ0(qi) ≥ −c(qi(0)) = −c(q(0)), and (6.28) impliesthat the upper limit of Iσ0(qi) − Iσ0([qi]T ), as i → ∞, is not positive, the identityIσ0([qi]T ) = Iσ0([q]T ) yields Iσ0([q]T ) ≥ −c(q(0)) for all T > 0, as desired.

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