domain: spi info math phys

text:
Pierre Apkarian

Dominikus Noll

MIXED L1/H∞-SYNTHESIS FOR L∞-STABILITY

Keywords: BIBO-stability, peak-to-peak norm, asymptotic constraint, boundary feedback control, wave equation, sector non-linearity

We consider stabilization and performance optimization of non-linear controlled systems, where the non-linearity satisfies a sector constraint asymptotically. This leads to optimization of the closed loop peak-to-peak system norm subject to H ∞performance constraints. Non-linear controlled systems tuned successfully by this novel approach are locally exponentially stable and globally BIBO-stable. Key Words.

Introduction

The peak-gain, or peak-to-peak norm of a BIBO-stable linear time-invariant system G, is the time-domain L ∞ operator norm, (1)

G pk_gn = sup{|G * w| ∞ : |w| ∞ ≤ 1},
where the signal norm on L ∞ ([0, ∞), R n ) is |x| ∞ = sup t≥0 max i=1,...,n |x i (t)|. In the SISO case it is also known as the system L 1 -norm. As opposed to the more standard H 2or H ∞ -norms, computation or optimization of G pk_gn has found only mild attention in the control literature, even though its importance e.g. for the rejection of persistent perturbations was recognized [START_REF] Munther | Robust controller design: Minimizing peak-to-peak gain[END_REF][START_REF] Dahleh | l 1 -optimal feedback controllers for MIMO discrete-time systems[END_REF][START_REF] Dahleh | Optimal and robust controllers for periodic and multirate systems[END_REF][START_REF] Diaz-Bobillo | Minimization of the maximum peak-to-peak gain: the general multiblock problem[END_REF][START_REF] Linnemann | Computing the L 1 -norm of continuous-time linear systems[END_REF][START_REF] Rutland | Computing the 1-norm of the impulse response of linear time-invariant systems[END_REF][START_REF] Sznaier | Mixed L 1 /H ∞ controllers for MIMO continuous-time systems[END_REF]. One of the reasons of this disesteem is probably the link of • pk_gn with the H ∞ -norm • ∞ , where in the chain

(2) m -1/2 G ∞ ≤ G pk_gn ≤ (2n + 1)p 1/2 G ∞ the right-hand estimate holds for real-rational systems G with n poles and p inputs, while the left-hand estimate is valid even for infinite dimensional well-posed BIBO-stable systems with m outputs. This may have been interpreted in the sense that optimizing G pk_gn offers nothing substantial over optimizing G ∞ . In the present work we show that optimizing G pk_gn has genuine scope.

For the purpose of motivation, we consider a possibly infinite-dimensional Lur'e system, where a tunable LTI (Linear Time-Invariant) block is in loop with a sector non-linearity. By the Small Gain theorem closed-loop L 2 -stability is assured if one succeeds in tuning the LTI-block to satisfy a suitable H ∞ -norm or frequency shape constraint. However, this sufficient conditions may be difficult, or even impossible, to achieve if the sector is too large. Here our new approach applies and replaces the large sector by a smaller one, which the non-linearity satisfies only asymptotically. Application of a small gain argument now requires working with the time-domain L ∞ -norm instead of the L 2 -norm. In consequence, the LTI-block is now tuned to satisfy a constraint in the peak-to-peak norm [START_REF] Apkarian | Nonsmooth µ-synthesis[END_REF]. If successful, the non-linear closed loop is BIBO stable. Due to the smaller primal sector, this is often easier to achieve than the original H ∞ -constraint, and it is one of the few remaining options for non-linear systems with different attraction regimes. This approach via asymptotic sectors may be combined with H ∞ -methods to guarantee local exponential stability along with global BIBO-stability. This leads to a novel type of mixed peak-gain/H ∞ -optimization program.

In order to demonstrate the potential of our method, we discuss feedback control of a wave equation with a non-linear anti-damping boundary causing instability. This model has been used to control slipstick vibrations in drilling systems [START_REF] Besselink | Analysis and control of stick-slip oscillations in drilling systems[END_REF][START_REF] Challamel | Rock destruction effect on the stability of a drilling structure[END_REF][START_REF] Roman | Robustness to in-domain viscous damping of a collocated boundary adaptive feedback law for an anti-damped boundary wave PDE[END_REF][START_REF] Saldivar | Suppressing axial-torsional vibrations in drillstrings[END_REF][START_REF] Saldivar | The control of drilling vibrations: a coupled PDE-ODE modeling approach[END_REF]. Our method allows to prove local exponential stability in tandem with global BIBO-stability for scenarios, where this was previously impossible, the challenge being to achieve this with finite-dimensional controllers of simple implementable structure. The second part of the paper extends the concept of asymptotic constraints to MIMO non-linearities, highlighting that applications are not limited to the SISO case.

The organization is as follows. In Section 2 we discuss the case of a sector non-linearity. An algorithm based on mixed H ∞ /H ∞ -and peak-gain/H ∞ -programs is presented in Section 2.2. In Section 2.3 we show how the aperture of the asymptotic sector may be optimized, a feature which is not possible with standard sectors. Section 3 discusses the application to the control of slipstick vibrations. Section 4 resumes theory and extends the asymptotic concept to MIMO non-linearity along with illustrations and applications. Properties of the peak-to-peak norm and implementation of the mixed programs are discussed in Section 5.

Mixed program for a Lur'e systems

For the purpose of motivation we consider a controlled Lur'e system with state x, control input u, measured output y, disturbances w, and regulated outputs z: G nl : ẋ = Ax + B p p + B w w + B u u q = C q x + D q u p = φ(t, q) z = C z x + D zw w + D zu u y = C y x [START_REF] Apkarian | Optimization-based control design techniques and tools[END_REF] where p(t) = φ(t, q(t)) is a non-linearity satisfying φ(t, 0) = 0, ∂φ ∂q (t, 0) = 0 and a sector constraint (φ(t, q) -aq) • (φ(t, q) -bq) ≤ 0 for all t ≥ 0 and all q, abbreviated φ ∈ sect(a, b). Since a ≤ ∂φ ∂q (t, 0) = 0 ≤ b, the linearized system is

G : ẋ = Ax + B w w + B u u z = C z x + D zw w + D zu u y = C y x (4)
In nominal H ∞ -synthesis, we might interpret the non-linearity as a mere disturbance and optimize a suitable closed-loop performance channel T zw (G, K) ∞ over a class K ∈ K of structured controllers [START_REF] Apkarian | Nonsmooth H ∞ synthesis[END_REF], with optimal H ∞ -controller K * ∈ K and gain γ ∞ = T zw (G, K * ) ∞ . Suppose this optimistic approach of representing the non-linearity by a disturbance w (as in Fig. 2 right) is too unspecific and K * is not entirely satisfactory. Then we have to target the sector non-linearity explicitly (as in Fig. 2 left). Putting c = (b + a)/2, r = (b -a)/2, and ψ(t, q) = φ(t, q) -cq, we have ψ ∈ sect(-r, r). The non-linear system (3) is now equivalently written as

G nl : ẋ = Ax + cB p C q x + B p p + B u u q = C q x + D q u p = ψ(t, q) y = C y x (5)
where the performance channel w → z is temporarily ignored for notational convenience. We introduce A ψ = A + cB p C q and G ψ :

ẋ = A ψ x + B p p + B u u q = C q x + D q u y = C y x (6) then (3) is equivalent to putting p → q of G ψ in loop with the centered non-linearity ψ(•) = φ(•) -cI.
Closing the controller loop u = Ky in G ψ leads to the channel q = T qp (G ψ , K)p. Suppose now we succeed in tuning

K ∈ K such that (G ψ , K ) is L 2 -stable and satisfies the estimate T qp (G ψ , K ) ∞ < r -1 .
Then by the small-gain theorem the non-linear loop

(T qp (G ψ , K ), ψ) is L 2 -stable, hence so is (G nl , K ). This is addressed by the structured mixed H ∞ /H ∞ -optimization program minimize T qp (G ψ , K) ∞ subject to T wz (G, K) ∞ ≤ (1 + τ )γ ∞ K stabilizes G, G ψ K ∈ K (7)
which optimizes stability of the non-linear system G nl under a constraint allowing a controlled loss of performance in the linearized channel w → z, where K ∈ K ranges over a class of structured controllers in the sense of [START_REF] Apkarian | Nonsmooth H ∞ synthesis[END_REF]. This is also known as multi-disk optimization [START_REF] Apkarian | Nonsmooth optimization for multidisk H ∞ synthesis[END_REF]. The algorithmic solution proposed in that reference is implemented in the systune package of [START_REF]Control System Toolbox 2020b[END_REF], which we use to solve [START_REF] Apkarian | Nonsmooth H ∞ synthesis[END_REF] algorithmically.

Proposition 1. Suppose the solution K ∈ K of (7) satisfies T qp (G ψ , K ) ∞ < r -1 .
Then the loop (G nl , K ) is stable in the L 2 -sense. That is, for every w ∈ L 2 [0, ∞) and every x 0 the solution of the non-homogenous Cauchy problem ẋcl

= A cl (K )x cl +B p φ(C q x)+ B w w, x cl (0) = x 0 is in L 2 [0, ∞). Moreover γ ∞ = T zw (G, K ) ∞ ≤ (1 + τ )γ ∞ .
Remark 1. Note that (G, K) and (G ψ , K) have different closed loop system matrices.

The A-matrix of G is A, that of G ψ is A ψ , so we have a structured simultaneous stabilization problem, which is known to be NP-hard for most structures.

2.1. Asymptotic sector constraint. Apart from the fact that optimization in ( 7) is over structured controllers K ∈ K , the method so far is standard. The situation changes if the sector sect(a, b) is too large, so that tuning K to achieve T qp (G ψ , K) ∞ < r -1 fails. Then we have to change strategy! What we propose in this work is to choose a different sector, also noted sect(a, b) for simplicity, which the non-linearity φ now satisfies only asymptotically, where Fig. 1 shows schematically what we have in mind.

Definition 1. (Asymptotic sector).

A non-linearity p = φ(t, q) satisfies a sector constraint asymptotically, noted φ ∼ sect(a, b), if there exist M, L > 0 such that for every t ≥ 0, (φ(t, q) -aq)(φ(t, q) -bq) ≤ 0 for |p| > M , and |φ(t, q)| ≤ L for |q| ≤ M .

Suppose we have identified a new typically smaller sector with φ ∼ sect(a, b). (See for instance Fig. 1 for some basic examples of asymptotic sector constraints). We center the non-linearity, now with the new c = (b + a)/2, r = (b -a)/2, which leads to a new ψ(t, p) = φ(t, p) -cp, now satisfying the sector constraint ψ ∼ sect(-r, r) asymptotically. With G ψ taken with regard to the new ψ, the non-linearity (3) is still equivalent to this modified loop (G ψ , ψ). Now we consider the mixed peak-gain/H ∞ -optimization program minimize

T pq (G ψ , K) pk_gn subject to T wz (G, K) ∞ ≤ (1 + τ )γ ∞ K stabilizes G, G ψ K ∈ K (8)
where T qp (G ψ , K) is the channel p → q of the modified G ψ in feedback with K. This optimizes the peak-gain norm of p → q subject to a controlled loss of H ∞ -performance in the channel w → z over the optimistic performance γ ∞ achieved by K * ∈ K . The algorithmic solution of this novel mixed synthesis program will be discussed in Section 5.2. We now have the following consequence of the Small-Gain theorem (compare [START_REF] Mareels | Monotone stability of non-linear feedback systems[END_REF][START_REF] Teel | On graphs, conic relations, and input-output stability of nonlinear feedback systems[END_REF]), see also Theorem 3:

Theorem 1. Let K ∈ K be a solution of program (8) satisfying T pq (G ψ , K ) pk_gn < r -1 . Then for every input w ∈ L ∞ [0, ∞) and all initial conditions x 0 the non-linear closed loop ẋcl = A cl (K )x cl +B p φ(C q x)+B w w has trajectories in (L ∞ [0, ∞), |•| ∞ ),
= T wz (G, K * ) ∞ . 3: Sector. Using φ ∈ sect(a 0 , b 0 ), let c 0 = (b 0 +a 0 )/2, r 0 = (b 0 -a 0 )/2, form ψ 0 = φ-c 0 I,
and represent non-linear system G nl as loop (G ψ 0 , ψ 0 ).

(G ψ 0 , K) ∞ subject to T wz (G, K) ∞ ≤ (1 + τ )γ ∞ K ∈ K If optimal solution K ∈ K satisfies T pq (G ψ 0 , K ) ∞ < r -1
0 , quit successfully. Otherwise continue with step 5.

T qp (G ψ , K) pk_gn subject to T wz (G, K) ∞ ≤ (1 + τ )γ ∞ K ∈ K If optimal solution K satisfies T qp (G ψ , K ) pk_gn < r -1 quit successfully.
Remark 2. By (2) we have G ∞ ≤ G pk_gn even for infinite dimensional systems, so that T pq (G ψ , K) pk_gn < 1 implies T pq (G ψ , K) ∞ < 1. Therefore it makes no sense to choose the asymptotic sector as a true sector. We need φ ∼ sect(a, b), but must have φ ∈ sect(a, b), as φ ∈ sect(a, b) would mean trying step 4 again, saddled with the even harder constraint T pq (G ψ , K) pk_gn < 1.

2.3.

Best asymptotic sector. Working with asymptotic sectors offers additional flexibility over conventional sectors, which we now exploit. Consider step 6 of the algorithm. Instead of choosing the asymptotic sector sect(a, b), which is the same as choosing c, r, we could in the first place only choose c. With ψ(t, q) = φ(t, q) -cq, we solve program

(8 ) minimize T pq (G φ-c , K) pk_gn subject to T wz (G, K) ∞ ≤ (1 + τ )γ ∞ K ∈ K
where r is not yet determined. The optimal controller K(c) ∈ K now depends on c, and as in step 6 of the algorithm, provides the value

(9) r(c) := 1/ T pq (G φ-c , K(c)) pk_gn .
This gives a curve r = r(c), and on putting a = c -r(c), b = c + r(c), it remains to check whether sect(a, b) is an asymptotic sector for φ.

An interesting case is when φ has a slope at infinity, i.e., when lim |x|→∞ φ(t,x) x = q ∞ exists independently of t. Then every choice a < q ∞ < b gives φ ∼ sect(a, b), so with [START_REF] Apkarian | IQC analysis and synthesis via nonsmooth optimization[END_REF] we have to check whether [START_REF] Apkarian | Nonsmooth methods for control design with integral quadratic constraints[END_REF] c -r(c) < q ∞ < c + r(c).

As soon as this holds, we have a posteriori found an asymptotic sector φ ∼ sect(a, b) as a = c -r(c) and b = c + r(c). We can then also determine the parameters L, M in the definition of an asymptotic sector. The smaller M , the closer the asymptotic sector comes to a true sector. Since the asymptotic sector is chosen in response to failure of the true sector sect(a 0 , b 0 ), we typically initialize the search for a, b by values c close to q ∞ , as this increases the chances of program [START_REF] Apkarian | Nonsmooth optimization for multidisk H ∞ synthesis[END_REF] to succeed. This highlights why an asymptotic sector typically will not satisfy 0 ∈ [a, b], whereas this is always satisfied for the true sector.

Remark 3. For every c the aperture 2r(c) of the candidate sector is maximized through program (8) due to [START_REF] Apkarian | IQC analysis and synthesis via nonsmooth optimization[END_REF]. Over the range of those c where [START_REF] Apkarian | Nonsmooth methods for control design with integral quadratic constraints[END_REF] holds, the resulting curve (c, r(c)) serves as a Pareto optimal front, from which we will pick our ultimate c. The decision will not just be based on the size of the aperture of sect(a, b), it may also matter how close sect(a, b) is to a true sector, how large the constant k = LM is, and ultimately, how G nl in loop with K(c) behaves in non-linear simulations. This method will be applied to the slipstick study in Section 3.

K u G G nl φ(•) y 1 y 2 p q - - r K u G y 1 y 2 w W u W y z y z u - - r 1 Figure 2.
Lur'e system G nl as loop between non-linearity φ and LTI system G (left). Nominal H ∞ -synthesis with G and non-linearity interpreted as disturbance w (right).

Application: BIBO-stable control of slipstick vibrations

We consider control of a damped wave equation with instability caused by non-linear boundary anti-damping dynamics,

x tt (ξ, t) = x ξξ (ξ, t) -2λx t (ξ, t), 0 < ξ < 1, t ≥ 0 G nl : x ξ (1, t) = -x t (1, t) + u(t)
αx tt (0, t) = x ξ (0, t) + qx t (0, t) + φ(x t (0, t)) [START_REF] Apkarian | Robustness via structured H ∞ /H ∞ -synthesis[END_REF] where (x, x t ) is the state, u(t) the boundary control, and the measured outputs are

(12) y 1 (t) = x t (0, t), y 2 (t) = x t (1, t).
The non-linearity satisfies φ(0) = 0, φ (0) = 0, so that the linearized system G in (4) is obtained by dropping the term φ(x t ). System G nl has among others been used to model slipstick vibrations in drilling systems, see [START_REF] Apkarian | Boundary feedback control of an anti-stable wave equation[END_REF] and the references given there. The challenge is to control G nl with a finite-dimensional controller u = Ky of simple, implementable structure such that slipstick caused by the non-linear boundary friction term φ(x t ) can be avoided or at least mitigated. In these applications λ ≥ 0, α ≥ 0, q ≥ 0 are typically positive, and the non-linearity derives from a frictional force depending on the angular velocity of the drill

T (ω) = γ 1 ω + γ 2 + γ 3 e -γ 4 |ω| sign(ω),
exhibiting a sharp jump at ω = 0, which based on experimental evidence in comparable situations [START_REF] Ravanbod-Shirazi | Friction identification using the Karnopp model applied to an electromagnetic actuator[END_REF], is slightly mollified around 0. With ω > 0, the nominal angular speed of the drill, step 1 of the algorithm leads to the centered non-linearity

φ(ω) = T (ω) -T (ω + ω) + T (ω) • ω
as in [START_REF] Apkarian | Optimization-based control design techniques and tools[END_REF], shown in Fig. 3 for two of the scenarios studied in [START_REF] Apkarian | Boundary feedback control of an anti-stable wave equation[END_REF]. This corresponds to step 1 of the algorithm. -ω for two scenarios labeled 'gray' and 'blue' in [START_REF] Apkarian | Boundary feedback control of an anti-stable wave equation[END_REF].

We continue to follow the pattern of the algorithm. By [4, sect. 3-4] the number of unstable open-loop poles of G is n p (q, α, λ) ∈ {0, 1, 2}, and several scenarios 'gray', 'blue', 'red', 'magenta' and 'green' were analyzed. The blue scenario, on which we focus here, has two unstable open-loop poles, the numerical parameters gathered in Table 1. Here the aperture of the sector sect(a 0 , b 0 ) in Fig. 3 (right) is extremely large, and step 4 of the algorithm fails even when a rather conservative τ is chosen. This is where we use the asymptotic sector of step 5 of the algorithm.

Numerical values for slipstick study q α λ ω γ 1 γ 2 γ 3 γ 4 gray 0.0019 0.7994 0.1957 3.7186 1.002e-4 11.0034 6.6020 2.4203 blue 0.9797 0.1828 0.5477 6.5044 1.002e-4 28.8697 17.3218 0.1537 Table 1 From [4, Lemma 4] we know that φ(ω) behaves asymptotically as φ(ω) ∼ a ± + φ (∞)ω for ω → ±∞ with φ (∞) = -0.9797. This means, every choice a < φ (∞) < b gives rise to an asymptotic sector φ ∼ sect(a, b). This can for instance be seen in Fig. 6. We now have to give the details of step 6 of the algorithm.

As in Fig. 2 (right), we consider a nominal linear model, where the non-linearity is interpreted as a disturbance w. We optimize the closed-loop H ∞ -channel T (r,w)→(zu,zy) (G, K), where a high pass filter W u is used for the control signal, and a low-pass for tracking of output y 1 , which corresponds to the rotational speed at the drill bit. The rationale is that attenuating the disturbance w should reduce the effect of the non-linearity. This leads to the nominal [START_REF] Apkarian | Boundary feedback control of an anti-stable wave equation[END_REF], exponential stabilizability and detectability of the linear open loop guarantee that the linear closed loop T (r,w)→(zu,zy) (G, K ∞ ) is not only H ∞ -stable, but even exponentially stable, and as a consequence, BIBO-stable.

H ∞ -performance γ ∞ = T (r,w)→(zu,zy) (G, K ∞ ) ∞ for an H ∞ -controller K ∞ ∈ K . As proved in
Since it is necessary to consider asymptotic sectors, we choose the parameter c, and obtain the loop transformed system G φ-c . This corresponds to representing the nonlinearity as a feedback loop as in Fig. 2 (left), and we now have to optimize the peak gain norm of the closed-loop channel p → q in G φ-c . This is the mixed peak-gain/H ∞

program minimize T qp (G φ-c , K) pk_gn subject to T (r,w)→(zu,zy) (G, K) ∞ ≤ (1 + τ )γ ∞ K stabilizes G, G φ-c K ∈ K (13)
which leads to the optimal K opt (c)

∈ K with value 1/r(c) = T qp (G φ-c , K opt (c)) pk_gn .
Then we compute a = c -r(c), b = c + r(c), and if a < q ∞ < b, then the non-linearity is asymptotically in the sector, so that the non-linear closed loop is BIBO-stable. In our experiment K designates 3rd order controllers, which due to n y = 2, n p = 1 leads to 18 optimization variables. Fig. 4 shows these curves for two scenarios τ = 0.1 and τ = 1.0, with c in the range c ∈ [-6.2, -0.5].

Remark 4. Application of our theory requires some preparation, as the system is now infinite dimensional and of boundary control type. We have to clarify the meaning of the impulse response ce A cl t b representing the closed-loop channel w → z. Following [START_REF] Curtain | An Introduction to Infinite-Dimensional Linear Systems Theory[END_REF]Sect. 3.3], [4, sect. 5], the linear wave equation and boundary feedback controller can after a change of variables be represented as an abstract boundary control system Remark 5. We recall that for linear systems BIBO-stability implies H ∞ -stability, and for finite-dimensional LTI-systems the two are equivalent. There exist infinite dimensional LTI-systems which are H ∞ -stable, but not BIBO stable. However, if the system is H ∞stable and exponentially stabilizable and detectable, then it is exponential stable [START_REF] Cheng | Well-posedness of boundary control systems[END_REF], and that implies BIBO stability. The latter because if the growth rate of A is < 0, then there exists a > 0 such that |ce At b| ≤ M e -at , which implies integrability of ce At b.

ẋ = A x, Px = u + w y = C x, z = C 1 x u = Ky
Results. Experiments with different tolerances τ = 1.0 and τ = 0.1 were performed. In each case the parameter c varied in the interval [-6.2, -0.5] and optimization led to an asymptotic sector, see Fig. 4. Smaller values of c lead to larger aperture in the sectors. Two scenarios were selected and underwent non-linear simulations with three types of disturbances shown in Figs. 7 and8. The resulting asymptotic sectors are shown in Fig. 6, and typical optimized impulse responses are shown in Fig. 5.

Extension to multi-dimensional non-linearity

In order to extend our algorithm to systems (3) with multi-dimensional non-linearity, we consider a feedback loop between an LTI-system G and the non-linearity ∆:

v = Gw + f w = ∆(v) + e (14)
as shown in Fig. 9 (left). Well posedness of ( 14) in the L 2 -sense means that G, ∆ are L 2 -bounded causal operators on L 2e , and that the map (v, w) → (e, f ) has a causal inverse on L 2e . The system is L 2 -stable if this inverse is bounded, i.e., if there exists a constant c > 0 with

|v| 2 2 + |w| 2 2 ≤ c |f | 2 2 + |e| 2
for any solution of ( 14). Since we are interested in BIBO-stability, we also need the corresponding notions in the time-domain L ∞ -sense.

Definition 2. The feedback connection (G, ∆) is well-posed in the time-domain L ∞ sense if ∆ and G both map L ∞e into L ∞e , and if the map (v, w) → (e, f ) from ( 14) has a causal inverse (e, f ) → (v, w) on the spaces L ∞e × L ∞e → L ∞e × L ∞e .

Definition 3. The L ∞ -well-posed feedback connection (G, ∆) is BIBO-stable, if in the setting of ( 14) there exist constants

k 1 > 0, k 2 > 0, such that |v| ∞ + |w| ∞ ≤ k 1 (|e| ∞ + |f | ∞ ) + k 2 for all e, f ∈ L ∞ ([0, ∞), R n ).
We now investigate ways in which the steps of the algorithm in Section 2.2 may be extended to MIMO non-linearity (14).

4.1.

Extending the mixed H ∞ /H ∞ -program. Extending step 4 to MIMO non-linearity leads to Integral Quadratic Constraints (IQC), where ∆ and G in loop as in [START_REF] Apkarian | Mixed H 2 /H ∞ -control via nonsmooth optimization[END_REF] satisfy the quadratic constraints induced by a multiplier Π = Π ∼ :

(15) u ∆(u) , Π u ∆(u) T ≤ 0, Gu u , Π Gu u T ≥ 0
for every u ∈ L 2e and every T ≥ 0. While [START_REF] Megretski | System analysis via integral quadratic constraints[END_REF] assures L 2 -stability of the loop if one of the inequalities is satisfied strictly, the crucial question is how the IQC for G may be verified algorithmically. In the literature these are traditionally transformed to LMIs, but in synthesis lead to BMIs, which are known to encounter numerical difficulties. This was recognized in [START_REF] Apkarian | IQC analysis and synthesis via nonsmooth optimization[END_REF][START_REF] Apkarian | Nonsmooth methods for control design with integral quadratic constraints[END_REF][START_REF] Apkarian | Nonsmooth µ-synthesis[END_REF], where non-differentiable optimization techniques in tandem with Hamiltonian tests for function evaluations [START_REF] Balakrishnan | On computing the worst case peak gain of linear systems[END_REF] were preferred instead. Recently this line has been further perfected in [START_REF] Cavalcanti | IQC-synthesis under structural constraints[END_REF][START_REF] Xia | Sector bounds in stability analysis and control design[END_REF].

In [START_REF] Xia | Sector bounds in stability analysis and control design[END_REF], the authors obtain a mixed H ∞ /H ∞ -program expanding on (8) for J-spectral factorable multipliers Π(s) = Ψ ∼ (s)JΨ(s), where J = [I p , 0; 0, -I m ] and Ψ(s) is a L 2bistable rational system. Defining processes ( 16)

Ψ 11 u + Ψ 12 ∆(u) Ψ 21 u + Ψ 22 ∆(u) =: u ∆( u) , Ψ 11 Gu + Ψ 12 u Ψ 21 Gu + Ψ 22 u =: G u u , L 2 -stability of the loop (G, ∆) is equivalent to L 2 -stability of the loop ( G, ∆). Assuming that ∆ is square, it follows from [69, Thm. 5] that G = (Ψ 11 G + Ψ 12 )(Ψ 21 G + Ψ 22 ) -1
is well-posed, and the IQC for G in ( 15) is transformed into G ∞ ≤ 1. Similarly, with

• denoting map or relation composition, ∆ = (Ψ 22 ∆ + Ψ 21 ) • (Ψ 12 ∆ + Ψ 11 ) -1
, and due to [START_REF] Apkarian | Time-domain control design: a non-differentiable approach[END_REF], left, this process is an L 2 contraction, i.e., signals

z 1 = Ψ 11 u + Ψ 12 ∆(u) and z 2 = Ψ 21 u + Ψ 22 ∆(u) satisfy T 0 |z 2 (t)| 2 dt ≤ T 0 |z 1 (t)| 2 dt for all T > 0.
What is not clear is whether ∆ is a mapping.

We now present an alternative way to obtain a mixed H ∞ /H ∞ -program, which gives an explicit loop transformation and, as we shall see, is also applicable to positivity type factorizations. Consider again IQC multipliers Π(s) factored as [START_REF] Bank | Non-linear parametric optimization[END_REF] Π(jω) = Ψ(jω) * P Ψ(jω)

for a bistable LTI-system Ψ(s) and a static invertible P = P T . Such factorizations exist for rational Π = Π ∼ if Π has neither poles nor zeros on jR and allows no equalizing vectors, i.e., no u ∈ H 2 , u = 0, with Πu ∈ H ⊥ 2 ; cf. [START_REF] Meinsma | J-spectral factorization and equalizing vectors[END_REF]. In particular, positivenegative multipliers satisfying Π 11 (jω)

I and Π 22 (jω) -I for some > 0 admit such factorizations [START_REF] Seiler | Stability analysis with dissipation inequalities and integral quadratic constraints[END_REF]. Following [START_REF] Seiler | A dissipation inequality formulation for stability analysis of integral quadratic constraints[END_REF]Thm. 6], there is no harm in assuming that both IQCs in [START_REF] Apkarian | Time-domain control design: a non-differentiable approach[END_REF] are satisfied strictly. Now define a new augmented interconnection (G a , ∆ a ) as ( 18)

G a = Ψ -I 2G 0 I Ψ -1 , ∆ a = Ψ • I 0 2∆ -I • Ψ -1 ,
then by [68, Thm. 2 (1)], L 2 -stability of (G, ∆) is equivalent to L 2 -stability of (G a , ∆ a ). Adopting Π 11 I and Π 22 -I for some > 0, it follows from [68, Thm. 2 [START_REF] Apkarian | Multi-model, multi-objective tuning of fixed-structure controllers[END_REF]] that ∆ a , G a satisfy IQCs for the passivity multiplier P a = [0, P ; P, 0] strictly, i.e.,

T 0 p a (t) T G a (P -1 p a )(t)dt ≤ - T 0 p a (t) T p a (t)dt T 0 p a (t) T P ∆ a (p a )(t)dt ≥ T 0 p a (t) T p a (t)dt (19)
for some > 0 and every T ≥ 0, where p a = (p, q) T .

The inequalities in [START_REF] Besselink | Analysis and control of stick-slip oscillations in drilling systems[END_REF] are now turned into bounded gain conditions using Möbius or bilinear transformations. We introduce [START_REF] Boyd | Linear controller design. Limits of performance[END_REF] G e := B G a P -1 = (G a P -1 -I) -1 (G a P -1 + I) and ( 21)

∆ e = P • ∆ a B = (I + P • ∆ a ) -1 • (I -P • ∆ a )
with

B := -I √ 2I - √ 2I I ,
where is the Redheffer star product [START_REF] Redheffer | On a certain linear fractional transformation[END_REF]. Here (I + P ∆ a ) -1 and (G a P -1 -I) -1 are welldefined and stable due to [START_REF] Besselink | Analysis and control of stick-slip oscillations in drilling systems[END_REF] and the passivity theorem, hence ∆ e , G e are well-defined and L 2 -stable. Indeed, (I + P • ∆ a ) -1 is the negative feedback loop between the upper block I and the lower block P •∆ a . Since P •∆ a is strictly passive by [START_REF] Besselink | Analysis and control of stick-slip oscillations in drilling systems[END_REF] and I is passive, stability follows from the passivity theorem [START_REF] Zames | On the input-output stability of time-varying non-linear feedback systems. Part I: Conditions derived using concepts of loop gain, conicity, and positivity[END_REF]. A similar argument applies to (G a P -1 -I)

for some > 0 and every T ≥ 0, where p e = (p, q) T . See [33, pp. 215-16] for a proof, which also applies to the non-linear case.

From ∆ e = P • ∆ a B we have

∆ e B = P • ∆ a B B = P • ∆ a I = P • ∆ a , hence ∆ a = P -1 • ∆ e B. That gives I 0 2∆ -I = Ψ -1 P -1 • ∆ e BΨ. Hence, (23)
∆ = I √ 2 I √ 2 T Ψ -1 (P -1 • ∆ e B)Ψ I √ 2 I √ 2
, which gives the inverse operation to ∆ → ∆ e in [START_REF] Boyd | Comparison of peak and RMS gains for discrete-time systems[END_REF]. What we have obtained is a parametrization of all non-linearities ∆ derived from L 2 -contractions ∆ e via the loop transformation through Ψ(s), P , or equivalently, all non-linearities satisfying IQCs with factorable multiplies Π = Ψ * P Ψ. For these ∆ the IQC-stability theorem can now be reduced to the small gain theorem [START_REF] Zames | On the input-output stability of time-varying non-linear feedback systems. Part I: Conditions derived using concepts of loop gain, conicity, and positivity[END_REF].

Theorem 2. (IQC as H ∞ constraint). Suppose (G, ∆) is loop transformed to (G e , ∆ e ), where ∆ satisfies the IQC with multiplier Π = Ψ * P Ψ factored with bistable Ψ(s) and invertible P = P T . Then G e ∞ < 1 implies L 2 -stability of the loop (G, ∆).

Applying the loop transformation (G, ∆) ∼ = (G e , ∆ e ) to the closed loop system F l (G, K) leads to (F l (G, K), ∆) ∼ = (F l (G, K) e , ∆ e ). This allows us now to extend step 4 of the algorithm to IQCs.

Corollary 1. Suppose the mixed H ∞ /H ∞ -synthesis program minimize F l (G, K) e ∞ subject to T wz (G, K) ∞ ≤ (1 + τ )γ ∞ K ∈ K (24)
admits an optimal solution K ∈ K satisfying F l (G, K ) e ∞ < 1. Then K stabilizes the loop (G, ∆) in the L 2 -sense, and linearized closed loop performance is degraded over nominal performance γ ∞ by no more than the factor 1 + τ .

Remark 6. Program ( 24) is now a natural MIMO-extension of [START_REF] Apkarian | Nonsmooth H ∞ synthesis[END_REF]. It can be efficiently solved by the method of [START_REF] Apkarian | Nonsmooth H ∞ synthesis[END_REF][START_REF] Apkarian | Nonsmooth optimization for multidisk H ∞ synthesis[END_REF] available in the systune package of [START_REF]Control System Toolbox 2020b[END_REF][START_REF] Apkarian | Multi-model, multi-objective tuning of fixed-structure controllers[END_REF][START_REF] Apkarian | Nonsmooth optimization for multidisk H ∞ synthesis[END_REF]. This is numerically preferable to transforming IQCs to BMIs. With the recent extension of non-smooth H ∞ -synthesis in [START_REF] Apkarian | Structured H ∞ -control of infinite dimensional systems[END_REF][START_REF] Apkarian | Boundary control of partial differential equations using frequency domain optimization techniques[END_REF][START_REF] Apkarian | Boundary feedback control of an anti-stable wave equation[END_REF] it becomes even possible to address [START_REF] Chellaboina | Induced convolution operator norms of linear dynamical systems[END_REF] for infinitedimensional systems with infinite-dimensional multipliers Ψ(s).

Remark 7. An advantage of this construction is that when (G, ∆) satisfies an IQC with positivity multiplier as in [START_REF] Besselink | Analysis and control of stick-slip oscillations in drilling systems[END_REF], then going from (G, ∆) to (G a , ∆ a ) can be skipped and we build (G e , ∆ e ) directly without the augmentation [START_REF] Bemporad | The explicit linear quadratic regulator for constrained systems[END_REF].

For multipliers Π(s) with a lower triangular factorizations both approaches ( 16) and the augmentation G → G a → G e lead to the same result. Suppose

(26) ∆ = Ψ 21 Ψ -1 11 + Ψ 22 • ∆ • Ψ -1 11 , G = Ψ 11 G(Ψ 22 + Ψ 21 G) -1 , give an equivalent loop (G, ∆) ∼ = ( G, ∆), where the IQC is transformed to a Small-Gain condition | ∆( v)| 2 ≤ | v| 2 , G ∞ < 1, now with ∆ and G of the same dimension as ∆, G.
Since G e ∞ < 1 is equivalent to G ∞ < 1 in the case (25), Theorem 2 implies:

Corollary 2. (Triangular transform). Suppose a non-linearity ∆ can be loop transformed to a L 2 -contraction ∆ in (26) using a lower triangular factorization [START_REF] Cheng | Well-posedness of boundary control systems[END_REF]. Suppose the transformed

LTI-system G = Ψ 11 G I + Ψ -1 22 Ψ 21 G -1 Ψ -1 22 is stable and satisfies G ∞ < 1. Then the loop (G, ∆) is stable in the L 2 -sense.
This may also be seen from [START_REF] Balakrishnan | On computing the worst case peak gain of linear systems[END_REF]; see also Fig. 9. For upper triangular Ψ we obtain the analogous result with

(27) ∆ = Ψ 22 ∆ • (Ψ 12 ∆ + Ψ 11 ) -1 , G = (Ψ 11 G + Ψ 12 )Ψ -1
22 . These results no longer require any reference to IQCs or multipliers.

4.2.

Extending the mixed peak-gain/H ∞ -program. The one-dimensional peak-gain norm [START_REF] Boyd | Linear controller design. Limits of performance[END_REF] allows several extensions to MIMO systems, because we can replace the absolute value |x|, x ∈ R, by any of the equivalent vector norms in R n . If we define a signal norm on L

∞ ([0, ∞), R n ) by |x| ∞,p = sup t≥0 |x(t)| p
with |v| p the p-norm of v ∈ R n , 1 ≤ p ≤ ∞, then with the notation adopted from [START_REF] Chellaboina | Induced convolution operator norms of linear dynamical systems[END_REF] any induced system norm ( 28)

G (∞,p),(∞,q) = sup x =0 |G * x| ∞,q |x| ∞,p
is a valid MIMO extension of • pk_gn . The peak-gain norm, to which we give preference here, is the special case G pk_gn = G (∞,∞),(∞,∞) , but all norms (28) are equivalent.

Theorem 3. Suppose the non-linear operator in [START_REF] Apkarian | Mixed H 2 /H ∞ -control via nonsmooth optimization[END_REF] satisfies |∆(t, x)| p ≤ |x| q for every |x| q > M and |∆(t, x)| p ≤ L for every |x| q ≤ M . If the LTI-system G satisfies G (∞,p),(∞,q) < 1, then the closed loop is BIBO-stable with

|v| ∞,q + |w| ∞,p ≤ k 1 (|e| ∞,p + |f | ∞,q ) + k 2 for all e, f ∈ L ∞e . Proof: Put ∆ 1 (t, x) = ∆(t, x)χ {|x|q≤M } (x), ∆ 2 (t, x) = ∆(t, x)χ {|x|q>M } (x), then we have sup t≥0 |∆ 1 (t, v(t))| p ≤ L, while sup t≥0 |∆ 2 (t, v(t))| p ≤ sup t≥0 |v(t)| q by hypothesis. Hence, assuming |Gx| ∞,q ≤ (1 -δ)|x| ∞,p for some 0 < δ < 1, |v| ∞,q ≤ |w | ∞,q + |f | ∞,q ≤ |G∆ 1 (v)| ∞,q + |G∆ 2 (v)| ∞,q + |Ge| ∞,q + |f | ∞,q ≤ G (∞,p)(∞,q) (|∆ 1 (v)| ∞,p + |∆ 2 (v)| ∞,p + |e| ∞,p ) + |f | ∞,q ≤ (1 -δ)L + (1 -δ)|v| ∞,p + (1 -δ)|e| ∞,p + |f | ∞,q hence δ|v| ∞,q ≤ (1 -δ)|e| ∞,p + |f | ∞,q + (1 -δ)L.
On the other hand

|w| ∞,p ≤ |∆ 1 (v)| ∞,p + |∆ 2 (v)| ∞,p + |e| ∞,p ≤ L + |v| ∞,q + |e| ∞,p .
Combining the two implies the estimate.

For this result compare the more general [START_REF] Mareels | Monotone stability of non-linear feedback systems[END_REF], [START_REF] Teel | On graphs, conic relations, and input-output stability of nonlinear feedback systems[END_REF]. The above proof is standard and included for convenience. This gives us now a clue how to extend asymptotic constraints as encountered in Section 2.1 to MIMO non-linearities.

Definition 4. (Asymptotic L ∞ -contraction). A non-linear operator ∆ : [0, ∞) × R n → R n is called an asymptotic L ∞ -contraction if there exist L, M > 0 such that |∆(t, x)| ∞ ≤ |x| ∞ for all |x| ∞ > M , t ≥ 0, and |∆(t, x)| ∞ ≤ L for all |x| ∞ ≤ M, t ≥ 0.
Remark 8. The proof of Theorem 3 shows that an asymptotic L ∞ -contraction satisfies

|∆(t, x)| ∞ ≤ |x| ∞ + k for all x. Conversely, suppose we have |∆(t, x)| ∞ ≤ |x| ∞ + k for all x.
Then for every > 0 there exists M > 0 such that |∆(t, x)| ∞ < (1 + )|x| ∞ for all |x| ∞ > M . For suppose on the contrary that there exist

x n with |x n | ∞ → ∞ such that for some > 0 |∆(t, x n )| ∞ ≥ (1 + )|x n | ∞ , then 1 + ≤ |∆(t, x n )| ∞ /|x n | ∞ ≤ 1 + k/|x n | ∞ → 1,
a contradiction. Since for the LTI-system we request the strict inequality G pk_gn < 1, both conditions for ∆ may be used indifferently in the small gain theorem. Proof: This refers to [START_REF] Curtain | An Introduction to Infinite-Dimensional Linear Systems Theory[END_REF] shown in Fig. 9, where (G, ∆) is loop transformed to ( G, ∆) in such a way that BIBO-stability of (G, ∆) is equivalent to BIBO-stability of ( G, ∆). But ( G, ∆) is amenable to Theorem 3, hence G pk_gn < 1 implies BIBO-stability of the loop.

G w ′ v f ∆ v ′ e w Ψ 11 G Ψ -1 22 Ψ -1 22 Ψ 21 w ′ ∆ Ψ 11 v Ψ 11 f - u w Ψ 22 e-Ψ 21 f G 1 Figure 9. Loop-transformation (G, ∆) to ( G, ∆).
We now extend program [START_REF] Apkarian | Nonsmooth optimization for multidisk H ∞ synthesis[END_REF] to the MIMO-case. In the case of Fig. 2, we transform the non-linearity ∆ to an asymptotic contraction ∆ via [START_REF] Curtain | An Introduction to Infinite-Dimensional Linear Systems Theory[END_REF]. Now consider a plant then the extension of ( 8) has the form of the mixed program

P :   q z y   =
(8 ) minimize F l (P 1 , K) ∼ pk_gn subject to F l (P 2 , K) ∞ ≤ (1 + τ )γ ∞ K ∈ K
where ∼ indicates that the loop transformation G of Fig. 9 is applied to the controlled system F l (P 1 , K). The program is successful as soon as a structured LTI controller K * ∈ K is found which stabilizes P 1 in the BIBO-sense, stabilizes P 2 exponentially, and achieves F l (P 1 , K * ) ∼ pk_gn < 1. Remark 9. We could also use the general loop transformation [START_REF] Challamel | Rock destruction effect on the stability of a drilling structure[END_REF]. Suppose ∆ is obtained from a | • | ∞ contraction ∆ e via [START_REF] Challamel | Rock destruction effect on the stability of a drilling structure[END_REF], where ∆ e B is L ∞ -well-posed. Then a sufficient condition for BIBO-stability of the loop is L ∞ -well-posedness of G e in [START_REF] Boyd | Linear controller design. Limits of performance[END_REF] with G e pk_gn < 1, and this includes both cases [START_REF] Curtain | An Introduction to Infinite-Dimensional Linear Systems Theory[END_REF], [START_REF] Munther | Robust controller design: Minimizing peak-to-peak gain[END_REF]. 4.3. Asymptotic L ∞ -contractions. In this section, we collect a variety of examples of MIMO non-linearities which may be assessed by way of asymptotic L ∞ -contractions.

Example 1. Consider a non-linearity ∆(t, q) in loop (G, ∆) as

G : ẋ = Ax + Bp q = Cx p(t) = ∆(t, q(t)).
We say that ∆ is asymptotically polyhedral, if there exist polyhedral norms | • | and | • | on R n and k > 0 such that |∆(t, q)| ≤ |q| + k for all q ∈ R n and all t ≥ 0. Now polyhedral norms are of the form

|x| = sup i=1,...,m n j=1 τ ij x j = |T x| ∞ , |y| = sup k=1,...,p n j=1 σ kj y j = |Sy| ∞ for certain T ∈ R m×n , S ∈ R p×n with {x ∈ R n : |T x| ∞ ≤ 1}, {y ∈ R n : |Sy| ∞ ≤ 1}
bounded. The latter means T, S are injective. Let T + , S + be left inverses, T + T p = p, S + Sq = q. Then we have

|T ∆(t, S + Sq)| ∞ = |T ∆(t, q)| ∞ ≤ |Sq| ∞ + k
so on introducing the new variables q 1 = Sq, p 1 = T p, we have a new non-linearity

∆(t, •) = T • ∆(t, •) • S + which satisfies | ∆(t, q 1 )| ∞ ≤ |q 1 | ∞ + k for all q 1 and t ≥ 0.
The non-linearity ∆ being in loop with G, we transform this to bring ∆ in loop with:

G : ẋ = Ax + BT + p 1 q 1 = SCx
This means the non-linear loop (G, ∆) is BIBO-stable if G pk_gn < 1. This is a special case of the transform [START_REF] Curtain | An Introduction to Infinite-Dimensional Linear Systems Theory[END_REF].

Example 2. The following is a concretization. We call ∆ differentiable at infinity if there exists a matrix

∆ ∞ := ∆ (∞) ∈ R n×n such that lim |x|→∞ |∆(t, x) -∆ (∞)x| |x| = 0
uniformly over t ≥ 0. Here we may choose arbitrary norms in numerator and denominator. Consider the case where ∆ (∞) = 0 and choose regular matrices T, S ∈ R n×n such that T ∆ (∞)S -1 = diag(1 -, . . . , 1 -, 0, . . . , 0) =: J , where the diagonal has rank(∆ (∞)) many entries 1 -. Now choose the norms |y| = |T y| ∞ and |x| = |Sx| ∞ , then

|∆(t, x)| |x| ≤ |∆(t, x) -∆ (∞)x| |x| + |∆ (∞)x| |x| = o(1) + |T ∆ (∞)S -1 Sx| ∞ |Sx| ∞ ≤ o(1) + |||J ||| ∞ |Sx| ∞ |Sx| ∞ ≤ o(1) + 1 -,
where

|||J ||| ∞ = 1 -is the maximum row sum norm. Choosing M > 0 such that |o(1)| < /2 for |x| > M , we arrive at |∆(t, x)| ≤ (1 -/2)|x| + sup{|∆(t, x)| : |x| ≤ M } =: (1 -/2)|x| + k.
This means every non-linearity which is differentiable at infinity admits asymptotic L ∞ -constraints.

Remark 10. Suppose a non-linearity

∆ : [0, ∞) × R n → R n satisfies |∆(t, q)| 2 ≤ |q| 2 + k
for some k ≥ 0 and all q ∈ R n , t ≥ 0. Then in Theorem 3 we would prefer the system norm G (∞,2),(∞,2) . Unfortunately, no computable expression is currently known for this norm, so its optimization is presently impossible.

Remark 11. For the case |∆(t, q)| 2 ≤ |q| 2 +k we have the following makeshift alternative. Choose approximations P 1 ⊂ B(0, 1) ⊂ P 2 by polytopes P 1 , P 2 , then ∆(P 1 ) ⊂ P 2 asymptotically, so we are in the situation of Example 1 and we may work with • pk_gn . In the case |∆(t, q)| 2 ≤ (1 -)|q| 2 + k we may even obtain this with P 1 = P 2 ⊂ {q : |q| 2 = 1}.

Example 3. MIMO sectors are defined via symmetric matrices

A, B ∈ S n satisfying A ≺ B. A mapping φ : R n → R n with φ(0) = 0 is in the sector sect(A, B), noted φ ∈ sect(A, B), if (φ(x) -Ax) T (φ(x) -Bx) ≤ 0 for all x ∈ R n . With the choice R = 1 2 (B -A) 0 and C = 1 2 (B + A) we find that ψ(x) = φ(x) -Cx satisfies ψ(x) T ψ(x) ≤ x T R T Rx, so we get a norm bound |ψ(x)| 2 ≤ |Rx| 2 ,

and if we define

∆ = ψ • R -1 = (φ -C) • R -1 , then |∆(y)| 2 ≤ |y| 2 .
When we allow |∆(y)| 2 ≤ |y| 2 + k for some k ≥ 0 and all y, this is a typical application of the two previous remarks, where we would like to apply Theorem 3 with • (∞,2),(∞,2) .

Example 4. As a concretization [START_REF] Ahmad | LMI-based stability criteria for discrete-time Lur'e systems with monotonic, sector-and slope-restricted nonlinearities[END_REF][START_REF] Bemporad | The explicit linear quadratic regulator for constrained systems[END_REF] consider a non-linearity φ generated by a convex quadratic program:

(29) φ(x) = argmin{ 1 2 v T Hv -v T x : Lv ≤ b} where H 0 and L ∈ R m×n , b ≥ 0, b ∈ R m are
fixed, and optimization is over v ∈ R n . Using the Kuhn-Tucker conditions one verifies that φ satisfies the MIMO sector bound φ(x) T (Hφ(x) -x) ≤ 0 for all x.

Then from the above ψ(

x) = 2H 1/2 φ(H 1/2 x) -x is a | • | 2 -contraction.
This means an asymptotic quadratic constraint for φ would lead to • (∞,2),(∞,2) , which is, however, not available for computations. A polyhedral approximation based on Remark 11 may be used instead.

Example 5. (Continued). In the above case we can bring in G pk_gn directly, because the solution mapping of a convex quadratic program with perturbation of the linear term or the constraints is known to be piecewise affine [START_REF] Bank | Non-linear parametric optimization[END_REF][START_REF] Bemporad | The explicit linear quadratic regulator for constrained systems[END_REF], so it maps polyhedra to polyhedra. This allows a construction as in Example 1.

Since H 0, (29) is equivalent to projecting H -1 x orthogonally on the polyhedron {v : Lv ≤ b} with regard to the Euclidean norm |x| 2 H = x T Hx. Let I ⊂ {1, . . . , m}, J = {1, . . . , m} \ I, so that

F = {v ∈ R n : L I v = b I , L J v ≤ b J }
is a face of the polyhedron, then projection of x on F is obtained as

φ(x) = H -1 x -L T I (L I H -1 L T I ) -1 L I H -1
x -b I using generalized least squares. This shows that φ is piecewise affine. Example 6. (Piecewise affine). A non-linearity φ on R n is piecewise affine if there exist finitely many non-overlapping polyhedra P 1 , . . . , P N with N i=1 P i = R n such that φ|P i is affine, i.e. there exist an affine mapping A i (x) = b i +L i x with A i |P i = φ|P i . Here L i is the linear part of A i . For each polyhedron P i choose a Motzkin decomposition Example 7. We consider a numerical example with a non-linearity based on [START_REF] Dahleh | Optimal and robust controllers for periodic and multirate systems[END_REF]. The parametric quadratic program

P i = Q i + C i with Q i a polytope and C i a polyhedral cone. Let B = {x : |x| ≤ 1} be the unit ball of a polyhedral norm, compute the polytopes B i = B ∩ C i and B i = L i (B i ). (Note that L i (C i ) is a polyhedral cone, so if it is bounded, it reduces to {0}, in which case B i = {0}, too. Therefore only unbounded L i (C i ) have to be considered). Finally let B be the convex hull of N i=1 B i , then B is a polytope containing 0. Let k 1 = max{|b | ∞ : b ∈ B }. We claim that there exists a constant k > 0 such that for every x ∈ R n , φ(x) ∈ |x| B + kB ∞ . Indeed, let x ∈ P i , x = y + z with y ∈ Q i , z ∈ C i . Then φ(x) = b i + L i x = b i + L i y + L i z = b i + L i y + |z| L i (z/|z| ) = b i + L i y i + |z| b for some b ∈ B i , using the fact that z/|z| ∈ B ∩ C i , hence L i (z/|z| ) ∈ L i (B ∩ C i ) = B i . Now |z| = |x -y| ≤ |x| + |y|
(P ) q minimize 1 2 x 2 1 + 1 2 x 2 2 -q 1 x 1 -q 2 x 2 subject to x 1 -x 2 ≤ 3 x 1 + x 2 ≥ 0 x 1 ≥ 0 x 2 ≥ -1
defines a non-linear operator φ : R 2 → R 2 via p = argmin(P ) q . Here p = φ(q) is the orthogonal projection of q on the polyhedron P = {x ∈ R 2 : Lx ≤ b}, where

L T = 1 -1 -1 0 -1 -1 0 -1 , b T = 3 0 0 1 . For every face F of P the set F = φ -1 (F ) is
a polyhedron and each φ|F : F → P is affine. Here P has 8 faces, three vertices, four facets, and P itself. The unbounded faces are

F 1 = {(2, -1)} + R + (1, 1) = Q 1 + C 1 , F 2 = R + (0, 1) = C 2
, and P = Q + 0 + P , with 0

+ P = {x : Lx ≤ 0}. Now F 1 = φ -1 (F 1 ) = {(2, -1)} + R + (1, 1) + R + (1, -1) = Q 1 + C 1 and φ 1 = φ|F 1 = A 1 |F 1 for the affine mapping A 1 (x) = L 1 x + b 1 = 1/2 1/2 1/2 1/2 x 1 x 2 + 3/2 -3/2 . Similarly, F 2 = φ -1 (F 2 ) = R + (-1, 0) + R + (0, 1) = C 2 ,
= co(B ∞ ∪ {(2, 0), (-2, 0)}). The polyhedral norm is |q| = |Sq| ∞ with S T = 0 1/2 1/2 1 1/2 -1/2 .
By example 1, BIBO-stability of

G : ẋ = Ax + Bp q = Cx p(t) = φ(q(t)).
can now be assessed via BIBO-stability of

G : ẋ = Ax + BT -1 p 1 q 1 = SCx p 1 (t) = T φ(S + q 1 (t)),
where

T • φ • S + is a | • | ∞ -contraction.
A sufficient condition for BIBO-stability of the loop is therefore G pk_gn < 1.

We compare this to the sector characterization of the non-linearity φ from example 3. With H = I 2 we get φ(q) T (φ(q) -q) ≤ 0, hence ψ(q) = 2φ(q) -q is an L 2 -contraction.

For instance, similar to [START_REF] Miranda-Villatoro | Analysis of Lur'e dominant systems in the frequency domain[END_REF]Example 4.6] we let Φ(q 1 , q 2 , q 3 ) = (φ(q 3 ), 0, 0) with φ(q) = a tanh(kq) + ρq, B = C = I 3 , and

A =   -(β 1 + β 2 + β 3 ) -(β 1 β 2 + β 1 β 3 + β 2 β 3 )/M -β 1 β 2 β 3 /M M 0 0 0 1 0   .
For M = a = k = 10, ρ = 0.3, β 1 = 2, β 2 = 3, β 3 = 5, the non-linearity ∆ has three steady states, the unstable 0, and two stable attractors (0, 0, ±2.963) (Fig. 10 left). The system is globally BIBO-stable, because φ has slope ρ at infinity, so integrating ρq 3 into the system H gives H 0 in loop with a non-linearity Φ 0 of bounded range. Then r = 1/3 gives H 0 pk_gn = 2.0759 < 3 = r -1 . The linearization of ∆ at 0 has system matrix A + E 1 with E 1 = 0, 0, φ (0); 0, 0, 0; 0, 0, 0 , which is unstable, but a stable linearization can readily be obtained by linearizing about one of the attractors, e.g. x = (0, 0, 2.963). A different case has a = -10, where the origin is an unstable steady state and an attracting limit cycle occurs (Fig. 10 middle). Global BIBO-stability of the loop follows again from H pk_gn < ∞, but a stable linearization now requires a modified LTI-system, where the limit cycle is subtracted from H to get a stable steady state.

In those cases, where z = ∆(w) is BIBO-stable, we can consider it in loop with a tunable LTI-system G as in Fig. 9. For unstable H we can still apply the results of section 4.2 when H is stabilized by feedback with a tunable G, now considering Φ in loop with F l (H, G): Proposition 2. Let z = ∆(w) in (30) be in upper feedback with an LTI-system G as in [START_REF] Apkarian | Mixed H 2 /H ∞ -control via nonsmooth optimization[END_REF]. Suppose Φ satisfies an asymptotic constraint

|Φ(q)| ∞ ≤ r|q| ∞ + k. Then a sufficient condition for global BIBO-stability of the loop (G, ∆) is F l (H, G) pk_gn < r -1 .
Example 12. (Attractors . . . continued). An interesting study in this line is Chua's circuit [START_REF] Matsumoto | The double scroll[END_REF], see [46, 5.4], where

A =   -α α 0 1 -1 1 0 -β 0   , B = C = I 3 ,
Φ(q) = (φ(q 1 ), 0, 0) φ(q 1 ) = α tanh(2q 1 ) + αρq 1 For α = 8.3, β = 16.5, ρ = 0.25 the double scroll attractor appears (Fig. 10 right). The non-linearity has slope αρ at infinity. Therefore global BIBO-stability of ∆ = (H, Φ) follow from stability of A + E 2 , E 2 = αρ, 0, 0; 0, 0, 0; 0, 0, 0 .

A common feature of these examples is that the sector non-linearity invites attempting L 2 -stability, which however fails due to the persistence of more than one attractor in feedback. This is where global L ∞ -stability is still in business.

Example 13. (Attractors . . . continued). We study this phenomenon in more detail through a feedback design example. Consider the MIMO Lur'e system

H : ẋ = Ax + Bp + B u u q = Cx, y = C y x, p = Φ(q) (31)
with Φ : R 3 → R 3 the MIMO static non-linearity

Φ(q) :=     q 2 1 a 1 +q 2 1 (tanh(c 1 q 1 ) + ρ 1 q 1 )) q 2 2 a 2 +q 2 2 (tanh(c 1 q 2 ) + ρ 2 q 1 )) q 2 3 a 3 +q 2 3 (tanh(c 3 q 3 ) + ρ 3 q 3 ))     , with a 1 = 0.1, a 2 = 0.2, a 3 = 0.3, ρ 1 = 0.1, ρ 2 = 0.2, ρ 3 = 0.3, c 1 = 2, c 2 = 3 and c 3 = 4.
State-space data are given as

A :=   -2 8.8 0 1 -1 1 0 -15 0   , B :=   5 0 0 0 0.1 0 0 0 0.3   , C := I 3 , B u :=   1 1 1   C y := 1 1 1 .
The uncontrolled linear dynamics show 2 unstable oscillating modes 0.1422 ± 3.0189i. Simulations of the uncontrolled non-linear system are shown in Fig. 11 (upper line) with a double-scroll regime close to the origin, and an escaping unstable spiral regime away from 0. Here we regard ∆ = (H, Φ) as mapping initial conditions x(0) to state x. Now we investigate whether the system may be stabilized by feedback u = Ky in the L ∞ -sense, using the techniques in sections 4.1 and 4.2. As can be seen each component Φ i belongs to the asymptotic sector sect(0, ρ i + ) for any > 0. Using section 4.2, we infer that the closed-loop system is L ∞ -stable whenever F l ( H, K) pk_gn < r -1 , where and r := max {ρ 1 /2, ρ 2 /2, ρ 3 /2} and H is obtained from H by centering the non-linearity. The latter amounts to shifting the A-matrix to A+BΓC with Γ := diag (ρ 1 /2, ρ 2 /2, ρ 3 /2).

Running program (8) over the class K of PID controllers leads to K * (s) := -0.796 + 0.000352 s + 0.097 940s + 1 , with the result F l ( H, K * ) pk_gn = 5.34 < r -1 = 6.67, affirming BIBO stability. Closedloop simulations in Fig. 11 bottom show co-existence of 3 stable equilibrium points at the origin (right) and away from the origin (bottom left) with state-space coordinates (±2.98, ∓0.0420, ∓2.94, ∓237.94, 0). Note that in this study step 4 of the algorithm fails, because Φ i ∈ sect(0, 1.17) tightly for every i. Therefore, to get a global L 2 -stability certificate it would have been required to determine a PID controller for which the H ∞ norm of the corresponding centered system was less than 1/r 0 = 1.71, and this value was not achievable in program [START_REF] Apkarian | Nonsmooth H ∞ synthesis[END_REF].

5. Peak-to-peak norm 5.1. Estimate. It is well-known [START_REF] Chellaboina | Induced convolution operator norms of linear dynamical systems[END_REF][START_REF] Munther | Robust controller design: Minimizing peak-to-peak gain[END_REF][START_REF] Desoer | Feedback Systems: Input-Output Properties[END_REF][START_REF] Boyd | Linear controller design. Limits of performance[END_REF][START_REF] Boyd | Comparison of peak and RMS gains for discrete-time systems[END_REF] that for real-rational systems G the peak-gain or peak-to-peak norm is [START_REF] Dao | Simultaneous plant and controller optimization based on nonsmooth techniques[END_REF] G pk_gn = max where

g ij (t) = c i e At b j + d ij δ(t) = g 0 ij (t) + d ij δ(t) with g 0 ij ∈ L 1 . A special case is the well-known expression |||A||| ∞ = max i=1,...,m p k=1 |a ik | = max i=1,...,m |row i (A)| 1 of the maximum row-sum-norm of A ∈ R m×p , i.e., the induced ∞ -∞ matrix norm.
Formula (32) holds also for infinite dimensional BIBO-stable systems and may be justified e.g. by the approach [START_REF] Unser | A note on BIBO-stability[END_REF][START_REF] Feichtinger | A novel mathematical approach to the theory of translation invariant linear systems[END_REF], which considers BIBO-stable systems as all those LTI-systems G, where G(s) is the Laplace transform of a matrix-valued Radon measure of bounded variation. While [START_REF] Unser | A note on BIBO-stability[END_REF] handles the SISO case, where the norm is referred to as the M-norm µ M = sup φ∈D(R + ), φ ∞≤1 R + φdµ, one easily generalizes this to the MIMO case and obtains the formula G pk_gn = sup i=1,...,m p k=1 µ ik M , which contains (32) as a special case. In particular, it was possible to apply it in the slipstick study, because the impulse response was an element of L 1 .

Estimate (2) is mentioned in [START_REF] Munther | Robust controller design: Minimizing peak-to-peak gain[END_REF] with non-specified constants, and the SISO case p = m = 1 is proved in [START_REF] Boyd | Comparison of peak and RMS gains for discrete-time systems[END_REF]Thm.] for discrete SISO systems, and in [35, pp. 11-12] for continuous SISO systems, where in the latter reference the idea of proof is attributed to I. Gohberg. The left hand estimate in (2) holds also for infinite dimensional systems, e.g. those where G(s) is the Laplace transform of a Radon measure of bounded variation, while the right hand estimate is true for finite-dimensional G. For strictly proper systems, the estimate G pk_gn ≤ 2np 1/2 G ∞ is valid. Details on computing these estimates will be published elsewhere. 5.2. Implementation. Stand alone computation of the peak-gain norm with high precision has been addressed in the literature [START_REF] Rutland | Computing the 1-norm of the impulse response of linear time-invariant systems[END_REF][START_REF] Linnemann | Computing the L 1 -norm of continuous-time linear systems[END_REF][START_REF] Boyd | Comparison of peak and RMS gains for discrete-time systems[END_REF][START_REF] Chellaboina | Induced convolution operator norms of linear dynamical systems[END_REF][START_REF] Munther | Robust controller design: Minimizing peak-to-peak gain[END_REF][START_REF] Dahleh | l 1 -optimal feedback controllers for MIMO discrete-time systems[END_REF][START_REF] Diaz-Bobillo | Minimization of the maximum peak-to-peak gain: the general multiblock problem[END_REF]. For optimization, due to non-smoothness of both norms in [START_REF] Apkarian | Nonsmooth optimization for multidisk H ∞ synthesis[END_REF], we need to supply subgradients of closedloop integral functionals φ ij : K → ∞ 0 |c i (K)e A(K)t b j (K)|dt, those for the H ∞ -norm being well-known [START_REF] Apkarian | Nonsmooth H ∞ synthesis[END_REF]. Putting f (K, t) = c(K)e A(K)t b(K) and F (K) = ∞ 0 |f (K, t)|dt for the generic terms, we need partial derivatives ∂f (K, t)/∂K µν , where K µν are the controller gains, which depend in turn on the tunable parameters x over which we ultimately optimize. Since dependence K µν (x) and f (K, t) are differentiable, non-smoothness occurs only when the absolute value |f (•, t)| is formed, and ultimately via the finite maximum over rows in [START_REF] Dao | Simultaneous plant and controller optimization based on nonsmooth techniques[END_REF]. Subgradients g(K, t) ∈ ∂| • | • f (K, t) are obtained as g µν (K, t) = ∂f (K, t)/∂K µν signf (K, t) for f (K, t) = 0, while those (K, t) where f (K, t) = 0 give the full set of subgradients g µν (K, t) ∈ ∂f (K, t)/∂K µν • [-1, 1]. Partial derivatives ∂f /∂K µν are obtained via algorithmic differentiation [START_REF] Apkarian | Time-domain control design: a non-differentiable approach[END_REF]. Finally, subgradients G ∈ ∂F (K) of integral functionals are by regularity simply integrals of pointwise subgradients G µν (K) = ∞ 0 ∂f (K, t)/∂K µν signf (K, t)dt [START_REF] Giner | Subdifferentiation of integral functionals[END_REF][START_REF] Noll | Second order differentiability of integral functionals on Sobolev spaces and L 2 -spaces[END_REF][START_REF] Noll | Graphical methods in first and second order differentiability theory of integral functionals[END_REF]. For mixed programs like [START_REF] Apkarian | Nonsmooth optimization for multidisk H ∞ synthesis[END_REF] it is possible to use a progress function approach as in [START_REF] Simões | A nonsmooth progress function algorithm for frequency shaping control design[END_REF][START_REF] Simões | Nonsmooth multi-objective synthesis with applications[END_REF][START_REF] Simões | Lateral flight control design for a highly flexible aircraft using nonsmooth optimization[END_REF][START_REF] Dao | Robust eigenstructure clustering by nonsmooth optimization[END_REF][START_REF] Dao | Minimizing the memory of a system[END_REF][START_REF] Dao | Simultaneous plant and controller optimization based on nonsmooth techniques[END_REF][START_REF] Gabarrou | Design of a flight control architecture using a nonconvex bundle method[END_REF][START_REF] Apkarian | Mixed H 2 /H ∞ -control via nonsmooth optimization[END_REF]. Here we rather follow the line [START_REF] Apkarian | Nonsmooth optimization for multidisk H ∞ synthesis[END_REF][START_REF] Apkarian | Robustness via structured H ∞ /H ∞ -synthesis[END_REF] suited for norm functionals, where an iteratively re-weighted maximum of several norms is minimized; cf. [START_REF] Apkarian | Optimization-based control design techniques and tools[END_REF] for an overview, where in particular the concept of hard and soft constraints is addressed. This approach is also used in the systune function [START_REF] Gahinet | Frequency-domain tuning of fixed-structure control systems[END_REF][START_REF] Apkarian | Multi-model, multi-objective tuning of fixed-structure controllers[END_REF][START_REF]Control System Toolbox 2020b[END_REF] based on [START_REF] Apkarian | Nonsmooth optimization for multidisk H ∞ synthesis[END_REF], and has been used in our experiments. For convergence issues of bundle and bundle trust-region techniques we refer to [START_REF] Apkarian | Nonsmooth bundle trust-region algorithm with applications to robust stability[END_REF][START_REF] Noll | Cutting plane oracles for non-smooth trust-regions[END_REF].

According to the line in [START_REF] Apkarian | Nonsmooth optimization for multidisk H ∞ synthesis[END_REF] program [START_REF] Apkarian | Nonsmooth optimization for multidisk H ∞ synthesis[END_REF] is addressed by minimizing a maximum min x∈R n max α T wz (G, K(x)) ∞ , β T pq (G ψ , K(x)) pk_gn , where the weights α, β are updated iteratively until the constraint of ( 8) is satisfied, from where on the objective is reduced. Here K(x) expresses dependence of K on the tunable parameters x. The first term splits into a semi-infinite maximum with c i (x)e A(x)t b j (x) + d ij (x)δ(t) the closed loop impulse response of the entry (i, j) of the channel T pq (G ψ , K(x)). For discretization we have used the method of [START_REF] Rutland | Computing the 1-norm of the impulse response of linear time-invariant systems[END_REF], which is readily extended to the MIMO case.

T wz (G, K(x)) ∞ = max ω∈[0,∞]
It is helpful to update the weights α, β in such a way that at the current iterate x the three branches are at least nearly active. Selecting a set of active and near active frequencies for the first and third criterion is explained in [8, sect. 4.4], and we proceed analogously for the second branch. This strategy to include near-active branches into local models has turned out highly effective, as it avoids stalling at non-optimal points.

Conclusion

We have presented a method for stabilization and performance optimization of nonlinear controlled systems, where the non-linearity satisfies a sector constraint asymptotically. This leads to global closed-loop BIBO-stability in tandem with local exponential stability in situations where global closed-loop L 2 -stability fails, either due to exceedingly large sectors, or more principally, due to persistence of several attracting regimes in closed loop. The new approach requires solving a mixed L 1 /H ∞ -synthesis program, and uses properties of the L 1 -or peak-gain system norm.

5 : 6 :

 56 Asymptotic sector. Find asymptotic sector φ ∼ sect(a, b). Form c = (b + a)/2, r = (b -a)/2, and ψ = φ -cI. Represent G nl as loop (G ψ , ψ). Complementary asymptotic sector. Attempt closed-loop BIBO-stability in tandem with local exponential stability by solving minimize

Figure 1 .

 1 Figure 1. Schematic view of true and asymptotic sectors for general sector non-linearity (upper left), positivity (upper right), saturation (lower left), and dead time (lower right). Asymptotic sector constraints are satisfied for large values |q| > M .

Figure 3 .

 3 Figure3. Sector non-linearity φ in slipstick model with strong jump at -ω for two scenarios labeled 'gray' and 'blue' in[START_REF] Apkarian | Boundary feedback control of an anti-stable wave equation[END_REF].

Figure 4 .

 4 Figure 4. Program (13) was run for c ∈ [-6.2, -0.5] and τ = 0.1 (right), τ = 1.0 (left). Upper row shows best achieved 1/r(c) as -o-, with r(c) ≈ -0.94 • c shown as --for τ = 1.0 and -0.92 for τ = 0.1, depending essentially linearly on c (magenta). Nominal H ∞ -norm is 3.03, and - *shows result after optimization. Lower row shows slopes a (blue), b (red) of asymptotic sectors with a < q as < b and b = c + r(c), a = c -r(c).

Figure 5 .

 5 Figure 5. Impulse response before and after optimization (8) for two cases c = -1.29, τ = 0.1 and c = -1.21, τ = 1.0.

Figure 6 .

 6 Figure 6. Asymptotic sectors for τ = 0.1 and c = -5.9, c = -2.79, with r(c) computed via[START_REF] Apkarian | Non-smooth optimization for robust control of infinitedimensional systems[END_REF]. Constraints are satisfied for large angular velocities.

Figure 7 .Figure 8 .

 78 Figure 7. c = -5.21, τ = 0.1. Initial value below steady state, causing slipstick. Controller switched on at t = 10. Uncontrolled system shows slipstick.

- 1 . 0 ∆

 10 Owing to B B = I = [0, I; I, 0], the unit of the star product, we get (stability) loop invariance (G a , ∆ a ) ∼ = (G a P -1 , P • ∆ a ) ∼ = (B G a P -1 , P • ∆ a B) = (G e , ∆ e ). This means the passivity-type conditions (19) are equivalent to bounded-gain conditions T 0 G e (p e )(t) 2 dt ≤ (1 -) T 0 p e (t) 2 dt T e (p e )(t) 2 dt ≤ (1 -) T 0 p e (t) 2 dt ,

with Ψ 11 ,

 11 Ψ -1 11 , Ψ 21 , Ψ 22 , Ψ -122 stable. Then the transformed non-linear operator and LTIsystem

Corollary 3 .

 3 (Triangular transform). Let (G, ∆) be L ∞ well-posed, and suppose ∆ can be loop-transformed via a lower triangular L ∞ -bistable Ψ to an asymptotic L ∞ -contraction ∆. Suppose G = Ψ 11 G(Ψ 22 + Ψ 21 G) -1 is L ∞ well-posed and satisfies G pk_gn < 1. Then the loop (14) is BIBO-stable.

  , hence |z| b = (|x| + |y| )ρb for some ρ ∈ [0, 1], and since 0 ∈ B i , we have ρb = b ∈ B i . Then |z| b = |y| b + |x| b . Altogether φ(x) = b i +L i y+|y| b +|x| b , and here the term b i +L i y+|y| b is bounded independently of |x| , because y ∈ Q i are bounded. We put k 2 = max i=1,...,N |b i | ∞ , k 3 = max i=1,...,N max y∈Q i |L i y| ∞ and k 4 = max i=1,...,N max y∈Q i |y| ∞ , then k = k 2 +k 3 +k 1 k 4 . Now let B = co(B ∪ (-B )), then B is a symmetric polytope, hence its Minkowski functional is a polyhedral norm, | • | , and we have shown |φ(x)| ≤ |x| + k for all x. The matrix T defining | • | can be obtained from the polyhedral representation B = {x : T x ≤ 1}, which for moderate dimensions of x can be pre-computed. The matrix S is obtained from |x| = |Sx| ∞ .

and φ 2 =-1 0 1 .

 21 φ|F 2 = A 2 |F 2 for the affine mapping L 2 x = A 2 (x) = 0 x 2 T , which is already linear. Clearly, φ -1 (P ) = P andφ 3 = φ|P = I|P = L 3 . Now consider the box B ∞ = {x ∈ R 2 : |x 1 | ≤ 1, |x 2 | ≤ 1}, then L 1 (B ∞ ∩ F 1 ) = [(0, 0), (1, 1)] is a segment. Similarly L 2 (B ∞ ∩ F 2 ) = [(0, 0), (0, 1)]. Moreover, L 3 (B ∞ ∩ 0 + P ) = B ∞ ∩ 0 + P .The convex hull of the union of these three polytopes is B = co{(0, 0), (0, 1), (1, 1)}. Hence we have B = co(B ∪(-B )) = co{(-1, -1), (0, 1), (1, 1), (0, -1)}. The construction shows |φ(q)| ≤ |q| ∞ + k for every q, where |x| = |T x| ∞ is the polyhedral norm generated by B, obtained with T = 2Now observe that instead of the | • | ∞ -unit ball B ∞ we can choose a larger polytope B , which still satisfies L i (B ∩ F i ) ⊂ B . Namely, we can take B

Figure 10 .

 10 Figure 10. Two attractors and unstable origin (left), limit cycle (middle), chaotic double scroll (right).

Figure 11 .

 11 Figure 11. Non-linearity ∆ = (H, Φ) with unstable H(s) shows 2-double scroll (top left) or diverging spiral (top right) for different initial values '•'. Peak-gain optimization with a PID controller gives BIBO-stability of (K, ∆) = ((K, H), Φ) with convergence to 3 stable equilibrium points (bottom left and right). '•'.

  σ(T wz (jω, G, K(x))), whereas the second term, after time-domain discretization, becomes a finite maximumT pq (G ψ , K(x)) pk_gn = max i=1,...,m p j=1 t∈T c i (x)e A(x)t b j (x) + |d ij (x)| ,

  and is locally exponentially stable.2.2.Algorithm. The findings of the previous sections lead to the following strategy:

Algorithm: Mixed peak-gain/H ∞ -control of Lur'e system G nl 1: Steady-state. Compute steady state of non-linear system G nl , shift it to origin, and obtain linearization G. 2: Nominal synthesis. Fix performance and robustness specifications and perform nominal synthesis for G, interpreting non-linearity as a disturbance. Optimal K * gives lower bound γ ∞

ONERA, Department of System Dynamics, Toulouse, France.

Institut de Mathématiques, Université de Toulouse, France.1

This leads to

G : ẋ = Ax + 1 2 BCx + 1 2 B p q = Cx p(t) = ψ( q(t)), whence a sufficient condition for L 2 -stability is G L∞ < 1, and this can also be obtained from the circle criterion. Choosing

In other words, if x(t) ∈ P , then the signal is unaffected by saturation, but if x(t) reaches the boundary of P , then along every ray ρx, ρ > 0, the magnitude of the signal is frozen at the value it had attained when crossing the boundary of P , while the direction of the signal is unchanged. This is now a special case of Example 1.

Example 9. A typical case is signal clipping, where y = σ(x) is given as

Here B = B ∞ . Indeed, consider for simplicity the case n = 2. Then σ is piecewise affine with 9 different polytopes P 1 , . . . , P 9 , where

So only the four facets among the 9 faces of B ∞ contribute to B . This immediately applies to systems like ẋ = σ(Ax + b) or ẋ = σ(Ax + Bu) etc. as for instance considered in [START_REF] Sontag | From linear to nonlinear: some complexity comparisons[END_REF].

Example 10. The authors of [START_REF] Anca | Model structure simplification of a biological reactor[END_REF] consider non-linear systems ẋ = N i=1 µ i (x, u)

This can be modeled by an operator p = φ(q) := N i=1 µ i (q)q i with µ i (q) a convex combination, so that |φ(q)| 1 ≤ |q| ∞ , which is a polyhedral non-linearity.

Example 11. (Attractors, limit cycles, chaotics). In [START_REF] Miranda-Villatoro | Analysis of Lur'e dominant systems in the frequency domain[END_REF] the authors generate MIMO non-linearities ∆ by putting LTI-systems H in feedback with a static non-linearities Φ. This leads to attractors, limit cycles, chaotic behavior, and much else. Some of these may be considered special cases of [START_REF] Munther | Robust controller design: Minimizing peak-to-peak gain[END_REF]. The out-set is a dynamic system z = ∆(w):

where Φ(0) = 0. If |Φ(q)| ∞ ≤ r|q| ∞ + k asymptotically, then by Theorem 3, BIBOstability of ∆ follows from H pk_gn < r -1 . In particular, if Φ has bounded range, then we can choose r > 0 arbitrarily, hence H pk_gn < ∞ gives BIBO-stability of ∆.


Saliha Chetouani
email: saliha.chetouani@irt-saintexupery.com

Alexandre Boyer
email: alexandre.boyer@laas.fr

Sonia Ben Dhia
email: sonia.ben.dhia@laas.fr

Sébastien Serpaud
email: sebastien.serpaud@irt-saintexupery.com

André Durier
email: andre.2.durier@continental-corporation.com

A Technique to Assess Conducted Immunity of an Electronic Equipment after an Obsolete Integrated Circuit Change

Keywords: Obsolescence, Integrated circuit, Conducted immunity, Direct Power Injection

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

A simple comparison of the commercial lifetime of electronic components (approximately 3 to 5 years) and that of electronic equipment and systems (15 to 40 years for an aircraft, for example) shows the inevitable problem of rapid obsolescence of integrated circuits (IC). To remain compliant with standards, any change of an obsolete component must lead the Original Equipment Manufacturers (OEM) to requalify their products to prove the non-regression of their EMC performances. The requalification usually requires timeconsuming tests, significant costs and delays. Currently, the standard EMC measurement method is the only authorized method to validate EMC of an equipment.

Different approaches have been deployed for managing consequences of electronic component obsolescence on EMC.

A common approach to avoid requalification proposes a complete and exact modeling of the equipment according to a bottom-up vision. It relies on the development of models at the component, board and equipment levels [START_REF] Durier | Modélisation l'immunité Électromagnétique des composants en vue de la gestion de l'obsolescence des systèmes et modules électroniques[END_REF] [2] [3] [START_REF] Ayed | Développement de méthodologies pour l'extraction et la construction des macromodèles d'immunité électromagnétique appliqués aux circuits intégrés[END_REF]. They are usually based on electromagnetic 3D modeling and full-wave simulations [START_REF] Durier | Using the EM simulation tools to predict EMC immunity behavior of an automotive electronic board after a component change[END_REF]. However, these approaches quickly find their limits due to the complexity of electronic equipment and the long modeling time. In addition, the exact information regarding each part of the equipment is not necessarily known for confidentiality reasons. These drawbacks severely limit the use of these methods and could be, in some cases, not envisaged due to the cost and delays.

In many cases, a complete requalification of the equipment is not essential to assess a risk of non-compliance, especially if the change concerns only one IC and introduces no modification of the equipment design. It could be advantageous to assess the impact of the change of this IC on the EMC performances of the equipment, by testing only the EMC of the IC, since EMC of IC tests remain cheaper than equipment-level tests. This is the aim of this paper, which proposes an approach based on EMC measurements at component level to estimate the non-regression of equipment in the context of the component obsolescence. Here, only conducted immunity (CI) along cable harness is considered, such as in Bulk Current Injection (BCI) test as defined by ISO11452-4 or RTCA/DO-160 standards. The proposed approach aims to decline the CI defined at the equipment input into a disturbance level defined at the boundary of the replacement IC, i.e. the component used to replace the obsolete IC. The residual level of disturbance that reaches the IC pins can be determined by S-parameter measurements made at the equipment input connector, according to the methodology presented in [START_REF] Chetouani | Fast and efficient approach to predict EMC immunity of complex equipment after a component change[END_REF]. The EMC risk assessment could be achieved by a conducted immunity test directly done at IC level, for example by the Direct Power Injection (DPI) method [START_REF]Integrated Circuits -Measurement of Electromagnetic Immunity 150 kHz to 1 GHz -Part 4: Direct RF power injection method[END_REF].

The paper is organized as follows: the approach is described in Section II. A validation case study, based on an op-amp dedicated to current measurement through a shunt resistor, is proposed and presented in Section III. The proposed approach relies on an equivalent model of the equipment and the DPI test-bench. Details about the modeling process are provided in Section IV. Section V illustrates the proposed methodology through the analysis of results on the case study.

II. PRESENTATION OF THE APPROACH

A. General Assumptions

The studied equipment is assumed reciprocal and linear. The non-linear behavior of the components (e.g. active circuits, protection structures, etc.) is neglected. The general ground of the system, which is usually mounted within a shielded enclosure, could be different from the local component ground (usually a local ground plane at PCB level). In this study, only conducted injection on the harness is considered and is common-mode type. The disturbance induced by the injection system and driving on the harness is modeled as an equivalent Thévenin generator placed at the input connector of the equipment. The method to determine this equivalent source in BCI test is not covered in this paper. Interested readers could find information about this topic in papers such as [START_REF] Grassi | Circuit Modeling of Injection Probes for Bulk Current Injection[END_REF] [9] [10] [START_REF] Miropolsky | Experimental Validation of the Generalized Accurate Modelling Method for System-Level BCI Setups up to 1 GHz[END_REF].

B. General approach

The purpose of the approach is to decline the normative constraint from the input of equipment to the input of the replacement component such as the common or differential voltage applied to the pins of the component to change. An equivalence between CI tests at equipment and IC levels has to be found, such as illustrated in Fig. 1. An important assumption has to be made: the same failure mechanism arises during both CI tests, with the same amount of residual disturbance VIC applied on IC pins. In Fig. 1-a, the equivalent model of CI test at equipment level is illustrated. The injection source, the coupling device and the cable harness are modeled by an equivalent Thévenin generator. If the transfer function of the equipment (i.e. all the interconnects and devices between the equipment input connector and the replacement IC) is known, the amplitude of the equivalent Thévenin generator 𝑉 𝑔 required to induce 𝑉 𝐼𝐶 at the IC pin can be determined. In this paper, the disturbance amplitude provided by the generator in equipment and IC tests is defined in terms of available power, as given by (1). In the equipment test, the power provided by the equivalent disturbance source 𝑉 𝑔 to the equipment input is noted 𝑃 𝑖𝑛𝑗_𝐸𝑞 and is given by [START_REF] Perdriau | Taking into Account Obsolescence in the EMC Performance of Integrated Circuits and Systems[END_REF]. It represents the amount of disturbance that the equipment has to fulfill.

𝑃 𝑔𝑒𝑛𝑒 = 𝑃 𝑚𝑎𝑥_𝑎𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒 = 𝑉 𝑔 2 4 𝑍 𝑔 ⁄   𝑃 𝑖𝑛𝑗_𝐸𝑞 (𝑊) = 𝑃 𝑔𝑒𝑛𝑒_𝐸𝑞 (1 -|Γ 𝑖𝑛_𝐸𝑞 | 2 )  
where 𝛤 𝑖𝑛_𝐸𝑞 is the reflection coefficient seen at the equipment input and is given by (3) for a two-port equipment characterized by its S parameters. 𝛤 𝐼𝐶 is the reflection coefficient at the replacement IC input. Similarly, Fig. 1-b shows the equivalent model of the CI test at IC level. If the transfer function of the CI test bench is known, the power 𝑃 𝑖𝑛𝑗_𝐼𝐶 provided by the disturbance source required to induce 𝑉 𝐼𝐶 at the IC pin can also be determined according to [START_REF] Ayed | Développement de méthodologies pour l'extraction et la construction des macromodèles d'immunité électromagnétique appliqués aux circuits intégrés[END_REF].

𝛤 𝑖𝑛_𝐸𝑞 = 𝑆 11𝐸𝑞 + 𝑆 12𝐸𝑞 𝑆 21𝐸𝑞 Γ 𝐼𝐶 1-𝑆 22𝐸𝑞 Γ 𝐼𝐶 (3) 𝑃 𝑖𝑛𝑗_𝐼𝐶 = 𝑃 𝑡𝑟𝑎𝑛𝑠_𝐼𝐶 (1-|Γ 𝑖𝑛𝐷𝑃𝐼 | 2 ) |1-𝑆 22𝐷𝑃𝐼 Γ 𝐼𝐶 | 2 |𝑆 21𝐷𝑃𝐼 | 2 (1-|Γ 𝐼𝐶 | 2 ) (4)
Where 𝛤 𝑖𝑛𝐷𝑃𝐼 is the reflection coefficient seen at the input of the DPI test bench. It is computed as (3) for a two-port equipment characterized by its S parameters. 𝑃 𝑡𝑟𝑎𝑛𝑠_𝐼𝐶 is the power transmitted to the IC and is given by ( 5), where ZIC is the input impedance of the replacement IC.

𝑃 𝑡𝑟𝑎𝑛𝑠_𝐼𝐶 = |𝑉 𝐼𝐶 |² 𝑍 𝐼𝐶 (5)
Thus, an equivalence between the disturbance sources in CI tests at equipment and IC levels can be found to induce the same effect on the considered IC. In other words, if the immunity level of the equipment is defined in terms of a power 𝑃 𝑖𝑛𝑗_𝐸𝑞 , this level can be translated into an equivalent power 𝑃 𝑖𝑛𝑗_𝐼𝐶 during the IC immunity test. If 𝑃 𝑖𝑛𝑗_𝐼𝐶 is applied during the IC test and if the IC is still operational, this IC should be still operational during the equipment test when 𝑃 𝑖𝑛𝑗_𝐸𝑞 is applied. For this demonstration, we used the DPI method as the immunity test because it is well suited for the conducted immunity. Also, for its simplicity in terms of test set-up and analysis, it is easy to measure the power and to calculate the voltage and the current at the IC level compared to other immunity tests (e.g. TEM cell or NFSi tests). The repeatability of DPI measurements is good. Furthermore, this test is widely used by most component manufacturers.

C. Detailed description

The different steps of the EMC risk assessment methodology are described in Fig. 2. In the following part, IC stands for the replacement IC. In a typical situation, an immunity level is defined at equipment level (e.g. the maximum amount of current induced in the equipment cable harness in a BCI test). The first step, not covered in this paper, consists in translating this immunity level into the available power of an equivalent disturbance source. In this study, the immunity level defined at equipment level is assumed the amount of power that the equipment has to fulfill (𝑃 𝑖𝑛𝑗_𝐸𝑞 ).

According to Fig. 1, the initial step is to create the equivalent black box model of the equipment and IC. This model is built from two measurements: the first one, described in Section IV, concerns the extraction of the equipment transfer function, which describes the filtering provided by connectors, interconnects and devices mounted between the connector input and IC. The second one aims at measuring the input impedance seen from the input IC pin (in terms of Sparameters). The next step consists in calculating the power 𝑃 𝑔𝑒𝑛𝑒_𝐸𝑞 , using (2), from 𝑃 𝑖𝑛𝑗_𝐸𝑞 and the parameter Γ 𝑖𝑛_𝐸𝑞 . Then, converting this power to an equivalent Thévenin generator, using [START_REF] Durier | Modélisation l'immunité Électromagnétique des composants en vue de la gestion de l'obsolescence des systèmes et modules électroniques[END_REF], where 𝑍 𝑔 = 50 𝛺. Therefore, the voltage 𝑉 𝐼𝐶_𝐸𝑞 applied on IC during the equipment immunity test can be easily extracted by simulation, the current as well. The power 𝑃 𝑡𝑟𝑎𝑛𝑠_𝐼𝐶_𝐸𝑞 can be calculated by [START_REF] Durier | Using the EM simulation tools to predict EMC immunity behavior of an automotive electronic board after a component change[END_REF].

The same failure mechanism occurs during an IC test with the same residual disturbance (𝑉 𝐼𝐶 = 𝑉 𝐼𝐶_𝐸𝑞 ) and with the same power that causes the failure (𝑃 𝑡𝑟𝑎𝑛𝑠_𝐼𝐶_𝐸𝑞 = 𝑃 𝑡𝑟𝑎𝑛𝑠_𝐼𝐶 ). Consequently, the DPI test level 𝑃 𝑖𝑛𝑗_𝐼𝐶 equivalent to the equipment test is computed from an equivalent model of the DPI test bench and the IC. Black-box modeling based on Sparameter measurements can be done on the different elements of the DPI test bench to accelerate the modeling process. Once 𝑃 𝑖𝑛𝑗_𝐼𝐶 has been determined, the setpoint DPI power 𝑃 𝑔𝑒𝑛𝑒_𝐼𝐶 can be calculated and the equivalent Thévenin generator that provides 𝑃 𝑖𝑛𝑗_𝐼𝐶 can be determined as explained previously. 𝑃 𝑔𝑒𝑛𝑒_𝐼𝐶 will be the reference of the evaluation of the replacement IC. DPI test can therefore be performed on IC and the EMC risk due to the IC change can be evaluated. This procedure requires only the development of a specific test board for the DPI test. If a generic test board dedicated to common IC package is developed, this board can be reused so that the overall cost of this procedure can be reduced. The risk assessment is based on a comparison between the DPI test results 𝑃 𝑡ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑_𝐼𝐶 , that represents the setpoint power of the generator during the test made on IC, and the limit 𝑃 𝑔𝑒𝑛𝑒_𝐼𝐶 . An error margin has to be accounted for in order to compensate the uncertainties related to equipment transfer function and DPI test bench measurement errors. Measurement test bench can be extracted from an uncertainty budget applied on the DPI test bench, for example based on standard method such as CISPR 16-4-2 [START_REF]Specification for radio disturbance and immunity measuring apparatus and methods -Uncertainties, statistics and limit modelling[END_REF]. An additional safety margin may also be introduced by the OEM. In this study, from several repeated measurements, the DPI test bench repeatability is evaluated to 1 dB and an arbitrary additional margin of 2 dB is introduced, so that the error margin is set to 3 dB. The risk assessment is done according to the following methodology: the susceptibility level 𝑃 𝑡ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑_𝐼𝐶 of the IC is determined at each test frequency. Then, it is compared to 𝑃 𝑔𝑒𝑛𝑒_𝐼𝐶 . The risk assessment is done according to the strategy of the three levels described in Fig. 2. If the IC fails for DPI power larger than 𝑃 𝑔𝑒𝑛𝑒_𝐼𝐶 plus the error margin, the introduction of the IC in the equipment will not compromise its immunity and the equipment CI requalification is not necessary. Moreover, the gain in terms of EMC margin provided by the IC can be computed. In contrary, if the IC fails for DPI power less than 𝑃 𝑔𝑒𝑛𝑒_𝐼𝐶 minus the error margin, the immunity of the equipment will regress with this new IC and a CI requalification of the equipment is not necessary. The requalification becomes necessary if the IC fails for DPI power comprised between 𝑃 𝑔𝑒𝑛𝑒_𝐼𝐶 plus or minus the error margin.

III. PRESENTATION OF THE CASE STUDY

A. Tested Circuit

The tested circuit is an operational amplifier, AD8515 from Analog Devices, mounted in a non-inverter configuration for a current sense application based on a shunt resistor, further information on this application are available in [START_REF] Durier | Study of the radiated immunity of a drain-source current sensor using Near Field Scan Immunity method[END_REF]. The electrical diagram of the tested application is presented in Fig. 3. The circuit is powered by a 5 V DC voltage source. In this study, the disturbance is applied on the power supply and leads to the generation of offset on the amplifier output voltage 𝑉 𝑂𝑈𝑇 . When the offset voltage exceeds 5 mV, a failure is detected. The differential amplifier is mounted on a daughter 4-layer board, which is also connected to a mother board (Fig. 4). For the sake of the validation of the proposed approach, the differential amplifier is considered as the IC to be replaced. The SMA connector Vcc is the IC input. The new IC version consists in the introduction of a RC filter at the noninverter input, as shown in Fig. 3. In the following part, the initial IC version is noted IC1 while the new version is called IC2. The mother board is supposed to be the equipment. It contains two microstrip lines routed on a FR4 substrate 4layers PCB connected to SMA connectors.

B. DPI Test Bench

The DPI test bench is illustrated in Fig. 5. It consists of an RF synthesizer (Rohde & Schwarz SMB 100A; 9 kHz-3.2 GHz), a 30 W RF power amplifier (BONN Elektronik BSA 1001-30D; 100 kHz-1 GHz), a bidirectional internal coupler and RF power meter (Rohde & Schwarz NRP-Z91) to monitor the forward power. The RF disturbance is superimposed to the 5 V power supply through a bias tee (10 MHz-6 GHz) available off-the-shelf from several vendors. The bias-tee output is connected to the Vcc of the daughter board through an SMA connector. The input (VIN) of this board is connected directly to a waveform generator (Keysight 33500B Series) which generates a ramp signal (10 V peak to peak with a frequency of 400 MHz). The DPI test is performed with 51 points linearly spaced between 10 MHz and 1 GHz. The maximum amount of forward power is limited to 40 dBm.

C. Equipment Conducted Immunity Test

A CI test is also realized at equipment level. The DPI testbench is reused. The bias-tee output is connected to the port 1 of the mother board through an SMA connector as shown in Fig. 4.

IV. MODELING OF THE EQUIPMENT AND THE TEST BENCH

As described in Fig. 2, the proposed methodology to assess EMC risk is based on modeling of equipment CI immunity and DPI test benches. In order to ensure a fast modeling process, a black-box modeling approach based on S-parameter measurements is applied. As all the equipment of the CI test bench and the devices mounted on the equipment behave linearly, this approach remains valid. The simulations are performed with the freeware QUCS [START_REF]Quite Universal Circuit Simulator (QUCS)[END_REF]. Fig. 7 describe the models of the equipment CI and DPI test benches. The Sparameters of the coupler, the bias tee, the cables, the equipment board and the IC input have been measured with a VNA and introduced in S-parameter boxes in QUCS.

In both models, the disturbance source takes into account the variable gain of the power amplifier. The QUCS component library does not include a variable voltage source that operates in frequency domain. We have created a simple model of an equivalent generator with variable gain. The voltage 𝑉 𝑔 is calculated from the power 𝑃 𝑔𝑒𝑛𝑒 according to [START_REF] Durier | Modélisation l'immunité Électromagnétique des composants en vue de la gestion de l'obsolescence des systèmes et modules électroniques[END_REF] where 𝑍 𝑔 = 50 Ω. This variable voltage (integrated in a touchstone file) is represented as an impedance in parallel with a 1A current source. The output of this circuit is connected to voltage-controlled voltage source.

A critical part of the modeling process is related to the measurement of the transfer function of the equipment, defined between the input connector of the equipment and the input pin of the IC to change. This transfer function is required to determine the amplitude of the residual disturbance that reaches the IC pins. This measurement could be easily done if proper RF ports exist both at input connector and at IC side. However, the component terminals are usually not accessible to place an S-parameter measurement port. Nevertheless, if the IC can be removed and replaced by some passive loads, the transfer function can be determined by an indirect Sparameter measurement, made at the input connector side. The method, initially proposed in [START_REF] Maeda | Mathematics of 2r-Port S-Parameter Estimation by the r-port Measurements[END_REF] and adapted to the case of IC change in [START_REF] Chetouani | Fast and efficient approach to predict EMC immunity of complex equipment after a component change[END_REF], is used to extract the S-parameters matrix between the input and output ports of the equipment board, as a demonstrator, only from S-parameter measurements done at the input ports. Fig. 6 shows a comparison between some Sparameters measured directly and those extracted by the indirect method. A. Immunity of the components Fig. 8 presents the susceptibility level of IC1 and IC2 during the DPI test. This result shows that both IC versions exhibit sufficiently different susceptibility levels that the proposed approach has to detect. Except between 386 and 426 MHz, the IC2 version is more immune than IC1. Note, due to forward power limitation, no default is detect for IC2 below 100 MHz.

B. Voltage and current applied on the components

An initial analysis is done to verify the general assumption on which relies the proposed approach: the equivalence of voltage level applied on IC pin during the CI equipment and the DPI test. From the DPI and equipment CI test results done on both equipment versions and the models of both test benches, the voltage and current at the input of the IC are computed. Fig. 9 compares the voltage and current at IC pin during both tests, when IC2 is mounted. Voltage and currents determined during both tests are nearly identical. Some differences are visible between 10 and 100 MHz. Actually, no failures arise on this frequency range and the injection power reaches the power limit. There are also some differences around 400 MHz, which are certainly related to repeatability issue of the DPI test bench. This result confirms the hypothesis about the equivalence of applied voltage on IC pin during both immunity tests at equipment and IC level.

C. Evaluation of the risk of EMC non-compliance

In this part, the risk assessment linked to a change of component is illustrated when IC2 replaces IC1. The purpose is to perform a DPI test on the "new" component IC2 to determine if no EMC risks are created by the introduction of this new component and if requalification test at equipment level is required. To this end, the methodology presented in Section II is used. The immunity target at equipment level 𝑃 𝑖𝑛𝑗_𝐸𝑞 is defined as the forward power measured during the CI test on the equipment with IC1. From the models of the CI test bench, the equipment and the input impedance of IC1, the power transmitted to IC1 is determined. Then, from the DPI test bench model, the setpoint DPI power 𝑃 𝑔𝑒𝑛𝑒_𝐼𝐶1 to reach this voltage level during the DPI test is computed. This level, shown in Fig. 10 constitutes the power target that IC2 has to withstand during the DPI test.

An immunity test is done on IC2 to measure its susceptibility threshold and compare it to the power target determined from the equipment CI level on IC1, in its obsolete version which has already been established. Both results are compared in Fig. 10. Two limits located at +/-3 dB around the power target curve are added to show the error margin. This result shows that the immunity threshold of IC2 is equal or larger than the power target, proving that the introduction of IC2 in the equipment should not reduce its immunity compared to the initial version. However, the EMC margin introduced by IC2 is less than the error margin between 350 and 640 MHz. The confidence in the lack of EMC regression is not sufficient, so that a CI requalification of the equipment should be considered, in particular is this frequency range.

VI. CONCLUSION -PERSPECTIVES

The paper has presented a method to assess the risk of regression of the conducted immunity of an electronic equipment due to the change of a circuit. Such a situation usually implies a complete requalification of the equipment with the replacement circuit. The approach described in this paper proposes to replace this costly test by a DPI test performed only on the replacement circuit in order to determine if:  The requalification test can be omitted since the replacement circuit introduces a sufficient EMC margin.

 The requalification test can be omitted because the replacement circuit is significantly more susceptible than the original one and an alternative solution should be identified.

 The requalification test is necessary on the frequency ranges where the EMC margin is not sufficient.

It is important to note that this approach allows estimating the real margin of a normative test from the DPI test on the component. Information that could really help the manufacturers in the specification of margins. This approach requires the development of two equivalent models about the equipment and the DPI test bench. Both models can be built from S-parameter measurements in order to limit the cost of the modeling process. The only extra cost of this approach is related to the development of a test board for the DPI test of the replacement circuit. Two issues are not addressed in this paper. The first concerns the modeling of the CI test bench in an equivalent disturbance source and the estimation of the error margins due to measurement errors. The second concerns the uncertainties of the equipment transfer function. Further studies are required to address these issues and confirm the validity of the approach on a component change scenario.

Fig. 1 .

 1 Fig. 1. Equivalence between conducted immunty test at equipment and component level

Fig. 2 .

 2 Fig. 2. Flowchart of the EMC risk assessment methodology

Fig. 3 .Fig. 4 .

 34 Fig. 3. Schematic of the tested application (IC1 board, IC2 board: IC1 board + additional RC filter)

Fig. 5 .

 5 Fig. 5. Simplified schematic diagram of the DPI test bench

Fig. 6 .Fig. 7 .

 67 Fig.6. Illustration of the indirect measurement method for S-parameters; example of some estimated S-parameters vs. fully measured S-parameters (Frequency: 100 kHz -4.5 GHz)

Fig. 8 .

 8 Fig. 8. Conducted immunity of IC1 and IC2 (forward power)

Fig. 9 .

 9 Fig. 9. Comparison of voltage (top) and current (bottom) at IC2 input during CI and DPI tests

Fig. 10 .

 10 Fig. 10. Comparaison between the measured generator power of the DPI test on IC2 and that calculated from the equipment conducted immunity test (top) -EMC risk assessment (bottom)

ACKNOWLEDGMENT

This study is supported by the IRT Saint Exupéry foundation (a French Institute of Technology) under the FELINE project. The authors would like to thank IRT, LAAS and all the project partners (Safran, Airbus, Continental, Nexio, Liebherr, Actia, Thales, etc.) for funding and supporting this work.