Anatolii Khalin

Rosane Ushirobira

Denis Efimov

On steady-state based reduced-order observer design for interlaced nonlinear systems

This paper proposes an analytical expression for a nonlinear mapping between steady-state solutions of certain types of nonlinear interconnected systems. This mapping is found using tools from the theory of output regulation for systems presented in lower-triangular or upper-triangular canonical forms. Next, this mapping helps design an excitation input and a corresponding reduced-order observer for interlaced systems, a combination of both upper-and lower-triangular subsystems. A proposed global observer is proved to be robust to additive disturbance and measurement noise by applying the Lyapunov function method. An example involving a massspring system demonstrates the efficiency of our approach.

I. INTRODUCTION

The moment theory for nonlinear systems [START_REF] Scarciotti | Data-driven model reduction by moment matching for linear and nonlinear systems[END_REF], [START_REF] Astolfi | Model reduction by moment matching for linear and nonlinear systems[END_REF], based on the theory of output regulation [START_REF] Francis | The internal model principle of control theory[END_REF], [START_REF] Isidori | Output regulation of nonlinear systems[END_REF] and center manifold theory, asserts that under certain assumptions, any nonlinear system must have a central manifold, an invariant subspace containing the steady-state solution of the system, that can be used to replicate its behavior (i.e., by frequency response). However, despite the solid theoretical background, the process of determining steady-state solutions of nonlinear systems is still dependent on a system's form.

This paper addresses the problem of finding the steadystate response for particular types of nonlinear systems inspired by recursive procedures, such as backstepping and forwarding [START_REF] Sepulchre | Constructive Nonlinear Control[END_REF]. We propose a way of determining steadystate solutions for systems based on the diagonal structure of its model.

The main idea of backstepping, introduced in [START_REF] Kokotovic | The joy of feedback: nonlinear and adaptive[END_REF], consists of using the lower-triangular form of a system to make a step-by-step output passivation through time-derivative, starting from the first equation to the last one, where the stabilizing input is designed (e.g. see [START_REF] Nizami | Design and implementation of a neuro-adaptive backstepping controller for buck converter fed pmdc-motor[END_REF], [START_REF] Krishnamurthy | Global high-gainbased observer and backstepping controller for generalized outputfeedback canonical form[END_REF], [START_REF] Haizhou | Experimental validation of a nonlinear backstepping liquid level controller for a state coupled two tank system[END_REF], [START_REF] Kim | Robust backstepping control for slew maneuver using nonlinear tracking function[END_REF]). Forwarding is a more recent method of stabilization [START_REF] Sepulchre | Integrator forwarding: A new recursive nonlinear robust design[END_REF], which utilizes the upper-triangular form of systems. It has a similar iterative idea to backstepping, which works in the opposite direction: it uses the integration starting from the input and going forward (e.g. see [START_REF] Zhang | Design of stabilizing controllers with a dynamic gain for feedforward nonlinear time-delay systems[END_REF], [START_REF] Boeren | Enhancing feedforward controller tuning via instrumental variables: with application to nanopositioning[END_REF]). The interlaced systems introduced in [START_REF] Sepulchre | Constructive Nonlinear Control[END_REF] combine both types of triangular dynamics in their subsystems. It is a large class of models that have restrictions only on the structure of their feedback and feedforward interconnections.

This paper uses techniques similar to backstepping or forwarding and derives the steady-state solutions of such an interlaced system. One of our goals is to design a robust reduced-order observer with the corresponding input to the system. This result could significantly improve the observer design for the considered classes of nonlinear systems without the hypothesis about uniform observability.

The outline of the paper is as follows. Notation and definitions used throughout the paper are introduced in Section II and the problem statement in Section III. The results, including the methods for analytical calculation of the steady-state solutions for two types of canonical forms of nonlinear systems, are presented in Section IV. The observer design is given in Section V. In Section VI, the efficiency of the proposed solutions is illustrated in the example.

II. PRELIMINARIES

A. Notation

• The set of real numbers is denoted by R and we write R + := {t ∈ R : t ≥ 0}.

→ R n such that u ∞ := u [0,+∞) < +∞. • For a vector ξ = (ξ 1 . . . ξ n ) ∈ R n and i ∈ {1, . . ., n}, the subvector (ξ 1 ξ 2 . . . ξ i ) ∈ R i is denoted by ξ i
, and correspondingly, the subvector

(ξ i ξ i+1 . . . ξ n ) ∈ R n-i+1 is denoted by ξ i .

B. Theory of moments

Let us consider a nonlinear continuous-time system:

x(t) = f (x(t), u(t)), ∀t ∈ R + , (1)
y(t) = h(x(t)),
where x : R + → R n is the state function, u : R + → R m is the input signal, and y : R + → R p is the output; f : R n ×R m → R n and h : R n → R p are smooth functions with f (0, 0) = 0 and h(0) = 0. Consider a signal generator system given by:

ω(t) = s(ω(t)), ∀t ∈ R + , (2)
u(t) = (ω(t)),
where ω : R + → R q is the state; s : R q → R q , : R q → R m are smooth functions with s(0) = 0 and (0) = 0. The interconnected system is given by ( 1) and (2)1 :

x = f (x, (ω)), ω = s(ω), y = h(x). (3)
We assume the following conditions: Assumption 1. [START_REF] Scarciotti | Nonlinear model reduction by moment matching[END_REF] The system (1) is minimal and the zero equilibrium is locally exponentially stable. The signal generator system (2) is observable and neutrally stable.

The minimal condition of the considered system implies its observability and accessibility. Lemma 1. [START_REF] Scarciotti | Nonlinear model reduction by moment matching[END_REF] Consider the system (1) and the signal generator system (2). Suppose Assumption 1 holds. Then there is a mapping π : R q → R n , locally defined in a neighborhood W ⊂ R q of the origin, with π(0) = 0, which solves the partial differential equation:

∂π ∂w (w)s(w) = f (π(w), (w)), (4)
for all w ∈ W. In addition, the steady-state response of the system (3) is x ss (t) = π(ω(t)) for any x(0) and ω(0) sufficiently small.

Note that other results and definitions accompanying the moment theory, along with the definitions of observability and accessibility for nonlinear systems, can be found in [START_REF] Scarciotti | Nonlinear model reduction by moment matching[END_REF] (see Def. 2.1-2.9).

C. Convergent systems

The expression ( 4) is a first-order quasi-linear partial differential equation. A solution of this equation represents an invariant trajectory for (3), but if, in addition, we ask for such a solution to be attractive for the surrounding trajectories in some region, then we come to the concept of convergent dynamics [START_REF] Pavlov | Convergent Systems: Analysis and Synthesis[END_REF].

Definition 1.

[1] The system (1) is said to be incrementally input-to-state stable (δISS) if there exist β ∈ KL and γ ∈ K ∞ such that for any t ≥ 0, any κ 1 , κ 2 ∈ R n and any couple of signals

d 1 , d 2 ∈ L m
∞ , the following holds:

x(t, κ 1 , d 1 ) -x(t, κ 2 , d 2 ) ≤ β( κ 1 -κ 2 , t) + γ( d 1 -d 2 ∞ ).
If this property is satisfied, all trajectories of (3) converge to a solution of (4) and stay close to it in the case of additive perturbations in the input channel of (1).

D. Existence of solutions for first order quasi-linear PDEs

Consider a quasi-linear partial differential equation of the first order [START_REF] Evan | Partial Differential Equations[END_REF] (a variant of (4)):

∂ χ ∂w σ( χ, w) = φ( χ, w), (5)
where χ : R q → R is a function, σ : R q+1 → R q and φ : R q+1 → R are real analytic functions (σ(0, 0) = 0 and φ(0, 0) = 0), with initial or boundary conditions

χ(w) = 0, ∀w ∈ Γ = {w ∈ W : w q = 0}, (6)
where W ⊂ R q is a neighborhood of the origin and w q denotes the q-th component of w ∈ R q . The Cauchy problem ( 5), ( 6) is called non-characteristic if

σ q (0, w) 0, ∀w ∈ Γ \ {0},
where the origin is excluded since in such a case the trivial solution χ(0) = 0 exists by construction. The conditions for the existence of a solution χ to ( 5), ( 6) are given by the Cauchy-Kowalevski Theorem:

Theorem 1. [21]
The Cauchy problem (5), ( 6) has a unique real analytic solution χ on W if the Cauchy problem is noncharacteristic.

In the case when σ and φ are continuously differentiable functions, analogues of this theorem can be found, for example, in [START_REF] Evan | Partial Differential Equations[END_REF], [START_REF] Han | Quasi-linear systems of PDE of first order with Cauchy data of higher codimensions[END_REF] providing a continuously differentiable solution χ.

III. PROBLEM STATEMENT

Consider an interconnection of two nonlinear systems one of which is in a form similar to (1) but with disturbance and measurement noise, and the second one being (2):

x = F(x) + G(x, u) + d, y = h(x) + v; (7)
where F : R n → R n and G : R n × R → R n are smooth or real analytic maps, and the remaining functions are as in ( 1) with m = 1;

F(0) = 0, G(•, 0) = 0; d ∈ L n ∞ is an unknown essentially bounded disturbance, v ∈ L p ∞ is a measurement noise signal.
This paper aims to design of a robust reduced-order observer for classes of lower-or upper-triangular functions F and G in (7) using tools from the moment theory for nonlinear systems. The derived expressions of steady-state solutions will be used next to design a robust reduced-order observer to reconstruct the state of ( 7), [START_REF] Astolfi | Model reduction by moment matching for linear and nonlinear systems[END_REF]. We also assume that the form of the input signal can be chosen to ensure observability or detectability of ( 7), [START_REF] Astolfi | Model reduction by moment matching for linear and nonlinear systems[END_REF].

IV. THE STEADY-STATE RESPONSE OF FEEDBACK AND FEEDFORWARD SYSTEMS IN A DISTURBANCE-FREE CASE

Two particular classes of F and G are studied in this section.

A. Feedback (lower-triangular form) case

A particular case of the system ( 7) is given by a nonlinear single-input system in the canonical feedback form (d = 0):

z = η(z, ξ 1 ), (8)
ξ 1 = f 1 (ξ 1 , z) + g 1 (ξ 1 , z)ξ 2 , (9) . . . ξ ν-1 = f ν-1 (ξ ν-1 , z) + g ν-1 (ξ ν-1 , z)ξ ν ( 10
)
ξ ν = f ν (ξ, z) + g ν (ξ, z)u, (11)
with the same signal generator system as in [START_REF] Astolfi | Model reduction by moment matching for linear and nonlinear systems[END_REF], where z : R + → R r is the zero dynamics state, ξ = (ξ 1 . . . ξ ν ) ∈ R ν and x = z ξ ∈ R ν+r , n = ν + r; and the functions η : R r+1 → R r , f i , g i : R r+i → R for i = 1, . . ., ν are smooth. The conditions providing a representation of [START_REF] Han | Quasi-linear systems of PDE of first order with Cauchy data of higher codimensions[END_REF] in the form ( 8)-( 11) can be checked in [START_REF] Sepulchre | Constructive Nonlinear Control[END_REF]. Remark that the input u of ( 8)- [START_REF] Kokotovic | The joy of feedback: nonlinear and adaptive[END_REF] appears solely in the state equation for ξ ν . Hence, inspired by the backstepping idea, if ξ 1 = π 1 (ω) is fixed, then an input function (ω) can be designed in such a way that it assigns π 1 as the steadystate solution of this interconnected system. Thus, using the theory of output regulation, we can formulate the conditions for the existence of a steady-state solution for (2)-( 11): Proposition 1. For any smooth function π 1 : R q → R, let a smooth function ρ : R q → R r be the steady-state solution of (8) under substitution ξ 1 = π 1 (ω). Then there always exists the input function : R q → R such that π(ω) = (π 1 (ω) . . . π ν (ω)) is the steady-state solution of the interconnected system (2)- [START_REF] Kokotovic | The joy of feedback: nonlinear and adaptive[END_REF] provided that g i (π i (ω), ρ(ω)) 0 for all i = 1, . . ., ν and ω ∈ R q . Proof. We can set ξ 1 = π 1 (ω), which is the first steadystate solution of the system [START_REF] Kokotovic | The joy of feedback: nonlinear and adaptive[END_REF], then z = ρ(ω) is the corresponding steady-state of the zero dynamics [START_REF] Isidori | Output regulation of nonlinear systems[END_REF]. To find the remaining components of π and the respective input function , let us use the output regulation theory, which states that equation ( 4) must be satisfied. Consequently, we obtain

∂π 1 (ω) ∂ω s(ω) = f 1 (π 1 (ω), ρ(ω)) + g 1 (π 1 (ω), ρ(ω))π 2 (ω), . . . ∂π ν-1 (ω) ∂ω s(ω) = f ν-1 (π ν-1 (ω), ρ(ω)) +g ν-1 (π ν-1 (ω), ρ(ω))π ν (ω), ∂π ν (ω) ∂ω s(ω) = f ν (π(ω), ρ(ω)) + g ν (π(ω), ρ(ω)) (ω).
Since the considered form of the system (11) is lowertriangular (feedback based), and only the last equation depends on the input, the components π i , and the input can be found step by step from the top system as follows:

π 2 (ω) = ∂π 1 (ω) ∂ω s(ω) -f 1 (π 1 (ω), ρ(ω)) g 1 (π 1 (ω), ρ(ω)) , . . . π ν (ω) = ∂π ν-1 (ω) ∂ω s(ω) -f ν-1 (π ν-1 (ω), ρ(ω)) g ν-1 (π ν-1 (ω), ρ(ω)) , u = (ω) = ∂π ν (ω) ∂ω s(ω) -f ν (π(ω), ρ(ω)) g ν (π(ω), ρ(ω)) ,
where g i (π i (ω), ρ(ω)) 0 according to the imposed conditions.

Note that the conditions of Proposition 1 do not imply any attractivity of the derived steady-state solution x(t) = π(ω(t)) for ( 2)- [START_REF] Kokotovic | The joy of feedback: nonlinear and adaptive[END_REF], only its existence is established. Remark 1. In Proposition 1, it is assumed that the zero dynamics has a well-defined respective steady-state solution ρ. Such an additional hypothesis has a counterpart in the backstepping method, where for the system stabilization, it is usually assumed that the zero dynamics is input-to-state stable [START_REF] Sepulchre | Constructive Nonlinear Control[END_REF]. Here, to find steady-state solutions for ξ, we need to settle the same property for z.

B. Feedforward (upper-triangular form) case

Let us consider a nonlinear single-input system in the feedforward canonical form (d = 0) [START_REF] Sepulchre | Constructive Nonlinear Control[END_REF]:

ζ 1 = f 1 (ζ 1 ) + g 1 (ζ 2 , z, u), (12)
. . .

ζ µ-1 = f µ-1 (ζ µ-1 ) + g µ-1 (ζ µ , z, u), (13)
ζ µ = f µ (ζ µ ) + g µ (u) (14) z = η(z, ζ µ , u) (15)
with the same signal generator [START_REF] Astolfi | Model reduction by moment matching for linear and nonlinear systems[END_REF], where z :

R + → R r is the zero dynamics state, ζ = ζ 1 . . . ζ µ ∈ R µ and x = z ζ
, n = µ + r; the functions η : R r+2 → R r , f i : R → R and g i : R µ-i+r+1 → R are real analytic for i = 1, . . ., µ taking zero value for zero argument. Due to the upper-triangular structure of the system ( 12)-( 15), the control propagates from the last equation for ζ µ till ζ 1 , allowing the steady-state solutions to be iteratively calculated. For instance, equation (4) for the variable ζ µ can be written as follows:

∂π µ (ω) ∂ω s(ω) = f µ (π µ (ω)) + g µ ( (ω)) (16)
provided that the control signal u = (ω) is fixed (for a real analytic function ). For a given s and any initial condition ω(0) ∈ Γ = {w ∈ W : w q = 0}, where W ⊂ R q is a neighborhood of the origin, the signal generator (2) has a solution ω that is defined at least locally in time. Assuming that

∀ω ∈ Γ \ {0} : s q (ω) 0, (17)
there is a real analytic solution π µ : W → R of the output regulation equation ( 16) according to Theorem 1 (since [START_REF] Sepulchre | Integrator forwarding: A new recursive nonlinear robust design[END_REF] implies that ( 16) can be completed by initial conditions making this Cauchy problem non-characteristic). The same procedure can be repeated/forwarded for the rest of the system:

Proposition 2. Let (17) be satisfied. For any real analytic : R q → R, let π µ be a solution of (16) and ρ : R q → R r be the steady-state solution of (15) under substitution ζ µ = π µ (ω) and u = (ω). Then there exists a real analytic steadystate mapping π(ω) = π 1 (ω) . . . π µ (ω) , ω ∈ W, of the interconnected system (2),( 12)-( 15) with π(0) = 0.

Proof. By choosing (ω) such that π µ (ω) satisfies ( 16), we obtain ρ(ω) from [START_REF] Scarciotti | Data-driven model reduction by moment matching for linear and nonlinear systems[END_REF]. For the remaining π i (ω), i = 1, . . ., µ-1, we have the following equations from (4):

∂π 1 (ω) ∂ω s(ω) = f 1 (π 1 (ω)) + g 1 (π 2 (ω), ρ(ω), (ω)), . . . ∂π µ-2 (ω) ∂ω s(ω) = f µ-2 (π µ-2 (ω)) + g µ-2 (π µ-1 (ω), ρ(ω), (ω)), ∂π µ-1 (ω) ∂ω s(ω) = f µ-1 (π µ-1 (ω)) + g µ-1 (π µ (ω), ρ(ω), (ω)),
which all yield the form of ( 16) and the same condition of being non-characteristic for the initial values π i (ω) = 0 for ω ∈ Γ (since the system is in the upper-triangular form, π 1 (ω), . . ., π µ-1 (ω) can be found iteratively from the last equation to the first). Then, according to Theorem 1, these first-order quasi-linear PDEs have unique analytic solutions

π i : W → R, i = 1, . . ., µ -1.
Here again, we need to assume the existence of the steadystate solution ρ for the zero dynamics [START_REF] Scarciotti | Data-driven model reduction by moment matching for linear and nonlinear systems[END_REF], while in the forwarding method, the input-to-state stability hypothesis is used (compare to Remark 1 for the lower-triangular case). Similarly, Proposition 2 establishes the existence of the steady-state solutions but not their attractiveness to surrounding solutions.

V. OBSERVER DESIGN FOR INTERLACED SYSTEMS

To show the utility of the presented results to the analytical derivation of a steady-state solution, consider the problem of state estimation for a nonlinear system. The considered model is a combination of both studied systems, the upperand the lower-triangular ones (with disturbances d i , v):

ζ 1 = f 1 (ζ 1 ) + g 1 (ζ 2 , z) + d 1 , . . . ζ µ-1 = f µ-1 (ζ µ-1 ) + g µ-1 (ζ µ , z) + d µ-1 , ζ µ = f µ (ζ µ ) + g µ (ξ 1 , z) + d µ , ξ 1 = f 1 (x µ+1 , z) + g 1 (x µ+1 , z)ξ 2 + d µ+1 , ξ 2 = f 2 (x µ+2 , z) + g 2 (x µ+2 , z)ξ 3 + d µ+2 , ( 18
) . . . ξ ν-1 = f ν-1 (x µ+ν-1 , z) + g ν-1 (x µ+ν-1 , z)ξ ν + d µ+ν-1 , ξ ν = f ν (x, z) + g ν (x, z)u + d µ+ν , z = η(z, ξ 1 ), y = h(z, x) + v.
A system in such a form is called interlaced. Variables ζ ∈ R µ correspond to the state of the upper-triangular part, variables ξ ∈ R ν describe the state of the lower-triangular part; x = ζ ξ ∈ R n and n = µ + ν, z ∈ R r is the state of zero dynamics; y ∈ R p is the output available for measurements. We assume that all nonlinear functions in [START_REF] Sepulchre | Constructive Nonlinear Control[END_REF] satisfy the requirements on regularity needed to use propositions 1 and 2, i.e., they all are real analytic and admit the zero solution, h is globally Lipschitz continuous. The signal generator is considered in the linear form to simplify the observer design:

ω = Aω, u = (ω), (19)
where as before ω : R + → R q , a neutrally stable matrix A ∈ R q×q is chosen in a way that the condition ( 17) is satisfied:

∀ω ∈ Γ \ {0} : [0 . . . 0 1]Aω 0. ( 20
)
Our goal is to design a robust reduced-order observer for ( 18), ( 19) using steady-state solutions π(ω) and ρ(ω), as established before, and assuming that they describe the steady-state solutions of this system. Note that there is no restriction imposed on the uniform observability of (18) (i.e., observability for all admissible inputs u). We assume below that the input function is set properly to ensure state estimation. In such a setup, it is enough to design an observer for the generator [START_REF] Sontag | On characterizations of input-to-state stability with respect to compact sets[END_REF] and calculate the asymptotic estimates of the steady-state solutions.

Thus, we start with the next hypotheses:

Assumption 2. Assume 1) the condition (20) holds;

2) for any real analytic π µ+1 : R q → R there is the corresponding steady-state solution ρ : R q → R r of the zero dynamics under substitution ξ 1 = π µ+1 (ω).

Under Assumption 2, the conditions of propositions 1 and 2 are satisfied for the system ( 18), [START_REF] Sontag | On characterizations of input-to-state stability with respect to compact sets[END_REF], with d = 0. Due to the structure of [START_REF] Sepulchre | Constructive Nonlinear Control[END_REF], for any π µ+1 and the related ρ, the steady-state solution π µ can be obtained using the forwarding guidelines in the proof of Proposition 2. Next, the steady-state solution π µ+2 can be calculated following the backstepping procedure of Proposition 1, together with the input function : R q → R. Thus, all steady-state solutions π : R q → R n can be derived, for which we assume the following: Assumption 3. Let 1) the function π µ+1 be chosen such that h(ρ(ω), π(ω)) = Cω, where the pair (A, C) is detectable;

2) the system (18) be δISS for u = (ω), for any solution ω of (19) with respect to the input d.

Under these restrictions, the following theorem formulates the main result of the paper: Theorem 2. Let assumptions 2, 3 be satisfied, u = (ω), and suppose there exist

P = P ∈ R q×q , Q = Q ∈ R q×q , Γ = Γ ∈ R p×p and U ∈ R q×p such that the linear matrix inequalities P > 0, Q > 0, Γ > 0, A P -C U + P A -UC + Q -U -U -Γ ≤ 0 have a solution.
Then

ω = A ω + L(y -C ω), x = π( ω), ẑ = ρ( ω) (21)
is a global robust asymptotic reduced-order observer for the system (18), ( 19) with L = P -1 U, where ω(t) ∈ R q , x(t) ∈ R n and ẑ(t) ∈ R r are the estimates of ω(t), x(t) and z(t), respectively.

Proof. According to Assumption 2, the system (18) has steady-state solutions π and ρ for any π µ+1 with the corresponding input (by propositions 1 and 2). Assumption 3 states that the function π µ+1 was selected in a way that the measured output y is a linear function of the generator state ω while being projected on the steady-state solution, and in addition, π(ω) and ρ(ω) are steady-state solutions to which x(t) and z(t) converge when t → +∞, respectively, for the assigned input u = (ω). Therefore, we can write that we measure y = Cω + ε + v, where ε : R + → R p is a bounded error representing the convergence of solutions of ( 18) to the steady-state ones, ε = h(z, x)h(ρ(ω), π(ω)). To analyze the behavior of e = ωω, whose dynamics take the form:

e = (A -LC)e + L(ε + v),
let us select a quadratic Lyapunov function candidate V(e) = e Pe, where P is given in a formulation of the theorem. Denote v = ε + v, then we have:

V = e v A P -C U + P A -UC + Q -U -U -Γ e v -e Qe + v Γ v.
According to the formulation of the theorem, the matrix above is non-positive. Then, we obtain:

V ≤ -e Qe + v Γ v.
After straightforward calculations, we can obtain the estimate on the behavior of e as follows:

e(t) ≤ λ min (P) λ max (P) e -0.5

λ min (Q) λmax(P) t e(0) + λ max (Γ)λ max (P) λ min (P) v ∞ .
The error ε possesses an estimate from Definition 1

ε(t) ≤ β( ε(0) , t) + γ( d ∞ ),
for any ε(0) ∈ R n and d, for some β ∈ KL and γ ∈ K and the disturbance d is considered to be essentially bounded. Therefore the error e is ISS with respect to v, [START_REF] Sontag | On characterizations of input-to-state stability with respect to compact sets[END_REF], [START_REF] Sontag | Lyapunov characterizations of input to output stability[END_REF]. Then, the estimation for x and z can be chosen as in [START_REF] Kowalevsky | Zur theorie der partiellen differentialgleichung[END_REF].

VI. EXAMPLES

Example 1. Mass-spring system. To demonstrate a rather practical application, consider a simple mechanical system of two masses connected vertically by a spring with nonlinear stiffness and a linear damper. The bottom mass is also attached to a fixed platform via spring with linear stiffness and a linear damper, and some perturbation u acts at bottom mass (Fig. 1). Then, the system dynamics will be as follows:

                 x 1 = x 2 , x 2 = -k 1 (x 1 -x 3 ) -k 2 (x 1 -x 3 ) 3 -a 1 (x 2 -x 4 ), x 3 = x 4 , x 4 = k 1 (x 1 -x 3 ) + k 2 (x 1 -x 3 ) 3 + a 1 (x 2 -x 4 ) -k 3 x 3 -a 2 x 4 + u, y = x 3 + v,
where x 1 , x 2 are the position and velocity of a top mass, x 3 , x 4 are the position and velocity of a second one, k 1 , k 2 , k 3 are the spring stiffness coefficients, a 1 , a 2 are damper coefficients and an input u is some external force or disturbance, acting on a second mass.

Similar simplified representation can be applied, for example, to active suspension models [START_REF] Kaddissi | Interlaced backstepping and integrator forwarding for nonlinear control of an electrohydraulic active suspension[END_REF], where the top mass represents a quarter-car, and the bottom mass represents a tire moving on some nonlinear (perturbed) road.

Task 2a). Let us assume we want our bottom mass (tire) to follow a certain reference trajectory produced by a simple signal generator and we want to obtain a representation of our dynamics in terms of generator states ω 1 , ω 2 . This can be done by finding the proper input u(ω), using Propositions 1 and 2.

However, the given system is not in the required interlaced form, and we need to do the substitution y

1 = x 1 , y 2 = x 2 , y 3 = x 1 -x 3 , y 4 = x 2 -x 4 .
Then, we have the following dynamics:

             y 1 = y 2 , y 2 = -k 1 y 3 -k 2 y 3 3 -a 1 y 4 , y 3 = y 4 , y 4 = +k 1 y 3 + k 2 y 3 3 + a 1 y 4 -k 3 (y 1 -y 3 ) -a 2 (y 2 -y 4 ) + u.
Now, we can see that the top subsystem is in a uppertriangular form, which allows us to find y 1 (ω) and y 2 (ω) using Proposition 2, having only y 3 (ω). Next logical step is to find u(ω) by a feedback-like procedure, using the Proposition 1.

Therefore, let us assign y 3 (ω) := ω 1 + ω 2 , then y 4 (ω) = -ω 1 + ω 2 . Let us substitute them in the equation for y 2 :

y 2 = -k 1 (ω 1 + ω 2 ) -k 2 (ω 1 + ω 2 ) 3 -a 1 (-ω 1 + ω 2 ).
Expressions for y 1 (ω) and y 2 (ω) can be found by integrating the linear part y lin 2 = y lin 1 = -k 1 (ω 1 + ω 2 )a 1 (-ω 1 + ω 2 ), and for the nonlinear part by finding the coefficients of cubic function y

i (ω) = y lin i + p i1 ω 3 1 + p i2 ω 2 1 ω 2 + p i3 ω 1 ω 2 2 + p i4 ω 3 2 , i = 1, 2.
For our case, the dynamics are as follows:

y 1 = y lin 1 -k 2 4 3 (ω 2 ω 2 1 -ω 1 ω 2 2 ) - 11 9 (ω 3 1 + ω 3 2 ) := y lin 1 -k 2 τ 1 , y 2 = y lin 2 + k 2 4 3 (ω 3 1 + ω 3 2 ) + (ω 2 ω 2 1 + ω 1 ω 2 2 ) := y lin 2 + k 2 τ 2 .
Next, we substitute y 1 (ω), y 2 (ω), y 3 (ω) in the equation for y 4 , and u(ω) can be simply found as: The final step to find π(ω) is to go back to substitution, we have a steady-state solution π(ω) as follows: Fig. 2 shows that the dynamics of x under chosen excitation input u indeed converges to a steady-state solution π(ω).

           π 1 = x 1 (ω) = φ 1 π 2 = x 2 (ω) =
Task 2b). Let us assume that the system receives some reference signal u, which we do not measure and the system dynamics follows it. Instead, we measure the output of the initial system affected by some white measurement noise: Y = (x 1x 3 )+ v. Because of damping, our mechanical system is asymptotically stable and the signal generator oscillates, by design.

The observer would take the form of ( 21), with the matrices A and C as follows:

A = 0 1 -1 0 , C = 1 1 .
And we can find such L, P, Q and Γ that A -LC is Hurwitz and the LMIs in theorem 2 are satisfied, for instance for L = 0.33 0.6 . Fig. 2 shows the convergence of an observer to the given steady-state solution of the system.

VII. CONCLUSION

In this paper, two particular forms of interconnected systems were investigated. The analytical expression for the steady-state solution in a disturbance-free scenario was found. The conditions of existence for such solutions were provided, the result is applied for a robust reduced-order observer design for an interlaced system. The main advantage of the suggested approach is the order reduction of the observer. An example was provided in the last section of the paper.

Fig. 1 .

 1 Fig. 1. Schematics for Example 1

Fig. 2 .

 2 Fig. 2. Simulation results for Example 1. Initial conditions:x i (0) = 0; ω 1 (0) = 0.1, ω 2 (0) = 0, ω = 0. Coefficients: k 1 = k 2 = 3, k 3 = 0.6, a 1 = 0.6, a 2 = 2.

φ 2 π 3 =

 3 x 3 = y 1 (ω)y 3 (ω) = y lin 1k 2 τ 1 -(ω 1 + ω 2 ) π 4 = x 4 = y 2 (ω)y 4 (ω) = y lin 2 + k 2 τ 2 -(ω 2 -ω 1 )

A. Khalin, R. Ushirobira and D. Efimov are with Inria, Univ. Lille, CNRS, UMR 9189 -CRIStAL, F-59000 Lille, France {Anatolii.Khalin, Rosane.Ushirobira, Denis.Efimov}@inria.fr

Throughout the paper, we simplify the notation by writing, for instance, x in place of x(t) when the time-dependency is clear.