STABILIZATION THROUGH COORDINATES TRANSFORMATION FOR SWITCHED SYSTEMS

on the occasion of his 70th birthday

STABILIZATION THROUGH COORDINATES TRANSFORMATION FOR SWITCHED SYSTEMS

ASSOCIATED TO ELECTROHYDRAULIC SERVOMECHANISMS

SILVIA BALEA, ANDREI HALANAY and IOAN URSU

Electrohydraulic servomechanisms considered in this paper are modeled by five- dimensional switched nonlinear systems of control differential equations. In the stability analysis of equilibria the critical case of a zero eigenvalue of the Jacobian matrix calculated in equilibria is encountered. Local coordinates transformation and the Lyapunov-Malkin approach provide conditions for synthesis of feedback control laws that stabilize all equilibria in the closed-loop system.

AMS 2000 Subject Classification: 34D20, 93D15, 93B27, 70K20.

Key words: switching control system, Lyapunov stability, feedback control law, local coordinates transformation.

1. INTRODUCTION

The present paper addresses the problem of control synthesis for an elec- trohydraulic servomechanism (EHS). The dynamics of EHS are subject to non- linearities due to directional change of valve opening, control saturation, dry friction, valve overlap and other. Control synthesis is supposed to cope with all these and assure the desired behaviour of the EHS. The classical approach in control design for a given EHS mathematical model starts by linearising the nonlinear dynamics around a specific equilibrium point and use linear design methodology to obtain the control law (see the pioneering books in the field [2], [5], [6], [18]). Nonlinear systems of differential equations are used in these models to describe valve-actuator-load dynamics. These systems are either three-dimensional ([12], [25]) or four-dimensional ([1], [8], [14], [21], [22], [27], [29]). As concerns linearization, realistic models lead to critical cases for sta- bility of equilibria ([8], [9], [10], [11]). The classical approach works well for particular settings of initial conditions, reference signals and perturbations.

MATH. REPORTS11(61),4 (2009), 279–292

The modern approach in the synthesis of control laws for EHSs is the de- velopment of strategies directly applicable to large classes of nonlinear models.

Among the most recent techniques applied in control synthesis for EHSs one finds those in [20] and [23] based on differential geometry methods (see [13] or [19]). It is this setting, precisely the use of local coordinates transfor- mations, that will provide controllers that stabilizes equilibria for a switching mathematical model of an aviation EHS. The characteristic features of the associated physical model are: a double-ended actuator, equal ram areas of the two chambers, well known and constant load inertia, negligible dry friction forces in the ram. Accordingly (see also [9], [10], [11], [26], [28]), the system of ordinary differential equations below describes the mathematical model for the EHS we study:

(1.1) x˙1 =x2, x˙2= (−kx₁−fνx2+Sx3−Sx4)/m,

˙ x₃ =

= B

V0+Sx1

C|x₅|sgn[ps(1+sgn(x5))−2x₃]

r|p_s(1+sgn(x5))−2x₃|

2 −Sx₂

! ,

˙ x₄ =

= B

V0−Sx₁ C|x₅|sgn[p_s(1−sgn(x₅))−2x₄]

r|p_s(1−sgn(x₅))−2x₄|

2 +Sx2

! ,

˙

x₅ =−1

τx₅+k_v

τ u, C:=c_dw r2

ρ.

Here, x1 is the load displacement, x2 is the load velocity, x3 and x4 are the pressures in the cylinder chambers, x₅ is the valve position and u is the con- trol variable, an input voltage. The constants involved are: m, the equivalent inertial load of primary control surface reduced to the actuator rod; fν, an equivalent viscous friction force coefficient;k, an equivalent aerodynamic elas- tic force coefficient; S, the effective area of the piston; V0, the cylinder half- volume; p_s, the supply pressure; B, the bulk modulus of oil;τ, the servovalve time-constant; cd, volumetric flow coefficient of the valve port; w, the valve port’s width; ρ, the oil volumetric density; k_v, a proportionality coefficient relating the input voltage to servovalve to valve displacement. Therefore, be- sides the four state variables defining the valve-actuator-load system, we have a first order dynamics of the valve; this is a realistic and satisfactory hypothesis considering the saturation in the velocity of the spool-valve.

Equations (1.1) are strongly nonlinear. In order to apply the stability analysis and control synthesis machineries, asimplifying hypothesisis assumed:

0 < xi < ps, i = 3,4. Among other things (for example, sudden reversal of load acceleration, instability of motion, ram working as a pump), cavitation in cylinder is so avoided, in other words, conditions for the prevention of cavitation are supposed.

System (1.1) is split now into two systems corresponding tox₅ ≥0 and tox5≤0, respectively. For x5 ≥0, system (1.1) becomes

˙

x₁ =x₂, x˙₂= (−kx₁−f_νx₂+Sx₃−Sx₄)/m, S₁ : x˙3 = B

V0+Sx1

(Cx5

√ps−x3−Sx2), (1.2)

˙

x₄ =− B

V₀−Sx₁(Cx₅√

x₄−Sx₂), x˙₅ =−1

τx₅+k_v τ u₁ and, for x5 ≤0,

˙

x1=x2, x˙2 = (−kx₁−fvx2+Sx3−Sx4)/m, S₂: x˙3= B

V0+Sx1

(Cx5

√x3−Sx2), (1.3)

x₄=− B

V₀−Sx₁(Cx₅√

p_s−x₄−Sx₂), x˙₅=−1

τx₅+k_v τ u₂. Systems (1.2) and (1.3) will be regarded as the two components of a switched system (see [15], [17], [24]) with state variables in the set

(1.4) G=

(x1, x2, x3, x4, x5) =x|xi ∈(0, ps), i= 3,4 and|x₁|< V0

S

. Consider a reference input x₀ to the system, |x₀| < ^V_S⁰. Let p ∈ (0,1) be such that

(1.5) ps(1−p)−kx0

S >0, psp+kx0

S >0 and define bx= (xb1,bx2,bx3,xb4,xb5) by

(1.6) xb₁=x₀, bx₂ = 0, bx₃ = kx₀

S +p_sp, xb₄ =p_sp, bx₅ = 0.

If u₁(x) = 0 andb u₂(bx) = 0, then (1.6) defines an equilibrium point for both (1.2) and (1.3), so for (1.1). Synthesis of controllers u1 and u2 is aimed to ensure stability of all bxin (1.6) and a good asymptotic behaviour:

(1.7) lim

t→∞[x₁(t)−x₀] = 0

for a solution x= (x1, . . . , x5) of the switched system (1.2), (1.3) that starts close tobx. The main tool for control synthesis will be the construction of local coordinates transformations.

2. CONTROL SYNTHESIS AND STABILITY ANALYSIS

Recall from [13] the definition of the relative degree Definition. A single-input single-output nonlinear system

(2.1) x˙ =f(x) +g(x)u,

y=h(x)

is said to have relative degree r at a pointx⁰ if

(1) (LgL^k_fh)(x) = 0 for every x in a neighbourhood of x⁰ for k = 0,1, . . . , r−2;

(2) (LgL^r−1_f h)(x⁰)6= 0.

Here L_fh denotes the Lie derivative

(2.2) Lfh=

n

X

j=1

∂h

∂xj

fj

of h along f = (f₁, . . . , f_n). For systems (1.2) and (1.3), g(x) = (0,0,0,0,1)^τ (τ stands for transpose),

f⁽¹⁾(x) = x2,−kx₁

m −f_νx₂

m +Sx₃

m −Sx₄

m ,BCx₅√

p_s−x₃−BSx₂ V0+Sx1

, (2.3)

−BCx₅√

x₄+BSx₂ V₀−Sx₁ ,−1

τx5

!

for (1.2) and

f⁽²⁾(x) = x₂,−kx₁

m −f_νx₂

m + Sx₃

m − Sx₄

m ,BCx5

√x3−BSx2

V₀+Sx₁ , (2.4)

−BCx₅√

ps−x4+BSx2

V0−Sx1

,−1 τx5

!

for (1.3). Relying also on (1.7) it is natural to consider

(2.5) h(x) =x₁−x₀

for (1.2) and (1.3). Then

L_f(1)h=x₂, L²_f(1)h=−k

mx₁−f_ν

mx₂+ S

mx₃− S mx₄, L³_f(1)h= fνk

m²x1+ f_ν²

m² − k m

x2− fνS

m x3+fν

mx4−

− 2BS²V₀x₂

m(V₀²−S²x²₁)+BCSx₅ m

√p_s−x₃ V₀+Sx₁ +

√x4

V₀−Sx₁

! .

So, Lgh=LgL_f(1)h=LgL²_f(1)h= 0 inGdefined by (1.4), and (L_gL³_f(1)h)(x) = BCS

m √

p_s−x₃ V0+Sx1

+

√x4

V0−Sx1

6= 0 ∀x∈G.

Also,

L_f(2)h=x₂, L²_f(2)h=−k

mx₁−f_ν

mx₂+ S

mx₃− S mx₄, L³_f(2)h= fνk

m²x1+ f_ν²

m² − k m

x2− 2BS²V0x2

m(V₀²−S²x²₁)−

−f_νS

m²(x₃−x₄) +BCSx₅ m

√ x3

V₀+Sx₁ +

√p_s−x₄ V₀−Sx₁

. Again,

L_gL_f(2)h=L_gL²_f(2)h= 0 ∀x∈G, (LgL³_f(2)h)(x) = BCS

m

√ x₃ V₀+Sx₁ +

√ps−x4

V₀−Sx₁

6= 0 ∀x∈G.

Consider then the coordinate transformation Φ⁽¹⁾₁ (x) =x₁−x₀, Φ⁽¹⁾₂ (x) =x₂, Φ⁽¹⁾₃ (x) =−k

mx₁−f_ν

mx₂+ S

m(x₃−x₄), Φ⁽¹⁾₄ (x) = fνk

m²x1+ f_ν²

m² − k m

x2− 2BS²V x2

m(V₀²−S²x²₁)− (2.6)

−f_νS

m² (x₃−x₄) +BCS m x₅

√ p_s−x₃ V₀+Sx₁ +

√x4

V₀−Sx₁

, Φ⁽¹⁾₅ (x) =x3−bx3−d⁽¹⁾₁ (x1−x0)−d⁽¹⁾₂ x2+

+d⁽¹⁾₃ k

mx₁+f_ν

mx₂− S

m(x₃−x₄)

with d⁽¹⁾₁ ,d⁽¹⁾₂ ,d⁽¹⁾₃ coefficients to be made precise below,

Φ⁽²⁾₁ (x) = Φ⁽¹⁾₁ (x), Φ⁽²⁾₂ (x) = Φ⁽¹⁾₂ (x), Φ⁽²⁾₃ (x) = Φ⁽¹⁾₃ (x), Φ⁽²⁾₄ (x) = fνk

m²x1+ f_ν² m² − k

m

!

x2− 2BS²V x2

m(V₀²−S²x²₁)− (2.7)

−f_νS

m² (x₃−x₄) +BCSx₅ m

√x3

V₀+Sx₁ +

√p_s−x₄ V₀−Sx₁

! ,

Φ⁽²⁾₅ (x) =x₃−xb₃−d⁽²⁾₁ (x₁−x₀)−d⁽²⁾₂ x₂+ +d⁽²⁾₃

k

mx1+fν

mx2− S

m(x3−x4)

,

where again d⁽²⁾₁ ,d⁽²⁾₂ ,d⁽²⁾₃ are coefficients that will be computed below accor- ding to a supplementary demand on the transformed systems.

The Jacobi matrices for Φ⁽¹⁾ and Φ⁽²⁾ calculated at equilibria xbare

(2.8) J^(l)=

















1 0 0 0 0

0 1 0 0 0

−_m^k −^f_m^ν _m^S −_m^S 0

fνh

m² j₄₂^(l) −^f_m^νS₂ ^f_m^νS₂ j₄₅^(l) d^(l)_{3 m}^k −d^(l)₁ d^(l)₃ ^f_m^ν −d^(l)₂ 1−d^(l)_{3 m}^S d^(l)_{3 m}^S 0

















, l= 1,2,

where j^(l)₄₅ =LgL³_f(l)h,l= 1,2, so that the Jacobians are detJ^(l)=−j₄₅^(l)S

m 6= 0

∀bxin (1.6), l= 1,2.

Remark that Φ^(l)(bx) = 0,l= 1,2, for everyxbin (1.6). As in [13], define u_l(x) = 1

(LgL³_f(l)h)(x)

h−(L⁴_f(l)h)(x) +c₁Φ^(l)₁ (x) +c₂Φ^(l)₂ (x)+

(2.9)

+c₃Φ^(l)₃ (x) +c₄Φ^(l)₄ (x)i

, l= 1,2, and introduce new coordinates for S_l,l= 1,2, through (2.10) zj = Φ^(l)_j (x), j= 1, . . . ,5, l= 1,2.

Then S₁,S₂ become

(2.11) Σ₁:

(ξ˙=Dξ

˙

z5=q⁽¹⁾(z), Σ₂ :

(ξ˙=Dξ

˙

z5 =q⁽²⁾(z), where ξ= (z1, z2, z3, z4)^τ,z= (z1, . . . , z5) = (ξ, z5),

(2.12) D=









0 1 0 0

0 0 1 0

0 0 0 1

c₁ c₂ c₃ c₄







 .

In order to computeq^(l),l= 1,2, the inverses of Φ^(l)have to be first calculated from (2.6), (2.7) and (2.10). So, for Φ^(l) the inverse Ψ^(l) is given by

x₁= Ψ⁽¹⁾₁ (z) = Ψ⁽²⁾₁ (z) =z₁+x₀, x2= Ψ⁽¹⁾₂ (z) = Ψ⁽²⁾₂ (z) =z2,

x₃= Ψ^(l)₃ (z) =z₅+xb₃+d^(l)₁ z₁+d^(l)₂ z₂+d^(l)₃ z₃, l= 1,2, x4= Ψ^(l)₄ (z) = Ψ^(l)₃ (z)−m

S

z3+ k

mz1+fν

mz2+ k mx0

, l= 1,2,

x₅= Ψ⁽¹⁾₅ (z) = m BCS



 q

p_s−Ψ⁽¹⁾₃ (z) V₀+Sz₁+Sx₀ +

q

Ψ⁽¹⁾₄ (z) V₀−Sz₁−Sx₀





−1

·

·

z4+fν

mz3+ k

mz2+ 2BS²V0z2

m[V₀²−S²(z₁+x₀)²] (2.13)

for S₁ and

x5 = Ψ⁽²⁾₅ (z) = m BCS



 q

Ψ⁽²⁾₃ (z) V0+Sz1+Sx0

+ q

ps−Ψ⁽²⁾₄ (z) V0−Sz1−Sx0





−1

(2.14) ·

· (

z4+fν

mz3+ k

mz2+ 2BS²V z2

m[V₀²−S²(z1+x0)²] )

for S₂. Remark that Ψ^(l)(0) =x, so thatb

(2.15) Ψ^(l)₅ (0) = 0, l= 1,2.

For further use it is convenient to define

R₁(z) = q

p_s−Ψ⁽¹⁾₃ (z) V0+Sz1+Sx0

+ q

Ψ⁽¹⁾₄ (z) V0−Sz1−Sx0

,

R2(z) = q

Ψ⁽²⁾₃ (z) V₀+Sz₁+Sx₀ +

q

ps−Ψ⁽²⁾₄ (z) V₀−Sz₁−Sx₀. (2.16)

By (2.10), (2.6) and (2.13), in Σ1 we have

˙

z₅ =q⁽¹⁾(z) = ˙x₃−d⁽¹⁾₁ x˙₁−d⁽¹⁾₂ x˙₂+d⁽¹⁾₃ k

mx˙₁+d⁽¹⁾₃ f_ν mx˙₂−

−d⁽¹⁾₃ S

m( ˙x3−x˙4) = B V0+Sz1+Sx0

CΨ⁽¹⁾₅ (z) q

ps−Ψ⁽¹⁾₃ (z)−Sz₂

+ +

d⁽¹⁾₃ f_ν

m −d⁽¹⁾₂

z₃+

d⁽¹⁾₃ k m −d⁽¹⁾₁

z₂− (2.17)

−d⁽¹⁾₃ S m

(

BCΨ⁽¹⁾₅ (z)



 q

ps−Ψ⁽¹⁾₃ (z) V0+Sz1+Sx0

+ q

Ψ⁽¹⁾₄ (z) V0−Sz1−Sx0



−

− 2BCV0z2

V₀²−S²(z₁+x₀)² )

. Since Ψ⁽¹⁾₅ (0) = 0, we have ^∂q_∂z⁽¹⁾

1 (0) = 0 independently of d⁽¹⁾₁ , d⁽¹⁾₂ and d⁽¹⁾₃ . The coefficients d⁽¹⁾₁ ,d⁽¹⁾₂ and d⁽¹⁾₃ will be determined such that

∂q⁽¹⁾

∂z₂ (0) = ∂q⁽¹⁾

∂z₃ (0) = ∂q⁽¹⁾

∂z₄ (0) = 0.

Now,

∂q⁽¹⁾

∂z2

(0) = BCp p_s−xb₃ V0+Sx0

∂Ψ⁽¹⁾₅

∂z2

(0)− BS V0+Sx0

+d⁽¹⁾₃ k

m −d⁽¹⁾₁ −

−d⁽¹⁾₃ R₁(0)∂Ψ⁽¹⁾₅

∂z₂ (0) +d⁽¹⁾₃ 2BS²V₀ m(V₀²−S²x²₀). Since ^∂Ψ

(1) 5

∂z2 (0) = _BCSR^m

1(0)

hk

m +_m(V^2BS2 ^2V0 0−S²x²₀)

i

, we have

∂q⁽¹⁾

∂z₂ (0) =

pps−xb3

V₀+Sx₀ 1 SR₁(0)

k+ 2BS²V0

V₀²−S²x²₀

− BS

V₀+Sx₀ +d⁽¹⁾₃ k m−

−d⁽¹⁾₁ −d⁽¹⁾₃ k

m + 2BS²V₀ m(V₀²−S²x²₀)

+d⁽¹⁾₃ 2BS²V₀

m(V₀²−S²x²₀) = 0 for

(2.18) d⁽¹⁾₁ =

pp_s−xb₃ R1(0)(V0+Sx0) ·

k+ 2BS²V₀ V₀²−S²x²₀

− BS V0+Sx0

. Also, ^∂Ψ

(1) 5

∂z3 (0) = _BCSR^fν

1(0) implies that

∂q⁽¹⁾

∂z₃ (0) = f_νp p_s−bx₃

SR₁(0)(V₀+Sx₀) +d⁽¹⁾₃ f_ν

m −d⁽¹⁾₂ −d⁽¹⁾₃ f_ν m = 0

for

(2.19) d⁽¹⁾₂ = fν

pps−bx3

SR₁(0)(V₀+Sx₀). Again, ^∂Ψ

(1) 5

∂z4 (0) = _BCSR^m

1(0) implies

∂q⁽¹⁾

∂z4

(0) = mp p_s−xb₃

SR1(0)(V0+Sx0) −d⁽¹⁾₃ = 0 for

(2.20) d⁽¹⁾₃ = mp

ps−bx3

SR₁(0)(V₀+Sx₀). Similarly, in Σ2,

˙

z5 =q⁽²⁾(z) = B

V₀+Sz₁+Sx₀[CΨ⁽²⁾₅ (z) q

Ψ⁽²⁾₃ (z)−Sz2]+

(2.21)

+

d⁽²⁾₃ fν

m −d⁽²⁾₂

z3+

d⁽²⁾₃ k m −d⁽²⁾₁

z2−

−d⁽²⁾₃ S m

(

BCΨ⁽²⁾₅ (z)



 q

Ψ⁽²⁾₃ (z) V₀+Sz₁+Sx₀+

+ q

p_s−Ψ⁽²⁾₄ (z) V₀−Sz₁−Sx₀



− 2BSV₀z₂ V₀²−S²(z1+x0)²

) .

As before, ^∂q_∂z⁽²⁾

1 (0) = 0 and the conditions ^∂q_∂z⁽²⁾

2 (0) = ^∂q_∂z⁽²⁾

3 (0) = ^∂q_∂z⁽²⁾

4 (0) = 0 yield the values of d⁽²⁾₁ ,d⁽²⁾₂ and d⁽²⁾₃ as

d⁽²⁾₁ =

√ xb3

SR₂(0)(V₀+Sx₀) k+ 2BS²V0

V₀²−S²x²₀

!

− BS V₀+Sx₀, d⁽²⁾₂ = f_ν√

xb₃

SR₂(0)(V₀+Sx₀), d⁽²⁾₃ = m√ xb₃ SR₂(0)(V₀+Sx₀). (2.22)

When d^(l)₁ ,d^(l)₂ and d^(l)₃ ,l= 1,2, are given in (2.18), (2.19), (2.20) and (2.22) the fifth equations in Σ_l,l= 1,2, have zero linear part (due also to ^∂q_∂z^(l)

1 (0) =

∂q^(l)

∂z5 (0) = 0,l = 1,2, since ^∂Ψ

(l) 5

∂z5 (0) = 0, l= 1,2). Since Ψ^(l)₅ (0) = 0, l= 1,2, one has

(2.23) q^(l)(0,0,0,0, z5) = 0, l= 1,2, for everyz5.

For switched systems stability of equilibria is defined (see [24]) as follows.

Definition. The equilibrium point xb is called uniformly stable for the switching system (1.1) if for every ε > 0 there exists δ_ε > 0 such that if kx(0)−xkb < δε thenkx(t)−xkb < εand every t∈[0,∞), for every solutionx of (1.1).

Here and in what followsk kdenotes the Euclidean norm.

Following [16], we can now prove

Theorem 2.1. If D in (2.12) is a Hurwitz matrix(i.e.,σ(D) ⊂C− = {ζ ∈C|Reζ < 0}), then every equilibrium point in (1.6) is uniformly stable for the switching system(1.1)and if initial data are close to such an equilibrium point, lim

t→∞[x₁(t)−x₀] = 0for a solution x= (x₁(t), . . . , x₅(t)) of (1.1).

Proof. SinceD is Hurwitz there exists a strictly positive definite matrix P, the unique solution of the Lyapunov equationD^τP+P D=−I. With dot denoting the scalar product in R⁴,

(2.24) V ξ =ξ·P ξ

is a Lyapunov function for ˙ξ =Dξ that satisfies ^∂V_∂ξ ·Dξ =−

4

P

j=1

z²_j (see, for example, [4]). Chooseα >0 such that

(2.25) −

4

X

j=1

z_j²+ 2αV(ξ) is negative definite

and perform the transformation on state variables in Σ_l,l= 1,2, given by (2.26) η_j(t) = e^αtz_j(t), j= 1, . . . ,4.

With I the identity matrix of order 4, (2.11) turn into

(2.27) Σ˜l:

(η˙= (D+αI)η

˙

z5=q^(l)(e^−αtη, z5) l= 1,2.

From (2.25) we infer that, for every solution of (2.27), _dt^dV[η(t)]<0, ∀t≥0.

Then V[η(t)]< V[η(0)], ∀t≥0 and sinceV is a Lyapunov function, we have kη(t)k ≤ K, ∀t≥0 (see, for example, [7]). If kη(0)k is small enough, K can be made as small as we want. It follows from (2.26) that

(2.28) kξ(t)k ≤Ke^−αt ∀t≥0.

Since q^(l)(0, . . . ,0, z5) = 0, l= 1,2 and q^(l) contains only powers of the varia- bles equal or greater than 2, there exists M > 0, independent of K ≤ 1, such that

|q^(l)(ξ(t), z5(t))|< K Me^−αt ∀t≥0, l= 1,2.

Since z5(t) =z5(0) +Rt

0q^(l)[ξ(s), z5(s)]dx,l= 1,2, we have (2.29) |z₅(t)| ≤ |z₅(0)|+KM

α ∀t≥0.

Letω >0 be given and letγ(ω)< ^ω₂ be so small thatkη(0)k< γ(ω),|z₅(0)|<

γ(ω) imply ^KM_α < ^ω₂. It follows from (2.28), (2.29) that kz(0)k< γ(ω) implies kz(t)k< ω,∀t≥0. Suppose ε >0 is given. From the continuity of the maps Ψ^(l), l = 1,2, there exists ωε > 0 such that kzk < ωε ⇒ kΨ^(l)(z)−xkb < ε.

Then, if kz(0)k< γ(ω_ε), we havekx(t)−bxk< ε,∀t≥0. By (2.10), (2.6) and (2.7) there exists δε>0 such that kx(0)−xkb < δε ⇒ kz(0)k< γ(ωε) and the stability of the equilibrium bx for the switching system (1.1) is proved. The last statement in Theorem 2.1 follows from (2.28) applied toz₁ =x₁−x₀. The characteristic polynomial ofDin (2.12) isP(λ) =λ⁴−c4λ³−c3λ²− c₂λ−c₁, so that in order thatP be Hurwitz one must have (see [3]), c₁ <0, c2<0,c3<0,c4<0,c3c4+c2>0,−c₂c3c4+c1c²₄−c²₂ >0 and one can take c₁ = −1, c₂ =−1, c₃ =−3, c₄ =−1. Then, by (2.9), the feedback laws for (1.2) and (1.3) are, respectively,

u₁(x) =− m BCS

√ p_s−x₃ V0+Sx1

+

√x4

V0−Sx1

−1"

−(L⁴_f(1)h)(x)−x₁+x₀−x₂+ +3

m(kx1+fνx2−Sx3+Sx4)−fνk m²x1−

f_ν² m² − k

m

x2+ 2BS²V0x2

m(V₀²−S²x²₁)+ +f_νS

m²(x₃−x₄)−BCSx₅ m

√ p_s−x₃ V₀+Sx₁ +

√x4

V₀−Sx₁ #

and

u2(x) =− m BCS

√ x₃ V0+Sx1

+

√ps−x4

V0−Sx1

−1"

−(L⁴_f(2)h)(x)−x1+x0−x2+ +3

m(kx₁+f_νx₂−Sx₃+Sx₄)−f_νk m²x₁−

f_ν² m² − k

m

x₂+ 2BS²V₀x₂ m(V₀²−S²x²₁)+ +f_νS

m²(x₃−x₄)− BCSx₅ m

√ x3

V0+Sx1

+

√p_s−x₄ V0−Sx1

# ,

and one can take u(x) =θ(x₅)u₁+θ(−x₅)u₂(x) in (1.1) withθthe Heaviside function.

3. CONCLUDING REMARKS

The research presented in this paper combines the theory of relative de- gree in [13] and the classical Lyapunov-Malkin approach for critical cases in stability theory ([16]) for the study of a mathematical model of an EHS. The strong nonlinearity due to the directional change of valve opening leads to a five dimensional switched system of control differential equations. The two systems that constitute the switching components have both relative degree 3 when the natural outputh(x) =x₁−x₀ is considered and can be transformed through local diffeomorphisms into systems of a special kind, with equilibria corresponding to the zero solution. When specific control laws are defined, these systems decouple into a four-dimensional linear system that is asymp- totically stable and that is common to both components and a fifth equation with no linear terms. The Quadratic (Common) Lyapunov Function yielded by the asymptotically stable (common) linear system ensures the uniform sim- ple stability of the zero solution in the transformed switched system. Thus, every equilibrium pointxbin the original system is stable and is asymptotically stable with respect to the first component (also with respect to the second one) (see [7]). This approach is different from those currently used in control syn- thesis of EHS. For the first time, in the case of a switched system exhibiting the critical case of a zero eigenvalue in the spectrum of the Jacobian matrix calculated at equilibria, a feeback stabilizing law is synthesized by this kind of geometric methods. As can be clearly seen from the paper, this construction is not unique: a large variety of control laws achieving the same task can be defined using the same method. This makes our approach suitable for optimal control synthesis.

Acknowledgements.This work is partially supported by CNCSIS, Grant 84.

REFERENCES

[1] R. Abott, T. McLain and R. Beard,Application of an optimal control synthesis strategy to an electrohydraulic positioning system. Trans. ASME J. of Dynamic Systems Mea- surement Control123(2001), 377–384.

[2] J.F. Blackburn, J.L. Shearer and G. Reethof,Fluid Power Control. Tech. Press of MIT and Wiley, New York, 1960.

[3] L. Brand,Differential and Difference Equations. Academic Press, New York, 1966.

[4] V. Dr˘agan and Aristide Halanay,Stabilization of Linear Systems. Birkh¨auser, Boston, 1999.

[5] N.S. Gamynin,Hydraulic Automatic Control Systems. Mashinostroienie, Moskow, 1972.

(Russian)

[6] M. Guillon,L’asservissement hydraulique et electrohydraulique. Dunod, Paris, 1972.

[7] Aristide Halanay, Differential Equations. Stability. Oscillations, Time Lags. Academic Press, New York, 1967.

[8] A. Halanay, C.A. Safta, I. Ursu and F. Ursu,Stability of equilibria in a four-dimensional nonlinear model of a hydraulic servomechanism. J. Engrg. Math.49(2004), 391–405.

[9] A. Halanay and C.A. Safta,Stabilization of some nonlinear controlled electrohydraulic servosystems. Appl. Math. Lett.18(2005), 911–915.

[10] A. Halanay, F. Popescu, C.A. Safta, F. Ursu and I. Ursu,Stability analysis and nonlinear control synthesis for hydraulic servos actuating primary flight controls. In: S. Sivasun- daram (Ed.),ICNPAA, 2004, pp. 243–251. Cambridge Scientific Publishers, 2005.

[11] A. Halanay and I. Ursu,Stability of equilibria in a model for electrohydraulic servosys- tems. Math. Rep. (Bucur.)9(2007),1, 47–54.

[12] R. Ingenbleedk and H. Schwarz,A nonlinear discrete-time observer based control scheme and its applications to hydraulic drives. Proc. 2nd European Control Conference, Gronin- gen, the Netherlands, pp. 215–219.

[13] A. Isidori, Nonlinear Control Systems, 2nd Edition. Springer, Berlin–Heidelberg–New York, 1995.

[14] M. Jelali and A. Kroll,Hydraulic Servosystems. Springer, Berlin–Heidelberg–New York, 2003.

[15] D. Liberzon,Switching in Systems and Control. Birkh¨auser, Boston, 2003.

[16] I.G. Malkin, Theory of Stability of Motion. Nauka, Moskow, 1966. (Russian; English translation: Atomic Energy Comm. Translation AEC–TR–3352)

[17] M. Margaliot, Stability analysis of switched systems using variational principles: an introduction. Automatica J. IFAC42(2006), 2059–2077.

[18] H.E. Meritt,Hydraulic Control Systems. Wiley, New York, 1967.

[19] H. Nijmeijer and A.J. Van der Schaft,Nonlinear dynamical control systems, 2nd Edition.

Springer, Berlin–Heidelberg–New York, 1995.

[20] R.M. Novak, K. Schlacher, A. Kugi and H. Frank, Nonlinear hydraulic gap control: a apractical approach. In: Proc. IFAC Control System Design, pp. 605–609. Bratislava, 2000.

[21] H.V. Panasian, Reduced order observers applied to state and parameter estimation of hydromechanical servoactuators. J. Guidance Control Dynamics9(1986), 249–251.

[22] A.R. Plummer,Feedback linearization for acceleration control of electrohydraulic actua- tors. Proc. Institution of Mechanical Engineers211(1997), 395–406.

[23] K. Schlacher, A. Kugi and R. Novak,Input to output linearization with constraint mea- surements. Preprints of 5th IFAC Symposium Nonlinear Control Systems NOLCOS’01 Saint-Petersburg, Russia, 2001, pp. 471–476.

[24] R. Shorten, F. Wirth, O. Mason, K. Wulff and C. King, Stability criteria for switched and hybrid systems, 2005. Available atwww.hamilton.ie/bob/switchedstability.pdf.

[25] I. Ursu, F. Ursu, M. Vladimirescu and T. Sireteanu,From robust control to antiwindup compensation of electrohydraulic servoactuators. Aircraft Engineering and Aerospace Technology70(1998), 259–264.

[26] I. Ursu, F. Ursu and F. Popescu, Backstepping design for controlling electrohydraulic servos. J. Franklin Inst.343(2006), 94–110.

[27] G. Vossoughi and M. Donath, Dynamic feedback linearization for electrohydraulically actuated control systems. Trans. ASME J. Dynamic Systems Measurements Control 117(1995), 468–477.

[28] P.K.C. Wang, Analytical design of electrohydraulic servomechanisms with near time- optimal response. IEEE Trans. Automat. Control Ac-8(1963), 15–27.

[29] H. Yamada and M. Shimahara,Sliding mode control of an electrohydraulic servo-motor using a gain scheduling type observer and controller. Proc. Institution of Mechanical Engineering211(1997), 407–416.

Received 15 March 2009 “Politehnica” University of Bucharest Department of Mathematics 2

313 Splaiul Independent¸ei 060042 Bucharest, Romania

silviabalea@yahoo.com

“Politehnica” University of Bucharest Department of Mathematics 1

313 Splaiul Independent¸ei 060042 Bucharest, Romania

halanay@mathem.pub.ro

“Elie Carafoli” National Institute for Aerospace Research

Bd. Iuliu Maniu 220 061126 Bucharest, Romania

iursu@aero.incas.ro