on the occasion of his 70th birthday
STABILITY OF EQUILIBRIA IN A MODEL FOR ELECTROHYDRAULIC SERVOMECHANISMS
ANDREI HALANAY and IOAN URSU
The mathematical model of an electrohydraulic servomechanism is developed and a theorem on the stability of equilibria is proved.
AMS 2000 Subject Classification: 70K20, 93D05, 93D15.
Key words: Lyapunov stability, critical case, Lyapunov-Malkin Theorem.
1. INTRODUCTION
(Electro)hydraulic servomechanisms (EHS) are used as actuators of mod- ern aircrafts control surfaces. The usual performance specifications for such devices are high accuracy of reference signals tracking and high bandwidth implying small servo time constants. But a property sine qua non of such vital system for aircraft safety is, first of all, the stability of its equilibria; this condition concerns the basis itself of the system approach, study and design.
Because of the complexity of EHS analysis and the nonlinearities in its dynam- ics, both the design and the control of EHS are still difficult and immature, although various methodologies of the automatic control theory were brought to the proof in this field; from the classical linearization [1], to the artificial intelligence paradigm ([17], [18]). In the last twenty years, a large amount of work in EHS control systems has been devoted to problems such as de- sign of observers ([3], [9], [12]), feedback linearization ([5], [13], [19]), feedback stabilization ([14]), high bandwidth control ([2]), robust and antichattering control ([16]).
During the qualification testing of the hydraulic servos for aircraft’s pri- mary flight controls actuation ([15]), a mass overload simulated on the test rig has generated, as a response to an accidentally impulse type disturbation introduced at the mechanical input, a strong self-excited motion regime of the entire test rig, induced, certainly, by the powered hydraulic servo. Thus, our belief is that modern linear control theory cannot explain all strange aircraft incidents or even catastrophes when the actuator is, for sure, involved; these misfortunes can originate in a problem of stability of the equilibrium point for
MATH. REPORTS9(59),1 (2007), 47–54
the loaded hydraulic servomechanism. So, the paper is a new attempt of anal- ysis and confirmation, using the classical approach of Lyapunov-Malkin, that such harmful phenomena in hydraulic servo’s operation can be foreseen and kept under by analytical calculus. Recently, a methodology of control synthe- sis was described in a series of authors works ([6]–[8]), ensuring the equilibria stability property of hydraulic servo. In fact, the approach can be subsumed to pseudoactive control paradigm ([18]).
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Detailed aproaches of different problems emerging in the process of math- ematical modeling of hydraulic servos, with emphasis on those used in air- craft control chains, have been presented in classical books ([4], [11]) but also in [6]. Herein, a basic mathematic model of EHS dynamics is succinctly rewritten. EHS is in fact a combination electrohydraulic servovalve (EHSV)- hydrocylinder (HC). The paradigm of EHSV as ideal spool valve ([11]) and having rectangular ports is assumed. An algebraic sign convention is chosen, that of positive incoming in cylinder flow and negative outgoing from cylin- der flow. Thus, the first equation of the EHSV-controlled piston is obtained based on the fact that the flow into and out of the cylinder is described by two components, one being due to the movement of the piston and one to compressibility effects: for the caseσ >0,
(1.1) cdwσ
2(ps−p1)
ρ =Sz˙+V+Sz
B p˙1, −cdwσ 2p2
ρ =−Sz˙+V−Sz B p˙2, and, similarly, ifσ <0,
(1.1) cdwσ 2
ρp1 =Sz˙+V+Sz
B p˙1, −cdwσ
2(ps−p2)
ρ =−Sz˙+V−Sz B p˙2. The parameters are cd – volumetric flow coefficient of the valve port; w – valve-port width; ps – supply pressure; ρ – volumetric density of oil; V – cylinder semivolume; B – bulk modulus of the oil. The variables arep1 and p2 – hydraulic pressures in the chambers of the HC;σ – relative displacement spool-sleeve;z – load displacement.
The equation of motion of the HC piston assembly is defined as a force balance equation relating to the displacement of the load generated by the hydraulic pressuresp1 andp2
(1.2) mz¨+f z+kz=S(p1−p2).
The components of the load are: m – equivalent inertial load of primary control surface reduced at the EHS’s rod; equivalent friction force, defined
by a coefficientfr; elastic force, defined by an equivalent aerodynamic elastic force coefficientk. S is the equivalent active area of the piston.
The linear displacement induced to spool valve by the torque motor of EHSV is described as a first order system
(1.3) πσ˙ +σ =kvu.
The parameters are: σ – time constant of the (servo)valve; kv – proportional valve displacement/voltage coefficient. The relative displacement spool-sleeve σ is induced in the presence of the control signalu.
For this system, the aim of control synthesis is to have a good tracking by the load displacementz of the specified desired position references signals x. The closed loop performance of the system can be measured by the actual (realized) servo time constant. Thus, a good tracking system is characterized by fast (little) servo time constant. In practice, this control objective is per- formed by implementing a compensator (controller) to generate the control u based on state feedback information given by transducers.
Using the notation
(1.4) C:=cdw
2 ρ and
(1.5) x1 :=z; x2:= ˙z; x3=p1; x4 =p2; x5 :=σ,
for the state variables, the canonical first order system derived from equations (1.1)–(1.3) is
(1.6)
x˙1 = x2, x˙2 = −k
mx1− fr
mx2+ S
mx3− S mx4, x˙3 = B
V0+Sx1
Cx5√
ps−x3−Sx2 , x˙4 = B
V0−Sx1 (−Cx5√
x4+Sx2), x˙5 = −1
τx5+kv
τ u(x1, x2, x3, x4, x5). Letp∈(0,1) and define, for ps(1−p)− kxS0 >0, (1.7) xˆ1 =x0, xˆ2 = 0, xˆ3 = k
Sx0+psp, xˆ4 =psp, xˆ5 = 0. Ifu(ˆx1,xˆ2,xˆ3,xˆ4,xˆ5) = 0, (1.7) is an equilibrium point of (1.6).
2. STABILITY ANALYSIS
When the equilibria defined in (1.7) are translated to zero through (2.1) y1 =x1−xˆ1, y2 =x2, y3=x3−xˆ3, y4 =x4−xˆ4, y5=x5
and ˜u is defined by
(2.2) u˜(y) = kv
τ u(y+ ˆx), system (1.1) becomes
(2.3)
y˙1 = y2, y˙2 = −k
my1−fr
my2+ S
my3− S my4,
y˙3 = BC
V0+Sy1+Sx0y5 ps−y3−xˆ3− BSy2
V0+Sy1+Sx0 :=f3(y), y˙4 = − BC
V0−Sy1−Sx0y5 y4+ ˆx4+ BSy2
V0−Sy1−Sx0 :=f4(y), y˙5 = −1
τy5+ ˜u(y). Denote by
(2.4) A=
0 1 0 0 0
−k m −fr
m S m −S
m 0
0 a32 0 0 a35
0 a42 0 0 a45
a51 a52 a53 a54 −1 τ
the Jacobian matrix at zero, where a32= ∂f3
∂y2(0), a35= ∂f3
∂y5(0), a42= ∂f4
∂y2(0), a45= ∂f4
∂y5(0), a51= ∂u˜
∂y1(0), a52= ∂u˜
∂y2(0), a53= ∂u˜
∂y3(0), a54= ∂u˜
∂y4(0).
It is easy to see that the characteristic polynomialQofAhas zero as a root, so (2.5) Q(λ) =λQ1(λ).
This corresponds to a critical case for stability theory that can be approached through the use of a theorem of Lyapunov and Malkin ([10]).
The following theorem settles the problem of stability for the zero solu- tion of the system (2.3).
Theorem2.1. Suppose Q1 in (2.5) is a stable polynomial (all its roots have strictly negative real parts). Then the zero solution of (3.2) is simple stable by Lyapunov. If initial data are small enough, then lim
t→∞y1(t) can be made arbitrary small, thus lim
t→∞[x1(t)−xˆ1]is arbitrary small.
Proof. Transform the linear part
(2.6) w˙ =Aw
of (2.3) throughη = 5
j=1djwj such that d1, . . . , d5 are not all zero and ˙η = 0, so 5
j=1djw˙j = 0. From (2.4) and (2.6) we deduce that
−d2k
m +d5a51= 0, d1−d2fr
m +d3a32+d4a42+d5a52= 0, d2S
m +d5a53= 0,
−d2S
m +d5a54= 0, d3a35+d4a45−1
τd5 = 0.
Ifa54+a53= 0 and mka53+mSa51= 0, then the last term ofQ1, namely, b5 =−k
ma35a53− k
ma45a54− S
ma35a51+ S
ma45a51−
−a35a42a53S
m+a32a45a53S
m −a35a42a54S
m +a32a45a54S m = 0,
which contradicts the hypotheses that Q1 is a stable polynomial. It follows thatd2 = 0, d5 = 0, and we take
(2.7) d4 = 1 so d3 =−a45
a35, d1= a32a45−a42a35 a35 . Introduce a new state variable through
(2.8) y4 =y−d1y1−d3y3.
The system that results is
(2.9)
y˙=d1y2+ d3B
V0+Sx0+Sy1(cy5 ps−y3−xˆ3−Sy2)+
+ B
V0−Sx0−Sy1(−Cy5 psp+y−d1y1−d3y3+Sy2) :=Y(y, y1, y2, y3, y5), y˙1 =y2,
y˙2 = Sd1
m − k m
y1−fr
my2+S(1 +d3)y3
m − Sy
m, y˙3 = BC
V0+Sx0+Sy1y5 ps−y3−xˆ3− BSy2 V0+Sx0+Sy1, y˙5 =−1
τy5+ ˜u(y1, y2, y3, y−d1y1−d3y3, y5) :=Y5(y, y1, y2, y3, y5). The linear part of the first equation in (2.9) is zero and the characteristic equation of the Jacobian matrix of (2.9) at zero has exactly one zero root, the other ones having strictly negative real parts. This implies that we can solve with respect toy1, y2, y3 and y5 the algebraic system
(2.10)
y2 = 0,
(Sd1−k)y1−fry2+S(1 +d3)y3−Sy= 0, Cy5√
ps−y3−xˆ3−Sy2 = 0, Y5(y, y1, y2, y3, y5) = 0.
Remark that y = 0, y1 =y2 =y3 =y5 = 0 verify (2.10). Thus, the Implicit Function Theorem implies that in a small neighborhood ofy= 0 there exists a unique solution
(2.11) y1 =ϕ1(y), y2 = 0, y3=ϕ3(y), y5= 0 of (2.10) withϕ1(0) =ϕ3(0) = 0.
The new variables
(2.12) ξ1 =y1−ϕ1(y), ξ3=y3−ϕ3(y) bring system (2.9) to
(2.13)
y˙=Y(y, ξ1+ϕ1(y), y2, ξ3+ϕ3(y), y5) := ˜Y(y, ξ1, y2, ξ3, y5), ξ˙1 =y2−ϕ1(y) ˜Y ,
y˙2 = Sd1
m − k m
ξ1−fr
my2+ S(1 +d3) m ξ3,
ξ˙3 = B
V0+Sx0+S[ξ1+ϕ1(y)][Cy5 ps−ξ3−ϕ3(y)−xˆ3−Sy2]−ϕ3(y) ˜Y , y˙5 =Y5(y, ξ1+ϕ1(y), y2, ξ3+ϕ3(y), y5).
Since ˜Y(y,0,0,0,0) = Y(y, ϕ1(y),0, ϕ3(y),0) = 0, system (2.13) belongs to the singular class covered by Lyapunov-Malkin Theorem in [10], Ch. IV, §34 (see also [7]). As proved in that theorem, (2.13) has the family of equilibria (2.14) y=c, ξ1 =ϕ1(c), y2 = 0, ξ3 =ϕ3(c), y5 = 0,
withca small constant, that are stable by Lyapunov. By (2.11),c= 0 in (2.14) gives the zero solution of (2.3). Solutions of (2.13) that start close to equilibria (2.14) satisfy lim
t→∞y(t) = α, lim
t→∞ξ1(t) = ϕ1(α), lim
t→∞y2(t) = 0, lim
t→∞ξ3(t) = ϕ3(α) and lim
t→∞y5(t) = 0. Here,α=y(0)+∞
0 Y˜[y(t), ξ1(t), y2(t), ξ3(t), y5(t)]dt the convergence of the integral resulting from the proof of Lyapunov-Malkin Theorem (see p. 116–117 in [10]). It is also proved that |α|<|y(0)|+K
γ for some positive K and that ∀ε >0,∃η >0 such that if |y(0)|< η, |ξ1(0)|< η,
|y2(0)| < η, |ξ3(0)| < η and |y5(0)| < η then one can choose K such that
|y(0)|+Kγ < εand so the last statement of Theorem 2.1 is proved.
3. CONCLUDING REMARKS
The methodology for treating the problem of equilibrium stability in (E)HS models developed in our recent works [6]–[8], as applied to the present study, concerns the covering of the following steps:
1) the choice of a feedback control lawu(x1, x2, x3, x4, x5) with u(ˆx1,xˆ2, xˆ3,xˆ4,xˆ5) = 0 (sometimes by a trial and error type methodology);
2) the use of the (p, x) – parametrization of the coefficients of the poly- nomialQ1(λ); the parameterp∈(0,1) refers to the initial conditions for state variablesx3 andx4and the parameter x∈(−xmin, xmax) is in fact the desired output;
3) the study of the Routh-Hurwitz conditions of stability relatively to polynomialQ1(λ).
Finally, it should be mentioned that the situation σ < 0 can be ap- proached in a similar way.
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Received 2 October 2006 “POLITEHNICA” University of Bucharest Department of Mathematics I
Splaiul Independent¸ei 313 060032 Bucharest, Romania
halanay@mathem.pub.ro and
“Elie Carafoli” National Institute for Aerospace Research Bd. Iuliu Maniu 220, 061126, Bucharest, Romania
iursu@aero.incas.ro
