New stabilization and tracking control laws for electrohydraulic servomechanisms

European Journal of Control 19 (2013) 65–80

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European Journal of Control journal homepage: www.elsevier.com/locate/ejcon

New stabilization and tracking control laws for electrohydraulic servomechanisms Ioan Ursu a,n, Adrian Toader a,1, Andrei Halanay b,2, Silvia Balea c,2 a

Elie Carafoli National Institute for Aerospace Research  INCAS, Systems Department, Bd. Iuliu Maniu 220, Bucharest 061126, Romania University Politehnica of Bucharest, Department of Mathematics 1, Splaiul Independentei 313, Bucharest 060042, Romania c University Politehnica of Bucharest, Department of Mathematics 2, Splaiul Independentei 313, Bucharest 060042, Romania b

a r t i c l e i n f o

abstract

Article history: Received 28 January 2012 Accepted 6 July 2012 Available online 6 March 2013 Recommended by M. Xiao / A.J. van der Schaft

Some nonlinear control laws for a fifth order mathematical model, representative for an electrohydraulic servomechanism (EHS), are presented in the paper. Intrinsically, the EHS mathematical model has several shortcomings: critical case for stability, relative degree defect, and switching type nonsmooth nonlinearity. First, the control synthesis is approached, in the framework of the so-called Malkin canonical form for a critical case in the stability theory, from the perspective of the two paradigms: the regulator, or stabilization problem, and the tracking problem. In the first part of the paper, the stabilization problem is solved and a stabilizing control law, of geometric type, is designed and then illustrated by numerical simulations. Further on, the solution of the stabilizing control is extended as a geometric control for the EHS tracking problem, but given the extreme difficulty of the problem, the proposed solution works only as a conjecture, well confirmed by numerical simulations. In this context, the importance of the electrohydraulic servovalve dynamic response, defined by the time constant, to ensure a reasonable parametric robustness of the control law, has been established. Leaving apart the geometric control approach, the EHS tracking problem was finally solved by appealing to the backstepping synthesis, also validated by numerical simulations. & 2013 European Control Association. Published by Elsevier Ltd. All rights reserved.

1. Introduction Electrohydraulic servomechanisms (EHSs) are the right choice for a variety of areas: civil engineering, machine tools, mobile equipment and robots, radar antenna, land vehicles, naval and aerospace systems. EHSs do not only allow the generation of large forces, but, thanks to modern control technology and sensors, are also capable of assuming important control tasks as, for example, highly precise positioning of heavy loads. The problem is that the control of an EHS is far from being trivial, due to such factors as the nonlinear hydraulic dynamics, the large uncertainties concerning the parameter variation and the switching and discontinuous nonlinearities due to control input saturation and directional change of valve ports opening. The basic EHS, servovalve controlled  classical system [4,15,36]  is a combination between an electrohydraulic servovalve (EHSV) and a hydrocylinder (Fig. 1).

n

Corresponding author. Tel.: þ40 21 434 00 83. E-mail addresses: [email protected] (I. Ursu), [email protected] (A. Toader), [email protected] (A. Halanay), [email protected] (S. Balea). URL: http://www.incas.ro (I. Ursu). 1 Tel.: þ40 21 434 00 83. 2 Tel.: þ40 21 402 94 89.

The modern approach in the synthesis of control laws for EHSs viewed as tracking systems [1] considers the development of strategies directly applicable to large classes of nonlinear models. So, in the notable work [62], a technique of adaptive robust control synthesis based on the discontinuous projection method developed in [61] is applied to an EHS mathematical model. Thereby, ‘‘the technical conflicts between the deterministic robust control design and the adaptive control design’’ for a class of nonlinear systems in the presence of uncertainties and nonsmooth model nonlinearities are solved. The resulting controller, not so simple, belongs to the general class of the synthesis via Control Lyapunov Functions (CLFs) and includes a constituent adaptation law. A closely related adaptive technique to account the parameter variations is that of the paper [9], this time based on the Lyapunov–Filippov apparatus of the differential inclusions in the class of the nonsmooth systems. Another way to tackle the EHS tracking problem is that of the input–output linearization technique, represented by the geometric control [23]. In the paper [16], such a technique provides a nonlinear compensator which is compared with a ‘‘standard linear multi-sensor controller’’. The EHS nonlinearities are handled so that, in a simplifying choice, the first order zero dynamics could be derived, but a strict proof is not offered. A third trenchant way of treating the problem is that of the artificial intelligence tools. Neural control and fuzzy logic techniques were

0947-3580/$ - see front matter & 2013 European Control Association. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ejcon.2012.07.001

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I. Ursu et al. / European Journal of Control 19 (2013) 65–80

x

C

u+

ps

TM

EHSV

x5

– T

S

fv +

x3

x4

m

k

L

HC

x1

Fig. 1. Block diagram of the servovalve controlled EHS. HC, hydraulic cylinder with piston; L, load; C, controller; T, transducer; TM, torque motor; EHSV, electrohydraulic servovalve.

developed in [26,46–48,51,55], and successfully implemented in [54,56], with the advantage that these strategies are rather independent of the EHS mathematical model, thus achieving certain robustness, reducing design complexity and acquiring antichattering properties. Remark also that a fourth EHS synthesis paradigm has recently emerged, the energy based approach [31]. The controller design considers the hydraulic oil to be isentropic. A proof is made concerning the structure of the control law which guarantees asymptotic tracking properties. The EHS equilibria stabilization is a problem less present in literature. In [35], successive Galerkin approximations of the solution of the Hamilton–Jacobi–Isaacs equation associated to H1 optimal stabilization control problem are used. The obtained controller, of stabilizing type, but difficult to implement given its complexity, is found to be robust in the presence of parametric variations. In [39], the feedback stabilization is developed in the framework of the Jurdjevic–Quinn approach [27], using an energy function as CLF candidate and Barbashin–Krasovskii–La Salle invariance principle [17]. To summarize this kind of approaches, as already remarked by several authors, just as the existence of a Lyapunov function is necessary and sufficient for the stability of a solution to a system without inputs, the existence of a CLF is necessary and sufficient for the stabilizability of a solution of a system with a control input [12,5]. Our interest was focused in a series of recent papers [2,3,18–22] on the EHS stability of equilibria problem. This problem was addressed with traditional mathematical tools, namely the Lyapunov–Malkin apparatus for critical cases in the stability theory [33], geometric methods in control synthesis [23] and switched systems theory [32]. The series of papers started in [18] with the superseding of a modeling hypothesis, usually assumed in classical textbooks [4,15] x3 þ x4 ¼ ps

the mathematical model has been left pending: the system might be unstable even if its components are stable [34]. Indeed, the mathematical model reveals a switching nonlinearity due to directional changes in the spool valve ports opening [36,47] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q L ¼ cd wx5 ðps pL signðx5 ÞÞ=r ð1:2Þ Q L is the load flow through hydraulic cylinder HC, a function of two variables: x5 , the EHSV spool valve displacement, and pL , the load pressure of the hydraulic cylinder. The other parameters in the equation are cd , the discharge coefficient; w, the valve port’s width; r, the hydraulic oil density. Recently, in [21], the problem of the stabilization of equilibria for the mathematical model obtained starting with Eq. (1.2) was addressed in the framework of the theory of switched systems. The approach combined both the apparatus of Lyapunov–Malkin stability theory in critical cases and CLF tool for the two linear systems corresponding to positive and respectively negative valve displacements x5 : However, the control laws were considered only as generic state feedback laws fulfilling some specific conditions. A further methodological progress marks the work [2], where the approach in [21] has been completed appealing to the apparatus of geometric control [23]. The CLF yielded by the common asymptotically stable linear system ensures the uniform simple stability of the zero solution in the switched system. A first concern of this paper is to exploit the approaches mentioned above in terms of formulating a nonstandard, but substantial problem of EHS control synthesis that of stabilizing equilibria in the presence of initial conditions perturbations. The EHSs are in fact position tracking systems. With the notation of Fig. 1, for such a system a qualitative statement of the standard tracking (or servomechanism) problem [1] is the following: 1.1. Tracking problem

ð1:1Þ

where x3 , x4 are the pressures in the hydraulic cylinder chambers and ps is the supply pressure (Fig. 1). In fact, Eq. (1.1) is not a direct consequence of the physical laws and mathematical inference on these. Giving up this equation allows us to consider a more realistic EHS model, with the consequence of taking into consideration two state variables x3 , x4 instead of pL : ¼ x3 x4 , the pressure drop across the load. So, in the seminal paper [18], a four dimensional mathematical model of the servovalve controlled EHS is developed, ignoring the valve dynamics. The stability of its equilibria is analyzed using a theorem of Lyapunov and Malkin to handle the critical case due to the presence of zero in the spectrum of the matrix of the linear part around equilibria. However, a problem related to the intrinsic switching structure of

Suppose the system’s regulated output y : ¼ x1 ðtÞ is required to track some desired output, the reference signal yd : ¼ x1d ðtÞ. Provide a state feedback control law uðtÞ : ¼ uðxðtÞÞ that guarantees the asymptotic convergence: limt-1 ðx1d ðtÞx1 ðtÞÞ ¼ 0. The stabilization problem, in which the desired outputs are the equilibria points of the system, is defined as a special case of the tracking problem. 1.2. Stabilization problem Suppose that the initial condition of the system’s state vector is different from a certain equilibrium point. Provide a state feedback control law uðtÞ : ¼ uðxðtÞÞ that brings the system solution to

I. Ursu et al. / European Journal of Control 19 (2013) 65–80

that equilibrium point. More specifically, the regulated output x1 ðtÞ is required to approach the desired constant value x1d , the first component of some equilibrium point: limt-1 ðx1d x1 ðtÞÞ ¼ 0. An eloquent example of the practical interest of this last problem is the altitude-hold autopilot synthesis [41], involving an EHS, where the target is the maintenance of the desired altitude of the aircraft, so allowing the pilot to perform other more important tasks. An earlier approach of the closely related EHS equilibria stability problem has been proposed in the classical framework of the absolute stability theory, as stated by Aizerman, Lurie, Lefschetz and Popov [45]. Two sufficient conditions for the absolute stability of an EHS nonlinear mathematical model are provided in the quoted paper, one of them representing an extended stochastic Popov–Morozan criterion [37]. Certainly, the two problems are perennial research branches in the theory of automatic control ([1]; see also [44,47] for Davison’s work and of coworkers included) and many powerful solutions were offered, especially in linear framework. A first interest of the present work is that for one of the most popular mathematical models of automation  the (electro)hydraulic servomechanism (remember the Norbert Wiener’s words ‘‘the present age is the age of servomechanisms’’)  the stabilization problem described above is solved in terms of a nonlinear geometric switching type control, and the validating numerical simulations are for the first time performed. After the general presentation of the EHS mathematical model in Section 1, both stabilizing control synthesis and the numerical simulations make up the subject to Section 3. A second concern of the paper is the treatment of the tracking problem, presented in detail in Sections 4 and 5. In Section 4, the solution of stabilizing control is extended as a geometric control for the EHS tracking problem, but given the extreme difficulty of the problem, the proposed solution works only as a conjecture, well confirmed by numerical simulations. In this context, the importance of the servovalve dynamic response, defined by the time constant, to ensure a reasonable parametric robustness of the control law, has been confirmed. A comparison with a proportional control law attests the more pronounced robustness of the geometric control. Leaving apart the geometric control approach, the EHS tracking problem was finally solved in Section 5 by appealing to the backstepping synthesis. The obtained control law guarantees asymptotic stability for the position tracking error in the case of step inputs. Also, bounded error tracking is obtained in the case of the sinusoidal inputs. Numerical simulations validate the proposed control. Some concluding remarks end the paper.

equivalent inertial load of primary control surface reduced to the actuator rod; f v , the combined coefficient of the damping and viscous friction forces on the load and the cylinder rod; k, an equivalent aerodynamic elastic force coefficient; S, the effective area of the piston; V 0 , the cylinder semivolume; B, the bulk modulus of hydraulic oil; t, the servovalve time constant; kv , a proportionality coefficient relating the input voltage to servovalve to valve displacement. Therefore, besides 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 [14]. Remark 2.1. EHS nonlinear mathematical model (2.1) can be expanded by adding additional states to servovalve model, or by considering the dry friction load (LuGre model [22]), or can be customized to the case of single-rod [25], etc. References to this subject include, for example, many chapters on calculating the equivalent coefficient f v , incorporating, among others, the dry friction [15]. The purpose of this paper is not to approach an exhaustive mathematical model (in fact, this does not exist!). We emphasize only that the model (2.1) follows a principle of parsimony, which means that is simultaneously simple and essential, thus ensuring the consistence and usefulness of the control synthesis that will be done next. In fact, it was shown that in terms of artificial intelligence paradigm applied to EHS synthesis has no great importance, for example, the presence or absence of a component of LuGre dry friction [46,48,51,54,56]. Instead, the presence of such component in the conventional synthesis of control considered in this paper can pose serious problems of adequacy between the mathematical theory and the mathematical model [50,57]. In order to apply the stability analysis and control synthesis techniques, a constitutive hypothesis is assumed (see Fig. 1) 0 o xi o ps ,

x_ 1 ¼ x2 ;

x_ 4 ¼ 

x_ 2 ¼ ðkx1 f v x2 þ Sx3 Sx4 Þ=m, x_ 3 ¼

x_ 1 ¼ x2 ; x_ 2 ¼ ðkx1 f v x2 þSx3 Sx4 Þ=m x_ 3 ¼

x_ 4 ¼

B V 0 þ Sx1

B V 0 Sx1

   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9ps ð1 þ sgnðx5 ÞÞ2x3 9=2Sx2 C9x5 9sgn ps ð1 þ sgnðx5 ÞÞ2x3

   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C9x5 9sgn ps ð1sgnðx5 ÞÞ2x4 9ps ð1sgnðx5 ÞÞ2x4 9=2 þ Sx2

pffiffiffiffiffiffiffiffiffi x_ 5 ¼ x5 =t þ kv u=t, C : ¼ cd w 2=r

pffiffiffiffiffi B ðCx5 x4 Sx2 Þ, x_ 5 ¼ x5 =t þkv u1 =t V 0 Sx1

x_ 2 ¼ ðkx1 f v x2 þ Sx3 Sx4 Þ=m, x_ 3 ¼

ð2:2Þ

pffiffiffiffiffi B ðCx5 x3 Sx2 Þ V 0 þ Sx1

pffiffiffiffiffiffiffiffiffiffiffiffiffi B ðCx5 ps x4 Sx2 Þ, x_ 5 ¼ x5 =t þkv u2 =t V 0 Sx1

ð2:3Þ

The systems (2.2) and (2.3) will be regarded as the two components of a switched system with state variables in the set: G ¼ fx : ¼ ðx1 ,x2 ,x3 ,x4 ,x5 Þ, xi A ð0, ps Þ, i ¼ 3,4, and 9x1 9 oV 0 =Sg ð2:4Þ The equilibrium points x^ ¼ ðx^ 1 , x^ 2 , x^ 3 , x^ 4 , x^ 5 Þare defined as a two-parameters ðx0 , pÞset x^ 1 ¼ x0 ,

x^ 2 ¼ 0,

x^ 3 ¼

kx0 þ ps p, S

x^ 4 ¼ ps p,

x^ 5 ¼ 0

ð2:5Þ

where x0 marks constant reference inputs to the system, 9x0 9 oV 0 =S and p A ð0, 1Þ is chosen such that

ð2:1Þ

In addition to the previously defined notations, x1 is the load displacement, x2 is the load velocity, and u is the control variable, an input voltage. The involved constants are (Fig. 1): m, the

pffiffiffiffiffiffiffiffiffiffiffiffiffi B ðCx5 ps x3 Sx2 Þ V 0 þ Sx1

and for x5 o0, respectively

x_ 4 ¼ 

A detailed presentation of the problems connected to the mathematical modeling as a first step towards the analysis and synthesis of the valve controlled EHS is available in the literature [4,6,7,11,15,18,20,24,25,30,36,40,42,43,47,60]. In this paper, we will consider as representative the following basic five dimensional mathematical model of a double-rod type EHS:

ð2:10 Þ

i ¼ 3,4

Among other things (for example, sudden reversal of load acceleration, instability of motion, ram working as a pump), this hypothesis implies that basic conditions for the prevention of cavitation are integrated in the model [7,18]. The EHS system is now split into two subsystems: for x5 Z0

x_ 1 ¼ x2 ;

2. Getting a switching type EHS mathematical model

67

ps ð1pÞ

kx0 4 0, S

ps p þ

kx0 40 S

ð2:6Þ

^ ¼ 0, then (2.5) defines an equilibrium If u1 ðx^ Þ ¼ 0 and u2 ðxÞ point for both (2.2) and (2.3), so for the switched system. EHS is a

68

I. Ursu et al. / European Journal of Control 19 (2013) 65–80

position control system, thus the synthesis of controls u1 and u2 is aimed to ensure stability of all x^ in (2.5) and a good asymptotic behavior of the first component:

nonlinear feedback and the classical Lyapunov–Malkin approach in a new configuration, that of the switched systems. Let us introduce the transform of coordinates

lim ½x1 ðtÞx0  ¼ 0

z ¼ FðlÞ ðxÞ, l ¼ 1, 2

ð2:7Þ

t-1

The system (2.1) will be now taken as defined for x5 A R. The two components of the system will be written as ð1Þ ð2Þ x_ ¼ f ðxÞ þ gðxÞu1 , x_ ¼ f ðxÞ þgðxÞu2

ð1Þ

ðxÞ ¼

x2 ,

T pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi kx1 f v x2 Sx3 Sx4 BCx5 ps x3 BSx2 BCx5 x4 þBSx2 1  þ  , , ,  x5 m m m m V 0 þSx1 V 0 Sx1 t

f

ð2Þ

ðxÞ ¼

x2 ,

T pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi kx1 f v x2 Sx3 Sx4 BCx5 x3 BSx2 BCx5 ps x4 þBSx2 1  þ  , , ,  x5 m m m m V 0 þ Sx1 V 0 Sx1 t



0,

0,

0,

0,

kv =t

ð1Þ ð2Þ ðlÞ ^ FðlÞ 5 ¼ x3 x3 d1 ðx1 x0 Þd1 x2 þ d3 ½kx1 =m þ f v x2 =mSðx3 x4 Þ=m

ð3:7Þ

ð2:8Þ

f

gðxÞ ¼

ðlÞ ðlÞ ðlÞ 2 3 FðlÞ 1 ¼ hðxÞ, F2 ¼ Lf ðlÞ ðxÞ, F3 ¼ Lf ðlÞ ðxÞ, F4 ¼ Lf ðlÞ ðxÞ

T

ð2:80 Þ

Relying on (2.5), it is natural to consider that the system output is the variable: y ¼ hðxÞ ¼ x1 x0

With from (3.7), from (3.2) and (3.3), and having available the tuning parameters ci , i ¼ 1, . . .,4, from (3.4) is clear that the following relation can be used as candidate control in the context of above mentioned: ul ðxÞ ¼

ðlÞ

L4f ð1Þ h

1

L4f ð2Þ h ¼

ðLg Lkf ðlÞ hÞðxÞ ¼ 0,

^ 8x A VðxÞ,

k ¼ 0, 1,2 and ðLg L3f ðlÞ hÞðxÞ a0

ð3:1Þ

ðLg L3f ð1Þ hÞðxÞ ¼

 pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi  ps x3 kv BCS x4 þ a 0, 8x A G V 0 þSx1 V 0 Sx1 mt

ð3:2Þ

ðLg L3f ð2Þ hÞðxÞ ¼

 pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ps x4 kv BCS x3 þ a0, 8x A G mt V 0 þSx1 V 0 Sx1

ð3:3Þ

The meaning of the concept is clear: the relative degree is the smallest order of the output derivative that explicitly depends on input: 4 3 & y ¼ Lf ðlÞ hðxÞ þLg Lf ðlÞ u,

l ¼ 1,2

ð3:4Þ

Next we point out a critical case for stability for the systems (2.8) and (2.80 ). Indeed, after the translation into zero of the equilibria: X 1 ¼ x1 x^ 1 ,

X 2 ¼ x2 ,

X 3 ¼ x3 x^ 3 ,

X 4 ¼ x4 x^ 4 ,

X 5 ¼ x5

ð3:5Þ

and denoting u~ 1 ðXÞ ¼ kv u1 ðX þ x^ Þ=t, u~ 2 ðXÞ ¼ kv u2 ðX þ x^ Þ=t, X ¼ ðX 1 ,X 2 ,X 3 ,X 4 ,X 5 Þ ð3:6Þ one can easily verify that the Jacobian matrix in zero has zero as a simple root of the characteristic polynomial. So the EHS system has three difficulties to cope with: a switched type mathematical model, a critical case for stability and defect of relative degree. The control synthesis will combine the geometric theory of

pffiffiffiffiffiffiffiffiffiffiffiffiffi  ps x3 ðV 0 þ Sx1 Þ2

þ

!

pffiffiffiffiffi x4 ðV 0 Sx1 Þ2

!  2 fv k 2BS2 V 0 k fv S  x ðx x þ   þ x Þ  1 2 3 4 m m m m2 m mðV 20 S2 x2 Þ 1   pffiffiffiffiffiffiffiffiffiffiffiffiffi BCx5 ps x3 BSx2 Sf BCSx5 þ  2v  pffiffiffiffiffiffiffiffiffiffiffiffiffi 2mðV 0 þ Sx1 Þ ps x3 V 0 þ Sx1 m   pffiffiffiffiffi Sf v BCSx5 BCx5 x4 þBSx2 þ þ pffiffiffiffiffi V 0 Sx1 m2 2mðV 0 Sx1 Þ x4  pffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffi  ps x3 BCS 1 x4 þ þ  x5 m V 0 þSx1 V 0 Sx1 t

x belongs to an open set U  R , f , g : U-R , l ¼ 1, 2 and h : U-R are smooth functions. Usually in modeling of EHS, the ðlÞ functions f , g are required to be at least locally Lipschitz, and in this situation the theorem of existence and uniqueness of solutions can be applied to each component. A rule for switching between the components is given by the conditions x5 Z 0, x5 o 0.

We will consider the stabilizing control synthesis for the systems (2.8) and (2.9) in the framework of Isidori’s geometric paradigm [23]. First, the relative degree r of the systems in (2.8) with the output (2.9) must be calculated. Denote by Vðx^ Þ a neighborhood of x^ and Lf h the Lie derivative of h along P f , Lf h ¼ nj¼ 1 ð@h=@xj Þf j , L2f h ¼ Lf ðLf ðhÞÞ,. . . . One can easily show that the relative degree associated to the output (2.9) is r ¼ 4:

i ðL4f ðlÞ hÞðxÞ þ c1 F1ðlÞ ðxÞ þ c2 F2ðlÞ ðxÞ þ c3 F3ðlÞ ðxÞ þ c4 F4ðlÞ ðxÞ , l ¼ 1,2

4BS4 V 0 x1 x22 f k BCS2 x2 x5 ¼ v 2 x2  þ 2 2 2 2 m m mðV 0 S x Þ

5

3. Construction of a stabilizing control for an EHS system in Malkin canonical form. Numerical simulations

h

1 ðLg L3f ðlÞ hÞðxÞ

ð2:9Þ 5

Lg L3f ðlÞ h

FiðlÞ

4BS4 V 0 x1 x22 f vk BCS2 x2 x5 x2  þ 2 2 2 2 m m2 mðV 0 S x1 Þ

pffiffiffiffiffi  x3

ðV 0 þ Sx1 Þ2

þ

pffiffiffiffiffiffiffiffiffiffiffiffiffi ! ps x4

ðV 0 Sx1 Þ2

!  k 2BS V 0 k f S  x1  v x2 þ ðx3 x4 Þ þ   2 2 m mðV 0 S x2 Þ m m m 1   pffiffiffiffiffi Sf BCSx5 BCx5 x3 BSx2 þ  2v  pffiffiffiffiffiffiffiffiffiffiffiffiffi 2mðV 0 þ Sx1 Þ ps x3 V 0 þ Sx1 m   pffiffiffiffiffiffiffiffiffiffiffiffiffi BCx5 ps x4 þ BSx2 Sf v BCSx5 þ  pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2mðV 0 þ Sx1 Þ ps x4 V 0 Sx1 m  pffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffi ps x4 BCS 1 x3 þ þ  x5 ð3:8Þ m V 0 þ Sx1 V 0 Sx1 t 2 fv m2

2

The systems (2.2) and (2.3) are now written in the form: 8 8 2 3 0 1 0 0 > > > > > > > > 6 7 < x_ ¼ Dx < x_ ¼ Dx 60 0 1 07 7 , S2 : , D¼6 S1 : 6 7 ð1Þ ð2Þ _ > > z_ 5 ¼ q ðzÞ > > 40 0 0 15 > z 5 ¼ q ðzÞ > > > : : c1 c2 c3 c4 ð3:9Þ T

where x ¼ ðz1 ,z2 ,z3 ,z4 Þ , z ¼ ðz1 ,. . .,z5 Þ ¼ ðx, z5 Þ Proposition 3.1. If we choose the key parameters ! pffiffiffiffiffiffiffiffiffiffiffiffiffi 2BS2 V 0 BS ps x^ 3 ð1Þ kþ 2 2  d1 ¼ , SR1 ð0ÞðV 0 þ Sx0 Þ V 0 þ Sx0 V 0 S x20 ! pffiffiffiffiffi 2BS2 V 0 BS x^ 3 ð2Þ kþ 2 2 d1 ¼  V 0 þ Sx0 SR2 ð0ÞðV 0 þ Sx0 Þ V 0 S x20 pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi f v ps x^ 3 f v x^ 3 ð1Þ ð2Þ ,d ¼ , d2 ¼ SR2 ð0ÞðV 0 þSx0 Þ SR1 ð0ÞðV 0 þ Sx0 Þ 2 pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi m ps x^ 3 m x^ 3 ð1Þ ð2Þ ,d ¼ d3 ¼ SR2 ð0ÞðV 0 þSx0 Þ SR1 ð0ÞðV 0 þ Sx0 Þ 3 pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x^ 3 kx0 =S x3 ps _ R1 ð0Þ : ¼ þ , V 0 Sx0 V 0 þ Sx0 pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ps x^ 3 þ kx0 =S x^ 3 þ R2 ð0Þ : ¼ V 0 Sx0 V 0 þSx0

ð3:10Þ

then qðlÞ ð0, 0, 0, 0, z5 Þ ¼ 0, l ¼ 1, 2, 8z5 and qðlÞ contains only powers of the variables Z 2.

I. Ursu et al. / European Journal of Control 19 (2013) 65–80

Proof (Sketch).. In order to compute qðlÞ , first is calculated from (3.7) the inverses CðlÞ of FðlÞ , l ¼ 1, 2. Then, the expressions of the i i ðlÞ key parameters di , i ¼ 1, 2, 3, l ¼ 1, 2 follow easily from the constraints on qðlÞ and its derivatives. & The form (3.9) obeying the conditions given in the conclusion of Proposition 3.1 is herein called the canonical Malkin form for the switched system (2.1) with a critical case for stability generated by a zero eigenvalue of the closed loop Jacobian matrix. For such systems the stability of equilibria is defined as follows [32]: Definition 3.1. The equilibrium point x^ is called uniformly stable for the switched system (2.1) if for every e 4 0, there exists de 4 0 such that if Jxð0Þx^ J o de then JxðtÞx^ J o e for every t A ½0, 1Þ and for every solution x of (2.1). Proposition 3.2. If: (a) D in (3.10) is a Hurwitz matrix (i.e., denoting by C the set of the complex numbers, sðDÞ  C  : ¼ fz A C9Rez o0g and (b) the key parameters involved in the functions qðlÞ (3.9) are chosen as in (3.10), then every equilibrium point (2.5) is uniformly stable for the switched system (2.1); moreover, if initial data are close to such an equilibrium point, then lim ½x1 ðtÞx0  ¼ 0 for a solution x ¼ ðx1 ðtÞ,. . .,x5 ðtÞÞ of (2.1). t-1

Proof (Sketch).. Since D is of Hurwitz type, using the properties of the functions qðlÞ described in Proposition 3.1, a common Lyapunov function is proved to be VðxÞ ¼ xP x, with DT P þ PD ¼ I. The stability and the asymptotic behavior lim ½x1 ðtÞx0  ¼ 0 follow; more details can be found in recent t-1 papers of the authors [2,3,21,22]. &

69

following physical constraints on the EHS were activated 9x1 9 o 3 cm, 9x2 9 o 7:75 cm=s, 1 baro xi o 200 bar,

i ¼ 3, 4, 9x5 9 r 0:1 cm, 9u9 r 10 V

ð3:13Þ In Fig. 2, the tuning parameters c1 ,c2 ,c3 ,c4 are indirectly expressed through eigðDÞ, the eigenvalues of D. Numerical simulations revealed that relatively slow eigenvalues prevent the control to saturate. On the contrary, the faster eigenvalues of the matrix D, even if stable, saturate the control u, thus producing chattering and an unacceptable and dangerous dynamic behavior, see Fig. 2(b)). It is clear now that, by carefully tuning the parameters involved in the matrix D described in (3.9), the goal of the control law (3.8), to stabilize the trivial equilibrium, is achieved; this preliminary process is usual in the control synthesis, either linear or nonlinear. Corollary 3.1. The control (3.8) is a solution of the EHS stabilization problem. 4. Numerical simulation of the geometric tracking control. Comparison with the classical proportional control In the previous section, a feedback control law (3.8) providing stabilizing conditions for the EHS system was given. We now address the problem of the asymptotic tracking of reference signals; in other words, we seek a state feedback uðxÞ that forces the regulated output: ð2:90 Þ

yðtÞ ¼ hðxðtÞÞ : ¼ x1 ðtÞ

to converge to a desired value yd ðtÞ as t-1. Define, therefore, the error eðtÞ

Remark 3.1. The existence of a common Lyapunov function for switched systems is a general well known sufficient condition of stability (see e.g. [59]). However, the two Propositions above are of particular interest, because: (1) the difficulty of the problem is amplified by the context of the critical case of stability and (2) even if quite complicate, the control calculation is performed to the final form. Furthermore, this control law is numerically simulated for the first time, as will be shown below.

eðtÞ ¼ yd ðtÞyðtÞ

Remark 3.2. The two propositions above provide together the stability of the so called zero dynamics [23] of the EHS system, described by the equations z_ 5 ¼ qðlÞ ð0, 0, 0, 0, z5 Þ, l ¼ 1, 2. As we will see in the next section, in the case of the tracking problem it is extremely difficult [28] to retrieve the result obtained previously by the same approach as in Proposition 3.1.

and similarly for l ¼2. The stabilizing control law (3.8) will be substituted by the tracking control:

ð4:1Þ

To remain in the framework of the canonical Malkin form for switched systems developed in the previous section, we consider the successive errors as new variables: e1 ¼ Fð1Þ 1 ðxÞ : ¼ yd hðxÞ ¼ yd x1 , _ _ e2 ¼ Fð1Þ 2 ðxÞ : ¼ y d Lf ð1Þ hðxÞ ¼ y d x2 ð1Þ 2 € e3 ¼ Fð1Þ yd L3f ð1Þ hðxÞ 3 ðxÞ : ¼ y d Lf ð1Þ hðxÞ, e4 ¼ F4 ðxÞ : ¼ &

ul ðxÞ ¼

ðlÞ L4f ðlÞ hðxÞ þ& yd þ c1 ðFðlÞ yd Þ 1 ðxÞyd Þ þ    þ c4 ðF3 ðxÞ&

Lg L3f ðlÞ hðxÞ

ð4:2Þ

,

l ¼ 1,2 ð4:3Þ

To illustrate the closed loop behavior of the EHS system with the control laws (3.8), consider the following design data, representing an EHS integrated in the aileron control chain of the Romanian jet fighter IAR99 [18,48–50,53]: m¼60 kg, f v ¼3000 N s/m, k¼105 N/m, S¼10  3 m2, cd ¼0.6, V0 ¼ 3  10  5 m3, ps ¼2  107 N/m2, B¼6  108 N/m2, r ¼850 kg/m3, kv ¼10  4 m/V (meaning an equivalent valve port width w¼0.85 mm and a maximal opening length of rectangular valve port x5max ¼ 1 mm at maximal valve input voltage umax ¼ 10 V) and t ¼ 1=573 s. The simulations were performed using the package Matlab with Simulink. A representative plot of the variables evolution is shown in Fig. 2, for initial conditions x10 ¼ 1:5 cm, x20 ¼ 1 cm=s, x30 ¼ ðkx0 =S þ ps pÞ bar, x40 ¼ ps p bar, x50 ¼ 0 cm

ð3:11Þ and with system’s parameters ðx0 ,pÞ ¼ ð1 cm,0:5Þ

ð3:12Þ

So, x1d ¼ x0 . The precise limits of the attraction basin of the equilibria are beyond the scope of this paper. To obtain a realistic portrait, during all the simulations presented in the paper, the

Indeed, after the substitution of (4.3) in (3.4), we obtain that yd c4& e    c1 e, & y ¼ L4f ðlÞ hðxÞ þ Lg L3f ðlÞ hðxÞu ¼ &

l ¼ 1, 2

ð4:4Þ

i.e. € 2 ec _ 1e ¼ 0 & ec4& ec3 ec

ð4:5Þ

The form of Eq. (4.5) is the same as in the stabilizing case (3.9) and its characteristic roots could be, as a first step, properly assigned to reach tracking performances for the EHS system. As in the stabilization problem, in the framework of the geometric control, a second condition related to the stability of zero dynamics must be fulfilled. Indeed, with the notations _ € xd ðtÞ ¼ ½yd ðtÞ, . . ., & yd ðtÞT , eðtÞ ¼ ½eðtÞ, eðtÞ, eðtÞ, & eðtÞT

ð4:6Þ

it can be seen that a system analogous to (3.9) is rewritten in the form: e_ ¼ De, z_ 5 ¼ qðlÞ ðxd ðtÞeðtÞ, z5 Þ,

l ¼ 1, 2

ð4:7Þ

The last equations in (4.7), referred to as internal dynamics [21], appear now as a time varying nonlinear system. For example, if the

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I. Ursu et al. / European Journal of Control 19 (2013) 65–80

Fig. 2. Time histories of the switched system (2.2)–(2.3), with initial condition (3.12) and the control given in (3.8): (a) stable evolution with eigðDÞ ¼ (  188 71.5896i;  618:9;  5); (b) violent chattering, with eigðDÞ ¼ (  9407 7948i;  3094;  25).

desired output yd ¼ sinðotÞ is chosen, then

xd ðtÞ ¼ ð sinðotÞ o cosðotÞ o2 sinðotÞ o3 cosðotÞ Þ

ð4:8Þ

The statement of Remark 3.2 becomes now clear: to repeat the procedure followed in Proposition 3.1 would mean an approach difficult, boring, perhaps unnecessarily. Nevertheless, the numerical simulations of the closed loop system (2.2), (2.3), (4.3) were very encouraging, and even showed a clear robustness of the geometric tracking control (4.3) to parametric uncertainties. First, in Fig. 3, the tracking behavior of EHS system is shown for nominal values of the parameters (see Table 1), and with suitable eigenvalues of the

matrix D, chosen based on the expertise gained in Section 3. Namely, the eigenvalues eigðDÞ ¼ ( 32972781.8i;1083:1; 8:8) are situated ‘‘between’’ slow eigenvalues in Fig. 2(a) and too fast eigenvalues in Fig. 2(b). Concerning the transitory regime properties, we remark that, naturally, a lower amplitude of desired output yd avoids insecure worsening of initial time history: compare Fig. 3(a) and (b). Fig. 4 confirms a good tracking of the step type reference (Fig. 4(a)) and also of the sinusoidal reference (Fig. 4(b)). Thus, we can conclude that the tracking accuracy of the desired output yd by the output y is notable. Next, an evaluation of the robustness of the geometric control law (4.3), in the context of parameter uncertainties is performed. S, the

I. Ursu et al. / European Journal of Control 19 (2013) 65–80

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10

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100 0

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t[s]

t[s]

0.05

10

0

0 -10 0

0.5

1 t[s]

1.5

2

0

0.5

1 t[s]

Fig. 3. Time histories of the switched system (2.2)–(2.3), with tracking control (4.3). Stable evolution based on eigðDÞ ¼ ( 329 7 2781.8i;  1083:1; 8:8); (a) yd ¼ sinð2ptÞ cm; (b) yd ¼ 0:2  sinð2ptÞ cm.

Table 1 Robustness evaluation of tracking control (4.3).

t

EHS parameters Nominal values

m 60 (kg)

fv 3000 (N s/m)

k 105 (N/m)

B 6  108 (N/m2)

C 2.47  10  5 (m5/2 kg  1/2)

kv 10  4 (m/V)

1/573 (s)

1st trial 2nd trial 3rd trial 4th trial 5th trial 6th trial 7th trial 8th trial 9th trial 10th trial

 0.94  3.41  33.40  36.02 16.72  26.41  1.48 9.51  46.85 14.22

8.18 2.85 17.86 14.28 29.77 35.38  25.27  25.23 10.70  6.39

4.37 26.67 40.59  10.00  30.06  20.13 15.36  0.49 22.39  9.44

 4.67 1.48  17.29 17.25  0.49 13.22 12.69  1.21 18.27  7.40

18.14  2.39 21.45 20.39  3.92 5.48 42.31 0.00 14.45  36.88

 14.71  20.81 31.35 17.80  40.10  23.05 14.78  7.95 1.01  5.85

54.58 7.36  39.84 32.26 6.43  54.27  16.09 27.38 16.93 2.96

Comment

20 GT GT  20 GT GT GT GT  30 GT

72

I. Ursu et al. / European Journal of Control 19 (2013) 65–80

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4

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2

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1.5

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0

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2

Fig. 4. Time histories of the switched system (2.2)–(2.3), same conditions as in Fig. 3; tracking of (a) yd ðtÞ ¼ 0:1  ð1et=0:03 Þ cm; (b)yd ¼ ð0:2  sinð2ptÞ1Þ cm.

effective area of the piston, and V 0 , the cylinder semivolume, has been fixed at nominal values, because they are precisely known from design. Table 1 shows the other EHS parameters m,f v , k, B, C, kv , t, over which hovers uncertainty, as they represent mathematical models based on simplifying assumptions, linearization, etc.; consequently, for simulations were generated as ten realizations (trials) of a discrete Gaussian white noise seven-dimensional vector process, with zero mean and root mean square (r.m.s.) ¼0.25. In the rows of table, parameter values are expressed as a percentage deviation from their nominal values. In the ninth column, the comment ‘‘GT’’ indicates a ‘‘good tracking’’  such as that manifested in Figs. 3 and 4  for the specified random distribution of parametric uncertainties, or, if numbers, the acceptable servovalve time constant t percent deviation

from nominal value to ensure good tracking performance. The idea is clearly expressed by Fig. 5: the alternative to GT shown in Fig. 5(b) is highlighted by a strong chattering, as in Fig. 5(a), what is to be avoided as dangerous in operation. The efficiency of the servovalve time constant in the tracking control is particularly emphasized: as expected, a servovalve with fast, i.e., small, time constant is beneficial for the control synthesis, given inherent parameters uncertainty; e.g., increased by only 20% from the nominal value (Fig. 5(b)), versus 54.58%, Fig. 5(a). Table 2 summarizes the conclusions of the analysis. By numerical simulations it was established that the control law (4.3) is clearly robust to wide parametric deviations, up to about 15% compared to the nominal values, while the usual deviations of

I. Ursu et al. / European Journal of Control 19 (2013) 65–80

10 x2 [cm/s]

y, yd [cm]

0.2 0 -0.2

0

0.5

1 t[s]

1.5

100

0

0.5

1 t[s]

1.5

50

2

u[V]

x5 [cm]

0

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2 x2 [cm/s]

y, yd [cm]

0.2 0

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2

110

110 x4 [bar]

x3 [bar]

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0

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x5 [cm]

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50

73

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Fig. 5. Robustness of the tracking control (4.3),yd ¼ 0:2  sinð2ptÞ cm (a) 1st trial in Table 1, chattering; (b) GT with random deviation of t reduced at þ 20% w.r.t. nominal value.

parameters estimation concerning the real data in EHS design can be considered to be about 5–10%. The relevance of these results is strengthened making a comparison with those given by a proportional law: eðtÞ ¼ yd ðtÞyðtÞ, yðtÞ ¼ kp x1 , kp ¼ 55 V=cm

Table 2 Robustness degree of tracking control (4.3) as depending on parameters uncertainty Trials number

r.m.s.

Comment

30 30 30

0.25 0.20 0.15

GT for 27 trials GT for 28 trials GT for all trials

0

ð4:1 Þ

The gain value kp was simply determined by a trial and error procedure, so as to be found, through simulation, the experimental time constant value of about 0.03 s [65]; see Fig. 6(a). For compliance, it is also depicted the response to sinusoidal reference (Fig. 6(b)). The inacceptable dynamics in the presence of a given random parameters configuration could not be corrected

on the basis of servovalve performance (Fig. 6(c) and (d)). Overall, the proportional control law proved to be much less robust than the geometric nonlinear control. For example, for 30 trials with

I. Ursu et al. / European Journal of Control 19 (2013) 65–80

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Fig. 6. Proportional tracking control: (a) parameters nominal values, step reference; (b) response to sinusoidal reference; (c) random combination of parameters, giving chattering; (d) decreased t, worse dynamics.

zero mean and root mean square (r.m.s.) ¼0.25 random deviations, only 21 have confirmed a satisfactory dynamic tracking (GT). It is remarkable that, despite the complexity, the nonlinear control law is proven to be more robust than a simple, schematic control, as the proportional control law.

e4 : ¼ Q L Q Ld , Q Ld : ¼ k3 e3 þSx2 ðg 1 þ g 2 Þ þ p_ Ld , kQ_ Lu : ¼ ðg 1 g 3 þ g 2 g 4 ÞCkv =t

This section presents a solution of the EHS tracking problem based on backstepping technique [25] that has been addressed in other works [48–50,52]. The novelty is that the present solution approaches the switching type EHS structure (2.1); the solutions to date have been partial since only considered the two components (2.2) and (2.3) separately, without ensuring the overall stability. Proposition 5.1. Let k3 4 0, k4 4 0, r3 4 0, r4 4 0be weighting parameters and let the desired output x1d be expressed as an usual sinusoidal signals ð5:1Þ

or as a response of a first order system to step inputs x1s x1d ðtÞ ¼ x1s ð1et=t1r Þ

u ¼ ðr4 Q_ Ld r3 e3 r4 Q_ Lx k4 e4 Þ=ðr4 kQ_ Lu Þ, pL ¼ : x3 x4 , e3 : ¼ pL pLd , pLd : ¼ kx1d =S

5. Solving of the EHS tracking problem by backstepping control

x1d ¼ x1a sinðotÞ

where t 1r stands for the associated desired time constant. Suppose also that the physical constraints (3.14) are valid. Then the control law u given by

ð5:2Þ

Q L : ¼ ðg 1 ðx1 Þg 3 ðx3 , x5 Þ þ g 2 ðx1 Þg4 ðx4 , x5 ÞÞCx5 , g 1 ðx1 Þ : ¼

B B , g ðx1 Þ : ¼ V 0 þ Sx1 2 V 0 Sx1

g 3 ðx3 , x5 Þ : ¼

( pffiffiffiffiffiffiffiffiffiffiffiffiffi ps x3 , for x5 Z 0 pffiffiffiffiffi , x3 , for x5 o 0

g 4 ðx4 , x5 Þ : ¼

( pffiffiffiffiffi x4 , for x5 Z 0 pffiffiffiffiffiffiffiffiffiffiffiffiffi ps x3 , for x5 o 0

    @g 1 @g @g @g g 3 þ 2 g 4 Cx2 x5 þ g 1 3 x_ 3 þ g 2 4 x_ 4 Cx5 ðg 1 g 3 þg 2 g 4 ÞCx5 =t Q_ Lx : ¼ @x1 @x1 @x3 @x4

ð5:3Þ when applied to system (2.1), guarantees asymptotic stability for the position tracking error e1 : ¼ x1 x1d ; that is, lim e1 ðtÞ ¼ 0 , in the t-1 case of step type input (5.2); also, bounded position tracking error is achieved for the sinusoidal input (5.1).

I. Ursu et al. / European Journal of Control 19 (2013) 65–80

Proof. Consider the first two equations in (2.1) reduced to one second order equation: mx€ 1 þf v x_ 1 þ kx1 ¼ SpL

ð5:4Þ

This equation and the last equation in the system (2.1) are linear and internal stable, thus a special care to stabilize the states x1, x2 and x5 it is not necessary. & From (5.4) it is clear that pL can be thought as a pressure defined control input of the first two states x1 and x2 . Therefore, we start the first step of backstepping by defining the Lyapunov like function related to this variable pL which has to track a desired value pLd 1 V 3 ¼ r3 e23 2

ð5:5Þ

Taking into account the corresponding relations in (5.3), the derivative of V 3 along the system’s (2.1) solutions is V_ 3 ¼ r3 e3 ðx_ 3 x_ 4 p_ Ld Þ ¼ r3 e3 ½Sx2 ðg 1 ðx1 Þ þg 2 ðx1 Þ þ Q L Þp_ Ld  ¼ r3 e3 ½Sx2 ðg 1 þ g 2 Þ þ e4 þ Q Ld p_ Ld  ¼ k3 r3 e23 þ r3 e3 e4

ð5:6Þ

By doing so, a flow defined control input Q L was introduced. In order to go one step ahead, a corresponding Lyapunov like function V 4 is introduced, such that Q L tracks the desired value Q Ld : V4 ¼ V3 þ

1 r e2 2 4 4

ð5:7Þ

The function Q L is differentiable everywhere, except for the singular point x5 ¼ 0, and is continuous everywhere. The partial derivatives @g 3 =@x3 , @g 4 =@x4 of the functions g 3 , g 4 , involved in the computation of Q_ Lx , are ( pffiffiffiffiffiffiffiffiffiffiffiffiffi 1=ð2 ps x3 Þ, for x5 Z 0 @g 3 ðx3 , x5 Þ pffiffiffiffiffi ¼ 1=ð2 x3 Þ, for x5 o0 @x3 ( pffiffiffiffiffi 1=ð2 x Þ, for x5 Z0 4 @g 4 ðx4 , x5 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1=ð2 ps x4 Þ, for x5 o0 @x4

75

To continue the proof we make use of Barbalat’s lemma [38] to show that the errors e3 and e4 tend to zero as time tends to infinity. Barbalat’s Lemma. If the function f(t) is differentiable and has a finite limit lim f ðtÞ , and if f_ is uniformly continuous, then lim f_ ðtÞ ¼ 0 . t-1

t-1

It is easily seen that when t-0, we have V 4 ð0Þ 40; indeed V 4 ð0Þ ¼ V 3 ð0Þ þ

1 r e2 ð0Þ 2 4 4

ð5:11Þ

and thus e4 ð0Þ a0 at least due to p_ Ld ð0Þ a0 (see (5.1)–(5.3)) e4 ð0Þ ¼ Q L ð0ÞQ Ld ð0Þ, Q Ld ð0Þ ¼ p_ Ld ð0Þ a0

ð5:12Þ

Since V_ 4 r 0 , it follows that 0 r V 4 ðtÞ r V 4 ð0Þ, ð8Þt 4 0, hence the positive function V 4 ðtÞ is bounded and therefore e3 and e4 are bounded. The derivative V€ 4 is V€ 4 ¼ 2k3 r3 e3 ðk3 e3 þ e4 Þ2k4 r4 e4 ðr3 e3 =r4 k4 e4 =r4 Þ Consequently, V€ 4 is bounded, so V_ 4 is uniformly continuous. Since V 4 is decreasing, there exists a finite limit lim V 4 ðtÞ. Then t-1 V_ 4 -0 and, thus, e3 and e4 tend to zero. Now, let us look again at the first two equations in (5.4), which can be rewritten as follows: rffiffiffiffiffi f k S , p~ ¼ p ð5:13Þ x€ 1 þ 2hx_ 1 þ r 2 x1 ¼ p~ L , h : ¼ v , r : ¼ m L m L 2m r is the undamped angular frequency, and p~ L is now seen as a bounded function of t, because both e3 and pLd are bounded. With initial conditions x1 ð0Þ ¼ x_ 1 ð0Þ ¼ 0, the solution of (5.13) is Z 1 ht t hu x1 ðtÞ ¼ e e p~ L ðsÞsinod ðtsÞds ð5:14Þ

od

0

The condition concerning under critical damping ratio z (z o 1) is specific to all EHSs, including those used in primary flight controls actuation systems: pffiffiffiffiffiffiffi z ¼ h=r ¼ f v =ð2 mkÞ, o2d : ¼ r2 ð1z2 Þ 4 0 ð5:15Þ Introducing

0 o x3 o ps , 0 o x4 o ps Thus, the control u can still be synthesized, even in the presence of a finite jump at x5 ¼ 0 due to these derivatives (for finite time, derivatives are finite). Hence, taking the derivative of V 4 along the system’s (2.1) solutions, we have V_ 4 ¼ k3 r3 e23 þ r3 e3 e4 þ r4 e4 ðQ_ L Q_ Ld Þ     @g 1 @g @g @g g 3 þ 2 g 4 Cx2 x5 þ g 1 3 x_ 3 þ g 2 4 x_ 4 Cx5 Q_ L ¼ @x1 @x1 @x3 @x4   1 kv þ ðg 1 g 3 þ g 2 g 4 ÞC  x5 þ u ¼ Q_ Lx þ kQ_ Lu u

t

t

ð5:8Þ

ð5:9Þ

r4 , k4 are the latest weighting parameters that have to be introduced; r3 ,k3 were already involved in the construction of V 3 . Note that as soon as the values of the state variables xi , i ¼ 1, . . ., 5 and the desired output x1d , x_ 1d , x€ 1d , & x1d are measured and, respectively, known, the control u can be calculated. Consequently, V 4 is a Lyapunov function for the closed loop error system: ð5:10Þ

ð5:16Þ

let us also consider Z 1 ht t hs x~ 1d ðtÞ : ¼ e e p~ Ld ðsÞsinod ðtsÞds

ð5:140 Þ

0

Since e3 -0 when t-1, it is clear that p~ L ðtÞ-p~ Ld ðtÞ, as t-1; this means: ð8Þe 4 0, (() d ðeÞ such that for t 4 d ðeÞ we have 9p~ L ðtÞp~ Ld ðtÞ9 o e. Then, if t 4 d ðeÞ 9x1 ðtÞx~ 1d ðtÞ9 r þ

r3 e3 þ r4 ðQ_ L Q_ Ld Þ ¼ k4 e4 1 u¼ ðr Q_ r e r Q_ k e Þ r4 kQ_ Lu 4 Ld 3 3 4 Lx 4 4

e_ 3 ¼ k3 e3 þ e4 , e_ 4 ¼ r3 e3 =r4 k4 e4 =r4

S k p ðtÞ ¼ x1d m Ld m

od

Summarizing, the control low given by (5.3), with a given x1d, renders the derivative V_ 4 negative semidefinite:

V_ 4 ¼ k3 r3 e23 k4 r4 e24

p~ Ld ðtÞ : ¼

1

od

eht

Z

t

d

1

od

eht

Z

d 0

ehs 9p~ L ðtÞp~ Ld ðtÞ99sinod ðtsÞ9ds

ehs 9p~ L ðtÞp~ Ld ðtÞ99sinod ðtsÞ9ds

ð5:17Þ

The first integral in (5.17) is, obviously, bounded and, due to the exponential function, the first term will tend to zero when t-1. Since for t 4 d ðeÞ, 9p~ L ðtÞp~ Ld ðtÞ9 o e, the second term will also converge to zero: Z Z 1 ht t hs e t hðtsÞ e e 9p~ L ðtÞp~ Ld ðtÞ99sinod ðtsÞ9ds r e ds

od

d

e 1eht -0 ¼ od h

od

0

ð5:18Þ

It results that x1 ðtÞ-x~ 1d ðtÞ as t-1

ð5:19Þ

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I. Ursu et al. / European Journal of Control 19 (2013) 65–80

Let us first take x1d given in (5.2). Then, successive calculations give Z 1 ht t hu S k x1s ð1eu=tr Þsinod ðtuÞ du x~ 1d ðtÞ ¼ e e mS od 0 Z x1s keht t hu e ð1eu=tr Þsinod ðtuÞdu ¼ od m 0 Z t ehu ð1eu=tr Þsinod ðtuÞdu 0

¼

ðh1=tr Þsinod tod ðeðh1=tr Þt cosod tÞ

o

2 þ ðh1=t Þ2 r d

þ

h sinod t þ od ðeht cosod tÞ

o2d þ h2

therefore x~ 1d ðtÞ-x1s as

t-1

ð5:20Þ

Thus, from (5.19) and (5.16), a standard procedure gives x1 ðtÞ-x1s as

t-1

ð5:21Þ

In the case of the desired output x1d (5.1) only boundedness of e1 can be ensured by applying the backstepping law (5.3). To facilitate the calculation of the response of the linear system (5.13) to the sine input p~ Ld ðtÞ ¼ ðk=mÞx1d ¼ ðkx1a =mÞ sinðotÞ, we will use the machinery of the transfer function instead of the convolution integral (5.14). For this purpose, Eq. (5.13) is rewritten as kx1a x€ 1 þ 2hx_ 1 þ r 2 x1 ¼ X~ 1a sinðotÞ, X~ 1a : ¼ m

ð5:130 Þ

The system’s solution has two components: the transient response, dependent on the initial conditions, and the so called steady state response, dependent herein on the sine excitation. As the homogenous part in (5.130 ) is a linear time invariant stable equation, the transient response disappears, in other words tends to zero. Thus, the steady state response x~ 1d to the sinusoidal input x1d given in (5.1) is obtained directly from the transfer function [13]: x~ 1d ðtÞ ¼ X~ 1a 9GðioÞ9sinðot þ argGðioÞÞ, GðioÞ ¼ ðr 2 o2 þ 2hioÞ1 ð5:22Þ Therefore, the steady state response to the sinusoidal input x1d is a scaled and shifted sine wave, even of the same frequency as the input. 9GðioÞ9 and arg GðioÞ are, respectively, the amplitude ratio and phase shift: 9GðioÞ9 ¼ 1=

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðr 2 o2 Þ2 þ 4z r 2 o2 , argGðioÞ ¼ arctanð2zro=ðr 2 o2 ÞÞ

ð5:23Þ As it is well known [36], EHSs have a low pass filter behavior, and those used in primary flight controls usually have a bandwidth of about 1–3 Hz [58]. Accordingly, appropriate little values of the desired inputs frequencies o are requested [29], [48]. Let us evaluate a bound of the desired sinusoidal input tracking error as defined by the steady state response: x~ 1d ðtÞx1d ðtÞ ¼ X~ 1a 9GðioÞ9sinðot þ argGðioÞÞx1a sinðotÞ ¼ x1a

k 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinðot þargGðioÞÞx1a sinðotÞ m 2 ðr 2 o2 Þ2 þ4z r 2 o2

ð5:24Þ pffiffiffi For moderately under damped order two system, with z o1= 2, the resonance effect, with a peak of transmissibility when driven by sinusoidal input, occurs at the frequency: qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð5:25Þ or ¼ r 12z2

The amplitude and phase shift of the desired sinusoidal input ratio at resonance are, respectively, 0 qffiffiffiffiffiffiffiffiffiffiffiffi1 2 1z 1 ~ A X 1a 9Gðior Þ9 ¼ qffiffiffiffiffiffiffiffiffiffiffiffi x1a , argGðior Þ ¼ arctan@ z 2 2z 1z ð5:26Þ It is easy to determine, by analyzing the sign of the derivative d9GðioÞ9=do that the amplitude ratio function 9GðioÞ9 is an increasing function on the interval½0, or , within which is located the interested frequency range of the desired sinusoidal inputs. At the limit, when o-0, X~ 1a 9GðioÞ9-x1a . Given the value of the damping ratio, a slight gain of the amplitude x1a at the resonance qffiffiffiffiffiffiffiffiffiffiffiffi 2 frequency, namely 1=ð2z 1z Þ ffi1:034, with z ¼ 0:61 herein, is reached. Thus, under the previous considerations on the frequency range, the tracking error (5.24) will result only from phase shift: x~ 1d ðtÞx1d ðtÞ ffix1a ðsinðot þ argGðioÞÞsinðotÞÞ

ð5:27Þ

Remark now that the phase shift jðoÞ : ¼ arg GðioÞ is a decreasing function of o. By analyzing the time-derivative of the function in brackets in (5.27) d ðsinðot þ jðoÞÞsinðotÞÞ dt

ð5:28Þ

it results that the solutions ot ¼ kpj=2, k integer, maximize the function in brackets and thus offer the following bound evaluation for the tracking error 9x~ 1d x1d 9 o M ¼ 2x1a sin j=2 , j ¼ arctanð2zr o=ðr 2 o2 ÞÞ ð5:29Þ Corroborating (5.29) and (5.16), the last assertion in Proposition 5.1 is verified. This consideration ends the proof. & With the data reported in Section 3, r ffi 40:82 rad=s ffi 6:5 Hz, z ffi0:61, od ffi 0:79r, or ¼ 20:64 rad=s ¼ 3:29 Hz . A value of the phase shift j, of about 0:9 rad, is reached at resonance frequency, meaning a bound M ¼ 0:86x1a ; at 1 Hz, an acceptable bound M ¼ 0:19x1a is evaluated. The function of the control (5.3) synthesized by backstepping is to have the EHS track the specified desired outputs x1d by the output x1 . Figs. 7 and 8 reveal a good tracking of sine (5.1) and, respectively, filtered step (5.2) desired outputs. The values of the tuning parameters are: k1 ¼400 s  1; k2 ¼4 s; k3 ¼400 cm6/ (daN2  s); r1 ¼400; r2 ¼0.033 s2; r3 ¼ 1 cm6/daN2. Good transient regimes can be achieved; however, attention must be paid to the monitorization of the control saturation during simulations (see the constraints (3.14)). To counteract the saturation effects, special antiwindup strategies can be used by the designer [44,47]. In the end, it is difficult to decide which of the two laws, (4.3) and (5.3), give better tracking performance. The powerful tool of the nonlinear synthesis, CLF, is present in both the constructions: indirectly, in the first machinery, directly in the second. The difference is that in the second case we have a rigorous demonstration of the solution (5.3). As for the robustness to parametric uncertainties, again was confirmed the role of fast servovalve dynamics in the prevention of the system states chattering.

6. Concluding remarks Noting the competition with other types of actuators – electromechanical actuators and hydrostatic actuators (see the so-called ‘‘green aircraft’’ and details in [64]) – we believe that the electrohydraulic servomechanisms (EHSs), also called electrohydraulic servo actuators, will still be for many years a right choice [10] in

I. Ursu et al. / European Journal of Control 19 (2013) 65–80

10 x2 [cm/s]

x1, xd [cm]

2 0 -2

0

0.5

1 t[s]

1.5

x3 [bar]

x4 [bar] 0

0.5

1 t[s]

1.5

0.5

1 t[s]

1.5

2

0

0.5

1 t[s]

1.5

2

0

0.5

1

1.5

2

1 t[s]

1.5

2

1

1.5

2

1.5

2

100 90

2

10 u[V]

0.1 x5 [cm]

0

110

110

0 -0.1

0 -10

2

120

100

0

0.5

1

1.5

0 -10

2

t[s]

t[s] 10 x2 [cm/s]

x1, xd [cm]

1 0 -1

0

0.5

1 t[s]

1.5

0

0.5

0

0.5

110 x4 [bar]

x3 [bar]

0 -10

2

110 105 100

0

0.5

1

1.5

105 100

2

t[s]

t[s] 10 u[V]

x5 [cm]

0.1 0 -0.1

77

0

0.5

1.5 1 t[s]

2

0 -10

0

0.5

1 t[s]

Fig. 7. Tracking of sine desired output x1d (5.1) by backstepping control (5.3); (a) x1d ¼ 1  sinð2p  2tÞ cm; (b) x1d ¼ 0:5  sinð2p  2tÞ cm.

many application domains; certainly, the above listing does not involve other types of actuators, for example the piezo actuators, characterized by very small ‘‘strokes’’ and present in completely different applications, such as the smart structures etc. Therefore, designing an EHS remained a theoretical and technical challenge during the 60–70 years of use in the actuation of aircraft primary flight controls. This challenge arises (a) primarily, from the following mathematical model shortcomings, highlighted throughout the study: critical case for stability, relative degree defect, nonsmooth nonlinearity and (b) second, from the rising demands on the system dynamic performances. To cope with these difficulties, some mathematical tools – Lyapunov–Malkin apparatus for critical

cases in the stability theory, geometric methods in control synthesis, backstepping synthesis – from classical and postmodern theory [63] of the automatic control were applied, from a control engineering practice perspective. Thus, the work is above all an attempt to reduce the well known schism between practitioners of feedback control design and its theoreticians [8]. As concrete results, it is important to note that the study presents a new approach, with a solution validated by numerical simulation, to the problem of equilibrium stability of the EHS actuated flight controls. As is done, the solution (3.8) is close to the solution proposed in [39], from which differs by a systematic construction, given by Propositions 3.1 and 3.2. Two tracking

78

I. Ursu et al. / European Journal of Control 19 (2013) 65–80

10 x2 [cm/s]

x1, xd [cm]

0.5

0

0

0.5

1 t[s]

1.5

0

2

x4 [bar]

x3 [bar]

110

0

0.5

1 t[s]

1.5

x5 [cm]

u[V] 0

0.5

1 t[s]

1.5

x1, xd [cm]

x2 [cm/s] 0

0.5

1 t[s]

1.5

0.5

1 t[s]

1.5

2

0

0.5

1 t[s]

1.5

2

0

0.5

1 t[s]

1.5

2

0

0.5

1 t[s]

1.5

2

0

0.5

1 t[s]

1.5

2

110 x4 [bar]

x3 [bar]

0

5 0

2

120 110

0

0.5

1 t[s]

1.5

100 90

2

10 u[V]

0.1 x5 [cm]

2

10

0.5

0.05 0

1.5

0 -10

2

1

100

1 t[s]

10

0.05

0

0.5

100 90

2

0.1

0

0

110

120

100

5

0

0.5

1 t[s]

1.5

2

0 -10

Fig. 8. Tracking of filtered step desired output x1d (5.2) by backstepping control (5.3); (a) x1d ðtÞ ¼ 0:5  ð1et=0:03 Þ cm; (b) x1d ðtÞ ¼ 1  ð1et=0:03 Þ cm.

control laws  (4.3), only conjectured, but successfully tested by simulations, and (5.3), mathematically proved and numerically tested  are also proposed. It is not surprising that the EHS tracking problem stated in Section 1 has not been rigorously solved, in the framework of the Malkin canonical form, this time either. As has been mentioned, to follow the procedure given in Proposition 3.1, for the construction sketched in Section 4, proved to be very difficult, even tedious; this inconvenience concerning the tracking solution is generally asserted in [25]. Following a suggestion from [16], a future work could consider an alternative approach to EHS nonlinearities, trying to find a satisfactory solution of the problem in geometric context.

As for the backstepping solution of the EHS tracking problem, it is to say that the control law (5.3) is a necessary complement to the solution given in [50]. At that time, the ‘‘bottle neck’’ of the problem was handling the EHS nonlinearities generated by Eq. (1.2). This time, the problem solving was possible due to some suggestions  in a broader view  coming from works such as [9,62], in which is involved the concept of differential inclusions, a paradigm of treating the discontinuous ordinary differential equations. Therefore, the calculation of control (5.3) can be done, with the idea that, really, the function Q_ Lx involves only finite jump at x5 ¼ 0. Thus, the proposed EHS tracking solution is significant in context. The solution also includes the special use of Barbalat’s Lemma, which allows

I. Ursu et al. / European Journal of Control 19 (2013) 65–80

bypassing of an alternate complicated demonstration, and a careful discussion on step or sinusoidal inputs. The algorithms associated to the proposed control laws are simple to implement, compared with other ones presented in the quoted literature. All the system states xi , i ¼ 1,. . .,5 are required in the feedback control laws, but this is only an apparently impediment, since the system states can be measured or, where applicable, calculated on line (for example, x2 and the state functions involved in (5.3)). A special mention must be made about the reference signal of the desired output x1d . The case closer to reality for tracking control algorithm is represented by arbitrary reference signals in time. In the online operation of the system, these signals are sampled as a sequence of step reference commands (5.2), on which Proposition 5.1 guarantees asymptotic stability for the position tracking error. Generally, the response time of the closed-loop system is short enough to be compared with the signals sampling period. Thus, the system can be assumed to approximately reach steady-state before the sampling period ends, when a new reference step command is sent to EHS input.

Acknowledgments The authors are very grateful to the anonymous referees for their valuable comments and suggestions. The first two authors acknowledge GRANT UEFISCDI-CNCS-ROMANIA No. PN-II-PTPCCA-2011-3.2-1212. The last two authors were partially supported by Grant CNCS-ROMANIA, PNII-ID-PCE-3-0198.

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