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Backstepping design for controlling electrohydraulic servos
ARTICLE IN PRESS
Journal of the Franklin Institute 343 (2006) 94–110 www.elsevier.com/locate/jfranklin
Backstepping design for controlling electrohydraulic servos Ioan Ursu, Felicia Ursu, Florica Popescu ‘‘Elie Carafoli’’ National Institute for Aerospace Research, B-dul Iuliu Maniu No. 220, 06 1126 Bucharest, Romania Received 25 January 2005; received in revised form 7 September 2005; accepted 13 September 2005
Abstract The backstepping technique is one of the tools which provides control Lyapunov functions (CLFs) and, therefore, control laws for automatic systems. In this paper, the backstepping design for position and force nonlinear electrohydraulic servos is investigated. Three control laws ensuring the asymptotic stability of references tracking are obtained by constructing CLFs on the errors concerning the state variables and theirs desired values. The possibility of converting a position servo into a force servo, with only minimal hard modification, is proved. An approach based on partitioning the state system into two subsystems—a first one stable, and a second one taken as framework of control synthesis by backstepping technique—was developed and used. Using as reference point a flight controls hydromechanical servo, numerical simulations were reported from viewpoint of servo time constant performance. Certain conjectures, concerning the behaviour of systems mathematical models, in connection with mathematical methodologies of control synthesis operating on them, are finally stressed. r 2005 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Nonlinear systems; Control Lyapunov functions; Backstepping; Electrohydraulic servo; Flight controls; Numerical simulation; Servo time constant
1. Introduction Mechanohydraulic and electrohydraulic servos (MHSs and EHSs) are widely used in various industries and advanced control technologies where heavy objects are manipulated and large forces or torques at high speeds are exerted. Features such as large processing Corresponding author. Tel.: +40 214340083; fax: +40 214340082.
E-mail address: [email protected] (I. Ursu). 0016-0032/$30.00 r 2005 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2005.09.003
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force and stiffness, high payload capabilities, good positioning and excellent power to weight ratio, make this type of actuation system appropriate for positioning of aircraft control surfaces (flight controls), high power industrial machinery, position control of military gun turrets and antennas, material handling, construction, mining and agricultural equipment. Beyond all doubt, ‘‘Industrial hydraulic technology is firmly entrenched in our global economy. The usage knows no boundary lines’’ [1]. The demanding performance specifications for such applications are high bandwidth implying small servo time constants, high accuracy and high fidelity control; such technical challenges have led the researchers to examine how to improve hydraulic control design. There are some factors that limit the applications of EHS, but the final factor limiting these applications is the designer’s understanding and knowledge of these systems. Because of the complexity of EHS analysis and the nonlinearities in its dynamics, 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 [2], to the artificial intelligence paradigm [3,4]. This paper develops control strategies for the EHS control synthesis using the concept of control Lyapunov function (CLF), concept introduced by Artstein [5] and Sontag [6], and based on the backstepping approach [7] of building these functions. In the beginning, Lyapunov’s stability theory deals with dynamic systems without inputs. For this reason, it has traditionally been applied only to closed loop control systems, that is, to systems for which the input has been eliminated through the substitution of a predetermined feedback control. Recently, some authors started using Lyapunov function candidates in feedback design itself by making the Lyapunov function derivative negative when choosing the control. Such idea has been made precise with the introduction of the CLF concept. The building of a CLF is not a unique matter, and the Jurdjevic–Quinn’s approach [8] and Sontag’s approach [9] are only two such examples. Another approach is just backstepping methodology [7], a recursive type control design brought out in 1990s. Its initial limitation to a class of pure feedback systems (lower triangular in the control variable) stimulated the development of various recursive procedures, applicable to more general nonlinear systems, as feedforward systems, adaptive systems and systems with structured uncertainty [7,10,11]. In the following, the backstepping is applied to position or force tracking control synthesis for an EHS whose equation belongs to a general class of nonlinear affine systems (f_means the derivative of the function f ðtÞ with respect to the time t) x_ ¼ FðxÞ þ GðxÞu,
(1)
where x 2 Rn is the state vector, u 2 Rm is the control vector and F and G are smooth vector fields of appropriate dimensions. The key idea of backstepping is simple. Each step of backstepping adds a new step in the construction of a CLF by the augmentation of the starting CLF with a term which penalises the error between a new state variable and its desired value. Therefore, the method will finally implicitly guarantee the references asymptotic stable tracking and a transient performance. It is worth noting that a particular feature of the backstepping is the construction of a CLF whose derivative can be made negative definite by a variety of control laws rather than by a specific control law. The remainder of the article is organised as follows. First of all, in Section 2, the backstepping is applied to perform the EHS control synthesis for position references
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tracking, using a simplified three state variables model. The constructed CLF involves all the errors concerning the state variables and theirs ‘‘desired’’ values, and the obtained control law, given in Proposition 1, leads to an equivalent, asymptotic stable, autonomous—in closed loop—dynamic system which describes the behaviour of these errors. Then, in Section 3, rendering evident the conjecture regarding mathematical model (MM) flexibility (see Section 6), the MM of EHS control synthesis for force references tracking is chosen as a (pseudo)extension of position references tracking case, by introducing a new state variable, as an alternative of simplifying the backstepping process. To vary and simplify the proof of Proposition 2, which provides the structure of force control law, the well-known Barbalat’s lemma is used to establish the asymptotic stability of the force references tracking, as an alternative call of Lyapunov stability standard theorem for the nonautonomous systems [12]. In Section 4, an authentic extension of the EHS MM concerns the ‘‘breaking’’ of the load pressure equation in two more realistic equations describing the behaviour of the pressures in the hydrocylinder chambers. Proposition 3 provides the form of position control law for this extended MM. Similar to the approach of Section 2, the control law was derived by splitting the state variable system into two subsystems, a first one stable, and a second one taken as framework of control synthesis by backstepping technique. Eloquent simulation results illustrating the working of different control laws for EHS, obtained in this paper and also in other related works of the authors, are described in Section 5. Key features of the present work are reported in a conclusive Section 6. 2. Backstepping position control design for EHS The differential equations governing the dynamics of the EHS are those given in [3] and are reported having as a reference point the MHS SMHR included in the aileron control chain of the Romanian military jet IAR 99: 1 ðkx1 fx2 þ Sx3 Þ, m rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p a x3 Sx2 k‘ x3 þ c x_ 3 ¼ kv u ; kc 2
x_ 1 ¼ x2 ;
x_ 2 ¼
sffiffiffi 2 . c :¼ cd w r
ð2Þ
The state variables are denoted by: x1 (cm)—EHS load displacement; x2 (cm/s)—EHS load velocity; x3 (da N/cm2)—pressure difference p1 p2 between the cylinder chambers, caused by load (load pressure); u (V)—control variable. To specialise, the load at EHS rod has three components: inertial, which is dominant, viscous and elastic loads. The nominal values of the parameters appearing in Eqs. (2) are: m ¼ 0:033 da N s2 =cm—equivalent inertial load of primary control surface reduced at the EHSs rod; S ¼ 10 cm2 —EHSs piston area; f ¼ 2:75 da N s=cm—equivalent viscous friction force coefficient; k ¼ 100 da N=cm—equivalent aerodynamic elastic force coefficient; w ¼ 0:05 cm—valveport width; pa ¼ 210 da N=cm2 —supply pressure; cd ¼ 0:63—volumetric flow coefficient of the valve port; k‘ ¼ 5=210 cm5 =ðda N sÞ—coefficient of internal leakage between the cylinder chambers; kv ¼ 0:17=10 cm=V ( ¼ valve maximum displacement/maximum voltage coefficient); kc ¼ 30=12 000 cm5 =da N ð:¼ V =ð2BÞÞ—coefficient involving the bulk modulus B of the used oil and the EHSs cylinder semivolume V; r ¼ 85=ð981 105 Þ da N s2 =cm4 —oil volumetric density. The valve dynamics is evaded in MM (2); a
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proportionality coefficient kv between the control u (input voltage to servovalve) and the valve displacement xv was considered. Clearly, system (2) is lower triangular and, therefore, suitable for application of backstepping. Taking all over article the nonadaptive case—the system parameters are assumed to be known—let us introduce the notations ei ¼ xi xid ;
i ¼ 1; . . . ; 3,
(3)
where xid stand for the ‘‘desired’’ values of the state variables. So, the control objective is to have the EHS track of a specified x1d position trajectory, in other words, to have e1 ! 0. Proposition 1. Let k1 ; k2 ; k3 ; r1 ; r2 ; r3 be strictly positive tuning parameters. Under the assumption x3 opa , which claims a process with nonsaturating load, the control u given by kc u¼ r3 ckv
x3d ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 r S S k‘ x2 þ x3 þ x_ 3d k3 e3 , 2 e 2 þ r3 p a x3 m kc kc
1 rm k2 m kx1 þ fx2 1 e1 þ mx_ 2d e2 , S r2 r2
x2d ¼ x_ 1d k1 e1
(4)
(5) (6)
when applied to Eq. (2), guarantees asymptotic stability for the position tracking error e1 ¼ x1 x1d ; more precisely, limt!1 e1 ðtÞ ¼ 0.
Proof. We start the backstepping by defining the Lyapunov like function V1 ¼
r1 2 e. 2 1
(7)
The derivative of V1 along system (a.s.) (2) is V_ 1 ¼ r1 e1 ðx_ 1 x_ 1d Þ ¼ r1 e1 ðx2 x_ 1d Þ ¼ r1 e1 ðe2 þ x2d x_ 1d Þ. With Eq. (6), V_ 1 becomes V_ 1 ¼ k1 r1 e2 þ r1 e1 e2 , 1
(8)
where r1, k1 stand for weighting parameters. Now, in order to go one step ahead, a new Lyapunov like function V2 is defined as r V 2 ¼ V 1 þ 2 e22 . (9) 2 Taking the derivative of function (9) a.s. (2) gives V_ 2 ¼ V_ 1 þ r2 e2 e_2 ¼ k1 r1 e21 þ e2 r1 e1 þ r2 ðx_ 2 x_ 2d Þ h i r ¼ k1 r1 e21 þ e2 r1 e1 2 ðkx1 þ fx2 Se3 Sx3d Þ r2 x_ 2d . m
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If x3d is substituted by Eq. (5), then a simplified V_ 2 can be rewritten as rS V_ 2 ¼ k1 r1 e21 k2 e22 þ 2 e2 e3 . (10) m Remark that the new weighting parameters r2, k2 were also introduced. Finally, V3 is defined as r V 3 ¼ V 2 þ 3 e23 (11) 2 and taking the derivative of function (11) a.s. (2) one may write rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rS r p a x3 V_ 3 ¼ k1 r1 e21 k2 e22 þ e3 2 e2 þ 3 Sx2 k‘ x3 þ c kv u kc x_ 3d . m kc 2 Finally, the control u is substituted by Eq. (4). Summarising, the control law given by Eqs. (4)–(6), with a given x1d (see, for instance, as in Eq. (38)), renders the derivative V_ 3 negative semidefinite V_ 3 ¼ k1 r1 e2 k2 e2 k3 e2 (12) 1
2
3
___
and r3, k3 were the latest weighting parameters introduced. Note that as soon as the values of the state variables x1, x2, x3 and desired x1d ; x_ 1d ; x€ 1d ; x1d are measured and, respectively, known, the control u can be calculated using Eqs. (2), (3), (5) and (6). Substituting Eqs. (3)–(6) in Eq. (2), an autonomous closed loop form, analogous to Eq. (1), occurs ~ e_ ¼ FðeÞ, (10 ) namely e_1 ¼ k1 e1 þ e2 ;
e_2 ¼
r1 k2 S e1 e2 þ e3 ; m r2 r2
e_3 ¼
r2 S k3 e2 e3 . r3 m r3
(100 )
Thus, V 3 is indeed a Lyapunov function for system (100 ): Lyapunov’s asymptotic stability theorem [13] holds; moreover, Barbashin–Krassovskii’s global asymptotic stability theorem [14] also holds. We conclude that all transient tracking errors ei, i ¼ 1; . . . ; 3, concerning system (2), are asymptotic convergent to zero (the unique equilibrium point of system (100 ) is the origin). & Remark 1. Choosing the ‘‘desired’’ state variable x1d defined in Eq. (38), from the convergence to zero of the errors ei ; i ¼ 1; . . . ; 3, it easily follows that x1 ! x1s , x3 ! kx1s =S, as t ! 1. Thus, starting from an initial equilibrium point of system (2), say (0, 0, 0), the control given in Eqs. (4)–(6), (38) guides the system to attain a new equilibrium point ðx1s ; 0; kx1s =SÞ. Certainly, this state transition cannot be carried out by ignoring just transient response; physical conditioning of this response must be pffiffiffiffiffiffiffiffiffiffiffi taken into account: 0ox3 ðtÞppa , jx2 ðtÞjpc=S ðpa =2Þkv umax , juðtÞjpumax , jx1 ðtÞjpx1 max , jx1d ðtÞjpx1 max , jx2d ðtÞjpvmax , jx3d jppa , ð8Þ t40. This is another problem, beyond the aim of the present work. Because of these necessary state limitations, the global asymptotic stability of tracking errors in Proposition 1 cannot be stipulated. At all events, choosing carefully the parameters t1r and x1s , in other words, a suitable x1d , the physical limitation of the problem will be preserved. Remark 2. Let us note that MM (2) supposes (and all the models in this work, too) the fulfilment of a conjecture: during the transitory regime of tracking positive references x1d, an initially determined positive u does not change its sign; otherwise, in the last equation
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pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2) must appear the factor ðpa x3 sgn uÞ=2 and the backstepping cannot work exactly in the described manner. In fact, the hypotheses ‘‘sign u40’’ and ‘‘nonsaturating load’’ will be assumed throughout the whole paper.
___
___
Remark 3. Consider system (2) at zero equilibrium. The calculation of starting control uð0Þ, derived from Eqs. (4)–(6), (38), involves the system parameters, the tuning parameters and the derivatives x_ 1d ð0Þ, x€ 1d ð0Þ, x1d ð0Þ: sffiffiffiffiffi kc 2 r2 S uð0Þ ¼ x_ 1d ð0Þ þ r3 x_ 3d ð0Þ k3 e3 ð0Þ , r3 ckv pa m 1 r1 m k2 m ðx€ 1d ð0Þ þ k1 x_ 1d ð0ÞÞ , x_ 1d ð0Þ þ mðx1d ð0Þ þ k1 x€ 1d ð0ÞÞ þ x_ 3d ð0Þ ¼ S r2 r 2 1 k2 m ð40 Þ e3 ð0Þ ¼ mðx€ 1d ð0Þ þ k1 x_ 1d ð0ÞÞ þ x_ 1d ð0Þ . S r2 Similar tedious—but not very difficult to obtain—relationships appear also in the next propositions, but will not be retained in text.
Remark 4. We will state, as a subsidiary feature of the proposed control design, the following. Consider x1d ðtÞ ¼ 0;
tX0.
(380 )
Then, Eqs. (4)–(6) can be rewritten as x2d ¼ k1 x1 , x3d
1 ¼ S
kc u¼ r3 ckv
r1 m k1 k2 m k2 m k x1 þ f mk1 x2 , r2 r2 r2
(60 ) (50 )
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 r k1 S r3 S r2 S x1 þ 2 x2 p a x3 m kc m
r3 k‘ k3 x3 þ k3 x3d þ r3 x_ 3d . þ kc The errors, with x2d and x3d given by Eq. (60 ) and, respectively, Eq. (50 ), will be e1 ¼ x1 ; e2 ¼ x2 x2d ; e3 ¼ x3 x3d .
ð400 Þ (30 )
The entire reasoning from the proof of Proposition 1 can be reproduced step by step, to assert that in fact the function r1 2 r2 r3 1 r1 m k1 k2 m 2 ~ V3 ¼ x þ ðx2 þ k1 x1 Þ þ x3 k x1 S r2 r2 2 1 2 2 2 k2 m ð110 Þ þ f mk1 x2 r2
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is a Lyapunov function for system (2), whose derivative a.s. (2) has the form V_~ ¼ k1 r1 x21 k2 ðx2 x2d Þ2 k3 ðx3 x3d Þ2 .
(120 )
Thus, the equilibrium point (0, 0, 0) of system (2) remains asymptotically stable in the presence of small perturbations involving ðx1 ; x2 ; x3 Þ and with assistance of the control (400 ), (50 ), (60 ).
Remark 5. Let us remember that all the Lyapunov functions determined in the paper are CLFs. 3. Backstepping force control design for EHS Consider the case of force control. It can be easily verified by inspecting system (2) that the internal states x1 and x2, as described by the first two equations in Eq. (2), are stable: the roots of the characteristic equation ml2 þ f l þ k ¼ 0 are ‘‘stable’’ complex roots with negative real parts, due to the viscous friction force in hydraulic cylinder. Thus, a special care to stabilise the states x1, x2 it is not necessary. On the other hand, the output of interest in force control is, of course, the load pressure x3. Thus, evading the first two equations in the backstepping procedure, this technique cannot be applied only about x3 itself. In these circumstances, we adapt the MM to the backstepping methodology (see the conjecture of MM flexibility, Section 6): system (2) will be completed by adding an equation of first order for the dynamics of the valve position xv :¼ x4 : 1 ðkx1 fx2 þ Sx3 Þ, m rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p a x3 x_ 3 ¼ Sx2 k‘ x3 þ c x4 ; kc 2
x_ 1 ¼ x2 ;
x_ 2 ¼
1 x_ 4 ¼ ðx4 þ kv uÞ, t
ð13Þ
where t is the time constant of the (servo)valve. Proposition 2. Let k3 ; k4 ; r3 ; r4 be strictly positive tuning parameters. Under the assumption x3 opa , which claims a process with nonsaturating load, the control u given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x t r3 c pa x3 4 e 3 þ r4 þ x_ 4d k4 e4 , u¼ (14) r4 kv kc 2 t x4d
1 ¼ c
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ½Sx2 þ k‘ x3 þ kc ðx_ 3d k3 e3 Þ p a x3
(15)
when applied to Eq. (13), guarantees asymptotic stability for the pressure (force) tracking error e3 ¼ x3 x3d ; more precisely, limt!1 e3 ðtÞ ¼ 0.
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Proof. Consider r V 3 ¼ 3 e23 ; e3 :¼ x3 x3d ; e4 :¼ x4 x4d 2 and differentiating a.s. (13) yields rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r3 p a x3 _ V 3 ¼ r3 e3 e_3 ¼ e3 Sx2 k‘ x3 þ c ðe4 þ x4d Þ r3 e3 x_ 3d . kc 2 If x4d is substituted by Eq. (15), then a simplified V_ 3 is obtained rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r3 c pa x3 2 _ V 3 ¼ k3 r3 e3 þ e3 e4 , kc 2
101
(16)
(17)
where r3 and k3 are weighting parameters. Now, in order to go one step ahead, define V4 as V4 ¼ V3 þ
r4 2 e 2 4
(18)
and taking again the derivative a.s. (13), we have rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r p a x3 x4 þ kv u x_ 4d . V_ 4 ¼ k3 r3 e23 þ e4 3 c e3 þ r 4 t kc 2 Now, the control u, synthesised as in Eq. (14), renders the derivative V_ 4 negative semidefinite V_ 4 ¼ k3 r3 e23 k4 e24 . Therefore, V4 is a Lyapunov function for the nonautonomous system rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c pa e3 x3d k4 r3 c pa e3 x3d e_3 ¼ k3 e3 þ e4 e4 ; e_4 ¼ e3 . kc r4 kc 2 r4 2
(19)
(1000 )
The conditions specified in Kalman and Bertram [12] (see Theorem 1) are fulfilled and thus the asymptotic stability of the errors e3 ; e4 holds. An alternative way to finish the proof makes use of Barbalat’s lemma [15]: Let f R: ð0; 1Þ ! R be an uniformly continuous function. If there exists the finite limit t limt!1 0 f ðtÞ dt, then limt!1 f ðtÞ ¼ 0. An equivalent version is the following: If the function f(t) is differentiable and has a finite limit limt!1 f ðtÞ, and if f_ is uniformly continuous, then limt!1 f_ðtÞ ¼ 0. Thus, Barbalat’s lemma will be applied to show that the errors e3 and e4 tend to zero as time tends to infinity. Making use of definitions (15), (16) and (38) for e3 and e4 , it is easily to see that when t ! 0; we have V 4 ð0Þ40; for instance, if the system is moved from zero initial equilibrium point, then sffiffiffiffiffi !2 r4 1 2 V 4 ð0Þ ¼ kc x_ 3d ð0Þ 40. c ra 2
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Since V_ 4 p0, it follows that 0pV 4 ðtÞ V 4 ð0Þ, ð8Þt40, hence the positive function V 4 ðtÞ is bounded and it follows that e3 and e4 are bounded. Now, taking the derivative of (19) yields rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k24 2 cr3 k4 p a x3 2 2 € V 4 ¼ 2r3 k3 e3 þ 2 e4 2 e3 e4 . k3 r4 kc r4 2 Under previous hypothesis on x3 (see Remarks 1 and 2), it follows that V€ 4 is bounded, so V_ 4 is uniformly continuous. Since V 4 is decreasing, there exists a finite limt!1 V 4 ðtÞ. Then pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V_ 4 ! 0 and, consequently, e3 and e4 tend to zero. The radical pa x3 is indeed bounded with finite x3d (see Eq. (38)). &
Remark 6. The partition of system (13) in two subsystems, a first one stable, and a second one taken as framework of control synthesis by backstepping technique, is once again pointed out as approach. An articulated basis of this procedure is furnished by the proof of Proposition 3. 4. Backstepping for an extended EHS MM The foundation for the mathematical modelling of the hydraulic servomechanisms was laid down by Blackburn et al. [16]. Important books in the field appear in the decade 1965–1975 [17–19]. The hydraulic servo is in fact a valve–piston combination and the usual treatment of the corresponding pressure-flow equation has as outcome a three-dimensional model like Eq. (2). The key-equation in deriving this model is p1 þ p2 ¼ pa .
(20)
However, Eq. (20) is rather a modelling hypothesis [20] than a direct consequence of physical laws and mathematical inference on these. The evading of this equation leads to an extended MM, involving the pressures p1 ; p2 as state variables; this MM was already introduced a long while ago by Wang [21], but only sporadically was taken back. A variant of the Wang’s model can be written in the form x_ 1 ¼ x2 ;
1 ½kx1 fx2 þ Sðx3 x4 Þ, m pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½cx5 pa x3 Sx2 k‘ ðx3 x4 Þ,
x_ 2 ¼
B V þ Sx1 pffiffiffiffiffi B x_ 4 ¼ ½cx5 x4 þ Sx2 þ k‘ ðx3 x4 Þ, V Sx1 x5 þ kv u . ð21Þ x_ 5 ¼ t This time, the variables x3 and x4 are the pressures p1 ; p2 in the cylinder chambers, and x5 stands for the valve position. x_ 3 ¼
Proposition 3. Let k1 40; k2 40 be tuning parameters. Under the (rather physical) assumptions 0ox3 opa ; 0ox4 opa ; jx1 joV =S, the control u given by u¼
1 ½x5d þ tðx_ 5d k2 g2 ep Þ, kv
(22)
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x5d ¼
1 ðk1 ep p_ d þ g1 Þ, g2 2BV ðSx2 þ k‘ pÞ ; g2 ¼ g2 ðx1 ; x3 ; x4 Þ V 2 S 2 x21 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi p a x3 x4 :¼ Bc þ , V þ Sx1 V Sx1
103
(23)
g1 ¼ g1 ðx1 ; x2 Þ :¼
ep :¼ p pd ; p :¼ x3 x4 ; e5 ¼ x5 x5d , pd ðtÞ :¼
k k x1d ðtÞ :¼ x1s ð1 et=t1r Þ, S S
ð24Þ (25) (26)
when applied to Eq. (21), guarantees asymptotic stability for the position tracking error e1 :¼ x1 x1d ; more precisely, limt!1 e1 ðtÞ ¼ 0. The notations x1s ; t1r define the expression of the time response x1d of a first-order stable system to step input x1s (see Eq. (38)); t1r is the time constant and x1d ð0Þ is taken to be zero. Proof. As in the previous propositions, consider Lyapunov like function: V1 ¼
1 2 e . 2 p
(27)
Then, its derivative a.s. (21), by using Eqs. (23)–(25), is V_ 1 ¼ k1 e2p þ g2 ep e5 .
(28)
Now, extend V 1 as V2 ¼ V1 þ
1 2 e 2k2 5
(29)
and, taking into account Eq. (22), the derivative a.s. (21) is 1 2 e. V_ 2 ¼ k1 e2p k2 t 5 The following equations for the errors ep, e5 hold: e5 e_p ¼ k1 ep þ g2 e5 ; e_5 ¼ k2 g2 ep . t € Performing the derivative V 2 a.s. (21), we obtain 1 2 1 2 2 € V 2 ¼ 2 k 1 ep þ e 2 k 1 g2 e p e 5 . k 2 t2 5 t
(30)
(31)
Repeating the arguments which end the proof of Proposition 2, it results that the errors ep and e5 are bounded; so, p ¼ pðtÞ is also bounded on Rþ ¼ ½0; 1Þ. Furthermore, V€ 2 is bounded, provided that g2 remains bounded during the dynamical process; this condition holds, having in view the assumptions involving the variables x1 ; x3 ; x4 . So, V_ 2 is uniformly continuous, and applying Barbalat’s lemma, V_ 2 ! 0. Consequently, ep and e5 tend to zero.
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Now, let us look at the first two equations in Eq. (21), which can be rewritten as follows: rffiffiffiffi f k S 2 ; r :¼ ; p1 ¼ p (32) x€ 1 þ 2hx_ 1 þ r x1 ¼ p1 ; h :¼ 2m m m and p is now seen as a bounded function of t, p :¼ pðtÞ ¼ x3 ðtÞ x4 ðtÞ. With initial conditions x1 ð0Þ ¼ x_ 1 ð0Þ ¼ 0, the solution of Eq. (32) is Z 1 ht t hu x1 ðtÞ ¼ e e p1 ðuÞ sin oðt uÞ du; (33) o 0 where the condition o2 :¼ r2 h2 40 is inherent to hydraulic servo systems owing to small viscous friction in cylinder (see Remark 7). Introducing S p ðtÞ m d let us also consider Z t 1 x~ 1d ðtÞ :¼ eht ehu p1d ðuÞ sin oðt uÞ du: o 0 p1d ðtÞ :¼
(34)
(3300 )
Since ep ! 0 when t ! 1, it is clear that p1 ðtÞ ! p1d ðtÞ, as t ! 1; this means: ð8Þe40, (() dðeÞ such that for t4dðeÞ we have jp1 ðtÞ p1d ðtÞjoe. Then, if t4dðeÞ Z t 1 jx1 ðtÞ x~ 1d ðtÞjp eht ehu jp1 ðtÞ p1d ðtÞjj sin oðt uÞj du o 0 Z e t hðtuÞ e 1 eht . p e du ¼ o 0 o h It results in x1 ðtÞ ! x~ 1d ðtÞ as t ! 1.
(35)
Then, successive calculations give Z 1 ht t hu S k x1s ð1 eu=tr Þ sin oðt uÞ du x~ 1d ðtÞ ¼ e e o mS 0 Z x1s keht t hu e ð1 eu=tr Þ sin oðt uÞ du, ¼ om 0 Z
t
ehu ð1 eu=tr Þ sin oðt uÞ du ¼ 0
ðh 1=tr Þ sin ot oðeðh1=tr Þt cos otÞ o2 þ ðh 1=tr Þ2 þ
h sin ot þ oðeht cos otÞ , o 2 þ h2
therefore x~ 1 ðtÞ ! x1s
as t ! 1.
(36)
Thus, from Eqs. (35) and (36), a standard proceeding gives x1 ðtÞ ! x1s
as t ! 1
and so ends the proof.
&
(37)
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Remark 7. The reasoning given above can be easily recomposed for the case of real negative roots of the characteristic equation l2 þ 2hl þ r2 ¼ 0. 5. Simulation results The EHS is in fact a tracking system. Therefore, for this system the aim of control synthesis is to have a good tracking of the specified desired position references introduced as electrical signals. Thus, the goal of the backstepping synthesised control is to have the EHS tracking of the specified x1d position or x3d pressure references. Such references can be expressed as a response of a first-order system to step inputs xis xid ¼ xis ð1 et=tir Þ;
i ¼ 1 or 3,
(38)
where x1s and x3s stand for stationary value of the states x1, respectively, x3 and t1r and t3r stand for associated desired time constants. As reference points of the numerical simulations we take both the experimental data and a three-dimensional MM of the MHS SMHR [3] 1 ðkx1 fx2 þ Sx3 Þ, m rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p a x3 x_ 3 ¼ Sx2 k‘ x3 þ cs ; kc 2 x_ 1 ¼ x2 ;
x_ 2 ¼
s :¼ lðr x1 Þ,
ð39Þ
where l ¼ 23 is the coefficient of the rigid feedback kinematics, r is the reference input at servo rigid feedback kinematics linkage point (cm) and s is the valve error signal. The analogous model with two pressure state variables is 1 ½kx1 fx2 þ Sðx3 x4 Þ, m pffiffiffiffiffi B B pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½cs pa x3 Sx2 ; x_ 4 ¼ ½cs x4 þ Sx2 . x_ 3 ¼ V þ Sx1 V Sx1
x_ 1 ¼ x2 ;
x_ 2 ¼
ð40Þ
Zero equilibrium points for systems (39) and (40) correspond to closed valve—s ¼ 0—in the case of zero reference input, r ¼ 0. Numerical simulations have been performed to investigate the performance of the proposed nonlinear controllers, in the three cases defined by the nonlinear control laws (4), (14) and (22). Zero initial conditions—corresponding to the zero equilibrium points—were chosen for a fourth-order Runge–Kutta integration (with integration step 0.001 s) of the systems: Eq. (39), with a simple, rigid mechanical position feedback; Eq. (2), with control law (4); (13), with control law (14); (40), with a simple, rigid mechanical position feedback; and (21), with control law (22); in the last two cases, initial conditions on the variables x3 and x4 were chosen 1 da N/cm2. A relatively good tracking of references r is possible with the MHS SMHR (39), but only in the absence of perturbations [22,23]. To have in this servo a concrete term of comparison for the position control technique developed in Section 2, we choose the step input r ¼ 0:17 cm=l ¼ 0:255 cm. Then, having this step reference, the entire valve port is in fact open to flow passing and, consequently, the resulting servo time constant characterises the best step input tracking with the ‘‘passive’’ system: tsffi0.034 s (Fig. 1a, with the error signal s represented; initial error: s0 ¼ 0:17 cm, the maximum length of the valve port).
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0.2
0.192 0.128
0.1
0
0.05
0.1
(a)
0.15
0.128
5
0.064
2.5 0
0 0.2
0
x 1d u
0.064
σ
0.192
7.5 u [V]
x1
0.05
x1
x 1 [cm]
σ [cm]
0.15
0.256
10
0
0.025
(b)
t [s]
0.05
0.075
x 1, x 1d [cm]
106
0 0.1
t [s]
Fig. 1. Comparison of the position step input tracking systems with load pressure state variable: (a) SMHR case: ts ffi 0:034 s; (b) EHS case with backstepping position control: ts ffi 0:022.
75
x3
4
50
2
25
0
8
0.1
0.2
0.3 t [s]
0.4
0.5
6
(b)
140 105
4
70
2
35 0
0.6
175 x3d x3
0
0 0
u
x3, x3d [daN/cm2]
u [V]
6
(a)
10
100 x3d
u [V]
u
x3, x3d [daN/cm2]
8
0 0.11 0.22 0.33 0.44 0.55 0.66 t [s]
Fig. 2. Backstepping pressure control: (a) x3s ¼ 100 da N=cm2 , t3r ¼ 0:1 s, ts ffi 0:16 s; (b) x3s ¼ 175 da N=cm2 , t3r ¼ 0:05 s, ts ffi 0:175 s.
The following definition of the servo time constant is used: the time necessary to system transient response to attain 66% of the value of stationary regime. An evidently better tracking of step references is ensured by backstepping position control synthesis with the values of the tuning parameters: k1 ¼ 400 s1 ; k2 ¼ 4 s; k3 ¼ 400 cm6 =ðda N2 sÞ; r1 ¼ 400; r2 ¼ 0:033 s2 ; r3 ¼ 1 cm6 =da N2 . The desired control objective, in terms of position reference defined by x1s ¼ 0:255 cm and t1r ¼ 0:01 s, is accomplished with faster servo time constant ts ffi 0:022 s; when t1r was chosen 0.005 s, the same time constant ts ffi 0:022 s was obtained (Fig. 1b). The transient regime is stable, irrespective of stationary regime values x1s (admissible) and t1r; however, the designer must be attentive to control saturation. To counteract this effect, special antiwindup strategies can be used [3,22,23]. Similar conclusions are valid in the case of backstepping force control (Fig. 2). The fit values for the tuning parameters were: k3 ¼ 400 s1 ; k4 ¼ 800 da N2 =ðcm6 sÞ; r3 ¼ 1010 ; r4 ¼ 1 da N2 =cm6 ; also, let note the value f ¼ 1 daNs=cm used here. A fit time constant 1 t ¼ 573 s of the valve was considered. Worthy noting, herein the coefficient kv ¼ 1:7 10 cm=V means a tenfold necessary valve port area. The desired control objective, in terms of pressure reference defined by x3s ¼ 100 da N=cm2 and t3r ¼ 0:1 s (Fig. 2a), and x3s ¼ 150 da N=cm2 and t3r ¼ 0:1 s, is accomplished with good servo time constants: ts ffi 0:16 s
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and, respectively, ts ffi 0:19 s; when x3s and t3r were chosen as 175 da N/cm2 and, respectively, 0.05 s, an also good servo time constant ts ffi 0:175 s was obtained (Fig. 2b). It can be seen that the state variables x1, in the first case, and x3, in the second case, come very close to the desired x1d and, respectively, x3d, values. The alternating MM (40) of the MHS SMHR promote a somewhat ‘‘stiffer’’ object; the same is the conclusion concerning system (21) governed by the control (22). Fig. 3 shows fit results obtained choosing r expressed in form (38), with x1s ¼ 0:255 cm and t1r ¼ 0:002 s; the HM SMHR MM produces the servo time constant ts ¼ 0:037 s (Fig. 3a) and, with the fit tuning parameters k1 ¼ 1000 s1 and k2 ¼ 105 da N2 cm6 , the corresponding Eqs. (21), (22) system denotes the same servo time constant ts ¼ 0:037 s. Compared with the systems having the load pressure as state variable, the systems with two pressures as state variables are dynamically slower. The time histories of the load pressures, depicted in Fig. 4, offer an explanation of the systems dynamic behaviour. The faster system is by all means that actuated by greater values of the external driving force; this force is derived from load pressure: Sx3, in the case of the first two equations in Eq. (2); and, respectively, S(x3x4), in the case of the first two equations in Eq. (21). Table 1 summarises the main quantifying results of the systems performance, expressed as servo time constants obtained in this work and related works of the authors. Slight
0.1
8
0.192
6
0.128
0.05 0 0
0.05
0.1
4
0.128
2
0 0.15
0.064
0 0
t [s]
(a)
0.192
u
0.064
σ
0.256
x1
x1d
0.05
0.1
x1,x1d [cm]
σ [cm]
0.15
0.256
u [V]
x1
x1 [cm]
0.2
0 0.15
t [s]
(b)
30
3
20
2.5
10 0 0 -10
0.05
0.1 t [s]
0.15
0.2
x3- x4 [daN/cm2]
x3 [daN/cm2]
Fig. 3. Comparison of the position step input tracking systems with two pressure state variables: (a) SMHR case: ts ffi 0:037 s; (b) EHS case with backstepping position control: ts ffi 0:037 s.
-20
1.5 1 0.5 0 0
-30 (a)
2
(b)
0.01
0.02 0.03 t [s]
0.04
0.05
Fig. 4. Comparison of the load pressure evolution: (a) system (2), (4), with load pressure state variable; (b) system (21), (22), with two pressure state variables.
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Table 1 Summary of the servo time constants for position references tracking: MHS SMHR case and related presumptive EHS with various control laws Synthesis strategy
Back-stepping, third-order model (2), (4), with load pressure state variable; step reference amplitude: r ¼ 0:255 cm Backstepping, fifth-order model (21), (22), two pressures as state variables; step reference amplitude: x1s ¼ 0:255 cm Neuro-fuzzy antiwindup synthesis [4], fourth-order Wang type model [21], two pressures as state variables; step reference amplitude r ¼ 0:1 cm, and easy modified l (l ¼ 0:55) and f (f ¼ 1 da N s=cm) Davison type robust servo synthesis [24], with antiwindup compensation [22,23], linear model [17–19], [3]; step reference amplitude r ¼ 0:3 cm (derived from position transducer gain) and f ¼ 1 da N s=cm, which compensates in some degree the increased r Measured servo time constant [25], step reference amplitude r ¼ 0:12 cm
Servo time constant ts (s) EHS
MHS SMHR
0.022
0.034 (system (39)) 0.037 (system (40)) 0.025
0.037 0.011
0.021
0.024 (theoretical servo time constant, derived from linear system) 0.02
changes in parameter values—involving the amplitude of step input (r or x1s), the equivalent viscous friction coefficient f, or the coefficient of rigid feedback kinematics l— do not affect the location of all calculated or determined from numerical simulation servo time constants in the close neighbourhood of the experimentally measured [25] servo time constant of the MHS SMHR. Indeed, roughly speaking, the servo time constant decreases with decreasing f, r or x1s; the faster time response in the case of neuro-fuzzy antiwindup synthesis can so be explained in Table 1. Therefore, the simulation studies attest good tracking performance in the presence of step signals. Based on sampled signals paradigm, a good tracking performance of arbitrary references can be also stipulated, having in view the decomposition of an arbitrary signal into a succession of step signals. 6. Concluding remarks The objectives of the research presented in this paper were twofold. First, we wanted to exploit a recently introduced technique of nonlinear control design—the backstepping synthesis—in the framework of the EHS design. The problem was a challenging one, considering the strong nonlinear feature of the MMs and the topical quality of these standard objects of the automatic systems. Certainly, if the reader is more interested in comprehending the backstepping machinery, she or he very well might find the EHS framework only a pretext. Second, we tried to offer new arguments that lend support to certain earlier enounced conjectures [3]. This time, several MMs of EHSs, obtained from various control strategies, were quantified and compared from the viewpoint of the servo time constant performance and all the values were found to be close to the experimental one [25]. Three case studies on
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backstepping synthesis for controlling EHS systems are presented. The full state information was considered available. We have illustrated how the main theory can be brought or adapted to design practice as defined by a given MM, and have shown that the backstepping controllers are able to work with a complex plant such as EHS. The performance of the position tracking controllers has been finally compared with that of several other techniques, such as antiwindup neuro-fuzzy synthesis [4,26] and classical robust synthesis [24] with antiwindup compensation [22,23]. A certain trend of servo time constants improvement by control synthesis strategies can be attested. Smaller such values can be obtained, but with the risk to compromise a stability-rapidity trade-off of the servo time response. Thus, our empirically derived conjectures [3] concern the behaviour of systems MMs in connection with mathematical methodologies of control synthesis operating on them; in Ref. [3], as basis of these conjectures has been served the field of active and semiactive road vehicle suspension control synthesis. First conjecture. All the MMs, describing the same object, will promote in the last analysis a certain ‘‘firmness’’ and ‘‘solidarity’’, in other words, will predict approximately the same object performance or behaviour improvement, in despite of the used mathematical methodology; see, herein, the results given in Table 1. In fact, there are always more profound structural-mechanical reasons explaining the limitation of achieved amount of improvement obtained by insertion of control in mechanical structures [3,27]. Second conjecture. The MM, as a thinking first result, derived from a physical reality, is more ‘‘flexible’’ than a methodology (i.e., a mathematical theory [28]), as a thinking second result, derived from a metareality (herein, the mathematical thinking). Thus, the first conjecture refers to certain conservative properties of MMs. The second conjecture advances a compensating property of MMs. This is a complemental, flexibility property: to be able to apply a certain mathematical construction, a mathematical model can be often ‘‘shaped’’ in view of coping with the constraints of a certain mathematical apparatus (see the approach in Section 3). Another result of the paper concerns the opportunity to convert a position servo into a force one with only minimal hard modification in valve port area. The backstepping has not been applied before to control a position tracking EHS used in aviation, to the best of the author’s knowledge. Acknowledgements The work described above was supported (partially) by the Romanian Space Agency (ROSA), Contract Nos. 85/2003 and 144/2004. The authors would like to thank Prof. Andrei Halanay for helpful discussions and his valuable advice, and Prof. Lucian Iorga, for suggesting many improvements of the text. References [1] J. Pippenger, T. Hicks, Industrial Hydraulics, third ed., McGraw-Hill, New York, 1979. [2] I. Ursu, F. Popescu, M. Vladimirescu, R. Costin, On some linearization methods of the generalized flow rate characteristic of the hydraulic servomechanisms, Rev. Roumaine. Sci. Tech. Se´r. Me´cani. Appl. 39 (2) (1994) 207–217.
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[3] I. Ursu, F. Ursu, Active and Semiactive Control, Romanian Academy Publishing House, Bucharest, 2002 (in Romanian). [4] I. Ursu, F. Ursu, L. Iorga, Neuro-fuzzy synthesis of flight control electrohydraulic servo, Aircraft Eng. and Aerospace Technol. 73 (5) (2001) 465–471. [5] Z. Artstein, Stabilisation with relaxed controls, Nonlinear Anal. 7 (1983) 1163–1173. [6] E.D. Sontag, A Lyapunov-like characterisation of asymptotic controllability, SIAM J. Control Optim. 21 (1983) 462–471. [7] M. Krstic, I. Kanellakopoulos, P.V. Kokotovic, Nonlinear and Adaptive Control Design, Wiley, New York, 1995. [8] V. Jurdjevic, J.P. Quinn, Controllability and stability, J. Differential Equations 28 (1978) 381–389. [9] E.D. Sontag, An universal construction of Artstein’s theorem on nonlinear stabilisation, Systems Control Lett. 13 (2) (1989) 117–123. [10] R. Sepulchre, M. Jankovic, P.V. Kokotovic, Constructive Nonlinear Control, Springer, London, 1997. [11] R.A. Freeman, P.V. Kokotovic, Robust Nonlinear Control Design, State-space and Lyapunov techniques, Birkha¨user, Boston, 1996. [12] R.E. Kalman, J.E. Bertram, Continuous system analysis and design via the second method of Lyapunov. I. Continuous-time systems, Transactions of the ASME, Journal of Basic Engineers 82 (Ser. D) (2) (1960) 371–393. [13] J.P. LaSalle, S. Lefschetz, Stability by Lyapunov’s Direct Method with Applications, Academic Press, New York, 1961 (Russian Ed.: Mir, 1964). [14] J.C. Hsu, A.U. Meyer, Modern Control Principles and Applications, McGraw-Hill, New York, 1968 (Russian Ed. Izd. Mashinostroenie, Moskow, 1972). [15] V.M. Popov, Hyperstability of Automatic Control Systems, Springer, Berlin, 1973. [16] J.E. Blackburn, J.L. Shearer, G. Reethof, Fluid Power Control, Technical Press of Massachusetts Institute of Technology, Wiley, New York, 1960. [17] H.E. Merritt, Hydraulic Control Systems, Wiley, New York, 1967. [18] N.S. Gamynin, Fluid Power Control, Mashinostroenie, Moskva, 1972 (in Russian). [19] M. Guillon, L’asse´rvissement Hydraulique et e´le´ctrohydraulique, Edition Dunod, Paris, 1972. [20] A. Halanay, C. Safta, I. Ursu, F. Ursu, Stability of the equilibrium in a four dimensional nonlinear model of hydraulic servomechanism, J. Eng. Math. 49 (4) (2004) 391–405. [21] P.K.C. Wang, Analytical design of electrohydraulic servomechanism with near time-optimal response, IEEE Trans. Automat. Control AC-8 (1963) 15–27. [22] I. Ursu, G. Tecuceanu, F. Ursu, T. Sireteanu, M. Vladimirescu, From robust control to antiwindup compensation of electrohydraulic servo actuators, Aircraft Eng. and Aerospace Technol. 70 (4) (1998) 259–264. [23] F. Ursu, I. Ursu, Robust synthesis with antiwindup compensation for electrohydraulic servomechanisms, 15th IFAC Symposium on Automatic Control in Aerospace, September 2–7, Bologna/Forli, Italy, 2001, pp. 197–202, preprints. [24] E.J. Davison, J. Ferguson, The design of controllers for the multivariable robust servo-mechanism problem using parameter optimization methods, IEEE Trans. Automat. Control AC-26 (1) (1981) 93–110. [25] I. Ursu, Theoretical and experimental data concerning qualification testing and flight clearance for aircraft servomechanism SMHR, National Institute for Scientific and Technical Creation-INCREST Report, N-5303, Bucharest, 1984. [26] I. Ursu, F. Ursu, New results in control synthesis for electrohydraulic servos, Int. J. Fluid Power 5 (3) (2004) 25–38 Fluid Power Net International FPNI and Tu Tech, TUHH Technologie Gmbh, 2004. [27] X. Moreau, A. Oustaloup, M. Nouillat, B. Bluteau, La suspension CRONE: une suspension active d’ordre non entier optimal, in: L. Je´ze´quel (Ed.), Active Control in Mechanical Engineering, Hermes, Paris, 1995, pp. 77–92. [28] H.G. Natke, About the role of mathematicians in engineering—thoughts of a scientist between both branches, GAMM-Mitteilungen 19 (2) (1996) pp. 121–131.
Journal of the Franklin Institute 343 (2006) 94–110 www.elsevier.com/locate/jfranklin
Backstepping design for controlling electrohydraulic servos Ioan Ursu, Felicia Ursu, Florica Popescu ‘‘Elie Carafoli’’ National Institute for Aerospace Research, B-dul Iuliu Maniu No. 220, 06 1126 Bucharest, Romania Received 25 January 2005; received in revised form 7 September 2005; accepted 13 September 2005
Abstract The backstepping technique is one of the tools which provides control Lyapunov functions (CLFs) and, therefore, control laws for automatic systems. In this paper, the backstepping design for position and force nonlinear electrohydraulic servos is investigated. Three control laws ensuring the asymptotic stability of references tracking are obtained by constructing CLFs on the errors concerning the state variables and theirs desired values. The possibility of converting a position servo into a force servo, with only minimal hard modification, is proved. An approach based on partitioning the state system into two subsystems—a first one stable, and a second one taken as framework of control synthesis by backstepping technique—was developed and used. Using as reference point a flight controls hydromechanical servo, numerical simulations were reported from viewpoint of servo time constant performance. Certain conjectures, concerning the behaviour of systems mathematical models, in connection with mathematical methodologies of control synthesis operating on them, are finally stressed. r 2005 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Nonlinear systems; Control Lyapunov functions; Backstepping; Electrohydraulic servo; Flight controls; Numerical simulation; Servo time constant
1. Introduction Mechanohydraulic and electrohydraulic servos (MHSs and EHSs) are widely used in various industries and advanced control technologies where heavy objects are manipulated and large forces or torques at high speeds are exerted. Features such as large processing Corresponding author. Tel.: +40 214340083; fax: +40 214340082.
E-mail address: [email protected] (I. Ursu). 0016-0032/$30.00 r 2005 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2005.09.003
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force and stiffness, high payload capabilities, good positioning and excellent power to weight ratio, make this type of actuation system appropriate for positioning of aircraft control surfaces (flight controls), high power industrial machinery, position control of military gun turrets and antennas, material handling, construction, mining and agricultural equipment. Beyond all doubt, ‘‘Industrial hydraulic technology is firmly entrenched in our global economy. The usage knows no boundary lines’’ [1]. The demanding performance specifications for such applications are high bandwidth implying small servo time constants, high accuracy and high fidelity control; such technical challenges have led the researchers to examine how to improve hydraulic control design. There are some factors that limit the applications of EHS, but the final factor limiting these applications is the designer’s understanding and knowledge of these systems. Because of the complexity of EHS analysis and the nonlinearities in its dynamics, 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 [2], to the artificial intelligence paradigm [3,4]. This paper develops control strategies for the EHS control synthesis using the concept of control Lyapunov function (CLF), concept introduced by Artstein [5] and Sontag [6], and based on the backstepping approach [7] of building these functions. In the beginning, Lyapunov’s stability theory deals with dynamic systems without inputs. For this reason, it has traditionally been applied only to closed loop control systems, that is, to systems for which the input has been eliminated through the substitution of a predetermined feedback control. Recently, some authors started using Lyapunov function candidates in feedback design itself by making the Lyapunov function derivative negative when choosing the control. Such idea has been made precise with the introduction of the CLF concept. The building of a CLF is not a unique matter, and the Jurdjevic–Quinn’s approach [8] and Sontag’s approach [9] are only two such examples. Another approach is just backstepping methodology [7], a recursive type control design brought out in 1990s. Its initial limitation to a class of pure feedback systems (lower triangular in the control variable) stimulated the development of various recursive procedures, applicable to more general nonlinear systems, as feedforward systems, adaptive systems and systems with structured uncertainty [7,10,11]. In the following, the backstepping is applied to position or force tracking control synthesis for an EHS whose equation belongs to a general class of nonlinear affine systems (f_means the derivative of the function f ðtÞ with respect to the time t) x_ ¼ FðxÞ þ GðxÞu,
(1)
where x 2 Rn is the state vector, u 2 Rm is the control vector and F and G are smooth vector fields of appropriate dimensions. The key idea of backstepping is simple. Each step of backstepping adds a new step in the construction of a CLF by the augmentation of the starting CLF with a term which penalises the error between a new state variable and its desired value. Therefore, the method will finally implicitly guarantee the references asymptotic stable tracking and a transient performance. It is worth noting that a particular feature of the backstepping is the construction of a CLF whose derivative can be made negative definite by a variety of control laws rather than by a specific control law. The remainder of the article is organised as follows. First of all, in Section 2, the backstepping is applied to perform the EHS control synthesis for position references
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tracking, using a simplified three state variables model. The constructed CLF involves all the errors concerning the state variables and theirs ‘‘desired’’ values, and the obtained control law, given in Proposition 1, leads to an equivalent, asymptotic stable, autonomous—in closed loop—dynamic system which describes the behaviour of these errors. Then, in Section 3, rendering evident the conjecture regarding mathematical model (MM) flexibility (see Section 6), the MM of EHS control synthesis for force references tracking is chosen as a (pseudo)extension of position references tracking case, by introducing a new state variable, as an alternative of simplifying the backstepping process. To vary and simplify the proof of Proposition 2, which provides the structure of force control law, the well-known Barbalat’s lemma is used to establish the asymptotic stability of the force references tracking, as an alternative call of Lyapunov stability standard theorem for the nonautonomous systems [12]. In Section 4, an authentic extension of the EHS MM concerns the ‘‘breaking’’ of the load pressure equation in two more realistic equations describing the behaviour of the pressures in the hydrocylinder chambers. Proposition 3 provides the form of position control law for this extended MM. Similar to the approach of Section 2, the control law was derived by splitting the state variable system into two subsystems, a first one stable, and a second one taken as framework of control synthesis by backstepping technique. Eloquent simulation results illustrating the working of different control laws for EHS, obtained in this paper and also in other related works of the authors, are described in Section 5. Key features of the present work are reported in a conclusive Section 6. 2. Backstepping position control design for EHS The differential equations governing the dynamics of the EHS are those given in [3] and are reported having as a reference point the MHS SMHR included in the aileron control chain of the Romanian military jet IAR 99: 1 ðkx1 fx2 þ Sx3 Þ, m rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p a x3 Sx2 k‘ x3 þ c x_ 3 ¼ kv u ; kc 2
x_ 1 ¼ x2 ;
x_ 2 ¼
sffiffiffi 2 . c :¼ cd w r
ð2Þ
The state variables are denoted by: x1 (cm)—EHS load displacement; x2 (cm/s)—EHS load velocity; x3 (da N/cm2)—pressure difference p1 p2 between the cylinder chambers, caused by load (load pressure); u (V)—control variable. To specialise, the load at EHS rod has three components: inertial, which is dominant, viscous and elastic loads. The nominal values of the parameters appearing in Eqs. (2) are: m ¼ 0:033 da N s2 =cm—equivalent inertial load of primary control surface reduced at the EHSs rod; S ¼ 10 cm2 —EHSs piston area; f ¼ 2:75 da N s=cm—equivalent viscous friction force coefficient; k ¼ 100 da N=cm—equivalent aerodynamic elastic force coefficient; w ¼ 0:05 cm—valveport width; pa ¼ 210 da N=cm2 —supply pressure; cd ¼ 0:63—volumetric flow coefficient of the valve port; k‘ ¼ 5=210 cm5 =ðda N sÞ—coefficient of internal leakage between the cylinder chambers; kv ¼ 0:17=10 cm=V ( ¼ valve maximum displacement/maximum voltage coefficient); kc ¼ 30=12 000 cm5 =da N ð:¼ V =ð2BÞÞ—coefficient involving the bulk modulus B of the used oil and the EHSs cylinder semivolume V; r ¼ 85=ð981 105 Þ da N s2 =cm4 —oil volumetric density. The valve dynamics is evaded in MM (2); a
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proportionality coefficient kv between the control u (input voltage to servovalve) and the valve displacement xv was considered. Clearly, system (2) is lower triangular and, therefore, suitable for application of backstepping. Taking all over article the nonadaptive case—the system parameters are assumed to be known—let us introduce the notations ei ¼ xi xid ;
i ¼ 1; . . . ; 3,
(3)
where xid stand for the ‘‘desired’’ values of the state variables. So, the control objective is to have the EHS track of a specified x1d position trajectory, in other words, to have e1 ! 0. Proposition 1. Let k1 ; k2 ; k3 ; r1 ; r2 ; r3 be strictly positive tuning parameters. Under the assumption x3 opa , which claims a process with nonsaturating load, the control u given by kc u¼ r3 ckv
x3d ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 r S S k‘ x2 þ x3 þ x_ 3d k3 e3 , 2 e 2 þ r3 p a x3 m kc kc
1 rm k2 m kx1 þ fx2 1 e1 þ mx_ 2d e2 , S r2 r2
x2d ¼ x_ 1d k1 e1
(4)
(5) (6)
when applied to Eq. (2), guarantees asymptotic stability for the position tracking error e1 ¼ x1 x1d ; more precisely, limt!1 e1 ðtÞ ¼ 0.
Proof. We start the backstepping by defining the Lyapunov like function V1 ¼
r1 2 e. 2 1
(7)
The derivative of V1 along system (a.s.) (2) is V_ 1 ¼ r1 e1 ðx_ 1 x_ 1d Þ ¼ r1 e1 ðx2 x_ 1d Þ ¼ r1 e1 ðe2 þ x2d x_ 1d Þ. With Eq. (6), V_ 1 becomes V_ 1 ¼ k1 r1 e2 þ r1 e1 e2 , 1
(8)
where r1, k1 stand for weighting parameters. Now, in order to go one step ahead, a new Lyapunov like function V2 is defined as r V 2 ¼ V 1 þ 2 e22 . (9) 2 Taking the derivative of function (9) a.s. (2) gives V_ 2 ¼ V_ 1 þ r2 e2 e_2 ¼ k1 r1 e21 þ e2 r1 e1 þ r2 ðx_ 2 x_ 2d Þ h i r ¼ k1 r1 e21 þ e2 r1 e1 2 ðkx1 þ fx2 Se3 Sx3d Þ r2 x_ 2d . m
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If x3d is substituted by Eq. (5), then a simplified V_ 2 can be rewritten as rS V_ 2 ¼ k1 r1 e21 k2 e22 þ 2 e2 e3 . (10) m Remark that the new weighting parameters r2, k2 were also introduced. Finally, V3 is defined as r V 3 ¼ V 2 þ 3 e23 (11) 2 and taking the derivative of function (11) a.s. (2) one may write rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rS r p a x3 V_ 3 ¼ k1 r1 e21 k2 e22 þ e3 2 e2 þ 3 Sx2 k‘ x3 þ c kv u kc x_ 3d . m kc 2 Finally, the control u is substituted by Eq. (4). Summarising, the control law given by Eqs. (4)–(6), with a given x1d (see, for instance, as in Eq. (38)), renders the derivative V_ 3 negative semidefinite V_ 3 ¼ k1 r1 e2 k2 e2 k3 e2 (12) 1
2
3
___
and r3, k3 were the latest weighting parameters introduced. Note that as soon as the values of the state variables x1, x2, x3 and desired x1d ; x_ 1d ; x€ 1d ; x1d are measured and, respectively, known, the control u can be calculated using Eqs. (2), (3), (5) and (6). Substituting Eqs. (3)–(6) in Eq. (2), an autonomous closed loop form, analogous to Eq. (1), occurs ~ e_ ¼ FðeÞ, (10 ) namely e_1 ¼ k1 e1 þ e2 ;
e_2 ¼
r1 k2 S e1 e2 þ e3 ; m r2 r2
e_3 ¼
r2 S k3 e2 e3 . r3 m r3
(100 )
Thus, V 3 is indeed a Lyapunov function for system (100 ): Lyapunov’s asymptotic stability theorem [13] holds; moreover, Barbashin–Krassovskii’s global asymptotic stability theorem [14] also holds. We conclude that all transient tracking errors ei, i ¼ 1; . . . ; 3, concerning system (2), are asymptotic convergent to zero (the unique equilibrium point of system (100 ) is the origin). & Remark 1. Choosing the ‘‘desired’’ state variable x1d defined in Eq. (38), from the convergence to zero of the errors ei ; i ¼ 1; . . . ; 3, it easily follows that x1 ! x1s , x3 ! kx1s =S, as t ! 1. Thus, starting from an initial equilibrium point of system (2), say (0, 0, 0), the control given in Eqs. (4)–(6), (38) guides the system to attain a new equilibrium point ðx1s ; 0; kx1s =SÞ. Certainly, this state transition cannot be carried out by ignoring just transient response; physical conditioning of this response must be pffiffiffiffiffiffiffiffiffiffiffi taken into account: 0ox3 ðtÞppa , jx2 ðtÞjpc=S ðpa =2Þkv umax , juðtÞjpumax , jx1 ðtÞjpx1 max , jx1d ðtÞjpx1 max , jx2d ðtÞjpvmax , jx3d jppa , ð8Þ t40. This is another problem, beyond the aim of the present work. Because of these necessary state limitations, the global asymptotic stability of tracking errors in Proposition 1 cannot be stipulated. At all events, choosing carefully the parameters t1r and x1s , in other words, a suitable x1d , the physical limitation of the problem will be preserved. Remark 2. Let us note that MM (2) supposes (and all the models in this work, too) the fulfilment of a conjecture: during the transitory regime of tracking positive references x1d, an initially determined positive u does not change its sign; otherwise, in the last equation
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pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2) must appear the factor ðpa x3 sgn uÞ=2 and the backstepping cannot work exactly in the described manner. In fact, the hypotheses ‘‘sign u40’’ and ‘‘nonsaturating load’’ will be assumed throughout the whole paper.
___
___
Remark 3. Consider system (2) at zero equilibrium. The calculation of starting control uð0Þ, derived from Eqs. (4)–(6), (38), involves the system parameters, the tuning parameters and the derivatives x_ 1d ð0Þ, x€ 1d ð0Þ, x1d ð0Þ: sffiffiffiffiffi kc 2 r2 S uð0Þ ¼ x_ 1d ð0Þ þ r3 x_ 3d ð0Þ k3 e3 ð0Þ , r3 ckv pa m 1 r1 m k2 m ðx€ 1d ð0Þ þ k1 x_ 1d ð0ÞÞ , x_ 1d ð0Þ þ mðx1d ð0Þ þ k1 x€ 1d ð0ÞÞ þ x_ 3d ð0Þ ¼ S r2 r 2 1 k2 m ð40 Þ e3 ð0Þ ¼ mðx€ 1d ð0Þ þ k1 x_ 1d ð0ÞÞ þ x_ 1d ð0Þ . S r2 Similar tedious—but not very difficult to obtain—relationships appear also in the next propositions, but will not be retained in text.
Remark 4. We will state, as a subsidiary feature of the proposed control design, the following. Consider x1d ðtÞ ¼ 0;
tX0.
(380 )
Then, Eqs. (4)–(6) can be rewritten as x2d ¼ k1 x1 , x3d
1 ¼ S
kc u¼ r3 ckv
r1 m k1 k2 m k2 m k x1 þ f mk1 x2 , r2 r2 r2
(60 ) (50 )
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 r k1 S r3 S r2 S x1 þ 2 x2 p a x3 m kc m
r3 k‘ k3 x3 þ k3 x3d þ r3 x_ 3d . þ kc The errors, with x2d and x3d given by Eq. (60 ) and, respectively, Eq. (50 ), will be e1 ¼ x1 ; e2 ¼ x2 x2d ; e3 ¼ x3 x3d .
ð400 Þ (30 )
The entire reasoning from the proof of Proposition 1 can be reproduced step by step, to assert that in fact the function r1 2 r2 r3 1 r1 m k1 k2 m 2 ~ V3 ¼ x þ ðx2 þ k1 x1 Þ þ x3 k x1 S r2 r2 2 1 2 2 2 k2 m ð110 Þ þ f mk1 x2 r2
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is a Lyapunov function for system (2), whose derivative a.s. (2) has the form V_~ ¼ k1 r1 x21 k2 ðx2 x2d Þ2 k3 ðx3 x3d Þ2 .
(120 )
Thus, the equilibrium point (0, 0, 0) of system (2) remains asymptotically stable in the presence of small perturbations involving ðx1 ; x2 ; x3 Þ and with assistance of the control (400 ), (50 ), (60 ).
Remark 5. Let us remember that all the Lyapunov functions determined in the paper are CLFs. 3. Backstepping force control design for EHS Consider the case of force control. It can be easily verified by inspecting system (2) that the internal states x1 and x2, as described by the first two equations in Eq. (2), are stable: the roots of the characteristic equation ml2 þ f l þ k ¼ 0 are ‘‘stable’’ complex roots with negative real parts, due to the viscous friction force in hydraulic cylinder. Thus, a special care to stabilise the states x1, x2 it is not necessary. On the other hand, the output of interest in force control is, of course, the load pressure x3. Thus, evading the first two equations in the backstepping procedure, this technique cannot be applied only about x3 itself. In these circumstances, we adapt the MM to the backstepping methodology (see the conjecture of MM flexibility, Section 6): system (2) will be completed by adding an equation of first order for the dynamics of the valve position xv :¼ x4 : 1 ðkx1 fx2 þ Sx3 Þ, m rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p a x3 x_ 3 ¼ Sx2 k‘ x3 þ c x4 ; kc 2
x_ 1 ¼ x2 ;
x_ 2 ¼
1 x_ 4 ¼ ðx4 þ kv uÞ, t
ð13Þ
where t is the time constant of the (servo)valve. Proposition 2. Let k3 ; k4 ; r3 ; r4 be strictly positive tuning parameters. Under the assumption x3 opa , which claims a process with nonsaturating load, the control u given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x t r3 c pa x3 4 e 3 þ r4 þ x_ 4d k4 e4 , u¼ (14) r4 kv kc 2 t x4d
1 ¼ c
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ½Sx2 þ k‘ x3 þ kc ðx_ 3d k3 e3 Þ p a x3
(15)
when applied to Eq. (13), guarantees asymptotic stability for the pressure (force) tracking error e3 ¼ x3 x3d ; more precisely, limt!1 e3 ðtÞ ¼ 0.
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Proof. Consider r V 3 ¼ 3 e23 ; e3 :¼ x3 x3d ; e4 :¼ x4 x4d 2 and differentiating a.s. (13) yields rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r3 p a x3 _ V 3 ¼ r3 e3 e_3 ¼ e3 Sx2 k‘ x3 þ c ðe4 þ x4d Þ r3 e3 x_ 3d . kc 2 If x4d is substituted by Eq. (15), then a simplified V_ 3 is obtained rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r3 c pa x3 2 _ V 3 ¼ k3 r3 e3 þ e3 e4 , kc 2
101
(16)
(17)
where r3 and k3 are weighting parameters. Now, in order to go one step ahead, define V4 as V4 ¼ V3 þ
r4 2 e 2 4
(18)
and taking again the derivative a.s. (13), we have rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r p a x3 x4 þ kv u x_ 4d . V_ 4 ¼ k3 r3 e23 þ e4 3 c e3 þ r 4 t kc 2 Now, the control u, synthesised as in Eq. (14), renders the derivative V_ 4 negative semidefinite V_ 4 ¼ k3 r3 e23 k4 e24 . Therefore, V4 is a Lyapunov function for the nonautonomous system rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c pa e3 x3d k4 r3 c pa e3 x3d e_3 ¼ k3 e3 þ e4 e4 ; e_4 ¼ e3 . kc r4 kc 2 r4 2
(19)
(1000 )
The conditions specified in Kalman and Bertram [12] (see Theorem 1) are fulfilled and thus the asymptotic stability of the errors e3 ; e4 holds. An alternative way to finish the proof makes use of Barbalat’s lemma [15]: Let f R: ð0; 1Þ ! R be an uniformly continuous function. If there exists the finite limit t limt!1 0 f ðtÞ dt, then limt!1 f ðtÞ ¼ 0. An equivalent version is the following: If the function f(t) is differentiable and has a finite limit limt!1 f ðtÞ, and if f_ is uniformly continuous, then limt!1 f_ðtÞ ¼ 0. Thus, Barbalat’s lemma will be applied to show that the errors e3 and e4 tend to zero as time tends to infinity. Making use of definitions (15), (16) and (38) for e3 and e4 , it is easily to see that when t ! 0; we have V 4 ð0Þ40; for instance, if the system is moved from zero initial equilibrium point, then sffiffiffiffiffi !2 r4 1 2 V 4 ð0Þ ¼ kc x_ 3d ð0Þ 40. c ra 2
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Since V_ 4 p0, it follows that 0pV 4 ðtÞ V 4 ð0Þ, ð8Þt40, hence the positive function V 4 ðtÞ is bounded and it follows that e3 and e4 are bounded. Now, taking the derivative of (19) yields rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k24 2 cr3 k4 p a x3 2 2 € V 4 ¼ 2r3 k3 e3 þ 2 e4 2 e3 e4 . k3 r4 kc r4 2 Under previous hypothesis on x3 (see Remarks 1 and 2), it follows that V€ 4 is bounded, so V_ 4 is uniformly continuous. Since V 4 is decreasing, there exists a finite limt!1 V 4 ðtÞ. Then pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V_ 4 ! 0 and, consequently, e3 and e4 tend to zero. The radical pa x3 is indeed bounded with finite x3d (see Eq. (38)). &
Remark 6. The partition of system (13) in two subsystems, a first one stable, and a second one taken as framework of control synthesis by backstepping technique, is once again pointed out as approach. An articulated basis of this procedure is furnished by the proof of Proposition 3. 4. Backstepping for an extended EHS MM The foundation for the mathematical modelling of the hydraulic servomechanisms was laid down by Blackburn et al. [16]. Important books in the field appear in the decade 1965–1975 [17–19]. The hydraulic servo is in fact a valve–piston combination and the usual treatment of the corresponding pressure-flow equation has as outcome a three-dimensional model like Eq. (2). The key-equation in deriving this model is p1 þ p2 ¼ pa .
(20)
However, Eq. (20) is rather a modelling hypothesis [20] than a direct consequence of physical laws and mathematical inference on these. The evading of this equation leads to an extended MM, involving the pressures p1 ; p2 as state variables; this MM was already introduced a long while ago by Wang [21], but only sporadically was taken back. A variant of the Wang’s model can be written in the form x_ 1 ¼ x2 ;
1 ½kx1 fx2 þ Sðx3 x4 Þ, m pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½cx5 pa x3 Sx2 k‘ ðx3 x4 Þ,
x_ 2 ¼
B V þ Sx1 pffiffiffiffiffi B x_ 4 ¼ ½cx5 x4 þ Sx2 þ k‘ ðx3 x4 Þ, V Sx1 x5 þ kv u . ð21Þ x_ 5 ¼ t This time, the variables x3 and x4 are the pressures p1 ; p2 in the cylinder chambers, and x5 stands for the valve position. x_ 3 ¼
Proposition 3. Let k1 40; k2 40 be tuning parameters. Under the (rather physical) assumptions 0ox3 opa ; 0ox4 opa ; jx1 joV =S, the control u given by u¼
1 ½x5d þ tðx_ 5d k2 g2 ep Þ, kv
(22)
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x5d ¼
1 ðk1 ep p_ d þ g1 Þ, g2 2BV ðSx2 þ k‘ pÞ ; g2 ¼ g2 ðx1 ; x3 ; x4 Þ V 2 S 2 x21 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi p a x3 x4 :¼ Bc þ , V þ Sx1 V Sx1
103
(23)
g1 ¼ g1 ðx1 ; x2 Þ :¼
ep :¼ p pd ; p :¼ x3 x4 ; e5 ¼ x5 x5d , pd ðtÞ :¼
k k x1d ðtÞ :¼ x1s ð1 et=t1r Þ, S S
ð24Þ (25) (26)
when applied to Eq. (21), guarantees asymptotic stability for the position tracking error e1 :¼ x1 x1d ; more precisely, limt!1 e1 ðtÞ ¼ 0. The notations x1s ; t1r define the expression of the time response x1d of a first-order stable system to step input x1s (see Eq. (38)); t1r is the time constant and x1d ð0Þ is taken to be zero. Proof. As in the previous propositions, consider Lyapunov like function: V1 ¼
1 2 e . 2 p
(27)
Then, its derivative a.s. (21), by using Eqs. (23)–(25), is V_ 1 ¼ k1 e2p þ g2 ep e5 .
(28)
Now, extend V 1 as V2 ¼ V1 þ
1 2 e 2k2 5
(29)
and, taking into account Eq. (22), the derivative a.s. (21) is 1 2 e. V_ 2 ¼ k1 e2p k2 t 5 The following equations for the errors ep, e5 hold: e5 e_p ¼ k1 ep þ g2 e5 ; e_5 ¼ k2 g2 ep . t € Performing the derivative V 2 a.s. (21), we obtain 1 2 1 2 2 € V 2 ¼ 2 k 1 ep þ e 2 k 1 g2 e p e 5 . k 2 t2 5 t
(30)
(31)
Repeating the arguments which end the proof of Proposition 2, it results that the errors ep and e5 are bounded; so, p ¼ pðtÞ is also bounded on Rþ ¼ ½0; 1Þ. Furthermore, V€ 2 is bounded, provided that g2 remains bounded during the dynamical process; this condition holds, having in view the assumptions involving the variables x1 ; x3 ; x4 . So, V_ 2 is uniformly continuous, and applying Barbalat’s lemma, V_ 2 ! 0. Consequently, ep and e5 tend to zero.
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Now, let us look at the first two equations in Eq. (21), which can be rewritten as follows: rffiffiffiffi f k S 2 ; r :¼ ; p1 ¼ p (32) x€ 1 þ 2hx_ 1 þ r x1 ¼ p1 ; h :¼ 2m m m and p is now seen as a bounded function of t, p :¼ pðtÞ ¼ x3 ðtÞ x4 ðtÞ. With initial conditions x1 ð0Þ ¼ x_ 1 ð0Þ ¼ 0, the solution of Eq. (32) is Z 1 ht t hu x1 ðtÞ ¼ e e p1 ðuÞ sin oðt uÞ du; (33) o 0 where the condition o2 :¼ r2 h2 40 is inherent to hydraulic servo systems owing to small viscous friction in cylinder (see Remark 7). Introducing S p ðtÞ m d let us also consider Z t 1 x~ 1d ðtÞ :¼ eht ehu p1d ðuÞ sin oðt uÞ du: o 0 p1d ðtÞ :¼
(34)
(3300 )
Since ep ! 0 when t ! 1, it is clear that p1 ðtÞ ! p1d ðtÞ, as t ! 1; this means: ð8Þe40, (() dðeÞ such that for t4dðeÞ we have jp1 ðtÞ p1d ðtÞjoe. Then, if t4dðeÞ Z t 1 jx1 ðtÞ x~ 1d ðtÞjp eht ehu jp1 ðtÞ p1d ðtÞjj sin oðt uÞj du o 0 Z e t hðtuÞ e 1 eht . p e du ¼ o 0 o h It results in x1 ðtÞ ! x~ 1d ðtÞ as t ! 1.
(35)
Then, successive calculations give Z 1 ht t hu S k x1s ð1 eu=tr Þ sin oðt uÞ du x~ 1d ðtÞ ¼ e e o mS 0 Z x1s keht t hu e ð1 eu=tr Þ sin oðt uÞ du, ¼ om 0 Z
t
ehu ð1 eu=tr Þ sin oðt uÞ du ¼ 0
ðh 1=tr Þ sin ot oðeðh1=tr Þt cos otÞ o2 þ ðh 1=tr Þ2 þ
h sin ot þ oðeht cos otÞ , o 2 þ h2
therefore x~ 1 ðtÞ ! x1s
as t ! 1.
(36)
Thus, from Eqs. (35) and (36), a standard proceeding gives x1 ðtÞ ! x1s
as t ! 1
and so ends the proof.
&
(37)
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Remark 7. The reasoning given above can be easily recomposed for the case of real negative roots of the characteristic equation l2 þ 2hl þ r2 ¼ 0. 5. Simulation results The EHS is in fact a tracking system. Therefore, for this system the aim of control synthesis is to have a good tracking of the specified desired position references introduced as electrical signals. Thus, the goal of the backstepping synthesised control is to have the EHS tracking of the specified x1d position or x3d pressure references. Such references can be expressed as a response of a first-order system to step inputs xis xid ¼ xis ð1 et=tir Þ;
i ¼ 1 or 3,
(38)
where x1s and x3s stand for stationary value of the states x1, respectively, x3 and t1r and t3r stand for associated desired time constants. As reference points of the numerical simulations we take both the experimental data and a three-dimensional MM of the MHS SMHR [3] 1 ðkx1 fx2 þ Sx3 Þ, m rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p a x3 x_ 3 ¼ Sx2 k‘ x3 þ cs ; kc 2 x_ 1 ¼ x2 ;
x_ 2 ¼
s :¼ lðr x1 Þ,
ð39Þ
where l ¼ 23 is the coefficient of the rigid feedback kinematics, r is the reference input at servo rigid feedback kinematics linkage point (cm) and s is the valve error signal. The analogous model with two pressure state variables is 1 ½kx1 fx2 þ Sðx3 x4 Þ, m pffiffiffiffiffi B B pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½cs pa x3 Sx2 ; x_ 4 ¼ ½cs x4 þ Sx2 . x_ 3 ¼ V þ Sx1 V Sx1
x_ 1 ¼ x2 ;
x_ 2 ¼
ð40Þ
Zero equilibrium points for systems (39) and (40) correspond to closed valve—s ¼ 0—in the case of zero reference input, r ¼ 0. Numerical simulations have been performed to investigate the performance of the proposed nonlinear controllers, in the three cases defined by the nonlinear control laws (4), (14) and (22). Zero initial conditions—corresponding to the zero equilibrium points—were chosen for a fourth-order Runge–Kutta integration (with integration step 0.001 s) of the systems: Eq. (39), with a simple, rigid mechanical position feedback; Eq. (2), with control law (4); (13), with control law (14); (40), with a simple, rigid mechanical position feedback; and (21), with control law (22); in the last two cases, initial conditions on the variables x3 and x4 were chosen 1 da N/cm2. A relatively good tracking of references r is possible with the MHS SMHR (39), but only in the absence of perturbations [22,23]. To have in this servo a concrete term of comparison for the position control technique developed in Section 2, we choose the step input r ¼ 0:17 cm=l ¼ 0:255 cm. Then, having this step reference, the entire valve port is in fact open to flow passing and, consequently, the resulting servo time constant characterises the best step input tracking with the ‘‘passive’’ system: tsffi0.034 s (Fig. 1a, with the error signal s represented; initial error: s0 ¼ 0:17 cm, the maximum length of the valve port).
ARTICLE IN PRESS I. Ursu et al. / Journal of the Franklin Institute 343 (2006) 94–110 0.256
0.2
0.192 0.128
0.1
0
0.05
0.1
(a)
0.15
0.128
5
0.064
2.5 0
0 0.2
0
x 1d u
0.064
σ
0.192
7.5 u [V]
x1
0.05
x1
x 1 [cm]
σ [cm]
0.15
0.256
10
0
0.025
(b)
t [s]
0.05
0.075
x 1, x 1d [cm]
106
0 0.1
t [s]
Fig. 1. Comparison of the position step input tracking systems with load pressure state variable: (a) SMHR case: ts ffi 0:034 s; (b) EHS case with backstepping position control: ts ffi 0:022.
75
x3
4
50
2
25
0
8
0.1
0.2
0.3 t [s]
0.4
0.5
6
(b)
140 105
4
70
2
35 0
0.6
175 x3d x3
0
0 0
u
x3, x3d [daN/cm2]
u [V]
6
(a)
10
100 x3d
u [V]
u
x3, x3d [daN/cm2]
8
0 0.11 0.22 0.33 0.44 0.55 0.66 t [s]
Fig. 2. Backstepping pressure control: (a) x3s ¼ 100 da N=cm2 , t3r ¼ 0:1 s, ts ffi 0:16 s; (b) x3s ¼ 175 da N=cm2 , t3r ¼ 0:05 s, ts ffi 0:175 s.
The following definition of the servo time constant is used: the time necessary to system transient response to attain 66% of the value of stationary regime. An evidently better tracking of step references is ensured by backstepping position control synthesis with the values of the tuning parameters: k1 ¼ 400 s1 ; k2 ¼ 4 s; k3 ¼ 400 cm6 =ðda N2 sÞ; r1 ¼ 400; r2 ¼ 0:033 s2 ; r3 ¼ 1 cm6 =da N2 . The desired control objective, in terms of position reference defined by x1s ¼ 0:255 cm and t1r ¼ 0:01 s, is accomplished with faster servo time constant ts ffi 0:022 s; when t1r was chosen 0.005 s, the same time constant ts ffi 0:022 s was obtained (Fig. 1b). The transient regime is stable, irrespective of stationary regime values x1s (admissible) and t1r; however, the designer must be attentive to control saturation. To counteract this effect, special antiwindup strategies can be used [3,22,23]. Similar conclusions are valid in the case of backstepping force control (Fig. 2). The fit values for the tuning parameters were: k3 ¼ 400 s1 ; k4 ¼ 800 da N2 =ðcm6 sÞ; r3 ¼ 1010 ; r4 ¼ 1 da N2 =cm6 ; also, let note the value f ¼ 1 daNs=cm used here. A fit time constant 1 t ¼ 573 s of the valve was considered. Worthy noting, herein the coefficient kv ¼ 1:7 10 cm=V means a tenfold necessary valve port area. The desired control objective, in terms of pressure reference defined by x3s ¼ 100 da N=cm2 and t3r ¼ 0:1 s (Fig. 2a), and x3s ¼ 150 da N=cm2 and t3r ¼ 0:1 s, is accomplished with good servo time constants: ts ffi 0:16 s
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and, respectively, ts ffi 0:19 s; when x3s and t3r were chosen as 175 da N/cm2 and, respectively, 0.05 s, an also good servo time constant ts ffi 0:175 s was obtained (Fig. 2b). It can be seen that the state variables x1, in the first case, and x3, in the second case, come very close to the desired x1d and, respectively, x3d, values. The alternating MM (40) of the MHS SMHR promote a somewhat ‘‘stiffer’’ object; the same is the conclusion concerning system (21) governed by the control (22). Fig. 3 shows fit results obtained choosing r expressed in form (38), with x1s ¼ 0:255 cm and t1r ¼ 0:002 s; the HM SMHR MM produces the servo time constant ts ¼ 0:037 s (Fig. 3a) and, with the fit tuning parameters k1 ¼ 1000 s1 and k2 ¼ 105 da N2 cm6 , the corresponding Eqs. (21), (22) system denotes the same servo time constant ts ¼ 0:037 s. Compared with the systems having the load pressure as state variable, the systems with two pressures as state variables are dynamically slower. The time histories of the load pressures, depicted in Fig. 4, offer an explanation of the systems dynamic behaviour. The faster system is by all means that actuated by greater values of the external driving force; this force is derived from load pressure: Sx3, in the case of the first two equations in Eq. (2); and, respectively, S(x3x4), in the case of the first two equations in Eq. (21). Table 1 summarises the main quantifying results of the systems performance, expressed as servo time constants obtained in this work and related works of the authors. Slight
0.1
8
0.192
6
0.128
0.05 0 0
0.05
0.1
4
0.128
2
0 0.15
0.064
0 0
t [s]
(a)
0.192
u
0.064
σ
0.256
x1
x1d
0.05
0.1
x1,x1d [cm]
σ [cm]
0.15
0.256
u [V]
x1
x1 [cm]
0.2
0 0.15
t [s]
(b)
30
3
20
2.5
10 0 0 -10
0.05
0.1 t [s]
0.15
0.2
x3- x4 [daN/cm2]
x3 [daN/cm2]
Fig. 3. Comparison of the position step input tracking systems with two pressure state variables: (a) SMHR case: ts ffi 0:037 s; (b) EHS case with backstepping position control: ts ffi 0:037 s.
-20
1.5 1 0.5 0 0
-30 (a)
2
(b)
0.01
0.02 0.03 t [s]
0.04
0.05
Fig. 4. Comparison of the load pressure evolution: (a) system (2), (4), with load pressure state variable; (b) system (21), (22), with two pressure state variables.
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Table 1 Summary of the servo time constants for position references tracking: MHS SMHR case and related presumptive EHS with various control laws Synthesis strategy
Back-stepping, third-order model (2), (4), with load pressure state variable; step reference amplitude: r ¼ 0:255 cm Backstepping, fifth-order model (21), (22), two pressures as state variables; step reference amplitude: x1s ¼ 0:255 cm Neuro-fuzzy antiwindup synthesis [4], fourth-order Wang type model [21], two pressures as state variables; step reference amplitude r ¼ 0:1 cm, and easy modified l (l ¼ 0:55) and f (f ¼ 1 da N s=cm) Davison type robust servo synthesis [24], with antiwindup compensation [22,23], linear model [17–19], [3]; step reference amplitude r ¼ 0:3 cm (derived from position transducer gain) and f ¼ 1 da N s=cm, which compensates in some degree the increased r Measured servo time constant [25], step reference amplitude r ¼ 0:12 cm
Servo time constant ts (s) EHS
MHS SMHR
0.022
0.034 (system (39)) 0.037 (system (40)) 0.025
0.037 0.011
0.021
0.024 (theoretical servo time constant, derived from linear system) 0.02
changes in parameter values—involving the amplitude of step input (r or x1s), the equivalent viscous friction coefficient f, or the coefficient of rigid feedback kinematics l— do not affect the location of all calculated or determined from numerical simulation servo time constants in the close neighbourhood of the experimentally measured [25] servo time constant of the MHS SMHR. Indeed, roughly speaking, the servo time constant decreases with decreasing f, r or x1s; the faster time response in the case of neuro-fuzzy antiwindup synthesis can so be explained in Table 1. Therefore, the simulation studies attest good tracking performance in the presence of step signals. Based on sampled signals paradigm, a good tracking performance of arbitrary references can be also stipulated, having in view the decomposition of an arbitrary signal into a succession of step signals. 6. Concluding remarks The objectives of the research presented in this paper were twofold. First, we wanted to exploit a recently introduced technique of nonlinear control design—the backstepping synthesis—in the framework of the EHS design. The problem was a challenging one, considering the strong nonlinear feature of the MMs and the topical quality of these standard objects of the automatic systems. Certainly, if the reader is more interested in comprehending the backstepping machinery, she or he very well might find the EHS framework only a pretext. Second, we tried to offer new arguments that lend support to certain earlier enounced conjectures [3]. This time, several MMs of EHSs, obtained from various control strategies, were quantified and compared from the viewpoint of the servo time constant performance and all the values were found to be close to the experimental one [25]. Three case studies on
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backstepping synthesis for controlling EHS systems are presented. The full state information was considered available. We have illustrated how the main theory can be brought or adapted to design practice as defined by a given MM, and have shown that the backstepping controllers are able to work with a complex plant such as EHS. The performance of the position tracking controllers has been finally compared with that of several other techniques, such as antiwindup neuro-fuzzy synthesis [4,26] and classical robust synthesis [24] with antiwindup compensation [22,23]. A certain trend of servo time constants improvement by control synthesis strategies can be attested. Smaller such values can be obtained, but with the risk to compromise a stability-rapidity trade-off of the servo time response. Thus, our empirically derived conjectures [3] concern the behaviour of systems MMs in connection with mathematical methodologies of control synthesis operating on them; in Ref. [3], as basis of these conjectures has been served the field of active and semiactive road vehicle suspension control synthesis. First conjecture. All the MMs, describing the same object, will promote in the last analysis a certain ‘‘firmness’’ and ‘‘solidarity’’, in other words, will predict approximately the same object performance or behaviour improvement, in despite of the used mathematical methodology; see, herein, the results given in Table 1. In fact, there are always more profound structural-mechanical reasons explaining the limitation of achieved amount of improvement obtained by insertion of control in mechanical structures [3,27]. Second conjecture. The MM, as a thinking first result, derived from a physical reality, is more ‘‘flexible’’ than a methodology (i.e., a mathematical theory [28]), as a thinking second result, derived from a metareality (herein, the mathematical thinking). Thus, the first conjecture refers to certain conservative properties of MMs. The second conjecture advances a compensating property of MMs. This is a complemental, flexibility property: to be able to apply a certain mathematical construction, a mathematical model can be often ‘‘shaped’’ in view of coping with the constraints of a certain mathematical apparatus (see the approach in Section 3). Another result of the paper concerns the opportunity to convert a position servo into a force one with only minimal hard modification in valve port area. The backstepping has not been applied before to control a position tracking EHS used in aviation, to the best of the author’s knowledge. Acknowledgements The work described above was supported (partially) by the Romanian Space Agency (ROSA), Contract Nos. 85/2003 and 144/2004. The authors would like to thank Prof. Andrei Halanay for helpful discussions and his valuable advice, and Prof. Lucian Iorga, for suggesting many improvements of the text. References [1] J. Pippenger, T. Hicks, Industrial Hydraulics, third ed., McGraw-Hill, New York, 1979. [2] I. Ursu, F. Popescu, M. Vladimirescu, R. Costin, On some linearization methods of the generalized flow rate characteristic of the hydraulic servomechanisms, Rev. Roumaine. Sci. Tech. Se´r. Me´cani. Appl. 39 (2) (1994) 207–217.
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