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Accurate state estimation for a hydraulic actuator via a SDRE nonlinear filter
Mechanical Systems and Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Accurate state estimation for a hydraulic actuator via a SDRE nonlinear filter Salvatore Strano, Mario Terzo n Department of Industrial Engineering, University of Naples Federico II, Italy
a r t i c l e i n f o
abstract
Article history: Received 17 March 2015 Received in revised form 7 September 2015 Accepted 8 December 2015
The state estimation in hydraulic actuators is a fundamental tool for the detection of faults or a valid alternative to the installation of sensors. Due to the hard nonlinearities that characterize the hydraulic actuators, the performances of the linear/linearization based techniques for the state estimation are strongly limited. In order to overcome these limits, this paper focuses on an alternative nonlinear estimation method based on the StateDependent-Riccati-Equation (SDRE). The technique is able to fully take into account the system nonlinearities and the measurement noise. A fifth order nonlinear model is derived and employed for the synthesis of the estimator. Simulations and experimental tests have been conducted and comparisons with the largely used Extended Kalman Filter (EKF) are illustrated. The results show the effectiveness of the SDRE based technique for applications characterized by not negligible nonlinearities such as dead zone and frictions. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Hydraulic actuator State Dependent Riccati Equation (SDRE) Nonlinear state estimation Extended Kalman Filter (EKF)
1. Introduction The ability of the hydraulic actuators of applying large forces with a fast response time, together with their size-to-power ratio, make them widely used in industrial applications. Typical usage concerns seismic test rig [1,2], positioning systems [3,4], vibration control [5,6]. The state estimation of such systems is fundamental for their operation [7]. Indeed, the hard working conditions and the complexity of the hydraulic systems strongly exhort to take under consideration their reliability and, so, the state observers become functional for a fast and economical detection of faults. At the same time, they are a valid alternative to the installation of sensors to give feedback to the control laws in case of prohibitive costs or harsh environments [8]. In any case, the highly nonlinear dynamics of the hydraulic actuators, due to the dead zone of the control valve, the pressure-flow rate relationship and frictions, makes the design of the state observers a challenging problem. Several approaches have been proposed in literature for the state estimation in hydraulic actuators. These methods include linear approaches, linearized model based techniques [9] and robust observers based on nonlinear models [10–12]. The high degree of parametric uncertainty that characterizes hydraulic actuators exhorts the researchers to follow dedicated approaches for the state estimation. Extended Kalman Filter (EKF) has been adopted to detect the fault in hydraulic actuators [13] by means of the state estimation. The approach is based on the local linearization of the nonlinear system and allows to closely track the state trajectories if compared with the linear approach of the Kalman filter [14]. Adaptive observers [15] and nonlinear adaptive robust observers [16,17] have been also developed. The last ones are based on robust
n
Corresponding author. E-mail address: [email protected] (M. Terzo).
http://dx.doi.org/10.1016/j.ymssp.2015.12.002 0888-3270/& 2015 Elsevier Ltd. All rights reserved.
Please cite this article as: S. Strano, M. Terzo, Accurate state estimation for a hydraulic actuator via a SDRE nonlinear filter, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.12.002i
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filter structure, to compensate the effect of the unmodeled dynamics, combined with on-line parameter adaptation, to account for the parametric uncertainty. This paper investigates on an alternative nonlinear technique for the state estimates of the hydraulic actuators. The approach is based on the State-Dependent-Riccati-Equation (SDRE) nonlinear filtering formulation. The SDRE techniques are recently emerging for optimal nonlinear control and filtering techniques. The SDRE filter (SDREF) originates from a suboptimal nonlinear regulator technique that uses parameterization to bring the nonlinear system into a linear-like structure with state-dependent coefficients (SDC) and is characterized by the structure of the steady state Kalman filter and the Kalman gain is obtained by solving an algebraic Riccati equation. The SDRE based techniques have been adopted in advanced guidance [18,19], in the control of hydraulic seismic isolator test rig [20], in sensor fusion applications [21], in output feedback nonlinear H2 autopilot [22]. Optimality, suboptimality, and stability properties have been discussed in [23], while the SDRE filter formulation has been used in [24]. A fundamental advantage of the SDRE filter technique consists of the possibility of employing a fully nonlinear nominal model, also including hard nonlinearities such as the dead zone, allowing to avoid linearization and unmodeled dynamics. Several hydraulic actuation systems present not negligible nonlinearities that have to be considered for the design of the state observers: this is common in the high power hydraulic actuation systems that employ, for example, control valves characterized by the lands of the spool greater than the annular parts of the valve body. Furthermore, differently from the adaptive-robust approaches [7], the SDRE based Kalman filter well manages the measurement noise [25] and allows to adopt a more faithful nominal model. Consequently, the SDREF implies the accurate knowledge of system in order to take into account the main dynamical characteristics. The paper is organized as follows: a fifth order nonlinear model is derived in Section 2 taking into account dead-zone, frictions and the nonlinear pressure-flow rate relationship; Sections 3 and 4 focus on the EKF and on the SDREF, respectively. Section 5 shows the simulation results, Section 6 the experimental ones, while conclusions are drawn in Section 7.
2. Dynamic model The hydraulic actuation system under consideration consists mostly of a double-rod hydraulic cylinder and a proportional valve. The hydraulic cylinder is linked to a mass that moves on a linear guide (Fig. 1). For the derivation of the mathematical model, some assumptions are made: the tank pressure PT is equal to zero, the fluid properties are not dependent on the temperature, the piston areas and the chamber volumes are equal, the internal and external fluid leakages are negligible. The dynamics of the movable mass displacement (y) is governed by: my€ þ σ y_ þF f ðy_ Þ ¼ Ap P L ;
ð1Þ
where m is the mass of the load, σ is the viscous friction coefficient, Ff is the friction force, Ap is the piston area, PL ¼PA–PB is the load pressure, PA and PB are the pressures inside the two chambers of the cylinder. The friction force is represented by the following equation: ( F c sgnðy_ Þ þ μmgsgnðy_ Þ y_ a0 F f ðy_ Þ ¼ ð2Þ y_ ¼ 0 Z jZ jr F c0 þ μ0 mg where Fc is the Coulomb friction force in the hydraulic actuator (assumed equal to its static value Fc0), μ is the Coulombian friction coefficient of the linear guide (assumed equal to its static value μ0), g is the gravitational acceleration and Z is the net tangential force that acts on the actuator when it is not moving [26]. The load pressure dynamics [27] is given by: V0 _ P L ¼ Ap y_ þ Q L 2β
ð3Þ
where V0 is the volume of each chamber for the centered position of the piston, QL ¼(QA þQB)/2 is the load flow and β is the effective Bulk modulus.
Fig. 1. Schematic diagram of the hydraulic actuation system.
Please cite this article as: S. Strano, M. Terzo, Accurate state estimation for a hydraulic actuator via a SDRE nonlinear filter, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.12.002i
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The load flow depends on the supply pressure, the load pressure and the valve spool position in accordance with the following: Q L ¼ Ψ ðve Þve
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P s jP L j
where ve is the displacement signal of the spool valve and proportional valve. The analytical expression of Ψ(ve) can be assumed as: 8 h i kq0 vep > > ve 4 vep > kqp 1 þ kqp 1 ve > < ven rve r vep Ψ ðve Þ ¼ kq0 h i > > kq0 > ven > ve o ven k 1 þ 1 : qn ve kqn
ð4Þ
Ψ(ve) is a gain that depends on the geometry of the adopted
ð5Þ
where ven, vep, kq0, kqn, and kqp are parameters that must be identified experimentally. The equations in (11) represent an asymmetric bilinear function of the valve spool position. If kq0 ¼kqp ¼kqn, the function Ψ(ve) reduces to a constant gain. If kq0 ¼0, the product Ψ(ve)ve describes a dead zone function, where ven and vep are the limits of the dead zone and kqn and kqp are the gains if ve is negative or positive respectively. The adoption of Ψ(ve) is particularly efficient to describe a dead zone function if an overlapped valve is adopted [20]. The proportional valve dynamics can be well represented by a second order differential equation [28]: 2ξ v€ e þ v v_ þ v ¼ ke u þ ve0 ω2nv ωnv e e
ð6Þ
where parameters ωnv and ξv are the natural frequency and the damping ratio of the valve respectively, ve0 is the spool position bias, ke is the input gain and u is the valve command. Finally, the equations governing the dynamics of the whole system (movable massþhydraulic system) are: 8 my€ þ σ y_ þ F f ðy_ Þ ¼ Ap P L > > > > > < V 0 P_ L ¼ Ap y_ þ Q L 2β ð7Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > Q L ¼ Ψ ðve Þve P s jP L j > > > > : v€ e ¼ ω2 ve 2ξ ωnv v_ e þ ω2 ðke u þ ve0 Þ v nv nv The developed fifth order model fully describes the nonlinear dynamical behavior of the hydraulic actuation system and takes the nonlinear friction forces and the nonlinear flow rate distribution into account. The nonlinear system (7) can be written in the following form: 8 _Þ A P f ðy σy _ Fmv > y€ ¼ m ve þ pm L > e > > pffiffiffiffiffiffiffiffiffiffiffiffiffi < 2βΨ ðve Þ P s jP L j 2βA P_ L ¼ V 0 p y_ þ ve ; V0 > > > > : v€ e ¼ 2ξv ωnv v_ e 1 vve0 ω2nv ve þ ω2nv ke u e
ð8Þ
The system (8), which state vector is given by x ¼ yL ve ve T , is nonlinear in the state and autonomous. It is possible to note in the first and in the third equation of (8) a division for the variable ve, as well as it happens in (5). In order to prevent divisions by zero, the variable v~ e ¼ ve þ ε has been introduced, where ε is given by: ( Δv e ve ¼ 0 ϵ¼ ð9Þ 0 otherwise where the value of Δve can be assumed less than the discrimination threshold of the valve spool position sensor. Replacing ve with v~ e , where ve is at the denominator of fractions in (5) and (8), it follows: 8 _ σy _ Fmf ðv~yÞve þ ApmPL > y€ ¼ m > e > > pffiffiffiffiffiffiffiffiffiffiffiffiffi < _P L ¼ 2βAp y_ þ 2βΨ ðv~ e Þ P s jP L jve : V0 V > 0 > > v > 2 2 e0 : v€ e ¼ 2ξv ωnv v_ e 1 v~ ωnv ve þ ωnv ke u e
ð10Þ
The nonlinear equations of the system (10) have been adopted in order to derive the EKF and the SDREF. Given the measurements of signals like the load displacement y, the load pressure PL and the spool valve displacement ve , the objective is the estimation of the states of the system. Please cite this article as: S. Strano, M. Terzo, Accurate state estimation for a hydraulic actuator via a SDRE nonlinear filter, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.12.002i
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3. The Extended Kalman Filter The linearization procedure is at the basis of the EKF approach, which is briefly recalled in the following since it has been adopted for a comparative analysis. The system and the measurement equations can be generically represented by: _ ¼ fðxðtÞ; uðtÞÞ þ ψk ; xðtÞ
ð11Þ
zðtÞ ¼ hðxðtÞ; uðtÞÞ þgk
ð12Þ
being x the state vector, f and h non-linear functions, u the input vector, ψk the process noise with covariance Q k , z the measurement vector, and gk the Gaussian white measurement noise with covariance R k . The estimator can be implemented in a discrete time form, integrating the system equation from time t k 1 to time t k . The EKF methodology is conceptually based on two fundamental steps, namely estimates and updates steps. Denoting the estimates as ð^ Þ, the following initializing conditions are applied to the state estimates (13) and to the error covariance (14): þ x^ 0 ¼ Eðx0 Þ
P 0þ ¼ E
ð13Þ
þ þ T x0 x^ 0 x0 x^ 0
ð14Þ
being E the expected value. The state estimates and the estimation of the error covariance are given by (15) and (16) respectively:
with
þ x^ k ¼ f k 1 ðx^ k 1 ; uk 1 Þ;
ð15Þ
Pk ¼ Ak 1 Pkþ 1 ATk 1 þ Lk 1 Q k 1 LTk 1 ;
ð16Þ
∂f k 1 Ak 1 ¼ ∂x^ ∂f k 1 Lk 1 ¼ ∂ψk
;
ð17Þ
:
ð18Þ
þ x^ k 1
þ x^ k 1
With the computation of the filter gain (19) and evaluating the measurement residual, the updates of the state estimates (20) and of the estimation of the error covariance (21) can be determined: Kk ¼ Pk HTk ðHk Pk HTk þ Mk R k MTk Þ 1 ;
ð19Þ
þ x^ k ¼ x^ k þ Kk zk hk ðx^ k ; uk Þ ;
ð20Þ
Pkþ ¼ ðI Kk Hk ÞPk ;
ð21Þ
where:
∂hk Hk ¼ ∂x^
;
ð22Þ
:
ð23Þ
x^ k
∂hk Mk ¼ ∂gk
x^ k
4. SDRE nonlinear filter SDRE techniques are used as control and filtering design methods and are based on state dependent coefficient (SDC) factorization [24]. Infinite-horizon nonlinear regulator problem is a generalization of time invariant infinite horizon linear quadratic regulator problem where all system coefficient matrices are state-dependent [29,30]. When the coefficient matrices are constant, the SDRE control method changes into the steady-state linear regulator. Filtering counterpart of the SDRE control algorithm is obtained by taking the dual system of the steady-state linear regulator and then allowing coefficient matrices of the dual system to be state-dependent. Starting from the nonlinear system (10), there are infinite Please cite this article as: S. Strano, M. Terzo, Accurate state estimation for a hydraulic actuator via a SDRE nonlinear filter, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.12.002i
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solutions to transform this nonlinear system into an SDC form as: _ ¼ FðxðtÞ; uðtÞÞx þ ψ; xðtÞ
ð24Þ
zðtÞ ¼ HðxðtÞ; uðtÞÞx þ g;
ð25Þ
where fðxðtÞ; uðtÞÞ ¼ FðxðtÞ; uðtÞÞx and hðxðtÞ; uðtÞÞ ¼ HðxðtÞ; uðtÞÞx;
ð26Þ
ψ is the process noise with covariance Q , and g is the Gaussian white measurement noise with covariance R. Starting from the SDC form, the derivative of the state estimate is given by: ðx ;uÞx þ Kf ðx Þ½zðxÞ Hðx ;uÞx x^ ^
^
^
^
^
ð27Þ
where
^ u R 1 Kf x^ ¼ PðxÞHT x;
ð28Þ
and P is the positive definite solution of the algebraic Riccati Eq. (29).
^ u P x^ þP x^ FT x; ^ u P x^ HT x; ^ u R 1 H x; ^ u P x^ þQ ¼ 0 F x; The highly nonlinear Eq. (10) has been parameterized in SDC form with the following choice [31]: 3 2 F ðy_ ̇ Þ A σ m 0 mp 0 mf v~ e 7 6 6 1 7 0 0 0 0 ffiffiffiffiffiffiffiffiffiffiffiffiffi 6 7 p 7 6 2 β Ap 2βΨ ðv~ e Þ P s jP L j 7 0 0 0 Fðx; uÞ ¼ 6 V0 V0 7 6 7 6 ω2nv ke u 7 ve0 6 0 2 0 0 2ξv ωnv 1 v~ e ωnv þ v~ e 5 4 0 2
0 6 Hðx; uÞ ¼ H ¼ 4 0 0
0
0
1
1
0
0
0
1
0
7 0 5:
0
0
0
1
0
ð29Þ
ð30Þ
0
3 ð31Þ
It is important to note that the EKF method, presented in the previous section, is based on the linearization of the nonlinear model (10) as indicated in (17), (18), (22), and (23). Differently, the mathematical derivation of the SDREF highlights that in this case no linearization has been performed as it can be noted in (24)–(31).
5. Simulation results with comparisons In order to evaluate the benefits of the proposed estimator, simulations have been carried out on the mathematical model of a hydraulic actuator and, successively, an experimental validation has been executed. The hydraulic actuator has been modeled by means of (10), in which the parameters have been determined by means of an identification procedure [20] and are: 3 m ¼ 440 kg, σ ¼ 23555 Nms, F C ¼ F C0 ¼ 950 N, μ ¼ μ0 ¼ 0:01, Ap ¼ 0:01 m2 , V 0 ¼ 0:004 m3 , β ¼ 1e9 Pa, kq0 ¼ 1e 12 m 1 , s V Pa2
kqp ¼ 6:15e 7 1 , kqn ¼ 5:86e 7 1 ,vep ¼ 0:43 V, ven ¼ 0:21 V, ve0 ¼ 0:01 V, ke ¼ 0:49, ωnv ¼ 152 rad=s, ξv ¼ 0:92, s V Pa2 s V Pa2 P S ¼ 6e6 Pa. m3
m3
Fig. 2. Input voltage (first simulation test).
Please cite this article as: S. Strano, M. Terzo, Accurate state estimation for a hydraulic actuator via a SDRE nonlinear filter, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.12.002i
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Fig. 3. a) Load velocity (first simulation test); b) zoom view.
Fig. 4. Load displacement (first simulation test).
The simulated experiments have been obtained by means of open loop tests employing sine and square waves as input. The performances of the SDRE filter will be evaluated by comparison with the EKF estimator and the real state. The observers have been designed taking into account as input the signal u and the measurements given by displacement y, the load pressure PL and the spool valve displacement ve . The first simulated result is based on a sine wave given as input (Fig. 2). The parameters adopted in the observers are the same of the hydraulic actuator model. The first important result is shown in Fig. 3, where the capability of the SDREF is easily visible: indeed, the comparison with the EKF technique highlights substantial differences due to the linearization process. The nonlinearities of the system are fully considered in the SDREF and, consequently, all the null speed zones, caused by the dead zone, are well estimated. This result finds an important application for fault detection algorithm and in velocity feedback control in which, for example, the speed sensor is not adopted. With the particular reference to the hydraulic actuators, the nonlinearities involve the load velocity equation and the load pressure one. As a consequence, the differences between the EKF and the SDREF are evident if the time histories of the estimates of the cited states are evaluated. In any case, with the aim of completeness of the discussion, the comparisons for all the states are shown in the following. Fig. 4 exhibits the comparison in terms of load displacement, Fig. 5 shows the comparison for the load pressure. Please cite this article as: S. Strano, M. Terzo, Accurate state estimation for a hydraulic actuator via a SDRE nonlinear filter, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.12.002i
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Fig. 5. a) Load pressure (first simulation test); b) zoom view.
Fig. 6. Spool velocity (first simulation test).
Fig. 7. Spool displacement (first simulation test).
Please cite this article as: S. Strano, M. Terzo, Accurate state estimation for a hydraulic actuator via a SDRE nonlinear filter, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.12.002i
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Fig. 8. Input voltage (second simulation test).
Fig. 9. Load velocity (second simulation test).
Fig. 10. Load displacement (second simulation test).
Fig. 11. Load pressure (second simulation test).
Please cite this article as: S. Strano, M. Terzo, Accurate state estimation for a hydraulic actuator via a SDRE nonlinear filter, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.12.002i
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Fig. 12. Spool velocity (second simulation test).
Fig. 13. Spool displacement (second simulation test).
While the estimates obtained with the two techniques are comparable for the load displacement, the differences are visible for the load pressure. Indeed, as for the load velocity, the SDREF gives an estimate for the load pressure that appears practically superimposed to the actual one. The result confirms the goodness of the technique in presence of not negligible nonlinearities and allows to consider it a valid alternative to the installation of sensors. Figs. 6 and 7 concern the spool valve velocity and the spool valve displacement respectively. A second simulated result concerns a square wave signal employed as input (Fig. 8). As in the previous test, the comparisons concerning the estimates of the five states are plotted. Also in this case, the goodness of the SDREF can be easily highlighted for the load velocity (Fig. 9) since the estimate is practically superimposed. Fig. 10 represents the load displacement with its estimates. The differences are lightly visible because of the absence of strong nonlinear effects. Differently, the nonlinear approach of the SDREF can be appreciated with reference to the load pressure (Fig. 11). As concerns the spool velocity (Fig. 12) and the spool displacement (Fig. 13), the performances of the two approaches are comparable. The illustrated simulation results are functional to validate the SDREF application for nonlinear system such as the hydraulic actuator and allow to consider the technique a valid tool for the fault detection algorithm and a functional alternative to the employment of sensors. The advantages of the SDREF are clear for the states dependent on hard nonlinearities, where the EKF approach fails because of the linearization procedure.
6. Experimental results with comparisons Experimental tests have been carried out on a hydraulic actuator, described in detail in previous works [1,2] and here omitted for brevity. The real time estimators have been implemented by means of a DS1103 controller board from dSPACE Inc. In order to better govern the load displacement, closed loop tests have been executed starting from sinusoidal laws for the target displacement of the load [2]. The first test consists of a sine wave of amplitude .05 m and a frequency of .1 Hz. Comparisons are illustrated in the following for the measured states, i.e. the load velocity, the load displacement, the load pressure and the spool valve displacement. Please cite this article as: S. Strano, M. Terzo, Accurate state estimation for a hydraulic actuator via a SDRE nonlinear filter, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.12.002i
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Fig. 14. Load velocity (first experimental test).
Fig. 15. Load displacement (first experimental test).
Fig. 16. Load pressure (first experimental test).
Fig. 17. Spool displacement (first experimental test).
Please cite this article as: S. Strano, M. Terzo, Accurate state estimation for a hydraulic actuator via a SDRE nonlinear filter, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.12.002i
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Fig. 18. Load velocity (second experimental test).
Fig. 19. Load displacement (second experimental test).
Fig. 20. Load pressure (second experimental test).
Fig. 21. Spool displacement (second experimental test).
Please cite this article as: S. Strano, M. Terzo, Accurate state estimation for a hydraulic actuator via a SDRE nonlinear filter, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.12.002i
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The capability of the SDREF appears clear in Fig. 14 in which the real load velocity is practically superimposed to the SDREF estimate. Moreover, the filtering of the measurement noise, differently from the adaptive-robust approaches [7], makes the SDREF particularly functional for applications characterized by noisy signals. The estimates obtained for the load displacement (Fig. 15) are, as described previously, fully comparable since nonlinearities are not involved. The SDREF can be appreciated for the estimate of the load pressure (Fig. 16), while the results for the spool displacement are comparable (Fig. 17). The second test has been realized by means of a sinusoidal law of amplitude .02 m and a frequency of 3 Hz. The estimate of the load velocity highlights the benefits of the SDREF (Fig. 18) able to fully capture nonlinearities. The advantages of the technique, less visible for the load displacement (Fig. 19), are strongly confirmed for the load pressure (Fig. 20). The performances of the two techniques are comparable for the spool displacement because of the already cited reasons (Fig. 21). The experimental results exhibit the SDREF performances and confirm the ability of the method for the state estimates in presence of nonlinearities and measurement noise.
7. Conclusions A State-Dependent-Riccati-equation based Kalman filter has been proposed for the estimates of the states of a hydraulic actuator. A fifth order nonlinear model has been derived and adopted to design the estimator. The effectiveness of the filter has been evaluated by means of simulation and experimental results that show a comparison with the EKF technique. The results show the advantages in terms of state estimates for all the states depending on nonlinearities. Consequently, the effectiveness of the SDREF is clearly visible for the load velocity and for the load pressure estimates. Moreover, the experimental test has highlighted the capability of the SDREF to strongly attenuate the noise, making the technique particularly functional for all the applications characterized by not negligible nonlinearities and noisy signal as measurements. Taking into account the performances of the technique, its main contributions can be found in fault detection algorithms or as alternative solution to the installation of velocity sensors. The SDREF can be considered a valid alternative to the EKF and to the adaptive-robust approaches for all the applications characterized by hydraulic actuators with hard nonlinearities and measurements affected by noise.
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[19] J.R. Cloutier, Time-to-go-less guidance with cross channel couplings, In: Proceedings of AIAA Missile Science Conference, Monterey, CA, December 1996. [20] S. Pagano, R. Russo, S. Strano, M. Terzo, Non-linear modelling and optimal control of a hydraulically actuated seismic isolator test rig, Mech. Syst. Signal Process. 35 (1–2) (2013) 255–278. [21] A. Nemra, N. Aouf, Robust INS/GPS sensor fusion for UAV localization using SDRE nonlinear filtering, IEEE Sensors J. 10 (4) (2010) 789–798. [22] C.P. Mracek, J.R. Cloutier, Missile longitudinal autopilot design using the state dependent Riccati equation method, In: Proceedings of International Conference on Nonlinear Problems in Aviation and Aerospace, Daytona Beach, FL, May 1996, pp. 387–396. [23] J.R. Cloutier, C.N. D’Souza, C.P. Mracek, Nonlinear regulation and nonlinear H2 control via the state-dependent Riccati equation technique; Part 1, Theory; Part 2, Examples, In: Proceedings of International Conference on Nonlinear Problems in Aviation and Aerospace, Daytona Beach, FL, May 1996. [24] C.P. Mracek, J.R. Cloutier, C.N. D’Souza, A new technique for nonlinear estimation, In: Proceedings of IEEE Conference on Control Applications, Dearborn, MI, Sep. 1996, pp. 338–343, DOI 10.1109/CCA.1996.558760. [25] D. Simon, Optimal State Estimation, Wiley-Interscience, Hoboken, New Jersey, 2006. [26] A. Bonchis, P. Corke, D. Rye, A pressure-based, velocity independent, friction model for asymmetric hydraulic cylinders, In: Proceedings of IEEE International Conference on Robotics and Automation, Detroit, MI, 1999, vol. 3, pp. 1746–1751. [27] H.E. Merritt, Hydraulic Control Systems, Wiley, New York, 1967. [28] L. Márton, S. Fodor, N. Sepehri, A practical method for friction identification in hydraulic actuators, Mechatronics 21 (1) (2011) 350–356. [29] J.R. Cloutier. Navigation and control branch U.S. air force armament directorate, in: Proceedings of American Control Conference, Albuquerque, NM, June 1997. [30] J.R. Cloutier, C.P. Mracek, D.B. Ridgely, K.D. Hammett, State dependent Riccati equation techniques: theory and applications, In: Proceedings of American Control Conference Workshop, 1998. [31] T. Çimen, Survey of state-dependent Riccati equation in nonlinear optimal feedback control synthesis, AIAA J. Guid. Control Dyn. 35 (4) (2012) 1025–1047.
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Accurate state estimation for a hydraulic actuator via a SDRE nonlinear filter Salvatore Strano, Mario Terzo n Department of Industrial Engineering, University of Naples Federico II, Italy
a r t i c l e i n f o
abstract
Article history: Received 17 March 2015 Received in revised form 7 September 2015 Accepted 8 December 2015
The state estimation in hydraulic actuators is a fundamental tool for the detection of faults or a valid alternative to the installation of sensors. Due to the hard nonlinearities that characterize the hydraulic actuators, the performances of the linear/linearization based techniques for the state estimation are strongly limited. In order to overcome these limits, this paper focuses on an alternative nonlinear estimation method based on the StateDependent-Riccati-Equation (SDRE). The technique is able to fully take into account the system nonlinearities and the measurement noise. A fifth order nonlinear model is derived and employed for the synthesis of the estimator. Simulations and experimental tests have been conducted and comparisons with the largely used Extended Kalman Filter (EKF) are illustrated. The results show the effectiveness of the SDRE based technique for applications characterized by not negligible nonlinearities such as dead zone and frictions. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Hydraulic actuator State Dependent Riccati Equation (SDRE) Nonlinear state estimation Extended Kalman Filter (EKF)
1. Introduction The ability of the hydraulic actuators of applying large forces with a fast response time, together with their size-to-power ratio, make them widely used in industrial applications. Typical usage concerns seismic test rig [1,2], positioning systems [3,4], vibration control [5,6]. The state estimation of such systems is fundamental for their operation [7]. Indeed, the hard working conditions and the complexity of the hydraulic systems strongly exhort to take under consideration their reliability and, so, the state observers become functional for a fast and economical detection of faults. At the same time, they are a valid alternative to the installation of sensors to give feedback to the control laws in case of prohibitive costs or harsh environments [8]. In any case, the highly nonlinear dynamics of the hydraulic actuators, due to the dead zone of the control valve, the pressure-flow rate relationship and frictions, makes the design of the state observers a challenging problem. Several approaches have been proposed in literature for the state estimation in hydraulic actuators. These methods include linear approaches, linearized model based techniques [9] and robust observers based on nonlinear models [10–12]. The high degree of parametric uncertainty that characterizes hydraulic actuators exhorts the researchers to follow dedicated approaches for the state estimation. Extended Kalman Filter (EKF) has been adopted to detect the fault in hydraulic actuators [13] by means of the state estimation. The approach is based on the local linearization of the nonlinear system and allows to closely track the state trajectories if compared with the linear approach of the Kalman filter [14]. Adaptive observers [15] and nonlinear adaptive robust observers [16,17] have been also developed. The last ones are based on robust
n
Corresponding author. E-mail address: [email protected] (M. Terzo).
http://dx.doi.org/10.1016/j.ymssp.2015.12.002 0888-3270/& 2015 Elsevier Ltd. All rights reserved.
Please cite this article as: S. Strano, M. Terzo, Accurate state estimation for a hydraulic actuator via a SDRE nonlinear filter, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.12.002i
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filter structure, to compensate the effect of the unmodeled dynamics, combined with on-line parameter adaptation, to account for the parametric uncertainty. This paper investigates on an alternative nonlinear technique for the state estimates of the hydraulic actuators. The approach is based on the State-Dependent-Riccati-Equation (SDRE) nonlinear filtering formulation. The SDRE techniques are recently emerging for optimal nonlinear control and filtering techniques. The SDRE filter (SDREF) originates from a suboptimal nonlinear regulator technique that uses parameterization to bring the nonlinear system into a linear-like structure with state-dependent coefficients (SDC) and is characterized by the structure of the steady state Kalman filter and the Kalman gain is obtained by solving an algebraic Riccati equation. The SDRE based techniques have been adopted in advanced guidance [18,19], in the control of hydraulic seismic isolator test rig [20], in sensor fusion applications [21], in output feedback nonlinear H2 autopilot [22]. Optimality, suboptimality, and stability properties have been discussed in [23], while the SDRE filter formulation has been used in [24]. A fundamental advantage of the SDRE filter technique consists of the possibility of employing a fully nonlinear nominal model, also including hard nonlinearities such as the dead zone, allowing to avoid linearization and unmodeled dynamics. Several hydraulic actuation systems present not negligible nonlinearities that have to be considered for the design of the state observers: this is common in the high power hydraulic actuation systems that employ, for example, control valves characterized by the lands of the spool greater than the annular parts of the valve body. Furthermore, differently from the adaptive-robust approaches [7], the SDRE based Kalman filter well manages the measurement noise [25] and allows to adopt a more faithful nominal model. Consequently, the SDREF implies the accurate knowledge of system in order to take into account the main dynamical characteristics. The paper is organized as follows: a fifth order nonlinear model is derived in Section 2 taking into account dead-zone, frictions and the nonlinear pressure-flow rate relationship; Sections 3 and 4 focus on the EKF and on the SDREF, respectively. Section 5 shows the simulation results, Section 6 the experimental ones, while conclusions are drawn in Section 7.
2. Dynamic model The hydraulic actuation system under consideration consists mostly of a double-rod hydraulic cylinder and a proportional valve. The hydraulic cylinder is linked to a mass that moves on a linear guide (Fig. 1). For the derivation of the mathematical model, some assumptions are made: the tank pressure PT is equal to zero, the fluid properties are not dependent on the temperature, the piston areas and the chamber volumes are equal, the internal and external fluid leakages are negligible. The dynamics of the movable mass displacement (y) is governed by: my€ þ σ y_ þF f ðy_ Þ ¼ Ap P L ;
ð1Þ
where m is the mass of the load, σ is the viscous friction coefficient, Ff is the friction force, Ap is the piston area, PL ¼PA–PB is the load pressure, PA and PB are the pressures inside the two chambers of the cylinder. The friction force is represented by the following equation: ( F c sgnðy_ Þ þ μmgsgnðy_ Þ y_ a0 F f ðy_ Þ ¼ ð2Þ y_ ¼ 0 Z jZ jr F c0 þ μ0 mg where Fc is the Coulomb friction force in the hydraulic actuator (assumed equal to its static value Fc0), μ is the Coulombian friction coefficient of the linear guide (assumed equal to its static value μ0), g is the gravitational acceleration and Z is the net tangential force that acts on the actuator when it is not moving [26]. The load pressure dynamics [27] is given by: V0 _ P L ¼ Ap y_ þ Q L 2β
ð3Þ
where V0 is the volume of each chamber for the centered position of the piston, QL ¼(QA þQB)/2 is the load flow and β is the effective Bulk modulus.
Fig. 1. Schematic diagram of the hydraulic actuation system.
Please cite this article as: S. Strano, M. Terzo, Accurate state estimation for a hydraulic actuator via a SDRE nonlinear filter, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.12.002i
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The load flow depends on the supply pressure, the load pressure and the valve spool position in accordance with the following: Q L ¼ Ψ ðve Þve
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P s jP L j
where ve is the displacement signal of the spool valve and proportional valve. The analytical expression of Ψ(ve) can be assumed as: 8 h i kq0 vep > > ve 4 vep > kqp 1 þ kqp 1 ve > < ven rve r vep Ψ ðve Þ ¼ kq0 h i > > kq0 > ven > ve o ven k 1 þ 1 : qn ve kqn
ð4Þ
Ψ(ve) is a gain that depends on the geometry of the adopted
ð5Þ
where ven, vep, kq0, kqn, and kqp are parameters that must be identified experimentally. The equations in (11) represent an asymmetric bilinear function of the valve spool position. If kq0 ¼kqp ¼kqn, the function Ψ(ve) reduces to a constant gain. If kq0 ¼0, the product Ψ(ve)ve describes a dead zone function, where ven and vep are the limits of the dead zone and kqn and kqp are the gains if ve is negative or positive respectively. The adoption of Ψ(ve) is particularly efficient to describe a dead zone function if an overlapped valve is adopted [20]. The proportional valve dynamics can be well represented by a second order differential equation [28]: 2ξ v€ e þ v v_ þ v ¼ ke u þ ve0 ω2nv ωnv e e
ð6Þ
where parameters ωnv and ξv are the natural frequency and the damping ratio of the valve respectively, ve0 is the spool position bias, ke is the input gain and u is the valve command. Finally, the equations governing the dynamics of the whole system (movable massþhydraulic system) are: 8 my€ þ σ y_ þ F f ðy_ Þ ¼ Ap P L > > > > > < V 0 P_ L ¼ Ap y_ þ Q L 2β ð7Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > Q L ¼ Ψ ðve Þve P s jP L j > > > > : v€ e ¼ ω2 ve 2ξ ωnv v_ e þ ω2 ðke u þ ve0 Þ v nv nv The developed fifth order model fully describes the nonlinear dynamical behavior of the hydraulic actuation system and takes the nonlinear friction forces and the nonlinear flow rate distribution into account. The nonlinear system (7) can be written in the following form: 8 _Þ A P f ðy σy _ Fmv > y€ ¼ m ve þ pm L > e > > pffiffiffiffiffiffiffiffiffiffiffiffiffi < 2βΨ ðve Þ P s jP L j 2βA P_ L ¼ V 0 p y_ þ ve ; V0 > > > > : v€ e ¼ 2ξv ωnv v_ e 1 vve0 ω2nv ve þ ω2nv ke u e
ð8Þ
The system (8), which state vector is given by x ¼ yL ve ve T , is nonlinear in the state and autonomous. It is possible to note in the first and in the third equation of (8) a division for the variable ve, as well as it happens in (5). In order to prevent divisions by zero, the variable v~ e ¼ ve þ ε has been introduced, where ε is given by: ( Δv e ve ¼ 0 ϵ¼ ð9Þ 0 otherwise where the value of Δve can be assumed less than the discrimination threshold of the valve spool position sensor. Replacing ve with v~ e , where ve is at the denominator of fractions in (5) and (8), it follows: 8 _ σy _ Fmf ðv~yÞve þ ApmPL > y€ ¼ m > e > > pffiffiffiffiffiffiffiffiffiffiffiffiffi < _P L ¼ 2βAp y_ þ 2βΨ ðv~ e Þ P s jP L jve : V0 V > 0 > > v > 2 2 e0 : v€ e ¼ 2ξv ωnv v_ e 1 v~ ωnv ve þ ωnv ke u e
ð10Þ
The nonlinear equations of the system (10) have been adopted in order to derive the EKF and the SDREF. Given the measurements of signals like the load displacement y, the load pressure PL and the spool valve displacement ve , the objective is the estimation of the states of the system. Please cite this article as: S. Strano, M. Terzo, Accurate state estimation for a hydraulic actuator via a SDRE nonlinear filter, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.12.002i
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3. The Extended Kalman Filter The linearization procedure is at the basis of the EKF approach, which is briefly recalled in the following since it has been adopted for a comparative analysis. The system and the measurement equations can be generically represented by: _ ¼ fðxðtÞ; uðtÞÞ þ ψk ; xðtÞ
ð11Þ
zðtÞ ¼ hðxðtÞ; uðtÞÞ þgk
ð12Þ
being x the state vector, f and h non-linear functions, u the input vector, ψk the process noise with covariance Q k , z the measurement vector, and gk the Gaussian white measurement noise with covariance R k . The estimator can be implemented in a discrete time form, integrating the system equation from time t k 1 to time t k . The EKF methodology is conceptually based on two fundamental steps, namely estimates and updates steps. Denoting the estimates as ð^ Þ, the following initializing conditions are applied to the state estimates (13) and to the error covariance (14): þ x^ 0 ¼ Eðx0 Þ
P 0þ ¼ E
ð13Þ
þ þ T x0 x^ 0 x0 x^ 0
ð14Þ
being E the expected value. The state estimates and the estimation of the error covariance are given by (15) and (16) respectively:
with
þ x^ k ¼ f k 1 ðx^ k 1 ; uk 1 Þ;
ð15Þ
Pk ¼ Ak 1 Pkþ 1 ATk 1 þ Lk 1 Q k 1 LTk 1 ;
ð16Þ
∂f k 1 Ak 1 ¼ ∂x^ ∂f k 1 Lk 1 ¼ ∂ψk
;
ð17Þ
:
ð18Þ
þ x^ k 1
þ x^ k 1
With the computation of the filter gain (19) and evaluating the measurement residual, the updates of the state estimates (20) and of the estimation of the error covariance (21) can be determined: Kk ¼ Pk HTk ðHk Pk HTk þ Mk R k MTk Þ 1 ;
ð19Þ
þ x^ k ¼ x^ k þ Kk zk hk ðx^ k ; uk Þ ;
ð20Þ
Pkþ ¼ ðI Kk Hk ÞPk ;
ð21Þ
where:
∂hk Hk ¼ ∂x^
;
ð22Þ
:
ð23Þ
x^ k
∂hk Mk ¼ ∂gk
x^ k
4. SDRE nonlinear filter SDRE techniques are used as control and filtering design methods and are based on state dependent coefficient (SDC) factorization [24]. Infinite-horizon nonlinear regulator problem is a generalization of time invariant infinite horizon linear quadratic regulator problem where all system coefficient matrices are state-dependent [29,30]. When the coefficient matrices are constant, the SDRE control method changes into the steady-state linear regulator. Filtering counterpart of the SDRE control algorithm is obtained by taking the dual system of the steady-state linear regulator and then allowing coefficient matrices of the dual system to be state-dependent. Starting from the nonlinear system (10), there are infinite Please cite this article as: S. Strano, M. Terzo, Accurate state estimation for a hydraulic actuator via a SDRE nonlinear filter, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.12.002i
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solutions to transform this nonlinear system into an SDC form as: _ ¼ FðxðtÞ; uðtÞÞx þ ψ; xðtÞ
ð24Þ
zðtÞ ¼ HðxðtÞ; uðtÞÞx þ g;
ð25Þ
where fðxðtÞ; uðtÞÞ ¼ FðxðtÞ; uðtÞÞx and hðxðtÞ; uðtÞÞ ¼ HðxðtÞ; uðtÞÞx;
ð26Þ
ψ is the process noise with covariance Q , and g is the Gaussian white measurement noise with covariance R. Starting from the SDC form, the derivative of the state estimate is given by: ðx ;uÞx þ Kf ðx Þ½zðxÞ Hðx ;uÞx x^ ^
^
^
^
^
ð27Þ
where
^ u R 1 Kf x^ ¼ PðxÞHT x;
ð28Þ
and P is the positive definite solution of the algebraic Riccati Eq. (29).
^ u P x^ þP x^ FT x; ^ u P x^ HT x; ^ u R 1 H x; ^ u P x^ þQ ¼ 0 F x; The highly nonlinear Eq. (10) has been parameterized in SDC form with the following choice [31]: 3 2 F ðy_ ̇ Þ A σ m 0 mp 0 mf v~ e 7 6 6 1 7 0 0 0 0 ffiffiffiffiffiffiffiffiffiffiffiffiffi 6 7 p 7 6 2 β Ap 2βΨ ðv~ e Þ P s jP L j 7 0 0 0 Fðx; uÞ ¼ 6 V0 V0 7 6 7 6 ω2nv ke u 7 ve0 6 0 2 0 0 2ξv ωnv 1 v~ e ωnv þ v~ e 5 4 0 2
0 6 Hðx; uÞ ¼ H ¼ 4 0 0
0
0
1
1
0
0
0
1
0
7 0 5:
0
0
0
1
0
ð29Þ
ð30Þ
0
3 ð31Þ
It is important to note that the EKF method, presented in the previous section, is based on the linearization of the nonlinear model (10) as indicated in (17), (18), (22), and (23). Differently, the mathematical derivation of the SDREF highlights that in this case no linearization has been performed as it can be noted in (24)–(31).
5. Simulation results with comparisons In order to evaluate the benefits of the proposed estimator, simulations have been carried out on the mathematical model of a hydraulic actuator and, successively, an experimental validation has been executed. The hydraulic actuator has been modeled by means of (10), in which the parameters have been determined by means of an identification procedure [20] and are: 3 m ¼ 440 kg, σ ¼ 23555 Nms, F C ¼ F C0 ¼ 950 N, μ ¼ μ0 ¼ 0:01, Ap ¼ 0:01 m2 , V 0 ¼ 0:004 m3 , β ¼ 1e9 Pa, kq0 ¼ 1e 12 m 1 , s V Pa2
kqp ¼ 6:15e 7 1 , kqn ¼ 5:86e 7 1 ,vep ¼ 0:43 V, ven ¼ 0:21 V, ve0 ¼ 0:01 V, ke ¼ 0:49, ωnv ¼ 152 rad=s, ξv ¼ 0:92, s V Pa2 s V Pa2 P S ¼ 6e6 Pa. m3
m3
Fig. 2. Input voltage (first simulation test).
Please cite this article as: S. Strano, M. Terzo, Accurate state estimation for a hydraulic actuator via a SDRE nonlinear filter, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.12.002i
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Fig. 3. a) Load velocity (first simulation test); b) zoom view.
Fig. 4. Load displacement (first simulation test).
The simulated experiments have been obtained by means of open loop tests employing sine and square waves as input. The performances of the SDRE filter will be evaluated by comparison with the EKF estimator and the real state. The observers have been designed taking into account as input the signal u and the measurements given by displacement y, the load pressure PL and the spool valve displacement ve . The first simulated result is based on a sine wave given as input (Fig. 2). The parameters adopted in the observers are the same of the hydraulic actuator model. The first important result is shown in Fig. 3, where the capability of the SDREF is easily visible: indeed, the comparison with the EKF technique highlights substantial differences due to the linearization process. The nonlinearities of the system are fully considered in the SDREF and, consequently, all the null speed zones, caused by the dead zone, are well estimated. This result finds an important application for fault detection algorithm and in velocity feedback control in which, for example, the speed sensor is not adopted. With the particular reference to the hydraulic actuators, the nonlinearities involve the load velocity equation and the load pressure one. As a consequence, the differences between the EKF and the SDREF are evident if the time histories of the estimates of the cited states are evaluated. In any case, with the aim of completeness of the discussion, the comparisons for all the states are shown in the following. Fig. 4 exhibits the comparison in terms of load displacement, Fig. 5 shows the comparison for the load pressure. Please cite this article as: S. Strano, M. Terzo, Accurate state estimation for a hydraulic actuator via a SDRE nonlinear filter, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.12.002i
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Fig. 5. a) Load pressure (first simulation test); b) zoom view.
Fig. 6. Spool velocity (first simulation test).
Fig. 7. Spool displacement (first simulation test).
Please cite this article as: S. Strano, M. Terzo, Accurate state estimation for a hydraulic actuator via a SDRE nonlinear filter, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.12.002i
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Fig. 8. Input voltage (second simulation test).
Fig. 9. Load velocity (second simulation test).
Fig. 10. Load displacement (second simulation test).
Fig. 11. Load pressure (second simulation test).
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Fig. 12. Spool velocity (second simulation test).
Fig. 13. Spool displacement (second simulation test).
While the estimates obtained with the two techniques are comparable for the load displacement, the differences are visible for the load pressure. Indeed, as for the load velocity, the SDREF gives an estimate for the load pressure that appears practically superimposed to the actual one. The result confirms the goodness of the technique in presence of not negligible nonlinearities and allows to consider it a valid alternative to the installation of sensors. Figs. 6 and 7 concern the spool valve velocity and the spool valve displacement respectively. A second simulated result concerns a square wave signal employed as input (Fig. 8). As in the previous test, the comparisons concerning the estimates of the five states are plotted. Also in this case, the goodness of the SDREF can be easily highlighted for the load velocity (Fig. 9) since the estimate is practically superimposed. Fig. 10 represents the load displacement with its estimates. The differences are lightly visible because of the absence of strong nonlinear effects. Differently, the nonlinear approach of the SDREF can be appreciated with reference to the load pressure (Fig. 11). As concerns the spool velocity (Fig. 12) and the spool displacement (Fig. 13), the performances of the two approaches are comparable. The illustrated simulation results are functional to validate the SDREF application for nonlinear system such as the hydraulic actuator and allow to consider the technique a valid tool for the fault detection algorithm and a functional alternative to the employment of sensors. The advantages of the SDREF are clear for the states dependent on hard nonlinearities, where the EKF approach fails because of the linearization procedure.
6. Experimental results with comparisons Experimental tests have been carried out on a hydraulic actuator, described in detail in previous works [1,2] and here omitted for brevity. The real time estimators have been implemented by means of a DS1103 controller board from dSPACE Inc. In order to better govern the load displacement, closed loop tests have been executed starting from sinusoidal laws for the target displacement of the load [2]. The first test consists of a sine wave of amplitude .05 m and a frequency of .1 Hz. Comparisons are illustrated in the following for the measured states, i.e. the load velocity, the load displacement, the load pressure and the spool valve displacement. Please cite this article as: S. Strano, M. Terzo, Accurate state estimation for a hydraulic actuator via a SDRE nonlinear filter, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.12.002i
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S. Strano, M. Terzo / Mechanical Systems and Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Fig. 14. Load velocity (first experimental test).
Fig. 15. Load displacement (first experimental test).
Fig. 16. Load pressure (first experimental test).
Fig. 17. Spool displacement (first experimental test).
Please cite this article as: S. Strano, M. Terzo, Accurate state estimation for a hydraulic actuator via a SDRE nonlinear filter, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.12.002i
S. Strano, M. Terzo / Mechanical Systems and Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Fig. 18. Load velocity (second experimental test).
Fig. 19. Load displacement (second experimental test).
Fig. 20. Load pressure (second experimental test).
Fig. 21. Spool displacement (second experimental test).
Please cite this article as: S. Strano, M. Terzo, Accurate state estimation for a hydraulic actuator via a SDRE nonlinear filter, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.12.002i
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S. Strano, M. Terzo / Mechanical Systems and Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎
The capability of the SDREF appears clear in Fig. 14 in which the real load velocity is practically superimposed to the SDREF estimate. Moreover, the filtering of the measurement noise, differently from the adaptive-robust approaches [7], makes the SDREF particularly functional for applications characterized by noisy signals. The estimates obtained for the load displacement (Fig. 15) are, as described previously, fully comparable since nonlinearities are not involved. The SDREF can be appreciated for the estimate of the load pressure (Fig. 16), while the results for the spool displacement are comparable (Fig. 17). The second test has been realized by means of a sinusoidal law of amplitude .02 m and a frequency of 3 Hz. The estimate of the load velocity highlights the benefits of the SDREF (Fig. 18) able to fully capture nonlinearities. The advantages of the technique, less visible for the load displacement (Fig. 19), are strongly confirmed for the load pressure (Fig. 20). The performances of the two techniques are comparable for the spool displacement because of the already cited reasons (Fig. 21). The experimental results exhibit the SDREF performances and confirm the ability of the method for the state estimates in presence of nonlinearities and measurement noise.
7. Conclusions A State-Dependent-Riccati-equation based Kalman filter has been proposed for the estimates of the states of a hydraulic actuator. A fifth order nonlinear model has been derived and adopted to design the estimator. The effectiveness of the filter has been evaluated by means of simulation and experimental results that show a comparison with the EKF technique. The results show the advantages in terms of state estimates for all the states depending on nonlinearities. Consequently, the effectiveness of the SDREF is clearly visible for the load velocity and for the load pressure estimates. Moreover, the experimental test has highlighted the capability of the SDREF to strongly attenuate the noise, making the technique particularly functional for all the applications characterized by not negligible nonlinearities and noisy signal as measurements. Taking into account the performances of the technique, its main contributions can be found in fault detection algorithms or as alternative solution to the installation of velocity sensors. The SDREF can be considered a valid alternative to the EKF and to the adaptive-robust approaches for all the applications characterized by hydraulic actuators with hard nonlinearities and measurements affected by noise.
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