# Model-Free Adaptive Control of Direct Drive Servo Valve of Electromagnetic Linear Actuator

Authors: Jianhui Zhu; Jianguo Dai; Cheng Wang
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

An electromagnetic linear actuator (EMLA) has a promising application in direct motion control. However, ELMA will inevitably inherit uncertainties in the face of load changes, system parameter perturbation, and inherent system nonlinearities, all of which constitute disturbances adversely affecting the precision and adaptability of the control system. A model-free adaptive control (MFAC) strategy based on full form dynamic linearization (FFDL) was proposed to reduce the sensitivity of the control system to the disturbances. An adaptive control of direct drive servo valve was achieved based on the online interaction of characteristic parameters and control algorithms. The feasibility and precision of the proposed algorithm were verified through simulation and experimental results. The results show that the proposed algorithm could achieve adaptive adjustment of the servo valve response at different openings of 0-3 mm without changing control parameters, with the response time controlled within 10ms and steady state error less than 0.04mm. Furthermore, the proposed algorithm had better robustness and capacity of resisting disturbance.


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## Body

## 1. Introduction

Electromagnetic linear actuator (EMLA) has a wide range of applications in the field of linear motion control, because of high-precision, high-response, and nonintermediate conversion [1, 2]. EMLA has been widely applied in electromagnetic engine valve, automatic transmission, and freedom motion platform of six degrees. It has also drawn more and more attention as a part of electrical-mechanical conversion device of direct drive servo valve [3–5].Currently, many algorithms on the moving control of electromagnetic actuators have been discussed [6]. Typical control algorithms include iterative learning control [7], inverse system method control [8], and sliding mode control [9]. However, for direct drive servo valve, the control algorithm is normally based on accurate modeling for control object, which is very difficult to be achieved; thus control effect is poor [10, 11].On the other hand, in order to meet the highly responsive requirements, data-driven algorithms, such as the classic PID control, have large differential coefficient and are more likely to cause system instability [12]. For discrete-time nonlinear systems, MFAC (Model-Free Adaptive Control) uses a new dynamic linearization method and a concept of pseudo-partial derivative. MFAC depends only on real-time measurement data of the control system, rather than depending on any mathematical model information of the control system. Through online interactions of characteristic parameters and control algorithms, the amendment of Pan model as well as the ideal control function of system can be achieved [13]. Since proposed by Hou Zhongsheng 20 years ago, MFAC has been used in permanent magnet linear motor [14, 15], freeway traffic systems, wind power plant, some complex industrial processes, etc. [16].In this paper, a model-free adaptive control strategy was proposed based on full format dynamic linearization to achieve better control effect and high-response. Simulation and test were carried out to verify the adaptive control of ELMA for direct drive servo valve.


## 2. System Description

The schematic diagram and three-dimensional model of ELMA in direct drive servo valve are shown in Figure1. It mainly consists of magneto-resistive displacement sensor, electromagnetic linear actuator, poppet valve, seat, and valve cover fixed to the central axis. The gas inlet is connected to the external air fuel supply line.Figure 1
The schematic diagram and three-dimensional model of EMLA in direct drive servo valve.
(a)
Schematic diagram (b)
Three-dimensional model diagramThe electromagnetic linear actuator controls the actuator current and position to achieve double-loop control spool motion through a certain control algorithm, thus controlling the opening of the servo valve connected to spool. Therefore, the spool movement is able to follow the predetermined motion rule.


## 3. Model-Free Adaptive Controller Design

### 3.1. Control Algorithm Design

Model-Free adaptive controller is based on “nonparametric model”, also known as “pan model”. It is worth noting that the term of “model-free” basically refers to the fact that the controller does not need to recognize the global dynamic model of controlled objects. With the help of dynamic linear technology, this model-free adaptive controller builds equivalent dynamic linear data near the work point at each sampling instant. Even though the data model is fictitious, the controller only needs to control the input and output (I/O) data of the object, which is a design method integrating identification and control as a whole [17].The control system of electromagnetic linear direct drive servo valve can be seen as a single-input and single-output system in discrete time. Input refers to controlled current or voltage signal, while output refers to lift range, that is, valve opening. The general discrete-time single-input single-output (SISO) nonlinear system is expressed as follows [18]:(1)yk+1=fyk,…,yk-ny,uk,…,uk-nuwhere u(k) and y(k) are system input and output at the instant k, ny and nu are two unknown positive integers, and f is an unknown nonlinear system.By utilizing the dynamic linearization method of MFAC theory [19, 20], the nonlinear discrete system can be equivalently converted to a dynamic linear data model based on input and output data increment. Data model based on FFDL (Full Form Dynamic Linearization) can be expressed as(2)Δyk+1=Φf,Ly,LuTkΔHLy,Luk(3)Φf,Ly,LuTk=Φ1k,…,ΦLyk,ΦLy+1k,…,ΦLy+LukT(4)ΔHLy,Luk=Δyk,…,Δyk-Ly+1,Δuk,…,Δuk-Lu+1Twhere Δy(k+1) is incremental output at two adjacent moments, Φf,Ly,LuT(k) is bounded pseudo-gradient, Ly and Lu are system pseudo-orders, Φf,Ly,LuT(k) is a time-varying parameter vector of pseudo-gradient, and ΔHLy,Lu(k) is change vector consisting of all input and output signals in the sliding time window.Based on this model, the control plan of FFDL-MFAC is(5)uk=uk-1+ρLy+1ΦLy+1ky∗k+1-ykλ+ΦLy+1k2-ΦLy+1k∑i=1LyρiΦikΔyk-i+1λ+ΦLy+1k2-ΦLy+1k∑i=Ly+2Ly+LuρiΦikΔuk-Ly-i+1λ+ΦLy+1k2where λ is the weighting factor as well as the penalty factor for controlling input change. The smaller the λ, the faster the system responses, overshoot, or instability that may be generated; on the contrary, the greater the λ is, the more stable the input and output are, and the smaller the overshoot is. ρi is step factor, ρi∈(0,1], i=1,2,⋯,Ly+Lu, which is set to make the control algorithm design more flexible. y∗(k+1) is the desired system output.


### 3.2. PG Vector

The traditional parameter estimation criterion function is to minimize the square of the difference between the system model output and the real output. However, when applying the parameter estimation algorithm derived from such a criterion function, the estimated value of the parameter often changes too fast, or some inaccurate sampled data of mutations (which may be caused by interference or sensor failure) are too sensitive. To this end, for the EMLA system in this paper, the following estimation criterion function is used.(6)Jϕf,Ly,Luk=yk-yk-1-ϕTf,Ly,Luk2+μϕf,Ly,Luk-ϕ^f,Ly,Luk2According to the optimal condition ∂J(ϕf,Ly,Lu(k))/ϕf,Ly,Lu(k)=0, the upper formula is used to find the extremum of ϕf,Ly,Lu(k), and the matrix inversion lemma is used to obtain the estimation algorithm of PG.(7)Φ^f,Ly,Luk=Φ^f,Ly,Luk-1+ηΔHLy,Luk-1yk-yk-1-Φ^f,Ly,LuTkΔHLy,Luk-1μ+ΔHLy,Luk-12where μ is a weighting factor, η is a step factor, η∈(0,2], and Φ^f,Ly,Lu(k-1) is the estimated value of Φ^f,Ly,Lu(k). If Φ^f,Ly,Lu(k)≤ε or ΔHLy,Lu(k-1)≤ε, or sign(Φ^Ly+1(k))≠sign(Φ^Ly+1(1)), Φ^f,Ly,Lu(k)=Φ^f,Ly,Lu(1), Φ^f,Ly,Lu(1) is the initial value of Φ^f,Ly,Lu(k). ε is sufficiently small positive integer, which is set to 0.0001 in simulation. Therefore, model-free adaptive control algorithm was established based on full format. With this established algorithm, the controller is designed using the online input and output data measured by closed-loop control system, which does not significantly or implicitly contain any information related to the control system dynamic model.


### 3.3. Control Scheme

By combining the control algorithm of the previous nonlinear chaotic system and the corresponding pseudo-partial derivative estimation algorithm, the model-free adaptive control scheme of chaotic system can be given as follows.(8)Φ^f,Ly,Luk=Φ^f,Ly,Luk-1+ηΔHLy,Luk-1yk-yk-1-Φ^f,Ly,LuTkΔHLy,Luk-1μ+ΔHLy,Luk-12(9)Φ^f,Ly,Luk≤εorΔHLy,Luk-1≤ε,orsignΦ^Ly+1k≠signΦ^Ly+11,Φ^f,Ly,Luk=Φ^f,Ly,Lu1(10)uk=uk-1+ρLy+1ΦLy+1ky∗k+1-ykλ+ΦLy+1k2-ΦLy+1k∑i=1LyρiΦikΔyk-i+1λ+ΦLy+1k2-ΦLy+1k∑i=Ly+2Ly+LuρiΦikΔuk-Ly-i+1λ+ΦLy+1k2In this scheme, the introduction of (9) is to make the pseudo-partial derivative estimation algorithm have stronger ability to track time-varying parameters. In this scheme, only one one-dimensional parameter, i.e., system pseudo-bias, needs to be adjusted online. It can be seen from the control scheme (8)-(10) that the proposed model-free adaptive control scheme only uses the I/O data of the controlled system for controller design, which is basically a data-driven control method. The proposed control scheme is independent of the explicit or implicit dynamic model and system structure information of the controlled system, so it is a control method typically for the modeless nonlinear system.


### 3.4. Stability Analysis of Algorithm

Before analyzing the robust stability of the above algorithm, the following assumptions are proposed.Assumption 1.
The partial derivative of the functionf(...) with respect to the (ny+2) variables is continuous except for a finite time point.Assumption 2.
Except for the finite time point, the system satisfies the generalized Lipschitz condition; that is, at any timek1≠k2, k1,k2≥0, and u(k1)≠u(k2), there is the following relationship:(11)yk1+1-yk2+1≤buk1-uk2y(ki+1)=f(y(ki),...,y(ki-ny),u(ki),...,u(ki-nu)),i=1,2;b>0 is a constant.
Assumption1 is a typical constraint on general nonlinear systems in control system design.
Assumption2 is a limitation on the upper bound of the system's output rate of change. A bounded input energy change should produce a bounded output energy change within the system.Assumption 3.
For a given bounded desired output signaly∗(k+1), there is always a bounded amount u∗(k). The system is driven by the control input signal and its output is equal to ∗(k+1).Assumption 4.
At any time k andΔu(k)≠0, the symbol of the system's PPD remains unchanged; i.e., ϕc(k)>ε_>0,orϕc(k)<-ε_, ε is a small positive number.
Assumption3 is a necessary condition for the design of the control problem; that is, the output of the system is controllable.
Assumption4 means that the corresponding output of controlled system should be undiminished when the control input is increased, which can be considered as a “quasi-linear” feature of the system.Theorem 5.
If system (1) satisfies assumptions 1–4, there is a positive number λmin>0, so that when λ>λmin,(1)
limk→∞⁡y∗-y(k+1)=0,(2)
y(k) and u(k) are bounded sequence.Proof.
Defineϕ~f,Ly,Lu(k)=ϕ^f,Ly,Lu(k)-ϕf,Ly,Lu(k) as the pseudo-partial derivative estimation error of the control algorithm. In the parameter estimation algorithm (8), by subtracting the terminal with ϕ^f,Ly,Lu(k), we can obtain the following.(12)ϕ~f,Ly,Luk=ϕ^f,Ly,Luk-1+ηΔuk-1μ+Δuk-12×Δyk-ϕ^f,Ly,Luk-1Δuk-1-ϕf,Ly,LukBy substituting Δy(k)=ϕf,Ly,Lu(k-1)Δu(k-1) into expressions:(13)ϕ~f,Ly,Luk=ϕ^f,Ly,Luk-1-ϕf,Ly,Luk-1+ηΔuk-1μ+Δuk-12×ϕf,Ly,Luk-1Δuk-1-ϕ^f,Ly,Luk-1Δuk-1+ϕf,Ly,Luk-1-ϕf,Ly,Lukdue to the expression ϕ~f,Ly,Lu(k-1)=ϕ^f,Ly,Lu(k-1)-ϕf,Ly,Lu(k-1), we can obtain the following.(14)ϕ~f,Ly,Luk=1-ηΔuk-12μ+Δuk-12ϕ~f,Ly,Luk-1+ϕf,Ly,Luk-1-ϕf,Ly,LukBy taking absolute values on both sides of (14), there is the following.(15)ϕ~f,Ly,Luk≤1-ηΔuk-12μ+Δuk-12ϕ~f,Ly,Luk-1+ϕf,Ly,Luk-1-ϕf,Ly,LukSince Δu(k)≠0, when 0<η≤1 0 and μ≥0, there is a constant d1, which satisfies the following.(16)0≤1-ηΔuk-12μ+Δuk-12≤d1<1On the other hand, based on Assumption 2, we get ϕf,Ly,Lu(k)≤b, so ϕf,Ly,Lu(k-1)-ϕf,Ly,Lu(k)≤2b. According to (15) and (16), the following recursive inequalities can be obtained.(17)ϕ~f,Ly,Luk≤d1ϕ~f,Ly,Luk-1+2b≤d12ϕ~f,Ly,Luk-2+2d1b+2b≤…≤d1k-1ϕ~f,Ly,Lu1+2b1-d1k-11-d1The above formula indicates that ϕ~f,Ly,Lu(k) is bounded. Since ϕf,Ly,Lu(k)≤b is bounded, ϕ^f,Ly,Lu(k) is bounded.
Define the tracking error of the system as(18)ek+1=y∗-yk+1(19)ek+1=y∗-yk+1=y∗-yk-ϕf,Ly,LukΔuk≤1-ρϕf,Ly,Lukϕ^f,Ly,Lukλ+ϕ^f,Ly,Luk2ekFrom assumption 4 and (9), ϕf,Ly,Lu(k)ϕ^f,Ly,Lu(k)≥0.
Letλmin=b2/4. Using the inequality α2+β2≥2αβ and choosing λ>λmin, there is a constant M1(0<M1<1) such that the following holds:(20)0<M1≤ϕf,Ly,Lukϕ^f,Ly,Lukλ+ϕ^f,Ly,Luk2≤bϕ^f,Ly,Lukλ+ϕ^f,Ly,Luk2≤bϕ^f,Ly,Luk2λϕ^f,Ly,Luk<b2λmin=1where b is the constant that satisfies the conclusion of theorem ϕf,Ly,Lu(k)≤b.
According to (20), 0<ρ<1 and λ<λmin, there must be a constant d2(d2<1), so that(21)1-ρϕf,Ly,Lukϕ^f,Ly,Lukλ+ϕ^f,Ly,Luk2=1-ρϕf,Ly,Lukϕ^f,Ly,Lukλ+ϕ^f,Ly,Luk2≤1-ρM1≜d2<1Combining (19) and (21), there is the following.(22)ek+1≤d2ek≤d22ek-1≤…≤d2ke1.The above formula means that Theorem 5(1) holds; that is, the control algorithm can control the ELMA to the desired output. In addition, since y∗(k) is a constant, e(k) converges to mean y(k) bounded. Using (λ)2+ϕ^f,Ly,Lu(k)2≥2λϕ^f,Ly,Lu(k) and λ>λmin, the following inequality is given by (10):(23)Δuk=ρϕ^f,Ly,Luky∗-ykλ+ϕ^f,Ly,Luk2≤ρϕ^f,Ly,Lukλ+ϕ^f,Ly,Luk2ek≤ρϕ^f,Ly,Luk2λϕ^f,Ly,Lukek≤ρ2λminek=M2ekwhere M2 = ρ is a bounded constant. Based on (22) and (23), the following equation can be obtained.(24)uk≤uk-uk-1+uk-1≤uk-uk-1+uk-1-uk-2+uk-2≤Δuk+Δuk-1+…+Δu2+u1≤M2ek+ek-1+…+e2+u1≤M2d2k-1e1+d2k-2e1+…+d2e2+u1<M2d21-d2e1+u1
Equation (24) indicates that Theorem 5(2) holds; that is, the system is stable.
In order to obtain input and output (I/O) data required by control algorithm, it is needed to establish a time-varying model. This time-varying model is only used to generate I/O data and to achieve online interaction of data, rather than participating in controller design. Therefore, the system parameters change has no effect on the controller. The nonlinear time-varying model for the motion equation of ELMA direct drive servo valve is described as:(25)x˙t=vt(26)ft=Kfit(27)ut=Kex˙t+Rit+Lditdt(28)ft=mx¨t+floadt+ffrictionx˙+ωt(29)ffrictionx˙=fc+fs-fce-x˙/xsδ+fvx˙sign⁡x˙where ffriction is friction (N), fload(t) is load force (N), u(t) is control voltage (v), f(t) is actuator force (N), x(t) is actuator position (m), v(t) is the velocity of actuator(m/s), Kf is actuator force constant (N/A), Ke is counter electromotive force constant (V/m/s), R is actuator coil resistance (Ω), sign is sign function, L is coil inductance (H), i(t) is coil current (A), m is moving mass (Kg), ω(t) is other disturbances, fs is static friction (N), fc is coulomb friction (N), xs is lubrication parameters (m/s), fv is dynamic friction coefficient (N/m/s), and δ is empirical parameters, which is set to 1 in simulation.
First of all, simulation was carried out to verify the feasibility of the proposed algorithm. The block diagram of controller is shown as Figure2. It can be seen that the controller mainly consists of model-free adaptive controller module for generating control quantity u(k), estimator module for estimating the algorithm in pseudo-gradient vector, the controlled object- direct drive servo valve of EMLA, and submodule constituted by the nonlinear model. On the basis of model framework, the model of control algorithm was established using software Matlab/Simulink, and simulation of the given fixed lift range and continuous lift range was carried out.
The simulation parameter values are shown in Table1. Parameters of control algorithm are set as follows: weighting factor μ=1, step factor η=2, ε=0.0001, and weighting factor λ=1.2e-6. Among them, μ is penalty factor for the control of estimated parameter variation, which is used to limit the linear dynamic range of alternatives; λ is to limit the change of controlled variables, so parameter μ and λ are critical adjustment parameters in controlling, which have a direct impact on the final control results. Step factor is ρ6=0.1,0.1,0.1,0.1,0.02,0.1,0.1, Ly=3 and Lu=3, the initial value of Φ^f,Ly,Lu(k) is Φ^f,Ly,Lu(1)=[0.03,0.03,0.03,0.03,0.03,0.03]T, and the sampling period of simulation is 0.0001s.Table 1
Prototype parameters.
Parameter Name / Unit Value Actuator Coil Resistance/Ω 0.45 Coil Inductance/mH 1.6 Moving Mass/Kg 0.036 Actuator Force Constant/(N/A) 10.8 Counter Electromotive Force Constant/(V/m/s) 10.8 Dynamic Friction Coefficient/(N/m/s) 10Figure 2
Structure diagram of controller.


## 3.1. Control Algorithm Design

Model-Free adaptive controller is based on “nonparametric model”, also known as “pan model”. It is worth noting that the term of “model-free” basically refers to the fact that the controller does not need to recognize the global dynamic model of controlled objects. With the help of dynamic linear technology, this model-free adaptive controller builds equivalent dynamic linear data near the work point at each sampling instant. Even though the data model is fictitious, the controller only needs to control the input and output (I/O) data of the object, which is a design method integrating identification and control as a whole [17].The control system of electromagnetic linear direct drive servo valve can be seen as a single-input and single-output system in discrete time. Input refers to controlled current or voltage signal, while output refers to lift range, that is, valve opening. The general discrete-time single-input single-output (SISO) nonlinear system is expressed as follows [18]:(1)yk+1=fyk,…,yk-ny,uk,…,uk-nuwhere u(k) and y(k) are system input and output at the instant k, ny and nu are two unknown positive integers, and f is an unknown nonlinear system.By utilizing the dynamic linearization method of MFAC theory [19, 20], the nonlinear discrete system can be equivalently converted to a dynamic linear data model based on input and output data increment. Data model based on FFDL (Full Form Dynamic Linearization) can be expressed as(2)Δyk+1=Φf,Ly,LuTkΔHLy,Luk(3)Φf,Ly,LuTk=Φ1k,…,ΦLyk,ΦLy+1k,…,ΦLy+LukT(4)ΔHLy,Luk=Δyk,…,Δyk-Ly+1,Δuk,…,Δuk-Lu+1Twhere Δy(k+1) is incremental output at two adjacent moments, Φf,Ly,LuT(k) is bounded pseudo-gradient, Ly and Lu are system pseudo-orders, Φf,Ly,LuT(k) is a time-varying parameter vector of pseudo-gradient, and ΔHLy,Lu(k) is change vector consisting of all input and output signals in the sliding time window.Based on this model, the control plan of FFDL-MFAC is(5)uk=uk-1+ρLy+1ΦLy+1ky∗k+1-ykλ+ΦLy+1k2-ΦLy+1k∑i=1LyρiΦikΔyk-i+1λ+ΦLy+1k2-ΦLy+1k∑i=Ly+2Ly+LuρiΦikΔuk-Ly-i+1λ+ΦLy+1k2where λ is the weighting factor as well as the penalty factor for controlling input change. The smaller the λ, the faster the system responses, overshoot, or instability that may be generated; on the contrary, the greater the λ is, the more stable the input and output are, and the smaller the overshoot is. ρi is step factor, ρi∈(0,1], i=1,2,⋯,Ly+Lu, which is set to make the control algorithm design more flexible. y∗(k+1) is the desired system output.


## 3.2. PG Vector

The traditional parameter estimation criterion function is to minimize the square of the difference between the system model output and the real output. However, when applying the parameter estimation algorithm derived from such a criterion function, the estimated value of the parameter often changes too fast, or some inaccurate sampled data of mutations (which may be caused by interference or sensor failure) are too sensitive. To this end, for the EMLA system in this paper, the following estimation criterion function is used.(6)Jϕf,Ly,Luk=yk-yk-1-ϕTf,Ly,Luk2+μϕf,Ly,Luk-ϕ^f,Ly,Luk2According to the optimal condition ∂J(ϕf,Ly,Lu(k))/ϕf,Ly,Lu(k)=0, the upper formula is used to find the extremum of ϕf,Ly,Lu(k), and the matrix inversion lemma is used to obtain the estimation algorithm of PG.(7)Φ^f,Ly,Luk=Φ^f,Ly,Luk-1+ηΔHLy,Luk-1yk-yk-1-Φ^f,Ly,LuTkΔHLy,Luk-1μ+ΔHLy,Luk-12where μ is a weighting factor, η is a step factor, η∈(0,2], and Φ^f,Ly,Lu(k-1) is the estimated value of Φ^f,Ly,Lu(k). If Φ^f,Ly,Lu(k)≤ε or ΔHLy,Lu(k-1)≤ε, or sign(Φ^Ly+1(k))≠sign(Φ^Ly+1(1)), Φ^f,Ly,Lu(k)=Φ^f,Ly,Lu(1), Φ^f,Ly,Lu(1) is the initial value of Φ^f,Ly,Lu(k). ε is sufficiently small positive integer, which is set to 0.0001 in simulation. Therefore, model-free adaptive control algorithm was established based on full format. With this established algorithm, the controller is designed using the online input and output data measured by closed-loop control system, which does not significantly or implicitly contain any information related to the control system dynamic model.


## 3.3. Control Scheme

By combining the control algorithm of the previous nonlinear chaotic system and the corresponding pseudo-partial derivative estimation algorithm, the model-free adaptive control scheme of chaotic system can be given as follows.(8)Φ^f,Ly,Luk=Φ^f,Ly,Luk-1+ηΔHLy,Luk-1yk-yk-1-Φ^f,Ly,LuTkΔHLy,Luk-1μ+ΔHLy,Luk-12(9)Φ^f,Ly,Luk≤εorΔHLy,Luk-1≤ε,orsignΦ^Ly+1k≠signΦ^Ly+11,Φ^f,Ly,Luk=Φ^f,Ly,Lu1(10)uk=uk-1+ρLy+1ΦLy+1ky∗k+1-ykλ+ΦLy+1k2-ΦLy+1k∑i=1LyρiΦikΔyk-i+1λ+ΦLy+1k2-ΦLy+1k∑i=Ly+2Ly+LuρiΦikΔuk-Ly-i+1λ+ΦLy+1k2In this scheme, the introduction of (9) is to make the pseudo-partial derivative estimation algorithm have stronger ability to track time-varying parameters. In this scheme, only one one-dimensional parameter, i.e., system pseudo-bias, needs to be adjusted online. It can be seen from the control scheme (8)-(10) that the proposed model-free adaptive control scheme only uses the I/O data of the controlled system for controller design, which is basically a data-driven control method. The proposed control scheme is independent of the explicit or implicit dynamic model and system structure information of the controlled system, so it is a control method typically for the modeless nonlinear system.


## 3.4. Stability Analysis of Algorithm

Before analyzing the robust stability of the above algorithm, the following assumptions are proposed.Assumption 1.
The partial derivative of the functionf(...) with respect to the (ny+2) variables is continuous except for a finite time point.Assumption 2.
Except for the finite time point, the system satisfies the generalized Lipschitz condition; that is, at any timek1≠k2, k1,k2≥0, and u(k1)≠u(k2), there is the following relationship:(11)yk1+1-yk2+1≤buk1-uk2y(ki+1)=f(y(ki),...,y(ki-ny),u(ki),...,u(ki-nu)),i=1,2;b>0 is a constant.
Assumption1 is a typical constraint on general nonlinear systems in control system design.
Assumption2 is a limitation on the upper bound of the system's output rate of change. A bounded input energy change should produce a bounded output energy change within the system.Assumption 3.
For a given bounded desired output signaly∗(k+1), there is always a bounded amount u∗(k). The system is driven by the control input signal and its output is equal to ∗(k+1).Assumption 4.
At any time k andΔu(k)≠0, the symbol of the system's PPD remains unchanged; i.e., ϕc(k)>ε_>0,orϕc(k)<-ε_, ε is a small positive number.
Assumption3 is a necessary condition for the design of the control problem; that is, the output of the system is controllable.
Assumption4 means that the corresponding output of controlled system should be undiminished when the control input is increased, which can be considered as a “quasi-linear” feature of the system.Theorem 5.
If system (1) satisfies assumptions 1–4, there is a positive number λmin>0, so that when λ>λmin,(1)
limk→∞⁡y∗-y(k+1)=0,(2)
y(k) and u(k) are bounded sequence.Proof.
Defineϕ~f,Ly,Lu(k)=ϕ^f,Ly,Lu(k)-ϕf,Ly,Lu(k) as the pseudo-partial derivative estimation error of the control algorithm. In the parameter estimation algorithm (8), by subtracting the terminal with ϕ^f,Ly,Lu(k), we can obtain the following.(12)ϕ~f,Ly,Luk=ϕ^f,Ly,Luk-1+ηΔuk-1μ+Δuk-12×Δyk-ϕ^f,Ly,Luk-1Δuk-1-ϕf,Ly,LukBy substituting Δy(k)=ϕf,Ly,Lu(k-1)Δu(k-1) into expressions:(13)ϕ~f,Ly,Luk=ϕ^f,Ly,Luk-1-ϕf,Ly,Luk-1+ηΔuk-1μ+Δuk-12×ϕf,Ly,Luk-1Δuk-1-ϕ^f,Ly,Luk-1Δuk-1+ϕf,Ly,Luk-1-ϕf,Ly,Lukdue to the expression ϕ~f,Ly,Lu(k-1)=ϕ^f,Ly,Lu(k-1)-ϕf,Ly,Lu(k-1), we can obtain the following.(14)ϕ~f,Ly,Luk=1-ηΔuk-12μ+Δuk-12ϕ~f,Ly,Luk-1+ϕf,Ly,Luk-1-ϕf,Ly,LukBy taking absolute values on both sides of (14), there is the following.(15)ϕ~f,Ly,Luk≤1-ηΔuk-12μ+Δuk-12ϕ~f,Ly,Luk-1+ϕf,Ly,Luk-1-ϕf,Ly,LukSince Δu(k)≠0, when 0<η≤1 0 and μ≥0, there is a constant d1, which satisfies the following.(16)0≤1-ηΔuk-12μ+Δuk-12≤d1<1On the other hand, based on Assumption 2, we get ϕf,Ly,Lu(k)≤b, so ϕf,Ly,Lu(k-1)-ϕf,Ly,Lu(k)≤2b. According to (15) and (16), the following recursive inequalities can be obtained.(17)ϕ~f,Ly,Luk≤d1ϕ~f,Ly,Luk-1+2b≤d12ϕ~f,Ly,Luk-2+2d1b+2b≤…≤d1k-1ϕ~f,Ly,Lu1+2b1-d1k-11-d1The above formula indicates that ϕ~f,Ly,Lu(k) is bounded. Since ϕf,Ly,Lu(k)≤b is bounded, ϕ^f,Ly,Lu(k) is bounded.
Define the tracking error of the system as(18)ek+1=y∗-yk+1(19)ek+1=y∗-yk+1=y∗-yk-ϕf,Ly,LukΔuk≤1-ρϕf,Ly,Lukϕ^f,Ly,Lukλ+ϕ^f,Ly,Luk2ekFrom assumption 4 and (9), ϕf,Ly,Lu(k)ϕ^f,Ly,Lu(k)≥0.
Letλmin=b2/4. Using the inequality α2+β2≥2αβ and choosing λ>λmin, there is a constant M1(0<M1<1) such that the following holds:(20)0<M1≤ϕf,Ly,Lukϕ^f,Ly,Lukλ+ϕ^f,Ly,Luk2≤bϕ^f,Ly,Lukλ+ϕ^f,Ly,Luk2≤bϕ^f,Ly,Luk2λϕ^f,Ly,Luk<b2λmin=1where b is the constant that satisfies the conclusion of theorem ϕf,Ly,Lu(k)≤b.
According to (20), 0<ρ<1 and λ<λmin, there must be a constant d2(d2<1), so that(21)1-ρϕf,Ly,Lukϕ^f,Ly,Lukλ+ϕ^f,Ly,Luk2=1-ρϕf,Ly,Lukϕ^f,Ly,Lukλ+ϕ^f,Ly,Luk2≤1-ρM1≜d2<1Combining (19) and (21), there is the following.(22)ek+1≤d2ek≤d22ek-1≤…≤d2ke1.The above formula means that Theorem 5(1) holds; that is, the control algorithm can control the ELMA to the desired output. In addition, since y∗(k) is a constant, e(k) converges to mean y(k) bounded. Using (λ)2+ϕ^f,Ly,Lu(k)2≥2λϕ^f,Ly,Lu(k) and λ>λmin, the following inequality is given by (10):(23)Δuk=ρϕ^f,Ly,Luky∗-ykλ+ϕ^f,Ly,Luk2≤ρϕ^f,Ly,Lukλ+ϕ^f,Ly,Luk2ek≤ρϕ^f,Ly,Luk2λϕ^f,Ly,Lukek≤ρ2λminek=M2ekwhere M2 = ρ is a bounded constant. Based on (22) and (23), the following equation can be obtained.(24)uk≤uk-uk-1+uk-1≤uk-uk-1+uk-1-uk-2+uk-2≤Δuk+Δuk-1+…+Δu2+u1≤M2ek+ek-1+…+e2+u1≤M2d2k-1e1+d2k-2e1+…+d2e2+u1<M2d21-d2e1+u1
Equation (24) indicates that Theorem 5(2) holds; that is, the system is stable.
In order to obtain input and output (I/O) data required by control algorithm, it is needed to establish a time-varying model. This time-varying model is only used to generate I/O data and to achieve online interaction of data, rather than participating in controller design. Therefore, the system parameters change has no effect on the controller. The nonlinear time-varying model for the motion equation of ELMA direct drive servo valve is described as:(25)x˙t=vt(26)ft=Kfit(27)ut=Kex˙t+Rit+Lditdt(28)ft=mx¨t+floadt+ffrictionx˙+ωt(29)ffrictionx˙=fc+fs-fce-x˙/xsδ+fvx˙sign⁡x˙where ffriction is friction (N), fload(t) is load force (N), u(t) is control voltage (v), f(t) is actuator force (N), x(t) is actuator position (m), v(t) is the velocity of actuator(m/s), Kf is actuator force constant (N/A), Ke is counter electromotive force constant (V/m/s), R is actuator coil resistance (Ω), sign is sign function, L is coil inductance (H), i(t) is coil current (A), m is moving mass (Kg), ω(t) is other disturbances, fs is static friction (N), fc is coulomb friction (N), xs is lubrication parameters (m/s), fv is dynamic friction coefficient (N/m/s), and δ is empirical parameters, which is set to 1 in simulation.
First of all, simulation was carried out to verify the feasibility of the proposed algorithm. The block diagram of controller is shown as Figure2. It can be seen that the controller mainly consists of model-free adaptive controller module for generating control quantity u(k), estimator module for estimating the algorithm in pseudo-gradient vector, the controlled object- direct drive servo valve of EMLA, and submodule constituted by the nonlinear model. On the basis of model framework, the model of control algorithm was established using software Matlab/Simulink, and simulation of the given fixed lift range and continuous lift range was carried out.
The simulation parameter values are shown in Table1. Parameters of control algorithm are set as follows: weighting factor μ=1, step factor η=2, ε=0.0001, and weighting factor λ=1.2e-6. Among them, μ is penalty factor for the control of estimated parameter variation, which is used to limit the linear dynamic range of alternatives; λ is to limit the change of controlled variables, so parameter μ and λ are critical adjustment parameters in controlling, which have a direct impact on the final control results. Step factor is ρ6=0.1,0.1,0.1,0.1,0.02,0.1,0.1, Ly=3 and Lu=3, the initial value of Φ^f,Ly,Lu(k) is Φ^f,Ly,Lu(1)=[0.03,0.03,0.03,0.03,0.03,0.03]T, and the sampling period of simulation is 0.0001s.Table 1
Prototype parameters.
Parameter Name / Unit Value Actuator Coil Resistance/Ω 0.45 Coil Inductance/mH 1.6 Moving Mass/Kg 0.036 Actuator Force Constant/(N/A) 10.8 Counter Electromotive Force Constant/(V/m/s) 10.8 Dynamic Friction Coefficient/(N/m/s) 10Figure 2
Structure diagram of controller.


## 4. Results and Analysis

### 4.1. Experimental System

To further verify the control effect of full format model-free adaptive control algorithm for the EMLA of direct drive servo valve system, the test platform was established, as shown in Figure3. It consists of DSP2812 controller, power driver circuit, H-bridge, current sensor, magneto-resistive displacement sensor, direct drive servo valve, host computer PC, and power source.Figure 3
Test platform.The composition of controller section is shown as Figure4. The PWM control signals, which are generated by algorithm program of DSP controller, are transmitted to the drive circuit to drive direct drive servo valve of EMLA. At the same time, signals collected by the displacement sensor and the current sensor are fed back to the controller section though A/D converter, forming a closed-loop control. Using the LAN interface of DSP Ethernet, the communication between the upper and slaver computer can be realized, so that functions such as signal input and output, display, and save can be realized.Figure 4
Block diagram of experiment bench and main controller.


### 4.2. Response and Precision Analysis

The experimental verification of control algorithm is conducted based on the built test platform, and test parameters are set as follows: control amountu is current, of which the initial state is set as u(1)=u(2)=⋯u(6)=0, output y is the valve opening y(1)=y(2)=⋯=y(6)=0, pseudo-order of full format model-free adaptive control algorithm is set as Ly=3 and Lu=3, step factor and weighting factors are separately set as λ=0.5, μ=1, η=1, ε=0.001, ρ[1]=[4;4;0.9;0.098;0.05;0.05], and the initial value of the pseudo-gradient vector is ϕ[1]=[1;1;1;1;-1;-1].During the test, the control parameters are adjusted within a fixed lift range of 3mm, until the output reaches satisfactory results. Then, without changing the parameters, lift range is adjusted in order to verify the algorithm’s adaptability to other target lift range. When the target position is 3mm, the response and current are shown as Figure5; the target location, the output position of actual control, and error are shown as Figure 6.Figure 5
Response and current when target location is 3mm.Figure 6
Target position, actual position, and tracking error.As can be seen from Figures5 and 6, within the lift range of 3mm, response time is 4-5ms, steady state phase can be reached after 10ms, the maximum error is 0.0655mm, the minimum error is -0.0149mm, and the average error is 0.0353mm. Error tends to exhibit constant decrease. When target location is 3mm, the change of control current and tracking error are shown as Figure 7.Figure 7
Target current, actual current, and tracking error.As can be seen from Figure7, the calculated control current is in good agreement with the actual current collected by sensor, the control peak is less than 1A, and current tracking error continues to decrease to almost zero.In order to verify the adaptability of algorithm to any other lift ranges, output results were investigated under the condition of different lift ranges. In order to facilitate the drawing, only the output results within lift range of 0.5-3mm were given, i.e., 6 major output results by lift range interval of 0.5mm (0.5, 1.0, 1.5, 2.0, 2.5, 3.0). In addition, tests were carried out by life range interval of 0.1mm. At the same time, in order to verify the control accuracy, the results were compared with laser displacement sensors. As shown in Figure8, the valve enters into steady state phase after 10ms in all cases, with average error less than 0.04mm.Figure 8
Response output within lift range of 0-3mm.


### 4.3. Adaptability Analysis of Disturbances

To further verify the robustness of the proposed algorithm, simulation system response under load and no-load, simulation verification was carried out within a lift range of 2mm. The simulation model was verified in our previous work [4]. The system response under load force of 5N and 10N is shown in Figure 9.Figure 9
Response and output of the system with the increase of load force.
(a)
The position response under different load (b)
The change of controlled amount under different load (c)
The change of velocity under different load (d)
The change of current under different loadFigure9(a) shows output response within lift range of 2mm. As can be seen from the figure a, with the increase of load, system moves a short distance in the opposite direction to the initial position, and system response becomes slow gradually. Figure 9(b) shows the change of control amount. With the increase of load, the required control amount was also increased to reach the designated target displacement. Figures 9(c) and 9(d) show corresponding speed and current changes. It can be seen that, with the increase in the load, the speed response became slow gradually, and the required current continued to increase. Upon reaching steady state, current and control amount were no longer zero without load, but maintained at a certain value, in order to overcome the load force in the opposite direction.From the simulation results, it can be known that the control algorithm has good adaptability with the increase of load on the servo valve. Under the same parameters, the proposed control algorithm can accurately reach the target position, with smaller overshoot and higher precision.In order to verify the suppression on interference, the system response with load disturbance forceω(t)=±100N within the lift range of 2mm and at motion time of 8-8.5ms was investigated, as shown in Figure 10. As can be seen from the figure, after applying the disturbing force, the system quickly recovered to a stable state, with overshoot less than 4%, indicating a strong anti-interference ability. Similar simulation results were observed within other lift range. The algorithm proposed in this paper for the direct drive servo valve of EMLA has better robustness and capacity of resisting disturbance than the algorithm in [4–6].Figure 10
System response with the increase of disturbance.


## 4.1. Experimental System

To further verify the control effect of full format model-free adaptive control algorithm for the EMLA of direct drive servo valve system, the test platform was established, as shown in Figure3. It consists of DSP2812 controller, power driver circuit, H-bridge, current sensor, magneto-resistive displacement sensor, direct drive servo valve, host computer PC, and power source.Figure 3
Test platform.The composition of controller section is shown as Figure4. The PWM control signals, which are generated by algorithm program of DSP controller, are transmitted to the drive circuit to drive direct drive servo valve of EMLA. At the same time, signals collected by the displacement sensor and the current sensor are fed back to the controller section though A/D converter, forming a closed-loop control. Using the LAN interface of DSP Ethernet, the communication between the upper and slaver computer can be realized, so that functions such as signal input and output, display, and save can be realized.Figure 4
Block diagram of experiment bench and main controller.


## 4.2. Response and Precision Analysis

The experimental verification of control algorithm is conducted based on the built test platform, and test parameters are set as follows: control amountu is current, of which the initial state is set as u(1)=u(2)=⋯u(6)=0, output y is the valve opening y(1)=y(2)=⋯=y(6)=0, pseudo-order of full format model-free adaptive control algorithm is set as Ly=3 and Lu=3, step factor and weighting factors are separately set as λ=0.5, μ=1, η=1, ε=0.001, ρ[1]=[4;4;0.9;0.098;0.05;0.05], and the initial value of the pseudo-gradient vector is ϕ[1]=[1;1;1;1;-1;-1].During the test, the control parameters are adjusted within a fixed lift range of 3mm, until the output reaches satisfactory results. Then, without changing the parameters, lift range is adjusted in order to verify the algorithm’s adaptability to other target lift range. When the target position is 3mm, the response and current are shown as Figure5; the target location, the output position of actual control, and error are shown as Figure 6.Figure 5
Response and current when target location is 3mm.Figure 6
Target position, actual position, and tracking error.As can be seen from Figures5 and 6, within the lift range of 3mm, response time is 4-5ms, steady state phase can be reached after 10ms, the maximum error is 0.0655mm, the minimum error is -0.0149mm, and the average error is 0.0353mm. Error tends to exhibit constant decrease. When target location is 3mm, the change of control current and tracking error are shown as Figure 7.Figure 7
Target current, actual current, and tracking error.As can be seen from Figure7, the calculated control current is in good agreement with the actual current collected by sensor, the control peak is less than 1A, and current tracking error continues to decrease to almost zero.In order to verify the adaptability of algorithm to any other lift ranges, output results were investigated under the condition of different lift ranges. In order to facilitate the drawing, only the output results within lift range of 0.5-3mm were given, i.e., 6 major output results by lift range interval of 0.5mm (0.5, 1.0, 1.5, 2.0, 2.5, 3.0). In addition, tests were carried out by life range interval of 0.1mm. At the same time, in order to verify the control accuracy, the results were compared with laser displacement sensors. As shown in Figure8, the valve enters into steady state phase after 10ms in all cases, with average error less than 0.04mm.Figure 8
Response output within lift range of 0-3mm.


## 4.3. Adaptability Analysis of Disturbances

To further verify the robustness of the proposed algorithm, simulation system response under load and no-load, simulation verification was carried out within a lift range of 2mm. The simulation model was verified in our previous work [4]. The system response under load force of 5N and 10N is shown in Figure 9.Figure 9
Response and output of the system with the increase of load force.
(a)
The position response under different load (b)
The change of controlled amount under different load (c)
The change of velocity under different load (d)
The change of current under different loadFigure9(a) shows output response within lift range of 2mm. As can be seen from the figure a, with the increase of load, system moves a short distance in the opposite direction to the initial position, and system response becomes slow gradually. Figure 9(b) shows the change of control amount. With the increase of load, the required control amount was also increased to reach the designated target displacement. Figures 9(c) and 9(d) show corresponding speed and current changes. It can be seen that, with the increase in the load, the speed response became slow gradually, and the required current continued to increase. Upon reaching steady state, current and control amount were no longer zero without load, but maintained at a certain value, in order to overcome the load force in the opposite direction.From the simulation results, it can be known that the control algorithm has good adaptability with the increase of load on the servo valve. Under the same parameters, the proposed control algorithm can accurately reach the target position, with smaller overshoot and higher precision.In order to verify the suppression on interference, the system response with load disturbance forceω(t)=±100N within the lift range of 2mm and at motion time of 8-8.5ms was investigated, as shown in Figure 10. As can be seen from the figure, after applying the disturbing force, the system quickly recovered to a stable state, with overshoot less than 4%, indicating a strong anti-interference ability. Similar simulation results were observed within other lift range. The algorithm proposed in this paper for the direct drive servo valve of EMLA has better robustness and capacity of resisting disturbance than the algorithm in [4–6].Figure 10
System response with the increase of disturbance.


## 5. Conclusion

A model-free adaptive control algorithm based on full format dynamic linearization was built. Through simulation analysis of system step response within different lift ranges as well as investigation of system outputs under increasing load force and disturbances, the application possibility of proposed algorithm in controlling direct drive servo valve of EMLA was verified. By using the full format model-free adaptive control algorithm, tests on actual control of the direct drive servo valve of EMLA were carried out. The simulation and test results show that the proposed algorithm could achieve adaptive adjustment of the servo valve response at different openings of 0-3mm, with the response time controlled within 10ms, and steady state error less than 0.04mm. Furthermore, the proposed algorithm has better robustness and capacity of resisting disturbance and achieves the purpose of servo control without changing controller parameters.



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*Source: 7105872-2018-10-22.xml*