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Data-driven control of hydraulic servo actuator: An event-triggered adaptive dynamic programming approach


  • Hydraulic servo actuators (HSAs) are often used in the industry in tasks that request great power, high accuracy and dynamic motion. It is well known that an HSA is a highly complex nonlinear system, and that the system parameters cannot be accurately determined due to various uncertainties, an inability to measure some parameters and disturbances. This paper considers an event-triggered learning control problem of the HSA with unknown dynamics based on adaptive dynamic programming (ADP) via output feedback. Due to increasing practical application of the control algorithm, a linear discrete model of HSA is considered and an online learning data driven controller is used, which is based on measured input and output data instead of unmeasurable states and unknown system parameters. Hence, the ADP-based data driven controller in this paper requires neither the knowledge of the HSA dynamics nor exosystem dynamics. Then, an event-based feedback strategy is introduced to the closed-loop system to save the communication resources and reduce the number of control updates. The convergence of the ADP-based control algorithm is also theoretically shown. Simulation results verify the feasibility and effectiveness of the proposed approach in solving the optimal control problem of HSAs.

    Citation: Vladimir Djordjevic, Hongfeng Tao, Xiaona Song, Shuping He, Weinan Gao, Vladimir Stojanovic. Data-driven control of hydraulic servo actuator: An event-triggered adaptive dynamic programming approach[J]. Mathematical Biosciences and Engineering, 2023, 20(5): 8561-8582. doi: 10.3934/mbe.2023376

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  • Hydraulic servo actuators (HSAs) are often used in the industry in tasks that request great power, high accuracy and dynamic motion. It is well known that an HSA is a highly complex nonlinear system, and that the system parameters cannot be accurately determined due to various uncertainties, an inability to measure some parameters and disturbances. This paper considers an event-triggered learning control problem of the HSA with unknown dynamics based on adaptive dynamic programming (ADP) via output feedback. Due to increasing practical application of the control algorithm, a linear discrete model of HSA is considered and an online learning data driven controller is used, which is based on measured input and output data instead of unmeasurable states and unknown system parameters. Hence, the ADP-based data driven controller in this paper requires neither the knowledge of the HSA dynamics nor exosystem dynamics. Then, an event-based feedback strategy is introduced to the closed-loop system to save the communication resources and reduce the number of control updates. The convergence of the ADP-based control algorithm is also theoretically shown. Simulation results verify the feasibility and effectiveness of the proposed approach in solving the optimal control problem of HSAs.



    Important properties of the HSA, such as fast and accurate responses, a high force/mass ratio and relatively good stiffness, have attracted great interest in the HSA and its applications. In the last two decades, high-performance controller design of the HSA has attracted increasing attention due to the expanded performance requirements of technical systems in the industry [1,2,3,4].

    A large number of machines driven by HSAs often work with high payloads in harsh and mostly external environments. As a result of variables of their environment, such as temperature, dust, humidity, wear, variable loads and disturbances, the HSA is usually subject to large uncertainties during operation. Hence, high-precision control of the HSA has always challenged researchers due to its unmodeled dynamics, large nonlinearities, parametric uncertainties, unmeasurable states in practice, etc. It is well known that it is impossible to determine most of the physical parameters of HSA components. While some HSA parameters are available only with certain accuracy, the other parameters are completely unknown. Dominant nonlinear sources existing in HSAs are impossibility of accurate determining parameters, which are very difficult to handle with high accuracy. These unknown parameters are caused by protection of proprietary data of individual manufacturers or indirect measuring and calculating, pressure losses, transient and turbulent flow conditions, friction, leakage characteristics, and generation discontinuous control signals to HSAs due to effects of saturation and changing the direction of servo valve. Furthermore, variable working conditions during operation, such as oil temperature, the bulk modulus, fluctuating supply pressure and pipe volume will lead to parameter changes, which worsen the existing control performances. These facts make it difficult to realize high-quality control of the HSA, which cannot be achieved without knowing the accurate HSA model [5,6,7,8].

    Further, direct measurement of the whole HSA state vector is not feasible for practical implementation and in addition would require very expensive measurement equipment. It is more convenient to use control algorithms which apply methods based on state reconstruction rather than to perform direct measurements of the states [9].

    In modern control theory, optimal control of the HSA plays a vital role in the controller design. Namely, the main challenge is to design optimal control algorithms that will affect the minimum energy consumption [10,11]. The optimal control design is an offline control technique that usually depends on perfect knowledge of the HSA model, which is not possible to obtain in most practical situations. Even if an approximated model of the HSA can be developed, the dynamic uncertainty, produced by the mismatch between the approximated model and the true HSA model, will degrade the control performance of the traditionally designed optimal controller [9,12]. Therefore, further research on the design of optimal controllers of HSAs is very important and our primary aim for this study.

    The practical applicability of control algorithms is enhanced by the fact that nonlinear systems can be very precisely represented by linear models with online estimated dynamics [13,14]. Many modern engineering applications such as intelligent vehicles [15,16], modernized microgrids [17], microphone sensing [18], strain prediction for fatigue [19], maintaining the security of cyber–physical systems [20], robotic manipulation tasks [21], 2-degree-of-freedom helicopter [22] and requests for online controller design which rely on linear systems.

    Adaptive dynamic programming (ADP) ensures an effective way to achieve high performance of the optimal controller which relies on adaptive control, optimal control and reinforcement learning [9,23,24,25,26]. ADP represents a kind of data-based control technique which can guarantee the stability of the feedback control system [9]. Recently, the field of ADP application has also been expanded to various research areas, including robotic systems, aerospace systems, guided missiles, spacecraft, etc. [27,28,29]. In circumstances of unknown system dynamics and unmeasurable states, of great interest is to use ADP techniques based on measured input/output data from linear systems, which are commonly called output feedback. A main benefit of the output feedback techniques is that knowledge of the HSA dynamics is not needed for their application. For an unknown HSA model, this indirect technique generates a sequence of suboptimal controllers which converge to the optimal control policy with an increasing number of iterations.

    The implementation of ADP algorithms is usually based on periodic selection [30]. In order to save limited communication and computational resources, event-triggered strategies have recently started to be applied in control algorithms based on ADP [31,32,33,34]. Moreover, the number of updates of the control inputs in this way will be smaller compared to the periodic update of the controller, because it is updated only when necessary (e.g., when the performance of the system deteriorates). The implementation of event-triggered algorithms is based on aperiodic sampling. Several event-based controllers have been proposed in the literature, most of which are state-feedback controllers [35,36,37,38,39]. In contrast, this paper will consider the event-triggered ADP-based control problem of HSAs in the case where only output feedback is available.

    This paper considers an online learning technique, where during operation, from measured input/output data, the controller learns to compensate unknown HSA dynamics, various disturbances and modeling errors, ensuring desired performances of the control system. The optimal control law is accomplished iteratively based on output feedback, state reconstruction and ADP. The unknown HSA model is first identified after which the algebraic Riccati equation (ARE) is iteratively solved. To ensure consistency of approximations and obtain unique solutions in each iteration step, some exploration noise must be added to control input to meet the requirements of the persistent excitation condition [40,41,42]. For exploration noise, some persistent excitation is usually applied such as white noise or pseudo random binary signals (PRBS). The selection of exploration noise is a non-trivial task for the most learning problems, as it can affect the accuracy of solutions, especially for large systems [43]. By applying the theory of experimental design, we will use the sum of sinusoidal signals as an exploration noise that will enable the output of the system to carry maximum information about the system, which will shorten the learning time, i.e., speed up the controller design process. Thus, the obtained input and output signals are used to reconstruct the state vector of the model, which is of great practical importance in relation to control techniques with direct state measurement that rely on a large number of sensors.

    Due to implementation of the ADP-based control techniques, it is easier to realize data acquisition for the discrete-time HSA model in relation to its continuous-time model. ADP-based control methodology for discrete-time systems is proposed in [44].

    We chose to use the measured input and output data to reconstruct the state vector of the discretized HSA model, after which ADP-based control can be implemented. The control law is learned iteratively and very efficiently provides solutions for optimal control of HSAs based only on measurements in real time. The main advantage of the proposed control methodology is avoiding the knowledge of system dynamics, which is very important under practical conditions.

    By applying an event-based control strategy, the number of control input updates will be reduced relative to periodic update of the controller, because the controller is only updated when certain conditions are met. In this way, energy, computing and communication resources will be significantly preserved.

    The rest of the paper is organized as follows. The problem of modelling an HSA with unknown dynamics is presented in Section 2. Event-triggered control based on ADP is shown in Section 3. Simulation results show the validity and effectiveness of the event-triggered ADP-based controller for HSAs in the presence of completely model uncertainty in Section 4. Finally, Section 5 gives the concluding remarks.

    The HSA under study is shown in Figure 1, and it consists of a servo valve and a hydraulic cylinder. The analysis of the properties of the HSA comes out from the dynamics of its components, which involves the piston motion dynamics, pressure dynamics at the cylinder and servo valve dynamics. Hence, the model of the HSA is derived from complex nonlinear equations that depend on many parameters which cannot be accurately obtained [7,8].

    Figure 1.  The HSA configuration.

    See Table 1 for the description of the HSA parameters. Using the notation in Figure 1, and defining the area ratio of the piston α=Ab/Aa as well as Va=Va0+yAa, Vb=Vb0+(Ly)αAa and qLi=cLi(papb), where cLi is the internal leakage flow coefficient; cvi>0 denote discharge coefficients, the sign function sg(x)={xx00x<0 and assuming an external leakage negligible, the considered model can be described by the following equations:

    mt¨y=AapaAbpbFf(˙y)KeyFext, (2.1)
    ˙pa=βeVa(y)(qaAa˙yqLiqLea), (2.2)
    ˙pb=βeVb(y)(qb+αAa˙y+qLiqLeb), (2.3)
    qa=qsv1qsv2=cv1sg(xv)sign(pspa)|pspa|cv2sg(xv)sign(pap0)|pap0|, (2.4)
    qb=qsv3qsv4=cv3sg(xv)sign(pspb)|pspb|cv4sg(xv)sign(pbp0)|pbp0|. (2.5)
    Table 1.  Parameters of the HSA.
    Notations Descriptions
    xv The spool valve displacement
    pa, pb Forward and return pressure
    qa, qb Forward and return flows
    y Piston displacement
    L Piston stroke
    Ke Load spring gradient
    pS, p0 Supply and tank pressure
    mt, mp, m total mass, piston mass, payload mass
    Ff Friction force
    Fext Disturbance force
    Aa, Ab Effective areas of the head and rod piston side
    Va, Vb, Va0, Vb0 Fluid volumes of the head and rod piston side and corresponding initial volumes
    qLi, qLe Internal and external leakage flow
    βe Bulk modulus of the fluid

     | Show Table
    DownLoad: CSV

    According to Eqs (2.1)–(2.5), and by defining the state and input variables as

    x(t)=[x1(t)x2(t)x3(t)x4(t)]T[y(t)˙y(t)pa(t)pb(t)]T, (2.6)
    u(t)=xv(t), (2.7)

    the governing nonlinear continuous-time dynamics of the HSA can be expressed in a state-space form as follows:

    ˙x(t)=f(x(t))+g(x(t),u(t))+h(t),y(t)=η(x(t)), (2.8)

    where f(x(t)) and g(x(t),u(t)) are the state dynamics and the input function, respectively:

    f(x(t))=[x21mt(Aax3αAax4Ff(x2)Kex1)βeAax1+Va0(Aax2+cLi(x3x4))βeαAa(Lx1)+Vb0(αAax2+cLi(x3x4))],
    g(x(t),u(t))=[00βeAax1+Va0(cv1sg(u)sign(psx3)|psx3|cv4sg(u)sign(x3p0)|x3p0|)βeαAa(Lx1)+Vb0(cv3sg(u)sign(psx4)|psx4|cv2sg(u)sign(x4p0)|x4p0|)],

    where output function η(x(t))=x1(t) and disturbance function h(t)=[h1(t)Fext/mt+h2(t)h3(t)h4(t)] include loads, unmodelled dynamics and parameter uncertainties.

    One of the main nonlinearities of the cylinder model is the nonlinear friction force Ff, which consists of static friction, Coulomb friction and Stribeck effect of velocity. An extensive study related to acting friction forces upon the HSA can be found in [7]. Further, we consider the linearized model of the HSA, whose parameters are experimentally identified for different working points of the HSA (i.e., different positions and external load conditions) [8]. Now, the model equations are expressed in a more suitable way in terms of the load pressure:

    pL=paαpb, (2.9)

    which leads to simplified dynamic equations. At last, using the new state vector [x1(t)x2(t)x3(t)]T[y(t)˙y(t)pL(t)]T allows us to express the HSA in a more compact form. Taking an operating point x0[y0˙y0pL0]T, and assuming dominance of the first order term from the Taylor series expansion, the linearized continuous-time description of the reduced order is stated as follows

    ˙x(t)=Ax(t)+B(t)u(t), (2.10)
    y(t)=Cx(t), (2.11)

    where A=[0100BCmtAamt0KdKp], B=[00Kx]T and C=[100]. The sensibility constants can be found as follows:

    Kd=Aa(βeVA+α2βeVB)1,Kp=βe(KpACLi(1+α2))VA(1+α2)αβe(KpBα2+CLi(1+α2))VB(1+α2),Kx=βeVAKxAαβeVBKxB,

    where the flow sensibility constants regarding the pressure at the cylinder chambers are stated as:

    KpA={cvxv0pspA0forxv>0cvxv0pA0p0forxv<0, (2.12)
    KpB={cvxv0pB0p0forxv>0cvxv0pspB0forxv<0, (2.13)

    and the flow sensibility constants regarding the spool position are stated as

    KxA={cvpspA0forxv>0cvpA0p0forxv<0, (2.14)
    KxB={cvpB0p0forxv>0cvpspB0forxv<0. (2.15)

    The previously mentioned valve sensibility constants are very significant in defining system stability and other dynamic characteristics [8]. Namely, the flow gain Kx has a direct impact on the stability of the HSA, because it directly affects the gain constant in the open loop of the HSA. Further, direct impact on the damping ratio of the HSA has the flow-pressure constant Kp. Hence, the pressure sensibility Kpx=Kx/Kp is quite high, which explains the ability of the HSA to transfer large friction loads with a small error.

    Let us consider a linear continuous-time model of the HSA with unknown dynamics, as follows:

    ˙x(t)=Ax(t)+Bu(t), (3.1)
    y(t)=Cx(t), (3.2)

    where x(t)Rn, u(t)Rm and y(t)Rr are the system state vector, the control input vector, and the output vector, respectively. ARn×n, BRn×m and CRr×n are unknown system matrices, assuming that (A,B) is controllable and (A,C) is observable.

    For the HSA described by (3.1) and (3.2), the performance index is stated as

    J(x0)=0[yT(τ)Qy(τ)+uT(τ)Ru(τ)]dτ, (3.3)

    where x0Rn is an initial state, Q=QT0 and R=RT>0, with (A,Q1/2C) being observable.

    A control law is also called a policy. The design objective is to find a linear optimal control policy in the form of

    u=Kx, (3.4)

    which minimizes the performance given by index (3.3). The optimal feedback gain matrix Kcan be determined as

    K=R1BTP, (3.5)

    where P=(P)T>0 is a unique symmetric positive definite solution of the well-known ARE

    ATP+PA+CTQCPBR1BTP=0, (3.6)

    under conditions that the system matrices are accurately known, as well as conditions that the pair (A,B) is controllable and the pair (A,Q1/2C) is observable [12]. It should be noted that this optimal control design is mainly applicable to low order simple linear systems. In fact, for high-order large scale systems, it is usually difficult to directly solve P from (3.6), which is nonlinear in P.

    Also, for practical implementation of the control system, it is easier to realize the data acquisition for discrete-time systems than for continuous-time systems. Consequently, we transform the continuous-time HSA into the following discrete-time HSA:

    xk+1=Adxk+Bduk, (3.7)
    yk=Cxk, (3.8)

    where Ad=eAh and Bd=h0(eAτdτ)B, where h>0 is a specific sampling period, assuming ωh=2π/h is the nonpathological sampling frequency whose existence is well known [45]. In other words, the controllability and observability of the original continuous-time HSA system is kept after discretization. Namely, if the state, input and output vectors at the sampled instant kh are xk, uk and yk, respectively, then (Ad,C) and (Ad,Q1/2C) are observable while (Ad,Bd) is controllable.

    As depicted in Figure 2, the ADP-based controller for the discretized HSA system consists of three parts: the state reconstruction, critic, and actor. The state reconstruction provides the relationship between the input/output data and the HSA states, which allows one to solve the optimal control problem of an HSA with unknown dynamics. Based on the input/output data, the critic part of the controller is designed to evaluate the performance of the control policy. The controller learns online in order to maximize its performance. Finally, the actor applies the improved control policy. The updates of the control actions are governed by an event-triggering mechanism to reduce the amount of data transmission from the controller to the HSA system.

    Figure 2.  Event-triggered ADP-based control algorithm for the discretized HSA system.

    The event-triggered design is based on a periodic sampling with a nonpathological h>0. We use ˆuk to represent the sampled value of uk, that is

    ˆuk=ukj,k[kj,kj+1), (3.9)

    where {kj}0 is a monotonically increasing sequence of the sampling time instants, and the control input is only updated at the discrete-time instants: k0,k1,k2, For the convenience of discussions, define the sampling error of the input data as

    Δk=ˆukuk. (3.10)

    Hence, the discrete-time system described by (3.7) and (3.8) can be rewritten as

    xk+1=Adxk+(Bduk+Δk), (3.11)
    yk=Cxk. (3.12)

    Further, the performance index for the discretized system described by (3.7) and (3.8) is

    Jd(x0)=j=0yTjQdyj+uTjRduj, (3.13)

    where Qd=Qh and Rd=Rh. The optimal control low minimizing (3.13) is

    uk=Kdxk, (3.14)

    where the discrete optimal feedback gain matrix is Kd=(R+BTdPdBd)1BTdPdAd, where Pd is the unique symmetric positive definite solution to

    ATdPdAdPd+CTQCATdPdBdKd=0. (3.15)

    Since (3.15) is nonlinear in Pd, it is difficult to directly solve Pd for high-order large-scale systems. Nevertheless, many efficient algorithms have been developed to numerically approximate the solution of (3.15). One of such algorithms was developed by Hewer [46], and it is introduced in the form of Lemma 3.1.

    Lemma 3.1. Let K0Rm×n be any stability feedback gain matrix and Pj be the symmetric positive definite solution of the Lyapunov equation

    (AdBdKj)TPj(AdBdKj)+CTQdC+KTjRdKj=0, (3.16)

    where Kj, j=1,2, can be updated as follows:

    Kj=(R+BTdPj1Bd)1BTdPj1Ad. (3.17)

    Then, it holds that

    1) AdBdKj is a stability matrix

    2) PdPj+1Pj

    3) limjKj=Kd, limjPj=Pd.

    By iteratively solving the Lyapunov equations given by (3.16), which is linear in Pj, and recursively updating the control policy Kj by (3.17), the solution to the nonlinear equation given by (3.15) is numerically approximated [46]. It has been concluded that the sequences {Pj}j=0 and {Kj}j=0, computed from this algorithm, converge to Pd and Kd, respectively. Moreover, for j=0,1,, AdBdKj is a Schur matrix. It should be noted that the method by Hewers involves a model-based policy iteration (PI) algorithm, which cannot be directly applied to the problem studied in this paper since it is an offline algorithm which depends on the system parameters. To apply this algorithm online for the discretized HSA described by (3.7) and (3.8), we will develop the control algorithm based on ADP via output feedback, which does not depend on the knowledge of HSA matrices.

    Motivated by [44,47], the discrete-time HSA described by (3.7) and (3.8) can be extended by using input/output sequences on the time horizon [kN,k1] as follows:

    xk=ANdxkN+V(N)ˉuk1,kN,ˉyk1,kN=U(N)xkN+T(N)ˉuk1,kN, (3.18)

    where

    ˉΔk=[ΔTk1ΔTk2ΔTkN]T,ˉuk1,kN=[ˆuTk1ˆuTk2ˆuTkN]T,ˉyk1,kN=[yTk1yTk2yTkN]T,V(N)=[BdAdBdAN1dBd]],U(N)=[(CAN1d)T(CAd)TCT]T,T(N)=[0CBdCAdBdCAN2dBd00CBdCAN3dBd000CBd0000],

    and the observability index is N=max(ρu,ρv) where ρu is the minimum integer which ensures that U(ρu) has full column rank and ρv is the minimum integer which ensures that V(ρv) has full row rank [44]. Therefore, there exists a left inverse of U(N), stated as U+(N)=[UT(N)U(N)]1UT(N). With the state reconstruction in (3.18), the idea of an ADP-based controller with output feedback can be applied to solve the optimal control problem of HSAs with unknown dynamics. A uniqueness of state reconstruction is stated in the form of Lemma 3.2 as follows [48].

    Lemma 3.2. If the conditions of observability and controllability of the system described by (3.7) and (3.8) are fulfilled, then the states of the HSA are uniquely received in terms of measured inputs and outputs signals as follows:

    xk=Θzk, (3.19)

    where Θ=[MuMy] has full row rank, Mu=V(N)MyT(N), My=ANdU+(N) and zk=[ˉuTk1,kNˉyTk1,kN]TRq, where q=N[dim(u)+dim(y)].

    Now, based on (3.16) and (3.17), an online learning strategy using output feedback can be introduced in the form of uk=¯Kdzk, providing suboptimal property of the closed-loop system. The discrete-time model (3.11) can be stated as follows:

    xk+1=Ajxk+Bd(Kjxk+ˆuk), (3.20)

    where Aj=AdBdKj. Setting ˉKj=KjΘ and ˉPj=ΘTPjΘ, from (3.16) and (3.20), it can be obtained

    zTk+1ˉPjzk+1zTkˉPjzk=(ˉKjzk+ˆuk)T[BTdˉPjBdBTdˉPjAd][ˉKjzk+ˆuk2zk](yTkQyk+zTkˉKTjRˉKjzk)=[ˆuTkˆuTk(zTkzTk)(ˉKTjˉKTj)]vec(ˉH1j)+2[(zTkzTk)(IqˉKTj)+(zTkuTk)]vec(ˉH2j)(yTkQyk+zTkˉKTjRˉKjzk)=ϕ1vec(ˉH1j)+ϕ2vec(ˉH2j)(yTkQyk+zTkˉKTjRˉKjzk), (3.21)

    where ˉH1j=BTdˉPjBd, ˉH2j=BTdˉPjAdΘ, ϕ1=ˆuTkˆuTk(zTkzTk)(ˉKTjˉKTj) and ϕ2=2[(zTkzTk)(IqˉKTj)+(zTkˆuTk)].

    The symbol is used to denote a Kronecker product operator. The vector function vec(V)=[vT1vT2vTm]T is stated as an mn-vector formed by stacking the columns of VRn×m on top of one another, where viRn denotes the columns of matrix V. For an arbitrary symmetric matrix MRn×n, vecs(M)=[m11,2m12,,2m1n,m22,2m23,,2mn1,n,mnn]TRn(n+1)/2 and for an arbitrary column vector vRn, ˜v=[v21,v1v2,,v1vn,v22,v2v3,,vn1vn,v2n]TRn(n+1)/2.

    The convergence of the online learning control using output feedback is guaranteed under the rank condition stated in the form of Lemma 3.3 [47]. Lemma 3.3 is about the condition of persistent excitation in adaptive control theory [49,50].

    Lemma 3.3. Let us suppose that for a sufficiently large sZ+, it holds that

    rank(Γ)=(dim(u)+dim(z))(dim(u)+dim(z)+1)/2, (3.22)

    where

    Γ=[ηk0ηk0,ηk1ηk1,,ηksηks],wherek0<k1<<ksZ+andηkj=[ˆuTkj,zTkj]T,j=¯0,s; (3.23)

    then (ˉPj,ˉH1j,ˉH2j) can be uniquely solved based on ˉKj and measurable online data during the period k[k0,ks]. Further, ˉKj+1 is obtained as follows:

    ˉKj+1=(R+ˉH1j)1ˉH2j. (3.24)

    Some exploration noise ek, which satisfies the persistent excitation condition, must be added into the input signal during the online learning phase due to the satisfaction of the rank condition given by (3.22), without affecting the convergence of the learning phase [43,51,52]. Note that (3.21) is called the policy evaluation, which is used to uniquely solve ˉPj, and (3.24) is the policy improvement, which is used to update the control gain ˉKj+1. Finally, we present the ADP-based online learning control algorithm in Figure 3.

    Figure 3.  Flowchart of ADP-based controller design.

    It should be noted that solving (3.21) instead of (3.16), completely eliminates the original request on the accurate knowledge of the HSA dynamics. Now, we only need to measure uk and yk. Namely, having in mind the expression for zk, we can see that the control policy ˆuk=¯Kkzk+Δk contains only the previously measured input-output data. With the event-triggered control law ˆuk, the system given by (3.20) is globally asymptotically stable (GAS) at the origin if

    Δk2αγyk2+λmin(Rd)ˆuk2η, (3.25)

    where α(0,1) and η is a positive constant satisfying ηλmax(Rd+BTdˉPdBd).

    The convergence of the ADP-based control algorithm is presented in the form of Theorem 3.4. For Hurwitz feedback matrix ABK, KRm×n is called stabilizing feedback gain matrix for a linear system ˙x=Ax+Bu.

    Theorem 3.4. If the condition of Lemma 3.3 is fulfilled, with some initial stabilizing feedback gain matrix ¯K0, then the sequences {¯Pj}j=0 and {¯Kj}j=0 received from this algorithm, converge to their optimal values ¯Pand ¯K, respectively [46,47].

    Proof. If Pj=PTj represents the solution of (3.16), under the stability feedback gain matrix ¯Kj, then Kj+1 is uniquely obtained from (3.17). It can be easily shown that ¯Pj and ¯Kj+1 fulfill (3.21) and (3.24). Now, setting ¯P and ¯K as solutions of (3.21) and (3.24), Lemma 3.3 provides that ¯Pj=¯P and ¯Kj+1=¯K are uniquely stated. Furthermore, from Lemma 3.1, we have that limj¯Kj=¯Kd and limj¯Pj=¯Pd. The proof of convergence is proved.

    The hybrid nature of the controller is shown in Figure 4. It is shown there that the feedback gain or policy is updated at discrete times by using (3.24) after the solution to (3.21) has been determined. On the other hand, the control input is a discrete time signal depending on the state z(k) at each time k. From Figure 4, it can be seen that the control gains are updated at discrete times, but the control signal is piecewise continuous.

    Figure 4.  Hybrid nature of control signal.

    In this section, we apply the proposed event-triggered ADP-based control design to the HSA. In the case of unknown dynamics and unmeasurable states of the HSA, it is meaningful to use the ADP-based method. Consequently, we conduct simulations on the HSA given by the linearized continuous-time description of (2.10) and (2.11) to show the effectiveness of the ADP-based control algorithm. A basic condition for energy savings in many hydraulically driven industrial systems is a high-quality design of event-triggered ADP control for the HSA.

    For this purpose, the HSA is discretized by applying the periodic sampling period h=0.1 s and the zero-order holder. The approximated optimal feedback gain and performance index for the discretized model of the HSA are iteratively obtained.

    The effectiveness of the ADP-based control algorithm will be considered for the HSA model described by (2.10) and (2.11) with the following parameters: the viscous friction BC=200 N s m1, the supply pressure pS=45 bar, the tank pressure p0=1.6 bar, the bulk modulus of the fluid βe=2×108 Pa, the total mass m=25 kg, the initial chamber volumes Va0=Vb0=8.2×106 m3, the load spring gradient Ke=101, the effective area of the head side of the piston Aa=4.91×104 m2, the effective area of the rod side of the piston Ab=2.43×104 m2, the internal leakage coefficient cLi=5×1014, the piston stroke L=1 m and discharge coefficients of valve orifices cvi=1.15, i=¯1,4.

    For the purpose of demonstrating the event-triggered ADP method with the HSA, the weight matrices, Q and R, are chosen to be identity matrices, the observability index is N=3, initial state vector is x0=[5510] and the convergence threshold ε is selected as 101.

    It should be noted that our event-driven ADP control design does not require exact knowledge of the HSA matrices. But, only for numerical verification via simulation, it is assumed that the system matrices in (2.10) and (2.11) are known.

    To verify the benefits of the ADP based online learning controller, Figure 5 depicts the errors between ˉPj and ˉPd and ˉKj and ˉKd, which indicate the convergence of ˉPj and ˉKj.

    Figure 5.  Convergence of ˉPj and ˉKj to their respective optimal values ˉP and ˉK during the learning process.

    The evolution of the maximum cost for HSA is shown in Figure 6(a), where V1 is the maximum cost by using the initial control policy, and V7 is the maximum cost by using the control policy after seven iterations. It can be seen that the approximated cost function V7 has been remarkably reduced relative to the initial cost V1. Figure 6(b) shows the 3D plot of the approximation error of the cost function. This error is close to zero which confirms that good approximation of the optimal cost function is achieved during the learning process.

    Figure 6.  (a) Comparison of the cost functions during learning; (b) error between the optimal and approximated cost function signal.

    The improved control policy and the initial control policy are compared in Figure 7(a). Further, Figure 7(b) shows the 3D plot of the difference between the approximated control obtained by using the online ADP-based control algorithm and the optimal control. This error is close to zero, which confirms that good approximation of the optimal input is also achieved during the learning process.

    Figure 7.  (a) Comparison of the control policies during the learning process; (b) error between the optimal and approximated input signal.

    Figure 8 shows the control input and the states of the HSA system desribed by (2.10) and (2.11) by using the ADP-based controller with periodic sampling.

    Figure 8.  Control input and states of the HSA model by using the ADP-based controller.

    To illustrate the benefits of the event-triggered ADP method, the control input and the states of the original HSA system described by (2.10) and (2.11), as obtained by using the event-triggered ADP-based controller is shown in Figure 9.

    Figure 9.  Control input and states of the HSA model by using the event-triggered ADP-based controller.

    The comparison of sampling numbers by using the event-triggered ADP controller versus the ADP controller with periodic sampling is shown in Figure 10.

    Figure 10.  Comparison of the total sampling numbers.

    It can be observed that similar control effects have been achieved by the two methods, however, for the event-triggered ADP method, the control input is updated only when the squared norm of the triggering error reaches the threshold, and it is kept constant otherwise. It is also shown that about 54% communication between the controller and the HSA is reduced by using the event-triggered ADP method instead of the ADP method. The sequence of steps of event-triggered sampling is depicted in Figure 11.

    Figure 11.  Sequence of steps of event-triggered sampling.

    This paper has considered the event-triggered data-driven optimal controller of the HSA with completely unknown dynamics as based on an ADP framework. A basic advantage of the presented control methodology is its ability to avoid the knowledge of entire system dynamics, which is very important in real conditions. By using the output feedback and the state reconstruction method an applied ADP-based control technique has been shown to be a useful tool for digital implementation in a real HSA. For that purpose, a discrete-time control policy was iteratively learned based on the discretized HSA model. The learned control policy very efficiently ensures online solutions to data-driven optimal control problems for the HSA. The presented online control policy only uses measured input/output data to learn the optimal control gain. Then, to reduce the communication between the controller and the HSA, an output feedback event-triggered ADP controller has been designed. The simulation results have shown the validity and effectiveness of the applied control approach for the HSA.

    This research was supported in part by the Serbian Ministry of Education, Science and Technological Development under grant 451-03-47/2023-01/200108, the National Natural Science Foundation of China under grants 61773181, 61976081, 62073001, 62103293 and 62203153,111 Project under grant B23008, the Fundamental Research Funds for the Central Universities under grant JUSRP51733B, the Anhui Provincial Key Research and Development Project under grant 2022i01020013 and the Natural Science Fund for Excellent Young Scholars of Henan Province under grant 202300410127.

    The authors declare that there is no conflict of interest.



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