Neuroadaptive control of information-poor servomechanisms with smooth and nonsmooth uncertainties

For motion control of uncertain servomechanisms, nonlinear dynamics including smooth and nonsmooth types, external disturbances, signal measurement noises, asymmetric input saturation, and so on seriously hinder the further development of high-performance closed-loop control algorithms. However, already existing control strategies cannot address the above-mentioned issues at the same time. It greatly increases the difficulty of controller design especially when some states are not measurable. Inspired by above motivations, this paper exploits neural networks to deal with nonlinear dynamics including discontinuous types, and combines extended state observers to estimate disturbances and unmeasurable states for uncertain nonlinear servomechanisms. Meanwhile, the desired-command-based model compensation approach is integrated into the controller design. It is worth noting that the neural network weights are updated by the combination of the estimation error and tracking error to acquire better approximation accuracy. According to above technologies, a novel extended-state-observer-based neural network adaptive motion control algorithm will be synthesized. The bounded stability of the whole closed-loop system is proved strictly. In addition, the comparisons of the application results on an electro-hydraulic servo system verify the availability and superiority of the developed control algorithm.


Introduction
Servo mechanical systems, such as servo motor systems [1][2][3][4], hydraulic servo systems [5][6][7][8], and aerocrafts [9,10], play an important role in industry and engineering applications, including processing, manufacturing, production, and so on. And they are often affected by unknown nonlinear dynamics, including continuous and discontinuous types, external disturbances, signal measurement noises, input saturation, and other factors. In addition, some signals in the system cannot be measured, because they are not equipped with corresponding sensors. Above-mentioned factors will seriously affect the motion accuracy of servo mechanical systems, leading it difficult to synthesize high-performance closed-loop controllers.
During the past few decades, the motion control for uncertain nonlinear servo systems has mainly revolved around B Hua Wang wanghua@njtech.edu.cn 1 School of Mechanical and Power Engineering, Nanjing Tech University, Nanjing 211816, China overcoming the modeling uncertainties therein. It is worth noting that a large number of leading control algorithms have been presented along with the development of control theory. Examples like adaptive robust control (ARC) [11], integral robust control (IRC) [12,13], sliding mode control (SMC) [14,15], disturbance-observer-based control [1,2,9,[16][17][18], neural network (NN)-based control [19][20][21][22][23][24], and so on. In [25], the authors have presented an integral retarded controller for velocity control of a direct current motor servo system. However, only a part of modeling uncertainties has been dealt with. In [26], for a servo motor actuated gantry system, the authors have developed a friction-compensationbased adaptive SMC control algorithm which utilizes the adaptive control technique to handle uncertain parameters that lie linearly in the system and employs the SMC to deal with external disturbances. However, the final control input contains a discontinuous term, which may cause severe chattering for the actual system. To handle large disturbances existing in a pneumatic servo system, Yuanqing Xia et al. have employed an extended state observer (ESO) to acquire the estimate of lumped nonlinearities and thus compensate for the total disturbances feedforwardly [27]. Furthermore, Daehee Won et al. have developed an output feedback integral sliding mode controller by incorporating a high gain observer to acquire the estimate of the unmeasured velocity as well as load pressure [16]. However, unknown nonlinear dynamics have not been considered in [16]. By taking advantage of the good approximation performance of NN for unknown functions, the authors in [7] and [23] have synthesized an adaptive NN (ANN) control algorithm to treat largely uncertain nonlinear dynamics existing in hydraulic cylinder actuated active suspension systems. Furthermore, by exploiting NNs to approximate unknown nonlinear functions and dead zone, the authors in [24] have proposed a ANN control strategy for a family of servo mechanisms. To simultaneously cope with unknown nonlinear dynamic uncertainties and exogenous disturbances, some control strategies which combine neural network with robust control methods have been proposed. In [21], a NN-based ARC control algorithm has been synthesized for a linear motor driven stage to obtain high-accuracy tracking effect and reject external disturbances. In [28], the authors have presented a recurrent NN-based SMC control structure for nonlinear systems to compensate for unknown nonlinear dynamics and suppress external disturbances. Moreover, a NN-based IRC control algorithm has been proposed by Jing Na et al. for uncertain servomechanisms. Nevertheless, the control strategies mentioned above exist certain conservatism when the considered system is faced with large external disturbances. Furthermore, influenced by external disturbances, the actual control input added to the system may be saturated due to the limitation of electrical hardware. Some control algorithms with saturation compensation [11,15,[29][30][31] have been developed for this working condition.
After careful analysis of the existing studies, it is known that scholars mainly deal with the continuous nonlinear dynamic uncertainties via NN, but pay little attention to the existence of discontinuous nonlinear dynamics in the system. As for servomechanisms, there prominently exist various discontinuous nonlinear dynamics, such as discontinuous nonlinear friction, discontinuous hydraulic fluid leakage, and so on. In addition, for feedback control, the measurement values of state feedback information are often needed, but they are usually not measurable for actual systems due to economic, structural, and other factors. As for speed feedback and acceleration feedback in the actual servo mechanical systems, these information are often not obtained by adding corresponding sensors, but by position information via filters. However, the dynamics of these filters are often not included in the closed-loop stability proof of the system, resulting in great conservatism in implementing the final controller. In addition, using the measured value of the signals directly when designing the controller causes a large amount of noises to be mixed, which may lead to unsatisfactory control effects. It is worth noting that most of the existing controllers still use the traditional backstepping method, which will lead to the issue of "explosion of complexity (EC)". By analyzing and summarizing the above studies, it can be found that unknown nonlinear dynamics in the system, especially the types of discontinuities, external interference, measurement noises, and input saturation, and so on, cannot simultaneously be handled under the condition that only some state signals can be measured. And it will also be affected by EC when designing the controller. The above factors bring great challenges to the design of the controller.
Stimulated by above analysis, we will propose a novel control algorithm to deal with the issues. Specially, the proposed control algorithm will employ NN adaptive control [32] to simultaneously estimate the continuous and discontinuous nonlinear dynamics that present in the system. And ESOs [33] will be exploited to estimate unmeasured states, mismatched, and matched exogenous disturbances in the system. By incorporating the desired-command-based technology into NN adaptive control, observer and controller to minimize the influence of signal measurement noises. More importantly, the effect of EC will be eliminated via the command-filtered backstepping technique. And the saturation compensation technology will be utilized to deal with the input saturation effect suffered by the system.
In summary, the proposed control algorithm has the following innovations: • The proposed control algorithm can simultaneously handle smooth and nonsmooth nonlinear dynamics, matched and mismatched exogenous disturbances, measurement signal noises, and input saturation. • The better neural network adaptive performance can be acquired, since the weight adaption laws are driven by the composite errors comprised of state prediction errors and system tracking errors. • The proposed control algorithm can be adapted in the working condition with unmeasurable information, which means that it has strong adaptability to working conditions. • The integrated control algorithm can not only eliminate the effects of "explosion of complexity", but also avoid the influence of filtering error.
For simplification, some notations are presented as follows:

Problem statement
A family of third-order uncertain nonlinear servomechanisms in the following state-space form is considered aṡ Here, χ i means the system state variable; ξ j (χ j ) denotes the unknown nonlinear dynamic with respect to χ j [χ 1 , …, χ j ] T ; ψ 3 (χ 3 ) means the unknown nonlinear dynamic with respect to χ 3 ; ε j (t) represents the lumped time-varying external disturbance; y o stands for the system output; and the input saturation nonlinearity u (ν) can be described by where ν indicates the control input; moreover, u > 0 and u < 0 denote known bounds for u (ν). Control Goal: To construct a bounded input ν with only partial state measurements, so that the y o χ 1 can follow any desired command χ 1d (t) ∈ Σ 1 closely in spite of various modeling uncertainties and input saturation nonlinearity.
Before synthesizing the control scheme to acquire the above-mentioned control objective, we firstly make the following several assumptions.

Assumption 1
The desired command χ 1d (t) belongs to C 2 . The system signals χ 1 and χ 3 are measurable, while the system signal χ 2 is immeasurable. The unknown nonlinear function ξ j (χ j ) belongs to C 1 . Specially, the unknown nonlinear function ψ 3 (χ 3 ) is bounded and continuous except at χ 3 c c where it exhibits a finite jump and is always continuous from the right, in which c c means a constant.

Neural network approximation
According to the powerful universal approximation characteristic of NN, for any C 1 nonlinear function F 1 (x), there exist ideal weights W * ∈ R L with L being the number of RBFNN nodes, so that [32] where x means the input vector, σ (x) indicates the approximation error, and η * (x) [η * 1 (x),η * 2 (x), …,η * L (x)] T with the function η * l (x), l 1, 2, …, L, usually being chosen as the Gaussian RBF [32] in which c l and w l are center vectors and widths of the RBF, respectively. Noting (5), the function F 1 (x) is able to be approximated bŷ Furthermore, for any continuously nonlinear function F 2 (x) except at x c c where it exhibits a jump and is also continuous from the right, there exist ideal weights G * ∈ R M with M being the number of NN nodes, so that where μ c (x) indicates the approximation error; and ψ * (x) [ψ * 1 (x),ψ * 2 (x), …,ψ * M (x)] T with the sigmoid jump function ψ * m (x), m 0, 1, …, M, usually being chosen as follows [35]: Based on (8), F 2 (x) can be estimated bŷ
First, let us introduce two variables χ D2 (t) σ 2 (χ 2d ) + ε 2 (t) and χ D3 (t) σ 3 (χ 3d ) + ε 3 (t) + ε 4 (t) as extended states; thus, the state-space expression (14) can be converted into the following form: Thus, the NN-based ESOs for (15) can be synthesized as where the adjustable gains ω o1 > 0 and ω o2 > 0 can be seen as the bandwidths of the designed observers; design parameters γ i and r b need to be chosen to guarantee that the characteristic polynomials γ 1 s 2 + γ 2 s + γ 3 and r 1 s + r 2 of the following matrices C o and R o are Hurwitz, respectively. It is worth noted from [36] that γ i and r b can be parameterized as γ 1 3, γ 2 3, γ 3 1 and r 1 2, r 2 1, respectively. Upon (15) and (16), the estimation error dynamics of the constructed observers are arranged as For facilitating the subsequent theoretical analysis, a set of state vectors ζ [ζ 1 , ζ 2 , can be rearranged aṡ in which As Furthermore, the theoretical analysis for the stability of the constructed NN-based ESOs which is included in that of the whole closed-loop system will be provided in Appendix A.

Command filtered model compensation controller with partial state feedback
To synthesize the subsequent control scheme, we define a set of tracking error variables z i and error compensation signals e i as follows: where v 1 and v 2 indicate virtual control functions; v 2,c denotes the filtered signal of v 2 via the presented command filter in (4). In addition, a set of compensating signals φ i is introduced as follows: where k i indicates positive feedback gains and u u (ν)-ν.
Step 1: Upon (1), (11), and (12), the time derivative of e 1 is produced aṡ Thus, the virtual control function v 1 is synthesized as After substituting (26) into (25), we havė Step 2: Noting (14), (23), and (24), we can get the time derivative of e 2 aṡ Aṡ We can achievė Therefore, we can construct the virtual control function v 2 as After substituting (31) into (30), one haṡ Step 3: Based on (14), (23), and (24), we are able to acquire the time derivative of e 3 aṡ Noting (33), the resulting control law ν can be synthesized by After substituting (34) into (33), one getṡ The framework of the developed controller is shown in Fig. 1.  Fig. 1 The framework of the developed control scheme

Theorem 1 With the NNs' weights online updated bẏ
in which Proj(•) indicates a smooth projection operator; U 2 , U 3 , and G mean positive definite and symmetric adjustable gain matrices; moreover, ρ 2 , ρ 3 , and ρ G indicate adjustable positive gains. By choosing design parameters w c , w f for the introduced command filter and other positive gains including ω o1 , ω o2 , k i felicitously, thus the integrated control algorithm can make sure that all signals in the closedloop servomechanism (1) with largely unknown mismatched, matched modeling uncertainties and input saturation nonlinearity are bounded.
Proof See Appendix A.

Remark 2
Notably, the developed control strategy can compensate for largely uncertain nonlinear dynamics and external disturbances with only partial state measurements. Specially, discontinuous nonlinear dynamics are also handled. And a command filter is introduced to remove the effect of EC. Moreover, by introducing a series of compensation functions, not only the input saturation effect is compensated, but also the filtering error is eliminated [38,39]. Furthermore, measurement noises are suppressed by introducing the desired compensation technique.

Application verification
To check the performance of the above-mentioned control strategy in different aspects, we will carry out numerical sim-ulations for a typical verification platform, i.e., a double-rod electric-hydraulic position servomechanism [8]. Moreover, random measurement noises are added. By introducing a set of state variables χ 1 y L , χ 2 ẏ L , and χ 3 A J P L /J, the nonlinear model of the double-rod hydraulic position servomechanism in [8] can be written aṡ where U(ν) g m u(ν) with g m being g m 4A J β oe k t k v /(JV t ). Specially, the definitions of the system variables are presented as follows. It is worth noted from (37) that an additional variable U (ν) is employed, which means that the actual control input ν can be calculated once U (ν) is synthesized out. Moreover, the central physical values of the system parameters for the employed hydraulic servomechanism are collected in Table  1.
To indicate the high-performance tracking effect of the developed algorithm, the following controllers are set for comparison in two working conditions.
(1) C1: This is the integrated control scheme in this paper, i.e., the NN controller with disturbance and nonlinear The bounds for u(ν) in working condition 1 The bounds for u(ν) in working condition 2 dynamics compensation via partial state information feedback. To make it clear, the design parameters of this controller are presented in Table 2. (2) C2: This is the controller which is same as the aforementioned C1 control scheme but without external disturbance compensation. The objective of setting such a controller is to test the disturbance compensation performance of the developed C1 controller. (3) C3: This is the controller which is same as the aforementioned C1 control algorithm but without NN adaptive and disturbance compensation. (4) C4: This is the integral robust controller whose control scheme is integrated as follows: The gains of this controller are presented as k r 80, β r2 80, and β r3 80, respectively.
(5) C5: This is the distinguished proportional integral controller with its proportional gain and integral gain being 1500 and 600, respectively.
For fair comparison, all corresponding design parameters of C2, C3, and C4 controllers are chosen as same as that of the C1 controller.

Working condition 1: normal motion
To indicate the high-performance tracking effects of the synthesized C1 controller, we employ the desired motion trajectory χ 1d (t) 38sin(πt)mm which is displayed in Fig. 2. Moreover, we can see from Fig. 2 that the output χ 1 under the C1 controller can track χ 1d closely. Furthermore, the tracking errors of the employed five controllers are shown in Fig. 3. From them, we can easily find that the synthesized C1 controller in this paper expresses best tracking performance than the other three controllers in terms of transient and steadystate performance. Moreover, the high tracking performance of C1 can be indicated via the presented performance indices in Figs. 4 and 5. It is noted that the reason why the C1 controller exhibits such excellent control performance is due to its model compensations for nonlinear dynamics and distur-  bances. In detail, the availability of the developed adaptive NN-based nonlinear dynamics compensation can be verified by comparing the tracking errors of C2 and C3. Furthermore, the validity of the developed ESO-based disturbance compensation performance can be proved by comparing the tracking errors of C1 and C2. Specially, the good state estimation performance and disturbance estimation performance from Figs. 6 and 7 illustrate the effectiveness of the developed ESOs. In addition, the estimation performance of nonlinear functions with the C1 controller in Fig. 8, who is regular and bounded, shows that the developed NN adaptive control has good function approximation performance. The effectiveness of combining nonlinear dynamics and disturbance compensation can be illustrated by comparing the tracking errors of C1 and C3. From Fig. 9, we can make out that the actual control input is always within the preset ranges and presents corresponding rules, even in the case of input saturation.

Working condition 2: high-frequency motion
To further indicate the high-performance tracking effects of the synthesized C1 controller, we employ the desired motion trajectory χ 1d (t) 10sin(6πt)mm which is displayed in Fig. 10. Moreover, we can see from Fig. 10 that the output χ 1 under the C1 controller can track χ 1d closely. Furthermore, the tracking errors and performance indices of the employed five controllers are shown from Figs. 11, 12, 13. All results indicate the high tracking performance of C1. In addition, from Fig. 14, we can make out that the actual control input is always within the preset ranges and presents corresponding rules, even in the case of input saturation.