
IEEE/CAA Journal of Automatica Sinica
Citation: | Jiankun Sun, Jun Yang and Shihua Li, "Reduced-Order GPIO Based Dynamic Event-Triggered Tracking Control of a Networked One-DOF Link Manipulator Without Velocity Measurement," IEEE/CAA J. Autom. Sinica, vol. 7, no. 3, pp. 725-734, May 2020. doi: 10.1109/JAS.2019.1911738 |
WITH the many applications of robot manipulators in different fields including advanced medical, space and defense, modern industries, etc., the control issues of manipulators have captured tremendous attention from industrial and academic communities [1], [2]. Meanwhile, the last two decades have witnessed a significant increase in interest in the area of networked control systems (NCS) due to the advances in network infrastructure, communication architecture and computer technology [3]–[8]. For the control issues of networked robot manipulators, many works have been published where the use of a network is essential for receiving the sensor signal and transmitting the control signal [9], [10]. For example, networked are used in the coordination control of multiple manipulators [11], telerobotic control systems [12], and so on.
Typically, a NCS is composed of five basic components including sensors, controllers, actuators, plants, and a shared communication network [13]–[15]. Those components need to exchange sensor and controller signals to achieve control tasks. For instance, the control input is transmitted and the sensor signal is received from a distance in telerobotic control systems [12]. The information is transported from one agent to another such that some complex tasks can be accomplished by multiple manipulators [11]. For NCSs, low energy consumption and computation are sought, with communication being costly due to the limited energy, computation and communication bandwidth. Even though many researchers have devoted themselves to the networked control manipulators [16]–[21], little attention has been paid to the communication constraint, which motivates us to develop a resource-efficient control method for the networked control manipulator.
To improve the resource efficiency while guaranteeing desirable control performance, event triggered control has been proposed in recent two decades as a kind of communication protocol where the control tasks are executed only when it is necessary [22]–[29]. In contrast to more commonly used periodic transmission schemes (i.e., time-triggering mechanism (TTM)), event triggered control tends to execute the control tasks that are sporadic in nature, rather than during a certain period of time as in the conventional TTMs [30]–[37]. Some experimental results have demonstrated that the event triggered control can significantly save the communication resource compared with the conventional TTMs with comparable performance [38]. Some survey papers on event triggered control can be found in [13], [39], [40].
In practical applications, it is hard to obtain the exact dynamics of robot manipulators due to the inevitable existence of lumped disturbances including external disturbances, load variation, friction, and system uncertainties [19], [20], [41]–[44]. Lumped disturbances not only deteriorate control properties, but also result in a waste of system resources, since even small disturbances may lead to increased transmission times [38], [45]. In addition, velocity measurements are generally unknown in most commercially available robot manipulators in order to decrease the manufacturing costs [16], [17], [21]. Therefore, designing a robust output feedback control method to attenuate the undesirable influence of lumped disturbances is essential and contributes to the improvement of both control properties and communication properties of event-triggered systems, which is another motivation of the current study.
In this paper, a novel dynamic event-triggered tracking control method is proposed for a one-degree of freedom (DOF) link manipulator subject to external disturbances and system uncertainties via the reduced-order generalized proportional-integral observer (GPIO) when only a sampled-data position measurement is available. By using a sampled-data position measurement and the control input, a new reduced-order GPIO is first proposed to estimate the velocity information and the lumped disturbance information, and the robust dynamic event-triggered controller is simultaneously designed by employing the technique of disturbance estimation/compensation to attenuate the undesirable influence of lumped disturbances on communication properties and tracking control properties. In the proposed control method, system information is transmitted via a communication network only when a well-designed dynamic event-triggering mechanism (DETM) is violated, such that a better tradeoff can be achieved between communication resource utilization and tracking control performance. In the proposed DETM, the threshold parameter is dynamically adjusted following an adaptive rule. Under the proposed event-triggering control method, it is shown that tracking errors asymptotically converge to a bounded region, and the bound can be set to be arbitrarily small by choosing appropriate parameters. The major merits of the proposed robust dynamic event-triggered tracking control method in this paper are fourfold:
1) The proposed tracking control method does not need the velocity measurement, and the values of the lumped disturbance and the velocity can be accurately estimated by the proposed reduced-order GPIO. Compared with the full-order GPIO, the reduced-order observer has one less parameter to be regulated.
2) The parameters of the proposed DETM can be adaptively updated according to a defined error, such that the communication resources can be significantly saved while guaranteeing a satisfactory tracking control performance.
3) By the virtue of the technique of disturbance estimation/compensation, the proposed robust control method can attenuate the undesirable influence of the lumped disturbance on communication properties and tracking control properties in the framework of the DETM.
4) Compared with some results on DETM [46], [47], where the triggering mechanisms are continuous-time, the proposed control method in the paper is more suitable for digital applications, since both the proposed robust output feedback tracking controller and the novel DETM are in discrete-time form.
The remainder of this paper is organized as follows: Section II describes the manipulator model and the problem statement. The proposed robust tracking controller and the novel DETM are shown in Section III. The stability analysis with some conditions on the existence of the proposed controller are given in Section IV. Then, Section V depicts the numerical simulation result to verify the efficiency of the proposed controller. Finally, the main conclusions are summarized in Section VI.
Throughout this paper, let
Consider the dynamics of a one-DOF link manipulator as follows:
D¨θ+C˙θ+G=τ+d | (1) |
where
In this paper, system uncertainties are taken into account since system parameters can not be accurately known in practical applications. We define
Defining a scale parameter
{˙x1(t)=Lx2(t)˙x2(t)=La0u(t)+w(t,x(t),u(t),d(t),˙θd(t))Ly(t)=x1(t) | (2) |
where
Inspired by most of the results on disturbance rejection control [42], [43], a common assumption on the lumped disturbance
Assumption 1: The lumped disturbance
Remark 1: It should be mentioned that the hypothesis on disturbances given in Assumption 1 is general and has been utilized in several existing works on disturbance rejection approaches [42], [43]. From a practical point of view, it is reasonable to assume that the lumped disturbance or its derivative are bounded since the external disturbance, the system uncertainties, and the desired velocity are all bounded in practice. The proposed method still works when the disturbance is piecewise continuous, since it can be viewed that the proposed observer is reset at every discontinuous instant.
Due to the finite rate digital communication channel between the sensor and the controller, the event-triggering mechanism is employed to reduce transmission times while a desirable trajectory tracking error can be guaranteed. In the presence of the lumped disturbance, trajectory tracking performance is inevitably deteriorated, and more communication times are probably generated if the disturbance is not properly handled. Therefore, to improve both the communication properties and trajectory tracking properties, this paper develops a new DETM for a robust output feedback controller via the reduced-order GPIO for the robot manipulator dynamics (1).
The structure of the proposed event-triggered method is shown in Fig. 1, where the signals are transmitted continuously along the solid lines, periodically along the dashed line, and intermittently based on the events along the dotted line. In the proposed control method, we only know the sampled-data position information at the sampling instant
Without the loss of generality, we assume the first event happens at
With the help of the sampled-data output, we design a new reduced-order GPIO for (2) in the time interval
{˙z1(t)=−Lβ1(z1(t)+β1y(tk+jT))+L(z2(t)+β2y(tk+jT))+La0u(t)˙z2(t)=−Lβ2(z1(t)+β1y(tk+jT))+L(z3(t)+β3y(tk+jT))˙z3(t)=−Lβ3(z1(t)+β1y(tk+jT))ˆx2(t)=z1(t)+β1y(t),ˆw(t)=L2(z2(t)+β2y(t))ˆ˙w(t)=L3(z3(t)+β3y(t)) | (3) |
where
It should be pointed out that the proposed reduced-order GPIO (3) is in continuous-time form, and such a design form is convenient for stability analysis, but not suitable for practical applications in digital computers. To cope with that, we give an accurate discretized version of (3). Firstly, (3) can be rewritten as follows:
˙Z(t)=L[−β110−β201−β300]⏟Π1Z(t)+L[a000]⏟Π2u(t)+L[−β21+β2−β1β2+β3−β1β3]⏟Π3y(tk+jT) | (4) |
where
Then, by integrating
Z(tk+(j+1)T)=eLΠ1TZ(tk+jT)+∫T0eLΠ1sds×(LΠ2u(t)+LΠ3y(tk+jT))=ˉΠ1Z(tk+jT)+ˉΠ2u(tk+jT)+ˉΠ3y(tk+jT) | (5) |
where
Let
{˙e1(t)=−Lβ1e1(t)+Le2(t)+L(β21−β2)(y(tk+jk)−y(t))˙e2(t)=−Lβ2e1(t)+Le3(t)+L(β1β2−β3)(y(tk+jk)−y(t))˙e3(t)=−Lβ3e1(t)+Lβ1β3(y(tk+jk)−y(t))+¨w(t)L3. | (6) |
Combining (3) and (5), we can get
u(t)=u(tk)=−1a0(k1x1(tk)+k2ˆx2(tk)+ˆw(tk)L2),t∈[tk,tk+1),k∈N | (7) |
where
The event-triggered controller (7) can be further redescribed by
u(t)=u(tk)=−1a0(k1x1(t)+k2x2(t)+w(t)L2)+1a0(k2e1(t)+e2(t))+k1a0(x1(t)−x1(tk+jT))+k2a0(x2(t)−x2(tk+jT))+1L2a0(w(t)−w(tk+jT))−k2a0(e1(t)−e1(tk+jT))−1a0(e2(t)−e2(tk+jT))+k1a0(x1(tk+jT)−x1(tk))+k2a0(ˆx2(tk+jT)−ˆx2(tk))+1L2a0(ˆw(tk+jT)−ˆw(tk)),t∈Ikj,k∈N,j=0,…,dk | (8) |
where the last five items
With the help of denotation
tk+1=min | (9) |
where
\begin{split} \sigma(qT) = & e^{-\alpha_{1} T}\sigma((q-1)T)+\dfrac{\alpha_{2}\left(1-e^{-\alpha_{1} T}\right)}{1+\bar{\varepsilon}((q-1)T)},\\ & t\in[(q-1)T,qT), \; q\in{\mathbb N}^{+}\end{split} | (10) |
with
From the proposed DETM (9), it can be easily deduced that
Remark 2: In the proposed DETM (9), there are three key parameters to be regulated. Firstly, we can choose a different weight matrix
Remark 3: It should be highlighted that the reduced-order GPIO (3) can be accurately discretized, the robust controller (7) is in the form of discrete-time, and the proposed DETM (9) is detected with a constant period as well. Hence, the proposed dynamic event-triggered control method is suitable for the implementation in digital computers.
Lemma 1: Consider the adaptive law (10) with a given initial condition
Proof: Firstly, by (10), we have that
\begin{split} \sigma(qT) &\leq e^{-\alpha_{1} T}\sigma((q-1)T)+\alpha_{2}\left(1-e^{-\alpha_{1}T}\right)\\ &\leq e^{-q\alpha_{1} T}\sigma(0)+\alpha_{2}\left(1-e^{-\alpha_{1}T}\right)\dfrac{1-e^{-q\alpha_{1}T}}{1-e^{-\alpha_{1}T}}\\ &= \left(\sigma(0)-\alpha_{2}\right)e^{-q\alpha_{1} T}+\alpha_{2}.\end{split} | (11) |
Therefore, it can be easily deduced from (11) that
Firstly, to develop the stability analysis of the event-triggered control systems, we introduce an important lemma as follows:
Lemma 2 [38]: Consider the following dynamics:
\dot{\zeta}(s) = F(\zeta(s), \zeta(s_{k})),\;\;\; \forall s\in [s_{k}, s_{k+1}),\; s_{k} = kT,\; k\in {\mathbb N} | (12) |
where F:
||F(\zeta(s), \zeta(s_{k}))||\leq \rho_{1}(||\zeta(s)-\zeta(s_{k})||+||\zeta(s_{k})||)+\rho_{2} | (13) |
where
\begin{split} &||\zeta(s)-\zeta(s_{k})|| \leq \left(||\zeta(s_{k})||+\dfrac{\rho_{2}}{\rho_{1}}\right)\left(e^{\rho_{1}(s-s_{k})}-1\right),\\ & \forall s\in[s_{k}, s_{k+1}),\; k\in {\mathbb N}. \end{split} |
Then, with the help of (2), (6) and (7), a closed-loop system can be finally obtained in the time interval
\left\{\!{\begin{aligned} {{\dot x}_1}(t) = & \,L{x_2}(t)\\ {{\dot x}_2}(t) = & - L{k_1}{x_1}(t) - L{k_2}{x_2}(t) + L{k_2}{e_1}(t) + L{e_2}(t)\\ & + L{k_1}({x_1}(t) \!-\! {x_1}({t_k} \!+\! jT)) \!+\! L{k_2}({x_2}(t) - {x_2}({t_k} + jT))\\ & + \dfrac{{(w(t) - w({t_k} + jT))}}{L} - L{k_2}({e_1}(t) - {e_1}({t_k} + jT))\\ & - L({e_2}(t) - {e_2}({t_k} + jT)) + L{k_1}({x_1}({t_k} + jT) - {x_1}({t_k}))\\ &+ L{k_2}({{\hat x}_2}({t_k} + jT) - {{\hat x}_2}({t_k})) + \dfrac{{(\hat w({t_k} + jT) - \hat w({t_k}))}}{L}\\ {{\dot e}_1}(t) = & - L{\beta _1}{e_1}(t) + {e_2}(t) + L(\beta _1^2 - {\beta _2})(y({t_k} + jk) - y(t))\\ {{\dot e}_2}(t) = & - L{\beta _2}{e_1}(t) \!+\! {e_3}(t) + L({\beta _1}{\beta _2}\! - \!{\beta _3})(y({t_k} \!+\! jk) - y(t))\\ {{\dot e}_3}(t) = & - L{\beta _3}{e_1}(t) + L{\beta _1}{\beta _3}(y({t_k} + jk) - y(t)) + \dfrac{{\ddot w(t)}}{{{L^3}}}. \end{aligned}} \right. | (14) |
Defining a new variable vector
\begin{array}{l} \dot \omega (t) = L\underbrace {\left[ {\begin{array}{*{20}{c}} 0&1&0&0&0\\ { - {k_1}}&{ - {k_2}}&{{k_2}}&1&0\\ 0&0&{ - {\beta _1}}&1&0\\ 0&0&{ - {\beta _2}}&0&1\\ 0&0&{ - {\beta _3}}&0&0 \end{array}} \right]}_A\omega (t)\\ \;\;\;\;\;\;\;\;\;\; + L\left[ {\begin{array}{*{20}{c}} 0\\ {\begin{array}{*{20}{l}} {{k_1}({x_1}(t) - {x_1}({t_k} + jT))}\\ { + {k_2}({x_2}(t) - {x_2}({t_k} + jT))}\\ { - {k_2}({e_1}(t) - {e_1}({t_k} + jT))}\\ { - ({e_2}(t) - {e_2}({t_k} + jT))} \end{array}}\\ {(\beta _1^2 - {\beta _2})(y({t_k} + jT) - y(t))}\\ {({\beta _1}{\beta _2} - {\beta _3})(y({t_k} + jT) - y(t))}\\ {{\beta _1}{\beta _3}(y({t_k} + jT) - y(t))} \end{array}} \right]\\ \;\;\;\;\;\;\;\;\;\;\; + \left[\!\!\!\!\!{\begin{array}{*{20}{c}} 0\\ {\begin{array}{*{20}{l}} {L{k_1}({x_1}({t_k} + jT) - {x_1}({t_k}))}\\ { + L{k_2}({{\hat x}_2}({t_k} + jT) - {{\hat x}_2}({t_k}))}\\ { + \dfrac{{(\hat w({t_k} + jT) - \hat w({t_k}))}}{L}} \end{array}}\\ 0\\ 0\\ 0 \end{array}}\!\!\!\!\right]+ \left[\!\!\!{\begin{array}{*{20}{c}} 0\\ {\dfrac{{w(t) - w({t_k} + jT))}}{L}}\\ 0\\ 0\\ {\dfrac{{\ddot w(t)}}{{{L^3}}}} \end{array}}\!\!\!\right]. \end{array} | (15) |
By Lemma 1, we have
\begin{split} &\left\| {\left[ {\begin{array}{*{20}{c}} 0\\ {\begin{array}{*{20}{l}} {L{k_1}({x_1}({t_k} + jT) - {x_1}({t_k}))}\\ { + L{k_2}({{\hat x}_2}({t_k} + jT) - {{\hat x}_2}({t_k}))}\\ { + \dfrac{{(\hat w({t_k} + jT) - \hat w({t_k}))}}{L}} \end{array}}\\ 0\\ 0\\ 0 \end{array}} \right]} \right\|\\ & \le L|{k_1}||{x_1}({t_k} + jT) - {x_1}({t_k})| + L|{k_2}||{{\hat x}_2}({t_k} + jT) - {{\hat x}_2}({t_k})|\\ & \quad + \dfrac{{|\hat w({t_k} + jT) - \hat w({t_k})|}}{L}\\ &\le {c_1}||\varepsilon({t_k} + jT)||\\ &\le \dfrac{{{c_1}}}{{\sqrt {{\lambda _m}(\varPhi )} }}(\bar \sigma |y({t_k} + jT)| + {\sigma _1}) \end{split} | (16) |
where
By Lemma 2 and Assumption 1, one gets
\begin{array}{l} \left\| {\left[ {\begin{array}{*{20}{c}} 0\\ {\begin{array}{*{20}{l}} {{k_1}({x_1}(t) - {x_1}({t_k} + jT))}\\ { + {k_2}({x_2}(t) - {x_2}({t_k} + jT))}\\ { - {k_2}({e_1}(t) - {e_1}({t_k} + jT))}\\ { - ({e_2}(t) - {e_2}({t_k} + jT))} \end{array}}\\ {(\beta _1^2 - {\beta _2})(y({t_k} + jT) - y(t))}\\ {({\beta _1}{\beta _2} - {\beta _3})(y({t_k} + jT) - y(t))}\\ {{\beta _1}{\beta _3}(y({t_k} + jT) - y(t))} \end{array}} \right]} \right\| \le {c_2}||\omega ({t_k} + jT) - \omega (t)|\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \le {c_2}(||\omega ({t_k} + jT)|| + {b_1})\\ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\times \left( {{e^{{b_2}T}} - 1} \right) \end{array} | (17) |
where
According to Assumption 1, one gets
\left\| {\left[ {\begin{array}{*{20}{c}} 0\\ {\dfrac{{(w(t) - w({t_k} + jT))}}{L}}\\ 0\\ 0\\ {\dfrac{{\ddot w(t)}}{{{L^3}}}} \end{array}} \right]} \right\| \le \tilde w | (18) |
where
Obviously, the system matrix A can be designed to be Hurwitz, when the parameters
Constructing a candidate Lyapunov function
\begin{split} \dot{V}(\omega(t))\leq\; & -L||\omega(t)||^{2}+2Lc_{2}||P||||\omega(t)||(||\omega(t_{k}+jT)||+b_{1})\\ & \times \left(e^{b_{2}T}-1\right)+\dfrac{2c_{1}}{\sqrt{\lambda_{m}(\varPhi)}} ||P||||\omega(t)||\\ & \times(\bar{\sigma}||\omega(t_{k}+jT)||+\sigma_{1})+2\tilde{w}||\omega(t)||.\end{split} | (19) |
Noticing that
\begin{split} \dot{V}(\omega(t)) \leq\; & -\gamma_{1}V(\omega(t))+\gamma_{2}\sqrt{V(\omega(t))V(\omega(t_{k}+jT))}\\ & +\gamma_{3}\sqrt{V(\omega(t))} \end{split} | (20) |
where
Since
\dfrac{d}{{dt}}\sqrt {V(\omega (t))} \le - \dfrac{{{\gamma _1}}}{2}\sqrt {V(\omega (t))} + \dfrac{{{\gamma _2}}}{2}\sqrt {V(\omega ({t_k} + jT))} + \dfrac{{{\gamma _3}}}{2}. | (21) |
Integrating
\begin{split} \sqrt {V(\omega ({t_k} + (j + 1)T))} \le\; & \left( {{e^{ - \dfrac{{{\gamma _1}}}{2}T}} + \left( {1 - {e^{ - \dfrac{{{\gamma _1}}}{2}T}}} \right)\dfrac{{{\gamma _2}}}{{{\gamma _1}}}} \right)\\ & \times \sqrt {V(\omega ({t_k} + jT))} + \left( {1 - {e^{ - \dfrac{{{\gamma _1}}}{2}T}}} \right)\dfrac{{{\gamma _3}}}{{{\gamma _1}}}\\ =\; & {\gamma _4}\sqrt {V(\omega ({t_k} + jT))} + \left( {1 - {e^{ - \dfrac{{{\gamma _1}}}{2}T}}} \right)\dfrac{{{\gamma _3}}}{{{\gamma _1}}} \end{split} | (22) |
where
Noticing that
\bar{\sigma}<\dfrac{L\lambda_{m}(P)\sqrt{\lambda_{m}(\varPhi)}}{2c_{1}\lambda_{M}^2(P)} | (23) |
can be satisfied by choosing appropriate parameters
T< \dfrac{1}{b_{2}}\ln\left(\dfrac{L\lambda_{m}(P)\sqrt{\lambda_{m}(\varPhi)}-2c_{1}\bar{\sigma}\lambda^2_{M}(P)}{2Lc_{2}\lambda^2_{M}(P)\sqrt{\lambda_{m}(\varPhi)}}+1\right) | (24) |
we have
{\cal{R}}_{1} = \left\{\omega \left|\; \right ||\omega|| \leq \dfrac{\gamma_{3}\left(1-e^{-\dfrac{\gamma_{1}}{2}T}\right)}{\sqrt{\lambda_{m}(P)}\gamma_{1}(1-\gamma_{4})} \right\}. | (25) |
Furthermore, by (17), it has
||\omega(t)|| \leq ||\omega(t_{k}+jT)||e^{b_{2}T}+b_{1}\left(e^{b_{2}T}-1\right). | (26) |
Since
{\cal{R}}_{2} = \left\{\omega \left|\; \right ||\omega|| \leq \dfrac{\gamma_{3}\left(1-e^{-\dfrac{\gamma_{1}}{2}T}\right)}{\sqrt{\lambda_{m}(P)}\gamma_{1}(1-\gamma_{4})}e^{b_{2}T}+b_{1}\left(e^{b_{2}T}-1\right) \right\} | (27) |
as
It can be observed that the bounded region
Theorem 1: Under the proposed dynamic event-triggered control method (3), (7) and (9). If the controller parameters are chosen to satisfy the conditions (23) and (24), then, the state variables of the closed-loop system (15) asymptotically converge to the bounded region
This section presents the simulation results on a single-link robot manipulator. For simplicity, the shorthand of the proposed event-triggered control method is denoted by DETM with compensation. In order to demonstrate the performance of the proposed control method, we conduct simulations under TTM. Meanwhile, we consider another simulation case where the lumped disturbance is not estimated and compensated in the dynamic event-triggered control method (we call the method as DETM without compensation), such that the benefit of disturbance estimation/compensation can be proven to improve both communication properties and tracking control performance.
The DETM without compensation means that the lumped disturbance is not estimated and compensated in the dynamic event-triggered control method. Specifically, for the DETM without compensation, the reduced-order observer is reduced to a first-order system as follows:
\left\{ \begin{aligned} & \dot{z}_{1}(t) = -L\beta_{1}(z_{1}(t)+\beta_{1}y(t_{k}+jT))+La_{0}u(t)\\ & \hat{x}_{2}(t) = z_{1}(t)+\beta_{1}y(t), \forall t\in I^{k}_{j}, k\in{\mathbb N}, j = 0,\ldots,d_{k} \end{aligned} \right. | (28) |
where
The event-triggered controller without disturbance compensation becomes
u(t) = u(t_{k}) = -\dfrac{1}{a_{0}}\left(k_{1}x_{1}(t_{k})+k_{2}\hat{x}_{2}(t_{k})\right), t\in[t_{k},t_{k+1}), k\in{\mathbb N} | (29) |
where
The nominal values of the parameters of (1) are chosen as follows:
\begin{split} & d(t) = \left\{ {\begin{array}{*{20}{l}} 1,&{t \in [0,10)}\\ 2,&{t \in [10,20)} \end{array}} \right.\\ & C(t) = \left\{ {\begin{array}{*{20}{l}} 2,&{t \in [0,5)}\\ {2 + 0.2(t - 5)},&{t \in [5,7)}\\ {2.2},&{t \in [7,20]} \end{array}} \right.\\ & m(t) = \left\{ {\begin{array}{*{20}{l}} 1,&{t \in [0,15)}\\ {1 + 0.2(t - 5)},&{t \in [15,17)}\\ {1.2},&{t \in [17,20]}. \end{array}} \right. \end{split} |
In the simulations, the sampling frequency is set as 100 Hz and the scale gain L is selected as 1. For the TTM and the DETM with compensation, the controller parameters and the initial states are the same, and set as follows:
Using each of the three different control methods, the response curves of the states, the control torque, and the event numbers are depicted in Fig. 2. From Fig. 2, we can see that the event numbers generated by the TTM and the DETMs with and without compensation are
The results of using variable
In this paper, we have considered the problem of robust output feedback tracking control design for a networked one-DOF robot manipulator without velocity measurement in the presence of disturbance/uncertainties and resource constraints. A novel reduced-order GPIO based dynamic event-triggered robust control method has been proposed to achieve a better balance of communication properties and tracking control performance. By using the sampled-data position measurement and the control signal, a reduced-order GPIO has been first proposed to estimate the velocity information and the lumped disturbance information. By the virtue of the disturbance estimation/compensation technique, the proposed control method can not only obtain the desired tracking performance, but also improve communication properties. The proposed control method is in the form of discrete-time, and only uses the sampled-data position signal, thereby being more suitable for practical applications. The results of the simulation of a one-DOF robot manipulator have been presented to demonstrate the effectiveness of the proposed control method.
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