Volume 16 Number 6
December 2019
Article Contents
Horacio Coral-Enriquez, Santiago Pulido-Guerrero and John Cortés-Romero. Robust Disturbance Rejection Based Control with Extended-state Resonant Observer for Sway Reduction in Uncertain Tower-cranes. International Journal of Automation and Computing, vol. 16, no. 6, pp. 812-827, 2019. doi: 10.1007/s11633-019-1179-6
Cite as: Horacio Coral-Enriquez, Santiago Pulido-Guerrero and John Cortés-Romero. Robust Disturbance Rejection Based Control with Extended-state Resonant Observer for Sway Reduction in Uncertain Tower-cranes. International Journal of Automation and Computing, vol. 16, no. 6, pp. 812-827, 2019. doi: 10.1007/s11633-019-1179-6

Robust Disturbance Rejection Based Control with Extended-state Resonant Observer for Sway Reduction in Uncertain Tower-cranes

Author Biography:
  • Horacio Coral-Enriquez received the B. Sc. degree in engineering in industrial automatica from the University of Cauca, Colombia in 2005, and the M. Sc. degree in automatica from the University of Valle, Colombia in 2010, the Ph. D. degree (Cum laude) in mechanical and mechatronics engineering from the National University of Colombia, Colombia in 2017. He is a research associate professor of the Faculty of Engineering at the University of San Buenaventura – Bogotá, Colombia. He is the author of over 29 technical papers in journals and international conference proceedings. His research interests include active disturbance rejection control, nonlinear control, wind turbine control, and applications of control theory. E-mail: hcoral@usbbog.edu.co (Corresponding author) ORCID iD: 0000-0002-1091-9112

    Santiago Pulido-Guerrero received the B. Sc. degree in mechatronics engineering from the San Buenaventura University, Colombia in 2017. He has published about 4 refereed journal and conference papers. His research interests include control theory, simulation and implementation of control systems for mechatronic systems. E-mail: sapulido@academia.usbbog.edu.co ORCID iD: 0000-0003-1676-8076

    John Cortés-Romero received the B. Sc. degree in electrical engineering, the M. Sc. degree in industrial automation and the Ph. D. degree in mathematics from the National University of Colombia, Colombia in 1995, 1999 and 2007, respectively. He is a research associate professor of Department of Electrical and Electronic Engineering at National University of Colombia, Colombia. He is the author of over 70 technical papers in journals and international conference proceedings. His research interests include nonlinear control applications, active disturbance rejection control and algebraic identification and estimation methods in feedback control systems. E-mail: jacortesr@unal.edu.co ORCID iD: 0000-0001-6991-4116

  • Received: 2018-10-08
  • Accepted: 2019-01-01
  • Published Online: 2019-05-21
  • In this paper, the problem of load transportation and robust mitigation of payload oscillations in uncertain tower-cranes is addressed. This problem is tackled through a control scheme based on the philosophy of active-disturbance-rejection. Here, a general disturbance model built with two dominant components: polynomial and harmonic, is stated. Then, a disturbance observer is formulated through state-vector augmentation of the tower-crane model. Thus, better performance of estimations for system states and disturbances is achieved. The control law is then formulated to actively reject the disturbances but also to accommodate the closed-loop system dynamics even under system uncertainty. The proposed control schema is validated via experimentation using a small-scale tower-crane, and compared with other relevant active disturbance rejection control (ADRC)-based techniques. The experimental results show that the proposed control scheme is robust under parametric uncertainty of the system, and provides improved attenuation of payload oscillations even under system uncertainty.
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Robust Disturbance Rejection Based Control with Extended-state Resonant Observer for Sway Reduction in Uncertain Tower-cranes

Abstract: In this paper, the problem of load transportation and robust mitigation of payload oscillations in uncertain tower-cranes is addressed. This problem is tackled through a control scheme based on the philosophy of active-disturbance-rejection. Here, a general disturbance model built with two dominant components: polynomial and harmonic, is stated. Then, a disturbance observer is formulated through state-vector augmentation of the tower-crane model. Thus, better performance of estimations for system states and disturbances is achieved. The control law is then formulated to actively reject the disturbances but also to accommodate the closed-loop system dynamics even under system uncertainty. The proposed control schema is validated via experimentation using a small-scale tower-crane, and compared with other relevant active disturbance rejection control (ADRC)-based techniques. The experimental results show that the proposed control scheme is robust under parametric uncertainty of the system, and provides improved attenuation of payload oscillations even under system uncertainty.

Horacio Coral-Enriquez, Santiago Pulido-Guerrero and John Cortés-Romero. Robust Disturbance Rejection Based Control with Extended-state Resonant Observer for Sway Reduction in Uncertain Tower-cranes. International Journal of Automation and Computing, vol. 16, no. 6, pp. 812-827, 2019. doi: 10.1007/s11633-019-1179-6
Citation: Horacio Coral-Enriquez, Santiago Pulido-Guerrero and John Cortés-Romero. Robust Disturbance Rejection Based Control with Extended-state Resonant Observer for Sway Reduction in Uncertain Tower-cranes. International Journal of Automation and Computing, vol. 16, no. 6, pp. 812-827, 2019. doi: 10.1007/s11633-019-1179-6
    • Tower cranes have an important task within the modern construction industry. This task is mainly focused on moving, lifting and positioning heavy loads in the safest and most effective way. Nevertheless, due to the mechanical configuration of tower cranes (see Fig. 1), these systems are prone to suffer dangerous oscillations on the load. This can be caused by environmental conditions such as wind gusts, earthquakes, or even the own inertia of the system. These oscillations have detrimental effects on the construction process, due to some problems that may occur like reduction of accuracy on the positioning of the load, the increase of load transportation time, the decrease of system integrity, among others.

      Figure 1.  Scale model of a tower-crane

      Under this scenario, the development and application of different control techniques have taken on great relevance in recent years in order to address such problems, but also to improve the performance in load transportation and mitigation of payload oscillations.

      The most common discrepancies between the theoretical model and the real system caused by both non-modeled dynamics (coulomb friction, fast dynamics) and the uncertainty in parameters (length of the rope, mass of the load), as well as external disturbances, non-linearity and the underactuated property of the system, have forced researchers to focus on two main control schemes. On one hand, control schemes which generate desired trajectories for the trolley in order to reduce oscillations on the load such as: task-oriented trajectory planning[1], minimum-time trajectory planning with constraints[2], and an energy-optimal solution[3]. On the other hand, control techniques focused on controlling both the payload oscillation and the trolley position of the tower-crane. These last control techniques are designed to reach fast and precise movements of the trolley with low/reduced oscillations of the payload[4, 5]. Also, nonlinear control has been applied to ship-mounted cranes[6], double-pendulum overhead cranes[7], and gantry cranes[8], but complex control laws are required and a detailed nonlinear model is needed. Sliding mode and nonlinear approaches[9, 10] for under-actuated systems have been also proposed. In a similar way, linear control techniques have been also applied such as $ H_{\infty} $[11] and linear quadratic regulator ($ LQR $)[12, 13]; also, robust controllers based on linear matrix inequalities (LMIs) have been designed to achieve robustness against parameter variations of the tower-crane[14, 15]. However, a vast variety of these techniques either require detailed mathematical models or do not use the knowledge of the nature of disturbances to improve the robustness and disturbance rejection capabilities of the control system.

      From a different point of view, Cai et al.[16] develop an active disturbance rejection controller (ADRC) to control the payload position of a tower-crane providing reduction of oscillations. Under this control methodology, the knowledge of disturbances affecting the system is taken into account in the design stage. Thus, equivalent disturbances at the system input are estimated by using a definition of an internal model approximation, to reject them using the control law[17, 18]. Today, ADRC is a vast field of research, developments for autonomous bicycles[19, 20], wind energy[21], and others[22, 23], but rarely explored and extended to tower-crane control. ADRC is characterized by its simple design and its great performance in terms of disturbance rejection properties while providing robustness to the control system. Some work in ADRC applied to tower cranes[16, 24] shows the advantages of using extended state observers to provide an active rejection of disturbances affecting the system. However, nowadays several works demonstrate that: 1) conventional tuning of ADRC schemes is not the best option for all systems[24, 25], 2) an internal model approximation such as $ \dfrac{{\rm{d}}}{{\rm{d}}t} \approx 0 $ may not be proper for harmonic disturbances (endogenous and exogenous) that affect the system[24, 26], and 3) the inherent uncertainty in the control gain (assumed to be known in ADRC and generalized proportional integral (GPI) control[16, 27]) should be taken into account from the design stage.

      Therefore, in this paper a robust disturbance rejection control scheme is proposed to solve the problem of payload oscillations in tower-cranes with uncertain payload mass and rope length. The proposed scheme differs from other ADRC schemes due to some facts: 1) it uses a set of LMIs to tune the controller, 2) the LMI tuning also provides (from the design stage) robustness to the control system against parameter variations (e.g., the rope length and payload mass), and 3) the scheme uses extended state observers designed, based on resonant and polynomial annihilators that provide better estimation/rejection of disturbances with harmonic components effecting the tower crane.

      The rest of the paper is organized as follows. In Section 2, the model of a tower-crane is obtained and describes the small-scale tower-crane used for validation purposes. In Section 3, the active disturbance rejection (ADR) philosophy-based control scheme is formulated. Section 4 shows the experimental results of the proposed control scheme compared with conventional ADR-based control approaches. Section 5, shows an experimental frequency-domain analysis of each implemented control system which measures the robustness under real conditions of the control systems. Finally, Section 6 establishes the conclusions of the work.

    • In this section, the mathematical model of a tower-crane is presented. First, the mechanical system is analyzed with the LaGrange method and subsequently is linearized around an equilibrium point. After that, the mechatronic model is obtained and then joined to the mechanical model.

      The model of a tower crane is composed by three main elements: the payload, the trolley, and the actuator. The payload with mass $ m $ is joined to the trolley of mass M by an inextensible rope of length L. The trolley is connected by a set of pulleys of radius $ r $ to the actuator (a direct current (DC)-motor). This motor has an input voltage $ V_{in} $, that generates a torque $ \tau _{m} $, which moves the trolley linearly along the X axis. Moreover, the DC-motor is controlled by a full-bridge driver, which is modeled as a gain $ k_{pwm} $ that converts the duty cycle signal to average voltage.

      In Fig. 2 a simplified model of the mechanical subsystem is presented. This subsystem shows the relationship between the elements and their variables. On one side, $ \theta _{p} $ describes the payload angle with respect to the vertical axis of the trolley. On the other side, the trolley is pulled by a tension $ T_{1} $ along the jib (see Fig. 1), generating a linear displacement in $ x $. This model considers the following assumptions:

      Figure 2.  Simplified model of the mechanical system of a tower crane

      1) The payload is modeled as a particle.

      2) Rope mass is neglected and considered as a rigid body.

      3) The movement of the system is restrained at the (X-Y) plane.

    • The mechanical model of the tower crane, is obtained with the following LaGrange formulation[28]:

      $\frac{{\rm{d}}}{{{\rm{d}}t}}\left( {\frac{{\partial {L_a}}}{{\partial {{\dot q}_i}}}} \right) - \left( {\frac{{\partial {L_a}}}{{\partial {q_i}}}} \right) + \left( {\frac{{\partial R}}{{\partial {{\dot q}_i}}}} \right) = {Q_i}$

      (1)

      where $ L_{a} = K-P $, with K and P as the kinetic and potential energy of the system, respectively; $ q_{i} $ with $ i = 1,2 $ is the generalized coordinate for the trolley position $ x $ and payload angle $ \theta _{p} $, respectively. Moreover, the term of viscous damping is given by the Rayleigh dissipation function $ R $. The term $ Q_{i} $ groups the conservative $ \delta{W_{c}} $ and non-conservative $ \delta{W_{nc}} $ forces on the system. Then, according to the LaGrange formulation mentioned above, the kinetic energy of the system (see Fig. 2) is defined as

      $K = \frac{1}{2}M{\dot x^2} + \frac{1}{2}m{v^2}$

      (2)

      where the vector $ v $ describes the payload velocity as

      ${v^2} = v_x^2 + v_y^2$

      (3)

      with $ v_{x} = \dot{x}+L\dot{\theta}_{p}{\rm{cos}}(\theta_{p}) $ and $ v_{y} = -L\dot{\theta}_{p}{\rm{sin}}(\theta _{p}) $. The total potential energy is represented by

      $P = mgL(1 - {\rm{cos}}({\theta _p}))$

      (4)

      where g is the gravitational acceleration. Furthermore, a linear damping component is considered in the trolley dynamics. Then, this term of viscous damping is represented by

      ${F_{vx}} = \frac{1}{2}{c_x}{\dot x^2}$

      (5)

      where $ c_{x} $ is the damping coefficient of the linear damper. Likewise, a torsional damping component is added to the pendulum dynamics, which is represented by

      ${F_{v\theta }} = \frac{1}{2}{c_\theta }\dot \theta _p^2$

      (6)

      where $ c_{\theta} $ is the damping coefficient of the torsional damper. Finally, a non-conservative Coulomb friction force, and a conservative external input force are considered for the trolley dynamics inside the term $ Q_{1} $. Subsequently, with the solution of (2)–(6) in the LaGrange (1), the nonlinear model of the tower crane, is stated

      $(m + M)\ddot x + mL({\ddot \theta _p}{\rm{cos}}({\theta _p}) - \dot \theta _p^2{\rm{sin}}({\theta _p})) + {c_x}\dot x = {F_x} - {F_c}$

      (7)

      $m\ddot xL{\rm{cos}}({\theta _p}) + m{L^2}{\ddot \theta _p} + mgL{\rm{sin}}({\theta _p}) + {c_\theta }{\dot \theta _p} = 0$

      (8)

      where $ F_{x} $ is a force input and $ F_{c} = N\mu {\rm{sgn}}(\dot{x}) $ is Coulomb friction, with N as the normal force and $ \mu $ is the friction coefficient. On the other hand, a linear model of a DC-motor with gear-box is considered and it is represented by

      ${{\tau _m}}= {J^\prime }{{\ddot \theta }_m} + {B^\prime }{{\dot \theta }_m}$

      (9)

      ${{V_{in}}}= \frac{{{\tau _m}{R_a}}}{{{k_b}}} + {{\dot \theta }_m}{k_b}$

      (10)

      where $ J^{\prime} = J_{g}+n^2J+J_{e}+J_{p} $, with $ J_{g} $ internal moment of inertia of the gear box, $ n $ is the gear ratio, $ J $ is the rotor inertia, $ J_{e} $ and $ J_{p} $ are the moment of inertia of the encoder disk and the pulley, respectively. Furthermore, $ B^{\prime} $ is the internal and external viscous damping of the DC-motor, $ k_{b} $ represents the mechanical constant of the motor, $ R_{a} $ is the armature resistance, $ V_{in} $ is the input voltage applied to the motor, $ \theta _{m} $ is the angular position and $ \tau _{m} $ is the motor torque. The DC-motor model (9) and (10), moreover, is linearly related with (7) and (8) through:

      $x= r{\theta _m}$

      (11)

      ${{\tau _m}}= r{F_x}$

      (12)

      where $ r $ is the pulley radius.

      The mechanical model (7)–(8) and the DC-motor model (9)–(10) are then unified with (11) and (12), obtaining the following nominal non-linear model:

      $\begin{split} {\ddot x} = &- \frac{{{F_c} + {F_{vx}} - m{\rm{sin}}({\theta _p})(g + L\dot \theta _p^2)}}{{{z_1}}} - \\ &\frac{{\dot x({B^\prime }{R_a} + k_b^2)}}{{{R_a}{r^2}{z_1}}} + \frac{{{V_{in}}{k_{pwm}}{k_b}}}{{{R_a}r{z_1}}} - \frac{{{F_{v\theta }}{\rm{cos}}({\theta _p})}}{{L{z_1}}} \end{split}$

      (13)

      $\begin{split} {{\ddot \theta }_m} = & - \frac{{({F_{v\theta }} + Lgm{\rm{sin}}({\theta _p})){z_2}}}{{{z_3}}} +\\ & \frac{{Lm{\rm{cos}}({\theta _p})\dot x(k_b^2 + {B^\prime }{R_a})}}{{{R_a}{z_3}}}+ \\ &\frac{{{r^2}({F_c} + {F_{vx}} - L\dot \theta _p^2{\rm{sin}}({\theta _p}))}}{{{z_3}}} - \\ &\frac{{{V_{in}}{k_{pwm}}{k_b}r}}{{{R_a}{z_3}}} \end{split}$

      (14)

      with

      ${{z_1}}= M + m - m{\rm{cos}}{{({\theta _p})}^2} + \frac{{{J^\prime }}}{{{r^2}}}$

      (15)

      ${{z_2}}= {J^\prime } + {r^2}(M + m)$

      (16)

      ${{z_3}}= {L^2}m({J^\prime } + {r^2}(M + m - m{\rm{cos}}{{({\theta _p})}^2})).$

      (17)

      An equilibrium-point $ (\dot{\theta}_{p} = 0,\;\theta _{p} = 0) $ is considered for the non-linear model (13) and (14). In this way, the state variables are defined as $ x_{1} = x $, $ x_{2} = \dot{x} $, $ x_{3} = \theta_{p} $, $ x_{4} = \dot{\theta}_{p} $ and a linear state-space model can be found:

      ${\dot x_s}(t) = {A_s}{x_s}(t) + {B_s}u(t)$

      (18)

      with

      $\begin{split} & {x_s}(t) = \left[ {\begin{aligned} {{x_1}(t)}\\ {{x_2}(t)}\\ {{x_3}(t)}\\ {{x_4}(t)} \end{aligned}} \right]\!,\;{A_s} = \left[\!\!{\begin{array}{*{20}{c}} 0&1&0&0\\ 0&{{a_{22}}}&{{a_{23}}}&{{a_{24}}}\\ 0&0&0&1\\ 0&{{a_{42}}}&{{a_{43}}}&{{a_{44}}} \end{array}}\!\!\right]\!\\ & {B_s} = \left[ {\begin{aligned} 0\;\\ {{b_{21}}}\\ 0\;\\ {{b_{41}}} \end{aligned}} \right]\end{split}$

      and

      ${{a_{22}}}= - \frac{{k_b^2 + {R_a}({c_x}{r^2} + {B^\prime })}}{{{R_a}({L^2}m(M{r^2} + J))}}$

      (19)

      ${{a_{23}}}= \frac{{mg{r^2}}}{{M{r^2} + {J^\prime }}}$

      (20)

      ${{a_{24}}}= - \frac{{{c_\theta }{r^2}}}{{L(M{r^2} + {J^\prime })}}$

      (21)

      ${{a_{42}}}= \frac{{k_b^2 + {c_x}{R_a}{r^2} + {B^\prime }{R_a}}}{{{R_a}(M{r^2} + {J^\prime })}}$

      (22)

      ${{a_{43}}}= - \frac{{g({r^2}(M + m) + {J^\prime })}}{{(L(M{r^2} + {J^\prime }))}}$

      (23)

      ${{a_{44}}}= - \frac{{{c_\theta }({r^2}(M + m) + {J^\prime })}}{{{L^2}m(M{r^2} + {J^\prime })}}$

      (24)

      ${{b_{21}}}= \frac{{{k_b}{k_{pwm}}r}}{{{R_a}(M{r^2} + {J^\prime })}}$

      (25)

      ${{b_{41}}}= - \frac{{{k_b}{k_{pwm}}r}}{{L{R_a}(M{r^2} + {J^\prime })}}$

      (26)

      where $ k_{pwm} $ converts the duty cycle signal to average voltage to the motor terminals, and $ u(t) $ is the control input.

    • The proposed control technique derived from the ADRC philosophy is built in two stages: the formulation of an extended state observer and the design of a robust control law. These stages allow estimating and rejecting disturbances in real-time, but also accommodate the dynamics of the closed-loop system under system uncertainty. Here, on one side, the conventional extended state observer used in ADRC is modified and formulated to properly address the load oscillations of the tower-crane, based on a polynomial and resonant combination of disturbance internal models. On the other side, the control scheme is proposed with a robust tuning methodology to handle parametric uncertainties, typical of tower-crane systems.

      In order to begin the formulation of the observer, the system model defined in (18) is modified to include a disturbance signal term $ \xi $ equivalent at the input. Then, the system takes the following form

      ${\dot x_s} = {A_s}{x_s} + {B_s}(u + \xi )$

      (27)

      where $ \xi $ is called, in ADRC terminology, the general disturbance signal that lumps together all exogenous and endogenous disturbances of the system including nonlinearities and un-modeled dynamics given by the difference between the real system (not known) and the linearized model of the system (used for design). Hence, as in ADRC, the main idea is as simple as providing an accurate estimation of $ \xi $ and then rejecting it.

    • For making a proper and accurate estimation of the disturbance signal $ \xi $, some knowledge of the signal should be assumed or stated. This concept comes from the internal model principle (IMP)[29] which states that in order to track/reject an exogenous signal, the model of such signal must be included in the control loop. Hence, for the observer case, a proper estimation of the signal is accomplished provided that the model of the signal is included in the observer. This model can be simple or complex depending on which components of the signal are needed, or should be accurately estimated.

      From this viewpoint, the concept of annihilator came up in the literature, and it was defined as the differential operator that cancels or suppresses the disturbance signal in time. This annihilator in the form of an internal model equation is used to formulate the disturbance signal dynamics as a state-space model which is added to the system model; this creates an extended state model that contains the states of the system and the states of the disturbance model. Under this scenario, if an unknown constant signal $ (\xi\approx const) $ is acting as a disturbance of the control system, an accurate estimation of the disturbance is developed by defining its annihilator as $ \dot{\xi} $. In consequence, this knowledge can be used in the observer design to provide some accurate estimations of the disturbance in real-time.

      Define a disturbance signal $ \xi $ with two dominant components: polynomial component and harmonic components, such as

      $\xi = {\xi _p} + {\xi _h}$

      (28)

      where $ {\xi _p} $ is the polynomial component and $ {\xi _h} $ contains harmonic components.

      For the case of the polynomial disturbance $ {\xi _p} $, a local approximation of this component is defined as

      ${\xi _p} \approx {\rho _0} + {\rho _1}t + {\rho _2}{t^2} + \cdots + {\rho _{m - 1}}{t^{m - 1}}$

      (29)

      where $ \{\rho_0,\rho_1,\cdots,\rho _{m - 1}\} $ are constants or slow-varying parameters. Then, the annihilator of the disturbance signal (29) can be stated as

      ${\varphi ^{{\xi _p}}} = \frac{{{{\rm{d}}^m}{\xi _p}}}{{{\rm{d}}{t^m}}} = \xi _p^{(m)}.$

      (30)

      In consequence, if the disturbance is an unknown constant signal $ \rho_0 $ (that is $ m = 1 $), the annihilator would be $ {{\dot \xi }_p} $; and if the disturbance is a ramp with bias (i.e., $ m = 2 $), the annihilator would be $ {{\ddot \xi }_p} $, etc. Note that the definition of the annihilator is not unique.

      Remark 1. Several authors[30, 31] have applied first order disturbance model approximation (first order annihilator $ {{\dot \xi }_p} $) to different areas; thus, the most usual model is a first order approximation mainly because the disturbances are taken locally as additive constant signals or model uncertainties/nonlinearities. Nevertheless, the disturbance internal model in (30) is a more generalized extension and representation of the disturbance signal which provides extra information and increases the ability to locally estimate different types of disturbances. Then, if it is assumed to be the case of completely unknown disturbances, it is more useful to use a generic polynomial disturbance model. Therefore, the degree of approximation of the disturbance internal model can be defined by analyzing the complexity of the signal, which in most of the cases is analyzed from its effects on the output of the system.

      Observe that a first-degree disturbance model approximation can estimate constant as well as more complex disturbances, but this is limited/given by the observer bandwidth. Then, a proper identification of the dominant disturbance components and consequent annihilator definition offers a better trade-off among disturbance estimation/performance, observer-bandwidth, and noise amplification.

      The general form of the annihilator defined in (30) can be described in state-space as

      ${\dot x_\xi ^p}= A_\xi ^px_\xi ^p + B_\xi ^p{\varphi ^{{\xi _p}}}$

      (31)

      ${{\xi _p}}= C_\xi ^px_\xi ^p$

      (32)

      with

      $A_\xi ^p = \left[\!\!{\begin{array}{*{20}{c}} 0&1&0& \cdots &0\\ 0&0&1& \cdots &0\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0&0&0& \cdots &1\\ 0&0&0& \cdots &0 \end{array}}\!\!\right]\!,\;B_\xi ^p = \left[ {\begin{aligned} 0\\ 0\\ \vdots \\ 0\\ 1 \end{aligned}} \right]$

      $C_\xi ^p = \left[ 1\quad 0\quad 0\quad \cdots \quad 0\right]\,,\;x_\xi ^p = \left[\!\!{\begin{array}{*{20}{c}} {{\xi _p}}\\ {{{\dot \xi }_p}}\\ {{{\ddot \xi }_p}}\\ \vdots \\ {\xi _p^{(m - 1)}} \end{array}}\!\!\right]$

      where $ x_1^p = {\xi _p} $, $ x_2^p = {{\dot \xi }_p} $, $ x_3^p = {{\ddot \xi }_p}, \cdots, $$ x_m^p = \xi _p^{(m - 1)} $.

      For the case of the disturbance with harmonic components, $ \xi_h $ describes the disturbance using the following formula:

      ${\xi _h} = \xi _1^h + \xi _2^h + \xi _3^h + \cdots + \xi _n^h$

      (33)

      where $ \xi _1^h $, $ \xi _2^h ,\;\cdots, $ $ \xi _n^h $ are the $ n $ dominant frequency components of the disturbance signal.

      Then, a general harmonic signal for a single frequency component can be stated as

      $\xi _1^h = {\beta _1}\sin \left( {{\omega _1}t + {\gamma _1}} \right)$

      (34)

      where $ \beta _1 $ is the magnitude, $ \omega _1 $ is the signal frequency and $ \gamma _1 $ is the phase. Under this formulation, the unique parameter of the disturbance that should be known is the signal frequency $ \omega _1 $. Hence, the annihilator for the disturbance signal (34) is stated as

      $\varphi _1^{{\xi _h}} = \ddot \xi _1^h + \omega _1^2\xi _1^h.$

      (35)

      A simple proof can be made to validate the annihilator (35) for the signal (34). Taking the first and second time-derivative of (34)

      $\begin{aligned} & \dot \xi _1^h = {\beta _1}{\omega _1}\cos \left( {{\omega _1}t + {\gamma _1}} \right)\\ & \ddot \xi _1^h = - {\beta _1}\omega _1^2\sin \left( {{\omega _1}t + {\gamma _1}} \right) \end{aligned}$

      and replacing them into (35), the annihilator is validated:

      $\ddot \xi _1^h + \omega _1^2\xi _1^h = {\beta _1}\omega _1^2\left( {\sin \left( {{\omega _1}t + {\gamma _1}} \right) - \sin \left( {{\omega _1}t + {\gamma _1}} \right)} \right) = 0.$

      Thus, a general annihilator for the $ i $-th harmonic disturbance signal $ \xi _i^h = {\beta _i}\sin \left( {{\omega _i}t + {\gamma _i}} \right) $ is

      $\varphi _i^{{\xi _h}} = \ddot \xi _i^h + \omega _i^2\xi _i^h$

      (36)

      and, the state-space description of (36) results in

      ${\dot x_i^h}= A_i^{{\xi _h}}x_i^h + B_i^{{\xi _h}}\varphi _i^{{\xi _h}}$

      (37)

      $\begin{array}{*{20}{l}} {\xi _i^h}\;{ =\; C_i^{{\xi _h}}x_i^h} \end{array}$

      (38)

      with

      $x_i^h = \left[ {\begin{aligned} {x_{i1}^h}\\ {x_{i2}^h} \end{aligned}} \right]\!,\;A_i^{{\xi _h}} = \left[\!\!{\begin{array}{*{20}{c}} 0&1\\ { - \omega _i^2}&0 \end{array}}\!\!\right]\!,\;B_i^{{\xi _h}} = \left[ {\begin{aligned} 0\\ 1 \end{aligned}} \right]$

      $C_i^{{\xi _h}} = \left[ {\begin{array}{*{20}{c}} 1&0 \end{array}} \right]$

      where $ x_{i1}^h = \xi _i^h $ and $ x_{i2}^h = \dot \xi _i^h $ are the first and second state variables of the $ i $-th disturbance component.

      If $ n $ harmonic components of the disturbance $ \xi_h $ are considered, its state-space description results in

      ${\dot x_\xi ^h}= A_\xi ^hx_\xi ^h + B_\xi ^h{\varphi ^{{\xi _h}}}$

      (39)

      ${{\xi _h}} = C_\xi ^hx_\xi ^h$

      (40)

      with

      $x_\xi ^h = \left[\!\!{\begin{array}{*{20}{c}} {x_1^h}\\ {x_2^h}\\ \vdots \\ {x_n^h} \end{array}}\!\!\right]\!,\;A_\xi ^h = \left[\!\!{\begin{array}{*{20}{c}} {A_1^{{\xi _h}}}&0&0&0\\ 0&{A_2^{{\xi _h}}}&0&0\\ 0&0& \ddots &0\\ 0&0&0&{A_n^{{\xi _h}}} \end{array}}\!\!\right]$

      $B_\xi ^h = \left[\!\!{\begin{array}{*{20}{c}} {B_1^{{\xi _h}}}&0&0&0\\ 0&{B_2^{{\xi _h}}}&0&0\\ 0&0& \ddots &0\\ 0&0&0&{B_n^{{\xi _h}}} \end{array}}\!\!\right]\!,\;{\varphi ^{{\xi _h}}} = \left[\!\!{\begin{array}{*{20}{c}} {\varphi _1^{{\xi _h}}}\\ {\varphi _2^{{\xi _h}}}\\ \vdots \\ {\varphi _n^{{\xi _h}}} \end{array}}\!\!\right]$

      $C_\xi ^h = \left[ {\begin{array}{*{20}{c}} {C_1^{{\xi _h}}}&{C_2^{{\xi _h}}}& \cdots &{C_n^{{\xi _h}}} \end{array}} \right].$

      Now, consider that disturbance signal $ \xi $ may contain both polynomial components and harmonic components as described in (28), then the complete description of the disturbance results by combining the disturbance models (31) and (39),

      $\begin{aligned} & \dot x_\xi ^p = A_\xi ^px_\xi ^p + B_\xi ^p{\varphi ^{{\xi _p}}};\;{\xi _p} = C_\xi ^px_\xi ^p\\ & \dot x_\xi ^h = A_\xi ^hx_\xi ^h + B_\xi ^h{\varphi ^{{\xi _h}}};\;{\xi _h} = C_\xi ^hx_\xi ^h. \end{aligned}$

      Thus, the complete model for $ \xi $ results as follows:

      ${{{\dot x}_\xi }}= {A_\xi }{x_\xi } + {B_\xi }{\varphi _\xi }$

      (41)

      $\xi = {C_\xi }{x_\xi }$

      (42)

      with

      ${A_\xi } = \left[\!\!{\begin{array}{*{20}{c}} {A_\xi ^p}&0\\ 0&{A_\xi ^h} \end{array}}\!\!\right]\!,\;{B_\xi } = \left[\!\!{\begin{array}{*{20}{c}} {B_\xi ^p}&0\\ 0&{B_\xi ^h} \end{array}}\!\!\right]$

      ${x_\xi } = \left[ {\begin{aligned} {x_\xi ^p}\\ {x_\xi ^h} \end{aligned}} \right]\!,\;{\varphi _\xi } = \left[ {\begin{aligned} {{\varphi ^{{\xi _p}}}}\\ {{\varphi ^{{\xi _h}}}} \end{aligned}} \right]\!,\;{C_\xi } = \left[ {C_\xi ^p}\quad {C_\xi ^h} \right].$

      It must be emphasized that the disturbance signal $ \xi $ is in fact unknown. However, there are some components that can be assumed to be partially known, for example in this specific application; the load of the tower-crane presents a harmonic oscillation at a defined frequency. Under that scenario, a disturbance model was proposed with polynomial and harmonic components. Polynomial components which come from a Taylor polynomial structure are helpful to approximately model local disturbances; this includes static and coulomb frictions, nonlinearities, parametric variations or neglected dynamics. That is one of the main strengths inherited of the active disturbance rejection technique (that is, lumping together all endogenous and exogenous disturbances inside a total disturbance signal equivalent at the input of the plant). Harmonic components, on the other hand, which come from Fourier series expansion, are used to model specific harmonic components of the disturbance. Therefore, the nature of disturbances that this application involves is a mixture of both polynomial and harmonic ones; and in consequence, the proposed disturbance model fits better and suitably exploits what is or is not known about the system.

    • Recall the system model with disturbance input:

      ${\dot x_s} = {A_s}{x_s} + {B_s}u + {B_s}\xi$

      and notice that the disturbance $ \xi $ is not an state-variable of this system. Hence, a Luenberger type conventional state observer would not be able to provide any estimations of the disturbance. However, if the disturbance signal $ \xi $ were part of the state-variables of the system, the estimations would be feasible. For this reason, the main philosophy behind an extended-state observer is to reformulate the system model for the disturbance to be an additional state of the system (also called extended-state) by means of the approximate knowledge of the disturbance (disturbance model).

      Given the above reasons, the system model (27) and the disturbance model (41) and (42) are combined to produce a model containing both the state-variables of the system and the state-variables of the disturbance, as follows:

      ${{{\dot x}_s}} = {A_s}{x_s} + {B_s}u + {B_s}{C_\xi }{x_\xi }$

      (43)

      ${{{\dot x}_\xi }}= {A_\xi }{x_\xi } + {B_\xi }{\varphi _\xi }$

      (44)

      which in compact form has the following description:

      ${{{\dot x}_a}} = {A_a}{x_a} + {B_a}u + {E_\xi }{\varphi _\xi }$

      (45)

      $y= {C_a}{x_a}$

      (46)

      with

      ${A_a} = \left[\!\!{\begin{array}{*{20}{c}} {{A_s}}&{{B_s}{C_\xi }}\\ 0&{{A_\xi }} \end{array}}\!\!\right],\;{B_a} = \left[ {\begin{aligned} {{B_s}}\\ 0\; \end{aligned}} \right],\;{E_\xi } = \left[{\begin{aligned} 0\;\\ {{B_\xi }} \end{aligned}}\right]$

      $y = \left[{\begin{aligned} x\;\\ {{\theta _p}} \end{aligned}}\right]\!,\;{C_a} = \left[ {{C_s}}\quad 0 \right]\!,\;{C_s} = \left[\!\!{\begin{array}{*{20}{c}} 1&0&0&0\\ 0&0&1&0 \end{array}}\!\!\right]$

      where $ {x_a} = {\left[ {{x_s}}\quad {{x_\xi }} \right]^{\rm{T}}} $ is the extended-state vector that contains the states of the system and the states of the disturbance, and the tower-crane has two measurements which are the trolley position $ x $ and the payload angle $ \theta_p $.

      Based on (45) and (46), a Luenberger type observer is formulated in order to provide estimations of $ x_a $ which contains tower-crane and disturbance state-variables:

      ${{{\dot {\hat x}}_a}}= {A_a}{{\hat x}_a} + {B_a}u + {L_a}\left( {y - \hat y} \right)$

      (47)

      ${\hat y}= {C_a}{{\hat x}_a}$

      (48)

      where $ \hat{x}_a = {\left[{{\hat{x}_s}}\quad {{\hat{x}_\xi }} \right]^{\rm{T}}} $ is the estimated version of the extended-state vector. Then, the extended-state observer (47) and (48) can be tuned by selecting $ L_a $ such that the matrix $ (A_a-L_aC_a) $ be Hurwitz. Useful tuning methodologies for extended-state observers are presented in Cortés-Romero et al.[27] and Madoński and Herman[25]. Note that the annihilator term $ {\varphi _\xi } $ has been eliminated from the observer formulation; this comes from the fact that the disturbance model (annihilator) is assumed to be valid, so that $ {\varphi _\xi } \approx 0 $.

      In order to provide an ultimate boundedness analysis for the observer estimations, first consider the following assumptions regarding the disturbance signal $ \xi $ and its annihilator $ \varphi_\xi $:

      1) The disturbance signal $ \xi $ is unknown.

      2) The disturbance signal $ \xi $ is absolutely uniformly bounded almost everywhere. This means that the annihilators, $ {\varphi ^{{\xi _p}}} $ and $ {\varphi ^{{\xi _h}}} $, are also absolutely uniformly bounded almost everywhere such that there exists a finite constant $ K_{\xi} $ that satisfies the following condition:

      ${\rm{max}}\mathop {\sup }\limits_t \left| {{\varphi _\xi }(t)} \right| \le {K_\xi }.$

      Under this approach, possible discontinuities of external disturbances are considered. The work presented by Xue and Huang[32] shows the necessity of this assumption and also proposes a relaxation where disturbances with discontinuities can be considered (including, e.g., unitary steps).

      Now, by subtracting the observer (47) from the augmented system (45), the following estimation error dynamics is found:

      ${{{\dot x}_a} - {{\dot {\hat x}}_a}}= {A_a}{x_a} - {A_a}{{\hat x}_a} + {E_\xi }{\varphi _\xi } - {L_a}\left( {y - \hat y} \right)$

      (49)

      ${{{\dot x}_a} - {{\dot {\hat x}}_a}}= {A_a}\left( {{x_a} - {{\hat x}_a}} \right) - {L_a}{C_a}\left( {{x_a} - {{\hat x}_a}} \right) + {E_\xi }{\varphi _\xi }$

      (50)

      ${{{\dot {\tilde e}}_{{x_a}}}}= {A_{\tilde e}}{{\tilde e}_{{x_a}}} + {E_\xi }{\varphi _\xi }$

      (51)

      where $ {{\tilde e}_{{x_a}}} = {x_a} - {{\hat x}_a} $ and $ {A_{\tilde e}} = {A_a} - {L_a}{C_a} $. Then, an ultimate bound for $ {{\tilde e}_{{x_a}}} $ can be found. Consider the following Lyapunov candidate function:

      $V\left( {{{\tilde e}_{{x_a}}}} \right) = \frac{1}{2}\tilde e_{{x_a}}^{\rm T}P{\tilde e_{{x_a}}}$

      where P is a positive definite symmetric matrix. Thus, its time derivative satisfies

      ${\dot V\left( {{{\tilde e}_{{x_a}}}} \right)}= \frac{1}{2}\left[ {\tilde e_{{x_a}}^{\rm{T}}P{{\dot {\tilde e}}_{{x_a}}} + {\dot {\tilde e}}_{{x_a}}^{\rm{T}}P{{\tilde e}_{{x_a}}}} \right]$

      (52)

      ${\dot V\left( {{{\tilde e}_{{x_a}}}} \right)}= \frac{1}{2}\tilde e_{{x_a}}^{\rm{T}}\left( {P{A_{\tilde e}} + A_{\tilde e}^{\rm{T}}P} \right){{\tilde e}_{{x_a}}} + \left( {\tilde e_{{x_a}}^{\rm{T}}P{E_\xi }{\varphi _\xi }} \right).$

      (53)

      Given that $ {A_{\tilde e}} $ is designed to be Hurwitz, then there exists a positive definite matrix P such that $ {P{A_{\tilde e}} + A_{\tilde e}^{\rm{T}}P} = -Q $ is satisfied, with $ Q $ a positive definite symmetric matrix. Thus, if $ Q = I $, then $ \dot V\left( {{{\tilde e}_{{x_a}}}} \right) $ satisfies

      ${\dot V\left( {{{\tilde e}_{{x_a}}}} \right)}= - \frac{1}{2}\tilde e_{{x_a}}^{\rm{T}}Q{{\tilde e}_{{x_a}}} + \tilde e_{{x_a}}^{\rm{T}}P{E_\xi }{\varphi _\xi }$

      (54)

      ${\dot V\left( {{{\tilde e}_{{x_a}}}} \right)}\le - \frac{1}{2}{{\left\| {{{\tilde e}_{{x_a}}}} \right\|}^2} + \left\| {\tilde e_{{x_a}}^{\rm{T}}} \right\|\cdot\left\| P \right\|\cdot\left\| {{E_\xi }} \right\|\cdot\left\| {{\varphi _\xi }} \right\|$

      (55)

      ${\dot V\left( {{{\tilde e}_{{x_a}}}} \right)}\le - \frac{1}{2}{{\left\| {{{\tilde e}_{{x_a}}}} \right\|}^2} + \left\| {\tilde e_{{x_a}}^{\rm{T}}} \right\|\cdot\left\| P \right\|\cdot\left\| {{E_\xi }} \right\|{K_\xi }.$

      (56)

      In consequence, $ \dot V\left( {{{\tilde e}_{{x_a}}}} \right) $ is strictly negative if

      ${\frac{1}{2}{{\left\| {{{\tilde e}_{{x_a}}}} \right\|}^2}}> \left\| {\tilde e_{{x_a}}^{\rm{T}}} \right\|\cdot\left\| P \right\|\cdot\left\| {{E_\xi }} \right\|{K_\xi }$

      (57)

      ${\left\| {{{\tilde e}_{{x_a}}}} \right\|} > 2{K_\xi }\left\| P \right\|\cdot\left\| {{E_\xi }} \right\|.$

      (58)

      Therefore, $ \dot V\left( {{{\tilde e}_{{x_a}}}} \right) $ is strictly negative outside the following disk:

      ${D_{\tilde e}} = \left\{ {{{\tilde e}_{{x_a}}} \in {{R}^{4 + m + 2n}},\;\left\| {{{\tilde e}_{{x_a}}}} \right\| \le 2{K_\xi }\left\| P \right\|\cdot\left\| {{E_\xi }} \right\|} \right\}.$

      Consequently, the ultimate boundedness of all estimation errors of the observer was obtained.

    • The proposed control scheme is depicted in Fig. 3. As it is shown, this scheme is composed of an integral control action, an observer-based state feedback control term and an active disturbance rejection injection. In the following, every component of the proposed scheme will be described.

      Figure 3.  Block diagram of the control scheme

      Define the controlled output of the system (27) to be the trolley position:

      ${{{\dot x}_s}}= {A_s}{x_s} + {B_s}u + {B_s}\xi$

      (59)

      $x= {C_x}{x_s}$

      (60)

      with $ {C_x} = \left[1\quad 0 ;\quad 0\quad 0\right] $. Then, the controller state (named $ x_i $ in Fig. 3) has the following dynamics:

      ${{{\dot x}_i}}= {x^*} - x$

      (61)

      ${{{\dot x}_i}}= {x^*} - {C_x}{x_s}$

      (62)

      which can be described in extended form along with the system dynamics, as

      $\begin{split} & {{{\dot x}_c} = {A_c}{x_c} + {B_c}\left( {u + \xi } \right) + {F_c}{x^*}}\\ & {x = {C_c}{x_c}} \end{split}$

      (63)

      with

      ${x_c} = \left[ {\begin{aligned} {{x_s}}\\ {{x_i}} \end{aligned}} \right]\!,\;{A_c} = \left[\!\!{\begin{array}{*{20}{c}} {{A_s}}&0\\ { - {C_x}}&0 \end{array}}\!\!\right]\!,\;{B_c} = \left[ {\begin{aligned} {{B_s}}\\ 0\;\end{aligned}} \right]$

      ${F_c} = \left[ {\begin{aligned} 0\\ 1 \end{aligned}} \right]\!,\;{C_c} = \left[{{C_x}}\quad 0 \right].$

      Accordingly, based on system (63), the control law is stated as

      $u = {k_i}{x_i} - {k_s}{\hat x_s} - \hat \xi$

      (64)

      which can also be written as

      $u= {k_i}{x_i} - {k_s}{{\hat x}_s} - \hat \xi + {k_s}{x_s} - {k_s}{x_s} + \xi - \xi$

      (65)

      $u= \underbrace {\left[ {\begin{aligned} { - {k_s}}\quad {{k_i}} \end{aligned}} \right]}_{{K_c}}\underbrace {\left[ {\begin{aligned} {{x_s}}\\ {{x_i}} \end{aligned}} \right]}_{{x_c}} - \xi + \left( {\xi - \hat \xi } \right) + {k_s}\left( {{x_s} - {{\hat x}_s}} \right)$

      (66)

      $u = {K_c}{x_c} - \xi + \left( {\xi - \hat \xi } \right) + {k_s}\left( {{x_s} - {{\hat x}_s}} \right).$

      (67)

      Then, replacing it into (63), the closed-loop system dynamics results in

      ${{{\dot x}_c}} = \left( {{A_c} + {B_c}{K_c}} \right){x_c} + {B_c}\left( {\underbrace {\xi - \hat \xi }_{{{\tilde e}_\xi }} + {k_s}\underbrace {\left( {{x_s} - {{\hat x}_s}} \right)}_{{{\tilde e}_{{x_s}}}}} \right) + {F_c}{x^*}$

      (68)

      $x= {C_c}{x_c}$

      (69)

      where $ {{{\tilde e}_\xi }} $ and $ {{\tilde e}_{{x_s}}} $ are estimation errors of disturbance signal and system state-variables, respectively.

      Then, given that the estimation errors $ {{{\tilde e}_\xi }} $ and $ {{\tilde e}_{{x_s}}} $ are the ultimately bounded, the closed-loop system dynamics can be accommodated by selecting the gain $ K_c $ such that the eigenvalues of $ ({{A_c} + {B_c}{K_c}}) $ are located in the left-half plane of the complex plane $ s $. Nevertheless, as the tower-crane system has parameters that change along with the normal operation of the system, e.g., the payload mass and the length of the rope, the tuning methodology is oriented to handle such system uncertainties.

      Describe the closed-loop system matrix as a function of 2 main parameters: the payload mass (m) and the length of the rope (L),

      ${A_{lc}} = {A_c}\left( {m,L} \right) + {B_c}\left( {m,L} \right){K_c}$

      (70)

      then, the eigenvalues of the closed-loop system matrix $ A_{lc} $ can be encapsulated in a $ 2^2 $-vertices polytope given by all the combinations of the maximum and minimum bounds of each varying parameter ($ m $ and L); thus, the closed-loop matrix is represented by

      ${A_{lc}} \in Co\left\{ {\begin{aligned} {{A_{c1}} + {B_{c1}}{K_c}}\\ {{A_{c2}} + {B_{c2}}{K_c}}\\ {{A_{c3}} + {B_{c3}}{K_c}}\\ {{A_{c4}} + {B_{c4}}{K_c}} \end{aligned}} \right\}$

      (71)

      where

      $\begin{aligned} & {{A_{c1}} = {A_c}\left( {{m_{\min }},{L_{\min }}} \right),\;{B_{c1}} = {B_c}\left( {{m_{\min }},{L_{\min }}} \right)}\\ & {{A_{c2}} = {A_c}\left( {{m_{\min }},{L_{\max }}} \right),\;{B_{c2}} = {B_c}\left( {{m_{\min }},{L_{\max }}} \right)}\\ & {{A_{c3}} = {A_c}\left( {{m_{\max }},{L_{\min }}} \right),\;{B_{c3}} = {B_c}\left( {{m_{\max }},{L_{\min }}} \right)}\\ & {{A_{c4}} = {A_c}\left( {{m_{\max }},{L_{\max}}} \right),\;{B_{c4}} = {B_c}\left( {{m_{\max }},{L_{\max }}} \right).} \end{aligned}$

      In consequence, the control gain $ K_c $ should be chosen such that the eigenvalues of $ ({A_{c1}} + {B_{c1}}{K_c}) $, $ ({A_{c2}} + {B_{c2}}{K_c}) $, $ ({A_{c3}} + {B_{c3}}{K_c}) $ and $ ({A_{c4}} + {B_{c4}}{K_c}) $ are placed on the left-half plane of the complex plane $ s $. Based on the well-known pole-placement techniques[33] for linear uncertain systems, the control gain $ K_c $ can be found using LMIs constraints. Therefore, $ K_c $ is found by solving the following LMI problem[34] described in terms of a quadratic cost

      ${J_{LQR}} = \int_0^\infty {\left({x_c^{\rm{T}}{Q_c}{x_c} + u{R_c}u}\right){\rm d}t}$

      where $ Q_c $ and $ R_c $ are given positive definite weighting matrices:

      minimize $ \left( { - {\rm{trace}}\left( P \right)} \right) $

      s.t.

      $\begin{aligned} & {P > 0}\\ &{\left[\!\!{\begin{array}{*{20}{c}} { - {A_{c1}}P - {B_{c1}}Z - {{\left( {{A_{c1}}P + {B_{c1}}Z} \right)}^{\rm{T}}}}&P&{{Z^{\rm{T}}}}\\ P&{Q_c^{ - 1}}&0\\ Z&0&{R_c^{ - 1}} \end{array}}\!\!\right]}> 0\\ & {\left[\!\!{\begin{array}{*{20}{c}} { - {A_{c2}}P - {B_{c2}}Z - {{\left( {{A_{c2}}P + {B_{c2}}Z} \right)}^{\rm{T}}}}&P&{{Z^{\rm{T}}}}\\ P&{Q_c^{ - 1}}&0\\ Z&0&{R_c^{ - 1}} \end{array}}\!\!\right]}> 0\\ & {\left[\!\!{\begin{array}{*{20}{c}} { - {A_{c3}}P - {B_{c3}}Z - {{\left( {{A_{c3}}P + {B_{c3}}Z} \right)}^{\rm{T}}}}&P&{{Z^{\rm{T}}}}\\ P&{Q_c^{ - 1}}&0\\ Z&0&{R_c^{ - 1}} \end{array}}\!\!\right]}> 0\\ & {\left[\!\!{\begin{array}{*{20}{c}} { - {A_{c4}}P - {B_{c4}}Z - {{\left( {{A_{c4}}P + {B_{c4}}Z} \right)}^{\rm{T}}}}&P&{{Z^{\rm{T}}}}\\ P&{Q_c^{ - 1}}&0\\ Z&0&{R_c^{ - 1}} \end{array}}\!\!\right]}> 0. \end{aligned}$

      If there exist a symmetric positive definite matrix P and a vector Z that fulfill all the design objectives for all four systems in the polytope $ ({A_{c1}} + {B_{c1}}{K_c}),\cdots,({A_{c4}} +$ $ {B_{c4}}{K_c}) $, then, the control gain can be calculated using:

      ${K_c} = Z{\left( P \right)^{ - 1}}.$

      (72)

      Remark 2. In this specific application, the tower-crane contains harmonic disturbances given by the nature of the mechanical system. In addition, its frequency of oscillation can be determined using parameters of the system such as load-mass and rope length. This frequency is not accurately known, but, there is a certain range in which this frequency will vary, and the control scheme and its tuning are developed to take into account this range.

    • In order to evaluate and analyze the proposed control scheme, three ADRC-based schemes are also designed and implemented to contrast and compare performance and robustness. The first one is a conventional ADRC[17, 18] scheme (labeled as ADRC). The second (labeled as PI+ADR+D) is an ADRC-based scheme with integral action and an active disturbance rejection term provided by an extended-state observer designed with a derivative annihilator $ \dot{\xi} \approx 0 $ (as it is usual in ADRC). Finally, the third scheme (labeled as PI+ADR+R) is an ADRC-based scheme[24] with integral action and an active disturbance rejection term provided by an extended-state observer designed with a resonant annihilator $ \ddot{\xi}+\omega^2_p{\xi}\approx 0 $, whose frequency $ \omega_p $ matches the frequency of the payload oscillation. See Appendix for details of the parameters of each control scheme.

      All control schemes are experimentally validated by using a small scale tower-crane (see Fig. 1), whose nominal parameters are listed in Table 1. The implementation was carried out by coding and embedding the controllers into the Texas Instruments® LaunchPad Development Kit LAUNCHXL-F28377S. All control schemes were discretized using the Tustin method with a sampling period of $ T_s = 1.0 $ ms.

      ParameterSymbolUnitValue
      Rope length$ {L}$$ {\rm{m}}$0.65
      Trolley mass$ {M}$$ {\rm{kg}}$0.5
      Payload mass$ {m}$$ {\rm{kg}}$1.2
      Armature resistance$ {R_{a}}$$ {\Omega}$6
      Viscosity coefficient$ {B^\prime}$$ {\rm{kg·m/s}}$0.003 122 4
      Gear ratio$ {n}$10:1
      Encoder disk inertia$ {J_{e}}$$ {\rm{kg·m^2}}$0.000 000 073 129
      Pulley inertia$ {J_{p}}$$ {\rm{kg·m^2}}$0.000 037 286
      Rotor inertia$ {J}$$ {\rm{kg·m^2}}$0.000 007 614 237
      Gear box inertia$ {J_{g}}$$ {\rm{kg·m^2}}$0.000 026 649 830 5
      Shaft total inertia$ {J^\prime}$$ {\rm{kg·m^2}}$0.000 825 432 69
      Mechanical constant$ {k_{b}}$$ {\rm{N·m/A}}$0.37
      Pulley radius$ {r}$$ {\rm{m}}$0.010
      Gravitational constant$ {g}$$ {\rm{m/s^2}}$9.806
      Conversion constant$ {k_{pwm}}$$ {{\rm{V}}/\%}$19/100

      Table 1.  Nominal parameters of the model

      The performance of each control scheme is measured by analyzing the main variables of the system through some performance indexes such as the maximum absolute Max(ABS) and the integral of square error (ISE). Signals such as the trolley position, the payload position and the control signal of each control scheme are analyzed and evaluated. In this way, a set of tests are developed to validate and asses the controllers. In the first test, a constant desired trolley position $ x^*(t) $ is applied to all control schemes and the position of the trolley, so the payload angle and the control signal are analyzed. A second test is performed to analyze the disturbance rejection properties of the control schemes. Finally, the rope length of the tower-crane is decreased $ 30\% $ from its nominal value and the first and second tests are repeated. Figs. 4 to 7 show the implementation results for all tests, and Table 2 shows the summary of all performance indexes.

      Figure 4.  Comparison of time-domain responses of all control systems for trolley positioning under nominal parameters of the tower-crane

      Figure 7.  Time-domain responses of the control systems for payload stabilization under a reduction of $30\%$ in the rope length of the tower-crane

      ${\max {\left| x \right|}, (\%) }$${\int {\left| {{x^*} - x} \right|^2{\rm{d}}t} }$, (%)${\max {\left| {{\theta _p}} \right|} }$, (%)${\int {\left| {{\theta _p}} \right|^2{\rm{d}}t} }$, (%)${\max {\left| u \right|} }$, (%)${\int {\left| u \right|^2{\rm{d}}t} , (\%)}$
      Test 1Proposed0.521 8, (100.0)0.119 2, (100.0)4.140 0, (100.0)4.914 4, (100.0)68.925 0, (100.0)3 622.395 3, (100.0)
      PI+ADR+R0.511 4, (98.0)0.118 7, (99.5)4.050 0, (97.8)5.567 2, (113.3)72.975 0, (105.9)3 923.830 3, (108.3)
      PI+ADR+D0.508 0, (97.3)0.117 6, (98.6)4.500 0, (108.7)6.483 2, (131.9)81.850 0, (118.8)3 704.431 6, (102.3)
      ADRC0.601 6, (115.3)0.130 9, (109.8)7.110 0, (171.7)36.160 6, (735.8)100.000 0, (145.1)8 128.647 1, (224.4)
      Test 2Proposed0.106 9, (100.0)0.752 8, (100.0)15.480 0, (100.0)63.224 9, (100.0)72.675 0, (100.0)2 408.881 6, (100.0)
      PI+ADR+R0.147 3, (137.8)0.759 3, (100.9)19.980 0, (129.1)96.683 7, (152.9)85.450 0, (117.6)2 945.362 0, (122.3)
      PI+ADR+D0.129 8, (121.5)0.754 3, (100.2)19.980 0, (129.1)109.418 3, (173.1)95.725 0, (131.7)2 956.106 0, (122.7)
      ADRC0.289 1, (270.4)0.998 7, (132.7)18.720 0, (120.9)195.899 7, (309.8)100.000 0, (137.6)8 463.870 3, (351.4)
      Test 3Proposed0.522 7, (100.0)0.116 9, (100.0)3.330 0, (100.0)4.121 9, (100.0)68.825 0, (100.0)3 824.644 9, (100.0)
      PI+ADR+R0.511 4, (97.8)0.118 7, (101.6)4.050 0, (121.6)5.567 2, (135.1)72.975 0, (106.0)3 923.830 3, (102.6)
      PI+ADR+D0.509 9, (97.6)0.117 1, (100.2)4.320 0, (129.7)6.222 7, (151.0)71.775 0, (104.3)3 361.384 6, (87.9)
      ADRC0.588 2, (112.5)0.131 1, (112.2)7.650 0, (229.7)33.356 5, (809.2)100.000 0, (145.3)7 743.982 2, (202.5)
      Test 4Proposed0.097 7, (100.0)0.747 7, (100.0)14.490 0, (100.0)50.969 5, (100.0)85.425 0, (100.0)2 885.062 6, (100.0)
      PI+ADR+R0.148 5, (152.1)0.748 2, (100.1)22.050 0, (152.2)97.073 8, (190.5)99.225 0, (116.2)3 339.595 2, (115.8)
      PI+ADR+D${\infty}$${\infty}$${\infty}$${\infty}$${\infty}$${\infty}$
      ADRC0.291 4, (298.3)0.958 6, (128.2)24.030 0, (165.8)265.206 6, (520.3)100.000 0, (117.1)9 225.998 9, (319.8)

      Table 2.  Performance indexes of the control systems. Boldface values indicate the best performance among the applied controllers

    • In this test, the objectives are positioning the trolley of the tower-crane at $ x^* = 0.5 $ m and maintaining the payload angle as close as possible to zero. Fig. 4 shows the time-domain responses of the trolley position, the payload oscillation angle and the control signal of all implemented control systems.

      The analysis of all responses and performance indexes evidence that the proposed control scheme forces the trolley to move between $ 0.5\% $ and $ 1.4\% $ more than the other control schemes (see ISE indexes), however this fact allowed us to improve reductions on load oscillations more than $ 13.3\% $ (ISE indexes) compared with the other controllers. In contrast, Fig. 4 shows that the proposed control scheme demands a lower maximum control signal (see Max(ABS) index) compared with the other controllers. On the other hand, Fig. 4 shows some oscillatory behavior of the load angle for the control scheme (PI+ADR+D), which is an indication of stability/performance issues. On the other hand, conventional ADRC shows constant error while positioning the trolley and some high frequency oscillations of the load angle, however ADRC was capable of reducing the load oscillations in steady-state.

    • This test consists in evaluating the disturbance rejection properties of each control system at the point $ x^*(t) = 0 $ and $ \theta_p = 0 ^\circ $. Thus, at the beginning of each experiment the load is positioned at $ \theta_p = 25 ^\circ $ and then released without any controller being active. Then, when the load passes through the point $ \theta_p = 0 ^\circ $, the controllers become active and should stabilize the system.

      Fig. 5 shows the time response of each control system to stabilize/reject the oscillations of the payload. Fig. 5 shows that the control scheme (PI+ADR+D) has difficulty eliminating the oscillations of the load (see load angle after 4 s), showing poor performance in rejecting these kind of harmonic oscillations. In contrast, both the proposed controller and the controller (PI+ADR+R) show the advantage of including a resonant element in the observer to provide high gain at certain frequencies. Table 2 shows that the proposed control scheme has the best performance indexes among all controllers, thus better stabilization and rejection of disturbances is achieved using less control energy. Note that the ADRC scheme presents a noisy control signal and has trouble returning the trolley to the origin. This steady-state offset in the trolley position may be solved by rising the bandwidth of the control/observer however vibrations of the structure limits its performance.

      Figure 5.  Time-domain responses of all control systems in Test 2 for payload stabilization under nominal parameters of the tower-crane

    • In this test, a reduction of $ 30\% $ in the rope length is applied to the tower-crane. This critical change in the system allows the evaluation of robustness and performance of all control systems under uncertainty. Thus, the objectives are positioning the trolley at $ 0.5 $ m and maintaining the pay-load angle close to zero (as in the first test). The results show (see Fig. 6 and Table 2) that the proposed scheme allowed reducing more than $ 21.6\% $ in peak load angle oscillation using almost the same trolley movements as the other controllers. Notice that the ADRC and the PI+ADR+D schemes showed high-frequency oscillations (see Fig. 6 around 1 s) in the control signal and in the payload angle; this had visible consequences in the structure of the tower-crane represented as strong vibrations.

      Figure 6.  Time-domain responses of all control systems for trolley positioning with a reduction of $30\%$ in the rope length of the tower-crane

    • In this last test, a reduction of $ 30\% $ in the rope length is applied to the tower-crane. Then, stabilization and disturbance rejection capabilities of the control systems are evaluated under system uncertainty. Thus, as in the second test, the same experimental setup is developed here. Fig. 7 shows the time-response of each control system and Table 2 shows the performance indexes obtained in the test.

      First, note that the control scheme PI+ADR+D resulted in instability as a consequence of system uncertainty. Except for this one, Fig. 7 shows that the rest of the controllers stabilize the load. However, the proposed scheme obtains better performance indexes in trolley movements, payload oscillations and peak control law (see Table 2). The proposed controller showed better performance in mitigating local oscillations when the system was submitted to critical uncertainty.

    • In this section, the robustness of the control schemes is evaluated and analyzed through experimental frequency domain Nyquist plots. To obtain the frequency response of each control system, a chirp signal is applied to the reference $ {x^*}(t) $:

      ${x^*}(t) = {R_x}\sin (2\pi \phi (t)t)$

      (73)

      with $ R_x = 0.1 $ m and $ \phi(t) $ defined as

      $\phi \left( t \right) = {f_0} + \left( {\frac{{{f_1} - {f_0}}}{{2{T_f}}}} \right)t$

      (74)

      where $ f_0 = 0 $ Hz is the initial frequency, $ f_1 = 46 $ Hz is the final frequency, and $ T_f = 420 $ s is the total time. The signal $ x^*(t) $ sweeps from $ f_0 $ to $ f_1 $. Then, the experimental data of both the tracking error ($ e(t) $, open-loop input) and the trolley position ($ x(t) $, output) are data-logged and analyzed to find the ratio of the output Fourier transform to the input Fourier transform (see details in Franklin and Powell[35]–Chapter 12). Finally, frequency-domain data are obtained from input/output time-domain data analysis, and the experimental Nyquist plot can be built for each control system under nominal and uncertain conditions of the tower-crane. Fig. 8 shows the general block diagram for the experimental setup.

      Figure 8.  Block diagram of the control schemes for robustness experimental setup

      As it is widely known, Nyquist plots are used to quantify stability margins of the control systems. In this case, the modulus margin of each control system is found experimentally. This margin is determined by means of measuring the minimum distance between the point $ [-1,0] $ of the plane and the Nyquist plot of the open-loop system. The larger this margin is, the larger the stability margin of the control system. Conventionally, a modulus margin of $ 0.5 $ is usually enough to say that a control system is robust.

      Fig. 9 shows the Nyquist plots of the open-loop control schemes (the experimental Nyquist plot for the ADRC scheme was omitted due to it presents the worst performance indexes in Table 2). Fig. 9 (left) shows the frequency response under nominal conditions, and Fig. 9 (right) shows the frequency response under a variation of $ -30\% $ in the rope length. The results shown in Fig. 9 (left) indicate that the control schemes have a modulus margin greater or equal to $ 0.7 $, therefore this is an indication of robustness of the control schemes. However, due to the rope length variation, Fig. 9 (right) shows that all modulus margins have been reduced, as expected. Nevertheless, it is interesting to note that the proposed control scheme maintains almost the same modulus margin (reduction of $ 1.4\% $), while the other schemes show reductions between $ 2.2\% $ and $ 13.1\% $.

      Figure 9.  Experimental Nyquist plots for the open-loop control systems. Left: Nominal case. Right: Reduction of $30\%$ in the rope length L

    • In this contribution, a disturbance rejection-based control scheme was proposed to address the problem of load transportation and load stabilization in tower-cranes with uncertainty. The proposed scheme was formulated based on the definition of the disturbance internal model, composed by polynomial and harmonic components, in such way that an extended-state observer can be formulated to provide improved estimations of system states and disturbances. The control law was proposed so that the closed-loop system is stable under the system uncertainty, based on a robust tuning methodology with a set of LMIs providing an LQR performance. Then, this proposal suitably combines two types of disturbance internal models in order to provide a wider capability of disturbance estimation/rejection, and the control scheme is proposed with a robust tuning methodology to handle parametric uncertainties typical of tower-crane systems.

      Comparison of the proposed robust ADR-based scheme with other ADR-based methods (including conventional ADRC) is provided using several experimental tests with a small-scale tower-crane. The proposed method shows multiple advantages such as, design simplicity, low order control design, and intuitive selection of the disturbance model. Moreover, the proposed scheme provides better robustness margins than conventional IMP-based controllers for disturbance rejection requirements. Experimental results validated the advantages of the proposed scheme in terms of robustness and disturbance rejection properties.

    • Consider the trolley dynamics of the tower-crane (18):

      $\ddot x = {b_{21}}{u_x} + {a_{22}}\dot x + {a_{23}}{\theta _p} + {a_{24}}{\dot \theta _p} + {d_x} \tag{A1}$

      where $ d_{x} $ is the external disturbance term in the trolley subsystem. Then, after lumping together some terms, this system is represented by

      $\ddot x = {b_{21}}{u_x} + {f_x}(\dot x,{\theta _p},{\dot \theta _p},{d_x})\tag{A2}$

      where $ f_{x}(\cdot) $ is the total disturbance term for the trolley subsystem and $ u_{x} $ is the control input. Then, based on (A2) and assuming that the model of the total disturbance is approximately given by $ \dot{f}_{x}\approx 0 $, an extended-state observer is developed:

      ${{{\dot{\hat x}}_x}}= {A_{ax}}{{\hat x}_x} + {B_{ax}}{u_x} + {L_x}(x - \hat x)\tag{A3}$

      ${\hat x}= {C_{ax}}{{\hat x}_x}\tag{A4}$

      with

      ${A_{ax}} = \left[\!\!{\begin{array}{*{20}{c}} 0&1&0\\ 0&0&1\\ 0&0&0 \end{array}}\!\!\right]\!,\;{B_{ax}} = \left[ {\begin{aligned} 0\;\\ {{b_{21}}}\\ 0\;\end{aligned}} \right]\!,\;{C_{ax}} = \left[1\quad 0\quad 0\right]$

      where $ {\hat x_x} = {\left[ {\hat x}\quad {\hat {\dot x}}\quad {{{\hat f}_x}}\right]^{\rm{T}}} $ is the estimated state vector and $ {L_x} = {\left[{{l_{1x}}}\quad {{l_{2x}}}\quad {{l_{3x}}}\right]^{\rm{T}}} $ is the observer gain vector.

      The control law for the trolley system, is defined as follows:

      ${u_x} = \frac{1}{{{b_{21}}}}({\ddot x^*} - {k_{1x}}(\hat{ \dot x }- {\dot x^*}) - {k_{2x}}(\hat x - {x^*}) - {\hat f_x})\tag{A5}$

      where $ x^* $ is the desired position of the trolley. Thus, after replacing the control law (A5) into the plant (A2), the following closed-loop dynamics is obtained:

      ${\ddot e_x} + {k_{1x}}{\dot e_x} + {k_{2x}}{e_x} \approx 0\tag{A6}$

      where $ e_{x} = x-x^* $. Thus, the tuning of the observer and the control gains is made by defining each desired dynamics, i.e., $ \left| {sI - {A_{ax}} + {L_{x}}{C_{ax}}} \right| = {\left( {s + {\omega _{ox}}} \right)^3} $ and $ {s^2} + {k_{1x}}s + {k_{2x}} = {\left( {s + {\omega _x}} \right)^2} $, with $ \omega _{ox} $ and $ \omega _{x} $ the desired bandwidths, respectively.

      On the other hand, the simplified dynamics of the pendulum of the tower-crane (18) is given by

      ${\ddot \theta _p} = {b_{41}}{u_p} + {f_p}(\dot x,{\theta _p},{\dot \theta _p},{d_p})\tag{A7}$

      where $ d_{p} $ is the external disturbance term for the pendulum dynamics, $ u_{p} $ is the control input and $ f_{p}(\cdot) $ is the total disturbance term. Then, based on (A7) and assuming that the model of the total disturbance is approximately given by $ \dot{f}_{p}\approx 0 $, an extended-state observer is developed:

      ${{{\dot{ \hat x}}_p}}= {A_{ap}}{{\hat x}_p} + {B_{ap}}{u_p} + {L_p}({\theta _p} - {{\hat \theta }_p})\tag{A8}$

      ${{{\hat \theta }_p}} = {C_{ap}}{{\hat x}_p}\tag{A9}$

      where $ {\hat x_p} = {\left[{{{\hat \theta }_p}}\quad {{{\hat {\dot \theta} }_p}}\quad {{{\hat f}_p}}\right]^{\rm{T}}} $ is the estimated state vector and $ L_{p} $ = $ \left[l_{1p}\quad l_{2p}\quad l_{3p}\right] $ is the observer gain vector and system matrices are

      ${A_{ap}} = \left[\!\!{\begin{array}{*{20}{c}} 0&1&0\\ 0&0&1\\ 0&0&0 \end{array}}\!\!\right],\;{B_{ap}} = \left[ {\begin{aligned} 0\;\\ {{b_{41}}}\\ 0\; \end{aligned}} \right],\;{C_{ap}} = \left[1\quad 0\quad 0\right].$

      Finally, the control law for the pendulum model is stated as

      ${u_p} = \frac{1}{{{b_{41}}}}\left( { - {k_{1p}}{{\hat{ \dot \theta} }_p} - {k_{2p}}{{\hat{ \theta }}_p} - {{\hat f}_p}} \right).\tag{A10}$

      Then, replacing (A10) in (A7), the closed-loop dynamics is given by

      ${\ddot \theta _p} + {k_{1p}}{\dot \theta _p} + {k_{2p}}{\theta _p} \approx 0.\tag{A11}$

      The tuning of the control gains is developed by selecting a desired bandwidth $ \omega _p $ such as: $ {s^2} + {k_{1p}}s + {k_{2p}} = $$ {(s + {\omega _{p}})^2} $, and the observer gain $ L_p $ is found by selecting a desired bandwidth $ \omega _{op} $ such that $ \left| {sI - {A_{ap}} + {L_{p}}{C_{ap}}} \right| = $${\left( {s + {\omega _{op}}} \right)^3} $. Unifying the (A10) and (A5), the final control law for the ADRC scheme is given by

      $u = {u_x} + {u_p}.\tag{A12}$

      The ADRC scheme was designed according to this conventional procedure[17, 18] and the bandwidth of the controllers and each extended-state observer were increased as much as possible (vibrations in the tower structure did not allow for more increases in bandwidth). The parameters were tuned as follows: $ \omega _{ox} = 17 $, $ \omega _{op} = 18 $, $ \omega _{x} = 9.8 $ and $ \omega _{p} = 14 $.

    • This control scheme is structured as shown in Fig. 3. Then, the control system can be described in open loop as

      ${{{\dot x}_c}}= {A_c}{x_c} + {B_c}\left( {u + \xi } \right) + {F_c}{x^*}\tag{A13}$

      $x= {C_c}{x_c}\tag{A14}$

      and the control law is formulated as follows:

      $u= {k_i}{x_i} - {k_s}{{\hat x}_s} - \hat \xi\tag{A15}$

      $u= \underbrace {\left[ {\begin{aligned} { - {k_s}}\quad {{k_i}} \end{aligned}} \right]}_{{K_c}}\left[ {\begin{aligned} {{{\hat x}_s}}\\ {{x_i}} \end{aligned}} \right] - \hat \xi .\tag{A16}$

      Thus, the closed-loop system dynamics results in

      ${\dot x_c} = \left( {{A_c} + {B_c}{K_c}} \right){x_c} + {B_c}\left( {{{\tilde e}_\xi } + {k_s}{{\tilde e}_{{x_s}}}} \right) + {F_c}{x^*}$

      where $ {{{\tilde e}_\xi }} $ and $ {{\tilde e}_{{x_s}}} $ are estimation errors. Then, the tuning of the control gain $ K_c $ was done such that the eigenvalues of $ ({{A_{c}} + {B_{c}}{K_{c}}}) $ are placed in $ [-2, -3, -5, -6, -7.5] $.

      Then, as is typical in the ADRC methodology, a model approximation of the disturbance signal is assumed to be $ \dot{\xi} \approx 0 $. Thus, an extended-state observer is formulated in order to provide estimations of $ x_s $ and $ \xi $:

      ${{{\dot {\hat x}}_a}} = {A_a}{{\hat x}_a} + {B_a}u + {L_a}\left( {y - \hat y} \right)\tag{A17}$

      ${\hat y} = {C_a}{{\hat x}_a}\tag{A18}$

      with

      ${A_a} = \left[\!\!{\begin{array}{*{20}{c}} {{A_s}}&{{B_s}}\\ 0&0 \end{array}}\!\!\right]\!,\;{B_a} = \left[ {\begin{aligned} {{B_s}}\\ 0\; \end{aligned}} \right]\!,\;{C_a} = \left[\!\!{\begin{array}{*{20}{c}} 1&0&0&0&0\\ 0&0&1&0&0 \end{array}}\!\!\right]$

      where $ {{\hat x}_a} = {\left[{\hat x}\quad {\hat {\dot x}}\quad {{{\hat \theta }_p}}\quad {{{\hat {\dot \theta} }_p}}\quad {\hat \xi }\right]^{\rm{T}}} $. The extended-state observer was tuned by selecting $ L_a $ such that the matrix $ (A_a-L_aC_a) $ has the following eigenvalues [–17, –19, –22, –25, –27].

    • This control scheme is also structured as shown in Fig. 3. Then, the control law is formulated as follows:

      $u = \underbrace {\left[ {\begin{array}{*{20}{c}} { - {k_s}}&{{k_i}} \end{array}} \right]}_{{K_c}}\left[ {\begin{aligned} {{{\hat x}_s}}\\ {{x_i}} \end{aligned}} \right] - \hat \xi .\tag{A19}$

      Thus, the tuning of the control gain $ K_c $ was done such that the eigenvalues of $ ({{A_{c}} + {B_{c}}{K_{c}}}) $ are placed in $ [-2, -3, -5, -6, -7.5] $.

      Then, a disturbance model approximation is assumed to be $ (\ddot{\xi}+\omega_r^2\xi) \approx 0 $, where $ \omega_r = 3.884\;9 $ $ {\rm{rad/s}} $ corresponds to the frequency of oscillation of the pendulum in nominal case. Thus, an extended-state observer is formulated:

      ${{{\dot {\hat x}}_a}}= {A_a}{{\hat x}_a} + {B_a}u + {L_a}\left( {y - \hat y} \right)\tag{A20}$

      ${\hat y}= {C_a}{{\hat x}_a}\tag{A21}$

      with

      ${A_a} = \left[\!\!{\begin{array}{*{20}{c}} {{A_s}}&{{B_s}{C_\xi }}\\ 0&{{A_\xi }} \end{array}}\!\!\right],\;{B_a} = \left[ {\begin{aligned} {{B_s}}\\ 0\; \end{aligned}} \right],\;{A_\xi } = \left[\!\!{\begin{array}{*{20}{c}} 0&1\\ { - \omega _r^2}&0 \end{array}}\!\!\right]$

      ${C_\xi } = \left[1\quad 0\right],\;{C_a} = \left[\!\!{\begin{array}{*{20}{c}} 1&0&0&0&0\\ 0&0&1&0&0 \end{array}}\!\!\right]$

      where $ {{\hat x}_a} = {\left[{\hat x}\quad {\hat {\dot x}}\quad {{{\hat \theta }_p}}\quad {{{\hat {\dot \theta} }_p}}\quad {\hat \xi }\quad {\hat {\dot \xi }}\right]^{\rm{T}}} $. The extended-state observer was tuned by selecting $ L_a $ such that the matrix $ (A_a-L_aC_a) $ has the following eigenvalues $ [-12, -15, -17, -20, -22, -25] $.

    • This control scheme is structured as shown in Fig. 3. Then, the control law was formulated as follows:

      $u = \underbrace {\left[ {\begin{array}{*{20}{c}} { - {k_s}}&{{k_i}} \end{array}} \right]}_{{K_c}}\left[ {\begin{aligned} {{{\hat x}_s}}\\ {{x_i}} \end{aligned}} \right] - \hat \xi \tag{A22}$

      and following the tuning procedure proposed in Section 3.3, the control gain $ K_c $ was found by solving[36] the set of LMIs with the following parameters: $ {L_{\min }} = 0.4 $, $ {L_{\max }} = $0.75, $ {m_{\min }} = 0.6 $, $ {m_{\max }} = 1.8 $, $ Q_c = {\rm{diag}}(1, 1, 500, 500, $ $ 36\,000\,000) $, $ R_c = 100 $. The results are

      $ {Z \!=\! \left[\!\!{\begin{array}{*{20}{c}} { - 0.678\;75}&{6.070\;90}&{1.287\;87}&{ - 9.057\;07}&{ - 0.028\;39} \end{array}}\!\!\right]\times{10^{ - 4}}}$

      $ {P\!=\!\left[{\begin{aligned} & {\;\;\;0.018\,48}\;\;{-0.035\,90}\;\;{-0.003\,29}\;\;{\;\;\;0.019\,67}\;\;{\;\;\;0.007\,43}\\ & {-0.035\,90}\;\;{\;\;\;0.439\,90}\;\;{\;\;\;0.075\,30}\;\;{-0.691\,89}\;\;{-0.000\,79}\\ & {-0.003\,29}\;\;{\;\;\;0.075\,30}\;\;{\;\;\;0.023\,00}\;\;{-0.096\,96}\;\;{\;\;\;0.001\,63}\\ & {\;\;\;0.019\,67}\;\;{-0.691\,89}\;\;{-0.096\,96}\;\;{\;\;\;1.366\,64}\;\;{-0.003\,19}\\ & {\;\;\;0.007\,43}\;\;{-0.000\,79}\;\;{\;\;\;0.001\,63}\;\;{-0.003\,19}\;\;{\;\;\;0.006\,40} \end{aligned}}\right]\times{10^{-5}}}$

      $ {{K_c} = \left[{ - 1.061\,31}\;\;{-0.335\,73}\;\;{0.744\,34}\;\;{ -0.166\,10}\;\;{0.872\,95}\right]\times{10^3}.}$

      A disturbance model was defined as follows:

      ${{{\dot x}_\xi }}= {A_\xi }{x_\xi } + {B_\xi }{\varphi _\xi }\tag{A23}$

      $\xi = {C_\xi }{x_\xi }\tag{A24}$

      with

      $\begin{array}{l}{A_\xi } = \left[\!\!{\begin{array}{*{20}{c}} {A_\xi ^p}&0\\ 0&{A_\xi ^h} \end{array}}\!\!\right]\!,\;{B_\xi } = \left[\!\!{\begin{array}{*{20}{c}} {B_\xi ^p}&0\\ 0&{B_\xi ^h} \end{array}}\!\!\right]\!,\;{C_\xi } = \left[{C_\xi ^p}\quad {C_\xi ^h}\right]\\ A_\xi ^h = \left[\!\!{\begin{array}{*{20}{c}} 0&1\\ { - \omega _r^2}&0 \end{array}}\!\!\right],\;B_\xi ^h = \left[ {\begin{aligned} 0\\ 1 \end{aligned}} \right],\;C_\xi ^h = \left[1\quad 0\right]\end{array}$

      $ A_\xi ^p = 0 $, $ B_\xi ^p = 1 $, $ C_\xi ^p = 1 $, and $ \omega_r = 3.884\;9$ ${\rm{rad/s}} $. Thus, an extended-state observer is formulated:

      ${{{\dot {\hat x}}_a}}= {A_a}{{\hat x}_a} + {B_a}u + {L_a}\left( {y - \hat y} \right)\tag{A25}$

      ${\hat y} = {C_a}{{\hat x}_a}\tag{A26}$

      with

      $ {{A_a} = \left[\!\!{\begin{array}{*{20}{c}} {{A_s}}&{{B_s}{C_\xi }}\\ 0&{{A_\xi }} \end{array}}\!\!\right],\;{B_a} = \left[ {\begin{aligned} {{B_s}}\\ 0\; \end{aligned}} \right],\;{C_a} = \left[\!\!{\begin{array}{*{20}{c}} 1&0&0&0&0\\ 0&0&1&0&0 \end{array}}\!\!\right]}$

      where $ {{\hat x}_a} \!=\! {\left[{\hat x}\quad {\hat {\dot x}}\quad {{{\hat \theta }_p}}\quad {{{\hat {\dot \theta} }_p}}\quad {{{\hat \xi }_p}}\quad {\hat \xi _1^h}\quad {\hat {\dot \xi }_1^h}\right]^{\rm{T}}} $ and $ \hat \xi \!=\! {{\hat \xi }_p} \!+\! \hat \xi _1^h $. The extended-state observer was tuned by selecting $ L_a $ such that the nominal matrix $ (A_a-L_aC_a) $ has the following eigenvalues $ [-4, -6, -11, -16, -20, -22, -25] $.

Reference (36)

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