Uniform ultimate bounded robust model reference adaptive PID control scheme for visual servoing

Author’s Accepted Manuscript UNIFORM ULTIMATE BOUNDED ROBUST MODEL REFERENCE ADAPTIVE PID CONTROL SCHEME FOR VISUAL SERVOING Raaja Ganapathy Subramanian, Vinodh Kumar Elumalai, Selvakumar Karuppusamy, Vamsi Krishna Canchi www.elsevier.com/locate/jfranklin

PII: DOI: Reference:

S0016-0032(16)30471-9 http://dx.doi.org/10.1016/j.jfranklin.2016.12.001 FI2826

To appear in: Journal of the Franklin Institute Received date: 26 January 2016 Revised date: 29 November 2016 Accepted date: 2 December 2016 Cite this article as: Raaja Ganapathy Subramanian, Vinodh Kumar Elumalai, Selvakumar Karuppusamy and Vamsi Krishna Canchi, UNIFORM ULTIMATE BOUNDED ROBUST MODEL REFERENCE ADAPTIVE PID CONTROL SCHEME FOR VISUAL SERVOING, Journal of the Franklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2016.12.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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UNIFORM ULTIMATE BOUNDED ROBUST MODEL REFERENCE ADAPTIVE PID CONTROL SCHEME FOR VISUAL SERVOING

Abstract This paper proposes a uniform ultimate bounded (UUB) controller framework using model reference adaptive control for visual servoing of the ball on plate system. To address the major challenges in designing a control scheme for visual servoing applications including inter-axis coupling, exogenous disturbances, and plant perturbations due to modelling errors, a robust model reference adaptive PID control scheme using e1 modification method is put forward. The key advantage of this methodology is its ability to yield asymptotic stability of the closed loop system without prior information on the plant perturbations. Moreover, exploiting the Erzberger’s perfect model following condition, the algorithm obtains the pseudo inverse of the system to make the system track different test trajectories. The stability and convergence of the proposed scheme are proved using Schwarz[U+2019]s inequality condition and Frobenius norms. To evaluate the tracking performance, two test cases namely reference following during exogenous disturbance and tracking under plant perturbations are validated. Simulation results accentuate that the proposed scheme yields satisfactory tracking response even during plant perturbation and exogenous disturbances.

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Keywords:

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MRAC, Visual servoing, Robust Adaptive PID, e1 -modification, Uniform ultimate boundedness

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(UUB), Ball on plate system.

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1. Introduction

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Visual servoing indicates the use of visual information obtained from the vision sensor as a

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feedback to control the dynamics of a robot or any mechanical system [1]. Visual servo control,

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a multidisciplinary field of research, spans across numerous disciplines ranging from image pro-

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cessing, kinematics, dynamics, control theory to real time systems. The interesting features which

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attract visual feedback for closed loop control are non-contact measurement, versatility, and ac-

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curacy. Moreover, the vision based servo control is insensitive to open loop non-linearities and

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calibration errors [2, 3, 4]. Hence, vision based control system has attracted considerable atten-

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tion in the last two decades because of its numerous real time applications including autonomous

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vehicle navigation, robot control and plant automation.

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The ball on plate system, an extension of classical two dimensional ball and beam system, is a

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typical benchmark system for visual servo control. This system is widely used in many universities

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to teach control engineering because it is a typical nonlinear, under actuated, multivariable, and

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open loop unstable system [5]. Moreover, the use of visual feedback enhances the complexity and

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challenges in designing a control scheme. The ball on plate system consists of a metal plate with

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two rotating axes and a digital camera to read the position of the ball on the plate. The control ob-

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jective of the system is to position the ball in any desired trajectory by controlling the inclination

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angle of the plate via DC motors. Several control algorithms have been reported on the stabi-

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lization and tracking control of ball on plate system. Utilizing the Euler estimator to determine

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the centre of the ball with interlaced-scanned image, Park and Lee [6] employed a sliding mode

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control to deal with the variations in surface characteristics of the plate. Hesar et al [7] compared

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the performance of the PID and sliding mode control schemes for low and high frame rates of

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visual tracking of a 2 DoF spherical parallel robots. Ho et al [8] implemented the visual servoing

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control of ball and plate system on a FPGA device to meet the real time constraints. To handle

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the after effects caused by the friction, Wang et al [9] proposed a novel disturbance observer based

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friction compensation scheme and compared their results with those of PID and direct compensa-

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tion control strategies. Fan et al [10] proposed a hierarchical fuzzy control scheme to control the

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movement of the ball from one point to another without hitting the obstacles. They also employed

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genetic algorithm to optimize the variables of the fuzzy planning controller. Moreno-Armendariz

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et al [11] proposed a fuzzy based indirect adaptive control to improve the tracking performance

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of ball and plate system. However, the tracking control of ball on plate system under exogenous

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disturbances with plant uncertainty has not been much explored. Hence, in this paper we aim 2

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to assess the effectiveness of robust adaptive control scheme for tracking control of ball on plate

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system.

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Even though, PID control scheme is widely used in many of the industries, the major problem

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with conventional PID method is its sensitivity to the plant uncertainties. The control performance

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of the conventional PID gets degraded when the system has uncertainty/modelling error. In that

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context, adaptive control has attracted considerable attention in the last few decades due to its ca-

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pability to handle uncertain dynamical system. Adaptive control schemes yield good convergence

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and stability characteristics as long as the exogenous disturbance is absent in the system [12].

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When the bounded external disturbance act on the system, adaptive control schemes do not guar-

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antee the convergence property and requires new control scheme to make the system robust against

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the disturbance. Hence, the robust adaptive control schemes have come in use to make the system

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not only adaptive for the change in plant characteristics but robust against the disturbances [13, 14].

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MRAC, a very common philosophy in the field of adaptive control, can asymptotically follow any

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given reference model as long as certain matching conditions on uncertainties are satisfied [15, 16].

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However, in reality, due to the presence of matched and unmatched plant dynamics, parameteri-

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zation errors and exogenous disturbances, these matching conditions do not hold good and need

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robustness modification of MRAC scheme [17, 18]. Two of the well-known MRAC robustness

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modification techniques include: σ modification and e1 modification. The major advantage of e1

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modification method over σ modification method is that it guarantees asymptotic stability of the

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error dynamics without prior information on plant perturbations[19]. Harnessing this robustness

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feature of MRAC technique using e1 modification method, we propose a novel robust adaptive

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PID (RAPID) control scheme for reference following applications of ball on plate system. One of

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the striking features of the proposed RAPID control scheme is that it guarantees uniform ultimate

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boundedness (UUB) of error signal without any prior information on the nature of disturbance.

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Moreover, for perfect model following, the proposed RAPID scheme uses Erzberger’s condition

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to obtain the pseudo inverse of model parameters. To the best of our knowledge, this is the first

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work which synthesizes the MRAC e1 modification method and the PID controller for enhancing

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the robustness of visual servoing system under exogenous disturbances and parameter uncertainty.

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The rigorous mathematical proof for stability of the closed loop system and UUB of error signal 3

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is given using Schwarz’s inequality and Lyapunov function.

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The remainder of the paper is structured as follows. Section 2 gives system description and

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mathematical modelling of the 2 DoF ball on plate system. Section 3 explains the proposed RAPID

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control scheme by synthesizing the model reference adaptive control using e1 modification tech-

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nique with PID controller. Section 4 explains the tracking performance of the RAPID control

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scheme during exogenous disturbance and model uncertainty. The paper ends with the concluding

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remarks in section 5.

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2. System description

Figure 1: 2 DoF vision based ball balancer.

Figure 1: 2 DoF vision based ball balancer.

Lplate

X Ball

Motor Gear

Balancing Plate

ra

rm

a

ql

Support Beam

Load Gear

Potentiometer Gear Bottom Support Plate Figure 2: Schematic diagram of ball on plate system.

Figure 2: Schematic diagram of ball on plate system.

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Ball on plate system is a typical multi variable, 1nonlinear, under actuated and open loop unsta-

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ble system. This under actuated system has four degrees of freedom (DoF) which are controlled 2

4

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by only two actuators. Hence, it is also referred as 2 DoF ball balancer. The Quanser 2 DoF

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ball balancer, shown in Figure. 1, is considered for assessing the efficacy of the proposed control

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scheme. The plant parameters of the ball balancer are given in Table 1. The 2 DoF vision based

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ball balancer test bed consists of two servo motors with load gears, digital camera, metal plate and

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a ball. Two rotary servo motors attached with load gears control the angular positions of the plate,

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and the digital camera mounted on the top of the plate captures the 2D images of the plate. The

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vision algorithm computes the X and Y coordinates of the ball from the input image read by the

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camera [20]. The control objective is to make the ball track the time varying reference trajectory

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by adjusting the plate angle through the X and Y axes servo motors.

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The X-Y coordinates of 2 DoF ball balancer have similar servo dynamics. Hence, for brevity,

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only modelling of X direction control is given here. Figure. 2 shows the X -axis control of ball on

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plate system. Assuming that the viscous damping and friction are absent, we write the following

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force balance equation.

mb 92

d2 x(t) = F x,t − F x,r dt2

(1)

where, F x,t is the force due to gravity and F x,r is the force due to inertia of the ball. F x,t = mb gsin α(t)

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Jb F x,r =

(2)

d2 x(t) dt2 rb2

(3)

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where mb is the mass of the ball, Jb is the moment of inertia of the ball, α is the plate angle and rb

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is the radius of the ball. Substituting (2) and (3) into (1), the equation of motion of the ball can be

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represented as, 2

mb 97

d x(t) = mb gsin α(t) − dt2

Jb

d2 x(t) dt2 rb2

(4)

The relationship between the plate angle and servo angle is given by, sin α(t) =

2sin θl (t)rarm Lt 5

(5)

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Hence, the nonlinear equation of motion of the system is, 2mb gθl rarm rb2 d2 x(t) = dt2 Lt (mb rb2 + Jb )

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100

(6)

Assuming θl as input and x as output, we obtain the following transfer function for ball balancer module. Pb (s) =

x(s) Kb = 2 θl (s) s

(7)

2mb gθl rarm rb2

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. The range of servo angle of the load gear is −30o ≤ θl ≤ 30o and the Lt (mb rb2 + Jb ) range of plate tilt angle is −5o ≤ α ≤ 5o . The transfer function of servo motor which controls the

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plate angle, is characterized by,

101

where Kb =

P s (s) =

θl (s) K = Vm (s) s(τs + 1)

(8)

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where K and τ are the static gain and time constant of the motor. As the servo motor is connected

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in cascade with the ball balancer module, the overall transfer function of the ball on plate system

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is given by, P(s) = P s (s)Pb (s) =

θl (s) K = 3 Vm (s) s (τs + 1)

(9)

Table 1: Nominal parameters of 2 DoF ball on plate system

Symbol Description wXd

Plate dimensions

Value

Unit

41.75 × 41.75

cm2

107

108

109

3. Robust Adaptive PID Controller Consider a second order nonlinear system described by the differential equation, y¨ p = a p1 y˙ p + a p0 y p + b p (u + f (y p )) + υ

(10)

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where a p0 , a p1 and b p are the system coefficients, u is the control input, υ is the un-modelled

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disturbance, y p is the process output and f (y p ) is the additive input uncertainty. The controller

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framework is formulated based on the following assumptions. 6

Lt

Table length

27.5

cm

rb

Radius of the ball

1.46

cm

mb

Mass of the ball

0.003

kg

Vm

Motor nominal voltage

6

V

Rm

Motor armature resistance

2.6

Ω

Lm

Motor armature inductance

0.18

mH

Kt

Motor torque current constant

7.68 × 10−3

Nm/A

Kgi

Internal gear box ratio

14

−

ωg

Maximum motor speed

628.3

rad/s

Command input

+

Closed Loop Reference Model

  yr y˙r

PID Parameter Update Mechanism

−

  e¯ e¯˙

+ + Robustness Parameter Update Mechanism

+

PI Controller

−

− +

Vm

Servo Module

θl

  ξ, Φ(xp1 )   yp y˙p Ball On Plate Module

Figure 3: Proposed RAPID controller framework for ball on plate system.

Figure 3: Proposed RAPID controller framework for ball on plate system. Command X (cm)

40

Reference RAPID

20 0

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Assumption 1. The exogenous perturbation υ ∈ Rn is introduced into the system to accomodate -20

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20 un-modelled bounded external disturbances or the model control Command failures,

0

20

40

Time (s)

60

80

Y (cm)

1

Reference RAPID

10

kυk ≤ υmax

0 -10 0

115

100

20

40 Time (s)60

(11) 80

100

with its known and constant upperbound υmax ≥ 0.

Figure 4: Tracking response during short term disturbance.

3

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Assumption 2. The non-linear vector function f (y p ) : Rn −→ Rm indicates the system matched

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uncertainty. Each component of f (y p ), as given in (12) , can be written as a linear combination of

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“n” known locally Lipchitz continuous basis functions ϕi (y p ) with unknown coefficients. f (y p ) = ζ > Φ(y p )

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(12)

where, ζ ∈ Rn×m is a constant matrix of unknown coefficient and Φ(y p ) = (ϕ1 (y p ), ϕ2 (y p ) . . . ϕn (y p ))> ∈ 7

120

121

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Rn is the known regression vector. The objective is to design a Robust adaptive PID control law such that the plant output y p , globally and asymptotically tracks the output yr of the reference model. y¨ r = ar1 y˙ r + ar0 yr + br uc (t)

(13)

123

where ar1 , ar0 and br are the system coefficients of the reference model and uc (t) is the external

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bounded reference command vector during tracking. For the system to track the reference signal,

125

all the signals should remain uniformly bounded. Hence, by defining the state vector as,      x     p1  def y p1  x p =   =   = y p  x˙  y˙  p2

(14)

p2

126

     x     r1  def yr1  xr =   =   = yr  x˙  y˙  r2 r2

(15)

127

and rewriting the equation (10) and (13) using the state vector defined by equation (14) and (15),

128

we write the following state space equations of the actual and reference models. x˙ p = A p x p + B p (u + f (x p1 )) + ξ

(16)

x˙r = Ar xr + Br uc

(17)

       0  0       ; B p =   ; ξ =   a p1 bp υ

(18)

129

130

  0  A p =  a

p0

1

131

  0  Ar =  a

r0

     0   1   ; Br =   b  ar1  r

(19)

132

A structural relationship between plant and model can be established if the system satisfies the

133

Erzberger’s perfect model following conditions [21] . Hence, the input matrix of the plant model

134

is transformed to its pseudo inverse as given in (20). B†p = [B>p B p ]−1 B>p 8

(20)

135

136

137

Using equation (20), the modified input model of the reference system is given by, Br = (B p B†p )br

(21)

Ar = (B p B†p )ar

(22)

ξ = (B p B†p )υ

(23)

The tracking error vector is chosen as,       e¯   x − x  y − y  r r  1  def  p  =  p  e¯ =   =  e˙¯   x˙ − x˙  y˙ − y˙  2 p r p r

(24)

For output of the system y p to globally asymptotically track yr , lim k¯e(t)k = 0

(25)

t→∞

138

From Figure. 3 , the conventional PID control law can be modified as given in (26) to accommodate

139

the uncertainty and modelling errors. ∞

Z u = K p e + Ki 140

0

e dt + Kd e˙ − ζ > Φ(x p1 )

(26)

where e = uc − y p is the tracking error. Hence Z ∞ u = K p (uc − y p ) + Ki (uc − y p ) dt + Kd (˙uc − y˙ p ) − ζ > Φ(y p )

(27)

Using the state vector given in (14), the control law is modified as, Z ∞ u = K p (uc − x p ) + Ki (uc − x p ) dt + Kd (˙uc − x˙ p ) − ζ > Φ(x p1 )

(28)

Substituting equation (28) into (16), Z ∞ Z    x˙ p = A p − B p K p x p + Ki x p dt + Kd x˙ p + B p K p uc + Ki

(29)

0

141

0

142

0

0

∞

 uc dt + Kd u˙ c + ξ

143

Comparing (29) with reference dynamics (17), it follows that for any bounded reference signal

144

uc , the ideal unknown control gains K p , Ki and Kd must satisfy the matching conditions given

9

145

in (30) and (31) to prevent unbounded growing of control signal (26) and provide global uniform

146

asymptotic error convergence. Z



∞

Ar xr = A p − B p K p x p + Ki x p dt + Kd x˙ p 0 Z ∞   Br uc = B p K p uc + Ki uc dt + Kd u˙ c



(30) (31)

0

147

However, there is no guarantee that ideal gains K p , Ki and Kd exist such that matching conditions

148

(30) and (31) are satisfied. Often in practice, the reference model matrices Ar and Br are chosen in

149

such a way that they have one ideal solution for K p , Ki and Kd . Assuming that K p , Ki and Kd do

150

exist, we consider the following control law, Z ∞ ˆ ˆ u = K p e + Ki e dt + Kˆ d e˙ − ζˆ > Φ(x p1 )

(32)

0

151

u = Kˆ p (uc − x p ) + Kˆ i 152

153

∞

Z 0

(uc − x p ) dt + Kˆ d (˙uc − x˙ p ) − ζˆ > Φ(x p1 )

where, Kˆ p , Kˆ i , Kˆ d and ζˆ are the estimates of the ideal unknown gains K p , Ki , Kd and ζ. Hence, to obtain the closed loop dynamics, (33) is substituted into (16). Z ∞ Z ∞      x˙ p = A p − B p K p x p + Ki x p dt + Kd x˙ p + B p K p uc + Ki uc dt + Kd u˙ c − B p (Kˆ p − K p )x p 0 0 Z ∞ Z ∞    ˆ ˆ ˆ ˆ ˆ +(Ki − Ki ) x p dt + (Kd − Kd ) x˙ p + B p (K p − K p )uc + (Ki − Ki ) uc dt + (Kd − Kd )˙uc 0

0

−B p (ζˆ − ζ)> Φ(x p1 ) + ξ 154

(33)

(34)

Rewriting (34) in terms of matching conditions (30) and (31), Z ∞    x˙ p = Ar x p + Br uc − B p (Kˆ p − K p )x p + (Kˆ i − Ki ) x p dt + (Kˆ d − Kd ) x˙ p + B p (Kˆ p − K p )uc 0 Z ∞  +(Kˆ i − Ki ) uc dt + (Kˆ d − Kd )˙uc − B p (ζˆ − ζ)> Φ(x p1 ) + ξ (35) 0

155

From (17) and (35) the closed loop dynamics of the tracking error vector e¯ (t) is formulated as, e˙¯ = x˙ p − x˙r Z ∞    = Ar x p + Br uc − B p (Kˆ p − K p )x p + (Kˆ i − Ki ) x p dt + (Kˆ d − Kd ) x˙ p + B p (Kˆ p − K p )uc 0 Z ∞  +(Kˆ i − Ki ) uc dt + (Kˆ d − Kd )˙uc − B p (ζˆ − ζ)> Φ(x p1 ) + ξ − Ar xr − Br uc 0 Z ∞   e dt + ∆Kd e˙ − B p ∆ζΦ(x p1 ) + ξ (36) e˙¯ = Ar e¯ + B p ∆K p e + ∆Ki 0

10

156

The parameter estimation errors are,

157

∆K p = (Kˆ p − K p )

158

∆Ki = (Kˆ i − Ki )

159

∆Kd = (Kˆ d − Kd )

160

∆ζ = (ζˆ − ζ)>

161

For the formulation of adaptive law, consider a globally radially unbounded quadratic Lyapunov

162

candidate in the form (37), V(¯e, ∆K p , ∆Ki , ∆Kd , ∆ζ) = e¯ P¯e + tr >

h

∆K p> Γ −1 p ∆K p

+

∆Ki> Γi−1 ∆Ki

+

∆Kd> Γd−1 ∆Kd

+ ∆ζ

>

Γζ−1 ∆ζ

i

(37) 163

with the rates of adaptation Γ p = Γ >p > 0, Γi = Γi> > 0, Γd = Γd> > 0 and Γζ = Γζ> > 0. For

164

satisfying the Lyapunov equation, P = P>  0, PAr + A>r P = −Q

165

and Q = Q>  0. Computing the time derivative of (37) along the trajectories results in, h i > > ˙ > −1 ˙ˆ > −1 ˙ˆ > −1 ˙ˆ > −1 ˙ˆ ˙ ˙ V = e¯ P¯e + e¯ Pe¯ + 2tr ∆K p Γ p K p + ∆Ki Γi Ki + ∆Kd Γd Kd + ∆ζ Γζ ζ

166

(38)

(39)

Substituting equation (36) into (39) gives, Z ∞   h ˙ˆ > > > > = e¯ + PAr e¯ − 2¯e PB ∆K p e + ∆Ki e dt + ∆Kd e˙ − ∆ζ Φ(x p1 ) + 2tr ∆K p> Γ −1 p Kp 0 i +∆Ki> Γi−1 K˙ˆ i + ∆Kd> Γd−1 K˙ˆ d + ∆ζ > Γζ−1 ζ˙ˆ + 2¯e> Pξ Z ∞ i h i h > −1 ˙ˆ > > > > > e dt + 2tr(∆Ki> Γi−1 K˙ˆ i ) = −¯e Q¯e + 2¯e PB p ∆K p e + 2tr(∆K p Γ p K p ) + 2¯e PB p ∆Ki 0 h i h i ˙ˆ + 2¯e> Pξ (40) > > > > > −1 ˙ˆ + 2¯e PB p ∆Kd e˙ + 2tr(∆Kd Γd Kd ) + − 2¯e PB p ∆ζ Φ(x p1 ) + 2tr(ζ > Γζ−1 ∆ζ) >



A>r P



>

11

167

Using vector state identity a> b = tr(ba> ) in (40), e¯ > PB p ∆K p> e = tr(∆K p> e e¯ > PB p ) | {z } |{z} |{z} | {z } a> a> b b Z ∞ Z ∞ e¯ > PB p ∆Ki> e dt = tr(∆Ki> e dt e¯ > PB p ) | {z } | {z } 0 0 | {z } | {z } a> a> b

e¯ = | {z } |{z} >

PB p ∆Kd> e˙

a>

b > tr(∆Kd e˙ e¯ > PB p )

b

(43)

a>

b >

>

a>

(42)

|{z} | {z }

e¯ PB p ∆ζ Φ(x p1 ) = tr(∆ζ Φ(x p1 ) e¯ > PB p ) | {z } | {z } | {z } | {z } >

(41)

b

b

(44)

a>

168

Z ∞  > > −1 ˙ˆ > > −1 ˙ˆ ˙ V = −¯e Q¯e + 2tr ∆K p (Γ p K p + e¯e PB p ) + ∆Ki (Γi Ki + e dt¯e> PB p ) + ∆Kd> (Γd−1 K˙ˆ d + e˙ e¯ > 0  (45) PB p ) + ∆ζ > (Γζ−1 ζ˙ˆ − Φ(x p1 )¯e> PB p ) + 2¯e> Pξ 169

If the adaptive laws are selected as given in (46)-(49), the Lyapunov stability holds good. K˙ˆ p = −Γ p e¯e> PB p Z ∞ ˙ ˆ Ki = −Γi e dt e¯ > PB p

(46)

K˙ˆ d = −Γd e˙ e¯ PB p

(48)

0 >

ζ˙ˆ = Γζ Φ(x p1 )¯e> PB p 170

(47)

(49)

Thereby, the time derivative of Lyapunov function (50) becomes globally negative semi-definite. V˙ = −¯e> Q¯e ≤ 0

(50)

171

The trajectories e¯ (t) of the error dynamics (36) enter a compact set (Ω0 ⊃ E0 ) ⊂ Rn in finite

172

time and will remain there for all future times. However, Ω0 is not compact in the V(e, ∆K p , ∆Ki , ∆Kd ,

173

∆ζ) space. Moreover, Ω0 is unbounded because the parameter estimation errors ∆K p , ∆Ki , ∆Kd

174

and ∆ζ are not restricted at all. Therefore, inside Ω0 , V˙ can become positive, and as a consequence,

175

the parameter errors ∆K p , ∆Ki , ∆Kd and ∆ζ can grow unbounded, even though the tracking error

176

norm remains finite at all times. This phenomenon is known as the “parameter drift” [22], which

177

is caused by the disturbance term ξ. This argument shows that the adaptive control laws (46-49)

178

are not robust to bounded disturbances, no matter how small the latter are. 12

179

3.1. e1 modification

180

To enhance the robustness of the plants with unknown parameters, Narendra and Annaswamy

181

[19], put forward a e1 modified MRAC technique, which gurantees uniformly bounded asymptotic

182

stability of the error equations without prior information on the plant perturbations. The key

183

feature of this technique is that it ensures boundedness of both output and parameter errors. Hence,

184

utilizing this technique to enhance the robustness of PID adaptive control laws, we introduce the

185

term called error dependent damping gain σ which is a linear combination of the system tracking

186

errors. The rational for introducing this damping gain is that it approaches zero when the regulated

187

output error diminishes. In addition, we have extended the methodology such that, the controller

188

still guarantees a uniformly bounded asymptotic stability in the presence of the model failures.

189

Consider the e1 modified adaptive laws, K˙ˆ p = −Γ p (e¯e> PB p − σk¯e> PB p kKˆ p ) Z ∞  ˙ > > ˆ ˆ Ki = −Γi e dt e¯ PB p − σk¯e PB p kKi

(51)

K˙ˆ d = −Γd (˙ee¯ PB p − σk¯e> PB p kKˆ d )

(53)

0 >

ˆ ζ˙ˆ = Γζ (Φ(x p1 )¯e> PB p − σk¯e> PB p kζ)

(52)

(54)

190

As seen from (51-54), the e1 -modification adds a tracking error-dependent damping σk¯e> PB p k to

191

the adaptive dynamics.

192

3.1.1. Proof for Uniform Ultimate Boundedness of error dynamics

193

Consider the time derivative of Lyapunov candidate given in (45), Z ∞  > > −1 ˙ˆ > > −1 ˙ˆ ˙ V = −¯e Q¯e + 2tr ∆K p (Γ p K p + e¯e PB p ) + ∆Ki (Γi Ki + e dt e¯ > PB p ) + ∆Kd> (Γd−1 K˙ˆ d + e˙ e¯ > 0  > −1 ˙ˆ > > PB p ) + ∆ζ (Γζ ζ − Φ(x p1 )¯e PB p ) + 2¯e Pξ ˆ + 2¯e> Pξ = −¯e> Q¯e + 2σk¯e> PB p ktr(∆K p> Kˆ p + ∆Ki> Kˆ i + ∆Kd> Kˆ d − ∆ζ > ζ)

194

195

where, Kˆ p = K p + ∆K p 13

(55)

196

Kˆ i = Ki + ∆Ki

197

Kˆ d = Kd + ∆Kd

198

ζˆ = ζ + ∆ζ V˙ = −¯e> Q¯e + 2σk¯e> PB p ktr(∆K p> ∆K p + ∆Ki> ∆Ki + ∆Kd> ∆Kd − ∆ζ > ∆ζ) + 2σk¯e> PB p k tr(∆K p> K p + ∆Ki> Ki + ∆Kd> Kd − ∆ζ > ζ) + 2¯e> Pξ

(56)

According to Frobenius norm of ∆x, >

tr(∆x ∆x) =

N m[U+200E] X X

PN Pm[U+200E] i=1

j=1

∆xi,2 j . The Schwarz’s inequality gives, (58)

|tr(∆x> x)| ≤ k∆x> xkF ≤ k∆xkF kxkF 199

(57)

j=1

i=1

where k∆xk2F =

∆xi,2 j ≥ k∆xk2F [U+200E][U+200E]

Substituting (57), (58) into (56) with appropriate modification of K p , Ki , Kd and ζ, we get, V˙ = −λmin (Q)kek2 + 2kekλmax (P)ξmax + 2σk¯e> PB p k(k∆K p k2F + k∆Ki k2F + k∆Kd k2F − k∆ζk2F ) +2σk¯e> PB p k(k∆K p kF kK p kF + k∆Ki kF kKi kF + k∆Kd kF kKd kF + k∆ζkF kζkF )

200

(59)

and using 2ab ≤ a2 + b2 for any a and b, (59) is written as, V˙ = −λmin (Q)k¯ek2 + 2kekλmax (P)ξmax + σk¯e> PB p k(k∆K p k2F + k∆Ki k2F + k∆Kd k2F − k∆ζk2F ) +σk¯e> PB p k(kK p k2F + kKi k2F + kKd k2F + kζk2F )

201

(60)

˙ e, ∆K p , ∆Ki , ∆Kd , ∆ζ) < 0 if, Hence V(¯ k¯ek2 − k¯ek |

λ

 kK k2   kK k2  p F i F − σk¯e> PB p k − σk¯e> PB p k λmin (Q) λ (Q) λ (Q) {z } | {z min } | {z min }

max (P)ξmax



C1

C2

C3

 kK k2   kζk2  d F F > > − σk¯e PB p k − σk¯e PB p k >0 λ (Q) λ (Q) | {z min } | {z min } C4

C5

14

(61)

202

or equivalently when, k¯ek > 2

203

λ

max (P)ξmax



= 2C1 λmin (Q) k∆K p k2F > kK p k2F = C6

(62)

k∆Ki k2F > kKi k2F = C7

(64)

k∆Kd k2F > kKd k2F = C8

(65)

k∆ζk2F > kζk2F = C9

(66)

Therefore, the compact and closed set is defined as,     λmax (P)ξmax     2 2      (¯e, ∆K p , ∆Ki , ∆Kd , ∆ζ) : k¯ek < 2 λmin (Q) ∧ k∆K p kF ≤ kK p kF ∧        Ω=         k∆Ki k2F ≤ kKi k2F ∧ k∆Kd k2F ≤ kKd k2F ∧ k∆ζk2F ≤ kζk2F 

(63)

(67)

204

For the given system with model uncertainty and unknown disturbance ξ and matched unknown

205

function f (x p1 ), the RAPID control scheme designed by (51)-(54), (60), (61) and (67) yields glob-

206

ally uniform asymptotic tracking performance of the reference model (17), driven by any bounded

207

time varying signal u(t). Furthermore, all signals in the corresponding closed loop system per-

208

sists to be uniformly bounded in time. Table 2 gives the design summary of the proposed RAPID

209

control scheme.

210

Theorem. As V˙ in (45) is globally negative semi-definite during uncertainties and bounded distur-

211

bances, the tracking error e(t) is ultimately uniformly stable and the parameter estimation errors

212

∆K p , ∆Ki , ∆Kd and ∆ζ are globally bounded. This guarantees that the parameter estimates, Kˆ p (t),

213

ˆ are bounded and V˙ in (60) to be uniformly continuous. Moreover, as V(t) is Kˆ i (t), Kˆ d (t) and ζ(t)

214

˙ = 0, the tracking error e(t) lower bounded, according to Barbalat[U+2019]s lemma, limt→∞ V(t)

215

is driven to zero, ensuring ultimate uniform boundedness.

216

4. Results and discussions

217

The efficacy of the RAPID control scheme is tested using Simulink. The overall closed loop

218

scheme comprises two loops: inner loop, which consists of PI controller, to adjust the servo angle;

219

outer loop, which consists of RAPID to control the position of the ball by controlling the plate 15

Table 2: Robust Adaptive PID design summary

Description

Expression

Open loop plant

x p = A p x p + B p (u + f (x p1 )) + ξ

Reference model

x˙r = Ar xr + Br uc

Tracking error vector

      e   x − x  y − y  r r  1  p  =  p  e¯ =   =    e   x˙ − x˙  y˙ − y˙  2

p

r

p

r

Derivative adaptive law

R∞ u = Kˆ p e + Kˆ i 0 e dt + Kˆ d e˙ − ζˆ > Φ(x p1 ) Kˆ˙ p = −Γ p (e¯e> PB p − σk¯e> PB p kKˆ p ) R∞ K˙ˆ i = −Γi ( 0 e dt e¯ > PB p − σk¯e> PB p kKˆ i ) K˙ˆ = −Γ (˙ee¯ > PB − σk¯e> PB kKˆ )

Disturbance adaptive law

ˆ ζ˙ˆ = Γζ (Φ(x p1 )¯e> PB p − σk¯e> PB p kζ)

Control input Proportional adaptive law Integral adaptive law

d

d

p

p

d

220

angle. The steady state gain and time constant of DC servo motor are 1.76 and 0.0285 respectively.

221

Since the DC servo module and ball balancer module are in cascade, the dynamics of the inner

222

loop must be faster than that of the outer loop. Moreover, the inner loop should result in zero

223

steady state error to ensure the close tracking of reference signal. Hence, the gains of the PI

224

controller, K p = 13.5 and Ki = 0.087, are determined using the pole placement technique to yield

225

a peaktime = 0.15 sec and an overshoot ≤ 5%. Similarly, the design requirement of outer loop

226

are e ss = 0.001 and t s = 2.5 sec. Using the pole assignment technique, we have determined the

227

initial gains of PID scheme as K p = 5.79, Ki = 3.67, and Kd = 2.90. In the following section, the

228

robustness of RAPID control scheme against the exogenous disturbance is assessed for two test

229

cases namely short term disturbance and persistent disturbance.

230

4.1. Short term disturbance

231

A sinusoidal disturbance with an amplitude of 25 cm at a frequency of 0.05 Hz is introduced

232

in both X and Y axes. The disturbance is introduced from t = 25 sec to 45 sec in X axis and from

233

t = 15 sec to 35 sec in Y axis. One can note from Figure. 4 that even though the disturbance

234

magnitude is 25 cm, the RAPID scheme maintains the deviation in state trajectories below 10 cm 16

Update Mechanism

−

  e¯ e¯˙

+ + Robustness Parameter Update Mechanism

+

Vm

PI Controller

−

− +

θl

Servo Module

  ξ, Φ(xp1 )   yp y˙p Ball On Plate Module

Figure 3: Proposed RAPID controller framework for ball on plate system.

X (cm)

40

Command Reference RAPID

20 0 -20 0

20

40

Time (s)

60

80

20

Command Reference RAPID

1

Y (cm)

100

10 0 -10 0

20

40 Time (s)60

80

100

Figure 4: Tracking response during short term disturbance.

3

Figure 4: Tracking response during short term disturbance.

V x (volts) V x (volts)

5

Reference RAPID Reference RAPID

5 0

-5

0 0

20

Vy (volts) Vy (volts)

-5 5 0

40

60

80

20

40

60

80

Reference RAPID

Time (s) 0

-5

100

Time (s)

5

100

Reference RAPID

0 0

20

40

60

80

100

Time (s)

-5 0

20

40

60

Figure 5: Control effort during short term disturbance.

80

100

Time (s)

Figure 5: Control effort short term disturbance. Figure 5: Control effortduring during short term disturbance. 6 46

Error (Y) Error (Y)

2 0 -2

4 2 0

-4 -2 -6 -4 -20

-10

0

10

20

30

Error (X)

-6 Figure 6: Phase-10 plot of tracking 0error during short -20 10 term disturbance. 20 Error (X)

30

4

Figure 6: Phase plot of tracking error during short term disturbance.

4 Figure 6: Phase plot of tracking error during short term disturbance.

235

by adaptively computing the disturbance gains. Figure. 5, the control inputs given to X and Y servo

236

motors, illustrates that during the disturbance the system requires a peak value of around 2 V to

237

bring back the X and Y trajectories to equilibrium point. Figure. 6, which shows the phase plot

238

of X and Y coordinates, highlights the asymptotic convergence of RAPID scheme under bounded

239

disturbance as claimed in Theorem 1. It can be noted that in spite of the initial deviation in state 17

240

trajectory due to exogenous disturbances, the state variables are brought back to equilibrium point

241

by adaptively tuning the gains of the controller.

242

4.2. Continuous Disturbance 40

Command Reference RAPID

X (cm)

20 0

40

Command Reference 100 RAPID

-20

Y (cm)

X (cm)

200

20

40 Time (s) 60

80

30

Command Reference RAPID

0

20

-20 10

0

20

40 Time (s) 60

0

80

100

30

Command 100 Reference RAPID

Y (cm)

-10

200

20

10

40 Time (s) 60

80

Figure 7: Tracking response during continuous disturbance.

0

Figure 7:-10Tracking during continuous 0 20 response 40 Time 80 100disturbance. (s) 60 20

Figure 7: Tracking response during continuous disturbance.

10

Error (Y)

0

20

-10

10 -20

Error (Y)

0 -30 -40 -10 -40

-30

-20

-20

-10 0 Error (X)

10

20

30

Figure 8: Phase plot of tracking error response during continuous disturbance.

-30 5

-40 -40

-30

-20

-10 0 Error (X)

10

20

30

Figure 8: Phase plot of tracking error response during continuous disturbance.

Figure 8: Phase plot of tracking error5 response during continuous disturbance.

V x (volts)

5

Reference RAPID

0

-5 0

20

40

60

80

100

Time (s)

Vy (volts)

5

Reference RAPID

0

-5 0

20

40

60

80

100

Time (s) Figure 9: Control effort during continuous disturbance.

Figure 9: Control effort during continuous disturbance. X (cm)

40

Command +15% +30%

18

20 0 -20 0

20

40

Time (s) 60

80

Y (cm)

40

100 Command +15% +30%

20 0 -20 0

20

40 Time (s) 60

80

100

243

A persistent disturbance with a magnitude of 10 cm at a frequency of 100 Hz is introduced

244

into the system to assess the performance of the closed loop scheme to minimize the effect of

245

continuous disturbance. From Figure. 7, the tracking response of X and Y axes during continuous

246

disturbance, illustrates that the RAPID scheme effectively attenuates the exogenous disturbance

247

in both X and Y axes by a factor of 20%. Thus, the deviation in X and Y position of the ball is

248

maintained below ±2.5 cm from the set point. It can be seen from Figure. 8 that the proposed

249

scheme drives the state trajectory to origin despite the persistent disturbance present in the system.

250

Thereby, the RAPID scheme ensures the global uniform asymptotic stability of the system even

251

during the persistent bounded external disturbance. From Figure. 9, which shows the control inputs

252

applied to X and Y motors, it is worth to note that the voltages are adaptively varied to ensure the

253

asymptotic convergence of the closed loop performance.

254

4.3. Parameter uncertainty

255

256

To evaluate the robustness of the RAPID scheme against parameter uncertainty, the internal states (output positions) are perturbed by the following nonlinear output dependent function (68).      −k, if x p1 < −uc          f (x p1 ) =  (68) 1 − |x p11 |2 kx p1 , if |x p1 | ≤ uc          if x p1 > uc k,

257

where k ∈ R is the percentage of uncertainty. Two levels of model uncertainty namely 15% and

258

30% are considered for validation. From Figure. 10, which shows the tracking response during

259

model uncertainty, it can be noted that in spite of the large initial oscillation during transient

260

state due to high level of model uncertainty, the RAPID scheme is less influenced by parameter

261

variations and yields smooth set point tracking. To highlight the convergence of the closed loop,

262

the error phase plot of X and Y coordinates is shown in Figure. 11. One can note that the phase

263

plot is driven to origin to ensure the global asymptotic stability of the closed loop system. From

264

Figure. 12, which illustrates the control inputs given to the X and Y motors, it is worth to note that

265

only during the initial transition, the system takes around 2.5 V to actuate the servo and requires

266

very less control effort once the system has reached the desired set point. 19

-5 0

20

40

60

80

100

Time (s)

Vy (volts)

5

Reference RAPID

0

-5 0

20

40

60

80

100

Time (s) Figure 9: Control effort during continuous disturbance.

40

Command +15% +30%

X (cm)

20 0 -20 0

20

40

Time (s) 60

80

100

40

Command +15% +30%

Y (cm)

20 0 -20 0

20

40 Time (s) 60

80

100

Figure 10: Tracking response during model uncertainty.

6 Figure 10: Tracking response during model uncertainty.

30 20

+15% +30%

20

0 10

Error (Y)

Error (Y)

10

+15% +30%

30

0

-10

-20 -10 -30 -20 -30

-20

-10

0

10

20

30

Error (X) -30 -30Phase plot-20 -10 response 0 10uncertainty.20 Figure 11: of tracking error during model

(s)

30

Error (X)

(s)

2 2 1 1 0 0 -1 -10

2

0

V x (volts)

V x (volts)

Phase plot of tracking error response during model uncertainty. Figure 11: Phase Figure plot11:of tracking error response during model uncertainty.

2

0

20

0

2 2 1 1 0 0 -1 -10

5

10 10

5

40

Time (s) 5

0

Reference

60

0

80

100

Reference

20

40

0

5

0

+15% +30%

10 10

5

2

2

Vy (volts)

Vy (volts)

0

Reference +15% +30%

10

Time (s)

60

+15% 80 +30%

Reference

2

2 0

0

20

40 Time (s)60 5

0

100

15

10

+15% 100 +30%

80 15

Figure 12: Control effort during model uncertainty.

0

20

40 Time (s)60

7

80

100

Figure 12: Control effort during model uncertainty.

7 during model uncertainty. Figure 12: Control effort

267

5. Conclusion

268

In this paper, we have presented a novel RAPID control scheme to address a couple of funda-

269

mental challenges in tracking applications of ball on plate system, namely, model inaccuracy and

270

exogenous disturbances. Harnessing the robustness feature of e1 modified MRAC technique, this

271

paper has put forward an uniform ultimate bounded controller framework for reference follow20

272

ing applications of ball on plate system. Moreover, using Erzberger’s model following condition,

273

the proposed RAPID scheme guarantess improved tracking even during model perturbation. The

274

mathematical proof of UUB of error signal is given using Schwarz[U+2019]s inequality condition

275

and Frobenius norms. Quanser 2 DoF ball balancer, a typical benchmark system for visual servo-

276

ing application, has been used to assess the performance of the proposed RAPID scheme for two

277

test cases namely tracking during disturbance and parameter uncertainties. The tracking and phase

278

plots of X and Y coordinates highlight that the the RAPID control scheme can yield satisfactory

279

response even during model variation and exogenous disturbance.

280

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