1-4244-0025-2/06/$20.00 ©2006 IEEE RAM 2006

Inversion-Based Nonlinear End-Tip Control of Flexible Arm in Presence of Large Model Uncertainties

Ehsan Soltani Graduate Student

Amirkabir University of Technology Tehran, Iran

ehsan_shs@aut.ac.ir

Mahyar Naraghi Department of Mechanical Engineering

Amirkabir University of Technology Tehran, Iran

Abstract—Achieving suitable performance in end-tip trajectory tracking while saving suitable margin of stability for flexible-arm system in presence of large model uncertainties is a very complicated task, because moving new output toward the end-tip, while using output-redefinition method, in order to increase tracking quality decreases system’s margin of stability. So large model uncertainties or large controller gains causes the system to leave the local region of stable internal dynamics. In this paper, power limiter system (PLS) is combined with time-delay controller (TDC) to guarantee stability of system in presence of large unstructured model uncertainties. The proposed controller is also compared with indirect adaptive linearizing controller.

Keywords— flexible-arm, inversion-based control, large model uncertainty, time-delay control, power limiter system

I. INTRODUCTION Over the past two decades, the study of flexible-arms

–lightweight and high-speed manipulators– has received a great deal of attention. Advantages over rigid manipulators such as lower energy consumption, higher speed of operation, smaller actuators, and enhanced load capacity are some of motivating aspects for further studies in this field [1]. Despite these advantages, modeling and control of this system suffers from some difficulties that make the control process complicated. The dynamic equations of a flexible-arm are a set of nonlinear PDE and ODE equations [2],[3]. Although some researchers have recently used distributed-parameter model to control this system [4], one commonly limits the number of degrees of freedom of the system by means of several methods such as assumed-mode method [5],[6]. The resulting equations are still highly nonlinear and form a coupled set of ODEs.

Also, it is well-known that the input-output relation for flexible arm system when the output is taken at the tip position exhibits, in general, a non-minimum phase behavior, namely its zero-dynamics is unstable. The source of this problem may be traced to the non-collocated nature of sensor and actuator in the system structure and causes the system’s input-output inversion to be unstable [7]-[10]. This is a generalization of the fact that the inverse of transfer function of a non-minimum phase linear system is unstable. Therefore, for such systems we should not look for causal and bounded control laws which achieve perfect or asymptotic tracking errors with output feedback. Instead, we should find controllers which lead to

small tracking errors for the desired trajectory of interest [11]. Thus, feedforward non-causal inputs in time and frequency

domains are used for linear systems requiring the whole trajectory to be known in advance [12],[13]. Output-redefinition method which is the most commonly used method and can be used for nonlinear systems as well is based on replacing system’s output with a new one so that the resulting input-output relation is minimum-phase. In [14], angular position of another suitable point along the link is used as a new output. In [7] general linear combination of deformation variables is used for this purpose.

Despite the good result reported, the majority of proposed controllers require knowledge about system’s parameters and its dynamical structure, while presence of uncertainty in parameters, e.g. in payload mass, or uncertainty in dynamical structure of system, e.g. effect of high-frequency ignored flexible modes, is usual. In [7] an indirect adaptive controller based on a discrete-time nonlinear model of system is considered to estimate payload mass. In [8] output-redefinition method is used together with sliding-mode controller. In [4] a controller is developed based on a distributed parameter model to eliminate spill-over effects. However, Adaptive methods commonly require dynamical structure of system’s uncertainties. Also sliding methods excite the flexible modes of system due to their switching behavior that leads to large errors in main output and large control effort. In [15] time delay control (TDC) is introduced to cope with unstructured dynamic uncertainties. A common difficulty in all of these controllers is to guarantee stability of system in presence of large uncertainties. In [7] a complicated lyapunov-based method is used to extract conditions for system stability which is very hard to generalize to a system with several unknown parameters. In addition, when we move new output toward end-tip to increase tracking quality, system’s margin of stability will decrease. Therefore a large uncertainty value simply leads to closed-loop instability. To avoid this problem, hub angle is used as new output in [15]. Some other researchers prefer to feedback flexible variables to increase damping of flexible modes instead of moving new output toward end-tip [10],[16]. However moving new output to main output has a significant role in tracking quality of fast reference commands and can not be neglected.

Hence, energy-based methods such as passivity approach

and power limiter system (PLS) are recently noted by many researchers because of their simple and reliable guaranty of stability [17]-[19]. These methods provide performance and stability requirements by combining two separate controllers such that added controller changes controller’s output suitably to save system’s stability when needed. In [20] a passivity observer (PO) is designed to observe the energy flow into controller. Based on observed energy, passivity controller (PC) changes system’s input to satisfy passivity conditions. In the studied case, combining PC with an end-tip PD controller is considered. However, tracking quality is not satisfactory in arms with extra flexibility and fast reference commands, because end-tip PD controller by itself leads to instability and PC only tries to save system stability. Therefore a suitable controller should be used instead of end-tip PD controller. In this paper a comparison between adaptive control and TDC in presence of large model uncertainties is considered and a hybrid method is presented to solve mentioned problems. In this method system stability is guaranteed by adding PLS to TDC. As a result, it is possible to move new output toward main output to increase tracking performance significantly. In PLS, which is an energy-based method, a conservative controller that has a great degree of stability in presence of large uncertainties is used in conjunction with main controller. The portion of each controller output in system’s control input is adjusted by energy observer, considering system’s degree of stability [21].

The organization of this paper is as follows. At first, dynamic modeling and inversion-based control of the system is investigated. In section two, parameter linearization for all parameters is performed and then using self-tuning approach and output-redefinition method, a controller is designed to track end-tip position in presence of time-varying parameters. Comparison of this controller with TDC is the subject of next section. Finally, the stability of TDC for large model uncertainties and outputs near end-tip is guaranteed using PLS method and the results are compared with previous results.

II. DYNAMIC MODELING AND INVERSION CONTROL A schematic diagram of a rotating Flexible arm is shown in

Fig. 1. Dynamic equations for flexible-arm system are derived in [2],[5],[6] using assumed-modes method where the essential assumption used is to approximate flexible deformation of link in terms of some known mode shapes.

[ ] [ ]1

1 1

( , ) ( ) ( ) ( ) ( )

( ) ( ) ( ) , ( ) ( ) ( )

mT

i ii

T Tm m

u x t x t x t where

x x x t t t

φ δ φ δ

φ φ φ δ δ δ=

= =

= =

∑L L

Assumed mode shapes, iφ , are commonly obtained from free vibration analysis of the system or similar systems. Although, using mode shapes of Euler-Bernoulli beams with pined-free and clamped-free boundary conditions are possible, it is shown in [5],[10] that clamped-free boundary conditions leads to better approximation in this case.

Consider the dynamic equations of a single-link flexible

Figure 1. Schematic diagram of a flexible-arm system.

manipulator derived by using the recursive Lagrangian assumed modes method.

( ) ( )( ) ( ) 1

, , , 00[ ]0 0[ ] [ ] , , ,

Trr rf

f mrf f f

H DM MGDM M H

θ

δ

θ θ δ δ τδ θ θδδ δθ θ δ δ ×

⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥+ + + =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

&

&

& &&& &

&& && &

Where, ( )tδ is deformation vector, m is the number of assumed flexible modes, M represents inertia matrix, H represents Coriolis and Centrifigual terms, D is structural damping matrix and G represents stiffness matrix. To present a physical meaningful version of model, deflection vector,δ , is stated in terms of end-tip deflection and its derivatives,η , [14]. Using new set of variables and neglecting damping we have:

( )( )

( ) ( ) ( ) ( ) ( )

1

10 1

0[ ] ( , , , )(1)

0[ ] [ ] ( , , , )

( ) [ ... ] , , , ... ,

Tr rf r

f mrf f f

TmTm

M M HGM M H

t u l t u l t u l t u l t

τη θ θ η ηθηθ θ η ηη

η ζ ζ

×

−−

⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤+ + =⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

⎡ ⎤′ ′′= = ⎣ ⎦

&&& &

& &&&

where, ( ) ( , )iu l t is the i-th derivative of deformation with respect to x at the end-tip i.e. x l= . Also, we have:

3

0 0

0 0

0 2 20

1 ( 2) 2 ( 1)

( 1) 2 ( 1) ( 1( 2) 1 ( 2) ( 2)

3(2)

00 0 0 0 0

10

0 00 0

l lT T T

l lT

l l

m ml

l lm m m

m m m

l dx x dx

M

x dx dx

I I Il l

m lI I

ρ η ρ ϕϕ η ρ ϕ

ρ ϕ ρ ϕϕ

ζ× − × −

− × − × −− × − × −

⎡ ⎤⎛ ⎞ ⎡ ⎤+⎢ ⎥⎜ ⎟ ⎢ ⎥

⎢ ⎥⎝ ⎠ ⎣ ⎦= ⎢ ⎥⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦⎡ ⎤+⎡ ⎤

⎡ ⎤+⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥+ + ⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦⎢ ⎥

⎢ ⎥⎣ ⎦

∫ ∫

∫ ∫

)

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

0 00

202

( 1) 10

2 2 (3)

0

lT T

r l

llT

fm

H dx m

mH dx

η ρ ϕϕ ηθ ζ ζ θ

ζ θρθ ϕϕ η

− ×

⎛ ⎞= +⎜ ⎟

⎝ ⎠⎡ ⎤ ⎡ ⎤⎛ ⎞

= − −⎢ ⎥ ⎢ ⎥⎜ ⎟⎢ ⎥⎢ ⎥⎝ ⎠ ⎣ ⎦⎣ ⎦

∫

∫

& & &&

&&

2 2

2 20

Tl d dG EI dx Kx xϕ ϕ η η

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥= =⎢ ⎥ ⎢ ⎥∂ ∂⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

∫

where, EI , lm , ρ , lJ and 0J are bending stiffness, payload mass, linear density of link, payload and hub inertias respectively. Also, ( )xϕ is the new resulting mode shape vector, after these changes. Relation between ( )tη and ( )tδ

cab be stated as follows: 1

1

( 1) ( 1)1

( ) ( )( ) ( )

( ) ( )

( ) ( )

m

m

m mm

l ll l

t E t where E

l l

φ φφ φ

η δ

φ φ− −

⎡ ⎤⎢ ⎥′ ′⎢ ⎥= =⎢ ⎥⎢ ⎥⎣ ⎦

L

L

M O M

L

So that ( )xϕ can be computed as follows: 1 1( , ) ( ) ( )T T T T Tu x t E x x Eφ δ φ η ϕ η ϕ φ− −= = = → =

System’s output is end-tip angular position (Fig. 1). 1 0 ( )( , ) ( , )( ) ( ) tan ( ) ( )

1( ) 0 0

Ttu l t u l tt t t t

l l l

t l

ζθ θ θ θ

θ η

− ⎛ ⎞= + ≈ + = +⎜ ⎟⎝ ⎠

⎡ ⎤= + ⎣ ⎦L

It is well-known that the input-output relation for flexible arm system when the output is taken at the tip position exhibits, in general, a non-minimum phase behavior. The source of this problem may be traced to the non-collocated nature of sensor and actuator in the system structure and causes the system’s input-output inversion to be unstable [7]-[10]. Therefore, for such systems, we should not look for causal and bounded control laws which achieve perfect or asymptotic tracking errors with output feedback. Instead, we should find controllers which lead to small tracking errors for the desired trajectory of interest [11]. Output-redefinition method which is the most commonly used method is based on replacing system output with a new one so that the new input-output relation is minimum-phase. General linear combination of deformation variables is used here as a new output:

[ ]1

( ) (4)n

m

y t l

l l

αθ η θ ση

α αασ

= + = +

= = L

where [ ]1 0α = L represents main output, [ ]1 0α = − L represents reflected tip position used in [22] and

[ ]0 0α = L is hub angle used in [10],[15]. To achieve relation between input and new output, (1) is rewritten in a suitable form. Hence, θ is replaced with ny ση− in (1).

1

( ) [ ] 0( , , , )(5)

[ ] [ ] ( ) 0( , , , )r rf r rn

T Trf f rf f mf

M M M HyM M M GH

η σ τθ θ η ησ ηθ θ η ηη ×

− ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤+ + =⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥− ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦

& &&&& &&&

Input-output relation of system is obtained by solving above equations for ny&& :

( )

1

1

(6)

[ ]

[ ]

nT

r rf r rf

r rf r f f

Tf rf

y

M M M M

H M M H G

M M

σ σ

σ

σ

τ

σ

σ

σ

−

−

Γ + Λ =

Γ = − − ∆

Λ = − − ∆ +

∆ = −

&&

Now, it is possible to find control input to force ny to track

the trajectory of interest, i.e. dy . For the SISO system given by (6), a controller which linearizes the system is:

( ) σστ Λ+−+−+Γ= )()( ndPndDd yyKyyKy &&&& Substituting this control law in (6) gives:

0n D n P n n d ne K e K e where e y y+ + = = −&& & Also internal dynamics equations are obtained, solving

equations (5) for η&& : τη σσσ

11 )( −− Γ−=++ΛΓ−+∆ Trfff

Trf MGHM&&

To achieve 0== nn yy& , control input becomes στ Λ= and related zero-dynamic equations are:

ησθ

η

&&

&&

−=

=++∆ 0ff GH

Substituting fH and fG from (3) we have:

ησσηη

ηη &&&& TTrMK∂∂

=+∆21

Linearizing above equation to analyze local stability and using TTTS ][ ηη &= , state-space model of zero-dynamics can be expressed as:

1

00I

S A S where AK−

⎡ ⎤= = ⎢ ⎥−∆⎣ ⎦

&

where K is the stiffness matrix. Non-positive real parts for eigenvalues of matrix A guarantees stability of zero-dynamics. Using first two flexible modes, Characteristic equation of A is:

4 21 00 ( ) ( ) 0A I C Cλ λ σ λ σ− = → + + =

which has four solutions as: 2

1 011,2,3,4

( ) 4 ( )( )2 2

C CC σ σσλ

−= ± − ±

These solutions show that imaginary eigenvalues are the only possible stable situation which in turn leads to a critical stability for zero-dynamics. The source of this property may be traced to lack of structural damping in system. In [16] feedback of deformation variables is used to inject damping to flexible modes:

( ) (7)d D n P n P Dy K e K e K Kσ δ δ στ η η= Γ + + + + + Λ&& & & or

( )[ ] [ ]

d D n P n

T T TP D

y K e K e K S

K K K and Sσ δ σ

δ δ δ

τ

η η

= Γ + + + + Λ

= =

&& &

&

So zero-dynamics equations changes to ( )

111

(8)

000

mTrf

S A B K S

IA B

MK

δ

×−−

= +

⎡ ⎤⎡ ⎤= = ⎢ ⎥⎢ ⎥ −∆−∆⎣ ⎦ ⎣ ⎦

&

One may compute ][ δδδ DP KKK = using Ackerman formula to push eigenvalues of zero-dynamics to a suitable location in left half complex plane. This in turn, enhances end-tip tracking quality due to added damping to flexible modes. In [14], angular position of another suitable point along the link is used for new output that is a special form of developed method. In Fig. 2 stable region for defining new output is compared for these two methods using parameters presented in table I. As is shown, to achieve stable zero-dynamics, while moving along link, we should depart from end-tip that in turn reduces tracking quality. However there is not any restriction to use physical meaningless outputs in this method. So hereafter we use the general form of developed output in this section.

Figure 2. Stable zero-dynamics region

III. INDIRECT ADAPTIVE CONTROL Parameter Linearization of (1) for EI , lm , ρ , lJ and 0J is

simply possible:

[ ]0

0

0

[ ]

l l

l l

T T

Tl l

l m J l J

m m

EI

q

P m J J EI

M q M q m M q J M q J M q

H H M H

G EI G

ρ

ρ

θ η

ρ

ρ

ρ

=

=

= + + +⎧⎪

= +⎨⎪ =⎩

&&&& &&

&& && && && &&

such that, none of the matrices iPM ,

iPH and iPG involve above

parameters. However model (1) can not be used directly for parameter estimation because it involves q&& terms which can not be simply measured. To avoid q&& terms in estimation,

/( )sλ λ+ which is a low-pass filter is used to filter (1). Assuming zero initial conditions we have:

{ }( )

( ) ( )

0 0

( )

0

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

t tt r t r

t t r

e T r dr M t q t e M r M r q r dr

e H r G r dr

λ λ

λ

λ− − − −

− −

= − +

+ +

∫ ∫∫

&& &

In order to use least square method for estimation, above equation is resorted in the ( ) ( ) ( )W t P t Y t= form.

( ) ( ) ( )

( ) ( )0 0

0

( )

0

( ) ( )

10 0

0

( )( ) ( )

0

l l l

l l

l l

tt r

m m m

J J J J EI

m J J

t tt r t r

m

W t e M M q H M M q H

M M q M M q G dr

M q M q M q M q

rY t e T r dr e dr

λρ ρ ρ

ρ

λ λ

λ λ

λ λ

τ

− −

− − − −

×

= − ⎡ + − + −⎣

⎤+ + − ⎦⎡ ⎤+ ⎣ ⎦

⎡ ⎤= = ⎢ ⎥

⎣ ⎦

∫

∫ ∫

& && &

& && &

& & & &

Cost function is selected as: ( ) ( ) ( ) 2

0 2ˆt

J Y r W r P t dr= −∫

It may be shown easily that the following parameter update law minimizes the selected cost function.

1

0

ˆ ˆ0 ( ) ( ) ( ( ) ( ) ( ) )ˆ

( ) [ ( ) ( ) ] ( )

T

t T T

J P t W t Y t W t P tP

t W r W r dr t W W−

∂= → = −Γ −

∂

Γ = → Γ = −Γ Γ∫

&

&

Simulation of output-redefinition method using parameters of Table I is shown in Fig. 3, assuming all parameters to be unknown. Command reference and new output are selected as

sin(2 )dy t= and [0.8 0]σ = respectively. 2P DK K= = and (0) 100 (1,1,1,1,1)diagΓ = × are also used. Kδ is also selected such

that moves eigenvalues of zero-dynamics 0.5 unit to LHP. New and main outputs track the fast reference command

suitably and all parameters converge to their real values. However, a larger (0)Γ is needed to estimate parameters in smaller time. Another approach to increase convergence speed is to use exponential forgetting for cost function which also helps to estimate time-varying parameters.

( ) ( )22( )

0 2ˆ( )

t t rJ e Y r W r P t drλ− −= −∫

This change does not affect parameter update law but ( )tΓ& changes to the following form [11].

2( ) Tt W WλΓ = Γ −Γ Γ& Fig. 4 shows simulation of system using 2 5λ = while

payload mass is the only unknown parameter and changes to 0.15kg at 5t s= . Also maximum allowable value for 2λ should be limited considering noise level in system.

IV. TIME DELAY CONTROL AND POWER LIMITER SYSTEM Despite the good result obtained, proposed controller

requires knowledge about system’s dynamical structure. Also several time-varying parameters cause the estimator to be confused while presence of unstructured uncertainties like friction which has abrupt changes during operation is usual. Therefore, time delay control (TDC) was introduced to cope with unstructured dynamic uncertainties, which is based on estimation of uncertainty value using previous step data. In [15], TDC is applied to hub angle output of flexible-arm system. To enhance tracking quality, extension of this method to a general redefined output is considered in this section. Hence, (6) is reshaped in the following form.

τττσσσ BAABAy n ++=+=Γ+ΛΓ−= −−21

11&& where 2A and B are unknown parts of input-output relation. Using time delay control, input-output linearization could be achieved when only system’s relative degree is known.

( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( ) ( )( )

11 1 1 1 1 1

ˆ ˆ

(9)

k k n k k k

D Pd k d k n k d k n k

B B y A A

y K y y K y y

τ τ−− − − −

⎡= − + −⎣⎤+ + − + − ⎦

&&

&& & &

where B̂ is estimation of B and ( ) ( ) ( )1 1 1 1ˆ( )k k kB y Aτ − − −− +&& is

estimation of 2A− using data of previous step. Obviously, this is a discrete method and index k shows value of variable at time t kL= . Also, L is time constant used in discretization namely ( ) ( )1k kt t −− .

Using theorem presented in [15], If there exists a positive integer N such that 1ˆ 1B B I α− − ≤ < for k N> then the

system (1) achieves input-output linearization using control law (9) for sufficiently small L and k →∞ i.e.:

( ) ( ) ( ) ( ) ( ) ( )( ) ( )

for and 0D Pn k d k d k n k d k n ky y K y y K y y

k L

= + − + −

→∞ →

&& && & &

Figure 3. Indirect adaptive control using output redefinition method (a). reference command-- redefined output▬ end-tip position — (b) control input (c) parameters estimation (d) redefined output error (rad) (e) main output error (rad) (f) bending stiffness estimation

Despite simple form of this controller, there are some difficulties in its implementation. First, reliable measurement of ny&& is not simply possible.

Another common difficulty in all of developed controllers is that the stability of internal dynamics in (8) is guaranteed locally whereas extracting conditions to guaranty stability of this system in presence of large model uncertainties is very hard. So large model uncertainties or large controller gains causes the system to leave the local region of stable internal dynamics and leads to overall instability at the first times of system operation. Also, when we move new output toward end-tip, to increase tracking quality, system’s margin of stability will decrease. To avoid this problem, hub angle is used as new output in [15]. Some other researchers prefer to feedback flexible variables to increase damping of flexible modes instead of moving new output toward end-tip. However moving new output to main output has a significant role in tracking quality of fast reference commands and can not be neglected.

So, energy-based methods such as passivity approach and power limiter system (PLS) are recently noted by many researchers because of their simple and reliable guaranty of stability [17]-[19]. These methods provide performance and stability requirements by combining two separate controllers such that added controller changes system’s input suitably to save system’s stability when needed. Power limiter system, which is a energy-based method, uses a conservative controller that has a great level of stability in presence of large uncertainties in conjunction with main controller. The portion of each controller output in system’s control input is adjusted by energy observer with respect to system’s degree of stability [21]. A similar approach is used here. Fig. 5 shows the main idea of this approach. So Control input gives the form:

1cs cs usr usr cs usrk k and k kτ τ τ= + + =

Figure 4. Indirect adaptive control in presence of time-varyingparameters(a). reference command-- redefined output▬ end-tip position — (b) control input. (c) end-tip error (d) estimation of payload mass .

where usrτ is the main control law, (7) or (9), and csτ is the conservative control law that is selected here as a PD controller based on hub position error

( ) ( )uc P d D dK y K yτ θ θ′ ′= − + − &&

which has a great degree of stability. usrk and csk which determine the portion of each controller in system’s input are computed in this paper based on system’s energy level, w, as follows:

0

00

10 0

1 0cs

w wk w

w w ww

⎧≥⎪

⎪= =⎨⎪− < <⎪⎩

∫ ∑=

≈+=t K

kkkLdrrrwtw

0 1)()()()()0()( θτθτ &&

where 0w is maximum allowable system energy which should be determined by designer. It should be selected more than maximum energy of system when perfectly tracks the command reference. Above equation states that when system energy is greater than 0w , inversion-based controller (TDC) has not any contribution in system input.

Figs. 6 and 7 show simulation of TDC, without and with PLS method respectively. The output is located at

[ ]0.9 0σ = and other controller parameters are as follows: 0.01L s= , 6p pK K ′= = , 9D DK K′ = = and 0 9w =

As is shown, unstable system is controlled successfully using developed method and tracking quality is excellent because of new output’s location. PO-PC method, used in [20], has a sudden effect on system’s input which causes the control input to become discontinuous. This in turn, leads to excitation of flexible modes while developed method has continuous effect on control input.

Figure 5. Power Limiter System

Figure 6. Time delay control of system without PLS (a) reference command-- redefined output▬ end-tip position— (b) control input.

Figure 7. Time delay control of system with PLS (a). reference command-- redefined output▬ end-tip position — (b) control input. (c) uncertainty value (d) main controller’s portion

TABLE I. PHYSICAL PROPERTIES OF FLEXIBLE MANIPULATOR

lρ EI ( )lm kg ( )2.l

J kg m ( )2

0.J kg m Parameter

Name

1 0.2772.43 0.3 41 10−× 0.3 Exact Value

-0.252 0.45 0.01 0.25 Initial Guess

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