Robust Output Control Algorithm for a Twin-Rotor Non-Linear

MIMO System

Sergey Vrazhevsky, Alexey Margun, Dmitry Bazylev, Konstantin Zimenko and Artem Kremlev

ITMO University, Kronverksky av. 49, Saint Petersburg, Russia

Keywords: Non-Linear MIMO System, Robust Control, Output Control, Consecutive Compensator, PID-Controller.

Abstract: This paper addresses to the problem of a non-linear MIMO systems control. A class of non-linear parameter

uncertain systems operating under unknown bounded disturbances is considered. It is assumed, that

mathematical model of such system can be decomposed on linear and non-linear dynamics. Proposed

control algorithm is based on the method of consecutive compensator. The only required parameter to be

known for the controller synthesis is a relative degree of linear part of plant. The effectiveness of the control

method is demonstrated experimentally using the laboratory platform named «Twin Rotor MIMO System».

The proposed method is compared with standard PID controller. Experimental results show that the

transient behaviour of the developed control algorithm provides higher accuracy and performance,

especially for the case of model parameters deviation from their nominal values.

1 INTRODUCTION

Modern technology development requires to

consider different complex mechatronic and robotic

systems as real technical objects, which are

described by systems of non-linear differential

equations, have uncertain parameters and

unaccounted dynamics in mathematical models,

operate under external and internal disturbances. For

such systems classical control methods (modal

control, PID control, etc.) are often inapplicable or

not capable to meet necessary technical

requirements. This problem is especially acute for a

class of nonlinear and multivariable systems with

sufficient couplings.

The main interest of development of advanced

control approaches satisfying these challenges is to

improve performance of such devices and extend

their application area.

This paper considers control problem for twin

rotor MIMO system (TRMS) (Feedback instruments

Ltd., 1998). There are a lot of articles proposing

different control methods applied in the area of non-

linear MIMO systems and applied to a TRMS

particularly, see (Rahideh et al, 2008).

Linear and nonlinear PID control algorithms are

analyzed in (Cajo, 2015).

Optimal controller using LQR technique is

proposed in (Pandey and Laxmi, 2015). Suboptimal

controller using iterative linearization algorithm is

proposed in (Vrazhevsky and Kremlev, 2015).

Suboptimal tracking controller using a linear

quadratic regulator (LQR) with integral action and

adaptive sliding mode controller is described in

(Phillips, 2014). Another control algorithm based

both on optimization method and on fuzzy logic

method was designed in (Allouani et al, 2012).

In (Juang et al, 2011) a fuzzy PID control

scheme with a real-valued genetic algorithm (RGA)

was proposed. A control technique based on

controller named «fuzzy-sliding and fuzzy-integral-

sliding controller» (FSFISC) is designed and

applied to the TRMS in (Tao et al, 2010). Fuzzy

controllers is a quite an intensive research area with

a set of result in non-linear MIMO systems

applications (Shi, 2014).

In (Basri et al, 2014) adaptive controller based

on on the backstepping technique was applied to

quadrotor that was described as non-linear MIMO

system.

A robust control solution is applied to a

linearizable non-linear MIMO systems in (Liu and

Söffker, 2014). It uses feedback linearization and

state feedback control with a disturbance rejection.

However, all considered solutions have a number

of disadvantages, such as complicated engineering

Vrazhevsky, S., Margun, A., Bazylev, D., Zimenko, K. and Kremlev, A.

Robust Output Control Algorithm for a Twin-Rotor Non-Linear MIMO System.

DOI: 10.5220/0005985004210427

In Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2016) - Volume 2, pages 421-427

ISBN: 978-989-758-198-4

Copyright

c

2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

421

realization, state vector knowledge, complicated

adjustment of control parameters. Some researches

describe modelling methods including identification

and linearization techniques, see (Nejjari et al,

2012). In (Radac et al, 2014.) iterative data-driven

algorithm for experiment-based tuning of controllers

for nonlinear systems was presented.

In this paper a robust output control method is

proposed to control the TRMS system. The control

algorithm is based on the consecutive compensator

method. Performance of proposed method is based

on its possibility to compensate a wide class of

external disturbances and save plant stability in

conditions of unaccounted internal dynamics. The

algorithm is simple to implement due to the fact, that

the only parameter required to build the controller is

a plant’s relative degree.

The paper is organized as follows. Section 2 is

devoted to a brief description of the non-linear twin

rotor MIMO system. Section 3 contains

mathematical model of TMRS bench. Designed

control algorithm is presented in Section 4. Finally,

Experimental results of the proposed control system

and its comparison with PID controller are shown in

Section 5. Finally, concluding remarks are given in

Section 6.

2 BENCH DESCRIPTION

Consider the non-linear twin rotor MIMO system

(TRMS) and obtain its mathematical model. TRMS

is a laboratory helicopter-like system with two

degrees of freedom and opportunity of independent

two-channel control. General view of TRMS is

shown on Fig. 1.

Figure 1: General view of Twin Rotor MIMO System.

Table 1: TRMS parameters.

Parameter description Value Units (SI)

I

1

Pitch inertia moment

6.12×10

-2

kg×m

2

I

2

Yaw inertia moment

2×10

-2

kg×m

2

M

g1

Gravity moment

coefficient

0.32

N×m

M

g2

Gravity moment

coefficient

0.48

N×m

a

1

Parameter of main

rotor static

charactetistic

1.35×10

-2

N/A

b

1

Parameter of main

rotor static

charactetistic

9.24×10

-2

N/A

a

2

Parameter of main

rotor static

charactetistic

2×10

-2

N/A

b

2

Parameter of main

rotor static

charactetistic

9×10

-2

N/A

B

1

ψ

Friction forces

moment parameter

6×10

-2

N×m×s/rad

B

2

ψ

Friction forces

moment parameter

1×10

-2

N×m×s/rad

B

1

ϕ

Friction forces

moment parameter

6×10

-3

N×m×s/rad

B

2

ϕ

Friction forces

moment parameter

1×10

-3

N×m×s/rad

K

gy

Gyroscopic forces

parameter

5×10

-2

s/rad

k

1

Main rotor gain

coefficient

1.1 N/A

k

2

Tail rotor gain

coefficient

0.8 N/A

T

11

Main rotor parameter 1.1 N/A

T

10

Main rotor parameter 1 N/A

T

21

Tail rotor parameter 1 N/A

T

20

Tail rotor parameter 1 N/A

k

c

Cross-reaction gain

coefficient

-0.2 N/A

T

p

Cross-reaction

moment parameter

2 N/A

T

0

Cross-reaction

moment parameter

3.5 N/A

Full list of its parameters is given on Table 1.

These include inertia moments, coefficients of

friction forces moment and gravity moments, cross-

reaction moment parameters, etc.

The system comprises two DC motors: one of

them provides movement in vertical plane (pitch

angle) and the other one is for motion in horizontal

plane (yaw angle). TRMS is controlled by

independent voltage levels on the armature of the

motors. The pitch and yaw angles are measurable

outputs.

ICINCO 2016 - 13th International Conference on Informatics in Control, Automation and Robotics

422

Maximum angles of the plant rotation are limited

by mechanical structure constraints. Input voltage

levels are limited within the [-2.5V; +2.5V].

Detailed description of the laboratory bench with

its mathematical model synthesis is given in

(Feedback Instruments, 1998).

3 TRMS MATHEMATICAL

MODEL

In this section we present mathematical model

description of the bench in form of two third-order

differential equations and in a vector-matrix form.

The TRMS works under the following forces:

DC motors torques, gravity forces, friction forces

and such coupling effects as gyroscopic moment and

cross-reaction force.

The plant can be represented as two coupled

subsystems. Given mathematical description of the

plant also include DC motor transfer functions and

friction forces moments.

Plant dynamics in vertical plane is described by

the following moment equation

,

(1)

where

ψ

is a plant pitch angle, I

1

is an inertia

moment, M

1

is a plant torque generated by the main

rotor torque

τ

1

, M

FG

is a gravity moment, M

B

ψ

is a

friction forces moment, M

G

is a gyroscopic moment.

These moments are represented as follows

sin

cos

cos

sign

(2)

DC motors dynamics approximated by first order

transfer function can be represented as

⇒

,

(3)

where

and

are

the torques generated by DC

motors,

and

are the input voltages of the

motors.

Taking into account (2) and (3), the dynamics in

vertical plane (1) can be represented as follows

,

1

1

sin

2

1

cos

sign

1

1

1

1

1

1

2

cos,

1

10

11

1

1

11

1

,

(4)

where

is a vertical angular speed of the plant.

Rewrite system (4) in the form

,

cos

sin

cos

sign

,

(5)

where

,

and

are transition coefficients,

is a function which contains all non-linear

components of the plant dynamic in vertical plane.

System (4) in vector-matrix form can be

represented as following

,

,

(6)

where

01 0

0

1

1

00

10

11

,

0

0

1

11

,

100

,

(7)

0

0

sin

0

0

cos

0

0

sign

0

0

0

0

cos,

(8)

where

is a vector-function of non-linear

components of the subsystem that related with

vertical plant dynamics.

Repeat analog calculations for plant dynamics in

horizontal plane. Moment equation is represented as

follows

,

(9)

where

ϕ

is a plant yaw angle, I

2

is an inertia

moment, M

2

is a plant torque generated by the tail

rotor torque

τ

2

, M

B

ϕ

is a friction forces moment, M

R

is a cross-reactions moment. These moments are

represented as shown

Robust Output Control Algorithm for a Twin-Rotor Non-Linear MIMO System

423

,

sign

,

.

(10)

Considering (3) and (10), the dynamics in

horizontal plane (9) can be represented as follows

,

sign

,

2

2

2

2

2

2

2

,

2

20

21

2

2

21

2

,

1

0

1

1

,

(11)

where

is a horizontal angular speed of the

system,

is a cross reaction variable.

System (11) can be represented as following:

,

2

1

1

2

sign

1

,

(12)

where

,

and

are transition coefficients,

is a function which contains all non-linear

components and cross-reactions of the plant

dynamic in vertical plane:

System (11) in vector-matrix form can be

represented as following

,

,

(13)

where

01 0

0

00

,

0

0

,

100

,

(14)

0

0

sign

0

0

0

0

,

(15)

where

is a vector-function of non-linear

components and cross-reactions of the subsystem

that related with horizontal dynamics of the plant.

As the result, TRMS mathematical model is

represented by two subsystems (5) and (12) with

couplings given as bounded external disturbances.

4 ROBUST CONTROL METHOD

In this section we propose control algorithm for

previously obtained mathematical model of TRMS.

The control law is based on consecutive

compensator (Bobtsov, A., 2002) method.

Consider multivariable control plant:

,,

,

,

,

(16)

where,1,2,1,2,

and

are linear

differential operators with dimensions

and

respectively,

∈ is an output signal,

∈

is an input signal,

,,

is a function of non-

linear components in each channel,

is a

functions of external unknown bound disturbances

in each channel,

and

are linear

differential operators of output and input couplings

respectively, / is a differential operator,

1 is a relative degree of the plant.

A reference model is described by equation:

,1,2

(17)

where

and

are linear differential

operators,

∈ is an output signal of

reference model,

∈ is a smooth bounded

reference signal.

It is necessary to provide tracking of plant’s

output for the reference model output.

Introduce decentralized consecutive compensator

control law in accordance with (Pyrkin, A., et al,

2015):

̂

,

(18)

where

is a positive number,

is a Hurwitz

polynomial with degree

1, is a complex

variable, ̂

is an estimation of tracking error

.

For error estimation the observer is used

Γ

,

̂

,

(19)

where

∈

is an observer state vector,

Γ

0

,00…

,

10…0

,

(20)

where Γ

is a Hurwitz matrix due to the choice of its

coefficients,

0 is a sufficiently large number.

In (Pyrkin, A., et al, 2015) it is proved that

control law (18) with observer (19) provides

convergence of output signal for the reference model

output with prespecified accuracy in the case of

ICINCO 2016 - 13th International Conference on Informatics in Control, Automation and Robotics

424

Lipshitz function

,,. Moreover, proposed

algorithm is robust with respect to the parameter

deviations. However, nonlinear functions in (5) and

(12) are not Lipshitz function due to the presence of

the function sign∙.

In (Margun, A., and Furtat, I., 2015a) it is proved

that controller (18)-(19) provides convergence of

tracking error of MIMO system to the limited area in

the case of quantized output measurement. The

function sign∙ is Lipshitz everywhere except at the

zero point. In the paper (Margun, A., and Furtat, I.,

2015a) the control problem is solved for the system

(16) in presence of quantized output. In this case the

system is exponentially stable with respect to area

around zero caused by the quantizer step and

disturbance value. Since dynamics of quantizer

function is similar to sign∙ at zero origin we can

conclude, that system (16) is exponentially stable

with respect to area around origin. Therefore,

controller (18) and estimation algorithm (19)

provide exponentially convergence of tracking error

of TRMS to the bounded area.

5 EXPERIMENTAL RESULTS

To verify the proposed technique experimentally we

use the following parameters of the consecutive

compensator

0.5,

5

2050,

1,

1,2.

(21)

Observer parameters are taken as

Γ

01

0.01 100

,

01

,

10

,1,2.

(22)

Linear part of TRMS within parameters from

Table 1 is represented in the following form

01 0

0

6∗10

2

1.51

000,91

,

0

0

1

,

100

,

(23)

010

0

6∗10

3

4.5

001

,

0

0

0.8

,

100

,

(24)

Presented control algorithm is experimentally

tested on the TRMS and compared with PID

controller. The plant is controlled simultaneously

and independently by each degree of freedom.

Therefore, all coupling effects are taken into account

during the experiments.

Experimental results (Fig. 2 - 9) for each plant

subsystem in tracking mode and in stabilization

mode are shown in Appendix.

Several scenarios for tracking (Fig. 2 - 5) and

stabilization (Fig. 6 - 9) modes are carried out for

both control systems. PID controller tuned by TRMS

developers is tested in the same modes under the

same conditions for evaluation of proposed

controller performance.

On Fig.4 and Fig.5 it is apparent that consecutive

compensator operates more accurate than PID

controller. On Fig.6 and Fig.7 consecutive

compensator operates with comparable with PID

controller accuracy. On Fig.6 - Fig.9 plant

trajectories in stabilization mode are demonstrated.

Transient processes of the consecutive compensator

and the PID controller are quite similar.

Figure 2: Plant yaw output in tracking mode under the

consecutive compensator work.

Figure 3: Plant yaw output in tracking mode under the PID

controller work.

Figure 4: Plant pitch output in tracking mode under the

consecutive compensator work.

Robust Output Control Algorithm for a Twin-Rotor Non-Linear MIMO System

425

Figure 5: Plant pitch output in tracking mode under the

PID controller work.

Figure 6: Plant yaw output in stabilization mode under the

consecutive compensator work.

Figure 7: Plant yaw output in stabilization mode under the

PID controller work.

Figure 8: Plant pitch output in stabilization mode under

the consecutive compensator work.

Figure 9: Plant pitch output in stabilization mode under

the PID controller work.

Experimental results analysis revealed that

robust output control was successfully applied to the

chosen plant and its quality exceeds the PID

controller.

6 CONCLUSIONS

In this paper we present control algorithm for non-

linear parameter uncertain MIMO systems operating

under unknown bounded disturbances. A key

assumption for application of the control law is that

mathematical model of the system can be

decomposed on linear and non-linear dynamics.

Relative degree of the linear part of plant is the only

required parameter for the proposed controller.

Introduced robust control algorithm is applied to

the laboratory platform named «Twin Rotor MIMO

System» and provides its exponential stability with

convergence to a bounded zero origin. Non-linear

part of the TRMS that contains cross-relations of the

plant is considered as unknown bounded

disturbance. Designed control system is tested

experimentally and compared with the standard PID

controller. Several scenarios of TMRS work are held

and experiments demonstrate that proposed control

algorithm provides higher performance and accuracy

of the laboratory bench. It should be noted that

presented control algorithm is robust with respect to

parametric disturbances that is supported by

experimental results.

ACKNOWLEDGEMENTS

This work was partially financially supported by

Government of Russian Federation, Grant 074-U01.

This work was supported by the Ministry of

Education and Science of Russian Federation

(Project 14.Z50.31.0031).

The work was supported by the Russian

Federation President Grant (No. MD-6325.2016.8).

This work was supported by the Russian

Federation President Grant №14.Y31.16.9281-НШ.

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