Balancing Control of Bicyrobo by Particle Swarm Optimization-based Structure-specifield Hoo Loop Shaping Control

8/4/2019 Balancing Control of Bicyrobo by Particle Swarm Optimization-based Structure-specifield Hoo Loop Shaping Control

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Balancing control of Bicyrobo by particle swarm optimization-based

structure-specified H∞ loop shaping control

Bui Trung Thanh, Manukid Parnichkun*

School of Engineering and Technology, Asian Institute of Technology,

P.O. Box 4, Klong Luang, Pathumthani 12120, Thailand *

Corresponding author. Tel.: +66 2 524 5229; Fax: +66 2 524 5697

E-mail address: [email protected] , [email protected]

Abstract

In this paper, structure-specified H∞ loop shaping control using particle swarm optimization

(PSO) is proposed to control balancing of Bicyrobo, a bicycle robot with gyroscopic stabilizer.

The structure-specified H∞ loop shaping controller design normally generates a complex

optimization problem. PSO is an efficient optimization method used to solve multi-objectives and

non-convex optimizations. A model-based systematic procedure for designing particle swarm

optimization-based structure-specified H∞ loop shaping controller is presented in this paper. The

controller obtained from this method is simple in structure but still robust. The proposed control

algorithm is applied to balance the bicycle robot, which is unstable system involved with sources

of uncertainties due to un-modeled dynamics, parameter variations and external disturbances.

Simulation and experimental results show the robustness and efficiency of the controller in

compared with the proportional plus derivative (PD) controller, and the conventional H∞ loop

shaping controller.

Keywords: Bicycle robot; Structure-specified controller; H∞ loop shaping control; Particle swarmoptimization; Gyroscopic stabilizer.

1. Introduction

Electrical bicycle is a good mean of transportation because of its advantages on environmental

friendly, light weight, and capability of traveling in narrow roads. However, bicycle is unstable in

nature. Without a proper control, it will easily fall down. Hence, development of a self-balancing

bicycle is an interesting topic for many researchers. An exciting example of bicycle robots is

Murata Boy robot developed in Japan in 2005 [1]. There are many methods used to control

balancing of this system such as flywheel balancing by Beznos et al. in 1998 [2], Gallaspy in1999 [3], and Suprapto in 2006 [4], mass balancing by Lee and Ham in 2002 [5], and steering

balancing by Tanaka and Murakami in 2004 [6]. Among these methods, flywheel balancing

method which uses a spinning wheel as a gyroscopic stabilizer is a good choice because the

response time is short and the system can be stable even at stationary position. The balancing

principle using flywheel can also be applied to many other systems which require dynamics

balancing during movement, for example, balancing of a biped robot [7]. Various balancing



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control algorithms have been proposed, such as nonlinear control by Beznos et al. in 1998 [2] and

Lee and Ham in 2002 [5], a compensator design using root locus approach by Gallaspy in 1999

[3], and PD control by Suprapto in 2006 [4]. However, these control algorithms are not robust,

the bicycles cannot carry loads with variable weights and cannot work in disturbance

environments. Therefore, a robust control algorithm is necessary in real applications of bicycle

robots.

H∞ loop shaping control is a well-known and effective method. It is a robust control technique

suiting for systems with unstructured uncertainties. This approach was firstly developed by

McFarlane and Glover in 1992 [8], and has been used successfully in many practical applications

[9-12]. However, in the conventional H∞ loop shaping control design, the obtained controllers are

normally high order, and difficult to implement in reality. Basically, low order H∞ loop shaping

controllers can be synthesized by either order reduction design method or structure-specified

design method. The former designs a full-order controller, and then approximates it by a lower-

order one as described by Ballois et al. in [10]. The latter determines parameters of a low order structure-specified controller such that the controller is admissible and the H∞ norm from

exogenous inputs to controlled outputs is minimized as presented in [12]. Because the latter is a

kind of direct design approach, a better solution can be achieved. However, the structure-

specified design method normally generates a complex and non-convex optimization problem,

which is difficult to solve analytically. Many searching algorithms are employed to deal with this

problem. However, most of the searching algorithms, the gradient-based searching algorithm, for

instance, require the parameters space being differentiable.

Evolutionary algorithm (EA) is a robust and optimization method that is able to find the solutionsof ill-behaved problems. EA has been widely applied in many control systems as reviewed in

[13]. A subset of EA, called genetic algorithm (GA), was previously proposed by Kaitwanidvilai

and Parnichkun to design a fixed-structure H∞ loop shaping controller for a pneumatic servo-

system [12]. Even though GA is a well-defined technique for optimization problems, and widely

used in controllers design, it still has limitations due to its stochastic searching characteristic and

complex computation that make it slow convergence to global optimum. Particle swarm

optimization (PSO) is one of the most recent developed evolutionary techniques initially

proposed by Kennedy and Eberhart in 1995 [14]. PSO is based on a model of social interaction

among independent particles. It uses social knowledge to find the global maximum or minimum

of a generic function. It is fast and easy to implement because of its oriented searching and

simple calculation [15,16]. PSO has recently been used to design advanced control algorithms.

Mukherjee and Ghoshal used PSO to design a fuzzy PID controller for an automatic voltage

regulator (AVR) system [15]. Kao et al. used PSO to design a self-tuning PID controller for a

slider-crank mechanism [16]. Chang used PSO to design a PID controller for chaotic

synchronization [17].



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In this paper, PSO is used to search for parameters of a structure-specified H∞ loop shaping

controller. In the method, a nominal model of the system is firstly shaped by a pre-compensator

and a post-compensator to achieve a desired open loop shape. A structure-specified controller is

then defined. Finally, PSO is used to search for parameters of the controller so that the controller

is admissible and the H∞ norm from exogenous inputs to controlled outputs is minimized. The

proposed algorithm is practically applied to the balancing control of Bicyrobo, a bicycle robot

with gyroscopic stabilizer. By neglecting forces generated from moving forward and steering, a

simplified dynamics model of Bicyrobo is derived using Lagrange method. The un-modeled

dynamics, parameter variations, and external disturbances make the system complicate and

require a robust controller. The simulation and experimental results show robustness and

efficiency of the designed controllers in compared with the PD, and the conventional H∞ loop

shaping controllers.

The remainder of this paper is organized as follows. In Section 2, a prototype of Bicyrobo which

is used as a platform to test control algorithm is described. Section 3 explains a systematic procedure for designing the proposed controller. Simulation and experimental results are

presented in Section 4 and Section 5. Section 6 finally concludes the paper.

2. Configuration and dynamics model of Bicyrobo

2.1 Configuration of Bicyrobo

A bicycle robot, Bicyrobo, has been developed at Mechatronics Laboratory, Asian Institute of

Technology (AIT), as a platform to test performance of the proposed control algorithm. The

system is modified from a regular size bicycle. Fig. 1 shows a photograph of Bicyrobo. The robot

is designed so that it can carry loads, go forward, and turn left or right without falling down.

Fig. 1. Photograph of Bicyrobo

Bicyrobo is equipped with the following components: a flywheel with weight of 8.1kg and

diameter of 380mm for creating precession torque; a 48V-200W-3600rpm DC servo motor for

rotating the flywheel around its spinning axis; a 48V-200W-3600rpm DC servo motor with 5:1



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chain transmission system for controlling the flywheel control axis; a 12V-15W-10rpm gear box

DC motor for steering Bicyrobo to turn left or right; a 12V-35 W-100rpm gear box DC motor for

driving Bicyrobo to move forward; a PCM-3350 embedded PC running at 300MHz as a central

controller with extended A/D and D/A boards; a VG400CC vertical-gyro sensor for measuring

the lean angle of Bicyrobo; an EB6-CWZ encoder for measuring angular position of the flywheel

control axis; and signal conditioning circuits. The dimension of Bicyrobo is about 1.65m in

length, 0.43m in width, and 1.14m in height. Its total weight is 51.2kg.

Fig. 2. Hardware configuration of balancing control system

The hardware configuration of the system is shown in Fig. 2. Program for implementing control

algorithms is written in C and run on PC/104 under DOS to ensure real time control.

2.2 Dynamics model of Bicyrobo

A complete dynamics model of a bicycle as derived by Sharp in 1971 [18] is complicated since

the system has many degrees of freedom, and not suitable for control purpose. Dynamics model

of a bicycle is basically based on equilibrium of gravity forces and centrifugal forces. In this

paper, a simplified dynamics model of the bicycle for balancing control purpose is derived using

Lagrange method by neglecting forces generated from moving forward and steering of the

bicycle. This model is based on the one proposed by Gallaspy in [3]. Several assumptions are

made to simplify the system as follows:

• The system is simplified to two rigid body links. The first link is the bicycle frame that has

one degree of freedom (DOF) on leaning angle only, rotation around Z axis. The second link

is the flywheel which has three DOFs including rotations around X1, Y1, and Z axes (Fig. 3).

• The flywheel is assumed to have constant speed ω. Center of gravity of the flywheel is fixed

relative to the bicycle frame.

The principle for balancing using flywheel is explained as follows: When the flywheel rotates

with a constant speed around Y1 axis, if we control angular position of the flywheel around X1



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axis, angular momentum on Z1 axis will generate a torque. This torque is called precession torque

generated by gyroscopic effect, and it is used to keep Bicyrobo balance.

Fig. 3. Diagram of reference coordinates of Bicyrobo: (a) side view; (b) front view

Diagram of the simplified reference coordinates is shown in Fig. 3, where B and F denote bicycle

and flywheel centers of gravity, respectively. The lean angle of the bicycle around Z axis is

defined as θ , and the angular position of the control axis of the flywheel around X1 axis is defined

as φ. The angular velocity of the bicycle around Z axis is defined as θ & , and the angular velocity

of the flywheel around its control axis (X1 axis) is defined as ϕ & . Since the flywheel center of

gravity does not move related to the bicycle center of gravity, absolute velocities of B and F are

given by

b bv hθ =&

(1)

f f v hθ = & (2)

where hb, h f are the height of bicycle center of gravity, and the height of flywheel center of

gravity, respectively. To derive the dynamics model of the system, Lagrange equation in [19] is

used

i

i i i

d T T V Q

dt q q q

⎧ ⎫∂ ∂ ∂− + =⎨ ⎬

∂ ∂ ∂⎩ ⎭&(3)

where T is system total kinetic energy, V is system total potential energy, Qi is external forces,

and qi is generalized coordinate. V and T are determined, and represented by the following

equations.

b b f f V m gh cos m gh cosθ θ = + (4)

2 2 2 2 2 21 1 1 1( ) ( ) ( ) ( os )

2 2 2 2b b f f b r p r T m v m v I I I sin I cθ ϕ θ ϕ θ ϕ ⎡ ⎤= + + + + +⎣ ⎦

& & &&



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2 2 2 2 2 2 2 21 1 1 1( ) ( ) ( ) ( os )

2 2 2 2b b f f b r p r T m h m h I I I sin I cθ θ θ ϕ θ ϕ θ ϕ ⎡ ⎤= + + + + +⎣ ⎦

& & & & && (5)

where I p is flywheel polar moment of inertia and I r is flywheel radial moment of inertia. m g and

m f are bicycle and flywheel masses, respectively. I b is bicycle moment of inertia.

Fig. 4. Moments of inertia assignment for flywheel: (a) side view; (b) front view

For qi = θ , the Lagrange equation becomes

d T T V Q

dt θ

θ θ θ

∂ ∂ ∂⎧ ⎫− + =⎨ ⎬

∂ ∂ ∂⎩ ⎭&(6)

Using equations (4)-(6), the following equation is derived.

2 2 2 2 2 ( )

( )

b b f f b p r p r

b b f f p

m h m h I I sin I cos sin cos I I

g m h m h sin I cos

θ ϕ ϕ ϕ ϕ θϕ

θ ωϕ ϕ

⎡ ⎤+ + + + + −⎣ ⎦

− + =

&& & &

&

(7)

For qi = φ, the Lagrange equation becomes

d T T V Q

dt ϕ

ϕ ϕ ϕ

⎧ ⎫∂ ∂ ∂− + =⎨ ⎬

∂ ∂ ∂⎩ ⎭&(8)

Using equations (4), (5), and (8), the following equation is derived.

2 ( )r p r m p m I I I sin cos T I cos Bϕ θ ϕ ϕ ωθ ϕ ϕ − − = − −& &&& & (9)

where Bm is DC motor viscosity coefficient. The dynamics of DC motor with a 5:1 ratio chain

transmission system follows the equations.

5m mT K i= (10)

e

diU L Ri K

dt ϕ = + + & (11)

where K m, K e are torque and back emf constants of the motor, respectively. R and L are resistance

and inductance of the motor, respectively. T m is torque generated by the motor. By substitution of



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equation (10) into equation (9), and linearization (7) and (9) around the equilibrium point, the

following equations are obtained.

2 2 ( ) 0b b f f b r b b f f pm h m h I I g m h m h I θ θ ωϕ ⎡ ⎤+ + + − + − =⎣ ⎦&& & (12)

5 0r p m m I I B K iϕ ωθ ϕ − + − =&&& & (13)

Define [ ]' x θ θ ϕ ϕ = & & , y θ = , and u U = . The dynamics model of the system in state-space

representation by combining (11), (12), and (13) is shown by the following equation.

x Ax Bu

y Cx Du

= +⎧⎨

= +⎩

&(14)

where

2 2 2 2

0 1 0 0

( ) 0 0

50

0 0

b b f f p

b b f f b r b b f f b r

p m m

r r r

e

g m h m h I m h m h I I m h m h I I

A I B K

I I I

K R

L L

ω

ω

⎡ ⎤⎢ ⎥

+⎢ ⎥⎢ ⎥+ + + + + +⎢ ⎥

= ⎢ ⎥− −⎢ ⎥

⎢ ⎥⎢ ⎥

− −⎢ ⎥⎣ ⎦

(15)

[ ]'

0 0 0 1/ B L= , [ ]1 0 0 0C = , and D = [0]. (16)

3.

Particle swarm optimization-based structure-specified H∞

loop shaping control

3.1 H ∞ loop shaping control

H∞ loop shaping control method is an effective approach for designing a robust controller. It is

based on the configuration shown in Fig. 5. Let define the nominal model of a system as P , and

the shaped plant with a pre-compensator, W 1, and a post-compensator, W 2, as P s, thus,

1

2 1 s P W PW M N −= = % % = s s

s s

A B

C D

⎡ ⎤⎢ ⎥⎣ ⎦

(17)

where s A , s B , sC , and s D are matrices of the shaped plant in state-space representation,%

and

N % are the normalized left coprime factors of P s. By assuming that the shaped plant is perturbed

by unstructured uncertainties Δ and N Δ , the perturbed plant, P Δ , thus becomes

1( ) ( ) P M M N N −Δ = + Δ + Δ% % (18)



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It is proved from the small gain theorem that the shaped plant, P s, is stable with all unknown but

bound uncertainties [ ]M N ε ∞

Δ Δ < if and only if there exists an admissible controller, K ∞,

such that

( )1 1 1/ zw s

I T I P K M

K γ ε

− −∞

∞ ∞ ∞

⎡ ⎤= + ≤ =⎢ ⎥

⎣ ⎦

% (19)

Fig. 5. Robust stabilization with respect to coprime factor uncertainties

Minimization of γ (maximization of ε) results in maximization of robustness of the system. A

procedure called H∞ loop shaping controller design proposed by McFarlane and Glover [8] and

further developed by Tang et al. [20] is summarized as followings.

Step1: The nominal plant, P , is shaped using a pre-compensator, W 1, and a post-compensator, W 2,

to achieve a desired open loop shape. W 1 is used to achieve tracking performance and disturbance

attenuation and W 2 is used to attenuate sensor noise. W 1 and W 2 are selected so that P s contains nohidden modes, and has the following properties:

• To achieve good tracking performance and good disturbance rejection, large open loop gain

at a low frequency range is required.

• To achieve good robust stability and sensor noise rejection, small open loop gain at a high

frequency range is required.

When W 1 and W 2 are selected, the value of opt γ is evaluated using equation (20) where maxλ is the

maximum eigenvalue.

[ ]1/ 2

max1 ( )opt ZX γ λ = + (20)

where Z and X are the solutions of the two following Riccati equations

1 1 1 1( ) ( ) 0T T T T T

s s s s s s s s s s s s A B S D C Z Z A B S D C ZC R C Z B S B− − − −− + − − + = (21)

1 1 1 1( ) ( ) 0T T T T T T

s s s s s s s s s s s s A B S D C X X A B S D C XB S B X C R C − − − −− + − − + = (22)



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where T

s s R I D D= + and T

s sS I D D= + (23)

W 1 and W 2 are adjusted until a satisfied opt γ is achieved. If opt γ is too large ( opt γ > 4), W 1 and W 2

are incompatible and should be adjusted.

Step 2: Select 1

opt opt ε ε γ −< = , and then synthesize a sub-optimal controller, K ∞, that satisfies (19).

A sub-optimal controller which satisfies this condition is

K ∞ =2 1 2 1( ) ( ) ( )T T T T

s s s s s s

T T

s s

A B F Q ZC C D F Q ZC

B X D

γ γ − −⎡ ⎤+ + +⎢ ⎥

−⎣ ⎦(24)

where 1( )T T

s s s F S D C B X −= − + and 2(1 )Q I XZ γ = − + (25)

Step 3: The final controller is

1 2 K W K W ∞= (26)

Fig. 6. Block diagram of H∞ loop shaping control

The procedure for designing H∞ loop shaping controller is straightforward, and easy to follow. It

is a very useful procedure for systems with unstructured uncertainties. However, the final

controller obtained by using this method is high order, which is difficult to implement in practical

applications. A procedure for designing a lower order but robust controller using PSO is

proposed.

3.2 Particle swarm optimization algorithm

PSO is one of the most recent evolutionary techniques. The method was developed by simulation

of simplified social model, where each population is called a swarm. In PSO, multiple solutions

are together and collaborate simultaneously. Each candidate, called a particle, flies through

problem space to look for the optimal position, similar to food searching of bird swarm. A

particle adapts its position based on its own knowledge, and knowledge of neighboring particles.



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The algorithm is initialized with a population of random particles. It searches for the optimal

solution by updating particles in generations. Fig. 7 shows the flowchart of PSO algorithm.

Fig. 7. Flowchart of PSO

Let the search space be N -dimensional, then the particle i is represented by an N -dimensional

position vector, 1 2( , ,..., )i i i iN x x x= . The velocity is represented also by an N -dimensional velocity

vector,1 2

( , ,..., )i i i iN v v v v= . The fitness of particles is evaluated by the objective function of the

optimization problem. The best previously visited position of particle i is noted as its individual

best position, 1 2( , ,..., )i i i iN P p p p= . The position of the best individual of the whole swarm is

noted as the global best position, 1 2( , ,..., ) N G g g g = . At each step of searching process, the

velocity of particle and its new position are updated according to the following two equations

[21].

1 1 2 2( 1) . ( ) . .( ( ) ( )) . .( ( ) ( ))i i i i iv k w v k c r P k x k c r G k x k + = + − + − (27)

( 1) ( ) ( )i i ik x k v k + = + (28)

where w, called inertia weight, controls the impact of previous velocity of the particle. 1r , 2r are

random variables in the range of [0,1]. 1c , 2c are positive constant parameters called acceleration

coefficients. The value of each component in v is limited to the rangemax max

[ , ]v v− to control

excessive roaming of particles outside the search space.

3.3 Structure-specified H ∞ loop shaping controller design



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In this part, we propose a new method using PSO to design a structure-specified H∞ loop shaping

controller for a single-input single-output (SISO) system. However, the algorithm can be

extended to multiple-input multiple-output (MIMO) systems. The procedure for control algorithm

design is described below.

3.3.1 Weighting functions selection

Since the algorithm is based on H∞ loop shaping method, the plant is firstly shaped by using pre-

compensator and post-compensator. In this paper, lead/lag type compensators are used for

weighting functions.

11 1

1

sW K

s

α

β

+=

+(29)

22 2

2

sW K

s

α

β

+=

+(30)

The shaped plant, thus, becomes

2 1 s P W PW = (31)

3.3.2 Structure-specified controller definition

The structure-specified controller, K(s), is defined as follows.

1

1 0

1

1 0

( ) ...( )

( ) ...

m m

k m m

n n

k n

N s a s a s a K s

D s s b s b

−−

−−

+ + += =

+ + +(32)

The structure-specified controller can be in any forms such as PID, first order, second order

controllers, etc., by selecting suitable values of m and n.

3.3.3 Objective function definition

The structure-specified H∞ loop shaping controller design problem can be defined as the problem

of finding the parameters of all admissible controllers represented by equation (32) such that the

H∞ norm presented by equation (19), zwT ∞

, is minimized.

Since 1 2( ) K s W K W ∞= , therefore 1 1

1 2( ) K W K s W − −∞ = . Then

[ ]1 1 1( ) ( ) zw s s s

I I T I P K M I P K I P

K K

− − −∞ ∞∞

∞ ∞∞ ∞

⎡ ⎤ ⎡ ⎤= + = +⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦

%

[ ]1 1 1

cos 1 21 1

1 2

( ( ) )( )

t zw s s

I J T I PW K s W I P

W K s W

− − −

− −∞

∞

⎡ ⎤= = +⎢ ⎥

⎣ ⎦(33)



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The equation (33) is defined as the objective function of the optimization problem and it can be

easily evaluated using robust control toolbox in MATLAB.

3.3.4 Particle swarm optimization-based design

Once an objective function and a structure of the controller are defined, the procedure, using PSO

to solve this optimization problem, is described as followings:

Step 1: Set particle i to 1 2 0 1 0 1( , ,..., ) ( , ,..., , ,..)i i i iN x x x x a a b b= = , the number of parameters of the

controller in equation (32) is the dimension of particle, N = m + n + 1. Define maximum number

of iterations as GenMax.

Step 2: Initialize a random swarm of H particles as[ ]1 2 ... H x x x , when the swarm size is set

to H .

Step 3: For each generation, evaluate objective function for each particle using the objective

function shown by equation (33), and determine individual best, ( )i P k , and global best, ( )G k .

Step 4: Update the velocity of particle and its new position using equations (27) and (28).

Step 5: When the maximum number of iterations is arrived, stop the algorithm. Otherwise go to

step 3.

4. Simulation results

The proposed algorithm is applied to design controllers to balance Bicyrobo, the bicycle robot

which has the analytical dynamics model as described in Section 2. The algorithm is

programmed, and run using MATLAB. The obtained controllers are simulated using MATLABSimulink. Parameters of Bicyrobo are identified as shown in Table 1.



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Table 1

Parameters of Bicyrobo

Parameters Value Unit

mf 8.1 kg

m b 43.1 kg

hf 0.86 m

h b 0.8 m

I b 27.584 kg.m2

I p 0.215926 kg.m2

Ir 0.112304 kg.m2

ω 157.08 rad/s

L 0.0006 H

R 0.41 Ω

Bm 0.000253 kg.m2/s

K m 0.119 Nm/A

K e 0.1184 V.s

g 9.81 m/s2

By substitution of these parameters into equations (14) – (16), the nominal transfer function of

Bicyrobo is described as

4 3 2

( ) 4887

( ) 683.3 1208 109700 6949

s P

U s s s s s

θ = =

+ + + −(34)

where U is input voltage to the DC motor that controls flywheel control axis, θ is output lean

angle of Bicyrobo. The weighting function, W 1, is selected by some trials for shaping the plant.

W 2 is selected as identity matrix with an assumption that sensor noise is negligible. W 1 and W 2 are

shown by the following equations.

1

0.0940.6

0.085

sW

s

+=

+and 2 1W = (35)

By substitution of W 1 and W 2 into equation (17), and using equations (20) – (23), finally

opt γ =1.5216 is obtained. The stability margin is opt ε = 0.6572. We select 0.6363 opt ε ε = < . Using

equation (24) – (26), the full order controller is obtained as

5 4 3 2

6 5 4 3 2

1275 8.695 5 5.151 5 1.359 8 2.435 7 1.091 6( )

715.7 2.355 4 2.789 5 3.802 6 6.591 5 2.872 4

s e s e s e s e s e K s

s s e s e s e s e s e

+ + + + +=

+ + + + + +(36)

The full order controller represented by equation (36) is sixth order, which is difficult to

implement in reality.



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4.1 First order controller design

A first order controller is selected as a structure-specified controller of the following form

01

0

( )a

K s s b

=+

(37)

The following parameters are selected: Swarm size = 20, the dimension of each particle in the

first order controller is two ( 0a , 0b ), 1 2 2c c= = , GenMax = 100. PSO is used to search for

parameters of the controller ( 0a , 0b ). In the PSO algorithm, the weight, w, is automatically

changed so that the algorithm converges slowly to the optimal solution at the end of searching

progress to avoid premature convergence. The initial weight is set to w = 0.95, and the final

weight is set to w = 0.4. Velocity limit max max[ , ]v v− is set to [-100,100].

The algorithm is run on ten trials, and in all cases the same value of cost function cost opt J γ = =

1.8365 ( opt ε = 0.5445) is obtained. The obtained controller is shown by equation (38). Fig. 8(a)

shows the convergence of the algorithm of three simulations. Fig. 8(b) shows the step response of

the closed loop system using the obtained controller.

1

135.2( )

4.63 K s

s=

+(38)

Fig. 8. (a) Cost function value versus generation number, (b) Step response of first order

controller

4.2 Second order controller design

A second order controller is selected as

1 02 2

1 0

( )a s a

K s s b s b

+=

+ +(39)



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The dimension of each particle in the second order controller is four ( 0a , 1a , 0b , 1b ). PSO is used

to search for parameters of the controller ( 0a , 1a , 0b , 1b ). The same parameters of PSO as

defined in the first order controller are used. The algorithm is run on ten trials, and the controller

as shown by equation (40) withcost opt J γ = = 1.798 ( opt ε = 0.55617) is obtained. Fig. 9(a) shows

the convergence of the algorithm of three simulations. Fig. 9(b) shows the step response of theclosed loop system using the obtained controller.

2 2

129.7 499.6( )

6.835 16.183

s K s

s s

+=

+ +(40)

Fig. 9. (a) Cost function value versus generation number, (b) Step response of second order

controller

4.3 Comparison

Step responses of the closed loop system using PD, first order, second order, and full order H∞

loop shaping controllers are compared in this Section. By tuning parameters K P and K D of PD

controller, a satisfied step response with the about same response time as the proposed structure-

specified controllers is obtained. This PD controller is expressed by

30 2.5( ) PD

s K s

s

+= (41)

The comparison is shown in Fig. 10. These simulations show that step responses of the systemusing conventional H∞ loop shaping controller and the proposed structure-specified H∞ loop

shaping controllers are similar. They are both better than the system using PD controller.



16

Fig. 10. (a) Step responses using PD, first order, and full order H∞ loop shaping controllers, (b)

Step responses using PD, second order, and full order H∞ loop shaping controllers

4.4 Robustness

Robustness of the proposed first order controller and PD controller is compared in this Section.

The nominal model expressed in equation (34) is obtained at a nominal mass of Bicyrobo

including flywheel at 51.2kg, and a nominal flywheel speed at 157.08rad/s. To test the robustness

of the proposed first order controller, varied load is applied on the bicycle, and the flywheel

speed is varied. The following cases are tested.

Case 1: The load is added with an additional 10kg, and the flywheel speed is reduced to

104.72rad/s. The system model thus becomes

1 4 3 22696

683.3 1102 37550 6857 P

s s s sΔ =

+ + + −(42)

Case 2: The load is added with an additional 20kg, and the flywheel speed is reduced to

104.72rad/s. The system model thus becomes

2 4 3 2

2299

683.3 1093 31400 6793 P

s s s sΔ =

+ + + −(43)

Step responses of the closed loop system using the PD, and the proposed first order controllers

from case one and case two are shown in Fig. 11(a) and Fig. 11(b), respectively. These

simulations show that the proposed controller is robust to the parameter variations in both cases

while the system becomes unstable by using PD controller.



17

Fig. 11. Step responses from PD, and proposed first order controllers with parameters variation:

(a) Case 1; (b) Case 2

5. Experimental Results

The experiments are conducted on Bicyrobo with the hardware setup as explained in Section 2.

The diagram for implementing the controllers on Bicyrobo is shown in Fig. 12. The program is

coded in C and implemented on PC/104, a 300MHz CPU embedded computer, with sampling

time of 12.5ms. The lean angle of Bicyrobo is read from vertical gyro-sensor VG400CC via an

A/D extended board. The calculated output voltage from a D/A extended board is sent to an

external PWM and Driver circuit to control DC motor of the flywheel control axis. The angular

position of the flywheel control axis (φ) is also read from an E6B-CWZ encoder. In case that the

measured lean angle is zero but Bicyrobo is not at the balancing position, the reference must be

changed to eliminate continuous rotation in one direction of the flywheel.

Fig. 12. Diagram for implementing controllers

Various experiments are conducted to evaluate balancing performance and robustness of the

proposed controllers. The first set of the experiments is tested on the system using the PD and the

proposed first order controllers at a zero forward speed of Bicyrobo without applied masses. The

controllers in equations (38) and (41) are converted to discrete forms (C(z)) and then coded in C



18

on the embedded PC for testing. The lean angles of the system are saved in a file while program

is running. Fig. 13 depicts balancing performance from the lean angle of Bicyrobo. The results

show that the proposed controller has better balancing performance than the conventional PD

controller.

Fig. 13. Balancing performance

Fig. 14. Balancing performance with load variations

In order to show that the proposed controllers is robust to parameter variations, iron masses of

4kg and 8kg are applied on the system at a zero forward speed of Bicyrobo. The experiments on

Bicyrobo using the proposed first order controller are tested. The experimental results are shown

in Fig. 14. In both cases, the system is stable against these parameter variations. Photographs of

Bicyrobo taken during the experiments with 8kg applied mass are shown in Fig. 15.



19

Fig. 15. Photographs during experiments with 8kg of applied mass: (a) left view; (b) right view;

(c) front view; (d) back view

6. Conclusion

This paper proposed a PSO-based structure-specified H∞ loop shaping controller design method

to control balancing of Bicyrobo. By using the proposed method, the first order and second order

controllers were designed with the stability margins, εopt , of 0.5445 and 0.55617, respectively.

These stability margins are quite closed to stability margin of the sixth order controller (εopt =

0.6363) designed using the conventional H∞ loop shaping method. The simulation results shown

that performance of the closed loop system using the proposed controllers and full order

controller were similar, they are both better than performance of the closed loop system using PD

controller. The simulation results also shown that the closed loop system was robustly stable to

parameter variations using the proposed controllers, while it was unstable using the PD

controller. The experimental results on Bicyrobo proved that the proposed first order controller

achieved better balancing performance with lean angle less than 0.5 degree in compared to thePD controller with maximum lean angle about 1 degree. Experimental results on applying masses

of 4kg and 8kg on the system using the proposed first order controller shown that the system had

good balancing performance and robustness with lean angle less than 1 degree against the load

variation.

In conclusion, the proposed controllers achieved two important requirements: simple in structure

and good robust performance against uncertainties. The design method is straightforward,



20

systematic, and simple by using PSO algorithm. Therefore, it is easy to apply to other systems.

The algorithm was demonstrated on a SISO unstable system. But it can be extended to MIMO

systems.



21

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Documents

Balancing Control of Bicyrobo by Particle Swarm Optimization-based Structure-specifield Hoo Loop Shaping Control