research article the lqr controller design of two-wheeled ...separates the wheel from the pendulum...

Research ArticleThe LQR Controller Design of Two-Wheeled Self-BalancingRobot Based on the Particle Swarm Optimization Algorithm

Jian Fang

Institute of Electrical Engineering, Jilin Teachers’ Institute of Engineering and Technology, Changchun 130052, China

Correspondence should be addressed to Jian Fang; [email protected]

Received 25 February 2014; Revised 5 May 2014; Accepted 7 May 2014; Published 11 June 2014

Academic Editor: Her-Terng Yau

Copyright © 2014 Jian Fang. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The dynamics model is established in view of the self-designed, two-wheeled, and self-balancing robot.This paper uses the particleswarm algorithm to optimize the parameter matrix of LQR controller based on the LQR control method to make the two-wheeledand self-balancing robot realize the stable control and reduce the overshoot amount and the oscillation frequency of the system atthe same time. The simulation experiments prove that the LQR controller improves the system stability, obtains the good controleffect, and has higher application value through using the particle swarm optimization algorithm.

1. Introduction

The two-wheeled and self-balancing robot belongs to amultivariable, nonlinear, high order, strong coupling, andunstable essential motion control system, and it is a typ-ical device of testing various control theories and controlmethods; therefore, the research has great theoretical andpractical significance. Because it has the advantages of simplestructure, stable running, high energy utilization rate, andstrong environmental adaption, it has the broad applicationprospects whether in the military field or in the civilian field.

Since 1980s, the scholars of various countries haveconducted the system research on the two-wheeled self-balancing robot. The two-wheeled and self-balancing robotcontrol system based on the fuzzy control can overcome theinstability and nonlinear nature of the system, but it relieson the expert’s experience too much [1, 2]. The optimal LQRcontroller is designed on the basis of establishing the systemstructuremodel; the correctness and effectiveness of the LQRcontroller are verified, but it is difficult to determine theweighted matrix 𝑄 and 𝑅 [3–5]. The genetic algorithm issuccessfully applied to the parameter optimization of the LQRcontroller of the inverted pendulum system, and it achievesthe good control effect. However, the parameters are difficultto adjust, and it is easy to fall into the local optimization [6].

This paper concerns the self-designed and two-roundself-balancing robot as the research object, which usesthe Newtonian mechanics equation method and the linearmethod near the balance point to establish the linearizedmathematical model of the system. In view of the mathemat-ical model of the system, LQR controller is designed basedon the particle swarm optimization and makes full use ofthe searching capability of the particle swarm algorithm tooptimize the matrix 𝑄 and matrix 𝑅 of the LQR controller. Itgains the global optimal solution of matrix 𝑄 and 𝑅 of theLQR controller so as to design the optimal state feedbackcontrol matrix 𝐾 and overcome the disadvantages of relyingon the experience and the trial and error in the selection ofmatrix𝑄 and𝑅 of the general LQR control design,making upfor the inadequacy of big workload. This method has bettercontrol effect by simulation tests and comparison.

2. The Dynamics Model of the Two-WheeledSelf-Balancing Robot

The two-wheeled and self-balancing robot structure ismainlycomposed of the body and the two wheels, and the robot isthe coaxial two wheels, driven by the independent motor;the parameters of quality, moment of inertia, and radius of

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 729095, 6 pageshttp://dx.doi.org/10.1155/2014/729095

2 Mathematical Problems in Engineering

Revolver

Hfl

Hl

Pl

Cl

MW

𝜃W

x

Figure 1: The force analysis of revolver.

the two wheels are regarded as the same, so the body centerof gravity is inverted above the axletree and it keeps balancethrough sports.

The two-wheeled and self-balancing robot can beregarded as the vehicle-mounted inverted pendulum, so thedynamic system analysis process is more complex.This paperseparates the wheel from the pendulum analysis first in theprocess of modeling, and then it deduces the dynamics stateequation of the two-wheeled self-balancing robot throughthe simultaneous two parts [7, 8].

The two wheels are regarded as the research object,Figure 1 is a diagram revolver force analysis. According tothe revolver force equation can be obtained in the followingaccording to the Newton’s law [9] and the rotational torqueformula [10]:

𝑀𝑤�� = 𝐻

𝑓𝑅− 𝐻𝑅,

𝐼𝜔

𝜃𝜔= 𝐶𝑅− 𝐻𝑓𝑅⋅ 𝑅.

(1)

The right wheel force equation is as follows:

𝑀𝑤�� = 𝐻

𝑓𝐿− 𝐻𝑅,

𝐼𝜔

𝜃𝜔= 𝐶𝐿− 𝐻𝑓𝐿⋅ 𝑅.

(2)

After finishing, it can obtain

2 (𝑀𝑤+𝐼𝜔

𝑅2) �� =

𝐶𝑅+ 𝐶𝐿

𝑅− (𝐻𝑅+ 𝐻𝐿) . (3)

Among them, 𝑀𝑤is the weight of the wheel; 𝐼

𝜔is the

moment of inertia of the wheel; 𝑅 is the radius of the wheel;�� is the wheel acceleration of𝑋 axis; 𝐶

𝑅and 𝐶

𝐿are the right

and left wheel torque; 𝐻𝐿and 𝐻

𝑅are the 𝑍 axis forces of

the left and right wheels with the car body; 𝐻𝑓𝑅

and 𝐻𝑓𝐿

are the interatomic forces of the right and left wheels withthe ground; and 𝜃

𝜔is the angle of wheel around the 𝑍 axis

direction.

HL + HR

PL + PR

l 𝜃p𝜃p

x

𝜃2

MPg ΣFxp

Figure 2: The analysis of car body force.

The car body of the two-wheeled and self-balancing robotis modelled as an inverted pendulum; the car body forceanalysis is shown in Figure 2. Using Newton’s second law, thehorizontal force is as follows:

∑𝐹𝑥= (𝐻𝑅+ 𝐻𝐿) − 𝑀

𝑝𝑙 𝜃𝑝sin 𝜃𝑝−𝑀𝑙 𝜃

𝑝cos 𝜃𝑝

= 𝑀𝑝⋅ ��𝑝= 𝑀𝑝(𝑥 + 𝑙 sin 𝜃

𝑝) .

(4)

Among them, 𝑥𝑝is the displacement of the car body

centre of gravity relative to the ground,

𝑥𝑝= 𝑥 + 𝑙 sin 𝜃

𝑝. (5)

Using Newton’s second law, the force of the verticaldirection of the car body is as follows:

∑𝐹𝑥𝑝= 𝑀𝑝𝑔 sin 𝜃 − (𝑃

𝑅+ 𝑃𝐿) sin 𝜃

𝑝+ (𝐻𝑅+ 𝐻𝐿) cos 𝜃

𝑝

= 𝑀𝑝(𝑥 + 𝑙 sin 𝜃

𝑝) cos 𝜃

𝑝.

(6)

The sum of the torques of the car body mass center is asfollows:

∑𝑀0= 𝐼𝑝

𝜃𝑝=(𝑃𝑅+ 𝑃𝑙)

sin 𝜃𝑝

−(𝐻𝐻+ 𝐻𝐿)

cos 𝜃𝑝

. (7)

On the small angle scope, 𝜃2𝑝≈ 0, sin 𝜃

𝑝≈ 𝜃𝑝, cos 𝜃

𝑝≈

1, the linearized equations are gotten after the linearizationprocess:

(𝑀𝑃+ 2𝑀𝑊

𝐼𝑊

𝑅2) �� =

𝐶𝑅+ 𝐶𝐿

𝑅− 2𝑀𝑃𝑙𝜃𝑝,

(2𝑀𝑃𝑙2

+ 𝐼𝑝) 𝜃𝑝= 𝑀𝑝𝑔𝑙𝜃𝑝−𝑀𝑝𝑙��.

(8)

Among them, 𝜃𝑝is the angle of the car body deviating

from the 𝑍 axis direction; 𝐼𝑝is the moment of inertia of the

car body;𝑀𝑃is the weight of the car body; and 𝑙 is the height

of which the car body is apart from the shaft. The output

Mathematical Problems in Engineering 3

torque of the wheel is 𝐶𝑅= 𝐶𝐿= 𝐼𝑅(𝑑𝜔/𝑑𝑡) = (𝑘

𝑚/𝑅)𝑈𝑎−

(−𝑘𝑚𝑘𝑒/𝑅) 𝜃𝜔; the state equation of the two-wheeled self-

balancing robot is obtained:

[[[

[

��

��𝜃𝑝

𝜃𝑝

]]]

]

=

[[[[[[[

[

0 1 0 0

02𝑘𝑚𝑘𝑒(𝑀𝑝𝑙𝑟 − 𝐼𝑝−𝑀𝑃𝑙2

)

𝑅𝑟2𝐴

𝑀2

𝑝𝑔𝑙2

𝐴0

0 0 0 1

02𝑘𝑚𝑘𝑒(𝑟𝐵 −𝑀

𝑝𝑙)

𝑅𝑟2𝐴

𝑀𝑝𝑔𝑙𝛽

𝐴0

]]]]]]]

]

[[[[[

[

𝑥

��

𝜃𝑝

𝜃𝑝

]]]]]

]

+

[[[[[[[

[

0

2𝑘𝑚(𝐼𝑝+𝑀𝑝𝑙2

−𝑀𝑝𝑙𝑟)

𝑅𝑟𝐴0

2𝑘𝑚(𝑀𝑝𝑙 − 𝑟𝐵)

𝑅𝑟𝐴

]]]]]]]

]

𝑈𝑎.

(9)

Among them, 𝐴 = [𝐼𝑝𝛽 + 2𝑀

𝑝𝑙2

(𝑀𝑤+ (𝐼𝑤/𝑟2

))]; 𝐵 =

(2𝑀𝑤+ (2𝐼𝑤/𝑟2

) + 𝑀𝑝).

The output equation is as follows:

𝑦 = [0 0 1 0]

[[[[[

[

𝑥

��

𝜃𝑝

𝜃𝑝

]]]]]

]

. (10)

3. The LQR Controller ParameterOptimization Based on the ParticleSwarm Optimization Algorithm

3.1. The Design of the Self-Balancing Robot LQR Controller.The LQR method is the most mature controller designmethod in the development of modern control theory [11];LQR optimal control is to seek the control amount 𝑢∗(𝑡) tomake the system reach the steady state and guarantee theperformance index 𝐽 to take the minimum value:

𝐽 = ∫

∞

0

(𝑋𝑇

𝑄𝑋 + 𝑢𝑇

𝑅𝑢) 𝑑𝑡. (11)

Among them, 𝑢∗ = −𝑅−1𝐵𝑇𝑃𝑥 = −𝐾𝑥, and 𝑃 is the solu-tion of the algebraic equation𝑃𝐴+𝐴𝑇𝑃+𝑄−𝑃𝐵𝑅−1𝐵𝑇𝑃 = 0 ofthe matrix Riccati [12]. In (13), the matrix𝑄 and matrix 𝑅 aremutually restricted, and the size of the𝑄 value is proportionalto the anti-interference ability of the system; increasing the𝑄value, the anti-interference ability of the system is enhanced,and the adjustment time of the system is shortened. However,at the same time, the oscillation of the system is strengthened,and the consumption of energy increases. The increase of 𝑅

value makes the energy consumed by the system less, butthe adjustment time increases. Therefore, the design key isto find the right weight matrix 𝑄 and matrix 𝑅. As longas we make sure of the matrix 𝑄 and matrix 𝑅, the statefeedback matrix 𝐾 is the only confirmation. However, theselection of the 𝑄 matrix and 𝑅 matrix entirely depends onthe experience and trial and error method in the process ofLQR controller design, so the subjectivity is larger, resultingin the imperfection of the controller design and affecting thecontrol effect.

3.2. The Parameter Optimization Principle of the LQR Con-troller Based on the Particle Swarm Optimization Algorithm.As a new optimization algorithm has been developed inrecent years, the particle swarm optimization algorithm isabbreviated as the PSO. The particle swarm optimizationalgorithm is a kind of evolutionary algorithms, and it startsfrom the random solutions and searches for the optimalsolution through the iterative algorithm [13].

Assuming 𝑋 particles are composed of a group in 𝑛dimensional space, among them, the position and velocityof 𝑖 particle in the space are 𝑥

𝑖= (𝑥𝑖1, 𝑥𝑖2, . . . , 𝑥

𝑖𝐷), V𝑖=

(V𝑖1, V𝑖2, V𝑖3, . . . , V

𝑖𝐷), 𝑖 = 1, 2, . . . , 𝑚, the best position that

the 𝑖 particle experiences is denoted by 𝑝bset(𝑖), and the bestposition that all the particles in the group experience isdenoted by 𝑔bset(𝑖). The whole particle swarm updates thevelocity and position through tracking the individual extremevalue and the optimal value [14]. The particle optimizationprocess is expressed in

V𝑘+1𝑖𝑑

= 𝑤V𝑘𝑖𝑑+ 𝑐1𝜕 (𝑝𝑘

𝑖𝑑− 𝑥𝑘

𝑖𝑑) + 𝑐2𝛽 (𝑝𝑘

𝑔𝑑− 𝑥𝑘

𝑖𝑑) ,

𝑥𝑘+1

𝑖𝑑(𝑡 + 1) = 𝑥

𝑘

𝑖𝑑(𝑡) + V𝑘

𝑖𝑑(𝑡) .

(12)

Among them, 𝜔 is the inertia weight; 𝑐1is the weight

coefficient of the optimal value that the particle tracks itshistory; 𝑐

2is weight coefficients that particle track the optimal

value; and 𝜕 and𝛽 are the randomnumbers changing in [0, 1].𝑝𝑘

𝑖𝑑is the individual optimal solution of the particle after the

𝑘 iterations; 𝑝𝑘𝑔𝑑

is the global optimal solution of the groupin the 𝑘 iterations. To make the algorithm a more accuratesearch scope, the movement speed of the particle is limitedin [−Vmax, Vmax]; if Vmax is too large, the particle will fly overthe optimal solution; if it is too small, it is easy to fall intothe local optimum. Assuming that the particle position isdefined as the interval [−Vmax, Vmax], the two-wheeled self-balancing robot state variables [𝑥, ��, 𝜃

𝑝, 𝜃𝑝]𝑇 are regarded as

the particles, and the particle’s position and the initial valueof the speed are produced at random in a certain range. Thefitness function is an important link in using the particleswarm optimization algorithm, and it is the standard of thewhole particle swarm algorithm iterative evolution. Becausewhat we have designed is the quadratic optimal controlregulator, we adopt the linear quadratic performance indexformula (10) as the fitness function. The 𝑄 is a symmetricpositive semidefinite matrix of 6 ∗ 6; 𝑅 is a constant positivedefinite matrix. In order to simplify the problem and makethe weighted matrix a clear physical meaning, we choose


Table 1: The parameter of the robot.

Symbol Actual value𝑘𝑚

0.0136Nm/A𝑘𝑒

0.01375V/(rad/s)𝑅 1.6Ω𝑀𝑃

0.52 kg𝑀𝑤

0.02 kg𝑙 0.16m𝐼𝑝

0.0038 kg⋅m2

𝑔 9.8m/s2

𝐼𝑤

0.0032 kg⋅m2

𝑟 0.025m

the weighted matrix 𝑄 as the diagonal matrix, so that theperformance index can be represented as

𝐽 = ∫

∞

0

(𝑞1𝑥2

1+ 𝑞2𝑥2

2+ 𝑞3𝑥2

3+ 𝑞4𝑥2

4+ 𝑅𝑢2

) 𝑑𝑡. (13)

Among them, 𝑞1, 𝑞2, 𝑞3, and 𝑞

4are the weights of the

position, speed, angle, and angular velocity of the two-wheeled self-balancing robot, respectively. 𝑅 is the squareweight of the control amount 𝑢 in the objective function.

3.3. The Parameter Optimization Steps of the LQR ControllerBased on the Practical Swarm Optimization Algorithm

Step 1. Initialize the particle swarm. Set the speed coefficients𝑐1, 𝑐2, the maximum evolution algebra gen, the size of the

group pop, and the location of the initial search point and itsspeed, and each particle has the value of the current position.

Step 2. Calculate the fitness value 𝐹𝑖𝑑[𝑖] of each particle.

Step 3. Compare the fitness value 𝐹𝑖𝑑[𝑖] with the individual

extremum 𝑝best(𝑖) of each particle; if 𝐹𝑖𝑑[𝑖] > 𝑝best(𝑖), use

𝐹𝑖𝑑[𝑖] to replace 𝑝best(𝑖).

Step 4. Compare the fitness value 𝐹𝑖𝑑[𝑖] with the global

extremum 𝑔best of each particle; if 𝐹𝑖𝑑[𝑖] > 𝑝best(𝑖), use 𝐹𝑖𝑡[𝑖]

to replace 𝑔best.

Step 5. Update the particle’s speed V𝑖and position 𝑥

𝑖accord-

ing to formula (12).

Step 6. If itmeets the end condition (the error is good enoughor it reaches the maximum cycle times), it will exit, or itreturns to Step 2.

4. The Simulation Experiment ofthe LQR Controller Based on the ParticleSwarm Algorithm

The parameter symbols, description, and the actual value ofthe two-wheeled self-balancing robot are shown in Table 1.

The actual parameters of this system are substituted intothe state equation; the actual state equation is obtained asfollows:

[[[

[

��

��𝜙𝜙

]]]

]

=[[[

[

0 1 0 0

0 −0.1038 25.5862 0

0 0 0 1

0 −0.5015 238.4685 0

]]]

]

[[[

[

𝑋

��

𝜙𝜙

]]]

]

+[[[

[

0

0.4891

0

2.3634

]]]

]

𝑉𝑎.

(14)

The initial state of self-balancing robot system is asfollows: [𝑥, ��, 𝜃

𝑝, 𝜃𝑝]𝑇

= [0, 0, 0, 0]𝑇, selecting 𝑐

1= 𝑐2= 1.2

to make experiment, at the same time, selecting the inertiaweight formula [15]:

𝜔𝑘

= 𝜔max −𝑘

gen(𝜔max − 𝜔min) . (15)

Among them, 𝜔max = 1, 𝜔min = 0.3, and gen = 30; itindicates the iterative number of the algorithm evolution, and𝑘 is the current evolution algebra.

For an initial population of 40 ∗ 8 matrix, the fourdimensions in the front represent particle updated locationand the four dimensions in the latter represent particleupdated speed. The particle swarm updated position curveis shown in Figure 3. The particle motion curve shown inFigure 3 is not lost regularity, so there is only one particleposition in critical condition. The PSO updated rate curveis shown in Figure 4. Figure 4 shows the particle movementspeed can be controlled, not beyond the intended scope.Selecting the inertia weight formula is appropriate.

PSO algorithm in this case has a total of 50 times iterativeand adaptive values. The number of iterations with the curveis shown in Figure 5.

Seen fromFigure 5,𝑄 and𝑅 parameters after 21 iterationsto achieve the optimization.

The global optimal solution can be gotten through theparticle swarm algorithm programming:

𝑄 =[[[

[

221.5326 0 0 0

0 169.2376 0 0

0 0 121.2542 1

0 0 0 187.6532

]]]

]

,

𝑅 = 1.7682.

(16)

With the aid of MATLAB function 𝐾 = lqr(𝐴, 𝐵, 𝑄, 𝑅)to work out the optimal feedback matrix: 𝐾 = [−11.1932,

−16.2145, −72.4045, −17.7329].The dynamic response curves of the two kinds of algo-

rithms of the LQR controller and the LQR controller basedon the particle swarm algorithm are shown, respectively, inFigure 6.

The simulation curves of Figure 6 show that when theinitial conditions of self-balancing robot are zero, the LQR

Mathematical Problems in Engineering 5

2

0

−2

2

0

−2

The first 1 dimensions The first 2 dimensions


0 5 10 15 20 25 30 35 40

Particle

0 5 10 15 20 25 30 35 40

Particle

0 5 10 15 20 25 30 35 40

Particle0 5 10 15 20 25 30 35 40

Particle

5

0

−5

5

0

−5

The i

nitia

lpo

sitio

n

The i

nitia

lpo

sitio

nTh

e ini

tial

posit

ion

The i

nitia

lpo

sitio

n

Figure 3: Particle swarm position initialization.



0 5 10 15 20 25 30 35 40

Particle0 5 10 15 20 25 30 35 40

Particle

0 5 10 15 20 25 30 35 40

Particle0 5 10 15 20 25 30 35 40

Particle

5

0

−5

5

0

−5

5

0

−5

5

0

−5Upd

ate r

ate

Upd

ate r

ate

Upd

ate r

ate

Upd

ate r

ate

Figure 4: Particle swarm speed initialization.

0 5 10 15 20 25 30 35 40 45 50

The number of iterations

1.5

2

2.5

3

3.5

4

4.5

5

Adap

tive v

alue

Figure 5: 𝑄 and 𝑅 iterative optimization Figure based on PSOalgorithm.

control and LQR control after the particle swarm optimiza-tion algorithm can make the system stable. However, thelatter algorithm has the advantages of stable short time, lessovershoot with fewer shocks.

In order to verify the LQR algorithm based on PSO opti-mization is better than that LQR algorithm in the literature[4] and literature [7]. LQR algorithm comparison and eachalgorithm control indicators are shown in Table 2.

0 2 4 6 8 10 12 14 16 18 20

Time (s)

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Ang

le

LQRParticle swarm

Figure 6: Based on self-balancing robot angle PSO optimizedresponse.

According to the above results, three kinds of algorithmscan all achieve the stability of the system. The proposedalgorithmon the overshoot and oscillation frequency is out ofthe more obvious advantages. That is because the algorithm


Table 2:The effective comparison table between othermethods andproposed methods.

Symbol Stability Overshoot Regulationtime

Oscillationfrequency

Literature [4] Stable 62% 1.8 s 1Literature [7] Stable 21% 3.2 s 3Proposed method Stable 18% 2.5 s 1

can find the optimal solution of the LQRmatrix𝑄 andmatrix𝑅.

5. Conclusion

Using the particle swarm algorithm to optimize the selectionof weighted matrix 𝑄 and matrix 𝑅 can overcome theblindness of selectingmatrix𝑄 andmatrix𝑅 in the traditionalLQR optimal control. This paper uses the characteristics thatthe particle swarm optimization algorithm can achieve anintelligent search, gradual optimization, and rapid conver-gence. Therefore, it is not easy to fall into local optimum,but easy to be implemented on the basis of the linear modelof the two-wheeled and self-balancing robot to obtain theglobal optimal solution of the 𝑄, 𝑅, achieving the optimalLQR controller design through the MATLAB simulationexperiments. It can be found that the design response speedof the LQR optimal controller is faster with less overshootamount, and it can keep the steady-state error zero, so thecontrol effect is better.

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper.

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research article the lqr controller design of two-wheeled ...separates the wheel from the pendulum...

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