research article the lqr controller design of two-wheeled ...separates the wheel from the pendulum...
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Research ArticleThe LQR Controller Design of Two-Wheeled Self-BalancingRobot Based on the Particle Swarm Optimization Algorithm
Jian Fang
Institute of Electrical Engineering, Jilin Teachersโ 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 ยฉ 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 expertโs 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 ๐ and ๐ [3โ5]. 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 ๐ and matrix ๐ of the LQR controller. Itgains the global optimal solution of matrix ๐ and ๐ of theLQR controller so as to design the optimal state feedbackcontrol matrix ๐พ and overcome the disadvantages of relyingon the experience and the trial and error in the selection ofmatrix๐ and๐ 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
๐W
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 Newtonโs law [9] and the rotational torqueformula [10]:
๐๐ค๏ฟฝ๏ฟฝ = ๐ป
๐๐ โ ๐ป๐ ,
๐ผ๐
๐๐= ๐ถ๐ โ ๐ป๐๐ โ ๐ .
(1)
The right wheel force equation is as follows:
๐๐ค๏ฟฝ๏ฟฝ = ๐ป
๐๐ฟโ ๐ป๐ ,
๐ผ๐
๐๐= ๐ถ๐ฟโ ๐ป๐๐ฟโ ๐ .
(2)
After finishing, it can obtain
2 (๐๐ค+๐ผ๐
๐ 2) ๏ฟฝ๏ฟฝ =
๐ถ๐ + ๐ถ๐ฟ
๐ โ (๐ป๐ + ๐ป๐ฟ) . (3)
Among them, ๐๐คis the weight of the wheel; ๐ผ
๐is the
moment of inertia of the wheel; ๐ is the radius of the wheel;๏ฟฝ๏ฟฝ is the wheel acceleration of๐ axis; ๐ถ
๐ and ๐ถ
๐ฟare the right
and left wheel torque; ๐ป๐ฟand ๐ป
๐ are the ๐ axis forces of
the left and right wheels with the car body; ๐ป๐๐
and ๐ป๐๐ฟ
are the interatomic forces of the right and left wheels withthe ground; and ๐
๐is the angle of wheel around the ๐ axis
direction.
HL + HR
PL + PR
l ๐p๐p
x
๐2
MPg ฮฃFxp
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 Newtonโs second law, thehorizontal force is as follows:
โ๐น๐ฅ= (๐ป๐ + ๐ป๐ฟ) โ ๐
๐๐ ๐๐sin ๐๐โ๐๐ ๐
๐cos ๐๐
= ๐๐โ ๏ฟฝ๏ฟฝ๐= ๐๐(๐ฅ + ๐ sin ๐
๐) .
(4)
Among them, ๐ฅ๐is the displacement of the car body
centre of gravity relative to the ground,
๐ฅ๐= ๐ฅ + ๐ sin ๐
๐. (5)
Using Newtonโs second law, the force of the verticaldirection of the car body is as follows:
โ๐น๐ฅ๐= ๐๐๐ sin ๐ โ (๐
๐ + ๐๐ฟ) sin ๐
๐+ (๐ป๐ + ๐ป๐ฟ) cos ๐
๐
= ๐๐(๐ฅ + ๐ sin ๐
๐) cos ๐
๐.
(6)
The sum of the torques of the car body mass center is asfollows:
โ๐0= ๐ผ๐
๐๐=(๐๐ + ๐๐)
sin ๐๐
โ(๐ป๐ป+ ๐ป๐ฟ)
cos ๐๐
. (7)
On the small angle scope, ๐2๐โ 0, sin ๐
๐โ ๐๐, cos ๐
๐โ
1, the linearized equations are gotten after the linearizationprocess:
(๐๐+ 2๐๐
๐ผ๐
๐ 2) ๏ฟฝ๏ฟฝ =
๐ถ๐ + ๐ถ๐ฟ
๐ โ 2๐๐๐๐๐,
(2๐๐๐2
+ ๐ผ๐) ๐๐= ๐๐๐๐๐๐โ๐๐๐๏ฟฝ๏ฟฝ.
(8)
Among them, ๐๐is the angle of the car body deviating
from the ๐ axis direction; ๐ผ๐is the moment of inertia of the
car body;๐๐is the weight of the car body; and ๐ 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 ๐ถ๐ = ๐ถ๐ฟ= ๐ผ๐ (๐๐/๐๐ก) = (๐
๐/๐ )๐๐โ
(โ๐๐๐๐/๐ ) ๐๐; the state equation of the two-wheeled self-
balancing robot is obtained:
[[[
[
๏ฟฝ๏ฟฝ
๏ฟฝ๏ฟฝ๐๐
๐๐
]]]
]
=
[[[[[[[
[
0 1 0 0
02๐๐๐๐(๐๐๐๐ โ ๐ผ๐โ๐๐๐2
)
๐ ๐2๐ด
๐2
๐๐๐2
๐ด0
0 0 0 1
02๐๐๐๐(๐๐ต โ๐
๐๐)
๐ ๐2๐ด
๐๐๐๐๐ฝ
๐ด0
]]]]]]]
]
[[[[[
[
๐ฅ
๏ฟฝ๏ฟฝ
๐๐
๐๐
]]]]]
]
+
[[[[[[[
[
0
2๐๐(๐ผ๐+๐๐๐2
โ๐๐๐๐)
๐ ๐๐ด0
2๐๐(๐๐๐ โ ๐๐ต)
๐ ๐๐ด
]]]]]]]
]
๐๐.
(9)
Among them, ๐ด = [๐ผ๐๐ฝ + 2๐
๐๐2
(๐๐ค+ (๐ผ๐ค/๐2
))]; ๐ต =
(2๐๐ค+ (2๐ผ๐ค/๐2
) + ๐๐).
The output equation is as follows:
๐ฆ = [0 0 1 0]
[[[[[
[
๐ฅ
๏ฟฝ๏ฟฝ
๐๐
๐๐
]]]]]
]
. (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 ๐ขโ(๐ก) tomake the system reach the steady state and guarantee theperformance index ๐ฝ to take the minimum value:
๐ฝ = โซ
โ
0
(๐๐
๐๐ + ๐ข๐
๐ ๐ข) ๐๐ก. (11)
Among them, ๐ขโ = โ๐ โ1๐ต๐๐๐ฅ = โ๐พ๐ฅ, and ๐ is the solu-tion of the algebraic equation๐๐ด+๐ด๐๐+๐โ๐๐ต๐ โ1๐ต๐๐ = 0 ofthe matrix Riccati [12]. In (13), the matrix๐ and matrix ๐ aremutually restricted, and the size of the๐ value is proportionalto the anti-interference ability of the system; increasing the๐value, 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 ๐
value makes the energy consumed by the system less, butthe adjustment time increases. Therefore, the design key isto find the right weight matrix ๐ and matrix ๐ . As longas we make sure of the matrix ๐ and matrix ๐ , the statefeedback matrix ๐พ is the only confirmation. However, theselection of the ๐ matrix and ๐ 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 ๐ particles are composed of a group in ๐dimensional space, among them, the position and velocityof ๐ particle in the space are ๐ฅ
๐= (๐ฅ๐1, ๐ฅ๐2, . . . , ๐ฅ
๐๐ท), V๐=
(V๐1, V๐2, V๐3, . . . , V
๐๐ท), ๐ = 1, 2, . . . , ๐, the best position that
the ๐ particle experiences is denoted by ๐bset(๐), and the bestposition that all the particles in the group experience isdenoted by ๐bset(๐). The whole particle swarm updates thevelocity and position through tracking the individual extremevalue and the optimal value [14]. The particle optimizationprocess is expressed in
V๐+1๐๐
= ๐คV๐๐๐+ ๐1๐ (๐๐
๐๐โ ๐ฅ๐
๐๐) + ๐2๐ฝ (๐๐
๐๐โ ๐ฅ๐
๐๐) ,
๐ฅ๐+1
๐๐(๐ก + 1) = ๐ฅ
๐
๐๐(๐ก) + V๐
๐๐(๐ก) .
(12)
Among them, ๐ is the inertia weight; ๐1is the weight
coefficient of the optimal value that the particle tracks itshistory; ๐
2is weight coefficients that particle track the optimal
value; and ๐ and๐ฝ are the randomnumbers changing in [0, 1].๐๐
๐๐is the individual optimal solution of the particle after the
๐ iterations; ๐๐๐๐
is the global optimal solution of the groupin the ๐ iterations. To make the algorithm a more accuratesearch scope, the movement speed of the particle is limitedin [โVmax, 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 [โVmax, Vmax], the two-wheeled self-balancing robot state variables [๐ฅ, ๏ฟฝ๏ฟฝ, ๐
๐, ๐๐]๐ are regarded as
the particles, and the particleโs 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 ๐ is a symmetricpositive semidefinite matrix of 6 โ 6; ๐ is a constant positivedefinite matrix. In order to simplify the problem and makethe weighted matrix a clear physical meaning, we choose
4 Mathematical Problems in Engineering
Table 1: The parameter of the robot.
Symbol Actual value๐๐
0.0136Nm/A๐๐
0.01375V/(rad/s)๐ 1.6ฮฉ๐๐
0.52 kg๐๐ค
0.02 kg๐ 0.16m๐ผ๐
0.0038 kgโ m2
๐ 9.8m/s2
๐ผ๐ค
0.0032 kgโ m2
๐ 0.025m
the weighted matrix ๐ as the diagonal matrix, so that theperformance index can be represented as
๐ฝ = โซ
โ
0
(๐1๐ฅ2
1+ ๐2๐ฅ2
2+ ๐3๐ฅ2
3+ ๐4๐ฅ2
4+ ๐ ๐ข2
) ๐๐ก. (13)
Among them, ๐1, ๐2, ๐3, and ๐
4are the weights of the
position, speed, angle, and angular velocity of the two-wheeled self-balancing robot, respectively. ๐ is the squareweight of the control amount ๐ข 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 coefficients๐1, ๐2, 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 ๐น๐๐[๐] of each particle.
Step 3. Compare the fitness value ๐น๐๐[๐] with the individual
extremum ๐best(๐) of each particle; if ๐น๐๐[๐] > ๐best(๐), use
๐น๐๐[๐] to replace ๐best(๐).
Step 4. Compare the fitness value ๐น๐๐[๐] with the global
extremum ๐best of each particle; if ๐น๐๐[๐] > ๐best(๐), use ๐น๐๐ก[๐]
to replace ๐best.
Step 5. Update the particleโs speed V๐and position ๐ฅ
๐accord-
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:
[[[
[
๏ฟฝ๏ฟฝ
๏ฟฝ๏ฟฝ๐๐
]]]
]
=[[[
[
0 1 0 0
0 โ0.1038 25.5862 0
0 0 0 1
0 โ0.5015 238.4685 0
]]]
]
[[[
[
๐
๏ฟฝ๏ฟฝ
๐๐
]]]
]
+[[[
[
0
0.4891
0
2.3634
]]]
]
๐๐.
(14)
The initial state of self-balancing robot system is asfollows: [๐ฅ, ๏ฟฝ๏ฟฝ, ๐
๐, ๐๐]๐
= [0, 0, 0, 0]๐, selecting ๐
1= ๐2= 1.2
to make experiment, at the same time, selecting the inertiaweight formula [15]:
๐๐
= ๐max โ๐
gen(๐max โ ๐min) . (15)
Among them, ๐max = 1, ๐min = 0.3, and gen = 30; itindicates the iterative number of the algorithm evolution, and๐ is the current evolution algebra.
For an initial population of 40 โ 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 5,๐ and๐ parameters after 21 iterationsto achieve the optimization.
The global optimal solution can be gotten through theparticle swarm algorithm programming:
๐ =[[[
[
221.5326 0 0 0
0 169.2376 0 0
0 0 121.2542 1
0 0 0 187.6532
]]]
]
,
๐ = 1.7682.
(16)
With the aid of MATLAB function ๐พ = lqr(๐ด, ๐ต, ๐, ๐ )to work out the optimal feedback matrix: ๐พ = [โ11.1932,
โ16.2145, โ72.4045, โ17.7329].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
โ2
2
0
โ2
The first 1 dimensions The first 2 dimensions
The first 3 dimensions The first 4 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
โ5
5
0
โ5
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.
The first 5 dimensions The first 6 dimensions
The first 7 dimensions The first 8 dimensions
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
โ5
5
0
โ5
5
0
โ5
5
0
โ5Upd
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: ๐ and ๐ 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)
โ0.2
โ0.1
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
6 Mathematical Problems in Engineering
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 LQRmatrix๐ andmatrix๐ .
5. Conclusion
Using the particle swarm algorithm to optimize the selectionof weighted matrix ๐ and matrix ๐ can overcome theblindness of selectingmatrix๐ andmatrix๐ 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 ๐, ๐ , 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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