2 stability and chaotic behavior of a PID controlled inverted pendulum

Introduction The problem of stability and dynamical behavior of an inverted pendulum subjected to harmonic vibrations in the pension point is related to many fields of physics and enginee

Stability and chaotic behavior of a PID controlled inverted

pendulum subjected to harmonic base excitations by using

the normal form theory

Departamento de Física, Ingeniería de Sistemas y Teoría de la Señal, Universidad de Alicante, Escuela Politécnica Superior, Campus de San Vicente,

Ó 2014 Elsevier Inc All rights reserved

1 Introduction

The problem of stability and dynamical behavior of an inverted pendulum subjected to harmonic vibrations in the pension point is related to many fields of physics and engineering, such as vibrations of oscillatory chains, control theory,bifurcations, normal form theory and chaos, among others The analysis of the dynamical behavior of a simple inverted pen-dulum has been studied in connection with stability problems, both from a theoretical and experimental viewpoint and withdelay[1–6] However, analytical solutions of the problem assuming oscillations in the suspension point are only consideredunder certain simplifications in the problem, as it appears in Ref.[2]

sus-On the other hand, the problem of swinging up and controlling a pendulum has been considered in the classical Refs.[7–8] Other more complex control strategies reveal the great interest of the inverted pendulum in the field of control, as it is thecase of control strategies based on space-state methods[9–11], control stabilization around homoclinic orbits[12], energymethods[13], passivity control[14]and bounded control[15]among others However, the use of a simple control law to

http://dx.doi.org/10.1016/j.amc.2014.01.102

⇑ Corresponding author.

E-mail addresses: manolo@dfists.ua.es (M.F Pérez-Polo), ma_perez_m@hotmail.com (M Pérez Molina), gil@dfists.ua.es (J Gil Chica), jberna@dfists.ua.es

(J.A Berna Galiano).

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Applied Mathematics and Computation

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / a m c

obtain chaotic behavior has been less used This is probably due to the difficulty in determining whether a pendulum withharmonic base excitations can exhibit chaotic dynamics[16–20].

The aim of this paper is to investigate the stability and dynamical behavior around the pointing-up position for a simpleinverted pendulum that is driven by a control torque and is harmonically excited in the vertical and horizontal directions It

is known that a dissipative pendulum subjected to vertical and horizontal harmonic disturbances of high frequency can bedriven to several equilibrium points apart from the stable pointing-down position (examples can be found in Refs.[21–26]).Consequently, for a pendulum with high-frequency vertical oscillations in the suspension point, the unstable pointing-upposition transforms into a stable one, whereas the pendulum can reach a stable tilt angle belowp/2 for high-frequency hor-izontal oscillations In these cases, the required forces in the suspension point to maintain these equilibriums can be verylarge What is more important, these equilibrium points depend on the initial conditions, and the presence of random noisecould to destroy them The previous reasoning will be used to justify the application of a simple control law based on a PIDcontroller to swing-up and maintain the pendulum in the pointing-up position

In the first part of this paper we assume that there are no harmonic disturbances, and the parameters values of the PIDcontroller which lead to a weak focus in the pointing-up position are deduced from the Routh stability criterion The stability

of this weak focus is studied through the normal form theory, from which it is possible to deduce the system equations innormal form and compare them with the numerical simulations of the system For this purpose, the method developed byBruno[27–32]will be applied (other approaches can be found in Refs.[33–36]) Once the system is reduced to its normalform, the stability properties associated to the parameter variations of the PID controller will be investigated

The conditions which result in chaotic behavior for the pendulum without control torque when a harmonic motion is plied to the suspension point have also been analyzed through the Melnikov function[33–38] The chaotic motion and theappearance of strange attractors are verified by means of the sensitive dependence, Lyapunov exponents, power spectraldensity and Poincaré sections[33–36,38] Taking into account the heteroclinic tangle in a strange attractor, there will always

ap-be trajectories in the phase plane that will ap-be very close to the upright position For such trajectories, the PID parameter ues are properly changed so that the chaotic motion is destroyed and the pendulum is maintained around the pointing-upequilibrium position with small oscillations, even in the presence of random noise

val-2 Mathematical model and statement of the problem

Fig 1shows the layout of the pendulum system as well as the notation used to deduce the Lagrangian of the system Thependulum is modeled by a mass m hanging at the end of a rod of negligible mass and length l, which is fixed to a support O

[4–6,7–16] Let O0XY be an inertial frame and Fx0ðtÞ, Fy0ðtÞ the forces applied at the suspension point O0 in the OX and OYdirections, which respectively produce the accelerations given by €x0ðtÞ and €y0ðtÞ

The kinetic and potential energies of the system can be written as follows:

x ¼ x0þ l sin h

y ¼ y0 l cos h

) T ¼1

2_h2þ ml _hð _x0cos h þ _y0sin hÞ  mgðl  l cos hÞ  mgy0: ð3Þ

Assuming a Rayleigh dissipation function Fr¼ b _h2=2 associated to the angular variable h, and taking into account the trol torque u(t) as well as the forces Fx0and Fy0applied at the suspension point O0, the mathematical model of the system can

con-be obtained from the Lagrange equations as:

ml2€hþ mgl sin h þ b _h þ mlð€x0cos h þ €y0sin hÞ ¼ uðtÞ; ð4Þ

where Axis the amplitude of the horizontal displacement of point O0,xxis the frequency of the horizontal motion component

of O0and / is an arbitrary phase shift, whereas Ayandxyare the vertical displacement and the frequency of the vertical tion component for O0respectively It should be noticed that Eq.(4)can be numerically solved from a specified control torqueu(t), and the forces Fx0and Fy0can be obtained from Eqs.(5)–(7) Next, it is assumed that the control torque is a PID control-ler, i.e it is defined by[39]:

mo-uðtÞ ¼ Kp ½hðtÞ p þ1

si

Z t 0

½hðsÞ pdsþsd

dhðtÞdt

where Kp,siandsdare the proportional action constant, the reset time and the derivative time respectively[39] From Eqs

(4) and (8), the mathematical model of the pendulum with harmonic base excitation and PID control can be written asfollows:

up position by varying the amplitude Ayand the frequencyxyof the external disturbances[25–26] To analyze this effect weshall consider a simplified case in which there are only vertical vibrations at O0, i.e x0= 0 Since the control torque is nowzero, from Eqs.(7) and (9)it is deduced that:

d2hðtÞ

dt2 þx2sin hðtÞ þ ddhðtÞ

dt 

Ayx2 y

Ayx2 y

A2yx2 y

where the angle h1accounts for the slow motions In the pointing-up equilibrium position of the pendulum, the Jacobian ofthe system (13) and the corresponding eigenvalues are given by:

2x0



ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid

2x0

 2

þ 1 12

A2yx2 y

l2x2

!v

u

Consequently, the pointing-up equilibrium position will be stable if Ayxy=lx0> ffiffiffi

2p Similarly, it can be shown that thepointing-down equilibrium position is stable for all values of Ayandxy To study this issue, Eqs.(9)have been simulatedassuming that Ax= 0, Ay= 0.5 m, l = 1 m,x0= 3.1305 m/s2 andxy= 10, so AyxLy=lx0¼ 1:597 > ffiffiffi

2p(xLy= 8.8544 rad/s isthe limit frequency) The initial conditions are h(0) = 2.9 rad, dh(0)/dt = 0 and u(0) = 0 (i.e they are close to the pointing-

up position) and the simulation results are shown in Fig 2 Fig 2(a) shows the time evolution of h(t), dh(t)/dt andu(t)  0 At t  15 s the position h =pis reached, whereas h(t) and dh(t)/dt show an irregular behavior that suggests chaoticbehavior for t < 15 s To analyze this behavior,Fig 2(b) depicts two simulations of Eqs.(9)whose initial conditions differ in

107, which allows to appreciate a clear sensitive dependence as a strong indicator of chaotic behavior.Fig 2(c) shows thevalues of the forces F0xand F0ydeduced from Eqs.(5) and (6), which can be regarded as acceptable On the other hand, weassume the presence of random noise in the system that is modeled as follows:

hðtÞ ¼ hðtÞ þ fna½X  0:5; dhðtÞ=dt ¼ dhðtÞ=dt þ fna½X  0:5; ð15Þ

where X is a random variable that is uniformly distributed between 0 and 1, and fna> 0 is an amplification factor to obtain auniformly distributed noise amplitude between fna/2 and fna/2 For h(0) = 2.9 rad, dh(0)/dt = 0, u(0) = 0 and fna= 0.2,Fig 2(d)shows that the desired set point he=pcannot be reached

To analyze the effect of the PID control law given by Eq.(8),Fig 3shows the simulation results of Eqs.(9)that have beenobtained with the previous values but taking Kp= 30 Nm, si= 1 s and sd= 103s, assuming the initial conditionsh(0) = 0.01 rad, dh(0)/dt = 0, u(0) = 0 and considering a noise factor fna= 0.4.Fig 3(a) and (b) show how the desired set point

he=pis reached even with strong noise and very disadvantageous initial conditions due to the PID controller action.Fig 3(c)shows that the magnitude of the forces deduced from Eqs.(5) and (6)are acceptable even in presence of the PID controltorque

It should be noticed that, although the PID parameters Kp,si, andsdhave been chosen arbitrarily, the derivative timesd

must be small enough to avoid an excessive value for u(t) due to the high values of the derivatives caused by the randomnoise[38–39] The effect of the high-frequency horizontal excitation can be analyzed in a similar way In this case, the aver-aged equation and the Jacobian of the pointing-down equilibrium position are given by[25]:

A2

xx2 x

Fig 2 Simulation results without control torque and with vertical harmonic disturbances The parameter values are x2 = 9.8 m 2 /s 4 , A y = 0.5 m,

xyL = 8.8544 rad/s andxy = 10 > 8.8544 rad/s The fourth-order Runge–Kutta integration method with a simulation step T = 0.002 s has been used (a) State variables h(t), dh(t)/dt and u(t) as a function of the time assuming initial conditions h(0) = 2.9 rad and dh(0)/dt = 0 (b) Sensitive dependence for two simulations of h(t) with initial conditions differing in 10 7

rad (c) Required forces F 0x and F 0y to produce the movements which appear in graphics (a) and



ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid

2x0

 2

 1 12

A2xx2 x

l2x2

!v

u

Therefore, the pointing-down equilibrium position will be unstable if Axxx=lx0> ffiffiffi

2p In this case, Eq.(16)allows to de-duce that two new equilibrium points appear in symmetric positions around h = 0, i.e.:

he¼ ar cos 2l

2

x2

A2xx2 x

!

where the sign of hedepends on the initial conditions[25–26] Taking he= 0.8 rad, l = 1 m,x0= 3.1305 rad/s and Ax= 0.035 m,

Eq.(18)allows to deduce thatxx= 151.54 rad/s As it can be observed inFig 4(a), the simulation results for Eqs.(9)indicatethat the tilt angle of the pendulum is he 0.8 rad, which is very close to the prescribed value The small amplitude aroundthe averaged value is a consequence of the large value forxx In accordance with Eqs.(5) and (6),Fig 4(b) shows the strongforces that must be applied to the suspension point O0to maintain the reference angle for the pendulum

The previous results allow to conclude that:

 It is possible to obtain chaotic behavior without PID control and with vertical excitations of moderate frequency

 It is possible to drive the pendulum to the set point he=pwithout PID control and with vertical excitation, although theinitial conditions must be close to the set point

 Without PID control and with vertical excitation, a small random noise or a small change in the initial conditions maydestroy the asymptomatically stable equilibrium point he=p(seeFig 2(a)), and the pendulum behavior may becomeoscillatory around he=pas shown inFig 2(d)

 Without PID control and with horizontal excitations of high frequency, the pendulum cannot be driven to the pointing-upposition, even with very high frequencies or strong forces applied at the suspension point O0

 Starting from arbitrary initial conditions, the pendulum can be driven to the pointing-up position with PID control andwith vertical excitations of high frequency, even in the presence of random noise

Once the need of a PID controller has been clarified, the following deviation variables are introduced:

1ðtÞ    , the mathematical model of the pendulum with harmonic oscillations in the axes OX and OY and withPID control can be written in matrix form (up to third order terms) as follows:

Fig 3 Simulation results with control torque and vertical harmonic disturbances The parameter values of the system arex2 = 9.8 m 2 /s 4 , A y = 0.5 m,

xyL = 8.8544 rad/s andxy = 10 > 8.8544 rad/s The fourth-order Runge–Kutta integration method with a simulation step T = 0.002 s has been used (a) State variables h(t), dh(t)/dt and u(t) as a function of the time (b) Pendulum position in the presence of noise (c) Required forces F 0x and F 0y to produce the movements that appear in graphics (a) and (b).

37

37

The eigenvalues of the matrix A associated to the linear part of Eq.(20)can be obtained from the Routh criterion[39], and

in addition we can investigate admissible values for the parameters Kp,siandsdof the PID controller Since the system is ofthird order, we pretend to obtain one real negative and two pure imaginary eigenvalues, so a weak focus appears around

h=p[34–37] If such weak focus is stable, the pendulum can be maintained around the pointing-up position with smoothoscillations The eigenvalues are obtained as the roots of the characteristic equation of matrix A, i.e.:

Consequently, if the PID parameters are chosen in accordance with Eq.(22), the inequality a1> 0 is fulfilled and the roots

of the characteristic equation are k0¼ ðd þaKpsdÞ and k1;1¼ ix(which can be verified substituting kifor i = 0, 1, 1 into

Eq.(21)) Therefore, in the unstable pointing-up position we have two pure imaginary eigenvalues as well as a real negativeone, whose stability will be analyzed in the next section

Fig 4 Simulation results without control torque and with horizontal harmonic disturbances of high frequency The parameter values of the system are

x2 = 9.8 m 2 /s 4 , h d = 0.8 rad (desired tilt angle for the pendulum), A xxx = 5.3040 m/s (value to achieve stability), A x = 0.035 m,xx = 151.5427 rad/s (frequency for stability) and t d = 3 s The fourth-order Runge–Kutta integration method with a simulation step T = 0.001 s has been used (a) State variable h(t) as a function of the time assuming initial conditions h(0) = 0.5 and dh(0)/dt = 0 (b) Required forces F 0x and F 0y to produce the movement that appears in graphic (a).

3 Stability analysis by using the normal form theory

In accordance with the results of the previous section, we shall determine the stability conditions for the weak focus as afunction of the parameters of the control law given by Eq.(8) The stability analysis is carried out by using the normal formtheory proposed by Bruno[27–31], since it provides a direct connection between the original and transformed equations ofthe system The first step consists of obtaining the (complex) Jordan canonical form of Eqs.(20)taking only up to third-orderterms The eigenvectors of the matrix given in Eq.(20)allow to find the matrix P associated to a linear transformation whichtransforms the matrix of the linear part into its complex Jordan canonical form, i.e.:

37

375;

3

7 _x_x1ðtÞ

1ðtÞ_x0ðtÞ

26

37

37

37

Assuming that there are no disturbances at the supporting point O0of the pendulum (seeFig 1), the system defined by Eq

(26)has the following general form:

being q0, q1and q1integer numbers that must satisfy the following relations[27,28]:

where the only non-null coefficients are the ones of the formvm

f m ;1;1g(where fm;1  1g denote any permutation of the mentsm, 1 and 1), since bmfm;1;1g¼ 0; m¼ 0; 1 as it follows from Eqs.(31)–(33) From the previous considerations, Eqs.(30)and (34)can be expanded as follows[27,28,32]:

where the asterisk denotes conjugate complex To calculate the normalizing transformation given by Eqs.(29) and (33)must

be applied taking into account that the coefficients bmlmp are deduced by identifying terms between Eqs.(26)–(28) For thispurpose, the following cases must be considered:

x xa2 deKpsd



 Kpx de sKi xp  deKpsd

and from Eq.(33)the successive values of bm

lmpare obtained as:

b1000¼ b

1 000

k0þ k1þ k1 k1¼ b

1

de 3xi; b

1 111¼ b

1 111

b0000¼ b

0 000

k0þ k1þ k1 k0¼  b

0

2xi; b

0 111¼ b

0 001

d e  x i



2664

377

d e  x i



26

37

37

solu-1 ¼ 2Reðg1Þ and introducing the variable p1ðtÞ ¼ y1aðtÞy1aðtÞ, the following differential equation can bededuced:

Eq (46) allows to deduce that the variable p1(t) will eventually be negative, which is impossible since

p1ðtÞ ¼ y1aðtÞy1aðtÞ ¼ jy1aðtÞj2 On the other hand, if 2Re½g1 < 0 for t ! 1 then p1ðtÞ ! 0 and therefore the system will bestable Consequently, taking into account the equations of the system in normal form as well as Eqs.(36) and (37), the sta-bility condition for the weak focus associated to the pointing-up position of the pendulum can be written as follows:

y1aðtÞ ¼xi þ g1p1ðtÞ

dt ) y1aðtÞ ¼ y1að0Þ exp ixt þ

Z t 0

g1p1ðsÞds

From Eqs.(46) and (49), the variable y1(t) can be written as:

y1aðtÞ ¼ y1að0Þ expðixtÞ 1  2jy1að0Þj2Re½g1

where y1a(t) is the conjugate complex of y1a(t) Following a similar procedure, the value of y0(t) can be obtained as[17,18]:

y0aðtÞ ¼ y0að0Þ expðdetÞ 1  2jy1að0Þj2Re½g1

Consequently, Eqs.(52) and (53)provide the general analytic solution for the normal form (see Eq.(35)) with accuracy up

to third-order terms The stability consideration deduced from the normal form can also be used to adjust the parameters Kp,

siandsdof the PID controller throughout the following steps:

1 From the condition for obtaining a stable weak focus (see Eqs.(22)–(24)) it is deduced that:

In accordance with the inequality given by Eq.(54), we take desi¼ f ; f > 1 Once the factor f has been chosen, the value of Kp

will be given by:

Kp¼ fx2

2 Next we choose a value for sd to obtain an appropriate value for the equivalent damping coefficient given by

de¼ d þaKpsd¼ b=ml2þaKpsd sdmust be large, and on the contrary,sdmust be small if b  1

3 Once the values for Kpandsdare known, the reset timesiof the PID controller is given by:

y 0a< /sub>tị ẳ y 0a< /small>0ị expdetị  2jy 1a< /sub>0ịj2< /sup>Reẵg1... The stability consideration deduced from the normal form can also be used to adjust the parameters Kp,

siand< h3>sdof the PID. .. sd to obtain an appropriate value for the equivalent damping coefficient given by

deẳ d ỵa< /h3>Kpsdẳ b=ml2< /sup>ỵa< /h3>Kpsd