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Tuning of a PID controlled gyro by using the bifurcation theory Manuel Pérez Polo a , Pedro Albertos b,∗ , José Ángel Berná Galiano a a Department of Física, Ingeniería de Sistemas y Teoría de la Señal, Universidad de Alicante, Escuela Politécnica Superior,

Campus de San Vicente, 03071, Alicante, Spain b Department of Systems Engineering and Control, E.T.S.I.I., Universidad Politécnica de Valencia, Camino de Vera s/n, Valencia, Spain

Received 26 January 2005; received in revised form 23 May 2006; accepted 13 January 2007

Abstract This paper concerns the control design of a PID controlled gimbals suspension gyro, whose parameters are determined by using bifurcation theory. The non-linear mathematical model of the gyro is deduced by using the nutation theory of gyroscopes. Considering a PID controller with constrained integral action, it is shown that depending on different values of the maximum allowed integral action a Poincaré–Andronov–Hopf bifurcation may appear. The analysis of the stability or instability of this bifurcation, from the first Lyapunov value, gives a procedure to adjust the parameters of the PID controller. The developed control methodology is evaluated through numerical simulations. © 2007 Elsevier B.V. All rights reserved. Keywords: Gyroscope; Non-linear dynamics; PID controller tuning; Bifurcation; Numerical simulation

1. Introduction It is known that the exact equations which describe the nutation motion of a gimbals suspension gyro, are non-linear [1,11]. Traditionally, the control of mechanical devices has mainly focused on the analysis and control of linear systems. However, for those non-linear mechanical devices with strong non-linearities, and for sufficiently large excursions from the equilibrium point, the controller design based on linear models can be inadequate. This situation is typical in the control of a gimbals suspension gyro, where the use of the local linearization, i.e. linearizing the modeling equations around an equilibrium point, is only used for simplified models of the gyro [2–4]. The non-linear behavior of simple-axis-gyro and more elaborated gyro models using the nutation theory, which consider the polar moment of inertia of the rotor, inner and outer gimbals to be non-null, have been analyzed in several papers [1,3,4]. These papers show that in a gyroscope in gimbals complicated motions such as self-oscillating and chaotic behaviors may appear. However, in all the previously cited papers, the design of a PID controller considering the nutation theory of the gyro ∗ Corresponding author. Tel.: +34 965 909870; fax: +34 963 879579.

E-mail addresses: manolo@dfists.ua.es (M.P. Polo), [email protected] (P. Albertos), jberna@dfists.us.es (J.A.B. Galiano).

with a disturbance external torque has not been researched up to the present time. This paper considers the motion of a heavy symmetrical gyro in gimbals with a PID controller. The mathematical model is taken from the paper [11]. It considers the total moment of inertia of the gyro as constant, with constrained integral action and negligible motor inductance [6,9]. The aim of this paper is to design a procedure to adjust the parameters of a PID controller by using the first Lyapunov value deduced from an Andronov–Poincaré–Hopf bifurcation type. It is shown in the paper that multiple equilibrium points can appear due to the integral action saturation. The analysis of these equilibrium points reveals that it is possible to use the Andronov–Poincaré–Hopf bifurcation to design an algorithm to tune the parameters of the PID controller. 2. Mathematical model The mathematical model of the gyro can be found taking into account the Euler–Lagrange’s equations [1,11]. These equations are the following: ⎫ d2 (t) d(t) ∗, ⎪ ⎬ J = M + H cos  + M  dt 2 dt . (1) 2 ⎪ d (t) d ⎭ Hr − H cos  = 0, dt 2 dt

0167-6911/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2007.06.007 Please cite this article as: M.P. Polo, et al., Tuning of a PID controlled gyro by using the bifurcation theory, Systems Control Lett. (2007), doi: 10.1016/j.sysconle.2007.06.007

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Gear: N2

Y1 ≡ Y2 ≡ Y

INNER GIMBAL

)

–

BODY SUBJECTED TO DISTURBANCES

OUTER GIMBAL dβ/dt

dα/dt

γ dγ/dt

α

X2

O

M*

ROTOR

Z1 ≡ Z

X1 ≡ X

β + Z2

SENSOR

β

-

TRANSDUCER INVERSOR

βr

R M(t)

i(t)

eb = K b

dϕ

AMPLIFIER PID CONTROLLER

u(t)

dt

Ka; Kp, τi, τd

ϕ

Gear: N1

V(t)

DC MOTOR: M(t) = Kmi(t)

Fig. 1. Schematic diagram of the controlled gyro. Parameters values: A1 = B1 = 0.045 kg m2 ; C1 = 0.05625 kg m2 ; A2 = 0.04875 kg m2 ; A = 0.01125 kg m2 ; C = 0.0225 kg m2 ; Hr = 0.05625 kg m2 ; J = 0.105 kg m2 ; Jm = 0.001 kg m2 ; j = 10; nr = 100.600 rad/s; H = 2.15 kg m2 rad/s; Km = 5 N m/A; Kb = 0.04 V s; R = 0.5 ; Ka = 10.

The parameter J is defined by the equation J = A2 + A1 + A = A2 + C1 where A2 is the moment of inertia of the external gimbal ring with respect to the OX 2 axis; A1 , B1 , C1 are the moments of inertia of the internal gimbal ring with respect to the principal axes (nutation theory of gyroscopes); A and C are the rotor equatorial and polar moments of inertia, respectively, and Hr = A + B1 . From the definition of parameter J it is assumed that A1 + A = C1 , and consequently the gyro total moment of inertia is constant [11]. H is the gyro kinetic moment, which is a constant obtained from the initial conditions of the motion (H = C.nr 2/60, where nr is the speed of the rotor in rpm). M is the torque applied to the OX 2 axis due to an external motor, and M ∗ is a disturbance torque in the axis of the outer gimbal, which is connected to the external body to stabilize it. Eqs. (1) represent the simplified gyroscope model without control. Fig. 1 shows a schematic representation of the whole system. The internal gimbal ring angle is sensed, while the output signal of the PID controller is applied to a DC motor. R, Kb and Km are the motor resistance, back-electromotive constant and the motor torque constant, respectively. Taking into account that the gear train is considered to be ideal, the following equations

from the feedback circuit can be written: M(t) = Jm

d 2  M , − dt 2 j

M = −j 2 Jm

M(t) = Km i(t),

d2  − j K m i(t), dt 2

Ri(t) = u(t) − Kb

j=

d d = −j , dt dt

N2 , N1

d , dt

(2)

where Jm is the moment of inertia of motor shaft, M(t) is the motor torque, M is the torque due to the feedback control system, j is the reduction rate of the gear train, and u(t) is the control signal. From Eqs. (1) and (2), the following equations can be deduced: ⎫ d2 (t) d(t) ∗, ⎪ = −j K + H cos (t) i(t) + M ⎪ m ⎪ ⎪ dt 2 dt ⎬ d2 (t) d(t) , Hr = 0, − H cos (t) ⎪ ⎪ dt 2 dt ⎪ ⎪ d(t) ⎭ Ri(t) = u(t) + j K b , dt

JT

(3)

Please cite this article as: M.P. Polo, et al., Tuning of a PID controlled gyro by using the bifurcation theory, Systems Control Lett. (2007), doi: 10.1016/j.sysconle.2007.06.007

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where JT = J + j 2 Jm . The Eqs. (3) show that  is an ignorable or cyclic coordinate, i.e. it does not appear in the Lagrangian of the system, so it is convenient to remove it from Eqs. (3). On the other hand, the PID controller equation can be written as follows:    d(t) 1 t u(t) = Ka Kp (t) − r + , ((t) − r ) dt + d i 0 dt (4) where Ka is the amplifier constant, Kp is the proportional constant, i is the reset time, d is the derivative time and r is the reference for the internal gimbal angle. Note that if the error signal ((t) − r ) is continuously applied to the integrator, the stored value will grow without limits (windup). The result can be a very large output of the PID controller. This problem can be overcome by using an anti-windup integrator, which maintains the integral action in a predetermined value Im [6,9]. Substituting Eq. (4) into Eqs. (3) and introducing the state variables: x1 (t) = (t), 

t

x4 (t) = 0

x2 (t) =

d(t) , dt

x3 (t) = i(t),

[(t) − r ] dt

(5)

the following equations can be deduced: x˙1 (t) = x2 (t),

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬



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the state variable x2 is always zero, and from Eqs. (6) different equilibrium points are possible [11]: • Point P1 (x1e , x2e = 0, x3e ): In this case, considering fixed values for x1r , M ∗ , Kp and i , the value of x1e and x3e can be obtained from the following equations: Rx 3e − Ka Kp (x1e − x1r ) − − c x3e +

K a Kp Im = 0, i

K a Kp j Kb ∗ (x1e − x1r ) + M = 0. Ri RJ T

(8) (9)

This point will be considered as the set point. • Points P2 (x1e = (4n + 1)/2 n = 0, ±1, ±2, . . . ; x2e = 0; x3e ): The value of x3e can be found from Eq. (9) substituting x1e by /2. • Points P3 (x1e = (4n + 3)/2 n = 0, ±1, ±2, . . . ; x2e = 0; x3e ): In this case, the value of x3e can be found from the same equation (9) but substituting x1e by 3/2. Note that since points P2 are all equivalent, only the value n = 0 is considered. Similar reasoning can be made to points P3 . So, taking x1e = /2 and x1e = 3/2 in Eqs. (8) and (9) it is possible to assign a value of the integral action to points P2 and P3 , respectively. For that purpose, by eliminating x3e between Eqs. (8) and (9), it yields:

x˙2 (t) = c cos x1 (t) Rx 3 (t) − Ka Kp (x1 (t) − x1r )  K a Kp −d Ka Kp x2 (t) − Im , i x˙3 (t) = −c x3 (t) − c x2 (t) cos x1 (t) + c cos x1 (t). , (6)  ⎪ ⎪ ⎪ ⎪ Rx 3 (t) − Ka Kp (x1 (t) − x1r ) − d Ka Kp x2 (t) ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ Ka Kp K a Kp ⎪ ⎪ − Im + [x1 (t) − x1r ] ⎪ ⎪ ⎪ i Ri ⎪ ⎪ ⎪ ⎪ Ka Kp ⎪ jKb ∗ ⎭ + x2 (t) + M Ri JT R

The values of I /2 and I 3/2 are constants only depending on the disturbance torque and the PID controller parameters. Note that from Eqs. (10) and (11), if x1r < /2 it is deduced that I 3/2 < I /2 . Once the equilibrium points are deduced, the bifurcations at these equilibrium points can be analyzed.

where c c c and c are the parameters defined by the following equations:

3.1. Andronov–Poincaré–Hopf bifurcation at point P3

c =

H , j K b Hr

c =

j 2 Km Kb , RJ T

c =

j K bH , RJ T

c =

 d Ka Kp H . j K b Hr R

(7)

Note that in Eqs. (6), the integral action has reached the maximum allowed value Im and the state variable x4 (t) has been changed to Im .

i RM ∗ 1  I = − i − − x1r , Ka Kp j K m c 2



3 i RM ∗ 1 I 3/2 = − i − − x1r . Ka Kp j K m c 2 /2

(10) (11)

The analysis of the system with a PID controller starts by considering first the point P3 since, as later on shown, this equilibrium point is the most important in order to understand the design of the PID controller. The analysis of the equilibrium point is carried out transforming the fixed point to the origin by the translation given by x1 (t) = x1 (t) − 3/2,

x2 (t) = x2 (t) − x2e ,

3. Equilibrium points and bifurcation analysis

x3 (t) = x3 (t) − x3e .

Let us consider Eqs. (6) when the integral action has reached the steady state value Im . It is clear that in the equilibrium state,

Substituting Eqs. (12) into Eqs. (6) and taking into account the equilibrium equation (9) the following equations are

(12)

Please cite this article as: M.P. Polo, et al., Tuning of a PID controlled gyro by using the bifurcation theory, Systems Control Lett. (2007), doi: 10.1016/j.sysconle.2007.06.007

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obtained: x˙2 (t) = c cos x1 (t)[Rx 3 (t) − Ka Kp x1 (t)

−d Ka Kp x2 (t) + p3 ],     x˙3 (t)=−c x3 (t)+c sin x1 (t) Rx 3 (t) − Ka Kp x1 (t) ⎪ ⎪

 ⎪ ⎪ c ⎪ ⎪  ⎪ − d Ka Kp + x2 (t) + p3 + ⎪ ⎪ c ⎪ 1 ⎪ ⎪ ⎪ ⎪ Ka Kp j Kb ∗ ⎪ ⎭ M , x2 (t) + JT R Ri

(13)

–

p3 = Rx 3e − Ka Kp

K a Kp 3 Im . − x1r − 2 i

(14)

b11 b02 c R a3 , + (15) 3 8 8 √ where  = c |p3 | and the coefficients b11 , b02 and a3 are expressed as a function of parameters of the system as follows:

L1 =

⎫ K a Kp K a Kp ,b = , c = c p3 , ⎪ ⎪ ⎬ Ri R

ac + b2 2c + 2

,

g=

(bc − a) 2c + 2

b11 = c (Rg − d Ka Kp ),

,

⎪ ⎪ ⎭

,

b02 = c (Rf − Ka Kp ),

(16)

(17)

c (42 + 2c )  2 (gc − c )[(Rf − Ka Kp ) × c  − (Rg − d Ka Kp ] − c 2 .



K K a p Im . − x1r − 2 i

(19)

Now, if Im = I /2 ± with >1, from a model similar to Eq. (13) and taking into account the Eq. (19), it is not difficult to show that

Now, if Im = I 3/2 is assumed, it is deduced from Eqs. (11) and (14) that p3 = 0, and consequently, from Eqs. (13) it is easy to deduce that the system has two zero eigenvalues. On the other hand if Im = I /2 + , with > 0, (i.e. when Im > I /2 ) then p3 0, the equilibrium point P3 will be unstable. When L1 < 0 the equilibrium point will be stable, and if L1 = 0 it is necessary to reduce the system (13) on the center manifold to the formal norm up to the fifth order terms [5,7,8,10,12–15]. Consequently, Eqs. (15)–(18), provide a criterion to know the stability or instability of point P3 as a function of the PID controller parameters.

(a) If Im = I /2 − ⇒ point P2 is a weak focus. (b) If Im = I /2 ⇒ a double zero eigenvalue appears at point P2 . (c) If Im = I /2 + ⇒ point P2 is a saddle. Note that case (a) can be considered as in the previous subsection, case (c) is trivial since point P2 is unreachable, and only case (b) is a problematic one, because a Bogdanov–Takens bifurcation appears. This difficulty can be overcome taking Im = I /2 + with > 0, so the analysis of this bifurcation is not necessary because point P2 will be always unstable. 3.3. Equilibrium point or set point P1 The analysis of the equilibrium point is carried out again by transforming the fixed points to the origin. At this point, the eigenvalues of the Jacobian matrix are defined by the following equations: ⎫

3 + b2 2 + b1 + b0 = 0, ⎪ ⎪ ⎪ ⎪ H2 ⎪ ⎪ ⎪ b2 = c ; b1 = cos x1e ⎪ ⎪ JT H r ⎬

(20) j K m Ka Kp d ⎪ , ⎪ × cos x1e + ⎪ HR ⎪ ⎪ ⎪

⎪ ⎪ H K a Kp 1 ⎪ b0 = c 1− cos x1e , ⎭ j K b Hr c i From the Routh–Hurwitz criterion it is deduced that point P1 will be stable if the coefficients b0 , b1 and b2 are positive and the inequality b1 b2 > b0 is fulfilled. This implies that −/2 < x1e < /2, with 1 > 1/c i . Note that c ? 1 (Eq. (7) and parameters values from legend of Fig. 1) and, consequently, the stability conditions for point P1 are established. 4. Procedure to determine admissible parameter values for the PID controller The analysis of the bifurcations at points P2 and P3 reveals that if the integral action is limited to a value given by Im = I /2 + and the first Lyapunov value L1 is positive, the equilibrium points P2 and P3 will be unreachable. On the other hand, taking into account Eqs. (20), the value of x1e must be

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in the interval −/2 < x1e < /2. So, considering Eqs (8) and (9) at point P2 , the following equations can be written:  K K ⎫ a p /2 /2 I = 0, ⎪ − x1r − Rx 3e − Ka Kp ⎬ 2 i . (21) jK ⎪ K a Kp   /2 b ⎭ ∗ −c x3e + M = 0, − x1r + Ri 2 RJ T Substituting Im = I /2 + in the Eq. (8) I /2 can be eliminated from Eqs. (8) and (21) to obtain: Ka Kp (x1e − x1r ) /2

= Rx 3e − Rx 3e + Ka Kp



K K a p − x1r − . 2 i

(22)

On the other hand, from Eqs. (9) and (21) the following equations can be considered:   ⎫ R Ka Kp j Kb ∗ Rx 3e = (x1e − x1r ) + M , ⎪ ⎪ ⎬ c Ri RJ T . (23)    j Kb ∗ ⎪ R Ka Kp  /2 ⎪ Rx 3e = − x1r + M ,⎭ c Ri 2 RJ T Subtracting Eqs. (23) it is deduced that K a Kp   /2 Rx 3e − Rx 3e = x1e − , i c 2

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(26) it is possible to determine the values of the first Lyapunov value as follows: for i = 1 : N Kp for j = 1 : N d L1 (i, j ) = f [i , p3 , Kp (i), d (j )]; end end 6. Plot L1 (i, j ) versus d taking Kp as parameter. 7. Choose values of Kp and d to obtain L1 > 0. Consequently, the equilibrium point P3 will be unreachable. It is possible to find different combinations of Kp and d giving L1 > 0. 8. Once Kp is selected compute I /2 from Eq. (10). The maximal integral action will be Im =I /2 + , so the equilibrium point P2 will be unreachable.

(24)

Substituting Eq. (24) into Eq. (22), the following relation between the reset time i of the PID controller and the desired equilibrium angle x1e is obtained: i =

 − 2x1e + 2c . c ( − 2x1e )

(25)

The value of p3 < 0 is defined in Eq. (14). This parameter is the responsible for two pure complex eigenvalues at the equilibrium point P3 . Taking into account that Im = I /2 + , from Eqs. (21) and (24), the following equation can be deduced:

K a Kp  p3 = −Ka Kp  − − . (26) i c

Fig. 2. First Lyapunov value L1 vs. Derivative time d taking the proportional constant Kp as parameter. Value of the reset time i = 1.2784 s.

From Eqs. (15) to (18), (21), (25) and (26) and the condition −/2 < x1e < /2 it is possible to deduce a procedure in order to obtain admissible values of the PID controller parameters. The design of the PID controller can be carried out by means of the following steps: 1. Consider a fixed value for the gyro kinetic moment H, disturbance torque M ∗ and the reference angle x1r . If −/2 < x1r < /2 it is possible to choose x1r ≡ x1e. 2. Choose a set point x1e such that −/2 < x1e < /2. 3. Taking into account Eq. (26), choose > /c in order to assure that p3 < 0. Note that the value of must not be very small or very high in order to obtain an admissible value of i . 4. Determine the reset time i from Eq. (25). Note that i must be positive. 5. Choose an appropriate interval of values Kp Kpmin , Kpmax  and d [dmin , dmax ]. Considering Eqs. (15)–(18) and (25),

Fig. 3. Design plot of the PID controller. Reset time i =1.2784 s. Proportional constant Kp vs. derivative time d taking L1 as parameter.

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9. If the value of Im is very small or negative it is necessary to repeat the previous steps until a suitable value for Im is reached. 10. Check that point P1 is stable from Eqs. (20). Note that the stability condition for point P1 is b1 b2 > b0 , which is verified without problems in almost all the practical cases taking into account the values of the system (see legend of Fig. 1). When all the previous steps are verified, the design of the PID controller is complete. An easy way to implement the previous steps is to choose small values for L1 > 0 and determine all combinations of Kp and d giving the same value of L1 . Then, the function Kp = f (d ) can be plotted. All the points of this plot are valid values to obtain the set point. Figs. 2 and 3 show the values of L1 and the function Kp = f (d ), respectively. 5. Numerical simulations and discussions of the gyro with the PID controller In this section, several examples are carried out to examine the design of the PID controller and the non-linear feedback control by means of numerical simulations. The Runge–Kutta method has been used in the simulations with integration steps between 0.0005 and 0.001 s. The dynamics of the gyro when the set point x1e and the reference x1r are the same is first examined. The simulation results are shown in Fig. 4. The values of Kp and d are chosen from Fig. 3 for a Lyapunov value of L1 = 0.5. Note that the value of L1 must not be very small, since Eqs. (15)–(18) have been deduced considering Taylor expansion only up to terms of third order. So the numerical method may not distinguish the instability of point P3 in case L1 is too small. For example, if the PID controller parameters are chosen

Fig. 4. Simulation results for transient and steady state response cases (a) and (b): nr = 2000 r.p.m.; M ∗ = 100 N m; x1r = x1e = 0 rad; = 2; i = 1.278 s; L1 = 0.5 > 0; Kp = 0.0997 V; d = 0.2524 s; I pi/2 = −0.7178; Im = 1.2822; initial conditions: (0)=0, d(0)/dt =0, i(0)=0.01 A, x4 (0)=0. Eigenvalues at the set point: [−95.8 ± 48.5j, −3.52].

Fig. 5. Simulation results when the first Lyapunov value is negative L1 = −0.5. nr = 2000 r.p.m.; M ∗ = 200 N m; x1r = /2; x1e = 0; = 2; i = 1.278 s; Kp = 0.0202 V; d = 0.3069 s; I pi/2 = 11.172; Im = 13.172; Initial conditions: (0) = 0, d(0)/dt = 0, i(0) = 0.01 A, x4 (0) = 0. Eigenvalues at the set point: [−169.05, −24.05, −2.02].

such that −0.001 < L1 < 0.001, nothing can be assured respect to the stability of point P3 . Nevertheless, this problem can be overcome by choosing an adequate positive value for L1 . Taking L1 =−0.5, Fig. 5 show how the gyroscope firstly evolves to point P2 . However, since Im = I /2 + = 11.1570 + 2 > I /2 = 11.1570 this point is an unstable saddle. Consequently, the system jumps to the equilibrium point P3 , which is a stable weak focus. Note that in this case, from the null initial conditions it is impossible to reach the desired set point. Considering fixed values for the reset time and the first Lyapunov value, if the proportional action decreases, the corresponding derivative time must increase in accordance with Fig. 3, and a more oscillating system is obtained. Note that the integral action I /2 is the responsible for the Kp value. Nevertheless, since the system is highly non-linear, its behavior will depend on the value of H, so it can be very difficult to intuitively predict the motion of the gyro. In the second case, a high value of H (i.e. a high value of the angular velocity of the rotor, such as 523.59 rad/s (5000 rpm)) is used. With L1 =0.5, choosing Kp =0.3 and the corresponding value d = 0.558, a maximum value of the integral action Im is obtained: Im = 0.5873. The simulation result is shown in Fig. 6. Note that in this case, a very fast response is obtained. This response appears when the value of I /2 is low respect to the value of , so the transient response depends on the integral action at point P2 . Fig. 7 show a very interesting case when L1 > 0 is small. The set point is reached after the gyro having a behavior nearly self-oscillating at point P3 . However, since point P3 is an unstable weak focus, the system leaves this point and jumps to the set point. Consequently, from a qualitative point of view, the transient response of the gyro depends on the value of the first Lyapunov value.

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Fig. 6. Simulation results for transient and steady state response from the following values: nr = 5000 r.p.m.; M ∗ = 200 N m; x1r = 0.5 rad; x1e = 1 rad; = 2; i = 3.509 s; L1 = 0.5 > 0; Kp = 0.3 V; d = 0.588 s; I pi/2 = −1.416; Im =0.584; Initial conditions: (0)=0, d(0)/dt =0, i(0)=0.01 A, x4 (0)=0. Eigenvalues at the set point: [−89.60 ± 48.75j, −15.91].

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PID feedback control system with constrained integral action, it has been shown that the equations of the whole system have multiple equilibrium points. Depending on the value of the integral action, a pair of pure imaginary eigenvalues at one of the equilibrium points may appear. Consequently, the appearance of an Andronov–Poincaré–Hopf bifurcation makes the dynamical behavior of the PID controlled gyro very difficult to understand. The paper shows that due to the constrained integral action, one of the equilibrium points is a weak focus. The stability of this point is determined from the sign of the first Lyapunov value. It is important to remark that the calculations are very cumbersome; nevertheless it is the only way to deduce a clear criterion in order to know the stability properties of the weak focus. So, the paper shows how to use a Poincaré–Andronov–Hopf bifurcation to deduce a safe procedure, from which the parameters of the PID controller can be selected. Note that even with more elaborated models of gimbals suspension gyro i.e. with structural flexibility and variable moment of inertia, the outlined method in this paper can be also applied. The difficulty only arises in the algebraic machinery, necessary to calculate the first Lyapunov value. An approach such the one presented in this letter clearly reveals the origin and the connection between complex dynamic phenomena and the design of the control systems. The current application of these methods, however, requires considerable efforts. This reflects an essential point: even with a simple controller such as a PID but with a non-linear model, it may be very difficult to determine the PID controller parameters to obtain a stable predetermined set point.

References

Fig. 7. Simulation results for transient and steady state response from the following values: nr =1200 r.p.m.; M ∗ =100 N m; x1r = /8; x1e =1 rad; =0.5; i = 0.8811 s; L1 = 0.01 > 0; Kp = 0.0237 V; d = 0.4343 s; I pi/2 = 2.6882; Im =3.1882; initial conditions: (0)=0, d(0)/dt=0, i(0)=0.01 A, x4 (0)=0. Eigenvalues at the set point: [−186.83 − 4.834, −3.456].

6. Conclusions In this paper, the non-linear dynamics of a PID controlled gimbals suspension gyro, used to stabilize an external body subjected to disturbance constant torque, has been investigated. By using the non-linear model of the gyroscope, and a simple

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Please cite this article as: M.P. Polo, et al., Tuning of a PID controlled gyro by using the bifurcation theory, Systems Control Lett. (2007), doi: 10.1016/j.sysconle.2007.06.007