Proceedings of the 2004 IEEE

International Conference on Control Applications

Taipei, Taiwan, September 2-4,2004

Synthesis of Absolutely Stabilizing PID Controllers and its

Application to a Ball and Wheel System
Ming-Tzu Ho and Jun-Ming Lu

Abstract-In this paper, we consider the problem of formulated by Lur’e, the absolute stability problem [71 is to
synthesizing proportional-integraI-deriva8ve (PID) controllers study stability of the origin of a Lur’e system, but not for a
that absolutely stahilize a given Lur’e system, Based on particular nonlinearity, rather for any nonlinearity in a given
the circle criterion and the stahility characterization of the
strictly pmitive real property, sufficient conditions for the sector. Several sufficient conditions for absolute stability of
existence of stabilizing PID conhollers are given in terms Lur’e systems have been developed (see [71, [81, [91 and
of simultaneous stabilization of complex polynomials. The references therein). Among these, the most celebrated ones
results from the earlier work are then used to solve the are the Popov criterion [ I O ] and the circle criterion [ I l l .
resulting complex polynomial stabilization problem. For a Based on the Popov criterion or the circle criterion, the
fixed proportional gain, and by sweeping over a variable,
the set of the stabilizing integral and derivative gain values problem of absolute stabilization of the Lur’e system can he
can be determined constructively using linear-programming reduced to synthesizing a linear time-invariant controller for
techniques. The proposed synthesis method is used to design a linear time-invariant system such that a given closed-loop
a stabilizing PID conholler for the hall and wheel system and transfer function is strictly positive real (SPR). Although
the experimental mulB are also presented. the SPR synthesis problem can be recast into the H ,
I. INTRODUCTION [I21 or LMI [I31 framework, the order of the resulting
controller is almost always quite high, being comparable to
In industrial practice, due to hardware and memory
that of the plant. Such high order controllers generally do
constraints, it is always desirable to control a complex
not cope with industrial hardware and memory constraints.
system by using a controller with the low-order and fixed
Thus, the aim of this paper is to develop a framework for
stmcture. Thus the proportional-integral-derivative (PID)
synthesizing PID controllers that absolutely stabilize a given
controller is used in a wide range of industrial applications
Lur’e system. We first show that based on the circle criterion
because of its simplicity and ability to effectively control
and the stability characterization of the SPR property [14],
many industrial processes. Despite the wide popularity of
the absolute stabilization problem can be converted into
PID control, it is unfortunate that currently there is not
simultaneous stabilization of a complex polynomial family.
much theory available for PID designs. Many of the PID
The results on complex PID stabilization developed in [IS]
design techniques are based on empirical evidence alone,
are then used to solve the resulting complex polynomial
with no theoretical justification. Recently, there has been
stabilization problem. For a fixed proportional gain, and by
substantial interest in the problems of PID stabilization
sweeping over a variable, the set of the stabilizing integral
and great progress has been made [1]-[61. Based on a
and derivative gain values can be determined constructively
generalization of the Hermite-Biehler Theorem, [I] pro-
using linear-programming techniques. Then we use the
vided a computational characterization of all stabilizing
proposed synthesis method to design a stabilizing PID
PID controllers for a given continuous-time linear time-
controller for the ball and wheel system to enhance the
invariant plant. Alternative approaches for constructing the
theoretical developments.
parametric space of all stabilizing PID gain values were
presented in [2], [1]. In [4], the characterization of all
stabilizing PID controllers was given for the first-order plant
with dead-time. The results of the discrete-time countelpart
of PID stabilization were presented in [ 5 ] , [6].
Motivated by these earlier works, in this paper we inves-
tigate PID stabilization of nonlinear systems. In particular,
The paper is organized as follows. Section 2 introduces
we focus on the problem of stabilizing Lur’e systems. The
notation and preliminaries. In Section 3, we reduce the
Lur’e system is an important and common class of nonlinear
problem of synthesizing stabilizing PID controllers for a
systems. The Lur’e system consists of a forward path
given Lur’e system to simultaneous stabilization of complex
containing a linear time-invariant subsystem and a feedback
polynomials. The resulting complex polynomial stahiliza-
path containing a sector bounded nonlinearity. Originally
tion problem can be solved by the earlier work. Section 4
mihis work was supponed by thc National Science Council of TGwm contains the design of stabilizing PID controllers for the
under Gmnt NSC 92-2213-E-Wh070. ball and wheel system and the experimental results are
Ming-Tzu Ho a d Jun-Miog Lu arc with the Depanmettt of Engineering
Science. National Cheng Kung University. I. University Road. Tailwn 701. also presented. Finally, Section 5 contains some concluding
Tarw~bbruceho~mail.ncku.edu.tw remarks.

0-7803-8633-7/04/$20.00 02004 IEEE 825

Authorized licensed use limited to: Slovak University of Technology Trial User. Downloaded on February 20,2023 at 11:02:41 UTC from IEEE Xplore. Restrictions apply.
 11. NOTATION
A N D PRELIMINARIES 111. AHSOLUTE
STABILIZATION USING PID
CONTROLLERS
Consider the Lur'e system described as
Consider the control configuration of the Lur'e system
x = Az+b (I) as shown in Fig. 2. Here P ( s ) is a 2 x 2 transfer function
y = m+du (2)
U = -4(&Y) (3)
where z E Rn, U, y E R,( A , b) is controllable, ( A , c ) is
observable, and 4 : [0, ea)x R i R is a memoryless, but
possibly time-varying, nonlinearity. This system is shown in
Fig. I. In the forward path, the single-input single-output
linear time-invariant system G(s) = c(sl-A)-'b+d. The

4-G-U L ' I
Fig. 2. The c o n n ~ configuration
l of the Lur'e system

Fig. 1. Lur'e system

Nonlinearity 4 lies in a given sector [a,01 and C(s) is

concepts of the sector bounded nonlinearity and absolute
the controller used for stabilizing the closed-loop system.
stability are introduced.
In this paper, the controller C(s) is chosen to be a PID
Definition 2.1: 4 : [0,a)x R + R,and a, 0 E R with
controller, i.e.
n < 0.Then 4 is said to belong to the sector [a,p] if
(1) 4(t, 0) = 0, vt t 0;
(2) [4(t>Y) -ayl[d(t, Y) -0Yl 5 0. 'Jt2 0%VY E
R. The interconnection given in Fig. 2 is defined by the
Definition 2.2: [I61 The Lur'e system (1)-(3) is said to following equations:
be absolutely stable if the equilibrium point z = 0 is glob-
ally uniformly asymptotically stable for any nonlinearity in
the given sector. It is absolutely stable with a finite domain
if the equilibrium point is uniformly asymptotically stable.
The circle criterion shows the SPR condition related to
absolute stability as follows: U1 = 4% Yd
Theorem 2.1: (Circle criterion) [ I l l 212 = C(S)Y?.
Consider the Lnr'e system as shown in Fig. 1 with The objective here is to synthesize the PID controller C ( s )
[a,p]. Then the system is absolutely stable if l+a~o
l+PC($
such that the system shown in Fig. 2 is absolutely stable
SPR. for 4 E [a,PI. Consider the transfer function from u1 to
The following lemma provides a stability characterization y1. It is given by
of proper stable real transfer functions satisfying the SPR
property. More precisely, let PIl(S) + PlZ(S)C(S)[l - Pzz(s)C(s)l-lP21(s).
By setting

G ( s ) = -[PII(s)
Piz(s)C(s) +
be a stable and real proper transfer function. N ( s ) and D ( s ) X P ~ ~22(s)c(s)1-~Pz1(s)1 (2)
are coprime polynomials.
the system shown in Fig. 2 can then be transformed into the
Lemma 2.1: [I41 G(s) is S P R if and only if the following
Lnr'e system as shown in Fig. 1. Furthermore, for C(s) to
conditions are satisfied
be a PID controller, G ( s ) can be presented in the following
(4 Re[G(O)l > 0; form:
(b) N ( s ) is Hunvitz stable;
(c) D(s)+jXN(s) is Hurwitz stable for all X E R. G(s) =
+
E ( s ) ( k d 2 k,s + +
ki)F(S)
+ + +
A(S) ( k d S 2 kpS k i ) B ( S )

826

Authorized licensed use limited to: Slovak University of Technology Trial User. Downloaded on February 20,2023 at 11:02:41 UTC from IEEE Xplore. Restrictions apply.
 where A ( s ) , B ( s ) , E ( s ) , and P ( s ) are some real polyno- set is denoted by S,. Then for a fixed k p , the stabilizing
mials. For a given sector [a, 01, we define the following ( k i , l i d ) region is given by
polynomials:
s = niZl,2, 3 ~ i .

Sz(s, IC,, ki, bd, A ) 5 IA(~)(l+jh)+E(s)(n+jhp)J By sweeping over kp in the necessary ranges and determin-
+(ICrs* + k,s + k,)IB(s)(l +?A) ing the corresponding S at each stage, we can obtain the
+F(.)(U + j A P ) l . stahilizing set of ( k p , ki, kd) values for the Lur'e system.
Remark 3.1: For the case of the nonlinearity locally be-
The next theorem reduces the PID absolute stabilization
longing in the sector, Theorem 3.1 could only lead to abso-
problem to the problem of simultaneously stabilizing a
lute stability with a finite domain. Once the stabilizing PID
complex polynomial family.
controller is determined, based on the Kalman-Yakubovich-
Theorem 3.1: Given the sector [a,p] and 4 E [a,01,
Popov lemma [16], a quadratic Lyapunov function can be
the Lur'e control system shown in Fig. 2 is absolutely
found for estimating the domain of attraction.
stahilizable by a PID controller if there exists (k,, ki, k d )
such that the following conditions hold Remark 3.2: It should he pointed out that Theorem 3.1
gives sufficient design conditions and thus may be conserva-
( I ) &(s, k p , Ai, k d ) is Hurwitz; tive. When G(s) is strictly proper and 4 is time-invariant,
(2) J2(s, k,, k i , k d , A) is Hurwitz for all X E E, the PID absolute stabilization problem can be treated in
a similar fashion based on the Popov criterion and it can
provide a less conservative design.
trol system shown in Fig. 2 is absolutely stable if
lA(a)+OE(*) + ( b s Z + k s+ki)lB(s)+4F(s)
n(,)+ua(,~+(IC~~~+k~ B~
( s++t a. jF ( a
is s p ~~h~~
, con.
&tion (I), 12). and (3) fohod immdiately from Lemma 1v. APPLICATION
TO THE BALLAND WHEEL SYSTEM
2.1.
Theorem 3.1 involves simultaneous stabilization of poly- In this section, we apply the preceding design technique
nomials &(s, k,> k i , k d ) and &(s, kp, k;, k d , A). With to the problem of balancing a hall on the periphery of a
a fixed A , both &(s, k,, k i , kd) and 62(s, k,, k6, k d , A) wheel as shown in Fig. 3. We first formulate the stabilization
are of the following form: problem in question as PILI absolute stabilization of a
Lur'e system. Then a set of stabilizing PID controllers is
6 ( S , kp, ki, kd) = L(S) + (kd? + kpS + k i ) M ( S ) determined and the experimental results are repolted.
(3)
where L ( s ) and M ( s ) are some given complex polynomials.
In [15], stabilization of (3) was referred to as complex PID
stabilization and a synthesis procedure was also provided
for determining all stabilizing (k,, k i , k d ) values, if any,
for which (3) is Hurwitz. It has been shown that for a fixed
k,. the stabilizing ( k i ; k d ) values are the feasible solutions
of a set of linear inequalities. Accordingly, by sweeping over
kp the linear-programming techniques are used to generate
the parametric space of the entire stabilizing (kp. k i , k d )
gain values for (3). Note that the necessary ranges of
stabilizing k, can he prescribed by using the root locus
method presented in [I], [171.
Now, we first determine the necessary ranges of kp such
that 61 and 62 are simultaneously stahilizable for all X E R.
Then with a fixed kp in the necessary ranges, using the re-
sults on complex PID stabilization, we are able to determine Fig. 3. Ball and wheel.
the entire admissible ( k i , k d ) region such that condition (1)
of Theorem 3.1 is satisfied. The resulting admissible set is
denoted by SI. With the same kp, sweeping over X E R,
and using the results on complex PlD stabilization again, A. Model
we can determine the entire admissible ( k , , kd) region such
that condition (2) of Theorem 3.1 is satisfied. Let S2 denote Assuming that the coefficient of friction is large enough
the resulting admissible set. Condition (3) of Theorem 3.1 such that the ball rolls on the wheel without slipping. Using
gives a set of linear inequalities in k; and the admissible Euler-Lagrange formulation [IS], the dynamic equations of

827

Authorized licensed use limited to: Slovak University of Technology Trial User. Downloaded on February 20,2023 at 11:02:41 UTC from IEEE Xplore. Restrictions apply.
 the system can be written as B. Controller Design
2 2 2 Now we will proceed to design a PID controller fur
(-,mST: - ;?7IbTw1.bj8; + (I, + 5 -mbT?")& = T
balancing the ball on the wheel around the equilibrium
(4) point. With the system given by (8)-(11), we first define
(-71.b - 7T,)il + 2r,& + 5gsino1 = 0 the output

(5) y = klXl +k3x3 (12)

where 81 is the angle made by the hall with respect to the where the sensor gain kl = 10 vlrad and the sensor gain
wheel, 02 is the wheel angular position, and r is the control k3 = -1 vlrad. Let the control voltage be
torque exerted on the wheel. I, is the inenia of the wheel,
m b is the mass of the ball, r, is the radius of the wheel, and
v = C(s)(r - y) (13)
rg is the radius of the ball. g is the gravitational acceleration. where C ( s ) is the PID controller to be designed, and r is
The system dynamic equations (4) and ( 5 ) are valid only the reference voltage that is set to be 0. From the system
when the centripetal force is,large enough to maintain the equations (8)-(13), we have that the origin is the equilibrium
circular motion of the hall on the wheel. Otherwise, the point of the closed-loop system. The resulting closed-loop
hall will leave the wheel. Thus tu maintain the hall on the system can also be represented in terms of block diagrams
wheel, the following condition must he satisfied with the transfer functions and the nonlinearity shown in
Fig. 4. Here
gCOS81 > (7, + r b ) 6 I 2 . (6)
A voltage signal is generated according to the designed
control law and it is supplied to an amplifier which drives c
a permanent magnet DC motor to control the wheel. The Hz(sj = -
s(s - d )
relation between the control torquer and the control voltage
u is given by H3(sj = e
s(s - d )
&(s) =
f
(7) ~

S(S-d)'
where R, is the motor m a t u r e resistance and K , is the The closed-loop feedback system shown in Fig. 4 can he
motor constant.
We define the state variables as follows:

1
. x2, 23, x41T = [OI, 61, 02, 621'.
With the physical parameters of the system given in Table
I, the state space representation of the system (4), (5) and
(7) is given by
dl = x2 (8) Fig. 4. Feedback convol of the ball and wheel system
i 2 = axr+bsinxl+cu (9)
13 = 24 (10) formulated as the control configuration of the Lur'e system
x4 = d x q + e s i n x l + fu (11) as shown in Fig. 2 with
where a = -0.1647, b = 53.047, c = 1.5913. d =
-0.6361, e = 2.1591, and f = 6.1459. The same system
will also be used for the experimental investigation in the
sequel.

TABLE I
THEPHYSICAI. PARAMETERS OF THE SYSTEM.

2, = inertla of the wheel=9.Q38 Y lo-' kg m'

vW = radius of the wheel = 0.121 m
mb = m s s of the bvll = 0.065 kg
y b =radius of the bvll = 0.0125 m
g = gnvitationvl acceleration = 9.8 I d s 2
R, = motor mature resislance = 1.6 n
K , = motor constant = 0.10352 N d A

828

Authorized licensed use limited to: Slovak University of Technology Trial User. Downloaded on February 20,2023 at 11:02:41 UTC from IEEE Xplore. Restrictions apply.
 To prevent noise amplification and intemal instability
induced by the pure derivative term, the PID controller is
implemented in the following form: .. :.. .......... .... ..... : . . . .

I
C(S) =
k d + k,s + k;
s(1 + 0.01s)
dl ..... ..:.........j .........I ...... /
/
I
With the physical parameters of the system given earlier
and (14), we obtain

Using the design technique proposed in this paper, we find

that there is no PID controller that can achieve absolute
stabilization for any global sector conditions. To improve
system robustness and to achieve regional stabilization. the
local sector condition is chosen to he [a;01 = [0.9549, 1.21
such that the nonlinearity sin(x1) lies in the chosen sector
for z1 E [-:, 51. Note that since the nonlinearity sin(z1) Fig. 5. Portion of the stabilizing set of (ki,k d ) values for k, = 35.
belongs locally to the chosen sector condition, we can
only achieve absolute stability with a finite domain. From
Theorem 3.1, we know that the stabilizing (kp, ki, k d ) ..... .. .. .
values exist if the following conditions hold:
+
(1) ( 0 . 0 1 ~ ~1 . 0 0 6 4 -~ ~0 . 0 0 0 5 ~ ~6 4 . 0 5 7 1 -
~ ~~
40.0F51s2)+(9.7671s2+387.1015)(kds~+k,s+
k ; ) is Hunvitz;
+ + +
( 2 ) ( 0 . 0 1 ~ ~1 . 0 0 6 4 ~ ~ n.636iS4)(i j x ) -
+ +
(0.5305~4 ~ ~ . u s o~3.3mS2)(o.9549 ~s~ +
+ + +
I.2jX) ( k d S 2 kp3 k;)[9.7671s2(1 j x ) + +
322.5846(0.9549+ l . Z j X ) ] is Hunvitz for all E
R.
Using the root locus method presented in [I], [171, the nec-
essary range of k, for the existence of stabilizing ( k ; , k d )
,,(a
values is that k , t [27.9102, 47.11973. Choosing a fixed
k, E 127.9102, 47.11971, for instance k, = 36, and using
the results on complex PID stabilization, we can obtain the
stabilizing ( k ; , k d ) region S.Portion of S is sketched in Fig. 6. The stabilizing se1 of (k,,, k i , kd) values
Fig. 5 . By sweeping over k, E [27.9102, 47.11973, and
determining the corresponding S at each stage, we can
obtain the stabilizing set of (kp, k i , k d ) values for this
of the finite domain of the sector condition: 1x11 S 2
ball and wheel system. This stabilizing set is sketched in
and (6), an estimate of the domain of attraction can then
Fig. 6.
he determined. With the initial condition chosen inside
C. Experimefiral Results the domain of attraction, the angular position responses of
In this subsection, we present the experimental results the ball and the wheel are shown in Fig. 8 and Fig. 9,
obtained on a hardware setup of the same system used in respectively. These results confirm that the resulting PID
the controller design. This experimental apparatus is shown controller stabilizes the system. Steady state errors in the
in Fig. 7. The rim of the wheel has a groove to prevent angular positions of both hall and wheel may be attributed to
the ball from slipping out. The angular position of the ball friction. To test robustness of this PID control, a disturbance
(81)is measured by an optical angular displacement sensor was added manually by pushing the hall away from the
and the angular position of the wheel (02) is measured by equilibrium state. Fig. 8 and Fig. 9 also show that this PID
a potentiometer. control responds robustly with the uncertain disturbance.
From the stabilizing region shown in Fig. 5, we choose
REMARKS
V. CONCLUDING
the stahilizina- PID gain
- values to be k , = 35, A i = 300,
and k d = 3. The resulting PID controller of the form In this paper, we have presented a synthesis method for
of (14) is implemented in the analog circuits. Using the determining the PlD controllers that absolutely stabilize a
Kalman-Yakubovich-Popovlemma, we can find a quadratic L u ' e system. First, the sufficient design conditions have
Lyapunov function for this system. Under the constraints been given in terms of simultaneous stabilization of a

829

Authorized licensed use limited to: Slovak University of Technology Trial User. Downloaded on February 20,2023 at 11:02:41 UTC from IEEE Xplore. Restrictions apply.
 ....... . . . . . . . . . . . 1 1
. .. ..: . . . . . .. .. .

. . . . . . . . . . . . .

. . . . . . . . . .

.............
II
-08 ..... ..... ....... . . . . . .
disturb-
-1
10 m
7,"'. ,Iri

Fig. 9. The angular position of the wheel.

REFERENCES
[I] A. Dum. M. T, Ho. and S. P. Bhuttacharyya, Struclure and Synthesis
~ i g7.. Photograph of the exptimntd apparatus balancbg the b d i on of PID Conrrollers. Springer-Vedag, London, U.K.. 2wO.
the wheel. 121 M. T. SBylemer, N. Munro, and H. B&, "Fasl Culmhtion of
Stabilizing PID Controllers," Automarica, Vol. 39, 121-126, 2003.
[31 I. Ackemann and D. Kacsbauer, "SLzble Polyhedra in Parameter
Space:' Automorica, Vol. 39. 937-943, 2003.
141 G.J. Silva. A. Datu. and S . P. Bhattacharyya, "New Results on the
Synthesis of PlD Controllers:' IEEE Trans. on A U I O ~ IContr,
. Vol.
............ 47, No. 2, 241-252, Feb. 2002.
[5] H. XU. A. Datta, and S. P. Bhattachiuyya, 'Computation of MI
Stabilizing PID Gains for Digital Control Systems," IEEE Trans. on
Automi. Conrr, Vol. 46, No. 4, 647-652, Apr. 2001.
[6] L. H. Keel, J. 1. Rego, and S. P. Bhatlachwyyu, "A New Approach
to Digital PID Controller Design," IEEE Tmns. on Aulomar. Contr,
Vol. 48, No. 4, 687-692, Apr. ZW3.
[71 M. A. Aluemwn and F. R. Guntmacher, Absolure Srability OfRegu-
lator Systems, Holden-Day, San Francisco, CA, 1964.
[XI K. S. Narendn and J. H. Taylor, Fmpuency Domain Criteria for
Absolute Srobility, Academic Press, New York, 1971.
191 C. A. Desoer and M. Vidyasagar. Feedback Systems: Inpur-Ourpur
Propenirs. Academic Press, New York. 1975.
[IO] V, M. Popov, "Ahsolute Stability of Nodimear Systems of Automatic
Control:' Auromotion Md Remore Control, Vol. 22, 857-875, 1961.
[ I11 G . Zumes. "On !he Input-Output Stability of Time-vazying Nonlincar
Fecdback Syssms," M s I and 11, IEEE Tmns. on Automat. Conrr,
Vol. I I, 228-238.465-476, 1966.
[I21 W. Sun, P. P. Khargonekar, and D. Shim, "Solution to !he Positive
Fig. 8. The angular position of the ball R w l ConmI Pmhlem for Linear lime-invariant Systems:' IEEE
Trans. on Aulomar. Conn. Vol. 39. No. 10, 20342046,Oct. 1994.
[IS] L. Turan, M. G. Safonav, and C. H. Huang, "Synthesis of Positive
Real Feedback Systems: A Simple Derivation via Parrott's Theorem:'
complex polynomial family. Then using the results from IEEE Trans. on Automat. Conlr. Vol. 42. No. 8, 1154-1157, Aug.
the earlier work, a linear-programming characterization of 1997.
[I41 S. P. Bhamchaqya, H. Chapellat and L. H. Keel, Robust Contml:
stabilizing PID controllers has been provided. The pro- 7%e Pammerric Approach. Prentice Hall, Upper Saddle River, NI,
posed method provides not just a single solution, hut the 1995.
parametric set of stabilizing PID gain values. This feature [IS] M. T. Ha. "Synthesis of H , PID Controllers: A ParameUic Ap-
proach:' Aulomtica, Vol. 39, 1069-1075, 2003.
possibly facilitates the robust and optimal control design 1161 H. K. m i l , ~ o n ~ i n e Sysrems,
or prentice all, upper saddlc River,
of Lur'e systems using PID control and it is the subject NJ, ZOM.
currently under investigation. In this paper, we also apply [I71 M. T. Ho md C. Y. Lin, " P D Controller Design far Robust
Perfomancc:' IEEE Trans. on Automat. Contr.. Vol. 48. No. 8. 14M-
the proposed method to design and implement PID control 1409, Aug. 2003.
on a physical device, the ball and wheel system. It has been [I81 M. W, Spong and M. Wdyaagar, Robor D y ~ m i c rand Conlml,
shown that the experimental investigations agree with the Wky, New York. 1989.
theoretical developments.

830

Authorized licensed use limited to: Slovak University of Technology Trial User. Downloaded on February 20,2023 at 11:02:41 UTC from IEEE Xplore. Restrictions apply.