Design of Control System Based on Functions of Catastrophe
Gulnara Abitova, Vladimir Nikulin, Binghamton University, USA
Mamyrbek Beisenbi, Aliya Ainagulova, Eurasian National University, Kazakhstan
Abstract: In this article a new approach for the choice of control laws for objects with
uncertain parameters in the class of structurally-stable mapping of catastrophe theory is
proposed, which allows construction of a control system with high capacity of
robust stability. The proposed control circuit uses function with three parameters of a
controller and two state variables. This paper presents conditions for equilibrium
states, and it explains how to create a practical design of a controller in order to increase
the potential of robust stability in a very wide range. Conclusions are confirmed by numerical
simulations.
Keywords: – robust stability, robust control, catastrophe function, system of the automation
control, mathematical model.
1. Introduction
The issues of maintaining and increasing robust stability are one of the most important issues
in control theory and are of great practical interest. The general formulation is to
specify restrictions on variations of system parameters, allowing maintenance
of stability. Clearly, these limits (restrictions) are determined by the area of stability for
uncertain parameters of an object and the given parameters of a control unit (controller).
Existing methods [1] of designing control systems with uncertain parameters mainly focus on
the assessment of robust stability of a system of a given structure with linear control
laws or instantaneous nonlinear (relay) characteristics and they do not allow designing a
control system with a sufficiently wide area of the robust stability provided that there is a
great uncertainty of parameters of the control object and the drift (driving) of their
characteristics in wide ranges.
In [2] particular, dynamic systems, which address the control development of the selforganization processes in physic-chemical and biological systems are considered. Models of
these systems are in the form of structurally stable mappings of the catastrophe
theory [3] and they are studied as a universal mathematical model of development and selforganization in nature. Therefore, it is of particular interest in conditions of great
uncertainty, to design a system of an automatic control in the class of structurally stable
mappings, with mathematical models of complex behavior related systems [4],
namely having a lot of both sustainable solutions.
This article discloses the essential issues of designing sustainable (stable) systems of
robust control of linear dynamic objects with uncertain parameters. In this paper an
approach for the choice of control laws in the class of structurally stable mappings
of catastrophe theory [4] proposes is proposed, which allows increasing the capacity of
robust stability and indicators of quality control. The study of the robust stability is shown on
the examples of automatic control systems for astatic objects of the first or second level.
One of the examples of such system is a manipulator that is typically described by a highly
nonlinear system which is prone to changes of dynamic characteristics at different payloads
and different operating conditions. As such, it requires a special control strategy that
assures stability over a wide range of parameter variations. Some
traditional robust control solutions are reported in [1], [2] and [3]. Our approach described in
the following sections is based on catastrophe functions [4-5].
2. Study of a Robust Stability of the System (by Examples
of Design of the Control System)
The possibility of increasing the capacity of robust stability can be shown on the examples of
designing automatic control systems for astatic objects of the first or second level. Control
systems with control laws in the class of structurally stable mappings preserve the stability
properties at a wide range of variations in the parameters of an object and a controller. The
study of robust stability of the system is based on the ideas of the linear
approximation and the linear principle of stability studies.
2.1 Design and analysis of the control system for oneparameter structurally stable mappings (catastrophe fold)
The integrating element with constant integrator T is used as a controlled object. A control
law in the form of one-parameter structurally stable mappings (catastrophe fold) is chosen for
specified object:
u = −x 3 + kx
(1).
The automatic control system (ACS) with the selected control law can be represented with a
diagram shown in the Figure 1.
Figure 1. Block diagram of the automatic control system.
where: F(x) = −x 3 .
Title
The equation of system state relative to the output variable x is written as:
dx
dt
1
= T (−x 3 + kx)
(2).
The system (2) with nonlinear control law (1) has steady state of the system x1S = 0 with a
negative k (k < 0) and it has two steady states points xS2 = √k , xS3 = −√k with a positive
k (k > 0). These pairs of steady states of the system converge to x1S when k = 0 and diverge
from it when k > 0, i.e. there is a bifurcation at k = 0.
As in this case a rather simple model is considered, the problem of studying the stability is
rather trivial, since the equation (2) permits the integration:
x(t) =
k
√
k − x02 −2kt
1+
e T
x02
where: x(t 0 ) = x0 is the initial value of state variable of the system.
It turns out, that the state x1S is asymptotically stable when k < 0 and it becomes unstable
when k>0; the other states xS2 and xS3 are also asymptotically stable. In other words,
derivations of steady states xS2 and xS3 are the result of a bifurcation at that time when the
state x1S = 0 loses its stability, and those derivations themselves are stable. Actually, the
connection between above-critical derivation and stability is not accidental. According to one
of the general results of the catastrophe theory [1] at the nonlinear function (1) above-critical
derivations are stable and under-critical derivations are unstable. Therefore it is obvious, that
steady states of the system (2) are directly determined by the strengthening coefficient k,
regardless of the constant of integration, and the nonlinear control law (1) enables the
stability of the system at any change of the parameter k within the limits (fig. 2).
T=0.5, k=2
T=0.5, k=-2
1.45
1.2
1.4
1
1.35
0.8
1.3
0.6
x1
x1
1.25
1.2
0.4
1.15
0.2
1.1
0
1.05
1
0
1
2
3
4
5
Time,s
6
7
8
9
10
0
1
2
3
4
5
Time,s
Figure 2. Simulation results: Experiment 1, x1-x2.
T=0.5, k=-2
1.2
1
0.8
x1
0.6
0.4
0.2
0
-0.2
-0.2
0
1
2
3
4
5
Time,s
6
7
8
9
10
6
7
8
9
10
2.2 Design and analysis of the control system with cascade (serial)
connection of astatic factors of the first level
The cascade (serial) connection of the integrating factor with integration constant T1 and
astatic actuating device with integration constant T2 (fig.3) is used as the controlled object.
Figure 3. Block diagram of the automatic control system.
It is generally known [2] the system with cascade (serial) connection of astatic factors of the
first level (fig.4) at the linear control law u = kx1 and at any values of strengthening
coefficient k is at the borderline of stability (k<0), or it is unstable (k>0). So, within the
linear control law it is not possible to maintain stability of the considered system.
2.3 Design and analysis of the control system for the astatic object
of the second level – one-parameter structure stable mappings
An astatic object of the second level is considered in this case. A control law in the form of
one-parameter structure stable mappings u = −x13 + kx of the system is selected for the
considered object, as it is represented on the Fig.4.
Figure 4. Block diagram of the automatic control system.
where: F(x1 ) = −x13 .
The block diagram of the system represented on the Fig.5 equals to the following state
equations:
Title
dx1
dt
dx2
dt
=
1
T1
(−x13 + kx1 + x2 ),
1
= T (−x13 + kx1 )
(3).
2
2,3
1
The system (3) has steady-states x1S
= x2S = 0 and x1S
= ±√k, x2S = 0.
2,3
1
Steady-states x1S
= ±√k, x2S = 0 converge with x1S
= x2S = 0 when strengthening
coefficient k = 0 and they diverge from it only when k>0.
The research of the system stability (3) proves that in order to reach stability of the steady1
state x1S
= x2S = 0 it is only necessary for the strengthening coefficient k to be less than 0
2,3
(k<0) , and to achieve the stability of the state x1S
= ±√k, x2S = 0 it is only necessary for
the k to be always more than 0 (k>0) (Fig. 5).
T1=0.2, T2=0.5, k1=-2
T1=0.2, T2=0.5, k1=-2
2
0.5
0.4
0.3
0.2
1.5
x2
x1
0.1
0
-0.1
1
-0.2
-0.3
-0.4
0.5
0
0.5
1
1.5
2
2.5
Time,s
3
3.5
4
4.5
5
-0.5
0
0.5
1
1.5
2
2.5
Time,s
Figure 5. Simulation results: Experiment 2, x1-x2.
3
3.5
4
4.5
5
2.4 Design and analysis of the control system for the astatic object
of the second level – two-parameter structure stable mappings
(catastrophe fitting-up /or catastrophe confection)
A control law for astatic object of the second level was selected as two-parameter structure
stable mappings catastrophe fitting-up (catastrophe confection) u1 = −x14 + k1 x12 + k 2 x1
(Fig.6).
Figure 6. Block diagram of the automatic control system.
where: F1 (x1 ) = x14 , F2 (x1 ) = x12 .
Block diagram of the system represented on the fig.9 is equal to state equations:
dx1
dt
dx2
dt
1
= T (−x14 + k1 x12 + k 2 x1 + x2 )
1
1
= T (−x14 + k1 x12 + k 2 x1 )
2
(4).
The steady-states of the system (4):
4
2
−x1S
+ k1 x1S
+ k 2 x1S = 0, x2S = 0.
(5).
The trivial solution of the equation (5) is:
1
x1S
= 0, x2S = 0
(6)
and the nontrivial solution defined by the solution of the equation is the following:
3
−x1S
+ k1 x1S + k 2 = 0
(7).
The solution of the equation (7) corresponds to the critical, twice singular (vacuous) points of
the catastrophe of setup.
Title
The position of the twice singular (vacuous) point leads to the following expression [3]:
k
3
k
2
( 31 ) + ( 22 ) = 0.
This equation shapes a parametric representation of the connection between coefficients k1
and k 2 . Hence:
2
3
3
k2
k
2
k1 = 3 ( √− 2 ) , x1S
= √− 22 , x2S = 0
(8).
The study of stability of the steady states (6) and (8) is based on the linear principle
(approach). That study shows that the steady state (6) is globally stable when any k1 > 0 and
k 2 < 0 , and another steady state (8), which emerge at k1 > 0 and k 2 > 0, is also stable.
Hence, points located on the folding line [4] are also stable. Therefore, the system (9) can
assure stability at any rates of uncertainty of values of the coefficient k 2 (fig. 7).
T1=0.2, T2=0.5, k1=2, k2=3
T1=0.2, T2=0.5, k1=2, k2=3
2.5
1
2
x2
x1
0.5
1.5
0
1
0.5
0
0.5
1
1.5
2
2.5
Time,s
3
3.5
4
4.5
-0.5
5
0
0.5
1
1.5
2
2.5
Time,s
3
3.5
4
4.5
5
Figure 7. Simulation results: Experiment 3, x1-x2.
T1=0.2, T2=0.5, k1=2, k2=3
1
2.5 Design and analysis of the control system for astatic object of
the second level - three-parameter structure stable mappings
(catastrophe of hyperbolic umbilic)
x2
0.5
0
If the control law for astatic object of the second level is in the form of three-parameter
structure stable mappings (catastrophe of hyperbolic umbilic):
-0.5
0
0.5
1
1.5
2
2.5
Time,s
3
3.5
−x13
4
4.5
x23
5
u=
− − k1 x1 x2 + k 2 x2 + k 3 x1 ,
then, the automatic control system can be represented with the following block diagram
(fig.8), where: F1 (x1 ) = −x 3 , F2 (x2 ) = −x23 , F3 (x1 , x2 ) = −k1 x1 x2 .
Figure 8. Block diagram of the automatic control system.
The equation of the system state with the variables x1 and x2 is written as follows [1, 5-6]:
dx1
dt
dx2
dt
1
= T x2 ,
1
1
= T (−x13 − x23 − k1 x1 x2 + k 2 x2 + k 3 x1 )
2
(9).
2,3
1
The steady states of the system (9) are the following values: x1S
= x2S = 0 and x1S
=
2,3
1
±√k 3 , x2S = 0. Steady states x1S = ±√k 3 , x2S = 0 converge with x1S = x2S = 0 when
k 3 = 0 and diverge from it when k 3 > 0 .
Steady states of the system (9) are calculated with the solutions of the following equation [7]:
3
3
−x1s
− x2s
− k1 x1s x2s + k 2 x2s + k 3 x1s = 0, x2S = 0
The trivial solution of the equation (10) is:
1
x1S
= 0, x2S = 0
(10).
(11)
and two other solutions are:
2
x1S
= √k 3 , x2S = 0
3
x1S
= −√k 3 , x2S = 0
(12),
(13).
The steady states (12) and (13) converge with (11) when k 3 = 0 and diverge from it when
k 3 > 0, and they both cannot exist at the same time.
The stability of the steady states (11) – (13) is studied using the principle of stability of the
linear system.
Title
The stability conditions of the steady state (11) when T1 > 0 and T2 > 0 are:
k2
2T2
k2
± √4T22 + T
2
k3
1 T2
<0
(14),
At the same time, the stability conditions of steady states (12) and (13) are:
−
k1 √k3 −k2
2T2
2
(k √k −k )
k
± √ 1 2T3 2 2 − 2 T T3 < 0,
2
(15)
1 2
and
k1 √k3 +k2
2T2
2
(k √k +k )
k
± √ 1 2T3 2 2 − 2 T T3 < 0 .
2
(16).
1 2
respectively.
Our analysis indicated that state (11) is asymptotically stable if k 2 < 0 and k 3 < 0; and it
becomes unstable when k 2 > 0, k 3 > 0. However, the steady states (12-13) appearing
when k 3 > 0 are also asymptotically stable if: k1 > 0, k 2 > 0, k 3 > 0 and as well as
under the conditions of (15-16) (fig. 9).
T1=0.2,T2=0.5, k1=2,k2=-3,k3=-5
T1=0.2,T2=0.5, k1=2,k2=3,k3=5
1.5
3
x1
x2
2.5
1
2
x1
x2
1.5
x
x
0.5
0
1
0.5
0
-0.5
-0.5
-1
0
1
2
3
4
5
Time,s
6
7
8
9
10
-1
0
1
2
3
4
5
Time,s
6
7
8
9
10
Figure 9. Simulation results: Experiment 4-5, x1-x2.
2.6 Design and analysis of the control system for the astatic object
of the second level – catastrophe of elliptic umbilic
The system with astatic object of the second level and with control law given in the form
elliptic umbilic catastrophe is considered in this case:
u = −x13 − 3x1 x22 − k1 (x22 +x12 ) + k 2 x1 + k 3 x2 ,
(Fig. 10),
where:
F1 (x1 ) = −x13 , F2 (x1 , x2 ) = − 3x1 x22 , F3 (x1 ) = −x12, F4 (x) = −x22 .
The control system with two successive integrating factors and given control law is written in
the following way:
dx1
dt
dx2
dt
1
= T x2 ,
1
1
= T (−x13 − 3x1 x22 − k1 (x22 + x12 ) + k 2 x2 + k 3 x1 ).
2
(17).
Figure 10. Block diagram of the automatic control system.
Steady states of the system (17) are calculated with the solution of the equations:
3
2
2
2 )
x1s
− 3x1s x2s
− k1 (x2s
+x1s
+ k 2 x2s + k 3 x1s = 0, x2S = 0 (18).
A trivial solution of the equation (18) is:
1
x1S
= 0, x2S = 0,
(19)
and two other steady states of the system are:
2
x1S
=−
and
k1
2
k2
+ √ 41 + k 3 ,
x2S = 0,
(20)
Title
2
x1S
=−
k1
2
k2
+ √ 41 + k 3 ,
x2S = 0
(21).
The study of the stability of the steady states (19) – (21) is based on the linear principle.
Steady state conditions (19) are illustrated in the following manner:
k2
k2
2T2
k3
± √4T2 2 + T
1 T2
2
< 0 , i.e. if T1 > 0, T2 > 0, then k 2 < 0 и k 3 < 0.
For positive values k1 and k 2 , provided that k12 + 4k 3 > 0 , the conditions of stability of
steady states (20) and (21) are: k 2 < 0, k 3 > 0 and k 2 < 0, k 3 < 0 respectively (fig. 11).
T1=0.2, T2=0.5, k1=2,k2=-3,k3=5
T1=0.2, T2=0.5, k1=2,k2=-3,k3=5
1
2
0.8
0.6
0.4
1.5
x1
x2
0.2
0
-0.2
1
-0.4
-0.6
-0.8
0.5
0
1
2
3
4
5
Time,s
6
7
8
9
10
-1
0
1
2
3
4
5
Time,s
6
7
8
9
10
Figure 11. Simulation results: Experiment 6, x1-x2.
3. Practical Example of Design of the System Control
3.1 Analysis of the Controlled Plant
As a practical case we can select a mechanical system whose operation can be described by a
set of nonlinear relationships. For example, dynamics of a robotic manipulator with n joints
can be presented by a second-order differential equation given in the matrix-vector form as
follows [7]:
M(θ)θ̈ + N(θ, θ̇) + G(θ) + H(θ̇) = U
(22),
where θ(t) is the vector of joint positions, U(t) is the vector of applied joint torques, M is the
inertia matrix, N(θ, θ̇) is the Coriolis and centrifugal torque vector, G(θ) is the gravity
loading vector, and H(θ̇) is the frictional torque vector.
Each element in M, N, G and H is a nonlinear function that describes manipulator dynamics
where all joints are interconnected by “coupling torques” [9]. Therefore, the model can be
represented by a set of n second-order nonlinear scalar differential equations in the following
form:
mii (θ)θ̈i + ∑nj=1 mij (θ)θ̈j + ni (θ, θ̇) + g i (θ) + (θ̇) = ui (23).
i≠j
By introducing a state vector
x1
θ
x = [x ] = [ ̇ ]
(24)
2
θ
equation (23) can be expressed as a nonlinear state equation (with subscript i omitted for
simplicity)
ẋ = f(x(t))
(25)
Our further discussion will concentrate on a model given by equation (25) and linearized
around a certain operating point. Since the state variables in vector x represent position and
velocity of manipulator joints, our linearization results in the controllable canonical form
0
1
0
]x + [ ]u
(26)
−a1 −a2
1
Variation of parameters in equation (26) can be expected within certain limits due to both
nonlinear nature of the process described by equations (23) and (25) and its time-varying
characteristics; therefore, a system that stabilizes the manipulator joints for a broad range of
parameter variations is highly desired.
ẋ = [
3.2 Design the control of the system
Our approach to designing a control system with robust stability is based on catastrophe
theory, which is a branch of bifurcation theory. Out of seven standard forms of functions [10]
we propose the hyperbolic umbilic catastrophe [11], which has three control factors and two
behavior axes. Combined with a second-order plant, our proposed control system applied to a
linearized version of the manipulator dynamics assumes the following form:
dx1
dt
dx2
dt
1
= T x2
1
1
= T (−x13 − x23 − k1 x1 x2 + k 2 x2 + k 3 x1 )
2
(27).
To identify a relationship between the above system and a linearized model of the controlled
plant given by (26), consider just the linear portion of equation (27):
dx1
dt
dx2
dt
1
= T x2 + ⋯
1
1
= T (k 2 x2 + k 3 x1 ) + ⋯
2
The above equations can be represented by the following block diagram:
(28).
Title
Figure 12. Block diagram of the linear part of the robust control system.
By introducing a new state variablex1∗ = x1 /T1 and performing a block diagram
transformation the following graphical representation of the linear part of the system can be
obtained:
Figure 13. Modified block diagram of the linear part of the robust control system.
It can be easily demonstrated that the above system has the same structure as that of a system
in controllable canonical form with a state-variable feedback. Indeed if the following control
law:
x1
u = WR − Fx = WR − [F1 F2 ] [x ]
2
(29)
is combined with the state equations given by (26), the following closed-loop description can
be obtained:
dx1
dt
dx2
dt
= x2
= −a0 x1 − a1 x2 + f1 x1 + f2 x2 = (−a0 + f1 )x1 + (−a1 + f2 )x2 (30).
By comparing (29) and (30) to the block diagram in Fig. 13, the following correspondences
can be identified:
W = 1/T1
k3
−a0 + f1 = T
1 T2
(31).
k
−a1 + f2 = T2
2
Hence, the linear part of equation (27) can be thought of as a linear control system with statevariable feedback. It is expected that, since the original system is nonlinear and can also
exhibit time-varying characteristics, coefficients a0 and a1 in its linearized version are going
to change within a certain range. This can be easily accounted for in our proposed system by
changing the coefficients k2 and k3, as follows from equations (31). Then the problem of
designing a system with robust stability can be formulated as a problem of finding a solution
that would not be susceptible to those variations in model parameters.
4. Simulation Results
Robust stability conditions derived in the previous section can be demonstrated by numerical
simulations. The following figure (Fig. 14) shows a Simulink block diagram of the system
with a feedback controller based on hyperbolic umbilic catastrophe function, as per equation
(27):
Figure 14. Simulation diagram of a control system with robust stability.
Title
Since we are only concerned about robust stability and do not consider robust performance,
the input to the above system should be set to zero and only natural motion to an equilibrium
point from arbitrary initial conditions is considered [12]. The following parameters of the
system are used in four different numerical simulations:
a) k1=2, k2=-3, k3=25, T1=0.5, T2=0.2, x1(0)=5, x2(0)=10.
б) k1=2, k2=3, k3=25, T1=0.5, T2=0.2, x1(0)=5, x2(0)=10.
c) k1= 5, k2=-3, k3=25 , T1=0.5, T2=0.2, x1(0)=5, x2(0)=10.
d) k1=2, k2=-3, k3=30, T1=0.5, T2=0.2, x1(0)=5, x2(0)=10.
The first set of parameters results in an asymptotically stable system whose convergence
process is shown in Fig. 15. If k2 is made positive, as in case b), the system loses its stability.
Figure 15. Numerical simulation results: experiment 1.
The waveforms presented in Fig. 16 – Fig. 17 show stable performance for a different set of
parameters and converge to equilibrium points.
Figure 16. Numerical simulation results: experiment 3.
Figure 17. Numerical simulation results: experiment 5.
The results presented above illustrate that it is possible for the system to converge to an
equilibrium point even if dynamics of the controlled process suddenly changes. This is
assured by having proper values of control factors, as was discussed in the previous section.
It can be easily demonstrated that, by solving (31) for k2 and k3, we have:
k 2 = T2 (−a1 + f2 )
k 3 = T1 T2 (−a0 + f1 )
(32)
and for any practical range of model parameter variations, e.g. a1, a2, T1 and T2, we can find
values of the state variable feedback controller coefficients f1 and f2 to assure that k2 and k3
Title
guarantee robust stability. In addition, the remaining control factor k1 can be used to improve
dynamic performance of the convergence process, but this property is not considered within
the scope of this paper.
5. CONCLUSION
Thus, the stability conditions of the automatic control system with the control law in the class
of the structurally stable mapping show:
- a system with linear control law for an astatic object of the first level, stable only within a
limited region, becomes robustly stable within very wide range of variations of uncertain
parameters of the object and given parameters of the regulator:
- a system with linear control law for an astatic object of the second level, unstable at any
values of parameters, not only becomes stable, but also does not have limits of the
changes of uncertain parameters of the object and given parameters of the regulator,
which allow maintaining stability.
Therefore, the proposed approach for the choice of control laws for linear dynamic objects in
the class of structurally stable mappings of catastrophe theory allows designing an automatic
control system with an increased capacity of robust stability.
This paper presents some results of our investigation on increasing robust stability of a
dynamic system with uncertain model parameters. As a practical case we can select a
mechanical system whose operation can be described by a set of nonlinear relationships. In
this case our approach is based on a controller in the form of a hyperbolic umbilic
catastrophe function with three control factors, which can be applied to a nonlinear dynamic
system described by a set of second-order nonlinear differential equations.
The synthesized control system preserves the stability properties within a very wide range of
variations of uncertain parameters of the object and given parameters of the control device.
Numerical experiments were carried out using MATLAB 7.12.0 (R2011a).
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