Active RLC Circuit Model of the Second-order Dynamical
System with Piecewise-linear Controlled Sources
Jaromír BRZOBOHATÝ, * Jiří POSPÍŠIL, Zdeněk KOLKA, Stanislav HANUS,
Department of Microelectronics, * Department of Radio Electronics
Faculty of Electrical Engineering and Communication, Brno University of Technology
Purkyňova 118, 612 00 Brno, CR
CZECH REPUBLIC
Abstract: - RLC active circuit model with voltage-controlled voltage-source (VCVS) and current-controlled
current-source (CCCS) for the second-order autonomous dynamical system realization is proposed. Its circuit
parameters are directly related to the state model parameters that lead to simple design formulas.
Key-Words: - Dynamical systems, State models, Active equivalent circuits, Piecewise-linear controlled sources
1 Introduction
Autonomous piecewise-linear (PWL) systems of
Class C can be described by the general state matrix
form [3], [4]
x& = Ax + b h ( w T x ) ,
(1)
where the normalized elementary PWL feedback
function (Fig. 1)
h ( w T x ) = 1  w T x+ 1 − w T x − 1 
2

(2)
contains the regions D0 and D+1 (D-1). The dynamical
behavior of the system is determined by two
characteristic polynomials associated to these
individual regions [3]. All systems of the Class C
having the same characteristic polynomials are
qualitatively equivalent and they are related by linear
topological conjugacy [4]. Typical systems of this
class are the Chua’s model, both its canonical forms
[3], and also recently derived optimized state model
having the minimum sum of relative eigenvalue
sensitivity squares with respect to the change of the
individual state matrix parameters [8]. Just this low-
sensitivity model is very useful as a prototype
h(w T x)
for practical chaotic system realization in the form of
electronic circuit. It provides the possibility to utilize
the block-decomposed form of the state matrix so
that the design procedure can be started from the
optimized second-order system and then extended by
a simple way to the optimized higher-order case [8].
State model can be used as a mathematical tool
for the numerical simulation of dynamical system
behavior as well as a prototype for the electronic
circuit realizat-ion using available circuit technique.
From the complete state equations can directly be
derived either the general integra-tor-based circuit
block-diagram (it is typical for both canonical forms)
or the corresponding RLC active circuit (it is typical
for Chua’s oscillator). In both cases only a single
PWL network element is used utilizing various types
of active electronic blocks operating in both the
voltage and current modes (op-amps, current
conveyors, transimpedance amplifiers, etc.).
For the optimized low-sensitivity model first the
corresponding integrator-based block diagram has
been derived for both second- and third-order cases
[8]. The intention of this contribution is to propose
the corresponding RLC active circuits where, unlike
the Chua’s model, the circuit parameters have a direct
relation to the model parameters.
1
2 Optimized Second-order State Models
1
D -1
D0
T
w x
D +1
Fig. 1. Simple memoryless PWL feedback function.
with Low Eigenvalue Sensitivities
The most frequently occurring autonomous
dynamical systems have their complex conjugate
eigenvalues in both regions of PWL function (Fig. 1),
i.e. for the inner region (D0) it is (µ1,2 = µ ′ ± jµ ′′) and
for the outer regions (D-1, D+1) it is (ν1,2 = ν ′ ± jν ′′) .
Then the associated characteristic polynomials are
defined as follows
(D0): P (s) = (s − µ 1 )(s − µ 2 ) = det (s 1− A 0 )
(3a)
(D-1, D+1): Q(s ) = (s − ν 1 )(s − ν 2 ) = det (s1− A )
(3b)
where relation between the state matrices can be
A0 = A + b w T ,
(4)
expressed as [3]
and 1 is the unity matrix. The optimized lowsensitivity state model (1) have been chosen in the
simplified and decomposed complex form [9], where
the corresponding state matrices are
ν ' - ν "
A=
,
ν " ν ' 
- µ "K 
 µ'
, (5a,b)
A0 = 
-1
µ' 
 µ "K
and the optimizing coefficient K is given as the real
root of the quadratic equation
K 2 − 2K( M + 1) + 1 = 0 , i.e. K = 1 + M ±
M (M + 2) ,
where the auxiliary parameter M is
M=
( µ ' - ν ' ) 2 + ( µ "-ν " ) 2
>0 ,
2 µ "ν "
In the vectors
(µ",ν" ≠ 0) .
w 1 
b 
b =  1  and w =   one of the
w 2 
b2 
para-meters can be chosen, e.g. w 1 = 1 , while the
others are obtained as [8]
b1 = µ' −ν ' , b 2 =
ν " − µ" K
( µ' −ν ' ) 2 ,
w2 =
ν " − µ" K
µ' −ν '
(6a,b,c)
Then the complete state equations of the optimized
__________________________________________________________
second-order PWL autonomous system can be then
written as
x& = ν ' [x − h ( x + w 2 y )] − ν " y + µ ' h ( x + w 2 y ) , (7)
y& = ν" x +ν' y + b2 h (x + w 2 y ) ,
(8)
where the parameters b2 and w 2 are given by the
formulas (6b,c). The corresponding integrator-based
circuit block diagram, suitable also as the prototype
for the practical realization, is shown in [8]. All the
sensitivity functions are obtained in the complex
form and the same functions, expressed separately
for the eigenvalues real and imaginary parts, can
easily be derived. Then the minimum sums of
relative eigenvalue sensitivity squares with respect to
the change of the individual state matrix parameters
can be expressed for both the real and imaginary
parts generally as
∑S
2
r (
λ ′, aij ) = ∑ Sr2 ( λ ′′, aij ) =
1
2
(9)
where in outer regions (D-1, D+1) λ' = ν ' , λ" = ν "
and in inner region (D0) λ' = µ' , λ" = µ" [9].
3 Equivalent active RLC circuits with
piecewise-linear controlled sources
3.1 Equivalent circuit utilizing VCVS
Consider the autonomous RLC circuit introduced
in Fig. 2 containing voltage-controlled voltage-source
(VCVS) with PWL transfer characteristic function
u 0 = f (u1 + R 2 i 2 ) having three segments (Fig. 3)
expressed as
u 0 = A1 (u1 + R 2 i 2 ) + (A0 − A1 )h(u1 + R 2 i 2 ) (10)
i2
R4
R1
L2
L2
di 2
dt
R
du
C1 1
dt
C
i
u
R3
A
A0
A1
0
u1 + R2i2
Fig. 2. Second-order autonomous circuit with PWL voltage
u0
can be rewritten into the normalized form
u0
A
k
k
x& = [(A1 − 1)g1 − g 3 ]x + (1 + A1g1r2 )y
α
+
-E
E
0
u1 + R2i2
D0
Fig. 3. Transfer PWL characteristic of VCVS.
Choosing state variables the capacitor voltage u1 and
the inductor current i2 two Kirchhoff’s equations of
this circuit can be written in the basic form
(11a)
di 2
+ R 2 i 2 − (u 0 − u1 ) = 0
dt
(11b)
and then rewritten to the complete (non-normalized)
state equation form, i.e.
du1 A1G1 − (G1 + G3 )
AG R +1
u1 + 1 1 2
i2
=
dt
C1
C1
A0 − A1
h(u1 + R 2 i 2 )
C1R1
A R − (R 2 + R 4 )
di 2 A1 − 1
u1 + 1 2
=
i2
L2
L2
dt
(12a)
(12b)
A − A1
+ 0
h(u1 + R 2 i 2 )
L2
Utilizing reference values of voltage E (Fig. 3),
resistance R0, and capacitance C0 the normalized state
variables including the time scaling can be given as
x=
u1
,
E
R 
y = i2  0  ,
E 
τ=
t
R0C0
r2 =
Denoting
R2
,
R0
β =
r3 =
L2
2
R 0 C0
,
k = sgn (R0 C 0 )
the following equations are obtained
a11 =
k
(14)
the state equations (12)
[(A1 − 1)g1 − g3 ] ,
α
k
b1 = (A0 − A1 )g1 ,
α
a22 =
k
a12 =
k
α
(1 + A1g1r2 ) ,
w1 = 1 ,
[(A1 − 1)r2 − r4 ] ,
β
k
b2 = (A0 − A1 ) ,
β
a21 =
k
β
(17a,b,c,d)
(A1 − 1) ,
w 2 = r2
(18a,b,c,d)
and then utilized as independent formulas for design
of the individual circuit parameters. For the case
when α , β and k are chosen as free parameters it
is summarized in the next design formulas where
both the general and optimized state model are
considered.
g1 =
1 R0 α b1 α ν ′′ − µ ′′K
=
=
=
r1 R1 β b2 β µ ′ − ν ′
r2 =
R2
ν ′′ − µ ′′K
= w2 =
R0
µ′ −ν ′
g3 =
1 R0 α

α  ν ′′ − µ ′′K
= (a21w 2 − a11 ) = ν ′′
=
− ν ′ 
r3 R 3 k
µ′ −ν ′
k

r4 =

β  ν ′′ − µ ′′K
β
R4
− ν ′ 
= (a21w 2 − a22 ) = ν ′′
µ′ −ν ′
k
k
R0

A1 = 1 +
1
R
r1 =
= 1,
g1 R0
1
R
R
= 3 , r4 = 4
g 3 R0
R0
(A0 − A1 )h(x + r 2 y )
a11 a12 
w1 
x
 b1 
 , b =   , w =   , (16)
x =  , A = 
a21 a22 
w 2 
y 
b2 
(13a,b,c)
Then the corresponding normalized capacitance,
inductance and all resistances are
C
α= 1,
C0
β
(15b)
Comparing them with general matrix form (1) for the
second-order system
D1
u 0 − u1
du
u
+ i 2 − C1 1 − 1 = 0
R1
dt
R3
β
k
+
D-1
(15a)
(A0 − A1 )g1h(x + r2 y )
β
A1
+
α
,
k
k
y& = (A1 − 1)x + [(A1 − 1)r 2 − r 4 ]y
A0
R 4 i 2 + L2
α
k
A0 = 1 +
β
k
β
k
a21 = 1 +
β
k
ν ′′

′
′ 2
(a21 + b2 ) = 1 + β ν ′′ + (µ − ν ) 
k 
ν ′′ − µ ′′K 
Any other details about realization conditions of the
individual circuit elements are introduced in [9].
3.2 Equivalent circuit utilizing CCCS
i0
Consider the autonomous RLC circuit introduced
in Fig. 4 containing current-controlled current-source
(CCCS) with PWL transfer characteristic function
i 0 = f (i1 + G2u 2 ) having three segments (Fig. 5)
expressed as
i 0 = B1 (i1 + G2u 2 ) + (B0 − B1 )h (i1 + G2u 2 ) (19)
-
di 1
− u 2 − R1 (i 0 − i1 ) = 0
dt
du
u
u
i 0 − i1 − C 2 2 − 2 − 2 = 0
dt
R4 R2
and then rewritten to the complete (non-normalized)
state equation form, i.e.
di 1 B1R1 − (R1 + R 3 )
B R G +1
=
i1 + 1 1 2
u2
dt
L1
L1
+
B1
D-1
B − B1
+ 0
h(i1 + G2u 2 )
C2
(21a)
(21b)
r1 =
R1
,
R0
R
R
1
= 2 , r3 = 3 ,
R0
g 2 R0
r4 =
1 R4
=
g 4 R0
2
R0 C0
β =
,
k
k
x& = [(B1 − 1)r1 − r3 ]x + (1 + B1r1g 2 )y
α
+
normalized state variables including the time scaling
can be established as
R 
x = i1  0  ,
 E 
C2
,
C0
L1
t
τ=
R0C0
α
k
α
k
k
y& = (B1 − 1)x + [(B1 − 1)g 2 − g 4 ]y
β
+
(22a,b,c)
β
k
β
(B0 − B1 )h(x + g 2 y )
R3
R1
L1
L1
di1
dt
B
R2
du 2
dt
C2
u2
R4
,
(24a)
(B0 − B1 )r1h(x + g 2 y )
i1
C2
(23)
Denoting k = sgn (R0 C 0 ) the state equations (21)
can be rewritten into the normalized form
Utilizing reference values of voltage E (in Fig. 5
I0=E/R0), resistance R0 , and capacitance C0 the
u
y= 2,
E
D1
Then the corresponding normalized inductance, capacitance, and all resistances are
r2 =
du 2 B1 − 1
B G − (G2 + G4 )
=
i1 + 1 2
u2
dt
C2
C2
D0
Fig. 5. Transfer PWL characteristic of CCCS.
α=
B 0 − B1
h(i 1 + G 2 u 2 )
L1G1
i1 +
B0
(20a)
(20b)
I0
0
Choosing state variables the inductor current i1 and
the capacitor voltage u2 two Kirchhoff’s equations of
this circuit can be written in the basic form
R3 i1 + L1
B
i1 + G2u2
Fig. 4. Autonomous 2nd order circuit with PWL current source
B0
B1
i0
(24b)
Comparing them with general matrix form (1) for the
second-order system
a11 a12 
 b1 
x 
w 1 
 , b =   , w =   , (25)
x =  , A = 
a 21 a 22 
b2 
y 
w 2 
the following equations are obtained
a11 =
b1 =
k
α
k
a12 =
(B0 − B1 )r1 ,
w1 = 1 ,
α
k
a22 =
b2 =
k
[(B1 − 1)r1 − r3 ] ,
[(B1 − 1)g 2 − g 4 ] ,
β
k
β
(B0 − B1 ) ,
α
(1 + B1r1g 2 ) ,
(26a,b,c,d)
a21 =
k
β
w 2 = g2
(B1 − 1) ,
(27a,b,c,d)
and then utilized as independent formulas for design
of the individual circuit parameters. For the case when
α , β and k are chosen as free parameters it is
summarized in the resultant design formulas where
both the general and optimized state model are
considered.
r1 =
R1 α b1 α ν ′′ − µ ′′K
=
=
R0
β b2 β µ ′ − ν ′
g2 =
ν ′′ − µ ′′K
1 R0
= w2 =
=
µ′ −ν ′
r 2 R2
r3 =
R3 α

α  ν ′′ − µ ′′K
= (a21w 2 − a11 ) = ν ′′
− ν ′ 
R0 k
k
µ′ −ν ′

g4 =

β  ν ′′ − µ ′′K
β
1 R0
− ν ′ 
= (a21w 2 − a22 ) = ν ′′
=
µ′ −ν ′
k
k
r4 R 4

B1 = 1 +
B0 = 1 +
β
k
β
k
a21 = 1 +
β
k
ν ′′

′ ′ 2
(a21 + b2 ) = 1 + β ν ′′ + (µ − ν ) 
k 
ν ′′ − µ ′′K 
Any other details about realization conditions of the
individual circuit elements are introduced in [9].
4 Conclusion
This contribution deals with the second-order
nonlinear dynamical systems and their realization
using active RLC circuit where the active elements the
VCVS and CCCS with three-segment PWL symmetric transfer characteristic is considered, i.e. suitable
especially for voltage- and current mode realization.
Dynamical behavior of such a system is determined
by two sets of complex conjugate state matrix eigenvalues associated with the corresponding regions.
The complete and normalized state equations are
introduced where simple relation between model and
circuit parameters entails also very simple design
formulas in the synthesis procedure either in general
or optimized (low eigenvalue sensitivities) forms. The
circuits proposed represent one possibility of the
second-order system realization and can be easily
extended also for the third-order system utilizing the
block decomposition of the state matrix. Such higherorder equivalent circuit can model also chaotic
behavior of the system.
This research is partially supported by the Grant
Agency of the Czech Republic, Grant projects No.
102/02/1312/A and No. 102/01/0229. It represents the
parts of the Research Programs of the Czech Ministry
of Education – CEZ: J22/98: 262200011 and …0022.
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