New Practical Integral Variable Structure Controllers for Uncertain Nonlinear Systems

229
( ) , 0
T
Vx xPx P
=
> (43)
The derivative of (43) becomes

00 0 0
() [ (,) (,)] (,) (,)
TT TT T
Vx x
f
xt P P
f
xt x u
g
xtPx xP
g
xtu=+++

(44)
By means of the Lyapunov control theory(Khalil, 1996), take the control input as

0
(,)
TT
u

then

{}
00
min
() (,)
[ ( , ) ]
[ ( , ) ( , )]
( , )
( , )
TTTTTT
TTTT
TT
cc
T
c
c
V x x Q x t x x C PBB Px x PBB PCx
x Q x t C PBB P PBB PC x
xfxtPPfxtx
xQ xtx
Qxt x
λ
=− − −
=− + +
=− +
=−
≤−

(48)

is a switching gain, respectively as

[ ] 1, ,
i
Gg i qΔ=Δ = (51)

{
}
{}
{}
{}
11
11 110
0
11
11 110
0
max ( ) ''( . ) ( ) ( , )
( ) 0
min

min ( ) ''( . ) ( ) ( , )
( ) 0
min
i
i
i
i
i
HCB HC f xt IHCB H f xt

230

{
}
2
max | ''( , )|
min{ }
dxt
G
II
=
+Δ
(55)
The real sliding dynamics by the proposed control (50) with the output feedback integral
sliding surface (35) is obtained as follows:

1
01 1 0
1
110 1 1 1 0
1
110 1 0
1
11 1
1
11
()[ ]
( )[ (,) (,) ( (,)) (,) ]
( ) [ ( , ) ( ) ]
( ) [ ( , ) ( , ) ( ) ]

11 110
( ( , ))( ( ) ( , )]
( ) [ ''( , ) ( , ) ( ) ] ( ) ( )
[( )( ( )) ''( , )]
( ) ''( , ) ( ) ( , ) ( )
Bgxt GyGSGsignS HCdxt
HCB HC f xtCx HC gxtG y y I I Gyy
IIGSGsignS dxt
HCBHCfxty IHCBHfxty I I
−
−−
+Δ −Δ − − +
=Δ+Δ−+ΔΔ
++Δ− − +
=Δ+Δ −+Δ
10 2 0
()
( )( ( )) ''( , )
Gyy
IIGSGsignS dxt
Δ
++Δ− − +
(56)
The closed loop stability by the proposed control input with the output feedback integral
sliding surface together with the existence condition of the sliding mode will be investigated
in next Theorem 1.
Theorem 2: If the output feedback integral sliding surface (35) is designed to be stable, i.e. stable
design of ()Gy , the proposed control input (50) with Assumption A1-A10 satisfies the existence
condition of the sliding mode on the output feedback integral sliding surface and closed loop
exponential stability.

GS I I
GSS
εε
ε
−−
=
=Δ+Δ −+ΔΔ
++Δ−− +
≤− = +Δ
=−


0
1
2 ( )GV y
ε
=−
(58)
From (58), the second requirement to get rid of the reaching phase is satisfied. Therefore, the
reaching phase is clearly removed. There are no reaching phase problems. As a result, the
real output dynamics can be exactly predetermined by the ideal sliding output with the
matched uncertainty. Moreover from (58), the following equations are obtained as

1
() 2 () 0Vy GVy
ε
+
≤

(59)

is bounded, which completes the proof of Theorem 2.
2.3.3 Continuous approximation of output feedback discontinuous control input
Also, the control input (50) with (35) chatters from the beginning without reaching phase.
The chattering of the discontinuous control input may be harmful to the real dynamic plant
so it must be removed. Hence using the saturation function for a suitable
0
δ
, one make the
part of the discontinuous input be continuous effectively for practical application as

0
01020
00
() { ( )}
||
c
S
uGyyGSGyGsignS
S
δ
=− − − Δ +
+
(62)
The discontinuity of control input of can be dramatically improved without severe output
performance deterioration.
3. Design examples and simulation studies
3.1 Example 1: Full-state feedback practical integral variable structure controller
Consider a second order affine uncertain nonlinear system with mismatched uncertainties
and matched disturbance
2

x
u
xxt
x
dxt
⎡
⎤
−+
⎡⎤
⎡⎤ ⎡⎤⎡ ⎤
=⋅++
⎢
⎥
⎢⎥
⎢⎥ ⎢⎥⎢ ⎥
+
+
⎣⎦ ⎣⎦⎣ ⎦
⎣⎦
⎣
⎦


(65)
where the nominal parameter
0
(,)
f
xt and
0

==Δ=
⎢
⎥
⎢⎥ ⎢⎥
⎣⎦ ⎣⎦
⎣
⎦
⎡⎤
Δ=
⎢⎥
⎣⎦
(66)

Recent Advances in Robust Control – Novel Approaches and Design Methods

232
To design the full-state feedback integral sliding surface, (,)
c
f
xt is selected as

00
11
(,) (,) (,)()
70 21
c
fxt fxt gxtKx
−
⎡
⎤


2650 670
(,) (,) 0
670 196
T
cc
fxtPPfxt
−−
⎡⎤
+
=<
⎢⎥
−−
⎣⎦
(70)
Hence, the continuous static feedback gain is chosen as

[
]
0
() (,) 35 11
T
Kx g xtP== (71)
Therefore, the coefficient of the sliding surface is determined as

[
]
[
]
11112

⎧
⎪
Δ=
⎨
−
<
⎪
⎩
(74a)

2
2
2
5.0 if 0
5.0 if 0
f
f
Sx
k
Sx
+
>
⎧
⎪
Δ=
⎨
−
<
⎪
⎩

phase, only the sliding exists from the initial condition. The one of the two main problems of
the VSS is removed and solved. The unmatched uncertainties influence on the ideal sliding
dynamics as in the case (iv). The sliding surface ( )
f
St (i) unmatched uncertainty and
matched disturbance is shown in Fig. 3. The control input (i) unmatched uncertainty and
matched disturbance is depicted in Fig. 4. For practical application, the discontinuous input
is made be continuous by the saturation function with a new form as in (32) for a
positive 0.8
f
δ
= . The output responses of the continuous input by (32) are shown in Fig. 5
for the four cases (i)ideal sliding output, (ii)no uncertainty and no disturbance (iii)matched
uncertainty/disturbance, and (iv)unmatched uncertainty and matched disturbance. There is
no chattering in output states. The four case trajectories (i)ideal sliding time trajectory, (ii)no
uncertainty and no disturbance (iii)matched uncertainty/disturbance, and (iv) unmatched
uncertainty and matched disturbance are depicted in Fig. 6. As can be seen, the trajectories
are continuous. The four case sliding surfaces are shown in fig. 7, those are continuous. The
three case continuously implemented control inputs instead of the discontinuous input in
Fig. 4 are shown in Fig. 8 without the severe performance degrade, which means that the
continuous VSS algorithm is practically applicable. The another of the two main problems of
the VSS is improved effectively and removed.
From the simulation studies, the usefulness of the proposed SMC is proven. Fig. 1. Four case
1
x and
2
x time trajectories (i)ideal sliding output, (ii) no uncertainty and
Fig. 5. Four case
1
x
and
2
x
time trajectories (i)ideal sliding output, (ii) no uncertainty and
no disturbance (iii)matched uncertainty/disturbance, and (iv)unmatched uncertainty and
matched disturbance by the continuously approximated input for a positive 0.8
f
δ
=

Recent Advances in Robust Control – Novel Approaches and Design Methods

236
Fig. 6. Four phase trajectories (i)ideal sliding trajectory, (ii)no uncertainty and no
disturbance (iii)matched uncertainty/disturbance, and (iv) unmatched uncertainty and
matched disturbance by the continuously approximated input

Fig. 7. Four sliding surfaces (i)ideal sliding surface, (ii)no uncertainty and no disturbance
(iii)matched uncertainty/disturbance, and (iv) unmatched uncertainty and matched

dxt
⎡
⎤
−−
⎡⎤
⎡⎤ ⎡⎤⎡ ⎤
⎢
⎥
⎢⎥
⎢⎥ ⎢⎥⎢ ⎥
=− ++
⎢
⎥
⎢⎥
⎢⎥ ⎢⎥⎢ ⎥
⎢
⎥
⎢⎥
⎢⎥ ⎢⎥⎢ ⎥
++ +
⎣⎦ ⎣⎦⎣ ⎦
⎣⎦
⎣
⎦



(75)

1

f
xt ,
0
(,)gxt B= and C , the unmatched system matrix
uncertainties and matched input matrix uncertainties and matched disturbance are
2
1
0
22
23
310 0 3sin()0 0
100
(,) 0 1 1, 0, C , 0 0 0
001
102 2 0.5sin()00.4sin()
x
fxt B f
xx
−−
⎡
⎤
⎡⎤⎡⎤
⎡⎤
⎢
⎥
⎢⎥⎢⎥
=− = = Δ=
⎢⎥
⎢
⎥

⎥
⎢⎥
⎢
⎥
⎢⎥
⎣⎦
⎣
⎦
. (78)
The eigenvalues of the open loop system matrix
0
(,)
f
xt are -2.6920, -2.3569, and 2.0489,
hence
0
(,)
f
xt is unstable. The unmatched system matrix uncertainties and matched input
matrix uncertainties and matched disturbance satisfy the assumption A3 and A8 as

2
1
1
22
23
3sin ( ) 0
1
" 0 0 , 0.15sin(2 ) 0.15 1, "( , ) ( , )
2

−
⎡
⎤
⎢
⎥
=− =−
⎢
⎥
⎢
⎥
−−
⎣
⎦
(80)

in order to assign the three stable pole to (,)
c
fxt at 30.0251
−
and 2.4875 0.6636i
−
± . The
constant feedback gain is designed as

[
]
{
}
1
() 2 [1 0 2] 19 0 30GyC

02
30h
=
. Hence
112
22HCB h
=
= is a non zero satisfying
A4. The resultant output feedback integral sliding surface becomes

[] [ ]
101
0
202
1
0 1 19 30
2
yy
S
yy
⎧
⎫
⎡
⎤⎡⎤
⎪
⎪
=+
⎨
⎬
⎢


The output feedback control gains in (50), (51)-(55) are selected as follows:

New Practical Integral Variable Structure Controllers for Uncertain Nonlinear Systems

239

01
1
01
1.6 if 0
1.6 if 0
Sy
g
Sy
+
>
⎧
Δ=
⎨
−
<
⎩
(87a)

02
2
02
1.7 if 0
1.7 if 0

2
y
(i)ideal sliding
output, (ii) with no uncertainty and no disturbance, (iii)with matched uncertainty and
matched disturbance, and (iv) with ummatched uncertainty and matched disturbance. The
each two output is insensitive to the matched uncertainty and matched disturbance, hence is
almost equal, so that the output can be predicted. The four case phase trajectories (i)ideal
sliding trajectory, (ii) with no uncertainty and no disturbance, (iii)with matched uncertainty
and matched disturbance, and (iv) with ummatched uncertainty and matched disturbance
are shown in Fig. 10. There is no reaching phase and each phase trajectory except the case
(iv) with ummatched uncertainty and matched disturbance is almost identical also. The
sliding surface is exactly defined from a given initial condition to the origin. The output
feedback integral sliding surfaces (i) with ummatched uncertainty and matched disturbance
is depicted in Fig. 11. Fig. 12 shows the control inputs (i)with unmatched uncertainty and
matched disturbance. For practical implementation, the discontinuous input can be made
continuous by the saturation function with a new form as in (32) for a positive
0
0.02
δ
=
. The
output responses by the continuous input of (62) are shown in Fig. 13 for the four cases
(i)ideal sliding output, (ii)no uncertainty and no disturbance (iii)matched
uncertainty/disturbance, and (iv)unmatched uncertainty and matched disturbance. There is
no chattering in output responses. The four case trajectories (i)ideal sliding time trajectory,
(ii)no uncertainty and no disturbance (iii)matched uncertainty/disturbance, and (iv)
unmatched uncertainty and matched disturbance are depicted in Fig. 14. As can be seen, the
trajectories are continuous. The four case sliding surfaces are shown in fig. 15, those are
continuous also. The three case continuously implemented control inputs instead of the
discontinuous input in Fig. 12 are shown in Fig. 16 without the severe performance loss,

disturbance (iii)matched uncertainty/disturbance, and (iv)unmatched uncertainty and
matched disturbance

New Practical Integral Variable Structure Controllers for Uncertain Nonlinear Systems

241
Fig. 11. Sliding surface
0
()St (i) unmatched uncertainty and matched disturbance

Fig. 12. Discontinuous control input (i) unmatched uncertainty and matched disturbance

Recent Advances in Robust Control – Novel Approaches and Design Methods

242 Fig. 13. Four case
Fig. 16. Three case continuous control inputs
0c
u
(i)no uncertainty and no disturbance
(ii)matched uncertainty/disturbance, and (iii) unmatched uncertainty and matched
disturbance by the continuously approximated input for a positive
0
0.02
δ
=Recent Advances in Robust Control – Novel Approaches and Design Methods

244
4. Conclusion
In this chapter, a new practical robust full-state(output) feedback nonlinear integral variable
structure controllers with the full-state(output) feedback integral sliding surfaces are
presented based on state dependent nonlinear form for the control of uncertain more affine
nonlinear systems with mismatched uncertainties and matched disturbance. After an affine
uncertain nonlinear system is represented in the form of state dependent nonlinear system, a
systematic design of the new robust integral nonlinear variable structure controllers with
the full-state(output) feedback (transformed) integral sliding surfaces are suggested for
removing the reaching phase. The corresponding (transformed) control inputs are proposed.
The closed loop stabilities by the proposed control inputs with full-state(output) feedback
integral sliding surface together with the existence condition of the sliding mode on the
selected sliding surface are investigated in Theorem 1 and Theorem 2 for all mismatched
uncertainties and matched disturbance. For practical application of the continuous

New Practical Integral Variable Structure Controllers for Uncertain Nonlinear Systems

245
Gutman, S. (1979). Uncertain dynamical Systems:A Lyapunov Min-Max Approach. IEEE
Trans. Autom. Contr, Vol. AC-24, no. 1, pp.437-443.
Horowitz, I. (1991). Survey of Quantitative Feedback Theory(QFT). Int. J. Control, vol.53,
no.2 pp.255-291.
Hu, X. & Martin, C. (1999). Linear Reachability Versus Global Stabilization. IEEE Trans.
Autom. Contr, AC-44, no. 6, pp.1303-1305.
Hunt, L. R., Su, R. & Meyer, G. (1987). Global Transformations of Nonlinear Systems," IEEE
Trans. Autom. Contr, Vol. AC-28, no. 1, pp.24-31.
Isidori, A., (1989). Nonlinear Control System(2e). Springer-Verlag.
Khalil, H. K. (1996). Nonlinear Systems(2e). Prentice-Hall.
Kokotovic, P. & Arcak, M. (2001). Constructive Nonlinear Control: a Historical Perspective.
Automatica, vol.37, pp.637-662.
Lee, J. H. & Youn, M. J., (1994). An Integral-Augmented Optimal Variable Structure control
for Uncertain dynamical SISO System, KIEE(The Korean Institute of Electrical
Engineers), vol.43, no.8, pp.1333-1351.
Lee, J. H. (1995). Design of Integral-Augmented Optimal Variable Structure Controllers, Ph.
D. dissertation, KAIST.
Lee, J. H., (2004). A New Improved Integral Variable Structure Controller for Uncertain
Linear Systems. KIEE, vol.43, no.8, pp.1333-1351.
Lee, J. H., (2010a). A New Robust Variable Structure Controller for Uncertain Affine
Nonlinear Systems with Mismatched Uncertainties," KIEE, vol.59, no.5, pp.945-949.
Lee, J. H., (2010b). A Poof of Utkin's Theorem for a MI Uncertain Linear Case," KIEE, vol.59,
no.9, pp.1680-1685.
Lee, J. H., (2010c). A MIMO VSS with an Integral-Augmented Sliding Surface for Uncertain
Multivariable Systems
," KIEE, vol.59, no.5, pp.950-960.
Lijun, L. & Chengkand, X., (2008). Robust Backstepping Design of a Nonlinear Output

Zheng, Q. & Wu, F. Lyapunov Redesign of Adpative Controllers for Polynomial Nonlinear
systems," (2009). Proceeding of IEEE ACC 2009, pp.5144-5149.
11
New Robust Tracking and Stabilization Methods
for Significant Classes of Uncertain
Linear and Nonlinear Systems
Laura Celentano
Dipartimento di Informatica e Sistemistica
Università degli Studi di Napoli Federico II, Napoli,
Italy
1. Introduction
There exist many mechanical, electrical, electro-mechanical, thermic, chemical, biological
and medical linear and nonlinear systems, subject to parametric uncertainties and non
standard disturbances, which need to be efficiently controlled. Indeed, e.g. consider the
numerous manufacturing systems (in particular the robotic and transport systems,…) and
the more pressing requirements and control specifications in an ever more dynamic society.
Despite numerous scientific papers available in literature (Porter and Power, 1970)-(Sastry,
1999), some of which also very recent (Paarmann, 2001)-(Siciliano and Khatib, 2009), the
following practical limitations remain:
1. the considered classes of systems are often with little relevant interest to engineers;
2. the considered signals (references, disturbances,…) are almost always standard
(polynomial and/or sinusoidal ones);
3. the controllers are not very robust and they do not allow satisfying more than a single
specification;
4. the control signals are often excessive and/or unfeasible because of the chattering.
Taking into account that a very important problem is to force a process or a plant to track
generic references, provided that sufficiently regular, e.g. the generally continuous
piecewise linear signals, easily produced by using digital technologies, new theoretical
results are needful for the scientific and engineering community in order to design control
systems with non standard references and/or disturbances and/or with ever harder

⎢⎥
⎣⎦
∑
     
, where tR
∈
is the time,
m
yR∈ is
the output,
r
uR∈ is the control input,
p
R
μ
∈℘⊂ is the vector of uncertain parameters,
with
℘
compact set,
1
mr
FR
×
∈ is limited and of rank m ,
2
mxm
i
FR∈ is limited and
m
f

xR∈ is the state, uR
∈
is the control signal, dR
∈
is the disturbance or, more in
general, the effect
d
y
of the disturbance d on the output,
y
R∈ is the
output,
,AAA
−+
≤≤ BBB
−
+
≤
≤ and CCC
−
+
≤≤ .
Suppose that this process is without zeros, is completely controllable and that the state is
measurable.
Moreover, suppose that the disturbance
d and the reference r are continuous signals with
limited first derivative (see Fig. 1).
A main goal is to design a linear and time invariant controller such that:
1. ,, ,, ,AAA BBB CCC
−+ −+ −+

satisfies relation
New Robust Tracking and Stabilization Methods
for Significant Classes of Uncertain Linear and Nonlinear Systems

249

[]
0,
1
ˆˆ
() , 0, (), (): max ( ) ( )
ˆ
rd rd rd
t
v
et t rt dt r d
K
σ
δ
δσσδ
−
−−
∈
≤∀≥∀ = −≤




, (2)


.
.
.
.

Fig. 1. Possible reference or disturbance signals with limited derivative.
where the maximum variation velocity
ˆ
rd
δ
−


of
() ()rt dt−
is a design specification.
Remark 1. Clearly if the initial state of the control system is not null and/or
(0) (0) 0rd−≠

(and/or, more in general,
() ()rt dt−
has discontinuities), the error
()et
in (2) must be
considered unless of a “free evolution“, whose practical duration can be made minus that a
preassigned settling time
ˆ
a
t .
Remark 2. If disturbance d does not directly act on the output

Gs CsI A B a a a a a a b b b
sas a
−
−+−+−+
−
=− = ≤≤ ≤≤ ≤≤
+++
, (3)
in the Laplace domain the considered control scheme is the one of Fig. 2.

1
+
n
k
s
1
1

nn
n
b
s
as a
−
+
++
12
12

nn

Finally, it is clear that, for the controllability of the process, the parameter b must be always
not null. In the following, without loss of generality, it is supposed that
0. b
−
>
Remark 4. In the following it will be proved that, by using the control scheme of Fig. 2, if (2)
is satisfied then the overshoot of the controlled system is always null.
From the control scheme of Fig. 2 it can be easily derived that

1
11
1
11 1
() ()
() (()())()(()())
() ()
nn
nn
nn
nn n
sabks abk
Ess RsDs SsRsDs
s a bk s a bk s bk
−
+
+
++ +++
=−=−
++ +++ +
. (4)

the sensitivity function
()Ss of the error and the constant gain
v
K turn out to be:

1
111
1
11

() , .

nn
nnn
v
nn
nn n n n
sds d d bk
Ss s K
sds dsd dabk
−
++
+
+
+++
===
++++ +
(7)
Moreover the sensitivity function
()Ws of the output is

1
()1
n
i
i
p
+
=
∏
−= is said to be set of reference poles.
Let be

1
11
()
nn
nn
ds s ds ds d
+
+
=+ +++ (9)
the polynomial whose roots are a preassigned set of reference poles
P . By choosing the
poles
P of the control system equal to P
ρ
, with
ρ
positive , it is


+
+
+++
==
++++
, (11)
from (4) and from the first of (7) it is
New Robust Tracking and Stabilization Methods
for Significant Classes of Uncertain Linear and Nonlinear Systems

251

()
1
0
() ()()(), () ()
t
ppp
e t s r t d t d where s t S s
ττ ττ
−
≤−−− =
∫


L
, (12)
from which, if all the poles of ( )
p
Ss have negative real part, it is

ττ
∞
−
∈
==−
∫




. (14)
Remark 5. Note that, while the constant gain
v
K allows to compute the steady-state
tracking error to a ramp reference signal,
v
H , denoted absolute constant gain, allows to obtain
t∀ an excess estimate of the tracking error to a generic reference with derivative. On this
basis, it is very interesting from a theoretical and practical point of view, to establish the
conditions for which
vv
HK= .
In order to establish the condition necessary for the equality of the absolute constant gain
v
H with the constant gain
v
K and to provide some methods to choose the poles
P
and
ρ

++++
(15)
and

1
1
1 1
1
11
0
1
,, ()

()
nn
n n
vv p
nn
n nn
p
dsdsd
K H where s t
dsdsdsd
sd
ττ
−
−
+
∞
+

Proof. By using the change of scale property of the Laplace transform, (8) and (10) it is

1
1
1
1
11
11
1
1
1
1
11
1
() () ()
1
= ( ).

n
n
nnnn
nn
n
nn
nn
td
w
s
sds dsd
d

++++
⎝⎠
L
L
(18)

Recent Advances in Robust Control – Novel Approaches and Design Methods

252
By using again the change of scale property of the Laplace transform, by taking into
account (10) and (11) it is

1
1
1
11
11
1
1
1
1
11
() ()
() () ()

= ( ),

nnn
n
p

⎜⎟
⎜⎟
++++
⎝⎠
⎝⎠
⎛⎞
+++
=
⎜⎟
++++
⎝⎠
L
L
(19)
from which

()
00 0
11
()
pp p
t
sd s dt stdt
ττ
ρρ ρ
∞∞ ∞
⎛⎞
==
⎜⎟
⎝⎠

ˆˆ
ˆˆ
,1,2, ,,
in
ii n
in
da d
kink
bb
ρρ
+
+
+
−
== =. (21)
Moreover the polynomial of the effective poles and the constant gain are:

ˆ
ˆ
() () () ()ds ds hns s
δ
=+ + (22)

1
11
ˆ
ˆ
ˆ
ˆ
11

ρρρ
+++
+
+
=+ ++ + =+ +++ (24)

1
1111
ˆˆˆ
ˆˆ ˆ ˆ
()
nnnn
nn nn
ns d s d s d ds ds d
ρρρ
+
+
+
=+++ =+++
(25)

1
1
1
111
,()
, , , .
n
n
n

ατ
+−
−= + = , (27)
New Robust Tracking and Stabilization Methods
for Significant Classes of Uncertain Linear and Nonlinear Systems

253
where ()ds is the polynomial (5) or (22), are given by using the affine transformation

1
1
2
2
2
12
1
1
1
.
1
ˆ
ˆ
10.0
1
ˆˆ
1
1.0
ˆ
2


α
αα χ
α
αααχ
−−
−
+
+
+
=+
−
+
−−
−
−
⎛⎞
⎡⎤
⎜⎟
⎝⎠
⎢⎥
⎛⎞
⎢⎥
⎜⎟
⎡⎤ ⎛ ⎞
⎢⎥
⎝⎠
⎜⎟
⎢⎥
⎝⎠
⎢⎥

α
+
+
+
⎡
⎤
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎛⎞
⎢

ˆ
12
11
ˆˆˆ
.1
11
nn
n
n
nn
n
k
k
nn
nn
k
nn
nn
α
χ
χ
αα
χ
ααα
−−
+
+
−
=
−

⎜⎟ ⎜⎟
⎣⎦ ⎝ ⎠ ⎝ ⎠
⎣
⎦
⎢⎥
⎢⎥
⎛⎞ ⎛ ⎞ ⎛⎞
⎢⎥
⎜⎟ ⎜ ⎟ ⎜⎟
⎢⎥
⎣⎝⎠ ⎝ ⎠ ⎝⎠ ⎦
(29)
Proof. The proof is obtained by making standard manipulations and for brevity it has been
omitted.
Now, as pre-announced, some preliminary results about the externally positive systems are
stated.
Theorem 4. Connecting in series two or more SISO systems, linear, time-invariant and
externally positive it is obtained another externally positive system.
Proof. If
1
()Ws and
2
()Ws are the transfer functions of two SISO externally positive systems
then
(
)
1
11
() () 0wt Ws
−

1
()
()
Ws
sp s
α
ω
=
−−+
⎡
⎤
⎣
⎦
, (31)
i.e. without zeros, with a real pole
p
and a couple of complex poles j
α
ω
±
, is externally
positive iff
p
α
≤ , i.e. iff the real pole is not on the left of the couple of complex poles.
Proof. By using the translation property of the Laplace transform it is