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Gain-scheduled PID controller design
Vojtech Veselý, Adrian Ilka
Institute of Control and Industrial Informatics,
Faculty of Electrical Engineering and Information Technology,
Slovak University of Technology in Bratislava,
Ilkovi£ova 3, 812 19 Bratislava
Abstract
Gain scheduling (GS) is one of the most popular approaches to nonlinear control design and it is
known that GS controllers have a better performance than robust ones. Following the terminology
of control engineering, linear parameter-varying (LPV) systems are time-varying plants whose
state space matrices are xed functions of some vector of varying parameters. Our approach
is based on considering that the LPV system, scheduling parameters and their derivatives with
respect to time lie in a priori given hyper rectangles. To guarantee the performance we use
the notion of guaranteed costs. The class of control structure includes centralized, decentralized
xed order output feedbacks like PID controller. Numerical examples illustrate the eectiveness
of the proposed approach.
Keywords:
Gain scheduled control, controller design, structured controller, decentralized
control, MIMO LPV systems.
1. Introduction
Linear parameter-varying systems are time-varying plants whose state space matrices are
xed functions of some vector of varying parameters
θ(t)
. Linear parameter varying (LPV)
systems have the following interpretations:
they can be viewed as linear time invariant (LTI) plants subject to time-varying known
parameters
θ(t)∈ hθ θi
,
they can be models of linear time-varying plants,
they can be LTI plant models resulting from linearization of the nonlinear plants along
trajectories of the parameter
θ(t)∈ hθ θi
which can be measured.
For the rst and third class of systems, parameter
θ
can be exploited for the control strategy
to increase the performance of closed - loop systems. Hence, in this paper the following LPV
system will be used:
˙x=A(θ(t))x+B(θ(t))u
y=Cx
(1)
Email addresses:
vojtech.vesely@stuba.sk
(Vojtech Veselý),
adrian.ilka@stuba.sk
(Adrian Ilka)
Preprint submitted to Journal of Process Control September 9, 2014
where for the ane case
A(θ(t)) = A0+A1θ1(t) + · · · +Apθp(t)
(2)
B(θ(t)) = B0+B1θ1(t) + · · · +Bpθp(t)
(3)
and
x∈Rn
is the state,
u∈Rm
is a control input,
y=Rl
is the measurement output vector,
A0
,
B0
,
Ai
,
Bi
,
i= 1,2. . . p
,
C
are constant matrices of appropriate dimension,
θ(t)∈ hθ θi ∈ Ω
and
˙
θ(t)∈ h ˙
θ˙
θi ∈ Ωt
are vectors of time-varying plant parameters which belong to the known
boundaries.
In the case of nonlinear dynamics a widely used idea among control engineers is to linearize
the plant around several operating points and to use linear control tools to design a controller for
each of these points. The actual controller is implemented using the gain scheduling approach.
Success of such an approach depends on establishing the relationship between a nonlinear system
and a family of linear ones. There are two main problems:
1. Stability results: stability of the closed-loop nonlinear system and of the closed-loop family
of linear systems, when scheduled parameters are changes.
2. Approximation results which provide a direct relationship between the solution of closed-
loop nonlinear systems and the solution of associated linear systems [1], [2]
The main motivation for our work lies in [3], [4], [5], [6], [7], [8], where in [3] the LPV controller
is designed using the bounded real lemma for continuous and discrete time LPV systems such as
to guarantee
H∞
performance.
Paper [4] discusses extensions of
H∞
synthesis techniques to allow for controller dependence
on time-varying but measured parameters. In this case a higher performance can be achieved by
control laws that incorporate measurements of
θ
to the control algorithm. Main results can be
formulated as follows: Find a control structure such that the LPV controller satises closed-loop
stability and minimizes of the induced
L2
norm of corresponding closed-loop systems. The au-
thor's approach [5] uses a bounding technique based on parameter-dependent Lyapunov function
for design of PD controllers. Note that if LPV synthesis problem is solvable, then the induced
L2
-norm of the closed-loop system is less than some given constant. The proposed approach rep-
resents generalization of the standard sub-optimal
H∞
control problem. In paper [6] the author
shows that the performance of LPV systems with LPV controller can be improved by combining
this LMI method with MPC techniques and optimizing the
H2
(
H∞
) norm. The author [8] tack-
les the design problem of gain scheduled controllers for LPV systems via parameter-dependent
Lyapunov function. The author proposed a new design method as a set LMIs with single line
search parameters. The author tackles two problems:
H∞
type problem and
H2
. Recently, [9]
proposed the design method for the gain scheduled problem using a similar technique to [8]. In
the above paper the LPV controller is given in time domain with the same or lower order than
the LPV systems using
H∞
optimization approach. The gain scheduling controller design for
discrete-time systems is given in [10]. Paper [11] presents the design of gain-scheduled PI con-
troller, when the uncertainty of the system is assumed to be the dierence between the nonlinear
model and the nominal linear model. PI controller is designed using quadratic Lyapunov
H∞
performance where index
γ
is
H∞
norm of closed-loop system, considered as closed-loop perfor-
mance measure. Minimizing
γ
via LMI the gain scheduled controller is obtained. In [12] the
authors design a novel gain scheduling controller for synchronous generator. Improved stability
analysis and gain scheduled controller synthesis for parameter-dependent systems are proposed
in [7]. Sucient conditions for robust stability as well as conditions for the existence of a gain-
scheduled controller are given in terms of a set of LMIs. The author's approach is based on
2
the notion of quadratic stability and linear fractional representation for parameter dependent
systems. The survey of scheduled controller analysis and synthesis can be found in excellent
papers [1] and [2].
In this paper our approach is based on:
A consideration of the LPV systems (1). The scheduling parameters
θi
,
i= 1,2,...p
and
their derivatives with respect to time are supposed to lie in a priori given hyper rectangles.
Ane quadratic stability (AQS) introduced by [13].
To guarantee the performance we use the notion of guaranteed cost to optimize the given
cost function.
The class of control structure includes centralized, decentralized xed order output feedback
like PID controller.
The gain-scheduled controller design procedure is in the form of BMI. A feasible solution
for closed-loop system ensures the ane quadratic stability [13] and guaranteed cost when the
performance is dened in
Q, R, S
structure (see eq. (10)).
Quadratic stability (one Lyapunov function with one constant positive denite matrix cover
all ane controller design procedure) is more conservative than AQS in general. AQS (Lyapunov
function has an ane structure like (2)) incorporates information about the rate of variation
˙
θ(t)
to reduce conservatism. As we mentioned, in this paper the AQS approach will be used.
Our notations are standard.
D∈Rm×n
denotes the set of real
m×n
matrices.
Im
is an
m×m
identity matrix. If the size can be determined from the context, we will omit the subscript.
P > 0
(
P≥0
) is a real symmetric, positive denite (semi-denite) matrix.
The paper is organized as follows. Section 2 brings preliminaries and problem formulation.
The main result is presented in Section 3. In Section 4, numerical examples illustrate the eec-
tiveness of the proposed approach.
2. Preliminaries and problem formulation
Suppose that the state-space representation of an LPV system with
p
independent scheduling
parameters is governed by (1). The scheduling parameters
θi
and their derivatives with respect
to time
˙
θi
are supposed to lie in given hyper rectangles
Ω
and
Ωt
, respectively. For design of
the I part of the controller system, equation (1) has to be augmented, see [14] and example 1.
Without change of notation the new augmented matrices dimensions are
A(θ)∈R(n+l)×(n+l)
,
B(θ)∈R(n+l)×m
,
C∈R2l×2l
and
Cd∈Rl×l
is the output matrix for D part of controller. The
output feedback gain-scheduled control law is considered for PID controller in the form
u(t) = F(θ)y+Fd(θ) ˙yd=F(θ)Cx +FdCd˙x
(4)
where
yd=Cdx
is the output feedback for the D part of the controller,
F(θ) = F0+
p
X
i=1
Fiθi∈Rm×2l
(5)
is the static output feedback gain scheduled matrix for the PI controller and
Fd(θ) = Fd0+
p
X
i=1
Fdiθi∈Rm×m
(6)
is the static output feedback gain scheduled matrix for the D part of controller.
3
Remark 1.
Since the reference signal does not inuence the closed-loop stability, we assume
that it is equal to zero.
Remark 2.
If the derivative part of the controller includes some lter, the model of this lter
can be included in the system model.
The closed-loop system is then
[I−B(θ)Fd(θ)Cd] ˙x= [A(θ) + B(θ)F(θ)C]x
(7)
Ad(θ) ˙x=Ac(θ)x
(8)
˙x=Acd(θ)x
(9)
where
Acd(θ) = Ad(θ)−1Ac(θ)x
Ad(θ) = I−B(θ)Fd(θ)Cd
Ac(θ) = A(θ) + B(θ)F(θ)C
It is well known that the xed order dynamic output feedback control design problem is a special
case of the static output feedback problem. To access the performance quality a quadratic cost
function [15] known from LQ theory is often used in the form
J=Z∞
0
(xTQx +uTRu + ˙xTS˙x)dt
(10)
with
Q=QT≥0
,
R > 0
and
S=ST≥0
. The guaranteed cost is dened in a standard way.
Denition 1.
Consider system (1) with control algorithm (4). If there exists a control law
u∗
and a positive scalar
J∗
such that the closed-loop system (7) is stable and the value of closed-loop
cost function (10) satises
J≤J∗
, then
J∗
is said to be a guaranteed cost and
u∗
is said to be
guaranteed cost control law for system (1).
Denition 2.
[13] The linear closed-loop system (7) for
θ∈Ω
and
˙
θ∈Ωt
is anely quadratically
stable if and only if there exist
p+ 1
symmetric matrices
P0, P1, . . . , Pp
such that
P(θ) = P0+
p
X
i=1
Piθi>0
(11)
and for the rst derivative of Lyapunov function
V(θ) = xTP(θ)x
along the trajectory of closed-
loop system (7) it holds
dV (x, θ)
dt =xTAcd(θ)TP(θ) + P(θ)Acd (θ) + dP (θ)
dt x < 0
(12)
where
dP (θ)
dt =
p
X
i=1
Pi˙
θi≤
p
X
i=1
Piρi
From LQ theory we introduce the well known results.
4
Lemma 1.
Consider the closed-loop system (7). Closed-loop system (7) is anely quadratically
stable with guaranteed cost if and only if the following inequality holds
Be= min
udV (θ)
dt +xTQx +uTRu + ˙xTS˙x≤0
(13)
for all
θ∈Ω
and
˙
θ∈Ωt
3. Main results
In this section the gain scheduled controller design procedure which guarantees the ane
quadratic stability and guaranteed cost for
θ∈Ω
and
˙
θ∈Ωt
is presented. The main results for
the case of gain scheduled closed-loop stability analysis reduce to LMI condition and for gain
scheduled controller synthesis to BMI one.
The main result of this section, the gain scheduled design procedure, relies in the concept of
multi-convexity, that is, convexity along each direction
θi
of the parameter space. The implica-
tions of multiconvexity for scalar quadratic functions are given in the next lemma [13].
Lemma 2.
Consider a scalar quadratic function of
θ∈Rp
.
f(θ1, . . . , θp) = a0+
p
X
i=1
aiθi+
p
X
i,j=1
bij θiθj+
p
X
i=1
ciθ2
i
(14)
and assume that
f(θ1, . . . , θp)
is multi-convex, that is
∂2f(θ)
∂θ2
i
= 2ci≥0
(15)
for
i= 1,2, . . . , p
. Then
f(θ)
is negative for all
θ∈Ω
if and only if it takes negative values at
the corners of
θ
.
Using Lemma 2 the following theorem is obtained
Theorem 1.
Closed-loop system
(7)
is AQS with guaranteed cost if there exist
p+ 1
de-
nite matrices
P0, P1, P2, . . . , Pp
such that
P(θ)
(11)
is positive dened for all
θ∈Ω
, matrices
N1, N2, Q, R, S
and controller gain scheduled matrices
F(θ)
and
Fd(θ)
, satisfying
M(θ)<0; θ∈Ω
(16a)
Mii ≥0; i= 1,2, . . . , p
(16b)
where
M(θ) = M0+
p
X
i=1
Miθi+
p
X
i=1
p
X
j=1
Mij θiθj
M0=W110W12 0
W120TW22 0
Mi=W11iW12 i
W12i
TW22i
Mij =W11ij W12 ij
W12ij
TW22ij
5
W110=N1Ad0+AdT
0NT
1+CT
dFdT
0RFd0Cd+S
W11i=N1Adi+AdT
iNT
1+CT
dFdT
0RFdiCd
+CT
dFdT
iRFd0Cd
W11ij =N1Adij +AdT
ij NT
1+CT
dFdT
iRFdjCd
W120=P0+AdT
0NT
2−N1Ac0+CT
dFdT
iRF0C
W12i=Pi+AdT
iNT
2−N1Aci+CT
dFdT
0RFiC
+CT
dFdT
iRF0C
W12ij =AdT
ij NT
2−N1Acij +CT
dFdT
iRFjC
W220=
p
X
k=1
Pkρk−N2Ac0−AcT
0NT
2+Q
+CTFT
0RF0C;ρk∈Ωt
W22i=−N2Aci−AcT
iNT
2+CTFT
0RFiC
+CTFT
iRF0C
W22ij =−N2Acij −AcT
ij NT
2+CTFT
iRFjC
Ac0=A0+B0F0C
Aci=Ai+B0FiC+BiF0C
Acij =BiFjC
Ad0=I−B0Fd0Cd
Adi=−B0FdiCd−BiFd0Cd
Adij =−BiFdjCd
Proof.
Proof is based on
Lemma 1
and
2
. From (8) and (12) we can obtain
[2N1˙x+ 2N2x]T[Ad(θ) ˙x−Ac(θ)x] = 0
(17)
and
dV
dt = ˙xTP(θ)x+xTP(θ) ˙x+xT˙
P(θ)x
(18)
Summarizing the above two equations, for the time derivative of Lyapunov function one obtains
dV
dt =zTN1Ad(θ) + Ad(θ)TNT
1−N1Ac(θ) + AT
dNT
2+P(θ)
∗ −N2Ac(θ)−AT
c(θ)NT
2+Pp
i=1 Piρiz
(19)
where
N1, N2∈Rn×n
are auxiliary matrices and
zT=˙xTxT
. When one substitutes
control algorithm (4) to the right hand side of (13) and then the obtained result is combined
with (19) and substituted to (13), after some manipulation, using
Lemma 2
we obtain (16),
which proofs the
Theorem 1
.
Let us denote
θm=Pp
i=1 θi
, multiplying (16a) with
Pp
i=1
θi
θm
assuming that
θm6= 0
and
θm∈ hθm, θmi
we obtain
M0
θ2
m
p
X
i=1
p
X
j=1
θiθj+
p
X
i=1
p
X
j=1
Mi
θm
θiθj+
p
X
i=1
p
X
j=1
Mij θiθj<0
(20)
6
After small manipulation
p
X
i=1
p
X
j=1 M0+Miθm+Mij θ2
mθiθj<0
(21)
The closed-loop system will be stable or (21) holds if
Kij +Kji <0, i = 1,2,· · · , p, j =i, i + 1,· · · , p
(22)
where
Kij =M0+Miθm+Mij θ2
m
Using stability conditions (22) and
Lemma 2
if the following inequalities are met, the closed-loop
system is ane quadratically stable
2M0+ (Mi+Mj)θm+ (Mij +Mji )θ2
m<0
2M0+ (Mi+Mj)θm+ (Mij +Mji )θ2
m<0
Mij +Mji ≥0
(23)
for
i= 1,2,· · · , p
,
j=i, i + 1,· · · , p
.
Lemma 3.
Closed-loop system (7) is
AQS
with guaranteed cost if there exist
p+ 1
denite
matrices
P0, P1,· · · , Pp
such that for all
θ∈Ω
,
P(θ)
(11) is positive denite, matrices
N1, N2
and gain scheduled matrices
F(θ)
and
Fd(θ)
are satisfying (23).
If the solution of
Theorem 1.
or
Lemma 3.
are feasible:
For the case of closed-loop system stability, with respect to matrices
N1
,
N2
and positive
denite matrix
P(θ)
the closed-loop system is ane quadratically stable with guaranteed
cost and for
θ∈Ω
,
˙
θ∈Ωt
. For this case gain matrices (4), (5) and (6) are known and (16),
(23) reduces to LMI.
For the gain-scheduled controller design with respect to matrices
F(θ)
,
Fd(θ)
,
N1
,
N2
and
positive denite matrix
P(θ)
, the closed-loop system is ane quadratically stable with
guaranteed cost and for
θ∈Ω
,
˙
θ∈Ωt
. For this case (16) and (23) are BMI.
4. Examples
The rst example is taken from paper [16]. Consider a simple linear time-varying plant with
parameter varying coecients
˙x(t) = a(α)x(t) + b(α)u(t)
y(t) = x(t)
(24)
where
α(t)∈R
is an exogenous signal that changes the parameters of the plant as follows
a(α) = −6−2
πarctan α
20
(25)
b(α) = 1
2+5
πarctan α
20
(26)
7
0 20 40 60 80 100
−7
−68
−66
−64
−62
−6
α
Amplitude
a(α)
0 20 40 60 80 100
1
2
3
α
Amplitude
b(α)
θ1θ2θ3
Figure 1: Exogenous signal
α(t)
Let the problem be the design of a gain scheduled PID controller which will guarantee the
closed-loop stability and guaranteed cost for
α∈<0,100 >
. We will demonstrate that with the
gain-scheduled controller we will obtain practically identical behaviour for closed-loop system.
To be able to demonstrate this feature, let us divide the working area to 3 sections with 4 transfer
functions in points
α= 0,15,50,100
so that in each area, where the plant parameter changes,
they are nearly linear (Fig. 1.).
In these working points the calculated transfer functions are:
Gs1|α=0 =0.5
s+ 6, Gs2|α=15 =1.5242
s+ 6.4097
Gs3|α=50 =2.0642
s+ 6.6257, Gs4|α=100 =2.6858
s+ 6.8743
(27)
We transform the above transfer functions to the time domain to obtain the scheduling model
in the form (1). The obtained model was extended for the gain-scheduled PID controller design.
The extended model is given as follows
A0=−6.4370 0
1 0 , A1=−0.2050 0
0 0
A2=−0.1080 0
0 0 , A3=−0.1240 0
0 0
B0=1.5930
0, B1=0.5120
0
B2=0.2700
0, B3=0.3110
0
C=1 0
0 1 , D = 0
8
Using
Theorem 1
for
θi∈ h−1,1i, i = 1,2,3
we obtain gain scheduled controller in the form:
GrGS =Gr0+Gr1θ1+Gr2θ2+Gr3θ3
(28)
where
Gr0=0.4386s2+ 2.8850s+ 4.4678
s
Gr1=−1.72 ×10−6s2+ 9.49 ×10−5s+ 5.26 ×10−5
s
Gr2=−0.0283s2+ 1.5645s+ 0.8676
s
Gr3=−0.0056s2+ 0.3085s+ 0.1711
s
Note that if plant models in all working points are equal, in this case we obtain
Gri= 0
,
i= 1,2, . . . , p
. If some of
Gri˙
≈0
it indicates that some parameters of plant model are close to
other ones.
Using (1) and control algorithm
u=F(θ) (Cx −w) + Fd(θ)Cd˙x
(29)
one obtains the structure for simulation of the closed-loop system with gain scheduled PID
controller.
Simulation results (Fig. 2.,3.) conrm that
Theorem 1
holds. Fig. 2 demonstrates that with
the gain-scheduled controller we have obtained practically identical behaviour for closed-loop
system even if
α
changes as shown in Fig. 3. In gures,
y(t)
is the output signal,
w(t)
is the
setpoint,
u(t)
is the controller output,
α(t)
is exogenous signal on which the system depends and
θ
is the gain scheduled parameter.
The second example is taken from paper [8]. The model in the form (1) is extended for gain
scheduled PID controller design. The extended model is given as follows for
θ1∈ h−1,1i
A0=
−4 3 5 0
0 7 −5 0
0.1−2−3 0
0 1 0 0
, A1=
1 0 1 0
2 0 −5 0
251.5 0
0 0 0 0
B0=
0
16
10
0
, B1=
1
−5
3.5
0
, C =0100
0001
Using
Lemma 3.
we have obtained the gain-scheduled controller in the form (4) which after small
manipulation can be transformed to the form
GrGS =Gr0+Gr1θ1
(30)
where
Gr0=0,139s2+ 2,0381s+ 0,2401
s
Gr1=−0,0027s2+ 0,014s+ 0,004
s
9
0 20 40 60 80 100 120
0.4
0.5
0.6
0.7
t[s]
Amplitude
Simulation results y(t), w(t) y
w
0 20 40 60 80 100 120
2
4
t[s]
Amplitude
u(t)
u
Figure 2: Simulation results
0 20 40 60 80 100 120
−1
−0.5
0
0.5
1
t[s]
Amplitude
θ(t)
θ1
θ2
θ3
0 20 40 60 80 100 120
0
20
40
60
80
100
t[s]
Amplitude
α(t)
Figure 3:
θ(t), α(t)
10
The simulation results (Fig. 4., 5., 6., 7) conrm, that
Lemma 3.
holds. Figs. 4., 5., 6
demonstrate that with the gain-scheduled controller designed using
Lemma 3
we are able to
stabilize and control system with such parameter changes. We can see in Fig. 4. that at
θ= 1
the system is slow, and the controller output is positive although when
θ= 0
or
θ=−1
the
system is rapidly fast and the controller output is negative as shown in Fig. 5., 6. Fig. 7. shows
a case, when
θ
is changing linearly in interval
h−1,1i
.
0 500 1,000 1,500 2,000
0
0.2
0.4
0.6
0.8
t[s]
Amplitude
Simulation results w(t), y(t) at θ= 1
w
y
0 500 1,000 1,500 2,000
0
5
10
t[s]
Amplitude
u(t)
Figure 4: Simulation results for
θ= 1
The third example is a realistic model from Humusoft (magnetic levitation, for more detail
see www.humusoft.com). The model consists of a coil and a steel ball levitating in a magnetic
eld. Position of the steel ball is aected by the intensity of the magnetic eld and is measured
by a linear induction sensor connected to A / D converter. In terms of system theory it is an
unstable nonlinear dynamic system with one input (amplier voltage for coil) and one output
(ball position).
We split the ball position (voltage converted by the data acquisition card and scaled to 0
÷
1
machine unit [MU]) into 3 operating points
1. Position:
0.3 MU
−→ θ1=−1, θ2=−1
2. Position:
0.5 MU
−→ θ1= +1, θ2=−1
3. Position:
0.7 MU
−→ θ1= +1, θ2= +1
11
0 20 40 60 80 100 120 140
0.4
0.6
0.8
t[s]
Amplitude
Simulation results w(t), y(t) at θ= 0
w
y
1.2
1.4
1.6
u(t)
−0.3
−0.2
−0.1
Amplitude
0 20 40 60 80 100 120 140
−2.8
−2.6
−2.4
t[s]
Figure 5: Simulation results for
θ= 0
In these working points identied
1
plant transfer functions are
Gs1=2.0921
0.000264s2+ 0.0004s−1
Gs2=2.2487
0.00027s2+ 0.0032s−1
Gs3=2.1205
0.000155s2+ 0.0065s−1
(31)
The above transfer functions are transformed to the time domain to obtain the scheduling
model in the form (1). The obtained model is extended for the gain-scheduled PID controller
design. The extended model is given as follows
1
Transfer functions were identied in closed-loop system
12
0 20 40 60 80 100 120 140
0.4
0.6
0.8
t[s]
Amplitude
Simulation results w(t), y(t) at θ= -1
w
y
1.2
1.4
1.6
u(t)
−0.3
−0.2
−0.1
Amplitude
0 20 40 60 80 100 120 140
−2.8
−2.6
−2.4
t[s]
Figure 6: Simulation results for
θ=−1
A0=
0 4.1667 ·1030
1−1 0
0 1 0
A1=
000
0−5.5 0
000
, A2=
0 833.3333 0
0 1.5 0
000
B0=
8.7083 ·103
0
0
, B1=
−805
0
0
B3=
2.7667 ·103
0
0
, C =010
001
Using
Lemma 3.
with weighting matrices
R=rI, r = 1, Q =qI , q = 1 ×10−1, S =sI , s =
1×10−3
(when increasing
q
or
s
with respect to
r
in the rst case dynamic behaviour of the
closed-loop system becomes faster and in the second case the overshoot of closed-loop system
is smaller, for more detail see [17]) we have obtained gain scheduled controller in the form (4)
which after small manipulation can be transformed to the form
13
0 500 1,000 1,500 2,000
−1
0
1
2
t[s]
Amplitude
u(t)
0 500 1,000 1,500 2,000
0
0.2
0.4
0.6
0.8
t[s]
Amplitude
Simulation results w(t), y(t)
w
y
0 500 1,000 1,500 2,000
−1
−0.5
0
0.5
1
t[s]
Amplitude
θ(t)
Figure 7: Simulation results for
θ∈ h−1,1i
GrGS =Gr0+Gr1θ1+Gr2θ2
(32)
where
Gr0=0.0926s2+ 2.2966s+ 1.7304
s
Gr1=−0.02s2+ 0.0103s−0.0007
s
Gr2=0.0017s2−0.0009s+ 0.0016
s
Simulation results are shown in Fig. 8.
14
0 20 40 60 80 100
0.2
0.4
0.6
0.8
t[s]
Amplitude
Simulation results w(t), y(t)
y
w
0 20 40 60 80 100
−1
−0.5
0
0.5
1
t[s]
Amplitude
θ(t)
θ1
θ2
1
2
3
u(t)
−0.4
−0.2
0
Amplitude
u
0 20 40 60 80 100
−3
−2
−1
t[s]
Figure 8: Simulation results for
R= 1, Q = 1 ·10−1, S = 1 ·10−3
5. Conclusion
The paper addresses the problem of the gain-scheduled controller design which ensures the
closed-loop stability and guaranteed cost for all scheduled parameter changes. The proposed
procedure is based on the Lyapunov theory of stability, guaranteed cost and BMI. In the gain-
scheduled controller design procedure one can include the maximal value of the rate of gain-
scheduled parameter changes, which allows to decrease conservativeness and obtain the controller
with a given performance. The obtained simulation results show that the gain-scheduled con-
15
troller may give a better performance of closed-loop system for all changes of scheduled parameter
than a classical one including robust controller. Another advantage of this method is the fact
that we can aect the quality and cost with weighting matrices
R, Q, S
. Numerical examples
illustrate the eectiveness of the proposed approach.
Acknowledgement
The work has been supported by the Slovak Scientic Grant Agency, Grant No. 1/1241/12
and by Slovak Research and Development Agency, Grant APVV-0211-10.
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