Partial Gain Scheduled Controller via Two Stage Design
M2015SC014 Akio SAMBAYASHI
Supervisor : Isao TAKAMI

Abstract
This paper proposes a new concept gain scheduled(GS) controller and its design method. For designing
this controller, we proposed Two-stage design method
and Pseudo ILQ. The eﬀectiveness of proposed method
is illustrated by simulations.

1

Introduction

It is natural for plants in the real world that dynamic
characteristics change due to parameter fluctuations.
Gain Scheduled (GS) controller gives the optimum control gain according to the fluctuation of parameters, andhigh controllability can be expected [1]. This method
can be applied to various controlled objects [2]. However, GS requires a large amount of computation, which
is a hindrance real-time control system design. Therefore, only the gain having the eﬀect of gain schedulingis set as a variable gain based on the scheduling parameter. We propose a GS control system with fixed gain
for a part of state variables. This is called Partial Gain
Scheduled (PGS) controller. This reduces the amountof calculation and realizes controllability equivalent tothat of the conventional GS. This paper shows the conditions of the plant to which this control system can beapplied, but it is applicable to many plants. Next, thecontrol system design method is shown. We adopt a twostage design method. This design method is derived asa theorem using Lienar Matrix Inequalities (LMI). At
this time, in order to guarantee control performance,
Inverse Optimum Control of Linear Quadratic Regulator design theory (ILQ) is applied [4]. Finally, we apply
the proposed method to the jib crane system and show
its eﬀectiveness.

2

The formulation of the target system

There are restrictions on the target plant which canapply the proposed controller. A form of the target plantis given in this section. Consider a following systemdescribed by:Φ : ẋ(t) = A(δ)x(t) + Bu(t)[A(δ) =

Ar (δ)0Aj (δ) = Aj0 +Asr (δ)As (δ)k∑

δi Aji

]

[
B=

(1)
Br
Bs

]

Remark 1 When the state variable is selected, such as
to satisfy the following assumptions, the proposed controller can be designed .
• The linear system has A(δ) such that the upper rightor the lower left of the block matrices are zero.
• The subsystem is controllable.

3

Two stage design for Partial GS controller

The obtained system in previous section has sometime-varying parameters which aﬀects largely the target
plant behavior. As a first stage for proposed controller,
in order to deal with this problem, a part of the target
plant which includes some or all the scheduling parameters is defined as a subsystem. A closed-loop system
is designed by using GS controller. As a second stage,a fixed controller which guarantees the robust stability and control performance is designed for target plantwith the closed-loop subsystem. Through the above twostages, varying gains (which are defined as Ks ) for thesubsystem and fixed gains (which are defined as Kcl )for the target system which include the closed-loop for
subsystem. The proposed controller is obtained by thesum of Ks (δ) and Kcl . As a result, the proposed controller has varying gains for a part of the state variable.
In the first stage and the second stage, the controller isdesigned such that the following cost functions, Eq.(3)and Eq.(4), are minimized, respectively.
∫ ∞
( T
)
J=
x (t)Qx(t) + uT (t)Ru(t) dt
(3)
∫0 ∞
( T
)
Js =
xs (t)Qs xs (t) + uTs (t)Rs us (t) dt (4)0These two cost functions need to be consistent in thewhole system. In the first-stage design, to derive thestabilizing feed back gain scheduled controller us =Ks (δ)xs (t) for the subsystem, the following conditions
are well known to minimize the cost function Eq. (4).
Ps (As (δ) + Bs Ks (δ)) + (As (δ) + Bs Ks (δ))T Ps(j ∈ {r, sr, s}) ,+KsT (δ)RKs (δ) < 0
Js = trace [Ps ] < γsi=1where A(δ) ∈ Rn×n , B ∈ Rn×m . This system has asubsystem defined as Eq. (2):Φs : ẋs (t) = As (δ)xs (t) + Bs us (t),The target system and the subsystem which don’t include δ by uniquely determining the value of δ are described as Φ(N ) and Φs (N ), respectively.

(2)

where As (δ) ∈ Rn1 ×n1 , Bs ∈ Rn1 ×m1 , (n1 < n,m1 < m). Then x is defined as x := [xr xs ]. (A(δ), B)
and (As (δ), Bs ) are controllable. Eq. (1) has aﬃne
perturbation in each coeﬃcient matrix, where δi ∈ R isperturbation element which satisfy |δi | ≤ 1. Let the target system and the subsystem be Φ and Φs , respectively.

(5)
(6)

If there exists Ps = (Ps )T > 0 satisfying above matrix inequality, the subsystem, Eq. (2), is stabilized by
us = Ks (δ)xs and J < γ. Now, Eq. (5) is formulated
as LMI by using congruence transformation, variabletransformation and Schur compliment [1]. Let Xs andYs be Xs := Ps−1 and Ys := Ks Xs . Then the following
matrix inequality is obtained by post-multiplying XsT
and using Schur complement.
trace[Xs−1 ] < γs(7)For (7), Zs is introduced as an upper bound for Xs−1
since it is diﬃcult to solve convex optimization problem.
Then (7) is trance formed to Eq. (8) by using Schurcomplement.
[]
Zs I> 0, trace[Zs ] < γs
(8)
I Xs
By using above conditions, the obtained class of gain
scheduled controller is a linear function of schedulingparameters. By defining both Lyapnov matrix Xs andthat of variable matrix Ys as linear functions of scheduling parameters, the obtained class of that is easily expanded to a polynomial rational function. These design
methods of GS controller will be described later. In thesecond stage, the proposed controller which is describedby the sum of Ks and Kcl such that the above the costfunction of Eq. (4) is minimized is designed. With theclosed-loop subsystem by using varying gains of Kse (δ),
the target plant is obtained as the following plant forthe subsystem of Eq. (9).
ẋ(t) = (A(δ) + BKse (δ)) x(t) + Bucl (t),Kse (δ) := [0 Ks (δ)] ,

(9)
(10)

Corollary 1 A. If there exists Xs > 0 and Ys (δ)
such that Eq. (12) and Eq. (13) are held, the closedloop system is stabilized by the state feedback us (t) =Ks (δ)xs (t) := Ys (δ)Xs xs := (Ys1 + δYs2 )Xs−1 x(t).
minimize γs , subject to :
He [As (δ)Xs + Bs Ys (δ)]∗∗
Qs (δ)Xs−Qs (δ) ∗  < 0,
1
2
R Ys (δ)
0
−I
(12)
[

Zs
I

I
Xs

]
> 0, trace[Zs ] < γs ,(13)where Qs (δ) > 0 and R > 0 are weighting matrices.
Furthermore, though maximising the trace of Xs , Js is
minimized.
If Qs is given as a parameter dependent quadratic func∑ktion of Qs (δ) = (Qs0 + i=1 δi Qsi )2 , the new corollary
is obtained by using Corollary 1 A. and Schur complement.

where Kse (δ) is defined by using Ks (δ) because the statevariables, xs and xcl , have diﬀerent dimension. The
fixed feedback controller ucl (t) = Kcl x(t) is designed
such that the cost function are minimized by Ks (δ) and
Kcl . The detailed design method is described from thenext section. It is described that the proposed methodsof many types are derived as following theorems andcorollaries with Proposition 1 as a base.

Case 1 R. The second type of GS controller has gains
given by polynomial functions of parameter variation.
The controller is given by following theorem. Theorem
description is omitted for want of space. [1]

Proposition 1 Consider the following system is givenby the following Eq. (11), where w(t) is an impulse
disturbance input.

Case 2 A. When Case 1 A. is used for the first-stage
design, the second-stage design of the proposed method
can be described as the following theorem by using the
Proposition 1.

{

ẋ(t) = (A(δ) + BKse (δ)) x(t)+Bucl (t) + Bw w(t)(11)z(t) = C(δ)x(t) + Ducl (t)
[
]
[
]
1
0
Q2
1
Bw = I, C(δ) =
, D=
1
R2
R 2 Kse (δ)

If this H2 state feedback controller, ucl = Kcl x is obtained, the proposed controller of u = use (t) + ucl (t) =(Kse (δ) + Kcl )x = (Kse (δ) + Kcl )x is minimized the thecost function of Eq. (3) for the target plant of (1) .
3.1

First stage design for subsystem

As described in the previous section, two typical examples are considered for designing the GS control system.
Case A. describes a design method in which the obtained gain is an aﬃne function with respect to the variation parameter. Case R. describes a design method
in which the obtained gain is a rational function of thevariation parameter.
Case 1 A. The first type of GS controller has gains
given by aﬃne functions of parameter variation. The
controller is given by following corollary.
If Qs is given as a parameter dependent linear function∑kof Qs (δ) = Qs0 + i=1 δi Qsi , Corollary 1 A. is obtained.

3.2

Second stage design for target system

Here, we explain the second stage after completing thefirst stage in the two-stage design.

Theorem 1 If there exist X > 0 and Y such thatEq. (14) and Eq. (15) held, the closed-loop systemis stabilized by the state feedback ucl (t) = Kcl x(t) :=Y X −1 x(t).
minimize γ subject to :[]He [A(δ)X + BY ] X T C T (δ) + Y T DT
< 0(14)
C(δ)X + DY
−I
[
]
Z I
> 0, trace[Z] < γ
(15)
I X
[
]
]
[
1
0
Q2
1
C(δ) =,, D=
1R2
R 2 Kse (δ)
where Q ≥ 0 and R > 0 are weighting matrices. Furthermore, though maximizing the trace of X, J is minimized.
Case 2 R. When Case 1 R. is used for the first-stage
design, Ks (δ) is given by rational functions. To solve
this diﬃculty, a new argument value, zδ , is introduced.
The equivalent system for the Eq.(11) is defined by using
a redundant variables. The second-stage design of the
proposed method can be described as a corollary by using
Theorem 1 and Linear Fractional Transformation. The
description of the corollary is omitted for want to space.

Remark 2 Ks (δ) which is described as rational functions of δ can also be applied to Theorem 1 by using
Aﬃne interpolation. The theorem used for the secondstage design is diﬀerent depending on whether the aﬃnefunctions give the gains or the rational functions givethe ones in the first-stage design. If you chose Case
1 A. in first-stage design, Case 2 A. in second-stage
design should be used.

Procedure 1 Following outline of the selection procedure of Qs for the proposed method are described in theorder of 1 to 3.
1. Design the LQ control system with stabilizing gains
for the target plant Φ̂.
2. Extract the part of the closed-loop subsystem fromthe closed-loop target plant. Note that the obtained
closed-loop system of the subsystem is stable.

4

3. The feedback gains K̂s that stabilizes this subsystemΦ̂s and the subsystem are applied to ILQ method orcorollaries.

The weighting matrices in the firststage design

The gains Ks (δ) obtained at the first stage in designing the control system is not always such as to minimize
the cost function at the second stage. The selection of
the weighting matrices in the first stage is the selectionof the free parameters in the design of proposed controller. The selection weighting matrices should ensure
the performance of the control system at least as highas the Robust Linear Quadratic(R-LQ) controller. The
systems in the ILQ method is not need to be considered
the same perturbation in the design. A new system isdefined for an ILQ method. Let Φ̂ and Φ̂s be the total
system and the subsystem for the ILQ method, respectively. When they include the perturbation δ explicitly,
they are described as Φ̂(δ) and Φ̂s (δ). The target systemand the subsystem which don’t include δ by uniquelydetermining the value of δ are described as Φ̂(N ) andΦ̂s (N ). Let K̂ and K̂s be gains for Φ̂ and for Φ̂s which
are extracted from K̂, respectively. When they include
the perturbation δ explicitly, they are described as K̂(δ)
and K̂s (δ), Let K̂(N ) and K̂s (N ) be fixed gains forΦ̂(N ) and Φ̂s (N ). Now, we propose Pseudo ILQ. ILQ
method [4] is naturally extend to the framework of suboptimal problem as following Corollary 2.

Considering the robustness of the controlled object ineach step of the above procedure, seven kinds of procedures are obtained.

5

The proposed controller is applied to the Jib cranesystem.
5.1

Mathematical model of Jib Crane

In this section, the dynamics of a crane is derived
as Eq. (17)-(18) from Euler-Lagrange equations. A
schematic diagram of the crane is shown in Figure 1.
The physical constants and variables are shown in Table 1.
(t)0Trolley

(t)

௣ (t)

Load

Corollary 2 F. If there exist P̂s = P̂s > 0 and ϵ >0 such that the following equation is hold, Qs (N ) =−P̂s Âs (N ) − Âs (N )P̂s + K̂s (N )K̂s (N ) is obtained. We
call this corollary Pseudo ILQ.
minimize ϵ, subject to :
]
[
S(N )
P̂s B̂s + K̂s (N )<0
B̂s P̂s + K̂s (N )−ϵNumerical example(t)Figure 1: Schematic diagram for crane system

Table 1: Physical parameters(16)

mp [m]
mt [kg]l [m]g [m/s2 ]
Kg,j
Kt,j

S(N ) = P̂s Âs (N ) + Âs (N )P̂s − K̂s (N )K̂s (N )where the K̂s (N ) is a stabilizing feedback gain and1(Qs2 (N ), Âs (N )) is detectable.
Although it is ideal if ϵ becomes 0, that can not always
be guaranteed due to numerical adverse conditions suchas parameter variation δ. However, what we should obtain is not a strict ILQ solution. Our aim is to obtainthe first stage Qs of the cost function which is convenient in the second stage. It is to propose an ILQ-likecorollary that gives priority to answering. However, thevalue of ϵ is important. The smaller the ϵ value is, thecloser it is to the strict ILQ solution. It can be said
that there is a high possibility that Qs of the first stage
which is convenient in the second stage is obtained. As
the ϵ value increases, it is more likely that the Qs isaway from the ILQ answer and the required Qs is not
obtained. The following the procedure gives an outlineof the selection procedure of the weighting matrix forthe proposed method using the above Corollary.

Mass of payload
Mass of trolley
lope length
Acceleration of gravity
Jib motor gear ratio
Jib motor torque constant

{

(
)
mp
¨l − g γ + Kg,j Kt,j Ijξ¨ = mj −m
mj −mpp(17)
yp = ξ − l sin γ(mj − mp )lγ̈ = (mp ¨l − mj g)γ − 2l˙γ̇(mj − mp )l + Kt,j Ij
(18)
5.2

Partial GS controller synthesis
The state variable x(t) = [wp ξ ξ˙ γ γ̇] and xs (t) = [γ γ̇]are defined. The descriptor representation can be described as Eq.(19) and Eq.(20).
E ẋ(t) = A(δ)x(t) + Bu(t)Es (δ)ẋs (t) = As (δ)xs (t) + Bs us (t)

(19)
(20)

δ1 and δ3 are treated as uncertain parameters. δ2 istreated as a scheduling parameter. To derive the stabilizing state feedback controller, the cost function (4)is considered. Let l, l˙ and ¨l are the lope length, theexpansion and contraction velocity and that of acceleration. The proposed controller is obtained by solving
LMI conditions shown in Theorem 2, Corollary 1 A.
and Corollary 2 F. at each vertex matrices
5.3

Table 2: The numerical value of physical parameters
mp [kg]mt [kg]

0.99
0.60

Kt,j
Kg,j

15.5
3.70

g [m/s2 ]9.81

0.1 [m] to 0.7 [m] with the velocity of 0.260 [m/s] andthe acceleration of 4 [m/s2 ] at 1 [s]. The trolley position moves from 0 [m] to 0.3 [m] with the velocity of 0.2[m/s]. The upper bounds and lower bounds are defined
as follows.
l ∈ [0.10, 0.70] , l˙ ∈ [−0.260, 0.2650] , ¨l ∈ [−5.0, 5.0]
The upper bound values of the cost function were obtained as shown in the following table. The Proposed
Table 3: The upper bound value of the cost functions
upper bound value

R-LQ
39.0773

SL-GS
24.8253

Proposed
26.1462

controller upper bounds of the cost function are nearly
identical to SL-GS, and it can be seen from this that thesame performance as SL-GS can be guaranteed. The left
graph of Figure 2 shows the horizontal load position of
crane. The proposed method has the same performanceas the conventional SL-GS, which shows that it is muchbetter performance than R-LQ, by the upper bound of
the cost function value and simulation. The cumulative

Input Current [A]

4

0.3
0.2

R-LQ
SL-GS
Proposed

0.1

×10-3 Input current

3
R-LQ
SL-GS
Proposed

2
1
0
-1

0

0

5

10
time [s]

15

-2

0

5

10

15

time[s]Figure 2: The load position (left) and the input current(right).
calculation time between input and output with eachcontroller is obtained. This cumulative calculation time
is obtained 1000 times for each controller. That of theaverage is obtained as Table 4. The proposed method
has been able to reduce the computational complexity
compared with SL-GS.
Table 4: Amount of the cumulative calculation time
Controller
times[s] ×10−2

Simulation result

In this study, the eﬀectiveness of the proposed controller is compared with robust LQ controller (R-LQ)
and GS controller via single Lyapunov function (SL-GS)
in the simulations. The R-LQ guarantees the robustness for the variation of the velocity of the lope length
perturbation by fixed controller gain. The SL-GS guarantees the robustness for the variation of the velocity of
the lope length perturbation by varying controller gain.
The physical constants of crane system are given as Table 2. In this study, the lope length is started from

Horizontal load Position

0.4
Position of Load [m]If the coeﬃcient matrix E is a regular matrix, section
3 - 4 can be expanded to the descriptor system, easily
from some previous studies [2] [3]. Es (δ), As (δ), Bs arecontrollable. The proposed controller can be designed
for Eq.(19). Hence, let xds (t) = [γ γ̇ γ̇ γ̈] as descriptor
variables. Most of the dynamical system can be designed with the argument of state variable, xd = [x ẋ].
However, It is reported that the argument of state variable can be reduced as xds (t) = [γ γ̇ γ̇ γ̈] ⇒ [γ γ̇ γ̈] [3].
The argument system Eq.(19) and Eq.(20) is obtained.
The lower bounds and upper bounds of l, l˙ and ¨l are
assumed as :
[ ]
[ ]
[ ]
δ1 = l ∈ l, l , δ2 = l˙ ∈ l,˙ l˙ , δ3 = ¨l ∈ ¨l, ¨l .

6

RLQ
3.12

SL-GS
5.17

Proposed
4.13

Conclusion

This paper proposed the new concept of the GS controller and the design methods. Several theorems, corollaries, methods, and procedures for that is obtained.
This controller can be applied to many systems which
has the certain structure. We made the structure clear.
Furthermore, by extending the theory of ILQ, an eﬀective method of determining free parameters included in
the proposed method was proposed. The proposed controller has varying gains for only the state variable ofthe subsystem. By the simulation, It is verified that the
proposed controller has almost the same performance as
the ordinary GS, and numerical results showing better
performance than robust LQ were obtained.

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Parametrically-Dependent Linear, Dynamic Feedback.”, Advances in Robust and Nonlinear Control
Systems DSC-Vol. 43, pp, 29-36, 1992
[2] A. Sanbayashi, M. Narita, G. Chen, I. Takami,“Gain scheduled control for active magnetic bearing system considering gyroscopic eﬀect, Information Technology and Electrical Engineering (ICITEE), 2015 7th International Conference on, pp.
297-303, 2015[3] T. Kanada, Y. Watanabe, G. Chen, “Robust H2Control for Two-Wheeled Inverted Pendulum Using LEGO Mindstorms”, Australian Control Conference, Engineers Australia, ThIT3.2, pp.136-141,
2011[4] T. Fujii, “The Inverse Problem of Optimal Regulator”, Journal of the Society of Instrument and Control Engineers, Vol. 27, No. 8 , pp.717-726, 1988 (inJapanese)