Gain Scheduling (GS)

Contents
1.1 Gain Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Design of Gain-Scheduling Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Nonlinear actuator (Non-linear valve problem) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Gain scheduling based on measurements of auxiliary variables (Tank system problem) . . . . . . . . . . . 8
1.2.3 Time scaling based on production rate (Concentration control problem) . . . . . . . . . . . . . . . . . . . 11
1.2.4 Nonlinear transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 MATLAB Codes and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.5 Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1

1.1. GAIN SCHEDULING Adaptive Control
1.1 Gain Scheduling
It is sometimes possible to find auxiliary variables that correlate well with the changes in process dynamics. It is then possible
to reduce the effects of parameter variations simply by changing the parameters of the controller as functions of the auxiliary
variables (see Figure 1.1). Gain scheduling can thus be viewed as a feedback control system in which the feedback gains are
adjusted by using feedforward compensation. The concept of gain scheduling originated in connection with the development of
flight control systems. In this application the Mach number and the dynamic pressure are measured by air data sensors and
used as scheduling variables.
Figure 1.1: Block diagram of a system in which influences of paramaer variations are reduced by gain scheduling.
A main problem in the design of systems with gain scheduling is to find suitable scheduling variables. This is normally done
on the basis of knowledge of the physics of a system. In process control the production rate can often be chosen as a scheduling
variable, since time constants and time delays are often inversely proportional to production rate.
When scheduling variables have been determined, the controller parameters are calculated at a number of operating conditions
by using some suitable design method. The controller is thus tuned or calibrated for each operating condition. The stability
and performance of the system are typically evaluated by simulation; particular attention is given to the transition between
different operating conditions. The number of entries in the scheduling tables is increased if necessary. Notice, however, that
there is no feedback from the performance of the closed-loop system to the controller parameters.
It is sometimes possible to obtain gain schedules by introducing nonlinear transformations in such a way that the transformed
system does not depend on the operating conditions. The auxiliary measurements are used together with the process measure-
ments to calculate the transformed variables. The transformed control variable is then calculated and retransformed before it is
applied to the process. The controller thus obtained can be regarded as being composed of two nonlinear transformations with
a linear controller in between. Sometimes the transformation is based on variables that are obtained indirectly through state
estimation. Examples are given in Sections 9.4 and 9.5.
One drawback of gain scheduling is that it is an open-loop compensation. There is no feedback to compensate for an incorrect
schedule. Another drawback of gain scheduling is that the design may be time-consuming. The controller parameters must be
determined for many operating conditions, and the performance must be checked by extensive simulations. This difficulty is
partly avoided if scheduling is based on nonlinear transformations.
Gain scheduling has the advantage that the controller parameters can be changed very quickly in response to process changes.
Since no estimation of parameters occurs, the limiting factors depend on how quickly the auxiliary measurements respond to
process changes.
1.2 Design of Gain-Scheduling Controllers
It is difficult to give general rules for designing gain-scheduling controllers. The key question is to determine the variables
that can be used as scheduling variables. It is clear that these auxiliary signals must reflect the operating conditions of the
plant. Ideally, there should be simple expressions for how the controller parameters relate to the scheduling variables. It is thus
necessary to have good insight into the dynamics of the process if gain scheduling is to be used. The following general ideas can
be useful:
• Linearization of nonlinear actuators,
• Gain scheduling based on measurements of auxiliary variables,
• Time scaling based on production rate, and
• Nonlinear transformations.
The ideas are illustrated by some examples.
Mohamed Mohamed El-Sayed Atyya Page 2 of 19

1.2. DESIGN OF GAIN-SCHEDULING CONTROLLERS Adaptive Control
1.2.1 Nonlinear actuator (Non-linear valve problem)
The ﬁgure below slows a process G(s) = 1
(s+1)3 with non-linear valve v = f(u) = u4, controlled by PI controller
Gc(s) = k 1 + 1
Tis
Figure 1.2: Block diagram of a ﬂow control loop with a PI controller and a nonlinear valve.
if k = 0.15 , Ti = 1 we get the following results

Valve Approximation
Because of non-linearity of the valve, so we split the valve characteristic line to two regions as shown,
as ˆf−1(c) =
0.433 c 0 ≤ c ≤ 3
0.0538 c + 1.139 3 ≤ c ≤ 16
Finally we get,
Figure 1.3: Compensation of a nonlinear actuator using an approximate inverse.

For the same values of k and Ti and same inputs we get,

Controller with Gain Change
Another solution of the problem is to change the controller parameter as I choose,
Ti = 0.05 , k =
uc if uc ≤ 1
1
uc
if uc ≥ 1
For the same inputs we get,

Controller with Parameter Change
Another solution of the problem is to change the controller parameter as I choose,
Ti =
0.05 if uc < 1
1
4uc
if uc ≥ 1
, k =
uc if uc ≤ 1
1
uc
if uc ≥ 1
For the same inputs we get,

1.2.2 Gain scheduling based on measurements of auxiliary variables (Tank system problem)
System Modeling
For the following tank,
• qin : Inlet volume ﬂow rate
• qout : Outlet volume ﬂow rate
• a : Cross sectional area of tap
• h : Height of the tank
• A(h) : Cross sectional area of tank at height h
• Ao : Cross sectional area of tank at height h
qin − qout =
dV
dt
=
dV
dh
dh
dt
dV = A(h) dh ⇒
dV
dh
= A(h)
qin − qout = A(h)
dh
dt
qout = ˙Vout = avout ;
v2
out = v2
in + 2gh , vin ≈ 0 ⇒ vout = 2gh
qout = a 2gh
qout − a 2gh = A(h)
dh
dt
Linearization :
qin = qo,in + ∆qin , h = ho + ∆h
qo,out + ∆qin − a 2g(ho + ∆h) = A(ho + ∆h)
d
dt
(ho + ∆h)
dho
dt
= 0 , A(ho + ∆h) ≈ A(ho)
2g(ho + ∆h) = 2gho 1 +
∆h
ho
≈ 2gho 1 +
1
2
∆h
ho
@ steady state qo,in − qo,out = 0 ⇒ qo,in − a 2gho = 0
∆qin → qin , ∆h → h
qo,in + qin − a 2gho − a 2gho
h
2ho
= A(ho)
dh
dt
Qin −
a
2
2g
ho
H = AoHs ; Ao = A(ho)
let β =
1
Ao
, α =
a
2Ao
2g
ho

G(s) =
H
Qin
(s) =
β
s + α
Use PI controller : Gc(s) = k 1 +
1
Tis
Closed loop TF :
H
Hd
=
Gc(s)G(s)
1 + Gc(s)G(s)
=
βk(Tis + 1)
Tis2 + (βkTi + αTi)s + βk
=
βk(s + 1
Ti
)
s2 + (βk + α)s + βk
Ti
2ζwn = βk + α ⇒ k =
2ζwn − α
β
w2
n =
βk
Ti
⇒ Ti =
βk
w2
n
=
2ζwn − α
w2
n
As : α << 2ζwn
k =
2ζwn
β
= 2ζwnAo , Ti =
2ζ
wn
Let :
A(h) = A(0) + h2
, A(0) = 20 , h = 7 , a = 0.1A(0) = 2 , ζ = 0.7 , wn = 4
Figure 1.4: Block diagram of tank system with PI gain scheduling controller

Some Simulation Results

1.2.3 Time scaling based on production rate (Concentration control problem)
System Modeling
• cin : The concentration at the inlet of the pipe
• Vd : The pipe volume
• Vm : Tank volume
• q : Flow rate
• c : The concentration in the tank and at the outlet
• Ts : Sampling Time

Vm
dc
dt
= q(t)[cin(t − τ) − c(t)]
τ =
Vd
q(t)
Vmcs = q[cine−τs
− c]
G(s) =
c
cin
=
qe−τs
Vms + q
=
e−τs
Ts + 1
; T =
Vm
q
G(z) = Z
1 − e−Tss
s
e−τs
Ts + 1
= z−d
(1 − z−1
)Z
1/T
s(s + 1/T)
= z−d
(1 − z−1
)
(1 − e−Ts/T )z−1
(1 − z−1)(1 − e−Ts/T z−1)
= z−d (1 − e−Ts/T )z−1
1 − e−Ts/T z−1
Where : τ = dTs
Let : a = e−Ts/T
= e−Tsq/Vm
= e−TsVd/(τVm)
= e−Vd/(Vmd)
G(z) = z−d (1 − a)z−1
1 − az−1
Closed Loop System Performance Without Gain Scheduling
Let : q = 1 , T = 1 , τ = 1 , ⇒ G(s) =
e−s
s + 1
And use PI controller Gc(s) = 0.5 1 + 1
1.1s

Closed Loop System Performance With Gain Scheduling
Use Ziegler-Nichols of PI controller of the open loop system,
Kc =
0.9τ
T
=
0.9Vd
Vm
, Ti =
L
0.3
=
Vd
0.3q

1.2.4 Nonlinear transformations
Examples
1. Nonlinear transformation of a pendulum
Consider the system
dx1
dt
= x2
dx2
dt
= − sin x1 + u cos x1
y = x1
which describes a pendulum, where the acceleration of the pivot point is the input and the output y is the angle from a
downward position. Introduce the transformed control signal
v(t) = − sin x1(t) + u cos x1(t)
This gives the linear equations
dx
dt
=
0 1
0 0
x +
0
1
v
Assume that x1 and x2 are measured, and introduce the control law
v(t) = −l1x1(t) − l2x2(t) + m uc(t)
The transfer function from uc to y is
m
s2 + l2s + l1
Let the desired characteristic equation be
s2
+ p1s + p2
which can be obtained with
l1 = p2 l2 = p1 m = p2

Transformation back to the original control signal gives
u(t) =
v(t) + sin x1(t)
cos x1(t)
=
1
cos x1(t)
(−p2x1(t) − p1x2(t) + p2uc(t) + sin x1(t))
Notice that the previous equation can be used for all angles except for x1 = ±π/2,that is, when the pendulum is
horizontal.
From Laplace transformation we get
x1
u
=
cos x1
s2 + sin x1
=
b
s2 + a
x2
u
=
s cos x1
s2 + sin x1
=
bs
s2 + a
Let :
p1 = 10, p2 = 25
For unit step input and linear controller
Figure 1.5: Block diagram of the system with linear controller

For unit step input and nonlinear controller
Figure 1.6: Block diagram of the system with nonlinear controller

2. Nonlinear transformation of a second-order system
Consider the system
dx1
dt
= f1(x1, x2)
dx2
dt
= f2(x1, x2, u)
y = x1
Assume that the state variables can be measured and that we want to find a feedback such that the response of the
variable x1 to the command signal is given by the transfer function
G(s) =
w2
s2 + 2ζws + w2
(1.1)
Introduce new coordinates z1 and z2, defined by
z1 = x1
z2 =
dx1
dt
= f1(x1, x2)
and the new control signal v, defined by
v = F(x1, x2, u) =
∂f1
∂x1
f1 +
∂f1
∂x2
f2 (1.2)
These transformations result in the linear system
dz1
dt
= z2
(1.3)
dz2
dt
= v
It is easily seen that the linear feedback
v = w2
(uc − z1) − 2ζwz2 (1.4)

1.3. MATLAB CODES AND SIMULATION Adaptive Control
gives the desired closed-loop transfer function of Eq. (1.1 ) from uc, to z1 = x1 for the linear system of Eqs. (1.4). It
remains to transform back to the original variables. It follows from Eqs. (1.2) and (1.4) that
F(x1, x2, u) =
∂f1
∂x1
f1 +
∂f1
∂x2
f2 = w2
(uc − x1) − 2ζwf1(x1, x2)
Solving this equation for u gives the desired feedback. It follows from the implicit function theorem that a condition for
local solvability is that the partial derivative ∂F/∂u is diﬀerent from zero.
1.3 MATLAB Codes and Simulation
1.2.1 http://goo.gl/NOPbhx
1.2.2 http://goo.gl/adJrkw
1.2.3 http://goo.gl/dJwTxB
1.2.4 http://goo.gl/mwWq7N
1.4 References
1. Karl Johan Astrom, Adaptive Control, 2nd
Edition.
1.5 Contacts
mohamed.atyya94@eng-st.cu.edu.eg

Gain Scheduling (GS)

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