The document describes designing an LQR full state feedback controller for an active suspension system. It includes: 1) Developing a quarter-car mathematical model and deriving the state-space representation. 2) Simulating an open-loop system in Simulink that has large body acceleration and suspension oscillation. 3) Implementing an LQR controller in Simulink and analyzing the closed-loop response for different weighting matrices Q. Larger Q weights reduce deviations from the desired state but increase control effort. 4) Increasing physical parameters up to 10% does not significantly change the closed-loop response, showing the controller is not robust to parameter variations.

1. PROJECT 2
DESIGN LQR FULL STATE FEEDBACK CONTROLLER
FOR HIGH BANDWIDTH ACTIVE SUSPENSION
SYSTEM
Name : IB.P.P.Mahartana
NRP : 211100177

2. Design LQR Full State Feedback Controller for High Bandwidth
Active Suspension System.
A. Mathematical Model
Conventional passive suspensions use a spring and damper between the car body and
wheel assembly. The spring-damper characteristics are selected to emphasize one of several
conflicting objectives such as passenger comfort, road handling, and suspension deflection.
Active suspensions allow the designer to balance these objectives using a feedback-controller
hydraulic actuator between the chassis and wheel assembly.
For this project we use a quarter-car model of the active suspension system shown in
figure 1.
Figure 1.1 Quarter-car mathematical model for active suspension system
The mass ms represent the car chassis (body) and the mass mu represents the wheel assembly.
The spring ks and damper cs represent the passive spring and shock absorber placed between
the car body and the wheel assembly. The spring kt and damper ct represent models
compressibility of the pneumatic tire. The variables zs, zu, and zr are the body travel, wheel
travel, and road disturbance, respectively. The force Fs applied between the body and the wheel
assembly is controlled by feedback and represents the active component of the suspension
system.

3. B. Free Body Diagram
To be able determine how the system move, we need to draw the free body diagram, shown
in figure 1.2 which are contain all the force work in the system,
zs
zu
Figure 1.2 Free body diagram for the active suspension system
B.1 Equation of motion.
The Equation of motion represent the motions each body through their acceleration, velocity
and displacement, in terms of differential equation.
∑ 𝐹𝑠 = 0
𝑚 𝑠 𝑧̈ 𝑠 + 𝑘 𝑠(𝑧𝑠 − 𝑧 𝑢) + 𝑐 𝑠(𝑧̇ 𝑠 − 𝑧̇ 𝑢) − 𝐹𝑠 = 0
∑ 𝐹𝑢 = 0
𝑚 𝑢 𝑧̈ 𝑢 − 𝑘 𝑠(𝑧𝑠 − 𝑧 𝑢) − 𝑐 𝑠(𝑧̇ 𝑠 − 𝑧̇ 𝑢) + 𝑘 𝑡(𝑧 𝑢 − 𝑧 𝑟)+ 𝑐𝑡(𝑧̇ 𝑢 − 𝑧̇ 𝑟) + 𝐹𝑠 = 0
Define
𝑥1 = 𝑧𝑠 − 𝑧 𝑢 𝑥2 = 𝑧 𝑢 − 𝑧 𝑟 𝑥3 = 𝑧̇ 𝑠 𝑥4 = 𝑧̇ 𝑢
𝜔 = 𝑧 𝑟
ms𝑚 𝑠 𝑧̈ 𝑠
𝑐 𝑠(𝑧̇ 𝑠 − 𝑧̇ 𝑢)
ms
𝑘 𝑠(𝑧𝑠 − 𝑧 𝑢) 𝐹𝑠
𝑘 𝑡(𝑧 𝑢 − 𝑧 𝑟) 𝑐𝑡(𝑧̇ 𝑢 − 𝑧̇ 𝑟)

4. The State Matrices
The Output Matrices
𝐶𝑧 = [
−𝑘𝑠/𝑚𝑠 0 −𝑐𝑠/𝑚𝑠
1 0 0
0 1 0
𝑐𝑠/𝑚𝑠
0
0
]
𝐷ℎ𝑢 = [
1/𝑚𝑠
0
0
]
𝐷ℎ𝑤 = [
0
0
0
]
C. Simulink Block Diagram
We will use conventional block diagram just for showing there is another way two describe
our system instead of state space block. The first open loop block diagram is shown in figure
1.4 below.
Figure 1.3 HBAS Open-loop system

5. C.1 Block Diagram Associated with the state and output matrices.
Figure 1.4 Block diagram associated with the state and output matrices
D. Matlab source code.
In this part, we will loads all the parameters, so we can run the simulink block.
%% Load all Phsyical Parameters
ms=300; %
mu=40;
ks=20000;
cs=1000;
kt=200000;
ct=10;
%% Derive State & output matrices
Ah=[0 0 1 -1; 0 0 0 1; -ks/ms 0 -cs/ms cs/ms; ks/mu -kt/mu cs/mu -
(cs+ct)/mu];
Bhw=[0; -1; 0; ct/mu];
Bhu=[0; 0; 1/ms; -1/mu];
Cz=[-ks/ms 0 -cs/ms cs/ms; 1 0 0 0; 0 1 0 0]
Dhu=[1/ms; 0; 0]
Dhw=[0; 0; 0]
%% Simulation control
dt=0.001;
t = 0:dt:3;
Am=0.5;
f=2;
T=1/f;
for ki=1:1:length(t);
tdata=t(ki);
if tdata>T;
ni(ki)=0;
else

6. ni(ki)=(Am)*(sin((2*pi*f)*tdata));
end
end
in=transpose([t; ni]);
[t,x]=sim('HBAS_LQR',t);
%% plotting open loop respon
figure(1)
plot(t,x3ddot(:,1),'r',t,x3ddot(:,2),'g',t,x3ddot(:,3),'b','linewidth',3)
legend ('Body Acceleration','Body travel','Wheel deflection');
% Obtain magnificion
axes ('position',[0.25 0.25 0.5 0.5]);
box on
h =x3ddot(:,2);
g = x3ddot(:,3);
my_index = 0 & 3;
plot(t,h,'g', t,g,'b','linewidth',3);
axis tight
grid on
E. Open Loop system Plot Result .
Figure 1.5 Open loop response .
As shown above, the open loop system has large body acceleration while the suspension has
oscillation amplitude range about -0.05 m to 0.05 . From this result , we need to add controller
which provides force to actuator to maintain the balance of passenger comfort due to body
acceleration, body travel and suspension deflection.

7. F. LQR Fullstate Feedback Controller .
F.1 Block Diagram associated
Figure 1.6 Associated block diagram for close loop system.
F.2 Subsystem blocks diagram
Figure 1.7 Subsystem.
F.3 Matlab source code.
We will vary the value of state cost by changing the weight of Q, also we will vary the physical
parameter of the system to see how the system behave when all parameters had changed. The
varying parameters listed in table 1.1.

8. Table 1.1 Vary weight of Q
Weight of Q
Q11 Q21 Q31
1 5000 1000
800 60000 18000
4000 100000 80000
12000 800000 120000
For varying the phsysical parameter we will improve each value by 10% change .
%% Load all Phsyical Parameters
ms=300; % up to 10
mu=40; % up to 10
ks=20000; % up to 10
cs=1000; % up to 10
kt=200000; % up to 10
ct=10; % up to 10
%% State & output matrices
Ah=[0 0 1 -1; 0 0 0 1; -ks/ms 0 -cs/ms cs/ms; ks/mu -kt/mu cs/mu -
(cs+ct)/mu];
Bhw=[0; -1; 0; ct/mu];
Bhu=[0; 0; 1/ms; -1/mu];
Cz=[-ks/ms 0 -cs/ms cs/ms; 1 0 0 0; 0 1 0 0]
Dhu=[1/ms; 0; 0]
Dhw=[0; 0; 0]
%% Obtain LQR Fullstate Feedback Gain
q1l=1; % vary based on tabel 1.1
q2l=5000; % vary based on tabel 1.1
q3l=1000; % vary based on tabel 1.1
Ql=diag([q1l q2l q3l]);
Qlq=Cz'*Ql*Cz
Rl=Dhu'*Ql*Dhu
N=Cz'*Ql*Dhu
[Klq,S,e] = lqr(Ah,Bhu,Qlq,Rl,N);
Klqr=-Klq
%% Simulation control
dt=0.001;
t = 0:dt:3;
Am=0.5;
f=2;
T=1/f;
for ki=1:1:length(t);
tdata=t(ki);
if tdata>T;
ni(ki)=0;
else
ni(ki)=(Am)*(sin((2*pi*f)*tdata));
end
end
in=transpose([t; ni]);
[t,x]=sim('HBAS_LQR',t);
%% plotting closed loop respon
figure(1) %changes the outputmatrix to closedloop
plot(t,x3ddot(:,1),'b',t,x2(:,1),'r','linewidth',3)

9. xlabel('Time
[s]','FontSize',20,'Fontname','SansSerif','Interpreter','latex')
ylabel('Amplitude
$ddot{z}_{s}$','FontSize',20,'Fontname','SansSerif','Interpreter','latex')
set(gca,'FontSize',18, 'Fontname','Times New Roman')
title ('Body Acceleration');
grid on
hold on
figure(2) %changes the outputmatrix to closedloop
plot(t,x3ddot(:,2),'k''linewidth',3)
xlabel('Time
[s]','FontSize',20,'Fontname','SansSerif','Interpreter','latex')
ylabel('Amplitude ${z}_{s}-
{z}_{u}$','FontSize',20,'Fontname','SansSerif','Interpreter','latex')
set(gca,'FontSize',18, 'Fontname','Times New Roman')
title ('Suspension Travel');
grid on
hold on
figure(3) %changes the outputmatrix to closedloop
plot(t,x3ddot(:,3),'k', 'linewidth',3)
xlabel('Time
[s]','FontSize',20,'Fontname','SansSerif','Interpreter','latex')
ylabel('Amplitude ${z}_{u}-
{z}_{r}$','FontSize',20,'Fontname','SansSerif','Interpreter','latex')
set(gca,'FontSize',18, 'Fontname','Times New Roman')
title ('Wheel travel');
grid on
hold on
F.4 Plotting Result
Figure 1.8 Body Acceleration Response due to vary weighting parameter

10. Figure 1.9 Suspension Travel Response due to vary weighting parameter
Figure 2.0 Wheel Travel Response due to vary weighting parameter

11. G. Analysis of changing the physical parameters and weighting matrices Q in LQR
Controller
The design matrices Q and R hold the penalties on the deviations of state
variables from their setpoint and the control actions, respectively. When element of Q
is increased, therefore the cost function increases the penalty associated with any
deviations from the desired setpoint of that state variable, and thus the specific control
gain will be large. When the values of the R matrix are increased, a large penalty is
applied to the aggressiveness of the control action, and the control gains are uniformly
decreased. From figure 1.8 until figure 2.0 we’d vary the diagonal element of Q matrices
into 3 step first we changed the q11 which is represented the body acceleration and then
q21 represented the suspension travel and then the last is changed 31. Every changed
of the Q element will change the gain K value. we’ve increased the value of each
weighting parameter and the response shown in figure 1.8 until 2.0. For body
acceleration the more value was given, the small body acceleration we’ve got so far.
Then if we take a look at suspension travel, the reason show large overshoot in negative
direction and then the system started to oscillate, the same behavior we’ve got for the
wheel travel, since we had added more weighting value, the system started to have more
oscillated.
G.1 Increased physical parameters up to 10%
Changing the physical parameter up until 10% won't change the system response
significantly as shown in figure 2.1 until 2.3.
Figure 2.1 Body Acceleration Response due to vary weighting parameter

12. Figure 2.2 Suspension Travel Responses due to vary weighting parameter
Figure 2.3 Wheel Travel Responses due to vary weighting parameter
From figure 2.1 until 2.3 as we can see all the state responded are still same. These meant that
our LQR controller won't worked anymore against the physical parameter changed. In
practiced we can see the physical car parameters always change, like the total mass of the car
will change because the presented of the passenger's load.