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8/20/2019 Prof.christenson July17 APSS2010
1/11
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Theory of Control: ITheory of Control: I
Richard ChristensonRichard Christenson
University of ConnecticutUniversity of Connecticut
Asia Asia‐‐Pacific Summer SchoolPacific Summer School
on Smart Structures Technology on Smart Structures Technology
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
OverviewOverview
Introduction to structural control
Control theory
Basic feedback control
Optimal control – state feedback control
Observers and LQG controllers
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Passive Control SystemsPassive Control Systems
StructureExcitation Response
Passive Device
M
Passive Damper
M
Base Isolation
M
m
Tuned Mass Damper
3
Types of Structural ControlTypes of Structural Control
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Active Control Systems Active Control Systems
StructureExcitation Response
4
Types of Structural ControlTypes of Structural Control
ActuatorsM
Active Bracing
M
m
Active Mass Damper
Sensor Sensor
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Active Control Systems Active Control Systems
StructureExcitation Response
Actuators
5
Types of Structural ControlTypes of Structural Control
Controller SensorsSensors
feedforward feedback
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Hybrid Control SystemsHybrid Control Systems
StructureExcitation Response
Actuators
6
Types of Structural ControlTypes of Structural Control
Controller SensorsSensors
feedforward feedback
Passive Device
M
Active Base Isolation
Sensor
Actuator
8/20/2019 Prof.christenson July17 APSS2010
2/11
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Semiactive Control SystemsSemiactive Control Systems
StructureExcitation Response
Actuators
7
Types of Structural ControlTypes of Structural Control
Controller SensorsSensors
Passive Device
feedforward feedback
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Functionally Upgraded Passive SystemsFunctionally Upgraded Passive Systems
StructureExcitation Response
Actuators
8
Types of Structural ControlTypes of Structural Control
Controller SensorsSensors
Passive Devicefeedforward feedback
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
StructureExcitation Response
Actuators
9
Types of Structural ControlTypes of Structural Control
Controller SensorsSensors
feedforward feedback
Our focus today Our focus today ……
Controller
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Introduction to Structural ControlIntroduction to Structural Control
Introduction to theory theory behind automatic
control systems – closed‐loop control
Control is used primarily for:
1. Reduce sensitivity to variations
2. Reduce sensitivity to output disturbance
3. Ability to control system bandwidth
4. Stabilization of an unstable system5. Control system transient response
Segway
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Introduction to Structural ControlIntroduction to Structural Control
Controlling the temperature of fluid in a tank
Open‐loop control
In open‐loop control the command signal alone is
selected to achieve the desired response
Controller
G(s)
Plant
H(s)
r(t)
reference
input
u(t)
control
input
y(t)
output
Example: Filling a bathtub with water*Example: Filling a bathtub with water*
*taken from Linear Control Systems, (*taken from Linear Control Systems, (RohrsRohrs, et al.), et al.)Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Introduction to Structural ControlIntroduction to Structural Control
Open‐loop control
Open hot water tap specified amount
Open cold water tap specified amount
If you have done this many time before, you might
know rather well the necessary settings
However, a number of factors might affect the
control of the output
Example: Filling a bathtub with waterExample: Filling a bathtub with water
8/20/2019 Prof.christenson July17 APSS2010
3/11
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Introduction to Structural ControlIntroduction to Structural Control
Closed‐loop control
In closed‐loop control, feedback measurements
are included to achieve the desired response
Controller
G(s)
Plant
H(s)
r(t)
reference
input
u(t)
control
input
y(t)
output
Example: Filling a bathtub with waterExample: Filling a bathtub with water
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Introduction to Structural ControlIntroduction to Structural Control
Closed‐loop control
In closed‐loop control feedback measurements are
included to achieve the desired response
Feel the water at several intervals while the tub is
filling
If water is not at right temperature, adjust hot or
cold water faucets
In this manner, the system output affects the control
of the system
Example: Filling a bathtub with waterExample: Filling a bathtub with water
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Introduction to Structural ControlIntroduction to Structural Control
Closed‐loop control
Use of the state of the output is termed feedback feedback
More measurements (temp. of each faucet, rate
of change of temp.) can achieve better results
Closed‐loop may be more complex than open‐loop,
but can provide better performance
Compromise between stability stability and performance performance
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Introduction to Structural ControlIntroduction to Structural Control
Closed‐loop control
Compromise between stability stability and performance performance
Controlling only hot water (cold predetermined
level) by turning fully on or fully off ;
Our slow response time with the dramatic
response may cause oscillations in temperature
Common causes of instability in automatic
control systems: (1) delay; and (2) high gain
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Introduction to Structural ControlIntroduction to Structural Control
Take the human out of the closed‐loop control
Automatic closed‐loop control
Sensor to measure the required variables
Actuator to adjust control valves
Controller Controller to interpret sensors and send control
signal (which would then be amplified) to actuator
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Introduction to Structural ControlIntroduction to Structural Control
Modeling the system is a crucial step in thedesign of a controller
The quality of the controller is linked to the
quality of the model used in the control design
Since no system can be perfectly modeled,
care must be taken in designing the controller
Parameter inaccuracies
Unmodeled dynamics
Nonlinearities
8/20/2019 Prof.christenson July17 APSS2010
4/11
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
OverviewOverview
Introduction to structural control
Control theory
Basic feedback control
Optimal control – state feedback control
Observers and LQG controllers
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Closed Loop ControlClosed Loop Control
General form of the closed‐loop control system
Controller
G(s)
Plant
H(s)
r(t)
reference
input
u(t)
control
input
y(t)
output
Feedback control can take many forms…
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Closed Loop ControlClosed Loop Control
Let’s begin with an example examining the
effect of control gains in the forward path:
Controller
G(s)
Plant
H(s)
r(t)
reference
input
u(t)
control
input
y(t)
output
When G(s) = K , this is called a proportional
controller with unity gain feedback
e(t)
error +-
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Closed Loop ControlClosed Loop Control
Let’s begin with an example examining the
effect of control gains in the forward path:
K H(s)r(t)
reference
input
u(t)
control
input
y(t)
output
When G(s) = K , this is called a proportional
controller with unity gain feedback
e(t)
error +-
Close loop system
H cl (s)
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Closed Loop ControlClosed Loop Control
The goal is to choose the control gain (K) to
stabilize the system and improve response time
Using the block diagram, we can write
K H(s)
r(t) u(t) y(t)e(t)+-
)()()( sU sH sY = )()()( sKE sH sY =
[ ])()()()( sY sR K sH sY −=[ ]
)()(1
)()( sR
sKH
sKH sY
+=
[ ])(1)(
)(
)()(
sKH
sKH
sR
sY sH cl +
==Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Closed Loop ControlClosed Loop Control
Consider a simple example of a dynamic system In the Laplace domain, the transfer function of the
plant is
Simple model of an electric motor or hydraulic actuator with theSimple model of an electric motor or hydraulic actuator with the
command/voltage as the input and the position/command/voltage as the input and the position/dispdisp. as the output. as the output
Note that this system is marginally stable because one pole is at the origin
)(
1)(
τ +=
sssH
8/20/2019 Prof.christenson July17 APSS2010
5/11
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Closed Loop ControlClosed Loop Control
The close loop system is:
[ ] K ssK
sKH
sKH sH cl ++
=+
=τ 2)(1
)()(
tao = 1;
K = 1;
num = K;
den = [1 tao K];
sys = tf(num,den);
[y,t]=step(sys,t);
plot(t,y)
K=1
K=0.1
K=10
uncontrolled system H(s)
improved response time
overshoot
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Closed Loop ControlClosed Loop Control
Let’s look at the closed loop poles
has poles at
Note:
The system is stable when
The system is underdamped when
2
42
2,1
K p
−±−=
τ τ
K ss
K sH cl
++
=
τ
2)(
τ τ K
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Closed Loop ControlClosed Loop Control
We can use the pole placement approach to
assign K values to achieve the specific behavior
K=1
K=0.1
K=10
K=1
K=0.1
K=10
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Closed Loop ControlClosed Loop Control
This system can equivalently be considered in
state space
)(
1
)(
)()(
τ +==
sssU
sY sH
( ) ( )sU sssY =+ )( 2 τ
)()()( t ut y t y =+ &&& τ
)()()( t ut y t y +−= &&& τ
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Closed Loop ControlClosed Loop Control
This system can equivalently be considered instate space
)()()(
)()()(
t Dut Cz t y
t But Az t z
+=
+=
[ ] [ ] )(0)(
)(01)(
)(1
0
)(
)(
0
10
)(
)(
t ut y
t y t y
t ut y
t y
t y
t y
+
=
+
−
=
&
&&&
&
τ
)()()( t ut y t y +−= &&& τ
−=
τ 0
10 A
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Closed Loop ControlClosed Loop Control
This system can equivalently be considered instate space
Poles of the transfer function are equal to the
eigenvalues of the state space A matrix
−
=τ 0
10 A
( )
+
−=
−−
=−
τ λ
λ
τ λ
λ λ
0
1
0
10
0
0det AI
λτ λ τ λ λ +=−−+= 2)0)(1()(
8/20/2019 Prof.christenson July17 APSS2010
6/11
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Closed Loop ControlClosed Loop Control
Let’s look at the poles and the step response of
a second order differential equation
The Laplace Transform (zero IC) is
Consider the poles of the system
)()()(2)( 22
t r t y t y t y nnn ω ω ζω =++ &&&
22
2
2)(
)(
nn
n
sssR
sY
ω ζω
ω
++=
( ) 22 1 nns ω ζ ζω −±−=Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Closed Loop ControlClosed Loop Control
Assuming the system is underdamped21 ζ ω ζω −±−= nn j s
real imaginary
21 ζ ω −n
nζω
( ) ( )222 1 ζ ω ζω −+ nnMagnitude:
( ) ( )222 1 ζ ω ζω −+ nn
nω =
22222
nnn ω ζ ω ω ζ −+
nω
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Closed Loop ControlClosed Loop Control
Assuming the system is underdamped21 ζ ω ζω −±−= nn j s
real imaginary
21 ζ ω −n
nζω
( ) ζ ω
ζω θ ==
n
nsin
Angle:
nω ζ=sinθ
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Closed Loop ControlClosed Loop Control
Consider the response of the system
To a step input
The step response can be determined as
)(2
)(22
2
sR ss
sY nn
n
ω ζω
ω
++=
ssR
1)( =
( ) ( )Ψ+−−−= −
t et y nt n 2
2 1sin1
1
1 ζ ω ζ
ζω
−=Ψ
ζ
ζ 21arctan
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Closed Loop ControlClosed Loop Control
The peak response occurs at
The peak value of y is then
And the overshoot is
Note: overshoot is only a function of dampingNote: overshoot is only a function of damping
21 ζ ω
π
−= nt
−−
+= 21
max 1)( ζ
ζπ
et y
−−
21 ζ
ζπ
e
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Closed Loop ControlClosed Loop Control
The settling time (defined as time required forresponse to remain within 5% of final value) is
Note: increasingNote: increasing wnwn decreases the rise timedecreases the rise time
05.0=− t ne ζω
3=t nζω
n
st ζω
3=
8/20/2019 Prof.christenson July17 APSS2010
7/11
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Closed Loop ControlClosed Loop Control
Let’s look at the poles and the step response
m a g
= ω
θ = ζ
Optimal poles move away from origin at desired damping
overshoot =
settling time = nst
ζω
3=
−−
21 ζ
ζπ
e
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Theory of Control: IITheory of Control: II
Richard ChristensonRichard Christenson
University of ConnecticutUniversity of Connecticut
Asia Asia‐‐Pacific Summer SchoolPacific Summer School
on Smart Structures Technology on Smart Structures Technology
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
OverviewOverview
Introduction to structural control
Control theory
Basic feedback control
Optimal control – state feedback control
Observers and LQG controllers
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Optimal ControlOptimal Control
Modern control theory uses the approach that
an optimal controller can be obtained for a plant
taking the form
Controller
G(s)
Plant
w(t)
excitationu(t)
control
y(t)
output)()()()(
)()()()(
t Fv t Dut Cz t y
t Ew t But Az t z
++=
++=&
v(t)
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
State Feedback ControlState Feedback Control
Assume that all states are measured and a fullstate feedback control law takes the form
The closed loop dynamics are given by
The poles of this system may be placedarbitrarily if the system is controllable
However, optimal placement is possible with aproperly chosen cost function
)()( t Kz t u −=
( ) )()()( t z At z BK At z cl =−=&
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
State Feedback ControlState Feedback Control
Consider the Linear Quadratic Regulator (LQR) We seek a state feedback controller (K) that
minimizes the cost function
Where Q is positive semidefinate, R is positive
definite, and subject to
dt RuuQz z J T t
T f
)(0
∫ +=
0)0( z z Bu Az z =+=&
8/20/2019 Prof.christenson July17 APSS2010
8/11
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
State Feedback ControlState Feedback Control
The solution to the LQR problem is given by
Where the control gain matrix K is given by
Where P is the Riccati matrix which is goverened
by the Riccati equation
)()( t Kz t u −=
0)(,)()()()( 1 =−++=− − f T T t P P BBR t P R At P t P At P &
P BR K T 1−=
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
State Feedback ControlState Feedback Control
As t f goes to infinity, we see that P becomes
constant and can be determined by solving the
algebraic Riccati equation (ARE)
We can use MATLAB to readily obtain this
solution
The matrices Q and R provide the mechanisms
to design an effective controller
P BBR t P R At P t P A T T 1)()()(0 −−++=
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
State Feedback ControlState Feedback Control
Let’s consider an example of a sdof building with
active bracing
M
Active Bracing
Sensor
wn = 1*2*pi; % rad/sec
xsi = 5/100; % damping 5%
M = 100;
K = M*wn^2;
C = 2*xsi*wn*M;
% State Space System
% dx = Ac*x + Bc*u + Ec*w
% y = Cc*x + Dc*u + Fc*w
Ac = [0 1;-inv(M)*K -inv(M)*C];
Bc = [0;1];
Ec = [0;inv(M)*1];
Cc = [eye(2);-inv(M)*K -inv(M)*C];
Dc = [1];
Fc = [0];
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
State Feedback ControlState Feedback Control
Plot the uncontrolled system’s response due to
Kobe earthquake [w(t)]
sys = ss(Ac,Ec,Cc,Fc);
t = linspace(0,30,1000);
load kobe
w = interp1(k(1,:),k(2,:),t);
y = lsim(sys,w,t);
figure(3);
subplot(311);plot(t,y(:,1),'g');
subplot(312);plot(t,y(:,2),'g');
subplot(313);plot(t,y(:,3),'g');
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
State Feedback ControlState Feedback Control
Plot the uncontrolled system’s response due toKobe earthquake [w(t)]
sys = ss(Ac,Ec,Cc,Fc);
t = linspace(0,30,1000);
load kobe
w = interp1(k(1,:),k(2,:),t);
y = lsim(sys,w,t);
figure(3);
subplot(311);plot(t,y(:,1),'g');
subplot(312);plot(t,y(:,2),'g');
subplot(313);plot(t,y(:,3),'g');
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
State Feedback ControlState Feedback Control
Design an LQR controller to weight displacementand velocity equally
Q = diag([1 1]);
R = 1e-2;
Klqr = lqr(Ac,Bc,Q,R,[]);
sys = ss(Ac-Bc*Klqr,Ec,Cc-Dc*Klqr,Fc);
y = lsim(sys,w,t);
figure(3);
subplot(311);hold on;plot(t,y(:,1),'b');
subplot(312);hold on;plot(t,y(:,2),'b');
subplot(313);hold on;plot(t,y(:,3),'b');
8/20/2019 Prof.christenson July17 APSS2010
9/11
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
State Feedback ControlState Feedback Control
Design an LQR controller to weight displacement
and velocity equally
Q = diag([1 1]);R = 1e-2;
Klqr = lqr(Ac,Bc,Q,R,[]);
sys = ss(Ac-Bc*Klqr,Ec,Cc-Dc*Klqr,Fc);
y = lsim(sys,w,t);
figure(3);
subplot(311);hold on;plot(t,y(:,1),'b');
subplot(312);hold on;plot(t,y(:,2),'b');
subplot(313);hold on;plot(t,y(:,3),'b');
dt RuuQz z J T t
T f
)(0
∫ +=
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
State Feedback ControlState Feedback Control
Let’s look at the poles
R decreases
R decreases
Q – displacement weighting Q – velocity weighting
dt RuuQz z J T t
T f
)(0
∫ +=
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
State ObserversState Observers
In practice, it is not feasible or practical to
measure all of the states of the system
Thus, feedback control design often requires
that one estimate the state variables
Controller
G(s)
Plant
w(t)
excitationu(t)
control
y(t)
output)()()()(
)()()()(
t Fv t Dut Cz t y
t Ew t But Az t z
++=
++=&
v(t)
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
State ObserversState Observers
An observer is a dynamic system with inputs u
(control input) and y (measured responses), and
output that estimates the state vector (called
xhat)
Observer (linear, continuous time) category:
Open loop observer
Full order observer
Kalman (Bucy) filter
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
Open Loop ObserversOpen Loop Observers
The objective is: Observer (linear, continuous time) category:
Linear invariant system:
Auxiliary dynamical system:
Estimation error:
If A is stable, then e(t) approaches zero
Drawbacks:
Unbounded error for unstable state matrix
Fails in the presence of modeling errors and disturbances
0)(ˆ)(lim =−∞→ t x t x t
00 )()()()()()( x t x t Cx t y t But Ax t x ==+=&
)()(ˆ)(ˆ t But x At x +=&
)(ˆ)()( t x t x t e −≡
( ) 0)()()(ˆ)()(ˆ)( t t t Aet But x At But x At e ≥=+−+=&
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
State ObserversState Observers
Full order observer – “Luenberger observer”
Linear invariant system:
Auxiliary dynamical system:
Estimation error:
If (A‐LC) is stable, then e(t) approaches zero
Drawbacks:
Still fails in the presence of modeling errors and disturbances
00 )()()()()()( x t x t Cx t y t But Ax t x ==+=&
( ))(ˆ)()()(ˆ)(ˆ t x C t y Lt But x At x −++=&
)(ˆ)()( t x t x t e −≡
( ) 0)()( t t t eLC At e ≥−=&Observer feedback
8/20/2019 Prof.christenson July17 APSS2010
10/11
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
State ObserversState Observers
Stochastic state observer, “Kalman‐Bucy filter”
Linear invariant system:
Auxiliary dynamical system:
)()()()()()()( t v t Cx t y t w t But Ax t x +=++=&
( ))(ˆ)()()(ˆ)(ˆ t x C t y K t But x At x −++=&
Process noise with
covariance Q(t)
Disturbance with
covariance R(t)
Both noise terms are assumed white, Gaussian and mutually independant
)()()()()()()( t K t R t K t Q At P t AP t P T T −++=&
)()()()( 1 t R t C t P t K T −=
Riccati equation:
Kalman Gain:
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
State ObserversState Observers
The Kalman filter provides the best estimate of
the states based on current available noisy
information
The solution P(t) to the associated differentialRiccati equation (DRE) is also the covariance of
estimation error
If only the steady‐state behavior is of interest,
the time derivative is eliminated in the DRE and
and results in an algebraic Riccati equation (ARE)
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
LQG ControlLQG Control
Kalman filter is often known as linear quadratic
estimation (LQE)
When we combine optimal state feedback with
estimator design, we realize a linear quadratic
Gaussian (LQG) controller
( ))(ˆ)(
ˆ)()(ˆ)(ˆ
t z K t u
z C y Lt But z At z
−=
−++=&
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
LQG ControlLQG Control
The closed loop system is thus
( ) ( )
)(ˆ)(ˆ)(
)()(ˆ)(ˆ
)()(ˆˆ)()(ˆ)(ˆ
t z C t z K t u
t y Bt z At z
t Ly t z BK LC Az C y Lt But z At z
e
ee
=−=+=
+−−=−++=
&
&
Controller
Plant
w(t)
excitationu(t)
control
y(t)
output)()()()(
)()()()(
t Fv t Dut Cz t y
t Ew t But Az t z
++=
++=&
v(t)
( ))(ˆ)(
ˆ)()(ˆ)(ˆ
t z K t u
z C y Lt But z At z
−=
−++=&
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
LQG ControlLQG Control
The closed loop system is thus
[ ]
=
+
+=
)(ˆ
)()(
)(0)(ˆ
)(
)(ˆ
)(
t z
t z DC C t y
t w E
t z
t z
DC B AC B
BC A
t z
t z
e
eeee
e
&
&
)(ˆ)()(
)()(ˆ)()(
t z DC t Cz t y
t Ew t z BC t Az t z
e
e
+=
++=&
)(ˆ)(ˆ)(
)()(ˆ)(ˆ
t z C t z K t u
t y Bt z At z
e
ee
=−=
+=&
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
State Feedback ControlState Feedback Control
Design an LQG controllerEww = 0.1;
Evv = 4e-5;
Lgain = lqe(Ac,Ec,Cc(3,:),Eww,Evv);
Ak = Ac-Bc*Klqr-Lgain*Cc(3,:);
Bk = Lgain;
Ck = -Klqr;
Dk = 0;
Acl = [Ac Bc*Ck;Bk*Cc(3,:)
Ak+Bk*Dc(3,:)*Ck];
Bcl = [Ec;zeros(2,1)];
Ccl = [Cc Dc*Ck];
Dcl = zeros(3,1);
sys = ss(Acl,Bcl,Ccl,Dcl);
y = lsim(sys,w,t);
8/20/2019 Prof.christenson July17 APSS2010
11/11
Advanced Hazards Mitigation LabAdvanced Hazards Mitigation Lab
Department of Civil & Environmental EngineeringDepartment of Civil & Environmental Engineering
State Feedback ControlState Feedback Control
Design an LQG controller
States (actual – blue; estimated – green) Response (lqr – blue; lqg– red)