D R . T A R E K A . T U T U N J I
A D V A N C E D C O N T R O L S Y S T E M S
M E C H A T R O N I C S E N G I N E E R I N G D E P A R T M E N T
P H I L A D E L P H I A U N I V E R S I T Y
J O R D A N
P R E S E N T E D A T
H O C H S C H U L E B O C H U M
G E R M A N Y
M A Y 1 9 - 2 1 , 2 0 1 5
State Space Control
State Space Description
Transfer functions concentrates on the input-output relationship only. But, it hides the details of the inner workings.
To get a better insight into the system’s behavior, variables ‘states’ are introduced.
State Space Description
State variables describe the complete dynamic behavior of a system
State variables change as a function of time and form a trajectory in dimensional space (referred to state-space)
Block Diagram Example
Properties of States
Memory.
The state summarizes the past.
Dynamics.
The effect of the input is directly connected to the derivative (the change) in the state vector.
Not unique.
The state representation is not unique.
Ordinary Differential Equations
The state of a system is a collection of variables that summarize the past of a system for the purpose of predicting the future
A system can be represented by the differential equation
x –state variable, u – input, y – output
f and h are functions
Linear Systems
where A, B, C and D are constant matrices. Such a system is said to be linear and time-invariant, or LTI for short.
Matrix A is called the dynamics (or system) matrix
Matrix B is called the control (or input-gain) matrix
Matrix C is called the sensor (or output-gain) matrix
Matrix D is called the direct term.
State Space Matrices
The system matrix captures the internal structure of the system and determines many fundamental properties.
The input-gain and output-gain matrices can be modified by adding, modifying or deleting some actuators (to control) or sensors (to measure) from the process.
State-Vector Differential Equation
Example: Spring-Mass with Damping
Example: Circuit
Two Mass Example
System Response using MATLAB
>> k1=1; k2=1; c=0.2; m1=5; m2=2;
>> A=[0 1 0 0;-(k1+k2)/m1 -c/m1 k2/m1 0;0 0 0 1;k2/m2 0 -k2/m2 0];
>> sys=ss(A,B,C,D);
>> B=[0 1/m1 0 0]'; >> C=[1 0 0 0;0 0 1 0]; >> D=0;
>> step(sys);
0
0.5
1
1.5
2
To:
Out(
1)
0 50 100 150 200 250 300 350 400 450 500-1
0
1
2
3T
o:
Out(
2)
Step Response
Time (seconds)
Am
plit
ude
Alternative Problem
Derive the state-space equations using three states 𝑥1 = 𝑦1, 𝑥2 = 𝑦 1, 𝑥3 = 𝑦2 and two outputs y1 and y2
State-Space and Transfer Functions
Direct Canonical Form
State Space and Transfer Functions
Example
Controllability
A System is controllable if a control vector u(t) exists that will transfer the system from any initial state x(t0) to some final state x(t)
Controllability Matrix MATLAB Command M=ctrb(A,B)
If Full Rank Controllable
Observability
A system is observable if the system states x(t) can be exactly determined from the measured output y(t)
Observability Matrix MATLAB Command N=obsv(A,C)
If Full Rank Observable
Controllability Flow
2
1
2
1
2
1
01
)(0
1
10
12
x
xy
tux
x
x
x
1s 1s 1
1 2
u y1x2x
s
x )0(2
s
x )0(1
1 1x2x
1
controllable
uncontrollable
Observability Flow
2
1
2
1
2
1
01
)(1
3
10
02
x
xy
tux
x
x
x
1s 1s 1
1 2
u y1x2x
s
x )0(2
s
x )0(1
1 1x2x
3 observable
unobservable
MIMO Example
>> M=ctrb(A,B) M = 0 0 1 3 3 9 1 3 3 9 7 21 0 1 1 6 11 42
>> r=rank(M) r = 3
Practical Example: Orbiting Satellite
Reference: Mauricio de Oliveira
State-Feedback Control
AND
Regulator Design via Pole Placement
MATLAB Command K=place(A,B,P)
Example
Motor State-Space Model
Motor Control Example
Motor Control Example
Motor Control Example
In general
If desired poles at -2, -3, -4
MATLAB k=place(A,B,[-2; -3 ;-4])
k = [6.0 4.5 2.0]
Full-State Observer
Example
Closed-Loop Control with Observer
AND
Reduced State-Observer
A full-order state observer estimates all state variables
In practice, some states are already measured. Then, we use a reduced-state observer.
Consider the case with three states: x1, x2, and x3
Assume x1 is measured. Then, need to estimate x2 and x3 only
Example
Desired Char. Eq. for the Controller
Desired Char. Eq. Observer
Controller-Observer
Controller-Observer
References
• Advanced Control Engineering (Chapter 8: State Space Methods for Control System Design) by Roland Burns 2001
• Modern Control Engineering (Chapters 9 and 10 Control System Analysis and Design in State Space) by Ogata 5th edition 2010
• Modern Control Engineering (Chapter 10: State Space Design Methods) by Paraskevopoulos 2002
• Feedback Systems: An Introduction to Scientists and Engineers (Chapter 8: System Models) by Astrom and Muray 2009