Automatic Control of State Space Systems - Pantelis Sopasakis
State Space Control - Philadelphia University. State Space...References •Advanced Control...
Transcript of State Space Control - Philadelphia University. State Space...References •Advanced Control...
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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
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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.
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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)
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Block Diagram Example
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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.
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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
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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.
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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.
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State-Vector Differential Equation
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Example: Spring-Mass with Damping
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Example: Circuit
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Two Mass Example
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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
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Alternative Problem
Derive the state-space equations using three states 𝑥1 = 𝑦1, 𝑥2 = 𝑦 1, 𝑥3 = 𝑦2 and two outputs y1 and y2
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State-Space and Transfer Functions
Direct Canonical Form
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State Space and Transfer Functions
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Example
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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
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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
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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
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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
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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
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Practical Example: Orbiting Satellite
Reference: Mauricio de Oliveira
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State-Feedback Control
AND
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Regulator Design via Pole Placement
MATLAB Command K=place(A,B,P)
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Example
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Motor State-Space Model
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Motor Control Example
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Motor Control Example
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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]
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Full-State Observer
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Example
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Closed-Loop Control with Observer
AND
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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
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Example
Desired Char. Eq. for the Controller
Desired Char. Eq. Observer
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Controller-Observer
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Controller-Observer
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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