State-Space: Controllably & Observability · • Controllability • Observability ELEC 3004:...
Transcript of State-Space: Controllably & Observability · • Controllability • Observability ELEC 3004:...
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State-Space:Controllably & Observability
ELEC 3004: Systems: Signals & ControlsDr. Surya Singh (with material from Dr. Paul Pounds)
Lecture 23
[email protected]://robotics.itee.uq.edu.au/~elec3004/
© 2013 School of Information Technology and Electrical Engineering at The University of Queensland
May 23, 2013
Today in Linear Systems…
ELEC 3004: Systems 24 May 2013 - 2
127-FebIntroduction1-MarSystems Overview
26-MarSignals & Signal Models8-MarSystem Models
313-MarLinear Dynamical Systems15-MarSampling & Data Acquisition
420-MarTime Domain Analysis of Continuous Time Systems22-MarSystem Behaviour & Stability
527-MarSignal Representation29-MarHoliday
610-AprFrequency Response12-Aprz-Transform
717-AprNoise & Filtering19-AprAnalog Filters
824-AprDiscrete-Time Signals26-AprDiscrete-Time Systems
91-MayDigital Filters & IIR/FIR Systems3-MayFourier Transform & DTFT
108-MayIntroduction to Digital Control
10-MayStability of Digital Systems
1115-MayPID & Computer Control17-MayApplications in Industry
1222-MayState-Space
24-MayControllability & Observability13
29-MayInformation Theory/Communications & Review31-MaySummary and Course Review
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Today:• State-Space
• Compensator Design
Friday:• Controllability
• Observability
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Goals for the Week
Affairs of state• Introductory brain-teaser:
– If you have a dynamic system model with history (ie. integration) how do you represent the instantaneous state of the plant?
Eg. how would you setup a simulation of a step response, mid-step?
t = 0t
start
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Introduction to state-space• Linear systems can be written as networks of simple dynamic
elements:
2
7 1224
13
7
1
12
2
u y
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Introduction to state-space• We can identify the nodes in the system
– These nodes contain the integrated time-history values of the system response
– We call them “states”
7
1
12
2
u yx1 x2
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• u: Input:
• x: State:
• y: Output
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State-Space Terminology
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State-Space Terminology
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• If the system is linear and time invariant, then A,B,C,D are constant coefficient
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LTI State-Space
• If the system is discrete, then x and u are given by difference equations
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Discrete Time State-Space
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• Series:
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Block Diagram Algebra in State Space
• Parallel:
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Block Diagram Algebra in State Space
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State-space representation• State-space matrices are not necessarily a unique
representation of a system– There are two common forms
• Control canonical form– Each node – each entry in x – represents a state of the system
(each order of s maps to a state)
• Modal form– Diagonals of the state matrix A are the poles (“modes”) of the
transfer function
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Control canonical form
• CCF matrix representations have the following structure:
⋯1 0 0 0 00 1⋮ ⋱ ⋮
1 0 00 0 ⋯ 0 1 0
Pretty diagonal!
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State variable transformation• Important note!
– The states of a control canonical form system are not the same as the modal states
– They represent the same dynamics, and give the same output, but the vector values are different!
• However we can convert between them:– Consider state representations, x and q where
x = Tq
T is a “transformation matrix”
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State variable transformation• Two homologous representations:
and
We can write:
Therefore, and
Similarly, and
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Controllability matrix
• To convert an arbitrary state representation in F, G, H and J to control canonical form A, B, C and D, the “controllability matrix”
⋯must be nonsingular.
Why is it called the “controllability” matrix?
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Controllability matrix
• If you can write it in CCF, then the system equations must be linearly independent.
• Transformation by any nonsingular matrix preserves the controllability of the system.
• Thus, a nonsingular controllability matrix means x can be driven to any value.
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Why is this “Kind of awesome”?• The controllability of a system depends on the particular set of
states you chose
• You can’t tell just from a transfer function whether all the states of x are controllable
• The poles of the system are the Eigenvalues of F, ( ).
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State evolution• Consider the system matrix relation:
The time solution of this system is:
If you didn’t know, the matrix exponential is:12!
13!
⋯
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Stability• We can solve for the natural response to initial conditions :
∴
Clearly, a system will be stable provided eig 0
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Characteristic polynomial
• From this, we can see or, I 0
which is true only when det I 0Aka. the characteristic equation!
• We can reconstruct the CP in s by writing:det I 0
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Great, so how about control?• Given , if we know and , we can design a
controller such thateig 0
• In fact, if we have full measurement and control of the states of , we can position the poles of the system in arbitrary locations!
(Of course, that never happens in reality.)
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Example: PID control• Consider a system parameterised by three states:
– , ,– where and
=1
12
0 1 0 0
is the output state of the system; is the value of the integral;
is the velocity.
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• We can choose to move the eigenvalues of the system as desired:
det1
12
All of these eigenvalues must be positive.
It’s straightforward to see how adding derivative gain can stabilise the system.
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Just scratching the surface
• There is a lot of stuff to state-space control
• One lecture (or even two) can’t possibly cover it all in depth
Go play with Matlab and check it out!
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Discretisation FTW!• We can use the time-domain representation to produce
difference equations!
Notice is not based on a discrete ZOH input, but rather an integrated time-series.We can structure this by using the form:
,
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Discretisation FTW!• Put this in the form of a new variable:
Then:
Let’s rename and
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Discrete state matricesSo,
1
Again, 1 is shorthand for
Note that we can also write as:
where
2! 3!⋯
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Simplifying calculation• We can also use to calculate
– Note that:
Γ1 !
itself can be evaluated with the series:
≅2 3
⋯1
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State-space z-transformWe can apply the z-transform to our system:
which yields the transfer function:
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State-space control design
• Design for discrete state-space systems is just like the continuous case.– Apply linear state-variable feedback:
such that detwhere is the desired control characteristic equation
Predictably, this requires the system controllability matrix ⋯ to be full-rank.
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• Final Exam:Saturday, June 15 at 9:30 AM (sorry!)
• Problem Set 2 is due this Friday!
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Announcements:
!
• AKA: I can see that. Yes, I can control that!
ELEC 3004: Systems 24 May 2013 - 34
Next Time in Linear Systems ….1
27-FebIntroduction1-MarSystems Overview
26-MarSignals & Signal Models8-MarSystem Models
313-MarLinear Dynamical Systems15-MarSampling & Data Acquisition
420-MarTime Domain Analysis of Continuous Time Systems22-MarSystem Behaviour & Stability
527-MarSignal Representation29-MarHoliday
610-AprFrequency Response12-Aprz-Transform
717-AprNoise & Filtering19-AprAnalog Filters
824-AprDiscrete-Time Signals26-AprDiscrete-Time Systems
91-MayDigital Filters & IIR/FIR Systems3-MayFourier Transform & DTFT
108-MayIntroduction to Digital Control
10-MayStability of Digital Systems
1115-MayPID & Computer Control17-MayApplications in Industry
1222-MayState-Space
24-MayControllability & Observability13
29-MayInformation Theory/Communications & Review31-MaySummary and Course Review