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State-space systems
Representations: From transfer function to state-space.
Consider a linear, shift invariant, system:
! P(z) !
y(k) u(k)
We can express this as a transfer function,
y(z) = P(z) u(z) =
b(z)
a(z)
u(z)
where a(z) and b(z) are polynomials, so,
P(z) =
b(z)
a(z)
=
b0zm + b1zm−1 + · · · + bm
zn + a1zn−1 + · · · + an
.
For causal systems the order of b(z) is less than or equal to the order of a(z). So m " n above.
Assume for now that m < n,
Roy Smith: ECE 147b 8: 2
State-space systems
Approach:
• Represent the plant, P(z) (or P(s)) as an nth order differential equation.
• Represent nth order differential equation as a 1st order matrix differential equation with
dimension n.
• Design methods now involve linear algebra.
• Easy to handle large systems (with Matlab).
• Easy to handle systems with multiple inputs and outputs.
• Easy to simulate systems.
x(k + 1) = Ax(k) + B u(k),
y(k) = C x(k) +Du(k).
A, B, C and D can be matrices. x(k) is a vector (state vector).
Roy Smith: ECE 147b 8: 1
State-space systems
Chain of delays:
Consider the first equation: !(z) =
1
a(z)
u(z)
We want to develop a chain of delay model to get !(k):
First step: Write the expression for n delays:
!(k) = z−1 !(k + 1)
...
...
!(k + n − 1) = z−1!(k + n)
In pictures ...
" z−1 ! ! z−1 ! z−1 "
!(k) !(k + 1) !(k + n − 2) !(k + n − 1) !(k + n)
Second step: Express !(k + n) in terms of !(k), . . . , !(k + n − 1) and u(k).
Roy Smith: ECE 147b 8: 4
State-space systems
Outline:
1. Draw the system as an interconnected “chain of delays”,
2. Relabel the signals in the system,
3. Rewrite the input/output equations in terms of the new signals,
4. Abbreviate the equations to a matrix form (state-space).
Drawing a digital system block diagram in terms of delays is exactly the same as drawing a
continuous system block diagram in terms of integrators.
Split the system,
b(z)
1
a(z)
" ! "
y(z) !(z) u(z)
!(z) =
1
a(z)
u(z) and y(z) = b(z) !(z).
Roy Smith: ECE 147b 8: 3
State-space systems
Block diagram of: !(z) =
1
a(z)
u(z)
!(k + n) = u(k) − (a1!(k + n − 1) + · · · + an!(k)) .
z−1 z−1
a1
an−1
an
! ! ! ! ! #$
%& ' +
" "
1. $
%& +
2. −
'
" "
3. $
%& +
4.
'
"
5.
!(k) !(k + 1) !(k + n − 1) !(k + n) u(k)
Roy Smith: ECE 147b 8: 6
State-space systems
Second step: Express !(k + n) in terms of !(k), . . . , !(k + n − 1) and u(k).
To do this, write this as: !(z) a(z) = u(z),
Expanding a(z) gives,
!zn + a1zn−1 + · · · + an" !(z) = u(z)
and expressing this in terms of the highest power of z gives:
zn!(z) = u(z) − !a1zn−1 + · · · + an"!(z).
Now write this is the time domain,
!(k + n) = u(k) − (a1!(k + n − 1) + · · · + an!(k)) .
This is the form we need to include in our block diagram.
Roy Smith: ECE 147b 8: 5
“Chain of delays” block diagram
bm
bm−1
b0
z−1 z−1
a1
an−1
an
( ! ! ! ! ! ( )*
+, + )*
+, + -
" "
)*
+, +
6. −
-
" "
)*
+, +
7.
-
"
8.
! !
)*
+, +
$
-
!
$
u(k)
!(k + n − 1) !(k + n)
!(k + 1)
!(k + m)
y(k) !(k)
This could actually be constructed from shift registers, summers and multipliers.
Roy Smith: ECE 147b 8: 8
State-space systems
Numerator term: y(z) = b(z) !(z)
Expanding the b(z) polynomial gives,
y(z) = b(z) !(z),
= !b0zm + b1zm−1 + · · · + bm" !(z)
and in the time domain this is,
y(k) = b0!(k + m) + b1!(k + m − 1) + · · · + bm!(k).
As m < n all of the signals !(k) to !(k + m) are available along to the top of the chain of
delays.
So y(k) is just a linear combination of signals from the previous block diagram.
Roy Smith: ECE 147b 8: 7
State-space representations
Developing the matrix equations
Relabel our intermediate variables,
x1(k) = !(k + n − 1)
x2(k) = !(k + n − 2)
...
...
xn(k) = !(k)
Now work out what happens to each of them at time k + 1:
x1(k + 1) = !(k + n) =u(k) − (a1!(k + n − 1) + · · · + an!(k))
= u(k) − (a1x1(n) + · · · + anxn(k))
x2(k + 1) = !(k + n − 2 + 1) = x1(k)
... ...
...
xn(k + 1) = !(k + 1) = xn−1(k)
Now look at this as a matrix multiplication.
Roy Smith: ECE 147b 8: 10
State-space representations
What if m = n ?
Remove the highest order terms by polynomial division,
b(z)
a(z)
= d +
ˆb
(z)
a(z)
.
Now d is a constant and ˆb(z) has order m − 1 < n.
We deal with
ˆb
(z)
a(z)
exactly as before and then add the constant, d, to the result.
In pictures . . .
ˆb
(z)
a(z)
d
! ! ! ' !
!
9. )*
+, +
y(k) u(k)
Roy Smith: ECE 147b 8: 9
State-space representations
Matrix equations
x(k + 1) = Ax(k) + B u(k),
where
A =
10. $$$%
−a1 −a2 · · · −an−1 −an
1 0 · · · 0 0
...
. . . . . . ...
0 · · · 1 0
&(
and B =
11. $$$%
1
0...
0
&(
This is the “state-update” equation (or sometimes just the “state”equation).
What about the output y(k)?
y(k) = b0!(k + m) +b1!(k + m − 1) · · · +bm!(k)
= b0xn−m(k) +b1xn−m−1(k) · · · +bmxn(k)
= C x(k)
where C = )0 . . . 0 b0 . . . bm*. If m = n − 1 then there are no leading zeros in C.
Roy Smith: ECE 147b 8: 12
State-space representations
Matrix equations
x1(k + 1) = −a1 x1(k) −a2 x2(k) · · · −an−1 xn−1(k) −an xn(k) + u(k)
x2(k + 1) = x1(k)
x3(k + 1) = x2(k)
...
xn(k + 1) = xn−1(k)
or . . .
12. $$$%
x1(k + 1)
x2(k + 1)
...
xn(k + 1)
&(
=
13. $$$%
−a1 −a2 · · · −an−1 −an
1 0 · · · 0 0
...
. . . . . . ...
0 · · · 1 0
&(
14. $$$%
x1(k)
x2(k)
...
xn(k)
&(
+
15. $$$%
1
0...
0
&(
u(k)
Define the state: x(k) =
16. $$$%
x1(k)
x2(k)
...
xn(k)
&'''(
to get the final equations.
Roy Smith: ECE 147b 8: 11
State-space systems
Other domains:
Discrete time,
time invariant:
x(k + 1) = Ax(k) + B u(k),
y(k) = C x(k) + Du(k)
Continuous time,
time invariant:
dx(t)
dt
= Ax(t) + B u(t),
y(t) = C x(t) + Du(t)
Nonlinear,
time invariant:
dx(t)
dt
= f(x(t), u(t)),
y(t) = g(x(t), u(t))
Nonlinear,
time varying:
dx(t)
dt
= f(t, x(t), u(t)),
y(t) = g(t, x(t), u(t))
Roy Smith: ECE 147b 8: 14
State-space systems
Equal numerator and denominator order case:
ˆb
(z)
a(z)
d
! ! ! ' !
!
17. )*
+, +
y(k) yˆ(k) u(k)
We can calcuate the state-space representation for
ˆb
(z)
a(z)
:
x(k + 1) = Ax(k) + B u(k),
ˆy(k) = C x(k),
and as, y(k) = ˆy(k) + du(k), we have,
x(k + 1) = Ax(k) + B u(k),
y(k) = C x(k) + du(k).
Roy Smith: ECE 147b 8: 13
State-space examples:
Thermal system:
Suppose that T1(t) is the output of interest and e(t) and T0(t) are both inputs.
Now select state variables which will allow us to put this into the generic state-space matrix
equation form.
Try x(t) = +T1(t)
T2(t),.
Then, rearranging gives,
dT1(t)
dt
=
−k12
m1c1
T1(t) +
k12
m1c1
T2(t)
dT2(t)
dt
=
k12
m2c2
T1(t) +
−k12 − k20
m2c2
T2(t) +
k20
m2c2
T0(t) +
1
m2c2
e(t)
So,
A =#
$$%
dT1(t)
dt
dT2(t)
dt
&(
= A +T1(t)
T2(t), + B +T0(t)
e(t) ,, where A =
18. $$$%
−k12
m1c1
k12
m1c1
k12
m2c2
−k12 − k20
m2c2
&'(
Roy Smith: ECE 147b 8: 16
State-space systems
Examples: A thermal control system:
Heater
terminals
e(t)
T (t)
T (t)
T (t)
2
0
1
Ti(t) temperature of volume i
e(t) energy into the volume 1
mi mass of volume i
ci specific heat for volume i
kij thermal conductance for i, j interface
We can derive the state-space representation directly from the thermal energy equations.
For volume 1: m1c1
dT1(t)
dt
= k12(T2(t) − T1(t))
For volume 2: m2c2
dT2(t)
dt
= −k12(T2(t) − T1(t)) − k20(T2(t) − T0(t)) + e(t)
Roy Smith: ECE 147b 8: 15
State-space examples:
DC Motor connected to a rotational load
L R
J
V Vemf in
+
+
−
−
"
i
19.
20. f
The “back emf” is proportional to motor speed: Vemf = K
d"
dt
.
The motor has a series resistance, R, and inductance, L, so Vin = Vemf + L
d i
dt
+ Ri.
The motor torque is proportional to the motor current: # = K" i.
There is a friction torque, #f opposing the motor and proportional to its speed: #f = b
d"
dt
.
The rotational load has inertia J so the torque balance equation is: J
d2 "
dt2 + #f = #.
Roy Smith: ECE 147b 8: 18
State-space examples
Exercises:
1. What is B?
2. What is C?
3. What is D?
4. What would C and D be if we had both T1(t) and T2(t) as outputs?
5. Calculate the transfer functions from e(t) and T0(t) to T1(t).
Roy Smith: ECE 147b 8: 17
State-space examples:
DC Motor connected to a rotational load
We are interested in the model from the input voltage, Vin, to the rotor angle, ".
Exercise:
1. How many states are required for this model?
2. Derive a state-space representation.
3. What are the poles of the system?
Roy Smith: ECE 147b 8: 19

Surface Preparation and Volume Meshing

* New "Repair Feature" mode in the Surface Repair Tool that makes it possible to fix problems in feature curves. This includes automatic detection of errors, a browser to move quickly between them, and automatic repair.
* Multi-Region Imprint mode for automatic interfacing
o Automatic detection of candidate pairs
o Option to browse and imprint in sequence, or to imprint all
o Imprint operation creates interfaces between regions
* New parameter, "layer reduction percentage", which enables automatic reduction of layers to improves mesh quality in tight cavities.
* Generalized Cylinder Mesher (optional with Polyhedral Mesher). This is a new volume mesher that will create extruded meshes in regions that can be characterized as generalized cylinders. Automatic detection of such regions is included.
* JT Open import format available on Windows
* The Select Objects dialog for selecting parts for regions etc has been improved to allow more advanced selection of objects, including better search capabilities
* New Outward Subsurface option in extruder mesher which allows an alternative method of generating an extruded volume mesh
* New interactive boundary selection method for trimmer wake refinement boundaries and wrapper contact prevention boundaries
* The thin mesher now allows a relative or absolute thickness to be specified and is now defaulted to polygonal prisms and not triangular ones
* A new cell set "invert" option has been added for volume meshes. The generation of a cell set also creates a field function for the set which can be useful when splitting a model etc.
* It is now possible to color face normals in the surface repair offset tool

Solvers

* DFBI (6-DOF) solver can now superimpose an additional rigid body motion on the 6-DOF motion. This enables modeling of propellers
* Significant improvement in parallel viewfactor calculations
* General performance increase on Windows platform

Physics

* Multiple Passive Scalars
* Equation of state models
o Real gas models: extended to include IAPWS-IF97 for gaseous steam
o IAPWS-IF97 added for modeling liquid water
o User defined density: extends range of variable density models; compressible liquids can be modeled with this feature
* Anisotropic porous energy: tensor profile input now possible for thermal conductivity and specific heat. Important for some heat exchanger applications
* Heat exchanger enthalpy source: can now be activated at a point specified by the user
* Large Eddy Simulation: Bounded Central Differencing solver provided for additional accuracy and stability
* Aeroacoustics: several broadband noise source models have been implemented to estimate the local acoustic power generated by a turbulent flow field. The models are:
o Curle model
o Goldstein axisymmetric model
o Linearized Euler Equation
o Lilley model
o Proudman model
* Combustion modeling
o New thermal NOx model that is not fuel dependent and does not require a pure fuel stream
o Coal combustion modeling: implemented in the Lagrangian Multiphase framework; includes phase models for vaporization, devolatilization and char oxidation.
* Lagrangian Multiphase
o Two new injector types: cone and surface. Cone injector accepts inner and outer angle specification; surface injector accepts flux specification
o General upgrade of injector values to accept field functions and tables
* Eulerian Multiphase
o Extended to turbulent flows with a multiphase form of the Standard K-Epsilon model. This includes high y+ wall functions
o Additional phase interaction model for turbulent dispersion
* Harmonic Balance method
o Now has blade flutter capability: this displaces the mesh to account for blade deformation
* Finite Volume Stress
o Implicit coupling with small displacement finite volume stress
o Now includes viscous shear traction load on solid from the fluid
o Large displacement theory added that allows solid vertices to automatically move with the predicted solid displacement field. This enables prediction of the apparent stiffening of members in bending when parts are placed in tension, and buckling phenomena when parts are under compression
* Convective velocity option for solid energy transport equation
* VOF Waves now independent of DFBI (6-DOF) model
* Cavitation model now coupled with standard saturation pressure model