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![Page 1: Dynamical Systems Analysis III: Phase Portraits By Peter Woolf (pwoolf@umich.edu) University of Michigan Michigan Chemical Process Dynamics and Controls.](https://reader031.fdocuments.in/reader031/viewer/2022032201/56649d455503460f94a21cb9/html5/thumbnails/1.jpg)
Dynamical Systems Analysis III:Phase Portraits
By Peter Woolf ([email protected])University of Michigan
Michigan Chemical Process Dynamics and Controls Open Textbook
version 1.0
Creative commons
![Page 2: Dynamical Systems Analysis III: Phase Portraits By Peter Woolf (pwoolf@umich.edu) University of Michigan Michigan Chemical Process Dynamics and Controls.](https://reader031.fdocuments.in/reader031/viewer/2022032201/56649d455503460f94a21cb9/html5/thumbnails/2.jpg)
Questions answered & questions remaining..
1) Create model of physical process and controllers2) Find fixed points3) Linearize your model around these fixed points4) Evaluate the stability around these fixed points
Questions:
• What about all of the other points? What happens when we are not at a fixed point?
• If there are multiple stable fixed points, how large are their ‘basins of attraction’?
• Is there a way to visualize this?• Is there a way to automatically do all of this?
![Page 3: Dynamical Systems Analysis III: Phase Portraits By Peter Woolf (pwoolf@umich.edu) University of Michigan Michigan Chemical Process Dynamics and Controls.](https://reader031.fdocuments.in/reader031/viewer/2022032201/56649d455503460f94a21cb9/html5/thumbnails/3.jpg)
€
dA
dt= 3A − A2 − AB
dB
dt= 2B − AB − 2B2
Nonlinear model
From last class…Linear approximation at A=0, B=0
€
′ A
′ B
⎡
⎣ ⎢
⎤
⎦ ⎥=
3 0
0 2
⎡
⎣ ⎢
⎤
⎦ ⎥A
B
⎡
⎣ ⎢
⎤
⎦ ⎥+
0
0
⎡
⎣ ⎢
⎤
⎦ ⎥
Linear approximation at A=0, B=1
€
′ A
′ B
⎡
⎣ ⎢
⎤
⎦ ⎥=
2 0
−1 −2
⎡
⎣ ⎢
⎤
⎦ ⎥A
B
⎡
⎣ ⎢
⎤
⎦ ⎥+
0
2
⎡
⎣ ⎢
⎤
⎦ ⎥
Linear approximation at A=3, B=0
€
′ A
′ B
⎡
⎣ ⎢
⎤
⎦ ⎥=
−3 −3
0 −1
⎡
⎣ ⎢
⎤
⎦ ⎥A
B
⎡
⎣ ⎢
⎤
⎦ ⎥+
9
0
⎡
⎣ ⎢
⎤
⎦ ⎥
Linear approximation at A=4, B=-1
€
′ A
′ B
⎡
⎣ ⎢
⎤
⎦ ⎥=
−4 −4
1 2
⎡
⎣ ⎢
⎤
⎦ ⎥A
B
⎡
⎣ ⎢
⎤
⎦ ⎥+
12
−2
⎡
⎣ ⎢
⎤
⎦ ⎥
unstable
unstablesaddle
stable
unstablesaddle
![Page 4: Dynamical Systems Analysis III: Phase Portraits By Peter Woolf (pwoolf@umich.edu) University of Michigan Michigan Chemical Process Dynamics and Controls.](https://reader031.fdocuments.in/reader031/viewer/2022032201/56649d455503460f94a21cb9/html5/thumbnails/4.jpg)
Linear approximation at A=0, B=0
€
′ A
′ B
⎡
⎣ ⎢
⎤
⎦ ⎥=
3 0
0 2
⎡
⎣ ⎢
⎤
⎦ ⎥A
B
⎡
⎣ ⎢
⎤
⎦ ⎥+
0
0
⎡
⎣ ⎢
⎤
⎦ ⎥
Linear approximation at A=0, B=1
€
′ A
′ B
⎡
⎣ ⎢
⎤
⎦ ⎥=
2 0
−1 −2
⎡
⎣ ⎢
⎤
⎦ ⎥A
B
⎡
⎣ ⎢
⎤
⎦ ⎥+
0
2
⎡
⎣ ⎢
⎤
⎦ ⎥
Linear approximation at A=3, B=0
€
′ A
′ B
⎡
⎣ ⎢
⎤
⎦ ⎥=
−3 −3
0 −1
⎡
⎣ ⎢
⎤
⎦ ⎥A
B
⎡
⎣ ⎢
⎤
⎦ ⎥+
9
0
⎡
⎣ ⎢
⎤
⎦ ⎥
Linear approximation at A=4, B=-1
€
′ A
′ B
⎡
⎣ ⎢
⎤
⎦ ⎥=
−4 −4
1 2
⎡
⎣ ⎢
⎤
⎦ ⎥A
B
⎡
⎣ ⎢
⎤
⎦ ⎥+
12
−2
⎡
⎣ ⎢
⎤
⎦ ⎥
unstable
unstablesaddle
stable
unstablesaddle
A
B ?
![Page 5: Dynamical Systems Analysis III: Phase Portraits By Peter Woolf (pwoolf@umich.edu) University of Michigan Michigan Chemical Process Dynamics and Controls.](https://reader031.fdocuments.in/reader031/viewer/2022032201/56649d455503460f94a21cb9/html5/thumbnails/5.jpg)
What happens at A=3, B=1
A
B ?
(Not steady state)Check derivatives of nonlinear model
€
dA
dt= 3A − A2 − AB
dB
dt= 2B − AB − 2B2
€
dA
dt= 3(3) − (3)2 − (3)(1) = −3
dB
dt= 2(1) − (3)(1) − 2(1)2 = −3
![Page 6: Dynamical Systems Analysis III: Phase Portraits By Peter Woolf (pwoolf@umich.edu) University of Michigan Michigan Chemical Process Dynamics and Controls.](https://reader031.fdocuments.in/reader031/viewer/2022032201/56649d455503460f94a21cb9/html5/thumbnails/6.jpg)
A
B
Trajectories
A
time
B
time
3
2
1
0
Phase Portrait
![Page 7: Dynamical Systems Analysis III: Phase Portraits By Peter Woolf (pwoolf@umich.edu) University of Michigan Michigan Chemical Process Dynamics and Controls.](https://reader031.fdocuments.in/reader031/viewer/2022032201/56649d455503460f94a21cb9/html5/thumbnails/7.jpg)
Fixed points
Vector field
Trajectory
Stable and unstable orbits
I: converge to fixed point
II: diverge
III: diverge
IV: diverge
![Page 8: Dynamical Systems Analysis III: Phase Portraits By Peter Woolf (pwoolf@umich.edu) University of Michigan Michigan Chemical Process Dynamics and Controls.](https://reader031.fdocuments.in/reader031/viewer/2022032201/56649d455503460f94a21cb9/html5/thumbnails/8.jpg)
Other possibilities
€
dx
dt= 2x − y + 3(x 2 − y 2) + 2xy
dx
dt= x − 3y − 3(x 2 − y 2) + 2xy
Another nonlinear system(Default example in PPLANE)
stable
unstable
Basin of attraction I
Basin of attraction II.1 Basin of
attraction II.2
![Page 9: Dynamical Systems Analysis III: Phase Portraits By Peter Woolf (pwoolf@umich.edu) University of Michigan Michigan Chemical Process Dynamics and Controls.](https://reader031.fdocuments.in/reader031/viewer/2022032201/56649d455503460f94a21cb9/html5/thumbnails/9.jpg)
Other possibilities
€
du
dt= u −
1
3u3 − w − 2
dw
dt= 0.1 1.5 + 2u − w( )
Another nonlinear system(FitzHugh-Nagumo model)
Limit cycle
unstable
Region I
Region II
Note: Locally unstable systems can be globally stable!
![Page 10: Dynamical Systems Analysis III: Phase Portraits By Peter Woolf (pwoolf@umich.edu) University of Michigan Michigan Chemical Process Dynamics and Controls.](https://reader031.fdocuments.in/reader031/viewer/2022032201/56649d455503460f94a21cb9/html5/thumbnails/10.jpg)
Other possibilities
€
dx
dt=10 x − y( )
dy
dt= 28x − y − xz
dz
dt= xy − 2.6667z
Another nonlinear system(Lorenz equations)
Chaotic system:3+ dimensionsNever converges to a point or cycle
Image from java app at http://www.geom.uiuc.edu/java/Lorenz/
Unstable fixed point
![Page 11: Dynamical Systems Analysis III: Phase Portraits By Peter Woolf (pwoolf@umich.edu) University of Michigan Michigan Chemical Process Dynamics and Controls.](https://reader031.fdocuments.in/reader031/viewer/2022032201/56649d455503460f94a21cb9/html5/thumbnails/11.jpg)
Other possibilities
€
dx
dt=10 x − y( )
dy
dt= 28x − y − xz
dz
dt= xy − 2.6667z
Image from java app at http://www.falstad.com/vector3d/
Unstable fixed point
(Same system shown in 3D with white balls following the trajectories)
![Page 12: Dynamical Systems Analysis III: Phase Portraits By Peter Woolf (pwoolf@umich.edu) University of Michigan Michigan Chemical Process Dynamics and Controls.](https://reader031.fdocuments.in/reader031/viewer/2022032201/56649d455503460f94a21cb9/html5/thumbnails/12.jpg)
Concepts from phase portraits extend to higher dimensions
• Fixed points, trajectories, limit cycles, chaos, basins of attraction
• Many real chemical engineering systems are high dimensional and very nonlinear.
€
dCA
dt=
F
VCAf − CA( ) − k1Exp
−ΔE1
RT
⎡ ⎣ ⎢
⎤ ⎦ ⎥CA
2
dCB
dt=
F
V0 − CB( ) + k1Exp
−ΔE1
RT
⎡ ⎣ ⎢
⎤ ⎦ ⎥CA
2 − k2Exp−ΔE2
RT
⎡ ⎣ ⎢
⎤ ⎦ ⎥CBCA
dCC
dt=
F
V0 − CB( ) + k2Exp
−ΔE2
RT
⎡ ⎣ ⎢
⎤ ⎦ ⎥CBCA
dT
dt=
F
VTf − T( ) +
−ΔH1
ρc p
⎡
⎣ ⎢
⎤
⎦ ⎥k1Exp
−ΔE1
RT
⎡ ⎣ ⎢
⎤ ⎦ ⎥CA
2 +−ΔH2
ρc p
⎡
⎣ ⎢
⎤
⎦ ⎥k2Exp
−ΔE2
RT
⎡ ⎣ ⎢
⎤ ⎦ ⎥CBCA −
UA
Vρc p
T − Tj( )
dTj
dt=
F j
V j
Tjin − Tj( ) +UA
V jρc p
T − Tj( )
Example: CSTR with cooling jacket, multiple reactions, and one PID controller
€
dF j
dt= F jss + Kc T − Tset( ) +
1
τ I
xI + τ D
d(T − Tset )
dt
dxI
dt= T − Tset
![Page 13: Dynamical Systems Analysis III: Phase Portraits By Peter Woolf (pwoolf@umich.edu) University of Michigan Michigan Chemical Process Dynamics and Controls.](https://reader031.fdocuments.in/reader031/viewer/2022032201/56649d455503460f94a21cb9/html5/thumbnails/13.jpg)
What does this have to do with controls?
• Control systems modify the dynamics of your process to:– Move fixed points to desirable places– Make unstable points stable– Modify boundaries between basins– Enlarge basins of attraction
![Page 14: Dynamical Systems Analysis III: Phase Portraits By Peter Woolf (pwoolf@umich.edu) University of Michigan Michigan Chemical Process Dynamics and Controls.](https://reader031.fdocuments.in/reader031/viewer/2022032201/56649d455503460f94a21cb9/html5/thumbnails/14.jpg)
–Move fixed points to desirable places–Make unstable points stable–Modify boundaries between basins–Enlarge basins of attraction
![Page 15: Dynamical Systems Analysis III: Phase Portraits By Peter Woolf (pwoolf@umich.edu) University of Michigan Michigan Chemical Process Dynamics and Controls.](https://reader031.fdocuments.in/reader031/viewer/2022032201/56649d455503460f94a21cb9/html5/thumbnails/15.jpg)
How can a control system change the dynamics?
• Adding new relationships between variables
• Adding new variables (I in PID control)
• Adding or countering nonlinearity
• Providing external information
![Page 16: Dynamical Systems Analysis III: Phase Portraits By Peter Woolf (pwoolf@umich.edu) University of Michigan Michigan Chemical Process Dynamics and Controls.](https://reader031.fdocuments.in/reader031/viewer/2022032201/56649d455503460f94a21cb9/html5/thumbnails/16.jpg)
Take Home Messages
• Phase portraits allow you to visualize the behavior of a dynamic system
• Control actions can be interpreted in the context of a phase portrait
• Local stability analysis works locally but can’t always be extrapolated for a nonlinear system.