Title: Section 3: Elements of Robust Control: Homeostasis and Integral Control
1Section 3 Elements of Robust Control
Homeostasis and Integral Control
2What is Robust Control?
- Control against worst-case uncertainty.
D
P
K
3Homeostasis is Fundamental to Life
- Homeostasis is dynamic self-regulation.
- Examples temperature, energy, key metabolites,
blood pressure, immune response, hormone balance,
neural functioning, etc. - Sensory adaptation is a type of homeostasis.
4Types of Feedback Control
P
u
y
C
- Proportional
- Integral Control
- Derivative Control
- PID
5Comparing the Controllers
(P)
(I)
(D)
6Bacterial chemotaxis signal transduction system.
7Bacterial chemotaxis signal transduction pathway
Attractant
Receptor Complex (MCP CheW CheA)
(CH3)
(-CH3)
CheB (demethylase)
CheR (methylase)
CheY-P
Only demethylates active receptor complexes.
Tumbling
8Perfect Adaptation in Bacterial Chemotaxis
Signaling
Segall, J. E., Block, S. M. Berg, H. E.
Temporal comparisons in bacterial
chemotaxis. Proc. Natl. Acad. Sci. USA 83,
8987-8991 (1986).
Adaptation precision
Yss
Y0
Asp
9Robust perfect adaptation.
Alon, U., Surette, M. G., Barkai, N. Leibler,
S. Robustness in bacterial chemotaxis. Nature
397, 168-171 (1998).
CheR
10Robust and non-robust perfect adaptation
- Two approaches to constructing a model that
exhibits perfect adaptation - (1) Fine-tune the parameters.
- (2) Design a specific structure that
inherently creates this property. - Bifurcation analysis can distinguish between
these two approaches.
11Non-robust perfect adaptation
Solid no attractant Dashed 1 mM
attractant Dashed-dot 1 mM attractant
0
1 mM
1 mM
Perfect Adaptation
12Robust perfect adaptationBarkai-Leibler Model
Solid no attractant Dashed 1 mM
attractant Dashed-dot 1 mM attractant
Perfect Adaptation
13Primer on Integral Feedback Control
- Common strategy for tracking a specific
steady-state output value. - The time integral of the system error is fed back
into the system. - Ensures that the steady-state error approaches
zero despite changes in the input or in the
system parameters. - Ubiquitous in complex engineered systems.
14Block diagram for integral control
15Simple derivation of integral control from
Barkai-Leibler Model
16Chemotaxis and integral control
A
Error
17Homeostasis and Integral Control Blood Calcium
Regulation
- Problem Parturient Hypocalcemia.
(PI controller)
H. El-Samad and M. Khammash JTB 21417-29 (2002).
18Integral Control and Biology
- Integral control may represent an important
strategy for ensuring homeostasis. - In complex man-made systems, integral control
loops are found at every level. - It is likely that integral control is important
not only for cellular homeostasis, but also for
homeostasis of the whole organism and even for
ecosystem balance.
19Necessity of Integral Control
- Integral feedback control is not only sufficient
but also necessary for robust perfect adaptation. - Other feedback strategies for achieving robust
perfect adaptation must be equivalent to integral
control. - If the Barkai-Leibler model is later
contradicted, another mechanism implementing
integral control is likely to be present.
20Internal Model Principle (IMP)
- Internal Model Principle is a generalization of
the necessity of integral control. - Robust tracking of an arbitrary signal requires a
model of that signal in the controller. - Intuitively, the internal model counteracts the
external signal.
21Simple (Linear) Mathematical Statement of IMP
U(s)
K
Y(s)
C(s)
- Consider the input
- contains no unstable poles.
- Then,
22Unstable RHP poles
23IMP in the Real World
- Biological systems are subjected to arbitrary,
changing disturbances. - Internal models of these disturbances must exist
within the biological system. - Homeostasis entails approximate internal models.