Transcript Special Topics in Computational Biology Lecture #9: Control Theory & Systems Biology ¦ Bud Mishra Professor of Computer Science and Mathematics (Courant, NYU) Professor (Watson School,

Special Topics in Computational Biology
Lecture #9:
Control Theory & Systems Biology
¦
Bud Mishra
Professor of Computer Science and Mathematics (Courant, NYU)
Professor (Watson School, CSHL)
3 ¦ 26 ¦ 2002
11/7/2015
©Bud Mishra, 2002
L9-1
A Digression into Classical
Control Theory
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©Bud Mishra, 2002
L9-2
Mass under Constant Force
v,
Velocity
f,
M,
Mass
Force
y,
Displacement
• f = M (dv/dt)
• v = dy/dt
• Solution
v(t) = v(t0) + (1/M) st0t f(t) d t
y(t) = y(t0) + v(t0)[t-t0]
+ (1/M) st0t dq st0q f(t) d t
• State Space of the System:
– velocity, v and position y.
• Also, called phase space:
– position, y and momentum, p = Mv.
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L9-3
Continuous Time System
• u 2 Rp = Input
u
(Input)
• y 2 Rq = Output
• Static or Zero-Memory System
Plant
(System)
y
(Output)
– y(t0) depends only on u(t0)…
– Output depends only on the current input..
• Dynamical System or System with Memory
– y(t0) depends on the trajectory of the input u(t),
t 2 [-1, t0].
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L9-4
State of the System
• x(t) = {u(t): t 2[-1, t]} 2 R1
• Usually, a finite dimensional state space:
x(t) Rn = State Space
• x(t) = F(t, t0, x0, u) ´ State Transition Function
• y(t) = g(t, x, u) ´ Output Equation
• x(t0) = x0
• dx(t)/dt = f(t, x(t), u(t))
´ State Evaluation Function
• y(t) = g(t, x(t), u(t)) ´ Output Equation
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L9-5
Time Invariant Systems
• Time Invariant Systems: Invariant under time-shift
operation:
dx(t)/dt = f(x(t), u(t), y(t) = g(x(t), u(t))
• Otherwise, it is a Time Varying System.
• Linearity:
– The state-evaluation function and output equation
satisfy principle of superposition…
– f and g are linear maps.
– Otherwise, it is a Nonlinear System.
– If f and g are nonlinear algebraic maps, then it is a Nonlinear
Differential Algebraic Equation (DAE) System
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L9-6
S-System Components
X2
X1
Reversible Reaction
X2
X1
X2
X1
X3
Divergence Branch Point:
Degradation processes of
X1 into X2 and X3 are
independent
X3
X1
X3
X1
X3
X2
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Convergence Branch Point:
Degradation processes of
X1 into X2 and X3 are
independent
X2
©Bud Mishra, 2002
Single splitting reaction
generating two products
X2 and X3, in stoichiometric
proportion.
Single synthetic reaction
involving two source
components X1 and X2, in
stoichiometric proportion.
L9-7
S-System Components
X3
X4
X2
X1
The reaction between X1 and X2
requires coenzyme X3 which is
converted to X4
X3
X2
X1
The conversion of X1 into X2 is
modulated by X3
X3
X1
-
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X2
The conversion of X1 into X2 is
modulated by an inhibitor X3
©Bud Mishra, 2002
L9-8
Glycolysis
Glycogen
P_i
Glucose
Glucose-1-P Phosphorylase a
Phosphoglucomutase
Glucokinase
Glucose-6-P
Phosphoglucose isomerase
Fructose-6-P
Phosphofructokinase
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L9-9
S-Systems
• Dependent Variables: Xi(t), i=1,…,n, 0 5 t.
• System is described in terms of the temporal
changes in dependent variables:
– E.g., Instantaneous product formation in response to
changes in the exogenous substrate, inhibitor or enzyme
concentration…
– Kinetic Laws: Relate a reaction rate to concentrations.
– Reaction Rate = Instantaneous temporal rate of change
in concentration of substrate or product.
• Nonlinear & Time-invariant System…
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L9-10
Systems of Differential Equations
•
dXi/dt
= (instantaneous) rate of change in Xi at time t
= Function of substrate concentrations, enzymes, factors
and products:
dXi/dt = f(S1, S2, …, E1, E2, …, F1, F2,…, P1, P2,…)
• E.g. Michaelis-Menten for substrate S & product
P:
1. dS/dt = - Vmax S/(KM + S)
2. dP/dt = Vmax S/(KM + S)
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L9-11
General Form
• dXi/dt = Vi+(X1, X2, …, Xn) – Vi-(X1, X2, …, Xn):
– Where Vi+(¢) term represents production (or
accumulation) rate of a particular metabolite and Vi-(¢)
represent s depletion rate of the same metabolite.
• Generalizing to n dependent variables and m
independent variables, we have:
dXi/dt =
Vi+(X1, X2, …, Xn, U1, U2, …, Um)
– Vi-(X1, X2, …, Xn, U1, U2, …, Um):
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L9-12
S-Systems
• S-systems result in Non-linear Time-Invariant DAE
System.
• Note that: Given a system of equations with f and g
being arbitrary rational functions, we can transform
the system into a set of Differential Binomial
Equation System with Linear Constraints:
dxi/dt = a x1a1L xnan - b x1b1L xnbn
& g1 x1 + L gn xn = 0
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L9-13
Transformation I
• Assume that an equation is given as
• dx/dt = p(x(t), u(t))/q(x(t), u(t))
– A rational function. p & q are polynomials
– p(x(t), u(t)) = a1 m1 + L + ak mk - b1 p1 - L - bl pl
– where m’s and p’s are power-products with arbitrary
power. a’s and b’s are positive-valued.
dx/dt = p(x(t), u(t)) y(t)-1,
dc/dt = q(x(t), u(t)) – y(t),
c = 0.
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L9-14
Transformation II
• dx/dt = a1 m1 + L + ak mk - b1 p1 - L - bl pl
= (a1 m1 – w(t)/k) + L + (ak mk – w(t)/k)
– (b1 p1 - w(t)/l) - L - (bl pl - w(t)/l)
• Equivalent System
x(t) - g1(t) - L - gk(t) + gk+1(t) + L + gk+l(t) = 0
dgi/dt = ai mi – w(t)/k , 1 · i · k
dgj/dt = b1 p1 - w(t)/l , k+1 · j · k+l
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L9-15
Linear Time-Invariant Systems
dx/dt = A x(t) + B u(t)
y(t) = C x(t)
A=n£n
B=n£p & C=q£n
• Goal of studying these systems:
• Open Loop Control:
– Generate appropriate inputs so as to constrain the
behavior of the dynamical system
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L9-16
Plant Behavior
• Behavior can be described in terms of the trajectory
in the state space.
Hx , { hx : T a X}
• Desired behavior of a system is given in terms of a
goal set:
G µ Hx.
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L9-17
Servo Problem
• Given a reference trajectory
g 2 Hx
G = { hx : |hx – g| · e }
• Repeat or track the trajectory as closely as possible.
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L9-18
Set Point Regulation Problem
• Given a set of states
CµX
G = { hx : 9 t’ 8 t > t’ hx(t) 2 C }
• Achieve and maintain the particular set of states
(C) starting from any initial state.
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L9-19
State-Avoidance Problem
• Given a state of states
QµX
G = { hx : 8 t hx(t)  Q }
• Starting from any initial state  Q, avoid Q.
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L9-20
¢ Break ¢
10 minutes
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L9-21
Controllability
• An event h t, x i 2 T £ X is said to be controllable
with respect to a set of target sets
C½X
• if and only if
(9 t > t) (9 v = control)
[ h t, x i !v { t : t ¸ t } £ C ´ [t, 1] £ C]
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L9-22
What controllability means…
• Completely Controllable:
C
h t, x’ i
h t, x i
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– A dynamic system is
completely controllable with
respect to C if and only if
every event in T £ X is
controllable with respect to C.
– A dynamical system is
completely controllable iff it
is possible to transfer any state
x(t0) 2 X to any other state in
finite amount of time t1 – t0.
©Bud Mishra, 2002
L9-23
Controllability of Linear
Systems
• Controllability Matrix:
• For the system dx/dt = A x(t) + B u(t),
C(A,B) = [ B | AB | L | An-1 B]
– an n £ np matrix
• Theorem
The system is completely controllable iff the rank of the
controllability matrix is n. (I.e., the rows of the
controllability matrix are linearly independent.)
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L9-24
Proof Idea
1. dx/dt = A x(t) + B u(t)
has a solution given by
x(t) = eAt x0 + s0t eA(t-t)B u(t} d t.
2. The system is controllable
iff the rows of e-At B are linearly independent
iff W(0,t) =
s0t e-AtB B* e-A*t dt = nonsingular 8 t > 0
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L9-25
Proof idea
• To get to x1, choose the following control:
u(t) = -B*e-A*tW-1(0,t1)[x0 – e-At1 x1]
• Note that the rows of e-AtB are linearly independent , eAtB
satisfies the same condition
eAt B = B + AB t + 1/(2!) A2 B t2 + L
has linearly independent rows for all t ,
[ B | AB | L | An-1 B | AnB| An+1B | L]
has linearly independent rows , The controllability matrix
[B | AB | L | An-1 B]
has linearly independent rows.
[Use Cayley-Hamilton Theorem.]
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L9-26
©Bud Mishra, 2002
Controllability
v,
Velocity
f,
M,
Mass
Displacement
Force
f = M dv/dt
v = dy/dt
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y,
 dydt   v  0 1  y   0 
.    1  f
 dv    f   

 dt   M  0 0  v   M 
0
0
 M1 
B   1 , AB   , [ B | AB]   1
0
M 
M

0 
1
M
• Controllability matrix
[B | AB]
• is of rank 2 and the system
is completely controllable.
©Bud Mishra, 2002
L9-27
Observability
• Goal is to recover the state by observing the output
trajectory.
• Observer:
– A system is said to be completely observable if it is
possible to identify any state
x(t0) 2 X
by observing the output y(t) 8 t0 · t · t1.
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L9-28
Principle of Duality
• Controllability & observability are dual concepts.
More formally:
– The linear time-invariant system represented by
dx/dt = A x(t) + B u(t)
y(t) = C x(t)
is controllable at time t0 iff its dual system (adjoint system)
dz/dt = -AT z(t) + CT v(t)
w(t) = BT z(t)
is observable at time t0.
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L9-29
Stability
• The property that small changes in input or initial conditions
do not result in a large change in the system’s behavior.
• Consider a linear time-invariant system:
dx/dt = A x(t) + B u(t)
y(t) = C x(t)
• Let u(t) = uc = constant input; If there exists a point xe 2 X
such that
dx/dt|x=xe, u=uc = A xe + B uc = 0
• Then we say that xe = an equilibrium point of the system
corresponding to the input uc.
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L9-30
Stability
• Assumptions:
– The system has only one equilibrium point.
– Without loss of generality, we may assume that the
origin of the state space is such an equilibrium point.
– The control uc = 0 a Zero-Control: ) dx/dt = A x(t)
• Defn: This system is stable in the sense of Lyapunov
at the origin if
8 e >0 9 d >0 such that
k x(t0) k · d ) 8 t ¸ t0 k x(t) k · e
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L9-31
Asymptotic Stability
•
The system is asymptotically stable at the origin if
1. It is stable in the sense of Lyapunov
2. 9 r > 0 (r 2 R) k x(t0) k · r )
x(t) + 0 as t " 1.
•
Theorem: The system described by the state
equation
dx/dt = A x(t) + B u(t)
is asymptotically stable iff all of the eigenvalues of
the matrix A have negative real parts.
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L9-32
Proof Idea
• Since dx/dt = A x(t) (with uc = 0), we have
• Thus A = U L UT,
x(t) = eAt x0.
– U = Eigenvectors (Orthonormal)
– L = diag(l0, l1, …, ln) = eigenvalues
eAt = I + At + ½ A2 t2 + L
= UUT + U L UT t + U ( ½ L2 t2) UT + L
=U eL t UT
• eL t ! 0 as t + 1 iff all eigenvalues of A have negative real
parts.
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L9-33
Nonlinear Systems
• Controllability…Becomes a much more difficult problem –
specially when the system is non-integrable.
– Assume that the system has a state space Q of dimension m.
– The local behavior of the system is described by an (m-k)distribution D [given by (m-k) independent vector fields] with local
basis X1(Q), …, Xm-k(Q) spanning D(Q) for all Q.
• The system dynamics may be assumed to be describable by
the following equation:
dQ/dt = i=1m-k Xi(Q) ui
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L9-34
Integral Manifold
• The set of vector fields D is called a distribution.
• Consider a manifold N such that
8 Q D(Q) = TQ(N)
– TQ(N) = Tangent Space of N at Q
• N = Integral Manifold of D
• Structure of N determines the controllability
properties.
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L9-35
Lie Derivative Operation
[X, Y] dt2
-Y dt
-X dt
X dt
Y dt
11/7/2015
• Let (X, Y) be a pair of independent
vector fields in D(Q).
• At the state Q, consider the following
cyclic motion – A motion in the
direction in X, followed by a motion in
the direction of Y, then –X and finally
–Y.
• The resulting motion is in the direction
[X, Y]…
• [X, Y] = Lie-bracket of X and Y.
©Bud Mishra, 2002
L9-36
Lie-Derivative
[ X , Y ]  DY  X  DX  Y


 
 Ym
 q1
Y1
q1
11/7/2015
L

L
 x1  
  
      

Ym 
X m



qm  xm   q1
Y1
qm
X 1
q1
©Bud Mishra, 2002
L

L
 y1 
 
   
X m 


y
qm  m 
X 1
qm
L9-37
Filtration
• D0 = D = Span (X1, … Xm)
• D1 = D0 + [D0, D0],
– where [D0, D0] = Span([X, Y]: X, Y 2 D0]
• Di = Di-1 + [Di-1, Di-1],
– where [Di-1, Di-1] = Span([X, Y]: X, Y 2 Di-1]
• Assuming a reasonable regularity condition on the
filtration: we see that
D0 µ D1 µ L µ Di µ L
• and that rank(Di+1) ¸ rank(Di) and if rank(Dp+1) = rank(Dp)
then
rank(Dp) = rank(Dp+1) = rank(Dp+2) = L
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©Bud Mishra, 2002
L9-38
Degree of Nonholonomy
• The smallest p satisfying
rank(Dp) = rank(Dp+1) = rank(Dp+2) = L
is called degree of nonholonomy of the
distribution D: dnh(D)
• If rank(Dp) = m = Dimension of the state space then
the system is controllable!
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©Bud Mishra, 2002
L9-39