1. Modeling and Simulation: Exploring Dynamic System Behaviour
Chapter 7
Modelling of Continuous Time Dynamic Systems

2. Continuous Time Dynamic Models
Synopsis
Distinctive features
Two examples
The canonical (standard) form
The decomposition problem
An M&S project with a CTDS SUI
Possible hazards on the road to successful solution of continuous M&S projects

3. CTDS Models Noteworthy Features
CM is formulated as a set of differential equations (ode or pde) possibly augmented with a set of algebraic equations
Frequently the SUI has its origins in the “physical world” hence behavior is governed by physical laws (physics, chemistry, heat transfer, etc); i.e., “deep knowledge”
Optimization studies are straightforward because (a) random effects rarely present and (b) behavior is ”smooth”
Time advance is not a separate issue because it is embedded in the equation solving mechanism

4. L q΄΄(t) + R q΄(t) + q(t)/C = E(t)
Some Examples
Electric Circuit
(An example of a deductive model formulated from the application of physical laws)
L q΄΄(t) + R q΄(t) + q(t)/C = E(t)
Initial conditions: q(t0) and q΄(t0) are required

5. 2. Population Dynamics (predator/prey).
(An example of an inductive model that is formulated from arguments based on observation and intuition)
without interaction:
P1΄(t) = – α1 P1(t)
P2΄(t) = α2 P2(t)

6. P1΄(t) = – α1 P1(t) + λ1 P1(t) P2(t)
with interaction:
P1΄(t) = – α1 P1(t) + λ1 P1(t) P2(t)
P2΄(t) = α2 P2(t) – λ2 P1(t) P2(t)
(these are the Lotka-Volterra equations )

7. typical predator/prey behavior

8. The Canonical Form
CTDS Models have the convenient feature of having a standard (general) representation:
x΄(t) = f(x(t), u(t), t)
with: x(t0) = x0
and y(t) = g(x(t))
Here x, u and y are vectors of dimension n, r and m respectively

9. State Variables are often not unique
Consider:
y΄΄(t) + a1 y΄(t) + a0 y(t) = b1 u΄(t) + b0 u(t)
with y(0) = α ; y’(0) = β
A possible “decomposition” (what does this mean?) is:
x1΄(t) = x2(t)
x2΄(t) = - a1 x2(t) - a0 x1(t) + b1 u΄(t) + b0 u(t)
y(t) = x1(t)
But this can have a problem !!

11. Consider an alternate decomposition:
x1΄(t) = x2(t)
x2΄(t) = –a0 x1(t) – a1 x2(t) + u(t)
 y(t) = b0 x1(t) + b1 x2(t)
Still a possible problem; namely the “initial condition
transformation” requires that:
b02 –a1 b0 b1 + a0 b12 ≠ 0

12. Yet another possible decomposition:
x1΄(t) = -a0 x2(t) + b0 u(t)
x2΄(t) = x1(t) – a1 x2(t) + b1 u(t)  
y(t) = x2(t)
Here the “initial condition transformation” does not introduce
any constraint.

13. The Safe Ejection Envelope

14. The Safe Ejection Relationship

15. Trajectory of Pilot/Seat

16. Trajectory of Pilot/Seat
X(t) = Xp(t) – Xa(t)  X’(t) = Xp’(t) – Xa’(t) ; X(0) = 0
Xa’(t) = Va (constant) ; hence Xa(t) = Va t
Xp’(t) = V(t) cos θ(t)
Y(t) = Yp(t) – Ya(t) Y’(t) = Yp’(t) – Ya’(t) ; Y(0) = 0
Ya’(t) = 0 (level flight assumption)
Yp’(t) = V(t) sin θ(t)

17. Configuration y vr Өr

18. Constrained Motion on Rails (Y ≤ Y1)
 V(t) = A ; V’(t) = 0
A = [(Vr cos( θr ))2 + (Va – Vr sin( θr ) )2 ]½
θ(t) = B ; θ’(t) = 0
B = tan-1 [Vr cos( θr ) /(Va - Vr sin( θr ) )]

19. Free Fall Motion (ballistic trajectory)
Xp’ (t) = Vx(t)
Yp’ (t) = Vy(t)
Vx’(t) = – (D/m) cos θ(t)
Vy’ (t) = – (D/m) sin θ(t) – g

20. But: D(t) = μ V2(t) and μ = ĈD ρ(H).
Thus (with some algebraic manipulation):
Vx’(t) = – Ψ(t) Vx
 Vy’(t) = – Ψ(t) Vy – g
where:Ψ(t) = [ĈD ρ(h + Yp) (Vx2 + Vy2)0.5] / m.

21. Summary Xp’(t) = Vx(t) ; Xp(0) = 0 Yp’(t) = Vy(t) ; Yp(0) = 0
On Rails:
Vx’(t) = 0; Vx(0) = Va – Vr sin θr
Vy’(t) = 0; Vy(0) = Va – Vr sin θr
Off Rails: 
Vx’(t) = – Ψ(t) Vx
Vy’(t) = – Ψ(t) Vy – g
with Ψ(t) = [ĈD ρ(h + Yp) (Vx2 + Vy2)0.5] / m.

22. Generating the Envelope Data
Va  Vstart
h  hstart
while (Va < Vlimit)
repeat
h  h + ∆1
solve ode’s up to t = tT where Xp(tT) = Va tT - BT
until (Yp(tT) > HT + Sf)
record [Va, h]
Va  Va + ∆2
endwhile
Plot the collected [Va, h] pairs

23. Bouncing Ball

24. Bouncing Ball Ball Dynamics y1(0) =1 y2(0) =V0 sinθ0
x1(0) = x2(0) =V0 cosθ0
y1(0) = y2(0) =V0 sinθ0

25. Bouncing Ball Collision Characterization x2(Tc+) = α x2(Tc)
y1(Tc+) = 0
y2(Tc+) = - α y2(Tc)