SIMULATION OF DYNAMICAL SYSTEMS
In practice we wish to study the behavior of complex
systems subject to general input disturbances. Then it
is not feasible to solve the (differential) equations which
describe the system analytically. Instead, it is necessary
to use numerical simulations.
In numerical differential equation solvers the systems are
usually represented in terms of coupled first-order differential equations. Then the system is characterized using
a number of state variables,
x1(t), x2(t), . . . , xn(t)
and the time derivative of each variable is expressed as a
function of the other variable and the input(s):
dxi(t)
= fi (x1(t), x2(t), . . . , xn(t), u(t)) , i = 1, 2, . . . , n
dt
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Example: Second-order system
d2y(t)
dy(t)
+
a
+ a2y(t) = bu(t)
1
dt2
dt
Define the state variables
x1(t) = y(t)
dy(t)
x2(t) =
dt
Then
dx1(t)
= x2(t)
dt
dx2(t)
= −a1x2(t) − a2x1(t) + bu(t)
dt
which has the desired form.
There are many good ordinary differential equation (ode)
solver available, for example in Matlab. The example below shows how a second-order system with a step change
in the input can be simulated.
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Example: Simulation of second-order system
The first program calls Matlab’s ode solver ode23. The
command
[T,X]=ode23(@sys2,[t_ini t_final],x_ini)
returns a vector T of time instants at which the values of
the state variables are given. The matrix X has the same
number of rows as T, and its columns contain the values of
the state variables, so that the kth column of X contains
the state variable xk (t) at time instants T(1), T(2), . . ..
The function subroutine sys2 returns the time derivatives of the state variables in the vector dxdt.
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% Simulation of 2nd order system
%
global a1 a2 b
%
a1=1; a2=1; b=1; % system parameters
t_ini=0; t_final=25; % initial and final times
x_ini=[0 0]; % initial values of state variables
%
[T,X]=ode23(@sys2,[t_ini t_final],x_ini) ;
function dxdt=sys2(t,x)
% Derivative of second-order systems
global a1 a2 b
%
% Introduce step input at t=10
if t<10
u=0;
else
u=1;
end
%
y=x(1); dydt=x(2);
%
dxdt(1)=dydt; dxdt(2)=-a1*dydt-a2*y+b*u ;
%
dxdt=dxdt’;
% end
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Example: Simulation of second-order system controlled by a
PID controller
The same approach can be applied to simulate more complex
system connections. For illustration, consider the second-order
system
d2 y(t)
dy(t)
+ a2 y(t) = bu(t)
+
a
1
dt2
dt
where u(t) is determined by the PID control law

Z t

dy(t) 
u(t) = Kp (r(t) − y(t))+Ki
(r(τ ) − y(τ )) dτ +Kd −
τ =0
dt
(Observe that the derivative action is applied only to y).
The we can define the state variables:
x1 (t) = y(t)
dy(t)
x2 (t) =
dt
Z t
x3 (t) =
(r(τ ) − y(τ )) dτ
τ =0
Then:
dx1 (t)
= x2 (t)
dt
dx2 (t)
= −a1 x2 (t) − a2 x1 (t) + bu(t)
dt
dx3 (t)
= r(t) − y(t)
dt
where
u(t) = Kp (r(t) − x1 (t)) + Ki x3 (t) + Kd (−x2 (t))
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The closed-loop system can be simulated with the following Matlab programs:
% Simulation of 2nd order system with PID control
%
global a1 a2 b Kp Ki Kd
%
a1=0.25; a2=1; b=1; % system parameters
%
Kp=1; Ki=1; Kd=1; % controller parameters
%
t_ini=0; t_final=25; % initial and final times
%
x_ini=[0 0 0]; % initial values of state variables
%
[T,X]=ode23(@sys2PID,[t_ini t_final],x_ini) ;
%
%
% end
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function dxdt=sys2PID(t,x)
% ---------------------------------% Derivative of second-order system
% under PID control
% ---------------------------------%
global a1 a2 b Kp Ki Kd
%
% Introduce step change in setpoint at t=10
%
if t<10
r=0;
else
r=1;
end
%
y=x(1);
dydt=x(2);
yi=x(3);
u=Kp*(r-y)+Ki*yi-Kd*dydt;
%
dxdt(1)=dydt;
dxdt(2)=-a1*dydt-a2*y+b*u ;
dxdt(3)=r-y;
%
dxdt=dxdt’;
% end
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Simulink
As seen from the above simple examples, when the number of
state variables required to describe a system increases, it may become rather tedious to construct programs which compute the
time derivatives of all state variables. Therefore, various tools
which facilitate the analysis and simulation of dynamical systems have been developed. One widely used tool of this kind is
Simulink for use with Matlab, in which the block diagram and
the individual blocks can be constructed using a graphical user
interface.
Below is shown an example of a Simulink program consisting of
a first-order system
G(s) =
K
Ts + 1
(s corresponds to p)
and a time delay controlled by a PID controller, where there is
a step change in setpoint. The block Scope plots the associated
variable.
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The parameters of the various blocks can be defined by opening
the individual blocks.
The block may have internal structure, for example the structure
of the PID block is shown below.
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