REFERENCE INPUTS:
The feedback strategies discussed in the previous sections were constructed
without consideration of reference inputs. We referred to the design of state
variable feedback compensators without reference inputs (i.e. r(t)=0) as
regulators. Since command following is also an important aspect of feedback
design, it is important to consider how we can introduce a reference signal into the
state variable feedback compensator. There are, in fact, many different techniques
that can be employed to permit the tracking of a reference input. We will explain
two common methods in this section.
The general form of the state variable feedback compensator is
x̂  A x̂  B ~u  L ~y  M r
u  ~u  N r  K x̂  N r
~y  y  C x̂
~
and u   K x̂
where
. The state variable compensator with
the reference input is illustrated in Figure 1.
Dorf and Bishop, Modern Control Systems
System model
u
r
N
x  A x  B u
+
+
~u
Observer
Control Law
-K
x̂
x̂  A x̂  B u  L~y  M r
x
y
C
~y  y  C x̂
+
-
C
Figure 1. State variable compensator with a reference input.
Notice that when M=0 and N=0, the compensator reduces to the regulator
described earlier.
The compensator’s key design parameters required to implement the command
tracking of the reference input are M and N. When the reference input is a
scalar signal (i.e., a single input), the parameter M is a column vector of length
n, where n is the length of the state vector x, and N is a scalar. Here, we
consider two possibilities for selecting M and N. In the first case, we select M
and N so that the estimation error e(t) is independent of the reference input r(t).
In the second case, we select M and N so that the tracking error y(t)-r(t) is
utilized as an input to the compensator. These two cases will result in
implementations wherein the compensator is in the feedback loop in the first
case and in the forward loop in the second case.
Employing the generalized compensator, the estimation error is found to be
described by the differential equation
e  x  x̂  A x  B u  A x̂  B ~u  L y  M r
or
e  A  L Ce  B N  Mr
Dorf and Bishop, Modern Control Systems
Suppose that we select
M  BN
Then the corresponding estimation error is given by
e   A  LC e
In this case, the estimation error is independent of the reference input r(t). This
is the identical result found in th previous section, where we considered the
observer design assuming no reference inputs. The remaining task is to
determine a suitable value of N, since the value of M follows from the equation
M=BN. For example, we might choose N to obtain a zero steady-state tracking
error to a step input r(t).
With M=BN, we find that the compensator is given by
x̂  A x̂  B ~u  L ~y
u  K x̂  N r
System model
r
N
+
+
u
Control Law
-K
x  A x  B u
y  Cx
y
Observer
x̂  A  LC  x̂  B u  L y
Compensator
Figure 2. State variable compensator with a reference input and M=BN.
Dorf and Bishop, Modern Control Systems
As an alternative approach, suppose that we select N=0 and M=-L. Then, the
compensator is given by
x̂  A x̂  B ~u  L ~y  L r
u  K x̂
which can be written as
x̂  A  BK  LC  x̂  L y  r 
u  K x̂
In this formulation, the observer is driven by the tracking error y-r. The reference
input tracking implementation is illustrated in Figure 3.
Compensator
Observer
r +
x̂  A  BK  LC  x̂  Ly  r 
-
System model
Control Law
-K
u
x  A x  B u
y  Cx
y
Figure 3. State variable compensator with
a reference input and N=0 and M=-L.
Notice that in the first implementation (with M=BN) the compensator is in the
feedback loop, whereas in the second implementation (N=0 and M=-L) the
compensator is in the forward path. These two implementations are
representative of possibilities open to control system designers when
considering reference inputs.
Dorf and Bishop, Modern Control Systems
Example:
Consider the system given in state variable form
x  A x  B u
where
 x1 
0 1
0
A
, B    , and x   

 2  3
1
x 2 
In the Simulink model, we will utilize full-state feedback, which means that the
entire state vector (x1 and x2) is available for feedback. Therefore the output is
y=x.
y  Cx  Du
where
1 0
0 
C
, D 

0 1 
0 
Dorf and Bishop, Modern Control Systems
The full-state feedback control law is
u  K x  r
where r is a reference input.
In this example, we use a sine wave as the reference input. We can choose
different input forms. The feedback gain matrix is chosen as
K  1 1
This choice places the closed-loop poles (that is, the eigenvalues of A-BK) at
s1=-1 and s2=-3. Using Simulink allows us easy access to the feedback gain
matrix so that simulation studies using varying feedback gain sets can be
readily performed. In the simulation presented here, the initial consitions are
chosen to be x1(0)=1 and x2(0)=0.
x1
x2
SCALING THE REFERENCE INPUT:
x  A x  B u
y  Cx
u  rKx
For good tracking performance we want
y( t )  r ( t ) as t  
Consider the performance issue in the frequency domanin. Use the final value
theorem
lim y( t )  lim s Y(s)
t 
s 0
MIT-Open Course Notes
Thus, for good performance, we want
Y(s)
s Y(s)  s R (s) as s  0 
1
R (s) s 0
Example: Consider the system
1 1 1
x    x    u , y  1 0x
1 2 0
u(t)  r  K x(t)
x ( t )  A x ( t )  B r  K x ( t ) 
x ( t )  A  B K  x ( t )  B r
y  C x(t)
The closed loop poles can be
located at s=-5 and s=-6
with K  14 57
MIT-Open Course Notes
The transfer function is
Y(s)
1
 CsI  A  BK  B
R (s)
s  13 56 
 1 0 

  1 s  2
s2
 2
s  11s  30
1
1 
0 
 
Assume that r(t) is a step, then by the final value theorem
Y(s)
2
  1
R (s) s 0
30
As seen from the result, our step
response is quite poor!
MIT-Open Course Notes
One solution is to scale the reference input r(t) so that
u  N r  K x(t )
N
is the extra gain used to scale the closed-loop transfer function.
Now we have
x ( t )  A  B K  x ( t )  B N r
y  C x(t)
So that
Y(s)
1
 C sI  A  BK  B N
R (s)
If we take
N  15 , then
Y(s)  15 s  2
 2
R (s) s  11s  30
So with a step input
y( t )  1 as t  
We can compute
Y(s)
1
 1  C sI  A  BK  B N
R (s)
s  0 for cons tan t r

N   C A  BK  B
1

1
Note that this development assumed that r was constant, but it could also be
used if r is a slowly time-varying command.
So the steady state step error is now zero, but is this OK?
clc;clear
a=[1 1;1 2];b=[1 0]';c=[1 0];d=0;
k=[14 57];
Nbar=-15;
sys1=ss(a-b*k,b,c,d);
sys2=ss(a-b*k,b*Nbar,c,d);
t=[0:.025:4];
[y,t,x]=step(sys1,t);
[y2,t2,x2]=step(sys2,t);
plot(t,y,'--',t2,y2,'Linewidth',2);axis([0 4 -1 1.2]);grid;
legend('u=r-Kx','u=Nbar*r-Kx','Location','SouthEast')
xlabel('time (sec)');ylabel('Y output');title('Step Response')
Step Response
1
0.8
0.6
Y output
0.4
0.2
As seen from the response, there is
a big improvement, but transient
part is a bit weird.
0
-0.2
-0.4
-0.6
u=r-Kx
u=Nbar*r-Kx
-0.8
-1
0
0.5
1
1.5
2
time (sec)
2.5
3
3.5
4
MIT-Open Course Notes
NONLINEAR SYSTEMS
When engineers analyze and design nonlinear dynamical systems in electrical
circuits, mechanical systems, control systems, and other engineering
disciplines, they need to absorb and digest a wide range of nonlinear analysis
tools.
Most of the nonlinear systems can be modeled by a finite number of first-order
ordinary differential equations.
x 1  f1 x1 ,, x n , t, u1 ,, u p 

x n  f n x1 ,, x n , t, u1 ,, u p 
Defining vectors
 x1 
 u1 
 f1 x, t, u 

x     , u     , f t, x, u   


x n 
u n 
f n x, t, u 
(1)
We can write equation (1) as follows
x  f x, t, u 
(2)
Equation (2) is the generalization of state-space respresentation to nonlinear
systems. The vector x is called the state vector of the system, and the function
u is the input. Similarly, the system output is obtained via the so-called read out
equation.
y  hx, t, u 
(3)
Equation (2) and (3) are referred to as the state space realization of the
nonlinear system.
Special Cases:
An important special case of equation (2) is when the input u is identically zero.
In this case, the equation takes the form
x  f x, t,0  f x, t 
This equation is referred to as the unforced state equation.
(4)
The second special case occurs when f(x,t) is not a function of time. In this
case we can write
x  f x 
(5)
in which case the system is said to be autonomous. Autonomous systems are
invariant to shifts in the time origin in the sense that changing the time variable
from t to τ=t-α does not change the right-hand side of the state equation.
m y   f orces
my  f ( t )  f k  f b
k(y)
b
In this example, we consider the more realistic
case of hardening spring in which the force
strengthens as y increases. We can approximate
this model by taking
m
f(t)
y

fk  k y 1 a y
2
2

Marquez JH, Nonlinear Control Systems
With this constant, the differential equation results in the following:
2 3



m y  b y  k y  k a y  f (t)
(6)
Defining state variables x1=y, x2=dy/dt results in the following state space
realization
x 1  x 2
k
k 2 3 b
f (t)
x 2   x1  a x1  x 2 
m
m
m
m
which is the form
x  f x, u 
In particular, if u=0, then
x 1  x 2
k
k 2 3 b
x 2   x1  a x1  x 2
m
m
m
or x  f x 
Marquez JH, Nonlinear Control Systems
Equilibrium Points (Singular points):
An important concept when dealing with the state equation is that of equilibrium
point.
Definition 1.1 A point x=xe in the state space is said to be an equilibrium point of
the autonomous system
x  f x 
if it has the property that whenever the state of the system starts at xe, it remains
at xe for all future time. According to this definition, the equilibrium points of (4)
are the real roots of the equation f(xe)=0. This is clear from equation (4). Indeed,
if
dx
x 
 f (x e )  0
dt
it follows that xe is constant and, by definition, it is an equilibrium point.
Example:
Consider the following first-order system
x  r  x 2
where r is a paremeter. To find the equilibrium pointsof the system, we solve
the equation r+x2=0 and we obtain
(i) If r<0, the system has two equilibrium points, namely x=±r0.5.
(ii) If r=0, both of the equilibrium points in (i) collapse into one and the same,
and the unique equilibrium point is x=0
(iii) Finally, if r>0, then the system has no equilibrium points.
x(0)=0.5
x(0)=-0.4
Example:
Consider the following nonlinear second-order system
x  0.6 x  3 x  x  0
2
x  0.6 x  3 x  x 2
At t=0
x(0)=1,
dx(0)/dt=2
Phase Plane
Initial values
(1,2)
x
x
Convergent solution
Phase Plane
(3,6)
x
x
Divergent solution.
x  0.6 x  3 x  x 2  0
dx1
x1  x , x 2 
dt
x 1  x 2
x  x 2  0.6 x 2  3 x1  x 21
At state of equilibrium
0  x2
dx1/dt=x2=0
0  0.6 x 2  3 x1  x 1
2
=0
x1=0
x1=-3
The system has two
singular points, one
at (0,0) and the
other at (-3,0)
The motion patterns of the system trajectories in the vicinity of the two
singular points have different natures. The trajectories move towards the
point x=0 while moving away from the point x=-3.
(0.1,0)
Stable
(-3.01,0)
Unstable
One may wonder why an equilibrium point of a second-order system is called a
singular point. To anser this, let us examine the slope of the trajectories. The
slope of the phase trajectory passing through a point (x1,x2) is determined by
dx 1
 f1 ( x 1 , x 2 )
dt
dx 2
 f 2 ( x1 , x 2 )
dt
dx 2 f 2 ( x1 , x 2 )

dx1 f1 ( x1 , x 2 )
Slotine and Weiping, Applied Nonlinear Control.
With the functions f1 and f2 assumed to be single
valued, there is usually a definite value for the slope
at any given point in phase plane. This implies that
the phase trajectories will not intersect. At singular
points, however, the value of the slope is 0/0, i.e.,
the slope is indeterminate. Many trajectories may
intersect at such points. The indeterminancy of the
slope accounts for the adjective “singular”.
Singular points are very important features in phase
plane. Examination of the singular points can reveal
a great deal of information about the properties of a
system. In fact, stability of linear systems in uniquely
characterized by the nature of their singular points.
For nonlinear systems, besides singular points, there
may be more complex features, such as limit cycles.
Figure 4. Phase portrait of the nonlinear
system.
Slotine and Weiping, Applied Nonlinear Control.