As it is well known, standard feedback control is based on generating the control signal
u by processing the error signal, e = r − y , that is, the difference between the reference input and the actual output. Therefore, the input to the plant is
u = K ( r − y) (1)
It is well known that in such a scenario the design problem has one degree of freedom (1-
DOF) which may be described in terms of the stable Youla parameter (Vidyasagar, 1985).
The error signal in the 1-DOF case, see figure 1, is related to the external input r and d by
means of the sensitivity function
1
S & (1 P K )−
= +
e = S r − d
o
, i.e.,
(
) .
d
r
u
y
K
Po
-
Fig. 1. Standard 1-DOF control system.
Disregarding the sign, the reference r and the disturbance d have the same effect on the
error e . Therefore, if r and d vary in a similar manner the controller K can be chosen to minimize e in some sense. Otherwise, if r and d have different nature, the controller has to be chosen to provide a good trade-off between the command tracking and the disturbance
rejection responses. This compromise is inherent to the nature of 1-DOF control schemes. To
allow independent controller adjustments for both r and d , additional controller blocks
have to be introduced into the system as in figure 2.
Two-degree-of-freedom (2-DOF) compensators are characterized by allowing a separate
processing of the reference inputs and the controlled outputs and may be characterized by
means of two stable Youla parameters. The 2-DOF compensators present the advantage of a
complete separation between feedback and reference tracking properties (Youla &
Bongiorno, 1985): the feedback properties of the controlled system are assured by a feedback
2
New Approaches in Automation and Robotics
Fig. 2. Standard 2-DOF control configuration.
controller, i.e., the first degree of freedom; the reference tracking specifications are
addressed by a prefilter controller, i.e., the second degree of freedom, which determines the
open-loop processing of the reference commands. So, in the 2-DOF control configuration
shown in figure 2 the reference r and the measurement y, enter the controller separately and
are independently processed, i.e.,
⎡ r ⎤
u = K
= K r − K y (2)
2
1
⎢ ⎥
⎣ y⎦
As it is pointed out in (Vilanova & Serra, 1997), classical control approaches tend to stress
the use of feedback to modify the systems’ response to commands. A clear example, widely
used in the literature of linear control, is the usage of reference models to specify the desired
properties of the overall controlled system (Astrom & Wittenmark, 1984). What is specified
through a reference model is the desired closed-loop system response. Therefore, as the
system response to a command is an open-loop property and robustness properties are
associated with the feedback (Safonov et al., 1981), no stability margins are guaranteed
when achieving the desired closed-loop response behaviour.
A 2-DOF control configuration may be used in order to achieve a control system with both a
performance specification, e.g., through a reference model, and some guaranteed stability
margins. The approaches found in the literature are mainly based on optimization problems
which basically represent different ways of setting the Youla parameters characterizing the
controller (Vidyasagar, 1985), (Youla & Bongiorno, 1985), (Grimble, 1988), (Limebeer et al.,
1993).
The approach presented in (Limebeer et al., 1993) expands the role of H optimization tools
∞
in 2-DOF system design. The 1-DOF loop-shaping design procedure (McFarlane & Glover,
1992) is extended to a 2-DOF control configuration by means of a parameterization in terms
of two stable Youla parameters (Vidyasagar, 1985), (Youla & Bongiorno, 1985). A feedback
controller is designed to meet robust performance requirements in a manner similar as in
the 1-DOF loop-shaping design procedure and a prefilter controller is then added to the
overall compensated system to force the response of the closed-loop to follow that of a
specified reference model. The approach is carried out by assuming uncertainty in the
normalized coprime factors of the plant (Glover & McFarlane, 1989). Such uncertainty
description allows a formulation of the H robust stabilization problem providing explicit
∞
formulae.
A frequency domain approach to model reference control with robustness considerations
was presented in (Sun et al., 1994). The design approach consists of a nominal design part
plus a modelling error compensation component to mitigate errors due to uncertainty.
A Model Reference Based 2-DOF Robust Observer-Controller Design Methodology
3
However, the approach inherits the restriction to minimum-phase plants from the Model
Reference Adaptive Control theory in which it is based upon.
In this chapter we present a 2-DOF control configuration based on a right coprime
factorization of the plant. The presented approach, similar to that in (Pedret C. et al., 2005),
is not based on setting the two Youla parameters arbitrarily, with internal stability being the
only restriction. Instead,
1. An observer-based feedback control scheme is designed to guarantee robust stability.
This is achieved by means of solving a constrained H optimization using the right
∞
coprime factorization of the plant in an active way.
2. A prefilter controller is added to improve the open-loop processing of the robust closed-
loop. This is done by assuming a reference model capturing the desired input-output
relation and by solving a model matching problem for the prefilter controller to make
the overall system response resemble as much as possible that of the reference model.
The chapter is organized as follows: section 2 introduces the Observer-Controller
configuration used in this work within the framework of stabilizing control laws and the
Youla parameterization for the stabilizing controllers. Section 3 reviews the generalized
control framework and the concept of H optimization based control. Section 4 displays the
∞
proposed 2-DOF control configuration and describes the two steps in which the associated
design is divided. In section 5 the suggested methodology is illustrated by a simple
example. Finally, Section 6 closes the chapter summarizing its content and drawing some
conclusions.
2. Stabilizing control laws and the Observer-Controller configuration
This section is devoted to introduce the reader to the celebrated Youla parameterization,
mentioned throughout the introduction. This result gives all the control laws that attain
closed-loop stability in terms of two stable but otherwise free parameters. In order to do so,
first a basic review of the factorization framework is given and then the Observer-Controller
configuration used in this chapter is presented within the aforementioned framework. The
Observer-Controller configuration constitutes the basis for the control structure presented in
this work.
2.1 The factorization framework
A short introduction to the so-called factorization or fractional approach is provided in this
section. The central idea is to factor a transfer function of a system, not necessarily stable, as
a ratio of two stable transfer functions. The factorization framework will constitute the
foundations for the analysis and design in subsequent sections. The treatment in this section
is fairly standard and follows (Vilanova, 1996), (Vidyasagar, 1985) or (Francis, 1987).
2.1.2 Coprime factorizations over RH∞
A usual way of representing a scalar system is as a rational transfer function of the form
n( s)
P ( s) =
(3)
o
m( s)
4
New Approaches in Automation and Robotics
where n( s) and m( s) are polynomials and (3) is called polynomial fraction representation of P ( s) . Another way of representing P ( s) is as the product of a stable transfer function o
o
and a transfer function with stable inverse, i.e.,
1
P ( s)
N ( s) M −
=
( s) (4)
o
where N ( s), M ( s) ∈ RH , the set of stable and proper transfer functions.
∞
In the Single-Input Single-Output (SISO) case, it is easy to get a fractional representation in
the polynomial form (3). Let δ ( s) be a Hurwitz polynomial such that
deg δ ( s) = deg m( s) and set
n( s)
m( s)
N ( s) =
M ( s) =
(5)
δ ( s)
δ ( s)
The factorizations to be used will be of a special type called Coprime Factorizations. Two
polynomials n( s) and m( s) are said to be coprime if their greatest common divisor is 1 (no common zeros). It follows from Euclid’s algorithm – see for example (Kailath, 1980) – that
n( s) and m( s) are coprime iff there exists polynomials x( s) and y( s) such that the following identity is satisfied:
x( s) m( s) + y( s) n( s) = 1 (6)
Note that if z is a common zero of n( s) and m( s) then x( z) m( z) + y( z) n( z) = 0 and therefore n( s) and m( s) are not coprime. This concept can be readily generalized to transfer functions N ( s), M ( s), X ( s), Y ( s) in RH . Two transfer functions M ( s), N ( s) in RH are
∞
∞
coprime when they do not share zeros in the right half plane. Then it is always possible to
find X ( s), Y ( s) in RH such that X ( s) M ( s) + Y ( s) N ( s) = 1.
∞
When moving to the multivariable case, we also have to distinguish between right and left
coprime factorizations since we lose the commutative property present in the SISO case.
The following definitions tackle directly the multivariable case.
Definition 1. (Bezout Identity) Two stable matrix transfer functions N and M are right
r
r
coprime if and only if there exist stable matrix transfer functions X and Y such that
r
r
⎡ M ⎤
[
r
X
Y
= X M + Y N = I (7)
r
r ] ⎢
⎥
r
r
r
r
⎣ Nr ⎦
Similarly, two stable matrix transfer functions N and M are left coprime if and only if
l
l
there exist stable matrix transfer functions X and Y such that
l
l
A Model Reference Based 2-DOF Robust Observer-Controller Design Methodology
5
⎡ X ⎤
[
l
M
N
= M X + N Y = I (8)
l
l ] ⎢
⎥
l
l
l l
⎣ Yl ⎦
The matrix transfer functions X , Y ( X , Y ) belonging to RH are called right (left) Bezout r
r
l
l
∞
complements.
Now let P ( s) be a proper real rational transfer function. Then,
o
Definition 2. A right (left) coprime factorization, abbreviated RCF (LCF), is a factorization
1
P ( s)
N M −
=
(
1
P ( s)
M −
=
N ), where N , M ( N , M ) are right (left) coprime over
o
r
r
o
l
l
r
r
l
l
RH .
∞
With the above definitions, the following theorem arises to provide right and left coprime
factorizations of a system given in terms of a state-space realization. Let us suppose that
⎡ A B⎤
P ( s) =& ⎢
⎥ (9)
o
C
D
⎣
⎦
is a minimal stabilisable and detectable state-space realization of the system P ( s) .
o
Theorem 1. Define
A + BF
B
− L
⎡
⎤
M
Y
−
⎡
⎤
r
l
⎢
⎥
=
F
I
0
⎢
⎥ &
N
X
⎢
⎥
⎣
⎦
r
l
⎢ C + DF − D I ⎥
⎣
⎦
(10)
A + LC
−( B + LD)
− L
⎡
⎤
⎡ X
Y ⎤
r
r
⎢
⎥
=
F
I
0
⎢
&
−
⎥
N
M
⎢
⎥
⎣
⎦
l
l
⎢ C
− D
I
⎥
⎣
⎦
where F and L are such that A + BF and A + LC are stable. Then, 1
P ( s)
N ( s) M −
=
( s)
o
r
r
(
1
P ( s)
M −
=
( s) N ( s) ) is a RCF (LCF).
o
l
l
Proof. The theorem is demonstrated by substituting (1.10) into equation (1.7).
Standard software packages can be used to compute appropriate F and L matrices
numerically for achieving that the eigenvalues of A + BF are those in the vector
T
p = ⎡ p L p ⎤ (11)
F
⎣ F
F ⎦
1
n
Similarly, the eigenvalues of A + LC can be allocated in accordance to the vector
T
p = ⎡ p L p ⎤ (12)
L
⎣ L
L ⎦
1
n
6
New Approaches in Automation and Robotics
By performing this pole placement, we are implicitly making active use of the degrees of
freedom available for building coprime factorizations. Our final design of section 4 will
make use of this available freedom for trying to meet all the controller specifications.
2.2 The Youla parameterization and the Observer-Controller configuration
A control law is said to be stabilizing if it provides internal stability to the overall closed-
loop system, which means that we have Bounded-Input-Bounded-Output (BIBO) stability
between every input-output pair of the resulting closed-loop arrangement. For instance, if
we consider the general control law u = K r − K y in figure 3a internal stability amounts to 2
1
being stable all the entries in the mapping ( r, d , d ) → ( u, y .
i
o
)
Let us reconsider the standard 1-DOF control law of figure 1 in which u = K ( r − y) . For this particular case, the following theorem gives a parameterization of all the stabilizing
control laws.
Theorem 2. (1-DOF Youla parameterization) For a given plant
1
P
N M −
= r r , let
C ( P) denote the set of stabilizing 1-DOF controllers K , that is,
stab
1
C
( P) =& { K : the control law u = K ( r − y) is stabiliz }
1
1
ing . (13)
stab
The set C ( P) can be parameterized by
stab
⎧ X + M Q
⎫
C
( P)
r
r y
= ⎨
: Q ∈ RH ⎬ (14)
stab
y
−
⎩ rY NrQ
∞ ⎭
y
As it was pointed out in the introduction of this chapter, the standard feedback control
configuration of figure 1 lacks the possibility of offering independent processing of
disturbance rejection and reference tracking. So, the controller has to be designed for
providing closed-loop stability and a good trade-off between the conflictive performance
objectives. For achieving this independence o