Output feedback adaptive command following and disturbance rejection for nonminimum phase uncertain dynamical systems

Abstract

In this paper, we develop an output feedback adaptive control framework for continuous-time nonminimum phase multivariable systems for output stabilization, command following, and disturbance rejection. The approach is based on a nonminimal state space realization that generates an expanded set of states using the filtered inputs and filtered outputs and their derivatives of the original system. Specifically, a direct adaptive controller for the nonminimal state space model is constructed using the expanded states of the nonminimal realization and is shown to be effective for multi-input, multi-output nonminimum phase systems with unstable dynamics. The adaptive controller does not require any model information nor does it require information of the system poles and system zeros or estimation of the system Markov parameters. Several illustrative numerical examples are provided to demonstrate the efficacy of the proposed approach. Copyright © 2010 John Wiley & Sons, Ltd.

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INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING
Int. J. Adapt. Control Signal Process. 2011; 25:352–373
Published online 28 August 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/acs.1202
Output feedback adaptive command following and disturbance
rejection for nonminimum phase uncertain dynamical systems
Tansel Yucelen, Wassim M. Haddad∗,†and Anthony J. Calise
School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150, U.S.A.
SUMMARY
In this paper, we develop an output feedback adaptive control framework for continuous-time nonminimum
phase multivariable systems for output stabilization, command following, and disturbance rejection. The
approach is based on a nonminimal state space realization that generates an expanded set of states using
the filtered inputs and filtered outputs and their derivatives of the original system. Specifically, a direct
adaptive controller for the nonminimal state space model is constructed using the expanded states of the
nonminimal realization and is shown to be effective for multi-input, multi-output nonminimum phase
systems with unstable dynamics. The adaptive controller does not require any model information nor does
it require information of the system poles and system zeros or estimation of the system Markov parameters.
Several illustrative numerical examples are provided to demonstrate the efficacy of the proposed approach.
Copyright ? 2010 John Wiley & Sons, Ltd.
Received 14 January 2010; Revised 19 April 2010; Accepted 29 June 2010
KEY WORDS:output feedback; direct adaptive control; nonminimum phase systems; nonminimal state
space realization; command following; disturbance rejection
1. INTRODUCTION
Mathematical models are critical in capturing and studying physical phenomena that undergo spatial
and temporal evolution arising in most applications of science and engineering. These models
are often based on first principles of physics and are derived using fundamental physical laws.
However, due to system complexity, nonlinearities, uncertainty, and disturbances, first principle
models are often based on simplifying approximations resulting in system modeling errors. For
systems where the system model does not adequately capture the physical system due to idealized
assumptions, model simplification, and model parameter uncertainty, adaptive control methods can
be used to achieve system performance without excessive reliance on system models.
Direct adaptive controllers require less system modeling information than robust controllers and
can address system uncertainties and system failures. These controllers adapt feedback gains in
response to system variations without requiring a parameter estimation algorithm. This property
distinguishes them from indirect adaptive controllers that employ an estimation algorithm to esti-
mate the unknown system parameters and adapt the controller gains. Direct adaptive controllers
can be classified as either full state feedback or output feedback designs.
Full state feedback designs assume knowledge of the state variables, and this assumption
leads to computationally simpler adaptive controller algorithms as compared with output feedback
∗Correspondence to: Wassim M. Haddad, School of Aerospace Engineering, Georgia Institute of Technology, Atlanta,
GA 30332-0150, U.S.A.
†E-mail: wassim.haddad@aerospace.gatech.edu
Copyright ? 2010 John Wiley & Sons, Ltd.

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OUTPUT FEEDBACK ADAPTIVE COMMAND FOR DYNAMICAL SYSTEMS
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algorithms. Output feedback direct adaptive controllers, however, are required for most applica-
tions that involve high-dimensional models, such as active noise suppression, active control of
flexible structures, fluid flow control systems, and combustion control processes. Models for these
applications vary from (reasonably) accurate low-frequency models in the case of structural control
problems, to less accurate low-order models in the case of active control of noise, vibrations, flows,
and combustion processes.
There has been a number of results in the recent decades focused on output feedback direct adap-
tive controllers (see [1–12], and references therein). These results require an observer for unknown
state variables, an observer for output tracking errors, an output predictor, and/or estimation of
Markov parameters that lead to adaptive control algorithms with varying sets of assumptions.
These assumptions include knowledge of the relative degree of the regulated system output and
the dimension of the system, as well as the requirement that the system be minimum phase or
passive.
Virtually all output feedback adaptive controllers are developed under a minimum-phase assump-
tion. The main reason for this is because direct adaptive controllers employ high-gain feedback
that can drive the system to instability. Notable expectations include [13–17]. In particular, to
circumvent the system minimum phase assumption, the author in [13] developed a zero annihilation
periodic law that uses lifting to move all system zeros to the origin. This framework, however,
requires an open-loop operation during alternating data windows. Alternatively, the adaptive control
algorithm in [14,15] utilizes a surrogate cost function involving a quadratic optimization step.
The proposed controller requires sign information of the high-frequency gain and knowledge of
the nonminimum-phase zeroes of the system. A key limitation of this approach, however, is that
the control law lacks a proof of stability. The approaches in [16,17] are limited to single-input,
single-output systems and assume that the sign of the high-frequency gain is known. The under-
lying assumptions in [16] are that the relative degree of the regulated output variable is known,
and that the zero-dynamics of the linear model for the plant captures the internal dynamics of
the nonminimum-phase system to sufficient accuracy. Reference [17] employs a backstepping
approach for a specified class of systems and assumes that the relative degree is known with a
linearly bounded unmatched uncertainty.
In this paper, we develop an output feedback adaptive control framework for continuous-time,
nonminimum phase multivariable systems for output stabilization, command following, and distur-
bance rejection. The approach is based on a nonminimal state space realization that generates an
expanded set of states using the filtered inputs and filtered outputs and their derivatives of the
original system. Specifically, a direct adaptive controller for the nonminimal state space model is
constructed using the expanded states of the nonminimal realization and is shown to be effective
for multi-input, multi-output nonminimum phase systems with unstable dynamics. The adaptive
controller does not require any model information nor does it require information of the system
poles and system zeros. In addition, the proposed adaptive adaptive controller does not require
knowledge about the system Markov parameters as in [14,15].
The notation used in this paper is fairly standard. Specifically, Rn(resp., Cn) denotes the set
of n×1 real (resp., complex) column vectors, Rn×m(resp., Cn×m) denotes the set of n×m real
(resp., complex) matrices, (·)Tdenotes transpose, (·)−1denotes inverse, and ? denotes equal by
definition. Furthermore, we write ?min(M) (resp., ?max(M)) for the minimum (resp., maximum)
eigenvalue of the Hermitian matrix M, ?·?2for the Euclidian norm, ?·?Ffor the Frobenius matrix
norm, spec(·) for the spectrum of a square matrix, tr(·) for the trace operator, vec(·) for the column
stacking operator, and (·)?for the Fr´ echet derivative.
2. NONMINIMAL STATE SPACE REALIZATION FORMULATION
In this section, we present a nonminimal state space realization architecture for continuous-time,
linear multivariable uncertain dynamical systems. The nonminimal state space realization involves
an expanded system state that consists entirely of the system filtered inputs and filtered outputs
Copyright ? 2010 John Wiley & Sons, Ltd.
Int. J. Adapt. Control Signal Process. 2011; 25:352–373
DOI: 10.1002/acs

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T. YUCELEN, W. M. HADDAD AND A. J. CALISE
and their derivatives, which allows us to cast an output feedback control problem as a full-state
feedback problem. Specifically, consider the controllable and observable linear uncertain dynamical
system given by
˙ xp(t)= Apxp(t)+Bpu(t),
xp(0)=xp0,
t?0,
(1)
y(t)=Cpxp(t),
(2)
where xp(t)∈Rn, t?0, is the state vector, u(t)∈Rm, t?0, is the control input, y(t)∈Rl, t?0,
is the system output, and Ap∈Rn×n, Bp∈Rn×m, and Cp∈Rl×nare unknown system matrices.
Here, we assume that although the system order is unknown, it is less than or equal to n. An
input–output equivalent nonminimal observer canonical state space model of (1) and (2) for l>1 is
given by [18]
˙ xo(t)= Aoxo(t)+Bou(t),
xo(0)=xo0,
t?0,
(3)
y(t)=Coxo(t),
(4)
where xo(t)∈Rln, t?0, is the state vector,
Ao=
⎡
⎢
⎣
⎢
⎢
⎢
⎢
0
Il
···
...
0
...
...
...
0
···
0
Il
−a0Il
−a1Il
⎡
⎢
⎢
···−an−1Il
⎤
⎥
⎥
⎤
⎥
⎦
⎥
⎥
⎥
⎥
∈Rln×ln,
(5)
Bo=
⎢
⎢
⎢
⎣
⎢
CpBp
CpApBp
...
CpAn−1
p
Bp
⎥
⎥
⎥
⎦
⎥
∈Rln×m,
(6)
and
Co=[Il 0 ··· 0]∈Rl×ln.
(7)
Note that ai, i =0,1,...,n−1, in (5) are the coefficients of the characteristic polynomial of the
matrix Apin (1).
Next, let
¯B0=Co(a1Iln+a2Ao+···+an−2An−3
¯B1=Co(a2Iln+a3Ao+···+an−1An−3
o
+an−1An−2
o
+An−1
)Bo,
o
)Bo,
(8)
o
+An−2
o
(9)
...
¯Bn−1=CoBo.
(10)
Now, an alternative input–output equivalent nonminimal controllable state space realization of (1)
and (2) is given by
˙ xf(t)= Afxf(t)+Bfu(t),
xf(0)=xf0,
t?0,
(11)
y(t)=Cfxf(t),
(12)
where xf(t)∈R(m+l)n, t?0, is the known filtered expanded state vector given by
xf(t)=[yT
f(t),...,yT(n−1)
f
(t), uT
f(t),...,uT(n−1)
f
(t)]T,
(13)
Copyright ? 2010 John Wiley & Sons, Ltd.
Int. J. Adapt. Control Signal Process. 2011; 25:352–373
DOI: 10.1002/acs

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OUTPUT FEEDBACK ADAPTIVE COMMAND FOR DYNAMICAL SYSTEMS
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where z(n)(t)?dnz(t)/dtnand where xf(t) is obtained by filtering u(t) and y(t) through the filter
1/?(s), where
?n
is a monic Hurwitz polynomial of degree n with ?>0,
⎡
⎢
⎢
⎢
⎢
⎣
⎡
⎢
⎣
and
?(s)=(s+?)n=
n ?
k=0
k
?
sn−k?k=sn+nsn−1?+···+?n
(14)
Af=
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
0
Il
0
··· ···
0
...
0
...
...
0
···
···
···
0
Il
0
¯B0
0
···
···
Im
···
···
0
−a0Il
0
···−an−1Il
···
¯Bn−1
0
...
...
...
...
0
···
···
0
Im
······
0
−?nIm
0
⎥
⎥
···−n?Im
⎤
⎥
⎥
⎥
⎥
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
∈R(m+l)n×(m+l)n,
(15)
Bf=
⎢
⎢
⎢
⎢
0
...
Im
⎤
⎥
⎦
⎥
⎥
∈R(m+l)n×m
(16)
Cf=[−a0Il+?nIl ··· ··· −an−1Il+n?Il
Theorem 2.1
System (1) and (2) is input–output equivalent to system (11) and (12).
¯B0 ··· ···
¯Bn−1]∈Rl×(m+l)n.
(17)
Proof
Using the input–output equivalence of (3) and (4) with (1) and (2), it follows that
a0y(t)=a0Coxo(t),
a1˙ y(t)=a1[CoAoxo(t)+CoBou(t)],
t?0,
(18)
(19)
...
an−1y(n−1)(t)=an−1[CoAn−1
y(n)(t)=CoAn
Now, adding the n+1 equations in (18)–(21) we obtain
y(n)(t)=−[a0Il a1Il ... an−1Il]Y(t)+[¯B0
+Co[An
where¯B0,¯B1,...,¯Bn−1are given in (8)–(10) and Y(t), t?0, and U(t), t?0, are defined as
Y(t)?[yT(t), ˙ yT(t),...,yT(n−1)(t)]T,
U(t)?[uT(t), ˙ uT(t),...,uT(n−1)(t)]T.
o
xo(t)+CoAn−2
o
Bou(t)+···+CoBou(n−2)(t)],
Bou(t)+···+CoBou(n−1)(t).
(20)
oxo(t)+CoAn−1
o
(21)
¯B1 ...
¯Bn−1]U(t)
o+an−1An−1
o
+···+a1Ao+a0Iln]xo(t),
(22)
(23)
(24)
Copyright ? 2010 John Wiley & Sons, Ltd.
Int. J. Adapt. Control Signal Process. 2011; 25:352–373
DOI: 10.1002/acs

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T. YUCELEN, W. M. HADDAD AND A. J. CALISE
Next, using the Cayley–Hamilton theorem [19], which states that every square matrix is a root
of its characteristic polynomial, and noting that ai, i =0,1,...,n−1, are the coefficients of the
characteristic polynomial of the matrix Aoin (3), it follows that
An
o+an−1An−1
o
+···+a1Ao+a0Iln=0.
(25)
Hence, (22) reduces to
y(n)(t)=−[a0Il a1Il ... an−1Il]Y(t)+[¯B0
Now, define the expanded state vector
¯B1 ...
¯Bn−1]U(t).
(26)
xnm(t)?[YT(t),UT(t)]T
=[yT(t), ˙ yT(t),...,yT(n−1)(t),uT(t), ˙ uT(t),...,uT(n−1)(t)]T
so that (26) can be written as
(27)
y(n)(t)=?xnm(t),
(28)
where
?=[−a0Il −a1Il ... −an−1Il
Next, consider the (m+l)n-th order nonminimal state space model given by
˙ xnm(t)= Anmxnm(t)+Bnmu(n)(t),
y(t)=Cnmxnm(t),
where
⎡
⎢
⎢
⎢
⎢
⎢
0
···
¯B0
¯B1 ...
¯Bn−1]∈Rl×(l+m)n.
(29)
xnm(0)=xnm0,
t?0,
(30)
(31)
Anm=
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0
Il
0
······
0
...
...
...
0
···
···
···
0
Il
0
···
···
Im
···
···
0
0
−a0Il
0
···−an−1Il
...
¯B0
0
¯Bn−1
0
...
...
...
...
···
0
Im
···
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
∈R(m+l)n×(m+l)n,
(32)
Bnm=
⎡
⎢
⎣
⎢
⎢
⎢
⎢
0
0
...
Im
⎤
⎥
⎦
⎥
⎥
⎥
⎥
∈R(m+l)n×m
(33)
and
Cnm=[I 0 ··· ··· 0]∈Rl×(m+l)n.
(34)
To eliminate differentiating the actual input and output signals in (30), we filter the input signals
in (30) and the output signals in (31) through the filter 1/?(s), where ?(s) is defined by (14). In
this case, the states xnm(t), t?0, become xf(t), t?0, given by (13).
Copyright ? 2010 John Wiley & Sons, Ltd.
Int. J. Adapt. Control Signal Process. 2011; 25:352–373
DOI: 10.1002/acs