Order reduction of n for robust adaptive control design of SISO linear systems

Abstract

In this paper, we propose an order-reduction design methodology to simplify the adaptive controller obtained in Z. Pan and T. Basar (2000) by n integrators. We study the same class of linear systems as Z. Pan and T. Basar (2000), make the same assumptions, and have the same formulation and approach to the problem. The main difference between our design methodology and that of Pan and T. Basar (2000) lies in the step O of the control design step. In this paper, we skip step O and immediately start the integrator backstepping procedure without stabilizing the filtered dynamics of the output. This relieves us from generating the reference trajectory for the filtered dynamics of the output and thus reducing the controller order by n. The trade-off for this order reduction Ls that the worst-case estimate for the expanded state vector has to be chosen as a suboptimal choice, rather than the optimal choice. Exactly the same robustness properties can be established for the reduced-order controllers as those of Z. Pan and T. Basar (2000). There is no definite performance comparison that can be made theoretically between the reduced-order controller and the full-order controller of Z. Pan and T. Basar (2000). Based on a few simulation examples, we observe that the reduced-order controller does not perform better than the full-order controller.

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Order Reduction of n for Robust Adaptive Control Design
of SISO Linear Systems
Qingrong Zhao Zigang Pan
Abstract—In this paper, we propose an order-reduction design
methodology to simplify the adaptive controller obtained in [1] by
n integrators. We study the same class of linear systems as [1],
make the same assumptions, and have the same formulation and
approach to the problem. The main difference between our design
methodology and that of [1] lies in the step 0 of the control design
step. In this paper, we skip step 0 and immediately start the integrator
backstepping procedure without stabilizing the filtered dynamics of
the output. This relieves us from generating the reference trajectory
for the filtered dynamics of the output and thus reducing the controller
order by n. The trade-off for this order reduction is that the worst-case
estimate for the expanded state vector has to be chosen as a suboptimal
choice, rather than the optimal choice. Exactly the same robustness
properties can be established for the reduced-order controllers as those
of [1]. There is no definite performance comparison that can be made
theoretically between the reduced-order controller and the full-order
controller of [1]. Based on a few simulation examples, we observe
that the reduced-order controller does not perform better than the
full-order controller.
Index Terms—adaptive control, nonlinear H∞control, cost-to-
come function, integrator backstepping.
I. INTRODUCTION
Adaptive control has been an important research topic in control
theory. Its early development since 1970s has been dominated by
certainty equivalence principle [2], [3], which decouples the pa-
rameter estimation design from the control design, by making use
of some standard parameter estimators and supplying the estimates
to control law as if they were true parameters. The certainty equiv-
alence based design simplifies the controller structure considerably
and leads to many successful applications [4] for linear systems.
On the other hand, early designs using this approach are shown
to be nonrobust when the system has unmodeled dynamics and
deterministic exogenous disturbance inputs [5]. Furthermore, it is
unsuccessful to generalize this approach to the nonlinear systems
with severe nonlinearity. All of these drawbacks motivate the study
of nonlinear adaptive control design in 1990s and robust adaptive
control design in 1980s and 1990s.
One of the major research focus for nonlinear adaptive control
design is in (partially) feedback linearizable systems which are
geometrically characterized in [6]. The introduction of integrator
backstepping methodology [7] provides a systematic design tool
to obtain adaptive control laws for the class of parametric strict-
(or pure-) feedback nonlinear systems. This method admits great
design flexibility evident in the selection of the value function and
the virtual control laws. See the book [8] for a complete list of
Research was supported in part by the National Science Foundation
under CAREER Award ECS-0296071.
Qingrong Zhao is with Department of Electrical and Computer Engi-
neering and Computer Science, University of Cincinnati, Cincinnati, OH
45221-0030. Tel: 513-556-1036; Email: zhaoqg@ececs.uc.edu.
Zigang Pan is with Department of Electrical and Computer Engineering
and Computer Science, 814 Rhodes Hall, Mail Loc. 0030, University of
Cincinnati, Cincinnati, OH 45221-0030. Tel: 513-556-1808; Fax: 513-556-
7326; Email: zpan@ececs.uc.edu. All correspondences should be addressed
to this author.
references. Adaptive control designs based on this method achieves
better system performance for linear systems if the system has no
disturbance input, in comparison with the certainty equivalence
based adaptive controller design [8]. However, it is shown that
such design may be nonrobust if the system is subject to exogenous
disturbance inputs.
The main objectives of robust adaptive control are to improve
transient performance, accommodate unmodeled dynamics, and
tolerate exogenous disturbance inputs, which are consistent with
the objectives of H∞-optimal control. One key feature of H∞-
optimal control is that all the above objectives can be achieved
by studying the disturbance attenuation property of the closed-
loop system. Therefore, we formulate the robust adaptive control
problem as a nonlinear H∞-optimal control problem under im-
perfect state measurements. The game-theoretic approach to H∞-
optimal control problems offer the most promising tool to address
nonlinear H∞-optimal control problems [9] that has led to many
successes [10], [11]. These motivate the worst-case analysis based
approach to robust adaptive control design [12], [13], [1], where
the measures of disturbance attenuation, asymptotic tracking, and
transient performance are all incorporated into a single soft-
constrained game theoretic cost function. In this approach, the
unknown parameters are treated as part of the expanded state
vector. An application of the cost-to-come function methodol-
ogy [14] to the nonlinear H∞-optimal control problem yields a
finite dimensional estimator for the expanded system, and converts
the H∞-optimal control problem with imperfect state measure-
ments into one with full-information measurements. Then, the
integrator backstepping methodology is applied to solve this full
information measurement problem. The above design paradigm
has been successfully applied to identification problems [12] and
robust adaptive control problems [13], [1], which indicates that the
resulted identifiers and adaptive controllers have strong robustness
properties. Encouraged by these successes, we continue to research
further on this topic.
Motivated by the result of [1], we studied reduced-order adap-
tive controller design in [15]. The key to the order reduction
is step 0 in the controller design step of [1]. In [15], instead
of generating reference trajectory for the entire state vector of
the filtered dynamics of the measured output as is done in [1],
we generate only the reference trajectory for a particular linear
combination of the state vector of the filtered dynamics. Thus,
the controller order is reduced by n − 1 or n − 2 depending
on the eigen structure of a feedback matrix, as compared with
the full-order controller design proposed in [1]. It is proved that
the closed-loop system, after order reduction, achieves the same
strong robustness properties as [1]. Simulation results demonstrate
a significant improvement in transient performance of the closed-
loop system with the reduced-order controller.
In this paper, we continue to study the reduced-order adaptive
controller design methodology, in comparison with the full-order
controller achieved in [1]. We study the same class of linear
2005 American Control Conference
June 8-10, 2005. Portland, OR, USA
0-7803-9098-9/05/$25.00 ©2005 AACC
ThC09.1
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systems as [1], make the same assumptions, and have the same
problem formulation and solution approach to the problem. We
assume the system under consideration has known upper bound
for its dynamic order, is observable, admits a transfer function that
is strictly minimum phase with a known relative degree. The true
system may be uncontrollable, as long as the uncontrollable part is
stable in the sense of Lyapunov, and the uncontrollable modes on
the imaginary axis are uncontrollable from the disturbance input.
Based on these assumptions, the unknown system is transformed
into the design model, where it is linear in all of the unknown
quantities. We assume that the measurement channel is noisy to
avoid singularity in the estimation design. We also assume that
the unknown parameter vector belongs to a convex compact set
characterized by a known smooth nonnegative radially unbounded
and strictly convex function P(¯θ). Furthermore, for any parameter
vector which belongs to the convex compact set, the corresponding
high frequency gain is never zero. The robust adaptive control
problem is then formulated as a nonlinear H∞-optimal control
problem under imperfect state measurements. We adopt a game
theoretic solution to this problem by separating it into estimation
design and controller design steps. In estimation design, we
apply the cost-to-come function methodology to obtain the finite
dimensional estimator. We apply a soft projection algorithm to
relieve the persistency of excitation assumption. The result of
this estimation step is exactly the same as [1]. Therefore, we
summarize the result of the estimation design in [1] for ease of
reference. The main difference between this paper and [1] starts
in the controller design step. As [1], we still apply the integrator
backstepping methodology to derive the control law. But, the
controller design begins from step 1, without first stabilizing the
filtered dynamics of the output as step 0 does in [1]. This relieves
us from generating the reference trajectory for state of the filtered
dynamics of the output to track. Therefore, the controller structure
can be simplified by n integrators, where n is the upper bound
of the order of the unknown system. The rest of the backstepping
design procedure is similar to that of [1]. The lack of step 0 results
in the lack of one nonpositive drift term in the derivative of the
closed-loop value function, which may degrade the performance
of the reduced-order controller. In addition, in order to guarantee
the boundness of the closed-loop signals, the worst-case estimate
for the expanded state has to be chosen suboptimally, rather than
optimally. Exactly the same robustness results are established for
the reduced-order controller as the full-order counterpart of [1].
It is shown that, whenever the disturbance input is bounded and
the reference trajectory and its derivatives up to rth order, where
r being the relative degree of the transfer function of the true
system, are bounded, then, all signals in the closed-loop system
are bounded. It is also shown that the reduced-order controller
achieves the desired disturbance attenuation level, whose ultimate
lower bound is the noise intensity in the measurement channel.
When the disturbance input is bounded and of finite energy, and
the reference trajectory and its derivatives up to rth order are
bounded, then we have asymptotic tracking. This completes the
preview of the results of this paper.
The organization of this paper is as follows. In Section II, we
list the notations to be used in the paper. We present the problem
formulation of the robust adaptive control problem in Section III.
Then, we present the summary of estimation design in Section
IV. In Section V, the controller design is presented. In Section
VI, we present the main robustness results in terms of a theorem.
An example is presented in Section VII. The paper ends with some
concluding remarks in Section VIII. Due to page limitation, some
details, for example, the detailed proof of the theorem, detailed
derivations, and simulation details of the example, are omitted in
this shortened version. For interested readers, please contact us for
a copy of the full version of the paper.
II. NOTATIONS
We denote the real line by I R, the set of natural numbers by I N.
We say that a function f belongs to C if it is continuous; we say
that it belongs to Ck if it is continuously (partial) differentiable
up to kth order; we say that it is smooth if it belongs to C∞. For
a vector or matrix A, A?denotes its transpose. For any z ∈ I Rn
and any n × n-dimensional symmetric matrix M, n ∈ I N, |z|2
denotes z?Mz and |z|2denotes z?z. For any b ∈ I R, sgn(b) =
?−1
1b > 0
stacking up its column vectors. For any symmetric matrix M, the
vector← −
M is formed by stacking up the column vectors of the
lower triangular part of M. For n × n -dimensional symmetric
matrices M1 and M2, where n ∈ I N, we write M1 > M2 if
M1−M2 is positive definite; we write M1 ≥ M2 if M1−M2 is
positive semi-definite. For n ∈ I N, the set of n × n-dimensional
positive definite matrices is denoted by S+n. Denote en,i to be
[01×(i−1)1 01×(n−i)]?, for i = 1,...,n and n ∈ I N. For any
matrix M, ?M?p denotes its p-induced norm, 1 ≤ p ≤ ∞. L2
denotes the set of square integrable functions, and L∞ denotes
the set of bounded functions.
M
b < 0
b = 00
. For any matrix M, the vector− →
M is formed by
III. PROBLEM FORMULATION
The linear system under consideration satisfies,
Assumption 1: The linear system is known to be at most n
dimensional, where n ∈ IN.
The true system dynamics are given by
?
˙` x=
` A` x +`Bu +`D ` w;
`C` x +`E ` w
` x(0) = ` x0
(1a)
(1b)
y=
where ` x is the ` n-dimensional state vector; ` n ∈ IN; u is the scalar
control input; y is the scalar system output; ` w is the ` q-dimensional
disturbance input, ` q ∈ IN; all signals in the system are assumed
to be continuous, i.e., in the space C; and the matrices` A,`B,`C,
`D, and`E are generally unknown or partially unknown. The true
system (1) satisfies the following assumption.
Assumption 2: The pair (` A, `C) is observable. The transfer
function H(s) = ` C (sI` n −` A)−1` B is known to have relative
degree r ∈ I N, and is strictly minimum phase. The uncontrollable
part (with respect to the control input u) of the unknown system
is stable in the sense of Lyapunov. Any uncontrollable mode
corresponding to an eigenvalue of the matrix` A on the jw-axis
are uncontrollable from the disturbance ` w.
Without loss of generality, we assume ` n = n ([1]).
By Assumption 2, there always exist a state transformation ` x =
`Tx and a disturbance transformation w =
system can be written as
?
`
M ` w, such that the
˙ x = Ax + (y¯ A211+ u¯ A212)θ + Bu + Dw; x(0) = x0(2a)
y = Cx + Ew
(2b)
where`T is an unknown real invertible matrix;`
real q × ` q-dimensional matrix, q ∈ IN; θ ∈ I Rσis the vector of
unknown parameters of the system, σ ∈ IN; and the matrices A,
M is an unknown
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¯ A211,¯ A212, B, D, C, and E are known and admit the following
structure: A = (aij)n×n, ai,i+1 = 1, aij = 0, for 1 ≤ i ≤ r − 1
and i+2 ≤ j ≤ n;¯ A212 = [0σ×(r−1)¯ A?
row vector; B = [01×(r−1)bp0 ···, bpn−r]?; C = [1 01×(n−1)];
bpi, i = 1,···,n−r, are constants. Denote the elements of x by
[x1 ... xn]?. The equation (2) is called the design model which
satisfies Assumptions 3–5 described below.
Assumption 3: Define ζ = (EE?)−1/2> 0 and L = DE?.
Because of the structures of A,¯ A212, and B, the high frequency
gain of the transfer function` H(s), b0, is equal to bp0+¯ A212 0θ.
The assumption on the parameter θ is given below.
Assumption 4: There exists a known smooth nonnegative radi-
ally unbounded strictly convex function P : I Rσ→ I R, such that
θ belongs to the set Θ := {¯θ ∈ I Rσ: P(¯θ) ≤ 1}. Furthermore,
∀¯θ ∈ Θ, we have sgn(b0)(bp0+¯ A212 0¯θ) > 0.
The following assumption is made on the reference trajectory yd.
Assumption 5: The reference trajectory yd is r times continu-
ously differentiable. The signal yd and the first r derivatives of
yd are available for feedback. Denote Yd:= [yd y(1)
and Yd0= [yd(0) y(1)
d
The control law is generated by u(t) = µ(y[0,t],Yd[0,t]).
Furthermore, it must satisfy the following condition. For any
uncertainty (x0,θ, ` w[0,∞),Yd0,y(r)
I Rr×C, there must be a unique solution ` x[0,∞)for the closed-loop
system, which results in a continuous control function u[0,∞). The
class of these admissible controllers is denoted by Mu.
The objective of the control design is to make the system output
Cx to track the reference trajectory ydasymptotically while atten-
uating the effect of the uncertainty (x0,θ, ` w[0,∞),Yd0,y(r)
where the exogenous input ` w can be taken to be any open-loop
time function, as in the case of H∞-optimal control problems. For
such uncertainty, we have (x0,θ,w[0,∞),Yd0,y(r)
the set W = I Rn× Θ × C × I Rr× C. Then, a precise definition
of the objective is further given below as that in [1].
Definition 1: A controller µ ∈ Mu is said to achieve distur-
bance attenuation level γ if there exists a nonnegative function
l(t,θ,x,y[0,t],Yd[0,t]) and a constant l0 ≥ 0, such that, for all
tf ≥ 0, sup(x0,θ, ` w[0,∞),Yd0,y(r)
212 0¯ A?
212r]?;¯ A212 0 is a
?
?
d
... y(r)
d]?
d(0) ··· y(r−1)
(0)]?.
?
d[0,∞)) ∈`
W := I Rn× Θ × C ×
d[0,∞)),
d[0,∞)) belongs to
d[0,∞))∈`
WJγtf≤ 0, where
Jγtf:=
?tf
−γ2|w(τ)|2)dτ − γ2|[θ?−ˇθ?
0
((x1(τ) − yd(τ))2+ l(τ,θ,x(τ),y[0,τ],Yd[0,τ])
0x?
0− ˇ x?
0]?|2
¯
Q0− l0 (3)
whereˇθ0 ∈ Θ is the initial guess of θ, ˇ x0 is the initial guess
of x0,¯Q0 > 0 is the weighting matrix, quantifying the level
of confidence in the estimate [ˇθ?
?
σ × σ- and n × n-dimensional, respectively.
The definition above aims to guarantee that, for any tf ≥ 0, the
squared L2 norm of the output tracking error x1− yd on [0,tf]
is bounded by γ2times the squared L2 norm of the transformed
disturbance input w[0,t]plus a constant that depends only on the
initial condition of the system.
The problem formulated above can be brought into the frame-
work of H∞-optimal control with imperfect state measurements
as in [1]. Let ξ denote the expanded state vector ξ = [θ?x?]?,
which satisfies the following dynamics:
0ˇ x?
?
0]?;¯ Q−1
0
admits the structure
Q−1
0
Φ0Q−1
Q−1
0Φ?
0
0Φ?
0
Π0+ Φ0Q−1
0
, where Q0 > 0 and Π0 > 0 are
?
˙ξ =
?
0σ×σ
0σ×n
Ay¯ A211+ u¯ A212
?
ξ +
?
0σ×1
B
?
u +
?
0σ×q
D
?
w
=:
¯ A(y,u)ξ +¯Bu +¯Dw;
?
The worst-case optimization of the cost function (3) can be carried
out in two steps with the following inequality:
(4a)
(4b)
y =
01×σ C?ξ + Ew =:¯ Cξ + Ew
sup
(x0,θ, ` w[0,∞),Yd0,y(r)
d[0,∞))∈`
sup
d[0,∞))∈W|y[0,∞),y(r)
W
Jγtf≤sup
Yd0∈I Rr,y[0,∞)∈C,y(r)
d[0,∞)∈C
(x0,θ,w[0,∞),Yd0,y(r)
d[0,∞),Yd0
Jγtf
(5)
The design procedure starts with the inner supremization, which
can be interpreted as the evaluation of the worst-case performance
for a known output waveform. As a function of the output, the con-
trol input waveform is independent of the actual disturbance input
waveform, and can be viewed as an open-loop time function. This
step is actually the estimation design step discussed next in Section
IV. The outer supremization can be interpreted as the computation
of the worst-case measurement waveform against a given control
law, which is crucial for the determination of achievability of the
control objective. This step is the control design step carried out in
Section V. The design function l(t,θ,x,y[0,t],Yd[0,t]) is selected
based on the same considerations as [1]: the existence of a solution
to the problem; the ease of analysis of stability and robustness of
the resulting closed-loop system.
This completes the formulation of the problem. Next, we turn
to the estimation and control design in the next two sections.
IV. ESTIMATION DESIGN
In order to set up an appropriate basis for the discussion of
control design in next section, we summarize the key results of
the estimation design obtained in [1].
Given Yd0, the measurement waveform y[0,∞), and the reference
trajectory y(r)
known. The cost function we consider in this step is
d[0,∞), then the control waveform u[0,∞) is also
Jiγtf=
?tf
−γ2|w(τ)|2+ 2(ξ − l1(τ,y[0,τ],Yd[0,τ])?
?
whereˆξ is the worst-case estimate for the expanded state ξ to
be designed later; l1 and l2 are functions to be introduced in
this section; the term 2(ξ − l1)?l2 is added to incorporate a soft-
projection algorithm, which keepsˇθ within a vicinity of Θ.¯Q
is the nonnegative-definite weighting function which exhibits a
special structure given by
0
?
|x1(τ) − yd(τ)|2+ |ξ(τ) −ˆξ(τ)|2
¯
Q(τ,y[0,τ],Yd[0,τ])
·l2(τ,y[0,τ],Yd[0,τ])dτ − γ2|[θ?−ˇθ?
0 x?
0− ˇ x?
0]?|2
¯
Q0(6)
¯Q(t,y[0,t],Yd[0,t]) = (¯Σ(t))−1
?
0σ×σ 0σ×n
0n×σ
∆
?
(¯Σ(t))−1
+
?
−(Φ(t))?
In
?
?(t)(Φ(t))?C?(γ2ζ2− 1)CΦ(t) 0σ×n
0n×σ
0n×n
??
?
=
?
+
?
γ4Π−1∆Π−1
?
−(Φ(t))?
In
?(t)(Φ(t))?C?(γ2ζ2− 1)CΦ(t) 0σ×n
0n×σ
0n×n
?
where¯Σ is the worst-case covariance matrix defined later in (7f); Φ
and Π are defined in (7e) and (7a), respectively, ∆ = γ−2β∆Π+
∆1with the constant β∆ ≥ 0 and the constant matrix ∆1 > 0; ? is
a scalar function defined by ?(t) = Tr((Σ(t))−1)/Kc, ∀t ∈ [0,tf]
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or by ?(t) = 1, ∀t ∈ [0,tf], Kc ≥ γ2Tr(Q0) is a constant; and
the matrix Σ will play the role of worst-case covariance matrix
of the parameter estimation error. With the choice of ? ≡ 1, we
observe, from (7b) and (7c), Σ and sΣ remains constant matrices
on [0,∞), which simplifies the controller structure by
integrators. In the following, we will considers¯ Q as a function
¯Q : I Rn×σ× I R → I R(n+σ)×(n+σ),¯Q(Φ,sΣ).
On the basis of Assumption 4, a soft projection algorithm is
applied. Define ρ := inf{P(¯θ) |¯θ ∈ Rσand bp0 +¯ A212 0¯θ =
0} then, ρ > 1. Fix any ρo ∈ (1,ρ). The design will try
to guarantee that the estimateˇθ lies in the open set Θo :=
{¯θ ∈ I Rσ| P(¯θ) < ρo}, then, we have that the estimateˇb0 :=
bp0 +¯ A212 0ˇθ is bounded away from 0. This soft projection
algorithm is incorporated into the cost function by setting l1 to
beˇξ and l2 to be [−Pr(ˇθ)?01×n]?, where
σ (σ+1)
2
+1
Pr(ˇθ) :=
?
exp((1−P(ˇθ))−1)
(ρo−P(ˇθ))3
0σ×1;
?∂P
∂θ(ˇθ)??;∀ˇθ ∈ Θo\Θ
∀ˇθ ∈ Θ
=:pr(ˇθ)
?∂P
∂θ(ˇθ)
??
It is obvious that Pr(ˇθ) and pr(ˇθ) are smooth on the set Θo, and
(θ −ˇθ)?Pr(ˇθ) ≤ 0, ∀ˇθ ∈ Θo.
Then the identifier dynamics are summarized as follows, which
is the result of [1].
(A − ζ2LC + β∆/2In)Π + Π(A − ζ2LC + β∆/2In)?
−ΠC?(ζ2− γ−2)CΠ + DD?− ζ2LL?+ γ2∆1 = 0
˙Σ = −(1 − ?)ΣΦ?C?(γ2ζ2− 1)CΦΣ; Σ(0) = γ−2Q−1
˙ sΣ = (γ2ζ2− 1)(1 − ?)CΦΦ?C?; sΣ(0) = γ2Tr(Q0)
Af = A − ζ2LC − ΠC?C (ζ2− γ−2)
˙Φ = AfΦ + y¯ A211+ u¯ A212;
?
¯Σ(t) =
Φ(t)Σ(t) γ−2Π + Φ(t)Σ(t)(Φ(t))?
˙ˇθ = −ΣPr(ˇθ) − ΣΦ?C?(yd− Cˇ x) −?Σ ΣΦ??¯Qξc
˙ˇ x = −ΦΣPr(ˇθ) + Aˇ x − (γ−2Π + ΦΣΦ?)C?(yd− Cˇ x)
−?ΦΣ γ−2Π + ΦΣΦ??¯Qξc+ Bu + (y¯ A211+ u¯ A212)ˇθ
W(ξ,ˇξ,¯Σ) = |ξ −ˇξ|2
+|θ −ˇθ|2
˙W(ξ,ˇξ,ˆξ,¯Σ,yd,u,w) = −|x1− yd|2+ 2(θ −ˇθ)?Pr(ˇθ)
−γ4|x − ˆ x − Φ(θ −ˆθ)|2
−?(γ2ζ2− 1)|θ −ˆθ|2
+γ2|w|2+ |Cˇ x − yd|2+ |ξc|2
(7a)
(7b)
(7c)
(7d)
(7e)
0
Φ(0) = Φ0
(¯Σ(t))−1=
(Σ(t))−1+ γ2(Φ(t))?Π−1Φ(t) −γ2(Φ(t))?Π−1
−γ2Π−1Φ(t)γ2Π−1
?
?
Σ(t) Σ(t)(Φ(t))?
?
;
(7f)
+γ2ζ2ΣΦ?C?(y − Cˇ x);ˇθ(0) =ˇθ0
(7g)
+ζ2(ΠC?+ γ2ΦΣΦ?C?+ L)(y − Cˇ x); ˇ x(0) = ˇ x0
¯Σ−1 = γ2|x − ˇ x − Φ(θ −ˇθ)|2
Σ−1; W : I Rn+σ× I Rn+σ× S+(n+σ)→ I R
(7h)
Π−1
(7i)
Π−1∆Π−1 − γ2ζ2|y − Cˇ x|2
Φ?C?CΦ− γ2|w − w∗(ξ,ˇξ,¯Σ,w)|2
¯
Q(Φ,sΣ)
(7j)
where sΣ(t) := Tr((Σ(t))−1), which is introduced to avoid the
computation of (Σ(t))−1on line; Φ is a filtered signal of y and
u;ˇθ is an estimate for θ; ˇ x is an estimate for x; W is the value
function for the estimation design step, which is defined when
Σ > 0; ξc =ˆξ −ˇξ andˇξ = [ˇθ?ˇ x?]?; w∗ : I Rn+σ× I Rn+σ×
S+(n+σ)× I Rq→ I Rqdenotes the worst-case disturbance for the
estimation step, (7j) holds as long as Σ > 0 andˇθ ∈ Θo.
Lemma 1: Consider the dynamic equation (7b) for the covari-
ance matrix Σ. Let γ ≥ ζ−1, Q0 > 0, with ? = K−1
c sΣ and
Kc ≥ γ2Tr(Q0). Then, the matrix Σ is upper and lower bounded:
K−1
∀t ∈ [0,tf], whenever it exists on [0,tf] and Φ is continuous on
[0,tf].
We list Assumptions 7 and 8 in [1] below.
Assumption 6: If the matrix A−ζ2LC is Hurwitz, then choose
the desired disturbance attenuation level γ ≥ ζ−1. In case γ =
ζ−1, choose β∆ ≥ 0 such that A − ζ2LC + β∆/2In is Hurwitz.
If the matrix A − ζ2LC is not Hurwitz, then choose the desired
disturbance attenuation level γ > ζ−1.
Assumption 7: The matrix Π0 is chosen as the unique positive-
definite solution to the algebraic Riccati equation (7a)
Assumption 6 makes the observations that the quantity ζ−1is
the ultimate lower bound on the achievable performance level for
the adaptive system using the proposed method. Assumption 7 sets
Π to a constant and simplifies the identifier. Under Assumption 7,
the matrix Af is Hurwitz.
To simplify the estimator dynamics, we may generate Φ by the
following 3n-dimensional prefiltering system for y and u: ˙ η =
Afη + pny, η(0) = η0;˙λ = Afλ + pnu, λ(0) = λ0;˙λo =
Afλo, λo(0) = pn, where pn is an n-dimensional vector such
that (Af,pn) is controllable. Then, Φ is given by
c Iσ ≤ Σ(t) ≤ Σ(0), γ2Tr(Q0) ≤ Tr((Σ(t))−1) ≤ Kc,
?
?
Φ =?An−1
where the matrix Mf =
η0 ∈ I Rn, λ0 ∈ I Rn, and Φo0 ∈ I Rn×σare such that (8) holds at
t = 0. This completes the summary of the identification design of
[1]. We now turn to the controller design step in the next section.
f
η ... Afη η?M−1
f
¯ A211+?An−1
An−1
f
pn
f
λ ... Afλ λ?
Afpn
pn
·M−1
f
¯ A212+?An−1
f
λo ···Afλo λo?M−1
?
f Φo0
(8)
...
?, and
V. CONTROL DESIGN
We describe in this section the reduced-order controller design
for the uncertain system. By (5) and (7j), we have
sup
(x0,θ, ` w[0,∞),Yd0,y(r)
?tf
+ˇl(t,y[0,t],Yd[0,t]) − γ2ζ2|y(τ) − Cˇ x(τ)|2)dτ − l0
d[0,∞))∈`
W
Jγtf≤ sup
Yd0∈I Rr,y[0,∞)∈C,y(r)
d[0,∞)∈C
0
(|Cˇ x(τ) − yd(τ)|2+ |ξc(τ)|2¯
Q(Φ(τ),sΣ(τ))
(9)
whereˇl = l − |ξ −ˆξ|2¯
closed-loop signals exist, Σ(t) > 0, andˇθ(t) ∈ Θo, on [0,tf]. The
control design aims to guarantee that the supremum is less than
or equal to zero for all measurement waveforms. A new variable
v := ζ (y − Cˇ x) is introduced, instead of considering y as the
maximizing variable. The control design problem now is an H∞-
optimal control problem with full information measurements.
We apply the integrator backstepping methodology to design the
control law u. Express the elements of ˇ x as?
non-negative definite weighting term on ξc in (9), the design for
ξcwill be carried out last. Therefore, we, for convenience, set ξcto
be zero in the backstepping procedure. We start the backstepping
procedure directly from Step 1 to stabilize ˇ x1 by viewing ˇ x2 as
the virtual control. Then, we view ˇ x3 as the virtual control to
stabilize ˇ x1 and ˇ x2. Continue in this fashion, until step r, where
u appears in the dynamics of ˇ xr, we complete the design for u.
As a result of the backstepping design, we have obtained
Xjo, Xja, Xjd, Djo, Dja, Djd, γj ∈ I R, α0 : D1o → I R,
βj : Djo × Dja × Djd → I R, j = 1,···,r, αj : Djo ×
Q+ 2(θ −ˇθ)?Pr(ˇθ), and (9) holds if all
ˇ x1
···ˇ xn
??.
Express row vectors of Φ as?
Φ?
1
···Φ?
n
??. In view of a
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Dja × Djd → I R, j = 1,···,r − 1, νr : Dro × Dra ×
Drd → I R, V : Dro × Dra → I R, and µ : Dro × Dra ×
Drd → I R such that Xj+1 o = [X?
Djo × Dja × Djd, j = 1,...,r − 1, αj, j = 0,···,r − 1,
νr, V , and µ are smooth, γj
cβj
> 0, ∀(Xjo,Xja,Xjd) ∈ Djo × Dja × Djd, Xja =
ˇ xj, Xjo = [ydˇθ?← −
Σ
Djo
:={Xjo
|
ˇθ ∈ Θo}, j
[y(r)
d
Φr ˇ xr+1 ··· ˇ xn Φr+1 ··· Φn]?, Drd= {Xrd}, and
joXja X?
jd]?, Dj+1 o =
> 0, βj(Xjo,Xja,Xjd) ≥
?sΣ ˇ x1 y(1)
d
Φ1 ··· ˇ xj−1 y(j−1)
=1,···,r, Xrd
d
Φj−1]?,
=
u = µ(Xro,Xra,Xrd)
(10)
V (Xro,Xra) =
r
?
j=1
γj(ˇ xj− αj−1(Xjo))2
˙V (Xro,Xra,Xrd,u,v)|u=µ(Xro,Xra,Xrd)= −(ˇ x1− α0(X1o))2
r
?
+γ2v2; ∀(Xro,Xrd) ∈ Dro× Drd,
−
j=1
βj(ˇ xj− αj−1(Xjo))2− γ2(v − νr(Xro,Xra,Xrd))2
∀Xra ∈ Dra,∀v ∈ I R
where ξc = 0 in the above. The closed-loop system admits the
state vector
X :=?θ?x?X?
which belongs to the set D := {X | Σ > 0,sΣ > 0,ˇθ ∈ Θo}.
Based on the value functions of the estimation design and control
design, we obtain value function for the closed-loop system: U =
V + W, where U : D → I R is smooth. Then,
roXra Φr ˇ xr+1 ··· ˇ xn Φr+1 ··· Φn
??
˙U(X,y(r)
−γ4|x − ˆ x − Φ(θ −ˆθ)|2
d,ˆξ,w) = −|x1− yd|2− ?(γ2ζ2− 1)|θ −ˆθ|2
Π−1∆Π−1 + 2(θ −ˇθ)?Pr(ˇθ)
Φ?C?CΦ
+|ξc+1
2ςr(Xro,Xra,Xrd)|2
¯
Q(Φ,sΣ)−
r
?
j=1
βjz2
j+ γ2|w|2
−γ2|w − wopt(X,y(r)
d)|2−1
4|ςr(Xro,Xra,Xrd)|2
∀ˆξ ∈ I Rσ+n,
¯
Q(Φ,sΣ)
∀X ∈ D,∀y(r)
d
∈ I R,∀w ∈ I Rq
(11)
where the worst-case disturbance wopt : D × I R → I Rqis
wopt(X,y(r)
d)=ζE?νr+ γ−2(Iq− ζ2E?E)¯D?¯Σ−1(ξ −ˇξ)
+ζ2E?C (ˇ x − x)
and ςr : Dro× Dra× Drd→ I Rn+σis smooth.
The design of ξcis based on two considerations: (1) to guarantee
[0 γ−2Π]¯Qξc is bounded, which is applied later in the proof of
Theorem 1; (2) to guarantee the terms |ξc+1
is non-positive. Then, we choose ξc to be
2ςr|2
¯
Q−1
4|ςr|2
¯
Q+|ξc|2
¯
Q
ξc:= µξ(Xro,Xra,Xrd)
= −1
2ε0(Xro,Xra,Xrd)ςr(Xro,Xra,Xrd)
(12)
where
ε0 :=
|kc|
?
k2
c+ |[0 γ−2Π]¯Qςr|2+ 1
and kc ∈ I R is a constant. Clearly, µξ: Dro×Dra×Drd→ I Rσ+n
is a smooth function. We observe that ξc ≡ 0 when kc = 0. This
special case will result in simpler control system structure.
This completes the design for the entire closed-loop system,
which involves state X. We shall turn to study the robustness and
tracking properties of the closed-loop system.
VI. MAIN RESULT
In this section, we investigate the robustness and tracking prop-
erties of the closed-loop system with the reduced-order controller.
The closed-loop system dynamics can be expressed as
˙X = F(X,y(r)
d) + G(X)`M ` w; X(0) = X0
(13)
where the function F and G are smooth mappings of D×I R and
D, respectively; and the initial state X0 satisfies X0 ∈ D0 :=
{X0 ∈ D | θ ∈ Θ,ˇθ0 ∈ Θ, Σ(0) = γ−2Q−1
γ2Tr(Q0) ≤ Kc}. As described in [1], the value function U
satisfies the Hamilton-Jacobi-Isaacs equation
0
> 0, sΣ(0) =
∂U
∂X(X)F(X,y(r)
+Q(X,y(r)
d) +
1
4γ2
∂U
∂X(X)G(X)(G(X))??∂U
d
∈ I R
∂X(X)
??
d) = 0; ∀X ∈ D,∀y(r)
(14)
where Q : D × I R → I R is smooth and given by
Q(X,y(r)
d) = |x1− yd|2+ γ4|x − ˆ x − Φ(θ −ˆθ)|2
+?(γ2ζ2− 1)|θ −ˆθ|2
Π−1∆Π−1
Φ?C?CΦ− 2(θ −ˇθ)?Pr(ˇθ)
r
?
−(ε2
0− 2ε0)|ςr|2
¯
Q/4 +
j=1
βjz2
j
The stability of the closed-loop system can not be deduced
directly from the value function U, which is not a positive-definite
function for the closed-loop system. The following theorem will
state the stability properties of the closed-loop system, it shows
that the closed-loop system admits strong robustness properties.
Theorem 1: Consider the robust adaptive control problem for-
mulated in Section III, with Assumptions 1–7 holding. Then, the
robust adaptive controller µ defined by (10), with ξcgiven by (12),
achieves the following strong robustness properties.
1) Given cw ≥ 0 and cd ≥ 0, there exists a constant cc ≥ 0
and a compact set Θc ⊂ Θo, such that, for any uncertainty
(x0,θ, ` w[0,∞),Yd0,y(r)
cw, |Yd(t)| ≤ cd, ∀t ∈ [0,∞), all closed-loop state variables
x, ˇ x,ˇθ, Σ, sΣ, and Φ are bounded as follows: ∀t ∈ [0,∞),
|x(t)| ≤ cc, |ˇ x(t)| ≤ cc,ˇθ(t) ∈ Θc, |− − →
Σ(t) ≤ γ−2Q−1
there is a compact set S ⊂ D such that X(t) ∈ S, ∀t ∈
[0,∞). Hence, there exists a constant cu ≥ 0 such that
|u(t)| ≤ cu, |ˆξ(t)| ≤ cu, |η(t)| ≤ cu, |λ(t)| ≤ cu, and
|λo(t)| ≤ cu, ∀t ∈ [0,∞).
2) The controller µ ∈ Muand achieves disturbance attenuation
level γ for any uncertainty (x0,θ, ` w[0,∞),Yd0,y(r)
`
W.
3) For any uncertainty (x0,θ, ` w[0,∞),Yd0,y(r)
` w[0,∞)∈ L2∩ L∞ and Yd[0,∞)∈ L∞, limt→∞(x1(t) −
yd(t)) = 0.
d[0,∞)) ∈`
W with |x0| ≤ cw, | ` w(t)| ≤
Φ(t)| ≤ cc, K−1
c Iσ ≤
0, γ2Tr(Q0) ≤ sΣ(t) ≤ Kc. Therefore,
d[0,∞)) ∈
d[0,∞)) ∈`
W with
VII. EXAMPLE
Consider a circuit shown in Figure 1(a). The resistance R is
1Ω. The capacitor C and the inductor L are linear and time
invariant, L = 1H. vi is a dependent voltage source. ve is
an unknown sinusoidal voltage source. vw1 is an unmeasured
exogenous voltage disturbance; is is an unmeasured exogenous
current disturbance. vo is the voltage output. Our objective is to
achieve the desired voltage output vo− vw1 by adjusting vi.
The simulation results are shown in Figures 1(b), (c), and (d).
The tracking error converge to zero and the parameter estimates
converge to the true value, which is consistent with our theoretical
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