Adaptive Controller Design and Disturbance Attenuation for SISO

Linear Systems with Zero Relative Degree under Noisy Output

Measurements∗

Sheng Zeng† Yu Chen‡ Zigang Pan§

Abstract

In this paper, we present robust adaptive controller design for SISO linear systems with zerorelative degree under noisy output measurements. We formulate the robust adaptive controlproblem as a nonlinear H∞-optimal control problem under imperfect state measurements, andthen solve it using game theory. By using the a priori knowledge of the parameter vector,we apply a soft projection algorithm, which guarantees the robustness property of the closed-loop system without any persistency of excitation assumption of the reference signal. Due toour formulation in state space, we allow the true system to be uncontrollable, as long as theuncontrollable part is stable in the sense of Lyapunov, and the uncontrollable modes on thejω-axis are uncontrollable from the exogenous disturbance input. This assumption allows theadaptive controller to asymptotically cancel out, at the output, the effect of exogenous sinusoidaldisturbance inputs with unknown magnitude, phase, and frequency. These strong robustnessproperties are illustrated by a numerical example.

Keywords Nonlinear H∞ control; cost-to-come function analysis; robust adaptive control.

1 Introduction

The design of adaptive controllers has been an important research topic since 1970s. The clas-sic adaptive control design, based on the certainty equivalence principle (Goodwin and Sin 1984,Goodwin and Mayne 1987), is to design the controller as if the system parameters are known andthen in implementation to supply the controller with estimates of the parameters, using standardidentifiers, as if the estimates are true values. This design method has been proven successful es-pecially for the linear systems with or without stochastic disturbance inputs (Morse 1980, Kumar1985, Narendra and Annaswamy 1989, Guo and Chen 1991, Ren and Kumar 1992, Naik et al. 1992,Guo 1996, Hsu et al. 1999). This approach leads to structurally simple adaptive controllers. Yet,

∗Manuscript completed December 21, 2004. Research supported in part by the National Science Foundation underCAREER Award ECS-0296071.

†Department of Electrical and Computer Engineering and Computer Science, University of Cincinnati, Cincinnati,OH 45221-0030. Tel: 513-556-1036; Email: zengs@ececs.uc.edu.

‡Department of Electrical and Computer Engineering and Computer Science, University of Cincinnati, Cincinnati,OH 45221-0030. Tel: 513-556-1036; Email: cheny4@ececs.uc.edu.

§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.

1

early designs based on this approach has been shown to be nonrobust (Ioannou and Kokotovic 1983,Rohrs et al. 1985) when the system is subject to exogenous disturbance inputs and unmodeled dy-namics. Then, the stability and the performance of a system under disturbance and/or uncertaintybecomes an important issue. This motivates the study of robust adaptive control which has at-tracted significant research attention since 1980s. Also, this approach fails to generalize to systemswith severe nonlinearities. This motivates the study of nonlinear adaptive control in 1990s.

Robust adaptive control has been an important research topic in late 1980s and early 1990s.Various adaptive controllers were modified to render the closed-loop systems robust (Datta andIoannou 1994, Ioannou and Sun 1996). Despite their successes, they fell short of directly addressingthe disturbance attenuation property of the closed-loop system.

The topic of nonlinear adaptive control has been widely studied in the last decade after the cel-ebrated characterization of feedback linearizable or partially feedback linearizable systems (Isidori1995). The introduction of the integrator backstepping methodology (Kanellakopoulos et al. 1991)allow us to design adaptive controllers for parametric strict-feedback and parametric pure-feedbacknonlinear systems systematically. Since then, a lot of important contributions were motivated bythis approach, and a complete list of references can be found in the book (Krstic et al. 1995).More recently, systems with unknown sign of the high frequency gain have been studied usingthis approach (Ye 2001). Moreover, this approach has been applied to linear systems to compareperformance with the certainty equivalence approach (Krstic et al. 1994). As to be expected, asystematically designed nonlinear adaptive control law leads to better closed-loop performance thanthat for the certainty equivalence based design when the system is free of disturbance. However,this approach has also been shown to be nonrobust (Basar et al. 1996) when the system is subjectto exogenous disturbance inputs.

H∞-optimal control has been proposed as a solution to the robust control problem. It achievesthe objectives of robust control, namely, improving transient response, accommodating unmodeleddynamics, and rejecting exogenous disturbance inputs, by studying only the disturbance attenuationproperty for the closed-loop system. The game-theoretic approach to H∞-optimal control (Basarand Bernhard 1995) developed for the linear quadratic problems, offers the most promising toolto generalize the results to nonlinear systems (van der Schaft 1991, 1992, Isidori and Astolfi 1992,Didinsky et al. 1993, Didinsky 1994, Marino et al. 1994, Isidori and Kang 1995). The worst-case analysis approach to adaptive control was proposed to address the disturbance attenuationproperties of the closed-loop system directly. In this approach, the robust adaptive control problemis formulated as a nonlinear H∞-optimal control problem under imperfect state measurements.Using cost-to-come function analysis, it can be converted into a problem under full informationmeasurements. This full information measurement problem is then solved for a suboptimal solutionusing the integrator backstepping methodology. This design paradigm has been applied to worst-case parameter identification problems (Didinsky et al. 1995, Pan and Basar 1996), which hasled to new classes of parametrized identifiers for linear and nonlinear systems. It has also beenapplied to adaptive control problems (Didinsky and Basar 1997, Pan and Basar 1998, Tezcan andBasar 1999, Pan and Basar 2000, Arslan and Basar 2001, Zeng and Pan 2003), which has led tonew classes of parametrized robust adaptive controllers for linear and nonlinear systems. Adaptivecontrol for strict-feedback systems was studied in Pan and Basar (1998) under noiseless full statemeasurements. More general class of nonlinear systems was studied in Arslan and Basar (2001),again with noiseless full state measurements. In Tezcan and Basar (1999), adaptive control forstrict-feedback nonlinear systems was considered under noiseless output measurements. In Pan

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and Basar (2000) and Zeng and Pan (2003), linear system was considered under noisy outputmeasurements. Zeng and Pan (2003) generalizes the results of Pan and Basar (2000), by assumingpart of the disturbance inputs are measured, which then lead to disturbance feedforword structurein the adaptive controller.

In this paper, we study the adaptive control design for SISO linear systems with zero relativedegree under noisy output measurements using a similar approach as that of Pan and Basar (2000).We assume that the linear system admits a known upper bound for its dynamic order, is observable,has a strictly minimum phase transfer function with relative degree 0. The linear system maybe uncontrollable, as long as the uncontrollable part is stable in the sense of Lyapunov, and alluncontrollable modes on the jω-axis are uncontrollable from the disturbance input. Under theseassumptions, the system may be transformed into the design model, which is linear in all of theunknown quantities. We formulate the robust adaptive control problem as a nonlinear H∞-optimalcontrol problem under imperfect state measurements, where the objectives of asymptotic tracking,transient performance, and disturbance attenuation are incorporated into a single game theoreticcost function. To avoid singularity is the estimation step, we assume that the measurement is noisy.Then, we can apply cost-to-come function methodology to derive the estimator, which has a finite-dimensional structure. To relieve the persistency of excitation condition for the closed-loop system,we assume that the true value of the parameter vector belongs to a convex compact set characterizedby a known smooth nonnegative radially unbounded and strictly convex function P (θ), and apply asoft projection algorithm for the estimator. Then, the closed-loop system is robust with or withoutthe persistently exciting signals. After the estimator is determined, the original problem becomesa nonlinear H∞-optimal control problem under full-information measurement, and the controllercan be obtained directly based on the cost function at that step. The closed-loop system admits aguaranteed disturbance attenuation level with respect to the exogenous disturbance inputs, wherethe ultimate attenuation lower bound for the achievable performance level is equal to the noiseintensity in the measurement channel. All closed-loop signals are bounded for bounded disturbanceinput and bounded reference trajectory. Furthermore, it achieves asymptotic tracking of uniformlycontinuous and bounded reference trajectories for all bounded disturbance inputs that are of finiteenergy. This result has significant impact on active noise cancellation problems. That is, when thetrue system is subject to disturbances generated by an unknown exogenous linear system, we canextend our system model to include the states of the exogenous system as part of the model, andthen asymptotically cancel out the effect of the noise at the output. This feature is illustrated byan example in the paper.

The balance of the paper is organized as follows. In Section 2, we list the notations to beused in this paper. In Section 3, we formulate adaptive control problem and discuss the generalsolution methodology. In Section 4, we present the estimation and control design using cost-to-come function methodology. In Section 5, we present the main result of the paper which statesthe robustness properties of the closed-loop system. The theoretical results are illustrated by anumerical example in Section 6. The paper ends with some concluding remarks in Section 7, andtwo appendices.

2 Notations

We denote IR to be the real line; IN to be the set of natural numbers; C to be the set of complexnumbers. For a function f , we say that it belongs to C if it is continuous; we say that it belongs to

3

Ck if it is k-times continuously (partial) differentiable. For any matrix A, A′ denotes its transpose.

For any b ∈ IR, sgn(b) =

−1 b < 00 b = 01 b > 0

. For any vector z ∈ IRn, where n ∈ IN, |z| denotes

(z′z)1/2. For any vector z ∈ IRn, and any n × n-dimensional symmetric matrix M , where n ∈ IN,|z|2M = z′Mz. For any matrix M , the vector

−→M is formed by stacking up its column vectors. For

any symmetric matrix M ,←−M denotes the vector formed by stacking up the column vector of the

lower triangular part of M . For n× n-dimensional symmetric matrices M1 and M2, where n ∈ IN,we write M1 > M2 if M1 −M2 is positive definite; we write M1 ≥ M2 if M1 −M2 is positivesemi-definite. For n ∈ IN, the set of n×n-dimensional positive definite matrices is denoted by S+n.For n ∈ IN ∪ 0, In denotes the n × n-dimensional identity matrix. For any matrix M , ‖M‖pdenotes its p-induced norm, 1 ≤ p ≤ ∞. L2 denotes the set of square integrable functions and L∞denotes the set of bounded functions. For any n, m ∈ IN, 0n×m denotes the n × m-dimensionalmatrix whose elements are zeros.

3 Problem Formulation

We consider the adaptive control problem for single-input and single-output (SISO) linear time-invariant systems. We make the following assumption on the unknown system.

Assumption 1 The linear system is known to be at most n dimensional, n ∈ IN. ⋄

We consider the following true system dynamics:

˙x = Ax + Bu + Dw; x(0) = x0 (1a)

y = Cx + b0u + Ew (1b)

where x is the n-dimensional state vector, n ∈ IN ∪ 0; u is the scalar control input; b0 ∈ IR andb0 6= 0; y is the scalar measurement output; w is the q-dimensional unmeasured disturbance inputvector, q ∈ IN; the state x has initial condition x0; and all input and output signals y, u, and w arecontinuous; the matrices A, B, C, D, and E are of the appropriate dimensions, generally unknownor partially unknown. The transfer function from u to y is H(s) = C (sIn − A)−1B + b0.

Assumption 2 1 The pair (A, C) is observable. The transfer function H(s) is known to haverelative degree 0, and is strictly minimum phase. Moreover, the uncontrollable part (with respect tou) of the unknown system is stable in the sense of Lyapunov. Any uncontrollable mode correspondingto the eigenvalues of the matrix A on the jω-axis are uncontrollable from w. ⋄

Remark 1 If the true system is of order n < n, we can add (n − n)-dimensional dynamics asoutlined in Lemma 6, such that the expanded system is of order n and satisfies the Assumption 2.Hence, without loss of generality, we will assume that the true system (1) is of order n.

Since (A, C) is observable, there always exists a state diffeomorphism x = T x, and a disturbancetransformation w = Mw, where T is an unknown real invertible matrix and M is a real q × q-dimensional unknown matrix with q ∈ IN, such that the system (1) can be transformed into the

1When n = 0, Assumption 2 is considered satisfied.

4

following form in the x coordinate with inputs u and w

x = Ax + (yA211 + uA212)θ + Bu + Dw; x(0) = x0 (2a)

y = Cx + uC1θ + bp0u + Ew (2b)

where θ is the σ-dimensional vector of unknown parameters of the system, σ ∈ IN; the matrices A,A211, A212, B, D, C, E, and C1 are of appropriate dimensions and completely known, and bp0 ∈ IRis also known. In addition, the high frequency gain of the transfer function H(s), b0, is equal tobp0 + C1θ. We will design the adaptive controller based on system (2), which is called the designmodel. By Assumption 2, the pair (A,C) is observable.

We have the following assumptions about the design model.

Assumption 3 EE′ > 0. ⋄

Define ζ := (EE′)−1/2 and L := DE′.

Remark 2 The above transformation matrix T always exists. One may choose the state diffeo-morphism x = T x to be the one which transforms the pair (A, C) into its observer canonical form.Next, we can choose a unknown matrix M such that the matrices D and E are completely known.

To guarantee the stability of the closed-loop system and the boundedness of the estimate of θ, wemake the following assumption on the parameter vector θ.

Assumption 4 The sign of the high-frequency gain b0 is known. There exists a known smoothnonnegative radially-unbounded strictly convex function P : IRσ → IR, such that the true value of θbelongs to the set Θ := θ ∈ IRσ | P (θ) ≤ 1. Furthermore, for any θ ∈ Θ, we have sgn(b0) (bp0 +C1θ) > 0. ⋄

We make the following assumption about the reference signal, yd.

Assumption 5 The reference trajectory, yd, is continuous, and available for the control design. ⋄

For the system (1), under Assumptions 1—5, the control law is generated by

u(t) = µ(y[0,t], yd[0,t]) (3)

Furthermore, it must satisfy the following condition. For any uncertainty (x0, θ, w[0,∞), yd[0,∞)) ∈

W := IRn ×Θ× C × C, which comprises the initial state, the true value of the unknown parametervector, the unknown disturbance input waveform, and the reference trajectory, there must bea unique solution x[0,∞) for the closed-loop system, which results in a continuous control inputwaveform u[0,∞). We denote the class of these admissible controllers byMu.

The objectives of our control design are to make the output of the system, Cx+ b0u, to asymp-totically track the reference trajectory yd, and guarantee the boundedness of all closed-loop signals,while rejecting the uncertainty (x0, θ, w[0,∞), yd[0,∞)) ∈ W. For design purposes, instead of attenu-ating the effect of w, we design the adaptive controller to attenuate the effect of w. We take theuncertainty (x0, θ, w[0,∞), yd[0,∞)) to belong to the setW := IRn×Θ×C×C. All of these objectivescan be captured by the optimization of a single game-theoretic cost function, defined as follows.

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Definition 1 A controller µ ∈Mu is said to achieve disturbance attenuation level γ if there exista nonnegative function l(t, θ, x, y[0,t], yd[0,t]) such that

sup(x0,θ,w[0,∞),yd[0,∞))∈W

Jγtf ≤ 0; ∀tf ≥ 0 (4)

where

Jγtf :=

∫ tf

0

(

(Cx(τ) + u(τ)C1θ + bp0u(τ)− yd(τ))2 + l(τ, θ, x(τ), y[0,τ ], yd[0,τ ])

−γ2|w(τ)|2)

dτ − γ2

∣

∣

∣

∣

[

θ′ − θ′0 x′0 − x′

0

]′∣

∣

∣

∣

2

Q0

(5)

where θ0 ∈ Θ is the initial guess of the unknown parameter vector θ; x0 is the initial guess of theunknown initial state x0; the (σ + n) × (σ + n)-dimensional matrix Q0 is the quadratic weighting

on the initial estimation error, quantifying the level of confidence in the estimate[

θ′0 x′0

]′; Q−1

0

admits the structure

[

Q−10 Q−1

0 Φ′0

Φ0Q−10 Π0 + Φ0Q

−10 Φ′

0

]

, where Q0 and Π0 are σ×σ- and n×n-dimensional

positive definite matrices, respectively.

Clearly, when the inequality (4) is achieved, the squared L2 norm of the output tracking errorCx+uC1θ + bp0u−yd is bounded by γ2 times the squared L2 norm of the transformed disturbanceinput w plus some constant. When the L2 norm of w is finite, the squared L2 norm of Cx+uC1θ+bp0u− yd is also finite, which implies lim

t→∞(Cx(t) + u(t)C1θ + bp0u(t)− yd(t)) = 0, under additional

assumptions.The following notation will be used throughout this paper. Let x denote the estimate of x,

x denote the state estimation error x − x, θ denote the estimate of θ, θ denote the parameterestimation error θ − θ.

We intend to solve this robust adaptive control problem by formulating it as an H∞ controlproblem with imperfect state measurements. To do this, we first expand the state space to includethe parameter θ as part of the state. Let ξ denote the expanded state vector ξ = [θ′ x′]′. Note thatθ = 0, we have the following expanded dynamics for system (2)

ξ =

[

0σ×σ 0σ×n

yA211 + uA212 A

]

ξ +

[

0σ×1

B

]

u +

[

0σ×q

D

]

w

=: A(u, y)ξ + Bu + Dw (6a)

y =[

uC1 C]

ξ + bp0u + Ew =: C(u)ξ + bp0u + Ew (6b)

The worst-case optimization of the cost function (5) can be carried out in two steps as depicted inthe following inequality.

sup(x0,θ,w[0,∞),yd[0,∞))∈W

Jγtf = supy[0,∞)∈C,yd[0,∞)∈C

sup(x0,θ,w[0,∞),yd[0,∞))∈W|y[0,∞),yd[0,∞)

Jγtf

≤ supy[0,∞)∈C,yd[0,∞)∈C

sup(x0,θ,w[0,∞),yd[0,∞))∈W|y[0,∞),yd[0,∞)

Jγtf (7)

6

The inner supremum operator will be carried out first. It is the estimation design step, whichwill be presented in Section 4. Succinctly stated, in this step, we will calculate the maximum costthat is consistent with the given measurement waveform.

The outer supremum operator will be carried out second. It is the control design step, whichwill be discussed after the estimation design. In this step we design the control input u, whichguarantees the robustness of the closed-loop system.

This completes the formulation of the robust adaptive control problem. Next, we turn to theestimation and control design in the next section.

4 Estimation and Control Design

In this section, we present the estimation and control design for the adaptive control problemformulated. The first step is estimation design. In this step, the measurement waveform y[0,∞)

and the reference trajectory yd[0,∞) are assumed to be known. Since the control input is a causalfunction of y and yd, then it is also known. We apply the cost-to-come function methodology. Setfunction l in (5) to |ξ− ξ|2

Q, where ξ is the worst-case estimate for the expanded state ξ, ξ = [θ′ x′]′,

and Q is a matrix-valued weighting function to be introduced later. The cost function becomes

Jγtf =

∫ tf

0

(

|Cx(τ) + u(τ)C1θ + bp0u(τ)− yd(τ)|2 + |ξ(τ)− ξ(τ)|2Q(τ,y[0,τ ],yd[0,τ ])

−γ2|w(τ)|2)

dτ − γ2

∣

∣

∣

∣

[

θ′ − θ′0 x′0 − x′

0

]′∣

∣

∣

∣

2

Q0

(8)

By the cost-to-come function analysis of Pan and Basar (2000), we have

˙Σ = (A(u, y)− ζ2LC(u))Σ + Σ (A(u, y) − ζ2LC(u))′ + γ−2DD′ − γ−2ζ2LL′

−Σ (γ2ζ2 (C(u))′C(u)− (C(u))′C(u)− Q(t, y[0,t], yd[0,t]))Σ; Σ(0) = γ−2Q−10 (9a)

˙ξ = A(u, y)ξ − ΣQ(t, y[0,t], yd[0,t])ξc + Σ (C(u))′ (C(u)ξ − (yd − bp0u)) + Bu

+ζ2 (γ2Σ (C(u))′ + L) (y − bp0u− C(u)ξ); ξ(0) =[

θ′0 x′0

]′(9b)

where L is defined as L =[

01×σ L′]′

and ξc := ξ − ξ.

Then, the cost function (8) can be equivalently written as, when Σ exists on [0, tf ] and Σ(t) ispositive definite ∀t ∈ [0, tf ].

Jγtf = −|ξ(tf )− ξ(tf )|2(Σ(tf ))−1 +

∫ tf

0

(

(C(u(τ))ξ(τ) + bp0u(τ)− yd(τ))2

−γ2ζ2 (y(τ)− bp0u(τ)− C(u(τ))ξ(τ))2 + |ξ(τ)− ξ(τ)|2Q(τ,y[0,τ ],yd[0,τ ])

−γ2|w(τ)− w∗(ξ(τ), ξ(τ), Σ(τ), u(τ), w(τ))|2)

dτ (10)

where w∗ : IRn+σ × IRn+σ × S+(n+σ) × IR× IRq → IRq is the worst-case disturbance for estimationstep, given by

w∗(ξ, ξ, Σ, u, w) = ζ2E′ (y − bp0u− C(u)ξ) + γ−2 (Iq − ζ2E′E)D′Σ−1 (ξ − ξ) (11)

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The following steps of derivation for the estimator closely resembles that in Pan and Basar(2000). Partition Σ(t) as

Σ(t) =

[

Σ(t) Σ12(t)Σ21(t) Σ22(t)

]

(12)

where Σ(t) is σ × σ-dimensional, and introduce Φ(t) := Σ21(t) (Σ(t))−1 and Π(t) := γ2 (Σ22(t) −

Σ21(t) (Σ(t))−1Σ12(t)). Also partition ξ compatibly as[

θ′ x′]

.

For the boundedness of Σ, the weighting matrix Q in (8) admits the following structure

Q(t, y[0,t], yd[0,t]) = (Σ(t))−1

[

0σ×σ 0σ×n

0n×σ ∆(t)

]

(Σ(t))−1

+

[

ǫ(t) (C1u(t) + CΦ(t))′ (γ2ζ2 − 1) (C1u(t) + CΦ(t)) 0σ×n

0n×σ 0n×n

]

(13)

=

[

−(Φ(t))′

In

]

γ4(Π(t))−1∆(t) (Π(t))−1

[

−(Φ(t))′

In

]′

+

[

ǫ(t) (C1u(t) + CΦ(t))′ (γ2ζ2 − 1) (C1u(t) + CΦ(t)) 0σ×n

0n×σ 0n×n

]

where ∆(t) = γ−2β∆Π(t)+∆1, with β∆ ≥ 0 being a constant and ∆1 being an n×n positive-definitematrix, and ǫ is a scalar function defined by

ǫ(t) := K−1c sΣ(t) := Tr((Σ(t))−1)/Kc t ∈ [0,∞) (14a)

or ǫ(t) := 1 (14b)

and Kc ≥ γ2Tr(Q0) is a constant. Because of this structure for Q, we will later treat Q as a functionQ : IRn×σ × IR× IR→ IR(n+σ)×(n+σ), Q(Φ, u, sΣ)2.

Then, we have the following differential equation for Σ, Φ, and Π

Σ = −(1− ǫ)Σ (C1u + CΦ)′ (γ2ζ2 − 1) (C1u + CΦ)Σ; Σ(0) = γ−2Q−10 (15a)

Φ = (A− ζ2LC −ΠC ′ (ζ2 − γ−2)C)Φ + yA211 + u (A212 − ζ2LC1 −ΠC ′ (ζ2 − γ−2)C1);

Φ(0) = Φ0 (15b)

Π = (A− ζ2LC + β∆/2In)Π + Π(A− ζ2LC + β∆/2In)′ −ΠC ′ (ζ2 − γ−2)CΠ + DD′

−ζ2LL′ + γ2∆1; Π(0) = Π0 (15c)

The matrix Σ will play the role of worst-case covariance matrix of the parameter estimation error.The choice of Q guarantees that Σ is bounded from above and bounded from below away from 0as depicted in the following Lemma, whose proof is given in Pan and Basar (2000).

Lemma 1 Consider the dynamic equation (15a) for the covariance matrix Σ. Let Kc ≥ γ2Tr(Q0),Q0 > 0, and γ ≥ ζ−1. Then, the matrix Σ is upper and lower bounded as follows; for either choiceof ǫ(t) as in (14), whenever Σ is defined on [0, tf ], and Φ and u are continuous on [0, tf ].

K−1c Iσ ≤ Σ(t) ≤ Σ(0) = γ−2Q−1

0 ; γ2Tr(Q0) ≤ Tr((Σ(t))−1) ≤ Kc; ∀t ∈ [0, tf ]

2Π(t) will be set as a constant by Assumption 7.

8

To avoid the inversion of Σ on-line, we define sΣ(t) := Tr((Σ(t))−1), and its time derivative is givenby

sΣ = (γ2ζ2 − 1) (1 − ǫ) (C1u + CΦ) (C1u + CΦ)′; sΣ(0) = γ2Tr(Q0) (16)

Then, ǫ(t) = K−1c sΣ(t), which does not require the inversion of Σ(t), when ǫ is defined by (14a).

Based on Lemma 1, we note that γ ≥ ζ−1. This means that the quantity ζ−1 is the ultimatelower bound on the achievable performance level for the adaptive system, using the design methodproposed in this paper.

Assumption 6 If the matrix A− ζ2LC is Hurwitz, then the desired disturbance attenuation levelγ ≥ ζ−1. In case γ = ζ−1, choose β∆ ≥ 0 such that A− ζ2LC + β∆/2In is Hurwitz. If the matrixA− ζ2LC is not Hurwitz, then the desired disturbance attenuation level γ > ζ−1. ⋄

Assumption 7 The initial weighting matrix Π0 in (15c) is chosen as the unique positive definitesolution to the algebraic Riccati equation:

(A−ζ2LC+β∆/2In)Π+Π(A−ζ2LC+β∆/2In)′−ΠC ′ (ζ2−γ−2)CΠ+DD′−ζ2LL′+γ2∆1 =0 (17)

⋄

Then, we note that the unique positive-definite solution of (15c) is time-invariant and equal to theinitial value Π0, and the matrix Af := A− ζ2LC −ΠC ′ (ζ2 − γ−2)C is Hurwitz.

To guarantee the boundedness of estimated parameters without persistently exciting signals,we introduce soft projection design on the parameter estimate, which is based on the a prioriinformation on the bounds of the true value of the parameter vector θ, i.e., Assumption 4.

Define

ρ := infP (θ) | θ ∈ IRσ and bp0 + C1θ = 0

By Assumption 4, we have 1 < ρ ≤ ∞. Fix any ρo ∈ (1, ρ), and define the open set Θo := θ ∈IRσ | P (θ) < ρo. Our control design will guarantee the estimate θ lies in Θo, which immediatelyimplies b0 := bp0 + C1θ ≥ c0 > 0 (See Zeng and Pan 2003). Moreover, the convexity of P impliesthe following inequality

∂P

∂θ(θ) (θ − θ) < 0 ∀θ ∈ IRσ\Θ

Add the term −Σ[(

Pr(θ))′

01×n

]′to the right-hand-side of the dynamics (9b), where

Pr(θ) :=

exp

(

11−P (θ)

)

(ρo−P (θ))3

(

∂P∂θ (θ)

)′∀θ ∈ Θo\Θ

0σ×1 ∀θ ∈ Θ

(18)

=: pr(θ)

(

∂P

∂θ(θ)

)′

(19)

9

and Pr(θ) and pr(θ) are smooth functions on Θo. Then, we have

˙ξ = −Σ[

(Pr(θ))′ 01×n

]′+ A(u, y)ξ − ΣQξc + Σ (C(u))′ (C(u)ξ

−(yd − bp0u)) + Bu + ζ2 (γ2Σ (C(u))′ + L) (y − bp0u− C(u)ξ);

ξ(0) =[

θ′0 x′0

]′(20)

Partition ξ into [θ′ x′]′ to obtain the dynamics for these individual estimates. We summarizethe equations below.

(A−ζ2LC+β∆/2In)Π+Π(A−ζ2LC+β∆/2In)′−ΠC ′ (ζ2−γ−2)CΠ+DD′−ζ2LL′+γ2∆1 =0 (21a)

Σ = −(1− ǫ)Σ (C1u + CΦ)′ (γ2ζ2 − 1) (C1u + CΦ)Σ; Σ(0) = γ−2Q−10 (21b)

sΣ = (γ2ζ2 − 1) (1 − ǫ) (C1u + CΦ) (C1u + CΦ)′; sΣ(0) = γ2Tr(Q0) (21c)

Af = A− ζ2LC −ΠC ′C (ζ2 − γ−2) (21d)

Φ = AfΦ + yA211 + u (A212 − ζ2LC1 −ΠC ′ (ζ2 − γ−2)C1); Φ(0) = Φ0 (21e)

˙θ = −ΣPr(θ)−[

Σ ΣΦ′]

Qξc − (ΣC ′1u + ΣΦ′C ′) (yd − bp0u− C1θu− Cx)

+γ2ζ2 (ΣC ′1u + ΣΦ′C ′) (y − bp0u− C1θu− Cx); θ(0) = θ0 (21f)

˙x = −ΦΣPr(θ) + Ax + (yA211 + uA212)θ + Bu−[

ΦΣ γ−2Π + ΦΣΦ′]

Qξc

+ζ2 (γ2 (ΦΣC ′1u + γ−2ΠC ′ + ΦΣΦ′C ′) + L) (y − bp0u− C1θu− Cx)

−(ΦΣC ′1u + γ−2ΠC ′ + ΦΣΦ′C ′) (yd − bp0u− C1θu− Cx); x(0) = x0 (21g)

To simplify the controller structure, the dynamics for Φ can be implemented with 3n integratorsinstead of the σn integrators. First, we observe that the pair (Af , C) is observable. Then weintroduce the matrix

Mf :=[

An−1f pn · · · Afpn pn

]

(22)

where pn is an n-dimensional vector such that the pair (Af , pn) is controllable3, which implies thatMf is invertible. Then the following 3n-dimensional prefiltering system for y and u generates theΦ online:

η = Afη + pny; η(0) = η0 (23a)

λ = Afλ + pnu; λ(0) = λ0 (23b)

λo = Afλo; λo(0) = pn (23c)

Φ =[

An−1f η · · · Afη η

]

M−1f A211 +

[

An−1f λ · · · Afλ λ

]

M−1f (A212

−ζ2LC1 −ΠC ′ (ζ2 − γ−2)C1) +[

An−1f λo · · · Afλo λo

]

M−1f Φo0 (23d)

where η0 ∈ IRn, λ0 ∈ IRn, and Φo0 ∈ IRn×σ are such that (23d) holds at t = 0.Associated with the above identifier, introduce the value function, W : IRn+σ × IRn+σ ×

S+(n+σ) → IR,

W (ξ, ξ, Σ) = |ξ − ξ|2Σ−1

= |θ − θ|2Σ−1 + γ2|x− x− Φ (θ − θ)|2Π−1 (24)

3The existence of pn is proven in Zhao and Pan (2003).

10

whose time derivative along the dynamics of ξ, ξ, and Σ are given by

W (ξ, ξ, Σ, sΣ, yd, ξ, u, w) = −|Cx + C1θu− (yd − bp0u)|2 + |Cx + C1θu− (yd − bp0u)|2

−|ξ − ξ|2Q + |ξ − ξ|2Q + γ2|w|2 − γ2|w − w∗|2

−γ2ζ2|y − bp0u− Cx− C1θu|2 + 2(θ − θ)′Pr(θ) (25)

which holds as long as Σ > 0, θ ∈ Θo. We note that the last term in W is nonpositive, zero onthe set Θ and approaches −∞ as θ approaches the boundary of the set Θo, which guarantees theboundness of θ.

Then the cost function (8) can be equivalently written as, assuming Σ(t) > 0 and θ(t) ∈ Θo,∀t ∈ [0, tf ],

Jγtf = Jγtf + W (ξ(0), ξ(0), Σ(0)) −W (ξ(tf ), ξ(tf ), Σ(tf )) +

∫ tf

0Wdτ

= −|ξ(tf )− ξ(tf )|2(Σ(tf ))−1 +

∫ tf

0

(

(Cx(τ) + bp0u(τ) + C1θ(τ)u(τ) − yd(τ))2

−γ2ζ2 (y(τ)− Cx(τ)− bp0u(τ)− C1θ(τ)u(τ))2 + |ξc(τ)|2Q(Φ(τ),u(τ),sΣ(τ))

−γ2|w(τ)− w∗(ξ(τ), ξ(τ), Σ(τ), u(τ), w(τ))|2 + 2(θ − θ(τ))′Pr(θ(τ)))

dτ (26)

This completes the estimation design step. Next, we consider the control design step.Based on the inequality (7) in Section 3, the controller design is to guarantee that the following

supremum is less than or equal to zero for all measurement waveforms,

sup(x0,θ,w[0,∞),yd[0,∞))∈W

Jγtf

≤ supy[0,∞)∈C,yd[0,∞)∈C

sup(x0,θ,w[0,∞),yd[0,∞))∈W|y[0,∞),yd[0,∞)

Jγtf

≤ supy[0,∞)∈C,yd[0,∞)∈C

∫ tf

0

(

(Cx(τ) + (bp0 + C1θ(τ))u(τ) − yd(τ))2

+|ξc(τ)|2Q(Φ(τ),u(τ),sΣ(τ)) − γ2ζ2 (y(τ)− Cx(τ)− (bp0 + C1θ(τ))u(τ))2)

dτ

(27)

By inequality (27), we observe that the cost function is expressed in terms of signals that we canmeasure or construct. This is then a nonlinear H∞-optimal control problem under full informationmeasurements. Instead of considering y as the maximizing variable, we can equivalently deal withthe transformed variable:

v := ζ (y − Cx− (bp0 + C1θ)u)

A clear choice for control input u, and the worst-case estimate ξ, which gurantees that theright-hand-side of (27) is nonpositive, is

u := µ(θ, x, yd) =yd − Cx

C1θ + bp0

(28a)

ξ = ξ (28b)

11

where µ : Θo × IRn × IR → IR is smooth. The value function for the closed-loop system is simplyW , whose time derivative along solutions of the dynamics for ξ, ξ, and Σ is

W = −(Cx + C1θu + bp0u− yd)2 − |ξ − ξ|2Q + γ2|w|2 − γ2|v|2 − γ2|w −w∗|

2 + 2(θ − θ)′Pr(θ)

= −(Cx + C1θu + bp0u− yd)2 − |ξ − ξ|2Q + γ2|w|2 − γ2|w − wopt(ξ, ξ, yd, Σ)|2

+2(θ − θ)′Pr(θ) (29)

where the worst-case disturbance with respect to the value function W is given by,4 wopt : IRn+σ ×IRn+σ × IR× S+(n+σ) → IRq,

wopt(ξ, ξ, yd, Σ) = −ζ2E′[

µ(θ, x, yd)C1 C]

(ξ − ξ) + γ−2 (Iq − ζ2E′E)D′Σ−1 (ξ − ξ) (30)

which holds as long as Σ > 0 and θ ∈ Θo. Clearly, the closed-loop system is dissipative with storagefunction W and supply rate

−(Cx + C1θu + bp0u− yd)2 + γ2|w|2 (31)

This completes the control design step. We will turn to present the main results in the nextsection.

5 Main Result

With the estimation and control design of the previous section, the state of the closed-loop systemis given by

X :=[

θ′ x′ ←−Σ′

sΣ θ′ x′ −→Φ′]′

which belongs to the open set

D := X | Σ > 0, sΣ > 0, θ ∈ Θo

The dynamics for X are

X = F (X, yd) + G(X, yd)w = F (X, yd) + G(X, yd)Mw; X(0) = X0 (32)

where F and G are smooth mappings of D × IR, respectively; and the initial condition

X0 ∈ D0 := X0 ∈ D | θ ∈ Θ, θ0 ∈ Θ,Σ(0) = γ−2Q−10 > 0, sΣ(0) = γ2Tr(Q0) ≤ Kc

Since (29) holds, by Lemma 7 in Pan and Basar (2000), the value function W satisfies a Hamilton-Jacobi-Isaacs equation.

∂W

∂X(X)F (X, yd) +

1

4γ2

∂W

∂X(X)G(X, yd) (G(X, yd))

′(

∂W

∂X(X)

)′

+ Q(X, yd) = 0; ∀X ∈ D,∀yd ∈ IR (33)

4See Appendix B for derivations.

12

where Q : D × IR→ IR is smooth and given by

Q(X, yd) = |Cx + (bp0 + C1θ)µ(θ, x, yd)− yd|2 +

∣

∣

∣ξ − ξ∣

∣

∣

2

Q(Φ,µ(θ,x,yd),sΣ)− 2 (θ − θ)′Pr(θ) (34)

The closed-loop adaptive system possesses a strong robustness property, which will be statedprecisely in the following theorem.

Theorem 1 Consider the robust adaptive control problem formulated in Section 3 with Assump-tions 1-7 holding. The robust adaptive controller µ defined by (28a), with the optimal choice (28b)for ξ, achieves the following strong robustness properties for the closed-loop system.

1. Given cw ≥ 0, and cd ≥ 0, there exists a constant cc ≥ 0 and a compact set Θc ⊂ Θo, suchthat for any uncertainty (x0, θ, w[0,∞), yd[0,∞)) ∈ W with

|x0| ≤ cw; |yd(t)| ≤ cd; |w(t)| ≤ cw; ∀t ∈ [0,∞)

all closed-loop state variables x, x, θ, Σ, sΣ, and Φ are bounded as follows, ∀t ∈ [0,∞),

|x(t)| ≤ cc; |x(t)| ≤ cc; θ(t) ∈ Θc; |−−→Φ(t)| ≤ cc

K−1c Iσ ≤ Σ(t) ≤ γ−2Q−1

0 ; γ2Tr(Q0) ≤ sΣ(t) ≤ Kc

Therefore, there is a compact set S ⊆ D such that X(t) ∈ S ∀t ∈ [0,∞). Hence, there existsa constant cu ≥ 0 such that |u(t)| ≤ cu, |ξ(t)| ≤ cu, |η(t)| ≤ cu, |λ(t)| ≤ cu, and |λo(t)| ≤ cu.

2. The controller µ ∈ Mu achieves disturbance attenuation level γ for any uncertainty (x0, θ,w[0,∞), yd[0,∞)) ∈ W.

3. For any uncertainty (x0, θ, w[0,∞), yd[0,∞)) ∈ W with w[0,∞) ∈ L2 ∩ L∞, yd[0,∞) ∈ L∞, andyd[0,∞) being uniformly continuous, the output of the system, Cx+(C1θ+bp0)u, asymptoticallytracks the reference trajectory, yd, i.e.,

limt→+∞

(Cx(t) + (C1θ + bp0)u(t)− yd(t)) = 0

Proof For the first statement, fix cw ≥ 0 and cd ≥ 0, consider any uncertainty (x0, θ, w[0,∞),

yd[0,∞)) ∈ W that satisfies

|x0| ≤ cw; |w(t)| ≤ cw; |yd(t)| ≤ cd; ∀t ∈ [0,∞)

We define [0, Tf ) to be the maximal length interval on which the close-loop system (32) has asolution that lies in D. By Lemma 1, we have Σ and sΣ are upper and lower bounded as desiredon [0, Tf ).

Introduce the vector of variables

Xe :=[

θ′ (x− Φθ)′]′

13

and two nonnegative and continuous functions defined on IRn+σ

UM (Xe) := Kc|θ|2 + γ2|x− Φθ|2Π−1

Um(Xe) := γ2|θ|2Q0+ γ2|x− Φθ|2Π−1

then, we have, if we interpret W as W (t,Xe),

Um(Xe) ≤ W (t,Xe) ≤ UM (Xe), ∀(t,Xe) ∈ [0, Tf )× IRn+σ

Since Um(Xe) is continuous, nonnegative and radially unbounded, then ∀α ∈ IR, the set S1α :=Xe ∈ IRn+σ : Um(Xe) ≤ α is compact or empty. Since |w(t)| ≤ cw, ∀t ∈ [0,∞), we have thefollowing inequality for the derivative of W :

W = −|Cx + (C1θ + bp0)u− yd|2 − γ4|x− x− Φ (θ − θ)|2Π−1∆Π−1

−ǫ (γ2ζ2 − 1)|θ − θ|2(uC1+CΦ)′ (uC1+CΦ) + 2(θ − θ)′Pr(θ) + γ2|w|2 − γ2|w − wopt|2

≤ −γ4|x− x− Φ (θ − θ)|2Π−1∆Π−1 + 2(θ − θ)′Pr(θ) + γ2‖M‖22c2w (35)

Since −γ4|x − x − Φ (θ − θ)|2Π−1∆Π−1 + 2(θ − θ)′Pr(θ) will tend to −∞ when Xe approaches the

boundary of Θo × IRn, then there exists a compact set Ω1(cw) ⊂ Θo × IRn, such that W < 0 for∀Xe ∈ (Θo × IRn)\Ω1. To apply the Lemma 9 in Pan and Basar (2000), we make the followingsubstitutions,

Um →W1; UM →W2; W → V ; S1α → S1α; IRn+σ → D; c1 → η

S1c1 → S1η; Xe → ξ; [0, Tf )→ [t0, t1); Ω1 → K; n + σ → n

Note that Xe(t) ∈ Θo × IRn, ∀t ∈ [0, Tf ). Then, W (t,Xe(t)) ≤ c1, and Xe(t) belongs to a compactset S1c1 ⊆ IRn+σ, ∀t ∈ [0, Tf ), for some c1 ∈ IR. It follows that the signal Xe is bounded, namely,θ and x− Φθ are bounded.

Based on the derivative of x − Φθ, 5 there is particular linear combination of the componentsof x− Φθ, denoted by ηL,

d(x(t)− Φ(t)θ(t))

dt= Af (x(t)− Φ(t)θ(t))− γ−2ΠC ′y(t) + γ−2ΠC ′yd(t) + (D + γ−2ΠC ′E

−ζ2 (ΠC ′ + L)E)Mw(t) (36a)

ηL = TL (x− Φθ) (36b)

which is strictly minimum phase and has relative degree 1 with respect to y. Then by Lemma8, the composite system of (1) and (36) has relative degree 1 from input u to output ηL; and itmay serve as a reference systems in the application of Lemma 11 in Pan and Basar (2000) in thefollowing proof.

By Lemma 7, the system (36) with input y and output ηL may serve as reference system in theapplication of Lemma 11 of Pan and Basar (2000).

Based on the dynamics of η, the relative degree for each element of η is at least 1 with respectto the input y. Taking ηL as the output and y as the input of the reference system, we conclude η

5See Appendix B for detailed derivations.

14

is bounded by Lemma 11 of Pan and Basar (2000). Based on the dynamics of λ, the relative degreefor each element of λ is at least 1 with respect to the input u. Taking ηL as the output and u asthe input of the reference system, we conclude λ is also bounded.

Based on the dynamics of λo (23c), we have λo is bounded. Based on the formula for Φ, (23d),we have Φ is bounded. Since x−Φθ, Φ, and θ are bounded, we conclude that x is bounded. Definethe following equations to separate x into two part:

x = xu + xy

xu = Afxu +

(

B + A212θ − (C1θ + bp0)

(

ζ2L + ΠC ′ (ζ2 −1

γ2)

))

u

xy = Afxy +

(

A211θ + ζ2L + ΠC ′ (ζ2 −1

γ2)

)

y +

(

D −

(

ζ2L + ΠC ′ (ζ2 −1

γ2)

)

E

)

Mw

We observe that the relative degree for each element of xu is at least 1 with respect to the input u,and the relative degree for each element of xy is at least 1 with respect to the input y. Taking ηL

as the output and u as the input of the reference system, we conclude xu is bounded by Lemma 11in Pan and Basar (2000). Similarly, Taking ηL as the output and y as the input of the referencesystem, we obtain that xy is bounded. Then, we have x is bounded. Further, we have x is bounded,because x and x are bounded and x = x− x. Since b0 is bounded, and bounded away from 0, weconclude that u is bounded.

Next, we need to prove the existence of a compact set Θc ⊂ Θo such that θ(t) ∈ Θc, ∀t ∈ [0, Tf ).First introduce the function Υ : [0, Tf )× (Θo × IRn)→ IR by

Υ(t,Xe) := W (t,Xe) + (ρo − P (θ))−1P (θ)

We notice that, when θ approaches the boundary of Θo, P (θ) approaches ρo. Then Υ approaches∞ as Xe approaches the boundary of Θo × IRn. We introduce two nonnegative and continuousfunctions defined on Θo × IRn:

ΥM(Xe) := UM (Xe) + (ρo − P (θ))−1P (θ)

Υm(Xe) := Um(Xe) + (ρo − P (θ))−1P (θ)

Then, by the previous analysis, we have

Υm(Xe) ≤ Υ(t,Xe) ≤ ΥM (Xe), ∀(t,Xe) ∈ [0, Tf )× (Θo × IRn)

Note that the set S2α := Xe ∈ Θo × IRn | Υm(Xe) ≤ α is a compact set or an empty set,∀α ∈ IR. Then, we consider the derivative of Υ,

Υ = ˙W + ρo (ρo − P (θ))−2 ∂P

∂θ(θ) ˙θ(t)

≤ −|Cx + (C1θ + bp0)u− yd|2 − γ4|x− x− Φ (θ − θ)|2Π−1∆Π−1 + 2(θ − θ)′Pr(θ)

−ǫ (γ2ζ2 − 1)|θ − θ|2(uC1+CΦ)′ (uC1+CΦ) + γ2‖M‖22c2w + ρo (ρo − P (θ))−2

(

−∂P

∂θ(θ)ΣPr(θ)

−∂P

∂θ(θ) (uΣC ′

1 + ΣΦ′C ′)(

yd −Cx− (C1θ + bp0)u)

+∂P

∂θ(θ)γ2ζ2 (uΣC ′

1 + ΣΦ′C ′)

·(

y − Cx− (C1θ + bp0)u)

)

15

which hold as long as Σ > 0 and θ ∈ Θo. We observe the following facts.

yd − Cx− u (C1θ + bp0) = 0

∂P

∂θ(θ)γ2ζ2 (uΣC ′

1 + ΣΦ′C ′) (y − Cx− (C1θ + bp0)u)

≤

∣

∣

∣

∣

∣

(

∂P

∂θ(θ)

)′

(ρo − P (θ))−1

∣

∣

∣

∣

∣

2

+

∣

∣

∣

∣

∣

(ρo − P (θ))

2γ2ζ2 (uΣC ′

1 + ΣΦ′C ′)(

y − Cx− (C1θ + bp0)u)

∣

∣

∣

∣

∣

2

Then,

Υ ≤ −γ4|x− x− Φ (θ − θ)|2Π−1∆Π−1 + 2(θ − θ)′Pr(θ) + γ2‖M‖22c2w

+ρo (ρo − P (θ))−4

−∂P

∂θ(θ)ΣPr(θ) (ρo − P (θ))2 +

∣

∣

∣

∣

∣

(

∂P

∂θ(θ)

)′∣

∣

∣

∣

∣

2

+ρo

4

∣

∣

∣γ2ζ2 (uΣC ′1 + ΣΦ′C ′)

(

y − Cx− (C1θ + bp0)u)∣

∣

∣

2

≤ −γ4|x− x− Φ (θ − θ)|2Π−1∆Π−1 + 2(θ − θ)′Pr(θ) + γ2‖M‖22c2w

−ρo

∣

∣

∣

∣

∣

(

∂P

∂θ(θ)

)′∣

∣

∣

∣

∣

2

(ρo − P (θ))−4(

K−1c pr(θ) (ρo − P (θ))2 − 1

)

+ c

where c ≥ 0 is a constant, which holds as long as Σ > 0 and θ ∈ Θo. Since Υ will tend to −∞when Xe approaches the boundary of Θo× IRn, then there exists a compact set Ω2(cw) ⊂ Θo× IRn,such that ∀Xe ∈ (Θo × IRn)\Ω2, Υ < 0. To use the Lemma 9 in Pan and Basar (2000), we makethe following substitutions,

Υm →W1; ΥM →W2; Υ→ V ; S2α → S1α; Θo × IRn → D; c2 → η

S2c2 → S1η; Xe → ξ; [0, Tf )→ [t0, t1); Ω2 → K; n + σ → n

Then, Υ(t,Xe(t)) ≤ c2, and Xe(t) belongs to the compact set S2c2 ⊂ Θo × IRn, ∀t ∈ [0, Tf ), forsome c2 ∈ IR. Moreover, there exists a compact set Θc ⊂ Θo, such that θ(t) ∈ Θc, ∀t ∈ [0, Tf ).

Then we can conclude that there exists a compact set S ⊂ D, such that X(t) ∈ S, ∀t ∈ [0, Tf ).Therefore, it follows that Tf =∞ and X(t) ∈ S, ∀t ∈ [0,∞).

For the second statement, we fix any uncertainty (x0, θ, w[0,∞), yd[0,∞)) ∈ W. Since yd and ware continuous functions, then, for any tf > 0, there exist constants cd ≥ 0 and cw ≥ 0 such that|yd(t)| ≤ cd and |w(t)| ≤ cw, ∀t ∈ [0, tf ]. By the first statement, there exists a solution X : [0, tf ]→D for the closed-loop system (32). Then, by causality of system (32) and its smoothness, it impliesthat the closed-loop system (32) admits a unique solution on [0,∞). We set

l(t, θ, x, y[0,t], yd[0,t]) := γ4|x− x(t)− Φ(t) (θ − θ(t))|2Π−1∆Π−1 − 2(θ − θ(t))′Pr(θ(t))

+ǫ (γ2ζ2 − 1)|θ − θ(t)|2(u(t)C1+CΦ(t))′ (u(t)C1+CΦ(t))

Then, we have

sup(x0,θ,w[0,∞),yd[0,∞))∈W

∫ tf

0

(

(Cx(τ) + (C1θ + bp0)u(τ)− yd(τ))2 + l(τ, θ, x(τ), y[0,τ ], yd[0,τ ])

16

−γ2|w(τ)|2)

dτ − γ2

∣

∣

∣

∣

[

θ′ − θ′0 x′0 − x′

0

]′∣

∣

∣

∣

2

Q0

≤ sup(x0,θ,w[0,∞),yd[0,∞))∈W

∫ tf

0

(

(Cx(τ) + (C1θ + bp0)u(τ)− yd(τ))2 + l(τ, θ, x(τ), y[0,τ ], yd[0,τ ])

−γ2|w(τ)|2)

dτ − γ2

∣

∣

∣

∣

[

θ′ − θ′0 x′0 − x′

0

]′∣

∣

∣

∣

2

Q0

+

∫ tf

0Wdτ −W (X(tf )) + W (X(0))

≤ −W (X(tf )) ≤ 0

This establishes the second statement.For the last statement, we consider the following inequality,

∫ tf

0Wdτ ≤

∫ tf

0(−|Cx(τ) + (C1θ + bp0)u(τ)− yd(τ)|2 + γ2|Mw(τ)|2)dτ ; ∀tf ≥ 0

Then,

∫ ∞

0|Cx(τ) + (C1θ + bp0)u(τ)− yd(τ)|2dτ ≤

∫ ∞

0

(

γ2|Mw(τ)|2)

dτ + W (X(0)) < +∞

By Lemma 2, X is uniformly continuous. Since yd − Cx is uniformly continuous and bounded,C1θ + bp0 is uniformly continuous and bounded away from 0, then u is uniformly continuous byLemmas 3 and 4. Then, Cx + (C1θ + bp0)u− yd is uniformly continuous. By Lemma 5, we have

limt→+∞

(Cx(t) + (C1θ + bp0)u(t)− yd(t)) = 0

This completes the proof of the theorem. 2

6 Example

In this section, we present one example to illustrate the main results of this paper. The designwas carried out using MATLABTM symbolic computation tools, and the closed-loop system wassimulated using SIMULINKTM.

Consider the following circuit problem in Figure 1, where vi is the input voltage source; vo isthe measured output; ve is an unknown sinusoidal voltage source; vw is an unmeasured exogenousvoltage source. The objective is to achieve asymptotic tracking of vo−vw to the reference trajectoryyd.

The equations that describe the circuit are obtained as

y = ve + b0u + w

where u = vi, y = vo, w = vw, and b0 is the ratio of vt to vi, which is assumed to be unknownand belong to the interval [0.1, 1], whose true value is set to 1 for illustration purposes. ve can bemodeled as the output of a second-order linear system as the following,

˙x1 = x2; x1(0) = 1

˙x2 = θ2x1; x2(0) = 1

17

Ve

− +

+

−

−

+

Vt

0

+ −

Vw

Vi

−

+

Vo

b

Figure 1: Diagram of the circuit.

where θ2 is assumed to be unknown and belong to the interval [−4, 0], whose true value is set to−4 for illustration purposes, and the initial conditions are set for illustration purposes. Note thatthe true system satisfies the Assumptions 1 and 2.

To normalize the parameters, we set bp0 = 0.55, and define θ = [θ1 θ2 θ3]′, where θ1 = (b0 −

bp0)/0.45, θ2 = (θ2+2)/2 and θ3 = θ1θ2. Then, the true value for parameter vector θ is [1 −1 −1]′,and θ1, θ2, and θ3 all belong to the interval [−1, 1].

Introduce the state transformation x = x, and disturbance transformations, w =[

1 θ2/3]′

w,

we obtain the design model for the adaptive controller

x =

[

0 1−2 0

]

x +

(

y

[

0 0 00 2 0

]

+ u

[

0 0 00 −1.1 −0.9

])

θ +

[

0 00 −6

]

w

y =[

1 0]

x + 0.55u +[

0.45 0 0]

θu +[

1 0]

w

For the adaptive control design, the ultimate lower bound for the achievable disturbance atten-uation level is 1 with respect to w. We set the desired disturbance attenuation level γ = 10.

The convex function P (θ) is chosen as

P (θ) = 0.8θ21 + 0.1(θ2

2 + θ23)

For other design and simulation parameters, we select

x0 =[

0 0]′

; θ0 =[

0 0 0]′

; Q0 = 0.005I3; ∆1 = I2; pn =[

0 1]′

Φ0 = 02×3; Kc = 1.5; ρo = 1.15; β∆ = 0; ǫ = K−1c sΣ; η0 = 02×1; λ0 = 02×1

Then, we obtain

Π =

[

10.998 9.8739.873 129.49

]

Set the reference trajectory, yd =√

|sin(3t)|, which is uniformly continuous on [0,∞). Wepresent two sets of simulation results for this example. The first set is to illustrate the regulatory

18

behaviour of the adaptive controller. We set w(t) ≡ 0. The result are shown in Figure 2. We observethat the parameter estimates converge to their true values, and the tracking error converges to 0.The transient of the system is well-behaved, and the control magnitude is upper bounded by 2.2.The convergence rate is clearly exponential. When t = 175, the magnitude of the tracking errorreduces to 10−14, and magnitude of the parameter estimation error reduces to 10−13.

The second set of simulation results is to illustrate the robustness property of the adaptivecontroller. We set w(t) = 0.4 sin(t). The simulation results are shown in the Figure 3. We observethat the parameter estimates no longer converge to the true values, but the output tracking errorsatisfies the targeted attenuation level as desired according to Figure 3(f). The transient of thesystem is well-behaved which lasted about 5 seconds, and the control magnitude is upper boundedby 2.3.

7 Conclusions

In this paper, we studied the adaptive control design for tracking and disturbance attenuation forSISO linear systems with zero relative degree under noisy output measurements. We assume thatthe linear system has a known upper bound of the dynamic order, has a strictly minimum phasetransfer function with known relative degree 0. We allow the system to be uncontrollable, as longas the uncontrollable part is stable in the sense of Lyapunov and those uncontrollable modes on thejω-axis are uncontrollable from the disturbance input. Under these assumptions, the system may betransformed into the design model, which is linear in all of the unknown quantities. The objectivesof the control design are to make the noiseless output of the system to asymptotically track thereference trajectory, and guarantee the boundedness of all closed-loop signals, while rejecting theuncertainties in the system, which comprises the initial state of the system, the true value of theunknown parameter vector, the waveform of the exogenous disturbance input, and the referencetrajectory. We use H∞-optimal control formulation and game theoretic approach to derive therobust adaptive controller. We treat the unknown parameter vector as part of the expanded statevector, and formulate this adaptive control problem as a nonlinear H∞-optimal control problemwith imperfect state measurements. For the design model, we assume that the measurement channelis noisy, such that the estimation step is a nonsingular optimization problem. We further assumethat the unknown parameter vector belongs to a convex compact set characterized by a knownsmooth nonnegative radially unbounded and strictly convex function P (θ). Furthermore, for anyparameter vector belonging to the set, the corresponding high frequency gain is never zero. Then,the cost-to-come function analysis is applied to derive the worst-case identifier and state estimator,which have a finite-dimensional structure. Using a priori information on the parameter vector, asmooth soft projection algorithm is applied in the estimation step, which relieves the persistencyof excitation condition for the closed-loop system. Then, the closed-loop system is robust withor without the persistently exciting signals. After the estimation step is completed, the originalproblem becomes a nonlinear H∞-optimal control problem under full-information measurements.Then, the controller can be obtained directly from the cost function in one step. The controller thenachieve the desired disturbance attenuation level, with the ultimate lower bound of the attenuationlevel being the noise intensity in the measurement channel. It guarantees the boundedness of allclosed-loop signals and achieves asymptotic tracking of uniformly continuous bounded referencetrajectories when the disturbance is of finite energy and bounded. Because of the assumptionswe made on the unknown system, the adaptive controller can asymptotically cancel out the effect

19

0 5 10 15 20 25 30−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

(a)

150 155 160 165 170 175 180 185 190 195 200−4

−3

−2

−1

0

1

2

3x 10

−13

(b)

0 5 10 15 20 25 30−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

(c)

150 155 160 165 170 175 180 185 190 195 200−5

−4

−3

−2

−1

0

1

2

3x 10

−12

(d)

0 5 10 15 20 25 30−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

(e)

0 10 20 30 40 50 60 70 80 90 1000

0.01

0.02

0.03

0.04

0.05

0.06

(f)

Figure 2: System response under reference trajectory yd(t) =√

|sin(3t)| and w(t) = 0.(a) Tracking error; (b) Tracking error (long term); (c) Parameter estimates; solid line for θ1; dashline for θ2; and dash dot line for θ3; (d) Parameter estimation error (long term); solid line forθ1 − θ1; dash line for θ2 − θ2; and dash dot line for θ3 − θ3; (e) Control input; (f) A performanceindex

∫ t0 ((Cx + b0u− yd)

2 − γ2|w|2)dτ .

20

0 5 10 15 20 25 30 35 40 45 50−1

−0.5

0

0.5

1

1.5

(a)

900 910 920 930 940 950 960 970 980 990 1000−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

(b)

0 5 10 15 20 25 30 35 40 45 50−1.5

−1

−0.5

0

0.5

1

1.5

(c)

900 910 920 930 940 950 960 970 980 990 1000−1.5

−1

−0.5

0

0.5

1

1.5

(d)

0 5 10 15 20 25 30 35 40 45 50−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

(e)

0 100 200 300 400 500 600 700 800 900 1000−9000

−8000

−7000

−6000

−5000

−4000

−3000

−2000

−1000

0

(f)

Figure 3: System response under reference trajectory yd(t) =√

|sin(3t)| and w(t) = 0.4sin(t).(a) Tracking error; (b) Tracking error (long term); (c) Parameter estimates; solid line for θ1; dashline for θ2; and dash dot line for θ3; (d) Parameter estimates (long term); solid line for θ1; dash linefor θ2; and dash dot line for θ3; (e) Control input; (f) A performance index

∫ t0 ((Cx + b0u− yd)

2 −γ2|w|2)dτ .

21

of exogenous sinusoidal inputs with unknown magnitudes, phases, and frequencies, as long aswe extend our system model to incorporate the knowledge of the existence of such sinusoidalinputs. This property of our adaptive controller has significant impact on active noise cancellationapplications. This feature is illustrated by a numerical example, which corroborates all of ourtheoretical findings.

Future research directions that are of interest are described as follows. One direction lies in thegeneralization of the results to nonlinear systems. Another fruitful direction lies in the extensionof the results to multiple-input and multiple-output systems. The class of MIMO systems understudy involves two subsystems, S1, and S2, interconnected to each other, where the connection isserial. Preliminary results have been obtained for this class of MIMO systems.

A Seven Lemmas

Lemma 2 Assume f : [0,∞)→ IR is differentiable in t, and satisfies

|f (1)(t)| ≤M <∞; ∀t ∈ [0,∞)

for some M ∈ IR, M ≥ 0. Then, f is uniformly continuous on [0,∞).

Proof Given a ǫ > 0, we take δ(ǫ) = ǫM+1 . Then, if x, y ∈ [0,∞) and |x − y| ≤ δ(ǫ), from the

Mean Value Theorem, we have

|f(x)− f(y)| = |f (1)(ξ)| · |x− y| ≤M · δ(ǫ) = M ·ǫ

M + 1< ǫ

This completes the proof of this Lemma. 2

Lemma 3 Assume g : [0,∞) → IR and h : [0,∞) → IR are uniformly continuous. If ∃ M > 0,such that |g(t)| ≤M and |h(t)| ≤M , ∀t ∈ [0,∞), then g · h is uniformly continuous on [0,∞).

Proof Since g is uniformly continuous, ∀ǫ > 0, ∃δg(ǫ) > 0 such that |g(x) − g(y)| < ǫ,∀x,y ∈ [0,∞) with |x − y| < δg(ǫ); since h is uniformly continuous, ∀ǫ > 0, ∃δh(ǫ) > 0 suchthat |h(x) − h(y)| < ǫ, ∀x,y ∈ [0,∞) with |x − y| < δh(ǫ). Then, given an ǫ > 0, we takeδ(ǫ) = minδh( ǫ

2M ), δg(ǫ

2M ), and we have

|g(x) · h(x)− g(y) · h(y)| = |g(x) · h(x)− g(x) · h(y) + g(x) · h(y)− g(y) · h(y)|

≤ |g(x)| · |h(x)− h(y)| + |h(y)| · |g(x)− g(y)|

≤ M |h(x) − h(y)|+ M |g(x) − g(y)|

< ǫ; ∀x, y ∈ [0,∞) with |x− y| < δ(ǫ)

This completes the proof of this Lemma. 2

Lemma 4 Assume h : [0,∞) → IR is uniformly continuous. If ∃ M > 0, such that |h(t)| ≥ M ,∀t ∈ [0,∞), then 1

h(t) is uniformly continuous on [0,∞).

22

Proof Since h is uniformly continuous, ∀ǫ > 0, ∃δh(ǫ) > 0 such that |h(x) − h(y)| < ǫ, ∀x,y ∈[0,∞) with |x− y| < δh(ǫ). Then, given an ǫ > 0, we take δ(ǫ) = δh( ǫ

M2 ), and we have

∣

∣

∣

∣

1

h(x)−

1

h(y)

∣

∣

∣

∣

=

∣

∣

∣

∣

h(x)− h(y)

h(x) · h(y)

∣

∣

∣

∣

≤ M2|h(x)− h(y)|

< ǫ; ∀x, y ∈ [0,∞) with |x− y| < δ(ǫ)

This completes the proof of this Lemma. 2

Lemma 5 If f : [0,∞)→ IR is uniformly continuous in t, and satisfies:∫ ∞

0|f(τ)|pdτ <∞; p ∈ (0,∞)

then limt→∞ f(t) = 0.

Proof We prove it by contradiction. Suppose limt→∞ f(t) 6= 0, then ∃ǫ0 > 0, ∀T > 0 existst0 > T such that |f(t0)| > ǫ0.

From the definition of uniformly continuous, take ǫ = ǫ02 > 0, there exists δ > 0 such that ∀ t1,

t2 ∈ [0,∞) with |t1 − t2| < δ, we have

|f(t1)− f(t2)| ≤ ǫ

For the above t0, it follows |f(t)| ≥ ǫ02 when t ∈ [t0 − δ, t0 + δ] ∩ [0,∞).

First we take T1 = δ, then we may obtain t1 > T1 and a time interval [t1 − δ, t1 + δ], such that|f(t)| ≥ ǫ0

2 , ∀t ∈ [t1 − δ, t1 + δ]. Next, take T2 = t1 + 2δ, then we obtain a t2 > T2 and a timeinterval [t2−δ, t2 +δ], such that |f(t)| ≥ ǫ0

2 , ∀t ∈ [t2−δ, t2 +δ]. Continue such procedure, we obtaina sequence tn

∞n=1 and corresponding disjoint time intervals [tn − δ, tn + δ], n = 1, 2 · · ·. Then

∫ ∞

0|f(τ)|pdτ ≥

∫ t1+δ

t1−δ|f(τ)|pdτ +

∫ t2+δ

t2−δ|f(τ)|pdτ + · · · +

∫ tn+δ

tn−δ|f(τ)|pdτ + · · ·

≥

∫ t1+δ

t1−δ|1

2ǫ0|

pdτ +

∫ t2+δ

t2−δ|1

2ǫ0|

pdτ + · · ·+

∫ tn+δ

tn−δ|1

2ǫ0|

pdτ + · · ·

≥ (1

2ǫ0)

p2δN ∀N ∈ IN

which contradicts∫∞0 |f(τ)|pdτ <∞.

Therefore limt→∞ f(t) = 0. 2

Lemma 6 Consider the system (1) under Assumption 2. Assume n < n. Then, we can always addadditional (n− n)-dimensional dynamics, such that the expanded system is of order n and satisfiesAssumption 2. Furthermore, the expanded system admits the same mapping from (x0, u[0,∞), w[0,∞))to y[0,∞) as system (1).

Proof We will discuss two exhaustive and mutually exclusive cases: Case 1: n = 0; Case 2:n > 0. First, consider Case 1: n = 0. We expand the system as following,

˙x =˜A22

˜x; ˜x(0) = 0 (37a)

y =˜C ˜x + b0u + Ew (37b)

23

where ˜x is the n-dimensional state vector;˜A22 is Hurwitz, and (

˜A22,

˜C) is observable. We notice

that, since (37a) is a stable system with zero initial conditions, the trajectory ˜x will always bezeros. Then, the trajectory y will remains the same for the expanded system (37) and the originalsystem (1) when the trajectories for u and w remain unchanged. Clearly, (37) satisfies Assumption2.

Next, we consider Case 2: n > 0. Since system (1) satisfies Assumption 2, we assume it is givenin observer canonical form. Then, we expand the original system to n dimensional by adding the˜x dynamics, which is (n− n)-dimensional,

[

˙x˙x

]

=

A˜A12

0˜A22

[

x˜x

]

+

[

B0

]

u +

[

D0

]

w;

[

x(0)˜x(0)

]

=

[

x0

0

]

(38a)

y =[

C 0]

[

x˜x

]

+ b0u + Ew (38b)

where˜A12 =

[

0 −˜C

′ ]′,

˜C is a row vector,

˜A22 is Hurwitz, and the pair (

˜A22,

˜C) is observable.

We observe that the trajectories x and y in (38) are as same as those in (1), when x0, u and wremain unchanged.

To check the observability of (38), fix any λ ∈ C, we observe

C 0

λI − A −˜A12

0 λI −˜A22

=

1 0 · · · 0∗ −1 · · · 0...

.... . .

...∗ ∗ · · · −1

0n×(n−n)

∗˜C

λI −˜A22

(39)

where ∗ stands for arbitrary scalars or matrices. Since the pair (˜A22,

˜C) is observable, the matrix

(39) is of full column rank ∀λ ∈ C. Then, system (38) is observable.Since the expanded dynamics, ˜x, is uncontrollable from u, the transfer function of (38) is the

same one of (1), and it is strictly minimum phase with relative degree 0.To check others conditions of Assumption 2, we follow the derivation below,First, we transform the system (1) into Jordan canonical form representation. We choose an

invertible complex matrix TJ . The transformation x = TJz will transform the dynamics (1) into

z =

J1

. . .

Jk

z +

BJ1...

BJk

u +

DJ1...

DJk

w (40a)

y =[

CJ1 · · · CJk

]

z + b0u + Ew (40b)

where Ji is a Jordan block associated with eigenvalue λi, i = 1, · · · , k, k ∈ IN; in the columnvectors of TJ , complex conjugate pairs appear together. Since a single-output Jordan canonicalform state space representation is observable only if there is only one Jordan block associated with

each distinct eigenvalue, then i 6= j ⇒ λi 6= λj , ∀i, j ∈ 1, · · · , k. Partition TJ =[

TJ1 · · · TJk

]

24

accordingly. For any i, j ∈ 1, · · · , k, λi is the complex conjugate of λj , implies that TJi is thecomplex conjugate of TJj . By Assumption 2, if λi is any uncontrollable mode from u, then the lastrow of Bi is zero; in addition, if Re(λi) = 0, then the last row of Di is zero. When Re(λi) = 0and λi is an uncontrollable mode, the second to last row of BJi must be nonzero, otherwise, theuncontrollable part of (1) with respect to u will contain a Jordan block associated with λi of order atleast 2, which is not stable in the sense of Lyapunov. Next, we separate the above Jordan Canonicalform into controllable and uncontrollable parts with respect to u, and we denote them as zc andzc, respectively. Then, we separate the uncontrollable part zc into Re(λ) < 0 and Re(λ) = 0 parts,and we denote them as z3 and z4, respectively. This corresponds to the existence of a permutationmatrix Tc, and the transformation z = Tc[z

′c z′3 z′4]

′ will transform (40) into

zc

z3

z4

=

Ac Aa

0(n−n1)×n1

A33 0

0 A44

zc

z3

z4

+

Bc

0

0

u +

Dc

D3

0

w (41a)

y =[

Cc C3 C4

]

zc

z3

z4

+ b0u + Ew (41b)

where zc is the n1-dimensional controllable part; z3 and z4 are n2- and n3-dimensional, respectively,and compose the uncontrollable part; ni ∈ IN ∪ 0, i = 1, 2, 3. The matrix Tc can be taken such

that Ac =

Jc1

. . .

Jck1

, A33 =

J31

. . .

J3k2

, and A44 =

J41

. . .

J4k3

, where

0 ≤ k1, k2, k3 ≤ k; Jci, i = 1, · · · , k1, J3j , j = 1, · · · , k2, J4l, l = 1, · · · , k3, are Jordan blocksassociated with eigenvalues λci, i = 1, · · · , k1, λ3j , j = 1, · · · , k2, λ4l, l = 1, · · · , k3, respectively.λci 6= λcj , if i 6= j; λ3i 6= λ3j , if i 6= j; and λ4i 6= λ4j , if i 6= j. Furthermore, ∀i ∈ 1, · · · , k3,J4i is of order 1, and λ4i has zero real part; ∀i ∈ 1, · · · , k2, λ3i has negative real part. Complexconjugate pairs appear together in each of the set λc1, · · · , λck1, λ31, · · · , λ3k2, λ41, · · · , λ4k3.

Partition TJTc =[

TJc1 · · · TJck1 TJ31 · · · TJ3k2 TJ41 · · · TJ4k3

]

accordingly. For any i,

j ∈ 1, · · · , k1, λci is the complex conjugate of λcj, implies TJci is the complex conjugate of TJcj.Similar statements can be made for λ3i and λ3j , and also for λ4i and λ4j .

It is easy to see that, we can choose three complex invertible matrices TJc, TJ3, TJ4, such thatTJ = block diagonal (TJc, TJ3, TJ4), whose partitioning is the compatible with that of [z′c z′3 z′4]

′,and T1 = TJTcTJ is a real invertible matrix. Then, the coordinate transformation x = T1[x

′c

¯x′3

¯x′4]′

will transform (1) into

xc

˙x3˙x4

=

Ac Ac,34

0(n−n1)×n1

A33 0

0 A44

xc¯x3¯x4

+

Bc

0

0

u +

Dc

D3

0

w (42a)

y =[

Cc C3 C4

]

xc¯x3¯x4

+ b0u + Ew (42b)

where all matrices are real, the triple (Ac, Bc, Cc) is controllable and observable, A33 is Hurwitz,A44 has all eigenvalues lying on the jω-axis and all Jordan blocks being of order 1.

25

Then, (38) admits the following state space representation.

xc

˙x3˙x4

˙x

=

Ac Ac,34˜Ac

0(n−n1)×n1

A33 0

0 A44

˜A3˜A4

0 0˜A

xc¯x3¯x4

˜x

+

Bc

0

0

0

u +

Dc

D3

0

0

w

y =[

Cc C3 C4 0]

xc¯x3¯x4

˜x

+ b0u + Ew

We define a coordinate transformation ˜x4 =˜T ˜x + ¯x4, such that

˜T

˜A − A44

˜T +

˜A4 = 0. Then,

the system (38) admits the following state space representation,

xc

˙x3˙x4

=

Ac Ac,34

0˜A33 0

0 A44

xc

˜x3˜x4

+

Bc

0

0

u +

Dc

˜D3

0

w (43a)

y =[

Cc˜C3 C4

]

xc˜x3˜x4

+ b0u + Ew (43b)

where ˜x3 =

[

¯x3˜x

]

, and˜A33 =

A33˜A3

0˜A

. Since A33 and˜A are Hurwitz, then

˜A33 is Hurwitz.

Based on (43), clearly the expanded system satisfies Assumption 2.This complete the proof of the lemma. 2

Lemma 7 Consider a linear time-invariant system with the following state space representation

x = Ax + Bu + Dw (44a)

y = Cx + Ew (44b)

where x is the n-dimensional state vector, n ∈ IN; w is q-dimensional disturbance input, q ∈ IN; uis the scalar input, and y is the scalar output; A is Hurwitz; and the transfer function from u to yis H(s) = C (sIn − A)−1B, which is strictly minimum phase and has relative degree r ∈ IN; Thenthe system (44) admits state space representation (70) in Pan and Basar (2000), which satisfiesAssumption 10 of Pan and Basar (2000), under some real invertible coordinate transformation,with index r1 = r.

Proof First, we introduce a linear state diffeomorphism for (44) to decompose the system intoobservable and unobservable part. We choose a real invertible matrix To, then the equivalencetransformation x = To[z

′o z′o]

′ will transform the dynamics of x into[

zo

zo

]

=

[

Ao 0

Aoo Ao

] [

zo

zo

]

+

[

Bo

Bo

]

u +

[

Do

Do

]

w (45a)

y =[

Co 0]

[

zo

zo

]

+ Ew (45b)

26

where zo is an n-dimensional vector, n ∈ IN and Ao is Hurwitz.Second, we introduce a linear state diffeomorphism for the zo subsystem of (45) to decompose

it into controllable and uncontrollable parts. We choose an invertible real matrix Tc, then thetransformation zo = Tc[z

′co z′co]

′ will transform the dynamics of zo into

[

zco

zco

]

=

[

Aco Acco

0 Aco

] [

zco

zco

]

+

[

Bco

0

]

u +

[

Dco

Dco

]

w (46a)

y =[

Cco Cco

]

[

zco

zco

]

+ Ew (46b)

where zco is n1-dimensional state vector, n1 ∈ IN, and the matrix Aco is Hurwitz. We note thatall matrices are real, and the triple (Aco, Bco, Cco) is controllable and observable, and the transferfunction of (44) from u to y is

H(s) = Cco (sIn1 −Aco)−1Bco =:

b0sn1−r + b1s

n1−r−1 + · · · + bn1−r

sn1 + a1sn1−1 + · · ·+ an1

with b0 6= 0.By Lemma 12 in Pan and Basar (2000), there exists a real invertible matrix T , such that

[

T−1 0n1×1

01×n1 1

] [

Aco Bco

Cco 0

] [

T 0n1×1

01×n1 1

]

=

A11 A12 B1

A21 A22 0

C1 0 0

where

A11 =

a1...

ar−1

Ir−1

ar 01×(r−1)

; A12 =

0

1

n1 > r

[ ] n1 = r

(47a)

A21 =

ar+1...

an1

0(n1−r)×(r−1)

; A22 =

−b1/b0...

−bn1−r−1/b0

In1−r−1

−bn1−r/b0 01×(n1−r−1)

n1 > r

[ ] n1 = r

(47b)

B1 =[

01×(r−1) b0

]′; C1 =

[

1 01×(r−1)

]

(47c)

Since H(s) is strictly minimum phase, A22 is Hurwitz.Hence the desired state transformation matrix is

T = To

[

Tc 0

0 In−n

]

T 0 0

0 In−n1 0

0 0 In−n

27

Then, the coordinate transformation x = T [z′1 z′2 z′co z′o]′ will transform system (44) into

z1

z2

zco

zo

=

A11 A12

A21 A22A12,3 0

0 Aco 0

Ao,12 Ao,3 Ao

z1

z2

zco

zo

+

B1

0

0

Bo

u +

D1

D2

Dco

Do

w (48a)

y =[

C1 0 Cco 0]

z1

z2

zco

zo

+ Ew (48b)

Let x1 = z1, x2 = z2, x3 = zco and x5 = zo. Then, in the [x′1 x′

2 x′3 x′

5]′ coordinate, the system (48)

admits the state space representation (70) as in Pan and Basar (2000), and satisfies Assumption10 in Pan and Basar (2000) with r1 = r.

This complete the proof of the lemma. 2

The next lemma establishes a result that system (1) composed with another linear system maystill satisfy Assumption 10 of (Pan and Basar 2000).

Lemma 8 Consider system (1) under Assumption 2 and a second linear system, which admits thefollowing state space representation

η = A2η + B2y + D2w (49a)

ηL = C2η + E2w (49b)

where η is the n2-dimensional state vector, n2 ∈ IN; ηL is the scalar output; y and w are thesame signals as in (1); A2 is Hurwitz; and the transfer function of (49) from y to ηL is H2(s) =C2 (sIn2 − A2)

−1B2, which is strictly minimum phase and has relative degree r2 ∈ IN. By Remark1, W.L.O.G., assume n = n ∈ IN.

Then the composite system of (1) and (49), which is given by the state space representation[

η˙x

]

=

[

A2 B2C

0n×n2 A

] [

ηx

]

+

[

B2b0

B

]

u +

[

B2E + D2

D

]

w (50a)

ηL =[

C2 01×n

]

[

ηx

]

+ E2w (50b)

admits state space representation (70) as in Pan and Basar (2000), which satisfies Assumption 10of Pan and Basar (2000) with r1 = r2, under some real invertible coordinate transformation.

Proof By analysis that is similar to that in the Case 2 in the proof of Lemma 6, there existsa real invertible matrix T1 such that system (1) admits the state space representation (42) in thecoordinates of T−1

1 x = [x′c¯x′3¯x′4], where xc is n1-dimensional, n1 ∈ IN ∪ 0; the triple (Ac, Bc, Cc)

is controllable and observable; the matrix A33 is Hurwitz; all of the eigenvalue of the matrix A44

are on the jω-axis and all Jordan blocks of A44 are of order 1.Now, we will discuss 2 exhaustive and mutually exclusive cases: Case 1: n1 > 0; Case 2: n1 = 0.

First, consider Case 1: n1 > 0. The transfer function of (42) from u to y is

H(s) = Cc (sIn1 −Ac)−1Bc + b0 =:

b1sn1−1 + · · ·+ bn1

sn1 + a1sn1−1 + · · ·+ an1

+ b0

28

where ai, i = 1, · · · , n1, bj, j = 0, · · · , n1, are some constants and b0 6= 0. Then, there exists ar0 ∈ 1, · · · , n1, such that br0 6= 0, and bj = 0, 1 ≤ j ≤ r0 − 1.

By Lemma 12 of Pan and Basar (2000), there exists a real invertible matrix T , which transformsthe triple (Ac, Bc, Cc) into

[

T−1 0n1×1

01×n1 1

] [

Ac Bc

Cc 0

] [

T 0n1×1

01×n1 1

]

=

A11 A12 B1

A21 A22 0

C1 0 0

where

A11 =

a1...

ar0−1

Ir0−1

ar0 01×(r0−1)

; A12 =

0

1

n1 > r0

[ ] n1 = r0

A21 =

ar0+1...

an1

0(n1−r0)×(r0−1)

; A22 =

−br0+1/br0

...−bn1−1/br0

In1−r0−1

−bn1/br0 01×(n1−r0−1)

n1 > r0

[ ] n1 = r0

B1 =[

01×(r0−1) br0

]′; C1 =

[

1 01×(r0−1)

]

Then, the coordinate transformation T−13 T−1

4 [η′ x′]′ = [η′ ¯x′1

¯x′2

¯x′3

¯x′4]′, where T3 = block diagonal

(In2 , T, In−n1), T4 = block diagonal (In2 , T1), will transform system (50) into

η˙x1˙x2

˙x3˙x4

=

A2 B2C1 0 B2C3 B2C4

0 A11 A12 A13 A14

0 A21 A22 A23 A24

0 0 0 A33 0

0 0 0 0 A44

η¯x1¯x2¯x3¯x4

+

B2b0

B1

0

0

0

u+

B2E + D2

D1

D2

D3

0

w (51a)

ηL =[

C2 0 0 0 0]

η¯x1¯x2¯x3¯x4

+E2w (51b)

We will focus on the [η′ ¯x′1

¯x′2]′ dynamics. First, we claim that any uncontrollable mode of the

following pair

A2 B2C1 0

0 A11 A12

0 A21 A22

,

b0B2

B1

0

29

is an eigenvalue of the matrix A2, which has negative real part. We show it as follows. Let λ ∈ Cbe a uncontrollable mode of the above pair, then the following matrix

λI −A2 −B2C1 0 b0B2

0 λI − A11 −A12 B1

0 −A21 λI − A22 0

is not full row rank. Since

[

λI − A11 −A12 B1

−A21 λI − A22 0

]

has full row rank for any λ, it implies that

the matrix λI −A2 is singular, i.e. λ is an eigenvalue of A2.Next, we claim that any unobservable mode of the following pair

A2 B2C1 0

0 A11 A12

0 A21 A22

,[

C2 0 0]

is an eigenvalue of A2 or is a zero of H2(s), which then has negative real part. We will prove it asfollows. Let λ ∈ C be a unobservable mode, then the following matrix

C2 0 0

λI −A2 −B2C1 0

0 λI − A11 −A12

0 −A21 λI − A22

(52)

is not full column rank. Since the submatrix of (52), consisting of the last n1 rows and the lastn1−1 columns of (52), has full column rank, and the submatrix of (52), consisting of the first n2+1

rows and the last n1 − 1 columns of (52), is zero matrix, then the submatrix

[

C2 0

λI −A2 −B2

]

must be singular, i.e.,

det

([

C2 0

λI −A2 −B2

])

= 0 = det

([

0 C2

−B2 λI −A2

])

There are two exhaustive and mutually exclusive cases. Case 1a: the matrix λI − A2 is singular,it implies the unobservable mode is an eigenvalue of A2. Case 1b: the matrix λI −A2 is invertible.Then the above determinant equality implies

det

([

C2(λI −A2)−1B2 0

0 λI −A2

])

= H2(λ) det(λI −A2) = 0

This further implies that λ is a zero of H2(λ). Therefore, this claim is proved.Next, we decompose the [η′ ¯x

′1

¯x′2]′ dynamics into observable and unobservable parts, and

then decompose the observable part into controllable and uncontrollable part. The real invertibletransformation [x′

co x′co x′

o]′ = T−1

co [η′ ¯x′1

¯x′2]′ will transform (51) into

xco

xco

xo˙x3˙x4

=

Aco ∗ 0 ∗ ∗0 Aco 0 ∗ ∗∗ ∗ Ao ∗ ∗

0 0 0 A33 0

0 0 0 0 A44

xco

xco

xo¯x3¯x4

+

Bco

0

Bo

0

0

u +

Dco

Dco

Do

D3

0

w (53a)

30

ηL =[

Cco Cco 0 0 0]

xco

xco

xo¯x3¯x4

+ E2w (53b)

where xco is the controllable and observable part, and is n3-dimensional, n3 ∈ IN; xco is theuncontrollable and observable part; and xo is the unobservable part; ∗’s denote arbitrary constantmatrices. Since the uncontrollable and unobservable modes of [η′ ¯x

′1

¯x′2]′ have negative real parts,

then Aco and Ao are Hurwitz. We note that the triple (Aco, Bco, Cco) is controllable and observable,and the transfer function for (53) from u to ηL is

Cco (sIn3 −Aco)−1Bco = H2(s)H(s)

which is strictly minimum phase and its relative degree is r2.By Lemma 12 in Pan and Basar (2000), there exists a real invertible matrix T5, and the coor-

dinate transformation xco = T5[x′1 x′

2]′ will transform system (53) into

˙x1

˙x2

xco

xo˙x3˙x4

=

A111 A112 ∗ 0 ∗ ∗A121 A122 ∗ 0 ∗ ∗0 0 Aco 0 ∗ ∗∗ ∗ ∗ Ao ∗ ∗

0 0 0 0 A33 0

0 0 0 0 0 A44

x1

x2

xco

xo¯x3¯x4

+

B11

0

0

Bo

0

0

u +

D11

D12

Dco

Do

D3

0

w (54a)

ηL =[

C11 0 Cco 0 0 0]

x1

x2

xco

xo¯x3¯x4

+ E2w (54b)

where ∗’s stand for some arbitrary constant matrices;

A111 =

a1...

ar2−1

Ir2−1

ar2 01×(r2−1)

; A112 =

0

1

n3 > r2

[ ] n3 = r2

A121 =

ar2+1...

an3

0(n3−r2)×(r2−1)

;A122 =

−br2+1/br2

...

−bn3−1/br2

In3−r2−1

−bn3/br2 01×(n3−r2−1)

n3 > r2

[ ] n3 = r2

B11 =[

01×(r2−1) br2

]′; C11 =

[

1 01×(r2−1)

]

where ai, i = 1, · · · , n3, bi, i = r2, · · · , n3, are some constants; br2 6= 0; the matrix A122 is Hurwitz,and the dimension of x1 is r2. We choose x3 = [x′

co¯x′3]′, x4 = ¯x4, and x5 = xo. Then the composite

31

system (54), in the coordinate of [x′1 x′

2 x′3 x′

4 x′5]′, admits the state space representation (70) as

in Pan and Basar (2000) and satisfies Assumption 10 of Pan and Basar (2000) with r1 = r2.Case 2: n1 = 0. Then, system (1) admits transfer function H(s) = b0 from u to y.The composite system (50) admits the following state space representation in the coordinates

of [η′ ¯x′3¯x′4]′,

η˙x3˙x4

=

A2 B2C3 B2C4

0 A33 0

0 0 A44

η¯x3¯x4

+

B2b0

0

0

u +

B2E + D2

D3

0

w (55a)

ηL =[

C2 0 0]

η¯x3¯x4

+ E2w (55b)

By an argument that is similar to that used in the proof of Lemma 7, there exists a real invertiblematrix T , such that the coordinate transformation η = T [z′1 z′2 z′co z′o]

′ will transform (49) into theform of (48). Then, the system (55) admits the following state space representation

z1

z2

zco

zo˙x3˙x4

=

A11 A12 ∗ 0 ∗ ∗A21 A22 ∗ 0 ∗ ∗0 0 Aco 0 ∗ ∗∗ ∗ ∗ Ao ∗ ∗

0 0 0 0 A33 0

0 0 0 0 0 A44

z1

z2

zco

zo¯x3¯x4

+

B1

0

0

Bo

0

0

u +

D1

D2

Dco

Do

D3

0

w (56a)

ηL =[

C1 0 Cco 0 0 0]

z1

z2

zco

zo¯x3¯x4

+ E2w (56b)

where ∗’s stand for some arbitrary constant matrices; the dimension of [z′1 z′2]′ is n4 ∈ IN; and A11,

A12, A21, A22, B1, and C1 admit structures as in (47) with n1 = n4 and r = r2. the matricesA22, Aco, and Ao are Hurwitz. We choose x1 = z1, x2 = z2, x3 = [z′co

¯x′3]′, x4 = ¯x4, and x5 = zo.

Then the composite system (56), in the coordinates of [x′1 x′

2 x′3 x′

4 x′5]′, admits the state space

representation (70) as in Pan and Basar (2000), which satisfies Assumption 10 of Pan and Basar(2000) with index r1 = r2.

This completes the proof of this Lemma. 2

B Some Derivations

d(x− Φθ)

dt

= x− ˙x− Φθ + Φ˙θ

= Ax + (yA211 + uA212)θ + Bu + Dw −(

− ΦΣPr(θ) + Ax + Bu− (uΦΣC ′1

32

+(γ−2Π + ΦΣΦ′)C ′) (yd − Cx− u (C1θ + bp0)) + (γ2uΦΣC ′1 + ΠC ′ + γ2ΦΣΦ′C ′ + L)

ζ2 (y − Cx− u (C1θ + bp0))−[

ΦΣ γ−2Π + ΦΣΦ′]

Qξc + (yA211 + uA212)θ)

−(

AfΦ + yA211 + (A212 + (γ−2 − ζ2)ΠC ′C1 − ζ2LC1)u)

θ + Φ(

− ΣPr(θ)− (uΣC ′1

+ΣΦ′C ′) (yd − Cx− u (C1θ + bp0))−[

Σ ΣΦ′]

Qξc + γ2ζ2 (uΣC ′1 + ΣΦ′C ′) (y − Cx

−u (C1θ + bp0)))

= Ax−AfΦθ − ((γ−2 − ζ2)ΦC ′C1 − ζ2LC1)uθ + Dw +[

0 γ−2Π]

Qξc + γ−2ΠC ′yd

+γ−2ΠC ′ (−Cx− u (C1θ + bp0))− ζ2 (ΠC ′ + L)y + ζ2 (ΠC ′ + L) (Cx + u (C1θ + bp0))

= Ax−AfΦθ + Dw +[

0 γ−2Π]

Qξc −(

(γ−2 − ζ2)ΠC ′C1θ − ζ2LC1θ

+γ−2ΠC ′ (C1θ + bp0)− ζ2 (ΠC ′ + L) (C1θ + bp0))

u +(

− γ−2ΠC ′C + ζ2 (ΠC ′ + L)C)

x

+γ−2ΠC ′yd − ζ2 (ΠC ′ + L)y + γ−2ΠC ′y − γ−2ΠC ′y

= Ax−AfΦθ +(

D − ζ2 (ΠC ′ + L)E + γ−2ΠC ′E)

w +[

0 γ−2Π]

Qξc + γ−2ΠC ′ (yd − y)

−(

(γ−2 − ζ2)ΠC ′C1θ − ζ2LC1θ + γ−2ΠC ′ (C1θ + bp0)− ζ2 (ΠC ′ + L) (C1θ + bp0)

+ζ2 (ΠC ′ + L) (C1θ + bp0)− γ−2ΠC ′ (C1θ + bp0))

u− (γ−2ΠC ′C − ζ2 (ΠC ′ + L)C)x

= (A + γ−2ΠC ′C − ζ2 (ΠC ′ + L)C)x−AfΦθ +(

D − ζ2 (ΠC ′ + L)E + γ−2ΠC ′E)

w

+[

0 γ−2Π]

Qξc + γ−2ΠC ′yd − γ−2ΠC ′y −(

(γ−2 − ζ2)ΠC ′C1θ − ζ2LC1θ

−γ−2ΠC ′C1θ + ζ2 (ΠC ′ + L)C1θ)

u

= Af (x− Φθ)+γ−2ΠC ′yd−γ−2ΠC ′y+(

D − ζ2 (ΠC ′ + L)E+γ−2ΠC ′E)

w+[

0 γ−2Π]

Qξc

Next we give the derivation of wopt. First, we notice that

v = ζ (y − Cx− (bp0 + C1θ)u)

= ζ (Cξ + bp0u + Ew − Cξ − bp0u)

= ζC (ξ − ξ) + ζEw

w∗ = ζ2E′ (y − bp0u− Cξ) + γ−2 (Iq − ζ2E′E)D′Σ−1 (ξ − ξ)

= ζ2E′Ew +

where = γ−2 (Iq − ζ2E′E)D′Σ−1 (ξ − ξ). Then, we have E = 0.

v2 = ζ2 (C (ξ − ξ) + Ew)′ (C (ξ − ξ) + Ew)

= ζ2|C (ξ − ξ)|2 + ζ2|Ew|2 + 2ζ2 (C (ξ − ξ))′Ew

|w − w∗|2 = |w|2 + |ζ2E′Ew +|2 − 2(ζ2E′Ew +)′w

= |w|2 + |ζ2E′Ew|2 + ||2 − 2′w − 2ζ2|Ew|2

v2 + |w − w∗|2 = |w|2 − 2(− ζ2E′C (ξ − ξ))′w + ||2 + ζ2|C (ξ − ξ)|2

= |w − (− ζ2E′C (ξ − ξ))|2 − ζ4 (C (ξ − ξ))′EE′C (ξ − ξ) + ζ2|C (ξ − ξ)|2

= |w − (− ζ2E′C (ξ − ξ))|2

= |w − wopt|2

33

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