# Adaptive Speed Control Design for Brushed Permanent Magnet DC Motor Based on Worst-Case Analysis Approach

Authors: Sheng Zeng
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper presents the adaptive controller design for brushed permanent magnet DC motor used in velocity-tracking applications based on worst-case approach. We first formulate the robust adaptive control problem as a nonlinearH∞-control problem under imperfect state measurement, and then solve it using game-theoretic approach. The controller guarantees the boundedness of closed-loop signals with bounded exogenous disturbances, and achieves desired disturbance attenuation level with respect to the unmeasured exogenous disturbance inputs and the measured disturbance inputs. The strong robustness properties are illustrated by a simulation example.


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## Body

## 1. Introduction

Permanent magnet brushed DC (PMBDC) motors are widely used in real world applications, and particularly in the high-volume commercial products, which is due to the PMBDC motors’ better cost-to-performance ratio than most other motors. However, the structure of these permanent magnetic motors induces several uncertainties such as unmodelled nonlinear dynamics and undesired commutation/dent torques. The magnets lose/degrade their magnetic properties over time, and the motor constant varies with the changes of temperature and operating conditions. Moreover, the ever increasing control demands for high-efficiency and low-cost required to run the PMBDC motor at its technical limit. All above design challenges call for a robust adaptive controller for motion control applications over a wide range of operating conditions.Adaptive control attracted a lot of research attention in control theory since 1970s. The classic adaptive control design based on the certainty equivalence approach leads to structurally simple adaptive controllers [1, 2], and its effectiveness for linear systems with or without stochastic disturbance inputs has been demonstrated when long-term asymptotic performance is considered [3]. However, early designs based on this approach were not robust to exogenous disturbance inputs and unmodeled dynamics [4]. Then, the stability and the performance of the closed-loop system becomes an important issue, which motivated the study of robust adaptive control in the 1980s and 1990s.Robust adaptive control design is one of the most important research topics in control theory, which addresses the design of controllers that are robust to model uncertainties, and insensitive to the exogenous disturbances. Various adaptive controllers were modified to render the closed-loop systems robust [5–10]. Despite their successes, they however fell short of addressing the disturbance attenuation property of the closed-loop system. Worst-case analysis-based robust adaptive-control design was motivated by the success of the game-theoretic approach to H∞-optimal control problems [11] in late 1990s, which addresses the disturbance attenuation property directly. In this approach, the robust adaptive control problem is formulated as a nonlinear H∞-control problem under imperfect state measurements. By cost-to-come function analysis, it is converted into an H∞-control problem with full information measurements. This full information measurements problem is then solved using nonlinear design tools for a suboptimal solution. This design paradigm has been applied to worst-case parameter identification problems [12], which has led to new classes of parametrized identifiers for linear and nonlinear systems [13–16]. It has also been applied to adaptive control problems [17–21], and offered a promising tool to system subjected to uncertainties.In this paper, we study the adaptive control design for permanent magnet DC motor based on worst-case analysis approach. We first model the permanent magnet DC motor servo system which is linear in all of the uncertainties. We then formulate the robust adaptive control problem as a nonlinearH∞-control problem under imperfect state measurements, and apply the cost-to-come function analysis to derive the worst-case identifier and state estimator. The control design of the plant subsystem follows [22], and the adaptive controller can be obtained by the integrator backstepping methodology. The robust adaptive controller achieves asymptotic tracking if the disturbances are bounded and of finite energy, and guarantees the stability of the closed-loop system with respect to the bounded disturbance inputs and the initial conditions. Furthermore, the closed-loop system admits a guaranteed disturbance attenuation level with respect to the exogenous disturbance inputs and the measured disturbances, where ultimate lower bound for the achievable attenuation performance level is only related to the noise intensity in the measurement channel of the plant system.The balance of the paper is organized as follows. In Section2, we present the formulation of the adaptive control problem and discuss the general solution methodology, then we obtain parameter identifier and state estimator using the cost-to-come function analysis in Section 3. In Section 4, we derive the adaptive control law based on backstepping method, and present the main results on the robustness of the system in Section 5. The effectiveness of the proposed approach is illustrated with an example in Section 6, and the paper ends with some concluding remarks in Section 7.


## 2. Problem Formulation

We consider a velocity-control problem of a brushed permanent magnet DC motor which is described by the following dynamics:(1a)x`˙=[-DlJKtJ-KeL-RL]x+[01L]u+[1J0]Twˇ+D`w`,(1b)y=[10]x+E`w`,where x`∈ℝ2 is the state vector and represents load shaft angular speed and motor current, respectively; u∈ℝ is the scalar control input; y∈ℝ is the load shaft angular speed measurement output; L∈ℝ is the motor inductor; R∈ℝ is the armature resistance; Ke is the back-emf constant; Kt is the motor torque constant; J is the motor system inertia; Dl∈ℝ is the motor-system damping factor; w`=[Tww`ωTf]′ is the arbitrary disturbance vector, which is composed of arbitrary disturbance torque, Tw∈ℝ, arbitrary disturbance at the output measurement channel, w`ω∈ℝ, and friction disturbance torque Tf∈ℝ; Twˇ∈ℝ is the measured or estimated disturbance torque. In this paper, we assume the variables Kt, Ke, J, Dl, R, and L are completely unknown or partially unknown.It is easy to check that the true system is observable, controllable, a minimal phase system, and the transfer function fromu to y has relative degree r=2. And then we can find a state diffeomorphism x=T`x` and a disturbance transformation w=M`w` by following [22], such that system (1a) and (1b) can be transformed into the following form in the x coordinates:(2a)x˙=Ax+(yA-21+uA-22+wˇA-23)θ+Bu+Dw+Dˇwˇ,(2b)y=Cx+Ew,where x∈ℝ2 is the state vector, x=[x1x2]′, wˇ=Twˇ, θ∈ℝσ, σ∈ℕ, is the σ-dimensional vector of unknown parameters of the true system; the matrices A, A-21, A-22, A-23, D, Dˇ, C, and E are of appropriate dimensions, known and have the structure as below,
(3)A=[a111a21a22],A-22=[01×σA-220],B=[0bp0],C=[10],
where A-220 is a σ-dimensional row vector and bp0∈ℝ. Then the high-frequency gain of (2a) and (2b) is b0=bp0+A-220θ.Since we consider a trajectory tracking control design problem, we make the following assumption about the reference signalyd.Assumption 1.
The reference trajectory,yd, is two times continuously differentiable. Define vector Yd:=[yd(0)yd(1)yd(2)]′ and Yd0:=[yd(0)yd(1)(0)]′. The signal Yd is available for control design.To guarantee the stability of the closed-loop system and the boundedness of the estimate ofθ, we assume there is an a priori convex compact set where the parameter vector θ lies in.Assumption 2.
There exists a known smooth nonnegative radially unbounded strictly convex functionP:ℝσ→ℝ, such that the true value θ∈Θ:={θ-∈ℝσ:P(θ-)≤1}. For all θ-∈Θ, b-0=bp0+A-220θ->0.The control objective is to design a robust adaptive controller for system (1), such that the outputy(t) tracks a desired reference signal yd(t), while rejecting the uncertainties (x(0),θ,w`[0,∞),wˇ[0,∞),Yd0,yd(2))∈𝒲:=ℝ2×Θ×𝒞×𝒞×ℝ2×𝒞 comprises the initial state, the true values of unknown parameters, unmeasured disturbances, and the measured disturbances, while all signals in the closed-loop system are uniformly bounded. We make the control objective precise as follows.Definition 3.
A controllerμ is said to achieve disturbance attenuation level γ if there exist nonnegative functions l(t,θ,x,y[0,t],Twˇ[0,t]) and l0(xˇ0,θˇ0) such that
(4)sup(x(0),θ,wω[0,∞),Tf[0,∞),Tw[0,∞),Twˇ[0,∞))Jt≤0,
where
(5)Jt=∫0t‍((Cx-yd)2+l-γ2|Tw|2-γ2|Tf|2-γ2|wω|2-γ2|Twˇ|2|Tf|2)dτ-γ2|θ-θˇ0|Q02-γ2|x(0)-xˇ0|Π0-12-l0,
where θˇ0∈Θ is the initial guess of the unknown parameter vector; Q0>0 is the quadratic weighting on the error between the true value of θ and the initial guess θˇ0 quantifying the level of confidence in the estimate θˇ0; xˇ0 is the initial guess of the unknown initial state x(0); Π0-1>0 is the weighting on the initial state-estimation error, quantifying the level of confidence in the estimate xˇ0; |z|Q denotes zTQz for any vector z and any symmetric matrix Q.The control law to system (2a), (2b) is generated by the following control law
(6)u(t)=μ(t,y[0,t],wˇ[0,t]),
where μ:[0,∞)×ℒ2×ℒ2→ℝ. We denote the class of these admissible controllers by ℳ.Clearly, when the inequality (4) is achieved, we have
(7)∥Cx-yd∥22(∥Tw∥22+∥Tf∥22+∥wω∥22+∥Twˇ∥22+|θ-θˇ0|Q02+|x(0)-xˇ0|Π0-12+C0)-1≤γ2,
where ∥·∥2 denotes ℒ2 norm, and C0 is a constant. When ∥w`∥2 and ∥wˇ∥2 are finite, ∥Cx-yd∥2 is also finite, which implies limt→∞|Cx-yd|=0, under additional assumptions.The following notation will be used throughout this paper:xˇ denotes the estimate of the current state of the system; x~ denotes the state-estimation error x-xˇ; θˇ denotes the estimate of the parameter vector θ; θ~ denotes the estimation error θ-θˇ; any function symbol with an ‘‘over bar’’ will denote a function defined in the terms of the transformed state variables, such as f-(z) being the identical function as f(x); for any matrix M, the vector M→ is formed by stacking up its column vectors; ej,i denotes a j-dimensional column vector, all of its elements are 0, except its ith row is 1, such as e2,2=[01]′.Letξ denote the expanded state vector ξ=[θ′x′]′, we have the following expanded dynamics for system (2a), (2b) in view of θ˙=0:(8a)ξ˙=[00yA-21+uA-22+A-23wˇA]ξ+[0B]u+[0D]w+[0Dˇ]wˇ=:A-ξ+B-u+D-w+Dˇ-wˇ,(8b)y=[0C]ξ+Ew=:C-ξ+Ew.The worst-case optimization of the cost function (4) can be carried out in two steps as depicted in the following equations:
(9)sup(x(0),θ,w[0,∞),wˇ[0,∞))∈𝒲Jt=supy[0,∞),wˇ[0,∞)sup(x(0),θ,w[0,∞))|y[0,∞),wˇ[0,∞)Jt.
The right-hand supremum operator will be carried out first. It is the identification design step, which will be presented in Section 3. Succinctly stated, in this step, we will calculate the maximum cost that is consistent with the given measurement waveform.The left-hand supremum operator will be carried out second. It is the controller design step, which will be discussed in Section4. In this step we use a backstepping method to design the control input u, and prove that all state variables of the closed-loop system are uniformly bounded in time for any uniformly bounded disturbance input waveforms.This completes the formulation of the robust adaptive control problem. Next, we turn to the identification design step in the next section.


## 3. Estimation Design

In this section, we present the identification design for the adaptive control problem formulated.In this step, the measurement waveformy[0,∞) and wˇ[0,∞) are assumed to be known. Since the control input is a causal function of y and wˇ, then it is known. We ignore terms considered to be constant in the estimation design step, and set l in (5) to be |ξ-ξ^|Q~-2+2(ξ-ξˇ)′l2+l^. The equivalent cost function of (5) is then given by,
(10)Jγtf∶=∫0tf‍(|ξ-ξ^|Q~-2+2(ξ-ξˇ)′l2+l^-γ2|w|2)dτ-γ2|ξ~0|Q~-02,
where Q~- is a matrix-valued weighting function, ξ^ is the worst-case estimates for the expanded state ξ, l2 is a design function, and l^ is considered to be constant in the estimation design step. The cost function is then of a linear quadratic structure, and the robust adaptive control problem becomes an H∞-control of affine quadratic problem, which admits a finite dimensional solution.We introduce the value functionW=|ξ-ξˇ|Σ--12, and then we can obtain the dynamics of state estimator ξˇ and worst-case covariance matrix Σ- as below(11a)Σ-˙=(A--ζ2L-C-)Σ-+Σ-(A--ζ2L-C-)′+1γ2D-D-′-1γ2ζ2L-L-′-Σ-(γ2ζ2C-′C--C-′C--Q~-)Σ-,Σ-(0)=1γ2[Q000n×σΠ0-1]-1,(11b)ξˇ˙=(A-+Σ-(C-′C-+Q~-))ξˇ+ζ2(γ2Σ-C-′+L-)(y-C-ξˇ)+B-u+Dˇ-wˇ-Σ-(C-′yd+Q~-ξ^),ξˇ(0)=[θˇ0xˇ0],where ζ:=1/(EE′)1/2 and L- is defined as L-=[0L′]′, where L:=DE′.Then, the cost function (5) can be equivalently written as
(12)Jt=-|ξ(t)-ξˇ(t)|Σ--1(t)2+∫0t(|C-ξˇ-yd|2+|ξc|Q~-2-γ2ζ2|y-C-ξˇ|2+∫0t‍-γ2|w-w*(u[0,τ],y[0,τ],wˇ[0,τ],ξ[0,τ],ξ^[0,τ],ξˇ[0,τ])|2)dτ,
where ξc:=ξ^-ξˇ, which will be determined later to improve the performance of the adaptive system; w* is the worst-case disturbance, given by
(13)w*(τ,u[0,τ],y[0,τ],wˇ[0,τ],ξ[0,τ],ξ^[0,τ],ξˇ[0,τ])=ζ2E′(y-C-ξ)+1γ2(I-ζ2E′E)D-′Σ--1(ξ-ξˇ).We partitionΣ- as
(14)Σ-=[ΣΣ-12Σ-21Σ-22],
and introduce Φ:=Σ-21Σ-1 and Π:=γ2(Σ-22-Σ-21Σ-1Σ-12).For the boundedness ofΣ, we make the following assumption on the weighting matrix Q~-.Assumption 4.
The weighting matrixQ~- of function l in (5) is given by
(15)Q~-=Σ--1[000Δ]Σ--1+[ϵΦ′C′(γ2ζ2-1)CΦ000],
where Δ is 2×2 positive-definite matrix, and ϵ is a scalar function defined by
(16)ϵ(τ):=Tr(Σ-1(τ))Kc,Kc≥γ2Tr(Q0),τ≥0.Then, we have the following differential equation ofΣ, Φ, and Π:(17a)Σ˙=-(1-ϵ)ΣΦ′C′(γ2ζ2-1)CΦΣ;Σ(0)=1γ2Q0-1,(17b)Φ˙=(A-ζ2LC-1γ2ΠC′(γ2ζ2-1)C)Φ+yA-21+uA-22+A-23wˇ;Φ(0)=0,(17c)Π˙=(A-ζ2LC)Π+Π(A-ζ2LC)′-ΠC′(ζ2-1γ2)CΠ+DD′-ζ2LL′+γ2Δ,Π(0)=Π0.The matrix Σ will play the role of worst-case covariance matrix of the parameter estimation error. Assumption 4 guarantees that Σ is uniformly bounded from above and uniformly bounded from below away from 0 as depicted in the following lemma, and its proof is given in [14].Lemma 5.
Consider the dynamic equation (17a) for the covariance matrix Σ. Let Assumption 4 hold and γ≥ζ-1. Then, Σ is uniformly upper and lower bounded as follows:
(18)1Kc≤Σ(τ)≤Σ(0)=1γ2Q0-1;γ2Tr(Q0)≤Tr(Σ-1(τ))≤Kc;∀τ∈[0,t].We definesΣ(t):=Tr((Σ(t))-1), and its dynamic is given by
(19)s˙Σ=γ2ζ2(1-ϵ)CΦΦ′C′;sΣ(0)=γ2Tr(Q0).
Then, ϵ(τ)=Kc-1sΣ-1(τ), which does not require the inversion of Σ.From Assumption4 and (17a), we note that γ≥ζ-1. This means the quantity ζ-1 is the ultimate lower bound on the achievable performance level for the adaptive system, using the design method proposed in this paper.Assumption 6.
If the matrixA-ζ2LC is Hurwitz, then the desired disturbance attenuation level γ≥ζ-1. If the matrix A-ζ2LC is not Hurwitz, then the desired disturbance attenuation level γ>ζ-1.Assumption 7.
The initial weighting matrixΠ0 in (17c) is chosen as the unique positive definite solution to the following algebraic Riccati equation:
(20)(A-ζ2LC)Π+Π(A-ζ2LC)′-ΠC′(ζ2-1γ2)CΠ+DD′-ζ2LL′+γ2Δ=0.Then, we note that the unique positive-definite solution of (17c) is time-invariant and equal to the initial value Π0.Remark 8.
To simplify the estimator structure, we can chooseϵ=1 so that Σ will be a constant positive-definite matrix, and sΣ will be a finite positive constant. To further simplify the identifier, the initial weighting matrix Π0 is chosen as the unique positive-definite solutions to its algebraic Riccati equation (17c), which also implies Σ->0 in view of Σ>0.To guarantee the boundedness of estimated parameters without persistently exciting signals, we introduce soft projection design on the parameter estimate. We define(21)ρ:=inf{P(θ-)∣θ-∈ℝσ,bp0+A-220θ-=0}.
By Assumption 2 and Lemma  2 in [23], we have 1<ρ≤∞. For any fixed ρo∈(1,ρ), we define the open set
(22)Θo:={θ-∈ℝσ∣P(θ-)<ρo}.
Our control design will guarantee that the estimate θˇ lies in Θo, which immediately implies |bp0+A-220θˇ|>c0>0, for some c0>0. Moreover, the convexity of P implies the following inequality:
(23)∂P∂θ(θˇ)(θ-θˇ)<0∀θˇ∈ℝσ∖Θ.
We set l2=[-(Pr(θˇ))′0]′, where
(24)Pr(θˇ):={e1/(1-P(θˇ))((∂P/∂θ)(θˇ))′(ρo-P(θˇ))3∀θ∈Θo∖Θ,0∀θ∈Θ.
Then, we obtain
(25)ξˇ˙=-Σ-[(Pr(θˇ))′0]′+A-ξˇ+B-u-Σ-Q~-ξc+ζ2(γ2Σ-C-′+L-)(y-C-ξˇ)+Dˇ-wˇ,ξˇ(0)=[θˇ0′xˇ0′]′,
where ξc=ξ^-ξˇ.We summarize the estimation dynamics equations below(26a)(A-ζ2LC)Π+Π(A-ζ2LC)-ΠC′(ζ2-1γ2)CΠ+DD′-ζ2LL′+γ2Δ=0,(26b)Σ˙=-(1-ϵ)ΣΦ′C′(γ2ζ2-1)CΦΣ;Σ(0)=1γ2Q0-1,(26c)s˙Σ=(γ2ζ2-1)(1-ϵ)CΦΦ′C′;sσ(0)=γ2Tr(Q0),(26d)ϵ=1KcsΣ,(26e)Af=A-ζ2LC-ΠC′C(ζ2-1γ2),(26f)Φ˙=AfΦ+yA-21+uA-22+wˇA-23;Φ(0)=0,(26g)θˇ˙=-ΣPr(θˇ)-ΣΦ′C′(yd-Cxˇ)-[ΣΣΦ′]Q~-ξc+γ2ζ2ΣΦ′C′(y-Cxˇ);θˇ(0)=θˇ0,(26h)xˇ˙=-ΦΣPr(θˇ)+Axˇ-(1γ2Π+ΦΣΦ′)C′(yd-Cxˇ)-[ΦΣ1γ2Π+ΦΣΦ′]Q~-ξc+(yA-21+uA-22+wˇA-23)θˇ+ζ2(ΠC′+γ2ΦΣΦ′C′+L)×(y-Cxˇ)+Dˇwˇ+Bu;xˇ(0)=xˇ0.For the controller structure simplification, the dynamics forΦ can be implemented as below. First, we observe the matrix Af has the same structure as the matrix A. Then, we introduce the matrix
(27)Mf:=[Afp2p2],
where p2 is a 2-dimensional vector such that the pair (Af,p2) is controllable, which implies that Mf is invertible. Then the following prefiltering system for y, u, and wˇ generates the Φ online:(28a)η˙=Afη+p2y;η(0)=0,(28b)λ˙=Afλ+p2u;λ(0)=0,(28c)η˙wˇ=Afηwˇ+p2wˇ;ηwˇ(0)=0,(28d)Φ=[Afηη]Mf-1A-21+[Afλλ]Mf-1A-22+[Afηwˇηwˇ]Mf-1A-23.Associated with the above identifier, introduce the value function(29)W(t,ξ(t),ξˇ(t),Σ-(t))=|ξ(t)-ξˇ(t)|Σ--1(t)2=|θ-θˇ(t)|Σ-1(t)2+γ2|x(t)-xˇ(t)-Φ(t)(θ-θˇ(t))|Π-12
whose time derivative is given by
(30)W˙=-|Cx-yd|2-γ4|x-x^-Φ(θ-θ^)|Π-1ΔΠ-12-ϵ(γ2ζ2-1)|θ-θ^|Φ′C′CΦ2+γ2|w|2+|Cxˇ-yd|2+|ξc|Q~-2-γ2ζ2|y-Cxˇ|2-γ2|w-w*(t,u[0,t],y[0,t],wˇ[0,t],ξ[0,t],ξ^[0,t],ξˇ[0,t])|2+2(θ-θˇ)′Pr(θˇ).
We note that the last term in W˙ is nonpositive, zero on the set Θ and approaches -∞ as θˇ approaches the boundary of the set Θ, which guarantees the boundness of θˇ.Then the cost function can be equivalently written as(31)Jt=Jt+W(0)-W(t)+∫0t‍W˙dτ=-|θ-θˇ(t)|Σ-1(t)2-γ2|x(t)-xˇ(t)-Φ(t)(θ-θˇ(t))|Π-12+∫0t‍(|Cxˇ-yd|2+|ξc|Q~-2-γ2ζ2|y-Cxˇ|2-γ2|(τ,u[0,τ],y[0,τ],wˇ[0,τ],ξ[0,τ],ξ^[0,τ],ξˇ[0,τ])w-w*×(τ,u[0,τ],y[0,τ],wˇ[0,τ],ξ[0,τ],ξ^[0,τ],ξˇ[0,τ])|2+l^-γ2|wˇ|2|Cxˇ-yd|2)dτ.This completes the identification design step.


## 4. Control Design

In this section, we describe the controller design for the uncertain system under consideration. Note that, we ignored some terms in the cost function (5) in the identification step, since they are constant when y and wˇ are given. In the control design step, we will include such terms. Then, based on the cost function (5) in the Section 2, the controller design is to guarantee that the following supremum is less than or equal to zero for all measurement waveforms:
(32)sup(x(0),θ,w[0,∞)wˇ[0,∞))∈𝒲Jt≤supy[0,∞),wˇ[0,∞){∫0t‍(|Cxˇ-yd|2+|ξc|Q~-2-γ2ζ2|y-Cxˇ|2+l^-γ2|wˇ|2|Cxˇ-yd|2)dτ-l0∫0t‍},
where function l^(τ,y[0,τ],wˇ[0,τ]) is part of the weighting function l(τ,θ,x,y[0,τ],wˇ[0,τ]) to be designed, which is a constant in the identifier design step, and is therefore neglected.By (32), we observe that the cost function is expressed in term of the states of the estimator we derived, whose dynamics are driven by the measurement y, input u, measured disturbance wˇ, and the worst-case estimate for the expanded state vector ξ^, which are signals we either measure or can construct. This is then a nonlinear H∞-optimal control problem under full information measurements. Instead of considering y and wˇ as the maximizing variable, we can equivalently deal with the transformed variable
(33)v:=[ζ(y-Cxˇ)wˇ].
Then, we have
(34)η˙=Afη+p2Cxˇ+p2(e2,1′vζ),(35)θˇ˙=-ΣPr(θˇ)-ΣΦ′C′(yd-Cxˇ)-[ΣΣΦ′]Q~-ξc+γ2ΣΦ′C′ζe2,1′v,(36)xˇ˙=Axˇ-(1γ2Π+ΦΣΦ′)C′(yd-Cxˇ)+A-21θˇCxˇ-ΦΣPr(θˇ)-[ΦΣ1γ2Π+ΦΣΦ′]Q~-ξc+Bu+A-22θˇu+((ζ-2A-21θˇ+ΠC′+γ2ΦΣΦ′C′+L)ζe2,1′+A-23θˇe2,2′+[0n×1Dˇ](ζ-2A-21θˇ+ΠC′+γ2ΦΣΦ′C′+L))v,(37)W˙=-|Cx-yd|2-γ4|x-x^-Φ(θ-θ^)|Π-1ΔΠ-12-ϵ(γ2ζ2-1)|θ-θ^|Φ′C′CΦ2+|Cxˇ-yd|2+|ξc|Q~-2+2(θ-θˇ)′Pr(θˇ)+γ2|wˇ|2+γ2|w|2-γ2|v|2-γ2|w-w*(t,u[0,t],y[0,t],wˇ[0,t],ξ[0,t],ξ^[0,t],ξˇ[0,t])|2.The variables to be designed at this stage includeu and ξc. The design for ξc will be carried out last. Note that the structure of A in the dynamics is in strict-feedback form, we will use the backstepping methodology [24] to design the control input u, which will guarantee the global uniform boundedness of the closed-loop system states and the asymptotic convergence of tracking error.Consider the dynamics ofΦ,
(38)Φ˙=AfΦ+yA-21+uA-22+wˇA-23;Φ(0)=0.
For ease of the ensuing study, we will separate Φ as the sum of several matrices as follows:(39a)Φ=Φu+Φy+Φwˇ,(39b)Φy=[Afηη]Mf-1A-21=[η′T1η′T2],(39c)Φ˙u=AfΦu+uA-22;Φu(0)=0,(39d)Φ˙wˇ=AfΦwˇ+wˇA-23;Φwˇ(0)=0,where Ti, i=1,2 are 2×1-dimensional constant matrices depending on Af, Mf, and A-21. Express Φu and Φwˇ in terms of their row vectors, Φu=[Φ1u′Φ2u′]′ and Φwˇ=[Φ1wˇ′Φ2wˇ′]′. Then, CΦy=η′T1, CΦu=Φ1u, and CΦwˇ=Φ1wˇ.We summarized the dynamics for backstepping design in the following, where we have emphasized the dependence of various functions on the independent variables:(40a)s˙Σ=(γ2ζ2-1)(1-ϵ)(η′T1+Φ1u+Φ1wˇ)×(η′T1+Φ1u+Φ1wˇ)′,(40b)ϵ=1KcsΣ,(40c)Σ˙=-(1-ϵ)Σ(η′T1+Φ1u+Φ1wˇ)′×(γ2ζ2-1)(η′T1+Φ1u+Φ1wˇ)Σ,(40d)θˇ˙=δ(yd-xˇ1,η,Φ1wˇ,Φ1u,θˇ,Σ→)+φ(η,Φ→wˇ,Φ→u,Σ→)Q~-ξc+κ(η,Φ1wˇ,Φ1u,Σ→)v,(40e)η˙=Afη+p2xˇ1+p2(e2,1′vζ),(40f)xˇ˙1=xˇ2+f1(yd-xˇ1,xˇ1,θˇ,η,Φ1wˇ,Φ1u,Σ→)+ϱ1(η,Φ→wˇ,Φ→u,Σ→)Q~-ξc+h1(θˇ,η,Φ1wˇ,Φ1u,Σ→)v,(40g)xˇ˙2=a22xˇ2+(bp0+A-220θˇ)u+f2(yd-xˇ1,xˇ1,xˇ2,θˇ,η,Φ1wˇ,Φ2wˇ,Φ1u,Φ2u,Σ→)+ϱ2(η,Φ→wˇ,Φ→u,Σ→)Q~-ξc+h2(θˇ,η,Φ1wˇ,Φ2wˇ,Φ1u,Φ2u,Σ→)v,(40h)Φ˙1u=ψ1(Φ1u)+Φ2u,(40i)Φ˙1wˇ=ϖ1(Φ1wˇ)+Φ2wˇ+e2,1′A-23e2,2′v,where the nonlinear functions δ, f1, and f2 are smooth as long as θˇ∈Θo; the nonlinear functions φ, κ, ϱ1, ϱ2, h1, h2, ψ1, and ϖ1 are smooth. Here, we use Φ1u, Φ2u, Φ1wˇ, and Φ2wˇ as independent variables, instead of λ, η1wˇ, for the clarity of ensuing analysis.We observe that the above dynamics is linear inξc, which will be optimatized after backstepping design. Σ, sΣ, Φwˇ, and θˇ will always be bounded by the design in Section 3, then they will not be stabilized in the control design. Φu is not necessary bounded, since the control input u appeared in their dynamics, it can not stabilzed in conjunction with xˇ using backstepping. Hence, we assume it is bounded, and prove later that it is indeed so under the derived control law.The following backstepping design will achieve theγ level of disturbance attenuation with respect to the disturbance v.Step 1.
In this step, we try to stabilizeη by virtual control law xˇ1=yd. Introduce variable ηd, as the desired trajectory of η, which satisfies the dynamics
(41)η˙d=Afηd+p2yd;ηd(0)=02×1.
Define the error variable η~=η-ηd. Then, η~ satisfies the dynamics
(42)η~˙=Afη~+p2(e2,1′vζ)+p2(xˇ1-yd).
By [14], the following holds.Lemma 9.
Given any Hurwitz matrixAf, there exists a positive-definite matrix Y such that the following generalized algebraic Riccati equation admits a positive-definite solution Z:
(43)Af′Z+ZAf+1γ2ζ2Zp2p2′Z+Y=0.Note thatAf in (42) is a Hurwitz matrix, then we define the following value function in terms of the positive-definite matrix Z:
(44)V0(η~):=|η~|Z2.
Then its time derivative is given by
(45)V˙0=-|η~|Y2+γ2|v|2-γ2|v-ν0|2+2η~′Zpn(xˇ1-yd),
where
(46)ν0(η~)=1γ2ζe2,1p2′Zη~.
If xˇ1 is control input, then we may choose the control law
(47)xˇ1=yd,
and the design achieves attenuation level γ from the disturbance v to the output Y1/2(η-ηd). This completes the virtual control design for the η dynamics.Step 2.
Define the transformed variable(48)z1:=xˇ1-yd,
which is the deviation of xˇ1 from its desired trajectory yd.Then the time derivative ofz1 is given by
(49)z˙1=f-1(z1,yd,θˇ,η,Φ1wˇ,Φ1u,Σ→)+xˇ2-yd(1)+ϱ1(η,Φ→wˇ,Φ→u,Σ→)Q~-ξc+h1(θˇ,η,Φ1wˇ,Φ1u,Σ→)v,
where the function f-1 is defined as
(50)f-1(z1,yd,θˇ,η,Φ1wˇ,Φ1u,Σ→)=f1(yd-xˇ1,xˇ1,θˇ,η,Φ1wˇ,Φ1u,Σ→).
Introduce the value function for this step
(51)V-1=V0+12z12
whose derivative is given by
(52)V-˙1=-|η~|Y2+γ2|v|2-γ2|v-ν0|2+2η~Zpnz1+z1(xˇ2-yd(1)+f-1+ϱ1Q~-ξc+h1v)=-z12-|η~|Y2-β-1z12+z1z2+ς-1′Q~-ξc+γ2|v|2-γ2|v-ν-1|2,
where(53a)z2=xˇ2-yd(1)-α-1,(53b)ν-1(z1,θˇ,η~,η,Φ1wˇ,Φ1u,Σ→)=ν0+12γ2h1′z1,(53c)α-1(z1,yd,θˇ,η~,η,Φ1wˇ,Φ1u,Σ→,s)=-z1-β-1z1-2pn′Zη~-f-1-h1ν0-14γ2h1h1′z1,(53d)β-1(z1,yd,θˇ,η~,η,Φ1wˇ,Φ1u,Σ→,s)≥cβ1>0,(53e)ς-1(z1,yd,η~,η,Φ1wˇ,Φu,Σ→)=ϱ1′z1,where cβ1 is any positive constant, and the nonlinear function β-1 is to be chosen by the designer. Note that the function α1 is smooth as long as θˇ∈Θo. If xˇ2 were the actual controls, then we would choose the following control law:
(54)xˇ2=yd(1)+α-1,
and set ξc=0, to guarantee the dissipation inequality with supply rate
(55)-|xˇ1-yd|2-|η~|Y2-β-1z12+γ2v2.This completes the second step of backstepping design.Step 3.
At this step the actual control appears in the derivative ofz2, which is given by
(56)z˙2=a22xˇ2+(bp0+A-220θˇ)u-yd(2)+χ-21+2γ2χ-22v+χ-23Q~-ξc,
where χ21, χ22, and χ23 are given as follows:
(57)χ21=f2-∂α1∂xˇ1(f1+xˇ2)-∂α1∂ydyd(1)-∂α1∂θˇδ-∂α1∂η~(Afη~+p2z1)-∂α1∂η(Afη+p2xˇ1)-∂α1∂Φ1wˇ(Φ2wˇ+ϖ1)′-∂α1∂Φ1u(Φ2u+ψ1)′-∂α1∂Σ→(ϵ-1)×(Σ(η′T1+Φ1wˇ+Φ1u)′(γ2ζ2-1)(η′T1+Φ1wˇ+Φ1u)Σ)→-∂α1∂sΣ(γ2ζ2-1)(1-ϵ)(η′T1+Φ1wˇ+Φ1u)×(η′T1+Φ1wˇ+Φ1u)′,χ22=12γ2(h2-∂α1∂xˇ1h1-∂α1∂θˇκ-∂α1∂η~p2e2,1′ζ-∂α1∂ηp2e2,1′ζ-∂α1∂Φ1wˇA-23′e2,2e2,2′),χ23=ϱ2-∂α1∂xˇ1ϱ1-∂α1∂θˇφ.
Introduce the following value function for this step:(58)V-2=V-1+12z22.
Its derivative can be written as
(59)V-˙2=-z12-|η~|Y2-∑j=12‍β-jzj2+γ2|v|2-γ2|v-ν-2|2+ς-2′Q~-ξc,
with the control law defined by
(60)u=μ-(z1,z2,xˇ1,xˇ2,yd,yd(1),θˇ,η~,η,Φ1wˇ,Φ2wˇ,Φ1u,…,Φru,wˇ,Σ→,sΣyd(1))∶=-1bp0+A-220θˇ(a22xˇ2-yd(2)-α-2),
where
(61)α-2=-χ-21-2γ2χ-22,2wˇ-2γ2χ-22,1e2,1′ν-1-γ2χ-22,12z2-β-2z2-z1,(62)ν-2=ν-1+e2,1χ-22,1z2,
where χ-22=[χ-22,1χ-22,2]. Clearly, the functions μ-, χ-21, χ-22, χ-23, ν-2, and ς-2 are smooth as long as θˇ∈Θo.
This completes the backstepping design procedure.For the closed-loop adaptive nonlinear system, we have the following value function:(63)U≔V2+W=|θ-θˇ|Σ-12+γ2|x-xˇ-Φ(θ-θˇ)|Π-12+|η~|Z2+12∑j=12‍(xˇj-yd(j-1)-αj-1)2,
where we have introduced α0=0 for notational consistency. The time derivative of this function is given by
(64)U˙=-|Cx-yd|2-γ4|x-x^-Φ(θ-θ^)|Π-1ΔΠ-12-ϵ(γ2ζ2-1)|θ-θ^|Φ′C′CΦ2+2(θ-θˇ)′Pr(θˇ)+|ξc|Q~-2+ς-r′Q~-ξc-|η~|Y2-∑j=12‍β-jzj2+γ2|w|2+γ2|wˇ|2-γ2|w-w*(t,u[0,t],y[0,t],wˇ[0,t],ξ[0,t],ξ^[0,t],ξˇ[0,t])|2-γ2|v-ν-2|2=-|Cx-yd|2-γ4|x-x^-Φ(θ-θ^)|Π-1ΔΠ-12-ϵ(γ2ζ2-1)|θ-θ^|Φ′C′CΦ2+2(θ-θˇ)′Pr(θˇ)+|ξc+12ς-2|Q~-2-14|ς-2|Q~-2-|η~|Y2-∑j=12‍β-jzj2+γ2|w|2+γ2|wˇ|2=-γ2|w-w*(t,u[0,t],y[0,t],wˇ[0,t],ξ[0,t],ξ^[0,t],ξˇ[0,t])|2=-γ2|v-ν-2|2.
Then the optimal choice for the variables ξc and ξ^ are
(65)ξc*=-12ς-2⟺ξ^*=ξˇ-12ς-2,
which yields that the closed-loop system is dissipative with storage function U and supply rate
(66)-|x1-yd|2+γ2|w|2+γ2|wˇ|2.
Furthermore, the worst case disturbance with respect to the value function U is given by
(67)wopt=ζE′e2,1′ν-2+1γ2(I-ζ2E′E)D-′Σ--1(ξ-ξˇ)+ζ2E′C(xˇ-x),(68)wˇopt=e2,2ν-2.


## 5. Main Result

For the adaptive control law withξc chosen according to (65), the closed-loop system dynamics are
(69)X˙=F(X,yd(2))+G(X)w+Gwˇ(X)wˇ.X is the state vector of the close-loop system and given by
(70)X:=[θ′x′sΣθˇ′xˇ′η′ηd′ηwˇ′Φu→′Σ←′ydyd(1)]′,
which belongs to the set 𝒟:={X|Σ>0,sΣ>0,θˇ∈Θo}; F and G are smooth mapping of 𝒟×ℝ and 𝒟, respectively, and with the initial condition X(0)=X0∈𝒟0:={X0∈𝒟|θ∈Θ,θˇ0∈Θ,Σ(0)=γ-2Q0-1>0,Tr((Σ(0))-1)≤Kc,sΣ(0)=γ2Tr(Q0)}.Since (64) holds, the value function U satisfies Hamilton-Jacobi-Isaacs equation, for all X∈𝒟, for all yd(2)∈ℝ. 						(71)∂U∂X(X)F(X,yd(2))+14γ2∂U∂X(X)[G(X)Gw(X)]·[G(X)Gw(X)]′(∂U∂X(X))′+Q(X,yd(2))=0,
where Q:𝒟×ℝ→ℝ is smooth and given by
(72)Q(X,yd(2))=|x1,1-yd|2+(∑j=12‍|ξ-ξˇ|Q~-2+Pr(θˇ)|η^|Y2-2(θ-θˇ)′×Pr(θˇ)+∑j=12‍βjzj2+14|ς2|Q-2).Although the value functionU satisfies an Hamilton-Jacobi-Isaacs equation, we cannot deduce the stability and robustness properties of the closed-loop system directly from (64), since U is not a positive-definite function of the closed-loop state vector X. We will use the following theorem to precisely state the strong stability properties of the closed-loop adaptive system.Theorem 10.
Consider the robust adaptive control problem formulated in Section2 with Assumptions 1–7 holding. The robust adaptive controller μ defined by (60) with the optimal choice for the worst-case estimate ξ^ defined by (65) achieves the following strong robustness properties for the closed-loop system:(1)
The controllerμ achieves disturbance attenuation level γ for any uncertainty quadruple (x(0),θ,w[0,∞),wˇ[0,∞),Yd0,yd(2))∈𝒲.(2)
Given acw>0, there exists a constant cc>0 and a compact set Θc⊂Θo, such that for any uncertainty (x(0),θ,w`[0,∞),wˇ[0,∞),Yd) with
(73)|x(0)|≤cw;|w`(t)|≤cw;|wˇ(t)|≤cw;|Yd(t)|≤cw;∀t∈[0,∞)
all closed-loop state variables x, xˇ, θˇ, Σ, sΣ, η, ηwˇ, ηd, and λ are bounded as follows, for all t∈[0,∞),
(74)|x(t)|≤cc,|xˇ(t)|≤cc,θˇ(t)∈Θc,|η(t)|≤cc,|ηd(t)|≤cc,|λ(t)|≤cc,|ηwˇ|≤cc,1KcI≤Σ(t)≤1γ2Q0-1,1Kc≤sΣ(t)≤1γ2Tr(Q0).(3)
For any uncertainty quadruple(x(0),θ,w`[0,∞),wˇ[0,∞),Yd[0,∞)) with w`[0,∞)∈ℒ2∩ℒ∞, wˇ[0,∞)∈ℒ2∩ℒ∞ and Yd[0,∞)∈ℒ∞, the output of the system, x1, asymptotically tracks the reference trajectory, yd, that is,
(75)limt→∞(x1(t)-yd(t))=0.Proof.
For the frits statement, if we define(76)l0(xˇ0,θˇ0):=V-2(0)=12∑j=12‍zj2(0),l(τ,θ,x,y[0,t],wˇ[0,t])≔|ξ-ξ^|Q~-2-2(θ-θˇ)′Pr(θˇ)+14|ς-2|Q~-2+|η~|Y2+∑j=12‍β-jzj2=γ4|(x-x^)-Φ(θ-θ^)|Π-1ΔΠ-12+ϵ(γ2ζ2-1)|θ-θ^|Φ′C′CΦ2-2(θ-θˇ)′Pr(θˇ)+14|ς-2|Q~-2+|η~|Y2+∑j=12‍β-jzj2,
then, we have
(77)Jt=Jt+∫0t‍U˙dτ-U(t)+U(0)≤-U(t)≤0.
It follows that
(78)sup(x(0),θ,w[0,∞),wˇ[0,∞))∈𝒲Jt≤0.
This establishes the first statement.
Next, we will prove the second statement. Define[0,tf) to be the maximal interval on which the closed-loop system admits a solution. We will show that tf is always ∞.
Fixcw≥0, and cd≥0, consider any uncertainty (x0,θ,w`[0,∞),wˇ[0,∞),Yd(t)) that satisfies:
(79)|x0|≤cw;|w`(t)|≤cw;|wˇ(t)|≤cw;|Yd(t)|≤cd∀t∈[0,∞).
We define [0,Tf) to be the maximal length interval on which for the closed system there exists a solution that lies in its definition. Furthemore, from the estiamtion design step, Σ and sΣ are uniformly upper bounded and uniformly bounded away from 0 as desired.
Introduce the vector of variables(80)Xe:=[θˇ′(x~-Φθ~)′η~′z1z2]′
and two nonnegtive and continuous functions defined on ℝ6+σ(81)UM(Xe):=Kc|θ~|2+γ2|x~-Φθ~|Π-12+|η~|Z2+∑j=12‍12zj2,Um(Xe):=γ2|θ~|Q02+γ2|x~-Φθ~|Π-12+|η~|Z2+∑j=12‍12zj2,
then, we have
(82)Um(Xe)≤U(t,Xe)≤UM(Xe),∀(t,Xe)∈[0,Tf)×ℝ6+σ.
Since Um(Xe) is continuous, nonnegative definite and radially unbounded, then for all α∈ℝ, the set S1α:={Xe∈ℝ6+σ∣Um(Xe)≤α} is compact or empty. Since |w`(t)|≤cw, and |wˇ(t)|≤cw, for all t∈[0,∞), we have the following inequality for the derivative of U:
(83)U˙≤-γ42|x-xˇ-Φ(θ-θˇ)|Π-1ΔΠ-12+2(θ-θˇ)′Pr(θˇ)-|η~|Y2-∑i=12‍β-izi2+γ2∥M`∥22cw2+γ2cw2.
Since -(γ4/2)|x-xˇ-Φ(θ-θˇ)|Π-1ΔΠ-12-|η~|Y2+2(θ-θˇ)′Pr(θˇ)-∑j=12‍β-jzj2 will tend to -∞ when Xe approaches the boundary of Θo×ℝ6, then there exists a compact set Ω1(cw)⊂Θo×ℝ6, such that U˙<0 for for all Xe∈Θo×ℝ6∖Ω1. Then, U(t,Xe(t))≤c1, and Xe(t) is in the compact set S1c1⊆ℝ6+σ, for all t∈[0,Tf). It follows that the signal Xe is uniformly bounded, namely, θ~, x~-Φθ~, η~, z1, and z2 are uniformly bounded.
Based on the dynamics ofηd, we have ηd is uniformly bounded. Since η~=η-ηd is uniformly bounded, then η is also uniformly bounded. Furthermore, there is a particular linear combination of the components of η, denoted by ηL,
(84)η˙=Afη+p2y,ηL=TLη,
which is strictly minimum phase and has relative degree 1 with respect to y. Then the signal ηL has relative degree 3 with respect to the input u, and is uniformly bounded.
NoteΦ=Φy+Φu+Φwˇ. Since Φy and Φwˇ are uniformly bounded, to prove Φ is bounded, we need to prove Φu is uniformly bounded. Define the following equations to separate Φu into two parts:
(85)Φu=Φus+λbA-220,λb=[λb1λb2],λ˙b=Afλb+e2,2u;λb(0)=02×1,Φus=[Φus1Φus2],Φ˙us=AfΦus;Φus(0)=Φu0.
Clearly Φus is uniformly bounded because Af is Hurwitz. The first-row element of x~-Φθ~ is
(86)x~1-Φus1θ~-λb1A-220θ~-Φ1wˇθ~-η′T1θ~.
We can conclude that x~1-λb1A-220θ~ is uniformly bounded in view of the boundedness of x~-Φθ~, θ~, Φus, Φwˇ, and η. Since z1=xˇ1-yd, and z1, yd are both uniformly bounded, xˇ1 is also uniformly bounded.
Notice thatAf=A-ζ2LC-ΠC′C(ζ2-γ-2) and b0=bp0+A-220θ, we generated the signal x1-b0λb1 by
(87)x˙-b0λ˙b=Af(x-b0λb)+A-21θy+DM`w`+wˇA-23θ+(ζ2L+ΠC′(ζ2-1γ2))(y-EM`w`)+Dˇwˇ,x1-b0λb1=C(x-b0λb).
Since x1-b0λb1 has relative degree at least 1 with respect to y, take ηL and y as output and input of the reference system, we conclude x1-b0λb1 is uniformly bounded by bounding lemma. It follows that xˇ1-λb1(bp0+A2120θˇ) is also uniformly bounded. Since xˇ1 is uniformly bounded and θˇ is uniformly bounded away from 0, we have λb1 is uniformly bounded. That further implies that Φ1, that is, CΦ, is uniformly bounded. Furthermore, since x1-b0λb1, wˇ and w` are bounded, we have that the signals of x1 and y are uniformly bounded. It further implies the uniform boundedness of x-b0λb since Af is a Hurwitz matrix. By a similar line of reasoning above, we have x2, λb,2 are uniformly bounded. Then we can conclude that Φus and Φ are uniformly bounded.
Next, we need to prove the existence of a compact setΘc⊂Θo such that θˇ(t)∈Θc, for all t∈[0,Tf). First introduce the function
(88)Υ:=U+(ρo-P(θˇ))-1P(θˇ).
We notice that, when θˇ approaches the boundary of Θo, P(θˇ) approaches ρo. Then Υ approaches ∞ as Xe approaches the boundary of Θo×ℝ6. We introduce two nonnegative and continuous functions defined on Θo×ℝ4:
(89)ΥM:=UM(Xe)+(ρo-P(θˇ))-1P(θˇ),Υm:=Um(Xe)+(ρo-P(θˇ))-1P(θˇ).
Then, by the previous analysis, we have
(90)Υm(Xe)≤Υ(t,Xe)≤ΥM(Xe),∀(t,Xe)∈[0,Tf)×Θo×ℝ6.
Note that the set S2α:={Xe∈Θo×ℝ6∣Υm(Xe)≤α} is a compact set or empty. Then, we consider the derivative of Υ as follows:
(91)Υ˙=U˙+(ρo-P(θˇ))-2ρo∂P∂θ(θˇ)θˇ˙≤-γ42|x-xˇ-Φ(θ-θˇ)|Π-1ΔΠ-12+2(θ-θˇ)′Pr(θˇ)-|η~|Y2-∑j=1r‍cβjzj2-|(∂P∂θ(θˇ))′|2(ρo-P(θˇ))-4×(Kc-1ρopr(θˇ)(ρo-P(θˇ))2-c)+c,
where c∈ℝ is a positive constant. Since Υ˙ will tend to -∞ when Xe approaches the boundary of Θo×ℝ4, there exists a compact set Ω2(cw)⊂Θo×ℝ4, such that for all Xe∈Θo×ℝ4∖Ω2, Υ˙(Xe)<0. Then, there exists a compact set Θc⊂Θo, such that θˇ(t)∈Θc, for all t∈[0,Tf). Moreover, Υ(t,Xe(t))≤c2, and Xe(t) is in the compact set S2c2⊆Θo×ℝ6, for all t∈[0,Tf). It follows that Pr(θˇ) is also uniformly bounded.
Also,η, λ are some stably filtered signals of u and y, they are uniformly bounded. Since ηwˇ is uniformly bounded, Φ is uniformly bounded. Then we can conclude xˇ is uniformly bounded from the boundedness of x~-Φθ~. This further implies that the control input u is uniformly bounded.
Then we can get the conclusion that the complete system states andu are uniformly bounded on [0,tf), Σ, sΣ are uniformly bounded and bounded away from 0, and θˇ is uniformly bounded away from the boundary of the set Θo. Therefore, it follows that tf=∞ and the complete system states are uniformly bounded on [0,∞).
Last, we will establish the third statement. By the following inequality:(92)∫0∞‍U˙dτ≤∫0∞‍(-|x1-yd|2+γ2|M`w`|2+γ2|wˇ|2)dτ,
it follows that
(93)∫0∞‍|x1-yd|2dτ≤∫0∞‍(γ2|M`w`|2+γ2|wˇ|2)dτ+U(0)<+∞.
By the second statement, we notice that
(94)sup0≤t<∞|x˙1-y˙d|<∞.
Then we have
(95)limt→∞|x1(t)-yd(t)|=0.
This complete the proof of the theorem.


## 6. Example

In this section, we present one example to illustrate the main results of this paper. The designs were carried out using MATLAB symbolic computation tools, and the closed-loop systems were simulated using SIMULINK.The example was based on a four-pole-permanent-magnet brushed DC motor. We assume that the nominal values ofKt, Ke, J, R, and L are given as below, and the variations can be lumped into the arbitrary disturbance w`.Kt=0.01; N-cm/Amp,Ke=1; Volt/rad/s,J=0.01; N-cm/rad/s2,R=1; Ohm,L=0.1; L.The value ofD is unknown and with true value 0.01 N-cm/rad/s. Then the true system is of the following state-space representation
(96)[ω˙i˙]=[θ1-10-10][ωi]+[010]u+[10]Twˇ+[101000][TwwωTf];[ω(0)i(0)]=[00],y=[10][ωi]+[010][TwwωTf],
where ω is the motor speed in rad/s, i is the motor current in amp, u is control input in volt, y is the motor speed measurement in rad/s, Twˇ is the estimated disturbance torque in N-cm, Tw is the arbitrary disturbance torque in N-cm, Tf is the friction torque in N-cm, wω is the measurement channel noise in rad/s, θ is the 1-dimensional unknown parameter with the true value θ*=-1 belonging to the interval [-2,0].The control objective is to have the system output tracking velocity reference trajectory,yd, which is generated by the following linear system:
(97)yd=ds3+2s2+2s+3,
where d is the command input signal.Introduce the following state and disturbance transformation:(98)x=[10101][ωi];w=[1-θ1010][TwwωTf].
We obtain the design model
(99)x˙=[-101-100]x+[110]yθ+[010]u+[110]wˇ+[10100]w,y=[10]x+[01]w.
The ultimate performance lower bound for this system is 1 with respect to w. For the adaptive control design, we set the desired disturbance attenuation level γ=2. The parameter θ is assumed to belong to the set [-2,0], with the projection function P(θ) chosen as
(100)P(θ)=(θ+1)2.For other design and simulation parameters, we select(101)xˇ0=[0.10.5];θˇ0=-0.5;Q0=1;Kc=100;Δ=[1001];β-1=β-2=0.5;Y=[159.2262-17.0150-17.01501.8786].
Then, we obtain
(102)Af=[-10.29931.0000-12.28820];Z=[8.8506-0.9393-0.93930.1229];Π=[0.59874.57644.576443.1208].We present two sets of simulation results in this example. In the first set of simulation, we setTf=0; N-cm,Tw=0; N-cm,wω=0; rad/s,Twˇ=0; N-cm.This simulation is to demonstrate the regulatory behaviour of the adaptive controller. The results are shown in Figures1(a)–1(f). We observe from Figure 1 that the parameter estimate of -D/J asymptotically converges to its true value −1, the output-tracking error and state-estimation error asymptotically converge to zeros and t within 20 second. The control input is bounded by 12, and the transient of the system is well behaved.System response for example under command inputd(t)=5sin(t), Tf=0, Tw=0, wω=0, and Twˇ=0. (a) Tracking error; (b) control input; (c) parameter estimate; (d) state-estimation error xˇ1-x1; (e) state-estimation error xˇ2-x2; (f) cost function ∫0t‍((x1-yd)2-γ2|w|2-γ2|wˇ|2d)τ.
(a)
(b)
(c)
(d)
(e)
(f)The second set of simulation results is to demonstrate the robustness of the adaptive controller to unmodeled exogenous disturbance inputs. We setTf=-0.01×sgn(ω); N-cm,Tw=0.04sin(t); N-cm,wω = White noise signal with power 0.01 sample d at 1 HZ; rad/s,Twˇ=0.05sin(4t); N-cm.The simulation results are presented in Figures2(a)–2(f). We observe that the the parameter estimate of -D/J no longer converges to the true value −1, but it’s stabilized around the true value. The output-tracking error and state-estimation error no longer converge to zeros, but output-tracking error satisfies the targeted attenuation level based on Figure 2(f), and the state-estimation errors asymptotically oscillate around zeros. The control input is again bounded by 12, and the transient of the system is well behaved as well.System response for example under command inputd(t)=5sin(t), Tf=-0.01×sgn(ω), and Tw=0.04sin(t), wω = white noise signal with power 0.01 sample d at 1 HZ, Twˇ=0.05sin(4t). (a) Tracking error; (b) control input; (c) parameter estimate; (d) state-estimation error xˇ1-x1; (e) state-estimation error xˇ2-x2; (f) cost function ∫0t‍((x1-yd)2-γ2|w|2-γ2|wˇ|2d)τ.
(a)
(b)
(c)
(d)
(e)
(f)


## 7. Conclusions

In this paper, we studied the permanent magnet brushed DC adaptive control design for velocity tracking applications. We formulate the robust adaptive control problem as a nonlinearH∞-control problem under imperfect state measurements, and then use cost-to-come function analysis and the integrator backstepping methodology to obtain the controller. The controller then achieves the desired disturbance attenuation level, with the ultimate lower bound of the attenuation level being the noise intensity in the measurement channel. It also guarantees the total stability of the closed-loop system and achieves asymptotic tracking of the reference trajectory when the disturbance is of finite energy and uniformly bounded.



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*Source: 698050-2013-02-25.xml*