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

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 nonlinear 𝐻^∞-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.

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 nonlinear 𝐻^∞-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 Section 2, 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:̇̀−𝐷𝑥=𝐽^𝑙𝐾_𝑡𝐽−𝐾_𝑒𝐿−𝑅𝐿01𝑥+𝐿1𝑢+𝐽0𝑇_̌‌𝑤+𝐷̀̀𝐸̀̀𝑤,𝑦=10𝑥+𝑤, (1a)(1b)where 𝑥̀∈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; 𝐾_𝑒 is the back-emf constant; 𝐾_𝑡 is the motor torque constant; 𝐽 is the motor system inertia; 𝐷_𝑙∈ℝ is the motor-system damping factor; 𝑤̀=[𝑇_𝑤𝑤̀_𝜔𝑇_𝑓]^′ is the arbitrary disturbance vector, which is composed of arbitrary disturbance torque, 𝑇_𝑤∈ℝ, arbitrary disturbance at the output measurement channel, 𝑤̀_𝜔∈ℝ, and friction disturbance torque 𝑇_𝑓∈ℝ; 𝑇_̌‌𝑤∈ℝ is the measured or estimated disturbance torque. In this paper, we assume the variables 𝐾_𝑡, 𝐾_𝑒, 𝐽, 𝐷_𝑙, 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 from 𝑢 to 𝑦 has relative degree 𝑟=2. And then we can find a state diffeomorphism 𝑥=𝑇̀𝑥̀ and a disturbance transformation 𝑤=𝑀̀𝑤̀ by following [22], such that system (1a) and (1b) can be transformed into the following form in the 𝑥 coordinates:𝑥̇=𝐴𝑥+𝑦𝐴₂₁+𝑢𝐴₂₂+𝑤^̌‌𝐴₂₃𝜃+𝐵𝑢+𝐷𝑤+𝐷^̌‌𝑤^̌‌,𝑦=𝐶𝑥+𝐸𝑤, (2a)(2b)where 𝑥∈ℝ² is the state vector, 𝑥=[𝑥₁𝑥₂]^′, 𝑤^̌‌=𝑇_̌‌𝑤, 𝜃∈ℝ^𝜎, 𝜎∈ℕ, is the 𝜎-dimensional vector of unknown parameters of the true system; the matrices 𝐴, 𝐴₂₁, 𝐴₂₂, 𝐴₂₃, 𝐷, 𝐷^̌‌, 𝐶, and 𝐸 are of appropriate dimensions, known and have the structure as below, 𝑎𝐴=¹¹1𝑎₂₁𝑎₂₂,𝐴₂₂=𝟎_1×𝜎𝐴₂₂₀,0𝑏𝐵=_𝑝0,𝐶=10, (3) where 𝐴₂₂₀ is a 𝜎-dimensional row vector and 𝑏_𝑝0∈ℝ. Then the high-frequency gain of (2a) and (2b) is 𝑏₀=𝑏_𝑝0+𝐴₂₂₀𝜃.

Since we consider a trajectory tracking control design problem, we make the following assumption about the reference signal 𝑦_𝑑.

Assumption 1. The reference trajectory, 𝑦_𝑑, is two times continuously differentiable. Define vector 𝑌_𝑑∶=[𝑦_𝑑⁽⁰⁾𝑦_𝑑⁽¹⁾𝑦_𝑑⁽²⁾]^′ and 𝑌_𝑑0∶=[𝑦_𝑑(0)𝑦_𝑑⁽¹⁾(0)]^′. The signal 𝑌_𝑑 is available for control design.

To guarantee the stability of the closed-loop system and the boundedness of the estimate of 𝜃, 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 function 𝑃∶ℝ^𝜎→ℝ, such that the true value 𝜃∈Θ∶={𝜃∈ℝ^𝜎∶𝑃(𝜃)≤1}. For all 𝜃∈Θ, 𝑏₀=𝑏_𝑝0+𝐴₂₂₀𝜃>0.

The control objective is to design a robust adaptive controller for system (1), such that the output 𝑦(𝑡) tracks a desired reference signal 𝑦_𝑑(𝑡), while rejecting the uncertainties (𝑥(0),𝜃,𝑤̀_[0,∞),𝑤^̌‌_[0,∞),𝑌_𝑑0,𝑦_𝑑⁽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 controller 𝜇 is said to achieve disturbance attenuation level 𝛾 if there exist nonnegative functions 𝑙(𝑡,𝜃,𝑥,𝑦_[0,𝑡],𝑇_{̌‌𝑤[0,𝑡]}) and 𝑙₀(𝑥^̌‌₀,𝜃^̌‌₀) such that sup_{(𝑥(0),𝜃,𝑤𝜔[0,∞),𝑇𝑓[0,∞),𝑇𝑤[0,∞),𝑇̌𝑤[0,∞))}𝐽^𝑡≤0, (4) where 𝐽^𝑡=^𝑡₀𝐶𝑥−𝑦_𝑑²+𝑙−𝛾²𝑇_𝑤²−𝛾²𝑇_𝑓²−𝛾²𝑤_𝜔²−𝛾²𝑇_̌‌𝑤²𝑑𝜏−𝛾²𝜃−𝜃^̌‌₀²_𝑄0−𝛾²𝑥(0)−𝑥^̌‌₀²_Π0−1−𝑙₀, (5) where 𝜃^̌‌₀∈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 𝜃^̌‌₀ quantifying the level of confidence in the estimate 𝜃^̌‌₀; 𝑥^̌‌₀ 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 𝑥^̌‌₀; |𝑧|_𝑄 denotes 𝑧^𝑇𝑄𝑧 for any vector 𝑧 and any symmetric matrix 𝑄.

The control law to system (2a), (2b) is generated by the following control law 𝑢(𝑡)=𝜇𝑡,𝑦_[0,𝑡],𝑤^̌‌_[0,𝑡], (6) 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 𝐶𝑥−𝑦_𝑑²₂𝑇_𝑤²₂+𝑇_𝑓²₂+𝑤_𝜔²₂+𝑇_̌‌𝑤²₂+𝜃−𝜃^̌‌₀²_𝑄0+𝑥(0)−𝑥^̌‌₀²_Π0−1+𝐶₀⁻¹≤𝛾², (7) where ‖⋅‖₂ denotes ℒ₂ norm, and 𝐶₀ is a constant. When ‖𝑤̀‖₂ and ‖𝑤^̌‌‖₂ are finite, ‖𝐶𝑥−𝑦_𝑑‖₂ is also finite, which implies lim_𝑡→∞|𝐶𝑥−𝑦_𝑑|=0, under additional assumptions.

The following notation will be used throughout this paper: 𝑥^̌‌ denotes the estimate of the current state of the system; 𝑥 denotes the state-estimation error 𝑥−𝑥^̌‌; 𝜃^̌‌ denotes the estimate of the parameter vector 𝜃; 𝜃 denotes the estimation error 𝜃−𝜃^̌‌; 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; 𝑒_𝑗,𝑖 denotes a 𝑗-dimensional column vector, all of its elements are 0, except its 𝑖th row is 1, such as 𝑒_2,2=[01]^′.

Let 𝜉 denote the expanded state vector 𝜉=[𝜃^′𝑥^′]^′, we have the following expanded dynamics for system (2a), (2b) in view of 𝜃̇=0:̇𝑦𝜉=𝟎𝟎𝐴₂₁+𝑢𝐴₂₂+𝐴^̌₂₃𝑤𝐴𝜉+𝟎𝐵𝟎𝐷𝟎𝑢+𝑤+_̌‌𝐷^̌‌𝑤=∶𝐴𝜉+𝐵𝑢+𝐷𝑤+𝐷^̌‌𝑤^̌‌,𝑦=𝟎𝐶𝜉+𝐸𝑤=∶𝐶𝜉+𝐸𝑤. (8a)(8b)

The worst-case optimization of the cost function (4) can be carried out in two steps as depicted in the following equations: sup_{(𝑥(0),𝜃,𝑤[0,∞),̌‌𝑤[0,∞))∈𝒲}𝐽^𝑡=sup_{𝑦[0,∞),̌‌𝑤[0,∞)}sup_{(𝑥(0),𝜃,𝑤[0,∞))|𝑦[0,∞),̌‌𝑤[0,∞)}𝐽^𝑡. (9) 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 Section 4. 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 waveform 𝑦_[0,∞) and 𝑤^̌‌_[0,∞) are assumed to be known. Since the control input is a causal function of 𝑦 and 𝑤^̌‌, then it is known. We ignore terms considered to be constant in the estimation design step, and set 𝑙 in (5) to be |𝜉−𝜉̂|²𝑄+2(𝜉−𝜉^̌‌)^′𝑙₂+𝑙̂. The equivalent cost function of (5) is then given by, 𝐽^𝛾𝑡𝑓∶=^𝑡𝑓₀𝜉−𝜉̂²𝑄+2𝜉−𝜉^̌‌′𝑙₂+𝑙̂−𝛾²|𝑤|²d𝜏−𝛾²𝜉̃₀²𝑄₀, (10) where 𝑄 is a matrix-valued weighting function, 𝜉̂ is the worst-case estimates for the expanded state 𝜉, 𝑙₂ is a design function, and 𝑙̂ 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 function 𝑊=|𝜉−𝜉^̌‌|²_Σ−1, and then we can obtain the dynamics of state estimator 𝜉^̌‌ and worst-case covariance matrix Σ as belowΣ̇‌=𝐴−𝜁²𝐿𝐶Σ+Σ𝐴−𝜁²𝐿𝐶^′+1𝛾²𝐷𝐷^′−1𝛾²𝜁²𝐿𝐿^′−Σ𝛾²𝜁²𝐶^′𝐶−𝐶^′𝐶−𝑄Σ,Σ(0)=1𝛾²𝑄₀𝟎𝟎_𝑛×𝜎Π₀⁻¹⁻¹,_̌‌̇𝜉=𝐴+Σ𝐶^′𝐶+𝑄𝜉^̌‌+𝜁²𝛾²Σ𝐶^′+𝐿𝑦−𝐶𝜉^̌‌+𝐵𝑢+𝐷^̌‌𝑤^̌‌−Σ𝐶^′𝑦_𝑑+𝑄𝜉̂,_̌‌𝜉(0)=^̌‌𝜃_0̌𝑥₀, (11a)(11b)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 𝐽^𝑡=−𝜉(𝑡)−𝜉^̌‌(𝑡)²_Σ−1(𝑡)+^𝑡₀𝐶𝜉^̌‌−𝑦_𝑑²+𝜉_𝑐²𝑄−𝛾²𝜁²𝑦−𝐶𝜉^̌‌2−𝛾²𝑤−𝑤^∗𝑢_[0,𝜏],𝑦_[0,𝜏],𝑤^̌‌_[0,𝜏],𝜉_[0,𝜏],𝜉̂_[0,𝜏],𝜉^̌‌_[0,𝜏]²𝑑𝜏, (12) where 𝜉_𝑐∶=𝜉̂−𝜉^̌‌, which will be determined later to improve the performance of the adaptive system; 𝑤^∗ is the worst-case disturbance, given by 𝑤^∗𝜏,𝑢_[0,𝜏],𝑦_[0,𝜏],𝑤^̌‌_[0,𝜏],𝜉_[0,𝜏],𝜉̂_[0,𝜏],𝜉^̌‌_[0,𝜏]=𝜁²𝐸^′𝑦−+1𝐶𝜉𝛾²𝐼−𝜁²𝐸^′𝐸𝐷^′Σ⁻¹𝜉−𝜉^̌‌. (13)

We partition Σ as ΣΣ=Σ₁₂Σ₂₁Σ₂₂, (14) and introduce Φ∶=Σ₂₁Σ⁻¹ and Π∶=𝛾²(Σ₂₂−Σ₂₁Σ⁻¹Σ₁₂).

For the boundedness of Σ, we make the following assumption on the weighting matrix 𝑄.

Assumption 4. The weighting matrix 𝑄 of function 𝑙 in (5) is given by 𝑄=Σ⁻¹𝟎𝟎𝟎ΔΣ⁻¹+𝜖Φ^′𝐶^′𝛾²𝜁²−1𝐶Φ𝟎𝟎𝟎, (15) where Δ is 2×2 positive-definite matrix, and 𝜖 is a scalar function defined by 𝜖(𝜏)Σ∶=Tr⁻¹(𝜏)𝐾_𝑐,𝐾_𝑐≥𝛾²Tr𝑄₀,𝜏≥0. (16)

Then, we have the following differential equation of Σ, Φ, and Π:Σ̇=−(1−𝜖)ΣΦ^′𝐶^′𝛾²𝜁²Σ(0)=1−1𝐶ΦΣ;𝛾²𝑄₀⁻¹,Φ̇=𝐴−𝜁²𝐿𝐶−1𝛾²Π𝐶^′𝛾²𝜁²−1𝐶Φ+𝑦𝐴₂₁+𝑢𝐴₂₂+𝐴^̌₂₃𝑤(0)̇;Φ=𝟎,Π=𝐴−𝜁²𝐿𝐶Π+Π𝐴−𝜁²𝐿𝐶^′−Π𝐶^′𝜁²−1𝛾²𝐶Π+𝐷𝐷^′−𝜁²𝐿𝐿^′+𝛾²Δ,Π(0)=Π₀. (17a)(17b)(17c)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: 1𝐾_𝑐(𝜏)(0)=1≤Σ≤Σ𝛾²𝑄₀⁻¹;𝛾²Tr𝑄₀≤TrΣ⁻¹(𝜏)≤𝐾_𝑐;∀𝜏∈[0,𝑡]. (18)

We define 𝑠_Σ(𝑡)∶=Tr((Σ(𝑡))⁻¹), and its dynamic is given by 𝑠_Σ̇=𝛾²𝜁²(1−𝜖)𝐶ΦΦ^′𝐶^′;𝑠_Σ(0)=𝛾²Tr𝑄₀. (19) Then, 𝜖(𝜏)=𝐾_𝑐⁻¹𝑠_Σ⁻¹(𝜏), which does not require the inversion of Σ.

From Assumption 4 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 matrix 𝐴−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 matrix Π₀ in (17c) is chosen as the unique positive definite solution to the following algebraic Riccati equation: 𝐴−𝜁²𝐿𝐶Π+Π𝐴−𝜁²𝐿𝐶^′−Π𝐶^′𝜁²−1𝛾²𝐶Π+𝐷𝐷^′−𝜁²𝐿𝐿^′+𝛾²Δ=0. (20)

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 choose 𝜖=1 so that Σ will be a constant positive-definite matrix, and 𝑠_Σ 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 𝜌∶=inf𝑃𝜃∣𝜃∈ℝ^𝜎,𝑏_𝑝0+𝐴₂₂₀𝜃=0. (21) By Assumption 2 and Lemma  2 in [23], we have 1<𝜌≤∞. For any fixed 𝜌_𝑜∈(1,𝜌), we define the open set Θ_𝑜∶=𝜃∈ℝ^𝜎∣𝑃𝜃<𝜌_𝑜. (22) Our control design will guarantee that the estimate 𝜃^̌‌ lies in Θ_𝑜, which immediately implies |𝑏_𝑝0+𝐴^̌₂₂₀𝜃|>𝑐₀>0, for some 𝑐₀>0. Moreover, the convexity of 𝑃 implies the following inequality: 𝜕𝑃𝜕𝜃^̌‌𝜃𝜃−𝜃^̌‌<0∀𝜃^̌‌∈ℝ^𝜎⧵Θ. (23) We set 𝑙₂=[−(𝑃_𝑟(𝜃^̌‌))^′𝟎]^′, where 𝑃𝜃_𝑟^̌‌𝑒∶=^{1/(1−𝑃(̌‌𝜃))}(𝜕𝑃/𝜕𝜃)𝜃^̌‌′𝜌_𝑜−𝑃𝜃^̌‌3∀𝜃∈Θ_𝑜⧵Θ,𝟎∀𝜃∈Θ. (24) Then, we obtain ̇‌_̌‌𝜉=−Σ𝑃𝜃_𝑟^̌‌′𝟎^′+𝐴𝜉^̌‌+𝐵𝑢−Σ𝑄𝜉_𝑐+𝜁²𝛾²Σ𝐶^′+𝐿𝑦−𝐶𝜉^̌‌+𝐷^̌‌𝑤^̌‌,_̌‌𝜉(0)=𝜃^̌‌𝑥^′₀^̌′₀^′, (25) where 𝜉_𝑐=𝜉̂−𝜉^̌‌.

We summarize the estimation dynamics equations below𝐴−𝜁²𝐿𝐶Π+Π𝐴−𝜁²𝐿𝐶−Π𝐶^′𝜁²−1𝛾²𝐶Π+𝐷𝐷^′−𝜁²𝐿𝐿^′+𝛾²̇()Δ=0,Σ=−1−𝜖ΣΦ^′𝐶^′𝛾²𝜁²(0)=1−1𝐶ΦΣ;Σ𝛾²𝑄₀⁻¹,𝑠_Σ̇=𝛾²𝜁²−1(1−𝜖)𝐶ΦΦ^′𝐶^′;𝑠_𝜎(0)=𝛾²Tr𝑄₀,1𝜖=𝐾_𝑐𝑠_Σ,𝐴_𝑓=𝐴−𝜁²𝐿𝐶−Π𝐶^′𝐶𝜁²−1𝛾²,Φ̇=𝐴_𝑓Φ+𝑦𝐴₂₁+𝑢𝐴₂₂+𝑤^̌‌𝐴₂₃(0)̇;Φ=𝟎,_̌‌𝜃=−Σ𝑃𝜃_𝑟^̌‌−ΣΦ^′𝐶^′𝑦_𝑑−𝐶𝑥^̌‌−ΣΣΦ^′𝑄𝜉_𝑐+𝛾²𝜁²ΣΦ^′𝐶^′𝑦−𝐶𝑥^̌‌;𝜃^̌‌(0)=𝜃^̌‌₀,_̌‌̇𝑥=−ΦΣ𝑃𝜃_𝑟^̌‌+𝐴𝑥^̌‌−1𝛾²Π+ΦΣΦ^′𝐶^′𝑦_𝑑−𝐶𝑥^̌‌−1ΦΣ𝛾²Π+ΦΣΦ^′𝑄𝜉_𝑐+𝑦𝐴₂₁+𝑢𝐴₂₂+𝑤^̌‌𝐴𝜃₂₃^̌‌+𝜁²Π𝐶^′+𝛾²ΦΣΦ^′𝐶^′×+𝐿𝑦−𝐶𝑥^̌‌+𝐷^̌‌𝑤^̌‌+𝐵𝑢;𝑥^̌‌(0)=𝑥^̌‌₀. (26a)(26b)(26c)(26d)(26e)(26f)(26g)(26h)

For the controller structure simplification, the dynamics for Φ can be implemented as below. First, we observe the matrix 𝐴_𝑓 has the same structure as the matrix 𝐴. Then, we introduce the matrix 𝑀_𝑓∶=𝐴_𝑓𝑝₂𝑝₂, (27) where 𝑝₂ is a 2-dimensional vector such that the pair (𝐴_𝑓,𝑝₂) is controllable, which implies that 𝑀_𝑓 is invertible. Then the following prefiltering system for 𝑦, 𝑢, and 𝑤^̌‌ generates the Φ online:𝜂̇=𝐴_𝑓𝜂+𝑝₂(0)̇𝑦;𝜂=0,𝜆=𝐴_𝑓𝜆+𝑝₂(0)𝜂̇𝑢;𝜆=0,_̌‌𝑤=𝐴_𝑓𝜂^̌‌𝑤+𝑝𝑤₂^̌;𝜂^̌‌𝑤(0)𝐴=0,Φ=_𝑓𝜂𝜂𝑀_𝑓⁻¹𝐴₂₁+𝐴_𝑓𝜆𝜆𝑀_𝑓⁻¹𝐴₂₂+𝐴_𝑓𝜂^̌‌𝑤𝜂^̌‌𝑤𝑀_𝑓⁻¹𝐴₂₃. (28a)(28b)(28c)(28d)

Associated with the above identifier, introduce the value function 𝑊𝑡,𝜉(𝑡),𝜉^̌‌(𝑡),Σ(𝑡)=𝜉(𝑡)−𝜉^̌‌(𝑡)²_Σ−1(𝑡)=𝜃−𝜃^̌‌(𝑡)²_Σ−1(𝑡)+𝛾²𝑥(𝑡)−𝑥^̌‌(𝑡)−Φ(𝑡)𝜃−𝜃^̌‌(𝑡)²_Π−1 (29) whose time derivative is given by 𝑊̇=−𝐶𝑥−𝑦_𝑑²−𝛾⁴𝑥−𝑥−Φ𝜃−𝜃²_{Π−1ΔΠ−1}−𝜖𝛾²𝜁²−1𝜃−𝜃²_{Φ′𝐶′𝐶Φ}+𝛾²|𝑤|²+𝐶𝑥^̌‌−𝑦_𝑑²+𝜉_𝑐²𝑄−𝛾²𝜁²𝑦−𝐶𝑥^̌‌2−𝛾²𝑤−𝑤^∗𝑡,𝑢_[0,𝑡],𝑦_[0,𝑡],𝑤^̌‌_[0,𝑡],𝜉_[0,𝑡],𝜉̂_[0,𝑡],𝜉^̌‌_[0,𝑡]²+2𝜃−𝜃^̌‌′𝑃𝜃_𝑟^̌‌. (30) We note that the last term in 𝑊̇ is nonpositive, zero on the set Θ and approaches −∞ as 𝜃^̌‌ approaches the boundary of the set Θ, which guarantees the boundness of 𝜃^̌‌.

Then the cost function can be equivalently written as 𝐽^𝑡=𝐽^𝑡+𝑊(0)−𝑊(𝑡)+^𝑡₀𝑊̇𝑑𝜏=−𝜃−𝜃^̌‌(𝑡)²_Σ−1(𝑡)−𝛾²𝑥(𝑡)−𝑥^̌‌(𝑡)−Φ(𝑡)𝜃−𝜃^̌‌(𝑡)²_Π−1+^𝑡₀𝐶𝑥^̌‌−𝑦_𝑑²+𝜉_𝑐²𝑄−𝛾²𝜁²𝑦−𝐶𝑥^̌‌2−𝛾²𝑤−𝑤^∗×𝜏,𝑢_[0,𝜏],𝑦_[0,𝜏],𝑤^̌‌_[0,𝜏],𝜉_[0,𝜏],𝜉̂_[0,𝜏],𝜉^̌‌_[0,𝜏]²+𝑙̂−𝛾𝑤^2̌‌2𝑑𝜏. (31)

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 𝑤^̌‌ 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: sup_{(0)(𝑥,𝜃,𝑤̌[0,∞)𝑤[0,∞))∈𝒲}𝐽^𝑡≤sup_{𝑦[0,∞),̌‌𝑤[0,∞)}^𝑡₀𝐶𝑥^̌‌−𝑦_𝑑²+𝜉_𝑐²𝑄−𝛾²𝜁²𝑦−𝐶𝑥^̌‌2+𝑙̂−𝛾𝑤^2̌‌2𝑑𝜏−𝑙₀, (32) where function 𝑙(̂𝜏,𝑦_[0,𝜏],𝑤^̌‌_[0,𝜏]) is part of the weighting function 𝑙(𝜏,𝜃,𝑥,𝑦_[0,𝜏],𝑤^̌‌_[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 𝑤^̌‌, and the worst-case estimate for the expanded state vector 𝜉̂, 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 𝑤^̌‌ as the maximizing variable, we can equivalently deal with the transformed variable 𝜁𝑣∶=𝑦−𝐶𝑥^̌‌_̌‌𝑤. (33) Then, we have 𝜂̇=𝐴_𝑓𝜂+𝑝₂𝐶𝑥^̌‌+𝑝₂𝑒^′_2,1𝑣𝜁,_̌‌̇𝜃=−Σ𝑃𝜃_𝑟^̌‌−ΣΦ^′𝐶^′𝑦_𝑑−𝐶𝑥^̌‌−ΣΣΦ^′𝑄𝜉_𝑐+𝛾²ΣΦ^′𝐶^′𝜁𝑒^′_2,1̇𝑣,_̌‌𝑥=𝐴𝑥^̌‌−1𝛾²Π+ΦΣΦ^′𝐶^′𝑦_𝑑−𝐶𝑥^̌‌+𝐴^̌₂₁𝜃𝐶𝑥^̌‌−ΦΣ𝑃𝜃_𝑟^̌‌−1ΦΣ𝛾²Π+ΦΣΦ^′𝑄𝜉_𝑐++𝐵𝑢𝐴^̌₂₂𝜃𝑢+𝜁⁻²𝐴^̌₂₁𝜃+Π𝐶^′+𝛾²ΦΣΦ^′𝐶^′+𝐿𝜁𝑒^′_2,1+𝐴^̌₂₃𝜃𝑒^′_2,2+𝟎^̌_𝑛×1𝐷̇𝑣,𝑊=−𝐶𝑥−𝑦_𝑑²−𝛾⁴𝑥−𝑥−Φ𝜃−𝜃²_{Π−1ΔΠ−1}−𝜖𝛾²𝜁²−1𝜃−𝜃²_{Φ′𝐶′𝐶Φ}+𝐶𝑥^̌‌−𝑦_𝑑²+𝜉_𝑐²𝑄+2𝜃−𝜃^̌‌′𝑃𝜃_𝑟^̌‌+𝛾𝑤^2̌‌2+𝛾²|𝑤|²−𝛾²|𝑣|²−𝛾²𝑤−𝑤^∗𝑡,𝑢_[0,𝑡],𝑦_[0,𝑡],𝑤^̌‌_[0,𝑡],𝜉_[0,𝑡],𝜉̂_[0,𝑡],𝜉^̌‌_[0,𝑡]². (34)(35)(36)(37)

The variables to be designed at this stage include 𝑢 and 𝜉_𝑐. The design for 𝜉_𝑐 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 of Φ, Φ̇=𝐴_𝑓Φ+𝑦𝐴₂₁+𝑢𝐴₂₂+𝑤^̌‌𝐴₂₃;Φ(0)=𝟎. (38) For ease of the ensuing study, we will separate Φ as the sum of several matrices as follows:Φ=Φ^𝑢+Φ^𝑦+Φ^̌‌𝑤,Φ^𝑦=𝐴_𝑓𝜂𝜂𝑀_𝑓⁻¹𝐴₂₁=𝜂^′𝑇₁𝜂^′𝑇₂Φ̇_𝑢,=𝐴_𝑓Φ^𝑢+𝑢𝐴₂₂;Φ^𝑢(0)Φ̇=𝟎,_̌‌𝑤=𝐴_𝑓Φ^̌‌𝑤+𝑤^̌‌𝐴₂₃;Φ^̌‌𝑤(0)=𝟎, (39a)(39b)(39c)(39d)where 𝑇_𝑖, 𝑖=1,2 are 2×1-dimensional constant matrices depending on 𝐴_𝑓, 𝑀_𝑓, and 𝐴₂₁. Express Φ^𝑢 and Φ^̌‌𝑤 in terms of their row vectors, Φ^𝑢=[Φ^𝑢′₁Φ^𝑢′₂]^′ and Φ^̌‌𝑤=[Φ^̌‌𝑤′₁Φ^̌‌𝑤′₂]^′. Then, 𝐶Φ^𝑦=𝜂^′𝑇₁, 𝐶Φ^𝑢=Φ^𝑢₁, and 𝐶Φ^̌‌𝑤=Φ^̌‌𝑤₁.

We summarized the dynamics for backstepping design in the following, where we have emphasized the dependence of various functions on the independent variables:𝑠_Σ̇=𝛾²𝜁²−1(1−𝜖)𝜂^′𝑇₁+Φ^𝑢₁+Φ^̌‌𝑤₁×𝜂^′𝑇₁+Φ^𝑢₁+Φ^̌‌𝑤₁^′,1𝜖=𝐾_𝑐𝑠_Σ,Σ̇=−(1−𝜖)Σ𝜂^′𝑇₁+Φ^𝑢₁+Φ^̌‌𝑤₁^′×𝛾²𝜁²−1𝜂^′𝑇₁+Φ^𝑢₁+Φ^̌‌𝑤₁̇Σ,_̌‌𝜃=𝛿𝑦_𝑑−𝑥^̌‌₁,𝜂,Φ^̌‌𝑤₁,Φ^𝑢₁,𝜃^̌‌,^→Σ+𝜑𝜂,Φ^→̌‌𝑤,Φ^→𝑢,^→Σ𝑄𝜉_𝑐+𝜅𝜂,Φ^̌‌𝑤₁,Φ^𝑢₁,^→Σ̇𝑣,𝜂=𝐴_𝑓𝜂+𝑝𝑥₂^̌₁+𝑝₂𝑒^′_2,1𝑣𝜁,_̌‌̇𝑥₁=𝑥^̌‌₂+𝑓₁𝑦_𝑑−𝑥^̌‌₁,𝑥^̌‌₁,𝜃^̌‌,𝜂,Φ^̌‌𝑤₁,Φ^𝑢₁,^→Σ+𝜚₁𝜂,Φ^→̌‌𝑤,Φ^→𝑢,^→Σ𝑄𝜉_𝑐+ℎ𝜃₁^̌‌,𝜂,Φ^̌‌𝑤₁,Φ^𝑢₁,^→Σ̇𝑣,_̌‌𝑥₂=𝑎^̌₂₂𝑥₂+𝑏_𝑝0+𝐴^̌₂₂₀𝜃𝑢+𝑓₂𝑦_𝑑−𝑥^̌‌₁,𝑥^̌‌₁,𝑥^̌‌₂,𝜃^̌‌,𝜂,Φ^̌‌𝑤₁,Φ^̌‌𝑤₂,Φ^𝑢₁,Φ^𝑢₂,^→Σ+𝜚₂𝜂,Φ^→̌‌𝑤,Φ^→𝑢,^→Σ𝑄𝜉_𝑐+ℎ𝜃₂^̌‌,𝜂,Φ^̌‌𝑤₁,Φ^̌‌𝑤₂,Φ^𝑢₁,Φ^𝑢₂,^→ΣΦ̇𝑣,_𝑢1=𝜓₁Φ^𝑢₁+Φ^𝑢₂,Φ̇^̌‌𝑤₁=𝜛₁Φ^̌‌𝑤₁+Φ^̌‌𝑤₂+𝑒^′_2,1𝐴₂₃𝑒^′_2,2𝑣, (40a)(40b)(40c)(40d)(40e)(40f)(40g)(40h)(40i)where the nonlinear functions 𝛿, 𝑓₁, and 𝑓₂ are smooth as long as 𝜃^̌‌∈Θ_𝑜; the nonlinear functions 𝜑, 𝜅, 𝜚₁, 𝜚₂, ℎ₁, ℎ₂, 𝜓₁, and 𝜛₁ are smooth. Here, we use Φ^𝑢₁, Φ^𝑢₂, Φ^̌‌𝑤₁, and Φ^̌‌𝑤₂ as independent variables, instead of 𝜆, 𝜂^̌‌𝑤₁, for the clarity of ensuing analysis.

We observe that the above dynamics is linear in 𝜉_𝑐, which will be optimatized after backstepping design. Σ, 𝑠_Σ, Φ^̌‌𝑤, and 𝜃^̌‌ will always be bounded by the design in Section 3, then they will not be stabilized in the control design. Φ^𝑢 is not necessary bounded, since the control input 𝑢 appeared in their dynamics, it can not stabilzed in conjunction with 𝑥^̌‌ 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 the 𝛾 level of disturbance attenuation with respect to the disturbance 𝑣.

Step 1. In this step, we try to stabilize 𝜂 by virtual control law 𝑥^̌‌₁=𝑦_𝑑. Introduce variable 𝜂_𝑑, as the desired trajectory of 𝜂, which satisfies the dynamics 𝜂̇_𝑑=𝐴_𝑓𝜂_𝑑+𝑝₂𝑦_𝑑;𝜂_𝑑(0)=𝟎_2×1. (41) Define the error variable 𝜂=𝜂−𝜂_𝑑. Then, 𝜂 satisfies the dynamics 𝜂̇=𝐴_𝑓𝜂+𝑝₂𝑒^′_2,1𝑣𝜁+𝑝𝑥₂^̌‌₁−𝑦_𝑑. (42) By [14], the following holds.

Lemma 9. Given any Hurwitz matrix 𝐴_𝑓, there exists a positive-definite matrix 𝑌 such that the following generalized algebraic Riccati equation admits a positive-definite solution 𝑍: 𝐴^′_𝑓𝑍+𝑍𝐴_𝑓+1𝛾²𝜁²𝑍𝑝₂𝑝^′₂𝑍+𝑌=0. (43)

Note that 𝐴_𝑓 in (42) is a Hurwitz matrix, then we define the following value function in terms of the positive-definite matrix 𝑍: 𝑉₀𝜂∶=𝜂²_𝑍. (44) Then its time derivative is given by 𝑉̇₀=−𝜂²_𝑌+𝛾²|𝑣|²−𝛾²𝑣−𝜈₀²+2𝜂^′𝑍𝑝𝑥_𝑛^̌‌₁−𝑦_𝑑, (45) where 𝜈₀𝜂=1𝛾²𝜁𝑒_2,1𝑝^′₂𝑍𝜂. (46) If 𝑥^̌‌₁ is control input, then we may choose the control law 𝑥^̌‌₁=𝑦_𝑑, (47) and the design achieves attenuation level 𝛾 from the disturbance 𝑣 to the output 𝑌^1/2(𝜂−𝜂_𝑑). This completes the virtual control design for the 𝜂 dynamics.

Step 2. Define the transformed variable 𝑧₁∶=𝑥^̌‌₁−𝑦_𝑑, (48) which is the deviation of 𝑥^̌‌₁ from its desired trajectory 𝑦_𝑑.

Then the time derivative of 𝑧₁ is given by 𝑧̇₁=𝑓₁𝑧₁,𝑦_𝑑,𝜃^̌‌,𝜂,Φ^̌‌𝑤₁,Φ^𝑢₁,^→Σ+𝑥^̌‌₂−𝑦_𝑑⁽¹⁾+𝜚₁𝜂,Φ^→̌‌𝑤,Φ^→𝑢,^→Σ𝑄𝜉_𝑐+ℎ𝜃₁^̌‌,𝜂,Φ^̌‌𝑤₁,Φ^𝑢₁,^→Σ𝑣, (49) where the function 𝑓₁ is defined as 𝑓₁𝑧₁,𝑦_𝑑,𝜃^̌‌,𝜂,Φ^̌‌𝑤₁,Φ^𝑢₁,^→Σ=𝑓₁𝑦_𝑑−𝑥^̌‌₁,𝑥^̌‌₁,𝜃^̌‌,𝜂,Φ^̌‌𝑤₁,Φ^𝑢₁,^→Σ. (50) Introduce the value function for this step 𝑉₁=𝑉₀+12𝑧²₁ (51) whose derivative is given by 𝑉̇‌₁=−𝜂²_𝑌+𝛾²|𝑣|²−𝛾²𝑣−𝜈₀²+2𝜂𝑍𝑝_𝑛𝑧₁+𝑧𝑥₁^̌‌₂−𝑦_𝑑⁽¹⁾+𝑓₁+𝜚₁𝑄𝜉_𝑐+ℎ₁𝑣=−𝑧²₁−𝜂²_𝑌−𝛽₁𝑧²₁+𝑧₁𝑧₂+𝜍^′₁𝑄𝜉_𝑐+𝛾²|𝑣|²−𝛾²𝑣−𝜈₁², (52) where𝑧₂=𝑥^̌‌₂−𝑦_𝑑⁽¹⁾−𝛼₁,𝜈₁𝑧₁,𝜃^̌‌,𝜂,𝜂,Φ^̌‌𝑤₁,Φ^𝑢₁,^→Σ=𝜈₀+12𝛾²ℎ^′₁𝑧₁,𝛼₁𝑧₁,𝑦_𝑑,𝜃^̌‌,𝜂,𝜂,Φ^̌‌𝑤₁,Φ^𝑢₁,^→Σ,𝑠=−𝑧₁−𝛽₁𝑧₁−2𝑝^′_𝑛𝑍𝜂−𝑓₁−ℎ₁𝜈₀−14𝛾²ℎ₁ℎ^′₁𝑧₁,𝛽₁𝑧₁,𝑦_𝑑,𝜃^̌‌,𝜂,𝜂,Φ^̌‌𝑤₁,Φ^𝑢₁,^→Σ,𝑠≥𝑐_𝛽1>0,𝜍₁𝑧₁,𝑦_𝑑,𝜂,𝜂,Φ^̌‌𝑤₁,Φ^𝑢,^→Σ=𝜚^′₁𝑧₁, (53a)(53b)(53c)(53d)(53e)where 𝑐_𝛽1 is any positive constant, and the nonlinear function 𝛽₁ is to be chosen by the designer. Note that the function 𝛼₁ is smooth as long as 𝜃^̌‌∈Θ_𝑜. If 𝑥^̌‌₂ were the actual controls, then we would choose the following control law: 𝑥^̌‌₂=𝑦_𝑑⁽¹⁾+𝛼₁, (54) and set 𝜉_𝑐=0, to guarantee the dissipation inequality with supply rate −𝑥^̌‌₁−𝑦_𝑑²−𝜂²_𝑌−𝛽₁𝑧²₁+𝛾²𝑣². (55)

This completes the second step of backstepping design.

Step 3. At this step the actual control appears in the derivative of 𝑧₂, which is given by 𝑧̇₂=𝑎^̌₂₂𝑥₂+𝑏_𝑝0+𝐴^̌₂₂₀𝜃𝑢−𝑦_𝑑⁽²⁾+𝜒₂₁+2𝛾²𝜒₂₂𝑣+𝜒₂₃𝑄𝜉_𝑐, (56) where 𝜒₂₁, 𝜒₂₂, and 𝜒₂₃ are given as follows: 𝜒₂₁=𝑓₂−𝜕𝛼₁𝜕𝑥^̌‌₁𝑓₁+𝑥^̌‌₂−𝜕𝛼₁𝜕𝑦_𝑑𝑦_𝑑⁽¹⁾−𝜕𝛼₁𝜕𝜃^̌‌𝛿−𝜕𝛼₁𝜕𝜂𝐴_𝑓𝜂+𝑝₂𝑧₁−𝜕𝛼₁𝐴𝜕𝜂^𝑓𝜂+𝑝𝑥₂^̌₁−𝜕𝛼₁𝜕Φ^̌‌𝑤₁Φ^̌‌𝑤₂+𝜛₁^′−𝜕𝛼₁𝜕Φ^𝑢₁Φ^𝑢₂+𝜓₁^′−𝜕𝛼₁𝜕^→Σ()×𝜖−1_→Σ𝜂^′𝑇₁+Φ^̌‌𝑤₁+Φ^𝑢₁^′𝛾²𝜁²−1𝜂^′𝑇₁+Φ^̌‌𝑤₁+Φ^𝑢₁Σ−𝜕𝛼₁𝜕𝑠_Σ𝛾²𝜁²−1(1−𝜖)𝜂^′𝑇₁+Φ^̌‌𝑤₁+Φ^𝑢₁×𝜂^′𝑇₁+Φ^̌‌𝑤₁+Φ^𝑢₁^′,𝜒₂₂=12𝛾²ℎ₂−𝜕𝛼₁𝜕𝑥^̌‌₁ℎ₁−𝜕𝛼₁𝜕𝜃^̌‌𝜅−𝜕𝛼₁𝜕𝜂𝑝₂𝑒^′_2,1𝜁−𝜕𝛼₁𝑝𝜕𝜂²𝑒^′_2,1𝜁−𝜕𝛼₁𝜕Φ^̌‌𝑤₁𝐴^′₂₃𝑒_2,2𝑒^′_2,2,𝜒₂₃=𝜚₂−𝜕𝛼₁𝜕𝑥^̌‌₁𝜚₁−𝜕𝛼₁𝜕𝜃^̌‌𝜑. (57)
Introduce the following value function for this step: 𝑉₂=𝑉₁+12𝑧²₂. (58) Its derivative can be written as 𝑉̇‌₂=−𝑧²₁−𝜂²_𝑌−²_𝑗=1𝛽_𝑗𝑧²_𝑗+𝛾²|𝑣|²−𝛾²𝑣−𝜈₂²+𝜍^′₂𝑄𝜉_𝑐, (59) with the control law defined by 𝑢=𝜇𝑧₁,𝑧₂,𝑥^̌‌₁,𝑥^̌‌₂,𝑦_𝑑,𝑦_𝑑⁽¹⁾,_̌‌𝜃,𝜂,𝜂,Φ^̌‌𝑤₁,Φ^̌‌𝑤₂,Φ^𝑢₁,…,Φ^𝑢_𝑟,𝑤^̌‌,^→Σ,𝑠_Σ1∶=−𝑏_𝑝0+𝐴^̌₂₂₀𝜃𝑎^̌₂₂𝑥₂−𝑦_𝑑⁽²⁾−𝛼₂, (60) where 𝛼₂=−𝜒₂₁−2𝛾²𝜒^̌_22,2𝑤−2𝛾²𝜒_22,1𝑒^′_2,1𝜈₁−𝛾²𝜒²_22,1𝑧₂−𝛽₂𝑧₂−𝑧₁,𝜈₂=𝜈₁+𝑒_2,1𝜒_22,1𝑧₂, (61)(62) where 𝜒₂₂=[𝜒_22,1𝜒_22,2]. Clearly, the functions 𝜇, 𝜒₂₁, 𝜒₂₂, 𝜒₂₃, 𝜈₂, and 𝜍₂ are smooth as long as 𝜃^̌‌∈Θ_𝑜.
This completes the backstepping design procedure.

For the closed-loop adaptive nonlinear system, we have the following value function: 𝑈∶=𝑉₂+𝑊=𝜃−𝜃^̌‌2_Σ−1+𝛾²𝑥−𝑥^̌‌−Φ𝜃−𝜃^̌‌2_Π−1+𝜂²_𝑍+12²_𝑗=1^̌‌𝑥_𝑗−𝑦_𝑑^(𝑗−1)−𝛼_𝑗−1², (63) where we have introduced 𝛼₀=0 for notational consistency. The time derivative of this function is given by 𝑈̇=−𝐶𝑥−𝑦_𝑑²−𝛾⁴𝑥−𝑥−Φ𝜃−𝜃²_{Π−1ΔΠ−1}−𝜖𝛾²𝜁²−1𝜃−𝜃²_{Φ′𝐶′𝐶Φ}+2𝜃−𝜃^̌‌′𝑃𝜃_𝑟^̌‌+𝜉_𝑐²𝑄+𝜍^′_𝑟𝑄𝜉_𝑐−𝜂²_𝑌−²_𝑗=1𝛽_𝑗𝑧²_𝑗+𝛾²|𝑤|²+𝛾𝑤^2̌‌2−𝛾²𝑤−𝑤^∗𝑡,𝑢_[0,𝑡],𝑦_[0,𝑡],𝑤^̌‌_[0,𝑡],𝜉_[0,𝑡],𝜉̂_[0,𝑡],𝜉^̌‌_[0,𝑡]²−𝛾²𝑣−𝜈₂²=−𝐶𝑥−𝑦_𝑑²−𝛾⁴𝑥−𝑥−Φ𝜃−𝜃²_{Π−1ΔΠ−1}−𝜖𝛾²𝜁²−1𝜃−𝜃²_{Φ′𝐶′𝐶Φ}+2𝜃−𝜃^̌‌′𝑃𝜃_𝑟^̌‌+𝜉_𝑐+12𝜍₂²𝑄−14𝜍₂²𝑄−𝜂²_𝑌−²_𝑗=1𝛽_𝑗𝑧²_𝑗+𝛾²|𝑤|²+𝛾𝑤^2̌‌2−𝛾²𝑤−𝑤^∗𝑡,𝑢_[0,𝑡],𝑦_[0,𝑡],𝑤^̌‌_[0,𝑡],𝜉_[0,𝑡],𝜉̂_[0,𝑡],𝜉^̌‌_[0,𝑡]²−𝛾²𝑣−𝜈₂². (64) Then the optimal choice for the variables 𝜉_𝑐 and 𝜉̂ are 𝜉^∗_𝑐1=−2𝜍₂⟺𝜉̂^∗=^̌‌1𝜉−2𝜍₂, (65) which yields that the closed-loop system is dissipative with storage function 𝑈 and supply rate −𝑥₁−𝑦_𝑑²+𝛾²|𝑤|²+𝛾𝑤^2̌‌2. (66) Furthermore, the worst case disturbance with respect to the value function 𝑈 is given by 𝑤_opt=𝜁𝐸^′𝑒^′_2,1𝜈₂+1𝛾²𝐼−𝜁²𝐸^′𝐸𝐷^′Σ⁻¹𝜉−𝜉^̌‌+𝜁²𝐸^′𝐶𝑥^̌‌−𝑥_̌‌,𝑤_opt=𝑒_2,2𝜈₂. (67)(68)

5. Main Result

For the adaptive control law with 𝜉_𝑐 chosen according to (65), the closed-loop system dynamics are 𝑋̇=𝐹𝑋,𝑦_𝑑⁽²⁾+𝐺(𝑋)𝑤+𝐺_̌‌𝑤(𝑋)𝑤^̌‌. (69)𝑋 is the state vector of the close-loop system and given by 𝑋∶=𝜃^′𝑥^′𝑠𝜃_Σ^̌𝑥^′̌′𝜂^′𝜂^′_𝑑𝜂Φ^′_̌‌𝑤^→_𝑢^′←Σ^′𝑦_𝑑𝑦_𝑑^(1)′, (70) which belongs to the set 𝒟∶={𝑋|Σ>0,𝑠_Σ>0,𝜃^̌‌∈Θ_𝑜}; 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,𝜃^̌‌₀∈Θ,Σ(0)=𝛾⁻²𝑄₀⁻¹>0,Tr((Σ(0))⁻¹)≤𝐾_𝑐,𝑠_Σ(0)=𝛾²Tr(𝑄₀)}.

Since (64) holds, the value function 𝑈 satisfies Hamilton-Jacobi-Isaacs equation, for all 𝑋∈𝒟, for all 𝑦_𝑑⁽²⁾∈ℝ. 𝜕𝑈(𝑋)𝐹𝜕𝑋𝑋,𝑦_𝑑⁽²⁾+14𝛾²𝜕𝑈(𝑋𝐺(𝑋)𝐺𝜕𝑋)_𝑤(𝑋)⋅𝐺(𝑋)𝐺_𝑤(𝑋)^′𝜕𝑈(𝑋)𝜕𝑋^′+𝑄𝑋,𝑦_𝑑⁽²⁾=0, (71) where 𝑄∶𝒟×ℝ→ℝ is smooth and given by 𝑄𝑋,𝑦_𝑑⁽²⁾=𝑥_1,1−𝑦_𝑑²+𝜉−𝜉^̌‌2𝑄+𝑃𝜃_𝑟^̌‌𝜂²_𝑌−2𝜃−𝜃^̌‌′×𝑃𝜃_𝑟^̌‌+²_𝑗=1𝛽_𝑗𝑧²_𝑗+14𝜍₂²_𝑄. (72)

Although the value function 𝑈 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 Section 2 with Assumptions 1–7 holding. The robust adaptive controller 𝜇 defined by (60) with the optimal choice for the worst-case estimate 𝜉̂ defined by (65) achieves the following strong robustness properties for the closed-loop system:(1)The controller 𝜇 achieves disturbance attenuation level 𝛾 for any uncertainty quadruple (𝑥(0),𝜃,𝑤_[0,∞),𝑤^̌‌_[0,∞),𝑌_𝑑0,𝑦_𝑑⁽²⁾)∈𝒲.(2)Given a 𝑐_𝑤>0, there exists a constant 𝑐_𝑐>0 and a compact set Θ_𝑐⊂Θ_𝑜, such that for any uncertainty (𝑥(0),𝜃,𝑤̀_[0,∞),𝑤^̌‌_[0,∞),𝑌_𝑑) with |𝑥(0)|≤𝑐_𝑤;|𝑤̀(𝑡)|≤𝑐_𝑤;𝑤^̌‌(𝑡)≤𝑐_𝑤;𝑌_𝑑(𝑡)≤𝑐_𝑤;∀𝑡∈[0,∞) (73) all closed-loop state variables 𝑥, 𝑥^̌‌, 𝜃^̌‌, Σ, 𝑠_Σ, 𝜂, 𝜂^̌‌𝑤, 𝜂_𝑑, and 𝜆 are bounded as follows, for all 𝑡∈[0,∞), |𝑥(𝑡)|≤𝑐_𝑐,𝑥^̌‌(𝑡)≤𝑐_𝑐,𝜃^̌‌(𝑡)∈Θ_𝑐,𝜂(𝑡)≤𝑐_𝑐,𝜂_𝑑(𝑡)≤𝑐_𝑐,|𝜆(𝑡)|≤𝑐_𝑐,𝜂^̌‌𝑤≤𝑐_𝑐,1𝐾_𝑐(𝑡)≤1𝐼≤Σ𝛾²𝑄₀⁻¹,1𝐾_𝑐≤𝑠_Σ(𝑡)≤1𝛾²Tr𝑄₀. (74)(3)For any uncertainty quadruple (𝑥(0),𝜃,𝑤̀_[0,∞),𝑤^̌‌_[0,∞),𝑌_𝑑[0,∞)) with 𝑤̀_[0,∞)∈ℒ₂∩ℒ_∞, 𝑤^̌‌_[0,∞)∈ℒ₂∩ℒ_∞ and 𝑌_𝑑[0,∞)∈ℒ_∞, the output of the system, 𝑥₁, asymptotically tracks the reference trajectory, 𝑦_𝑑, that is, lim_𝑡→∞𝑥₁(𝑡)−𝑦_𝑑(𝑡)=0. (75)

Proof. For the frits statement, if we define 𝑙𝑥₀^̌‌₀,𝜃^̌‌₀∶=𝑉₂(0)=12²_𝑗=1𝑧²_𝑗(0),𝑙𝜏,𝜃,𝑥,𝑦_[0,𝑡],𝑤^̌‌_[0,𝑡]∶=𝜉−𝜉̂²𝑄−2𝜃−𝜃^̌‌′𝑃𝜃_𝑟^̌‌+14𝜍₂²𝑄+𝜂²_𝑌+²_𝑗=1𝛽_𝑗𝑧²_𝑗=𝛾⁴(𝑥−𝑥)−Φ𝜃−𝜃²_{Π−1ΔΠ−1}+𝜖𝛾²𝜁²−1𝜃−𝜃²_{Φ′𝐶′𝐶Φ}−2𝜃−𝜃^̌‌′𝑃𝜃_𝑟^̌‌+14𝜍₂²𝑄+𝜂²_𝑌+²_𝑗=1𝛽_𝑗𝑧²_𝑗, (76) then, we have 𝐽^𝑡=𝐽^𝑡+^𝑡₀̇(𝑡)(0)(𝑡)𝑈𝑑𝜏−𝑈+𝑈≤−𝑈≤0. (77) It follows that sup_{(𝑥(0),𝜃,𝑤[0,∞),̌‌𝑤[0,∞))∈𝒲}𝐽^𝑡≤0. (78) This establishes the first statement.
Next, we will prove the second statement. Define [0,𝑡_𝑓) to be the maximal interval on which the closed-loop system admits a solution. We will show that 𝑡_𝑓 is always ∞.
Fix 𝑐_𝑤≥0, and 𝑐_𝑑≥0, consider any uncertainty (𝑥₀,𝜃,𝑤̀_[0,∞),𝑤^̌‌_[0,∞),𝑌_𝑑(𝑡)) that satisfies: 𝑥₀≤𝑐_𝑤;|𝑤̀(𝑡)|≤𝑐_𝑤;𝑤^̌‌(𝑡)≤𝑐_𝑤;𝑌_𝑑(𝑡)≤𝑐_𝑑∀𝑡∈[0,∞). (79) We define [0,𝑇_𝑓) 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 𝑠_Σ are uniformly upper bounded and uniformly bounded away from 0 as desired.
Introduce the vector of variables 𝑋_𝑒∶=𝜃^̌‌′𝑥−Φ𝜃^′𝜂^′𝑧₁𝑧₂^′ (80) and two nonnegtive and continuous functions defined on ℝ^6+𝜎𝑈_𝑀𝑋_𝑒∶=𝐾_𝑐𝜃²+𝛾²𝑥−Φ𝜃²_Π−1+𝜂²_𝑍+²_𝑗=112𝑧²_𝑗,𝑈_𝑚𝑋_𝑒∶=𝛾²𝜃²_𝑄0+𝛾²𝑥−Φ𝜃²_Π−1+𝜂²_𝑍+²_𝑗=112𝑧²_𝑗, (81) then, we have 𝑈_𝑚𝑋_𝑒≤𝑈𝑡,𝑋_𝑒≤𝑈_𝑀𝑋_𝑒,∀𝑡,𝑋_𝑒∈0,𝑇_𝑓×ℝ^6+𝜎. (82) Since 𝑈_𝑚(𝑋_𝑒) is continuous, nonnegative definite and radially unbounded, then for all 𝛼∈ℝ, the set 𝑆_1𝛼∶={𝑋_𝑒∈ℝ^6+𝜎∣𝑈_𝑚(𝑋_𝑒)≤𝛼} is compact or empty. Since |𝑤̀(𝑡)|≤𝑐_𝑤, and |𝑤^̌‌(𝑡)|≤𝑐_𝑤, for all 𝑡∈[0,∞), we have the following inequality for the derivative of 𝑈: 𝑈̇≤−𝛾⁴2𝑥−𝑥^̌‌−Φ𝜃−𝜃^̌‌2_{Π−1ΔΠ−1}+2𝜃−𝜃^̌‌′𝑃𝜃_𝑟^̌‌−𝜂²_𝑌−²_𝑖=1𝛽_𝑖𝑧²_𝑖+𝛾²𝑀̀²₂𝑐²_𝑤+𝛾²𝑐²_𝑤. (83) Since −(𝛾⁴/2)|𝑥−𝑥^̌‌−Φ(𝜃−𝜃^̌‌)|²_{Π−1ΔΠ−1}−|𝜂|²_𝑌+2(𝜃−𝜃^̌‌)^′𝑃_𝑟(𝜃^̌‌)−∑²_𝑗=1𝛽_𝑗𝑧²_𝑗 will tend to −∞ when 𝑋_𝑒 approaches the boundary of Θ_𝑜×ℝ⁶, then there exists a compact set Ω₁(𝑐_𝑤)⊂Θ_𝑜×ℝ⁶, such that 𝑈̇<0 for for all 𝑋_𝑒∈Θ_𝑜×ℝ⁶⧵Ω₁. Then, 𝑈(𝑡,𝑋_𝑒(𝑡))≤𝑐₁, and 𝑋_𝑒(𝑡) is in the compact set 𝑆_1𝑐1⊆ℝ^6+𝜎, for all 𝑡∈[0,𝑇_𝑓). It follows that the signal 𝑋_𝑒 is uniformly bounded, namely, 𝜃, 𝑥−Φ𝜃, 𝜂, 𝑧₁, and 𝑧₂ are uniformly bounded.
Based on the dynamics of 𝜂_𝑑, we have 𝜂_𝑑 is uniformly bounded. Since 𝜂=𝜂−𝜂_𝑑 is uniformly bounded, then 𝜂 is also uniformly bounded. Furthermore, there is a particular linear combination of the components of 𝜂, denoted by 𝜂_𝐿, 𝜂̇=𝐴_𝑓𝜂+𝑝₂𝜂𝑦,_𝐿=𝑇_𝐿𝜂, (84) which is strictly minimum phase and has relative degree 1 with respect to 𝑦. Then the signal 𝜂_𝐿 has relative degree 3 with respect to the input 𝑢, and is uniformly bounded.
Note Φ=Φ^𝑦+Φ^𝑢+Φ^̌‌𝑤. Since Φ^𝑦 and Φ^̌‌𝑤 are uniformly bounded, to prove Φ is bounded, we need to prove Φ^𝑢 is uniformly bounded. Define the following equations to separate Φ^𝑢 into two parts: Φ^𝑢=Φ_𝑢𝑠+𝜆_𝑏𝐴₂₂₀,𝜆_𝑏=𝜆_𝑏1𝜆_𝑏2,𝜆̇_𝑏=𝐴_𝑓𝜆_𝑏+𝑒_2,2𝑢;𝜆_𝑏(0)=𝟎_2×1,Φ_𝑢𝑠=Φ_𝑢𝑠1Φ_𝑢𝑠2,Φ̇_𝑢𝑠=𝐴_𝑓Φ_𝑢𝑠;Φ_𝑢𝑠(0)=Φ_𝑢0. (85) Clearly Φ_𝑢𝑠 is uniformly bounded because 𝐴_𝑓 is Hurwitz. The first-row element of 𝑥−Φ𝜃 is 𝑥₁−Φ_𝑢𝑠1𝜃−𝜆_𝑏1𝐴₂₂₀𝜃−Φ^̌‌𝑤₁𝜃−𝜂^′𝑇1𝜃. (86) We can conclude that 𝑥₁−𝜆_𝑏1𝐴₂₂₀𝜃 is uniformly bounded in view of the boundedness of 𝑥−Φ𝜃, 𝜃, Φ_𝑢𝑠, Φ^̌‌𝑤, and 𝜂. Since 𝑧₁=𝑥^̌‌₁−𝑦_𝑑, and 𝑧₁, 𝑦_𝑑 are both uniformly bounded, 𝑥^̌‌₁ is also uniformly bounded.
Notice that 𝐴_𝑓=𝐴−𝜁²𝐿𝐶−Π𝐶^′𝐶(𝜁²−𝛾⁻²) and 𝑏₀=𝑏_𝑝0+𝐴₂₂₀𝜃, we generated the signal 𝑥₁−𝑏₀𝜆_𝑏1 by 𝑥̇−𝑏₀𝜆̇_𝑏=𝐴_𝑓𝑥−𝑏₀𝜆_𝑏+𝐴₂₁𝜃𝑦+𝐷𝑀̀𝑤̀+𝑤^̌‌𝐴₂₃𝜃+𝜁²𝐿+Π𝐶^′𝜁²−1𝛾²𝑦−𝐸𝑀̀𝑤̀+𝐷^̌‌𝑤^̌‌,𝑥₁−𝑏₀𝜆_𝑏1=𝐶𝑥−𝑏₀𝜆_𝑏. (87) Since 𝑥₁−𝑏₀𝜆_𝑏1 has relative degree at least 1 with respect to 𝑦, take 𝜂_𝐿 and 𝑦 as output and input of the reference system, we conclude 𝑥₁−𝑏₀𝜆_𝑏1 is uniformly bounded by bounding lemma. It follows that 𝑥^̌‌₁−𝜆_𝑏1(𝑏_𝑝0+𝐴^̌₂₁₂₀𝜃) is also uniformly bounded. Since 𝑥^̌‌₁ is uniformly bounded and 𝜃^̌‌ is uniformly bounded away from 0, we have 𝜆_𝑏1 is uniformly bounded. That further implies that Φ₁, that is, 𝐶Φ, is uniformly bounded. Furthermore, since 𝑥₁−𝑏₀𝜆_𝑏1, 𝑤^̌‌ and 𝑤̀ are bounded, we have that the signals of 𝑥₁ and 𝑦 are uniformly bounded. It further implies the uniform boundedness of 𝑥−𝑏₀𝜆_𝑏 since 𝐴_𝑓 is a Hurwitz matrix. By a similar line of reasoning above, we have 𝑥₂, 𝜆_𝑏,2 are uniformly bounded. Then we can conclude that Φ_𝑢𝑠 and Φ are uniformly bounded.
Next, we need to prove the existence of a compact set Θ_𝑐⊂Θ_𝑜 such that 𝜃^̌‌(𝑡)∈Θ_𝑐, for all 𝑡∈[0,𝑇_𝑓). First introduce the function Υ∶=𝑈+𝜌_𝑜−𝑃𝜃^̌‌−1𝑃𝜃^̌‌. (88) We notice that, when 𝜃^̌‌ approaches the boundary of Θ_𝑜, 𝑃(𝜃^̌‌) approaches 𝜌_𝑜. Then Υ approaches ∞ as 𝑋_𝑒 approaches the boundary of Θ_𝑜×ℝ⁶. We introduce two nonnegative and continuous functions defined on Θ_𝑜×ℝ⁴: Υ_𝑀∶=𝑈_𝑀𝑋_𝑒+𝜌_𝑜−𝑃𝜃^̌‌−1𝑃𝜃^̌‌,Υ_𝑚∶=𝑈_𝑚𝑋_𝑒+𝜌_𝑜−𝑃𝜃^̌‌−1𝑃𝜃^̌‌. (89) Then, by the previous analysis, we have Υ_𝑚𝑋_𝑒≤Υ𝑡,𝑋_𝑒≤Υ_𝑀𝑋_𝑒,∀𝑡,𝑋_𝑒∈0,𝑇_𝑓×Θ_𝑜×ℝ⁶. (90) Note that the set 𝑆_2𝛼∶={𝑋_𝑒∈Θ_𝑜×ℝ⁶∣Υ_𝑚(𝑋_𝑒)≤𝛼} is a compact set or empty. Then, we consider the derivative of Υ as follows: Υ̇=𝑈̇+𝜌_𝑜−𝑃𝜃^̌‌−2𝜌_𝑜𝜕𝑃𝜕𝜃^̌‌𝜃_̌‌̇𝜃𝛾≤−⁴2𝑥−𝑥^̌‌−Φ𝜃−𝜃^̌‌2_{Π−1ΔΠ−1}+2𝜃−𝜃^̌‌′𝑃𝜃_𝑟^̌‌−𝜂²_𝑌−^𝑟_𝑗=1𝑐_𝛽𝑗𝑧²_𝑗−𝜕𝑃𝜕𝜃^̌‌𝜃^′2𝜌_𝑜−𝑃𝜃^̌‌−4×𝐾_𝑐⁻¹𝜌_𝑜𝑝𝜃_𝑟^̌‌𝜌_𝑜−𝑃𝜃^̌‌2−𝑐+𝑐, (91) where 𝑐∈ℝ is a positive constant. Since Υ̇ will tend to −∞ when 𝑋_𝑒 approaches the boundary of Θ_𝑜×ℝ⁴, there exists a compact set Ω₂(𝑐_𝑤)⊂Θ_𝑜×ℝ⁴, such that for all 𝑋_𝑒∈Θ_𝑜×ℝ⁴⧵Ω₂, Υ̇(𝑋_𝑒)<0. Then, there exists a compact set Θ_𝑐⊂Θ_𝑜, such that 𝜃^̌‌(𝑡)∈Θ_𝑐, for all 𝑡∈[0,𝑇_𝑓). Moreover, Υ(𝑡,𝑋_𝑒(𝑡))≤𝑐₂, and 𝑋_𝑒(𝑡) is in the compact set 𝑆_2𝑐2⊆Θ_𝑜×ℝ⁶, for all 𝑡∈[0,𝑇_𝑓). It follows that 𝑃_𝑟(𝜃^̌‌) is also uniformly bounded.
Also, 𝜂, 𝜆 are some stably filtered signals of 𝑢 and 𝑦, they are uniformly bounded. Since 𝜂^̌‌𝑤 is uniformly bounded, Φ is uniformly bounded. Then we can conclude 𝑥^̌‌ 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 and 𝑢 are uniformly bounded on [0,𝑡_𝑓), Σ, 𝑠_Σ are uniformly bounded and bounded away from 0, and 𝜃^̌‌ is uniformly bounded away from the boundary of the set Θ_𝑜. Therefore, it follows that 𝑡_𝑓=∞ and the complete system states are uniformly bounded on [0,∞).
Last, we will establish the third statement. By the following inequality: 𝑈̇^∞₀𝑑𝜏≤^∞₀−𝑥₁−𝑦_𝑑²+𝛾²𝑀̀𝑤̀²+𝛾𝑤^2̌‌2𝑑𝜏, (92) it follows that ^∞₀𝑥₁−𝑦_𝑑²≤𝑑𝜏_∞0𝛾²𝑀̀𝑤̀²+𝛾𝑤^2̌‌2𝑑𝜏+𝑈(0)<+∞. (93) By the second statement, we notice that sup_0≤𝑡<∞𝑥̇₁−𝑦̇_𝑑<∞. (94) Then we have lim_𝑡→∞𝑥₁(𝑡)−𝑦_𝑑(𝑡)=0. (95) 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 of 𝐾_𝑡, 𝐾_𝑒, 𝐽, 𝑅, and 𝐿 are given as below, and the variations can be lumped into the arbitrary disturbance 𝑤̀. 𝐾_𝑡=0.01; N-cm/Amp, 𝐾_𝑒=1; Volt/rad/s, 𝐽=0.01; N-cm/rad/s2, 𝑅=1; Ohm, 𝐿=0.1; L.

The value of 𝐷 is unknown and with true value 0.01 N-cm/rad/s. Then the true system is of the following state-space representation 𝜔̇𝑖̇=𝜔𝑖+010𝑇𝜃1−10−1010𝑢+_̌‌𝑤+𝑇101000^𝑤𝑤_𝜔𝑇_𝑓;𝜔(0)𝑖(0)=00,𝜔𝑖+𝑇𝑦=10010𝑤^𝑤_𝜔𝑇_𝑓, (96) 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, 𝑇_̌‌𝑤 is the estimated disturbance torque in N-cm, 𝑇_𝑤 is the arbitrary disturbance torque in N-cm, 𝑇_𝑓 is the friction torque in N-cm, 𝑤_𝜔 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, 𝑦_𝑑, which is generated by the following linear system: 𝑦_𝑑=𝑑𝑠³+2𝑠²+2𝑠⁺3, (97) where 𝑑 is the command input signal.

Introduce the following state and disturbance transformation: 𝜔𝑖𝑇𝑥=10101;𝑤=1−𝜃1010^𝑤𝑤_𝜔𝑇_𝑓. (98) We obtain the design model ̇1+01𝑥=−101−100𝑥+10𝑦𝜃10𝑢+10^̌‌𝑤+10100𝑤,𝑦=10𝑥+01𝑤. (99) 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 𝑃(𝜃)=(𝜃+1)². (100)

For other design and simulation parameters, we select 𝑥^̌‌₀=;0.10.5^̌‌𝜃₀𝑄=−0.5;₀=1;𝐾_𝑐=100;Δ=;1001𝛽₁=𝛽₂=0.5;𝑌=.159.2262−17.0150−17.01501.8786 (101) Then, we obtain 𝐴_𝑓=;.−10.29931.0000−12.28820;𝑍=8.8506−0.9393−0.93930.1229Π=0.59874.57644.576443.1208 (102)

We present two sets of simulation results in this example. In the first set of simulation, we set  𝑇_𝑓=0; N-cm,  𝑇_𝑤=0; N-cm,  𝑤_𝜔=0; rad/s,  𝑇_̌‌𝑤=0; N-cm.

This simulation is to demonstrate the regulatory behaviour of the adaptive controller. The results are shown in Figures 1(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.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1 

System response for example under command input 𝑑(𝑡)=5sin(𝑡), 𝑇𝑓=0, 𝑇𝑤=0, 𝑤𝜔=0, and 𝑇̌‌𝑤=0. (a) Tracking error; (b) control input; (c) parameter estimate; (d) state-estimation error 𝑥̌‌1−𝑥1; (e) state-estimation error 𝑥̌‌2−𝑥2; (f) cost function ∫𝑡0((𝑥1−𝑦𝑑)2−𝛾2|𝑤|2−𝛾2|𝑤̌‌|2d)𝜏.

The second set of simulation results is to demonstrate the robustness of the adaptive controller to unmodeled exogenous disturbance inputs. We set 𝑇_𝑓=−0.01×sgn(𝜔); N-cm, 𝑇_𝑤=0.04sin(𝑡); N-cm, 𝑤_𝜔 = White noise signal with power 0.01 sample 𝑑 at 1 HZ; rad/s, 𝑇_̌‌𝑤=0.05sin(4𝑡); N-cm.

The simulation results are presented in Figures 2(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.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2 

System response for example under command input 𝑑(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, 𝑇̌‌𝑤=0.05sin(4𝑡). (a) Tracking error; (b) control input; (c) parameter estimate; (d) state-estimation error 𝑥̌‌1−𝑥1; (e) state-estimation error 𝑥̌‌2−𝑥2; (f) cost function ∫𝑡0((𝑥1−𝑦𝑑)2−𝛾2|𝑤|2−𝛾2|𝑤̌‌|2d)𝜏.

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 nonlinear 𝐻^∞-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.