#### 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 -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 -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 -control problem under imperfect state measurements. By *cost-to-come function* analysis, it is converted into an -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 -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:where is the state vector and represents load shaft angular speed and motor current, respectively; is the scalar control input; is the load shaft angular speed measurement output; is the motor inductor; is the armature resistance; is the back-emf constant; is the motor torque constant; is the motor system inertia; is the motor-system damping factor; is the arbitrary disturbance vector, which is composed of arbitrary disturbance torque, , arbitrary disturbance at the output measurement channel, , and friction disturbance torque ; is the measured or estimated disturbance torque. In this paper, we assume the variables , , , , , and 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 to has relative degree . And then we can find a state diffeomorphism and a disturbance transformation by following [22], such that system (1a) and (1b) can be transformed into the following form in the coordinates:where is the state vector, , , , , is the -dimensional vector of unknown parameters of the true system; the matrices , , , , , , , and are of appropriate dimensions, known and have the structure as below, where is a -dimensional row vector and . Then the high-frequency gain of (2a) and (2b) is .

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

*Assumption 1. *The reference trajectory, , is two times continuously differentiable. Define vector and . The signal is available for control design.

To guarantee the stability of the closed-loop system and the boundedness of the estimate of , we assume there is an *a priori* convex compact set where the parameter vector lies in.

*Assumption 2. *There exists a known smooth nonnegative radially unbounded strictly convex function , such that the true value . For all , .

The control objective is to design a robust adaptive controller for system (1), such that the output tracks a desired reference signal , while rejecting the uncertainties 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 is said to achieve disturbance attenuation level if there exist nonnegative functions and such that
where
where is the initial guess of the unknown parameter vector; is the quadratic weighting on the error between the true value of and the initial guess quantifying the level of confidence in the estimate ; is the initial guess of the unknown initial state ; is the weighting on the initial state-estimation error, quantifying the level of confidence in the estimate ; denotes for any vector and any symmetric matrix .

The control law to system (2a), (2b) is generated by the following control law where . We denote the class of these admissible controllers by .

Clearly, when the inequality (4) is achieved, we have where denotes norm, and is a constant. When and are finite, is also finite, which implies , under additional assumptions.

The following notation will be used throughout this paper: denotes the estimate of the current state of the system; denotes the state-estimation error ; denotes the estimate of the parameter vector ; denotes the estimation error ; any function symbol with an ‘‘over bar’’ will denote a function defined in the terms of the transformed state variables, such as being the identical function as ; for any matrix , the vector is formed by stacking up its column vectors; denotes a -dimensional column vector, all of its elements are , except its th row is , such as .

Let denote the expanded state vector , we have the following expanded dynamics for system (2a), (2b) in view of :

The worst-case optimization of the cost function (4) can be carried out in two steps as depicted in the following equations: 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 , 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 and are assumed to be known. Since the control input is a causal function of and , then it is known. We ignore terms considered to be constant in the estimation design step, and set in (5) to be . The equivalent cost function of (5) is then given by, where is a matrix-valued weighting function, is the worst-case estimates for the expanded state , is a design function, and 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 -control of affine quadratic problem, which admits a finite dimensional solution.

We introduce the value function , and then we can obtain the dynamics of state estimator and worst-case covariance matrix as belowwhere and is defined as , where .

Then, the cost function (5) can be equivalently written as where , which will be determined later to improve the performance of the adaptive system; is the worst-case disturbance, given by

We partition as and introduce and .

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

*Assumption 4. *The weighting matrix of function in (5) is given by
where is positive-definite matrix, and is a scalar function defined by

Then, we have the following differential equation of , , and :The matrix will play the role of worst-case covariance matrix of the parameter estimation error. Assumption 4 guarantees that is uniformly bounded from above and uniformly bounded from below away from as depicted in the following lemma, and its proof is given in [14].

Lemma 5. *Consider the dynamic equation (17a) for the covariance matrix . Let Assumption 4 hold and . Then, is uniformly upper and lower bounded as follows:
*

We define , and its dynamic is given by Then, , which does not require the inversion of .

From Assumption 4 and (17a), we note that . This means the quantity 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 is Hurwitz, then the desired disturbance attenuation level . If the matrix is not Hurwitz, then the desired disturbance attenuation level .

*Assumption 7. *The initial weighting matrix in (17c) is chosen as the unique positive definite solution to the following algebraic Riccati equation:

Then, we note that the unique positive-definite solution of (17c) is time-invariant and equal to the initial value .

*Remark 8. *To simplify the estimator structure, we can choose so that will be a constant positive-definite matrix, and will be a finite positive constant. To further simplify the identifier, the initial weighting matrix is chosen as the unique positive-definite solutions to its algebraic Riccati equation (17c), which also implies in view of .

To guarantee the boundedness of estimated parameters without persistently exciting signals, we introduce soft projection design on the parameter estimate. We define By Assumption 2 and Lemma 2 in [23], we have . For any fixed , we define the open set Our control design will guarantee that the estimate lies in , which immediately implies , for some . Moreover, the convexity of implies the following inequality: We set , where Then, we obtain where .

We summarize the estimation dynamics equations below

For the controller structure simplification, the dynamics for can be implemented as below. First, we observe the matrix has the same structure as the matrix . Then, we introduce the matrix where is a 2-dimensional vector such that the pair is controllable, which implies that is invertible. Then the following prefiltering system for , , and generates the online:

Associated with the above identifier, introduce the value function whose time derivative is given by We note that the last term in is nonpositive, zero on the set and approaches as approaches the boundary of the set , which guarantees the boundness of .

Then the cost function can be equivalently written as

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 and 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: where function is part of the weighting function 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 , input , measured disturbance , and the worst-case estimate for the expanded state vector , which are signals we either measure or can construct. This is then a nonlinear -optimal control problem under full information measurements. Instead of considering and as the maximizing variable, we can equivalently deal with the transformed variable Then, we have

The variables to be designed at this stage include and . The design for will be carried out last. Note that the structure of in the dynamics is in strict-feedback form, we will use the backstepping methodology [24] to design the control input , which will guarantee the global uniform boundedness of the closed-loop system states and the asymptotic convergence of tracking error.

Consider the dynamics of , For ease of the ensuing study, we will separate as the sum of several matrices as follows:where , are -dimensional constant matrices depending on , , and . Express and in terms of their row vectors, and . Then, , , and .

We summarized the dynamics for backstepping design in the following, where we have emphasized the dependence of various functions on the independent variables:where the nonlinear functions , , and are smooth as long as ; the nonlinear functions , , , , , , , and are smooth. Here, we use , , , and as independent variables, instead of , , for the clarity of ensuing analysis.

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

*Step 1. *In this step, we try to stabilize by virtual control law . Introduce variable , as the desired trajectory of , which satisfies the dynamics
Define the error variable . Then, satisfies the dynamics
By [14], the following holds.

Lemma 9. * Given any Hurwitz matrix , there exists a positive-definite matrix such that the following generalized algebraic Riccati equation admits a positive-definite solution :
*

Note that in (42) is a Hurwitz matrix, then we define the following value function in terms of the positive-definite matrix : Then its time derivative is given by where If is control input, then we may choose the control law and the design achieves attenuation level from the disturbance to the output . This completes the virtual control design for the dynamics.

*Step 2. *Define the transformed variable
which is the deviation of from its desired trajectory .

Then the time derivative of is given by where the function is defined as Introduce the value function for this step whose derivative is given by wherewhere is any positive constant, and the nonlinear function is to be chosen by the designer. Note that the function is smooth as long as . If were the actual controls, then we would choose the following control law: and set , to guarantee the dissipation inequality with supply rate

This completes the second step of backstepping design.

*Step 3. *At this step the actual control appears in the derivative of , which is given by
where , , and are given as follows:

Introduce the following value function for this step:
Its derivative can be written as
with the control law defined by
where
where . Clearly, the functions , , , , , and are smooth as long as .

This completes the backstepping design procedure.

For the closed-loop adaptive nonlinear system, we have the following value function: where we have introduced for notational consistency. The time derivative of this function is given by Then the optimal choice for the variables and are which yields that the closed-loop system is dissipative with storage function and supply rate Furthermore, the worst case disturbance with respect to the value function is given by

#### 5. Main Result

For the adaptive control law with chosen according to (65), the closed-loop system dynamics are is the state vector of the close-loop system and given by which belongs to the set ; and are smooth mapping of and , respectively, and with the initial condition .

Since (64) holds, the value function satisfies Hamilton-Jacobi-Isaacs equation, for all , for all . where is smooth and given by

Although the value function satisfies an Hamilton-Jacobi-Isaacs equation, we cannot deduce the stability and robustness properties of the closed-loop system directly from (64), since is not a positive-definite function of the closed-loop state vector . 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 defined by (60) with the optimal choice for the worst-case estimate defined by (65) achieves the following strong robustness properties for the closed-loop system:*(1)*The controller achieves disturbance attenuation level for any uncertainty quadruple .*(2)*Given a , there exists a constant and a compact set , such that for any uncertainty with
all closed-loop state variables , , , , , , , , and are bounded as follows, for all ,
*(3)*For any uncertainty quadruple with , and , the output of the system, , asymptotically tracks the reference trajectory, , that is,
*

*Proof. *For the frits statement, if we define