Closed-Loop Identification Applied to a DC Servomechanism: Controller Gains Analysis

Abstract

Usually, when parameter identification is applied, there are some gains related to the identification algorithm whose value must be carefully adjusted in order to obtain a good performance of the algorithm. However, when performing closed loop identification, there are some other constants that in general are not taken into account for the identification algorithm: the controller gains, which may appear inside the identification algorithm, specifically in the regressor vector, which is very important for the parameter convergence according to the persistence of excitation condition. Therefore, the effect of these gains on the estimated parameters should be analyzed so that better estimates can be obtained. This paper addresses the behavior of the parameter estimates for a closed-loop identification methodology applied to a DC servomechanism with a bounded perturbation signal and a PD controller. It is shown that, with this perturbation, the parameter estimates converge to a region whose size can be modified not only by varying the identification algorithm gains but also by modifying the P and D controller gains in a suitable way.

1. Introduction

It is well known that servomechanisms are fundamental in modern robotic and mechatronic systems for industrial applications where high speed and high precision are of prime importance. On the other hand, in order to apply model-based tuning methods it is necessary to apply an identification algorithm to the servomechanism and, if it is required, to design a high performance controller where a small steady state tracking error and high precision are necessary. Besides, the system parameters are also important to know if the system itself behaves as it should, for example, by evaluating its performance with some specified test signals. Thus, parameter identification plays a very important role for designing a controller as pointed out from several points of view [1–4].

For a DC servomechanism, there are several techniques that may be applied for identifying its parameters, when the servomechanism operates in open loop and in closed loop. However, it is important to note that if the variable of interest is the servo position, then a linear model of a servomechanism contains a pole on the imaginary axis, thus making the system not BIBO stable, that is, a bounded input applied to the servomechanism would not produce a bounded position and this may lead to undesired effects, for example, in robotic applications. Therefore, for security reasons, parameter identification should be performed in closed loop.

One way to characterize the system identification algorithms is [5, 6] (i) direct methods and (ii) indirect methods. Many of the identification procedures using a linear controller [7–9] fall into the category of direct methods; that is, the parameter identification procedures are applied without regard to the controller being used to close the loop. On the other hand, if a technique takes into account the controller structure, then it falls into the category of indirect methods.

As mentioned before, for security reason it is better to estimate the system parameters with the system working in closed loop. One way to perform closed loop parameter identification of a servomechanism is with the relay-based techniques [10–12], which are widespread for servo identification. The idea with this methodology is to close the loop through a relay in order to obtain a sustained oscillation and then the values of the frequency and amplitude allows identifying a model of the system. The relay based techniques may work well; however, the reference signal used for exciting the system becomes constant in many time instants, and then no control is being applied in that time and then the effect of the disturbances may affect the estimates obtained with this methodology. Besides, tuning of the relay controller can be cumbersome and no one of the reviewed methods takes explicitly into account the disturbances affecting the servomechanism.

References [7–9, 13–15] propose methods for closed loop identification of position-controlled servos where the loop is closed using a linear controller. The approach proposed in [13, 14] uses a PD controller to close the loop and an on-line gradient algorithm allows estimating a linear and nonlinear model of a servomechanism. Also, in [15] a two-step identification method for estimating the four parameters of a nonlinear model of a position control servomechanism is presented, where the proposed identification scheme relies on the theory of operational calculus. However, the effect of perturbations is considered only for constant values. On the other hand, a common technique used for identifying DC servomechanisms is based on the least squares algorithm. In [16], an approach for identifying a DC motor is presented considering the Coulomb friction and dead zone using a recursive discrete time least squares algorithm with forgetting factor using a Hammerstein model for the motor. In [17], second-order sliding modes approach is applied for estimating the parameters of a DC motor operating in open-loop and closed loop, while in [18] the inertia of a second-order model of a DC servomechanism driving a human assistive system is identified using Bode plots obtained from input-output measurements. Finally, another interesting approach is presented in [2, 4], where an online algebraic identification method is proposed. Besides, [19, 20] show how this methodology can be used for parameter identification of a DC motor and also this methodology was used and modified in [21] introducing a batch least squares algorithm for estimating the parameters of an induction motor.

All the previous works consider in general the importance of some gains related with the identification algorithm for closed-loop identification. However, in some cases (as that presented in [13, 14]) the controller gains get involved in with some important elements for the identification scheme, for example, in the regressor vector. In this case, it has not been analyzed which effect actually has such gains in the parameter identification procedure and how the parameter estimates will behave when a disturbance is affecting the system. Then, this work presents the analysis of a closed loop identification algorithm for a perturbed position-controlled DC servomechanism where a PD controller closes the loop and achieves stability without knowledge about the servomechanism parameters. Theoretical results show that, when the perturbation signal is identically zero, exponential convergence can be claimed, and in the presence of a bounded perturbation it is possible to obtain a region Ω_𝛿 where the parameter estimates belong. What’s more, it will be shown that this region can be made arbitrarily small if the controller gains are tuned properly. Some simulations depict the behavior of the region Ω_𝛿, showing the effect that the controller gains have on the estimated parameters. The paper is organized as follows. Section 2 is devoted to present preliminary theoretical results of passivity based control, adaptive control, and the general description of the DC servomechanism. Section 3 presents the analysis for closed loop identification of the servomechanism when no disturbances are affecting the system and the exponential stability of the overall identification scheme is shown. Section 4 extends the analysis for the perturbed case and Section 5 deals with the analysis of the results. Finally, Section 6 gives some concluding remarks.

2. Preliminary Results

The closed loop identification analysis that will be presented is based on some results related with passivity theory and adaptive control, and thus the following results are worth presenting.

2.1. Passivity and Adaptive Control

As pointed out before, the aim of this work is to show the convergence properties of the identification algorithm with respect to the value of the controller gains. To this end it will be necessary to use the observability property given in [22] and some results of passivity and stability that will be described in the following.

Let us consider the linear time-varying system [𝐶(𝑡),𝐴(𝑡)] defined by ̇(𝑡)𝑥(𝑡),𝑦(𝑡)(𝑡)𝑥(𝑡),𝑥=𝐴=𝐶 (1) where 𝑥(𝑡)∈𝑅^𝑛, 𝑦(𝑡)∈𝑅^𝑚, while 𝐴(𝑡)∈𝑅^𝑛×𝑛 and 𝐶(𝑡)∈𝑅^𝑚×𝑛 are piecewise continuous functions. A system [𝐶(𝑡),𝐴(𝑡)] is said to be uniformly completely observable (UCO) [22] if there exist strictly positive constants {𝛽₁,𝛽₂,𝛿} such that, for all 𝑡₀≥0,𝛽₁𝐼≤𝑁(𝑡₀,𝑡₀+𝛿)≤𝛽₂𝐼, where 𝑁(𝑡₀,𝑡₀+𝛿)∈𝑅^𝑛×𝑛 is the so-called observability Gramian as 𝑁𝑡₀,𝑡₀+𝛿=^𝑡0_𝑡^+𝛿₀Φ^𝑇𝜏,𝑡₀𝐶^𝑇(𝜏)𝐶(𝜏)Φ𝜏,𝑡₀𝑑𝜏, (2) where Φ(𝜏,𝑡₀) is the state transition matrix associated with 𝐴(𝑡). Now, assume that, for all 𝛿>0, there exists 𝑘_𝛿>0 such that, for all 𝑡₀≥0,∫^𝑡0_𝑡^+𝛿₀‖𝐾(𝜏)‖²𝑑𝜏≤𝑘_𝛿, where the matrix 𝐾 corresponds to the system under output injection as ̇{𝐴(𝑡)(𝑡)𝐶(𝑡𝑤(𝑡),𝑧(𝑡)(𝑡)𝑤(𝑡),𝑤=+𝐾)}=𝐶 (3) with 𝑤(𝑡)∈𝑅^𝑛, 𝐾(𝑡)∈𝑅^𝑛×𝑚, and 𝑧(𝑡)∈𝑅^𝑚. Then, the system [𝐶,𝐴] is UCO if and only if [𝐶,𝐴+𝐾𝐶] is UCO, where the term 𝐾𝐶 stands for the output injection term as indicated in [22]. Moreover, if the observability Gramian of the system [𝐶,𝐴] satisfies the previous upper and lower bounds on 𝑁, then the observability Gramian of the system [𝐶,𝐴+𝐾𝐶] satisfies these inequalities with identical 𝛿 and 𝛽^′₁=𝛽₁1+𝑘_𝛿𝛽₂,𝛽^′₂=𝛽₂exp𝑘_𝛿𝛽₂. (4) Now assume that 𝑓(𝑡,𝑥)∶𝑅⁺×𝑅^𝑛→𝑅^𝑛 has continuous and bounded first partial derivatives in 𝑥(𝑡) and is piecewise continuous in 𝑡 for all 𝑥∈𝐵_ℎ, 𝑡≥0. Then, the following statements are equivalent [22].(1)𝑥=0 is an exponentially stable equilibrium point of 𝑥̇=𝑓(𝑡,𝑥), 𝑥(𝑡₀)=𝑥₀.(2)There exists a function 𝑉(𝑡,𝑥) and some strictly positive constants ℎ^′, 𝛼₁, 𝛼₂, 𝛼₃, and 𝛼₄ such that, for all 𝑥∈𝐵_ℎ′ and 𝑡≥0, 𝛼₁‖𝑥‖²≤𝑉(𝑡,𝑥)≤𝛼₂‖𝑥‖², 𝑉̇(𝑡,𝑥)|_{̇𝑥=𝑓(𝑡,𝑥)}≤−𝛼₃‖𝑥‖², and |𝜕𝑉(𝑡,𝑥)/𝜕𝑥|≤𝛼₄‖𝑥‖.

Now, if there exists a function 𝑉(𝑡,𝑥) and strictly positive constants 𝛼₁, 𝛼₂, 𝛼₃, and 𝛿, such that, for all 𝑥∈𝐵_ℎ, 𝑡≥0: 𝛼₁‖𝑥‖²≤𝑉(𝑡,𝑥)≤𝛼₂‖𝑥‖², 𝑉̇(𝑡,𝑥)|_{̇𝑥=𝑓(𝑡,𝑥)}≤0 and ∫_𝑡^𝑡+𝛿𝑑𝑉(𝜏,𝑥(𝜏))/𝑑𝜏|_{̇𝑥=𝑓(𝑡,𝑥)}𝑑𝜏≤−𝛼₃‖𝑥‖² then, 𝑥(𝑡) converges exponentially to zero.

Finally [23], the state equation [𝐴,𝐵,𝐶,𝐷] is a minimal realization of a proper rational function 𝑔(𝑠) if and only if (𝐴,𝐵) is controllable, (𝐴,𝐶) is observable, or if and only if dim(𝐴)=deg(𝑔(𝑠)), where 𝑔(𝑠)=𝑁(𝑠)/𝐷(𝑠) and deg(𝑔(𝑠))=deg(𝐷(𝑠)).

In the next definitions, let us consider the system Π defined as 𝑥̇(𝑡)(),Π∶𝑥(0)=𝑓𝑥,𝑢=𝑥₀∈𝑅^𝑛,𝑦(𝑡)=𝐻(𝑥,𝑢), (5) where 𝑥∈𝑅^𝑛 is the state vector, 𝑢∈𝑅^𝑚 the input, and 𝑦∈𝑅^𝑚 the system output. Consider the set Ξ of 𝑛 dimensional real valued functions 𝑓(𝑡)∶𝑅⁺→𝑅^𝑛 and define the set 𝐿₂={𝑥∈Ξ∶‖𝑓‖²₂=∫^∞₀‖𝑓(𝑡)‖²𝑑𝑡<∞}, with ‖⋅‖₂ the Euclidean norm. This set constitutes a normed vector space with the field 𝑅 and norm ‖⋅‖₂. Let us introduce the extended space 𝐿_2𝑒 as 𝐿_2𝑒=𝑥∈Ξ∶𝑓²_2𝑇=^𝑇₀𝑓(𝑡)²𝑑𝑡<∞,∀𝑇, (6) where 𝐿₂⊂𝐿_2𝑒. In the same way, let us introduce the inner product and the truncated inner product of functions 𝑢 and 𝑦 as 𝑢,𝑦=^∞₀𝑢^𝑇(𝑡)𝑦(𝑡)𝑑𝑡,𝑢,𝑦_𝑇=^𝑇₀𝑢^𝑇(𝑡)𝑦(𝑡)𝑑𝑡. (7) The system (5) is dissipative [24] with respect to the supply rate 𝜔(𝑢,𝑦)∶𝑅^𝑚×𝑅^𝑚→𝑅 if and only if there exists a storage function: 𝑅^𝑛→𝑅_≥0, such that 𝐻(𝑥(𝑇))≤𝐻(𝑥(0))+∫^𝑇₀𝜔(𝑢(𝑡),𝑦(𝑡))𝑑𝑡, for all 𝑢(𝑡), all 𝑇≥0, and all 𝑥₀∈𝑅^𝑛. The system is passive if its supply rate is given by (𝑢,𝑦)=𝑢^𝑇𝑦. It is input strictly passive (ISP) if there exists a positive constant 𝛿_𝑖 such that the supply rate can be expressed as (𝑢,𝑦)=𝑢^𝑇𝑦−𝛿_𝑖‖𝑢‖², 𝛿_𝑖>0. Finally, it is output strictly passive (OSP) if there exists a positive constant 𝛿₀ such that the supply rate can be expressed as (𝑢,𝑦)=𝑢^𝑇𝑦−𝛿_𝑜‖𝑦‖², 𝛿_𝑜>0. The system (5) is called 𝐿₂ finite gain stable if there exists a positive constant 𝛾 such that for any initial condition 𝑥₀ there exists a finite constant 𝛽(𝑥₀)such that ‖𝑦‖_2𝑇≤𝛾‖𝑢‖_2𝑇+𝛽(𝑥₀) and one important result from [24] is that if Π∶𝑢→𝑦 is OSP, then it is 𝐿₂ stable.

A state space system 𝑥̇(𝑡)=𝑓(𝑥), 𝑥∈𝑅^𝑛, is zero state observable from the output 𝑦(𝑡)=ℎ(𝑥) if, for all initial conditions 𝑥(0)∈𝑅^𝑛, the output 𝑦(𝑡)≡0 implies that 𝑥(𝑡)≡0. It is zero state detectable if 𝑦(𝑡)≡0 implies that lim𝑥(𝑡)=0 as 𝑡→∞.

Regarding the identification algorithms, the persistence excitation (PE) condition is very important and its definition is given now according to [22]: a vector 𝜙∶𝑅⁺→𝑅^2𝑛 is persistently exciting (PE) if there exist constants {𝛼₁,𝛼₂,𝛿}>0 such that 𝛼₁𝐼≤^𝑡0_𝑡^+𝛿₀𝜙(𝜏)𝜙^𝑇(𝜏)𝑑𝜏≤𝛼₂𝐼,∀𝑡₀≥0. (8) Besides, let 𝑤∶𝑅⁺→𝑅^2𝑛. If 𝑤 is PE, the signals {𝑤,𝑤̇} belong to the space 𝐿_∞, and 𝑈 is a rational stable strictly proper minimum phase transfer function, then 𝑈(𝑤) is PE.

Theorem 1 (small signal I/O stability, [22]). Consider the perturbed system 𝑥̇=𝑓(𝑡,𝑥,𝑢), 𝑥(0)=𝑥₀ and the unperturbed system 𝑥̇=𝑓(𝑡,𝑥,0), 𝑥(0)=𝑥₀, where 𝑡≥0, 𝑥∈𝑅^𝑛, and 𝑢∈𝑅^𝑚. Let 𝑥=0 be an equilibrium point of the unperturbed system, that is, 𝑓(𝑡,0,0)=0, for all 𝑡≥0. Let 𝑓 be piecewise continuous in 𝑡 and have continuous and bounded first partial derivatives in 𝑥 for all 𝑡≥0, 𝑥∈𝐵_ℎ, 𝑢∈𝐵_𝑐. Let 𝑓 be Lipchitz in u, with Lipchitz constant 𝑙_𝑢, for all 𝑡≥0, 𝑥∈𝐵_ℎ, 𝑢∈𝐵_𝑐. Let 𝑢∈𝐿_∞. If 𝑥=0 is an exponentially stable equilibrium point of the unperturbed system, then the perturbed system is small-signal 𝐿_∞ stable; that is, there exist 𝛾_∞, 𝑐_∞>0, such that ‖𝑢‖_∞<𝑐_∞ implies that ‖𝑥‖_∞≤𝛾_∞‖𝑢‖_∞<ℎ, where 𝑥 is the solution of 𝑓(𝑡,𝑥,𝑢) starting at 𝑥₀=0. There exists 𝑚≥1 such that, for all ‖𝑥₀‖<ℎ/𝑚, 0<‖𝑢‖_∞<c_∞ implies that 𝑥(𝑡) converges to a 𝐵_𝛿 ball of radius 𝛿=𝛾_∞‖𝑢‖_∞<ℎ; that is, for all 𝜑>0, there exists 𝑇≥0 such that ‖𝑥(𝑡)‖≤(1+𝜑)𝛿 for all 𝑡≥𝑇, along the solutions of 𝑓(𝑡,𝑥,𝑢) starting at 𝑥₀. Also for 𝑡≥0, ‖𝑥(𝑡)‖<ℎ.

2.2. Model of the DC Servomechanism

The closed loop identification analysis that will be presented considers the model of a perturbed DC servomechanism described by the following equation: 𝐽𝑞̈+𝑓𝑞(̇𝑡)=𝑘𝑢(𝑡)+𝜈₁(𝑡)=𝜏(𝑡)+𝜈₁(𝑡), (9) where 𝐽,𝑓,𝑘,𝑢,𝜈₁, and 𝜏(𝑡)=𝑘𝑢(𝑡) are the inertia, viscous friction coefficient, amplifier gain, input voltage, perturbation signal, and torque input, respectively. The perturbation signal may be due to possibly constant disturbances or parasitic voltages appearing inside the amplifier. The model (9) can be rewritten as 𝑞𝑣̈(𝑡)=−𝑎𝑞(̇𝑡)+𝑏𝑢(𝑡)+𝜈(𝑡), (10) where 𝑎=𝑓/𝐽 and 𝑏=𝑘/𝐽 are positive constants and 𝜈=𝜈₁/𝐽 and {𝑞(𝑡),𝑞(̇𝑡)} are the servo angular position and velocity, respectively. It is also assumed that the perturbation signal is bounded, that is: ‖𝜈₁(𝑡)‖≤𝛽, 𝛽∈𝑅⁺. In the following it will be shown that there exists a region Ω_𝛿 where the parameter estimates converge, even in the presence of the perturbation signal 𝜈(𝑡). Furthermore, it will be shown that the size of this region can be arbitrarily reduced by increasing the controller gains employed for stabilizing the servomechanism. This analysis will consider two cases.(i)Closed loop identification of the unperturbed DC servomechanism, where the main objective will be to get a formula to estimate the system parameters and show the condition under which the system is exponentially stable.(ii)Closed loop identification of the perturbed DC servomechanism where the existence of the region Ω_𝛿 will be shown where the parameter estimates converge, when a perturbation signal affects the system.

3. Closed-Loop Identification Algorithm: Nonperturbed Case

This section describes the closed loop identification approach and the way the parameter updating law is constructed. Besides, it is proved that under some assumptions the overall system is exponentially stable. Then, first the stability analysis will be presented to ensure the asymptotic stability of the closed loop system and the parameter error dynamics. Later, the system exponential stability property will be established.

3.1. Stability Analysis and Parameter Convergence

The closed loop identification analysis to be described is similar to that presented in [14] and depicted in Figure 1. For this (unperturbed) case, it will be considered that 𝜈(𝑡) is identically zero, leading to the following unperturbed servo model: 𝑞(̈𝑡)=−𝑎𝑞(̇𝑡)+𝑏𝑢(𝑡). (11) The closed loop identification methodology consists of selecting a model of (11) with the following dynamics: 𝑞̈_𝑒(𝑡)=−𝑎𝑞̇_𝑒(𝑡)+𝑏̂𝑢_𝑒(𝑡), (12) where 𝑞_𝑒(𝑡) is the estimated servo position, 𝑞̇_𝑒(𝑡) the estimated velocity, {𝑎,𝑏̂} are the estimates of {𝑎,𝑏}, and 𝑢_𝑒 the control law for the estimated model. Then, the loop is closed around the real servomechanism and its model by using two PD controllers, where the same gains are used for both of them. Now let us define the output error 𝜀(𝑡)=𝑞(𝑡)−𝑞_𝑒(𝑡), and then this error and its time derivative are employed to feed an identification algorithm which estimates the system parameters and updates them in the estimated model. All this procedure is now theoretically summarized.

Figure 1 

Blocks diagram for the identification algorithm.

For closing the loop around the unperturbed system and its model, let us consider the PD controllers u(t) and 𝑢_𝑒(𝑡) as follows: 𝑢(𝑡)=𝑘_𝑝𝑒(𝑡)−𝑘_𝑑𝑞(̇𝑡) and 𝑢_𝑒(𝑡)=𝑘_𝑝𝑒_𝑒(𝑡)−𝑘_𝑑𝑞̇_𝑒(𝑡), where 𝑒(𝑡)=(𝑞_𝑑(𝑡)−𝑞(𝑡)), 𝑒_𝑒(𝑡)=(𝑞_𝑑(𝑡)−𝑞_𝑒(𝑡)), 𝑞_𝑑(𝑡) is the reference signal, and 𝑘_𝑝>0, 𝑘_𝑑>0 the proportional and derivative controller gains which are used in both controllers. Then, the system (11) in closed loop with 𝑢(𝑡) leads to the next closed-loop dynamics (for the sake of simplicity, from now on the time argument will be omitted) as Σ₁∶𝑞̈=−𝑎+𝑏𝑘_𝑑𝑞̇+𝑏𝑘_𝑝𝑒, (13) while the system (12) in closed loop with 𝑢_𝑒(𝑡) leads to the following dynamics: Σ₂∶𝑞̈_𝑒=−𝑎+𝑏̂𝑘_𝑑𝑞̇_𝑒+𝑏̂𝑘_𝑝𝑒_𝑒. (14) By using the Routh-Hurwitz criterion, it is easy to show that the control law 𝑢(𝑡) stabilizes the unperturbed servo model (11). However, even when the system (13) is stable, the same conclusion cannot be drawn for (14), because the coefficients of this estimated model (i.e., 𝑎,  𝑏̂) are updated by one identification algorithm to be described later, and then these parameters are time varying; therefore, it is necessary to analyze the stability of (14). From the definition of 𝜀(𝑡), it is possible to evaluate its second time derivative and employing (13) and (14), the error dynamics is established as follows: 𝜀+𝑐𝜀̇^..+𝑏𝑘_𝑝𝜀=𝜃^𝑇𝜙, (15) where 𝑐=(𝑎+𝑏𝑘_𝑑)>0, 𝜃(𝑡) is the parameter error vector, and 𝜙(𝑡) is the so-called regressor vector, where 𝜃=𝜃−𝜃=(𝑎−𝑎,𝑏̂−𝑏)^𝑇, 𝜙(𝑡)=(𝑞̇_𝑒,−𝑢_𝑒)^𝑇, with 𝜃 being the estimate of 𝜃. In order to analyze the behavior of the signals involved in the error dynamics (15), passivity based arguments will be used [24, 25]. To this end, let 𝑥=(𝜀,𝜀)̇^𝑇 be the state vector of (15) and consider the following storage function: 𝐻₁(𝑥)=12𝑥^𝑇𝑏𝑘_𝑝𝜇𝜇1𝑥=𝑥^𝑇𝑀𝑥, (16) where 𝜇∈𝑅⁺. The choice of 𝐻₁ will lead to concluding the OSP property of (15). Then, taking the time derivative of 𝐻₁ along the trajectories of (15) leads to 𝐻̇₁=𝜃^𝑇𝜙𝜇𝜀+𝜀̇−𝑐2𝜇𝜀+𝜀̇²−𝑐2−𝜇𝜀²̇−𝜇𝑏𝑘_𝑝−𝜇𝑐2𝜀². (17) It is easy to prove that 𝐻₁>0 if 𝜇<𝑏𝑘_𝑝. Besides, if 𝜇<min{𝑐/2,2𝑏𝑘_𝑝/𝑐}, then the time derivative of 𝐻₁ along the trajectories of (15) yields 𝐻̇₁≤𝜃^𝑇𝜙(𝜇𝜀+𝜀)̇−𝑐(𝜇𝜀+𝜀)̇²/2; that is, (15) defines an OSP operator 𝜃^𝑇𝜙→(𝜇𝜀+𝜀)̇. Moreover, it is well known that the feedback interconnection of passive subsystems is passive [24], thus making an intuitive to consider the parameter error dynamics as follows: Σ₃∶̇𝜃=−Γ𝜙(𝜇𝜀+𝜀)̇, with matrix Γ=Γ^𝑇>0. Then, by considering the storage function 𝐻₂(𝜃)=𝜃^𝑇Γ⁻¹𝜃/2, its time derivative along the trajectories of Σ₃ leads to 𝐻̇₂=𝜃^𝑇Γ⁻¹𝜃̇=(𝜇𝜀+𝜀)̇(−𝜃^𝑇𝜙), so that Σ₃ defines the passive operator (𝜇𝜀+𝜀)̇→(−𝜃^𝑇𝜙). Finally, let us consider the feedback interconnection of (15) with Σ₃ given by Σ^..𝜀(𝑡)+𝑐𝜀(̇𝑡)+𝑏𝑘_𝑝𝜀(𝑡)=𝜃^𝑇𝜙(𝑡),𝜃̇=−Γ𝜙𝜇𝜀+𝜀̇. (18) Then, by considering the storage function for (18) as the sum of 𝐻₁ and 𝐻₂, it is easy to prove that Σ is still OSP; thus, it follows that (𝜇𝜀+𝜀)̇∈𝐿₂. Let us define the signal 𝑦(𝑡)=(𝜇𝜀+𝜀)̇ as the output for the interconnected system (18). Then, note that 𝜀(𝑡) corresponds to the output of an exponentially stable filter whose input belongs to the 𝐿₂ space; therefore, 𝜀(𝑡) tends to zero as 𝑡→∞ [26]. Therefore, it has been proved that 𝑞(𝑡)∈𝐿_∞ and 𝜀(𝑡)→0 as 𝑡→∞, and thus 𝑞_𝑒(𝑡)→𝑞(𝑡) as 𝑡→∞; that is, 𝑞_𝑒(𝑡)∈𝐿_∞, and then it turns out that the time varying system (14) is stable, as desired, although it remains to prove that the parametric error converges to zero, that is, that 𝜃(𝑡) converges to the real parameter vector 𝜃. To this end, let us consider the state-space description of (15) as 𝑥̇=𝐴𝑥+𝐵𝑈,𝑦=𝐶𝑥, (19) with: 𝐴=01−𝑏𝑘_𝑝01,𝐶−𝑐,𝐵=_𝑇=𝜇1,𝑈=𝜃^𝑇𝜙. (20) As described in [22], a necessary and sufficient condition for parameter convergence in linear systems is the PE condition of the regressor vector 𝜙(𝑡). However, the results presented in [22] assume that all the signals in 𝜙(𝑡) come from a linear time invariant system, which is not the case here because here the regressor 𝜙(𝑡) has signals from the estimated model (14), which is a time varying system. To overcome this technical difficulty it is possible to consider the real regressor vector 𝜙^𝑇_𝑟(𝑡)=(𝑞,̇−𝑢)^𝑇 which consists of signals from the real servomechanism (13). Following the same procedure as that presented in [22], it is not a difficult task to show that if 𝑐>𝜇, then 𝜙_𝑟(𝑡) will be PE. Now, let us consider the difference 𝜙_𝑟𝑞̇−𝑞̇−𝜙=−𝑢−𝑢^𝑒_𝑒=𝜀̇𝑘_𝑝𝜀+𝑘_𝑑𝜀̇, (21) and consider the Lyapunov function candidate 𝑉₁=𝐻₁+𝐻₂. Clearly 𝑉₁>0 if 𝜇<𝑏𝑘_𝑝 and 𝑉̇₁≤−𝛽𝜀², 𝛽=(2/𝛼)(𝜇𝑏𝑘_𝑝𝛼/2−𝜇²𝑐²/8), with 𝛼=(𝑐−𝜇)>0. Therefore, if 𝜇≤4𝑏𝑘_𝑝𝑐/(4𝑏𝑘_𝑝+𝑐²), then 𝜀(𝑡)∈𝐿₂, and considering the fact that 𝑦(𝑡)=(𝜇𝜀+𝜀)̇∈𝐿₂ permits concluding that 𝜀(̇𝑡)∈𝐿₂, which makes clear that (𝜙_𝑟−𝜙)∈𝐿₂. Now, given a PE signal 𝜔(𝑡) and a signal 𝑧(𝑡)∈𝐿₂, the sum (𝜔+𝑧) is still PE [22]. Thus, 𝜙=𝜙_𝑟−(𝜙_𝑟−𝜙) is PE as desired and the convergence of 𝜃(𝑡) to zero can be claimed, that is, the estimated parameters converge to the real ones. Then, based on the previous analysis, the following proposition follows.

Proposition 2. Consider the system (13) and assume that 𝜇≤min{𝑏𝑘_𝑝,2𝑏𝑘_𝑝/𝑐,4𝑏𝑘_𝑝𝑐/(4𝑏𝑘_𝑝+𝑐²)}. Then 𝜀,𝜀̇ and the parameter error vector 𝜃(t) converge to zero as time tends to infinity.

From the previous result, asymptotic stability has been proved for the closed loop system and identification approach. However, in the identification framework usually it is preferable to have a more robust stability condition, that is, exponential stability to guarantee that the parameter identification scheme will be robust against disturbances which may affect the system, such as unmodeled dynamics and measurement noise. Therefore, the next analysis will establish the necessary conditions for the servomechanism to be an exponentially stable system.

3.2. Exponential Convergence

As mentioned above, one appealing and important property of the parameter identification schemes is the exponential stability, because it guarantees robustness of the identification algorithm in presence of external disturbances or unmodeled dynamics. Thus, in the following it will be proved which conditions must be satisfied in order to make the system to be exponentially stable. First recall the state space description given by (19). Clearly the system (𝐴,𝐵,𝐶) is observable and controllable [23]; that is, it is a minimal realization, if 𝜇<𝑐/2. Let us consider the systems Σ₄∶̇𝑦𝜃=0₂=𝜙^𝑇(𝑡)𝜃(𝑡),Σ₅∶̇𝜀̇𝑦𝜃=−Γ𝜙𝜇𝜀+₂=𝜙^𝑇(𝑡)𝜃(𝑡). (22) The last analysis proved that 𝜙(𝑡) is PE which turns out to be equivalent to the UCO condition on the system Σ₄ and so do Σ₅. Then, the main result of this section can be established as follows.

Theorem 3. Given the system (18), assume that μ fulfills the bound conditions of Proposition 2. Let 𝐰=(𝑥,𝜃) be an equilibrium point of Σ. Then 𝐰 is an exponentially stable equilibrium point.

Proof. Let 𝐰(𝑡)=(𝑥,𝜃)^𝑇=(𝜀,𝜀,̇𝜃)^𝑇 be the state of (18) and consider the following Lyapunov function candidate: 𝑉₂(𝐰(𝑡))=𝐻₂𝜃(𝑡)+𝐻₃(𝑥(𝑡)), (23) with 𝐻₂ as previously mentioned and 𝐻₃ defined as follows: 𝐻₃(𝑥(𝑡=1))2𝑥^𝑇𝑀₂𝑥,𝑀₂=𝑏𝑘_𝑝+𝜇𝑐𝜇𝜇1. (24) Clearly, 𝐻₃>0 if the inequality 𝜇<(𝑐+𝑐²+4𝑏𝑘_𝑝)/2 holds. Besides, it is possible to show that 12𝜆_min𝑀₂‖𝑥‖²+12𝜆_min(Γ)𝜃²≤𝑉₂(𝐰)≤12𝜆_max𝑀₂‖𝑥‖²+12𝜆_max(Γ)𝜃². (25) Taking the time derivative of 𝑉₂ along the trajectories of (18) yields 𝑉̇₂=−𝑥^𝑇𝜇𝑏𝑘_𝑝00𝑐−𝜇𝑥. (26) Let us define the diagonal matrix 𝑄=diag{𝜇𝑏𝑘_𝑝,𝑐−𝜇}; then we have that 𝑉̇₂≤−𝑥^𝑇𝑄𝑥≤0, where 𝑄=𝑄^𝑇>0 if 𝜇<𝑐. Now assume that there exist strictly positive constants {𝛼₃,𝛿} such that _𝑡^𝑡+𝛿𝑑𝑉𝑑𝜏²(𝐰(𝜏))₍₁₈₎𝑑𝜏≤−𝛼₃‖𝐰(𝑡)‖². (27) First note from (18) and the time derivative of 𝑉₂ that _𝑡^𝑡+𝛿𝑑𝑉𝑑𝜏²(𝐰(𝜏))₍₁₈₎𝜆≤−^min(𝑄)𝜇²+1_𝑡^𝑡+𝛿𝑦(𝜏)²𝑑𝜏, (28) and then (27) will be valid if, for 𝛼₃>0, the following inequality holds: 𝜆_min(𝑄)𝜇²+1^𝑡0_𝑡^+𝛿₀𝑦(𝜏)²𝑑𝜏≥𝛼₃𝑥𝑡₀²+𝜃𝑡₀², (29) for all 𝑡₀≥0 and 𝐰(𝑡₀). Now consider the following system: 𝑥̇=𝐴𝑥+𝐵𝑈,𝜃̇=0,𝑦=𝐶𝑥. (30) Let us consider the UCO property of Σ₅ with 𝐾=−Γ(𝜇𝜀+𝜀)̇. It is clear that, for the UCO condition, 𝑘_𝛿=^𝑡0_𝑡^+𝛿₀Γ^𝑇𝜇𝜀+𝜀̇²Γ𝑑𝜏≤^𝑡0_𝑡^+𝛿₀𝜆²_max(Γ)𝜇𝜀+𝜀̇²𝑑𝜏. (31) Also we know that (𝜇𝜀+𝜀)̇∈𝐿₂; thus, there exists a positive constant 𝜅>0 such that 𝑘_𝛿≤𝜅𝜆²_max(Γ). What’s more, the output of (19) is given by 𝑦(𝑡)=𝐶^𝑇𝑒^{𝐴(𝑡−𝑡0)}𝑥𝑡₀+^𝑡_𝑡0𝐶^𝑇𝑒^{𝐴(𝑡−𝜏)}𝐵𝜙(𝜏)𝑑𝜏𝜃𝑡₀ (32) because from (30) we note that the vector 𝜃 is constant. Let us define the signals 𝑧₁(𝑡) and 𝑧₂(𝑡) as follows: 𝑧₁(𝑡)=𝐶^𝑇𝑒^{𝐴(𝑡−𝑡0)}𝐱𝑡₀,𝑧₂(𝑡)=^𝑡_𝑡0𝐶^𝑇𝑒^{𝐴(𝑡−𝜏)}𝐵𝜙(𝜏)𝑑𝜏𝜃𝑡₀. (33) From the stability analysis of Section 3 it was concluded that {𝜙(𝑡),𝜙̇(𝑡)}∈𝐿_∞ and that 𝜙(𝑡) is PE. Thus, it is clear that [22] 𝜙_𝑓(𝑡)=∫^𝑡_𝑡0𝐶^𝑇𝑒^{𝐴(𝑡−𝜏)}𝐵𝜙(𝜏)𝑑𝜏 is PE. Therefore, there exist positive constants {𝛼₁,𝛼₂,𝜎} such that 𝛼₁‖𝜃(𝑡₀)‖²≤∫^𝑡1_𝑡^+𝜎₁𝑧²₂(𝜏)𝑑𝜏≤𝛼₂‖𝜃(𝑡₀)‖² for all 𝑡₁≥𝑡₀≥0 and 𝜃(𝑡₀). Moreover, because matrix 𝐴 is stable there exist positive constants {𝛾₁, 𝛾₂} such that ∫^∞_{𝑡0+𝑚𝜎}𝑧²₁(𝜏)𝑑𝜏≤𝛾₁‖𝑥(𝑡₀)‖²𝑒^{−𝛾2𝑚𝜎} for all 𝑡₀≥0, 𝑥(𝑡₀) and an integer 𝑚>0 to be defined later. Because (𝐴,𝐶) is observable there exists 𝛾₃(𝑚𝜎)>0 with 𝛾₃(𝑚𝜎) increasing such that ∫^𝑡0_𝑡^+𝑚𝜎₀𝑧²₁(𝜏)𝑑𝜏≤𝛾₃(𝑚𝜎)‖𝑥(𝑡₀)‖² for all 𝑡₀≥0, 𝑥(𝑡₀) and 𝑚>0. Let 𝑛>0 be another integer to be defined and 𝛿=(𝑚+𝑛)𝜎. Then ^𝑡0_𝑡^+𝛿₀𝑦(𝜏)²≥𝑑𝜏_{𝑡0𝑡+𝑚𝜎0}𝑧²₁(𝜏)𝑑𝜏−^𝑡0_𝑡^+𝛿_0+𝑚𝜎𝑧²₁(𝜏)+𝑑𝜏_{𝑡0𝑡+𝑚𝜎0}𝑧²₂(𝜏)𝑑𝜏−^𝑡0_𝑡^+𝛿_0+𝑚𝜎𝑧²₂(𝜏)𝑑𝜏≥𝛾₃(𝑚𝜎)𝑥𝑡₀²−𝛾₁𝑒^{−𝛾2𝑚𝜎}𝑥𝑡₀²+𝑛𝛼₁𝜃𝑡₀²−𝑚𝛼₂𝜃𝑡₀²=𝑥𝑡₀²𝛾₃(𝑚𝜎)−𝛾₁𝑒^{−𝛾2𝑚𝜎}+𝜃𝑡₀²𝑛𝛼₁−𝑚𝛼₂. (34) Let both 𝑚 and 𝑛 be large enough such that 𝑛𝛼₁−𝑚𝛼₂≥𝛼₁>0 and 𝛾₃(𝑚𝜎)−𝛾₁𝑒^{−𝛾2𝑚𝜎}≥𝛾₃(𝑚𝜎)/2>0; then, ^𝑡0_𝑡^+𝛿₀𝑦(𝜏)²𝛾𝑑𝜏≥³(𝑚𝜎)2𝑥𝑡₀²+𝛼₁𝜃𝑡₀². (35) Similarly, it is possible to conclude that ^𝑡0_𝑡^+𝛿₀𝑦(𝜏)²𝑑𝜏≤2𝛾₁𝑥𝑡₀²𝑒^{−𝛾2𝑚𝜎}+2(𝑚+𝑛)𝛼₂𝜃𝑡₀². (36) Now we define 𝛽₁=min(𝛼₁,𝛾₃(𝑚𝜎)/2) and 𝛽₂=max(2𝛾₁,2(𝑚+𝑛)𝛼₂). Then, it is possible to obtain the following bounds: 𝛽₁𝑥𝑡₀²+𝜃𝑡₀²≤^𝑡0_𝑡^+𝛿₀𝑦(𝜏)²𝑑𝜏≤𝛽₂𝑥𝑡₀²+𝜃𝑡₀², (37) and therefore _𝑡^𝑡+𝛿𝑑𝑉𝑑𝜏²(𝐰(𝜏))₍₁₈₎𝜆≤−^min(𝑄)𝜇²𝛽+1¹𝑥𝑡₀²+𝜃𝑡₀²≤−𝛼₃𝐰₀, (38) where 𝛼₁=min(𝜆_min(𝑀₂), 𝜆_min(Γ))/2, 𝛼₂=max(𝜆_max=(𝑀₂),𝜆_max(Γ))/2, and 𝛼₃=𝜆_min(𝑄)min(𝛼₁,𝛾₃(𝑚𝜎)/2)/(𝜇²+1). Then we conclude that the system (18) is exponentially stable.

The last result established the exponential stability of (18) for the unperturbed case. However, in practice there are several disturbance sources (e.g., unmodeled dynamics, noise from the environment or from the measuring devices, etc.) which may affect the system performance of both the system and the identification algorithm. Therefore, the next analysis shows the extension of the last analysis to those scenarios.

4. Closed-Loop Identification Algorithm: Perturbed Case

Till now exponential stability of the equilibrium of (18) has been established. Now, it will be considered the perturbed system (10) in closed loop with the PD controller 𝑢(𝑡) and it will be proved that this closed loop system is 𝐿_∞ stable in the sense of Theorem 1. Let us consider the system (10) in closed loop with 𝑢(𝑡), the system Σ₂, and the output error 𝜀(𝑡) defined as before. Then, it is possible to obtain the following error dynamics: 𝜀^..(𝑡)=−𝑐𝜀(̇𝑡)−𝑏𝑘_𝑝𝜀(𝑡)+𝜃^𝑇𝜙(𝑡)+𝜈(𝑡). (39) For the system (39) it is important to know if it is still stable in the presence of the perturbation signal 𝜈(𝑡). If it is the case, then the system will be robust even in the presence of perturbations, which is desirable in the identification context because reliable parameter estimates can be obtained even in the perturbed case. The next result obtained from [22] allows concluding that the perturbed system (39) is 𝐿_∞ stable.

Theorem 4. Consider the perturbed system (39) and the unperturbed system (18). If the equilibrium 𝐰₀ of (18) is exponentially stable, then one has the following.(1)The perturbed system (39) is small signal 𝐿_∞-stable; that is, there exists 𝛾_∞ such that ‖𝐰(𝑡)‖≤𝛾_∞𝛽<ℎ, where 𝐰(𝑡) is the solution of (39) starting at 𝐰₀.(2)There exists 𝑚≥1, such that ‖𝐰₀‖<ℎ/𝑚 implies that 𝐰(𝑡) converges to a ball Ω_𝛿 of radius 𝛿=𝛾_∞𝛽<ℎ; that is, for all 𝜀>0, there exists 𝑇≥0 such that ‖𝐰(𝑡)‖≤𝛿(1+𝜀) for all 𝑡≥𝑇, along the solutions of (39) starting at 𝐰₀. Also for all 𝑡≥0, ‖𝐰₀‖<ℎ.

Proof. Consider again the Lyapunov function 𝑉₂(𝐰(𝑡)). Assuming that inequalities for 𝜇 given in Proposition 2, it is possible to obtain 𝑉̇₂≤−𝜆_min(𝑄)‖𝑥‖²+𝛽𝜇²+1‖𝑥‖. Let us define the constants 𝛼₃=𝜆_min(𝑄), 𝛼₄=𝜇²+1, 𝛾_∞=𝛼₄/𝛼₃𝛼₂/𝛼₁,   𝛿=𝛾_∞𝛽, and 𝑚=𝛼₂/𝛼₁≥1. Then, the time derivative of 𝑉₂(𝐰(𝑡)) along the trajectories of (39) yields 𝑉̇₂≤−𝛼₃‖𝑥‖[‖𝑥‖−𝛿/𝑚] and two cases may be considered.(1)First it will be proved that ‖𝐰(𝑡)‖≤𝛾_∞𝛽<ℎ. To this end consider the case where ‖𝑥₀‖≤𝛿/𝑚. Note that 𝛿/𝑚≤𝛿 because of 𝑚≥1, which implies that 𝐰(𝑡)∈Ω_𝛿 for all 𝑡≥0. Suppose that it is not true, and then, by continuity of the solutions there exists 𝑇₀ and 𝑇₁, with 𝑇₁>𝑇₀≥0 such that ‖𝑥(𝑇₀)‖=𝛿/𝑚, ‖𝑥(𝑇₁)‖>𝛿 and for all 𝑡∈[𝑇₀,𝑇₁]: ‖𝐰(𝑡)‖≥𝛿/𝑚, and then, from the time derivative of 𝑉₂ it is clear that in [𝑇₀,𝑇₁]: 𝑉̇₂≤0, but in this case 𝑉₂(𝑇₀,𝑥(𝑇₀))≤𝛼₂(𝛿/𝑚)²=𝛼₁𝛿² and 𝑉₂(𝑇₁,𝑥(𝑇₁))>𝛼₁𝛿², which is a contradiction; therefore, for all 𝐰(𝑡) with initial condition 𝐰₀, a solution for (39) remains on Ω_𝛿.(2)Assume that ‖𝐰₀‖>𝛿/𝑚 and that for all 𝜀>0 there exists 𝑇≥0 such that ‖𝐰(𝑇)‖≤𝛿(1+𝜀)/𝑚 and suppose that it is not true; then, for some 𝜀>0 and for all 𝑡≥0: ‖𝐰(𝑡)‖>𝛿(1+𝜀)/𝑚 and from the time derivative of 𝑉₂ given previously it is possible to obtain 𝑉̇₂≤−𝛼₃(𝛿/𝑚)²𝜀(1+𝜀), which is a strictly negative constant. However, this contradicts with the fact that 𝑉₂(0,𝑥₀)≤𝛼₂‖𝐱₀‖²<𝛼₂ℎ²/𝑚²,𝑉₂(𝑡,𝐱(𝑡))≥0 for all 𝑡≥0, because the inequality must be strict. On the other hand, let us assume that for all 𝑡≥𝑇 the following inequality holds: ‖𝑥(𝑡)‖≤𝛿(1+𝜀) and it is possible to probe this affirmation in the same way that the first step of this proof. Thus, we conclude that 𝐰(𝑡) converges to Ω_𝛿, as desired, and the proof is completed.

5. Analysis of the Results

The previous analysis showed that even in the presence of a bounded perturbation signal the parameter estimates remain in a region Ω_𝛿=𝛾_∞𝛽. Now it is important to analyze the behavior of this region and to see that how the controller gains influence the way this region behaves. To this end, recall from Theorem 4 that the state of (39) is bounded by Ω_𝛿=𝛾_∞𝛽=𝜇²+1𝛽²max𝜆_max𝑀₂,𝜆_max(Γ)𝜆_min(Q)²min𝜆_min𝑀₂,𝜆_min(Γ). (40) Besides, it is easy to obtain the eigenvalues 𝜆_𝑖 of 𝑀₂ as 𝜆_𝑖=(𝑏𝑘_𝑝+𝜇𝑐+1±(𝑏𝑘_𝑝+𝜇𝑐−1)²+4𝜇²)/2. Thus, the minimal (𝜆_min) and maximum (𝜆_max) eigenvalues for 𝑀₂ are given by 𝜆_min𝑀₂=𝑏𝑘_𝑝+𝜇𝑐+1−𝑏𝑘_𝑝+𝜇𝑐−1²+4𝜇²2,𝜆_max𝑀₂=𝑏𝑘_𝑝+𝜇𝑐+1+𝑏𝑘_𝑝+𝜇𝑐−1²+4𝜇²2. (41) Besides, for the diagonal matrix 𝑄, note that 𝜆_min(𝑄)=min{𝜇𝑏𝑘_𝑝,𝑐−𝜇},𝜆_max(𝑄)=max{𝜇𝑏𝑘_𝑝,𝑐−𝜇}.Then, note that the region Ω_𝛿 depends on several parameters, among which are the controller gains 𝑘_𝑝 and 𝑘_𝑑. In order to see the effect of the controller gains, let us consider the minimal and maximum eigenvalues of 𝑀₂. From (40), it is possible to note that the region Ω_𝛿 can be reduced by minimizing the value of 𝜆_max(𝑀₂) and maximizing that of 𝜆_min(𝑀₂); that is, the value of (𝑏𝑘_𝑝+𝜇𝑐−1)²+4𝜇² must be the minimum possible. Then, let us consider the function 𝑓_𝑚(𝜇,𝑘_𝑝,𝑘_𝑑)=(𝑏𝑘_𝑝+𝜇𝑐−1)²+4𝜇² and its first and second partial derivatives with respect to 𝜇, 𝑘_𝑝, and 𝑘_𝑑 as follows: 𝜕𝑓_𝑚𝜕𝜇=2𝑐𝑏𝑘_𝑝𝜕+𝜇𝑐−1+8𝜇,²𝑓_𝑚𝜕𝜇²=2𝑐²+8,𝜕𝑓_𝑚𝜕𝑘_𝑝=2𝑏𝑏𝑘_𝑝,𝜕+𝜇𝑐−1²𝑓_𝑚𝜕𝑘²_𝑝=2𝑏²,𝜕𝑓_𝑚𝜕𝑘_𝑑=2𝜇𝑏𝑏𝑘_𝑝,𝜕+𝜇𝑐−1²𝑓_𝑚𝜕𝑘²_𝑑=2𝜇²𝑐. (42) If we set the first partial derivatives to zero, we obtain the equilibrium points {𝜇^∗,𝑘^∗_𝑝,𝑘^∗_𝑑} with the following expressions: 𝑘^∗_𝑝=1−𝜇𝑐𝑏,𝑘^∗_𝑑=1−𝑏𝑘_𝑝+𝜇𝑎𝜇𝑏,𝜇^∗=𝑐1−𝑏𝑘_𝑝𝑐².+4 (43) Now, let us analyze the qualitative behavior of the region Ω_𝛿 for different values of 𝜆_min(𝑄), 𝜆_min(Γ), 𝜆_max(𝑄), 𝜆_max(Γ), 𝑘_𝑝, 𝑘_𝑑, and 𝜇. Figure 2 depicts the behavior of region Ω_𝛿 for different values of 𝑘_𝑝 and 𝑘_𝑑 (the controller gains) and Figure 3 shows also the Ω_𝛿 behavior for different values of 𝜆_min(Γ) and 𝜆_max(Γ) (the matrix related with the identification algorithm). From this set of graphics and (40), note the following.(i)For a given constant value of 𝑘_𝑑 and 𝜇, if 𝑘_𝑑 is big, the value of 𝑘^∗_𝑝 is reduced and the region Ω_𝑑 is also reduced. For a given constant value of 𝑘_𝑝, if 𝑘_𝑝 is big, the values of 𝑘_𝑑^̈∗ and 𝜇^∗ are reduced, which turns out to reduce again the size of the region Ω_𝛿.(ii)The region Ω_𝛿 can be made arbitrarily small if the values for 𝜆_min(𝑄) and min(𝜆_min(𝑀₂),𝜆_min(Γ)) are as large as possible and the value for 𝜆_min(𝑄) can be increased if the controller gains 𝑘_𝑑 and 𝑘_𝑝 have large values, which is the case if a high gain PD controller is used. Then, the size of Ω_𝛿 may be reduced and therefore the quality of the parameter estimates may be improved by means of the controller gains.(iii)By increasing the value of 𝑘_𝑝 and 𝑘_𝑑, the value of 𝜆_min(𝑀₂) is increased, which may reduce the size of Ω_𝛿 and by increasing the values of the gain matrix Γ the value of 𝜆_min(Γ) is also increased, reducing then the size of Ω_𝛿.(iv)As it can be seen from Figure 4, for the controller gains, the effect of the gain 𝑘_𝑝 is more important for reducing the size of region Ω_𝛿 than that of gain 𝑘_𝑑, because large values of 𝑘_𝑑 do not influence significantly in the reduction of the region Ω_𝛿. What’s more, the effect of gain 𝑘_𝑑 is almost imperceptible if the gain 𝑘_𝑝 is large enough.(v)Finally, as expected, if the values of the maximum or minimum eigenvalues for the gain matrix Γ dominate, the size of Ω_𝛿 will also be reduced.

Figure 2 

Behavior of the region Ω𝑑 when the values of 𝑘𝑝 and 𝑘𝑑 are taken as variables.

Figure 3 

Behavior of the region Ω𝑑 when the values of 𝜆min(Γ) and 𝜆max (Γ) are taken as variables.

(a)

(b)

(c)

Figure 4 

Behavior of the region Ω𝑑 for different cases: (a) variation of 𝑘𝑝; (b) variation of 𝑘𝑑; (c) variation of 𝜇.

From the last analysis, it is possible to conclude that, for robustness of the identification algorithm, it is important to pay attention not only to the setup of the gain matrix Γ and 𝜇 from the identification algorithm, but also to the value of the controller gains 𝑘_𝑝 and 𝑘_𝑑. Then, the region Ω_𝑑 is effectively reduced if a high gain 𝑘_𝑝 is employed, and also if large eigenvalues are assigned to matrix Γ.

Another subject to be considered is the controller structure, because it is possible to note that not all the gains have the same effect of reducing region Ω_𝛿. Therefore, when the parameter identification algorithm is designed, it is important to note that the structure of the controller should be an important issue that has been underestimated and that will be important in real applications where noisy measurements or unmodeled dynamics may be affecting the system.

6. Conclusion

This paper analyzes a method for on-line closed loop identification of a linear DC servomechanism with a bounded perturbation signal and a PD controller. Theoretical results show that, without perturbation signals, the system is exponentially stable and the parameter error converges to zero, but when a perturbation signal is present, the parameter estimates converge to a region Ω_𝛿 and the overall system is 𝐿_∞ stable; that is, the system is robust in face of perturbation signals. Besides, by increasing the controller and adaptation gains the region Ω_𝑑 can be arbitrarily reduced, which means that the accuracy of the parameter estimates can be increased, which gives some insight into the importance of the controller structure in the identification process. Future work will be devoted to design more complicated controllers and verify the controller structure needed to obtain a better estimate of the system parameters, that is, a controller which reduces Ω_𝛿 as much as possible and ensures the overall exponential stability.