A new optimal control approach to uncertain Euler-Lagrange equations: $H_\infty$ disturbance estimator and generalized $H_2$ tracking controller

1.   Introduction

Uncertain elements could make real systems unstable if their effects on the systems are not concerned with. They occur from various sources such as modeling errors, actual parameter variations, changes in operating points, unexpected external disturbances, and so on, as discussed in ^[1,2,3]. The issue of dealing with unknown elements is also a significant problem in Euler-Lagrange systems ^[4,5,6], and thus there have been a number of studies on developing robust ^[7,8,9] and/or optimal ^[10,11,12] control approaches to uncertain Euler-Lagrange equations. These studies could be also classified by two approaches in the following aspects.

1.1.   Related studies on uncertain Euler-Lagrange systems

The first approach is concerned with ensuring the stability ^[13,14,15] and/or the optimality ^[16,17,18] with respect to norm-bounded uncertain elements. Even though this first approach can be applied extensively to Euler-Lagrange systems, it is still quite difficult to characterize norm properties of uncertain elements in real systems. To put it another way, determining a set of all possible uncertain elements in real Euler-Lagrange systems is a nontrivial task, and, thus, an intrinsically different approach might be required.

To tackle these limitations in the first approach, the second approach aims at estimating and compensating uncertain elements at every moment, not obtaining all the possible ranges of them. One of the most representative methods in the second approach is to design disturbance observers (DOBs) ^[19,20,21], in which uncertain elements are obtained by using the real control input for the difference of the outputs between the real system and the nominal system. Because the inverse of the nominal system is used in the DOB-based method, the so-called Q-filter is proposed to deal with the case of strictly proper nominal systems. To avoid some involved arguments on designing Q-filters as in ^[22,23,24] (e.g., high-order requirements and direct current-gain constraints), relatively straightforward methods for estimating uncertain elements are also developed in ^[8,9] by using the differences between nominal parameters from their filtered and delayed values, respectively. However, all the estimation methods mentioned above do depend on the prespecified tracking controllers. In other words, the corresponding DOBs and estimators are determined after the closed-loop systems consisting of the nominal systems and the tracking controllers are obtained, thus, enhancing both the estimation and tracking accuracy at the same time are quite difficult in the above methods.

On the other hand, as a novel study on dealing with model uncertainties for nonlinear systems in the presence of dynamic model uncertainties and input constraints, the so-called triple event trigger mechanism is recently introduced in ^[25] to establish the finite-time convergence. However, the relevant performance measures are confined to quadratic forms and it should also be prespecified norm properties of uncertain elements.

To put it another way, there is no discussion on reducing both the magnitudes of uncertain elements and the trajectory tracking errors at the same time in all the aforementioned methods. With respect to this, it would be also worthwhile to note that the tracking errors are often desired to be suppressed in the time-domain bounds for the safety issue such as the collision avoidance. In the same line, the $L_\infty$ norm is taken in ^[26,27,28] as the measure for evaluating the output of various systems, but no discussion on extending this norm to the second approach is provided in the literature. To summarize, solving the following issues would lead to significant improvements of treating uncertain Euler-Lagrange equations:

● A new framework for dealing with estimating uncertain elements and reducing tracking errors in an independent fashion.

● Adequate performance measures for both the estimation and tracking errors.

● Optimal synthesis procedures for both the corresponding estimator and the tracking controller.

● Analysis and synthesis in terms of the time-domain bounds on signals (i.e., the $L_\infty$ norms) for the safety issue.

1.2.   Contributions and organization of this paper

To facilitate the application of robust and optimal control techniques ^[1,2,3], we initially employ the inverse dynamics control (IDC) approach ^[29,30,31], which transforms the nonlinear Euler-Lagrange equation into a linear time-invariant system with unknown signals. To tackle the two tasks of estimating uncertain elements and trajectory tracking control in an independent fashion, we propose a new structural framework based on the IDC approach. More precisely, we design a disturbance estimator for minimizing the $L_2$-induced norm (i.e., the $H_\infty$ norm) from unknown signals to estimation errors by noting the fact that the unknown signals can be regarded as involved in the $L_2$ space. Because it is required to suppress the time-domain bounds of the trajectory tracking errors for the disturbance estimation errors in the $L_2$ space, we consider a tracking controller for minimizing the induced norm from $L_2$ to $L_\infty$ (i.e., the generalized $H_2$ norm) of the mapping from the estimation errors to the tracking errors. Beyond the aforementioned practical aspects, the rationale taking these two performance measures can be interpreted as that the minimization problems are feasible and the corresponding controller synthesis procedures can be carried out through the linear matrix inequality (LMI) approach. In connection with this, the contributions of this paper can be summarized as follows:

● The new structure proposed in this paper allows us to design a disturbance estimator and a tracking controller independently of each other.

● The disturbance estimation accuracy is optimized in terms of the $L_2$-induced norm in the time domain, i.e., the $H_\infty$ norm in the frequency domain.

● The tracking accuracy of uncertain Euler-Lagrange systems is optimized in the time-domain bounds for the first time, through the generalized $H_2$ optimal tracking controller minimizing the induced norm from $L_2$ to $L_\infty$.

This paper is structured as follows: The IDC approach to uncertain Euler-Lagrange systems and motivating issues are reviewed in Section 2. The main results of this paper, i.e., the new control framework with the $H_\infty$ optimal estimator and the generalized $H_2$ optimal tracking controller, are derived in Section 3. Some simulations and experiments are provided in Section 4 to demonstrate the effectiveness and validity of the overall arguments. Finally, conclusions are given in Section 5.

The notations used in this paper are as follows: The notations ${{\mathbb R}}^\nu$ and $I_\nu$ are taken to denote the set of $\nu$-dimensional real vectors and the $\nu$-dimensional identity matrix, respectively. For a real vector or a real matrix, the notation $|\cdot|_{2}$ denotes its $2$-norm, i.e.,

where $\lambda_{\rm max}(\cdot)$ means the maximum eigenvalue of a real symmetric matrix $(\cdot)$. For a real vector-valued signal, we use the notations ${\lVert \cdot \rVert}_{2}$ and ${\lVert \cdot \rVert}_{\infty}$ to mean the ${\mathcal{L}}_{2}$ and ${\mathcal{L}}_{\infty}$ norms, respectively, i.e.,

For symmetric matrices $A$ and $B$, the notation $A\succ B$ implies that $A-B$ is positive definite.

2.   Motivating issues and problem formulation

Consider the Euler-Lagrange equation described as

where $M(q)\in\mathbb{R}^{n\times n}$ is the inertia matrix, $V(q, \dot{q})\in {{\mathbb R}}^{n}$ is the Coriolis and centrifugal torque vector, $G(q)$ is the gravity vector, $\tau \in {{\mathbb R}}^n$ is the torque input, and $q\in {{\mathbb R}}^n$ is the vector of generalized coordinates.

For a predetermined reference trajectory $q_{d}(t)\in\mathbb{R}^{n}$, this paper aims at reducing the corresponding error in the time-domain bound. We define the position error as

and are concerned with reducing the $L_\infty$ norm of a tracking error function associated with both the error $e$ and its rate $\dot{e}$ as small as possible. With respect to minimizing such $L_\infty$ norms in practical environments, it would be required to establish an optimal control framework. However, it is quite difficult to ensure optimality in the nonlinear differential equation of (2.1) itself. Thus, we adopt the IDC approach ^[29,30,31], which reformulates the Euler-Lagrange equation into a linear time-invariant system through a specific torque input given by

where $\hat{(\cdot)}$ implies the nominal value of $(\cdot)$^* and $u$ serves as an auxiliary input (which will be determined by outer control loops). More precisely, substituting (2.3) into (2.1) leads to

^*No gravity compensation error can be often assumed in various real Euler-Lagrange systems.

with the model uncertainties defined as

We call $w$ in (2.4) the total uncertainty throughout the paper.

For the ideal case of the above IDC without model uncertainties (i.e., $w\equiv 0$), we can readily make the tracking error $e$ converge to $0$ through a simple proportional-derivative controller, e.g.,

with $K_p, K_d\succ 0$. However, there are often model uncertainties and their effects on the tracking error cannot be ignored in many real Euler-Lagrange systems. In this sense, one could raise another approach to adequately defining the regulated output $z$ and reducing an induced norm from $w$ to $z$ in a certain level. With this in mind, let us note the following example:

Example 1. We first take an IDC approach to a 6-dimensional Euler-Lagrange systems with model uncertainties generated by random signals establishing

as shown in Figure 1.

More precisely, the simulations are conducted in MATLAB Simulink such that the nominal values $\hat{M}(q)$ and $\hat{V}(q, \dot{q})$ are set to be in the range satisfying

by which the above condition is ensured. Based on Eq (2.4), we next design the generalized $H_2$ optimal controller by using the arguments in ^[32,33,34] (the details will also be discussed in Subsection 3.2) with respect to the measured output

and the regulated output

Then, the tracking error $e$ is not bounded as shown in Figure 2, although the generalized $H_2$ controller is implemented to ensure the internal stability for the nominal system.

As clarified from Example 1, there could exist a case such that the generalized $H_2$ optimal controller cannot lead to stabilizing uncertain Euler-Lagrange systems. This can be interpreted as occurring from the fact that the generalized $H_2$ optimal controller ^[32,33,34] aims at minimizing the induced norm from $w(\in L_2)$ to $z(\in L_\infty)$, assuming that $w$ is an external signal independent of the internal signals. However, $w$ in (2.4) is a function of $(q, \dot{q}, \ddot{q})$. To put it another way, treating $w$ as an external disturbance and simply designing a sort of disturbance rejection controller might not lead to suppressing the position error $e$ in a desired level, and, thus, a new control framework for dealing with model uncertainties in Euler-Lagrange systems should be established. With respect to this, this paper proposes a direct consideration of the total uncertainty $w$ to improve tracking accuracy in terms of the $L_\infty$ norm, and formulates the problem definition as follows.

Problem 1. In the IDC approach to uncertain Euler-Lagrange systems described by (2.4), construct a new optimal control framework with the $H_\infty$ optimal disturbance estimator and the generalized $H_2$ optimal tracking controller.

3.   Main results

This section develops a new optimal control framework for uncertain Euler-Lagrange systems by developing the $H_\infty$ optimal disturbance estimator and the generalized $H_2$ optimal tracking controller.

3.1.   Structure of new optimal control approach

As a preliminary step to establish such a new control framework, we first introduce some important properties introduced in ^[4,5,6] associated with the Euler-Lagrange systems in (2.1) as follows:

Property 1. The following assertions hold:

(1) $M(q)$ is a positive definite matrix for all $q\in {{\mathbb R}}^n$.

(2) There exist positive constants $\underline{m}$ and $\overline{m}$ such that

(3) There exists a positive constant $v$ such that

Because the conditions in Property 1 are ensured for the parameters in real Euler-Lagrange equations, i.e., $M(q)$ and $V(q, \dot{q})$ in (2.1), their nominal values $\hat{M}(q)$ and $\hat{V}(q, \dot{q})$ should be determined to establish the same conditions. In connection with this, we provide the following property:

Property 2. There exist positive constants $\underline{\hat{m}}, \ \overline{\tilde{m}}$, and $\tilde{v}$ such that

On the other hand, we remark that all the signals in the Euler-Lagrange equation, especially for robot manipulators, could be regarded as bounded functions without loss of generality because there should exist hardware limitations, and, thus, all the signals have their upper and lower bounds. In addition, the trajectory tracking process is carried out in a finite-time interval and/or robot manipulators are required to achieve their stationary states at a sufficiently large time $t > > 0$. In connection with this, all the signals of uncertain robot manipulators are assumed in ^[35,36,37] to be bounded in the $L_2$ sense, and we provide the following assumption:

Assumption 1. All the signals are bounded in terms of the $L_2$ norm, i.e.,

Based on Property 2 and Assumption 1, we can readily obtain the following result.

Lemma 1. The total uncertainty $w$ (defined in (2.4)) is bounded in terms of the $L_2$ norm.

Proof. Note from (2.4) that

This completes the proof. □

It is obvious from Lemma 1 that taking the $L_2$ space as the underlying space for the total uncertainty $w$ is theoretically meaningful. By noting this together with the fact that the $L_\infty$ norm is taken for the tracking accuracy, the following parts are devoted to providing a new framework for dealing with the total uncertainty $w$.

We propose a two-step optimal control approach to uncertain Euler-Lagrange systems as shown in Figure 3, and the details can be described as follows:

$({\rm{i}})$ Design the $H_\infty$ optimal disturbance estimator minimizing the $L_2$-induced norm from the total disturbance $w$ to an associated estimation error function.

$({\rm{ii}})$ Design the generalized $H_2$ optimal tracking controller minimizing the induced norm from $L_2$ to $L_\infty$ associated with from the estimation error function to a tracking error function.

In the context of the above framework, the equation of error dynamics in (2.4) can be reinterpreted by

where $u_o$ is obtained from the generalized $H_2$ optimal tracking controller $C$ and $\hat{w}$ is an estimated value of $w$ obtained through the $H_\infty$ optimal estimator $K$.

The rationale behind taking these two different system norms can be described in the following aspects. Because $w$ is in $L_2$, its estimate as well as an associated estimation error should be regarded as elements in $L_2$. In this sense, it should be required for the associated optimal estimator to minimize the $L_2$-induced norm (i.e., the $H_\infty$ norm) in the Step (ⅰ). After the $H_\infty$ optimal disturbance estimation, the tracking accuracy can be further improved by designing an optimal tracking controller minimizing the effect of the estimation error function to a relevant tracking error function. In connection with this, if we note that the estimation error function is in $L_2$ while the tracking error function is regarded as an element in $L_\infty$, it is natural to design the optimal tracking controller minimizing the induced norm from $L_2$ to $L_\infty$, i.e., the generalized $H_2$ norm, for the Step (ⅱ).

On the other hand, it is important to note that the design processes for the $H_\infty$ optimal disturbance estimator and the generalized $H_2$ optimal tracking controller considered in the Steps (ⅰ) and (ⅱ), respectively, do not depend on each other, although the latter is suggested to be designed after the former is obtained. The details could be clarified in the following subsection, which addresses the associated synthesis procedures in depth.

3.2.   Optimal synthesis for estimator and controller

With respect to a disturbance estimator, we consider the estimation error function defined as

where $F(s)$ is an $n\times n$-dimensional transfer function matrix with the real coefficients and $H$ is an $n\times n$-dimensional weighting matrix, i.e., $H\in {{\mathbb R}}^{n\times n}$, and they are determined according to desired performance specifications. Based on the representation (3.5), we aim at designing the $H_\infty$ optimal disturbance estimator minimizing the $L_2$-induced norm defined as

Regarding a tracking controller, on the other hand, the tracking error function is given by

with some weighting matrices $E_{1}, E_2,$ and $D_{12}$, and their selection does depend on required specifications of tracking performances. For the disturbance estimation error defined as

and the tracking error function given by (3.7), we design the generalized $H_2$ optimal tracking controller minimizing the induced norm from $d$ to $z_o$ defined as

On the other hand, it would be worthwhile to discuss the rationale behind taking $F(s)$ and $H$ as mentioned above. First of all, it is often required to consider a specific frequency range of $w$ and not the total range $[0, \infty)$ in practical systems. Hence, it is reasonable to take $F(s)$ by a strictly proper transfer function matrix. This also makes the synthesis problem of the $H_\infty$ disturbance estimator to be practically meaningful, and the details will be also discussed in the arguments around Theorem 2. For a selection of $H$, if $H$ is set to $0$, then the $H_\infty$ disturbance estimator often leads to a high-gain, which is not desired in practical systems. In this sense, we have taken a nonzero matrix $H$, and such a selection does not affect the feasibility of the synthesis of the $H_\infty$ optimal disturbance estimator.

For the synthesis of an $H_\infty$ optimal disturbance estimator, we first derive the relevant plant. To simplify the arguments, assume that the measured output $y$ and $F(s)$ in (3.5) are taken by $\begin{bmatrix} e^T & \dot{e}^T \end{bmatrix}^T$ and $\dfrac{1}{1+\tau s}I_n$, respectively (other cases can also be proceeded in an equivalent fashion to the following arguments). With the state vector $f(t)$ of $F(s)$, defining

leads to the state-space representation of $P_f$ in Figure 3 described by

where

With respect to the representation (3.10), we are concerned with designing an $H_\infty$ optimal disturbance estimator such that

where $K(s)$ is described by the state-space representation

To put it another way, we aim at obtaining the estimation parameters

in (3.12) corresponding to the optimal problem given by (3.11). In connection with this, taking the arguments in ^[34] leads to the following theorem.

Theorem 1. For a prespecified $\gamma_{2/2} > 0$, there exists an estimation parameter $(A_k, B_k, C_k, D_k)$ ensuring the internal stability of the closed-loop system and the $H_\infty$ performance given by

if and only if there exist decision variables $X, \ Y, \ J, \ L, \ M, \ N$ such that the LMIs

are feasible, where

Furthermore, if these LMI-based conditions are feasible, then there exist non-singular matrices $U$ and $V$ such that

and the estimation parameter $(A_k, B_k, C_k, D_k)$ is determined by

where

Remark 1. The first LMI in (3.14) (i.e., $\mathcal{X} \succ 0$) corresponds to a necessary and sufficient condition for the internal stability of the closed-loop system, while the second LMI in (3.14) is equivalent to a necessary and sufficient condition for ensuring that the $H_\infty$ norm of the closed-loop systems is less than $\gamma_{2/2}$. Furthermore, it immediately follows that $S_1$ and $S_3$ are invertible since $U$ and $V$ are non-singular.

Theorem 1 provides a solution to the optimal control problem of (3.11) since this theorem establishes a necessary and sufficient condition for the existence of an $H_\infty$ estimator $K(s)$ with respect to a pre-given performance level $\gamma_{2/2}$. To put it another way, minimizing $\gamma_{2/2}$ in the LMI-based constraints (3.14) leads to the $H_\infty$ optimal disturbance estimator. In a practical aspect, it would be also worthwhile to note that the order of the proposed $H_\infty$ disturbance estimator coincides with that of the plant $P_f$, independently of that of a tracking controller.

In connection with taking $F(s)$ by a strictly proper transfer function matrix, we remark that there exists a nonzero feedthrough term $D_{11}^{[f]}(\in {{\mathbb R}}^{n\times n})$ in (3.10) if $F(s)$ is not strictly proper. This leads to modifying the second inequality in (3.14) by

In terms of (3.16), we develop the following results.

Theorem 2. The LMI of (3.16) is feasible only if

Proof. Note that

This completes the proof. □

Theorem 2 implies that the $L_2$-induced norm defined as

cannot be smaller than $|D_{11}^{[f]}|_2$ if $F(s)$ is not strictly proper. In contrast, no such a constraint is required for solving the LMIs given in Eq (3.14), and, thus, taking a strictly proper transfer function matrix $F(s)$ allows us to obtain a wider range of achievable $H_\infty$ performance with respect to $K(s)$.

After the $H_\infty$ optimal estimator $K(s)$ is obtained via the arguments in Theorem 1, regardless of the choice of $K(s)$, we next consider the synthesis of an outer-loop tracking controller $C(s)$. The main objective in Step (ⅱ) is to minimize the effect of the residual disturbance (i.e., the estimation error defined as (3.8)) on the tracking error function $z_o$ (defined as (3.7)). With respect to this, defining

results in the form for the generalized plant given by

where

Based on the representation described by (3.18), we aim at designing the generalized $H_2$ controller such that

where $C(s)$ is described by the state-space representation

This paper considers designing the control parameters

and

in (3.20) for ensuring the optimal problem described by (3.19). With taking the arguments in ^[34], such a control parameter can be characterized by the following theorem.

Theorem 3. For a given

there exists a control parameter $(A_c, B_c, C_c, D_c)$ ensuring the internal stability of the closed-loop system and the generalized $H_2$ performance given by

if, and only if, there exist decision variables $X, \ Y, \ J, \ L, \ M, \ N$ such that the LMIs

are feasible, where $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$, $\mathcal{X}$ are defined as in Theorem 1 by replacing $(\cdot)^{[f]}$ with $(\cdot)^{[o]}$. Furthermore, if these LMIs are feasible, then the control parameter $(A_c, B_c, C_c, D_c)$ is also determined as in (3.15), i.e.,

Remark 2. If the second LMI in (3.22) is established, then we obtain that $\mathcal{X} \succ 0$. This ensures that the closed-loop system is internally stable as discussed in Remark 1. However, in contrast to Theorem 1, establishing both the first and second LMIs in (3.22) leads to a necessary and sufficient condition for the generalized $H_2$ norm of the closed-loop system to be less than a prespecified $\gamma_{\infty/2}$.

Similarly to Theorem 1, we can obtain the generalized $H_2$ optimal controller with respect to (3.19) by minimizing $\gamma_{\infty/2}$ in the LMI-based constraints (3.22) since this theorem leads to a necessary and sufficient condition for the existence of a generalized $H_2$ tracking controller $C(s)$ achieving a pre-given performance level $\gamma_{\infty/2}$. The order of the proposed generalized $H_2$ tracking controller also is equal to that of the plant $P_o$. More importantly, it should be stressed that the LMI-based conditions in Theorem 3 do not depend on the $H_\infty$ optimal disturbance estimator obtained in Theorem 1 and thus we can confirm once again that the synthesis procedures in these theorems are independent of each other.

On the other hand, it would be worthwhile to note that the feedthrough term $D_{11}^{[o]}$ is assumed to be zero in (3.18), and this assumption is necessary for the generalized $H_2$ norm (i.e., the induced norm from $L_2$ to $L_\infty$) to be well-defined and bounded. Thus, regardless of the choice of $C(s)$, the transfer function matrix from $d$ to $z_o$ should be strictly proper whenever the generalized $H_2$ norm is taken as the corresponding performance measure. This turns out that the transfer function matrix from $w$ to $z_o$ is also strictly proper and thus its frequency response gain decreases as the considered frequency becomes larger. In other words, the generalized $H_2$ optimal control might also be interpreted in the frequency domain as dealing with low ranges rather than the overall bound $(-\infty, \infty)$. In this sense, it is natural to take a strictly proper transfer function matrix $F(s)$ when we are concerned with reducing $w$ itself.

To summarize, the overall process of the proposed control framework and its characteristics can be described as follows:

● The total uncertainty $w$ (given in (2.4)) is estimated and its magnitude is reduced, especially for low frequency ranges, by designing the $H_\infty$ optimal disturbance estimator.

● The effect of the estimation error

on the tracking function $z_o$ (defined as (3.7)) is minimized in terms of the induced norm from $L_2$ to $L_\infty$ by designing the generalized $H_2$ optimal tracking controller.

● Even though the generalized $H_2$ tracking controller is proposed to be designed after the $H_\infty$ disturbance estimator is obtained, their LMI-based synthesis conditions do not depend on each other.

4.   Illustrative examples

This section evaluates the validity of the overall arguments presented in the preceding section by means of simulation and experimental results. We employ the $6$-degree of freedom robot manipulators shown in Figure 4 with the reference trajectories shown in Figure 5 for both simulations and experiments.

Comparative simulations and experiments to the recent results of the enhanced unknown system dynamics estimator-based sliding mode control (EUSDE-SMC) ^[8] and the time-delay based disturbance estimation (TDE) ^[9] are conducted. The rationale behind taking these two existing studies is that they are sorts of the second approach mentioned in Subsection 1.1 and their control architectures are similar to that of the proposed method (PM); this fact makes the comparison between the PM to the EUSDE-SMC and the TDE fair.

More precisely, the superiority of the PM over the existing two methods in estimating uncertainties is demonstrated through simulations, while that in trajectory tracking is validated through experiments. For both simulations and experiments, we take the following parameters for $P_f$ and $P_o$:

and

4.1.   Simulation results

Similarly to Example 1, let us consider the case such that model uncertainties are generated by embedding random signals as shown in Figure 1, and comparative analysis on estimating uncertain elements is considered with respect to the PM, EUSDE-SMC, and TDE; all the three methods derive time-series signals for all the joints in real-time as the estimated values of uncertainties. The overall simulations are conducted through MATLAB, and the LMI conditions required for obtaining the $H_\infty$ optimal disturbance estimator and the generalized $H_2$ tracking controller are tackled by using the function "mincx" in the MATLAB LMI solver.

With respect to solving the LMIs in (3.14), the minimum value of $\gamma_{2/2}$ can also be obtained by taking the arguments in ^[15,18] relevant to the $H_\infty$ optimal controller synthesis, and such a value is given by $0.5010$.

On the other hand, we consider the total uncertainty $w$ as shown in Figure 6, and the simulation results for the estimation errors with respect to the three methods are shown in Figure 7. First of all, we can observe from Figure 7 that the estimation error with the PM is closer to $0$ than those with the existing EUSDE-SMC and TDE methods under the same joint. Here, these estimation errors are expected to show significant oscillations because the total uncertainty $w$ consists of high frequency elements as observed from Figure 6. However, the estimation errors with the PM are quite smoother than those with the existing two methods for all the $6$ joints. This observation implies that the strictly proper transfer function matrix $F(s)$ in (3.5) is adequately taken to reduce the high frequency elements in $w$.

To make the comparison between the PM and the two existing methods clearer in a quantitative aspect, the simulation results for the root mean square (RMS) value of the estimation errors are shown in Table 1.

We can observe from Table 1 that the RMS values obtained through the PM are quite smaller than those of the two existing methods for all the $6$ joints, and, thus, the PM can be regarded as outperforming the existing EUSDE-SMC and TDE methods in estimating uncertain elements of robot manipulators.

These observations validate the effectiveness of the developed arguments relevant to designing the $H_\infty$ optimal disturbance estimator.

4.2.   Experiment results

The experimental results related to the joint position errors are shown in Figure 8. We can observe from Figure 8 that the maximum magnitude of the tracking error achieved by PM is always lower compared to those of the EUSDE-SMC and TDE methods for the same joint. This finding indicates that the PM exhibits superior tracking performance relative to the two conventional approaches.

For a more quantitative analysis relevant to the experiment results, we are in a position to compare the $L_\infty$ norm (i.e., the maximum magnitude) of the joint position errors, as shown in Table 2. We can see from Table 2 that the $L_\infty$ norms of the joint position errors obtained through the PM are quite smaller than those through the EUSDE-SMC and the TDE, with respect to all the $6$ joints.

Regarding a different aspect of the practical effectiveness, the experiment results for the total torque powers are shown in Table 3. We can observe from Table 3 that the torque power required by the PM is lower than that required by both the EUSDE-SMC and TDE.

To summarize, the experimental analysis above clearly highlights the practical advantages of the PM over the existing EUSDE-SMC and TDE in reducing tracking errors for uncertain robot manipulators. Specifically, it shows that the PM achieves the smallest $L_\infty$ norms across all six joints while requiring the least torque power among the three control methods. These experiment observations also clearly validate the effectiveness of designing the generalized $H_2$ optimal tracking controller for reducing the tracking error in the time-domain bounds, and, thus, the safety issue on robot manipulators can be expected to be solved by taking the PM.

5.   Conclusions

Motivated by the fact that estimating uncertain elements at every moment is practically useful and it is required to suppress the relevant tracking errors in the time-domain bounds for the safety issue such as the collision avoidance, this paper developed a new control framework for uncertain Euler-Lagrange systems. Based on this new control framework, we next proposed two dynamic compensators called the $H_\infty$ optimal disturbance estimator and the generalized $H_2$ optimal tracking controller, respectively. More precisely, the $L_2$-induced norm of the mapping from the unknown disturbance to the corresponding estimation error function is minimized through the $H_\infty$ optimal disturbance estimator, while the generalized $H_2$ norm (i.e., the induced norm from $L_2$ to $L_\infty$) of the mapping from the estimation error function to the relevant tracking error function is minimized by the generalized $H_2$ optimal tracking controller. It was also shown in this paper that the $H_\infty$ disturbance estimator and the generalized $H_2$ optimal tracking controller can be readily obtained through the linear matrix inequality-based approach. Even though the $H_\infty$ optimal disturbance estimator and the generalized $H_2$ optimal tracking controller were explained in the step (ⅰ) and the step (ⅱ), respectively, it should be remarked that their synthesis procedures could be carried out independently of each other. The effectiveness of the $H_\infty$ optimal disturbance estimator was demonstrated through some comparative simulations to the conventional methods in ^[8,9]. The practical superiority of the proposed method, which combines the $H_\infty$ optimal disturbance estimator and the generalized $H_2$ optimal tracking controller, over conventional methods was also validated through comparative experimental results.

Finally, it would be worthwhile to note that the arguments developed in this paper (i.e., the new structural framework, the $H_\infty$ optimal disturbance estimator, and the generalized $H_2$ optimal tracking controller) could be applied in a parallel fashion to different linearized systems such as electric motors, power systems, and so on, although this paper dealt with only uncertain Euler-Lagrange equations. However, it is still unclear how to obtain performance limitations (i.e., lower bounds on the $H_\infty$ norm and the generalized $H_2$ norm) before the optimal estimator and controller synthesis. Thus, it might be possible to require quite long computation times for minimizing $\gamma_{2/2}$ and $\gamma_{\infty/2}$ in Theorems 1 and 3, respectively, when the initial conditions are very far from the optimal values. This issue is left for an interesting but difficult future study.

Author contributions

Taewan Kim: original draft preparation, methodology, investigation, and formal analysis; Jung Hoon Kim: supervision, validation, manuscript-review, project administration, funding acquisition, and revisions. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

This work was supported by project for Smart Manufacturing Innovation R & D funded Korea Ministry of SMEs and Startups in 2022 (Project No. RS–202200141122).

Conflict of interest

The authors declare no conflicts of interest.