Majorant-Based Control Methodology for Mechatronic and Transportation Processes

This paper provides a unified approach via majorant systems, which allows one to easily design a family of robust, smooth and effective control laws of proportional - h order integral - k order derivative (PIhDk)-type for broad classes of uncertain nonlinear multi-input multi-output (MIMO) systems, including mechatronic and transportation processes with ideal or real actuators, subject to bounded disturbances and measurement errors. The proposed control laws are simple to design and implement and are used, acting on a single design parameter, to track a sufficiently smooth but generic reference signal, yielding a tracking error norm less than a prescribed value, with a good transient phase and feasible control signals, despite the presence of disturbances, parametric and structural uncertainties, measurement errors, and in case of real actuators and amplifiers. Moreover, some guidelines to easily design the proposed controllers are given. Finally, the stated unified methodology and various performance comparisons are illustrated and validated in two case studies.


I. INTRODUCTION
Nowadays, in a deeply mechanized and computerized global society, one of the challenging problems is to develop reliable control techniques for mechatronic and transportation processes, that can be easily implemented using modern digital and wireless technologies to force them to behave like skilled workers who work quickly, accurately, and cheaply, despite parametric variations, nonlinearities, and persistent disturbances. However, many engineering control problems still remain unsolved, especially for mechatronic and transportation systems, under the following realistic hypotheses: parametric and/or structural uncertainties, fast-varying references, measurement errors, real amplifiers and actuators, and/or finite online computation time of the control signal. Furthermore, to reduce the gap between theory and practical feasibility, the designed control laws should be easy to design and implement with smart sensors, power supplies, and intelligent actuators.
Noting that the computation time for feedback linearization or control signal generation by the inverse model or MPC techniques is non-negligible, these simplified hypotheses make the above-mentioned techniques not always reliable (see e.g., [3]- [4], [7], [9], [11]- [12], [15], [19], [23], [36]). In some cases, the control system can even be unstable, as it can be confirmed by simple counter-examples (see, for example, Appendix A in [21]). In addition, the control techniques based on high-frequency and high-amplitude control signals do not always yield good performance (see also Appendix A in [21]).
With the aim to give a solution to the above criticisms, some adjustments of the inverse model and feedback linearization techniques have been proposed, such as, for instance, a computed-torque-like control with variablestructure compensation (see, e.g., [10], [22]), which is, however, difficult to design and implement and also has chattering phenomenon. On the other hand, some popular adaptive control techniques (see e.g., [13], [24], [42]) and linear matrix inequality (LMI) approaches often turn out to be quite complex and arise criticisms from a practical point of view (see e.g., [2], [12], [29]). Recently, also fuzzy control methods deal with the above matters in specific cases (see e.g., [14], [18], [25], [31]).
Consequently, there is still a high demand for smooth robust controllers that allow a plant belonging to a broad class of uncertain nonlinear systems, including mechatronic and transportation systems, to track a sufficiently smooth reference signal with a tracking error norm less than a prescribed value, despite the presence of disturbances, parametric and structural uncertainties, measurement errors, and using real actuators and/or amplifiers. Concerning the above issues, for uncertain linear MIMO systems with bounded additional nonlinearities, a systematic method has recently been presented in [38]. Instead, this paper presents a comprehensive and unified approach via majorant systems that can be successfully applied to numerous engineering systems (e.g., control of rolling mills, conveyor belts, automatic guided vehicles (AGVs), unicycles, cars, trains, ships, airplanes, drones, missiles, satellites, manufacturing and surgical robots). The stated methodology allows one to easily design a family of robust, smooth and effective control laws of hk PI D -type ( 4 , 1; 1,...,4, 0,..., 1) h i j k i j i j i          for a broad class of uncertain nonlinear MIMO systems, including mechatronic and transportation processes with ideal or real actuators, subject to bounded disturbances and measurement errors. The proposed controllers are used to track a reference signal with a bounded i-th derivative ( 1,..., 4 i  ), yielding a tracking error norm less than a prescribed value, with a good transient phase, and feasible control signals.
Note that the classes of systems considered in the present manuscript are broader in comparison to [34], new controllers more efficient than the ones in [21] and [27] are proposed, and the treatment is unified. Moreover, some guidelines to easily design all the provided controllers are given.
Finally, the obtained theoretical results are illustrated and validated in two case studies: the first one illustrates the design methodology by considering a simple underwater robot, and the second one deals with the kinematic inversion problem and the tracking one of an industrial robot both in the joint space and in the workspace. It is worth noting that peculiarities of the proposed results are as follows:  They are basic for the control of mechanical systems with real actuators and/or real amplifiers, and when high performance are required, also without using compensation signals (difficult to obtain and also to implement in real time).  The provided controllers are also useful for the control of thermal processes and fluid dynamic ones, to solve a nonlinear equation, etc. .  Moreover, the proposed results can also be used to make the kinematic inversion, when the knowledge of the position, velocity and acceleration is required to ease the implementation of the controller, and also, in some cases, of the acceleration derivative to compute a more effective compensation signal. The result that allows, in some cases, controlling a robot in the workspace without using the transpose of the Jacobian matrix, which requires in any case the knowledge of the angular coordinates of the robot, is surely a basic matter.  The proposed control laws are simple to design and implement, providing good performance also in the case of real actuators and bounded velocity and acceleration measurement errors.  The provided properties of filters are major since they allow evaluating quickly the reduction of the velocity, acceleration and of the first and second derivatives of the acceleration, which is a basic issue to reduce the control effort and, hence, the power of the actuators. The paper is organized as follows. In Section II, the classes of uncertain nonlinear MIMO systems to be controlled are introduced, the robust tracking problem is stated in a general way, and the structures of the proposed controllers with the performance of the related control systems are reported. In Section III, the main theorems are established to design various robust controllers for the considered systems. In Section IV, some guidelines to easily design the proposed controllers are given. Section V includes two case studies to illustrate and validate the obtained theoretical results. Finally, Section VI outlines the main advantages of the provided results and presents the ongoing research.

A. SYSTEM DESCRIPTION
This paper deals with the control of the significant and broad classes of uncertain nonlinear MIMO systems described by fR  is a nonlinear vector function satisfying the following conditions 1) and 2).

1) Positivity condition:
There exists a matrix ( , ) 2) Bounding condition of class There exists a continuous non decreasing function , .
For the above mentioned systems, the paper provides a unified approach, via majorant systems, which allows one to easily design several robust and effective hk PI D

B. CLASSES OF THE CONSIDERED SYSTEMS
There exist several classes of systems whose models, after a) the use of a possible appropriate compensation signal dependent on ( 1) , ,..., i t y y  and/or b) possible appropriate mathematical manipulations by using the derivative of Lie, are of the type (1). In the following, nine significant classes of the above systems are reported. Class 1. Note that many systems (e.g., mechanical, thermal, fluid dynamic) are described by the equation where It is easy to verify, taking into account the structure of , , , , , ,    Class 6. Consider the system (6) or (11), actuated by real actuators powered with real amplifiers, described by where m R   is the input vector of the amplifiers,

M K K T T g A A B C C
that the resulting system is of the type (1) and can satisfy the conditions (2), (3).
e.g., the coordinate transformation between the joint space and the workspace of a robot, or consider the nonlinear equation ( ) x=0, cq  e.g., with the aim to compute the equilibrium points of a nonlinear system represented by where () J q c q    is the Jacobian matrix of () cq .
and taking into account that pp p J q q q dJ q q dt  .
and taking into account that . y u f t y y y    (21) d) If the fourth derivative of x is bounded, deriving the equation (20), posing (4) 1 () q J q u   , and taking into account that 1 There are other classes of systems of the type (1). They include, e.g., the following class 8.

Notations
is a symmetric and positive definite (p.d.) matrix, and T x is the transpose of Definitions Definition. Given the system ( ) ( , ( ), ( ), ), and a ..
xX  is said to be a majorant system of the system (33).

Formulation of the main results
Now, let ( ) : be a generic reference signal with bounded i-th derivative. For the various classes of systems (1), it will be proven that, using the following control laws: where e r y , and c u is a possible compensation signal satisfying the condition .
.  1 a   , it will be proven that the following increases hold: This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.  4 3 2     Note, now, that the control system, composed of one of the systems S1),…,S10) and one of the corresponding proposed controllers, can be described by a model of the type , , 1,..., 4, 1,.., ,    [17], [21], [34].
The proof of theorem derives from (67) and Lemma 2.
Theorem 1 allows one to establish the following main theorem.  Proof. The proof easily follows by taking into account Remark 3 and Fig. 4.     [17], [34] T V x P x   (see also [16], [21]).
If, instead, the control law The first increase in (42) follows from (80) and (72), the second one derives from (82) and (69).
From Theorems 2, 3, 4, and 5 it derives the following theorem of robustness of the tracking error with respect to the measurement errors. tracking error norm () et is inversely proportional to a or to 2 a or to 3 a .
Proof. If () yt is affected by a bounded measurement error (1) y n then w has an increase equal to 1) On the other hand, note that () et is 1) inversely proportional to 2 a in case S2, 3 a in case S6, 4 a in case S9; 2) inversely proportional to 3 a in case S3, 4 a in case S7; 3) inversely proportional to 4 a in case S4.
If () yt is affected by a bounded measurement error ( 2 ) y n then w has an increase equal to 1) On the other hand, () et is 1) inversely proportional to 3 a in case S3, 4 a in case S7; 2) inversely proportional to 4 a in case S4.
If () yt is affected by a bounded measurement error (3) y n then, in case S4, w has an increase equal to (3) 4, y aHn and c  has an increase not greater than (3)

IV. GUIDELINES FOR THE DESIGN AND IMPLEMENTATION OF THE PROPOSED CONTROL LAWS
In the following, some guidelines useful to easily design and implement the various proposed control laws are provided.

Determination of G (K).
To design the proposed controllers, it is basic to determine a matrix , not necessarily a p.d. matrix, satisfies the inequality (2). Since the matrix F , in many practical cases, is ratio-multi-affine with respect to uncertain parameters and functions of ( 1) ,..., i yy  , the determination of G () K can be easily made by using Lemmas in [8] and [20] and the MATLAB command "fmincon", and/or by using randomized algorithms, with i) . If the measurement of y is affected by a large bandwidth error, especially at medium frequency, instead of using a real derivative action to obtain y () y it is appropriate to directly measure y or, alternately, measure y and estimate y using an optimal estimator (see e.g., [37]). In this regard, note that nowadays, in many cases, the direct measurement of speed, and even acceleration, can be easily achieved with accurate economic sensors.

Remark 5.
To reduce the gains of the controller and the control signal, above all during the transient phase, it can:

Smooth the references () rt
with appropriate filters and suitable initial conditions; 2. Better identify the process parameters and the disturbances, and use a compensation signal encountering instability problems due to unavoidable delays of digital controllers. 2. The proposed control laws are continuous and assure tracking errors as small as desired both in norm and of the first derivative norm, and, in some cases, also of the second derivative norm. From this fact it also follows that the control signals are without the chattering phenomenon. 3. For some significant classes of systems (for which such that is a symmetric . . G FG p d  matrix, e.g., robots and transportation systems), the design and performance of the control laws depend only on two parameters: the first one 0 g  to satisfy the positivity condition (2), i.e., min ( ) 1, g FG   the second one a>0 to obtain the desired precision. This peculiarity is of a noteworthy practical importance, since the design of the proposed controllers can be made also by technicians or simple expert systems. 4. The stated theoretical results are useful for numerous applications and to obtain other results, significant both from a theoretical and practical point of view.

V. EXAMPLES
The following simple example illustrates the proposed methodology and shows that, if a system of the class (27) can be transformed into one of the class (29), the performance of the control system with controllers having an integral action improves.
in accordance with the second of (38). If, instead, the reference () rt is the one in Fig. 6  in accordance with the first of (42).
A similar treatment can be made if the fourth derivative (4) () rt of () rt is not bounded.
The following example shows applicability, utility, and efficiency of the numerous results provided in the previous sections. Fig. 7. The objective is to realize a cat as in Fig. 8 with the desired time histories in the workspace shown in Fig. 9, which is obtained by interpolating 83 points with cubic splines and using a fourth-order Bessel filter with 10 . to the left (red line in Fig. 8) and start from initial conditions different from the "exact" ones, in such a way "eliminating" the transient phase.

Kinematic inversion
In the following, it is shown how to make the kinematic inversion easier and more efficient in comparison to other methods (e.g., see [3], [4], [9], [10] and the related references therein), also obtaining the acceleration.

Control Design
In the following, it is shown how it is easy to design, by using the proposed method, simple, robust, efficient and without chattering controllers.
In Fig. 15, some possible control schemes for a robot are reported.

VI. CONCLUSIONS AND FUTURE DEVELOPMENTS
In conclusion, the main advantages of the obtained results can be summarized as follows.
1) The considered classes of uncertain nonlinear MIMO systems are broad and include numerous mechatronic and transportation processes.
2) A unified and comprehensive approach based on the concept of majorant system is provided to design robust, effective and smooth hk PI D -type control laws.
3) The proposed control laws are simple to design, have no high gains, are free from discontinuities, and, in many cases, can be realized by simple analog circuits.
4) The provided control laws allow one, acting on a single design parameter, to track a generic reference signal with a bounded i-th derivative, with a tracking error norm less than a prescribed value, with a good transient phase and feasible control signals, also in presence of disturbances, parametric and structural uncertainties, measurement errors, real actuators and amplifiers. 5) The new proposed controllers are more general and efficient than the ones in [21], [27], and [34]. The ongoing research aims at extending the proposed results to more general systems with respect to the considered ones and/or in the hypothesis that () , 1, k yk  is not measurable, and/or providing new control laws to satisfy more than a single design specification.