A Robust Tracking Method for MIMO Uncertain Discrete-Time Systems: Mechatronic Applications

This paper develops a systematic method to design robust tracking controllers for multi-input multi-output (MIMO) uncertain discrete-time systems with bounded parametric uncertainties, in particular of rational multi-affine type, and generic discrete reference signals with bounded first or second discrete derivatives, also in presence of generic disturbances with bounded first or second discrete derivatives. Theoretical tools and systematic methodologies are provided to effectively design robust innovative controllers for the considered systems. Applicability and efficiency of the proposed methods are validated in two examples via simulation and experimental tests.


I. INTRODUCTION
There exist numerous discrete and continuous-time uncertain systems, subject to non-standard disturbances, which need to be efficiently regulated with discrete-time controllers, whose main feature is to be versatile and easily realizable using digital technologies. Examples of such systems can be found among mechatronic, demographic, economic, traffic management, environmental, agricultural, biological, medical, and other systems (see, e.g., [1], [4], [11], [16], [20]- [24], [36]).
Note that for a continuous-time system the dependence of its corresponding discrete-time representation matrices on parameters is quite complex. The performance and/or control design specifications are usually given by gains, settling time, bandwidth, stability margins, mean-square error, or a combination of the control signal energy with the mean-square error.
This paper provides a systematic method for the robust tracking design of MIMO uncertain discrete-time systems with bounded parametric uncertainties, in particular, of rational multi-affine type, and generic discrete reference signals with bounded first or second discrete derivatives, also in presence of generic disturbances with bounded first or second discrete derivatives. Similar research for a class of MIMO continuous-time systems has been conducted in [28] and [40]. Some results on robust tracking controller design with generic reference signals for continuous and discrete singular-input singular-output (SISO) uncertain linear systems are provided VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see http://creativecommons.org/licenses/by/4.0/ in [21]. On the other hand, some results have been obtained in [27] for MIMO uncertain discrete-time systems with multi-linear structures with respect to parameters and controllers without a proportional action, using the majorant systems approach. In this paper, MIMO uncertain discrete-time systems or sampled-data plants are regulated with proportional-integral (PI) and proportional-second order integral (PI 2 ) discretetime controllers to track non-standard reference signals. The provided results are particularly useful for mechatronic systems (e.g., rigid and flexible Cartesian robots, rolling mills, AGVs, conveyor belts, active suspension systems, printing machines), whose reference signals and disturbances are represented by non-standard waveforms (see, e.g., [22], [23], [36]). A hardware-software prototype has been constructed and used to validate utility of the proposed results from the engineering point of view.
The paper contribution can be summarized as follows.
• The systematic control design approach is proposed for generic uncertain LTI discrete-time or sampled-data plants and reference signals and/or disturbances with bounded first or second discrete derivatives.
• The proposed approach allows one to design a controller minimizing the tracking error. Alternatively, the proposed method allows one to design a controller minimizing the maximum time constant.
• The innovative structure controllers with PI/PI 2 control action have been developed. Comparisons of the designed controllers to a classical feedback one are presented.
• The obtained results can be considered as a pseudogeneralization of the Kharitonov's results and stability margins for the discrete-time systems. The paper is organized as follows. In Section II, the considered class of MIMO uncertain discrete-time systems is introduced, the synthesis problem is stated, and a theoretical background is provided. Section III presents the main analysis and synthesis results. Section IV provides a method to effectively design robust controllers for the considered systems. In Section V, the main proposed results are validated in two examples via simulation and experimentally. Comparisons of the designed controllers to a classical feedback one are presented in the second example. Section VI concludes this study.

II. PROBLEM STATEMENT AND THEORETICAL BACKGROUND
Consider an uncertain discrete-time MIMO plant described by where x k ∈ R n is the system state, u k ∈ R r is the control input, d k ∈ R l is a disturbance, y k ∈ R m is the system output, p ∈ ℘ ⊂ R v is the vector of uncertain parame- are satisfied for each p ∈ ℘. Remark 1: The plant (1) can also represent a sampled-data model of the continuous-time procesṡ In such a case, ifĀ(p) is a nonsingular matrix, then where T is the sampling time.

Remark 2:
The condition (2) implies that rank C = m ≤ n and rank B ≥ m, i.e., the m outputs of the plant are independent and the number of the independent control inputs is at least equal to the number of the outputs to be controlled.
The main objective of the paper is to control the plant (1) to track any reference signal r k with bounded first discrete derivative δ 1 r k = r k+1 − r k or bounded second discrete derivative δ 2 r k = δ 1 (δ 1 r k ) = r k+2 − 2r k+1 + r k (see Fig. 1) in presence of a disturbance d k with bounded first discrete derivative δ 1 d k = d k+1 − d k or bounded second discrete derivative δ 2 d k = d k+2 − 2d k+1 + d k . Note that generic reference signals with bounded first or second discrete derivatives are commonly encountered in practice and easily realizable by digital technologies. In case of manufacturing systems, the first discrete derivative of is proportional to the working velocity, while the discrete second derivative is proportional to the acceleration.
In the following, for simplicity of notation, the explicit dependence of A(p), B(p), E(p), C(p), D(p) on p is omitted when unnecessary.
If the objective is to track reference signals with bounded discrete derivatives, the plant (1) can be controlled using the state feedback control scheme with a PI controller shown in Fig. 2. Accordingly, in order to track reference signals with bounded second discrete derivatives, the plant (1) can be controlled using the state feedback control scheme with a PI 2 controller shown in Fig. 3.  The control scheme in Fig. 2 is represented as Hence, Similarly, the control scheme in Fig. 3 is represented as where The preliminary notation and definitions are introduced as follows.
R + 0 denotes the set of non-negative real numbers.
is an upper estimate ofα, and τ ≥τ is an upper estimate ofτ .
To design the proposed controllers, the following preliminary results are stated.
and the pair (A, B) is reachable, then the pairs are also reachable.
where C is the space of complex numbers, the pair (F, G) is reachable [7]. Hence, the pair ( the equality (14) follows from (15) and (10), if λ = 1, and from the reachability condition for the pair (A, B) and (15), the equality (16) follows from (17) and (10), if λ = 1, and from the reachability condition for the pair (A, B) and (17), if λ = 1. Lemma 2: If r 0 = 0 and d 0 = 0, the controlled system (6) can be represented as or, equivalently, after applying the Zeta-transform, as where it follows that Given a matrix ∈ R m×m , it is easy to prove that Hence, Similarly, using Symbolic Math Toolbox yields Lemma 3. If r 0 = r 1 = 0 and d 0 = d 1 = 0, the controlled system (8) can be represented as or, equivalently, after applying the Zeta-transform, as where . The proof follows upon verifying (26) via Symbolic Math Toolbox.
Lemma 4: If the pair of matrices A ∈ R n×n , C ∈ R m×n is observable, then the pairs of matrices are also observable. Proof: The proof easily follows by noting that 33116 VOLUME 8, 2020 Lemma 5: Consider a nonsingular matrix function F(p) ∈ Rn ×n , p ∈ ℘ = p − , p + ⊂ R ν , defined as a ratio of a multiaffine matrix function to a multi-affine polynomial where , is achieved at one of the 2 ν vertices of ℘. Proof: Note that for constants p j , j = i, it follows that x T Qx yields (31), as shown at the bottom of this page. Therefore, definingp i ,x as the maximum points of function (32), as shown at the bottom of this page, and taking into account (33) hold, as shown at the bottom of this page.
Note that if the matrix A has distinct eigenvalues, the matrix P given by (36) is always p.d. and the equality α = λ max (QP −1 ) = λ max (A) always holds in (37), even if not all eigenvalues of A have magnitudes less than one.

III. MAIN RESULTS
Theorem 1: Suppose that the dependence of the dynamic matrix of the discrete-time system on uncertain parameters p is of rational multi-affine type.
If for a givenp ∈ ℘ the eigenvalues of the matrix A = A(p) are distinct and all with magnitude less than one, whereP is the matrix obtained from (36) with A =Â, and λ max (A T (p)PA(p)P −1 ) ≤ 1 (m s ) 2 < 1 in the 2 v vertices of ℘, then the system (38) is asymptotically stable for each p ∈ ℘, with stability margin m s = 1 λ max (A) ≥m s . Proof. The proof follows from Lemmas 5 and 6. Theorem 1 can be used to compute the stability margin m s and considered as a pseudo-generalization of the Kharitonov's results to discrete-time systems.
Theorem 2: Given the system where and P ∈ Rn i ×n i ,n 1 = n + m,n 2 = n + 2m is a symmetric p.d. matrix. Assuming that the plant matrices have rational multiaffine structures with respect to parameters ℘ = [p − , p + ], the following equalities hold: where b cij 1 is the j 1 -th column of B ci , h ij 2 is the j 2 -th row of H ci , e cij 1 is the j 1 -th column of E ci (p), and V p is the set of the 2 ν vertices of ℘. Proof: The proof follows from [27] and Lemmas 2, 3, and 6.
A more general method to obtain the maximum time constant and the maximum absolute values of the system (39) outputs is based on the following theorem.

IV. CONTROLLER DESIGN
To design the proposed controllers, note that Hence, since in view of Lemma 1 the pairs (A 1 , B 1 ) and (A 2 , B 2 ) are reachable, the eigenvalues of A c1 and A c2 for a fixed p can be assigned at will. Therefore, since in view of Lemma 4 the control system, whose output coincides with the tracking error e k , is observable with suitably chosen matrices K and K p , it is possible to stabilize the control system and optimize a performance index related to the tracking error.
In view of Theorem 2, if the plant matrices are rational multi-affine with respect to parameters, upper estimates of the maximum time constant τ i of A ci (p) and/or the gains g rij 1 j 2 , g dij 1 j 2 can be obtained, upon covering ℘ with a finite number of N hyper-rectangles as follows: where P j is obtained from (36) with A = A ci (p) computed at the midpoint of the interval [p − j , p + j ] or a close point, provided that con(P j ) 1, and V pj is the set of 2 ν vertices of ℘ j . If the matrices of the plant are not rational multi-affine with respect to parameters, time constants τ i and gains g rij 1 j 2 , g dij 1 j 2 can be obtained using the equations (43), (44).
It is well-known that the proportional action makes the control system faster and results in reducing the error e k . On the other hand, the control magnitude may increase, for instance, due to sudden variations of r k and/or d k . For example, if ζ 0 = 0, then u 0 = K p (r 0 −Dd 0 ). Therefore, it is appropriate to make the matrix K p bounded, K p ≤K p . Note that once the matrix K (and, therefore, the matrix K t ) is computed and the matrix K p is fixed, the relation K t = K s − K p C implies K s = K p C + K t . Now, it is possible to design the proposed controllers by solving optimization problems. For instance, taking into account Lemma 8, if a desired maximum valueê d is chosen for the maximum errorê = max(G riri + G didi ), the design algorithm consists in solving the following min-max conditioned problem: This problem can be solved by using Matlab commands fmincon and place (see e.g., [2]). Note that ifê d = 0, then (53) provides the controller minimizingê.
Furthermore, it is possible to design a controller to minimize min d:Fd≤c whereτ d is a desired maximum time constant, and then to compute G ri and G id (and, therefore, the maximum values of r i andd i ) to obtain a prefixed maximum value ofê. VOLUME 8, 2020 Finally, it is also possible to design a controller minimizing a quality index of the following type: The following examples demonstrate applicability and efficiency of the results obtained in the previous sections. Example 1: Consider an uncertain plant The closed-loop control system is given by It is difficult to establish the asymptotic stability of the control system (58) and even more difficult to calculatê α = max p 2 )). Numerically, it is computed asâ = 0.8742. By settingp 1 = 0.5,p 2 = 0.5 and using the first equality of (52) with N = 1, an upper estimate ofα is calculated as α = 0.9510. By using the first equality of (52) with N = 4 (four rectangles), an upper estimate is found as a = 0.9047.
The objective is to design a discrete-time controller with sampling time T = 0.05s.
The sampled-data model of the plant is given by a) Using the classical control theory, a sub-optimal P controller with Butterworth cutoff angular frequency ω n = 20rad/s under the constraint K p ∈ [0, 10] and state feedback minimizing the steady-state error corresponding to the unit step input is obtained as b) Using Theorem 3, a sub-optimal PI controller with Butterworth cutoff angular frequency ω n = 20rad/s under the constraint K p ∈ [0, 10] and state feedback minimizing G r1 is obtained as c) Using again Theorem 3, a sub-optimal controller PI 2 with Butterworth poles for ω n = 20rad/s under the constraint K p ∈ [0, 10] and state feedback minimizing G r2 is obtained as (63) Figure 4 shows the errors corresponding to the unit step input obtained with the designed control laws, assuming p 1 = 10, p 2 = 7.     Figure 6 shows the tracking errors for a filtered square wave signal obtained with the designed control laws, assuming p 1 = 9, p 2 = 7.7. Note that the tracking errors e k1 and e k2 are almost equal to zero in the intervals where the reference is close to constant.   Figure 7 shows the tracking errors for a filtered sawtooth wave signal obtained with the designed control laws, assuming p 1 = 9, p 2 = 7.7. Note that the tracking error e k2 is VOLUME 8, 2020  almost equal to zero in the intervals where the reference is close to linear. Figure 8 shows the tracking errors for a reference signal with max |δ 1 r k | = 0.149 and max |δ 2 r k | = 0.0038 obtained with the designed control laws, assuming p 1 = 9, p 2 = 7.7. It is theoretically obtained from (42) that G r1 max |δ 1 r k | = 0.3025 and G r2 max |δ 2 r k | = 0.0147, while it follows from Fig. 8 that |e k1 | ≤ 0.1667 and |e k2 | ≤ 0.0071.
The last two cases have been experimentally validated by using an industrial HP PC equipped with a 12-bit input/output data acquisition board (National Instruments) and a positive-feedback RC circuit (see Fig. 9). The Matlab Real-Time Windows Target has been used with a 20 Hz sampling frequency.
Using the controller (62), Figure 10 shows the time histories of the experimental error e s and theoretical error e t .
Using the controller (63), Figure 11 shows the time histories of the experimental error e s and theoretical error e t .  Finally, if the reference ''velocity'' is halved, then the tracking errors obtained with the controllers PI and PI 2 are reported in Figs. 11 and 12, respectively. Note that after the transient phase the obtained errors are respectively the half and the one-fourth of those in the previous case, in accordance with Remark 4.
To reduce the tracking errors or increase the reference ''velocity'' without reducing the errors, it is possible to increase ω n . However, this approach may result in higher control signals during the transient phase.

VI. CONCLUSION
This paper provides a novel systematic method to design robust tracking controllers for MIMO uncertain discrete-time systems, with bounded parametric uncertainties, in particular, of rational multi-affine type, and discrete reference signals with bounded first or second discrete derivatives, also in presence of disturbances with bounded first or second discrete derivatives. The ongoing research is being conducted on robust tracking methods and fault detection techniques for MIMO uncertain nonlinear discrete-time systems, in particular, with unmeasurable states.
LAURA CELENTANO (Senior Member, IEEE) received the Ph.D. degree in automation and computer science engineering (with major in automatic control and system analysis) from the University of Naples Federico II, Naples, Italy, in 2006.
She has been an Assistant Professor of automation and control with the University of Naples Federico II, and a Professor of fundamentals of dynamical systems, modeling and simulation, and automation of navigation systems, since 2006. She has also taught classes at the University of Naples Parthenope and the Italian Air Force Academy, Naples. She has taken an active part in activities co-funded by the European Union, the Italian Ministry of Education, University and Research, Region Campania, public and/or private corporations, and industrial companies. She is an author/reviewer of the IEEE, ASME, Elsevier, and AIP journals and conferences. She has authored or coauthored scientific and educational books, presented by the IEEE Control Systems Society and international experts. Her current research interests include the design of versatile, fast, precise, and robust control systems of linear and nonlinear uncertain systems, methods for the analysis of stability and for the stabilization of linear and nonlinear uncertain systems (as well as MIMO and discrete-time systems), modeling and control of rigid and flexible mechanical systems, multivalued control design methodologies, modeling and control of aeronautical, naval, and structural systems, rescue and security robotics, and tele-monitoring/control systems.
Dr. Celentano has also been the Chair and an Organizer of conference sessions, also of Women in Control session. She is currently an Associate Editor of the Journal of the Franklin Institute. She is also a Guest Editor of a special issue in the Asian Journal of Control. She is also a Science Journalist and has cooperated with radio programs and journals on the dissemination of scientific matters.
MICHAEL V. BASIN (Senior Member, IEEE) received the Ph.D. degree in physical and mathematical sciences with major in automatic control and system analysis from Moscow Aviation University (MAI), in 1992. He is currently a Full Professor with the Autonomous University of Nuevo León, Mexico, and a Leading Researcher with ITMO University, St. Petersburg, Russia. Since 1992, he has published more than 300 research articles in international refereed journals and conference proceedings. He is the author of the monograph New Trends in Optimal Filtering and Control for Polynomial and Time-Delay Systems (Springer). His works are cited more than 5000 times (H index = 40). His research interests include optimal filtering and control problems, stochastic systems, time-delay systems, identification, sliding mode control, and variable structure systems. He is also a Regular Member of the Mexican Academy of Sciences. He has supervised 15 Doctoral and nine Master's theses. He has served as the Editor-in-Chief and has serves as a Senior Editor in control for the Journal of The Franklin Institute, a Technical Editor for the IEEE/ASME TRANSACTIONS ON