An Approach to Design Robust Tracking Controllers ...

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An Approach to Design Robust Tracking Controllers for Nonlinear Uncertain Systems Laura Celentano, Member, IEEE, and Michael V. Basin , Senior Member, IEEE

Abstract—This paper provides an approach to design robust smooth controllers that allow a plant belonging to a broad class of nonlinear uncertain systems, with possible real actuators and subject to bounded or rate-bounded disturbances, to track a sufficiently smooth reference signal with an error norm smaller than a prescribed value. The proposed control laws are based on the concept of majorant systems and allow one to establish asymptotic bounds for the tracking error and its first and second derivatives. The proposed controller design is based on two parameters: the first is related to the minimum eigenvalue of an appropriate matrix, which the practical stability depends on, and the second is determined by the desired maximum norm of the tracking error and its convergence velocity. If the trajectories to be tracked are not sufficiently smooth, suitable filtering laws are proposed to facilitate implementation of the control laws and reduce the control magnitude, especially during the transient phase. The obtained theoretical results are validated in two case studies. The first one presents a tracking control design for an industrial robot, both in the joint space and workspace, with and without real actuators or velocity measurement noise. The second one deals with tracking control design for a complex uncertain nonlinear system. Index Terms—Control of robots with real actuators, nonlinear uncertain systems, practical robust stability, robust control of mechanical uncertain systems, robust tracking control.

I. I NTRODUCTION OWADAYS, it is of great importance to develop control algorithms, which can be easily implemented using modern digital and wireless technologies, to force electromechanical systems (AGVs, cars, trains, ships, airplanes, drones, robots, etc.) to behave like skilled workers, who work fast, accurately, and at low cost, despite parametric variations, nonlinearities, and persistent disturbances. In particular, special

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Manuscript received March 6, 2018; accepted April 30, 2018. This work was supported by the Italian Ministry of University Education and Research and the Mexican National Science and Technology Council (CONACyT) under Grant 250611. This paper was recommended by Associate Editor C. J. Ferreira Silvestre. (Corresponding author: Michael V. Basin.) L. Celentano is with the Department of Electrical Engineering and Information Technology, University of Naples Federico II, 80125 Naples, Italy (e-mail: [email protected]). M. V. Basin is with the Department of Physical and Mathematical Sciences, Autonomous University of Nuevo Leon, San Nicolás de los Garza 66450, Mexico, and also with the Department of Physical and Mathematical Sciences, ITMO University, Saint Petersburg 197101, Russia (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSMC.2018.2834908

attention has been paid to solving robust tracking control problems, which resulted in a considerable number of publications (see [1]–[46]). However, there exist many results obtained under the following simplified hypotheses, which are not always realistic and feasible. 1) Exact knowledge of the controlled system is assumed. 2) Actuators are considered ideal. 3) Signals are assumed measurable without noises, therefore, ideal or almost ideal derivative actions can be used. Taking also into account that the computation time for feedback linearization or control signal generation by the inverse model or model predictive control techniques is nonnegligible, these simplified hypotheses make the above mentioned techniques not always reliable (see [5], [8], [10], [11], [13], [19], [25], [26], [32], [36], [42], [43]). In some cases the control system can even be unstable. Furthermore, the control techniques using high-frequency and high-amplitude control signals in the presence of real actuators do not yield good performance or can result in unstable closed-loop systems. To avoid the aforementioned problems, various modifications of the inverse model and feedback linearization techniques have been proposed, for instance, a computedtorque-like control with variable-structure compensation (see [10], [25], [26]), which is, however, difficult to design and implement and also presents high-frequency oscillations. Thus, there is still a demand for smooth robust controllers that allow a plant belonging to a broad class of nonlinear uncertain systems, including electromechanical ones, to track a sufficiently smooth reference signal with a tracking error norm smaller than a prescribed value despite the presence of disturbances and parametric and structural uncertainties, using real actuators and real derivative actions. In the previous authors’ contributions (see [20], [38], [39]) some control methodologies of multi-input multioutput (MIMO) uncertain linear systems and of a class of MIMO nonlinear uncertain systems have been provided in order to design simple controllers: state feedback controller with integral action, pseudo-PD, and pseudo-proportional-integralderivative (pseudo-PID) in order to track sufficiently smooth trajectories with an a priori given maximum error. This paper provides an approach to design robust controllers for three broad classes of uncertain nonlinear MIMO systems, including the manufacturing systems and the land, sea and air transportation systems, with possible real actuators

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and subject to bounded or rate-bounded disturbances, thus extending the results of [20], [38], and [39], and presents background theorems for establishing the developed control methodology. This allows one to design control laws to track a sufficiently smooth reference signal with an error norm smaller than a prescribed value. The developed approach can be applied to engineering and transportation systems either when the information used by the controller is related to the joint space (obtainable with encoders, resolvers, tachometric dynamometers, etc.) or the workspace (obtainable with cameras, sonar, global positioning system, etc.). The proposed control laws are based on the concept of majorant systems and result in establishing asymptotic bounds for the tracking error and its first and second derivatives. The proposed controller design is based on two parameters: the first is related to the minimum eigenvalue of a matrix, which the practical stability depends on, and the second is determined by the maximum desired norm of the tracking error and its convergence velocity. If the tracked trajectories are not sufficiently smooth, suitable filtering laws are proposed to facilitate implementation of the control laws and reduce the control magnitude, especially, during the transient phase. The designed controllers using the developed majorant systems technique allow one to obtain less conservative results, i.e., the actual tracking error is quite close to the error computed by the proposed methodology, unlike many other approaches based on the Lyapunov functions theory. This enables one to avoid significant technological problems, such as oversizing amplifiers and actuators. The designed control laws have no high gains and are free from discontinuities. Finally, the obtained theoretical results are validated in two case studies. The first one presents a tracking control design for an industrial robot, both in the joint space and workspace, with and without real actuators or velocity measurement noise. The second one deals with tracking control design for a complex uncertain nonlinear system. This paper is organized as follows. The problem statement and theoretical background including the majorant systems technique are given in Section II. Section III provides the main results, including the tracking control law and the corresponding tracking error bounds. Two case studies of an industrial robot and a complex uncertain nonlinear system are presented in Section IV. Section V outlines the main contributions, thus concluding this paper. II. P ROBLEM S TATEMENT AND P RELIMINARIES Consider an uncertain nonlinear dynamic plant y¨ = F(t, y, y˙ , p)u + f (t, y, y˙ , d, p)

(1)

where t ∈ T = [0 , tf ] ⊆ R is the time, y ∈ Rm is the output, u ∈ U ⊂ Rr is the input, d ∈ D ⊂ Rh is a disturbance, p ∈ ℘ ⊂ Rμ is the vector of uncertain parameters, f ∈ Rm is a nonlinear vector function, and F ∈ Rm×r , with rank(F) = m, is a nonlinear matrix function. Remark 1: Note that many mechanical systems with m degrees of freedom (cars, trains, conveyor belts, other

Fig. 1.

Control scheme with a pseudo-PID controller.

transportation systems, manufacturing machineries, Cartesian robots, etc.) are modeled using the equation M(p)¨q + Ka (p)˙q + Ke (p)q = Tu (q)u + g(q, p) + Td (q)d (2) where q ∈ Rm is the vector of generalized coordinates, u ∈ Rr is the vector of generalized control forces, g ∈ Rm is the vector of generalized gravity forces, d ∈ Rh is the vector of generalized disturbance forces, M, Ka , Ke ∈ Rm×m , Tu ∈ Rm×r , Td ∈ Rm×h are the inertia, damping, stiffness, and transmission matrices of the generalized forces u and d, respectively, and p ∈ ℘ ⊂ Rμ is the vector of uncertain parameters. It can be readily checked that the model (2) is of the type (1). Remark 2: Note that many robots, satellites, drones, ships, etc. with m degrees of freedom are modeled using the equation M(q, p)¨q + Ka (q, q˙ , p)˙q + Ke (q, p)y = Tu (q)u + g(q, p) + Td (q)d

(3)

where q ∈ Rm is the vector of generalized coordinates, u ∈ Rr is the vector of generalized control forces, g ∈ Rm is the vector of generalized gravity forces, d ∈ Rh is the vector of generalized disturbance forces, M(q, p) ∈ Rm×m is the inertia matrix, Ka (q, q˙ , p)˙q is the vector of damping, centrifugal, and Coriolis forces, Ke (q, p)q is the vector of possible stiffness forces, Tu (q) ∈ Rm×r , Td (q) ∈ Rm×h are transmission matrices of the generalized forces u and d, p ∈ ℘ ⊂ Rμ is the vector of uncertain parameters. It can also be readily checked that the model (3) is of the type (1). For brevity, assuming that Tu (q) = Td (q) = I (a very recurring case), the systems (2) and (3) are represented as M¨y = u + γ .

(4)

A possible control scheme for the plant (1) to track a reference signal r(t) ∈ Rm with a bounded second derivative is shown in Fig. 1. It uses a pseudo-PID controller with a possible compensation signal given by  t u(t) = Kp (t, y, y˙ )e(t) + Ki (t, y, y˙ ) e(τ )dτ + Kd (t, y, y˙ )˙e(t) 0

+ uc (t, y, y˙ , r, r˙ , r¨ )    t e(τ )dτ + Hd e˙ (t) = G(t, y, y˙ ) Hp e(t) + Hi + uc (t, y, y˙ , r, r˙ , r¨ )

0

(5)

where e(t) = r(t) − y(t), G, Hp , Hi , Hd are matrices of appropriate dimensions and uc is a possible compensation signal.

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Remark 3: To facilitate the implementation of the proposed control law, a reference signal r(t) with bounded second derivative can be obtained by interpolating a set of given points (tk , rk ), k = 0, 1, . . . , n, with cubic splines or by filtering any piecewise constant or piecewise linear signal r˜ (t) with the following MIMO filter: ⎡ ⎤ ⎡ ⎤ 0 I 0 0 0 I ⎦ζ + ⎣ 0 ⎦r˜ ζ˙ = ⎣ 0 −f1 I −f2 I −f3 I f1 I ⎡ ⎤ ⎡ ⎤ r I 0 0 ⎣ r˙ ⎦ = ⎣ 0 I 0 ⎦ζ. (6) 0 0 I r¨ In this way, since r˙ (t) is available, the derivative action of the controller can be avoided, if y˙ (t) is measurable. Furthermore, upon suitably choosing initial conditions of the filter and its cutoff frequency, it is possible to improve the transient phase and reduce the control magnitude. It is important to note that if the filter is a Bessel one with cutoff angular frequency ωb greater or equal to the angular frequency of r˜ (t), then r(t) ∼ = r˜ (t − tr ), tr = 3π/(4ωb ) (see [38], [39]). 1) Setting:  e = r − y, x1 = edτ, x2 = e

T x3 = e˙ , x = x1T x2T x3T (7) the plant (1) controlled with the PID (5) is described by the equations ⎡ ⎤ ⎡ ⎤ 0 I 0 0 0 I ⎦x + ⎣ 0 ⎦w = Ax + Bδ x˙ = ⎣ 0 −HHi −HHp −HHd I



e = 0 I 0 x = C2 x, e˙ = 0 0 I x = C3 x (8) where H = FG, δ = r¨ − f − Fuc .

(9)

and, therefore,  ... y = H Hi e + Hp e˙ + Hd e¨ + fp

H = M −1 G  d  d −1  −1 M G G M y¨ − M −1 γ + M −1 γ fp = dt

dt −1 ˙ −1 −1 −1 = −M MM GG M y¨ − M γ ˙ −1 γ + M −1 γ˙ − M −1 MM  −1 ˙y . = M γ˙ − M¨

(14)

In this case, the plant (1) controlled with the PID controller (5) is described by the equations ⎤ ⎡ ⎤ ⎡ 0 0 I 0 0 I ⎦x + ⎣ 0 ⎦w x˙ = ⎣ 0 I −HHi −HHp −HHd = Ax + Bδp



e = I 0 0 x = C1 x, e˙ = 0 I 0 x = C2 x

e¨ = 0 0 I x = C3 x (15) where

T ... x = eT e˙ T e¨ T , δp = r − fp .

(16)

3) Consider the following real actuator of the plant (1): u˙ = Aa u + Ba υ + Ca y˙

(17)

where υ ∈ Rr is the actuator input (e.g., the actuator supply voltage), Aa , Ba , Ca are matrices of appropriate dimensions, and the matrix Ba is nonsingular. Combining (1) and (17) yields  ... ˙ + f˙ = FBa υ + F˙ + FAa u + FCa y˙ + f˙ y = F u˙ + Fu u = F † (¨y − f )

(18)

where F † denotes the pseudo-inverse of F. The first equation in (18) can be recast in the form ... y = Fa υ + fa (19) where

(20)

(10)

2) If the matrix H = FG is nonsingular and Hp , Hi , and Hd are constant, for the system (1) one obtains  y¨ = Hυ + Fuc + f , υ = Hp e + Hi edτ + Hd e˙ = H −1 (¨y − f − Fuc ) ... ˙ + d(Fuc )/dt + f˙ y = H υ˙ + Hυ

For the system (4), if G = const. and uc = 0, the equalities (13) turn to

 Fa = FBa , fa = F˙ + FAa F † (¨y − f ) + FCa y˙ + f˙ .

For the system (4), the equalities (9) become H = M −1 G, δ = r¨ − M −1 γ − M −1 uc .

3

(11)

(12)

where ˙ −1 (¨y − f − Fuc ) + d(Fuc )/dt + f˙ . H = FG, fp = HH (13)

For the system (4), the equalities (20) turn to Fa = M −1 Ba  ˙ y + Ca y˙ + γ˙ . (21) fa = M −1 Aa (M¨y − γ ) − M¨ If the system (19) is controlled with the second order PD controller (referred to as PD2 )  (22) υ = Ga Hp e + Hd e˙ + Hd2 e¨ + υc the closed-loop system is described by the equations ⎤ ⎡ ⎤ ⎡ 0 0 I 0 ⎦x + ⎣ 0 ⎦w 0 I x˙ = ⎣ 0 I −HHp −HHd −HHd2 = Ax + Bδa



e = I 0 0 x = C1 x, e˙ = 0 I 0 x = C2 x

e¨ = 0 0 I x = C3 x (23)

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where ... H = Ha = Fa Ga , δa = r − fa − Fa υc .

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Remark 4: If not only y˙ but y¨ is also measurable [e.g., by using an accelerometer for the plant (4)], the controller (22) ... can be implemented generating r˙ and r¨ (and also r , if necessary) by the filter (6) (or a similar fourth-order filter, if the ... compensation of r is also assumed). If y¨ is not measurable, the controller (22) can be realized as the parallel of a proportional P controller of e and a proportional-derivative controller PD of e˙ . Making the substitutions y = r − e,

y˙ = r˙ − e˙ ,

y¨ = r¨ − e¨

(25)

in the expressions of H and of w (case 1: w = δ, case 2: w = δp , and case 3: w = δa ) leads to study the practical stability of the system ⎤ ⎡ ⎤ ⎡ 0 I 0 0 0 I ⎦ε + ⎣ 0 ⎦w ε˙ = ⎣ 0 I −H4a3 −H6a2 −H3a ⎡ ⎤ ⎡ ⎤ ε1 ε1 = Aε + Bw, ε = ⎣ ε2 ⎦ = ⎣ ε˙ 1 ⎦ ε3 ε¨ 1



ε1 = I 0 0 ε = C1 ε, ε2 = 0 I 0 ε = C2 ε

ε3 = 0 0 I ε = C3 ε (26) in order to design and implement robust smooth control algorithms for tracking reference signals with bounded second derivatives with an error norm smaller than a prescribed value, despite the presence of parametric and/or structural uncertainties of the plant (1) and/or real actuators. The following preliminary notations and definitions are introduced: √ √ xP = xT Px, x = xI = xT x SP,ρ = {x : xP ≤ ρ}, ρ ≥ 0 CP,ρ = {x : xP = ρ}, Cˆ P,ρ ⊇ CP,ρ (27) where P ∈ Rn×n is a symmetric and positive definite (p.d.) matrix, xT is the transpose of x ∈ Rn , and Cˆ P,ρ is a compact set; λmax (W)(λmin (W)) denotes the maximum (minimum) eigenvalue of a square matrix W ∈ Rn×n with real eigenvalues. Definition 1: Given the system x˙ = f (t, x, u, p), y = Cx

t ∈ T = 0 , tf ⊆ R, x ∈ Rn , u ∈ Rr p ∈ ℘ ⊂ Rμ , y ∈ Rm , C ∈ Rm×n

(28)

a constant U ≥ 0 and a p.d. symmetric matrix P ∈ Rn×n , a positive first-order system ρ˙ = ϕ(ρ, U), ρ0 = x0 P , Y = cρ

(29)

where ρ(t) = x(t)P , such that y(t) ≤ Y(t) ∀t ∈ T ∀u : u ≤ U and ∀p ∈ ℘, is said to be a majorant system of the system (28). Lemma 1: Let P ∈ Rn×n be a symmetric p.d. matrix, Q(t, x, p) ∈ Rn×n be a symmetric matrix, t ∈ [0, tf ] = T ⊆ R,

x ∈ Rn , p ∈ ℘, ℘ be a compact subset of Rμ , w(t, x, p) ∈ Rm be a vector continuous with respect to t ∈ T, x ∈ Rn and p ∈ ℘, B ∈ Rn×r be a matrix of rank r, and C ∈ Rm×n be a matrix of rank m. Then, ∀ρ ≥ 0 and ∀x ∈ CP,ρ , the following inequalities hold: max

t∈T, x∈CP,ρ ,p∈℘

xT Q(t, x, p)x

 λmax Q(t, x, p)P−1 ρ 2 t∈T, x∈CP,ρ ,p∈℘

 ≤ max λmax Q(t, x, p)P−1 ρ 2

(30)

  w(t, x, p) λmax BT PB ρ max t∈T, x∈CP,ρ , p∈℘   w(t, x, p) max ≤ λmax BT PB ρ

(31)

≤

max

t∈T, x∈Cˆ P,ρ , p∈℘ max xT PBw(t, x, p) t∈T, x∈CP,ρ , p∈℘

≤

Cx ≤



t∈T, x∈Cˆ P,ρ , p∈℘

 λmax CP−1 CT ρ.

(32)

Proof: See [20, Lemmas 1 and 3]. Lemma 2: Consider a matrix function H(p) ∈ Rm×m multiaffine with respect to the parameter vector p = [p1 p2 · · · pμ ]T ∈ ℘ ⊂ Rμ , i.e.,  i H(p) = Hi1 ,i2 ,...,iμ pi11 pi22 . . . pμμ i1 ,i2 ,...,iμ ∈{0,1}

Hi1 ,i2 ,...,iμ ∈ Rm×m .

(33)

Suppose that ℘ is a hyper-rectangle given by

−

−

+ + + ℘ = p− 1 , p1 × p2 , p2 × · · · × pμ , pμ

(34) = p− , p+ . Then minp∈℘ λmin (H+H T ) (maxp∈℘ λmax (H+H T )) is attained in one of the 2μ vertices Vp of ℘. Proof: It follows from [20, Lemma 4] by setting P = I and Q = H + HT .  Lemma 3: Let H(g(π )) = i1 ,i2 ,...,iμ ∈{0,1} Hi1 ,i2 ,...,iμ g1 (π )i1 g2 (π )i2 · · · gμ (π )iμ be a m × m real matrix, where g = [g1 g1 · · · gμ ]T , π ∈  ⊂ Rν ,  is a compact set, and each function gi , i = 1, 2, . . . , μ, is continuous with respect to π ; and let P ∈ Rn×n be a symmetric p.d. matrix. Then, a lower estimate of minπ ∈ λmin (H(g(π ))+H T (g(π ))) [an upper estimate of maxπ ∈ λmax (H(g(π )) + H T (g(π )))] is given by minp∈℘ λmin (H(p) + H T (p)) (maxp∈℘ λmax (H(p) + H T (p))), which is attained in one of the vertices of ℘ = {p ∈ Rμ : [ min g1 · · · min gμ ]T ≤ p ≤ [ max g1 . . . max gμ ]T }. Proof: The proof easily follows from Lemma 2 by making the change of variables p = g(π ) = [g1 (π )g2 (π ) · · · gμ (π )]T .

III. M AIN R ESULTS This section presents the main theorems providing a general approach to design robust smooth controllers for a system (1), in order to track a sufficiently smooth reference signal with an error norm smaller than a prescribed value.

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Theorem 1: Consider an uncertain nonlinear dynamic system ⎡ ⎤ ⎡ ⎤ 0 I 0 0 ⎦ε + ⎣ 0 ⎦w 0 0 I ε˙ = ⎣ I −FG4a3 −FG6a2 −FG3a ⎡ ⎤ ⎡ ⎤ ε1 ε1 = Aε + Bw, ε = ⎣ ε2 ⎦ = ⎣ ε˙ 1 ⎦ ε3 ε¨ 1



ε1 = I 0 0 ε = C1 ε, ε2 = 0 I 0 ε = C2 ε

ε3 = 0 0 I ε = C3 ε (35)

Case 2:

  √ √ √ 3 2 w1 6 w2 √ ρ˙ = 3w0 − a − − − 3w3 ρ 4 a2 2 a  √ √ √ 3 w12 4 3w22 + 3 8w13 3 3 w11 + + + 8 a4 4 a3 8 a2  √ 6 w23 √ + + 3w33 ρ 2 2 a

= α0 − α1 ρ + α2 ρ 2 √ √ 2ρ 6 ρ ε1  =  ε = , , 2 4 a2 2 a

where εi ∈ = 1, 2, 3, F(t, ε, p) ∈ ∈ Rm , t ∈ T = [0 , tf ] ⊂ R, d ∈ D ⊂ Rh is a disturbance, p ∈ ℘ ⊂ Rμ is the vector of uncertain parameters, G(t, ε) ∈ Rr×m is a design matrix, and a > 0 is a design parameter. Assume that there exists a matrix G(t, ε) ∈ Rr×m such that  λmin H + H T ≥ 2, H = H(t, ε, p) = FG (36) Rm , i

Rm×r , w(t, ε, d, p)

w ≤ μ1 (ε1 , ε2 , ε3 ) = μ1 (ε1 , ˙ε1 , ¨ε1 ) (37) = w0 + w1 ε1  + w2 ε2  + w3 ε3 . 2. Quadratically Bounded Growth

(38)

3. Cubically Bounded Growth w ≤ μ3 (ε1 , ε2 , ε3 ) = w0 + w1 ε1  + w2 ε2  + w3 ε3  + w11 ε1 2 + w22 ε2 2 + w33 ε3 2 + w12 ε1 ε2  + w13 ε1 ε3  + w23 ε2 ε3  + w111 ε1 3

(41)

  √ √ √ 6 w2 √ 3 2 w1 − 3w3 ρ ρ˙ = 3w0 − a − − 4 a2 2 a  √ √ √ 3 3 w11 3 w12 4 3w22 + 3 8w13 + + + 8 a4 4 a3 8 a2  √ 6 w23 √ + + 3w33 ρ 2 2 a  √ √ √ 9 2 w111 3 6 w112 3 3 w113 + + + 32 a6 16 a5 8 a4 √ √ √ 3w123 + 6w222 2 3w223 + 3 2w133 + + 4a3 4a2  √ 6 w233 √ + + 3w333 ρ 3 2 a

1. Linearly Bounded Growth

+ w23 ε2 ε3 .

ε3  = ρ.

Case 3:

and that w satisfies one of the following conditions.

w ≤ μ2 (ε1 , ε2 , ε3 ) = w0 + w1 ε1  + w2 ε2  + w3 ε3  + w11 ε1 2 + w22 ε2 2 + w33 ε3 2 + w12 ε1 ε2  + w13 ε1 ε3 

5

= α0 − α1 ρ + α 2 ρ 2 + α3 ρ 3 √ √ 6 ρ 2ρ ε1  = , ε2  = , ε3  = ρ 2 4 a 2 a where ρ = εP , with ⎡ ⎤ 16a4 I 16a3 I 4a2 I P = ⎣ 16a3 I 20a2 I 6aI ⎦, a > 0. 2 4a I 6aI 3I

(42)

(43)

Proof: Let us choose the quadratic form εT Pε = ε2P = as Lyapunov function. Its derivative satisfies the following inequality for ε belonging to the generic hyper-circumference CP,ρ :  2ρ ρ˙ ≤ max εT Qε + 2εT PBw   ≤ max εT Qε + max 2εT PBw , Q = AT P + PA (44) ρ2

+ w222 ε2 3 + w333 ε3 3 + w123 ε1 ε2 ε3  + w112 ε1 2 ε2  + w113 ε1 2 ε3  + w122 ε1 ε2 2 + w133 ε1 ε3 2 + w223 ε2 2 ε3  + w233 ε2 ε3 2 (39) where w0 , w1 , w2 , wij , and wijk are appropriate non-negative constants. Then a majorant system for the system (35) is given as follows. Case 1:   √ √ √ 6 w2 √ 3 2 w1 − 3w3 ρ − ρ˙ = 3w0 − a − 4 a2 4 a = α0 − α1 ρ √ √ 6 ρ 2ρ ε1  = (40) , ε3  = ρ. , ε2  = 4 a2 2 a

which implies, in view of Lemma 1, that 

  ρ˙ ≤ max λ QP−1 ρ/2 + λmax BT PB max w. Since P−1

⎡ 3I 1 ⎣ = 4 −3aI 8a 2a2 I

−3aI 4a2 I −4a3 I

⎤ 2a2 I −4a3 I ⎦ 8a4 I

(45)

(46)

if λmin (H + H T ) ≥ 2, the following equality takes place:

 max λ QP−1  ⎤⎞ ⎛⎡ −2aI 0 4a3 2I − H − H T T ⎦⎠ = −2a. = max λ⎝⎣ 0 −2aI 6a2 2I − H − H T 0 0 −3a H + H + 4aI (47)

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Furthermore,   λmax BT PB =   λmax C1 P−1 C1T =   λmax C3 P−1 C3T =

√ 3 √ 6 1 , 4 a2



 λmax C2 P−1 C2T =

√

1.

21 2 a (48)

Hence, a majorant system for the system (35) is given by √ ρ˙ = −aρ + 3 maxw √ √ 6 ρ 2ρ ε1  = (49) , ε2  = , ε3  = ρ. 2 4 a 2 a The rest of proof follows upon taking into account (37), (38), or (39) and the inequalities: √ √ 6 ρ 2ρ ε1  ≤ , ε3  ≤ ρ (50) , ε2  ≤ 4 a2 2 a which are valid for ε belonging to the generic hypercircumferenceCP,ρ . Theorem 1 allows one to establish the following theorems. Theorem 2: Consider the system (35) with G satisfying (36). Then, for ∀a ≥ a˘ , the following inequalities hold: lim ρ(t) ≤ ρ1 √ 6 ρ1 lim ε1 (t) ≤ 2 t→∞ 4 √ a 2 ρ1 lim ε2 (t) ≤ t→∞ 2 a lim ε3 (t) ≤ ρ1

(a)

(b)

(c)

t→∞

Fig. 2. Graphical representations of majorant systems: (a) case 1, (b) case 2, and (c) case 3.

(51)

t→∞

where: for the case 1 a˘ is such that α1 > 0, ρ1 = α0 /α1 and ∀ρ0 ∈ [0, ∞) [see Fig. 2(a)]; for the case 2 a˘ is such that b = α12 − 4α2 α0 > 0, ρ1 , ρ2 , ρ1 < ρ2 , are the roots of the algebraic equation α2 ρ 2 − α1 ρ + α0 = 0 and ∀ρ0 ∈ [0, ρ2 ) [see Fig. 2(b)]; for the case 3 a˘ is the smallest value such that for ∀a ≥ a˘  c =

α¯ 23 α¯ 1 α¯ 2 α¯ 0 + + 27 6 2



2 −

α¯ 2 α¯ 1 + 2 3 9

3 0 the equation α2 ρ 2 −α1 ρ +α0 = 0 has two distinct real positive roots, and Fig. 2(b). Case 3 follows from the formula for the roots of a third degree equation, the fact that the root locus of the equation α3 ρ 3 + α2 ρ 2 − α1 ρ + α0 = 0 with respect to a ≥ a˘ always follows the pattern in Figs. 2(c) and 3 and Theorem 1. Theorem 3: Consider the system (35) with G satisfying (36). For sufficiently large a, the following approximations hold: √ ∼ 3w0 , cases 1, 2, 3 (53) ρ1 = a α¯ i =

Fig. 3. Root locus of equation α3 ρ 3 + α2 ρ 2 − α1 ρ + α0 = 0 with respect to a ≥ a˘ .

a ρ2 ∼ , case 2 = √ 3w33 √ a ρ2 ∼ , case 3 with w333 > 0 = √ 3w333

(54)

and, therefore, the last three inequalities of (51) turn to √ √ 6 ρ1 ∼ 3 2 w0 lim ε1 (t) ≤ 2 = 4 a3 t→∞ √4 a √ 2 ρ1 ∼ 6 w0 lim ε2 (t) = ε˙ 1  ≤ = t→∞ 2 a 2 a2 √ w0 ∼ (55) lim ε3 (t) = ε¨ 1  ≤ ρ1 = 3 t→∞ a and the time constant τl of the model linearized around the equilibrium point ρ1 is τl ∼ = 1/a.

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Finally, if ε(0)P ≤ ρ1 then √ √ 6 ρ1 ∼ 3 2 w0 ε1 (t) ≤ ∀t ≥ 0 2 = 4 a3 √4 a √ 2 ρ1 ∼ 6 w0 ε2 (t) = ε˙ 1  ≤ ∀t ≥ 0 = 2 a 2 a2 √ w0 ε3 (t) = ε¨ 1  ≤ ρ1 ∼ ∀t ≥ 0. (56) = 3 a Proof: Note that for sufficiently large a, the following inequalities hold: √ α0 − α1 ρ ∼ case 1) = 3w0 − aρ, √ √ 2 ∼ 2 α0 − α1 ρ + α2 ρ = 3w0 − aρ + 3w33 ρ , case 2) √ √ α0 − α1 ρ + α2 ρ 2 + α3 ρ 3 ∼ = 3w0 − aρ + 3w33 ρ 2 √ + 3w333 ρ 3 , case 3). (57) Proof in case √ 1 is trivial. In case 2, if a  a˘ , the relations ρ1 √ + ρ2 = a/ √ 3w33 , ρ1 ρ2 = w0 /w33 , ρ1  ρ2 , yield ρ2 ∼ = ∼ a/ 3w33 , ρ1 = 3w0 /a. In case 3, if a  a˘ , the√ root locus in Fig. 3 shows that ρ1  1 and, therefore, ρ1 ∼ = 3w0 /a. Furthermore, the relations ρ1 ρ2 ρ3 = −w √ 0 /w333 , ρ1 + ρ2 + ρ3 = −w33 /w333 , yield ρ2 ρ3∼ = −a/ 3w333 , ρ2 + ρ3 ∼ = −w33 /w333 , therefore, ρ2 ∼ = √ √ 3w333 , ρ3 ∼ a/ . −ρ = 2 The remaining part of the proof readily follows. Remark 5: It is worth noting that if y˙ is affected by a bounded measurement noise ny˙ , it has be added to w0 the term 3aHny˙  in case 1 and the term 6a2 Hny˙  in cases 2 and 3. From (7), (16), (23), (26), and (56), for sufficiently large a, e is inversely proportional to a2 in the case 1 and inversely proportional to a3 in cases 2 and 3. Hence, the rate of e due to the noise ny˙ , for sufficiently large a, is inversely proportional to a. Remark 6: To design the controller, it is essential to find a matrix G satisfying (36). Since in many practical cases the matrix F is multilinear with respect to uncertain parameters, a matrix G can be easily found using Lemmas 2 and 3. There are some examples. 1) G = gI, g > 0, if F is a p.d. matrix [in case of the system (4)]. 2) G = gJ T , g > 0, where J is the Jacobian matrix of a suitable coordinate transformation (in case of robots in their workspace or equipped with cameras used as sensors, as well as ships, drones, satellites, etc.). 3) G = F T (FF T )−1 , if F is independent of p (in case of kinematic inversion, etc.) or if variations of p around the nominal value pˆ are uniformly bounded. 4) Other suitable matrices G (see, for instance, Example 1). Since a is the only parameter, which determines how fast the tracking error and its first and second derivatives decrease as a increases, a value of the parameter a can be calculated via simulations or experimentally for an assigned class of reference signals r. Remark 7: Taking into account the relation 2ρ ρ˙ = 2εT P˙ε , a less conservative majorant system of type ρ˙ = g(ρ), providing smaller values of ρ1 and larger values of ρ2 , can be obtained through the equation ρ˙ = f (ρ) =

max

ε∈CP,ρ ,p∈℘,t∈T

εT P(Aε + Bw)/ρ

(58)

Fig. 4.

Three-link planar robot.

Fig. 5.

Welding phase of a coil with a bar.

7

which can be solved using the MATLAB procedure “fmincon” and/or employing randomized algorithms (see [21]). Remark 8: To reduce the controller gains and control signal during the transient phase, if they are very high, it is possible to: 1) smooth the trajectories with appropriate filters and suitable initial conditions (see Remark 3); 2) better identify process parameters and disturbances and use a compensation signal uc to reduce the norm of w; 3) use a connection trajectory, if the initial error is excessive (see [39] for more details); 4) slow down r(t), i.e., replace the reference signal r(t) by r(ct), with c < 1. IV. C ASE S TUDIES The results given in the preceding sections are validated through two examples. Example 1: This example presents control design for a three-link planar robot designated to periodical industrial processing, e.g., making cages for concrete piles (see Figs. 4 and 5). The example shows how it is easy to design simple, robust, and efficient controllers, using the developed technique. Suppose, for simplicity, that the ith link of the robot is a straight line of constant section and a mass density mi with the tip two masses Mmi and Mpi of inertia moments Imi and Ipi . If Hb = 0.30 m, L1 = 0.75 m, L2 = 0.75 m, L3 = 0.30 m m1 = 1 Kg/m, m2 = 1 Kg/m, m3 = 1 Kg/m Mm1 = 0, Mm2 = 0, Mm3 = 0

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Fig. 6.

Time histories of r(t) and r˙ (t).

Fig. 8. Workspace region where matrix (64) satisfies condition  λmin JM −1 Gc + (JM −1 Gc )T ≥ 2.

Fig. 7.

Time histories of rβ (t) and r˙β (t).

Mp1 = 0.25 Kg, Mp2 = 0.25 Kg, Mp3 = p1 ∈ [0, 0.50] Kg Im1 = 0.10 Kgm2 , Im2 = 0.10 Kgm2 , Im3 = 0.10 Kgm2 Ip1 = 0, Ip2 = 0, Ip3 = p2 ∈ [0, 0.10] Kgm2 Ka1 = 1.00 Nms/rad, Ka2 = 1.00 Nms/rad Ka3 = 1.00 Nms/rad

(59)

then the inertia matrix M can be computed using the following relations (see [9]): M(p, β) = M00 + M01 p1 + M02 p2 + (M10 + M11 p1 + M12 p2 ) cos β2 + (M20 + M21 p1 + M22 p2 ) cos β3 + (M30 + M31 p1 + M32 p2 ) cos(β2 + β3 )

(60)

where the matrices Mij (and the corresponding model of the robot) can be found using the algorithm from [9]. The Jacobian matrix is computed as ⎞ L1 cos β1 + L2 cos(β1 + β2 ) + L3 cos(β1 + β2 + β3 ) ∂ ⎜ ⎟ J= ⎝Hb + L1 sin β1 + L2 sin(β1 + β2 ) + L3 sin(β1 + β2 + β3 )⎠. ∂β β1 + β2 + β3 ⎛

(61) Suppose that y1c = 1.00 m, y2c = 0.30 m, and R = 0.30 m and that the reference signals r(t) for y(t) = [y1 y2 y3 ]T in the workspace are the ones shown in Fig. 6. The corresponding reference signals in the joint space are reported in Fig. 7. These reference signals are obtained applying a third-order Bessel filter with ωb = 10 rad/s and kinematic inversion to the primary reference signal r˜ (t). Several controllers are designed as follows.

1) Using Lemmas 2 and 3, a matrix Ga of the type Ga = gI, g > 0, which satisfies condition λmin (M −1 Ga + (M −1 Ga )T ) = 2g/λmax (M) ≥ 2, is assigned as ⎡ ⎤ 1 0 0 Ga = λmax (M)I = 5.9495⎣ 0 1 0 ⎦ 0 0 1 ⎡ ⎤ 5.9495 0 0 5.9495 0 ⎦. =⎣ 0 (62) 0 0 5.9495 2) Using Lemmas 2 and 3, a matrix Gb of the type Gb = g diag([3 2 1]), g > 0, satisfying condition λmin (M −1 Gb +(M −1 Gb )T ) = λmin (M −1 Gb +Gb M −1 ) ≥ 2, is assigned as ⎡ ⎤ 3 0 0 Gb = 2.9199⎣ 0 2 0 ⎦ 0 0 1 ⎡ ⎤ 8.7596 0 0 5.8397 0 ⎦. =⎣ 0 (63) 0 0 2.9199 3) Assuming that βi ∈ [rβi − 15π/180,rβi + 15π/180], i = 1, 2, β3 ∈ [rβ3 − 25π/180,rβ3 + 25π/180], and using Lemmas 2 and 3, it can be easily verified that ¯ −1 (β, ¯ p¯ ))−1 = the constant matrix Gc = g(J(β)M −1 ¯ p¯ )J (β), ¯ g = 4, β¯ = [π/2 −π/2 −π/2]T , p¯ = gM(β, [¯p1 , p¯ 2 ] = [0.50, 0.10], equal to ⎡ ⎤ −1.6633 1.1083 0.6068 1.1083 0.1955 ⎦ Gc = 4⎣ 0.1950 0.1950 0 0.1955 ⎡ ⎤ −6.6533 4.4333 2.4270 4.4333 0.7820 ⎦ = ⎣ 0.7800 (64) 0.7800 0 0.7820 satisfies condition λmin (JM −1 Gc + (JM −1 Gc )T ) ≥ 2 in the workspace region shown in Fig. 8.

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Fig. 9. Time histories of the control torques obtained with the control law (66).

9

Fig. 11. Time history of the error norm produced by the control law (66) with integrator initial conditions xoI .

Fig. 12. Time histories of y˙ + n˙y, e produced by the control law (66) with integrator initial conditions xoI in case of velocity measurement noise. Fig. 10.

Time history of the error norm produced by the control law (66).

4) Assuming that β1 ∈ [−π, π ], β2 ∈ [−11π/12, −π/12], β3 ∈ [−11π/12, −π/12] and using Lemmas 2 and 3, it can be easily verified that a value of g such that a matrix Gd = gJ T (β) satisfying condition λmin (JM −1 Gd +(JM −1 Gd )T ) = 2gλmin (JM −1 J T ) ≥ 2 is given by g = 19.39 ⇒ Gd = 19.39J T (β).

(65)

I. Assume that the actuators are ideal and Mp3 = 0.25 Kg and Ip3 = 0.05 Kgm2 . The classic MIMO PID control law in the workspace    t 2 3 e(τ )dτ + 3a˙e(t) u(t) = Gc 4a e(t) + 6a 0

a = 5, e(t) = r(t) − y(t)

(66)

yields the control torques u = [u1 u2 u3 ]T and the norm of the error e shown in Figs. 9 and 10, respectively. It is significant to note the following. 1) The error during the starting phase can be reduced by compensating the gravity torques Cg (t) at t = 0 with initial conditions of the integrators x0I = (4a3 Gc )−1 Cg (0) (see Fig. 11).

Fig. 13. Time histories of n˙y, y˙ + n˙y, e produced by the control law (66) with integrator initial conditions xoI in case of velocity measurement noise.

2) The errors during the stopping phases at the points P1 – P4 (to do the welding) are virtually zero, due to integral action. 3) The control law in the workspace (66) is independent of β. If velocity y˙ (t) is affected by a measurement noise ny˙ uniformly distributed in [−0.05, 0.05] (see Fig. 12), the error e is shown in Fig. 13, in accordance with Remark 5. On the other hand, if the velocity y˙ (t) is affected by a measurement noise ny˙ 1 = −0.1 + 0.05 sin t + 0.04 cos 4t, ny˙ 2 =

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Fig. 14. Time history of the error norm produced by the control law (66), if the gravity torques are compensated.

Fig. 15.

Fig. 16. Time history of e in presence of real actuator (68) for R = 2 , using PD control law (69).

Time history of the error norm produced by the control law (67).

0.05 + 0.05 sin 2t sin t, ny˙ 3 = −0.07 + 0.05 sin 2t sin 3t the corresponding error e is shown in Fig. 13. If the gravity torques are compensated, the error norm of the error e is the one shown in Fig. 14. II. The control law consisting of three classic single-inputsingle-output PD controllers in the joint space, designed with the methodology presented in [38]

 u(t) = Gb 2a2 e(t) + 2a˙e(t) , a = 10, e(t) = rβ (t) − β(t) (67) yields the error norm of the error e shown in Fig. 15. Now assume that torques u are provided by three identical dc motors described by the equation ⎡ ⎤ ⎡ ⎤ 1 0 0 1 0 0 u˙ = −10R⎣ 0 1 0 ⎦u + 10ga ⎣ 0 1 0 ⎦υ 0 0 1 0 0 1 ⎡ ⎤ 1 0 0 (68) − 10⎣ 0 1 0 ⎦β˙ 0 0 1

Fig. 17. Time history of e in presence of real actuator (68) for R = 1 , using PD control law (69).

where υ is the vector of the supply voltages of the motors, R is the armature resistance of the motors, and ga is the gain of the power amplifier (which is set equal to R to make the static gain of each motor equal to one). Using the control law (67) with u = υ, i.e.,

 υ(t) = Gb 2a2 e(t) + 2a˙e(t) , a = 10, e(t) = rβ (t) − β(t) (69) the time histories of e for R = 2, 1, 0.50  are, respectively, shown in Figs. 16–18. Note that the performance of the control system becomes worse as R decreases. For R = 0.50 , the control system is even unstable.

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Fig. 18. Time history of e in presence of real actuator (68) for R = 0.5 , using PD control law (69).

Fig. 19. Time history of e in presence of real actuator (68) for R = 2 , using PD2 controller (70).

11

Fig. 20. Time history of e in presence of real actuator (68) for R = 1 , using PD2 controller (70).

Fig. 21. Time history of e in presence of real actuator (68) for R = 0.5 , using PD2 controller (70).

If controller PD2 defined in (22) is employed

 Gb 3 4a e(t) + 6a2 e˙ (t) + 3a¨e(t) 10 min R ⎡ ⎤ 1.7519 0 0

 =⎣ 0 1.1680 0 ⎦ 4a3 e(t) + 6a2 e˙ (t) + 3a¨e(t) 0 0 0.5840 (70) a = 10, e(t) = rβ (t) − β(t)

υ(t) =

the time histories of e for R = 2, 1, 0.50  are shown in Figs. 19–21, respectively. If the term 6a2 e˙ (t) + 2a¨e(t) is realized with a real PD controller of e˙ (t) with time constant of the derivative action τ = 0.01 s, the time histories of e for R = 2, 1, 0.50  are practically equal to the ones in Figs. 19–21. Example 2: Consider the plant y¨ = p1 p2 y˙ y/(1 + |y|) + sin(y)y˙y/4 + atan(t/5) sin t

 + p2 / 2 + sin2 (t/50) u, p1 ∈ [0.3, 0.5], p2 ∈ [1, 1.2] 3

(71) which does not belong to the class of systems (4).

Fig. 22. Time histories of the reference signal and its first and second derivatives.

The desired reference signal is the one shown in Fig. 22, which is obtained from an easily generable piecewise-linear signal filtered by a third-order Bessel filter with ωb = 2 rad/s.

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Majorant systems obtained with (58) for a = 2, 5, 10.

Fig. 24.

Time histories of u and e for a = 2.

Fig. 25.

Time histories of u and e for a = 5.

The following PID controller is applied:  2 3 u(t) = G(6a e + 4a e(τ )dτ + 3a˙e),

Fig. 26.

Time histories of u and e for a = 10.

Fig. 27. Time history of y˙ + n˙y and e produced by the control law (72) with a = 5.

Fig. 28. Time history of y˙ + n˙y and e produced by the control law (72) with a = 10.

e=r−y

(72)

with G = 3 satisfying condition (36). In Fig. 23, the graphical models of the majorant systems obtained with (58) are reported for a = 2, 5, 10.

The time histories of u and e are shown for a = 2, 5, 10 in Figs. 24–26, respectively. Note that for a = 2 the majorant system has no equilibrium points. For a = 5, instead, the majorant system has two equilibrium points. The corresponding value of the maximum of |e| is max |e| = 0.0525, which is not much larger than the

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actual value 0.0251 (see Figs. 23 and 25). For a = 10, the majorant system still has two equilibrium points. The corresponding value of the maximum of |e| is max |e| = 0.00575, which is not much larger than the actual value 0.00344 (see Figs. 23 and 26). Note also that max |e|a=5 / max |e|a=10 = 7.299 = 1.9403 ∼ = 23 , in accordance with the fact that, for a sufficiently large a, the maximum of |e| is inversely proportional to a3 . Finally, if y˙ (t) is affected by a measurement noise ny˙ uniformly distributed in [−0.10, 0.10], the control u and the absolute value of the error |e| are shown in Figs. 27 and 28 for a = 5 and a = 10, respectively. V. C ONCLUSION The main advantages of the results provided in this paper can be summarized as follows. 1) The considered class of nonlinear uncertain MIMO systems is quite general, including also an important class of uncertain linear systems. 2) A comprehensive approach based on the concept of majorant systems is provided to design smooth robust controllers to track a sufficiently smooth reference signal with an error norm smaller than a prescribed value. 3) The obtained control laws are easy to design and implement. They have no high gains and are free from discontinuities. 4) The trajectories to be tracked are considered generic, provided that they are sufficiently smooth. 5) Suitable filtering laws are proposed for tracked trajectories to facilitate the implementation of the control laws and reduce the control magnitude, especially, during the transient phase. 6) The maximum tracking error can be fixed a priori despite bounded parametric uncertainties, disturbances, and velocity measurement noise. 7) There is a simple relation between a single design parameter of the controller and the maximum tracking error, which is useful to obtain the desired tracking precision. The established theoretical results can be used to obtain further analysis and synthesis results and study complex systems with parametric and structural uncertainties. The treated cases are readily applicable to other engineering and/or transportation complex nonlinear uncertain systems. R EFERENCES [1] P. Dorato, Ed., Robust Control. New York, NY, USA: IEEE Press, 1987. [2] C. T. Abdallah, D. Dawson, P. Dorato, and M. Jamshidi, “Survey of robust control for rigid robots,” IEEE Control Syst. Mag., vol. 11, no. 2, pp. 24–30, Feb. 1991. [3] J. Ackermann, Robust Control: Systems With Uncertain Physical Parameterss. New York, NY, USA: Springer-Verlag, 1993. [4] M. Zhihong and M. Palaniswami, “Robust tracking control for rigid robotic manipulators,” IEEE Trans. Autom. Control, vol. 39, no. 1, pp. 154–159, Jan. 1994. [5] R. J. Adams, J. M. Buffington, A. G. Sparks, and S. S. Banda, Robust Multivariable Flight Control. London, U.K.: Springer-Verlag, 1994.

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Laura Celentano (M’04) received the Ph.D. degree in automation and computer science engineering (with a 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 Italian Air Force Academy, Naples (Fundamentals of Dynamical Systems). She is a science journalist and has cooperated with radio programs and journals on the popularization of scientific matters. She has taken an active part in activities co-funded by the European Union, the Italian Ministry of University and Research and the Economic Development, Region Campania, and public and/or private corporations and industries. She has authored/reviewed IEEE, ASME, Elsevier, and AIP journals and conferences. She has authored and co-authored 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 telemonitoring and/or telecontrol systems. Ms. Celentano has been the Chair and an Organizer for conference sessions. Michael V. Basin (M’95–SM’07) received the Ph.D. degree in physical and mathematical sciences (with major in automatic control and system analysis) from Moscow Aviation University, Moscow, Russia, in 1992. He is currently a Full Professor with the Autonomous University of Nuevo Leon, San Nicolás de los Garza, Mexico, and a Leading Researcher with ITMO University, Saint Petersburg, Russia. He has published over 300 research papers in international referred journals and conference proceedings with over 3000 citations and an H-index of 32. He has authored the monograph entitled New Trends in Optimal Filtering and Control for Polynomial and Time-Delay Systems (Springer). He has supervised 14 doctoral and seven master’s theses. His current research interests include optimal filtering and control problems, stochastic systems, time-delay systems, identification, sliding mode control, and variable structure systems. Dr. Basin was a recipient of the Highly Cited Researcher Award by Thomson Reuters, the publisher of Science Citation Index, in 2009. He has served as the Editor-in-Chief and serves as the Senior Editor-inControl for the Journal of the Franklin Institute, a Technical Editor for the IEEE/ASME T RANSACTIONS ON M ECHATRONICS, and an Associate Editor for Automatica, the IEEE T RANSACTIONS ON S YSTEMS , M AN AND C YBERNETICS : S YSTEMS , IET-Control Theory and Applications, the International Journal of Systems Science, and Neural Networks. He is a regular member of the Mexican Academy of Sciences.