Design of controllers with arbitrary convergence time

Automatica 112 (2020) 108710

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Automatica journal homepage: www.elsevier.com/locate/automatica

Brief paper

Design of controllers with arbitrary convergence time✩ ∗

Anil Kumar Pal a , Shyam Kamal a , , Shyam Krishna Nagar a , Bijnan Bandyopadhyay b ,1 , Leonid Fridman c a

Department of Electrical Engineering, Indian Institute of Technology (BHU), Varanasi, India Control Engineering Group, Technische Universität Ilmenau, 98684, Ilmenau, Germany Department of Control Engineering and Robotics, Facultad de Ingeniería, Universidad Nacional Autónoma de México (UNAM), Coyoacán D.F., 04510, Mexico b c

article

info

Article history: Received 6 December 2018 Received in revised form 26 July 2019 Accepted 5 November 2019 Available online xxxx Keywords: Nonlinear control Nonautonomous system Backstepping Lyapunov stability Free-will arbitrary time stability

a b s t r a c t In this paper, a method for new controller design ensuring the arbitrary time of convergence is proposed. The sufficient condition for Lyapunov stability is also given for this arbitrary chosen time stable system. Disturbances have been taken care of by introducing sliding mode control in the design approach. Finally, the efficacy of the proposed method is illustrated through a practical system, viz., magnetic suspension system. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Over the years there has been much research about rated convergence. Recent researches are focused on obtaining finite and fixed time convergence. A detailed analysis of stability notions, including finite time and fixed time stability can be found in Polyakov and Fridman (2014). A rigorous analysis of finitetime stability of continuous autonomous systems is presented, for example, in Bhat and Bernstein (2000). In Levant and Yu (2018), a novel kth-order differentiator was developed and applied for solving the classical SMC problem of the equivalent control estimation. The most important application of the finite-time convergence is a problem of differentiation, allowing the designer to be sure that after some time moment a controller is using correct information (Levant & Yu, 2018). That is why the upper bound of the settling time function for any initial ✩ This work is supported by the project titled "Construction of Non-monotonic Lyapunov Function for the Dynamical Systems governed by Differential Inclusions": Mathematical Research Impact Centric Support (MATRICS) to the Science and Engineering Research Board (SERB), India, 2019–2021 (3 years) Project’s reference no. MTR/2018/000799. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Luca Zaccarian under the direction of Editor Daniel Liberzon. ∗ Corresponding author. E-mail addresses: [email protected] (A.K. Pal), [email protected] (S. Kamal), [email protected] (S.K. Nagar), [email protected] (B. Bandyopadhyay), [email protected] (L. Fridman). 1 Bijnan Bandyopadhyay is currently visiting Technische Universität Ilmenau, Germany as an Alexander von Humboldt Fellow, on leave from IIT Bombay. https://doi.org/10.1016/j.automatica.2019.108710 0005-1098/© 2019 Elsevier Ltd. All rights reserved.

conditions is crucial. It was a reason to introduce the notion of fixed-time convergence (Angulo, Moreno, & Fridman, 2013; CruzZavala, Moreno, & Fridman, 2011; Levant, 2013; Polyakov, 2011). It is to be noted that for the case of fixed time stability, the upper bound of the settling time needs to have certain uniformity with respect to the initial conditions (Polyakov, 2011; Polyakov & Fridman, 2014). The algorithms based on stabilizing polynomial feedbacks and modified second order SMC were developed in Polyakov (2011). These algorithms allow adjusting guaranteed settling time independently of initial conditions. However, the problem of the design of fixed time convergent controllers/differentiators was just partially solved. The authors of Angulo et al. (2013), Corradini and Cristofaro (2018), CruzZavala et al. (2011) and Polyakov (2011) have just estimated a convergence time but did not offer a design of the controller parameters allowing to ensure desired convergence time. Although the existing approaches (Polyakov, Efimov, & Perruquetti, 2015; Polyakov & Fridman, 2014) allow attaining convergence within desired time by properly choosing design parameters, the proper tuning of these parameters to attain arbitrary time convergence may not be an easy task. The local asymptotic stability of a nonlinear vector field can be established by studying the stability of its homogeneous approximation. The concept of homogeneity also finds application in finite time stabilization (Bhat & Bernstein, 2005). Analysis of homogeneity in bi-limit (Andrieu, Praly, & Astolfi, 2008) provides an insight into the fixed time behaviour of an asymptotically stable system. Unfortunately, an explicit relation that allows the

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A.K. Pal, S. Kamal, S.K. Nagar et al. / Automatica 112 (2020) 108710

designer to adjust the settling time is not facilitated even in the homogeneity approach. Fixed time controller design utilizes the knowledge of global behaviour of the system, which at times may not be available. Another constraint in the design is the unavailability of actuators with enormous capabilities. A sophisticated time-varying control formulation for regulation in prescribed finite time has been given in Song, Wang, Holloway, and Krstic (2017). This approach has been successful in overcoming the design constraints of fixed time controllers by using the concept of time-varying scaling function. In most of the approaches, it is observed that settling time function depends on initial conditions. In Wang and Song (2018), the authors have been successful in achieving settling time independent of initial conditions and any other design parameters. However, their prespecified finite time approach lacks simplicity due to the deployment of two different time-varying scaling functions. In missile guidance (Zarchan, 2012), the impact time control guidance (ITCG) laws demand stabilization in a given prespecified time. The ITCG laws are useful in salvo attack scenario of multiple missiles (Jeon, Lee, & Tahk, 2006; Kim, Jung, Han, Lee, & Kim, 2015). The design of controllers with arbitrary convergence time would be helpful for this purpose and other applications.

Notations Throughout the paper, the notations used are standard. R, R≥k and Rn denote the fields of real numbers, real numbers greater than or equal to k and n-dimensional Euclidean space respectively. The notation ∥ · ∥ refers to the Euclidean norm. min refers to the least element of the given set. sup refers to the supremum of the given set. If S1 , S2 and S3 are any sets and g1 : S1 → S2 and g2 : S2 → S3 are functions, then the function g2 o g1 : S1 → S3 , defined by (g2 o g1 )(·) = g2 (g1 (·)) is called the composition of g1 and g2 (Khalil, 2002). A continuous function g : [0, c) → [0, ∞) is said to belong to class K if it is strictly increasing and g(0) = 0 (Khalil, 2002). An n-dimensional open ball Bδ of radius δ is the collection of points of distance less than δ from a fixed point in n-dimensional Euclidean space i.e. Bδ = {x ∈ Rn : ∥x − x0 ∥ < δ}, where x0 is the fixed point. Lf g is the Lie derivative of g with respect to the vector field f . The acronym s.t. means such that. 2. Preliminaries Consider the nonautonomous nonlinear system x˙ = f (t , x; ϕ ), x(t0 ) = x0

Example In this paper, a unique and novel phenomenon captured by the following nonautonomous differential equation is explored

{ x˙ =

−η(ex −1) , ex (tf −t)

0,

if t0 ≤ t < tf

(1)

otherwise

where η ∈ R≥1 and tf is the settling time moment independent of any system parameters or initial conditions. The solution to (1) is x = ln(C (tf − t)η + 1) where integration constant C =

(2) ex(t0 ) −1 (tf −t0

. The motivation for our )η

work lies in the special features of the differential equation (1). It can be easily realized that both x and x˙ tends to zero as t → tf . Therefore, it implies that x remains at the origin for any time t ≥ tf . It can be observed that the right-hand side of (1) remains bounded even if x grows large. This special feature makes the system (1) perfectly suitable to function as a controller. Settling time bound based characterization for this class of nonautonomous systems is proposed. It is to be observed that the upper bound on settling time is independent of initial conditions and any system parameter and can be set as per our will. Such class of systems is said to be free-will arbitrary time stable. The notion of free-will weak arbitrary time stable and free-will strong arbitrary time stable is also proposed. The control being invoked in the proposed approach is simple, elegant, and may be reduced to involve only one tuning parameter. This tuning parameter is used to alter the rate of convergence. This constructive design process is proposed for first, second, third-order systems and is realizable for higher-order systems as well. The rest of the paper is structured as follows: Section 2 provides the basic definitions for the nonautonomous case. Section 3 contains the main results. Section 4 includes the proposed constructive ideology for first-order system and its subsequent extension to second-order and third-order systems using the backstepping technique. The simulation results are also attached. Section 5 contains the application of this approach to the magnetic suspension system and the corresponding simulation results. Finally, Section 6 concludes the paper.

(3)

where x ∈ R is the system state, ϕ ∈ R represents the constant parameters of the system, and f : R≥0 × Rn → Rn is a nonlinear function such that f (t , 0; ϕ ) = 0, i.e., origin x = 0 is an equilibrium point of (3), t0 ∈ R≥0 is the initial time. n

p

Definition 1 (Polyakov, 2011 Global Finite Time Stability). The origin of the system (3) is said to be globally finite time stable if it is globally asymptotically stable and any solution x(t , t0 , x0 ) of (3) converges to the origin at some finite time, i.e., ∀ t ≥ t0 + T (t0 , x0 ), x(t , t0 , x0 ) = 0, where T : R≥0 × Rn → R≥0 , is the settling time function. Definition 2 (Polyakov, 2011 Fixed Time Stability). The origin of system the (3) is called fixed time stable if it is globally finite time stable and the settling time function is bounded, i.e., ∃ Tmax > 0 : ∀ x0 ∈ Rn and ∀ t0 ∈ R≥0 , T (t0 , x0 ) ≤ Tmax . 3. Main results In this section, two definitions are introduced to highlight an entirely new and exciting property of the proposed class of nonautonomous systems. Definition 3 (Arbitrary Time Stable). The origin of system (3) is called Arbitrary time stable if (i) it is fixed time stable, (ii) ∃ Ta > 0, which depends on the known system parameters ϕ and which can be evaluated in advance for some given ϕ, (iii) it is possible to adjust Ta arbitrarily by making variations in system parameters ϕ under the assumption that such variations are permitted for the design purpose and (iv) for any given ϕ , either of the following can be established (a) Ta ≥ Ttf (Weak arbitrary time stable) (b) Ta = Ttf (Strong arbitrary time stable) where Ttf is the true fixed time. Definition 4 (Free-will Arbitrary Time Stable). The origin of the system (3) is called free-will arbitrary time stable if (i) it is fixed time stable,

A.K. Pal, S. Kamal, S.K. Nagar et al. / Automatica 112 (2020) 108710

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(ii) ∃ Ta > 0, which is independent of any system parameters and initial conditions and can be arbitrarily chosen in advance, and (iii) either of the following can be established (a) Ta ≥ Ttf (Free-will weak arbitrary time stable) (b) Ta = Ttf (Free-will strong arbitrary time stable) Remark 1. Free-will is a strong notion since it allows us to have ultimate control over the system and it makes the system trajectories to converge to the equilibrium point at a given time as per our own will without concerning to initial conditions and the system parameters. Hence, the nomenclature is coined as free-will. Remark 2. True fixed time is the actual or exact time in which the system trajectories converge to the origin. The following theorem provides a Lyapunov characterization for free-will arbitrary time stable systems. Theorem 1. Consider the system (3) and let D ⊂ Rn be a domain containing the equilibrium point x = 0. Let α1 (x) and α2 (x) be two continuous positive definite functions on D. Assume that there exist a real-valued continuously differentiable function V : I × D → R≥0 (I is a finite time interval, i.e., I = [t0 , tf ]) and a real number η ≥ 1 such that

Fig. 1. The plot of ζ with time.

1 ∥x(t)∥ ≤ λ− 1 (λ2 (∥x(t0 )∥)) establishes the uniform stability of the

origin (Khalil, 2002). Since the boundedness of the solution and uniform stability of the origin is ensured, let us proceed to prove the free-will arbitrary time stability. In the following, we prove the free-will strong arbitrary time stability, which will then imply the free-will weak arbitrary time stability, due to the comparison lemma in Khalil (2002). Suppose that V (t , x) satisfies the differential inequality V˙ ≤ −η(eV −1) , V (0, x0 ) = V (t0 , x0 ). Let ζ (t) be the solution of the eV (t −t) f

following differential equation

ζ˙ =

−η(eζ − 1) , eζ (tf − t)

(6)

• α1 (x) ≤ V (t , x) ≤ α2 (x), ∀ t ∈ I , ∀ x ∈ D \ {0}

(4a)

• V (t , 0) = 0, ∀ t ∈ I −η(eV − 1) • V˙ ≤ V ∀V ̸= 0, ∀t ∈ I , e (tf − t)

where η ∈ R≥1 . The solution to (6) is

(4b)

ζ = ln(C (tf − t)η + 1)

(4c)

then the equilibrium point x = 0 is Free-will weak arbitrary time stable and Ta = tf − t0 ≥ Ttf (Ttf is the true fixed time in which the system stabilizes and tf is the settling time moment which is independent of system parameters and initial conditions). Finally, if V˙ =

−η(eV − 1) eV (tf − t)

∀V ̸= 0, ∀t ∈ I ,

(5)

then the equilibrium point x = 0 is free-will strong arbitrary time stable and Ta = tf − t0 = Ttf . Proof. Let us first establish the uniform stability of the dynamical system (3). Using (4c) one can say that V˙ (t , x) ≤ 0 for all x and t ∈ I. Let us choose δ > 0 and β > 0 s.t. Bδ ⊂ D and β < min∥x∥=δ α1 (x). Then {x ∈ Bδ : α1 (x) ≤ β} is in the interior of Bδ . Let ωt ,β := {x ∈ Bδ : V (t , x) ≤ β} be a time dependent set. The set {x ∈ Bδ : α2 (x) ≤ β} is a subset of ωt ,β since α2 (x) ≤ β ⇒ V (t , x) ≤ β . Similarly, since V (t , x) ≤ β ⇒ α1 (x) ≤ β , therefore it is inferred that ωt ,β is a subset of {x ∈ Bδ : α1 (x) ≤ β}. Thus, overall one obtains {x ∈ Bδ : α2 (x) ≤ β} ⊂ ωt ,β ⊂ {x ∈ Bδ : α1 (x) ≤ β} ⊂ Bδ ⊂ D ∀t ≥ 0. Since V˙ (t , x) ≤ 0 on D ∀t0 ≥ 0 and ∀x0 ∈ ωt0 ,β , the solution starting at (t0 , x0 ) stays in ωt0 ,β ∀t ≥ t0 . Therefore, we can also deduce that any solution starting in {x ∈ Bδ : α2 (x) ≤ β} stays in ωt ,β and consequently in {x ∈ Bδ : α1 (x) ≤ β} ∀t ≥ t0 . Hence, the solution is bounded and defined ∀t ≥ t0 . Moreover, since V˙ ≤ 0, therefore V (t , x(t)) ≤ V (t0 , x(t0 )) ∀t ≥ t0 . For the given conditions, there exist class K functions λ1 and λ2 , defined on [0, δ] (see Khalil, 2002) s.t. λ1 (∥x∥) ≤ α1 (x) ≤ V (t , x) ≤ α2 (x) ≤ λ2 (∥x∥). Combining the preceding two inequalities: 1 −1 −1 ∥x(t)∥ ≤ λ− 1 (V (t , x(t))) ≤ λ1 (V (t0 , x(t0 ))) ≤ λ1 (λ2 (∥x(t0 )∥)). −1 Since λ1 ◦ λ2 is a class K function, therefore the inequality

(7)

where integration constant C is

ζ˙ =

eζ (t0 ) −1 . (t −t )η f

0

From (7), we have

−ηC (tf − t)η−1 . C (tf − t)η + 1

(8)

From (8), we see that when t = tf , ζ˙ = 0. Further, from (7), at t = tf , we have ζ = 0. Therefore, ∀t ≥ tf , ζ = 0 is maintained. The same result is valid for V (t , x) for the free-will strong arbitrary time. Therefore, Ta = tf − t0 = Ttf . This completes the proof. □ The graph of the function ζ that satisfies Eq. (6) with t0 = 0, ζ (t0 ) = 5, η = 2 and tf = 5 seconds is plotted in Fig. 1. 4. Free-will arbitrary time stabilization Consider the single-input–single-output affine in control system: x˙ = f (x) + g(x)ν

(9a)

y = h(x)

(9b)

where x ∈ Rn is the state, ν ∈ R is the control input, y ∈ R is the controlled output, and the functions f , g and h are sufficiently smooth in x. Assume that the system has relative degree n in x ∈ D ⊂ Rn i.e., Lg h(x) = · · · = Lg Lnf −2 h(x) = 0 and Lg Lfn−1 h(x) ≥ a > 0 ∀x ∈ D. Under this assumption the system (9a), (9b) can be written as (n)

(n−1)

y(n) = Lf h(x) + Lg Lf

h(x)ν.

(10)

Our goal is to design a time varying state feedback control

ν :=

1 (n−1)

Lg Lf

[ h(x)

−L(n) f h(x) + u(t , x)

]

(11)

where u(t , x) is some specific function of time t and x, to be designed later such that y and its derivatives up to order n are free-will arbitrary time stable.

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A.K. Pal, S. Kamal, S.K. Nagar et al. / Automatica 112 (2020) 108710

Theorem 2. Consider the system (10). Let ν be a time varying state feedback control law for (10) with ν (t , 0) = 0 and V (t , x) be a Lyapunov function satisfying (4a), (4b) with some continuous positive definite functions α1 (x) and α2 (x) on D ⊂ Rn . Then

ν=

⎧ ⎨

[

1 (n−1) Lg Lf h(x)

] −L(n) f h(x) + u(t , x) , if t0 ≤ t < tf

⎩0,

(12)

otherwise

stabilizes the origin of (10) within time tf − t0 , provided V˙ (t , x) ≤ −η(eV −1) , eV (tf −t)

∀t ∈ [t0 , tf ) and V˙ (t , x) = 0, otherwise. Here, tf is the

desired convergence time. Proof. Proof of the above theorem is done by the method of mathematical induction. The following subsections construct the proof. 4.1. First order system Let us consider the following first order system, x˙ = u, x(t0 ) = x0

(13)

where x ∈ R is the system state, u: R≥0 × R → R is a control, origin x = 0 is an equilibrium point of (13) and t0 ∈ R≥0 is the initial time. Let the control be

{ u=

−η(ex −1) ex (tf −t)

, if t0 ≤ t < tf

0,

(14)

otherwise

Fig. 2. State and control for first order system.

Taking the time derivative, we have z˙2 = x˙ 2 +

∂ψ1 ∂ψ1 x˙ 1 + . ∂ x1 ∂t

So, the transformed system is

Let us select the Lyapunov function V (x) = x2 ⇒ |x| =

√ V

(15)

V˙ = 2xx˙ =

−2η|x|(e − 1) −2ηx(e − 1) ≤ ex (tf − t) e|x| (tf − t)

Using (15) we have V˙ ≤

Let ξ =

√ V then time

which leads to free-will

arbitrary time dynamics given by ξ˙ ≤

Therefore, using

V˙ √ 2 V

≤

−η (e √

−η(eξ −1) . eξ (tf −t)

√

= 2x1 (z2 − ψ1 ) + 2z2 (u +

Then,

4.2. Second order system

V˙ 2 = −2x1 ψ1 − 2z2 ψ2 .

(16)

In the following, the concept of integrator backstepping is applied (see Khalil, 2002). Let us set the desired value x2d as x1

−η1 (e − 1) = −ψ1 , say. ex1 (tf − t)

(17)

To backstep, let us change the variable by z2 = x2 − x2d = x2 + ψ1 .

∂ψ1 ∂ψ1 (z2 − ψ1 ) + ) ∂ x1 ∂t

Let the control be

{ u=

−x1 − 0,

where,

∂ψ1 ∂t

ψ2 =

1 − x2 ∂ψ − ψ2 , if t0 ≤ t < tf ∂ x1 otherwise,

(21)

η2 (ez2 − 1) ez2 (tf − t)

(22)

η2 ≥ 1 .

(23)

From (20) we have

Let us consider the second order system,

x2d =

(20)

where V1 (x1 ) = is a Lyapunov function corresponding to the first order system x˙ 1 = u. Taking the time derivative of V2 , we have ∂ V1 x˙ 1 + 2z2 z˙2 = 2x1 x˙ 1 + 2z2 z˙2 V˙ 2 = ∂ x1

(6) and (7), one obtains ξ = 0 ∀ t ≥ tf . Further, since ξ = V , we have V = 0 ∀ t ≥ tf . Thus, from (15) we have x = 0 ∀ t ≥ tf . Therefore, with the given control, the system (13) represents a free-will weak arbitrary time stable system. The simulation of this first-order system (13) has been carried out with x(t0 ) = 5, η = 2 and tf = 7. Fig. 2 shows the convergence of the state and the corresponding control effort. It can be seen that the state converges to the origin within the desired time tf with a very nominal control effort requirement.

x˙ 1 = x2 , x˙ 2 = u.

(19)

x21

V −1) e V (tf −t)

derivative is ξ˙ =

∂ψ1 ∂ψ1 x˙ 1 + . ∂ x1 ∂t

V2 (x1 , z2 ) = V1 (x1 ) + z2 2 ,

|x|

√ √ −2η√ V (e V −1) . e V (tf −t) √

z˙2 = u +

Let us take the composite Lyapunov function as

The time derivative of the Lyapunov function is given by x

x˙ 1 = z2 − ψ1 ,

(18)

V2 ≥ x 1 2 ⇒

√

V2 ≥ |x1 |, V2 ≥ z2 2 ⇒

√

V2 ≥ |z2 |

From (23) we have V˙ 2 = −

2η1 x1 (ex1 − 1) ex1 (t

−

2η2 z2 (ez2 − 1)

− t) ez2 (tf − t) |x1 | 2η1 |x1 |(e − 1) 2η2 |z2 |(e|z2 | − 1) ≤− − e|x1 | (tf − t) e|z2 | (tf − t) √ √ V2 2(η1 + η2 ) V2 (e − 1) √ ≤− , using (24). e V2 (tf − t) f

(24)

A.K. Pal, S. Kamal, S.K. Nagar et al. / Automatica 112 (2020) 108710

5

The time differentiation of (29) leads to

∂ x3d ∂ x3d ∂ x3d x˙ 1 − z˙2 − . ∂ x1 ∂ z2 ∂t

z˙3 = x˙ 3 − x˙ 3d = u −

Therefore, the transformed dynamics is x˙ 1 = z2 − ψ1 , z˙2 = z3 − ψ2 − x1 ∂ x3d ∂ x3d ∂ x3d z˙3 = u − (z2 − ψ1 ) − (z3 − ψ2 − x1 ) − . ∂ x1 ∂ z2 ∂t Let the Lyapunov function be V3 = V2 + z3 2 .

(30)

It follows that V3 ≥ x 1 2 ⇒ 2

V3 ≥ z 3 ⇒

√ √

V3 ≥ |x1 |, V3 ≥ z2 2 ⇒

√

V3 ≥ |z2 |,

(31)

V3 ≥ |z3 |

The time differentiation of Eq. (30) leads to

∂ V2 ∂ V2 x˙ 1 + z˙2 + 2z3 z˙3 = 2x1 x˙ 1 + 2z2 z˙2 + 2z3 z˙3 ∂ x1 ∂ z2 ( ∂ x3d = −2x1 ψ1 + 2z2 z3 − 2z2 ψ2 + 2z3 u − (z2 − ψ1 ) ∂ x1 ) ∂ x3d ∂ x3d − (z3 − ψ2 − x1 ) − . ∂ z2 ∂t

V˙ 3 = Fig. 3. States and control for second order system.

Using the approach analogous to the first-order system, it can be said that V2 = 0 at some time t ≤ tf . Therefore, this control input makes the system (19) to achieve x1 = 0 and z2 = 0 within a desired time tf and consequently it also leads to the stabilization of the origin of system (16) within the desired time tf . Thus, with this control structure (21), the second-order system (16) becomes free-will arbitrary time stable system. The simulation of this second order system (12) has been carried out with x1 (t0 ) = 5, x2 (t0 ) = 2, η1 = η2 = 2.12 and tf = 5 seconds. Fig. 3 shows the convergence of the state to the origin and the corresponding control effort. Here, it can be seen that the origin is stabilized within a desired time tf with a very nominal control effort requirement.

Let the control be

⎧( ∂ x3d ⎪ ⎪ ⎨ −z2 + ∂ x1 (z2 − ψ1 ) u = + ∂ x3d (z3 − ψ2 − x1 ) + ∂ z2 ⎪ ⎪ ⎩ 0, where

x˙ 1 = x2 , x˙ 2 = x3 , x˙ 3 = u.

(25)

=−

2η1 x1 (ex1 − 1)

≤−

2η1 |x1 |(e

−

∂ψ1 ∂ψ1 x˙ 1 + . ∂ x1 ∂t

(27)

From (26) we have x2 = z2 − ψ1 ⇒ x˙ 1 = z2 − ψ1 . Treating x3 as control input and considering the results obtained in the case of second order system, the desired value of x3 is given by: x3d (x1 , z2 , t) = −x1 −

∂ψ1 ∂ψ1 − (z2 − ψ1 ) − ψ2 , ∂t ∂ x1

(28)

where ψ1 and ψ2 are as given in (17) and (22), respectively. To backstep, let us apply change of variables: z3 = x3 − x3d z˙2 = z3 − ψ2 − x1

|x1 |

− 1)

e|x1 | (tf − t)

(29)

(33)

ez2 (tf − t) 2η2 |z2 |(e

|z 2 |

−

≤−

−

2η3 z3 (ez3 − 1) ez3 (tf − t)

− 1)

e|z2 | (tf − t)

2η3 |z3 |(e|z3 | − 1) e|z3 | (tf − t) 2(η1 + η2 + η3 ) V3 (e

√

e

Taking time derivative, we have

˙ 1 = x3 + z˙2 = x˙ 2 + ψ

ex1 (tf − t)

2η2 z2 (ez2 − 1)

−

√

(26)

η3 ≥ 1

V˙ 3 = − 2x1 ψ1 − 2z2 ψ2 − 2z3 ψ3

As in the previous case, we take z2 = x2 − x2d = x2 + ψ1

(32)

otherwise,

η3 (ez3 − 1) ez3 (tf − t)

ψ3 =

) − ψ3 , if t0 ≤ t < tf

Then,

4.3. Third order system Let us consider the third order system

∂ x3d ∂t

V3

√ V3

− 1)

.

(tf − t)

Using a similar approach as in the previous cases, we observe that the origin of the system (25) can be stabilized within a time, tf . The simulation of the third order system (25) has been carried out with x1 (t0 ) = 3, x2 (t0 ) = 5, x3 (t0 ) = 1, η1 = η2 = η3 = 6 and tf = 6 seconds. Fig. 4 shows the convergence of the states to the origin and the corresponding control effort. It can be observed that the origin is stabilized within a desired time tf . In order to prove the free-will arbitrary time stabilization for an nth order system, for any n ∈ N, let us use the principle of mathematical induction. The case of the first order system works as the basis for the induction. Assume that there exists a control un which stabilizes the nth order system in a free-will arbitrary time. Then, for (n + 1)th order system, the composite Lyapunov function turns out to be Vn+1 = Vn + zn2+1 . The time derivative yields V˙ n+1 =

∂ Vn ∂ Vn ∂ Vn x˙ 1 + z˙2 + · · · + z˙n + 2zn+1 z˙n+1 . ∂ x1 ∂ z2 ∂ zn

6

A.K. Pal, S. Kamal, S.K. Nagar et al. / Automatica 112 (2020) 108710

disturbance, i.e., |d(t)| ≤ d0 . System (35) can be further reduced to i = 1, 2, . . . , n − 1

x˙ i = xi+1

(36)

x˙ n = u + d(t)

where ν := G−1 (x, t)(u−F(x, t)). System (36) is a convenient chain of integrators form representation for system (34). Designing a control u to stabilize system (36) suffices to show that a particular control ν exists which stabilizes system (34). To design robust free-will arbitrary time convergent controller, we follow the approach which is akin to the one given in Chalanga et al. (2015). Consider the system (34) which undergoes transformation to find a form given by (36). To undermine the effect of disturbance d(t), let the control u be made up of nominal control (unom ) and discontinuous control (udisc ). The udisc brings the sliding mode into the picture. Let us consider Integral sliding mode control (ISMC) which restricts the occurrence of the reaching phase, and the trajectory directly starts on the sliding phase (Edwards & Spurgeon, 1998). The sliding surface s ∈ R is assumed to be

∫

Fig. 4. States and control for third order system.

t

unom dτ .

s = xn − xn0 −

(37)

t0

Then the time derivative of the sliding surface is given by

After some simplification we obtain:

s˙ = x˙ n − unom = u + d(t) − unom

V˙ n+1 = 2x1 x˙ 1 + 2z2 z˙2 + · · · + 2zn z˙n + 2zn+1 z˙n+1

= unom + udisc + d(t) − unom

Then one can define control un+1 such that

= udisc + d(t)

V˙ n+1 = −2x1 ψ1 − 2z2 ψ2 · · · − 2zn ψn − 2zn+1 ψn+1 where ψn = V˙ n+1

−ηn (ezn −1) ezn (tf −t)

; ηn ≥ 1. This finally leads to √ √ −2(η1 + η2 + · · · + ηn+1 ) Vn+1 (e Vn+1 − 1) ≤ √ e Vn+1 (tf − t)

Then by Theorem 1 this (n + 1)th order system becomes freewill arbitrary time stable. Thus by mathematical induction, it is concluded that the same results hold ∀n ∈ N. Therefore, without loss of generality, it is established that the nth order system is free-will arbitrary time stabilizable with control un . This completes the proof. □ 5. Robust free-will arbitrary time stabilization It can be observed that the aforementioned control u (14), (21), (32) is a nominal stabilizing control that stabilizes the plant within the desired convergence time. The appearance of the disturbances through the input channel can be handled with the inclusion of sliding mode control (SMC) approach (Chalanga, Kamal, & Bandyopadhyay, 2015). Consider the system

σ˙ = F (σ , δ (t)) + G(σ , δ (t))ν, y = h(σ ),

(34)

where σ represents the states, y is the output, ν is the control input and δ (t) represents external matched perturbations or model uncertainties. Assume that the system has relative degree n in σ ∈ D ⊂ Rn . Then (34) can be transformed into the following normal form (see Isidori, 2013): x˙ i = xi+1 , i = 1, 2, . . . .n − 1 x˙ n = F(x, t) + G(x, t)ν + d(t)

(38)

From (37) one gets s(t0 ) = 0, i.e., the trajectory is initially on the sliding surface. The persistence of the sliding phase can be ensured by the inclusion of the additional condition s˙ = 0, i.e., udisc + d(t) = 0. Let us select udisc = −Ksgn(s) where K ≥ d0 and sgn(.) is the signum function. This results in [−Ksgn(s)] + d(t) = 0, which shows that the disturbance is being rejected right from the start. Therefore, the system behaves as if being free from the disturbance and the motion is governed by the unom part of the control. Thus the idea of designing a robust free-will arbitrary time convergent controller can be accomplished by selecting unom as a free-will arbitrary time stabilizable control (14), (21), (32). 6. An example Let us consider the magnetic suspension system (Khalil, 2002). The motion of the ball can be expressed mathematically as my¨ = −ky˙ + mg + F (y, i), where m is the mass of the ball, y ≥ 0 is the vertical (downward) position of the ball measured from a reference point (y = 0 when the ball is next to the coil), k is the viscous friction coefficient, g is the acceleration due to gravity, F (y, i) is the force generated by the electromagnet, and i is its electric current. The inductance of the ball is modelled as L L(y) = L1 + 1+y0/a where L1 , L0 and a are positive constants. The energy stored in the electromagnet is E(y, i) =

force F (y, i) =

∂E ∂y

L0 i2

= − 2a(1+y/a)2 . When the electric circuit of the

x˙ 1 = x2

where x ∈ D0 ⊂ D ⊂ Rn are the states, F(x, t) and G(x, t) are known nonlinear functions, d(t) represents bounded matched

Thus the

coil is energized with voltage V , we have V = φ˙ + iR, where R is the series resistance of the circuit and φ = L(y)i is the magnetic flux linkage. Assuming x1 = y, x2 = y˙ and x3 = i as the state variables and u = V as the control input, the state equations can be framed as:

(35)

y = x1 ,

1 L(y)i2 . 2

x˙ 2 = g − x˙ 3 =

k m

1 L(x1 )

x2 −

[

L0 ax23 2m(a + x1 )2 L0 ax2 x3

−Rx3 +

(a + x1 )2

] +u

A.K. Pal, S. Kamal, S.K. Nagar et al. / Automatica 112 (2020) 108710

7

We assume the following numerical data: m = 0.1 kg, k = 0.001 N/m/s, g = 9.81 m/s2 , a = 0.05 m, L0 = 0.01 H, L1 = 0.02 H and R = 1 . Our objective is to design timevarying state feedback control law to stabilize the ball at y = 0.05 m. We assume that the initial position of the ball is y = 0.09 m. The required control objective can be attained with the aforementioned free-will arbitrary time control. Let us set the desired time tf = 0.5 s. With the help of feedback linearization one can obtain:

( T (x) =

x1 , x2 , g −

k m

x2 −

)⊤

L0 ax23 2m(a + x1 )2

One can easily verify that T (x) is a diffeomorphism in D = {x1 + a > 0 and x3 > 0}. The change of variables q = T (x) transforms the system into q˙ 1 = q2 , q˙ 2 = q3 q˙ 3 = −

k m

q3 −

[ ] L1 x 2 x 3 − Rx − + u 3 mL(x1 )(a + x1 )2 a + x1 aL0 x3

Let the control be: u = Rx3 +

L1 x 2 x 3 a + x1

−

Fig. 5. States and control for magnetic suspension system.

mL(x1 )(a + x1 ) aL0 x3

2

[

k m

q3 + v

] References

Thus the system in q coordinate transforms to: q˙ 1 = q2 , q˙ 2 = q3 , q˙ 3 = v + d(t) where d(t) = 0.1 + 0.1 sin(4π t) represents matched disturbances and/or perturbations due to model uncertainties. For this system, v can be assumed to be consisting of two parts: Nominal control (vnom ) and Discontinuous control (vdisc ). vnom can be designed as a free-will arbitrary time control in a similar manner as given in Section 4.3. Thus using (32) we have:

vnom = −z2 +

∂ q3d ∂ q3d ∂ q3d (z2 − ψ1 ) + (z3 − ψ2 − q1 ) + − ψ3 , ∂ q1 ∂ z2 ∂t

where the variables on the right hand side are to be defined in terms of q1 , q2 and q3 following the similar approach as given in Section 4. And the discontinuous part vdisc = −Ksgn(s) is integral sliding mode control with the sliding surface s and gain K selected as per the discussion in 5. For simulations, K = 0.3 and η = 8 has been selected. To stabilize the ball at arbitrary position x1 = y = r, the q1 variable has to be replaced by q1 − r in the control law v . In an attempt to stabilize the ball at y = 0.05 m within time tf = 0.5 s the simulation results as depicted in Fig. 5 have been observed. The control is discontinuous as is evident from the chattering behaviour Fig. 5(b). The simulation results justify the fulfilment of the control design objective. 7. Conclusions A new dimension in the category of prespecified time stable systems, known as free-will arbitrary time stable system has been introduced. This was achieved by designing controllers with arbitrary convergence time. This control scheme is capable of stabilizing the origin of a system within a desired time independent of any system parameters. The control is simple and elegant. Robustness has also been incorporated with the help of sliding mode control. The effectiveness of this scheme is well illustrated through simulations for first, second and third-order systems. The given control scheme has also performed satisfactorily when implemented on a magnetic suspension system. With time being a vital factor, this control strategy would be of great importance in many industrial and engineering applications.

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A.K. Pal, S. Kamal, S.K. Nagar et al. / Automatica 112 (2020) 108710 Anil Kumar Pal received the B.E. in Electronics and Telecommunication engineering from Bhilai Institute of Technology, Durg, Chhattisgarh, India in 2008. He received M.Tech with specialization in Control Systems from Indian Institute of Technology (BHU), Varanasi, India in 2013. He is currently working towards his Ph.D. from the same institute. His research interests include Nonlinear Control, Sliding Mode Control, Contraction Analysis, Finite and Fixed Time Control Design.

Shyam Kamal received his Bachelor’s degree in Electronics and Communication Engineering from the Gurukula Kangri Vishwavidyalaya Haridwar, Uttrakhand, India in 2009, and Ph.D. in Systems and Control Engineering from the Indian Institute of Technology Bombay, India in 2014. From 2014 to 2016, he was with the Department of Systems Design and Informatics, Kyushu Institute of Technology, Japan as a Project Assistant Professor. Currently, he is an assistant professor in the Department of Electrical Engg., Indian Institute of Technology (BHU) Varanasi, India. He has published one monograph and 64 journal articles and conference papers. His research interests include the areas of fractional-order systems, contraction analysis, discrete and continuous higher-order sliding mode control. He has received excellence in Ph.D. thesis award from IIT Bombay in 2015 for his thesis "Sliding Mode Control of Fractional-order Systems" and INAE Young Engineer Award in 2019 for his research work. He is now an INAE Young Associate.

Shyam Krishna Nagar was born in Varanasi, India, on May 10, 1955. He studied at Institute of Technology (I.T) B.H.U and received the B.Tech and M.Tech degree in Electrical Engineering in 1976 and 1978 respectively. He completed Ph.D. degree in Electrical Engineering from University of Roorkee, Roorkee, India under Quality Improvement Programme (QIP). Dr. Nagar joined as lecturer at I.T, B.H.U in 1980 and promoted to Reader in 1993. He joined as professor of Electrical Engineering at I.T, B.H.U, in 2001 and is continuing as professor in the department of Electrical Engineering, IIT, BHU. His field of interest is model reduction, digital control and discrete event systems. He has supervised 5 Ph.D. thesis in the area of model reduction and controller design. Dr. Nagar has published 20 papers in regular journals and presented 40 papers in National and International Conferences. He was National Advisory Committee member for the National System Conference in 2008 (NSC-2008) and Technical Program Committee member for the Student’s Conference on Engineering and Systems (SCES-2014). He received Best Paper award at National System Conference (NSC-2011). He is life member of System Society of India and Fellow of Institution of Engineers.

Bijnan Bandyopadhyay received his B.E. degree in Electronics and Telecommunication Engineering from the University of Calcutta, Calcutta, India in 1978, and Ph.D. in Electrical Engineering from IIT Delhi, India in 1986. In 1987, he joined the Systems and Control Engineering group , IIT Bombay, India, as a faculty member, where he is currently a chair professor. In 1996, he was with the Lehrstuhl fur Eleck-trische Steuerung und Regelung, RUB, Bochum, Germany, as an Alexander von Humboldt Fellow. He was awarded Distinguished Visiting Fellow by the Royal Academy of Engineering, London in 2009 and 2012. Professor Bandyopadhyay is a Fellow of Indian National Academy of Engineering and Fellow of National Academy of Science. He has 400 publications which include monographs, book chapters, journal articles and conference papers. He has guided 36 Ph.D. theses at IIT Bombay. His research interests include the areas of multirate output feedback based discretetime sliding mode control, event-triggered sliding mode control and nuclear reactor control. Prof. Bandyopadhyay served as Co-Chairman of the International Organization Committee and as Chairman of the Local Arrangements Committee for the IEEE ICIT, Goa, India, in 2000. He also served as one of the General Chairs of IEEE ICIT conference, Mumbai, India in 2006. Prof. Bandyopadhyay has served as General Chair for IEEE International Workshop on VSSSMC, Mumbai, 2012. Prof. Bandyopadhyay is currently serving as Technical Editor of IEEE/ASME Transactions on Mechatronics, Associate Editor of IEEE Transaction on Industrial Electronics and Associate Editor IET Control Theory and Application. Prof. Bandyopadhyay has been awarded IEEE Distinguished Lecturer of IEEE IES society in 2019. Professor Bandyopadhyay is an IEEE Fellow. Leonid M. Fridman received the M.S. degree in mathematics from Kuibyshev (Samara) State University, Samara, Russia, in 1976, the Ph.D. degree in applied mathematics from the Institute of Control Science, Moscow, Russia, in 1988, and the Dr. Sc. degree in control science from Moscow State University of Mathematics and Electronics, Moscow, Russia, in 1998. From 1976 to 1999, he was with the Department of Mathematics, Samara State Architecture and Civil Engineering University, Samara, Russia. From 2000 to 2002, he was with the Department of Postgraduate Study and Investigations at the Chihuahua Institute of Technology, Chihuahua, Mexico. In 2002, he joined the Department of Control Engineering and Robotics, Division of Electrical Engineering of Engineering Faculty at National Autonomous University of Mexico (UNAM), Mexico City, Mexico. He also worked as an Invited Professor with more than 20 universities and research laboratories of Argentina, Australia, Austria, China, France, Germany, Italy, Israel, and Spain. He has authored and has been an Editor for 10 books and 17 special issues devoted to the sliding mode control. His research interest includes variable structure systems. He is an International Chair of INRIA(France) and High-Level Foreign Expert of the Secretary of Education of China for 2017–2021. Dr. Fridman was the recipient of a Scopus Prize as the best cited Mexican Scientists in Mathematics and Engineering 2010 and UNAM Prize as the best researcher in exact sciences in 2019.