Continuous Fixed-Time Convergent Regulator for

Accepted Manuscript

Continuous Fixed-Time Convergent Regulator for Dynamic Systems with Unbounded Disturbances Michael Basin, Pablo Rodriguez-Ramirez, Steven X. Ding, Tim Daszenies, Yuri Shtessel PII: DOI: Reference:

S0016-0032(18)30070-X 10.1016/j.jfranklin.2018.01.010 FI 3301

To appear in:

Journal of the Franklin Institute

Received date: Revised date: Accepted date:

28 April 2017 11 October 2017 7 January 2018

Please cite this article as: Michael Basin, Pablo Rodriguez-Ramirez, Steven X. Ding, Tim Daszenies, Yuri Shtessel, Continuous Fixed-Time Convergent Regulator for Dynamic Systems with Unbounded Disturbances, Journal of the Franklin Institute (2018), doi: 10.1016/j.jfranklin.2018.01.010

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ACCEPTED MANUSCRIPT 1

Continuous Fixed-Time Convergent Regulator for Dynamic Systems with Unbounded Disturbances

Abstract

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Michael Basin, Pablo Rodriguez-Ramirez, Steven X. Ding, Tim Daszenies, Yuri Shtessel

This paper presents a novel continuous fixed-time convergent control law for dynamic systems in the presence of unbounded disturbances. A continuous fixed-time convergent control is designed to drive all states of a multidimensional integrator chain at the origin for a finite pre-established (fixed) time, using a scalar input. The fixed-time convergence is established and the uniform upper bound of the settling time is computed. The designed control algorithm is applied to fixed-time stabilization of two mechatronic systems, a cart inverted pendulum and a single machine infinite bus turbo generator with main steam valve control.

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I. I NTRODUCTION Nowadays, design of finite-time convergent control laws and estimation of their convergence (settling) times has become an attractive and popular research area. The first results yielding convergence time estimates for control algorithms in two-dimensional systems, such as twisting [1] and super-twisting [2], can be found in [3] and [4]. Convergence time estimates for the super-twisting algorithm were obtained in [5], [6], and [7], based on an explicit Lyapunov function or a geometric approach. Another challenging problem is to design a continuous control law driving system states at the origin for a finite pre-established (fixed) settling time. A generalization of the super-twisting algorithm resulting in fixed-time convergence was given in [8]. The consistent study of fixed-time convergent control laws was initiated in [9]. A comprehensive survey summarizing existing results on finite and fixed-time convergence can be found in [10]. However, a continuous fixed-time convergent control law is not yet known for a multi-dimensional integrator chain subject to unbounded disturbances. The contribution of this paper is in designing a novel continuous fixed-time convergent control law for dynamic systems in the presence of unbounded disturbances. A critical advantage of the obtained continuous fixed-time convergent control law consists in assuring convergence of all system states to the origin for a fixed (pre-established finite) time for any initial condition, whereas a linear feedback or a control law using only terms with exponents less than one cannot guarantee this important feature: the corresponding convergence times are infinite or tend to infinity as initial conditions increase. The fixedtime convergence property means that the tracking problem for all system states can be solved for any unknown initial conditions for a pre-established (fixed) time. Furthermore, the obtained continuous fixedtime control is robust with respect to disturbances satisfying the Lipschitz condition, including unbounded ones. As already examined applications of continuous fixed-time convergent control laws, we can mention the DC motor stabilization problem treated in [11] and the angular rate commands tracking problem for an F-16 fighter [12]. In this paper, a continuous fixed-time convergent control is applied to stabilizing The authors thank the German Academic Exchange Service (DAAD) and the Mexican National Science and Technology Council (CONACyT) for financial support under Grants 57064906, 207608 and 250611. Michael Basin and Pablo Rodriguez-Ramirez are with Department of Physical and Mathematical Sciences, Autonomous University of Nuevo Leon, San Nicolas de los Garza, Nuevo Leon, Mexico [email protected], [email protected] Michael Basin is also with ITMO University, St. Petersburg, Russia. Steven X. Ding and Tim Daszenies are with Institute for Automatic Control and Complex Systems, University of Duisburg-Essen, Duisburg, Germany [email protected], [email protected] Yuri Shtessel is with Department of Electrical and Computer Engineering, University of Alabama in Huntsville, United States

[email protected]


II. P ROBLEM S TATEMENT Consider a multi-dimensional chain of integrators

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two mechatronic systems, a cart inverted pendulum and a single machine infinite bus turbo generator with main steam valve control. The paper is organized as follows. The control problem statement is given in Section 2. Section 3 designs as a novel continuous fixed-time convergent control law driving all states of a multi-dimensional integrator chain at the origin. Section 4 and 5 apply the developed technique to fixed-time stabilization of a cart inverted pendulum and a single machine infinite bus turbo generator, respectively. Section 6 concludes this paper. Proof of Theorem 1 is given in Appendix. The conference version of this paper can be found in [13]. This journal version includes the proof of the main result, Theorem 1, and a case study of turbo generator.

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x˙ 1 (t) = x2 (t), x1 (t0 ) = x10 , x˙ 2 (t) = x3 (t), x2 (t0 ) = x20 , .. .. .. .. . . . . 00 x˙ n (t) = u (t) + ζ(t), xn (t0 ) = xn0 ,

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where x(t) = [x1 (t), . . . , xn (t)] ∈ Rn is the vector of system states, u00 (t) ∈ R is the control input, and ζ(t) is an unbounded external disturbance satisfying the Lipschitz condition with a constant L, so that its ˙ is uniformly bounded by L almost everywhere. derivative ζ(t) The control objective is to design continuous fixed-time control law that drives all states of the system (1) at the origin. The fixed-time convergence is understood in the sense of the following definition [12]. Definition 1. A dynamic control system χ(t) ˙ = f (χ(t), v(t), t),

χ(t0 ) = χ0 ,

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where χ(t) ∈ Rn , v(t) ∈ Rm , is called fixed-time convergent to the origin, if there exists a time moment T such that the system state χ(t) ∈ Rn is equal to zero, χ(t) = 0, for all t ≥ T , starting from any initial condition χ0 ∈ Rn . The next section presents continuous fixed-time control design for the system (1) and an upper estimate for the corresponding fixed convergence time.

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III. C ONTROL D ESIGN Consider the following control input for the system (1) u00 (t) = u1 (t) + u2 (t),

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where the control law u1 (t) is given by u1 (t) = v1 (t) + v2 (t) + · · · + vn (t) + w1 (t) + w2 (t) + · · · + wn (t),

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vi (t) = −ki | xi (t) |γi sign(xi (t)), wi (t) = −Ki | xi (t) |βi sign(xi (t)), i = 1, . . . , n.

Here, the exponents γi and βi , i = 1, . . . , n, are selected in accordance with [14]: γi ∈ (0, 1), i = 1, . . . , n, satisfy the recurrent relations γi−1 = γi γi+1 /(2γi+1 − γi ), i = 2, . . . , n, γn+1 = 1 and γn = γ; βi > 1, i = 1, . . . , n satisfy the recurrent relations βi−1 = βi βi+1 /(2βi+1 − βi ), i = 2, . . . , n, βn+1 = 1 and βn = β, where γ belongs to an interval (1 − , 1) and β belongs to an interval (1, 1 + 1 ), for sufficiently


small > 0 and 1 > 0. Control gains ki and Ki , i = 1, . . . , n, are assigned such that sn +kn sn−1 +· · ·+k1 and sn + Kn sn−1 + · · · + K1 are Hurwitz polynomials. On the other hand, the control law u2 (t) is assigned as Z t 1/2 p sign(s(τ ))dτ. (4) u2 (t) = −λ1 | s(t) | sign(s(t)) − λ2 | s(t) | sign(s(t)) − α t0

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Here, the control gains λ1 , λ2 , α > 0, the sliding surface s(t) is defined as s(t) = xn (t) − ϑ(t), ˙ = u00 (t) − u2 (t), and p > 1. The resulting closed-loop system (1) with a control input (2) takes the ϑ(t) form

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x˙ 1 (t) = x2 (t), x1 (t0 ) = x10 , x˙ 2 (t) = x3 (t), x2 (t0 ) = x20 , .. .. .. .. . . . . x˙ n−1 (t) = xn (t), xn−1 (t0 ) = x(n−1)0 , x˙ n (t) = v1 (t) + v2 (t) + · · · + vn (t) + w1 (t) + w2 (t) + · · · + wn (t) −λ1 | s(t) |1/2 sign(s(t)) − λ2 | s(t) |p sign(s(t)) + y(t), xn (t0 ) = xn0 , y(t) ˙ = −αsign(s(t)) + ξ(t), y(t0 ) = 0,

where the disturbance ξ(t) = dζ(t)/dt exists almost everywhere and is bounded by the constant L. Let symmetric positive definite matrices P and P1 satisfy Lyapunov equations P A + AT P = −Q,

symmetric positive definite matrices, the matrices A and  1 0 ... 0 0 1 ... 0  .. .. ..  .. . . . .   0 0 ... 1 

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respectively, where Q, Q1 ∈ Rn×n are certain A1 are defined as  0  0  . A=  ..  0 −k1

P1 A1 + AT1 P1 = −Q1 ,

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and



  A1 =   

−k2 −k3 . . .

0 1 0 0 0 1 .. .. .. . . . 0 0 0 −K1 −K2 −K3

... ... .. . ... ...

−kn

0 0 .. . 1 −Kn



  ,  

λmin (Q) and λmin (Q1 ) are the minimum eigenvalues of the matrices Q and Q1 , respectively, and λmax (P ) and λmax (P1 ) are the maximum eigenvalues of the matrices P and P1 , respectively. The main result for the fixed-time convergent control law (2) is given as follows. Theorem 1. Consider a dynamic system (5) in the presence of a disturbance ξ(t) bounded by a constant L. Then, both states x(t) and y(t) converge to the origin uniformly in fixed time 1 λρmax (P ) + r0 ρ r1 σΥσ  1/2 1 2 1 + 1 + + p−1 λ2 (p − 1) λ1 m M1 −

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Tf ≤

h(λ1 ) λ1



 ,

ACCEPTED MANUSCRIPT 4 min (Q) min (Q1 ) , σ = β−1 , r0 = λλmax , r1 = λλmax , Υ ≤ λmin (P1 ) is a positive number, > 0, where ρ = 1−γ γ β (P ) (P1 ) 1/3 M = α + L, m = α − L, h(λ1 ) = 1/λ1 + (2e/mλ1 ) , and e is the base of natural logarithms, provided that the following conditions hold for control gains: α > L, λ1 h−1 (λ1 ) > M . The minimum value of 1 Tf () is reached for = (λ1 /λ2 ) p+1/2 .

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Proof of this theorem is given in the Appendix. The next sections present applications of the designed fixed-time convergent control law to stabilization problems for a cart inverted pendulum and a single machine infinite turbo generator with main steam valve control. IV. C ONTINUOUS F IXED -T IME R EGULATOR FOR C ART I NVERTED P ENDULUM The inverted pendulum system is governed [15] by the following nonlinear second-order differential equations ¨ + C Φ˙ − M1 lS g sin Φ + M1 lS r¨ cos Φ = 0, ΘΦ (7) ¨ cos Φ − (Φ) ˙ 2 sin Φ) = F. M r¨ + Fr r˙ + M1 lS (Φ

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Here, r(t) represents the cart position on the horizontal axis with respect to a fixed point, Φ(t) is the ˙ rotational angle of the pendulum, r˙ is the cart velocity, Φ(t) is the angular velocity, M0 is the cart mass, M1 is the pendulum mass, Fr is the cart friction coefficient, C is the pendulum friction coefficient, lS is the pendulum length, ΘS is the pendulum inertia moment, F is the force applied to the cart, Θ = ΘS + M1 lS2 , and M = M0 + M1 . Upon introducing the state vector T ˙ z(t) = [z1 (t), z2 (t), z3 (t), z4 (t)]T = [r(t), Φ(t), r(t), ˙ Φ(t)]

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and the control input u(t) = F , a linear model is obtained by linearizing the nonlinear system at the operating point z1 = Ω1 and z2 = z3 = z4 = 0 corresponding to the unstable equilibrium of the pendulum at an arbitrarily fixed cart position z(t) ˙ = Az(t) + Bu(t), (9) where

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 0 0 1 0  0 0 0 1   A=  0 a32 a33 a34  , 0 a42 a43 a44

 0  0   B=  b3  . b4 

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2 2 2 2 2 Here, a32 = −(N 2 g)/N01 , a33 = −(ΘFr )/N01 , a34 = (N C)/N01 , a42 = (M N g)/N01 , a43 = (N Fr )/N01 , 2 2 2 2 2 a44 = −(M C)/N01 , b3 = Θ/N01 , b4 = −N/N01 . N = M1 lS , and N01 = ΘM − N . Numerical values of the system coefficients calculated according to the technical specifications given in Table I are: a32 = −0.88 m/s2 , a33 = −1.915 1/s, a34 = 0.0056 m/s, a42 = 21.473 1/s2 , a43 = 3.85 1/(ms), a44 = −0.136 1/s, a45 = −0.08968 rad, b3 = 0.30882 1/Kg, b4 = −0.62032 1/(Kg m). Finally, the canonical form of the linear system (9) is obtained using the transformation

x(t) = T −1 z(t) =

−6.0854 0 0 0

0.0385 0.0010 −6.0854 0

0.3088 −0.6203 0.0385 0.0010

0 0 0.3088 −0.6203

!−1

z(t),


Value 3.2 0.329 3.529 0.44 0.072 0.1446 0.23315 6.2 0.009 Table I

Unit Kg Kg Kg m Kgm2 Kgm Kg2 m2 Kg/s Kgm2 /s

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Constant M0 M1 M lS Θ N 2 N01 Fr C

taking into account numerical values of the parameters aij , i, j = 2, 3, 4 and bi , i = 3, 4. The resulting canonical form takes the form = = = =

x2 (t), x3 (t), x4 (t), 37.7328x2 (t) + 21.2341x3 (t) − 2.0510x4 (t) + u(t) + ζ(t).

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x˙ 1 (t) x˙ 2 (t) x˙ 3 (t) x˙ 4 (t)

Here, ζ(t) is an unbounded external disturbance satisfying the Lipschitz condition with a constant L, so ˙ is uniformly bounded by L almost everywhere. that its derivative ζ(t) The scalar control input u(t) is split into two terms, u(t) = u0 (t) + u00 (t), where u0 (t) = −37.7328x2 (t) − 21.2341x3 (t) + 2.0510x4 (t).

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As a result, the system (10) is represented as an 4-dimensional integrator

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x˙ 1 (t) x˙ 2 (t) x˙ 3 (t) x˙ 4 (t)

= = = =

x2 (t), x3 (t), x4 (t), u00 (t) + ζ(t).

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The control law u00 (t) in (11) is given according to (2) as follows:

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u00 (t) = −k1 | x1 (t) |γ1 sign(x1 (t)) − k2 | x2 (t) |γ2 sign(x2 (t)) −k3 | x3 (t) |γ3 sign(x3 (t)) − k4 | x4 (t) |γ4 sign(x4 (t)) − K1 | x1 (t) |β1 sign(x1 (t)) −K2 | x2 (t) |β2 sign(x2 (t)) − K3 | x3 (t) |β3 sign(x3 (t)) − K4 | x4 (t) |β4 sign(x4 (t)) Z t 1/2 p −λ1 | s(t) | sign(s(t)) − λ2 | s(t) | sign(s(t)) − α sign(s(τ ))dτ.

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t0

Here, the exponents are assigned as γ4 = 19/20, γ3 = 19/21, γ2 = 19/22, γ1 = 19/23, β4 = 21/20, β3 = 21/19, β2 = 21/18, β1 = 21/17, and p = 3/2. The control gains are set to k1 = K1 = 1, k2 = K2 = 3, k3 = K3 = 3, k4 = K4 = 3, λ1 = 5, λ2 = 1, and α = 20. The matrices Q and Q1 are chosen as four-dimensional identity matrices, with all eigenvalues equal to 1. Accordingly, P = P1 =   3 4 2.5 0.5  4 9.5 7 1.5    , λ (P ) = λmin (P1 ) = 0.2194, λmax (P ) = λmax (P1 ) = 18.1729, ρ = 0.0526,  2.5 7 9 2.5  min 0.5 1.5 2.5 1 σ = 0.0476, r0 = r1 = 0.055. The value of is calculated as = 2.2361. The external disturbance is assigned as ζ(t) = 1 + 0.1 sin(t) + 0.01 sin(1000t). The simulation step is set to 10−3 seconds. The examined initial conditions [x1 (0), x2 (0), x3 (0), x4 (0)] and the corresponding convergence times are given


x1 (0), x2 (0), x3 (0), x4 (0) 0, π/6, 0, 0 10, π/2, 10, 10 100, π, 100, 100 103 , 4π/3, 103 , 103 104 , 3π/2, 104 , 104 105 , 2π, 105 , 105

Convergence time via simulation (sec) 50 70 80 92 100 110 Table II

Convergence time estimate (sec) 827.79 827.79 827.79 827.79 827.79 827.79

Rate 16.56 11.83 10.35 9 8.28 7.52

0.4

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x1(t) x (t) 2

x3(t)

0.2

x (t) 4

0.1 0 −0.1 −0.2 −0.3

0

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Canonical form states

0.3

100

150

Time (sec)

Figure 1. States [x1 (t), x2 (t), x3 (t), x4 (t)] in the entire simulation interval.

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in Table II. The upper estimate of the fixed convergence time is calculated using (6) as Tf = 827.79 seconds. The simulation data in Table II show that the actual finite convergence times are about 10–15 times less than the convergence time upper estimate obtained in Theorem 1. The simulation graphs corresponding to the initial values [x1 (0) = 0.2, x2 (0) = 0.1, x3 (0) = 0.1, x4 (0) = −0.1] are presented in Figs. 1–7. Figure 1 shows the time history of the state trajectories [x1 (t), x2 (t), x3 (t), x4 (t)] of the system (11) in the entire simulation interval. Figure 2 presents their zoomed time histories around the convergence time separately for each variable. Figure 3 shows the complete control law u(t) = u0 (t) + u00 (t) for the canonical form system (10) against the disturbance ζ(t). Figure 4 demonstrates the control law components u0 (t) and u00 (t) against the disturbance ζ(t). Figure 5 shows the control law subcomponents u1 (t) and u2 (t). The simulation graphs corresponding to the states of the linearized system (9) are presented in Fig. 6 and 7, which display the state trajectories of [z1 (t), z2 (t), z3 (t), z4 (t)] in the entire simulation interval and zoomed around the convergence time separately for each variable, respectively.

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V. C ONTINUOUS F IXED -T IME R EGULATOR FOR S INGLE M ACHINE I NFINITE B US T URBO G ENERATOR W ITH M AIN S TEAM VALVE C ONTROL Consider the single machine infinite bus turbo generator with main steam valve control governed by the following nonlinear differential equations [16], [17]: δ˙ = ω − ω0 , ω0 Eq0 VS D ω0 ω0 ω˙ = − (ω − ω0 ) + PM L0 − sin(δ) + PH , (13) 0 H H HXdΣ H 1 CH CH P˙H = − PH + Pm0 + u + ζ(t), THΣ THΣ THΣ where δ and ω is the angle and angular velocity of the generator rotor, respectively, Pm is the mechanical power of the prime motor, H is the moment of inertia of the generator rotor; D is the damping coefficient,


−6

1

x 10

x1(t)

0.5

0

−1

40

60

80 100 Time (sec)

60

80 100 Time (sec)

−6

1

x 10

0

−0.5

−1

40 −5

x 10

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140

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0

−0.5

120

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x3(t)

0.5

140

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x2(t)

0.5

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−0.5

40

60

80 100 Time (sec)

120

140

60

80 100 Time (sec)

120

140

−5

x 10

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1

x4(t)

0.5

0

−0.5

−1

40

Figure 2. States [x1 (t), x2 (t), x3 (t), x4 (t)] zoomed around the convergence time.


Control and Disturbance

10 u(t)=u'(t)+u''(t) Disturbance

5

0

-10

0

50

100

Time (sec)

8

u’’(t) u’(t) Disturbance

6 4 2 0 −2

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Figure 3. The complete control law u(t) = u0 (t) + u00 (t) and disturbance ζ(t).

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−4

−8

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−6 0

50

100

150

Time (sec)

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Figure 4. The control law components u0 (t) and u00 (t) and disturbance ζ(t).

2

u 1 (t) u 2 (t)

Control

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0

-1

-2

0

50

100

Time (sec)

Figure 5. The control law subcomponents u1 (t) and u2 (t) according to (2).

150


1.5

0.5 0 −0.5

z (t) 1

−1

z2(t)

−1.5

z3(t) z4(t)

−2 −2.5

0

50

100 Time (sec)

Figure 6. States [z1 (t), z2 (t), z3 (t), z4 (t)] in the entire simulation interval.

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Constant Value H 10 δ0 57 degrees ω0 314 deg/s Pm0 0.8 THΣ 0.4 CM L 0.7 CH 0.3 VS 1 Eq0 1.08 D 5 0 XdΣ 0.9 Table III

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Linearized system states

1

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Eq0 is the generator q axis transient electric potential, VS is the infinite bus voltage, THΣ is the equivalent 0 time constant of the high-pressure cylinder, XdΣ is the equivalence reactance between the generator and the infinite bus power system, CM L is the power distribution coefficient of the low-pressure cylinder, CH is the power distribution coefficient of the high-pressure cylinder, PH is the mechanical power of the high-pressure cylinder, and u is the steam valve control input. The technical specifications of the single machine infinite turbo generator system can be found in Table III. To obtain the state equations in the backstepping form, the new system states are introduced as x1 = δ − δ0 , x2 = ω − ω0 and x3 = PH − CH Pm0 . Hence, the system (13) can be written as x˙ 1 = x2 , x˙ 2 = θx2 + a0 x3 + b0 − k sin(x1 + δ0 ), 1 CH x˙ 3 = − x3 + u + ζ(t), THΣ THΣ

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ω E0 V

0 q S D where a0 = ωH0 , b0 = (CH − CM L ) ωH0 Pm0 , k = HX and θ = − H . Upon introducing another state 0 dΣ variable, x03 = x˙ 2 , the system (14) is transformed into the integrator chain for the states x1 , x2 , and x03 :

x˙ 1 = x2 , x˙ 2 = x03 ,

−1 (x03 − θx2 − b0 + k sin(x1 + δ0 ))+ THΣ a0 CH + u + ζ(t) − k cos(x1 + δ0 )x2 . THΣ

x˙ 03 = θx03 + a0

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−5

1

x 10

z1(t)

0.5

0

−1

40

60

80 100 Time (sec)

60

80 100 Time (sec)

−5

1

x 10

0

−0.5

−1

40 −5

x 10

CE −1

140

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0

−0.5

120

PT

z3(t)

0.5

140

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z2(t)

0.5

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−0.5

40

60

80 100 Time (sec)

120

140

60

80 100 Time (sec)

120

140

−5

x 10

AC

1

z4(t)

0.5

0

−0.5

−1

40

Figure 7. States [z1 (t), z2 (t), z3 (t), z4 (t)] zoomed around the convergence time.


x1 (0), x2 (0), x03 (0) π/3, 1, 1 π/2, 10, 10 π, 100, 100 4π/3, 103 , 103 3π/2, 104 , 104 2π, 105 , 105

Convergence time via simulation (sec) 25 28 30 33 35 40 Table IV

Convergence time estimate (sec) 216.32 216.32 216.32 216.32 216.32 216.32

Rate 8.65 7.72 7.21 6.55 6.18 5.4

v1 =

THΣ v CH a0

and

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The control input u(t) is represented as u(t) = v1 (t) + v2 (t), where

THΣ (−θx03 + k cos(x1 + δ0 )x2 C H a0 1 + (x0 − θx2 − b0 + k sin(x1 + δ0 )). THΣ a0 3

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v2 =

As a result, the system (15) is reduced to a 3-dimensional integrator x˙ 1 = x2 , x˙ 2 = x03 , x˙ 03 = v + ζ(t).

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˙ is bounded The unbounded disturbance is assigned as ζ = t + sin(t) + 0.01 sin(1000t), so that ξ(t) = ζ(t) by L = 12. The control law v in (16) is given according to (2) as follows: v(t) = u0 (t) + u00 (t)

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where

PT

u0 (t) = −k1 | x1 (t) |γ1 sign(x1 (t)) − k2 | x2 (t) |γ2 sign(x2 (t)) −k3 | x03 (t) |γ3 sign(x03 (t)) − K1 | x1 (t) |β1 sign(x1 (t)) −K2 | x2 (t) |β2 sign(x2 (t)) − K3 | x03 (t) |β3 sign(x03 (t))

CE

and

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00

1/2

u (t) = λ1 | s(t) |

p

sign(s(t)) − λ2 | s(t) | sign(s(t)) − α

Z

(18)

t

sign(s(τ ))dτ.

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t0

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Here, the exponents are assigned as γ3 = 19/20, γ2 = 19/21, γ1 = 19/22, β3 = 21/20, β2 = 21/19, β1 = 21/18 and p = 3/2. The control gains are set to k1 = K1 = 1, k2 = K2 = 3, k3 = K3 = 3, λ1 = 5, λ2 = 1 and α = 15. The simulation step is equal to 10−3 seconds. The examined initial conditions [x1 (0); x2 (0); x03 (0)] and the corresponding simulation times are given in Table IV for the control law (17). The upper estimate of the fixed convergence time calculated using (6) is Tf = 216.3267 seconds. The simulation data in Table IV show that the actual fixed convergence time exceeds no more than 9 times the convergence time upper estimate obtained using Theorem 1. The simulation graphs for state variables [x1 (t), x2 (t), x03 (t)] corresponding to the initial values [x1 (0) = π/3; x2 (0) = 1; x03 (0) = 1] are presented in Fig. 8. The simulation graphs for the original states [δ(t), ω(t); PH (t)] are given in Fig. 9. Figure 10 shows the complete control law u(t) = u0 (t) + u00 (t) in the entire simulation interval [0, 30]. Figure 11 demonstrates the control law components u0 (t) and u00 (t) against the disturbance ζ(t).

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Figure 8. The single machine infinite bus turbo generator system states [x1 (t), x2 (t), x03 (t)] in the entire simulation interval.

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Figure 9. The single machine infinite bus turbo generator system states [δ(t), ω(t), PH (t)] in the entire simulation interval.

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Figure 11. The control law components u0 (t) and u00 (t) and disturbance ζ(t) in the entire simulation interval.

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VI. C ONCLUSIONS This paper presented a novel continuous fixed-time convergent control law driving all states of a multidimensional chain of integrators at the origin using a scalar control input for a finite pre-established (fixed) time. The designed control algorithm was employed for stabilizing two mechatronic systems, a cart inverted pendulum and a single machine infinite bus turbo generator with main steam valve control. The cart inverted pendulum was linearized at the unstable upright equilibrium and then transformed into a 4-dimensional chain of integrators. Using the designed continuous fixed-time convergent control law, the cart inverted pendulum was stabilized exactly at the origin. Similarly, the single machine infinite bus turbo generator was linearized at an operating point, transformed into a 3-dimensional chain of integrators, and stabilized exactly at the equilibrium in the presence of an unbounded disturbance. Conducted numerical simulations confirmed validity of the theoretical results. An extension of the presented results to dynamic systems with partially available states is expected.

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VII. A PPENDIX : P ROOF OF T HEOREM 1 ˙ Consider the dynamics of the sliding variable s(t) defined in (4) as s(t) = xn (t) − ϑ(t), ϑ(t) = 00 u (t) − u2 (t), where u(t) is obtained as (2) and xn (t) satisfies the last equation in (1). The derivative s(t) ˙ satisfies the relation

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˙ = u00 (t) + ζ(t) − u00 (t) + u2 (t) = u2 (t) + ζ(t) = s(t) ˙ = x˙ n (t) − ϑ(t) Z t 1/2 p = −λ1 | s(t) | sign(s(t)) − λ2 | s(t) | sign(s(t)) − α sign(s(τ ))dτ + ζ(t). t0

In accordance with [7], [12], the last equation provides fixed-time convergence of both, s(t) and s(t), ˙ to the origin for a time no greater than   1 21/2  1  , + 1+ T1 = 1) λ2 (p − 1)p−1 λ1 m M1 − h(λ λ1

independently of initial conditions for s(t). This expression corresponds to the last term in formula (6). In addition, since s(t) ˙ = 0 is reached in finite time, it yields 0 = u2 (t)+ζ(t), t ≥ T1 , i.e., u2 (t) = −ζ(t) holds for all t ≥ T1 . Next, substituting u00 (t) = u1 (t) + u2 (t) defined by (2) into (1) compensates for the


disturbance ζ(t) and results in the equations x˙ 1 (t) = x2 (t), x1 (t0 ) = x10 , x˙ 2 (t) = x3 (t), x2 (t0 ) = x20 , .. .. .. .. . . . . x˙ n−1 (t) = xn (t), xn−1 (t0 ) = x(n−1)0 , x˙ n (t) = u1 (t) = v1 (t) + v2 (t) + · · · + vn (t) + w1 (t) + w2 (t) + · · · + wn (t) xn (t0 ) = xn0 .

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(P ) 1 + r1 σΥ Fixed-time convergence of the system (20) for a time no greater than T2 = λmax σ follows r0 ρ from Theorem 2 in [11]. The expression for T2 corresponds to the first two terms in formula (6). Finally, summing up the convergence time upper estimates T1 and T2 yields the complete formula (6), Tf = T1 + T2 , which presents the settling time upper estimate for the system (5), independently of its initial conditions, in the presence of a disturbance ξ(t). Theorem 1 is proven.

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[1] A. Levant, “Sliding order and sliding accuracy in sliding mode control,” International Journal of Control, vol. 58, no. 6, pp. 1247–1263, 1993. [2] A. Levant, “Robust exact differentiation via sliding mode technique,” Automatica, vol. 34, no. 3, pp. 379–384, 1998. [3] A. Polyakov and A. Poznyak, “Lyapunov function design for finite-time convergence analysis: Twisting controller for second-order sliding mode realization,” Automatica, vol. 45, no. 2, pp. 444–448, 2009. [4] A. Polyakov and A. Poznyak, “Reaching time estimation for super-twisting second order sliding mode controller via Lyapunov function designing,” Automatic Control, IEEE Transactions on, vol. 54, pp. 1951–1955, Aug 2009. [5] J. Moreno, “Lyapunov approach for analysis and design of second order sliding mode algorithms,” in Sliding Modes after the First Decade of the 21st Century (L. Fridman, J. Moreno, and R. Iriarte, eds.), vol. 412 of Lecture Notes in Control and Information Sciences, pp. 113–149, Springer Berlin Heidelberg, 2012. [6] J. Moreno and M. Osorio, “Strict Lyapunov functions for the super-twisting algorithm,” Automatic Control, IEEE Transactions on, vol. 57, pp. 1035–1040, April 2012. [7] Y. Shtessel, C. Edwards, L. Fridman, and A. Levant, Sliding Mode Control and Observation. Springer, 2014. [8] E. Cruz-Zavala, J. Moreno, and L. Fridman, “Uniform robust exact differentiator,” Automatic Control, IEEE Transactions on, vol. 56, pp. 2727–2733, Nov 2011. [9] A. Polyakov, “Nonlinear feedback design for fixed-time stabilization of linear control systems,” Automatic Control, IEEE Transactions on, vol. 57, pp. 2106–2110, Aug 2012. [10] A. Polyakov and L. Fridman, “Stability notions and Lyapunov functions for sliding mode control systems,” Journal of the Franklin Institute, vol. 351, no. 4, pp. 1831–1865, 2014. Special Issue on 2010-2012 Advances in Variable Structure Systems and Sliding Mode Algorithms. [11] M. Basin, Y. Shtessel, and F. Aldukali, “Continuous finite- and fixed-time high-order regulators,” Journal of The Franklin Institute, vol. 353, no. 18, pp. 5001–5012, 2016. [12] M. Basin, C. B. Panathula, and Y. Shtessel, “Continuous second-order sliding mode control: convergence time estimation,” in Proceedings of the 54th IEEE Conference on Decision and Control, pp. 5408–5413, 2015. [13] M. Basin, P. Rodriguez-Ramirez, S. X. Ding, T. Daszenies, and Y. Shtessel, “Continuous fixed-time control for cart inverted pendulum stabilization,” in Proceedings of the 55th IEEE Conference on Decision and Control, pp. 6440–6445, 2016. [14] S. P. Bhat and D. S. Bernstein, “Geometric homogeneity with applications to finite-time stability,” Mathematics of Control, Signals and Systems, vol. 17, pp. 101–127, Jun 2005. [15] S. X. Ding, Model-Based Fault Diagnosis Techniques: Design Schemes, Algorithms and Tools. London: Springer, 2013. [16] N. Jiang, B. Liu, and Y. W. Jing, “Nonlinear steam valve adaptive controller design for the power systems,” International Journal of Intelligent Control and Automation, vol. 42, pp. 31–37, 2011. [17] Y. Wan and J. Zhao, “Extended backstepping method for single-machine infinite-bus power systems with SMES,” IEEE Transactions on Control Systems Technology, vol. 21, no. 3, pp. 915–923, 2013.