Second Order Sliding Mode Output Feedback Control: Impulsive Gain ...
51st IEEE Conference on Decision and Control December 10-13, 2012. Maui, Hawaii, USA
Second order sliding mode output feedback control: impulsive gain and extension with adaptation Antonio Estrada, Franck Plestan Abstract— This paper proposes a discontinuous-impulsive output feedback which is a sampled based second order sliding mode controller. First, constructive convergence conditions are established for the controller, in the unperturbed and perturbed case. An adaptation mechanism is based on the aforementioned convergence conditions and allows to reduce gain magnitude.
I. I NTRODUCTION Sliding mode (SM) and high order sliding modes (HOSM) are nonlinear control strategies that provides robustness with respect to uncertainties and perturbations. The main common properties of SMC/HOSM control laws are robustness (insensitivity) to the bounded disturbances matched by control, and finite convergence time. The main drawback of the The main drawback of HOSM control laws is the necessity of high order derivatives of the sliding variable [5], [3], which are obtained using HOSM differentiators [5]. Differentiation yields a performance degradation of the controlled system due to the presence of measurement noise. Then, there is a real interest to propose high order sliding mode controllers with a reduced number of time derivatives of the sliding variable. Concerning the second order sliding mode controllers, a popular second order sliding mode output feedback is the supertwisting algorithm [5]. In fact, this controller allows the establishment of a second order sliding mode with respect to the sliding variable, in a finite time, by using only the measurement of the sliding variable. However, this algorithm is intended to be applied to systems whose relative degree of the sliding variable equals 1. An alternative strategy, which do not have the previous drawback, has been proposed in [8]: it was a discrete-continuous output feedback control algorithm that requires finite sampling time for its analysis and implementation. The basic idea consists in change the gain magnitude in an impulsive manner, at some precise time instants. The gain is then evolving between a small value (fixed in order to counteract the effects of uncertainties and perturbations) and a larger one, which defines the size of the impulsion, this latter having a duration equal to a sampling period. These gain “jumps” allow to ensure the compensation of delays induced by the sampling period, when the sliding variable is changing of sign. A drawback of this approach is that there is no constructive way to fix the value of the larger gain. The result reported in [8] has motivated [4], in which impulsive sliding mode state feedback control laws Antonio Estrada and Franck Plestan are with LUNAM Université, Ecole Centrale de Nantes, IRCCyN UMR CNRS 6597, Nantes, France. E-mail: [email protected] and [email protected]
978-1-4673-2064-1/12/$31.00 ©2012 IEEE
are proposed for a larger class of system (sampling period can be equal to 0). In the current paper, the first result consists in improving the main result of [8], by given a formal condition on the larger gain, in order to get a constructive way to design the second order sliding mode output feedback controller. The approach is presented firstly for unperturbed systems, then for perturbed ones. For this latter class of systems, the output feedback controller is proposed in an adaptive version which allows to reduce the gain when system trajectories are close from the origin. Note that the adaptation of gain in sliding mode context [9], [10] allows to reduce the magnitude of the control input, allowing a reduction of the chattering. II. P ROBLEM
STATEMENT
Second order sliding mode control of uncertain nonlinear system is equivalent to the finite time stabilization of the following system z˙1 =z2 z˙2 =u + ω
(1)
where z = [z1 z2 ]T is the state vector, u the control input and |ω| ≤ δ (0 ≤ δ) a bounded perturbation. The system (1) is viewed as an uncertain nonlinear continuous one, whereas the control law u is supposed to be evaluated at a sampling period Te which yields u = − K(t)sign(z1 (kTe ))
(2)
where K(t) > 0 and k ∈ IN (k can be viewed as a time counter). The gain K(t) is constant on the time interval t ∈ [k · Te , (k + 1) · Te ], and k(0) = 0. Note that the control input depends only on the output z1 . Then, the objective is to propose an output feedback (2) which allows to reach, in a finite time, a vicinity of the origin of (1) in spite of perturbation ω. III. S ECOND
ORDER SLIDING MODE OUTPUT FEEDBACK CONTROL OF THE UNPERTURBED CASE
The result displayed in the sequel allows to ensure the finite time establishment of a second order sliding mode in system (1)-(2), when there is no perturbation (ω = 0). Theorem 1: Consider system (1) with ω = 0 and controlled by (2). The gain K(t) is defined as Km if t 6∈ T (3) K(t) = KM if t ∈ T
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with T = {t | sign(z1 (kT e)) 6= sign(z1 ((k − 1)T e))} and Km > 0. If KM is such that KM ≥ γKm
with γ > 3,
(4)
then the control law (2) ensures that there exists a time tF such that for all t ≥ tF 1 2 |z1 | < Km [¯ η (γ) − 1] Te2 , |z2 | < Km η¯(γ)Te 2 (5) with γ2 − γ − 2 (6) η¯ = 2 (γ − 3) Proof. For the sake of clarity, consider Figure 1. A delay exists, due to the discrete sampling, between the time Tsr at which z1 crosses de z2 -axis, and the time Ts at which this crossing is detected. It has been shown in [8] that, due to this delay, switching gain K(t) is required in order to ensure convergence. If the gain of control law u = −Ksign(z1 (kTe )) is kept constant, this control law is not able to ensure the convergence of the closed-loop system trajectories: the system trajectories diverges. Defining the aforementioned delay as Td = Ts − Tsr , one has 0 ≤ Td < Te
time, the initial conditions are defines as z = [z1◦ z2◦ ]T . For the sake of simplicity but without loss of generality, suppose that z1◦ > 0 and z2◦ > 0. Then, for t ∈ [t◦ , Ts ], system trajectories belong to the parabola defined by (t − t◦ )2 + z2◦ (t − t◦ ) + z1◦ 2 z2 (t) = −Km (t − t◦ ) + z2◦ .
z1 (t) = −Km
(9b)
Furthermore, for t ∈ [Ts , Ts + Te ]
(t − Ts )2 + z2 (Ts )(t − Ts ) + z1 (Ts ) (10a) 2 z2 (t) = KM (t − Ts ) + z2 (Ts ) (10b)
z1 (t) = KM
In the sequel the trajectories evolving according to equations (9) and (10) will be referred as Km -trajectories and KM trajectories respectively, regardless of its particular initial condition. For an arbitrary Km -trajectory, let z(Tm ) be the point at which the z1 -axis is crossed, i.e. where z1 has its maximum, z(Tm ) = [z1m 0]T . Let us express the time interval Ts − Tm in terms of the sampling time, Te , and the gain KM in terms of Km ηTe = Ts − Tm KM = γKm
(7)
According to Theorem 1, the gain K(t) equals KM for
(9a)
(11) (12)
where η, γ are positive parameters. From (10b), one gets z2 (Ts + Te ) = KM Te + z2 (Ts ) = γKm Te + z2 (Ts )
z2
(13) (14)
By supposing t◦ = Ts , from (9b) it is obtained. z◦
z2 (Ts ) = − Km (T s − Tm ) + z2 (Tm ) = − ηKm Te
z(Tm ) z(Ts + Te)
Then, it yields z2 (Ts + Te ) = =
z1 z(Ts − γTe )
KM Te = γKm Te
z(Tsr ) z(Ts ) Fig. 1.
(z1 , z2 )-phase portrait of unperturbed system (1).
t ∈ [Ts , Ts + Te ]. In order to ensure convergence, KM must have a sufficient magnitude in order to reach parabolic trajectories which are closer from the origin, when one detects that the single measured output z1 has changed of sign. The proof consists, first-of-all, in determining the formal expression of z(Ts + Te ) and z(Ts − γTe ). Then, one has to prove that the following inequalities (see Figure 1) |z1 (Ts + Te )| ≤ |z2 (Ts + Te )|
3, since |z2 (Ts + Te )| < |z2 (Ts − Td − 2Te )| must be fulfilled. From (10a), one has 1 (18) z1 (Ts + Te ) = KM Te2 + z2 (Ts ) · Te + z1 (Ts ) 2 From (9a), and by supposing t◦ = Tsr Km (Ts − Tsr )2 + z2 (Tsr ) · (Ts − Tsr ) 2 +z1 (Tsr ) Km = − · Td2 + z2 (Tsr ) · Td 2 Given that z2 (Tsr ) = −Km (Tsr − Tm ), then
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z1 (Ts ) =
−
z1 (Ts ) = −
Km · Td2 − Km (Tsr − Tm ) · Td 2
(19)
Finally, z1 (Ts + Te ) can be expressed as z1 (Ts + Te )
=
Km 2 1 KM Te2 − ηKm Te2 − T 2 2 d −Km (Tsr − Tm ) · Td
(20)
It is obvious that |z1 (Ts )| is reaching a maximum value (which is the worst case) when Td = Te , which gives Tsr − Tm
= Ts − Td − Tm = (η − 1)Te .
Then, z1 (Ts + Te )
=
Km 2 1 γKm Te2 − 2ηKm Te2 + T 2 2 e
Example. Figure 2 depicts two limit cycles of system (1) (with ω = 0) trajectories, with different initial conditions (located on the Figure by black arrows). System (1) with ω = 0, has been simulated with a control law (2)-(3) with Km = 1, γ = 4 and Te = 0.01 sec, using the Euler integration method with a step equal to 10−5 sec. The dotted lines of Figure 2 display the evolution of the trajectories system close from the origin. The solid line displays the attractive domain defined by (5), i.e. |z1 | ≤ 8.10−4 and |z2 |0.05. Once the trajectories are evolving inside this domain, they do not leave it: a real sliding mode is established.
(21) Now, in order to analyze (8), it is necessary to evaluate z1 (Ts −γTe ). Considering t◦ = Ts −γTe , with Td = Ts −Tsr , one has 1 z1 (Tsr ) = − Km (γTe − Td )2 + z1 (Ts − γTe ) 2 +(γTe − Td ) · z2 (Ts − γTe ) with (from (9) and with t◦ = Tm ) z2 (Ts − γTe ) = −Km (Ts − γTe − Tm ) = −Km (ηTe − γTe )
(22)
it gives z1 (Ts − γTe ) =
1 Km (γTe − Td )2 2 +Km (ηTe − γTe )(γTe − Td ).
(23)
Fig. 2. Phase portrait (z1 , z2 ) of system (1) with ω = 0. The black arrows show two different initial conditions. The dotted lines display the system trajectories for these two different initial conditions. The solid line displays the limit of the convergence domain.
As previously, by considering Td = Te , one gets
1 Km [γ − 1]2 Te2 + Km [η − γ][γ − 1]Te2 . 2 (24) From inequality (8), one has z1 (Ts − γTe ) =
|z1 (Ts − γTe )| − |z1 (Ts + Te )| > 0. By using (21)-(24) in (25) and multiplying by next expression is obtained 2η[γ − 3] − γ 2 + γ + 2 > 0.
(25) 2 , the Km Te2 (26)
Then, inequality (26) allows to conclude that, given a fixed γ, there exists some η¯(γ) such that (26) holds for any η > η¯(γ); furthermore, given the definition of η, η > 0. It means that if Ts − Tm > Te · η¯(γ) then trajectories will be forced to converge to a vicinity of the origin, in a finite time. By supposing that the system trajectories are initially far from origin, it is obvious that η is large. It yields a convergence of the trajectories to a parabola closer from the origin, until η becomes lower than η¯. Of course, a small η means that the system is evolving around the origin: the system has converged to a domain defined as 1 Km [¯ η (γ) − 1]2 · Te2 , 2 γ2 − γ − 2 with η¯(γ) = . 2(γ − 3) z1 ≤
Now, suppose that system (1) with ω = 0 is initialized at z◦ = [−0.6 1]T , and the control law is tuned such that γ = 4, Km = 1 and Te = 0.01 (the integration method is Euler scheme wih a step 0.0001). Figures 3 and 4 display the (z1 , z2 )-phase portrait and its respective zoom: these both figures show that the system converges to a vicinity of the origin. Note also that the gain switching is clearly visible on these Figures. Figures 5 displays evolution of η: at the beginning, there is a value below 50 due to the particular initial conditions selected. The final value has converged, as expected, to η = 3 < e¯ta(4) = 5. 1.5 1 0.5 0 −0.5 −1 −1.5 −1.5
−1
−0.5
0
0.5
1
z2 ≤ Km η¯(γ) · Te Fig. 3.
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(z1 , z2 )-phase portrait.
1.5
z2 0.4 0.3 0.2 0.1
z(Ts + Te)
0
(Km − δ)
−0.1 −0.2
z (Tsrp )
z(Tm )
z2p
−0.3
z1
−0.4 −0.04
−0.02
0
0.02
0.04
|z2b − z2p | Fig. 4.
(Km + δ)
z (Tsrb)
(z1 , z2 )-phase portrait.
z2b
150
Fig. 6. 100
50
0 0
50
Fig. 5.
IV. A
100
150
200
250
Parameter η versus time (sec).
SOLUTION FOR PERTURBED CASE
In this section, a robust output feedback controller is proposed for system (1) with ω 6= 0. Nevertheless, the global validity of Theorem 1, for a single lower bound condition on the γ which relates KM to Km , cannot be resembled for the perturbed case. The reason is that, when perturbation is different from zero in system (1), the minimum required value for KM that ensures convergence, increases when trajectories are far from the origin. Theorem 2: Consider system (1) controlled by (2)-(3). If the next inequalities are fulfilled (with γ > 3) ω ≤ δ < Km
γKm + (γ + 1) δ < KM < ∞
(27) (28)
then, a real second order sliding mode with respect to z1 is established provided that initial conditions are inside the region of the (z1 , z2 )-phase plane delimited by z22 (t) =
2(Km − δ)(z1m + ν · z1 (t))
(29)
with ν
=
and
1 for z1 ∈ [−z1m , 0) −1 for z1 ∈ [0, z1m ] 2
z1m
=
(KM − 3Km ) · Te2 . p 2 − δ2 4 Km − Km
(30)
Phase portrait of perturbed system (1).
Proof. Without loss of generality, consider a Km trajectory which starts at a point z(Tm ) = [z1m , 0]T (see Figure 6). In order to ensure convergence, the KM -trajectory should finish above the symmetric of the inner parabola, even if (the “so-called” worst case) the mentioned trajectory was evolving on the extern parabola. Furthermore, by a similar way than the unperturbed case, one will consider the worst case with respect to the delay of the z1 -sign change detection, i.e. Td = Te , for both the extern and inner bounding parabolas. These two latter parabola delimit the domain in which the system trajectories are going to evolve: for each of these two parabolas, z2 (t) is defined as (as previously, t◦ is defining the considered initial time) Inner parabola. z2 (t)
=
−(Km − δ) · (t − t◦ ) + z2 (t◦ ),
(31)
External parabola. z2 (t)
=
−(Km + δ) · (t − t◦ ) + z2 (t◦ ).
(32)
Define Tsrp and Tsrb as the time intervals, for the inner and extern parabolas respectively, between Tm and the time instant for which z1 = 0 (a similar notation has been used in Figure 1). It is obvious that the lower value taken by z2 (t), denoted z2b , is obtained from the external parabola by supposing that the delay Td to detect z1 -sign change is maximum, i.e. Td = Te . By choosing t◦ = Tm , one gets z2b
=
−(Km + δ) · (Tsrb + Te )
Then, from (32), one derives 1 2 z1 (Tsrb ) = − (Km + δ) Tsrb + z1 (Tm ) = 0 2 It yields s 2z1 (Tm ) Tsrb = Km + δ
(33)
(34)
(35)
Finally, the minimal value z2 , denoted z2b , obtained for a given z1 (Tm ) reads as s 2z1 (Tm ) (36) + Te ) z2b = −(Km + δ) · ( Km + δ
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The variable z2p (see Figure 6) is defined as the value over which the system trajectories need to “jump” when KM is applied, because it ensures that the system has reached a parabola closer from the origin. In order to fulfill this latter constraint (even in the worst case), it is trivial that z2p reads as (from (32 )with t◦ = Tm ) z2p
= −(Km − δ)(Tsrp − 2Te ),
(37)
the time Tsrp being derived by the similar way than Tsrp ; one has s 2z1 (Tm ) (38) Tsrp = Km − δ As depicted by Figure 6, the KM -gain “jump” has to be sufficient in order to reach a parabola closer, which yields (KM − δ)Te
> |z2b − z2p |
(39)
From equations (36)-(37)-(39), the next condition is obtained hp i p √ 2z1m Km + δ − Km − δ (KM − δ)Te > +3KmTe − δTe
(40) From the above inequality, it yields that the right hand side √ is proportional to z1m . Next, one gives an estimation of the region of attraction for a given KM ; solving (40) versus z1m , one gets z1m
γ(Km + δ) ⇒ KM > γKm + (γ + 1) δ (44) 5486
ηi
1 i i∗ 2 i∗ = 0 = (Km + ω)[Tsr − Tm ] + z1 (Tm ) 2 1 i∗ 2z1 (Tm ) 2 ≤ Km − δ
=
s
floor
i∗ 2z1 (Tm ) Km − δ +1 Te
(49)
Then, one gets
2
i Tsr
i∗ Tm
−
i
≤ η · Te
1.5
(50)
1
From this latter inequality, it comes i∗ Tsi − Tm
0.5
(η i + 1) · Te
(51)
(Km + δ) · (η i + 1) · Te
(52)
≤
0 −0.5
From (47), one has |z2 (Tsi )| ≤
−1 −1.5
Then, from (46), one has i KM
−2 −1.5
i
= (η + 1)(Km + δ) + δ.
−1
−0.5
0
0.5
1
1.5
(53)
From Theorem 2, by defining KMin as
Fig. 8.
(z1 , z2 )-phase plan of system (1) with ω = 0.6 sin(z2 ).
KMin = γKm + (γ + 1) δ with Km > δ and by supposing that the gain has an upper bound equal to KMax (due to practical constraints - saturation of the control input applied to an actuator, for instance), the KM -gain adaptation law of the controller (2)-(3) reads as i KM ≥ KMax KMax if i i KM if KMin ≤ KM < KMax KM = (54) i KMin if KM < KMin
Example. Simulations have been made with system (1) such that ω = 0.6sign(z2 ), KMax = 100, Km = 1 and Te = 0.01 sec. The initial conditions are z = [1.1 0.1]T . Figure (7) shows the behavior of the control signal with the above adaptation for KM , whereas Figure (8) depicts the corresponding phase portrait and shows the convergence to a vicinity of the origin. Note that the gain KM is starting with large values and, when trajectories become closer from the origin, it starts to be reduced and goes to its minimum value. 100
50
0
−50
−100 0
5
10
15
20
25
Fig. 7. Control input u versus time (sec) for a system (1) with ω = 0.6 sin(z2 ).
ACKNOWLEDGMENT This work has been made when Antonio Estrada was in postdoctoral position at IRCCyN; this position was financially supported by Ecole Centrale de Nantes. Furthermore, this work takes place in ANR project "ChaSliM" between IRCCyN, INIRIA Rhone-Alpes (Grenoble, France) and INRIA North-Europe (Lille, France). R EFERENCES [1] C. Edwards, and S. Spurgeon, Sliding Mode Control: Theory and Applications, Francis and Taylor, 1998. [2] A. Levant Sliding order and sliding accuracy, International Journal of Control, vol. 58, pp.1247-1263, 1993. [3] L. Fridman Chattering analysis in sliding mode systems with inertial sensors, International Journal of Control, vol.76, no.9, pp.906-912, 2003. bibitemGua05 Z.H Guan, D. J. Hill and X. Shen On hybrid impulsive and switching systems and application to nonlinear control, IEEE Trans. Autom. Control, vol.50, no.7, pp.1058-1062, 2005. [4] A. Glumineau, Y. Shtessel, and F. Plestan Control of a double integrator via discontinuous-impulsive feedback with adaptation IFAC World Congress, Milano, Italy, 2011. [5] A. Levant Higher order sliding mode, differentiation and output feedback control, International Journal of Control, vol.76, pp.924-941, 2003. [6] Z. G. Li, C. Y. Wen, and Y. C. Soh Analysis and Design of Impulsive Control Systems, IEEE Trans. Autom. Control, vol.46, no.6, pp.894897, 2001. [7] Y. Orlov. Discontinuous Systems: Lyapunov Analysis and Robust Synthesis Under uncertainty Conditions. Springer, London, 2008. [8] F. Plestan, E. Moulay, A. Glumineau and T. Cheviron Robust output feedback sampling control based on second order sliding mode, Automatica, vol.46, no.6, pp.1096-1100, 2010. [9] F. Plestan, Y. Shtessel, V. Brégeault, and A. Poznyak New methodologies for adaptive sliding mode control, International Journal of Control, vol.83, no.9, pp.1907-1919, 2010. [10] Y. Shtessel, M. Taleb, and F. Plestan, A novel adaptive-gain supertwisting sliding mode controller: methodology and application, Automatica, vol.48, no.5, pp. 759-769, 2012. [11] T. Yang Impulsive control, IEEE Trans. Autom. Control, vol.44, no. 5, pp.1081-1083, 1999.
VI. C ONCLUSION This paper proposed output feedback scheme based on second order sliding mode theory and impulsive gain. The solutions have been presented in case of certain and uncertain systems, and a first adaptation law for the impulsion magnitude has been given. Future works consist in applying this class of controllers on real systems, and in the extension for systems with unknown bounds of uncertainties and perturbations. 5487
Second order sliding mode output feedback control: impulsive gain and extension with adaptation Antonio Estrada, Franck Plestan Abstract— This paper proposes a discontinuous-impulsive output feedback which is a sampled based second order sliding mode controller. First, constructive convergence conditions are established for the controller, in the unperturbed and perturbed case. An adaptation mechanism is based on the aforementioned convergence conditions and allows to reduce gain magnitude.
I. I NTRODUCTION Sliding mode (SM) and high order sliding modes (HOSM) are nonlinear control strategies that provides robustness with respect to uncertainties and perturbations. The main common properties of SMC/HOSM control laws are robustness (insensitivity) to the bounded disturbances matched by control, and finite convergence time. The main drawback of the The main drawback of HOSM control laws is the necessity of high order derivatives of the sliding variable [5], [3], which are obtained using HOSM differentiators [5]. Differentiation yields a performance degradation of the controlled system due to the presence of measurement noise. Then, there is a real interest to propose high order sliding mode controllers with a reduced number of time derivatives of the sliding variable. Concerning the second order sliding mode controllers, a popular second order sliding mode output feedback is the supertwisting algorithm [5]. In fact, this controller allows the establishment of a second order sliding mode with respect to the sliding variable, in a finite time, by using only the measurement of the sliding variable. However, this algorithm is intended to be applied to systems whose relative degree of the sliding variable equals 1. An alternative strategy, which do not have the previous drawback, has been proposed in [8]: it was a discrete-continuous output feedback control algorithm that requires finite sampling time for its analysis and implementation. The basic idea consists in change the gain magnitude in an impulsive manner, at some precise time instants. The gain is then evolving between a small value (fixed in order to counteract the effects of uncertainties and perturbations) and a larger one, which defines the size of the impulsion, this latter having a duration equal to a sampling period. These gain “jumps” allow to ensure the compensation of delays induced by the sampling period, when the sliding variable is changing of sign. A drawback of this approach is that there is no constructive way to fix the value of the larger gain. The result reported in [8] has motivated [4], in which impulsive sliding mode state feedback control laws Antonio Estrada and Franck Plestan are with LUNAM Université, Ecole Centrale de Nantes, IRCCyN UMR CNRS 6597, Nantes, France. E-mail: [email protected] and [email protected]
978-1-4673-2064-1/12/$31.00 ©2012 IEEE
are proposed for a larger class of system (sampling period can be equal to 0). In the current paper, the first result consists in improving the main result of [8], by given a formal condition on the larger gain, in order to get a constructive way to design the second order sliding mode output feedback controller. The approach is presented firstly for unperturbed systems, then for perturbed ones. For this latter class of systems, the output feedback controller is proposed in an adaptive version which allows to reduce the gain when system trajectories are close from the origin. Note that the adaptation of gain in sliding mode context [9], [10] allows to reduce the magnitude of the control input, allowing a reduction of the chattering. II. P ROBLEM
STATEMENT
Second order sliding mode control of uncertain nonlinear system is equivalent to the finite time stabilization of the following system z˙1 =z2 z˙2 =u + ω
(1)
where z = [z1 z2 ]T is the state vector, u the control input and |ω| ≤ δ (0 ≤ δ) a bounded perturbation. The system (1) is viewed as an uncertain nonlinear continuous one, whereas the control law u is supposed to be evaluated at a sampling period Te which yields u = − K(t)sign(z1 (kTe ))
(2)
where K(t) > 0 and k ∈ IN (k can be viewed as a time counter). The gain K(t) is constant on the time interval t ∈ [k · Te , (k + 1) · Te ], and k(0) = 0. Note that the control input depends only on the output z1 . Then, the objective is to propose an output feedback (2) which allows to reach, in a finite time, a vicinity of the origin of (1) in spite of perturbation ω. III. S ECOND
ORDER SLIDING MODE OUTPUT FEEDBACK CONTROL OF THE UNPERTURBED CASE
The result displayed in the sequel allows to ensure the finite time establishment of a second order sliding mode in system (1)-(2), when there is no perturbation (ω = 0). Theorem 1: Consider system (1) with ω = 0 and controlled by (2). The gain K(t) is defined as Km if t 6∈ T (3) K(t) = KM if t ∈ T
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with T = {t | sign(z1 (kT e)) 6= sign(z1 ((k − 1)T e))} and Km > 0. If KM is such that KM ≥ γKm
with γ > 3,
(4)
then the control law (2) ensures that there exists a time tF such that for all t ≥ tF 1 2 |z1 | < Km [¯ η (γ) − 1] Te2 , |z2 | < Km η¯(γ)Te 2 (5) with γ2 − γ − 2 (6) η¯ = 2 (γ − 3) Proof. For the sake of clarity, consider Figure 1. A delay exists, due to the discrete sampling, between the time Tsr at which z1 crosses de z2 -axis, and the time Ts at which this crossing is detected. It has been shown in [8] that, due to this delay, switching gain K(t) is required in order to ensure convergence. If the gain of control law u = −Ksign(z1 (kTe )) is kept constant, this control law is not able to ensure the convergence of the closed-loop system trajectories: the system trajectories diverges. Defining the aforementioned delay as Td = Ts − Tsr , one has 0 ≤ Td < Te
time, the initial conditions are defines as z = [z1◦ z2◦ ]T . For the sake of simplicity but without loss of generality, suppose that z1◦ > 0 and z2◦ > 0. Then, for t ∈ [t◦ , Ts ], system trajectories belong to the parabola defined by (t − t◦ )2 + z2◦ (t − t◦ ) + z1◦ 2 z2 (t) = −Km (t − t◦ ) + z2◦ .
z1 (t) = −Km
(9b)
Furthermore, for t ∈ [Ts , Ts + Te ]
(t − Ts )2 + z2 (Ts )(t − Ts ) + z1 (Ts ) (10a) 2 z2 (t) = KM (t − Ts ) + z2 (Ts ) (10b)
z1 (t) = KM
In the sequel the trajectories evolving according to equations (9) and (10) will be referred as Km -trajectories and KM trajectories respectively, regardless of its particular initial condition. For an arbitrary Km -trajectory, let z(Tm ) be the point at which the z1 -axis is crossed, i.e. where z1 has its maximum, z(Tm ) = [z1m 0]T . Let us express the time interval Ts − Tm in terms of the sampling time, Te , and the gain KM in terms of Km ηTe = Ts − Tm KM = γKm
(7)
According to Theorem 1, the gain K(t) equals KM for
(9a)
(11) (12)
where η, γ are positive parameters. From (10b), one gets z2 (Ts + Te ) = KM Te + z2 (Ts ) = γKm Te + z2 (Ts )
z2
(13) (14)
By supposing t◦ = Ts , from (9b) it is obtained. z◦
z2 (Ts ) = − Km (T s − Tm ) + z2 (Tm ) = − ηKm Te
z(Tm ) z(Ts + Te)
Then, it yields z2 (Ts + Te ) = =
z1 z(Ts − γTe )
KM Te = γKm Te
z(Tsr ) z(Ts ) Fig. 1.
(z1 , z2 )-phase portrait of unperturbed system (1).
t ∈ [Ts , Ts + Te ]. In order to ensure convergence, KM must have a sufficient magnitude in order to reach parabolic trajectories which are closer from the origin, when one detects that the single measured output z1 has changed of sign. The proof consists, first-of-all, in determining the formal expression of z(Ts + Te ) and z(Ts − γTe ). Then, one has to prove that the following inequalities (see Figure 1) |z1 (Ts + Te )| ≤ |z2 (Ts + Te )|
3, since |z2 (Ts + Te )| < |z2 (Ts − Td − 2Te )| must be fulfilled. From (10a), one has 1 (18) z1 (Ts + Te ) = KM Te2 + z2 (Ts ) · Te + z1 (Ts ) 2 From (9a), and by supposing t◦ = Tsr Km (Ts − Tsr )2 + z2 (Tsr ) · (Ts − Tsr ) 2 +z1 (Tsr ) Km = − · Td2 + z2 (Tsr ) · Td 2 Given that z2 (Tsr ) = −Km (Tsr − Tm ), then
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z1 (Ts ) =
−
z1 (Ts ) = −
Km · Td2 − Km (Tsr − Tm ) · Td 2
(19)
Finally, z1 (Ts + Te ) can be expressed as z1 (Ts + Te )
=
Km 2 1 KM Te2 − ηKm Te2 − T 2 2 d −Km (Tsr − Tm ) · Td
(20)
It is obvious that |z1 (Ts )| is reaching a maximum value (which is the worst case) when Td = Te , which gives Tsr − Tm
= Ts − Td − Tm = (η − 1)Te .
Then, z1 (Ts + Te )
=
Km 2 1 γKm Te2 − 2ηKm Te2 + T 2 2 e
Example. Figure 2 depicts two limit cycles of system (1) (with ω = 0) trajectories, with different initial conditions (located on the Figure by black arrows). System (1) with ω = 0, has been simulated with a control law (2)-(3) with Km = 1, γ = 4 and Te = 0.01 sec, using the Euler integration method with a step equal to 10−5 sec. The dotted lines of Figure 2 display the evolution of the trajectories system close from the origin. The solid line displays the attractive domain defined by (5), i.e. |z1 | ≤ 8.10−4 and |z2 |0.05. Once the trajectories are evolving inside this domain, they do not leave it: a real sliding mode is established.
(21) Now, in order to analyze (8), it is necessary to evaluate z1 (Ts −γTe ). Considering t◦ = Ts −γTe , with Td = Ts −Tsr , one has 1 z1 (Tsr ) = − Km (γTe − Td )2 + z1 (Ts − γTe ) 2 +(γTe − Td ) · z2 (Ts − γTe ) with (from (9) and with t◦ = Tm ) z2 (Ts − γTe ) = −Km (Ts − γTe − Tm ) = −Km (ηTe − γTe )
(22)
it gives z1 (Ts − γTe ) =
1 Km (γTe − Td )2 2 +Km (ηTe − γTe )(γTe − Td ).
(23)
Fig. 2. Phase portrait (z1 , z2 ) of system (1) with ω = 0. The black arrows show two different initial conditions. The dotted lines display the system trajectories for these two different initial conditions. The solid line displays the limit of the convergence domain.
As previously, by considering Td = Te , one gets
1 Km [γ − 1]2 Te2 + Km [η − γ][γ − 1]Te2 . 2 (24) From inequality (8), one has z1 (Ts − γTe ) =
|z1 (Ts − γTe )| − |z1 (Ts + Te )| > 0. By using (21)-(24) in (25) and multiplying by next expression is obtained 2η[γ − 3] − γ 2 + γ + 2 > 0.
(25) 2 , the Km Te2 (26)
Then, inequality (26) allows to conclude that, given a fixed γ, there exists some η¯(γ) such that (26) holds for any η > η¯(γ); furthermore, given the definition of η, η > 0. It means that if Ts − Tm > Te · η¯(γ) then trajectories will be forced to converge to a vicinity of the origin, in a finite time. By supposing that the system trajectories are initially far from origin, it is obvious that η is large. It yields a convergence of the trajectories to a parabola closer from the origin, until η becomes lower than η¯. Of course, a small η means that the system is evolving around the origin: the system has converged to a domain defined as 1 Km [¯ η (γ) − 1]2 · Te2 , 2 γ2 − γ − 2 with η¯(γ) = . 2(γ − 3) z1 ≤
Now, suppose that system (1) with ω = 0 is initialized at z◦ = [−0.6 1]T , and the control law is tuned such that γ = 4, Km = 1 and Te = 0.01 (the integration method is Euler scheme wih a step 0.0001). Figures 3 and 4 display the (z1 , z2 )-phase portrait and its respective zoom: these both figures show that the system converges to a vicinity of the origin. Note also that the gain switching is clearly visible on these Figures. Figures 5 displays evolution of η: at the beginning, there is a value below 50 due to the particular initial conditions selected. The final value has converged, as expected, to η = 3 < e¯ta(4) = 5. 1.5 1 0.5 0 −0.5 −1 −1.5 −1.5
−1
−0.5
0
0.5
1
z2 ≤ Km η¯(γ) · Te Fig. 3.
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(z1 , z2 )-phase portrait.
1.5
z2 0.4 0.3 0.2 0.1
z(Ts + Te)
0
(Km − δ)
−0.1 −0.2
z (Tsrp )
z(Tm )
z2p
−0.3
z1
−0.4 −0.04
−0.02
0
0.02
0.04
|z2b − z2p | Fig. 4.
(Km + δ)
z (Tsrb)
(z1 , z2 )-phase portrait.
z2b
150
Fig. 6. 100
50
0 0
50
Fig. 5.
IV. A
100
150
200
250
Parameter η versus time (sec).
SOLUTION FOR PERTURBED CASE
In this section, a robust output feedback controller is proposed for system (1) with ω 6= 0. Nevertheless, the global validity of Theorem 1, for a single lower bound condition on the γ which relates KM to Km , cannot be resembled for the perturbed case. The reason is that, when perturbation is different from zero in system (1), the minimum required value for KM that ensures convergence, increases when trajectories are far from the origin. Theorem 2: Consider system (1) controlled by (2)-(3). If the next inequalities are fulfilled (with γ > 3) ω ≤ δ < Km
γKm + (γ + 1) δ < KM < ∞
(27) (28)
then, a real second order sliding mode with respect to z1 is established provided that initial conditions are inside the region of the (z1 , z2 )-phase plane delimited by z22 (t) =
2(Km − δ)(z1m + ν · z1 (t))
(29)
with ν
=
and
1 for z1 ∈ [−z1m , 0) −1 for z1 ∈ [0, z1m ] 2
z1m
=
(KM − 3Km ) · Te2 . p 2 − δ2 4 Km − Km
(30)
Phase portrait of perturbed system (1).
Proof. Without loss of generality, consider a Km trajectory which starts at a point z(Tm ) = [z1m , 0]T (see Figure 6). In order to ensure convergence, the KM -trajectory should finish above the symmetric of the inner parabola, even if (the “so-called” worst case) the mentioned trajectory was evolving on the extern parabola. Furthermore, by a similar way than the unperturbed case, one will consider the worst case with respect to the delay of the z1 -sign change detection, i.e. Td = Te , for both the extern and inner bounding parabolas. These two latter parabola delimit the domain in which the system trajectories are going to evolve: for each of these two parabolas, z2 (t) is defined as (as previously, t◦ is defining the considered initial time) Inner parabola. z2 (t)
=
−(Km − δ) · (t − t◦ ) + z2 (t◦ ),
(31)
External parabola. z2 (t)
=
−(Km + δ) · (t − t◦ ) + z2 (t◦ ).
(32)
Define Tsrp and Tsrb as the time intervals, for the inner and extern parabolas respectively, between Tm and the time instant for which z1 = 0 (a similar notation has been used in Figure 1). It is obvious that the lower value taken by z2 (t), denoted z2b , is obtained from the external parabola by supposing that the delay Td to detect z1 -sign change is maximum, i.e. Td = Te . By choosing t◦ = Tm , one gets z2b
=
−(Km + δ) · (Tsrb + Te )
Then, from (32), one derives 1 2 z1 (Tsrb ) = − (Km + δ) Tsrb + z1 (Tm ) = 0 2 It yields s 2z1 (Tm ) Tsrb = Km + δ
(33)
(34)
(35)
Finally, the minimal value z2 , denoted z2b , obtained for a given z1 (Tm ) reads as s 2z1 (Tm ) (36) + Te ) z2b = −(Km + δ) · ( Km + δ
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The variable z2p (see Figure 6) is defined as the value over which the system trajectories need to “jump” when KM is applied, because it ensures that the system has reached a parabola closer from the origin. In order to fulfill this latter constraint (even in the worst case), it is trivial that z2p reads as (from (32 )with t◦ = Tm ) z2p
= −(Km − δ)(Tsrp − 2Te ),
(37)
the time Tsrp being derived by the similar way than Tsrp ; one has s 2z1 (Tm ) (38) Tsrp = Km − δ As depicted by Figure 6, the KM -gain “jump” has to be sufficient in order to reach a parabola closer, which yields (KM − δ)Te
> |z2b − z2p |
(39)
From equations (36)-(37)-(39), the next condition is obtained hp i p √ 2z1m Km + δ − Km − δ (KM − δ)Te > +3KmTe − δTe
(40) From the above inequality, it yields that the right hand side √ is proportional to z1m . Next, one gives an estimation of the region of attraction for a given KM ; solving (40) versus z1m , one gets z1m
γ(Km + δ) ⇒ KM > γKm + (γ + 1) δ (44) 5486
ηi
1 i i∗ 2 i∗ = 0 = (Km + ω)[Tsr − Tm ] + z1 (Tm ) 2 1 i∗ 2z1 (Tm ) 2 ≤ Km − δ
=
s
floor
i∗ 2z1 (Tm ) Km − δ +1 Te
(49)
Then, one gets
2
i Tsr
i∗ Tm
−
i
≤ η · Te
1.5
(50)
1
From this latter inequality, it comes i∗ Tsi − Tm
0.5
(η i + 1) · Te
(51)
(Km + δ) · (η i + 1) · Te
(52)
≤
0 −0.5
From (47), one has |z2 (Tsi )| ≤
−1 −1.5
Then, from (46), one has i KM
−2 −1.5
i
= (η + 1)(Km + δ) + δ.
−1
−0.5
0
0.5
1
1.5
(53)
From Theorem 2, by defining KMin as
Fig. 8.
(z1 , z2 )-phase plan of system (1) with ω = 0.6 sin(z2 ).
KMin = γKm + (γ + 1) δ with Km > δ and by supposing that the gain has an upper bound equal to KMax (due to practical constraints - saturation of the control input applied to an actuator, for instance), the KM -gain adaptation law of the controller (2)-(3) reads as i KM ≥ KMax KMax if i i KM if KMin ≤ KM < KMax KM = (54) i KMin if KM < KMin
Example. Simulations have been made with system (1) such that ω = 0.6sign(z2 ), KMax = 100, Km = 1 and Te = 0.01 sec. The initial conditions are z = [1.1 0.1]T . Figure (7) shows the behavior of the control signal with the above adaptation for KM , whereas Figure (8) depicts the corresponding phase portrait and shows the convergence to a vicinity of the origin. Note that the gain KM is starting with large values and, when trajectories become closer from the origin, it starts to be reduced and goes to its minimum value. 100
50
0
−50
−100 0
5
10
15
20
25
Fig. 7. Control input u versus time (sec) for a system (1) with ω = 0.6 sin(z2 ).
ACKNOWLEDGMENT This work has been made when Antonio Estrada was in postdoctoral position at IRCCyN; this position was financially supported by Ecole Centrale de Nantes. Furthermore, this work takes place in ANR project "ChaSliM" between IRCCyN, INIRIA Rhone-Alpes (Grenoble, France) and INRIA North-Europe (Lille, France). R EFERENCES [1] C. Edwards, and S. Spurgeon, Sliding Mode Control: Theory and Applications, Francis and Taylor, 1998. [2] A. Levant Sliding order and sliding accuracy, International Journal of Control, vol. 58, pp.1247-1263, 1993. [3] L. Fridman Chattering analysis in sliding mode systems with inertial sensors, International Journal of Control, vol.76, no.9, pp.906-912, 2003. bibitemGua05 Z.H Guan, D. J. Hill and X. Shen On hybrid impulsive and switching systems and application to nonlinear control, IEEE Trans. Autom. Control, vol.50, no.7, pp.1058-1062, 2005. [4] A. Glumineau, Y. Shtessel, and F. Plestan Control of a double integrator via discontinuous-impulsive feedback with adaptation IFAC World Congress, Milano, Italy, 2011. [5] A. Levant Higher order sliding mode, differentiation and output feedback control, International Journal of Control, vol.76, pp.924-941, 2003. [6] Z. G. Li, C. Y. Wen, and Y. C. Soh Analysis and Design of Impulsive Control Systems, IEEE Trans. Autom. Control, vol.46, no.6, pp.894897, 2001. [7] Y. Orlov. Discontinuous Systems: Lyapunov Analysis and Robust Synthesis Under uncertainty Conditions. Springer, London, 2008. [8] F. Plestan, E. Moulay, A. Glumineau and T. Cheviron Robust output feedback sampling control based on second order sliding mode, Automatica, vol.46, no.6, pp.1096-1100, 2010. [9] F. Plestan, Y. Shtessel, V. Brégeault, and A. Poznyak New methodologies for adaptive sliding mode control, International Journal of Control, vol.83, no.9, pp.1907-1919, 2010. [10] Y. Shtessel, M. Taleb, and F. Plestan, A novel adaptive-gain supertwisting sliding mode controller: methodology and application, Automatica, vol.48, no.5, pp. 759-769, 2012. [11] T. Yang Impulsive control, IEEE Trans. Autom. Control, vol.44, no. 5, pp.1081-1083, 1999.
VI. C ONCLUSION This paper proposed output feedback scheme based on second order sliding mode theory and impulsive gain. The solutions have been presented in case of certain and uncertain systems, and a first adaptation law for the impulsion magnitude has been given. Future works consist in applying this class of controllers on real systems, and in the extension for systems with unknown bounds of uncertainties and perturbations. 5487
