Uniform sliding mode controllers and uniform ... - Semantic Scholar
IMA Journal of Mathematical Control and Information (2012), 491−505 doi:10.1093/imamci/dns005 Advance Access publication on March 8, 2012
Uniform sliding mode controllers and uniform sliding surfaces
[Received on 11 March 2011; revised on 19 November 2011; accepted on 5 January 2012] In this note, we propose two uniform sliding mode controllers (SMCs) for a second-order uncertain system. They provide convergence to an arbitrarily small vicinity centred at the origin in finite time, which can be bounded by some constant independent of the initial conditions and uncertainties. Towards this aim, a non-linear sliding surface is designed ensuring convergence, during the sliding motion, of every trajectory to an arbitrarily small vicinity of the origin in a finite time bounded by some constant, independent of the initial condition on the surface. A first-order SMC is designed providing convergence of every trajectory to the sliding surface in a finite time, bounded by another constant independent of the initial conditions and uncertainties/disturbances of a special class. In order to adjust chattering, a super twisting-based controller is suggested providing convergence of the trajectories to the sliding surface in a finite time bounded by some constant independent of any initial condition and uncertainties/disturbances of another class. Keywords: sliding mode control; Lyapunov methods.
1. Introduction The problem of robust prescribed time stabilization is one of the actual tasks in modern control theory. For example, in hybrid systems control with strictly positive dwell time, it is preferable that the control action provide for robust exact system stabilization before the next switching or impulse take place. A reasonable class of controllers providing both, finite-time convergence and insensitivity with respect to matched uncertainties/disturbances, are sliding mode controllers (SMCs, see, e.g. Utkin, 1992). Traditional SMC design consists of two steps (Luk´yanov & Utkin, 1981; Utkin, 1992): (a) design of a sliding surface to ensure the desired behaviour of the system without uncertainties/disturbances and (b) design of discontinuous controllers to enforce the sliding motion and compensation of the matched uncertainties/disturbances. The main disadvantage of such methodology is the so-called ‘chattering’ phenomenon, restricting the possibilities of implementing first-order SMCs in hardware, where switching is a natural mode of work. A super-twisting controller (STC; Levant, 1993) allows adjusting the chattering problem in systems with Lipschitz continuous (in time) uncertainties/disturbances. It is necessary to remark that in both, classical and super-twisting control design methodologies, the convergence time grows together with the initial conditions, i.e. (a) the convergence time to the sliding surface grows together with the initial conditions and (b) even if the trajectory starts on the sliding surface, the convergence time to a given vicinity around the origin on the sliding surface also grows together with the initial conditions on the surface. A so-called uniform super twisting-based differentiator and an observer for mechanical systems has been designed in Cruz Zavala et al. (2011). The differentiator/observer convergence is guaranteed to c The author 2012. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
E MMANUEL C RUZ -Z AVALA , JAIME A. M ORENO AND L EONID F RIDMAN∗ Instituto de Ingenier´ıa, Universidad Nacional Aut´onoma de M´exico UNAM, 04510 M´exico D.F., Mexico and Departamento de Control Autom´atico, CINVESTAV-IPN, AP-14-740 M´exico, D.F., Mexico ∗ Corresponding author: [email protected]
492
E. CRUZ-ZAVALA ET AL.
•
a non-linear sliding surface is suggested such that, during the sliding motion, the convergence of every trajectory to any arbitrarily small vicinity of the origin will be ensured in a finite time bounded by some constant independent of the initial conditions on the surface.
•
a first-order SMC is designed, providing for the convergence of the system trajectories to the sliding surface in a finite time bounded by some constant independent of the initial conditions and uncertainties/disturbances of a special class.
•
an absolutely continuous super twisting-based controller is suggested guaranteeing convergence of the system trajectories to the sliding surface in a finite time bounded by some constant independent of the initial conditions and another uncertainties/disturbances class and
•
the convergence time for all system trajectories is estimated for both proposed sliding mode enforcement algorithms in terms of the radius of a vicinity centred at the origin and upper bounds of uncertainties/disturbances.
The methodology used in this paper is based on Lyapunov functions proposed in Moreno (2011), Cruz Zavala et al. (2011) and Gonzalez et al. (2011). 2. Problem statement Consider a controllable single input uncertain linear system in regular form: x˙1 = a11 x1 + a12 x2 , x˙2 = a21 x1 + a22 x2 + u + w(x, t),
(2.1)
where x = [x 1 , x2 ] ∈ R 2 is the state vector, u ∈ R1 is the control input, the parameters a11 , a12 = 0, a21 , a22 are constants, w(x, t) ∈ W0 is an uncertainty/disturbance acting on the system and is assumed to belong to the class W0 = {w(x, t): |w(x, t)| 0 , 0 > 0} of signals uniformly bounded in t, ∀ t 0. Note that w(x, t) may be non-vanishing at the origin x = 0 and thus a continuous static feedback law is not able to stabilize the origin of the system. However, classical SMC is able to obtain this objective. In this case, all solutions of the differential equations and inclusion are defined in the sense of Filippov (1988). For its design, it is first necessary to propose the sliding surface. Usually, a linear sliding surface (Utkin, 1992) for system (2.1) is considered as s = x 2 + c1 x 1 .
(2.2)
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
be in a finite time bounded by some constant independent of the initial conditions error. This property is studied in Andrieu et al (2008) for designing a recursive observer. In the case of SMC design, the methodology presented in Gonzalez et al. (2011) can ensure convergence uniform with respect to (w.r.t) initial conditions to only the sliding surface. However, it does not ensure convergence uniform w.r.t any initial condition for the system trajectories on the sliding surface to the origin. On the other hand, the use of non-linear sliding surfaces instead of linear surfaces in classical SMC design has been proved to enhance the desired performance in closed-loop systems with SMC algorithms, which cannot always be achieved using only linear switching surfaces (see Bartoszewicz & Nowacka-Leverton, 2009, Bandyopadhyay et al., 2009; Shtessel et al., 2002 and references there in). In this paper, we suggest two uniform SMCs for a second-order system providing convergence of the system trajectories to any arbitrarily small vicinity of the origin in a finite time bounded by some constant ‘independent from initial conditions and uncertainties/disturbances of a special class’. Towards this aim:
UNIFORM SLIDING MODE CONTROLLERS AND UNIFORM SLIDING SURFACE
493
The sliding mode s(x(t)) = 0 can be ensured in finite time by some discontinuous control law (see, e.g. Utkin, 1992) and, when the sliding motion takes place, the system dynamics are described by the linear system x˙1 = (a11 − a12 c1 )x1 ,
(2.3)
u = −(c1 a11 + a21 )x1 − (c1 a12 + a22 )x2 − Q sign(s),
(2.4)
where Q is chosen from the knowledge of a certain upper bound of the uncertainties, i.e. Q > ρ0 . Note that the classical SMC described above reaches the origin asymptotically, and it enters any neighbourhood of the origin in finite time. However, the further initial conditions are from the origin, the larger the reaching time to a neighbourhood of the origin will be. In fact, when the initial condition tends to infinity, the convergence time also tends to infinity. This means that it is impossible to prescribe the time to reach a neighbourhood of the origin independently of the value of the initial condition. Our objective here is, using the main idea of the SMC design and considering that uncertainties/disturbances belong to some class Wnv of time functions, to design a controller capable of reaching, in a prescribed time, a neighbourhood of the origin of the linear system (2.1) independently of the initial conditions and of the acting (matched) perturbations belonging to class Wnv . To attain this objective we will: 1. Design a (non-linear) sliding surface such that when the motion is restricted to the manifold s = 0, the trajectories converge ‘asymptotically and uniformly w.r.t. initial conditions’ to a neighbourhood of the origin (Section 3). That is, any trajectory on the sliding surface will reach a neighbourhood of radius μ, centred at the origin, before a certain (prescribed) time Tμ > 0, which depends on μ but which is independent of the initial conditions x0 of the trajectory. 2. Design a control law to enforce sliding motion in prescribed time. In this sense, any trajectory will be forced to converge ‘uniformly w.r.t. initial condition and in finite time’ from any arbitrary initial condition to the sliding surface. Therefore, any system trajectory driven by the input control will reach the sliding surface in a finite time from any initial condition x0 in spite of a certain class of uncertainties/disturbances before some certain time, which can be upper bounded uniformly by a constant independent of the initial conditions. We will propose two types of controllers that also correspond to two different classes Wnv of perturbations. One of them is based on a relay-type controller (Section 4.1), while the other is based on a generalized STC (Section 4.2). 3. Uniform sliding surface Let us take a uniform sliding surface described by s = x2 + c1 x1 + c2 |x1 |q sign(x1 ),
(3.1)
where c1 , c2 > 0 and q > 0 are positive scalars. We will show that if q > 1, the convergence rate becomes so strong that the convergence time to a neighbourhood of the origin is uniform w.r.t. initial
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
where the constant c1 is chosen to ensure the desired eigenvalue for (2.3). Once the sliding surface has been selected, a control input must be obtained to reach the sliding surface in finite time from an arbitrary initial condition and to ensure the sliding motion despite of (matched) perturbations. The standard SMC design suggests that the control law that enforces the sliding mode should be designed such that the sufficient condition s s˙ < −n|s| is satisfied (Utkin, 1992). It guarantees the sliding mode existence. For system (2.1), a control law that satisfies this condition is designed as
494
E. CRUZ-ZAVALA ET AL.
conditions, i.e. the convergence time can be upper bounded by a positive constant. Note that when q = 1, the usual linear sliding surface is obtained. Introducing (s, x1 ) as new state variables (where s is defined as in (3.1)) and applying the control input u = u eq + v,
(3.2)
−qc2 |x1 |q−1 [(a11 − a12 c1 )x1 − a12 c2 |x1 |q sign(x1 ) + a12 s],
(3.3)
the system (2.1) may be represented as x˙1 = (a11 − a12 c1 )x1 − a12 c2 |x1 |q sign(x1 ) + a12 s, s˙ = v + w(x, t).
(3.4)
The term u eq compensates the nominal (known) dynamics in the system. It is easy to see that the dynamics of system (2.1), in the sliding mode (i.e. s = 0), are governed by the differential equation x˙1 = (a11 − a12 c1 )x1 − a12 c2 |x1 |q sign(x1 ).
(3.5)
|q
The high degree term |x1 sign(x1 ) (when q > 1) accelerates the convergence rate to the origin since it is stronger the further the initial condition is. As a consequence, the convergence time of any trajectory to a vicinity of the origin can be bounded by a constant independent of the initial condition! T HEOREM 3.1 If c1 and c2 are selected such that a11 − a12 c1 < 0 and a12 c2 > 0, the reduced system (3.5) is asymptotically and uniformly stable w.r.t. initial conditions, i.e. it will reach the origin asymptotically but, furthermore, every trajectory will reach a neighbourhood of the origin of radius μ > 0 before 1 α Tμ = ln 1 + , (3.6) (q − 1)β βμq−1 where α = −a11 + a12 c1 and β = a12 c2 . Proof. When a11 − a12 c1 < 0 and a12 c2 > 0, the equilibrium point is asymptotically stable. This can be easily demonstrated using the Lyapunov function Vs = |x1 |. The time derivative along the trajectories q of (3.5) is given by V˙s = −αVs − βVs , Vs (x1 (0)) = v 0 0 . Then, V˙s is negative definite. For q > 1, the solution of the differential equation for Vs is given by α 1−q (3.7) Vs (t)1−q = exp(−(1 − q)βt)v 0 − exp(−(1 − q)βt)[exp((1 − q)βt) − 1]. β The time Ts required for a trajectory starting at the initial state x1 (0) to reach a level set Vs = μ, where 0 < μ < Vs (x1 (0)), is determined by q−1 βv 0 1 βμq−1 T1 (v 0 , μ) = ln − ln . q−1 (q − 1)β α + βμq−1 α + βv 0
The time T1 (v 0 , μ) depends on the parameters c1 , c2 , μ and is upper bounded by a constant independent of the initial conditions x1 (0) ∈ Vs (x0 ). Moreover, since limv 0 →∞ T1 (v 0 , μ) = Tμ , the convergence time T1 (v 0 , μ) of every trajectory is uniformly upper bounded by (3.6), i.e. T1 (v 0 , μ) Tμ . Note that Tμ does not depend on the initial conditions and that limμ→0 Tμ = ∞. This is due to the fact that the convergence to the origin is exponential.
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
u eq = −[c1 (a11 − a12 c1 ) + (a21 − a22 c1 )]x1 − (a12 c1 + a22 )s + (c1 a12 + a22 )c2 |x1 |q sign(x1 )
UNIFORM SLIDING MODE CONTROLLERS AND UNIFORM SLIDING SURFACE
495
4. Uniform controllers with respect to the initial conditions
4.1 First-order uniform sliding mode based control A control law should be chosen in such a way that, from any initial condition, the system trajectories are attracted towards the sliding surface uniformly and in finite time even in presence of uncertainties/disturbances, i.e. the system trajectories will reach the sliding surface in a finite time bounded by a constant and then slide along the uniform sliding surface. Once there, the system is invariant against parameter variations and disturbances satisfying the matching condition. As the uncertainties/disturbances of a special class supported by the system depend on the control law selected, we consider for this first-order uniform sliding controller (FOUSC) that w(x, t) belongs to the class W1 = {w: |w(x, t)| 0 + 02 |s| + 03 |s|qv , 0 , 02 , 03 > 0, qv > 1}. Note that these functions are uniformly bounded in t, ∀t > 0. T HEOREM 4.1 Suppose that the uncertainty/disturbance w(x, t) ∈ W1 . Then, the control law (3.2), with u eq as in (3.3) and v = −Q 0 sign(s) − K 2 s − K 3 |s|qv sign(s),
qv > 1,
(4.1)
where Q 0 , K 2 and K 3 are positive constants, enforces every trajectory of (3.4) to move from any initial condition to the sliding surface in a finite time and thereafter ensures it remains on the sliding surface. Moreover, the sliding manifold s = 0 is ‘uniformly finite-time stable w.r.t initial conditions x0 and uncertainties/disturbances’, i.e. every trajectory reaches it in spite of any uncertainty/disturbance w(x, t) ∈ W1 in a finite time smaller than κ02 12 2 2 − 1 (qv −1) μs 2 + ln μs + 1 , (4.2) Tf = κ01 (qv − 1)κ03 κ02 where κ01 = root of
√
2(Q 0 − 0 ), κ02 = 2(K 2 − 02 ), κ03 = 2(qv +1)/2 (K 3 − 03 ) and μs is the positive real 1
qv
μs2 + (κ01 /κ02 ) = (κ03 /κ02 )μs2 .
(4.3)
Proof. To show that the proposed control law ensures that the state trajectories reach the manifold s = 0 before a constant time T f > 0, we use the quadratic function V = 0.5s 2 . Since V˙ = s(v + w(x, t)),
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
The previous theorem has shown that for every x0 on the sliding surface, the trajectories of system (3.5) will be in the interior of a ball Bμ = {x1 : |x1 | < μ} centred at the origin after a time Tμ , independent of the initial condition. Once the sliding surface has been specified, subject to some performance requirements for the system, the new control input v must be designed to reach the sliding surface in a prescribed finite time, independent of the initial conditions, while also guaranteeing the sliding mode. There exist some control laws based on the classical design that enforce the sliding mode (see, e.g. Hung et al., 1993). These control laws can be obtained by the so-called ‘reaching law’ approach in which the switching function dynamics are specified a priori; they are basically based on a first-order sliding mode (FOSM). However, they are not able to achieve the property of convergence uniform in the initial conditions we are interested in. In the following two subsections, we will describe two controllers to assure this property.
496
E. CRUZ-ZAVALA ET AL.
applying the control (4.1) and considering the fact that w(x, t) ∈ W1 , leads to V˙ s(−Q 0 sign(s) − K 2 s − K 3 |s|qv sign(s) + 0 + 02 |s| + 03 |s|qv ) −(Q 0 − 0 )|s| − (K 2 − 02 )|s|2 − (K 3 − 03 )|s|qv +1 ,
1
1
v(t) 2 = exp(−κ02 t/2)v 02 −
κ01 exp(−κ02 t/2)[exp(κ02 t/2) − 1], κ02
(4.4)
and the solution of the differential equation v˙ = −κ03 v (qv +1)/2 , v(0) = v 0 0, for qv > 1, is given by 1 − 2 qv −1 − (qv −1) v(t) = v 0 2 + (qv − 1)κ03 t/2 . It follows from the comparison principle (Khalil, 2002) that V (s(t)) v(t) when V0 = V (s(x0 )) v 0 , and therefore, 1
2 κ01 1 1 2 V0 − , κ02 t − 1 exp V (t) min V1 = exp − κ02 t κ02 2 2 − 2 qv −1 − 12 (qv −1) ×V2 = V0 + (qv − 1)κ03 t/2 . This expression allows estimating the convergence time. First, consider a trajectory starting at point x0 at an energy level V0 . An upper bound T1 (x0 , μs ) of the time at which it reaches the level set V = μs , for some 0 < μs < V0 , can be calculated from V2 = μs as 1 2 − (qv −1) − 1 (qv −1) − V0 2 μs 2 . T1 (x0 , μs ) = (qv − 1)κ03 Now, starting from the level set V = μs , an upper bound T2 (μs ) of the time at which it reaches the point s = 0 can be calculated from V1 = 0 as κ02 12 2 ln μs + 1 , T2 (μs ) = κ01 κ02 and therefore, the convergence time of every trajectory from any x0 ∈ V0 to the sliding surface can be estimated as 1 κ02 12 2 2 − (qv −1) − 1 (qv −1) − V0 2 ln μs + 1 . μs 2 + T (x0 , μs ) = T1 (x0 , μs ) + T2 (μs ) = κ01 (qv − 1)κ03 κ02
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
√ which may be rewritten as V˙ −κ01 V 1/2 − κ02 V − κ03 V (qv +1)/2 , where κ01 = 2(Q 0 − 0 ), κ02 = 2(K 2 − 02 ), κ03 = 2(qv +1)/2 (K 3 − 03 ). This implies V˙ < 0, for Q 0 > 0 , K 2 > 02 , K 3 > 03 . It is clear that every trajectory converges to s = 0 in a finite time. After this, the trajectory will slide along the sliding surface. Finally, we will show that there exits some constant T f > 0, independent of the initial conditions, such that ∀ t T f the state trajectories x(t) cannot leave s = 0. Since V satisfies both differential inequalities V˙ −κ01 V 1/2 − κ02 V and V˙ −κ03 V (qv +1)/2 , the value of V is below the solution of both inequalities. Note that the solution of the differential equation v˙ = 1 −κ01 v 2 − κ02 v, v(0) = v 0 0 is given by
UNIFORM SLIDING MODE CONTROLLERS AND UNIFORM SLIDING SURFACE
497
The convergence time T (x0 , μs ) is uniformly upper bounded by the constant T f , i.e. T (x0 , μs ) T f . Choosing μs as the positive real root of (4.3) ensures the best estimation of the time T f .
4.2
Uniform super twisting-based sliding mode enforcement
One disadvantage of using a relay (or unit controllers) sliding mode in the control law to enforce the sliding motion is that it produces the so-called ‘chattering effect’. To eliminate the high frequency component of these controllers, a low-pass filter is frequently used with the FOSM. Another approach to deal with this problem is using high-order sliding modes. Up until now, only a few two sliding controllers have been proposed (Levant, 2007). A second-order sliding mode, called the ‘Super-Twisting Algorithm’ (STA), has been widely used as an ‘absolutely continuous’ controller because it depends only on the sliding constraint. The STA is effective in alleviating the chattering effect and it ensures all the main properties of a first-order SMC for a system with Lipschitz continuous matched uncertainties/disturbances with bounded gradients. Taking into account this advantages, we will introduce a controller based on the generalized STA (Moreno, 2011). The super-twisting uniform sliding controller (STUSC) is given by
t
v = −k1 φ1 (s) − k2
φ2 (s)dt,
(4.5)
0
where k1 and k2 are the gains to be designed and 1
3
φ1 (s) = μ1 |s| 2 sign(s) + μ2 s + μ3 |s| 2 sign(s), 1 3 1 3 5 3 φ2 (s) = μ21 sign(s) + μ1 μ2 |s| 2 sign(s) + (μ22 + 2μ1 μ3 )s + μ2 μ3 |s| 2 sign(s) + μ23 |s|2 sign(s), 2 2 2 2
with μ1 , μ2 and μ3 being positive constants. The STUSC is based on the standard STC that may be recovered by setting μ2 = μ3 = 0 and the gains k1 , k2 > 0. Also, the new control law v inherits the robustness properties of an STC. Recall that the FOUSC can deal with a great variety of uncertainties/disturbances, including bounded discontinuous functions belonging to the class W1 . The STUSC, however, is able to compensate uncertainties/disturbances w(x,t) in the class W2 , defined
by W2 = w: |g1 (x1 , x2 , t)| 1 |φ1 (s)|, | dtd g2 (x1 , x2 , t)| 2 |tφ2 (s)| , for some known constants 1 , 2 0, where w(x1 , x2 , t) = g1 (x1 , x2 , t) + g2 (x1 , x2 , t),
(4.6)
is a decomposition of the signal. Note that the classes W1 and W2 are disjoint. For example, bounded discontinuous functions belong to W1 but not to W2 , whereas continuous unbounded functions belong to W2 but not to W1 .
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
The control law (4.1) was derived using a quadratic Lyapunov function and it is equivalent to satisfying the reachability condition s s˙ < −η|s|. Alternatively, different switching function dynamics can be obtained from the control law proposed (see Hung et al., 1993). Furthermore, K 3 > 0 allows for a faster convergence and deals with uncertainties/disturbances growing with a degree larger than a linear term in s. K 2 > 0 allows to deal with uncertainties/disturbances growing with a linear term in s.
498
E. CRUZ-ZAVALA ET AL.
System (3.4) along with controller (4.5) and the disturbance w ∈ W2 can be extended as follows: x˙1 = (a11 − a12 c1 )x1 − a12 c2 |x1 |q sign(x1 )) + a12 s, s˙ = −k1 φ1 (s) + ς + g1 (x1 , x2 , t), d g2 (x1 , x2 , t). dt
(4.7)
T HEOREM 4.2 The control law (3.2), with u eq as in (3.3) and v as in (4.5), enforces every trajectory of (3.4) to move from any initial condition to the sliding surface in a finite time and to remain on it thereafter. Moreover, the sliding manifold s = 0 is uniformly finite-time stable w.r.t initial conditions x 0 and uncertainties/disturbances, i.e. every trajectory reaches it in spite of any uncertainty/disturbance w(x, t) ∈ W2 in a finite time smaller than κ2 12 6 −1 2 TSTC = μss 6 + ln μss + 1 , (4.8) κ1 κ3 κ2 where 1 1 14 v min C3 3 2 , v min = μ33 , 7 2 2 (λmax {P} + C2 ) 6 (4.9) and , λmax {P}, C2 and C3 are positive scalars depending on the selection of the parameters k1 , k2 , 1 , 2 and μss is the positive real root of μ21
μ2
, κ2 = , κ3 = κ1 = 1 (λmax {P} + C2 ) 2(λmax {P} + C2 ) 2
1
2
3 2 μss + (κ1 /κ2 ) = (κ3 /κ2 )μss .
(4.10)
Proof. See Section 5.
Both uniform controllers enforce the sliding mode for every trajectory of system (2.1), taking into account their respective uncertainties/disturbances. Actually, the uniform controllers enforce the system trajectories to the uniform sliding surface before a time T f or TSTC , which are positive constants. Once in the sliding motion, the trajectories converge within a time Tμ to an arbitrary neighbourhood of the origin of radius μ > 0. As a result, the system trajectories reach an arbitrary ball Bμ (μ > 0) in a time bounded by another constant. Therefore, we can conclude that a system driven by FOUSC or STUSC is ‘asymptotically uniformly stable w.r.t. initial conditions’ for any uncertainty/disturbance w(x, t) belonging to the class W1 or W2 , respectively. 5. Proof of Theorem 4.1 We propose a Lyapunov-based approach to show the uniform and finite-time convergence of the system trajectories from any initial condition to the non-linear sliding surface. Consider the continuous function W (s, ς) = V1 (ζ ) + V2 (s, ς),
(5.1)
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
ς˙ = −k2 φ2 (s) +
499
UNIFORM SLIDING MODE CONTROLLERS AND UNIFORM SLIDING SURFACE
as a candidate Lyapunov function. The function V1 (ζ ) = ζ Pζ is quadratic in the vector ζ = φ1 (s) ς and the definite positive symmetric matrix P = P > 0 is the solution of a linear matrix inequality (LMI)
A P + P A + R + I PB < 0, (5.2) B P −Θ B=
0 , 1
1 0
C= 1
T
0
,
Θ=
θ1 0 , 0 θ2
2
(5.3)
4
∀ θ1 , θ2 0 and with > 0. The function V2 (s, ς) = δk2 |φ1 (s)|2 −|φ1 (s)| 3 sign(s)|ς| 3 sign(ς)+δ|ς|2 and δ is a positive constant which satisfies 1
3
7
7
7
7
δ > (4/27k2 ) 3 , 7δk2 (k1 − 1 ) > 4(3) 2 24 δ 4 + (k1 + 1 ) 3 + 4(k2 + 2 ) 6 .
(5.4)
The following result shows that the subsystem (s, ζ ) of (4.7) is robust against uncertainties/ disturbances w(x, t) ∈ W2 when the gains (k1 , k2 ) and δ are selected properly. P ROPOSITION 5.1 The Lyapunov function (5.1) is a ‘strong, robust Lyapunov function for the subsystem (s, ς) of (4.7). Moreover, the time derivative W˙ of the Lyapunov function taken along the trajectories of the subsystem satisfies the differential inequality W˙ (s, ς) −κ1 W 2 (s, ς) − κ2 W (s, ς) − κ3 W 6 (s, ς), 1
7
(5.5)
with κ1 , κ2 and κ3 as in (4.9). Proof. First, we establish an inequality derived from the classical Young’s inequality. It is used extensively to show the positive and negative definiteness of V2 (s, ς). The proof can be consulted in Cruz Zavala et al. (2011). L EMMA 5.1 For every real numbers a > 0, b > 0, c > 0, p > 1, q > 1, with inequality is satisfied:
1 p
+
1 q
= 1 the following
ab c p a p / p + c−q bq /q. To show positive definiteness of V2 (s, ς), we use Lemma 5.1, and let γ00 and γ01 be constants that satisfy 2
4
2
4
−3
|φ1 (s)| 3 sign(s)|ς| 3 sign(ς) |φ1 (s)| 3 |ς| 3 γ0i3 |φ1 (s)|2 /3 + 2γ0i 2 |ς|2 /3,
∀ γ0i > 0, i = 0, 1,
therefore, ∀(s, ς), the function V2 can be bounded as α1 (γ00 )|φ1 (s)|2 + α2 (γ00 )|ς|2 V2 (s, ς) α2 (γ01 )|φ1 (s)|2 + α4 (γ01 )|ς |2 , 3 3 /3), α (γ ) = δ − 2γ − 2 /3 , α (γ ) = (δk + γ 3 /3) and α (γ ) = where α1 (γ00 ) = (δk2 − γ00 2 00 3 01 2 4 01 00 01 − 32 δ + 2γ01 /3 . Also, V2 (s, ς) is positive definite if and only if α1 (γ00 ) > 0 and α2 (γ00 ) > 0, which is always possible if δ satisfies (5.4). Moreover, V2 can be bounded as
C1 ζ 22 = C1 (|φ1 (s)|2 + |ς |2 ) V2 (s, ς) C2 (|φ1 (s)|2 + |ς |2 ) = C2 ζ 22 ,
(5.6)
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
where R = (θ1 12 + θ2 22 )C C,
−k1 1 A= , −k2 0
500
E. CRUZ-ZAVALA ET AL. 2
2 /3 and C = δk + z 2 /3 are positive constants, z = γ 3 is the positive real root where C1 = δk2 − z m 2 2 m M 00 2
Using the following inequalities derived from Lemma (5.1) 4
7
−7
7
∀γ1 > 0,
7 3
7
− 74
7
∀γ2 = (k1 + 1 ) 7 > 0,
7
7
7
4
φ1 (s) 3 ς |φ1 (s)| 3 |ς| 4γ14 |φ1 (s)| 3 /7 + 3γ1 3 |ς | 3 /7, 4
4
4
φ1 (s)|ς | 3 sign(ς) |φ1 (s)||ς | 3 3γ2 |φ1 (s)| 3 /7 + 4γ2 |ς | 3 /7, |φ1 (s)|2 |ς| 3 6γ36 |φ1 (s)| 3 /7 + γ3−7 |ς | 3 /7, 1
we obtain
7
1
∀γ3 = (k2 + 2 ) 7 > 0,
7 7 V˙2 (s, ς) = −v 1 (s) ψ1 |φ1 (s)| 3 + ψ2 |ς | 3 ,
where v 1 (s) = |φ1 (s)| 7
−1 3
−7
7
φ1 (s), ψ1 = 2(Υ1 − 42 δγ14 /7), ψ2 = 2(1 − 92 δγ1 3 )/21, Υ1 = δk2 (k1 − 7
4
3
1 )−(k1 +1 ) 3 /7−4(k2 +2 ) 6 /7 and γ1 always exits iff it satisfies (7Υ1 /82 δ) 7 > γ1 > (92 δ) 7 . For negative definiteness of V˙2 , it is required that ψ1 , ψ2 > 0, which is always possible if (5.4) is satisfied. Note that ∀s ∈ R, the function v 1 (s) > 0 and −1
−1
μ1 1 −1 + μ |s| 2 + 3 μ μ1 |s| 2 + μ2 + 32 μ3 s 2 2 2 3 2 |s| lim v 1 (s) = 2 = 1 1 3 1 −3 |s|→∞ −1 μ1 |s| 2 + μ2 |s| + μ3 |s| 2 3 μ1 |s| + μ2 |s| 2 + μ3 3 1
2
3 3 μ 3 2 = μ33 , lim v(s) = 2 21 = ∞. |s|→0 2 |s| 3
We immediately establish that the minimum v min = mins∈R v 1 (s) exists and is positive. Then, this leads to 7 7 7 7 V˙2 (s, ς) −v 1 (s) ψ1 |φ1 (s)| 3 + ψ2 |ς| 3 −v min C3 |φ1 (s)| 3 + |ς | 3 , 7
where C3 = 2(1 − 92 δ/ym )/21 and ym = γ13 is the positive real root of 32 δ + 7(Υ1 − 1/21)ym = 7
42 δym4 . Using the following standard inequality (see Hardy et al., 1951) 1
1
(5.7) (α|x1 |s + (1 − α)|x2 |s ) s (α|x1 |r + (1 − α)|x2 |r ) r , 0 α 1, ∀x ∈ R 2 , s < r, 7 7 7 1 clearly we can find that (1/2) 14 (|φ1 (s)|2 + |ς|2 ) 6 |φ1 (s)| 3 + |ς | 3 . Finally, using this inequality along with the inequality (5.6), we find that the derivative of the Lyapunov function V2 (s, ς) satisfies 7 7 1 7 1 7 V˙2 −v min C3 |φ1 (s)| 3 + |ς | 3 −(1/2) 14 v min C3 (|φ1 (s)|2 + |ς |2 ) 6 −(1/2) 14 v min C3 ( ζ 22 ) 6 .
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
3 and z = γ 3 is the positive real root of 2 + 3δ(1 − k )z = z 3 . Since the of 2 + 3δ(k2 − 1)z m = z m M 2 M M 01 uncertainty/disturbance term w(x, t) ∈ W2 , the time derivative of V2 is given by −1 7 4 4 2 ˙ 3 V2 −|φ1 (s)| φ1 (s) 2δk2 (k1 − g1 )|φ1 (s)| 3 − 2g2 δ|φ1 (s)| 3 ς − (k1 + g1 )φ1 (s)|ς | 3 sign (ς ) 3 1 4 2 7 2 3 3 − (k2 + g2 )|φ1 (s)| |ς| + |ς| . 3 3
501
UNIFORM SLIDING MODE CONTROLLERS AND UNIFORM SLIDING SURFACE
On other hand, for the following analysis, we will present subsystem (4.7) in a convenient form. Using the vector ζ , subsystem (4.7) can be written as ⎤ ⎡ g1 (x1 , x2 , t) −k1 φ1 (s) + ς + g1 (x1 , x2 , t) d d ⎦ ζ˙ = φ1 (s) = φ1 (s)(Aζ + B ρ), ˜ ρ˜ = ⎣ , g (x ,x ,t) dt g2 (x 1 ,x 2 ,t) 1 3 −k2 φ2 (s) + dt 2 1 2 1 3 φ1 (s)
2 μ1 |s|
2 +μ2 +
2 μ3 |s|
2
s=φ −1 (ζ )
ωi (ρ˜i , ζ ) =
−ρ˜i2 (t, ζ ) + i2 ζ12
ρ˜ = i ζi
−1 0 0 i2 C C
ρ˜i ζi
0,
where C is defined in (5.3). It means that the uncertainty/disturbance w(x, t) ∈ W2 can be expressed as a sector condition (in ζ coordinates), i.e. |ρ˜i (t, ζ )| i |φi (s)| in the original variables, which is equivalent to |ρ˜i (t, ζ )| i |ζi |, with i > 0, in transformed coordinates. It follows that ω(ρ, ˜ ζ) = θ1 ω1 (ρ˜1 , ζ ) + θ2 ω2 (ρ˜2 , ζ ) 0, ∀θ1 , θ2 0 and hence ω(ρ, ˜ ζ) =
ρ(t, ˜ ζ) ζ
−Θ 0
0 R
ρ(t, ˜ ζ) , ζ
(5.8)
with Θ and R as in (5.3). The derivative of the Lyapunov function V1 (ζ ) is
ζ A P + PA PB A P + P A P B ζ ζ ζ ˙ V1 = φ1 φ1 + ω(ρ, ˜ ζ) ρ˜ ρ˜ ρ˜ ρ˜ B P B P 0 0 = φ1
ζ A P + PA + R PB ζ −φ1 ζ 22 . ρ˜ ρ˜ B P −Θ
For negative definiteness of V˙1 , the feasibility of the LMI (5.2) is required. Recall the standard inequality from the quadratic forms λmin {P} ζ 22 ζ Pζ λmax {P} ζ 22 , where ζ 22 = φ12 (s) + 5
3
ς 2 = μ22 |s|2 + μ23 |s|3 + 2μ2 μ3 |s| 2 + μ21 |s| + 2μ1 μ2 |s| 2 + 2μ1 μ3 |s|2 + ς22 is the Euclidean norm of 1 −1 1 ( ζ 2 )−1 is always satisfied. Then, as φ1 (s) = μ1 |s| 2 /2 + ζ . From this, we can see that μ1 |s| 2 1
μ2 + 3μ3 |s| 2 /2, we immediately establish V˙1 −φ1 (s) ζ 22 −(μ21 /2) ζ 2 − μ2 ζ 22 − (3μ3 /2) |s| 2 ζ 22 . 1
Thanks to the negative definiteness of V˙1 (ζ ) and V˙2 (s, ς), 1
W˙ (s, ς) = V˙1 + V˙2 −(μ21 /2) ζ 22 − μ2 ζ 22 − (3μ3 /2) |x1 | 2 ζ 22 − (1/2) 14 v min C3 ( ζ 22 ) 6 . 1
1
7
As both Lyapunov functions satisfy λmin {P} ζ 22 V1 (ζ ) λmax {P} ζ 22 and C1 ζ 22 V2 (s, ς) C2 ζ 22 , the function W (s, ς) can be upper bounded as W (s, ς) (λmax {P} + C2 ) ζ 22 , which allows us to obtain (5.5).
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
with A and B as in (5.3). To take into account a bigger variety of disturbances, it will be assumed that the components of the (transformed) uncertainty/disturbance term ρ(t, ˜ ζ ) satisfy the sector conditions (for i = 1, 2, ∀t 0 and ∀ζ ∈ R 2 )
502
E. CRUZ-ZAVALA ET AL.
− 16
v(t) = (v 0
+ κ3 t/6)−6 .
From the comparison principle (Khalil, 2002), it follows that W (s(t), ς(t)) v(t) when we have W0 = W (s(x0 ), ς(x0 )) v 0 . Then 1
2 −6 κ1 1 1 − 16 2 W0 − , W0 + κ3 t/6 κ2 t − 1 . exp W (t) min exp − κ2 t κ2 2 2 The convergence time can be estimated in similar way as in the proof of Theorem 4.1. First, calculate an upper bound T1 (x0 , μss ) of the convergence time of a trajectory starting at point x0 at an energy level W0 at which it reaches the level set W (s, ς) = μss (for some 0 < μss < W0 ). Afterwards, calculate an upper bound T2 (μss ) at which s = 0 is reached (starting from this level set W = μss ). Then, every trajectory reaches the sliding surface within a time 6 − 16 2 κ2 12 −1 ln μss + 1 . μss − W0 6 + T (x0 , μss ) = T1 (x0 , μss ) + T2 (μss ) = κ1 κ3 κ2 The convergence time T (x0 , μss ) is uniformly upper bounded by the constant TSTC , i.e. T (x0 , μss ) TSTC . To ensure the best estimation of the time TSTC , choose μss as the positive real root of (4.10). 6. Simulation results of an academic example This section is devoted to illustrating the effectiveness of the proposed controllers for the control of a hybrid system with strictly positive dwell time. Consider the following hybrid linear system modelled by the equation x˙ = Aq x + b(u + w(t)), where x ∈ R 2 is the continuous state, q ∈ [1, 2] is the discrete state that indexes the subsystems
0 1 1 1 , A2 = . A1 = 2 −1 2 1
Both A1 and A2 are unstable, b = 0 1 , w(t) = 0.5 sin(2t) + 0.5 cos(5t) and the control input u is defined by (3.2). The control task is to stabilize the origin x = 0 of the system, with convergence to a neighbourhood of the origin in a prescribed time for every trajectory of the system. The system dynamics change every 4 s. If only the state x1 is measured, the pairs [C, A1 ] and [C, A2 ] are observable with the output vector C = 1 0 . In this case, it is possible to design a uniform differentiator (Cruz Zavala et al., 2011), which provides the system states after at most 1 s. Therefore, we can turn on the controller after 1 s. The sliding surfaces 3 3 for each system are designed as s A1 = x2 + x1 + 0.5|x1 | 2 sign(x1 ) and s A2 = x2 + 2x1 + |x1 | 2 sign(x1 ),
the initial conditions x(0) = 1.5 1 and the sampling time (for the numerical simulation) is τ = 0.001. For the simulation, we design and compare three controllers (3.2), where:
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
It is clear that every trajectory converges to zero in a finite time. The proof of the uniform convergence follows immediately from Proposition 5.1. Since W satisfies both differential inequalities 7 1 W˙ (s, ς) −κ1 W 2 (s, ς) − κ2 W (s, ς) and W˙ (s, ς) −κ3 W 6 (s, ς), the value of W is below the 1 solution of any of the two inequalities. Note that the solution of the differential equation v˙ = −κ1 v 2 − 7 κ2 v, v(0) = v 0 0 is similar to (4.4) and that the solution of the differential equation v˙ = −κ3 v 6 , v(0) = v 0 0 is given by
UNIFORM SLIDING MODE CONTROLLERS AND UNIFORM SLIDING SURFACE
503
1. v = −M0 sign(s), the first-order sliding controller (FOSC), with M0 = 4. 3
2. v = −Q 0 sign(s) − K 2 s − K 3 |s| 2 sign(s), the FOUSC, with Q 0 = 4, K 2 = 2, K 3 = 1. t 3. v = −k1 φ1 (s) − k2 0 φ2 (s)dt, the super-twisting uniform sliding controller (STUSC), with μ1 = 2, μ2 = 1, μ3 = 0.5.
FIG. 1. (a) and (b) show the states x1 and x2 with the: FOSC (dashed line), FOUSC (dotted line) and STUSC (solid line); (c) Surfaces and (d) Control input.
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
Note that with the FOUSC, the disturbance w(t) ∈ W1 = {w(t): |w(x, t)| 0 + 02 |s| +
03 |s|qv , 0 = 1, 02 = 0, 03 = 0, qv > 1} while w(t) ∈ W2 = w(t): |g1 (x1 , s, t)| 1 |φ1 (s)|, |w(t)| ˙ = dtd g2 (x1 , t) 2 |φ2 (s)| , with 1 = 0 and 2 = 1 when the STUSC is used. For this case, when 1 = 0 (see Moreno, 2011), the gains k1 and k2 should be selected in the set κ = {(k1 , k2 ) ∈ √ √ R 2 |0 < k1 2 ρ2 , k2 > 0.25k12 + 4ρ2 /k12 } {(k1 , k2 ) ∈ R 2 |k1 > 2 ρ2 , k2 > 22 }. We choose k1 = 2 and k2 = 4. The simulation results are shown in the Fig. 1. Clearly, both the FOUSC and the STUSC enforce the sliding mode but not the FOSC. Moreover, the STUSC provides chattering alleviation. Nevertheless, the value of the control signal during the transient response for both controllers, FOUSC and STUSC, is highly increased (see Fig. 1(d)). This is a natural consequence of the property of uniform convergence w.r.t. initial conditions since the control action has to be very strong in order to attract the system trajectories, starting far away from the origin, to a neighbourhood of the origin in a time bounded by a constant. Figure 2(a) shows a linear sliding surface and the uniform surfaces s A1 = 0 and s A2 = 0, which are clearly non-linear. Considering that the control action starts since t = 0 at the first operation mode, during the first 4 s, Fig. 2(b) puts in evidence that the convergence time to the surfaces grows to infinity with the growth of the initial conditions for the FOSC while the convergence time using the FOUSC and the STUSC is uniformly bounded by a constant.
504
E. CRUZ-ZAVALA ET AL.
In order to estimate the convergence time to the sliding surfaces using the FOUSC, it is required calculating κ1 , κ2 , κ3 and obtaining μss from (4.3) and then substituting all the corresponding values in expression (4.2). In a similar way, when the STUSC is used, for the convergence time estimation the following steps are required: 1. Find P = P > 0 and > 0 such that the LMI (5.2) is satisfied. Choose δ in such a way that the inequalities (5.4) are satisfied. Then, find z M and ym to obtain C2 and C3 and compute κ1 , κ2 , κ3 . 2. Calculate μss from (4.10) and substitute all the corresponding values in expression (4.8). Finally, Fig. 2(c) shows the real convergence time applying the FOUSC in the system with its estimation. The upper bound of the estimation is approximately twice the real upper bound of the real convergence time. On the other hand, Tmax = 301.34 s was the best convergence time estimation using the STUSC. However, the time estimation for the STUSC is cumbersome. 7. Concluding remarks A non-linear sliding surface has been suggested ensuring, during the sliding motion, convergence of the system trajectories to any arbitrary small vicinity of the origin in a finite time bounded by some constant independent of the initial conditions on the surface. Two types of uniform sliding controllers have been proposed. At the beginning, a FOUSC controller is designed as a generalization of the FOSM algorithm inheriting all its properties. This controller also provides finite-time convergence uniform w.r.t. initial conditions of the trajectories to the sliding surface in presence of a special class of uncertainties/disturbances. To adjust the chattering effect produced by
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
FIG. 2. (a) A linear sliding surface sL = x2 + x1 and the uniform sliding surfaces s A1 and s A2 , (b) convergence time to the sliding surface during the first operation mode (simulation data) and (c) comparison between the real convergence time and its estimation.
UNIFORM SLIDING MODE CONTROLLERS AND UNIFORM SLIDING SURFACE
505
the first controller, an absolutely continuous super-twisting uniform controller is suggested providing for finite-time convergence uniform w.r.t. initial conditions of the trajectories to the sliding surface in presence of another uncertainty/disturbance class. A Lyapunov-based approach is used in the paper to estimate the convergence time to the sliding surface and to an arbitrary small vicinity of origin.
The authors gratefully acknowledge the financial support form: (i) Consejo Nacional de Ciencia y Tecnolog´ıa (51244, 132125) and CVU (267513), (ii) Programa de Apoyo a Proyectos de Investigaci´on e Innovaci´on Tecnol´ogica, UNAM (111012, 117610 and 117211) and (iii) Fondo de Colaboraci´on del II-FI, UNAM (IISGBAS-165-2011). R EFERENCES A NDRIEU , V., P RALY, L. & A STOLFI , A. (2008) Homogeneous approximation, recursive observer design and output feedback. SIAM J. Control Optim., 47, 1814–1850. BANDYOPADHYAY, B., D EEPAK , F. & K IM , K. S. (2009) Sliding Mode Control Using Novel Sliding Surfaces. Lecture Notes in Control and Information Sciences, vol. 392. Berlin: Springer, 136 p. BARTOSZEWICZ , A. & N OWACKA -L EVERTON , A. (2009) Time-Varying SlidingModes for Second and Third Order Systems. Lecture Notes in Control and Information Sciences, vol. 382. Berlin: Springer, 200 p. C RUZ Z AVALA , E., M ORENO , J. A. & F RIDMAN , L. (2011) Uniform robust exact differentiator. IEEE Trans. Automat. Control, 56, 2727–2733. F ILIPPOV, A. F. (1988) Differential Equations with Discontinuous Right Hand Side. Dordrecht, The Netherlands: Kluwer, 304 p. G ONZALEZ , T., A., M ORENO , J. A. & F RIDMAN , L. (2011) Variable gains super-twisting algorithm. IEEE Trans. Automat. Control. doi: 10.1109/TAC.2011.2179878. ´ H ARDY, G. H., L ITTLEWOOD , J. E. & P OLYA , G. (1951) Inequalities. Cambridge University Press: London. H UNG , J. Y., G AO , W. & H UNG , J. C. (1993) Variable structure control: a survey. IEEE Trans. Ind. Electron., 40, 2–22. K HALIL , H. K. (2002) Nonlinear Systems, 3rd edn. Upsaddle River, NJ: Prentice–Hall, 750 p. L EVANT, A. (1993) Sliding order and sliding accuracy in sliding mode control. Int. J. Control, 58, 1247–1263. L EVANT, A. (2007) Construction principles of 2-sliding mode design. Automatica, 43, 576–586. ´ , A. G. & U TKIN , V. I. (1981) Methods of reducing equations for dynamic systems to a regular form. L UK YANOV Automat. Remote Control, 42, 413–420. M ORENO , J. A. (2011) Lyapunov approach for analysis and design of second order sliding mode algorithms. Sliding Modes after the First Decade of the 21st Century (L. Fridman, J. Moreno & R. Iriarte eds). Lecture Notes in Control and Information Sciences, vol. 412. Berlin: Springer, pp. 113–150. S HTESSEL , Y. B., Z INOBER , A. S. I. & S HKOLNIKOV, I. A. (2002) Boost and buck-boost power converters control via sliding modes using dynamic sliding manifold. 41th IEEE Conference on Decision and Control, Las Vegas, NV, pp. 2456–2461. U TKIN , V. I. (1992) Sliding Modes in Control and Optimization. Berlin: Springer.
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
Acknowledgments
Uniform sliding mode controllers and uniform sliding surfaces
[Received on 11 March 2011; revised on 19 November 2011; accepted on 5 January 2012] In this note, we propose two uniform sliding mode controllers (SMCs) for a second-order uncertain system. They provide convergence to an arbitrarily small vicinity centred at the origin in finite time, which can be bounded by some constant independent of the initial conditions and uncertainties. Towards this aim, a non-linear sliding surface is designed ensuring convergence, during the sliding motion, of every trajectory to an arbitrarily small vicinity of the origin in a finite time bounded by some constant, independent of the initial condition on the surface. A first-order SMC is designed providing convergence of every trajectory to the sliding surface in a finite time, bounded by another constant independent of the initial conditions and uncertainties/disturbances of a special class. In order to adjust chattering, a super twisting-based controller is suggested providing convergence of the trajectories to the sliding surface in a finite time bounded by some constant independent of any initial condition and uncertainties/disturbances of another class. Keywords: sliding mode control; Lyapunov methods.
1. Introduction The problem of robust prescribed time stabilization is one of the actual tasks in modern control theory. For example, in hybrid systems control with strictly positive dwell time, it is preferable that the control action provide for robust exact system stabilization before the next switching or impulse take place. A reasonable class of controllers providing both, finite-time convergence and insensitivity with respect to matched uncertainties/disturbances, are sliding mode controllers (SMCs, see, e.g. Utkin, 1992). Traditional SMC design consists of two steps (Luk´yanov & Utkin, 1981; Utkin, 1992): (a) design of a sliding surface to ensure the desired behaviour of the system without uncertainties/disturbances and (b) design of discontinuous controllers to enforce the sliding motion and compensation of the matched uncertainties/disturbances. The main disadvantage of such methodology is the so-called ‘chattering’ phenomenon, restricting the possibilities of implementing first-order SMCs in hardware, where switching is a natural mode of work. A super-twisting controller (STC; Levant, 1993) allows adjusting the chattering problem in systems with Lipschitz continuous (in time) uncertainties/disturbances. It is necessary to remark that in both, classical and super-twisting control design methodologies, the convergence time grows together with the initial conditions, i.e. (a) the convergence time to the sliding surface grows together with the initial conditions and (b) even if the trajectory starts on the sliding surface, the convergence time to a given vicinity around the origin on the sliding surface also grows together with the initial conditions on the surface. A so-called uniform super twisting-based differentiator and an observer for mechanical systems has been designed in Cruz Zavala et al. (2011). The differentiator/observer convergence is guaranteed to c The author 2012. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
E MMANUEL C RUZ -Z AVALA , JAIME A. M ORENO AND L EONID F RIDMAN∗ Instituto de Ingenier´ıa, Universidad Nacional Aut´onoma de M´exico UNAM, 04510 M´exico D.F., Mexico and Departamento de Control Autom´atico, CINVESTAV-IPN, AP-14-740 M´exico, D.F., Mexico ∗ Corresponding author: [email protected]
492
E. CRUZ-ZAVALA ET AL.
•
a non-linear sliding surface is suggested such that, during the sliding motion, the convergence of every trajectory to any arbitrarily small vicinity of the origin will be ensured in a finite time bounded by some constant independent of the initial conditions on the surface.
•
a first-order SMC is designed, providing for the convergence of the system trajectories to the sliding surface in a finite time bounded by some constant independent of the initial conditions and uncertainties/disturbances of a special class.
•
an absolutely continuous super twisting-based controller is suggested guaranteeing convergence of the system trajectories to the sliding surface in a finite time bounded by some constant independent of the initial conditions and another uncertainties/disturbances class and
•
the convergence time for all system trajectories is estimated for both proposed sliding mode enforcement algorithms in terms of the radius of a vicinity centred at the origin and upper bounds of uncertainties/disturbances.
The methodology used in this paper is based on Lyapunov functions proposed in Moreno (2011), Cruz Zavala et al. (2011) and Gonzalez et al. (2011). 2. Problem statement Consider a controllable single input uncertain linear system in regular form: x˙1 = a11 x1 + a12 x2 , x˙2 = a21 x1 + a22 x2 + u + w(x, t),
(2.1)
where x = [x 1 , x2 ] ∈ R 2 is the state vector, u ∈ R1 is the control input, the parameters a11 , a12 = 0, a21 , a22 are constants, w(x, t) ∈ W0 is an uncertainty/disturbance acting on the system and is assumed to belong to the class W0 = {w(x, t): |w(x, t)| 0 , 0 > 0} of signals uniformly bounded in t, ∀ t 0. Note that w(x, t) may be non-vanishing at the origin x = 0 and thus a continuous static feedback law is not able to stabilize the origin of the system. However, classical SMC is able to obtain this objective. In this case, all solutions of the differential equations and inclusion are defined in the sense of Filippov (1988). For its design, it is first necessary to propose the sliding surface. Usually, a linear sliding surface (Utkin, 1992) for system (2.1) is considered as s = x 2 + c1 x 1 .
(2.2)
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
be in a finite time bounded by some constant independent of the initial conditions error. This property is studied in Andrieu et al (2008) for designing a recursive observer. In the case of SMC design, the methodology presented in Gonzalez et al. (2011) can ensure convergence uniform with respect to (w.r.t) initial conditions to only the sliding surface. However, it does not ensure convergence uniform w.r.t any initial condition for the system trajectories on the sliding surface to the origin. On the other hand, the use of non-linear sliding surfaces instead of linear surfaces in classical SMC design has been proved to enhance the desired performance in closed-loop systems with SMC algorithms, which cannot always be achieved using only linear switching surfaces (see Bartoszewicz & Nowacka-Leverton, 2009, Bandyopadhyay et al., 2009; Shtessel et al., 2002 and references there in). In this paper, we suggest two uniform SMCs for a second-order system providing convergence of the system trajectories to any arbitrarily small vicinity of the origin in a finite time bounded by some constant ‘independent from initial conditions and uncertainties/disturbances of a special class’. Towards this aim:
UNIFORM SLIDING MODE CONTROLLERS AND UNIFORM SLIDING SURFACE
493
The sliding mode s(x(t)) = 0 can be ensured in finite time by some discontinuous control law (see, e.g. Utkin, 1992) and, when the sliding motion takes place, the system dynamics are described by the linear system x˙1 = (a11 − a12 c1 )x1 ,
(2.3)
u = −(c1 a11 + a21 )x1 − (c1 a12 + a22 )x2 − Q sign(s),
(2.4)
where Q is chosen from the knowledge of a certain upper bound of the uncertainties, i.e. Q > ρ0 . Note that the classical SMC described above reaches the origin asymptotically, and it enters any neighbourhood of the origin in finite time. However, the further initial conditions are from the origin, the larger the reaching time to a neighbourhood of the origin will be. In fact, when the initial condition tends to infinity, the convergence time also tends to infinity. This means that it is impossible to prescribe the time to reach a neighbourhood of the origin independently of the value of the initial condition. Our objective here is, using the main idea of the SMC design and considering that uncertainties/disturbances belong to some class Wnv of time functions, to design a controller capable of reaching, in a prescribed time, a neighbourhood of the origin of the linear system (2.1) independently of the initial conditions and of the acting (matched) perturbations belonging to class Wnv . To attain this objective we will: 1. Design a (non-linear) sliding surface such that when the motion is restricted to the manifold s = 0, the trajectories converge ‘asymptotically and uniformly w.r.t. initial conditions’ to a neighbourhood of the origin (Section 3). That is, any trajectory on the sliding surface will reach a neighbourhood of radius μ, centred at the origin, before a certain (prescribed) time Tμ > 0, which depends on μ but which is independent of the initial conditions x0 of the trajectory. 2. Design a control law to enforce sliding motion in prescribed time. In this sense, any trajectory will be forced to converge ‘uniformly w.r.t. initial condition and in finite time’ from any arbitrary initial condition to the sliding surface. Therefore, any system trajectory driven by the input control will reach the sliding surface in a finite time from any initial condition x0 in spite of a certain class of uncertainties/disturbances before some certain time, which can be upper bounded uniformly by a constant independent of the initial conditions. We will propose two types of controllers that also correspond to two different classes Wnv of perturbations. One of them is based on a relay-type controller (Section 4.1), while the other is based on a generalized STC (Section 4.2). 3. Uniform sliding surface Let us take a uniform sliding surface described by s = x2 + c1 x1 + c2 |x1 |q sign(x1 ),
(3.1)
where c1 , c2 > 0 and q > 0 are positive scalars. We will show that if q > 1, the convergence rate becomes so strong that the convergence time to a neighbourhood of the origin is uniform w.r.t. initial
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
where the constant c1 is chosen to ensure the desired eigenvalue for (2.3). Once the sliding surface has been selected, a control input must be obtained to reach the sliding surface in finite time from an arbitrary initial condition and to ensure the sliding motion despite of (matched) perturbations. The standard SMC design suggests that the control law that enforces the sliding mode should be designed such that the sufficient condition s s˙ < −n|s| is satisfied (Utkin, 1992). It guarantees the sliding mode existence. For system (2.1), a control law that satisfies this condition is designed as
494
E. CRUZ-ZAVALA ET AL.
conditions, i.e. the convergence time can be upper bounded by a positive constant. Note that when q = 1, the usual linear sliding surface is obtained. Introducing (s, x1 ) as new state variables (where s is defined as in (3.1)) and applying the control input u = u eq + v,
(3.2)
−qc2 |x1 |q−1 [(a11 − a12 c1 )x1 − a12 c2 |x1 |q sign(x1 ) + a12 s],
(3.3)
the system (2.1) may be represented as x˙1 = (a11 − a12 c1 )x1 − a12 c2 |x1 |q sign(x1 ) + a12 s, s˙ = v + w(x, t).
(3.4)
The term u eq compensates the nominal (known) dynamics in the system. It is easy to see that the dynamics of system (2.1), in the sliding mode (i.e. s = 0), are governed by the differential equation x˙1 = (a11 − a12 c1 )x1 − a12 c2 |x1 |q sign(x1 ).
(3.5)
|q
The high degree term |x1 sign(x1 ) (when q > 1) accelerates the convergence rate to the origin since it is stronger the further the initial condition is. As a consequence, the convergence time of any trajectory to a vicinity of the origin can be bounded by a constant independent of the initial condition! T HEOREM 3.1 If c1 and c2 are selected such that a11 − a12 c1 < 0 and a12 c2 > 0, the reduced system (3.5) is asymptotically and uniformly stable w.r.t. initial conditions, i.e. it will reach the origin asymptotically but, furthermore, every trajectory will reach a neighbourhood of the origin of radius μ > 0 before 1 α Tμ = ln 1 + , (3.6) (q − 1)β βμq−1 where α = −a11 + a12 c1 and β = a12 c2 . Proof. When a11 − a12 c1 < 0 and a12 c2 > 0, the equilibrium point is asymptotically stable. This can be easily demonstrated using the Lyapunov function Vs = |x1 |. The time derivative along the trajectories q of (3.5) is given by V˙s = −αVs − βVs , Vs (x1 (0)) = v 0 0 . Then, V˙s is negative definite. For q > 1, the solution of the differential equation for Vs is given by α 1−q (3.7) Vs (t)1−q = exp(−(1 − q)βt)v 0 − exp(−(1 − q)βt)[exp((1 − q)βt) − 1]. β The time Ts required for a trajectory starting at the initial state x1 (0) to reach a level set Vs = μ, where 0 < μ < Vs (x1 (0)), is determined by q−1 βv 0 1 βμq−1 T1 (v 0 , μ) = ln − ln . q−1 (q − 1)β α + βμq−1 α + βv 0
The time T1 (v 0 , μ) depends on the parameters c1 , c2 , μ and is upper bounded by a constant independent of the initial conditions x1 (0) ∈ Vs (x0 ). Moreover, since limv 0 →∞ T1 (v 0 , μ) = Tμ , the convergence time T1 (v 0 , μ) of every trajectory is uniformly upper bounded by (3.6), i.e. T1 (v 0 , μ) Tμ . Note that Tμ does not depend on the initial conditions and that limμ→0 Tμ = ∞. This is due to the fact that the convergence to the origin is exponential.
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
u eq = −[c1 (a11 − a12 c1 ) + (a21 − a22 c1 )]x1 − (a12 c1 + a22 )s + (c1 a12 + a22 )c2 |x1 |q sign(x1 )
UNIFORM SLIDING MODE CONTROLLERS AND UNIFORM SLIDING SURFACE
495
4. Uniform controllers with respect to the initial conditions
4.1 First-order uniform sliding mode based control A control law should be chosen in such a way that, from any initial condition, the system trajectories are attracted towards the sliding surface uniformly and in finite time even in presence of uncertainties/disturbances, i.e. the system trajectories will reach the sliding surface in a finite time bounded by a constant and then slide along the uniform sliding surface. Once there, the system is invariant against parameter variations and disturbances satisfying the matching condition. As the uncertainties/disturbances of a special class supported by the system depend on the control law selected, we consider for this first-order uniform sliding controller (FOUSC) that w(x, t) belongs to the class W1 = {w: |w(x, t)| 0 + 02 |s| + 03 |s|qv , 0 , 02 , 03 > 0, qv > 1}. Note that these functions are uniformly bounded in t, ∀t > 0. T HEOREM 4.1 Suppose that the uncertainty/disturbance w(x, t) ∈ W1 . Then, the control law (3.2), with u eq as in (3.3) and v = −Q 0 sign(s) − K 2 s − K 3 |s|qv sign(s),
qv > 1,
(4.1)
where Q 0 , K 2 and K 3 are positive constants, enforces every trajectory of (3.4) to move from any initial condition to the sliding surface in a finite time and thereafter ensures it remains on the sliding surface. Moreover, the sliding manifold s = 0 is ‘uniformly finite-time stable w.r.t initial conditions x0 and uncertainties/disturbances’, i.e. every trajectory reaches it in spite of any uncertainty/disturbance w(x, t) ∈ W1 in a finite time smaller than κ02 12 2 2 − 1 (qv −1) μs 2 + ln μs + 1 , (4.2) Tf = κ01 (qv − 1)κ03 κ02 where κ01 = root of
√
2(Q 0 − 0 ), κ02 = 2(K 2 − 02 ), κ03 = 2(qv +1)/2 (K 3 − 03 ) and μs is the positive real 1
qv
μs2 + (κ01 /κ02 ) = (κ03 /κ02 )μs2 .
(4.3)
Proof. To show that the proposed control law ensures that the state trajectories reach the manifold s = 0 before a constant time T f > 0, we use the quadratic function V = 0.5s 2 . Since V˙ = s(v + w(x, t)),
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
The previous theorem has shown that for every x0 on the sliding surface, the trajectories of system (3.5) will be in the interior of a ball Bμ = {x1 : |x1 | < μ} centred at the origin after a time Tμ , independent of the initial condition. Once the sliding surface has been specified, subject to some performance requirements for the system, the new control input v must be designed to reach the sliding surface in a prescribed finite time, independent of the initial conditions, while also guaranteeing the sliding mode. There exist some control laws based on the classical design that enforce the sliding mode (see, e.g. Hung et al., 1993). These control laws can be obtained by the so-called ‘reaching law’ approach in which the switching function dynamics are specified a priori; they are basically based on a first-order sliding mode (FOSM). However, they are not able to achieve the property of convergence uniform in the initial conditions we are interested in. In the following two subsections, we will describe two controllers to assure this property.
496
E. CRUZ-ZAVALA ET AL.
applying the control (4.1) and considering the fact that w(x, t) ∈ W1 , leads to V˙ s(−Q 0 sign(s) − K 2 s − K 3 |s|qv sign(s) + 0 + 02 |s| + 03 |s|qv ) −(Q 0 − 0 )|s| − (K 2 − 02 )|s|2 − (K 3 − 03 )|s|qv +1 ,
1
1
v(t) 2 = exp(−κ02 t/2)v 02 −
κ01 exp(−κ02 t/2)[exp(κ02 t/2) − 1], κ02
(4.4)
and the solution of the differential equation v˙ = −κ03 v (qv +1)/2 , v(0) = v 0 0, for qv > 1, is given by 1 − 2 qv −1 − (qv −1) v(t) = v 0 2 + (qv − 1)κ03 t/2 . It follows from the comparison principle (Khalil, 2002) that V (s(t)) v(t) when V0 = V (s(x0 )) v 0 , and therefore, 1
2 κ01 1 1 2 V0 − , κ02 t − 1 exp V (t) min V1 = exp − κ02 t κ02 2 2 − 2 qv −1 − 12 (qv −1) ×V2 = V0 + (qv − 1)κ03 t/2 . This expression allows estimating the convergence time. First, consider a trajectory starting at point x0 at an energy level V0 . An upper bound T1 (x0 , μs ) of the time at which it reaches the level set V = μs , for some 0 < μs < V0 , can be calculated from V2 = μs as 1 2 − (qv −1) − 1 (qv −1) − V0 2 μs 2 . T1 (x0 , μs ) = (qv − 1)κ03 Now, starting from the level set V = μs , an upper bound T2 (μs ) of the time at which it reaches the point s = 0 can be calculated from V1 = 0 as κ02 12 2 ln μs + 1 , T2 (μs ) = κ01 κ02 and therefore, the convergence time of every trajectory from any x0 ∈ V0 to the sliding surface can be estimated as 1 κ02 12 2 2 − (qv −1) − 1 (qv −1) − V0 2 ln μs + 1 . μs 2 + T (x0 , μs ) = T1 (x0 , μs ) + T2 (μs ) = κ01 (qv − 1)κ03 κ02
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
√ which may be rewritten as V˙ −κ01 V 1/2 − κ02 V − κ03 V (qv +1)/2 , where κ01 = 2(Q 0 − 0 ), κ02 = 2(K 2 − 02 ), κ03 = 2(qv +1)/2 (K 3 − 03 ). This implies V˙ < 0, for Q 0 > 0 , K 2 > 02 , K 3 > 03 . It is clear that every trajectory converges to s = 0 in a finite time. After this, the trajectory will slide along the sliding surface. Finally, we will show that there exits some constant T f > 0, independent of the initial conditions, such that ∀ t T f the state trajectories x(t) cannot leave s = 0. Since V satisfies both differential inequalities V˙ −κ01 V 1/2 − κ02 V and V˙ −κ03 V (qv +1)/2 , the value of V is below the solution of both inequalities. Note that the solution of the differential equation v˙ = 1 −κ01 v 2 − κ02 v, v(0) = v 0 0 is given by
UNIFORM SLIDING MODE CONTROLLERS AND UNIFORM SLIDING SURFACE
497
The convergence time T (x0 , μs ) is uniformly upper bounded by the constant T f , i.e. T (x0 , μs ) T f . Choosing μs as the positive real root of (4.3) ensures the best estimation of the time T f .
4.2
Uniform super twisting-based sliding mode enforcement
One disadvantage of using a relay (or unit controllers) sliding mode in the control law to enforce the sliding motion is that it produces the so-called ‘chattering effect’. To eliminate the high frequency component of these controllers, a low-pass filter is frequently used with the FOSM. Another approach to deal with this problem is using high-order sliding modes. Up until now, only a few two sliding controllers have been proposed (Levant, 2007). A second-order sliding mode, called the ‘Super-Twisting Algorithm’ (STA), has been widely used as an ‘absolutely continuous’ controller because it depends only on the sliding constraint. The STA is effective in alleviating the chattering effect and it ensures all the main properties of a first-order SMC for a system with Lipschitz continuous matched uncertainties/disturbances with bounded gradients. Taking into account this advantages, we will introduce a controller based on the generalized STA (Moreno, 2011). The super-twisting uniform sliding controller (STUSC) is given by
t
v = −k1 φ1 (s) − k2
φ2 (s)dt,
(4.5)
0
where k1 and k2 are the gains to be designed and 1
3
φ1 (s) = μ1 |s| 2 sign(s) + μ2 s + μ3 |s| 2 sign(s), 1 3 1 3 5 3 φ2 (s) = μ21 sign(s) + μ1 μ2 |s| 2 sign(s) + (μ22 + 2μ1 μ3 )s + μ2 μ3 |s| 2 sign(s) + μ23 |s|2 sign(s), 2 2 2 2
with μ1 , μ2 and μ3 being positive constants. The STUSC is based on the standard STC that may be recovered by setting μ2 = μ3 = 0 and the gains k1 , k2 > 0. Also, the new control law v inherits the robustness properties of an STC. Recall that the FOUSC can deal with a great variety of uncertainties/disturbances, including bounded discontinuous functions belonging to the class W1 . The STUSC, however, is able to compensate uncertainties/disturbances w(x,t) in the class W2 , defined
by W2 = w: |g1 (x1 , x2 , t)| 1 |φ1 (s)|, | dtd g2 (x1 , x2 , t)| 2 |tφ2 (s)| , for some known constants 1 , 2 0, where w(x1 , x2 , t) = g1 (x1 , x2 , t) + g2 (x1 , x2 , t),
(4.6)
is a decomposition of the signal. Note that the classes W1 and W2 are disjoint. For example, bounded discontinuous functions belong to W1 but not to W2 , whereas continuous unbounded functions belong to W2 but not to W1 .
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
The control law (4.1) was derived using a quadratic Lyapunov function and it is equivalent to satisfying the reachability condition s s˙ < −η|s|. Alternatively, different switching function dynamics can be obtained from the control law proposed (see Hung et al., 1993). Furthermore, K 3 > 0 allows for a faster convergence and deals with uncertainties/disturbances growing with a degree larger than a linear term in s. K 2 > 0 allows to deal with uncertainties/disturbances growing with a linear term in s.
498
E. CRUZ-ZAVALA ET AL.
System (3.4) along with controller (4.5) and the disturbance w ∈ W2 can be extended as follows: x˙1 = (a11 − a12 c1 )x1 − a12 c2 |x1 |q sign(x1 )) + a12 s, s˙ = −k1 φ1 (s) + ς + g1 (x1 , x2 , t), d g2 (x1 , x2 , t). dt
(4.7)
T HEOREM 4.2 The control law (3.2), with u eq as in (3.3) and v as in (4.5), enforces every trajectory of (3.4) to move from any initial condition to the sliding surface in a finite time and to remain on it thereafter. Moreover, the sliding manifold s = 0 is uniformly finite-time stable w.r.t initial conditions x 0 and uncertainties/disturbances, i.e. every trajectory reaches it in spite of any uncertainty/disturbance w(x, t) ∈ W2 in a finite time smaller than κ2 12 6 −1 2 TSTC = μss 6 + ln μss + 1 , (4.8) κ1 κ3 κ2 where 1 1 14 v min C3 3 2 , v min = μ33 , 7 2 2 (λmax {P} + C2 ) 6 (4.9) and , λmax {P}, C2 and C3 are positive scalars depending on the selection of the parameters k1 , k2 , 1 , 2 and μss is the positive real root of μ21
μ2
, κ2 = , κ3 = κ1 = 1 (λmax {P} + C2 ) 2(λmax {P} + C2 ) 2
1
2
3 2 μss + (κ1 /κ2 ) = (κ3 /κ2 )μss .
(4.10)
Proof. See Section 5.
Both uniform controllers enforce the sliding mode for every trajectory of system (2.1), taking into account their respective uncertainties/disturbances. Actually, the uniform controllers enforce the system trajectories to the uniform sliding surface before a time T f or TSTC , which are positive constants. Once in the sliding motion, the trajectories converge within a time Tμ to an arbitrary neighbourhood of the origin of radius μ > 0. As a result, the system trajectories reach an arbitrary ball Bμ (μ > 0) in a time bounded by another constant. Therefore, we can conclude that a system driven by FOUSC or STUSC is ‘asymptotically uniformly stable w.r.t. initial conditions’ for any uncertainty/disturbance w(x, t) belonging to the class W1 or W2 , respectively. 5. Proof of Theorem 4.1 We propose a Lyapunov-based approach to show the uniform and finite-time convergence of the system trajectories from any initial condition to the non-linear sliding surface. Consider the continuous function W (s, ς) = V1 (ζ ) + V2 (s, ς),
(5.1)
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
ς˙ = −k2 φ2 (s) +
499
UNIFORM SLIDING MODE CONTROLLERS AND UNIFORM SLIDING SURFACE
as a candidate Lyapunov function. The function V1 (ζ ) = ζ Pζ is quadratic in the vector ζ = φ1 (s) ς and the definite positive symmetric matrix P = P > 0 is the solution of a linear matrix inequality (LMI)
A P + P A + R + I PB < 0, (5.2) B P −Θ B=
0 , 1
1 0
C= 1
T
0
,
Θ=
θ1 0 , 0 θ2
2
(5.3)
4
∀ θ1 , θ2 0 and with > 0. The function V2 (s, ς) = δk2 |φ1 (s)|2 −|φ1 (s)| 3 sign(s)|ς| 3 sign(ς)+δ|ς|2 and δ is a positive constant which satisfies 1
3
7
7
7
7
δ > (4/27k2 ) 3 , 7δk2 (k1 − 1 ) > 4(3) 2 24 δ 4 + (k1 + 1 ) 3 + 4(k2 + 2 ) 6 .
(5.4)
The following result shows that the subsystem (s, ζ ) of (4.7) is robust against uncertainties/ disturbances w(x, t) ∈ W2 when the gains (k1 , k2 ) and δ are selected properly. P ROPOSITION 5.1 The Lyapunov function (5.1) is a ‘strong, robust Lyapunov function for the subsystem (s, ς) of (4.7). Moreover, the time derivative W˙ of the Lyapunov function taken along the trajectories of the subsystem satisfies the differential inequality W˙ (s, ς) −κ1 W 2 (s, ς) − κ2 W (s, ς) − κ3 W 6 (s, ς), 1
7
(5.5)
with κ1 , κ2 and κ3 as in (4.9). Proof. First, we establish an inequality derived from the classical Young’s inequality. It is used extensively to show the positive and negative definiteness of V2 (s, ς). The proof can be consulted in Cruz Zavala et al. (2011). L EMMA 5.1 For every real numbers a > 0, b > 0, c > 0, p > 1, q > 1, with inequality is satisfied:
1 p
+
1 q
= 1 the following
ab c p a p / p + c−q bq /q. To show positive definiteness of V2 (s, ς), we use Lemma 5.1, and let γ00 and γ01 be constants that satisfy 2
4
2
4
−3
|φ1 (s)| 3 sign(s)|ς| 3 sign(ς) |φ1 (s)| 3 |ς| 3 γ0i3 |φ1 (s)|2 /3 + 2γ0i 2 |ς|2 /3,
∀ γ0i > 0, i = 0, 1,
therefore, ∀(s, ς), the function V2 can be bounded as α1 (γ00 )|φ1 (s)|2 + α2 (γ00 )|ς|2 V2 (s, ς) α2 (γ01 )|φ1 (s)|2 + α4 (γ01 )|ς |2 , 3 3 /3), α (γ ) = δ − 2γ − 2 /3 , α (γ ) = (δk + γ 3 /3) and α (γ ) = where α1 (γ00 ) = (δk2 − γ00 2 00 3 01 2 4 01 00 01 − 32 δ + 2γ01 /3 . Also, V2 (s, ς) is positive definite if and only if α1 (γ00 ) > 0 and α2 (γ00 ) > 0, which is always possible if δ satisfies (5.4). Moreover, V2 can be bounded as
C1 ζ 22 = C1 (|φ1 (s)|2 + |ς |2 ) V2 (s, ς) C2 (|φ1 (s)|2 + |ς |2 ) = C2 ζ 22 ,
(5.6)
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
where R = (θ1 12 + θ2 22 )C C,
−k1 1 A= , −k2 0
500
E. CRUZ-ZAVALA ET AL. 2
2 /3 and C = δk + z 2 /3 are positive constants, z = γ 3 is the positive real root where C1 = δk2 − z m 2 2 m M 00 2
Using the following inequalities derived from Lemma (5.1) 4
7
−7
7
∀γ1 > 0,
7 3
7
− 74
7
∀γ2 = (k1 + 1 ) 7 > 0,
7
7
7
4
φ1 (s) 3 ς |φ1 (s)| 3 |ς| 4γ14 |φ1 (s)| 3 /7 + 3γ1 3 |ς | 3 /7, 4
4
4
φ1 (s)|ς | 3 sign(ς) |φ1 (s)||ς | 3 3γ2 |φ1 (s)| 3 /7 + 4γ2 |ς | 3 /7, |φ1 (s)|2 |ς| 3 6γ36 |φ1 (s)| 3 /7 + γ3−7 |ς | 3 /7, 1
we obtain
7
1
∀γ3 = (k2 + 2 ) 7 > 0,
7 7 V˙2 (s, ς) = −v 1 (s) ψ1 |φ1 (s)| 3 + ψ2 |ς | 3 ,
where v 1 (s) = |φ1 (s)| 7
−1 3
−7
7
φ1 (s), ψ1 = 2(Υ1 − 42 δγ14 /7), ψ2 = 2(1 − 92 δγ1 3 )/21, Υ1 = δk2 (k1 − 7
4
3
1 )−(k1 +1 ) 3 /7−4(k2 +2 ) 6 /7 and γ1 always exits iff it satisfies (7Υ1 /82 δ) 7 > γ1 > (92 δ) 7 . For negative definiteness of V˙2 , it is required that ψ1 , ψ2 > 0, which is always possible if (5.4) is satisfied. Note that ∀s ∈ R, the function v 1 (s) > 0 and −1
−1
μ1 1 −1 + μ |s| 2 + 3 μ μ1 |s| 2 + μ2 + 32 μ3 s 2 2 2 3 2 |s| lim v 1 (s) = 2 = 1 1 3 1 −3 |s|→∞ −1 μ1 |s| 2 + μ2 |s| + μ3 |s| 2 3 μ1 |s| + μ2 |s| 2 + μ3 3 1
2
3 3 μ 3 2 = μ33 , lim v(s) = 2 21 = ∞. |s|→0 2 |s| 3
We immediately establish that the minimum v min = mins∈R v 1 (s) exists and is positive. Then, this leads to 7 7 7 7 V˙2 (s, ς) −v 1 (s) ψ1 |φ1 (s)| 3 + ψ2 |ς| 3 −v min C3 |φ1 (s)| 3 + |ς | 3 , 7
where C3 = 2(1 − 92 δ/ym )/21 and ym = γ13 is the positive real root of 32 δ + 7(Υ1 − 1/21)ym = 7
42 δym4 . Using the following standard inequality (see Hardy et al., 1951) 1
1
(5.7) (α|x1 |s + (1 − α)|x2 |s ) s (α|x1 |r + (1 − α)|x2 |r ) r , 0 α 1, ∀x ∈ R 2 , s < r, 7 7 7 1 clearly we can find that (1/2) 14 (|φ1 (s)|2 + |ς|2 ) 6 |φ1 (s)| 3 + |ς | 3 . Finally, using this inequality along with the inequality (5.6), we find that the derivative of the Lyapunov function V2 (s, ς) satisfies 7 7 1 7 1 7 V˙2 −v min C3 |φ1 (s)| 3 + |ς | 3 −(1/2) 14 v min C3 (|φ1 (s)|2 + |ς |2 ) 6 −(1/2) 14 v min C3 ( ζ 22 ) 6 .
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
3 and z = γ 3 is the positive real root of 2 + 3δ(1 − k )z = z 3 . Since the of 2 + 3δ(k2 − 1)z m = z m M 2 M M 01 uncertainty/disturbance term w(x, t) ∈ W2 , the time derivative of V2 is given by −1 7 4 4 2 ˙ 3 V2 −|φ1 (s)| φ1 (s) 2δk2 (k1 − g1 )|φ1 (s)| 3 − 2g2 δ|φ1 (s)| 3 ς − (k1 + g1 )φ1 (s)|ς | 3 sign (ς ) 3 1 4 2 7 2 3 3 − (k2 + g2 )|φ1 (s)| |ς| + |ς| . 3 3
501
UNIFORM SLIDING MODE CONTROLLERS AND UNIFORM SLIDING SURFACE
On other hand, for the following analysis, we will present subsystem (4.7) in a convenient form. Using the vector ζ , subsystem (4.7) can be written as ⎤ ⎡ g1 (x1 , x2 , t) −k1 φ1 (s) + ς + g1 (x1 , x2 , t) d d ⎦ ζ˙ = φ1 (s) = φ1 (s)(Aζ + B ρ), ˜ ρ˜ = ⎣ , g (x ,x ,t) dt g2 (x 1 ,x 2 ,t) 1 3 −k2 φ2 (s) + dt 2 1 2 1 3 φ1 (s)
2 μ1 |s|
2 +μ2 +
2 μ3 |s|
2
s=φ −1 (ζ )
ωi (ρ˜i , ζ ) =
−ρ˜i2 (t, ζ ) + i2 ζ12
ρ˜ = i ζi
−1 0 0 i2 C C
ρ˜i ζi
0,
where C is defined in (5.3). It means that the uncertainty/disturbance w(x, t) ∈ W2 can be expressed as a sector condition (in ζ coordinates), i.e. |ρ˜i (t, ζ )| i |φi (s)| in the original variables, which is equivalent to |ρ˜i (t, ζ )| i |ζi |, with i > 0, in transformed coordinates. It follows that ω(ρ, ˜ ζ) = θ1 ω1 (ρ˜1 , ζ ) + θ2 ω2 (ρ˜2 , ζ ) 0, ∀θ1 , θ2 0 and hence ω(ρ, ˜ ζ) =
ρ(t, ˜ ζ) ζ
−Θ 0
0 R
ρ(t, ˜ ζ) , ζ
(5.8)
with Θ and R as in (5.3). The derivative of the Lyapunov function V1 (ζ ) is
ζ A P + PA PB A P + P A P B ζ ζ ζ ˙ V1 = φ1 φ1 + ω(ρ, ˜ ζ) ρ˜ ρ˜ ρ˜ ρ˜ B P B P 0 0 = φ1
ζ A P + PA + R PB ζ −φ1 ζ 22 . ρ˜ ρ˜ B P −Θ
For negative definiteness of V˙1 , the feasibility of the LMI (5.2) is required. Recall the standard inequality from the quadratic forms λmin {P} ζ 22 ζ Pζ λmax {P} ζ 22 , where ζ 22 = φ12 (s) + 5
3
ς 2 = μ22 |s|2 + μ23 |s|3 + 2μ2 μ3 |s| 2 + μ21 |s| + 2μ1 μ2 |s| 2 + 2μ1 μ3 |s|2 + ς22 is the Euclidean norm of 1 −1 1 ( ζ 2 )−1 is always satisfied. Then, as φ1 (s) = μ1 |s| 2 /2 + ζ . From this, we can see that μ1 |s| 2 1
μ2 + 3μ3 |s| 2 /2, we immediately establish V˙1 −φ1 (s) ζ 22 −(μ21 /2) ζ 2 − μ2 ζ 22 − (3μ3 /2) |s| 2 ζ 22 . 1
Thanks to the negative definiteness of V˙1 (ζ ) and V˙2 (s, ς), 1
W˙ (s, ς) = V˙1 + V˙2 −(μ21 /2) ζ 22 − μ2 ζ 22 − (3μ3 /2) |x1 | 2 ζ 22 − (1/2) 14 v min C3 ( ζ 22 ) 6 . 1
1
7
As both Lyapunov functions satisfy λmin {P} ζ 22 V1 (ζ ) λmax {P} ζ 22 and C1 ζ 22 V2 (s, ς) C2 ζ 22 , the function W (s, ς) can be upper bounded as W (s, ς) (λmax {P} + C2 ) ζ 22 , which allows us to obtain (5.5).
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
with A and B as in (5.3). To take into account a bigger variety of disturbances, it will be assumed that the components of the (transformed) uncertainty/disturbance term ρ(t, ˜ ζ ) satisfy the sector conditions (for i = 1, 2, ∀t 0 and ∀ζ ∈ R 2 )
502
E. CRUZ-ZAVALA ET AL.
− 16
v(t) = (v 0
+ κ3 t/6)−6 .
From the comparison principle (Khalil, 2002), it follows that W (s(t), ς(t)) v(t) when we have W0 = W (s(x0 ), ς(x0 )) v 0 . Then 1
2 −6 κ1 1 1 − 16 2 W0 − , W0 + κ3 t/6 κ2 t − 1 . exp W (t) min exp − κ2 t κ2 2 2 The convergence time can be estimated in similar way as in the proof of Theorem 4.1. First, calculate an upper bound T1 (x0 , μss ) of the convergence time of a trajectory starting at point x0 at an energy level W0 at which it reaches the level set W (s, ς) = μss (for some 0 < μss < W0 ). Afterwards, calculate an upper bound T2 (μss ) at which s = 0 is reached (starting from this level set W = μss ). Then, every trajectory reaches the sliding surface within a time 6 − 16 2 κ2 12 −1 ln μss + 1 . μss − W0 6 + T (x0 , μss ) = T1 (x0 , μss ) + T2 (μss ) = κ1 κ3 κ2 The convergence time T (x0 , μss ) is uniformly upper bounded by the constant TSTC , i.e. T (x0 , μss ) TSTC . To ensure the best estimation of the time TSTC , choose μss as the positive real root of (4.10). 6. Simulation results of an academic example This section is devoted to illustrating the effectiveness of the proposed controllers for the control of a hybrid system with strictly positive dwell time. Consider the following hybrid linear system modelled by the equation x˙ = Aq x + b(u + w(t)), where x ∈ R 2 is the continuous state, q ∈ [1, 2] is the discrete state that indexes the subsystems
0 1 1 1 , A2 = . A1 = 2 −1 2 1
Both A1 and A2 are unstable, b = 0 1 , w(t) = 0.5 sin(2t) + 0.5 cos(5t) and the control input u is defined by (3.2). The control task is to stabilize the origin x = 0 of the system, with convergence to a neighbourhood of the origin in a prescribed time for every trajectory of the system. The system dynamics change every 4 s. If only the state x1 is measured, the pairs [C, A1 ] and [C, A2 ] are observable with the output vector C = 1 0 . In this case, it is possible to design a uniform differentiator (Cruz Zavala et al., 2011), which provides the system states after at most 1 s. Therefore, we can turn on the controller after 1 s. The sliding surfaces 3 3 for each system are designed as s A1 = x2 + x1 + 0.5|x1 | 2 sign(x1 ) and s A2 = x2 + 2x1 + |x1 | 2 sign(x1 ),
the initial conditions x(0) = 1.5 1 and the sampling time (for the numerical simulation) is τ = 0.001. For the simulation, we design and compare three controllers (3.2), where:
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
It is clear that every trajectory converges to zero in a finite time. The proof of the uniform convergence follows immediately from Proposition 5.1. Since W satisfies both differential inequalities 7 1 W˙ (s, ς) −κ1 W 2 (s, ς) − κ2 W (s, ς) and W˙ (s, ς) −κ3 W 6 (s, ς), the value of W is below the 1 solution of any of the two inequalities. Note that the solution of the differential equation v˙ = −κ1 v 2 − 7 κ2 v, v(0) = v 0 0 is similar to (4.4) and that the solution of the differential equation v˙ = −κ3 v 6 , v(0) = v 0 0 is given by
UNIFORM SLIDING MODE CONTROLLERS AND UNIFORM SLIDING SURFACE
503
1. v = −M0 sign(s), the first-order sliding controller (FOSC), with M0 = 4. 3
2. v = −Q 0 sign(s) − K 2 s − K 3 |s| 2 sign(s), the FOUSC, with Q 0 = 4, K 2 = 2, K 3 = 1. t 3. v = −k1 φ1 (s) − k2 0 φ2 (s)dt, the super-twisting uniform sliding controller (STUSC), with μ1 = 2, μ2 = 1, μ3 = 0.5.
FIG. 1. (a) and (b) show the states x1 and x2 with the: FOSC (dashed line), FOUSC (dotted line) and STUSC (solid line); (c) Surfaces and (d) Control input.
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
Note that with the FOUSC, the disturbance w(t) ∈ W1 = {w(t): |w(x, t)| 0 + 02 |s| +
03 |s|qv , 0 = 1, 02 = 0, 03 = 0, qv > 1} while w(t) ∈ W2 = w(t): |g1 (x1 , s, t)| 1 |φ1 (s)|, |w(t)| ˙ = dtd g2 (x1 , t) 2 |φ2 (s)| , with 1 = 0 and 2 = 1 when the STUSC is used. For this case, when 1 = 0 (see Moreno, 2011), the gains k1 and k2 should be selected in the set κ = {(k1 , k2 ) ∈ √ √ R 2 |0 < k1 2 ρ2 , k2 > 0.25k12 + 4ρ2 /k12 } {(k1 , k2 ) ∈ R 2 |k1 > 2 ρ2 , k2 > 22 }. We choose k1 = 2 and k2 = 4. The simulation results are shown in the Fig. 1. Clearly, both the FOUSC and the STUSC enforce the sliding mode but not the FOSC. Moreover, the STUSC provides chattering alleviation. Nevertheless, the value of the control signal during the transient response for both controllers, FOUSC and STUSC, is highly increased (see Fig. 1(d)). This is a natural consequence of the property of uniform convergence w.r.t. initial conditions since the control action has to be very strong in order to attract the system trajectories, starting far away from the origin, to a neighbourhood of the origin in a time bounded by a constant. Figure 2(a) shows a linear sliding surface and the uniform surfaces s A1 = 0 and s A2 = 0, which are clearly non-linear. Considering that the control action starts since t = 0 at the first operation mode, during the first 4 s, Fig. 2(b) puts in evidence that the convergence time to the surfaces grows to infinity with the growth of the initial conditions for the FOSC while the convergence time using the FOUSC and the STUSC is uniformly bounded by a constant.
504
E. CRUZ-ZAVALA ET AL.
In order to estimate the convergence time to the sliding surfaces using the FOUSC, it is required calculating κ1 , κ2 , κ3 and obtaining μss from (4.3) and then substituting all the corresponding values in expression (4.2). In a similar way, when the STUSC is used, for the convergence time estimation the following steps are required: 1. Find P = P > 0 and > 0 such that the LMI (5.2) is satisfied. Choose δ in such a way that the inequalities (5.4) are satisfied. Then, find z M and ym to obtain C2 and C3 and compute κ1 , κ2 , κ3 . 2. Calculate μss from (4.10) and substitute all the corresponding values in expression (4.8). Finally, Fig. 2(c) shows the real convergence time applying the FOUSC in the system with its estimation. The upper bound of the estimation is approximately twice the real upper bound of the real convergence time. On the other hand, Tmax = 301.34 s was the best convergence time estimation using the STUSC. However, the time estimation for the STUSC is cumbersome. 7. Concluding remarks A non-linear sliding surface has been suggested ensuring, during the sliding motion, convergence of the system trajectories to any arbitrary small vicinity of the origin in a finite time bounded by some constant independent of the initial conditions on the surface. Two types of uniform sliding controllers have been proposed. At the beginning, a FOUSC controller is designed as a generalization of the FOSM algorithm inheriting all its properties. This controller also provides finite-time convergence uniform w.r.t. initial conditions of the trajectories to the sliding surface in presence of a special class of uncertainties/disturbances. To adjust the chattering effect produced by
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
FIG. 2. (a) A linear sliding surface sL = x2 + x1 and the uniform sliding surfaces s A1 and s A2 , (b) convergence time to the sliding surface during the first operation mode (simulation data) and (c) comparison between the real convergence time and its estimation.
UNIFORM SLIDING MODE CONTROLLERS AND UNIFORM SLIDING SURFACE
505
the first controller, an absolutely continuous super-twisting uniform controller is suggested providing for finite-time convergence uniform w.r.t. initial conditions of the trajectories to the sliding surface in presence of another uncertainty/disturbance class. A Lyapunov-based approach is used in the paper to estimate the convergence time to the sliding surface and to an arbitrary small vicinity of origin.
The authors gratefully acknowledge the financial support form: (i) Consejo Nacional de Ciencia y Tecnolog´ıa (51244, 132125) and CVU (267513), (ii) Programa de Apoyo a Proyectos de Investigaci´on e Innovaci´on Tecnol´ogica, UNAM (111012, 117610 and 117211) and (iii) Fondo de Colaboraci´on del II-FI, UNAM (IISGBAS-165-2011). R EFERENCES A NDRIEU , V., P RALY, L. & A STOLFI , A. (2008) Homogeneous approximation, recursive observer design and output feedback. SIAM J. Control Optim., 47, 1814–1850. BANDYOPADHYAY, B., D EEPAK , F. & K IM , K. S. (2009) Sliding Mode Control Using Novel Sliding Surfaces. Lecture Notes in Control and Information Sciences, vol. 392. Berlin: Springer, 136 p. BARTOSZEWICZ , A. & N OWACKA -L EVERTON , A. (2009) Time-Varying SlidingModes for Second and Third Order Systems. Lecture Notes in Control and Information Sciences, vol. 382. Berlin: Springer, 200 p. C RUZ Z AVALA , E., M ORENO , J. A. & F RIDMAN , L. (2011) Uniform robust exact differentiator. IEEE Trans. Automat. Control, 56, 2727–2733. F ILIPPOV, A. F. (1988) Differential Equations with Discontinuous Right Hand Side. Dordrecht, The Netherlands: Kluwer, 304 p. G ONZALEZ , T., A., M ORENO , J. A. & F RIDMAN , L. (2011) Variable gains super-twisting algorithm. IEEE Trans. Automat. Control. doi: 10.1109/TAC.2011.2179878. ´ H ARDY, G. H., L ITTLEWOOD , J. E. & P OLYA , G. (1951) Inequalities. Cambridge University Press: London. H UNG , J. Y., G AO , W. & H UNG , J. C. (1993) Variable structure control: a survey. IEEE Trans. Ind. Electron., 40, 2–22. K HALIL , H. K. (2002) Nonlinear Systems, 3rd edn. Upsaddle River, NJ: Prentice–Hall, 750 p. L EVANT, A. (1993) Sliding order and sliding accuracy in sliding mode control. Int. J. Control, 58, 1247–1263. L EVANT, A. (2007) Construction principles of 2-sliding mode design. Automatica, 43, 576–586. ´ , A. G. & U TKIN , V. I. (1981) Methods of reducing equations for dynamic systems to a regular form. L UK YANOV Automat. Remote Control, 42, 413–420. M ORENO , J. A. (2011) Lyapunov approach for analysis and design of second order sliding mode algorithms. Sliding Modes after the First Decade of the 21st Century (L. Fridman, J. Moreno & R. Iriarte eds). Lecture Notes in Control and Information Sciences, vol. 412. Berlin: Springer, pp. 113–150. S HTESSEL , Y. B., Z INOBER , A. S. I. & S HKOLNIKOV, I. A. (2002) Boost and buck-boost power converters control via sliding modes using dynamic sliding manifold. 41th IEEE Conference on Decision and Control, Las Vegas, NV, pp. 2456–2461. U TKIN , V. I. (1992) Sliding Modes in Control and Optimization. Berlin: Springer.
Downloaded from http://imamci.oxfordjournals.org/ at Universidad Nacional Autónoma de México on February 7, 2013
Acknowledgments
