CONTROLLING OUTPUT VARIABLES VIA HIGHER ORDER SLIDING MODES A. Levant Institute for Industrial Mathematics, 4/24 Yehuda Ha-Nachtom St., Beer-Sheva 84311, Israel Fax: +972-7-231211 and E-mail:
[email protected]
Keywords: Nonlinear Control; Sliding Mode; Tracking.
which impose an r-dimensional condition on the state of the dynamic system (Fig. 1).
Abstract. Featuring exactness and robustness, standard sliding mode may also cause the so-called chattering effect. Having hidden the switching in higher derivatives of output variables, higher order sliding modes preserve or generalize the main properties of standard sliding mode and remove the chattering effect. With finite convergence time, the presented controllers provide for full real-time control of the output variable if the relative degree of the dynamic system is known. Used together with recently developed real-time robust exact differentiators, these controllers actually need only measurements of the controlled output variable.
1 Introduction Generally speaking, sliding mode is a mode of motions on the discontinuity set of a discontinuous dynamic system, understood mostly in the Filippov sense [4] and realized due to theoretically-infinite frequency of control switching. Sliding modes are used in order to keep a dynamic system accurately within a given constraint. Their implementation is based on their insensitivity to external and internal disturbances [13, 14]. Unfortunately, standard sliding modes may cause possibly dangerous system vibrations (the socalled chattering effect). Another restriction is that control has to appear explicitly already in the first total derivative of the controlled output variable (the relative degree is to be 1). While successively differentiating output variable σ along trajectories of a discontinuous system, a discontinuity will be encountered sooner or later in the general case. Thus, sliding modes σ ≡ 0 may be classified by the number r of (r) the first successive total derivative σ which is not a continuous function of the state space variables or does not exist due to some reason like trajectory nonuniqueness. That number is called sliding order (see [5, 7] for the formal definitions). Hence, the r-th order sliding mode is determined by the equalities (r-1) = ... = σ σ = σ = σ =0 (1)
Fig. 1: 2-sliding mode The standard sliding mode used in the most variable structure systems (VSS) is of the first order ( σ is discontinuous). Higher order sliding modes (HOSM) generalize the sliding mode notion and its main features. While the standard modes feature finite time convergence, convergence to HOSM may be asymptotic as well. HOSM provides for up-to-its-order precision with respect to the measurement time step [7] and, properly used, completely removes the chattering effect. Any relative degree is admissible. Words “r-th order sliding” are often shortened for brevity to “r-sliding”. Trivial cases of asymptotically stable HOSM are easily found in many classic VSSs. For example there is an asymptotically stable 2-sliding mode with respect to the constraint x = 0 at the origin x = [ = 0 (at one point only) of a 2-dimensional VSS keeping the constraint x + [ = 0 in a standard 1-sliding mode. Asymptotically stable or unstable HOSMs inevitably appear in VSSs with fast actuators [5]. Stable HOSM reveals itself in that case by spontaneous disappearance of the chattering effect. Thus, examples of asymptotically stable or unstable sliding modes of any order are well known [2, 3, 5, 11]. However, till recently r-sliding modes attracting in finite time were known for r = 1 (which is trivial), for r = 2 [1, 3, 7, 9] and for r = 3 [5]. A family of r-sliding controllers with finite time convergence for any natural number r is presented in this paper. These controllers provide actually for full real-time arbitrarily smooth control of the output variable when the relative degree of the dynamic system is known.
(k)
r-k
(0)
Implementation of the proposed r-sliding controllers requires real-time observation of the successive derivatives (r-2) , ..., σ σ , σ . The possibility of successive real-time robust exact differentiation is demonstrated, providing for control using only observation of the output variable σ. The corresponding best possible accuracy is estimated in the presence of infinitesimally small measurement errors. Thus the above-mentioned control of the output variable may be realized when only that variable is available.
|σ | ~ τ , k = 0, 1, .... r, are satisfied at the same time (σ = σ). Thus, in order to achieve the r-th order of sliding precision in discrete time realization, the sliding mode order in the continuous-time VSS has to be at least r. The standard sliding modes provide for first order real sliding only. The second order of real sliding was achieved by discrete switching modifications of the 2-sliding algorithms [1, 3, 7, 9] and by a special discrete switching algorithm [12]. Real sliding of the third order is demonstrated in [5].
A regularization method is presented removing the complicated discontinuity structure of the proposed controllers. 3-sliding controller usage for kinematic control of a car is demonstrated. 4-sliding control is shown on a model example.
In practice the final sliding accuracy is always achieved in finite time. With asymptotically stable modes, however, it is not observable at any fixed moment, for the convergence time tends to infinity with the rise in accuracy. There are also no known examples of the higher-order accuracy with infinite-time-convergence sliding mode realization.
2. Preliminaries
3. The problem statement (r-1)
Examples. Let σ, , ..., σ be the coordinates, the origin being the r-sliding set. It is assumed in this paper that it is (j) clear from the context when σ is an independent variable and when a derivative. r-sliding modes appear in the simplest systems like σ(r) = sign σ as an unstable Filippov solution σ ≡ 0. Consider another example of r-sliding mode σ
(r)
σ = - sign [Pr-1( GWG )σ], (r-1)
(2)
(r-2)
where Pr-1(λ) = λ +α1λ +...+αr-1 is a stable polynomial, αi ∈R. If r = 1, P0 = 1, (2) defines a standard 1-sliding mode. Let r > 1. An r-sliding mode exists here at the origin and is asymptotically stable. There is also a 1-sliding mode on the manifold Pr-1( GWG )σ = 0 in some vicinity of the origin. Trajectories transfer in finite time into the 1-sliding mode Pr-1( GWG )σ = 0 and then exponentially converge to the r-
Consider a dynamic system of the form
[ = a(t,x) + b(t,x)u,
σ = σ(t, x),
(3)
n
where x ∈ R , a, b, σ are smooth functions, u ∈ R. The relative degree r of the system is assumed to be constant and known. That means, in a simplified way, that u first appears (r) G σ ≠ explicitly only in the r-th total derivative of σ and GX 0 at the given point [6]. The task is to fulfill the constraint σ(t, x) = 0 in finite time and to keep it exactly by discontinuous feedback control. Introduce new local coordinates y = (y1,..., yn), where y1= σ, (r-1) y2 = σ , ..., yr = σ . Then (r)
σ = h(t,y) + g(t,y)u, (r-1)
g(t,y) ≠ 0, (r-1)
sliding mode. Controllers [2, 11] are based on such modes.
ξ = η(t, σ ,...,σ
When stable, the HOSMs in the above examples are only asymptotically stable. Controllers providing for appearance of a 2-sliding mode attracting in finite time are listed in [3, 7, 9]. Following are two simple examples:
Let a trivial controller u = - K sign σ be chosen with K > sup|ueq|, ueq =-h(t,y)/g(t,y) (equivalent control [13]). Then the substitution u = ueq defines a differential equation on the r-sliding manifold (1). Its solution provides for the r-sliding motion. Usually, however, such a mode is not stable.
= - αM sign σ - αm sign σ ; σ ρ
σ = - λ |σ| sign σ + ξ,
ξ = - α sign σ.
Here αM > αm > 0, λ, α > 0, 0 < ρ ≤ 0.5. Corresponding controllers are listed in [7].
, ξ) + γ(t, σ ,...,σ
(4)
Thus, the r-sliding motion is described by the equivalent control method [13], on the other hand, that dynamics coincides with the zero-dynamics [6]. r-1
Real sliding. In reality, switching imperfections being present, ideal sliding σ ≡ 0 could not be attained. The simplest switching imperfection is discrete switching caused by discrete measurements. It was proved [7] that the best possible sliding accuracy attainable with discrete switching (r) r in σ is given by the relation |σ| ~ τ , where τ > 0 is the minimal switching time interval. Moreover, the relations
,ξ)u, ξ = (yr+1, ..., yn).
It is easy to check that g = LbLa σ =
∂ ∂X
(r)
σ . Obviously, h=
r
La σ is the r-th total time derivative of σ calculated with u = 0. In other words, functions h and g may be defined in terms of input-output relations. The problem is to find a discontinuous feedback u = U(t, x) causing finite-time convergence to an r-sliding mode. That
controller has to generalize the 1-sliding relay controller u = - K sign σ. Hence, g(t,y) and h(t,y) in (4) are to be bounded, h > 0. Thus, we require that for some Km , KM, C > 0 0 < Km ≤
∂ ∂X
(r)
r
σ ≤ KM,
| La σ | ≤ C.
(5)
4. Arbitrary-order sliding controllers Let p be a positive number. Denote (r- 1)/ r
N1,r = |σ| , p/r p/(r-1) (i-1) p/( r-i+1) (r- i)/p Ni,r = (|σ| + | σ | + ... + |σ | ) , i = 1,..., r-1, p/r p/(r-1) (r-2) p/2 1/p Nr-1,r = (|σ| + | σ | + ... + |σ | ) . φ0,r = σ, φ1,r = σ + β1 N1,r sign(σ), (i) φi,r = σ + βi Ni,r sign(φi-1,r ), i = 1,..., r-1,
(6) (Fig. 2). That sliding mode is described by the differential equation φr-1,r = 0 providing in its turn for the existence of a 1-sliding mode φr-2,r = 0. But the primary sliding mode disappears at the moment when the secondary one is to appear. The resulting movement takes place in some vicinity of the subset of Γ satisfying φr-2,r = 0, transfers in finite time into some vicinity of the subset satisfying φr-3,r = 0 and so on. While the trajectory approaches the r-sliding set, set Γ retracts to the origin in the coordinates σ, σ , ..., (r-1) σ . Set Γ with r = 3 is shown in Fig. 3. (r-1) σ (r-2)
σ, σ , ...,σ
Γ
where β1,..., βr-1 are positive numbers.
Fig. 2: The idea of the r-sliding controller
Theorem 1. Let system (3) have relative degree r with respect to the output function σ and (5) be fulfilled. Then with properly chosen positive parameters β1,..., βr-1, α controller (r-1)
u = - α sign(φr-1,r(σ, σ , ..., σ
)).
(6)
provides for the appearance of r-sliding mode σ≡0 attracting trajectories in finite time. The positive parameters β1,..., βr-1 are to be chosen sufficiently large in the index order. Each choice determines a controller family applicable to all systems (3) of relative degree r. Parameter α > 0 is to be chosen specifically for any fixed C, Km, KM. The proposed controller is easily generalized: coefficients of Ni,r may be any positive numbers etc. Obviously, α is to be negative with
∂ ∂X
(r)
σ < 0.
Certainly, the number of choices of βi is infinite. Here are a few examples with βi tested for r ≤ 4, p being the least common multiple of 1, 2, ..., r. The first is the relay controller, the second is listed in [3, 7]. u = - α sign σ, 1/2 u = - α sign( σ + |σ| sign σ), 3 2 1/6 2/3 + 2 (| σ | +|σ| ) u = - α sign( σ sign( σ + |σ| sign σ)), 6 4 3 1/12 + σ +|σ| ) + u = - α sign{ σ + 3 ( σ sign[ σ 4 3 1/6 3/4 ( σ +|σ| ) sign( σ +0.5 |σ| sign σ )]}, (4) 12 15 20 30 1/60 | + | σ | ) 5. u = -α sign (σ + β4 (|σ| + | σ | + | σ 12 15 20 1/30 12 | ) + β (|σ| + sign( σ +β3 (|σ| + | σ | + | σ sign( σ 2 15 1/20 4/5 | σ | ) sign( σ +β1|σ| sign σ )))) 1. 2. 3. 4.
The idea of the controller is that a 1-sliding mode is established on the smooth parts of the discontinuity set Γ of
Fig. 3: The discontinuity set of the 3-sliding controller (r-1)
Controller (6) requires the availability of σ, σ , ..., σ . The needed information may be reduced if the measurements are carried out at times ti with constant step τ > 0. Consider the controller (r-2)
u(t) = -α sign(∆σi + (r-2) (r-2) βr-1τNr-1,r(σi , σ ,...,σi )sign(φr-2,r(σi , σ ,...,σi ))), L
where σi ti+1).
(j)
L
(j)
= σ (ti , x(ti )), ∆σi
(r-2)
= σi
(r-2)
- σi-1
(7)
(r-2)
, t ∈ [ti ,
Theorem 2. Under conditions of Theorem 1 with discrete measurements both algorithms (6) and (7) provide in finite time for some positive constants a0, a1, ..., ar-1 for fulfillment of inequalities r
r-1
(r-1)
|σ| < a0τ , | σ | < a1τ , ..., |σ
| < ar-1τ.
That is the best possible accuracy attainable with (r) discontinuous σ . Following are some remarks on the usage of the proposed controllers.
Convergence time may be reduced by changing coefficients -j (j) (j) r βj. Another way is to substitute λ σ for σ , λ α for α and λτ for τ in (6) and (7), λ > 0, causing convergence time to be diminished approximately by λ times. Implementation of r-sliding controller when the relative degree is less than r. Introducing successive time (r-k-1) derivatives u, X , ..., u as new auxiliary variables and (r-k) u as a new control, achieve different modifications of each r-sliding controller intended to control systems with relative degrees k = 1, 2, ..., r. The resulting control is (r-k1)-smooth function of time with k< r, a Lipschitz function with k = r - 1 and a bounded “infinite-frequency switching” function with k = r. Chattering removal. The same trick removes the chattering (r-1) effect. For example, substituting u for u in (7), receive a local r-sliding controller to be used instead of the relay controller u = - sign σ and attain r-th order sliding precision with respect to τ by means of (r - 2)-smooth control with Lipschitz (r-2)-th time derivative. It has to be modified like in [3, 7] for global usage. Controlling systems nonlinear on control. Consider a (i) system [ = f(t,x,u) nonlinear on control. Let ∂∂ σ (t,x,u) = X
0 for i = 1, ..., r-1, (r+1)
σ
∂ ∂X
(r)
σ (t,x,u) > 0. It is easy to check that
= Λu σ + ∂∂ σ r+1
(r)
X
X , Λu(⋅) = ∂∂ (⋅) + ∂∂ (⋅) f(t,x,u). W
[
The problem is now reduced to that considered above with relative degree r+1 by introducing a new auxiliary variable u and a new control v = X . Discontinuity regularization. The complicated discontinuity structure of the above-listed controllers may be smoothed by replacing the discontinuities under the sign-function with their finite-slope approximations. As a result, the transient process becomes smoother. Consider, for example, the above-listed 3-sliding controller. The function sign( σ + 2/3 |σ| sign σ) may be replaced by the function max[-1, min(1, -2/3 2/3 |σ| ( σ + |σ| sign σ)/ε)] for some sufficiently small ε > 0. For ε = 0.1 the resulting tested controller is 3
2 1/6
+ 2 (| σ | +|σ| ) u = - α sign( σ max[-1, min(1, -2/3 2/3 10|σ| ( σ + |σ| sign σ))]).
(8)
Controller (8) provides for existence of a standard 1-sliding mode on the corresponding continuous piece-wise smooth surface. Theorem 3. Theorems 1, 2 remain valid for controller (8).
Thus, theoretically no model of the controlled process needs to be known. Only the relative degree and 3 constants are needed in order to adjust the controller. Unfortunately, the problem of successive real-time exact differentiation is usually considered to be practically unsolvable. Nevertheless, under some assumptions the real-time exact robust differentiation is possible. Indeed, let input signal η(t) be a Lebesgue-measurable locally bounded function defined on [0, ∞) and let it consist of a base signal η0(t) having a derivative with Lipschitz’s constant C > 0 and a bounded measurable noise N(t). Then the following system realizes a real-time differentiator [8]: ς = v, v = ζ1-λ |ζ - η(t)| sign(ζ - η(t)), ς = -µ sign(ζ-η(t)) 1/2
where µ, λ > 0. Here v(t) is the output of the differentiator. Solutions of the system are understood in the Filippov sense. Parameters may be chosen in the form µ = 1.1C, λ = 1/2 1.5C , for example (it is only one of possible choices). That differentiator provides for finite-time convergence to the exact derivative of η0(t) if N(t) = 0. Otherwise, if sup 1/2 1/2 N(t) = ε it provides for accuracy proportional to C ε . Therefore, having been implemented k times successively, that differentiator will provide for k-th order differentiation (2 k)
accuracy of the order of ε . Thus, full local real-time robust control of output variables is possible, using only output variable measurements and knowledge of the relative degree. It is proved in [8] that when the base signal η0(t) has (r-1)th derivative with Lipschitz’s constant C > 0, the best k/r (r-k)/r possible k-th order differentiation accuracy is dkC ε , where dk > 1 may be estimated [8] (this asymptotics may be improved with additional restrictions on η0(t)). Moreover, it is proved that such a robust exact differentiator really exists. Unfortunately, that is only a pure existence theorem. Hence, the following result is rather abstract. Theorem 4. An optimal k-th order differentiator [8] having been applied, r-sliding controller (6) provides locally for (i) (r-i)/r the sliding accuracy |σ | ≤ ci ε , i = 0, 1, ..., r-1, where ε is the maximal possible error of real-time measurements of σ and ci are some positive constants. Theorem 3 probably determines the best sliding asymptotics attainable, when only σ is available. Following is the first known second order asymptotically-optimal robust exact differentiator: 2/3
[ = v, v = - λ1(x - η(t)) sign(x - η(t)) + v1 + z, 1/3 Y = - α1(x - η(t)) , 1/2
Real-time control of output variables. The implementation of the above-listed r-sliding controllers requires real-time (r-1) , ..., σ observation of the successive derivatives σ , σ .
] = z1, z1 = - λ2|z - v| sign(z - v) + z2, ] = - α2 sign(z - v) .
Here λ2, α2 > 0 are chosen with respect to given Lipschitz constant C of η , λ1, α1 > 0 are chosen in advance; v and z1 are estimations of η and η with accuracies proportional 1/3 2/3
to C ε
2/3 1/3
and C ε
respectively.
5. Examples Car control. Consider a simple kinematic model of car control [10] [ = v cos ϕ, \ = v sin ϕ, ϕ = v/l tan δ, δ = u,
where x and y are Cartesian coordinates of the rear-axle middle point, ϕ is the orientation angle, v is the longitudinal velocity, l is the length between the two axles and δ is the steering angle. The task is to steer the car from a given initial position to the trajectory y = g(x), while x, y and ϕ are assumed to be measured in real time. Define σ = y - g(x), Let v = const = 10 m/s, l = 5 m, g(x) = 10 sin(0.05x) + 5, x = y = ϕ = δ = 0 at t = 0. The relative degree of the system is 3 and both 3-sliding controller N°3 and its regularized form (8) may be applied here. It was taken α = 20. The corresponding trajectories are the same, but the performance is different. The trajectory and function y = -4 g(x) with measurement step τ = 2⋅10 are shown in Fig. 4. are shown in Fig. 5, 6 for regularized Graphs of σ, σ , σ and not regularized controllers respectively.
Fig. 6: Standard 3-sliding controller 4-sliding control. Consider a model example of a tracking system. Let input z(t) and the control system satisfy equations (4) z + 3 ] + 2z = 0, (4) x = u. The task is to track z by x, σ = x - z, the 4-th controller with α = 40 is used. Initial conditions for z and x at time t = 0 are ] = 0, ] (0) = 0, ] (0) = 2, ] (0) = 0; x(0) = 1, [ (0) = 1, [ (0) = 1, [ (0) = 1.
A mutual graph of x and z with τ = 0.01 is shown in Fig. 7. A mutual graph of ] and [ with τ = 0.001 is shown in , σ with τ = 0.001 are Fig. 8. Mutual graphs of σ, σ , σ demonstrated in Fig. 9. The attained accuracies are |σ| ≤ -4 -12 1.33⋅10 with τ = 0.01 and |σ| ≤ 1.49⋅10 with τ = 0.0001.
Fig. 7: 4-sliding tracking
Fig. 4: Car trajectory
Fig. 8: Third derivative tracking Fig. 5: Regularized 3-sliding controller
References
Fig. 9: Tracking deviation and its three derivatives
5. Conclusions A family of r-sliding controllers with finite time convergence are presented, providing for full real-time control of any output variable if its relative degree r is known. Being globally applicable when the stated conditions hold, these controllers are still locally applicable in the general case of a smooth system. Instead of the rsliding controller providing for discontinuous switching control, a higher order controller may be used, featuring any desired control smoothness, finite-time arbitrarily fast transient process, and chattering elimination. The controller parameters may be chosen in advance, so that only a single scalar parameter is to be adjusted in order to control any system with a given relative degree. While the r-sliding controller uses measurements of σ, σ , (r-1) ..., σ and keeps σ ≡ 0, its modification, corresponding to (r-2) discrete measurements of σ, σ , ..., σ , provides for the (j) r-j accuracy |σ | ~ τ , j = 0, 1, ..., r with measurement step τ. (r) That accuracy is the best possible when σ is discontinuous [7]. A discontinuity regularization method is proposed additionally improving the control features. Robust exact differentiators with finite-time convergence allow real-time exact successive differentiation of any order, , ..., provided higher derivatives are bounded. Thus, with σ (r) σ bounded, the only needed real-time information is the current value of σ. In principle, an accuracy is attainable proportional to the accuracy of σ measurements. At the same time, the differentiation accuracy inevitably deteriorates rapidly with the growth of the differentiation order in the presence of measurement noises, and direct observation of the derivatives is preferable. The author is grateful to A. Stotsky for helpful discussions.
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