Robust Exact Uniformly Convergent Arbitrary ... - Semantic Scholar
CONFIDENTIAL. Limited circulation. For review only
Robust Exact Uniformly Convergent Arbitrary Order Differentiator Marco Tulio Anguloa,∗, Jaime A. Morenoc , Leonid Fridmanb a Facultad de Ingeniería, UAQ, México
(e-mail: [email protected]) b Departamento de Control y Robótica, UNAM, México
(e-mail: [email protected]) c Eléctrica y Computación, Instituto de Ingeniería, UNAM, México
(e-mail: [email protected])
Abstract An arbitrary order differentiator that, in the absence of noise, converges to the true derivatives of the signal after a finite time independent of the initial differentiator error is presented. The only assumption on a signal to be differentiated ( n − 1)-times is that its n-th derivative is uniformly bounded by a known constant. The proposed differentiator switches from a newly designed uniform differentiator to the classical High-Order Sliding Mode (HOSM) differentiator. The Uniform part drives the differentiation error trajectories into a compact neighborhood of the origin in a time that is independent of the initial differentiation error. Then, the HOSM differentiator is used to bring the differentiation error to zero in finite-time. Keywords: differentiator; robustness; sliding-mode control.
not be satisfactory for switched or hybrid systems. To avoid this requirement, [10] proposed a fixed-time conReal-time differentiators are used to solve a wide vergent first-order differentiator based on the supervariety of problems: from the construction of PID regtwisting algorithm. ulators, to the construction of observers and fault diagThis Note considers the application of the HOSM nosis algorithms [1, 2, 3]. differentiator to systems with some type of (strictly Conventional (first order) sliding mode differentiapositive) dwell-time. This kind of behavior naturally tors (see for example [4]) use filters in order to recover arises in hybrid and switching systems, but is not limthe desired derivative. In the presence of measureited to them. A “dwell-time” is also present in nonlinment noise, their precision is limited by the filter’s time ear dynamics with finite-time escape and, in general, constant. In contrast, Levant’s first order differentiain any model of a real system whose validity is limtor [5] based on second-order sliding modes do not reited to certain bounded time interval. If the HOSM quire any filtration and, more importantly, provides differentiator would be applied in any of the situations the best possible asymptotic precision in the sense of described above, it should provide an estimate of the [6]. However, the step-by-step application of first-order required derivatives during the dwell-time of the sysdifferentiators to obtain higher-order derivatives [7] do tem independently of the initial differentiation error. not ensure the best possible asymptotic accuracy in the The HOSM differentiator can fulfill this last consense of [6]. dition only by increasing its gain as the initial differTo reach such precision, Levant introduced the HOSM entiation error increases. This is obviously impossidifferentiator [8], see also [9]. However, its converble since the initial differentiation error is unknown. gence time grows unboundedly together with the iniTherefore, it becomes necessary to assume a known tial differentiation error. This means that when the bound for it [11, 1, 2]. system has some kind of dwell-time1 , the HOSM differSolving this problem without making additional asentiator requires the additional knowledge of a bound sumptions requires that the convergence time of the in its initial differentiation error to ensure its converdifferentiator is uniform with respect to the initial difgence during the dwell-time of the system. This could ferentiation error. It means that its convergence time from an arbitrary initial condition must be uniformly bounded. An exact first-order differentiator with this ∗ Corresponding author: [email protected], April 10, 2013 property has been recently presented in [10]. It was 1 In an hybrid or switching system, the so-called “dwell-time” is designed by adding high-degree terms to the the difthe time the system takes to switch between modes of operation. 1. Introduction
1
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CONFIDENTIAL. Limited circulation. For review only 2
Given a signal σ( t), the real-time differentiation problem consists in obtaining an estimate of its successive derivatives σ( i) ( t), i = 1, . . . , n − 1. The only assumption made on the signal to be differentiated is that |σ(n) ( t)| ≤ L for t ≥ 0, with L a known constant. Defining x1 = σ, x2 = σ˙ , . . . , xn = σ(n−1) yields
x˙ 2 = x3 ,
···
x˙ n = −w,
x˙ˆ i x˙ˆ n
n− i n
+ xˆ i+1 ,
=
− k i d x˜ 1 c
=
− k n sign( x˜ 1 ),
i = 1, . . . , n − 1, (2)
where d xc p = | x| p sign( x). Due to the discontinuities in its right-hand side, its solutions will be understood in Filippov’s sense [14]. The gains of the HOSM differentiator { k i }ni=1 can be selected based on its recursive form, as shown in [8], providing for its finite-time exact convergence. This procedure is illustrated in Section 5. However, the HOSM differentiator does not converge uniformly with respect to the initial differentiation error. Indeed, its convergence time grows as an homogeneous norm of the initial differentiation error [15]. Therefore, the convergence time of the HOSM differentiator grows unboundedly as the initial differentiation error grows, as illustrated by Figure 1.
initial differentiation error Figure 1: Picture of the qualitative behavior of the maximal
2. Problem Statement and Main Result
x˙ 1 = x2 ,
Rn , the convergence time of the differentiator is uniformly bounded by a constant. A precise definition of this last property will be given in Definition 1. The HOSM differentiator [8] is finite-time exact. It takes the following (non-recursive) form
convergence time
ferentiator [5], and using a Lyapunov-based analysis. The contribution of this paper is the construction of an exact arbitrary order differentiator with uniform convergence with respect to the initial differentiation error. In contrast to [10], we use homogeneity properties in the differentiator design, similarly to [12]. It is shown that by simply reversing the sign of the homogeneity degree characterizing the finite-time convergence of the HOSM differentiator, uniform convergence with respect to the initial condition is obtained. In this sense, both properties are complementary. This consideration allows designing a practically uniformly convergent differentiator that can be combined with the HOSM differentiator. Unlike [10], the stability properties of the new uniform convergent part are deduced by applying a simple quadratic Lyapunov function on a linear system, and using the properties of homogeneity and continuity in the same spirit as [13]. In this sense, the newly designed nonlinear part of the differentiator is totally designed by using linear methods. The remainder of the paper is organized as follows. Section 2 introduces the problem statement and presents the main result of the paper: the Uniform HOSM differentiator. Section 3 formally introduces the concept of uniform convergence with respect to the initial condition and its characterization using homogeneity. Section 4 presents the stability analysis of the new uniform convergent part of the differentiator. Section 5 presents a simulation example and, finally, Section 6 summarizes the paper. All the proofs of the Lemmas and Theorems, except for Lemma 2, are collected in the Appendices.
(1)
and the problem of constructing an ( n − 1)-th order differentiator for σ( t) can be transformed into the construction of an observer for system (1), based on the measured output x1 = σ, despite the bounded disturbance w( t) = −σ(n) ( t). Let xˆ ( t) denote the estimate of x( t), and x˜ = xˆ − x the differentiation error. We are interested in two main properties of the differentiator: finite-time exact convergence and uniform convergence with respect to the initial differentiation error. Finite-time exactness means that the differentiator provides the exact value of the derivatives after a finite-time, despite w. Stated informally, the uniform convergence property means that, for any initial differentiation error x˜ (0) ∈
convergence time of a differentiator as a function of the norm of the initial differentiation error. The differentiator plotted in solid black line has uniform convergence with respect to the initial differentiation error. In contrast, the differentiator plotted in dashed blue is not uniform.
The focus of this paper is in the design of an “uniformly finite-time exact differentiator”, which has the two properties: finite-time exact convergence despite perturbations and uniform convergence w.r.t. the initial differentiation error. Definition 1. Assume that the disturbance w belongs to certain functional class W . A differentiator is said to be uniformly finite-time exact, if there exists time T such that for all w ∈ W , and any initial differentiation errors x˜ (0) ∈ Rn the condition xˆ ( t) = x( t), ∀ t ≥ T is satisfied. In this paper, the class W consists of all measurable functions of time bounded by the given number L in their absolute value. In what follows, and without loss of generality, the initial time instant for all differential equations is assumed to be zero.
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CONFIDENTIAL. Limited circulation. For review only 3 In contrast to the HOSM differentiator, the convergence time (from an arbitrary initial differentiation error) of an uniformly finite-time exact differentiator is uniformly bounded. As we have motivated in the introduction, this feature is useful when the differentiator is applied to systems with dwell-time. Our main result is: Theorem 1. The ( n − 1)-th order differentiator
x˙ˆ i
=
−κ i θ d x˜ 1 c
n− i n
− k i (1 − θ )d x˜ 1 c
n+α i n
+ xˆ i+1 ,
i = 1, . . . , n − 1; x˙ˆ n
=
3. Homogeneity and Uniform Convergence with respect to the initial condition The design of the HOSM differentiator was based on homogeneity properties [12]. We briefly recall three basic concepts: Definition 2 (see [12]). a) The family of dilations Λλ , associated with the “weight vector” ( r 1 , . . . , r n ) ∈ n R+ , is the linear map
Λλ : ( x1 , x2 , . . . , xn ) 7→ (λr 1 x1 , λr 2 x2 , . . . , λr n xn ),
(3) 1+α
−κn θ sign( x˜ 1 ) − k n (1 − θ )d x˜ 1 c
,
is uniformly finite-time exact when its parameters are selected as follows: (i) {κ i }ni=1 are selected based on the bound of the perturbation L using the formulas for the original HOSM differentiator [8]; (ii) α > 0 is chosen small enough and { k i }ni=1 are selected such that the following matrix in Hurwitz: −k1 1 0 · · · 0 −k2 0 1 · · · 0 (4) A= . ; .. .. . −k n 0 · · · 0 0 (iii) the function θ : [0, ∞) → {0, 1} is selected as ½ 0 if t ≤ T u , θ ( t) = 1 otherwise, with some arbitrarily chosen T u > 0.
P ROOF. See Appendix A. ( i −1)
Theorem 1 shows that the equalities xˆ i = σ , i= 1, . . . , n, will be stablished after a finite-time transient that is upper bounded by a number that is independent of the initial differentiation error. In what follows, we will refer to the differentiator (3) as the Uniform HOSM differentiator. The Uniform HOSM differentiator has two kinds of terms. The n− i terms d x˜ 1 c n have degrees lower than one. They are inherited from the HOSM differentiator and are responsible for the finite-time exact convergence of the n+α i differentiator. The terms d x˜ 1 c n have degrees larger than one, and are responsible for the uniform convergence with respect to the initial differentiation error. The Uniform HOSM is constructed by switching from the high to the low order terms (i.e. the HOSM differentiator) after an arbitrary positive amount of time T u . The parameter T u can be arbitrarily chosen using any method, as long as it is positive. The rest of the paper provides the ground to prove Theorem 1. Indeed, we shall show that the design of the new “uniform part” is closely related to the design of the original HOSM differentiator using homogeneity properties: it is just necessary to change the sign of its homogeneity degree.
for λ > 0, where r i is the weight (or degree) of x i and is denoted as deg x i = r i . b) The vector field f (respectively the vector-set field F ) are called homogeneous of the homogeneity degree p with dilation Λλ if
f (Λλ x) = λ p Λλ f ( x),
∀λ > 0,
(respectively F (Λλ x) = λ p Λλ F ( x)) and is denoted as deg f = p (respectively deg F = p). In particular, note that a differential equation x˙ = f ( x) with deg f = p (or inclusion x˙ ∈ F ( x) with deg F = p) is invariant with respect to the time-coordinate transformation ( t, x) 7→ (λ− p t, Λλ x). This fact suggests to define deg t = − p, and call the differential equation itself (or inclusion) homogenous with degree p. The structure of the HOSM differentiator (2) can be derived based on the homogeneity properties introduced above. Consider a differentiator with the following general form
x˙ˆ i = − f i ( x˜ 1 ) + xˆ i+1 , i = 1, . . . , n − 1;
x˙ˆ n = − f n ( x˜ 1 ),
and restrict the functions f i to be the simplest homogeneous functions f i ( x˜ 1 ) = k i d x˜ 1 cm i , k i , m i ∈ R. Set deg f = α or, equivalently, deg t = −α. Then, due to the structure of the chain of integrators, the weight of each variable x˜ i will be fixed once the weight of one variable is selected. Picking deg x˜ 1 = n yields deg x˜ i = n + α( i − 1), which means that the differentiator takes the form
x˙ˆ i ˙xˆ n
− k i d x˜ 1 c
=
n+α i n
1+α
− k n d x˜ 1 c
=
+ xˆ i+1 ,
i = 1, . . . , n − 1;
,
(5)
and the differentiation error is given by
x˙˜ 1
x˙˜ n−1 x˜˙ n
= .. . = =
n+α n
− k 1 d x˜ 1 c
+ x˜ 2 ,
(6) − k n−1 d x˜ 1 c
n+( n−1)α n
1+α
− k n d x˜ 1 c
+ x˜ n ,
+ w,
where { k i }ni=1 are the gains of the differentiator. Homogeneous Asymptotically stable Filippov differential inclusions (and hence, Filippov differential
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CONFIDENTIAL. Limited circulation. For review only 4 equations) with negative homogeneity degree converge in finite-time [12]. The HOSM differentiator (2) is obtained from (5) by setting deg f = α = −1. Its homogeneity degree α is negative to guarantee finite-time stability. More importantly, this choice yields a discontinuous algorithm ensuring its exactness against the bounded disturbance w, see [12]. Note that any other choice of α ∈ (−1, 0) results in a continuous differentiator that is finite-time convergent, but can not be exact against a bounded disturbance. 3.1. Uniform Convergence with respect to the Initial Condition. In analogy to [12], uniform convergence with respect to the initial condition can also be characterized in terms of homogeneity. Indeed, we shall see that simply reversing the homogeneity degree from negative to positive yields uniform convergence w.r.t. the initial condition instead of finite-time stability. In this sense, both properties are complementary. To see this, let us rewrite system (6) in the form ˙˜ = f ( x˜ ) + g( x, ˜ w), x
x˜ (0) = x˜ 0 ,
(7)
P ROOF. See Appendix B.
Lemma 1 shows that the general homogeneous differentiator (5) is practically uniformly convergent when α > 0 and the parameters { k i }ni=1 are selected to provide asymptotic stability when w ≡ 0. A method to determine the parameters { k i }ni=1 is presented in Section 4. Moreover, as shown in Theorem 1, the HOSM differentiator can be combined with the practically uniformly convergent differentiator to obtain a uniform exact finite-time convergent differentiator. For this, it is sufficient to start with the practically uniformly convergent differentiator, given by (5) with α > 0, and then switch to the HOSM differentiator (2) after an arbitrary positive amount of time. Note that using the practically uniformly convergent differentiator for any positive amount of time is enough to bring the trajectories that started at “infinity” into a compact. From that compact, the convergence time of the HOSM differentiator is uniformly bounded. In addition, the convergence of the HOSM differentiator can be further accelerated by using a time-varying gain without deteriorating its asymptotic precision [9].
n+α i
˜ w) = 0 for i = where f i ( x˜ ) = − k i d x˜ 1 c n + x˜ i+1 , g i ( x, ˜ w) = w. The 1, . . . , n − 1; and f n ( x˜ ) = − k n d x˜ 1 c1+α , g n ( x, vector field g can be regarded as a disturbance to the nominal part f . Moreover, recall that the perturbation w belongs to the class of functions uniformly bounded by a constant, that we denote as W . Definition 3 distinguishes two kinds of uniform convergence with respect to the initial condition.
Remark 1. When the signal σ( t) is measured with additive noise uniformly bounded by δ, the Uniform HOSM differentiator has the same precision as the HOSM differentiator. It means that the following accuracies i
| xˆ i − x i | = O (L n+1 δ
n− i +1 n+1
),
i = 1, . . . , n,
are obtained after finite-time [8].
Definition 3. System (7) is said to be: i) assume that g ≡ 0. The system is globally asymptotically stable, uniformly in the initial condition, if for any r > 0, there exists T > 0 such that for all x˜ 0 ∈ Rn we have k x˜ ( t)k ≤ r if t ≥ T ; ii) practically uniformly convergent with respect to the initial condition if there exists T > 0 and r > 0 such that for all w ∈ W and x˜ 0 ∈ Rn we have k x˜ ( t)k ≤ r if t ≥ T . For the case of an undisturbed system ( g ≡ 0) with continuous f , uniform convergence has been characterized based on its positive homogeneity degree [16]. Practical uniform convergence can be characterized for general disturbed systems of the form (7), by the properties that the disturbance g need to satisfy with respect to the nominal part f , as shown below. Lemma 1. System (7) is practically uniformly convergent with respect to the initial condition x˜ 0 if the following three conditions hold: (i) its origin is globally asymptotically stable when g ≡ 0; (ii) f is a continuous vector field and deg f = α > 0; (iii) w is uniformly bounded.
4. Stability Analysis of the Uniform part To guarantee the practical uniform convergence w.r.t. the initial condition of the differentiation error (6), it is necessary to appropriately select its parameters α and { k i }ni=1 . As shown in Lemma 1, it is enough to select them to guarantee that system (6) with w ≡ 0 is asymptotically stable for α > 0. Note that when α = 0, the nonlinear system (6) is reduced to a linear one, whose stability is completely determined by the stability of the A matrix of equation (4) in Theorem 1. This fact can be easily proved using a quadratic Lyapunov function. Nevertheless, in such a case, the designed differentiator does not converge uniformly since α is not positive. To analyze the stability when α > 0, the idea is to use the information of the same quadratic Lyapunov function on the nonlinear system, together with the continuity of its derivative with respect to the parameter α. This classical idea can be found in [13]. Lemma 2. If α > 0 is sufficiently small and the parameters { k i }ni=1 are selected such that the A matrix is Hurwitz, then the differentiation error (6) is asymptotically stable if w ≡ 0.
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CONFIDENTIAL. Limited circulation. For review only 5 P ROOF. Consider V ( x˜ ) = x˜ T P x˜ with P positive definite and solution to AT P + PA ≺ 0. Let us denote by ˜ the derivative of V ( x˜ ) along system (6) with V˙ (α, x) w ≡ 0 and the parameter α. Then
V˙ (0, x˜ ) < 0,
∀ x˜ ∈ S,
for any set S ⊆ Rn \ {0}, since it indeed is a Lyapunov function for the linear system obtained when α = 0. Moreover, note that V˙ (α, x˜ ) is continuous in both of its arguments, α and x˜ . In particular, if S is compact, V˙ (α, x˜ ) is uniformly continuous in the set {(α, x˜ ) ∈ R×Rn |α = 0, x˜ ∈ S }. This means that there exists vicinities of S and α = 0, denoted as N S and Nα=0 respectively, such that V˙ (α, x˜ ) < 0 for all (α, x˜ ) ∈ Nα=0 × N S . Now, let us choose S as an arbitrary level curve of V , i.e. S = V −1 (δ), δ > 0. Therefore, there exists α > 0 and a vicinity N S , such that the trajectories of the system cross from the outside of N S to its inside. The rest of the proof follows the same argument as in the proof of Lemma 4 shown in the Appendix. By the homogeneity of the system, the trajectories of the system also cross from the outside to the inside of any dilated versions of N S . Moreover, since Rn \ {0} can be covered by dilated version of N S , this ensures that the system is globally asymptotically stable.
starting with α = 0 and increasing its value while2 V −1 (δ) ∩ V˙α−1 (0) = ;. When α = 0, i.e. a linear system, there is no intersection for any δ > 0, since V −1 (δ) is an ellipsoid and V˙0−1 (0) is just the point x˜ = 0. Let us illustrate the procedure described above for our particular example. First, selecting the matrix P as a solution to A T P + P A = Q = − I yields 71/196 57/28 1/2 P = 57/28 9965/686 757/196 . 1/2 757/196 87/28 The surface V˙α−1 (0) for α = 0.06 is shown in Fig. 2. It consists of two parts: a central lobe and two exterior hyperboloids. As α tends to zero, the central lobe shrinks and the exterior hyperboloids tend to retreat to infinity; in the limit α = 0, it consists of only one point x˜ = 0. The ellipsoids V −1 (δ) turned out to be horizontally aligned, irrespectively of the selection of Q in the Lyapunov equation. For α = 0.06, the ellipsoid V −1 (10) does not intersect neither the interior lobes, nor the 1 exterior hyperboloids of V˙0−.06 (0), see Fig. 3. According to the discussion above, this shows that the system is stable for α = 0.06 and the selected gains { k i }3i=1 , and, therefore, the combined differentiator is uniformly exact convergent. In Figure 4, the convergence of the trajectories of the differentiator is shown.
For the case of n = 3, the conditions for stability are α > 0 small enough and k 1 > 0, k 3 > 0, k 1 k 2 > k 3 . 5. Simulation example A second order differentiator was tested using the signal σ( t) = 5 t + sin( t) + 0.01 cos(10 t). For n = 3, the Uniform HOSM differentiator takes the form 2
α
1
2α 3
x˙ˆ 1 = −κ1 θ d xˆ 1 − σc 3 − k 1 (1 − θ )d xˆ 1 − σc1+ 3 + xˆ 2 , x˙ˆ 2 = −κ2 θ d xˆ 1 − σc 3 − k 2 (1 − θ )d xˆ 1 − σc1+
+ xˆ 3 ,
1+α
x˙ˆ 3 = −κ3 θ sign( xˆ 1 − σ) − k 3 (1 − θ )d xˆ 1 − σc
,
where T u = 0.8 determines the form of the function θ ( t). The parameters {κ i }3i=1 can be selected based on the gains for the recursive HOSM differentiator as p κ1 = 2L1/3 , κ2 = 1.5 2L1/2+1/6 , κ3 = 1.1L, with L such that |σ(3) ( t)| ≤ L, see [8]. The gain L = 30 was used. The parameters { k i }3i=1 are selected as k 1 = 7, k 2 = 1/7 + 2, k 3 = 1. Then, Lemma 2 ensures the existence of a small enough parameter α > 0 that guarantees uniform convergence w.r.t. the initial differentiation error using these gains. Let us show, additionally, that it is possible to compute an explicit value for parameter α > 0. Using the proof of Lemma 2, one picks a level curve S = V −1 (δ), and then checks if V˙ (α, x˜ ) < 0 on that level curve for the given α. The largest value of α that satisfies the last condition can be computed iteratively,
Figure 2: The surface V˙α−1 (0) for α = 0.06.
Figure 3: V˙ = 0 for α = 0.06 (yellow) and the ellipsoid V = 10
(magenta). There is no intersection of V˙ = 0 with V = 10, therefore V˙ < 0 in V = 10.
To confirm the property of uniform convergence w.r.t. the initial differentiation error, several initial conditions xˆ 0 were tested and the convergence time of the 2 We use the notation V˙ −1 (0) := { x ˜ ∈ Rn |V˙ (α, x˜ ) = 0}. α
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CONFIDENTIAL. Limited circulation. For review only 6 in the example a simple method to compute all the required parameters of this new part of the differentiator. Future work may include using a Norm Observer to make the switching in the differentiator.
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Acknowledgements
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The authors gratefully acknowledge the financial support from projects PAPIIT 17211 and IN111012, CONACyT 56819, 132125, 51244 and CVU 229959, Fondo de Colaboración del II-FI, UNAM, IISGBAS-165-2011.
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Appendix A. Proof of Theorem 1
−1 −2
When t ≤ T u , the Uniform HOSM differentiator is reduced to the general homogeneous differentiator (5) Figure 4: True derivatives: black, estimated derivatives: blue. with its parameters selected using condition (ii). ThereThe initial condition for the differentiator was chosen as upon, Lemmas 1 and 2 immediately show that it is xˆ (0) = (100, 200, 300). practically uniformly convergent. Moreover, using it a positive amount of time T u brings any trajectory of differentiator measured. The initial conditions were the differentiation error into a sufficiently large but chosen as multiples of the vector [1, 2, 3]. Figure 5 presents compact set containing the origin, see Lemma 3. The a graph between the convergence time of the Uniform size of this compact set only depends on T u . When HOSM differentiator versus the norm of the initial dift > T u , the Uniform HOSM differentiator is reduced ferentiation error. As expected, it shows the presence to the original HOSM differentiator with proper gains of an asymptote in the convergence time as the norm (i), whose convergence time starting from a compact is of the initial condition increases. uniformly bounded.
convergence time
16
Appendix B. Proof of Lemma 1
14
Several preliminary results are required for the proof of Lemma 1. Denote by x˜ ( t, x˜ 0 ) the solution of (7) with initial condition x˜ 0 at time t. For an undisturbed system g ≡ 0 with deg f = p, its homogeneity provides for the following scaling property (see, e.g., [12] or [17]):
12 10 8 6 asymptote linear interpolation measurements
4 2 0 0 10
2
10
4
10
6
10
8
10
10
10
12
10
norm of the initial differentiation error Figure 5: Convergence time of the Uniform HOSM differen-
tiator as a function of the norm of the initial condition.
6. Conclusions We have presented an arbitrary-order differentiator that converges to the true derivatives of the signal after a finite-time independent of the initial differentiation error. The differentiator was constructed by combining the HOSM differentiator with a new part that is practically uniformly convergent. For this, uniform convergence w.r.t. the initial condition was characterized in terms of the homogeneity of the vector fields, and latter shown to be robust (in the sense of practical uniform convergence) to a disturbance that gets eventually dominated by the nominal part. The stability of the new uniform part of the differentiator was analyzed by using a quadratic “Lyapunov function”, together with the continuity and homogeneity of the differentiation error. Using this analysis, we illustrated
Λλ x˜ ( t, x˜ 0 ) = x˜ (λ− p t, Λλ x˜ 0 ).
(B.1)
From this, we can prove the following preliminary result about the uniform convergence without inputs. It is an extension of the proof in [16] that does not require continuity of the vector field f . Lemma 3. Consider (7) with g ≡ 0. Assume that its origin is asymptotically stable and that deg f = p > 0. Then the system is also uniformly convergent in the initial condition. In particular, this implies that for any given time T u > 0, there exists a ball B r such that x˜ ( t, x˜ 0 ) ∈ B r for all t ≥ T u and x˜ 0 ∈ Rn . P ROOF. To prove the first part of the claim, let B0 be an arbitrary compact ball of Rn . Consider the sequence of balls obtained by dilation B i = Λλ B i−1 for all i ≥ 1, with λ > 1 fixed large enough so B i+1 ⊃ B i . Due to the asymptotic stability of the system, every trajectory starting in B1 enters B0 before certain time T ; in symbols x˜ (T, B1 ) ∈ B0 . Applying dilation to both sides of this last expression and using (B.1) yields
Λλ x˜ (T, B1 ) = x˜ (λ− p T, Λλ B1 ) = x˜ (λ− p T, B2 ) ∈ Λλ B0 = B1 ,
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CONFIDENTIAL. Limited circulation. For review only 7 i.e., the trajectories starting in B2 enter B1 before time λ− p T . Repeating the same procedure for each i ≥ 1 and summing each time interval, obtain that for any x˜ 0 ∈ Rn the trajectory of the system enters into B0 after at most ∞ ¡ X ¢i 1 λ− p = T tr = T , 1 − λ− p i =0 where the convergence of the geometric series occurs by observing that λ− p < 1, since λ > 1 and p > 0. Since B0 was arbitrary, the first part of the claim of the Lemma is complete. For the second part of the claim consider two cases. If T u ≥ t r , then one can select B r = B0 . If not, remove a finite number N of terms (λ− p ) i until t reach − PN −p i i =0 (λ ) < T u and pick B r = B N .
This means that if k x˜ kis taken large enough, then V˙ | pert = const · V˙ |nopert . The indexes “pert” and “no pert” denote evaluating the derivative of V along the trajectories of system (7) with and without the perturbation w, respectively. Thus the perturbed system inherits the behavior of the nominal system if k x˜ k is large enough. By hypothesis (ii), p > 0 and using Lemma 3 obtain the desired claim. BIBLIOGRAPHY.
[1] F. Bejarano and L. Fridman, “State exact reconstruction for switched linear systems via a super-twisting algorithm,” International Journal of Systems Science, vol. 42, no. 5, pp. 717–724, 2011. [2] J. Davila, A. Pisano, and E. Usai, “Continuous and discrete state reconstruction for nonlinear switched systems via highorder sliding-mode observers,” International Journal of Systems Science, vol. 42, no. 5, pp. 725 – 735, 2011. Note that the proof of the last lemma is also valid [3] D. Efimov, A. Zolghadri, and T. Raissi, “Actuator fault detection for differential inclusions. Indeed, an analogous arguand compensation under feedback control,” Automatica, vol. 47, no. 8, pp. 1699 – 1705, 2011. ment was used in [12] to prove finite-time exact stabil[4] M. Djemai, N. Manamanni, and J. P. Barbot, “Sliding mode obity when p < 0. This fact evidences the close relation server for triangular input hybrid system,” in IFAC Proceedthat exists between both properties. ings of 16th Triennial World Congress of International Federation of Automatic Control, vol. 16, no. 1, 2005, pp. 181–186. For the proof of Lemma 1, we will also need a pre[5] A. Levant, “Robust exact differentiation via sliding mode techliminary result from [17] that we state as follows: nique,” Automatica, vol. 34, no. 3, pp. 379–384, 1998. [6] A. N. Kolmogorov, “On inequalities between upper bounds of Theorem 2. [17, Thm. 5.8] Let f be a continuous vecconsecutive derivatives of an arbitrary function defined on an tor field such that the origin of (7) with g ≡ 0 is locally infinite interval,” Amer. Math. Soc. Transl, vol. 2, no. 9, pp. 233–242, 1962. asymptotically stable. Assume that deg f = p for some [7] J. P. Barbot and T. Floquet, “Super twisting algorithm-based r ∈ (0, ∞)n . Then for any s ∈ N∗ and any m > s · max r i step-by-step sliding mode observers for nonlinear systems with there exists a strong Lypunov function V ∈ C p which is unknown inputs,” Intern. J. Syst. Sci., vol. 38, no. 10, pp. 803– ˙ homogeneous with deg V = m. Therefore V = 〈∇V , f 〉 is 815, 2007. [8] A. Levant, “High-order sliding modes: differentiation and outhomogeneous of degree m + p. put feedback control,” International Journal of Control, vol. 76, no. 9-10, pp. 1924–041, nov 2003. n Analogously to Definition 2, a function V : R → R [9] ——, “Non-homogeneous finite-time-convergent differentiator,” is said to be homogeneous of degree m if V (Λλ x) = in Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. Proceedings of λm V ( x), ∀λ > 0, and is written as deg V = m. Now we the 48th IEEE Conference on, dec. 2009, pp. 8399 –8404. are ready for the proof of Lemma 1: [10] E. Cruz-Zavala, J. Moreno, and L. Fridman, “Uniform robust exact differentiator,” Automatic Control, IEEE Transactions on, P ROOF (P ROOF OF L EMMA 1). On the one hand, by Hyvol. 56, no. 11, pp. 2727 –2733, nov. 2011. pothesis (i) and (ii), Theorem 2 shows that there exist a [11] A. Pisano and E. Usai, “Globally convergent real-time differentiation via second order sliding modes,” International Journal homogeneous function V with deg V = m and 〈∇V ( x˜ ), f ( x˜ )〉 < of Systems Science, vol. 38, no. 10, pp. 833–844, 2007. 0. On the other hand, using hypothesis (iii) and the [12] A. Levant, “Homogeneity approach to high-order sliding mode structure of f , it is easy to verify that for each ε > 0 design,” Automatica, vol. 41, no. 5, pp. 823–830, 2005. ˜ w)k ≤ εk f ( x˜ )k if k x˜ k ≥ there exists c ≥ 0 such that k g( x, [13] S. Bhat and D. Bernstein, “Geometric homogeneity with applications to finite-time stability,” Math. Control Signals Systems, c. Therefore, in the ( n − 1) dimensional sphere S , then pp. 17: 101–127 DOI 10.1007/s00 498–005–0151–x, 2005. ∗ ∀ε > 0 there exists λ large enough such that [14] A. Filippov, Differential equations with discontinuous righthand side. London: Kruwler, 1960. k g(Λλ S, w)k ≤ εk f (Λλ S )k, ∀λ ≥ λ∗ . [15] M. T. Angulo, L. Fridman, and A. Levant, “Output-feedback finite-time stabilization of disturbed lti systems,” Automatica, vol. 48, no. 4, pp. 606 – 611, 2012. Now compute the derivative of V along the per[16] V. Andrieu, L. Praly, and A. Astolfi, “Homogeneous approxi˜ w)〉. turbed system V˙ ( x˜ ) = 〈∇V ( x˜ ), f ( x˜ )〉 + 〈∇V ( x˜ ), g( x, mation, recursive observer design and output feedback,” SIAM Taking x˜ ∈ S and fixing 0 < ε < 1, with its correspondJournal on Control and Optimization, vol. 47, no. 4, pp. 1814– ing λ∗ , yields 1850, 2008. [17] A. Bacciotti and L. Rosier, Liapunov Functions and Stability in ˜ w) V˙ (Λλ x˜ ) = 〈∇V , f 〉(Λλ x˜ ) + 〈∇V , g〉(Λλ x, Control Theory, 2nd ed. London: Springer, 2005.
≤
˜ w) −|〈∇V , f 〉|(Λλ x˜ ) + k∇V kk gk(Λλ x,
≤
−|〈∇V , f 〉|(Λλ x˜ ) + εk∇V kk f k(Λλ x˜ ),
≤
−|〈∇V , f 〉|(Λλ x˜ ) + ε|〈∇V , f 〉|(Λλ x˜ )
=
−(1 − ε)|〈∇V , f 〉|(Λλ x˜ ),
∀λ ≥ λ∗ .
Preprint submitted to Automatica Received April 10, 2013 07:44:46 PST
Robust Exact Uniformly Convergent Arbitrary Order Differentiator Marco Tulio Anguloa,∗, Jaime A. Morenoc , Leonid Fridmanb a Facultad de Ingeniería, UAQ, México
(e-mail: [email protected]) b Departamento de Control y Robótica, UNAM, México
(e-mail: [email protected]) c Eléctrica y Computación, Instituto de Ingeniería, UNAM, México
(e-mail: [email protected])
Abstract An arbitrary order differentiator that, in the absence of noise, converges to the true derivatives of the signal after a finite time independent of the initial differentiator error is presented. The only assumption on a signal to be differentiated ( n − 1)-times is that its n-th derivative is uniformly bounded by a known constant. The proposed differentiator switches from a newly designed uniform differentiator to the classical High-Order Sliding Mode (HOSM) differentiator. The Uniform part drives the differentiation error trajectories into a compact neighborhood of the origin in a time that is independent of the initial differentiation error. Then, the HOSM differentiator is used to bring the differentiation error to zero in finite-time. Keywords: differentiator; robustness; sliding-mode control.
not be satisfactory for switched or hybrid systems. To avoid this requirement, [10] proposed a fixed-time conReal-time differentiators are used to solve a wide vergent first-order differentiator based on the supervariety of problems: from the construction of PID regtwisting algorithm. ulators, to the construction of observers and fault diagThis Note considers the application of the HOSM nosis algorithms [1, 2, 3]. differentiator to systems with some type of (strictly Conventional (first order) sliding mode differentiapositive) dwell-time. This kind of behavior naturally tors (see for example [4]) use filters in order to recover arises in hybrid and switching systems, but is not limthe desired derivative. In the presence of measureited to them. A “dwell-time” is also present in nonlinment noise, their precision is limited by the filter’s time ear dynamics with finite-time escape and, in general, constant. In contrast, Levant’s first order differentiain any model of a real system whose validity is limtor [5] based on second-order sliding modes do not reited to certain bounded time interval. If the HOSM quire any filtration and, more importantly, provides differentiator would be applied in any of the situations the best possible asymptotic precision in the sense of described above, it should provide an estimate of the [6]. However, the step-by-step application of first-order required derivatives during the dwell-time of the sysdifferentiators to obtain higher-order derivatives [7] do tem independently of the initial differentiation error. not ensure the best possible asymptotic accuracy in the The HOSM differentiator can fulfill this last consense of [6]. dition only by increasing its gain as the initial differTo reach such precision, Levant introduced the HOSM entiation error increases. This is obviously impossidifferentiator [8], see also [9]. However, its converble since the initial differentiation error is unknown. gence time grows unboundedly together with the iniTherefore, it becomes necessary to assume a known tial differentiation error. This means that when the bound for it [11, 1, 2]. system has some kind of dwell-time1 , the HOSM differSolving this problem without making additional asentiator requires the additional knowledge of a bound sumptions requires that the convergence time of the in its initial differentiation error to ensure its converdifferentiator is uniform with respect to the initial difgence during the dwell-time of the system. This could ferentiation error. It means that its convergence time from an arbitrary initial condition must be uniformly bounded. An exact first-order differentiator with this ∗ Corresponding author: [email protected], April 10, 2013 property has been recently presented in [10]. It was 1 In an hybrid or switching system, the so-called “dwell-time” is designed by adding high-degree terms to the the difthe time the system takes to switch between modes of operation. 1. Introduction
1
Preprint submitted to Automatica Received April 10, 2013 07:44:46 PST
CONFIDENTIAL. Limited circulation. For review only 2
Given a signal σ( t), the real-time differentiation problem consists in obtaining an estimate of its successive derivatives σ( i) ( t), i = 1, . . . , n − 1. The only assumption made on the signal to be differentiated is that |σ(n) ( t)| ≤ L for t ≥ 0, with L a known constant. Defining x1 = σ, x2 = σ˙ , . . . , xn = σ(n−1) yields
x˙ 2 = x3 ,
···
x˙ n = −w,
x˙ˆ i x˙ˆ n
n− i n
+ xˆ i+1 ,
=
− k i d x˜ 1 c
=
− k n sign( x˜ 1 ),
i = 1, . . . , n − 1, (2)
where d xc p = | x| p sign( x). Due to the discontinuities in its right-hand side, its solutions will be understood in Filippov’s sense [14]. The gains of the HOSM differentiator { k i }ni=1 can be selected based on its recursive form, as shown in [8], providing for its finite-time exact convergence. This procedure is illustrated in Section 5. However, the HOSM differentiator does not converge uniformly with respect to the initial differentiation error. Indeed, its convergence time grows as an homogeneous norm of the initial differentiation error [15]. Therefore, the convergence time of the HOSM differentiator grows unboundedly as the initial differentiation error grows, as illustrated by Figure 1.
initial differentiation error Figure 1: Picture of the qualitative behavior of the maximal
2. Problem Statement and Main Result
x˙ 1 = x2 ,
Rn , the convergence time of the differentiator is uniformly bounded by a constant. A precise definition of this last property will be given in Definition 1. The HOSM differentiator [8] is finite-time exact. It takes the following (non-recursive) form
convergence time
ferentiator [5], and using a Lyapunov-based analysis. The contribution of this paper is the construction of an exact arbitrary order differentiator with uniform convergence with respect to the initial differentiation error. In contrast to [10], we use homogeneity properties in the differentiator design, similarly to [12]. It is shown that by simply reversing the sign of the homogeneity degree characterizing the finite-time convergence of the HOSM differentiator, uniform convergence with respect to the initial condition is obtained. In this sense, both properties are complementary. This consideration allows designing a practically uniformly convergent differentiator that can be combined with the HOSM differentiator. Unlike [10], the stability properties of the new uniform convergent part are deduced by applying a simple quadratic Lyapunov function on a linear system, and using the properties of homogeneity and continuity in the same spirit as [13]. In this sense, the newly designed nonlinear part of the differentiator is totally designed by using linear methods. The remainder of the paper is organized as follows. Section 2 introduces the problem statement and presents the main result of the paper: the Uniform HOSM differentiator. Section 3 formally introduces the concept of uniform convergence with respect to the initial condition and its characterization using homogeneity. Section 4 presents the stability analysis of the new uniform convergent part of the differentiator. Section 5 presents a simulation example and, finally, Section 6 summarizes the paper. All the proofs of the Lemmas and Theorems, except for Lemma 2, are collected in the Appendices.
(1)
and the problem of constructing an ( n − 1)-th order differentiator for σ( t) can be transformed into the construction of an observer for system (1), based on the measured output x1 = σ, despite the bounded disturbance w( t) = −σ(n) ( t). Let xˆ ( t) denote the estimate of x( t), and x˜ = xˆ − x the differentiation error. We are interested in two main properties of the differentiator: finite-time exact convergence and uniform convergence with respect to the initial differentiation error. Finite-time exactness means that the differentiator provides the exact value of the derivatives after a finite-time, despite w. Stated informally, the uniform convergence property means that, for any initial differentiation error x˜ (0) ∈
convergence time of a differentiator as a function of the norm of the initial differentiation error. The differentiator plotted in solid black line has uniform convergence with respect to the initial differentiation error. In contrast, the differentiator plotted in dashed blue is not uniform.
The focus of this paper is in the design of an “uniformly finite-time exact differentiator”, which has the two properties: finite-time exact convergence despite perturbations and uniform convergence w.r.t. the initial differentiation error. Definition 1. Assume that the disturbance w belongs to certain functional class W . A differentiator is said to be uniformly finite-time exact, if there exists time T such that for all w ∈ W , and any initial differentiation errors x˜ (0) ∈ Rn the condition xˆ ( t) = x( t), ∀ t ≥ T is satisfied. In this paper, the class W consists of all measurable functions of time bounded by the given number L in their absolute value. In what follows, and without loss of generality, the initial time instant for all differential equations is assumed to be zero.
Preprint submitted to Automatica Received April 10, 2013 07:44:46 PST
CONFIDENTIAL. Limited circulation. For review only 3 In contrast to the HOSM differentiator, the convergence time (from an arbitrary initial differentiation error) of an uniformly finite-time exact differentiator is uniformly bounded. As we have motivated in the introduction, this feature is useful when the differentiator is applied to systems with dwell-time. Our main result is: Theorem 1. The ( n − 1)-th order differentiator
x˙ˆ i
=
−κ i θ d x˜ 1 c
n− i n
− k i (1 − θ )d x˜ 1 c
n+α i n
+ xˆ i+1 ,
i = 1, . . . , n − 1; x˙ˆ n
=
3. Homogeneity and Uniform Convergence with respect to the initial condition The design of the HOSM differentiator was based on homogeneity properties [12]. We briefly recall three basic concepts: Definition 2 (see [12]). a) The family of dilations Λλ , associated with the “weight vector” ( r 1 , . . . , r n ) ∈ n R+ , is the linear map
Λλ : ( x1 , x2 , . . . , xn ) 7→ (λr 1 x1 , λr 2 x2 , . . . , λr n xn ),
(3) 1+α
−κn θ sign( x˜ 1 ) − k n (1 − θ )d x˜ 1 c
,
is uniformly finite-time exact when its parameters are selected as follows: (i) {κ i }ni=1 are selected based on the bound of the perturbation L using the formulas for the original HOSM differentiator [8]; (ii) α > 0 is chosen small enough and { k i }ni=1 are selected such that the following matrix in Hurwitz: −k1 1 0 · · · 0 −k2 0 1 · · · 0 (4) A= . ; .. .. . −k n 0 · · · 0 0 (iii) the function θ : [0, ∞) → {0, 1} is selected as ½ 0 if t ≤ T u , θ ( t) = 1 otherwise, with some arbitrarily chosen T u > 0.
P ROOF. See Appendix A. ( i −1)
Theorem 1 shows that the equalities xˆ i = σ , i= 1, . . . , n, will be stablished after a finite-time transient that is upper bounded by a number that is independent of the initial differentiation error. In what follows, we will refer to the differentiator (3) as the Uniform HOSM differentiator. The Uniform HOSM differentiator has two kinds of terms. The n− i terms d x˜ 1 c n have degrees lower than one. They are inherited from the HOSM differentiator and are responsible for the finite-time exact convergence of the n+α i differentiator. The terms d x˜ 1 c n have degrees larger than one, and are responsible for the uniform convergence with respect to the initial differentiation error. The Uniform HOSM is constructed by switching from the high to the low order terms (i.e. the HOSM differentiator) after an arbitrary positive amount of time T u . The parameter T u can be arbitrarily chosen using any method, as long as it is positive. The rest of the paper provides the ground to prove Theorem 1. Indeed, we shall show that the design of the new “uniform part” is closely related to the design of the original HOSM differentiator using homogeneity properties: it is just necessary to change the sign of its homogeneity degree.
for λ > 0, where r i is the weight (or degree) of x i and is denoted as deg x i = r i . b) The vector field f (respectively the vector-set field F ) are called homogeneous of the homogeneity degree p with dilation Λλ if
f (Λλ x) = λ p Λλ f ( x),
∀λ > 0,
(respectively F (Λλ x) = λ p Λλ F ( x)) and is denoted as deg f = p (respectively deg F = p). In particular, note that a differential equation x˙ = f ( x) with deg f = p (or inclusion x˙ ∈ F ( x) with deg F = p) is invariant with respect to the time-coordinate transformation ( t, x) 7→ (λ− p t, Λλ x). This fact suggests to define deg t = − p, and call the differential equation itself (or inclusion) homogenous with degree p. The structure of the HOSM differentiator (2) can be derived based on the homogeneity properties introduced above. Consider a differentiator with the following general form
x˙ˆ i = − f i ( x˜ 1 ) + xˆ i+1 , i = 1, . . . , n − 1;
x˙ˆ n = − f n ( x˜ 1 ),
and restrict the functions f i to be the simplest homogeneous functions f i ( x˜ 1 ) = k i d x˜ 1 cm i , k i , m i ∈ R. Set deg f = α or, equivalently, deg t = −α. Then, due to the structure of the chain of integrators, the weight of each variable x˜ i will be fixed once the weight of one variable is selected. Picking deg x˜ 1 = n yields deg x˜ i = n + α( i − 1), which means that the differentiator takes the form
x˙ˆ i ˙xˆ n
− k i d x˜ 1 c
=
n+α i n
1+α
− k n d x˜ 1 c
=
+ xˆ i+1 ,
i = 1, . . . , n − 1;
,
(5)
and the differentiation error is given by
x˙˜ 1
x˙˜ n−1 x˜˙ n
= .. . = =
n+α n
− k 1 d x˜ 1 c
+ x˜ 2 ,
(6) − k n−1 d x˜ 1 c
n+( n−1)α n
1+α
− k n d x˜ 1 c
+ x˜ n ,
+ w,
where { k i }ni=1 are the gains of the differentiator. Homogeneous Asymptotically stable Filippov differential inclusions (and hence, Filippov differential
Preprint submitted to Automatica Received April 10, 2013 07:44:46 PST
CONFIDENTIAL. Limited circulation. For review only 4 equations) with negative homogeneity degree converge in finite-time [12]. The HOSM differentiator (2) is obtained from (5) by setting deg f = α = −1. Its homogeneity degree α is negative to guarantee finite-time stability. More importantly, this choice yields a discontinuous algorithm ensuring its exactness against the bounded disturbance w, see [12]. Note that any other choice of α ∈ (−1, 0) results in a continuous differentiator that is finite-time convergent, but can not be exact against a bounded disturbance. 3.1. Uniform Convergence with respect to the Initial Condition. In analogy to [12], uniform convergence with respect to the initial condition can also be characterized in terms of homogeneity. Indeed, we shall see that simply reversing the homogeneity degree from negative to positive yields uniform convergence w.r.t. the initial condition instead of finite-time stability. In this sense, both properties are complementary. To see this, let us rewrite system (6) in the form ˙˜ = f ( x˜ ) + g( x, ˜ w), x
x˜ (0) = x˜ 0 ,
(7)
P ROOF. See Appendix B.
Lemma 1 shows that the general homogeneous differentiator (5) is practically uniformly convergent when α > 0 and the parameters { k i }ni=1 are selected to provide asymptotic stability when w ≡ 0. A method to determine the parameters { k i }ni=1 is presented in Section 4. Moreover, as shown in Theorem 1, the HOSM differentiator can be combined with the practically uniformly convergent differentiator to obtain a uniform exact finite-time convergent differentiator. For this, it is sufficient to start with the practically uniformly convergent differentiator, given by (5) with α > 0, and then switch to the HOSM differentiator (2) after an arbitrary positive amount of time. Note that using the practically uniformly convergent differentiator for any positive amount of time is enough to bring the trajectories that started at “infinity” into a compact. From that compact, the convergence time of the HOSM differentiator is uniformly bounded. In addition, the convergence of the HOSM differentiator can be further accelerated by using a time-varying gain without deteriorating its asymptotic precision [9].
n+α i
˜ w) = 0 for i = where f i ( x˜ ) = − k i d x˜ 1 c n + x˜ i+1 , g i ( x, ˜ w) = w. The 1, . . . , n − 1; and f n ( x˜ ) = − k n d x˜ 1 c1+α , g n ( x, vector field g can be regarded as a disturbance to the nominal part f . Moreover, recall that the perturbation w belongs to the class of functions uniformly bounded by a constant, that we denote as W . Definition 3 distinguishes two kinds of uniform convergence with respect to the initial condition.
Remark 1. When the signal σ( t) is measured with additive noise uniformly bounded by δ, the Uniform HOSM differentiator has the same precision as the HOSM differentiator. It means that the following accuracies i
| xˆ i − x i | = O (L n+1 δ
n− i +1 n+1
),
i = 1, . . . , n,
are obtained after finite-time [8].
Definition 3. System (7) is said to be: i) assume that g ≡ 0. The system is globally asymptotically stable, uniformly in the initial condition, if for any r > 0, there exists T > 0 such that for all x˜ 0 ∈ Rn we have k x˜ ( t)k ≤ r if t ≥ T ; ii) practically uniformly convergent with respect to the initial condition if there exists T > 0 and r > 0 such that for all w ∈ W and x˜ 0 ∈ Rn we have k x˜ ( t)k ≤ r if t ≥ T . For the case of an undisturbed system ( g ≡ 0) with continuous f , uniform convergence has been characterized based on its positive homogeneity degree [16]. Practical uniform convergence can be characterized for general disturbed systems of the form (7), by the properties that the disturbance g need to satisfy with respect to the nominal part f , as shown below. Lemma 1. System (7) is practically uniformly convergent with respect to the initial condition x˜ 0 if the following three conditions hold: (i) its origin is globally asymptotically stable when g ≡ 0; (ii) f is a continuous vector field and deg f = α > 0; (iii) w is uniformly bounded.
4. Stability Analysis of the Uniform part To guarantee the practical uniform convergence w.r.t. the initial condition of the differentiation error (6), it is necessary to appropriately select its parameters α and { k i }ni=1 . As shown in Lemma 1, it is enough to select them to guarantee that system (6) with w ≡ 0 is asymptotically stable for α > 0. Note that when α = 0, the nonlinear system (6) is reduced to a linear one, whose stability is completely determined by the stability of the A matrix of equation (4) in Theorem 1. This fact can be easily proved using a quadratic Lyapunov function. Nevertheless, in such a case, the designed differentiator does not converge uniformly since α is not positive. To analyze the stability when α > 0, the idea is to use the information of the same quadratic Lyapunov function on the nonlinear system, together with the continuity of its derivative with respect to the parameter α. This classical idea can be found in [13]. Lemma 2. If α > 0 is sufficiently small and the parameters { k i }ni=1 are selected such that the A matrix is Hurwitz, then the differentiation error (6) is asymptotically stable if w ≡ 0.
Preprint submitted to Automatica Received April 10, 2013 07:44:46 PST
CONFIDENTIAL. Limited circulation. For review only 5 P ROOF. Consider V ( x˜ ) = x˜ T P x˜ with P positive definite and solution to AT P + PA ≺ 0. Let us denote by ˜ the derivative of V ( x˜ ) along system (6) with V˙ (α, x) w ≡ 0 and the parameter α. Then
V˙ (0, x˜ ) < 0,
∀ x˜ ∈ S,
for any set S ⊆ Rn \ {0}, since it indeed is a Lyapunov function for the linear system obtained when α = 0. Moreover, note that V˙ (α, x˜ ) is continuous in both of its arguments, α and x˜ . In particular, if S is compact, V˙ (α, x˜ ) is uniformly continuous in the set {(α, x˜ ) ∈ R×Rn |α = 0, x˜ ∈ S }. This means that there exists vicinities of S and α = 0, denoted as N S and Nα=0 respectively, such that V˙ (α, x˜ ) < 0 for all (α, x˜ ) ∈ Nα=0 × N S . Now, let us choose S as an arbitrary level curve of V , i.e. S = V −1 (δ), δ > 0. Therefore, there exists α > 0 and a vicinity N S , such that the trajectories of the system cross from the outside of N S to its inside. The rest of the proof follows the same argument as in the proof of Lemma 4 shown in the Appendix. By the homogeneity of the system, the trajectories of the system also cross from the outside to the inside of any dilated versions of N S . Moreover, since Rn \ {0} can be covered by dilated version of N S , this ensures that the system is globally asymptotically stable.
starting with α = 0 and increasing its value while2 V −1 (δ) ∩ V˙α−1 (0) = ;. When α = 0, i.e. a linear system, there is no intersection for any δ > 0, since V −1 (δ) is an ellipsoid and V˙0−1 (0) is just the point x˜ = 0. Let us illustrate the procedure described above for our particular example. First, selecting the matrix P as a solution to A T P + P A = Q = − I yields 71/196 57/28 1/2 P = 57/28 9965/686 757/196 . 1/2 757/196 87/28 The surface V˙α−1 (0) for α = 0.06 is shown in Fig. 2. It consists of two parts: a central lobe and two exterior hyperboloids. As α tends to zero, the central lobe shrinks and the exterior hyperboloids tend to retreat to infinity; in the limit α = 0, it consists of only one point x˜ = 0. The ellipsoids V −1 (δ) turned out to be horizontally aligned, irrespectively of the selection of Q in the Lyapunov equation. For α = 0.06, the ellipsoid V −1 (10) does not intersect neither the interior lobes, nor the 1 exterior hyperboloids of V˙0−.06 (0), see Fig. 3. According to the discussion above, this shows that the system is stable for α = 0.06 and the selected gains { k i }3i=1 , and, therefore, the combined differentiator is uniformly exact convergent. In Figure 4, the convergence of the trajectories of the differentiator is shown.
For the case of n = 3, the conditions for stability are α > 0 small enough and k 1 > 0, k 3 > 0, k 1 k 2 > k 3 . 5. Simulation example A second order differentiator was tested using the signal σ( t) = 5 t + sin( t) + 0.01 cos(10 t). For n = 3, the Uniform HOSM differentiator takes the form 2
α
1
2α 3
x˙ˆ 1 = −κ1 θ d xˆ 1 − σc 3 − k 1 (1 − θ )d xˆ 1 − σc1+ 3 + xˆ 2 , x˙ˆ 2 = −κ2 θ d xˆ 1 − σc 3 − k 2 (1 − θ )d xˆ 1 − σc1+
+ xˆ 3 ,
1+α
x˙ˆ 3 = −κ3 θ sign( xˆ 1 − σ) − k 3 (1 − θ )d xˆ 1 − σc
,
where T u = 0.8 determines the form of the function θ ( t). The parameters {κ i }3i=1 can be selected based on the gains for the recursive HOSM differentiator as p κ1 = 2L1/3 , κ2 = 1.5 2L1/2+1/6 , κ3 = 1.1L, with L such that |σ(3) ( t)| ≤ L, see [8]. The gain L = 30 was used. The parameters { k i }3i=1 are selected as k 1 = 7, k 2 = 1/7 + 2, k 3 = 1. Then, Lemma 2 ensures the existence of a small enough parameter α > 0 that guarantees uniform convergence w.r.t. the initial differentiation error using these gains. Let us show, additionally, that it is possible to compute an explicit value for parameter α > 0. Using the proof of Lemma 2, one picks a level curve S = V −1 (δ), and then checks if V˙ (α, x˜ ) < 0 on that level curve for the given α. The largest value of α that satisfies the last condition can be computed iteratively,
Figure 2: The surface V˙α−1 (0) for α = 0.06.
Figure 3: V˙ = 0 for α = 0.06 (yellow) and the ellipsoid V = 10
(magenta). There is no intersection of V˙ = 0 with V = 10, therefore V˙ < 0 in V = 10.
To confirm the property of uniform convergence w.r.t. the initial differentiation error, several initial conditions xˆ 0 were tested and the convergence time of the 2 We use the notation V˙ −1 (0) := { x ˜ ∈ Rn |V˙ (α, x˜ ) = 0}. α
Preprint submitted to Automatica Received April 10, 2013 07:44:46 PST
CONFIDENTIAL. Limited circulation. For review only 6 in the example a simple method to compute all the required parameters of this new part of the differentiator. Future work may include using a Norm Observer to make the switching in the differentiator.
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Acknowledgements
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The authors gratefully acknowledge the financial support from projects PAPIIT 17211 and IN111012, CONACyT 56819, 132125, 51244 and CVU 229959, Fondo de Colaboración del II-FI, UNAM, IISGBAS-165-2011.
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Appendix A. Proof of Theorem 1
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When t ≤ T u , the Uniform HOSM differentiator is reduced to the general homogeneous differentiator (5) Figure 4: True derivatives: black, estimated derivatives: blue. with its parameters selected using condition (ii). ThereThe initial condition for the differentiator was chosen as upon, Lemmas 1 and 2 immediately show that it is xˆ (0) = (100, 200, 300). practically uniformly convergent. Moreover, using it a positive amount of time T u brings any trajectory of differentiator measured. The initial conditions were the differentiation error into a sufficiently large but chosen as multiples of the vector [1, 2, 3]. Figure 5 presents compact set containing the origin, see Lemma 3. The a graph between the convergence time of the Uniform size of this compact set only depends on T u . When HOSM differentiator versus the norm of the initial dift > T u , the Uniform HOSM differentiator is reduced ferentiation error. As expected, it shows the presence to the original HOSM differentiator with proper gains of an asymptote in the convergence time as the norm (i), whose convergence time starting from a compact is of the initial condition increases. uniformly bounded.
convergence time
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Appendix B. Proof of Lemma 1
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Several preliminary results are required for the proof of Lemma 1. Denote by x˜ ( t, x˜ 0 ) the solution of (7) with initial condition x˜ 0 at time t. For an undisturbed system g ≡ 0 with deg f = p, its homogeneity provides for the following scaling property (see, e.g., [12] or [17]):
12 10 8 6 asymptote linear interpolation measurements
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6. Conclusions We have presented an arbitrary-order differentiator that converges to the true derivatives of the signal after a finite-time independent of the initial differentiation error. The differentiator was constructed by combining the HOSM differentiator with a new part that is practically uniformly convergent. For this, uniform convergence w.r.t. the initial condition was characterized in terms of the homogeneity of the vector fields, and latter shown to be robust (in the sense of practical uniform convergence) to a disturbance that gets eventually dominated by the nominal part. The stability of the new uniform part of the differentiator was analyzed by using a quadratic “Lyapunov function”, together with the continuity and homogeneity of the differentiation error. Using this analysis, we illustrated
Λλ x˜ ( t, x˜ 0 ) = x˜ (λ− p t, Λλ x˜ 0 ).
(B.1)
From this, we can prove the following preliminary result about the uniform convergence without inputs. It is an extension of the proof in [16] that does not require continuity of the vector field f . Lemma 3. Consider (7) with g ≡ 0. Assume that its origin is asymptotically stable and that deg f = p > 0. Then the system is also uniformly convergent in the initial condition. In particular, this implies that for any given time T u > 0, there exists a ball B r such that x˜ ( t, x˜ 0 ) ∈ B r for all t ≥ T u and x˜ 0 ∈ Rn . P ROOF. To prove the first part of the claim, let B0 be an arbitrary compact ball of Rn . Consider the sequence of balls obtained by dilation B i = Λλ B i−1 for all i ≥ 1, with λ > 1 fixed large enough so B i+1 ⊃ B i . Due to the asymptotic stability of the system, every trajectory starting in B1 enters B0 before certain time T ; in symbols x˜ (T, B1 ) ∈ B0 . Applying dilation to both sides of this last expression and using (B.1) yields
Λλ x˜ (T, B1 ) = x˜ (λ− p T, Λλ B1 ) = x˜ (λ− p T, B2 ) ∈ Λλ B0 = B1 ,
Preprint submitted to Automatica Received April 10, 2013 07:44:46 PST
CONFIDENTIAL. Limited circulation. For review only 7 i.e., the trajectories starting in B2 enter B1 before time λ− p T . Repeating the same procedure for each i ≥ 1 and summing each time interval, obtain that for any x˜ 0 ∈ Rn the trajectory of the system enters into B0 after at most ∞ ¡ X ¢i 1 λ− p = T tr = T , 1 − λ− p i =0 where the convergence of the geometric series occurs by observing that λ− p < 1, since λ > 1 and p > 0. Since B0 was arbitrary, the first part of the claim of the Lemma is complete. For the second part of the claim consider two cases. If T u ≥ t r , then one can select B r = B0 . If not, remove a finite number N of terms (λ− p ) i until t reach − PN −p i i =0 (λ ) < T u and pick B r = B N .
This means that if k x˜ kis taken large enough, then V˙ | pert = const · V˙ |nopert . The indexes “pert” and “no pert” denote evaluating the derivative of V along the trajectories of system (7) with and without the perturbation w, respectively. Thus the perturbed system inherits the behavior of the nominal system if k x˜ k is large enough. By hypothesis (ii), p > 0 and using Lemma 3 obtain the desired claim. BIBLIOGRAPHY.
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≤
˜ w) −|〈∇V , f 〉|(Λλ x˜ ) + k∇V kk gk(Λλ x,
≤
−|〈∇V , f 〉|(Λλ x˜ ) + εk∇V kk f k(Λλ x˜ ),
≤
−|〈∇V , f 〉|(Λλ x˜ ) + ε|〈∇V , f 〉|(Λλ x˜ )
=
−(1 − ε)|〈∇V , f 〉|(Λλ x˜ ),
∀λ ≥ λ∗ .
Preprint submitted to Automatica Received April 10, 2013 07:44:46 PST
