On Observer Design for a Class of Continuous ... - Semantic Scholar
53rd IEEE Conference on Decision and Control December 15-17, 2014. Los Angeles, California, USA
On observer design for a class of continuous-time affine switched or switching systems Laura Menini, Corrado Possieri and Antonio Tornambè Abstract— The main goal of this paper is to design a state observer for a class of affine switched or switching dynamical systems, without requiring the knowledge of the switching signal. To reach such a goal some tools, taken from Algebraic Geometry, are used to express the switching signal as a function of the output and its time derivatives. Then, similar tools are used to design an observer to estimate both the switching signal and the state of the system. A physically motivated example of application is reported.
I. I NTRODUCTION A classic problem in control theory is to estimate, in real time, unmeasurable state variables of a dynamical system from the available measured outputs [1]–[5]. Here continuous-time affine switched or switching systems (for simplicity, without inputs), which change their behavior according to an external switching signal or according to their state [6]–[11], respectively, are considered. The approach, presented in this paper, to solve the mode and state estimation problem, is to define a polynomial vector field f (σ, x), which generally depends on the switching signal, σ, and on the state, x, such that, for each time interval (ti , ti+1 ), in which the value of σ is constant, the dynamics of the system are x˙ = f (σ, x). Since f (σ, x) is polynomial, it is possible to apply the techniques of Algebraic Geometry, to analyze this kind of systems. Such techniques have been used for control problems, including observability studies [12]–[17]. Two different approaches are presented to estimate the state of the system. The first one is based on finding a polynomial (or rational) formula, expressing the switching variable σ as a function of the output and of its time derivatives, up to a finite order. Hence, it is possible to design a switched observer, which changes its dynamical behavior according to an estimate of the switching signal obtained by such a formula. The second approach is based on finding a σ-independent polynomial (respectively, rational) expression of the state of the system, as a function of the output and of its time derivatives, up to a finite order. In this paper, to estimate the output time derivatives, a high-gain observer is used [18]–[20], but, in principle, the proposed techniques can be applied also in combination with other methods, capable of estimating the output derivatives. The time derivatives of the output are used to estimate the switching signal also in [21], [22].
Even if the computations seem to be complex, thanks to some specific CAS software, as Macaulay2 [23], they can be effectively carried out, in many cases of practical interest. Moreover, the computations needed for observer realization need not be executed in real time, but can be carried out in the observer design process. Hence, the observers presented in this paper, can be effectively applied in real time control implementations, because they only need a real time estimation of the output time derivatives. II. S WITCHING SIGNAL ESTIMATION Consider the affine continuous-time dynamical system = Aσ(t) x(t) + Bσ(t) , t ∈ (ti , ti+1 ),
(1a)
y(t)
= Cσ(t) x(t),
(1b)
where x = [ x1 · · · xn ]> ∈ Rn , y ∈ R, Aσ ∈ Rn×n , Bσ ∈ Rn×1 , Cσ ∈ R1×n , for σ = 1, . . . , m, σ(t) ∈ {1, . . . , m} is a switching signal [24], which can be state-dependent (i.e., the state space Rn is partitioned into a set of regions, and, in each region, the function σ(t) assumes a certain constant value) or time-dependent (i.e., σ(t) is an externally generated piecewise constant function σ : [0, ∞) → {1, . . . , m}). Let t0 be the initial time and, for i ∈ N , let ti be the i-th time instant in which the function σ(t) is discontinuous. Let (ti , ti+1 ) be any time interval such that σ(t) is constant, i.e., σ(t) = σ ¯, σ ¯ ∈ {1, . . . , m}, for all + t ∈ (ti , ti+1 ), and σ(t− ) = 6 σ(t i+1 i+1 ). In addition, assume that there exists a dwell time td > 0, such that ti+1 − ti ≥ td + and that x(t− i+1 ) = x(ti+1 ), i.e., that no jumps are imposed to the state variables. Note that the times ti and the function σ(t) are assumed to be unknown. Define the polynomial Qm functions qi (σ) ∈ R[σ], i = (σ−j)
j6=i 1, . . . , m, as qi (σ) := Qj=1, , which are such that m j=1, j6=i (i−j) 1, if σ = i, qi (σ) = , the vector 0, if σ ∈ {1, . . . , m}, σ 6= i, n field f (σ, x) ∈ R [σ, x] and the function h(σ, x) ∈ R[σ, x]:
Dipartimento di Ingegneria Civile e Ingegneria Informatica, Università di Roma Tor Vergata, Via del Politecnico 1, 00133 Roma, Italy L. Menini: [email protected] C. Possieri: [email protected] A. Tornambè: [email protected]
978-1-4673-6088-3/14/$31.00 ©2014 IEEE
x(t) ˙
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f (σ, x) h(σ, x)
= =
m X j=1 m X
(Aj x + Bj )qj (σ),
(2a)
(Cj x)qj (σ).
(2b)
j=1
System (1), for all t ∈ (ti , ti+1 ), can be rewritten as x(t) ˙
= f (σ(t), x(t)),
(3a)
y(t)
= h(σ(t), x(t)).
(3b)
The observability map of order k + 1, k ∈ Z, k ≥ 0, is de> fined as ye,k := Ok (σ, x), where ye,k = y0 · · · yk , > Ok (σ, x) := L0f h(σ, x) · · · Lkf h(σ, x) , ∂Lif h(σ,x) f (σ, x) and L0f h(σ, x) := ∂x di y(t) h(σ, x). Letting yi (t) = dti , and defining, ∀t ∈ (ti , ti+1 ), k ye,k (t) := [ y(t) dy(t) ]> , one has · · · d dty(t) k dt
where Li+1 f h(σ, x) :=
ye,k (t) = Ok (σ(t), x(t)), ∀t ∈ (ti , ti+1 ),
where yˆe,N (t) = [ yˆ0 (t) · · · yˆN (t) ]> is an estimate of ye,N (t). Thus, using the “practical” observer (5), it is possible to obtain an estimate of the output of the dynamical system (3) and of its time derivatives, in the sense that, since ti+1 − ti ≥ td by assumption, where td is the dwell time, − the error y˜e,N (t− ˆe,N (t− i+1 ) := ye,N (ti+1 )− y i+1 ) can be made arbitrarily small, by taking sufficiently small. To obtain a solution of Problem 1, a possible approach is to eliminate the x variables from the equations
where (x(t), y(t)) is any solution of (3). Problem 1. Let system (3) and an integer N ≥ 0 be given. Find, if any, a polynomial p(σ, ye,N ) ∈ R[σ, ye,N ], such that p ◦ ON (σ, x) = 0, ∀x ∈ Rn , ∀σ = 1, . . . , m. A polynomial p(σ, ye,N ), solution of Problem 1 and dependent on the unknown σ, is called a σ-dependent embedding of system (3); whereas, a polynomial p(σ, ye,N ), solution of Problem 1 and independent of σ, is called a σindependent embedding of (3). Remark 1. Let p(σ, ye,N ) be a solution of Problem 1. A > 0 0 point [σ , ye,N ] is said to be σ-regular for p(σ, ye,N ) if ∂p 6= 0. By the Implicit Function Theorem 0 0 ∂σ (σ,ye,N )=(σ ,ye,N )
[25], about any σ-regular point for p(σ, ye,N ), there exists a function φ(ye,N ) such that, locally, p(σ, ye,N ) = 0 ⇔ σ = φ(ye,N ), where φ(ye,N ) need not be polynomial nor rational. Hence, by Remark 1, if one is able to compute a σdependent embedding of (3), then it may be possible to compute the value of the switching signal σ from the output of system (3) and its time derivatives. Hence, from a practical point of view, an instrument is needed, to estimate the vector ye,N (t) from the measurement of y(t). Remark 2. Let p¯(σ, ye,N −1 ) be a polynomial or rational funcN tion, such that d dty(t) = p¯(σ(t), ye,N −1 (t)), ∀t ∈ (ti , ti+1 ), N where y(t) is the output of (3). Such a differential equation, for each t ∈ (ti , ti+1 ), can be rewritten as y˙ 0 (t)
y˙ N (t) y(t)
= y1 (t), .. .
(4a)
= p¯(σ(t), ye,N −1 (t)),
(4b)
= y0 (t).
(4c)
Let the polynomial λN +1 + κ1 λN + · · · + κN +1 be Hurwitz and let 0 < 1 be a sufficiently small parameter. Under the assumptions of Theorems 1 and 2 of [20] (essentially, boundedness of p¯(σ(t), ye,N −1 (t)), as a function of t ∈ (ti , ti+1 )), a high-gain “practical” observer for (4) is: κ1 (5a) yˆ˙ 0 (t) = yˆ1 (t) + (y0 (t) − yˆ0 (t)), .. . ˙yˆN (t) = κN +1 (y0 (t) − yˆ0 (t)), (5b) N +1
y0 − L0f h(σ, x)
= .. .
0,
(6a)
yN − LN f h(σ, x)
=
0,
(6b)
which involve the entries of the observability map of order N +1. The Elimination Theorem [26] seems to be the correct instrument to obtain such a result. Thus, let IN ⊆ R[σ, ye,N ] be the set of all polynomials which are solution of Problem 1. Define the ideal JN ⊆ R[x, σ, ye,N ] as follows: JN := hy0 − L0f h(σ, x), . . . , yN − LN f h(σ, x)i.
(7)
Next theorem is wholly similar to Theorem 3 of [27]. Theorem 1. i) IN is an ideal in R[σ, ye,N ]. ii) IN = JN ∩ R[σ, ye,N ]. According to Theorem 1, if one fixes the lex ordering, with x1 >l x2 >l · · · >l xn >l σ >l yN >l yN −1 >l · · · >l y0 and GN is a Groebner basis of JN , by the Elimination Theorem [26], a Groebner basis of IN is given by GN ∩R[σ, ye,N ]. Just for this section, consider the following assumption. Assumption 1. The lex ordering, with x1 >l x2 >l · · · >l xn >l σ >l yN >l yN −1 >l · · · >l y0 is fixed. Note that each polynomial p ∈ IN can be expressed as a linear combination, with coefficients in R[σ, ye,N ], of the polynomials in the Groebner basis of the ideal IN . Proposition 1. Let GN be a Groebner basis of JN , under Assumption 1, and let IN be the set of all the solutions of Problem 1. If GN ∩ R[σ, ye,N ] = GN ∩ R[ye,N ], then there exist no σ-regular point, for all p ∈ IN , along the trajectories of system (3). Let p be a σ-dependent solution of Problem 1 (when the conditions of Proposition 1 do not hold). By Remark 1, there exists, about any σ-regular point for p, a function φ(ye,N ), such that σ = φ(ye,N ), locally. Note that φ(ye,N ) need not be polynomial nor rational. Thus, the following restriction of Problem 1 is considered in this paper. Problem 2. Let system (3) and an integer N ≥ 0 be given. Find, if any, a polynomial p(σ, ye,N ) ∈ R[σ, ye,N ], which is a solution of Problem 1 and is such that either: (a) there exists a polynomial function φ(ye,N ) ∈ R[ye,N ], such that p(σ, ye,N ) = σ − φ(ye,N ) ; (b) there exist two polynomial functions γ(ye,N ), ρ(ye,N ) ∈ R[ye,N ], such that p(σ, ye,N ) = γ(ye,N )σ + ρ(ye,N ).
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If one is able to find a solution of Problem 2 (a) (respectively, to Problem 2 (b)), then the switching signal can be expressed as a polynomial (respectively, rational) function of the entries of ye,N . A solution of Problem 2 is given in the following two propositions. Proposition 2. Let system (3) and an integer N ≥ 0 be given. Let GN be a Groebner basis of the ideal JN under Assumption 1. There exists a solution of Problem 2 (a) if and only if there is a polynomial b(σ, ye,N ) ∈ GN ∩ R[σ, ye,N ], such that the leading term [26] of b(σ, ye,N ) is σ. Proposition 3. Let system (3) and an integer N ≥ 0 be given. Let GN be a Groebner basis of the ideal JN under Assumption 1. There exists a solution of Problem 2 (b) if and only if there is a polynomial b(σ, ye,N ) ∈ GN ∩ R[σ, ye,N ], such that the leading term [26] of b(σ, ye,N ) is η(ye,N )σ, where η(ye,N ) is a monomial in R[ye,N ]. Hence, if the conditions of Proposition 2 (respectively, of Proposition 3) hold, then one is able to compute a polynomial (respectively, rational) expression of the switching variable σ, as a function of ye,N . Remark 3. Let φ(ye,N (t)) be the polynomial, or rational, expression of the switching variable σ(t), dependent only on ye,N (t). A “practical” estimate σ ˆ (t) of the switching signal σ(t) can be computed as σ ˆ (t) = φ(ˆ ye,N (t)), where yˆe,N (t) is the estimate of the output of system (3) and of its time derivatives, obtained using the high-gain “practical” observer (5). Since the high-gain observer yields relevant peaking phenomena and the estimates of the higher order derivatives are quite noisy, one has that, typically, the values of the estimate σ ˆ (t) are non-integer. Therefore, considering that the signal σ(t) ∈ {1, . . . , m} ⊂ Z, it may be necessary to filter and quantize the estimate σ ˆ . In the examples reported in this paper a Butterworth filter [28] is used to filter the estimate σ ˆ , obtaining σ ˆf . Then, σ ˆf is quantized to the nearest admissible value of σ, which has to be in {1, . . . , m}, thus obtaining σ ˆf q . III. A STATE OBSERVER FOR SWITCHED OR SWITCHING AFFINE SYSTEMS
Let system (3) be given. Such a system is said to be (N+1)differentially observable if the observability map of order N , in which σ is considered as a constant parameter, is injective, −1 i.e., if there exists a left inverse x = ON (σ, ye,N ), such that −1 ON ◦ ON (σ, x) = x. One of the main objectives of this section is formalized in the following problem. Problem 3. Let system (3) and an integer N ≥ 0 be given. −1 (a) Find, if any, a polynomial left inverse ON (ye,N ) ∈ −1 n R [ye,N ], independent of σ, such that ON ◦ON (σ, x) = x, for σ = 1, . . . , m. −1 (b) Find, if any, a rational left inverse ON (ye,N ) ∈ −1 n R (ye,N ), independent of σ, such that ON ◦ON (σ, x) = x for σ = 1, . . . , m.
Remark 4. Suppose that the following polynomial equations ψ1,N (x1 , ye,N )
= .. .
0,
(8a)
ψn,N (xn , ye,N )
=
0,
(8b)
where ψi,N ∈ R[xi , ye,N ], for i = 1, . . . , n, hold whenever ye,N = ON (σ, x). By the Implicit Function Theorem [25], about any xi -regular point for the function ψi,N (xi , ye,N ), there exists a function νi (ye,N ), such that, locally, ψi,N = 0 ⇔ xi = νi (ye,N ). Thus, the function νi (ye,N ) is, locally, an inverse of the observability map. Hence, a possible approach to solve Problem 3 is to search for polynomials ψi,N as in (8), and use them to compute a solution of Problem 3. Let Pi,N be the set of all the polynomials ψi,N such that the i-th equation of (8) holds, whenever ye,N = ON (σ, x). Define the ideal JN as in (7). By a little modification of the proof given for Theorem 1, the following theorem can be easily proved. Theorem 2. i) Pi,N is an ideal in R[xi , ye,N ]. ii) Pi,N = JN ∩ R[xi , ye,N ]. Therefore, according to Theorem 2, if one fixes the lex ordering, with σ >l x1 >l · · · >l xi−1 >l xi+1 >l · · · >l xn >l xi >l yN >l · · · >l y0 , and Gi,N is the Groebner basis of JN , then, a Groebner basis of Pi,N is given by Gi,N ∩ R[xi , ye,N ]. In the remainder of this section the following assumption is done, which allows to compute the set of all the polynomials, which solve the i-th equation of (8). Assumption 2. Let i ∈ {1, . . . , n} be given. Fix the lex ordering, with σ >l x1 >l x2 >l · · · >l xi−1 >l xi+1 >l · · · >l xn >l xi >l yN >l · · · >l y0 . By a little modification of the proofs given for Proposition 2 and 3, the following two propositions can be proved. Proposition 4. Let system (3) and an integer N ≥ 0 be given. For each i = 1, . . . , n, let Gi,N be a Groebner basis of the ideal JN under Assumption 2. There exists a solution of Problem 3 (a) if and only if there is a polynomial b(xi , ye,N ) ∈ Gi,N ∩ R[xi , ye,N ] such that the leading term [26] of b(xi , ye,N ) is xi , for each i = 1, . . . , n. Proposition 5. Let system (3) and an integer N ≥ 0 be given. For each i = 1, . . . , n, let Gi,N be a Groebner basis of the ideal JN under Assumption 2. There exists a solution of Problem 3 (b) if and only if there is a polynomial b(xi , ye,N ) ∈ Gi,N ∩ R[xi , ye,N ] such that, the leading term [26] of b is γ(ye,N )xi , where γ(ye,N ) is a monomial in R[ye,N ], for each i = 1, . . . , n. Hence, if the conditions of Proposition 4 (respectively, of Proposition 5) hold, then one is able to find a polynomial (respectively, rational) left inverse of the observability map ON (σ, x), independent of the switching signal σ(t).
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Remark 5. Consider system (1) and let σ ˆf q (t) be the estimate of σ(t) described in Remark 3. Define N := {σ ∈ {1, . . . , m} : the pair [Aσ , Cσ ] is not observable} [29]. Since system (1) is affine, the state x(t), solution of (1a), can be estimated, for each t such that σ(t) ∈ / N , from the output y(t), with the classical Luenberger observer [30]: x ˆ˙ (t) = Aσˆf q (t) x ˆ(t) + Bσˆf q (t) +
assumed that the length at rest of both springs, l, and the initial conditions are such that no other impact occurs and the linear behavior of the springs is maintained. Hence, such a system has two possible configurations: in the first one, reported in Figure 1a, the mass M2 is outside the obstacle; whereas, in the second one, reported in Figure 1b, the mass M2 is inside the obstacle, and the additional force applied to this mass is modeled by a spring.
+ Lσˆf q (t) (y(t) − Cσˆf q (t) x ˆ(t)), (9a)
q1
ˆ1 · · · x ˆn ]> is an estimate of x(t) and where x ˆ = [ x n×1 Lσ ∈ R is such that the matrix Aσ − Lσ Cσ is Hurwitz, i.e., has all its eigenvalues with negative real part, ∀σ ∈ / N; whereas, for each t such that σ ˆf q (t) ∈ N , an estimate x ˆ(t) of the state can be obtained always with observer (9a), but with Lσ such that the observable part [29] of Aσ − Lσ Cσ is Hurwitz, or just with a copy of the system: x ˆ˙ (t) = Aσˆf q (t) x ˆ(t) + Bσˆf q (t) .
k1 M1 M2 q2
(9b)
(a) Configuration for q2 > 0.
Define as (τi,b , τf,b ) the maximum length time interval such that σ(t) ∈ N , ∀t ∈ (τi,b , τf,b ) and as (τi,g , τf,g ) the minimum length time interval such that σ(t) ∈ / N , ∀t ∈ (τi,g , τf,g ). Assume that ∃ Tb > 0 : τf,b − τi,b < Tb and that ∃ Tg > 0 : τf,g − τi,g > Tg . If these assumptions hold, observer (9) can estimate the state of system (1). As a matter of fact, since τf,b − τi,b < Tb , the estimation error, that can grow in the intervals in which the system is not detectable, is upper bounded. In the intervals in which the system is observable, such an error can be arbitrarily reduced by the choice of Lσ , which therefore depends on Tg and Tb . Remark 6. Let ν(ye,N ) = [ ν1 (ye,N ) · · · νn (ye,N ) ]> := −1 ON (ye,N ) be the left inverse, independent of the switching signal σ(t), of the observability map. An estimate x ˘(t) of the state x(t) can be computed as x ˘(t) = ν(ˆ ye,N (t)), where yˆe,N is the estimate of the output of system (3) and of its time derivatives, obtained using the high-gain “practical” observer of (5). Since the high-gain observer yields peaking phenomena, it may be needed to filter the estimate x ˘(t). In the first example reported in this paper a Butterworth filter is used to filter each entry of the vector x ˘(t), obtaining x ˘f (t). Hence, using this tool and the one presented in Remark 3, one may be able to estimate both the switching signal and the state of system (3). IV. E XAMPLE
k1 M1 M2
k2
k3
(b) Configuration for q2 ≤ 0.
Fig. 1: Mechanical switching system. A state-space model of such a system is the following one: x(t) ˙ = Aσ(t) x(t),
(10)
where x = [ x1 x2 x3 x4 ]> = > [ q1 − l q˙1 q2 q˙2 ] , q1 is the position of the mass M1 , q2 is the position of the mass M2 , 0 1 0 0 k k +k − 1M 2 0 M2 0 1, if x3 (t) > 0 σ(t) = , A1 = 0 1 0 01 1 , 2, otherwise k2 k 0 − M2 0 M2 2 0 1 0 0 A2 =
−
k1 +k2 M1
0 k2 M2
k
2 0 0 M1 0 0 1 k2 +k3 0− M 0
and assume that the only
2
available output is y = h(x) := x1 = q1 − l. Thus, defining the following parameters:
Here, a physically motivated example illustrates how Propositions 2, 3, 4, 5 and Remarks 5 and 6 can be applied. Example 1. Consider the mechanical system represented in Figure 1, where the two bodies having mass M1 and M2 move on an horizontal line. The mass M1 is connected to a fixed obstacle by a spring having stiffness k1 and with the mass M2 by a spring having stiffness k2 . The only impact that is considered is between the mass M2 and the obstacle, and is modeled as an elastic impact, so that, when the mass M2 is inside the obstacle, it receives a force proportional to the obstacle deformation, through a stiffness k3 . It is
k2
a :=
k1 k2 k2 k3 , b := , c := , d := , M1 M1 M2 M2
the vector field f and the function h in (2) can be written as follows: (−(σ − 2)A1 + (σ − 1)A2 )x
f (σ, x)
=
h(σ, x)
:= x1 .
(11a) (11b)
Let I5 be the set of all the solutions of Problem 1, with respect to system (10) and N = 5, and let t ∈ (ti , ti+1 ). Computing L0f h(σ, x), L1f h(σ, x), L2f h(σ, x), L3f h(σ, x), L4f h(σ, x), L5f h(σ, x), J5 = hy0 − L0f h, y1 − L1f h, y2 −
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L2f h, y3 −L3f h, y4 −L4f h, y5 −L5f hi and, under Assumption 1, the Groebner basis G5 of J5 , one has that, within the polynomials in G5 ∩ R[σ, ye,5 ], which, by Theorem 1, is the Groebner basis of the ideal I5 , there are the two following polynomials which respect the conditions of Proposition 3: b5,1 (σ, ye,5 ) = b6,1 (σ, ye,5 ) =
γ1 (ye,5 ) + γ2 (ye,5 )σ, γ3 (ye,5 ) + γ4 (ye,5 )σ,
where γ1 (ye,4 ) = y4 + (a + b + c − d)y2 + (ac − ad − bd)y0 , γ2 (ye,4 ) = dy2 + (ad + bd)y0 , γ3 (ye,5 ) = y5 + (a + b + c − d)y3 + (ac − ad − bd)y1 and γ4 (ye,5 ) = dy3 + (ad + bd)y1 . Thus, consider the polynomial p(σ, ye,5 ) = γ2 (ye,5 )b5,1 (σ, ye,5 ) + γ4 (ye,5 )b5,2 (σ, ye,5 ). It can be easily proved that p ∈ I5 and that it satisfies the conditions of Proposition 3. Hence, for each t such that γ22 (ye,5 (t)) + γ42 (ye,5 (t)) does not vanish, the switching signal σ can be computed as σ(t) = φ(ye,5 (t)), where γ1 (ye,5 )γ2 (ye,5 ) + γ3 (ye,5 )γ4 (ye,5 ) . φ(ye,5 ) = − γ22 (ye,5 ) + γ42 (ye,5 )
in B4,3 = G4,3 ∩R[x4 , ye,3 ], the following polynomial, which respects the conditions of Proposition 4: y3 + (a + b)y1 . b Hence, assuming that the constant b is not zero, the state x4 can be computed as a polynomial expression of the output and its time derivatives. y3 (t) + (a + b)y1 (t) x4 (t) = ν4 (ye,3 (t)) = . (14d) b Thus, using (14), one is able to compute a polynomial left inverse of the observability map of order 4, O3 (σ, x), which is independent of σ. b4,3,1 (x4 , ye,3 ) = x4 −
3
13.7
Standard computations show that it is impossible that γ2 (ye,5 (t)) = 0 and γ4 (ye,5 (t)) = 0 for all times t in any open subset of (ti , ti+1 ). System (10) is observable for σ = 1, 2. Hence, the method described in Remark 5 can be used to obtain an estimate of the state of the system, defining the following observer:
1.5
(14a)
x2 (t) = ν2 (ye,3 (t)) = y1 (t).
(14b)
5
13.8
10
13.9
14
15
t [s]
20
Fig. 2: Estimate σ ˆf q of the switching signal σ(t). Figure 2 shows the results of a simulation (where the N model parameters are assumed to be k1 = 1 m , k2 = N N 1 m , M1 = 1kg, M2 = 1kg, k3 = 100 m and x(0) = [ 2 0 4 0 ]> ) in which σ ˆf q (t) has been computed as in Remark 3 using (12), the high-gain “practical” observer (5), with κ1 = 1.31, κ2 = 0.81, κ3 = 0.31, κ4 = 0.07, κ5 = 0.01, κ6 = 0.007, = 2 · 10−4 and ye,5 (0) = > 0.01 0.01 0.01 0.01 0.01 0.01 , and a Butterworth filter of order 1, with ωn = 100 rad . s
Being G3,3 the Groebner basis of ideal J3 , computed according to lex ordering, with σ >l x1 >l x2 >l x4 >l x3 >l y3 >l y2 >l y1 >l y0 , one has, within the polynomials in B3,3 = G3,3 ∩R[x3 , ye,3 ], the following polynomial, which respects the conditions of Proposition 4: y2 + (a + b)y0 . b
4
[cm]
x1 (t) = ν1 (ye,3 (t)) = y0 (t),
1 0
x 1 (t) x ˆ 1(t) x ˘ 1,f(t)
2 0 −2
[cm/s]
(13)
ˆ1 x ˆ2 x ˆ3 x ˆ4 ]> is an estimate of the state where x ˆ=[ x x of system (10), f is given in (11a), σ ˆf q is an estimate of the switching variable, obtained as in Remark 3, L1 ∈ R4 is such that A1 − L1 [ 1 0 0 0 ] is Hurwitz and L2 ∈ R4 is such that A2 − L2 [ 1 0 0 0 ] is Hurwitz. The state of system (10) can be estimated also without computing σ(t). The state x1 and its derivative x2 = x˙ 1 , can be trivially computed as:
2
x 2 (t) x ˆ 2(t) x ˘ 2,f(t)
0 −2
Being G4,3 the Groebner basis of ideal J3 , computed according to lex ordering, with σ >l x1 >l x2 >l x3 >l x4 >l y3 >l y2 >l y1 >l y0 , one has, within the polynomials 6238
x 3 (t) x ˆ 3(t) x ˘ 3,f(t)
2 0
[cm/s]
(14c)
[cm]
4
Hence, assuming that the constant b is not zero, the state x3 can be computed as a polynomial expression of the output and its time derivatives. y2 (t) + (a + b)y0 (t) . x3 (t) = ν3 (ye,3 (t)) = b
1.5 1
(12)
b3,3,1 (x3 , ye,3 ) = x3 −
2
2.5 2
x ˆ˙ (t) = f (ˆ σf q (t), x(t)) + Lσˆf q (y(t) − x ˆ1 (t)),
σ ˆ f q(t) σ(t)
2
x 4 (t) x ˆ 4(t) x ˘ 4,f(t)
0 −2 0
5
10
t
15
20
[s]
Fig. 3: Estimates x ˆ(t) and x ˘f (t) of the state x(t).
Figure 3 shows, from the same simulation, the estimate x ˆ(t) of the state x(t), obtained using (13), with L1 and L2 such that the eigenvalues of the matrix Aσ − Lσ Cσ are {−4, −5, −6, −7}, for σ = {1, 2}. and the estimate x ˘(t), obtained using (14) as in Remark 6, with the same high-gain observer as above and with a Butterworth filter of order 1, ˘f (t) practically with ωn = 100 rad s . Note that the entries of x coincide with the true state variables.
[cm]
0.5
eˆ1(t) e˘1(t)
0
[cm/s]
−0.5 2
eˆ2(t) e˘2(t)
0 −2
[cm]
10
eˆ3(t) e˘3(t)
5
[cm/s]
0 10
eˆ4(t) e˘4(t)
5 0 −5 0
5
t
10
[s]
15
20
Fig. 4: Estimation errors of the state x. Figure 4 shows, from the same simulation, the estimation errors eˆ = [ eˆ1 eˆ2 eˆ3 eˆ4 ]> = x − x ˆ and e˘ = [ e˘1 e˘2 e˘3 e˘4 ]> = x − x ˘f . As it can be seen in such a figure, these estimation errors go to zero, after an initial transient behavior. V. C ONCLUSIONS In this paper, two methods to design state observers for a class of affine switched or switching dynamical systems, without requiring the knowledge of the switching signal, are given. The estimation goals are reached using some tools taken from Algebraic Geometry to express the switching signal as a function of the output and its time derivatives. Then, similar tools are used to design an observer to estimate both the switching signal and the state of the system. R EFERENCES [1] D. Aeyels, “Generic observability of differentiable systems,” SIAM Journal on Control and Optimization, vol. 19, no. 5, pp. 595–603, 1981. [2] J.-P. Gauthier and I. A. Kupka, “Observability and observers for nonlinear systems,” SIAM Journal on Control and Optimization, vol. 32, no. 4, pp. 975–994, 1994. [3] E. G. Gilbert, “Controllability and observability in multivariable control systems,” Journal of the Society for Industrial & Applied Mathematics, Series A: Control, vol. 1, no. 2, pp. 128–151, 1963. [4] A. Isidori, Nonlinear control systems, vol. 1. Springer, 1995. [5] A. Tornambe, “Use of asymptotic observers having-high-gains in the state and parameter estimation,” in Decision and Control, 1989., Proceedings of the 28th IEEE Conference on, pp. 1791–1794, IEEE, 1989.
[6] J. Barbot, H. Saadaoui, M. Djemai, and N. Manamanni, “Nonlinear observer for autonomous switching systems with jumps,” Nonlinear Analysis: Hybrid Systems, vol. 1, no. 4, pp. 537–547, 2007. [7] F. Bejarano, A. Pisano, and E. Usai, “Finite-time converging jump observer for switched linear systems with unknown inputs,” Nonlinear Analysis: Hybrid Systems, vol. 5, no. 2, pp. 174–188, 2011. [8] W. Kang, J.-P. Barbot, and L. Xu, “On the observability of nonlinear and switched systems,” in Emergent Problems in Nonlinear Systems and Control, pp. 199–216, Springer, 2009. [9] S. Pettersson, “Observer design for switched systems using multiple quadratic lyapunov functions,” in Intelligent Control, 2005. Proceedings of the 2005 IEEE International Symposium on, Mediterrean Conference on Control and Automation, pp. 262–267, IEEE, 2005. [10] A. Tanwani, H. Shim, and D. Liberzon, “Observability for switched linear systems: characterization and observer design,” Automatic Control, IEEE Transactions on, vol. 58, no. 4, pp. 891–904, 2013. [11] W. Xiang, M. Che, C. Xiao, and Z. Xiang, “Observer design and analysis for switched systems with mismatching switching signal,” in Intelligent Computation Technology and Automation (ICICTA), 2008 International Conference on, vol. 1, pp. 650–654, IEEE, 2008. [12] S. Diop, “Elimination in control theory,” Mathematics of control, signals and systems, vol. 4, no. 1, pp. 17–32, 1991. [13] D. Neši´c and I. M. Y. Mareels, “Dead beat controllability of polynomial systems: symbolic computation approaches,” IEEE Transactions on Automatic Control, vol. 43, no. 2, pp. 162–175, 1998. [14] B. Tibken, “Observability of nonlinear systems-an algebraic approach,” in Decision and Control, 2004. CDC. 43rd IEEE Conference on, vol. 5, pp. 4824–4825, IEEE, 2004. [15] L. Menini and A. Tornambe, “Immersion and Darboux polynomials of Boolean networks with application to the Pseudomonas syringae hrp regulon,” in 52th IEEE Conference on Decision and Control, Firenze, Italy, December 2013. [16] L. Menini and A. Tornambe, “Observability and dead-beat observers for Boolean networks modeled as polynomial discrete-time systems,” in 52th IEEE Conference on Decision and Control, Firenze, Italy, December 2013. [17] L. Menini and A. Tornambe, “On a Lyapunov equation for polynomial continuous-time systems,” International Journal of Control, vol. 87, no. 2, pp. 393–403, 2014. [18] G. Bornard and H. Hammouri, “A high gain observer for a class of uniformly observable systems,” in Decision and Control, 1991., Proceedings of the 30th IEEE Conference on, pp. 1494–1496, IEEE, 1991. [19] F. Esfandiari and H. K. Khalil, “Output feedback stabilization of fully linearizable systems,” International Journal of Control, vol. 56, no. 5, pp. 1007–1037, 1992. [20] A. Tornambe, “High-gain observers for non-linear systems,” International Journal of Systems Science, vol. 23, no. 9, pp. 1475–1489, 1992. [21] R. Vidal, A. Chiuso, S. Soatto, and S. Sastry, “Observability of linear hybrid systems,” in Hybrid systems: Computation and control, pp. 526–539, Springer, 2003. [22] M. Babaali and G. J. Pappas, “Observability of switched linear systems in continuous time,” in Hybrid systems: Computation and control, pp. 103–117, Springer, 2005. [23] D. R. Grayson and M. E. Stillman, “Macaulay2, a software system for research in algebraic geometry.” Available at http://www.math.uiuc.edu/Macaulay2/. [24] D. Liberzon, Switching in systems and control. Springer, 2003. [25] W. Fleming, Functions of several variables. Springer, 1977. [26] D. Cox, J. Little, and D. O’Shea, “Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra. undergraduate texts in mathematics,” Springer Verlag, vol. 1, no. 5, pp. 7–3, 1992. [27] L. Menini and A. Tornambe, “On the use of algebraic geometry for the design of high-gain observers for continuous-time polynomial systems.” To appear in IFAC World Congress 2014. [28] L. Zhongshen, “The design of butterworth lowpass filter based on MATLAB [J],” Heilongjiang Electronic Technology, vol. 3, p. 017, 2003. [29] T. Kailath, Linear systems, vol. 1. Prentice-Hall Englewood Cliffs, NJ, 1980. [30] J. O’Reilly, Observers for linear systems, vol. 170. Academic Press, 1983.
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On observer design for a class of continuous-time affine switched or switching systems Laura Menini, Corrado Possieri and Antonio Tornambè Abstract— The main goal of this paper is to design a state observer for a class of affine switched or switching dynamical systems, without requiring the knowledge of the switching signal. To reach such a goal some tools, taken from Algebraic Geometry, are used to express the switching signal as a function of the output and its time derivatives. Then, similar tools are used to design an observer to estimate both the switching signal and the state of the system. A physically motivated example of application is reported.
I. I NTRODUCTION A classic problem in control theory is to estimate, in real time, unmeasurable state variables of a dynamical system from the available measured outputs [1]–[5]. Here continuous-time affine switched or switching systems (for simplicity, without inputs), which change their behavior according to an external switching signal or according to their state [6]–[11], respectively, are considered. The approach, presented in this paper, to solve the mode and state estimation problem, is to define a polynomial vector field f (σ, x), which generally depends on the switching signal, σ, and on the state, x, such that, for each time interval (ti , ti+1 ), in which the value of σ is constant, the dynamics of the system are x˙ = f (σ, x). Since f (σ, x) is polynomial, it is possible to apply the techniques of Algebraic Geometry, to analyze this kind of systems. Such techniques have been used for control problems, including observability studies [12]–[17]. Two different approaches are presented to estimate the state of the system. The first one is based on finding a polynomial (or rational) formula, expressing the switching variable σ as a function of the output and of its time derivatives, up to a finite order. Hence, it is possible to design a switched observer, which changes its dynamical behavior according to an estimate of the switching signal obtained by such a formula. The second approach is based on finding a σ-independent polynomial (respectively, rational) expression of the state of the system, as a function of the output and of its time derivatives, up to a finite order. In this paper, to estimate the output time derivatives, a high-gain observer is used [18]–[20], but, in principle, the proposed techniques can be applied also in combination with other methods, capable of estimating the output derivatives. The time derivatives of the output are used to estimate the switching signal also in [21], [22].
Even if the computations seem to be complex, thanks to some specific CAS software, as Macaulay2 [23], they can be effectively carried out, in many cases of practical interest. Moreover, the computations needed for observer realization need not be executed in real time, but can be carried out in the observer design process. Hence, the observers presented in this paper, can be effectively applied in real time control implementations, because they only need a real time estimation of the output time derivatives. II. S WITCHING SIGNAL ESTIMATION Consider the affine continuous-time dynamical system = Aσ(t) x(t) + Bσ(t) , t ∈ (ti , ti+1 ),
(1a)
y(t)
= Cσ(t) x(t),
(1b)
where x = [ x1 · · · xn ]> ∈ Rn , y ∈ R, Aσ ∈ Rn×n , Bσ ∈ Rn×1 , Cσ ∈ R1×n , for σ = 1, . . . , m, σ(t) ∈ {1, . . . , m} is a switching signal [24], which can be state-dependent (i.e., the state space Rn is partitioned into a set of regions, and, in each region, the function σ(t) assumes a certain constant value) or time-dependent (i.e., σ(t) is an externally generated piecewise constant function σ : [0, ∞) → {1, . . . , m}). Let t0 be the initial time and, for i ∈ N , let ti be the i-th time instant in which the function σ(t) is discontinuous. Let (ti , ti+1 ) be any time interval such that σ(t) is constant, i.e., σ(t) = σ ¯, σ ¯ ∈ {1, . . . , m}, for all + t ∈ (ti , ti+1 ), and σ(t− ) = 6 σ(t i+1 i+1 ). In addition, assume that there exists a dwell time td > 0, such that ti+1 − ti ≥ td + and that x(t− i+1 ) = x(ti+1 ), i.e., that no jumps are imposed to the state variables. Note that the times ti and the function σ(t) are assumed to be unknown. Define the polynomial Qm functions qi (σ) ∈ R[σ], i = (σ−j)
j6=i 1, . . . , m, as qi (σ) := Qj=1, , which are such that m j=1, j6=i (i−j) 1, if σ = i, qi (σ) = , the vector 0, if σ ∈ {1, . . . , m}, σ 6= i, n field f (σ, x) ∈ R [σ, x] and the function h(σ, x) ∈ R[σ, x]:
Dipartimento di Ingegneria Civile e Ingegneria Informatica, Università di Roma Tor Vergata, Via del Politecnico 1, 00133 Roma, Italy L. Menini: [email protected] C. Possieri: [email protected] A. Tornambè: [email protected]
978-1-4673-6088-3/14/$31.00 ©2014 IEEE
x(t) ˙
6234
f (σ, x) h(σ, x)
= =
m X j=1 m X
(Aj x + Bj )qj (σ),
(2a)
(Cj x)qj (σ).
(2b)
j=1
System (1), for all t ∈ (ti , ti+1 ), can be rewritten as x(t) ˙
= f (σ(t), x(t)),
(3a)
y(t)
= h(σ(t), x(t)).
(3b)
The observability map of order k + 1, k ∈ Z, k ≥ 0, is de> fined as ye,k := Ok (σ, x), where ye,k = y0 · · · yk , > Ok (σ, x) := L0f h(σ, x) · · · Lkf h(σ, x) , ∂Lif h(σ,x) f (σ, x) and L0f h(σ, x) := ∂x di y(t) h(σ, x). Letting yi (t) = dti , and defining, ∀t ∈ (ti , ti+1 ), k ye,k (t) := [ y(t) dy(t) ]> , one has · · · d dty(t) k dt
where Li+1 f h(σ, x) :=
ye,k (t) = Ok (σ(t), x(t)), ∀t ∈ (ti , ti+1 ),
where yˆe,N (t) = [ yˆ0 (t) · · · yˆN (t) ]> is an estimate of ye,N (t). Thus, using the “practical” observer (5), it is possible to obtain an estimate of the output of the dynamical system (3) and of its time derivatives, in the sense that, since ti+1 − ti ≥ td by assumption, where td is the dwell time, − the error y˜e,N (t− ˆe,N (t− i+1 ) := ye,N (ti+1 )− y i+1 ) can be made arbitrarily small, by taking sufficiently small. To obtain a solution of Problem 1, a possible approach is to eliminate the x variables from the equations
where (x(t), y(t)) is any solution of (3). Problem 1. Let system (3) and an integer N ≥ 0 be given. Find, if any, a polynomial p(σ, ye,N ) ∈ R[σ, ye,N ], such that p ◦ ON (σ, x) = 0, ∀x ∈ Rn , ∀σ = 1, . . . , m. A polynomial p(σ, ye,N ), solution of Problem 1 and dependent on the unknown σ, is called a σ-dependent embedding of system (3); whereas, a polynomial p(σ, ye,N ), solution of Problem 1 and independent of σ, is called a σindependent embedding of (3). Remark 1. Let p(σ, ye,N ) be a solution of Problem 1. A > 0 0 point [σ , ye,N ] is said to be σ-regular for p(σ, ye,N ) if ∂p 6= 0. By the Implicit Function Theorem 0 0 ∂σ (σ,ye,N )=(σ ,ye,N )
[25], about any σ-regular point for p(σ, ye,N ), there exists a function φ(ye,N ) such that, locally, p(σ, ye,N ) = 0 ⇔ σ = φ(ye,N ), where φ(ye,N ) need not be polynomial nor rational. Hence, by Remark 1, if one is able to compute a σdependent embedding of (3), then it may be possible to compute the value of the switching signal σ from the output of system (3) and its time derivatives. Hence, from a practical point of view, an instrument is needed, to estimate the vector ye,N (t) from the measurement of y(t). Remark 2. Let p¯(σ, ye,N −1 ) be a polynomial or rational funcN tion, such that d dty(t) = p¯(σ(t), ye,N −1 (t)), ∀t ∈ (ti , ti+1 ), N where y(t) is the output of (3). Such a differential equation, for each t ∈ (ti , ti+1 ), can be rewritten as y˙ 0 (t)
y˙ N (t) y(t)
= y1 (t), .. .
(4a)
= p¯(σ(t), ye,N −1 (t)),
(4b)
= y0 (t).
(4c)
Let the polynomial λN +1 + κ1 λN + · · · + κN +1 be Hurwitz and let 0 < 1 be a sufficiently small parameter. Under the assumptions of Theorems 1 and 2 of [20] (essentially, boundedness of p¯(σ(t), ye,N −1 (t)), as a function of t ∈ (ti , ti+1 )), a high-gain “practical” observer for (4) is: κ1 (5a) yˆ˙ 0 (t) = yˆ1 (t) + (y0 (t) − yˆ0 (t)), .. . ˙yˆN (t) = κN +1 (y0 (t) − yˆ0 (t)), (5b) N +1
y0 − L0f h(σ, x)
= .. .
0,
(6a)
yN − LN f h(σ, x)
=
0,
(6b)
which involve the entries of the observability map of order N +1. The Elimination Theorem [26] seems to be the correct instrument to obtain such a result. Thus, let IN ⊆ R[σ, ye,N ] be the set of all polynomials which are solution of Problem 1. Define the ideal JN ⊆ R[x, σ, ye,N ] as follows: JN := hy0 − L0f h(σ, x), . . . , yN − LN f h(σ, x)i.
(7)
Next theorem is wholly similar to Theorem 3 of [27]. Theorem 1. i) IN is an ideal in R[σ, ye,N ]. ii) IN = JN ∩ R[σ, ye,N ]. According to Theorem 1, if one fixes the lex ordering, with x1 >l x2 >l · · · >l xn >l σ >l yN >l yN −1 >l · · · >l y0 and GN is a Groebner basis of JN , by the Elimination Theorem [26], a Groebner basis of IN is given by GN ∩R[σ, ye,N ]. Just for this section, consider the following assumption. Assumption 1. The lex ordering, with x1 >l x2 >l · · · >l xn >l σ >l yN >l yN −1 >l · · · >l y0 is fixed. Note that each polynomial p ∈ IN can be expressed as a linear combination, with coefficients in R[σ, ye,N ], of the polynomials in the Groebner basis of the ideal IN . Proposition 1. Let GN be a Groebner basis of JN , under Assumption 1, and let IN be the set of all the solutions of Problem 1. If GN ∩ R[σ, ye,N ] = GN ∩ R[ye,N ], then there exist no σ-regular point, for all p ∈ IN , along the trajectories of system (3). Let p be a σ-dependent solution of Problem 1 (when the conditions of Proposition 1 do not hold). By Remark 1, there exists, about any σ-regular point for p, a function φ(ye,N ), such that σ = φ(ye,N ), locally. Note that φ(ye,N ) need not be polynomial nor rational. Thus, the following restriction of Problem 1 is considered in this paper. Problem 2. Let system (3) and an integer N ≥ 0 be given. Find, if any, a polynomial p(σ, ye,N ) ∈ R[σ, ye,N ], which is a solution of Problem 1 and is such that either: (a) there exists a polynomial function φ(ye,N ) ∈ R[ye,N ], such that p(σ, ye,N ) = σ − φ(ye,N ) ; (b) there exist two polynomial functions γ(ye,N ), ρ(ye,N ) ∈ R[ye,N ], such that p(σ, ye,N ) = γ(ye,N )σ + ρ(ye,N ).
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If one is able to find a solution of Problem 2 (a) (respectively, to Problem 2 (b)), then the switching signal can be expressed as a polynomial (respectively, rational) function of the entries of ye,N . A solution of Problem 2 is given in the following two propositions. Proposition 2. Let system (3) and an integer N ≥ 0 be given. Let GN be a Groebner basis of the ideal JN under Assumption 1. There exists a solution of Problem 2 (a) if and only if there is a polynomial b(σ, ye,N ) ∈ GN ∩ R[σ, ye,N ], such that the leading term [26] of b(σ, ye,N ) is σ. Proposition 3. Let system (3) and an integer N ≥ 0 be given. Let GN be a Groebner basis of the ideal JN under Assumption 1. There exists a solution of Problem 2 (b) if and only if there is a polynomial b(σ, ye,N ) ∈ GN ∩ R[σ, ye,N ], such that the leading term [26] of b(σ, ye,N ) is η(ye,N )σ, where η(ye,N ) is a monomial in R[ye,N ]. Hence, if the conditions of Proposition 2 (respectively, of Proposition 3) hold, then one is able to compute a polynomial (respectively, rational) expression of the switching variable σ, as a function of ye,N . Remark 3. Let φ(ye,N (t)) be the polynomial, or rational, expression of the switching variable σ(t), dependent only on ye,N (t). A “practical” estimate σ ˆ (t) of the switching signal σ(t) can be computed as σ ˆ (t) = φ(ˆ ye,N (t)), where yˆe,N (t) is the estimate of the output of system (3) and of its time derivatives, obtained using the high-gain “practical” observer (5). Since the high-gain observer yields relevant peaking phenomena and the estimates of the higher order derivatives are quite noisy, one has that, typically, the values of the estimate σ ˆ (t) are non-integer. Therefore, considering that the signal σ(t) ∈ {1, . . . , m} ⊂ Z, it may be necessary to filter and quantize the estimate σ ˆ . In the examples reported in this paper a Butterworth filter [28] is used to filter the estimate σ ˆ , obtaining σ ˆf . Then, σ ˆf is quantized to the nearest admissible value of σ, which has to be in {1, . . . , m}, thus obtaining σ ˆf q . III. A STATE OBSERVER FOR SWITCHED OR SWITCHING AFFINE SYSTEMS
Let system (3) be given. Such a system is said to be (N+1)differentially observable if the observability map of order N , in which σ is considered as a constant parameter, is injective, −1 i.e., if there exists a left inverse x = ON (σ, ye,N ), such that −1 ON ◦ ON (σ, x) = x. One of the main objectives of this section is formalized in the following problem. Problem 3. Let system (3) and an integer N ≥ 0 be given. −1 (a) Find, if any, a polynomial left inverse ON (ye,N ) ∈ −1 n R [ye,N ], independent of σ, such that ON ◦ON (σ, x) = x, for σ = 1, . . . , m. −1 (b) Find, if any, a rational left inverse ON (ye,N ) ∈ −1 n R (ye,N ), independent of σ, such that ON ◦ON (σ, x) = x for σ = 1, . . . , m.
Remark 4. Suppose that the following polynomial equations ψ1,N (x1 , ye,N )
= .. .
0,
(8a)
ψn,N (xn , ye,N )
=
0,
(8b)
where ψi,N ∈ R[xi , ye,N ], for i = 1, . . . , n, hold whenever ye,N = ON (σ, x). By the Implicit Function Theorem [25], about any xi -regular point for the function ψi,N (xi , ye,N ), there exists a function νi (ye,N ), such that, locally, ψi,N = 0 ⇔ xi = νi (ye,N ). Thus, the function νi (ye,N ) is, locally, an inverse of the observability map. Hence, a possible approach to solve Problem 3 is to search for polynomials ψi,N as in (8), and use them to compute a solution of Problem 3. Let Pi,N be the set of all the polynomials ψi,N such that the i-th equation of (8) holds, whenever ye,N = ON (σ, x). Define the ideal JN as in (7). By a little modification of the proof given for Theorem 1, the following theorem can be easily proved. Theorem 2. i) Pi,N is an ideal in R[xi , ye,N ]. ii) Pi,N = JN ∩ R[xi , ye,N ]. Therefore, according to Theorem 2, if one fixes the lex ordering, with σ >l x1 >l · · · >l xi−1 >l xi+1 >l · · · >l xn >l xi >l yN >l · · · >l y0 , and Gi,N is the Groebner basis of JN , then, a Groebner basis of Pi,N is given by Gi,N ∩ R[xi , ye,N ]. In the remainder of this section the following assumption is done, which allows to compute the set of all the polynomials, which solve the i-th equation of (8). Assumption 2. Let i ∈ {1, . . . , n} be given. Fix the lex ordering, with σ >l x1 >l x2 >l · · · >l xi−1 >l xi+1 >l · · · >l xn >l xi >l yN >l · · · >l y0 . By a little modification of the proofs given for Proposition 2 and 3, the following two propositions can be proved. Proposition 4. Let system (3) and an integer N ≥ 0 be given. For each i = 1, . . . , n, let Gi,N be a Groebner basis of the ideal JN under Assumption 2. There exists a solution of Problem 3 (a) if and only if there is a polynomial b(xi , ye,N ) ∈ Gi,N ∩ R[xi , ye,N ] such that the leading term [26] of b(xi , ye,N ) is xi , for each i = 1, . . . , n. Proposition 5. Let system (3) and an integer N ≥ 0 be given. For each i = 1, . . . , n, let Gi,N be a Groebner basis of the ideal JN under Assumption 2. There exists a solution of Problem 3 (b) if and only if there is a polynomial b(xi , ye,N ) ∈ Gi,N ∩ R[xi , ye,N ] such that, the leading term [26] of b is γ(ye,N )xi , where γ(ye,N ) is a monomial in R[ye,N ], for each i = 1, . . . , n. Hence, if the conditions of Proposition 4 (respectively, of Proposition 5) hold, then one is able to find a polynomial (respectively, rational) left inverse of the observability map ON (σ, x), independent of the switching signal σ(t).
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Remark 5. Consider system (1) and let σ ˆf q (t) be the estimate of σ(t) described in Remark 3. Define N := {σ ∈ {1, . . . , m} : the pair [Aσ , Cσ ] is not observable} [29]. Since system (1) is affine, the state x(t), solution of (1a), can be estimated, for each t such that σ(t) ∈ / N , from the output y(t), with the classical Luenberger observer [30]: x ˆ˙ (t) = Aσˆf q (t) x ˆ(t) + Bσˆf q (t) +
assumed that the length at rest of both springs, l, and the initial conditions are such that no other impact occurs and the linear behavior of the springs is maintained. Hence, such a system has two possible configurations: in the first one, reported in Figure 1a, the mass M2 is outside the obstacle; whereas, in the second one, reported in Figure 1b, the mass M2 is inside the obstacle, and the additional force applied to this mass is modeled by a spring.
+ Lσˆf q (t) (y(t) − Cσˆf q (t) x ˆ(t)), (9a)
q1
ˆ1 · · · x ˆn ]> is an estimate of x(t) and where x ˆ = [ x n×1 Lσ ∈ R is such that the matrix Aσ − Lσ Cσ is Hurwitz, i.e., has all its eigenvalues with negative real part, ∀σ ∈ / N; whereas, for each t such that σ ˆf q (t) ∈ N , an estimate x ˆ(t) of the state can be obtained always with observer (9a), but with Lσ such that the observable part [29] of Aσ − Lσ Cσ is Hurwitz, or just with a copy of the system: x ˆ˙ (t) = Aσˆf q (t) x ˆ(t) + Bσˆf q (t) .
k1 M1 M2 q2
(9b)
(a) Configuration for q2 > 0.
Define as (τi,b , τf,b ) the maximum length time interval such that σ(t) ∈ N , ∀t ∈ (τi,b , τf,b ) and as (τi,g , τf,g ) the minimum length time interval such that σ(t) ∈ / N , ∀t ∈ (τi,g , τf,g ). Assume that ∃ Tb > 0 : τf,b − τi,b < Tb and that ∃ Tg > 0 : τf,g − τi,g > Tg . If these assumptions hold, observer (9) can estimate the state of system (1). As a matter of fact, since τf,b − τi,b < Tb , the estimation error, that can grow in the intervals in which the system is not detectable, is upper bounded. In the intervals in which the system is observable, such an error can be arbitrarily reduced by the choice of Lσ , which therefore depends on Tg and Tb . Remark 6. Let ν(ye,N ) = [ ν1 (ye,N ) · · · νn (ye,N ) ]> := −1 ON (ye,N ) be the left inverse, independent of the switching signal σ(t), of the observability map. An estimate x ˘(t) of the state x(t) can be computed as x ˘(t) = ν(ˆ ye,N (t)), where yˆe,N is the estimate of the output of system (3) and of its time derivatives, obtained using the high-gain “practical” observer of (5). Since the high-gain observer yields peaking phenomena, it may be needed to filter the estimate x ˘(t). In the first example reported in this paper a Butterworth filter is used to filter each entry of the vector x ˘(t), obtaining x ˘f (t). Hence, using this tool and the one presented in Remark 3, one may be able to estimate both the switching signal and the state of system (3). IV. E XAMPLE
k1 M1 M2
k2
k3
(b) Configuration for q2 ≤ 0.
Fig. 1: Mechanical switching system. A state-space model of such a system is the following one: x(t) ˙ = Aσ(t) x(t),
(10)
where x = [ x1 x2 x3 x4 ]> = > [ q1 − l q˙1 q2 q˙2 ] , q1 is the position of the mass M1 , q2 is the position of the mass M2 , 0 1 0 0 k k +k − 1M 2 0 M2 0 1, if x3 (t) > 0 σ(t) = , A1 = 0 1 0 01 1 , 2, otherwise k2 k 0 − M2 0 M2 2 0 1 0 0 A2 =
−
k1 +k2 M1
0 k2 M2
k
2 0 0 M1 0 0 1 k2 +k3 0− M 0
and assume that the only
2
available output is y = h(x) := x1 = q1 − l. Thus, defining the following parameters:
Here, a physically motivated example illustrates how Propositions 2, 3, 4, 5 and Remarks 5 and 6 can be applied. Example 1. Consider the mechanical system represented in Figure 1, where the two bodies having mass M1 and M2 move on an horizontal line. The mass M1 is connected to a fixed obstacle by a spring having stiffness k1 and with the mass M2 by a spring having stiffness k2 . The only impact that is considered is between the mass M2 and the obstacle, and is modeled as an elastic impact, so that, when the mass M2 is inside the obstacle, it receives a force proportional to the obstacle deformation, through a stiffness k3 . It is
k2
a :=
k1 k2 k2 k3 , b := , c := , d := , M1 M1 M2 M2
the vector field f and the function h in (2) can be written as follows: (−(σ − 2)A1 + (σ − 1)A2 )x
f (σ, x)
=
h(σ, x)
:= x1 .
(11a) (11b)
Let I5 be the set of all the solutions of Problem 1, with respect to system (10) and N = 5, and let t ∈ (ti , ti+1 ). Computing L0f h(σ, x), L1f h(σ, x), L2f h(σ, x), L3f h(σ, x), L4f h(σ, x), L5f h(σ, x), J5 = hy0 − L0f h, y1 − L1f h, y2 −
6237
L2f h, y3 −L3f h, y4 −L4f h, y5 −L5f hi and, under Assumption 1, the Groebner basis G5 of J5 , one has that, within the polynomials in G5 ∩ R[σ, ye,5 ], which, by Theorem 1, is the Groebner basis of the ideal I5 , there are the two following polynomials which respect the conditions of Proposition 3: b5,1 (σ, ye,5 ) = b6,1 (σ, ye,5 ) =
γ1 (ye,5 ) + γ2 (ye,5 )σ, γ3 (ye,5 ) + γ4 (ye,5 )σ,
where γ1 (ye,4 ) = y4 + (a + b + c − d)y2 + (ac − ad − bd)y0 , γ2 (ye,4 ) = dy2 + (ad + bd)y0 , γ3 (ye,5 ) = y5 + (a + b + c − d)y3 + (ac − ad − bd)y1 and γ4 (ye,5 ) = dy3 + (ad + bd)y1 . Thus, consider the polynomial p(σ, ye,5 ) = γ2 (ye,5 )b5,1 (σ, ye,5 ) + γ4 (ye,5 )b5,2 (σ, ye,5 ). It can be easily proved that p ∈ I5 and that it satisfies the conditions of Proposition 3. Hence, for each t such that γ22 (ye,5 (t)) + γ42 (ye,5 (t)) does not vanish, the switching signal σ can be computed as σ(t) = φ(ye,5 (t)), where γ1 (ye,5 )γ2 (ye,5 ) + γ3 (ye,5 )γ4 (ye,5 ) . φ(ye,5 ) = − γ22 (ye,5 ) + γ42 (ye,5 )
in B4,3 = G4,3 ∩R[x4 , ye,3 ], the following polynomial, which respects the conditions of Proposition 4: y3 + (a + b)y1 . b Hence, assuming that the constant b is not zero, the state x4 can be computed as a polynomial expression of the output and its time derivatives. y3 (t) + (a + b)y1 (t) x4 (t) = ν4 (ye,3 (t)) = . (14d) b Thus, using (14), one is able to compute a polynomial left inverse of the observability map of order 4, O3 (σ, x), which is independent of σ. b4,3,1 (x4 , ye,3 ) = x4 −
3
13.7
Standard computations show that it is impossible that γ2 (ye,5 (t)) = 0 and γ4 (ye,5 (t)) = 0 for all times t in any open subset of (ti , ti+1 ). System (10) is observable for σ = 1, 2. Hence, the method described in Remark 5 can be used to obtain an estimate of the state of the system, defining the following observer:
1.5
(14a)
x2 (t) = ν2 (ye,3 (t)) = y1 (t).
(14b)
5
13.8
10
13.9
14
15
t [s]
20
Fig. 2: Estimate σ ˆf q of the switching signal σ(t). Figure 2 shows the results of a simulation (where the N model parameters are assumed to be k1 = 1 m , k2 = N N 1 m , M1 = 1kg, M2 = 1kg, k3 = 100 m and x(0) = [ 2 0 4 0 ]> ) in which σ ˆf q (t) has been computed as in Remark 3 using (12), the high-gain “practical” observer (5), with κ1 = 1.31, κ2 = 0.81, κ3 = 0.31, κ4 = 0.07, κ5 = 0.01, κ6 = 0.007, = 2 · 10−4 and ye,5 (0) = > 0.01 0.01 0.01 0.01 0.01 0.01 , and a Butterworth filter of order 1, with ωn = 100 rad . s
Being G3,3 the Groebner basis of ideal J3 , computed according to lex ordering, with σ >l x1 >l x2 >l x4 >l x3 >l y3 >l y2 >l y1 >l y0 , one has, within the polynomials in B3,3 = G3,3 ∩R[x3 , ye,3 ], the following polynomial, which respects the conditions of Proposition 4: y2 + (a + b)y0 . b
4
[cm]
x1 (t) = ν1 (ye,3 (t)) = y0 (t),
1 0
x 1 (t) x ˆ 1(t) x ˘ 1,f(t)
2 0 −2
[cm/s]
(13)
ˆ1 x ˆ2 x ˆ3 x ˆ4 ]> is an estimate of the state where x ˆ=[ x x of system (10), f is given in (11a), σ ˆf q is an estimate of the switching variable, obtained as in Remark 3, L1 ∈ R4 is such that A1 − L1 [ 1 0 0 0 ] is Hurwitz and L2 ∈ R4 is such that A2 − L2 [ 1 0 0 0 ] is Hurwitz. The state of system (10) can be estimated also without computing σ(t). The state x1 and its derivative x2 = x˙ 1 , can be trivially computed as:
2
x 2 (t) x ˆ 2(t) x ˘ 2,f(t)
0 −2
Being G4,3 the Groebner basis of ideal J3 , computed according to lex ordering, with σ >l x1 >l x2 >l x3 >l x4 >l y3 >l y2 >l y1 >l y0 , one has, within the polynomials 6238
x 3 (t) x ˆ 3(t) x ˘ 3,f(t)
2 0
[cm/s]
(14c)
[cm]
4
Hence, assuming that the constant b is not zero, the state x3 can be computed as a polynomial expression of the output and its time derivatives. y2 (t) + (a + b)y0 (t) . x3 (t) = ν3 (ye,3 (t)) = b
1.5 1
(12)
b3,3,1 (x3 , ye,3 ) = x3 −
2
2.5 2
x ˆ˙ (t) = f (ˆ σf q (t), x(t)) + Lσˆf q (y(t) − x ˆ1 (t)),
σ ˆ f q(t) σ(t)
2
x 4 (t) x ˆ 4(t) x ˘ 4,f(t)
0 −2 0
5
10
t
15
20
[s]
Fig. 3: Estimates x ˆ(t) and x ˘f (t) of the state x(t).
Figure 3 shows, from the same simulation, the estimate x ˆ(t) of the state x(t), obtained using (13), with L1 and L2 such that the eigenvalues of the matrix Aσ − Lσ Cσ are {−4, −5, −6, −7}, for σ = {1, 2}. and the estimate x ˘(t), obtained using (14) as in Remark 6, with the same high-gain observer as above and with a Butterworth filter of order 1, ˘f (t) practically with ωn = 100 rad s . Note that the entries of x coincide with the true state variables.
[cm]
0.5
eˆ1(t) e˘1(t)
0
[cm/s]
−0.5 2
eˆ2(t) e˘2(t)
0 −2
[cm]
10
eˆ3(t) e˘3(t)
5
[cm/s]
0 10
eˆ4(t) e˘4(t)
5 0 −5 0
5
t
10
[s]
15
20
Fig. 4: Estimation errors of the state x. Figure 4 shows, from the same simulation, the estimation errors eˆ = [ eˆ1 eˆ2 eˆ3 eˆ4 ]> = x − x ˆ and e˘ = [ e˘1 e˘2 e˘3 e˘4 ]> = x − x ˘f . As it can be seen in such a figure, these estimation errors go to zero, after an initial transient behavior. V. C ONCLUSIONS In this paper, two methods to design state observers for a class of affine switched or switching dynamical systems, without requiring the knowledge of the switching signal, are given. The estimation goals are reached using some tools taken from Algebraic Geometry to express the switching signal as a function of the output and its time derivatives. Then, similar tools are used to design an observer to estimate both the switching signal and the state of the system. R EFERENCES [1] D. Aeyels, “Generic observability of differentiable systems,” SIAM Journal on Control and Optimization, vol. 19, no. 5, pp. 595–603, 1981. [2] J.-P. Gauthier and I. A. Kupka, “Observability and observers for nonlinear systems,” SIAM Journal on Control and Optimization, vol. 32, no. 4, pp. 975–994, 1994. [3] E. G. Gilbert, “Controllability and observability in multivariable control systems,” Journal of the Society for Industrial & Applied Mathematics, Series A: Control, vol. 1, no. 2, pp. 128–151, 1963. [4] A. Isidori, Nonlinear control systems, vol. 1. Springer, 1995. [5] A. Tornambe, “Use of asymptotic observers having-high-gains in the state and parameter estimation,” in Decision and Control, 1989., Proceedings of the 28th IEEE Conference on, pp. 1791–1794, IEEE, 1989.
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