Interval observer design for continuous-time linear parameter-varying systems

Systems & Control Letters 134 (2019) 104541

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Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle

Interval observer design for continuous-time linear parameter-varying systems ∗

Jitao Li a , Zhenhua Wang a , , Wenhan Zhang a , Tarek Raïssi b , Yi Shen a a b

School of Astronautics, Harbin Institute of Technology, Harbin 150001, PR China Conservatoire National des Arts et Metiers (CNAM), Cedric-lab 292, Rue Saint-Martin, 75141 Paris Cedex 03, France

article

info

Article history: Received 21 November 2018 Received in revised form 6 May 2019 Accepted 12 September 2019 Available online xxxx Keywords: Interval observer Linear parameter-varying systems L∞ norm

a b s t r a c t This paper studies interval observer design for linear parameter-varying systems subject to uncertainties in parameters, input and output. Based on a general assumption that these uncertainties are unknown but bounded by known intervals, a novel interval observer is presented to estimate the bounds of state. To attenuate the effect of uncertainties and obtain tight state intervals, a novel L∞ norm based design method is proposed. Numerical simulations are conducted to demonstrate the performance of the proposed method. © 2019 Elsevier B.V. All rights reserved.

1. Introduction For safety critical systems, e.g. autonomous driving and flight control systems, to ensure system reliability, it is very important to know current system state. However, in practice, not all state variables are measurable due to the lack of economic and/or technical support. To deal with this difficulty, too many efforts have been made on observer and filter design in the control systems community [1–3]. In general, according to the treatment of uncertainties, two frameworks can be found in the literature: stochastic and deterministic ones [4]. The former estimates state by minimizing the variance of the state error, e.g. [5]. However, some assumptions on probabilistic distribution of uncertainties are required. These hypotheses can be neglected by utilizing set-membership techniques, which belong to the deterministic framework. Set-membership techniques assume that the uncertainties are unknown but bounded, which is easily determinable from the datasheets of equipments [6]. There are two categories of set-membership techniques: the first one is based on the mechanism ‘‘prediction/correction’’ and the second one (known as interval observer) is based on monotone systems theory. The interval observer is considered in this paper due to its high computational efficiency. Interval observers consist of two sub-observers that provide the upper and lower bounds of the real state using available information. Interval observers were first proposed in [7] and have received considerable attention recently [8–10]. The fundamental idea of this method is to design the observer such that the ∗ Corresponding author. E-mail address: [email protected] (Z. Wang). https://doi.org/10.1016/j.sysconle.2019.104541 0167-6911/© 2019 Elsevier B.V. All rights reserved.

error state matrix is Hurwitz and Metzler (i.e. all its off-diagonal elements are nonnegative) for linear continuous-time case [11]. Note that conventional observers only require that the error state matrix is Hurwitz. Compared with the design of a conventional observer, the extra constraint of Metzler property leads to more theoretical difficulty and computational complexity in interval observer design. To cope with this difficulty, interval observer design methods based on coordinate transformation have been proposed to obtain relaxed design conditions [9,12]. Although these methods can relax design conditions, the performance of interval observer is heavily influenced by the choice of the target matrix [13]. Moreover, the interval may be enlarged in the process of reconstructing the state estimation using the inverse coordinate transformation. Interval observer design for linear parameter-varying (LPV) systems subject to parametric uncertainty has been studied in a few literatures [14–16]. However, these methods are only dedicated to the case of LPV systems with unmeasured parameters, which will be less representative when some parameters are measurable. To fill this gap, [6] and [17] have done some work on interval observer design for LPV systems with both measurable and unmeasurable parameters. [6] neglects the performance optimization whereas [17] fails to consider the output uncertainty. We also note that the method in [17] is based on H∞ technique, which requires the uncertainties are energy bounded. However, in the field of interval observer, the uncertainties should be peak-bounded. Most recently, a novel interval observer structure has been proposed in [18] for discrete-time linear systems. Note that the method in [18] is straightforward and does not depend on coordinate transformation. Motivatived by [18], in this paper, we

2

J. Li, Z. Wang, W. Zhang et al. / Systems & Control Letters 134 (2019) 104541

develop a novel interval observer design method for continuoustime linear parameter-varying systems subject to uncertainties in parameters, input and output. The main contributions are two folds: first, the structure of the interval observer in [18] is extended to the case of continuous-time LPV systems with both measurable and unmeasurable parameters. In contrast to conventional Luenberger-like form in [14] and [17], the merit of the observer structure is that it provides more design parameters, which can be used to broaden the application scope and improve the accuracy of interval estimation. Second, a novel L∞ norm based design method is presented in this paper to attenuate the effect of uncertainties.

A+ x − A− x ≤ Ax ≤ A+ x − A− x. (2) If A ∈ Rn×m is a matrix variable satisfying A ≤ A ≤ A, then +

− −

+ +

−

A+ x+ − A x− − A− x + A x ≤ Ax ≤ A x − A+ x − A x+ + A− x− . +

−

Lemma 3 ([19]). Given a function F (z , θ ) = M(z , θ )z, if there exist functions F , F , M and M such that

{

M ≤ M(z , θ ) ≤ M , ∀θ ∈ Θ , z ∈ Z = {z ∈ Rn : z ≤ z ≤ z }, F (z , z) ≤ F (z , θ ) ≤ F (z , z),

then, the following inequalities hold:

∥F (z , z) − F (z , θ )∥ ≤ lF ∥z − z ∥ + lF ∥z − z ∥ + lF ,

2. Preliminaries The notations used in this paper are standard. Rn and Rn×m denote the n–dimensional Euclidean space and the set of n × m real matrices, respectively. R+ = {τ ∈ R : τ ≥ 0}. 0 and I denote the zero and identity matrices with compatible dimensions, respectively. For a matrix A, AT stands for its transposition, A† stands for its Moore–Penrose inverse, and He(A) is used to denote He(A) := A + AT . The absolute value operator |·| and max{·} on vectors and matrices should be understood elementwise. ∥ · ∥ denotes the 2- norm. For a measurable and locally essentially bounded input u : R+ → Rp , its L∞ norm is defined as the supremum over all time, i.e.

∥u∥∞ = sup{∥u(t)∥, t ∈ R+ }, p

We will denote L∞ as the set of all inputs u with the property ∥u∥∞ < ∞. For a symmetric matrix P, P ≻ 0 and P ≺ 0 indicate that P is positive definite and negative definite, respectively. In a symmetric matrix, we use ⋆ to represent a term induced by symmetry. 2.1. Cooperativity Definition 1. A matrix A ∈ Rn×n is said to be a Metzler matrix if all its off-diagonal elements are nonnegative, i.e. aij ≥ 0, ∀i ̸ = j, where aij is the (i, j)-element of A.

x˙ (t) = A(t)x(t) + B(t)w (t),

q

where +

lF = lF = ∥M ∥ + ∥M + ∥, −

lF = lF = ∥M ∥ + ∥M − ∥, lF =

max {∥(M

θ ∈Θ ,z ∈Z −

w : R+ → R+ , w ∈ L∞ ,

with x(t) ∈ R and a Metzler matrix A ∈ Rn×n , is elementwise n×q nonnegative for all t ≥ 0 provided that x(0) ≥ 0 and B(t) ∈ R+ . Such dynamical systems are called cooperative. 2.2. Interval relations

+

− M + (z , θ ))z + ∥ + ∥(M + − M + (z , θ ))z − ∥

+∥(M − M − (z , θ ))z + ∥+ ∥(M − − M − (z , θ ))z − ∥}, + lF = max {∥(M − M + (z , θ ))z − ∥ + ∥(M + − M + (z , θ ))z + ∥ θ ∈Θ ,z ∈Z −

+∥(M − M − (z , θ ))z − ∥+ ∥(M − − M − (z , θ ))z + ∥}. 3. Problem statement Consider a class of continuous-time uncertain LPV systems as follows x˙ (t) = [A(ρ (t)) + ∆A(ρ (t))]x(t) + B(ρ (t))u(t), y(t) = Cx(t) + v (t),

{

(1)

where x(t) ∈ Rnx is the state vector, y(t) ∈ Rny is the output vector, u(t) ∈ Rnu is the input vector that contains the input uncertainty and v (t) ∈ Rny denotes the output uncertainty, A(ρ (t)), ∆A(ρ (t)) and B(ρ (t)) are parameter-varying matrices scheduled by measurable vector ρ (t), and can be expressed as follows A(ρ (t)) =

s ∑

hi (ρ (t))Ai , B(ρ (t)) =

i=1 s

∑

s ∑

hi (ρ (t))Bi ,

i=1

(2)

hi (ρ (t))∆Ai ,

i=1

q

n

where s is the number of vertices of polytopic set M that bounds ρ (t), and hi (ρ (t)), i ∈ I = {1, . . . , s} are the polytopic coordinates satisfying s ∑

hi (ρ (t)) = 1, hi (ρ (t)) ≥ 0, i ∈ I .

i=1

For two vectors x1 , x2 ∈ Rn or matrices A1 , A2 ∈ Rn×n , the relations x1 Rx2 and A1 RA2 , R ∈ {<, >, ≤, ≥} are understood elementwise. Definition 2. For a vector x, we define x = max{0, x}, x = x − x. −

∥F (z , z) − F (z , θ )∥ ≤ lF ∥z − z ∥ + lF ∥z − z ∥ + lF ,

∆A(ρ (t)) =

Lemma 1 ([11]). Any solution of the system

+

(1) If A ∈ Rn×m is a constant matrix, then

+

For a matrix A, A+ = max{0, A}, A− = A+ − A, Lemma 2 ([15]). Let x ∈ Rm be a vector variable that satisfies x ≤ x ≤ x. Then,

Ai ∈ Rnx ×nx , C ∈ Rny ×nx are known constant matrices, while the matrix ∆Ai is unknown and denotes the parametric uncertainty of Ai , Bi ∈ Rnx ×nu is unknown but bounded by known matrices. The following assumptions are used in this paper. Assumption 1. The initial state x(0) and the uncertainties of system (1) in parameters, input and output are unknown but bounded as follows: x(0) ≤ x(0) ≤ x(0), u(t) ≤ u(t) ≤ u(t), v (t) ≤ v (t) ≤ v (t),

∆Ai ≤ ∆Ai ≤ ∆Ai , Bi ≤ Bi ≤ Bi , where x(0), x(0), u(t), u(t), v (t), v (t), ∆Ai , ∆Ai , Bi and ∆Bi are known.

J. Li, Z. Wang, W. Zhang et al. / Systems & Control Letters 134 (2019) 104541

Assumption 2. The states of system (1) are bounded, i.e. x(t) ∈ X ⊂ Ln∞x , where X is a given compact set. The objective of this paper is to compute signals x(t) and x(t) such that x(t) ≤ x(t) ≤ x(t) for all t ≥ 0. In addition, the interval x(t) − x(t) should be as tight as possible to obtain an accurate state estimation. In the following, the time index t in some variables may be omitted for brevity if no ambiguity is introduced. 4. Main results

provided that T + NC = I

Proof. Let eς (t) = ς (t) − Tx(t), exˆ (t) = xˆ (t) − x(t), eς (t) = ς (t) − Tx(t) and exˇ (t) = xˇ (t) − x(t). Combining (1) and (3), we have

(7)

Substituting (7) into (6), we obtain (3)

e˙ ς = [TA(ρ ) − L(ρ )C ]eς +

3 ∑

di

(8)

i=1

where

δ1 = T δA − T δA ,

d1 = δ1 − T ∆A(ρ )x d2 = δ2 − TB(ρ )u d3 = δ3 + [(TA(ρ ) − L(ρ )C )N + L(ρ )]v. Similarly, we have e˙ ς = [TA(ρ ) − L(ρ )C ]eς +

δ2 = T + δB − T − δB ,

3 ∑

di

(9)

i=1

δ3 = −[(TA(ρ ) − L(ρ )C )N + L(ρ )]+ v

where d1 = δ1 − T ∆A(ρ )x d2 = δ2 − TB(ρ )u d3 = δ3 + [(TA(ρ ) − L(ρ )C )N + L(ρ )]v.

+ [(TA(ρ ) − L(ρ )C )N + L(ρ )] v, −

δ1 = T + δA − T − δA , δ2 = T + δB − T − δB ,

Using Lemma 2, it is easy to show that

δ3 = −[(TA(ρ ) − L(ρ )C )N + L(ρ )] v

δ1 ≤ T ∆A(ρ )x ≤ δ1 , δ2 ≤ TB(ρ )u ≤ δ2 , δ3 ≤ −[(TA(ρ ) − L(ρ )C )N + L(ρ )]v ≤ δ3 .

+

+ [(TA(ρ ) − L(ρ )C )N + L(ρ )] v, −

+

It follows that

−

δA = ∆A(ρ ) x+ − ∆A(ρ )+ x− − ∆A(ρ ) x+ + ∆A(ρ )− x− , +

3 ∑

−

δA = ∆A(ρ )+ x+ − ∆A(ρ ) x− − ∆A(ρ )− x+ + ∆A(ρ ) x− , − +

− −

i=1

−

3 ∑

δB = B(ρ ) u − B(ρ ) u − B(ρ ) u + B(ρ ) u , + − +

δB = B(ρ )+ u+ − B(ρ ) u− − B(ρ )− u+ + B(ρ ) u− . In (3), N ∈ Rnx ×ny , T ∈ Rnx ×nx and L(ρ ) ∈ Rnx ×ny are the matrices to be designed. L(ρ ) has the following form: L(ρ ) = ∑ s i=1 hi (ρ (t))Li . Remark 1. If we choose T = I and N = 0, observer (3) reduces to the commonly used interval observer form, e.g. [17]. Obviously, the proposed method contains more design degrees of freedom by introducing matrices T and N. In fact, the merit of introducing matrices T and N lies in that it offers a solution when we cannot find a matrix L(ρ ) such that A(ρ ) − L(ρ )C is a stable and Metzler matrix. Therefore, in addition to its performances in terms of tight bounds, the proposed approach is less restrictive than those in [14] and [17]. To design interval observer (3), we propose the following theorem. Let Assumptions 1 and 2 hold, the solution of (3)

x(t) ≤ x(t) ≤ x(t),

(6)

Using (5), we have

−

Theorem 1. satisfies

δi − T ∆A(ρ )x − TB(ρ )u + Lv.

exˆ = ς + Ny − (T + NC )x = eς + N v

where ς , ς, xˆ and xˇ are intermediate variables, δi and δi , i = 1, 2, 3 are as follows

+ +

3 ∑ i=1

For system (1), we propose the following interval observer:

+

(5)

and TA(ρ ) − L(ρ )C is a Metzler matrix.

e˙ ς = [TA(ρ ) − L(ρ )C ]exˆ +

⎧ 3 ∑ ⎪ ⎪ ⎪ ⎪ ς˙ = TA(ρ )xˆ + L(ρ )(y − C xˆ ) + δi , ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ xˆ = ς + Ny, ⎪ ⎪ ⎨ x = xˆ + N + v − N − v, 3 ∑ ⎪ ⎪ ⎪ ˇ ˇ ς ˙ = TA( ρ ) x + L( ρ )(y − C x ) + δi , ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎪ xˇ = ς + Ny, ⎪ ⎪ ⎪ ⎩ x = xˇ + N + v − N − v,

3

(4)

δi − T ∆A(ρ )x − TB(ρ )u + [(TA(ρ ) − L(ρ )C )N + L(ρ )]v ≥ 0, δi − T ∆A(ρ )x − TB(ρ )u + [(TA(ρ ) − L(ρ )C )N + L(ρ )]v ≤ 0.

i=1

Now, according to Lemma 1, ς ≤ Tx ≤ ς holds for all t ≥ 0, provided that ς (0) ≤ Tx(0) ≤ ς (0) and TA(ρ ) − L(ρ )C is Metzler. If ς ≤ Tx ≤ ς holds, it can be readily obtained that (4) holds. □ Theorem 1 guarantees the cooperativity of error systems (8) and (9), i.e. inclusion property (4) is ensured for ∀t ≥ 0. To further ensure the boundness of x(t), x(t) (i.e. x(t) ∈ Ln∞x and x(t) ∈ Ln∞x ), error systems (8) and (9) should be stable, similar strategy has been widely applied in the domain of interval observer design, see e.g. [9,10,20]. More step further, to obtain a tight interval of estimation, in this paper, a novel L∞ norm based criterion is established to attenuate the effect of uncertainties. Sufficient design conditions are given in the following theorem. Theorem 2. Suppose that ς (0) and ς (0) in (3) satisfy ς (0) ≤ Tx(0) ≤ ς (0). For given scalars γ > 0, β2 > 0 and β3 > 0, if there exist scalars α > 0, ϵ > 0, a vector ξ ≥ 0 ∈ Rnx , a diagonal matrix P ≻ 0 ∈ Rnx ×nx and a matrix Wi ∈ Rnx ×ny such that P ≻ αI ,

(10)

4

J. Li, Z. Wang, W. Zhang et al. / Systems & Control Letters 134 (2019) 104541

PTAi − Wi C + diag(ξ ) ≥ 0, ⎡ ⎤ Φ11 P P P −γ β2 0 0 ⎥ ⎢ ⋆ ≺ 0, ⎣ ⋆ ⋆ −γ β3 0 ⎦ ⋆ ⋆ ⋆ −ϵ I

T

(12)

2

2

where Φ11 = He{PTAi − Wi C }+ 9ϵ (ld + ld )I +γ P, then (3) is a robust interval observer for the LPV system (1), i.e. x(t) and x(t) in (3) are, respectively, the lower and upper bounds of x(t) with the following L∞ performance:

∥eς (t)∥ ≤

√

1/α (V (0)e−γ t + β2 ∥d2 ∥2∞ + β3 ∥d3 ∥2∞ ),

where eς =

[

eTς

eTς

]T

, d2 =

[

T

d2 T

d2 V (0) is a quadratic function of eς (0).

]T

(13)

[

, d3 =

d3

T

d3 T

]T

,

(14)

(10)–(12)

If (14) is solvable, matrix L(ρ ) in (3) can be determined by L(ρ ) = P −1

s ∑

hi (ρ (t))Wi .

Proof. According to [21], it can be verified that the general solution of (5) is T = GZ1 + S(I − GG† )Z1 , N = GZ2 + S(I − GG† )Z2 ,

(15)

[ ]

2

d1 T d1 ≤ 3(ld ∥e∥2 + l2d ∥e∥2 + l2d ) 2

≤ 3{ld (3∥eς ∥2 + 3v T N T N v + 3∥N v∥2 ) + l2d (3∥eς ∥2 + 3v T N T N v + 3∥N v∥2 ) + l2d } Using the fact that ld = ld , ld = ld , we have

2

+ 9ϵ (ld + ld )]eς + eTς [P(TA(ρ ) − L(ρ )C ) + (TA(ρ ) − L(ρ )C )T P + ϵ −1 PP 2

2

+ 9ϵ (ld + ld )]eς

= eTς [P(TA(ρ ) − L(ρ )C ) + (TA(ρ ) − L(ρ )C )T P + ϵ −1 PP 2

2

+ 9 ϵ (ld + ld )]eς + 2eTς Pd2 + 2eTς Pd3 + ϵη. 2

where η = 9(ld + l2d )(2∥N v∥2 + ∥N v∥2 + ∥N v∥2 ) + 3(l2 + l2d ). d Letting Wi = PLi and using Schur complement, from (12), we have

where S ∈ Rnx ×(nx +ny ) is a freely chosen matrix and

[ ]

+ 3∥N v∥2 ) + ld2 }

+ 2eTς Pd2 + 2eςT Pd3 + 2eTς Pd2 + 2eTς Pd3 + ϵη

i=1

{

2

≤ 3{ld (3∥eς ∥2 + 3v T N T N v + 3∥N v∥2 ) + l2d (3∥eς ∥2 + 3v T N T N v

2

α

s.t .

2

d1 d1 ≤ 3(ld ∥e∥2 + l2d ∥e∥2 + l2d )

V˙ ≤ eTς [P(TA(ρ ) − L(ρ )C ) + (TA(ρ ) − L(ρ )C )T P + ϵ −1 PP

An optimal solution can be found by solving max

According to (17), we have

(11)

[ ]

I I 0 G= , Z1 = , Z2 = . C 0 I Note that the matrix P is constrained to have a diagonal form, then P ≻ 0 indicates P > 0. Consequently, TA(ρ ) − L(ρ )C is Metzler if P(TA(ρ ) − L(ρ )C ) is Metzler, which is ensured by (11). Let us consider the following Lyapunov function: V = eςT Peς + eTς Peς .

(16)

The derivative of V is governed by

[ Γ11 ⋆ ⋆

P

−γ β2 ⋆

P 0

≺ 0,

−γ β3

(18) 2

2

where Γ11 = He{PTA(ρ ) − PL(ρ )C } + 9ϵ (ld + ld )I + γ P + ϵ −1 PP. [ By pre-multiplying] and post-multiplying inequality (18) with eTς (t) dT2 (t) dT3 (t) and its transpose, we have V˙ ≤ −γ V + γ (β2 dT2 d2 + β3 dT3 d3 ) + ϵη

V˙ = eTς [P(TA(ρ ) − L(ρ )C ) + (TA(ρ ) − L(ρ )C )T P ]eς

(19)

Since ϵη is a positive constant, (19) is implied by

+ 2eTς Pd1 + 2eςT Pd2 + 2eTς Pd3 + eTς [P(TA(ρ ) − L(ρ )C ) + (TA(ρ ) − L(ρ )C )T P ]eς + 2eTς Pd1 + 2eTς Pd2 + 2eTς Pd3 .

V˙ ≤ −γ V + γ (β2 dT2 d2 + β3 dT3 d3 )

According to Lemma 3, d1 and d1 are globally Lipschitz and there always exist positive constants ld = ld , ld = ld , ld , and ld such that

∥d1 ∥ ≤ ld ∥e∥ + ld ∥e∥ + ld , ∥d1 ∥ ≤ ld ∥e∥ + ld ∥e∥ + ld ,

(17)

≤ −γ V + γ (β2 ∥d2 ∥2∞ + β3 ∥d3 ∥2∞ ). It follows that V (t) ≤ V (0)e−γ t + γ β2 ∥d2 ∥2∞

t

∫

e−γ (t −τ ) dτ 0

where

+ γ β3 ∥d3 ∥2∞

e = x − x = eς + N v + N v,

Nv = N v − N v

e = x − x = eς + N v + N v,

N v = N + v − N − v.

+

−

2eς Pd1 ≤ eς P ϵ T

−1

T

Peς + d1 ϵ d1 ,

2eς Pd1 ≤ eς P ϵ T

T

−1

t

∫

e−γ (t −τ ) dτ 0

= V (0)e−γ t + (1 − e−γ t )β2 ∥d2 ∥2 + (1 − e−γ t )β3 ∥d3 ∥2∞

Completing the squares as T

]

Peς + d1 ϵ d1 , T

we obtain V˙ ≤ eςT [P(TA(ρ ) − L(ρ )C ) + (TA(ρ ) − L(ρ )C )T P + ϵ −1 PP ]eς

≤ V (0)e−γ t + β2 ∥d2 ∥2∞ + β3 ∥d3 ∥2∞ . In addition, (10) implies V (t) ≥ α eTς eς , which follows (13). □ 5. Numerical examples

+ eTς [P(TA(ρ ) − L(ρ )C ) + (TA(ρ ) − L(ρ )C )T P + ϵ −1 PP ]eς T

+ 2eTς Pd2 + 2eTς Pd3 + 2eTς Pd2 + 2eTς Pd3 + ϵ d1 d1 + ϵ d1 T d1 .

In this section, two examples are used to compare the proposed method with the existing methods.

J. Li, Z. Wang, W. Zhang et al. / Systems & Control Letters 134 (2019) 104541

5

Example 1. A benchmark from [14] is considered in this paper to demonstrate the superiority of the proposed method:

[ ⎧ 0.01 cos t 1 + 0.01 sin x3 ⎪ ⎪ ⎪ x˙ = 0.001 sin x3 −0.5 + 0.01 sin t ⎪ ⎪ ⎪ ⎨ 0.001 sin x2 0.3 + 0.001 cos 2t [ ] 6 cos x1 ⎪ ⎪ sin t + 0.1 sin x3 , + ⎪ ⎪ ⎪ ⎪ −cos 3t + 0.1 sin 2x2 ⎩ y = x1 + v (t).

0.001 sin x2 1 + 0.01 cos 2t −1 + 0.01 sin t

]

This system can be reformulated as

{

x˙ (t) = (A + ∆A)x(t) + Bu(t), y(t) = Cx(t) + v (t),

where

[ A=

0 0 0

1 −0.5 0.3

0 1 , ∆A = −1

]

∆A = −∆A, [ ] 1 0 0

B=

0 1 0

[

0.01 0.001 0.001

⎡

0.01 0.01 0.001

0.001 0.01 , 0.01

]

Fig. 1. Real state x1 (t) and its interval estimation.

⎤

0 6f (y) 0 , u(t) = ⎣ sin t + 0.1 ⎦ , 1 −cos t + 0.1

⎤

⎡

6f (y) u(t) = ⎣ sin t − 0.1 ⎦ , −cos t − 0.1

{

cos y + |sin y| sin 0.1 if cos y ≥ 0, cos y cos 0.1 + |sin y| sin 0.1 if cos y < 0,

{

cos y cos 0.1 − |sin y| sin 0.1 if cos y ≥ 0, cos y − |sin y| sin 0.1 if cos y < 0,

f (y) = f (y) =

[

C = 1

0

0 , v (t) = 0.1(sin 5t + cos 3t)/2, v (t) = 0.1,

]

v (t) = −0.1, Fig. 2. Real state x2 (t) and its interval estimation.

which is a special case of (1) by dropping the measurable vector ρ (t). 1 0 0

[ Choosing S = 0.5 −0.5 0

[ T =

0 1 0

0 1 0

0 0 1

0 0 ,N = 1

]

1 1 , according to (15), we have 0

]

0.5 0.5 . 0

[

]

Following Lemma 3, we have ld = ld = 2.1213 × 10−4 . By setting γ = 0.1,[β2 = 1, β3 = 1 and ]T solving (14), we obtain α = 0.0272 and L = 58.3333 0 0 . In the simulation, the initial state is x(0) = and the bounds of initial state are x(0) =

0.1

[

[

0

5

]T

0

0

]T

and

5

]T

x(0) = −0.1 −5 −5 . The results obtained by the proposed method and the method in [14] are depicted in Figs. 1–3. From Figs. 1–3, we can see that, under same simulation conditions, the proposed interval observer provides more accurate interval estimation than the method in [14]. This is reasonable since the proposed method contains more degrees of freedom than the method in [14] by introducing matrices T and N. This illustrates the superiority of the proposed design method. In the following, we use a numerical example from [6] to further demonstrate the viability and validity of the proposed method.

[

Fig. 3. Real state x3 (t) and its interval estimation.

Example 2. The system is described as follows:

(

x˙ 1 (t) x˙ 2 (t)

)

( =

a11 (t) a21 (t)

(

x1 (t) y(t) = 0 1 · x2 (t) ( ρ (t) = a11 (t) a12 (t)

(

)

) ( ) ( ) x (t) b · 1 + 1 u(t),

a12 (t) a22 (t)

)

x2 (t)

b2

= x2 (t), a21 (t)

a22 (t)

b1

b2 .

)

6

J. Li, Z. Wang, W. Zhang et al. / Systems & Control Letters 134 (2019) 104541

Fig. 5. State x1 (t) estimation by the proposed method and that by [6].

Fig. 4. Envelope of input signal u(t).

In [6], the time-varying parameter vector is chosen as ρ (t) = 2 − 11 − 3 − 3 1). Considering the same uncertainties as in [6] and using parameter interpolation technique, this system can be expressed as a polytopic LPV form with 2 vertices as follows:

(−8.5 + 6.5sin(t)

{

x˙ 1 = (A1 + ∆A1 )x(t) + B1 u(t), y(t) = Cx(t) + v (t)

{

x˙ 2 = (A2 + ∆A2 )x(t) + B2 u(t), y(t) = Cx(t) + v (t)

Vertex 1: Vertex 2: where

[

−2 −11

A1 =

]

[

]

2 −15 , A2 = −3 −11

[ 2 ,C = 0 −3

1 ,

]

[ ] 0.2 0 ∆A1 = ∆A2 = −∆A1 = −∆A2 = , 0.5 0.2 [ ] [ ] −2.9 −3.1 B1 = B2 = , B1 = B2 = , −0.1 ≤ v (t) ≤ 0.1 1.1 0.9 and the envelope of input u(t) is depicted in Fig. 4. The polytopic coordinates are h1 ( ρ ) =

1 + sin(t) 2

, h2 ( ρ ) =

1 − sin(t) 2

.

Note that we will meet infeasible problem for (14) if we set T = I and N = 0. In fact, for this LPV systems, we cannot find gain matrices L1 and L2 such that A(ρ ) − L(ρ )C is stable and Metzler simultaneously, due to the existence of value −11. Thus, the method in [17] fails to be applied. Conventional interval observer design methods [9,10,22] solve this problem by the way of coordinate transformation, which may lead to some conservatism. In this paper, we solve this problem by introducing matrices T and N.[ ] 1 0 0 Choosing S = , according to (15), we have 0 1 2

[ T =

1 0

]

[ ]

0 0 ,N = . 0 1

Following Lemma 3, we have ld = 0.2 and ld = 0.2. By setting γ = 1, β2 = 1, β3 = 1 and solving (14), we obtain α = 0.6515 with the following gain matrices 2.0000 −0.8052 , L2 = . 4.8610 3.6174

[ L1 =

]

[

]

Fig. 6. State x2 (t) estimation by the proposed method and that by [6].

]T 0 and the envelope of initial state [ ]T and x(0) = −1 −1 . [

The initial state is x(0) = 0

[

]T

is x(0) = 1 1 The results obtained by the proposed method and the method of [6] are depicted in Figs. 5 and 6. From Figs. 5 and 6, we can see that, under same simulation conditions, the proposed interval observer provides more accurate interval estimation than that by [6]. This is reasonable because the effect of uncertainties is attenuated by incorporating the technique based on L∞ norm. Additionally, the introduced matrices T and N can also make a contribution to the performance improvement. This illustrates the superiority of the proposed design method. 6. Conclusions This paper considers interval observer design for a class of continuous-time LPV systems. Based on a general assumption that uncertainties are unknown but bounded by intervals, we propose a novel observer structure and develop a novel direct design method based on L∞ norm. Sufficient design conditions of the proposed method are derived as linear matrix inequalities. Finally, simulation results are presented to illustrate the viability and validity of the proposed method. Note that the constraints on state are present in the real systems. However, to the best of our knowledge, currently there is no results on interval observer design with state constraints. This interesting point will be considered in our further works.

J. Li, Z. Wang, W. Zhang et al. / Systems & Control Letters 134 (2019) 104541

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