Exponential Nonlinear Observer Based on the ... - Springer Link

International Journal of Automation and Computing

9(4), August 2012, 358-368 DOI: 10.1007/s11633-012-0656-y

Exponential Nonlinear Observer Based on the Differential State-dependent Riccati Equation Hossein Beikzadeh1 1 2

Hamid D. Taghirad2

Department of Electrical and Computer Engineering, University of Alberta, Edmonton T6G 2V4, Canada

Faculty of Electrical and Computer Engineering, K. N. Toosi University of Technology, Tehran 16315-1355, Iran

Abstract: This paper presents a novel nonlinear continuous-time observer based on the differential state-dependent Riccati equation (SDRE) filter with guaranteed exponential stability. Although impressive results have rapidly emerged from the use of SDRE designs for observers and filters, the underlying theory is yet scant and there remain many unanswered questions such as stability and convergence. In this paper, Lyapunov stability analysis is utilized in order to obtain the required conditions for exponential stability of the estimation error dynamics. We prove that under specific conditions, the proposed observer is at least locally exponentially stable. Moreover, a new definition of a detectable state-dependent factorization is introduced, and a close relation between the uniform detectability of the nonlinear system and the boundedness property of the state-dependent differential Riccati equation is established. Furthermore, through a simulation study of a second order nonlinear model, which satisfies the stability conditions, the promising performance of the proposed observer is demonstrated. Finally, in order to examine the effectiveness of the proposed method, it is applied to the highly nonlinear flux and angular velocity estimation problem for induction machines. The simulation results verify how effectively this modification can increase the region of attraction and the observer error decay rate. Keywords: Detectability, direct method of Lyapunov, exponential stability, nonlinear observer, region of attraction, state-dependent Riccati equation (SDRE) technique.

1

Introduction

The problem of estimating the state of a nonlinear dynamical system has attracted considerable attention since the development of linear observers[1, 2] . Numerous design methods exist for various classes of systems: feedback linearization, variable structure techniques, extended Kalman filter, high gain observers, Lyapunov-based observer design, state-dependent Riccati equation (SDRE) technique, etc, among others (see [3 – 6] and the references cited therein). The most popular reported method is the extended Kalman filter (EKF)[7, 8] . In spite of its satisfactory results, there is documented evidence of erratic filter behavior such as the premature collapse of the error covariance[9] and the loss of system observability[10] . The state-dependent Riccati equation (SDRE) techniques are rapidly emerging as general design and synthesis methods of nonlinear feedback controllers and estimators for a broad class of nonlinear problems[11] . Essentially, the SDRE filter, developed over the past several years, is formulated by constructing the dual problem of the SDRE-based nonlinear regulator design technique[10] . The resulting observer has the same structure as the continuous steady state linear Kalman filter. In contrast to the EKF which uses the Jacobian of the nonlinearity in the system dynamics, the SDRE filter is based on parameterization that brings the nonlinear system to a linear-like structure with statedependent coefficients (SDC). As shown in [12], in the multivariable case, the SDC parameterization is not unique. Consequently, this method creates additional degrees of freedom that can be used to overcome the limitations such as low performance, singularities and loss of observability in a traditional estimation Manuscript received September 13, 2010; revised November 2, 2011

method[10] . Indeed, an algebraic Riccati equation is solved at every time step to obtain the SDRE observer gain. Hence in [13 – 15], this method is called the state-dependent algebraic Riccati equation (SDARE) observer. One drawback of the SDARE is its high computational load for large scale systems. The other is that if loss of observability occurs during certain time intervals, then the algebraic Riccati equation may not have a solution and the SDARE cannot be used during these time-intervals. Although the SDRE observer has been empirically implemented in a number of applications[13,16−21] , its practical usefulness is accompanied with a heuristic theoretical derivation. In fact, there is no mathematically rigorous stability analysis regarding this estimating method except, the theoretical results which are provided in a recent paper[21] . In [21, 22], by splitting the state-dependent matrices of the SDC form into a constant element and a state-dependent incremental element, it is shown that the estimator is locally asymptotically stable under some observability and Lipschitzian conditions. However, the provided results are essentially local and require certain simplicity and boundedness assumptions on the incremental matrices in a neighborhood of the origin, which may be violated for many practical systems. Recently an alternative form of this estimator, namely the state-dependent differential Riccati equation (SDDRE) observer, which has the same structure as the linear Kalman filter has been offered[15, 23−25] . This alternatively addresses the issues of high computational load and the potentially overly restrictive observability requirement in the SDARE. The key idea underlying this new method is to remove the infinite-time horizon assumption and to use differential rather than algebraic Riccati equation. In [23, 25], a

H. Beikzadeh and H. D. Taghirad / Exponential Nonlinear Observer Based on the Differential · · ·

discrete-time form of the SDDRE observer is considered and two distinct sufficient conditions set for its asymptotic stability are obtained. However, these conditions can be verified only based on the simulation results and are confined to unforced systems. The stochastic stability and robust synthesis of the SDDRE state estimator for nonlinear systems exposed to disturbance inputs are addressed in [26, 27], respectively. In this paper, motivated by the exponential stability results provided in [28, 29] regarding the widely used EKF, the continuous-time SDDRE formulation is modified in order to have an exponentially stable observer, i.e., the estimation error goes to zero exponentially[30] . The suggested methodology is based on the regulation theory with prescribed degree of stability[31] and exponential data weighting for linear stochastic systems[32] . Using the Lyapunov stability analysis, a set of sufficient conditions is obtained which ensures the local exponential stability of the proposed observer. These conditions do not implicate any particular splitting or simplicity limitations on the SDC factorization like those imposed in [21], and can be satisfied easier than those conditions in [23, 25] which are completely simulationbased. Furthermore, a new definition of a detectable SDC parameterization is introduced, which is derived from interesting results presented in [33], and is closely related to the existence of bounded and positive definite solutions for the state-dependent differential Riccati equations. By this means, a new SDRE-based observer with guaranteed exponential stability is obtained, which inherits the elaborated properties of the SDRE technique. Moreover, this modification allows an effective treatment of the nonlinearities and under a specific condition grants in advance assignment of the degree of stability[31] . Comparative studies reveal the preferable performance and an increased region of attraction compared to the common SDRE-based observers. This paper is organized as follows. In Section 2, we state the necessary preliminaries and introduce the proposed observer. Then in Section 3, by choosing an appropriate Lyapunov function, we show that the proposed observer is an exponential observer. The role of uniform detectability in this context is discussed in Section 4. Section 5 which contains two simulation examples, states the estimation of a simple nonlinear system and an induction machine respectively, illustrates the recent SDRE-based observer in a deterministic setting. We also compare the results of the proposed observer to that of the SDARE and SDDRE observers given in the literature. Finally, the conclusions are given in Section 6. Throughout this paper, · denotes the Euclidian norm of real vectors or the induced norm of real matrices. Moreover, Rq is the real q-dimensional vector space and C 1 is the space of continuously differentiable functions.

2

Proposed SDRE observer and some preliminaries

Consider a nonlinear continuous-time system affine in the input that is represented by x(t) ˙ = f (x(t)) + g(x(t))u(t)

(1)

y(t) = h(x(t))

(2)

359

where x(t) ∈ Rn is the state, u(t) ∈ Rp the input, and y(t) ∈ Rm the output. Assume that by direct parameterization, the nonlinear dynamics can be rewritten in the following state-dependent coefficient (SDC) form: x(t) ˙ = A(x(t))x(t) + B(x(t))u(t)

(3)

y(t) = C(x(t))x(t)

(4)

where f (x(t)) = A(x(t))x(t) g(x(t)) = B(x(t)),

h(x(t)) = C(x(t))x(t).

(5)

The former parameterization for a continuous A(x) is possible if f (0) = 0 and f (x) ∈ C 1[34, 35] . However, as discussed in [17], even if f is only continuous, i.e., f (x) ∈ C 0 , finding a continuous factorization is still possible (but not guaranteed). Also see [11] for effective handling of some situations which may prevent a straightforward parameterization. It is also shown in [12] that in multivariable case, the SDC parameterization is not unique. We first make the following assumptions on the chosen SDC form, the control input u(t), and the signal to be estimated x(t). Assumption 1. The SDC parameterization is chosen such that A(x), B(x) and C(x) are at least locally Lipschitz (see [21, 27]). i.e., there exist constants kA , kB , kC > 0 such that A(x1 ) − A(x2 ) kA x1 − x2

(6)

B(x1 ) − B(x2 ) kB x1 − x2

(7)

C(x1 ) − C(x2 ) kC x1 − x2

(8)

for x1 , x2 ∈ Rn with x1 − x2 εA , x1 − x2 εB , and x1 − x2 εC , respectively. It should be mentioned that if the SDC form fulfills the Lipschitz condition globally in Rn , then all the results in this and the ensuing sections will be valid globally. Assumption 2. The time varying state-dependent matrix C(x(t)) is bounded by C(x(t)) c¯

(9)

where c¯ > 0 is a real number. Assumption 3. Assume that there exist σ, ρ > 0 such that for all t 0 x(t) σ,

u(t) ρ.

(10)

These assumptions will be used in our stability analysis in the next section. Remark 1. If A1 (x) and A2 (x) are two distinct parameterization of f (x), then ˜ A(x) = M (x)A1 (x) + (I − M (x))A2 (x)

(11)

is also a parameterization of f (x) for all matrix valued functions M (x) ∈ Rn×n (see [10, 12, 25]). This is also valid for the output matrix C(x). These additional degrees of freedom provided by nonuniqueness of the SDC parameterization can be used to either enhance the observer performance and avoid loss of observability, or particularly satisfy (6) – (8).

360

International Journal of Automation and Computing 9(4), August 2012

Remark 2. The Lipschitz conditions (6) – (8) are also considered in [21, 25]. Consequently, compared to the previous studies on the SDRE observers, we have not imposed any new restrictive assumptions on the chosen SDC form. Remark 3. Inequalities (9) – (10) are mild conditions. In particular, for many applications, the state variables, which often represent physical quantities, are bounded. Boundedness of the control input seems also to be a trivial hypothesis. Thus, (10) are satisfied easily. Besides, if C(x) fulfills (9) for every reasonable value of the state vector x(t), we may suppose without loss of generality that (9) is also satisfied. Now, the proposed observer is developed. For the dynamical system given by (3) – (4), let us introduce an observer as follows

Adding and subtracting A(ˆ x(t))x(t) to the whole equation together with adding and subtracting C(ˆ x(t))x(t) into the bracket lead to e(t) ˙ =A(ˆ x(t))x(t) − A(ˆ x(t))ˆ x(t) + A(x(t))x(t)− A(ˆ x(t))x(t) + [B(x(t)) − B(ˆ x(t))]− K(t)[C(ˆ x(t))x(t) − C(ˆ x(t))ˆ x(t)+ C(x(t))x(t) − C(ˆ x(t))x(t)]. So the error dynamics are given by e(t) ˙ = [A(ˆ x(t)) − K(t)C(ˆ x(t)] e(t)+ φ(x(t), x ˆ(t), u(t)) − K(t)χ(x(t), x ˆ(t))

φ(x(t), x ˆ(t), u(t)) = [A(x(t)) − A(ˆ x(t))] x(t)+

(13)

where P (x(t)) ∈ Rn×n is symmetric and computed through the following state-dependent differential Riccati equation: P˙ (x(t)) = (A(ˆ x(t)) + αI) P (x(t)) + P (x(t))(AT(ˆ x(t))+ x(t))R−1 C(ˆ x(t))P (x(t)) + Q αI) − P (x(t))C T (ˆ (14) with a positive real number α > 0, and symmetric positive definite matrices Q ∈ Rn×n and R ∈ Rm×m . It should be noted that (14) renders to the Riccati equation of the usual SDDRE[15] for α = 0. Moreover, the observer gain K and the solution P of (14) are state-dependent. However, their state dependency is omitted thereafter just for notational convenience. Remark 4. Note that (14) is indeed a modified version of the Riccati equation with an additional term 2αP (x(t)). The system dynamics is still incorporated in obtaining P (x(t)) and our stability analysis in the next section. Remark 5. The scalar α is a design parameter, which indirectly indicates the error decay rate of the proposed exponential observer. This verity as well as the feasibility of α will be clarified in the next section. Remark 6. In a stochastic framework, a common choice for the matrices Q and R are the covariances of the corrupting noise signals. However, this is not the only possibility. Especially, for a deterministic estimation problem that is tackled in this paper, any other positive definite matrices can be chosen as well. Define the estimation error by e(t) = x(t) − x ˆ(t).

(15)

By subtracting (12) from (3), the error dynamics is obtained e(t) ˙ =A(x(t))x(t) + B(x(t))u(t) − A(ˆ x(t))ˆ x(t)− B(ˆ x(t))u(t) + K(t) [y(t) − C(ˆ x(t))ˆ x(t)] .

[B(x(t)) − B(ˆ x(t))]u(t)

(19)

χ(x(t), x ˆ(t)) = [C(x(t)) − C(ˆ x(t))] x(t).

(20)

(12)

where x ˆ(t) denotes the state estimate and the observer gain K(t) is a time varying n×m matrix. We define the observer gain by K(x(t)) = P (x(t))C T (ˆ x(t))R−1

(18)

where

x ˆ˙ (t) =A(ˆ x(t))ˆ x(t) + B(ˆ x(t))u(t)+ K(x(t)) [y(t) − C(ˆ x(t))ˆ x(t)]

(17)

(16)

In order to analyze the error dynamics, we make use of the following two definitions. Definition 1. The equilibrium point e(t) = 0 of (18) is locally exponentially stable, if there exist constants ε, η, θ > 0 such that t e(t) η e(0) exp(− ) (21) θ holds for every t 0 and for every solution e(·) of equation (18), originating from an initial state inside Bε = {e ∈ Rn | e < ε}[36] . Definition 2. The observer given by (12) – (14) is an exponential observer, if the differential (18) of the estimation error has a locally exponentially stable equilibrium point at e(t) = 0[30] .

3

Stability analysis

In this section, we determine sufficient conditions that guarantee exponential stability of the proposed observer. Theorem 1 states the main result of this paper. Note that the matrix inequalities Ω Δ and further Ω − Δ 0 mean that the matrix Ω − Δ is negative semidefinite. Theorem 1. Consider the nonlinear continuous-time system (1) – (2) put into the SDC form (3) – (4), along with the proposed SDRE-based observer (12) – (14). Let Assumptions 1 – 3 hold and the solution P (t) of differential Riccati equation (14) is bounded via pI P (t) p¯I

(22)

for some positive real numbers p, p¯ > 0. Then the proposed observer is an exponential observer in the sense of Definition 2, provided that the design parameter α satisfies α>

qp c¯kC σp + k A σ + kB ρ − 2 r 2¯ p

(23)

where q = λmin (Q) and r = λmin (R). Remark 7. Inequality (22) which seems the key condition in our stability analysis, is closely related to observability and detectability properties of the system to be observed. These relations will be discussed in Section 4.


Remark 8. The Lipschitz constants kA , kB , and kC are derived analytically from the system dynamics. The bounds r and q can also be obtained from the weighting matrices Q and R, respectively. Likewise, ρ can be determined by considering the actuators saturation limit, although conservatively, and values of σ, c¯ are calculated analytically. The bounds p and p¯ for the matrix P (t) are related to the detectability of the nonlinear system (see the next section). Accordingly, (23) can be verified in advance. Also, see the following two remarks. Remark 9. Inequality (23) roughly means that α should be chosen sufficiently large. Surprisingly, this is in accordance with the purpose of performance improvement which calls for a smaller time constant (see the proof of Theorem 1 and also Remark 12). Remark 10. It can be shown that, (23) is obviated while the estimation error remains still exponentially stable, provided that (6) – (8) are replaced by more restricted Lipschitz conditions with exponent two, e.g., A(x1 ) − A(x2 ) kA x1 − x2 2 (see [26]). The proof of this theorem can be modified easily for this case. To prove Theorem 1, we state the following preparatory lemma. Lemma 1. Consider a positive-definite m × m matrix R with R rI for some r > 0. Assume that the matrix K(t) and the nonlinearities φ(x(t), x ˆ(t), u(t)) and χ(x(t), x ˆ(t)) are given by (13), (19), and (20), respectively. Then under the assumptions of Theorem 1, there exist real numbers ε, κ > 0 such that Π(t) = P −1 (t) satisfies the inequality T

T

T

K

1 1 e(t)2 V (e(t), t) e(t)2 p¯ p

(30)

which imply that V (e(·), ·) is positive definite and decrescent, thus this function is an appropriate Lyapunov function candidate. Taking time derivative of the Lyapunov function, we get ˙ V˙ (e(t), t) =e˙ T (t)Π(t)e(t) + eT (t)Π(t)e(t)+ ˙ eT (t)Π(t)e(t).

(31)

Inserting e(t) ˙ according to differential equation (18) yields with a few rearrangements ˙ V˙ (e(t), t) =eT (t)Π(t)e(t)+ eT (t) [A(ˆ x(t)) − K(t)C(ˆ x(t))]T e(t)+ x(t)) − K(t)C(ˆ x(t))] e(t)+ eT (t)Π(t) [A(ˆ ˆ(t), u(t))− 2eT (t)Π(t)[φ(x(t), x K(t)χ(x(t), x ˆ(t))].

(32)

Applying Lemma 1 and considering (13) yields

By the hypothesis that A(x), B(x), and C(x) are locally Lipschitz and the use of (10), we have

ˆ . [B(x) − B(ˆ x)] u (kA σ + kB ρ) x − x (26) 1 , r

we

ˆ, u) − (x − x ˆ)T ΠKχ(x, x ˆ) (x − x ˆ)T Πφ(x, x (kA σ + kB ρ) c¯kC σ x − x ˆ + x − x ˆ x − x ˆ p r (27)

for x − x ˆ ε with ε = min(εA , εB , εC ). Thus (24) follows immediately with (28)

(33)

for e(t) ε, where κ is given by (28) and ε = min(εA , εB , εC ). Considering ˙ Π(t) = −Π(t)P˙ (t)Π(t)

(34)

and the differential Riccati equation (14) leads to V˙ (e(t), t) −2αeT (t)Π(t)e(t) − eT (t) [Π(t) × x(t))R−1 C(ˆ x(t)) e(t) + 2κ e(t)2 . QΠ(t) + C T (ˆ

φ(x, x ˆ, u) [A(x) − A(ˆ x)] x + χ(x, x ˆ) = [C(x) − C(ˆ x)] x kC σ x − x ˆ . Considering (26), Π 1p , C c¯, and R−1 obtain

x(t))Π(t) − 2C T (ˆ x(t))R−1 C(ˆ x(t))]e(t)+ AT (ˆ 2κ e(t)2

(x − x ˆ)T Πφ(x, x ˆ, u) + (x − x ˆ)T C(ˆ x)T R−1 χ(x, x ˆ). (25)

c¯kC σ (kA σ + kB ρ) + . p r

(29)

with Π(t) = P −1 (t). Because of (22), we have the following bounds for the Lyapunov function:

=

T

κ=

V (e(t), t) = eT (t)Π(t)e(t)

˙ V˙ (e(t), t) eT (t)[Π(t) + Π(t)A(ˆ x(t))+

ˆ, u) − (x − x ˆ) ΠKχ(x, x ˆ) (x − x ˆ) Πφ(x, x

x − x ˆ

Proof of Theorem 1. We consider differential equation (18) of the estimation error and prove its exponential stability by choosing the Lyapunov function

2

ˆ, u) − (x − x ˆ) ΠKχ(x, x ˆ) κ x − x ˆ (x − x ˆ) Πφ(x, x (24) for every x − x ˆ ε. Proof. Applying the triangular inequality, P C T R−1 and ΠP = I yield to

361

(35)

Denote the smallest eigenvalue of the positive-definite matrix Q by q, then we have qI < Q. Together with bounds (22) for P (t) we obtain q − 2κ e(t)2 . (36) V˙ (e(t), t) −2αV (e(t), t) − p¯2 According to (30), we get p − e(t)2 −pV (e(t), t) − e(t)2 p¯ hence

qp V˙ (e(t), t) − 2α + 2 − 2κp V (e(t), t) p¯

(37)

(38)

362


for e(t) ε. Therefore, if 2α + qp p¯2 − 2κp > 0, it then follows that V˙ (e(t), t) is locally negative definite. By applying standard results concerning the direct method of Lyapunov (see [36]), we conclude that the differential equation (18) has a uniformly asymptotic stable equilibrium at 0. Moreover, by separating variables and integration we have qp V (e(t), t) V (e(0), 0) exp − 2α + 2 − 2κp t . (39) p¯ Together with (30) lead to qp p¯ e(0) exp − α + 2 − κp t e(t) p 2¯ p

(40)

2 p −κp. i.e., (21) is valid with η= p¯/p and θ−1=α + qp 2¯ Remark 11. For α = 0, we have the standard differential SDRE observer[15] which, according to this theorem, is an exponential observer if qp − κp > 0. 2¯ p2

(41)

It is clear that, satisfaction of (23) for the proposed observer is much easier than (41) for the SDDRE. In fact, the free parameter α can be chosen such that (23) holds. On the other hand, (41) depends on the parameters which are not under our control, and it may be violated. thus Remark 12. If qp 2¯ p2 − κp > 0, then the time constant θ for the exponential error decay in (21) satisfies θ < α−1 . In this situation, selecting an appropriate α > 0 and by using θ < α−1 , it can be seen that the constant θ in (21) can be assigned in advance, i.e., we obtain an observer with a prescribed degree of stability (see [28, 31]).

4

Relation to detectable SDC parameterization

According to (22), bounds are required for the solution P (t) of the differential Riccati equation (14) to prove the convergence of the estimation error. Interesting results between the observability of the nonlinear system and existence of positive definite solutions for the differential Riccati equations are given in [33]. Lower and upper bounds for the error covariances, P (t), have been also obtained using dual optimal control problems. In [25], a discrete-time SDRE-based observer is considered and using the results presented in [37 – 38], a priori bounds on the Riccati equation are obtained. These bounds follow from a uniform observability condition. However, the nonlinear system is treated like a frozen-in-time linear system, and the state-dependency fact is ignored. In this section, based on the results given in [33], we discuss how the following uniform detectability notation is related to the boundedness of the solution P (t) of the continuous-time state-dependent differential Riccati equation (14). Definition 3. The pair {C(x), A(x)} is called a uniformly detectable SDC parameterization of the nonlinear system (1) – (2), if there exist a bounded matrix valued

function Λ(x) and a real number γ > 0 such that ω T [A(x) + Λ(x)C(x)] ω −γ ω2

(42)

holds for all x ∈ Rn . Another detectability definition associated with the SDC form, has been proposed in the literature (see [21, 22, 34]) where A(x) is called as a detectable (observable) parameterization if the pair {C(x), A(x)} is pointwise detectable (observable) in the linear sense for all x in the domain of interest. This definition is clearly equivalent to Definition 3. The reason is that, this pointwise detectability implies that there exists a matrix Λ(x) such that the matrix A(x) + Λ(x)C(x) has all eigenvalues in the open left half plane and therefore, inequality (42) will be satisfied. Remark 13. Note that the pointwise detectability condition is not necessarily equivalent to nonlinear detectability. Due to the nonuniqueness of the SDC form, different choices may yield different state-dependent observability matrices[21] and thus different pointwise observability characteristics. The uniform detectability condition provided in Definition 3 will inherit this property as well (see [39] for rigorous establishment of connections between the pointwise controllability and true nonlinear controllability). Now it is possible to state the following lemma. Lemma 2. Consider the nonlinear system (3) – (4), the solution P (t) of the differential Riccati equation (14) and assume that the following conditions hold: 1) The system matrix A(x) is norm bounded, i.e., A < ∞; 2) The SDC parameterization is chosen such that the pair {C(x), A(x) + αI} is uniformly detectable according to Definition 3; 3) The initial condition P (0) of the differential Riccati equation (14) is positive definite. Then P (t) satisfies (22) in Theorem 1. The proof of Lemma 2 is in the Appendix. This lemma indicates that (23) can be replaced by the meaningful uniform detectability condition, which incorporates an important notion into picture. Remark 14. If the nonlinear system given by (3) – (4) is uniformly detectable in the sense of (42), then the pair {C(x), A(x) + αI} will be also uniformly detectable for the same bounded matrix value function provided that α < γ. Clearly, the reverse is always true.

5

Simulation examples

5.1

Second order nonlinear model

Consider the following continuous-time unforced nonlinear system with state x = [x1 (t) x2 (t)]T x˙ 1 (t) = 0.01x1 (t) − x2 (t) (43) x˙ 2 (t) = x1 (t) − 0.003x22 (t) y(t) = x1 (t).

(44)

This model can be parameterized as follows: x(t) ˙ = A(x)x y(t) = C(x)x. In which

(45)


A(x) =

0.01 1

−1 −0.003x2

363

(46)

and C(x) = [1 0] is a constant matrix. The statedependent observability matrix is

C(x) 1 0 O(x) = = . (47) C(x)A(x) 0.01 −1

function (29), when the proposed SDRE-based estimator is used. We see that V (e(t), t) is positive definite and decreasing for all t > 0. This verifies the convergence and stability of the proposed observer.

Since O(x) has full rank throughout R2 , the system is observable. It can be verified through some algebraic calculations that the SDC form fulfills the uniform detectability condition of Definition 3 with

−(0.01 + a) + 0.003x2 (48) Λ(x) = (b − 1) + 0.003(−a + 0.003x2 ) for some positive real numbers a and b. The pair {C(x), A(x) + αI} is also uniformly detectable for the same matrix value function given in (48), if the following inequality holds α2 + b . (49) α √ Note that it is necessary that α < b. For a certain parameter α > 0, it is always possible to choose a, b > 0 such that the foregoing inequality is met. Therefore, according to Lemma 2 the differential Riccati equation (14) has a bounded and positive definite solution. Obviously, the output matrix C is a Lipschitz matrix and satisfies (8) with any positive real number as kC . It follows from (46) that for all x, x ˜ ∈ R2

0 0 . (50) A(x) − A(˜ x) = ˜2 ) 0 −0.003(x2 − x 2α < a
0 ω T A(x) for all x ∈ Rn . Consider the dual control problem ¯T (x)ω + H T (x)v, −ω˙ = A

ω(T ) = h

(A2)

where T is the finite time horizon, h is given and v is the control signal. Applying a feedback control v(t) = ΛT (x)ω in (A2) yields ¯T (x) + H T (x)ΛT (x) ω, ω(T ) = h (A3) −ω˙ = A

(A7)

Thus, from (A4) it follows that

hT P (T )h hT

P (0) +

Q + Λ(x)2 2γ

h

(A8)

which indicates that for every T 0, the matrix P (T ) fulfills P (T ) P (0) +

Q + Λ2 = p¯ 2γ

(A9)

with Λ = supx∈Rn Λ(x), and thus P(t) is bounded. ¯ < ∞, the pair Similarly,! since Q is full rank and A(x) 1 ¯ A(x), Q 2 is uniformly controllable. Hence, there exists a bounded matrix valued function Γ(x) such that

1

λT A(x) + Q 2 Γ(x) λ δ λ2 ,

δ>0

(A10)

for all x ∈ Rn . Using an optimal control problem similar to (A2) along with (A10), it can be concluded that Π(t), i.e., the inverse of P (t), satisfies Π(T ) Π(0) +

Proof of Lemma 2

h2 . 2γ

H2 + Γ2 =p 2δ

(A11)

with Γ = supx∈Rn Γ(x). This implies the positive definiteness of the solution P (t).

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(A4)

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0

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(A5)

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Hossein Beikzadeh received his B. Sc. and M. Sc. degrees in electrical engineering from K. N. Toosi University of Technology, Iran in 2006 and 2009, respectively. He is currently a Ph. D. candidate in the Department of Electrical and Computer Engineering University of Alberta, Canada. His research interests include nonlinear systems analysis, nonlinear observer design, robust control, and sampled-data control. E-mail: [email protected] (Corresponding author) Hamid D. Taghirad received his B. Sc. degree in mechanical engineering from Sharif University of Technology, Iran in 1989, his master degree in mechanical engineering in 1993, and his Ph. D. degree in electrical engineering in 1997, both from McGill University, Canada. He is currently an associate professor with the Department of Electrical Engineering, and the Director of the Advanced Robotics and Automated System, ARAS Research Center at K. N. Toosi University of Technology, Iran. He was appointed as the director of the office of international scientific cooperation of the university in 2007. He is a senior member of the IEEE, and his publications include 2 books and more than 90 papers in international journals and conference proceedings. His research interests include robust and nonlinear control applied on robotic systems. E-mail: [email protected]