Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
ThB07.1
Adaptive Observer Design for Nonlinear Systems with Parametric Uncertainties in Unmeasured State Dynamics Øyvind Nistad Stamnes, Jing Zhou, Ole Morten Aamo and Glenn-Ole Kaasa Abstract— This paper presents an adaptive observer design for nonlinear systems that have parametric uncertainties in the unmeasured state dynamics. Without persistency of excitation (PE) the convergence of the state estimation error to zero is proved. Under PE conditions, uniform global asymptotic stability of the origin of the state and parameter error is shown. Furthermore, if the regressor is sufficiently smooth, uniform local asymptotic stability of the origin is proved. Two simulation examples are shown to illustrate the results.
I. I NTRODUCTION It is often the case that states/signals needed for control purposes are not measured. In these cases observers can be design to estimate the unmeasured states/signals. For deterministic linear time-invariant systems the Luenberger observer is a well known tool [1]. In the stochastic setting the Kalman filter [2] is a popular tool. If in addition to unmeasured states there are parametric uncertainties an observer can be made robust by adapting to these uncertainties. These observers are often called adaptive observers. Methods for the design of adaptive observers are often based on Lyapunov like functions containing a parameter error term. Under a strictly positive real (SPR) condition [3] the derivative of such functions can be found to be negative semidefinite, further use of Barb˘alat’s lemma guarantees that the state estimation error converges to zero. For linear time invariant systems, early results can be found in [4], [5]. A complete theory on adaptive observers for linear systems can be found in e.g. [3] or [6]. For nonlinear systems, paralleling the work done in the non-adaptive case [7], [8], [9], [10], adaptive observers have been developed using transformation techniques and exploiting certain structures/canonical forms. An early result can be found in [11] where adaptive observers were developed for systems in the so called adaptive observer canonical form. It was also shown that several useful systems can be transformed into this form. Building on these results, [12] characterized all systems that are transformable into the adaptive observer canonical form through the use of filtered transformations (possibly dependent on unknown parameters). If the filtered transformations depend on unknown parameters, persistency of excitation (PE) is needed to reconstruct the actual state estimate. These approaches are well presented in [13]. Ø. Stamnes, J. Zhou and O.M. Aamo are with Department of Engineering Cybernetics, Norwegian University of Science and Technology
[email protected]
Without relying on geometric techniques the results in [14] apply to a class of multiple-input multiple-output nonlinear system satisfying an SPR-like condition and a Lipschitz condition on the nonlinearities. In the spirit of [11] and [13], [15] collects most of the existing results into one adaptive observer form for which design procedures are available. If parameter identification is desired it is necessary to impose PE conditions on the regressor. When proper PE conditions are assumed, uniform global asymptotic and/or uniform local exponential stability can be concluded. As pointed out in the introduction of [16], these stability results imply robustness towards modelling errors. For linear timevarying systems results on PE can be found in [6]. More recently [16], [17] give PE conditions and stability results for nonlinear time-varying systems. In this paper we derive an adaptive observer for a class of systems with parametric uncertainty in the unmeasured state dynamics. The class of systems considered cannot be dealt with using any of the general schemes found in [13], [14], [15]. The only general approach found, that can deal with such systems, is an adaptive high-gain observer found in [18] which requires a priori known bounds on the states so that the nonlinearities can be dominated using high-gains. Opposed to this, the observer presented here takes into account that if the nonlinearities satisfy a sector condition, they need not be dominated using high gains. A key part of the design is a coordinate change that makes it possible to implement the adaptation law even when it is driven by an unmeasured error signal. II. O BSERVER D ESIGN We consider the following class of systems (1a)
z˙ = A22 z + A21 y + g (t, u, y) + BθT φ (t, y, z) .
(1b)
Where y ∈ Rny is measured and z ∈ Rnz is not measured. θ ∈ Rnθ is a vector of unknown parameters. We make the following assumptions are bounded. Assumption 1: φ (t, y, z) , dφ(t,y,z) dt Assumption 2: The right hand side of (1) is locally Lipschitz in y, z uniformly in t. Assumption 3: y, z are bounded. Consider the change of variables
G-O. Kaasa is with Modelling, Control and Flow Assurance, StatoilHydro Research Centre Porsgrunn
[email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
y˙ = A11 y + A12 z + f (t, u, y) ,
4414
ξ = z + Ly,
(2)
ThB07.1 where L ∈ Rnz ×ny . We get y˙ = (A11 − A12 L) y + A12 ξ + f (t, u, y) , (3) ˙ξ = (A21 + LA11 − A22 L − LA12 L) y + (A22 + LA12 ) ξ + g (t, u, y) + Lf (t, u, y) + BθT φ (t, y, ξ − Ly) .
(4)
At this point, design an observer as yˆ˙ = (A11 − A12 L) yˆ + A12 ξˆ + f (t, u, y) + K1 (y − yˆ) , (5) ˙ ξˆ = (A21 + LA11 − A22 L − LA12 L) yˆ + (A22 + LA12 ) ξˆ + g (t, u, y) + Lf (t, u, y) + B θˆT φ t, y, ξˆ − Ly + K2 (y − yˆ) ,
(6)
ˆ the parameter where K1 ∈ Rny ×ny , K2 ∈ Rnz ×ny and θ, estimate, will be defined later. Define the state estimation errors as y˜ = y − yˆ, ξ˜ = ξ − ξˆ for which the error dynamics are ˜ y˜˙ = (A11 − A12 L − K1 ) y˜ + A12 ξ, ˙ ξ˜ = (A21 + LA11 − A22 L − LA12 L − K2 ) y˜ + (A22 + LA12 ) ξ˜ + BθT φ (t, y, ξ − Ly) − B θˆT φ t, y, ξˆ − Ly .
(7a)
(7b)
Furthermore, motivated by [19], assume that the sector condition ξ˜T Pξ BθT φ (t, y, ξ − Ly) − φ t, y, ξˆ − Ly ≤ 0, (15) is satisfied. Now, using (14), (15) and (11) we get
2
V˙ 1 ≤ −λ1 y˜, ξ˜ + 2ξ˜T Pξ B θ˜T φ t, y, ξˆ − Ly . (16)
For the case without parametric uncertainties (θ˜ = 0), we summarize our findings in the following theorem. Theorem 4: Suppose that θ˜ = 0 and that there exists (Qξ , Pξ , L) that satisfy (13) and (15). Select K1 such that (12) and (14) are satisfied and K2 as defined in (8). Then, the origin of the error dynamics (7) is uniformly globally exponentially stable. Proof: The proof follows by applying [20, Th. 4.10] to (10) and (16) with θ˜ = 0. Remark 5: Detectability of (A22 , A12 ) is a necessary condition for the existence of a solution (Pξ , Qξ ) of (13) that satisfies (15), but not sufficient. Remark 6: Given (Qξ , Pξ , L) that satisfy (13) and (15), let K1 = A11 − A12 L − γK with K Hurwitz and γ > 0. With this choice, it is possible to select a Qy , and adjust γ until Py given by (12) is such that (14) is satisfied.
Selecting
K2 = A21 + LA11 − A22 L − LA12 L,
III. A DAPTIVE L AW (8)
we obtain ˜ y˜˙ = (A11 − A12 L − K1 ) y˜ + A12 ξ, ˙ ξ˜ = (A22 + LA12 ) ξ˜ + B θ˜T φ t, y, ξˆ − Ly + BθT φ (t, y, ξ − Ly) − φ t, y, ξˆ − Ly ,
(9a)
(9b)
To make the observer robust towards parametric uncertainties we will extend the design with an adaptation law. Based on a Lyapunov analysis we design an adaptation law that is ˜ Then, we show that driven by the unmeasured state error ξ. the adaptation law can in fact be implemented using a change of coordinates. Using V1 from (10), we append a term that is quadratic in the parameter error θ˜ which gives 1 ˜ V2 = V1 + θ˜T Γ−1 θ, 2
ˆ Consider the Lyapunov function candidate where θ˜ = θ − θ. ˜ V1 = y˜T Py y˜ + ξ˜T Pξ ξ,
(10)
where P∗ = P∗T > 0. Differentiating with respect to time gives V˙ 1 = −˜ y T Qy y˜ − ξ˜T Qξ ξ˜ + 2˜ y T Py A12 ξ˜ + 2ξ˜T Pξ B θ˜T φ t, y, ξˆ − Ly + 2ξ˜T Pξ BθT φ (t, y, ξ − Ly) − φ t, y, ξˆ − Ly , (11) where Q∗ = QT∗ > 0 satisfy the Lyapunov equations
with Γ = ΓT > 0. Using (16) we find
2
V˙ 2 ≤ −λ1 y˜, ξ˜ + 2ξ˜T Pξ B θ˜T φ t, y, ξˆ − Ly ˙ ˜ + 2θ˜T Γ−1 θ.
(17)
(18)
By choosing
we get
˙ ˙ θˆ = −θ˜ = Γφ t, y, ξˆ − Ly B T Pξ ξ˜
2
V˙ 2 ≤ −λ1 y˜, ξ˜ .
(19)
(20) T −Qy = (A11 − A12 L − K1 ) Py + Py (A11 − A12 L − K1 ) , (12) The stability result for this case is summarized in the theorem T −Qξ = (A22 + LA12 ) Pξ + Pξ (A22 + LA12 ) . (13) below. Theorem 7: The origin of the error dynamics (7) and As Qy can be chosen arbitrarily large (12)–(13) imply that (19) is globally uniformly bounded and solutions satisfy there exist λ1 > 0 such that limt→∞ y˜, ξ˜ = 0. Proof: The result follows by application of [21, Th. Qy −Py A12 ≥ λ1 I. (14) A.8], using (17) and (20). −AT12 Py Qξ
4415
ThB07.1 The adaptive law (19) cannot be implemented as-is, since ξ˜ is an unknown signal. We will solve this problem now. Define σ = θ + η t, y, ξˆ , (21) where η t, y, ξˆ is a function of known/measured signals to be designed to give σ desired dynamics. Differentiating with respect to time and inserting (3) gives ∂η ∂η ∂η ˆ˙ + ((A11 − A12 L) y + A12 ξ + f (t, u, y))+ ξ. ∂t ∂y ∂ ξˆ (22) Now, let the parameter estimate be given as θˆ = σ ˆ − η t, y, ξˆ , (23) σ˙ =
Theorem 9: For the system, observer and adaptive law summarized in Table I we have the following properties. ˜ = 0, (y − yˆ, z − zˆ) = 0 is UGES. • For θ ˜ ˜ θ˜ = 0 is globally uniformly bounded • For θ 6= 0, y ˜, ξ, and limt→∞ (y − yˆ, z − zˆ) = 0. Furthermore the origin is UGAS if and only if the PE condition (27) is satisfied. dφ(t,p) exists and is • If in addition φp (t, z(t)) = dp p=z(t)
bounded, then the origin is ULES. Proof: The proof can be found in the appendix.
TABLE I S UMMARY OF OBSERVER AND SYSTEM y˙ = A11 y + A12 z + f (t, u, y) , z˙ = A22 z + A21 y + g (t, u, y) + BθT φ (t, y, z) .
System
with
∂η ∂η ∂η ˆ˙ σ ˆ˙ = + (A11 − A12 L) y + A12 ξˆ + f (t, u, y) + ξ. ˆ ∂t ∂y ∂ξ (24) Denoting σ ˜ = σ−σ ˆ , and subtracting (24) from (22) we get ∂η ˜ A12 ξ. σ ˜˙ = ∂y
Observer
(25) Adaptive Law
In view of (21) and (23), θ˜ and σ ˜ are identical, so from (25) we see that by finding η t, y, ξˆ such that ∂η t, y, ξˆ A12 = −Γφ t, y, ξˆ − Ly B T Pξ , (26) ∂y
θ˜ gets the desired dynamics of (19). Equations (23) and (24) represent an implementable version of (19). Remark 8: To the authors best knowledge, using (23) and (24) to implement (19) is a new result that enables adaptive observers to be designed for a larger class of systems than previously dealt with. The proposed adaptive observer is summarized in Table I. Note that the final unmeasured state estimate, zˆ, is generated using yˆ, thus giving an estimate that is more robust to measurement noise than a typical reduced order observer (where zˆ would be generated by directly using the noise affected measurement of y).
Initial Values Design Variables
yˆ˙ = (A11 − A12 L) yˆ + A12 ξˆ + f (t, u, y) +K1 (y − yˆ) zˆ = ξˆ − Lˆ y ˙ ξˆ = (A21 + LA11 − A22 L − LA12 L) yˆ ˆ g (t, u, y) + Lf (t, u, y) + (A22 + LA12 ) ξ + +B θˆT φ t, y, ξˆ − Ly + K2 (y − yˆ) , θˆ = σ ˆ − η t, y, ξˆ σ ˆ˙ =
∂η ∂t
+
˙ ∂η ˆ ξ ∂ ξˆ
(A11 − A12 L) y + A12 ξˆ + f (t, u, y) + ∂η ∂y η t, y, ξˆ s.t. ∂η (t,y,ξˆ) A12 = −Γφ t, y, ξˆ − Ly B T Pξ ∂y yˆ(0) = y0 , ξˆ (0) = z0 + Ly0 , σ ˆ (0) = θ0 + η (t0 , y0 , z0 + Ly0 ) K1 , K2 , L to satisfy Theorem 4, Γ = ΓT > 0.
V. E XAMPLES The following examples illustrate the use of the adaptive observer in cases that do not fall into the general design procedures found in [13], [14], [15]. A. Example 1, Drilling System In [23] a reduced order adaptive observer was derived for the model of a managed pressure drilling system. The system is described by
IV. P ERSISTENCY OF E XCITATION Based on results found in [22] it is possible to prove uniform global asymptotic stability (UGAS) and uniform local exponential stability (ULES) of the origin of (9) and (19) under the PE condition
Vd p˙p = qpump − qbit , βd Va p˙c = qbit + qback − qchoke , βa M q˙bit = pp − pc − Fd |qbit | qbit − Fa |qbit | qbit + (ρd − ρa ) ghbit ,
(28a) (28b)
(28c)
where pp and pc are the pump and choke pressures. qpump , q t+T bit , qback and qchoke are the volume flows through the mud Z T T pump, bit, back pressure pump and the choke. V∗ , β∗ , F∗ φ(τ, y (τ ) , z(τ ))B Bφ (τ, y (τ ) , z(τ ))dτ ≥ µI ∀t ∈ R. and ρ∗ are volume, bulk modulus, lumped friction factor t (27) and density. The subscripts d and a refer to drill string and annulus respectively. g is gravitational acceleration and To be precise the UGAS definition we will use can be found hbit is the vertical depth of the bit. M can be viewed as in [17, Definition 1 and 2], while the ULES definition can a geometric density coefficient, see [23] for details about be found in [20, Def. 4.5]. The main result is stated in the the model. The objective is to estimate the states under the following theorem. following assumptions:
4416
ThB07.1 qpump , qback ,qchoke , Vd , Va , hbit , h˙ bit are bounded and measured external signals. • M, Fa , ρa are unknown positive constants. • pp , pc and qbit are bounded. The system (28) can be put into the form (1) by defining pp 0 0 y= , z = qbit , A11 = , pc 0 0 # " − Vβdd , A21 = 0 0 , A22 = 0, A12 = βa
•
,
bar
bar l min
0 0 2000 0 0
, φ (t, y, z) =
Applying Theorem 9 condition (15) reduces to
1 −1 y . − |z| z hbit (t)
−˜ z Pξ Bθ2 (|z| z − |ˆ z | zˆ) ≤ 0 =⇒ Pξ > 0 as θ2 ≥ 0. From (13) we get the condition βd βa −LA12 = − −l1 > 0. + l2 Vd Va
(29)
(30)
Constraining K1 to K1 = −A12 L +
k1 0
0 k2
,
Py = λ y
1 2k1
0
0 1 2k2
,
(31)
(33)
is a solution. Finally the condition (14) can now be written as λy βd
2k1 Vd λ β
λy I
λy βd 2k1 Vd
λ β
− 2ky2 Vaa
− 2ky2 Vaa Qξ
≥ λ1 I,
(34)
which can be satisfied for any k1 , k2 , λ1 > 0 by choosing λy > λ1 and Qξ sufficiently large. Let Γ = diag (γ1 , γ2 , γ3 ) , a solution η t, y, ξˆ satisfying (26) is γ P V Va 2 d 2 1 ξ (35) y1 − y2 , η1 t, y, ξˆ = 2 βd βa 3 γ2 Pξ ξˆ − Ly , (36) η2 t, y, ξˆ = 3 l1 Vβdd − l2 Vβaa Vd η3 t, y, ξˆ = −γ3 Pξ α − y1 + (37) β d Va (1 − α) y2 hbit (t) , α ∈ [0, 1] . (38) βa
100
200
300
100
200
300 400 q pump q choke − q back
100
200 s
300
M ˆ M 100
200
300
400 Fa Fˆa
100
200
300
400 ρa
0.4 −0.1 0 0.02
400
States, state estimates and inputs, drilling example.
5
0 0 0.8
400 q bit qˆbit
10
ρˆa 0.01 0
such that (A12 L + K1 ) and Py in (12) is diagonal and Qy = λy I gives k1 0 λy I = 2 Py , (32) 0 k2 for which
Fig. 1.
10 kg m4
θ=
8
1 M Fd +Fa M (ρd −ρa )g M
400 pc pˆc
1000
B = 1,
300
2000
106 bar s2 m6
g (t, u, y) = 0,
(qback (t) − qchoke (t))
200
20
0 0 4000
l min
βa Va
100
105 kg m3
f (t, u, y) =
#
βd Vd qpump (t)
pˆp
100 0 0 40
Va
"
pp
200
Fig. 2.
100
200 s
300
400
Parameters and parameter estimates, drilling example.
A simulation was performed using the parameter values Vd = 3 28.3 [m3 ], Va = h 96.1 [m i], βa = βd = 14000 h [bar], ρa2 = i kg s 5 , ρd = 0.0125 10 × m3 , Fd = 0.1650 106 × bar 6 m i h i h kg bar s2 8 6 Fa = 0.0208 10 × m6 , Ma = 1.6009 10 × m4 , h i kg Md = 5.7296 108 × m 4 , hbit = 2000 [m] . The design
variables of the observer where chosen to be k1 = 10−2 Vβdd , k2 = 10−2 Vβaa , L = 10−3 10−3 , K2 = −LA12 L, Γ = P1ξ diag( 10−4 5 × 106 10−7 ), α = 0.5, and Pξ = 1.4326 × 104 . At ts = 100s the observer was started with ˆ (ts ) = 1.5M , yˆ(ts ) = 1.5y(ts ), qˆbit (ts ) = 1.5qbit (ts ), M ˆ Fa (ts ) = 2Fa and ρˆa (ts ) = 1.5ρa . From Fig. 1–2 we see that the state estimation error converges to zero while the parameter error does not. This is in accordance with Theorem 7. B. Example 2, Nonlinear Mass-Spring-Damper Consider the nonlinear mass-spring-damper system y˙ = z,
(39)
T
(40)
z˙ = θ φ (t, y, z) ,
where y ∈ R is position and z ∈ R is velocity. The regressor and unknown parameters are k −y m b , (41) φ (t, y, z) = − |z| z , θ = m 1 u (t) m
4417
ThB07.1
5 0 −5 0 5
10
20
30
40
0 −5 0 50
10
20
10
20
y
2
yˆ
0
50 z zˆ
30
40
50 u
30
40
50
−2 0 3
10
20
30
40
θ2 θˆ2
2 1 0 5
10
20
0 0
10
20
30
40
50 θ3 θˆ3
30
40
50
0 −50 0
s
s Fig. 3.
θ1 ˆ1 θ50
Fig. 4. Parameters and parameter estimates, mass-spring-damper example.
States, state estimates and input, mass-spring-damper example.
with positive constants k, m, b, being the spring stiffness, mass, and nonlinear damping coefficient. This system obviously fits into the form (1) thus an adaptive observer can be designed. An adaptive observer is yˆ˙ = −Lˆ y + ξˆ + K1 (y − yˆ) , (42) zˆ = ξˆ − Lˆ y, (43) ˙ ξˆ = −L2 yˆ + Lξˆ + θˆT φ t, y, ξˆ − Ly + K2 (y − yˆ) , (44) θˆ = σ ˆ − η t, y, ξˆ , (45) ∂η ∂η ∂η ˆ˙ ξ, (46) σ ˆ˙ = −Ly + ξˆ + + ∂t ∂y ∂ ξˆ
Considering the cascaded structure of(9) we will first ˜ θ˜ = 0 and then analyse the stability properties of ξ, ˜ θ˜ = 0. We conclude with the stability properties of y˜, ξ, will first prove UGAS based on [22, Section 6], later ULES follows from analysis of the linearized system. A. UGAS ξ˜ Defining x = ˜ we can write (9b) and (19) as θ A(t, x) + B(t, x) , x˙ = F (t, x) := C(t, x)
with
A(t, x) = (A22 + LA12 ) ξ˜ + BθT φ (t, z (t)) − φ t, z (t) − ξ˜ , ˜ B(t, x) = BφT t, z (t) − ξ˜ θ,
with
γ P 1 ξ 2 η1 t, y, ξˆ = y , η3 t, y, ξˆ = −γ3 uyPξ , (47) 2 3 −γ2 ξˆ − Ly Pξ . (48) η2 t, y, ξˆ = 3L Fig. 3–4 show simulation results using the parameter values k = 0.5, m = 1, b = 1, the observer gains L = −0.1, K1 = 0.9, K2 = −L2 , γ1 = 0.2000, γ2 = 0.0100, γ3 = 0.0020 and Pξ = 5. The input was u(t) = 20 sin(2πt) + π ). The observer was started at ts = 10s 10 sin(πt + 2.5 with initial conditions yˆ(ts ) = 1.5y(ts ), zˆ(ts ) = 1.5z(ts ), ˆ s ) = 1.2θ. In this case the input is PE and we see that θ(t in addition to the state estimation error converging to zero the parameter estimates converge to their true values as well. This is in accordance with Theorem 9. VI. C ONCLUSIONS An adaptive observer design for a class of nonlinear system with parametric uncertainties in the unmeasured state dynamics has been presented. Stability and convergence of the error has been analysed with and without PE. A PPENDIX Given that the PE condition (27) is satisfied we will prove UGAS and ULES of the origin of (9) and (19). Note that (27) is equivalent, and follows, from the Uδ-PE property given in [17, Property 3, p191].
(49)
˜ ˜ C(t, x) = −Γφ(t, z(t) − ξ)B Pξ ξ, T
(50) (51) (52)
where we have used φ (t, y (t) , ξ (t) − Ly (t)) = φ (t, z (t)) , φ t, y (t) , ξˆ (t) − Ly (t) = φ t, z (t) − ξ˜ .
(53) (54)
The system is now in the form found in [22]. To prove UGAS we first note the following. • From the previous analysis based on (17) and (20), x = 0 is UGS. • A(t, 0) ≡ 0, B(t, 0) = 0, C(t, 0) = 0. ˜ := B(t, x)| ˜ = BφT (t, z(t))θ˜ • B0 (t, θ) ξ=0 • Equations (17) and (20) satisfies [22, Assumption 12]. • By Assumption 1, 3 and Theorem 7 we have that for each ( ∆ ≥ 0 there exists bM > 0 such that )
∂B (t, θ) ˜
0 ˜ ∂B0 (t, θ)
˜ max B0 (t, θ)
≤ bm .
,
, ˜ ˜
∂t ∂θ |θ|≤∆
(55)
From (51) and the definition of B0 we have for θ˜ ≤ ∆
˜
B(t, x) − B0 (t, θ)
≤
∆ kBk φ t, z(t) − ξ˜ − φ(t, z(t)) . (56)
4418
ThB07.1 Note that as ξ˜ and φ(t, z(t)) are bounded there exists a non-decreasing function ρ ξ˜ such that
˜ t − φ(x(t), t) ≤ ρ ξ˜ . Inserting
φ x(t) − ξ,
this into (56) gives ˜ ≤ ρ1 ξ˜ , (57) |B(t, x) − B0 (t, θ)| where ρ1 ξ˜ = ∆ kBk ρ ξ˜ . From (50), (52), Assumption 3 and Theorem 7 we have ˜ max {|A(t, x)| , |C(t, x)|} ≤ ρ2 (|ξ|).
˜ |θ|≤∆
(58)
where ρ2 is non-decreasing. Equations (55), (57) and (58) together with the fact that A, B, C are locally Lipschitz in x, uniformly in t, by Assumption 2, satisfy [22, Assumption 13]. Applying [22, Theorem 3] to the system (49) we conclude with the following theorem. Theorem 10: The origin of (49) is UGAS if and only if the PE condition (27) is satisfied. B. ULES The proof of ULES of the origin is based on analysis of the linearization of (49) around the origin. For the linearized system to exist we assume that φp (t, z(t)) = dφ(t,p) dp exists and is bounded. The linearized system is
p=z(t)
∆x˙ = D(t)∆x, (59) A22 + LA12 + BθT φp (t, z(t)) BφT (t, z (t)) D(t) = . −Γφ(t, z(t), t)B T Pξ 0
Theorem 11: The origin of (59) is uniformly globally exponentially stable if and only if condition (27) is satisfied. Proof: Consider the positive definite function
Pξ with P∆x = 0 time we have
V = ∆xT P∆x ∆x (60) 0 , differentiating with respect to Γ−1
˜ V˙ = −∆ξ˜T Qξ ∆ξ˜ + 2∆ξ˜T Pξ BθT φp (t, z(t))∆ξ, T ˜ ≤ −∆ξ˜ Qξ ∆ξ, (61) where Qξ follows from equation (13). Note that BθT φp (t, z(t)) ≤ 0 from condition (15). From (60) and (61) we conclude that ∆x = 0 is UGS. Noticing that the system (59) is of the same structure as (49) and that all the steps in the UGAS proof are the same we can conclude UGAS for ∆x = 0 under the same PE condition (27) as before. Since UGAS of a linear time-varying system implies UGES, [20, Theorem 4.11] we have that the origin of (59) is UGES. Lemma 12: If φp (t, z(t)) exists and is bounded. The origin of (49) is ULES if and only if condition (27) is satisfied. (t,x) Proof: Noticing that the Jacobian matrix ∂F∂x of (49) exists and is bounded and Lipschitz on E = {x ∈ Rnz +nθ | kxk2 < r} we apply [20, Theorem 4.13] together with Theorem 11.
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