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Proceedings of the 23rd CANCAM
NONLINEAR ADAPTIVE OBSERVER DESIGN FOR AN ELECTROMECHANICAL ROTATIVE PLANT
K. Khayati Department of Mechanical & Aerospace Engineering Royal Military College of Canada Kingston, ON, Canada [email protected]
J. Zhu Department of Mechanical & Industrial Engineering Concordia University Montreal, QC, Canada z [email protected] server. This observer estimates the full-state variables in the presence of uncertain parameters possibly depending on the input and state variables. The stability of the algorithm is guaranteed when at least some of the measured outputs are such that the transfer matrix from the unknown parameters to these outputs is dissipative [9]. This particular condition on the dynamic representation remains very weak and is rarely reached in general. Moreover, there is no straightforward way of choosing the observer gains to satisfy this condition [10]. In this paper, inspired by [9], we intend to improve the procedure by introducing a new adaptation law that eliminates in particular the dissipativity condition that [9] requires. The stability condition of the proposed adaptive observer will be presented using the LMI framework.
ABSTRACT
This paper deals with the design of an adaptive observer that can estimate both the states and the parameters of a class of nonlinear systems. The purpose of the proposed method is to eliminate the restrictive dissipativity condition from the stability proof that is present in many previous works presented in the literature. The observer gain calculus is cast as a linear matrix inequality (LMI) feasibility problem. We aim to apply the proposed algorithm to estimate the unknown parameters of a second order electromechanical rotary model. The appeal of this proven theoretical design is further demonstrated numerically with a comparative design. Keywords: Nonlinear systems, Adaptive observer, Parameter estimation, LMI.
The design of the adaptive observer is applied to estimate the full states and to identify all parameters for a class of nonlinear dynamic systems based on a set of feasible assumptions. In particular, we target the estimation of the model parameters of a second order electromechanical system which are considered unknown a priori. This paper is organized as follows. In Section 2, we describe the nonlinear model statement setting forth the required assumptions that are used in the estimation procedure. Then, we describe the observer dynamics and the adaptation law and study the convergence of the estimated model states and parameters in Section 3. Section 4 is dedicated to the presentation of the electromechanical model to which we intend to apply the proposed adaptive observer. Finally, numerical and comparative results are presented in Section 5, while Section 6 concludes with recommendations.
INTRODUCTION The observer design for linear and nonlinear systems has been widely investigated during the last decades (e.g. [1, 2] and references cited therein). In [3, 4], observers of global exponential convergence have been introduced. Adaptive observers estimating both the states and parameters of linear systems [5] and nonlinear systems [6, 7, 8] were developed using the input and output signals. The authors provided constructive methods for the design of global adaptive observers enabling the estimation of the unknown parameters with a common condition of persistent excitation [7, 9]. In [9], the authors considered nonlinear dynamics with linearly dependent parameters to design a systematic approach of an adaptive ob-
631
PROBLEM STATEMENT AND ASSUMPTIONS
A. Theorem
Consider the nonlinear multiple-input and multiple-output (MIMO) model described by the following adaptive observer form:
Under assumptions 1)–5), if there exist matrices P = P T > 0 in R2n×2n and W ∈ R2n×n such that P A + AT P − W C − C T W T + δI2n P < 0 (12) P − δ1 I2n
x˙ 1 x˙ 2 y
= x2 + F01 (u, x1 ) = A1 x1 + A2 x2 + F02 (u, x) + F (u, x)θ
(1) (2)
= x1
(3)
x1 , u ∈ R m , θ ∈ R q , A1 , x2 n A2 in Rn×n , F01 (u, x1 ), F02 (u, x) nonlinear functions in R 0n and F (u, x) ∈ Rn×q , respectively. Let B = , where In In and 0n are the identity and null-square matrices of order n respectively. For the forthcoming analysis, we need the following assumptions:
with x1 , x2 and y in Rn , x =
then the adaptive observer (9)–(11), with the observer gain matrix L = P −1 W , is asymptotically stable, x˜ = x − xˆ → 0 and θ˜ = θ − θˆ → 0 as t → ∞. B. Proof The estimation error dynamics are obtained from (1)–(3) and (9): x ˜˙ =
(A − LC)˜ x + B[F02 (u, x) − F02 (u, x ˆ) + ˜ F (u, x)θ − F (u, x ˆ)θ + F (u, x ˆ)θ] (13)
1. The vector of unknown constant parameters θ is bounded, with kθk ≤ α (4)
Using the adaptation law (10)–(11), we have
2. The functions F02 and F are Lipschitz in x, with
As CB = 0, we have
kF02 (u, x) − F02 (u, x ˆ)k ≤ β0 kx − x ˆk
(5)
kF (u, x) − F (u, x ˆ)k ≤ βkx − x ˆk
(6)
and F (u, x) is continuously bounded; 3. A gain matrix L ∈ R2n×n can be chosen such that P (A − LC) + (A − LC)T P + δ(P P + In ) < 0 (7) 0 n In for any matrix P = P T > 0, with A = , A1 A2 C = In 0n and δ = β0 + αβ > 0;
4. The input vector u is continuously differentiable;
5. There exist positive scalars α0 , α1 and t0 , such that ∀t Z t0 +t α0 I q ≤ F T (u, x)B T BF (u, x)dτ ≤ α1 Iq (8) t
These hypotheses are widely acceptable in practice and can characterize adequately most real devices.
˙ θˆ = ΓF T (u, x ˆ)B T P [(C T + BCL)C x ˜ + BC x ˜˙ ]
Cx ˜˙ = C(A − LC)˜ x
˙ θ˜ = −ΓF T (u, x ˆ)B T P [C T C + BCA]˜ x = −ΓF T (u, x ˆ)B T P x ˜
(16)
To investigate the stability, given P = P T > 0 and Γ = Γ > 0, consider the Lyapunov candidate function V (t) = ˜ We have x ˜T P x ˜ + θ˜T Γ−1 θ. T
V˙
=
2˜ xT P x ˜˙ + 2θ˜T Γ−1 θ˜˙
=
2˜ xT P {(A − LC)˜ x + B[F02 (u, x) − F02 (u, x ˆ) + T −1 ˜ ˜ F (u, x)θ − F (u, x ˆ)θ + F (u, x ˆ)θ]} − 2θ Γ ·
[ΓF T (u, x ˆ)B T P x ˜] T ≤ x ˜ [P (A − LC) + (A − LC)T P ]˜ x+ (17)
Notice the inequality
Consider the following full-order observer xˆ˙ = Aˆ x + F0 (u, x1 , x ˆ) + BF (u, x ˆ)θˆ + L(y − C x ˆ)
(15)
˙ ˆ˙ Thus, we obtain Based on assumption 1), we have θ˜ = −θ. the adaptation error dynamics
2β0 kBkkP x ˜kk˜ xk + 2αβkBkkP x ˜kk˜ xk ADAPTIVE OBSERVER DESIGN
(14)
2kP x ˜kk˜ xk ≤ x˜T P P x ˜+x ˜T x˜ (9)
and adaptation law ˙ θ˘ = Γ[F T (u, x ˆ)B T P (C T + BCL) − F˙ T (u, u, ˙ x ˆ)B T P B] · (y − C x ˆ) (10) T T ˆ ˘ θ = θ + ΓF (u, x ˆ)B P B(y − C x ˆ) (11) F01 (u, x1 ) where F0 (u, x1 , x ˆ) = , F˙ (u, u, ˙ x ˆ) is the total F02 (u, x ˆ) T q×q time derivative of F (u, x ˆ) and Γ = Γ > 0 in R .
632
(18)
and δ = β0 + αβ > 0. Thus, we obtain V˙ ≤ x ˜T [P (A − LC) + (A − LC)T P + δ(P P + I2n )]˜ x (19) The right member of the inequality (19) is negative definite if P (A − LC) + (A − LC)T P + δ(P P + I2n ) < 0
(20)
that is, ∃Q = QT > 0 such that P (A − LC) + (A − LC)T P + δ(P P + I2n ) = −Q (21)
Then, V˙ < −λmin (Q)k˜ xk2 . This implies x˜ ∈ L∞ (i.e. timefunctions of finite ∞-norm) and θ˜ ∈ L∞ , and then V (t) ∈ L∞ . Integrating (19) V (t) ≤ V (0) − λmin (Q)
Z
Note that the total time derivative of the function F (u, x ˆ) given by:
t 2
k˜ xk dt
(22)
0
Since V (0) is finite, we obtain x ˜ ∈ L2 (i.e. finite 2-norm vector-function). From (13), we have x˜˙ ∈ L∞ . Therefore, by applying theorem 8.4 of [11] based on Barbalat’s Lemma, ˆ → x ˜ → 0. Also, from (13), we have B[F (u, x)θ − F (u, x ˆ)θ] 0. Using the inequality (6), in assumption 2), and noting that ˆ → 0 as t → ∞. x ˆ → x will lead to BF (u, x)(θ − θ) Hence, F T (u, x)B T BF (u, x)θ˜ → 0 as t → ∞. Define Rt G(t0 ) = t 0 F T (u, x)B T BF (u, x)dτ . We apply the analysis of persistency of excitation, as given in [7], to obtain ˜ → 0. Using integration by parts, we have θ(t) Z
t0 +t t
˜ )dτ = G(t0 + t) · F T (u, x)B T BF (u, x)θ(τ Z t0 +t ˜ 0 + t) − G(t)θ(t) ˜ − ˜˙ )dτ θ(t G(τ )θ(τ
(23)
As G(t) = 0, we obtain t0 +t
˜ )dτ = G(t0 + t) · F T (u, x)B T BF (u, x)θ(τ
t
˜ 0 + t) − θ(t
Z
t0 +t
˜˙ )dτ G(τ )θ(τ
(24)
t
Since F T (u, x)B T BF (u, x)θ˜ → 0, then, for any finite t, we have Z t0 +t ˜ )dτ → 0 F T (u, x)B T BF (u, x)θ(τ (25) t
Based on assumption 2) and since x ˜ → 0, from (16), we have ˙θ → 0, and then ˜ Z
t0 +t
˜˙ )dτ → 0 as t → ∞ G(τ )θ(τ
(26)
˜ 0 + t) → 0 as t → ∞ G(t0 + t)θ(t
(27)
t
Thus, we obtain
By assumption 5), i.e. γ0 Iq ≤ G(t0 + t) ≤ γ1 Iq for γ0 > 0 ˜ 0 + t) → 0, implying θ(t) ˜ → 0 [7]. and γ1 > 0, we obtain θ(t Finally, we use the Schur complement [12] and the change of variable W = P L to transform the nonlinear inequality (20) into the LMI (12) in decision variables P and W . C. Discussions The condition (8) is necessary to guarantee the convergence of the parameter estimates given by the adaptive observer scheme (9)–(11) to the true parameter values [7, 8].
633
(28)
can be implemented by using any continuously bounded input u and the definition of x ˆ˙ given by the right member of (9). In general, the synthesis of the adaptive observer gain, stated by the problem (12), can be solved by any efficient convex feasibility algorithm [12] using e.g. the LMI control toolbox in Matlab. ELECTROMECHANICAL ROTARY DYNAMICS We address the problem of designing a global adaptive observer for a second order nonlinear electromechanical dynamics. We consider the model of a helicopter-based flight engine simulator mounted on a fixed base with one propeller that is driven by a DC motor. The propeller controls the elevation of the helicopter nose about the pitch axis. The pitch dynamics is described by: Je ψ¨ = kp u − bp ψ˙ − mgl cos ψ
t
Z
dF ∂F ∂F (u, u, ˙ xˆ) = (u, x ˆ)x ˆ˙ + (u, x ˆ)u˙ dt ∂x ˆ ∂u
(29)
where ψ is the pitch angle, u the input motor voltage, Je the moment of inertia about pitch pivot, kp the thrust torque constant acting on pitch axis from the motor-propeller, bp the viscous damping coefficient about the pitch axis, m the total moving mass of the helicopter, l the center of mass length along the helicopter body from the pitch axis and g the gravity acceleration. All parameters of (29) are unknown. The nonlinear term, that is mgl cos ψ, represents the gravitational torque. The remaining terms in (29) are the acceleration torque, the torque generated by the motor on the pitch axis and the viscous rotary friction acting about the pitch axis respectively. Only the pitch angle is measurable. Using the states x1 = ψ ˙ the rotary dynamics can be written in the state and x2 = ψ, 0 1 space representation (1)–(3), with A = , B = 0 0 0 , C = 1 0 . The nonlinear terms are F0 (u, x) = 1 0 and F (u, x) = − cos x1 −x2 u . The components bp kp of the unknown vector θ are θ1 = mgl Je , θ2 = Je and θ3 = Je respectively. In the following section, we intend to identify these parameters. SIMULATION RESULTS Computer simulations of the adaptive observer algorithm for the second order rotary system are developed. The parameters characterizing the simulated pitch dynamics (29) are bp = 0.80 NV−1 , g = 9.81 ms−2 , Je = 0.09 kgm2 , kp = 0.20 NmV−1 , l = 0.19 m and m = 1.39 kg respectively. The 375 observer and adaptation law parameters are L = 189 and Γ = diag(103 , 103 ), respectively. Consider the input signal u = 3.5 + 5.0 sin 0.4πt + 6.0 sin 1.1πt + 7.0 sin 1.7πt
Parameter θ
(see Fig. 1). The simulation results for the estimation of the states and parameters are shown in Fig. 2–6. The proposed estimation design exhibits a satisfactory convergence of both the states and the parameters to the actual values.
1
34
Actual value Estimated value
33.5 33 32.5
1
32 20
θ
Motor voltage vs. time
31 30.5
15
30
10
Input signal in volt
31.5
29.5
5
29 0
5
10
15
0
−10
2
4
Time in sec
6
8
40
2
Actual value Estimated value
10 11.5 11
θ
2
Pitch angle vs. time
0
angular displacement in rad
35
Parameter θ 12
Fig.1: Voltage input
10.5
−20
10
−40
9.5
−60
9 0
5
10
15
−80
20 25 Time in sec
30
35
40
Fig.5: Parameter θ2 - proposed adaptive observer
−100 −120 −140 0
30
Fig.4: Parameter θ1 - proposed adaptive observer
−5
−15 0
20 25 Time in sec
Actual angle Estimated angle 2
4
Time in sec
6
8
Parameter θ
3
3.2
10
Actual value Estimated value
3
Fig.2: Pitch angle
2.8
θ
200
3
Pitch angle variation vs. time
150
2.6 2.4
angular rate in rad
100 2.2
50 0
2 0
−50 −100
10
15
20 25 Time in sec
30
35
40
Fig.6: Parameter θ3 - proposed adaptive observer
−150 −200 −250 0
5
Actual angular velocity Estimated angular velocity 2
4
Time in sec
6
8
10
Fig.3: Pitch speed In addition, to motivate the scheme of parameters estimate developed in this paper, we propose to compare it with the nonlinear observer for estimating parameters in nonlinear dynamics x˙ = f (u, x, θ) discussed in [8]. It is based on the following estimation algorithm [8]: ˙ ˆ − Ψ(u, x)u˙ θ˘ = −Φ(u, x)f (u, x, θ) θˆ = θ˘ + φ(u, x)
(30) (31)
634
i (u,x) i (u,x) with Φ(u, x) = [ ∂φ∂x ]ij and Ψ(u, x) = [ ∂φ∂u ]ij . We j j T 2 select φ(u, x) = −0.1x2 cos x1 + u −0.5x2 + u x2 u . The results of estimates, based on the input signal u and the known state vector x (see Fig. 1–3 respectively), are shown in Fig. 7–9. The selection of the φ(u, x) is quite difficult, as there is no general rule to obtain it or to take advantage of it (in particular to guarantee the convergence and reduce the time of the latter). In fact, the proposed adaptation law, introduced in this paper, shows a better convergence rate of the estimated parameters to their actual values than the one of the alternative scheme despite the fact that the latter has less parameters to estimate (all the states are supposed to be measurable).
Parameter θ
REFERENCES
1
34
Actual value Estimated value
33.5
[1] Zemouche, A., Boutayeba, M., 2009, ”A Unified Adaptive Observer Synthesis Method For A Class Of Systems With Both Lipschitz And Monotone Nonlinearities,” Systems & Control Letters, 58(4), pp. 282-288.
33 32.5
θ
1
32 31.5 31
[2] Gevers, M., Bastin, G., 1986, A Stable Adaptive Observer For A Class Of Nonlinear Second Order Systems, Analysis And Optimization Of Systems, Springer-Verlag, 83, pp. 143-155.
30.5 30 29.5 29 0
500
1000
1500
2000 2500 Time in sec
3000
3500
4000
[3] Mishkov, R.L., 2005, ”Nonlinear Observer Design By Reduced Generalized Observer Canonical Form,” Proc. of the 2005 International Journal of Control, 78(3), pp. 172-185.
Fig.7: Parameter θ1 - alternative parameter estimation CONCLUSION
[4] Farza, M., M’Saad, M., Rossignol, L., 2003, ”Observer Design For A Class Of MIMO Nonlinear Systems,” Automatica, 40, pp. 135-143.
An adaptive observer is designed for a class of nonlinear dynamic systems with first-order states being measured. The observer estimates the full states and identifies the full parameters that join the functions depending on input and state variables. An example with simulation results successfully demonstrates the theorem. A comparative method is also investigated. It should be mentioned that the parameter identification requires persistent excitation of the signals within both techniques. For future work, the parameter estimation concurrently with the disturbance and noise rejection is expected to be a consequent challenging topic. Finally, experiments will be achieved to validate the practical utility of the proposed estimation scheme.
[5] Zhang, Q., 2001, Adaptive Observer For MIMO Linear Time Varying Systems, Technical Report, Institut National de Recherche en Informatique et en Automatique, Rennes, France. [6] Maatoug, T., Farza, M., M’Saad, M., Koubaa, Y., Kamoun, M., 2008, ”Adaptive Observer Design For A Class Of Nonlinear Systems With Coupled Structures,” International Journal of Sciences and Techniques of Automatic Control & Computer Engineering, 2(1), pp. 484499.
Parameter θ
2
12
[7] Dong, Y.-L., Mei, S.-W., 2007, ”Adaptive Observer For A Class Of Nonlinear Systems,” Acta Automatica Sinica, 33(10), pp. 1081-1084.
Actual value Estimated value
11.5
θ
2
11
[8] Friedland, B., 1997, ”A Nonlinear Observer For Estimating Parameters In Dynamic Systems,” Automatica, 33(8), pp. 1525-1530.
10.5 10
[9] Cho, Y.M., Rajamani, R., 1997, ”A Systematic Approach To Adaptive Observer Synthesis For Nonlinear Systems,” IEEE Transactions on Automatic Control, 42(4), pp. 534-537.
9.5 9 0
500
1000
1500
2000 2500 Time in sec
3000
3500
4000
Fig.8: Parameter θ2 - alternative parameter estimation
[10] Khayati, K., Bigras, P., Dessaint, L.-A., 2006, ”A MultiStage Position/Force Control For Constrained Robotic Systems With Friction: Joint-Space Decomposition, Linearization And Multi-objective Observer/Controller Synthesis Using LMI Formalism,” IEEE Transactions on Industrial Electronics, 53(5), pp. 1698-1712.
Parameter θ
3
3.2
Actual value Estimated value
3
θ3
2.8
[11] Khalil, H., 2002, Nonlinear Systems, 3rd ed., PrenticeHall, NY.
2.6 2.4
[12] Boyd, S., ElGhaoui, L., Feron, E., Balakrishnan, V., 1994, Linear Matrix Inequalities In Systems And Control Theory, Society for Industrial and Applied Mathematics, Philadelphia.
2.2 2 0
500
1000
1500
2000 2500 Time in sec
3000
3500
4000
Fig.9: Parameter θ3 - alternative parameter estimation
635
NONLINEAR ADAPTIVE OBSERVER DESIGN FOR AN ELECTROMECHANICAL ROTATIVE PLANT
K. Khayati Department of Mechanical & Aerospace Engineering Royal Military College of Canada Kingston, ON, Canada [email protected]
J. Zhu Department of Mechanical & Industrial Engineering Concordia University Montreal, QC, Canada z [email protected] server. This observer estimates the full-state variables in the presence of uncertain parameters possibly depending on the input and state variables. The stability of the algorithm is guaranteed when at least some of the measured outputs are such that the transfer matrix from the unknown parameters to these outputs is dissipative [9]. This particular condition on the dynamic representation remains very weak and is rarely reached in general. Moreover, there is no straightforward way of choosing the observer gains to satisfy this condition [10]. In this paper, inspired by [9], we intend to improve the procedure by introducing a new adaptation law that eliminates in particular the dissipativity condition that [9] requires. The stability condition of the proposed adaptive observer will be presented using the LMI framework.
ABSTRACT
This paper deals with the design of an adaptive observer that can estimate both the states and the parameters of a class of nonlinear systems. The purpose of the proposed method is to eliminate the restrictive dissipativity condition from the stability proof that is present in many previous works presented in the literature. The observer gain calculus is cast as a linear matrix inequality (LMI) feasibility problem. We aim to apply the proposed algorithm to estimate the unknown parameters of a second order electromechanical rotary model. The appeal of this proven theoretical design is further demonstrated numerically with a comparative design. Keywords: Nonlinear systems, Adaptive observer, Parameter estimation, LMI.
The design of the adaptive observer is applied to estimate the full states and to identify all parameters for a class of nonlinear dynamic systems based on a set of feasible assumptions. In particular, we target the estimation of the model parameters of a second order electromechanical system which are considered unknown a priori. This paper is organized as follows. In Section 2, we describe the nonlinear model statement setting forth the required assumptions that are used in the estimation procedure. Then, we describe the observer dynamics and the adaptation law and study the convergence of the estimated model states and parameters in Section 3. Section 4 is dedicated to the presentation of the electromechanical model to which we intend to apply the proposed adaptive observer. Finally, numerical and comparative results are presented in Section 5, while Section 6 concludes with recommendations.
INTRODUCTION The observer design for linear and nonlinear systems has been widely investigated during the last decades (e.g. [1, 2] and references cited therein). In [3, 4], observers of global exponential convergence have been introduced. Adaptive observers estimating both the states and parameters of linear systems [5] and nonlinear systems [6, 7, 8] were developed using the input and output signals. The authors provided constructive methods for the design of global adaptive observers enabling the estimation of the unknown parameters with a common condition of persistent excitation [7, 9]. In [9], the authors considered nonlinear dynamics with linearly dependent parameters to design a systematic approach of an adaptive ob-
631
PROBLEM STATEMENT AND ASSUMPTIONS
A. Theorem
Consider the nonlinear multiple-input and multiple-output (MIMO) model described by the following adaptive observer form:
Under assumptions 1)–5), if there exist matrices P = P T > 0 in R2n×2n and W ∈ R2n×n such that P A + AT P − W C − C T W T + δI2n P < 0 (12) P − δ1 I2n
x˙ 1 x˙ 2 y
= x2 + F01 (u, x1 ) = A1 x1 + A2 x2 + F02 (u, x) + F (u, x)θ
(1) (2)
= x1
(3)
x1 , u ∈ R m , θ ∈ R q , A1 , x2 n A2 in Rn×n , F01 (u, x1 ), F02 (u, x) nonlinear functions in R 0n and F (u, x) ∈ Rn×q , respectively. Let B = , where In In and 0n are the identity and null-square matrices of order n respectively. For the forthcoming analysis, we need the following assumptions:
with x1 , x2 and y in Rn , x =
then the adaptive observer (9)–(11), with the observer gain matrix L = P −1 W , is asymptotically stable, x˜ = x − xˆ → 0 and θ˜ = θ − θˆ → 0 as t → ∞. B. Proof The estimation error dynamics are obtained from (1)–(3) and (9): x ˜˙ =
(A − LC)˜ x + B[F02 (u, x) − F02 (u, x ˆ) + ˜ F (u, x)θ − F (u, x ˆ)θ + F (u, x ˆ)θ] (13)
1. The vector of unknown constant parameters θ is bounded, with kθk ≤ α (4)
Using the adaptation law (10)–(11), we have
2. The functions F02 and F are Lipschitz in x, with
As CB = 0, we have
kF02 (u, x) − F02 (u, x ˆ)k ≤ β0 kx − x ˆk
(5)
kF (u, x) − F (u, x ˆ)k ≤ βkx − x ˆk
(6)
and F (u, x) is continuously bounded; 3. A gain matrix L ∈ R2n×n can be chosen such that P (A − LC) + (A − LC)T P + δ(P P + In ) < 0 (7) 0 n In for any matrix P = P T > 0, with A = , A1 A2 C = In 0n and δ = β0 + αβ > 0;
4. The input vector u is continuously differentiable;
5. There exist positive scalars α0 , α1 and t0 , such that ∀t Z t0 +t α0 I q ≤ F T (u, x)B T BF (u, x)dτ ≤ α1 Iq (8) t
These hypotheses are widely acceptable in practice and can characterize adequately most real devices.
˙ θˆ = ΓF T (u, x ˆ)B T P [(C T + BCL)C x ˜ + BC x ˜˙ ]
Cx ˜˙ = C(A − LC)˜ x
˙ θ˜ = −ΓF T (u, x ˆ)B T P [C T C + BCA]˜ x = −ΓF T (u, x ˆ)B T P x ˜
(16)
To investigate the stability, given P = P T > 0 and Γ = Γ > 0, consider the Lyapunov candidate function V (t) = ˜ We have x ˜T P x ˜ + θ˜T Γ−1 θ. T
V˙
=
2˜ xT P x ˜˙ + 2θ˜T Γ−1 θ˜˙
=
2˜ xT P {(A − LC)˜ x + B[F02 (u, x) − F02 (u, x ˆ) + T −1 ˜ ˜ F (u, x)θ − F (u, x ˆ)θ + F (u, x ˆ)θ]} − 2θ Γ ·
[ΓF T (u, x ˆ)B T P x ˜] T ≤ x ˜ [P (A − LC) + (A − LC)T P ]˜ x+ (17)
Notice the inequality
Consider the following full-order observer xˆ˙ = Aˆ x + F0 (u, x1 , x ˆ) + BF (u, x ˆ)θˆ + L(y − C x ˆ)
(15)
˙ ˆ˙ Thus, we obtain Based on assumption 1), we have θ˜ = −θ. the adaptation error dynamics
2β0 kBkkP x ˜kk˜ xk + 2αβkBkkP x ˜kk˜ xk ADAPTIVE OBSERVER DESIGN
(14)
2kP x ˜kk˜ xk ≤ x˜T P P x ˜+x ˜T x˜ (9)
and adaptation law ˙ θ˘ = Γ[F T (u, x ˆ)B T P (C T + BCL) − F˙ T (u, u, ˙ x ˆ)B T P B] · (y − C x ˆ) (10) T T ˆ ˘ θ = θ + ΓF (u, x ˆ)B P B(y − C x ˆ) (11) F01 (u, x1 ) where F0 (u, x1 , x ˆ) = , F˙ (u, u, ˙ x ˆ) is the total F02 (u, x ˆ) T q×q time derivative of F (u, x ˆ) and Γ = Γ > 0 in R .
632
(18)
and δ = β0 + αβ > 0. Thus, we obtain V˙ ≤ x ˜T [P (A − LC) + (A − LC)T P + δ(P P + I2n )]˜ x (19) The right member of the inequality (19) is negative definite if P (A − LC) + (A − LC)T P + δ(P P + I2n ) < 0
(20)
that is, ∃Q = QT > 0 such that P (A − LC) + (A − LC)T P + δ(P P + I2n ) = −Q (21)
Then, V˙ < −λmin (Q)k˜ xk2 . This implies x˜ ∈ L∞ (i.e. timefunctions of finite ∞-norm) and θ˜ ∈ L∞ , and then V (t) ∈ L∞ . Integrating (19) V (t) ≤ V (0) − λmin (Q)
Z
Note that the total time derivative of the function F (u, x ˆ) given by:
t 2
k˜ xk dt
(22)
0
Since V (0) is finite, we obtain x ˜ ∈ L2 (i.e. finite 2-norm vector-function). From (13), we have x˜˙ ∈ L∞ . Therefore, by applying theorem 8.4 of [11] based on Barbalat’s Lemma, ˆ → x ˜ → 0. Also, from (13), we have B[F (u, x)θ − F (u, x ˆ)θ] 0. Using the inequality (6), in assumption 2), and noting that ˆ → 0 as t → ∞. x ˆ → x will lead to BF (u, x)(θ − θ) Hence, F T (u, x)B T BF (u, x)θ˜ → 0 as t → ∞. Define Rt G(t0 ) = t 0 F T (u, x)B T BF (u, x)dτ . We apply the analysis of persistency of excitation, as given in [7], to obtain ˜ → 0. Using integration by parts, we have θ(t) Z
t0 +t t
˜ )dτ = G(t0 + t) · F T (u, x)B T BF (u, x)θ(τ Z t0 +t ˜ 0 + t) − G(t)θ(t) ˜ − ˜˙ )dτ θ(t G(τ )θ(τ
(23)
As G(t) = 0, we obtain t0 +t
˜ )dτ = G(t0 + t) · F T (u, x)B T BF (u, x)θ(τ
t
˜ 0 + t) − θ(t
Z
t0 +t
˜˙ )dτ G(τ )θ(τ
(24)
t
Since F T (u, x)B T BF (u, x)θ˜ → 0, then, for any finite t, we have Z t0 +t ˜ )dτ → 0 F T (u, x)B T BF (u, x)θ(τ (25) t
Based on assumption 2) and since x ˜ → 0, from (16), we have ˙θ → 0, and then ˜ Z
t0 +t
˜˙ )dτ → 0 as t → ∞ G(τ )θ(τ
(26)
˜ 0 + t) → 0 as t → ∞ G(t0 + t)θ(t
(27)
t
Thus, we obtain
By assumption 5), i.e. γ0 Iq ≤ G(t0 + t) ≤ γ1 Iq for γ0 > 0 ˜ 0 + t) → 0, implying θ(t) ˜ → 0 [7]. and γ1 > 0, we obtain θ(t Finally, we use the Schur complement [12] and the change of variable W = P L to transform the nonlinear inequality (20) into the LMI (12) in decision variables P and W . C. Discussions The condition (8) is necessary to guarantee the convergence of the parameter estimates given by the adaptive observer scheme (9)–(11) to the true parameter values [7, 8].
633
(28)
can be implemented by using any continuously bounded input u and the definition of x ˆ˙ given by the right member of (9). In general, the synthesis of the adaptive observer gain, stated by the problem (12), can be solved by any efficient convex feasibility algorithm [12] using e.g. the LMI control toolbox in Matlab. ELECTROMECHANICAL ROTARY DYNAMICS We address the problem of designing a global adaptive observer for a second order nonlinear electromechanical dynamics. We consider the model of a helicopter-based flight engine simulator mounted on a fixed base with one propeller that is driven by a DC motor. The propeller controls the elevation of the helicopter nose about the pitch axis. The pitch dynamics is described by: Je ψ¨ = kp u − bp ψ˙ − mgl cos ψ
t
Z
dF ∂F ∂F (u, u, ˙ xˆ) = (u, x ˆ)x ˆ˙ + (u, x ˆ)u˙ dt ∂x ˆ ∂u
(29)
where ψ is the pitch angle, u the input motor voltage, Je the moment of inertia about pitch pivot, kp the thrust torque constant acting on pitch axis from the motor-propeller, bp the viscous damping coefficient about the pitch axis, m the total moving mass of the helicopter, l the center of mass length along the helicopter body from the pitch axis and g the gravity acceleration. All parameters of (29) are unknown. The nonlinear term, that is mgl cos ψ, represents the gravitational torque. The remaining terms in (29) are the acceleration torque, the torque generated by the motor on the pitch axis and the viscous rotary friction acting about the pitch axis respectively. Only the pitch angle is measurable. Using the states x1 = ψ ˙ the rotary dynamics can be written in the state and x2 = ψ, 0 1 space representation (1)–(3), with A = , B = 0 0 0 , C = 1 0 . The nonlinear terms are F0 (u, x) = 1 0 and F (u, x) = − cos x1 −x2 u . The components bp kp of the unknown vector θ are θ1 = mgl Je , θ2 = Je and θ3 = Je respectively. In the following section, we intend to identify these parameters. SIMULATION RESULTS Computer simulations of the adaptive observer algorithm for the second order rotary system are developed. The parameters characterizing the simulated pitch dynamics (29) are bp = 0.80 NV−1 , g = 9.81 ms−2 , Je = 0.09 kgm2 , kp = 0.20 NmV−1 , l = 0.19 m and m = 1.39 kg respectively. The 375 observer and adaptation law parameters are L = 189 and Γ = diag(103 , 103 ), respectively. Consider the input signal u = 3.5 + 5.0 sin 0.4πt + 6.0 sin 1.1πt + 7.0 sin 1.7πt
Parameter θ
(see Fig. 1). The simulation results for the estimation of the states and parameters are shown in Fig. 2–6. The proposed estimation design exhibits a satisfactory convergence of both the states and the parameters to the actual values.
1
34
Actual value Estimated value
33.5 33 32.5
1
32 20
θ
Motor voltage vs. time
31 30.5
15
30
10
Input signal in volt
31.5
29.5
5
29 0
5
10
15
0
−10
2
4
Time in sec
6
8
40
2
Actual value Estimated value
10 11.5 11
θ
2
Pitch angle vs. time
0
angular displacement in rad
35
Parameter θ 12
Fig.1: Voltage input
10.5
−20
10
−40
9.5
−60
9 0
5
10
15
−80
20 25 Time in sec
30
35
40
Fig.5: Parameter θ2 - proposed adaptive observer
−100 −120 −140 0
30
Fig.4: Parameter θ1 - proposed adaptive observer
−5
−15 0
20 25 Time in sec
Actual angle Estimated angle 2
4
Time in sec
6
8
Parameter θ
3
3.2
10
Actual value Estimated value
3
Fig.2: Pitch angle
2.8
θ
200
3
Pitch angle variation vs. time
150
2.6 2.4
angular rate in rad
100 2.2
50 0
2 0
−50 −100
10
15
20 25 Time in sec
30
35
40
Fig.6: Parameter θ3 - proposed adaptive observer
−150 −200 −250 0
5
Actual angular velocity Estimated angular velocity 2
4
Time in sec
6
8
10
Fig.3: Pitch speed In addition, to motivate the scheme of parameters estimate developed in this paper, we propose to compare it with the nonlinear observer for estimating parameters in nonlinear dynamics x˙ = f (u, x, θ) discussed in [8]. It is based on the following estimation algorithm [8]: ˙ ˆ − Ψ(u, x)u˙ θ˘ = −Φ(u, x)f (u, x, θ) θˆ = θ˘ + φ(u, x)
(30) (31)
634
i (u,x) i (u,x) with Φ(u, x) = [ ∂φ∂x ]ij and Ψ(u, x) = [ ∂φ∂u ]ij . We j j T 2 select φ(u, x) = −0.1x2 cos x1 + u −0.5x2 + u x2 u . The results of estimates, based on the input signal u and the known state vector x (see Fig. 1–3 respectively), are shown in Fig. 7–9. The selection of the φ(u, x) is quite difficult, as there is no general rule to obtain it or to take advantage of it (in particular to guarantee the convergence and reduce the time of the latter). In fact, the proposed adaptation law, introduced in this paper, shows a better convergence rate of the estimated parameters to their actual values than the one of the alternative scheme despite the fact that the latter has less parameters to estimate (all the states are supposed to be measurable).
Parameter θ
REFERENCES
1
34
Actual value Estimated value
33.5
[1] Zemouche, A., Boutayeba, M., 2009, ”A Unified Adaptive Observer Synthesis Method For A Class Of Systems With Both Lipschitz And Monotone Nonlinearities,” Systems & Control Letters, 58(4), pp. 282-288.
33 32.5
θ
1
32 31.5 31
[2] Gevers, M., Bastin, G., 1986, A Stable Adaptive Observer For A Class Of Nonlinear Second Order Systems, Analysis And Optimization Of Systems, Springer-Verlag, 83, pp. 143-155.
30.5 30 29.5 29 0
500
1000
1500
2000 2500 Time in sec
3000
3500
4000
[3] Mishkov, R.L., 2005, ”Nonlinear Observer Design By Reduced Generalized Observer Canonical Form,” Proc. of the 2005 International Journal of Control, 78(3), pp. 172-185.
Fig.7: Parameter θ1 - alternative parameter estimation CONCLUSION
[4] Farza, M., M’Saad, M., Rossignol, L., 2003, ”Observer Design For A Class Of MIMO Nonlinear Systems,” Automatica, 40, pp. 135-143.
An adaptive observer is designed for a class of nonlinear dynamic systems with first-order states being measured. The observer estimates the full states and identifies the full parameters that join the functions depending on input and state variables. An example with simulation results successfully demonstrates the theorem. A comparative method is also investigated. It should be mentioned that the parameter identification requires persistent excitation of the signals within both techniques. For future work, the parameter estimation concurrently with the disturbance and noise rejection is expected to be a consequent challenging topic. Finally, experiments will be achieved to validate the practical utility of the proposed estimation scheme.
[5] Zhang, Q., 2001, Adaptive Observer For MIMO Linear Time Varying Systems, Technical Report, Institut National de Recherche en Informatique et en Automatique, Rennes, France. [6] Maatoug, T., Farza, M., M’Saad, M., Koubaa, Y., Kamoun, M., 2008, ”Adaptive Observer Design For A Class Of Nonlinear Systems With Coupled Structures,” International Journal of Sciences and Techniques of Automatic Control & Computer Engineering, 2(1), pp. 484499.
Parameter θ
2
12
[7] Dong, Y.-L., Mei, S.-W., 2007, ”Adaptive Observer For A Class Of Nonlinear Systems,” Acta Automatica Sinica, 33(10), pp. 1081-1084.
Actual value Estimated value
11.5
θ
2
11
[8] Friedland, B., 1997, ”A Nonlinear Observer For Estimating Parameters In Dynamic Systems,” Automatica, 33(8), pp. 1525-1530.
10.5 10
[9] Cho, Y.M., Rajamani, R., 1997, ”A Systematic Approach To Adaptive Observer Synthesis For Nonlinear Systems,” IEEE Transactions on Automatic Control, 42(4), pp. 534-537.
9.5 9 0
500
1000
1500
2000 2500 Time in sec
3000
3500
4000
Fig.8: Parameter θ2 - alternative parameter estimation
[10] Khayati, K., Bigras, P., Dessaint, L.-A., 2006, ”A MultiStage Position/Force Control For Constrained Robotic Systems With Friction: Joint-Space Decomposition, Linearization And Multi-objective Observer/Controller Synthesis Using LMI Formalism,” IEEE Transactions on Industrial Electronics, 53(5), pp. 1698-1712.
Parameter θ
3
3.2
Actual value Estimated value
3
θ3
2.8
[11] Khalil, H., 2002, Nonlinear Systems, 3rd ed., PrenticeHall, NY.
2.6 2.4
[12] Boyd, S., ElGhaoui, L., Feron, E., Balakrishnan, V., 1994, Linear Matrix Inequalities In Systems And Control Theory, Society for Industrial and Applied Mathematics, Philadelphia.
2.2 2 0
500
1000
1500
2000 2500 Time in sec
3000
3500
4000
Fig.9: Parameter θ3 - alternative parameter estimation
635
