Contraction Based Adaptive Control of a Class of Nonlinear Systems

2009 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 10-12, 2009

WeB04.5

Contraction Based Adaptive Control of a Class of Nonlinear Systems B. B. Sharma and I. N. Kar, Member IEEE

Abstract— Adaptive control problem of nonlinear systems having dynamics in parametric strict feedback form is addressed here. Effort is made to derive adaptive methodology for controller design in contraction framework. General results and conditions for stabilization are derived using backstepping. At each step of recursive design, system is made contracting by suitable selection of control inputs. As contraction property is not intrinsic to the systems, so proposed strategy helps in identifying a coordinate transformation along with controller to establish contracting nature of the system. Contracting dynamics ensures exponential convergence of state trajectories to each other. Results are further extended to address control problem of systems having uncertain parameters. Tracking control problem of single link manipulator with actuator dynamics is addressed using the proposed scheme. Numerical simulations justify the effectiveness of the proposed methodology.

I. INTRODUCTION Adaptive control is one of the main approach to deal with uncertain nonlinear systems in practice. Global stability and tracking results for a large class of nonlinear systems has been presented in [1]. The backstepping technique is one of the nonlinear technique which is prominently used to obtain a single controller for a particular class of nonlinear systems. This technique offers a systematic design procedure for systems which can be transformed into parametric strict feedback form [2]. This method is based on Lyapunov stability theory and has been explored widely in [3]-[6]. In backstepping, stability of each subsystem is ensured by constructing suitable Lyapunov functions at each stage. Then overall asymptotic stability of closed loop system is ensured by a combined Lyapunov function constructed by summing up the individual Lyapunov functions of each step [1]. Traditionally, Lyapunov function based approach has been widely used for stability analysis. Later on, incremental stability based approaches for stability analysis were proposed [7]-[10]. Lyapunov based techniques analyze the behavior of system trajectories with respect to origin or some given nominal motion. On the other hand, incremental stability based approaches analyze the behaviour of nonlinear system trajectories with respect to each other. Recently introduced contraction theory framework is based on incremental stability concepts. In this approach, system description in terms of differential equations is used to carry out stability analysis[11]-[13]. This approach does not require selection of energy like function as is the case in Lyapunov based stability B. B. Sharma is with Department of Electrical Engineering, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi- 110016, INDIA

[email protected] I. N. Kar is faculty with the Department of Electrical Engineering, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi- 110016, INDIA

[email protected]

978-1-4244-4524-0/09/$25.00 ©2009 AACC

analysis. Selection of such function may be formidable task sometimes due to lack of general guidelines for this purpose. Here in this paper, methodology of applying contraction framework to a strict feedback class of nonlinear systems is presented. The main advantage of this approach is that it will ensure the exponential stability of the system trajectories with respect to each other. So knowledge of equilibrium point is not required at all. Hence, it provides a stronger notion of stability in comparison to Lyapunov based approach where stability is analyzed w.r.t. equilibrium point. Such recursive techniques based on incremental stability framework are still not explored much. Construction of integrator backstepping under the framework of contraction theory for a simple class of system is proposed by [14]. A particular case of this this approach is presented in [15]. However, in present paper a more general class of nonlinear systems is considered and a generalized recursive procedure is proposed for stabilizing controller design. At each step of the proposed recursive procedure, system is made contracting by suitable selection of virtual control inputs. This input is chosen so as to make individual system contracting and to bring overall system to feedback combination form of contracting subsystems [11]. As contraction property is not intrinsic to the systems, so proposed contraction theory based methodology helps in identifying a coordinate transformation along with controller to establish contracting nature of the system. For known parameter case, exponential stability is established using the given approach. These results are extended to address those systems which are having uncertainty in system parameters. Asymptotic stability of states is shown through virtual system concept for this case. So our main contribution is to develop contraction based recursive approach to design controller for strict feedback class of systems and to identify a coordinate transformation to establish contracting nature of the system. The proposed strategy is applied to address tracking control of single link manipulator system with actuator dynamics. The paper is outlined as follows: Section II presents brief overview of contraction theory. Section III elaborates contraction based recursive backstepping procedure for strict feedback class of systems. Section IV presents tracking control problem for single link manipulator system with actuator dynamics. Finally, section V presents concluding remarks of the paper. II. BASICS OF C ONTRACTION T HEORY Contraction is a property regarding the convergence between two arbitrary system trajectories. A nonlinear dynamic system is called contracting if trajectories of the perturbed system return to their nominal behavior with an exponential

808

convergence rate [11]-[13]. Consider a nonlinear system as x˙ = f(x,t)

(1)

where x ∈ Rm×1 is a state vector and f(x,t) is an m × 1 continuously differentiable vector function. Let G x is infinitesimal virtual displacement in the state x at fixed time. Hence, first variation of system in (1) will be

w f(x,t) Gx wx From this equation, we can further write d T wf G x G x = 2G xT G x˙ = 2G xT G x dt wx ≤ 2Om (x,t)G xT G x G x˙ =

(2)

(3)

Here, the Jacobian matrix is denoted as J (= ww xf ) and the largest eigenvalue of the symmetric part of Jacobian is represented by O m (x,t). If Om (x,t) is strictly uniformly negative, then any infinitesimal length G x converges exponentially to zero. Here G xT G x represents the squared distance between the neighbouring trajectories. By carrying out integration in (3), it is assured that all the solution trajectories of the system in (1) converge exponentially to single trajectory, independently of the initial conditions. Definition 1: Given the system x˙ = f(x,t), a region of state space is called a contracting region if the Jacobian ww xf is uniformly negative definite (UND) in that region. Definition 2: Uniformly negative definiteness of Jacobian w f(x,t) means that there exists a scalar D > 0, ∀x, ∀t ≥ 0 wx s.t. ww xf ≤ −D I < 0. As all matrix inequalities will refer to symmetric part of the square matrix involved, so we can T further write, 12 ww xf + wwfx ≤ −D I < 0. The basic results (without proof) related to exponential convergence of the trajectories can be stated as follows [11][13]: Lemma 1: Given the system dynamics (1), any trajectory which starts in a ball of constant radius centered about a given trajectory and contained at all times in a contraction region, remains in that ball and converges exponentially to the given trajectory. Further, global exponential convergence is guaranteed if the whole state space region is contracting. To represent the above results in more general way, consider a coordinate transformation

Gz = TGx

(4)

is UND. So all the solution trajectories of the system (1) converge exponentially to single trajectory using the nature of transformation in (4). The absolute value of largest eigenvalue of the symmetric part of F is called contracting rate of the system w.r.t. the uniformly positive definite metric M = T T T . These results are stated in the form of following lemma as given in [11]-[13]. Lemma 2: For a dynamic system x˙ = f(x,t), if there exists a uniformly positive definite metric M(x,t) = T T (x,t)T (x,t) such that the associated generalized Jacobian matrix

wf F = (T˙ + T )T −1 (8) wx is UND, then all system trajectories converge exponentially to a single trajectory, with convergence rate |O m (x,t)|, where Om (x,t) is the largest eigenvalue of the symmetric part of F. Then the system is said to be contracting. For system (1), the Jacobian matrix J = wwxf can be represented in matrix form as   J1,1 J1,2 J1,3 . . . J1,n−1 J1,n  J2,1 J2,2 J2,3 . . . J2,n−1 J2,n    J= . (9) .. .. .. .. ..   .. . . . . .  Jn,1

T

G z G z = G x MG x

(5)

TTT

where M = is a uniformly positive definite metric. Taking the time derivative of (5), we get wf d T G z G z = 2G zT G z˙ = 2G zT (T˙ + T )T −1 G z (6) dt wx From above, it is clear that exponential convergence of G z to zero is guaranteed if the generalized Jacobian matrix

wf F = (T˙ + T )T −1 wx

(7)

Jn,3

...

Jn,n−1

Jn,n

For such Jacobian matrix, following lemma can be stated to ensure the contracting behaviour of the system (1): Lemma 3: The system with dynamics in (1) will be contracting if the Jacobian in (9) satisfies following conditions: (i) All the diagonal elements J i,i are uniformly negative definite for i = 1, 2, . . . , n. (ii) All off diagonal elements satisfy Ji, j = −J j,i condition for i, j = 1, 2, . . . , n, ∀ i = j. For systems in (1), if Jacobian matrix ww xf turns out to be negative semi-definite then such system are called semicontracting systems. For these systems, asymptotic stability can be ensured using the contraction theory results. Contraction theory results are also extended to various combinations of systems. An important combination is feedback combination which is discussed here, briefly. A. Feedback Combination of Systems Consider that two systems of different dimensions and possibly contracting with different metrics to be having following dynamics

where T (x,t) is a uniformly invertible matrix. Squared distance in transformed domain can be written as T

Jn,2

x˙1

=

f1 (x1 , x2 ,t)

x˙2

=

f2 (x1 , x2 ,t)

(10)

Let the transformation G z = T G x is used such that the dynamics of above systems can be written in terms of virtual displacements in transformed domain as d G z1 G G z1 F1 = (11) −GT F2 G z2 dt G z2 Then the augmented system will be contracting. This concept can be generalized to any number of systems. Other combinations of contracting systems can be found in the work cited in [11]-[16]. The present paper will utilize the

809

results presented above to design the required controller for the proposed class of systems.

So for the combination of the first two subsystems, virtual displacement in differential framework can be represented as G x˙1 G x1 J1,1 J1,2 0 G z2 (19) = + J2,1 J2,2 G z˙1 G z1 g2

III. BACKSTEPPING BASED C ONTROLLER D ESIGN Consider a class of systems having dynamics as x˙i

=

fi (¯x) + gi xi+1 ; i = 1, 2, . . . , n − 1

x˙n

=

fn (x) + u

i

(12) n

where ¯x ∈ i.e. ¯x = (x1 , x2 , . . . , xi ) and x ∈ is the state vector. f is n × 1 smooth vector function defined as f : n → 1 , and gi is a non-zero constant for i = 1, 2, . . . , n − 1. Here u is control input and f i for i = 1, 2, . . . , n are linear/nonlinear functions. Main objective here is to design backstepping based controller for stabilization of system in (12).

Here, Jacobian matrix J modifies as J1,1 J1,2 J= J2,1 J2,2

having following additional entries

A. Controller for Systems without Parametric Uncertainty To develop the basic notion of contraction based backstepping, case without any uncertainty in system parameters is taken initially. In this regard, following lemma is stated: Lemma 4: For the system having dynamics given in (12), there exists a controller u = u([ ,t), with [ = (x 1 , z1 , . . . , zn−1 ) s.t. the closed loop system is contracting with its Jacobian J. Here, auxiliary variables z i are defined as zi−1 = xi − Di−1 , for i = 2, 3, . . . , n and D i represents the virtual control function used to make the subsystem contracting at i-th step. Proof: First subsystem of parametric strict feedback system given in (12) can be represented as x˙1 = f1 (x1 ) + g1x2

(13)

Let D1 (x1 ) is the virtual control input which makes first subsystem contracting w.r.t. x 1 . Defining an auxiliary variable z1 = x2 − D1 (x1 ), dynamics in (13) is written as x˙1 = f1 (x1 ) + g1 D1 (x1 ) + g1z1

=

J2,1

=

J2,2

=

J1,3

=

J3,1

=

J3,2

=

J3,3

=

(15)

where (1 × 1) Jacobian matrix J is represented by

w (16) [ f1 (x1 ) + g1 D1 (x1 )] w x1 This Jacobian J is UND in nature by careful selection of D 1 and by considering G z 1 to be a bounded external input with a constant coefficient column vector J = J11 =

D = [g1 ]

(17)

g1

w w D1 x˙1 f2 (x1 , x2 ) + g2D2 (x1 , z1 ) − w x1 w x1

w w D1 x˙1 (21) g2 D2 (x1 , z1 ) − w z1 w x1

Now taking time derivative of z 2 and using (12) while defining a new auxiliary variable z 3 = x4 − D3 (x1 , z1 , z2 ), we can develop the transformed dynamics in differential framework following the procedure of previous steps as        G x˙1 G x1 J1,1 J1,2 J1,3 0  G z˙1  =  J2,1 J2,2 J2,3   G z1  +  0  G z3 G z˙2 G z2 J3,1 J3,2 J3,3 g3 (23) where new (3 × 3) Jacobian matrix is having following additional entries to the matrix in (20):

(14)

= J G x1 + DG z1

J1,2

which is UND in nature by suitable selection of D 2 (x1 , z1 ) and by considering G z 2 to be bounded external input. The constant coefficient column vector in (17) gets modified as T (22) D = 0 g2

For this system, virtual displacement in differential framework can be represented as ⇒ G x˙1

(20)

0; J2,3 = g2 w ( f3 (x1 , x2 , x3 ) + R(x1 , z1 , z2 )) w x1 w w D2 w D2 g3 D3 (x1 , z1 , z2 ) − x˙1 − z˙1 w z1 w x1 w z1

w w D2 (24) g3 D3 (x1 , z1 , z2 ) − z˙1 w z2 w z1

where function R(x 1 , z1 , z2 ) = g3 D3 (x1 , z1 , z2 ) − w D2 w z1 z˙1 .

w D2 w x1 x˙1

−

The above Jacobian matrix is again UND in nature while considering G z 3 to be bounded external input with coefficient column matrix in (22) modified to T (25) D = 0 0 g3 In general for i-th stage, the transformed dynamics is

By taking time derivative of z 1 and using (12), we get

i−2

(18)

w Di−1 w Di−1 z˙i−1 = fi (¯x) + gi zi + gi Di ([¯ ) − x˙1 − ¦ z˙k (26) w x1 k=1 w zk

Define new virtual control input D 2 (x1 , z1 ) to make (18) contracting w.r.t. z 1 . It also ensures feedback interconnection of subsystems in (14) and (18). Defining new auxiliary variable z2 = x3 − D2 (x1 , z1 ), (18) can be represented as

where ¯x = (x1 , x2 , . . . , xi ) and [¯ = (x1 , z1 , . . . , zi−1 ), for index i = 2, 3, . . . , n − 1. For n-th stage, the dynamics is simplified by defining an auxiliary variable as z n−1 = xn − Dn−1 . The time derivative of z n−1 can be given as

w D1 x˙1 z˙1 = x˙2 − D˙ 1 (x1 ) = f2 (x1 , x2 ) + g2x3 − w x1

z˙1 = f2 (x1 , x2 ) + g2z2 + g2 D2 (x1 , z1 ) −

w D1 x˙1 w x1

z˙n−1 = fn (x) + u −

810

n−2 w Dn−1 w Dn−1 x˙1 − ¦ z˙k w x1 k=1 w zk

(27)

Suitable selection of control u([ ,t), ensures contracting nature of final subsystem w.r.t. z n−1 . It also ensures feedback combination of contracting subsystems. Now, dynamics of transformed system can be represented as

[˙ = h([ ,t)

(28)

This dynamics is obtained using coordinate transformation T (x,t) along with a feedback control u([ ,t), which ensures contraction behaviour of the overall system. The transformation is indirectly obtained here using backstepping procedure. System (28) in differential framework can be written as

w h([ ,t) G[ = J G[ G [˙ = w[

(29)

where (n × n) Jacobian matrix denoted by J contracting. The structure of Jacobian matrix J  0 ... 0 0 J1,1 J1,2  J2,1 J2,2 J2,3 . . . 0 0  J= . .. .. .. .. .. .  . . . . . . Jn,1

Jn,2

Jn,3

...

Jn,n−1

Jn,n

= is 

w h([ ,t) w[

  . 

is

(30)

Different elements of Jacobian matrix can be represented in general form as follows: a) Diagonal Elements: J1,1

=

J2,2

=

Ji,i

=

Jn,n

=

w ( f1 + g1D1 ) w x1 w w D1 (g2 D2 − x˙1 ) w z1 w x1 w w Di−1 (gi Di − z˙i−2 ); i = 3, . . . , n − 1 w zi−1 w zi−2 w w Dn−1 (u − z˙n−2 ) w zn−1 w zn−2 = =

j = 1.

x˙n

=

fn (x) + E T q(x) + u.

(32)

Here x ∈ n is state vector, E ∈ p is uncertain parameter vector and q(x) is a (p × 1) linear or nonlinear vector function. Functions f i for i = 1, 2, . . . , n are linear or nonlinear functions in nature. Here, control function u is to be designed along with suitable update laws for uncertain parameters. This choice is driven by the fact that overall system should be semi-contracting so that stability of system states could be ascertained. Again, the transformed system is obtained by using equations (13)-(26). The dynamics of final stage as shown in (27) modifies as

\˙ E˙¯

1 < j < i.

n−2 w w Dn−1 w Dn−1 (u − x˙1 − ¦ z˙k ); w z j−1 w x1 k=1 w zk

fi (¯x) + gi xi+1 ; i = 1, 2, . . . , n − 1

n−2 w Dn−1 w Dn−1 x˙1 − ¦ z˙k + E T q(x) (33) w x1 w z k k=1

E˜ = E − E

(34)

[˙ = h([ ,t) + W(x,t)(E − E )

(35)

Virtual system1 for system in (35) & (36) is defined as

n−2 w w Dn−1 w Dn−1 ( fn + u − x˙1 − ¦ z˙k ) Jn,1 = w x1 w x1 k=1 w zk

Jn, j =

=

where W(x,t) is regression vector and the parameter updation law is defined as ˙ E˙˜ = E = −WT (x,t)[ (36)

i−2 w w Di−1 w Di−1 (gi Di − x˙1 − ¦ z˙k ); w z j−1 w x1 k=1 w zk

for i = 3, 4, . . . , n − 1;

x˙i

In compact form, dynamics of transformed system will be

i−2 w w Di−1 w Di−1 ( fi + giDi − x˙1 − ¦ z˙k ); Ji, j = wxj w x1 k=1 w zk

Ji, j =

Consider the dynamics of a system with parametric uncertainty as

As parameter vector E is uncertain, so the controller u is selected as u = u1 ([ ,t) − E T q(x) so that overall system is in feedback combination form of contracting subsystems. Control function u 1 ([ ,t) is obtained exactly as the control obtained in earlier case for the system without E T q(x) term. Once u1 ([ ,t) is designed, complete u can be used in transformed system. Here E is estimate of uncertain parameter set and the parametric error is defined as

gi ; i = 1, 2, 3, . . . , n − 1 0; i = 1, 2, 3, . . . , n and i + 2 ≤ j ≤ n

for i = 2, 3, 4, . . . , n − 1;

B. Control of Uncertain Systems

z˙n−1 = fn (x) + u −

b) Off-diagonal Elements: Ji, i+1 Ji, j

Jacobian matrix is contracting, then the dynamics in (28) will be contracting. Using definition of auxiliary variables z i for i = 1, 2, . . . , n − 1, it can be shown that state vector x is contracting. So all the states of the system converge to each other i.e. exponential stability of system states is obtained. ⋄

(31)

for 1 < j < n. The Jacobian matrix J is to be UND in nature to show contracting behaviour of overall system. Selection of control function u([ ,t) and virtual control D i , i = 1, 2, . . . , n − 1 is made so that conditions of lemma 3 are satisfied. If the

= h(\ ,t) + W(x,t)(E¯ − E ) = −WT (x,t)\

(37)

For above system in (37), defining the virtual increments by G \ & G E¯ in \ & E¯ , respectively, the above dynamics can be written in differential framework as w h(\ ,t) G \˙ G\ W(x,t) w \ = (38) G E¯ G E˙¯ −WT (x,t) 0 1 Actual system given in (35) and (36) is a particular solution of system in (37) i.e. if we replace (\ ,E¯ ) pair by ([ , E), we get the actual system. So system in (37) is called virtual system for the actual system.

811

The Jacobian matrix J can be represented as J11 J12 J= J21 J22

(39)

where n × n contracting submatrix J 11 = w h(w \\ ,t) is is given by equation (30), submatrix J 12 of size n × p is given as   0 0 ... 0  0 0 ... 0     0 0 ... 0  (40) J12 =  ;   .. .. .. ..   . . . . q1 (x) q2 (x) . . . q p (x)

submatrix J21 = −JT12 and p× p sub-matrix J 22 = 0. Different elements of Jacobian matrix J 11 are same as given in (31) except with the difference in following entries: Jn,n

=

Jn,1

=

Jn, j

=

w w Dn−1 (u + E T q(x) − z˙n−2 ) w zn−1 w zn−2 w ( fn + u + E T q(x) − F ) w x1 w (u + E T q(x) − F ); 1 < j < n (41) w z j−1

function F is given as F = . For ensuring the UND nature of sub-matrix J 11 , u([ ,t) and Di , i = 1, 2, . . . , n − 1 is selected so that conditions of lemma 3 are satisfied. As the system in (38) is semi-contracting system, so using the results shown in [16], asymptotic stability of above system can be established. The actual system in (35) and (36) is a particular solution of this virtual system, so the state vector [ is contracting. Hence, asymptotic stability of state variables can be ensured though estimates of uncertain parameters may not converge to true value. where

nonlinear

w Dn−1 n−2 w Dn−1 w x1 x˙1 + ¦k=1 w zk z˙k

IV. T RACKING C ONTROLLER FOR M ANIPULATOR S YSTEM WITH ACTUATOR DYNAMICS To analyze the proposed strategy, let the dynamics of single link manipulator system with actuator is given by Dq¨ + Bq˙ + N sin(q) = W M W˙ + H W + Km q˙ = u

(42)

Selecting state variables as x1 = q, x2 = q˙ and x3 = W , respectively, we get x˙1 x˙2

= x2 = f1 (x1 , x2 ) + g1x3

x˙3

=

f2 (x1 , x2 , x3 ) + g2 u

(43)

Different functions involved in (43) are given as: f 1 (x1 , x2 ) = 1 f2 (x1 , x2 , x3 ) = − M1 (Km x2 + Hx3 ) D (−N sin(x1 ) − Bx2 ), 1 1 and g1 = D ; g2 = M . In tracking control problem, objective is to design a controller u so that the output y = q converges to a desired trajectory y d with all other signals remaining bounded. Defining tracking error e 1 = y − yd and selecting auxiliary variables as z 1 = x2 − D1 , z2 = x3 − D2 , respectively, the controller u can be designed. Here, virtual

control inputs are selected as D 1 = −e1 + y˙d and D2 = D ND sin(e1 + yd ) + DB x2 − k1 z1 + y¨d , respectively. With these choices along with controller function as

1 u = M k1 De1 − k1 D(1 − k1) − z1 − k2z2 D

2 B Km − +M Nx2 cos(x1 ) − x2 D M ... BN B H y + sin(x1 ) + D d +M (44) x3 − D M D the dynamics of the system (43) in transformed domain is e˙1

=

−e1 + z1

z˙1

=

−e1 + (1 − k1)z1 +

1 z2 D

1 (45) − z1 − (k2 − k1 )z2 D The controller in (44) is derived using the proposed recursive procedure. The transformed system (45) in differential framework will be      −1 1 0 G e˙1 G e1 1  G z˙1  =  −1   G z1  1 − k1 D 1 G z˙2 G z2 −(k2 − k1) 0 −D (46) The matrix T (x,t) used for transformation G z = T (x,t)G x comes out to be   1 0 0 1 1 0  (47) T (x,t) =  k1 D − B 1 k1 D − Ncos(x1) z˙2

=

which is uniformly invertible matrix. The transformation metric M = T (x,t)T T (x,t) will be   2 + N2 M12 M13 M =  M21 1 − (k1D − B)2 M23  (48) M31 M32 1

where M12 = M21 = 1 + (k1D − B)(k1 D − Ncos(x1 )), M13 = M31 = k1 D − Ncos(x1 ) and M23 = M32 = k1 D − B, respectively and N = k 1 D − Ncos(x1 ). This symmetric metric is uniformly positive definite for suitable selection of gains k 1 and k2 . The Jacobian w f(x,t) w x of the closed loop system with controller in (44) can be given as   0 1 0 w f(x,t)  N 1  (49) = − D cos(x1 ) − DB D wx r31 r32 r33 where r31 = k12 D − k1 k2 D − D1 + Nk2 − BN D cos(x1 ) − 2 Nx2 sin(x1 ), r32 = Bk2 − k1 D − BD − D1 + Ncos(x1 ) + k12 D − k1 k2 D and r33 = DB − k2 , respectively. Matrix F = T m T −1 with Tm = T˙ + T ww xf comes out to be   −1 1 0 1  1 − k1 F =  −1 (50) D 1 0 −D −(k2 − k1) which is UND in nature. So convergence of G e 1 → 0 is ensured using lemma 1 and 2. So output y tracks the

812

(a)

(b)

1.5

0.5

(a)

y yd

(b)

3

2 y yd

0

f0.5

f0.5

2

1.5 Tracking Error

0

Tracking Error

d

y and y states

0.5

y and yd states

1

1 0

1 0.5 0

f1 f1

5

10 15 t (time in seconds)

20

f1 0

5


20

20

Fig. 1. Control of single link manipulator with actuator dynamics (without uncertainty): (a) tracking of desired state yd and (b) output tracking error.


80

e˙1

=

−e1 + z1

1 z2 D K˜m H˜ 1 x2 + x3 (51) z˙2 = − z1 − (k2 − k1 )z2 + D M M In this case, the controller structure is same as proposed in (44) except that uncertain parameters K m & H are replaced respectively. Here, estimates of m & H, by their estimates K uncertain parameters are represented as x2 ˙ ˙ = − x3 z (52) K H m = − z2 ; 2 M M For numerical simulation, the estimates for uncertain parameters are initialized as [0 0] T and simulations are run for 80 seconds. The tracking performance and the boundedness of parameter estimates is shown in fig. 2. The approach is also applied to design controller for chaotic systems belonging to the proposed class of systems but the results are omitted here due to lack of space. z˙1

=

−e1 + (1 − k1)z1 +

V. C ONCLUSION Contraction based recursive approach for designing controller for a class of nonlinear systems is proposed. The conditions for contracting behavior of closed loop system are derived analytically and are presented in terms of conditions on elements of Jacobian matrix. Further, adaptive control problem of the systems is addressed with uncertainty in some of the parameters. The stability results shown here are achieved in quite comprehensive manner. The coordinate transformation obtained through backstepping approach helps in establishing contracting behaviour of the systems. Numerical simulations for adaptive tracking control of single link manipulator system with actuator dynamics is presented

20

(c)


80

(d)

100

30 estimate of K

Control U

50

desired trajectory y d as per the requirement. The results showing the tracking performance are presented in fig. 1. The desired trajectory is defined as y d = sin(t). Different parameters of system are taken as D = 1, M = 0.01, B = 1, Km = 10, H = 0.5 and N = 10. The initial conditions are taken as [0.1 6.28 0] T . Controller is switched on at time t = 2 seconds. The response shows the effectiveness of the proposed controller in meeting out the tracking performance. In case of uncertainty in parameters K m and H, the approach given in section III-B is used to develop estimates m and H, respectively. System after transformation can be K written as

f0.5 0

Parameter Estimates

f1.5 0

f2 0

0

f50 f100 0

20


80

m

estimate of H

20 10 0

f10 0

20


80

Fig. 2. Control of single link manipulator with actuator dynamics (with uncertainty): (a) tracking of desired state yd , (b) output tracking error, (c) time variation of control function and (d) estimates of uncertain parameters.

for the system with uncertainty in parameters. These results may be extended further to address stabilization and tracking control of systems with unmatched uncertainty. R EFERENCES [1] M. Krstic, I. Kanellakapoulous and P. V. Kokotovic, Nonlinear and Adaptive Control Design, Wiley Interscience,NY, 1995. [2] I. Kanellakopoulos, P. V. Kokotovic and A. S. Morse, ”Systematic design of adaptive controllers for feedback linearizable systems,” IEEE Trans. Aut. Control, Vol. 36, pp.1241-1253, 1991. [3] M. Krstic, I. Kanellakopoulos and P. V. Kokotovic, ”Adaptive nonlinear control without overparametrization,” System and Control Letters, vol. 19, pp.177-185, 1992. [4] H. K. Khalil, Non-linear systems (3rd Edition). New Jersey: Prentice Hall; 2002. [5] M. Rios-Bolivar and A. S. I. Zinober, ”Dynamical adaptive sliding mode output tracking control of a class of nonlinear systems,” Int. J. Robust and Nonlinear Control, vol. 7, pp.387-405, 1997. [6] B. Yao and M. Tomizuka, ”Adaptive robust control of SISO nonlinear systems in a semi-strict feedback form,” Automatica, vol. 33, pp.893900, 1997. [7] D. Angeli, ”A Lyapunov approach to incremental stability properties,” IEEE Trans. Aut. Control, vol. 47, no. 3, pp.410-421, 2002. [8] A. Pavlov, A. Pogromsky, N. Wouw and H. Nimeijer, ”Convergent dynamics, a tribute to Boris Pavlovich Demidovich,” System and Control Letters, vol. 52, pp.257-261, 2004. [9] B. Ingalls and E.D. Sontag, ”A small-gain lemma with applications to input/output systems, incremental stability, detectability, and interconnections,” Journal of Franklin Inst., vol. 339, pp.211-229, 2002. [10] V. Fromian, G. Scorletti and G. Ferreres, ”Nonlinear performance of a PI controlled missile: an explanation,” Int. Journal of Robust and Nonlinear Control, vol. 9 no. 8, pp.485-518, 1999. [11] W. Lohmiller and J.J.E. Slotine, ”On contraction analysis for nonlinear systems,” Automatica, vol. 34, no. 6, pp.683-696, 1998. [12] W. Lohmiller, Contraction Analysis of Nonlinear Systems, Ph.D. Thesis, Department of Mechanical Engineering, MIT, 1999. [13] W. Lohmiller and J.J.E. Slotine, ”Control system design for mechanical systems using contraction theory,” IEEE Trans. Aut. Control, vol. 45, no. 5, pp.884-889, 2000. [14] J. Jouffroy and J. Lottin, ”Integrator backstepping using contraction theory: a brief technological note,” Proc. of the IFAC World Congress, Barcelona, Spain, 2002. [15] B.B. Sharma and I. N. Kar, ”Adaptive control of wing rock system in uncertain environment using contraction theory”, IEEE American Contr Conf, Washington, pp. 2963-2968, June 11-13, 2008. [16] J. Jouffroy and J.J.E. Slotine, ”Methodological remarks on contraction theory,” 43rd IEEE Conf. On Decision and Control, Atlantis, Bahamas, pp.2537-43, 2004.

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