Contraction Based Adaptive Control of a Class of Nonlinear Systems

Transcription

1 9 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June -, 9 WeB4.5 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. INTROUCTION 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 []. 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 []. 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 []. Traditionally, Lyapunov function based approach has been widely used for stability analysis. Later on, incremental stability based approaches for stability analysis were proposed [7]-[]. 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[]-[3]. This approach does not require selection of energy like function as is the case in Lyapunov based stability B. B. Sharma is with epartment of Electrical Engineering, Indian Institute of Technology, elhi, Hauz Khas, New elhi- 6, INIA bbs.iit@gmail.com I. N. Kar is faculty with the epartment of Electrical Engineering, Indian Institute of Technology, elhi, Hauz Khas, New elhi- 6, INIA ink@ee.iitd.ac.in 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 [4]. A particular case of this this approach is presented in [5]. 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 []. 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 CONTRACTION THEORY 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 /9/$5. 9 AACC 88

2 convergence rate []-[3]. Consider a nonlinear system as ẋ = fx,t) ) where x R m is a state vector and fx,t) is an m continuously differentiable vector function. Let is infinitesimal virtual displacement in the state x at fixed time. Hence, first variation of system in ) will be ẋ = fx,t) ) From this equation, we can further write d T ) dt = T ẋ = T f 3) m x,t) T Here, the Jacobian matrix is denoted as J = f ) and the largest eigenvalue of the symmetric part of Jacobian is represented by m x,t). If m x,t) is strictly uniformly negative, then any infinitesimal length converges exponentially to zero. Here T ) 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 ) converge exponentially to single trajectory, independently of the initial conditions. efinition : Given the system ẋ = fx,t), a region of state space is called a contracting region if the Jacobian f is uniformly negative definite UN) in that region. efinition : Uniformly negative definiteness of Jacobian fx,t) means that there exists a scalar >, x, t f s.t. I <. As all matrix inequalities will refer to symmetric part of the square ) matrix involved, so we can further write, f ft + I <. The basic results without proof) related to exponential convergence of the trajectories can be stated as follows []- [3]: Lemma : Given the system dynamics ), 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 z = 4) where x,t) is a uniformly invertible matrix. Squared distance in transformed domain can be written as z T z = T M 5) where M = T is a uniformly positive definite metric. Taking the time derivative of 5), we get d z T z ) = z T ż = z T + f dt ) z 6) From above, it is clear that exponential convergence of z to zero is guaranteed if the generalized Jacobian matrix F = + f ) 7) is UN. So all the solution trajectories of the system ) 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. These results are stated in the form of following lemma as given in []-[3]. Lemma : For a dynamic system ẋ = fx,t), if there exists a uniformly positive definite metric Mx,t)= T x,t)x,t) such that the associated generalized Jacobian matrix F = + f ) 8) is UN, then all system trajectories converge exponentially to a single trajectory, with convergence rate m x,t), where m x,t) is the largest eigenvalue of the symmetric part of F. Then the system is said to be contracting. For system ), the Jacobian matrix J = f can be represented in matrix form as J, J, J,3... J,n J,n J, J, J,3... J,n J,n J = ) J n, J n, J n,3... J n,n J n,n For such Jacobian matrix, following lemma can be stated to ensure the contracting behaviour of the system ): Lemma 3: The system with dynamics in ) 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 =,,...,n. ii) All off diagonal elements satisfy J i, j = J j,i condition for i, j =,,...,n, i j. For systems in ), if Jacobian matrix f 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. 89 A. Feedback Combination of Systems Consider that two systems of different dimensions and possibly contracting with different metrics to be having following dynamics ẋ = f x,x,t) ẋ = f x,x,t) ) Let the transformation z = is used such that the dynamics of above systems can be written in terms of virtual displacements in transformed domain as d dt [ z ] = [ F G T ][ ] G z F ) 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 []-[6]. The present paper will utilize the

3 results presented above to design the required controller for the proposed class of systems. III. BACKSTEPPING BASE CONTROLLER ESIGN Consider a class of systems having dynamics as ẋ i = f i x)+g i x i+ ; i =,,...,n ẋ n = f n x)+u ) where x i i.e. x =x,x,...,x i ) and x n is the state vector. f is n smooth vector function defined as f : n,and g i is a non-zero constant for i =,,...,n. Here u is control input and f i for i =,,...,n are linear/nonlinear functions. Main objective here is to design backstepping based controller for stabilization of system in ). 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 ), there exists a controller u = u,t), with =x,z,...,z n ) s.t. the closed loop system is contracting with its Jacobian J. Here, auxiliary variables z i are defined as z i = x i i, for i =,3,...,n and 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 ) can be represented as ẋ = f x )+g x 3) Let x ) is the virtual control input which makes first subsystem contracting w.r.t. x.efining an auxiliary variable z = x x ), dynamics in 3) is written as ẋ = f x )+g x )+g z 4) For this system, virtual displacement in differential framework can be represented as ẋ = J + 5) where ) Jacobian matrix J is represented by J = J = [ f x )+g x )] 6) This Jacobian J is UN in nature by careful selection of and by considering to be a bounded external input with a constant coefficient column vector =[g ] 7) By taking time derivative of z and using ), we get ż = ẋ x )= f x,x )+g x 3 ẋ 8) efine new virtual control input x,z ) to make 8) contracting w.r.t. z. It also ensures feedback interconnection of subsystems in 4) and 8). efining new auxiliary variable z = x 3 x,z ), 8) can be represented as ż = f x,x )+g z + g x,z ) ẋ So for the combination of the first two subsystems, virtual displacement in differential framework can be represented as [ ] [ ][ ] [ ] ẋ J, J =, + z ż J, 9) g J, Here, Jacobian matrix J modifies as [ ] J, J J =, J, J, having following additional entries J, = g J, = J, = ) f x,x )+g x,z ) ) ẋ g x,z ) ) ẋ ) which is UN in nature by suitable selection of x,z ) and by considering to be bounded external input. The constant coefficient column vector in 7) gets modified as = [ g ] T ) Now taking time derivative of z and using ) while defining a new auxiliary variable z 3 = x 4 3 x,z,z ), we can develop the transformed dynamics in differential framework following the procedure of previous steps as ẋ ż ż = + z 3 J, J, J,3 J, J, J,3 J 3, J 3, J 3,3 g 3 3) where new 3 3) Jacobian matrix is having following additional entries to the matrix in ): J,3 = ; J,3 = g J 3, = f 3 x,x,x 3 )+Rx,z,z )) J 3, = J 3,3 = g 3 3 x,z,z ) ẋ ) ż z g 3 3 x,z,z ) ) ż 4) where function Rx,z,z )=g 3 3 x,z,z ) ẋ z ż. The above Jacobian matrix is again UN in nature while considering z 3 to be bounded external input with coefficient column matrix in ) modified to = [ g 3 ] T 5) In general for i-th stage, the transformed dynamics is ż i = f i x)+g i z i + g i i ) i i i ẋ ż k 6) k= where x =x,x,...,x i ) and =x,z,...,z i ), for index i =,3,...,n. For n-th stage, the dynamics is simplified by defining an auxiliary variable as z n = x n n. The time derivative of z n can be given as ż n = f n x)+u n ẋ n k= n ż k 7) 8

4 Suitable selection of control u,t), ensures contracting nature of final subsystem w.r.t. z n. It also ensures feedback combination of contracting subsystems. Now, dynamics of transformed system can be represented as = h,t) 8) This dynamics is obtained using coordinate transformation 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 8) in differential framework can be written as = h,t) = J 9) h,t) where n n) Jacobian matrix denoted by J = is contracting. The structure of Jacobian matrix J is J, J,... J, J, J,3... J = ) J n, J n, J n,3... J n,n J n,n ifferent elements of Jacobian matrix can be represented in general form as follows: a) iagonal Elements: J, = f + g ) J, = g ẋ ) J i,i = g i i i ż i ); z i z i i = 3,...,n J n,n = u n ż n ) z n z n b) Off-diagonal Elements: J i, i+ = g i ; i =,,3,...,n J i, j = ; i =,,3,...,n and i+ j n J i, j = f i + g i i i i i ẋ j ż k ); k= for i =,3,4,...,n ; j =. J i, j = g i i i i i ẋ z j ż k ); k= for i = 3,4,...,n ; < j < i. J n, = f n + u n ẋ J n, j = u n ẋ z j n k= n k= n ż k ) n ż k ); 3) for < j < n. The Jacobian matrix J is to be UN in nature to show contracting behaviour of overall system. Selection of control function u,t) and virtual control i, i =,,...,n is made so that conditions of lemma 3 are satisfied. If the Jacobian matrix is contracting, then the dynamics in 8) will be contracting. Using definition of auxiliary variables z i for i =,,...,n, 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. B. Control of Uncertain Systems Consider the dynamics of a system with parametric uncertainty as ẋ i = f i x)+g i x i+ ; i =,,...,n ẋ n = f n x)+ T qx)+u. 3) Here x n is state vector, p is uncertain parameter vector and qx) is a p ) linear or nonlinear vector function. Functions f i for i =,,...,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 3)-6). The dynamics of final stage as shown in 7) modifies as ż n = f n x)+u n ẋ n k= n ż k + T qx) 33) As parameter vector is uncertain, so the controller u is selected as u = u,t) T qx) so that overall system is in feedback combination form of contracting subsystems. Control function u,t) is obtained exactly as the control obtained in earlier case for the system without T qx) term. Once u,t) is designed, complete u can be used in transformed system. Here is estimate of uncertain parameter set and the parametric error is defined as = 34) In compact form, dynamics of transformed system will be = h,t)+wx,t) ) 35) where Wx,t) is regression vector and the parameter updation law is defined as = = W T x,t) 36) Virtual system for system in 35) & 36) is defined as = h,t)+wx,t) ) = W T x,t) 37) For above system in 37), defining the virtual increments by & in &, respectively, the above dynamics can be written in differential framework as [ ] = [ h,t) Wx,t) W T x,t) ] [ ] 38) Actual system given in 35) and 36) is a particular solution of system in 37) i.e. if we replace, ) pair by, ), we get the actual system. So system in 37) is called virtual system for the actual system. 8

5 The Jacobian matrix J can be represented as [ ] J J J = J J h,t) 39) where n n contracting submatrix J = is is given by equation 3), submatrix J of size n p is given as J =... ; 4) q x) q x)... q p x) submatrix J = J T and p p sub-matrix J =. ifferent elements of Jacobian matrix J are same as given in 3) except with the difference in following entries: J n,n = J n, = J n, j = u+ T qx) n ż n ) z n z n f n + u+ T qx) ) u+ T qx) ); < j < n 4) z j where nonlinear function is given as = n ẋ + n n k= z ż k k ). For ensuring the UN nature of sub-matrix J, u,t) and i, i =,,...,n 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 [6], 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. IV. TRACKING CONTROLLER FOR MANIPULATOR SYSTEM WITH ACTUATOR YNAMICS To analyze the proposed strategy, let the dynamics of single link manipulator system with actuator is given by q+ B q+ N sinq) = M + H+ K m q = u 4) Selecting state variables as x = q, x = q and x 3 =, respectively, we get ẋ = x ẋ = f x,x )+g x 3 ẋ 3 = f x,x,x 3 )+g u 43) ifferent functions involved in 43) are given as: f x,x )= N sinx ) Bx ), f x,x,x 3 )= M K mx + Hx 3 ) and g = ; g = 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. efining tracking error e = y y d and selecting auxiliary variables as z = x, z = x 3, respectively, the controller u can be designed. Here, virtual control inputs are selected as = e + ẏ d and = [ N sine + y d )+ B x ] k z + ÿ d, respectively. With these choices along with controller function as [ u = M k e k k ) ] )z k z B +M [Nx cosx ) x K )] m M [ B +M + H ) x 3 BN M sinx )+... ] y d 44) the dynamics of the system 43) in transformed domain is ė = e + z ż = e + k )z + z ż = z k k )z 45) The controller in 44) is derived using the proposed recursive procedure. The transformed system 45) in differential framework will be ė ż ż = k k k ) e 46) The matrix x,t) used for transformation z = x,t) comes out to be x,t)= k Ncosx ) k B 47) which is uniformly invertible matrix. The transformation metric M =x,t) T x,t) will be M = + M M 3 M k B) M 3 48) M 3 M 3 where M = M = +k B)k Ncosx )), M 3 = M 3 = k Ncosx ) and M 3 = M 3 = k B, respectively and = k Ncosx ). This symmetric metric is uniformly positive definite for suitable selection of gains k fx,t) and k. The Jacobian of the closed loop system with controller in 44) can be given as fx,t) = N cosx ) B 49) r 3 r 3 r 33 where r 3 = k k k + Nk BN ) cosx ) Nx sinx ), r 3 = Bk k B + Ncosx )+k k k and r 33 = B k, respectively. Matrix F = m with m = + f comes out to be F = k k k ) 5) which is UN in nature. So convergence of e is ensured using lemma and. So output y tracks the 8

6 5 5 y and y d states a) t time in seconds) y y d Tracking Error.5.5 b) 5 5 t time in seconds) Fig.. Control of single link manipulator with actuator dynamics without uncertainty): a) tracking of desired state y d and b) output tracking error. desired trajectory y d as per the requirement. The results showing the tracking performance are presented in fig.. The desired trajectory is defined as y d = sint). ifferent parameters of system are taken as =, M =., B =, K m =, H =.5 and N =. The initial conditions are taken as [. 6.8 ] T. Controller is switched on at time t = 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 K m and Ĥ, respectively. System after transformation can be written as ė = e + z ż = e + k )z + z ż = z k k )z + K m M x + H M x 3 5) In this case, the controller structure is same as proposed in 44) except that uncertain parameters K m & H are replaced by their estimates K m & Ĥ, respectively. Here, estimates of uncertain parameters are represented as K m = x M z ; Ĥ = x 3 M z 5) For numerical simulation, the estimates for uncertain parameters are initialized as [ ] T and simulations are run for 8 seconds. The tracking performance and the boundedness of parameter estimates is shown in fig.. 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. V. CONCLUSION 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 y and y d states Control U 3 a) y y d t time in seconds) t time in seconds) c) Tracking Error.5.5 b) t time in seconds) Parameter Estimates 3 d) estimate of K m estimate of H t time in seconds) Fig.. 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