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Design of Contractive Output Feedback Controllers For Filtered Backstepping
5th 5th International International Conference Conference on on Advances Advances in in Control Control and and Optimization of Dynamical Dynamical Systems 5th International Conference on Advances in Control Optimization of Systems Available onlineand at www.sciencedirect.com 5th International Conference on Advances in Control and February 18-22, 2018. Hyderabad, India Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India February 18-22, 2018. Hyderabad, India
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IFAC PapersOnLine 51-1 (2018) 30–35
Design Design of of Contractive Contractive Output Output Feedback Feedback Design of Contractive Output Feedback Controllers For Filtered Backstepping Design of Contractive Output Feedback Controllers For Filtered Backstepping Controllers For Filtered Backstepping Controllers For Filtered Backstepping ∗∗ M. M. Rayguru ∗∗∗ I. N. Kar ∗∗ ∗∗
M. M. Rayguru ∗∗ I. N. Kar ∗∗ ∗∗ M. M. Rayguru ∗ I. N. Kar ∗∗ M. M. Rayguru I. N. Kar ∗ ∗ Indian Institute of Technology Delhi, New Delhi, India, Pin:110016 ∗ Indian Institute of Technology Delhi, New Delhi, India, Pin:110016 ∗ ∗ Indian Institute of Technology Delhi, New Delhi, India, Pin:110016 (e-mail: [email protected]). (e-mail: [email protected]). ∗ ∗∗ Indian Institute of Technology Delhi, New Delhi, India, Pin:110016 ∗∗ Indian Institute of Technology Delhi, (e-mail: [email protected]). ∗∗ Indian Pin:110016 Institute of Technology Delhi, New New Delhi, Delhi, India, India, Pin:110016 ∗∗ ∗∗ (e-mail: [email protected]). (e-mail: [email protected]) Indian Institute of Technology Delhi, New Delhi, India, Pin:110016 (e-mail: [email protected]) ∗∗ Indian Institute of Technology Delhi, New Delhi, India, Pin:110016 (e-mail: [email protected]) (e-mail: [email protected]) Abstract: Abstract: This This paper paper proposes proposes a a novel novel observer observer based based filtered filtered backstepping backstepping controller controller for for aa class class Abstract: This paper proposes a novel observer based filtered backstepping controller for adesign class of nonlinear systems. For this purpose, the contraction theory based tools are exploited to of nonlinear systems. For this purpose, the contraction theory based tools are exploited to design Abstract: This paper proposes a novel observer based filtered backstepping controller for adesign class an exponentially convergent linear time varying observer. The proposed observer satisfies the of nonlinear systems. For this purpose, the contraction theory based tools are exploited to an exponentially convergent time observer. The proposed observer satisfies the of nonlinear systems. For this linear purpose, thevarying contraction theory based tools are exploited to design sufficient observability condition and the resulting output feedback controller assures ultimate an exponentially convergent linear time varying observer. The proposed observer satisfies the sufficient observability condition and the resulting output feedback controller assures ultimate an exponentially convergent linear time varying observer. The proposed observer satisfies the boundedness of closed loop trajectories. The use of partial contraction concept for convergence sufficient observability condition and the resulting output feedback controller assures ultimate boundedness of closed loop trajectories. The use of output partial feedback contraction conceptassures for convergence sufficient observability condition and the resulting controller ultimate boundedness of closed loop trajectories. The use of partial contraction concept for convergence analysis serves serves a a dual dual purpose. purpose. It It relaxes relaxes the the choice choice of of high high gain gain filter filter parameter, parameter, whose whose analysis boundedness of closed loop trajectories. The use of partial contraction concept for convergence magnitude needs be constrained aa conservative range Lyapunov based approaches. analysis serves a to dual purpose. It inside relaxes the choice of highin gain filter parameter, whose magnitude needs to be constrained inside conservative range in Lyapunov based approaches. analysis serves a todual purpose.between It inside relaxes theloop choice of highingain filter parameter, whose Moreover, the exact dependency closed system performance and design parameters magnitude needs be constrained a conservative range Lyapunov based approaches. Moreover, the exact dependency between closed loop system performance and based designapproaches. parameters magnitude needs to be constrained inside a conservative range in Lyapunov are quantified in of constants. The developed technique successfully simulated Moreover, the exact dependency closed system performance design parameters are quantified in terms terms of known knownbetween constants. Theloop developed technique is is and successfully simulated Moreover, exact dependency closed system performance design parameters are in terms ofmanipulator knownbetween constants. Theloop developed technique is and successfully simulated for aquantified D.C. the motor driven manipulator system. for a D.C. motor driven system. are in terms known constants. for aquantified D.C. motor drivenofmanipulator system.The developed technique is successfully simulated © 2018, IFAC (International Federation ofsystem. Automatic Control) Hosting by Elsevier Ltd. All rights reserved. for a D.C. motor driven manipulator Keywords: Keywords: Filtered Filtered Backstepping, Backstepping, Time Time Varying Varying Observer, Observer, Singular Singular Perturbation, Perturbation, Contraction Contraction Keywords: Filtered Backstepping, Time Varying Observer, Singular Perturbation, Contraction Theory, Convergence Convergence Analysis Theory, Analysis Keywords: Filtered Backstepping, Time Varying Observer, Singular Perturbation, Contraction Theory, Convergence Analysis Theory, Convergence Analysis 1. INTRODUCTION loop system system is is guaranteed (Swaroop (Swaroop et et al., 2000; 2000; Farrell 1. loop 1. INTRODUCTION INTRODUCTION loop system is guaranteed guaranteed (Swaroop et al., al., 2000; Farrell Farrell et al., 2009; Pan and Yu, 2015) using either Tikhonov theoet al., 2009; Pan and Yu, 2015) using either Tikhonov theo1. INTRODUCTION loop system is guaranteed (Swaroop et al., 2000; Farrell et al., 2009; Pan and Yu, 2015) using either Tikhonov theorem or Lyapunov indirect method. In both these methods, rem or Lyapunov indirect method. In both these methods, Backstepping controllers controllers are are efficiently efficiently utilized utilized for for solvsolv- et al., 2009; Pan and Yu, 2015) using either Tikhonov theoBackstepping or Lyapunov indirect method.for Inaboth the stability stability can only only be assured assured verythese smallmethods, range of of Backstepping controllers are problems efficiently inutilized for engisolv- rem ing stabilization stabilization and tracking tracking numerous the can be very small range rem stability orparameter. Lyapunov indirect method.for Inaaboth these methods, ing and problems in numerous engithe can only be assured for very small range of filter Backstepping controllers are efficiently utilized for solving stabilization and involving tracking problems in numerous engi- filter neering applications electro-mechanical systems, parameter. thenew stability can only be assured for a very small of neering applications involving electro-mechanical systems, filter parameter. A approach to stability stability analysis, based onrange virtual ing stabilization and trackingetc problems in numerous engineering applications involving electro-mechanical systems, robotics, chemical reactions (Khalil, 2002; Slotine and A new approach to analysis, based on virtual filter parameter. robotics, chemical reactions etc (Khalil, 2002; Slotine and A new approach to stability analysis, based on virtual displacements and referred as called contraction theory, neering applications involving systems, robotics, reactions etcelectro-mechanical (Khalil, 2002; Slotine and displacements andtoreferred as called contraction theory, Li, 1991). The involves repetitive calculation Li, 1991).chemical The methodology methodology involves repetitive calculation A proposed new approach stability based virtual referred as analysis, called theory, is in and (Lohmiller and Slotine,contraction 1998), on which exrobotics, reactions etc (Khalil, 2002; Slotine and displacements Li, 1991).chemical Thefor methodology involves repetitive calculation of derivatives a number of signals called virtual control is proposed in (Lohmiller and Slotine, 1998), which exdisplacements and referred as called contraction theory, of derivatives for a number of signals called virtual control is proposed in (Lohmiller and Slotine, 1998), which examines the convergence behavior of system trajectories Li, 1991). The methodology involves repetitive calculation of derivatives for a numberetof al., signals called virtual inputs (Kanellakopoulos 1992; Zhao and control Kanel- amines the behavior of trajectories is proposed in and 1998), whichdoes exinputs (Kanellakopoulos et 1992; Zhao and Kanelamines the convergence convergence behavior of system system trajectories with respect to (Lohmiller each other. other. TheSlotine, differential analysis of derivatives for aZamani numberand signals called virtual inputs (Kanellakopoulos etof al., al., 1992; Zhao and control Kanelwith respect to each The differential analysis does lakopoulos, 1998; Tabuada, 2011; Jouffroy and amines the convergence behavior of system trajectories lakopoulos, 1998; Zamani and Tabuada, 2011; Jouffroy and with respect to each other. The differential analysis does not necessitate a priori presence of any attractor and has inputs (Kanellakopoulos et al., 1992; 2011; Zhao Jouffroy and Kanellakopoulos, 1998;These Zamani and Tabuada, necessitate priori presence of any attractor anddoes has Lottin, 2002a). analytical calculations become unLottin, 2002a). These analytical calculations become and un- not withnecessitate respect to aaeach other. The differential analysis priori presence ofsee any attractor andForni has strong interconnection properties, (Angeli, 2002; lakopoulos, 1998; Zamani and Tabuada, 2011; Jouffroy and Lottin, 2002a). These analytical calculations become un- not reliable with increase in the system order and this problem strong interconnection properties, see (Angeli, 2002; Forni not necessitate a priori presence ofsee any attractor andForni has reliable with increase in the order and problem interconnection properties, (Angeli, and Sepulchre, 2014; Wang and Slotine, Slotine, 2005).2002; This techLottin, Theseof become un- strong reliable with increase inanalytical the system systemcalculations order and this this problem is called2002a). as explosion explosion complexity (Swaroop et al., 2000). and Sepulchre, 2014; Wang and 2005). This techstrong interconnection properties, see (Angeli, 2002; Forni is called as of complexity (Swaroop et al., 2000). and Sepulchre, 2014; Wang and Slotine, 2005). This technique is is effectively effectively utilized utilized for for many many control control problems problems like like reliable with increase of inofcomplexity the order and this problem is called asphilosophy explosion (Swaroop et al., 2000). The core thesystem filtered backstepping meth- nique and Sepulchre, 2014; Wang and Slotine, 2005). This techThe core philosophy the filtered backstepping methnique is effectively utilized for many control problems like observer design (Jouffroy and Lottin, 2002b), estimator deis called aspass explosion ofof complexity (Swaroop et al.,through 2000). The core philosophy of the filtered backstepping methobserver design (Jouffroy and Lottin, 2002b), estimator deods is to the variables to be differentiated ods iscore to pass the variables to be differentiated through nique(Sharma is effectively utilized forLottin, many 2002b), control problems like observer design (Jouffroy and estimator design and Kar, 2008), synchronization (Wang and The philosophy of the filtered backstepping methods is to passorthe variables to high be differentiated through a first order second order gain filter (Pan and sign (Sharma and Kar, 2008), synchronization (Wang and observer design (Jouffroy anddesign Lottin, 2002b),and estimator dea order or second orderto high gain filter (Pan and (Sharma and Kar, 2008), synchronization (Wang and Slotine, 2005), backstepping (Zamani Tabuada, odsfirst is to pass variables be differentiated through aYu, first order orthe second gain (Pan and sign 2015; Swaroop et al., al.,order 2000;high Song andfilter Hedrick, 2004; Slotine, 2005), backstepping design (Zamani and Tabuada, sign (Sharma and Kar, 2008), synchronization (Wang and Yu, 2015; Swaroop et 2000; Song and Hedrick, 2004; Slotine, 2005), backstepping design (Zamani and Tabuada, 2011; Rayguru Rayguru and and Kar, Kar, 2015) 2015) etc. etc. Recently Recently aa contraction contraction a first or second high gain and Yu, 2015; et al.,order Song andfilter Hedrick, 2004; Wu et order al.,Swaroop 2014) which in2000; turn circumvent the(Pan need of 2011; Wu et al., 2014) which in turn circumvent the need of Slotine, 2005),approach backstepping design and Tabuada, 2011; Rayguru and Kar,is2015) etc. (Zamani Recently a contraction theory based proposed for analysis analysis of singusinguYu, 2015; Swaroop et al.,in 2000; Song and Hedrick, 2004; Wu et al., 2014) which turn circumvent the need of analytic differentiations. Dynamic surface control (DSC) theory based approach is proposed for of analytic differentiations. Dynamic surface control (DSC) 2011;perturbed Rayguru and Kar,is 2015) etc. Recently a contraction theory based approach proposed for analysis of singularly systems in (Del Vecchio and Slotine, 2013) Wu et al., 2014) which in turn circumvent the need of analytic Dynamic surface control techniquedifferentiations. (Swaroop et et al., al., 2000) uses uses first order (DSC) filters larly perturbed systems in (Del Vecchio and Slotine, 2013) theory based approach is proposed forclass analysis of singutechnique (Swaroop 2000) first order filters larly perturbed systems in (Del Vecchio and Slotine, 2013) and extended to stabilization of same of systems by analytic differentiations. Dynamic surface control (DSC) technique (Swaroopfiltered et al., backstepping 2000) uses first order et filters whereas command command (Farrell al., and to stabilization same systems by larlyextended perturbed systems in (Delof and of Slotine, 2013) whereas filtered backstepping (Farrell et al., and extended to stabilization ofVecchio same class class of systems by (Rayguru and Kar, 2017). technique (Swaroop et al., 2000) uses first order filters whereas command filtered backstepping (Farrell et al., (Rayguru and Kar, 2017). 2009) uses uses second second order order filters filters for for this this purpose. purpose. The The high high and extended to stabilization of same class of systems by 2009) and Kar, 2017). whereas command filtered backstepping (Farrell al., (Rayguru 2009) uses second order filters for this purpose. Theethigh gain filters not only act as approximate differentiators but Motivation and Contribution The work in in this this paper paper gain filters not only act as approximate differentiators but Motivation (Rayguru andand Kar,Contribution 2017). The 2009) uses second order filters for this purpose. The high gain filterscertain not only act as approximate differentiators also relax relax differentiability assumptions requiredbut for Motivation and Contribution The work workin inthe thiscurrent paper is motivated by the following limitations also certain differentiability assumptions required for is motivated by the following limitations in the current gain relax filterscertain notcontroller only act as approximate differentiators Motivation and Contribution The work thiscurrent paper also differentiability assumptions required for backstepping design. Various extensions of but fil- is motivated by the inin the literature (Swaroop etfollowing al., 2000; 2000;limitations Pan and Yu, 2015; Farrell backstepping controller design. Various extensions of filliterature (Swaroop et al., Pan and Yu, 2015; Farrell also relax certain differentiability assumptions required for is motivated by the following limitations in the current backstepping controller design. Various extensions of filtered backstepping backstepping are are proposed proposed by by the the authors authors of of (Gerdes (Gerdes literature (Swaroop et al., 2000; Pan and Yu, 2015; Farrell et al., al., 2009) 2009) on filtered backstepping. tered on backstepping. backstepping controller design. Various extensions ofand fil- et literature (Swaroop et al., 2000;with Pan filtered and Yu, backstepping 2015; Farrell tered backstepping proposed by the authors of (Gerdes and Hedrick, 2002;are Song and Hedrick, Hedrick, 2004; Wang et al., 2009) on filtered filtered backstepping. (i) Most of the results dealing and Hedrick, 2002; Song and 2004; Wang and Most of the results dealing with filtered backstepping teredHedrick, backstepping proposed by Mehraeen the authors ofal., (Gerdes et al., 2009) on filtered backstepping. and 2002;are Song and2008; Hedrick, 2004;etWang and (i) Huang, 2005; Zhang and Ge, 2011; (i) Most of the results dealing with filtered backstepping assume full state measurement which are not suitable Huang, 2005; Zhang and Ge, Mehraeen et al., 2011; assume full stateresults measurement whichfiltered are notbackstepping suitable for for and Hedrick, 2002; Song and2008; Hedrick, 2004; and (i) Mostpractical of the dealing with Huang, 2005; Zhang and Ge, 2008; Mehraeen etWang al.,et 2011; Zhou and Yin, 2014; Huang et al., 2015; Peimani al., assume full state measurement which are not suitable for various applications. Zhou and Yin, 2014; Huang et al., 2015; Peimani et al., various practical applications. Huang,and 2005; and Ge, 2008; Mehraeen et in al.,et 2011; assume full statefiltered measurement which are not do suitable for Zhou Yin,Zhang 2014; Huang 2015; Peimani al., various 2016) to different type uncertainties applications. (ii) The practical existing backstepping designs not concon2016) to handle handle different typeetof ofal., uncertainties in system system The existing filtered backstepping designs do not Zhou and Yin,to2014; Huang 2015; Peimani et al., (ii) various practical applications. 2016) to handle different typeetpractical ofal., uncertainties in system dynamics and suit different applications. (ii) The existing filtered backstepping designs do not consider time time dependent dependent nonlinearities nonlinearities in in system system dynamics. dynamics. dynamics and to different suit different practical applications. 2016)use toofhandle type of uncertainties in system sider (ii) The existing filtered backstepping do not condynamics and suitfilters different practical applications. The highto gain give rise rise to time time scale separation separation sider time dependent nonlinearities in designs system dynamics. (iii) The restriction on filter parameters to be inside a The use of high gain filters give to scale (iii) The restriction on filter parameters to be inside a small small dynamics and to suitfilters different practical applications. siderThe time dependent nonlinearities in noise system dynamics. The use ofloop highdynamics gain give rise to time scalea separation in closed and transform it into singularly (iii) restriction on filter parameters to be inside a small bound may not hold in presence of and sampling in closed dynamics and transform it into singularly bound not in of sampling The use ofloop high gain filters give rise to time scalea (iii) Themay restriction on filter parameters to beand inside a small in closed loop dynamics and transform into a separation singularly perturbed form. Semi-global practical stability of may not hold hold in presence presence of noise noise and sampling perturbed form. Semi-global practical it stability of closed closed bound in closed loop dynamics and transform it into a singularly perturbed form. Semi-global practical stability of closed bound may not hold in presence of noise and sampling perturbed Semi-global practical stability of closed 2405-8963 © form. 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2018 30 Copyright 2018 IFAC IFAC 30 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2018 IFAC 30 10.1016/j.ifacol.2018.05.006 Copyright © 2018 IFAC 30
5th International Conference on Advances in Control and M.M. Rayguru et al. / IFAC PapersOnLine 51-1 (2018) 30–35 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India
restrictions. (iv) The dependence of closed loop system performance on design parameters are not rigorously discussed in current literature. In order to overcome the above mentioned limitations, this paper develops a time varying observer based filtered backstepping controller for a class of nonlinear systems. The utilization of contraction theory ensures a exponentially convergent observer unlike (Song and Hedrick, 2004) and relaxes the constraint on the magnitude of filter parameter. Furthermore the tracking error bound is quantified in terms of design parameters which can be helpful in tuning the controller performance. Notations:The following notations are used throughout the paper. Rm denotes a m-dimensional real vector space. For real vectors v, |v| denotes the Euclidean norm and for real matrix |E| denotes induced matrix norm. Lipschitz constant of the functions are denoted as Li where i denotes a positive scalar. A function is a C k function if its derivatives upto kth order are continuous. A functions belongs to L∞ if it is essentially bounded. The maximum and minimum eigen value of a matrix is denoted by λmax (.) and λmin (.) respectively. A metric Θ denotes a symmetric positive definite matrix and In is an n × n identity matrix.
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χd (4) λ where |d(xp , t)| ≤ d and χ is the condition number of contraction metric Θ. Another useful tool called partial contraction (Wang and Slotine, 2005) is used in this paper and briefly summarized in the form of a lemma. Definition: Consider a system expressed as: ||xp (t) − x(t)|| ≤ χe−λt ||xp (0) − x(0)|| +
x˙ = f (x, x, t).
(5)
Another system defined as y˙ = f (x, y, t)
(6)
is called a virtual/auxiliary system of (5) if the solution x is also a particular solution of (6). In other words the dynamics (6) can be regarded as a copy of (5). Lemma 3: A System x˙ = f (x , y, t) is said to be partially contracting in x if an virtual system defined as z˙ = f (z , y, t) is contracting in z for any value of y, ∀t > 0 . If the auxiliary system verifies a smooth specific property, then the trajectories of original system will also verify that property exponentially (Wang and Slotine, 2005). Furthermore if a system x˙ = f (x, y) is partially contracting in x, then there exists a unique smooth global mapping x = p(y) such that f (x, y) = 0. The proof can be found in (Del Vecchio and Slotine, 2013).
1.1 Prerequisites From Contraction Theory Let’s consider a nonlinear dynamical system expresses as x˙ = f (x, t). Further assume the following relation is satisfied for the nonlinear system in a region: (1) F + F T ≤ −λI ∂f −1 ˙ where F = (Θ + Θ ∂x )Θ , Θ is a nonsingular metric and λ > 0 is a positive constant. The region in which (1) is satisfied is called a contraction region (Lohmiller and Slotine, 1998) and every trajectory inside the contraction region exponentially converge towards each other irrespective of their initial conditions. The system for which F is negative definite is contracting with an associated metric Θ and a contraction rate λ. For brevity, the system is said to be contracting with (Θ, λ). The inequality (1) can be generalized as: ∂f ∂f T (M˙ + M + M ) ≤ −2λM (2) ∂x ∂x where M (x, t) = ΘT Θ is called a contraction metric. If (2) is relaxed to a negative semidefinite condition, the system is semi-contracting. The following lemmas discuss some important properties of contraction taken from Lohmiller and Slotine (1998), Sontag (2010), Wang and Slotine (2005) and Rayguru and Kar (2017) etc.
2. SYSTEM DESCRIPTION Consider a class of nonlinear system in the form of: x˙ 1 = f1 (x1 ) + b1 (x1 )x2 x˙ 2 = f2 (x1 , x2 ) + b2 (x1 , x2 )x3 ... x˙ n = fn (x1 , x2 , x3 ....xn ) + bn (x1 , x2 , . . . , xn )u.
(7)
y = x1 The functions fi (.), bi (.) are smooth, the terms bi (.) > 0 ∀x = [x1 , x2 , ..., xn ]T ∈ Rn , u ∈ R is the control input and y is the system output. The objective of the controller is to ensure that, the output x1 (t) tracks a desired trajectory xd (t). Following assumptions hold true for (7). Assumption 1 : The reference signal xd (t) and x˙ d (t) are bounded. xi ) xi ) i (¯ i (¯ Assumption 2: The terms, fi (¯ xi ), bi (¯ xi ), ∂f∂x and ∂b∂x i i T are bounded ∀ i ∈ 1, . . . , n where (¯ xi = [x1 , x2 , . . . , xi ] ). Above assumptions are standard in the literature dealing with filtered backstepping technique (Swaroop et al., 2000; Farrell et al., 2009; Pan and Yu, 2015) and can be satisfied for numerous engineering applications (Gerdes and Hedrick, 2002; Huang et al., 2015; Pan and Yu, 2015).
Lemma 1: Suppose the autonomous dynamical system defined as x˙ = f (x) is contracting with (Θ, λ) in some region of the state space. Then there exists an unique equilibrium point inside that region to which all the trajectories converge exponentially The following lemma discusses the robustness property of contraction. Lemma 2: Consider a dynamical system expressed as: (3) x˙ p = f (xp , t) + d(xp , t). Moreover assume that x˙ = f (x, t) is contracting with (Θ, λ). The trajectories of (3) verify the following bounds:
3. OUTPUT FEEDBACK FOR FILTERED BACKSTEPPING The system (7) can be rewritten as x˙ = F (x) + B(x)u y = Cx 31
(8)
5th International Conference on Advances in Control and 32 M.M. Rayguru et al. / IFAC PapersOnLine 51-1 (2018) 30–35 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India
T f1 (x1 ) + b1 (x1 )x2 1 f2 (x1 , x2 ) + b2 (x1 , x2 )x3 .. where F (x) = .. , C = . . 0 fn (x1 , x2 , . .. , xn ) 0 .. and B(x) = . A filtered backstepping . bn (x1 , x2 , ..., xn ) control law is proposed as: 1 u= [−fn (ˆ x) − bn−1 zn−1 − χn (ˆ zn ) + α˙ (n)f ] bn (ˆ x) ˆi − xi , (i = 1, 2......n) zi = xi − αif + x α1f = α1 = xd (t) (9) 1 [−fi−1 (ˆ x1 , .., x ˆi−1 ) − χi−1 (zi−1 ) α ˆi = bi−1 − bi−2 zi−2 + α˙ (i−1)f ] for i ∈ [2 . . . n]
µα˙ f = g(αf , α ˆ ) = −αf + α. ˆ
0 1 FT = ... 0
... ... .. . ...
0 −ao 0 −a1 . 0 .. 1 −an
(15)
where the positive scalars ai , i = (1, . . . , n) are selected such that, they constitute the coefficients of a Hurwitz polynomial. The equation (15) can be simplified in terms of column vectors θi of the transformation matrix Θ as: θ˙i + (A − LC)θi = θi+1 , i = 1, . . . , n − 1 a0 a1 θ˙i + (A − LC)θi = −[θ1 , . . . , θn ] ... .
(16)
Cθ1 = L0 Cθ1 = 0, Cθ2 = L1 Cθ1 = 0, Cθ3 = L2 Cθ1 = 0, .. . Cθn = Ln−1 Cθ1 = det(|L0 C . . . Ln−1 C|) = D
(17)
an
It is important to make the procedure of selecting θi independent of the choice of control input. Hence the following constraints are recursively imposed on the selection of θi .
(10)
T
ˆn ] denotes the estimate of x, αf = where x ˆ = [ˆ x1 , . . . , x [α1f , . . . , αnf ], µ ∈ (0, 1) is the high gain filter parameter and α ˆ = [ˆ α1 , . . . , α ˆ n ]T . It can be noted that, the high gain filters (20) circumvent the repeated analytical calculation of derivatives of virtual control signals α˙ i . 3.1 Observer Design
where Li C are Lie derivatives for which L0 C = C, d Li+1 C = Li CA + dt (Li C) and the symbol det(.) denotes the determinant of a matrix. The equations (17) can be solved for choosing a smooth θ1 (t). Then the other columns θi of Θ(t) can be obtained recursively through the relation given by:
The idea is to design a linear observer which is contracting inside the region of interest containing the desired trajectory xd (t). In order for the trajectory x1 (t) to track the desired signal xd (t), other state variables xi (t) have to track αi (t), i : (1, . . . n) respectively. As the terms fi (¯ xi ), bi (¯ xi ) and its first partial derivatives with respect to its arguments are bounded, there exist time varying matrices A(t), B(t) such that, ∂F (x) (11) A(t) = ∂x x1 =xd ,x2 =α2 (.),...,xn =αn (.) . (12) B(t) = B(x)
θi+1 = −θ˙i + Aθi .
(18)
The observer gain matrix can be obtained by solving a0 a1 ˙ LD = [θ1 , . . . , θn ] (19) ... − θn + Aθn . an
x1 =xd ,x2 =α2 (.),...,xn =αn (.)
An observer is proposed as x ˆ˙ = A(t)ˆ x + B(t)u + L(t)(y − C x ˆ)
0 0 .. . 0
If the contraction metric and observer gain are chosen according to (18) and (19) respectively, then the observer dynamics (13) is contracting with respect to x ˆ. Therefore the differential displacement vector δ x ˆ exponentially converges to zero. The assumptions 1 and 2 assure that, the matrix M = Θ−T Θ is uniformly bounded for the choice of observer gain (19) and the metric Θ. Therefore the sufficient observability condition (Θ−T Θ is bounded) is satisfied and the estimate x ˆ converges to x exponentially (Lohmiller and Slotine, 1998). Remark: Theorem 1 proves that, the LTV observer proposed in (13) is capable in estimating the states of the nonlinear system defined in (7). It is not generally true for other classes of nonlinear systems because the sufficient observability condition may not hold for them. However for the class of systems defined in (7), it is always possible to select a contraction metric Θ(t) and an observer gain L(t) such that the estimation error (ˆ x − x) exponentially converges to zero.
(13)
where the observer gain L(t) is decided later. The convergence of observer dynamics is summarized in the following theorem. Theorem 1: Let the assumptions 1 and 2 are satisfied for the system (7) and an observer is designed in the form of (13). If the observer gain is selected as given in (19), then the estimation error (ˆ x − x) exponentially converges to zero. Proof: Let there exists a smooth transformation matrix Θ(t) such that the generalized Jacobian of (13) with resepct to x ˆ can be expressed as ˙ + (A − LC)Θ . (14) FT = Θ−1 − Θ
The observer gain L(t) and the contraction metric Θ has to be selected such that, (13) is contracting with respect to x ˆ (Lohmiller and Slotine, 1998). The observer dynamics (13) is contracting in x ˆ, if the generalized Jacobian FT is a Hurwitz matrix, i.e,
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4. DERIVATION OF TRACKING ERROR BOUND
ˆ ≤ µc1 . lim ||αf (t) − α(t)||
t→∞
z˙iro = −bi−1 z(i−1)ro − χi (ziro ) + bi z(i+1)ro + 2bi α ¯ i+1 + x ˜˙ i ˙ z˙nro = −bn−1 z(n−1)ro − χn (znro ) + x ˜n (27)
z˙i = bi−1 zi−1 − χi (zi ) + bi zi+1 + bi α ˜ i+1 + bi α ¯ i+1 + x ˜˙ i z˙n = −bn−1 zn−1 − χn (zn ) + x ˜˙ n (21) ˙x ˆ = A(t)ˆ x + B(t)u + L(t)(y − C x ˆ). (22)
where zro = [z1ro , ..., znro ]T denote the states of the reduced slow system. Rewriting (27) in a compact form z˙ro = f (zro , α, α)+2¯b+F (ˆ x)−F (x)+(B(ˆ x)−B(x))u (28) where −χ1 (z1ro ) + b1 z2ro .. . f (zro , α, α) = b n−2 z(n−2)ro − χn−1 (z(n−1)ro ) + bn−1 znro −bn−1 z(n−1)ro − χn (znro )
The dynamics (21) can be expressed in a compact form as: z˙ = f (z, αf , α) + ¯b + F (ˆ x) − F (x) + (B(ˆ x) − B(x))u. (23) ¯ 1 , b2 α ¯ 2 , . . . , bn−1 α ¯ n−1 , 0]T and where ¯b = [b1 α −χ1 (z1 ) + b1 z2 + b1 α ˜2 . .. . f (.) = b z −χ (z )+b z +b α ˜ n−1
n−1
n−1 n
(26)
As the fast dynamics (20) is partially contracting in αf , there exists a smooth slow manifold (α + α ¯ ) inside its contraction region. Therefore αf → (α + α) ¯ ⇒α ˜→α ¯. The reduced slow system can be obtained can be expressed as: ¯2 + x ˜˙ 1 z˙1ro = −χ1 (z1ro ) + b1 z2ro + 2b1 α
The overall closed loop system comprises of the dynamics (7), (13), (20) and the control law (9). This section intends to prove ultimate boundedness of closed loop trajectories and derive staedy state tracking error bound. Define x ˜= ¯=α ˆ − α, the fast filter dynamics x ˆ − x, α ˜ = αf − α and α (20) can be expressed as: µα˙ f = g(αf , α) + α ¯ = −αf + α + α. ¯ (20) Using the control law (9),(20), the closed loop dynamics can be expressed as: ˜ 2 + b1 α ¯2 + x ˜˙ 1 z˙1 = −χ1 (z1 ) + b1 z2 + b1 α
n−2 n−2
33
The nominal system of (28) can be expressed as: z˙r = f (zr , α, α) (29) where zr denotes the states of (28) when the perturbation terms are absent. As the Jacobian of f (z, α, α) is uniformly negative definite, the nominal system is contracting with an identity matrix in the absence of perturbation terms. Exploiting assumptions 1 and 2, the perturbation terms α ¯i, x ˜˙ i are bounded and hence the perturbation terms (2¯b+ F (ˆ x) − F (x) + (B(ˆ x) − B(x))u) in (27) are bounded. Exploiting lemma 2, the following can be obtained. 2||¯b|| + (L4 + L5 )||ˆ x − x|| lim ||zr (t) − zro (t)|| ≤ (30) t→∞ β where β is the contraction rate of the nominal system (29) and L4 , L5 are Lipschitz constant for functions F (.) and B(.) respectively. From the assumptions 1 and 2, the term ¯b is also Lipschitz in its arguments, i.e ||¯b|| ≤ L2 ||ˆ α − α||. Therefore ˆ − α|| + (L4 + L5 )||ˆ x − x|| 2L2 ||α . lim ||zr (t) − zro (t)|| ≤ t→∞ β (31) As the estimation error converges to zero, lim ||zr (t) − zro (t)|| ≤ 0. (32)
n−1 n
−bn−1 zn−1 − χn (zn ) The overall closed loop system is in singularly perturbed form and the convergence analysis follows similar steps as given in (Rayguru and Kar, 2017). Define an auxiliary system as: µα˙ des = g(αdes , α) + µQ(z, α ˆ ) + g(αdes , α ˆ ) − g(αdes , α) (24) ˆ˙ . As µα ˆ˙ = g(α ˆ , α) + µQ(z, α ˆ) where Q(z, α) = ∂∂zαˆ z˙ + ∂∂αxˆˆ x and g(α, α) = 0), the signal αf is a particular solution of (24). In similar manner, it can be argued that the variable αf is also a particular solution of (24) when the perturbation term µQ(z, α) is absent. The observer (13) is contracting in x ˆ and hence starting from any initial condition x ˆ˙ is always bounded inside a finite region of interest. Exploiting assumptions (12), the terms α(.), α ˆ (.) and their derivatives are also bounded. Therefore the terms Q(z, α ˆ ) can be assumed to be uniformly bounded. Therefore there exists a positive ˆ )|| ≤ c1 inside (Bz × Bα ). scalar c1 > 0 such that ||Q(z, α Exploiting Lipschitz property of g(.) and α, ||g(αdes , α ˆ ) − g(αdes , α)|| ≤ L1 ||α ˆ − α||. Hence (24) can be treated as a perturbed virtual system of (20). The unperturbed dynamics of (24) is partially contracting in αdes with a contraction metric M = I. Exploiting lemma 2, the steady state convergence bound between the variables αf and α can be derived as: ˆ ≤ µ(c1 + L1 ||α lim ||αf (t) − α(t)|| ˆ − α||). (25)
t→∞
It can be noted that, (23) is a combination of nominal contracting system and extra perturbation terms i.e z˙ = f (z, α, α) + f (z, αf , α ˆ ) − f (z, α, α) + ¯b
+ F (ˆ x) − F (x) + (B(ˆ x) − B(x))u. (33) The dynamics (33) can be argued to be a perturbed virtual system of (28). Exploiting lemma 2 and Lipschitz inequality, the following result can be obtained: L6 c1 + ||¯b|| (34) lim ||z(t) − zr (t)|| ≤ µ t→∞ β where β is the contraction rate of the reduced slow system (28) and L6 is the Lipschitz constant for the function f (.). As the observer dynamics is contracting, the estimation
t→∞
As α(.) is Lipschitz in its arguments therefore, x − x||. ||ˆ α − α|| ≤ L3 ||ˆ The observer dynamics is contracting and therefore the estimation error (ˆ x − x) exponentially converges to zero and lim ||α ˆ − α|| → 0. t→∞
33
5th International Conference on Advances in Control and 34 M.M. Rayguru et al. / IFAC PapersOnLine 51-1 (2018) 30–35 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India
The variables x1 , x2 , x3 correspond to position, velocity and armature current respectively and the control input voltage is denoted as u. The system parameters are given by: M = J + mL20 /3 + M0 L20 + 2M0 R02 /5/Kt , N = mL0 g/2 + M0 L0 g/Kt and B = B0 /Kt . J = 1.625103 Kg m2, m = 0.506 Kg, M0 = 0.434 Kg, L0 = 0.305 m, R0 = 0.023 m, B0 = 16.25163 N m s/rad, L = 25.0103 H, R = 5.0 , Kt = Kb = 0.90 N m/A and g = 9.8 m/s2. The controller design objective is position tracking of a desired signal xd = (π/2)sin(8πt/5). The tuning functions are selected as χi (.) = 5zi , i = 1, 2, 3, the filter parameter is selected as µ = 0.1 and the virtual control inputs αi are calculated as α1 = xd , α2 = −5z1 + N B sin(x1 ) + M x2 + α˙ 2 − 5z2 − z1 ). Following x˙ d , α3 = M ( M the observer design method, a time varying observer is constructed as (13). The observer parameters are selected as a0 = 2, a1 = 3, a2 = 5. The contraction metric for the observer is derived as: 1 0, 0, M R 1 −B and the determinant is calacΘ= , − 0, M M2 M L2 −R −Kb R 1, , + 2 L ML L 1 ulated as D = M . The observer gain L(t) is obtained using the expression (19). The simulation is done with an initial condition [2π, 0, 0]T . Figure 1 confirms the boundedness of closed loop trajec-
error (ˆ x − x) exponentially converges to zero. As α(.) is Lipschitz in its arguments and bi (x) is bounded lim ||¯b|| ⇒ lim L7 ||ˆ x − x|| → 0 t→∞
t→∞
where L7 is the Lipschitz constant. Exploiting lemma 1 and triangle inequality L6 c1 . (35) lim ||z1 (t)|| = lim ||x(t) − xd (t)|| ≤ µ t→∞ t→∞ β The analysis can be summarized in the form of a theorem. Theorem 2:Let Bz ⊂ Rn , Bα ⊂ Rn and Bxˆ ⊂ Rn are compact sets and z(0), αf (0), x ˆ(0) ∈ Bz × Bα × Bxˆ . Suppose the assumptions (1-2) are satisfied for the system (7). Let an observer is constructed as (13) where the gain L(t) is chosen as (19) and a filtered backstepping controller is chosen as (9), (20). Then the trajectories of closed loop system are uniformly ultimate bounded ∀ t > 0 and the steady state tracking error bound is given by L6 c1 . lim ||z1 (t)|| = lim ||x(t) − xd (t)|| ≤ µ t→∞ t→∞ β 4.1 Advantages of The Proposed Design Controller Tuning: The computation of steady state error bounds obtained in (4), (26) require the knowledge of Lipschitz constants which may be difficult to obtain. However it can be noted that, the steady state tracking error under the influence of the controller (9), (20) is inversely proportional to contraction rate of reduced slow system (β) and high gain filter parameter (µ). Therefore the steady state error bounds can be reduced by increasing either β or decreasing µ, which can be done without the knowledge of Lipschitz constants. The parameter β can be increased by increasing the magnitudes of tuning functions χi (zi ) and µ can be decreased by decreasing the magnitude of filter parameters. Trade-off on Perturbation Parameter : The realization of controller depends on the selection of µ. Conventional Lyapunov based analysis requires µ to be very small so that overall closed system is stable. However the controller parameter can’t be reduced arbitrarily because of noise and sampling effects. One important attribute of contraction theory is that, the bound (4), does not depend on the smallness of µ. These bounds are valid for all values of µ ∈ (01] and hence relaxes the conservatism of Lyapunov based analysis and design. Therefore the magnitude of µ can be increased such that it is not too restrictive for controller realization and the performance is satisfactory. Moreover it can be pointed out that it is not necessary to decrease µ to reduce steady state error bounds of the trajectories. Using a moderate value of µ, the bounds can be reduced by proper selection of β.
TrackingError
40
Tracking Error (z1) z2 z3
20 0 -20 -40 0
1
2
3
Simulation Time
4
5
Fig. 1. Tracking Error and Closed loop Trajectories tories as proved in theorem 2. However it can be noted that, the the closed loop response is oscillatory. As the magnitude of tuning functions are not chosen sufficiently high, the contraction rate β is small and consequently the steady state tracking error bound is quite large. The performance is improved by selecting χi (.) = 20zi and thereby ensuring higher magnitude of contraction rate β. The closed loop simulation is shown in figure 2.
5. SIMULATION RESULTS Consider the dynamics of a D.C motor driven manipulator system which can be expressed as Slotine and Li (1991). x˙ 1 = x2 −N B 1 sinx1 − x2 + x3 x˙ 2 = (36) M M M Kb R 1 x˙ 3 = − x2 − x3 + u. L L L
6. CONCLUSION A contraction analysis based procedure is proposed to design a output feedback controller for filtered backstepping. The proposed observer is exponentially convergent and can deal with time varying nonlinearities. The convergence error bounds for closed loop trajectories are quantified in 34
5th International Conference on Advances in Control and M.M. Rayguru et al. / IFAC PapersOnLine 51-1 (2018) 30–35 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India
Pan, Y. and Yu, H. (2015). Dynamic surface control via singular perturbation analysis. Automatica, 57, 29 – 33. Peimani, M., Yazdanpanah, M.J., and Khaji, N. (2016). Adaptive dynamic surface control of bouc–wen hysteretic systems. Journal of Dynamic Systems, Measurement, and Control, 138(9), 091007. Rayguru, M.M. and Kar, I.N. (2015). Contraction based stabilization of approximate feedback linearizable systems. Proceedings of Europian Control Conference, 587– 592. Rayguru, M.M. and Kar, I.N. (2017). Contraction-based stabilisation of nonlinear singularly perturbed systems and application to high gain feedback. International Journal of Control, 90(8), 1778–1792. Sharma, B, ..B. and Kar, I, ..N. (2008). Design of asymptotically convergent frequency estimator using contraction theory. IEEE Trans. on Auto. Control, 53(8), 1932–1937. Slotine, J. and Li, W. (1991). Applied Nonlinear Control. Prentice Hall. Song, B. and Hedrick, J.K. (2004). Observer-based dynamic surface control for a class of nonlinear systems: an lmi approach. IEEE Trans. Auto. Control, 49(11), 1995–2001. Sontag, E.D. (2010). Contractive systems with inputs. In J. Willems, S. Hara, Y. Ohta, and H. Fujioka (eds.), Perspectives in Mathematical System Theory, Control, and Signal Processing, volume 398 of Lecture Notes in Control and Information Sciences, 217–228. Springer Berlin Heidelberg. Swaroop, D., Gerdes, J., Yip, P., and Hedrick, J. (2000). Dynamic surface control for a class of nonlinear systems. IEEE Trans. on Auto. Control, 45(10), 1893–1899. Wang, D. and Huang, J. (2005). Neural network-based adaptive dynamic surface control for a class of uncertain nonlinear systems in strict-feedback form. IEEE Trans. on Neural Networks, 16(1), 195–202. Wang, W. and Slotine, J.J.E. (2005). On partial contraction analysis for coupled nonlinear oscillators. Biological Cybernetics, 92(1), 38–53. Wu, J., Huang, J., Wang, Y., and Xing, K. (2014). Nonlinear disturbance observer-based dynamic surface control for trajectory tracking of pneumatic muscle system. IEEE Trans. on Control Systems Technology, 22(2), 440–455. Zamani, M. and Tabuada, P. (2011). Backstepping design for incremental stability. IEEE Trans. on Auto. Control, 56(9), 2184–2189. Zhang, T. and Ge, S. (2008). Adaptive dynamic surface control of nonlinear systems with unknown dead zone in pure feedback form. Automatica, 44(7), 1895 – 1903. Zhao, J. and Kanellakopoulos, I. (1998). Flexible backstepping design for tracking and disturbance attenuation. International Journal of Robust and Nonlinear Control, 8(4-5), 331–348. Zhou, L. and Yin, L. (2014). Dynamic surface control based on neural network for an air-breathing hypersonic vehicle. Optimal Control Applications and Methods.
TrackingError (z1)
6 4 2 0 -2 -4 -6 0
1
2
3
Simulation Time
4
35
5
Fig. 2. Tracking Error (After Tuning) terms of known constants which are helpful in tuning the closed loop system performance. The procedure relaxes the constraint on magnitude of perturbation parameter and also circumvent the restrictions of quadratic Lyapunov based analysis. The method is validated using a simulation example. REFERENCES Angeli, D. (2002). A lyapunov approach to incremental stability properties. IEEE Trans. on Auto. Control, 47(3), 410–421. Del Vecchio, D. and Slotine, J.J.E. (2013). A contraction theory approach to singularly perturbed systems. IEEE Trans. on Auto. Control, 58(3), 752–757. Farrell, J., Polycarpou, M., Sharma, M., and Dong, W. (2009). Command filtered backstepping. Automatic Control, IEEE Transactions on, 54(6), 1391–1395. Forni, F. and Sepulchre, R. (2014). A differential lyapunov framework for contraction analysis. IEEE Tran. on Auto. Control, 59(3), 614–628. Gerdes, J.C. and Hedrick, J.K. (2002). loop-at-a-time design of dynamic surface controllers for nonlinear systems. Journal of dynamic systems, measurement, and control, 124(1), 104–110. Huang, J., Ri, S., Liu, L., Wang, Y., Kim, J., and Pak, G. (2015). Nonlinear disturbance observer-based dynamic surface control of mobile wheeled inverted pendulum. IEEE Trans. on Control Systems Technology. Jouffroy, J. and Lottin, J. (2002a). Integrator backstepping using contraction theory: a brief technological note. Proceedings of the 15th IFAC World Congres, Barcelona, Spain, 15, 238–244. Jouffroy, J. and Lottin, J. (2002b). On the use of contraction theory for the design of nonlinear observers for ocean vehicles. Proceedings of American Control Conference, 2647–2652 vol.4. Kanellakopoulos, I., Kokotovic, P., and Morse, A. (1992). A toolkit for nonlinear feedback design. Systems & Control Letters, 18(2), 83 – 92. Khalil, H. (2002). Nonlinear Systems. Prentice Hall; Third edition. Lohmiller, W. and Slotine, J.J.E. (1998). On contraction analysis for non-linear systems. Automatica, 34(6), 683– 696. Mehraeen, S., Jagannathan, S., and Crow, M. (2011). Power system stabilization using adaptive neural network-based dynamic surface control. IEEE Trans. on Power Systems, 26(2), 669–680. 35
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IFAC PapersOnLine 51-1 (2018) 30–35
Design Design of of Contractive Contractive Output Output Feedback Feedback Design of Contractive Output Feedback Controllers For Filtered Backstepping Design of Contractive Output Feedback Controllers For Filtered Backstepping Controllers For Filtered Backstepping Controllers For Filtered Backstepping ∗∗ M. M. Rayguru ∗∗∗ I. N. Kar ∗∗ ∗∗
M. M. Rayguru ∗∗ I. N. Kar ∗∗ ∗∗ M. M. Rayguru ∗ I. N. Kar ∗∗ M. M. Rayguru I. N. Kar ∗ ∗ Indian Institute of Technology Delhi, New Delhi, India, Pin:110016 ∗ Indian Institute of Technology Delhi, New Delhi, India, Pin:110016 ∗ ∗ Indian Institute of Technology Delhi, New Delhi, India, Pin:110016 (e-mail: [email protected]). (e-mail: [email protected]). ∗ ∗∗ Indian Institute of Technology Delhi, New Delhi, India, Pin:110016 ∗∗ Indian Institute of Technology Delhi, (e-mail: [email protected]). ∗∗ Indian Pin:110016 Institute of Technology Delhi, New New Delhi, Delhi, India, India, Pin:110016 ∗∗ ∗∗ (e-mail: [email protected]). (e-mail: [email protected]) Indian Institute of Technology Delhi, New Delhi, India, Pin:110016 (e-mail: [email protected]) ∗∗ Indian Institute of Technology Delhi, New Delhi, India, Pin:110016 (e-mail: [email protected]) (e-mail: [email protected]) Abstract: Abstract: This This paper paper proposes proposes a a novel novel observer observer based based filtered filtered backstepping backstepping controller controller for for aa class class Abstract: This paper proposes a novel observer based filtered backstepping controller for adesign class of nonlinear systems. For this purpose, the contraction theory based tools are exploited to of nonlinear systems. For this purpose, the contraction theory based tools are exploited to design Abstract: This paper proposes a novel observer based filtered backstepping controller for adesign class an exponentially convergent linear time varying observer. The proposed observer satisfies the of nonlinear systems. For this purpose, the contraction theory based tools are exploited to an exponentially convergent time observer. The proposed observer satisfies the of nonlinear systems. For this linear purpose, thevarying contraction theory based tools are exploited to design sufficient observability condition and the resulting output feedback controller assures ultimate an exponentially convergent linear time varying observer. The proposed observer satisfies the sufficient observability condition and the resulting output feedback controller assures ultimate an exponentially convergent linear time varying observer. The proposed observer satisfies the boundedness of closed loop trajectories. The use of partial contraction concept for convergence sufficient observability condition and the resulting output feedback controller assures ultimate boundedness of closed loop trajectories. The use of output partial feedback contraction conceptassures for convergence sufficient observability condition and the resulting controller ultimate boundedness of closed loop trajectories. The use of partial contraction concept for convergence analysis serves serves a a dual dual purpose. purpose. It It relaxes relaxes the the choice choice of of high high gain gain filter filter parameter, parameter, whose whose analysis boundedness of closed loop trajectories. The use of partial contraction concept for convergence magnitude needs be constrained aa conservative range Lyapunov based approaches. analysis serves a to dual purpose. It inside relaxes the choice of highin gain filter parameter, whose magnitude needs to be constrained inside conservative range in Lyapunov based approaches. analysis serves a todual purpose.between It inside relaxes theloop choice of highingain filter parameter, whose Moreover, the exact dependency closed system performance and design parameters magnitude needs be constrained a conservative range Lyapunov based approaches. Moreover, the exact dependency between closed loop system performance and based designapproaches. parameters magnitude needs to be constrained inside a conservative range in Lyapunov are quantified in of constants. The developed technique successfully simulated Moreover, the exact dependency closed system performance design parameters are quantified in terms terms of known knownbetween constants. Theloop developed technique is is and successfully simulated Moreover, exact dependency closed system performance design parameters are in terms ofmanipulator knownbetween constants. Theloop developed technique is and successfully simulated for aquantified D.C. the motor driven manipulator system. for a D.C. motor driven system. are in terms known constants. for aquantified D.C. motor drivenofmanipulator system.The developed technique is successfully simulated © 2018, IFAC (International Federation ofsystem. Automatic Control) Hosting by Elsevier Ltd. All rights reserved. for a D.C. motor driven manipulator Keywords: Keywords: Filtered Filtered Backstepping, Backstepping, Time Time Varying Varying Observer, Observer, Singular Singular Perturbation, Perturbation, Contraction Contraction Keywords: Filtered Backstepping, Time Varying Observer, Singular Perturbation, Contraction Theory, Convergence Convergence Analysis Theory, Analysis Keywords: Filtered Backstepping, Time Varying Observer, Singular Perturbation, Contraction Theory, Convergence Analysis Theory, Convergence Analysis 1. INTRODUCTION loop system system is is guaranteed (Swaroop (Swaroop et et al., 2000; 2000; Farrell 1. loop 1. INTRODUCTION INTRODUCTION loop system is guaranteed guaranteed (Swaroop et al., al., 2000; Farrell Farrell et al., 2009; Pan and Yu, 2015) using either Tikhonov theoet al., 2009; Pan and Yu, 2015) using either Tikhonov theo1. INTRODUCTION loop system is guaranteed (Swaroop et al., 2000; Farrell et al., 2009; Pan and Yu, 2015) using either Tikhonov theorem or Lyapunov indirect method. In both these methods, rem or Lyapunov indirect method. In both these methods, Backstepping controllers controllers are are efficiently efficiently utilized utilized for for solvsolv- et al., 2009; Pan and Yu, 2015) using either Tikhonov theoBackstepping or Lyapunov indirect method.for Inaboth the stability stability can only only be assured assured verythese smallmethods, range of of Backstepping controllers are problems efficiently inutilized for engisolv- rem ing stabilization stabilization and tracking tracking numerous the can be very small range rem stability orparameter. Lyapunov indirect method.for Inaaboth these methods, ing and problems in numerous engithe can only be assured for very small range of filter Backstepping controllers are efficiently utilized for solving stabilization and involving tracking problems in numerous engi- filter neering applications electro-mechanical systems, parameter. thenew stability can only be assured for a very small of neering applications involving electro-mechanical systems, filter parameter. A approach to stability stability analysis, based onrange virtual ing stabilization and trackingetc problems in numerous engineering applications involving electro-mechanical systems, robotics, chemical reactions (Khalil, 2002; Slotine and A new approach to analysis, based on virtual filter parameter. robotics, chemical reactions etc (Khalil, 2002; Slotine and A new approach to stability analysis, based on virtual displacements and referred as called contraction theory, neering applications involving systems, robotics, reactions etcelectro-mechanical (Khalil, 2002; Slotine and displacements andtoreferred as called contraction theory, Li, 1991). The involves repetitive calculation Li, 1991).chemical The methodology methodology involves repetitive calculation A proposed new approach stability based virtual referred as analysis, called theory, is in and (Lohmiller and Slotine,contraction 1998), on which exrobotics, reactions etc (Khalil, 2002; Slotine and displacements Li, 1991).chemical Thefor methodology involves repetitive calculation of derivatives a number of signals called virtual control is proposed in (Lohmiller and Slotine, 1998), which exdisplacements and referred as called contraction theory, of derivatives for a number of signals called virtual control is proposed in (Lohmiller and Slotine, 1998), which examines the convergence behavior of system trajectories Li, 1991). The methodology involves repetitive calculation of derivatives for a numberetof al., signals called virtual inputs (Kanellakopoulos 1992; Zhao and control Kanel- amines the behavior of trajectories is proposed in and 1998), whichdoes exinputs (Kanellakopoulos et 1992; Zhao and Kanelamines the convergence convergence behavior of system system trajectories with respect to (Lohmiller each other. other. TheSlotine, differential analysis of derivatives for aZamani numberand signals called virtual inputs (Kanellakopoulos etof al., al., 1992; Zhao and control Kanelwith respect to each The differential analysis does lakopoulos, 1998; Tabuada, 2011; Jouffroy and amines the convergence behavior of system trajectories lakopoulos, 1998; Zamani and Tabuada, 2011; Jouffroy and with respect to each other. The differential analysis does not necessitate a priori presence of any attractor and has inputs (Kanellakopoulos et al., 1992; 2011; Zhao Jouffroy and Kanellakopoulos, 1998;These Zamani and Tabuada, necessitate priori presence of any attractor anddoes has Lottin, 2002a). analytical calculations become unLottin, 2002a). These analytical calculations become and un- not withnecessitate respect to aaeach other. The differential analysis priori presence ofsee any attractor andForni has strong interconnection properties, (Angeli, 2002; lakopoulos, 1998; Zamani and Tabuada, 2011; Jouffroy and Lottin, 2002a). These analytical calculations become un- not reliable with increase in the system order and this problem strong interconnection properties, see (Angeli, 2002; Forni not necessitate a priori presence ofsee any attractor andForni has reliable with increase in the order and problem interconnection properties, (Angeli, and Sepulchre, 2014; Wang and Slotine, Slotine, 2005).2002; This techLottin, Theseof become un- strong reliable with increase inanalytical the system systemcalculations order and this this problem is called2002a). as explosion explosion complexity (Swaroop et al., 2000). and Sepulchre, 2014; Wang and 2005). This techstrong interconnection properties, see (Angeli, 2002; Forni is called as of complexity (Swaroop et al., 2000). and Sepulchre, 2014; Wang and Slotine, 2005). This technique is is effectively effectively utilized utilized for for many many control control problems problems like like reliable with increase of inofcomplexity the order and this problem is called asphilosophy explosion (Swaroop et al., 2000). The core thesystem filtered backstepping meth- nique and Sepulchre, 2014; Wang and Slotine, 2005). This techThe core philosophy the filtered backstepping methnique is effectively utilized for many control problems like observer design (Jouffroy and Lottin, 2002b), estimator deis called aspass explosion ofof complexity (Swaroop et al.,through 2000). The core philosophy of the filtered backstepping methobserver design (Jouffroy and Lottin, 2002b), estimator deods is to the variables to be differentiated ods iscore to pass the variables to be differentiated through nique(Sharma is effectively utilized forLottin, many 2002b), control problems like observer design (Jouffroy and estimator design and Kar, 2008), synchronization (Wang and The philosophy of the filtered backstepping methods is to passorthe variables to high be differentiated through a first order second order gain filter (Pan and sign (Sharma and Kar, 2008), synchronization (Wang and observer design (Jouffroy anddesign Lottin, 2002b),and estimator dea order or second orderto high gain filter (Pan and (Sharma and Kar, 2008), synchronization (Wang and Slotine, 2005), backstepping (Zamani Tabuada, odsfirst is to pass variables be differentiated through aYu, first order orthe second gain (Pan and sign 2015; Swaroop et al., al.,order 2000;high Song andfilter Hedrick, 2004; Slotine, 2005), backstepping design (Zamani and Tabuada, sign (Sharma and Kar, 2008), synchronization (Wang and Yu, 2015; Swaroop et 2000; Song and Hedrick, 2004; Slotine, 2005), backstepping design (Zamani and Tabuada, 2011; Rayguru Rayguru and and Kar, Kar, 2015) 2015) etc. etc. Recently Recently aa contraction contraction a first or second high gain and Yu, 2015; et al.,order Song andfilter Hedrick, 2004; Wu et order al.,Swaroop 2014) which in2000; turn circumvent the(Pan need of 2011; Wu et al., 2014) which in turn circumvent the need of Slotine, 2005),approach backstepping design and Tabuada, 2011; Rayguru and Kar,is2015) etc. (Zamani Recently a contraction theory based proposed for analysis analysis of singusinguYu, 2015; Swaroop et al.,in 2000; Song and Hedrick, 2004; Wu et al., 2014) which turn circumvent the need of analytic differentiations. Dynamic surface control (DSC) theory based approach is proposed for of analytic differentiations. Dynamic surface control (DSC) 2011;perturbed Rayguru and Kar,is 2015) etc. Recently a contraction theory based approach proposed for analysis of singularly systems in (Del Vecchio and Slotine, 2013) Wu et al., 2014) which in turn circumvent the need of analytic Dynamic surface control techniquedifferentiations. (Swaroop et et al., al., 2000) uses uses first order (DSC) filters larly perturbed systems in (Del Vecchio and Slotine, 2013) theory based approach is proposed forclass analysis of singutechnique (Swaroop 2000) first order filters larly perturbed systems in (Del Vecchio and Slotine, 2013) and extended to stabilization of same of systems by analytic differentiations. Dynamic surface control (DSC) technique (Swaroopfiltered et al., backstepping 2000) uses first order et filters whereas command command (Farrell al., and to stabilization same systems by larlyextended perturbed systems in (Delof and of Slotine, 2013) whereas filtered backstepping (Farrell et al., and extended to stabilization ofVecchio same class class of systems by (Rayguru and Kar, 2017). technique (Swaroop et al., 2000) uses first order filters whereas command filtered backstepping (Farrell et al., (Rayguru and Kar, 2017). 2009) uses uses second second order order filters filters for for this this purpose. purpose. The The high high and extended to stabilization of same class of systems by 2009) and Kar, 2017). whereas command filtered backstepping (Farrell al., (Rayguru 2009) uses second order filters for this purpose. Theethigh gain filters not only act as approximate differentiators but Motivation and Contribution The work in in this this paper paper gain filters not only act as approximate differentiators but Motivation (Rayguru andand Kar,Contribution 2017). The 2009) uses second order filters for this purpose. The high gain filterscertain not only act as approximate differentiators also relax relax differentiability assumptions requiredbut for Motivation and Contribution The work workin inthe thiscurrent paper is motivated by the following limitations also certain differentiability assumptions required for is motivated by the following limitations in the current gain relax filterscertain notcontroller only act as approximate differentiators Motivation and Contribution The work thiscurrent paper also differentiability assumptions required for backstepping design. Various extensions of but fil- is motivated by the inin the literature (Swaroop etfollowing al., 2000; 2000;limitations Pan and Yu, 2015; Farrell backstepping controller design. Various extensions of filliterature (Swaroop et al., Pan and Yu, 2015; Farrell also relax certain differentiability assumptions required for is motivated by the following limitations in the current backstepping controller design. Various extensions of filtered backstepping backstepping are are proposed proposed by by the the authors authors of of (Gerdes (Gerdes literature (Swaroop et al., 2000; Pan and Yu, 2015; Farrell et al., al., 2009) 2009) on filtered backstepping. tered on backstepping. backstepping controller design. Various extensions ofand fil- et literature (Swaroop et al., 2000;with Pan filtered and Yu, backstepping 2015; Farrell tered backstepping proposed by the authors of (Gerdes and Hedrick, 2002;are Song and Hedrick, Hedrick, 2004; Wang et al., 2009) on filtered filtered backstepping. (i) Most of the results dealing and Hedrick, 2002; Song and 2004; Wang and Most of the results dealing with filtered backstepping teredHedrick, backstepping proposed by Mehraeen the authors ofal., (Gerdes et al., 2009) on filtered backstepping. and 2002;are Song and2008; Hedrick, 2004;etWang and (i) Huang, 2005; Zhang and Ge, 2011; (i) Most of the results dealing with filtered backstepping assume full state measurement which are not suitable Huang, 2005; Zhang and Ge, Mehraeen et al., 2011; assume full stateresults measurement whichfiltered are notbackstepping suitable for for and Hedrick, 2002; Song and2008; Hedrick, 2004; and (i) Mostpractical of the dealing with Huang, 2005; Zhang and Ge, 2008; Mehraeen etWang al.,et 2011; Zhou and Yin, 2014; Huang et al., 2015; Peimani al., assume full state measurement which are not suitable for various applications. Zhou and Yin, 2014; Huang et al., 2015; Peimani et al., various practical applications. Huang,and 2005; and Ge, 2008; Mehraeen et in al.,et 2011; assume full statefiltered measurement which are not do suitable for Zhou Yin,Zhang 2014; Huang 2015; Peimani al., various 2016) to different type uncertainties applications. (ii) The practical existing backstepping designs not concon2016) to handle handle different typeetof ofal., uncertainties in system system The existing filtered backstepping designs do not Zhou and Yin,to2014; Huang 2015; Peimani et al., (ii) various practical applications. 2016) to handle different typeetpractical ofal., uncertainties in system dynamics and suit different applications. (ii) The existing filtered backstepping designs do not consider time time dependent dependent nonlinearities nonlinearities in in system system dynamics. dynamics. dynamics and to different suit different practical applications. 2016)use toofhandle type of uncertainties in system sider (ii) The existing filtered backstepping do not condynamics and suitfilters different practical applications. The highto gain give rise rise to time time scale separation separation sider time dependent nonlinearities in designs system dynamics. (iii) The restriction on filter parameters to be inside a The use of high gain filters give to scale (iii) The restriction on filter parameters to be inside a small small dynamics and to suitfilters different practical applications. siderThe time dependent nonlinearities in noise system dynamics. The use ofloop highdynamics gain give rise to time scalea separation in closed and transform it into singularly (iii) restriction on filter parameters to be inside a small bound may not hold in presence of and sampling in closed dynamics and transform it into singularly bound not in of sampling The use ofloop high gain filters give rise to time scalea (iii) Themay restriction on filter parameters to beand inside a small in closed loop dynamics and transform into a separation singularly perturbed form. Semi-global practical stability of may not hold hold in presence presence of noise noise and sampling perturbed form. Semi-global practical it stability of closed closed bound in closed loop dynamics and transform it into a singularly perturbed form. Semi-global practical stability of closed bound may not hold in presence of noise and sampling perturbed Semi-global practical stability of closed 2405-8963 © form. 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
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restrictions. (iv) The dependence of closed loop system performance on design parameters are not rigorously discussed in current literature. In order to overcome the above mentioned limitations, this paper develops a time varying observer based filtered backstepping controller for a class of nonlinear systems. The utilization of contraction theory ensures a exponentially convergent observer unlike (Song and Hedrick, 2004) and relaxes the constraint on the magnitude of filter parameter. Furthermore the tracking error bound is quantified in terms of design parameters which can be helpful in tuning the controller performance. Notations:The following notations are used throughout the paper. Rm denotes a m-dimensional real vector space. For real vectors v, |v| denotes the Euclidean norm and for real matrix |E| denotes induced matrix norm. Lipschitz constant of the functions are denoted as Li where i denotes a positive scalar. A function is a C k function if its derivatives upto kth order are continuous. A functions belongs to L∞ if it is essentially bounded. The maximum and minimum eigen value of a matrix is denoted by λmax (.) and λmin (.) respectively. A metric Θ denotes a symmetric positive definite matrix and In is an n × n identity matrix.
31
χd (4) λ where |d(xp , t)| ≤ d and χ is the condition number of contraction metric Θ. Another useful tool called partial contraction (Wang and Slotine, 2005) is used in this paper and briefly summarized in the form of a lemma. Definition: Consider a system expressed as: ||xp (t) − x(t)|| ≤ χe−λt ||xp (0) − x(0)|| +
x˙ = f (x, x, t).
(5)
Another system defined as y˙ = f (x, y, t)
(6)
is called a virtual/auxiliary system of (5) if the solution x is also a particular solution of (6). In other words the dynamics (6) can be regarded as a copy of (5). Lemma 3: A System x˙ = f (x , y, t) is said to be partially contracting in x if an virtual system defined as z˙ = f (z , y, t) is contracting in z for any value of y, ∀t > 0 . If the auxiliary system verifies a smooth specific property, then the trajectories of original system will also verify that property exponentially (Wang and Slotine, 2005). Furthermore if a system x˙ = f (x, y) is partially contracting in x, then there exists a unique smooth global mapping x = p(y) such that f (x, y) = 0. The proof can be found in (Del Vecchio and Slotine, 2013).
1.1 Prerequisites From Contraction Theory Let’s consider a nonlinear dynamical system expresses as x˙ = f (x, t). Further assume the following relation is satisfied for the nonlinear system in a region: (1) F + F T ≤ −λI ∂f −1 ˙ where F = (Θ + Θ ∂x )Θ , Θ is a nonsingular metric and λ > 0 is a positive constant. The region in which (1) is satisfied is called a contraction region (Lohmiller and Slotine, 1998) and every trajectory inside the contraction region exponentially converge towards each other irrespective of their initial conditions. The system for which F is negative definite is contracting with an associated metric Θ and a contraction rate λ. For brevity, the system is said to be contracting with (Θ, λ). The inequality (1) can be generalized as: ∂f ∂f T (M˙ + M + M ) ≤ −2λM (2) ∂x ∂x where M (x, t) = ΘT Θ is called a contraction metric. If (2) is relaxed to a negative semidefinite condition, the system is semi-contracting. The following lemmas discuss some important properties of contraction taken from Lohmiller and Slotine (1998), Sontag (2010), Wang and Slotine (2005) and Rayguru and Kar (2017) etc.
2. SYSTEM DESCRIPTION Consider a class of nonlinear system in the form of: x˙ 1 = f1 (x1 ) + b1 (x1 )x2 x˙ 2 = f2 (x1 , x2 ) + b2 (x1 , x2 )x3 ... x˙ n = fn (x1 , x2 , x3 ....xn ) + bn (x1 , x2 , . . . , xn )u.
(7)
y = x1 The functions fi (.), bi (.) are smooth, the terms bi (.) > 0 ∀x = [x1 , x2 , ..., xn ]T ∈ Rn , u ∈ R is the control input and y is the system output. The objective of the controller is to ensure that, the output x1 (t) tracks a desired trajectory xd (t). Following assumptions hold true for (7). Assumption 1 : The reference signal xd (t) and x˙ d (t) are bounded. xi ) xi ) i (¯ i (¯ Assumption 2: The terms, fi (¯ xi ), bi (¯ xi ), ∂f∂x and ∂b∂x i i T are bounded ∀ i ∈ 1, . . . , n where (¯ xi = [x1 , x2 , . . . , xi ] ). Above assumptions are standard in the literature dealing with filtered backstepping technique (Swaroop et al., 2000; Farrell et al., 2009; Pan and Yu, 2015) and can be satisfied for numerous engineering applications (Gerdes and Hedrick, 2002; Huang et al., 2015; Pan and Yu, 2015).
Lemma 1: Suppose the autonomous dynamical system defined as x˙ = f (x) is contracting with (Θ, λ) in some region of the state space. Then there exists an unique equilibrium point inside that region to which all the trajectories converge exponentially The following lemma discusses the robustness property of contraction. Lemma 2: Consider a dynamical system expressed as: (3) x˙ p = f (xp , t) + d(xp , t). Moreover assume that x˙ = f (x, t) is contracting with (Θ, λ). The trajectories of (3) verify the following bounds:
3. OUTPUT FEEDBACK FOR FILTERED BACKSTEPPING The system (7) can be rewritten as x˙ = F (x) + B(x)u y = Cx 31
(8)
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T f1 (x1 ) + b1 (x1 )x2 1 f2 (x1 , x2 ) + b2 (x1 , x2 )x3 .. where F (x) = .. , C = . . 0 fn (x1 , x2 , . .. , xn ) 0 .. and B(x) = . A filtered backstepping . bn (x1 , x2 , ..., xn ) control law is proposed as: 1 u= [−fn (ˆ x) − bn−1 zn−1 − χn (ˆ zn ) + α˙ (n)f ] bn (ˆ x) ˆi − xi , (i = 1, 2......n) zi = xi − αif + x α1f = α1 = xd (t) (9) 1 [−fi−1 (ˆ x1 , .., x ˆi−1 ) − χi−1 (zi−1 ) α ˆi = bi−1 − bi−2 zi−2 + α˙ (i−1)f ] for i ∈ [2 . . . n]
µα˙ f = g(αf , α ˆ ) = −αf + α. ˆ
0 1 FT = ... 0
... ... .. . ...
0 −ao 0 −a1 . 0 .. 1 −an
(15)
where the positive scalars ai , i = (1, . . . , n) are selected such that, they constitute the coefficients of a Hurwitz polynomial. The equation (15) can be simplified in terms of column vectors θi of the transformation matrix Θ as: θ˙i + (A − LC)θi = θi+1 , i = 1, . . . , n − 1 a0 a1 θ˙i + (A − LC)θi = −[θ1 , . . . , θn ] ... .
(16)
Cθ1 = L0 Cθ1 = 0, Cθ2 = L1 Cθ1 = 0, Cθ3 = L2 Cθ1 = 0, .. . Cθn = Ln−1 Cθ1 = det(|L0 C . . . Ln−1 C|) = D
(17)
an
It is important to make the procedure of selecting θi independent of the choice of control input. Hence the following constraints are recursively imposed on the selection of θi .
(10)
T
ˆn ] denotes the estimate of x, αf = where x ˆ = [ˆ x1 , . . . , x [α1f , . . . , αnf ], µ ∈ (0, 1) is the high gain filter parameter and α ˆ = [ˆ α1 , . . . , α ˆ n ]T . It can be noted that, the high gain filters (20) circumvent the repeated analytical calculation of derivatives of virtual control signals α˙ i . 3.1 Observer Design
where Li C are Lie derivatives for which L0 C = C, d Li+1 C = Li CA + dt (Li C) and the symbol det(.) denotes the determinant of a matrix. The equations (17) can be solved for choosing a smooth θ1 (t). Then the other columns θi of Θ(t) can be obtained recursively through the relation given by:
The idea is to design a linear observer which is contracting inside the region of interest containing the desired trajectory xd (t). In order for the trajectory x1 (t) to track the desired signal xd (t), other state variables xi (t) have to track αi (t), i : (1, . . . n) respectively. As the terms fi (¯ xi ), bi (¯ xi ) and its first partial derivatives with respect to its arguments are bounded, there exist time varying matrices A(t), B(t) such that, ∂F (x) (11) A(t) = ∂x x1 =xd ,x2 =α2 (.),...,xn =αn (.) . (12) B(t) = B(x)
θi+1 = −θ˙i + Aθi .
(18)
The observer gain matrix can be obtained by solving a0 a1 ˙ LD = [θ1 , . . . , θn ] (19) ... − θn + Aθn . an
x1 =xd ,x2 =α2 (.),...,xn =αn (.)
An observer is proposed as x ˆ˙ = A(t)ˆ x + B(t)u + L(t)(y − C x ˆ)
0 0 .. . 0
If the contraction metric and observer gain are chosen according to (18) and (19) respectively, then the observer dynamics (13) is contracting with respect to x ˆ. Therefore the differential displacement vector δ x ˆ exponentially converges to zero. The assumptions 1 and 2 assure that, the matrix M = Θ−T Θ is uniformly bounded for the choice of observer gain (19) and the metric Θ. Therefore the sufficient observability condition (Θ−T Θ is bounded) is satisfied and the estimate x ˆ converges to x exponentially (Lohmiller and Slotine, 1998). Remark: Theorem 1 proves that, the LTV observer proposed in (13) is capable in estimating the states of the nonlinear system defined in (7). It is not generally true for other classes of nonlinear systems because the sufficient observability condition may not hold for them. However for the class of systems defined in (7), it is always possible to select a contraction metric Θ(t) and an observer gain L(t) such that the estimation error (ˆ x − x) exponentially converges to zero.
(13)
where the observer gain L(t) is decided later. The convergence of observer dynamics is summarized in the following theorem. Theorem 1: Let the assumptions 1 and 2 are satisfied for the system (7) and an observer is designed in the form of (13). If the observer gain is selected as given in (19), then the estimation error (ˆ x − x) exponentially converges to zero. Proof: Let there exists a smooth transformation matrix Θ(t) such that the generalized Jacobian of (13) with resepct to x ˆ can be expressed as ˙ + (A − LC)Θ . (14) FT = Θ−1 − Θ
The observer gain L(t) and the contraction metric Θ has to be selected such that, (13) is contracting with respect to x ˆ (Lohmiller and Slotine, 1998). The observer dynamics (13) is contracting in x ˆ, if the generalized Jacobian FT is a Hurwitz matrix, i.e,
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4. DERIVATION OF TRACKING ERROR BOUND
ˆ ≤ µc1 . lim ||αf (t) − α(t)||
t→∞
z˙iro = −bi−1 z(i−1)ro − χi (ziro ) + bi z(i+1)ro + 2bi α ¯ i+1 + x ˜˙ i ˙ z˙nro = −bn−1 z(n−1)ro − χn (znro ) + x ˜n (27)
z˙i = bi−1 zi−1 − χi (zi ) + bi zi+1 + bi α ˜ i+1 + bi α ¯ i+1 + x ˜˙ i z˙n = −bn−1 zn−1 − χn (zn ) + x ˜˙ n (21) ˙x ˆ = A(t)ˆ x + B(t)u + L(t)(y − C x ˆ). (22)
where zro = [z1ro , ..., znro ]T denote the states of the reduced slow system. Rewriting (27) in a compact form z˙ro = f (zro , α, α)+2¯b+F (ˆ x)−F (x)+(B(ˆ x)−B(x))u (28) where −χ1 (z1ro ) + b1 z2ro .. . f (zro , α, α) = b n−2 z(n−2)ro − χn−1 (z(n−1)ro ) + bn−1 znro −bn−1 z(n−1)ro − χn (znro )
The dynamics (21) can be expressed in a compact form as: z˙ = f (z, αf , α) + ¯b + F (ˆ x) − F (x) + (B(ˆ x) − B(x))u. (23) ¯ 1 , b2 α ¯ 2 , . . . , bn−1 α ¯ n−1 , 0]T and where ¯b = [b1 α −χ1 (z1 ) + b1 z2 + b1 α ˜2 . .. . f (.) = b z −χ (z )+b z +b α ˜ n−1
n−1
n−1 n
(26)
As the fast dynamics (20) is partially contracting in αf , there exists a smooth slow manifold (α + α ¯ ) inside its contraction region. Therefore αf → (α + α) ¯ ⇒α ˜→α ¯. The reduced slow system can be obtained can be expressed as: ¯2 + x ˜˙ 1 z˙1ro = −χ1 (z1ro ) + b1 z2ro + 2b1 α
The overall closed loop system comprises of the dynamics (7), (13), (20) and the control law (9). This section intends to prove ultimate boundedness of closed loop trajectories and derive staedy state tracking error bound. Define x ˜= ¯=α ˆ − α, the fast filter dynamics x ˆ − x, α ˜ = αf − α and α (20) can be expressed as: µα˙ f = g(αf , α) + α ¯ = −αf + α + α. ¯ (20) Using the control law (9),(20), the closed loop dynamics can be expressed as: ˜ 2 + b1 α ¯2 + x ˜˙ 1 z˙1 = −χ1 (z1 ) + b1 z2 + b1 α
n−2 n−2
33
The nominal system of (28) can be expressed as: z˙r = f (zr , α, α) (29) where zr denotes the states of (28) when the perturbation terms are absent. As the Jacobian of f (z, α, α) is uniformly negative definite, the nominal system is contracting with an identity matrix in the absence of perturbation terms. Exploiting assumptions 1 and 2, the perturbation terms α ¯i, x ˜˙ i are bounded and hence the perturbation terms (2¯b+ F (ˆ x) − F (x) + (B(ˆ x) − B(x))u) in (27) are bounded. Exploiting lemma 2, the following can be obtained. 2||¯b|| + (L4 + L5 )||ˆ x − x|| lim ||zr (t) − zro (t)|| ≤ (30) t→∞ β where β is the contraction rate of the nominal system (29) and L4 , L5 are Lipschitz constant for functions F (.) and B(.) respectively. From the assumptions 1 and 2, the term ¯b is also Lipschitz in its arguments, i.e ||¯b|| ≤ L2 ||ˆ α − α||. Therefore ˆ − α|| + (L4 + L5 )||ˆ x − x|| 2L2 ||α . lim ||zr (t) − zro (t)|| ≤ t→∞ β (31) As the estimation error converges to zero, lim ||zr (t) − zro (t)|| ≤ 0. (32)
n−1 n
−bn−1 zn−1 − χn (zn ) The overall closed loop system is in singularly perturbed form and the convergence analysis follows similar steps as given in (Rayguru and Kar, 2017). Define an auxiliary system as: µα˙ des = g(αdes , α) + µQ(z, α ˆ ) + g(αdes , α ˆ ) − g(αdes , α) (24) ˆ˙ . As µα ˆ˙ = g(α ˆ , α) + µQ(z, α ˆ) where Q(z, α) = ∂∂zαˆ z˙ + ∂∂αxˆˆ x and g(α, α) = 0), the signal αf is a particular solution of (24). In similar manner, it can be argued that the variable αf is also a particular solution of (24) when the perturbation term µQ(z, α) is absent. The observer (13) is contracting in x ˆ and hence starting from any initial condition x ˆ˙ is always bounded inside a finite region of interest. Exploiting assumptions (12), the terms α(.), α ˆ (.) and their derivatives are also bounded. Therefore the terms Q(z, α ˆ ) can be assumed to be uniformly bounded. Therefore there exists a positive ˆ )|| ≤ c1 inside (Bz × Bα ). scalar c1 > 0 such that ||Q(z, α Exploiting Lipschitz property of g(.) and α, ||g(αdes , α ˆ ) − g(αdes , α)|| ≤ L1 ||α ˆ − α||. Hence (24) can be treated as a perturbed virtual system of (20). The unperturbed dynamics of (24) is partially contracting in αdes with a contraction metric M = I. Exploiting lemma 2, the steady state convergence bound between the variables αf and α can be derived as: ˆ ≤ µ(c1 + L1 ||α lim ||αf (t) − α(t)|| ˆ − α||). (25)
t→∞
It can be noted that, (23) is a combination of nominal contracting system and extra perturbation terms i.e z˙ = f (z, α, α) + f (z, αf , α ˆ ) − f (z, α, α) + ¯b
+ F (ˆ x) − F (x) + (B(ˆ x) − B(x))u. (33) The dynamics (33) can be argued to be a perturbed virtual system of (28). Exploiting lemma 2 and Lipschitz inequality, the following result can be obtained: L6 c1 + ||¯b|| (34) lim ||z(t) − zr (t)|| ≤ µ t→∞ β where β is the contraction rate of the reduced slow system (28) and L6 is the Lipschitz constant for the function f (.). As the observer dynamics is contracting, the estimation
t→∞
As α(.) is Lipschitz in its arguments therefore, x − x||. ||ˆ α − α|| ≤ L3 ||ˆ The observer dynamics is contracting and therefore the estimation error (ˆ x − x) exponentially converges to zero and lim ||α ˆ − α|| → 0. t→∞
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The variables x1 , x2 , x3 correspond to position, velocity and armature current respectively and the control input voltage is denoted as u. The system parameters are given by: M = J + mL20 /3 + M0 L20 + 2M0 R02 /5/Kt , N = mL0 g/2 + M0 L0 g/Kt and B = B0 /Kt . J = 1.625103 Kg m2, m = 0.506 Kg, M0 = 0.434 Kg, L0 = 0.305 m, R0 = 0.023 m, B0 = 16.25163 N m s/rad, L = 25.0103 H, R = 5.0 , Kt = Kb = 0.90 N m/A and g = 9.8 m/s2. The controller design objective is position tracking of a desired signal xd = (π/2)sin(8πt/5). The tuning functions are selected as χi (.) = 5zi , i = 1, 2, 3, the filter parameter is selected as µ = 0.1 and the virtual control inputs αi are calculated as α1 = xd , α2 = −5z1 + N B sin(x1 ) + M x2 + α˙ 2 − 5z2 − z1 ). Following x˙ d , α3 = M ( M the observer design method, a time varying observer is constructed as (13). The observer parameters are selected as a0 = 2, a1 = 3, a2 = 5. The contraction metric for the observer is derived as: 1 0, 0, M R 1 −B and the determinant is calacΘ= , − 0, M M2 M L2 −R −Kb R 1, , + 2 L ML L 1 ulated as D = M . The observer gain L(t) is obtained using the expression (19). The simulation is done with an initial condition [2π, 0, 0]T . Figure 1 confirms the boundedness of closed loop trajec-
error (ˆ x − x) exponentially converges to zero. As α(.) is Lipschitz in its arguments and bi (x) is bounded lim ||¯b|| ⇒ lim L7 ||ˆ x − x|| → 0 t→∞
t→∞
where L7 is the Lipschitz constant. Exploiting lemma 1 and triangle inequality L6 c1 . (35) lim ||z1 (t)|| = lim ||x(t) − xd (t)|| ≤ µ t→∞ t→∞ β The analysis can be summarized in the form of a theorem. Theorem 2:Let Bz ⊂ Rn , Bα ⊂ Rn and Bxˆ ⊂ Rn are compact sets and z(0), αf (0), x ˆ(0) ∈ Bz × Bα × Bxˆ . Suppose the assumptions (1-2) are satisfied for the system (7). Let an observer is constructed as (13) where the gain L(t) is chosen as (19) and a filtered backstepping controller is chosen as (9), (20). Then the trajectories of closed loop system are uniformly ultimate bounded ∀ t > 0 and the steady state tracking error bound is given by L6 c1 . lim ||z1 (t)|| = lim ||x(t) − xd (t)|| ≤ µ t→∞ t→∞ β 4.1 Advantages of The Proposed Design Controller Tuning: The computation of steady state error bounds obtained in (4), (26) require the knowledge of Lipschitz constants which may be difficult to obtain. However it can be noted that, the steady state tracking error under the influence of the controller (9), (20) is inversely proportional to contraction rate of reduced slow system (β) and high gain filter parameter (µ). Therefore the steady state error bounds can be reduced by increasing either β or decreasing µ, which can be done without the knowledge of Lipschitz constants. The parameter β can be increased by increasing the magnitudes of tuning functions χi (zi ) and µ can be decreased by decreasing the magnitude of filter parameters. Trade-off on Perturbation Parameter : The realization of controller depends on the selection of µ. Conventional Lyapunov based analysis requires µ to be very small so that overall closed system is stable. However the controller parameter can’t be reduced arbitrarily because of noise and sampling effects. One important attribute of contraction theory is that, the bound (4), does not depend on the smallness of µ. These bounds are valid for all values of µ ∈ (01] and hence relaxes the conservatism of Lyapunov based analysis and design. Therefore the magnitude of µ can be increased such that it is not too restrictive for controller realization and the performance is satisfactory. Moreover it can be pointed out that it is not necessary to decrease µ to reduce steady state error bounds of the trajectories. Using a moderate value of µ, the bounds can be reduced by proper selection of β.
TrackingError
40
Tracking Error (z1) z2 z3
20 0 -20 -40 0
1
2
3
Simulation Time
4
5
Fig. 1. Tracking Error and Closed loop Trajectories tories as proved in theorem 2. However it can be noted that, the the closed loop response is oscillatory. As the magnitude of tuning functions are not chosen sufficiently high, the contraction rate β is small and consequently the steady state tracking error bound is quite large. The performance is improved by selecting χi (.) = 20zi and thereby ensuring higher magnitude of contraction rate β. The closed loop simulation is shown in figure 2.
5. SIMULATION RESULTS Consider the dynamics of a D.C motor driven manipulator system which can be expressed as Slotine and Li (1991). x˙ 1 = x2 −N B 1 sinx1 − x2 + x3 x˙ 2 = (36) M M M Kb R 1 x˙ 3 = − x2 − x3 + u. L L L
6. CONCLUSION A contraction analysis based procedure is proposed to design a output feedback controller for filtered backstepping. The proposed observer is exponentially convergent and can deal with time varying nonlinearities. The convergence error bounds for closed loop trajectories are quantified in 34
5th International Conference on Advances in Control and M.M. Rayguru et al. / IFAC PapersOnLine 51-1 (2018) 30–35 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India
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TrackingError (z1)
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Fig. 2. Tracking Error (After Tuning) terms of known constants which are helpful in tuning the closed loop system performance. The procedure relaxes the constraint on magnitude of perturbation parameter and also circumvent the restrictions of quadratic Lyapunov based analysis. The method is validated using a simulation example. REFERENCES Angeli, D. (2002). A lyapunov approach to incremental stability properties. IEEE Trans. on Auto. Control, 47(3), 410–421. Del Vecchio, D. and Slotine, J.J.E. (2013). A contraction theory approach to singularly perturbed systems. IEEE Trans. on Auto. Control, 58(3), 752–757. Farrell, J., Polycarpou, M., Sharma, M., and Dong, W. (2009). Command filtered backstepping. Automatic Control, IEEE Transactions on, 54(6), 1391–1395. Forni, F. and Sepulchre, R. (2014). A differential lyapunov framework for contraction analysis. IEEE Tran. on Auto. Control, 59(3), 614–628. Gerdes, J.C. and Hedrick, J.K. (2002). loop-at-a-time design of dynamic surface controllers for nonlinear systems. Journal of dynamic systems, measurement, and control, 124(1), 104–110. Huang, J., Ri, S., Liu, L., Wang, Y., Kim, J., and Pak, G. (2015). Nonlinear disturbance observer-based dynamic surface control of mobile wheeled inverted pendulum. IEEE Trans. on Control Systems Technology. Jouffroy, J. and Lottin, J. (2002a). Integrator backstepping using contraction theory: a brief technological note. Proceedings of the 15th IFAC World Congres, Barcelona, Spain, 15, 238–244. Jouffroy, J. and Lottin, J. (2002b). On the use of contraction theory for the design of nonlinear observers for ocean vehicles. Proceedings of American Control Conference, 2647–2652 vol.4. Kanellakopoulos, I., Kokotovic, P., and Morse, A. (1992). A toolkit for nonlinear feedback design. Systems & Control Letters, 18(2), 83 – 92. Khalil, H. (2002). Nonlinear Systems. Prentice Hall; Third edition. Lohmiller, W. and Slotine, J.J.E. (1998). On contraction analysis for non-linear systems. Automatica, 34(6), 683– 696. Mehraeen, S., Jagannathan, S., and Crow, M. (2011). Power system stabilization using adaptive neural network-based dynamic surface control. IEEE Trans. on Power Systems, 26(2), 669–680. 35