Adaptive Control of Time-Varying Parameter Systems with Asymptotic Tracking
Omkar Sudhir Patil, Runhan Sun, Shubhendu Bhasin, Warren E. Dixon
AAdaptive Control of Time-Varying ParameterSystems with Asymptotic Tracking
Omkar Sudhir Patil, Runhan Sun, Shubhendu Bhasin and Warren E. Dixon
Abstract —A continuous adaptive control design is developedfor nonlinear dynamical systems with linearly parameterizableuncertainty involving time-varying uncertain parameters. Thekey feature of this design is a robust integral of the signof the error (RISE)-like term in the adaptation law whichcompensates for potentially destabilizing terms in the closed-loop error system arising from the time-varying nature of un-certain parameters. A Lyapunov-based stability analysis ensuresasymptotic tracking, and boundedness of the closed-loop signals.
I. I
NTRODUCTION
Adaptive control of nonlinear dynamical systems withtime-varying uncertain parameters is an open and practicallyrelevant problem. It has been well established that tradi-tional gradient-based update laws can compensate for con-stant unknown parameters yielding asymptotic convergence.Moreover, the development of robust modifications of suchadaptive update laws result in uniformly ultimately bounded(UUB) results for slowly varying parametric uncertaintyusing a Lyapunov-based analysis, under the assumption ofbounded parameters and their time-derivatives (cf. [1]).More recent results focus on tracking and parameter esti-mation performance improvement, though still limited to aUUB result, using various adaptive control approaches forsystems with unknown time-varying parameters. One suchapproach involves a fast adaptation law [2], where a matrix oftime-varying learning rates is utilized to improve the trackingand estimation performance under a finite excitation condi-tion. Another approach uses a set-theoretic control architec-ture [3]–[5] to reject the effects of parameter variation, whilerestricting the system error within a prescribed performancebound. While the aforementioned approaches can potentiallyyield improved transient response, the results still yield UUBerror systems.
Omkar Sudhir Patil, Runhan Sun, and Warren E. Dixon are withthe Department of Mechanical and Aerospace Engineering, Universityof Florida, Gainesville FL 32611-6250 USA. Email: {patilomkarsud-hir,runhansun,wdixon}@ufl.edu.Shubhendu Bhasin is with the Department of Electrical Engineering,Indian Institute of Technology Delhi, New Delhi, India (e-mail: [email protected]).This research is supported in part by NSF award number 1762829,Office of Naval Research Grant N00014-13-1-0151, AFOSR award numberFA9550-18-1-0109 and FA9550-19-1-0169. Any opinions, findings andconclusions or recommendations expressed in this material are those of theauthor(s) and do not necessarily reflect the views of the sponsoring agency.
Motivation exists to obtain asymptotic convergence of thetracking error to zero, despite the time-varying nature ofthe uncertain parameters. Robust adaptive control approachessuch as [6] yield asymptotic adaptive tracking for systemswith time-varying uncertain parameters; however, such ap-proaches exploit high-gain feedback based on worst-caseuncertainty, rather than an adaptive control approach thatscales to compensate for the uncertainty without using worst-case gains. In [7], the iterative learning control result in [6]is extended to yield asymptotic tracking for systems withperiodic time-varying parameters with known periodicity.To the best of our knowledge, an asymptotic tracking resulthas not been achieved for a generalized class of nonlinearsystems with unknown time-varying parameters, where theparameters are not necessarily periodic. Asymptotic trackingis difficult to achieve for the time-varying parameter casebecause the time-derivative of the parameter acts like anunknown exogenous disturbance in the parameter estimationdynamics, which is difficult to cancel with an adaptive updatelaw in a Lyapunov-based stability analysis.To illustrate this problem, consider the scalar dynamicalsystem ˙ x ( t ) = a ( t ) x ( t ) + u ( t ) , (1)with the controller u ( t ) = − kx ( t ) − ˆ a ( t ) x ( t ) , where k isa positive constant gain, a ( t ) is the unknown time-varyingparameter, ˆ a ( t ) is the parameter estimate of a ( t ) and theparameter estimation error ˜ a ( t ) is defined as ˜ a ( t ) (cid:44) a ( t ) − ˆ a ( t ) . The traditional stability analysis approach for suchproblems is to consider the Lyapunov function candidate V ( x ( t ) , ˜ a ( t )) = x ( t ) + γ ˜ a ( t ) , where γ is a positiveconstant gain. The given definitions and controller yieldthe following time-derivative of the candidate Lyapunovfunction: ˙ V ( t ) = − kx ( t )+˜ a ( t ) x ( t )+ ˜ a ( t ) γ ( ˙ a ( t ) − ˙ˆ a ( t )) . Forthe constant parameter case, i.e., ˙ a ( t ) = 0 , the well-knownadaptive update law ˙ˆ a ( t ) = γx ( t ) will cancel the cross term ˜ a ( t ) x ( t ) in ˙ V ( t ) . However, when the parameters are time-varying, it is unclear how to cancel or dominate ˙ a ( t ) via anupdate law such that ˙ V ( t ) becomes at least negative semi-definite. It would be desirable to have a sliding mode-liketerm based on ˜ a ( t ) in the adaptation law, but ˜ a ( t ) is unknown. Note that the system ( ) is considered only for illustrative purpose. Thispaper presents result for a general system with a vector state and a linearlyparameterizable uncertainty with time-varying parameters. a r X i v : . [ ee ss . S Y ] J u l nother approach could be to use a robust controller, e.g., u ( t ) = − kx ( t ) − ¯ ax ( t ) , where ¯ a is a known constant upperbound on the norm of parameter | a ( t ) | , or an adaptive robustcontroller which involves certainty equivalence in terms ofthe unknown bound ¯ a . Either of these approaches wouldyield an asymptotic tracking result (cf., [6]), but, as statedearlier, these approaches are based on a high-gain worst casescenario, rather than an adaptive control approach that scalesto compensate for the uncertainty without using worst-casegains.A popular approach to design adaptive controllers for thetime-varying parameter case is to consider robust modifica-tion of the update law and assume upper bounds of | a ( t ) | and ˙ a ( t ) to obtain a UUB result. For instance, consider astandard gradient update law with sigma-modification [8], ˙ˆ a ( t ) = γx ( t ) − γσ ˆ a ( t ) , which yields ˙ V ( t ) = − kx ( t ) − σ ˜ a ( t ) + ˜ a ( t )( ˙ a ( t ) γ + σa ( t )) , implying a UUB result whenthe parameter a ( t ) and its time-derivative ˙ a ( t ) are bounded.Moreover, the approaches developed in results such as [2]and [4] can be used to improve the transient response of theUUB error system.The major challenge to achieve asymptotic stability isthe derivative of the time-varying parameter term in theLyapunov analysis, which is addressed in this paper witha Lyapunov-based design approach, that is inspired by themodular adaptive control approach in [9]. This approachincludes higher order dynamics which appear after takingthe time-derivative of (1). Since these higher order dynamicscontain the time-derivative of the parameter estimate ˙ˆ a ( t ) , it is possible to design ˙ˆ a ( t ) to facilitate the subsequentstability analysis. With this motivation, a continuous adaptivecontrol algorithm is developed for nonlinear dynamical sys-tems with linearly parameterized uncertainty involving time-varying parameters, where a semi-global asymptotic trackingresult is achieved. A key feature of the proposed methodis a robust integral of the sign of the error (RISE)-like(see [9]–[12]) update law, i.e., the update law contains asignum function of the tracking error term multiplied bysome desired regressor based terms. The update law alsoinvolves a projection algorithm to ensure that the parameterestimates stay within a bounded set. However, the projectionalgorithm introduces a potentially destabilizing term in thetime-derivative of the Lyapunov function candidate, leadingto an additional technical obstacle to obtain asymptotic track-ing. This challenge is resolved by using an auxiliary term inthe control input, which facilitates stability by providing astabilizing term and canceling the aforementioned potentiallydestabilizing term in the time-derivative of the candidateLyapunov function. With the proposed method, the closed-loop system dynamics have the same structure as previousRISE controllers [9]–[12], for which the stability analysistools are well established, yielding asymptotic convergenceof the tracking error to zero, boundedness of the parameterestimation error, and boundedness of the closed-loop signals. II. D YNAMIC M ODEL
Consider a control affine system with the nonlinear dy-namics ˙ x ( t ) = h ( x ( t ) , t ) + d ( t ) + u ( t ) , (2)where x : [0 , ∞ ) → R n denotes the state, h : R n × [0 , ∞ ) → R n denotes a continuously differentiable function, d : [0 , ∞ ) → R n represents an exogenous disturbance actingon the system, and u : [0 , ∞ ) → R n represents the controlinput. The function h ( x ( t ) , t ) in ( ) is assumed to be linearlyparameterized as h ( x ( t ) , t ) (cid:44) Y h ( x ( t ) , t ) θ f ( t ) , (3)where Y h : R n × [0 , ∞ ) → R n × m is a known regressionmatrix, and θ f : [0 , ∞ ) → R m is a vector of time-varyingunknown parameters.The disturbance parameter vector d ( t ) can be appendedto the θ f ( t ) vector, yielding an augmented parameter vector θ : [0 , ∞ ) → R n + m as θ ( t ) (cid:44) (cid:20) θ f ( t ) d ( t ) (cid:21) , (4)and the augmented regressor Y : R n × [0 , ∞ ) → R n × ( n + m ) can be designed as Y ( x ( t ) , t ) (cid:44) (cid:2) Y h ( x ( t ) , t ) I n (cid:3) . (5)The parameterization in (4) and ( ) yields h ( x ( t ) , t )+ d ( t ) = Y ( x ( t ) , t ) θ ( t ) , so the dynamics in ( ) can be rewritten as ˙ x ( t ) = Y ( x ( t ) , t ) θ ( t ) + u ( t ) . (6) Assumption 1.
The time-varying augmented parameter θ ( t ) and its time-derivatives, i.e., ˙ θ ( t ) , ¨ θ ( t ) are bounded by knownconstants, i.e., (cid:107) θ ( t ) (cid:107) ≤ ¯ θ , (cid:13)(cid:13)(cid:13) ˙ θ ( t ) (cid:13)(cid:13)(cid:13) ≤ ζ , and (cid:13)(cid:13)(cid:13) ¨ θ ( t ) (cid:13)(cid:13)(cid:13) ≤ ζ ,where ¯ θ, ζ , ζ ∈ R > are known bounding constants, and (cid:107)·(cid:107) denotes the Euclidean norm.III. C ONTROL D ESIGN
A. Control Objective
The objective is to design a controller such that thestate tracks a smooth bounded reference trajectory, despitethe time-varying nature of the uncertain parameters. Theobjective is quantified by defining the tracking error e :[0 , ∞ ) → R n as e (cid:44) x − x d , (7)where x d : [0 , ∞ ) → R n is a reference trajectory. All function dependencies are suppressed equation (7) onward; assumeall variables to be time dependent unless stated otherwise. ssumption 2.
The reference trajectory x d ( t ) is boundedand smooth, such that (cid:107) x d ( t ) (cid:107) ≤ ¯ x d , (cid:107) ˙ x d ( t ) (cid:107) ≤ δ , and (cid:107) ¨ x d ( t ) (cid:107) ≤ δ , where ¯ x d , δ , δ ∈ R > are known boundingconstants.Substituting ( ) into the time-derivative of ( ) yields ˙ e = Y θ + u − ˙ x d . (8)To facilitate the subsequent analysis, a filtered tracking error r : [0 , ∞ ) → R n is defined as r (cid:44) ˙ e + αe, (9)where α ∈ R > is a constant control gain. Substituting ( ) into ( ) yields r = Y θ + u − ˙ x d + αe. (10) B. Control and Update Law Development
From the subsequent stability analysis, the continuouscontrol input is designed as u (cid:44) − Y d ˆ θ − αe + ˙ x d + µ, (11)where Y d (cid:44) Y ( x d ( t ) , t ) is the desired regression matrix, µ : [0 , ∞ ) → R n is a subsequently defined auxiliary controlterm, and ˆ θ : [0 , ∞ ) → R n + m denotes the parameter estimateof θ ( t ) . Substituting the control input in ( ) into the open-loop error system in ( ) yields the following closed-loopsystem r = Y θ − Y d ˆ θ + µ. (12)Adding and subtracting Y d θ in (12) yields r = ( Y − Y d ) θ + Y d ˜ θ + µ, (13)where ˜ θ : [0 , ∞ ) → R n + m denotes the parameter estimationerror, i.e., ˜ θ ( t ) (cid:44) θ ( t ) − ˆ θ ( t ) . Taking the time-derivative of ( ) yields ˙ r = ( ˙ Y − ˙ Y d ) θ + ( Y − Y d ) ˙ θ + ˙ Y d ˜ θ + Y d ˙ θ − Y d ˙ˆ θ + ˙ µ. (14)The control variables ˙ˆ θ ( t ) and ˙ µ ( t ) now appear in the higherorder dynamics in ( ) , and these control variables aredesigned with the use of a continuous projection algorithm[13, Appendix E]. The projection algorithm constrains ˆ θ ( t ) tolie inside a bounded convex set B = { θ ∈ R ( n + m ) | (cid:107) θ (cid:107) ≤ ¯ θ } by switching the adaptation law to its component tangential tothe boundary of the set B when ˆ θ ( t ) reaches the boundary. Acontinuously differentiable convex function f : R ( n + m ) → R is used to describe the boundaries of the bounded convexset B such that f ( θ ( t )) < ∀ (cid:107) θ ( t ) (cid:107) < ¯ θ and f ( θ ( t )) =0 ∀ (cid:107) θ ( t ) (cid:107) = ¯ θ . The adaptation law is then designed as ˙ˆ θ (cid:44) proj (Λ ( t ))= (cid:40) Λ , || ˆ θ || < ¯ θ ∨ ( ∇ f (ˆ θ )) T Λ ≤ , || ˆ θ || ≥ ¯ θ ∧ ( ∇ f (ˆ θ )) T Λ > , (15)where || ˆ θ (0) || < ¯ θ, ∨ , ∧ denote the logical ‘or’, ‘and’operators, respectively, ∇ represents the gradient operator,i.e., ∇ f (ˆ θ ) = (cid:104) ∂f∂φ . . . ∂f∂φ n + m (cid:105) Tφ =ˆ θ , and Λ : [0 , ∞ ) → R n + m and Λ : [0 , ∞ ) → R n + m are designed as Λ (cid:44) Γ Y Td ( Y d Γ Y Td ) − [ β sgn ( e )] , (16) Λ (cid:44) (cid:32) I m + n − ( ∇ f (ˆ θ ))( ∇ f (ˆ θ )) T ||∇ f (ˆ θ ) || (cid:33) Λ , (17)respectively. In ( ) and ( ) , β ∈ R > is a constant gain,and Γ ∈ R ( n + m ) × ( n + m ) is a positive-definite matrix with ablock diagonal structure, i.e., Γ (cid:44) (cid:20) Γ m × n n × m Γ (cid:21) , with Γ ∈ R m × m , Γ ∈ R n × n . The continuous auxiliary term µ ( t ) , used in the control input in ( ) , acts as a stabilizingterm in the Lyapunov analysis to account for the side effectsof the projection, and is designed as a generalized solutionto ˙ µ (cid:44) (cid:40) µ , || ˆ θ || < ¯ θ ∨ ( ∇ f (ˆ θ )) T Λ ≤ ,µ || ˆ θ || ≥ ¯ θ ∧ ( ∇ f (ˆ θ )) T Λ > , (18)where µ (0) = 0 , and µ : [0 , ∞ ) → R n and µ : [0 , ∞ ) → R n are defined as µ (cid:44) − Kr and µ (cid:44) µ − Y d (Λ − Λ ) , respectively. Substituting (15) and (18) in (14) , the closed-loop dynamics can be rewritten as ˙ r = ( ˙ Y − ˙ Y d ) θ +( Y − Y d ) ˙ θ + ˙ Y d ˜ θ + Y d ˙ θ − β sgn ( e ) − Kr, (19)for both cases, i.e., when || ˆ θ || < ¯ θ ∨ ( ∇ f (ˆ θ )) T Λ ≤ or || ˆ θ || ≥ ¯ θ ∧ ( ∇ f (ˆ θ )) T Λ > . To facilitate the subsequentanalysis, ( ) can be rewritten as ˙ r = (cid:101) N + N B − β sgn ( e ) − Kr − e, (20)where the variables (cid:101) N : [0 , ∞ ) → R n and N B : [0 , ∞ ) → R n are defined as (cid:101) N (cid:44) ( ˙ Y − ˙ Y d ) θ + ( Y − Y d ) ˙ θ + e, and N B (cid:44) Y d ˙ θ + ˙ Y d θ − ˙ Y d ˆ θ, respectively. The Mean Value Theorem (MVT) can be usedto develop the following upper bound on the term (cid:101) N ( t ) From Lemma 1 in the Appendix section, Y d Γ Y Td is proven to beinvertible, therefore it is reasonable to include ( Y d Γ Y Td ) − in the updatelaw. | (cid:101) N || ≤ ρ ( || z || ) || z || , (21)where z (cid:44) (cid:2) r T e T (cid:3) T ∈ R n and ρ : R n → R is apositive, globally invertible and non-decreasing function. ByAssumption 1, Assumption 2, Corollary 1 in the Appendix,and the bounding effect of projection algorithm on ˆ θ ( t ) ,the term N B ( t ) and its time-derivative ˙ N B ( t ) can be upperbounded by some constants γ ,γ ∈ R > as || N B ( t ) || ≤ γ , || ˙ N B ( t ) || ≤ γ , (22)respectively. IV. S TABILITY A NALYSIS
Theorem 1.
The controller designed in (11) along with theadaptation laws designed in (15) and ( ) ensure the closed-loop system is bounded and the tracking error (cid:107) e ( t ) (cid:107) → as t → ∞ , provided that the gains α, β are selected suchthat the following condition is satisfied β > γ + γ α . (23) Proof:
Let
D ⊂ R n +1 be an open connected set contain-ing y ( t ) = 0 , where y : [0 , ∞ ) → R n +1 is defined as y ( t ) (cid:44) (cid:2) z T ( t ) (cid:112) P ( t ) (cid:3) T . Let V L : D × [0 , ∞ ) → R ≥ be a positive-definite candidateLyapunov function defined as V L ( y ( t ) , t ) (cid:44) r T r + 12 e T e + P, where P : [0 , ∞ ) → R is a generalized solution to thedifferential equation ˙ P ( t ) (cid:44) − L ( t ) , (24)where P (0) (cid:44) β n (cid:80) i =1 | e i (0) | − e (0) T N B (0) and L (cid:44) r T ( N B − β sgn ( e )) . (25) Remark . Provided that the gain condition in ( ) is satis-fied, P ( t ) ≥ . Hence it is valid to use P ( t ) in the candidateLyapunov function as function of the variable (cid:112) P ( t ) .From ( ) , (20) and ( ) , the differential equations describingthe closed-loop system are ˙ e = r − αe, (26) ˙ r = (cid:101) N + N B − β sgn ( e ) − Kr − e, (27) ˙ P = − r T ( N B − β sgn ( e )) . (28)Let g : R n +1 × [0 , ∞ ) → R n +1 denote the right-handside of ( ) - ( ) . Since g ( y ( t ) , t ) is continuous almost See [10] for details. everywhere, except in the set { ( y ( t ) , t ) | e = 0 } , an absolutecontinuous Filippov solution y ( t ) exists almost everywhere(a.e.), so that ˙ y ( t ) ∈ K [ g ]( y ( t ) , t ) a.e., except at the pointsin the set { ( y ( t ) , t ) | e = 0 } , where the Filippov set-valuedmap includes unique solutions. Using a generalized Lyapunovstability theory under the framework of Filippov solutions,a generalized time-derivative of the Lyapunov function V L exists and ˙ V L ( y, t ) ∈ ˙ (cid:101) V L ( y, t ) , where ˙ (cid:101) V L ( y, t ) = (cid:92) ξ ∈ ∂V L ( y,t ) ξ T K (cid:2) ˙ e T ˙ r T P − ˙ P (cid:3) T = ∇ V TL K (cid:2) ˙ e T ˙ r T P − ˙ P (cid:3) T ⊂ (cid:2) e T r T P (cid:3) × K (cid:2) ˙ e T ˙ r T P − ˙ P (cid:3) T , (29)where ∂V L ( y, t ) denotes Clarke’s generalized gradient [14].Substituting ( ) - ( ) into ( ) yields ˙ (cid:101) V L a.e. ⊂ r T ( (cid:101) N + N B − β sgn ( e ) − Kr − e )+ e T ( r − αe ) − r T ( N B − β sgn ( e )) (30)where K [ sgn ( e )] = SGN ( e ) such thatSGN ( e i ) = { } , e i > − , , e i = 0 {− } , e i < . Using (21) , the expression in ( ) can be upper bounded as ˙ (cid:101) V L a.e. ≤ ρ ( (cid:107) z (cid:107) ) (cid:107) z (cid:107) (cid:107) r (cid:107) − K (cid:107) r (cid:107) − αe . Using Young’s Inequality on ρ ( (cid:107) z (cid:107) ) (cid:107) z (cid:107) (cid:107) r (cid:107) yields ρ ( (cid:107) z (cid:107) ) (cid:107) z (cid:107) (cid:107) r (cid:107) ≤ ρ ( (cid:107) z (cid:107) ) (cid:107) z (cid:107) + (cid:107) r (cid:107) . Therefore, ˙ (cid:101) V L a.e. ≤ ρ ( (cid:107) z (cid:107) ) (cid:107) z (cid:107) − ( K −
12 ) (cid:107) r (cid:107) − αe a.e. ≤ − (cid:18) λ − ρ ( (cid:107) z (cid:107) )2 (cid:19) (cid:107) z (cid:107) , (31)where λ (cid:44) min { α, K − } ∈ R > is a known constant.The expression in (
31) can be rewritten as ˙ V L a.e. ≤ − c (cid:107) z (cid:107) ∀ y ∈ D , (32)for some constant c ∈ R > , where D (cid:44) (cid:110) y ∈ R n +1 | (cid:107) y (cid:107) ≤ ρ − (cid:16)(cid:112) λ (cid:17)(cid:111) . In this region, λ > ρ ( (cid:107) z (cid:107) )2 , so a constant c satisfies (32),and larger values of λ expand the size of D . Furthermore,the relationship in (32) implies that V L ( y ( t ) , t ) ∈ L ∞ , hence e ( t ) , r ( t ) , P ( t ) ∈ L ∞ . These facts along with thexpression in ( ) , indicate that µ ( t ) ∈ L ∞ . The param-eter estimate ˆ θ ( t ) ∈ L ∞ due to the projection operation.The state and its time-derivative, i.e., x ( t ) , ˙ x ( t ) ∈ L ∞ ,because e ( t ) , r ( t ) , x d ( t ) , ˙ x d ( t ) ∈ L ∞ . Further the regressionmatrix Y ( x ( t ) , t ) ∈ L ∞ since its a bounded function fora bounded argument x ( t ) . Similarly, Y d ( t ) ∈ L ∞ , hence ˙ˆ θ ∈ L ∞ by Corollary 1. From the expression in (11), since ˆ θ ( t ) , e ( t ) , ˙ x d ( t ) , µ ( t ) ∈ L ∞ , u ( t ) ∈ L ∞ . Hence all theclosed-loop signals are bounded.Consider λ (cid:107) y (cid:107) ≤ V L ≤ λ (cid:107) y (cid:107) , where λ , λ ∈ R > . To ensure (cid:107) z (cid:107) ≤ ρ − ( √ λ ) , it is sufficientto obtain the result from (cid:107) y (cid:107) ≤ ρ − ( √ λ ) . Since (cid:113) V L λ ≤ (cid:107) y (cid:107) , then (cid:113) V L λ ≤ ρ − ( √ λ ) , and V L isnon-increasing, so V L ( t ) ≤ V L (0) . Hence it sufficientto show that (cid:113) V L (0) λ ≤ ρ − ( √ λ ) to ensure that (cid:113) V L λ ≤ ρ − ( √ λ ) . Since λ (cid:107) y (0) (cid:107) ≤ V L (0) implies (cid:107) y (0) (cid:107) ≤ (cid:113) V L (0) λ ≤ (cid:113) λ λ ρ − ( √ λ ) , so y ∈ S (cid:44) (cid:110) y ( t ) ∈ D| y ( t ) ≤ (cid:113) λ λ ρ − ( √ λ ) (cid:111) is the region where y (0) should lie for guaranteed asymptotic stability. Thegain condition λ = min { α, K − } ≥ ρ (cid:16)(cid:113) λ λ (cid:107) y (0) (cid:107) (cid:17) needs to be satisfied according to the initial condition forasymptotic stability and the region of attraction can be madearbitrarily large to include any initial condition by increasingthe gains α and K accordingly. By the extension of LaSalle-Yoshizawa theorem for non-smooth systems in [14] and [15], c (cid:107) z ( t ) (cid:107) → and hence (cid:107) e (cid:107) → as t → ∞ ∀ y (0) ∈ S , sothe closed-loop error system is semi-globally asymptoticallystable. (cid:4) V. C
ONCLUSION
A continuous adaptive control design was presented toachieve semi-global asymptotic tracking for linearly pa-rameterizable nonlinear systems with time-varying uncertainparameters. The key feature of this design is a RISE-likeparameter update law along with a projection algorithm,which allows the system to compensate for potentiallydestabilizing terms in the closed-loop error system, arisingdue to the time-varying nature of parameters. Semi-globalasymptotic tracking for the error system is guaranteed via aLyapunov-based stability analysis. Future work will involveimprovement of the parameter estimation performance oftime-varying parameter systems and its extension to thesystem identification problem.R
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Lemma 1.
Consider a positive-definite matrix Γ ∈ R ( n + m ) × ( n + m ) such that Γ has the block diagonal structureas Γ (cid:44) (cid:20) Γ m × n n × m Γ (cid:21) , where Γ ∈ R m × m and Γ ∈ R n × n . The matrix Y ( x ( t ) , t )Γ Y T ( x ( t ) , t ) is positive-definite,and hence invertible. Furthermore, the inverse of this ma-trix satisfies the property (cid:13)(cid:13)(cid:13)(cid:0) Y ( x ( t ) , t )Γ Y T ( x ( t ) , t ) (cid:1) − (cid:13)(cid:13)(cid:13) ≤ λ min { Γ } , where (cid:107)·(cid:107) denotes the spectral norm and λ min {·} denotes the minimum eigenvalue of {·} .Proof : Substituting the definitions for Y ( x ( t ) , t ) and Γ in ( x ( t ) , t )Γ Y T ( x ( t ) , t ) yields Y ( x ( t ) , t )Γ Y T ( x ( t ) , t ) = (cid:2) Y h ( x ( t ) , t ) I n (cid:3) (cid:20) Γ m × n n × m Γ (cid:21) (cid:20) Y h ( x ( t ) , t ) I n (cid:21) = Y h ( x ( t ) , t )Γ Y h ( x ( t ) , t ) + Γ . Since Γ is selected to be a positive-definite matrix, the blockmatrices Γ and Γ are both positive-definite, so the first term Y h ( x ( t ) , t )Γ Y h ( x ( t ) , t ) in this expression is positive semi-definite while the second term Γ is positive-definite, hencethe sum of these two terms, i.e., Y ( x ( t ) , t )Γ Y T ( x ( t ) , t ) ispositive-definite, and therefore invertible. Furthermore, thespectral norm satisfies the property, (cid:107) A (cid:107) = (cid:112) λ max { A T A } for some A ∈ R p × q with p, q ∈ Z > , where λ max {·} denotesthe maximum eigenvalue of {·} . Utilizing this property with (cid:13)(cid:13)(cid:13)(cid:0) Y Γ Y T (cid:1) − (cid:13)(cid:13)(cid:13) yields (cid:13)(cid:13)(cid:13)(cid:0) Y Γ Y T (cid:1) − (cid:13)(cid:13)(cid:13) = (cid:115) λ max (cid:26)(cid:16) ( Y Γ Y T ) − (cid:17) T ( Y Γ Y T ) − (cid:27) = λ max (cid:110)(cid:0) Y Γ Y T (cid:1) − (cid:111) . (33)The eigenvalues of the inverse of a positive definite matrix B satisfy the property, λ max (cid:8) B − (cid:9) = λ min { B } . Applying thisproperty with the right-hand side of ( ) yields (cid:13)(cid:13)(cid:13)(cid:0) Y Γ Y T (cid:1) − (cid:13)(cid:13)(cid:13) = 1 λ min { Y Γ Y T }≤ λ min { Γ } . (cid:4) Corollary 1.
The norm of time-derivative of the parameterestimate, (cid:13)(cid:13)(cid:13) ˙ˆ θ (cid:13)(cid:13)(cid:13) can be upper bounded by a constant γ ∈ R > ,i.e., (cid:13)(cid:13)(cid:13) ˙ˆ θ (cid:13)(cid:13)(cid:13) ≤ γ .Proof: Based on ( ) (cid:13)(cid:13)(cid:13) ˙ˆ θ (cid:13)(cid:13)(cid:13) = (cid:107) proj (Λ ) (cid:107) ≤ (cid:107) Λ (cid:107) = (cid:13)(cid:13) Γ Y Td ( Y d Γ Y Td ) − β sgn ( e ) (cid:13)(cid:13) ≤ (cid:13)(cid:13) Γ Y Td ( Y d Γ Y Td ) − β (cid:13)(cid:13) . (34)Applying Holder’s inequality to the right-hand side of ( ) yields (cid:13)(cid:13)(cid:13) ˙ˆ θ (cid:13)(cid:13)(cid:13) ≤ β (cid:107) Γ (cid:107) (cid:107) Y d (cid:107) (cid:13)(cid:13) ( Y d Γ Y Td ) − (cid:13)(cid:13) . (35)Using Lemma 1 with the right-hand side of ( ) yields (cid:13)(cid:13)(cid:13) ˙ˆ θ (cid:13)(cid:13)(cid:13) ≤ β (cid:107) Γ (cid:107) (cid:107) Y d (cid:107) λ min { Γ } . Given a bounded reference x d ( t ) , such that (cid:107) x d ( t ) (cid:107) ≤ ¯ x d , thespectral norm of the desired regressor may be upper-bounded by a constant ¯ Y d ∈ R > , i.e., (cid:107) Y d (cid:107) ≤ ¯ Y d , because Y d is acontinuously differentiable function. Therefore, (cid:13)(cid:13)(cid:13) ˙ˆ θ (cid:13)(cid:13)(cid:13) ≤ β (cid:107) Γ (cid:107) ¯ Y d λ min { Γ } = γ3