1
A Corollary for Switched Nonsmooth Systems with Applications to Switching in Adaptive Control
arXiv:1609.05880v1 [cs.SY] 19 Sep 2016
Rushikesh Kamalapurkar, Joel A. Rosenfeld, Anup Parikh, Andrew R. Teel, Warren E. Dixon
Abstract—This paper generalizes the Lasalle-Yoshizawa Theorem to switched nonsmooth systems. It is established that a common candidate Lyapunov function with a negative semidefinite derivative is sufficient for boundedness of the system state and convergence of a positive semidefinite function of the system state to the origin. The developed generalization is motivated by adaptive control of switched systems where the derivative of the candidate Lyapunov function is typically negative semidefinite.
I. I NTRODUCTION Switching in adaptive systems can occur due to intermittent feedback or abrupt changes in the plant parameters. Switching is also utilized as a tool to improve transient response of adaptive controllers by selecting between multiple estimated models of stable plants (cf. [1]–[10]). Lyapunov-based stability analysis of switched adaptive system is challenging because adaptive update laws typically result in negative semi-definite (NSD) derivatives of the candidate Lyapunov functions for the individual subsystems. For each subsystem, convergence of the error signal to the origin is typically established using Barbalat’s lemma [11], [12] (or one of its variants). However, since Barbalat’s lemma provides no information about the decay rate of the candidate Lyapunov function, the stability of the overall switched system cannot be inferred from the stability of the subsystems using traditional dwell-time approaches. Approaches based on common Lyapunov functions (CLFs) have been developed for systems with negative definite Lyapunov derivatives; however, CLF-based approaches do not trivially extend to systems with NSD derivatives of the candidate Lyapunov function [13, Example 2.1]. Hence, generalizations to Barbalat’s Lemma that result in CLF theorems for systems with NSD derivatives of the candidate Lyapunov function are necessary to analyze the stability of switched system. Switched systems with NSD derivatives of the candidate Lyapunov functions have been studied in results such as [11], [14]–[17]. However, the objective in the aforementioned results is to achieve asymptotic stability (i.e., in the context of Rushikesh Kamalapurkar is with the School of Mechanical and Aerospace Engineering, Oklahoma State University, Stillwater, OK, USA. Email:
[email protected]. Joel A. Rosenfeld, Anup Parikh and Warren E. Dixon are with the Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL, USA. Email: {joelar, anuppari, wdixon}@ufl.edu. Andrew R. Teel is with the Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA, USA. Email:
[email protected] This research is supported in part by NSF award number 1509516, ONR grant number N00014-13-1-0151, AFOSR Award Number FA9550-14-10399, and a contract with the AFRL, Munitions Directorate at Eglin AFB. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the sponsoring agency.
adaptive control, asymptotic convergence of the error states and the parameters). Hence, further assumptions such as persistence of excitation (PE) [11], [15], Matrosov conditions [14]–[17], etc, are required. However, for adaptive methods, such assumptions are often difficult to verify (especially for nonlinear systems). For example, while Matrosov conditions have numerous applications in the analysis of time-varying systems, they are not specifically developed for adaptive systems, and it is not clear how they could be satisfied in general for adaptive systems without further development. Moreover, the U -PE (or U δ-PE) condition developed in [15], [18], which is applicable for adaptive systems, still suffers from the same drawbacks of traditional PE conditions, i.e., when system dynamics are nonlinear, PE cannot generally be guaranteed. In this paper a weaker result that does not require PE-like conditions is targeted. The objective of this paper is to establish boundedness of the system state (i.e., tracking errors and parameter estimates), and convergence of the error signal to the origin. Because of the complications resulting from a negative semi-definite Lyapunov derivative, few results are available in literature that examine adaptive control of uncertain nonlinear switched systems (i.e., where an adaptive update law is designed to compensate for uncertainty). An adaptive controller for switched nonlinear systems that utilizes a generalization of Barbalat’s lemma [19] is developed in [20]. The controller can asymptotically stabilize a switched system, where each subsystem has nonlinearly parameterized uncertainties. Multiple Lyapunov functions are utilized to analyze the stability of the switched system. However, the generalized Barbalat’s Lemma in [19] requires a minimum dwell time, and in general, statedependent switching conditions cannot guarantee a minimum dwell time. This paper generalizes the Lasalle-Yoshizawa Theorem ( [12, Theorem 8.4]) and its nonsmooth extension in [21] to switched systems. Boundedness of the system state and convergence of a positive semidefinite function of the system state to the origin is established under arbitrary switching between nonsmooth nonlinear systems provided a CLF with a NSD derivative is available. II. M AIN
RESULTS
Consider a switched system of the form x˙ (t) = fρ(x(t),t) (x (t) , t) ,
(1)
where ρ : Rn × R≥0 → N denotes a piece-wise continuous switching signal, and x : R≥0 → Rn denotes the system
2
state trajectory.1 The functions fσ : Rn × R≥0 → Rn are Lebesgue measurable for all σ ∈ N. Let f : Rn × R≥0 → Rn be a function defined as f (x, t) , fρ(x,t) (x, t) . Since the functions fσ are essentially locally bounded, uniformly in t, and σ, the function f is essentially locally bounded, uniformly in t. The objective of this paper is to establish asymptotic properties of the system x˙ (t) = f (x (t) , t) ,
x˙ (t) = fσ (x (t) , t) .
(3)
Filippov’s generalized solutions are utilized to analyze the nonsmooth systems in (1) and (2) using the Clarke gradientbased Lyapunov methods in [22]. The asymptotic properties of the system in (2) and the individual subsystems in (3) can be analyzed using the set-valued maps 2 \ K [f ] (x, t) (4) ξT V˜˙ (x, t) , {1} ξ∈∂V (x,t)
V˜˙ σ (x, t) ,
\
ξ
ξ∈∂V (x,t)
T
Claim 1. Fix some (x, t) ∈ D × R≥0 . Provided A (x, t) ≤ −W (x), then E (x, t) ≤ −W (x) , (6)
(2)
using asymptotic properties of the individual subsystems
and
Proposition 1. [25, Page 103] If P ⊂ Rn and3 x ∈ co (P ) then there exists m ∈ N with m ≤ n + 1, p1 , · · · , pm ∈ P , Pm and a , · · · , a ∈ R with 1 m >0 i=1 ai = 1 such that x = Pm i=1 ai pi .
K [fσ ] (x, t) , {1}
(5)
respectively, where ∂V denotes the Clarke-gradient of V [22], [24]. Thus, the objective of this paper to find a bound on the set-valued map in (4) using the bounds on the set-valued maps in (5). Based on the definition of the Filippov differential inclusion, \ \ co {f (y, t) | y ∈ B (x, δ) \ N } , K [f ] (x, t) =
Proof of Claim 1 : Let z ∈ E (x, t) and for the sake of contradiction, suppose that z > −W (x) + ǫ for some ǫ > 0. Then, z can be expressed as z = ξ1T ζ + ξ2 , for some ζ ∈ co ∪σ∈N AN,δ,σ (x, t). Since the set co ∪σ∈N AN,δ,σ (x, t) is the closure of co ∪σ∈N AN,δ,σ (x, t), there exists a sequence of points in co ∪σ∈N AN,δ,σ (x, t) converging to ζ. Hence, there exists a point z ∈ co ∪σ∈N AN,δ,σ (x, t) arbitrarily close to ζ. Hence, for any ξ ∈ ∂V (x, t), δ > 0, and measurable set N such that µ (N ) = 0, there exists a point z ∈ co ∪σ∈N AN,δ,σ (x, t) such that z − ξ1T z + ξ2 < ǫ/2. From Proposition 1, there is a collection of m ≤ n + 1 points {Z1 , · · P · , Zm } ⊂ Rn , positive real numbers {a1 , · · · , am } for m which i=1 ai = 1, and integers {σ1 , · · · , σm } P ∈ N, such m (y, t) | y ∈ B (x, δ) \ N } and z = i=1 ai Zi . that Zi ∈ {fσi P m T < ǫ/2, and therefore Hence, z − i=1 ai ξ1 Zi + ξ2 P m T a ξ Z + ξ > −W (x) + ǫ/2. Since ai > 0 for all i i 2 1 i=1 Pm i = 1, · · · , m and i=1 ai = 1, there exists j ∈ {1, · · · , m} for which ξ1T Zj + ξ2 > −W (x) + ǫ/2. However, ξ1T Zj + ξ2 ∈ AN,δ,σj ,ξ (x, t) and AN,δ,σj ,ξ (x, t) ≤ −W (x) by hypothesis. This is a contradiction. Claim 2. The Filippov differential inclusion K [f ] satisfies
δ>0 µ(N )=0
=
\
\
δ>0 µ(N )=0
co fρ(y,t) (y, t) | y ∈ B (x, δ) \ N .
T and for every To simplify the notation, let ξ = ξ1T , ξ2 measurable set N with µ (N ) = 0, δ > 0, σ ∈ N, x ∈ Rn , t ∈ R≥0 , and ξ ∈ ∂V (x, t) let AN,δ,σ (x, t) , {fσ (y, t) | y ∈ B (x, δ) \ N } , AN,δ,σ,ξ (x, t) , ξ1T coAN,δ,σ (x, t) + {ξ2 } , EN,δ,ξ (x, t) , ξ1T co ∪σ∈N AN,δ,σ (x, t) + {ξ2 } , [ \ \ \ AN,δ,σ,ξ (x, t) , A (x, t) , σ∈N ξ∈∂V (x,t) δ>0 µ(N )=0
E (x, t) ,
\
\
\
EN,δ,ξ (x, t) .
ξ∈∂V (x,t) δ>0 µ(N )=0
Before stating the main result, a property of convex hulls, due to Carathéodory, and two supporting claims are stated to facilitate the proof. 1 For a ∈ R, the notation R ≥a denotes the interval [a, ∞) and the notation R>a denotes the interval (a, ∞). 2 For a function g : D × R n ≥0 → R , essentially locally bounded, uniformly in t, K [g] (·) is an upper semi-continuous, nonempty, compact, convex,T and set-valued map on D × R≥0 , defined as KT [f ] (x, t) = T denotes δ>0 µ(N)=0 co {f (y, t) | y ∈ B (x, δ) \ N } , where µ(N)=0
the intersection over sets N of Lebesgue measure zero [23, Page 50]. For (x, y) ∈ Rn × R>0 , the notation B (x, y) denotes the open ball {υ ∈ Rn | kx − υk < y}
K [f ] (x, t) ⊂
\
\
co
δ>0 µ(N )=0
[
AN,δ,σ (x, t) .
(7)
σ∈N
for all (x, t) ∈ D × R≥0 . Proof of Claim 2: For any given (x, t), K [f ] (x, t) =
\
\
coAN,δ,σ (x, t) ,
δ>0 µ(N )=0
for some σ ∈ ρ (B (x, δ) \ N, t). Hence, \ \ [ K [f ] (x, t) ⊂ co δ>0 µ(N )=0
AN,δ,σ (x, t) ,
σ∈ρ(B(x,δ)\N,t)
for all (x, t) ∈ D × R≥0 . Since [ [ AN,δ,σ (x, t) ⊆ AN,δ,σ (x, t) , σ∈ρ(B(x,δ)\N,t)
σ∈N
the claim follows. Theorem 1 generalizes the Lasalle-Yoshizawa Theorem (cf. [12, Theorem 8.4]) and its nonsmooth extension (cf. [21]) to switched systems Theorem 1. Let D ⊂ Rn be an open and connected set containing the origin, and assume that the map x 7→ fσ (x, t) 3 The notation co {A} denotes the convex hull, and the notation co {A} denotes the closed convex hull of the set A.
3
is essentially locally bounded, uniformly in t and σ. Consider a locally Lipschitz continuous and regular function V : D × R≥0 → R such that
where W : D → R and W : D → R are continuous positive definite functions. Let W : D → R be a positive semidefinite function such that V˜˙ (x, t) ≤ −W (x) , ∀ (x, t) ∈ D × R , ∀σ ∈ N. (8) ≥0
For r > 0 and c > 0 such that B (0, r) ⊂ D and c < minkxk=rW (x), all the Filippov solutions of (2) that satisfy x (0) ∈ x ∈ B (0, r) | W (x) ≤ c are bounded, and W (x (t)) → 0, as t → ∞.
Proof: From the hypothesis of Theorem 1, \ T K [fσ ] (x, t) ≤ −W (x) , ∀σ ∈ N, ξ {1}
ξ∈∂V (x,t)
(9) Then, the inequality in (9) can be expressed as A (x, t) ≤ −W (x) , ∀ (x, t) ∈ D × R≥0 . Thus, using Claim 1, if ( ) \ \ \ [ T ξ1 co z∈ AN,δ,σ (x, t) + {ξ2 } , σ∈N
then z ≤ −W (x). Hence, S T T \ δ>0 µ(N )=0 co σ∈N AN,δ,σ (x, t) ξT {1} ξ∈∂V (x,t)
≤ −W (x) . (10)
Using Claim 2, (10) implies \ T K [f ] (x, t) ≤ −W (x) . ξ {1} ξ∈∂V (x,t)
where θˆ : R≥0 → RL denotes an estimate of the vector of unknown parameters, θ, k, β ∈ R>0 are positive constant control gains, and sgn (·) is the signum function. The estimate, ˆ is updated using the update law θ, ˙ T (x (t)) x (t) . θˆ (t) = Yρ(x(t),t) For each σ ∈ N, the closed-loop error system can then be expressed as x˙ (t) = −kx (t) + Yσ (x (t)) θ˜ (t) + d (t) − β sgn (x (t)) , (12) ˙˜ θ = −YσT (x (t)) x (t) , (13) where θ˜ , θ − θˆ denotes the parameter estimation error. The closed-loop system in (12) and (13) is discontinuous, and hence, does not admit classical solutions. Thus, the analysis will focus on the Filippov solutions to (12) and (13). B. Stability Analysis
≥0 .
Using [21, Corollary 2], Theorem 1 follows. Remark 1. The geometric condition in (8) can be relaxed to the following trajectory-based condition. Let the subsystems in (3) satisfy a.e. V˜˙ σ (xσ (t) , t) ≤ −W (xσ (t)) ,
A. Control Design
u (t) = −kx (t) − Yρ(x(t),t) (x (t)) θˆ (t) − β sgn (x (t)) ,
that is, for all σ ∈ N, T T \ δ>0 µ(N )=0 coAN,δ,σ (x, t) ξT ≤ −W (x) . {1}
Using the definition of V˜˙ from (4), V˜˙ (x, t) ≤ −W (x) , ∀ (x, t) ∈ D × R
where x : R≥0 → Rn denotes the state, u : R≥0 → Rn denotes the control input, d : R≥0 → Rn denotes an unknown disturbance, ρ : Rn × R≥0 → N denotes a known piece-wise continuous switching signal, Yσ : Rn → Rn×L , for each σ ∈ N, is a known function, and θ ∈ RL is the vector of constant unknown parameters. The control objective is to regulate the system state to the origin. The disturbance is assumed to be bounded, with a known bound d such that kdk∞ ≤ d.
The following nonlinear robust adaptive controller is designed to satisfy the control objective.
ξ∈∂V (x,t)
ξ∈∂V (x,t) δ>0 µ(N )=0
Consider the following nonlinear dynamical system. x˙ (t) = Yρ(x(t),t) (x) θ + u (t) + d (t) ,
W (x) ≤ V (x, t) ≤ W (x) , ∀ (x, t) ∈ D × R≥0 ,
σ
III. E XAMPLE
(11)
for all σ ∈ N, and for all Filippov solutions xσ : R≥0 → Rn to (3), where the qualifier a. e. implies that the inequality holds for almost all t ∈ R≥0 . In addition, let the set {t | ρ (x∗ (·) , ·) is discontinuous at t} be countable for a specific Filippov solution x∗ : R≥0 → Rn to (2). Then, a weaker version of Theorem 1 that establishes the convergence of W (x∗ (t)) to the origin as t → ∞ can be proved using [21, Corollary 1].
Consider the candidate Lyapunov function V : Rn × RL → R≥0 , defined as 1 1 ˜ V x, θ˜ , xT x + θ˜T θ. (14) 2 2 Since the candidate Lyapunov function is smooth, the Clarke gradient reduces to the standard gradient, i.e, ∂V = T θ˜T . Hence, using the calculus of K [·] from [26], x the set-valued maps in (5) can be computed as n o ˜ t = xT V˜˙ σ x, θ, −kx + Yσ (x) θ˜ + d (t) − xT (β K [sgn] (x)) + θ˜T −YσT (x) x , n o 2 = −k kxk + xT d (t) − β |x| . Provided β > d,
˜ t ≤ −W x, θ˜ , V˜˙ σ x, θ,
(15)
4
˜ t ∈ Rn × RL × R≥0 and σ ∈ N, where for all x, θ, 2 W x, θ˜ = k kxk is a positive semidefinite function. Using (14), (15), and Theorem 1, all the Filippov trajectories of the switched nonsmooth system in (12) and (13) are bounded and satisfy kxk → 0 as t → ∞. IV. C ONCLUSION Motivated by applications in switched adaptive control, the generalized Lasalle-Yoshizawa corollary in [21] is extended to switched nonsmooth systems. The extension facilitates the analysis of the asymptotic characteristics of a switched system based on the asymptotic characteristics of the individual subsystems where candidate Lyapunov functions with negative semidefinite derivatives can be constructed for the subsystems. Application of the developed extension to a switched adaptive system is demonstrated through a simple example. The developed method requires a strong convergence result for the subsystems. That is, all the Filippov solutions to the subsystems are required to converge. Future research will focus on the development of results for switched nonsmooth systems where only weak convergence results are available for the subsystems. R EFERENCES [1] J. P. Hespanha, D. Liberzon, and A. S. Morse, “Overcoming the limitations of adaptive control by means of logic-based switching,” Syst. Control Lett., vol. 49, no. 1, pp. 49–65, 2003. [2] J. Hespanha, “Stabilization of nonholonomic integrators via logic-based switching,” in Proc. World Cong. Int. Fed. Autom. Control, 1996. [3] J. Hespanha, D. Liberzon, and A. S. Morse, “Logic-based switching control of a nonholonomic system with parametric modeling uncertainty,” Systems & Control Letters, vol. 38, pp. 167–177, 1999. [4] J. Hespanha, D. Liberzon, A. Stephen Morse, B. Anderson, T. S. Brinsmead, and F. De Bruyne, “Multiple model adaptive control. part 2: switching,” Int. J. Robust Nonlinear Control, vol. 11, no. 5, pp. 479–496, 2001. [5] B. Anderson, T. Brinsmead, D. Liberzon, and A. Stephen Morse, “Multiple model adaptive control with safe switching,” Int. J. Adapt. Control Signal Process., vol. 15, no. 5, pp. 445–470, 2001. [6] L. Vu, D. Chatterjee, and D. Liberzon, “Input-to-state stability of switched systems and switching adaptive control,” Automatica, vol. 43, no. 4, pp. 639–646, 2007. [7] J. P. Hespanha, D. Liberzon, and A. Morse, “Supervision of integralinput-to-state stabilizing controllers,” Automatica, vol. 38, no. 8, pp. 1327 – 1335, 2002. [8] J. P. Hespanha, D. Liberzon, and A. S. Morse, “Hysteresis-based switching algorithms for supervisory control of uncertain systems,” Automatica, vol. 39, no. 2, pp. 263–272, 2003. [9] A. S. Morse, “Supervisory control of families of linear set-point controllers part I. exact matching,” IEEE Transactions on Automatic Control, vol. 41, no. 10, pp. 1413–1431, 1996. [10] ——, “Supervisory control of families of linear set-point controllers part II. robustness,” IEEE Trans Autom. Contol, vol. 42, no. 11, pp. 1500– 1515, 1997. [11] J. Slotine and W. Li, Applied Nonlinear Control. Prentice Hall, 1991. [12] H. K. Khalil, Nonlinear Systems, 3rd ed. Upper Saddle River, NJ: Prentice Hall, 2002. [13] D. Liberzon, Switching in Systems and Control. Birkhauser, 2003. [14] V. M. Matrosov, “On the stability of motion,” J. Appl. Math. Mech., vol. 26, pp. 1337–1353, 1962. [15] A. Loría, E. Panteley, D. Popovic, and A. R. Teel, “A nested Matrosov theorem and persistency of excitation for uniform convergence in stable nonautonomous systems,” IEEE Trans. Autom. Control, vol. 50, no. 2, pp. 183–198, 2005. [16] R. Sanfelice and A. Teel, “Asymptotic stability in hybrid systems via nested Matrosov functions,” IEEE Trans. on Autom. Control, vol. 54, no. 7, pp. 1569–1574, Jul. 2009.
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