Adaptive Control Allocation
Adaptive Control Allocation Johannes Tjønnås and Tor A. Johansen Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway
Abstract In this work we address the control allocation problem for a nonlinear over-actuated time-varying system where parameters a¢ ne in the e¤ector model may be assumed unknown. Instead of optimizing the control allocation at each time instant, a dynamic approach is considered by constructing update-laws that represent asymptotically optimal allocation search and adaptation. Using Lyapunov analysis for cascaded set-stable systems, uniform global/local asymptotic stability is guaranteed for the sets described by the system, the optimal allocation update-law and the adaptive update-law. Key words: Control allocation; Adaptive control; Nonlinear systems; Cascaded systems; Optimization.
1
Introduction
This work is motivated by the over-actuated control allocation problem. The problem is described by a nonlinear system, divided into a dynamic and a static part. The main contribution of this work is to show that the instantaneous control allocation problem, optimizing the desired input of the dynamic part of the system, not necessarily needs to be solved exactly at each time instant. In order to imply convergence and stability properties of the whole system, stability and attractivity of the optimal set, described by the control allocation problem, is pursued. Optimizing control allocation solutions have been derived for certain classes of over-actuated systems, such as aircraft, automotive vehicles and marine vessels, (Enns 1998, Bu¢ ngton, Enns & Teel 1998, Sørdalen 1997, Bodson 2002, Härkegård 2002, Luo, Serrani, Yurkovich, Doman & Oppenheimer 2004, Luo, Serrani, Yurkovich, Doman & Oppenheimer 2005, Poonamallee, Yurkovich, Serrani, Doman & Oppenheimer 2005, Johansen, Fossen & Berge 2004) and (Johansen, Fossen & Tøndel 2005). The control allocation problem is, in (Enns 1998, Bu¢ ngton et al. 1998, Sørdalen 1997, Johansen et al. 2005, Bodson 2002) and (Härkegård 2002), viewed as a static or quasi-dynamic optimization problem that is solved independently of the dynamic control problem considering non-adaptive linear e¤ector models of the form = Gu. In (Luo et al. 2004) and (Luo et al. 2005) a dynamic model predictive approach is considered to solve the allocation problem with linear time-varying dynamic in the actuator/e¤ector model, T u+u _ = uc . In (Poonamallee et al. 2005) and (Johansen et al. 2004) sequential quadratic programming tech-
Preprint submitted to Automatica
niques are used to cope with nonlinearities in the control allocation problem due to singularity avoidance and actuator failure. The main advantage of the control allocation approach is in general modularity and the ability to handle redundancy and constraints. Consider the system x_ = fx (t; x) + gx (t; x) (1) = (t; x; u; ) := (t; x; u) + 0 (t; x; u) (2) where t 0; x 2 Rn is the state vector; u 2 Rr is control input vector; 2 Rd is a vector of virtual controls and 2 Rm is a constant vector that contains parameters that either will be assumed known, or viewed as uncertain parameters to be adapted. The state x; is assumed known and typically represents the tracking error. The mapping (2) represents an actuator and e¤ector model, where the control signals u; are mapped to the virtual controls . We limit our study to the over-actuated control allocation problem, where m d r: Problem: Assume that there exist a virtual control c := k(t; x); with k(t; 0) = 0; that is di¤erentiable and uniformly globally asymptotically stabilizes the origin of (1), when = k(t; x). i) If is known, de…ne an update law u_ := fu (t; x; u), such that the stability property of the closed loop is conserved and u(t) converges to an optimal solution with respect to the static minimization problem min J(t; x; u) s:t (t; x; u; ) = 0: (3) c u ^ ii) If is unknown, solve problem i) with = ; u_ := _ fu (t; x; u; ^) and de…ne an adaptive update law ^ := f^ (t; x; u; ^); where ^ 2 Rm is an estimate of : 24 January 2008
and F : D ! Rq is locally Lipschitz with D Rq . In the following, if referred to a set, it has the properties of being nonempty. De…nition 1 The system (4) is said to be forward complete if, for each z0 2 D, the solution z( ; z0 ) 2 D is de…ned on R t0 . De…nition 2 The system (4) is said to be …nite escape time detectable through j jA , if any solution, z(t; z0 ) 2 D, which is right maximally de…ned on a bounded interval [t0 ; T ); satis…es limt%T jz(t; z0 )jA = 1. De…nition 3 If the system (4) is forward complete, then the closed set A D is: Uniformly Stable (US), if there exists a function, 2 K, and a constant, c > 0; such that, 8 jz0 jA < c, (6) jz(t; z )j (jz j ); 8t 0 :
In (Johansen 2004) it was shown that it is not necessary to solve the optimization problem (3) exactly at each time instant. Further a control Lyapunov function was used to derive an exponentially convergent updatelaw for u (similar to a gradient or Newton-like optimization) such that the control allocation problem (3) could be solved dynamically. And …nally it was shown that convergence and asymptotic optimality of the system, composed by the dynamic control allocation and a uniform globally exponentially stable trajectory-tracking controller, guarantees uniform boundedness and uniform global exponential convergence to the optimal solution of the system. The advantage of this approach is computational e¢ ciency and simplicity of implementation, since the optimizing control allocation algorithm is implemented as a dynamic nonlinear controller. Solving (3) online at each sampling instant requires a computationally more expensive numerical solution. In (Tjønnås & Johansen 2005) the results from (Johansen 2004) were extended by allowing uncertain parameters, associated with an adaptive law, in the e¤ector model, and by applying set-stability analysis in order to also conclude asymptotic stability of the optimal solution set. In what follows we will extend the ideas from (Johansen 2004) and (Tjønnås & Johansen 2005) by utilizing the set-stability result for cascaded systems established in (Tjønnås, Chaillet, Panteley & Johansen 2006). This result enables us to relax the assumptions in (Johansen 2004) and (Tjønnås & Johansen 2005) where fx (t; x); gx (t; x) and (t; x; u; ) are assumed to be globally Lipschitz in x: Further the virtual controller c does only need to render equilibrium of (1) UGAS, not UGES as assumed in (Johansen 2004) and (Tjønnås & Johansen 2005). The implementation of the adaptive law presented in (Tjønnås & Johansen 2005) depends directly on the Lyapunov function used in the analysis. In this work the analysis and implementation are separated, and although the Lyapunov functions in the analysis need to satisfy certain requirements, the adaptive implementation does not assume knowledge of these Lyapunov functions. 2
0
A
0 A
Uniformly Globally Stable (UGS), when D = Rq ; if (6) is satis…ed with, 2 K1 ; and for any z0 2 Rq : Uniformly Attractive (UA) if there exists a constant c > 0 such that for all jz0 jA < c and any > 0 there exists T = T ( ) > 0; such that jz0 jA c; t T ) jz(t; z0 )jA ; (7)
Uniformly Globally Attractive (UGA), when D = Rq , if for each pair of strictly positive numbers (c; ) there exists T = T ( ) > 0 such that for all z0 2 Rq , (7) holds. Uniformly Asymptotically Stable (UAS) if it is US and UA. Uniformly Globally Asymptotically Stable (UGAS), when D = Rq , if it is UGS and UGA. From (Khalil 1996), we adapt the de…nition of uniform boundedness of solutions to the case when A is not reduced to f0g.
De…nition 4 With respect to the closed set A Rq ; the solutions of system (4) are said to be: Uniformly Bounded (UB) if there exist a positive constant c; such that for every positive constant r < c there is a positive constant = (r); such that jz0 jA r ) jz(t; z0 )jA ; 8t 0 : (8)
Notation, de…nitions and preliminary results
Uniformly Globally Bounded (UGB), when D = Rq ; if for every r 2 R 0 , there is a positive constant = (r) such that (8) is satis…ed.
j j denotes the Euclidian norm and j jA0 : Rq ! R 0 denotes the distance from a point z 2 Rq to a set A0 Rq ; jzjA0 := inf fjz yj : y 2 A0 g : The solution of an autonomous dynamic system z_ = F (z) is denoted by z(t; z0 ) at t t0 ; where z0 = z(t0 ; z0 ) is the initial state and t0 is the initial time.
De…nition 5 A smooth Lyapunov function for (4) with respect to a non-empty, closed forward invariant set A D is a function V : D ! R that satis…es: i) there exists two K functions 1 and 2 such that for any z 2 D; V (z) 1 (jzjA ) 2 (jzjA ): ii) There exists a continuous and a positive de…nite or semide…nite function 3 such that for any z 2 DnA: @V 3 (jzjA ). @z (z)F (z)
2.1 De…nitions The de…nitions that follows are either motivated by, or can be found in (Teel, Panteley & Loria 2002) and (Lin, Sontag & Wang 1996). They pertain to systems of the form z_ = F (z) ; (4) where z := (p; xT )T ; p is the time-state p_ = 1; p0 = t0 : (5)
2.2 Preliminary results In order to conclude stability of a set of points, rather than a single equilibrium point, we have the following result from (Skjetne 2005).
2
UGS, when D = Rq2 ; if O2 is UGS with respect to the system (12) and that the solution of system (11)-(12) is UGB with respect to A. UAS, if O2 is UAS with respect to the system (12). US, if O2 is US with respect to the system (12).
Theorem 1 Assume that the system (4) is …nite escapetime detectable through jzjA : If there exists a smooth Lyapunov function for the system (4) with respect to a nonempty, closed, forward invariant set A, then A is UGS with respect to (4). Furthermore, if 3 is a positive definite function, then A is UGAS with respect to (4), and r if i (jzjA ) = ci jzjA for i = 1; 2; 3 where ci and r are strictly positive values and r > 1; then A is UGES with respect to (4).
3
The adaptive control allocation approach is based on three modular steps, see also Figure 1.
Consider the cascaded system x_ 1 = fc1 (t; x1 ) + gc (t; x1 ; x2 ) x_ 2 = fc2 (t; x2 ):
(9) (10)
(1) The high level control algorithm. The virtual control is treated as an available input to system (1), and a virtual control law c := k(t; x) is designed such that the origin of (1) is UGAS. (2) The control allocation algorithm. Based on the minimization problem (3) where J is a cost function that incorporates objectives such as minimum power consumption and actuator constraints (implemented as barrier functions), the Lagrangian function
When the functions fc1 , fc2 and gc are locally Lipschitz in all arguments, this class of nonlinear time-varying systems can be represented by the following autonomous system z_1 = F1 (z1 ) + G(z) z_2 = F2 (z2 )
Optimizing adaptive control allocation
(11) (12)
T T 2 Rq2 , where z1 := (p; xT 2 Rq1 , z2 := (p; xT 2) 1) T T q T T T z := (p; x ) 2 R ; x := (x1 ; x2 ) ; F1 (z1 ) := (1; fc1 (p; x1 )T )T , G(z) := (0; gc (p; x)T )T and F2 (z2 ) := (1; fc2 (p; x2 )T )T . Based on the cascaded system formulation we will make use of the results from (Tjønnås et al. 2006).
(t; x; u; ))T (17) is introduced. Update laws for the control input u and the Lagrangian parameter are then de…ned such that u and converges to a set de…ned by the time-varying optimality condition. (3) The adaptive algorithm. In order to cope with an unknown parameter vector in the e¤ector model, an adaptive law is de…ned. The parameter estimate is used in the control allocation algorithm and a certainty equivalent adaptive optimal control allocation can be de…ned. L(t; x; u; ) = J(t; x; u) + (k(t; x)
Lemma 1 Let O1 and O2 be some closed subsets of Rq1 and Rq2 respectively. Then, under the following assumptions, the set A := O1 O2 is UGAS with respect to the cascade (11)-(12). A 1 The set O2 is UGAS with respect to the system (12) and that the solution of the system (11)-(12) is UGB with respect to A: A 2 The functions F1 , F2 and G are locally Lipschitz. A 3 The cascade (11)-(12) is forward complete. A 4 There exist a continuous function G1 : R 0 ! R and a class K function G2 such that, for all z 2 Rq , jG(z)j
G1 (jzjA )G2 (jz2 jO2 ):
0
(13)
A 5 There exists a continuously di¤ erentiable function V 1 : Rq1 ! R 0 , class K1 functions 1 , 2 and 3 , and a continuous function & 1 : R 0 ! R 0 such that, for all z1 2 Rq1 , 1 (jz1 jO1 )
@V 1 (z1 )F1 (z1 ) @z1 @V 1 (z1 ) @x1
V 1 (z1 )
2 (jz1 jO1 )
3 (jz1 jO1 )
& jz1 jO1 :
(14)
(15) (16)
Corollary 1 Let O1 and O2 D be closed subsets of Rq1 and Rq2 respectively, and the assumptions A 2 - A5 be satis…ed. Then, with respect to the cascade (11)-(12), the set A is:
Fig. 1. Adaptive control allocation design philosophy
3
3.1
High level control algorithm
the set
From Assumption 2 a) there exists a Lyapunov function Vx : R 0 Rn ! R 0 and K1 functions x1 ; x2 ; x3 and x4 such that
Assumption 1 (Plant) a) There exists a continuous function Gf : R 0 ! R>0 and a constant K; such that jfx (t; x)j Gf (jxj) and jgx (t; x)j K for all t and x. Moreover fx is locally Lipschitz in t and x, and fx (t; 0) = 0: b) There exist a continuous function & @gx : R 0 ! R 0 such that for all t and x; gx is di¤ erentiable and
x1 (jxj)
+
T
T
%2 I: (18)
T @ @ @u uc @u uc
1
y;
of the equation, y = u1 ; which exist for any uc : Thus for every y there exists a solution (t; x; u1 ; ) = (t; x; 0; ) + y by the Mean Value Theorem, where uc 2 (0; u1 ): ii) follows from the Implicit Function Theorem by i) and Assumption 1 d).
:R
0
x3 (jxj)
!R
0;
such (25)
Certainty equivalent adaptive control allocation algorithm
In order to account for the unknown, but bounded, parameter vector ; an adaptive mechanism is include in the optimal control allocation design. We use a Lyapunov based indirect parameter estimation scheme (see for example (Krstic, Kanellakopoulos & Kokotovic 1995) for a systematic Lyapunov based procedure). Since the state vector of (1) is assumed to be known, we consider the estimation model: x ^_ = A(x x ^) + fx (t; x) + gx (t; x) (t; x; u; ^) (26)
Assumption 2 (High Level Control Algorithm) a) There exists a feedback control c := k(t; x) & k (jxj); where & k : R 0 7! R 0 is a continuous function, that render the equilibrium of (1) UGAS for = c . The @k function k is di¤ erentiable, @k & @k (jxj); @t + @x where & @k : R 0 7! R 0 is continuous, and k(t; 0) = 0 for all t.
in the construction of the adaptive law. This estimation model has the same structure as a typical series parallel model (SP) (Ioannou & Sun 1996), (Landau 1979). For analytical purpose, the …ltered error estimate of the unknown parameter vector:
If we rewrite (1) such that (t; x; u; ))
k
k
In this section we …rst establish update-laws for the control input u, the Lagrangian multipliers , and the parameter estimate ^; such that stability and convergence with respect to the time-varying …rst order optimal set de…ned by the optimization problem (3) can be concluded. Then we show that the stabilizing properties of the virtual controller from Assumption 2, are conserved for the closed loop system.
@ @u uc
x_ = f (t; x) + g(t; x)(k(t; x)
(24)
Remark 2 If the origin of x_ = f (t; x) is UGES, then Assumption 2 is generally satis…ed, with for example x3 quadratic, x4 linear and k sublinear, and Vx does not need to be known explicitly in order to verify Assumption 2 b) . 3.2
T @ @u uc
(23)
x4 (jxj)
x4
Remark 1 Assumption 1 d) can be viewed as a controllability assumption in the sense that: i) the mapping (t; x; ; ) : Rr ! Rd is surjective for all t; x and and ii) for all t; x and there exist a continuous function fu (t; x; ) such that (t; x; fu (t; x; ); ) = k(t; x): The surjective property can be seen by the Moore-Penrose pseudoinverse solution, u1 =
(22)
x3 (jxj)
Assumption 2 (Continued) b) There exists a K1 function that (jxj) (jxj)
xT ; uT j j + j 0j G for all t; x; and u. Further there exist a continuous function & @ (jxj) @ @ @ @ 0 @ 0 @ 0 @t + @t + @x + @x + @u + @u where & @ : R 0 ! R 0: d) There exists constants %2 > %1 > 0, such that 8t, x, u and @ (t; x; u; ) @u
x2 (jxj)
for the system x_ = f (t; x) with respect to its origin.
& @gx (jxj). c) The function is twice di¤ erentiable and there exist a continuous function G : R 0 ! R 0 ; such that
@ (t; x; u; ) @u
Vx (t; x)
@Vx @Vx + f (t; x) @t @x @Vx @x
@gx @x
%1 I
k1 > 0, such that 8 t; x; and (uT ; T )T 2 = Ou0 @2L k1 I < (t; x; u; ; ) k2 I; (32) @u2 where Ou0 is the interior of Ou : If (uT ; T )T 2 Ou0 2 then the lower bound is replaced by @@uL2 0.
(28)
:= r + d + n + m; zxu
@ L @t@
Ou (t; x):=
Based on the perturbing system, we consider the estimated optimal solution set: n o Ou ~ (t; x):= (u; ; ; ~)2Rnu ~ fOu ~ (t; zxu ~)=0 (29) ~
1
2
In order to see the cascaded coupling between the system (19) and the adaptive and optimal control allocation update-laws, equation (19) can be rewritten by: x_ = f (t; x) + g(t; x) k(t; x) (t; x; u; ^)
where nu
@ 2 L^ @ @u
@L^ : (31) @
if det( @@uL2 ) > or uf f ^ := 0 if det( @@uL2 ) < ; where r (k1 ) > > 0: k1 is de…ned in the following assumption which both guarantee existence of the proposed update laws and the time-varying …rst order optimal set,
Stability of the optimal certainty equivalent control allocation
(t; x; u)~:
^
ff
is that, the adaptive law is independent of Vx . Remark 3 A di¤ erent approach would be to expand the perturbed system ( 1 ) with an adaptive law. In this case the adaptive law will be dependent on the initial Lyapunov function (similar to (Tjønnås & Johansen 2005)), but convergence results like, x(t) ! 0 as t ! 1; may be concluded even if a persistence of excitation condition is not satis…ed.
+gx (t; x)
@ 2 L^ @u2 @ 2 L^ @u@
T
@ 2 L^ @x@
A ; the matrices ; and 0 ^ ; are symmetric positive de…nite and uf f ^ is a feedforward like term given by: 0 0 2 11 ! ! @ L^ @2L @2L _ + @x@u f (t; x)+@ @ 2^@u A^A u ^ : =H 1 @ @t@u 2 2 where H^ : = @
3.2.1
T
(t; x; u)) (gx (t; x))
Remark 4 The second order su¢ cient conditions in Theorem 12.6 in (Nocedal & Wright 1999) are satis…ed for all t; x; u; and by Assumption 1 and 3, thus the set Ou (t; x) describes global optimal solutions of the problem (3).
x; u; ; ; ~ ,
T
and the esti(t; x; u; ^))T :
Lemma 2 If Assumptions 1, 2 a) and 3 are satis…ed, then Ou is non-empty for all t; x and : Further for all t; x; and (u; ) 2 Ou ; there exist a continuous function & Ou : R 0 ! R 0 such that:
We will in what follows prove that the time and statevarying optimal set Ou ~ (t; x), in a certain sense is
(uT ;
5
T T
)
& Ou
xT ;
T
Ozdaglar 2003) we have that G : Rq ! RqG is continuous i¤ G 1 (U ) is closed in Rq for every closed U in RqG . From the de…nition of Ou ~ , U = f0g ; and since G is continuous (by Assumption 1 - 3), Ou ~ is a closed set. The set is forward invariant if at t1 ; G(t1 ; x(t1 ); u(t1 ); (t1 ); ~(t1 )) = 0 and d(G(t;x;u; ;~)) = 0 8t t1 with respect to (27), dt (30), (31) and (28). Since there exist a continuous solution of (30) as long as x exist, we only need to check this condition on the boundary of Ou ~ (t; x(t)) (Note that det(H^ ) 6=0 on the boundary of Ou ~ (t; x(t)) by Assumption 1d) and 3b)). We _ get G(t1 ; x(t1 ); u(t1 ); (t1 ); ~(t1 )) = 0 ) (_ ; ^) = 0 d@L d@L = 0: and it remains to prove that dt@ ^ ; dt@u^ d@L^ @ 2 L^ @ 2 L^ @ 2 L^ @ 2 L^ ^ _ = @t@ + @x@ x_ + @u@ u_ + @ ^@ and dt@ @ 2 L^ @ 2 L^ @ 2 L^ @ 2 L^ _ @ 2 L^ ^ d@L^ _ _ + @u@u u_ + @ @u + @ ^@u ; thus dt@u = @t@u + @x@u x ! ! ! @L^ d@L^ u_ dt@u @u +H^ uf f ^ = H^ H^ @L^ = 0: =H^ d@L^ _
PROOF. See Appendix A.
The idea of proving stability and convergence of the set Ou ~ (t; x); relies on the construction of the Lyapunovlike function: ! T T @L @L @L @L 1 ^ ^ ^ ^ Vu ~ (t; x; u; ; ; ~) : = + 2 @u @u @ @ ~T
T
+
2
+
^
2
~ :
(33)
Its time derivative along the trajectories of the considered system is: ! T 2 T 2 @L @ L @ L @L ^ ^ ^ ^ + u_ V_ u ^ = @u @u2 @ @u@ ! @L^ T @ 2 L^ _ @L^ T @ 2 L^ @L^ T @ 2 L^ + + + x_ @u @ @u @u @x@u @ @x@ ! @L^ T @ 2 L^ ~_ @L^ T @ 2 L^ + @u @ @ ^@u @ ^@ +
T
@L^ @u
@ 2 L^ + @t@u
@L^ @
T
@ 2 L^ + @t@
T
T
_+~
^
dt@
@
ii) Since x(t) is assumed to be forward complete, it follows from Lemma 2 that there always exists a T ~ that bounded by x and ^ = pair u T ; T
~_ :
satisfy the time-varying …rst order optimal condiFurther by (27), (30) and (31), we get ! @L^ T @L^ T @L^ T @L^ _ Vu ^ = ; H^ H^ ; @u @ @u @ T
c^
A @L^ @u
T
@L^ @L^ + @u @
T
@L^ @
!
T
tions
! T T
A
(34)
and it follows that there exists constants such that
Proposition 1 Let Assumptions 1, 2 a) and 3 be satis…ed. Then if x(t) exists for all t, the set Ou ~ (t; x(t)) is UGS with respect to the system (27), (30) and (31), and @L @L ( ; @u^ ; @ ^ ) converges asymptotically to zero.
~
; ~;
@L^ T @L^ T @u ; @
@L^ @L^ @u + @
2
u
2 1 2
zu
~ O u
; where ~
1
:=
1 2
min(
1;
;
^)
2
:=
max( 2 ; ; ^ ); such that Vu ~ (t; zu ~ ) is a radially unbounded Lyapunov function.
iii) Vu ~ (t; zu ~ ) := Vu ~ (t; x(t); u; ; ; ~); is a radially unbounded Lyapunov function. :=
jzu jOu
(35) 0
1 > T @L^ @
jzu jOu since HT^ H^ is positive de…nite and bounded from Assumption 1 and 3. By a similar @2L argument, in (A.4) change P c with @u2^ , H^ is non-singular, hence the control allocation law (30) always exist and, 1 zu ~ O ~ Vu ~ (t; zu ~ )
u
i) De…ne G(t; x; u; ; ~)
T
@L^ @u
2
1
2
2
PROOF. In order to prove this result we show that i) Ou ~ (t; x(t)) is a closed forward invariant set with respect to system (27), (30) and (31), ii) the system (27), (30) and (31) is …nite escape time detectable through T T z ~ , where z ~ := uT ; T ; T ; ~ , and that Ou
= 0, and thus the system (27),
(30) and (31) is …nite escape time detectable through zu ~ O ~ : u iii) In the proof of Lemma, change L with L^ such that (A.3) become !T ! u u u u @L^ T@L^ @L^ T@L^ + = HT^ H^ @u @u @ @
where c^ := inf t min H^ H^ > 0: We formalize the result, based on set-stability.
u
@L^ T @L^ T @u ; @
Since (34) is negative semide…nite, UGS of the set Ou ~ can be concluded by Theorem 1. The convergence result follows by Barbalat’s lemma
T
:
Corollary 2 Let Assumptions 1, 2 a) and 3 be satis…ed and zu ~0 O ~ < r where r > 0: If x(t) exists
From Proposition 1.1.9 b) in (Bertsekas, Nedic &
u
6
for t 2 [t0 ; T ) where T t0 , then there exists a positive constant B(r) > 0 such that for all t 2 [t0 ; T ); zu ~ (t) O ~ (t;x(t)) B(r).
3.2.2
u
The optimal set for the combined control and certainty equivalent control allocation problem is de…ne by:
PROOF. Since V_ u ~ 0 and Vu ~ (t; zu ~ (t)) Vu ~ (t0 ; zu ~ (t0 )) for all t 2 [t0 ; T ): From iii) in the proof of Proposition 1 1 zu ~ O ~ Vu ~ (t; zu ~ )
Oxu ~ (t) := Ox (t)
2
that q
1
~ O u
~
zu ~ (t)
for all t
Ou
~ (t;x(t))
Vu ~ (t0 ; zu ~ (t0 )) 1
2
q
[t0 ; T ); thus it follows q 1 Vu ~ (t; zu ~ (t)) 1 zu
2 1
~0 O u
~
Ou ~ (t; 0):
(36)
In the framework of cascaded systems, we consider (30), (31) together with (27) to be the perturbing system ( 2 ); while (28) represents the perturbed system ( 1 ): Loosely explained, we will use Lemma 1 to conclude UGAS of the set Oxu ~ if, Ox and Ou ~ individually are UGAS (which we already have established) and the combined system is UGB with respect to Oxu ~ .
u
zu
Stability of the combined control and certainty equivalent control allocation
=: B(r)
Remark 5 Provided that the gain matrix, > 0; is bounded away from zero, may be chosen time-varying. 1 If for example = H^ H^ for some > 0; then ! ! @L^ u_ 1 @u = H^ H ^ 1 uf f @L^ _
Before establishing the main results of this section, we need to state an assumption on the interconnection term between the two systems. We start by stating the following property: Lemma 3 By Assumption 1 and Lemma 2 there exists continuous functions & x ; & xu , & u : R 0 ! R 0 ; such that
@
where the …rst term is the Newton direction when L^ is considered the cost function to be minimized. In case H^ H^ is poorly conditioned one may choose 1 = H^ H^ + I for some > 0; to avoid c^ in (34) from being small.
j
(t; x; u)j
& x (jxj)& xu (jxj) + & x (jxj)& u ( zu
~ O u
~
):
PROOF. This can bee seen by applying (Mazenc & Praly 1996)’s lemma B.1 Assumption 2 (Continued) c) The function, k , from Assumption 2 b) has the following additional property:
Assumption 4 (Persistence of Excitation) The signal matrix g (t) := gx (t; x(t)) (t; x(t); u(t)) is Persistently Excited (PE), which means that there exist constants T and > 0; such that R t+T T I ; 8t > t0 : g( ) g ( )d t
1 k
(jxj)
x3 (jxj)
x4 (jxj)& x max (jxj)
,
(37)
where & x max (jxj) := max(1; & x (jxj); & x (jxj)& xu (jxj)): Remark 7 If there exists a constant K > 0 such that & x max (jxj) K 8x; which is common in a mechanical system, Assumption 2 c) is satis…ed.
Remark 6 The system trajectories x(t) de…ned by (1) typically represents the tracking error i.e. x := xs xd ; where xs is states vector of the system, and xd represents the desired reference for these states. This means that PE assumption on g (t) is dependent on the reference trajectory xd ; in addition to disturbances and noise, and imply that some "richness" properties of these signals are satis…ed.
Next we consider the closed loop of the plant, the virtual controller and the adaptive dynamic control allocation law: Proposition 3 If Assumptions 1-3 are satis…ed, then the set Oxu ~ is UGS, with respect to system (28),(30), (31) and (27). If in addition Assumption 4 is satis…ed, then Oxu ~ is UGAS with respect to (28),(30), (31) and (27).
Proposition 2 Let x(t) be UGB, then if Assumption 4 and the assumptions of Proposition 1 are satis…ed, the set Ou ~ is UGAS with respect to system (26), (27), (30) and (31).
PROOF. The main part of this proof is to prove boundedness, completeness and invoke Lemma 1. Let zxu ~0 O ~ r; where r > 0, and assume xu that jzx (t)jOx escapes to in…nity at T: Then for any constant M (r) there exists a t 2 [t0 ; T ) such that M (r) jzx (t)jOx . In what follows we show that M (r) can not be chosen arbitrarily. De…ne v(t; zx ) := Vx (t; x) such that
PROOF. See Appendix B
Unless the PE condition is satis…ed for g ; only stability of the optimal set is guaranteed. Thus in the sense of achieving asymptotic optimality, parameter convergence is of importance.
7
v_ +
@Vx g(t; x) k(t; x) @x (t; x; u)~
x3 (jzx jOx )
@Vx gx (t; x) @x x3 (jzx jOx )
+
+
Proposition 3 implies that the time-varying optimal set Oxu ~ (t) is uniformly stable, and in addition uniformly attractive if Assumption 4 is satis…ed. Thus optimal control allocation is achieved asymptotically for the closed loop. A local version of this result can be proven using Corollary 1.
(t; x; u; ^)
+
@L^ @Vx jg(t; x)j (t; x; u; ^) @x @
+
@Vx jgx (t; x)j @x x3 (jzx jOx )
(t; x; u)~
x4 (jzx jOx )K( 2 +j
(t; x; u)j) zu
~ O u
~
:
Corollary 3 If for u 2 U Rr there exist constant cx > 0 such that for jxj cx the domain Uz R 0 Rn U Rd+n+m contain Oxu ~ ; then if the Assumptions 1-3 are satis…ed, the set Oxu ~ is US with respect to the system (28), (30), (31) and (27). If in addition Assumption 4 is satis…ed, Oxu ~ is UAS with respect to the system (28), (30), (31) and (27). PROOF. Since Oxu ~ Uz , there exist a positive constants r zxu ~0 O ~ such that jzx (t)jOx cx0 where xu 0 < cx0 < cx ; hence the domain Uu ~ R 0 U d+n+m R contain Ou ~ : US/UAS of Ou ~ follows from the Lyapunov-like function Vu ~ and the PE Assumption. UB follows from zxu ~0 O r, and the US/UAS xu property of Oxu ~ follows from Corollary 1
(38)
From Corollary 2, there exists a positive constant B(r) 0; for all t 2 [t0 ; T ); such that for zu ~0 O ~ r, u
zu ~ (t) O ~ B( zu ~0 O ~ ) B(r). From Assumpu u tion 1 and 2, v_ x3 (jzx jOx ) + x4 (jzx jOx )K ( 2 + j (t; x; u)j) B(r) k (jzx jOx ) x4 (jzx jOx )& x max ((jzx jOx ) + x4 (jzx jOx )K ( 2 + j (t; x; u)j) B(r) x4 (jzx jOx ) k (jzx jOx )& x max (jzx jOx ) + x4 (jzx jOx )K 2 B(r) + x4 (jzx jOx )K& x (jzx jOx )& xu (jzx jOx )B(r) + x4 (jzx jOx )K& x (jzx jOx )& u (B(r))B(r) (39)
Corollary 4 If is known and the Assumptions 1-3 are satis…ed then the set Oxu := Ox (t) Ou ~ (t; 0): is UGAS with respect to the system (19) and (30). PROOF. This result follows by working through the above arguments, neglecting the adaptive dynamic and considering a Lyapunov like function of the form
1 x4 (jzx jOx ) k ( x2
(v))& x max (jzx jOx ) + x4 (jzx jOx )K 2 B(r) + x4 (jzx jOx )K& x (jzx jOx )& xu (jzx jOx )B(r) + x4 (jzx jOx )K& x (jzx jOx )& u (B(r))B(r):
Thus, if jzx0 jOx >
1 k
(K (
2
1 k
and from (40), v(t; zx (t)) such that jzx (t)jOx By choosing M (r) := max 1 x1
x2
1 k
(K ( 1
x2
k
1 x1 1 x1
(
(K (
2
(K ( 2+1+& u (B(r))) B(r))
x2 (r))
T @L^ @
Since the set with respect to system 2 may only be US, due to actuator/e¤ector constraints and parameter uncertainty, only US of the cascade, may be concluded. But if the PE property is satis…ed on g , a UAS result may be achieved with both optimal and adaptive convergence. 4 Example In this section simulation results of an over-actuated scaled-model ship, manoeuvred at low-speed, is presented. A 3DOF horizontal plane model described by: _ e = R( ) _ = M 1 D + M 1 ( + b) (41) = (u) ;
:
;
(K ( 2+1+& u (B(r))) B(r))
@L^ @
(v(zx0 ))
+ 1 + & u (B(r))) B(r)))
1
T @L^ @u +
Corollary 5 If for u 2 U Rr there exist constant cx > 0 such that for jxj cx the domain Uz R 0 n d R U R contain Oxu ; then if the Assumptions 13 are satis…ed, the set Oxu is UAS with respect to the system (19) and (30).
+ 1 + & u (B(r))) B(r))
2
x2 ( k
1 x1
@L^ @u
(40)
+ 1 + & u (B(r))) B(r))
then from (39), v(t0 ; zx0 ) v(t; zx (t)) and jzx (t)jOx 1 ( (r)) ; else, x2 x1 jzx0 jOx
Vu (t; x; u; ) := 12
;
the assumption of jzx (t)jOx escaping to in…nity is contradicted, since M (r) > jzx (t)jOx and j jOx is …nite escape time detectable. Further more Ox is UGB. From Propositions 1 and 2 and the assumptions of these propositions, the assumptions of Lemma 1 and Corollary 1 are satis…ed and the result is proved
T
is considered, where e := (xe ; ye ; e ) := (xp xd ; yp T yd ; p d ) is the north and east positions and compass heading deviation. Subscript p and d denotes the
8
actual and desired states. := (u; ; r)T is the body…xed velocities, surge, sway and yaw, is the generalized force vector, b := (b1 ; b2 ; b3 )T is a disturbance due to wind and current and R( ) is the rotation matrix function between the body …xed and the earth …xed coordinate frame. The example we present here is based on (Lindegaard & Fossen 2003), and is also studied in (Johansen 2004) and (Tjønnås & Johansen 2005). In the considered model there are …ve force producing devices; the two main propellers aft of the hull, in conjunction with two rudders, and one tunnel thruster going through the hull of the vessel. ! i denotes the propeller angular velocity and i denotes the rudder de‡ection. i = 1; 2 denotes the aft actuators, while i = 3 denotes the tunnel thruster. (41) can be rewritten in the form of (1) and (2) by: x := ( ;
e;
T
) ;
:= ( 1 ;
T 2)
The gain matrices are chosen as follows: Kp := M diag(3:13; 3:13; 12:5)10 2 ; Kd := M diag(3:75; 3:75; 7:5)10 1 ; KI := M diag(0:2; 0:2; 4)10 3 ; A := I9 9 ; Q := diag(1; 1); T Q := diag(a; 150 (100; 100; 1)T ); a := (1; 1; 1; 1; 1; 1)
:= ( 1 ; 2 ; 3 ) ; u := (! 1 ; ! 2 ; ! 3 ; 0 1 X1 + X2 0 B C C (u) := B @ Y1 + Y2 T3 A ; l3;x T3 13
T
1; 2)
where
W := diag (1; 1; 1; 1; 1; 0:6; 0:6; 0:6) and " := 10 9 . The weighting matrix W is chosen such that the deviation @L of @ ^ = k(t; x) (t; x; u; ^) from zero, is penalized more then the deviation of
@L^ @u
^_ =
;
8
0.
Remark 8 In order to satisfy Assumption 1 we replace in (41) by = (u; ) = (u) + 0 (u); where T 0 (u) := & (! 1 ; ! 2 ; ! 3 ) and & > 0; in the control allocation algorithm. Since & may be chosen arbitrarily small, this actuator/e¤ ector mapping is practically similar to (41), but ensures that the allocation algorithm is well conditioned. Assumption 3 is satis…ed locally since for 2 bounded u and ; ki2 ensures that @@uL2 > 0: It is also worth noticing that the optimal control allocation solution in this example, at any time, is a single point. Based on the wind and current disturbance vector b := 0:05(1; 1; 1)T and the parameter vector := (0:8; 0:9)T ; the simulation results are presented in the Figures 3-7. Due to the disturbance and reference change, the transients excite the parameter update-law at t 0 as well as at t 200 and t 400, and the estimated parameters converges to the true values. The control objective is satis…ed and the commanded virtual controls are tracked closely, by the forces generated by the adaptive control allocation law, except for t 0; t 200 and t 400; where the control allocation is suboptimal due to initial conditions and actuator saturation: see Figure 7. The simulations are carried out in the MATLAB environment with a sampling rate of 10Hz:
Kd ;
proposed in (Lindegaard & Fossen 2003), stabilizes the system (41) augmented with: _ = := b e , where and _ = e ; uniformly and exponentially, for some physically limited yaw rate. The cost function used is: J(u) :=
(u) gx
n S := ^ 2 R2
The unknown parameter vector represents thrust losses. 2 is also related to the parameters kT p3 and kT n3 in a multiplicative way. This suggest that the estimate of 2 gives a direct estimate of the tunnel thruster loss factor. The virtual controller :=
T
1
2 (S) and
T
c
from zero, in the
search direction. In order to keep ^ from being zero, due to a physical consideration, a projection algorithm can be used. i.e. from (30)
;
T
1
:= HT ^ W H^ + "I
and
qi 2i ;
i=1
q1 = q2 = 500;
9
5
1
0
1 [deg]
0.5
η
0 -0.5
0
100
200
300
400
500
δ
1 [m]
1.5
600
-5 -10 -15 -20
1
0
100
200
0
100
200
300
400
500
600
300
400
500
600
30 0
100
200
300
400
500
600
20
2 [deg]
-0.5
0.3
0.1
0 -10
0 -0.1
10
δ
0.2
η
3 [deg]
0
0.5
η
2 [m]
1.5
0
100
200
300
400
500
-20
600
t [s]
t [s]
Fig. 3. Simulation results - the solid lines represent positions while the dashed lines represent references
Fig. 5. Simulation results - computed rudder de‡ection by the allocation algorithm 1.1
ω
1 [Hz]
1.05 0
1
-5
0.95 0.9
-10
0.85
-15
0.8 -20
0
100
200
300
400
500
600
0.75
0
100
200
100
200
300
400
500
600
300
400
500
600
ω
2 [Hz]
20 10
0.9
0
0.8
ω
3 [Hz]
-10
0
100
200
300
400
500
0.7
600
20
0.6
10
0.5
0
0.4 0
-10 -20
t [s] 0
100
200
300
400
500
600
t [s]
Fig. 6. Simulation results - the parameter adaption
Fig. 4. Simulation results - computed propeller angular velocities by the allocation algorithm
6
5
The authors are grateful for insightful comments from Antoine Chaillet, Elena Panteley and Alexey Pavlov. The work is sponsored by the Research Council of Norway through the Strategic University Programme on Computational Methods in Nonlinear Motion Control.
Conclusion
Based on a control-Lyapunov design approach, an adaptive optimizing nonlinear control allocation algorithm is derived. Under certain assumptions on the system (actuator/e¤ector model) and the control design (growth rate conditions on the Lyapunov function), a cascade result is used to prove, closed-loop, stability and attractivity of a set representing the optimal actuator con…guration. Typical applications for the control allocation algorithm are over-actuated mechanical systems, especially systems that exhibits fast dynamics since the algorithm is computational e¢ cient. An automotive example is presented in (Tjønnås & Johansen 2006).
Acknowledgements
References Abrahamson, H., Marsden, J. E. & Ratiu, T. (1988), Manifolds, Tensor Analysis and applications, 2’nd edn, Springer-Verlag. Bertsekas, D. P., Nedic, A. & Ozdaglar, A. E. (2003), Convex Analysis and Optimization, Athena Scienti…c. Bodson, M. (2002), ‘Evaluation of optimization methods for control allocation’, J. Guidance, Control and Dynamics 25, 703–711. Bu¢ ngton, J. M., Enns, D. F. & Teel, A. R. (1998), ‘Control allocation and zero dynamics’, J. Guidance, Control and Dynamics 21, 458–464.
10
tracking of time-varying control trajectories’, In Proceedings of the 2005 American Control Conference, Portland, OR .
0.1 0
Mazenc, F. & Praly, L. (1996), ‘Adding integrations, saturated controls, and stabilization for feedforward systems’, IEEE Transactions on Automatic Control 41, 1559–1578.
τ
1 [N]
0.2
-0.1 -0.2
0
100
200
300
400
500
600
Nocedal, J. & Wright, S. J. (1999), Numerical Optimization, Springer.
0.2 0
Pantely, E., Loria, A. & Teel, A. R. (2001), ‘Relaxed persistency of exitation for uniform asymptotic stability’, IEEE Trans. Automat. Contr. 46, 1874 – 1886.
τ
2 [N]
0.4
-0.2 -0.4
0
100
200
300
400
500
600
Poonamallee, V., Yurkovich, S., Serrani, A., Doman, D. & Oppenheimer, M. (2005), ‘Dynamic control allocation with asymptotic tracking of time-varying control trajectories’, In Proceedings of the 2004 American Control Conference, Boston, MA .
τ
3 [Nm]
0 -0.02 -0.04 -0.06 -0.08
0
100
200
300
400
500
600
Skjetne, R. (2005), The Maneuvering Problem, Phd thesis, NTNU, Trondheim, Norway.
t [s]
Fig. 7. Simulation results - the solid lines represent actual forces generated by the actuators, the dashed lines represent the virtual forces generated by the control law.
Sørdalen, O. J. (1997), ‘Optimal thrust allocation for marine vessels’, Control Engineering Practice 5, 1223–1231. Teel, A., Panteley, E. & Loria, A. (2002), ‘Integral characterization of uniform asymptotic and exponential stability with applications’, Maths. Control Signals and Systems 15, 177–201.
Enns, D. (1998), Control allocation approaches, in ‘Proc. AIAA Guidance, Navigation and Control Conference and Exhibit, Boston MA’, pp. 98–108.
Tjønnås, J., Chaillet, A., Panteley, E. & Johansen, T. A. (2006), ‘Cascade lemma for set-stabile systems’, 45th IEEE Conference on Decision and Control, San Diego, CA .
Härkegård, O. (2002), E¢ cient active set algorithms for solving constrained least squares problems in aircraft control allocation, in ‘Proc. IEEE Conf. Decision and Control, Las Vegas NV’.
Tjønnås, J. & Johansen, T. A. (2005), ‘Adaptive optimizing nonlinear control allocation’, In Proc. of the 16th IFAC World Congress, Prague, Czech Republic .
Ioannou, P. A. & Sun, J. (1996), Robust Adaptive Control, Prentice Hall PTR, New Jersey.
Tjønnås, J. & Johansen, T. A. (2006), ‘Adaptive optimizing dynamic control allocation algorithm for yaw stabilization of an automotive vehicle using brakes’, 14th Mediterranean Conference on Control and Automation, Ancona, Italy .
Johansen, T. A. (2004), ‘Optimizing nonlinear control allocation’, Proc. IEEE Conf. Decision and Control. Bahamas pp. 3435– 3440. Johansen, T. A., Fossen, T. I. & Berge, S. P. (2004), ‘Constrained nonlinear control allocation with singularity avoidance using sequential quadratic programming’, IEEE Trans. Control Systems Technology 12, 211–216.
A
Proof of Lemma 2
The boundary of the set Ou and the time-varying …rst order optimal solution is described by:
Johansen, T. A., Fossen, T. I. & Tøndel, P. (2005), ‘E¢ cient optimal constrained control allocation via multi-parametric programming’, AIAA J. Guidance, Control and Dynamics 28, 506–515.
@J @ @L (t; x; u; ; ) = (t; x; u) (t; x; u; ) @u @u @u @L (t; x; u; ; ) = k(t; x) (t; x; u; ) @
Khalil, H. K. (1996), Nonlinear Systems, Prentice-Hall, Inc, New Jersey. Krstic, M., Kanellakopoulos, I. & Kokotovic, P. (1995), Nonlinear and Adaptive Control Design, John Wiley and Sons, Inc, New York.
T
By using theorem 2.4.7 in (Abrahamson, Marsden & Ratiu 1988), it can be shown that
Landau, H. (1979), Adaptive Control: The Model Reference Approach, Marcel Dekker, Inc., New York. Lin, Y., Sontag, E. D. & Wang, Y. (1996), ‘A smooth converse lyapunov theorem for robust stability’, SIAM Journal on Control and Optimization 34, 124–160.
@L (t; x; u; ; ) @u @L (t; x; uc ; ; ) @u @L (t; x; uc ; ; ) @u @L (t; x; u; ; ) @ @L (t; x; u ; ; ) @
Lindegaard, K. P. & Fossen, T. I. (2003), ‘Fuel-e¢ cient rudder and propeller control allocation for marine craft: Experiments with a model ship’, IEEE Trans. Control Systems Technology 11, 850–862. Luo, Y., Serrani, A., Yurkovich, S., Doman, D. & Oppenheimer, M. (2004), ‘Model predictive dynamic control allocation with actuator dynamics’, In Proceedings of the 2004 American Control Conference, Boston, MA . Luo, Y., Serrani, A., Yurkovich, S., Doman, D. & Oppenheimer, M. (2005), ‘Dynamic control allocation with asymptotic
11
@L (t; x; uc ; @u @L (t; x; uc ; @u @L (t; x; u ; @u @L (t; x; u ; @ @L (t; x; u ; @
@2L (u uc ) @u2 c @2L ; )= ( ) @ @u c @2L ; ) = 2 (uc u ) @u @2L ; )= (u u ) @u@ ; )=
; ) = 0:
R1
fOu ~ (t) := fOu ~ (t; x(t); u(t); (t); (t); ~(t));
@2L (t; x; (1 s)uc + su; ; )ds; 0 @u2 c R 2 2 2 1 @ L L := := 0 @@uL2 (t; x; (1 s)u + suc ; ; )ds; @@ @u @u2 c R 1 @2L T (t; x; uc ; (1 s) + s ; )ds = @@u (t; x; uc ; ) 0 @ @u R 1 @2L 2 @ L := 0 @u@ and @u@ (t; x; (1 s)u + su; ; )ds = @ @u (t; x; uc ; ): Since @L ; ) = 0 and @L ; ) = 0; we get @ (t; x; u ; @u (t; x; u ;
where
@2L @u2
:=
@L @ L (t; x; u; ; ) = @u @u2
@2L @u2 = (1
(u
uc ) +
c T
@ L @u2
@ (t; x; uc ; )( @u @ (t; x; uc ; )(u @u
and from uc := #(u 0 < #i < 1 :
_ g (t) = @gx (t; x) (t; x; u) + gx (t; x) @ (t; x; u) @t @t @ (t; x; u) @gx (t; x) (t; x; u)+gx (t; x) x_ + @x @x @ (t; x; u) +gx (t; x) u; _ @u
2
2
@L (t; x; u; ; ) = @
is bounded, by using the PE property; and ii) use the integral bound to prove UGA by contradiction. The proof is based on ideas from (Pantely, Loria & Teel 2001) and (Teel et al. 2002). i) First we establish a bound on _ g (t). We have
(uc
u )
)
(A.1)
u );
(A.2) thus from Assumption 1, system (28) and update& 1x _ g (jxj) + law (30), there exists a bound _ g (t)
u ) + u where # := diag(#i ) and
@2L (u uc ) + (uc u ) @u2 c @2L @2L #) (u u ) + # (u @u2 c @u2
From Assumption 3,
@2L @u2
c
and
@2L @u2
& 1x _ g (jxj)& u _ g ( zu ~ O ~ ) where & 1x _ g ; & 1x _ g ,& 1x _ g : u R 0 ! R 0 are continuous functions. This can be seen by following the same approach as in Lemma 3. From the assumption that x is uniformly bounded, T we use jxj Bx : The integrability of ~ ~ is investigated by considering the auxiliary function: V aux1 := T ~ V aux1 where g (t) ; bounded by BV aux1 (Bx ; r) r zu ~0 O ~ and r > 0: Its derivative along the u solutions of (27) and (31) is given by T V_ aux1 = ~ g (t)T g (t)~ + T AT g (t) _ g (t) ~
u ):
are positive de…2
2
nite matrices, such that P c := (1 #) @@uL2 + # @@uL2 c is also a positive de…nite matrix. Further !T ! T T u u u u @L @L @L @L + = HT H @u @u @ @
where H =
P
@ T @u (t; x; uc ;
c
@ @u (t; x; uc ;
) 0 sumption 3 and 1 it can be seen that: jdet(H )j= det(P c )det
)
!
+ +
(A.3) : By As-
where
1 c
@ (t; x;uc ; ) @u
T
!
> 0;
T
(A.4)
is non-singular and there exists constants > 0 such that 1 I HT H 2 I: Furthermore choose u = 0 and = 0; such that @L @L (t; x; 0; 0; ) & (jxj) and @J @u @ (t; x; 0; 0; ) G (jxj) (1 + j j) + & k (jxj) from Assumption 1, 2 and 3. 1
u
T
;
T
T
& Ou
xT ;
T
2 @x L^ @u (t):=
+
AT T
and H
Thus from (A.3)
g (t) ^
T g (t)
(B.1)
2 T @x L^
1
g (t)
@ 2 L^ @x@u @L^ @u
:=
1
T
; T
;
@u
@ 2 L^ @x@
@L^ @
T
4T
@ L^ (t); @u
(B.2)
T T
(t; x(t); u(t); (t);^(t)): 1 T nb b
2 bT a , we
_ g (t) ~
g (t)
AT
(t)
(t; x(t); u(t); (t); ^(t))
From Young’s inequality, naT a + have
T 2
g (t) ^
T
@ L^ @u (t)
and @ (t; x;uc ; )P @u
T
g (t)
_ g (t)
AT
g (t)
_ g (t)
~T ~;
T
(B.3)
where a = ~; b = T AT g (t) _ g (t) and n = 2T such that and T are positive scalars of desirable choice. Next
T
2
where & Ou (jsj) := 1 1 (G (jsj) (1 + jsj) + & k (jsj)) + 1 2 1 & @J (jsj) B Proof of Proposition 2
T
The proof is divided into two parts: i) show that the integral of fOu ~ (t)T fOu ~ (t); where
T
12
@x2 L^ @ L^ (t) (t) @u @u T @ L^ @ L^ T (t) KL (t) g (t) g (t) + @u @u
g (t) ^
1
T g (t)
T
for n = 1, a = where KL Hence V_
g (t) and b 2 @x L^
:=
@u
=
(t)
1 ^
g (t)
2 @ L^ T @x L^ g (t) @u (t) @u (t), 2 T @x L^ T 1 T g (t) @u (t): ^ ^
T
T
AT
+
T
1 T ~T g (t) ^ g (t) ~T ~ + T g (t) T (t) g
aux1
+ +
4T @ LT ^ @u
_ g (t) AT
g (t)
_ g (t)
g (t) T g (t)
T
T g (t)
g (t)
T
T
AT
g (t)
+
T
g (t)
^
1
@ LT ^
+
aux1
@u
4T
AT
1
(B.4)
_ g (t)
g (t)
@ L^ (t) . @u
(t)KL
~( )T = ~(t)T
g(
)T
g(
T
g(
)
g(
T
+R (t; ) 2~(t)T
)
Z
g(
T
4T ( g )
g(
)
)
)T
g(
)
T
(B.5)
)T
R (t; )T
g(
g(
)R (t; )
)T
g(
^
2
4T
4T
R (t; )T
g(
)T
g(
g(
)
)
(1 + ) I R (t; ) (1 )4T ! 1+ 1 I ~(t) (B.7) 1 1 4T
(s) we get
T g (s)
Z
(s) ds
1
2
T g (s)
^
t
+2
Z
1~ T (t)
g(
)T
(s) ds (
t)
2
t) ;
g(
4T g ( )R (t; )
)
T g (s)
(s)ds (
@ L^ (s)ds( @u
(s)KL
T (Bx ; r; T )
)~(t)
+V
:=
max
t)
t):
T g (t)
g (t)
(1+ ) (1 ) 4T
I
aux1 (t)
)R (t; )
T
@u
1 ^
(1+ ) be the maximal eigenvalue of g ( )T g ( ) (1 ) 4T I and max KL be the maximal eigenvalue of KL ; then from combining (B.5) and (B.7) and investigating the integral over , we get ! Z 1+ 1 1 ~ T t+T T I d ~(t) 1 (t) g( ) g( ) 1 1 4T t Z t+T Z B T (s)T g (s) T t)d g (s) (s)ds ( t t ! Z t+T Z @ LT @ L^ ^ +B T max KL (s)T (s)ds ( t) d @u @u t t
(B.6)
g(
@ LT ^
t
Let B
I R (t; ) 4T I R (t; ):
~(t)T
)T
t
and ~(t)T R (t; )
)T
2 1
+ max KL 1
g(
R (t; )T R (t; ) Z 2 (s)T g (s) ^ T
1 T Again by Young’s inequality, n1 aT 1 a1 n 1 b 1 b1 1 2 bT n2 aT bT b 2 2 bT 1 a1 and 2 a2 2 a2 with p n2 2 ~ ~ a1 = ( ) ; a2 = g ( )R (t; ); 4T ; b1 = p g b2 = 4T R (t; ) and n1 = n2 = 1 ; such that g(
g(
I ~( )
such that
I ~(t)
4T
4T
and by similar arguments Z @x2 L^ @ L^ 1 T (s) (s) (s) ds g ^ @u @u t Z 2 @x2 L^ @ L^ 1 T (s) ds ( (s) (s) g ^ @u @u t
I ~( )
4T
~(t)T
T g (s)
^
T g (t)
+I
)
for > 1: From Holder’s inequality, 1 1 R R 2 2 2 2 R jgj ; with g = 1 and f = jf gj jf j
I ~(t)
_ g (t)
g(
~
The solution of (30) can be represented by: ~( ) = ~(t) R (t; ), where 2 R @x L^ @L 1 T (s) + @u R (t; ) := (s) @u ^ (s) ds, g (s) ^ t such that
2~(t)T
1
t
~(t)T
)T
) R (t; )T
+ 1
and thus
V_
g(
(1
g (t)
@ L^ (t); @u
(t)KL
~( )T
Z +
~ g ( ) (t)
t
+
T
t+T
( )T Z
( )T AT
Further
13
g(
)
1 ^
+I
T g(
) ( )d
t+T
t
which gives
V aux1 (t + T ) ! Z t+T @ LT @ L^ ^ (s) (s) ds @u @u t
g(
) _ g ( ) AT
g(
)
_ g( )
T
( )d :
Z
Z
t+T
t
Z
T2
(s)T
g (s) ^
t t+T
(s)T
1
T g (s)
^
T
g (s) ^
t
T
1
T g (s)
^
(s)ds (
Thus ~( )T ~( ) is integrable. Let min V be the minimum eigenvalue of (1; HT ^ H^ ; A); then there exists a constant BfO ~ (Bx ; r; T ) > 0; such that u Z t0 +TT fOu ~ ( )T fOu ~ ( )d t0 Z t fOu ~ ( )T fOu ~ ( )d lim
t)d
(s)ds;
such that for B (Bx ; r; T ) :=
T
A
max
_ g (t)
g (t) g (t)
it follows that Z 1 ~ T (t) 1
B
TT
T
A 2
g(
T
)
aux1 (t)
Z
g(
)
aux1 (t + T ) T @ L^ ( ) HT ^ ( ) H^ (
t+T
+BV
V
@u
t
1 ^
1+ 1
t
V
g (t)
+1
t+T
_ g (t)
1
4T
~(t)T
Z
t+T g(
)T
g(
1+ 1
)
t
1
(V aux1 (t) 1 1 +BV Vu ^ (T + t)
V
aux1 (t
1
2T + B
!
B fO
I d ~(t)
1
4T
I
!
B ~(t)T ~(t) Z t+T ~(t)T
g(
)T
g(
t
=
V
aux1 (
aux1 (
)
)d
t0
2T BV
+ T ))
aux1 (
Z
aux1 (Bx ; r)
(B.8)
Ou
( zu
~
~0 O u
~
);
8t
1
4T
I
!
zu ~ (t0 ; zu
d ~(t)
~0 ) O u
~
>
0
!
zu ~ (t ; zu
~
~0 ) O u
("):
~
zu ~ (t ; zu
1
u
(") 8t0 2 [t0 ; TT ]. Thus
~0 ) O u
~
8t0 2 [t0 ; TT ]
;
which from radial unboundedness of Vu ^ ; related to (33), imply that there exist positive constants ! 1 ; and 1 such that !1
fOu ~ (t0 )T fOu ~ (t0 )
Then Z t0 +TT t0
= ! 1 TT = 2BfO
t0 +TT +T aux1 (
0
1
~0 ) A
( (")) = " for zu ~0 O ~ r and t T T + t0 ; u which satis…es de…nition 3: Suppose the assumption is r such that not true, i.e., there exists zu ~0 O ~
+ T )) d V
~0 ) O u
1
2
1
:= (r) and is given by ~
u
Thus zu ~ (t; zu
1+ 1
u
t0
u ~
ists t0 2 [t0 ; TT ] such that zu ~ (t0 ; zu
)d
t0 +TT
and Z t0 +TT ~( )T ~( )d t0
V
;
; where ! 1 is speci…ed ! and : De…ne TT = !1 later, and assume that for all zu ~0 O ~ r; there ex-
t0 +TT
(V
+ BV Vu ^ (t0 )
2BfO
V aux1 (t) V aux1 (t + T ) 1 1 + BV Vu ^ (T + t) Vu ^ (t) ;
t0 Z t0 +T
u ~
!
where 2 K1 : Fix r > 0; " > 0. De…ne ! := min ; 1 (") and note that BfO
1
Z
1
zu ~ (t)
d ~(t)
(t;T )+
)
aux1 (Bx ; r) 1
ii) From The UGS property we have
max KL (B T T +1) : (1 1 ) min A By the PE assumption we can choose T; ; > 0 and (1+ 1 ) > 1 such that, B < , where B > 0 i.e. (1 1 ) 4 max B
BV
Vu ^ (1)
and the integral of fOu ~ (t)T fOu ~ (t) is bounded.
Vu ^ (t) ;
where BV (Bx ; r; T ) :=
Vu ^ (t0 )
min V
! @ L^ T ( )+ ( ) A ( ) d ) @u
and
t0
1
+
T g (t)
+I
1
t!1
T
u ~
T
fOu ~ ( ) fOu ~ ( )d
8t0 2 [t0 ; TT ]
1;
Z
t0 +TT
!1 d
t0
;
which contradicts (B.8), and the proposition is proved.
! 2T BV aux1 (Bx ; r) +BV Vu ^ (t0 ) : B 1 1
14
Abstract In this work we address the control allocation problem for a nonlinear over-actuated time-varying system where parameters a¢ ne in the e¤ector model may be assumed unknown. Instead of optimizing the control allocation at each time instant, a dynamic approach is considered by constructing update-laws that represent asymptotically optimal allocation search and adaptation. Using Lyapunov analysis for cascaded set-stable systems, uniform global/local asymptotic stability is guaranteed for the sets described by the system, the optimal allocation update-law and the adaptive update-law. Key words: Control allocation; Adaptive control; Nonlinear systems; Cascaded systems; Optimization.
1
Introduction
This work is motivated by the over-actuated control allocation problem. The problem is described by a nonlinear system, divided into a dynamic and a static part. The main contribution of this work is to show that the instantaneous control allocation problem, optimizing the desired input of the dynamic part of the system, not necessarily needs to be solved exactly at each time instant. In order to imply convergence and stability properties of the whole system, stability and attractivity of the optimal set, described by the control allocation problem, is pursued. Optimizing control allocation solutions have been derived for certain classes of over-actuated systems, such as aircraft, automotive vehicles and marine vessels, (Enns 1998, Bu¢ ngton, Enns & Teel 1998, Sørdalen 1997, Bodson 2002, Härkegård 2002, Luo, Serrani, Yurkovich, Doman & Oppenheimer 2004, Luo, Serrani, Yurkovich, Doman & Oppenheimer 2005, Poonamallee, Yurkovich, Serrani, Doman & Oppenheimer 2005, Johansen, Fossen & Berge 2004) and (Johansen, Fossen & Tøndel 2005). The control allocation problem is, in (Enns 1998, Bu¢ ngton et al. 1998, Sørdalen 1997, Johansen et al. 2005, Bodson 2002) and (Härkegård 2002), viewed as a static or quasi-dynamic optimization problem that is solved independently of the dynamic control problem considering non-adaptive linear e¤ector models of the form = Gu. In (Luo et al. 2004) and (Luo et al. 2005) a dynamic model predictive approach is considered to solve the allocation problem with linear time-varying dynamic in the actuator/e¤ector model, T u+u _ = uc . In (Poonamallee et al. 2005) and (Johansen et al. 2004) sequential quadratic programming tech-
Preprint submitted to Automatica
niques are used to cope with nonlinearities in the control allocation problem due to singularity avoidance and actuator failure. The main advantage of the control allocation approach is in general modularity and the ability to handle redundancy and constraints. Consider the system x_ = fx (t; x) + gx (t; x) (1) = (t; x; u; ) := (t; x; u) + 0 (t; x; u) (2) where t 0; x 2 Rn is the state vector; u 2 Rr is control input vector; 2 Rd is a vector of virtual controls and 2 Rm is a constant vector that contains parameters that either will be assumed known, or viewed as uncertain parameters to be adapted. The state x; is assumed known and typically represents the tracking error. The mapping (2) represents an actuator and e¤ector model, where the control signals u; are mapped to the virtual controls . We limit our study to the over-actuated control allocation problem, where m d r: Problem: Assume that there exist a virtual control c := k(t; x); with k(t; 0) = 0; that is di¤erentiable and uniformly globally asymptotically stabilizes the origin of (1), when = k(t; x). i) If is known, de…ne an update law u_ := fu (t; x; u), such that the stability property of the closed loop is conserved and u(t) converges to an optimal solution with respect to the static minimization problem min J(t; x; u) s:t (t; x; u; ) = 0: (3) c u ^ ii) If is unknown, solve problem i) with = ; u_ := _ fu (t; x; u; ^) and de…ne an adaptive update law ^ := f^ (t; x; u; ^); where ^ 2 Rm is an estimate of : 24 January 2008
and F : D ! Rq is locally Lipschitz with D Rq . In the following, if referred to a set, it has the properties of being nonempty. De…nition 1 The system (4) is said to be forward complete if, for each z0 2 D, the solution z( ; z0 ) 2 D is de…ned on R t0 . De…nition 2 The system (4) is said to be …nite escape time detectable through j jA , if any solution, z(t; z0 ) 2 D, which is right maximally de…ned on a bounded interval [t0 ; T ); satis…es limt%T jz(t; z0 )jA = 1. De…nition 3 If the system (4) is forward complete, then the closed set A D is: Uniformly Stable (US), if there exists a function, 2 K, and a constant, c > 0; such that, 8 jz0 jA < c, (6) jz(t; z )j (jz j ); 8t 0 :
In (Johansen 2004) it was shown that it is not necessary to solve the optimization problem (3) exactly at each time instant. Further a control Lyapunov function was used to derive an exponentially convergent updatelaw for u (similar to a gradient or Newton-like optimization) such that the control allocation problem (3) could be solved dynamically. And …nally it was shown that convergence and asymptotic optimality of the system, composed by the dynamic control allocation and a uniform globally exponentially stable trajectory-tracking controller, guarantees uniform boundedness and uniform global exponential convergence to the optimal solution of the system. The advantage of this approach is computational e¢ ciency and simplicity of implementation, since the optimizing control allocation algorithm is implemented as a dynamic nonlinear controller. Solving (3) online at each sampling instant requires a computationally more expensive numerical solution. In (Tjønnås & Johansen 2005) the results from (Johansen 2004) were extended by allowing uncertain parameters, associated with an adaptive law, in the e¤ector model, and by applying set-stability analysis in order to also conclude asymptotic stability of the optimal solution set. In what follows we will extend the ideas from (Johansen 2004) and (Tjønnås & Johansen 2005) by utilizing the set-stability result for cascaded systems established in (Tjønnås, Chaillet, Panteley & Johansen 2006). This result enables us to relax the assumptions in (Johansen 2004) and (Tjønnås & Johansen 2005) where fx (t; x); gx (t; x) and (t; x; u; ) are assumed to be globally Lipschitz in x: Further the virtual controller c does only need to render equilibrium of (1) UGAS, not UGES as assumed in (Johansen 2004) and (Tjønnås & Johansen 2005). The implementation of the adaptive law presented in (Tjønnås & Johansen 2005) depends directly on the Lyapunov function used in the analysis. In this work the analysis and implementation are separated, and although the Lyapunov functions in the analysis need to satisfy certain requirements, the adaptive implementation does not assume knowledge of these Lyapunov functions. 2
0
A
0 A
Uniformly Globally Stable (UGS), when D = Rq ; if (6) is satis…ed with, 2 K1 ; and for any z0 2 Rq : Uniformly Attractive (UA) if there exists a constant c > 0 such that for all jz0 jA < c and any > 0 there exists T = T ( ) > 0; such that jz0 jA c; t T ) jz(t; z0 )jA ; (7)
Uniformly Globally Attractive (UGA), when D = Rq , if for each pair of strictly positive numbers (c; ) there exists T = T ( ) > 0 such that for all z0 2 Rq , (7) holds. Uniformly Asymptotically Stable (UAS) if it is US and UA. Uniformly Globally Asymptotically Stable (UGAS), when D = Rq , if it is UGS and UGA. From (Khalil 1996), we adapt the de…nition of uniform boundedness of solutions to the case when A is not reduced to f0g.
De…nition 4 With respect to the closed set A Rq ; the solutions of system (4) are said to be: Uniformly Bounded (UB) if there exist a positive constant c; such that for every positive constant r < c there is a positive constant = (r); such that jz0 jA r ) jz(t; z0 )jA ; 8t 0 : (8)
Notation, de…nitions and preliminary results
Uniformly Globally Bounded (UGB), when D = Rq ; if for every r 2 R 0 , there is a positive constant = (r) such that (8) is satis…ed.
j j denotes the Euclidian norm and j jA0 : Rq ! R 0 denotes the distance from a point z 2 Rq to a set A0 Rq ; jzjA0 := inf fjz yj : y 2 A0 g : The solution of an autonomous dynamic system z_ = F (z) is denoted by z(t; z0 ) at t t0 ; where z0 = z(t0 ; z0 ) is the initial state and t0 is the initial time.
De…nition 5 A smooth Lyapunov function for (4) with respect to a non-empty, closed forward invariant set A D is a function V : D ! R that satis…es: i) there exists two K functions 1 and 2 such that for any z 2 D; V (z) 1 (jzjA ) 2 (jzjA ): ii) There exists a continuous and a positive de…nite or semide…nite function 3 such that for any z 2 DnA: @V 3 (jzjA ). @z (z)F (z)
2.1 De…nitions The de…nitions that follows are either motivated by, or can be found in (Teel, Panteley & Loria 2002) and (Lin, Sontag & Wang 1996). They pertain to systems of the form z_ = F (z) ; (4) where z := (p; xT )T ; p is the time-state p_ = 1; p0 = t0 : (5)
2.2 Preliminary results In order to conclude stability of a set of points, rather than a single equilibrium point, we have the following result from (Skjetne 2005).
2
UGS, when D = Rq2 ; if O2 is UGS with respect to the system (12) and that the solution of system (11)-(12) is UGB with respect to A. UAS, if O2 is UAS with respect to the system (12). US, if O2 is US with respect to the system (12).
Theorem 1 Assume that the system (4) is …nite escapetime detectable through jzjA : If there exists a smooth Lyapunov function for the system (4) with respect to a nonempty, closed, forward invariant set A, then A is UGS with respect to (4). Furthermore, if 3 is a positive definite function, then A is UGAS with respect to (4), and r if i (jzjA ) = ci jzjA for i = 1; 2; 3 where ci and r are strictly positive values and r > 1; then A is UGES with respect to (4).
3
The adaptive control allocation approach is based on three modular steps, see also Figure 1.
Consider the cascaded system x_ 1 = fc1 (t; x1 ) + gc (t; x1 ; x2 ) x_ 2 = fc2 (t; x2 ):
(9) (10)
(1) The high level control algorithm. The virtual control is treated as an available input to system (1), and a virtual control law c := k(t; x) is designed such that the origin of (1) is UGAS. (2) The control allocation algorithm. Based on the minimization problem (3) where J is a cost function that incorporates objectives such as minimum power consumption and actuator constraints (implemented as barrier functions), the Lagrangian function
When the functions fc1 , fc2 and gc are locally Lipschitz in all arguments, this class of nonlinear time-varying systems can be represented by the following autonomous system z_1 = F1 (z1 ) + G(z) z_2 = F2 (z2 )
Optimizing adaptive control allocation
(11) (12)
T T 2 Rq2 , where z1 := (p; xT 2 Rq1 , z2 := (p; xT 2) 1) T T q T T T z := (p; x ) 2 R ; x := (x1 ; x2 ) ; F1 (z1 ) := (1; fc1 (p; x1 )T )T , G(z) := (0; gc (p; x)T )T and F2 (z2 ) := (1; fc2 (p; x2 )T )T . Based on the cascaded system formulation we will make use of the results from (Tjønnås et al. 2006).
(t; x; u; ))T (17) is introduced. Update laws for the control input u and the Lagrangian parameter are then de…ned such that u and converges to a set de…ned by the time-varying optimality condition. (3) The adaptive algorithm. In order to cope with an unknown parameter vector in the e¤ector model, an adaptive law is de…ned. The parameter estimate is used in the control allocation algorithm and a certainty equivalent adaptive optimal control allocation can be de…ned. L(t; x; u; ) = J(t; x; u) + (k(t; x)
Lemma 1 Let O1 and O2 be some closed subsets of Rq1 and Rq2 respectively. Then, under the following assumptions, the set A := O1 O2 is UGAS with respect to the cascade (11)-(12). A 1 The set O2 is UGAS with respect to the system (12) and that the solution of the system (11)-(12) is UGB with respect to A: A 2 The functions F1 , F2 and G are locally Lipschitz. A 3 The cascade (11)-(12) is forward complete. A 4 There exist a continuous function G1 : R 0 ! R and a class K function G2 such that, for all z 2 Rq , jG(z)j
G1 (jzjA )G2 (jz2 jO2 ):
0
(13)
A 5 There exists a continuously di¤ erentiable function V 1 : Rq1 ! R 0 , class K1 functions 1 , 2 and 3 , and a continuous function & 1 : R 0 ! R 0 such that, for all z1 2 Rq1 , 1 (jz1 jO1 )
@V 1 (z1 )F1 (z1 ) @z1 @V 1 (z1 ) @x1
V 1 (z1 )
2 (jz1 jO1 )
3 (jz1 jO1 )
& jz1 jO1 :
(14)
(15) (16)
Corollary 1 Let O1 and O2 D be closed subsets of Rq1 and Rq2 respectively, and the assumptions A 2 - A5 be satis…ed. Then, with respect to the cascade (11)-(12), the set A is:
Fig. 1. Adaptive control allocation design philosophy
3
3.1
High level control algorithm
the set
From Assumption 2 a) there exists a Lyapunov function Vx : R 0 Rn ! R 0 and K1 functions x1 ; x2 ; x3 and x4 such that
Assumption 1 (Plant) a) There exists a continuous function Gf : R 0 ! R>0 and a constant K; such that jfx (t; x)j Gf (jxj) and jgx (t; x)j K for all t and x. Moreover fx is locally Lipschitz in t and x, and fx (t; 0) = 0: b) There exist a continuous function & @gx : R 0 ! R 0 such that for all t and x; gx is di¤ erentiable and
x1 (jxj)
+
T
T
%2 I: (18)
T @ @ @u uc @u uc
1
y;
of the equation, y = u1 ; which exist for any uc : Thus for every y there exists a solution (t; x; u1 ; ) = (t; x; 0; ) + y by the Mean Value Theorem, where uc 2 (0; u1 ): ii) follows from the Implicit Function Theorem by i) and Assumption 1 d).
:R
0
x3 (jxj)
!R
0;
such (25)
Certainty equivalent adaptive control allocation algorithm
In order to account for the unknown, but bounded, parameter vector ; an adaptive mechanism is include in the optimal control allocation design. We use a Lyapunov based indirect parameter estimation scheme (see for example (Krstic, Kanellakopoulos & Kokotovic 1995) for a systematic Lyapunov based procedure). Since the state vector of (1) is assumed to be known, we consider the estimation model: x ^_ = A(x x ^) + fx (t; x) + gx (t; x) (t; x; u; ^) (26)
Assumption 2 (High Level Control Algorithm) a) There exists a feedback control c := k(t; x) & k (jxj); where & k : R 0 7! R 0 is a continuous function, that render the equilibrium of (1) UGAS for = c . The @k function k is di¤ erentiable, @k & @k (jxj); @t + @x where & @k : R 0 7! R 0 is continuous, and k(t; 0) = 0 for all t.
in the construction of the adaptive law. This estimation model has the same structure as a typical series parallel model (SP) (Ioannou & Sun 1996), (Landau 1979). For analytical purpose, the …ltered error estimate of the unknown parameter vector:
If we rewrite (1) such that (t; x; u; ))
k
k
In this section we …rst establish update-laws for the control input u, the Lagrangian multipliers , and the parameter estimate ^; such that stability and convergence with respect to the time-varying …rst order optimal set de…ned by the optimization problem (3) can be concluded. Then we show that the stabilizing properties of the virtual controller from Assumption 2, are conserved for the closed loop system.
@ @u uc
x_ = f (t; x) + g(t; x)(k(t; x)
(24)
Remark 2 If the origin of x_ = f (t; x) is UGES, then Assumption 2 is generally satis…ed, with for example x3 quadratic, x4 linear and k sublinear, and Vx does not need to be known explicitly in order to verify Assumption 2 b) . 3.2
T @ @u uc
(23)
x4 (jxj)
x4
Remark 1 Assumption 1 d) can be viewed as a controllability assumption in the sense that: i) the mapping (t; x; ; ) : Rr ! Rd is surjective for all t; x and and ii) for all t; x and there exist a continuous function fu (t; x; ) such that (t; x; fu (t; x; ); ) = k(t; x): The surjective property can be seen by the Moore-Penrose pseudoinverse solution, u1 =
(22)
x3 (jxj)
Assumption 2 (Continued) b) There exists a K1 function that (jxj) (jxj)
xT ; uT j j + j 0j G for all t; x; and u. Further there exist a continuous function & @ (jxj) @ @ @ @ 0 @ 0 @ 0 @t + @t + @x + @x + @u + @u where & @ : R 0 ! R 0: d) There exists constants %2 > %1 > 0, such that 8t, x, u and @ (t; x; u; ) @u
x2 (jxj)
for the system x_ = f (t; x) with respect to its origin.
& @gx (jxj). c) The function is twice di¤ erentiable and there exist a continuous function G : R 0 ! R 0 ; such that
@ (t; x; u; ) @u
Vx (t; x)
@Vx @Vx + f (t; x) @t @x @Vx @x
@gx @x
%1 I
k1 > 0, such that 8 t; x; and (uT ; T )T 2 = Ou0 @2L k1 I < (t; x; u; ; ) k2 I; (32) @u2 where Ou0 is the interior of Ou : If (uT ; T )T 2 Ou0 2 then the lower bound is replaced by @@uL2 0.
(28)
:= r + d + n + m; zxu
@ L @t@
Ou (t; x):=
Based on the perturbing system, we consider the estimated optimal solution set: n o Ou ~ (t; x):= (u; ; ; ~)2Rnu ~ fOu ~ (t; zxu ~)=0 (29) ~
1
2
In order to see the cascaded coupling between the system (19) and the adaptive and optimal control allocation update-laws, equation (19) can be rewritten by: x_ = f (t; x) + g(t; x) k(t; x) (t; x; u; ^)
where nu
@ 2 L^ @ @u
@L^ : (31) @
if det( @@uL2 ) > or uf f ^ := 0 if det( @@uL2 ) < ; where r (k1 ) > > 0: k1 is de…ned in the following assumption which both guarantee existence of the proposed update laws and the time-varying …rst order optimal set,
Stability of the optimal certainty equivalent control allocation
(t; x; u)~:
^
ff
is that, the adaptive law is independent of Vx . Remark 3 A di¤ erent approach would be to expand the perturbed system ( 1 ) with an adaptive law. In this case the adaptive law will be dependent on the initial Lyapunov function (similar to (Tjønnås & Johansen 2005)), but convergence results like, x(t) ! 0 as t ! 1; may be concluded even if a persistence of excitation condition is not satis…ed.
+gx (t; x)
@ 2 L^ @u2 @ 2 L^ @u@
T
@ 2 L^ @x@
A ; the matrices ; and 0 ^ ; are symmetric positive de…nite and uf f ^ is a feedforward like term given by: 0 0 2 11 ! ! @ L^ @2L @2L _ + @x@u f (t; x)+@ @ 2^@u A^A u ^ : =H 1 @ @t@u 2 2 where H^ : = @
3.2.1
T
(t; x; u)) (gx (t; x))
Remark 4 The second order su¢ cient conditions in Theorem 12.6 in (Nocedal & Wright 1999) are satis…ed for all t; x; u; and by Assumption 1 and 3, thus the set Ou (t; x) describes global optimal solutions of the problem (3).
x; u; ; ; ~ ,
T
and the esti(t; x; u; ^))T :
Lemma 2 If Assumptions 1, 2 a) and 3 are satis…ed, then Ou is non-empty for all t; x and : Further for all t; x; and (u; ) 2 Ou ; there exist a continuous function & Ou : R 0 ! R 0 such that:
We will in what follows prove that the time and statevarying optimal set Ou ~ (t; x), in a certain sense is
(uT ;
5
T T
)
& Ou
xT ;
T
Ozdaglar 2003) we have that G : Rq ! RqG is continuous i¤ G 1 (U ) is closed in Rq for every closed U in RqG . From the de…nition of Ou ~ , U = f0g ; and since G is continuous (by Assumption 1 - 3), Ou ~ is a closed set. The set is forward invariant if at t1 ; G(t1 ; x(t1 ); u(t1 ); (t1 ); ~(t1 )) = 0 and d(G(t;x;u; ;~)) = 0 8t t1 with respect to (27), dt (30), (31) and (28). Since there exist a continuous solution of (30) as long as x exist, we only need to check this condition on the boundary of Ou ~ (t; x(t)) (Note that det(H^ ) 6=0 on the boundary of Ou ~ (t; x(t)) by Assumption 1d) and 3b)). We _ get G(t1 ; x(t1 ); u(t1 ); (t1 ); ~(t1 )) = 0 ) (_ ; ^) = 0 d@L d@L = 0: and it remains to prove that dt@ ^ ; dt@u^ d@L^ @ 2 L^ @ 2 L^ @ 2 L^ @ 2 L^ ^ _ = @t@ + @x@ x_ + @u@ u_ + @ ^@ and dt@ @ 2 L^ @ 2 L^ @ 2 L^ @ 2 L^ _ @ 2 L^ ^ d@L^ _ _ + @u@u u_ + @ @u + @ ^@u ; thus dt@u = @t@u + @x@u x ! ! ! @L^ d@L^ u_ dt@u @u +H^ uf f ^ = H^ H^ @L^ = 0: =H^ d@L^ _
PROOF. See Appendix A.
The idea of proving stability and convergence of the set Ou ~ (t; x); relies on the construction of the Lyapunovlike function: ! T T @L @L @L @L 1 ^ ^ ^ ^ Vu ~ (t; x; u; ; ; ~) : = + 2 @u @u @ @ ~T
T
+
2
+
^
2
~ :
(33)
Its time derivative along the trajectories of the considered system is: ! T 2 T 2 @L @ L @ L @L ^ ^ ^ ^ + u_ V_ u ^ = @u @u2 @ @u@ ! @L^ T @ 2 L^ _ @L^ T @ 2 L^ @L^ T @ 2 L^ + + + x_ @u @ @u @u @x@u @ @x@ ! @L^ T @ 2 L^ ~_ @L^ T @ 2 L^ + @u @ @ ^@u @ ^@ +
T
@L^ @u
@ 2 L^ + @t@u
@L^ @
T
@ 2 L^ + @t@
T
T
_+~
^
dt@
@
ii) Since x(t) is assumed to be forward complete, it follows from Lemma 2 that there always exists a T ~ that bounded by x and ^ = pair u T ; T
~_ :
satisfy the time-varying …rst order optimal condiFurther by (27), (30) and (31), we get ! @L^ T @L^ T @L^ T @L^ _ Vu ^ = ; H^ H^ ; @u @ @u @ T
c^
A @L^ @u
T
@L^ @L^ + @u @
T
@L^ @
!
T
tions
! T T
A
(34)
and it follows that there exists constants such that
Proposition 1 Let Assumptions 1, 2 a) and 3 be satis…ed. Then if x(t) exists for all t, the set Ou ~ (t; x(t)) is UGS with respect to the system (27), (30) and (31), and @L @L ( ; @u^ ; @ ^ ) converges asymptotically to zero.
~
; ~;
@L^ T @L^ T @u ; @
@L^ @L^ @u + @
2
u
2 1 2
zu
~ O u
; where ~
1
:=
1 2
min(
1;
;
^)
2
:=
max( 2 ; ; ^ ); such that Vu ~ (t; zu ~ ) is a radially unbounded Lyapunov function.
iii) Vu ~ (t; zu ~ ) := Vu ~ (t; x(t); u; ; ; ~); is a radially unbounded Lyapunov function. :=
jzu jOu
(35) 0
1 > T @L^ @
jzu jOu since HT^ H^ is positive de…nite and bounded from Assumption 1 and 3. By a similar @2L argument, in (A.4) change P c with @u2^ , H^ is non-singular, hence the control allocation law (30) always exist and, 1 zu ~ O ~ Vu ~ (t; zu ~ )
u
i) De…ne G(t; x; u; ; ~)
T
@L^ @u
2
1
2
2
PROOF. In order to prove this result we show that i) Ou ~ (t; x(t)) is a closed forward invariant set with respect to system (27), (30) and (31), ii) the system (27), (30) and (31) is …nite escape time detectable through T T z ~ , where z ~ := uT ; T ; T ; ~ , and that Ou
= 0, and thus the system (27),
(30) and (31) is …nite escape time detectable through zu ~ O ~ : u iii) In the proof of Lemma, change L with L^ such that (A.3) become !T ! u u u u @L^ T@L^ @L^ T@L^ + = HT^ H^ @u @u @ @
where c^ := inf t min H^ H^ > 0: We formalize the result, based on set-stability.
u
@L^ T @L^ T @u ; @
Since (34) is negative semide…nite, UGS of the set Ou ~ can be concluded by Theorem 1. The convergence result follows by Barbalat’s lemma
T
:
Corollary 2 Let Assumptions 1, 2 a) and 3 be satis…ed and zu ~0 O ~ < r where r > 0: If x(t) exists
From Proposition 1.1.9 b) in (Bertsekas, Nedic &
u
6
for t 2 [t0 ; T ) where T t0 , then there exists a positive constant B(r) > 0 such that for all t 2 [t0 ; T ); zu ~ (t) O ~ (t;x(t)) B(r).
3.2.2
u
The optimal set for the combined control and certainty equivalent control allocation problem is de…ne by:
PROOF. Since V_ u ~ 0 and Vu ~ (t; zu ~ (t)) Vu ~ (t0 ; zu ~ (t0 )) for all t 2 [t0 ; T ): From iii) in the proof of Proposition 1 1 zu ~ O ~ Vu ~ (t; zu ~ )
Oxu ~ (t) := Ox (t)
2
that q
1
~ O u
~
zu ~ (t)
for all t
Ou
~ (t;x(t))
Vu ~ (t0 ; zu ~ (t0 )) 1
2
q
[t0 ; T ); thus it follows q 1 Vu ~ (t; zu ~ (t)) 1 zu
2 1
~0 O u
~
Ou ~ (t; 0):
(36)
In the framework of cascaded systems, we consider (30), (31) together with (27) to be the perturbing system ( 2 ); while (28) represents the perturbed system ( 1 ): Loosely explained, we will use Lemma 1 to conclude UGAS of the set Oxu ~ if, Ox and Ou ~ individually are UGAS (which we already have established) and the combined system is UGB with respect to Oxu ~ .
u
zu
Stability of the combined control and certainty equivalent control allocation
=: B(r)
Remark 5 Provided that the gain matrix, > 0; is bounded away from zero, may be chosen time-varying. 1 If for example = H^ H^ for some > 0; then ! ! @L^ u_ 1 @u = H^ H ^ 1 uf f @L^ _
Before establishing the main results of this section, we need to state an assumption on the interconnection term between the two systems. We start by stating the following property: Lemma 3 By Assumption 1 and Lemma 2 there exists continuous functions & x ; & xu , & u : R 0 ! R 0 ; such that
@
where the …rst term is the Newton direction when L^ is considered the cost function to be minimized. In case H^ H^ is poorly conditioned one may choose 1 = H^ H^ + I for some > 0; to avoid c^ in (34) from being small.
j
(t; x; u)j
& x (jxj)& xu (jxj) + & x (jxj)& u ( zu
~ O u
~
):
PROOF. This can bee seen by applying (Mazenc & Praly 1996)’s lemma B.1 Assumption 2 (Continued) c) The function, k , from Assumption 2 b) has the following additional property:
Assumption 4 (Persistence of Excitation) The signal matrix g (t) := gx (t; x(t)) (t; x(t); u(t)) is Persistently Excited (PE), which means that there exist constants T and > 0; such that R t+T T I ; 8t > t0 : g( ) g ( )d t
1 k
(jxj)
x3 (jxj)
x4 (jxj)& x max (jxj)
,
(37)
where & x max (jxj) := max(1; & x (jxj); & x (jxj)& xu (jxj)): Remark 7 If there exists a constant K > 0 such that & x max (jxj) K 8x; which is common in a mechanical system, Assumption 2 c) is satis…ed.
Remark 6 The system trajectories x(t) de…ned by (1) typically represents the tracking error i.e. x := xs xd ; where xs is states vector of the system, and xd represents the desired reference for these states. This means that PE assumption on g (t) is dependent on the reference trajectory xd ; in addition to disturbances and noise, and imply that some "richness" properties of these signals are satis…ed.
Next we consider the closed loop of the plant, the virtual controller and the adaptive dynamic control allocation law: Proposition 3 If Assumptions 1-3 are satis…ed, then the set Oxu ~ is UGS, with respect to system (28),(30), (31) and (27). If in addition Assumption 4 is satis…ed, then Oxu ~ is UGAS with respect to (28),(30), (31) and (27).
Proposition 2 Let x(t) be UGB, then if Assumption 4 and the assumptions of Proposition 1 are satis…ed, the set Ou ~ is UGAS with respect to system (26), (27), (30) and (31).
PROOF. The main part of this proof is to prove boundedness, completeness and invoke Lemma 1. Let zxu ~0 O ~ r; where r > 0, and assume xu that jzx (t)jOx escapes to in…nity at T: Then for any constant M (r) there exists a t 2 [t0 ; T ) such that M (r) jzx (t)jOx . In what follows we show that M (r) can not be chosen arbitrarily. De…ne v(t; zx ) := Vx (t; x) such that
PROOF. See Appendix B
Unless the PE condition is satis…ed for g ; only stability of the optimal set is guaranteed. Thus in the sense of achieving asymptotic optimality, parameter convergence is of importance.
7
v_ +
@Vx g(t; x) k(t; x) @x (t; x; u)~
x3 (jzx jOx )
@Vx gx (t; x) @x x3 (jzx jOx )
+
+
Proposition 3 implies that the time-varying optimal set Oxu ~ (t) is uniformly stable, and in addition uniformly attractive if Assumption 4 is satis…ed. Thus optimal control allocation is achieved asymptotically for the closed loop. A local version of this result can be proven using Corollary 1.
(t; x; u; ^)
+
@L^ @Vx jg(t; x)j (t; x; u; ^) @x @
+
@Vx jgx (t; x)j @x x3 (jzx jOx )
(t; x; u)~
x4 (jzx jOx )K( 2 +j
(t; x; u)j) zu
~ O u
~
:
Corollary 3 If for u 2 U Rr there exist constant cx > 0 such that for jxj cx the domain Uz R 0 Rn U Rd+n+m contain Oxu ~ ; then if the Assumptions 1-3 are satis…ed, the set Oxu ~ is US with respect to the system (28), (30), (31) and (27). If in addition Assumption 4 is satis…ed, Oxu ~ is UAS with respect to the system (28), (30), (31) and (27). PROOF. Since Oxu ~ Uz , there exist a positive constants r zxu ~0 O ~ such that jzx (t)jOx cx0 where xu 0 < cx0 < cx ; hence the domain Uu ~ R 0 U d+n+m R contain Ou ~ : US/UAS of Ou ~ follows from the Lyapunov-like function Vu ~ and the PE Assumption. UB follows from zxu ~0 O r, and the US/UAS xu property of Oxu ~ follows from Corollary 1
(38)
From Corollary 2, there exists a positive constant B(r) 0; for all t 2 [t0 ; T ); such that for zu ~0 O ~ r, u
zu ~ (t) O ~ B( zu ~0 O ~ ) B(r). From Assumpu u tion 1 and 2, v_ x3 (jzx jOx ) + x4 (jzx jOx )K ( 2 + j (t; x; u)j) B(r) k (jzx jOx ) x4 (jzx jOx )& x max ((jzx jOx ) + x4 (jzx jOx )K ( 2 + j (t; x; u)j) B(r) x4 (jzx jOx ) k (jzx jOx )& x max (jzx jOx ) + x4 (jzx jOx )K 2 B(r) + x4 (jzx jOx )K& x (jzx jOx )& xu (jzx jOx )B(r) + x4 (jzx jOx )K& x (jzx jOx )& u (B(r))B(r) (39)
Corollary 4 If is known and the Assumptions 1-3 are satis…ed then the set Oxu := Ox (t) Ou ~ (t; 0): is UGAS with respect to the system (19) and (30). PROOF. This result follows by working through the above arguments, neglecting the adaptive dynamic and considering a Lyapunov like function of the form
1 x4 (jzx jOx ) k ( x2
(v))& x max (jzx jOx ) + x4 (jzx jOx )K 2 B(r) + x4 (jzx jOx )K& x (jzx jOx )& xu (jzx jOx )B(r) + x4 (jzx jOx )K& x (jzx jOx )& u (B(r))B(r):
Thus, if jzx0 jOx >
1 k
(K (
2
1 k
and from (40), v(t; zx (t)) such that jzx (t)jOx By choosing M (r) := max 1 x1
x2
1 k
(K ( 1
x2
k
1 x1 1 x1
(
(K (
2
(K ( 2+1+& u (B(r))) B(r))
x2 (r))
T @L^ @
Since the set with respect to system 2 may only be US, due to actuator/e¤ector constraints and parameter uncertainty, only US of the cascade, may be concluded. But if the PE property is satis…ed on g , a UAS result may be achieved with both optimal and adaptive convergence. 4 Example In this section simulation results of an over-actuated scaled-model ship, manoeuvred at low-speed, is presented. A 3DOF horizontal plane model described by: _ e = R( ) _ = M 1 D + M 1 ( + b) (41) = (u) ;
:
;
(K ( 2+1+& u (B(r))) B(r))
@L^ @
(v(zx0 ))
+ 1 + & u (B(r))) B(r)))
1
T @L^ @u +
Corollary 5 If for u 2 U Rr there exist constant cx > 0 such that for jxj cx the domain Uz R 0 n d R U R contain Oxu ; then if the Assumptions 13 are satis…ed, the set Oxu is UAS with respect to the system (19) and (30).
+ 1 + & u (B(r))) B(r))
2
x2 ( k
1 x1
@L^ @u
(40)
+ 1 + & u (B(r))) B(r))
then from (39), v(t0 ; zx0 ) v(t; zx (t)) and jzx (t)jOx 1 ( (r)) ; else, x2 x1 jzx0 jOx
Vu (t; x; u; ) := 12
;
the assumption of jzx (t)jOx escaping to in…nity is contradicted, since M (r) > jzx (t)jOx and j jOx is …nite escape time detectable. Further more Ox is UGB. From Propositions 1 and 2 and the assumptions of these propositions, the assumptions of Lemma 1 and Corollary 1 are satis…ed and the result is proved
T
is considered, where e := (xe ; ye ; e ) := (xp xd ; yp T yd ; p d ) is the north and east positions and compass heading deviation. Subscript p and d denotes the
8
actual and desired states. := (u; ; r)T is the body…xed velocities, surge, sway and yaw, is the generalized force vector, b := (b1 ; b2 ; b3 )T is a disturbance due to wind and current and R( ) is the rotation matrix function between the body …xed and the earth …xed coordinate frame. The example we present here is based on (Lindegaard & Fossen 2003), and is also studied in (Johansen 2004) and (Tjønnås & Johansen 2005). In the considered model there are …ve force producing devices; the two main propellers aft of the hull, in conjunction with two rudders, and one tunnel thruster going through the hull of the vessel. ! i denotes the propeller angular velocity and i denotes the rudder de‡ection. i = 1; 2 denotes the aft actuators, while i = 3 denotes the tunnel thruster. (41) can be rewritten in the form of (1) and (2) by: x := ( ;
e;
T
) ;
:= ( 1 ;
T 2)
The gain matrices are chosen as follows: Kp := M diag(3:13; 3:13; 12:5)10 2 ; Kd := M diag(3:75; 3:75; 7:5)10 1 ; KI := M diag(0:2; 0:2; 4)10 3 ; A := I9 9 ; Q := diag(1; 1); T Q := diag(a; 150 (100; 100; 1)T ); a := (1; 1; 1; 1; 1; 1)
:= ( 1 ; 2 ; 3 ) ; u := (! 1 ; ! 2 ; ! 3 ; 0 1 X1 + X2 0 B C C (u) := B @ Y1 + Y2 T3 A ; l3;x T3 13
T
1; 2)
where
W := diag (1; 1; 1; 1; 1; 0:6; 0:6; 0:6) and " := 10 9 . The weighting matrix W is chosen such that the deviation @L of @ ^ = k(t; x) (t; x; u; ^) from zero, is penalized more then the deviation of
@L^ @u
^_ =
;
8
0.
Remark 8 In order to satisfy Assumption 1 we replace in (41) by = (u; ) = (u) + 0 (u); where T 0 (u) := & (! 1 ; ! 2 ; ! 3 ) and & > 0; in the control allocation algorithm. Since & may be chosen arbitrarily small, this actuator/e¤ ector mapping is practically similar to (41), but ensures that the allocation algorithm is well conditioned. Assumption 3 is satis…ed locally since for 2 bounded u and ; ki2 ensures that @@uL2 > 0: It is also worth noticing that the optimal control allocation solution in this example, at any time, is a single point. Based on the wind and current disturbance vector b := 0:05(1; 1; 1)T and the parameter vector := (0:8; 0:9)T ; the simulation results are presented in the Figures 3-7. Due to the disturbance and reference change, the transients excite the parameter update-law at t 0 as well as at t 200 and t 400, and the estimated parameters converges to the true values. The control objective is satis…ed and the commanded virtual controls are tracked closely, by the forces generated by the adaptive control allocation law, except for t 0; t 200 and t 400; where the control allocation is suboptimal due to initial conditions and actuator saturation: see Figure 7. The simulations are carried out in the MATLAB environment with a sampling rate of 10Hz:
Kd ;
proposed in (Lindegaard & Fossen 2003), stabilizes the system (41) augmented with: _ = := b e , where and _ = e ; uniformly and exponentially, for some physically limited yaw rate. The cost function used is: J(u) :=
(u) gx
n S := ^ 2 R2
The unknown parameter vector represents thrust losses. 2 is also related to the parameters kT p3 and kT n3 in a multiplicative way. This suggest that the estimate of 2 gives a direct estimate of the tunnel thruster loss factor. The virtual controller :=
T
1
2 (S) and
T
c
from zero, in the
search direction. In order to keep ^ from being zero, due to a physical consideration, a projection algorithm can be used. i.e. from (30)
;
T
1
:= HT ^ W H^ + "I
and
qi 2i ;
i=1
q1 = q2 = 500;
9
5
1
0
1 [deg]
0.5
η
0 -0.5
0
100
200
300
400
500
δ
1 [m]
1.5
600
-5 -10 -15 -20
1
0
100
200
0
100
200
300
400
500
600
300
400
500
600
30 0
100
200
300
400
500
600
20
2 [deg]
-0.5
0.3
0.1
0 -10
0 -0.1
10
δ
0.2
η
3 [deg]
0
0.5
η
2 [m]
1.5
0
100
200
300
400
500
-20
600
t [s]
t [s]
Fig. 3. Simulation results - the solid lines represent positions while the dashed lines represent references
Fig. 5. Simulation results - computed rudder de‡ection by the allocation algorithm 1.1
ω
1 [Hz]
1.05 0
1
-5
0.95 0.9
-10
0.85
-15
0.8 -20
0
100
200
300
400
500
600
0.75
0
100
200
100
200
300
400
500
600
300
400
500
600
ω
2 [Hz]
20 10
0.9
0
0.8
ω
3 [Hz]
-10
0
100
200
300
400
500
0.7
600
20
0.6
10
0.5
0
0.4 0
-10 -20
t [s] 0
100
200
300
400
500
600
t [s]
Fig. 6. Simulation results - the parameter adaption
Fig. 4. Simulation results - computed propeller angular velocities by the allocation algorithm
6
5
The authors are grateful for insightful comments from Antoine Chaillet, Elena Panteley and Alexey Pavlov. The work is sponsored by the Research Council of Norway through the Strategic University Programme on Computational Methods in Nonlinear Motion Control.
Conclusion
Based on a control-Lyapunov design approach, an adaptive optimizing nonlinear control allocation algorithm is derived. Under certain assumptions on the system (actuator/e¤ector model) and the control design (growth rate conditions on the Lyapunov function), a cascade result is used to prove, closed-loop, stability and attractivity of a set representing the optimal actuator con…guration. Typical applications for the control allocation algorithm are over-actuated mechanical systems, especially systems that exhibits fast dynamics since the algorithm is computational e¢ cient. An automotive example is presented in (Tjønnås & Johansen 2006).
Acknowledgements
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10
tracking of time-varying control trajectories’, In Proceedings of the 2005 American Control Conference, Portland, OR .
0.1 0
Mazenc, F. & Praly, L. (1996), ‘Adding integrations, saturated controls, and stabilization for feedforward systems’, IEEE Transactions on Automatic Control 41, 1559–1578.
τ
1 [N]
0.2
-0.1 -0.2
0
100
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Nocedal, J. & Wright, S. J. (1999), Numerical Optimization, Springer.
0.2 0
Pantely, E., Loria, A. & Teel, A. R. (2001), ‘Relaxed persistency of exitation for uniform asymptotic stability’, IEEE Trans. Automat. Contr. 46, 1874 – 1886.
τ
2 [N]
0.4
-0.2 -0.4
0
100
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600
Poonamallee, V., Yurkovich, S., Serrani, A., Doman, D. & Oppenheimer, M. (2005), ‘Dynamic control allocation with asymptotic tracking of time-varying control trajectories’, In Proceedings of the 2004 American Control Conference, Boston, MA .
τ
3 [Nm]
0 -0.02 -0.04 -0.06 -0.08
0
100
200
300
400
500
600
Skjetne, R. (2005), The Maneuvering Problem, Phd thesis, NTNU, Trondheim, Norway.
t [s]
Fig. 7. Simulation results - the solid lines represent actual forces generated by the actuators, the dashed lines represent the virtual forces generated by the control law.
Sørdalen, O. J. (1997), ‘Optimal thrust allocation for marine vessels’, Control Engineering Practice 5, 1223–1231. Teel, A., Panteley, E. & Loria, A. (2002), ‘Integral characterization of uniform asymptotic and exponential stability with applications’, Maths. Control Signals and Systems 15, 177–201.
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Tjønnås, J., Chaillet, A., Panteley, E. & Johansen, T. A. (2006), ‘Cascade lemma for set-stabile systems’, 45th IEEE Conference on Decision and Control, San Diego, CA .
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Tjønnås, J. & Johansen, T. A. (2006), ‘Adaptive optimizing dynamic control allocation algorithm for yaw stabilization of an automotive vehicle using brakes’, 14th Mediterranean Conference on Control and Automation, Ancona, Italy .
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A
Proof of Lemma 2
The boundary of the set Ou and the time-varying …rst order optimal solution is described by:
Johansen, T. A., Fossen, T. I. & Tøndel, P. (2005), ‘E¢ cient optimal constrained control allocation via multi-parametric programming’, AIAA J. Guidance, Control and Dynamics 28, 506–515.
@J @ @L (t; x; u; ; ) = (t; x; u) (t; x; u; ) @u @u @u @L (t; x; u; ; ) = k(t; x) (t; x; u; ) @
Khalil, H. K. (1996), Nonlinear Systems, Prentice-Hall, Inc, New Jersey. Krstic, M., Kanellakopoulos, I. & Kokotovic, P. (1995), Nonlinear and Adaptive Control Design, John Wiley and Sons, Inc, New York.
T
By using theorem 2.4.7 in (Abrahamson, Marsden & Ratiu 1988), it can be shown that
Landau, H. (1979), Adaptive Control: The Model Reference Approach, Marcel Dekker, Inc., New York. Lin, Y., Sontag, E. D. & Wang, Y. (1996), ‘A smooth converse lyapunov theorem for robust stability’, SIAM Journal on Control and Optimization 34, 124–160.
@L (t; x; u; ; ) @u @L (t; x; uc ; ; ) @u @L (t; x; uc ; ; ) @u @L (t; x; u; ; ) @ @L (t; x; u ; ; ) @
Lindegaard, K. P. & Fossen, T. I. (2003), ‘Fuel-e¢ cient rudder and propeller control allocation for marine craft: Experiments with a model ship’, IEEE Trans. Control Systems Technology 11, 850–862. Luo, Y., Serrani, A., Yurkovich, S., Doman, D. & Oppenheimer, M. (2004), ‘Model predictive dynamic control allocation with actuator dynamics’, In Proceedings of the 2004 American Control Conference, Boston, MA . Luo, Y., Serrani, A., Yurkovich, S., Doman, D. & Oppenheimer, M. (2005), ‘Dynamic control allocation with asymptotic
11
@L (t; x; uc ; @u @L (t; x; uc ; @u @L (t; x; u ; @u @L (t; x; u ; @ @L (t; x; u ; @
@2L (u uc ) @u2 c @2L ; )= ( ) @ @u c @2L ; ) = 2 (uc u ) @u @2L ; )= (u u ) @u@ ; )=
; ) = 0:
R1
fOu ~ (t) := fOu ~ (t; x(t); u(t); (t); (t); ~(t));
@2L (t; x; (1 s)uc + su; ; )ds; 0 @u2 c R 2 2 2 1 @ L L := := 0 @@uL2 (t; x; (1 s)u + suc ; ; )ds; @@ @u @u2 c R 1 @2L T (t; x; uc ; (1 s) + s ; )ds = @@u (t; x; uc ; ) 0 @ @u R 1 @2L 2 @ L := 0 @u@ and @u@ (t; x; (1 s)u + su; ; )ds = @ @u (t; x; uc ; ): Since @L ; ) = 0 and @L ; ) = 0; we get @ (t; x; u ; @u (t; x; u ;
where
@2L @u2
:=
@L @ L (t; x; u; ; ) = @u @u2
@2L @u2 = (1
(u
uc ) +
c T
@ L @u2
@ (t; x; uc ; )( @u @ (t; x; uc ; )(u @u
and from uc := #(u 0 < #i < 1 :
_ g (t) = @gx (t; x) (t; x; u) + gx (t; x) @ (t; x; u) @t @t @ (t; x; u) @gx (t; x) (t; x; u)+gx (t; x) x_ + @x @x @ (t; x; u) +gx (t; x) u; _ @u
2
2
@L (t; x; u; ; ) = @
is bounded, by using the PE property; and ii) use the integral bound to prove UGA by contradiction. The proof is based on ideas from (Pantely, Loria & Teel 2001) and (Teel et al. 2002). i) First we establish a bound on _ g (t). We have
(uc
u )
)
(A.1)
u );
(A.2) thus from Assumption 1, system (28) and update& 1x _ g (jxj) + law (30), there exists a bound _ g (t)
u ) + u where # := diag(#i ) and
@2L (u uc ) + (uc u ) @u2 c @2L @2L #) (u u ) + # (u @u2 c @u2
From Assumption 3,
@2L @u2
c
and
@2L @u2
& 1x _ g (jxj)& u _ g ( zu ~ O ~ ) where & 1x _ g ; & 1x _ g ,& 1x _ g : u R 0 ! R 0 are continuous functions. This can be seen by following the same approach as in Lemma 3. From the assumption that x is uniformly bounded, T we use jxj Bx : The integrability of ~ ~ is investigated by considering the auxiliary function: V aux1 := T ~ V aux1 where g (t) ; bounded by BV aux1 (Bx ; r) r zu ~0 O ~ and r > 0: Its derivative along the u solutions of (27) and (31) is given by T V_ aux1 = ~ g (t)T g (t)~ + T AT g (t) _ g (t) ~
u ):
are positive de…2
2
nite matrices, such that P c := (1 #) @@uL2 + # @@uL2 c is also a positive de…nite matrix. Further !T ! T T u u u u @L @L @L @L + = HT H @u @u @ @
where H =
P
@ T @u (t; x; uc ;
c
@ @u (t; x; uc ;
) 0 sumption 3 and 1 it can be seen that: jdet(H )j= det(P c )det
)
!
+ +
(A.3) : By As-
where
1 c
@ (t; x;uc ; ) @u
T
!
> 0;
T
(A.4)
is non-singular and there exists constants > 0 such that 1 I HT H 2 I: Furthermore choose u = 0 and = 0; such that @L @L (t; x; 0; 0; ) & (jxj) and @J @u @ (t; x; 0; 0; ) G (jxj) (1 + j j) + & k (jxj) from Assumption 1, 2 and 3. 1
u
T
;
T
T
& Ou
xT ;
T
2 @x L^ @u (t):=
+
AT T
and H
Thus from (A.3)
g (t) ^
T g (t)
(B.1)
2 T @x L^
1
g (t)
@ 2 L^ @x@u @L^ @u
:=
1
T
; T
;
@u
@ 2 L^ @x@
@L^ @
T
4T
@ L^ (t); @u
(B.2)
T T
(t; x(t); u(t); (t);^(t)): 1 T nb b
2 bT a , we
_ g (t) ~
g (t)
AT
(t)
(t; x(t); u(t); (t); ^(t))
From Young’s inequality, naT a + have
T 2
g (t) ^
T
@ L^ @u (t)
and @ (t; x;uc ; )P @u
T
g (t)
_ g (t)
AT
g (t)
_ g (t)
~T ~;
T
(B.3)
where a = ~; b = T AT g (t) _ g (t) and n = 2T such that and T are positive scalars of desirable choice. Next
T
2
where & Ou (jsj) := 1 1 (G (jsj) (1 + jsj) + & k (jsj)) + 1 2 1 & @J (jsj) B Proof of Proposition 2
T
The proof is divided into two parts: i) show that the integral of fOu ~ (t)T fOu ~ (t); where
T
12
@x2 L^ @ L^ (t) (t) @u @u T @ L^ @ L^ T (t) KL (t) g (t) g (t) + @u @u
g (t) ^
1
T g (t)
T
for n = 1, a = where KL Hence V_
g (t) and b 2 @x L^
:=
@u
=
(t)
1 ^
g (t)
2 @ L^ T @x L^ g (t) @u (t) @u (t), 2 T @x L^ T 1 T g (t) @u (t): ^ ^
T
T
AT
+
T
1 T ~T g (t) ^ g (t) ~T ~ + T g (t) T (t) g
aux1
+ +
4T @ LT ^ @u
_ g (t) AT
g (t)
_ g (t)
g (t) T g (t)
T
T g (t)
g (t)
T
T
AT
g (t)
+
T
g (t)
^
1
@ LT ^
+
aux1
@u
4T
AT
1
(B.4)
_ g (t)
g (t)
@ L^ (t) . @u
(t)KL
~( )T = ~(t)T
g(
)T
g(
T
g(
)
g(
T
+R (t; ) 2~(t)T
)
Z
g(
T
4T ( g )
g(
)
)
)T
g(
)
T
(B.5)
)T
R (t; )T
g(
g(
)R (t; )
)T
g(
^
2
4T
4T
R (t; )T
g(
)T
g(
g(
)
)
(1 + ) I R (t; ) (1 )4T ! 1+ 1 I ~(t) (B.7) 1 1 4T
(s) we get
T g (s)
Z
(s) ds
1
2
T g (s)
^
t
+2
Z
1~ T (t)
g(
)T
(s) ds (
t)
2
t) ;
g(
4T g ( )R (t; )
)
T g (s)
(s)ds (
@ L^ (s)ds( @u
(s)KL
T (Bx ; r; T )
)~(t)
+V
:=
max
t)
t):
T g (t)
g (t)
(1+ ) (1 ) 4T
I
aux1 (t)
)R (t; )
T
@u
1 ^
(1+ ) be the maximal eigenvalue of g ( )T g ( ) (1 ) 4T I and max KL be the maximal eigenvalue of KL ; then from combining (B.5) and (B.7) and investigating the integral over , we get ! Z 1+ 1 1 ~ T t+T T I d ~(t) 1 (t) g( ) g( ) 1 1 4T t Z t+T Z B T (s)T g (s) T t)d g (s) (s)ds ( t t ! Z t+T Z @ LT @ L^ ^ +B T max KL (s)T (s)ds ( t) d @u @u t t
(B.6)
g(
@ LT ^
t
Let B
I R (t; ) 4T I R (t; ):
~(t)T
)T
t
and ~(t)T R (t; )
)T
2 1
+ max KL 1
g(
R (t; )T R (t; ) Z 2 (s)T g (s) ^ T
1 T Again by Young’s inequality, n1 aT 1 a1 n 1 b 1 b1 1 2 bT n2 aT bT b 2 2 bT 1 a1 and 2 a2 2 a2 with p n2 2 ~ ~ a1 = ( ) ; a2 = g ( )R (t; ); 4T ; b1 = p g b2 = 4T R (t; ) and n1 = n2 = 1 ; such that g(
g(
I ~( )
such that
I ~(t)
4T
4T
and by similar arguments Z @x2 L^ @ L^ 1 T (s) (s) (s) ds g ^ @u @u t Z 2 @x2 L^ @ L^ 1 T (s) ds ( (s) (s) g ^ @u @u t
I ~( )
4T
~(t)T
T g (s)
^
T g (t)
+I
)
for > 1: From Holder’s inequality, 1 1 R R 2 2 2 2 R jgj ; with g = 1 and f = jf gj jf j
I ~(t)
_ g (t)
g(
~
The solution of (30) can be represented by: ~( ) = ~(t) R (t; ), where 2 R @x L^ @L 1 T (s) + @u R (t; ) := (s) @u ^ (s) ds, g (s) ^ t such that
2~(t)T
1
t
~(t)T
)T
) R (t; )T
+ 1
and thus
V_
g(
(1
g (t)
@ L^ (t); @u
(t)KL
~( )T
Z +
~ g ( ) (t)
t
+
T
t+T
( )T Z
( )T AT
Further
13
g(
)
1 ^
+I
T g(
) ( )d
t+T
t
which gives
V aux1 (t + T ) ! Z t+T @ LT @ L^ ^ (s) (s) ds @u @u t
g(
) _ g ( ) AT
g(
)
_ g( )
T
( )d :
Z
Z
t+T
t
Z
T2
(s)T
g (s) ^
t t+T
(s)T
1
T g (s)
^
T
g (s) ^
t
T
1
T g (s)
^
(s)ds (
Thus ~( )T ~( ) is integrable. Let min V be the minimum eigenvalue of (1; HT ^ H^ ; A); then there exists a constant BfO ~ (Bx ; r; T ) > 0; such that u Z t0 +TT fOu ~ ( )T fOu ~ ( )d t0 Z t fOu ~ ( )T fOu ~ ( )d lim
t)d
(s)ds;
such that for B (Bx ; r; T ) :=
T
A
max
_ g (t)
g (t) g (t)
it follows that Z 1 ~ T (t) 1
B
TT
T
A 2
g(
T
)
aux1 (t)
Z
g(
)
aux1 (t + T ) T @ L^ ( ) HT ^ ( ) H^ (
t+T
+BV
V
@u
t
1 ^
1+ 1
t
V
g (t)
+1
t+T
_ g (t)
1
4T
~(t)T
Z
t+T g(
)T
g(
1+ 1
)
t
1
(V aux1 (t) 1 1 +BV Vu ^ (T + t)
V
aux1 (t
1
2T + B
!
B fO
I d ~(t)
1
4T
I
!
B ~(t)T ~(t) Z t+T ~(t)T
g(
)T
g(
t
=
V
aux1 (
aux1 (
)
)d
t0
2T BV
+ T ))
aux1 (
Z
aux1 (Bx ; r)
(B.8)
Ou
( zu
~
~0 O u
~
);
8t
1
4T
I
!
zu ~ (t0 ; zu
d ~(t)
~0 ) O u
~
>
0
!
zu ~ (t ; zu
~
~0 ) O u
("):
~
zu ~ (t ; zu
1
u
(") 8t0 2 [t0 ; TT ]. Thus
~0 ) O u
~
8t0 2 [t0 ; TT ]
;
which from radial unboundedness of Vu ^ ; related to (33), imply that there exist positive constants ! 1 ; and 1 such that !1
fOu ~ (t0 )T fOu ~ (t0 )
Then Z t0 +TT t0
= ! 1 TT = 2BfO
t0 +TT +T aux1 (
0
1
~0 ) A
( (")) = " for zu ~0 O ~ r and t T T + t0 ; u which satis…es de…nition 3: Suppose the assumption is r such that not true, i.e., there exists zu ~0 O ~
+ T )) d V
~0 ) O u
1
2
1
:= (r) and is given by ~
u
Thus zu ~ (t; zu
1+ 1
u
t0
u ~
ists t0 2 [t0 ; TT ] such that zu ~ (t0 ; zu
)d
t0 +TT
and Z t0 +TT ~( )T ~( )d t0
V
;
; where ! 1 is speci…ed ! and : De…ne TT = !1 later, and assume that for all zu ~0 O ~ r; there ex-
t0 +TT
(V
+ BV Vu ^ (t0 )
2BfO
V aux1 (t) V aux1 (t + T ) 1 1 + BV Vu ^ (T + t) Vu ^ (t) ;
t0 Z t0 +T
u ~
!
where 2 K1 : Fix r > 0; " > 0. De…ne ! := min ; 1 (") and note that BfO
1
Z
1
zu ~ (t)
d ~(t)
(t;T )+
)
aux1 (Bx ; r) 1
ii) From The UGS property we have
max KL (B T T +1) : (1 1 ) min A By the PE assumption we can choose T; ; > 0 and (1+ 1 ) > 1 such that, B < , where B > 0 i.e. (1 1 ) 4 max B
BV
Vu ^ (1)
and the integral of fOu ~ (t)T fOu ~ (t) is bounded.
Vu ^ (t) ;
where BV (Bx ; r; T ) :=
Vu ^ (t0 )
min V
! @ L^ T ( )+ ( ) A ( ) d ) @u
and
t0
1
+
T g (t)
+I
1
t!1
T
u ~
T
fOu ~ ( ) fOu ~ ( )d
8t0 2 [t0 ; TT ]
1;
Z
t0 +TT
!1 d
t0
;
which contradicts (B.8), and the proposition is proved.
! 2T BV aux1 (Bx ; r) +BV Vu ^ (t0 ) : B 1 1
14
