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An adaptive control scheme for systems with unknown actuator failures
Automatica 38 (2002) 1027 – 1034
www.elsevier.com/locate/automatica
Brief Paper
An adaptive control scheme for systems with unknown actuator failures Gang Taoa; ∗ , Shuhao Chena , Suresh M. Joshib a Department
of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22903, USA b NASA Langley Research Center, Mail Stop 161, Hampton, VA 23681, USA
Received 27 October 2000; received in revised form 19 October 2001; accepted 6 November 2001
Abstract A state feedback output tracking adaptive control scheme is developed for plants with actuator failures characterized by the failure pattern that some inputs are stuck at some unknown 4xed values at unknown time instants. New controller parametrization and adaptive law are developed under some relaxed system conditions. All closed-loop signals are bounded and the plant output tracks a given reference output asymptotically, despite the uncertainties in actuator failures and plant parameters. Simulation results verify the desired adaptive control system performance in the presence of actuator failures. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Actuator failure; Adaptive control; Plant-model output matching; State feedback; Output tracking
1. Introduction Actuator failure may cause major problems in many critical control systems such as :ight control systems. It is often not known when an actuator fails and how much the failure is but the remaining actuation could be still enough to accomplish a desired control task such as emergency landing of an aircraft. The question is whether a control system is intelligent enough to use remaining actuation in the presence of unknown actuator failures. An adaptive approach, which is capable of controlling systems with uncertainties, is thus of interest in developing control schemes which are e=cient for handling unknown actuator failures. While many open issues still exist in the area, control of systems with actuator or component failures has been studied with di?erent methods including multiple models, switching and tuning designs (Boskovic & Mehra, 1998, 1999), adaptive designs with focus on indirect adaptive LQ control to accommodate failures in the pitch control channel or the horizontal stabilizer (Ahmed-Zaid, Ioannou, Gousman, & Rooney, 1991), indirect or direct adaptive designs for
This paper was not presented at any IFAC meeting. This paper was recommended for Publication in revised form by Associate Editor Bernard Brogliato under the direction of Editor Frank L. Lewis. ∗ Corresponding author. E-mail addresses: [email protected] (G. Tao), [email protected] (S. Chen), [email protected] (S.M. Joshi).
an aircraft system with a locked left horizontal tail surface (Bodson & Groszkiewicz, 1997), and adaptive actuator failure compensation with known plant dynamics (Boskovic, Yu, & Mehra, 1998). Fault diagnosis designs (Vemuri & Polycarpou, 1997; Wang, Huang, & Daley, 1997) were also used for control of systems with actuator failures. In Tao, Joshi, and Ma (2001b), we developed direct adaptive state feedback control schemes for linear time-invariant plants with actuator failures characterized by inputs stuck at some values not in:uenced by control action. For a controlled plant x(t)=Ax(t)+Bu(t) ˙ whose input u(t) may have failed components, to achieve desired plant-model state dynamics matching in the presence of actuator failures, it ∗ is necessary that there exist constant vectors ks1i ∈ Rn and ∗ non-zero constant scalars ks2i ∈ R; i = 1; : : : ; m, such that ∗T ∗ A + bi ks1i = AM ; bi ks2i = bM , where bi ; i = 1; : : : ; m, is the ith column of B, and AM ; bM are a pair of reference model matrices independent of A; B. For many applications when only output tracking is the control objective, less restrictive conditions on the system matrices A and B are needed for achieving desired system performance. In Tao, Tang, and Joshi (2001c), an adaptive output tracking control design, relaxing the above conditions, is proposed for systems with unknown actuator failures, and its performance is studied only by simulations. It is the goal of this paper to present a new adaptive control scheme for systems with unknown parameters and unknown
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G. Tao et al. / Automatica 38 (2002) 1027–1034
failures, achieving closed-loop system stability and asymptotic output tracking, with both of the above conditions relaxed. In Section 2, we formulate the adaptive actuator failure compensation problem and derive plant-model matching conditions in the presence of actuator failures where some plant inputs are stuck at some 4xed values, such as a hydraulic failure. In Section 3, we develop an adaptive actuator compensation control scheme and analyze its performance, proving signal boundedness and asymptotic output tracking. In Section 4, we present simulation results to verify the desired performance of our adaptive control design.
2. Control objective and output matching In this section, we formulate the actuator failure compensation control problem for plant-model output matching 4rst (the adaptive control problem will be solved in Section 3), choose a suitable controller structure, and derive the needed design conditions.
(2.1)
where A ∈ Rn×n ; B ∈ Rn×m ; C ∈ R1×n are unknown constant parameter matrices, the state vector x(t) ∈ Rn is available for measurement, u(t) = [u1 ; : : : ; um ]T ∈ Rm is the input vector whose components may fail during system operation, and y(t) ∈ R is the plant output. One type of actuator failure (Boskovic et al., 1998) under consideration in this paper is uj (t) = u j ;
t ¿ tj ;
j ∈ {1; 2; : : : ; m};
(2.3)
where v(t) is an applied control input to be designed, and T
u = [u 1 ; u 2 ; : : : ; u m ] ;
= diag{1 ; 2 ; : : : ; m };
if the ith actuator fails; i:e:; ui = u i ;
0
otherwise:
(2.5)
The control task is to design a feedback control v(t) for plant (2.1) with actuator failures (2.2) under Assumption (A1) such that despite the control error, all closed-loop signals are bounded and the plant output y(t) asymptotically tracks a given reference output ym (t) from ym (t) = Wm (s)[r](t);
Wm (s) =
1 ; Pm (s)
(2.6)
where Pm (s) is a stable monic polynomial of degree n∗ , and r(t) is bounded and piecewise continuous. To derive suitable parametrizations for a plant-model output matching controller, we denote bi ∈ Rn ; i = 1; : : : ; m:
B = [b1 ; : : : ; bm ];
(2.7)
2.2. A plant-model output matching controller
(2.4)
(2.8)
∗ ∗ ∗ ∗ T with K1∗ = [k11 ; : : : ; k1m ] ∈ Rn×m ; k2∗ = [k21 ; : : : ; k2m ] ∈ Rm to be de4ned for plant-model output matching, and k ∗ = [k1∗ ; : : : ; km∗ ]T ∈ Rm for compensation of the actuation error. If the system has p failed actuators, that is, uj (t) = uQ j ; j = j1 ; : : : ; jp , we have the closed-loop system as
x(t) ˙ = A +
+
∗T x(t) + bj k1j
j=j1 ;:::; jp
(2.2)
where the constant value uQ j and the failure time instant tj are unknown. The basic assumption for a class of actuator failure compensation problems is (A1) system (2.1) is so designed that for any up to m − 1 actuator failures, the remaining actuators can still achieve a desired control objective. The key task of adaptive control is to adjust the remaining controls to achieve the desired system performance when there are up to m − 1 actuator failures whose parameters are unknown. In the presence of actuator failures, u(t) can be expressed as u(t) = v(t) + (u − v(t));
1
v(t) = v∗ (t) = K1∗T x(t) + k2∗ r(t) + k ∗
Consider a linear time-invariant plant y(t) = Cx(t);
i =
For the failure model (2.2), we consider the controller structure in Tao et al. (2001b),
2.1. Problem statement
x(t) ˙ = Ax(t) + Bu(t);
∗ bj k2j r(t)
j=j1 ;:::; jp
bj kj∗ +
bj u j ;
j=j1 ;:::; jp
j=j1 ;:::; jp
y(t) = Cx(t):
(2.9)
For this system to match reference system (2.6), we need to choose K1∗ ; k2∗ and k ∗ to satisfy −1 ∗T C sI − A − bj k1j j=j1 ;:::; jp
×
∗ bj k2j = Wm (s);
(2.10)
j=j1 ;:::; jp
C sI − A −
−1 ∗T bj k1j
j=j1 ;:::; jp
×
j=j1 ;:::; jp
bj kj∗ +
j=j1 ;:::; jp
bj u j = 0:
(2.11)
G. Tao et al. / Automatica 38 (2002) 1027–1034
If the columns bi ; i = 1; : : : ; m, of B are parallel to each other, the above conditions can be easily met. When bi ; i = 1; : : : ; m, are not parallel to each other, there may be di?erent parametrization schemes suitable for adaptive actuator failure compensation designs. In this paper, as a solution to the stated actuator failure compensation control problem, we choose the following controller structure: ∗ ∗T ∗ v1∗ (t) = · · · = vm (t) = k11 x(t) + k21 r(t) + k1∗
(2.12)
∗ for v∗ (t) = [v1∗ (t); : : : ; vm (t)]T in (2.8). This choice implies that the controller parameters in (2.8) have the special forms: ∗ ∗ k1i∗ = k11 ; k2i∗ = k21 , and ki∗ = k1∗ , for i = 2; 3; : : : ; m. Such a controller structure is suitable for plant-model output matching with known parameters as well as for adaptive control for closed-loop stability and asymptotically output tracking with parameter estimates. For this design, the following assumptions are needed: (A1a) (A; j=j1 ;:::;jp bj ); p ∈ {0; : : : ; m − 1}, are controllable; (A1b) (C; A; j=j1 ;:::;jp bj ); p ∈ {0; : : : ; m − 1}, have the ∗ same relative degree n ; (A1c) (C; A; j=j1 ;:::;jp bj ); p ∈ {0; : : : ; m − 1}, are minimum phase; and ∗ (A1d) CAn −1 j=j1 ;:::;jp bj , p ∈ {0; : : : ; m − 1}, have the same sign which is known: ∗ ∗ (2.13) ] = sign CAn −1 bj = constant: sign[k21 j=j1 ;:::; jp
Under Assumptions (A1a) – (A1c), for each actuator failure pattern, uj (t)= uQ j ; j =j1 ; : : : ; jp , the matching condition (2.10) can be satis4ed by −1 ∗T C sI − A − bj k11 j=j1 ;:::; jp
×
∗ bj k21 = Wm (s)
(2.14)
j=j1 ;:::; jp ∗ ∗ for some k11 ∈ Rn and k21 ∈ R, that is, under (A1a) and ∗ (A1b), the vector gain k11 places the closed-loop poles at the poles of Wm (s) and the zeros of the open-loop system (C; A; j=j1 ;:::;jp bj ), which are stable by (A1c), and the ∗ ∗ scalar gain k21 = kp−1 , where kp = CAn −1 j=j1 ;:::;jp bj . For condition (2.11), treating k1∗ ∈ R as an input and uQ j as disturbances, we have −1 ∗T fp (t) , C sI − A − bj k11 j=j1 ;:::; jp
×
j=j1 ;:::; jp
bj k1∗ +
j=j1 ;:::; jp
bj u j (t)
= Wm (s) ×
1029
k1∗ ∗ k21
(t) + C sI − A −
−1 ∗T bj k11
j=j1 ;:::; jp
bj u j (t)
(2.15)
j=j1 ;:::; jp
whose s-domain expression is −1 ∗T Fp (s) = C sI − A − bj k11 j=j1 ;:::; jp
×
j=j1 ;:::; jp
uj k1∗ + bj bj s s j=j ;:::; j 1
p
−1 k1∗ ∗T = Wm (s) ∗ + C sI − A − bj k11 k21 s j=j1 ;:::; jp
×
j=j1 ;:::; jp
bj
uj : s
(2.16)
∗T Since all zeros of det(sI − A − j=j1 ;:::; jp bj k11 ) are stable, ∗ there exists k1 such that lims→0 sFp (s) = 0; that is, in the time-domain, limt→∞ fp (t) = 0 exponentially. This asymptotic property is crucial for the parametrization of an actuator failure compensation design for the control law (2.12) which ensures that limt→∞ (y(t) − ym (t)) = 0, as we just veri4ed. As in Tao et al. (2001b), we let (Ti ; Ti+1 ); i =0; 1; : : : ; m0 , with T0 = 0, be the time intervals on which the actuator failure pattern is 4xed, that is, actuators only fail at time Ti ; i = 1; : : : ; m0 . Since there are m actuators, at least one of them does not fail, we have m0 ¡ m and Tm0 +1 =∞. Then, at time Tj ; j = 1; : : : ; m0 , the unknown plant-model matching ∗ ∗ ; k21 and k1∗ change their values such that parameters k11 ∗ ∗ k11 = k11(i) ;
∗ ∗ k21 = k21(i) ;
t ∈ (Ti ; Ti+1 )
∗ k1∗ = k1(i) ;
(2.17)
for i = 0; 1; : : : ; m0 , that is, the plant-model matching param∗ ∗ eters k11 ; k21 and k1∗ are piecewise constant, because the plant has di?erent characterizations under di?erent failure conditions so that the plant-model matching parameters are also di?erent. At each time when the piecewise constant parameters change due to the change of the actuator failure pattern, a transient component occurs in the system response and the e?ect of such a transient response converges to zero exponentially, which follows from the analysis in Tsakalis (1992). Remark 2.1. When there are p failed actuators; the remaining m − p actuators need to deliver desired controls to the closed-loop system for output matching or tracking as well
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G. Tao et al. / Automatica 38 (2002) 1027–1034
as for failure compensation. The design of the m − p control signals is crucial for this task; given that in the adaptive control case it is not known which m − p actuators of the total m actuators are alive and how much the failure is. The equal-control design in (2.12) is a chosen design which is able to ful4ll the task. For the design (2.8); Assumptions (A1a) – (A1c) are su=cient for a non-adaptive plant-model output matching control scheme; as well as for an adaptive plant-model output tracking control scheme; in the presence of up to m−1 actuator failures; as shown in this paper. These assumptions are in fact also necessary for desired system matching performance; under the controller structure (2.12); even in the case when the actuator failure pattern (when; how many and how much failure) is known; that is; in order to satisfy the desired output matching and failure compensation equations (2.10) and (2.11); Assumptions (A1a) – (A1c) have to be satis4ed.
variations. Based on the analysis of Tsakalis (1992), it can be shown that limt→∞ 0 (t) = 0 and limt→∞ t (t) = 0, both exponentially. Ignoring these exponentially decaying terms, we have the tracking error equation e(t) = y(t) − ym (t) = Wm (s)
1 ˜T !](t); ∗ [ k21
T ; k21 ; k1 ]T is the estimate where ˜(t) = (t) − ∗ ; (t) = [k11 ∗ ∗T ∗ ∗ T T of = [k11 ; k21 ; k1 ] , and ! = [x ; r; 1]T . Introducing the auxiliary signals
"(t) = Wm [!](t); #(t) =
T
(3.5)
(t)"(t) − Wm (s)[
T
!](t);
(t) = e(t) + $(t)#(t);
(3.7)
∗
Now we develop an adaptive control scheme for system (2.1) with unknown parameters A and B, and with unknown ∗ ∗ actuator failures (2.2). In this case, the parameters k11 ; k21 ∗ and k1 are unknown. As an adaptive version of (2.12), we use the controller structure v1 (t) = v2 (t) = · · · = vm (t) T = k11 (t)x(t) + k21 (t)r(t) + k1 (t);
(3.1)
where k11 (t) ∈ Rn ; k21 (t) ∈ R and k1 (t) ∈ R are the estimates ∗ ∗ ∈ Rn ; k21 ∈ R and k1∗ ∈ R. The of the unknown parameters k11 resulting closed-loop system is ∗T ∗ x+ x(t) ˙ = A + bj k11 bj k21 r j=j1 ;:::; jp
+
j=j1 ;:::; jp
+
j=j1 ;:::; jp
bj u j +
j=j1 ;:::; jp
bj k˜21 r +
(#(t) (t) ; 1 + " T " + #2
( ¿ 0:
(3.8) (3.9)
∗ Note that k21 is the scalar gain de4ned in (2.14), and by As∗ sumption (A1d), the sign of k21 is 4xed for di?erent failure patterns and is known. To analyze the stability and tracking performance of the adaptive control system, we de4ne the positive de4nite function T 1 V ( ˜; $) ˜ = (|$∗ | ˜ &−1 ˜ + (−1 $˜2 ); 2
˜= −
∗
;
$˜ = $ − $∗ :
(3.10)
˜ + p (t); (t) = $∗ ˜ (t)"(t) + $(t)#(t) t ∈ (Ti ; Ti+1 ); i = 0; 1; : : : ; m0 ; (3.2)
j=j1 ;:::; jp
∗ ∗ ; k˜21 = k21 − k21 , and k˜1 = k1 − k1∗ . where k˜11 = k11 − k11 In view of (2.9), (2.14), (2.15) and (2.17), the closed-loop system output can be expressed as
1 ˜T ˜ ˜ y(t) = ym (t) + Wm (s) ∗ (k 11 x + k 21 r + k 1 ) (t) k21
+ fp (t) + 0 (t) + t (t);
$(t) ˙ =−
& = &T ¿ 0;
T
T
bj k˜11 x
j=j1 ;:::; jp
bj k˜1 ;
˙(t) = − sign[k21 ]&"(t) (t) ; 1 + " T " + #2
From (3.4) – (3.7), we obtain
j=j1 ;:::; jp
bj k1∗ +
(3.6)
∗ , we choose the where $(t) is the estimate of $∗ = 1=k21 adaptive laws as
3. Adaptive control design
(3.4)
(3.3)
where fp (t) is de4ned in (2.15) such that limt→∞ fp (t)=0 exponentially, 0 (t) is related to the system initial conditions and t (t) is related to the transient system response equivalent to the e?ect of the piecewise constant parameter
(3.11)
where p (t) = p1 (t) + p2 (t); p1 (t) = $∗ (
∗T
(t)"(t) − Wm (s)[ T
(3.12) ∗T
!](t)); T
p2 (t) = Wm (s)$∗ [ ˜ !](t) − $∗ Wm (s)[ ˜ !](t):
(3.13) (3.14)
Before evaluating the time derivative of V (t), we 4rst show that lim p (t) = 0 exponentially:
t→∞
(3.15)
Note that p = m0 for t ¿ Tm0 , that is, there are m0 failed actuators for t ∈ (Tm0 ; ∞). Let the impulse response function
G. Tao et al. / Automatica 38 (2002) 1027–1034
of Wm (s) be wm (t). Then, for t ∈ (Ti ; Ti+1 ), we can express (3.13) as t ∗T p1 (t) = $∗ (t) wm (t − +)!(+) d+ 0
−
t
wm (t − +)
0
∗
∗T
=$
−
Ti
0
(t)
Ti
0
∗T
(+)!(+) d+
∗T wm (t − +) (+)!(+) d+ :
(3.16)
−
∗
(t)
Ti
t
∗T wm (t − +) (+)!(+) d+ = 0:
2 (t) 1+
t ∈ (Ti ; Ti+1 );
For the second equality of (3.16), we used the fact that is constant for t ∈ (Ti ; Ti+1 ), that is, t ∗ ∗T $ (t) wm (t − +)!(+) d+
so that | p2 (t)| 6 a2 ,e−-t . Therefore, limt→∞ p (t) = 0 exponentially. With (3.5) – (3.9) and (3.11), ignoring the exponentially decaying term p (t) which does not destabilize a gradient adaptive law (3.8) and (3.9), the time-derivative of V along (3.8) and (3.9) is V˙ (t) = −
wm (t − +)!(+) d+
(3.17)
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"T (t)"(t)
+ #2 (t)
6 0;
i = 0; 1; : : : ; m0 :
(3.23)
It is important to note that V (·) as a function of t is not ∗T ∗ ∗T ∗ ∗ T ; k21 ; k1∗ ]T =[k11(i) ; k21(i) ; k1(i) ] continuous because ∗ =[k11 is a piecewise constant parameter vector as described in (2.17). Since there are only 4nite number of failures in the system, it follows that Tm0 is 4nite and V˙ (t) = −
2 (t) 1+
"T (t)"(t)
+ #2 (t)
6 0;
t ∈ (Tm0 ; ∞) (3.24)
(+))T !(+) d+: (3.18)
which implies that (t); $(t) ∈ L∞ ; [ (t)= 1 + "T (t)"(t) + #2 (t)] ∈ L2 ∩ L∞ ; ˙(t) ∈ L2 ∩ L∞ , and $(t) ˙ ∈ L2 ∩ L∞ . Based on this, closed-loop stability and asymptotic tracking can be proved as the following result.
Since Wm (s) is stable, we have |wm (t − +)| 6 ,e−-(t−+) for some - ¿ 0; , ¿ 0, so that Tm0 ∗ | p1 (t)| 6 ,|$∗ |e−-t e-+ |( (m − ∗ (+))T !(+)| d+: 0)
Theorem 3.1. Adaptive controller (3:1); with the adaptive law (3:8) and (3:9); applied to system (2:1) with actuator failures (2:2); guarantees that all closed-loop signals are bounded and the tracking error e(t) = y(t) − ym (t) goes to zero as t goes to in9nity.
Ti
Therefore, for t ¿ Tm0 , we have Tm 0 ∗ ∗ p1 (t) = $ wm (t − +)( (m − 0) 0
∗
0
(3.19) ∗
∗ m0
∗
As (t) = is constant for t ¿ Tm0 ; (t) is piecewise constant in (0; Tm0 ); Tm0 is 4nite and !(t) is bounded in (0; Tm0 ), there exist a constant a1 ¿ 0 such that Tm0 ∗ e-+ |( (m − ∗ (+))T !(+)| d+ 6 a1 (3.20) 0) 0
so that | p1 (t)| 6 a1 ,|$∗ |e−-t : Similarly, we have Tm 0 T p2 (t) = wm (t − +)($∗ (+) − $∗(m0 ) ) ˜ (+)!(+) d+ 0
−-t
6 ,e
Tm0
0
!(+)| d+:
T
e-+ |($∗(m0 ) − $∗ (+)) ˜ (+)
4. Simulation study (3.21)
As $∗ (t) = $∗(m0 ) is constant for t ¿ Tm0 ; $∗ (t) is piecewise T constant in (0; Tm0 ); Tm0 is 4nite and ˜ (t)!(t) is bounded in (0; Tm0 ), there exists a constant a2 such that Tm 0 T (3.22) e-+ |($∗(m0 ) − $∗ (+)) ˜ (+)!(+)| d+ 6 a2 0
Proof (outline). The standard properties of the adaptive laws: (t); $(t) ∈ L∞ ; [ (t)= 1 + "T (t)"(t) + #2 (t)] ∈ L2 ∩ L∞ , and ˙(t) ∈ L2 ∩ L∞ , ensure that the closed-loop system has a small loop gain so that it is stable in the sense that all signals are bounded (Tao & KokotoviUc, 1996, Appendix A). Then, it follows that (t) ∈ L2 ∩ L∞ , and that #(t) ∈ L2 ∩ L∞ as ˙(t) ∈ L2 ∩ L∞ , so that e(t) = (t) − $(t)#(t) ∈ L2 ∩ L∞ . Finally, it follows from (3.4) that e(t) ˙ is bounded so that limt→∞ e(t) = 0 as from Barbalat’s lemma. For details of the proof, see Tao, Chen, and Joshi (2001a).
As an illustrative example, we use the linearized lateral dynamic model of a Boeing 747 airplane (Franklin, Powell, & Emami-Naeini, 1994, p. 686) as the controlled plant, with two augmented actuation vectors b2 u2 and b3 u3 representing two additional pieces to augment a rudder for the study of actuator failure compensation, to which the adaptive control scheme consisting of (3.1), (3.5) – (3.9) is applied.
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G. Tao et al. / Automatica 38 (2002) 1027–1034
Fig. 1. Tracking error and control input for r(t) = 0:025.
The plant with augmented actuation vectors is described by x(t) ˙ = Ax(t) + Bu(t); y(t) = Cx(t); −0:0558 −0:9968 0:0802 0:598 −0:115 −0:0318 A= −3:05 0:388 −0:465 0 0:0805 1 B = [b1 ; b2 ; b3 ]; 0:00729 −0:475 b1 = 0:153 ; 0 0:005 −0:3 b3 = 0:1 ; 0
0:0415 0 ; 0 0
C = [0 1 0 0]: 0:01 −0:5 b2 = 0:2 ; 0
where x(t) = [,; r; p; 0]T ; , is the side-slip angle, r is the yaw rate, p is the roll rate, 0 is the roll angle, y is the plant output which is the yaw rate in this case, and u is the control input vector which contains three control signals u=[u1 ; u2 ; u3 ]T to represent three rudder servos: 1r1 , 1r2 ; 1r3 , from a three-piece rudder for achieving compensation in the presence of actuator failures. In this study, we simulate the case of two actuator failures and consider the failure pattern u2 (t) = 0:02 rad; t ¿ 50, and u3 (t) = −0:03 rad; t ¿ 100. The simulation parameters
are: & = 10I; ( = 1; Wm = 1=(s + 3); ym (0) = 0; y(0) = −0:02; (0) = 0:5 ∗ ( ∗ is the true matching parameter vector when there is no actuator failure), $(0) = 0. The simulation results, including the plant output y(t), reference output ym (t), tracking error e(t) = y(t) − ym (t), and control input v1 (t) = v2 (t) = v3 (t), are shown in Fig. 1 for the reference input r(t) = 0:025, and in Fig. 2 for r(t) = 0:02 sin(0:1t). The system responses are as expected, which indicate that at the time instant when one of the actuators fails, there is a transient response in the tracking errors. As the time goes on, the tracking errors become smaller. The controller parameters k11 ; k21 , and k1 (not shown) also change their values at the time instants when actuator failures occur. All signals in the adaptive control system are bounded, and stability and output tracking are ensured. Similar results were also observed in the simulations for the failure pattern u2 (t)=u2 (50); t ¿ 50, and u3 (t)=u3 (100); t ¿ 100, which means the actuators were stuck at the positions where they failed.
5. Conclusions We have developed an adaptive control scheme for systems with unknown actuator failures. This scheme, when implemented with true parameters, ensures desired plant-model matching in the presence of actuator failures, and, when implemented with adaptive parameter estimates, ensures asymptotic output tracking in the presence of unknown plant parameters and unknown actuator failure parameters.
G. Tao et al. / Automatica 38 (2002) 1027–1034
1033
Fig. 2. Tracking error and control input for r(t) = 0:02 sin(0:1t).
The design conditions on the plant matrices (A; B) are relaxed as compared with a state tracking design, so that adaptive actuator failure compensation is applicable to a larger class of systems. Simulation results veri4ed the desired performance of the developed adaptive actuator failure compensation design. Adaptive failure compensation using output feedback is currently under extensive investigation. Acknowledgements This research was partially supported by NASA Langley Research Center under grant NCC-1-342. Related discussion with Xiaoli Ma and Xidong Tang was bene4cial to this work. References Ahmed-Zaid, F., Ioannou, P., Gousman, K., & Rooney, R. (1991). Accommodation of failures in the F-16 aircraft using adaptive control. IEEE Control Systems Magazine, 11(1), 73–78. Bodson, M., & Groszkiewicz, J. E. (1997). Multivariable adaptive algorithms for recon4gurable :ight control. IEEE Transactions on Control Systems Technology, 5(2), 217–229. Boskovic, J. D., Yu, S. -H., & Mehra, R. K. (1998). A stable scheme for automatic control recon4guration in the presence of actuator failures. Proceedings of the 1998 ACC (pp. 2455 –2459). Philadelphia, Pennsylvania. Boskovic, J. D., & Mehra, R. K. (1998). A multiple model-based recon4gurable :ight control system design. Proceedings of the 37th IEEE Conference on Decision and Control (pp. 4503– 4508). Tampa, Florida. Boskovic, J. D., & Mehra, R. K., (1999). Stable multiple model adaptive :ight control for accommodation of a large class of control e?ector failures. Proceedings of the 1999 ACC (pp. 1920 –1924). San Diego, California.
Franklin, G. F., Powell, J. D., & Emami-Naeini, A. (1994). Feedback control of dynamic systems (3rd ed). Reading, MA: Addison-Wesley. Tao, G., Chen, S. H., & Joshi, S. M. (2001a). An adaptive control scheme for systems with unknown actuator failures. Technical Report UVA-ECE-ASC-01-03-0, Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA, March 2001. Tao, G., Joshi, S. M., & Ma, X. L. (2001b). Adaptive state feedback and tracking control of systems with actuator failures. IEEE Transactions on Automatic Control, 46(1), 78–95. Tao, G., & KokotoviUc, P. V. (1996). Adaptive control of systems with actuator and sensor nonlinearities. New York: Wiley. Tao, G., Tang, X. T., & Joshi, S. M. (2001c). Output tracking actuator failure compensation control. Proceedings of the 2001 ACC (pp. 1821–1826). Arlington, Virginia. Tsakalis, K. S. (1992). Model reference adaptive control of linear time-varying plants: the case of “jump” parameter variations. International Journal of Control, 56(6), 1299–1345. Vemuri, A. T., & Polycarpou, M. M. (1997). Robust nonlinear fault diagnosis in input–output systems. International Journal of Control, 68(2), 343–360. Wang, H., Huang, Z. J., & Daley, S. (1997). On the use of adaptive updating rules for actuator and sensor fault diagnosis. Automatica, 33(2), 217–225. Gang Tao received his Ph.D. degree in Electrical Engineering in 1989, from University of Southern California. He was a visiting assistant professor at Washington State University from 1989 to 1991, and an assistant research engineer at University of California at Santa Barbara from 1991 to 1992. He joined Department of Electrical Engineering at University of Virginia in 1992, where he is now an associate professor. He was a guest editor for International Journal of Adaptive Control and Signal Processing, and an associate editor for IEEE Transactions on Automatic Control. He was a program committee member for numerous international conferences, and was the organizer and chair of 2001 International Symposium on Adaptive and Intelligent Systems and
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Control, held in Charlottesville, Virginia, USA. He co-edited one book, authored or co-authored one book, over 45 journal papers and 5 book chapters, and 115 conference papers=presentations on adaptive control, nonlinear control, multivariable control, optimal control, control applications and robotics.
Shuhao Chen received his B.S. degree in automatic control from Tsinghua University, Beijing, China, and his M.S. degree in industrial automation from Xi’an Jiaotong University, Xi’an, China, in 1993 and 1998, respectively. He is now working toward his Ph.D. degree at the University of Virginia. He was an engineer at the Automation Research Institute of the Ministry of Metallurgical Industry, Beijing, China, from 1993 to 1999. His main research interest is adaptive control of systems with actuator failures, for aircraft and industrial applications.
Suresh M. Joshi received his Ph.D. in electrical engineering from Rensselaer Polytechnic Institute, Troy, NY, in 1973. He is Senior Scientist for Control Theory at NASA-Langley Research Center in Hampton, Virginia. His research interests include multivariable robust control, adaptive control, nonlinear systems, and applications to advanced aircraft and spacecraft.Dr. Joshi is a Fellow of the IEEE, the AIAA, and the ASME. He served on numerous editorial boards, technical committees, and organizing committees, including the IEEE-Control Systems Society’s Board of Governors (1989 –94). His publications include several articles and two books, “Control of Large Flexible Space Structures” (Berlin: Springer-Verlag, 1989) and “Control of Nonlinear Multibody Flexible Space Structures” (London: Springer-Verlag, 1996). He is the recipient of the IEEE Control Systems Technology Award, as well as a number of awards from NASA-Langley Research Center. He is also an amateur cartoonist and contributed the “Out of Control” cartoons to the IEEE Control Systems Magazine from 1985 until 1993.
www.elsevier.com/locate/automatica
Brief Paper
An adaptive control scheme for systems with unknown actuator failures Gang Taoa; ∗ , Shuhao Chena , Suresh M. Joshib a Department
of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22903, USA b NASA Langley Research Center, Mail Stop 161, Hampton, VA 23681, USA
Received 27 October 2000; received in revised form 19 October 2001; accepted 6 November 2001
Abstract A state feedback output tracking adaptive control scheme is developed for plants with actuator failures characterized by the failure pattern that some inputs are stuck at some unknown 4xed values at unknown time instants. New controller parametrization and adaptive law are developed under some relaxed system conditions. All closed-loop signals are bounded and the plant output tracks a given reference output asymptotically, despite the uncertainties in actuator failures and plant parameters. Simulation results verify the desired adaptive control system performance in the presence of actuator failures. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Actuator failure; Adaptive control; Plant-model output matching; State feedback; Output tracking
1. Introduction Actuator failure may cause major problems in many critical control systems such as :ight control systems. It is often not known when an actuator fails and how much the failure is but the remaining actuation could be still enough to accomplish a desired control task such as emergency landing of an aircraft. The question is whether a control system is intelligent enough to use remaining actuation in the presence of unknown actuator failures. An adaptive approach, which is capable of controlling systems with uncertainties, is thus of interest in developing control schemes which are e=cient for handling unknown actuator failures. While many open issues still exist in the area, control of systems with actuator or component failures has been studied with di?erent methods including multiple models, switching and tuning designs (Boskovic & Mehra, 1998, 1999), adaptive designs with focus on indirect adaptive LQ control to accommodate failures in the pitch control channel or the horizontal stabilizer (Ahmed-Zaid, Ioannou, Gousman, & Rooney, 1991), indirect or direct adaptive designs for
This paper was not presented at any IFAC meeting. This paper was recommended for Publication in revised form by Associate Editor Bernard Brogliato under the direction of Editor Frank L. Lewis. ∗ Corresponding author. E-mail addresses: [email protected] (G. Tao), [email protected] (S. Chen), [email protected] (S.M. Joshi).
an aircraft system with a locked left horizontal tail surface (Bodson & Groszkiewicz, 1997), and adaptive actuator failure compensation with known plant dynamics (Boskovic, Yu, & Mehra, 1998). Fault diagnosis designs (Vemuri & Polycarpou, 1997; Wang, Huang, & Daley, 1997) were also used for control of systems with actuator failures. In Tao, Joshi, and Ma (2001b), we developed direct adaptive state feedback control schemes for linear time-invariant plants with actuator failures characterized by inputs stuck at some values not in:uenced by control action. For a controlled plant x(t)=Ax(t)+Bu(t) ˙ whose input u(t) may have failed components, to achieve desired plant-model state dynamics matching in the presence of actuator failures, it ∗ is necessary that there exist constant vectors ks1i ∈ Rn and ∗ non-zero constant scalars ks2i ∈ R; i = 1; : : : ; m, such that ∗T ∗ A + bi ks1i = AM ; bi ks2i = bM , where bi ; i = 1; : : : ; m, is the ith column of B, and AM ; bM are a pair of reference model matrices independent of A; B. For many applications when only output tracking is the control objective, less restrictive conditions on the system matrices A and B are needed for achieving desired system performance. In Tao, Tang, and Joshi (2001c), an adaptive output tracking control design, relaxing the above conditions, is proposed for systems with unknown actuator failures, and its performance is studied only by simulations. It is the goal of this paper to present a new adaptive control scheme for systems with unknown parameters and unknown
0005-1098/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 2 ) 0 0 0 1 8 - 3
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G. Tao et al. / Automatica 38 (2002) 1027–1034
failures, achieving closed-loop system stability and asymptotic output tracking, with both of the above conditions relaxed. In Section 2, we formulate the adaptive actuator failure compensation problem and derive plant-model matching conditions in the presence of actuator failures where some plant inputs are stuck at some 4xed values, such as a hydraulic failure. In Section 3, we develop an adaptive actuator compensation control scheme and analyze its performance, proving signal boundedness and asymptotic output tracking. In Section 4, we present simulation results to verify the desired performance of our adaptive control design.
2. Control objective and output matching In this section, we formulate the actuator failure compensation control problem for plant-model output matching 4rst (the adaptive control problem will be solved in Section 3), choose a suitable controller structure, and derive the needed design conditions.
(2.1)
where A ∈ Rn×n ; B ∈ Rn×m ; C ∈ R1×n are unknown constant parameter matrices, the state vector x(t) ∈ Rn is available for measurement, u(t) = [u1 ; : : : ; um ]T ∈ Rm is the input vector whose components may fail during system operation, and y(t) ∈ R is the plant output. One type of actuator failure (Boskovic et al., 1998) under consideration in this paper is uj (t) = u j ;
t ¿ tj ;
j ∈ {1; 2; : : : ; m};
(2.3)
where v(t) is an applied control input to be designed, and T
u = [u 1 ; u 2 ; : : : ; u m ] ;
= diag{1 ; 2 ; : : : ; m };
if the ith actuator fails; i:e:; ui = u i ;
0
otherwise:
(2.5)
The control task is to design a feedback control v(t) for plant (2.1) with actuator failures (2.2) under Assumption (A1) such that despite the control error, all closed-loop signals are bounded and the plant output y(t) asymptotically tracks a given reference output ym (t) from ym (t) = Wm (s)[r](t);
Wm (s) =
1 ; Pm (s)
(2.6)
where Pm (s) is a stable monic polynomial of degree n∗ , and r(t) is bounded and piecewise continuous. To derive suitable parametrizations for a plant-model output matching controller, we denote bi ∈ Rn ; i = 1; : : : ; m:
B = [b1 ; : : : ; bm ];
(2.7)
2.2. A plant-model output matching controller
(2.4)
(2.8)
∗ ∗ ∗ ∗ T with K1∗ = [k11 ; : : : ; k1m ] ∈ Rn×m ; k2∗ = [k21 ; : : : ; k2m ] ∈ Rm to be de4ned for plant-model output matching, and k ∗ = [k1∗ ; : : : ; km∗ ]T ∈ Rm for compensation of the actuation error. If the system has p failed actuators, that is, uj (t) = uQ j ; j = j1 ; : : : ; jp , we have the closed-loop system as
x(t) ˙ = A +
+
∗T x(t) + bj k1j
j=j1 ;:::; jp
(2.2)
where the constant value uQ j and the failure time instant tj are unknown. The basic assumption for a class of actuator failure compensation problems is (A1) system (2.1) is so designed that for any up to m − 1 actuator failures, the remaining actuators can still achieve a desired control objective. The key task of adaptive control is to adjust the remaining controls to achieve the desired system performance when there are up to m − 1 actuator failures whose parameters are unknown. In the presence of actuator failures, u(t) can be expressed as u(t) = v(t) + (u − v(t));
1
v(t) = v∗ (t) = K1∗T x(t) + k2∗ r(t) + k ∗
Consider a linear time-invariant plant y(t) = Cx(t);
i =
For the failure model (2.2), we consider the controller structure in Tao et al. (2001b),
2.1. Problem statement
x(t) ˙ = Ax(t) + Bu(t);
∗ bj k2j r(t)
j=j1 ;:::; jp
bj kj∗ +
bj u j ;
j=j1 ;:::; jp
j=j1 ;:::; jp
y(t) = Cx(t):
(2.9)
For this system to match reference system (2.6), we need to choose K1∗ ; k2∗ and k ∗ to satisfy −1 ∗T C sI − A − bj k1j j=j1 ;:::; jp
×
∗ bj k2j = Wm (s);
(2.10)
j=j1 ;:::; jp
C sI − A −
−1 ∗T bj k1j
j=j1 ;:::; jp
×
j=j1 ;:::; jp
bj kj∗ +
j=j1 ;:::; jp
bj u j = 0:
(2.11)
G. Tao et al. / Automatica 38 (2002) 1027–1034
If the columns bi ; i = 1; : : : ; m, of B are parallel to each other, the above conditions can be easily met. When bi ; i = 1; : : : ; m, are not parallel to each other, there may be di?erent parametrization schemes suitable for adaptive actuator failure compensation designs. In this paper, as a solution to the stated actuator failure compensation control problem, we choose the following controller structure: ∗ ∗T ∗ v1∗ (t) = · · · = vm (t) = k11 x(t) + k21 r(t) + k1∗
(2.12)
∗ for v∗ (t) = [v1∗ (t); : : : ; vm (t)]T in (2.8). This choice implies that the controller parameters in (2.8) have the special forms: ∗ ∗ k1i∗ = k11 ; k2i∗ = k21 , and ki∗ = k1∗ , for i = 2; 3; : : : ; m. Such a controller structure is suitable for plant-model output matching with known parameters as well as for adaptive control for closed-loop stability and asymptotically output tracking with parameter estimates. For this design, the following assumptions are needed: (A1a) (A; j=j1 ;:::;jp bj ); p ∈ {0; : : : ; m − 1}, are controllable; (A1b) (C; A; j=j1 ;:::;jp bj ); p ∈ {0; : : : ; m − 1}, have the ∗ same relative degree n ; (A1c) (C; A; j=j1 ;:::;jp bj ); p ∈ {0; : : : ; m − 1}, are minimum phase; and ∗ (A1d) CAn −1 j=j1 ;:::;jp bj , p ∈ {0; : : : ; m − 1}, have the same sign which is known: ∗ ∗ (2.13) ] = sign CAn −1 bj = constant: sign[k21 j=j1 ;:::; jp
Under Assumptions (A1a) – (A1c), for each actuator failure pattern, uj (t)= uQ j ; j =j1 ; : : : ; jp , the matching condition (2.10) can be satis4ed by −1 ∗T C sI − A − bj k11 j=j1 ;:::; jp
×
∗ bj k21 = Wm (s)
(2.14)
j=j1 ;:::; jp ∗ ∗ for some k11 ∈ Rn and k21 ∈ R, that is, under (A1a) and ∗ (A1b), the vector gain k11 places the closed-loop poles at the poles of Wm (s) and the zeros of the open-loop system (C; A; j=j1 ;:::;jp bj ), which are stable by (A1c), and the ∗ ∗ scalar gain k21 = kp−1 , where kp = CAn −1 j=j1 ;:::;jp bj . For condition (2.11), treating k1∗ ∈ R as an input and uQ j as disturbances, we have −1 ∗T fp (t) , C sI − A − bj k11 j=j1 ;:::; jp
×
j=j1 ;:::; jp
bj k1∗ +
j=j1 ;:::; jp
bj u j (t)
= Wm (s) ×
1029
k1∗ ∗ k21
(t) + C sI − A −
−1 ∗T bj k11
j=j1 ;:::; jp
bj u j (t)
(2.15)
j=j1 ;:::; jp
whose s-domain expression is −1 ∗T Fp (s) = C sI − A − bj k11 j=j1 ;:::; jp
×
j=j1 ;:::; jp
uj k1∗ + bj bj s s j=j ;:::; j 1
p
−1 k1∗ ∗T = Wm (s) ∗ + C sI − A − bj k11 k21 s j=j1 ;:::; jp
×
j=j1 ;:::; jp
bj
uj : s
(2.16)
∗T Since all zeros of det(sI − A − j=j1 ;:::; jp bj k11 ) are stable, ∗ there exists k1 such that lims→0 sFp (s) = 0; that is, in the time-domain, limt→∞ fp (t) = 0 exponentially. This asymptotic property is crucial for the parametrization of an actuator failure compensation design for the control law (2.12) which ensures that limt→∞ (y(t) − ym (t)) = 0, as we just veri4ed. As in Tao et al. (2001b), we let (Ti ; Ti+1 ); i =0; 1; : : : ; m0 , with T0 = 0, be the time intervals on which the actuator failure pattern is 4xed, that is, actuators only fail at time Ti ; i = 1; : : : ; m0 . Since there are m actuators, at least one of them does not fail, we have m0 ¡ m and Tm0 +1 =∞. Then, at time Tj ; j = 1; : : : ; m0 , the unknown plant-model matching ∗ ∗ ; k21 and k1∗ change their values such that parameters k11 ∗ ∗ k11 = k11(i) ;
∗ ∗ k21 = k21(i) ;
t ∈ (Ti ; Ti+1 )
∗ k1∗ = k1(i) ;
(2.17)
for i = 0; 1; : : : ; m0 , that is, the plant-model matching param∗ ∗ eters k11 ; k21 and k1∗ are piecewise constant, because the plant has di?erent characterizations under di?erent failure conditions so that the plant-model matching parameters are also di?erent. At each time when the piecewise constant parameters change due to the change of the actuator failure pattern, a transient component occurs in the system response and the e?ect of such a transient response converges to zero exponentially, which follows from the analysis in Tsakalis (1992). Remark 2.1. When there are p failed actuators; the remaining m − p actuators need to deliver desired controls to the closed-loop system for output matching or tracking as well
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G. Tao et al. / Automatica 38 (2002) 1027–1034
as for failure compensation. The design of the m − p control signals is crucial for this task; given that in the adaptive control case it is not known which m − p actuators of the total m actuators are alive and how much the failure is. The equal-control design in (2.12) is a chosen design which is able to ful4ll the task. For the design (2.8); Assumptions (A1a) – (A1c) are su=cient for a non-adaptive plant-model output matching control scheme; as well as for an adaptive plant-model output tracking control scheme; in the presence of up to m−1 actuator failures; as shown in this paper. These assumptions are in fact also necessary for desired system matching performance; under the controller structure (2.12); even in the case when the actuator failure pattern (when; how many and how much failure) is known; that is; in order to satisfy the desired output matching and failure compensation equations (2.10) and (2.11); Assumptions (A1a) – (A1c) have to be satis4ed.
variations. Based on the analysis of Tsakalis (1992), it can be shown that limt→∞ 0 (t) = 0 and limt→∞ t (t) = 0, both exponentially. Ignoring these exponentially decaying terms, we have the tracking error equation e(t) = y(t) − ym (t) = Wm (s)
1 ˜T !](t); ∗ [ k21
T ; k21 ; k1 ]T is the estimate where ˜(t) = (t) − ∗ ; (t) = [k11 ∗ ∗T ∗ ∗ T T of = [k11 ; k21 ; k1 ] , and ! = [x ; r; 1]T . Introducing the auxiliary signals
"(t) = Wm [!](t); #(t) =
T
(3.5)
(t)"(t) − Wm (s)[
T
!](t);
(t) = e(t) + $(t)#(t);
(3.7)
∗
Now we develop an adaptive control scheme for system (2.1) with unknown parameters A and B, and with unknown ∗ ∗ actuator failures (2.2). In this case, the parameters k11 ; k21 ∗ and k1 are unknown. As an adaptive version of (2.12), we use the controller structure v1 (t) = v2 (t) = · · · = vm (t) T = k11 (t)x(t) + k21 (t)r(t) + k1 (t);
(3.1)
where k11 (t) ∈ Rn ; k21 (t) ∈ R and k1 (t) ∈ R are the estimates ∗ ∗ ∈ Rn ; k21 ∈ R and k1∗ ∈ R. The of the unknown parameters k11 resulting closed-loop system is ∗T ∗ x+ x(t) ˙ = A + bj k11 bj k21 r j=j1 ;:::; jp
+
j=j1 ;:::; jp
+
j=j1 ;:::; jp
bj u j +
j=j1 ;:::; jp
bj k˜21 r +
(#(t) (t) ; 1 + " T " + #2
( ¿ 0:
(3.8) (3.9)
∗ Note that k21 is the scalar gain de4ned in (2.14), and by As∗ sumption (A1d), the sign of k21 is 4xed for di?erent failure patterns and is known. To analyze the stability and tracking performance of the adaptive control system, we de4ne the positive de4nite function T 1 V ( ˜; $) ˜ = (|$∗ | ˜ &−1 ˜ + (−1 $˜2 ); 2
˜= −
∗
;
$˜ = $ − $∗ :
(3.10)
˜ + p (t); (t) = $∗ ˜ (t)"(t) + $(t)#(t) t ∈ (Ti ; Ti+1 ); i = 0; 1; : : : ; m0 ; (3.2)
j=j1 ;:::; jp
∗ ∗ ; k˜21 = k21 − k21 , and k˜1 = k1 − k1∗ . where k˜11 = k11 − k11 In view of (2.9), (2.14), (2.15) and (2.17), the closed-loop system output can be expressed as
1 ˜T ˜ ˜ y(t) = ym (t) + Wm (s) ∗ (k 11 x + k 21 r + k 1 ) (t) k21
+ fp (t) + 0 (t) + t (t);
$(t) ˙ =−
& = &T ¿ 0;
T
T
bj k˜11 x
j=j1 ;:::; jp
bj k˜1 ;
˙(t) = − sign[k21 ]&"(t) (t) ; 1 + " T " + #2
From (3.4) – (3.7), we obtain
j=j1 ;:::; jp
bj k1∗ +
(3.6)
∗ , we choose the where $(t) is the estimate of $∗ = 1=k21 adaptive laws as
3. Adaptive control design
(3.4)
(3.3)
where fp (t) is de4ned in (2.15) such that limt→∞ fp (t)=0 exponentially, 0 (t) is related to the system initial conditions and t (t) is related to the transient system response equivalent to the e?ect of the piecewise constant parameter
(3.11)
where p (t) = p1 (t) + p2 (t); p1 (t) = $∗ (
∗T
(t)"(t) − Wm (s)[ T
(3.12) ∗T
!](t)); T
p2 (t) = Wm (s)$∗ [ ˜ !](t) − $∗ Wm (s)[ ˜ !](t):
(3.13) (3.14)
Before evaluating the time derivative of V (t), we 4rst show that lim p (t) = 0 exponentially:
t→∞
(3.15)
Note that p = m0 for t ¿ Tm0 , that is, there are m0 failed actuators for t ∈ (Tm0 ; ∞). Let the impulse response function
G. Tao et al. / Automatica 38 (2002) 1027–1034
of Wm (s) be wm (t). Then, for t ∈ (Ti ; Ti+1 ), we can express (3.13) as t ∗T p1 (t) = $∗ (t) wm (t − +)!(+) d+ 0
−
t
wm (t − +)
0
∗
∗T
=$
−
Ti
0
(t)
Ti
0
∗T
(+)!(+) d+
∗T wm (t − +) (+)!(+) d+ :
(3.16)
−
∗
(t)
Ti
t
∗T wm (t − +) (+)!(+) d+ = 0:
2 (t) 1+
t ∈ (Ti ; Ti+1 );
For the second equality of (3.16), we used the fact that is constant for t ∈ (Ti ; Ti+1 ), that is, t ∗ ∗T $ (t) wm (t − +)!(+) d+
so that | p2 (t)| 6 a2 ,e−-t . Therefore, limt→∞ p (t) = 0 exponentially. With (3.5) – (3.9) and (3.11), ignoring the exponentially decaying term p (t) which does not destabilize a gradient adaptive law (3.8) and (3.9), the time-derivative of V along (3.8) and (3.9) is V˙ (t) = −
wm (t − +)!(+) d+
(3.17)
1031
"T (t)"(t)
+ #2 (t)
6 0;
i = 0; 1; : : : ; m0 :
(3.23)
It is important to note that V (·) as a function of t is not ∗T ∗ ∗T ∗ ∗ T ; k21 ; k1∗ ]T =[k11(i) ; k21(i) ; k1(i) ] continuous because ∗ =[k11 is a piecewise constant parameter vector as described in (2.17). Since there are only 4nite number of failures in the system, it follows that Tm0 is 4nite and V˙ (t) = −
2 (t) 1+
"T (t)"(t)
+ #2 (t)
6 0;
t ∈ (Tm0 ; ∞) (3.24)
(+))T !(+) d+: (3.18)
which implies that (t); $(t) ∈ L∞ ; [ (t)= 1 + "T (t)"(t) + #2 (t)] ∈ L2 ∩ L∞ ; ˙(t) ∈ L2 ∩ L∞ , and $(t) ˙ ∈ L2 ∩ L∞ . Based on this, closed-loop stability and asymptotic tracking can be proved as the following result.
Since Wm (s) is stable, we have |wm (t − +)| 6 ,e−-(t−+) for some - ¿ 0; , ¿ 0, so that Tm0 ∗ | p1 (t)| 6 ,|$∗ |e−-t e-+ |( (m − ∗ (+))T !(+)| d+: 0)
Theorem 3.1. Adaptive controller (3:1); with the adaptive law (3:8) and (3:9); applied to system (2:1) with actuator failures (2:2); guarantees that all closed-loop signals are bounded and the tracking error e(t) = y(t) − ym (t) goes to zero as t goes to in9nity.
Ti
Therefore, for t ¿ Tm0 , we have Tm 0 ∗ ∗ p1 (t) = $ wm (t − +)( (m − 0) 0
∗
0
(3.19) ∗
∗ m0
∗
As (t) = is constant for t ¿ Tm0 ; (t) is piecewise constant in (0; Tm0 ); Tm0 is 4nite and !(t) is bounded in (0; Tm0 ), there exist a constant a1 ¿ 0 such that Tm0 ∗ e-+ |( (m − ∗ (+))T !(+)| d+ 6 a1 (3.20) 0) 0
so that | p1 (t)| 6 a1 ,|$∗ |e−-t : Similarly, we have Tm 0 T p2 (t) = wm (t − +)($∗ (+) − $∗(m0 ) ) ˜ (+)!(+) d+ 0
−-t
6 ,e
Tm0
0
!(+)| d+:
T
e-+ |($∗(m0 ) − $∗ (+)) ˜ (+)
4. Simulation study (3.21)
As $∗ (t) = $∗(m0 ) is constant for t ¿ Tm0 ; $∗ (t) is piecewise T constant in (0; Tm0 ); Tm0 is 4nite and ˜ (t)!(t) is bounded in (0; Tm0 ), there exists a constant a2 such that Tm 0 T (3.22) e-+ |($∗(m0 ) − $∗ (+)) ˜ (+)!(+)| d+ 6 a2 0
Proof (outline). The standard properties of the adaptive laws: (t); $(t) ∈ L∞ ; [ (t)= 1 + "T (t)"(t) + #2 (t)] ∈ L2 ∩ L∞ , and ˙(t) ∈ L2 ∩ L∞ , ensure that the closed-loop system has a small loop gain so that it is stable in the sense that all signals are bounded (Tao & KokotoviUc, 1996, Appendix A). Then, it follows that (t) ∈ L2 ∩ L∞ , and that #(t) ∈ L2 ∩ L∞ as ˙(t) ∈ L2 ∩ L∞ , so that e(t) = (t) − $(t)#(t) ∈ L2 ∩ L∞ . Finally, it follows from (3.4) that e(t) ˙ is bounded so that limt→∞ e(t) = 0 as from Barbalat’s lemma. For details of the proof, see Tao, Chen, and Joshi (2001a).
As an illustrative example, we use the linearized lateral dynamic model of a Boeing 747 airplane (Franklin, Powell, & Emami-Naeini, 1994, p. 686) as the controlled plant, with two augmented actuation vectors b2 u2 and b3 u3 representing two additional pieces to augment a rudder for the study of actuator failure compensation, to which the adaptive control scheme consisting of (3.1), (3.5) – (3.9) is applied.
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Fig. 1. Tracking error and control input for r(t) = 0:025.
The plant with augmented actuation vectors is described by x(t) ˙ = Ax(t) + Bu(t); y(t) = Cx(t); −0:0558 −0:9968 0:0802 0:598 −0:115 −0:0318 A= −3:05 0:388 −0:465 0 0:0805 1 B = [b1 ; b2 ; b3 ]; 0:00729 −0:475 b1 = 0:153 ; 0 0:005 −0:3 b3 = 0:1 ; 0
0:0415 0 ; 0 0
C = [0 1 0 0]: 0:01 −0:5 b2 = 0:2 ; 0
where x(t) = [,; r; p; 0]T ; , is the side-slip angle, r is the yaw rate, p is the roll rate, 0 is the roll angle, y is the plant output which is the yaw rate in this case, and u is the control input vector which contains three control signals u=[u1 ; u2 ; u3 ]T to represent three rudder servos: 1r1 , 1r2 ; 1r3 , from a three-piece rudder for achieving compensation in the presence of actuator failures. In this study, we simulate the case of two actuator failures and consider the failure pattern u2 (t) = 0:02 rad; t ¿ 50, and u3 (t) = −0:03 rad; t ¿ 100. The simulation parameters
are: & = 10I; ( = 1; Wm = 1=(s + 3); ym (0) = 0; y(0) = −0:02; (0) = 0:5 ∗ ( ∗ is the true matching parameter vector when there is no actuator failure), $(0) = 0. The simulation results, including the plant output y(t), reference output ym (t), tracking error e(t) = y(t) − ym (t), and control input v1 (t) = v2 (t) = v3 (t), are shown in Fig. 1 for the reference input r(t) = 0:025, and in Fig. 2 for r(t) = 0:02 sin(0:1t). The system responses are as expected, which indicate that at the time instant when one of the actuators fails, there is a transient response in the tracking errors. As the time goes on, the tracking errors become smaller. The controller parameters k11 ; k21 , and k1 (not shown) also change their values at the time instants when actuator failures occur. All signals in the adaptive control system are bounded, and stability and output tracking are ensured. Similar results were also observed in the simulations for the failure pattern u2 (t)=u2 (50); t ¿ 50, and u3 (t)=u3 (100); t ¿ 100, which means the actuators were stuck at the positions where they failed.
5. Conclusions We have developed an adaptive control scheme for systems with unknown actuator failures. This scheme, when implemented with true parameters, ensures desired plant-model matching in the presence of actuator failures, and, when implemented with adaptive parameter estimates, ensures asymptotic output tracking in the presence of unknown plant parameters and unknown actuator failure parameters.
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Fig. 2. Tracking error and control input for r(t) = 0:02 sin(0:1t).
The design conditions on the plant matrices (A; B) are relaxed as compared with a state tracking design, so that adaptive actuator failure compensation is applicable to a larger class of systems. Simulation results veri4ed the desired performance of the developed adaptive actuator failure compensation design. Adaptive failure compensation using output feedback is currently under extensive investigation. Acknowledgements This research was partially supported by NASA Langley Research Center under grant NCC-1-342. Related discussion with Xiaoli Ma and Xidong Tang was bene4cial to this work. References Ahmed-Zaid, F., Ioannou, P., Gousman, K., & Rooney, R. (1991). Accommodation of failures in the F-16 aircraft using adaptive control. IEEE Control Systems Magazine, 11(1), 73–78. Bodson, M., & Groszkiewicz, J. E. (1997). Multivariable adaptive algorithms for recon4gurable :ight control. IEEE Transactions on Control Systems Technology, 5(2), 217–229. Boskovic, J. D., Yu, S. -H., & Mehra, R. K. (1998). A stable scheme for automatic control recon4guration in the presence of actuator failures. Proceedings of the 1998 ACC (pp. 2455 –2459). Philadelphia, Pennsylvania. Boskovic, J. D., & Mehra, R. K. (1998). A multiple model-based recon4gurable :ight control system design. Proceedings of the 37th IEEE Conference on Decision and Control (pp. 4503– 4508). Tampa, Florida. Boskovic, J. D., & Mehra, R. K., (1999). Stable multiple model adaptive :ight control for accommodation of a large class of control e?ector failures. Proceedings of the 1999 ACC (pp. 1920 –1924). San Diego, California.
Franklin, G. F., Powell, J. D., & Emami-Naeini, A. (1994). Feedback control of dynamic systems (3rd ed). Reading, MA: Addison-Wesley. Tao, G., Chen, S. H., & Joshi, S. M. (2001a). An adaptive control scheme for systems with unknown actuator failures. Technical Report UVA-ECE-ASC-01-03-0, Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA, March 2001. Tao, G., Joshi, S. M., & Ma, X. L. (2001b). Adaptive state feedback and tracking control of systems with actuator failures. IEEE Transactions on Automatic Control, 46(1), 78–95. Tao, G., & KokotoviUc, P. V. (1996). Adaptive control of systems with actuator and sensor nonlinearities. New York: Wiley. Tao, G., Tang, X. T., & Joshi, S. M. (2001c). Output tracking actuator failure compensation control. Proceedings of the 2001 ACC (pp. 1821–1826). Arlington, Virginia. Tsakalis, K. S. (1992). Model reference adaptive control of linear time-varying plants: the case of “jump” parameter variations. International Journal of Control, 56(6), 1299–1345. Vemuri, A. T., & Polycarpou, M. M. (1997). Robust nonlinear fault diagnosis in input–output systems. International Journal of Control, 68(2), 343–360. Wang, H., Huang, Z. J., & Daley, S. (1997). On the use of adaptive updating rules for actuator and sensor fault diagnosis. Automatica, 33(2), 217–225. Gang Tao received his Ph.D. degree in Electrical Engineering in 1989, from University of Southern California. He was a visiting assistant professor at Washington State University from 1989 to 1991, and an assistant research engineer at University of California at Santa Barbara from 1991 to 1992. He joined Department of Electrical Engineering at University of Virginia in 1992, where he is now an associate professor. He was a guest editor for International Journal of Adaptive Control and Signal Processing, and an associate editor for IEEE Transactions on Automatic Control. He was a program committee member for numerous international conferences, and was the organizer and chair of 2001 International Symposium on Adaptive and Intelligent Systems and
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Control, held in Charlottesville, Virginia, USA. He co-edited one book, authored or co-authored one book, over 45 journal papers and 5 book chapters, and 115 conference papers=presentations on adaptive control, nonlinear control, multivariable control, optimal control, control applications and robotics.
Shuhao Chen received his B.S. degree in automatic control from Tsinghua University, Beijing, China, and his M.S. degree in industrial automation from Xi’an Jiaotong University, Xi’an, China, in 1993 and 1998, respectively. He is now working toward his Ph.D. degree at the University of Virginia. He was an engineer at the Automation Research Institute of the Ministry of Metallurgical Industry, Beijing, China, from 1993 to 1999. His main research interest is adaptive control of systems with actuator failures, for aircraft and industrial applications.
Suresh M. Joshi received his Ph.D. in electrical engineering from Rensselaer Polytechnic Institute, Troy, NY, in 1973. He is Senior Scientist for Control Theory at NASA-Langley Research Center in Hampton, Virginia. His research interests include multivariable robust control, adaptive control, nonlinear systems, and applications to advanced aircraft and spacecraft.Dr. Joshi is a Fellow of the IEEE, the AIAA, and the ASME. He served on numerous editorial boards, technical committees, and organizing committees, including the IEEE-Control Systems Society’s Board of Governors (1989 –94). His publications include several articles and two books, “Control of Large Flexible Space Structures” (Berlin: Springer-Verlag, 1989) and “Control of Nonlinear Multibody Flexible Space Structures” (London: Springer-Verlag, 1996). He is the recipient of the IEEE Control Systems Technology Award, as well as a number of awards from NASA-Langley Research Center. He is also an amateur cartoonist and contributed the “Out of Control” cartoons to the IEEE Control Systems Magazine from 1985 until 1993.