Multivariable State Feedback for Output Tracking. MRAC for Piecewise Linear Systems with Relaxed Design Conditions. Qian Sang, Gang Tao, and Jiaxing Guo.
2014 American Control Conference (ACC) June 4-6, 2014. Portland, Oregon, USA
Multivariable State Feedback for Output Tracking MRAC for Piecewise Linear Systems with Relaxed Design Conditions Qian Sang, Gang Tao, and Jiaxing Guo Department of Electrical and Computer Engineering University of Virginia Charlottesville, VA 22904-4743 Abstract—The adaptive state feedback control problem for multivariable piecewise linear systems with switched parameters is considered in this paper. A direct state feedback model reference adaptive control (MRAC) scheme is developed based on high frequency gain matrix decompositions. This scheme relaxes a key design condition, namely the high frequency gain matrices sharing a common L matrix in their LDS decompositions, required by an existing adaptive control scheme. Under the usual slow system parameter switching condition, closed-loop stability (signal boundedness) and small output tracking error in the mean square sense are achieved. The proposed scheme is simulated on linearized NASA GTM models to demonstrate its effectiveness in tracking performance improvement.
I. I NTRODUCTION Adaptive control of multi-input, multi-output (MIMO) linear time-invariant (LTI) systems have been approached by many researchers with different methods [1], [6], [7], [14], [17]. The effort was extended to expand the capabilities of the adaptive control approach to linear time-varying (LTV) systems (e.g., [15], [16]). Smoothness and slowness of system parameter variations are typical requirements of such designs, which limits their applicability to a wide class of systems, the socalled piecewise linear systems. A piecewise linear system is a dynamical system with dynamics switching among a set of continuous-time subsystems according to certain switching criteria. A nonlinear system may be represented as a global piecewise linear system, as an extension of local linearization based approximations, for wider ranges of system operating conditions and suitable for more applications [9]. Additionally, piecewise linear system models find applications in many practical systems of a hybrid nature. Despite the recent growth of research interest in stability analysis and control design of such systems (see [8] and the references therein), adaptive control of piecewise linear systems is still open for research. An adaptive scheme was proposed in [2] for piecewise affine systems in controllable canonical form based on the so-called minimal control synthesis algorithms. Such a design is extended to the discretetime case in [3]. Piecewise linear systems with time dependent subsystem switches and dynamic offsets were approached in [10]–[13] by directly expanding the capacity of conventional MRAC schemes. It has been shown that under the usual slow subsystem switching condition, closed-loop signal bounded978-1-4799-3274-0/$31.00 ©2014 AACC
ness and small tracking error in the mean square sense are achieved. With the existence of a common Lyapunov function, asymptotic tracking performance is ensured. Compared with the state tracking designs [10], [11], state feedback for output tracking MRAC, to be considered in this paper, is of interest due to a simpler controller structure and arelaxed plant-model matching condition [4]. For the multivariable state feedback for output tracking MRAC design presented in [9], a key design condition is the assumption of a common L matrix shared by the high frequency gain matrices of the subsystems in their LDS decompositions. Such an assumption may not be satisfied without further structural knowledge of the high frequency gain matrices. It is the main purpose of this paper to present a new multivariable state feedback MRAC design to relax such an otherwise restrictive condition. The contributions of the current paper are summarized as follows: (i) A key design condition in [9] is relaxed. In particular, the Ls matrices in the LDS decompositions of the high frequency gain matrices can be different. (ii) A new estimation error model is derived based on the relaxed design conditions. (iii) New adaptive laws are proposed for updating controller parameter estimates. The reminder of the paper is organized as follows. The formulation of the adaptive control problem is presented in section II. In section III, the non-adaptive model reference control problem is considered, and an MRAC design is proposed in Section IV. An illustrative example is provided in Section V, and concluding remarks are given in Section VI. II. P ROBLEM S TATEMENT Consider the following piecewise linear system ˙ x(t) = A(t)x(t) + B(t)u(t), y(t) = C T x(t)
(1)
with x(t0 ) = x0 , where x(t) ∈ Rn is the state, u(t) ∈ RM and y(t) ∈ RM are the system input and output, respectively. During different time periods, (A(t), B(t)) takes on different values as specified by the parameter matrix sets (Ai , Bi ), i ∈ I , {1, 2, . . . , l}, where Ai and Bi are unknown but constant parameter matrices representing the ith subsystem, and l is the total number of subsystems.
703
Indicator functions χi (t), i ∈ I, assumed to be known, are introduced to characterize system parameter discontinuities: 1, if (A(t), B(t)) = (Ai , Bi ), χi (t) = (2) 0, otherwise. Pl It follows that i=1 χi (t) = 1, χj (t)χk (t) = 0, j 6= k, and the P time-varying parameter matrices Pl can be expressed as l A(t) = i=1 Ai χi (t) and B(t) = i=1 Bi χi (t). Control objective. The control objective is to develop a state feedback control law u(t) for the piecewise linear system (1) such that the closed-loop system is stable, and the output y(t) tracks a reference signal ym (t) as close as possible, with ym (t) generated from a reference model system −1 ym (t) = Wm (s)[r](t), Wm (s) = ξm (s),
−0.6137 −0.6630 0.2412 0 0 −0.6870 −0.7292 0.2736 0 0
0.0959 −6.7565 −0.3162 1.0000 0 8.0057 −7.5625 −0.3078 1.0000 0
−0.0274 −0.9009 B1 = −0.0252 0 0 −0.0396 −1.1653 B2 = −0.0313 0 0
0.4892 0.2963 −0.4683 0 0 0.6160 0.3827 −0.5953 0 0
A1 =
(3)
where ξm (s) is a common modified left interactor matrix for the transfer matrix of each subsystem, i.e., Gi (s) = C T (sI − Ai )−1 Bi , of the piecewise linear system (1), and r(t) ∈ RM is a bounded, piecewise continuous reference input signal. Assumptions. For adaptive control design, the following assumptions are made for i ∈ I: (A1) (C T , Ai , Bi ) is stabilizable and detectible; (A2) All zeros of Gi (s) are stable with their real parts less than −δ for some known δ > 0; (A3) Gi (s) is strictly proper, has full rank and its modified left interactor matrix ξm (s) is known; (A4) All leading principal minors ∆ij , j = 1, 2, . . . , M , of the high frequency gain matrix, defined as Kpi = lims→∞ ξm (s)Gi (s), are nonzero with known signs. Remark 1: The knowledge of δ in Assumption (A2) is for the design of a suitable dynamic normalizing signal for robust parameter update laws, to ensure the desired closedloop stability properties in the presence of piecewise constant (jumping) parameters [16]. Equivalent to Assumption (A4), we may assume: (A4a) The LDS decomposition of Kpi [14] is such that Kpi = Lsi Dsi Si ,
straight, wings-level flight conditions at 80 knots and 90 knots at 800 ft., respectively, to obtain a piecewise linear lateral system model as in (1), where x = [v, p, r, φ, ψ]T with the elements being the perturbed aircraft velocity component along the y-body-axis (fps), angular velocity along the xand z-body-axis (rad/s), roll angle (rad), and yaw angle (rad), respectively. The outputs are chosen as y = [v, ψ]T , while the control inputs are the perturbed aileron deflection δa and rudder deflection δr , i.e., u = [δa , δr ]T , and the nominal parameter matrices are
(4)
where Lsi ∈ RM ×M is unity lower triangular, Si ∈ RM ×M is symmetric, positive definite, and Dsi = diag{sign[∆i1 ]γi1 , . . . , sign[∆iM /∆iM −1 ]γiM } with γij > 0, j = 1, 2, . . . , M . This assumption will be used to specify the conditions for error system parametrization and adaptive law design in Section IV, when the controlled system is unknown. A benchmark piecewise linear aircraft model. As an illustration of the design conditions, we consider a piecewise linear model of the lateral dynamics of the NASA generic transport model (GTM). It is obtained by trimming and subsequently linearizing the NASA GTM at multiple operating points. The same model will also be used to simulate the proposed adaptive control design in Section V. We choose the number of operating points as l = 2 for simplicity of presentation, and trim the GTM at steady-state,
A2 =
−134.5378 1.8813 −1.4992 0.0691 1.0000 −151.5286 1.8623 −1.6865 0.0513 1.0000
32.0974 0 0 0.0002 0 32.1318 0 0 0.0001 0
0 0 0 0 0 0 0 0 0 0
, ,
,
, C =
1 0 0 0 0
0 0 0 0 1
.
For this piecewise linear system model, we have M = 2 and n = 5, and the transfer matrices for the two subsystem models can be obtained. The modified left interactor matrix can be chosen as ξm (s) = diag{s + 1, (s + 1)2 }, based on which we can compute the system high frequency gain matrices to be Kp1 =
−0.0274 −2.5200
0.4892 −46.8300
, Kp2 =
−0.0396 −3.1300
0.6160 −59.5300
.
The sign information of their leading principal minors are as follows: sign[∆11 ] = −1, sign[∆12 ] = 1, sign[∆21 ] = −1, sign[∆22 ] = 1. It can be verified that all design conditions (Assumptions (A1)–(A4a)) are satisfied. In particular, with Ds1 = Ds2 = −10I, we have h i h i 1 0 1 0 Ls1 = 18.7737 1 , Ls2 = 16.3460 1 . Remark 2: The above choice of Dsi , i = 1, 2, is in contrast to that in [9], which is made specifically to satisfy a key design condition: Ls1 = Ls2 . Since in this case, Ls1 6= Ls2 , the design in [9] is not applicable here. Note that in general it is not straightforward (and possibly restrictive) to specify Dsi for Lsi to be the same to each other. It is important that the common Ls assumption in [9] be relaxed for the development of an adaptive control design applicable to a wider class of piecewise linear systems. III. N OMINAL C ONTROL S CHEME A nominal controller is an ideal design based on the exact knowledge of the controlled plant, which has the desired features (structure and parameters) to ensure the desired system performance.
704
A. Nominal Controller Structure If the system parameter matrices Ai and Bi , i ∈ I, are known, the following state feedback control law can be applied u(t) = Kx∗T (t)x(t) + Kr∗ (t)r(t)
(5)
the controller parameter P matrices Kx∗ (t) = l ∗ ∗ ∗ χ (t), where χ (t), K (t) = K K i i r ri xi i=1 i=1 ∗ ∗ Kxi ∈ Rn×M and Kri ∈ RM ×M are computed from the piecewise plant-model matching condition:
with Pl
∗−1 ∗T ∗ C T (sI −Ai −Bi Kxi )Bi Kri = Wm (s), Kri = Kpi . (6)
When the ith subsystem is active (as indicated by χi (t) = 1), the controller (5) leads to an input/output transfer matrix matching of the closed-loop system (C T , Ami , Bmi ) with ∗T ∗ Ami , Ai + Bi Kxi , Bmi , Bi Kri
Let (C T , Am(k−1) , Bm(k−1) ) ∈ {(C T , Ami , Bmi )}li=1 denote the active subsystem over [tk−1 , tk ), k ∈ Z+ , Φ(t, τ ) the state transition matrix associated with (10), and T (Cm , Am , Bm ) a state space realization of the reference model system (3) with the associated state transition matrix Φm (t, τ ). For the nominal closed-loop system, we have the following results: Theorem 1 [9]. All signals in the nominal closed-loop system are bounded if the minimum switching time interval satisfies (12). The tracking error e(t) = y(t)−ym (t) is e(t) = η0 (t)+ ε0 (t) with ε0 (t) an initial condition related term T ε0 (t) = C T Φ(t, t0 )x(t0 ) − Cm Φm (t, t0 )xm (t0 ),
which is exponentially decaying, and for t ∈ [tk−1 , tk ),
(7)
η0 (t) =
−1
C (sI − Ami )
x˙ =
Bmi = Wm (s).
(Ami χi x + Bmi χi r) , y = C T x.
(8)
(9)
i=1
B. Stability Properties Let the increasing sequence {ti }∞ i=1 denote the time instants at which subsystem switches occur. It is well known [5] that the exponential stability of the homogeneous system of (9): ˙ z(t) = Am (t)z(t),
(10)
Pl
with Am (t) = i=1 Ami χi (t) and Ami stable is sufficient for the stability of (9). Let T0 denote the minimum switching time interval, i.e., T0 = mink∈Z+ {tk −tk−1 }, where Z+ stands for all positive integers, and Pmi , Qmi ∈ Rn×n be symmetric, positive definite satisfying AT mi Pmi
+ Pmi Ami = −Qmi , i ∈ I.
(11)
Due to the stability of Ami , there exist ami , λmi > 0 such that keAmi t k ≤ ami e−λmi t . Define am = maxi∈I ami , λm = mini∈I λmi , α = maxi∈I λmax [Pmi ], and β = mini∈I λmin [Pmi ], where λmin [·] and λmax [·] denote the minimum and maximum eigenvalues of a matrix. The following lemma gives a lower bound on T0 that ensures exponential stability of (10), thus the stability of (9): Lemma 1 [9]. The homogeneous system (10) is exponentially stable with decay rate σ ∈ (0, 1/2α) if the minimum switching time interval T0 satisfies
h
i T C T Φ(t, τ )Bm(j−2) − Cm Φm (t, τ )Bm r(τ )dτ
tj−2
a2m
which is small in the mean square sense, under the slow switching condition (12). Furthermore, there exist positive constants C1 and K1 , such that e(t) satisfies Z t+T ke(τ )k2 dτ ≤ C1 + K1 nT , ∀t ≥ t0 , ∀T ≥ 0, (15) t
where nT is the number of subsystem switches over [t, t + T ]. Note that as a part of the tracking error e(t), η0 (t) in (14) captures the deviations of y(t) from ym (t) caused by subsystem switches. IV. A DAPTIVE C ONTROL D ESIGN In this section, we consider the case when the plant parameter matrices Ai and Bi (i ∈ I) are unknown. In this case, the nominal controller (5) cannot be implemented as its parameters are unknown, and an adaptive control scheme is proposed to handle the parameter uncertainties. A. Controller Structure The adaptive version of (5) is applied u(t) = KxT (t)x(t) + Kr (t)r(t),
(16)
×M where KxP (t) ∈ Rn×M and Kr (t) ∈ RM P are defined as l l Kx (t) = i=1 Kxi (t)χi (t) and Kr (t) = i=1 Kri (t)χi (t). The parameter matrices Kxi (t) ∈ Rn×M and Kri (t) ∈ ∗ ∗ RM ×M are the adaptive estimates of Kxi and Kri , defined to satisfy the matching condition (6), and are updated from some adaptive laws to be developed as follows. By applying the adaptive control law (16) to the plant (1), ˜ x (t) = Kx (t) − K ∗ (t), K ˜ r (t) = Kr (t) − and defining K x ˜ xi (t) = Kxi (t) − K ∗ , K ˜ ri (t) = Kri (t) − K ∗ , we Kr∗ (t), K xi ri obtain the closed-loop system
x˙ =
α ln(1 + µ∆Am ), µ = maxkPmi k, (12) 1 − 2σα λm β i∈I where ∆Am stands for the largest difference between any two subsystem matrices of Am (t), i.e., ∆Am = maxi,j∈I kAmi − Amj k. T0 ≥
tj−1
(14)
∗ ∗ The existence of Kxi , Kri is guaranteed by Assumption (A1)–(A2) [4]. Furthermore, the matrices Ami satisfying the matching equation (8) are all stable. With (5) in (1), the closed-loop system becomes l X
k Z X j=2
to the reference model system, i.e., T
(13)
705
l X
(Ami χi x + Bmi χi r) +
i=1
l X i=1
∗−1 ˜ T χi x + K ˜ ri χi r Bmi Kri K xi (17)
In view of (3) and (6), the tracking error equation is " l # X ˜ T χi ω + η + η0 + ε0 e = Wm (s) Kpi Θ (18) i i=1
˜ i (t) = Θi (t) − Θ∗ (t), Θi (t) = [K T (t), Kri (t)]T , with Θ i xi ∗ T ∗T ∗ ] , ω(t) = [xT (t), r(t)]T , where , Kri Θi = [Kxi η0 (t) and ε0 (t) are as in (14) and (13), respectively, and = R tj−1is Tin the same form T of η0 (t): η(t) Pk η(t) C Φ(t, τ )B − C Φ (t, τ )B ∆(τ )dτ m m m(j−2) m j=2 tj−2 ˜ (j−2) (τ )ω(τ ) with (j − 2) in the where ∆(τ ) = Kp(j−2) Θ subscript denoting the active subsystem over [tj−2 , tj−1 ). Remark 3: Similar to η0 (t), η(t) captures the unparameterized part of the output tracking error e(t) due to controller parameter errors, in the presence of subsystem switches. They are treated as perturbations to the parameterizable part of e(t) in subsequent analysis. A property of η(t) is that with the controller parameter estimates generated by parameter projection adaptive laws (developed next), a normalized version of η(t) is small in the mean square sense, which is critical to closed-loop system stability analysis. B. Error Model We now proceed to derive an estimation error model from the tracking error equation (18), for the development of adaptive laws. From (3) and (18), it follows that ξm (s)[e] =
l X
is a sum of exponentially decaying terms starting at the switching time instants and with initial values being those of e(t) and its up to (nm −1)th derivatives at those time instants. It can also be verified that l X i=1
=
Example. We consider h(s) = 1/(s + 1)2 with one indicator function χ(t) which has a switch at t = t1 from 0 to 1, for the scalar case with e(t) ∈ R. We simply examine η1 (t) = h(s)[χξm (s)[e]](t) − h(s)[ξm (s)[χe]](t). For ξm (s) = a2 s2 + a1 s + a0 , we have
˜ T χi ω + ξm (s)[η + η0 + 0 ]. Kpi Θ i
˜ T χi ω+ξm (s)[η+η0 +0 ]. (19) Lsi Dsi Si Θ i
h(s)[ξm (s)[χe]] =h(s)[a2 (s[χe] ˙ + χ˙ e˙ + χ¨ e) + a1 (χe ˙ + χe) ˙ + a0 χe] =h(s)[a2 (s[χe] ˙ + χ˙ e) ˙ + a1 χe] ˙ + h(s)[χ(a2 e¨ + a1 e˙ + a0 e)].
Since h(s)[χ(a2 e¨ + a1 e˙ + a0 e)](t) = h(s)[χξm (s)[e]](t), we have
i=1
Multiplying both sides of this equation by the indicator function χi (t) and using the properties of the indicator functions, we obtain ˜ T χi ω + χi ξm (s)[η + η0 + 0 ]. χi ξm (s)[e] = Lsi Dsi Si Θ i By multiplying both sides of this equation by L−1 si and conducting a summation from i = 1 to i = l, we have l X
L−1 si χi ξm (s)[e] =
l X
+
l X
η1 (t) = −h(s)[a2 (s[χe] ˙ + χ˙ e) ˙ + a1 χe](t). ˙
It can be verified that h(s)[s[χe]](t) ˙ and h(s)[χe](t) ˙ are proportional to e−(t−t1 ) e(t1 ) for t > t1 , and h(s)[χ˙ e](t) ˙ is proportional to e−(t−t1 ) e(t ˙ 1 ) for t > t1 , e.g., Z t e−(t−τ ) χ(τ ˙ )e(τ )dτ = e−(t−t1 ) e(t1 ), 0
L−1 si χi ξm (s)[η + η0 + 0 ].
For h(s) = 1/f (s) with a stable polynomial f (s) of the same degree nm as that of ξm (s), filtering both sides of the above equation by h(s) leads to L−1 si h(s)[ξm (s)[χi e]] + η1 =
i=1
l X
˜ T χi ω] Dsi Si h(s)[Θ i
i=1
+
l X
−1 Lsi h(s)[χi ξm (s)[η + η0 + 0 ]]
l X
i=1
L−1 si h(s)[χi ξm (s)[e]] −
=
l X
˜ T χi ω] (22) Dsi Si h(s)[Θ i
i=1
as the basic equation (which contains L−1 si , i = 1, 2, . . . , l) for the estimation error derivation and for the adaptive law design Based on (22), we write (20) as l X
(20)
i=1
η1 =
L−1 si h(s)[ξm (s)[χi e]]
i=1
∗T ∗T ∗T [0, θ(i)2 η(i)2 , θ(i)3 η(i)3 . . . , θ(i)M η(i)M ]T + e¯(i)
i=1
where the “swapping term” l X
∇
The effect of η1 (t) and η2 (t) is small if the switching time intervals are large. This analysis implies that based on (20) we can consider
i=1
l X
(21)
for χ(t) ˙ being a δ-function centered at t = t1 .
˜ T χi ω Dsi Si Θ i
i=1
i=1
L−1 si h(s)ξm (s)[χi (η + η0 + 0 )] + η2
for some η2 (t) has some similar properties as those of Pwhich l η1 (t), where i=1 L−1 si h(s)ξm (s)[χi (η+η0 +0 )](t) is welldefined with a proper transfer matrix h(s)ξm (s). To illustrate the properties of η1 (t) (and similarly for η2 (t)), we consider the following example.
i=1
l X
l X i=1
Under Assumption (A4a) with Kpi = Lsi Dsi Si and different Lsi in their LDS decompositions, we have ξm (s)[e] =
L−1 si h(s)[χi ξm (s)[η + η0 + 0 ]]
l X
= L−1 si h(s)[ξm (s)[χi e]]
l X i=1
i=1
706
˜ T χi ω] + d Dsi Si h(s)[Θ i (23)
∗ where θ(i)j ∈ Rj−1 , j = 2, . . . , M , denotes the jth row elements (non-zero part) of Θ∗(i)0 = L−1 si − I, and
e¯(i) = ξm (s)h(s)[χi e] = [¯ e(i)1 , e¯(i)2 , . . . , e¯(i)M ]T , η(i)j = [¯ e(i)1 , e¯(i)2 , . . . , e¯(i)j−1 ]T ∈ Rj−1 , d=
l X
−1 Lsi h(s)[χi ξm (s)[η + η0 + 0 ]] − η1 .
updated by the adaptive laws (26)–(28), there exists T0∗ > 0 such that for T0 = mink∈Z + {tk − tk−1 } ≥ T0∗ , all signals in the closed-loop system are bounded, and for some positive constants C2 and K2 , the output tracking error e(t) = y(t) − ym (t) satisfies Z t+T ke(τ )k2 dτ ≤ C2 + K2 nT , ∀t ≥ t0 , ∀T ≥ 0, (29) t
i=1
Define the estimation error signal =
l X
T T T [0, θ(i)2 η(i)2 , θ(i)3 η(i)3 . . . , θ(i)M η(i)M ]T + e¯(i) + Ψi ξi
i=1
(24)
∗ where θ(i)j , j = 2, 3, . . . , M , is the estimate of θ(i)j , Ψi (t) ∗ T is the estimate of Ψi = Dsi Si , ξi (t) = Θi (t)ζi (t) − h(s)[ΘT i χi ω](t), and ζi (t) = h(s)[χi ω](t). Based on (23) and (24), we can obtain the error model: l X T T T = [0, θ˜(i)2 η(i)2 , θ˜(i)3 η(i)3 , . . . , θ˜(i)M η(i)M ]T i=1
+
l X
(25) ˜ i ξi + Dsi Si Θ ˜ T ζi + d, Ψ i
i=1 ∗ ˜ i (t) = Ψi (t) − Ψ∗ (t). with θ˜(i)j (t) = θ(i)j (t) − θ(i)j and Ψ i Similarly, the error term d(t) is the unparameterized part of (t), due to subsystem switches and initial conditions.
C. Adaptive Laws and Stability Properties For the estimation error model (25) with the estimation error (t) in (24), we propose the following gradient adaptive laws to update θ(i)j (t), Θi (t), and Ψi (t), i ∈ I, j = 2, 3, . . . , M : Γθ(i)j η(i)j (t)j (t) + f(i)j (t) m2 (t) T ˙ T (t) = − Dsi (t)ζi (t) + Fi (t) Θ i 2 m (t) T ˙ i (t) = − Γi (t)ξi (t) + Hi (t), Ψ 2 m (t)
θ˙ (i)j (t) = −
(26) (27) (28)
where (t) = [1 (t), 2 (t), . . . , M (t)]T , fj (t), Fi (t), and Hi (t) are the parameter projection terms so designed as to confine the parameter estimates within their respective known bounds at all time, assuming certain knowledge of the nominal controller parameter bounds. The initial parameter estimates are chosen to be within these bounds. The adaptation gain matrices Γθ(i)j and Γi are positive definite and diagonal, and the normalizing signal is m2 (t) = 1 + ms (t) with ms (t) also generated from 2
2
m ˙ s (t) = −2δ0 ms (t)+ku(t)k +ky(t)k , ms (0) = 0, δ0 < δ where δ0 < δ for δ in Assumption (A2). For the developed adaptive control scheme, we have the following result: Theorem 2. For the closed-loop system consisting of the plant (1), the reference model (3), and the adaptive controller (16)
where nT is the number of subsystem switches over [t, t + T ]. For closed-loop stability, we first note that the parameter estimates from the adaptive laws of have the desired properties that they are bounded and within their respective parameter bounds. By considering the positive definite function l M 1 X X ˜T −1 ˜ −1 ˜ TΓ Ψ ˜ i ] + tr[Θ ˜ i Si Θ ˜ T ] θ Γ V = θj + tr[Ψ i i i 2 i=1 j=2 j θ(i)j and its time derivative along the adaptive laws (26)–(28), toT ˜ T −1 gether with the fact that θ˜(i)j Γ−1 θ(i)j f(i)j ≤ 0, tr[Ψi Γi Hi ] ≤ ˜ i Si Fi ] ≤ 0, we can obtain that for z(t) = 0, and tr[Θ (t) ˙ ˙ i (t), Ψ ˙ i (t), and some c, k > 0 , θ (t), Θ (i)j m(t) Z t+T Z t+T kd(τ )k2 dτ, ∀t ≥ t0 , ∀T ≥ 0. kz(τ )k2 dτ ≤ c + k m2 (τ ) t t Then, closed-loop signal boundedness can be proved by deriving a feedback structure in terms of some instrumental signals, for the control system whose loop gain is inversely proportional to the minimum switching interval of the piecewise linear system. Under sufficiently slow subsystem switches (T0 ≥ T0∗ for some T0∗ > 0), the boundedness of those instrumental signals can be concluded, which in turn implies the closed-loop signal boundedness. The mean square tracking performance can be obtained by first dividing the integration time interval [t, t + T ] into corresponding switching time intervals. The integral of ke(t)k2 /m2 (t) over each interval is upper bounded by a sum ˙ i (t)k, and kΨ ˙ i (t)k. of those of k(t)k2 /m2 (t), kθ˙ j (t)k, kΘ With their mean square properties, it can be shown that t+T
Z
ke(τ )k2 dτ ≤ c2 + c3
t
t+T
Z
kd(τ )k2 dτ
(30)
t
for some c2 , c3 > 0, from which the tracking performance (29) can be established. Remark 4: If the closed-loop system ceases to switch after some time, η(t) and η0 (t) will decay to zero exponentially. We can then conclude from (29) that limt→∞ e(t) = 0 and e(t) ∈ L2 . V. I LLUSTRATIVE E XAMPLES The proposed adaptive control design was simulated on the piecewise linear system model of the NASA GTM, presented at the end of Section II. With the interactor matrix ξm (s) = diag{s + 1, (s + 1)2 } and the system high frequency gain matrices Kpi in (5), the reference model system is specified by (3), and plant model matching condition (6) is satisfied.
707
We choose the initial system state as [2, 0, 0, 0, −10]T , while that for the reference model system is set to zero. The initial parameter estimates are chosen to be 70% of their nominal values. The adaptation gains are chosen as Γθ(i)2 = 10, Γi = 10I. The signal filter used in the derivation of the error model is h(s) = 1/(s + 6)2 . The simulation results presented here are for a switching time interval of T = 10 with the sinusoidal reference input signal r(t) = sin(0.014t)[1, 0.1]T , corresponding to desired fluctuations of the lateral velocity in between ±1 fps and of the yaw angle in between ±5.73◦ , despite the switches of operating points. Figure 1 shows the output tracking error e(t). For comparison, the tracking performance with the scheme proposed in [9] under the same simulation settings is presented in Figure 2. Note that, the key design condition of a common Ls matrix assumed in [9] is violated due to the (arbitrary) choice of Ds1 = Ds2 = 10I. The aileron and rudder deflections for all simulations are within their respective limits (±20◦ and ±30◦ ), and are not shown here due to space limitation.
tainties and the presence of repetitive subsystem switches, the proposed adaptive controls scheme can make the system output to track the reference trajectories closely. Furthermore, as compared with the design in [9], it achieves a better transient tracking performance, due to an improved parameterization of the estimation error system. VI. C ONCLUSIONS The adaptive state feedback control problem for multivairiable piecewise linear systems was revisited in this paper. By employing the knowledge of the time instant indicator functions of system parameter switching, a direct model reference adaptive control scheme was developed based on the LDS decomposition of high frequency gain matrices of such systems. Compared with an existing adaptive control scheme, this scheme relaxes a key design condition, namely the high frequency gain matrices sharing a common L matrix in their LDS decompositions. A new estimation error model was derived under the relaxed design conditions. R EFERENCES
e 1 (t) = v(t) − v m (t) [fps]
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Fig. 1. Tracking error e(t) for T = 10s.
e 1 (t) = v(t) − v m (t) [fps]
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Fig. 2. Tracking error e(t) for T = 10s with the scheme in [9].
From Figure 1, we can see that despite the parametric uncer-
[1] M. de Mathelin and M. Bodson, “Multivariable model reference adaptive control without constraints on the high-frequency gain matrix,” Automatica, vol. 31, no. 4, pp. 597–604, 1995. [2] M. di Bernardo, U. Montanaro, and S. Santini, “Hybrid model reference adaptive control of piecewise affine systems,” IEEE Transactions on Automatic Control, vol. 58, no. 2, pp. 304–316, 2013. [3] ——, “Model reference adaptive control of discrete-time piecewise linear systems,” International Journal of Robust and Nonlinear Control, vol. 23, no. 7, pp. 709–730, 2013. [4] J. X. Guo, Y. Liu, and G. Tao, “Multivariable MRAC with state feedback for output tracking,” in Proceedings of the 2009 American Control Conference, St. Louis, MO, Jun. 2009, pp. 592–597. [5] J. P. Hespanha and A. S. Morse, “Stability of switched systems with average dwell-time,” University of Southern California/Lab. for Control Science & Eng., Yale University, Santa Barbara, Tech. Rep., Jan. 1999. [6] A. K. Imai, R. R. Costa, L. Hsu, G. Tao, and P. V. Kokotovic, “Multivariable adaptive control using high-frequency gain matrix factorization,” IEEE Transactions on Automatic Control, vol. 49, no. 7, pp. 1152–1156, Jul. 2004. [7] P. Ioannou and J. Sun, Robust Adaptive Control. Prentice Hall, 1996. [8] D. Liberzon, Switching in Systems and Control. Boston, MA: Birkh¨auser, 2003. [9] Q. Sang and G. Tao, “Multivariable state feedback for output tracking MRAC for piecewise linear systems,” in Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, FL, Dec. 2011. [10] ——, “Adaptive control of piecewise linear systems: the state tracking case,” IEEE Transactions on Automatic Control, vol. 57, no. 2, pp. 522– 528, 2012. [11] ——, “Adaptive control of piecewise linear systems with output feedback for output tracking,” in Proceedings of the 51st IEEE Conference on Decision and Control, Maui, HI, Dec. 2012, pp. 5422–5427. [12] ——, “Multivariable adaptive control of piecewise linear systems with applications to NASA Generic Transport Model,” AIAA Journal of Guidance, Control, and Dynamics, vol. 35, no. 5, Sep./Oct. 2012. [13] ——, “Adaptive control of piecewise linear systems with state feedback for output tracking,” Asian Journal of Control, vol. 15, no. 4, pp. 933– 943, Jul. 2013. [14] G. Tao, Adaptive Control Design and Analysis. New York: Wiley, 2003. [15] K. S. Tsakalis and P. A. Ioannou, “Adaptive control of linear timevarying plants,” Automatica, vol. 23, pp. 459–468, Jul. 1987. [16] ——, Linear Time-Varying Systems: Control and Adaptation. Englewood Cliffs, NJ: Prentice-Hall, 1993. [17] S. Weller and G. Goodwin, “Hysteresis switching adaptive control of linear multivariable systems,” IEEE Transactions on Automatic Control, vol. 39, no. 7, pp. 1360 –1375, Jul. 1994.
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