Ensemble Controllability of Time-Invariant Linear ... - Semantic Scholar
52nd IEEE Conference on Decision and Control December 10-13, 2013. Florence, Italy
Ensemble Controllability of Time-Invariant Linear Systems Ji Qi and Jr-Shin Li
Abstract— In this paper, we study the control of an ensemble of structurally similar time-invariant linear systems. In particular, we derive explicit necessary and sufficient controllability conditions for such systems in terms of the rank of the system matrices. We present examples to demonstrate these rank conditions, and construct optimal controls for steering a linear ensemble system between states of interest by using an optimization-free computational method based on the singular value decomposition. This work extends our previous results in ensemble control of time-varying linear systems, where the established controllability conditions are implicit and are defined by the singular system of the linear operator that characterizes the system dynamics.
I. INTRODUCTION Robust and sensorless manipulation of a collection of structurally similar systems with variation in common parameters, or of a single system with uncertainty in the parameters, is compelling in various areas of science and engineering. Premier examples range from the application of optimal pulses to produce a desired evolution of a large quantum ensemble in quantum control [1], [2], [3], and the use of external stimuli to desynchronize a population of neurons in the treatment of neurological disorders [4], [5], to the implementation of open-loop controls for approximate steering of robots under bounded model perturbation in robotics [6]. Such practical control designs give rise to challenging problems involving the guidance of a large number or a continuum of structurally similar dynamical systems using a common open-loop control input, which arises because measurements for the state of each individual system of the ensemble is impractical and hence state feedback is unavailable. The research in the control of ensemble systems has been active in both theoretical and computational aspects. The controllability for an ensemble of systems evolving on the Lie group SO(3) has been investigated through a conversion of the analysis to polynomial approximation [7]. The necessary and sufficient controllability characterization of an ensemble of finite-dimensional time-varying linear systems was provided in terms of the singular system of the input-to-state operator of the system [8]. Controllability for an ensemble of neuron oscillators described by nonlinear phase models was also analyzed [9]. Recently, a unified This work was supported by the NSF Career Award #0747877 and the AFOSR YIP FA9550-10-1-0146. J. Qi is with the Department of Electrical and Systems Engineering, Washington University in St. Louis, St. Louis, MO 63130, USA
[email protected] J.-S. Li is with the Faculty of Electrical and Systems Engineering, Washington University in St. Louis, St. Louis, MO 63130, USA
[email protected] 978-1-4673-5716-6/13/$31.00 ©2013 IEEE
computational method for solving optimal ensemble control problems based on multidimensional pseudospectral approximations has been developed [10] and successfully employed to design optimal pulses for protein NMR spectroscopy [2]. An optimization-free computational algorithm based on the singular value decomposition (SVD) was also established, and can be used to compute minimum-energy controls for steering linear ensemble systems [11], such as quantum transport systems [12]. Numerous work in ensemble control has also emerged in the biological domain, with the aim of understanding the coordination of the movement of flocks [13], and in controlling large-scale complex networks [14]. Although intensive work has been conducted to characterize the controllability of ensemble systems [7], [8], [9], [15], explicit controllability conditions have yet to be provided. For example, the controllability of an ensemble of finitedimensional time-varying linear systems is determined by the growth rate of the singular values of the input-to-state operator, which is intractable to verify. In this paper, we study a class of time-invariant linear ensemble systems and derive controllability conditions in terms of the rank of the system matrices. In Section II, we review some preliminary results on the ensemble control of linear systems, which motivate this work. In Section III, we construct the explicit necessary and sufficient controllability conditions for a class of timeinvariant linear ensemble systems, which are of practical importance. Examples and simulations of optimal controls for steering such linear ensemble systems are illustrated in Section IV. II. P RELIMINARY R ESULTS A typical goal of ensemble control is to steer a family of structurally similar dynamical systems between states of interest by the use of a common open-loop control applied to each system. In this section, we review the basic results on the ensemble controllability of finite-dimensional timevarying linear systems, through which we define the mathematical settings and notations for linear ensemble systems. Consider an ensemble of linear systems indexed by a parameter β varying over a compact set K, given by ˙ β) = A(t, β)X(t, β) + B(t, β)u(t), X(t,
(1)
where X ∈ M ⊂ Rn is the state , β ∈ K ⊂ R, and u ∈ n×n Lm and B(t, β) ∈ 2 [0, T ] is a control, and A(t, β) ∈ R Rn×m have elements that are real L∞ and L2 functions, respectively, defined on a compact set D = [0, T ] × K, and n×n are denoted A ∈ L∞ (D) and B ∈ L2n×m (D). We say that the system (1) is uniformly ensemble controllable if there exists a finite time T > 0 and an open-loop control function
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u : [0, T ] → Rm that steers the system from an initial state X0 into an ε-neighborhood of a target state XF in time T , i.e., if sup kX(T, β) − XF (β)k < ε
(2)
β∈K
holds [7]. Remark 1: Ensemble controllability can also be defined according to the Lp -norms for 1 ≤ p < ∞, namely, the system is ensemble controllable if Z p1 p kX(T, β) − XF (β)k dβ < ε. K
Necessary and sufficient ensemble controllability conditions for the system (1) in a Hilbert space setting have been derived and are based on the solvability of the Fredholm integral operator that characterizes the system dynamics, given by [8] Z T (Lu)(β) = Φ(0, σ, β)B(σ, β)u(σ)dσ = ξ(β), (3) 0
where Φ(t, 0, β) is the transition matrix for the system ˙ β) = A(t, β)X(t, β) and ξ(β) = Φ(0, T, β)XF (β) − X(t, X0 (β). The controllability conditions are represented in terms of the singular system of the operator L as in (3), and are given by [8] (i)
∞ X |hξ, νn iK |2 < ∞, σn2 n=1
(ii) ξ ∈ R(L),
(4)
where (σn , µn , νn ) is a singular system [16] of L and R(L) denotes the closure of the range space of L. The characterization of ensemble controllability as stated in (4) is in terms of the growth rate of the singular values of L defined in (3), and hence is intractable to verify even numerically although numerical computations of the singular values and singular vectors can be efficient [11]. As a result, constructing controllability conditions that are practically checkable is compelling. In this paper, we study a class of time-invariant linear ensemble systems and establish ensemble controllability conditions with respect to the rank of the system matrices, which are direct and easy to examine.
where Φ1 (t, 0, β) = eA(β)t is the transition matrix, ξ1 (β) = Φ1 (0, T, β)XF (β) − X0 (β), and P C m and C n denote the spaces of m-tuples of piecewise continuous functions and ntuples of continuous functions, respectively. It is known that the reachable set of the system (5) starting from X0 can be characterized by the range space of L1 , denoted as R(L1 ), that is, RT (X0 ) = Φ1 (T, 0, β)(R(L1 ) + X0 ).
In the following, we derive the relation between R(L) and the Lie algebra generated by the vector fields A(β) and B(β), which is essential to the characterization of ensemble controllability. Proposition 1: The closure of the range space of operator L1 coincides with that of the Lie algebra L0 generated by A(β) and B(β), which is given by L0 = span Ak (β)bj , j = 1, ..., m; k = 0, 1, ... . (8)
That is, R(L1 ) = L0 . Proof: See Appendix VI-A. By Proposition 1, we can express the reachable set in terms of L0 as RT (X0 ) = Φ1 (T, 0, β)(h + X0 ),
(9)
ξ1 (β) = Φ1 (0, T, β)XF (β) − X0 (β) ∈ L0 . Proof: The systems (5) is ensemble controllable, by definition and by (7), if and only if for any X0 (β), XF (β) ∈ C n (K) and any ε > 0, there exists some h ∈ R(L1 ) and T ∈ (0, ∞) such that kXF − Φ1 (T, 0, β)(h + X0 )kK < ε1 .
(10)
where k · kK is a well-defined norm on the function space C n (K). Therefore, kΦ1 (0, T, β)XF − X0 − hkK = kΦ1 (0, T, β) XF − Φ1 (T, 0, β)(h + X0 ) kK
≤ kΦ1 (0, T, β)kK kXF − Φ1 (T, 0, β)(h + X0 )kK < ε1 kΦ1 (0, T, β)kK = ε,
Consider the time-invariant ensemble system indexed by a parameter β varying on a compact set K, given by (5)
where X ∈ M ⊂ Rn , β ∈ K ⊂ R, U : [0, T ] → Rm is a piecewise continuous function of t, and the matrices n×n n×m A ∈ L∞ (D) and B ∈ L∞ (D) are defined over the compact set D = [0, T ]×K. Then, the input-to-state operator L1 : P C m [0, T ] → C n (K) of the system (5) is given by Z T (L1 u)(β) = Φ1 (0, σ, β)B(β)U (σ)dσ = ξ1 (β), (6) 0
h ∈ L0 .
Proposition 2: The system (5) is ensemble controllable if and only if for any given initial, X0 (β) = X(0, β) ∈ C n (K), and target state, XF (β) ∈ C n (K), there exists a finite time T > 0 such that
III. C ONTROLLABILITY C ONDITIONS FOR T IME -I NVARIANT L INEAR S YSTEMS
˙ β) = A(β)X(t, β) + B(β)U (t), X(t,
(7)
where the last inequality is due to (10) and the fact that kΦ1 (t, 0, β)kK is bounded for all t ∈ [0, T ] and for all β ∈ K. We conclude that ξ1 ∈ R(L1 ) = L0 . Remark 2: The result of Proposition 2 is evident since ξ1 (β) ∈ L0 = R(L1 ) implies the existence of a solution u ∈ P C m [0, T ] to the integral equation (6). In the following, we study the class of time-invariant linear ensemble systems whose system matrix is linear in the parameter, and construct explicit ensemble controllability conditions. Theorem 1: (Main Result) Consider the time-invariant lin-
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ear ensemble system d X(t, β) = βAX(t, β) + BU, (11) dt where X ∈ M ⊂ Rn , β ∈ K = [−β1 , β2 ] ⊂ R with β1 , β2 > 0, the control U : [0, T ] → Rm is a piecewise continuous function, and A ∈ Rn×n and B ∈ Rn×m are constant matrices. This system is uniformly ensemble controllable if and only if (i) rank(A) = n, (ii) rank(B) = n.
(12) (13)
Note that condition (ii) implies that the number of control inputs m is no less than the dimension of the system n. Proof: We first rewrite the system (11) as m X d u i bi , X(t, β) = βAX(t, β) + dt i=1
results in a transformed system given by
where U = (u1 , . . . , um )′ and bi is the ith column of B. (Sufficiency) Suppose that the conditions (i) and (ii) hold. The Lie algebra L0 defined as in (8) can be easily computed and is given by L0 = span β k Ak bi : i = 1, . . . , m and k = 0, 1, . . . .
It is then sufficient to show, according to Proposition 2, that for any given respective initial and target states, X0 (β) and . XF (β), ξ2 (β) = e−βAT XF (β) − X0 (β) ∈ L0 for some T > 0. In other words, it is equivalent to showing that for any given ε > 0, there exists an η ∈ L0 such that kξ2 (β) − ηk ≤ ε. Since η ∈ L0 , it can be represented as a linear combination, η= =
∞ X m X
αjk β k Ak bj
k=0 j=1 ∞ X
(α1k Ak b1 + . . . + αmk Ak bm )β k ,
(Case I): Suppose that rank(B) < n and that B has a row of zeros, say without loss of generality the last row ℓn . Thus, for the system with β = 0, the state that can be reached is of the form η = αj0 bj ∈ L0 , j = 1, . . . , m, as in (14) with k = 0. Since ℓn = 0, the last entry of η is zero. Therefore, the system (11) is not ensemble controllable, because any given function ξ2 (β) = e−βAT XF (β) − X0 (β) with ξ2 (0) having a nonzero last entry cannot be uniformly approximated by η. Alternatively, if B has no rows of zeros, then we express P the last row as a linear combination of the n−1 others, i.e., ℓn = i=1 αi ℓi , where αi ∈ R, i = 1, 2, ...n−1, and at least one of them is nonzero. A simple row operation T applied to (11), where 1 0 0 ... 0 0 1 0 ... 0 T = . (15) , .. −α1 −α2 . . . −αn−1 1 d (T X) = β(T AT −1 )(T X) + (T B)U. (16) dt The last row of T B contains all zeros, which is equivalent to the previous case. (Case II): Suppose that rank(A) < n and that A has a row of zeros, say the last row. Thus, the last row of the matrices Ak B, k = 1, 2, . . . contains only zeros. As a result, the last entry of any η ∈ L0 as in (14) is a constant function, and hence the system (11) is not ensemble controllable. The case when A has no rows of zeros can be proved in the same fashion as the case discussed in Case I. We now present two examples to demonstrate the rank conditions derived in Theorem 1. Example 1: Consider steering a harmonic oscillator with uncertainty in its frequency ω from an initial state X0 (ω) = X(0, ω) to a desired target state XF (ω), modeled by
(14)
d X(t, ω) = A(ω)X(t, ω) + Bu, dt
k=0
where αjk ∈ R for j = 1, . . . , m and k = 0, 1, . . .. Also, because ξ2 ∈ C n (K), it can be uniformly approximated by a PN vector-valued polynomial of order N , pN (β) = k=0 ck β k , such that kξ2 (β) − pN (β)k ≤ ε, where ck ∈ Rn are coefficient vectors for k = 0, . . . , N . It remains to show that the coefficients αjk as in (14) can be chosen so that . α1k Ak b1 + . . . + αmk Ak bm = ck = (c1k , c2k , . . . , cnk )′ for all k = 0, . . . , N . This is possible because A and B are of full rank, so the underdetermined system of linear equations c1k α1k k A b1 . . . Ak bm ... = ... cnk αmk
has a solution for all k = 0, 1, . . . , N .
(Necessity) We will show that if either of the conditions in (12) or (13) fails to hold, then the system (11) is not ensemble controllable.
(17)
where A(ω) = ωA = ω
0 −1 1 0
,B=
1 0 0 1
, u=
u1 u2
and the frequency is known to be in the range ω ∈ K = [−1, 1]. Since rank(A) = 2 and rank(B) = 2, this system is ensemble controllable according to the conditions (12) and (13). This can be illustrated using the concept of polynomial approximation. The Lie algebra of A(ω) and B is given by 0 1 k k , k = 0, 1, 2, . . .}, ,ω L0 = span{ω 1 0
and it follows that for any ξ ∈ C n [0, T ], it can be uniformly approximated by a linear combination of the vector fields in −ωAT L0 . Namely, for a given XF (ω) − X0 (ω), PN ξ(ω) k= e there exists P (ω) = k=1 ck ω , where ck ∈ R2 , such that supω∈K kξ(ω) − P (ω)k < ε. If, however, there is only one control available, say u2 = 0,
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,
then rank(B) = 1 and 0 1 2k+1 2k , k = 0, 1, 2, . . .}. ,ω L0 = span{ω 1 0
L0 . This ensemble controllability indicates that the condition (13) is not necessary when the parameter 0 ∈ / K.
In this case, a uniform approximation for a given ξ(ω) = (ξ1 (ω), ξ2 (ω)) for ω ∈ [−1, 1] by an η ∈ L0 , i.e., kξ − ηk < ε, is possible only when ξ1 (ω) and ξ2 (ω) are an even and an odd function, respectively. This verifies that the condition (13) is necessary. Remark 3: The characterization of ensemble controllability stated in Theorem 1 is for ensemble systems with parameter variation on a set that includes zero. The situation becomes different if the zero parameter value is not included. For the ensemble system (11), if the parameter satisfies β ∈ [β1 , β2 ] with β1 , β2 > 0 or β1 , β2 < 0, the rank conditions (12) and (13) are sufficient as shown in the proof of Theorem 1. However, the condition rank(B) = n is not necessary in this case, which is illustrated by the following examples. Example 2: Consider a linearized lateral-directional model that describes aircraft dynamics in the presence of uncertainty [17],
In this section, we present several controllable ensemble systems and construct optimal controls using a robust computational method based on the SVD developed in our previous work [11]. Example 4: Consider steering an ensemble of harmonic oscillators modeled in (17) with their frequencies distributed in ω ∈ K = [−1, 1] from the initial state X0 (ω) = X(0, ω) = (5, 3)′ to the target state XF (ω) = (ω, 2ω)′ , which is a linear function in ω over K, at time T = 1. The system matrices A and B are both full rank, so the system is ensemble controllable. We employed the SVD method [11] to synthesize the minimum-energy ensemble control law that achieves the desired transfer, where the number of discretization points in time, t ∈ [0, 1], is N = 40000 and in the frequency domain, K, is 400. In order to avoid numerical conditioning errors, we set the ratio of the largest and the smallest singular values used to s1 /smq < 104 and q denotes the number of singular values used to synthesize the ensemble control. The ensemble control function is shown in Figure 1(a). The terminal states for ω ∈ [−1, 1] using this control are displayed in Figure 1(b), yielding errors less than 10−3 for all ω ∈ [−1, 1]. Five sample trajectories for frequencies uniformly distributed within [−1, 1] are shown in Figure 1(c). Example 5: Consider manipulating an aircraft system with the system dynamics described as in Example 2 in the presence of uncertainty, where ǫ ∈ [0.8, 1.2]. We wish to design a robust open-loop control that is insensitive to such parameter uncertainty and that drives the system from the initial state (2π, 1, 1)′ to the target state (π, 0, 0)′ at T = 3. We employed the SVD method [11] to synthesize the minimum-energy ensemble control law that achieves the desired transfer, where the number of discretization points in t ∈ [0, 3] is N = 40000 and in ǫ = [0.8, 1.2] is 40. The ratio of the largest and the smallest singular values is set at s1 /smq < 105 for the control synthesis. The ensemble control function is shown in Figure 2(a). Some sample trajectories for ǫ ∈ [0.8, 1.2] are illustrated in Figure 2(b).
IV. E NSEMBLE C ONTROL S YNTHESIS
d X(t, ǫ) = A(ǫ)X(t, ǫ) + Bu, dt where
0 0.1 −1 A(ǫ) = ǫA = ǫ 10 0.1 0 , 4 0 0.1
1 B= 0 0
0 0 1 0 . 0 1
The state X = (γ, ps , rs )′ , where γ denotes the angle of sideslip, ps and rs denote the stability axis roll and yaw rate, respectively, and u = (u1 , u2 , u3 )′ is the control input. Unpredictable perturbations in the environment of the aircraft may result in dispersion in system dynamics, which we model ǫ ∈ K = [0.8, 1.2]. Because rank(A) = 3 and rank(B) = 3, these are sufficient for the system to be ensemble controllable. An optimal ensemble control for steering this uncertain system is illustrated in Section IV. Example 3: Consider the lateral-directional aircraft model introduced in Example 2 with different dynamics in A, given by 0 0.1 0 A(ǫ) = ǫ 0 0 1 , 4 0 0
and with only one control u1 available, i.e., u2 = u3 = 0. In this case, rank(A) = 3 but rank(B) = 1 < 3, which is not of full rank. However, this system is ensemble controllable. Observe that 0 0 n ǫ3k o L0 = span 0 , 0 , ǫ3k+2 , 0 ǫ3k+1 0 where k = 0, 1, 2, . . .. Then, by the M¨untz−Sz´asz theorem (see Appendix VI-B), for any given ξ(ǫ) = (ξ1 (ǫ), ξ2 (ǫ), ξ3 (ǫ))′ with ǫ ∈ K ⊂ R+ , it can be uniformly approximated by a linear combination of the vector fields in
AND
S IMULATIONS
V. CONCLUSIONS In this paper, we investigated the ensemble control of timeinvariant linear systems and derived controllability conditions for the ensemble with system dynamics of linear dependence on the parameter, i.e., A(β) = βA. The conditions are in terms of the rank of the system matrices, which are easy to examine. We also discussed the cases when the parameter set K ⊂ R+ or K ⊂ R− , and provided examples to address the difference between these cases and the main result in Theorem 1 with 0 ∈ K. Simulations of several controllable ensemble systems driven by minimum-energy controls between desired states were presented to illustrate our results. We plan to establish explicit controllability conditions for the
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50 2
100
1.5
0
u1,2,3
150
−50
1
50
−100
0.5
(a) 0
X2(1,ω)
0 −50
u1 0.5
1
1.5 t
u2 2
u3 2.5
3
0 −0.5
−100 −1 2
−150
−1.5
u1
1
−2
u
2
0
0
0.5
1
(a)
−2.5 −1
0
1
2
s
−250
r
−200
1
−2
6
−3
ω=−1 ω=−0.5 ω=0 ω=0.5 ω=1
4 2 0 2
X (t,ω)
−1
X (1,ω)
(b)
−4 10 5
8 6
0
4 2
−5
−2
p
−10
s
(b)
0 −2
γ
−4
Fig. 2. Controlling an uncertain aircraft system. (a) The minimum-energy ensemble control function that drives the ensemble from (2π, 1, 1)′ to (π, 0, 0)′ at T = 3. (b) Sample trajectories for ǫ ∈ [0.8, 1.2] following the optimal control shown in (a).
−6 −8 −10 −10
−5
0
5
10
15
X1(t,ω)
(c)
Fig. 1. Steering an ensemble of harmonic oscillators. (a) The minimumenergy ensemble control function that steers the ensemble from (5, 3)′ to (ω, 2ω) at T = 1. (b) The final states X(1, ω) for ω ∈ [−1, 1] following the ensemble control law. (c) Sample trajectories for ω = −1, −0.5, 0, 0.5, and 1.
general time-invariant linear ensemble system as described in (5). VI. APPENDIX A. Proof of Proposition 1 1) (R(L1 ) ⊆ L0 ): Suppose that ξ ∈ R(L1 ). Then, there exists some U ∈ P C m [0, T ] such that ξ = L1 U , and we have by (6) Z T ξ= Φ1 (0, σ, β)BU (σ)dσ 0 Z TX ∞ (−σ)n n = A (β)BU (σ)dσ (18) n! 0 n=0 ∞ Z T X (−σ)n n = A (β)BU (σ)dσ (19) n! n=0 0 "Z # ∞ T n X (−σ) = An (β)B U (σ)dσ . n! 0 n=0
and (19)) because ∞ Z T X (−σ)n n dσ A (β)BU (σ) n! n=0 0 ∞ Z T X (−σ)n n · |BU |dσ ≤ A (β) n! n=0 0 n+1 ∞ X T < T An (β) M, (n + 1)! n=0 ≤ T2
≤ T2
∞ X |T A(β)|n M, (n + 1) ! n=0
∞ X |T A(β)|n M = T 2 e|A(β)T | M, n ! n=0
which is finite, where M = (m, m, . . . , m)′ ∈ Rn with m equal to the largest entry of supt∈[0,T ] |BU (t)| and the inequalities above are pointwise. 2) (L0 ⊆ R(L1 )): Now, suppose that η0 = αik Ak (β)bi ∈ L0 , where bi is the ith column of B and αik ∈ R. By the mean value theorem, the transition matrix of the system (5) can be represented as
This implies that ξ ∈ span{An (β)bi }, n = 0, 1, . . . and i = 1, . . . , m, and hence ξ ∈ L0 . This follows that R(L1 ) ⊆ L0 . Note that the infinite sum and integration commute (see (18) 2713
A2 (β)t2 + ... 2! n n n+1 (n+1) A (β)t A (β)t + + Φ1 (r, 0, β), (20) n! (n + 1) !
Φ1 (t, 0, β) = I + A(β)t +
where r ∈ (0, T ). Let Sn (t, 0, β) be a partial sum defined by n X Ak (β)tk Sn (t, 0, β) = . k! k=0
We can now construct polynomial control functions, ui , so that, for k ≤ n and 1 ≤ i ≤ m, Z T p t ui (t)dt = αik , if p = k, (21) 0 p! Z T p t ui (t)dt = 0, if p 6= k. (22) 0 p! This yields Z T Sn (t, 0, β)bi ui (t)dt = αik Ak (β)bi ,
1 ≤ i ≤ m. (23)
0
Thus, there exists ξ = LU ∈ R(L1 ), such that
Z T
kξ − η0 kK = Φ1 (0, t, β)BU (t)dt − αik Ak (β)bi K 0
Z T
≤ Φ1 (t, 0, β) − Sn (t, 0, β) BU (t)dt K 0
Z T
+ Sn (t, 0, β)BU (t)dt − αik Ak (β)bi
[7] J.-S. Li and N. Khaneja, “Ensemble control of bloch equations,” IEEE Transactions on Automatic Control, vol. 54, pp. 528–536, 2009. [8] J.-S. Li, “Ensemble control of finite-dimensional time-varying linear systems,” IEEE Transastions on Automatic Control, vol. 56, pp. 345– 357, 2011. [9] J.-S. Li, I. Dasanayake, and J. Ruths, “Control and synchronization of neuron ensembles,” IEEE Transactions on Automatic Control, vol. 58, no. 8, pp. 1919–1930, 2013. [10] J. Ruths and J.-S. Li, “Optimal control of inhomogeneous ensembles,” IEEE Transactions on Automatic Control: Special Issue on Control of Quantum Mechanical Systems, vol. 57, no. 8, pp. 2021–2032, 2012. [11] A. Zlotnik and J.-S. Li, “Synthesis of optimal ensemble controls for linear systems using the singular value decomposition,” in American Control Conference, (Montreal), pp. 5849–5854, 2012. [12] D. Stefanatos and J.-S. Li, “Minimum-time quantum transport with bounded trap velocity,” IEEE Transactions on Automatic Control (in press). [13] R. Brockett, “On the control of a flock by a leader,” Proceedings of the Steklov Institute of Mathematics, vol. 268, no. 1, pp. 49–57, 2010. [14] Y.-Y. Liu, J.-J. Slotine, and A.-L. Barabasi, “Controllability of complex networks,” Nature, vol. 473, no. 1, pp. 167–173, 2011. [15] J.-S. Li, “Control of a network of spiking neurons,” in 8th IFAC Symposium on Nonlinear Control Systems, (Italy), Sep. 2010. [16] I. Gohberg, S. Goldberg, and M. A. Kaashoek, Basic Classes of Linear Operators. Boston, MA: Birkh¨auser Verlag, 2003. [17] E. Lavretsky and K. Wise, Robust and Adaptive Control: With Aerospace Applications (Advanced Textbooks in Control and Signal Processing). Springer-Verlag, 2012. ¨ [18] C. H. M¨untz, “Uber den approximationssatz von weierstrass,” H. A. Schwarz’s Festschrift, pp. 303–312, 1914.
K
0
≤ ǫM T,
(24)
by the convergence of Φ1 (t, 0, β) and by (23). The same procedures can be applied to obtain as in P∞ Pma similark result k (24) for any element η = α β A b ∈ L0 . ik i k=0 i=1 Consequently, we conclude that L0 ⊆ R(L1 ). B. M¨untz−Sz´asz Theorem [18] Let {λi }∞ i=1 be a sequence with inf i λi > 0. Then span{1, xλ1 , xλ2 , . . .}, is dense in C[0, 1] if and only if ∞ X 1 = ∞. λ i=1 i
R EFERENCES [1] J.-S. Li and N. Khaneja, “Control of inhomogeneous quantum ensembles,” Physical Review A, vol. 73, p. 030302, 2006. [2] J.-S. Li, J. Ruths, T.-Y. Yu, H. Arthanari, and G. Wagner, “Optimal pulse design in quantum control: A unified computational method,” Proceedings of the National Academy of Sciences, vol. 108, no. 5, pp. 1879–1884, 2011. [3] S. J. Glaser, T. Schulte-Herbr¨uggen, M. Sieveking, N. C. N. O. Schedletzky, O. W. Sørensen, and C. Griesinger, “Unitary control in quantum ensembles, maximizing signal intensity in coherent spectroscopy,” Science, vol. 280, pp. 421–424, 1998. [4] A. L. Benabid, P. Pollak, D. Haffmann, C. Gervason, M. Hommek, J. Perret, J. de Rougemont, and D. Gao, “Long-term suppression of tremor by chronic stimulation of the ventral intermediate thalamic nucleus,” The Lancet, vol. 337, pp. 403–406, 1991. [5] A. Zlotnik and J.-S. Li, “Optimal entrainment of neural oscillator ensembles,” Journal of Neural Engineering, vol. 9, no. 4, p. 046015(12), 2012. [6] A. Becker and T. Bretl, “Approximate steering of a unicycle under bounded model perturbation using ensemble control,” IEEE Transactions on Robotics, vol. 28, no. 3, pp. 580–591, 2012.
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Ensemble Controllability of Time-Invariant Linear Systems Ji Qi and Jr-Shin Li
Abstract— In this paper, we study the control of an ensemble of structurally similar time-invariant linear systems. In particular, we derive explicit necessary and sufficient controllability conditions for such systems in terms of the rank of the system matrices. We present examples to demonstrate these rank conditions, and construct optimal controls for steering a linear ensemble system between states of interest by using an optimization-free computational method based on the singular value decomposition. This work extends our previous results in ensemble control of time-varying linear systems, where the established controllability conditions are implicit and are defined by the singular system of the linear operator that characterizes the system dynamics.
I. INTRODUCTION Robust and sensorless manipulation of a collection of structurally similar systems with variation in common parameters, or of a single system with uncertainty in the parameters, is compelling in various areas of science and engineering. Premier examples range from the application of optimal pulses to produce a desired evolution of a large quantum ensemble in quantum control [1], [2], [3], and the use of external stimuli to desynchronize a population of neurons in the treatment of neurological disorders [4], [5], to the implementation of open-loop controls for approximate steering of robots under bounded model perturbation in robotics [6]. Such practical control designs give rise to challenging problems involving the guidance of a large number or a continuum of structurally similar dynamical systems using a common open-loop control input, which arises because measurements for the state of each individual system of the ensemble is impractical and hence state feedback is unavailable. The research in the control of ensemble systems has been active in both theoretical and computational aspects. The controllability for an ensemble of systems evolving on the Lie group SO(3) has been investigated through a conversion of the analysis to polynomial approximation [7]. The necessary and sufficient controllability characterization of an ensemble of finite-dimensional time-varying linear systems was provided in terms of the singular system of the input-to-state operator of the system [8]. Controllability for an ensemble of neuron oscillators described by nonlinear phase models was also analyzed [9]. Recently, a unified This work was supported by the NSF Career Award #0747877 and the AFOSR YIP FA9550-10-1-0146. J. Qi is with the Department of Electrical and Systems Engineering, Washington University in St. Louis, St. Louis, MO 63130, USA
[email protected] J.-S. Li is with the Faculty of Electrical and Systems Engineering, Washington University in St. Louis, St. Louis, MO 63130, USA
[email protected] 978-1-4673-5716-6/13/$31.00 ©2013 IEEE
computational method for solving optimal ensemble control problems based on multidimensional pseudospectral approximations has been developed [10] and successfully employed to design optimal pulses for protein NMR spectroscopy [2]. An optimization-free computational algorithm based on the singular value decomposition (SVD) was also established, and can be used to compute minimum-energy controls for steering linear ensemble systems [11], such as quantum transport systems [12]. Numerous work in ensemble control has also emerged in the biological domain, with the aim of understanding the coordination of the movement of flocks [13], and in controlling large-scale complex networks [14]. Although intensive work has been conducted to characterize the controllability of ensemble systems [7], [8], [9], [15], explicit controllability conditions have yet to be provided. For example, the controllability of an ensemble of finitedimensional time-varying linear systems is determined by the growth rate of the singular values of the input-to-state operator, which is intractable to verify. In this paper, we study a class of time-invariant linear ensemble systems and derive controllability conditions in terms of the rank of the system matrices. In Section II, we review some preliminary results on the ensemble control of linear systems, which motivate this work. In Section III, we construct the explicit necessary and sufficient controllability conditions for a class of timeinvariant linear ensemble systems, which are of practical importance. Examples and simulations of optimal controls for steering such linear ensemble systems are illustrated in Section IV. II. P RELIMINARY R ESULTS A typical goal of ensemble control is to steer a family of structurally similar dynamical systems between states of interest by the use of a common open-loop control applied to each system. In this section, we review the basic results on the ensemble controllability of finite-dimensional timevarying linear systems, through which we define the mathematical settings and notations for linear ensemble systems. Consider an ensemble of linear systems indexed by a parameter β varying over a compact set K, given by ˙ β) = A(t, β)X(t, β) + B(t, β)u(t), X(t,
(1)
where X ∈ M ⊂ Rn is the state , β ∈ K ⊂ R, and u ∈ n×n Lm and B(t, β) ∈ 2 [0, T ] is a control, and A(t, β) ∈ R Rn×m have elements that are real L∞ and L2 functions, respectively, defined on a compact set D = [0, T ] × K, and n×n are denoted A ∈ L∞ (D) and B ∈ L2n×m (D). We say that the system (1) is uniformly ensemble controllable if there exists a finite time T > 0 and an open-loop control function
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u : [0, T ] → Rm that steers the system from an initial state X0 into an ε-neighborhood of a target state XF in time T , i.e., if sup kX(T, β) − XF (β)k < ε
(2)
β∈K
holds [7]. Remark 1: Ensemble controllability can also be defined according to the Lp -norms for 1 ≤ p < ∞, namely, the system is ensemble controllable if Z p1 p kX(T, β) − XF (β)k dβ < ε. K
Necessary and sufficient ensemble controllability conditions for the system (1) in a Hilbert space setting have been derived and are based on the solvability of the Fredholm integral operator that characterizes the system dynamics, given by [8] Z T (Lu)(β) = Φ(0, σ, β)B(σ, β)u(σ)dσ = ξ(β), (3) 0
where Φ(t, 0, β) is the transition matrix for the system ˙ β) = A(t, β)X(t, β) and ξ(β) = Φ(0, T, β)XF (β) − X(t, X0 (β). The controllability conditions are represented in terms of the singular system of the operator L as in (3), and are given by [8] (i)
∞ X |hξ, νn iK |2 < ∞, σn2 n=1
(ii) ξ ∈ R(L),
(4)
where (σn , µn , νn ) is a singular system [16] of L and R(L) denotes the closure of the range space of L. The characterization of ensemble controllability as stated in (4) is in terms of the growth rate of the singular values of L defined in (3), and hence is intractable to verify even numerically although numerical computations of the singular values and singular vectors can be efficient [11]. As a result, constructing controllability conditions that are practically checkable is compelling. In this paper, we study a class of time-invariant linear ensemble systems and establish ensemble controllability conditions with respect to the rank of the system matrices, which are direct and easy to examine.
where Φ1 (t, 0, β) = eA(β)t is the transition matrix, ξ1 (β) = Φ1 (0, T, β)XF (β) − X0 (β), and P C m and C n denote the spaces of m-tuples of piecewise continuous functions and ntuples of continuous functions, respectively. It is known that the reachable set of the system (5) starting from X0 can be characterized by the range space of L1 , denoted as R(L1 ), that is, RT (X0 ) = Φ1 (T, 0, β)(R(L1 ) + X0 ).
In the following, we derive the relation between R(L) and the Lie algebra generated by the vector fields A(β) and B(β), which is essential to the characterization of ensemble controllability. Proposition 1: The closure of the range space of operator L1 coincides with that of the Lie algebra L0 generated by A(β) and B(β), which is given by L0 = span Ak (β)bj , j = 1, ..., m; k = 0, 1, ... . (8)
That is, R(L1 ) = L0 . Proof: See Appendix VI-A. By Proposition 1, we can express the reachable set in terms of L0 as RT (X0 ) = Φ1 (T, 0, β)(h + X0 ),
(9)
ξ1 (β) = Φ1 (0, T, β)XF (β) − X0 (β) ∈ L0 . Proof: The systems (5) is ensemble controllable, by definition and by (7), if and only if for any X0 (β), XF (β) ∈ C n (K) and any ε > 0, there exists some h ∈ R(L1 ) and T ∈ (0, ∞) such that kXF − Φ1 (T, 0, β)(h + X0 )kK < ε1 .
(10)
where k · kK is a well-defined norm on the function space C n (K). Therefore, kΦ1 (0, T, β)XF − X0 − hkK = kΦ1 (0, T, β) XF − Φ1 (T, 0, β)(h + X0 ) kK
≤ kΦ1 (0, T, β)kK kXF − Φ1 (T, 0, β)(h + X0 )kK < ε1 kΦ1 (0, T, β)kK = ε,
Consider the time-invariant ensemble system indexed by a parameter β varying on a compact set K, given by (5)
where X ∈ M ⊂ Rn , β ∈ K ⊂ R, U : [0, T ] → Rm is a piecewise continuous function of t, and the matrices n×n n×m A ∈ L∞ (D) and B ∈ L∞ (D) are defined over the compact set D = [0, T ]×K. Then, the input-to-state operator L1 : P C m [0, T ] → C n (K) of the system (5) is given by Z T (L1 u)(β) = Φ1 (0, σ, β)B(β)U (σ)dσ = ξ1 (β), (6) 0
h ∈ L0 .
Proposition 2: The system (5) is ensemble controllable if and only if for any given initial, X0 (β) = X(0, β) ∈ C n (K), and target state, XF (β) ∈ C n (K), there exists a finite time T > 0 such that
III. C ONTROLLABILITY C ONDITIONS FOR T IME -I NVARIANT L INEAR S YSTEMS
˙ β) = A(β)X(t, β) + B(β)U (t), X(t,
(7)
where the last inequality is due to (10) and the fact that kΦ1 (t, 0, β)kK is bounded for all t ∈ [0, T ] and for all β ∈ K. We conclude that ξ1 ∈ R(L1 ) = L0 . Remark 2: The result of Proposition 2 is evident since ξ1 (β) ∈ L0 = R(L1 ) implies the existence of a solution u ∈ P C m [0, T ] to the integral equation (6). In the following, we study the class of time-invariant linear ensemble systems whose system matrix is linear in the parameter, and construct explicit ensemble controllability conditions. Theorem 1: (Main Result) Consider the time-invariant lin-
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ear ensemble system d X(t, β) = βAX(t, β) + BU, (11) dt where X ∈ M ⊂ Rn , β ∈ K = [−β1 , β2 ] ⊂ R with β1 , β2 > 0, the control U : [0, T ] → Rm is a piecewise continuous function, and A ∈ Rn×n and B ∈ Rn×m are constant matrices. This system is uniformly ensemble controllable if and only if (i) rank(A) = n, (ii) rank(B) = n.
(12) (13)
Note that condition (ii) implies that the number of control inputs m is no less than the dimension of the system n. Proof: We first rewrite the system (11) as m X d u i bi , X(t, β) = βAX(t, β) + dt i=1
results in a transformed system given by
where U = (u1 , . . . , um )′ and bi is the ith column of B. (Sufficiency) Suppose that the conditions (i) and (ii) hold. The Lie algebra L0 defined as in (8) can be easily computed and is given by L0 = span β k Ak bi : i = 1, . . . , m and k = 0, 1, . . . .
It is then sufficient to show, according to Proposition 2, that for any given respective initial and target states, X0 (β) and . XF (β), ξ2 (β) = e−βAT XF (β) − X0 (β) ∈ L0 for some T > 0. In other words, it is equivalent to showing that for any given ε > 0, there exists an η ∈ L0 such that kξ2 (β) − ηk ≤ ε. Since η ∈ L0 , it can be represented as a linear combination, η= =
∞ X m X
αjk β k Ak bj
k=0 j=1 ∞ X
(α1k Ak b1 + . . . + αmk Ak bm )β k ,
(Case I): Suppose that rank(B) < n and that B has a row of zeros, say without loss of generality the last row ℓn . Thus, for the system with β = 0, the state that can be reached is of the form η = αj0 bj ∈ L0 , j = 1, . . . , m, as in (14) with k = 0. Since ℓn = 0, the last entry of η is zero. Therefore, the system (11) is not ensemble controllable, because any given function ξ2 (β) = e−βAT XF (β) − X0 (β) with ξ2 (0) having a nonzero last entry cannot be uniformly approximated by η. Alternatively, if B has no rows of zeros, then we express P the last row as a linear combination of the n−1 others, i.e., ℓn = i=1 αi ℓi , where αi ∈ R, i = 1, 2, ...n−1, and at least one of them is nonzero. A simple row operation T applied to (11), where 1 0 0 ... 0 0 1 0 ... 0 T = . (15) , .. −α1 −α2 . . . −αn−1 1 d (T X) = β(T AT −1 )(T X) + (T B)U. (16) dt The last row of T B contains all zeros, which is equivalent to the previous case. (Case II): Suppose that rank(A) < n and that A has a row of zeros, say the last row. Thus, the last row of the matrices Ak B, k = 1, 2, . . . contains only zeros. As a result, the last entry of any η ∈ L0 as in (14) is a constant function, and hence the system (11) is not ensemble controllable. The case when A has no rows of zeros can be proved in the same fashion as the case discussed in Case I. We now present two examples to demonstrate the rank conditions derived in Theorem 1. Example 1: Consider steering a harmonic oscillator with uncertainty in its frequency ω from an initial state X0 (ω) = X(0, ω) to a desired target state XF (ω), modeled by
(14)
d X(t, ω) = A(ω)X(t, ω) + Bu, dt
k=0
where αjk ∈ R for j = 1, . . . , m and k = 0, 1, . . .. Also, because ξ2 ∈ C n (K), it can be uniformly approximated by a PN vector-valued polynomial of order N , pN (β) = k=0 ck β k , such that kξ2 (β) − pN (β)k ≤ ε, where ck ∈ Rn are coefficient vectors for k = 0, . . . , N . It remains to show that the coefficients αjk as in (14) can be chosen so that . α1k Ak b1 + . . . + αmk Ak bm = ck = (c1k , c2k , . . . , cnk )′ for all k = 0, . . . , N . This is possible because A and B are of full rank, so the underdetermined system of linear equations c1k α1k k A b1 . . . Ak bm ... = ... cnk αmk
has a solution for all k = 0, 1, . . . , N .
(Necessity) We will show that if either of the conditions in (12) or (13) fails to hold, then the system (11) is not ensemble controllable.
(17)
where A(ω) = ωA = ω
0 −1 1 0
,B=
1 0 0 1
, u=
u1 u2
and the frequency is known to be in the range ω ∈ K = [−1, 1]. Since rank(A) = 2 and rank(B) = 2, this system is ensemble controllable according to the conditions (12) and (13). This can be illustrated using the concept of polynomial approximation. The Lie algebra of A(ω) and B is given by 0 1 k k , k = 0, 1, 2, . . .}, ,ω L0 = span{ω 1 0
and it follows that for any ξ ∈ C n [0, T ], it can be uniformly approximated by a linear combination of the vector fields in −ωAT L0 . Namely, for a given XF (ω) − X0 (ω), PN ξ(ω) k= e there exists P (ω) = k=1 ck ω , where ck ∈ R2 , such that supω∈K kξ(ω) − P (ω)k < ε. If, however, there is only one control available, say u2 = 0,
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,
then rank(B) = 1 and 0 1 2k+1 2k , k = 0, 1, 2, . . .}. ,ω L0 = span{ω 1 0
L0 . This ensemble controllability indicates that the condition (13) is not necessary when the parameter 0 ∈ / K.
In this case, a uniform approximation for a given ξ(ω) = (ξ1 (ω), ξ2 (ω)) for ω ∈ [−1, 1] by an η ∈ L0 , i.e., kξ − ηk < ε, is possible only when ξ1 (ω) and ξ2 (ω) are an even and an odd function, respectively. This verifies that the condition (13) is necessary. Remark 3: The characterization of ensemble controllability stated in Theorem 1 is for ensemble systems with parameter variation on a set that includes zero. The situation becomes different if the zero parameter value is not included. For the ensemble system (11), if the parameter satisfies β ∈ [β1 , β2 ] with β1 , β2 > 0 or β1 , β2 < 0, the rank conditions (12) and (13) are sufficient as shown in the proof of Theorem 1. However, the condition rank(B) = n is not necessary in this case, which is illustrated by the following examples. Example 2: Consider a linearized lateral-directional model that describes aircraft dynamics in the presence of uncertainty [17],
In this section, we present several controllable ensemble systems and construct optimal controls using a robust computational method based on the SVD developed in our previous work [11]. Example 4: Consider steering an ensemble of harmonic oscillators modeled in (17) with their frequencies distributed in ω ∈ K = [−1, 1] from the initial state X0 (ω) = X(0, ω) = (5, 3)′ to the target state XF (ω) = (ω, 2ω)′ , which is a linear function in ω over K, at time T = 1. The system matrices A and B are both full rank, so the system is ensemble controllable. We employed the SVD method [11] to synthesize the minimum-energy ensemble control law that achieves the desired transfer, where the number of discretization points in time, t ∈ [0, 1], is N = 40000 and in the frequency domain, K, is 400. In order to avoid numerical conditioning errors, we set the ratio of the largest and the smallest singular values used to s1 /smq < 104 and q denotes the number of singular values used to synthesize the ensemble control. The ensemble control function is shown in Figure 1(a). The terminal states for ω ∈ [−1, 1] using this control are displayed in Figure 1(b), yielding errors less than 10−3 for all ω ∈ [−1, 1]. Five sample trajectories for frequencies uniformly distributed within [−1, 1] are shown in Figure 1(c). Example 5: Consider manipulating an aircraft system with the system dynamics described as in Example 2 in the presence of uncertainty, where ǫ ∈ [0.8, 1.2]. We wish to design a robust open-loop control that is insensitive to such parameter uncertainty and that drives the system from the initial state (2π, 1, 1)′ to the target state (π, 0, 0)′ at T = 3. We employed the SVD method [11] to synthesize the minimum-energy ensemble control law that achieves the desired transfer, where the number of discretization points in t ∈ [0, 3] is N = 40000 and in ǫ = [0.8, 1.2] is 40. The ratio of the largest and the smallest singular values is set at s1 /smq < 105 for the control synthesis. The ensemble control function is shown in Figure 2(a). Some sample trajectories for ǫ ∈ [0.8, 1.2] are illustrated in Figure 2(b).
IV. E NSEMBLE C ONTROL S YNTHESIS
d X(t, ǫ) = A(ǫ)X(t, ǫ) + Bu, dt where
0 0.1 −1 A(ǫ) = ǫA = ǫ 10 0.1 0 , 4 0 0.1
1 B= 0 0
0 0 1 0 . 0 1
The state X = (γ, ps , rs )′ , where γ denotes the angle of sideslip, ps and rs denote the stability axis roll and yaw rate, respectively, and u = (u1 , u2 , u3 )′ is the control input. Unpredictable perturbations in the environment of the aircraft may result in dispersion in system dynamics, which we model ǫ ∈ K = [0.8, 1.2]. Because rank(A) = 3 and rank(B) = 3, these are sufficient for the system to be ensemble controllable. An optimal ensemble control for steering this uncertain system is illustrated in Section IV. Example 3: Consider the lateral-directional aircraft model introduced in Example 2 with different dynamics in A, given by 0 0.1 0 A(ǫ) = ǫ 0 0 1 , 4 0 0
and with only one control u1 available, i.e., u2 = u3 = 0. In this case, rank(A) = 3 but rank(B) = 1 < 3, which is not of full rank. However, this system is ensemble controllable. Observe that 0 0 n ǫ3k o L0 = span 0 , 0 , ǫ3k+2 , 0 ǫ3k+1 0 where k = 0, 1, 2, . . .. Then, by the M¨untz−Sz´asz theorem (see Appendix VI-B), for any given ξ(ǫ) = (ξ1 (ǫ), ξ2 (ǫ), ξ3 (ǫ))′ with ǫ ∈ K ⊂ R+ , it can be uniformly approximated by a linear combination of the vector fields in
AND
S IMULATIONS
V. CONCLUSIONS In this paper, we investigated the ensemble control of timeinvariant linear systems and derived controllability conditions for the ensemble with system dynamics of linear dependence on the parameter, i.e., A(β) = βA. The conditions are in terms of the rank of the system matrices, which are easy to examine. We also discussed the cases when the parameter set K ⊂ R+ or K ⊂ R− , and provided examples to address the difference between these cases and the main result in Theorem 1 with 0 ∈ K. Simulations of several controllable ensemble systems driven by minimum-energy controls between desired states were presented to illustrate our results. We plan to establish explicit controllability conditions for the
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50 2
100
1.5
0
u1,2,3
150
−50
1
50
−100
0.5
(a) 0
X2(1,ω)
0 −50
u1 0.5
1
1.5 t
u2 2
u3 2.5
3
0 −0.5
−100 −1 2
−150
−1.5
u1
1
−2
u
2
0
0
0.5
1
(a)
−2.5 −1
0
1
2
s
−250
r
−200
1
−2
6
−3
ω=−1 ω=−0.5 ω=0 ω=0.5 ω=1
4 2 0 2
X (t,ω)
−1
X (1,ω)
(b)
−4 10 5
8 6
0
4 2
−5
−2
p
−10
s
(b)
0 −2
γ
−4
Fig. 2. Controlling an uncertain aircraft system. (a) The minimum-energy ensemble control function that drives the ensemble from (2π, 1, 1)′ to (π, 0, 0)′ at T = 3. (b) Sample trajectories for ǫ ∈ [0.8, 1.2] following the optimal control shown in (a).
−6 −8 −10 −10
−5
0
5
10
15
X1(t,ω)
(c)
Fig. 1. Steering an ensemble of harmonic oscillators. (a) The minimumenergy ensemble control function that steers the ensemble from (5, 3)′ to (ω, 2ω) at T = 1. (b) The final states X(1, ω) for ω ∈ [−1, 1] following the ensemble control law. (c) Sample trajectories for ω = −1, −0.5, 0, 0.5, and 1.
general time-invariant linear ensemble system as described in (5). VI. APPENDIX A. Proof of Proposition 1 1) (R(L1 ) ⊆ L0 ): Suppose that ξ ∈ R(L1 ). Then, there exists some U ∈ P C m [0, T ] such that ξ = L1 U , and we have by (6) Z T ξ= Φ1 (0, σ, β)BU (σ)dσ 0 Z TX ∞ (−σ)n n = A (β)BU (σ)dσ (18) n! 0 n=0 ∞ Z T X (−σ)n n = A (β)BU (σ)dσ (19) n! n=0 0 "Z # ∞ T n X (−σ) = An (β)B U (σ)dσ . n! 0 n=0
and (19)) because ∞ Z T X (−σ)n n dσ A (β)BU (σ) n! n=0 0 ∞ Z T X (−σ)n n · |BU |dσ ≤ A (β) n! n=0 0 n+1 ∞ X T < T An (β) M, (n + 1)! n=0 ≤ T2
≤ T2
∞ X |T A(β)|n M, (n + 1) ! n=0
∞ X |T A(β)|n M = T 2 e|A(β)T | M, n ! n=0
which is finite, where M = (m, m, . . . , m)′ ∈ Rn with m equal to the largest entry of supt∈[0,T ] |BU (t)| and the inequalities above are pointwise. 2) (L0 ⊆ R(L1 )): Now, suppose that η0 = αik Ak (β)bi ∈ L0 , where bi is the ith column of B and αik ∈ R. By the mean value theorem, the transition matrix of the system (5) can be represented as
This implies that ξ ∈ span{An (β)bi }, n = 0, 1, . . . and i = 1, . . . , m, and hence ξ ∈ L0 . This follows that R(L1 ) ⊆ L0 . Note that the infinite sum and integration commute (see (18) 2713
A2 (β)t2 + ... 2! n n n+1 (n+1) A (β)t A (β)t + + Φ1 (r, 0, β), (20) n! (n + 1) !
Φ1 (t, 0, β) = I + A(β)t +
where r ∈ (0, T ). Let Sn (t, 0, β) be a partial sum defined by n X Ak (β)tk Sn (t, 0, β) = . k! k=0
We can now construct polynomial control functions, ui , so that, for k ≤ n and 1 ≤ i ≤ m, Z T p t ui (t)dt = αik , if p = k, (21) 0 p! Z T p t ui (t)dt = 0, if p 6= k. (22) 0 p! This yields Z T Sn (t, 0, β)bi ui (t)dt = αik Ak (β)bi ,
1 ≤ i ≤ m. (23)
0
Thus, there exists ξ = LU ∈ R(L1 ), such that
Z T
kξ − η0 kK = Φ1 (0, t, β)BU (t)dt − αik Ak (β)bi K 0
Z T
≤ Φ1 (t, 0, β) − Sn (t, 0, β) BU (t)dt K 0
Z T
+ Sn (t, 0, β)BU (t)dt − αik Ak (β)bi
[7] J.-S. Li and N. Khaneja, “Ensemble control of bloch equations,” IEEE Transactions on Automatic Control, vol. 54, pp. 528–536, 2009. [8] J.-S. Li, “Ensemble control of finite-dimensional time-varying linear systems,” IEEE Transastions on Automatic Control, vol. 56, pp. 345– 357, 2011. [9] J.-S. Li, I. Dasanayake, and J. Ruths, “Control and synchronization of neuron ensembles,” IEEE Transactions on Automatic Control, vol. 58, no. 8, pp. 1919–1930, 2013. [10] J. Ruths and J.-S. Li, “Optimal control of inhomogeneous ensembles,” IEEE Transactions on Automatic Control: Special Issue on Control of Quantum Mechanical Systems, vol. 57, no. 8, pp. 2021–2032, 2012. [11] A. Zlotnik and J.-S. Li, “Synthesis of optimal ensemble controls for linear systems using the singular value decomposition,” in American Control Conference, (Montreal), pp. 5849–5854, 2012. [12] D. Stefanatos and J.-S. Li, “Minimum-time quantum transport with bounded trap velocity,” IEEE Transactions on Automatic Control (in press). [13] R. Brockett, “On the control of a flock by a leader,” Proceedings of the Steklov Institute of Mathematics, vol. 268, no. 1, pp. 49–57, 2010. [14] Y.-Y. Liu, J.-J. Slotine, and A.-L. Barabasi, “Controllability of complex networks,” Nature, vol. 473, no. 1, pp. 167–173, 2011. [15] J.-S. Li, “Control of a network of spiking neurons,” in 8th IFAC Symposium on Nonlinear Control Systems, (Italy), Sep. 2010. [16] I. Gohberg, S. Goldberg, and M. A. Kaashoek, Basic Classes of Linear Operators. Boston, MA: Birkh¨auser Verlag, 2003. [17] E. Lavretsky and K. Wise, Robust and Adaptive Control: With Aerospace Applications (Advanced Textbooks in Control and Signal Processing). Springer-Verlag, 2012. ¨ [18] C. H. M¨untz, “Uber den approximationssatz von weierstrass,” H. A. Schwarz’s Festschrift, pp. 303–312, 1914.
K
0
≤ ǫM T,
(24)
by the convergence of Φ1 (t, 0, β) and by (23). The same procedures can be applied to obtain as in P∞ Pma similark result k (24) for any element η = α β A b ∈ L0 . ik i k=0 i=1 Consequently, we conclude that L0 ⊆ R(L1 ). B. M¨untz−Sz´asz Theorem [18] Let {λi }∞ i=1 be a sequence with inf i λi > 0. Then span{1, xλ1 , xλ2 , . . .}, is dense in C[0, 1] if and only if ∞ X 1 = ∞. λ i=1 i
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