JOURNAL OF LATEX CLASS FILES, VOL. 11, NO. 4, DECEMBER 20121

Maximum Hands-Off Control: A Paradigm ofControl Effort Minimization
Masaaki Nagahara, Senior Member, IEEE,
Daniel E. Quevedo, Senior Member, IEEE,
Dragan Nešić, Fellow, IEEE

Abstract—In this paper, we propose a paradigm of control,
called a maximum hands-off control. A hands-off control isdefined as a control that has a short support per unit time.
The maximum hands-off control is the minimum support (orsparsest) per unit time among all controls that achieve control objectives. For finite horizon continuous-time control, weshow the equivalence between the maximum hands-off control
and L1 -optimal control under a uniqueness assumption called
normality. This result rationalizes the use of L1 optimality incomputing a maximum hands-off control. The same result isobtained for discrete-time hands-off control. We also proposean L1 /L2 -optimal control to obtain a smooth hands-off control.
Furthermore, we give a self-triggered feedback control algorithm
for linear time-invariant systems, which achieves a given sparsity
rate and practical stability in the case of plant disturbances. An
example is included to illustrate the effectiveness of the proposed
control.
Index Terms—Hands-off control, sparsity, L1 -optimal control,
self-triggered control, stability, nonlinear systems

I. I NTRODUCTION
In practical control systems, we often need to minimize
the control effort so as to achieve control objectives underlimitations in equipment such as actuators, sensors, and networks. For example, the energy (or L2 -norm) of a control
signal can be minimized to prevent engine overheating or to
reduce transmission cost by means of a standard LQ (linear
quadratic) control problem; see e.g., [1]. Another example isthe minimum fuel control, discussed in e.g., [2], [3], in which
the total expenditure of fuel is minimized with the L1 normof the control.
Alternatively, in some situations, the control effort can bedramatically reduced by holding the control value exactly zero
over a time interval. We call such control a hands-off control.
A motivation for hands-off control is a stop-start system
in automobiles. It is a hands-off control; it automaticallyshuts down the engine to avoid it idling for long periodsof time. By this, we can reduce CO or CO2 emissions aswell as fuel consumption [14]. This strategy is also used in
A preliminary version of parts of this work was presented in [37].
This research is supported in part by JSPS KAKENHI Grant Numbers
26120521, 15K14006, and 15H02668.
M. Nagahara is with Graduate School of Informatics, Kyoto University,
Kyoto, 606-8501, Japan; email: nagahara@ieee.org
D. E. Quevedo is with the Department of Electrical Engineering (EIM-E),
The University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany;
email: dquevedo@ieee.org
D. Nešić is with Department of Electrical and Electronic Engineering, The University of Melbourne, Victoria 3010 Australia; email:dnesic@unimelb.edu.auelectric/hybrid vehicles [8]; the internal combustion engine isstopped when the vehicle is at a stop or the speed is lower
than a preset threshold, and the electric motor is alternatively
used. Thus hands-off control also has potential for solving environmental problems. In railway vehicles, hands-off control,
called coasting, is used to reduce energy consumption [34].
Furthermore, hands-off control is desirable for networked andembedded systems since the communication channel is not
used during a period of zero-valued control. This property isadvantageous in particular for wireless communications [28],[32] and networked control systems [36], [26], [38], [31].
Motivated by these applications, we propose a paradigm ofcontrol, called maximum hands-off control that maximizes thetime interval over which the control is exactly zero.
The hands-off property is related to sparsity, or the L0“norm” (the quotation marks indicate that this is not a norm;
see Section II below) of a signal, defined by the total length of
the intervals over which the signal takes non-zero values. The
maximum hands-off control, in other words, seeks the sparsest(or L0 -optimal) control among all admissible controls. The
notion of sparsity has been recently adapted to control systems,
including works on model predictive control [36], [19], [22],[39], [38], system gain analysis [41], sparse controller design[17], state estimation [9], to name a few. The maximum handsoff control is also related to the minimum attention control[5], and also to the approach by Donkers et al. [12], which
maximizes the time between consecutive execution of thecontrol tasks. The minimum attention control minimizes thenumber of switching per unit time. In contrast, the maximum
hands-off control does not necessarily minimize the numberof switching, although we show this number is bounded forlinear systems.
The maximum hands-off control (or L0 -optimal control)
problem is hard to solve since the cost function is non-convex
and discontinuous.1 To overcome the difficulty, one can adopt
L1 optimality as a convex relaxation of the problem, as often
used in compressed sensing [13], [6]. Compressed sensing has
shown by theory and experiments that sparse high-dimensionalsignals can be reconstructed from incomplete measurements
by using `1 optimization; see e.g., [15], [16], [23] for details.
Interestingly, a finite horizon L1 -optimal (or minimum fuel)control has been known to have such a sparsity property, traditionally called ”bang-off-bang” [3]. Based on this, L1 -optimal
1 Very recently, Lp control with p ∈ [0, 1) has been investigated in [27],which introduces regularization terms to guarantee the existence of optimal
solutions.

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JOURNAL OF LATEX CLASS FILES, VOL. 11, NO. 4, DECEMBER 2012

control has been recently investigated for designing sparsecontrol [33], [7], [29]. Although advantage has implicitly beentaken of the sparsity property for minimizing the L1 norm, weare not aware of results on the theoretical connection between
sparsity and L1 optimality of the control. In the present
manuscript, we prove that a solution to an L1 -optimal control
problem gives a maximum hands-off control, and vice versa.
As a result, the sparsest solution (i.e., the maximum handsoff control) can be obtained by solving an L1 -optimal control
problem. The same result is obtained for discrete-time handsoff control. We also propose L1 /L2 -optimal control to avoid
the discontinuous property of ”bang-off-bang” in maximum
hands-off control. We show that the L1 /L2 -optimal control is
an intermediate control between the maximum hands-off (orL1 -optimal) control and the minimum energy (or L2 -optimal)
control, in the sense that the L1 and L2 controls are thelimiting instances of the L1 /L2 -optimal control.
We also extend the maximum hands-off control to feedback
control for linear time-invariant, reachable, and nonsingularsystems by a self-triggering approach [42], [35], [24], [4].
For this, we define sparsity of infinite horizon control signals
by the sparsity rate, the L0 norm per unit time. We givea self-triggered feedback control algorithm that achieves agiven sparsity rate and practical stability in the presenceof plant disturbances. Simulations studies demonstrate theeffectiveness of the proposed control method.
The present manuscript expands upon our recent conference
contribution [37] by incorporating feedback control into theformulation.
The remainder of this article is organized as follows: In Section II, we give mathematical preliminaries for our subsequentdiscussion. In Section III, we formulate the maximum handsoff control problem. Section IV is the main part of this paper,in which we introduce L1 -optimal control as relaxation of the
maximum hands-off control, and establish the theoretical connection between them. We also analyze discrete-time handsoff control in this section. In Section V, we propose L1 /L2 optimal control for a smooth hands-off control in this section.
In Section VI, we address the feedback hands-off control.
Section VII presents control design examples to illustrate
the effectiveness of our method. In Section VIII, we offer
concluding remarks.
II. M ATHEMATICAL P RELIMINARIESFor a vector x ∈ Rn , we define its norm by√
kxk , x> x,

and for a matrix A ∈ Rn×n ,
kAk ,

max
n

x∈R ,kxk=1

kAxk.

For a continuous-time signal u(t) over a time interval [0, T ],we define its Lp norm with p ∈ [1, ∞) by
!1/p
Z
T

kukp ,

0|u(t)|p dt,(1)

and let Lp [0, T ] consist of all u for which kukp < ∞. Notethat we can also define (1) for p ∈ (0, 1), which is not anorm (It fails to satisfy the triangle inequality.). We define thesupport set of u, denoted by supp(u), the closure of the set{t ∈ [0, T ] : u(t) 6= 0}.
Then we define the L0 “norm” of measurable function u as
the length of its support, that is,
kuk0 , mL supp(u) ,where mL is the Lebesgue measure on R. Note that the L0“norm” is not a norm since it fails to satisfy the positive
homogeneity property, that is, for any non-zero scalar α suchthat |α| =6 1, we havekαuk0 = kuk0 6= |α|kuk0 ,∀u 6= 0.

The notation k · k0 may be however justified from the fact that
if u ∈ L1 [0, T ], then kukp < ∞ for any p ∈ (0, 1) andlim kukpp = kuk0 ,p→0which can be proved by using Lebesgue’s monotone convergence theorem [40]. For more details of Lp when p ∈ [0, 1),see [30]. For a function f = [f1 , . . . , fn ]> : Rn → Rn , theJacobian f 0 is defined by
 ∂f1
∂f1 
. . . ∂x∂x1n..  ,..
f 0 (x) ,  ...
.
. 
∂fn
∂fn
. . . ∂xn
∂x1
where x = [x1 , . . . , xn ]> . For functions f and g, we denoteby f ◦ g the composite function f (g(·)).
III. M AXIMUM H ANDS -O FF C ONTROL P ROBLEM
In this section, we formulate the maximum hands-off control
problem. We first define the sparsity rate, the L0 norm of asignal per unit time, of finite-horizon continuous-time signals.
Definition 1 (Sparsity rate): For measurable function u on
[0, T ], T > 0, the sparsity rate is defined by1
kuk0 .
(2)
T
Note that for any measurable u, 0 ≤ RT (u) ≤ 1. If
RT (u)  1, we say u is sparse.2 The control objective is,roughly speaking, to design a control u which is as sparse
as possible, whilst satisfying performance criteria. For that
purpose, we will first focus on finite T and then, in Section VI,
study the infinite horizon case, where T → ∞.
To formulate the control problem, we consider nonlinear
multi-input plant models of the form
RT (u) :=

m
 X

dx(t)
= f x(t) +g i x(t) ui (t),dti=1t ∈ [0, T ],(3)

where x(t) ∈ Rn is the state, u1 , . . . , um are the scalar control
inputs, f and g i are functions on Rn . We assume that f (x),g i (x), and their Jacobians f 0 (x), g 0i (x) are continuous. We
use the vector representation u , [u1 , . . . , um ]> .
2 This is analogous to the sparsity of a vector. When a vector has a small
number of non-zero elements relative to the vector size, then it is called sparse.
See [15], [16], [23] for details.

SHELL et al.: BARE DEMO OF IEEETRAN.CLS FOR JOURNALS

3

The control {u(t) : t ∈ [0, T ]} is chosen to drive the statex(t) from a given initial statex(0) = ξ,(4)

to the origin at a fixed final time T > 0, that is,x(T ) = 0.

(5)

Also, the components of the control u(t) are constrained inmagnitude by
max |ui (t)| ≤ 1,
(6)i

for all t ∈ [0, T ]. We call a control {u(t) : t ∈ [0, T ]} ∈L1 [0, T ] admissible if it satisfies (6) for all t ∈ [0, T ], andthe resultant state x(t) from (3) satisfies boundary conditions(4) and (5). We denote by U(T, ξ) the set of all admissible
controls.
To consider control in U(T, ξ), it is necessary that U(T, ξ) is
non empty. This property is basically related to the minimumtime control formulated as follows:
Problem 2 (Minimum-time control): Find a control u ∈L1 [0, T ] that satisfies (6), and drives x from initial stateξ ∈ Rn to the origin 0 in minimum time.
Let T ∗ (ξ) denote the minimum time (or the value function)of Problem 2. Also, we define the reachable set as follows:3
Definition 3 (Reachable set): We define the reachable set attime t ∈ [0, ∞) by
R(t) , {ξ ∈ Rn : T ∗ (ξ) ≤ t} .

(7)hands-off control problem, which gives the exact solution of
Problem 4 under some assumptions.
Remark 5: The input constraint (6) is necessary. Let us
consider the integrator ẋ(t) = u(t) and remove the constraint(6). Then for any  > 0, the following control is an admissible
control(
ξ/, t ∈ [0, ),u (t) =0,t ∈ [, T ],which has arbitrarily small L0 norm. But lim→0 u is not afunction, so called Dirac’s delta, and hence is not in L1 . In
this case, the maximum hands-off problem has no solution.
IV. S OLUTION TO M AXIMUM H ANDS -O FF C ONTROLP ROBLEM
In this section we will show how the maximum hands-off
control can be solved in closed form.
A. Convex Relaxation
Here we consider convex relaxation of the maximum handsoff control problem. We replace kui k0 in (9) with L1 norm
kui k1 , and obtain the following L1 -optimal control problem,
also known as minimum fuel control discussed in e.g. [2], [3].
Problem 6 (L1 -optimal control): Find an admissible controlu ∈ U(T, ξ) on [0, T ] that minimizes

and the reachability set
R,

[

t≥0

R(t).

(8)

To guarantee that U(T, ξ) is non-empty, we introduce thestanding assumptions:1) ξ ∈ R,
2) T > T ∗ (ξ).
Now let us formulate our control problem. The maximum
hands-off control is a control that is the sparsest among all
admissible controls in U(T, ξ). In other words, we try to find acontrol that maximizes the time interval over which the control
u(t) is exactly zero. 4 We state the associated optimal control
problem as follows:
Problem 4 (Maximum hands-off control): Find an admissible
control on [0, T ], u ∈ U(T, ξ), that minimizes the sum ofsparsity rates:
J0 (u) ,

m
X
i=1

m

λi RT (ui ) =1X
λi kui k0 ,
T i=1

(9)

where λ1 > 0, . . . , λm > 0 are given weights.
This control problem is quite difficult to solve since theobjective function is highly nonlinear and non-smooth. In thenext section, we discuss convex relaxation of the maximum
3 For linear systems, the reachable set is known to have nice properties suchas convexity and compactness [25], [20].
4 More precisely, the maximum hands-off control minimizes the Lebesgue
measure of the support. Hence, the values on the sets of measure zero are
ignored and treated as zero in this setup.

J1 (u) ,

Z T
m
m
1X
1X
λi kui k1 =λi|ui (t)|dt,
T i=1
T i=1
0

(10)

where λ1 > 0, . . . , λm > 0 are given weights.
The objective function (10) is convex in u and this control
problem is much easier to solve than the maximum handsoff control problem (Problem 4). The main contribution ofthis section is that we prove the solution set of Problem 6
is equivalent to that of Problem 4, under the assumption of
normality. Before proving this property, we review L1 -optimal
control in the next subsection.
B. Review of L1 -Optimal Control
Here we briefly review the L1 -optimal control problem(Problem 6) based on the discussion in [3, Section 6-13].
Let us first form the Hamiltonian function for the L1 optimal control problem as


m
m
 X
1X
>
H(x, p, u) =λi |ui | + p f x +g i (x)ui ,
T i=1
i=1
(11)
where p is the costate (or adjoint) vector [3, Section 5-7].
Assume that u∗ = [u∗1 , . . . , u∗m ]> is an L1 -optimal control andx∗ is the resultant state trajectory. According to the minimum
principle, there exists a costate p∗ such that the optimal controlu∗ satisfiesH x∗ (t), p∗ (t), u∗ (t) ≤ H x∗ (t), p∗ (t), u(t) ,4

JOURNAL OF LATEX CLASS FILES, VOL. 11, NO. 4, DECEMBER 2012

Dλ (w)

1
−λ

0

w
λ

−1

Fig. 1. Dead-zone function Dλ (w)for for all t ∈ [0, T ] and all admissible u. The optimal statex∗ and costate p∗ satisfies the canonical equationsm Xdx∗ (t)= f x∗ (t) +g i x∗ (t) u∗i (t),
dt
i=1

>
dp∗ (t)= −f 0 x∗ (t) p∗ (t)
dt
m
X
>
u∗i (t)g 0i x∗ (t) p∗ (t),−i=1with boundary conditionsx∗ (0) = ξ,x∗ (T ) = 0.

The minimizer u∗ = [u∗1 , . . . , u∗m ]> of the Hamiltonian in (11)is given by>u∗i (t) = −Dλi /T g i x∗ (t) p∗ (t) , t ∈ [0, T ],where Dλ (·) : Rn → [−1, 1] is the dead-zone (set-valued)
function defined by
−1, if w < −λ,Dλ (w) = 0,
if − λ < w < λ,1,if λ < w,(12)
Dλ (w) ∈ [−1, 0], if w = −λ,Dλ (w) ∈ [0, 1], if w = λ.

See Fig. 1 for the graph of Dλ (·).
If g i (x∗ )> p∗ is equal to −λi /T or λi /T over a non-zerotime interval, say [t1 , t2 ] ⊂ [0, T ], where t1 < t2 , then thecontrol ui (and hence u) over [t1 , t2 ] cannot be uniquelydetermined by the minimum principle. In this case, the interval[t1 , t2 ] is called a singular interval, and a control problem thathas at least one singular interval is called singular. If there isno singular interval, the problem is called normal:
Definition 7 (Normality): The L1 -optimal control problem
stated in Problem 6 is said to be normal if the set∗> ∗Ti , {t ∈ [0, T ] : |T λ−1i g i (x (t)) p (t)| = 1}

is countable for i = 1, . . . , m. If the problem is normal, theelements t1 , t2 , · · · ∈ Ti are called the switching times for thecontrol ui (t).

If the problem is normal, the components of the L1 -optimal
control u∗ (t) are piecewise constant and ternary, taking values±1 or 0 at almost all5 t ∈ [0, T ]. This property, named ”bangoff-bang,” is the key to relate the L1 -optimal control with the
maximum hands-off control as discussed in the next section.
In general, it is difficult to check if the problem is normal
without solving the canonical equations [3, Section 6-22]. For
linear plants, however, a sufficient condition for normality is
obtained [3, Theorem 6-13].
C. Maximum Hands-Off Control and L1 Optimality
In this section, we study the relation between maximum
hands-off control stated in Problem 4 and L1 -optimal control
stated in Problem 6. The theorem below rationalizes the use of
L1 optimality in computing the maximum hands-off control.
Theorem 8: Assume that the L1 -optimal control problem(Problem 6) is normal and has at least one solution. Let U0∗and U1∗ be the sets of the optimal solutions of Problem 4 (maximum hands-off control problem) and Problem 6, respectively.
Then we have U0∗ = U1∗ .
Proof: By assumption, U1∗ is non-empty, and so isU(T, ξ), the set of all admissible controls. Also we haveU0∗ ⊂ U(T, ξ). We first show that U0∗ is non-empty, and thenprove that U0∗ = U1∗ .
First, for any u ∈ U(T, ξ), we have
Z T
m
1X
λi
|ui (t)| dt
J1 (u) =
T i=1
0
Z
m
1X
=
λi
|ui (t)| dt
(13)
T i=1
supp(ui )
Z
m
1X
≤
λi
1 dt = J0 (u).
T i=1
supp(ui )Now take an arbitrary u∗1 ∈ U1∗ . Since the problem is normal
by assumption, each control u∗1i (t) in u∗1 (t) takes values −1,0, or 1, at almost all t ∈ [0, T ]. This implies thatZ T
m
1X
J1 (u∗1 ) =λi|u∗1i (t)| dt
T i=1
0
(14)
Z
m
1X
∗
λi
1 dt = J0 (u1 ).
=T i=1
supp(u∗1i )From (13) and (14), u∗1 is a minimizer of J0 , that is, u∗1 ∈ U0∗ .
Thus, U0∗ is non-empty and U1∗ ⊂ U0∗ .
Conversely, let u∗0 ∈ U0∗ ⊂ U(T, ξ). Take independently∗
u1 ∈ U1∗ ⊂ U(T, ξ). From (14) and the optimality of u∗1 , wehave
J0 (u∗1 ) = J1 (u∗1 ) ≤ J1 (u∗0 ).
(15)

On the other hand, from (13) and the optimality of u∗0 , wehave
J1 (u∗0 ) ≤ J0 (u∗0 ) ≤ J0 (u∗1 ).
(16)

It follows from (15) and (16) that J1 (u∗1 ) = J1 (u∗0 ), and henceu∗0 achieves the minimum value of J1 . That is, u∗0 ∈ U1∗ andU0∗ ⊂ U1∗ .

5 Throughout this paper, “almost all” means “all but a set of Lebesgue
measure zero.”SHELL et al.: BARE DEMO OF IEEETRAN.CLS FOR JOURNALS

5Theorem 8 suggests that L1 optimization can be used for the
maximum hands-off (or the L0 -optimal) solution. The relation
between L1 and L0 is analogous to the situation in compressedsensing, where `1 optimality is often used to obtain the sparsest(i.e. `0 -optimal) vector; see [15], [16], [23] for details.
Finally, we show that when the system is linear, the numberof switching in the maximum hands-off control is bounded.
Proposition 9: Suppose that the plant is given by a linear
systemmXdx(t)= Ax(t) +bi ui (t),dt
i=1where A ∈ Rn×n and b1 , . . . , bm ∈ Rn . Assume that(A, b1 ), . . . , (A, bm ) are all controllable and A is nonsingular.
Assume also that the horizon length T > 0 (for given initialstate x(0) = ξ ∈ R) is chosen such that an L1 -optimal control
exists. Let ω be the largest imaginary part of the eigenvaluesof A. Then, the maximum hands-off control is a piecewise
constant signal, with values −1, 0, and 1, with no switchesfrom +1 to −1 or −1 to +1, and with 2nm(1 + T ω/π)
discontinuities at most.
Proof: Since (A, b1 ), . . . , (A, bm ) are controllable and A
is nonsingular, the L1 -optimal control problem is normal [3,Theorem 6-13]. Then, by Theorem 8, the maximum hands-off
control is identical to the L1 -optimal control. Combining thiswith Theorem 3.2 of [21] gives the results.
D. Discrete-time hands-off control
Here we consider discrete-time hands-off control. We assume the plant model is given by

x[k + 1] = f x[k] +mXg i x[k] ui [k], k = 0, 1, . . . , N − 1,

where kvk`0 denotes the number of the nonzero elements ofv ∈ RN . The associated `1 -optimal control problem is givenbyminimize J1 (u),u∈U [N,ξ]J1 (u) ,m
m N −11 XX
1 X
λi kui k`1 =λi |ui [k]|.
N i=1
N i=1

(20)

k=0

For the `1 -optimal control problem, we define the Hamiltonian
H(x, p, u) by
!
m
m
X


1 X
H(x, p, u) ,λi |ui | + p> f x +g i x ui ,
N i=1
i=1
where p denotes the costate for the `1 -optimal control problem. Let u∗ be an `1 -optimal control, and x∗ and p∗ are theassociated state and costate, respectively. Then the discretetime minimum principle [18] gives
H(x∗ [k], p∗ [k + 1], u∗ [k]) ≤ H(x∗ [k], p∗ [k + 1], u[k]),for k = 0, 1, . . . , N − 1 and all admissible u ∈ U[N, ξ]. From
this, the `1 -optimal control u∗i (if it exists) satisfies


> ∗
∗
∗
ui [k] = −Dλi /N g i x [k] p [k + 1] ,where Dλ (·) is the dead-zone function defined in (12) (see also
Fig. 1). Based on this, we define the discrete-time normality.
Definition 10 (Discrete-time normality): The discrete-time`1 -optimal control problem is said to be normal if
> ∗
∗
N λ−1
p [k + 1] 6= 1,i g i x [k](17)where x[k] ∈ Rn is the discrete-time state, u1 [k], . . . , um [k]are the discrete-time scalar control inputs, f and g i arefunctions on Rn . We assume that f (x), g i (x), f 0 (x), andg 0i (x) are continuous. We use the vector notation u[k] ,[u1 [k], . . . , um [k]]> .
The control {u[0], u[1], . . . , u[N − 1]} is chosen to drive
the state x[k] from a given initial state x[0] = ξ to the originx[N ] = 0. The components of the control u[k] are constrainedin magnitude byfor k = 0, 1, . . . , N − 1.
Then we have the following result:
Theorem 11: Assume that the discrete-time `1 -optimal control problem described in (20) is normal and has at least
one solution. Let U0∗ and U1∗ be the sets of the solutions of
the maximum hands-off control problem in (19) and the `1 optimal control problem in (20), respectively. Then we have
U0∗ = U1∗ .
Proof: The theorem can be proved using the same ideas
used in the proof of Theorem 8. Details are omitted for sakeof brevity.

max |ui [k]| ≤ 1, k = 0, 1, . . . , N − 1.

V. L1 /L2 -O PTIMAL C ONTROL

i

i=1

(18)

We call a control {u[0], . . . , u[N − 1]} admissible (as in thecontinuous-time case) if it satisfies (18) and the resultant statex[k] from (17) satisfies x[0] = ξ and x[N ] = 0. We denoteby U[N, ξ] the set of all admissible controls. We assume that
N is sufficiently large so that the set U[N, ξ] is non-empty.
For the admissible control, we consider the discrete-time
maximum hands-off control (or `0 -optimal control) definedbym
1 X
λi kui k`0 ,(19)minimize J0 (u), J0 (u) ,N i=1u∈U [N,ξ]In the previous section, we have shown that the maximum
hands-off control problem can be solved via L1 -optimal control. From the ”bang-off-bang” property of the L1 -optimal
control, the control changes its value at switching times
discontinuously. This is undesirable for some applications inwhich the actuators cannot move abruptly. In this case, one
may want to make the control continuous. For this purpose,
we add a regularization term to the L1 cost J1 (u) defined in(10). More precisely, we consider the following mixed L1 /L2 optimal control problem.

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
sat Sλ/θ (v)Sλ/θ (v)

1
−λ/θ

v
0

λ/θ

−λ/θ

v
0

λ/θ
−1


Fig. 3. Saturated shrinkage function sat Sλ/θ (v)Fig. 2. Shrinkage function Sλ/θ (v)Problem 12 (L1 /L2 -optimal control): Find an admissible
control on [0, T ], u ∈ U(T, ξ), that minimizes

m 
θi
1X
λi kui k1 + kui k22
J12 (u) ,
T i=1
2
(21)


Z
m
1X T
θi
=
λi |ui (t)| + |ui (t)|2 dt,
T i=1 02
where λi > 0 and θi > 0, i = 1, . . . , m, are given weights.
To discuss the optimal solution(s) of the above problem, wenext give necessary conditions for the L1 /L2 -optimal control
using the minimum principle of Pontryagin.
The Hamiltonian function associated to Problem 12 is given
by

m 
X
θi
λi |ui | + |ui |2
H(x, p, u) =
2
i=1


m
X
g i (x)ui+ p> f (x) +i=1

where p is the costate vector. Let u∗ denote the optimal
control and x∗ and p∗ the resultant optimal state and costate,
respectively. Then we have the following result.
Lemma 13: The i-th element u∗i (t) of the L1 /L2 -optimal
control u∗ (t) satisfies
n

o
>
u∗i (t) = − sat Sλi /θi θi−1 g i x∗ (t) p∗ (t) ,(22)
where Sλ/θ (·) is the shrinkage function defined by
v + λ/θ if v < −λ/θ,
Sλ/θ (v) , 0,
if − λ/θ ≤ v ≤ λ/θ,v − λ/θ, if λ/θ < v,

and sat(·) is the saturation function defined by
−1, if v < −1,sat(v) , v,
if − 1 ≤ v ≤ 1,
1,
if 1 < v.

See Figs. 2 and 3 for the graphs of Sλ/θ (·) and sat Sλ/θ (·) ,respectively.

Proof: The result is easily obtained upon noting thatθ− sat Sλ/θ θ−1 a = arg min λ|u| + |u|2 + au,2|u|≤1

for any λ > 0, θ > 0, and a ∈ R.
From Lemma 13, we have the following proposition.
Proposition 14 (Continuity): The L1 /L2 -optimal control∗
u (t) is continuous in t over [0, T ].
Proof: Without loss of generality, we assume m = 1 (asingle input plant), and omit subscripts for u, θ, λ, and so on.
Letū(x, p) , − sat Sλ/θ θ−1 g(x)> p .
Since functions sat ◦Sλ/θ (·) and g(·) are continuous,ū(x, p) is also continuous in x and p. It follows from
Lemma 13 that the optimal control u∗ given in (22) is
continuous in x∗ and p∗ . Hence, u∗ (t) is continuous, if x∗ (t)and p∗ (t) are continuous in t over [0, T ].
The canonical system for the L1 /L2 -optimal control is givenbydx∗ (t)= f x∗ (t) + g x∗ (t) ū x∗ (t), p∗ (t) ,dt>dp∗ (t)= −f 0 x∗ (t) p∗ (t)dt>− ū x∗ (t), p∗ (t) g 0 x∗ (t) p∗ (t).

Since f (x), g(x), f 0 (x), and g 0 (x) are continuous in x byassumption, and so is ū(x, p) in x and p, the right hand sideof the canonical system is continuous in x∗ and p∗ . From acontinuity theorem of dynamical systems, e.g. [3, Theorem 314], it follows that the resultant trajectories x∗ (t) and p∗ (t)are continuous in t over [0, T ].
Proposition 14 motivates us to use the L1 /L2 optimizationin Problem 12 for continuous hands-off control.
In general, the degree of continuity (or smoothness) and thesparsity of the control input cannot be optimized at the sametime. The weights λi or θi can be used for trading smoothness
for sparsity. Lemma 13 suggests that increasing the weight
λi (or decreasing θi ) makes the i-th input ui (t) sparser (seealso Fig. 3). On the other hand, decreasing λi (or increasingθi ) smoothens ui (t). In fact, we have the following limiting
properties.

SHELL et al.: BARE DEMO OF IEEETRAN.CLS FOR JOURNALS

7

Proposition 15 (Limiting cases): Assume the L1 -optimal
control problem is normal. Let u1 (λ) and u12 (λ, θ) be
solutions to respectively Problems 6 and 12 with parametersλ , (λ1 , . . . , λm ),Optimal Control
1

θ , (θ1 , . . . , θm ).

L1/L2 optimal
max hands−off

0.5

lim u12 (λ, θ) = u1 (λ).

θ→0

u(t)

1) For any fixed λ > 0, we have02) For any fixed θ > 0, we have
lim u12 (λ, θ) = u2 (θ),

−0.5

λ→0

where u2 (θ) is an L2 -optimal (or minimum energy)
control discussed in [3, Chap. 6], that is, a solution to acontrol problem where J1 (u) in Problem 6 is replacedwithZm
1 X θi T|ui (t)|2 dt.
(23)
J2 (u) =T i=1 2 0

−1
0

4
6
time (sec)810

Fig. 4. Maximum hands-off control (dashed) and L1 /L2 -optimal control(solid)Proof: The first statement follows directly from the factthat for any fixed λ > 0, we have
lim sat Sλ/θ (θ−1 w) = Dλ (w), ∀w ∈ R \ {±λ},state variables xi(t)
6
x1
x2
x3
x4
u

5

θ→0

4

3

xi(t)

where Dλ (·) is the dead-zone function defined in (12). The
second statement derives from the fact that for any fixed θ > 0,
we have
lim sat Sλ/θ (v) = sat(v), ∀v ∈ R.

2

2

λ→0

In summary, the L1 /L2 -optimal control is an intermediate
control between the L1 -optimal control (or the maximum
hands-off control) and the L2 -optimal control.
Example 16: Let us consider the following linear system

 

2
0 −1 0 00
dx(t) 
1
0
0
0

 
=
0 1 0 0 x(t) + 0 u(t).
dt
0
0 0 1 0We set the final time T = 10, and the initial and final states
asx(0) = [1, 1, 1, 1]> , x(10) = 0.

Fig. 4 shows the L1 /L2 optimal control with weights λ1 =
θ1 = 1. The maximum hands-off control is also illustrated.
We can see that the L1 /L2 -optimal control is continuous
but sufficiently sparse. Fig. 5 shows the state trajectories ofxi (t), i = 1, 2, 3, 4. By the sparse L1 /L2 control, each state
approaches zero within time T = 10.
VI. S ELF -T RIGGERED H ANDS -O FF F EEDBACK C ONTROLIn the previous section, we have shown that the maximum
hands-off control is given by the solution to an associatedL1 -optimal control problem. The L1 -optimal control can becomputed, for example, via convex optimization after time discretization. However, it is still difficult to give optimal control
as a function of the state variable x(t). This is a drawback if
there exist uncertainties in the plant model and disturbances

1

0

−1

−2

0

1

2

3

4

5
time (sec)

6

7

8

9

10

Fig. 5. State trajectory by L1 /L2 -optimal control

added to the signals. Therefore, we extend maximum handsoff control to feedback control. In this section, we assume
the controlled plant model is given by a single-input, linear
time-invariant systemdx(t)= Ax(t) + bu(t) + d(t),dtt ∈ [0, ∞),(24)

where A ∈ Rn×n and b ∈ Rn are given constants, and d(t) ∈Rn denotes an unknown plant disturbance. For a nonlinear
plant, one can use (24) as a linearized model and d(t) as thelinearization error (see Section VII). We assume that
1) (A, b) is reachable,
2) A is nonsingular.
This is a sufficient condition so that the L1 -optimal control problem with the single-input linear system (24) in the
disturbance-free case where d ≡ 0 is normal for any horizon
length T > 0 and any initial condition x(0) ∈ R [3, Theorem
6-13].

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JOURNAL OF LATEX CLASS FILES, VOL. 11, NO. 4, DECEMBER 2012

A. Sparsity Rate for Infinite Horizon Signals
Before considering feedback control, we define the sparsity
rate for infinite horizon signals (cf. Definition 1).
Definition 17 (Sparsity rate): For infinite horizon signal u ={u(t) : t ∈ [0, ∞)}, we define the sparsity rate by1(25)
u|[0,T ] 0 ,T →∞ T
where u|[0,T ] is the restriction of u to the interval [0, T ]. Notethat
1) If kuk0 < ∞, then R∞ (u) = 0.
2) If |u(t)| > 0 for almost all t ∈ [0, ∞), then R∞ (u) = 1.
3) For any measurable function u on [0, ∞), we have 0 ≤R∞ (u) ≤ 1.
We say again that an infinite horizon signal u is sparse if thesparsity rate R∞ (u)  1.
Lemma 18: Let u be a measurable function on [0, ∞). If
there exist time instants t0 , t1 , t2 , . . . such thatR∞ (u) , lim

t0 = 0, tk+1 = tk + Tk , Tk > 0,
RTk u|[tk ,tk+1 ] ≤ r, ∀k ∈ {0, 1, 2, . . . },then R∞ (u) ≤ r.
Proof: The following calculation proves the statement.
1
u|[0,T ] 0
R∞ (u) = lim
T →∞ TN −1
1 X
u|[tk ,tk+1 ] 0= lim
N →∞ tN
k=0
N −1


1 X
= lim
(tk+1 − tk )RTk u|[tk ,tk+1 ]N →∞ tN
k=0

1

≤ lim

N →∞ tN(tN − t0 )r=r

Fix a bound on the sparsity rate R∞ (u) ≤ r with r ∈ (0, 1).
We here propose a feedback control algorithm that achieves
the sparsity rate r of the resultant control input. Our method
involves applying maximum hands-off control over finite horizons, and to use self-triggered feedback to compensate for
disturbances. In self-triggered control, the next update time isdetermined by the current plant state.
First, let us assume that an initial state x(0) = x0 ∈ Rn is
given. For this, we compute the minimum-time T ∗ (x0 ), thesolution of the minimum-time control. Then, we define thefirst sampling period (or the first horizon length) byT0 , max Tmin , r−1 T ∗ (x0 ) ,where Tmin is a given positive time length that prevents the
sampling period from zero (thereby avoiding Zeno executions[43]). For this horizon length, we compute the maximum
hands-off control on the interval [0, T0 ]. Let this optimal
control be denoted u0 (t), t ∈ [0, T0 ], that isu∈U (T0 ,x0 )uk = arg min kuk0 .

t ∈ [0, T0 ],(27)u∈U (Tk ,xk )Apply u(t) = uk (t − tk ), t ∈ [tk , tk+1 ] to the plant.
end for

where U(T0 , x0 ) is the set of admissible control on timeinterval [0, T0 ] with initial state x0 ; see Section III. Apply
this control, u0 (t), to the plant (24) from t = 0 to t = T0 . If
d ≡ 0 (i.e. no disturbances), then x(T0 ) = 0 by the terminal
constraint, and applying u(t) = 0 for t ≥ T0 gives x(t) = 0
for all t ≥ T0 .
However, if d 6≡ 0, then x(T0 ) will in general not be exactly
zero. To steer the state to the origin, we should again apply
a control to the plant. Let x1 , x(T0 ), and t1 , T0 . We
propose to compute the minimum time T ∗ (x1 ) and let
T1 , max Tmin , r−1 T ∗ (x1 ) .

For this horizon length T1 , we compute the maximum handsoff control, u1 (t), t ∈ [t1 , t1 + T1 ], as well, which is applied
to the plant on the time interval [t1 , t1 + T1 ].
Continuing this process gives a self-triggered feedback
control algorithm, described in Algorithm 1, which results in
an infinite horizon control
u(t) = uk (t − tk ),B. Control Algorithmu0 (t) = arg min kuk0 ,Algorithm 1 Self-triggered Hands-off Control
Given initial state x0 and minimum inter-sampling time
Tmin .
Let x(0) = x0 and t0 = 0.
for k = 0, 1, 2, . . . do
Measure xk := x(tk ).
Compute T ∗ (xk).
Put Tk := max Tmin , r−1 T ∗ (xk ) .
Put tk+1 := tk + Tk .
Compute max hands-off controlt ∈ [tk , tk+1 ],k = 0, 1, 2, . . . , (26)where uk is defined in (27). For this control, we have thefollowing proposition.
Proposition 19 (Sparsity rate): For the infinite horizon
control u in (26), the sparsity rate R∞ (u) is less than r.
Proof: Fix k ∈ {0, 1, 2, . . . }. Let xk , x(tk ). The k-th
horizon length Tk is given by
Tk = max Tmin , r−1 T ∗ (xk ) .
(28)

Let us first consider the case when Tmin ≤ r−1 T ∗ (xk ), or
Tk = r−1 T ∗ (xk ). Let u∗k (t) denote the minimum-time control
for initial state xk , and define(u∗k (t), t ∈ [0, T ∗ (xk )],ũk (t) :=(29)0,t ∈ (T ∗ (xk ), r−1 T ∗ (x)].
Note that T ∗ (xk ) < r−1 T ∗ (xk ) since r ∈ (0, 1). Clearly thisis an admissible control, that is, ũk ∈ U(Tk , xk ), andkũk k0 = ku∗k k0 = T ∗ (xk ),for which see also Fig. 6. On the other hand, let uk denote

SHELL et al.: BARE DEMO OF IEEETRAN.CLS FOR JOURNALS

92) T ∗ (x) ≤ α(kxk), ∀x ∈ R, where R is the reachable
set defined in Definition 3

minimum-time control u∗k

Proof: For v ≥ 0, define

1α(v) , max T ∗ (ξ).
kξk≤v

0
−1

T ∗ (xk )Tkadmissible control ũkFig. 6. Minimum-time control u∗k (t) and admissible control ũk (t) definedin (29).

the maximum hands-off control on time interval [0, Tk ] withinitial state xk . Since uk has the minimum L0 norm, we have∗

kuk k0 ≤ kũk k0 = T (xk ).
It follows that the sparsity rate of uk (t − tk ), t ∈ [tk , tk + Tk ]is
T ∗ (xk )1= r.
kuk k0 ≤ −1 ∗RTk (uk ) =Tkr T (xk )Next, for the case when Tmin ≥ r−1 T ∗ (xk ), we have Tmin >T ∗ (x). It follows that RTk (uk ) ≤ r by a similar argument. In
either case, we have RTk (uk ) ≤ r for k = 0, 1, 2, . . . . Finally,
Lemma 18 gives the result.
Remark 20 (Minimum time computation): Algorithm 1 includes computation of the minimum time T ∗ (xk ). For singleinput, linear time-invariant system, an efficient numericalalgorithm has been proposed in [10], which one can use forthe computation. Also, this can be used to check whether the
initial state x0 lies in the reachable set R.

By this definition, it is easy to see that if v1 ≥ v2 then α(v1 ) ≥α(v2 ). Since T ∗ (ξ) is continuous on R (see [20]), α(v) is
continuous. The first statement is a result from T ∗ (0) = 0.
Then, setting v = kxk for x ∈ R gives the second statement.
Now, we have the following stability theorem.
Theorem 23: Assume that the plant noise is bounded by
δ > 0, that is, kd(t)k ≤ δ for all t ≥ 0. Assume also that theinitial state x(0) = x0 is in the reachable set R, and let
T0 , max{Tmin , r−1 T ∗ (x0 )}.

(31)Ω , {x ∈ Rn : kxk ≤ γ} ,δ  µ(A)T0
e
−1 ,
γ,
µ(A)

(32)

α(γ) ≤ rT0 ,(33)

Defineand assume Ω ⊂ R. Choose a function α which satisfies theconditions in Lemma 22. Ifthen the feedback control with Algorithm 1 achieves practicalstability in the sense that1) x(t) is bounded for t ∈ [0, t1 ].
2) xk , x(tk ) ∈ Ω, ∀k ∈ {1, 2, . . . }.
3) For t ∈ [tk , tk+1 ], k ∈ {1, 2, . . . }, we have kx(t)k ≤ h,where if µ(A) < 0
h=γ+C. Practical Stability
By the feedback control algorithm (Algorithm 1), the statex(t) is sampled at sampling instants tk , k = 1, 2, . . . , andbetween sampling instants the system acts as an open loop
system. Since there exists disturbance d(t), it is impossible toasymptotically stabilize the feedback system to the origin. We
thus focus on practical stability of the feedback control system
under bounded disturbances. The following are fundamental
lemmas to prove the stability.
Lemma 21: For A ∈ Rn×n , we haveeAt ≤ eµ(A)t ,∀t ∈ [0, ∞),where µ(A) is the maximum eigenvalue of (A + A> )/2, thatis,

A + A>.
(30)µ(A) = λmax
2
Proof: This can be easily proved by a general theorem of
the matrix measure [11, Theorem II.8.27].
Lemma 22: There exists a scalar-valued, continuous, andnon-decreasing function α : [0, ∞) → [0, ∞) such that1) α(0) = 0,

kbk + δ, h1 ,|µ(A)|and if µ(A) > 0
h = h1 eµ(A) max{Tmin ,r

−1

α(γ)}

−

kbk + δ.
µ(A)Proof: Since the system is linear time-invariant and u(t)and d(t) are bounded, the state x(t) is also bounded on [0, t1 ].
For t = t1 , we have
Z T0
kx1 k = kx(t1 )k ≤eA(T0 −τ ) δdτ0Z T0≤eµ(A)(T0 −τ ) δdτ0δ  µ(A)T0
=
e
−1 ,
µ(A)
and hence x1 = x(t1 ) ∈ Ω. Note that since x0 ∈ R, we have
T0 < ∞. Note also that since A is nonsingular, µ(A) 6= 0. Fix
k ∈ {1, 2, . . . }, and assume xk = x(tk ) ∈ Ω. Then we haveδ  µ(A)Tk
kxk+1 k ≤
e
−1 ,
µ(A)

10

JOURNAL OF LATEX CLASS FILES, VOL. 11, NO. 4, DECEMBER 2012

where Tk is as in (28). Note that Tk < ∞ since xk ∈ Ω ⊂ R.
If Tk = Tmin thenδ  µ(A)Tmine−1
kxk+1 k ≤µ(A)δ  µ(A)T0
−1 =γ
≤
e
µ(A)
since T0 ≥ Tmin . On the other hand, if Tk = r−1 T ∗ (xk ) thenδ  µ(A)r−1 T ∗ (xk )
e
−1
µ(A)

δ  µ(A)r−1 T ∗ (xk )=γ+e− eµ(A)T0 .
µ(A)kxk+1 k ≤(34)

Lemma 22, assumption xk ∈ Ω, and equation (33) giveT ∗ (xk ) ≤ α(kxk k) ≤ α(γ) ≤ rT0 ,eµ(A)r∗T (xk )− eµ(A)T0 ≤ 0.

From (34), we have kxk+1 k ≤ γ. In each case, we havexk+1 = x(tk+1 ) ∈ Ω.
Then, let us consider the intersample behavior of x(t), t ∈[tk , tk+1 ] for k = 1, 2, . . . . As proved above, we have xk =x(tk ) ∈ Ω. This gives
kx(t)k ≤ e+A(t−tk )Z te

kxk k +A(t−τ )tk=eeA(t−τ )tk

kbk|uk (t)|dτ

kd(τ )kdτ

≤ eµ(A)(t−tk ) kxk k +µ(A)(t−tk )

Z t

Z t

eµ(A)(t−τ ) dτ (kbk + δ)tk

kbk + δ  µ(A)(t−tk )kxk k +
e
−1 .
µ(A)

If µ(A) < 0 then x(t) is bounded as
kx(t)k ≤ kxk k +kbk + δkbk + δ≤γ+.
|µ(A)||µ(A)|

If µ(A) > 0 then x(t) is again bounded as
kx(t)k ≤ eµ(A)Tk kxk k +First, let us consider a simple example with a 1-dimensional
stable plant modeldx(t)= ax(t) + au(t) + d(t),(36)
dtwhere a < 0. We assume bounded disturbance, that is, there
exists δ > 0 such that |d(t)| ≤ δ for all t ≥ 0. The plant isnormal and hence the maximum hands-off control is given by
L1 -optimal control thanks to Theorem 8. In fact, the optimal
control uk in (27) is computed via the minimum principle forL1 -optimal control [3, Section 6.14] as(
0,t ∈ [0, τ ),uk (t) =− sgn(x(tk )), t ∈ [τ, Tk ],
where

(35)

and hence−1VII. E XAMPLE
kbk + δ  µ(A)Tk
e
−1
µ(A)

−1

≤ eµ(A) max{Tmin ,r α(γ)} γ
kbk + δ  µ(A) max{Tmin ,r−1 α(γ)}
+
e
−1 .
µ(A)
From (31) and (32), we conclude that the larger the sparsity
rate r, the smaller the upper bound γ. This shows there
is a tradeoff between the sparsity rate of control and theperformance. The analysis is deterministic and the bound isfor the worst-case disturbance, but this is reasonably tight insome cases when a worst-case disturbance is applied to thesystem, as shown in the example below.

τ,



1
log e|a|Tk − |x(tk )| .
|a|Also, the minimum time function T ∗ (x) is computed as (see[3, Example 6-4])
T ∗ (x) =1
log(1 + |x|),|a|x ∈ R.

It follows that the reachable set R = R, and the conditionΩ ⊂ R in Theorem 23 always holds. Since A = a ∈ R, wehave µ(A) = a by (30). Then, for any x ∈ R, we haveT ∗ (x) =1|x|log(1 + |x|) ≤
,
|a|
|a|

and hence we can choose α(v) = v/|a| for Lemma 22 andTheorem 23. The stability condition (33) becomes


δ
aT0
(1 − e ) ≤ rT0α|a|

or rT0 a2 ≥ δ(1 − eaT0 ).
For example, with a = −1, δ = 1, x0 = 1, and if we choose
Tmin < T ∗ (x0 ) = log(1 + |x0 |) = log 2, then rT0 = log 2 andthe condition becomeslog 2≈ 0.587.
r≥−log(1 − log 2)

We set r = 0.6 and simulate the feedback control with
disturbance d(t) as uniform noise with mean 0 and boundδ = 1. Fig. 7 shows the maximum hands-off control obtained by Algorithm 1. We can observe that the control issufficiently sparse. In fact, the sparsity rate for this control
is R∞ (u) = 0.148, which is smaller than the upper boundr = 0.6.
Since the plant is asymptotically stable, one can choose thezero control, that is, u ≡ 0, to achieve stability, which is thesparsest. Fig. 8 shows the state x(t) for the maximum handsoff control and the zero control. Due to the time optimalityof the hands-off control, the state approaches to 0 faster thanthat of the zero control.
Then let us consider the influence of disturbances. The
bound γ in (32) is computed as γ = 1 − exp(−r−1 log 2)with r = 0.6, and the set Ω becomesΩ = x ∈ R : |x| ≤ 1 − exp(−r−1 log 2) .

SHELL et al.: BARE DEMO OF IEEETRAN.CLS FOR JOURNALS

11state x(t)hands−off control input u(t)11.1
hands−off control
zero control: u(t)=0

0.8
1
0.6
0.9

0.4

amplitude

amplitude

0.2
0

0.8

0.7

−0.2
−0.4

0.6

−0.6
0.5
−0.8
−1

0

0.2

0.4

0.6

0.8

1
time (sec)

1.2

1.4

1.6

1.8

0.4

2

Fig. 7. Hands-off feedback control with sparsity rate R∞ (u) = 0.148.

0

1

2

3

4

5
time (sec)

6

7

8

9

10

Fig. 9. State trajectory with worst-case disturbance: hands-off control (solid)
and zero control (dots).

state x(t)1.2state x(t)hands−off control
zero control: u(t)=0
sampled state

1.2
hands−off control
zero control: u(t)=0
sampled state

1
1
0.8

amplitude

amplitude

0.8
0.6

0.4

0.6

0.4

0.2
0.2
0
0
−0.2

0

0.2

0.4

0.6

0.8

1
time (sec)

1.2

1.4

1.6

1.8

2

Fig. 8. State trajectory: hands-off control (solid) and zero control (dots).
Sampled states x(tk ) are also shown (circles)

This bound is obtained in a deterministic manner, and hence
the bound is for the worst-case disturbance. In fact, let us apply
a worst-case disturbance d(t) = 1 for all t ≥ 0 to the feedback
system. Fig. 9 shows the state trajectories. The trajectory by
the zero control remains 1 and do not approache 0, while thatby the maximum hands-off control still approaches 0, and we
can see that the bound is reasonably tight.
Next, let us consider a nonlinear plant modeldx(t)= sin ax(t) + au(t).
(37)
dt
We linearize this nonlinear plant to obtain the linear
plant(36), with the linearization error d(t) , sin ax(t) − ax(t).
Assume a = −1 (i.e. stable). We adopt the control law given
as above to the nonlinear plant (37). Fig. 10 shows the result.
This figure shows that the hands-off control works well for the
nonlinear plant (37). The sparsity rate of the hands-off control
is R∞ (u) = 0.0717, which is sufficiently small.
On the other hand, let us consider the nonlinear plant (37)with a = 1 (i.e. unstable). For the linearized plant (37), the

−0.2

0

0.2

0.4

0.6

0.8

1
time (sec)

1.2

1.4

1.6

1.8

2

Fig. 10. State trajectory of nonlinear plant (37) with a = −1 (stable): andsoff control (solid), zero control (dots), and sampled states x(tk ) (circles).

hands-off control law is given by(− sgn(x(tk )), t ∈ [0, τ ),uk (t) =0,t ∈ [τ, Tk ],
where τ , −a−1 log 1−|x(tk )| . The minimumtime function
T ∗ (x) is given by T ∗ (x) = −a−1 log 1 − |x| for x ∈ R,
where R = (−1, 1). We set the initial state x0 = 0.25 andthe sparsity rate r = 0.6, and simulate the feedback control
with the nonlinear plant (37). Fig. 11 shows the obtained
state trajectory of (37). Obviously, the zero control cannot
stabilize the unstable plant and hence the state diverges, while
the hands-off control keeps the state close to the origin. The
sparsity rate is R∞ (u) = 0.1135, which is sufficiently small.
VIII. C ONCLUSION
In this paper, we have proposed maximum hands-off control.
It has the minimum support per unit time, or is the sparsest,
among all admissible controls. Under normality assumptions,

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JOURNAL OF LATEX CLASS FILES, VOL. 11, NO. 4, DECEMBER 2012

state x(t)0.5
hands−off control
zero control: u(t)=0
sampled state
0.4

amplitude

0.3

0.2

0.1

0

−0.1

−0.2

0

0.2

0.4

0.6

0.8

1
time (sec)

1.2

1.4

1.6

1.8

2

Fig. 11. State trajectory of the nonlinear plant (37) with a = 1 (unstable):hands-off control (solid), zero control (dots), and sampled states x(tk )(circles).

the maximum hands-off control can be computed via L1 optimal control. For linear systems, we have also proposed a
feedback control algorithm, which guarantees a given sparsity
rate and practical stability. An example has illustrated theeffectiveness of the proposed control. Future work includes
the development of an effective computation algorithm for
maximum hands-off control, for situations when the control
problem does not satisfy normality conditions, and also when
the plant is nonlinear.
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SHELL et al.: BARE DEMO OF IEEETRAN.CLS FOR JOURNALS

Masaaki Nagahara (S’00–M’03–SM’14) received
the Bachelor’s degree in engineering from Kobe University in 1998, the Master’s degree and the Doctoral
degree in informatics from Kyoto University in 2000
and 2003. He is currently a Senior Lecturer at Graduate School of Informatics, Kyoto University. His
research interests include digital signal processing,
networked control, and sparse modeling. He received
Young Authors Award in 1999 and Best Paper Awardin 2012 from SICE, Transition to Practice Award
from IEEE Control Systems Society in 2012, andBest Tutorial Paper Award from IEICE Communications Society in 2014. He
is a senior member of IEEE, and a member of SIAM, SICE, ISCIE and IEICE.

Daniel E. Quevedo (S’97–M’05–SM’14) holds thechair in Automatic Control (Regelungs- und Automatisierungstechnik) at the University of Paderborn,
Germany. He received Ingeniero Civil Electrónicoand M.Sc. degrees from the Universidad Técnica
Federico Santa Marı́a, Chile, in 2000. In 2005, he
was awarded the Ph.D. degree from The Universityof Newcastle, Australia, where he subsequently held
various research academic positions. His researchinterests include several areas within automatic control, signal processing, and power electronics. Dr.
Quevedo was supported by a full scholarship from the alumni association
during his time at the Universidad Técnica Federico Santa Marı́a and received several university-wide prizes upon graduating. He received the IEEE
Conference on Decision and Control Best Student Paper Award in 2003 andwas also a finalist in 2002. In 2009 he was awarded a five-year Research
Fellowship from the Australian Research Council. Prof. Quevedo is Editorof the International Journal of Robust and Nonlinear Control and serves aschair of the IEEE Control Systems Society Technical Committee on Networks& Communication Systems.

Dragan Nešić (F’08) is a Professor at the Department of Electrical and Electronic Engineering(DEEE) at The University of Melbourne, Australia.
He received his B.E. degree in Mechanical Engineering from The University of Belgrade,Yugoslavia
in 1990, and his Ph.D. degree from Systems Engineering, RSISE, Australian National University,
Canberra, Australia in 1997. Since February 1999
he has been with The University of Melbourne.
His research interests include networked control systems, discrete-time, sampled-data and continuoustime nonlinear control systems, input-to-state stability, extremum seeking
control, applications of symbolic computation in control theory, hybrid control
systems, and so on. He was awarded a Humboldt Research Fellowship(2003) by the Alexander von Humboldt Foundation, an Australian Professorial
Fellowship (2004–2009) and Future Fellowship (2010–2014) by the Australian
Research Council. He is a Fellow of IEEE and a Fellow of IEAust. He iscurrently a Distinguished Lecturer of CSS, IEEE (2008–). He served as an
Associate Editor for the journals Automatica, IEEE Transactions on AutomaticControl, Systems and Control Letters and European Journal of Control.

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