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sensor guidance for a linear diffusion process
Available online at www.sciencedirect.com
Journal of the Franklin Institute 356 (2019) 7246–7262 www.elsevier.com/locate/jfranklin
Integrated design of switching control and mobile actuator/sensor guidance for a linear diffusion process Huai-Ning Wu∗, Xiao-Wei Zhang The Science and Technology on Aircraft Control Laboratory, School of Automation Science and Electrical Engineering, Beihang University (Beijing University of Aeronautics and Astronautics), Beijing 100191, China Received 25 September 2018; received in revised form 28 May 2019; accepted 27 June 2019 Available online 2 July 2019
Abstract This paper is concerned with the exponential stabilization problem for a class of diffusion processes described by a linear parabolic partial differential equation (PDE) using mobile collocated actuators and sensors. Each collocated actuator/sensor pair is capable of moving within the respective spatial domain divided in advance and a mode indicator function is employed to indicate the different modes for all actuator/sensor pairs according to whether each actuator/sensor pair is static or mobile. By utilizing Lyapunov direct method, an integrated design of switching controllers and mobile actuator/sensor guidance laws for the diffusion process is developed such that the resulting closed-loop system is exponentially stable. Finally, numerical simulations are presented to illustrate the effectiveness of the proposed design method. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction It is well known that many industrial processes are inherently distributed parameter systems (DPSs), such as diffusion, thermal, fluid flow and chemical reactor processes, which are usually described by partial differential equations (PDEs). Over the past two decades, extensive research has been carried out for the analysis and control design of DPSs [1–4]. In particular, ∗
Corresponding author. E-mail addresses: [email protected], [email protected] (H.-N. Wu), [email protected] (X.-W. Zhang). https://doi.org/10.1016/j.jfranklin.2019.06.045 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
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there has been a great amount of work dealing with the optimal placement issue of actuators and sensors in DPSs for improving the control performance [5–8]. All kinds of metrics can be adopted for this optimization, such as enhancing the system theoretic properties of observability and controllability [5], improving the state response speed of the closed-loop system [6,7] and minimizing the given performance index of the closed-loop system [8]. The past few years have witnessed rapidly growing interest in the use of mobile actuators and sensors in DPSs because they can provide a better effect on reducing power consumption and improving performance in many applications [9–12]. In [9], the Central Voronoi Tessellations technique was employed to allocate the mobile actuators into the desired positions for the control of a diffusion process. However, the result in [9] did not address the effect of the planned paths or desired trajectories of the mobile actuators on the stability of the diffusion process in theory. In view of this shortcoming, some researchers have devoted considerable effort toward dealing with the guidance problem of mobile actuators and sensors for DPSs based on the Lyapunov stability theory [10–12]. In [10], a stable guidance scheme for each mobile collocated actuator/sensor pair in linear DPSs was developed in an abstract framework via Lyapunov stability arguments and used to improve the control performance. In [11], a Lyapunov-based combined control law-plus-guidance law was further provided in an abstract framework for a class of 2-D linear diffusion PDE systems using mobile collocated actuator/sensor pairs with augmented vehicle dynamics. More recently, a guidance scheme for non-collocated actuators and sensors in linear DPSs has been given [12]. But, it is worth pointing out that the results in [10–12] require that the open-loop operator is coercive, which limits their practical applications. In this paper, we propose a linear matrix inequality (LMI) approach [13,14] to the integrated design of switching control and mobile actuator/sensor guidance for exponentially stabilizing a class of diffusion processes described by a linear parabolic PDE. The novelty and main contribution of this paper can be summarized as follows: (i) The Lyapunov functional depending on the rest and motion modes of each actuator/sensor pair is constructed to deal with the integrated design; (ii) The switching control laws and mobile actuator/sensor guidance laws are designed simultaneously to exponentially stabilize the diffusion process without the coercivity restriction for the open-loop operator; (iii) Compared with static actuators and sensors, the employment of mobile actuators and sensors can improve the state response speed of the closed-loop diffusion process. Notations: R denotes the set of all real numbers. The superscript T is used for the transpose of a matrix or a vector. For a matrix Q = QT , Q < ( ) 0 means that it is negative definite (negative semidefinite). λmax (·) and λmin (·) represent the maximal and minimal eigenvalues of a matrix, respectively. | · | indicate the absolute value. Given a constant L > 0, for some scalar function ω(x) ∈ R, x ∈ [0, L] ⊂ R, we denote ω(·){ω(x), x ∈ [0, L]}. X L2 ((0, L); R ) is a real Hilbert space of square integrable scalar functions ω(x) : [0, L] → R with the inner L √ product ω1 (·), ω2 (·) 0 ω1 (x)ω2 (x)dx and norm ω1 (·)2 ω1 (·), ω1 (·), where ω1 (·) and ω2 (·) are two elements of X. H(·) denotes the Heaviside step function, defined as 0, x < H (x − ) = 1, x . The Dirac delta function δ(·) on [x1 , x2 ] is defined as follows: x = 0, δ(x − ) , x = ∞ x2 δ(x − ) dx = 1 , ∈ (x1 , x2 ). x1
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The symbol ∗ is used as an ellipsis for terms in matrix expressions that are induced by symmetry, e.g., S + [M + ∗] X S + [M + M T ] X . ∗ Y XT Y 2. Problem formulation and preliminaries This paper considers a class of diffusion processes described by the following linear parabolic PDE: zt (t, x) = γ zxx (t, x) + θ z(t, x) +
m
b(x ; xia (t ))ui (t )
(1)
i=1
L
yi (t ) = 0
c(x ; xis (t ))z(t , x )dx , i ∈ S {1, 2, . . . , m}
(2)
subject to the Dirichlet boundary conditions z(t, 0) = z(t, L) = 0
(3)
and the initial condition z(0, x) = z0 (x)
(4)
where γ > 0 and θ > 0 are the diffusion parameter and reaction term coefficient, respectively; z(t, x) ∈ R denotes the state; x ∈ [0, L] ⊂ R and t ∈ [0, ∞) stand for the spatial and temporal coordinate, respectively; the subscripts x and t stand for the partial derivatives with respect to x and t, respectively; b(x ; xia (t )) (i ∈ S) denotes the spatial distribution of the ith mobile actuator, which describes how the control input ui (t ) ∈ R is distributed in the interval [0, L]; xia (t ) ∈ [0, L] indicates the spatial position of the ith mobile actuator; yi (t ) ∈ R is the measurement output of the ith mobile sensor; c(x ; xis (t )) (i ∈ S) is a known function determined by the shape (point or distributed) of the ith mobile sensor and xis (t ) ∈ [0, L] indicates the spatial position of the ith mobile sensor; z0 (x) is the initial value. In this paper, the mobile actuators and sensors are collocated (i.e., xia (t ) = xis (t ), i ∈ S), and the spatial distribution functions b(x ; xia (t )) and c(x ; xia (t )) (i ∈ S) are taken as b(x ; xia (t )) = c(x ; xia (t )) = δ(x − xia (t )), i ∈ S.
(5)
To avoid the mobile actuator/sensor pairs colliding with each other and achieve the control design objective, we give the following assumption: Assumption 2.1. Let the points l1 , l2 , . . . , lm+1 divide [0, L] into m subintervals, denoted by [li , li+1 ], i ∈ S, where 0 = l1 < l2 < · · · < lm+1 = L. Define the sets iε [li + κε, li+1 − κε] ⊂ (li , li+1 ), i ∈ S, where κ > 1 and ε is a known small positive scalar. The actuator/sensor pair Ai (Ai is the identity of the ith actuator/sensor pair and i ∈ S) only can move in the space domain iε (i.e., xia (t ) ∈ iε ). Remark 2.1. In this work, the integrated design of switching control and mobile actuator/sensor guidance is under the premise that xia (t ) ∈ iε , i ∈ M, which will be taken into account in the guidance design. Without considering this premise, the designed guidance laws
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may make the ith actuator/sensor pair move out of the space domain iε divided in advance, which may lead to the mobile actuator/sensor pairs colliding with each other and the ineffectiveness of the control design. Further, in order to implement the mobile actuator/sensor guidance, another assumption is made on the ability of the actuator/sensor pairs as follows: Assumption 2.2. The ith actuator/sensor pair can also provide the state information at the two edges of its range, in the form of z(t, xia (t ) − ε) and z(t, xia (t ) + ε) (i ∈ S). Note that for each i ∈ S, x˙ia (t ) = 0 corresponds to the mode of the ith actuator/sensor pair Ai rest, and x˙ia (t ) = 0 corresponds to the mode of Ai motion. Obviously, there exist 2m possible modes for m actuator/sensor pairs. Let us define the following mode indicator function for all Ai , i ∈ S: m g(t ) 2i−1 πi (t ) (6) i=1
where π i (t) is the characteristic function of Ai defined by 1, x˙ia (t ) = 0 πi (t ) = 0, x˙ia (t ) = 0. From Eq. (6), it follows that g(t ) ∈ M {0; 1; 2; . . . ; 2m − 1}. Obviously, the value of g(t) indicates the present mode of the m actuator/sensor pairs. When g(t ) = h, h ∈ M, then h we have πi (t ) = mod([ 2i−1 ], 2), i ∈ S, where mod(a, b) denotes the remainder from dividing a by b and [ξ ] indicates the biggest integer which is no more than ξ . For example, when g(t ) = 1, it means that π1 (t ) = 1 and πi (t ) = 0, i ∈ S/{1}. At some time instance tw , if g(tw ) changes from one value to another, it implies that one mode of the m actuator/sensor pairs is switched to another. Although a simple fixed static output feedback (SOF) control scheme ui (t ) = ki yi (t ), i ∈ S
(7)
where ki , i ∈ S are the control gains to be determined, can be employed to control the PDE system, the main drawback of such scheme is that the fixed control gain ki of the ith actuator/sensor pair Ai must work for all possible modes of the m actuator/sensor pairs, which can lead to a very conservative controller design. To avoid using the fixed control gain, we will adopt different control gains for different modes of the m actuator/sensor pairs. Considering the fact that one mode may be switched to any other at some time instance, we assume the switching time instances are t1 , . . . , tw , . . . , tN , where 0 N < ∞ is a integer. Let 0 = t0 < t1 < · · · < tw < · · · < tN < tN+1 = ∞. Then, we consider the following switching control laws: IF g(tw ) = h, w ∈ N {0, 1, 2, . . . , N } THEN ui (t ) = kh,i yi (t ) = kh,i z(t , xia (t )) i ∈ S, t ∈ [tw , tw+1 )
(8)
where kh,i , h ∈ M, i ∈ S are the control gains to be determined. Remark 2.2. In fact, if the number of switches reaches N¯ , we can set x˙ia (t ) = 0, i ∈ S for any t tN¯ , where N¯ is a predetermined upper limit, which guarantees that the number of switches is finite. Therefore, we can assume the number of switches is finite.
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Therefore, the design problem under consideration is to find the switching controllers of the form (8) and mobile actuator/sensor guidance laws of the form x˙ia (t ) = fi (t ), i ∈ S, where fi (t ), i ∈ S are the nonlinear functions to be designed, such that the resulting closed-loop PDE system is exponentially stable. To this end, the following definition and lemma are useful: Definition 2.1 [15]. The unforced system (1)–(4) (i.e., ui (t ) ≡ 0, i ∈ S) is said to be exponentially stable in the sense of L2 -norm, if there exist constants σ 1 > 0 and σ 2 > 0 such that the solution z(t, x) of the system satisfies z(t, ·)2 σ1 z0 (·)2 exp(−σ2t ). Lemma 2.1. (Wirtinger’s Inequality [16]) Let z(t, ·) ∈ L2 ([a, b]; R ) be an absolutely continuous scalar function with square integrable derivative zx (t, x). Then, if z(t, a) = 0 or z(t, b) = 0, the following inequality holds: z(t, ·)22 4(b − a)2 π −2 zx (t, ·)22 . 3. Main results In this section, we present an LMI-based approach to the integrated design of switching control laws and mobile actuator/sensor guidance laws for exponentially stabilizing the PDE system. Let us choose the mode-dependent Lyapunov functional candidate for the PDE system (1)–(4) as m 1 li+1 Vh (t ) = Ph (x ; xia (t ))z2 (t , x)dx 2 l i i=1 t ∈ [tw , tw+1 ), w ∈ N, h ∈ M where Ph (x ; xia (t ))
β (d (x ; xia (t )) + 2), = h 2βh ,
(9)
πi (t ) = 1 πi (t ) = 0, i ∈ S, h ∈ M
(10)
in which βh > 0, h ∈ M are the constants to be determined and d (x ; xia (t )) = H (x − (xia (t ) + ε)) − H (x − (xia (t ) − ε)), i ∈ S. In fact, according to the definition of H(·), d (x ; xia (t )) can be rewritten as −1, x ∈ [xia (t ) − ε, xia (t ) + ε) d (x ; xia (t )) = 0, otherwise.
(11)
Thus, we have βh Ph (x ; xia (t )) 2βh , i ∈ S, h ∈ M
(12)
which guarantees that the Lyapunov functional in Eq. (9) is positive definite. For t ∈ [tw , tw+1 ), computing the time derivative of Vh (t) along the solution of the system (1)–(4) yields m li+1 ˙ Vh (t ) = Ph (x ; xia (t ))z(t , x) γ zxx (t , x) i=1
li
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+ θ z(t, x) + +
m 1
2
i=1
m
7251
b(x ; x aj (t ))u j (t ) dx
j=1 li+1
li
x˙ia (t )
∂Ph (x ; xia (t )) 2 z (t, x )dx . ∂xia (t )
(13)
Utilizing integration by parts, we have li+1 Ph (x ; xia (t ))z(t , x)zxx (t , x)dx li
=Ph (x ; xia (t ))z(t , x)zx (t , x )|lli+1 i
li+1
− li
− li
+ mod([
li+1 li
Ph (x ; xia (t ))zx2 (t , x)dx
∂Ph (x ; xia (t )) z(t, x)zx (t, x)dx ∂x
=Ph (x ; xia (t ))z(t , x)zx (t , x)|lli+1 − i
−
li+1
h
2
], 2) i−1
xia (t )+ε xia (t )−ε
li
li+1
2βh zx2 (t , x)dx
βh zx2 (t, x)dx
∂Ph (x ; xia (t )) z(t, x)zx (t, x )dx . ∂x
(14)
a From Assumption 2.1, one has xia (t ) ∈ iε and xi+1 (t ) ∈ (i+1)ε , i ∈ S/{m} which ina a a a dicate li+1 ∈ / [xi (t ) − ε, xi (t ) + ε] ∪ [xi+1 (t ) − ε, xi+1 (t ) + ε], i ∈ S/{m}. Hence, it follows a from Eq. (11) that d (li+1 ; xia (t )) = d (li+1 ; xi+1 (t )) = 0, i ∈ S/{m}, and thus Ph (li+1 ; xia (t )) = a Ph (li+1 ; xi+1 (t )) = 2βh , i ∈ S/{m}, which implies that m
Ph (x ; xia (t ))z(t , x)zx (t , x)|lli+1 i
i=1 m−1 a Ph (li+1 ; xia (t )) − Ph (li+1 ; xi+1 = (t )) z(t , li+1 )zx (t , li+1 ) i=1
=0.
(15)
Using Eqs. (5), (8) and the following property of function δ(·) [17]: x2
f (x)δ(x − )dx = f (), ∈ (x1 , x2 ).
(16)
x1
we obtain li+1 Ph (x ; xia (t ))b(x ; xia (t ))ui (t )z(t , x)dx li
h , 2 βh kh,i z2 (t, xia (t )) = 2 − mod 2i−1
li+1 h 2βh kh,i 2 a , 2 βh kh,i z2 (t, xia (t )). = z (t, xi (t ))dx − mod li+1 − li 2i−1 li
(17)
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By defining z¯i (t, x) z(t, x) − z(t , xia (t ))
(18)
it is clear that z¯i (t, xia (t )) = 0, z¯i,x (t, x) = zx (t, x).
(19)
Using Lemma 2.1, Eqs. (18) and (19) yields li+1 − zx2 (t, x)dx
− − −
li
xia (t )
π2 z¯2 (t, x)dx a 4(xi (t ) − li )2 i li li+1 π2 z¯2 (t, x)dx a 2 i xia (t ) 4(li+1 − xi (t )) li+1 π2 z¯2 (t, x )dx . 4(li+1 − li )2 i li
(20)
From the definitions of H(·) and δ(·), it is immediate that x 0, x< δ(s − )ds = x2 δ(s − ) ds = 1 , x x1 x1 = H (x − ) which implies that dH (x − ) = δ(x − ). dx Using variable substitution, we can also easily get dH (−x − ) = −δ(−x − ). dx By Eqs. (21) and (22), it follows respectively that for x ∈ [li , li+1 ], i ∈ S
h ∂Ph (x ; xia (t )) , 2 βh δ(x − (xia (t ) + ε)) − δ(x − (xia (t ) − ε)) = mod i−1 ∂x 2 and
h ∂Ph (x ; xia (t )) , 2 βh δ(x − (xia (t ) − ε)) − δ(x − (xia (t ) + ε)) . = mod a i−1 ∂xi (t ) 2 Thus, by Eq. (16), we have from Eqs. (23) and (24), respectively li+1 ∂Ph (x ; xia (t )) z(t, x)zx (t, x)dx ∂x li
h , 2 βh z(t, xia (t ) + ε)zx (t, xia (t ) + ε) − z(t, xia (t ) − ε)zx (t, xia (t ) − ε) = mod i−1 2 and li
li+1
∂Ph (x ; xia (t )) 2 z (t, x)dx ∂xia (t )
(21)
(22)
(23)
(24)
(25)
=mod([ li
h 2i−1
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], 2)βh z2 (t, xia (t ) − ε) − z2 (t, xia (t ) + ε)) .
(26)
Using inequality (12), it is immediate that li+1 li+1 Ph (x ; xia (t ))z2 (t , x)dx 2βh z2 (t , x )dx .
(27)
li
Furthermore, from (13)–(15), (17), (20) and (25)–(27), we have m li+1 V˙h (t ) z˘iT (t, x)h,i z˘i (t, x)dx i=1
+
li
m i=1
mod
h 2i−1
,2
βh × x˙ia (t ) (z2 (t, xia (t ) − ε) − z2 (t, xia (t ) + ε)) 2 + γ βh z(t, xia (t ) − ε)zx (t, xia (t ) − ε) − γ βh z(t, xia (t ) + ε)zx (t, xia (t ) + ε) xia (t )+ε + βh zx2 (t, x)dx xia (t )−ε
− βh kh,i z2 (t, xia (t )) + ηi βh (li+1 − li )z2 (t, xia (t ))
(28)
where ηi 0, i ∈ S are adjustable parameters, z˘i (t, x) [z (t, x) z (t , xia (t ))]T , and
βh π 2 h,i11 2(lγi+1 2 −li ) , i ∈ S h,i ∗ h,i22 with γ βh π 2 + 2θ βh 2(li+1 − li )2 2βh kh,i h γ βh π 2 − + − mod([ i−1 ], 2)ηi βh . 2(li+1 − li )2 li+1 − li 2
h,i11 − h,i22
On the basis of Assumption 2.1 and inequality (28), we give the actuator/sensor guidance laws as follows: ⎧ 0, if z2 (t , xia (t ) − ε) − z2 (t , xia (t ) + ε) χc ⎪ ⎪ ⎨ or if xia (t ) li+1 − (κ + κc )ε and vh,i > 0 x˙ia (t ) = (29) or if xia (t ) li + (κ + κc )ε and vh,i < 0 ⎪ ⎪ ⎩ vh,i , otherwise, i ∈ S ¯ iε [li + (κ + κc )ε, li+1 − (κ + κc )ε], i ∈ S, where κ c > 0 with the initial positions xia (0) ∈ is an adjustable small positive scalar, χ c is the predetermined critical value and vh,i = − μi z2 (t, xia (t ) − ε) − z2 (t, xia (t ) + ε) 2βh−1 − 2 z (t, xia (t ) − ε) − z2 (t, xia (t ) + ε) × γ βh z(t, xia (t ) − ε)zx (t, xia (t ) − ε)
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− γ βh z(t, xia (t ) + ε)zx (t, xia (t ) + ε) xia (t )+ε + βh zx2 (t, x)dx xia (t )−ε
− βh kh,i z2 (t, xia (t )) + ηi βh (li+1 − li )z2 (t, xia (t )) , i ∈ S in which μi > 0, i ∈ S are adjustable parameters. Then, we have the following result. Theorem 3.1. Consider the PDE system (1)–(4). Given scalars μi > 0 and ηi ≥ 0, i ∈ S, for each h ∈ M, suppose there exist scalars β h > 0 and k¯ h,i , i ∈ S such that the following LMIs are satisfied:
2 hγ π h,i11 2(lβi+1 −li )2 < 0, i ∈ S (30) ¯ h,i22 ∗ with
h 2k¯ h,i βh γ π 2 + − mod , 2 ηi βh . 2(li+1 − li )2 li+1 − li 2i−1
¯ h,i22 −
Then the switching controllers (8) with control gains kh,i = k¯ h,i βh−1 , i ∈ S, h ∈ M
(31)
and the actuator/sensor guidance laws (29) with (31) can ensure that the resulting closed-loop system is exponentially stable. Proof. Set k¯ h,i = kh,i βh , i ∈ S, h ∈ M.
(32)
Assume that the LMIs in (30) hold. Then, from Eq. (32), we have h,i < 0, i ∈ S
(33)
which implies h,i −λmin (−h,i )I , i ∈ S
(34)
where I stands for the identity matrix with appropriate dimension. Consequently, using (28), (29) and (34), we obtain m li+1 ˙ Vh (t ) −λmin (−h,i )z2 (t, x)dx i=1
li
βh χc2 mod 2Vh (t ) i=1 m
−
λh βh χc2 − − 2βh 2Vh (tw )
, 2 μiVh (t ) i−1 h
2
m
mod
i=1
, 2 μi Vh (t ) i−1 h
2
t ∈ [tw , tw+1 )
(35)
where λh = mini∈S {λmin (−h,i )} > 0. From inequality (35), it follows that V˙h (t ) −
λh Vh (t ), t ∈ [tw , tw+1 ) 2βh
(36)
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which means that
λh Vh (t ) exp − (t − tw ) Vh (tw ). 2βh
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(37)
Without loss of generality, at tw , w ∈ N, we assume h = hw , where hw ∈ M. Then, referring max to (9) and (12), we can get Vhw (tw ) 2β V (t − ), where Vhw−1 (tw− ) denotes the left limit βmin hw−1 w of the Vhw−1 (t ) at tw , βmax = maxh∈M {βh } and βmin = minh∈M {βh }. Therefore, for t ∈ [tw , tw+1 ), w ∈ N, we have
λh Vhw (t ) exp − w (t − tw ) Vhw (tw ) 2βhw
λh 2βmax exp − w (t − tw ) Vhw−1 (tw− ) βmin 2βhw
λh 2βmax exp − w (t − tw ) βmin 2βhw
λh × exp − w−1 (tw − tw−1 ) Vhw−1 (tw−1 ) 2βhw−1 .. . ¯ 2βmax N λ ( ) exp − t Vh0 (0) βmin 2βmax ¯ = minh∈M {λh }. Then, for t ∈ [tw , tw+1 ), w ∈ N, where λ βmin z(t, ·)22 Vhw (t ) 2 ¯ (2βmax )N βmax λ z0 (·)22 exp(− t) N (βmin ) 2βmax i.e.,
z(t, ·)2
2βmax βmin
N+1 2
¯ λ z0 (·)2 exp − t . 4βmax
(38)
¯
max Letting σ1 = ( 2β ) 2 and σ2 = 4βλmax , from Definition 2.1, we can conclude from βmin inequality (38) that the closed-loop PDE system is exponentially stable. Moreover, from Eq. (32), we have Eq. (31). The proof is complete. N+1
Remark 3.1. It is worth mentioning that although the main result in Theorem 3.1 is developed for the case of Dirichlet boundary conditions (3), the similar results can be obtained for the cases of Neumann boundary conditions or mixed boundary conditions. Remark 3.2. From inequality (35), we can get that when g(tw ) = h, h ∈ M, the exponential βh χc2 m λh h m decay rate at [tw , tw+1 ) is 2β + i=1 mod ([ 2i−1 ], 2)μi . Especially, if h = 2 − 1, the 2V h h (tw ) β m −1 χc2 m λ2m −1 decay rate is 2β + 2V22m −1 i=1 μi and if h = 0, all of the adjustable parameters μi , i ∈ S (tw ) 2m −1 are absent. Therefore, provided the parameters μi , i ∈ S are chosen appropriately, compared with any other mode, the exponential decay rates for the modes h = 0 and h = 2m − 1 are the
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smallest and biggest, respectively. Namely, the employment of mobile actuator/sensor pairs can improve the state response speed of the closed-loop system. Remark 3.3. To implement the mobile actuator/sensor guidance, some state information in Eq. (29) can be approximated as follows: z(t, xia (t )) − z(t, xia (t ) − ε) ε z(t, xia (t ) + ε) − z(t, xia (t )) a zx (t, xi (t ) + ε) ≈ ε xia (t )+ε 2 a 2 a z (t, x (t i ) − ε) + zx (t, xi (t ) + ε) zx2 (t, x)dx ≈(2ε) x 2 xia (t )−ε zx (t, xia (t ) − ε) ≈
=ε(zx2 (t, xia (t ) − ε) + zx2 (t, xia (t ) + ε)). Obviously, in order to achieve the proposed switching control scheme, it is required to monitor the modes for the actuator/sensor pairs in real time, which increases the difficulty of controller implementation. If the modes are not available for switching due to the practical constraints, then we can turn to the fixed SOF control scheme given by Eq. (7). An LMI approach to the fixed SOF control design can be easily obtained from Theorem 3.1 by taking βh = β and k¯ h,i = k¯ i , h ∈ M, i ∈ S. In this case, the LMIs in (30) can be simplified as
π2 i11 2(lβγ 2 i+1 −li ) < 0, i ∈ S (39) ¯ i22 ∗ with i11 = −
2 2k¯ i βγ π 2 ¯ i22 = − βγ π + 2θ β, + 2 2 2(li+1 − li ) 2(li+1 − li ) li+1 − li
and the actuator/sensor guidance laws (29) become ⎧ 0, if z2 (t , xia (t ) − ε) − z2 (t , xia (t ) + ε) χc ⎪ ⎪ ⎨ or if xia (t ) li+1 − (κ + κc )ε and vi > 0 x˙ia (t ) = or if xia (t ) li + (κ + κc )ε and vi < 0 ⎪ ⎪ ⎩ vi , otherwise, i ∈ S ¯ iε , and with xia (0) ∈ 2 vi = − μi z (t, xia (t ) − ε) − z2 (t, xia (t ) + ε) 2β −1 − 2 z (t, xia (t ) − ε) − z2 (t, xia (t ) + ε) × γ βz(t, xia (t ) − ε)zx (t, xia (t ) − ε) − γ βz(t, xia (t ) + ε)zx (t, xia (t ) + ε) xia (t )+ε +β zx2 (t, x)dx xia (t )−ε
− βki z2 (t, xia (t )) + ηi β(li+1 − li )z2 (t, xia (t )) , i ∈ S in which μi > 0 and ηi 0, i ∈ S. Then, we have the following corollary.
(40)
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Fig. 1. Open-loop profile of evolution of z(t, x).
Corollary 3.1. Consider the PDE system (1)–(4). Given scalars μi > 0 and ηi 0, i ∈ S, if there exist scalars β > 0 and k¯ i , i ∈ S satisfying the LMIs in (39), then the fixed SOF controllers (7) with control gains ki = k¯ i β −1 , i ∈ S and actuator/sensor guidance laws (40) can ensure that the resulting closed-loop system is exponentially stable. Remark 3.4. It is easily seen that the LMI conditions in (39) correspond to the mode that all actuator/sensor pairs are in rest. Therefore, Corollary 3.1 can also provide an exponential stabilization condition for the PDE system (1)–(4) with the static actuator/sensor pairs (i.e., ¯ iε , i ∈ S). x˙ia (t ) ≡ 0 with the initial positions xia (0) ∈ Remark 3.5. Note that in this paper, the integrated design method is developed for a class of DPSs with deterministic system parameters. However, as is well known, many physical systems have variable structures and parameters subject to random changes. As a special class of hybrid systems, Markovian jump systems has attracted much attention [18–20], because they can represent various processes in different fields, such as manufacturing systems, fault-tolerant systems, communication systems, and economic systems. In the future, we will investigate the switching control design for DPSs with Markovian jump parameters under mobile actuators and sensors. Remark 3.6. It is worth pointing out that the design method in [10] requires that the openloop operator is coercive, which implies the zero equilibrium point of the open-loop system is stable. The main goal of [10] is to design mobile actuator/sensor guidance laws for improving the transient response of the closed-loop state. However, for the open-loop unstable parabolic PDE system, the design method of [10] is ineffective in theory and may not work well. In this paper, the proposed design method has not coercivity requirement and thus has a wider range of applications. 4. Numerical simulations To illustrate the effectiveness of the proposed design method, this section will provide numerical simulations for the PDE system (1)–(4) with γ = 0.02, θ = 0.23, L = 1, m = 3 and 2 z0 (x) = 0.32sin(π x)e−8x + 0.32sin(π x). With the above settings, it is obvious from Fig. 1 that the equilibrium point z(t, ·) = 0 of the open-loop system is unstable, which implies that the open-loop operator is not coercive.
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Table 1 Switching control gains. h
Gain of A1
0 1 2 3 4 5 6 7
k0,1 k1,1 k2,1 k3,1 k4,1 k5,1 k6,1 k7,1
= −0.2927 = −0.2777 = −0.2927 = −0.2777 = −0.2927 = −0.2777 = −0.2927 = −0.2777
Gain of A2 k0,2 k1,2 k2,2 k3,2 k4,2 k5,2 k6,2 k7,2
= −0.4862 = −0.4862 = −0.4562 = −0.4562 = −0.4862 = −0.4862 = −0.4562 = −0.4562
Gain of A3 k0,3 k1,3 k2,3 k3,3 k4,3 k5,3 k6,3 k7,3
= −0.2927 = −0.2927 = −0.2927 = −0.2927 = −0.2627 = −0.2627 = −0.2627 = −0.2627
Fig. 2. Closed-loop profile of evolution of z(t, x) with mobile actuator/sensor pairs under switching control scheme.
In this simulation study, we will use the following approximation to the delta function in Eq. (5): 1 if x ∈ [xia (t ) − ζ , xia (t ) + ζ ] δ(x − xia (t )) ≈ 2ζ 0, otherwise, i = 1, 2, 3 where ζ = 0.015. Set l1 = 0, l2 = 0.3, l3 = 0.7, l4 = 1, ε = 0.05, κ = 1.12, κc = 0.3 χc = 0.01, μ1 = 8, μ2 = 10, μ3 = 8, η1 = 0.1, η2 = 0.15, η3 = 0.2. Then we have 1ε = [0.056, 0.244], 2ε = [0.356, 0.644], 3ε = [0.756, 0.944] and ¯ 1ε = [0.071, 0.229], ¯ 2ε = [0.371, 0.629], ¯ 3ε = [0.771, 0.929]. Hence, the initial positions of the three mobile actuator/sensor pairs are chosen as ¯ 1ε , x2a (0) = 0.48 ∈ ¯ 2ε , x3a (0) = 0.88 ∈ ¯ 3ε . x1a (0) = 0.10 ∈
(41)
Solving the LMIs in (30) via the LMI Toolbox in Matlab, the switching control gains are presented in Table 1. Fig. 2 shows the state response of closed-loop system with mobile actuator/sensor pairs, which implies the resulting switching controllers and mobile actuator/sensor
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Fig. 3. Switching control actions of closed-loop system with mobile actuator/sensor pairs.
Fig. 4. Trajectories of mobile actuator/sensor pairs under switching control scheme.
guidance laws can stabilize the PDE system. Figs. 3 and 4 show the corresponding switching control actions and the trajectories of the mobile actuator/sensor pairs, respectively. In order to be more persuasive, we apply Corollary 3.1 to make the control performance comparison between two cases of using mobile and static actuator/sensor pairs. It is known from Corollary 3.1 that the resulting control gains for the two cases are the same. For the sake of comparisons, the actuator/sensor pairs are assumed to be fixed at the positions in Eq. (41) for the static case. Solving the LMIs in (39) via the LMI Toolbox in Matlab, we have the control gains k1 = −0.2927, k2 = −0.4862, k3 = −0.2927. Fig. 5 depicts the L2 -norm of the closed-loop system state using static and mobile actuator/sensor pairs with the initial positions in Eq. (41) under the fixed control scheme. It can be observed obviously that the mobile case exhibits faster state response speed than the
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Fig. 5. State L2 -norm for static and mobile cases under fixed control scheme.
Fig. 6. Spatial distributions of closed-loop state at different time instances for static (blue solid line) and mobile (black dotted line) cases under fixed control scheme. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
static one. The pointwise-in-space convergence of the state distribution at four different time instances is presented in Fig. 6. Clearly, the distributed state converges to zero much faster when the actuator/sensor pairs are mobile. 5. Conclusion This paper has investigated the exponential stabilization problem of a class of linear diffusion processes using mobile actuators and sensors. By utilizing the Lyapunov direct approach, the integrated design of switching controllers and mobile actuator/sensor guidance laws is de-
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veloped to exponentially stabilize the diffusion process. Finally, numerical simulations indicate that the proposed design method is effective and using mobile actuators/sensors can achieve faster state response speed. In the future work, we will focus on the switching control design for DPSs via mobile actuators and sensors in the presence of external disturbances, stochastic noises or Markovian jump parameters. Moreover, the mobile non-collocated actuators and sensors will be employed to deal with the control design issue of the PDE systems. Acknowledgments This work was supported in part by the National Natural Science Foundation for Distinguished Young Scholars of China under Grant 61625302, and in part by the National Natural Science Foundation of China under Grant 61473011 and Grant 61721091. The authors would like to thank the Associate Editor and the anonymous reviewers for their valuable comments and suggestions which have improved the presentation of this paper. References [1] W.H. Ray, Advanced Process Control, McGraw-Hill, New York, 1981. [2] R.F. Curtain, H. Zwart, An Introduction to Infinite-dimensional Linear Systems Theory, Springer-Verlag, New York, 1995. [3] W. He, T. Meng, D. Huang, X. Li, Adaptive boundary iterative learning control for an Euler-Bernoulli beam system with input constraint, IEEE Trans. Neural Netw. Learn. Syst. 29 (2018) 1539–1549. [4] J.M. Wang, L.L. Su, H.X. Li, Stabilization of an unstable reaction-diffusion PDE cascaded with a heat equation, Syst. Control Lett. 76 (2015) 8–18. [5] A. Armaoua, M.A. Demetriou, Optimal actuator/sensor placement for linear parabolic PDEs using spatial H2 norm, Chem. Eng. Sci. 61 (2006) 7351–7367. [6] P. Hébrard, A. Henrot, Optimal shape and position of the actuators for the stabilization of a string, Syst. Control Lett. 48 (2003) 199–209. [7] G. Zheng, B.Z. Guo, M.M. Ali, Continuous dependence of optimal control to controlled domain of actuator for heat equation, Syst. Control Lett. 79 (2015) 30–38. [8] C. Antoniades, P.D. Christofides, Integrated optimal actuator/sensor placement and robust control of uncertain transport-reaction processes, Comput. Chem. Eng. 26 (2002) 187–203. [9] Y.Q. Chen, Z. Wang, J. Liang, Actuation scheduling in mobile actuator networks for spatial-temporal feedback control of a diffusion process with dynamic obstacle avoidance, in: Proceedings of the IEEE International Conference on Mechatronics and Automation, Niagara Falls, Canada, 2005. [10] M.A. Demetriou, Guidance of mobile actuator-plus-sensor networks for improved control and estimation of distributed parameter systems, IEEE Trans. Autom. Control 55 (2010) 1570–1584. [11] M.A. Demetriou, Adaptive control of 2-D PDEs using mobile collocated actuator/sensor pairs with augmented vehicle dynamics, IEEE Trans. Autom. Control 57 (2012) 2979–2993. [12] W. Mu, B. Cui, W. Li, Z. Jiang, Improving control and estimation for distributed parameter systems utilizing mobile actuator-sensor network, ISA Trans. 53 (2014) 1087–1095. [13] S. Boyd, L. El-Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, PA, 1994. [14] P. Gahinet, A. Nemirovski, A.J. Laub, M. Chilali, LMI Control Toolbox for Use with Matlab, MathWorks, Natick, MA, 1995. [15] J.-W. Wang, H.-N. Wu, Lyapunov-based design of locally collocated controllers for semi-linear parabolic PDE systems, J. Frankl. Inst. 351 (2014) 429–441. [16] G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, Cambridge University Press, Cambridge, 1959. [17] V. Balakrishnan, All about the Dirac delta function (?), Resonance 8 (2003) 48–58. [18] J.W. Wang, H.N. Wu, H.X. Li, Stochastically exponential stability and stabilization of uncertain linear hyperbolic PDE systems with Markov jumping parameters, Automatica 48 (2012) 569–576.
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[19] L. Wu, X. Su, P. Shi, Sliding mode control with bounded L2 gain performance of Markovian jump singular time-delay systems, Automatica 48 (2012) 1929–1933. [20] F. Li, C. Du, C. Yang, W. Gui, Passivity-based asynchronous sliding mode control for delayed singular Markovian jump systems, IEEE Trans. Autom. Control 63 (2018) 2715–2721.
Journal of the Franklin Institute 356 (2019) 7246–7262 www.elsevier.com/locate/jfranklin
Integrated design of switching control and mobile actuator/sensor guidance for a linear diffusion process Huai-Ning Wu∗, Xiao-Wei Zhang The Science and Technology on Aircraft Control Laboratory, School of Automation Science and Electrical Engineering, Beihang University (Beijing University of Aeronautics and Astronautics), Beijing 100191, China Received 25 September 2018; received in revised form 28 May 2019; accepted 27 June 2019 Available online 2 July 2019
Abstract This paper is concerned with the exponential stabilization problem for a class of diffusion processes described by a linear parabolic partial differential equation (PDE) using mobile collocated actuators and sensors. Each collocated actuator/sensor pair is capable of moving within the respective spatial domain divided in advance and a mode indicator function is employed to indicate the different modes for all actuator/sensor pairs according to whether each actuator/sensor pair is static or mobile. By utilizing Lyapunov direct method, an integrated design of switching controllers and mobile actuator/sensor guidance laws for the diffusion process is developed such that the resulting closed-loop system is exponentially stable. Finally, numerical simulations are presented to illustrate the effectiveness of the proposed design method. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction It is well known that many industrial processes are inherently distributed parameter systems (DPSs), such as diffusion, thermal, fluid flow and chemical reactor processes, which are usually described by partial differential equations (PDEs). Over the past two decades, extensive research has been carried out for the analysis and control design of DPSs [1–4]. In particular, ∗
Corresponding author. E-mail addresses: [email protected], [email protected] (H.-N. Wu), [email protected] (X.-W. Zhang). https://doi.org/10.1016/j.jfranklin.2019.06.045 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
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there has been a great amount of work dealing with the optimal placement issue of actuators and sensors in DPSs for improving the control performance [5–8]. All kinds of metrics can be adopted for this optimization, such as enhancing the system theoretic properties of observability and controllability [5], improving the state response speed of the closed-loop system [6,7] and minimizing the given performance index of the closed-loop system [8]. The past few years have witnessed rapidly growing interest in the use of mobile actuators and sensors in DPSs because they can provide a better effect on reducing power consumption and improving performance in many applications [9–12]. In [9], the Central Voronoi Tessellations technique was employed to allocate the mobile actuators into the desired positions for the control of a diffusion process. However, the result in [9] did not address the effect of the planned paths or desired trajectories of the mobile actuators on the stability of the diffusion process in theory. In view of this shortcoming, some researchers have devoted considerable effort toward dealing with the guidance problem of mobile actuators and sensors for DPSs based on the Lyapunov stability theory [10–12]. In [10], a stable guidance scheme for each mobile collocated actuator/sensor pair in linear DPSs was developed in an abstract framework via Lyapunov stability arguments and used to improve the control performance. In [11], a Lyapunov-based combined control law-plus-guidance law was further provided in an abstract framework for a class of 2-D linear diffusion PDE systems using mobile collocated actuator/sensor pairs with augmented vehicle dynamics. More recently, a guidance scheme for non-collocated actuators and sensors in linear DPSs has been given [12]. But, it is worth pointing out that the results in [10–12] require that the open-loop operator is coercive, which limits their practical applications. In this paper, we propose a linear matrix inequality (LMI) approach [13,14] to the integrated design of switching control and mobile actuator/sensor guidance for exponentially stabilizing a class of diffusion processes described by a linear parabolic PDE. The novelty and main contribution of this paper can be summarized as follows: (i) The Lyapunov functional depending on the rest and motion modes of each actuator/sensor pair is constructed to deal with the integrated design; (ii) The switching control laws and mobile actuator/sensor guidance laws are designed simultaneously to exponentially stabilize the diffusion process without the coercivity restriction for the open-loop operator; (iii) Compared with static actuators and sensors, the employment of mobile actuators and sensors can improve the state response speed of the closed-loop diffusion process. Notations: R denotes the set of all real numbers. The superscript T is used for the transpose of a matrix or a vector. For a matrix Q = QT , Q < ( ) 0 means that it is negative definite (negative semidefinite). λmax (·) and λmin (·) represent the maximal and minimal eigenvalues of a matrix, respectively. | · | indicate the absolute value. Given a constant L > 0, for some scalar function ω(x) ∈ R, x ∈ [0, L] ⊂ R, we denote ω(·){ω(x), x ∈ [0, L]}. X L2 ((0, L); R ) is a real Hilbert space of square integrable scalar functions ω(x) : [0, L] → R with the inner L √ product ω1 (·), ω2 (·) 0 ω1 (x)ω2 (x)dx and norm ω1 (·)2 ω1 (·), ω1 (·), where ω1 (·) and ω2 (·) are two elements of X. H(·) denotes the Heaviside step function, defined as 0, x < H (x − ) = 1, x . The Dirac delta function δ(·) on [x1 , x2 ] is defined as follows: x = 0, δ(x − ) , x = ∞ x2 δ(x − ) dx = 1 , ∈ (x1 , x2 ). x1
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The symbol ∗ is used as an ellipsis for terms in matrix expressions that are induced by symmetry, e.g., S + [M + ∗] X S + [M + M T ] X . ∗ Y XT Y 2. Problem formulation and preliminaries This paper considers a class of diffusion processes described by the following linear parabolic PDE: zt (t, x) = γ zxx (t, x) + θ z(t, x) +
m
b(x ; xia (t ))ui (t )
(1)
i=1
L
yi (t ) = 0
c(x ; xis (t ))z(t , x )dx , i ∈ S {1, 2, . . . , m}
(2)
subject to the Dirichlet boundary conditions z(t, 0) = z(t, L) = 0
(3)
and the initial condition z(0, x) = z0 (x)
(4)
where γ > 0 and θ > 0 are the diffusion parameter and reaction term coefficient, respectively; z(t, x) ∈ R denotes the state; x ∈ [0, L] ⊂ R and t ∈ [0, ∞) stand for the spatial and temporal coordinate, respectively; the subscripts x and t stand for the partial derivatives with respect to x and t, respectively; b(x ; xia (t )) (i ∈ S) denotes the spatial distribution of the ith mobile actuator, which describes how the control input ui (t ) ∈ R is distributed in the interval [0, L]; xia (t ) ∈ [0, L] indicates the spatial position of the ith mobile actuator; yi (t ) ∈ R is the measurement output of the ith mobile sensor; c(x ; xis (t )) (i ∈ S) is a known function determined by the shape (point or distributed) of the ith mobile sensor and xis (t ) ∈ [0, L] indicates the spatial position of the ith mobile sensor; z0 (x) is the initial value. In this paper, the mobile actuators and sensors are collocated (i.e., xia (t ) = xis (t ), i ∈ S), and the spatial distribution functions b(x ; xia (t )) and c(x ; xia (t )) (i ∈ S) are taken as b(x ; xia (t )) = c(x ; xia (t )) = δ(x − xia (t )), i ∈ S.
(5)
To avoid the mobile actuator/sensor pairs colliding with each other and achieve the control design objective, we give the following assumption: Assumption 2.1. Let the points l1 , l2 , . . . , lm+1 divide [0, L] into m subintervals, denoted by [li , li+1 ], i ∈ S, where 0 = l1 < l2 < · · · < lm+1 = L. Define the sets iε [li + κε, li+1 − κε] ⊂ (li , li+1 ), i ∈ S, where κ > 1 and ε is a known small positive scalar. The actuator/sensor pair Ai (Ai is the identity of the ith actuator/sensor pair and i ∈ S) only can move in the space domain iε (i.e., xia (t ) ∈ iε ). Remark 2.1. In this work, the integrated design of switching control and mobile actuator/sensor guidance is under the premise that xia (t ) ∈ iε , i ∈ M, which will be taken into account in the guidance design. Without considering this premise, the designed guidance laws
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may make the ith actuator/sensor pair move out of the space domain iε divided in advance, which may lead to the mobile actuator/sensor pairs colliding with each other and the ineffectiveness of the control design. Further, in order to implement the mobile actuator/sensor guidance, another assumption is made on the ability of the actuator/sensor pairs as follows: Assumption 2.2. The ith actuator/sensor pair can also provide the state information at the two edges of its range, in the form of z(t, xia (t ) − ε) and z(t, xia (t ) + ε) (i ∈ S). Note that for each i ∈ S, x˙ia (t ) = 0 corresponds to the mode of the ith actuator/sensor pair Ai rest, and x˙ia (t ) = 0 corresponds to the mode of Ai motion. Obviously, there exist 2m possible modes for m actuator/sensor pairs. Let us define the following mode indicator function for all Ai , i ∈ S: m g(t ) 2i−1 πi (t ) (6) i=1
where π i (t) is the characteristic function of Ai defined by 1, x˙ia (t ) = 0 πi (t ) = 0, x˙ia (t ) = 0. From Eq. (6), it follows that g(t ) ∈ M {0; 1; 2; . . . ; 2m − 1}. Obviously, the value of g(t) indicates the present mode of the m actuator/sensor pairs. When g(t ) = h, h ∈ M, then h we have πi (t ) = mod([ 2i−1 ], 2), i ∈ S, where mod(a, b) denotes the remainder from dividing a by b and [ξ ] indicates the biggest integer which is no more than ξ . For example, when g(t ) = 1, it means that π1 (t ) = 1 and πi (t ) = 0, i ∈ S/{1}. At some time instance tw , if g(tw ) changes from one value to another, it implies that one mode of the m actuator/sensor pairs is switched to another. Although a simple fixed static output feedback (SOF) control scheme ui (t ) = ki yi (t ), i ∈ S
(7)
where ki , i ∈ S are the control gains to be determined, can be employed to control the PDE system, the main drawback of such scheme is that the fixed control gain ki of the ith actuator/sensor pair Ai must work for all possible modes of the m actuator/sensor pairs, which can lead to a very conservative controller design. To avoid using the fixed control gain, we will adopt different control gains for different modes of the m actuator/sensor pairs. Considering the fact that one mode may be switched to any other at some time instance, we assume the switching time instances are t1 , . . . , tw , . . . , tN , where 0 N < ∞ is a integer. Let 0 = t0 < t1 < · · · < tw < · · · < tN < tN+1 = ∞. Then, we consider the following switching control laws: IF g(tw ) = h, w ∈ N {0, 1, 2, . . . , N } THEN ui (t ) = kh,i yi (t ) = kh,i z(t , xia (t )) i ∈ S, t ∈ [tw , tw+1 )
(8)
where kh,i , h ∈ M, i ∈ S are the control gains to be determined. Remark 2.2. In fact, if the number of switches reaches N¯ , we can set x˙ia (t ) = 0, i ∈ S for any t tN¯ , where N¯ is a predetermined upper limit, which guarantees that the number of switches is finite. Therefore, we can assume the number of switches is finite.
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Therefore, the design problem under consideration is to find the switching controllers of the form (8) and mobile actuator/sensor guidance laws of the form x˙ia (t ) = fi (t ), i ∈ S, where fi (t ), i ∈ S are the nonlinear functions to be designed, such that the resulting closed-loop PDE system is exponentially stable. To this end, the following definition and lemma are useful: Definition 2.1 [15]. The unforced system (1)–(4) (i.e., ui (t ) ≡ 0, i ∈ S) is said to be exponentially stable in the sense of L2 -norm, if there exist constants σ 1 > 0 and σ 2 > 0 such that the solution z(t, x) of the system satisfies z(t, ·)2 σ1 z0 (·)2 exp(−σ2t ). Lemma 2.1. (Wirtinger’s Inequality [16]) Let z(t, ·) ∈ L2 ([a, b]; R ) be an absolutely continuous scalar function with square integrable derivative zx (t, x). Then, if z(t, a) = 0 or z(t, b) = 0, the following inequality holds: z(t, ·)22 4(b − a)2 π −2 zx (t, ·)22 . 3. Main results In this section, we present an LMI-based approach to the integrated design of switching control laws and mobile actuator/sensor guidance laws for exponentially stabilizing the PDE system. Let us choose the mode-dependent Lyapunov functional candidate for the PDE system (1)–(4) as m 1 li+1 Vh (t ) = Ph (x ; xia (t ))z2 (t , x)dx 2 l i i=1 t ∈ [tw , tw+1 ), w ∈ N, h ∈ M where Ph (x ; xia (t ))
β (d (x ; xia (t )) + 2), = h 2βh ,
(9)
πi (t ) = 1 πi (t ) = 0, i ∈ S, h ∈ M
(10)
in which βh > 0, h ∈ M are the constants to be determined and d (x ; xia (t )) = H (x − (xia (t ) + ε)) − H (x − (xia (t ) − ε)), i ∈ S. In fact, according to the definition of H(·), d (x ; xia (t )) can be rewritten as −1, x ∈ [xia (t ) − ε, xia (t ) + ε) d (x ; xia (t )) = 0, otherwise.
(11)
Thus, we have βh Ph (x ; xia (t )) 2βh , i ∈ S, h ∈ M
(12)
which guarantees that the Lyapunov functional in Eq. (9) is positive definite. For t ∈ [tw , tw+1 ), computing the time derivative of Vh (t) along the solution of the system (1)–(4) yields m li+1 ˙ Vh (t ) = Ph (x ; xia (t ))z(t , x) γ zxx (t , x) i=1
li
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+ θ z(t, x) + +
m 1
2
i=1
m
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b(x ; x aj (t ))u j (t ) dx
j=1 li+1
li
x˙ia (t )
∂Ph (x ; xia (t )) 2 z (t, x )dx . ∂xia (t )
(13)
Utilizing integration by parts, we have li+1 Ph (x ; xia (t ))z(t , x)zxx (t , x)dx li
=Ph (x ; xia (t ))z(t , x)zx (t , x )|lli+1 i
li+1
− li
− li
+ mod([
li+1 li
Ph (x ; xia (t ))zx2 (t , x)dx
∂Ph (x ; xia (t )) z(t, x)zx (t, x)dx ∂x
=Ph (x ; xia (t ))z(t , x)zx (t , x)|lli+1 − i
−
li+1
h
2
], 2) i−1
xia (t )+ε xia (t )−ε
li
li+1
2βh zx2 (t , x)dx
βh zx2 (t, x)dx
∂Ph (x ; xia (t )) z(t, x)zx (t, x )dx . ∂x
(14)
a From Assumption 2.1, one has xia (t ) ∈ iε and xi+1 (t ) ∈ (i+1)ε , i ∈ S/{m} which ina a a a dicate li+1 ∈ / [xi (t ) − ε, xi (t ) + ε] ∪ [xi+1 (t ) − ε, xi+1 (t ) + ε], i ∈ S/{m}. Hence, it follows a from Eq. (11) that d (li+1 ; xia (t )) = d (li+1 ; xi+1 (t )) = 0, i ∈ S/{m}, and thus Ph (li+1 ; xia (t )) = a Ph (li+1 ; xi+1 (t )) = 2βh , i ∈ S/{m}, which implies that m
Ph (x ; xia (t ))z(t , x)zx (t , x)|lli+1 i
i=1 m−1 a Ph (li+1 ; xia (t )) − Ph (li+1 ; xi+1 = (t )) z(t , li+1 )zx (t , li+1 ) i=1
=0.
(15)
Using Eqs. (5), (8) and the following property of function δ(·) [17]: x2
f (x)δ(x − )dx = f (), ∈ (x1 , x2 ).
(16)
x1
we obtain li+1 Ph (x ; xia (t ))b(x ; xia (t ))ui (t )z(t , x)dx li
h , 2 βh kh,i z2 (t, xia (t )) = 2 − mod 2i−1
li+1 h 2βh kh,i 2 a , 2 βh kh,i z2 (t, xia (t )). = z (t, xi (t ))dx − mod li+1 − li 2i−1 li
(17)
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By defining z¯i (t, x) z(t, x) − z(t , xia (t ))
(18)
it is clear that z¯i (t, xia (t )) = 0, z¯i,x (t, x) = zx (t, x).
(19)
Using Lemma 2.1, Eqs. (18) and (19) yields li+1 − zx2 (t, x)dx
− − −
li
xia (t )
π2 z¯2 (t, x)dx a 4(xi (t ) − li )2 i li li+1 π2 z¯2 (t, x)dx a 2 i xia (t ) 4(li+1 − xi (t )) li+1 π2 z¯2 (t, x )dx . 4(li+1 − li )2 i li
(20)
From the definitions of H(·) and δ(·), it is immediate that x 0, x< δ(s − )ds = x2 δ(s − ) ds = 1 , x x1 x1 = H (x − ) which implies that dH (x − ) = δ(x − ). dx Using variable substitution, we can also easily get dH (−x − ) = −δ(−x − ). dx By Eqs. (21) and (22), it follows respectively that for x ∈ [li , li+1 ], i ∈ S
h ∂Ph (x ; xia (t )) , 2 βh δ(x − (xia (t ) + ε)) − δ(x − (xia (t ) − ε)) = mod i−1 ∂x 2 and
h ∂Ph (x ; xia (t )) , 2 βh δ(x − (xia (t ) − ε)) − δ(x − (xia (t ) + ε)) . = mod a i−1 ∂xi (t ) 2 Thus, by Eq. (16), we have from Eqs. (23) and (24), respectively li+1 ∂Ph (x ; xia (t )) z(t, x)zx (t, x)dx ∂x li
h , 2 βh z(t, xia (t ) + ε)zx (t, xia (t ) + ε) − z(t, xia (t ) − ε)zx (t, xia (t ) − ε) = mod i−1 2 and li
li+1
∂Ph (x ; xia (t )) 2 z (t, x)dx ∂xia (t )
(21)
(22)
(23)
(24)
(25)
=mod([ li
h 2i−1
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], 2)βh z2 (t, xia (t ) − ε) − z2 (t, xia (t ) + ε)) .
(26)
Using inequality (12), it is immediate that li+1 li+1 Ph (x ; xia (t ))z2 (t , x)dx 2βh z2 (t , x )dx .
(27)
li
Furthermore, from (13)–(15), (17), (20) and (25)–(27), we have m li+1 V˙h (t ) z˘iT (t, x)h,i z˘i (t, x)dx i=1
+
li
m i=1
mod
h 2i−1
,2
βh × x˙ia (t ) (z2 (t, xia (t ) − ε) − z2 (t, xia (t ) + ε)) 2 + γ βh z(t, xia (t ) − ε)zx (t, xia (t ) − ε) − γ βh z(t, xia (t ) + ε)zx (t, xia (t ) + ε) xia (t )+ε + βh zx2 (t, x)dx xia (t )−ε
− βh kh,i z2 (t, xia (t )) + ηi βh (li+1 − li )z2 (t, xia (t ))
(28)
where ηi 0, i ∈ S are adjustable parameters, z˘i (t, x) [z (t, x) z (t , xia (t ))]T , and
βh π 2 h,i11 2(lγi+1 2 −li ) , i ∈ S h,i ∗ h,i22 with γ βh π 2 + 2θ βh 2(li+1 − li )2 2βh kh,i h γ βh π 2 − + − mod([ i−1 ], 2)ηi βh . 2(li+1 − li )2 li+1 − li 2
h,i11 − h,i22
On the basis of Assumption 2.1 and inequality (28), we give the actuator/sensor guidance laws as follows: ⎧ 0, if z2 (t , xia (t ) − ε) − z2 (t , xia (t ) + ε) χc ⎪ ⎪ ⎨ or if xia (t ) li+1 − (κ + κc )ε and vh,i > 0 x˙ia (t ) = (29) or if xia (t ) li + (κ + κc )ε and vh,i < 0 ⎪ ⎪ ⎩ vh,i , otherwise, i ∈ S ¯ iε [li + (κ + κc )ε, li+1 − (κ + κc )ε], i ∈ S, where κ c > 0 with the initial positions xia (0) ∈ is an adjustable small positive scalar, χ c is the predetermined critical value and vh,i = − μi z2 (t, xia (t ) − ε) − z2 (t, xia (t ) + ε) 2βh−1 − 2 z (t, xia (t ) − ε) − z2 (t, xia (t ) + ε) × γ βh z(t, xia (t ) − ε)zx (t, xia (t ) − ε)
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− γ βh z(t, xia (t ) + ε)zx (t, xia (t ) + ε) xia (t )+ε + βh zx2 (t, x)dx xia (t )−ε
− βh kh,i z2 (t, xia (t )) + ηi βh (li+1 − li )z2 (t, xia (t )) , i ∈ S in which μi > 0, i ∈ S are adjustable parameters. Then, we have the following result. Theorem 3.1. Consider the PDE system (1)–(4). Given scalars μi > 0 and ηi ≥ 0, i ∈ S, for each h ∈ M, suppose there exist scalars β h > 0 and k¯ h,i , i ∈ S such that the following LMIs are satisfied:
2 hγ π h,i11 2(lβi+1 −li )2 < 0, i ∈ S (30) ¯ h,i22 ∗ with
h 2k¯ h,i βh γ π 2 + − mod , 2 ηi βh . 2(li+1 − li )2 li+1 − li 2i−1
¯ h,i22 −
Then the switching controllers (8) with control gains kh,i = k¯ h,i βh−1 , i ∈ S, h ∈ M
(31)
and the actuator/sensor guidance laws (29) with (31) can ensure that the resulting closed-loop system is exponentially stable. Proof. Set k¯ h,i = kh,i βh , i ∈ S, h ∈ M.
(32)
Assume that the LMIs in (30) hold. Then, from Eq. (32), we have h,i < 0, i ∈ S
(33)
which implies h,i −λmin (−h,i )I , i ∈ S
(34)
where I stands for the identity matrix with appropriate dimension. Consequently, using (28), (29) and (34), we obtain m li+1 ˙ Vh (t ) −λmin (−h,i )z2 (t, x)dx i=1
li
βh χc2 mod 2Vh (t ) i=1 m
−
λh βh χc2 − − 2βh 2Vh (tw )
, 2 μiVh (t ) i−1 h
2
m
mod
i=1
, 2 μi Vh (t ) i−1 h
2
t ∈ [tw , tw+1 )
(35)
where λh = mini∈S {λmin (−h,i )} > 0. From inequality (35), it follows that V˙h (t ) −
λh Vh (t ), t ∈ [tw , tw+1 ) 2βh
(36)
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which means that
λh Vh (t ) exp − (t − tw ) Vh (tw ). 2βh
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(37)
Without loss of generality, at tw , w ∈ N, we assume h = hw , where hw ∈ M. Then, referring max to (9) and (12), we can get Vhw (tw ) 2β V (t − ), where Vhw−1 (tw− ) denotes the left limit βmin hw−1 w of the Vhw−1 (t ) at tw , βmax = maxh∈M {βh } and βmin = minh∈M {βh }. Therefore, for t ∈ [tw , tw+1 ), w ∈ N, we have
λh Vhw (t ) exp − w (t − tw ) Vhw (tw ) 2βhw
λh 2βmax exp − w (t − tw ) Vhw−1 (tw− ) βmin 2βhw
λh 2βmax exp − w (t − tw ) βmin 2βhw
λh × exp − w−1 (tw − tw−1 ) Vhw−1 (tw−1 ) 2βhw−1 .. . ¯ 2βmax N λ ( ) exp − t Vh0 (0) βmin 2βmax ¯ = minh∈M {λh }. Then, for t ∈ [tw , tw+1 ), w ∈ N, where λ βmin z(t, ·)22 Vhw (t ) 2 ¯ (2βmax )N βmax λ z0 (·)22 exp(− t) N (βmin ) 2βmax i.e.,
z(t, ·)2
2βmax βmin
N+1 2
¯ λ z0 (·)2 exp − t . 4βmax
(38)
¯
max Letting σ1 = ( 2β ) 2 and σ2 = 4βλmax , from Definition 2.1, we can conclude from βmin inequality (38) that the closed-loop PDE system is exponentially stable. Moreover, from Eq. (32), we have Eq. (31). The proof is complete. N+1
Remark 3.1. It is worth mentioning that although the main result in Theorem 3.1 is developed for the case of Dirichlet boundary conditions (3), the similar results can be obtained for the cases of Neumann boundary conditions or mixed boundary conditions. Remark 3.2. From inequality (35), we can get that when g(tw ) = h, h ∈ M, the exponential βh χc2 m λh h m decay rate at [tw , tw+1 ) is 2β + i=1 mod ([ 2i−1 ], 2)μi . Especially, if h = 2 − 1, the 2V h h (tw ) β m −1 χc2 m λ2m −1 decay rate is 2β + 2V22m −1 i=1 μi and if h = 0, all of the adjustable parameters μi , i ∈ S (tw ) 2m −1 are absent. Therefore, provided the parameters μi , i ∈ S are chosen appropriately, compared with any other mode, the exponential decay rates for the modes h = 0 and h = 2m − 1 are the
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smallest and biggest, respectively. Namely, the employment of mobile actuator/sensor pairs can improve the state response speed of the closed-loop system. Remark 3.3. To implement the mobile actuator/sensor guidance, some state information in Eq. (29) can be approximated as follows: z(t, xia (t )) − z(t, xia (t ) − ε) ε z(t, xia (t ) + ε) − z(t, xia (t )) a zx (t, xi (t ) + ε) ≈ ε xia (t )+ε 2 a 2 a z (t, x (t i ) − ε) + zx (t, xi (t ) + ε) zx2 (t, x)dx ≈(2ε) x 2 xia (t )−ε zx (t, xia (t ) − ε) ≈
=ε(zx2 (t, xia (t ) − ε) + zx2 (t, xia (t ) + ε)). Obviously, in order to achieve the proposed switching control scheme, it is required to monitor the modes for the actuator/sensor pairs in real time, which increases the difficulty of controller implementation. If the modes are not available for switching due to the practical constraints, then we can turn to the fixed SOF control scheme given by Eq. (7). An LMI approach to the fixed SOF control design can be easily obtained from Theorem 3.1 by taking βh = β and k¯ h,i = k¯ i , h ∈ M, i ∈ S. In this case, the LMIs in (30) can be simplified as
π2 i11 2(lβγ 2 i+1 −li ) < 0, i ∈ S (39) ¯ i22 ∗ with i11 = −
2 2k¯ i βγ π 2 ¯ i22 = − βγ π + 2θ β, + 2 2 2(li+1 − li ) 2(li+1 − li ) li+1 − li
and the actuator/sensor guidance laws (29) become ⎧ 0, if z2 (t , xia (t ) − ε) − z2 (t , xia (t ) + ε) χc ⎪ ⎪ ⎨ or if xia (t ) li+1 − (κ + κc )ε and vi > 0 x˙ia (t ) = or if xia (t ) li + (κ + κc )ε and vi < 0 ⎪ ⎪ ⎩ vi , otherwise, i ∈ S ¯ iε , and with xia (0) ∈ 2 vi = − μi z (t, xia (t ) − ε) − z2 (t, xia (t ) + ε) 2β −1 − 2 z (t, xia (t ) − ε) − z2 (t, xia (t ) + ε) × γ βz(t, xia (t ) − ε)zx (t, xia (t ) − ε) − γ βz(t, xia (t ) + ε)zx (t, xia (t ) + ε) xia (t )+ε +β zx2 (t, x)dx xia (t )−ε
− βki z2 (t, xia (t )) + ηi β(li+1 − li )z2 (t, xia (t )) , i ∈ S in which μi > 0 and ηi 0, i ∈ S. Then, we have the following corollary.
(40)
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Fig. 1. Open-loop profile of evolution of z(t, x).
Corollary 3.1. Consider the PDE system (1)–(4). Given scalars μi > 0 and ηi 0, i ∈ S, if there exist scalars β > 0 and k¯ i , i ∈ S satisfying the LMIs in (39), then the fixed SOF controllers (7) with control gains ki = k¯ i β −1 , i ∈ S and actuator/sensor guidance laws (40) can ensure that the resulting closed-loop system is exponentially stable. Remark 3.4. It is easily seen that the LMI conditions in (39) correspond to the mode that all actuator/sensor pairs are in rest. Therefore, Corollary 3.1 can also provide an exponential stabilization condition for the PDE system (1)–(4) with the static actuator/sensor pairs (i.e., ¯ iε , i ∈ S). x˙ia (t ) ≡ 0 with the initial positions xia (0) ∈ Remark 3.5. Note that in this paper, the integrated design method is developed for a class of DPSs with deterministic system parameters. However, as is well known, many physical systems have variable structures and parameters subject to random changes. As a special class of hybrid systems, Markovian jump systems has attracted much attention [18–20], because they can represent various processes in different fields, such as manufacturing systems, fault-tolerant systems, communication systems, and economic systems. In the future, we will investigate the switching control design for DPSs with Markovian jump parameters under mobile actuators and sensors. Remark 3.6. It is worth pointing out that the design method in [10] requires that the openloop operator is coercive, which implies the zero equilibrium point of the open-loop system is stable. The main goal of [10] is to design mobile actuator/sensor guidance laws for improving the transient response of the closed-loop state. However, for the open-loop unstable parabolic PDE system, the design method of [10] is ineffective in theory and may not work well. In this paper, the proposed design method has not coercivity requirement and thus has a wider range of applications. 4. Numerical simulations To illustrate the effectiveness of the proposed design method, this section will provide numerical simulations for the PDE system (1)–(4) with γ = 0.02, θ = 0.23, L = 1, m = 3 and 2 z0 (x) = 0.32sin(π x)e−8x + 0.32sin(π x). With the above settings, it is obvious from Fig. 1 that the equilibrium point z(t, ·) = 0 of the open-loop system is unstable, which implies that the open-loop operator is not coercive.
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Table 1 Switching control gains. h
Gain of A1
0 1 2 3 4 5 6 7
k0,1 k1,1 k2,1 k3,1 k4,1 k5,1 k6,1 k7,1
= −0.2927 = −0.2777 = −0.2927 = −0.2777 = −0.2927 = −0.2777 = −0.2927 = −0.2777
Gain of A2 k0,2 k1,2 k2,2 k3,2 k4,2 k5,2 k6,2 k7,2
= −0.4862 = −0.4862 = −0.4562 = −0.4562 = −0.4862 = −0.4862 = −0.4562 = −0.4562
Gain of A3 k0,3 k1,3 k2,3 k3,3 k4,3 k5,3 k6,3 k7,3
= −0.2927 = −0.2927 = −0.2927 = −0.2927 = −0.2627 = −0.2627 = −0.2627 = −0.2627
Fig. 2. Closed-loop profile of evolution of z(t, x) with mobile actuator/sensor pairs under switching control scheme.
In this simulation study, we will use the following approximation to the delta function in Eq. (5): 1 if x ∈ [xia (t ) − ζ , xia (t ) + ζ ] δ(x − xia (t )) ≈ 2ζ 0, otherwise, i = 1, 2, 3 where ζ = 0.015. Set l1 = 0, l2 = 0.3, l3 = 0.7, l4 = 1, ε = 0.05, κ = 1.12, κc = 0.3 χc = 0.01, μ1 = 8, μ2 = 10, μ3 = 8, η1 = 0.1, η2 = 0.15, η3 = 0.2. Then we have 1ε = [0.056, 0.244], 2ε = [0.356, 0.644], 3ε = [0.756, 0.944] and ¯ 1ε = [0.071, 0.229], ¯ 2ε = [0.371, 0.629], ¯ 3ε = [0.771, 0.929]. Hence, the initial positions of the three mobile actuator/sensor pairs are chosen as ¯ 1ε , x2a (0) = 0.48 ∈ ¯ 2ε , x3a (0) = 0.88 ∈ ¯ 3ε . x1a (0) = 0.10 ∈
(41)
Solving the LMIs in (30) via the LMI Toolbox in Matlab, the switching control gains are presented in Table 1. Fig. 2 shows the state response of closed-loop system with mobile actuator/sensor pairs, which implies the resulting switching controllers and mobile actuator/sensor
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Fig. 3. Switching control actions of closed-loop system with mobile actuator/sensor pairs.
Fig. 4. Trajectories of mobile actuator/sensor pairs under switching control scheme.
guidance laws can stabilize the PDE system. Figs. 3 and 4 show the corresponding switching control actions and the trajectories of the mobile actuator/sensor pairs, respectively. In order to be more persuasive, we apply Corollary 3.1 to make the control performance comparison between two cases of using mobile and static actuator/sensor pairs. It is known from Corollary 3.1 that the resulting control gains for the two cases are the same. For the sake of comparisons, the actuator/sensor pairs are assumed to be fixed at the positions in Eq. (41) for the static case. Solving the LMIs in (39) via the LMI Toolbox in Matlab, we have the control gains k1 = −0.2927, k2 = −0.4862, k3 = −0.2927. Fig. 5 depicts the L2 -norm of the closed-loop system state using static and mobile actuator/sensor pairs with the initial positions in Eq. (41) under the fixed control scheme. It can be observed obviously that the mobile case exhibits faster state response speed than the
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Fig. 5. State L2 -norm for static and mobile cases under fixed control scheme.
Fig. 6. Spatial distributions of closed-loop state at different time instances for static (blue solid line) and mobile (black dotted line) cases under fixed control scheme. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
static one. The pointwise-in-space convergence of the state distribution at four different time instances is presented in Fig. 6. Clearly, the distributed state converges to zero much faster when the actuator/sensor pairs are mobile. 5. Conclusion This paper has investigated the exponential stabilization problem of a class of linear diffusion processes using mobile actuators and sensors. By utilizing the Lyapunov direct approach, the integrated design of switching controllers and mobile actuator/sensor guidance laws is de-
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veloped to exponentially stabilize the diffusion process. Finally, numerical simulations indicate that the proposed design method is effective and using mobile actuators/sensors can achieve faster state response speed. In the future work, we will focus on the switching control design for DPSs via mobile actuators and sensors in the presence of external disturbances, stochastic noises or Markovian jump parameters. Moreover, the mobile non-collocated actuators and sensors will be employed to deal with the control design issue of the PDE systems. Acknowledgments This work was supported in part by the National Natural Science Foundation for Distinguished Young Scholars of China under Grant 61625302, and in part by the National Natural Science Foundation of China under Grant 61473011 and Grant 61721091. The authors would like to thank the Associate Editor and the anonymous reviewers for their valuable comments and suggestions which have improved the presentation of this paper. References [1] W.H. Ray, Advanced Process Control, McGraw-Hill, New York, 1981. [2] R.F. Curtain, H. Zwart, An Introduction to Infinite-dimensional Linear Systems Theory, Springer-Verlag, New York, 1995. [3] W. He, T. Meng, D. Huang, X. Li, Adaptive boundary iterative learning control for an Euler-Bernoulli beam system with input constraint, IEEE Trans. Neural Netw. Learn. Syst. 29 (2018) 1539–1549. [4] J.M. Wang, L.L. Su, H.X. Li, Stabilization of an unstable reaction-diffusion PDE cascaded with a heat equation, Syst. Control Lett. 76 (2015) 8–18. [5] A. Armaoua, M.A. Demetriou, Optimal actuator/sensor placement for linear parabolic PDEs using spatial H2 norm, Chem. Eng. Sci. 61 (2006) 7351–7367. [6] P. Hébrard, A. Henrot, Optimal shape and position of the actuators for the stabilization of a string, Syst. Control Lett. 48 (2003) 199–209. [7] G. Zheng, B.Z. Guo, M.M. Ali, Continuous dependence of optimal control to controlled domain of actuator for heat equation, Syst. Control Lett. 79 (2015) 30–38. [8] C. Antoniades, P.D. Christofides, Integrated optimal actuator/sensor placement and robust control of uncertain transport-reaction processes, Comput. Chem. Eng. 26 (2002) 187–203. [9] Y.Q. Chen, Z. Wang, J. Liang, Actuation scheduling in mobile actuator networks for spatial-temporal feedback control of a diffusion process with dynamic obstacle avoidance, in: Proceedings of the IEEE International Conference on Mechatronics and Automation, Niagara Falls, Canada, 2005. [10] M.A. Demetriou, Guidance of mobile actuator-plus-sensor networks for improved control and estimation of distributed parameter systems, IEEE Trans. Autom. Control 55 (2010) 1570–1584. [11] M.A. Demetriou, Adaptive control of 2-D PDEs using mobile collocated actuator/sensor pairs with augmented vehicle dynamics, IEEE Trans. Autom. Control 57 (2012) 2979–2993. [12] W. Mu, B. Cui, W. Li, Z. Jiang, Improving control and estimation for distributed parameter systems utilizing mobile actuator-sensor network, ISA Trans. 53 (2014) 1087–1095. [13] S. Boyd, L. El-Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, PA, 1994. [14] P. Gahinet, A. Nemirovski, A.J. Laub, M. Chilali, LMI Control Toolbox for Use with Matlab, MathWorks, Natick, MA, 1995. [15] J.-W. Wang, H.-N. Wu, Lyapunov-based design of locally collocated controllers for semi-linear parabolic PDE systems, J. Frankl. Inst. 351 (2014) 429–441. [16] G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, Cambridge University Press, Cambridge, 1959. [17] V. Balakrishnan, All about the Dirac delta function (?), Resonance 8 (2003) 48–58. [18] J.W. Wang, H.N. Wu, H.X. Li, Stochastically exponential stability and stabilization of uncertain linear hyperbolic PDE systems with Markov jumping parameters, Automatica 48 (2012) 569–576.
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