Intell Ind Syst (2016) 2:119–132DOI 10.1007/s40903-016-0043-z

ORIGINAL PAPER

Control of the Nonlinear Wave-Type Dynamics Using theDerivative-Free Nonlinear Kalman Filter

G. Rigatos1 · P. Siano2

Received: 24 December 2015 / Accepted: 28 February 2016 / Published online: 21 April 2016© Springer Science+Business Media Singapore 2016

Abstract The objective of the paper is to develop a point-wise control method for a 1D nonlinear wave equation and afiltering approach for estimating the dynamics of such a sys-tem from measurements provided from a small number ofsensors. It is shown that the numerical solution of the associ-ated partial differential equation results into a set of nonlinearordinary differential equations. With the application of a dif-feomorphism that is based on differential flatness theory itis shown that an equivalent description of the system in thelinear canonical (Brunovsky) form is obtained. This transfor-mation enables to obtain estimates about the state vector ofthe system through the application of the standard KalmanFilter recursion. For the local subsystems, intowhich the non-linear wave equation is decomposed, it becomes possible toapply pointwise state estimation-based feedback control. Theefficiency of the proposed filtering and control approach fornonlinear systems described by 1D partial differential equa-tions of the wave type (e.g. sine-Gordon PDE) is confirmedthrough simulation experiments. It is shown that, by apply-ing feedback control, the nonlinear wave-type dynamics canbe made to track any reference setpoint.

Keywords Differential flatness theory · Derivative-freenonlinear Kalman Filter · Distributed parameter systems ·Nonlinear feedback control · Nonlinear wave equations

B G. Rigatosgrigat@ieee.org

P. Sianopsiano@unisa.it

1 Unit of Industrial Automation, Industrial Systems Institute,26504 Rion Patras, Greece

2 Department of Industrial Engineering, University of Salerno,84084 Fisciano, Italy

1 Introduction

The problem of state estimation and control of distributedparameter systems described by partial differential equationsof the parabolic, hyperbolic and elliptic type is of advanceddifficulty comparing to control of lumped parameter systems,the latter being modeled in the form of ordinary differen-tial equations [1–3]. The problem becomes more difficultin case that the distributed parameter systems are character-ized by nonlinearities [4–9]. The paper treats the problemof piecewise control of nonlinear wave-type partial differ-ential equations. Control of nonlinear wave dynamics ismet in several applications. For instance, nonlinear wave-type differential equations is met in communication systems(transmission lines, optical fibers and electromagnetic wavespropagation), in electronics (Josephson junctions), in fluidflow models, in structural engineering (dynamic analysisof buildings under seismic waves, mechanical structuressubjected to vibrations, pendulum chains), in biomedicalsystems (voltage propagation and variations in neuron’smembrane), etc. Solving nonlinear estimation and controlproblems for such systems is important for modifying theirdynamics and for succeeding their functioning according tospecifications [10–17].

Following the procedure for numerical solution of thenonlinear PDE of the wave-type dynamics, a set of cou-pled nonlinear ordinary differential equations is obtained andwritten in a state-space form [18]. For the latter state-spacedescription, differential flatness properties are proven. Thus,it is shown that all state variables and the control inputs ofthe state-space model can be written as functions of a vec-tor of algebraic variables that constitute the flat output ändalso as functions of the flat output’s derivatives [19–23]. Byapplying a change of coordinates (diffeomorphism) which isbased on differential flatness theory it is shown that the state-

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120 Intell Ind Syst (2016) 2:119–132

space model of the wave-type PDE can be written in thelinear canonical (Brunovsky) form, in which the previouslynoted nonlinear ordinary differential equations are now trans-formed into linear ones. Next, pointwise feedback control isapplied to the wave-type PDE. For each local linear model ofthe aforementioned differential equations the state feedbackcontrol is selected such that asymptotic stability is assured.This can be done using for instance pole-placement methods.

Another objective of the article is to implement state-feedback control of the nonlinear wave dynamics usingmeasurements from a small number of sensors [24,25]. Thisimplies that for state vector elements of the PDE’s state-spacedescription which cannot be measured directly, state estima-tion with filtering methods has to be be applied. Filtering fornonlinear distributed parameter systems is again a non-trivialproblem [26–29]. Both observer-based and Kalman Filter-based approaches have been proposed [30–34]. To this end, inthis paper, a new nonlinear filtering method, under the nameDerivative-free nonlinear Kalman Filtering, is proposed. Thefilter consists of the standard Kalman Filter recursion appliedto the linear equivalent state-space model of the wave PDE[35–37]. Moreover, an inverse transformation which is basedon differential flatness properties enables to obtain estimatesof the state variables of the initial system’s description.

The structure of the paper is as follows: in “Nonlinear1D wave-type partial differential equations” section, modelsof nonlinear 1D wave-type partial differential equations areexamined, while emphasis is given to the sine-Gordon PDEmodel. In “Differential flatness theory and transformationinto canonical forms” section, the use of differential flat-ness theory for transforming nonlinear dynamical systemsinto the linear and canonical state-space form is explained.In “State-space description of the nonlinear wave dynamics”section a state-space description is obtained for the model ofthe nonlinear wave-type dynamics. In “Solution of the con-trol and estimation problem for nonlinear wave dynamics”section, the previous linearization transformation is appliedto control and filtering of distributed parameter systems. In“Simulation tests” section simulation tests are carried out toevaluate the proposed control and filtering approach for dis-tributed parameter systems. Finally, in “Conclusions” sectionconcluding remarks are stated.

2 Nonlinear 1D Wave-Type Partial DifferentialEquations

2.1 Sine-Gordon Nonlinear PDE in Coupled NonlinearPendula

Nonlinear 1Dwave-type partial differential equations appearin models of coupled oscillators. One can consider for exam-ple the forced damped sine-Gordon equation [10]

∂2φ

∂t2+ c

∂φ

∂t− k

∂2φ

∂x2+ sin(φ) = l (1)

where c and l are constants. This type of PDE appears inmany physical phenomena, such nonlinear resonant opticsand Josephson junctions, or as a dynamic model of elec-trons in a crystal lattice. Equation (1) describes the motionof an array of pendula each of which is coupled to its nearestneighbors by a torsional spring with a coupling coefficientk. Each pendulum is subject to a constant torque l and to aviscous drag force with coefficient c. The angle xi of the i-thpendulum and the vertical axis evolves according to Eq. (1).

By considering a one dimensional grid of N sample pointsand by computing the second-order derivative

∂2φ

∂x2� 1

h2(φi+1 − 2φi + φi−1) (2)

one has the decomposition of the nonlinear partial differ-ential equation into a set of nonlinear ordinary differentialequations of the form

xi +cxi +εsin(xi ) = k(xi+1−2xi +xi−1+l) i = 1, 2, . . .

(3)

One can also use a periodicity condition

xi+N = xi + 2π (4)

The periodicity condition reduces the system from infinitemany pendula into one with N degrees of freedom

x1 + cx1 + εsin(x1) = x2 + xN − 2x1 + l − 2π

x2 + cx2 + εsin(x2) = x3 + x1 − 2x2 + l

· · ·xN + cxN + εsin(xN ) = x1 + xN − 2xN + l + 2π

(5)

2.2 Sine-Gordon Nonlinear PDE in the Model of theJosephson Junction

A transmission line is considered where transverse electro-magnetic waves propagate as shown in Fig. 1. The trans-mission line consists of inductors, capacitors and Josephsonjunctions, such that for a length dx of the transmission linethe capacitance is dC = Cdx , the inductance is dL = Ldxand the critical current is d I0 = I0dx [10]. Dividing the dif-ferential voltage drop dV and the shunt current d I by dx ,one obtains

∂V

∂x= −L

∂ I

∂t∂ I

∂x= −C

∂V

∂t− I0sin(φ)

(6)

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Intell Ind Syst (2016) 2:119–132 121

Fig. 1 Josephson transmission line described by Josephson junctions(top), a capacitance per unit length C , an inductance per unit length Land a critical current per unit length I0

where φ is the difference φ = φupper − φlower of the super-conducting phases. The voltage is related to the rate of changeof φ through the Josepshon equation

∂φ

∂t= 2eV

h(7)

and equivalently

∂φ

∂x= −2eV

hI (8)

Thus, one arrives at the equation

1

c2∂2φ

∂t2− ∂2φ

∂x2+ 1

λ2jsin(φ) = 0 (9)

where c = (LC)−1/2 is the Swihart velocity and λ j =(h/2eL I0)1/2 is the Josephson length. By rescaling lengthsλ j and λ j/c one arrives at the sine-Gordon equation

∂2φ

∂t2+ c

∂φ

∂t− ∂2φ

∂x2+ sin(φ) = l (10)

2.3 Current Equation in a Josepshon Transmission Line

A wire of length l is considered which is interruptedby identical Josephson junctions at positions denoted asx1, x2, . . . , xN . The partial differential equation governingthe dynamics of the Josephson transmission line can be deter-mined by considering thefinite element representation shownin Fig. 2. Each inductor–capacitor segment represents a short

Fig. 2 Finite element representation of the Josephson transmission line(× denotes a Josephson junction)

length�x of the wire. Some segments also contain a Joseph-son junction in series with the inductor.

At the i-th segment it holds that

Ii−1 = qi + Ii (11)

where qi is the charge on the i-th capacitor and Ii is thecurrent in the i-th inductor. Equating voltage drops of thetwo paths from node i to ground it holds

qiC�x

= L�x Ii + qi+1

C�x+ h

2eφ jδi,i j (12)

where h is Planck’s constant divided by 2π , e is the magni-tude of the electronic charge and φ j is the difference in thephase of the macroscopic quantum wave function across thej-th junction. The Kronecker δ term denotes the voltage dropacross the j-th Josephson junction if it appears in segment iof the transmission line. From Eqs.(11) and (12) one obtains

v2Ii+1 − 2Ii + Ii−1

�x2= Ii + h

2eL�xφ jδi,i j (13)

v2 = 1/(LC). Setting the limit �x→0 yields an equationfor the current I (x, t)

∂2 I

∂t2− v2

∂2 I

∂x2+

N∑

i=1

h

2eLφ jδ(x − x j ) = 0 (14)

The associated boundary conditions are I (0, t) = I (l, t)= Ib. Moreover, about the dynamics of phase φ at the j-thJosephson junction one has

hCJ

2eφ j + h

2eR jφ j + Icsin(φ) j = I (x j , t) (15)

where I (x j , t) is the current through the j-th junction. Theparameters of the junction are the capacitance C j , the resis-tance R j and the current Ic.

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122 Intell Ind Syst (2016) 2:119–132

3 Differential Flatness Theory and Transformationinto Canonical Forms

3.1 Definition of Differentially Flat Systems

Differential flatness theory will be used to obtain a linearizedstate-space description of the nonlinearwave-type PDE. Sub-sequently, it will be used for solving the associated stateestimation and control problems. Differential flatness the-ory is currently a main direction in the analysis of nonlineardynamical systems. To conclude if a dynamical system isdifferentially flat, the following should be examined: (i) theexistence of the so-called flat output, i.e. a newvariablewhichis expressed as a function of the system’s state variables. Itshould hold that the flat output and its derivatives should notbe coupled in the form of an ordinary differential equation,(ii) the components of the system (i.e. state variables andcontrol input) should be expressed as functions of the flatoutput and its derivatives [20–23].

The main principles of differential flatness theory are asfollows [20–23]: A finite dimensional system is considered.This can be written in the general form of an ordinary dif-ferential equation (ODE), i.e. Si (w, w, w, . . . , w(i)), i =1, 2, . . . , q. The term w denotes the system variables (thesevariables are for instance the elements of the system’s statevector and the control input) while w(i), i = 1, 2, . . . , qare the associated derivatives. Such a system is said to bedifferentially flat if there is a collection of m functionsy = (y1, . . . , ym) of the system variables and of theirtime-derivatives, i.e. yi = φ(w, w, w, . . . , w(αi )), i =1, · · · ,m satisfying the following two conditions [20–23]:(1) There does not exist any differential relation of the formR(y, y, . . . , y(β)) = 0 which implies that the derivatives ofthe flat output are not coupled in the sense of an ODE, orequivalently it can be said that the flat output is differen-tially independent, (2) All system variables (i.e. the elementsof the system’s state vector w and the control input) can beexpressed using only the flat output y and its time derivativeswi = ψi (y, y, . . . , y(γi )), i = 1, . . . , s.

3.2 Conditions for Applying Differential FlatnessTheory

Next, a new filtering and control method for distributedparameter systems will be developed, in accordance to thedifferential flatness theory. It will be shown that the proposedfiltering and control methods can be efficiently applied to themodel of the nonlinear wave-type PDE.

It will be shown that following established discretizationprocedures the model of the nonlinear wave-type PDE canbe written as a state-space model of the form x = f (x, u).Moreover, the previous state-space description can be trans-formed to the form of an affine in-the-input system by adding

an integrator to each input [21,23]. In the latter case oneobtains

x = f (x) +m∑

i=1

gi (x)ui (16)

The following definitions are now used [24]:

(i) Lie derivative L f h(x) stands for the Lie derivativeL f h(x) = (∇h) f and the repeated Lie derivatives arerecursively defined as L0

f h = h for i = 0, Lif h =

L f Li−1f h = ∇Li−1

f h f for i = 1, 2, . . ..

(ii) Lie Bracket adif g stands for a Lie Bracket which is

defined recursively as adif g = [ f, adi−1f ]g with ad0f g =

g and ad f g = [ f, g] = ∇g f − ∇ f g.

If the system of Eq. (16) can be linearized by a diffeomor-phism z = φ(x) and a static state feedback u = α(x)+β(x)vinto the following form

zi, j = zi+1, j for 1≤ j≤m and 1≤i≤v j − 1

zvi, j = v j(17)

with∑m

j=1v j = n, then y j = z1, j for 1≤ j≤m are the 0-flatoutputswhich canbewritten as functions of only the elementsof the state vector x . To define conditions for transformingthe system of Eq. (16) into the canonical form described inEq. (17) the following theorem holds [23].

Theorem For nonlinear systems described by Eq. (16) thefollowing variables are defined: (i) G0 = span[g1, . . . , gm],(ii) G1 = span[g1, . . . , gm, ad f g1, . . . , ad f gm], . . . (k)

Gk = span{ad jf gi for 0≤ j≤k, 1≤i≤m}. Then, the lin-

earization problem for the system of Eq. (16) can be solvedif and only if: (1) The dimension of Gi , i = 1, . . . , k is con-stant for x∈X⊆Rn and for 1≤i≤n − 1, (2) The dimensionof Gn−1 if of order n, (3). The distribution Gk is involutivefor each 1≤k≤n − 2.

3.3 Transformation of the Nonlinear Wave-Type PDEinto a Canonical State-Space Form

It is assumed now that after defining the flat outputs of theprevious state-space description of the nonlinear wave-typePDE and after expressing the system state variables and con-trol inputs as functions of the flat output and of the associatedderivatives, the system can be transformed in the Brunovskycanonical form:

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Intell Ind Syst (2016) 2:119–132 123

x1 = x2

· · ·xr1−1 = xr1

xr1 = f1(x) +p∑

j=1

g1 j (x)u j + d1

xr1+1 = xr1+2

· · ·x p−1 = xp

x p = f p(x) +p∑

j=1

gp j (x)u j + dp

y1 = x1

· · ·yp = xn−rp+1

(18)

where x = [x1, . . . , xn]T is the state vector of the trans-formed system (according to the differential flatness for-mulation), u = [u1, . . . , u p]T is the set of control inputs,y = [y1, . . . , yp]T is the output vector, fi are the driftfunctions and gi, j , i, j = 1, 2, . . . , p are smooth func-tions corresponding to the control input gains, while d j isa variable associated to external disturbances. It holds thatr1 + r2 + · · · + rp = n. Having written the initial nonlinearsystem into the canonical (Brunovsky) form it holds

y(ri )i = fi (x) +

p∑

j=1

gi j (x)u j + d j (19)

Next the following vectors and matrices can be defined:f (x) = [ f1(x), . . . , fn(x)]T , g(x) = [g1(x), . . . , gn(x)]T ,with gi (x) = [g1i (x), . . . , gpi (x)]T , A = diag[A1, . . . , Ap],B = diag[B1, . . . , Bp], CT = diag[C1, . . . ,Cp], d =[d1, . . . , dp]T , where matrix A has the MIMO canonicalform, i.e. with block-diagonal elements

Ai =

⎛

⎜⎜⎜⎜⎜⎝

0 1 · · · 00 0 · · · 0...

... · · · ...

0 0 · · · 10 0 · · · 0

⎞

⎟⎟⎟⎟⎟⎠

ri×ri

BTi = (

0 0 · · · 0 1)1×ri

Ci = (1 0 · · · 0 0

)1×ri

(20)

Thus, Eq. (19) can be written in state-space form

x = Ax + Bv + Bd

y = Cx(21)

where the control input is written as v = f (x) + g(x)u. Thesystem of Eqs. (20) and (21) is in controller and observercanonical form.

3.4 Derivative-Free Nonlinear Kalman Filtering

Asmentioned above, for the system of Eq. (21), state estima-tion is possible by applying the standard Kalman Filter. Thesystem is first turned into discrete-time form using commondiscretization methods and then the recursion of the linearKalman Filter described in Eqs. (40) and (41) is applied.

If the derivative-free Kalman Filter is used in place ofthe Extended Kalman Filter then in the EKF equationsthe following matrix substitutions should be performed:Jφ(k)→Ad , Jγ (k)→Cd , where matrices Ad and Cd are thediscrete-time equivalents of matrices A and C which havebeen defined Eq. (21) and which appear also in the measure-ment and timeupdate of the standardKalmanFilter recursion.Matrices Ad and Cd can be computed using established dis-cretizationmethods.Moreover, the covariancematrices P(k)and P−(k) are the ones obtained from the linear Kalman Fil-ter update equations.

4 State-Space Description of the Nonlinear WaveDynamics

The following nonlinear wave equation is considered

∂2φ

∂t2= k

∂2φ

∂x2+ f (φ) + u(x, t) (22)

where u(x, t) is assumed to be the external control input.Using the approximation for the partial derivative

∂2φ

∂x2� = φi+1 − 2φi + φi−1

�x2(23)

and considering spatial measurements of variable φ alongaxis x at points x0 + i�x, i = 1, 2, . . . , N one has

∂2φi

∂t2= K

�x2φi+1 − 2K

�x2φi + K

�x2φi−1 + f (φi )+u(xi , t)

(24)

By considering the associated samples of φ given byφ0, φ1, . . . , φN , φN+1 one has

∂2φ1

∂t2= K

�x2φ2 − 2K

�x2φ1 + K

�x2φ0 + f (φ1) + u(x1, t)

∂2φ2

∂t2= K

�x2φ3 − 2K

�x2φ2 + K

�x2φ1 + f (φ2) + u(x2, t)

∂2φ3

∂t2= K

�x2φ4 − 2K

�x2φ3 + K

�x2φ2 + f (φ3) + u(x3, t)

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124 Intell Ind Syst (2016) 2:119–132

· · ·∂2φN−1

∂t2= K

�x2φN − 2K

�x2φN−1 + K

�x2φN−2 + f (φN−1)

+ u(xN1 , t)

∂2φN

∂t2= K

�x2φN+1 − 2K

�x2φN + K

�x2φN−1

+ f (φN ) + u(xN , t) (25)

By defining the following state vector

xT = (φ1, φ2, . . . , φN

)(26)

one obtains the following state-space description

x1 = K

�x2x2 − 2K

�x2x1 + K

�x2φ0 + f (x1) + u(x1)

x2 = K

�x2x3 − 2K

�x2x2 + K

�x2x1 + f (x2) + u(x2)

x3 = K

�x2x4 − 2K

�x2x3 + K

�x2x2 + f (x3) + u(x3)

· · ·xN−1 = K

�x2xN − 2K

�x2xN−1 + K

�x2xN−2

+ f (xN−1) + u(xN−1)

xN = K

�x2φN+1 − 2K

�x2xN + K

�x2xN−1

+ f (xN ) + u(xN )

(27)

Next, the following state variables are defined

y1,i = xi

y2,i = xi(28)

and the state-space description of the system becomes asfollows

y1,1 = y2,1

y2,1 = K

�x2y1,2 − 2K

�x2y1,1 + K

�x2φ0 + f (y1,1) + u(y1,1)

y1,2 = y2,2

y2,2 = K

�x2y1,3 − 2K

�x2y1,2 + K

�x2y1,1 + f (y1,2) + u(y1,2)

y1,3 = y2,3

y2,3 = K

�x2y1,4 − 2K

�x2y1,3 + K

�x2y1,2 + f (y1,3) + u(y1,3)

· · ·· · ·y1,N−1 = y2,N−1

y2,N−1 = K

�x2y1,N − 2K

�x2y1,N−1 + K

�x2y1,N−2 + f (y1,N−1)

+ u(y1,N−1)

y1,N = y2,N

y2,N = K

�x2φN+1 − 2K

�x2y1,N + K

�xy1,N−1

+ f (y1,N ) + u(y1,N ) (29)

The dynamical system described in Eq. (29) is a differen-tially flat one with flat output defined as the vector y =[y1,1, y1,2, . . . , y1,N ]. Indeed all state variables can be writ-ten as functions of the flat output and its derivatives.

Moreover, by defining the new control inputs

v1 = K

�x2y1,2 − 2K

�x2y1,1 + K

�x2φ0 + f (y1,1) + u(y1,1)

v2 = K

�x2y1,3 − 2K

�x2y1,2 + K

�x2y1,1 + f (y1,2) + uy1,2

v3 = K

�x2y1,4 − 2K

�x2y1,3 + K

�x2y1,2 + f (y1,3) + u(y1,3)

· · ·vN−1 = K

�x2y1,N − 2K

�x2y1,N−1 + K

�x2y1,N−2

+ f (y1,N−1) + u(y1,N−1)

vN = K

�x2φN+1 − 2K

�x2y1,N + K

�x2y1,N−1

+ f (y1,N ) + u(y1,N ) (30)

the following state-space description is obtained

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

y1,1y2,1y1,2y2,2· · ·

y1,N−1

y2,N−1

y1,Ny2,N

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 1 0 0 · · · 0 0 0 00 0 0 0 · · · 0 0 0 00 0 0 1 · · · 0 0 0 00 0 0 0 · · · 0 0 0 00 0 0 0 · · · 0 0 0 00 0 0 0 · · · 0 0 0 0

· · · · · · · · · · · · · · · · · ·0 0 0 0 · · · 0 1 0 00 0 0 0 · · · 0 0 0 00 0 0 0 · · · 0 0 0 10 0 0 0 · · · 0 0 0 0

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

y1,1y2,1y1,2y2,2· · ·

y1,N−1

y2,N−1

y1,Ny2,N

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

+

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 · · · 0 01 0 0 · · · 0 00 0 0 · · · 0 00 1 0 · · · 0 00 0 0 · · · 0 00 0 1 · · · 0 0· · · · · · · · · · · ·0 0 0 · · · 0 00 0 0 · · · 1 00 0 0 · · · 0 00 0 0 · · · 0 1

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛

⎜⎜⎜⎜⎜⎜⎝

v1v2v3· · ·

vN−1

vN

⎞

⎟⎟⎟⎟⎟⎟⎠(31)

By selectingmeasurements froma subset of points x j j∈[1, 2,. . . ,m], the associated observation (measurement) equationbecomes

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Intell Ind Syst (2016) 2:119–132 125

⎛

⎜⎜⎝

z1z2· · ·zm

⎞

⎟⎟⎠ =

⎛

⎜⎜⎜⎜⎝

1 0 0 · · · 0 00 0 0 · · · 0 0· · · · · · · · · · · ·0 0 0 · · · 1 00 0 0 · · · 0 0

⎞

⎟⎟⎟⎟⎠

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

y1,1y2,1y1,2y2,2· · ·y1,Ny2,N

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

(32)

Thus, in matrix form one has the following state-spacedescription of the system

˙y = Ay + Bv

z = C y(33)

Moreover, denoting a = K�x2

and b = − 2K�x2

, the initialdescription of the system given in Eq. (31) is rewritten asfollows

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

y1,1y2,1y1,2y2,2· · ·

y1,N−1

y2,N−1

y1,Ny2,N

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 1 0 0 0 0 0 · · · 0 0 0 0 0 0b 0 a 0 0 0 0 · · · 0 0 0 0 0 00 0 0 1 0 0 0 · · · 0 0 0 0 0 0a 0 b 0 a 0 0 · · · 0 0 0 0 0 00 0 0 0 0 1 0 · · · 0 0 0 0 0 00 0 a 0 b 0 a · · · 0 0 0 0 0 0· · · · · · · · · · · · · · · · · · · · ·0 0 0 0 0 0 0 · · · 0 0 0 1 0 00 0 0 0 0 0 0 · · · a 0 b 0 a 00 0 0 0 0 0 0 · · · 0 0 0 0 0 10 0 0 0 0 0 0 · · · 0 0 a 0 b 0

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

y1,1y2,1y1,2y2,2· · ·

y1,N−1

y2,N−1

y1,Ny2,N

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

+

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 0 · · · 0 01 0 0 · · · 0 00 0 0 · · · 0 00 1 0 · · · 0 00 0 0 · · · 0 00 0 1 · · · 0 0· · · · · · · · · · · ·0 0 0 · · · 0 00 0 0 · · · 1 00 0 0 · · · 0 00 0 0 · · · 0 1

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛

⎜⎜⎜⎜⎜⎜⎝

v1v2v3· · ·

vN−1

vN

⎞

⎟⎟⎟⎟⎟⎟⎠(34)

The associated control inputs are defined as

v1 = K

�x2φ0 + f (y1,1) + u(y1,1)

v2 = f (y1,2) + u(y1,2)

v3 = f (y1,3) + u(y1,3)

· · ·vN−1 = f (y1,N−1) + u(y1,N−1)

vN = K

�x2φN+1 + f (y1,N ) + u(y1,N )

(35)

5 Solution of the Control and Estimation Problemfor Nonlinear Wave Dynamics

5.1 Solution of the Control Problem

It holds that the dynamics of the linearized equivalent modelof the nonlinear wave-type PDE takes the form

y1,1 = by1,1 + ay1,2 + v1

y1,2 = ay1,1 + by1,2 + ay1,3 + v2

y1,3 = ay1,2 + by1,3 + ay1,4 + v3

· · ·· · ·y1,N−1 = ay1,N−2 + by1,N−1 + ay1,N + vN−1

y1,N = ay1,N−1 + by1,N + vN

(36)

For the dynamics given in Eq. (36), the feedback controllaw that assures tracking of the reference setpoint yd =[yd1,1, yd1,2, yd1,3, . . . , yd1,N−1, y

d1,N ]T is

v1 = yd1,1 − by1,1 − ay1,2 − Kd1,1(yd1,1 − yd1,1)

− Kp1,1(yd1,1 − yd1,1)

v2 = yd1,2 − ay1,1 − by1,2 − ay1,3 − Kd1,2(yd1,2 − yd1,2)

− Kp1,2(yd1,2 − yd1,2)

v3 = yd1,3 − ay1,2 − by1,3 − ay1,4 − Kd1,3(yd1,3 − yd1,3)

− Kp1,3(yd1,3 − yd1,3)

· · ·· · ·vN−1 = yd1,N−1 − ay1,N−2 − by1,N−1 − ay1,N

− Kd1,N−1(yd1,N−1 − yd1,N−1)

− Kp1,N−1(yd1,N−1 − yd1,N−1)

vN = yd1,N − ay1,N−1 − by1,N − Kd1,N (yd1,N − yd1,N )

− Kp1,N (yd1,N − yd1,N )

(37)

Next, using Eq. (37) one can compute the control action thatis applied to the wave-type dynamics

v1 = K

�x2φ0 + f (y1,1) + u(y1,1)

⇒ u(y1,1) = v1 − K

�x2φ0 − f (y1,1)

v2 = f (y1,2) + u(y1,2)⇒, u(y1,2) = v2 − f (y1,2)

v3 = f (y1,3) + u(y1,3)⇒ u(y1,3) = v3 − f (y1,3)

· · ·· · ·

123

126 Intell Ind Syst (2016) 2:119–132

vN−1 = f (y1,N−1) + u(y1,N−1)

⇒ u(y1,N−1) = vN−1 − f (y1,N−1)

vN = K

�x2φN+1 + f (yN ) + u(y1,N )

⇒ u(y1,N ) = vN − K

�x2φN+1 − f (y1,N ) (38)

5.2 Solution of the Estimation Problem

By selecting measurements from a subset of pointsx j j∈[1, 2, . . . ,m], the associated observation (measure-ment) equation remains as in Eq. (32), i.e.

⎛

⎜⎜⎝

z1z2· · ·zm

⎞

⎟⎟⎠ =

⎛

⎜⎜⎜⎜⎝

1 0 0 · · · 0 00 0 0 · · · 0 0· · · · · · · · · · · ·0 0 0 · · · 1 00 0 0 · · · 0 0

⎞

⎟⎟⎟⎟⎠

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

y1,1y2,1y1,2y2,2· · ·y1,Ny2,N

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

(39)

For the description of the system in the form of Eq. (33) onecan perform estimation using the Derivative-free nonlinearKalman Filter recursion. In the filter’s algorithm, the previ-ously defined matrices A,B and C are substituted by theirdiscrete-time equivalents Ad ,Bd and Cd . The discrete-timeKalman filter can be decomposed into two parts: (i) timeupdate (prediction stage), and (ii) measurement update (cor-rection stage).

measurement update

K (k) = P−(k)CTd [C ·P−(k)CT

d + R]−1

y(k) = y−(k) + K (k)[z(k) − Cd y−(k)]

P(k) = P−(k) − K (k)Cd P−(k)

(40)

time update

P−(k + 1) = Ad(k)P(k)ATd (k) + Q(k)

y−(k + 1) = Ad(k)y(k) + Bd(k)u(k)(41)

Therefore, by taking measurements of φ(x, t) at timeinstant t at a small number ofmeasuring points j = 1, . . . , n1it is possible to estimate the complete state vector, i.e. to getvalues of φ in a mesh of points that covers efficiently thevariations of φ(x, t). By processing a sequence of outputmeasurements of the system, one can obtain local estimatesof the state vector y. The measuring points (active sensors)can vary in time provided that the observability criterion forthe state-space model of the PDE holds.

According to Eqs. (40) and (41) an estimate of the sys-tem’s state vector y is obtained. Should one want to obtain an

Fig. 3 Grid points for measuring φ(x, t)

estimate of the state vector x of the initial state-space descrip-tion of the nonlinear wave-type dynamics he should apply theinverse differential flatness transformation connecting xi ’s toyi ’s. In the examined model of the wave PDE it holds that

x1 = y1,1 x2 = y1,2 x3 = y1,3· · · · · · · · ·xN−2 = y1,N−2 xN−1 = y1,N−1 xN = y1,N

(42)

Remark The proposed derivative-free nonlinear Kalman Fil-ter is of improved precision because unlike other nonlinearfiltering schemes, e.g. unlike the Extended Kalman Fil-ter it does not introduce cumulative numerical errors dueto approximative linearization of the system’s dynamics.Besides it is computationally more efficient (faster) becauseit does not require to calculate Jacobian matrices and partialderivatives (Fig. 3).

If state estimation-based control is applied to the state-space model of the nonlinear wave-PDE then the associatedcontrol inputs are

v1 = K

�x2φ0 + f (y1,1) + u(y1,1)

⇒ u(y1,1) = v1 − K

�x2φ0 − f (y1,1)

v2 = f (y1,2) + u(y1,2)⇒ u(y1,2) = v2 − f (y1,2)

v3 = f (y1,3) + u(y1,3)⇒ u(y1,3) = v3 − f (y1,3)

· · ·· · ·vN−1 = f (y1,N−1) + u(y1,N−1)

⇒ u(y1,N−1) = vN−1 − f (y1,N−1)

123

Intell Ind Syst (2016) 2:119–132 127

Fig. 4 a Nonlinear wave dynamics φ(x, t) without control. b Nonlinear wave dynamics φ(x, t) made to track a sinusoidal setpoint after applyingcontrol

Fig. 5 State estimation-based control with the use of the derivative-free nonlinear Kalman Filter: a nonlinear wave dynamics φ(x, t) withoutcontrol, b nonlinear wave dynamics φ(x, t) made to track a sinusoidal setpoint after applying control

vN = K

�x2φN+1 + f (yN ) + u(y1,N )

⇒ u(y1,N ) = vN − K

�x2φN+1 − f (y1,N ) (43)

6 Simulation Tests

The performance of the proposed control scheme has beentested in the model of the sine-Gordon nonlinear wave-typepartial differential equation. A discretization grid of the PDEconsisting of N = 50 points was considered. At each grid

point the local control input u(xi , t) was exerted to thesystem. The extended state-space description of the wave-type PDE comprised a state vector of dimension y∈R100×1.The number of measurement points was m≤N were m wasselected such that the observability of the linearized state-space model of the PDE is preserved. The obtained resultsare presented in Figs. 4, 5, 6 and 7. In Figs. 4 and 6 results arepresented about control of the PDE dynamics using feedbackof the entire state vector, while in Figs. 5 and 7 results aboutthe state estimation-based control of the PDE are given. It canbe observed that through the application of suitable controlinputs thewave functionφ(x, t) and its derivative φ(x, t) can

123

128 Intell Ind Syst (2016) 2:119–132

Fig. 6 a First time derivative of the nonlinear wave dynamics φ(x, t) without control. b First time derivative of the nonlinear wave dynamicsφ(x, t) made to track a sinusoidal setpoint after applying control

Fig. 7 State estimation-based control with the use of the derivative-free nonlinear Kalman Filter: a first time derivative of the nonlinear wavedynamics φ(x, t) without control, b first time derivative of the nonlinear wave dynamics φ(x, t) made to track a sinusoidal setpoint after applyingcontrol

bemade to track the desirable sinusoidal setpoints.Moreover,in Figs. 8, 9, 10, 11 and 12 diagrams are provided about thetracking accuracy of the proposed control method at specificpoints of the discretization grid. It can be noticed again thatthe state estimation-based control method resulted in fast andaccurate tracking of the reference setpoints.

7 Conclusions

A new nonlinear control and filtering method has been pro-posed for wave-type nonlinear partial differential equations.

After applying numerical methods for the discretization ofthe wave equation a state-space description of the associatedPDE was obtained. It has been shown that this state spacemodel of the wave-type nonlinear PDE is a differentiallyflat one. By proving that differential flatness properties holdand by applying a state variables transformation (diffeomor-phism) it has been shown that the nonlinear wave-type PDEcan be written as a state space-model of the linear canon-ical (Brunovsky) form. Thus, the model of the nonlinearwave-type PDE is represented by an equivalent set of linearordinary differential equations (ODE). For the latter descrip-tion the solution of the pointwise control problem becomes

123

Intell Ind Syst (2016) 2:119–132 129

0 10 20 30 400

1

2

3

4

time

5

0 10 20 30 40−1

−0.5

0

0.5

1

time6

0 10 20 30 400

2

4

6

time

7

0 10 20 30 40−1

−0.5

0

0.5

1

time

8

0 10 20 30 40−4

−2

0

2

4

time

5

0 10 20 30 40−1

0

1

2

time

6

0 10 20 30 40−4

−2

0

2

4

time

7

0 10 20 30 40−2

−1

0

1

2

time

8

(a) (b)

Fig. 8 Control and estimation of wave dynamics at grid points 3 (first row of diagrams) to 4 (second row of diagrams) a function φ(x, t) and itsderivative when no control input is applied, b function φ(x, t) and its derivative under state feedback control

0 10 20 30 400

2

4

6

time

17

0 10 20 30 40−1

0

1

2

time

18

0 10 20 30 400

2

4

6

time

19

0 10 20 30 40−2

−1

0

1

2

time

20

0 10 20 30 40−4

−2

0

2

4

time

17

0 10 20 30 40−1

0

1

2

time

18

0 10 20 30 40−4

−2

0

2

4

time

19

0 10 20 30 40−2

−1

0

1

2

time

20

(a) (b)

Fig. 9 Control and estimation of wave dynamics at grid points 9 (first row of diagrams) to 10 (second row of diagrams) a function φ(x, t) and itsderivative when no control input is applied, b function φ(x, t) and its derivative under state feedback control

easy by applying a pole placement technique to each localODE.

Another objective of the article has been to implementstate-feedback control of the nonlinear wave dynamics usingmeasurements from a small number of sensors. This has

become possible through the application of a new nonlinearfiltering method under the name Derivative-free nonlinearKalman Filter. The filter comprises the standard KalmanFilter recursion applied to the linear equivalent state-spacemodel of the wave PDE and of an inverse transformation

123

130 Intell Ind Syst (2016) 2:119–132

0 10 20 30 400

5

10

15

time

57

0 10 20 30 40−2

−1

0

1

2

time

58

0 10 20 30 400

5

10

15

time

59

0 10 20 30 40−2

−1

0

1

2

time

60

0 10 20 30 40−4

−2

0

2

4

time

57

0 10 20 30 40−1

0

1

2

time

58

0 10 20 30 40−4

−2

0

2

4

time

59

0 10 20 30 40−2

−1

0

1

2

time

60

(a) (b)

Fig. 10 Control and estimation of wave dynamics at grid points 29 (first row of diagrams) to 30 (second row of diagrams) a function φ(x, t) andits derivative when no control input is applied, b function φ(x, t) and its derivative under state feedback control

0 10 20 30 400

2

4

6

time

85

0 10 20 30 40−4

−2

0

2

4

time

86

0 10 20 30 400

2

4

6

time

87

0 10 20 30 40−4

−2

0

2

4

time

88

0 10 20 30 40−4

−2

0

2

4

time

85

0 10 20 30 40−1

0

1

2

time

86

0 10 20 30 40−4

−2

0

2

4

time

87

0 10 20 30 40−2

−1

0

1

2

time

88

(a) (b)

Fig. 11 Control and estimation of wave dynamics at grid points 43 (first row of diagrams) to 44 (second row of diagrams) a function φ(x, t) andits derivative when no control input is applied, b function φ(x, t) and its derivative under state feedback control

which is based on differential flatness properties which pro-vides estimates for the state variables of the initial nonlineardescription of the system. The efficiency of the proposedcontrol and filtering methods for nonlinear wave-type PDEs

has been tested through simulation experiments. It is alsonoted that the method can be also applied to other distributedparameter systems of the parabolic, elliptic or hyperbolictype.

123

Intell Ind Syst (2016) 2:119–132 131

0 10 20 30 40−5

0

5

10

time

93

0 10 20 30 40−10

−5

0

5

10

time94

0 10 20 30 40−5

0

5

10

time

95

0 10 20 30 40−10

0

10

20

time

96

0 10 20 30 40−4

−2

0

2

4

time

93

0 10 20 30 40−1

0

1

2

time

94

0 10 20 30 40−4

−2

0

2

4

time

95

0 10 20 30 40−2

−1

0

1

2

time

96

(a) (b)

Fig. 12 Control and estimation of wave dynamics at grid points 47 (first row of diagrams) to 48 (second row of diagrams) a function φ(x, t) andits derivative when no control input is applied, b function φ(x, t) and its derivative under state feedback control

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