Control Engineering Practice · Pulse-width predictive control for LTV systems with application to...

Control Engineering Practice ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Contents lists available at ScienceDirect

Control Engineering Practice

http://d0967-06

n CorrE-m

eduardo

Pleas

journal homepage: www.elsevier.com/locate/conengprac

Pulse-width predictive control for LTV systems with application tospacecraft rendezvous

R. Vazquez a,n, F. Gavilan a, E.F. Camacho b

a Departamento de Ingeniería Aeroespacial, Escuela Superior de Ingenieros, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spainb Departamento de Ingeniería de Sistemas y Automática, Escuela Superior de Ingenieros, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092Sevilla, Spain

a r t i c l e i n f o

Article history:Received 16 October 2015Received in revised form23 June 2016Accepted 24 June 2016

Keywords:Spacecraft autonomySpace roboticsPulse-width modulationTrajectory planningOptimal trajectoryLinear time-varying systems

x.doi.org/10.1016/j.conengprac.2016.06.01761/& 2016 Elsevier Ltd. All rights reserved.

esponding author.ail addresses: [email protected] (R. Vazquez), [email protected] (E.F. Camacho).

e cite this article as: Vazquez, R., et

a b s t r a c t

This work presents a Model Predictive Controller (MPC) that is able to handle Linear Time-Varying (LTV)plants with Pulse-Width Modulated (PWM) control. The MPC is based on a planner that employs a Pulse-Amplitude Modulated (PAM) or impulsive approximation as a hot-start and then uses explicit linear-ization around successive PWM solutions for rapidly improving the solution by means of quadraticprogramming. As an example, the problem of rendezvous of spacecraft for eccentric target orbits isconsidered. The problem is modeled by the LTV Tschauner–Hempel equations, whose state transitionmatrix is explicit; this is exploited by the algorithm for rapid convergence. The efficacy of the method isshown in a simulation study.

& 2016 Elsevier Ltd. All rights reserved.

1. Introduction

Frequently in applications, systems (and in particular, aero-space systems) need to be controlled by using Pulse-WidthModulated (PWM) actuators, i.e., actuators whose output level isfixed and can only be turned on and off, such as spacecraftthrusters. It would be therefore desirable to use control designmethods that directly take into account this type of actuation.However, most feedback design and motion planning methodsignore variable-width pulses and approximate the control inputseither by impulses (which produce instantaneous changes in somecombination of the states) or Pulse-Amplitude Modulated (PAM)control. However, neither impulsive actuation nor PAM actuationcaptures with precision the behavior of PWM actuators, whichcannot produce arbitrary control forces, but instead can only beswitched on (producing a fixed amount of control) or off (produ-cing no control). These switching times are the only signals thatcan be controlled. Control design with PWM actuation poses achallenge because the system becomes nonlinear in the switchingtimes, even if the dynamics are linear.

One can find in the literature several procedures to find anequivalent PWM solution starting from a PAM solution (see for

[email protected] (F. Gavilan),

al. (2016), http://dx.doi.org/

instance Bernelli-Zazzera, Mantegazza, & Nurzia, 1998; Ieko, Ochi,& Kanai, 1999; Shieh, Wang, & Sunkel, 1996). These methods allowus to, given the PAM inputs of a system, compute PWM inputs thatproduce a system output which optimally approximates the out-put of the system when driven by the PAM signals. The results arebased on the so-called Principle of Equivalent Areas, which com-putes the PWM signal so that its integral (area covered by thesignal as time advances) during a sample time is same as the in-tegral computed with the PAM signal. However, while these pro-cedures are quite effective in the sense that the output producedby the approximate PWM signals is very similar to the one pro-duced by PAM signals, they assume that the plant is linear time-invariant in order to prove the optimality of the approximation.

In this paper, Model Predictive Control (MPC) is used to directlyfind PWM signals to control the system. MPC (see, e.g., Camacho &Bordons, 2004) is a family of methods that originated in the lateseventies and has considerably developed since then. In MPC, aprocess model is used to predict the future plant outputs, based onpast and current values and on the proposed optimal future con-trol actions. These actions are calculated by an optimizer takinginto account a cost function as well as constraints. Since the plantis nonlinear in the control signals (ON–OFF times), the underlyingoptimization problem is nonlinear and possibly non-convex. Tosolve the problem, the algorithm starts from an initial guesscomputed by solving an optimal linear program with PAM or im-pulsive actuation, approximate the solution with ON–OFF thrus-ters, and then iteratively linearize around the obtained solutions to

10.1016/j.conengprac.2016.06.017i

www.sciencedirect.com/science/journal/09670661

www.elsevier.com/locate/conengprac

http://dx.doi.org/10.1016/j.conengprac.2016.06.017



mailto:[email protected]







Fig. 1. PWM variables.

R. Vazquez et al. / Control Engineering Practice ∎ (∎∎∎∎) ∎∎∎–∎∎∎2

improve the PWM solution. While the idea of linearization tospecifically compute optimal PWM control signals in the context ofMPC is, to the best knowledge of the authors, original, it must benoted that local linearization techniques have been used for op-timal trajectory problems in other contexts (see e.g. Kim, Shim, &Sastry, 2002) and are standard to solve general optimization pro-blems (a popular linearization-based method is the SequentialQuadratic Programming technique, see, e.g. Nocedal & Wright,2006, p. 529).

As an application, the problem of rendezvous of spacecraft isconsidered, i.e., the controlled close encounter of two space ve-hicles. Autonomous spacecraft rendezvous capabilities are be-coming a necessity as access to space continues increasing. Thefield has become very active in recent years, with a rapidlygrowing literature. Among others, approaches based on trajectoryplanning and optimization (such as Arzelier, Louembet, Ronde-pierre, & Kara-Zaitri, 2011, 2013; Breger & How, 2008; D'Amicoet al., 2013; Deaconu, Louembet, & Théron, 2014, 2015; Gaias,D'Amico, & Ardaens, 2014; Louembet, Arzelier, & Deaconu, 2015)and predictive control, (see for instance Asawa, Nagashio, & Kida,2006; Gavilan, Vazquez, & Camacho, 2009, 2012; Hartley, Trodden,Richards, & Maciejowski, 2012; Jewison, Erwin, & Saenz-Otero,2015; Larsson, Berge, Bodin, & Jönsson, 2006; Leomanni, Rogers, &Gabriel, 2014; Richards & How, 2003; Rossi & Lovera, 2002; Weiss,Kolmanovsky, Baldwin, & Erwin, 2012) are emerging.

Classically, in these approaches the problem of rendezvous ismodeled by using impulsive maneuvers; one computes a sequenceof impulses (possibly optimal, according to a defined metric thatmay take into account fuel consumption and state errors), usuallyreferred to as ΔV 's, to achieve rendezvous.

Recently, Vazquez, Gavilan, and Camacho (2011, 2014), in-troduced a trajectory planning algorithm for spacecraft rendez-vous that was able to incorporate PWM control signals. The formerconsidered the linear time-invariant Clohessy–Wiltshire model(target orbiting in a circular Keplerian orbit, see Clohessy & Wilt-shire, 1960). The latter extended the approach to elliptical targetorbits by using the linear time-varying Tschauner–Hempel model(see Tschauner & Hempel, 1965). Both methods start from an in-itial guess computed by solving an optimal linear program withPAM or impulsive actuation, approximate the solution with ON–OFF thrusters, and then iteratively linearize around the obtainedsolutions to improve the PWM solution. For both circular and el-liptical target orbits the algorithms are simple and reasonably fast,and simulations favorably compare with an impulsive-only ap-proach. These results were extended in Vazquez, Gavilan, andCamacho (2015) to a decreasing-horizon model predictive con-troller able to take into account orbital perturbations, disturbancesor model errors.

In this paper, a receding-horizon model predictive controllerwith PWM inputs is formulated for general Linear Time-Varying(LTV) plants and both alternatives (PAM or impulsive startingguess) are discussed in detail, with an application to rendezvousgiven at the end of the paper. The advantage of a receding-horizonformulation is that it allows a softer formulation of constraints,avoiding the need for e.g. a fixed rendezvous time, thus improvingfeasibility, and in addition it can be run indefinitely, allowing us toconsider properties such as stability that cannot even be properlydefined for decreasing-horizon algorithms.

It must be noted that, while this paper gives a very detailedapplication to the problem of rendezvous of spacecraft, there aremany other potential applications for the methodology developedin this paper, both in the aerospace domain and in other fields,such as current control in electrical servo-motors for movingcontrol surfaces or stabilization of coupled thermoelastic vibra-tions in space structures, see e.g. Bernelli-Zazzera et al. (1998).

The structure of the paper is as follows. In Section 2 the plant

Please cite this article as: Vazquez, R., et al. (2016), http://dx.doi.org/

model is introduced. Three types of inputs are considered: PWM,PAM and impulsive. Section 3 follows with a formulation of theunderlying optimization problem. Section 4 describes a methodthat solves the planning problem using PWM signals. Section 5develops the model predictive controller. Next, Section 6 describesthe application to spacecraft rendezvous. Section 7 presents a si-mulation study of the method applied to spacecraft rendezvous.The paper finishes with some remarks in Section 8.

2. System model

Consider a linear time-varying system given as

= ( ) + ( ) ( )x A t x B t u, 1

where ∈x n is the state, ∈u m is the input (control) vector, andA(t) and B(t) are, respectively, ×n n and ×n m matrices dependingon time ≥t 0.

Considering that, for some time ≥t 0k , initial conditions( ) ∈x tk

n are given and the input is known, the solution to (1) for>t tk is given by

∫Φ Φ( ) = ( ) ( ) + ( ) ( ) ( )( )

x t t t x t t s B s u s s, , d ,2k k

t

t

k

where Φ( )t t, k is the system transition matrix, see for instance Rugh(1996). This matrix can be computed numerically (or analytically ifpossible) as the unique solution to the linear matrix differentialequation

Φ Φ( ) = ( ) ( ) > ( )t t A t t t t t, , , 3k k k

Φ( ) = ( )t t, I. 4k k

To obtain a unified notation in terms of the inputs, denote byBi(t) the i-th column of B(t), corresponding to the i-th input ui(t),for = …i m1, , . In this paper, time intervals starting at some initialtime tk and ending at = ++t t Tk k1 are considered, where T will bean adequate sample time. Then Eq. (2) can be written as

∫∑Φ Φ( ) = ( ) ( ) + ( ) ( ) ( )( )

+ +=

++

x t t t x t t s B s u s s, , d ,5

k k k ki

m

t

t

k i i1 11

1k

k 1

The objective is solving the problem with PWM inputs. In ad-dition, two other types of inputs are considered; they will be usedas an intermediate step towards computing PWM inputs by thealgorithm. All types of input are analyzed in the following sections.

2.1. Pulse width-modulated (PWM) control

In the PWM case, for each sampling interval, each input ui is apulse starting at time τk i, with pulse width κk i, , with constant

magnitude =u umax k iW, , as shown in Fig. 1 (notice that the subscript

10.1016/j.conengprac.2016.06.017i




R. Vazquez et al. / Control Engineering Practice ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3

k is used to denote the sampling interval of each term), i.e.,

⎧⎨⎪⎪

⎩⎪⎪

⎡⎣⎡⎣⎡⎣ ⎤⎦

))

τ

τ τ κ

τ κ

( ) =

∈ +

∈ + + +

∈ + + ( )+

u t

t t t

u t t t

t t t

0, , ,

, , ,

0, , , 6

i

k k k i

k iW

k k i k k i k i

k k i k i k

,

, , , ,

, , 1

with κ > 0k i, , τ > 0k i, and τ κ+ < Tk i k i, , , where the last constraintprevents the PWM signal from spilling over to the next time in-terval. Then, substituting ui(t) in (5) one obtains

⎛⎝⎜⎜

⎞⎠⎟⎟∫∑Φ Φ( ) = ( ) ( ) + ( ) ( )

( )τ

τ κ

+ += +

+ +

+x t t t x t t s B s s u, , d ,7

k k k ki

m

t

t

k i k iW

1 11

1 ,k k i

k k i k i

,

, ,

and denoting

∫τ κ Φ( ) = ( ) ( )( )τ

τ κ

+

+ +

+B t s B s s, , d ,8k i

Wk i k i

t

t

k i, , , 1k k i

k k i k i

,

, ,

one can write the solution as

∑Φ τ κ( ) = ( ) ( ) + ( )( )

+ +=

x t t t x t B u, , .9

k k k ki

m

k iW

k i k i k iW

1 11

, , , ,

There is an important difference between a PWM input and thePAM or impulsive inputs that will be subsequently introduced.While the latter can in principle take positive or negative values atdifferent times, the former is fixed either as positive or negativefor all time. Thus, typically a PWM model has twice the number ofinputs than a PAM/impulsive model. To make this explicit in themodel (9), denote with a plus or minus super-index the positive ornegative inputs, as follows:

⎡⎣⎢⎢

⎤⎦⎥⎥∑

Φ

τ κ τ κ

( ) = ( ) ( )

+ ( ) − ( )( )

+ +

=

+ + + − − −

x t t t x t

B u B u

,

, , ,10

k k k k

i

m

k iW

k i k i k iW

k iW

k i k i k iW

1 1

1, , , , , , , ,

with τ κ+ + +u , ,k iW

k i k i, , , and τ κ− − −u , ,k iW

k i k i, , , denoting, respectively, themagnitude, start, and width of the positive and negative i-th inputpulses.

2.2. Pulse amplitude-modulated (PAM) control

In this case, each control ui(t) in (5) is constant inside the in-terval [ )+t t,k k 1 , and equal to uk i

A, . Then, substituting ui(t) in (5) one

obtains

⎛⎝⎜⎜

⎞⎠⎟⎟∫∑Φ Φ( ) = ( ) ( ) + ( ) ( )

( )+ +

=+

+x t t t x t t s B s s u, , d ,

11k k k k

i

m

t

t

k i k iA

1 11

1 ,k

k 1

and denoting by

∫ Φ= ( ) ( )( )+

+B t s B s s, d ,

12k iA

t

t

k i, 1k

k 1

the solution can be written as

∑Φ( ) = ( ) ( ) +( )

+ +=

x t t t x t B u, .13

k k k ki

m

k iA

k iA

1 11

, ,

2.3. Impulsive control

In this case, δ τ( ) = ( − ( + ))u t u t ti k iI

k k i, , , where δ( )t is Dirac's deltafunction, τ+tk k i, is the instant at which the impulse is given, anduk i, is the magnitude of the impulse. Then, assuming τ< < T0 k i, forall i (all the impulses are given inside the considered time interval)and substituting ui(t) in (5) one obtains


∑Φ Φ τ τ( ) = ( ) ( ) + ( + ) ( + )( )

+ +=

+x t t t x t t t B t u, , ,14

k k k ki

m

k k i i k i k iI

1 11

1 ,

and denoting by τ Φ τ τ( ) = ( ) ( + )+B t B t,k iI

k i k i i k i, , 1 ,

∑Φ τ( ) = ( ) ( ) + ( )( )

+ +=

x t t t x t B u, .15

k k k ki

m

k iI

k i k iI

1 11

, , ,

2.4. Discretization and compact notation

Consider now a sequence of time instants = +t t kTk 0 , = …k 0, ,and denote = ( )x x tk k . Then, it is possible to write, for both PAMand impulsive control,

= + ( )+x A x B U , 16k k k k k1

where Φ= ( )+A t t,k k k1 , and Bk and Uk depend on the input type.Note that, in general, Ak and Bk are matrices that must be com-puted numerically. In the PWM case, write

= + − ( )++ + − −x A x B U B U , 17k k k k k k k1

where +Bk is a matrix whose i-th column is τ κ( )+ +B ,k iW

k i k i, , , and +Uk , τ+k ,

and κ +k are column vectors whose i-th entries are, respectively, uk i

W, ,

τ+k i, and κ +

k i, . The same definitions (with minus super-index) areused for the negative pulses. Then, to reach model (16), define

⎡⎣⎢⎢

⎤⎦⎥⎥

⎡⎣⎢⎢

⎤⎦⎥⎥

⎡⎣⎢⎢

⎤⎦⎥⎥

⎡⎣⎢⎢

⎤⎦⎥⎥τ

ττ

κκκ

= − =

= =( )

+ −+

−

+

−

+

−

B B B UU

U, ,

, .18

k k k kk

k

kk

kk

k

k

The definitions of Bk are simpler for the other types of actua-tion. In the PAM case, Bk is a matrix whose i-th column is Bk i

A, and

Uk a column vector whose i-th entry is uk iA, . In the impulsive case, Bk

is a matrix whose i-th column is τ( )Bk iI

k i, , , and Uk and τk are column

vectors whose i-th entries are, respectively, uk iI, and τk i, .

Next a compact formulation is developed to simplify the no-tation of the problem. The state at time + +tk j 1, given the state xk attime tk, and the input signals from tk to time +tk j, is computed byapplying recursively, in the PAM and impulsive cases, Eq. (16). Oneobtains

∑= +( )

+ + + +=

+

+x A x A B U ,19

k j k j k ki k

k j

k j i k i k i1 1, , , ,

where the definition =A Ik k, , =+A Ak k k1, , and if >j 0 then= …+ + + + −A A A Ak j k k j k j k1, 1 has been used. Define now k and k as a

stack vector of Np state and input vectors, respectively, spanningfrom time +tk 1 to time +tk Np

for the state and from time tk to time

+ −tk N 1pfor the controls, where Np is the planning horizon:

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

= ⋮ = ⋮+

+ + −

x

x

U

U, .k

k

k N

k

k

k N

1

1p p

Similarly, for the impulsive and PWM cases, define

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎡

⎣⎢⎢

⎤

⎦⎥⎥Γ

τ

τΛ

κ

κΥ

ΓΛ

= ⋮ = ⋮ =+ − + −

, ,k

k

k Nk

k

k Nk

k

k1 1p p

Then one can write

= + ( )F x G , 20k k k k k

where Gk is a block lower triangular matrix of size ×nN mNp p,defined as

10.1016/j.conengprac.2016.06.017i





⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

⋯⋯

⋮ ⋮ ⋱ ⋮… ( )

+ + +

+ + + + + + −

G

B

A B B

A B A B B

0 0

0,

21

k

k

k k k k

k N k k k N k k k Np

2, 1 1

, 1 , 2 1 1p p

this is, its non-null blocks are defined by ( ) = + + + −G A Bk jl k j k l k l, 1, andthe matrix Fk is defined as:

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=⋮

( )

+

+

+

F

A

A

A

.

22

k

k k

k k

k N k

1,

2,

,p

Note that, in the impulsive case, Gk is a (nonlinear) function ofΓk, whereas in the PWM case it is a (nonlinear) function of Υk. Toavoid lengthy expressions this dependence has been omitted. Inthe PWM case, k is fixed whereas in the other cases it is the inputvariable.

3. Formulation of the planning problem

Next the planning problem is formulated, introducing theconstraints and the objective function. The formulation is done forthe three types of control signals.

3.1. Constraints on the problem

First constraints on the state and input are introduced. Whileonly inequalities are considered, equality constraints would betreated similarly.

3.1.1. Inequality constraints on the stateThe state can be subject to inequality constraints along the

planning horizon, which can vary as time advances. These can beformulated in general as ≤C dS k k S k, , , and using (20), one reaches aexpression in terms of inputs, namely

≤ − ( )C G d C F x . 23S k k k S k S k k k, , ,

3.1.2. Input constraintsInput constraints are different depending on the type of input.In the PWM case, the inputs k are fixed, but the start time of

impulse, τ+tk k, and its end, τ κ+ +tk k k, must be within the timeinterval both for negative and positive pulses. Thus

Γ≤ ( )0 , 24k

Λ≤ ( )0 , 25k

Γ Λ+ ≤ ( ), 26k k

with being a stack vector of the same size than Γk and Λk,containing the sampling time T in each component. These in-equalities can be summarized as

Υ ≤ ( )C d . 27W k W

In the PAM case, the inputs are limited above and below. Thus

≤ ≤ ( )U U , 28PAM k PAM

where U PAM and U PAM are stacked vectors containing the lowerand upper bounds of the control forces exerted by each thruster ateach sampling time.


In the impulsive case, the inputs uk are limited above and be-low, but also the times of impulse, τ+tk k, must be within the timeinterval. Thus

≤ ≤ ( )U U , 29IMP k IMP

Γ≤ ≤ ( )0 , 30k

where, again U IMP and U IMP are stacked vectors containing thelower and upper bounds of the impulses exerted by each thrusterat each sampling time.

3.2. Objective function

At each sampling time k, the objective function to be mini-mized is a combination of the 1-norm of the control signal, whichis denoted as J k, , (which gives an estimation of fuel consumptionin case the control signal is thrust, see Section 6) and a weightedsquared 2-norm of the state, which is denoted as J k, , both takenover the planning horizon. Thus,

α= + ( )J J J , 31k k k, ,

where α is a positive constant that allows us to give a relativeweight between input cost and state error. Note that α could beincluded inside any two of the state or control cost indexes;however it is convenient to have a single parameter controlling therelative weight between the two. J k, is computed as

= ( )J Q , 32k kT

k k,

for >Q 0k . Written in terms of the inputs and the starting point xk,an equivalent objective function reads

′ = + ( )J x F Q G G Q G2 , 33k kT

kT

k k k kT

kT

k k k,

an expression in which the constant term x F Q F xkT

kT

k k k—which doesnot play a role in the planning optimization as it is constant, givenxk—is neglected (notice the prime in ′J k, introduced to reflect thischange).

The value of J k, does, however, depend on the control type. Itsdefinition is shown in the next sections according to the type ofcontrol input.

3.2.1. PWM control inputsFor the case of PWM control inputs, using definition (6) it can

be seen that the objective function J k, is given by

⎡

⎣⎢⎢

⎤

⎦⎥⎥∑ κ κ Λ Υ= ( ) + ( ) = =

( )=

+ −+ + − −J U U ,

34k

j k

k N

jT

j jT

j kT

k kJ

k,

1p

with kJ defined by blocks as

⎡⎣ ⎤⎦= ( )0 . 35kJ

kT

The times Γk where inputs start does not play a role in the costfunction (only their duration Λk).

3.2.2. PAM control inputsFor the case of PAM control inputs, it can be seen that the

objective function JU(k) is given by

∑= ∥ ∥ = ∥ ∥( )=

+ −

J T U T .36

kj k

k N

j k,

1

1 1

p

3.2.3. Impulsive control inputsFor the case of impulsive control inputs, J k, is given by

10.1016/j.conengprac.2016.06.017i





∑= ∥ ∥ = ∥ ∥( )=

+ −

J U ,37

kj k

k N

j k,

1

1 1

p

where it should be noticed that, as in the PWM case, the locationτk i, of the impulses does not play a role in the cost function.

3.3. Planning optimization problem

Now, for each of the input types, one can formulate a planningoptimization problem starting from initial condition xk at time tk,with a planning horizon of Np, as follows.

3.3.1. PWM control inputsFor PWM control inputs, the planning optimization problem is

formulated as

Υ Υ Υ α Υ

Υ

Υ

( ) + ( ) ( ) +

( ) ≤ −

≤ ( )

Υx F Q G G Q G

C G d C F x

C d

min 2 ,

s. t. ,

. 38

kT

kT

k k k k kT

kT

k k k k k kJ

k

S k k k k S k S k k k

W k W

, , ,

k

Notice that k is known, and one has to compute the start andwidth of the pulses, contained in Υk (start and duration of pulses),which enter nonlinearly in the optimization problem. The de-pendence of Gk on Υk has been made explicit, because it is thevariable with respect to which the optimization is carried out.

3.3.2. PAM control inputsFor PAM control inputs, the planning optimization problem is

formulated as

α+ + ∥ ∥

≤ −

≤ ≤ ( )

x F Q G G Q G T

C G d C F x

U U

min 2 ,

s.t. ,

. 39

kT

kT

k k k kT

kT

k k k k

S k k k S k S k k k

PAM k PAM

1

, , ,

k

3.3.3. Impulsive control inputsFor impulsive control inputs, the planning optimization pro-

blem is formulated as

Γ Γ Γ α

Γ

Γ

( ) + ( ) ( ) + ∥ ∥

( ) ≤ −

≤ ≤

≤ ≤ ( )

Γx F Q G G Q G

C G d C F x

U U

min 2 ,

s. t. ,

,

0 . 40

kT

kT

k k k k kT

kT

k k k k k k

S k k k k S k S k k k

IMP k IMP

k

,1

, , ,

k k

The dependence of Gk onΓk (location of impulses) has been madeexplicit to emphasize that the optimization problem is nonlinear.

4. PWM planning algorithm

In this section the subindex k is kept even though it does notplay any role. For a “pure” planning problem, it could be set to zero.However, k will be useful when defining the MPC algorithm inSection 5.

Consider now the problem (38), given xk and k. The planningalgorithm proposed in this paper follows the next steps (whichwill be further explained in the next sections).

Step 1 (Section 4.1): Solve either the PAM optimization problem(39), or the impulsive problem (40) with a fixed Γk, toprovide an initial guess of the PWM solution.

Step 2 (Section 4.2): The PAM or impulsive control inputs re-sulting from the optimization algorithm in Step 1 are


converted to a sequence of PWM inputs. Denote thisinitial sequence by Υk

0. Set i¼0.Step 3 (Section 4.3): The trajectory of the system with the PWM

inputs Υki is computed analytically (if possible) or nu-

merically by using Eq. (20). Denote the trajectory by ki .

Step 4 (Section 4.4): The system with PWM inputs is linearizedaround k

i , thus obtaining a linear, explicit plant withrespect to increments, denoted as Δk

i, in the PWM inputs.Then a quadratic program can be posed and solved tofind the increments that improve the cost function.

Step 5 (Section 4.4): The resulting solution Δki is used to im-

prove the approximation towards the real solution, bysetting Υ Υ Δ= ++

ki

ki

ki1 . Increase i by one and go back to

Step 3. The process is iterated until the solution con-verges or time is up. Note that if there is no convergence,or if Step 4 cannot be performed (there is no solution),the initial sequence Υk

0 can be used as a backup.

Note that an alternative solution would be to use a “generic”SQP algorithm (see, e.g. Nocedal & Wright, 2006, p. 529), or using adedicated MPC Toolkit like ACADO (described in Houska, Ferreau,& Diehl, 2011). There are several key features that differentiatethese generic approaches from the tailored procedure presentedhere. First, the problem formulation itself (in the continuous-timedomain) is used in the linearization (Step 4). This allows us toexplicitly find the derivatives in terms of PWM inputs. The hot-start approach (Step 1), based on impulsive or PAM signals, is alsoan important feature of the algorithm. Finally, for specific appli-cations (see e.g. Section 6), this procedure can easily use explicitclosed-form solutions of the system dynamics. This might be ad-vantageous for applications with less computational resources. Inany case, other methods that could be tailored to use these ele-ments would obviously benefit from them.

In the following sections, all the steps in the algorithm are described.

4.1. Step 1: Computation of PAM/impulsive control input

First, one has to choose whether to find an initial guess using aPAM approach or an impulsive approach. The PAM guess is moresuitable if one expects wide pulses, whereas the impulsive guess isbest when the pulses are rather short.

If a PAM guess is chosen, it is computed from (39), setting= +U TPAM k and = −U TPAM k . The maximum and minimum is set

in this way, so that the solution can always be converted to PWMfollowing the procedure of Section 4.2. On the other hand, theimpulsive guess is computed from (40), setting = +UIMP k and

= −U IMP k . The impulsive guess also requires to set the impulselocation Γk to some pre-determined value, so only the impulsemagnitude (which appears linearly in (40)) is unknown. Typicalpositions would be the middle of the interval (all entries of Γk

equal to T/2) or start of the interval (Γ = 0k ).

4.2. Step 2: Initial PWM solution: adapting the PAM/impulsive solution

The PAM/impulsive solution from Section 4.1, , is trans-formed to a PWM sequence of inputs, as follows:

1. From extract uj iA, (or uj i

I, if the initial solution is of impulsive

type) for = … + −j k k N, , 1p and = …i m1, , . Also extract τj i, ifthe initial solution is of impulsive type.

2. If the initial solution is of PAM type, set

⎧⎨⎪⎪

⎩⎪⎪

⎧⎨⎪⎪

⎩⎪⎪

τ τ( ) =>

≤

( ) =− <

≥ ( )

+ ± − ±t

Tu

uu

u

t

Tu

uu

u

, 0,

0, 0,

, 0,

0, 0, 41

j i

j iA

j iW j i

A

j iA

j i

j iA

j iW j i

A

j iA

,

,

,,

,

,

,

,,

,

10.1016/j.conengprac.2016.06.017i





and if the initial solution is of impulsive type,

⎧⎨⎪⎪

⎩⎪⎪

⎧⎨⎪⎪

⎩⎪⎪

τ τ=>

≤

=− <

≥ ( )

+ ± − ±

u

uu

u

u

uu

u

, 0,

0, 0,

, 0,

0, 0. 42

j i

j iI

j iW j i

A

j iI

j i

j iI

j iW j i

A

j iI

,

,

,,

,

,

,

,,

,

3. In the PAM case, the PWM input should be centered in the in-

terval: κ =τ+ − +

j i

T

, 2j i, , κ =

τ− − −

j i

T

, 2j i, . In the impulsive case, the PWM

input should be centered around the chosen τj i, (corrected if

necessary to avoid spillover), i.e.

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

κ

ττ

τ ττ

ττ

=

− <

− + >

−( )

+

+

++

+

T T

0,2

0,

,2

,

2, otherwise,

43

j i

j ij i

j i j ij i

j ij i

,

,,

, ,,

,,

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

κ

ττ

τ ττ

ττ

=

− <

− + >

−( )

−

−

−−

−

T T

0,2

0,

,2

,

2, otherwise,

44

j i

j ij i

j i j ij i

j ij i

,

,,

, ,,

,,

4. From τ+j i, , τ−

j i, , κ+j i, , and κ−

j i, , construct Γk and Λk and thus Υk.

The PWM signals Γ Λ,k k constructed by this method produce amoderately similar (but not equal) output to the system driven byPAM or impulsive signals, but as time advances the output mightconsiderably differ. See Shieh et al. (1996), Ieko et al. (1999) andBernelli-Zazzera et al. (1998) for more details and other methods.In addition, these approximated PWM controls are not optimal(with respect to the real systemwith PWM inputs), and they mightnot even satisfy the constraints. However, this solution is onlyused as an initialization for the optimization algorithm proposednext. Denote as Υk

0 the found solution and set i¼0.

4.3. Step 3: Computation of trajectories under PWM inputs

For the current iteration i, apply (20) to compute the states ofthe system k

i at all times, with PWM inputs Υki. The matrix Gk

might need to be computed numerically if an explicit solution forthe integrals (8) is not known.

4.4. Steps 4 and 5: Refined PWM solution: an optimization algorithm

To linearize (20) around inputs Υki,

notice from (8) that

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

∫ττ κ

τΦ

Φ τ κ τ κ

Φ τ τ

∂∂

= ∂∂

= + + + +

− + +( )

τ

τ κ

+

+ +

+

+

+

B t s B s s

t t B t

t t B t

, , d ,

,

, ,45

k ik iW

k i k ik i t

t

k i

k k k i i i k k i k i

k k k i i k k i

,, , ,

,1

1 , , ,

1 , ,

k k i

k k i k i

,

, ,


and

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

∫κτ κ

κΦ

Φ τ κ τ κ

∂∂

= ∂∂

= + + + +( )

τ

τ κ

+

+ +

+

+

B t s B s s

t t B t

, , d ,

, .46

k ik iW

k i k ik i t

t

k i

k k k i i i k k i k i

,, , ,

,1

1 , , ,

k k i

k k i k i

,

, ,

Thus, (16) can be explicitly linearized around some given τ+k , τ−

k

and κ +k , κ −

k , reaching

δτ δκ= + + + ( )τ κ+

Δ Δx A x B U B B , 47k k k k k k k k k1

where Bk is computed with τ+k , τ−

k , κ +k , and κ −

k (notice that the twofirst terms of the right-hand-side part correspond to the nominalterms of the Taylor series expansion), and define

⎡⎣ ⎤⎦ ⎡⎣ ⎤⎦= = ( )τ δτ δτ κ δκ δκΔ Δ+ − + +

B B B B B B, , 48k k k k k k

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥δτ

τ ττ τ

δκκ κκ κ

= ¯ −¯ −

= ¯ −¯ − ( )

+ +

− −

+ +

− −, ,49

kk k

k kk

k k

k k

where the i-th entries of the Bkδ matrices in (48) are given, re-

spectively, by

( ) ( )( ) ( )

Φ τ κ τ κ

Φ τ τ

( ) = + + ¯ + ¯ × + ¯ + ¯

− + + ¯ + ¯ ( )

δτ + + + + +

+ + +

+B t T t B t u

t T t B t u

,

, 50

k i k k k i k i i k k i k i k iW

k k k i i k k i k iW

, , , , ,

, , ,

( ) ( )( ) ( )

Φ τ κ τ κ

Φ τ τ

( ) = − + + ¯ + ¯ × + ¯ + ¯

+ + + ¯ + ¯ ( )

δτ − − − − −

− − −

−B t T t B t u

t T t B t u

,

, 51

k i k k k i k i i k k i k i k iW

k k k i i k k i k iW

, , , , ,

, , ,

Φ τ κ τ κ( ) = ( + + ¯ + ¯ ) × ( + ¯ + ¯ ) ( )δκ + + + + ++B t T t B t u, , 52k i k k k i k i i k k i k i k i

W, , , , ,

Φ τ κ τ κ( ) = − ( + + ¯ + ¯ ) × ( + ¯ + ¯ ) ( )δκ − − − − −−B t T t B t u, 53k i k k k i k i i k k i k i k i

W, , , , ,

Thus, defining stack vectors with the increments in the PWMvariables at step i as

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

Γδτ

δτΛ

δκ

δκΔ = ⋮ Δ = ⋮

+ − + −

, ,ki

k

k N

ki

k

k N1 1p p

and grouping all increments as

⎡⎣⎢⎢

⎤⎦⎥⎥

⎡⎣⎢⎢

⎤⎦⎥⎥Δ

Γ

Λ=

Δ

Δ=Δ

τ

κ

Δ

ΔB

B

B, ,k

i ki

ki k

k

k

and defining ΔGk as in (21), i.e.,

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

=

⋯

⋯⋮ ⋮ ⋱ ⋮

… ( )

Δ

Δ

Δ Δ

Δ Δ Δ

+ + +

+ + + + + + −

G

B

A B B

A B A B B

0 0

0,

54

k

k

k k k k

k N k k k N k k k Np

2, 1 1

, 1 , 2 1 1p p

10.1016/j.conengprac.2016.06.017i





one can write

Υ Υ Δ≈ + ( ) + ( ) ( )ΔF x G G , 55ki

k k k ki

k k ki

ki

The state constraints (23) become

Υ Δ Υ( ) ≤ − − ( ) ( )ΔC G d C F x G . 56S k k ki

ki

S k S k k k k ki

k, , ,

The constraints on ΓΔ ki and ΛΔ k

i are as follows:

Δ Υ− ≤ − ( )C d C , 57W ki

W W ki

Δ Δ Δ− ≤ ≤ ( ), 58max ki

max

where the last constraint (58) is used to avoid large variations thatmight make the linearization approximation to fail. All theseconstraints are might be summarized as

Δ ≤ ( )Δ ΔC d . 59ki

Finally, the objective function can be rewritten in terms of Δki

as Υ Υ Δ= ( ) + ( )ΔJ J J ,ki

ki

ki

k ki

ki , where

Υ Υ Υ α Υ= ( ) + ( ) ( ) + ( )J x F Q G G Q G2 , 60ki

kT

kT

k k ki

k kT

kT

ki

k k ki

k kJ

ki

( ( )) ( ) ( ) ( )Υ Υ Δ Δ Υ Υ Δ

α Δ

= + + ( )

+ ( )

Δ Δ ΔΔJ x F G Q G G Q G2

. 61

k kT

kT

kT

kT

ki

k k ki

ki

ki T

kT

ki

k k ki

ki

kJ

ki

Noting that JkΔ is quadratic in Δk

i, a quadratic optimization pro-gramwith linear restriction, formulated on the output increments,can be posed as follows:

Υ Δ

Υ Δ Υ

Δ

( )

( ) ≤ − − ( )

≤ ( )

Δ

Δ

Δ

Δ Δ

J

C G d C F x G

C d

min ,

s. t.: ,

. 62

k ki

ki

S k k ki

ki

S k S k k k k ki

k

ki

, , ,

ki

The solution Δk is used to recompute new PWM inputs,Υ Υ Δ= ++

ki

ki

ki1 . Then the linearization process can be repeated

around the new Υ +ki 1, refining the solution in each iteration.

5. Model predictive control with PWM inputs

In this section, building upon the trajectory planning algorithmof Section 4, which is open-loop and has a finite time-horizon, aclosed-loop algorithm is developed based on the ideas of modelpredictive control (also known as receding horizon control). Modelpredictive control closes the loop by simply re-planning themaneuver at each time step, after applying just the set of controlinputs corresponding to the first time step, and keeps lookingahead Np time steps. Thus, the algorithm starts at k¼0 and is re-peated for each k. The re-planning is done from the actual positionat each time step, which seldom coincides with the planned po-sition due to disturbances, thus effectively closing the loop.

However, except at the start, it is not necessary to repeat all thesteps of Section 4. Since the new position should be close to theplanned one, one can apply the linearization scheme of the plan-ning algorithm starting from the last available linearization. TheMPC algorithm is summarized next:

Step 1: At time step k¼0 and starting from x0 apply the Planningalgorithm of Section 4, obtaining a set of PWM inputs Υ0

that would optimize the planning problem (38) for thenext Np time steps, if there were no disturbances.


Step 2: Apply pulses corresponding to the first time instant; savethe rest of them. Set k¼1.

Step 3: One arrives at xk, which probably is not the intendedvalue of the state at time k but close. Thus re-planning isnecessary.

Step 4: For re-planning, apply the planning algorithm of Section4. However, to avoid the initial step of having to use aPAM or impulsive model and compute an initial guess,use instead as an initial guess the pulses of Υk�1 thatwere not used (all of them except for those corre-sponding to time −k 1) and guess the remaining pulses(at the end) for instance by using some simple controllaw. In this work, for the rendezvous example, a terminallaw based on the impulsive solution that would cancelthe residual velocity at the last step is used. In this way,form an initial guess Υ0

k.Step 5: Apply the linearization algorithm of Section 4.4 using Υ0

k

as initial guess to obtain, after iterating, a new set ofpulses Υk. Apply the set of pulses corresponding to timek. Save the rest of pulses.

Step 6: Repeat step 3.

6. Example application: spacecraft rendezvous

Rendezvous of spacecraft is the controlled close encounter oftwo (or more) space vehicles. This work assumes just two vehicles,one of which is the target vehicle (which is in a known orbit, andconsidered passive) and the other is the chaser spacecraft, whichbegins from a known position and maneuvers until it approachestarget. As explained in Fehse (2003), only close-range rendezvousis considered, which starts at hundreds of meters and ends whenthe chaser is very close to target (just several meters, with speedsof centimeters per second).

There are numerous mathematical models for spacecraft ren-dezvous; which one should be used depends on the parameters ofthe scenario. In Carter (1998) a survey of numerous mathematicalmodels for spacecraft rendezvous can be found.

For instance, if the target is orbiting in a circular Keplerian orbit,the general equations of the relative movement between an activechaser spacecraft close to a passive target vehicle are the lineartime-invariant Hill–Clohessy–Wiltshire (HCW) equations (in-troduced in Hill, 1878; Clohessy & Wiltshire, 1960). While theseequations are frequently used in the literature and they are suffi-ciently accurate for many applications, it must be noted that, insome situations, the HCW equations might not be a good enoughapproximation—for instance, if the target vehicle is moving in aKeplerian eccentric orbit (see Inalhan, Tillerson, & How, 2002) or ifsome orbital perturbations are taken into account (see for exampleHumi & Carter, 2008). A more complex model, the Tschauner–Hempel model (see Tschauner & Hempel, 1965; Carter, 1998) as-sumes that the target vehicle is passive and moving along an el-liptical orbit with semi-major axis a and eccentricity e. The systemequations are linear time-varying and cannot be exactly integratedin time to obtain a discrete transition model; however, if onesubstitutes the time t by the eccentric anomaly of the target orbit,E, it is possible to obtain explicit expressions for the system evo-lution in the PWM, impulsive, and PAM actuation cases. This is themodel considered in this work.

The rendezvous model can be expressed in cartesian co-ordinates, but also in the so-called relative orbital elements (see,e.g. Gaias et al., 2014; Sinclair et al., 2014), which are also useful topose a model predictive control problem for spacecraft applica-tions (see for instance the reference Breger & How, 2005, in which

10.1016/j.conengprac.2016.06.017i




Fig. 2. LVLH frame.


the formation flying maneuver is handled using this technique).For simplicity, the cartesian approach has been chosen in thiswork.

Let us first establish some notation. Define the orbital meanmotion = μn

a3, where μ is the gravitational parameter of the

central body around which the target spacecraft is orbiting.Now, note that t and E are related in a one-to-one fashion by

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

ρρ ρ ρ ρ

ρ ρ

αρ α ρ αρ

α ρ α αρ ρ α ρα α

=

− −( + ) − ( + )

( − − )

( − − ) ( − ) ( + + )− ( + ) ( )

Y

s esJ c

c J s

c s

c es e Jc s

s es es J c e c

s c e

0 0 2/ 3 0

1 1/ 1/ 0 3 1 1/ 0

0 0 / 0 0 /

0 0 3 0

1 0 3 3 00 0 0 0

,

64

E 2 2 2

2 2

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

( )

( )

( )

( ) ( )

( ) ( )

ρ ρ ρ α ρ α

ρ ρ ρ α ρ α

ρ ρ ρ ρα

ρρα

ρ ρ ρα

ρρα

αρ

ρ ρ ρα

ρα

ρ ρ ρα

ραρ

αρ

=−

( + ) −

( + ) −

+−

− + + ( + )− − ( + )

− ( + ) ( − ) + − − ( + )

( + ) −− −

( + ) −

( + ) − − ( + ) − ( + ) +

−−

( )

−YJ

e

e e s e s e

e s es

e

s e esc e s

es ce e sec es

c e es e

eses

c e e s esc e ss c e

s ec e

3

1

1 0 / / 0

1 0 / / 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

1

1

2 1 02 / 1

0

1 1 02/ 1

0

0 0 1 0 01

1 0 0

3 1 01

0

0 0 1 0 01

65

E1

2

2 2 2 2

2 2

2

2 2 2

2 2 2 2

22

2 2

2 2

22

using Kepler's equation

( − ) = − ( )n t t E e Esin , 63p

where tp is the time at periapsis used as a starting point to


measure the eccentric anomaly E. The time tp is chosen such that itis equal or less than the starting time which is denoted as t0(subtracting, if necessary, any number of orbital periods). Kepler'sequation is not analytically invertible, but its inverse can be foundnumerically with any desired degree of precision (see any OrbitalMechanics reference, such as Wie, 1998). Denote its inverse ex-plicitly as a function of time, namely = ( )E E t . Denote by E0 theeccentric anomaly corresponding to t0, this is, = ( )E E t0 0 . Then,

= ( ) = ( + )E E t E t kTk k 0 , where T is the sampling time (not to beconfused with the orbital period). Call as rx k, , ry k, , and rz k, theposition of the chaser in a local-vertical/local-horizontal (LVLH)frame of reference fixed on the center of gravity of the target ve-hicle (see Fig. 2) at time tk. In the (elliptical) LVLH frame, x refers tothe radial position, z to the out-of-plane position (in the directionof the orbital angular momentum), and y is perpendicular to thesecoordinates (not necessarily aligned with the target velocity giventhat its orbit is not circular). The velocity and inputs of the chaserin the LVLH frame at time tk are denoted, respectively, by vx k, , vy k, ,and vz k, , and by ux k, , uy k, , and uz k, .

If there is no actuation (i.e. = = =u u u 0x k y k z k, , , ), the resultingevolution equation can be obtained in a simple form (see Yama-naka & Ankersen, 2002). Define first the followingmatrices YE and YE

-1 appearing in Eqs. (64)-(65),where the fol-lowing symbols are used (expressed in terms of E)

ρ = −−

= −−

= −− ( )

ee E

se Ee E

cE ee E

11 cos

,1 sin1 cos

,cos

1 cos, 66

2 2

10.1016/j.conengprac.2016.06.017i





α α= − ^ − ( − ^)( − )

=( − ) ( )

JE E e E E

e

n

e

sin sin1

,1

,672 3/2 2 3/2

where E in (67) can be substituted by zero or any other desiredreference value of E. For instance, if when evaluating (70) one

chooses ^ = = ( )E E E tk k , then for ( )−YE t

1k

one gets J¼0 and the firstmatrix in (65) becomes zero.

Then, denoting

⎡⎣ ⎤⎦= ( )x r r r v v v , 68k x k y k z k x k y k z kT

, , , , , ,

the transition matrix is expressed as

Φ= ( ) ( )+ +x t t x, , 69k k k k1 1

where Φ is obtained from the previously-defined matrices (64)and (65) as

Φ( ) = ( )+ ( ) ( )−

+t t Y Y, . 70k k E t E t1

1k k1

The matrix Ytkis the fundamental matrix solution of the Tschau-

ner–Hempel model, which is expressed in Yamanaka and Anker-sen (2002),1 as a function of true anomaly θ. However there is aone-to-one relation between E and θ given by

θ = +− ( )

ee

Etan

211

tan2

,71

which is exploited in the sequel.Using (70), one gets Ak in (16) explicitly, as well as Bk in the

impulsive case (explicitly defined in terms of Φ( )+t t,k k1 ). To obtainthe Bk matrix in the PWM and PAM cases, one needs to solve (8) or(12), respectively, which involves an integral. Defining

∫ Φ( ) = ( ) ( )( )

b r r r r s B s s, , , d ,72i

r

r

i1 2 3 31

2

one has that, from (8),

τ κ τ τ κ( ) = ( + + + ) ( )+B b t t t, , , , 73k iW

k i k i i k k i k k i k i k, , , , , , 1

and, from (12),

= ( ) ( )+ +B b t t t, , . 74k iA

i k k k, 1 1

To compute the bi's, the following integral is needed

∫ ( )( )−

+Y td , 75E t i1

3

where i is a 6-element column vector of zeros with a value of oneat row i. For the computation, define the functions fi(t), for

=i 1, 2, 3, as the indefinite integrals of (75), in terms of eccentricanomaly

∫( ) = −( )

−+f E Y

e En

E1 cos

d . 76i E i1

3

Once the fi's are computed, one finds the bi's as

( )( ) = ( ( )) − ( ( )) ( )( )b r r r Y f E r f E r, , 77i E r i i1 2 3 2 13

Inserting the expression of (65) in (76) and integrating, one ob-tains

1 These expressions slightly differ from Yamanaka and Ankersen (2002) be-cause the two transformation matrices that appear in that paper have been pre-multiplied; also, the reference axes are not the same.


⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

( )( ) ( )

( )

( )( )

α=

~

+ − + + + ^

− − + − + + ^

− −

− −

( )

−f

e

e S e e E e Ch e S

e e S e e E eCh e S

eC e

C e

2

2 1 6 3 2 /2

2 8 4 7 2 /2

0

2 1

2 1

0

,

78

1

7/2

2

2 2 2 3

2 2 4 2

2 3/2

2 3/2

⎜ ⎟

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎛⎝

⎞⎠

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥( )

( )( )( )

( )α=

~

+ − − −

− − − −

−

− − + ^

( )

−f

e

C e eC eEh eE

eC e eC Eh E

E e

e S e E S

2

4 1 3

10 2 3

0

2 1

1 4 3 /2

0

,

79

2

3

2

2 2

2 2

2 3/2

2

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥( )( )

α=

~ − ( − )

+ − + ^( )

−f

e e C eC

e S e E S

4

00

2 1 200

4 1 6

,

80

3

5/2

2

2

2

where ~ = − = ( )e e S E1 , sin2 , ^ = ( )S Esin 2 , =C Ecos ,

α= (^ − − (^))h E E e E6 sin . Similar expressions for the B matrices canbe found in Ankersen (2010), however using a slightly differentdefinition of reference axes.

Note that, using these formulas, it is possible to express (16)explicitly for all actuation types. This greatly speeds up thealgorithms.

6.1. Constraints for the rendezvous problem

Besides the input constraints (which were given in Section3.1.2), the inequality state constraints which were genericallyspecified in Section 3.1.1 are, in general, related to safety andsensing considerations (see e.g. Breger & How, 2008). In this work,it is considered that during rendezvous the chaser vehicle has toremain inside a line of sight (LOS) area. To simplify the constraint,2

in this work a 2-D LOS area is used as shown in Fig. 3. This LOSregion is the intersection of a cone, given by the equations

≥ ( − )r c r ry LOS x x0and ≥ − ( + )r c r ry LOS x x0

, and the region ≥r 0y .The LOS constraint is ≤C x dLOS k LOS , where

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

=−−

− −

=

( )

C c

c

d c r

c r

0 1 0 0 0 01 0 0 0 0

1 0 0 0 0,

0.

81

LOS LOS

LOS

LOS LOS x

LOS x

0

0

Using the compact formulation that was developed in Section

2 More complicated constraints could be considered, see Gavilan et al. (2012)for examples including a rotating LOS constraint.

10.1016/j.conengprac.2016.06.017i




Fig. 3. Line of sight region.

Fig. 4. System trajectories in the target orbital plane: open-loop PWM inputscomputed from impulsive solution (dashed), closed-loop Model Predictive Controlwith PWM inputs using impulsive model (dot-dashed), and closed-loop ModelPredictive Control with PWM inputs using the PWM planning algorithm (solid).


2, the constraints equations for the state can be rewritten as:

≤ ( )C d , 82S S

where CS and dS are given by:

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

= ⋱ = ⋮

( )

CC

Cd

d

d, .

83S

LOS

LOS

S

LOS

LOS

Notice that the subscript k is no longer needed in CS and dS, sincethe state constraints do not vary along the prediction horizon.

Then, using Eq. (20), one can reformulate the LOS constraints asconstraints for the control signals, starting at time step tk, in thefollowing way:

≤ − ( )C G d C F x . 84S k k S S k 0

7. Simulation results

For simulations the following values have been used: Np¼50 asplanning horizon, T¼60 s, and ¯ = −u 10 N/kg1 . The target orbit hase¼0.7 and perigee altitude =h 500 kmp . Initial conditions wereθ = °450 ,

= [ − ] = [ − − ]r v0.25 0.4 0.2 km, 0.005 0.005 0.005 km/sT T0 0 . TheLOS constraint is defined by =x 0.001 km0 and = °c tan30LOS . Forthe cost function, α has been set to 103 and Qk as

⎡

⎣⎢⎢⎢

⎤

⎦⎥⎥⎥

= ⋱

( )

+

+

Q

R

R,

85k

k

k N

1

p

where Rk is defined as

= ( − ) ( )×R h k k I 86k a 6 6

where, h is the step function and ka is the desired arrival time forrendezvous. Notice that the dimensions of each block matrix areshown for disambiguation. The reason for choosing (86) is that it isdesired to arrive at the origin at time ka (and remain there) and atthe same time minimize the control effort. In the sequel ka¼Np, sothe spacecraft is expected to rendezvous after about 50 min.

In the simulations three algorithms were considered: first, an


impulsive open-loop trajectory planner, as described in Section 4.1.Next, closed-loop simulations using MPC but considering im-pulsive instead of PWM actuation in the model (this algorithm isdenoted as impulsive MPC). Finally, closed-loop simulations usingMPC, based on the PWM algorithms as explained in Section 5, withinitial solutions computed using the impulsive hot-start (the PAMhot-start produces similar solutions with slightly worse cost). Allalgorithms are simulated on the more realistic model with PWMactuation. Since the first and second methods use the impulsivemodels but are simulated with PWM actuation, the impulsivesignals are subsequently transformed to PWM inputs using thealgorithm of Section 4.2.

Compare first the algorithms without disturbances. The tra-jectories (projected on the target orbital plane) are shown in Fig. 4.The open-loop impulsive solution does not achieve rendezvousand drifts away, which is to be expected given that the planningmodel and the simulation model are not the same, whereas theother solutions successfully reach the origin. At the predefinedrendezvous time (50 min), the chaser vehicle is about 5 m awayfrom the target. Taking advantage of the receding-horizon for-mulation, the algorithm can be left running for longer and thenchaser vehicle keeps asymptotically approaching the target, inparticular after two more minutes, the chaser is at less than half ameter from the target, which is sufficiently close to start the nextphase of rendezvous (docking or berthing). On the other hand, it isremarkable that the impulsive MPC is able to mostly compensateits imperfect thruster model and reached the target in 50 min,however it must be taken into account that for very small impulsesboth models do not disagree by much, so it is mainly at the be-ginning, when more strong actuation is necessary, that both al-gorithms differ. The PWM-MPC algorithm had a cost of 15.2 m/sand the impulsive MPC had a cost of 16 m/s. Thus, while a basic(impulsive) MPC is able to rendezvous, the use of an imperfectmodel has some fuel costs. In addition, the impulsive MPC doesnot satisfy the line-of-sight constraints for a period of time. It isinteresting to compare the resulting pulses of both algorithms,whose duration is represented in Fig. 5. It can be seen that bothalgorithms start with a strong pulse in the y direction to com-pensate for the initial velocity; soon, the impulsive MPC uses manycorrections of small magnitude whereas the PWM-MPC algorithmonly has a few of them. This justifies its smaller cost.

10.1016/j.conengprac.2016.06.017i




Fig. 5. Duration of pulses in the target orbital plane (x and y directions): closed-loop Model Predictive Control with PWM inputs using impulsive model (triangles),and closed-loop Model Predictive Control with PWM inputs using the PWMplanning algorithm (circles). Pulses lasting less than 2 ms have not been re-presented. Negative values correspond to negative pulses.

Fig. 6. System trajectories in the target orbital plane, with inexact orbit model:open-loop PWM inputs computed with the planning algorithm (dashed), closed-loop Model Predictive Control with PWM inputs using impulsive model (dot-da-shed), and closed-loop Model Predictive Control with PWM inputs using the PWMplanning algorithm (solid).


Next, Fig. 6 shows a simulation where the real orbit is differentfrom the reference orbit used in the model (the real eccentricity ise¼0.83, the real perigee altitude is =h 525 kmp , and the realθ = °600 ). Both MPC algorithms asymptotically approach the origin(as in the previous scenario, the chaser vehicle approaches thetarget to less than 0.5 m in 52 min). The impulsive MPC again exitsthe line-of-sight region. In this case both algorithms had imperfectmodels and the costs were similar, in particular the cost for thePWM MPC algorithm was 15.3 m/s, whereas the impulsive MPChad the same cost of 16 m/s as in the perfectly-known orbit case.The simulations were performed by using MATLAB and the Gurobioptimization package (see Gurobi Optimization, Inc., 2014), whichuses a barrier (interior point) method, even though other methodsthat provide more guarantees in the optimization process could beimplemented, see e.g. Guiggiani, Kolmanovsky, Patrinos, andBemporad (2015). In each step of the PWM algorithm, the opti-mizer has to solve a quadratic programming problem with 600


decision variables and 150 constraints, and Gurobi took between16 and 26 iterations. A maximum number of 6 steps were suffi-cient to achieve convergence in the case under study.

It must be noted that, while the PWM MPC verifies the line-of-sight constraints in the two considered cases, the proposedmethod (based on linearization) does not offer any guarantees thatthis will be the case. A possible idea to enforce constraint sa-tisfaction would be including an estimate of linearization error inthe model. Taylor's theorem offers such a bound based on thesecond derivative of the actuation term with respect to PWMvariables. This term can be computed analytically in the samefashion that the first-order derivatives were computed. Such abound could be used to robustify the constraints with respect tothe linearization error. However, since the bound would containthe maximum possible linearization error, the resulting con-straints might be too conservative and even lead to an unfeasibleoptimization problem. Thus, further research is necessary to con-firm the validity of this idea, which will be explored in futureworks.

8. Concluding remarks

This paper has presented a MPC algorithm that computes op-timal PWM inputs for LTV systems. The algorithm is based on aninitial approximation with either PAM or impulsive inputs, fol-lowed by iterative explicit linearization. As an application, theproblem of rendezvous in elliptical orbits has been considered. Inparticular, the algorithm might be particularly useful for satelliteswith small specific thrust. The algorithm improves the fuel cost ofan impulsive-only MPC (with the impulses posteriorly trans-formed to PWM inputs), and is able to satisfy safety constraintsand handle disturbances such as imperfect knowledge of the tar-get's orbit. This algorithm would help avoiding having to include aPWM approximation term in the “uncertainty budget” and there-fore save costs. However, inclusion of real-life constraints, acomputational study including real spacecraft hardware, and morerealistic simulations are needed to validate the method.

Possible future lines of research include the following. First, itwill be of great interest to study the convergence of the planningalgorithm (by analyzing other SQP schemes, possibly under addi-tional, more restrictive assumptions). Another possible line wouldbe guaranteeing constraint satisfaction as explained at the end ofSection 7. Another option would be to guarantee recursive feasi-bility of the MPC optimization problem, by converting hard con-straints into soft ones by using, e.g. a distance-from-set function.Finally, the formulation of the cost function in the solution meansthe control is attractive, but more analysis is necessary to provestability of the closed-loop system, which should be feasible byusing previous MPC results (see for instance Grune & Pannek,2011; Goodwin, Seron, & De Dona, 2005). Addressing this issuewould also help to analyze the convergence of the method.

Acknowledgments

The authors acknowledge financial support from the SpanishMinisterio de Economía y Competitividad under grants DPI2008–05818 and MTM2015-65608-P.

References

Ankersen, F. (2010). Guidance, navigation, control and relative dynamics for spacecraftproximity maneuvers (Ph.D. thesis).

Arzelier, D., Kara-Zaitri, M., Louembet, C., & Delibasi, A. (2011). Using polynomial

10.1016/j.conengprac.2016.06.017i

http://refhub.elsevier.com/S0967-0661(16)30138-1/sbref1





optimization to solve the fuel-optimal linear impulsive rendezvous problem.Journal of Guidance, Control, and Dynamics, 34, 1567–1572.

Arzelier, D., Louembet, C., Rondepierre, A., & Kara-Zaitri, M. (2013). A new mixediterative algorithm to solve the fuel-optimal linear impulsive rendezvous pro-blem. Journal of Optimization Theory and Applications, 159, 210–230.

Asawa, S., Nagashio, T., & Kida, T. (2006). Formation flight of spacecraft in earthorbit via MPC. In SICE-ICASE international join conference.

Bernelli-Zazzera, F., Mantegazza, P., & Nurzia, V. (1998). Multi-pulse-width modu-lated control of linear systems. Journal of Guidance, Control, and Dynamics, 21(1),64–70.

Breger, L., & How, J. (2005). J2-modified gve-based mpc for formation flyingspacecraft. In Collection of technical papers—AIAA guidance, navigation, andcontrol conference (Vol. 1, pp. 158–169).

Breger, L., & How, J. P. (2008). Safe trajectories for autonomous rendezvous ofspacecraft. Journal of Guidance, Control, and Dynamics, 31(5), 1478–1489.

Camacho, E., & Bordons, C. (2004). Model predictive control (2nd edition). London:Springer-Verlag.

Carter, T. E. (1998). State transition matrices for terminal rendezvous studies: Briefsurvey and new example. Journal of Guidance, Control, and Dynamics, 21(1),148–155.

Clohessy, W. H., & Wiltshire, R. S. (1960). Terminal guidance systems for satelliterendezvous. Journal of Aerospace Science, 27(9), 653–658.

D'Amico, S., Ardaens, J.-S., Gaias, G., Benninghoff, H., Schlepp, B., & Jörgensen, J. L.(2013). Noncooperative rendezvous using angles-only optical navigation: Sys-tem design and flight results. Journal of Guidance, Control, and Dynamics, 36(6),1576–1595.

Deaconu, G., Louembet, C., & Théron, A. (2014). Minimizing the effects of the na-vigation uncertainties on the spacecraft rendezvous precision. Journal of Gui-dance, Control, and Dynamics, 37(2), 695–700.

Deaconu, G., Louembet, C., & Théron, A. (2015). Designing continuously constrainedspacecraft relative trajectories for proximity operations. Journal of Guidance,Control, and Dynamics, 38(7), 1208–1217.

Fehse, W. (2003). Automated rendezvous and docking of spacecraft. Cambridge, UK:Cambridge University Press.

Gaias, G., D'Amico, S., & Ardaens, J.-S. (2014). Angles-only navigation to a non-cooperative satellite using relative orbital elements. Journal of Guidance, Con-trol, and Dynamics, 37(2), 439–451.

Gavilan, F., Vazquez, R., & Camacho, E. F. (2009). Robust model predictive control forspacecraft rendezvous with online prediction of disturbance bound. In Pro-ceedings of AGNFCS'09, Samara, Russia.

Gavilan, F., Vazquez, R., & Camacho, E. F. (2012). Chance-constrained model pre-dictive control for spacecraft rendezvous with disturbance estimation. ControlEngineering Practice, 20(2), 111–122.

Goodwin, G., Seron, M. M., & De Dona, J. (2005). Constrained control and estimation:An optimisation approach. London: Springer-Verlag.

Grune, L., & Pannek, J. (2011). Nonlinear model predictive control: Theory and algo-rithms. London: Springer-Verlag.

Guiggiani, A., Kolmanovsky, I., Patrinos, P., & Bemporad, A. (2015). Fixed-pointconstrained model predictive control of spacecraft attitude. In Proceedings of2015 American control conference (pp. 2317–2322).

Gurobi Optimization, Inc. (2014). Gurobi optimizer reference manual. URL ⟨http://www.gurobi.com⟩.

Hartley, E. N., Trodden, P. A., Richards, A. G., & Maciejowski, J. M. (2012). Modelpredictive control system design and implementation for spacecraft rendez-vous. Control Engineering Practice, 20(7), 695–713.

Hill, G. (1878). Researches in lunar theory. American Journal of Mathematics, 1(3),5–26 129–147, 245–260.


Houska, B., Ferreau, H., & Diehl, M. (2011). ACADO toolkit–An open-source frame-work for automatic control and dynamic optimization. Optimal Control Appli-cations and Methods, 32(3), 298–312.

Humi, M., & Carter, T. (2008). Orbits and relative motion in the gravitational field ofan oblate body. Journal of Guidance, Control, and Dynamics, 31(3), 522–532.

Ieko, T., Ochi, Y., & Kanai, K. (1999). New design method for pulse-width modula-tion control systems via digital redesign. Journal of Guidance, Control, and Dy-namics, 22(1), 123–128.

Inalhan, G., Tillerson, M., & How, J. P. (2002). Relative dynamics and control ofspacecraft formations in eccentric orbits. Journal of Guidance, Control, and Dy-namics, 25(1), 48–59.

Jewison, C., Erwin, R. S., & Saenz-Otero, A. (2015). Model predictive control withellipsoid obstacle constraints for spacecraft rendezvous. In ACNAAV 2015 IFACworkshop.

Kim, H. J., Shim, D. H., & Sastry, S. (2002). Nonlinear model predictive trackingcontrol for rotorcraft-based unmanned aerial vehicles. In Proceedings of ACC2002.

Larsson, R., Berge, S., Bodin, P., & Jönsson, U. (2006). Fuel efficient relative orbitcontrol strategies for formation flying and rendezvous within prisma. In Pro-ceedings of the 29th AAS guidance and control conference.

Leomanni, M., Rogers, E., & Gabriel, S. B. (2014). Explicit model predictive controlapproach for low-thrust spacecraft proximity operations. Journal of Guidance,Control, and Dynamics, 37(6), 1780–1790.

Louembet, C., Arzelier, A., & Deaconu, G. (2015). Robust rendezvous planning undermaneuvering errors. Journal of Guidance, Control, and Dynamics, 38(1), 76–93.

Nocedal, J., & Wright, S. (2006). Numerical optimization (2nd edition). New York:Springer.

Richards, A. G. & How, J. (2003). Performance evaluation of rendezvous using modelpredictive control. AIAA paper 2003-5507.

Rossi, M. & Lovera, M. (2002). A multirate predictive approach to orbit control ofsmall spacecraft. In Proceedings of ACC 2002.

Rugh, W. J. (1996). Linear system theory (2nd edition). Upper Saddle River, NJ, USA:Prentice-Hall, Inc.

Shieh, L.-S., Wang, W.-M., & Sunkel, J. (1996). Design of PAM and PWM controllersfor sampled-data interval systems. Journal of Dynamic Systems, Measurement,and Control, 118(4), 673–681.

Sinclair, A. J., Sherrill, R. E., & Lovell, T. A. (2014). Calibration of linearized solutionsfor satellite relative motion. Journal of Guidance, Control, and Dynamics, 37(4),1362–1367.

Tschauner, J., & Hempel, P. (1965). Rendevous zu einem in elliptischer bahn um-laufenden. Ziel Acta Astronaut II, 2, 104–109.

Vazquez, R., Gavilan, F., & Camacho, E. F. (2011). Trajectory planning for spacecraftrendezvous with on/off thrusters. In 2011 Proceedings of IFAC world congress.

Vazquez, R., Gavilan, F., & Camacho, E. F. (2014). Trajectory planning for spacecraftrendezvous in elliptical orbits with on/off thrusters. In IFAC world congress,Cape Town.

Vazquez, R., Gavilan, F., & Camacho, E. F. (2015). Model predictive control forspacecraft rendezvous in elliptical orbits with on/off thrusters. In ACNAAV 2015IFAC workshop.

Weiss, A., Kolmanovsky, I., Baldwin, M., & Erwin, R. S. (2012). Model predictivecontrol of three dimensional spacecraft relative motion. In 2012 AmericanControl Conference (ACC). IEEE, pp. 173–178.

Wie, B. (1998). Space vehicle dynamics and control. AIAA, AIAA Education Series,Reston, VA.

Yamanaka, K., & Ankersen, F. (2002). New state transition matrix for relative motionon an arbitrary elliptical orbit. Journal of Guidance, Control, and Dynamics, 25(1),60–66.

10.1016/j.conengprac.2016.06.017i



















































http://www.gurobi.com

http://www.gurobi.com




















































Date post:	31-Aug-2020
Category:	Documents
Upload:	others
View:	2 times
Download:	0 times

Control Engineering Practice · Pulse-width predictive control for LTV systems with application to...

Documents