A Generalized PI Sliding Mode and PWM Control of Switched Fractional Systems

Colonia Residencial Zacatenco AP 14740, 07300 M´exico D.F.,M´exico based Σ − Δ modulation implementation of an average model based designed feedback controller.. For an interesting accou

Control of Switched Fractional Systems

Hebertt Sira Ram´ırez1and Vicente Feliu Battle2

1 Cinvestav IPN, Av IPN No 2508, Departamento de Ingenier´ıa El´ectrica, Secci´on

de Mecatr´onica Colonia Residencial Zacatenco AP 14740, 07300 M´exico D.F.,M´exico

based Σ − Δ modulation implementation of an average model based designed

feedback controller Alternatively, a Pulse Width Modulation (PWM) duty ratiosynthesis approach is also developed for the approximate discontinuous control ofthe same class of systems A fractional order GPI controller is proposed whichtransforms the average model of the system into a pure, integer order, chain

of integrations with desired closed loop dynamics achieved through a classicalcompensation network robustly acting in the presence of constant load pertur-

bations A sliding mode based Σ − Δ modulation and a PWM based Σ − Δ

modulation implementation of the continuous, bounded, dynamic average put feedback control signal is adopted for the switched system An illustrativesimulation example dealing with an electric radiator system is presented.The implications of fractional calculus in the modeling and control of physicalsystems of various kinds is well known and documented in the control systemsand applied mathematics literature The reader may benefit from the books by

out-Oustaloup [6], Polubny [9], and the articles by Vinagre, et al [17] and by Polubny

[10] For an interesting account of classical control and state based control of(non-switched) FOS see the interesting article by Hartley and Lorenzo [4].The design of feedback controllers for linear fractional order systems has beenapproached from the viewpoint of absolute stability aided with generalizations

of some classical design methods, such as the Nyquist stability criterion andgraphical frequency domain analysis methods As such, the control design tech-niques available for this class of ubiquitous systems suffer from a lack of directsystematic approaches based on ideas related to pole placement, observer design,

and some other popular modern controller design techniques.

G Bartolini et al (Eds.): Modern Sliding Mode Control Theory, LNCIS 375, pp 201–221, 2008 springerlink.com  Springer-Verlag Berlin Heidelberg 2008c

Switched fractional order systems (SFOS), i.e., systems whose mathematicaldescription entitles Fractional Order Systems (FOS) including the presence ofideal switches acting as control input variables, have been little studied in thefractional order control systems domain Some interesting examples, which areindeed of switched nature or have applications in switched environments, havebeen treated without the benefit of systematic average feedback controller designtechniques and pole placement oriented synthesis of feedback control laws (See

Petras et al [7] and Riu et al [11])

In this chapter, we propose a systematic fractional order dynamic outputfeedback controller design method for a special class of switched fractional order

systems, known as the benchmark model (see Poinot and Trigeassou [8] and,

also, Melchior, Poty and Oustaloup [5]) whose average description is available incomplex variable fractional power transfer function representation We propose

a combination of the Generalized Proportional Integral (GPI) feedback control

design technique and Σ − Δ modulation for the switched implementation of

average designed feedback control laws We also explore a PWM implementation

of an average duty ratio design based on the same average technique combinedwith GPI control

Section 2 deals with the definitions and notation used in this chapter tion 3 formulates and solves, in substantial generality -within the benchmarkmodel perspective- the trajectory tracking problem for the time invariant SISO

Sec-switched benchmark linear system model Section 4 presents a Σ − Δ based

sliding mode controller design example for a heating system along with digitalcomputer simulations Section 5 is devoted to the PWM approach to discontinu-ous feedback control of FOS of the class here treated Section 6 gives an outline

of how to implement a PWM Σ − Δ based controller to the benchmark

sys-tem Section 7 contains some conclusions and suggestions for further research

The appendices collect some facts and generalities about Σ − Δ modulation and

PWM-Σ − Δ modulation as efficient means of implementing average feedback

control designs in a given switched dynamical system

A Switched Fractional Order System (SFOS) is a FOS system where the control

input u is restricted to take values in the discrete set {0, 1} Generally speaking,

such systems are also addressed as “ ON-OFF” systems

An average model of a SFOS is obtained from the description of the system by simply replacing the discrete-valued control input variable u by the continuous valued control input, u av , taking values in the open interval (0, 1) of the real line.

When it is clear from the context to which system we are referring we use thesame symbols to denote average states and outputs as in the switched version

of the system The following example stresses this particular notation point

Example 1 Let W be a strictly positive real number The fractional order

switched system

s γ+ 1



ν av (s)

We address, in this chapter, the following benchmark class (see Poinot and

Trige-assou [8]) of switched SISO-FOS where n represents an arbitrary strictly positive

integer:

y (α)=−ay + bν, ν ∈ {λ0, λ1}, [α] = n − 1. (1)

with α being a strictly positive real number and where [ · ] stands for the integer

part of the bracketed real number The constants λ0 < λ1 are two arbitrary

real numbers representing the extreme constant values of the switched inputs ν.

We define the auxiliary switch position function input, u, as

ν = λ0+ u(λ1− λ0), u ∈ {0, 1} (2)The average switched system is denoted, in transfer function representation as

y(s) = b

s α + a ν av (s) (3)The corresponding signal ν av (t) takes values, continuously in the open interval (λ0, λ1) Similarly the average switch position function, u av ∈ (0, 1), is defined

by means of ν av = λ0+ u av (λ1− λ0)

Sliding Mode Approach

Notice that, in our particular case, the variable y fractionally differentially

pa-rameterizes the inputs of the system This means that the average input ν av

as well as the average switch position function, u av, can be written as linear

functions of y and of a finite number of its fractional derivatives Indeed, from

the system equations we have:

3.1 Problem Formulation

Given a desired smooth output reference trajectory y ∗ (t), such that the

corre-sponding nominal, continuous, control input trajectory u ∗ (t), given by

In the context of sliding mode based Σ − Δ modulation (see Appendix A), we

have the following result:

Theorem 1 Given a desired output smooth reference trajectory y ∗ (t), to be

tracked by the output of the system (1), with u ∗ (t) ∈ [0, 1] for all t, then, the following switched feedback controller, with 0 < β = n − α < 1:

semi globally renders the origin of the tracking error space e y = y − y ∗ (t) as an

exponentially asymptotic equilibrium point of the closed loop system, provided the coefficients of the classical compensator network, {k 2n −1 , k 2n −2 , , k1, k0}, are chosen so that the closed loop characteristic polynomial of the average system, given by

p(s) = s 2n + k 2n −1 s 2n −1+· · · + k1s + k0

is a Hurwitz polynomial The extra integer integral action in C(s) guarantees robustness with respect to constant perturbation inputs.

Proof

First note, that according to the theorem in the Appendix, the Σ −Δ modulator

renders the zero dynamics of the underlying sliding motion, occurring in finite

time in the artificially extended state dimension e, into the average closed loop

system of the form:

Let e y = y − y ∗ (t) Rearranging the previous equations we obtain, in the

abusive, but customary, time domain-frequency domain mixed notation, the lowing average closed loop dynamics for the output tracking error,

is based on integral reconstructors and additional iterated integral error sation The details and interesting connections with flatness and module theory

compen-may be found in Fliess et al [2] see also Sira-Ram´ırez [13].

Once we understand we are dealing with hard constraints in average controlinput amplitudes, not every initial state of the system will render controlledtrajectories as described by the above equations Under saturation conditions,the feedback loop is actually broken For those initial states that do not lead

to saturation conditions on the controller, the closed loop system evolves asdescribed above The stability features of the origin of the tracking error spaceare only semi-global The result follows



Heating Radiator System

Consider the following fractional order switched model of the Electric Radiator

system (see [7])

D t 1.26 y + 0.0150y = ν

= 0.0252 {W min + u[W max − W min]} (6)

where u ∈ {0, 1} and, hence,

ν ∈ {λ0, λ1} = {0.0252W min , 0.0252W max }

while W min = 0, W max = 220, i.e., ν ∈ {0, 5.544}

In average fractional order transfer function representation:

ν av = 0.0150 y +

1

where the set of constant gains:{k3, k2, k1, k0}, are chosen so that the closed loop

characteristic polynomial, governing the average tracking error system dynamics

e = y − y ∗ (t), given by,

p(s) = s4+ k3s3+ k2s2+ k1s + k0

is a Hurwitz polynomial The controller gains can be obtained, for instance, by

forcing p(s) to be identical to the desired fourth order polynomial:

u av = 0.1803 ν av The Σ − Δ modulation implementation of the average feedback controller for

the switched system is simply accomplished by means of,

˙e = u av (t) − u, u = 1

2[1 + signe]

Figure 1 depicts the fractional order GPI control scheme achieving tory tracking for the fractional order system describing the heating radiatorsystem

trajec-systemHeating

loop performance of the discontinuous feedback controller At time t = 40 sec.

the step perturbation input is seen to affect the tracking with excellent robustrecovery features

Figure 3 depicts the nature of the average feedback control inputs (nominalfeed-forward plus feedback generated) as well as the discontinuous (switched)

control input entering the system after the Σ −Δ modulator At time t = 40 sec.

the step perturbation input of magnitude 10 is allowed to affect the controlledsignals

The control of switched systems via PWM actuators constitutes a vast area ofapplications with a sufficient number of theoretical contributions The reader isinvited to browse the book by Gelig and Churilov [3] for a number of interesting

Fig 2 Closed loop performance in trajectory tracking task for the heating system

using a GPI-Σ − Δ modulation switched feedback controller (Step perturbation input

at t = 40 sec)

theoretical and practical issues as well as a complete historical perspective of thearea In the following paragraphs we adopt the PWM formulation found in [15]

We have the following problem fomulation:

Given a desired, smooth, output reference trajectory y ∗ (t), for the switched

FOS (1), such that the nominal duty ratio control input function u ∗ (t), given by

PWM-5.1 Main Result

In the context of PWM based Σ − Δ modulation of the Appendix, we have the

following result:

Theorem 2 Given a desired output smooth reference trajectory y ∗ (t), such that

u ∗ (t) ∈ [0, 1], then, the following switched feedback controller, with 0 < β =

n − α < 1:

Fig 3 Average and actual control inputs for trajectory tracking task in the heating

system using a combination of GPI-Σ −Δ modulation discontinuous feedback controller

(Step perturbation input at t = 40 sec)

{k 2n −1 , k 2n −2 , , k1, k0}, are chosen so that the closed loop characteristic nomial of the average system, given by

poly-p(s) = s 2n + k 2n −1 s 2n −1+· · · + k1s + k0

is a Hurwitz polynomial and T and are chosen to satisfy: T < 2

Proof

First note that, according to theorems 3 and 6 in the Appendix, the PWM based

Σ −Δ modulator ideally renders, under infinite sampling frequency assumptions,

a zero dynamics -corresponding to the underlying Σ − Δ modulator induced

sliding motion- described by an average closed loop system of the form:

Let e y = y − y ∗ (t) Rearranging the previous equations, we obtain, in the

abusive, but customary, time domain-frequency domain mixed notation, the lowing average closed loop dynamics for the output tracking error,

constitute uniform approximations to the ideal sliding mode Σ − Δ equivalent

responses

Since we are dealing with hard constraints in average control input tudes, it is clear that not every initial state of the system will render closedloop controlled trajectories as described by the above equations Under satu-ration conditions, the feedback loop is actually broken For those initial statesthat do not lead to permanent or semi-permanent saturation conditions on the

ampli-controller, the closed loop system -approximately evolves as described by the

average behavior explained above The stability features around the origin of thetracking error space are only semi-global The result follows 

6 PWM Control Trajectory Tracking for a Switched Heating Radiator System

Consider the same fractional order switched model of the Electric Radiator

sys-tem treated in Section 4

The average feedback control law, for the control input ν av, is given, as before,

by the fractional order GPI compensator network:

controller for the switched system is simply accomplished by means of

˙e = u av (t) − u, u = P W M μ (e(t k))with the PWM operator chosen as indicated in equations (12) and (16) in theAppendix

Figure 1 also qualifies as the fractional order GPI PWM based control schemeachieving approximate trajectory tracking for the fractional order switched heat-

ing radiator system when the Σ −Δ modulator block is replaced by a PWM-Σ−Δ

modulator block

Simulations would yield very similar, almost identical, results to those already

obtained via the sliding mode Σ − Δ modulation approach For this reason

they are not included here This fact was theoretically justified in Theorem 6 ofAppendix B

In this chapter, we have presented a systematic feedback controller design forswitched fractional order linear systems whose leading exponent is of fractionalorder The feedback controller design is greatly facilitated by resorting to theaverage system model In the average description of the system, a GeneralizedProportional Integral controller is readily designed in a systematic manner using

a recent extension of the GPI controller methodology which boils down to an

“ educated ” classical compensation network controller design The design taskdoes not resort to traditional graphical methods, such as Nyquist, or Bode plots,and it results in a rather robust, easy to implement, dynamical output feedbackcompensation network allowing both trajectory tracking and stabilization.The average control implementation restriction, represented by the binarynature of the switched input, may also be handled by using a PWM based

Σ − Δ modulator coding block accepting the designed average feedback control

Σ... follows