A Generalized PI Sliding Mode and PWM Control of Switched Fractional Systems Hebertt Sira Ram´ırez 1 and Vicente Feliu Battle 2 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 hsira@cinvestav.mx 2 Escuela T´ecnica Superior de Ingenieros Industriales, Universidad de Castilla La Mancha, Av. Camilo Jos´e Cela S/N, 13005 Ciudad Real, Espa˜na Vicente.Feliu@uclm.es 1 Introduction The control of a special class of Single Input Single Output (SISO) switched fractional order systems (SFOS) is addressed from the viewpoints of the Gener- alized Proportional Integral (GPI) feedback control approach and a sliding mode based Σ − Δ modulation implementation of an average model based designed feedback controller. Alternatively, a Pulse Width Modulation (PWM) duty ratio synthesis approach is also developed for the approximate discontinuous control of the same class of systems. A fractional order GPI controller is proposed which transforms the average model of the system into a pure, integer order, chain of integrations with desired closed loop dynamics achieved through a classical compensation 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 out- put feedback control signal is adopted for the switched system. An illustrative simulation example dealing with an electric radiator system is presented. The implications of fractional calculus in the modeling and control of physical systems of various kinds is well known and documented in the control systems and applied mathematics literature. The reader may benefit from the books by 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 been approached from the viewpoint of absolute stability aided with generalizations of some classical design methods, such as the Nyquist stability criterion and graphical frequency domain analysis methods. As such, the control design tech- niques available for this class of ubiquitous systems suffer from a lack of direct systematic 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 c  Springer-Verlag Berlin Heidelberg 2008 202 H.S. Ram´ırez and V.F. Battle Switched fractional order systems (SFOS), i.e., systems whose mathematical description entitles Fractional Order Systems (FOS) including the presence of ideal switches acting as control input variables, have been little studied in the fractional order control systems domain. Some interesting examples, which are indeed of switched nature or have applications in switched environments, have been treated without the benefit of systematic average feedback controller design techniques 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 output feedback 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 in complex 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 combined with GPI control. Section 2 deals with the definitions and notation used in this chapter. Sec- tion 3 formulates and solves, in substantial generality -within the benchmark model perspective- the trajectory tracking problem for the time invariant SISO switched benchmark linear system model. Section 4 presents a Σ − Δ based sliding mode controller design example for a heating system along with digital computer 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. 2 Definitions and Notation 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 the same 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 A Generalized PI Sliding Mode and PWM Control 203 y (γ) = −y + W (2u − 1),u∈{0, 1}, 0 <γ<1 has an average representation as y (γ) = −y + W (2u av − 1),u av ∈ (0, 1) By letting ν av = W (1 − 2u av ), we may also represent the average system in classical transfer function fractional degree representation y(s)=  1 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 ∈{λ 0 ,λ 1 }, [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 λ 0 <λ 1 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 = λ 0 + u(λ 1 − λ 0 ),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 (λ 0 ,λ 1 ). Similarly the average switch position function, u av ∈ (0, 1), is defined by means of ν av = λ 0 + u av (λ 1 − λ 0 ). 3 Problem Formulation, Assumptions and Main Results: 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: ν av = 1 b  y (α) + ay  , u av = 1 λ 1 − λ 0  y (α) + ay b − λ 0  The output variable y may be addressed as the flat output of the system (See Fliess et al. [1] and, also, Sira-Ram´ırez and Agrawal [14]). 204 H.S. Ram´ırez and V.F. Battle 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 u ∗ (t)= 1 λ 1 − λ 0  [y ∗ (t)] (α) + ay ∗ (t) b − λ 0  satisfies the restriction u ∗ (t) ∈ [0, 1] for all t, find a dynamical output feedback controller for the SISO-SFOS (1) such that, on the average, the output, y(t), of the switched system (1), asymptotically exponentially tracks the given smooth reference trajectory y ∗ (t) while being robust with respect to additive constant perturbation inputs. 3.2 Main Result 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),tobe 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: u = 1 2 [1 + sign e] ˙e = u av − u u av = 1 λ 1 − λ 0 [ν av − λ 0 ] ν av = ν ∗ av (t)+ 1 b [−a + C(s)] (y ∗ (t) − y) C(s)= k n s n + k n−1 s n−1 + ···+ k 1 s + k 0 s β+1 (s n−1 + k 2n−1 s n−2 + ···+ k n−1 ) ν ∗ av (t)= 1 b  [y ∗ (t)] (α) + ay ∗ (t)  (4) 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 , ,k 1 ,k 0 },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 + ···+ k 1 s + k 0 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 A Generalized PI Sliding Mode and PWM Control 205 time in the artificially extended state dimension e, into the average closed loop system of the form: y (α) = −ay + b [λ 0 + u av (λ 1 − λ 0 )] u av = 1 λ 1 − λ 0 [ν av − λ 0 ] ν av = ν ∗ av (t)+ 1 b [−a + C(s)] (y ∗ (t) − y) C(s)= k n s n + k n−1 s n−1 + ···+ k 1 s + k 0 s β+1 (s n−1 + k 2n−1 s n−2 + ···+ k n−1 ) ν ∗ av (t)= 1 b  [y ∗ (t)] (α) + ay ∗ (t)  (5) Let e y = y − y ∗ (t). Rearranging the previous equations we obtain, in the abusive, but customary, time domain-frequency domain mixed notation, the fol- lowing average closed loop dynamics for the output tracking error, e (α) y (t)=−C(s)e y (t) whose governing characteristic equation is given by,  s 2n + k 2n−1 s 2n−1 + ···+ k 1 s + k 0  e y =0 The systematic design of classical compensation networks, via pole placement, can be shown to be equivalent to the recently introduced Generalized Propor- tional Integral (GPI) feedback controller design methodology. This methodology is based on integral reconstructors and additional iterated integral error compen- sation. The details and interesting connections with flatness and module theory 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 control input amplitudes, not every initial state of the system will render controlled trajectories 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 as described above. The stability features of the origin of the tracking error space are only semi-global. The result follows.  4 Sliding Mode Trajectory Tracking for a Switched Heating Radiator System Consider the following fractional order switched model of the Electric Radiator system (see [7]) D 1.26 t y +0.0150y = ν =0.0252 {W min + u[W max − W min ]} (6) where u ∈{0, 1} and, hence, 206 H.S. Ram´ırez and V.F. Battle ν ∈{λ 0 ,λ 1 } = {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: G(s)= y(s) ν av (s) = 1 s 1.26 +0.0150 (7) The FOS is flat, with flat output given by y. The average control input tra- jectory, ν ∗ av , and the average switch position function u ∗ av (t) corresponding to a desired nominal reference output signal y ∗ (t)aregivenby ν ∗ av (t)=D 1.26 t y ∗ (t)+0.0150y ∗ (t), u ∗ av (t)=0.1803  D 1.26 t y ∗ (t)+0.0150y ∗ (t)  The state-dependent input coordinate transformation ν av =0.0150 y +  1 s 0.74  ϑ av =0.0150 y + D −0.74 t ϑ av transforms the average system of (6) into the integer order system: ¨y = ϑ av A classical GPI controller with integral control action completes the output feedback tracking controller design ϑ av =¨y ∗ (t) −  k 2 s 2 + k 1 s + k 0 s(s + k 3 )  (y −y ∗ (t)) where the set of constant gains: {k 3 ,k 2 ,k 1 ,k 0 }, are chosen so that the closed loop characteristic polynomial, governing the average tracking error system dynamics e = y −y ∗ (t), given by, p(s)=s 4 + k 3 s 3 + k 2 s 2 + k 1 s + k 0 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: p d (s)=(s 2 +2ξω n s + ω 2 n ) 2 = s 4 +4ξω n s 3 +(4ξ 2 ω 2 n +2ω 2 n )s 2 +4ξω 3 n s + ω 4 n The average feedback control law, for the auxiliary control input v av ,isthus given by the fractional order GPI compensator network: ν av = ν ∗ av (t)+  0.0150 −  k 2 s 2 + k 1 s + k 0 s 1.74 (s + k 3 )  (y −y ∗ (t)) A Generalized PI Sliding Mode and PWM Control 207 and 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 trajec- tory tracking for the fractional order system describing the heating radiator system. system Heating à 0:0150 + s 1:74 (s+k 3 ) k 2 s 2 +k 1 s+k 0 0:1803 ÷ av modulator ÎàÉ Fig. 1. Average fractional GPI feedback control scheme via Σ − Δ implementation of switched control Simulations were performed for the given plant with an output trajectory tracking task involving a smooth temperature rise from the initial value of zero to the final value of 50 in 30 sec. An un-modelled step input perturbation of amplitude 10 was set to be triggered at time 40 sec. Figure 2 depicts the closed loop performance of the discontinuous feedback controller. At time t = 40 sec. the step perturbation input is seen to affect the tracking with excellent robust recovery features. Figure 3 depicts the nature of the average feedback control inputs (nominal feed-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 controlled signals. 5APWMΣ − Δ Modulation GPI Control Approach The control of switched systems via PWM actuators constitutes a vast area of applications with a sufficient number of theoretical contributions. The reader is invited to browse the book by Gelig and Churilov [3] for a number of interesting 208 H.S. Ram´ırez and V.F. Battle 0 10 20 30 40 50 60 70 −10 0 10 20 30 40 50 60 reference and output y(t), y ∗ (t) t [sec] unmodelled step perturbation input 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 =40sec) theoretical and practical issues as well as a complete historical perspective of the area. 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 u ∗ (t)= 1 λ 1 − λ 0  [y ∗ (t)] (α) + ay ∗ (t) b − λ 0  satisfies the restriction u ∗ (t) ∈ [0, 1] for all t, find a dynamical output, PWM- based, feedback controller such that, on the average, the output of the switched system, y(t), asymptotically exponentially tracks the given smooth reference tra- jectory y ∗ (t). 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: A Generalized PI Sliding Mode and PWM Control 209 u = PWM μ (e(t k )) ˙e = u av − u PWM μ (e(t k )) = ⎧ ⎨ ⎩ 1fort k ≤ t<t k + μ(e(t k ))T 0fort k + μ(e(t k ))T ≤ t<t k + T k =0, 1, 2, μ(e)= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1fore ≥  1  e for 0 <e< 0fore ≤ 0 u av = 1 W 1 − W 0 [ν av − W 0 ] ν av = ν ∗ av (t)+ 1 b  −a + k n s n + k n−1 s n−1 + ···+ k 1 s + k 0 s β+1 (s n−1 + k 2n−1 s n−2 + ···+ k n−1 )  (y ∗ (t) − y) ν ∗ av (t)= 1 b  [y ∗ (t)] (α) + ay ∗ (t)  (8) semi-globally renders the trajectories of the controlled system to oscillate above the origin of the tracking error space e y = y − y ∗ (t), within a bounded interval of magnitude , provided the coefficients of the classical compensator network, 0 10 20 30 40 50 60 70 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 average control signals 0 10 20 30 40 50 60 70 −1 −0.5 0 0.5 1 1.5 2 switch control signal u ∗ (t), u av (t) u(t) ∈ {0,1} t [sec] t [sec] 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 =40sec) 210 H.S. Ram´ırez and V.F. Battle {k 2n−1 ,k 2n−2 , ,k 1 ,k 0 }, are chosen so that the closed loop characteristic poly- nomial of the average system, given by p(s)=s 2n + k 2n−1 s 2n−1 + ···+ k 1 s + k 0 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: y (α) = −ay + b [W 0 + u av (W 1 − W 0 )] u av = 1 W 1 − W 0 [ν av − W 0 ] ν av = ν ∗ av (t)+ 1 b  −a + k n s n + k n−1 s n−1 + ···+ k 1 s + k 0 s β+1 (s n−1 + k 2n−1 s n−2 + ···+ k n−1 )  (y ∗ (t) − y) ν ∗ av (t)= 1 b  [y ∗ (t)] (α) + ay ∗ (t)  (9) Let e y = y − y ∗ (t). Rearranging the previous equations, we obtain, in the abusive, but customary, time domain-frequency domain mixed notation, the fol- lowing average closed loop dynamics for the output tracking error, e (α) y (t)=−  k n s n + k n−1 s n−1 + ···+ k 1 s + k 0 s β+1 (s n−1 + k 2n−1 s n−2 + ···+ k n−1 )  e y (t) (10) whose governing characteristic equation is given by,  s 2n + k 2n−1 s 2n−1 + ···+ k 1 s + k 0  e y =0 Under the realistic finite sampling frequency assumption, theorem 5 in the Appendix guarantees, for a given >0, the existence of a sampling period T , (T<2), for which the closed loop responses of the PWM switched system constitute uniform  approximations to the ideal sliding mode Σ −Δ equivalent responses. Since we are dealing with hard constraints in average control input ampli- tudes, it is clear that not every initial state of the system will render closed loop controlled trajectories as described by the above equations. Under satu- ration conditions, the feedback loop is actually broken. For those initial states that do not lead to permanent or semi-permanent saturation conditions on the controller, the closed loop system -approximately evolves as described by the average behavior explained above. The stability features around the origin of the tracking error space are only semi-global. The result follows.  [...].. .A Generalized PI Sliding Mode and PWM Control 211 6 PWM Control Trajectory Tracking for a Switched Heating Radiator System Consider the same fractional order switched model of the Electric Radiator system treated in Section 4 The average feedback control law, for the control input av , is given, as before, by the fractional order GPI compensator network: av = (t) + 0.015 av and uav = k2... Proportional Integral Control of Linear Systems Asian Journal of Control 5, 467475 (2003) A Generalized PI Sliding Mode and PWM Control 213 13 Sira-Ram rez, H., Marquez, R., Fliess, M.: Sliding Mode Control of DC-to-DC Power Converters using Integral Reconstructors International Journal of Robust and Nonlinear Control 12, 11731186 (2002) 14 Sira-Ram rez, H., Agrawal, S.: Dierentially Flat Systems Marcel Dekker,... binary nature of the switched input, may also be handled by using a PWM based modulator coding block accepting the designed average feedback control 212 H.S Ram rez and V.F Battle signal as an input and yielding, as an output, a pulsed signal with the required average control properties to be processed by the plant Some other well-known fractional order system examples of physical avor can be tackled... (2004) 15 Sira-Ram rez, H.: A Geometrical Approach to PWM Control in Nonlinear Dynamical Systems IEEE Transactions on Automatic Control 34, 184187 (1989) 16 Steele, R.: Delta Modulation Systems Pentech Press, London (1975) 17 Vinagre, B., Polubny, I., Hernandez, A. , Feliu, V.: Some approximations of fractional order operators in control theory and applications Fractional Calculus & Applied Analysis 3,... this chapter, we have presented a systematic feedback controller design for switched fractional order linear systems whose leading exponent is of fractional order The feedback controller design is greatly facilitated by resorting to the average system model In the average description of the system, a Generalized Proportional Integral controller is readily designed in a systematic manner using a recent... P.: Flatness and defect of nonlinear systems: introductory theory and examples International Journal of Control 61, 13271361 (1995) 2 Fliess, M., Marquez, R., Delaleau, E., Sira-Ram rez, H.: Correcteurs Proportionnels-Int`graux Gnraliss ESAIM: Control, Optimization and e e e e Calculus of Variations 7, 2341 (2002), http://www.emath.fr/cocv/ 3 Gelig, A. K., Churilov, A. N.: Stability and Oscillations of. .. Academic Press, San Diego (1999) 10 Polubny, I.: Fractional order systems and PI D -controllers IEEE Transactions on Automatic Control 44, 208214 (1999) 11 Riu, D., Reti`re, N., Linzen, D.: Half-order modelling of supercapacitors In: Proc e 39th IEEE Industry Applications Society Annual Meeting IAS-2004, Seattle, US (2004) 12 Sira-Ram rez, H.: Sliding Modes, -modulators, and Generalized Proportional... 1 [ av 0.0252W0] = 0.1803 av 0.0252 (W1 W0 ) The PWM based modulation implementation of the average feedback controller for the switched system is simply accomplished by means of e = uav (t) u, u = P W M (e(tk )) with the PWM operator chosen as indicated in equations (12) and (16) in the Appendix Figure 1 also qualies as the fractional order GPI PWM based control scheme achieving approximate... Nonlinear Pulse Modulated Systems Birkhăuser, Boston (1998) a 4 Hartley, T.T., Lorenzo, C.F.: Dynamics and Control of Initialized Fractional- Order Systems Nonlinear Dynamics 29, 201233 (2002) 5 Melchior, P., Potty, A. , Oustaloup, A. : Motion control by ZV shaper synthesis extended for fractional systems and its applications ot CRONE control Nonlinear dynamics 38, 401416 (2004) 6 Oustaloup, A. : La Drivation... = uav (t) u 1 for e > u = sat (e) = 1 e for e [0, ] 0 for e < 0 (19) A Generalized PI Sliding Mode and PWM Control 221 The high gain modulation also represents a soft switch approximation to the average behavior of sliding mode modulation Using the Lyapunov function candidate: V (e) = 1 e2 , it is easy to establish that the trajectories of the 2 modulator encoding error e in (19) remain . the controlled signals. 5APWMΣ − Δ Modulation GPI Control Approach The control of switched systems via PWM actuators constitutes a vast area of applications with a sufficient number of theoretical. implications of fractional calculus in the modeling and control of physical systems of various kinds is well known and documented in the control systems and applied mathematics literature. The reader. A Generalized PI Sliding Mode and PWM Control of Switched Fractional Systems Hebertt Sira Ram´ırez 1 and Vicente Feliu Battle 2 1 Cinvestav IPN, Av. IPN No. 2508, Departamento de Ingenier´ıa