Computationally Efficient Control System Based on Digital Dynamic Pulse Frequency Modulation for Microprocessor Implementation

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1 IJCSI International Journal of Couter Science Issues, Vol. 0, Issue 3, No 2, May 203 ISSN (Print): ISSN (Online): Coutationally Efficient Control Syste Based on Digital Dynaic Pulse Frequency Modulation for Microrocessor Ileentation Bekurza H. Aitchanov, Vladiir V. Nikulin 2, Olizhon A. Baiuratov and Zhylduz B. Аitzhanova 3 K.I. Sataev Kazakh National Technical University, Alaty, 05003, Kazakhstan 2 State University of New York at Binghaton, 4400 Vestal Parkway East, Binghaton NY, 3902, USA 3 Kazakhstan Institute of Technical Develoents, Alaty, , Kazakhstan Abstract This aer is focused on digital dynaic ulse-frequency odulation (DPFM) control systes that can be ileented on a icrocontroller. We describe a structure of a discrete nonlinear closed-loo syste that is equivalent to DPFM. A general-case odel of a digital odulator of l-th order is obtained and an exale of ileentation and coutational erforance is deonstrated. Keywords: Рulse-frequency odulation, controlled lant, discrete filter, ulse generator, icrocontroller Digital control syste resented below includes a DPFM, a ulse-foring generator (PFG), and a controlled lant. Functions of both the DPFM and PFG are ileented on a icrocontroller that generates a sequence of ulses, each with the width h [6].. Introduction Digital control systes based on dynaic ulse-frequency odulation are widely acceted. The ain reason for their oularity is a set of features, such as the low suscetibility to noise and silicity in icrorocessor-based ileentation. Theory of deterinistic control systes based on dynaic ulse-frequency odulation is well-established and their ractical use is becoing ore coon in the industrial environent. However, any technological rocesses cannot be controlled in a trivial way, as they are characterized by the resence of transort delays. In addition, it is exected that oeration of technological lines reains stable even in the resence of internal and external rando variations of both signals and araeters of the syste. Hence, these factors ust be taken into consideration on the stages of design, ileentation, and integration of controls [] [3]. Fig.. Dynaic ulse-frequency odulation control syste The DPFM roosed in this aer is digital, hence its ileentation requires secial treatent of its descrition. First, we can forulate atheatical definition of a closedloo digital control syste that is equivalent to the DPFM. By using analogy with an analog version of a ulse-frequency odulator [9] we can reresent a digital DPFM as a cobination of a discrete filter (DF) and a ulse generator (PG), as shown in Fig Syste Overview DPFM devices are widely used in control systes [4] [7]. They are based on the rincile of dynaic ulse-frequency odulation [8] [6] and their structure can be illustrated as shown in Fig.. A discrete filter DF erfors certain transforation of a ulse signal x[nt] into signal y[nt]. At a tie instance nt= nνt, Fig. 2. Equivalent structure of a digital DPFM when the value of у[nt] is greater or equal than soe threshold Δ, a ulse generator PG roduces a single ulse that Coyright (c) 203 International Journal of Couter Science Issues. All Rights Reserved.

2 IJCSI International Journal of Couter Science Issues, Vol. 0, Issue 3, No 2, May 203 ISSN (Print): ISSN (Online): can be described by a delta n n T with a roer sign, which resets all eory registers of the discrete filter DF as follows n n, T 0, if if n n, n n. () By using the concets fro [9], [4] [6], a odel of a digital DPFM can be fored as a closed-loo syste shown in Fig. 3, which has a block of reset (BоR) that includes a nonlinear function A[x,y*,γ ] and a relay eleent (RE). Fig. 3. Configuration of a nonlinear equivalent syste Dynaic equations that describe the above control syste and all its coonents, including the DPFM and controlled lant can be suarized as follows [6], [7]: x[ nt] f [ nt] z[ nt], (2) z[nt] is the outut signal of a controlled lant that can be described by a certain oerator z[nt] H[, 0,y *[nt] 0 T] (3) Here γ 0 is a araeter that denotes transort delay of the lant. In addition, y *[nt] [(n n )T] (4) n sign y[n T 0] (5) Also we have y[n T 0] y[n and (6) T 0] 0 (7) y [nt] Ф[y (rt),x(rt), T n r n ] (8) x[nt] x(t) is the error signal; y*[nt] is the t nt outut of DPFM, y[nt] is the outut of discrete filter DF; γ is a araeter of the odulator that is used to account for transort delay of the controlled lant such that γ γ Detailed Analysis of the Control Syste One of the ain tasks is finding the for of oerator A[x,y*,γ ] such that the syste shown in Fig. 3 generates the sae sequence of ulses as a digital DPFM. Let a linear digital filter DF be described by an l-th order difference equation. a n T a ynt a2 yn T... a yn T b xnt b2xn T..... b xn T 0 y By alying z-transfor to (9) we obtain: z X a0 z a a2 z... a z z b b2 z... b z a0 z y0 2 a 2 a3z... a z y T a 3 a4 z... a z y 2T... a y T ( b b z... (9) b z x T... b x T ) (0) Let us introduce a new notation G ( z) a0 z a a2 z... a z () R(z) b b2z... b z (2) i Q (z) aiz ; (,2,..., ). (3) i i F (z) biz ; (, 2,..., ). (4) i Coyright (c) 203 International Journal of Couter Science Issues. All Rights Reserved.

3 IJCSI International Journal of Couter Science Issues, Vol. 0, Issue 3, No 2, May 203 ISSN (Print): ISSN (Online): Then, by taking into consideration exressions () (4), equation (0) can be re-written as follows z z R z X z Q( z) y[( ) T ] G R z F ( z) x[( ) T] (5) R z 2 Let the initial tie be n v T. Let us assue that y(n v T 0) 0, then a solution to equations (9) and (0) becoes nt n T xt xt y 2 0 k n f g kn k T k n T yn T n n n (6) k T k n T xn T, R z T Z Gz ; Q z g T Z ; Rz F z f T Z ; Rz When controlling lants with a transort delay [7], equation (6) can be re-written as nt n T xt y 0 g k n k T k n T y n kt kt f 2 k n k T k n Tx n T T (7) By coaring equations (6) and (7), we conclude that yn T y n T if coonents of the signal are in the following for. T g k And k n T *... k n T yn * f 2 k n k T *... k n T x n T * T (8) x T, если n n T (9) 0, если n n To obtain the reset signal η[t] let us consider transforation of the signal y[nt] erfored by a relay eleent RE, which can be given by the following nonlinearity φ(y). y nt y nt nt, if, u nt ynt 0, if, (20), if y, Let us introduce a new signal n n, if y T, 2 s nt u nt (2) 0, if y T. By analyzing exression (7) we conclude that snt n n T, n n n (22) In order to for additional signal μ[nt], let us use the existing outut signal s[qt]. Then signal μ[nt] can be exressed in the for of a roduct as follows. [nt] x[nt] [nt] (23) r [nt] q [(n )T] s[ T] (24) 0 Fig. 4. Structure of an equivalent odel of a digital ulse-frequency odulator Coyright (c) 203 International Journal of Couter Science Issues. All Rights Reserved.

4 IJCSI International Journal of Couter Science Issues, Vol. 0, Issue 3, No 2, May 203 ISSN (Print): ISSN (Online): Then the iulse resonse transfer function can be found as follows. - q [nt] Z {G (z)} z - G (z) [- z z - Equations (7) (8) and (22) (24) give a colete descrition of a nonlinear oerator A[x,y*,γ ]. Using these equations we can coe u with a structural odel of a digital DPFM shown in Fig. 4. ] By coaring the derived odel of a digital DPFM resented in this aer with a siilar odel discussed in [9] we can see significant differences. First, we are not using a discrete-tie integrator block. Also, instead of a bi-stable relay function with hysteresis we are using an unabiguous nonlinear relay-tye function that has a sensitivity dead-zone. Both features allow significant silification of the analysis of the odel of the digital DPFM. 4. Coutational Perforance of the Design It should also be understood that the designed controller can be intended for fast oeration that ioses soe feasibility constraints on the aount of real-tie coutations. Many adative control techniques used for controlled lants with transort delay erfor coutationally intensive tasks. The aroach resented in this aer relies only on sile oerations with individual inut/outut signals. In order to assess the colexity of the roosed algorith, consider the nuber of floating oint oerations (flos) necessary to erfor one ste of calculations. For coarison, we also estiated the nuber of flos for such a well-known coutational task as atrix inversion. It is very coon that controller colexity deends on the order of the syste. Inversion is erfored on the atrix that has the size equal to this order. Fig. 3 resents the results of coutational colexity evaluation for the systes with the order being varied fro to 5. Fig. 5. Coutational efficiency of the algorith The results shown in the above figure have been obtained using MATLAB. The control law given by the equations in the revious section been ileented for an l-th order dynaic syste, as shown in Fig. 4. A standard MATLAB function has been used for atrix inversion. It can be seen fro Fig. 5 that as the syste order increases, flos nuber for atrix inversion grows in an exonential fashion. At the sae tie, for our control rocedure one can observe virtually linear relationshi. This exale shows that the suggested algorith is advantageous esecially for higher order systes. 5. Conclusions In this aer, we resented a control syste based on a digital dynaic ulse-frequency odulator. Its behavior is described by difference equations augented with additional logic relationshis. To exclude the logic conditions of the DPFM, we develoed an equivalent discrete-tie nonlinear control syste araetric relationshis were relaced with equivalent signals. Our siulation exerients deonstrate that the nonlinear control syste offers a feasible solution for the alications that require good coutational efficiency. Since the icrocontroller ileentation is the riary concern addressed in this aer, its scoe does not extend beyond the coutational efficiency of the roosed algorith. At the sae tie, control syste erforance of the DPFM has been deonstrated in a different aer of this research grou [8]. References [] J. Wu, S. Wo, and G. Lu, Asytotic stability and stabilization for a class of nonlinear descritor systes with delay, Asian Journal of Control, Vol. 3, , 20. [2] F.M. Malik, M. B. Malik, and K. Munawar, Saled-data state feedback stabilization of a class of nonlinear systes based on Euler aroxiation, Asian Journal of Control,Vol. 3, 86 97, 20. [3] A.S. Ghaffari, S. H. Tabatabaei Oreh, R. Kazei, and M. A. R. Karbalaei, An intelligent aroach to the lateral forces usage in Coyright (c) 203 International Journal of Couter Science Issues. All Rights Reserved.

5 IJCSI International Journal of Couter Science Issues, Vol. 0, Issue 3, No 2, May 203 ISSN (Print): ISSN (Online): controlling the vehicle yaw rate, Asian Journal of Control, Vol. 3, 23 23, 20. [4] B.H. Aitchanov, B. K. Kuranov, and T. F. Uarov Dynaic ulse-frequency odulation in objects control with delay, Asian Journal of Control, Vol. 4, No. 5, 7, 20 [5] Y.S. Pokov, A.A. Ashiov, and K.S. Asaubaev, Statistical theory of autoatic systes with dynaic ulse-frequency odulation, Nauka, Moscow, 988. [6] B.H. Aitchanov, Tie-doain quantization in dynaic ulsefrequency systes with transort delay. Poisk, Natural and technical sciences series, No. 3, , Alaty, 2003 [7] B.H. Aitchanov, On the analysis of stochastic digital ulsefrequency odulation systes with transort delay, Poisk, Natural and technical sciences series, No. 2, , Alaty, 2004 [8] A.A. Ashiov, K.S. Asaubaev, and B.H. Aitchanov, Statistical analysis of digital integral ulse-frequency odulation control systes of objects with transort delay. Develoent of autoatic control systes for technological rocesses. 3-0, KazPTI, Alaty, 983. [9] B.H. Aitchanov, Pulse-frequency odulation control systes with transort delay, Proc. 3rd International Science and Practice Conference Science and Innovations, 54-58, GRAD-BG, Sofia, [0] M. Schwartz, Inforation transission, odulation, and noise, McGraw-Hill, 970. [] R.W. Rochelle, Pulse-Frequency Modulation, IRE Transactions on Sace Electronics and Teleetry, 07-, 962. [2] G. Gestri, Pulse-frequency odulation in neural systes, Biohysical Journal, Vol., 98-09, 97. [3] B. Friedland, Design of Observer-Based Pulse-Frequency- Modulated Feedback Control Systes, Proc. IECON 200, 36th Annual Conference of IEEE Industrial Society, 87-92, 200 [4] Y.Z. Tsykin, Theory of nonlinear ulse systes, Nauka, 973. [5] K.S. Asaubaev, A.A. Ashiov, and Y.S. Pokov, Modeling of autoatic control systes with ulse-frequency odulation, Autoatics and Teleechanics, No. 2, 52-63, 977. [6] Y.A. Li, Modeling of digital control systes with dynaic ulse-frequency odulation. Autoatics and Teleechanics, 72-76, 978. [7] H. Guretskiy, Analysis and synthesis of control systes with delay, Mashinostroenie, 974. [8] B. Aitchanov, V. Nikulin, O. Baiuratov, Analysis and Modeling of a Control Syste Based on Digital Dynaic Pulse- Frequency Modulation for Objects with Transort Delay, Proc. IEEE, ASCC, 203 First Author Bekurza H. Aitchanov Dr.Sci.Tech., the rofessor, the acadeician of the International acadey of inforation and acadeician of the Kazakhstan National acadey of natural sciences. He is ublished ore than scientific 65 works. He conducts the occuations on the following discilines: «Inforation technologies», «Theory of the inforation», «Designing of inforation systes». His exerience of scientific and edagogical activity is 37 years. Olizhon A. Baiuratov PhD student, Kazakh National Technical University after K.I. Sataev. He is ublished ore than scientific 2 works. His exerience of scientific and edagogical activity is 3 years. Second Author Vladiir V. Nikulin rofessor, State University of New York at Binghaton. He is ublished ore than scientific 70 works. His exerience of scientific and edagogical activity is 2 years. Third Author Zhylduz B. Аitzhanova aster, Kazakhstan Institute of Technical Develoents. She is ublished ore than scientific 0 works. Coyright (c) 203 International Journal of Couter Science Issues. All Rights Reserved.