ON OPTIMAL CONTROL OF DYNAMIC SYSTEMS USING THE ALGORITHM OF NUMERICAL METHODS

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1 ON OPTIMAL CONTROL OF DYNAMIC SYSTEMS USING THE ALGORITHM OF NUMERICAL METHODS HRUBINA Kaml JADLOVSKÁ Anna, SR Abstract The paper solves the problem of the dynamc system optmal control or process. Behavour of the dynamc system s descrbed by the system of dfferental equatons. In order to solve the defned task of optmal control, the author presents the created algorthm based on the successve approxmatons method. The algorthm was realsed wthn MS Excel envronment. Key words: optmal control, algorthm of the successve approxmatons method, Hamlton functon 1 Introducton Regardng the development of computer and nformaton technology, requrements for creatng qualtatvely new control systems, especally that of optmal due to the selected qualty ndex eventually more ndexes have been extended. That s why there has been assumed the development of the theory of processes or systems optmal control. Orgnal works of Athans, Bellman, Butkovskj, Lons, Mesarovč, Pontrjagn etc, are consdered to be sgnfcant n ths feld of scence. [1,, 3, 7, 8]. The followng authors n ther monographs and papers deal wth the theory and creaton of the algorthm for numercal smulaton of mathematcal models of dynamc processes. [3, 4, 5, 6]. The am of the paper s to present theoretcal suppostons for creaton and realsaton of the algorthm usng the successve approxmatons method to the numercal soluton of the mathematcal model of the dynamc process. Mathematcal formulaton of the problem In ths paper we are gong to consder the dynamc system control. For the part of realty that we are gong to control t s necessary to create the adequate model of the object. Thus, we defne the system on the object. We create the dynamc system of the model n order to predct the behavour of the object n the future. In fact, there exst two methods how to create the system on the gven object: analytcal and expermental. Wth the analytcal method we use dfferent physcal, economcal and other prncples based on whch we are searchng for the relatonshps between varables. We defne ths method of model creaton as mathematcal-physcal analyss. The other method of system creaton follows from measurements made on real object as well as from analyss of measured data n order to determne the relatonshps between the varables. We call ths method of model creaton of the nvestgated object expermental dentfcaton. Wth ths method of model creaton we have to consder the fact that measurement has always lmted accuracy, and consequently, model created by ths method s often a stochastc system. Because the model of the object or process s a system and we are observng tme contnuance of ts varables, we say that we are nvestgatng the dynamc system. 33

2 The structure (S, R) wll be called a system, where S s a set of elements and R s a set of relatons (relatonshps) between them. Takng nto consderaton Mesarovč s defnton of a system, let us ntroduce the equvalent defnton: system s a propostonal formula S(x) producng true statement V 1 (x), V (x),..., V m (x) on mutual relatonshps of elements and system envronment ncludng state of the system. Presented defnton of the system s of a great practcal mportance especally for our consderatons because n many systems only formal statements or functons, dfferental equatons, ntegral equatons etc are nvestgated. We dstngush dynamc systems and statc systems accordng to the state of the system: whether t changes or does not change n tme. Dynamc system, state of whch can be nfluenced, s called controllable. If t does not satsfy the mentoned condtons, t s called non-controllable. We are gong to deal wth the dynamc systems and controllable systems. Wth the mathematcal models, we cannot work drectly wth the state of the system but only wth the certan nformaton on ts state. That s why we gong to suppose that the state of the system can be descrbed n any nstant by n-number of ordered real T numbers whch we are gong to call the state vector: x ( t) = ( x1 ( t), x ( t),..., xn ( t) ) The set of all feasble states determned by the condtons of certan task wll be called state space. Controllablty of dynamc system can be descrbed as follows: for each concrete functon u(t) from the set U R m t s possble accordng to gven dependence defntely to assgn the state vector x(t+δt), [4, 6]. Theory of automated control nvestgates methods by means of whch we can nfluence the system n order the controlled system behaves accordng to our requrements. Requrements for control can be as follows: 1. Compensaton of falure varables nfluence. Problem of regulators (proposton of control structure to stablze the system) 3. Problem of montorng (mnmsaton of regulatng devaton) 4. Optmal control (n the modern theory of control the requrements towards control are compled n qualty control crteron and the problem of control s transformed nto optmsaton problem of mnmsaton of qualty control crteron) Let the controlled process or system be descrbed by the set of dfferental equatons wth the ntal condtons and constrants of the followng form: dx() t = f (x(t), u(t), t) (1) x(t 0 ) = x 0, u(t) U, t t 0 where x = (x 0, x 1,, x n ) s n-dmensonal vector of the phase coordnates, u = (u 1,, u m ), m-dmensonal vector of control functons, t s tme, f = (f 1,, f n ) s a gven vector functon, x 0 s a constrant vector, t 0 the ntal tme, U s a closed set of m- dmensonal space. Partally contnuous functons u(t) whch satsfy the constrants (1) wll be called the feasble controls. The task s to determne the feasble control u(t) whch mnmzes the followng form of the functonal: J(u) = (c, x(t 1 )), t 1 > t 0 () 34

3 where t 1 s a gven moment of tme, c = (c 1,, c n ) s non-zero constant vector; the dot product s ntroduced by brackets (, ). We wll assume that the defned problem has ts soluton n the doman of feasble controls u(t) and we wll call them the optmal control. Let us defne n-dmensonal vector p = (p 1,, p n ) of the adjont varables and the Hamlton functon H. Let us ntroduce the adjont system and the condtons of transversalty: H(x(t), u(t), p(t), t) = (p(t), f(x(t), u(t), t) (3) d p d t H = x = n j p j j= 1 f ( x,u, t) x ; p (t 1 ) = - c (4) Accordng to the maxmum prncple the searched optmal control mnmzes the functon H n the relaton (3) for u U when t t 0, t 1 s arbtrary, f x, p satsfy the equatons and constrants (1), (4). 3 The methods of soluton and algorthms For the defned problem optmal control soluton let us ntroduce at frst the smpler varant of the successve approxmatons method [1]: The k- th teraton contans these steps (k = 1,, ): 1. usng control u = u (k) (t) we can solve the Cauchy problem (1) n order to defne the trajectory x = x (k) (t) at the nterval t 0, t 1 ;. we solve the Cauchy problem n reverse tme from the tme t = t 1 to t = t 0 when u = u (k) (t), x = x (k) (t) and we wll defne the adjont varables p (k) (t) at the nterval t 0, t 1 ; 3. we wll defne the control u (k + 1) (t) at the nterval t 0, t 1 from the condton H(x (k) (t), u (k + 1) (t), p (k) (t), t) = max H(x (k) (t), u (k) (t), p (k) (t), t) (5) If the condton (5) defnes u (k+1) (t) by multvalent way, we wll chose the arbtrary possble value and we wll go to the followng teraton accordng to the algorthm of successve approxmatons: Step 1. Readng of the output values Step. Soluton of the set of dfferental equatons Step 3. Calculaton of the optmal crteron value Step 4. Soluton of the adjont set of dfferental equatons Step 5. If the condton of convergence s satsfed, we wll fnsh the calculatng process. On the other hand, we wll determne the gan of control and contnue wth the step. 35

4 If the process of successve approxmatons converges, we contnue untl the condton of u(t), x(t) defned accordng to the accuracy s satsfed. Thus, we wll obtan the soluton whch satsfes the condton of convergence, t satsfes the maxmum prncple. 4 Solvng the Optmal Control problem The task s to determne u(t) and x(t) so that the cost functon 1 t J( u(t), x(t) ) = 1 t0 ( (t) + (t) + ( (t) + (t)) ) x 1 under the condtons and constrants fulfllment of the followng form: d 1 x () t () t = - x (t) u 1 (t) u (t) d x = x 1 (t) u 1 (t) + u (t) u(t) 1 ; t 1.57, 6.8 ; x x T (0) = (x 1 (0), x (0)) = (1.50, 1.78), obtans the mnmum value. The soluton of the presented task, whch s the mathematcal model of the process optmum control, wll use the programmng system MS EXCEL SOLVER, [4, 5, 6]. 5 Concluson The artcle presents a mathematcal formulaton of the problem of the process or system optmal control. The Pontryagn s prncple of maxmum and the approxmaton method appled to determne the components of the control vector u(t) were used to solve the defned problem. The artcle also presents the algorthm of the successve approxmatons method as well as the ways (quckenng) of the teraton process convergence mprovement. The successve approxmatons method and ts modfcatons can be used to solve another group of problems whch termnal condtons for the control process and also correspondng cost functon are defned. Fnally, the artcle presents the soluton of the problem of the process optmal control whch s expressed by the mathematcal model,.e. by a system of dfferental equatons and the optmal crteron s represented by the cost functon. The program system MS EXCEL was used to solve the defned problem. Príspevok bol vytvorený v rámc rešena nšttuconálnej úlohy FVT TU č. 1/008 a projektu VEGA MŠ SR č. 1/4077/07. u 1 u 36

5 6 References 1. ATHANS, M. et al: Neccessery and Suffcent Condtons for Dfferentable Nonscalar Valeced Funtons to Atlan Extrema IEEE Trans. On Aut. Control, 1973, Vol. AC-18, No. BELLMAN, R.: Dynamc programmng, 1967, New York, Prncenton Unversty Press 3. BUTKOVSKIJ, A. G.: Metody upravlenja sstemam s raspredelenym parametram, Nauka Moskva, HRUBINA, K.: Algorthm Applcatons to Determne optmum Trajectores of the Controlled Process, IFAC workshop on Programmable Devces and Systems, PDS 003, TU Ostrava, 003, pp HRUBINA, K. JADLOVSKÁ, A.: Optmal control problems solved by the Applcaton of Algorthms of Numercal Method, Chapter 6, DAAAM Internatonal Scentfc Book 004, pp. 65-8, Venna, Austra, ISBN , ISSN HRUBINA, K. JADLOVSKÁ, A.: Optmal control and Approxmaton of Varatonal Inequaltus, Kybernetes, The Internatonal Yournal of Systems and Cybernetcs, MCB Unversty Press of England, Vol. 31, No 9/10, 00, pp JADLOVSKÁ, A. et al: Algorthms for Optmal Decson Makng and Processes Control, Chapter 1, DAAAM Internatonal Scentfc Book 005, pp , Venna Austra, ISBN , ISSN LIONS, J. L.: Contrôle optmal de systémes gouvernés par des aquatons aux derves partelles, Dunod Gauter-Vllars, Pars, PONTRYAGIN, L. S. et al: Matematčeskaja teorja optmaľnch procesov, Nauka Moskva, 1983 Lektoroval: Ivan Taufer, prof. Ing., DrSc. Kontaktní adresa: Department of Process Control and Computer Technques, Unversty of Pardubce, Nám. Čs. legí 565, Pardzbce, Czech Republc, Tel.: , e-mal: van.taufer@upce.cz Kaml Hrubna, doc. Ing. Mgr., PhD. Katedra matematky, nformatky a kybernetky Fakulta výrobných technológí TU v Košcach so sídlom v Prešove Bayerova 1, Prešov, SR, tel , fax , e-mal: hrubna.kaml@fvt.sk Anna Jadlovská, doc. Ing., PhD. Katedra kybernetky a umelej ntelgence Fakulta elektrotechnky a nformatky TU Letná 9, Košce, SR e-mal: anna.jadlovska@tuke.sk 37